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Springer
Handbook Atomic, Molecular,
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and Optical Physics Drake Editor 2nd Edition
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Springer Handbooks
Springer Handbooks provide a concise compilation of approved key information on methods of research, general principles, and functional relationships in physical and applied sciences. The world’s leading experts in the fields of physics and engineering will be assigned by one or several renowned editors to write the chapters comprising each volume. The content is selected by these experts from Springer sources (books, journals, online content) and other systematic and approved recent publications of scientific and technical information. The volumes are designed to be useful as readable desk book to give a fast and comprehensive overview and easy retrieval of essential reliable key information, including tables, graphs, and bibliographies. References to extensive sources are provided.
Gordon W. F. Drake Editor
Springer Handbook of Atomic, Molecular, and Optical Physics 2nd Edition With 328 Figures and 125 Tables
Editor Gordon W. F. Drake Dept. of Physics University of Windsor Windsor, Canada
ISSN 2522-8692 ISBN 978-3-030-73892-1 https://doi.org/10.1007/978-3-030-73893-8
ISSN 2522-8706 (electronic) ISBN 978-3-030-73893-8 (eBook)
1st edition: © Springer Science+Business Media New York 2006 2nd edition: © Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Foreword
Atomic, molecular, and optical physics (AMO physics) encompasses the foundational knowledge of atomic physics and essential concepts and data for molecular and optical physics. AMO physics provides the tools for understanding the full gamut of atomic, ionic, and molecular collisions that is essential for astronomy and astrophysics. Knowledge from AMO physics overflows into neighboring fields: nuclear physics, material science, biophysics, geo-physics, and atmospheric science. AMO science is crucial to the understanding of climate change. AMO physics has undergone dramatic advances since the last edition of the AMO Handbook. The discovery of ultracold atoms has led to new tools for many-body physics. Cavity quantum electrodynamics has opened avenues into field theory and quantum information processing. Advances in experimental techniques have revolutionized metrology and enabled a new generation of atomic frequency standards that have reached precision of parts in 1019 . A metrological revolution has taken place: all the basic units of science have been redefined in terms of quantum measurements. Fundamental tests continue to flourish. Atoms now serve as tools for manipulating photons, quantum information theory and quantum communication have advanced, and the search for physics beyond the Standard Model has been carried forward. The discovery of gravitational waves by LIGO was enabled by advances in quantum optics from the AMO community, and the observation of radiation by matter as it falls through the event horizon of a black hole was made possible by synchronizing signals from observatories around the world using atomic clocks at their limit of accuracy The Table of Contents of this new edition of the Springer Handbook of Atomic, Molecular, and Optical Physics reflects the broad scope of AMO physics today. Its 91 chapters, written by experts and carefully reviewed, are an essential resource for physicists in academic, industrial, and federal research laboratories. With the changing style in scientific publishing and library practice, browsing has become increasingly difficult. All the more reason to have access to the Springer Handbook of Atomic, Molecular and Optical Physics, for which browsing is almost irresistible. January, 2021
Daniel Kleppner
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Foreword to the First Edition by Herbert Walther
The Handbook of Atomic, Molecular and Optical (AMO) Physics gives an in-depth survey of the present status of this field of physics. It is an extended version of the first issue to which new and emerging fields have been added. The selection of topics thus traces the recent historic development of AMO physics. The book gives students, scientists, engineers, and other interested people a comprehensive introduction and overview. It combines introductory explanations with descriptions of phenomena, discussions of results achieved, and gives a useful selection of references to allow more detailed studies, making the handbook very suitable as a desktop reference. AMO physics is an important and basic field of physics. It provided the essential impulse leading to the development of modern physics at the beginning of the last century. We have to remember that at that time not every physicist believed in the existence of atoms and molecules. It was due to Albert Einstein, whose work we commemorate this year with the world year of physics, that this view changed. It was Einstein’s microscopic view of molecular motion that led to a way of calculating Avogadro’s number and the size of molecules by studying their motion. This work was the basis of his PhD thesis submitted to the University of Zurich in July 1905 and after publication became Einstein’s most quoted paper. Furthermore, combining kinetic theory and classical thermodynamics led him to the conclusion that the displacement of a microparticle in Brownian motion varies as the square root of time. The experimental demonstration of this law by Jean Perrin three years later finally afforded striking proof that atoms and molecules are a reality. The energy quantum postulated by Einstein in order to explain the photoelectric effect was the basis for the subsequently initiated development of quantum physics, leading to a revolution in physics and many new applications in science and technology. The results of AMO physics initiated the development of quantum mechanics and quantum electrodynamics and as a consequence led to a better understanding of the structure of atoms and molecules and their respective interaction with radiation and to the attainment of unprecedented accuracy. AMO physics also influenced the development in other fields of physics, chemistry, astronomy, and biology. It is an astonishing fact that AMO physics constantly went through periods where new phenomena were found, giving rise to an enormous revival of this area. Examples are the maser and laser and their many applications, leading to a better understanding of the basics and the detection of new phenomena, and new possibilities such as laser cooling of atoms, squeezing, and other nonlinear behaviour. Recently, coherent interference effects allowed slow or fast light to be produced. Finally, the achievement of Bose–Einstein condensation in dilute media has opened up a wide range of new phenomena for study. Special quantum phenomena are leading to new applications for transmission of information and for computing. Control of photon emission through specially designed cavities allows controlled and deterministic generation of photons opening the way for a secure information transfer. Further new possibilities are emerging, such as the techniques for producing attosecond laser pulses and laser pulses with known and controlled phase relation between the envelope and carrier wave, allowing synthesis of even shorter pulses in a controlled manner. Furthermore, laser pulses may soon be available that are sufficiently intense to allow polarization of the vacuum field. Another interesting development is the generation of artificial atoms, e.g., vii
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quantum dots, opening a field where nanotechnology meets atomic physics. It is thus evident that AMO physics is still going strong and will also provide new and interesting opportunities and results in the future. Herbert Walther
Preface
The field of atomic, molecular, and optical physics is the oldest part of modern physics, and yet it provides the foundation upon which all the rest is built. The previous edition of the Springer Handbook of Atomic, Molecular, and Optical Physics was published in 2005, which was the International Year of Physics celebrating the Annus mirabilis of Einstein exactly 100 years earlier. 2005 also marked another in a long sequence of Nobel Prizes awarded for work in the area of AMO physics, this time to Roy Glauber, John Hall, and Ted Hänsch for their work on quantum optics and high-precision measurement. The sequence continued with the 2012 prize being awarded to Serge Haroche and David Wineland for their work on measuring and manipulating individual quantum systems, and the 2018 prize to Arthur Askin for the development of optical tweezers, and Gérard Mourou and Donna Srickland for high-intensity ultrashort laser pulses. Most recently, the Nobel Prize in Physics for 2022 was awarded jointly to Alain Aspect, John F. Clauser and Anton Zeilinger for experiments with entangled photons, establishing the violation of Bell inequalities, and pioneering quantum information science. The 2015 Prize for neutrino oscillations and the 2017 Prize for gravitational waves also made essential use of the techniques of AMO physics for their success. 2015 was also the International Year of Light, where the ever-expanding power and versatility of lasers played a prominent role; this continues to provide a unifying theme for much of AMO physics. The intent of this Handbook is to provide a ready reference for the principal ideas, techniques, and results that are common to all these areas of research where exciting advances continue apace. The success of the previous 2005 edition and advances in the field provide the motivation for the current revised edition, with the expectation that it will continue as a standard reference source for the field. Many of the topics have not changed significantly since the previous edition was published, such as angular momentum algebra (Chap. 2), perturbation theory (Chap. 5), second quantization (Chap. 6), and the properties of hydrogenic wave functions (Chap. 9), and so these chapters are still the same (except for minor corrections). However, most of the other chapters have been extensively revised, and in some cases completely rewritten. In the search for new physics beyond the Standard Model, it has become increasingly evident that high-precision atomic physics measurements at low energy can be used to supplement, or in some cases extend, what can be learned from high-energy particle accelerators. Entirely new chapters are included on searches for dark matter and an electron electric dipole moment (EDM), and tests of Lorentz invariance and general relativity. The major previous sections on lasers, laser interactions with matter, and quantum optical tests of the foundations of physics remain of central importance in the revised edition, along with the various forms of spectroscopy. The sections on scattering phenomena, both theoretical and experimental, have been augmented to include a new chapter on ultracold Rydberg atom collisions, and another on quantum defect theory. The section on applications remains as important and timely as ever, with extensively updated chapters on aeronomy and global change. Chapter 1 on units and constants has been completely transformed by the redefinition of Planck’s constant, Avogadro’s number, Boltzmann’s constant, and the elementary charge as defined physical constants with zero uncertainty. The preparation of the original Handbook and the current revision would not have been possible without the help and advice of many people throughout the AMO physics commuix
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nity. To name just a few, I am particularly indebted to Thomas Kirchner, Klaus Bartschat, and Jonathan Tennyson for their help and advice on revisions to the section on scattering theory, and to Pierre Meystre and Bruce Shore for their advice and guidance in extending and restructuring the section on quantum optics. Marianna Safronova provided valuable input on searches for new physics beyond the Standard Model. We regret that the following authors, all giants in the field, from the previous edition are no longer with us. In order of their appearance in the previous edition, they are James D. Louck, Robert N. Hill, William C. Martin, Anthony F. Starace, M. Raymond Flannery, Philip Burke, Bernd Crasemann, Stig Stenholm, Alexander Dalgarno, Alan Garscadden, Hans Bichsel, Mitio Inokuti, John M. Brown, Lorenzo J. Curtis, Kenneth Evenson, Gordon Feldman, Thomas Fulton, Axel Schenzle, Paul Feldman, and Michael R. Strayer. Work on the current revision project began in February, 2016, and so it has been nearly seven years in the making. Much of the day-to-day work involved corresponding with over 100 authors and keeping track of the status of each of the 91 chapters as they passed through an external review process followed by revisions. In many cases, it also involved actively helping authors solve problems with their LATEX source code and establishing good lines of communication with each author concerning the content of other related chapters. All of this would not have been possible without the very capable help of the three Editorial Assistants who worked on various stages of the project: Jacklyn Bizarre, Shawn Steven, and Fatemeh Hamdizadeh. Their tireless efforts and energy contributed greatly to the successful completion of the project. I would also like to acknowledge the very capable assistance of the Springer Handbook Coordinators: Veronika Hamm, Heather King, and Judith Hinterberg. I owe a great debt of gratitude to all of these people in making this revised edition of the Springer Handbook of Atomic, Molecular and Optical Physics a reality. I am especially grateful for the love and support of my wife, Mary Louise, and my children Susan and Peter. I hope that the present edition will continue to serve as a valuable resource for many years to come. November, 2022
Gordon W.F. Drake
Preface to the First Edition
The year 2005 has been officially declared by the United Nations to be the International Year of Physics to commemorate the three famous papers of Einstein published in 1905. It is a fitting tribute to the impact of his work that the Springer Handbook of Atomic, Molecular, and Optical Physics should be published in coincidence with this event. Virtually all of AMO Physics rests on the foundations established by Einstein in 1905 (including a fourth paper on relativity and his thesis) and his subsequent work. In addition to the theory of relativity, for which he is best known, Einstein ushered in the era of quantum mechanics with his explanation of the photoelectric effect, and he demonstrated the influence of molecular collisions with his explanation of Brownian motion. He also laid the theoretical foundations for all of laser physics with his discovery (in 1917) of the necessity of the process of stimulated emission, and his discussions of the Einstein–Podolsky–Rosen Gedanken experiment (in 1935) led, through Bell’s inequalities, to current work on entangled states and quantum information. The past century has been a Golden Age for physics in every sense of the term. Despite this history of unparalleled progress, the field of AMO Physics continues to advance more rapidly than ever. At the time of publication of an earlier Handbook published by AIP Press in 1996 I wrote “The ever increasing power and versatility of lasers continues to open up new areas for study.” Since then, two Nobel Prizes have been awarded for the cooling and trapping of atoms with lasers (Steven Chu, Claude Cohen-Tannoudji, William D. Phillips in 1997), and for the subsequent achievement of Bose–Einstein condensation in a dilute gas of trapped atoms (Eric A. Cornell, Wolfgang Ketterle, Carl E. Wieman in 2001). Although the topic of cooling and trapping was covered in the AIP Handbook, Bose–Einstein condensation was barely mentioned. Since then, the literature has exploded to nearly 2500 papers on Bose–Einstein condensation alone. Similarly, the topics of quantum information and quantum computing barely existed in 1995, and have since become rapidly growing segments of the physics literature. Entirely new topics such as “fast light” and “slow light” have emerged. Techniques for both high precision theory and measurement are opening the possibility to detect a cosmological variation of the fundamental constants with time. All of these topics hold the promise of important engineering and technological applications that come with advances in fundamental science. The more established areas of AMO Physics continue to provide the basic data and broad understanding of a great wealth of underlying processes needed for studies of the environment, and for astrophysics and plasma physics. These changes and advances provide more than sufficient justification to prepare a thoroughly revised and updated Atomic, Molecular and Optical Physics Handbook for the Springer Handbook Program. The aim is to present the basic ideas, methods, techniques and results of the field at a level that is accessible to graduate students and other researchers new to the field. References are meant to be a guide to the literature, rather than a comprehensive bibliography. Entirely new chapters have been added on Bose–Einstein condensation, quantum information, variations of the fundamental constants, and cavity ring-down spectroscopy. Other chapters have been substantially expanded to include new topics such as fast light and slow light. The intent is to provide a book that will continue to be a valuable resource and source of inspiration for both students and established researchers. xi
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I would like to acknowledge the important role played by the members of the Advisory Board in their continuing support of this project, and I would especially like to acknowledge the talents of Mark Cassar as Assistant Editor. In addition to keeping track of the submissions and corresponding with authors, he read and edited the new material for every chapter to ensure uniformity in style and scientific content, and he composed new material to be added to some of the chapters, as noted in the text. February 2005
Gordon W.F. Drake
Editorial Board
Editor Gordon W. F. Drake Department of Physics, University of Windsor, Windsor, Ontario N9B 3P4, Canada [email protected] Assistant Editors Jacklyn Bizarre, Shaun Steven, and Fatemeh Hamdizadeh Department of Physics, University of Windsor, Windsor, ON N9B 3P4, Canada
Advisory Board Klaus Bartschat Department of Physics, Drake University, DeMoines, Iowa, USA [email protected] Thomas Kirchner Department of Physics and Astronomy, York University, Toronto, ON M3J 1P3, Canada [email protected] Pierre Meystre Optical Sciences Center, The University of Arizona, Tucson, Arizona, USA [email protected] Marianna Safronova Department of Physics and Astronomy, University of Delaware, Newark, DE 19716 USA [email protected] Jonathan Tennyson Department of Physics and Astronomy, University College London, London WC1E 6BT, UK [email protected]
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Contents
Part A Mathematical Methods 1
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Units and Constants . . . . . Eite Tiesinga 1.1 Introduction . . . . . . . 1.2 Atomic Units . . . . . . 1.3 Natural Units . . . . . . 1.4 Fundamental Constants References . . . . . . . . . . . .
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Angular Momentum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . James D. Louck 2.1 Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Cartesian Representation . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Spherical Polar Coordinate Representation . . . . . . . . . . . 2.2 Abstract Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Representation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Parametrizations of the Groups SU(2) and SO(3,R) . . . . . 2.3.2 Explicit Forms of Representation Functions . . . . . . . . . . 2.3.3 Relations to Special Functions . . . . . . . . . . . . . . . . . . 2.3.4 Orthogonality Properties . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Symmetry Relations . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Group and Lie Algebra Actions . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Matrix Group Actions . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Lie Algebra Actions . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Relation to Angular Momentum Theory . . . . . . . . . . . . 2.5 Differential Operator Realizations of Angular Momentum . . . . . . 2.6 The Symmetric Rotor and Representation Functions . . . . . . . . . . 2.7 Wigner–Clebsch–Gordan and 3–j Coefficients . . . . . . . . . . . . . . 2.7.1 Kronecker Product Reduction . . . . . . . . . . . . . . . . . . . 2.7.2 Tensor Product Space Construction . . . . . . . . . . . . . . . 2.7.3 Explicit Forms of WCG Coefficients . . . . . . . . . . . . . . 2.7.4 Symmetries of WCG Coefficients in 3–j Symbol Form . . . 2.7.5 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . 2.7.6 Limiting Properties and Asymptotic Forms . . . . . . . . . . 2.7.7 WCG Coefficients as Discretized Representation Functions 2.8 Tensor Operator Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Conceptual Framework . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Universal Enveloping Algebra of J . . . . . . . . . . . . . . . 2.8.3 Algebra of Irreducible Tensor Operators . . . . . . . . . . . .
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2.8.4 Wigner–Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.5 Unit Tensor Operators or Wigner Operators . . . . . . . . . . . . . 2.9 Racah Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Basic Relations Between WCG and Racah Coefficients . . . . . . 2.9.2 Orthogonality and Explicit Form . . . . . . . . . . . . . . . . . . . . 2.9.3 The Fundamental Identities Between Racah Coefficients . . . . . 2.9.4 Schwinger–Bargmann Generating Function and Its Combinatorics 2.9.5 Symmetries of 6–j Coefficients . . . . . . . . . . . . . . . . . . . . . 2.9.6 Further Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 The 9–j Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1 Hilbert Space and Tensor Operator Actions . . . . . . . . . . . . . 2.10.2 9–j Invariant Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.3 Basic Relations Between 9–j Coefficients and 6–j Coefficients . 2.10.4 Symmetry Relations for 9–j Coefficients and Reduction to 6–j Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.5 Explicit Algebraic Form of 9–j Coefficients . . . . . . . . . . . . . 2.10.6 Racah Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.7 Schwinger–Wu Generating Function and Its Combinatorics . . . 2.11 Tensor Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.1 Spinor Spherical Harmonics as Matrix Functions . . . . . . . . . . 2.11.2 Vector Spherical Harmonics as Matrix Functions . . . . . . . . . . 2.11.3 Vector Solid Harmonics as Vector Functions . . . . . . . . . . . . . 2.12 Coupling and Recoupling Theory and 3n–j Coefficients . . . . . . . . . . . 2.12.1 Composite Angular Momentum Systems . . . . . . . . . . . . . . . 2.12.2 Binary Coupling Theory: Combinatorics . . . . . . . . . . . . . . . 2.12.3 Implementation of Binary Couplings . . . . . . . . . . . . . . . . . 2.12.4 Construction of All Transformation Coefficients in Binary Coupling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12.5 Unsolved Problems in Recoupling Theory . . . . . . . . . . . . . . 2.13 Supplement on Combinatorial Foundations . . . . . . . . . . . . . . . . . . . 2.13.1 SU(2) Solid Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.2 Combinatorial Definition of Wigner–Clebsch–Gordan Coefficients 2.13.3 Magic Square Realization of the Addition of Two Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . 2.13.4 MacMahon’s and Schwinger’s Master Theorems . . . . . . . . . . 2.13.5 The Pfaffian and Double Pfaffian . . . . . . . . . . . . . . . . . . . . 2.13.6 Generating Functions for Coupled Wave Functions and Recoupling Coefficients . . . . . . . . . . . . . . . . . . . . . . . 2.14 Author’s Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Group Theory for Atomic Shells . . . . . . . . . . . . Brian R. Judd 3.1 Generators . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Group Elements . . . . . . . . . . . . . . 3.1.2 Conditions on the Structure Constants 3.1.3 Cartan–Weyl Form . . . . . . . . . . . . 3.1.4 Atomic Operators as Generators . . . . 3.2 Classification of Lie Algebras . . . . . . . . . . . 3.2.1 Introduction . . . . . . . . . . . . . . . . . 3.2.2 The Semisimple Lie Algebras . . . . . .
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Irreducible Representations . . . . . . . . . . . . . . . . . . 3.3.1 Labels . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Dimensions . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Casimir’s Operator . . . . . . . . . . . . . . . . . . 3.4 Branching Rules . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 U.n/ SU.n/ . . . . . . . . . . . . . . . . . . . . . 3.4.3 Canonical Reductions . . . . . . . . . . . . . . . . . 3.4.4 Other Reductions . . . . . . . . . . . . . . . . . . . 3.5 Kronecker Products . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Outer Products of Tableaux . . . . . . . . . . . . . 3.5.2 Other Outer Products . . . . . . . . . . . . . . . . . 3.5.3 Plethysms . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Atomic States . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Shell Structure . . . . . . . . . . . . . . . . . . . . . 3.6.2 Automorphisms of SO(8) . . . . . . . . . . . . . . 3.6.3 Hydrogen and Hydrogen-Like Atoms . . . . . . . 3.7 The Generalized Wigner–Eckart Theorem . . . . . . . . . 3.7.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 The Theorem . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Calculation of the Isoscalar Factors . . . . . . . . 3.7.4 Generalizations of Angular Momentum Theory . 3.8 Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Dynamical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . Josef Paldus 4.1 Noncompact Dynamical Groups . . . . . . . . . . . . . . . 4.1.1 Realizations of so(2,1) . . . . . . . . . . . . . . . . 4.1.2 Hydrogenic Realization of so(4,2) . . . . . . . . . 4.2 Hamiltonian Transformation and Simple Applications . . 4.2.1 N-Dimensional Isotropic Harmonic Oscillator . 4.2.2 N-Dimensional Hydrogenic Atom . . . . . . . . . 4.2.3 Perturbed Hydrogenic Systems . . . . . . . . . . . 4.3 Compact Dynamical Groups . . . . . . . . . . . . . . . . . . 4.3.1 Unitary Group and Its Representations . . . . . . 4.3.2 Orthogonal Group O(n) and Its Representations 4.3.3 Clifford Algebras and Spinor Representations . . 4.3.4 Bosonic and Fermionic Realizations of U(n) . . 4.3.5 The Vibron Model . . . . . . . . . . . . . . . . . . . 4.3.6 Many-Electron Correlation Problem . . . . . . . . 4.3.7 Clifford Algebra Unitary Group Approach . . . . 4.3.8 Spin-Dependent Operators . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Perturbation Theory . . . . . . . . . . . Josef Paldus 5.1 Matrix Perturbation Theory (PT) 5.1.1 Basic Concepts . . . . . . 5.1.2 Level-Shift Operators . . 5.1.3 General Formalism . . . 5.1.4 Nondegenerate Case . .
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5.2
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Time-Independent Perturbation Theory . . . . . . . . . . . . . . . . 5.2.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Brillouin–Wigner and Rayleigh–Schrödinger PT (RSPT) 5.2.3 Bracketing Theorem and RSPT . . . . . . . . . . . . . . . . 5.3 Fermionic Many-Body Perturbation Theory (MBPT) . . . . . . . . 5.3.1 Time-Independent Wick’s Theorem . . . . . . . . . . . . . 5.3.2 Normal Product Form of PT . . . . . . . . . . . . . . . . . . 5.3.3 Møller–Plesset and Epstein–Nesbet PT . . . . . . . . . . . 5.3.4 Diagrammatic MBPT . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Vacuum and Wave Function Diagrams . . . . . . . . . . . 5.3.6 Hartree–Fock Diagrams . . . . . . . . . . . . . . . . . . . . 5.3.7 Linked and Connected Cluster Theorems . . . . . . . . . . 5.3.8 Coupled Cluster Theory . . . . . . . . . . . . . . . . . . . . 5.4 Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . 5.4.1 Evolution Operator PT Expansion . . . . . . . . . . . . . . 5.4.2 Gell-Mann and Low Formula . . . . . . . . . . . . . . . . . 5.4.3 Potential Scattering and Quantum Dynamics . . . . . . . 5.4.4 Born Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Variation of Constants Method . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Second Quantization . . . . . . . . . . . . . . . . . . Brian R. Judd 6.1 Basic Properties . . . . . . . . . . . . . . . . . 6.1.1 Definitions . . . . . . . . . . . . . . . 6.1.2 Representation of States . . . . . . . 6.1.3 Representation of Operators . . . . . 6.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Construction . . . . . . . . . . . . . . 6.2.2 Coupled Forms . . . . . . . . . . . . . 6.2.3 Coefficients of Fractional Parentage 6.3 Quasispin . . . . . . . . . . . . . . . . . . . . . 6.3.1 Fermions . . . . . . . . . . . . . . . . 6.3.2 Bosons . . . . . . . . . . . . . . . . . . 6.3.3 Triple Tensors . . . . . . . . . . . . . 6.3.4 Conjugation . . . . . . . . . . . . . . . 6.3.5 Dependence on Electron Number . 6.3.6 The Half-Filled Shell . . . . . . . . . 6.4 Complementarity . . . . . . . . . . . . . . . . . 6.4.1 Spin–Quasispin Interchange . . . . . 6.4.2 Matrix Elements . . . . . . . . . . . . 6.5 Quasiparticles . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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Density Matrices . . . . . . . . . . . . . . . . . . . Klaus Bartschat 7.1 Basic Formulae . . . . . . . . . . . . . . . . 7.1.1 Pure States . . . . . . . . . . . . . 7.1.2 Mixed States . . . . . . . . . . . . 7.1.3 Expectation Values . . . . . . . . 7.1.4 The Liouville Equation . . . . . . 7.1.5 Systems in Thermal Equilibrium 7.1.6 Relaxation Processes . . . . . . .
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7.2
Spin and Light Polarizations . . . . . . . . . . . . . . . . . . . 7.2.1 Spin-Polarized Electrons . . . . . . . . . . . . . . . . 7.2.2 Light Polarization . . . . . . . . . . . . . . . . . . . . 7.3 Atomic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Scattering Amplitudes . . . . . . . . . . . . . . . . . 7.3.2 Reduced Density Matrices . . . . . . . . . . . . . . . 7.4 Irreducible Tensor Operators . . . . . . . . . . . . . . . . . . . 7.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Transformation Properties . . . . . . . . . . . . . . . 7.4.3 Symmetry Properties of State Multipoles . . . . . . 7.4.4 Orientation and Alignment . . . . . . . . . . . . . . 7.4.5 Coupled Systems . . . . . . . . . . . . . . . . . . . . 7.5 Time Evolution of State Multipoles . . . . . . . . . . . . . . 7.5.1 Perturbation Coefficients . . . . . . . . . . . . . . . . 7.5.2 Quantum Beats . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Time Integration over Quantum Beats . . . . . . . . 7.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Generalized STU-Parameters . . . . . . . . . . . . . 7.6.2 Radiation from Excited States: Stokes Parameters 7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
9
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Computational Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . David Schultz and Michael R. Strayer 8.1 Representation of Functions . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Approximating Integrals . . . . . . . . . . . . . . . . . . . . . 8.1.5 Approximating Derivatives . . . . . . . . . . . . . . . . . . . 8.2 Differential and Integral Equations . . . . . . . . . . . . . . . . . . . . 8.2.1 Ordinary Differential Equations . . . . . . . . . . . . . . . . 8.2.2 Differencing Algorithms for Partial Differential Equations 8.2.3 Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Computational Linear Algebra . . . . . . . . . . . . . . . . . . . . . . 8.4 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Distributions of Random Numbers . . . . . . . . . . . . . . . 8.4.3 Monte Carlo Integration . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Hydrogenic Wave Functions . . . . . . . . . . . . . . . . . . . . . . . Robert N. Hill 9.1 Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . 9.1.2 Parabolic Coordinates . . . . . . . . . . . . . . . . . . 9.1.3 Momentum Space . . . . . . . . . . . . . . . . . . . . . 9.2 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The Coulomb Green’s Function . . . . . . . . . . . . . . . . . . 9.3.1 The Green’s Function for the Schrödinger Equation 9.3.2 The Green’s Function for the Dirac Equation . . . .
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9.4
Special Functions . . . . . . . . . . . . . . . . . 9.4.1 Confluent Hypergeometric Functions 9.4.2 Laguerre Polynomials . . . . . . . . . 9.4.3 Gegenbauer Polynomials . . . . . . . . 9.4.4 Legendre Functions . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Part B 11
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Software for Computational Atomic and Molecular Physics . . . Edmund J. Mansky II 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Software for Atomic Physics . . . . . . . . . . . . . . . . . . . . 10.2.1 Atomic Structure Programs . . . . . . . . . . . . . . . 10.2.2 Electron-Atom/Ion Scattering Programs . . . . . . . 10.3 Software for Molecular Physics . . . . . . . . . . . . . . . . . . 10.3.1 Molecular Structure Programs . . . . . . . . . . . . . 10.3.2 Molecular and Heavy Particle Scattering Programs 10.4 Software Libraries and Repositories . . . . . . . . . . . . . . . 10.4.1 The CPC Journal and Library . . . . . . . . . . . . . . 10.4.2 Perl and CPAN . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Python and PyPI . . . . . . . . . . . . . . . . . . . . . . 10.4.4 C++ and Boost . . . . . . . . . . . . . . . . . . . . . . . 10.4.5 GitHub . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 General Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Configuration: GNU Autotools . . . . . . . . . . . . . 10.5.2 C compilers: gcc . . . . . . . . . . . . . . . . . . . . . . 10.5.3 FORTRAN Compilers . . . . . . . . . . . . . . . . . . 10.5.4 Shell Scripting Languages . . . . . . . . . . . . . . . . 10.5.5 Mixing Languages . . . . . . . . . . . . . . . . . . . . . 10.5.6 Third-Party Libraries and Unit Tests . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Atoms Atomic Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . William C. Martin, Wolfgang L. Wiese, and Alexander Kramida 11.1 Frequency, Wavenumber, Wavelength . . . . . . . . . . . . 11.2 Atomic States, Shells, and Configurations . . . . . . . . . 11.3 Hydrogen and Hydrogen-Like Ions . . . . . . . . . . . . . . 11.4 Alkalis and Alkali-Like Spectra . . . . . . . . . . . . . . . . 11.5 Helium and Helium-Like Ions; LS Coupling . . . . . . . . 11.6 Hierarchy of Atomic Structure in LS Coupling . . . . . . . 11.7 Allowed Terms or Levels for Equivalent Electrons . . . . 11.7.1 LS Coupling . . . . . . . . . . . . . . . . . . . . . . 11.7.2 jj Coupling . . . . . . . . . . . . . . . . . . . . . . 11.8 Notations for Different Coupling Schemes . . . . . . . . . 11.8.1 LS Coupling (Russell–Saunders Coupling) . . . . 11.8.2 jj Coupling of Equivalent Electrons . . . . . . . 11.8.3 J1 j or J1 J2 Coupling . . . . . . . . . . . . . . . . . 11.8.4 J1 l or J1 L2 Coupling (J1 K Coupling) . . . . . . 11.8.5 LS1 Coupling (LK Coupling) . . . . . . . . . . . . 11.8.6 Coupling Schemes and Term Symbols . . . . . .
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11.9 11.10 11.11 11.12 11.13
Eigenvector Composition of Levels . . . . . . . . . . . . . . . . . . . . . . . Ground Levels and Ionization Energies for Neutral Atoms . . . . . . . . . Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Term Series, Quantum Defects, and Spectral-Line Series . . . . . . . . . . Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.13.1 Isoelectronic Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 11.13.2 Isoionic, Isonuclear, and Homologous Sequences . . . . . . . . . . 11.14 Spectral Wavelength Ranges, Dispersion of Air . . . . . . . . . . . . . . . . 11.15 Wavelength Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.16 Spectral Lines: Selection Rules, Intensities, Transition Probabilities, f Values, and Line Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.16.1 Emission Intensities (Transition Probabilities) . . . . . . . . . . . . 11.16.2 Absorption f -values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.16.3 Line Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.16.4 Relationships Between A, f, and S . . . . . . . . . . . . . . . . . . . 11.16.5 Relationships Between Line and Multiplet Values . . . . . . . . . 11.16.6 Relative Strengths for Lines of Multiplets in LS Coupling . . . . 11.17 Atomic Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.18 Regularities and Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.18.1 Transitions in Hydrogenic (One-Electron) Species . . . . . . . . . 11.18.2 Systematic Trends and Regularities in Atoms and Ions with Two or More Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.19 Tabulations of Transition Probabilities . . . . . . . . . . . . . . . . . . . . . . 11.20 Spectral Line Shapes, Widths, and Shifts . . . . . . . . . . . . . . . . . . . . 11.20.1 Doppler Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.20.2 Pressure Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.21 Spectral Continuum Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.21.1 Hydrogenic Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.21.2 Many-Electron Systems . . . . . . . . . . . . . . . . . . . . . . . . . 11.22 Sources of Spectroscopic Data . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
High Precision Calculations for Helium . . . . . . . . . . . . . . Gordon W. F. Drake 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Three-Body Schrödinger Equation . . . . . . . . . . . 12.2.1 Formal Mathematical Properties . . . . . . . . . . 12.3 Computational Methods . . . . . . . . . . . . . . . . . . . . 12.3.1 Variational Methods . . . . . . . . . . . . . . . . . . 12.3.2 Construction of Basis Sets . . . . . . . . . . . . . . 12.3.3 Calculation of Matrix Elements . . . . . . . . . . . 12.3.4 Other Computational Methods . . . . . . . . . . . 12.4 Variational Eigenvalues . . . . . . . . . . . . . . . . . . . . . 12.4.1 Expectation Values of Operators and Sum Rules 12.5 Total Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Quantum Defect Extrapolations . . . . . . . . . . 12.5.2 Asymptotic Expansions . . . . . . . . . . . . . . . 12.6 Radiative Transitions . . . . . . . . . . . . . . . . . . . . . . 12.6.1 Basic Formulation . . . . . . . . . . . . . . . . . . . 12.6.2 Oscillator Strength Table . . . . . . . . . . . . . . . 12.7 Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Atomic Multipoles . . . . . . . . . . . . . . . . . . . . William E. Baylis 13.1 Polarization and Multipoles . . . . . . . . . . 13.2 The Density Matrix in Liouville Space . . . 13.3 Diagonal Representation: State Populations 13.4 Interaction with Light . . . . . . . . . . . . . . 13.5 Extensions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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Atoms in Strong Fields . . . . . . . . . . . . . . S. Pedro Goldman and Mark M. Cassar 14.1 Electron in a Uniform Magnetic Field . 14.1.1 Nonrelativistic Theory . . . . . 14.1.2 Relativistic Theory . . . . . . . 14.2 Atoms in Uniform Magnetic Fields . . . 14.2.1 Anomalous Zeeman Effect . . 14.2.2 Normal Zeeman Effect . . . . . 14.2.3 Paschen–Back Effect . . . . . . 14.3 Atoms in Very Strong Magnetic Fields 14.4 Atoms in Electric Fields . . . . . . . . . 14.4.1 Stark Ionization . . . . . . . . . 14.4.2 Linear Stark Effect . . . . . . . 14.4.3 Quadratic Stark Effect . . . . . 14.4.4 Other Stark Corrections . . . . 14.5 Recent Developments . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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Rydberg Atoms . . . . . . . . . . . . . . . . . . . . . . . Thomas F. Gallagher 15.1 Wave Functions and Quantum Defect Theory 15.2 Optical Excitation and Radiative Lifetimes . . 15.3 Electric Fields . . . . . . . . . . . . . . . . . . . . 15.4 Magnetic Fields . . . . . . . . . . . . . . . . . . 15.5 Microwave Fields . . . . . . . . . . . . . . . . . 15.6 Collisions . . . . . . . . . . . . . . . . . . . . . . 15.7 Autoionizing Rydberg States . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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Rydberg Atoms in Strong Static Fields . . . . . . . . . . John B. Delos, Thomas Bartsch, and Turgay Uzer 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Semiclassical Approximations . . . . . . . . . . . . . 16.3 Regular Trajectories and Regular Wave Functions 16.4 Chaotic Trajectories and Irregular Wave Functions 16.4.1 Small-Scale Structure . . . . . . . . . . . . . 16.4.2 Large-Scale Structure of Energy Spectra . 16.4.3 Chaotic Ionization . . . . . . . . . . . . . . . 16.5 Nuclear-Mass Effects . . . . . . . . . . . . . . . . . . 16.6 Quantum Theories . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Hyperfine Structure . . . . . . . . . . . . . . . . . . . . . . Guy T. Emery 17.1 Splittings and Intensities . . . . . . . . . . . . . . . 17.1.1 Angular Momentum Coupling . . . . . . 17.1.2 Energy Splittings . . . . . . . . . . . . . . 17.1.3 Intensities . . . . . . . . . . . . . . . . . . . 17.2 Isotope Shifts . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Normal Mass Shift . . . . . . . . . . . . . 17.2.2 Specific Mass Shift . . . . . . . . . . . . . 17.2.3 Field Shift . . . . . . . . . . . . . . . . . . . 17.2.4 Separation of Mass Shift and Field Shift 17.3 Hyperfine Structure . . . . . . . . . . . . . . . . . . 17.3.1 Electric Multipoles . . . . . . . . . . . . . 17.3.2 Magnetic Multipoles . . . . . . . . . . . . 17.3.3 Hyperfine Anomalies . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Precision Oscillator Strength and Lifetime Measurements . . . . . . . . . . Lorenzo J. Curtis 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Oscillator Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 Absorption and Dispersion Measurements . . . . . . . . . . . . 18.2.2 Emission Measurements . . . . . . . . . . . . . . . . . . . . . . . 18.2.3 Combined Absorption, Emission and Lifetime Measurements 18.2.4 Branching Ratios in Highly Ionized Atoms . . . . . . . . . . . 18.3 Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 The Hanle Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2 Time-Resolved Decay Measurements . . . . . . . . . . . . . . . 18.3.3 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.4 Multiplexed Detection . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Spectroscopy of Ions Using Fast Beams and Ion Traps . . Elmar Träbert and Eric H. Pinnington 19.1 Spectroscopy Using Fast Ion Beams . . . . . . . . . . . 19.1.1 Beam–Foil Spectroscopy . . . . . . . . . . . . . 19.1.2 Beam–Gas Spectroscopy . . . . . . . . . . . . . 19.1.3 Beam–Laser Spectroscopy . . . . . . . . . . . . 19.1.4 Other Techniques of Ion–Beam Spectroscopy 19.2 Spectroscopy Using Ion Traps . . . . . . . . . . . . . . . 19.2.1 Electron Beam Ion Traps . . . . . . . . . . . . . 19.2.2 Heavy-Ion Storage Rings . . . . . . . . . . . . 19.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Line Shapes and Radiation Transfer . . . . . . . . Alan Gallagher 20.1 Collisional Line Shapes . . . . . . . . . . . . . 20.1.1 Voigt Line Shape . . . . . . . . . . . 20.1.2 Interaction Potentials . . . . . . . . . 20.1.3 Classical Oscillator Approximation 20.1.4 Impact Approximation . . . . . . . . 20.1.5 Examples: Line Core . . . . . . . . .
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20.1.6 and c Characteristics . . . . . . . . . . . 20.1.7 Quasistatic Approximation . . . . . . . . . 20.1.8 Satellites . . . . . . . . . . . . . . . . . . . . . 20.1.9 Bound States and Other Quantum Effects 20.1.10 Einstein A and B Coefficients . . . . . . . . 20.2 Radiation Trapping . . . . . . . . . . . . . . . . . . . 20.2.1 Holstein–Biberman Theory . . . . . . . . . 20.2.2 Additional Factors . . . . . . . . . . . . . . . 20.2.3 Measurements . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
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Thomas-Fermi and Other Density-Functional Theories . . . . . . . . . . . . . John D. Morgan III 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Thomas–Fermi Theory and Its Extensions . . . . . . . . . . . . . . . . . . . 21.2.1 Thomas–Fermi Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Thomas–Fermi–von Weizsäcker Theory . . . . . . . . . . . . . . . 21.2.3 Thomas–Fermi–Dirac Theory . . . . . . . . . . . . . . . . . . . . . . 21.2.4 Thomas–Fermi–von Weizsäcker–Dirac Theory . . . . . . . . . . . 21.2.5 Thomas–Fermi Theory with Different Spin Densities . . . . . . . 21.3 Nonrelativistic Energies of Heavy Atoms . . . . . . . . . . . . . . . . . . . . 21.4 General Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . 21.4.1 The Hohenberg–Kohn Theorem for the One-Electron Density . . 21.4.2 The Kohn–Sham Method for Including Exchange and Correlation Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.3 Density Functional Theory for Excited States . . . . . . . . . . . . 21.4.4 Locality of Density Functional Theory . . . . . . . . . . . . . . . . 21.4.5 Relativistic and Quantum Field Theoretic Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atomic Structure: Variational Wave Functions and Properties . . . . . . . . . Charlotte Froese Fischer and Michel Godefroid 22.1 Nonrelativistic and Relativistic Hamiltonians . . . . . . . . . . . . . . . . . 22.1.1 Schrödinger’s Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 22.1.2 Dirac–Coulomb(–Breit) Hamiltonian . . . . . . . . . . . . . . . . . 22.1.3 Breit-Pauli Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Many-Electron Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2.1 Nonrelativistic Orbitals and LS Configuration State Functions . 22.2.2 Relativistic Dirac Orbitals and jj Configuration State Functions 22.3 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Hartree–Fock and Dirac–Hartree–Fock Methods . . . . . . . . . . . . . . . 22.4.1 Diagonal Energy Parameters and Koopmans’ Theorem . . . . . . 22.4.2 Hartree–Fock Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4.3 Brillouin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4.4 The Dirac–Hartree–Fock Equations . . . . . . . . . . . . . . . . . . 22.4.5 Numerical Solution of Variational Equations . . . . . . . . . . . . 22.4.6 Properties of Hartree–Fock Solutions . . . . . . . . . . . . . . . . . 22.5 Multiconfiguration (Dirac)-Hartree–Fock Method . . . . . . . . . . . . . . . 22.5.1 Z-Dependent Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5.2 The MC(D)HF Approximation . . . . . . . . . . . . . . . . . . . . . 22.5.3 Systematic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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22.5.4 Excited States . . . . . . . . . . . . . . . . . . . . . 22.5.5 Autoionizing States . . . . . . . . . . . . . . . . . . 22.6 Configuration Interaction Methods . . . . . . . . . . . . . . 22.7 Atomic Properties . . . . . . . . . . . . . . . . . . . . . . . . 22.7.1 Transition Data—Allowed and Forbidden Lines 22.7.2 Electron Affinities . . . . . . . . . . . . . . . . . . . 22.7.3 Metastable States and Autoionization . . . . . . . 22.7.4 Nuclear Effects . . . . . . . . . . . . . . . . . . . . . 22.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
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Relativistic Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . Ian Grant 23.1 Mathematical Basics . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1.1 Relativistic Notation: Minkowski Space-Time . . . . . 23.1.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . 23.1.3 Classification of Lorentz Transformations . . . . . . . 23.1.4 Contravariant and Covariant Vectors . . . . . . . . . . . 23.1.5 Poincaré Transformations . . . . . . . . . . . . . . . . . 23.2 Dirac’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.1 Characterization of Dirac States . . . . . . . . . . . . . 23.2.2 The Charge-Current Four-Vector . . . . . . . . . . . . . 23.3 QED: Relativistic Atomic and Molecular Structure . . . . . . . 23.3.1 The QED Equations of Motion . . . . . . . . . . . . . . 23.3.2 The Quantized Electron–Positron Field . . . . . . . . . 23.3.3 Quantized Electromagnetic Field . . . . . . . . . . . . . 23.3.4 QED Perturbation Theory . . . . . . . . . . . . . . . . . 23.3.5 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3.6 Effective Interaction of Electrons . . . . . . . . . . . . . 23.4 Many-Body Theory For Atoms . . . . . . . . . . . . . . . . . . . 23.4.1 Effective Hamiltonians . . . . . . . . . . . . . . . . . . . 23.4.2 Nonrelativistic Limit: Breit–Pauli Hamiltonian . . . . 23.4.3 Perturbation Theory: Nondegenerate Case . . . . . . . 23.4.4 Perturbation Theory: Open-Shell Case . . . . . . . . . 23.4.5 Perturbation Theory: Algorithms . . . . . . . . . . . . . 23.5 Spherical Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 23.5.1 Eigenstates of Angular Momentum . . . . . . . . . . . 23.5.2 Dirac Hamiltonian in Spherical Coordinates . . . . . . 23.5.3 Solutions Near the Nucleus . . . . . . . . . . . . . . . . 23.5.4 Bound states: r ! 1 . . . . . . . . . . . . . . . . . . . . 23.5.5 Hydrogenic Atoms . . . . . . . . . . . . . . . . . . . . . 23.5.6 The Free Electron Problem in Spherical Coordinates 23.6 Numerical Methods for the Radial Dirac Equation . . . . . . . 23.6.1 Dirac’s Equation on a Finite Domain . . . . . . . . . . 23.6.2 Expansion Methods . . . . . . . . . . . . . . . . . . . . . 23.6.3 Finite Basis Set Formalism . . . . . . . . . . . . . . . . 23.6.4 Physically Acceptable Basis Sets . . . . . . . . . . . . . 23.6.5 Basis Sets Defined on R3 . . . . . . . . . . . . . . . . . 23.6.6 B-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.7 Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . .
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23.8
Many-Electron Atoms . . . . . . . . . . . . . . . . . . . . . 23.8.1 Atomic States . . . . . . . . . . . . . . . . . . . . 23.8.2 Atomic Orbitals . . . . . . . . . . . . . . . . . . . 23.8.3 Slater Determinants . . . . . . . . . . . . . . . . . 23.8.4 Configurational States . . . . . . . . . . . . . . . 23.8.5 ASFs and CSF . . . . . . . . . . . . . . . . . . . . 23.8.6 Matrix Construction . . . . . . . . . . . . . . . . . 23.8.7 Dirac–Hartree–Fock–Breit . . . . . . . . . . . . . 23.8.8 Electron Correlation . . . . . . . . . . . . . . . . 23.8.9 Electron Self-Energy and Vacuum Polarization 23.8.10 Radiative Transitions . . . . . . . . . . . . . . . . 23.9 GRASP – Information and Software . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
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Many-Body Theory of Atomic Structure and Processes . . . . . . . . . Miron Y. Amusia 24.1 Diagrammatic Technique . . . . . . . . . . . . . . . . . . . . . . . . . 24.1.1 Basic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 24.1.2 Construction Principles for Diagrams . . . . . . . . . . . . 24.1.3 Correspondence Rules . . . . . . . . . . . . . . . . . . . . . 24.1.4 Higher-Order Corrections and Summation of Sequences 24.2 Calculation of Atomic Properties . . . . . . . . . . . . . . . . . . . . 24.2.1 Electron Correlations in Ground State Properties . . . . . 24.2.2 Characteristics of One-Particle States . . . . . . . . . . . . 24.2.3 Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . 24.2.4 Two-Electron and Two-Vacancy States . . . . . . . . . . . 24.2.5 Electron–Vacancy States . . . . . . . . . . . . . . . . . . . . 24.2.6 Photoionization in RPAE and Beyond . . . . . . . . . . . . 24.2.7 Photon Emission and Bremsstrahlung . . . . . . . . . . . . 24.2.8 RPAE in the Magnetic Channel . . . . . . . . . . . . . . . . 24.2.9 Consideration of “Big Atoms” . . . . . . . . . . . . . . . . 24.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photoionization of Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anthony F. Starace 25.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 25.1.1 The Interaction Hamiltonian . . . . . . . . . . . . . . . . . 25.1.2 Alternative Forms for the Transition Matrix Element . . 25.1.3 Selection Rules for Electric Dipole Transitions . . . . . . 25.1.4 Boundary Conditions on the Final State Wave Function . 25.1.5 Photoionization Cross Sections . . . . . . . . . . . . . . . . 25.2 An Independent Electron Model . . . . . . . . . . . . . . . . . . . . 25.2.1 Central Potential Model . . . . . . . . . . . . . . . . . . . . 25.2.2 High Energy Behavior . . . . . . . . . . . . . . . . . . . . . 25.2.3 Near-Threshold Behavior . . . . . . . . . . . . . . . . . . . 25.3 Particle–Hole Interaction Effects . . . . . . . . . . . . . . . . . . . . 25.3.1 Intrachannel Interactions . . . . . . . . . . . . . . . . . . . . 25.3.2 Virtual Double Excitations . . . . . . . . . . . . . . . . . . . 25.3.3 Interchannel Interactions . . . . . . . . . . . . . . . . . . . . 25.3.4 Photoionization of Ar . . . . . . . . . . . . . . . . . . . . . .
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Theoretical Methods for Photoionization . . . . . . . . . . . . . . . . . 25.4.1 Calculational Methods . . . . . . . . . . . . . . . . . . . . . . . 25.4.2 Other Interaction Effects . . . . . . . . . . . . . . . . . . . . . . 25.5 Related Photoionization Processes . . . . . . . . . . . . . . . . . . . . . 25.6 Applications to Other Processes . . . . . . . . . . . . . . . . . . . . . . . 25.6.1 Applications to Two and Three-Photon Ionization Processes 25.6.2 Application to High-Order Harmonic Generation . . . . . . . 25.7 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Autoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anand K. Bhatia and Aaron Temkin 26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.1.1 Auger Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 26.1.2 Autoionization, Autodetachment, and Radiative Decay 26.1.3 Formation, Scattering, and Resonances . . . . . . . . . . 26.2 Projection Operator Formalism . . . . . . . . . . . . . . . . . . . . 26.2.1 Optical Potential . . . . . . . . . . . . . . . . . . . . . . . . 26.2.2 Expansion of Vop : The QHQ Problem . . . . . . . . . . 26.3 Forms of P and Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3.1 The Feshbach Form . . . . . . . . . . . . . . . . . . . . . . 26.3.2 Reduction for the N D 1 Target . . . . . . . . . . . . . . 26.3.3 Alternative Projection and Projection-Like Operators . 26.4 Width, Shift, and Shape Parameter . . . . . . . . . . . . . . . . . . 26.4.1 Width and Shift . . . . . . . . . . . . . . . . . . . . . . . . 26.4.2 Shape Parameter . . . . . . . . . . . . . . . . . . . . . . . . 26.4.3 Relation to Breit–Wigner Parameters . . . . . . . . . . . 26.5 Other Calculational Methods . . . . . . . . . . . . . . . . . . . . . 26.5.1 Complex Rotation Method . . . . . . . . . . . . . . . . . . 26.5.2 Pseudopotential Method . . . . . . . . . . . . . . . . . . . 26.5.3 Exact Bound States in Continuum . . . . . . . . . . . . . 26.6 Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Green’s Functions of Field Theory . . Gordon Feldman and Thomas Fulton 27.1 Introduction . . . . . . . . . . . . . 27.2 The Two-Point Green’s Function 27.3 The Four-Point Green’s Function 27.4 Radiative Transitions . . . . . . . 27.5 Radiative Corrections . . . . . . . References . . . . . . . . . . . . . . . . . .
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390 390 390 391 391 392 392 392 393
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Quantum Electrodynamics . . . . . . . . . . . . . Jonathan R. Sapirstein 28.1 Introduction . . . . . . . . . . . . . . . . . . . 28.2 Basic QED Formalism . . . . . . . . . . . . 28.3 Perturbation Theory with Green Functions 28.4 Two-Particle Bound States . . . . . . . . . . 28.5 Many-Electron Bound States . . . . . . . . 28.6 Recoil Corrections at High Z . . . . . . . . 28.7 Concluding Remarks . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
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Tests of Fundamental Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eite Tiesinga and Peter J. Mohr 29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2 Consistency of Fundamental Physics . . . . . . . . . . . . . . . . . . . . . 29.3 Topics in This Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.4 Electron g-Factor Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . 29.5 Atom Recoil Experiments and Mass Spectrometry . . . . . . . . . . . . 29.6 Mass-Ratio Measurements Using the g-Factor of Hydrogen-Like Ions 29.6.1 Theory for the g-Factor of Hydrogen-Like Ions . . . . . . . . . 29.7 Hydrogen Atom Energy levels . . . . . . . . . . . . . . . . . . . . . . . . . 29.7.1 Theory for the Hydrogen Energy Levels . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Atomic Clocks and Constraints on Variations of Fundamental Constants . . Savely G. Karshenboim, Victor Flambaum, and Ekkehard Peik 30.1 Atomic Clocks and Frequency Standards . . . . . . . . . . . . . . . . . . . . 30.1.1 Caesium Atomic Fountain . . . . . . . . . . . . . . . . . . . . . . . . 30.1.2 Trapped Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.1.3 Laser-Cooled Neutral Atoms . . . . . . . . . . . . . . . . . . . . . . 30.1.4 Two-Photon Transitions and Doppler-Free Spectroscopy . . . . . 30.1.5 Optical Frequency Measurements . . . . . . . . . . . . . . . . . . . 30.1.6 Limitations on Frequency Variations . . . . . . . . . . . . . . . . . 30.2 Atomic Spectra and Their Dependence on the Fundamental Constants . . 30.2.1 The Spectrum of Hydrogen and Nonrelativistic Atoms . . . . . . 30.2.2 Hyperfine Structure and the Schmidt Model . . . . . . . . . . . . . 30.2.3 Atomic Spectra: Relativistic Corrections . . . . . . . . . . . . . . . 30.3 Laboratory Constraints on Temporal Variations of Fundamental Constants 30.3.1 Constraints from Absolute and Relative Optical Measurements . 30.3.2 Constraints from Microwave Clocks . . . . . . . . . . . . . . . . . . 30.3.3 A Model-Independent Constraint on the Variation of the Electron-to-Proton Mass Ratio from Molecular Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3.4 Other Model-Dependent Constraints . . . . . . . . . . . . . . . . . 30.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Searches for New Particles Including Dark Matter with Atomic, Molecular, and Optical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Victor Flambaum and Yevgeny Stadnik 31.1 Nongravitational Interactions of Spinless Bosons . . . . . . . . . . . . . . . 31.2 New Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Macroscopic-Scale Experiments . . . . . . . . . . . . . . . . . . . . 31.2.2 Atomic-Scale Experiments . . . . . . . . . . . . . . . . . . . . . . . 31.3 Laboratory Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Astrophysical Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Cosmological Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5.1 Haloscope Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 31.5.2 Spin-Precession Experiments . . . . . . . . . . . . . . . . . . . . . . 31.5.3 Time-Varying Physical Constants . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Searches for New Physics . . . . . . . . . . . . . . . . . . . . . Marianna S. Safronova 32.1 Parity Nonconserving Effects in Atoms . . . . . . . . . 32.1.1 Nuclear-Spin-Independent PNC Effects . . . . 32.1.2 Nuclear-Spin-Dependent PNC Effects . . . . 32.1.3 PNC in Cesium and Implications for Particle and Hadronic Physics . . . . . . . . . . . . . . . 32.1.4 Current PNC Experiments . . . . . . . . . . . . 32.2 Electric Dipole Moments and Related Phenomena . . 32.2.1 Experiments with Paramagnetic Systems . . . 32.2.2 Experiments with Diamagnetic Systems . . . 32.2.3 Impact on Particle Physics . . . . . . . . . . . . 32.3 Tests of the CPT Symmetry . . . . . . . . . . . . . . . . 32.4 Lorentz Symmetry Tests . . . . . . . . . . . . . . . . . . 32.4.1 Electron–Photon Sector of SME . . . . . . . . 32.4.2 Proton and Neutron Sectors of SME . . . . . . 32.5 AMO Tests of General Relativity . . . . . . . . . . . . . 32.5.1 Tests of the Einstein Equivalence Principle . 32.5.2 Other AMO Tests of Gravity . . . . . . . . . . 32.5.3 Detection of Gravitational Waves . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Part C Molecules 33
Molecular Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . David R. Yarkony 33.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1.1 Nonadiabatic Ansatz: Born–Oppenheimer Approximation . . 33.1.2 Born–Oppenheimer Potential Energy Surfaces and Their Topology . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1.3 Classification of Interstate Couplings: Adiabatic and Diabatic Bases . . . . . . . . . . . . . . . . . . . . 33.1.4 Surfaces of Intersection of Potential Energy Surfaces . . . . . 33.2 Characterization of Potential Energy Surfaces . . . . . . . . . . . . . . . 33.2.1 The Self-Consistent Field (SCF) Method . . . . . . . . . . . . . 33.2.2 Electron Correlation: Wave Function Based Methods . . . . . 33.2.3 Electron Correlation: Density Functional Theory . . . . . . . . 33.2.4 Weakly Interacting Systems . . . . . . . . . . . . . . . . . . . . . 33.3 Intersurface Interactions: Perturbations . . . . . . . . . . . . . . . . . . . 33.3.1 Derivative Couplings . . . . . . . . . . . . . . . . . . . . . . . . . 33.3.2 Breit–Pauli Interactions . . . . . . . . . . . . . . . . . . . . . . . 33.3.3 Surfaces of Intersection . . . . . . . . . . . . . . . . . . . . . . . 33.4 Nuclear Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . 33.4.2 Rotational-Vibrational Structure . . . . . . . . . . . . . . . . . . 33.4.3 Coupling of Electronic and Rotational Angular Momentum in Weakly Interacting . . . . . . . . . . . . . . . . . . . . . . . . . 33.4.4 Reaction Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.5 Reaction Mechanisms: A Spin-Forbidden Chemical Reaction . . . . . 33.6 Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Molecular Symmetry and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . William G. Harter and Tyle C. Reimer 34.1 Dynamics and Spectra of Molecular Rotors . . . . . . . . . . . . . . . . . . 34.1.1 Rigid Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.1.2 Molecular States Inside and Out . . . . . . . . . . . . . . . . . . . . 34.1.3 Rigid Asymmetric Rotor Eigensolutions and Dynamics . . . . . . 34.2 Rotational Energy Surfaces and Semiclassical Rotational Dynamics . . . 34.3 Symmetry of Molecular Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . 34.3.1 Asymmetric Rotor Symmetry Analysis . . . . . . . . . . . . . . . . 34.4 Tetrahedral-Octahedral Rotational Dynamicsand Spectra . . . . . . . . . . 34.4.1 Semirigid Octahedral Rotors and Centrifugal Tensor Hamiltonians . . . . . . . . . . . . . . . . . 34.4.2 Octahedral and Tetrahedral Rotational Energy Surfaces . . . . . . 34.4.3 Octahedral and Tetrahedral Rotational Fine Structure . . . . . . . 34.4.4 Octahedral Superfine Structure . . . . . . . . . . . . . . . . . . . . . 34.5 High-Resolution Rovibrational Structure . . . . . . . . . . . . . . . . . . . . 34.5.1 Tetrahedral Nuclear Hyperfine Structure . . . . . . . . . . . . . . . 34.5.2 Superhyperfine Structure and Spontaneous Symmetry Breaking . 34.5.3 Large Spherical Top Molecules: Cubic-Octahedral Symmetry Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.5.4 Icosahedral Spherical Top Molecules: Extreme Spin Symmetry Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.5.5 12 C60 versus 13 C60 : a World of Difference in Spin- 12 Hyperfine Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.6 Composite Rotors and Multiple RES . . . . . . . . . . . . . . . . . . . . . . . 34.6.1 3-D-Rotor and 2-D-Oscillator Analogy . . . . . . . . . . . . . . . . 34.6.2 Gyro-Rotors and 2-D-Local Mode Analogy . . . . . . . . . . . . . 34.6.3 Multiple Gyro-Rotor RES and Rotational Energy Eigensurfaces (REES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.6.4 Multi-Quantum CF4 Rovibrational Polyads and REES Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Radiative Transition Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . David L. Huestis 35.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.1.1 Intensity Versus Line-Position Spectroscopy . . . . . . . . . . . 35.2 Molecular Wave Functions in the Rotating Frame . . . . . . . . . . . . . 35.2.1 Symmetries of the Exact Wave Function . . . . . . . . . . . . . 35.2.2 Rotation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.2.3 Transformation of Ordinary Objects into the Rotating Frame 35.3 The Energy–Intensity Model . . . . . . . . . . . . . . . . . . . . . . . . . . 35.3.1 States, Levels, and Components . . . . . . . . . . . . . . . . . . 35.3.2 The Basis Set and Matrix Hamiltonian . . . . . . . . . . . . . . 35.3.3 Fitting Experimental Energies . . . . . . . . . . . . . . . . . . . 35.3.4 The Transition Moment Matrix . . . . . . . . . . . . . . . . . . . 35.3.5 Fitting Experimental Intensities . . . . . . . . . . . . . . . . . . 35.4 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.4.1 Symmetry Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.4.2 Rotational Branches and Parity . . . . . . . . . . . . . . . . . . . 35.4.3 Nuclear Spin, Spatial Symmetry, and Statistics . . . . . . . . . 35.4.4 Electron Orbital and Spin Angular Momenta . . . . . . . . . .
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Absorption Cross Sections and Radiative Lifetimes . 35.5.1 Radiation Relations . . . . . . . . . . . . . . . 35.5.2 Transition Moments . . . . . . . . . . . . . . . 35.6 Vibrational Band Strengths . . . . . . . . . . . . . . . . 35.6.1 Franck–Condon Factors . . . . . . . . . . . . 35.6.2 Vibrational Transitions . . . . . . . . . . . . . 35.7 Rotational Branch Strengths . . . . . . . . . . . . . . . 35.7.1 Branch Structure and Transition Type . . . . 35.7.2 Hönl–London Factors . . . . . . . . . . . . . . 35.7.3 Sum Rules . . . . . . . . . . . . . . . . . . . . 35.7.4 Hund’s Cases . . . . . . . . . . . . . . . . . . . 35.7.5 Symmetric Tops . . . . . . . . . . . . . . . . . 35.7.6 Asymmetric Tops . . . . . . . . . . . . . . . . 35.8 Forbidden Transitions . . . . . . . . . . . . . . . . . . . 35.8.1 Spin-Changing Transitions . . . . . . . . . . 35.8.2 Orbitally-Forbidden Transitions . . . . . . . 35.9 Recent Developments . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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Molecular Photodissociation . . . . . . . . . . . . . . . . . . . . Abigail J. Dobbyn, David H. Mordaunt, and Reinhard Schinke 36.1 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.1.1 Scalar Properties . . . . . . . . . . . . . . . . . . . 36.1.2 Vector Correlations . . . . . . . . . . . . . . . . . 36.2 Experimental Techniques . . . . . . . . . . . . . . . . . . . 36.3 Theoretical Techniques . . . . . . . . . . . . . . . . . . . . 36.4 Concepts in Dissociation . . . . . . . . . . . . . . . . . . . 36.4.1 Direct Dissociation . . . . . . . . . . . . . . . . . 36.4.2 Vibrational Predissociation . . . . . . . . . . . . 36.4.3 Electronic Predissociation . . . . . . . . . . . . . 36.5 Recent Developments . . . . . . . . . . . . . . . . . . . . . 36.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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545 545 545 546 546 547 547 547 548 548 549 550 550 551 551 551 552 552
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Time Resolved Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . Volker Engel and Patrick Nuernberger 37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.2 The Principle of Time-Resolved Spectroscopy . . . . . . . . . . . . . . . 37.3 Pump-Probe Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.4 Transient Absorption in the Liquid Phase . . . . . . . . . . . . . . . . . . 37.5 Further Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.5.1 Coherent Two-Dimensional Spectroscopy . . . . . . . . . . . . 37.5.2 Ultrafast Dynamics Studied with X-Ray and Electron Pulses 37.5.3 Dynamics and Control . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Nonreactive Scattering . . . . . . . . . . . . . . . . . . . . Alexandre Faure, Francois Lique, and David R. Flower 38.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . 38.2 Quantal Method . . . . . . . . . . . . . . . . . . . . 38.3 Symmetries and Conservation Laws . . . . . . . . 38.4 Coordinate Systems . . . . . . . . . . . . . . . . . . 38.5 Scattering Equations . . . . . . . . . . . . . . . . . .
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557 557 557 558 559 561 561 561 562 562 563 564
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38.6
Matrix Elements . . . . . . . . . . . . 38.6.1 Interaction Potential . . . . . 38.6.2 Centrifugal Potential . . . . 38.7 Semi and Quasi-Classical Methods . 38.8 Example: CO–H2 . . . . . . . . . . . . 38.9 New Directions . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . 39
40
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Gas Phase Reactions . . . . . . . . . . . . . . . . . . . Eric Herbst 39.1 Introduction . . . . . . . . . . . . . . . . . . . . . 39.2 Normal Bimolecular Reactions . . . . . . . . . 39.2.1 Capture Theories . . . . . . . . . . . . 39.2.2 Phase Space Theories . . . . . . . . . . 39.2.3 Short-Range Barriers . . . . . . . . . . 39.2.4 Complexes Followed by Barriers . . . 39.2.5 The Role of Tunneling . . . . . . . . . 39.3 Association Reactions . . . . . . . . . . . . . . . 39.3.1 Radiative Stabilization . . . . . . . . . 39.3.2 Complex Formation and Dissociation 39.3.3 Competition with Exoergic Channels 39.4 Concluding Remarks . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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578 578 579 579 579 580 580
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Gas Phase Ionic Reactions Abstract . . . . . . . . . . . . . . . . Nigel G. Adams 40.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.2 Reaction Energetics . . . . . . . . . . . . . . . . . . . . . . . 40.3 Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . 40.4 Reaction Processes . . . . . . . . . . . . . . . . . . . . . . . . 40.4.1 Binary Ion–Neutral Reactions . . . . . . . . . . . . 40.4.2 Ternary Ion–Molecule Reactions . . . . . . . . . . 40.5 Electron Attachment . . . . . . . . . . . . . . . . . . . . . . . 40.6 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . 40.6.1 Electron–Ion Recombination . . . . . . . . . . . . 40.6.2 Ion–Ion Recombination (Mutual Neutralization) References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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583 585 585 587 588 589 590 591 591 592 593 593 594
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597 598 599 600 600 603 604 604 604 605 606
Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mary L. Mandich 41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Metal Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Geometric Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Electronic and Magnetic Properties . . . . . . . . . . . . . . . . . . 41.2.3 Chemical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.4 Stable Metal Cluster Molecules and Metallocarbohedrenes . . . . 41.3 Carbon Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 Small Carbon Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.3 Giant Carbon Clusters: Tubes, Capsules, Onions, Russian Dolls, Papier Mâché . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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41.4
Ionic Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.1 Geometric Structures . . . . . . . . . . . . . . . . . . . . . 41.4.2 Electronic and Chemical Properties . . . . . . . . . . . . 41.5 Semiconductor Clusters . . . . . . . . . . . . . . . . . . . . . . . . . 41.5.1 Silicon and Germanium Clusters . . . . . . . . . . . . . . 41.5.2 Group III–V and Group II–VI Semiconductor Clusters 41.6 Noble Gas Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6.1 Geometric Structures . . . . . . . . . . . . . . . . . . . . . 41.6.2 Electronic Properties . . . . . . . . . . . . . . . . . . . . . 41.6.3 Doped Noble Gas Clusters . . . . . . . . . . . . . . . . . . 41.6.4 Helium Clusters . . . . . . . . . . . . . . . . . . . . . . . . 41.7 Molecular Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7.1 Geometric Structures and Phase Dynamics . . . . . . . . 41.7.2 Electronic Properties: Charge Solvation . . . . . . . . . 41.8 Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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Infrared Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Henry Buijs 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Historical Evolution of Infrared Spectroscopy Practice . . . . . . . . 42.3 Quantitative Analysis by Infrared Spectroscopy . . . . . . . . . . . . 42.4 Molecular Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.6 The Evolution of Fourier Transform Infrared Spectroscopy (FTIR) 42.7 Laser-Based Infrared Spectroscopy . . . . . . . . . . . . . . . . . . . . 42.8 Intensities of Infrared Radiation . . . . . . . . . . . . . . . . . . . . . . 42.9 Sources for IR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 42.10 Relationship Between Source Spectrometer Sample and Detector . 42.11 Simplified Principle of FTIR Spectroscopy . . . . . . . . . . . . . . . 42.11.1 Interferogram Generation: The Michelson Interferometer . 42.11.2 Description of Wavefront Interference with Time Delay . . 42.11.3 Operation of Spectrum Determination . . . . . . . . . . . . . 42.11.4 Optical Aspects of FTIR Technology . . . . . . . . . . . . . 42.12 The Scanning Michelson Interferometer . . . . . . . . . . . . . . . . . 42.13 Infrared Spectroscopy Application Activity 2020 . . . . . . . . . . . 42.13.1 Analytical Chemistry Laboratories . . . . . . . . . . . . . . . 42.13.2 Biomedical and Pharmaceutical Laboratories . . . . . . . . 42.13.3 Forensic Investigation . . . . . . . . . . . . . . . . . . . . . . 42.13.4 Infrared Spectroscopy in Quality Assurance Laboratories . 42.13.5 Process Monitoring by Infrared Spectroscopy . . . . . . . . 42.13.6 Environmental Monitoring . . . . . . . . . . . . . . . . . . . . 42.13.7 Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . 42.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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615 615 616 616 616 617 618 618 618 619 619 620 620 621 621 622
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Laser Spectroscopy in the Submillimeterand Far-Infrared Regions . . . . . . Kenneth Evenson and John M. Brown 43.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Experimental Techniques Using Coherent SM-FIR Radiation . . . . . . . 43.2.1 Tunable FIR Spectroscopy with CO2 Laser Difference Generation in a MIM Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
626 626 627 627 628 628 629 629 630 630 630 630 631 631 632 633 634 634 635 635 636 636 637 637 638 639 641 641 642 642
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43.2.2 Laser Magnetic Resonance 43.2.3 TuFIR and LMR Detectors 43.3 Submillimeter and FIR Astronomy . 43.4 Upper Atmospheric Studies . . . . . References . . . . . . . . . . . . . . . . . . . . 44
45
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Spectroscopic Techniques: Lasers . . . . . . Paul Engelking 44.1 Laser Basics . . . . . . . . . . . . . . . . 44.1.1 Stimulated Emission . . . . . 44.1.2 Laser Configurations . . . . . 44.1.3 Gain . . . . . . . . . . . . . . . 44.1.4 Laser Light . . . . . . . . . . . 44.2 Laser Designs . . . . . . . . . . . . . . . 44.2.1 Cavities . . . . . . . . . . . . . 44.2.2 Pumping . . . . . . . . . . . . . 44.3 Interaction of Laser Light with Matter 44.3.1 Linear Absorption . . . . . . . 44.3.2 Multiphoton Absorption . . . 44.3.3 Level Shifts . . . . . . . . . . . 44.3.4 Hole Burning . . . . . . . . . . 44.3.5 Nonlinear Optics . . . . . . . . 44.3.6 Raman Scattering . . . . . . . 44.4 Recent Developments . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
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644 645 645 646 646
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649 649 649 650 650 651 651 652 652 652 654 654 654 655 655 656 656
Spectroscopic Techniques: Cavity-Enhanced Methods . . . . . . . . . . . . . . Barbara Paldus and Alexander A. Kachanov 45.1 Limitations of Traditional Absorption Spectrometers . . . . . . . . . . . . . 45.2 Cavity Ring-Down Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 45.2.1 Pulsed Cavity Ring-Down Spectroscopy . . . . . . . . . . . . . . . 45.2.2 Continuous-Wave Cavity Ring-Down Spectroscopy (CW-CRDS) 45.3 Cavity-Enhanced Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 45.3.1 Cavity-Enhanced Transmission Spectroscopy (CETS) . . . . . . . 45.3.2 Locked Cavity-Enhanced Transmission Spectroscopy (L-CETS) 45.4 Extensions to Solids and Liquids . . . . . . . . . . . . . . . . . . . . . . . . . 45.4.1 Evanescent-Wave Cavity Ring-Down Spectroscopy (EW-CRDS) 45.4.2 Cavity Ring-Down of Condensed Media for Analytical Chemistry 45.4.3 Cavity Ring-Down Spectroscopy Using Waveguides . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
657 657 658 659 659 660 661 662 663 663 663 663 664
Spectroscopic Techniques: Ultraviolet . . . . . . . . Glenn Stark, Nelson de Oliveira, and Peter L. Smith 46.1 Light Sources . . . . . . . . . . . . . . . . . . . . 46.1.1 Synchrotron Radiation . . . . . . . . . 46.1.2 Laser-Produced Plasmas . . . . . . . . 46.1.3 Arcs, Sparks, and Discharges . . . . . 46.1.4 Supercontinuum Radiation . . . . . . 46.2 VUV Lasers . . . . . . . . . . . . . . . . . . . . . 46.2.1 Primary Lasers . . . . . . . . . . . . . . 46.2.2 Nonlinear Techniques . . . . . . . . . .
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46.3
Spectrometers . . . . . . . . . . . . . . . . . . . . . . . . 46.3.1 Grating Spectrometers . . . . . . . . . . . . . 46.3.2 Fourier Transform Spectrometers . . . . . . 46.4 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.4.1 Photomultiplier Tubes . . . . . . . . . . . . . 46.4.2 Microchannel Plates . . . . . . . . . . . . . . 46.4.3 Silicon Photodiodes . . . . . . . . . . . . . . . 46.4.4 Charge-Coupled Devices . . . . . . . . . . . . 46.5 Optical Materials . . . . . . . . . . . . . . . . . . . . . . 46.5.1 Windows . . . . . . . . . . . . . . . . . . . . . 46.5.2 Capillaries . . . . . . . . . . . . . . . . . . . . . 46.5.3 Thin Films . . . . . . . . . . . . . . . . . . . . 46.5.4 Coatings . . . . . . . . . . . . . . . . . . . . . . 46.5.5 Interference Filters and Multilayer Coatings 46.5.6 Polarizers . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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673 673 674 675 675 675 675 676 676 676 676 676 677 677 677 677
Part D Scattering Theory 47
Classical, Quantal, and Semiclassical Propagators and Applications to Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 Jan-Michael Rost 47.1 What Is Semiclassics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 47.2 Quantum, Classical, and Semiclassical Propagators . . . . . . . . . . . . . . 686 47.2.1 The Quantum Propagator as a Feynman Path Integral . . . . . . . 686 47.2.2 The Classical Propagator . . . . . . . . . . . . . . . . . . . . . . . . . 687 47.2.3 The Semiclassical Propagator . . . . . . . . . . . . . . . . . . . . . . 687 47.3 Advantages and Disadvantages of Semiclassics . . . . . . . . . . . . . . . . 689 47.4 Applications to Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 689 47.4.1 Central Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 47.4.2 Center-of-Mass to Laboratory Coordinate Conversion . . . . . . . 689 47.5 Quantal Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690 47.5.1 Scattering Amplitude from the Propagator . . . . . . . . . . . . . . 690 47.5.2 Partial Wave Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 690 47.5.3 Phase Shift and Cross Sections . . . . . . . . . . . . . . . . . . . . . 691 47.5.4 Identical Particles: Symmetry Oscillations . . . . . . . . . . . . . . 692 47.5.5 Low-Energy E ! 0 Scattering . . . . . . . . . . . . . . . . . . . . . 693 47.5.6 Nonspherical Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 695 47.5.7 Reactive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 47.6 Classical Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 47.6.1 Elastic Scattering Cross Section . . . . . . . . . . . . . . . . . . . . 696 47.6.2 Deflection Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 47.6.3 Glory and Rainbow Scattering . . . . . . . . . . . . . . . . . . . . . 697 47.6.4 Orbiting and Spiraling Collisions . . . . . . . . . . . . . . . . . . . 698 47.6.5 Deflection Function and Time Delay from Action . . . . . . . . . 698 47.6.6 Approximate Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 698 47.7 Semiclassical Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 698 47.7.1 Semiclassical Amplitudes and Cross Sections . . . . . . . . . . . . 698 47.7.2 Diffraction and Glory Amplitudes . . . . . . . . . . . . . . . . . . . 700 47.7.3 Small-Angle (Diffraction) Scattering . . . . . . . . . . . . . . . . . 701 47.7.4 Small-Angle (Glory) Scattering . . . . . . . . . . . . . . . . . . . . 701
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47.7.5 Oscillations in Elastic Scattering . . . . . . . . . . . . . . . 47.7.6 Scattering Amplitude in Poisson Sum Representation . . 47.7.7 Semiclassical Phase Shifts . . . . . . . . . . . . . . . . . . . 47.7.8 Semiclassical Amplitudes: Integral Representation . . . . 47.8 Coulomb Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . 47.8.1 Quantum Phase Shift, Rutherford and Mott Scattering . . 47.8.2 Classical Coulomb Scattering . . . . . . . . . . . . . . . . . 47.8.3 Semiclassical Coulomb Scattering . . . . . . . . . . . . . . 47.8.4 Born Approximation for Coulomb Scattering . . . . . . . 47.9 Results for Model Potentials . . . . . . . . . . . . . . . . . . . . . . . 47.9.1 Classical Deflection Functions and Cross Sections . . . . 47.9.2 Amplitudes and Cross Sections in Born Approximation . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
49
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702 702 702 704 705 705 706 706 706 707 707 710 711
Orientation and Alignment in Atomic and Molecular Collisions . . . . . . . . Nils Andersen 48.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.2 Collisions Involving Unpolarized Beams . . . . . . . . . . . . . . . . . . . . 48.2.1 The Fully Coherent Case . . . . . . . . . . . . . . . . . . . . . . . . . 48.2.2 The Incoherent Case with Conservation of Atomic Reflection Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.2.3 The Incoherent Case Without Conservation of Atomic Reflection Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.3 Collisions Involving Spin-Polarized Beams . . . . . . . . . . . . . . . . . . . 48.3.1 The Fully Coherent Case . . . . . . . . . . . . . . . . . . . . . . . . . 48.3.2 The Incoherent Case with Conservation of Atomic Reflection Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.3.3 The Incoherent Case Without Conservation of Atomic Reflection Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.4.1 The First Born Approximation . . . . . . . . . . . . . . . . . . . . . 48.5 Further Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.5.1 S ! D Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.5.2 P ! P Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.5.3 Relativistic Effects in S ! P Excitation . . . . . . . . . . . . . . . 48.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
713
Electron–Atom, Electron–Ion, and Electron–Molecule Collisions . . . . . . . Klaus Bartschat, Jonathan Tennyson, and Philip Burke 49.1 Electron–Atom and Electron–Ion Collisions . . . . . . . . . . . . . . . . . . 49.1.1 Low-Energy Elastic Scattering and Excitation . . . . . . . . . . . . 49.1.2 Relativistic Effects for Heavy Atoms and Ions . . . . . . . . . . . 49.1.3 Multichannel Resonance Theory . . . . . . . . . . . . . . . . . . . . 49.1.4 Solution of the Coupled Integro-Differential Equations . . . . . . 49.1.5 Intermediate and High-Energy Elastic Scattering and Excitation 49.1.6 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.2 Electron–Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.2.1 Laboratory Frame Representation . . . . . . . . . . . . . . . . . . . 49.2.2 Molecular Frame Representation . . . . . . . . . . . . . . . . . . . . 49.2.3 Inclusion of the Nuclear Motion . . . . . . . . . . . . . . . . . . . . 49.2.4 Electron Collisions with Polyatomic Molecules . . . . . . . . . . .
713 714 714 717 717 718 718 718 720 721 721 722 722 722 722 722 722 725 725 725 728 729 731 735 737 741 741 742 742 744
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49.3
Electron–Atom Collisions in a Laser Field . . . . 49.3.1 Potential Scattering . . . . . . . . . . . . . 49.3.2 Scattering by Complex Atoms and Ions . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
51
52
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744 744 746 747
Quantum Defect Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chris H. Greene 50.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.2 Conceptual Foundation of QDT . . . . . . . . . . . . . . . . . . . . . . . . 50.2.1 Single-Channel Nonrelativistic Quantum Defect Theory . . . 50.2.2 Multichannel QDT . . . . . . . . . . . . . . . . . . . . . . . . . . 50.2.3 Molecular Ionization Channels Treated by the Rovibrational Frame Transformation . . . . . . . . . . . . . . . . . . . . . . . . 50.2.4 Ultracold Atom–Atom Collisions in MQDT . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 751
. . 756 . . 757 . . 759
Positron Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joshua R. Machacek, Robert P. McEachran, and Allan D. Stauffer 51.1 Scattering Channels . . . . . . . . . . . . . . . . . . . . . . . . 51.1.1 Positronium Formation . . . . . . . . . . . . . . . . . 51.1.2 Annihilation . . . . . . . . . . . . . . . . . . . . . . . 51.2 Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Particular Applications . . . . . . . . . . . . . . . . . . . . . . 51.3.1 Atomic Hydrogen . . . . . . . . . . . . . . . . . . . . 51.3.2 Noble Gases . . . . . . . . . . . . . . . . . . . . . . . 51.3.3 Other Atoms . . . . . . . . . . . . . . . . . . . . . . . 51.3.4 Molecular Hydrogen . . . . . . . . . . . . . . . . . . 51.3.5 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.6 Other Molecules . . . . . . . . . . . . . . . . . . . . . 51.4 Binding of Positrons to Atoms . . . . . . . . . . . . . . . . . 51.5 Positronium Scattering . . . . . . . . . . . . . . . . . . . . . . 51.6 Antihydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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751 751 752 753
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Adiabatic and Diabatic Collision Processes at Low Energies . . . . . . . . . Evgeny Nikitin and Alexander Kandratsenka 52.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 Slow Quasi-Classical Collisions . . . . . . . . . . . . . . . . . . . 52.1.2 Adiabatic and Diabatic Electronic States . . . . . . . . . . . . . . 52.1.3 Nonadiabatic Transitions: The Massey Parameter . . . . . . . . 52.2 Two-State Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 Relation Between Adiabatic and Diabatic Basis Functions . . . 52.2.2 Coupled Equations and Transition Probabilities in the Common Trajectory Approximation . . . . . . . . . . . . . 52.2.3 Selection Rules for Nonadiabatic Coupling . . . . . . . . . . . . 52.3 Single-Passage Transition Probabilities in Common Trajectory Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.1 Transitions Between Noncrossing Adiabatic Potential Energy Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.2 Transitions Between Crossing Adiabatic Potential Curves . . .
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761 762 762 763 765 765 765 766 766 767 767 768 768 768 768 769
. 773 . . . . . .
773 773 774 774 775 775
. 775 . 777 . 778 . 778 . 779
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52.4
53
54
55
Double-Passage Transition Probabilities . . . . . . . . . . . . . . . 52.4.1 Transition Probabilities in the Classically-Allowed and Classically-Forbidden WKB Regimes: Interference and Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4.2 Nonadiabatic Transitions near Turning Points . . . . . . . 52.5 Multiple-Passage Transition Probabilities . . . . . . . . . . . . . . . 52.5.1 Multiple Passage in Atomic Collisions . . . . . . . . . . . 52.5.2 Surface Hopping . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Ion–Atom and Atom–Atom Collisions . . . . . . . . . . . . . . . Tom Kirchner, A. Lewis Ford, and John F. Reading 53.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 General Considerations and Formulation of the Problem 53.3 Approximate Versus Full Many-Electron Treatments . . . 53.4 Calculational Techniques . . . . . . . . . . . . . . . . . . . . 53.4.1 Coupled-Channel Methods . . . . . . . . . . . . . 53.4.2 Numerical Grid Methods . . . . . . . . . . . . . . . 53.5 Description of the Ionization Continuum . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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780 781 782 782 782 783
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Ultracold Rydberg Atom–Atom Interaction . . . . . . . . . . . . . . . . . . . Hossein Sadeghpour 54.1 Zero/Short-Range Neutral Collisions . . . . . . . . . . . . . . . . . . . . 54.2 Low-Energy Phase Shift and Zero-Energy Scattering Length . . . . . . 54.3 Ultralong-Range Rydberg Molecules: Fermi’s Idea Redux . . . . . . . 54.4 Fermi Extended: Do Elastic Collisions Result in Inelastic Chemical Reactions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.5 Ion-Pair Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.6 Few-Body Short-Range Scattering . . . . . . . . . . . . . . . . . . . . . . 54.6.1 Additivity of Binary Interactions . . . . . . . . . . . . . . . . . . 54.6.2 Three-Body Effective Interactions . . . . . . . . . . . . . . . . . 54.7 Collective Quantum Many-Body Effects . . . . . . . . . . . . . . . . . . 54.7.1 Bose and Fermi Polarons . . . . . . . . . . . . . . . . . . . . . . . 54.7.2 Linear Response Absorption . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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785 785 787 789 789 790 791 793
. . 795 . . 796 . . 797 . . 798 . . . . . . . . .
Ion–Atom Charge Transfer Reactions at Low Energies . . . . . . . . . . . . . Bernard Zygelman, Phillip C. Stancil, Muriel Gargaud, and Ronald McCarroll 55.1 Classical and Semiclassical Treatments . . . . . . . . . . . . . . . . . . . . 55.1.1 Over the Barrier Models . . . . . . . . . . . . . . . . . . . . . . . . 55.1.2 Langevin Orbiting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.1.3 Landau–Zener–Rosen Theories . . . . . . . . . . . . . . . . . . . . 55.2 The Molecular Orbital Approach . . . . . . . . . . . . . . . . . . . . . . . . 55.2.1 A Gauge Theoretic Framework for the PSS Equations . . . . . . 55.3 Cold and Ultracold Charge Exchange and Association . . . . . . . . . . . 55.4 New Developments and Future Prospects . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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798 799 800 800 800 800 800 801 802
. 805 . . . . . . . . .
806 806 806 807 808 809 810 811 811
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Continuum Distorted Wave and Wannier Methods . . . . . . . . . . . . . . . Roberto D. Rivarola, Omar A. Fojón, Marcelo Ciappina, and Derrick Crothers 56.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.2 Continuum Distorted Wave Method . . . . . . . . . . . . . . . . . . . . . . 56.2.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.2.2 Relativistic Continuum Distorted Waves . . . . . . . . . . . . . . 56.2.3 Variational CDW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.2.4 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.2.5 Ionization of Molecular Targets . . . . . . . . . . . . . . . . . . . . 56.3 Wannier Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.3.1 The Wannier Threshold Law . . . . . . . . . . . . . . . . . . . . . 56.3.2 Peterkop’s Semiclassical Theory . . . . . . . . . . . . . . . . . . . 56.3.3 The Quantal Semiclassical Approximation . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 813 . . . . . . . . . . . .
Basic Atomic Processes in High-Energy Ion–Atom Collisions . . . Alexander Voitkiv 57.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57.2 Atomic Ionization and Projectile-Electron Loss . . . . . . . . . 57.2.1 Atomic Ionization in Collisions with a Bare Nucleus 57.2.2 Projectile-Electron Loss . . . . . . . . . . . . . . . . . . 57.3 Electron Transfer Processes . . . . . . . . . . . . . . . . . . . . . 57.3.1 Nonradiative Electron Capture . . . . . . . . . . . . . . 57.3.2 Radiative Electron Capture . . . . . . . . . . . . . . . . 57.4 Electron–Positron Pair Production . . . . . . . . . . . . . . . . . 57.4.1 Free Pair Production . . . . . . . . . . . . . . . . . . . . 57.4.2 Bound–Free Pair Production . . . . . . . . . . . . . . . 57.4.3 Bound–Bound Pair Production . . . . . . . . . . . . . . 57.4.4 More Advanced Theoretical Methods . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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829 830 832 834 836 836 839 840 841 842 843 844 844
Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes . . . Edmund J. Mansky II and M. Raymond Flannery 58.1 Recombination Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.1.1 Electron–Ion Recombination . . . . . . . . . . . . . . . . . . . . . . 58.1.2 Positive–Ion Negative–Ion Recombination . . . . . . . . . . . . . . 58.1.3 Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.1.4 Neutral–Neutral Recombination . . . . . . . . . . . . . . . . . . . . 58.1.5 N-Body Recombination . . . . . . . . . . . . . . . . . . . . . . . . . 58.2 Collisional-Radiative Recombination . . . . . . . . . . . . . . . . . . . . . . 58.2.1 Saha and Boltzmann Distributions . . . . . . . . . . . . . . . . . . . 58.2.2 Quasi-Steady State Distributions . . . . . . . . . . . . . . . . . . . . 58.2.3 Ionization and Recombination Coefficients . . . . . . . . . . . . . 58.2.4 Working Rate Formulae . . . . . . . . . . . . . . . . . . . . . . . . . 58.2.5 Computer Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.3 Macroscopic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.3.1 Resonant Capture-Stabilization Model: Dissociative and Dielectronic Recombination . . . . . . . . . . . . 58.3.2 Reactive Sphere Model: Three-Body Electron–Ion and Ion–Ion Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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813 814 814 816 816 817 819 822 822 822 823 827
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846 846 846 846 847 848 849 849 850 850 850 850 850 850 851
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58.3.3 Working Formulae for Three-Body Collisional Recombination at Low Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.3.4 Recombination Influenced by Diffusional Drift at High Gas Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.4 Zero-Range Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.5 Hyperspherical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.5.1 Three-Body Recombination Rate (Identical Particles) . . . . . . . 58.5.2 N-Body Recombination Rate (Identical Particles) . . . . . . . . . 58.6 Field-Assisted Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.7 Dissociative Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.7.1 Curve-Crossing Mechanisms . . . . . . . . . . . . . . . . . . . . . . 58.7.2 Quantal Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . 58.7.3 Noncrossing Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 58.8 Mutual Neutralization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.8.1 Landau–Zener Probability for Single Crossing at RX . . . . . . . . 58.8.2 Cross Section and Rate Coefficient for Mutual Neutralization . . 58.9 One-Way Microscopic Equilibrium Current, Flux, and Pair Distributions 58.10 Microscopic Methods for Termolecular Ion–Ion Recombination . . . . . . 58.10.1 Time-Dependent Method: Low Gas Density . . . . . . . . . . . . . 58.10.2 Time-Independent Methods: Low Gas Density . . . . . . . . . . . 58.10.3 Recombination at Higher Gas Densities . . . . . . . . . . . . . . . 58.10.4 Master Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.10.5 Recombination Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.11 Radiative Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.11.1 Detailed Balance and Recombination-Ionization Cross Sections 58.11.2 Kramers Cross Sections, Rates, Electron Energy-Loss Rates, and Radiated Power for Hydrogenic Systems . . . . . . . . . . . . . . . 58.11.3 Basic Formulae for Quantal Cross Sections . . . . . . . . . . . . . 58.11.4 Bound–Free Oscillator Strengths . . . . . . . . . . . . . . . . . . . . 58.11.5 Radiative Recombination Rate . . . . . . . . . . . . . . . . . . . . . 58.11.6 Gaunt Factor, Cross Sections, and Rates for Hydrogenic Systems 58.11.7 Exact Universal Rate Scaling Law and Results for Hydrogenic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.12 Useful Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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Dielectronic Recombination . . . . . . . . . . . . . . . . . . Michael Pindzola, Nigel Badnell, and Donald Griffin 59.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 59.2 Theoretical Formulation . . . . . . . . . . . . . . . . . 59.3 Comparisons with Experiment . . . . . . . . . . . . . 59.3.1 Low-Z Ions . . . . . . . . . . . . . . . . . . . . 59.3.2 High-Z Ions and Relativistic Effects . . . . . 59.4 Radiative-Dielectronic Recombination Interference . 59.5 Dielectronic Recombination in Plasmas . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Rydberg Collision Theories . . . . . . . . . . Edmund J. Mansky II 60.1 Rydberg Collision Processes . . . . . . 60.2 General Properties of Rydberg States 60.2.1 Dipole Moments . . . . . . . . 60.2.2 Interaction Potentials . . . . .
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853 853 855 856 856 857 857 857 857 858 860 860 860 861 861 862 862 863 864 865 865 866 867 867 869 871 871 872 873 873 873
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875 876 877 877 878 878 879 879
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60.2.3 Radial Integrals . . . . . . . . . . . . . . . . . 60.2.4 Line Strengths . . . . . . . . . . . . . . . . . . 60.2.5 Form Factors . . . . . . . . . . . . . . . . . . . 60.2.6 Impact Broadening . . . . . . . . . . . . . . . 60.2.7 Effective Lifetimes and Depopulation Rates 60.3 Correspondence Principles . . . . . . . . . . . . . . . . 60.3.1 Bohr–Sommerfeld Quantization . . . . . . . 60.3.2 Bohr Correspondence Principle . . . . . . . . 60.3.3 Heisenberg Correspondence Principle . . . . 60.3.4 Strong Coupling Correspondence Principle 60.3.5 Equivalent Oscillator Theorem . . . . . . . . 60.4 Distribution Functions . . . . . . . . . . . . . . . . . . . 60.4.1 Spatial Distributions . . . . . . . . . . . . . . 60.4.2 Momentum Distributions . . . . . . . . . . . 60.5 Classical Theory . . . . . . . . . . . . . . . . . . . . . . 60.6 Universality Properties . . . . . . . . . . . . . . . . . . 60.6.1 Equal Mass Case . . . . . . . . . . . . . . . . . 60.6.2 Unequal Mass Case . . . . . . . . . . . . . . . 60.7 Many-Body and Multiparticle Effects . . . . . . . . . 60.7.1 Many-Body Theory . . . . . . . . . . . . . . . 60.7.2 Dilute Bose Gas . . . . . . . . . . . . . . . . . 60.8 Working Formulae for Rydberg Collisions . . . . . . 60.8.1 Inelastic n,`-Changing Transitions . . . . . . 60.8.2 Inelastic n ! n0 Transitions . . . . . . . . . . 60.8.3 Quasi-Elastic `-Mixing Transitions . . . . . 60.8.4 Elastic n` ! n`0 Transitions . . . . . . . . . 60.8.5 Inelastic n` ! n`0 Transitions . . . . . . . . 60.8.6 Fine Structure n`J ! n`J 0 Transitions . . 60.9 Impulse Approximation . . . . . . . . . . . . . . . . . . 60.9.1 Quantal Impulse Approximation . . . . . . . 60.9.2 Classical Impulse Approximation . . . . . . 60.9.3 Semiquantal Impulse Approximation . . . . 60.10 Binary Encounter Approximation . . . . . . . . . . . . 60.10.1 Differential Cross Sections . . . . . . . . . . 60.10.2 Integral Cross Sections . . . . . . . . . . . . . 60.10.3 Classical Ionization Cross Section . . . . . . 60.10.4 Classical Charge Transfer Cross Section . . 60.11 Born Approximation . . . . . . . . . . . . . . . . . . . . 60.11.1 Form Factors . . . . . . . . . . . . . . . . . . . 60.11.2 Hydrogenic Form Factors . . . . . . . . . . . 60.11.3 Excitation Cross Sections . . . . . . . . . . . 60.11.4 Ionization Cross Sections . . . . . . . . . . . 60.11.5 Capture Cross Sections . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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Mass Transfer at High Energies: Thomas Peak . . . . . . . . . Jack C. Straton and James H. McGuire 61.1 The Classical Thomas Process . . . . . . . . . . . . . . . . . 61.2 Quantum Description . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Uncertainty Effects . . . . . . . . . . . . . . . . . . 61.2.2 Conservation of Overall Energy and Momentum 61.2.3 Conservation of Intermediate Energy . . . . . . . 61.2.4 Example: Proton–Helium Scattering . . . . . . . .
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885 885 886 887 887 887 887 887 888 888 888 888 888 889 889 890 891 891 891 891 891 892 892 893 894 894 894 895 896 896 900 901 903 903 904 906 906 907 907 907 908 909 909 910
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913 914 914 914 914 915
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61.3 Off-Energy-Shell Effects . . . . . . . . . 61.4 Dispersion Relations . . . . . . . . . . . . 61.5 Destructive Interference of Amplitudes 61.6 Recent Developments . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . 62
63
Part E 64
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916 916 916 917 917
Classical Trajectory and Monte Carlo Techniques . . . . . . . . . . . . . . . . . Marcelo Ciappina, Raul O. Barrachina, Francisco Navarrete, and Ronald E. Olson 62.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Hydrogenic Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Nonhydrogenic One-Electron Models . . . . . . . . . . . . . . . . . 62.1.3 Multiply Charged Projectiles and Many-Electron Targets . . . . . 62.2 Region of Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.1 Hydrogenic Atom Targets . . . . . . . . . . . . . . . . . . . . . . . . 62.3.2 Pseudo-One-Electron Targets . . . . . . . . . . . . . . . . . . . . . . 62.3.3 State-Selective Electron Capture . . . . . . . . . . . . . . . . . . . . 62.3.4 Exotic Projectiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.5 Heavy-Particle Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 62.3.6 Ion-Molecule Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.7 Strong Laser Field Ionization . . . . . . . . . . . . . . . . . . . . . . 62.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
919 919 919 920 920 921 921 921 922 922 923 923 923 924 925 925
Collisional Broadening of Spectral Lines . . . . . . . . . . . . Gillian Peach 63.1 Impact Approximation . . . . . . . . . . . . . . . . . . . . 63.2 Isolated Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.1 Semiclassical Theory . . . . . . . . . . . . . . . . 63.2.2 Simple Formulae . . . . . . . . . . . . . . . . . . . 63.2.3 Perturbation Theory . . . . . . . . . . . . . . . . . 63.2.4 Broadening by Charged Particles . . . . . . . . . 63.2.5 Empirical Formulae . . . . . . . . . . . . . . . . . 63.3 Overlapping Lines . . . . . . . . . . . . . . . . . . . . . . . 63.3.1 Transitions in Hydrogen and Hydrogenic Ions . 63.3.2 Infrared and Radio Lines . . . . . . . . . . . . . . 63.4 Quantum-Mechanical Theory . . . . . . . . . . . . . . . . 63.4.1 Impact Approximation . . . . . . . . . . . . . . . 63.4.2 Broadening by Electrons . . . . . . . . . . . . . . 63.4.3 Broadening by Atoms . . . . . . . . . . . . . . . 63.5 One-Perturber Approximation . . . . . . . . . . . . . . . . 63.5.1 General Approach and Utility . . . . . . . . . . . 63.5.2 Broadening by Electrons . . . . . . . . . . . . . . 63.5.3 Broadening by Atoms and Molecules . . . . . . 63.6 Unified Theories and Conclusions . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
928 928 928 929 930 931 931 932 932 933 934 934 935 936 936 936 937 938 939 939
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Scattering Experiment Photodetachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943 David Pegg and Dag Hanstorp 64.1 Negative Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943 64.2 Photodetachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944
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64.2.1 Photodetachment Cross Sections . . . . . . . 64.2.2 Threshold Behavior . . . . . . . . . . . . . . . 64.2.3 Structure in Continuum . . . . . . . . . . . . . 64.2.4 Photoelectron Angular Distributions . . . . . 64.2.5 Higher-Order Processes . . . . . . . . . . . . 64.3 Experimental Procedures . . . . . . . . . . . . . . . . . 64.3.1 Production of Negative Ions . . . . . . . . . . 64.3.2 Interacting Beams . . . . . . . . . . . . . . . . 64.3.3 Light Sources . . . . . . . . . . . . . . . . . . . 64.3.4 Detection Schemes . . . . . . . . . . . . . . . 64.4 Measuring Properties of Negative Ions . . . . . . . . 64.4.1 Electron Affinities . . . . . . . . . . . . . . . . 64.4.2 Bound States . . . . . . . . . . . . . . . . . . . 64.4.3 Continuum Processes . . . . . . . . . . . . . . 64.5 Investigation of Fundamental Processes . . . . . . . . 64.5.1 Threshold Studies . . . . . . . . . . . . . . . . 64.5.2 Photodetachment Using Short Laser Pulses 64.6 Observations and Applications of Negative Ions . . . 64.6.1 Natural Occurrence of Negative Ions . . . . 64.6.2 Applications of Negative Ions . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Photon–Atom Interactions: Low Energy . . Denise Caldwell and Manfred Krause 65.1 Theoretical Concepts . . . . . . . . . . . 65.1.1 Differential Analysis . . . . . . 65.1.2 Electron Correlation Effects . . 65.2 Experimental Methods . . . . . . . . . . 65.2.1 Synchrotron Radiation Source 65.2.2 Photoelectron Spectrometry . . 65.2.3 Resolution and Natural Width . 65.3 Additional Considerations . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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944 945 946 946 946 947 947 947 948 948 948 948 949 949 950 950 950 950 950 951 951
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Photon–Atom Interactions: Intermediate Energies Michael W. J. Bromley 66.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . 66.2 Scattering Cross Sections . . . . . . . . . . . . . . 66.2.1 Applications of Atomic Cross Sections 66.3 Experimental Progress . . . . . . . . . . . . . . . 66.3.1 Inelastic Processes . . . . . . . . . . . . . 66.4 Theory, Computation, and Data . . . . . . . . . . 66.4.1 Databases . . . . . . . . . . . . . . . . . . 66.4.2 Bulk Modeling . . . . . . . . . . . . . . . 66.5 Future Directions . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Electron–Atom and Electron–Molecule Collisions Isik Kanik, William McConkey, and Sandor Trajmar 67.1 Basic Concepts . . . . . . . . . . . . . . . . . . . 67.1.1 Electron Impact Processes . . . . . . . 67.1.2 Definition of Cross Sections . . . . . . 67.1.3 Scattering Measurements . . . . . . .
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955 956 958 960 961 961 963 964 965
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967 968 970 971 972 974 974 974 975 975
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67.2
Collision Processes . . . . . . . . . . . . . . . . 67.2.1 Total Scattering Cross Sections . . . . 67.2.2 Elastic Scattering Cross Sections . . . 67.2.3 Momentum Transfer Cross Sections . 67.2.4 Excitation Cross Sections . . . . . . . 67.2.5 Dissociation Cross Sections . . . . . . 67.2.6 Ionization Cross Sections . . . . . . . 67.3 Coincidence and Superelastic Measurements . 67.4 Experiments with Polarized Electrons . . . . . 67.5 Electron Collisions with Excited Species . . . 67.6 Electron Collisions in Traps . . . . . . . . . . . 67.7 Current Applications . . . . . . . . . . . . . . . 67.8 Emerging Applications . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . 68
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Ion–Atom Scattering Experiments: Low Energy . . . . . . . . . . . Charles Havener, Ruitian Zhang, and Ronald Phaneuf 68.1 Low-Energy Ion–Atom Collision Processes . . . . . . . . . . . 68.2 Experimental Methods for Total Cross Section Measurements 68.2.1 Gas-Target Beam Attenuation Method . . . . . . . . . 68.2.2 Gas-Target Product Growth Method . . . . . . . . . . . 68.2.3 Crossed Ion and Thermal Beams Method . . . . . . . . 68.2.4 Fast Merged Beams Method . . . . . . . . . . . . . . . . 68.2.5 Trapped Ion Method . . . . . . . . . . . . . . . . . . . . 68.2.6 Swarm Method . . . . . . . . . . . . . . . . . . . . . . . . 68.3 Methods for State-Selective Measurements . . . . . . . . . . . . 68.3.1 Photon Emission Spectroscopy . . . . . . . . . . . . . . 68.3.2 Translational Energy Spectroscopy . . . . . . . . . . . 68.3.3 Electron Emission Spectroscopy . . . . . . . . . . . . . 68.3.4 Recoil-Ion Momentum Spectroscopy . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Ion–Atom Collisions – High Energy . . . . . . . . . . . . . . . Michael Schulz and Lew Cocke 69.1 Basic One-Electron Processes . . . . . . . . . . . . . . . . 69.1.1 Perturbative Processes . . . . . . . . . . . . . . . 69.1.2 Nonperturbative Processes . . . . . . . . . . . . . 69.2 Multielectron Processes . . . . . . . . . . . . . . . . . . . . 69.3 Electron Spectra in Ion–Atom Collisions . . . . . . . . . 69.3.1 General Characteristics . . . . . . . . . . . . . . . 69.3.2 High-Resolution Measurements . . . . . . . . . 69.4 Quasi-Free Electron Processes in Ion–Atom Collisions 69.4.1 Radiative Electron Capture . . . . . . . . . . . . 69.4.2 Resonant Transfer and Excitation . . . . . . . . 69.4.3 Excitation and Ionization . . . . . . . . . . . . . 69.5 Some Exotic Processes . . . . . . . . . . . . . . . . . . . . 69.5.1 Molecular Orbital X-Rays . . . . . . . . . . . . . 69.5.2 Positron Production from Atomic Processes . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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984 984 984 985 985 986 986 987 989 990 990 990 991 991
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995 997 997 997 998 998 999 999 999 999 999 1000 1000 1000
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1003 1003 1009 1009 1012 1012 1013 1013 1014 1014 1014 1014 1014 1015 1015
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Reactive Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . Hongwei Li, Arthur G. Suits, and Yuan T. Lee 70.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 70.2 Experimental Methods . . . . . . . . . . . . . . . . . . . . 70.2.1 Molecular Beam Sources . . . . . . . . . . . . . 70.2.2 Reagent Preparation . . . . . . . . . . . . . . . . . 70.2.3 Detection of Neutral Products . . . . . . . . . . . 70.2.4 A Typical Signal Calculation . . . . . . . . . . . 70.3 Experimental Configurations . . . . . . . . . . . . . . . . 70.3.1 Crossed-Beam Rotatable Detector . . . . . . . . 70.3.2 Laboratory to Center-of-Mass Transformation 70.3.3 Product Imaging . . . . . . . . . . . . . . . . . . . 70.4 Elastic and Inelastic Scattering . . . . . . . . . . . . . . . 70.4.1 The Differential Cross Section . . . . . . . . . . 70.4.2 Rotationally Inelastic Scattering . . . . . . . . . 70.4.3 Vibrationally Inelastic Scattering . . . . . . . . . 70.4.4 Electronically Inelastic Scattering . . . . . . . . 70.5 Reactive Scattering . . . . . . . . . . . . . . . . . . . . . . 70.5.1 Harpoon and Stripping Reactions . . . . . . . . 70.5.2 Rebound Reactions . . . . . . . . . . . . . . . . . 70.5.3 Long-Lived Complexes . . . . . . . . . . . . . . . 70.6 Recent Developments . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Ion–Molecule Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . James M. Farrar 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Specification of Cross Sections . . . . . . . . . . . . . . . . . . . . . . 71.3 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Reactant Ion Preparation . . . . . . . . . . . . . . . . . . . . . 71.3.2 Reactant Mass Selection . . . . . . . . . . . . . . . . . . . . . 71.3.3 The Collision Region . . . . . . . . . . . . . . . . . . . . . . . 71.3.4 Product Detection . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.5 Imaging Methods in Velocity Space . . . . . . . . . . . . . . 71.4 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Recent Examples of State-Resolved Measurements . . . . . . . . . . 0 ; jVrel / . 71.5.1 Velocity-Angle Differential Cross Sections .Vrel 0 71.5.2 State-Resolved Cross Sections .n jn; Vrel / and Rate Constants k.n0 jn; T / . . . . . . . . . . . . . . . . . 71.6 The Future of the Field . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1019 1020 1020 1020 1021 1023 1024 1024 1025 1026 1028 1028 1028 1029 1029 1029 1029 1030 1030 1031 1032
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1035 1036 1037 1037 1037 1038 1038 1039 1039 1040 1040
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Part F Quantum Optics 72
Light-Matter Interaction . . . . . . . . . . . . . . . . Pierre Meystre 72.1 Multipole Expansion . . . . . . . . . . . . . . . 72.1.1 Electric Dipole (E1) Interaction . . . 72.1.2 Electric Quadrupole (E2) Interaction 72.1.3 Magnetic Dipole (M1) Interaction . .
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72.2
Lorentz Atom . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Complex Notation . . . . . . . . . . . . . . . 72.2.2 Index of Refraction . . . . . . . . . . . . . . 72.2.3 Beer’s Law . . . . . . . . . . . . . . . . . . . 72.2.4 Slowly Varying Envelope Approximation 72.3 Two-Level Atoms . . . . . . . . . . . . . . . . . . . . 72.3.1 Hamiltonian . . . . . . . . . . . . . . . . . . 72.3.2 Rotating Wave Approximation . . . . . . . 72.3.3 Rabi Frequency . . . . . . . . . . . . . . . . 72.3.4 Dressed States . . . . . . . . . . . . . . . . . 72.3.5 Optical Bloch Equations . . . . . . . . . . . 72.4 Relaxation Mechanisms . . . . . . . . . . . . . . . . . 72.4.1 Relaxation Toward Unobserved Levels . . 72.4.2 Relaxation Toward Levels of Interest . . . 72.4.3 Optical Bloch Equations with Decay . . . 72.4.4 Density Matrix Equations . . . . . . . . . . 72.5 Rate Equation Approximation . . . . . . . . . . . . . 72.5.1 Steady State . . . . . . . . . . . . . . . . . . . 72.5.2 Saturation . . . . . . . . . . . . . . . . . . . . 72.5.3 Einstein A and B Coefficients . . . . . . . . 72.6 Light Scattering . . . . . . . . . . . . . . . . . . . . . 72.6.1 Rayleigh Scattering . . . . . . . . . . . . . . 72.6.2 Thomson Scattering . . . . . . . . . . . . . . 72.6.3 Resonant Scattering . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Absortion and Gain Spectra . . . . . . . . . . . . . . . . . Stig Stenholm 73.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Index of Refraction . . . . . . . . . . . . . . . . . . . 73.3 Density Matrix Treatment of the Two-Level Atom 73.4 Line Broadening . . . . . . . . . . . . . . . . . . . . . 73.5 The Rate Equation Limit . . . . . . . . . . . . . . . . 73.6 Two-Level Doppler-Free Spectroscopy . . . . . . . 73.7 Three-Level Spectroscopy . . . . . . . . . . . . . . . 73.8 Special Effects in Three-Level Systems . . . . . . . 73.9 Summary of the Literature . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Laser Principles . . . . . . . . . . . . . . . . . . . . . . Ralf Menzel and Peter W. Milonni 74.1 Gain, Threshold, and Matter–Field Coupling . 74.2 Continuous Wave, Single-Mode Operation . . 74.3 Laser Resonators and Transverse Modes . . . 74.4 Photon Statistics . . . . . . . . . . . . . . . . . . 74.5 Multimode and Pulsed Operation . . . . . . . . 74.6 Instabilities and Chaos . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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Types of Lasers . . . . . . . . . . . . . . . . . . . Richard S. Quimby and Richard C. Powell 75.1 Introduction . . . . . . . . . . . . . . . . . 75.2 Single-Atom Transitions . . . . . . . . . 75.2.1 Neutral Atom Gas Lasers . . . 75.2.2 Ion Lasers . . . . . . . . . . . . . 75.2.3 Metal Vapor Lasers . . . . . . . 75.2.4 Rare-Earth Ion Lasers . . . . . 75.2.5 Transition Metal Ion Lasers . . 75.3 Molecular Transitions . . . . . . . . . . . 75.3.1 Molecular Vibrational Lasers . 75.3.2 Dye Lasers . . . . . . . . . . . . 75.3.3 Excimer Lasers . . . . . . . . . . 75.4 Solid-State Transitions . . . . . . . . . . 75.4.1 Semiconductor Lasers . . . . . 75.4.2 Quantum Cascade Lasers . . . 75.5 Free Electron Lasers . . . . . . . . . . . . 75.6 Nonlinear Optical Processes . . . . . . . 75.6.1 Raman Lasers . . . . . . . . . . 75.6.2 Optical Parametric Oscillators 75.6.3 Frequency Conversion . . . . . 75.6.4 High Harmonic Generation . . References . . . . . . . . . . . . . . . . . . . . . .
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Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . Robert W. Boyd, Alexander L. Gaeta, and Enno Giese 76.1 Nonlinear Susceptibility . . . . . . . . . . . . . . . . . 76.1.1 Tensor Properties . . . . . . . . . . . . . . . . 76.1.2 Nonlinear Refractive Index . . . . . . . . . . 76.1.3 Quantum Mechanical Expression for .n/ . 76.1.4 Hyperpolarizability . . . . . . . . . . . . . . . 76.2 Wave Equation in Nonlinear Optics . . . . . . . . . . 76.2.1 Coupled-Amplitude Equations . . . . . . . . 76.2.2 Phase Matching . . . . . . . . . . . . . . . . . 76.2.3 Manley–Rowe Relations . . . . . . . . . . . . 76.2.4 Pulse Propagation . . . . . . . . . . . . . . . . 76.3 Second-Order Processes . . . . . . . . . . . . . . . . . 76.3.1 Sum-Frequency Generation . . . . . . . . . . 76.3.2 Second Harmonic Generation . . . . . . . . . 76.3.3 Parametric Amplification and Oscillation . . 76.3.4 Difference-Frequency Generation . . . . . . 76.3.5 Two-Mode Squeezing . . . . . . . . . . . . . 76.3.6 Spontaneous Parametric Down-Conversion 76.3.7 Focused Beams . . . . . . . . . . . . . . . . . 76.4 Third-Order Processes . . . . . . . . . . . . . . . . . . . 76.4.1 Third-Harmonic Generation . . . . . . . . . . 76.4.2 Self-Phase and Cross-Phase Modulation . . 76.4.3 Four-Wave Mixing . . . . . . . . . . . . . . . 76.4.4 Self-Focusing and Self-Trapping . . . . . . . 76.4.5 Saturable Absorption . . . . . . . . . . . . . . 76.4.6 Two-Photon Absorption . . . . . . . . . . . . 76.4.7 Nonlinear Ellipse Rotation . . . . . . . . . . .
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76.5
Stimulated Light Scattering . . . . . . . . . . . . . . . . . . . . . 76.5.1 Stimulated Raman Scattering . . . . . . . . . . . . . . . 76.5.2 Stimulated Brillouin Scattering . . . . . . . . . . . . . . 76.6 Other Nonlinear Optical Processes . . . . . . . . . . . . . . . . . 76.6.1 High-Order Harmonic Generation . . . . . . . . . . . . 76.6.2 Electro-Optic Effect . . . . . . . . . . . . . . . . . . . . . 76.6.3 Photorefractive Effect . . . . . . . . . . . . . . . . . . . . 76.7 New Regimes of Nonlinear Optics . . . . . . . . . . . . . . . . . 76.7.1 Ultrafast and Intense-Field Nonlinear Optics . . . . . 76.7.2 Nonlinear Plasmonics and Epsilon-Near-Zero Effects References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Coherent Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joseph H. Eberly and Carlos R. Stroud Jr. 77.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77.2 Origin of Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77.3 State Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77.4 Numerical Estimates of Parameters . . . . . . . . . . . . . . . . . . . . 77.5 Homogeneous Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 77.5.1 Rabi Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 77.5.2 Bloch Vector and Bloch Sphere . . . . . . . . . . . . . . . . . 77.5.3 Pi Pulses and Pulse Area . . . . . . . . . . . . . . . . . . . . . 77.5.4 Adiabatic Following . . . . . . . . . . . . . . . . . . . . . . . 77.6 Inhomogeneous Relaxation . . . . . . . . . . . . . . . . . . . . . . . . 77.6.1 Free Induction Decay . . . . . . . . . . . . . . . . . . . . . . . 77.6.2 Photon Echoes . . . . . . . . . . . . . . . . . . . . . . . . . . . 77.7 Resonant Pulse Propagation . . . . . . . . . . . . . . . . . . . . . . . . 77.7.1 Maxwell–Bloch Equations . . . . . . . . . . . . . . . . . . . . 77.7.2 Index of Refraction and Beer’s Law . . . . . . . . . . . . . . 77.7.3 The Area Theorem and Self-Induced Transparency . . . . 77.8 Multilevel Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 77.8.1 Rydberg Packets and Intrinsic Relaxation . . . . . . . . . . 77.8.2 Multiphoton Resonance and Two-Photon Bloch Equations 77.8.3 Pump-Probe Resonance and Dark States . . . . . . . . . . . 77.8.4 Induced Transparency . . . . . . . . . . . . . . . . . . . . . . 77.9 Disentanglement and “Sudden Death” of Coherent Transients . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Multiphoton and Strong-Field Processes . . . . . . . . . . . . . Marcelo Ciappina, Alexis A. Chacon S., and Maciej Lewenstein 78.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78.2 Weak-Field Multiphoton Processes . . . . . . . . . . . . . . 78.2.1 Perturbation Theory . . . . . . . . . . . . . . . . . . 78.2.2 Resonant-Enhanced Multiphoton Ionization . . . 78.2.3 Multielectron Effects . . . . . . . . . . . . . . . . . 78.2.4 Autoionization . . . . . . . . . . . . . . . . . . . . . 78.2.5 Coherence and Statistics . . . . . . . . . . . . . . . 78.2.6 Effects of Field Fluctuations . . . . . . . . . . . . 78.2.7 Excitation with Multiple Laser Fields . . . . . . . 78.2.8 Waveforms . . . . . . . . . . . . . . . . . . . . . . . 78.3 Strong-Field Multiphoton Processes . . . . . . . . . . . . . 78.3.1 Nonperturbative Multiphoton Ionization . . . . . 78.3.2 Tunneling Ionization . . . . . . . . . . . . . . . . .
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78.3.3 Multiple Ionization . . . . . . . . . . . . . . . . . 78.3.4 Above Threshold Ionization . . . . . . . . . . . . 78.3.5 High-Order Harmonic Generation . . . . . . . . 78.3.6 Stabilization of Atoms in Intense Laser Fields . 78.3.7 Molecules in Intense Laser Fields . . . . . . . . 78.3.8 Microwave Ionization of Rydberg Atoms . . . . 78.4 Strong-Field Calculational Techniques . . . . . . . . . . 78.4.1 Floquet Theory . . . . . . . . . . . . . . . . . . . . 78.4.2 Direct Integration of the TDSE . . . . . . . . . . 78.4.3 Volkov States . . . . . . . . . . . . . . . . . . . . . 78.4.4 Strong-Field Approximations . . . . . . . . . . . 78.4.5 phase-space Averaging Method . . . . . . . . . . 78.5 Atto-Nano Physics . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Cooling and Trapping . . . . . . . . . . . . . . . . . . . . . . . Juha Javanainen 79.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79.2 Control of Atomic Motion by Light . . . . . . . . . . . 79.2.1 General Theory . . . . . . . . . . . . . . . . . . 79.2.2 Two-State Atoms . . . . . . . . . . . . . . . . . 79.2.3 Multistate Atoms . . . . . . . . . . . . . . . . . 79.3 Magnetic Trap for Atoms . . . . . . . . . . . . . . . . . . 79.3.1 Evaporative Cooling . . . . . . . . . . . . . . . 79.4 Trapping and Cooling of Charged Particles . . . . . . . 79.4.1 Paul Trap . . . . . . . . . . . . . . . . . . . . . . 79.4.2 Penning Trap . . . . . . . . . . . . . . . . . . . . 79.4.3 Collective Effects in Ion Clouds . . . . . . . . 79.5 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . 79.5.1 Free Particles . . . . . . . . . . . . . . . . . . . . 79.5.2 Trapped Particles . . . . . . . . . . . . . . . . . 79.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 79.6.1 Cold Molecules . . . . . . . . . . . . . . . . . . 79.6.2 Quantum Systems of Internal and CM States References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Quantum Degenerate Gases . . . . . . . . . . . . . . . . . . . Juha Javanainen 80.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 80.2 Elements of Quantum Field Theory . . . . . . . . . . . 80.2.1 Bosons . . . . . . . . . . . . . . . . . . . . . . . . 80.2.2 Fermions . . . . . . . . . . . . . . . . . . . . . . 80.2.3 Bosons Versus Fermions . . . . . . . . . . . . . 80.3 Basic Properties of Degenerate Gases . . . . . . . . . . 80.3.1 Atoms Are Trapped . . . . . . . . . . . . . . . . 80.3.2 Atom–Atom Interactions . . . . . . . . . . . . . 80.3.3 Model Hamiltonian . . . . . . . . . . . . . . . . 80.3.4 Bosons . . . . . . . . . . . . . . . . . . . . . . . . 80.3.5 Meaning of the Macroscopic Wave Function 80.3.6 Fermions . . . . . . . . . . . . . . . . . . . . . .
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Experimental . . . . . . . . . . . . . . . . . . 80.4.1 Preparing a BEC . . . . . . . . . . . 80.4.2 Preparing a Degenerate Fermi Gas 80.4.3 Monitoring Degenerate Gases . . . 80.5 BEC Superfluid . . . . . . . . . . . . . . . . . 80.5.1 Vortices . . . . . . . . . . . . . . . . 80.5.2 Superfluidity . . . . . . . . . . . . . 80.6 Optical Lattice as Quantum Simulator . . . 80.6.1 Basics of the Optical Lattice . . . 80.6.2 Strongly Correlated Systems . . . 80.6.3 Topological Phases of Matter . . . References . . . . . . . . . . . . . . . . . . . . . . . .
81
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De Broglie Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carsten Henkel and Martin Wilkens 81.1 Wave-Particle Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 The Hamiltonian of de Broglie Optics . . . . . . . . . . . . . . . . 81.2.1 Gravitation and Rotation . . . . . . . . . . . . . . . . . . . 81.2.2 Charged Particles . . . . . . . . . . . . . . . . . . . . . . . 81.2.3 Magnetic Moments . . . . . . . . . . . . . . . . . . . . . . 81.2.4 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Evolution of De Broglie Waves . . . . . . . . . . . . . . . . . . . . 81.3.1 Light Optics Analogy . . . . . . . . . . . . . . . . . . . . . 81.3.2 WKB Approximation . . . . . . . . . . . . . . . . . . . . . 81.3.3 Phase and Group Velocity . . . . . . . . . . . . . . . . . . 81.3.4 Paraxial Approximation . . . . . . . . . . . . . . . . . . . 81.3.5 Raman–Nath Approximation . . . . . . . . . . . . . . . . 81.4 Refraction and Reflection . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Atomic Mirrors . . . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Atomic Cavities . . . . . . . . . . . . . . . . . . . . . . . . 81.4.3 Atomic Lenses . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.4 Atomic Waveguides . . . . . . . . . . . . . . . . . . . . . . 81.5 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5.1 Fraunhofer Diffraction . . . . . . . . . . . . . . . . . . . . 81.5.2 Fresnel Diffraction . . . . . . . . . . . . . . . . . . . . . . 81.5.3 Near-Resonant Kapitza–Dirac Effect . . . . . . . . . . . 81.5.4 Atom Beam Splitters . . . . . . . . . . . . . . . . . . . . . 81.6 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6.1 Interference Phase Shift . . . . . . . . . . . . . . . . . . . 81.6.2 Internal-State Interferometry . . . . . . . . . . . . . . . . 81.6.3 Manipulation of Cavity Fields by Atom Interferometry 81.7 Coherence of Scalar Matter Waves . . . . . . . . . . . . . . . . . . 81.7.1 Atomic Sources . . . . . . . . . . . . . . . . . . . . . . . . 81.7.2 Atom Decoherence . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Quantum Properties of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Da-Wei Wang and Girish S. Agarwal 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Quantization of the Electromagnetic Field . . . . . . . . . . . . . . . . . . 82.3 Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Field Observables: Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Phase-Space Representations of the Light: P, Q, and Wigner Functions
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82.6 Squeezed State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 Detection of Quantum Light by Array Detectors . . . . . . . 82.8 Two-Mode Squeezed States . . . . . . . . . . . . . . . . . . . . 82.9 Quantum Entanglement . . . . . . . . . . . . . . . . . . . . . . . 82.10 Non-Gaussian Nonclassical States . . . . . . . . . . . . . . . . 82.11 Beam Splitter, Interferometer, and Measurement Sensitivity References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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Entangled Atoms and Fields: Cavity QED . . . . . . . . . . . . . . . . . . Qiongyi He, Wei Zhang, Dieter Meschede, and Axel Schenzle 83.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Atoms and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.1 Two-Level Atoms . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.2 Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . 83.2.3 Dipole Coupling of Fields and Atoms . . . . . . . . . . . . . 83.3 Weak Coupling in Cavity QED . . . . . . . . . . . . . . . . . . . . . . 83.3.1 Radiating Atoms in Waveguides . . . . . . . . . . . . . . . . 83.3.2 Trapped Radiating Atoms and Their Mirror Images . . . . 83.3.3 Radiating Atoms in Resonators . . . . . . . . . . . . . . . . . 83.3.4 Radiative Shifts and Forces . . . . . . . . . . . . . . . . . . . 83.3.5 Experiments on Weak Coupling . . . . . . . . . . . . . . . . 83.3.6 Cavity QED and Dielectrics . . . . . . . . . . . . . . . . . . . 83.4 Strong Coupling in Cavity QED . . . . . . . . . . . . . . . . . . . . . 83.4.1 The Jaynes-Cummings Model . . . . . . . . . . . . . . . . . 83.4.2 Fock States, Coherent States, and Thermal States . . . . . . 83.4.3 Vacuum Splitting . . . . . . . . . . . . . . . . . . . . . . . . . 83.4.4 Strong Coupling in Experiments . . . . . . . . . . . . . . . . 83.5 Micromasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5.1 Maser Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5.2 Nonclassical Features of the Field . . . . . . . . . . . . . . . 83.5.3 Trapping States . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5.4 Atom Counting Statistics . . . . . . . . . . . . . . . . . . . . 83.6 Cavity Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6.1 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6.2 Cavity Cooling Experiments . . . . . . . . . . . . . . . . . . 83.7 Cavity QED for Cold Atomic Gases . . . . . . . . . . . . . . . . . . . 83.7.1 Atomic Ensembles in a Cavity . . . . . . . . . . . . . . . . . 83.7.2 Bose–Einstein Condensate in a Cavity . . . . . . . . . . . . 83.7.3 Cavity Optomechanics with Cold Atoms . . . . . . . . . . . 83.8 Applications of Cavity QED . . . . . . . . . . . . . . . . . . . . . . . . 83.8.1 Quantum Nondemolition (QND) Counting of Photons . . 83.8.2 Detecting and Trapping Atoms Through Strong Coupling 83.8.3 Single-Photon Sources . . . . . . . . . . . . . . . . . . . . . . 83.8.4 Generation of Entanglement . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Quantum Optical Tests of the Foundations of Physics . . . . . . . . . . . . . . . 1231 L. Krister Shalm, Aephraim M. Steinberg, Paul G. Kwiat, and Raymond Y. Chiao 84.1 Introduction: The Photon Hypothesis . . . . . . . . . . . . . . . . . . . . . . 1232 84.2 Quantum Properties of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232 84.2.1 Vacuum Fluctuations: Cavity QED . . . . . . . . . . . . . . . . . . 1232 84.2.2 Two-Photon Light Sources . . . . . . . . . . . . . . . . . . . . . . . 1233 84.2.3 Squeezed States of Light . . . . . . . . . . . . . . . . . . . . . . . . . 1233
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Nonclassical Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3.1 Single-Photon and Matter-Wave Interference . . . . . . . . . . . 84.3.2 “Nonlocal” Interference Effects and Energy-Time Uncertainty 84.3.3 Two-Photon Interference . . . . . . . . . . . . . . . . . . . . . . . . 84.4 Complementarity and Coherence . . . . . . . . . . . . . . . . . . . . . . . . 84.4.1 Wave-Particle Duality . . . . . . . . . . . . . . . . . . . . . . . . . 84.4.2 Quantum Eraser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.4.3 Vacuum-Induced Coherence . . . . . . . . . . . . . . . . . . . . . . 84.4.4 Suppression of Spontaneous Parametric Downconversion . . . 84.5 Measurements in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 84.5.1 Quantum (Anti-)Zeno Effect . . . . . . . . . . . . . . . . . . . . . 84.5.2 Quantum Nondemolition . . . . . . . . . . . . . . . . . . . . . . . . 84.5.3 Quantum Interrogation . . . . . . . . . . . . . . . . . . . . . . . . . 84.5.4 Weak Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 84.5.5 Direct Measurements of a Wave Function . . . . . . . . . . . . . 84.6 The EPR Paradox and Bell’s Inequalities . . . . . . . . . . . . . . . . . . . 84.6.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.6.2 Bell’s Inequalities as a Game . . . . . . . . . . . . . . . . . . . . . 84.6.3 Loopholes in Bell Tests . . . . . . . . . . . . . . . . . . . . . . . . . 84.6.4 Closing the Loopholes in Bell Tests . . . . . . . . . . . . . . . . . 84.6.5 Polarization-Based Entangled Sources . . . . . . . . . . . . . . . 84.6.6 Other Entanglement Sources for Bell Tests . . . . . . . . . . . . . 84.6.7 Advanced Experimental Tests of Nonlocality . . . . . . . . . . . 84.6.8 Nonlocality Without Inequalities . . . . . . . . . . . . . . . . . . . 84.6.9 Connection to Quantum Information . . . . . . . . . . . . . . . . 84.7 Single-Photon Tunneling Time . . . . . . . . . . . . . . . . . . . . . . . . . 84.7.1 An Application of EPR Correlations to Time Measurements . . 84.7.2 Superluminal Tunneling Times . . . . . . . . . . . . . . . . . . . . 84.7.3 Tunneling Delay in a Multilayer Dielectric Mirror . . . . . . . . 84.7.4 Interpretation of Tunneling Time . . . . . . . . . . . . . . . . . . . 84.7.5 Other Fast and Slow Light Schemes . . . . . . . . . . . . . . . . . 84.8 Gravity and Quantum Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.8.1 Gravitational Wave Detection . . . . . . . . . . . . . . . . . . . . . 84.8.2 Gravity and Quantum Information . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Quantum Information . . . . . . . . . . . . . . . . . . . . . Daniel F. V. James, Peter L. Knight, and Stefan Scheel 85.1 Entanglement and Information . . . . . . . . . . . . 85.1.1 Testing for and Quantifying Entanglement 85.2 Simple Quantum Protocols . . . . . . . . . . . . . . . 85.2.1 Quantum Key Distribution . . . . . . . . . . 85.2.2 Quantum Teleportation . . . . . . . . . . . . 85.3 Quantum Logic . . . . . . . . . . . . . . . . . . . . . . 85.3.1 Single-Qubit Operations . . . . . . . . . . . 85.3.2 Two-Qubit Operations . . . . . . . . . . . . 85.3.3 Multiqubit Gates and Networks . . . . . . . 85.3.4 Cluster-State Quantum Computing . . . . . 85.4 Quantum Algorithms . . . . . . . . . . . . . . . . . . 85.4.1 Deutsch–Jozsa Algorithm . . . . . . . . . . 85.4.2 Grover’s Search Algorithm . . . . . . . . . 85.4.3 Shor’s Factor-Finding Algorithm . . . . . . 85.4.4 Other Algorithms . . . . . . . . . . . . . . .
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1234 1234 1235 1235 1236 1236 1236 1237 1237 1238 1238 1238 1239 1239 1240 1240 1240 1241 1242 1242 1243 1244 1245 1246 1246 1247 1247 1247 1248 1249 1250 1250 1250 1251 1251
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85.5 85.6
Error Correction . . . . . . . . . . . . . . . . . The DiVincenzo Checklist . . . . . . . . . . . 85.6.1 Qubit Characterization, Scalability . 85.6.2 Initialization . . . . . . . . . . . . . . 85.6.3 Long Decoherence Times . . . . . . 85.6.4 Universal Set of Quantum Gates . . 85.6.5 Qubit-Specific Measurement . . . . 85.7 Physical Implementations . . . . . . . . . . . 85.7.1 Linear Optics . . . . . . . . . . . . . . 85.7.2 Trapped Ions . . . . . . . . . . . . . . 85.8 Outlook . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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1266 1267 1267 1267 1268 1268 1268 1268 1268 1269 1269 1270
Part G Applications 86
87
Applications of Atomic and Molecular Physicsto Astrophysics Stephen Lepp, Phillip C. Stancil, and Alexander Dalgarno 86.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86.2 Photoionized Gas . . . . . . . . . . . . . . . . . . . . . . . . . . 86.3 Collisionally Ionized Gas . . . . . . . . . . . . . . . . . . . . . 86.4 Diffuse Molecular Clouds . . . . . . . . . . . . . . . . . . . . 86.5 Dark Molecular Clouds . . . . . . . . . . . . . . . . . . . . . . 86.6 Circumstellar Shells and Stellar Atmospheres . . . . . . . . 86.7 Supernova Ejecta . . . . . . . . . . . . . . . . . . . . . . . . . . 86.8 Shocked Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86.9 The Early Universe . . . . . . . . . . . . . . . . . . . . . . . . 86.10 Atacama Large Millimeter/Submillimeter Array . . . . . . . 86.11 Recent Developments . . . . . . . . . . . . . . . . . . . . . . . 86.12 Other Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Comets . . . . . . . . . . . . . . . . . . . . . . . . Paul D. Feldman 87.1 Introduction . . . . . . . . . . . . . . . . . 87.2 Observations . . . . . . . . . . . . . . . . 87.3 Excitation Mechanisms . . . . . . . . . . 87.3.1 Basic Phenomenology . . . . . 87.3.2 Fluorescence Equilibrium . . . 87.3.3 Swings and Greenstein Effects 87.3.4 Bowen Fluorescence . . . . . . 87.3.5 Electron Impact Excitation . . 87.3.6 Prompt Emission . . . . . . . . 87.3.7 OH Level Inversion . . . . . . . 87.4 Cometary Models . . . . . . . . . . . . . 87.4.1 Photolytic Processes . . . . . . 87.4.2 Density Models . . . . . . . . . 87.4.3 Radiative Transfer Effects . . . 87.5 Summary . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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1275 1276 1278 1278 1280 1281 1282 1283 1283 1284 1285 1285 1286
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1289 1290 1291 1291 1292 1293 1294 1294 1294 1295 1295 1295 1296 1296 1297 1297
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90
Contents
Aeronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jane L. Fox 88.1 Basic Structure of Atmospheres . . . . . . . . . . . . . . . . . . . . . . . 88.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88.1.2 Atmospheric Regions . . . . . . . . . . . . . . . . . . . . . . . . 88.2 Density Distributions of Neutral Species . . . . . . . . . . . . . . . . . 88.2.1 The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . 88.2.2 Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . 88.3 Interaction of Solar Radiation with the Atmosphere . . . . . . . . . . 88.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88.3.2 The Interaction of Solar Photons with Atmospheric Gases . 88.3.3 Interaction of Energetic Electrons with Atmospheric Gases 88.4 Ionospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88.4.1 Ionospheric Regions . . . . . . . . . . . . . . . . . . . . . . . . 88.4.2 Sources of Ionization . . . . . . . . . . . . . . . . . . . . . . . . 88.4.3 Nightside Ionospheres . . . . . . . . . . . . . . . . . . . . . . . 88.4.4 Ionospheric Density Profiles . . . . . . . . . . . . . . . . . . . 88.4.5 Ion Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88.5 Neutral, Ion, and Electron Temperatures . . . . . . . . . . . . . . . . . 88.6 Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88.7 Planetary Escape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications of Atomic and Molecular Physics to Global Change Gonzalo González Abad, Kelly Chance, and Kate P. Kirby 89.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89.1.1 Global Change Issues . . . . . . . . . . . . . . . . . . . . 89.1.2 Structure of the Earth’s Atmosphere . . . . . . . . . . . 89.2 Atmospheric Models and Data Needs . . . . . . . . . . . . . . . 89.2.1 Modeling the Thermosphere and Ionosphere . . . . . 89.2.2 Heating and Cooling Processes . . . . . . . . . . . . . . 89.2.3 Atomic and Molecular Data Needs . . . . . . . . . . . 89.3 Tropospheric Warming/Upper Atmosphere Cooling . . . . . . 89.3.1 Incoming and Outgoing Energy Fluxes . . . . . . . . . 89.3.2 Tropospheric “Global” Warming . . . . . . . . . . . . . 89.3.3 Upper Atmosphere Cooling . . . . . . . . . . . . . . . . 89.4 Stratospheric Ozone . . . . . . . . . . . . . . . . . . . . . . . . . . 89.4.1 Production and Destruction . . . . . . . . . . . . . . . . 89.4.2 The Antarctic Ozone Hole . . . . . . . . . . . . . . . . . 89.4.3 Arctic Ozone Loss . . . . . . . . . . . . . . . . . . . . . . 89.4.4 Global Ozone Depletion . . . . . . . . . . . . . . . . . . 89.5 Atmospheric Measurements . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1299 1299 1300 1304 1304 1305 1305 1305 1306 1309 1311 1311 1312 1317 1320 1322 1323 1325 1332 1334
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Surface Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Erik T. Jensen 90.1 Low Energy Electrons and Surface Science . . . . . . . . . . . . . . . . . 90.2 Electron–Atom Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 90.2.1 Elastic Scattering: Low Energy Electron Diffraction (LEED) 90.2.2 Inelastic Scattering: Electron Energy LossSpectroscopy . . . 90.2.3 Auger Electron Spectroscopy . . . . . . . . . . . . . . . . . . . .
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1337 1337 1338 1339 1339 1339 1340 1340 1340 1340 1341 1342 1342 1343 1344 1344 1344 1345
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1349 1350 1350 1351 1351
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90.3
Photon–Atom Interactions . . . . . . . . . . . . . . . . . . 90.3.1 Ultraviolet Photoelectron Spectroscopy (UPS) 90.3.2 Inverse Photoemission Spectroscopy (IPES) . . 90.3.3 X-Ray Photoelectron Spectroscopy (XPS) . . . 90.3.4 X-Ray Absorption Methods . . . . . . . . . . . . 90.4 Atom–Surface Interactions . . . . . . . . . . . . . . . . . . 90.4.1 Physisorption . . . . . . . . . . . . . . . . . . . . . 90.4.2 Chemisorption . . . . . . . . . . . . . . . . . . . . 90.5 Recent Developments . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1352 1352 1353 1353 1355 1357 1357 1357 1357 1358
Interface with Nuclear Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359 James S. Cohen and John D. Morgan III 91.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359 91.2 Nuclear Size Effects in Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1360 91.2.1 Nuclear Size Effects on Nonrelativistic Energies . . . . . . . . . . 1360 91.2.2 Nuclear Size Effects on Relativistic Energies . . . . . . . . . . . . 1361 91.2.3 Nuclear Size Effects on QED Corrections . . . . . . . . . . . . . . 1362 91.3 Electronic Structure Effects in Nuclear Physics . . . . . . . . . . . . . . . . 1362 91.3.1 Electronic Effects on Closely Spaced Nuclear Energy Levels . . 1362 91.3.2 Electronic Effects on Tritium Beta Decay . . . . . . . . . . . . . . 1362 91.3.3 Electronic Screening of Low-Energy Nuclear Reactions . . . . . 1362 91.3.4 Atomic and Molecular Effects in Relativistic Ion–Atom Collisions1363 91.4 Muon-Catalyzed Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363 91.4.1 The Catalysis Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365 91.4.2 Muon Atomic Capture . . . . . . . . . . . . . . . . . . . . . . . . . . 1365 91.4.3 Muonic Atom Deexcitation and Transfer . . . . . . . . . . . . . . . 1367 91.4.4 Muonic Molecule Formation . . . . . . . . . . . . . . . . . . . . . . 1368 91.4.5 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1370 91.4.6 Sticking and Stripping . . . . . . . . . . . . . . . . . . . . . . . . . . 1370 91.4.7 Prospectus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1372 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1372
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377
List of Tables
Table 1.1 Table 1.2
Table 2.1 Table 2.2 Table 2.3 Table 2.4 Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 10.1 Table 10.2 Table 10.3 Table 10.4 Table 10.5 Table 10.6 Table 10.7 Table 10.8 Table 10.9 Table 11.1 Table 11.2 Table 11.3 Table 11.4 Table 11.5 Table 11.6
Table 11.7
Exact quantities and their symbols, numerical values, and units in the SI . Selected constants as well as atomic and natural units based on the 2018 CODATA adjustment of the fundamental constants [5]. The first two columns describe the quantity and its mathematical symbol. The third and fourth columns give its numerical value and unit. For quantity X the number in parenthesis in the numerical value is the combined statistical and systematic one-standard-deviation uncertainty u.X/ in the last two digits of the numerical value. Finally, the last column gives the relative standard uncertainty ur .X/ D u.X/=jXj. The unit u is the atomic mass unit, one-twelfth of the mass of a 12 C atom . . . . . . . . . . . . . . . . . . . The solid and spherical harmonics Ylm and the tensor harmonics T k (labeled by k D l and D m) for l D 0; 1; 2; 3, and 4 . . . . . . . . . . . . The 3j coefficients for all M ’s D 0 or J3 D 0; 12 . . . . . . . . . . . . . . The 3j coefficients for J3 D 1; 32 ; 2 . . . . . . . . . . . . . . . . . . . . . . . The 6j coefficients for d D 0; 12 ; 1; 32 ; 2, with s D a C b C c . . . . . . . Generators of the Lie groups for the atomic l shell. The subscripts i and j run over all 4l C 2 states of a single electron . . . . . . . . . . . . . . . . . . Dimensions D of the irreducible representations (IR’s) of various Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eigenvalues of Casimir’s operator C for groups used in the atomic l shell The states of the d shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atomic structure programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atomic scattering programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular structure programs . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular and heavy particle scattering programs . . . . . . . . . . . . . . . GNU autotools table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of FORTRAN compilers . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of mixing techniques of languages . . . . . . . . . . . . . . . . . . . . Types of third-party libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing harnesses (frameworks) . . . . . . . . . . . . . . . . . . . . . . . . . . Atomic structure hierarchy in LS coupling and names for groups of transitions between structural entities . . . . . . . . . . . . . . . . . . . . . . . Allowed J -values for ljN equivalent electrons (jj ) coupling . . . . . . . . . Ground levels and ionization energies for neutral atoms . . . . . . . . . . . Selection rules for discrete transitions . . . . . . . . . . . . . . . . . . . . . . Conversion relations between S and Aki for forbidden transitions . . . . . Wavelengths , upper energy levels Ek , statistical weights gi and gk of lower and upper levels, and transition probabilities Aki for persistent spectral lines of neutral atoms. Many tabulated lines are resonance lines (marked “g”), where the lower energy level belongs to the ground term . . Some transitions of the main spectral series of hydrogen . . . . . . . . . . .
4
6 65 65 65 66 73 74 74 76 166 166 167 167 169 170 171 172 172 179 180 184 187 188
190 194 lvii
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List of Tables
Values of Stark-broadening parameter ˛1=2 of the Hˇ line of hydrogen (486:1 nm) for various temperatures and electron densities . . . . . . . . . . 195 c ˛r1 ˇr2 e i and Table 12.1 Formulas for the radial integrals I0 .a; b; cI ˛; ˇ/ D hr1a r2b r12 Pn1rad 1 log a b c ˛r1 ˇr2 irad . .n/ D C kD1 k I0 .a; b; cI ˛; ˇ/ D hr1 r2 r12 ln r12 e is the digamma function, 2 F1 .a; bI cI z/ is the hypergeometric function, and s D a C b C c C 5. Except as noted, the formulas apply for a 1, b 1, c 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Table 12.2 Nonrelativistic eigenvalue coefficients "0 and "1 for helium . . . . . . . . . 206 Table 12.3 Eigenvalue coefficients "2 for helium . . . . . . . . . . . . . . . . . . . . . . . 207 Table 12.4 Values of the reduced electron mass ratio =M , including binding energy corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Table 12.5 Nonrelativistic eigenvalues E D "0 C .=M /"1 C .=M /2 "2 for helium-like ions (in units of e 2 =a ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Table 12.6 Expectation values of various operators for He-like ions for the case M D 1 (in a.u.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Table 12.7 Total ionization energies for 4 He, calculated with RM D 3 289 391 006:600 MHz . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Table 12.8 QED corrections to the ionization energy included in Table 12.7 for the S and P -states of helium (in MHz) . . . . . . . . . . . . . . . . . . . . . . . . . 210 Table 12.9 Quantum defects for the total energies of helium with the Wn term subtracted [Eq. (12.56)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Table 12.10 Formulas for the hydrogenic expectation value hr j i hnljr j jnli in Table 11.8
p
p
Z .2lpC2/Š , fpl D .lCp/Š terms of Gpnl D 2npC1 .lp/Š . . . . . . . . . . . . . . . . . . . . 212 .2lCp1/Š Table 12.11 Oscillator strengths for helium. The factor in brackets gives the finite mass correction, with y D =M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Table 12.12 Singlet-triplet mixing angles for helium . . . . . . . . . . . . . . . . . . . . . 215 Table 14.1 Relativistic ground state binding energy Egs =Z 2 and finite nuclear size correction ıEnuc =Z 2 (in a.u.) of hydrogenic atoms for various magnetic fields B (in units of 2:35 105 T). ıEnuc should be added to Egs . . . . . . 226 Table 14.2 Relativistic binding energy E2S;1=2 for the 2S1=2 .mj D 12 / and E2P;1=2 for the 2P1=2 .mj D 12 / excited states of hydrogen (in a.u.) in an intense magnetic field B (in units of 2:35 105 T) . . . . . . . . . . . . . 226 Table 14.3 Relativistic corrections ıE D .E ENR /=jER j to the nonrelativistic energies ENR for the ground state and n D 2 excited states of hydrogen in an intense magnetic field B (in units of 2:35 105 T). The numbers in brackets denote powers of 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Table 14.4 Relativistic dipole polarizabilities for the ground state of hydrogenic atoms 228 Table 18.1 Measured np 2 PJ lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Table 22.1 The effective quantum number and quantum defect parameters of the 2snd Rydberg series in Be . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Table 22.2 Observed and Hartree–Fock ionization potentials for the ground states of neutral atoms, in eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Table 22.3 Classification of replacements from the 1s2 2s ground state of Li. The 1s and 2s orbitals are occupied, and all the other orbitals are virtual orbitals, vl . 319 Table 22.4 Comparison of theoretical and experimental energies for Be 1s2 2s2 1 S in hartrees. All theoretical values include some form of extrapolation . . . . . 322 Table 22.5 Comparison of CAS-MCHF and PCFI total energies (in hartrees) for the ground state of beryllium with increasing orbital active sets. NCSF corresponds to the size M of the CSF expansion Eq. (22.17) . . . . . . . . 323 Table 22.6 Convergence of transition data for the 1s2 2s2 2p 2 Pı ! 1s2 2s2p2 2 D transition in boron with increasing active set . . . . . . . . . . . . . . . . . . 324
List of Tables
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Table 22.7
Specific mass shift parameter and electron density at the nucleus as a function of the active set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Table 22.8 MCHF hyperfine constants (in MHz) for the 1s2 2s2p 1 Pı state of B II . . . 328 Table 23.1 Relativistic angular density functions . . . . . . . . . . . . . . . . . . . . . . . 344 Table 23.2 Nonrelativistic angular density functions . . . . . . . . . . . . . . . . . . . . . 344 Table 23.3 Spectroscopic labels and angular quantum numbers . . . . . . . . . . . . . . 345 Table 23.4 Radial moments h s i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Table 23.5 j N configurational states in the seniority scheme. The multiplicity of each unresolved degenerate state is indicated by a superscript . . . . . . . . . . . 357 Table 26.1 Test of sum rule in Eq. (26.15) for the lowest He .1s2s2 2 S/ autodetachment state [5]. Projection operators are based on a 4 term Hylleraas 0 and the variational form of v˛ given below. Values of other constants are given in [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Table 26.2 Comparison of methods for calculating the energy of the lowest He .1s2s2 2 S/ autodetachment state. The QHQ results are denoted .EO E0 /Quasi for the quasi-projection method and (E E0 /Complete for the complete projection method [5]. The entries labeled Other results give the full resonant energy. Units are eV . . . . . . . . . . . . . . . . . . . . . . . . . 397 Table 26.3 Energies Es of the He(2s2p 1 P0 ) autoionization states below HeC .n D 2/ threshold from the variational calculations of O’Malley and Geltman [11]. Units are Ry; N is the number of terms in the trial function . . . . . . . . . 397 Table 26.4 Comparison of high-precision calculations with the experiment for the resonance parameters of the He(1 P0 ) resonances below the n D 2 threshold. For photoabsorption, the appropriate Rydberg constant is RM D R1 =.1 C m=M / [19]. The value used here is RM D 13:603833 eV, and E0 D 79:0151 eV [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Table 26.5 Comparison of resonance parameters (in eV) obtained from different methods for calculating 1 De states in H . . . . . . . . . . . . . . . . . . . . . 401 Table 26.6 Singlet autoionization states of He of S, P , D, and F angular momentum below the n D 2 threshold of HeC . Results are in eV relative to the ground state energy of He D 5:8074475 Ry . . . . . . . . . . . . . . . . . . . . . . 401 Table 26.7 Triplet autoionization states of He of S, P , D, and F angular momentum below the n D 2 threshold of HeC . Results are in eV relative to the ground state energy of He D 5:8074475 Ry . . . . . . . . . . . . . . . . . . . . . . 401 Table 26.8 Singlet and triplet autoionization states of Ps . Results are in eV relative to the ground state energy of Ps D 0:5 Ry . . . . . . . . . . . . . . . . . . . . . . 401 Table 26.9 3 Pe states of Ps . Results are in eV relative to the ground state energy of Ps D 0:5 Ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 Table 26.10 Odd parity 1;3 P states of Ps . Resonance positions are with respect to the ground state energy of Ps D 0:5 Ry . . . . . . . . . . . . . . . . . . . . . . . . 402 Table 26.11 Resonance energies EF (Ry) and widths (eV) for 1 P states of He below the n D 2 threshold (1 Ry) of HeC . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Table 26.12 3 Pe shape resonances in Ps . Results are in eV relative to the ground state energy of Ps D 0:5 Ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Table 29.1 Coefficients for the QED contributions to the electron anomaly. The .2n/ .2n/ coefficients A1 and functions A2 .x/, evaluated at x D xe me =m and xe£ me =m£ for the muon and tau leptons, respectively, are listed with .2n/ two significant digits for ease of comparison; summed values Ce , based on the 2014 CODATA adjustment, are listed as accurately as needed for the tests described in this article. Missing values indicate that their contribution to the electron anomaly is negligible . . . . . . . . . . . . . . . . . . . . . . . 437
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Table 29.2
Table 29.3
Table 29.4
Table 29.5 Table 29.6 Table 29.7
Table 29.8 Table 29.9 Table 30.1
Table 30.2
Table 30.3
Table 30.4 Table 30.5
Table 30.6
Table 34.1 Table 34.2 Table 34.3
List of Tables
Theoretical contributions and total value for the g-factor of hydrogenic carbon 12 based on the 2014 CODATA recommended values of the constants and corresponding theory. The total g-factor has a relative uncertainty of 1:3 1011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 Theoretical contributions and total value for the g-factor of hydrogenic silicon 28 based on the 2014 CODATA recommended values of the constants and corresponding theory. The total g-factor has a relative uncertainty of 8:5 1010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 List of contributions and their main dependence on fundamental constants to the hydrogen transition frequencies ordered by appearance in the text. The first two columns give the section in Sec. 29.7.1 in which the contribution is described in detail and the name of the contribution, respectively. The fundamental-constant dependence of a contribution in the third column is given in terms of the Hartree energy Eh D me c 2 ˛ 2 and four dimensionless variables with values small compared to one: the fine-structure constant ˛, the proton charge radius divided by the reduced Compton wavelength rp =C , and the mass ratios me =m and me =mp . The last column gives the order of magnitude of each of the contributions in Hz for the 1S1=2 –2S1=2 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Values of the Bethe logarithms ln k0 .n; `/. Missing entries correspond to states for which no experimental measurements are available . . . . . . . . 443 Values of the function GSE .˛/. Missing entries correspond to states for which no experimental measurements are available . . . . . . . . . . . . . . 445 .1/ Values of the function GVP .˛/. No experimental data is available for missing entries. Zero values indicate that their contributions are negligibly small . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Values of B61 used in the 2014 CODATA adjustment. Zero values indicate that their contributions are negligibly small . . . . . . . . . . . . . . . . . . . 445 Values of B60 used in the 2014 CODATA adjustment. The uncertainties of B60 for S-states are explained in the text . . . . . . . . . . . . . . . . . . . . . 445 Accurately measured atomic transition frequencies and their fractional uncertainty ([9, 12] and references therein). These transitions have been used in evaluations of temporal variation of fundamental constants . . . . 453 Magnetic moments and relativistic corrections for atoms involved in microwave standards. The relativistic sensitivity is defined in Sec. 30.2.3. Here, is an actual value of the nuclear magnetic moment, N is the nuclear magneton, and S stands for the Schmidt value of the nuclear magnetic moment; the nucleon g factors are gp =2 ' 2:79 and gn =2 ' 1:91 454 The frequencies of different transitions in neutral atoms and ions, applied or tried for precision spectroscopy, and their sensitivity to variations in ˛ due to relativistic corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Model-independent laboratory constraints on the possible time variations of natural constants [41, 42] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Progress in constraining a possible time variation of the fundamental constants: the results of 2004 (following [43, 44]), 2008 (following [45]), and of 2014 using data summarized in [42] but following [43, 44] (see [41] for details). Here, is the correlation coefficient .@ ln ˛=@t; @ lnfR1 g=@t/ 457 Model-dependent laboratory constraints on possible time variations of fundamental constants. The uncertainties here do not include uncertainties from the application of the Schmidt model . . . . . . . . . . . . . . . . . . . 458 Tunneling energy eigensolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 512 Character table for symmetry group C2 . . . . . . . . . . . . . . . . . . . . . . 513 Character table for symmetry group D2 . . . . . . . . . . . . . . . . . . . . . 514
List of Tables
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Table 34.4 Table 34.5 Table 34.6 Table 40.1 Table 42.1 Table 44.1 Table 44.2 Table 46.1 Table 47.1 Table 48.1
Table 48.2
Table 52.1 Table 52.2 Table 58.1 Table 60.1 Table 60.2 Table 60.3 Table 60.4 Table 60.5 Table 70.1
Table 75.1 Table 79.1
Table 85.1
Table 86.1 Table 88.1 Table 88.2 Table 88.3 Table 88.4 Table 88.5 Table 88.6
Character table for symmetry group O . . . . . . . . . . . . . . . . . . . . . . 516 HKD28 matrix solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 Hyperfine spin states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 Examples illustrating the range of ionic reactions that can occur in the gas phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 Summary of reproducibility, accuracy, and precision of measured components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 Fixed frequency lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 Approximate tuning ranges for tunable lasers . . . . . . . . . . . . . . . . . . 653 Representative third-order frequency conversion schemes for generation of tunable coherent VUV light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 Model interaction potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 Summary of cases of increasing complexity and the orientation and alignment parameters necessary for unpolarized beams; Np is the number of independent parameters, and Nd is the number of observation directions required . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718 Summary of cases of increasing complexity for spin-polarized beams. The number of independent dimensionless parameters Np is listed, along with NOA , the number determined from orientation and alignment only; Nd is the number of observation directions required . . . . . . . . . . . . . . . . . . . . 721 Selection rules for the coupling between diabatic and adiabatic states of a diatomic quasi molecule (w D g; u; D C; ) . . . . . . . . . . . . . . . 777 Selection rules for dynamic coupling between adiabatic states of a system of three atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 778 Recombination computer codes . . . . . . . . . . . . . . . . . . . . . . . . . . 851 General n-dependence of characteristic properties of Rydberg states. After [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883 Multipoles QL for the Stark states j˙i of H(n) . . . . . . . . . . . . . . . . . 884 Fit parameters for Cs-Cs and Rb-Rb pseudo-potentials (from [12]) . . . . . 885 Coefficients C.ni `i ! nf `f / in the Born capture cross section formula in Eq. (60.330) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910 Functions F .ni `i ! nf `f I x/ in the Born capture cross section formula in Eq. (60.330) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 910 Collision numbers for coupling between different modes. V, R, and T refer to vibrational, rotational, and translational energy, respectively. Each entry is the typical range of ZAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1020 Overview of laser types, organized by the nature of the lasing transition . 1082 Laser cooling parameters for the lowest S1=2 –P3=2 transition of hydrogen and most alkalis (the D2 line). Also shown are the nuclear spin I and the ground state hyperfine splitting hfs . . . . . . . . . . . . . . . . . . . . . . . 1142 BB84 protocol for secret key distribution. “Alice” sends information encoded in either of two basis sets. “Bob” randomly chooses a measurement basis that is publicly communicated. For those cases when sender and receiver chose the same basis, the receiver’s measurement yields a secure bit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1262 Molecules observed in interstellar clouds . . . . . . . . . . . . . . . . . . . . 1280 Homopause characteristics of planets and other solar system bodies . . . . 1301 Molecular weights and fractional composition of dry air in the terrestrial atmospherea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301 Composition of the lower atmospheres of Mars and Venus . . . . . . . . . . 1302 Composition of the lower atmospheres of Jupiter and Saturn . . . . . . . . 1303 Composition of the lower atmospheres of Uranus and Neptunea . . . . . . 1303 Composition of the lower atmosphere of Titana . . . . . . . . . . . . . . . . . 1303
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Composition of the atmosphere of Tritona . . . . . . . . . . . . . . . . . . . . 1304 Number densities of neutral and ion species at the surface of Mercurya . . 1304 Composition of the lower atmosphere of Plutoa . . . . . . . . . . . . . . . . . 1304 Ionization potentials (IP ) of common atmospheric species (computed with data taken from [56], except as noted) . . . . . . . . . . . . . . . . . . . . . . 1313 Table 88.11 Exobase properties of solar system bodies . . . . . . . . . . . . . . . . . . . . 1333 Table 91.1 Resonant (quasiresonant if negative) collision energies res (in meV) calculated using Eq. (91.35) a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369 Table 91.2 Comparison of sticking valuesa . . . . . . . . . . . . . . . . . . . . . . . . . . 1372 Table 88.7 Table 88.8 Table 88.9 Table 88.10
List of Tables
About the Authors
Nigel G. Adams Girish S. Agarwal Institute for Quantum Science and Engineering, Texas A&M University, College Station, USA Miron Y. Amusia Nils Andersen The Niels Bohr Institute, Univerity of Copenhagen, Copenhagen, Denmark Nigel Badnell Dept. of Physics, University of Strathclyde Glasgow, Glasgow, UK Raul O. Barrachina Bariloche Atomic Centre, National Atomic Energy Commission, Bariloche, Argentina Thomas Bartsch Mathematical Sciences, Loughborough University, Leicestershire, UK Klaus Bartschat Dept. of Physics & Astronomy, Drake University, Des Moines, USA William E. Baylis Department of Physics, University of Windsor, Windsor, ON, Canada Anand K. Bhatia Laboratory for Astronomy & Solar Physics, NASA Goddard Space Flight Center, Greenbelt, USA Robert W. Boyd Dept. of Physics, University of Ottawa, Ottawa, Canada, Institute of Optics, University of Rochester, Rochester, USA Michael W. J. Bromley University of Southern Queensland, Toowoomba, Australia John M. Brown Henry Buijs FTS Consulting, Québec, Canada Philip Burke Denise Caldwell Physics Division, National Science Foundation, Alexandria, VA, USA Mark M. Cassar Zekelman School of Information Technology, St. Clair College, Windsor, Canada Alexis A. Chacon S. Dept. of Physics, Pohang University of Science and Technology and Max Planck Research Initiative/Korea, Pohang, Korea, Republic of lxiii
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Kelly Chance Harvard-Smithsonian Center for Astrophysics, Cambridge, USA Raymond Y. Chiao Departments of Physics and Engineering, University of California Merced, Merced, USA Marcelo Ciappina Physics Program, Guangdong Technion – Israel Institute of Technology, Shantou, Guangdong, China Lew Cocke Dept. of Physics, Kansas State University, Manhattan, USA James S. Cohen Atomic and Optical Theory, Los Alamos National Laboratory, Los Alamos, USA Derrick Crothers Lorenzo J. Curtis Alexander Dalgarno Nelson de Oliveira Ligne DESIRS, Synchrotron Soleil, Gif-sur-Yvette Cedex, France John B. Delos Butterfly Dynamics LLC, Williamsburg, USA Abigail J. Dobbyn Göttingen, Germany Joseph H. Eberly Dept. of Physics and Astronomy, University of Rochester, Rochester, USA Guy T. Emery Dept. of Physics, Bowdoin College, Brunswick, USA Volker Engel Institute of Physical and Theoretical Chemistry, Universität Würzburg, Würzburg, Germany Paul Engelking Dept. of Chemistry and Chemical Physics Institute, University of Oregon, Eugene, USA Kenneth Evenson James M. Farrar Dept. of Chemistry, University of Rochester, Rochester, USA Alexandre Faure Institute of Planetology and Astrophysics of Grenoble (IPAG), University of Grenoble Alpes (CNRS), Grenoble, France Gordon Feldman Baltimore, USA Paul D. Feldman Victor Flambaum Dept. of Physics, University of New South Wales, Sydney, Australia M. Raymond Flannery David R. Flower Dept. of Physics, University of Durham, Durham, UK
About the Authors
About the Authors
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Omar A. Fojón Instituto de Fisica Rosario, Rosario, Argentina A. Lewis Ford Dept. of Physics, Texas A&M University, College Station, USA Jane L. Fox Dept. of Physics, Wright State University, Dayton, USA Charlotte Froese Fischer Dept. of Computer Science, University of British Columbia, Vancouver, Canada Thomas Fulton Alexander L. Gaeta Applied Physics and Applied Mathematics, Columbia University, New York, USA Alan Gallagher Quantum Physics Division, University of Colorado, Boulder, USA Thomas F. Gallagher Dept. of Physics, University of Virginia, Charlottesville, VA, USA Muriel Gargaud Observatoire Aquitain des Sciences de l’Univers, Floirac, France Enno Giese Institut für Angewandte Physik, Technische Universität Darmstadt, Darmstadt, Germany Michel Godefroid Chemistry Department (SQUARES), Université libre de Bruxelles (ULB), Brussels, Belgium S. Pedro Goldman Dept. of Physics, Toronto Metropolitan University, Toronto, Canada Gonzalo González Abad Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, USA Ian Grant Oxford Mathematical Institute, University of Oxford, Oxford, UK Chris H. Greene Dept. of Physics & Astronomy, Purdue University, West Lafayette, USA Donald Griffin Dept. of Physics, Rollins College, Winter Park, USA Dag Hanstorp Dept. of Physics, University of Gothenburg, Gothenburg, Sweden William G. Harter Dept. of Physics, University of Arkansas, Fayetteville, USA Charles Havener Physics Division, Oak Ridge National Laboratory, Oak Ridge, USA Qiongyi He School of Physics, Peking University, Beijing, China Carsten Henkel Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany Eric Herbst Dept. of Chemistry and Dept. of Astronomy, University of Virginia, Charlottsville, USA Robert N. Hill
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David L. Huestis Molecular Physics Laboratory, SRI International, Menlo Park, USA Daniel F. V. James Dept. of Physics, University of Toronto, Toronto, Canada Juha Javanainen Department of Physics, University of Connecticut, Storrs, USA Erik T. Jensen Dept. of Physics, University of Northern British Columbia, Prince George, Canada Brian R. Judd Dept. of Physics & Astronomy, The Johns Hopkins University, Baltimore, USA Alexander A. Kachanov Picarro, Inc., Sunnyvale, USA Alexander Kandratsenka Dynamics at Surfaces, Max-Planck-Institute for Multidisciplinary Sciences, Göttingen, Germany Isik Kanik Jet Propulsion Laboratory, California Institute of Technology, Pasadena, USA Savely G. Karshenboim Faculty of Physics, Ludwig-Maximilians-University, Munich, Germany Kate P. Kirby American Physical Society, College Park, USA Tom Kirchner Dept. of Physics & Astronomy, York University, Toronto, Canada Peter L. Knight Dept. of Physics, Imperial College London, London, UK Alexander Kramida Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD, USA Manfred Krause Oak Ridge National Laboratory, Oak Ridge, USA Paul G. Kwiat Dept. of Physics, University of Illinois at Urbana-Champaign, Urbana, USA Yuan T. Lee Institute of Atomic and Molecular Science, Academia Sinica, Taipai, Taiwan Stephen Lepp Dept. of Physics & Astronomy, University of Nevada, Las Vegas, Las Vegas, USA Maciej Lewenstein The Barcelona Institute of Science and Technology, Quantum Optics Theory, ICFO and ICREA, Castelldefels, Spain Hongwei Li State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian, Liaoning, China Francois Lique CNRS, IPR (Institut de Physique de Rennes), Université de Rennes 1, Rennes, France James D. Louck Joshua R. Machacek Research School of Physics, Australian National University, Canberra, Australia
About the Authors
About the Authors
lxvii
Mary L. Mandich NOKIA Corp. (retired), Martinsville, USA Edmund J. Mansky II Eikonal Research Institute, Bend, OR, USA William C. Martin Ronald McCarroll Laboratoire de Chimie Physique, Université Pierre et Marie Curie, Paris, France William McConkey Dept. of Physics, University of Windsor, Windsor, Canada Robert P. McEachran Research School of Physics, Australian National University, Canberra, Australia James H. McGuire Dept. of Physics, Tulane University, New Orleans, USA Ralf Menzel Dept. of Physics, University of Potsdam, Potsdam, Germany Dieter Meschede Inst. f. Applied Physics, Rheinische Friedrich-Wilhelm Universität Bonn, Bonn, Germany Pierre Meystre University of Arizona, Tucson, USA Peter W. Milonni Theoretical Division, Los Alamos National Laboratory, Los Alamos, USA Peter J. Mohr National Institute of Standards and Technology, Gaithersburg, USA David H. Mordaunt Göttingen, Germany John D. Morgan III Dept. of Physics & Astronomy, University of Delaware, Newark, USA Francisco Navarrete Institute of Physics, University of Rostock, Rostock, Germany Evgeny Nikitin Dept. of Physics, Technion – Israel Institute of Technology, Haifa, Israel Patrick Nuernberger Institute of Physical and Theoretical Chemistry, Universität Regensburg, Regensburg, Germany Ronald E. Olson Physics Dept., University of Missouri-Rolla, Rolla, USA Barbara Paldus Skymoon Ventures, Palo Alto, USA Josef Paldus Dept. of Applied Mathematics, University of Waterloo, Waterloo, Canada Gillian Peach Dept. of Physics and Astronomy, University College London, London, UK David Pegg Dept. of Physics, University of Tennessee, Knoxville, USA Ekkehard Peik Physikalisch-Technische Bundesanstalt, Braunschweig, Germany Ronald Phaneuf Dept. of Physics, University of Nevada, Reno, Reno, USA Michael Pindzola College of Sciences and Mathematics, Auburn University, Auburn, USA
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Eric H. Pinnington Dept. of Physics, University of Alberta, Edmonton, Alberta, Canada Richard C. Powell Optical Sciences Center, University of Arizona, Tuscon, USA Richard S. Quimby Dept. of Physics, Worcester Polytechnic Institute, Worcester, USA John F. Reading Dept. of Physics, Texas A&M University, College Station, USA Tyle C. Reimer University of Arkansas, Fayetteville, USA Roberto D. Rivarola Instituto de Física Rosario, CONICET – UNR, Rosario, Argentina Jan-Michael Rost Max Planck Institute for the Physics of Complex Systems, Dresden, Germany Hossein Sadeghpour Institute for Theoretical Atomic Molecular and Optical Physics (ITAMP), Harvard University, Cambridge, USA Marianna S. Safronova Dept. of Physics & Astronomy, University of Delaware, Newark, USA Jonathan R. Sapirstein Department of Physics, University of Notre Dame, Notre Dame, USA Stefan Scheel Institute of Physics, University of Rostock, Rostock, Germany Axel Schenzle Reinhard Schinke Max-Planck-Institute for Dynamics and Self-Organization, Göttingen, Germany David Schultz Department of Astronomy and Planetary Science, Northern Arizona University, Flagstaff, AZ, USA Michael Schulz Dept. of Physics, Missouri University of Science & Technology, Rolla, USA L. Krister Shalm Dept. of Physics, University of Colorado at Boulder, Boulder, USA Peter L. Smith retired, Cathedral City, USA Yevgeny Stadnik School of Physics, The University of Sydney, Sydney, Australia Phillip C. Stancil Dept. of Physics & Astronomy, The University of Georgia, Athens, USA Anthony F. Starace Glenn Stark Dept. of Physics, Wellesley College, Wellesley, USA Allan D. Stauffer Dept. of Physics & Astronomy, York University, Toronto, Canada Aephraim M. Steinberg Dept. of Physics, University of Toronto, Toronto, Canada Stig Stenholm
About the Authors
About the Authors
lxix
Jack C. Straton Dept. of Physics, Portland State University, Portland, OR, USA Michael R. Strayer Carlos R. Stroud Jr. The Institute of Optics, University of Rochester, Rochester, USA Arthur G. Suits Dept. of Chemistry, University of Missouri, Columbia, USA Barry N. Taylor Quantum Measurement Division, National Institute of Standards and Technology, Gaithersburg, USA Aaron Temkin Laboratory for Solar and Space Physics, NASA Goddard Space Flight Center, Greenbelt, USA Jonathan Tennyson Dept. of Physics & Astronomy, University College London, London, UK Eite Tiesinga National Institute of Standards and Technology, Gaithersburg, MD, USA Elmar Träbert Astronomical Institute, Ruhr University Bochum, Bochum, Germany Sandor Trajmar Jet Propulsion Laboratory, California Institute of Technology, Redwood City, USA Turgay Uzer School of Physics, Georgia Institute of Technology, Atlanta, USA Alexander Voitkiv The Institute of Theoretical Physics I, Heinrich-Heine University Düsseldorf, Düsseldorf, Germany Da-Wei Wang Department of Physics, Zhejiang University, Hangzhou, China Wolfgang L. Wiese National Institute of Standards and Technology, Gaithersburg, MD, USA Martin Wilkens Institute of Physics and Astronomy, University of Potsdam, Potsdam, Germany David R. Yarkony Dept. of Chemistry, The Johns Hopkins University, Baltimore, USA Ruitian Zhang Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China Wei Zhang Dept. of Physics, Renmin University of China, Beijing, China Bernard Zygelman Dept. of Physics & Astronomy, University of Nevada, Las Vegas, Las Vegas, USA
Part A Mathematical Methods
2
Part A gathers together the mathematical methods applicable to a wide class of problems in atomic, molecular, and optical physics. The application of angular momentum theory to quantum mechanics is presented. The basic tenet that isolated physical systems are invariant to rotations of the system is thereby implemented into physical theory. The powerful methods of group theory and second quantization show how simplifications arise if the atomic shell is treated as a basic structural unit. The well-established symmetry groups of quantum mechanical Hamiltonians are extended to the larger compact and noncompact dynamical groups. Perturbation theory is introduced as a bridge between an exactly solvable problem and a corresponding real one, allowing approx-
imate solutions of various systems of differential equations. The consistent manner in which the density matrix formalism deals with pure and mixed states is developed, showing how the preparation of an initial state, as well as the details regarding the observation of the final state, can be treated in a systematic way. The basic computational techniques necessary for accurate and efficient numerical calculations essential to all fields of physics are outlined, and a summary of relevant software packages is given. The ever-present one-electron solutions of the nonrelativistic Schrödinger equation and the relativistic Dirac equation for the Coulomb potential are then summarized. A summary of the computer software available for atomic and molecular physics calculations is included.
1
Units and Constants Eite Tiesinga
Contents
1.1 Introduction
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Atomic Units . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Natural Units . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.4
Fundamental Constants . . . . . . . . . . . . . . . . . . .
5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Abstract
We describe atomic and natural units relevant for the atomic, molecular, and optical physics described in this book. For nonrelativistic models of atoms and molecules absorbing and emitting photons atomic units are appropriate. In this system, energies and lengths are expressed in terms of the Hartree energy and Bohr radius, respectively. Relativistic models are required for precision quantum electrodynamics-based determinations of energy levels of hydrogen and other light atoms and molecules. Natural units are then the most appropriate, and energies and length are expressed in the rest energy of the electron and the reduced Compton wavelength, respectively. Finally, we present values for an abbreviated list of fundamental constants taken from the CODATA (Committee on Data for Science and Technology) adjustment of fundamental constants based on data published or made available before the end of 2018. Keywords
atomic units fine structure constant fundamental constants International System of Units (SI) natural units
E. Tiesinga () National Institute of Standards and Technology Gaithersburg, MD, USA e-mail: [email protected]
In science we communicate measurements of quantities or observables in terms of a product of a numerical value and a unit, where the unit is a particular example of the quantity. For example, the spin of an electron is .1=2/ „, where 1/2 is the numerical value, and the reduced Planck constant „ is the unit or example angular momentum. Equivalently, the spin of the electron is 0:527 285 : : : 1034 J s in the International System of Units (SI) [1]. More digits for the numerical value can be easily found now that the Planck constant is exactly defined in the SI. In fact, seven exact constants define the seven SI base units second, meter, kilogram, Coulomb, kelvin, and lumen [1]. The names and values of these constants are given in Table 1.1. It was only on World Metrology Day, 20 May 2019, that the Planck constant, elementary charge, Boltzmann constant, and the Avogadro constant were given their exact value in the SI. See [2] for a review of the ideas that led to this redefinition. The example in the previous paragraph shows that some units are more convenient than others. We tend to prefer units where the numerical value of the measured quantity is of order 1. Preferred units for a system follow from its Hamiltonian H in a process that makes H as well as all operators and constants appearing in H dimensionless. Operators can correspond to particle spins, positions, and momenta but also fields in quantum field theories. In this chapter, we derive atomic and natural units as they appear in atomic, molecular, and optical physics. For a different view of units in the field of electromagnetism, we recommend the appendix on units and dimensions of [3]. We also present a list of relevant fundamental constants.
1.2 Atomic Units We derive atomic units starting from the nonrelativistic Hamiltonian for an electron with charge e and mass me
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_1
3
4
E. Tiesinga
Table 1.1 Exact quantities and their symbols, numerical values, and units in the SI Quantity Hyperfine transition frequency of 133 Cs Speed of light in vacuum Planck constanta Elementary charge Boltzmann constant Avogadro constant Luminous efficacy
Sym. Cs c h „ e k NA Kcd
Value 9 192 631 770 299 792 458 6:626 070 15 1034 1:054 571 817 : : : 1034 1:602 176 634 1019 1:380 649 1023 6:022 140 76 1023 683
Unit Hz m s1 J Hz1 Js C J K1 mol1 lm W1
a
The energy of a photon with frequency expressed in unit Hz is E D h in J. The unitary time evolution of the state of this photon is given by exp.iEt =„/j'i, where j'i is the photon state at time t D 0, and time is expressed in unit s. The ratio Et =„ is a phase.
bound to an infinitely heavy point-like source with charge CZe, possibly absorbing and emitting transverse photons. Following [4] we use the Coulomb gauge and assume that the charges are contained in a large cubic box with periodic boundary conditions and length L on each side. Here, e is the (positive) elementary charge, and Z is a positive integer. Hence, we have 1 Ze 2 X 1 .pE C e AE? .Er //2 „!j aj aj C C ; H D 2me 4 0 r 2 j (1.1) where rE and pE are the position and momentum operators of the electron, respectively, and the commutation relations Œri ; pj D i„ıij for vector components i and j hold. The E 0 is the vacuum electric permitsource is located at rE D 0, tivity, and ıij is the Kronecker delta. For simplicity, we have omitted the Zeeman interaction of the electron spin coupling to a magnetic field. The last term of Eq. (1.1) describes the photon-field Hamiltonian, where operators aj and aj annihilate and create photons, respectively. For index j , photons are specified by polarization Ej and wavevector kEj . The components of the wavevector are integer multiples of 2 =L. By construction Ej and kEj are perpendicular to each other, and the frequency of photon j is !j D cjkj j, where c is the speed of light in vac uum. Operators aj and aj satisfy the commutation relations
stants 0 , c, and the vacuum magnetic permeability 0 are not independent. They satisfy c2 D
1 : 0 0
(1.3)
We are now ready to make the Hamiltonian dimensionless. We assume that there exists a convenient length scale a0 and write for the position operator of the electron rE D xE a0 , where operator xE is dimensionless. An equally valid interpretation of xE a0 is that xE is a vector of numerical values, and a0 is the unit for position. The commutation relation for rE and pE imply pE D i„rr , where rr is the gradient or nabla vector differential operator in rE, and thus pE D irx „a01 . The wavevector and frequency of the photon field are made dimensionless with kEj D qEj a01 and !j D jqj j ca01 . The components of qEj are integer multiples of 2 =`, where L D ` a0 . Thus, the units of electron momentum, photon wavevector, and frequency are „a01 , a01 , and ca01 , respectively. To summarize, Eq. (1.1) becomes H D
2 ea0 E e2 Z „2 1 C . x E a / ir A x ? 0 „ 4 0 a0 x me a02 2 X 1 jqj j aj aj C : C „ca01 2 j
(1.4) Œai ; aj D ıij . Finally, the transverse vector field operator A comparison of the energies of the first two terms suggests that we choose a0 such that AE? .Er / 1 is given by AE? .Er / D
X j
s
„2 e2 D Eh ; 2 4 0 a0 me a 0
h i „ i kEj Er i kEj Er a : (1.2) E e C a E e j j j j 20 !j L3
(1.5)
and, thus, Similar expressions for the transverse electric EE? .Er / and E r / field operators can be written down. The conmagnetic B.E A transverse vector field FE? .Er / in position space rE is a field such that E is perpendicular to kE for all wavevecits Fourier representation FE? .k/ E Magnetic fields are transverse vector fields in any gauge and tors k. subscript ? is often dropped. 1
a0 D
4 0 „2 : me e 2
(1.6)
The reference quantities Eh and a0 or the atomic units of energy and length are better known as the Hartree energy and the Bohr radius, respectively. The Rydberg frequency cR1 is defined by cR1 D Eh =2h.
1
Units and Constants
5
In atomic units the vector potential is made dimensionless with the choice „ E ? .x/ AE? .xE a0 / A E ; ea0
and, by comparing the energies of the first two terms, choose C such that (1.12) me c 2 D „c1 C ;
(1.7) or C D
so that the Hamiltonian reads 2 Z 1 E ? .x/ irx C A H D E 2 x X1 1 C jqj j aj aj C Eh ; ˛ 2 j
„ : me c
(1.13)
The reference quantities me c 2 and C or the natural units of energy and length are the energy equivalent of the electron rest mass and the reduced Compton wavelength, respectively. (1.8) We also note that C D ˛a0 and Eh D ˛ 2 me c 2 . In natural units the transverse vector potential is not made dimensionless from its appearance in the Hamiltonian but where the dimensionless fine-structure constant ˛ is rather from rewriting its definition in Eq. (1.2) in terms of the reduced Compton wavelength. In fact, we choose e2 ˛D ; (1.9) 4 0 „c p „ E ? .x/ AE? .xE C / 4 ˛ A E ; (1.14) eC and 1=˛ is the numerical value of the speed of light in vacuum in atomic units. Specifically, the unit of velocity is where s „=me a0 . i X 1 h i qEj xE i qEj xE E ? .x/ a (1.15) A E D E e C a E e j j j j 2jqj j`3 j
1.3 Natural Units
E ? .x/ by inspection. The transverse vector field A E only depends on geometric quantities related to the periodic boundNatural units follow from Dirac’s relativistic description of and not ary conditions in the cubic box of length L D ` C the negatively-charged electron bound to an infinitely-heavy point charge in the presence of transverse photons. Thus, in on ˛ or the charge of the electron. Finally, the Hamiltonian in Eq. (1.11) reads the Coulomb gauge, we have p 2 1 Ze E ? .x/ H D ˇ C ˛E irx C 4 ˛ A E 2 rel Hrel D ˇme c C c ˛E .pE C e AE? .Er // I4 4 0 r X Z˛ 1 X 1 I jq j a a C (1.16) C me c 2 ; 4 j j j C „!j aj aj C ; x 2 2 j j
(1.10) where ˛E and ˇ are four dimensionless mutually anticommuting four-component Dirac matrices with ˛i2 D ˇ 2 D I4 , and I4 is the four-component identity matrix. The definition for the transverse vector potential AE? .Er / remains that of Eq. (1.2). For our purpose of defining natural units, we can ignore interpretations of the positive and negative energy solutions of the Dirac equation as well as effects such as virtual electron-positron pairs, which will be present in the complete quantum electrodynamic (QED) theory. This relativistic Hamiltonian can be made dimensionless by assuming the length unit C and rE D xE C with similar expressions for box size L, momentum p, E and photon E wavevector and frequency kj and !j . Then we find eC E 1 2 A? .xE C / Hrel D ˇme c C „cC ˛E irx C „ X e2 Z 1 1 jqj j aj aj C C „cC ; 4 0 C x 2 j
where the fine-structure constant now appears in the strength E ? .x/ of the minimal coupling ˛E A E of the electron with the photons and in the Coulomb term of the Hamiltonian. The numerical value of the speed of light in vacuum is one in natural units. In fact, the unit of velocity is „=me C D c.
1.4 Fundamental Constants
In Table 1.2 we present an abbreviated list of values for fundamental constants based on the 2018 adjustment of fundamental constants as published by the CODATA taskgroup [5]. The table also gives values for atomic and natural units for a range of quantities, such as charge, speed, and time. The two most accurately known constants are the g-factor of a free electron and the Rydberg frequency cR1 (or, equivalently, the Hartree energy) with relative uncertainties of 1:7 1013 and 1:9 1012 , respectively. The least well-known quantity in the table is the proton root-mean(1.11) square (rms) charge radius rp with a relative uncertainty of
1
6
E. Tiesinga
Table 1.2 Selected constants as well as atomic and natural units based on the 2018 CODATA adjustment of the fundamental constants [5]. The first two columns describe the quantity and its mathematical symbol. The third and fourth columns give its numerical value and unit. For quantity X the number in parenthesis in the numerical value is the combined statistical and systematic one-standard-deviation uncertainty u.X/ in the last two digits of the numerical value. Finally, the last column gives the relative standard uncertainty ur .X/ D u.X/=jXj. The unit u is the atomic mass unit, one-twelfth of the mass of a 12 C atom Quantity Non-SI units accepted for use within the SI Electron volt: (e=C) J (Unified) atomic mass unit: m.12 C/=12 General Vacuum magnetic permeability 4 ˛„=e 2 c 0 =.4 107 / Vacuum electric permittivity 1=0 c 2 Fine-structure constant e 2 =4 0 „c Inverse fine-structure constant Rydberg frequency ˛ 2 me c 2 =2h D Eh =2h Energy equivalent Rydberg constant Bohr magneton e„=2me Nuclear magneton e„=2mp Electron, e Electron mass Energy equivalent Electron–muon mass ratio Reduced Compton wavelength „=me c D ˛a0 Compton wavelength Classical electron radius ˛ 2 a0 Electron magnetic moment to Bohr magneton ratio to nuclear magneton ratio Electron magnetic moment anomaly je j=B 1 Electron g-factor 2.1 C ae / Electron–proton magnetic moment ratio Proton, p Proton mass Energy equivalent Proton–electron mass ratio Proton rms charge radius Proton magnetic moment to Bohr magneton ratio to nuclear magneton ratio Proton g-factor 2p =N
Symbol
Numerical value
Unit
Relative std. uncert. ur
eV u
1:602 176 634 1019 1:660 539 066 60.50/ 1027
J kg
exact 3:0 1010
0
1:256 637 062 12.19/ 106 1:000 000 000 55.15/ 8:854 187 8128.13/ 1012 7:297 352 5693.11/ 103 137:035 999 084.21/ 3:289 841 960 2508.64/ 1015 2:179 872 361 1035.42/ 1018 13:605 693 122 994.26/ 10 973 731:568 160.21/ 9:274 010 0783.28/ 1024 1:399 624 493 61.42/ 1010 5:050 783 7461.15/ 1027 7:622 593 2291.23/
N A2 N A2 F m1
1:5 1010 1:5 1010 1:5 1010 1:5 1010 1:5 1010 1:9 1012 1:9 1012 1:9 1012 1:9 1012 3:0 1010 3:0 1010 3:1 1010 3:1 1010
9:109 383 7015.28/ 1031 5:485 799 090 65.16/ 104 8:187 105 7769.25/ 1014 0:510 998 950 00.15/ 4:836 331 69.11/ 103 3:861 592 6796.12/ 1013 2:426 310 238 67.73/ 1012 2:817 940 3262.13/ 1015 9:284 764 7043.28/ 1024 1:001 159 652 181 28.18/ 1838:281 971 88.11/ 1:159 652 181 28.18/ 103 2:002 319 304 362 56.35/ 658:210 687 89.20/
kg u J MeV
1:672 621 923 69.51/ 1027 1:007 276 466 621.53/ 1:503 277 615 98.46/ 1010 938:272 088 16.29/ 1836:152 673 43.11/ 8:414.19/ 1016 1:410 606 797 36.60/ 1026 1:521 032 202 30.46/ 103 2:792 847 344 63.82/ 5:585 694 6893.16/
kg u J MeV
0 ˛ ˛ 1 cR1 hc R1 R1 B B = h N N = h me me c 2 me =m C C re e e =B e =N ae ge e =p mp mp c 2 mp =me rp p p =B p =N gp
Hz J eV [m1 ]a J T1 Hz T1 J T1 MHz T1
m [m]a m J T1
m J T1
3:0 1010 2:9 1011 3:0 1010 3:0 1010 2:2 108 3:0 1010 3:0 1010 4:5 1010 3:0 1010 1:7 1013 6:0 1011 1:5 1010 1:7 1013 3:0 1010 3:1 1010 5:3 1011 3:1 1010 3:1 1010 6:0 1011 2:2 103 4:2 1010 3:0 1010 2:9 1010 2:9 1010
The full description of m1 is cycles or periods per meter and that of m is meter per cycle (m/cycle). The scientific community is aware of the implied use of these units. It traces back to the conventions for phase and angle and the use of unit Hz versus cycles/s. No solution has been agreed upon.
a
1
Units and Constants
7
Table 1.2 (Continued) Quantity Atomic units (a.u.) a.u. of charge a.u. of mass a.u. of action a.u. of length: Bohr radius (bohr) „=˛me c a.u. of energy: Hartree energy (Hartree) ˛ 2 me c 2 D e 2 =4 0 a0 D 2hc R1 a.u. of time a.u. of force a.u. of velocity: ˛c a.u. of momentum a.u. of current a.u. of charge density a.u. of electric potential a.u. of electric field a.u. of electric dipole moment a.u. of electric quadrupole moment a.u. of electric polarizability a.u. of magnetic flux density a.u. of magnetic dipole moment: 2B a.u. of magnetizability a.u. of permittivity Natural units (n.u.) n.u. of velocity n.u. of action
Symbol
Numerical value
Unit
Relative std. uncert. ur
e me „ a0 Eh
1:602 176 634 1019 9:109 383 7015.28/ 1031 1:054 571 817 : : : 1034 5:291 772 109 03.80/ 1011 4:359 744 722 2071.85/ 1018
C kg Js m J
exact 3:0 1010 exact 1:5 1010 1:9 1012
„=Eh Eh =a0 a0 Eh =„ „=a0 eEh =„ e=a03 Eh =e Eh =ea0 ea0 ea02 e 2 a02 =Eh „=ea02 „e=me e 2 a02 =me e 2 =a0 Eh
2:418 884 326 5857.47/ 1017 8:238 723 4983.12/ 108 2:187 691 263 64.33/ 106 1:992 851 914 10.30/ 1024 6:623 618 237 510.13/ 103 1:081 202 384 57.49/ 1012 27:211 386 245 988.53/ 5:142 206 747 63.78/ 1011 8:478 353 6255.13/ 1030 4:486 551 5246.14/ 1040 1:648 777 274 36.50/ 1041 2:350 517 567 58.71/ 105 1:854 802 015 66.56/ 1023 7:891 036 6008.48/ 1029 1:112 650 055 45.17/ 1010
s N m s1 kg m s1 A C m3 V V m1 Cm C m2 C2 m2 J1 T J T1 J T2 F m1
1:9 1012 1:5 1010 1:5 1010 1:5 1010 1:9 1012 4:5 1010 1:9 1012 1:5 1010 1:5 1010 3:0 1010 3:0 1010 3:0 1010 3:0 1010 6:0 1010 1:5 1010
c „
299 792 458 1:054 571 817 : : : 1034 6:582 119 569 : : : 1016 197:326 980 4 : : : 9:109 383 7015.28/ 1031 8:187 105 7769.25/ 1014 0:510 998 950 00.15/ 2:730 924 530 75.82/ 1022 0:510 998 950 00.15/ 3:861 592 6796.12/ 1013 1:288 088 668 19.39/ 1021
m s1 Js eV s MeV fm kg J MeV kg m s1 MeV/c m s
exact exact exact exact 3:0 1010 3:0 1010 3:0 1010 3:0 1010 3:0 1010 3:0 1010 3:0 1010
n.u. of mass n.u. of energy
„c me me c 2
n.u. of momentum
me c
n.u. of length: „=me c n.u. of time
C „=me c 2
2:2 103 . In fact, rp extracted from precision spectroscopy References on the hydrogen atom and on muonic-hydrogen, where the electron is replaced by a muon, are marginally discrepant. 1. Bureau International des Poids et Mesures: The official description The fine-structure constant ˛, vacuum electric permittivof the International System of Units (2019). https://www.bipm. org/en/measurement-units ity 0 , and vacuum magnetic permeability 0 are dependent. 2. Mills, I.M., Mohr, P.J., Quinn, T.J., Taylor, B.N., Williams, E.R.: They are related through Eqs. (1.3) and (1.9) and, since Adapting the international system of units to the twenty-first cenin the SI the values for „, e, and c are exact, one of ˛, tury. Phil. Trans. R. Soc. A 369, 3907–3924 (2011) 0 , or 0 fixes the other two. In current state-of-the-art ex3. Jackson, J.D.: Classical Electrodynamics. Wiley, New York, London (1962) periments that constrain these constants, the dimensionless 4. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Atomfine-structure constant is measured or the most conveniently Photon Interactions. Wiley, New York, London (1992) extracted. In fact, the CODATA adjustment uses ˛ as an ad5. NIST: CODATA internationally recommended values of the justed constant, and values for 0 and 0 are derived from fundamental physical constants (2019). https://physics.nist.gov/ constants Eqs. (1.3) and (1.9). Finally, the mass of the electron follows 2 2 from me D Eh =˛ c . Its relative uncertainty is twice that of ˛ as the relative uncertainty of the Hartree energy is much better known.
1
8
E. Tiesinga Eite Tiesinga Dr Eite Tiesinga received his PhD from Eindhoven University of Technology in The Netherlands in 1993. He works at the National Institute of Standards and Technology and the University of Maryland in the US. His research focuses on developing clocks and sensors using laser-cooled atoms. He maintains a web-based database of fundamental constants, an initiative by the Committee on Data of the International Science Council (CODATA).
2
Angular Momentum Theory James D. Louck
Contents
2.9.3 2.9.4
2.1 2.1.1 2.1.2
Orbital Angular Momentum . . . . . . . . . . . . . . . . Cartesian Representation . . . . . . . . . . . . . . . . . . . Spherical Polar Coordinate Representation . . . . . . . .
12 12 14
2.2
Abstract Angular Momentum . . . . . . . . . . . . . . .
15
2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6
Representation Functions . . . . . . . . . . . . . . . Parametrizations of the Groups SU(2) and SO(3,R) Explicit Forms of Representation Functions . . . . . Relations to Special Functions . . . . . . . . . . . . . Orthogonality Properties . . . . . . . . . . . . . . . . Recurrence Relations . . . . . . . . . . . . . . . . . . . Symmetry Relations . . . . . . . . . . . . . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
17 17 19 19 20 21 21
2.4 2.4.1 2.4.2 2.4.3 2.4.4
Group and Lie Algebra Actions . . . . Matrix Group Actions . . . . . . . . . . . Lie Algebra Actions . . . . . . . . . . . . Hilbert Spaces . . . . . . . . . . . . . . . . Relation to Angular Momentum Theory
. . . . .
. . . . .
. . . . .
24 24 24 25 25
2.5
Differential Operator Realizations of Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.6
The Symmetric Rotor and Representation Functions
27
2.7 2.7.1 2.7.2 2.7.3 2.7.4 2.7.5 2.7.6 2.7.7
Wigner–Clebsch–Gordan and 3–j Coefficients . . . Kronecker Product Reduction . . . . . . . . . . . . . . . Tensor Product Space Construction . . . . . . . . . . . . Explicit Forms of WCG Coefficients . . . . . . . . . . . Symmetries of WCG Coefficients in 3–j Symbol Form Recurrence Relations . . . . . . . . . . . . . . . . . . . . . Limiting Properties and Asymptotic Forms . . . . . . . WCG Coefficients as Discretized Representation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . .
29 30 31 31 33 33 34
.
34
2.8 2.8.1 2.8.2 2.8.3 2.8.4 2.8.5
Tensor Operator Algebra . . . . . . . . . . Conceptual Framework . . . . . . . . . . . . Universal Enveloping Algebra of J . . . . . Algebra of Irreducible Tensor Operators . . Wigner–Eckart Theorem . . . . . . . . . . . Unit Tensor Operators or Wigner Operators
. . . . . .
35 35 36 36 37 37
2.9 2.9.1 2.9.2
Racah Coefficients . . . . . . . . . . . . . . . . . . . . . . Basic Relations Between WCG and Racah Coefficients Orthogonality and Explicit Form . . . . . . . . . . . . . .
40 40 40
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The Fundamental Identities Between Racah Coefficients Schwinger–Bargmann Generating Function and Its Combinatorics . . . . . . . . . . . . . . . . . . . . . Symmetries of 6–j Coefficients . . . . . . . . . . . . . . . Further Properties . . . . . . . . . . . . . . . . . . . . . . . .
41
The 9–j Coefficients . . . . . . . . . . . . . . . . . . . . . . Hilbert Space and Tensor Operator Actions . . . . . . . . 9–j Invariant Operators . . . . . . . . . . . . . . . . . . . . Basic Relations Between 9–j Coefficients and 6–j Coefficients . . . . . . . . . . . . . . . . . . . . . . Symmetry Relations for 9–j Coefficients and Reduction to 6–j Coefficients . . . . . . . . . . . . . . . . . . . . . . . Explicit Algebraic Form of 9–j Coefficients . . . . . . . . Racah Operators . . . . . . . . . . . . . . . . . . . . . . . . Schwinger–Wu Generating Function and Its Combinatorics . . . . . . . . . . . . . . . . . . . . .
44 44 44
2.11 2.11.1 2.11.2 2.11.3
Tensor Spherical Harmonics . . . . . . . . . . . . Spinor Spherical Harmonics as Matrix Functions Vector Spherical Harmonics as Matrix Functions Vector Solid Harmonics as Vector Functions . . .
48 49 49 50
2.12 2.12.1 2.12.2 2.12.3 2.12.4
Coupling and Recoupling Theory and 3n–j Coefficients Composite Angular Momentum Systems . . . . . . . . . Binary Coupling Theory: Combinatorics . . . . . . . . . . Implementation of Binary Couplings . . . . . . . . . . . . Construction of All Transformation Coefficients in Binary Coupling Theory . . . . . . . . . . . . . . . . . . Unsolved Problems in Recoupling Theory . . . . . . . . .
51 51 52 53
Supplement on Combinatorial Foundations . . . . . SU(2) Solid Harmonics . . . . . . . . . . . . . . . . . . . Combinatorial Definition of Wigner–Clebsch–Gordan Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . Magic Square Realization of the Addition of Two Angular Momenta . . . . . . . . . . . . . . . . . . MacMahon’s and Schwinger’s Master Theorems . . . . The Pfaffian and Double Pfaffian . . . . . . . . . . . . . Generating Functions for Coupled Wave Functions and Recoupling Coefficients . . . . . . . . . . . . . . . .
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56 56
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57
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59 60 61
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61
2.14
Author’s Comments . . . . . . . . . . . . . . . . . . . . . .
64
2.15
Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
2.9.5 2.9.6 2.10 2.10.1 2.10.2 2.10.3 2.10.4 2.10.5 2.10.6 2.10.7
2.12.5 2.13 2.13.1 2.13.2 2.13.3 2.13.4 2.13.5 2.13.6
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_2
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41 42 43
45 46 46 46 47
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9
10
J. D. Louck
Abstract
Angular momentum theory is presented from the viewpoint of the group SU.1/ of unimodular unitary matrices of order 2. This is the basic quantum mechanical rotation group for implementing the consequences of rotational symmetry into isolated complex physical systems and gives the structure of the angular momentum multiplets of such systems. This entails the study of representation functions of SU.2/, the Lie algebra of SU.2/ and copies thereof, and the associated Wigner–Clebsch–Gordan coefficients, Racah coefficients, and 3nj coefficients, with an almost boundless set of interrelations, and presentations of the associated conceptual framework. The relationship of SU.2/ to the rotation group in physical 3-space R3 is given in detail. Formulas are often given in a compendium format with brief introductions on their physical and mathematical content. A special effort is made to interrelate the material to the special functions of mathematics and to the combinatorial foundations of the subject. Keywords
Presentation of a point in R 3 : x D col .x1 ; x2 ; x3 /
column matrix ;
T
x D .x1 ; x2 ; x3 / XD
x3 x1 C ix2
x1 ix2 x3
!
row matrix ;
2 2 traceless Hermitian matrix I Cartan’s representation : A one-to-one correspondence between the set R 3 of points in 3-space and the set H 2 of 2 2 traceless Hermitian matrices is obtained from xi D 12 Tr .i X/, where the i denote the matrices (Pauli matrices) 1 D
0 1 1 0
! ; 2 D
0 i i 0
! ; 3 D
! 1 0 : 0 1 (2.1)
Mappings of R 3 onto itself: x ! x 0 D Rx ;
X ! X 0 D UXU ; angular momentum algebra invariant operator coupling scheme symmetry relation tensor operator Wigner–Eckart theorem Lie algebra Racah coefficients where denotes Hermitian conjugation of a matrix or an operator. Clebsch-Gordan coefficients rotation group Two-to-one homomorphism of SU.2/ onto SO.3; R/: Angular momentum theory in its quantum mechanical applications, which is the subject of this section, is the study of the group of 22 unitary unimodular matrices and its irreducible representations. It is the mathematics of implementing into physical theory the basic tenet that isolated physical systems are invariant to rotations of the system in physical 3-space, denoted R 3 , or, equivalently, to the orientation of a Cartesian reference system used to describe the system. That it is the group of 2 2 unimodular matrices that is basic in quantum theory in place of the more obvious group of 3 3 real, orthogonal matrices representing transformations of the coordinates of the constituent particles of the system, or of the reference frame, is a consequence of the Hilbert space structure of the state space of quantum systems and the impossibility of assigning overall phase factors to such states because measurements depend only on the absolute value of transition amplitudes. The exact relationship between the group SU.2/ of 2 2 unimodular unitary matrices and the group SO.3; R/ of 3 3 real, proper, orthogonal matrices is an important one for keeping the quantum theory of angular momentum, with its numerous conventions and widespread applications across all fields of quantum physics, free of ambiguities. These notations and relations are fixed at the outset.
1
Rij D Rij .U / D Tr i Uj U ; 2 1 0 1 0 0 0 ! ! B C C B0 C B D DB x R.U / C x0 A @0 0 ! D A .U U /A ; x
(2.2)
(2.3)
where is an indeterminate, A is the unitary matrix given by 0
1 B 1 B 0 AD p B B 2 @0 1
1 0 0 1 C 1 i 0 C C; 1 i 0C A 0 0 1
U U denotes the matrix direct product, and denotes complex conjugation. There is a simple unifying theme in almost all the applications. The basic mathematical notions that are implemented over and over again in various contexts are: group action on the underlying coordinates and momenta of the physical system and the corresponding group action
2
Angular Momentum Theory
11
in the associated Hilbert space of states; the determination of those subspaces that are mapped irreducibly onto themselves by the group action; the Lie algebra and its actions as derived from the group actions, and conversely; the construction of composite objects from elementary constituents, using the notion of tensor product space and Kronecker products of representations, which are the basic precepts in quantum theory for building complex systems from simpler ones; the reduction of the Kronecker product of irreducible representations into irreducibles with the associated Wigner– Clebsch–Gordan and Racah coefficients determining not only this reduction but also having a dual role in the construction of the irreducible state spaces themselves; and, finally, the repetition of this process for many-particle systems with the attendant theory of 3nj coefficients. The universality of this methodology may be attributed to being able, in favorable situations, to separate the particular consequences of physical law (e.g., the Coulomb force) from the implications of symmetry imposed on the system by our underlying conceptions of space and time. Empirical models based on symmetry that attempt to identify the more important ingredients underlying observed physical phenomena are also of great importance. The group actions in complex systems are often modeled after the following examples for the actions of the groups SO.3; R/ and SU.2/ on functions defined over the twosphere S 2 R 3 : Hilbert space: V D ff jf is a polynomial satisfying r 2 f .x/ D 0g : Inner or scalar product:
R ! OR is a unitary representation of SO.3; R/; that is, OR1 OR2 D OR1 R2 .
U ! TU is a unitary representation of SU.2/; that is, TU1 TU2 D TU1 U2 .
OR.U / D TU D TU is an operator identity on the space V. One parameter subgroups: Uj .t/ D exp.itj =2/ ;
t 2R;
j D 1; 2; 3 I
Rj .t/ D R.Uj .t// D exp.itMj / ; t 2R;
j D 1; 2; 3 I
where 0
0 B M1 D i@0 0 0 0 B M3 D i@1 0
1 0 0 C 0 1A ; 1 0 1 1 0 C 0 0A : 0
0
1 0 0 1 B C M2 D i@ 0 0 0A ; 1 0 0 (2.4)
0
Infinitesimal generators: Lj D i.dORj .t / =dt/ t D0 ; Lj D i.dTUj .t / =dt/ t D0 ; @ @ .Lj f /.x/ D i xk xl f .x/ ; @xl @xk j; k; l cyclic in 1, 2, 3 :
(2.5)
Z
f; f 0 D
Historically, the algebra of angular momentum came about through the quantum rule of replacing the linear momentum unit sphere p of a classical point particle, which is located at position r, by p ! i„r, thus replacing the classical angular momenwhere f .x/ D f .X/ for x presented in the Cartan matrix tum r p about the origin of a chosen Cartesian inertial form X. system by the angular momentum operator: Group actions: f .x/f 0 .x/ dS ;
.OR f /.x/ D f R1 x ;
L D i r r
.in units of „/ :
(2.6)
each f 2 V ; each x 2 R 3 ;
The quantal angular momentum properties of this simple one-particle system are then to be inferred from the propeach f 2 V ; .TU f /.X/ D f U XU ; erties of these operators and their actions in the associated each X 2 H 2 : Hilbert space. This remains the method of introducing angular momentum theory in most textbooks because of its Operator properties: simplicity and historical roots. It also leads to focusing the developments of the theory on the algebra of operators in
OR is a unitary operator on V ; that is, (OR f; OR f 0 / D contrast to emphasizing the associated group transformations .f; f 0 /. of the Hilbert space, although the two viewpoints are inti TU is a unitary operator on V ; that is, .TU f; TU f 0 / D mately linked, as illustrated above. Both perspectives will be .f; f 0 /. presented here.
2
12
J. D. Louck
2.1
Orbital Angular Momentum
Action of angular momentum operators:
1 The model provided by orbital angular momentum operators L˙ Ylm .x/ D Œ.l m/.l ˙ m C 1/ 2 Yl;m˙1 .x/ ; is the paradigm for standardizing many of the conventions L3 Ylm .x/ D mYlm .x/ ; and relations used in more abstract and general treatments. L2 Ylm .x/ D l.l C 1/Ylm : These basic results for the orbital angular momentum operator L D i r r acting in the vector space V are given in this section both in Cartesian coordinates x D col .x1 ; x2 ; x3 / Highest weight eigenfunction: and spherical polar coordinates: LC Yl l .x/ D 0 ; L3 Yl l .x/ D l Yl l .x/ ; x D .r sin cos ; r sin sin ; r cos / ; 1 1 .2l C 1/Š 2 Yl l .x/ D l .x1 ix2 /l : 0 r < 1 ; 0 < 2 ; 0 : 2 lŠ 4
2.1.1
Cartesian Representation
Generation from highest weight:
Commutation relations: Cartesian form:
Ylm .x/ D
ŒL1 ; L2 D iL3 ;
ŒL2 ; L3 D iL1 ;
ŒL3 ; L1 D iL2 :
.l C m/ .2l/Š.l m/Š
12
Llm Yl l .x/ :
Relation to Gegenbauer and Jacobi polynomials: Ylm .x/ D r ljmj Ym .x1 ; x2 / 1
Cartan form:
Œ.2l C 1/.l C m/Š.l m/Š=2 2
ŒL3 ; LC D LC ;
ŒL3 ; L D L ;
ŒLC ; L D 2L3 : Squared orbital angular momentum: L2 D L21 C L22 C L23 D L LC C L3 .L3 C 1/ D LC L C L3 .L3 1/ D r 2 r 2 C .x r/2 C .x r/ : L2 ; L3 form a complete set of commuting Hermitian operators in V with eigenfunctions
12 2l C 1 Ylm .x/ D .l C m/Š.l m/Š 4 X .x1 ix2 /kCm .x1 ix2 /k x lm2k 3 ; 22kCm .k C m/ŠkŠ.l m 2k/Š k
Hl;jmj .x3 =r/ ; .2/Š C 1 .l C /Š .;/ Pl .z/ ; Hl .z/ D Cl 2 .z/ D 2 Š 2 lŠ 0 l D 0; 1; 2; : : : ; where the Ym .x1 ; x2 / are homogeneous polynomial solutions of degree jmj of Laplace’s equation in 2-space, R 2 : ( p .x1 ix2 /m = 2 ; m 0 ; p Ym .x1 ; x2 / D .x1 ix2 /m = 2 ; m 0 : (Sect. 2.1.2 for the definition of Gegenbauer and Jacobi polynomials.) Orthogonal group action: .OR Ylm /.x/ D Ylm .R1 x/ D
X
Dlm0 m .R/Ylm0 .x/ ;
m0
where l D 0; 1; 2; : : : ; I m D l; l 1; : : : ; l. l l Homogeneous polynomial solutions of Laplace’s equa- where the functions Dm0 m .R/ D Dm0 m .U.R// are defined in Sect. 2.3 for various parametrizations of R. tion: Unitary group action: l Ylm .x/ D Ylm .x/ ; .TU Ylm /.X/ D Ylm .U XU / .x r/Ylm .x/ D l Ylm .x/ ; X l D Dm 0 m .U /Ylm0 .X/ ; r 2 Ylm .x/ D 0 : m0
Complex conjugate: Ylm .x/
m
D .1/ Yl;m .x/ :
l where the functions Dm 0 m .U / are defined in Sects. 2.2 and 2.3.
2
Angular Momentum Theory
13
Legendre polynomials:
Orthogonality on the unit sphere: Z Yl0 m0 .x/Ylm .x/dS D ıl 0 l ım0 m : unit sphere
Product of solid harmonics: X lkl 0 Yk .x/Ylm .x/ D hl 0 jjYk jjliCm;;mC Yl 0 ;mC .x/ l0
D
X l0
4 X .1/m Yl;m .y/ O Ylm .x/ O ; 2l C 1 m ! ! 12 2l 2k 4 1 X k l Yl0 .x/ D l .1/ 2l C 1 2 l k k Pl .cos / D
! l k l0 .l jjYk jjl/ m m
x3l2k .x x/k D r l Pl .x3 =r/ :
0
0
.1/l CmC Yl 0 ;mC .x/ ; 12 0 lCkl 0 .2l C 1/.2k C 1/ lkl 0 C000 ; hl jjYk jjli D r 0 4.2l C 1/ 12 0 0 lCkl 0 .2l C 1/.2k C 1/.2l C 1/ .l jjYk jjl/ D r 4 ! 0 l k l 0 .1/l ; 0 0 0
Rayleigh plane wave expansion:
eikx D 4
l2 m2
jl .kr/ D
0
Ylm .z C z / D
X k
j k;k;l Cm;;m
4.2l C 1/Š .2l 2k C 1/Š.2k C 1/Š
12
lk;k;l Cm;;m
Ylk;m .z/Yk .z0 / ; z; z0 2 C 3 ; !# 12 ! ! " 2l l Cm l m D : 2k kC k
Rotational invariant in two vectors: ! ! 1 X 2l 2k k l .1/ Il .x; y/ D l 2 k l k
.1=2/
.z/ is a Gegenbauer polynomial (Sect. 2.1.2),
xO D x=jxj ;
yO D y=jyj ;
cos D xO yO :
JlC1=2 .kr/ :
lD0
Il .x; y/=r 2lC1 D
For R D x y ;
1=R D
X
1 .1/l .y r/l : lŠ r 1
1
s D . y y/ 2 ,
r D .x x/ 2 ;
Pl .cos /
l
sl r lC1
;
s r; cos D xO yO :
Rotational invariants in three vectors:
I.l1 l2 l3 / x 1 ; x 2 ; x 3 D
.1=2/
where Cl and
! 12
1 .1/l .2l/Š Ylm .x/ Ylm .r/ ; D r 2l lŠ r 2lC1 1 X 1=R D Il .x; y/=r 2lC1 ;
.x y/l2k .x x/k .y y/k D .x x/l=2 .y y/l=2 Cl .xO y/ O 4 X .1/m Ylm .x/Yl;m .y/ ; D 2l C 1 m
2kr
Relations in potential theory:
! l .1/l1 l2 Cm l1 l2 l Cm 1 m 2 m D p m 2l C 1
denote Wigner–Clebsch–Gordan coefficients and 3j coefficients, respectively (Sect. 2.7). Vector addition theorem for solid harmonics:
O il jl .kr/Ylm O ; k Ylm .x/
lD0 mDl
2l and where Cml11lm 2m
l1 m1
1 X l X
l1 where m1
l2 m2
.4/3=2 1
Œ.2l1 C 1/.2l2 C 1/.2l3 C 1/ 2 ! X l2 l3 l1 m1 m2 m3 m1 m2 m3
Yl1 m1 x 1 Yl2 m2 x 2 Yl3 m3 x 3 ;
! l3 is a 3j coefficient (Sect. 2.7). m3
I.l1 l2 l3 / x 1 ; x 2 ; 0
1
ı
D ıl1 l2 ıl3 0 .1/l1 Il1 x 1 ; x 2 2l1 C 1 2 :
2
14
J. D. Louck
Rotational invariant for vectors of zero length:
Product law: I.l/ .x/I.k/ .x/ ! " 3 X Y l˛ k˛ j˛ j˛ D .1/ .2j˛ C 1/ 0 0 0 .j / ˛D1 9 8 #ˆ l1 l2 l3 > = < .x ˛ x ˛ /.l˛ Ck˛ j˛ /=2 k1 k2 k3 I.j / .x/ ; > ˆ ; : j1 j2 j3
.˛ x/ D
4 2l C 1
12
2l lŠ
X
Plm .z/Ylm .x/ :
m
Spinorial invariant under zi ! U zi .i D 1; 2; 3/: X m1 m2 m3
where l D .l1 ; l2 ; l3 /, etc., x D x 1 ; x 2 ; x 3 . Coplanar vectors:
I.l/ x 1 ; x 2 ; ˛x 1 C ˇx 2 1 X .2l3 C 1/Š 2 D .2l3 2k/Š.2k/Š
j1 m1
j2 m2
j3 m3
!
Pj1 m1 z1 Pj2 m2 z2 Pj3 m3 z3
D Œ.j1 C j2 C j3 C 1/Š1=2
12 j1 Cj2 j3 31 j3 Cj1 j2 23 j2 Cj3 j1 z12 z12 z12 ; 1 Œ.j1 C j2 j3 /Š.j3 C j1 j2 /Š.j2 C j3 j1 /Š 2 ij
j
j
z12 D z1i z2 z1 z2i :
kl
˛ l3 k ˇ k .1/l1 Cl3 Ck .2l C 1/ ! !( k l2 l l3 k l3 l3 k l1 l 0 0 0 0 0 0 l2 l
1 1 .l1 Cl3 lk/=2 2 2 .l2 Ckl/=2 x x x x
1 2 Il x ; x :
l
This relation is invariant under the transformation k l1
)
z ! U z D .U z/1 ; .U z/2 D .u11 z1 C u12 z2 ; u21 z1 C u22 z2 / ; where U 2 SU.2/. Transformation properties of vectors of zero length:
The bracket symbols in these relations are 6j and 9j co˛ ! R˛; ˛ D col.˛1 ; ˛2 ; ˛3 / ; efficients (Sects. 2.9, 2.10). Cartan’s vectors of zero length: where z ! U z and R is given in terms of U in the beginning
2 of this chapter. Simultaneous eigenvectors of L2 and J 2 : 2 2 2 ˛ D z1 C z2 ; i z1 C z2 ; 2z1 z2 ; ˛ ˛ D ˛12 C ˛22 C ˛32 D 0 ; 2
z D .z1 ; z2 / 2 C :
L2 .˛ x/l D l.l C 1/.˛ x/l ;
l D 0; 1; : : : ;
J 2 .˛ x/l D l.l C 1/.˛ x/l ;
l D 0; 1; : : : :
Solutions of Laplace’s equation using vectors of zero length: r 2 .˛ x/l D 0 ;
l D 0; 1; : : : :
Solid harmonics for vectors of zero length: .1/m Yl;m .˛/ D
1 .2l/Š 2l C 1 2 Plm .z1 ; z2 / ; lŠ 4
z1lCm z2lm : Plm .z1 ; z2 / D p .l C m/Š.l m/Š Orbital angular momentum operators for vectors of zero length:
2.1.2 Spherical Polar Coordinate Representation The results given in Sect. 2.1.1 may be presented in any system of coordinates well-defined in terms of Cartesian coordinates. The principal relations for spherical polar coordinates are given in this section, where a vector in R 3 is now given in the form x D r xO D r.sin cos ; sin sin ; cos / ; 0 ;
0 < 2 :
Orbital angular momentum operators: J D i.˛ r˛ / ; @ @ ; J D z2 ; JC D z1 @z2 @z1 @ @ 1 J3 D z2 z1 : 2 @z1 @z2
@ @ C i sin ; @ @ @ @ L2 D i sin cot i cos ; @ @
L1 D i cos cot
2
Angular Momentum Theory
15
@ ; @ @ @ ; L˙ D e˙i ˙ C i cot @ @ 1 @ @ 1 @2 2 L D : sin sin @ @ sin2 @ 2 L3 D i
Laplacian: @ ; @r" # 2 1 @ @ Cr L2 : r2 D 2 r r @r @r
xr Dr
Spherical harmonics (solid harmonics on the unit sphere S 2 ): 12 m 2l C 1 .l C m/Š.l m/Š eim Ylm .; / D .1/ 4 X .1/k .sin /2kCm .cos /l2km : 22kCm .k C m/ŠkŠ.l 2k m/Š
k
Relations between Jacobi polynomials for n C ˛, n C ˇ, n C ˛ C ˇ nonnegative integers: .n C ˛/Š.n C ˇ/Š x C 1 ˇ .˛;ˇ/ PnCˇ .x/ ; nŠ.n C ˛ C ˇ/Š 2 .n C ˛/Š.n C ˇ/Š x 1 ˛ .˛;ˇ/ PnC˛ .x/ ; Pn˛;ˇ .x/ D nŠ.n C ˛ C ˇ/Š 2 x 1 ˛ x C 1 ˇ .˛;ˇ/ ˛;ˇ PnC˛Cˇ .x/ : Pn .x/ D 2 2 Pn˛;ˇ .x/ D
Nonstandard form (˛ arbitrary): Pn.˛;˛/ .x/ D
X .1/s .˛ C s C 1/ns 1 x 2 s x n2s ; 22s sŠ.n 2s/Š s
.z/k D z.z C 1/ .z C k 1/ ; .z/0 D 1 : Gegenbauer polynomials .˛ > 1=2/: Cn.˛/ .x/
Orthogonality on the unit sphere: Z2
Z
˛ 12 ;˛ 12 .2˛/n D Pn .x/ .˛ C 1=2/n X .1/s .˛/ns .2x/n2s D : sŠ.n 2s/Š s
d sin Yl0 m0 .; / Ylm .; / D ıl 0 l ım0 m :
d 0
k D 1; 2; : : : I
0
2.2 Abstract Angular Momentum
Relation to Legendre, Jacobi, and Gegenbauer polynomials: Ylm .; / D .1/m
.2l C 1/.l m/Š 4.l C m/Š
12
Plm .cos / eim ; .l C m/Š sin m .m;m/ Plm .cos / D Plm .cos / : lŠ 2 Jacobi polynomials: Pn.˛;ˇ/ .x/
! ! X nC˛ nCˇ D ns s s ns x1 xC1 s ; 2 2
Abstract angular momentum theory addresses the problem of constructing all finite Hermitian matrices, up to equivalence, that satisfy the same commutation relations ŒJ1 ; J2 D iJ3 ;
ŒJ2 ; J3 D iJ1 ;
ŒJ3 ; J1 D iJ2
(2.7)
as some set of Hermitian operators J1 ; J2 ; J3 appropriately defined in some Hilbert space; that is, of constructing all finite Hermitian matrices Mi such that under the correspondence Ji ! Mi .i D 1; 2; 3/ the commutation relations are still obeyed. If M1 ; M2 ; M3 is such a set of Hermitian matrices, then AM1 A1 ; AM2 A1 ; AM3 A1 , is another such set, where A is an arbitrary unitary matrix. This defines what is meant by equivalence. The commutation relations Eq. (2.7) may also be formulated as:
n D 0; 1; : : : ; ŒJ3 ; J˙ D ˙J˙ ; where ˛; ˇ are arbitrary parameters and 8 ˆz.z 1/ .z k C 1/=kŠ ˆ ! ˆ ˆ < for k D 1; 2; : : : z D ˆ 1 for k D 0 k ˆ ˆ ˆ :0 for k D 1; 2; : : :
J˙ D J1 ˙ iJ2 ;
ŒJC ; J D 2J3 ;
JC D J :
(2.8)
The squared angular momentum J 2 D J12 C J22 C J32 D J JC C J3 .J3 C 1/ D JC J C J3 .J3 1/
(2.9)
2
16
J. D. Louck
commutes with each Ji , and J3 is, by convention, taken with J 2 as a pair of commuting Hermitian operators to be diagonalized. Examples of matrices satisfying relations Eq. (2.7) are provided by Ji ! i =2 (the 2 2 Hermitian Pauli matrices defined in Eq. (2.1)) and Ji ! Mi (the 3 3 matrices defined in Eq. (2.4)), these latter matrices being equivalent to those obtained from the matrices of the orbital angular momentum operators for l D 1. One could determine all Hermitian matrices solving Eqs. (2.7) and (2.8) by using only matrix theory, but it is customary in quantum mechanics to formulate the problem using Hilbert space concepts appropriate to that theory. Thus, one takes the viewpoint that the Ji are linear Hermitian operators with an action defined in a separable Hilbert space H such that Ji W H ! H . One then seeks to decompose the Hilbert space into a direct sum of subspaces that are irreducible with respect to this action; that is, subspaces that cannot be further decomposed as a direct sum of subspaces that all the Ji leave invariant (map vectors in the space into vectors in the space). In this section, the solution of this fundamental problem for angular momentum theory is given. These results set the notation and phase conventions for all of angular momentum theory, in all of its varied realizations, and the relations are, therefore, sometimes referred to as standard. The method most often used to solve the posed problem is called the method of highest weights. The solution of this problem is among the most important in quantum theory because of its generality and applicability to a wide range of problems. The space H can be written as a direct sum H D
X
˚ nj Hj ;
fjj mijm D j; j C 1; : : : ; j g : 0
hj m j j mi D ı
j ¤ j0 ;
(2.10)
m0 ;m
:
(2.11) (2.12)
Simultaneous eigenvectors: J 2 j j mi D j.j C 1/j j mi ;
J3 j j mi D mj j mi : (2.13)
Action of angular momentum operators: JC j j mi D Œ.j m/.j C m C 1/1=2 j j m C 1i ; J j j mi D Œ.j C m/.j m C 1/1=2 j j m 1i :
(2.14)
Defining properties of highest weight vector: JC j jj i D 0 ;
J3 j jj i D j j jj i :
Generation of general vector from highest weight: j j mi D
.j C m/Š .2j /Š.j m/Š
1=2 Jj m j jj i :
Necessary property of lowest weight vector: J j j; j i D 0 ;
J3 j j; j i D j j j; j i :
Operator in H corresponding to a rotation by angle direction nO in R 3 : TU.
j D0; 12 ;1;:::
each Hj ? Hj 0 ;
ous physical systems constitutes spectroscopy in the broadest sense. The characterization of the space Hj with respect to angular momentum properties is given by the following results, where basis vectors are denoted in the Dirac bra-ket notation. Orthonormal basis:
;n/ O
about
D exp.i nO J / ;
nO nO D n21 C n22 C n23 D 1 ; nO J D n1 J1 C n2 J2 C n3 J3 ;
U. ; n/ O D exp.i nO =2/ in which Hj denotes a vector space of dimension 2j C 1
that is invariant and irreducible under the action of the set D 0 cos 12 i.nO / sin 12
!
of operators Ji ; i D 1; 2; 3, and where the direct sum is over in3 sin 12 .in1 n2 / sin 12 cos 12 all half-integers j D 0; 12 ; 1; : : : . There may be multiple oc
D .in1 C n2 / sin 12 cos 12 C in3 sin 12 currences, nj in number, of the same space Hj for given j , or no such space, nj D 0, in the direct sum. Abstractly, in 0 2 ; (2.15) so far as angular momentum properties are concerned, each repeated space Hj is identical. Such spaces may, however, where 0 denotes the 2 2 unit matrix. be distinguished by their properties with respect to other Action of TU. ;n/ O on Hj : physical observables, but not by the angular action of moX j mentum operators themselves. The result, Eq. (2.10), applies Dm0 m .U /j j m0 i ; (2.16) TU j j mi D 0 to any physical system, no matter how complex, in which rom tational symmetry, hence SU.2/ symmetry, is present, even j O and Dm0 m .U / denotes a homogein situations of higher symmetry where SU.2/ is a subgroup. in which U D U. ; n/ Indeed, the resolution of the terms in Eq. (2.10) for vari- neous polynomial of degree 2j defined on the elements
2
Angular Momentum Theory
17
uij D Uij . ; n/ O in row i and column j of the matrix U. ; n/ O 2.3 given by Eq. (2.15). The explicit form of this polynomial is
Representation Functions
2.3.1 Parametrizations of the Groups SU(2) and SO(3,R)
j
Dm0 m .U /
2
1
D Œ.j C m/Š.j m/Š.j C m0 /Š.j m0 /Š 2 X .u11 /˛11 .u12 /˛12 .u21 /˛21 .u22 /˛22 : ˛11 Š˛12 Š˛21 Š˛22 Š ˛
The irreducible representations of the quantal rotation group, (2.17) SU.2/, are among the most important quantities in all of angular momentum theory. These are the unitary matrices of dimension 2j C 1, denoted by D j .U /, where this notation The notation ˛ symbolizes a 2 2 array of nonnegative in- is jused to signify that the elements of this matrix, denoted Dm0 m .U /, are functions of the elements uij of the 2 2 tegers with certain constraints: unitary unimodular matrix U 2 SU.2/. It has become standard to enumerate the rows and columns of these matrices ˛11 ˛12 j C m0 in the order j; j 1; : : : ; j as read from top to bottom ˛21 ˛22 j m0 : down the rows and from left to right across the columns (see also Eq. (2.18)). These matrices may be presented in a vaj Cm j m riety of parametrizations, all of which are useful. In order In this array, the ˛ij are nonnegative integers subject to the to make comparisons between the group SO.3; R/ and the row and column constraints (sums) indicated by the (nonneg- group SU.2/, it is most useful to parametrize these groups so that they are related according to the two-to-one homoative) integers j ˙ m; j ˙ m0 . Explicitly, morphism given by Eq. (2.2). ˛11 C ˛12 D j C m0 ; ˛21 C ˛22 D j m0 ; The general parametrization of the group SU.2/ is given in terms of the Euler–Rodrigues parameters corresponding to ˛11 C ˛21 D j C m ; ˛12 C ˛22 D j m : points belonging to the surface of the unit sphere S 3 in R 4 , The summation is over all such arrays. Any one of the ˛ij ˛02 C ˛12 C ˛22 C ˛32 D 1 : (2.19) may serve as a single summation index if one wishes to eliminate the redundancy inherent in the square-array notation. The form Eq. (2.17) is very useful for obtaining symmetry Each U 2 SU.2/ can be written in the form: ! relations for these polynomials (Sect. 2.3.6). ˛0 i˛3 i˛1 ˛2 Unitary property on H : U.˛0 ; ˛/ D i˛1 C ˛2 ˛0 C i˛3 hTU jTU i D h j i ; each 2 H : D ˛0 0 i˛ : Irreducible unitary matrix representation of SU.2/: j
.D j .U //j m0 C1;j mC1 D Dm0 m .U / ; m0 D j; j 1; : : : ; j I
m D j; j 1; : : : ; j ; (2.18)
denotes the element in row j m0 C 1 and column j m C 1. Then, dimension of D j .U / D 2j C 1 and
(2.20)
The R 2 SO.3; R/ corresponding to this U in the two-to-one homomorphism given by Eq. (2.2) is: R.˛0 ; ˛/ 0 2
˛0 C ˛12 ˛22 ˛32 D @ 2˛1 ˛2 C 2˛0 ˛3 2˛1 ˛3 2˛0 ˛2
2˛1 ˛2 2˛0 ˛3 ˛02 C ˛22 ˛32 ˛12 2˛2 ˛3 C 2˛0 ˛1
1
2˛1 ˛3 C 2˛0 ˛2 2˛2 ˛3 2˛0 ˛1 A : ˛02 C ˛32 ˛12 ˛22
(2.21) The procedure of parametrization is implemented uniformly by first parametrizing the points on the unit sphere S 3 so U 2 SU.2/ ; U 0 2 SU.2/ ; as to cover the points in S 3 exactly once, thus obtaining .D j .U // D .D j .U //1 D D j .U / : a parametrization of each U 2 SU.2/. Equation (2.21) is then used to obtain the corresponding parametrization of each Kronecker (direct) product representation: R 2 SO.3; R/, where one notes that R.˛0 ; ˛/ D R.˛0 ; ˛/. Because of this two-to-one correspondence ˙U ! R, the doD j1 .U / D j2 .U / main of the parameters that cover the unit sphere S 3 exactly is a .2j1 C 1/.2j2 C 1/ dimensional reducible representation once will cover the group SO.3; R/ exactly twice. This is of SU.2/. One can also effect the reduction of this represen- taken into account uniformly by redefining the domain for tation into irreducible ones by abstract methods. The results SO.3; R/ so as to cover only the upper hemisphere .˛0 0/ of S 3 . are given in Sect. 2.7. D j .U /D j .U 0 / D D j .U U 0 / ;
18
J. D. Louck
In the active viewpoint (reference frame fixed with points Euler angle parametrization: being transformed into new points), an arbitrary vector x D i˛3 =2 iˇ2 =2 i3 =2 3 e e to the new vector x 0 D U.˛ˇ/ D e col.x
10 ; x20 ; x30 / 2 R is transformed !
1 i=2
i˛=2 col x1 ; x2 ; x3 by the rule x 0 D Rx, or, equivalently, in terms cos 2 ˇ e ei˛=2 sin 12 ˇ ei=2 e
D ; of the Cartan matrix X 0 D UXU . In the passive viewei˛=2 sin 12 ˇ ei=2 ei˛=2 cos 12 ˇ ei=2 point, the basic inertial reference system, which is taken to 0 ˛ < 2 ; 0 ˇ or 2 ˇ 3 ; be a right-handed triad of unit vectors .eO 1 ; eO 2 ; eO 3 /, is trans O O O formed by R to a new right-handed triad f 1 ; f 2 ; f 3 by 0 < 2 ; the rule U.˛; ˇ C 2; / D U.˛ˇ/ I X O fj D Rij eO i ; i D 1; 2; 3 ; R.˛ˇ/ D ei˛M3 eiˇM2 eiM3 0 10 1 i cos ˛ sin ˛ 0 cos ˇ 0 sin ˇ B CB C so that eO i fOj D Rij . In this viewpoint, the coordinates of D @ sin ˛ cos ˛ 0A@ 0 1 0 A one and the same point P undergo a redescription under the 0 0 1 sin ˇ 0 cos ˇ change of frame. If the coordinates of P are .x1 ; x2 ; x3 / rel0 1 ative to the frame .eO 1 ; eO 2 ; eO 3 / and x10 ; x20 ; x30 relative to the cos sin 0
B C frame fO 1 , fO 2 , fO 3 , then @ sin cos 0A 0 0 1 x1 eO 1 C x2 eO 2 C x3 eO 3 D x10 fO 1 C x20 fO 2 C x30 fO 3 ; 1 0 cos ˛ cos ˇ cos cos ˛ cos ˇ sin cos ˛ sin ˇ 0 T so that x D R x. C B sin ˛ cos C B sin ˛ sin Rotation about direction nO 2 S 2 by positive angle C B B D B sin ˛ cos ˇ cos sin ˛ cos ˇ sin sin ˛ sin ˇ C (right-hand rule): C C B C cos ˛ cos A @ C cos ˛ sin 1 1 .˛0 ; ˛/ D cos ; nO sin ; 0 2 ; sin ˇ cos sin ˇ sin cos ˇ 2 2 1 0 ˛ < 2 ; 0 ˇ ; 0 < 2 : U. ; n/ O D exp i nO 2 ! This matrix corresponds to the sequence of frame rotations cos 12 in3 sin 12 .in1 n2 / sin 12 D ; given by .in1 C n2 / sin 12 cos 12 C in3 sin 12 rotate by about eO 3 D .0; 0; 1/ ; R. ; n/ O D exp.i nO M / ; 0
R11
D I3 i sin .nO M / .nO M /2 .1 cos / 1 0 R11 R12 R13 C B D @R21 R22 R23 A ; R31 R32 R33
2 D n1 C 1 n21 cos ;
rotate by ˇ about eO 2 D .0; 1; 0/ ;
rotate by ˛ about eO 3 D .0; 0; 1/ : Equivalently, it corresponds to the sequence of frame rotations given by rotate by ˛ about nO 1 D .0; 0; 1/ ;
R21 D n1 n2 .1 cos / C n3 sin
;
rotate by ˇ about nO 2 D . sin ˛; cos ˛; 0/ ;
R31 D n1 n3 .1 cos / n2 sin
;
rotate by about nO 3 D .cos ˛ sin ˇ; sin ˛ sin ˇ; cos ˇ/ :
R12 D n1 n2 .1 cos / n3 sin
R22 D n22 C 1 n22 cos ;
;
R32 D n2 n3 .1 cos / C n1 sin
;
R13 D n1 n3 .1 cos / C n2 sin
;
R23 D n2 n3 .1 cos / n1 sin
R33 D n23 C 1 n23 cos :
;
This latter sequence of rotations is depicted in Fig. 2.1 in obtaining the frame fO 1 ; fO 2 ; fO 3 from .eO 1 ; eO 2 ; eO 3 /. The four complex numbers .a; b; c; d / D .˛0 C i˛3 ; i˛1 ˛2 ; i˛1 C ˛2 ; ˛0 i˛3 /
The unit vector nO 2 S 2 can be further parametrized in terms are called the Cayley–Klein parameters, whereas the four real numbers .˛0 ; ˛/ defining a point on the surface of the of the usual spherical polar coordinates: unit sphere in four-space, S 3 , are known as the Euler– nO D .sin cos ; sin sin ; cos / ; Rodrigues parameters. The three ratios ˛i =˛0 form the ho0 ; 0 < 2 : mogeneous or symmetric Euler parameters.
2
Angular Momentum Theory
19
Euler angle parametrization:
eˆ 3 = nˆ 1 nˆ 3 = fˆ3
0
j
j
Dm0 m .˛ˇ/ D eim ˛ dm0 m .ˇ/ eim ; j
dm0 m .ˇ/ D hj m0 j eiˇJ2 jj mi 0
fˆ2
α α
nˆ 2
eˆ 2 η
Explicit matrices: 0
fˆ1
Fig. 2.1 Euler angles. The three Euler angles (˛ˇ ) are defined by a sequence of three rotations. Reprinted with the permission of Cambridge University Press, after [1]
2.3.2
1 2
D Œ.j C m /Š.j m /Š.j C m/Š.j m/Š 0
X .1/m0 mCs cos 1 ˇ 2j Cmm 2s 2 .j C m s/ŠsŠ.m0 m C s/Š s
1 m0 mC2s sin 2 ˇ : (2.23) .j m0 s/Š
η
eˆ 1
2 0
Explicit Forms of Representation Functions
1 1 1 ˇ sin ˇ cos 1 B 2 2 C d 2 .ˇ/ D @ 1 A; 1 cos ˇ sin ˇ 2 2 0 1 1 C cos ˇ sin ˇ 1 cos ˇ p B C 2 2 2 B C B C sin ˇ sin ˇ 1 B C: cos ˇ p d .ˇ/ D B p 2 2 C B C @ 1 cos ˇ 1 C cos ˇ A sin ˇ p 2 2 2
The general form of the representation functions is given in its most basic and symmetric form in Eq. (2.17). This form applies to every parametrization, it being necessary only to introduce the explicit parametrizations of U 2 SU.2/ Formal polynomial form (zij are indeterminates): or R 2 SO.3; R/ given in Sect. 2.3.1 to obtain the explicit 1 j Dm0 m .Z/ D Œ.j C m0 /Š.j m0 /Š.j C m/Š.j m/Š 2 results given in this section. A choice is also made for the single independent summation parameter in the ˛-array. The 2 X Y notation for functions is abused by writing .zij /˛ij =.˛ij /Š ; (2.24) ˛
D j .!/ D D j .U.!// ;
D j .Z 0 /D j .Z/ D D j .Z 0 Z/ :
! D set of parameters of U 2 SU.2/ :
Euler–Rodrigues representation .˛0 ; ˛/ 2 S 3 : 1
j
Dm0 m .˛0 ; ˛/ D Œ.j C m0 /Š.j m0 /Š.j C m/Š.j m/Š 2 X .˛0 i˛3 /j Cms .i˛1 ˛2 /m0 mCs .j C m s/Š.m0 m C s/Š s
i;j D1
Boson operator form: j j Put ai D zij .i; j D 1; 2/ in Eq. (2.24). Let aN i denote the Hermitian conjugate boson so that h i h i h i j j j alk ; ai D 0 ; aN lk ; aN i D 0 ; aN lk ; ai D ı kj ıli :
0 Then the boson polynomials are orthogonal in the boson in.i˛1 C ˛2 /s .˛0 C i˛3 /j m s : (2.22) ner product: sŠ.j m0 s/Š 0
Quaternionic multiplication rule for points on the sphere S 3 :
0 0
˛0 ; ˛ .˛0 ; ˛/ D ˛000 ; ˛00 ; ˛000 D ˛00 ˛0 ˛0 ˛ ; ˛00 D ˛00 ˛ C ˛0 ˛0 C ˛0 ˛ I
D ˛00 ; ˛0 D j .˛0 ; ˛/ D D j ˛000 ; ˛00 :
j
2.3.3 Relations to Special Functions Jacobi polynomials (Sect. 2.1.2):
j dm0 m .ˇ/
The ( ; n) O parameters: ˛0 D cos
j N j 0 .A/ j 0i D .2j /Šıj 0j ı0 m0 ım : h0 j D0 .A/D mm
1 2
;
˛ D nO sin
1 2
:
0 1 .j C m/Š.j m/Š 2 1 mm D ˇ sin .j C m0 /Š.j m0 /Š 2 m0 Cm 1 .mm0 ;mCm0 / cos ˇ Pj m .cos ˇ/ ; 2
20
J. D. Louck j
0
j
0
j
d˝ D d! sin2 d ;
dm0 m .ˇ/ D .1/m m dm0 ;m .ˇ/
d! D d sin ;
j
D .1/m m dmm0 .ˇ/ D dmm0 .ˇ/ :
d D invariant surface measure for S 2 I
Legendre polynomials:
Z2
1 .l m/Š 2 m l .ˇ/ D .1/m Pl .cos ˇ/ Dm0 .l C m/Š 1 .l C m/Š 2 m D Pl .cos ˇ/ : .l m/Š
Z 0
0 2
2 ıjj 0 ım0 0 ım : 2j C 1
D
Coordinates . ; n/ O for S 3 :
Spherical harmonics: Ylm .ˇ˛/ D
2l C 1 4
12
.˛0 ; ˛/ D cos
; nO sin ; 2 2 0 2 ; nO nO D 1 ; d ; d˝ D dS.n/ O sin2 2 2 dS.n/ O D d! for nO D .sin cos ; sin sin ; cos / ;
l eim˛ dm0 .ˇ/
1 2l C 1 2 l Dm0 .˛ˇ/ ; 4 .ˇ˛/ D .1/m Yl;m .ˇ˛/ : Ylm D
Z2
Z
Gegenbauer polynomials:
dS.n/ O 1
l .ˇ/ D .1/m Œ.l C m/Š.l m/Š 2 dm0 8 9m sin ˇ > .2m/Š ˆ .mC1=2/ ˆ > .cos ˇ/ ; : ; Clm mŠ 2
j
r42 Dm0 m .x0 ; x/ D 0 ;
D
m0:
sin
2
ˇ 1 ˇ 1 cos . C ˛/; sin sin . ˛/; 2 2 2 2 ˇ 1 ˇ 1 sin cos . ˛/; cos sin . C ˛/ ; 2 2 2 2
.˛0 ; ˛/ D cos
2.3.4 Orthogonality Properties
1 d˛d sin ˇdˇ ; 8 Z2 Z2 Z 1 j j0 d˛ d dˇ sin ˇDm0 m .˛ˇ/D0 .˛ˇ/ 8 d˝ D
0
1 C 8
Inner (scalar) product: Z .; ˚/ D d˝ .x/˚.x/ ;
0
0
Z2
Z2 d˛
0
Z3
0
d˝ D invariant surface measure for S ;
(2.25)
j0
j
dˇ sin ˇDm0 m .˛ˇ/D0 .˛ˇ/
d 2
2 2 D ıjj 0 ım0 0 ım : 2j C 1
3
(2.26)
Euler angles for hemisphere of S 3 .SO.3; R/I j 0 and j both integral):
d˝ D 2 2 : S3
Z2
Spherical polar coordinate for S 3 :
Z2 d˛
Z j
0
.cos ; cos sin sin ; sin sin sin ; cos sin / ;
8 2 D ıjj 0 ım0 0 ım : 2j C 1
0 ;
0
j0
dˇ sin ˇDm0 m .˛ˇ/D0 .˛ˇ/
d
.˛0 ; ˛/ D 0 < 2 ;
j0
j
Dm0 m . ; n/D O 0 . ; n/ O
2 2 ıjj 0 ım0 0 ım ; 2j C 1
.x0 ; x/ 2 R 4 ;
3 X @2 : @x2 D0
0 ;
2
Euler angles for S 3 .SU.2//:
Replace the Euler–Rodrigues parameters .˛0 ; ˛/ in Eq. (2.22) by an arbitrary point .x0 ; x/ 2 R 4 .
Z
d 2
0
Solutions of Laplace’s equation in R 4 (Sect. 2.5):
r42 D
j0
j
d sin2 Dm0 m .˛0 ; ˛/D0 .˛0 ; ˛/
d sin
d 0
Z
0
(2.27)
2
Angular Momentum Theory
21
Formal polynomials Eq. (2.24):
j j0 Dm0 m ; D0 D .2j /Šıjj 0 ım0 0 ım ;
1 j 1=2 .j C m/ sin ˇ dm0 C1=2;m1=2 .ˇ/ 2 1 1 j 1=2 2 C .j m/ cos ˇ dm0 C1=2;mC1=2 .ˇ/ 2 0
D .j m /
with inner product .P; P 0 / D P
@ P 0 .Z/jZD0 ; @Z
1 2
j dm0 m .ˇ/
:
Two useful relations implied by the above are: 1
j
Œ.j m/.j C m C 1/ 2 sin ˇ dm0 ;mC1 .ˇ/
@ where P @Z is the complex conjugate polynomial P of P in which each zij is replaced by @z@ij . Boson polynomials:
D
1 2
ˇ E j ˇ j0 Dm0 m ˇD0 D .2j /Šıjj 0 ım0 0 ım ;
with inner product hP jP 0 i D h0jP AN P 0 .A/j0i.
1
j
C Œ.j C m/.j m C 1/ 2 sin ˇ dm0 ;m1 .ˇ/ j
D 2.m cos ˇ m0 /dm0 m .ˇ/ ; 1
j
Œ.j C m/.j m C 1/ 2 dm0 ;m1 .ˇ/ 1 j
C .j C m0 /.j m0 C 1/ 2 dm0 1;m .ˇ/ 1 j 0 D .m m / cot ˇ dm0 m .ˇ/ : 2 By considering
2.3.5 Recurrence Relations Many useful relations between the representation functions may be derived as special cases of general relations between these functions and the WCG coefficients given in Sect. 2.7.1. The simplest of these are obtained from the Kronecker reduction 1
D j D 2 D D j C1=2 ˚ D j 1=2 :
D j D 1 D D j C1 ˚ D j ˚ D j 1 ; one can also readily derive the matrix elements of the direction cosines specifying the orientation of the body-fixed frame fO 1 ; fO 2 ; fO 3 of a symmetric rotor relative to the inertial frame .eO 1 ; eO 2 ; eO 3 /: j
; m;m0 D
1 X 2j C 1 2 j1j 0 j1j 0 j0 CmmC Cm0 m0 C mC;m0 C ; 0 2j C 1 j0
Such relations are usually presented in terms of the Euler angle realization of U , leading to the following relations be- where the wave functions are those defined for integral j by j Eq. (2.37), for half-integral j by Eq. (2.36), and tween the functions dm0 ;m .ˇ/: 1 j C1=2 .j m C 1/ cos ˇ dm0 1=2;m1=2 .ˇ/ 2 1 1 j C1=2 C .j C m C 1/ 2 sin ˇ dm0 1=2;mC1=2 .ˇ/ 2 1 2
1
j
D .j m0 C 1/ 2 dm0 m .ˇ/ ; 1 1 j C1=2 .j m C 1/ 2 sin ˇ dm0 C1=2;m1=2 .ˇ/ 2 1 1 j C1=2 C .j C m C 1/ 2 cos ˇ dm0 C1=2;mC1=2 .ˇ/ 2 1
j
D .j C m0 C 1/ 2 dm0 m .ˇ/ ; 1 1 j 1=2 2 .j C m/ cos ˇ dm0 1=2;m1=2 .ˇ/ 2 1 1 j 1=2 2 .j m/ sin ˇ dm0 1=2;mC1=2 .ˇ/ 2 1
j
D .j C m0 / 2 dm0 m .ˇ/ ;
1 ; ; D eO fO D D; p eO C1 D .eO 1 C ieO 2 /= 2 ; p eO 1 D .eO 1 ieO 2 /= 2 ;
p fOC1 D fO 1 C ifO 2 = 2 ;
p fO1 D fO 1 ifO 2 = 2 :
; D 1; 0; C1 I eO 0 D eO 3 ; fO 0 D fO 3 ;
2.3.6 Symmetry Relations Symmetry relations for the representation functions j Dm0 m .Z/ defined by Eq. (2.24) are associated with the action of a finite group G on the set M.2; 2/ of complex 2 2 matrices: g W M.2; 2/ ! M.2; 2/; g 2 G. Equivalently, if Z 2 M.2; 2/ is parametrized by a set ˝ of parameters ! 2 ˝ (parameter space), then g may be taken to act directly in the parameter space g W ˝ ! ˝. The action, denoted
2
22
J. D. Louck
, of a group G D fe; g; g 0 ; : : :g (e D identity) on a set X D fx; x 0 ; : : :g must satisfy the rules
Each element of this group is an element of U.2/. The action of the group P may, therefore, be defined on the group U.2/ by left and right actions, as discussed g W X ! X e x D x; all x 2 X ; in Sect. 2.4.1. g 0 .g x/ D .g 0 g/ x; all g 0 ; g 2 G; all x 2 X : 2. Symmetric group S4 : (2.28) S4 D fpjp is a permutation ˇof the ˇ four Euler–Rodrigues parameters .˛0 ; ˛1 ; ˛2 ; 3 /g; ˇ S4 ˇD 24. Points in S 3 are Using to denote the action of G on M.2; 2/ and to denote mapped to distinct points in S 3 ; hence, one can take the action of G on ˝, one has the relation: Z 2 SU.2/ and define the group action directly from U.˛0 ; ˛/ in Eq. (2.20). It is simpler, however, to define .g Z/.!/ D Z.g 1 !/ : the action of the group directly on the representation functions Eq. (2.22). Not all elements of this group define Only finite subgroups G of the unitary group U.2/ (group of a symmetry in the sense defined by Eq. (2.29) (see be2 2 unitary matrices) are considered here: G U.2/. low). Generally, when G acts on M.2; 2/, it ˚effects a unitary j 3. Abelian group T : linear transformation of the set of functions Dm0 m (j fixed) defined over Z 2 M.2; 2/. For certain groups or for some ˚ G, j j T D f.t0 ; t1 ; t2 ; t3 /j each t D ˙1g ; elements of G, a single function D0 2 Dm0 m occurs in jT j D 16 : the transformation, so that j j Group multiplication is component-wise multiplication, g Dm0 m .Z/ D Dm0 m .g 1 Z/ and the identity is (1,1,1,1). The action of an element of j (2.29) D gm0 m D0 .Z/ ; T is defined directly on the Euler–Rodrigues parameters by component-wise multiplication, thus mapping points .0 / 2 f.0 m0 ; m/; .m; 0 m0 /j0 D ˙1; D ˙1g ; in S 3 to points in S 3 ; hence, one can take Z 2 SU.2/. where gm0 m is a complex number of unit modulus. Relation This group is isomorphic to the direct product group j Eq. (2.29) is called a symmetry relation of Dm0 m with respect S2 S2 S2 S2 ; S2 D symmetric group on two distinct to g. Usually, not all elements in G correspond to symmetry objects. relations. In a symmetry relation, the action of the group is 4. Group G: effectively transferred to the discrete quantum labels themselves: G D hR; C ; T ; K i; jGj D 32 ; g W m0 ! 0 D m0 .g/ ; m ! D m.g/ :
(2.30)
In terms of a parametrization ˝ of M.2; 2/, relation Eq. (2.29) is written j j gDm0 m .!/ D Dm0 m .g 1 !/ j
D gm0 m D0 .!/ :
(2.31)
In practice, action symbols such as and are often dropped in favor of juxtaposition, when the context is clear. Moreover, the set of complex matrices M.2; 2/ may be replaced by U.2/ or SU.2/ whenever the action conditions Eq. (2.28) are satisfied. Relations (Eqs. (2.29)–(2.31)) are illustrated below by examples. There are several finite subgroups of interest with various group–subgroup relations between them: 1. Pauli group: P D f ; ; i ; i j D 0; 1; 2; 3g ; jP j D 16 :
where R; C ; T ; K denote the operations of row interchange, column interchange, transposition, and conjugation (see below) of an arbitrary matrix. a ZD c
b d
!
The notation h i designates that the enclosed elements generate G. It is impossible to give here all the interrelationships among the groups defined in (1)–(4). Instead, some relations are j listed as obtained directly from either Dm0 m .Z/ defined by j Eq. (2.24) or Dm0 m .˛0 ; ˛/ defined by Eq. (2.22). The actions of the groups T and G defined in (3) and (4) are fully given. Abelian group T of order 16. Generators: T D ht0 ; t1 ; t2 ; t3 i ;
t0 D .1; 1; 1; 1/ ;
t1 D .1; 1; 1; 1/ ;
t2 D .1; 1; 1; 1/ ;
t3 D .1; 1; 1; 1/ :
2
Angular Momentum Theory
23 j
Group action:
These function relations are valid for Dm0 m defined over the arbitrary matrix Z defined by Eq. (2.24). They are also true for Z D U 2 SU.2/, but now the operations R and C change the sign of the determinant of the matrix Z, so that the transformed matrix no longer belongs to SU.2/. It does, however, belong to U.2/, the group of all 2 2 unitary matrices. The special irreducible representation functions of U.2/ defined by Eq. (2.24),
t a D .t0 ˛0 ; t1 ˛1 ; t2 ˛2 ; t3 ˛3 / ; each t D .t0 ; t1 ; t2 ; t3 / 2 T ; each a D .˛0 ; ˛1 ; ˛2 ; ˛3 / 2 S 3 ; .tF /.a/ D F .t a/ ; j
0
j
j
0
j
t0 Dm0 m D .1/m m Dmm0 ; t1 Dm0 m D .1/m m Dmm0 ; j t2 Dm0 m j t3 Dm0 m
D D
j Dmm0 ; j Dmm0
j
Dm0 m .U / ;
possess each of the 32 symmetries corresponding to the operations in the group G. (There exist other irreducible representations of U.2/, involving det U .) The operation K is closely related to complex conjugation, since for each U 2 U.2/; U D .uij /, one can write ! u u 22 21 ; U D .det U /1 u12 u11 j j K Dm0 m .U / D .det U /2j Dm0 m .U /
:
Group G of order 32. Generators: G D h R; C ; T ; K i : Generator actions: a .RF / c .C F /
a c
a .T F / c a .K F / c
! b DF d ! b DF d ! b DF d ! b DF d
! d ; b ! b a ; d c ! a c ; b d c a
d b
c a
row interchange
j
D .det U /2j Dm0 m .U /
column interchange
0
j
D .1/m m Dm0 m .U / :
(2.33)
Relations Eqs. (2.32) and (2.33) are valid in an arbitrary parametrization of U 2 U.2/. In terms of the parametrization
transposition ! ;
U 2 U.2/ ;
conjugation
U.; ˛0 ; ˛/ D ei=2 U.˛0 ; ˛/ ;
0 2 ;
where U.˛0 ; ˛/ 2 SU.2/ is the Euler–Rodrigues parametrization, the actions of R, C , T , and K correspond to the following transformations in parameter space:
Subgroup H : H D h R; C ; T i D f1; R; C ; T ; RC D CR; T R D CT ;T C
RW
D RT ; RCT g ; CW
with relations in H given by R2 D C 2 D T 2 D 1 ;
T W
T RC D T CR D RCT D CRT ; RT C D CT R D T :
CW
! C ; .˛0 ; ˛1 ; ˛2 ; ˛3 / ! .˛1 ; ˛0 ; ˛3 ; ˛2 / ; ! C ; .˛0 ; ˛1 ; ˛2 ; ˛3 / ! .˛1 ; ˛0 ; ˛3 ; ˛2 / ; ! ; .˛0 ; ˛1 ; ˛2 ; ˛3 / ! .˛0 ; ˛1 ; ˛2 ; ˛3 / ; ! ; .˛0 ; ˛1 ; ˛2 ; ˛3 / ! .˛0 ; ˛1 ; ˛2 ; ˛3 / :
Adjoining the idempotent element K to H gives the group The new angle 0 D C is to be identified with the correG of order 32: sponding point on the unit circle so that these mappings are always in the parameter space, which is the sphere S 3 toG D fH; H K ; H KR; H KRK g : gether with the unit circle for . Observe that the following identities hold for functions over SU.2/; hence, over U.2/: Symmetry relations: j
j
j
j
j
j
C D T t1 T t3 ;
RDm0 m D Dm0 m ; C Dm0 m D Dm0 m ; T Dm0 m D Dmm0 ; j
0
j
K Dm0 m D .1/m m Dm0 m :
T D T t2 :
Abelian subgroup of S4 . Generators: K D h.0; 3/; .1; 2/i, where .0; 3/ and .1; 2/ denote trans(2.32) positions in S4 ; jKj D 4.
2
24
J. D. Louck
2.4.1 Matrix Group Actions
Group action in parameter space: .0; 3/.˛0 ; ˛1 ; ˛2 ; ˛3 / D .˛3 ; ˛1 ; ˛2 ; ˛0 / ; .1; 2/.˛0 ; ˛1 ; ˛2 ; ˛3 / D .˛0 ; ˛2 ; ˛1 ; ˛3 / :
Left and right translations of Z 2 M.n; m/: Lg Z D gZ ;
each g 2 G ;
each Z 2 M.n; m/ ;
T
each h 2 H ;
each Z 2 M.n; m/ :
Rh Z D Zh ;
Symmetry relations: j
0
j
j
0
j
.0; 3/Dm0 m D .i/m Cm Dmm0 ; .1; 2/Dm0 m D .i/m m Dmm0 : Diagonal subgroup ˙ of the direct product group P P (P D Pauli group). Group elements: ˙ D f.; /j 2 P g ;
Z 0 D Lg .Rh Z/ D Rh .Lg Z/ ;
each g 2 G; h 2 H ;
Z 2 M.n; m/ : Equivalent form as a transformation on z 2 C nm :
j˙ j D 16 :
z0 D .g h/z ;
each 2 P ;
where denotes the direct product of g and h; the column matrix z (respectively, z0 ) is obtained from the columns of Z (respectively, Z 0 ), z˛ ; ˛ D 1; 2; : : : ; m, of the n m matrix Z as successive entries in a single column vector z 2 C nm . Left and right translations in function space:
Group action: .; / W U ! U T ;
(T denotes matrix transposition.) Left and right translations commute:
T
Œ.; /F .U / D F . U / : Example: D i2 : .; / W .˛0 ; ˛1 ; ˛2 ; ˛3 / ! .˛0 ; ˛1 ; ˛2 ; ˛3 / ; Œ.; /F .˛0 ; ˛1 ; ˛2 ; ˛3 / D F .˛0 ; ˛1 ; ˛2 ; ˛3 / ; .; / D t1 t2 on functions over U.2/ :
.Lg f /.Z/ D f .g T Z/ ;
each g 2 G ;
.Rh f /.Z/ D f .Zh/ ;
each h 2 H ;
where f .Z/ D f zi˛ , and the commuting property holds for all well-defined functions f :
The relations presented above barely touch on the interLg .Rh f / D Rh .Lg f / : relations among the finite groups introduced in (1)–(4). Symmetry relations Eqs. (2.32) and (2.33), however, give the j symmetries of the dm0 m .ˇ/ given in Sect. 2.3.3 in the Eu- 2.4.2 Lie Algebra Actions ler angle parametrization. In general, it is quite tedious to present the above symmetries in terms of Euler angles, with Lie algebra of left and right translations: adjoined when necessary, because the Euler angles are not uniquely determined by the points of S 3 . d T .DX f /.Z/ D i f eitX Z j t D0 ; dt d
Y .D f /.Z/ D i f Z eit Y j t D0 I 2.4 Group and Lie Algebra Actions dt
DX D Tr Z T X@=@Z ; each X 2 L.G/ ;
The concept of a group acting on a set is fundamental to D Y D Tr Y T Z T @=@Z ; each Y 2 L.H / ; applications of group theory to physical problems. Because L.G/ D Lie algebra of G ; of the unity that this notion brings to angular momenL.H / D Lie algebra of H : tum theory, it is well worth a brief review in a setting in which a matrix group acts on the set of complex matrices. Thus, let G GL.n; C / and H GL.m; C / denote ar- Linear derivations: bitrary subgroups, respectively, of the general linear groups D˛X CˇX 0 D ˛DX C ˇDX 0 ; ˛; ˇ 2 C ; of n n and m m nonsingular complex matrices, and let M.n; m/ denote the set of n m complex matrices. ŒDX ; DX 0 D DŒX;X 0 ; ma A trix Z 2 M.n; m/ has row and column entries zi˛ ; i D D Y obeys these same rules. 1; 2; : : : ; nI ˛ D 1; 2; : : : ; m.
2
Angular Momentum Theory
25
Commuting property of left and right derivations:
DX ; D Y D 0 ; X 2 L.G/ ; Y 2 L.H / : Basis set:
.RU f /.z/ D f .U T z/ ; Di =2 D .zT i @=@z/=2 ; J˙ D D1 =2 ˙ iD2 =2 ; JC D z1 @=@z2 ;
DX D
n X
xij Dij ;
X D .xij / ;
i;j D1
DY D
m X
y ˛ˇ D ˛ˇ ;
Y D .y ˛ˇ / ;
Dij D D ˛ˇ D
˛D1 n X
zi˛
.P; P 0 / D P .@=@z1 ; @=@z2/P .z1 ; z2 /jz1 Dz2 D0 : Orthonormal basis:
zi˛
i D1
j Cm j m z2 =Œ.j
Pj m.z1 ; z2 / D z1
@ ; @zj˛ @ ˇ @zi
2
J D z2 @=@z1 ;
J3 D .1=2/.z1 @=@z1 z2 @=@z2 / ;
˛;ˇD1 m X
J3 D D3 =2 ;
j D 0; 1=2; 1; 3=2; : : : I
1
C m/Š.j m/Š 2 ;
m D j; j 1; : : : ; j :
Standard action:
:
J 2 Pj m.z/ D j.j C 1/Pj m.z/ ;
Commutation rules:
J3 Pj m.z/ D mPj m.z/ ;
ŒDij ; Dkl D ıj k Di l ıi l Dkj ;
1
J˙ Pj m.z/ D Œ.j m/.j ˙ m C 1/ 2 Pj;m˙1 .z/ :
ŒD ˛ˇ ; D D ı ˇ D ˛ ı ˛ D ˇ ; Group transformation:
ŒDij ; D ˛ˇ D 0 ; where i; j; k; l D 1; 2; : : : ; n, and ˛; ˇ; ; D 1; 2; : : : ; m. The operator sets fDij g and fD ˛ˇ g are realizations of the Weyl generators of GL.n; C / and GL.m; C /, respectively.
.RU Pj m /.z/ D
Space of polynomials with inner product: .P; P 0 / D P .@=@Z/P 0.Z/jZD0 : Bargmann space of entire functions with inner product: Z hF; F 0 i D F .Z/F 0 .Z/d.Z/ ; ! X Y zi˛ zi˛ dxi˛ dyi˛ ; d.Z/ D nm exp i;˛
i;˛
zi˛ D xi˛ C iyi˛ ; i D 1; 2; : : : ; n I ˛ D 1; 2; : : : ; m :
where the representation functions are given by Eq. (2.17).
G D H D SU .2/ ; Z 2 M.2; 2/ ;
z˛ D col z1˛ z2˛ ; Z D z1 z2 ; X D Y D set of 2 2 traceless ; Hermitian matrices ; .RU f /.Z/ D f .U T Z/ ;
Di =2 D Tr.Z T i @=@Z/=2 ; D i =2 D Tr.i Z T @=@Z/=2 : M˙ D D1 =2 ˙ iD2 =2 ; K˙ D D MC D
.P; P 0 / D hP; P 0 i :
1 =2
2 X
˙ iD
2 =2
z1˛ @=@z2˛
;
M3 D D3 =2 ;
;
K3 D D 3 =2 ;
M D
˛D1
Relation to Angular Momentum Theory
Spinorial realization of Sects. 2.4.2 and 2.4.3: H D .1/ ;
.LV f /.Z/ D f .ZV / ;
U; V 2 SU.2/ ;
Numerical equality of inner products:
G D SU .2/ ;
j
Dm0 m .U /Pj m0 .z/ ;
m0
The 2-Spinorial Realization of Sects. 2.4.2 and 2.4.3:
2.4.3 Hilbert Spaces
2.4.4
X
X D set of 2 2 traceless, Hermitian matrices,
z2˛ @=@z1˛ ;
˛D1
2 1 X ˛ M3 D z1 @=@z1˛ z2˛ @=@z2˛ ; 2 ˛D1
KC D
2 X
zi1 @=@zi2 ;
i D1
Z 2 M.2; 1/ ;
z D col.z1 ; z2 / ;
2 X
K3 D
1 2
2 X i D1
K D
2 X
zi2 @=@zi1 ;
i D1
1 zi @=@zi1 zi2 @=@zi2 :
26
J. D. Louck
Generating functions:
Mutual commutativity of Lie algebras: ŒMi ; Kj D 0 ;
i; j D 1; 2; 3 :
.x T Zy/2j =.2j /Š D
Inner product: T
0
exp.tx Zy/ D
0
.P; P / D P .Z/P .@=@Z/jZD0 ;
j
j
mm0
t 2j
j
Pj m.x/Dmm0 .Z/Pj m0 . y/ ; X
j
Pj m .x/Dmm0 .Z/Pj m0 . y/ ;
mm0
x D col.x1 x2 / ; y D col.y1 y2 / ;
Z D zi˛ ; i; ˛ D 1; 2 I all indeterminates :
Orthogonal basis Eq. (2.24): Dmm0 .Z/ ;
X
X
j D 0; 1=2; 1; 3=2; : : : ;
m D j; j 1; : : : ; j I m0 D j; j 1; : : : ; j I j j0 Dmm0 ; D0 D .2j /Šıjj 0 ım ım0 0 :
2.5 Differential Operator Realizations of Angular Momentum
Equality of Casimir operators: M 2 D K 2 D M12 C M22 C M32 D K12 C K22 C K32 : Standard actions: j
j
j
M 2 Dmm0 .Z/ D K 2 Dmm0 .Z/ D j.j C 1/Dmm0 .Z/ ; j
j
M3 Dmm0 .Z/ D mDmm0 .Z/ ; j
j
K3 Dmm0 .Z/ D m0 Dmm0 .Z/ ; 1
j
j
M˙ Dmm0 .Z/ D Œ.j m/.j ˙ m C 1/ 2 Dm˙1;m0 .Z/ ; 1
j
K˙ Dmm0 .Z/ D Œ.j m0 /.j ˙ m0 C 1/ 2 j
Differential operators realizing the standard commutation relations Eqs. (2.7) and (2.8) can be obtained from the 2spinorial realizations given in Sect. 2.4.4 by specializing the matrix Z to the appropriate unitary unimodular matrix U 2 SU.2/ and using the chain rule of elementary calculus. j Similarly, one obtains the explicit functions Dmm0 simply by substituting for Z the parametrized U in Eq. (2.24). This procedure is used in this section to obtain all the realizations listed. The notations M D .M1 ; M2 ; M3 / and K D .K1 ; K2 ; K3 / and the associated M˙ and K˙ refer to the differential operators given by the 2-spinorial realization now transformed to the parameters in question. Euler angles with Z D U.˛ˇ/ (Sect. 2.3.1):
Dm;m0 ˙1 .Z/ : M3 D i@=@˛ ; K3 D i@=@ ; 1 i˛ e MC ei˛ M 2 @ 1 i e K ei KC D ; D 2 @ˇ 1 i˛ e MC C ei˛ M 2 D .cot ˇ/M3 C .sin ˇ/1 K3 ; 1 i e K C ei KC D .cot ˇ/K3 .sin ˇ/1 M3 ; 2 MC D eia Œ@=@ˇ .cot ˇ/M3 C .sin ˇ/1 K3 ;
Special values: ! 1 0 D I2j C1 D unit matrix ; Dj 0 1 ! z1 0 j D ıj m0 Pj m.z1 ; z2 / ; Dmm0 z2 0 ! 0 z1 j D ıj mPj m0 .z1 ; z2 / ; Dmm0 0 z2 ! z1 0 j j Cm j m ; D ımm0 z1 z2 Dmm0 0 z2
2j j Djj .Z/ D z11 :
M D ei˛ Œ@=@ˇ .cot ˇ/M3 C .sin ˇ/1 K3 ; KC D ei Œ@=@ˇ C .cot ˇ/K3 .sin ˇ/1 M3 ; K D ei Œ@=@ˇ C .cot ˇ/K3 .sin ˇ/1 M3 :
Symmetry relation:
T
D j .Z/ D D j Z T :
Euler angles with Z D U .˛ˇ/ (replace i by i in the above relations):
Generation from highest weight: j Dmm0 .Z/
D
.j C m/Š .j C m0 /Š .2j /Š.j m/Š .2j /Š.j m0 /Š 0
j
Mj m Kj m Djj .Z/ :
12
(2.34) M3 D i@=@˛ ; K3 D i@=@ ;
˙i˛ 1 ˙@=@ˇ .cot ˇ/M3 C .sin ˇ/ K3 ; M˙ D e
K˙ D e˙i @=@ˇ C .cot ˇ/K3 .sin ˇ/1 M3 :
2
Angular Momentum Theory
27
j Since D j .U / D .D j .U // , which is denoted D j .U /, Dmm 0 .x0 ; x/ defined by Eq. (2.22) (Replace ˛0 by x0 and ˛ these operators have the standard action on the complex con- by x.) Additional relations: j jugate functions Dmm0 .U /. 1 Quaternionic variables. .x0 ; x/ 2 R 4 : K 2 D M 2 D R2 r42 C K02 C K0 ; 4
0 0 x0 ;x .x0 ; x/ D x00 x0 x 0 x; x00 x C x0 x 0 C x 0 x I R2 D x02 C x x ; ! ! X @2 @2 x0 ix3 ix1 x2 z11 z12 2 r42 D C r D ; D I ZD @x2 @x02 z21 z22 ix1 C x2 x0 C ix3 ! 3 @ @=@z11 @=@z12 @ @ 1 1X T D x I K0 D Tr z D @Z @=@z21 @=@z22 2 @Z 2 D0 @x ! j j 1 @=@x0 C i@=@x3 i@=@x1 @=@x2 K0 Dmm0 .x0 ; x/ D jDmm0 .x0 ; x/ ; D I 2 i@=@x1 C @=@x2 @=@x0 i@=@x3 j r42 Dmm0 .x0 ; x/ D 0 I ! 1
1
X X X Mi D Tr Z T i @=@Z ; Ki D Tr i Z T @=@Z : 2 2 Ri1 Ki ; Ri 2 Ki ; Ri 3 Ki ; .M1 ; M2 ; M3 / D i
(The form of @=@Z is determined by the requirement .@=@zij /zlk D ıi l ıj k ; for example, 12 .@=@x0 C i@=@x3 /.x0 ix3 / D 1). Define the six orbital angular momentum operators in R 4 by
Rij D
i
i
2 x0 x x ıij 2eij k x0 xk C 2xi xj x02 C x x
;
each .x0 ; x/ 2 R 4 :
The relation Rij D Rij .x0 ; x/ is a mapping of all points of four-space R 4 (except the origin) onto the group of proper, orthogonal matrices; for x02 C x x D 1, it is just the Euler– which may be written as the orbital angular momentum L in Rodrigues parametrization, Eq. (2.21). The operators R D .M1 ; M2 ; M3 / and K D R 3 together with the three operators A given by .K1 ; K2 ; K3 / have the standard action on .1/j Cm j Dm;m0 .x0 ; x/, so that the orbital angular momentum in L D ix r ; L1 D L23 ; R 3 is given by the addition L D R C K . Thus, one finds: L DL ; L DL ; Lj k D i.xj @=@xk xk @=@xj / ;
2
31
3
j < k D 0; 1; 2; 3 ;
12
A D .A1 ; A2 ; A3 / D .L01 ; L02 ; L03 / :
K2 D .L2 A2 /=2 ;
D A2j;L R2j L YLM .x/C2j L .x0 =R/ ; 1 4.2j L/Š 2 L 2j A2j;L D .2i/ .1/ LŠ : .2j C L C 1/Š
M2 D .L2 C A2 /=2 ;
M3 D .L3 C A3 /=2 :
2.6 The Symmetric Rotor and Representation Functions
Commutation rules: ŒMj ; Kk D 0 ;
j
.LC1/
K3 D .L3 A3 /=2 I M1 D .L1 C A1 /=2 ;
jjL
Cmm0 M .1/j Cm Dmm0 .x0 ; x/
mm0
Then, we have the following relations: K1 D .L1 A1 /=2 ;
X
j; k D 1; 2; 3 ;
M M D iM ;
K K D iK ;
L L D iL ;
A A D iL ;
ŒLj ; Ak D iej kl Al ; where ej kl D 1 for j; k; l an even permutation of 1; 2; 3; ej kl D 1 for an odd permutation of 1; 2; 3; ej kl D 0, otherwise. The M D .M1 ; M2 ; M3 / and K D .K1 ; K2 ; K3 / operators have the standard action given in Sect. 2.2 on the functions
The rigid rotor is an important physical object, and its quantum description enters into many physical theories. This description is an application of angular momentum theory with subtleties that need to be made explicit. It is customary to describe the classical rotor in terms of a right-handed triad
of unit vectors fO 1 ; fO 2 ; fO 3 fixed in the rotor and constituting a principal axes system located at the center of mass. The instantaneous orientation of this body-fixed frame relative to a right-handed triad of unit vector .eO 1 ; eO 2 ; eO 3 / specifying an inertial frame, also located at the center of mass, is then given, say, in terms of Euler angles (for this purpose, one
2
28
J. D. Louck
could use any parametrization of a proper, orthogonal matrix). For Euler angles, the relationship is X fOj D Rij .˛ˇ/eO i : (2.35)
J2 D i sin ˛ cot ˇ J3 D i
i
The Hamiltonian for the rigid rotor is then of the form H D AP12 C B P22 C C P32 ; J where A; B, and C are physical constants related to the reciprocals of the principal moments of inertia, and the angular momenta Pj .j D 1; 2; 3/ are the components of the total angular momentum J referred to the body-fixed frame: X Pj D fOj J D Rij .˛ˇ/Ji ; i
J D eO 1 J1 C eO 2 J2 C eO 3 J3 :
@ @ sin ˛ @ i cos ˛ i ; @˛ @ˇ sin ˇ @
@ : @˛
momentum J with components referred to
Physical angular fO 1 ; fO 2 ; fO 3 : @ @ cos @ i sin Ci ; @ @ˇ sin ˇ @˛ @ @ sin @ P2 D i sin cot ˇ i cos i ; @ @ˇ sin ˇ @˛ @ P3 D i : @ P1 D i cos cot ˇ
Standard commutation of the Ji :
ŒJi ; Jj D iJk ; For the symmetric rotor (taking A D B), the Hamiltonian can Ji can stand on either side: be written in the form X H D aP2 C b P32 : Pj D Rij .˛ˇ/Ji ; It is in the interpretation of this Hamiltonian for quantum mechanics that the subtleties already enter, since the nature of angular momentum components referred to a moving reference system must be treated correctly. Relation Eq. (2.35) shows that the body-fixed axes cannot commute with the components of the total angular momentum J referred to the frame (eO 1 ; eO 2 ; eO 3 ). A position vector x and the orbital angular momentum L, with components both referred to an inertial frame, satisfy the commutation relations ŒLj ; xk D iej kl xl , and for a rigid body thought of as a collection of point particles rotating together, the same conditions are to be enforced. Relative to the body-fixed frame, the vector x is expressed as X X xk eO k D ah fO h ; each ah D constant ; k
xk D
Ji D
X
i
Ji Rij .˛ˇ/ :
i
The famous Van Vleck factor of i in the commutation of the Pi : ŒPi ; Pj D iPk ;
i; j; k cyclic .
Mutual commutativity of the Jj and Pi : ŒPi ; Jj D 0 ;
i; j D 1; 2; 3 :
Same invariant (squared) total angular momentum: P12 C P22 C P32 D J12 C J22 C J32 D J 2
@2 @2 @2 C 2 2 cos ˇ 2 @˛ @ @˛@ 2 @ @ 2 cot ˇ : @ˇ @ˇ
D csc2 ˇ
h
X
i; j; k cyclic :
ah Rkh .˛ˇ/ :
h
Standard actions: The direction cosines Rkh D Rkh .˛ˇ/ D eO k fO h and the j j physical total angular momentum components referred to an J 2 Dmm0 .˛ˇ/ D j.j C 1/Dmm0 .˛ˇ/ ; inertial frame must satisfy j j J3 Dmm0 .˛ˇ/ D mDmm0 .˛ˇ/ ;
j j Jj ; Rkh D iej kl Rlh ; each h D 1; 2; 3 ; P D .˛ˇ/ D m0 D .˛ˇ/ I 3
in complete analogy to ŒLj ; xk D iej kl xl . The description of the angular momentum associated with a symmetric rigid rotor and the angular momentum states is summarized as follows [Eq. (2.35)]: Physical total angular momentum J with components referred to .eO 1 ; eO 2 ; eO 3 /: J1 D i cos ˛ cot ˇ
@ @ cos ˛ @ C i sin ˛ i ; @˛ @ˇ sin ˇ @
mm0 j J˙ Dmm0 .˛ˇ/
mm0
1
D Œ.j m/.j ˙ m C 1/ 2 j
Dm˙1;m0 .˛ˇ/ ; 1
j
.P1 iP2 /Dmm0 .˛ˇ/ D Œ.j m0 /.j C m0 C 1/ 2 j
Dm;m0 C1 .˛ˇ/ ; 1
j
.P1 C iP2 /Dmm0 .˛ˇ/ D Œ.j C m0 /.j m0 C 1/ 2 j
Dm;m0 1 .˛ˇ/ :
2
Angular Momentum Theory
29
Normalized wave functions. Integral or half-integral j (SU.2/ solid body):
of (2) to operators, recognizing that the set of mappings of a vector space into itself is also a vector space. The subject of tensor operator algebra is considered in the next section * + r ˇ ˇ j because of its special importance for physical applications. 2j C 1 j Dmm0 .˛ˇ/ ; (2.36) ˛ˇ ˇˇ D 2 0 This section summarizes formulas relating to the first three 2 mm viewpoints, also giving the explicit mathematical expression of the coefficients in their several forms. with inner product Either viewpoint, (1) or (2), may be taken as an interZ pretation of the Clebsch–Gordan series, which abstractly 0 0 hF jF i D d˝ h˛ˇjF i h˛ˇjF i ; expresses the reduction of a Kronecker product of matrices (denoted ) into a direct sum (denoted ˚) of matrices: where d˝ is defined by Eq. (2.25), and the integration exj1 Cj2 tends over all ˛; ˇ; given by Eq. (2.26). X ˚Œj Œj1 Œj2 D Integral j (collection of rigid point particles): r j
mm0 .˛ˇ/ D
j Djj1 j2 j
2j C 1 j Dmm0 .˛ˇ/ ; 8 2
(2.37)
with inner product Z2 0
.; / D
Z
d F .˛ˇ/F 0 .˛ˇ/ :
dˇ sin ˇ
d˛ 0
Z2
0
0
D Œjj1 j2 j ˚ Œjj1 j2 j C 1 ˚ ˚ Œj1 C j2 : (2.38) Given two angular momenta j1 2 f0; 12 ; 1; : : :g and j2 2 f0; 12 ; 1; : : :g, the Clebsch–Gordan (CG) series also expresses the rule of addition of two angular momenta: j D j1 C j2 ; j1 C j2 1; : : : ; jj1 j2 j :
The concept of a solid (impenetrable) body is conceptually The integers j1 j2 j defined by distinct from that of a collection of point particles moving 8 ˆ collectively together in translation and rotation. 0 and define V D .2E/1=2 VQ , obtaining an so(3,1) where, again, realization.] The last identity of Eq. (4.23) now becomes (4.18) X˙ D X1 ˙ iX2 ; X D J or V :
V 2 D L2 C 1 Z2 =2En ; (4.26) For so(3,1) and e(3), only infinite dimensional nontrivial irreps are possible, while for so(4), only finite dimensional which immediately implies Bohr’s formula, since V 2 C L2 D ones arise. To unirreps,we require J and V to be Hermi- 4M 2 D 1 Z2 =2En , so that ˚ get 2 tian. Using J ; J3 ; C1 ; C2 as a complete set of commuting En D Z2 =2.2j1 C 1/2 D Z2 =2n2 ; (4.27) operators for so(4), we label the basis vectors by the four quantum numbers as jj mi j.j0 ; /j mi, so that where n D 2j1 C 1 and j1 is the angular momentum quan2 tum number for M , Eq. (4.16). In terms of the irrep labels J jj mi D j.j C 1/jj mi ; Eq. (4.20), we have that j0 D 0, D n, so that j`mi D J3 jj mi D mjj mi ; j.0; n/`mi jn`mi, ` D 0; 1; : : : ; n 1.
C1 jj mi D j02 2 1 jj mi ; Using the stepwise merging of so(4) and so(2,1) [adding C2 jj mi D j0 jj mi ; (4.19) first T2 , which leads to so(4,1), and subsequently T1 and T3 ],
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J. Paldus
we arrive at the hydrogenic realization of so(4,2) having 15 4.2 generators L; A; B; ; T1 ; T2 ; T3 , namely ([2, 14, 15, 17]) LDR^P ; ) A 1 D RP 2 P.R P/ R ; 2 B D RP ; ) T1 1
1
D RP 2 R D RPR2 C L2 R1 R ; 2 2 T3 T2 D R P iI D RPR :
(4.28)
Hamiltonian Transformation and Simple Applications
The basic idea is to transform the relevant Schrödinger equation into an eigenvalue problem for one of the operators from the complete set of commuting operators in our realizations, e.g., T3 for so(2,1). Instead of using a rather involved “tilting” transformation ([1, p. 20], and [2, 14, 15]), we can rely on a simple scaling transformation [16, 25] r D R ; where
p D 1 P ;
r D R ; 0
N X
pr D 1 PR ; (4.33)
11=2
Relabeling these generators by the elements of an antisymxj2 A ; (4.34) rD@ metric 6 6 matrix according to the scheme j D1 1 0 and 0 L3 L2 A1 B1 1 C B C B 0 L A B 1 2 2 2C 2 2 2 1 2 B D p C r p .N 1/.N 3/ C L ; (4.35) B r 0 A3 B3 3 C 4 C B (4.29) Lj k $ B C B 0 T2 T1 C with pr defined analogously to PR in Eq. (4.7); r and p are C B B 0 T3 C the physical operators in terms of which the Hamiltonian of A @ the studied system is expressed. Recall that L2 has eigenval0 ues [26] we can write the commutation relations in the following stan`.` C N 2/ ; ` D 0; 1; 2; : : : for N 2 ; (4.36) dard form
and we can set ` D 0 for N D 1 (the angular momentum term Lj k ; L`m D
vanishes in the one-dimensional case). The units in which i gj ` Lkm C gkm Lj ` gk` Lj m gj mLk` ; (4.30) m D e D „ D 1; c 137 are used throughout. with the diagonal metric tensor gj k defined by the matrix G D diagŒ1; 1; 1; 1; 1; 1. The matrix form Eq. (4.29) also 4.2.1 implies the subalgebra structure so.4; 2/ so.4; 1/ so.4/ so.3/ ;
N-Dimensional Isotropic Harmonic Oscillator
(4.31)
Considering the Hamiltonian with so(4,1) generated by L; A; B and T2 , and so(4) by 1 1 L; A. [L; B also generate so(3,1).] (4.37) H D p2 C ! 2 r 2 ; 2 2 The three independent Casimir operators (quadratic, cubic, and quartic) are [24] (summation over all indices is with p 2 in the form Eq. (4.35), transforming the correspondimplied) ing Schrödinger equation using the scaling transformation Eq. (4.33) and multiplying by 14 2 , we get for the radial com1 jk ponent Q2 D Lj k L 2 1 1 2 1 1 D L2 C A2 B 2 2 C T32 T12 T22 ; PR CR2 C ! 2 4 R2 2 E R .R/ D 0 ; (4.38) 2 4 4 2 1 "ij k`mn Lij Lk` Lmn Q3 D 48 with D T1 .B L/ C T2 . L/ C T3 .A L/ 1 1 D .N 1/.N 3/ C `.` C N 2/ : (4.39) C A .B ^ / ; 16 4
(4.32) Choosing such that !=2 2 4 D 1, we can rewrite Q4 D Lj k Lk` L`m Lmj : Eq. (4.38) using the so(2,1) realization Eq. (4.9) with D 2 For our hydrogenic realization Q2 D 3, Q3 D Q4 D 0. Thus, as our hydrogenic realization implies a single unirrep of so(4,2) 1 (4.40) T3 2 E R .R/ D 0 : adapted to the chain Eq. (4.31). 4
4
Dynamical Groups
85
Thus, using the second equation of Eq. (4.6) we can interrelate R .R/ with jkqi and set 14 2 E D q, so that E D 4q=2 D 2q! ;
Choosing the upper sign [since ` 0 and k > 1], so that q0 D `C 12 .N 1/, and identifying q with the principal quantum number n, we finally have that
(4.41) E En D
with q given by Eqs. (4.12) and (4.13), i.e., q D q0 C ; and q0 D k C 1 D
D 0; 1; 2; : : : ;
(4.42)
1 1 1˙ `C N 1 : 2 2 1 4
Z2 1 D : 22 2n2
(4.51)
The N -dimensional relativistic hydrogenic atom can be treated in the same way, using either the Klein–Gordon or Dirac–Coulomb equations [2, 14–17].
(4.43)
4.2.3 Perturbed Hydrogenic Systems 3 4,
Now, for N D 1, we set ` D 0 so that q0 D and
1 q0 D 1 yielding for E D 2q! the values 2 C 2 ! and 2 C 2 C 1 !, D 0; 1; 2; : : : . Combining both sets we thus get for N D 1 the well-known result 1 ! ; n D 0; 1; 2; : : : : (4.44) E En D n C 2
The so(4,2) based Lie algebraic formalism can be conveniently exploited to carry out large-order perturbation theory ([27–29] and Chap. 5) for hydrogenic systems described by the Schrödinger equation ŒH0 C "V .r/ .r/ D .E0 C E/ .r/ ;
(4.52)
Similarly, for the general case N 2, we choose the upper with H0 given by Eq. (4.22) and E0 by E of Eq. (4.51). Apsign in Eq. (4.43) [so that k > 1] and get plying transformation Eq. (4.33), using Eqs. (4.49), (4.51), and multiplying on the left-hand side by 2 R, we get 1 E En D n C N !; n D 0; 1; 2; : : : ; (4.45) 1 1 2 RP 2 q C R C "2 RV .R/ 2 RE .R/ D 0 ; 2 2 where we identified .` C 2/ with the principal quantum (4.53) number n. where we set .R/ .R/. For the important case of a three-dimensional hydrogenic atom ŒN D 3; D `.` C 1/; q n, using the so(4,2) realization Eq. (4.28) [or so(2,1) 4.2.2 N-Dimensional Hydrogenic Atom realization Eq. (4.9) with D 1] we get Applying the scaling transformation Eq. (4.33) to the hydrogenic Hamiltonian Eq. (4.22) in N -dimensions, we get for the radial component (after multiplying from the left by 2 R) with 1
RPR2 C R1 22 ER Z R .R/ D 0 ; (4.46) 2 where now D
1 .N 1/.N 3/ C `.` C N 2/ : 4
(4.47)
In this case, we must set 22 E D 1 and use realization Eq. (4.9) with D 1 to obtain .T3 Z/
R .R/
D0:
(4.48)
This immediately implies that Z D q ;
(4.49)
and q0 D k C 1 D
1 Œ1 ˙ .2` C N 2/ : 2
(4.50)
.K C "W SE/ .R/ D 0 ;
(4.54)
K D T3 n ; W D 2 RV .R/ ; S D 2 R :
(4.55)
We also have that D n=Z and for the ground state case n D q D 1. Although Eq. (4.54) has the form of a generalized eigenvalue problem requiring perturbation theory formalism with a nonorthogonal basis (where S represents an overlap), T3 is Hermitian with respect to a .1=R/ scalar product, and the required matrix elements can, therefore, be evaluated easily [2, 14, 15, 17, 27–29]. For central field perturbations, V .r/ D V .r/, the problem reduces to one dimension, and since R D T3 T1 , the so(2,1) hydrogenic realization . D 1/ can be employed. For problems of a hydrogenic atom in a magnetic field (Zeeman effect) [27–30] or a one-electron diatomic ion [31], the so(4,2) formalism is required (note, however, that the LoSurdo– Stark effect can also be treated as a one-dimensional problem using parabolic coordinates [32]).
4
86
J. Paldus
The main advantage of the LA approach stems from the fact that the spectrum of T3 is discrete, so that no integration over continuum states is required. Moreover, the relevant perturbations are closely packed around the diagonal in this representation, so that infinite sums are replaced by small finite sums. For example, for the LoSurdo–Stark problem, when V .r/ D F z, where F designates electric field strength in the z-direction, we get Eq. (4.54) with " D F and W D .n=Z/3 RZ, S D .n=Z/2 R. Since both R and Z are readily expressed in terms of so(4,2) generators, Z D B3 A3 ;
R D T3 T1 ;
(4.56)
3. “Nothing of algebraic import is lost by the unitary restriction” [36]. All U(n) irreps have finite dimension and are, thus, relevant to problems involving a finite number of bound states [3–6, 10–12, 36–45].
4.3.1 Unitary Group and Its Representations The unitary group U(n) has n2 generators Eij spanning its LA and satisfying the commutation relations
Eij ; Ek` D ıj k Ei ` ıi ` Ekj
we can easily compute all the required matrix elements [2, and the Hermitian property 14, 15, 17]. Similarly, considering the Zeeman effect with Eij D Ej i :
(4.59)
(4.60)
1
1 BL 3 C B2 r 2 z 2 ; 2 8
(4.57) They are classified as raising .i < j /, lowering .i > j /, and weight .i D j / generators according to whether they raise, where B designates magnetic field strength in the lower, and preserve the weight, respectively. The weight vector is a vector of the carrier space of an irrep that is z-direction, we have for the ground state when n D 1, ` D a simultaneous eigenvector of all weight generators Ei i of 1 2 4 2 2 m D 0 that " D 8 B , K D T3 1, W D Z R.R Z / U(n) (comprising its Cartan subalgebra), and the vector 2 and S D Z R. Again, the matrix elements of W and S are m D .m1 ; m2 ; : : : ; mn / with integer components, consisting obtained from those of Z and R, Eq. (4.56), by matrix mulof corresponding eigenvalues, is called a weight. The highest tiplication (for tables and programs, see [17]). weight mn (in lexical ordering), One can treat one-electron diatomic ions [2, 14, 15, 31] and screened Coulomb potentials, including charmonium mn D .m1n ; m2n ; : : : ; mnn / ; (4.61) and harmonium [10–12, 17, 33, 34], in a similar way. Note, finally, that we can also formulate the perturbed with problem Eq. (4.54) in a standard form not involving the m1n m2n mnn ; (4.62) “overlap” by defining the scaling factor as D .2E/1=2 , where E is now the exact energy E D E0 C E; Eq. (4.54) uniquely labels U(n) irreps, .mn /, and may be represented by a Young pattern. Subducing .mr / of U(r) to U.r 1/, then becomes embedded as U.r 1/ ˚ 1 in U(r), gives [41] M (4.58) .T3 C "W Z/ .R/ D 0 ; .mr1 / ; (4.63) .mr / # U.r 1/ D V .r/ D
with the eigenvalue Z. In this case, any conventional perturbation formalism applies, but the desired energy has to be where the sum extends over all U.r 1/ weights mr1 D .m1;r1 ; m2;r1 ; : : : mr1;r1 / satisfying the so-called “befound from Z [35]. tweenness conditions” [38]
4.3
Compact Dynamical Groups
mi r mi;r1 mi C1;r
.i D 1; : : : ; r 1/ :
(4.64)
Two irreps .mn / and .m0n / of U(n) yield the same irUnitary groups U(n) and their LAs often play the role of rep when restricted to SU(n) if mi D m0 C h; i D 1; : : : ; n. i (compact) dynamical groups since The SU(n) irreps are thus labeled with highest weights with mnn D 0. The dimension of .mn / of U(n) is given by the 1. Quantum mechanical observables are Hermitian, and the Weyl dimension formula [36] LA of U(n) is comprised of Hermitian operators [under Y
. the exp.iA/ mapping]. mi n mj n C j i 1Š2Š .n 1/Š : dim .mn / D 2. Any compact Lie group is isomorphic to a subgroup of i 2. A double covering of SO(m) leads to spin groups Spin theory has a similar structure. The suitable generators are (m). The best way to proceed is, however, to construct the so-called Clifford algebras Cm , whose multiplicative group Fij D Eij Ej i ; Fj i D Fij ; (consisting of invertible elements) contains a subgroup that Fij D Fj i (4.72) provides a double cover of SO(m). The key fact is that C2k is Fi i D 0 ;
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J. Paldus
isomorphic with gl.2k / and C2kC1 with gl.2k / ˚ gl.2k /. The The first-order Casimir operator, Eq. (4.66) with k D 1, then represents the total number operator reps of Cm thus provide the required spinor reps. A Clifford algebra Cm is an associative algebra generated n n X X by Clifford numbers ˛i satisfying the anticommutation relaU.n/ NO C1 D Gi i D bi bi ; (4.86) tions i D1
f˛i ; ˛j g D 2ıij
i D1
.i; j D 1; : : : ; m/ :
(4.80) and the physically relevant states, being totally symmetric, P .N 0 0/. carry single row irreps .N 0/ Since ˛i2 D 1; dim Cm D 2m , and a general element of Cm Similarly, for fermion creation (annihilation) operam is a product of Clifford numbers ˛11 ˛22 ˛m with i D 0 tors XI .XI / that are associated with some orthonoror 1. mal spin-orbital set fjI ig; I D ˚ and satisfy ˚ 1; 2; : : : ; 2n, To see the relation with so.m C 1/ note that ; X D XI ; XJ D 0, the anticommutation relations X I J ˚ XI ; XJ D ıIJ , the operators 1 1 1 F0k D i˛k ; Fj k D ˛j ; ˛k D ˛j ˛k ; 2 4 2 (4.87) eIJ D XI XJ .j ¤ k/ (4.81) satisfy the commutation relations Eq. (4.73). As an example, C2 can be realized by Pauli matrices by setting ! ! 0 1 0 i ; ˛2 D 2 D : (4.82) ˛1 D 1 D 1 0 i 0
again represent the U(2n) generators satisfying Eqs. (4.59) and (4.60). The first-order Casimir then represents the total P number operator NO D I XI XI , while the possible physical states are characterized by totally antisymmetric single P .11 1 0 0/. column irreps .1N 0/
Clearly, the four matrices 12 ; ˛1 ; ˛2 , and ˛1 ˛2 are linearly independent (note that 3 D i 1 2 /, so that C2 is isomorphic to gl.2; C/. Similarly, considering Dirac–Pauli matrices ! i12 0 0 D D i4 ; 0 i12 ! 0 i k ; .k D 1; 2; 3/ ; (4.83) k D i k 0
4.3.5 The Vibron Model
Similar to the unified description of nuclear collective rovibrational states using the interacting boson model [49–51], one can build an analogous model for molecular rotationvibration spectra [8]. For diatomics, an appropriate dynamical group is U.4/ [8, 52–54] and, generally, for rotationvibration spectra in r-dimensions, one requires U.r C 1/. For triatomics, the U.4/ generating algebra is generalized to U.4/ ˝ U.4/, and for the .k C 1/ atomic molecule to U.1/ .4/ ˝ ˝ U.k/ .4/ [8, 52–54]. we have that For the bosonic realization of U.4/, we need four cre ation .bi ; i D 1; : : : ; 4/ and four annihilation .bi / operators (4.84) f i ; j g D 2ıij ; .i; j D 1; : : : ; 4/ ; (Sect. 4.3.4). The Hamiltonian may be generally expressed so that i .i D 1; : : : ; 4/ or .i D 0; : : : ; 3/ represent Clifford as a multilinear
form in terms of boson number preserving numbers for C4 and 14 ; i ; i j .i < j /, i j k .i < j < k/ products bi bj , so that using Eq. (4.85) we can write and 5 1 2 3 4 D i 0 1 2 3 form an additive basis X .1/ 1 X .2/ for gl.4; C/ (the i themselves are said to form a multiplicahij Gij C hij k` Gij Gk` C : (4.88) H D h.0/ C 2 tive basis). For general construction of Cm Clifford numbers ij ij k` in terms of direct products of Pauli matrices see [47, 48]. The energy levels (as a function of 0; 1; 2; : : : -body matrix .1/ .2/ elements h0 ; hij ; hij k` , etc.) are then determined by diag4.3.4 Bosonic and Fermionic Realizations of U(n) onalizing H in an appropriate space, which is conveniently provided by the carrier space of the totally symmetric irrep P of U.4/. Designating by bi .bi / the boson creation (annihilation) op- .N 0 0 0/ .N 0/ The requirement that the resulting states be charactererators (Sect. 6.1.1) satisfying the commutation relations ized by angular momentum J and parity P quantum numŒbi ; bj D Œbi ; bj D 0, Œbi ; bj D ıij , we obtain a possible 2 bers necessitates that the boson operators involved have U(n) realization by defining its n generators as follows definite transformation properties under rotations and re Gij D bi bj : (4.85) flections [8]. The boson operators are thus subdivided into
4
Dynamical Groups
89
the scalar operators ; , J D 0, and vectorJ operators ; I D 0; ˙1 , J D 1 with parity P D ./ . All commutators vanish except for
; 0 D ı0 : (4.89) ; D 1;
In addition to handling diatomic and triatomic systems, the vibron model was also applied to the overtone spectrum of acetylene [55], intramolecular relaxation in benzene and its dimers [56, 57], octahedral molecules of the XF6 type (X D S, W, and U) [58], and to linear polyatomics [59]. Most recently, the experimental (dispersed fluorescence and Since H preserves the total number of vibrons N D n C n , stimulated emission pumping) vibrational spectra of H2 O the second-order Hamiltonian Eq. (4.88) within the irrep P can be expressed in terms of four independent pa- and SO2 in their ground states, representing typical local .N 0/ mode and normal-mode molecules, respectively, have been rameters (apart from an overall constant) as analyzed, including highly excited levels, by relying on the
.0/ U(2) algebraic effective Hamiltonian approach [60–62]. The H D e .0/ C e .1/ Q 0 U(2) algebraic scheme [63] also enabled the treatment of h
.0/ i.0/ .2/ .0/ Franck–Condon transition intensities [64, 65] in rovibronic C e1 Q Q 0 spectra. The attempts at a similar heuristic phenomenologih .2/ i.0/ .2/ .2/ cal description of electronic spectra have so far met with only C e2 Q Q 0 limited success [66]. h .0/ .0/ .2/ C e3 Q Q
.0/ i.0/ .0/ C Q Q C ; (4.90) 4.3.6 Many-Electron Correlation Problem 0
where Q D ; Q D ./1 and square brackets indicate In atomic and molecular electronic structure calculations one employs a spin-independent model Hamiltonian the SU(2) couplings. In special cases, the eigenvalue problem for H can be 2 X X solved analytically, assuming that H can be expressed in h Xi Xj H D ij terms of Casimir operators of a complete chain of subgroups i;j D1 of U.4/ (referred to as dynamical symmetries). Requiring 2 X that the chain contains the physical rotation group O.3/, one 1 X C v Xi Xj X` Xk ; (4.94) ij;k` has two possibilities 2 i;j;k;`
.I/
;D1
U.4/ O.4/ O.3/ O.2/ ;
(4.91) where XI Xi .XI / designate the creation (annihilation) operators associated with the orthonormal spin orbitals These imply labels (quantum numbers): N [total vibron jI i ji i D jii ˝ j iI i D 1; : : : ; nI D 1; 2 D 1; 2 lanumber defining a totally symmetric irrep of U.4/], ! D beling the spin-up and spin-down eigenstates of Sz , and O i; vij;k` D hi.1/j.2/jvjk.1/`.2/i N; N 2; N 4; : : : ; 1 or 0 [defining a totally symmetric hij D hijhjj O are the one and irrep of O.4/] and n D N; N 1; : : : ; 0 [defining the U.3/ two-electron integrals in the orbital basis fjiig. As stated irrep], in addition to the O.3/ O.2/ labels J; M I jM j J . in Sect. 4.3.4, eIJ ei ;j D Xi Xj may then be regarded In terms of these labels one finds for the respective Hamilto- as U.2n/ generators, and the appropriate U.2n/ irrep for N
nians electron states is 1N 0P . Similar to the nuclear many-body problem [67], one O.4/ O.3/ H .I/ D F C A C2 C BC2 ; defines mutually commuting partial traces of spin-orbital U.3/ U.3/ O.3/ (4.92) generators eIJ , Eq. (4.87), H .II/ D F C "C1 C ˛C2 C ˇC2 ; .II/ U.4/ U.3/ O.3/ O.2/ :
U.k/
O.k/
where F , A, B, ", ˛, ˇ are free parameters, and Ci , Ci are relevant Casimir operators, the following expressions [8, 52–54] for their eigenvalues E .I/ .N; !; J; M / D F C A!.! C 2/ C BJ.J C 1/; E .II/ .N; n ; J; M / D F C n C ˛ n .n C 3/
Eij D E D
2 X
ei ;j D
2 X
D1
D1
n X
n X
i D1
ei ;i D
Xi Xj ;
Xi Xi ;
(4.95)
i D1
(4.93) which again satisfy the unitary group commutation relations Eq. (4.59) and property Eq. (4.60), and may thus be considThe limit (I) is appropriate for rigid diatomics and limit (II) ered as the generators of the orbital group U.n/ and the spin for nonrigid ones [8, 52–54]. group U.2/. The Hamiltonian Eq. (4.94) is thus expressible C ˇJ.J C 1/ :
4
90
J. Paldus
in terms of orbital U.n/ generators HD
X
hij Eij C
i;j
1 X vij;k` .Ei k Ej ` ıj k Ei ` / : (4.96) 2 i;j;k;`
We can thus achieve an automatic spin adaptation by exploiting the chain U.2n/ U.n/ ˝ U.2/ (4.97) and diagonalize H within the carrier space of two-column U.n/ irreps .2a 1b 0c / .a; b; c/ with [39, 68] aD
1 N S ; 2
b D 2S ;
1 c Dnab Dn N S ; 2
(4.98)
considering the states of multiplicity .2S C1/ involving n orbitals and N electrons. The dimension of each spin-adapted subproblem equals [39, 68] ! ! bC1 nC1 nC1 ; dim .2 1 0 / D nC1 a c a b c
(4.99)
where mn designate binomial coefficients. Exploiting simplified irrep labeling by triples of integers .a; b; c/, Eq. (4.98), at each level of the canonical chain Eq. (4.67), one achieves more efficient state labeling by replacing Gel’fand tableaux Eq. (4.68) by n 3 ABC [68] or Paldus or Gel’fand–Paldus tableaux [40, 69–77] ŒP D Œai bi ci ;
(4.100)
where ai C bi C ci D i. Another convenient labeling uses the ternary step numbers di ; 0 di 3 [68–70, 78, 79] di D 1 C 2.ai ai 1 / .ci ci 1 / :
(4.101)
A compact representation of a spin-adapted basis of configuration state functions (CSFs) spanning a given U(n) irrep .2a 1b 0c / is given by a distinct row table (DRT) ([69, 70], cf. also [10–12, 39, 71]). The entire basis is then conveniently described by a Shavitt graph [11, 69, 70], providing a compact and transparent representation of the entire basis of CSFs, and the corresponding formalism is referred to as the graphical UGA (GUGA). This facilitates the design of computer codes not only for the full configuration interaction (FCI) calculations ([80]) but mainly for a limited or truncated CI. The Shavitt graph is a two-rooted graph whose vertices represent distinct rows of the ABC tableaux for each subgroup U(i) (i D 1; : : : ; n) with the top root representing the highest weight .a; b; c/ .an ; bn ; cn / for U(n) and the bottom root the trivial row .0; 0; 0/ .a0 ; b0 ; c0 /. The entire graph is placed on a grid so that the slope of its edges im-
plies the relevant step numbers. Each basis vector or CSF is thus represented by a path interconnecting both roots. Considering the generator matrix elements, we note that the bra and ket state paths coincide outside the range given by the generator indices, thus forming a loop within this range that can be split into the segment values for each elementary step. The explicit form of segment values has been derived in several different ways [10–12, 68–71, 75–77, 79, 81–85] (but not within GUGA per se [82]). However, this factorization into a product of segment values is not unique and can be done in several ways ([79, 84, 85]). The most efficient way for a derivation of explicit expressions for segment values exploits the graphical methods of spin algebras employing Jucys-type angular-momentum diagrams ([86]) while relying on the fact that the U(n) electronic CSFs correspond to the Yamanouchi–Kotani coupling scheme [67]. This approach is particularly efficient for matrix elements involving generator products [79, 87, 88]. Yet another useful approach [84, 85, 89] introduced spinadapted creation and annihilation operators that in a certain way mimic those of the standard second-quantization formalism ([82]), which also proved to be very useful for spin-dependent formalism (Sect. 4.3.8). A recent development that exploits GUGA representation of the electronic CSF U(n) basis is the so-called graphically contracted function (GCF) method [90, 91], which is based on the graph density concept. This approach enables a comparison of CSFs in terms of nodes and arcs (edges) of the Shavitt graph, thus opening a possibility of handling extremely large CIs exceeding traditional ones by many orders of magnitude. For the most recent development that also introduced a multifacet GCF (MFGCF), see [92]. Finally, we note that the unitary group formalism that is based either on U(n) or on the universal enveloping algebra of U(n) proved to be of great usefulness in various post-Hartree– Fock approaches to molecular electronic structure [93], especially in large-scale CI calculations (cf., the C OLUMBUS Program System [94–97]; see also [24–31] in [12]), in MCSCF approaches [98–101], propagator methods [102], and coupled cluster (CC) methods of both single-reference [103– 118] and multi-reference [119–122] type, as well as in various other investigations, such as quantum dots [123], in handling of composite systems [89, 124–126], valence-bond (VB) approaches [127–132], reduced density matrices (RDMs) [105, 133], nuclear magnetic resonance (NMR) [134], or charge migration in fragmentation of peptide ions [135, 136]; see also [10–12] for other references. We should also mention numerous other developments and extensions, such as Clifford algebra UGA (CAUGA) (Sect. 4.3.7), bonded tableau UGA [137], parafermi algebras [138], and related developments ([139]). Very recently there have been a new, very promising, exploitation of UGA (see recent critical reviews [140–
4
Dynamical Groups
91
142]) and GUGA in the FCI-QMC (Quantum Monte Carlo) method [143–145], enabling a very efficient handling of relatively large, complex systems [146–151], as well as an exploitation [152] of ecCCSD (externally corrected Coupled Cluster methods with Singles and Doubles) ([153] and Sect. 5.3.8).
4.3.7 Clifford Algebra Unitary Group Approach
can then be represented as a linear combination of twobox states, labeled by the Weyl tableaux Œijj i j . In particular, the highest weight state of .a; b; c/ is represented by 2c j2bCc . Once this representation is available, it is straightforward to compute explicit representations of U(n) generators, since Epq act trivially on Œijj [125]. Defining unnormalized states .ijj / as .ijj / D
q 1 C ıij
Œijj ;
(4.105)
The Clifford algebra unitary group approach (CAUGA) ex- we have ploits a realization of the spinor algebra of the rotation group (4.106) Epq .ijj / D ıqi .pjj / C ıqj .ijp/ : SO(2n C 1) in the covering algebra of U(2n ) to obtain explicit representation matrices for the U(n) [or SO(2n C 1) or The main features of CAUGA may, thus, be summarized SO(2n)] generators in the basis adapted to the chain [125, as follows: CAUGA 154–157] U.2n / Spin.m/ SO.m/ U.n/ ; .m D 2n C 1 or m D 2n/
(4.102)
supplemented, if desired, by the canonical chain Eq. (4.67) for U(n). To realize the connection with the fermionic Grassmann algebra generated by the creation XI and annihilation .XI / operators, I D 1; : : : ; 2n, note that it is isomorphic with the Clifford algebra C4n when we define [12, 25]
˛I D XI C XI ;
˛I C2n D i XI XI ;
.I D 1; : : : ; 2n/ :
1. Effectively reduces an N -electron problem to a number of two-boson problems 2. Enables an exploitation of an arbitrary coupling scheme (being particularly suited for the valence bond method) 3. Can be applied to particle-number nonconserving operators 4. Easily extends to fermions with an arbitrary spin 5. Drastically simplifies evaluation of explicit representations of U(n) generators and of their products 6. Can be exploited in other than shell-model approaches [127–129, 131, 158–160].
(4.103)
4.3.8 Spin-Dependent Operators For practical applications, the most important is the final imbedding U.2n / U.n/ (for the role of intermediate groups, see [154–156]). All states of an n-orbital model, regardless the electron number N and the total spin S, are P contained in a single two-box totally symmetric irrep h20i n of U.2 / [125, 157]. To simplify the notation, one employs the one-to-one correspondence between the Clifford algebra monomials, labeled by the occupation numbers mi D 0 or mi D 1 .i D 1; : : : ; n/, and “multiparticle” single-column U(n) states labeled by p pfmi g D 2n .m1 m2 mn /2 ;
(4.104)
where the occupation number array .m1 mn / is interpreted as a binary integer, which we then regard as one-box states jp/ of U(2n ). The orbital U(n) generators ij may then be expressed as simple linear combinations of U(2n ) generators Epq D jp/.qj with coefficients equal to ˙1 [125, 157]. Generally, any p-column U(n) irrep is contained at least once in the totally symmetric p-box irrep of U(2n ). For many-electron problems, one thus requires a two-box irP Any state arising in the U(n) irrep .a; b; c/ rep h20i.
The spin-adapted U(n)-based UGA is entirely satisfactory in most investigations of molecular electronic structure. However, when exploring the fine structure in high-resolution spectra, the intersystem crossings, phosphorescent lifetimes, molecular predissociation, spin–orbit interactions in transition metals, and like phenomena, the explicitly spindependent terms must be included in the Hamiltonian. Since in most cases the total spin S represents a good approximate quantum number, it is advantageous to employ U(n) spinadapted N -electron states as a point of departure and consider the matrix elements of general spin-orbital U(2n) generators in terms of the U.2n/ U.n/ ˝ SU.2/ basis. Such an approach is generally referred to as a spin-dependent UGA and was first considered in the context of the symmetric group and Racah algebra by Drake and Schlesinger [83] and later on in terms of the Gel’fand–Paldus tableaux [82, 161–165]. Now, the segmentation of the matrix elements of spindependent one-body operators may be shown to be very similar to that for the two-body operators, which makes it possible to exploit the existing UGA or GUGA codes. However, this necessitates that we employ a larger group, say
4
92
J. Paldus
U(n C 1), or even U(n C 2) [82, 165], since U(n C 1) generators can change the spin at the U(n) level, as is the case for spin-dependent one-body operators. In fact, the advantage of a large U(n C 1) group follows immediately when we employ the graphical methods of spin algebras ([79]), as was done by Drake and Schlesinger [83]. This fact was also exploited by Yabushita et al. [166] and implemented in GUGA codes. However, since this approach [166] relies on Racah–Wigner calculus, it requires six additional steps in order to conform with the standard UGA-based formalism, including the introduction of an additional phase factor and a transformation to a “real spherical” spin function in order to ascertain a real-valued form of the relevant matrix elements. These steps may be avoided using an approach that is based on the spin-free analogues of creation and annihilation operators [84, 85]. Indeed, this approach leads to a simple formalism [82] that enables the evaluation of the required one-electron spin-dependent terms in much the same way as the standard spin-independent two-electron (i.e., Coulomb) matrix elements. Only a few additional expressions had to be derived and all the required segment values are given in [82]. In general, the U(2n) generators ei ;j eIJ may be resolved into the spin-shift components e.˙/IJ that increase (+) or decrease () the total spin S by one unit and the zerospin component e.0/IJ that preserves S. The relevant matrix elements can then be expressed in terms of the matrix elements of a single U(n) adjoint tensor operator , which is given by the following second-degree polynomial in U(n) generators, D E .E C N=2 n 2/;
E D kEij k ;
(4.107)
and by the well-known matrix elements of U(2) or SU(2) generators in terms of the pure spin states [161, 162] ([167, 168]). The operator Eq. (4.107), referred to as the Gould–Paldus operator [169], also plays a key role in the determination of reduced density matrices [105, 133] and has recently been exploited in the multireference spin-adapted variant of the density functional theory [169]. Very recently Robb et al. used spin dependent UGA to evaluate spin densities for CI in a basis of S 2 -adapted CSFs [170].
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94 120. Maitra, R., Sinha, D., Mukherjee, D.: J Chem Phys 139, 229903 (2013). Erratum 121. Sen, S., Shee, A., Mukherjee, D.: J. Chem. Phys. 137, 074104 (2012) ˇ ˇ 122. Paldus, J., Cársky, P., Pittner, J.: In: Cársky, P., Paldus, J., Pittner, J. (eds.) Recent Progress in Coupled Cluster Methods: Theory and Applications, pp. 455–489. Springer, Berlin (2010) 123. Remacle, F., Levin, R.D.: Chem. Phys. Chem. 2, 20 (2001) 124. Gould, M.D., Paldus, J.: Int. J. Quantum Chem. 30, 327 (1986) 125. Paldus, J., Gao, M.-J., Chen, J.-Q.: Phys. Rev. A 35, 3197 (1987) 126. Kent, R.D., Schlesinger, M.: Phys. Rev. A 48, 4156 (1993) 127. Li, X., Paldus, J.: Int. J. Quantum Chem. 41, 117 (1992) 128. Paldus, J., Planelles, J.: Theor. Chim. Acta 89, 13 (1994) 129. Planelles, J., Paldus, J., Li, X.: Theor. Chim. Acta 89(33), 59 (1994) 130. Paldus, J., Li, X.: In: Frank, A., Seligman, T.H., Wolf, K.B. (eds.) Group Theory in Physics AIP Conference Proceedings, vol. 266, pp. 159–178. American Institute of Physics, New York (1992) 131. Paldus, J., Li, X.: In: Gruber, B. (ed.) Symmetries in Science VI: From the Rotation Group to Quantum Algebras, pp. 573–592. Plenum, New York (1993) 132. Paldus, J.: J. Math. Chem. 56, 1595 (2018) 133. Gould, M.D., Paldus, J., Chandler, G.S.: J. Chem. Phys. 93, 4142 (1990) 134. Kent, R.D., Schlesinger, M., Ponnapalli, P.S.: Phys. Rev. B 31, 1264 (1985) 135. Remacle, F., Levin, R.D.: J. Chem. Phys. 110, 5089 (1999) 136. Remacle, F., Levin, R.D.: J. Phys. Chem. A. 104, 2341 (2000) 137. Li, X., Zhang, Q.: Int. J. Quantum Chem. 36, 599 (1989) 138. Gould, M.D., Paldus, J.: Phys. Rev. A 34, 804 (1986) 139. Harter, W.G., Patterson, C.W.: A Unitary Calculus for Atomic Orbitals. Lecture Notes in Physics, vol. 49. Springer, Berlin (1976) 140. Paldus, J.: J. Math. Chem. 59, 1 (2021) 141. Paldus, J.: J. Math. Chem. 59, 37 (2021) 142. Paldus, J.: J. Math. Chem. 59, 72 (2021) 143. Dobrautz, W., Smart, S.D., Alavi, A.: J. Chem. Phys. 151, 094104 (2019) 144. Dobrautz, W.: Development of Full Configuration Interaction Quantum Monte Carlo Methods for Strongly Correlated Electron Systems. PhD Thesis, University of Stuttgart (2019) 145. Li Manni, G., Guther, K., Ma, D., Dobrautz, W.: In: González, L., Lindh, R. (eds.): Quantum Chemistry and Dynamics of Excited States: Methods and Applications, Chap. 6, 133—204. Wiley, New York (2021) 146. Li Manni, G., Dobrautz, W., Alavi, A.: J. Chem. Theory Comput. 16, 2202 (2020) 147. Guther, K., Anderson, R.J., Blunt, N.S., Bogdanov, N.A., Cleland, D., Dattani, N., Dobrautz, W., Ghanem, K., Jeszenszki, P., Liebermann, N., Li Manni, G., Lozovoi, A.Y., Luo, H., Ma, D., Merz, F., Overy, C., Rampp, M., Samantha, P.K., Schwarz, L.R., Shepherd, J.J., Smart, S.D., Vitale, E., Weser, O., Booth, G.H., Alavi, A.: J. Chem. Phys. 153, 034107 (2020)
J. Paldus 148. Ghanem, K., Guther, K., Alavi, A.: J. Chem. Phys. 153, 224115 (2020) 149. Li Manni, G., Dobrautz, W., Bogdanov, N.A., Guther, K., Alavi, A.: J. Phys. Chem. A 125, 4727 (2021) 150. Dobrautz, W., Weser, O., Bogdanov, N.A., Alavi, A., Li Manni, G.: J. Chem. Theory Comput. 17, 5684 (2021) 151. Yun, S., Dobrautz, W., Luo, H., Alavi, A.: Phys. Rev. B 104, 235102 (2021) 152. Vitale, E., Alavi, A., Kats, D.S.: J. Chem. Theory Comput. 16(9), 5621 (2020) 153. Paldus, J.: J. Math. Chem. 55, 477 (2017) 154. Nikam, R.S., Sarma, C.R.: J. Math. Phys. 25, 1199 (1984) 155. Sarma, C.R., Paldus, J.: J. Math. Phys. 26, 1140 (1985) 156. Gould, M.D., Paldus, J.: J. Math. Phys. 28, 2304 (1987) 157. Paldus, J., Sarma, C.R.: J. Chem. Phys. 83, 5135 (1985) 158. Paldus, J.: In: Malli, G.L. (ed.) Relativistic and Electron Correlation Effects in Molecules and Solids NATO ASI Series B. pp. 207–282. Plenum, New York (1994) 159. Li, X., Paldus, J.: J. Chem. Phys. 102, 8059 (1995) 160. Piecuch, P., Li, X., Paldus, J.: Chem. Phys. Lett. 230, 377 (1994) 161. Gould, M.D., Paldus, J.: J. Chem. Phys. 92, 7394 (1990) 162. Gould, M.D., Battle, J.S.: J. Chem. Phys. 99, 5961 (1993) 163. Kent, R.D., Schlesinger, M.: Phys. Rev. A 42, 1155 (1990) 164. Kent, R.D., Schlesinger, M.: Phys. Rev. A 50, 186 (1994) 165. Kent, R.D., Schlesinger, M., Shavitt, I.: Int. J. Quantum Chem. 41, 89 (1992) 166. Yabushita, S., Zhang, Z., Pitzer, R.M.: J. Phys. Chem. A 103, 5791 (1999) 167. Gould, M.D., Battle, J.S.: J. Chem. Phys. 98, 8843 (1993) 168. Gould, M.D., Battle, J.S.: J. Chem. Phys. 99, 5983 (1993) 169. Khait, Y.G., Hoffmann, M.R.: J. Chem. Phys. 120, 5005 (2004) 170. Polyak, I., Bearpark, M.J., Robb, M.A.: Int. J. Quantum Chem. 18(12), e.25559 (2017); https://doi.org/10.1002/qua.25559
Josef Paldus Josef Paldus is Distinguished Professor Emeritus at the Department of Applied Mathematics of the University of Waterloo, Canada. He received his PhD from the Czechoslovak Academy of Sciences and his RNDr and DrSc from the Faculty of Mathematics and Physics of Charles University, Prague, Czech Republic. His research interests are the methodology of quantum chemistry and the electronic structure of molecular systems. He has published over 340 papers, reviews, and chapters.
5
Perturbation Theory Josef Paldus
Contents 5.1 5.1.1 5.1.2 5.1.3 5.1.4
Matrix Perturbation Theory (PT) . Basic Concepts . . . . . . . . . . . . . Level-Shift Operators . . . . . . . . . General Formalism . . . . . . . . . . . Nondegenerate Case . . . . . . . . . .
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95 96 96 96 97
5.2 5.2.1 5.2.2 5.2.3
Time-Independent Perturbation Theory . . . . . . . . General Formulation . . . . . . . . . . . . . . . . . . . . . . Brillouin–Wigner and Rayleigh–Schrödinger PT (RSPT) Bracketing Theorem and RSPT . . . . . . . . . . . . . . .
97 97 98 98
5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.3.6 5.3.7 5.3.8
Fermionic Many-Body Perturbation Theory (MBPT) Time-Independent Wick’s Theorem . . . . . . . . . . . . . Normal Product Form of PT . . . . . . . . . . . . . . . . . Møller–Plesset and Epstein–Nesbet PT . . . . . . . . . . . Diagrammatic MBPT . . . . . . . . . . . . . . . . . . . . . Vacuum and Wave Function Diagrams . . . . . . . . . . . Hartree–Fock Diagrams . . . . . . . . . . . . . . . . . . . . Linked and Connected Cluster Theorems . . . . . . . . . Coupled Cluster Theory . . . . . . . . . . . . . . . . . . . .
99 99 99 100 100 101 101 102 103
5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5
Time-Dependent Perturbation Theory . . Evolution Operator PT Expansion . . . . . . . Gell-Mann and Low Formula . . . . . . . . . . Potential Scattering and Quantum Dynamics Born Series . . . . . . . . . . . . . . . . . . . . . Variation of Constants Method . . . . . . . . .
105 105 105 106 106 107
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enabling physicists and engineers to obtain approximate solutions of various systems of differential equations [1– 5]. For the problems of atomic and molecular structure and dynamics, the perturbed problem is usually given by the time-independent or time-dependent Schrödinger equation [6–10]. Keywords
perturbation theory (PT) level shift operators BrillouinWigner PT Rayleigh-Schrödinger PT many-body PT (MBPT) diagrammatic MBPT bracketing theorem Wick’s theorem Møller-Plesset PT Epstein-Nesbet PT linked and connected cluster theorems coupled cluster (CC) theory multireference CC (MRCC) methods state universal and valence universal MRCC internally and externally corrected CC time-dependent PT GellMann and Low formula Tomonaga-Schwinger equation Born series potential scattering
5.1
Matrix Perturbation Theory (PT)
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A prototype of a time-independent PT considers an eigenvalue problem for the Hamiltonian H of the form Abstract 1 X Perturbation theory (PT) represents one of the bridges that "i Vi ; (5.1) H D H0 C V ; V D takes us from a simpler, exactly solvable (unperturbed) i D1 problem to a corresponding real (perturbed) problem by expressing its solutions as a series expansion in a suit- acting in a (finite-dimensional) Hilbert space Vn , assuming ably chosen “small” parameter " in such a way that the that the spectral resolution of the unperturbed operator H0 is problem reduces to the unperturbed problem when " D 0. known; i.e., It originated in classical mechanics and eventually develX X !i Pi ; Pi Pj D ıij Pj ; Pi D I ; (5.2) H0 D oped into an important branch of applied mathematics i
J. Paldus () Dept. of Applied Mathematics, University of Waterloo Waterloo, ON, Canada e-mail: [email protected]
i
where !i are distinct eigenvalues of H0 , the Pi form a complete orthonormal set of Hermitian idempotents, and I is the identity operator on Vn . The PT problem for H can then be
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_5
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formulated within the Lie algebra A (Sect. 3.2) generated by we find from Eq. (5.8) that H0 and V [11, 12]. 1 E C F D V C ŒG; V C E 2 1 X 5.1.1 Basic Concepts .kŠ/1 Bk .ad G/k .V E/ ; C
(5.13)
kD2
Define the diagonal part hXi of a general operator X 2 A by where we used the identity [14] X hXi D Pi XPi (5.3) ! 1 ! 1 X i Bk k X 1 X Xk D I ; kŠ .k C 1/Š and recall that the adjoint action of X 2 A, ad XWA ! A is kD0 kD0 defined by and Bk designates the Bernoulli numbers [15] (5.4) ad X.Y / D ŒX; Y ; .8 Y 2 A/ ; 1 1 B0 D 1 ; B1 D ; B2 D ; where the square bracket denotes the commutator. The key 2 6 problem of PT is the “inversion” of this operation, i.e., the B2kC1 D 0 .k 1/ ; solution of the equation [11–13] 1 1 ; B6 D ; etc. B4 D 30 42 (5.5) ad H .X/ ŒH ; X D Y : 0
(5.14)
(5.15)
0
Assuming that hY i D 0, then
5.1.3 General Formalism
X D R.Y / C hAi ;
(5.6) Introducing the PT expansion for relevant operators,
where A 2 A is arbitrary, and X R.Y / D 1 ij Pi YPj ;
(5.7)
XD
1 X
" i Xi ;
X D E; F; G I
Fi D ŒH0 ; Gi ; (5.16)
i D1
i ¤j
with ij D !i !j , represents the solution of Eq. (5.5) with Eq. (5.13) leads to the following system of equations the vanishing diagonal part hR.Y /i D 0. E1 C F1 D V1 ; 1 E2 C F2 D V2 C ŒG1 ; V1 C E1 ; 2 5.1.2 Level-Shift Operators 1 E3 C F3 D V3 C ŒG1 ; V2 C E2 2 To solve the PT problem for H , Eq. (5.1) we search for a uni1 tary level-shift transformation U [11, 12], U U D U U D I , C ŒG2 ; V1 C E1 2 1 (5.8) UH U D U.H0 C V /U D H0 C E ; (5.17) C ŒG1 ; ŒG1 ; V1 E1 ; etc. ; 12 where the level-shift operator E satisfies the condition which can be solved recursively for Ei and Gi by taking their E D hEi : (5.9) diagonal part and applying operator R, Eq. (5.7), since To guarantee the unitarity of U we express it in the form U D eG ;
G D G ;
hGi D 0 :
(5.10)
1 X
hGi i D hFi i D 0 ;
RFi D Gi ;
REi D 0 :
(5.18)
We thus get
Using the Hausdorff formula eA B eA D
hEi i D Ei ;
.kŠ/1 .ad A/k B
E1 D hV1 i ; (5.11)
kD0
and defining the operator F D ŒH0 ; G ;
(5.12)
1 E2 D hV2 i C hŒRV1 ; V1 i ; 2 E3 D hV3 i C hŒRV1 ; V2 i 1 C hŒRV1 ; ŒRV1 ; 2V1 C E1 i ; 6
etc. ;
(5.19)
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97
5.2 Time-Independent Perturbation Theory
and G1 D RV1 ;
For stationary problems, particularly those arising in atomic etc. (5.20) and molecular electronic structure studies relying on ab initio model Hamiltonians, the PT of Sect. 5.1 can be given a more Since explicit form that avoids a priori the nonphysical, size inextensive terms [7–10, 16–18]. hR.X/i D RhXi D 0 ; hR.X/Y i D hXR.Y /i ; 1 G2 D RV2 C RŒRV1 ; V1 C E1 ; 2
R.XhY i/ D R.X/hY i;
R.hXiY / D hXiR.Y / ;
(5.21) 5.2.1 General Formulation these relationships can be transformed to a more conventional form We wish to find the eigenvalues and eigenvectors of the full E2 D hV2 i hV1 RV1 i ; (perturbed) problem E3 D hV3 i hV1 RV2 i hV2 RV1 i (5.24) Kji i .K0 C W /ji i D ki ji i ; 1 C hR.V1 /R.V1 /Œ2V1 C hV1 ii 6 assuming we know those of the unperturbed problem 1 hR.V1 /Œ2V1 C hV1 iR.V1 /i 3 (5.25) K0 j˚i i D i j˚i i; h˚i j˚j i D ıij : 1 (5.22) C hŒ2V1 C hV1 iR.V1 /R.V1 /i ; etc. 6 For simplicity, we restrict ourselves to the nondegenerate However, in this way certain nonphysical terms arise that case ( i ¤ j if i ¤ j ) and consider only the first-order exactly cancel when the commutator form is employed perturbation (Eq. (5.1), "V1 W; Vi D 0 for i 2). Of (Sect. 5.3.7). course, K and K0 are Hermitian operators acting in a Hilbert space, which, in ab initio applications, is finite-dimensional. Using the intermediate normalization for ji i,
5.1.4
Nondegenerate Case
(5.26) hi j˚i i D 1 ; In the nondegenerate case, when Pi D jiihij, with jii representing the eigenvector of H0 associated with the eigenvalue the asymmetric energy formula gives !i , the level-shift operator is diagonal, and its explicit PT expansion (as well as that for the corresponding eigenvectors) (5.27) ki D i C h˚i jW ji i : is readily obtained from Eqs. (5.19) and (5.20). Writing xij for the matrix element hijXjj i, we get The idempotent Hermitian projectors .e1 /i i D .v1 /i i ; X j˚j ih˚j j (5.28) Pi D j˚i ih˚i j; Qi D Pi? D 1 Pi D X0 .v1 /ij .v1 /j i ; .e2 /i i D .v2 /i i j.¤i / j i j .e3 /i i D .v3 /i i C
X0 .v1 /ij .v2 /j i C .v2 /ij .v1 /j i j i j
X0 .v1 /ij .v1 /j k .v1 /ki j i ki
j;k
.v1 /i i
X0 .v1 /ij .v1 /j i j
j2i
;
etc.;
commute with K0 , so that . K0 /Qi ji i D Qi . ki C W /ji i ;
(5.29)
being an arbitrary scalar (note that we write I simply as ). Since the resolvent . K0 /1 of K0 is nonsingular on (5.23) the orthogonal complement of the i-th eigenspace, we get
Qi ji i D ji i j˚i i D Ri ./. ki C W /ji i ; (5.30) the prime on the summation symbols indicating that the terms with the vanishing denominator are to be deleted. where Note that in contrast to PT expansions, which directly ex2 pand the level-shift transformation U; U D 1C"U1 C" U2 C Ri Ri ./ D . K0 /1 Qi , the above Lie algebraic formulation has the advantage X j˚j ih˚j j that U stays unitary in every order of PT. This is particularly D Qi . K0 /1 D ; (5.31) j useful in spectroscopic applications, such as line broadening. j.¤i /
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assuming . ¤ j /. Iterating this relationship, we get proto- 5.2.3 Bracketing Theorem and RSPT types of the desired PT expansion for ji i, Expressions Eqs. (5.37) and (5.38) are not explicit, since they 1 X n ji i D ŒRi . ki C W / j˚i i ; (5.32) involve the exact eigenvalues ki on the right-hand side. To achieve an order-by-order separation, set nD0 and, from Eq. (5.27), for ki , ki D i C
1 X
n
h˚i jW ŒRi . ki C W / j˚i i :
ki k D (5.33)
1 X
k .j / ;
ji i j i D
j D0
1 X
j .j / i ;
(5.40)
j D0
nD0
where the superscript .j / indicates the j -th-order in the perturbation W . We only consider the eigenvalue expressions, 5.2.2 Brillouin–Wigner and since the corresponding eigenvectors are readily recovered Rayleigh–Schrödinger PT (RSPT) from them by removing the bra state and the first interaction W (Eqs. (5.37) and (5.38)). We also simplify the mean So far, the parameter was arbitrary as long as ¤ j .j ¤ value notation writing for a general operator X, i/. The following two choices lead to the two basic types of many-body perturbation theory (MBPT): (5.41) hXi h˚i jXj˚i i : Brillouin–Wigner (BW) PT Setting D ki gives ki D i C
1 X
n .BW/ h˚i jW Ri W j˚i i ;
Substituting the first expansion Eq. (5.40) into Eq. (5.37) and collecting the terms of the same order in W , we get (5.34) k .0/ D i ;
nD0 1 n X .BW/ Ri W j˚i i ; ji i D
(5.35)
k .1/ D hW i ;
(5.36)
k .2/ D hWRW i ; ˝ ˛ ˛ ˝ k .3/ D W .RW /2 hW i WR2 W ; ˝ ˛ k .4/ D W .RW /3 ˛ ˛ ˝
˝ hW i WR.RW /2 C .WR/2 RW ˛ ˝ ˛ ˝ C hW i2 WR3 W hWRW i WR2 W ;
nD0
where .BW/
Ri
D
X j˚j ih˚j j : ki j
j.¤i /
Rayleigh–Schrödinger (RS) PT Setting D i gives ki D i C
1 X
h in .RS/ h˚i jW Ri . i ki C W / j˚i i ;
nD0
(5.37) 1 h in X .RS/ Ri . i ki C W / j˚i i ; ji i D nD0
where now .RS/
Ri Ri
D
X j˚j ih˚j j :
i j
etc. (5.42)
The general expression has the form ˝ ˛ k .n/ D W .RW /n1 C R.n/ ;
(5.43)
(5.38) the first term on the right-hand side being referred to as the principal n-th-order term, while R.n/ designates the so-called renormalization terms that are obtained by the bracketing theorem [16, 19] as follows: (5.39)
j.¤i /
The main distinction between these two PTs lies in the fact that the BW form has the exact eigenvalues appearing in the denominators and, thus, leads to polynomial expressions for ki . Although these are not difficult to solve numerically, since the eigenvalues are separated, the resulting energies are never size extensive and, thus, unusable for extended systems. They are also unsuitable for finite systems when the particle number changes, as in various dissociation processes. From now on, we thus investigate only the RSPT, which yields a fully size-extensive theory.
1. Insert the bracketings h i around the W , WRW , : : :, WR RW operator strings of the principal term in all possible ways. 2. Bracketings involving the rightmost and/or the leftmost interaction vanish. 3. The sign of each bracketed term is given by .1/nB , where nB is the number of bracketings. 4. Bracketings within bracketings˝ are ˛allowed, e.g., 2 hWRhWRhW iRW iRW i D hW i WR2 W . 5. The total number of bracketings (including the principal term) is .2n 2/Š=ŒnŠ.n 1/Š.
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5.3
99
Fermionic Many-Body Perturbation Theory (MBPT)
and a corresponding unperturbed problem H0 j˚i i D "i j˚i i ;
5.3.1 Time-Independent Wick’s Theorem
H0 D Z C U ;
The development of an explicit MBPT formalism is greatly facilitated by the exploitation of the time-independent version of Wick’s theorem. This version of the theorem ex presses an arbitrary product of creation (a ) and annihilation (a ) operators (Chap. 6) as a normal product (relative to j˚0 i) and as normal products with all possible contractions of these operators [16–18],
h˚i j˚j i D ıij ;
(5.48)
with U representing some approximation to V . In the case that U is also a one-electron operator, U D ˙i u.i/, the unperturbed problem Eq. (5.48) is separable and reduces to a one-electron problem, .z C u/ji D ! ji ;
(5.49)
which is assumed to be solved. Choosing the orthonormal x1 x2 xk D N Œx1 x2 xk C ˙ N Œx1 x2 xk ; spin orbitals fjig as a basis of the second quantization rep .xi D ai or xi D ai / ; (5.44) resentation Sect. 6.1, the N -electron solutions of Eq. (5.48) can be represented as where (5.50) j˚i i D a 1 a 2 a N j0i ; a a D a a D 0 ; N X "i D !j ; (5.51) a a D h./ı ; a a D p./ı ; (5.45)
j D1
and
the state label i representing the occupied spin orbital set f1 ; 2 ; : : : ; N g, while the one and two-body operators take the form
h./ D 1 ; p./ D 0 if ji is occupied in j˚0 i (hole states), h./ D 0 ; p./ D 1 if ji is unoccupied in j˚0 i (particle states) .
XD (5.46)
The N -product with contractions is defined as a product of individual contractions times the N -product of uncontracted operators (defining N Œ; 1 for an empty set) with the sign given by the parity of the permutation reordering the operators into their final order. Note that the Fermi vacuum mean value of an N -product vanishes unless all operators are contracted. Thus, hx1 x2 xk i is given by the sum over all possible fully contracted terms (vacuum terms). Similar rules follow for the expressions of the type .x1 x2 xk /j˚i. Moreover, if some operators on the left-hand side of Eq. (5.44) are already in the N -product form, all the terms involving contractions between these operators vanish.
5.3.2
V D
H DZCV D
i
z.i/ C
X i N , the sign being determined by the parity of the permutation p W j 7! j .) Defining K D H hH i;
K0 D H0 hH0 i D H0 "0 ; (5.57)
we can return to Eqs. (5.24) and (5.25), where now ki D Ei hH i; "0 D
N X
i D "i "0 ;
! ;
(5.58)
D1
and W D K K0 D V U hV U i :
(5.59)
With this choice, hW i D 0, so that for the reference state j˚i, Eq. (5.42) simplify to (we drop the subscript 0 for simplicity) k .0/ D 0;
k .1/ D 0 ;
assuming that j˚j i is a -times excited configuration relative to j˚i obtained through excitations i ! i ; i D 1; : : : ; . Using the Slater rules (or the second quantization algebra), we can express the second-order contribution in terms of the two-electron integrals and HF orbital energies as k .2/ D
a;b;r;s
where the summations over a; b .r; s/ extend over all occupied (unoccupied) spin orbitals in j˚i. Obtaining the corresponding higher-order corrections becomes more and more laborious and, beginning with the fourth-order, important cancellations arise between the principal and renormalization terms, even when the N -product form is employed. These will be addressed in Sect. 5.3.7. The above outlined PT with H0 given by the HF operator is often referred to as the Møller–Plesset PT [20] and, when truncated to the n-th order, is designated by the acronym MPn, n D 2; 3; : : :. In this version, the two-electron integrals enter the denominators only through the HF orbital energies. In an alternative, less often employed variant, referred to as the Epstein–Nesbet PT [21, 22], the whole diagonal part of H is used as the unperturbed Hamiltonian, i.e.,
k .2/ D hWRW i ; k
.3/
k
.4/
1 X habjvjrsi hrsjvjabi hrsjvjbai ; (5.64) 2 !a C !b !r !s
H0 D
D hWRWRW i ; ˝ ˛ ˛ ˝ D W .RW /3 hWRW i WR2 W ;
X
h˚i jH j˚i iPi :
(5.65)
i
etc.
(5.60) With this choice, the denominators are given as differences of the diagonal elements of the configuration interaction matrix.
Note that W is also in the N -product form, W D W1 C W2 ;
W1 D GN UN ;
W2 D VN : (5.61) 5.3.4
5.3.3 Møller–Plesset and Epstein–Nesbet PT Choosing U D G we have H0 D Z C G F ;
(5.62)
so that Eq. (5.49) represent Hartree–Fock (HF) equations, and ! and ji the canonical HF orbital energies and
P spin orbitals, respectively. Since hH i D N D1 hjzji C 1 hjgji is the HF energy, k D k0 gives directly the ground 2 state correlation energy. (Note, however, that the N -product form of PT eliminates the first-order contribution k .1/ D hW i in any basis, even when F is not diagonal.) With this choice, W1 D 0, W D VN , and the denominators in Eq. (5.39) are given by the differences of HF orbital energies
0 j D
X i D1
.!i !i / hfi gI fj gi ;
(5.63)
Diagrammatic MBPT
To facilitate the evaluation of higher-order terms and, especially, to derive the general properties and characteristics of the MBPT, it is useful to employ a diagrammatic representation [7–10, 16–18]. Representing all the operators in Eqs. (5.42) and (5.43) or (5.60) in the second quantized form, we have to deal with the reference state (i.e., the Fermi vacuum) mean values of the strings of annihilation and creation operators (or with these strings acting on the reference in the case of a wave function). This is efficiently done using Wick’s theorem and its diagrammatic representation via a special form of Feynman diagrams. In this representation, we associate with various operators suitable vertices with incident oriented lines representing the creation (outgoing lines) and annihilation (ingoing lines) operators that are involved in their second quantization form. A few typical diagrams representing operators .U /; W1 and V are shown in Fig. 5.1a, Fig. 5.1b and Fig. 5.1c, Fig. 5.1d, respectively. Using the N -product form of PT with HF orbitals (Sect. 5.3.3), we only need the two-electron operator V or VN , which
5
Perturbation Theory
a
b ν
101
c ν
d ν
σ τ
ν
σ τ
Fig. 5.3 Hugenholtz diagrams for the third-order energy contribution Fig. 5.1 Diagrammatic representation of one and two-electron operators
on each vertex define the bra (ket) states of a given matrix element, and for the Goldstone diagrams, the oriented lines attached to the same node are associated with the same electron number, (e.g., for the leftmost vertex in diagram (a) of Fig. 5.2, we have habjvjrsi O ha.1/b.2/jvjr.1/s.2/i). 2. Associate a denominator, Eq. (5.63), or its appropriate power, with every neighboring pair of vertices (and, for 5.3.5 Vacuum and Wave Function Diagrams the wave function diagrams, also with the free lines extending to the left of the leftmost vertex; with each pair Applying Wick’s theorem to the strings of operators inof such free lines associate also the corresponding pair of volved, we represent the individual contractions, Eq. (5.45), particle creation and hole annihilation operators). by joining corresponding oriented lines. To obtain a nonvan- 3. Sum over all hole and particle labels. ishing contribution, only contractions preserving the orienta- 4. Multiply each diagram contribution by the weight factor tion need be considered (Eq. (5.45)). The resulting internal given by the reciprocal value of the order of the group lines have either the left–right orientation (hole lines) or the of automorphisms of the diagram (stripped of summation right–left one (particle lines). Only fully contracted terms, labels) and by the sign .1/hC` , where h designates the represented by the so-called vacuum diagrams (having only number of internal hole lines, and ` gives the number of internal lines), can contribute to the energy, while those repclosed loops of oriented lines (for Hugenholtz diagrams, resenting wave function contributions have uncontracted or use any of its Goldstone representatives to determine the free lines extending to the left. When the operators involved correct phase). are in the N -product form, no contractions of oriented lines Applying these rules to diagrams (a) and (b) of Fig. 5.2 we issuing from the same vertex are allowed. The projectionlike operators R, Eq. (5.39), or their powers, lead to the clearly recover Eq. (5.64) or, using the Hugenholtz diagram denominators, Eq. (5.63), given by the difference of hole of Fig. 5.2, the equivalent expression and particle orbital energies associated with, respectively, 1 X hole and particle lines passing through the interval sepahabjvjrsiA hrsjvjabiA 1 .a; bI r; s/ : (5.66) k .2/ D 4 rating the corresponding two neighboring vertices. Clearly, a;b;r;s there must always be at least one pair of such lines lest the denominator vanish. Thus, for example, the second-order The possible third-order Hugenholtz diagrams are shown contribution hWRW i is represented either by the two Gold- in Fig. 5.3 with the central vertex involving particle–particle, stone diagrams [23] (Fig. 5.2a,b) or by the single Hugenholtz hole–hole, and particle–hole interaction [16–18]. diagram [24] (Fig. 5.2c). The rules for the energy (vacuum) diagram evaluation are as follows: can be represented using either nonantisymmetrized vertices (Fig. 5.1c), leading to the Goldstone diagrams [23], or antisymmetrized vertices (Fig. 5.1d), associated with antisymmetrized two-electron integrals Eq. (5.56) and yielding the Hugenholtz diagrams [24].
5.3.6 Hartree–Fock Diagrams 1. Associate appropriate matrix elements with all vertices and form their product. The outgoing (ingoing) lines In the general case (non-HF orbitals and/or not normal product form of PT), the one-electron terms, as well as the contractions between operators associated with the same r a b c r r vertex, can occur (the latter are always the hole lines). Reps a resenting the W1 and .U / operators as shown in Fig. 5.1, a a b s the one-body perturbation W1 represents in fact the three dib s agrams as shown in Fig. 5.4. The second-order contribution b of this type is then represented by the diagrams in Fig. 5.5, Fig. 5.2 The second-order Goldstone (a), (b) and Hugenholtz (c) dia- which in fact represents nine diagrams that result when each grams W1 vertex is replaced by three vertices as shown in Fig. 5.4.
5
102
J. Paldus
a
b
c ν
ν =
d ν
A
a
+
+ a
ν
B
Fig. 5.4 Schematic representation of W1 D GN UN Fig. 5.5 The second-order, oneparticle contribution
Fig. 5.6 The fourth-order unlinked diagrams
r a
Using HF orbitals, all these terms mutually cancel out, as can be seen above. For this reason, the diagrams involving contractions of lines issuing from the same vertex are referred to as Hartree–Fock diagrams. Note, however, that even when not employing the canonical HF orbitals, it is convenient to introduce W1 vertices of the normal product form PT and replace all nine HF-type diagrams by a single diagram of Fig. 5.5. Clearly, this feature provides even greater efficiency in higher orders of PT.
5.3.7 Linked and Connected Cluster Theorems
top and the bottom part by A and B, respectively, we find for the overall contribution 1 1 B ACB 1 D C B
1 1 1 1 C B A ACB B 1 1 1 : D A .A C B/B AB 2
Thus, the contribution from these terms˝ exactly ˛cancels that from the renormalization term hWRW i WR2 W . Generalizing Eq. (5.67) we obtain the factorization lemma of Frantz and Mills [27], which implies the cancellation of renormalization terms by the unlinked terms originating from the principal term. This result holds for the energy as well as for the wave function contributions in every order of PT, as ascertained by the linked cluster theorem, which states that E D k D
1 X
h˚jW j .n/ i D
nD0
Using the N -product form of PT, the first nonvanishing renormalization term occurs in the fourth-order (Eq. (5.60)). For a system consisting of N noninteracting species, the energy given by this nonphysical term is proportional to N 2 and, thus, violates the size extensivity of the theory. It was first shown by Brueckner [25] that in the fourth order, these terms are in fact exactly canceled by the corresponding contributions originating in the principal term. A general proof of this cancellation in an arbitrary order was then given by Goldstone [23] and Hubbard [26] using a time-dependent PT based on Gell-Mann and Low adiabatic theorem and evolution operator (Sect. 5.4) and by Hugenholtz [24] employing a time-independent PT, relying on Green’s function or the resolvent operator. To comprehend this cancellation, consider the fourthorder energy contribution arising from the so-called unlinked diagrams (no such contribution can arise in the second or third order) shown in Fig. 5.6. An unlinked diagram is defined as a diagram containing a disconnected vacuum diagram (for the energy diagrams, the terms unlinked and disconnected are synonymous). The numerators associated with both diagrams being identical, we only consider the denominators. Designating the denominator associated with the
(5.67)
j i D
1 X nD0
j .n/ i D
1 X
hW .RW /n iL ; (5.68)
nD0 1 X
˚
.RW /n j˚i
L
;
(5.69)
nD0
where the subscript L indicates that only linked diagrams (or terms) are to be considered. This enables us to obtain general, explicit expressions for the n-th-order PT contributions by first constructing all possible linked diagrams involving n vertices and by converting them into the explicit algebraic expressions using the rules of Sect. 5.3.5. Note that linked energy diagrams are always connected, but the linked wave function diagrams are either connected or disconnected, each disconnected component possessing at least one pair of particle–hole free lines extending to the left. To reveal a deeper structure of the result Eq. (5.69) define the cluster operator T that generates all connected wave function diagrams, T j˚i D
1 X ˚ .RW /n j˚i C ;
(5.70)
nD1
the subscript C indicating that only contributions from connected diagrams are to be included. Since the general component with r disconnected parts can be shown to be
5
Perturbation Theory
103
represented by the term .rŠ/1 T r j˚i, the general structure of of the full CC equations is equivalent to the exact soluthe exact wave function j i is given by the connected cluster tion of the Schrödinger equation, we must—in all practical theorem [24, 26], which states that applications—introduce a suitable truncation scheme, which implies that only diagrams of certain types are summed. (For T (5.71) different possible truncation schemes, see [44, Sect. 2].) j i D e j˚i ; Generally, using the cluster expansion Eq. (5.71) in the assuming an intermediate normalization h˚j i D 1. In other N -product form of the Schrödinger equation, words, the wave operator W that transforms the unperturbed HN j i .H hH i/j i D Ej i ; independent particle model wave function j˚i into the exact one according to
E D E E0 ; j i D W j˚i
premultiplying with the inverse of the wave operator, and using the Hausdorff formula Eq. (5.11) yields
is given by the exponential of the cluster operator T , W D eT ;
(5.75)
(5.72)
(5.73)
which, in turn, is given by the connected wave function diagrams. This is in fact the basis of the coupled cluster methods [28–33] (Sect. 5.3.8). For recent reviews, see, e.g., [10, 17, 18, 34–41]; for a historical perspective, see [42, 43]. The contributions to T may be further classified by their excitation rank i, N X Ti ; (5.74) T D i D1
where Ti designates connected diagrams with i pairs of free particle–hole lines, producing i-times excited components of j i when acting on j˚i.
5.3.8 Coupled Cluster Theory To motivate coupled cluster (CC) approaches we recall that summing all HF diagrams (Sect. 5.3.6) is equivalent to solving the HF equations. Likewise, depending on the average electron density of the system, it may be essential to sum certain types of PT diagrams to infinite order at the post-HF level, namely the ring diagrams (random phase approximation) in the case of high-density electron gas and ladder diagrams for low-density systems. An ideal approach that can sum different kinds of diagrams and their combinations, and is thus capable of recovering a large part of the electronic correlation energy, is based on the connected cluster theorem (Sect. 5.3.7), referred to in this context as the exponential cluster ansatz Eq. (5.71) for the wave operator. Using this ansatz, one derives a system of energy-independent nonlinear coupled cluster (CC) equations [10, 17, 18, 30–33] determining the cluster amplitudes of T . These CC equations can be regarded as recurrence relations generating the MBPT series [17, 18], so that by solving these equations one, in fact, implicitly generates all the relevant MBPT diagrams and sums them to infinite order. Since the solution
T
e
1 X Œad .T /n HN HN e j˚i D D Ej˚i : (5.76) nŠ nD0 T
In fact, this expansion terminates, so that using Eq. (5.74) and projecting onto j˚i we obtain the energy expression E D hHN T2 i C
˛ 1˝ HN T12 ; 2
(5.77)
while the projection onto the manifold of excited states fj˚i ig relative to j˚i j˚0 i gives the system of CC equations 1 h˚i jHN C ŒHN ; T C ŒŒHN ; T ; T C j˚i D 0 : (5.78) 2 Approximating, e.g., T by the most important pair cluster component T T2 gives the so-called CCD (coupled clusters with doubles) approximation ˝ .2/ ˇ 1 ˚i ˇHN C ŒHN ; T2 C ŒŒHN ; T2 ; T2 j˚i D 0 ; (5.79) 2 the superscript (2) indicating pair excitations relative to j˚i. Equivalently, Eqs. (5.77) and (5.78) can be written in the form ˝ ˛ E D HN eT C ; ˇ ˛ ˝ ˇ
˚i ˇ HN eT ˇ˚ D 0 ; C
(5.80) (5.81)
the subscript C again indicating that only connected diagrams are to be considered. The general explicit form of CC equations is ai C
X j
bij tj C
X
cij k tj tk C D 0 ;
(5.82)
j k
where ai D h˚i jHN j˚0 i; bij D h˚i jHN j˚j iC , cij k D h˚i jHN j˚j ˝ ˚k iC , etc. Writing the diagonal linear term bi i in the form (5.83) bi i D i C bi0 i ;
5
104
J. Paldus
this system can be solved iteratively by rewriting it in the a proper description of such states requires a multireference form (MR) generalization based on the effective Hamiltonian X formalism [7, 18, 36, 52–54]. Unfortunately, such a gener.nC1/ .n/ .n/ 0 ai C bi0 i ti C ti D 1 bij tj i alization is not unambiguous. The two viable formulations, j the so-called valence universal (VU) [7, 53] and state uni X .n/ .n/ C cij k tj tk C : (5.84) versal (SU) [54] MR CC methods have enjoyed continued j k following (see [55] for recent developments), yet are computationally demanding and often plagued with the intruder .0/ Starting with the zeroth approximation ti D 0, the first iterstate and other problems [17, 52]. For these reasons, no ation is general-purpose codes have yet been developed, and only .1/ (5.85) ti D 1 i ai ; a few actual applications have been carried out [18, 36] (see, which yields the second-order PT energy when used in however, the recently formulated SU CC approach for genEq. (5.77). Clearly, the successive iterations generate higher eral model spaces [56–59]). and higher orders of the PT. At any truncation level, a size In view of the ambiguity, complexity, and computational extensive result is obtained. demands of genuine MR CC approaches it is tempting to The single reference (SR) CC methods continue to be design SR CC-type methods that are capable of accommoremarkably efficient in handling the dynamic correlation dating both the nondynamic and dynamic correlation effects. effects and are, thus, widely exploited in computations This may be achieved by exploiting the complementarity of of molecular electronic structure. Several general-purpose perturbative (i.e., CC) and variational (CI, CAS SCF, etc.)codes are available for this purpose (for reviews: [10, 18, 34– type approaches, leading to the so-called externally corrected 36, 38]). The basic workhorse that provides the required (ec) ecCCSD or ecCCSD(T) methods (for an overview, T1 and T2 cluster amplitudes results by setting Ti D 0 for see [44]; see also [60, 61] and the so-called state selective i > 2 in Eq. (5.74) (the CC with singles and doubles (CCSD) or state specific (SS) approaches [62]). The basic idea here is .0/ .0/ method [45]). A small, but often important, T3 component to extract a suitable approximation T3 and T4 of, respecis usually accounted for perturbatively via the CCSD(T) tively, the T3 and T4 clusters from some independent source method [46, 47], which is often referred to as the “gold stan- and account for them in the CCD or CCSD equations, i.e., dard” of quantum chemistry, since it provides very accurate 1 2 .0/ .0/ and reliable results. Unfortunately, this is no longer the case C C T C T T T h˚i jHN 2 j˚iC D 0 ; (5.86) 3 4 2 2 in the presence of quasidegeneracy when handling stronglycorrelated systems or when breaking chemical bonds as in thus achieving a more meaningful decoupling of the CC .0/ .0/ generating potential energy curves or surfaces. This can to chain by simply evaluating the T3 and T4 -dependent some extent be avoided by employing one of the renor- terms and adding them to the absolute term. Note that malized versions of CCSD(T) [48] but, in general, requires using the exact (i.e., full configuration interaction (FCI)) T3 and T4 amplitudes the ecCCSD equations will recover multireference-type approaches (see below). In order to compute the excitation, ionization, or elec- the exact FCI energy. The most promising version of such tron attachment energies one employs the equation-of-motion an ecCCSD approach employs the three- and four-body am(EOM) and the linear-response formalisms relying on the plitudes obtained by a cluster analysis of a small MRCI generalized cluster ansatz [49] j i D R eT j˚i, where R wave function and is referred to as the reduced MR CCSD yields an appropriate zero (or first)-order wave function of the (RMR CCSD) method [63–65] (see [44] for a list of numeropen-shell system considered when acting on j˚i. In general, ous applications). Very recently, the required approximate this leads to a non-Hermitian eigenvalue problem, which re- three and four-body clusters were extracted by relying on the quires a computation of the right and left eigenvectors [10]. (G)UGA based ((graphical) UGA) (see [66–68] for recent These developments also enabled calculation of other proper- reviews) FCI quantum Monte Carlo (FCI-QMC) or i-FCIties than the energy (dipole and quadrupole moments, polariz- QMC codes [69–71] yielding encouraging results [61, 72– abilities, etc.), [10, 18, 34–36]. Here, we should also mention 74]. Very recently, Chan et al. [75] exploited for the same a possibility of using the cluster ansatz based on the unitary purpose the density matrix renormalization group (DMRG) group approach (UGA) [50, 51] (Sect. 4.3.6). approach [76–78] in the ecCCSD formalism [44]. At this stage it is important to recall that the standard The same goal, which simultaneously leads to more efSR MBPT and CC approaches pertain to nondegenerate, ficient SR CC formalisms, may also be achieved via an lowest-lying closed-shell states of a given symmetry species. effective implicit account of higher-order clusters by relying Although the CC methods are often used even for open-shell on the role of the EPV (exclusion principle violating) diastates by relying on the unrestricted HF (UHF) reference grams that are separable over the hole lines, leading to the (of the different-orbitals-for-different-spins (DODS) type), so-called internally corrected (ic) methods, such as approx-
5
Perturbation Theory
105
imate coupled pair (ACP-D45) approach, approximate CC with doubles (ACCD) or with approximate quads (ACPQ) methods (see [44] for relevant references). Very recently, similar ideas to the icCCSD approaches [44] were exploited by Kats et al. [79–81] in the so-called distinguished cluster (DC) approximations, such as DC with doubles (DCD), DC with singles and doubles (DCSD) and DCSD corrected for triples (DCSD(T)) methods, leading to numerous exploitations in actual systems [82–85], including transcorrelated CC methods [86]. A very recent development [87] exploits the quantum Monte Carlo (MC) formalism in solving the CC equations, resulting in a stochastic CCMC method [87, 88], which was also extended to the equation of motion CC (EOMCC) type approaches [89] enabling handling the excited states energetics. This way of solving the CC equations should allow applications to larger systems than do the currently available standard codes. Finally, note that the CC approach has also been used to handle bosonic-type problems of the vibrational structure in molecular spectra and, generally, multimode dynamics [90].
the Schrödinger equation becomes i„
@ j .t/iI D V .t/I j .t/iI : @t
This is known as the Tomonaga–Schwinger equation [91]. Analogously, the evolution operator in this picture (we drop the subscript I from now on) satisfies i„
@ U.t; t0 / D V .t/U.t; t0 / ; @t
i U.t; t0 / D 1 „
Zt
D
Evolution Operator PT Expansion
Zt1
becomes i„
@ U.t; t0 / D H U.t; t0 / : @t
(5.87)
If the Hamiltonian is time-independent, then i U.t; t0 / D exp H.t t0 / : „ In the interaction picture (subscript I), i j .t/iI D exp H0 t j .t/i ; „
dt1
dtn T ŒV .t1 / V .tn / ;
(5.97)
t0
(5.88)
5.4.2
Gell-Mann and Low Formula
(5.89) For a time-independent perturbation, one introduces the socalled adiabatic switching by writing H˛ .t/ D H0 C e˛jt j V;
˛>0;
(5.98)
so that H˛ .t ! ˙1/ D H0 and H˛ .t ! 0/ D H D H0 CV . (5.90) Then (5.99) j .t/iI D U˛ .t; 1j/j˚0 i ; with U˛ .t; 1j/ obtained with V˛ .t/ D e˛jt j V (all in the (5.91) interaction picture). The desired energy is then given by the Gell-Mann and Low formula [94] E D lim i„˛ ˛!0C
@ lnh˚0 jU˛ .0; 1j/j˚0 i ; (5.100a) @
(5.92) or E D
where now H D H0 C V ;
Zt
where T Œ designates the time-ordering or chronological operator.
U.t0 ; t0 / D 1 ; U.t; t0 /1 D U.t0 ; t/ D U .t; t0 / :
dtn V .t1 /V .t2 / V .tn / t0
nŠ
t0
Clearly, U.t; t0 / D U.t; t 0 /U.t 0 ; t0 / ;
t0
1 X .i=„/n
the time-dependent Schrödinger equation @ j .t/i D H j .t/i @t
(5.96)
Ztn1 dt2
dt1
nD0 Zt
By introducing the evolution operator U.t; t0 /
i„
V .t1 /U.t1 ; t0 / dt1 :
Iterating we get [92, 93] 1 X i n U.t; t0 / D „ nD0
Time-Dependent Perturbation Theory
j .t/i D U.t; t0 /j .t0 /i ;
5
Zt t0
t0
5.4.1
(5.95)
with the initial condition U.t0 ; t0 / D 1. This differential equation is equivalent to an integral equation
5.4
(5.94)
(5.93)
@ 1 lim i„˛ lnh˚0 jU˛ .1; 1j/j˚0 i ; 2 ˛!0C @ (5.100b)
106
J. Paldus
which result from the asymmetric energy formula Eq. (5.27). where the zero-order time-dependent Green’s function now One can similarly obtain the perturbation expansion for the satisfies the equation one or two-particle Green’s functions, e.g.,
@ ˝ ˚ ˛ 0 0 0 0 0 i„ H 0 G0 r; r I t; t D ı r r ı t t : (5.107)
0 T a .t/a .t /U˛ .1; 1j/ @t G t; t D lim ; ˛!0C hU˛ .1; 1j/i (5.101) Again, for causal propagation one chooses the time-retarded with the operators in the interaction representation and the or causal Green’s function or propagator G .C/ .r; r 0 I t; t 0 /. 0 expectation values in the noninteracting ground state j˚0 i. Analogous expressions result for G.rt; r 0 t 0 /, etc., when the creation and annihilation operators are replaced by the corre5.4.4 Born Series sponding field operators. Iteration of Eq. (5.106) gives the Born sequence
5.4.3
Potential Scattering and Quantum Dynamics
The Schrödinger equation for a free particle of energy E D „2 k 2 =2m, moving in the potential V .r/,
2 r C k 2 .k; r/ D v.r/ .k; r/ ;
(5.102) v.r/ D 2m=„2 V .r/ ; has the formal solution .k; r/ D ˚.k; r/ C
Z
G0 k; r; r 0 v r 0 k; r 0 dr 0 ;
(5.103) where ˚.k; r/ is a solution of the homogeneous equation Œv.r/ 0, and G0 .k; r; r 0 / is a classical Green’s function
2
r C k 2 G0 k; r; r 0 D ı r r 0 : (5.104)
0 .r; t/
D ˚.r; t/ ;
Z .C/
.r; t/ D ˚.r; t/ C G0 r; r 0 I t; t 0 1
V r 0 ; t 0 ˚ r 0 ; t 0 dr 0 dt 0 ; Z .C/
G0 r; r 0 I t; t 0 2 .r; t/ D ˚.r; t/ C
V r 0 ; t 0 1 r 0 ; t 0 dr 0 dt 0 ;
(5.108a)
(5.108b)
and, generally Z .C/
.r; t/ D ˚.r; t/ C G0 r; r 0 I t; t 0 n
V r 0 ; t 0 n1 r 0 ; t 0 dr 0 dt 0 :
(5.109)
Summing individual contributions gives the Born series for .r; t/ .C/.r; t/, For an in-going plane wave ˚.k; r/ ˚ki .r/ D 3=2 .2/ exp.iki r/ with the initial wave vector ki and appro1 X priate asymptotic boundary conditions (outgoing spherical .r; t/ D n .r; t/ ; (5.110) 0 wave with positive phase velocity), when G0 .k; r; r / 0 .C/ nD0 G0 .jr r 0 j/ D .4jr r 0 j/1 ei kjrr j , Eq. (5.103) is referred to as the Lippmann–Schwinger equation [95]. It can where be equivalently transformed into the integral equation for Green’s function 0 .r; t/ D ˚.r; t/ ; Z Z
00 .C/
.C/
.C/ 0 0 00 r; r D G0 r; r C G0 r; r v r G (5.111) n .r; t/ D Gn r; r 0 I t; t 0 ˚ r 0 ; t 0 dr 0 dt 0 ;
.C/ 00 0 00 (5.105) r ; r dr ; G with representing a special case of the Dyson equation. Z In the time-dependent case, considering the scattering of
0 0 G I t; t G1 r; r 00 I t; t 00 r; r D n a spinless massive particle by a time-dependent potential
V .r; t/, we get similarly Gn1 r 00 ; r 0 I t 00 ; t 0 dr 00 dt 00 ; .n > 1/
.C/
.r; t/ D ˚.r; t/ G1 r; r 0 I t; t 0 D G0 r; r 0 I t; t 0 V r 0 ; t 0 : Z (5.112)
0 0 0 0 r ; t dr dt ; C G0 r; r 0 I t; t 0 V r 0 ; t 0 In a similar way we obtain the Born series for the scattering (5.106) amplitudes or transition matrix elements.
5
Perturbation Theory
107
.0/ with the initial condition CPj D 0, which implies that .0/ .0/ Cj are time independent, so that Cj D cj , obtaining An alternative way of formulating the time-dependent PT Eq. (5.114) in the zeroth order. The system Eq. (5.120) can is the method of variation of the constants [96, 97]. Start be integrated to any prescribed order. For example, if the sysagain with the time-dependent Schrödinger equation (5.88) tem is initially in a stationary state j˚i i, then set with H D H0 C V and assume that H0 is time-independent, ( while V is a time-dependent perturbation. Designating the for discrete states ; ıj i .0/ (5.121) Cj D eigenvalues and eigenstates of H0 by "i and j˚i i, respecı.j i/ for continuous states ; tively (Eq. (5.48)), the general solution of the unperturbed time-dependent Schrödinger equation so that
5.4.5 Variation of Constants Method
i„
@ j0 i D H0 j0 i @t
(5.113)
Zt .1/ Cj.i /.t/
1
D .i„/
Vj i exp .i=„/ j i t 0 dt 0 ;
(5.122)
1
has the form X
i cj j˚j i exp "j t ; „
.1/ Cj.i /.1/
.1/
assuming D 0. Clearly, jCj.i /.t/j2 gives the first(5.114) order transition probability for the transition from the initial j0 i D j state j˚i i to a particular state j˚j i. These in turn will yield the first-order differential cross sections [98]. with cj representing arbitrary constants, and the sum indicating both the summation over the discrete part and the integration over the continuum part of the spectrum of H0 . References In the spirit of the general variation of constants procedure, write the unknown perturbed wave function j .t/i, 1. Kato, T.: Perturbation Theory for Linear Operators. Springer, Eq. (5.88), in the form Berlin, Heidelberg (1966) 2. Baumgärtel, H.: Analytic Perturbation Theory for Matrices and X i Operators. Akademie, Berlin (1984) Cj .t/j˚j i exp "j t ; (5.115) j .t/i D 3. Hinch, E.J.: Perturbation Methods. Cambridge Univ. Press, Cam„ j where the Cj .t/ are now functions of time. Substituting this ansatz into the time-dependent Schrödinger equation (5.88) gives X Ck .t/Vj k expŒ.i=„/ j k t ; (5.116) CPj .t/ D .i„/1 k
where j k D "j "k ;
Vj k D h˚j jV j˚k i :
(5.117)
Introducing again the “small” parameter by writing the Hamiltonian H in the form H D H0 C V .t/
(5.118)
and expanding the “coefficients” Cj .t/ in powers of , Cj Cj .t/ D
1 X
.k/
Cj .t/k ;
(5.119)
kD0
gives the system of first-order differential equations X .n/
.nC1/ D .i„/1 Ck Vj k exp .i=„/ j k t ; CPj k
n D 0; 1; 2; : : : ;
(5.120)
bridge (1991) 4. Bogaevski, V.N., Povzner, A.: Algebraic Methods in Nonlinear Perturbation Theory. Springer, Berlin, Heidelberg (1991) 5. Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory. Springer, New York (1999) 6. Corson, E.M.: Perturbation Methods in the Quantum Mechanics of n-Electron Systems. Blackie & Son, London (1951) 7. Lindgren, I., Morrison, J.: Atomic Many-Body Theory. Springer, Berlin, Heidelberg (1982) 8. Gross, E.K.U., Runge, E., Heinonen, O.: Many-Particle Theory. Hilger, New York (1991) 9. Harris, F.E., Monkhorst, H.J., Freeman, D.L.: Algebraic and Diagrammatic Methods in Many-Fermion Theory. Oxford Univ. Press, Oxford (1992) 10. Shavitt, I., Bartlett, R.J.: Many-Body Methods in Chemistry and Physics: MBPT and Coupled Cluster Theory. Cambridge Univ. Press, Cambridge (2009) 11. Primas, H.: Helv. Phys. Acta 34, 331 (1961) 12. Primas, H.: Rev. Mod. Phys. 35, 710 (1963) 13. Rosenblum, M.: Duke Math. J. 23, 263 (1956) 14. Arfken, G.: Mathematical Methods for Physicists. Academic Press, New York, p. 327 (1985) 15. Iyanaga, S., Kawada, Y. (eds.): Encyclopedic Dictionary of Mathematics. MIT Press, Cambridge, p. 1494 (1980). Appendix B, Table 3 ˇ 16. Paldus, J., Cížek, J.: Adv. Quantum Chem. 9, 105 (1975) 17. Paldus, J.: In: Wilson, S., Diercksen, G.H.F. (eds.) Methods in Computational Molecular Physics. NATO ASI Series B, vol. 293, pp. 99–194. Plenum, New York (1992) 18. Paldus, J.: Chap. 19. In: Wilson, S. (ed.): Handbook of Molecular Physics and Quantum Chemistry, vol. 2, pp. 272–313. Wiley, Chichester (2003)
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108 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.
36. 37.
38. 39.
40. 41. 42.
43.
44. 45. 46. 47.
48. 49. 50. 51. 52.
53. 54. 55.
56. 57.
J. Paldus Silverstone, H.J., Holloway, T.T.: J. Chem. Phys. 52, 1472 (1970) Møller, C., Plesset, M.S.: Phys. Rev. 46, 618 (1934) Epstein, P.S.: Phys. Rev. 28, 695 (1926) Nesbet, R.K.: Proc. R. Soc. Lond. A 250, 312 (1955) Goldstone, J.: Proc. R. Soc. Lond. A 239, 267 (1957) Hugenholtz, H.M.: Physica (Utrecht) 23, 481 (1957) Brueckner, K.A.: Phys. Rev. 100, 36 (1955) Hubbard, J.: Proc. R. Soc. Lond. A 240, 539 (1957) Frantz, L.M., Mills, R.L.: Nucl. Phys. 15, 16 (1960) Coester, F.: Nucl. Phys. 7, 421 (1958) Coester, F., Kümmel, H.: Nucl. Phys. 17, 477 (1960) ˇ Cížek, J.: J. Chem. Phys. 45, 4256 (1966) ˇ Cížek, J.: Adv. Chem. Phys. 14, 35 (1969) ˇ Cížek, J., Paldus, J.: Int. J. Quantum Chem. 5, 359 (1971) ˇ Paldus, J., Cížek, J., Shavitt, I.: Phys. Rev. A 5, 50 (1972) Bartlett, R.J.: Part I. In: Yarkony, D.R. (ed.) Modern Electronic Structure Theory, pp. 47–108. World Scientific, Singapore (1995) Bartlett, R.J. (ed.): Recent Advances in Coupled-Cluster Methods. Recent Advances in Computational Chemistry, vol. 3. World Scientific, Singapore (1997) Paldus, J., Li, X.: Adv. Chem. Phys. 110, 1 (1999) Crawford, T.D., Schaefer III, H.F.: In: Lipkowitz, K.B., Boyd, D.B. (eds.) Reviews of Computational Chemistry, vol. 14, pp. 33– 136. Wiley, New York (2000) Bartlett, R.J., Musiał, M.: Rev. Mod. Phys. 79, 291 (2007) ˇ Cársky, P., Paldus, J., Pittner, J. (eds.): Recent Progress in Coupled Cluster Methods: Theory and Applications. Challenges and Advances in Computational Chemistry and Physics, vol. 11. Springer, Berlin (2010) Bartlett, R.J.: Mol. Phys. 108, 2905 (2010) Bartlett, R.J.: Wiley. Interdiscip. Rev. Comput. Mol. Sci. 2, 126– 138 (2012) Paldus, J.: Chap. 7. In: Dykstra, C.E., Frenking, G., Kim, K.S., Scuseria, G.E. (eds.) Theory and Applications of Computational Chemistry: The First Forty Years, pp. 115–147. Elsevier, Amsterdam (2005) Bartlett, R.J.: Chap. 42. In: Dykstra, C.E., Frenking, G., Kim, K.S., Scuseria, G.E. (eds.) Theory and Applications of Computational Chemistry: The First Forty Years, pp. 1191–1221. Elsevier, Amsterdam (2005) Paldus, J.: J. Math. Chem. 55, 477 (2017) Bartlett, R.J., Purvis, G.D.: Int. J. Quantum Chem. 14, 561 (1978) Raghavachari, K., Trucks, G.W., Pople, J.A., Head-Gordon, M.: Chem. Phys. Lett. 157, 479 (1989) Lee, T.J., Scuseria, G.E.: In: Langhoff, S.R. (ed.) Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, pp. 47–108. Kluwer, Dordrecht (1995) Kowalski, K., Piecuch, P.: J. Chem. Phys. 120, 1715 (2004) ˇ Paldus, J., Cížek, J., Saute, M., Laforgue, A.: Phys. Rev. A 17, 805 (1978) Li, X., Paldus, J.: J. Chem. Phys. 101, 8812 (1994) Jeziorski, B., Paldus, J., Jankowski, P.: Int. J. Quantum Chem. 56, 129 (1995) Paldus, J.: In: Malli, G.L. (ed.) Relativistic and Electron Correlation Effects in Molecules and Solids. NATO ASI Series B: Physics, vol. 318, pp. 207–282. Plenum, New York (1994) Lindgren, I., Mukherjee, D.: Phys. Rep. 151, 93 (1987) Jeziorski, B., Monkhorst, H.J.: Phys. Rev. A 24, 1686 (1981) ˇ ˇ Paldus, J., Cársky, P., Pittner, J.: Chap. 17. In: Cársky, P., Paldus, J., Pittner, J. (eds.) Recent Progress in Coupled-Cluster Methods: Theory and Applications, pp. 455–489. Springer, Berlin (2010) Li, X., Paldus, J.: J. Chem. Phys. 119, 5320 (2003) Li, X., Paldus, J.: J. Chem. Phys. 119, 5334 (2003)
58. Li, X., Paldus, J.: J. Chem. Phys. 119, 5343 (2003) 59. Li, X., Paldus, J.: J. Chem. Phys. 120, 5890 (2004) 60. Margoulas, I., Gururangan, K., Piecuch, P., Deustua, J.E., Shen, J.: J. Chem. Theory Comput. 17, 4006 (2021) 61. Margoulas, I., Shen, J., Piecuch, P.: Addressing Strong Correlation by Approximate Coupled-Pair Methods with Active-Space Full Treatment of Three-Body Clusters, arXiv:2111.13787v1 [physics.chem-ph] (2021) 62. Chattopadhyay, S., Pahari, D., Mukherjee, D., Mahapatra, U.S.: J. Chem. Phys. 120, 5968 (2004) 63. Li, X., Paldus, J.: J. Chem. Phys. 107, 6257 (1997) 64. Li, X., Paldus, J.: J. Chem. Phys. 108, 637 (1998) 65. Li, X., Paldus, J.: J. Chem. Phys. 129, 054104 (2004) 66. Paldus, J.: J. Math. Chem. 59, 1 (2021) 67. Paldus, J.: J. Math. Chem. 59, 37 (2021) 68. Paldus, J.: J. Math. Chem. 59, 72 (2021) 69. Dobrautz, W., Smart, S.D., Alavi, A.: J. Chem. Phys. 151, 94104 (2019) 70. Dobrautz, W.: Development of Full Configuration Interaction Quantum Monte Carlo Methods for Strongly Correlated Electron Systems. PhD Thesis, University of Stuttgart (2019) 71. Li Manni, G., Guther, K., Ma, D., Dobrautz, W.: In: González, L., Lindh, R. (eds.): Quantum Chemistry and Dynamics of Excited States: Methods and Applications, Chap. 6, pp. 133–204. Wiley, New York (2021) 72. Deustua, J.E., Magoulas, I., Shen, J., Piecuch, P.: J. Chem. Phys. 149, 151101 (2018) 73. Deustua, J.E., Shen, J., Piecuch, P.: J. Chem. Phys. 154, 124103 (2021) 74. Gururangan, K., Deustua, J.E., Shen, J., Piecuch, P.: J. Chem. Phys. 155, 174114 (2021) 75. Lee, S., Zhai, H., Sharma, S., Umrigar, C.J., Chan, G.K.-L.: J. Chem. Theory Comput. 17, 3414 (2021) 76. Chan, G.K.-L., Head-Gordon, M.: J. Chem. Phys. 116, 4462 (2001) 77. Chan, G.K.-L.: J. Chem. Phys. 120, 3172 (2003) 78. Olivares-Amaya, R., Hu, W., Nakatani, N., Sharma, S., Yang, J., Chan, G.K.-L.: J. Chem. Phys. 142, 034102 (2015) 79. Kats, D., Manby, F.R.: J. Chem. Phys. 139, 021102 (2013) 80. Kats, D.: J. Chem. Phys. 141, 061101 (2014) 81. Kats, D.: J. Chem. Phys. 144, 044102 (2016) 82. Kats, D., Kreplin, D., Werner, H.-J., Manby, F.R.: J. Chem. Phys. 142, 064111 (2015) 83. Kats, D., Köhn, A.: J. Chem. Phys. 150, 151101 (2019) 84. Vitale, E., Alavi, A., Kats, D.: J. Chem. Theory Comput. 16(9), 5621 (2020) 85. Schraivogel, T., Kats, D.: J. Chem. Phys. 155, 064101 (2021) 86. Schraivogel, T., Cohen, A.J., Alavi, A., Kats, D.: J. Chem. Phys. 155, 191101 (2021) 87. Thom, A.J.W.: Phys. Rev. Lett. 105, 263004 (2010) 88. Deustua, J.E., Shen, J., Piecuch, P.: Phys. Rev. Lett. 119, 223003 (2017) 89. Deustua, J.E., Yuwono, S.H., Shen, J., Piecuch, P.: J. Chem. Phys. 150, 111101 (2019) 90. Christiansen, O.: J. Chem. Phys. 120, 2149 (2004) 91. Dyson, F.J.: Phys. Rev. 75, 486 (1949) 92. Tomonaga, S.: Prog. Theor. Phys. (Kyoto) 1, 27 (1946) 93. Schwinger, J.: Phys. Rev. 74, 1439 (1948) 94. Gell-Mann, M., Low, F.: Phys. Rev. 84, 350 (1951) 95. Lippmann, B.A., Schwinger, J.: Phys. Rev. 79, 469 (1950) 96. Dirac, P.A.M.: Proc. R. Soc. Lond. A 112, 661 (1926) 97. Dirac, P.A.M.: Proc. R. Soc. Lond. A 114, 243 (1926) 98. Joachain, C.J.: Quantum Collision Theory. Elsevier, New York (1975)
5
Perturbation Theory
109 Josef Paldus Josef Paldus is Distinguished Professor Emeritus at the Department of Applied Mathematics of the University of Waterloo, Canada. He received his PhD from the Czechoslovak Academy of Sciences and his RNDr and DrSc from the Faculty of Mathematics and Physics of Charles University, Prague, Czech Republic. His research interests are the methodology of quantum chemistry and the electronic structure of molecular systems. He has published over 340 papers, reviews, and chapters.
5
6
Second Quantization Brian R. Judd
Contents 6.1 6.1.1 6.1.2 6.1.3
Basic Properties . . . . . . . Definitions . . . . . . . . . . . Representation of States . . . Representation of Operators
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
111 111 112 112
6.2 6.2.1 6.2.2 6.2.3
Tensors . . . . . . . . . . . . . . . . . . Construction . . . . . . . . . . . . . . . Coupled Forms . . . . . . . . . . . . . Coefficients of Fractional Parentage
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
112 112 112 113
6.3 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.3.6
Quasispin . . . . . . . . . . . . . . Fermions . . . . . . . . . . . . . . . Bosons . . . . . . . . . . . . . . . . Triple Tensors . . . . . . . . . . . . Conjugation . . . . . . . . . . . . . Dependence on Electron Number The Half-Filled Shell . . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
113 113 113 114 114 114 115
6.4 6.4.1 6.4.2
Complementarity . . . . . . . . . . . . . . . . . . . . . . . 115 Spin–Quasispin Interchange . . . . . . . . . . . . . . . . . 115 Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . 115
6.5
Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . 116
. . . . . . .
. . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Abstract
in all configurations may be determined from the knowledge of its matrix elements in but one. This can be viewed as an extension of the usual Wigner-Eckart theorem. The basic concepts of quasispin and quasiparticle are also introduced within this context. Keywords
second quantization intrinsic shell structure coefficients of fractional parentage Wigner–Eckart theorem quasispin
6.1 Basic Properties 6.1.1 Definitions
The creation operator a creates the quantum state . The annihilation (or destruction) operator a annihilates the quantum state . The vacuum (or reference) state j0i satisfies the equation (6.1) a j0i D 0 :
In second quantization, the characteristic properties of Bosons satisfy the commutation relations eigenfunctions are transferred to operators. This approach
has the advantage of treating the atomic shell as the basic a ; a D 0 ; (6.2) unit, as opposed to the electron configuration. The cre(6.3) Œa ; a D 0 ; ation and annihilation operators allow one to move from
(6.4) a ; a D ı.; / ; configuration to configuration, exposing an intrinsic shell structure. The introduction of coefficients of fractional parentage (cfp) then allows the calculation of the matrix where ŒA; B AB BA. Fermions satisfy the anticommuelements of an operator in one configuration to be ex- tation relations pressed in terms of those of the same operator in another
(6.5) a ; a C D 0 ; configuration; hence the matrix elements of an operator (6.6) Œa ; a C D 0 ;
B. R. Judd () (6.7) a ; a C D ı.; / ; Dept. of Physics & Astronomy, The Johns Hopkins University Baltimore, MD, USA e-mail: [email protected]
where ŒA; BC AB C BA.
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_6
111
112
6.1.2
B. R. Judd
Representation of States
t m t t, satisfy the commutation relations of Racah [2] for an irreducible spherical tensor of rank t with respect to For an electron in an atom, characterized by the quantum the total angular momentum T , given by number quartet .n ` ms m` /, the identification .n ` ms m` / X (6.14) T D a hjtjia : for fermions is made. For normalized Slater determinants f˛ˇ : : : g characterized by the electron states ˛; ˇ; : : : ; , That is, with the phase conventions of Condon and Shortthe equivalences ley [3], (6.8) a˛ aˇ : : : a j0i f˛ˇ : : : g ;
Tz ; a D m t a ; (6.15) (6.9) h0ja : : : aˇ a˛ f˛ˇ : : : g
1 Tx ˙ iTy ; a D Œt.t C 1/ m t .m t ˙ 1/ 2 a ; (6.16) are valid, where the asterisk denotes the complex conjugate. For a normalized boson state f g in which the label where .t; m t ˙ 1/. A spherical tensor a constructed from annihilation operaappears N times, the additional factor tors possesses the components aQ , which satisfy 1
ŒN˛ ŠNˇ Š : : : N Š 2
(6.10)
aQ D .1/p a ;
(6.17)
must be included on the left-hand sides of the equivalences with p D t m t and .t; m t /. Eqs. (6.8) and (6.9). The 4` C 2 components of the creation operator for an electron in the atomic ` shell form a double tensor of rank 12 with respect to the total spin S , and rank ` with respect to 6.1.3 Representation of Operators the total angular momentum L. For an operator F , consisting of the sum of operators fi acting on the single electron i, F
X
a hjf
jia :
(6.11)
;
For an operator G, consisting of the sum of operators gij acting on the pair of electrons i and j , G
1 X a a h1 2 jg12 j1 2 ia a : 2 ;;;
For an N -particle system j i, X
a a j i D N j i :
6.2.2 Coupled Forms Tensors formed from annihilation and creation operators can be coupled by means of the usual rules of angular momentum theory [4]. The double tensor defined for electrons in the ` shell by
. k/ ; (6.18) W . k/ D a a
(6.12) possesses a rank with respect to S , and rank k with respect to L. Its reduced matrix element, defined here as in (5.4.1) of Edmonds [4], for a single electron in both the spin and orbital spaces, is given by (6.13)
1 s ` jj W . k/ jj s ` D Œ.2 C 1/.2k C 1/ 2 :
(6.19)
The connections to tensors whose matrix elements have been The representations of single-particle and two-particle optabulated [5, 6] are erators for bosons are identical to those given above for 1 fermions [1]. W .0k/ D Œ.2k C 1/=2 2 U .k/ ; (6.20) 1
W .1k/ D Œ2.2k C 1/ 2 V .k1/ :
6.2 Tensors 6.2.1
Construction
For terms with common spin S, say
jjW .0k/ jj
0
and
(6.21) 0
,
1
If the description for a single fermion or boson state inD Œ.2S C 1/.2k C 1/=2 2 jjU .k/jj 0 : (6.22) cludes an angular momentum quantum number t and the corresponding magnetic quantum number m t , then the 2t C1 This result is obtained because the ranks assigned to the ten components of a creation operator a , where .t; m t / and sors imply that W .0k/ is to be reduced with respect to both
6 Second Quantization
113
If, through successive applications of the two-particle opthe spin S and the orbit L, while U .k/ is to be reduced only erators .aa/.00/ , a state of `N can be reduced to `v , but no with respect to L. The following relations hold for electrons with azimuthal further, then v is the seniority number of Racah [8]. If the ranks s and ` of a are coupled to SN and LN of N , the quantum numbers ` [7]: term 1
ˇ ˛.SL/ S D Œ.2` C 1/=2 2 W .10/ ; (6.23) (6.32) a ˇN 1
L D Œ2`.` C 1/.2` C 1/=3 2 W .01/ ; (6.24) either vanishes, or is a term of `N characterized by S and L. 1 .1/ Such a term is said to possess the godparent N . RedX `.` C 1/.2` C 1/ 2 .12/1 .2/ si C i D W ; mond [11] has used the notion of godparents to generate an 10.2` 1/.2` C 3/ i explicit formula for the single particle cfp [7]. (6.25) X .si ` i / D Œ`.` C 1/.2` C 1/=2W .11/0 ; (6.26)
6.3
i
Quasispin
k
where the tensor C of Racah [2] is related to the spherical 6.3.1 harmonics by
6
Fermions
1
(6.27) For electrons, the components Q˙ . Qx ˙ iQy / and Qz of the quasispin Q are defined by [7, 12] and where the tensors of the type W . k/K indicate that the .00/ 1
; (6.33) QC D Œ.2` C 1/=2 2 a a spin and orbital ranks are coupled to a resultant K. Cq.k/ D Œ4=.2k C 1/ 2 Ykq ;
1
6.2.3 Coefficients of Fractional Parentage Let and N denote terms of `N and `N 1 characterized by N L/. N The coefficients of fractional parentage .S; L/ and .S; N (cfp) . fj / of Racah [8] allow one to calculate an antisymmetrized function by vector-coupling N to the spin and orbit of the N th electron: Xˇ ˛
ˇ N ; 2`; SL N jg ; (6.28) j iD
Q D Œ.2` C 1/=2 2 .aa/.00/ ; .00/ .00/ 1
Qz D Œ.2` C 1/=8 2 a a C aa :
(6.34) (6.35)
The term quasispin comes from the fact that the components of Q satisfy the commutation relations of an angular momentum vector. The eigenvalues MQ of Qz , for a state of `N , are given by (6.36) MQ D .2` C 1 N /=2 :
The shift operators QC and Q connect states of the ` shell possessing the same value of the seniority v of Racah [8]. N A string of such connected states defines the extrema of MQ , N and any other quantum from which it follows that where the sum over N includes SN ; L, numbers necessary to define the spectroscopic terms of `N 1 . Q D .2` C 1 v/=2 : (6.37) The cfp’s are given by 1
jja jj N D .1/N ŒN.2S C 1/.2L C 1/ 2 fj N ; (6.29)
1
g N jjajj D .1/ ŒN.2S C 1/.2L C 1/ 2 N jg ; (6.30)
Rudzikas has placed special emphasis on quasispin in his reworking of atomic shell theory, and he has also introduced isospin to embrace electrons differing in their principal quantum numbers n [13]. Concise tables of one-electron cfp with their quasispin dependence factored out have been where g D N C SN C LN s S ` L. A tabulation for the given [14], as have the algebraic dependences on and S of two-electron cfp [15]. p, d, and f shells has been given by Nielson and Koster [5]. Two-electron cfp are given by
. k/ 1 6.3.2 Bosons jj Q D ŒN.N 1/.2S C 1/.2L C 1/ 2 jj a a
fj Q ; `2 . k/ ; (6.31) For real vibrational modes created by a . D 1; 2; : : : ; d /, where Q denotes a term of `N 2 , and the symbols and k the analogs of Eqs. (6.33)–(6.35) are stand for the S and the L of a term of `2 . A tabulation for the 1X p, d, and f shells has been given by Donlan [9]. An extension a a ; (6.38) PC D 2 to all multielectron cfp has been carried out by Velkov [10].
114
B. R. Judd
1X a a ; 2 1 X Pz D a a C a a ; 4
P D
(6.39) (6.40)
and P is an angular momentum vector [16]. The eigenvalues MP for an n-boson state are given by
where .mq ms m` / and ..mq /ms m` /. In terms of the tensors a and a, C a C 1 D a ;
C aC 1 D a :
(6.50)
C 1 D .1/KMK X.K k/ ;
(6.51)
Furthermore, .K k/
CX
MP D .2n C d /=4 ;
(6.41) where .MK M Mk / and Œ.MK /M Mk , and ˇ ˇ ˛ ˛ and can therefore be quarter-integral. Successive application (6.52) C ˇQMQ D .1/QMQ ˇQ MQ : of the operator PC to a state jn0 i, for which P jn0 i D 0, generates an infinite ladder of states characterized by Thus, from Eq. (6.36), the action of C takes N into 4`C 2 N ; that is, C interchanges electrons and holes. When the P D .2n C d 4/=4 : (6.42) case D k D 0 is excluded, application of Eqs. (6.51) and (6.52) yields
N 6.3.3 Triple Tensors jj W . k/ jj `N 0 `
(6.53) D .1/y ` 4`C2N jjW . k/ jj` 4`C2N 0 ; The creation and annihilation operators a and a for a given state can be regarded as the two components of a tensor of where y D C k C 1 .v 0 v/ C 1, and where the seniori2 rank 12 with respect to quasispin (either Q or P). For elec- ties v and v 0 are implied by and 0 . A similar application 1 .qs`/ (for which q D s D 2 ) to reduced matrix elements of a and a gives the following trons, this leads to triple tensors a satisfying relation between cfp: .qs`/ .qs`/ a
.qs`/
C a.qs`/ a
D .1/xC1 ı mq ; m0q ı ms ; m0s ı m` ; m0` ; (6.43) where .mq ms m` /, .m0q m0s m0` /, and x D q C s C ` C mq C ms C m` . In terms of the coupled tensor a
X .K k/ D .a.qs`/ a.qs`/ /.K k/ ; the angular momenta Q, S , and L are given by 1
Q D Œ.2` C 1/=4 2 X .100/ ; 1
S D Œ.2` C 1/=4 2 X .010/ ; 1
L D Œ`.` C 1/.2` C 1/=3 2 X .001/ :
N C1 fj`N 0 `
D .1/z ` 4`C1N jg` 4`C2N 0 1 .4` C 2 N /.2S 0 C 1/.2L0 C 1/ 2 ; .N C 1/.2S C 1/.2L C 1/
(6.54)
(6.44) where z D S CS 0 s CLCL0 `C 12 .v Cv 0 1/. The phases y and z stem from the conventions of angular momentum theory, which enter via quasispin. Racah [2, 8] did not use this concept, and his phase choices are slightly different from (6.45) the ones above. (6.46) For a Cartesian component Qu of the quasispin Q, (6.47)
CQu C 1 D Qu :
(6.55)
.K k/
Furthermore, the components of X for which MK D 0 Thus, C is the analog of the time-reversal operator T , for 1 are identical to the corresponding components of 2 2 .a a/. k/ which when K C C k is odd; and 1 (6.56) T Lu T 1 D Lu ; (6.48) X .K k/ D .2` C 1/ 2 ı.K; 0/ı. ; 0/ı.k; 0/ 1 (6.57) T Su T D Su : when K C C k is even. Both C and T are antiunitary; thus,
6.3.4 Conjugation
C iC 1 D i :
(6.58)
Creation and annihilation operators can be interchanged by the operation of the conjugation operator C [7, 17]. For elec- 6.3.5 Dependence on Electron Number trons in the atomic ` shell, Application of the Wigner–Eckart theorem to matrix ele.qs`/ (6.49) ments whose component parts have well-defined quasispin C a C 1 D .1/qmq a.qs`/ ;
6 Second Quantization
115
ranks yields the dependence of the matrix elements on the where the quasispin of the ket on the right is S and the spin electron number N [18, 19]. For C k even and nonzero, is Q. The phase factor t depends on S and Q and on phase choices made for the coefficients of fractional parentage. The the quasispin rank of W . k/ is 1, and symbol denotes the additional labels necessary to com N ` jjW . k/ jj`N 0 pletely define the state in question, including L and ML . For every , Racah [20] observed that there are two pos.2` C 1 N / v jj W . k/ jj ` v 0 : (6.59) sible pairs .v ; S / and .v ; S / satisfying ` D 1 1 2 2 .2` C 1 v/ v1 C 2S2 D v2 C 2S1 D 2` C 1 : For C k odd, W . k/ is necessarily a quasispin scalar, and the matrix elements are diagonal with respect to the seniority From Eq. (6.37) it follows that and independent of N . These properties were first stated in Eqs. (69) and (70) of [8]. S 2 D Q1 : S 1 D Q2 ; Application of these ideas to single-electron cfp yields, for states and N with seniorities v and v C 1, respectively,
`N fj`N 1 N
(6.64)
(6.65)
6.4.2 Matrix Elements
6
ˇ 1
D Œ.N v/.v C 2/=2N 2 ` vC2 ˇ ` vC1 N :
(6.60) Application of the complementarity operator R to the component parts of a matrix element leads to the equation ˝ ˛ .K k/ 0 0 QMQ SMS jX j Q MQ0 S 0 MS0 6.3.6 The Half-Filled Shell ˝ ˛ D .1/y SMS QMQ jX. Kk/ j 0 S 0 MS0 Q0 MQ0 ; (6.66) Selection rules for operators of good quasispin rank K, taken between states of the half-filled shell (for which MQ D 0), where and have the same significance as in Eq. (6.62), and where y, like t of Eq. (6.63), depends on the spins and can be found by inspecting the 3–j symbol quasispins but not on the associated magnetic quantum num! bers. Equation (6.66) leads to a useful special case when Q K Q0 ; MK D M D 0 and the tensors X are converted to those of 0 0 0 type W , defined in Eq. (6.18). The sum K C C k is taken which appears when the Wigner-Eckart theorem is applied in to be odd, with the scalars D k D 0 and K D k D 0 exquasispin space. This 3–j symbol vanishes unless Q C K C cluded. Application of the Wigner-Eckart theorem to the spin Q0 is even. An equivalent result can be obtained for W . k/ and orbital spaces yields by referring to Eq. (6.53) and insisting that y be even. QMQ SjjW . k/ jj 0 Q0 MQ0 S 0
SMS QjjW .Kk/ jj 0 S 0 MS0 Q0 ! 6.4 Complementarity Q K Q0 MQ 0 MQ 6.4.1 Spin–Quasispin Interchange ! ; (6.67) D .1/z S
S0 The operator R formally interchanges spin and quasispin. MS 0 MS The result for the creation and annihilation operators for electrons can be expressed in terms of triple tensors: where z D y C Q MQ S C MS . An equivalent form is .qs`/
Ra
R1 D a.qs`/ ;
(6.61)
where .mq ms m` / and .ms mq m` /. For the tensors X .K k/ defined in Eq. (6.44), we get .K k/
RX
R1 D X. Kk/ ;
(6.62)
where .MK M Mk / and .M MK Mk /. For states of the ` shell, RjQMQ SMS i D .1/t jSMS QMQ i ;
N ` v1 S1 jjW . k/ jj`N 0 v10 S10
0 `N v2 S2 jjW .Kk/ jj`N 0 0 v20 S20
D .1/z
1 2 .2` 1 2 .2`
C 1 v1 / K C 1 N/ 0
1 2 .2` C 1 v2 / 1 0 2 .2` C 1 N /
0
!
1 0 2 .2` C 1 v1 / 1 2 .N 2` 1/
!;
1 0 2 .2` C 1 v2 / 1 0 2 .N 2` 1/
(6.68) where Eq. (6.64) is satisfied both for the unprimed and (6.63) primed quantities.
116
6.5
B. R. Judd
Quasiparticles
where the are Dirac matrices satisfying
Sets of linear combinations of the creation and annihilation operators for electrons in the ` shell can be constructed such that every member of one set anticommutes with a member of a different set. To preserve the tensorial character of these quasiparticle operators with respect to L, it is convenient to define [21, 22] h i 1 (6.69) q D 2 2 a 1 ;q C .1/`q a 1 ;q ; 2 2 h i 1 q D 2 2 a 1 ;q .1/`q a 1 ;q ; (6.70) 2 2 h i 1 q D 2 2 a 1 ;q C .1/`q a 1 ;q ; (6.71) 2 2 h i 1 q D 2 2 a 1 ;q .1/`q a 1 ;q : (6.72)
C D 2ı.; / ;
(6.82)
and the are phases, to some extent dependent on the definitions Eqs. (6.69)–(6.72) [25]. The superscript .10 : : : 0/ indi cates that q and q each of which belongs to the elementary 11 spinor . 2 2 : : : 12 / of SO .2` C 1/, are to be coupled to the resultant .10 : : : 0/, which matches the group label for . In the quark model, the 24`C2 states of the atomic ` shell are given by
q q q q j0ipp0 ;
(6.83)
0
where p and p are parity labels that distinguish the four reference states j0i corresponding to the evenness and oddness of the number of spin-up and spin-down electrons. The scalar nature of (and hence of q ) with respect to S C Q can be used to derive relations between spin-orbit matrix el2 2 ements that go beyond those expected from an application of The four tensors . , , , or / anticommute with the Wigner–Eckart theorem [26]. each other; the first two act in the spin-up space, the second two in the spin-down space. The tensors , whose components Qq are defined as in Eq. (6.17) with t D ` and m t D q, References are related to their adjoints by the equations D ;
D ;
(6.73)
D ;
D :
(6.74)
Under the action of the complementarity operator R (see Eq. (6.61)) [23], R R1 D ;
R R1 D ;
(6.75)
1
1
(6.76)
R R
D;
R R
D :
The tensors , , and , for a given component q, form a vector with respect to S C Q. Every component of is scalar with respect to S C Q [24]. The compound quasiparticle operators defined by [21, 22] 1 (6.77) #q D 2 2 q ; 0 ; where q > 0 and , , , or satisfy the anticommutation relations
(6.78) #q ; #q 0 C D 0 ;
(6.79) #q ; #q 0 C D 0 ;
0 (6.80) #q ; #q 0 C D ı.q; q / ;
for q; q 0 > 0. The #q with q > 0 can thus be regarded as the creation operators for a fermion quasiparticle with ` components. The connection between the creation and annihilation operators for quasiparticles and for quarks (appearing in the last two rows of Table 3.1) is
.10:::0/ ; ! 2.`1/=2 q q
(6.81)
1. Gross, E.K.U., Runge, E., Heinonen, O.: Many-Particle Theory. Hilger, New York (1991) 2. Racah, G.: Phys. Rev. 62, 438 (1942) 3. Condon, E.U., Shortley, G.H.: The Theory of Atomic Spectra. Cambridge Univ. Press, New York (1935) 4. Edmonds, A.R.: Angular Momentum in Quantum Mechanics. Princeton Univ. Press, Princeton (1957) 5. Nielson, C.W., Koster, G.F.: Spectroscopic Coefficients for the pn , dn , and fn Configurations. MIT Press, Cambridge (1963) 6. Karazija, R., Vizbarait˙e, J., Rudzikas, Z., Jucys, A.P.: Tables for the Calculation of Matrix Elements of Atomic Operators. Computing Centre of the Academy of Sciences, Moscow (1967) 7. Judd, B.R.: Second Quantization and Atomic Spectroscopy. Johns Hopkins, Baltimore (1967) 8. Racah, G.: Phys. Rev. 63, 367 (1943) 9. Donlan, V.L.: Air Force Materials Laboratory Report No. AFMLTR-70-249. Wright-Patterson Air Force Base, Ohio (1970) 10. Velkov, D.D.: Multi-Electron Coefficients of Fractional Parentage for the p, d, and f Shells. Ph.D. Thesis. The Johns Hopkins University, Baltimore (2000). http://www.pha.jhu.edu/groups/cfp/ 11. Redmond, P.J.: Proc. R. Soc. A 222, 84 (1954) 12. Flowers, B.H., Szpikowski, S.: Proc. Phys. Soc. 84, 673 (1964) 13. Rudzikas, Z.: Theoretical Atomic Spectroscopy. Cambridge Univ. Press, New York (1997) 14. Gaigalas, G., Rudzikas, Z., Froese Fischer, C.: At. Data Nucl. Data Tables 70, 1 (1998) 15. Judd, B.R., Lo, E., Velkov, D.: Mol. Phys. 98, 1151 (2000). Table 4 16. Judd, B.R.: J. Phys. C 14, 375 (1981) 17. Bell, J.S.: Nucl. Phys. 12, 117 (1959) 18. Watanabe, H.: Prog. Theor. Phys. 32, 106 (1964) 19. Lawson, R.D., Macfarlane, M.H.: Nucl. Phys. 66, 80 (1965) 20. Racah, G.: Phys. Rev. 76, 1352 (1949). Table I 21. Armstrong, L., Judd, B.R.: Proc. R. Soc. A 315, 27 (1970) 22. Armstrong, L., Judd, B.R.: Proc. R. Soc. A 315, 39 (1970) 23. Judd, B.R., Li, S.: J. Phys. B 22, 2851 (1989) 24. Judd, B.R., Lister, G.M.S., Suskin, M.A.: J. Phys. B 19, 1107 (1986) 25. Judd, B.R.: Phys. Rep. 285, 1 (1997) 26. Judd, B.R., Lo, E.: Phys. Rev. Lett. 85, 948 (2000)
6 Second Quantization
117 Brian Judd Brian Judd has had a lifelong interest in applying group theory to the spectroscopic properties of the rare earths. He held appointments at Oxford, Chicago, Paris and Berkeley before joining the Physics Department of the Johns Hopkins University in 1966. He received the Spedding Award for Rare-Earth Research in 1988 and is an Honorary Fellow of Brasenose College, Oxford.
6
7
Density Matrices Klaus Bartschat
Contents
Abstract
7.1 7.1.1 7.1.2 7.1.3 7.1.4 7.1.5 7.1.6
Basic Formulae . . . . . . . . . . . Pure States . . . . . . . . . . . . . . Mixed States . . . . . . . . . . . . . Expectation Values . . . . . . . . . The Liouville Equation . . . . . . Systems in Thermal Equilibrium Relaxation Processes . . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
7.2 7.2.1 7.2.2
Spin and Light Polarizations . . . . . . . . . . . . . . . . 121 Spin-Polarized Electrons . . . . . . . . . . . . . . . . . . . 121 Light Polarization . . . . . . . . . . . . . . . . . . . . . . . . 122
7.3 7.3.1 7.3.2
Atomic Collisions . . . . . . . . . . . . . . . . . . . . . . . 122 Scattering Amplitudes . . . . . . . . . . . . . . . . . . . . . 122 Reduced Density Matrices . . . . . . . . . . . . . . . . . . 122
7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.4.5
Irreducible Tensor Operators . . . . . . Definition . . . . . . . . . . . . . . . . . . . . Transformation Properties . . . . . . . . . . Symmetry Properties of State Multipoles Orientation and Alignment . . . . . . . . . Coupled Systems . . . . . . . . . . . . . . .
. . . . . .
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. . . . . .
. . . . . .
. . . . . .
. . . . . .
123 123 123 124 124 124
7.5 7.5.1 7.5.2 7.5.3
Time Evolution of State Multipoles . Perturbation Coefficients . . . . . . . . Quantum Beats . . . . . . . . . . . . . . Time Integration over Quantum Beats
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
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. . . .
125 125 125 125
7.6 7.6.1 7.6.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Generalized STU-Parameters . . . . . . . . . . . . . . . . . 126 Radiation from Excited States: Stokes Parameters . . . . 127
7.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
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120 120 120 120 120 121 121
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
K. Bartschat () Dept. of Physics & Astronomy, Drake University Des Moines, IA, USA e-mail: [email protected]
The density operator was first introduced by J. von Neumann [1] in 1927 and has since been widely used in quantum statistics. Over the past decades, however, the application of density matrices has spread to many other fields of physics. Density matrices have been used to describe, for example, coherence and correlation phenomena, alignment and orientation and their effect on the polarization of emitted radiation, quantum beat spectroscopy, optical pumping, and scattering processes, particularly when spinpolarized projectiles and/or targets are involved. A thorough introduction to the theory of density matrices and their applications, with an emphasis on atomic physics, can be found in the book by Blum [2], from which many equations have been extracted for use in this chapter. Keywords
density matrix density operator reduce density matrix Stokes parameter tensor operator
The main advantage of the density matrix formalism is its ability to deal with pure and mixed states in the same consistent manner. The preparation of the initial state as well as the details regarding the observation of the final state can be treated in a systematic way. In particular, averages over quantum numbers of unpolarized beams in the initial state and incoherent sums over nonobserved quantum numbers in the final state can be accounted for via the reduced density matrix. Furthermore, expansion of the density matrix in terms of irreducible tensor operators and the corresponding state multipoles allows for the use of advanced angular momentum techniques, as outlined in Chaps. 2, 3 and 13. More details can be found in textbooks such as [3, 4].
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_7
119
120
7.1
K. Bartschat
Basic Formulae
of is Trf g D
7.1.1
Pure States
X
wn D 1 :
(7.9)
n
Since the trace is invariant under unitary transformations of Consider a system in a quantum state that is represented by the basis functions, Eq. (7.9) also holds if the jn i states a single wave function j i. The density operator for this sit- themselves are expanded in terms of basis functions as uation is defined as in Eq. (7.4). For a pure state and the normalization Eq. (7.9), one finds in an arbitrary basis
D j ih j : (7.1) ˚ (7.10) Trf g D Tr 2 D 1 : If j i is normalized to unity, i.e., if h j i D 1 ;
(7.2) 7.1.3
Expectation Values
then
The density operator contains the maximum available infor(7.3) mation about a physical system. Consequently, it can be used Equation (7.3) is the basic equation for identifying pure to calculate expectation values for any operator A that reprequantum mechanical states represented by a density opera- sents a physical observable. In general, tor. hA i D TrfA g=Trf g ; (7.11) Next, consider the expansion of j i in terms of a complete orthonormal set of basis functions fj˚n ig, i.e., where Trf g in the denominator of Eq. (7.11) ensures the X correct result even for a normalization that is different cn j˚n i : (7.4) from Eq. (7.9). The invariance of the trace operation ensures j i D n the same result—independently of the particular choice of the basis representation. The density operator then becomes X
D cn cm j˚n ih˚m j D nm j˚n ih˚m j ; (7.5)
2 D :
7.1.4
n;m
The Liouville Equation
where the star denotes the complex conjugate quantity. Note Suppose Eq. (7.8) is valid for a time t D 0. If the functions that the density matrix elements nm D h˚n j j˚m i depend jn .r; t/i obey the Schrödinger equation, i.e., on the choice of the basis and that the density matrix is Her@ mitian, i.e., (7.12) i jn .r; t/i D H.t/ jn .r; t/i ; @t D nm : (7.6)
mn the density operator at the time t can be written as Finally, if j i D j˚ i is one of the basis functions, then i
mn D ıni ımi ;
(7.7)
.t/ D U.t/ .0/ U .t/ :
(7.13)
In Eq. (7.13), U.t/ is the time evolution operator that relates where ıni is the Kronecker ı. Hence, the density matrix is the wave functions at times t D 0 and t according to diagonal in this representation with only one nonvanishing element. (7.14) jn .r; t/i D U.t/ jn .r; 0/i ; and U .t/ denotes its adjoint. Note that
7.1.2
Mixed States
U.t/ D eiHt ;
(7.15)
The above concepts can be extended to treat statistical en- if the Hamiltonian H is time independent. sembles of pure quantum states. In the simplest case, such Differentiation of Eq. (7.13) with respect to time and inmixed states can be represented by a diagonal density matrix serting Eq. (7.14) into the Schrödinger equation (7.12) yields of the form the equation of motion X wn jn ihn j ; (7.8)
D @ n (7.16) i .t/ D ŒH.t/; .t/ ; @t where the weight wn is the fraction of systems in the pure quantum state jn i. The standard normalization for the trace where ŒA; B denotes a commutator.
7
Density Matrices
121
representation for the reservoir at any time t is the same as the representation for t D 0. Another frequently made assumption is the Markov approximation. In this approximation, one assumes that the system forgets all knowledge of the past, so that the density matrix elements at the time t C t depend only on the values of these elements, and their first derivatives, at the time t. (7.17) When Eq. (7.19) is put back into Eq. (7.18), the result in the H.t/ D H0 C V .t/ ; Markov approximation can be rewritten as the interaction picture is preferably used. The Liouville equa
@ tion then becomes
SI .t/ D i TrR VI .t/; SI .0/ R .0/ @t
@ Zt
i I .t/ D VI .t/; I .t/ ; (7.18) dTrR VI .t/;ŒVI ./; SI .t/ R .0/ ; (7.22) @t
The Liouville equation (7.16) can be used to determine the density matrix and to treat transitions from nonequilibrium to equilibrium states in quantum mechanical systems. Especially for approximate solutions in the presence of small time-dependent perturbation terms in an otherwise timeindependent Hamiltonian, i.e., for
0 where the subscript I denotes the operator in the interaction picture. In first-order perturbation theory, Eq. (7.18) can be where TrR denotes the trace with regard to all variables of the integrated to yield reservoir. Note that the integral over d contains the system density matrix in the interaction picture, SI , at the time t, Zt rather than at all times that are integrated over (the Markov
I .t/ D I .0/ i ŒVI ./; I .0/ d ; (7.19) approximation), and that the density matrix for the reservoir is taken as R .0/ at all times. For more details, see Chap. 7 0 of Blum [2] and references therein. and higher-order terms can be obtained through subsequent Equations such as Eq. (7.22) are the basis for the master or iterations. rate equation approach used, for example, in quantum optics for the theory of lasers and the coupling of atoms to cavity modes. For more details, see Chaps. 72, 74, 77 and 82.
7.1.5 Systems in Thermal Equilibrium
According to quantum statistics, the density operator for a system that is in thermal equilibrium with a surrounding reservoir R at a temperature T (canonical ensemble) can be expressed as exp.ˇH / ; (7.20)
D Z where H is the Hamiltonian, and ˇ D 1=kB T with kB being the Boltzmann constant. The partition sum ˚ Z D Tr exp.ˇH /
7.2 Spin and Light Polarizations Density matrices are frequently used to describe the polarization state of spin-polarized particle beams as well as light. The latter can either be emitted from excited atomic or molecular ensembles or can be used, for example, for laser pumping purposes.
(7.21) 7.2.1
Spin-Polarized Electrons
ensures the normalization condition Eq. (7.9). Expectation The spin polarization of an electron beam with respect to values are calculated according to Eq. (7.11), and extensions a given quantization axis nO is defined as [5] to other types of ensembles are straightforward. N" N# ; (7.23) PnO D N" C N#
7.1.6
Relaxation Processes
where N" .N# / is the number of electrons with spin up (down) with regard to this axis. An arbitrary polarization Transitions from nonequilibrium to equilibrium states can state is described by the density matrix also be described within the density matrix formalism. One ! of the basic problems is to account for irreversibility in Px iPy 1 1 C Pz ; (7.24)
D the energy (and sometimes particle) exchange between the 2 Px C iPy 1 Pz system of interest, S, and the reservoir, R. This is usually achieved by assuming that the interaction of the system with where Px;y;x are the Cartesian components of the spin polarthe reservoir is negligible and, therefore, the density matrix ization vector. The individual components can be obtained
7
122
K. Bartschat
where T is the transition operator. Furthermore, J0 .J1 / is the total electronic angular momentum in the initial (7.25) (final) state of the target and M .M / its corresponding Pi D Trfi g ; 0 1 .k / is the initial (final) momentum z-component, while k 0 1 where i (i D x; y; z) are the standard Pauli spin matrices. of the projectile and m0 .m1 / its spin component. from the density matrix as
7.2.2
Light Polarization
7.3.2 Reduced Density Matrices
Another important use of the density matrix formalism is the description of light polarization in terms of the so-called Stokes parameters [6]. For a given direction of observation, the general polarization state of light can be fully determined by the measurement of one circular and two independent linear polarizations. Using the notation of Born and Wolf [7], the density matrix is given by ! P1 iP2 1 P3 Itot ; (7.26)
D 2 P1 C iP2 1 C P3
While the scattering amplitudes are the central elements in a theoretical description, some restrictions usually need to be taken into account in a practical experiment. The most important ones are: (i) there is no pure initial state, and (ii) not all possible quantum numbers are simultaneously determined in the final state. The solution to this problem can be found by using the density matrix formalism. First, the complete density operator after the collision process is given by [2]
(7.34)
out D T in T ; where P1 and P2 are linear light polarizations, while P3 is the circular polarization (also Sect. 7.6). In Eq. (7.26), the where in is the density operator before the collision. The density matrix is normalized in such a way that corresponding matrix elements are given by Trf g D Itot ;
(7.27)
where Itot is the total light intensity. Other frequently used names for the various Stokes parameters are P1 D
3 D M ;
(7.28)
P2 D
1 D C ;
(7.29)
P3 D 2 D S :
(7.30)
k ;M 0 M1
. out /m10 m 1 1
1
D
X
m00 m0 M00 M0
m00 m0 M00 M0
f M10 m01 I M00 m00
f M1 m 1 I M0 m 0 ;
(7.35)
where the term m00 m0 M00 M0 describes the preparation of the initial state (i). Secondly, reduced density matrices account The Stokes parameters of electric dipole radiation can be re- for (ii). For example, if only the scattered projectiles are oblated directly to the charge distribution of the emitting atomic served, the corresponding elements of the reduced density ensemble. As discussed in detail in Chap. 48, one finds, for matrix are obtained by summing over the atomic quantum numbers as follows: example, (7.31) L? D P3 X 1 M1 . out /km10 ;M : (7.36) . out /km10 m D for the angular momentum transfer perpendicular to the scat1 1 1 m1 M1 tering plane in collisional (de)excitation, and D
1 argfP1 C iP2 g 2
The differential cross section for unpolarized projectile and (7.32) target beams is given by
for the alignment angle.
X d . out /km11 m1 ; DC d˝ m
(7.37)
1
7.3
Atomic Collisions
7.3.1
Scattering Amplitudes
Transitions from an initial state jJ0 M0 I k0 m0 i to a final state jJ1 M1 I k1 m1 i are described by scattering amplitudes f .M1 m1 I M0 m0 / D hJ1 M1 I k1 m1 jT jJ0 M0 I k0 m0 i ; (7.33)
where C is a constant that depends on the normalization of the continuum waves in a numerical calculation. On the other hand, if only the atoms are observed (for example, by analyzing the light emitted in optical transitions), the elements Z X k ;M 0 M (7.38) . out /M10 M1 D d3 k1 . out /m11 m11 1 m1
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123
determine the integrated Stokes parameters [8, 9], i.e., the where the irreducible tensor operators are defined in terms of polarization of the emitted light. They contain information 3–j -symbols as about the angular momentum distribution in the excited tarX
0 0p T J 0 J KQ D .1/J M 2K C 1 get ensemble. M 0M Finally, for electron–photon coincidence experiments ! J0 J K without spin analysis in the final state, the elements jJ 0 M 0 ihJM j ; 0 M M Q X k ;M 0 M k1 . out /m11 m11 1 (7.39) . out /M 0M D (7.43) 1 1 m1
and the state multipoles or statistical tensors are given by simultaneously contain information about the projectiles and D
X E 0 0p .1/J M 2K C 1 T J 0 J KQ D the target. This information can be extracted by measuring M 0M the angle-differential Stokes parameters. In particular, for un! polarized electrons and atoms, the natural coordinate system, J0 J K hJ 0 M 0 j jJM i : where the quantization axis coincides with the normal to the M 0 M Q scattering plane, allows for a simple physical interpretation (7.44) of the various parameters [10] (Chap. 48). The density matrix formalism outlined above is very use- Hence, the selection rules for the 3–j -symbols imply that ful for obtaining a qualitative description of the geometrical jJ J 0 j K J C J 0 ; (7.45) and sometimes also of the dynamical symmetries of the 0 M M D Q : (7.46) collision process [11]. Two explicit examples are discussed in Sect. 7.6. Equation (7.44) can be inverted through the orthogonality condition of the 3–j -symbols to give X 7.4 Irreducible Tensor Operators 0 0p hJ 0 M 0 j jJM i D .1/J M 2K C 1 The general density matrix theory can be formulated in a very elegant fashion by decomposing the density operator in terms of irreducible components whose matrix elements then become the state multipoles. In such a formulation, full advantage can be taken of the most sophisticated techniques developed in angular momentum algebra (Chap. 2). Many explicit examples can be found in [3, 4]. 7.4.2
KQ
J0 M0
J M
! K D 0 E T J J KQ : Q (7.47)
Transformation Properties
Suppose a coordinate system .X2 ; Y2 ; Z2 / is obtained from another coordinate system .X1 ; Y1 ; Z1 / through a rotation by a set of three Euler angles .; ˇ; ˛/ as defined in EdThe density operator for an ensemble of particles in quantum monds [12]. The irreducible tensor operators Eq. (7.43) states labeled as jJM i, where J and M are the total angular defined in the .X1 ; Y1 ; Z1 / system are then related to the opmomentum and its magnetic component, respectively, can be erators hT .J 0 J / i in the .X2 ; Y2 ; Z2 / system by KQ written as X
0 T J 0 J Kq D.; ˇ; ˛/K (7.48) T J J KQ D X 0 qQ ; J J 0 0
M 0 M jJ M ihJM j ; (7.40)
D q
7.4.1
Definition
J 0 JM 0 M
where
0
J J 0 0
M 0 M D hJ M j jJM i
where 0
(7.41)
J iM J iM˛ D.; ˇ; ˛/M d.ˇ/M 0M D e 0M e
(7.49)
are the matrix elements. (For simplicity, interactions out- is a rotation matrix (Chap. 2). Note that the rank K of the side the single manifold of momentum states jJM i are tensor operator is invariant under such rotations. Similarly, D
E XD 0 E neglected.) Alternatively, one may write T J J Kq D.; ˇ; ˛/K (7.50) T J 0 J KQ D qQ E
X D
q T J 0 J KQ T J 0 J KQ ; C D (7.42) 0 holds for the state multipoles. J JKQ
7
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K. Bartschat
˝
˛ The irreducible tensor operators fulfill the orthogonality that only state multipoles with Q D 0, i.e., T J 0 J K0 , condition can be different from zero in such a situation. n
o
3. For planar symmetric systems with fixed J 0 D J , Tr T J 0 J KQ T J 0 J K 0 Q0 D ıK 0 K ıQ0 Q : (7.51) D E D E T .J /KQ D .1/K T .J /KQ (7.59) With
1 if the system properties are invariant under reflection in T J 0 J 00 D p ıJ 0 J 1 (7.52) the xz-plane. Hence, state multipoles with even rank K 2J C 1 are real numbers, while those with odd rank are purely being proportional to the unit operator 1, it follows that all imaginary in this case. tensor operators have vanishing trace, except for the monopole T .J 0 J /00 . The above results can be applied immediately to the deReduced tensor operators fulfill the Wigner–Eckart theo- scription of atomic collisions where the incident beam axis rem (Sect. 2.8.4) is the quantization axis (the so-called collision system). For E D
example, impact excitation of unpolarized targets by unJ 0 M 0 jT J 0 J KQ jJM polarized projectiles without observation of the scattered ! projectiles is symmetric both with regard to rotation around ˛ K J ˝ 0 J0 J 0 M 0 D .1/ k T kJ ; (7.53) J K the incident beam axis and with regard to reflection in any M 0 Q M plane ˝containing this axis. Consequently, the state multi ˛ ˝ ˛ ˝ ˛ , T .J / , T .J / ; : : : fully characterize the poles T .J / 00 20 40 where the reduced matrix element is simply given by atomic ensemble of interest. Using Eq. (7.50), similar rela˛ ˝ 0 1 tionships can be derived for state multipoles defined with : (7.54) J k TK kJ D p regard to other coordinate systems, such as the natural sys2K C 1 tem where the quantization axis coincides with the normal vector to the scattering plane (Chap. 48).
7.4.3
Symmetry Properties of State Multipoles
The Hermiticity condition for the density matrix implies 7.4.4 Orientation and Alignment D
E D E 0 T J 0 J KQ D .1/J J CQ T .JJ 0 /KQ ; (7.55) From the above discussion, it is apparent that the description of systems that do not exhibit spherical symmetry requires which, for sharp angular momentum J 0 D J , yields the knowledge of state multipoles with rank K ¤ 0. Frequently, the multipoles with K D 1 and K D 2 are determined D E D E T .J /KQ D .1/Q T .J /KQ : (7.56) via the angular correlation and the polarization of radiation emitted from an ensemble of collisionally excited targets. ˝ ˛ The state multipoles with K D 1 are proportional to the Hence, the state multipoles T .J /K0 are real numbers. Furthermore, the transformation property Eq. (7.50) of the spherical components of the angular momentum expectation state multipoles imposes restrictions on nonvanishing state value and, therefore, give rise to a nonvanishing circular light multipoles to describe systems with given symmetry proper- polarization (Sect. 7.6). This corresponds to a sense of rotation or an orientation in the ensemble, which is, therefore, ties. In detail, one finds: called oriented (Sect. 48.1). On the other hand, nonvanishing multipoles with rank 1. For spherically symmetric systems, K D 2 describe the alignment of the system. Some authors, D
E D
E (7.57) however, use the terms alignment or orientation synonyT J 0 J KQ D T J 0 J KQ rot mously for all nonvanishing state multipoles with rank for all sets of Euler angles. This implies that only the K ¤ 0, thereby describing any system with anisotropic oc˝ ˛ monopole term T .J /00 can be different from zero. cupation of magnetic sublevels as aligned or oriented. For 2. For axially symmetric systems, details on alignment and orientation, see Chap. 48 and [3, 4]. D
E D
E (7.58) T J 0 J KQ D T J 0 J KQ rot
7.4.5 Coupled Systems
for all Euler angles that describe a rotation around the z-axis. Since this angle enters via a factor exp.iQ/ into Tensor operators and state multipoles for coupled systems the general transformation formula Eq. (7.50), it follows are constructed as direct products (˝) of the operators for
7
Density Matrices
125
the individual systems. For example, the density operator for two subsystems in basis states jL; ML i and jS; MS i is constructed as [2] X ˝ ˛
D T .L/KQ ˝ T .S/kq T .L/KQ ˝ T .S/kq : KQkq
(7.60) If the two systems are uncorrelated, the state multipoles factor as ED E D ˝ ˛ (7.61) T .L/KQ ˝ T .S/kq D T .L/KQ T .S/kq : More generally, irreducible representations of coupled operators can be defined in terms of a 9–j -symbol as X
T J 0 ; J K 0 Q0 D KO kO JO JO 0 KQ; kqjK 0 Q0 KQkq
8 ˆ
= J 0 T .L/KQ ˝ T .S/kq ; > ; J (7.62)
p where xO 2x C 1 and .j1 m1 ; j2 m2 jj3 m3 / is a standard Clebsch–Gordan coefficient.
7.5 7.5.1
Time Evolution of State Multipoles Perturbation Coefficients
sequently decay by optical transitions. Such an excitation may be performed, for example, in beam-foil experiments or electron–atom collisions where the energy width of the electron beam is too large to resolve the fine structure (or hyperfine structure) of the target states. Suppose, for instance, that explicitly relativistic effects, such as the spin–orbit interaction between the projectile and the target, can be neglected during a collision process between an incident electron and a target atom. In that case, the orbital angular momentum (L) system of the collisionally excited target states may be oriented, depending on the scattering angle of the projectile. On the other hand, the spin (S) system remains unaffected (unpolarized), provided that both the target and the projectile beams are unpolarized. During the lifetime of the excited target states, however, the spin– orbit interaction within the target produces an exchange of orientation between the L and the S systems, which results in a net loss of orientation in the L system. This effect can be observed directly through the intensity and the polarization of the light emitted from the excited target ensemble. The perturbation coefficients for the finestructure interaction are found to be [2, 13]
exp. t/ X 0 2J C 1 2J C 1 G.LI t/K D 2S C 1 J 0 J )2 (
L J0 S cos !J 0 !J t ; (7.66) J L K
Hence, these coefficients relate the state multipoles at time t to those at t D 0.
where !J 0 !J corresponds to the (angular) frequency difference between the various multiplet states with total electronic angular momenta J 0 and J , respectively. Also, is the natural width of the spectral line; for simplicity, the same lifetime was assumed in Eq. (7.66) for all states of the multiplet. Note that the perturbation coefficients are independent of the multipole component Q in this case, and that there is no mixing between different multipole ranks K. Similar results can be derived [2, 13] for the hyperfine interaction and also to account for the combined effect of fine and hyperfine structure. The cosine terms represent correlation between the signal from different fine-structure states, and they lead to oscillations in the intensity as well as the measured Stokes parameters in a time-resolved experiment. Finally, generalized perturbation coefficients have been derived for the case where both the L and the S systems may be oriented and/or aligned during the collision process [14]. This can happen when spin-polarized projectiles and/or target beams are prepared.
7.5.2
7.5.3
From the general expansions XD
E
T j 0 j I t kq T j 0 j kq
.t/ D
(7.63)
j 0 j kq
in terms of irreducible components, together with Eq. (7.42) for time t D 0 and Eq. (7.13) for the time development of the density operator, it follows that D
X D
E E
Qq T J 0 J I 0 KQ G J 0 J; j 0 j I t Kk ; T j 0 j I t kq D J 0 JKQ
(7.64) where the perturbation coefficients are defined as Qq
G J 0 J; j 0 j I t Kk o n
D Tr U.t/T J 0 J KQ U.t/ T .j 0 j /kq :
Quantum Beats
(7.65)
Time Integration over Quantum Beats
An important application of the perturbation coefficients is If the excitation and decay times cannot be resolved in the coherent excitation of several quantum states, which sub- a given experimental setup, the perturbation coefficients need
7
126
K. Bartschat
to be integrated over time. As a result, the quantum beats These parameters can be expressed in terms of the elements disappear, but a net effect may still be visible through a de- Eq. (7.36). To analyze this problem explicitly, one defines the quanpolarization of the emitted radiation. For the case of atomic tities fine structure interaction discussed above, one finds [2, 13] Z1 N G.L/ K D 0
D
˝ G.LI t/K dt
X 1 2S C 1 0 J J ( L J0 J L
2J 0 C 1 2J C 1
S K
)2
; 2 C !J2 0 J
(7.67)
where !J 0 J D !J 0 !J . Note that the amount of depolarization depends on the relationship between the fine structure splitting and the natural line width. For j!J 0 J j (if J 0 ¤ J ), the terms with J 0 D J dominate and cause the maximum depolarization; for the opposite case j!J 0 J j , the sum rule for the 6–j -symbols can be applied and no depolarization is observed. Similar depolarizations can be caused through hyperfine structure effects, as well as through external fields. An important example of the latter case is the Hanle effect (Sect. 18.2.1).
7.6
Examples
In this section, two examples of the reduced density formalism are discussed explicitly. These are: (i) the change of the spin polarization of initially polarized spin- 12 projectiles after scattering from unpolarized targets, and (ii) the Stokes parameters describing the angular distribution and the polarization of light as detected in projectile-photon coincidence experiments after collisional excitation. The recent book by Andersen and Bartschat [4] provides a detailed introduction to these topics, together with a thorough discussion of benchmark studies in the field of electronic and atomic collisions, including extensions to ionization processes, as well as applications in plasma, surface, and nuclear physics. Even more extensive compilations of such studies can be found in a review series dealing with unpolarized electrons colliding with unpolarized targets [10], heavy-particle collisions [15], and the special role of projectile and target spins in such collisions [16].
7.6.1
˛ m01 m00 I m1 m0 D
Generalized STU-Parameters
For spin-polarized projectile scattering from unpolarized targets, the generalized STU-parameters [11] contain information about the projectile spin polarization after the collision.
X
1 f M1 m01 I M0 m00 2J0 C 1 M M 1 0
f M1 m 1 I M0 m 0 ; (7.68)
which contain the maximum information that can be obtained from the scattering process, if only the polarization of the projectiles is prepared before the collision and measured thereafter. Next, the number of independent parameters that can be determined in such an experiment needs to be examined. For spin- 21 particles, there are 2 2 2 2 D 16 possible combinations of fm01 m00 I m1 m0 g and, therefore, 16 complex or 32 real parameters (in the most general case of spin-S particles, there would be .2S C 1/4 combinations). However, from the definition Eq. (7.68) and the Hermiticity of the reduced density matrix contained therein, it follows that ˝
˛ ˝ ˛ m01 m00 I m1 m0 D m1 m0 I m01 m00 :
(7.69)
Furthermore, parity conservation of the interaction or the equivalent reflection invariance with regard to the scattering plane yields the additional relationship [11]
1 1 f M1 m1 I M0 m0 D .1/J1 M1 C 2 m1 CJ0 M0 C 2 m0
˘1 ˘0 f M1 m1 IM0 m0 ; (7.70) where ˘1 and ˘0 are ˙1, depending on the parities of the atomic states involved. Hence, ˛ ˝ 0 0 0 0 m1 m0 I m1 m0 D .1/m1 m1 Cm0 m0 ˝ ˛ m01 m00 I m1 m0 :
(7.71)
Note that Eqs. (7.70) and (7.71) hold for the collision frame, where the quantization axis (Oz) is taken as the incident beam axis and the scattering plane is the xz-plane. Similar formulas can be derived for the natural frame (Sect. 7.3.2). Consequently, eight independent parameters are sufficient to characterize the reduced spin density matrix of the scattered projectiles. These can be chosen as the absolute differential cross section u D
˛ 1 X ˝ m1 m0 I m1 m0 ; 2 m ;m 1
0
(7.72)
7
Density Matrices
127
for the scattering of unpolarized projectiles from unpolarized The following geometries are particularly suitable for the targets and the seven relative parameters experimental determination of the individual parameters; u and SP can be measured with unpolarized incident projec tiles. A transverse polarization component perpendicular to 2 1 1 11 ; (7.73) the scattering plane P D P yO is needed to obtain S Im I SA D y A u 2 2 22 and T . Finally, the measurement of T , U , T , and U y x zx z xz 11 11 2
SP D Im I ; (7.74) requires both transverse Px xO and longitudinal Pz zO prou 22 22 jectile polarization components in the scattering plane. 1 11 1 1 1 1 11 Ty D I ; (7.75) I u 2 2 22 22 2 2 11 1 1 1 1 11 1 7.6.2 Radiation from Excited States: C I ; (7.76) I Tx D u 2 2 22 22 2 2 Stokes Parameters 11 11 1 11 11 Tz D ; (7.77) The state-multipole description is also widely used for the I I u 2 2 2 2 22 22 parametrization of the Stokes parameters that describe the 2 11 11 Uxz D ; (7.78) polarization of light emitted in optical decays of excited Re I u 22 22 atomic ensembles. The general case of excitation by spin1 1 11 2 Uzx D Re I ; (7.79) polarized projectiles has been treated by Bartschat and u 2 2 22 collaborators [8]. The basic experimental setup for electron– photon coincidence experiments and the definition of the where Refxg and Imfxg denote the real and imaginary parts Stokes parameters are illustrated in Figs. 7.2 and 7.3. of the complex quantity x, respectively. Note that normalizaFor impact excitation of an atomic state with total election constants have been omitted in Eq. (7.72) to simplify the tronic angular momentum J and an electric dipole transition notation. to a state with Jf , the photon intensity in a direction nO D Therefore, the most general form for the polarization vec- .# ; ˚ / is given by tor after scattering, P 0 , for an initial polarization vector P D E 2 .1/J Jf D T .J /00 I.# ; ˚ / D C p .Px ; Py ; Pz / is given by 3 2J C 1 ) (
1 1 2 SP CTy Py yO C Tx Px CUxz Pz xO C Tz Pz Uzx Px zO : J J Jf 1 C SA Py Eo nD (7.80) Re T .J /22 sin2 # cos 2˚ The physical meaning of the above relation is illustrated Eo nD in Fig. 7.1. Re T .J /21 sin 2# cos ˚ r D E 1 2 T .J / C y 20 .3 cos # 1/ 6 nD Eo x
Py
P
Im T .J /22 sin2 # sin 2˚ Eo i nD ; (7.81) C Im T .J /21 sin 2# sin ˚
σ u (1 + SA Py) P'
Pz Px k0
SP + Ty + Py 1 + SA + Py
k1 Θ Tz Pz – UzxPz 1 + SA Py
z
y
Tx Px + UxzPz 1 + SA Py
Fig. 7.1 Physical meaning of the generalized STU-parameters: the polarization function SP gives the polarization of an initially unpolarized projectile beam after the collision, while the asymmetry function SA determines a left–right asymmetry in the differential cross section for the scattering of a spin-polarized beam. Furthermore, the contraction parameters (Tx ; Ty ; Tz ) describe the change of an initial polarization component along the three Cartesian axes, while the parameters Uxz and Uzx determine the rotation of a polarization component in the scattering plane
hv
x
Θγ Φγ
e , k0 –
Θ
e–, k1 z
Fig. 7.2 Geometry of electron–photon coincidence experiments
7
128
K. Bartschat
to deal, for example, with depolarization effects due to internal or external fields. General formulas for P1 D 3 and P 2 D 1 can be found in [8] and, for both the natural and the Photon detector collision systems, in [4]. As was pointed out before, some of the state multipoles may vanish, depending on the experimental arrangement. A detailed analysis of the information contained in the state nˆ multipoles and the generalized Stokes parameters (which are eˆ2 defined for specific values of the projectile spin polarizaΦγ n tion) has been given by Andersen and Bartschat [4, 17, 18]. eˆ1 They reanalyzed the experiment performed by Sohn and Hanne [19] and showed how the density matrix of the excited atomic ensemble can be determined by a measurement of the Θγ y generalized Stokes parameters. In some cases, this will allow for the extraction of a complete set of scattering amplitudes x for the collision process. Such a perfect scattering experiFig. 7.3 Definition of the Stokes parameters: photons are observed in ment was called for by Bederson many years ago [20] and a direction nO with polar angles .# ; ˚ / in the collision system. The is within reach even for fairly complex excitation processes. three unit vectors .n; O eO 1 ; eO 2 / define the helicity system of the photons, The most promising cases were discussed by Andersen and eO 1 D .# C 90ı ; ˚ / lies in the plane spanned by nO and zO and is perı Bartschat [4, 17, 21]. pendicular to n, O while eO 2 D .# ; ˚ C 90 / is perpendicular to both nO z
and eO 1 . In addition to the circular polarization P3 , the linear polarizations P1 and P2 are defined with respect to axes in the plane spanned by eO 1 and eO 2 . Counting from the direction of eO 1 , the axes are located at (0ı ; 90ı ) for P1 and at (45ı ; 135ı ) for P2 , respectively
7.7 Summary
The basic formulas dealing with density matrices in quantum mechanics, with particular emphasis on reduced matrix theory and its applications in atomic physics, have been sumˇ2 e 2 ! 4 ˇˇ J Jf ˇ marized. More details are given in the introductory textbooks hJ krkJ i .1/ (7.82) C D f 2c 3 by Blum [2], Balashov et al. [3], Andersen and Bartschat [4], and the references listed below. is a constant containing the frequency ! of the transition as well as the reduced radial dipole matrix element. Similarly, the product of the intensity I and the circular light polarization P3 can be written in terms of state multi- References poles as where
(
)
1 1 1 J J Jf nD Eo Im T .J /11 2sin # sin˚
I P3 # ; ˚ D C
Eo nD Re T .J /11 2sin # cos˚ E p D C 2 T .J /10 cos # ; (7.83) so that P3 can be calculated as
P3 # ; ˚ D I P3 # ; ˚ =I # ; ˚ :
(7.84)
Note that each state multipole gives rise to a characteristic angular dependence in the formulas for the Stokes parameters, and that perturbation coefficients may need to be applied
1. von Neumann, J.: Göttinger Nachr. 1927, 245–272 (1927) 2. Blum, K.: Density Matrix Theory and Applications, 3rd edn. Springer, New York (2012) 3. Balashov, V.V., Grum-Grzhimailo, A.N., Kabachnik, N.M.: Polarization and Correlation Phenomena in Atomic Collisions. A Practical Theory Course. Plenum, New York (2000) 4. Andersen, N., Bartschat, K.: Polarization, Alignment, and Orientation in Atomic Collisions, 2nd edn. Springer, New York (2017) 5. Kessler, J.: Polarized Electrons. Springer, New York (1985) 6. Baylis, W.E., Bonenfant, J., Derbyshire, J., Huschilt, J.: Am. J. Phys. 61, 534 (1993) 7. Born, M., Wolf, E.: Principles of Optics. Pergamon, New York (1970) 8. Bartschat, K., Blum, K., Hanne, G.F., Kessler, J.: J. Phys. B 14, 3761 (1981) 9. Bartschat, K., Blum, K.: Z. Phys. A 304, 85 (1982) 10. Andersen, N., Gallagher, J.W., Hertel, I.V.: Phys. Rep. 165, 1 (1988) 11. Bartschat, K.: Phys. Rep. 180, 1 (1989) 12. Edmonds, A.R.: Angular Momentum in Quantum Mechanics. Princeton Univ. Press, Princeton (1957) 13. Fano, U., Macek, J.H.: Rev. Mod. Phys. 45, 553 (1973)
7
Density Matrices 14. Bartschat, K., Andrä, H.J., Blum, K.: Z. Phys. A 314, 257 (1983) 15. Andersen, N., Broad, J.T., Campbell, E.E., Gallagher, J.W., Hertel, I.V.: Phys. Rep. 278, 107 (1997) 16. Andersen, N., Bartschat, K., Broad, J.T., Hertel, I.V.: Phys. Rep. 279, 251 (1997) 17. Andersen, N., Bartschat, K.: Adv. At. Mol. Phys. 36, 1 (1996) 18. Andersen, N., Bartschat, K.: J. Phys. B 27, 3189 (1994). corrigendum: J. Phys. B 29, 1149 (1996) 19. Sohn, M., Hanne, G.F.: J. Phys. B 25, 4627 (1992) 20. Bederson, B.: Comments At. Mol. Phys. 1, 41, 65 (1969) 21. Andersen, N., Bartschat, K.: J. Phys. B 30, 5071 (1997)
129 Klaus Bartschat Klaus Bartschat is the Levitt Distinguished Professor of Physics at Drake University. His research focuses on the general theory and computational treatment of electron and photon collisions with atoms and ions, including the direct solution of the time-dependent Schrödinger equation for ultrashort intense laser–matter interactions and the use of the R-matrix (close-coupling) method for electron collisions with complex targets.
7
8
Computational Techniques David Schultz
and Michael R. Strayer
Contents 8.1 8.1.1 8.1.2 8.1.3 8.1.4 8.1.5
Representation of Functions Interpolation . . . . . . . . . . . Fitting . . . . . . . . . . . . . . . Fourier Analysis . . . . . . . . . Approximating Integrals . . . . Approximating Derivatives . .
. . . . . .
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8.2 8.2.1 8.2.2 8.2.3 8.2.4 8.2.5
Differential and Integral Equations . . . . . . . . . . . 137 Ordinary Differential Equations . . . . . . . . . . . . . . . 137 Differencing Algorithms for Partial Differential Equations 138 Variational Methods . . . . . . . . . . . . . . . . . . . . . . 140 Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . 140 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . 142
8.3
Computational Linear Algebra . . . . . . . . . . . . . . 143
8.4 8.4.1 8.4.2 8.4.3
Monte Carlo Methods . . . . . . . Random Numbers . . . . . . . . . . . Distributions of Random Numbers Monte Carlo Integration . . . . . . .
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132 132 133 135 135 136
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Abstract
Essential to atomic, molecular, and optical physics is the ability to perform numerical computations accurately and efficiently. Whether the specific approach involves perturbation theory, close coupling expansion, solution of classical equations of motion, or fitting and smoothing of data, basic computational techniques such as integration, differentiation, interpolation, matrix and eigenvalue manipulation, Monte Carlo sampling, and solution of differential equations must be among the standard tool kit. This chapter outlines a portion of this tool kit with the aim of giving guidance and organization to a wide array D. Schultz () Department of Astronomy and Planetary Science, Northern Arizona University Flagstaff, AZ, USA e-mail: [email protected]
of computational techniques. After having digested the present overview, the reader is then referred to detailed treatments given in many of the large number of texts existing on numerical analysis and computational techniques [1–6], mathematical functions [7–9], and mathematical physics [10–18]. In addition to these excellent general references, in the age of the internet, many resources are also available through free publishing projects or research laboratory resources made public. Many of these resources seek to provide techniques and computer codes of high accuracy, portability, robustness, and efficiency, and often take advantage of modern structured programming and computational parallelism, going beyond the highly accessible, broadly applicable, but simple numerical recipes and codes described in the classic texts. A list of such numerical analysis software is given on the Wikipedia, providing very brief descriptions of the packages available [19], and the journal Computer Physics Communications (CPC) publishes computational physics research and applications software with many codes applicable to atomic, molecular, and optical physics (see the CPC program library maintained at Queen’s University Belfast [20]). Especially in the sections that follow on differential equations and computational linear algebra, mention is made of the role of software packages readily available to aid in implementing practical solutions. Finally, in this brief introduction to computational techniques, we note the existence of commercial packages for mathematics, including those for computer algebra, performing numerical calculations and visualizing results through proprietary programming languages, and even performing simulations through such tools as finiteelement analysis, including Mathematica, Maple, MATLAB, Mathcad, and COMSOL, for example.
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_8
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Lagrange Interpolation The polynomial of degree n 1 that passes through all n numerical methods numerical analysis computational points Œx ; f .x /; Œx ; f .x /; : : : ; Œx ; f .x / is given by 1 1 2 2 n n methods numerical approximation numerical integran n tion random numbers linear algebra differential X Y x xi P .x/ D f .x / (8.3) k equations interpolation Fourier analysis xk xi kD1 Keywords
i D1;i ¤k
D
n X
f .xk /Lnk .x/ ;
(8.4)
kD1
8.1
Representation of Functions
where Lnk .x/ are the Lagrange interpolating polynomials. Perhaps the most familiar example is that of linear interpolaThe ability to represent functions in terms of polynomials tion between the points Œx ; y f .x / and Œx ; y f .x /, 1 1 1 2 2 2 or other basic functions is the key to interpolating or fitting namely, x x2 x x1 data, and to approximating numerically the operations of inP .x/ D y1 C y2 : (8.5) tegration and differentiation. In addition, using methods such x1 x2 x2 x1 as Fourier analysis, knowledge of the properties of functions In practice, it is difficult to estimate the formal error bound beyond even their intermediate values, derivatives, and an- for this method, since it depends on knowledge of the .nC1/tiderivatives may be determined (e.g., the spectral properties). th derivative. Alternatively, one uses iterated interpolation,
8.1.1 Interpolation Given the value of a function f .x/ at a set of points x1 ; x2 ; : : : ; xn , the function is often required at some other values between these abscissas. The process known as interpolation seeks to estimate these unknown values by adjusting the parameters of a known function to approximate the local or global behavior of f .x/. One of the most useful representations of a function for these purposes utilizes the algebraic polynomials, Pn .x/ D a0 Ca1 x C Can x n , where the coefficients are real constants, and the exponents are nonnegative integers. The utility stems from the fact that given any continuous function defined on a closed interval, there exists an algebraic polynomial that is as close to that function as desired (Weierstrass theorem). One simple application of these polynomials is the power series expansion of the function f .x/ about some point, x0 , i.e., 1 X f .x/ D ak .x x0 /k : (8.1) kD0
A familiar example is the Taylor expansion, in which the coefficients are given by ak D
f .k/ .x0 / ; kŠ
(8.2)
where f .k/ indicates the k-th derivative of the function. This form, although quite useful in the derivation of formal techniques, is not very useful for interpolation, since it assumes the function and its derivatives are known and since it is guaranteed to be a good approximation only very near the point x0 about which the expansion has been made.
in which successively higher-order approximations are tried until appropriate agreement is obtained. Neville’s algorithm defines a recursive procedure to yield an arbitrary order interpolant from polynomials of lower order. This method and subtle refinements of it form the basis for most recommended polynomial interpolation schemes [3–5]. One important word of caution to bear in mind is that the more points used in constructing the interpolant, and therefore the higher the polynomial order, the greater will be the oscillation in the interpolating function. This highly oscillating polynomial most likely will not correspond more closely to the desired function than polynomials of lower order. As a general rule of thumb, fewer than six points should be used.
Cubic Splines By dividing the interval of interest into a number of subintervals and in each using a polynomial of only modest order, one may avoid the oscillatory nature of high-order (many-point) interpolants. This approach utilizes piecewise polynomial functions, the simplest of which is just a linear segment. However, such a straight line approximation has a discontinuous derivative at the data points – a property that one may wish to avoid, especially if the derivative of the function is also desired – and which clearly does not provide a smooth interpolant. The solution, therefore, is to choose the polynomial of lowest order that has enough free parameters (the constants a0 ; a1 ; : : : ) to satisfy the constraints that the function and its derivative are continuous across the subintervals, as well as specifying the derivative at the endpoints x0 and xn . Piecewise cubic polynomials satisfy these constraints and have a continuous second derivative as well. Cubic spline interpolation does not, however, guarantee that the derivatives of the interpolant agree with those of the function at the data
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points, much less globally. The cubic polynomial in each in- is zero for i ¤ j and positive for i D j . In this case, for any polynomial P .x/ of degree at most n, there exists unique terval has four undetermined coefficients, constants ˛k such that Pi .x/ D ai C bi .x xi / C ci .x xi /2 C di .x xi /3 ; (8.6) n X P .x/ D ˛k k .x/ : (8.8) for i D 0; 1; : : : ; n 1. Applying the constraints, a system kD0 of equations is found that may be solved once the endpoint derivatives are specified. If the second derivatives at the endAmong the more commonly used orthogonal polynomials points are set to zero, then the result is termed a natural are Legendre, Laguerre, and Chebyshev polynomials. spline, and its shape is like that which a long flexible rod would take if forced to pass through all the data points. Chebyshev Interpolation A clamped spline results if the first derivatives are specified The significant advantages of employing a representation of at the endpoints, and is usually a better approximation since a function in terms of Chebyshev polynomials, Tk .x/, i.e., it incorporates more information about the function (if one has a reasonable way to determine or approximate these first 1 X derivatives). f .x/ D ak Tk .x/ ; (8.9) The set of equations in the unknowns, along with the kD0 boundary conditions, constitute a tridiagonal system or matrix, and is therefore amenable to solution by algo- stems from the fact that (i) the expansion rapidly converges, rithms designed for speed and efficiency for such systems (ii) the polynomials have a simple form, and (iii) the polyno(Sect. 8.3; [1–5]). Other alternatives of potentially significant mial approximates the solution of the minimax problem very utility are schemes based on the use of rational functions and closely. This latter property refers to the requirement that the expansion minimizes the maximum magnitude of the error orthogonal polynomials. of the approximation. In particular, the Chebyshev series expansion can be truncated, so that for a given n it yields the Rational Function Interpolation most accurate approximation to the function. Thus, ChebyIf the function that one seeks to interpolate has one or more poles for real x, then polynomial approximations are not shev polynomial interpolation is essentially as good as one good, and a better method is to use quotients of polynomi- can hope to do. Since these polynomials are defined on the als, so-called rational functions. This occurs since the inverse interval [1; 1], if the endpoints of the interval in question powers of the dependent variable will fit the region near the are a and b, the change of variable pole better if the order is large enough. In fact, if the function is free of poles on the real axis but its analytic continuation in the complex plane has poles, the polynomial approximation may also be poor. It is this property that slows or prevents the convergence of power series. Numerical algorithms very similar to those used to generate iterated polynomial interpolants exist [1, 3–5] and can be useful for functions that are not amenable to polynomial interpolation. Rational function interpolation is related to the method of Padé approximation used to improve convergence of power series. This is a rational function analog of Taylor expansion.
Orthogonal Function Interpolation Interpolation using functions other than the algebraic polynomials can be defined and are often useful. Particularly worthy of mention are schemes based on orthogonal polynomials since they play a central role in numerical quadrature. A set of functions 1 .x/; 2 .x/; : : : ; n .x/ defined on the interval [a; b] is said to be orthogonal with respect to a weight function W .x/, if the inner product defined by ˛ ˝ i jj D
Zb i .x/j .x/W .x/dx a
yD
x 12 .b C a/ 1 2 .b
a/
(8.10)
will affect the proper transformation. Press et al. [3–5], for example, give convenient and efficient routines for computing the Chebyshev expansion of a function. See [7, 10] for tabulations, recurrence formulas, orthogonality properties, etc., of these polynomials.
8.1.2 Fitting
Fitting of data stands in distinction from interpolation in that the data may have some uncertainty. Therefore, simply determining a polynomial that passes through the points may not yield the best approximation of the underlying function. In fitting, one is concerned with minimizing the deviations of some model function from the data points in an optimal or best-fit manner. For example, given a set of data points, even a low-order interpolating polynomial might have significant oscillation. In fact, if one accounts for the statistical uncer(8.7) tainties in the data, the best fit may be obtained simply by considering the points to lie on a line.
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In addition, most of the traditional methods of assigning this quality of best fit to a particular set of parameters of the model function rely on the assumption that the random deviations are described by a Gaussian (normal) distribution. Results of physical measurements, for example the counting of events, is often closer to a Poisson distribution, which tends (not necessarily uniformly) to a Gaussian in the limit of a large number of events, or may even contain outliers that lie far outside a Gaussian distribution. In these cases, fitting methods might significantly distort the parameters of the model function in trying to force these different distributions to the Gaussian form. Thus, the least-squares and chi-square fitting procedures discussed below should be used with this caveat in mind. Other techniques, often termed robust fitting [3–5, 21], should be used when the distribution is not Gaussian or replete with outliers.
Least Squares In this common approach to fitting, we wish to determine the m parameters al of some function f .xI a1 ; a2 ; : : : ; am /, in this example depending on one variable, x. In particular, we seek to minimize the sum of the squares of the deviations n X kD1
Œy.xk / f .xk I a1 ; a2 ; : : : ; am /2
(8.11)
goodness of fit. If there are n data points and m adjustable parameters, then the probability that 2 should exceed a particular value purely by chance is
n m 2 ; QDQ 2 2
;
(8.15)
where Q.a; x/ D %.a; x/=%.a/ is the incomplete gamma function. For small values of Q, the deviations of the fit from the data are unlikely to be by chance, and values close to one are indications of better fits. In terms of the chi-square, reasonable fits often have 2 n m. Other important applications of the chi-square method include simulation and estimating standard deviations. For example, if one has some idea of the actual (i.e., nonGaussian) distribution of uncertainties of the data points, Monte Carlo simulation can be used to generate a set of test data points subject to this presumed distribution, and then the fitting procedure may be performed on the simulated data set. This allows one to test the accuracy or applicability of the model function chosen. In other situations, if the uncertainties of the data points are unknown, one can assume that they are all equal to some value, say , fit using the chi-square procedure, and solve for the value of . Thus, some measure of the uncertainty from this statistical point of view can be provided.
by adjusting the parameters, where the y.xk / are the n data points. In the simplest case, the model function is just General Least Squares a straight line, f .xI a1 ; a2 / D a1 x C a2 . Elementary multi- The least-squares procedure can be generalized, usually by allowing any linear combination of basis functions to detervariate calculus implies that a minimum occurs if mine the model function n n n X X X 2 m xi C a2 xi D xi yi ; (8.12) a1 X al l .x/ : (8.16) f .xI a1 ; a2 ; : : : ; am / D kD1 kD1 kD1 n n lD1 X X xi C a2 n D yi ; (8.13) a1 The basis functions need not be polynomials. Similarly, the kD1 kD1 formula for chi-square can be generalized and normal equawhich are called the normal equations. Solution of these tions determined through minimization. The equations may equations is straightforward, and an error estimate of the fit be written in compact form by defining a matrix A with elecan be found [3–5]. In particular, variances may be computed ments for each parameter, as well as measures of the correlation bej .xi / ; (8.17) Ai;j D tween uncertainties and an overall estimate of the goodness i of fit of the data. and a column vector B with elements yi Chi-Square Fitting Bi D : i If the data points each have associated with them a different standard deviation, k , the least-squares principle is modified Then the normal equations are [3–5] by minimizing the chi-square, defined as m X n X yk f .xk I a1 ; a2 ; : : : ; am / 2 2 ˛kj aj D ˇk ; : (8.14) k j D1 kD1 Assuming that the uncertainties in the data points are nor- where mally distributed, the chi-square value gives a measure of the
Œ˛ D A T A ;
Œˇ D A T B ;
(8.18)
(8.19)
(8.20)
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and aj are the adjustable parameters. These equations may be solved using standard methods of computational linear algebra such as Gauss–Jordan elimination. Difficulties involving sensitivity to round-off errors can be avoided by using carefully developed codes to perform this solution [3–5]. We note that elements of the inverse of the matrix ˛ are related to the variances associated with the free parameters and to the covariances relating them.
Statistical Analysis of Data Data generated by an experiment, or from a Monte Carlo simulation, have uncertainties due to the statistical, or random, character of the processes by which they are acquired. Therefore, one must be able to describe certain features of the data statistically, such as their mean, variance and skewness, and the degree to which correlations exist, either between one portion of the data and another, or between the data and some other standard or model distribution. A very readable introduction to this type of analysis was given by Young [22], while more comprehensive treatments are also available [23].
8.1.3 Fourier Analysis The Fourier transform takes, for example, a function of time into a function of frequency, or vice versa, namely 1 '.!/ Q Dp 2 1 '.t/ D p 2
Z1 '.t/ ei!t dt ;
(8.21)
'.!/ Q ei!t d! :
(8.22)
1 Z1
1
In this case, the time history of the function '.t/ may be termed the signal and '.!/ Q the frequency spectrum. Also, if the frequency is related to the energy by E D „!, one obtains an energy spectrum from a signal, and thus the name spectral methods for techniques based on the Fourier analysis of signals. The Fourier transform also defines the relationship between the spatial and momentum representations of wave functions, i.e., 1 .x/ D p 2 Q .p/ D p1 2
Z1 Q .p/ eipx dp ;
(8.23)
convolution of functions, filtering, and analysis of correlation. Good introductions to these techniques with particular attention to applications in physics can be found in [10, 16, 24]. To implement the Fourier transform numerically, the integral transform pair can be converted to sums 1
'.! Q j/ D p 2N 2
2N 1 X
'.tk / ei!j tk ;
(8.25)
kD0
2N 1 X 1 '.tk / D p '.! Q j / ei!j tk ; 2N 2 j D0
(8.26)
where the functions are sampled at 2N points. These equations define the discrete Fourier transform (DFT). Two cautions in using the DFT are as follows. First, if a continuous function of time is sampled at, for simplicity, uniformly spaced intervals, (i.e., ti C1 D ti C ), then there is a critical frequency !c D = , known as the Nyquist frequency, which limits the fidelity of the DFT of this function in that it is aliased. That is, components outside the frequency range !c to !c are falsely transformed into this range due to the finite sampling. This effect can be remediated by filtering or windowing techniques. If, however, the function is bandwidth limited to frequencies smaller than !c , then the DFT does not suffer from this effect, and the signal is completely determined by its samples. Second, implementing the DFT directly from the equations above would require approximately N 2 multiplications to perform the Fourier transform of a function sampled at N points. A variety of fast Fourier transform (FFT) algorithms have been developed (e.g., the Danielson–Lanczos and Cooley–Tukey methods) that require only on the order of .N=2/ log2 N multiplications. Thus, for even moderately large sets of points, the FFT methods are, indeed, much faster than the direct implementation of the DFT. Issues involved in sampling, aliasing, and selection of algorithms for the FFT are discussed in great detail, for example, in [3–5, 15, 25]. In addition to basic computer codes with which to implement the FFT and related tasks, given, for example, in Numerical Recipes [3–5], codes for real and complex valued FFTs that have been implemented and benchmarked on a variety of platforms including parallel computer systems are available, for example, the Fastest Fourier Transform in the West (FFTW) [26].
1 Z1
.x/ eipx dx :
(8.24)
8.1.4 Approximating Integrals
1
Along with the closely related sine, cosine, and Laplace Polynomial Quadrature transforms, the Fourier transform is an extraordinarily pow- Definite integrals may be approximated through a procedure erful tool in the representation of functions, spectral analysis, known as numerical quadrature by replacing the integral by
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an appropriate sum, i.e., Zb f .x/dx a
n X kD0
ak f .xk / :
consideration, it can be shown that the coefficients may be optimally chosen by solving a simple set of linear equations. Thus, a Gaussian quadrature scheme approximates the definite integral of a function multiplied by the weight func(8.27) tion appropriate to the orthogonal polynomial being used as
Zb n Most formulas for such approximations are based on the inX W .x/f .x/dx ak f .xk / ; (8.31) terpolating polynomials described in Sect. 8.1.1, especially kD1 the Lagrange polynomials, in which case the coefficients ak a are given by where the function is to be evaluated at the abscissas given Zb by the roots of the orthogonal polynomial, xk . In this case, (8.28) ak D Lnk .xk /dx : the coefficients ak are often referred to as weights but should a not be confused with the weight function W .x/ (Sect. 8.1.1). If first or second degree Lagrange polynomials are used with Since the Legendre polynomials are orthogonal over the ina uniform spacing between the data points, one obtains the terval [1; 1] with respect to the weight function W .x/ 1, trapezoidal and Simpson’s rules, i.e., this equation has a particularly simple form, leading immediately to the Gauss–Legendre quadrature. If f .x/ contains Zb the weight function of another of the orthogonal polynomials
ı f .x/dx Œf .a/ C f .b/ C O ı 3 f .2/ ./ ; (8.29) as a factor, the corresponding Gauss–Laguerre or Gauss– 2 Chebyshev quadrature should be used. a The roots and coefficients have been tabulated [7] for Zb
5 .4/ many common choices of the orthogonal polynomials (e.g., ı ı f .x/dx f .a/ C 4f C f .b/ C O ı f ./ ; 3 2 Legendre, Laguerre, Chebyshev) and for various orders. a (8.30) Simple computer subroutines are also available that conveniently compute them [3–5]. Since the various orthogonal polynomials are defined over different intervals, use of the respectively, with ı D b a, and for some in Œa; b. Other commonly used formulas based on low-order poly- change of variables such as that given in Eq. (8.10) may be nomials, generally referred to as Newton–Cotes formulas, required. So, for Gauss–Legendre quadrature we make use are described and discussed in detail in numerical analysis of the transformation texts [1, 2]. Since potentially unwanted rapid oscillations in interpolants may arise, it is generally the case that increasing the order of the quadrature scheme too greatly does not generally improve the accuracy of the approximation. Dividing the interval Œa; b into a number of subintervals and summing the result of application of a low-order formula in each subinterval is usually a much better approach. This procedure, referred to as composite quadrature, may be combined with choosing the data points at a nonuniform spacing, decreasing the spacing where the function varies rapidly, and increasing the spacing for economy where the function is smooth to construct an adaptive quadrature.
Gaussian Quadrature If the function whose definite integral is to be approximated can be evaluated explicitly, then the data points (abscissas) can be chosen in a manner in which significantly greater accuracy may be obtained than using Newton–Cotes formulas of equal order. Gaussian quadrature is a procedure in which the error in the approximation is minimized owing to this freedom to choose both data points (abscissas) and coefficients. By utilizing orthogonal polynomials and choosing the abscissas at the roots of the polynomials in the interval under
Zb Z1 .b a/ .b a/y C b C a dy : (8.32) f .x/ dx f 2 2 a
1
Other Methods Especially for multidimensional integrals that cannot be reduced analytically to separable or iterated integrals of lower dimension, Monte Carlo integration may provide the only means of finding a good approximation. This method is described in Sect. 8.4.3. Also, a convenient quadrature scheme can be devised based on the cubic spline interpolation described in Sect. 8.1.1, since in each subinterval, the definite integral of a cubic polynomial of known coefficients is evident.
8.1.5 Approximating Derivatives Numerical Differentiation The calculation of derivatives from a numerical representation of a function is generally less stable than the calculation of integrals, because differentiation tends to enhance fluctuations and worsen the convergence properties of power series.
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For example, if f .x/ is twice continuously differentiable on 8.2.1 Ordinary Differential Equations [a; b], then differentiation of the linear Lagrange interpolation formula Eq. (8.5) yields An ordinary differential equation is an equation involving an unknown function and one or more of its derivatives that
f .x0 C ı/ f .x0 / f .1/ .x0 / D C O ıf .2/ ./ ; (8.33) depend on only one independent variable [27]. The order of ı a differential equation is the order of the highest derivative appearing in the equation. A solution of a general differential for some x0 and in [a; b], where ı D b a. In the limit equation of order n, ı ! 0, Eq. (8.33) coincides with the definition of the deriva
tive. However, in practical calculations with finite precision (8.36) f t; y; y; P : : : ; y .n/ D 0 ; arithmetic, ı cannot be taken too small because of numerical cancellation in the calculation of f .a C ı/ f .a/. In practice, increasing the order of the polynomial used is a real-valued function y.t/ having the following properdecreases the truncation error, but at the expense of in- ties: (1) y.t/ and its first n derivatives exist, so y.t/ and creasing round-off error, the upshot being that three and its first n 1 derivatives must be continuous, and (2) y.t/ five-point approximations are usually the most useful. Var- satisfies the differential equation for all t. A unique soluious three and five-point formulas are given in standard tion requires the specification of n conditions on y.t/ and texts [2, 7, 9]. Two common five-point formulas (centered its derivatives. The conditions may be specified as n initial conditions at a single t to give an initial-value problem, or at and forward=backward) are the end points of an interval to give a boundary value problem.
1 f .x0 2ı/ 8f .x0 ı/ f .1/ .x0 / D First consider solutions to the simple equation 12ı C 8f .x0 C ı/ f .x0 C 2ı/
yP D f .t; y/ ; y.a/ D A : (8.37) (8.34) C O ı 4 f .5/ ./ 1 The methods discussed below can be extended to systems 25f .x0 / C 48f .x0 C ı/ f .1/ .x0 / D 12ı of first-order differential equations and to higher-order dif 36f .x0 C 2ı/ C 16f .x0 C 3ı/ ferential equations. The methods are referred to as discrete
4 .5/ 3f .x0 C 4ı/ C O ı f ./ : (8.35) variable methods and generate a sequence of approximate values for y.t/; y1 ; y2 ; y3 ; : : : at points t1 ; t2 ; t3 ; : : : . For simThe second formula is useful for evaluating the derivative plicity, the discussion here assumes a constant spacing h at the left or right endpoint of the interval, depending on between t points. We shall first describe a class of methods known as one-step methods [28]. They have no memory whether ı is positive or negative, respectively. of the solutions at past times; given yi , there is a recipe for yi C1 that depends only on information at ti . Errors enter into Derivatives of Interpolated Functions An interpolating function can be directly differentiated to numerical solutions from two sources. The first is the disobtain the derivative at any desired point. For example, if cretization error and depends on the method being used. The f .x/ a0 C a1 x C a2 x 2 , then f .1/ .x/ D a1 C 2a2 x. How- second is the computational error that includes such things ever, this approach may fail to give the best approximation as round-off error. For a solution on the interval [a; b], let the t points be to f .1/ .x/ if the original interpolation was optimized to give equally spaced; so for some positive integer n and h D the best possible representation of f .x/. .b a/=n, ti D a C ih, i D 0; 1; : : : ; n. If a < b, h is positive, and the integration is forward; if a > b, h is negative, and the integration is backward. The latter case could occur in solv8.2 Differential and Integral Equations ing for the initial point of a solution curve given the terminal The subject of differential and integral equations is immense point. A general one-step method can then be written in the in both richness and scope. The discussion here focuses on form techniques and algorithms, rather than the formal aspect of y0 D y.t0 / ; (8.38) yi C1 D yi C h .ti ; yi / ; the theory. Further information can be found elsewhere under the broad categories of finite-element and finite-difference methods. The Numerov method, which is particularly useful where is a function that characterizes the method. Difin integrating the Schrödinger equation, is described in great ferent functions are displayed next, giving rise to the Taylor-series methods and the Runge–Kutta methods. detail in [18].
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An attractive feature of the approach is the form of the Taylor-Series Algorithm P because To obtain an approximate solution of order p on Œa; b, gen- underlying polynomial approximation, P .t/, to y.t/ it can be used to approximate y.t/ between mesh points erate the sequence hp1 ; yi C1 D yi C h f .ti ; yi / C C f .p1/ .ti ; yi / pŠ (8.39) ti C1 D ti C h; i D 0; 1; : : : ; n 1 ;
Zt y.t/ Š yi C
P .t/ dt :
(8.42)
ti
The lowest-order Adams–Bashforth formula arises from inwhere t0 D a, and y0 D A. The Taylor method of order p D 1 terpolating the single value fi D f .ti ; yi / by P .t/. The interpolating polynomial is constant, so its integration from ti is known as Euler’s method: to ti C1 results in hf .ti ; yi /, and the first-order Adams– Bashforth formula: yi C1 D yi C hf .ti ; yi / ; ti C1 D ti C h :
(8.40)
yi C1 D yi C hf .ti ; yi / :
(8.43)
Taylor-series methods can be quite effective if the total This is just the forward Euler formula. For constant step derivatives of f are not too difficult to evaluate. Software size h, the second-order Adams–Bashforth formula is packages are available that perform exact differentiation 3 1 D y C h ; y / ; y / : f .t f .t y i C1 i i i i 1 i 1 (ADIFOR, Maple, Mathematica, etc.), facilitating the use of 2 2 this approach. (8.44)
Runge–Kutta Methods Runge–Kutta methods are designed to approximate Taylorseries methods [29] but have the advantage of not requiring explicit evaluations of the derivatives of f .t; y/. The basic idea is to use a linear combination of values of f .t; y/ to approximate y.t/. This linear combination is matched up as closely as possible with a Taylor series for y.t/ to obtain methods of the highest possible order p. Euler’s method is an example using one function evaluation. To obtain an approximate solution of order p D 2, let h D .b a/=n and generate the sequences yi C1 D yi C h .1 /f .ti ; yi / h h C f ti C ; yi C f .ti ; yi / ; 2 2
The lowest-order Adams–Moulton formula involves interpolating the single value fi C1 D f .xi C1 ; yi C1 / and leads to the backward Euler formula yi C1 D yi C hf .ti C1 ; yi C1 / ;
(8.45)
which defines yi C1 implicitly. From its definition it is clear that it has the same accuracy as the forward Euler method; its advantage is vastly superior stability. The second-order Adams–Moulton method also does not use previously computed solution values; it is called the trapezoidal rule, because it generalizes the trapezoidal rule for integrals to differential equations: yi C1 D yi C
h Œf .ti C1 ; yi C1 / C f .ti ; yi / : 2
(8.46)
The Adams–Moulton formula of order p is more accurate than the Adams–Bashforth formula of the same order. Hence, ti C1 D ti C h ; i D 0; 1; : : : ; n 1 ; (8.41) it can use a larger step size; the Adams–Moulton formula is also more stable. A code based on such methods is more where ¤ 0, t0 D a, y0 D A. complex than a Runge–Kutta code, because it must cope with Euler’s method is the special case, D 0, and has order 1; the difficulties of starting the integration and changing the the improved Euler method has D 1=2, and the Euler– step size. Modern Adams codes attempt to select the most Cauchy method has D 1. efficient formula at each step, as well as to choose an optimal step size h to achieve a specified accuracy. The Adams–Bashforth and Adams–Moulton Formulas These formulas furnish important and widely used examples 8.2.2 Differencing Algorithms for Partial of multistep methods [30]. On reaching a mesh point ti with Differential Equations approximate solution yi Š y.ti /, (usually) approximate solutions yi C1j Š y.ti C1j / for j D 2; 3; : : : ; p are available. Differencing schemes, based on flux conservation methFrom the differential equation itself, approximations to the ods [31], are the modern approach to solving partial differderivatives y.t P i C1j / can be obtained. ential equations describing the evolution of physical systems.
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One begins by writing the balance equations for a single cell and (ii) right-end-point quadrature and subsequently applying quadratures and interpolation forZtj C1 mulas. Such approaches have been successful for the full dt Œq.n1 ; t/ q.n ; t/ spectrum of hyperbolic, elliptic, and parabolic equations. For tj simplicity, we begin by discussing systems involving only
one space variable. q.n1 ; tj C1 / q.n ; tj C1 / k : (8.52) As a prototype, consider the parabolic equation Combining Eq. (8.48) with the respective approximations @2 @ (8.47) yields: from (i) an explicit method c u.x; t/ D 2 u.x; t/ ; @t @x
c u.xn ; tj C1 / u.xn ; tj / h
where c and are constants, and u.x; t/ is the solution. We q. ; t / q. ; t / k; (8.53) n1 j n j begin by establishing a grid of points on the xt-plane with step size h in the x direction and step size k in the t-direction. and from (ii) an implicit method Let spatial grid points be denoted by xn D x0 C nh and time
grid points by tj D t0 C j k, where n and j are integers, and c u.xn ; tj C1 / u.xn ; tj / h
.x0 ; t0 / is the origin of the space–time grid. The points n1 q.n1 ; tj C1 / q.n ; tj C1 / k : (8.54) and n are introduced to establish a control interval. We begin with a conservation statement Using centered-finite-difference formulas to approximate the fluxes at the control points n1 and n yields Zn
u.xn ; tj / u.xn1 ; tj / dx r.x; tj C1 / r.x; tj / q.n1 ; tj / D ; (8.55) h n1 Ztj C1 dt Œq.n1 ; t/ q.n ; t/ : D
and (8.48) q.n ; tj / D
tj
u.xnC1 ; tj / u.xn ; tj / ; h
(8.56)
This equation states that the change in the field density on the interval .n1 ; n / from time t D tj to time t D tj C1 is given by the flux into this interval at n1 minus the flux out of the interval at n from time tj to time tj C1 . This expresses the conservation of material in the case that no sources or sinks are present. We relate the field variable u to the physical variables (the density r and the flux q). We consider the case in which the density is assumed to have the form r.x; t/ D cu.x; t/ C b ; with c and b constants; thus Zn
c dx u.x; tj C1 / u.x; tj /
where is a constant. We also obtain similar formulas for the fluxes at time tj C1 . We have used a lower case u to denote the continuous field variable, u D u.x; t/. Note that all of the quadrature and difference formulas involving u are stated as approximate equalities. In each of these approximate equality statements, the amount by which the right-hand side differs from the left-hand side is called the truncation error. If u is a wellbehaved function (has enough smooth derivatives), then it (8.49) can be shown that these truncation errors approach zero as the grid spacings, h and k, approach zero. j If Un denotes the exact solution on the grid, from (i) we have the result
j j c Unj C1 Unj h2 D k Un1 C UnC1 2Unj : (8.57)
n1
(8.50) This is an explicit method, since it provides the solution to the difference equation at time tj C1 , knowing the values at When developing conservation-law equations, there are two time tj . If we use the numerical approximations (ii), we obtain the commonly used strategies for approximating the right-handresult side of Eq. (8.48): (i) left-end-point quadrature
j C1 j C1 c Unj C1 Unj h2 D k Un1 C UnC1 2Unj C1 : (8.58) Ztj C1 dt Œq.n1 ; t/ q.n ; t/ Note that this equation defines the solution at time tj C1 imtj plicitly, since a system of algebraic equations is required to
q.n1 ; tj / q.n ; tj / k ; (8.51) be satisfied. cŒu.xn ; tj C1 / u.xn ; tj /h :
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8.2.3 Variational Methods
8.2.4 Finite Elements
A common problem in atomic, molecular, and optical physics is to find the extrema or the stationary values of functionals. For example, one might seek the eigenvalues and eigenvectors of a Hamiltonian system, such as via the minimization of the expectation value of the energy of a trial wave function to determine the ground state of an atom or molecule, via so-called variational methods. We shall outline in detail the Rayleigh–Ritz method [32]. This method is limited to boundary value problems that can be formulated in terms of the minimization of a functional J Œu. For definiteness we consider the case of a differential operator defined by Lu.x/ D f .x/ ; (8.59)
As discussed in Sect. 8.2.2, in the finite-difference method for classical partial differential equations, the solution domain is approximated by a grid of uniformly spaced nodes. At each node, the governing differential equation is approximated by an algebraic expression that references adjacent grid points. A system of equations is obtained by evaluating the previous algebraic approximations for each node in the domain. Finally, the system is solved for each value of the dependent variable at each node. The finite-element method evolved from computational approaches to implementation of the variational method and of potentially greater accuracy and flexibility. In the finite-element method [33], the solution domain can be discretized into a number of uniform or nonuniform finite elements that are connected via nodes. The change of the dependent variable with regard to location is approximated within each element by an interpolation function. The interpolation function is defined relative to the values of the variable at the nodes associated with each element. The original boundary value problem is then replaced with an equivalent integral formulation. The interpolation functions are substituted into the integral equation, integrated, and combined with the results from all other elements in the solution domain. The results of this procedure can be reformulated into a matrix equation of the form
with x D xi ; i D 1; 2; 3 in R, for example, and with u D 0 on the boundary of R. The function f .x/ is the source. It is assumed that L is always nonsingular, and in addition, for the Ritz method L is Hermitian. The real-valued functions u are in the Hilbert space ˝ of the operator L. We construct the functional J Œu defined as Z (8.60) J Œu D dx Œu.x/Lu.x/ 2u.x/f .x/ : ˝
The variational ansatz considers a subspace of ˝, ˝n , spanned by a class of functions n .x/, and we construct the function un u as n
u .x/ D
n X
n X
ci i .x/ :
(8.61)
i D1
We solve for the coefficients ci by minimizing J Œun: @ci J Œun D 0 ;
i D 1; : : : ; n :
(8.62)
These equations are cast into a set of well-behaved algebraic equations n X
Ai;j cj D gi ;
Ai;j cj D gi ;
i D 1; : : : ; n ;
(8.64)
j D1
i D 1; : : : ; n ;
(8.63)
j D1
R R with Ai;j D ˝ dxi .x/Lj .x/, and gi D ˝ dxi .x/f .x/. Under very general conditions, the functions un converge uniformly to u. The main drawback of the Ritz method lies in the assumption of hermiticity of the operator L. For the Galerkin method we relax this assumption with no other changes. Thus, we obtain an identical set of equations as above, with the exception that the function g is no longer symmetric. The convergence of the sequence of solutions un to u is no longer guaranteed, unless the operator can be separated into a symmetric part L0 , L D L0 C K, so that L1 0 K is bounded.
R R with Ai;j D ˝ dxi .x/Lj .x/, and gi D ˝ dxi .x/f .x/ exactly as obtained in Sect. 8.2.3. The only difference arises in the definitions of the support functions i .x/. In general, if these functions are piecewise polynomials on some finite domain, they are called finite elements or splines. Finite elements make it possible to deal in a systematic fashion with regions having curved boundaries of an arbitrary shape. Also, one can systematically estimate the accuracy of the solution in terms of the parameters that label the finite-element family, and the solutions are no more difficult to generate than more complex variational methods. In one space dimension, the simplest finite-element family begins with the set of step functions defined by ( 1 xi 1 x xi (8.65) i .x/ D 0 otherwise : The use of these simple hat functions as a basis does not provide any advantage over the usual finite-difference schemes. However, for certain problems in two or more dimensions, finite-element methods have distinct advantages
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over other methods. Generally, the use of finite elements requires complex, sophisticated computer programs for implementation. The use of higher-order polynomials, commonly called splines, as a basis has been extensively used in atomic and molecular physics. An extensive literature is available [34, 35]. We illustrate the use of the finite-element method by applying it to the Schrödinger equation. In this case, the linear operator L is H E, where, as usual, E is the energy, and the Hamiltonian H is the sum of the kinetic and potential energies, that is, L D H E D T C V E and Lu.x/ D 0. We define the finite elements through support points, or knots, given by the sequence fx1 ; x2 ; x3 ; : : :g, which are not necessarily spaced uniformly. Since the hat functions have vanishing derivatives, we employ the next more complex basis, that is, tent functions, which are piecewise linear functions given by 8 x xi 1 ˆ ˆ xi 1 x xi ˆ ˆ xi xi 1 ˆ < i .x/ D xi C1 x xi x xi C1 (8.66) ˆ ˆ xi C1 xi ˆ ˆ ˆ :0 otherwise ; and for which the derivative is given by 8 1 ˆ ˆ xi 1 x xi ˆx x ˆ ˆ i i 1 < d 1 i .x/ D xi x xi C1 ˆ dx ˆ x i C1 xi ˆ ˆ ˆ : 0 otherwise :
D
1 .xi xi 1 / I 6
(8.71)
and Oij D 0 otherwise. The potential energy is represented by the matrix Z1 Vij D
dx i .x/V .x/j .x/ ;
(8.72)
1
which may be well approximated by Z1 Vij V .xi /
dx i .x/j .x/
(8.73)
1
D V .xi /Oij ;
8
if xj xi is small. The kinetic energy, T D 12 d2 =dx 2 , is similarly given by Z1 dxi .x/
d2 j .x/ ; dx 2
(8.74)
1 Tij D 2
Z1
d d dx i .x/ j .x/ ; dx dx
(8.75)
1
which in turn is evaluated to yield dx i .x/j .x/ :
.x xi 1 /2 dx C .xi xi 1 /2
(8.68)
xi C1 Z
xi
.x xi /2 dx .xi C1 xi /2
1 D .xi C1 xi 1 / I 3 if i D j 1, xi C1 Z .x xi /.xi C1 x/ Oij D dx .xi C1 xi /2 xi
D
xi 1
which we compute by integrating by parts, since the tent functions have a singular second derivative
Thus, if i D j ,
xi 1
.x xi 1 /.xi x/ .xi xi 1 /2
dx
(8.67)
1
Oi i D
Oij D
1
Z1
Zxi
Zxi
1 Tij D 2
The functions have a maximum value of 1 at the midpoint of the interval Œxi 1 ; xi C1 , with partially overlapping adjacent elements. In fact, the overlaps may be represented by a matrix O with elements Oij D
if i D j C 1,
1 .xi C1 xi / I 6
8 xi C1 xi 1 ˆ ˆ ˆ 2.x xi 1 /.xi C1 xi / ˆ i ˆ ˆ ˆ ˆ 1 ˆ < 2.x xi C1 / i Tij D ˆ ˆ 1 ˆ ˆ ˆ ˆ 2.xi 1 xi / ˆ ˆ ˆ : 0
i Dj i Dj 1 (8.76) i Dj C1 otherwise :
(8.69) Finally, since the Hamiltonian matrix is Hij D Tij C Vij , the solution vector ui .x/ may be found by solving the eigenvalue equation (8.77) ŒHij EOij ui .x/ D 0 : Going beyond this simple example, discrete variable representation (DVR) methods, also known as pseudospectral (8.70) methods, and direct solution of the Schrödinger equation
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on numeral grids via finite-difference, finite-element, and high-order interpolant methods, have been adopted broadly in atomic, molecular, and optical physics as the power of computational resources has grown in recent decades. An introduction to DVR methods applied to solving the timedependent Schrödinger equation for quantum dynamics of molecules, as an example, has been given by Light [36, 37].
and Hilbert [13] give other examples, as well as being excellent references for the application of integral equations and Green’s functions in mathematical physics. In their analytic form, these transform methods provide a powerful method of solving integral equations for special cases. In addition, they may be implemented by performing the transform numerically.
Power Series Solution For an equation of the form (in one dimension for simplicity) Z Central to much of practical and formal scattering theory .r/ D .r/ C dr 0 K.r; r 0 / .r 0/ ; (8.82) is the integral equation and techniques of its solution. For example, in atomic collision theory, the Schrödinger differ- a solution may be found by iteration. That is, as a first apential equation proximation, set .r/ D .r/ so that
8.2.5 Integral Equations
0
ŒE H0 .r/ .r/ D V .r/ .r/ ;
(8.78)
Z 1 .r/
D .r/ C
dr 0 K.r; r 0 /.r 0 / :
(8.83)
where the Hamiltonian H0 .„2 =2m/r 2 C V0 may be solved by exploiting the solution for a delta function source, This may be repeated to form a power series solution, i.e., i.e., n X (8.79) .E H0 /G.r; r 0 / D ı.r r 0 / : .r/ D k Ik .r/ ; (8.84) n kD0
In terms of this Green’s function G.r; r 0 /, and any solution .r/ of the homogeneous equation (i.e., with V .r/ D 0), the where general solution is I0 .r/ D .r/ ; (8.85) Z Z 0 0 0 0 (8.80) I .r/ D dr 0 K.r; r 0 /.r 0 / ; .r/ D .r/ C dr G.r; r /V .r / .r / ; (8.86) 1 Z Z for which, given a choice of the functions G.r; r 0 / and .r/, I2 .r/ D dr 00 dr 0 K.r; r 0 /K.r 0 ; r 00 /.r 00/ ; (8.87) particular boundary conditions are determined. This integral Z Z equation is the Lippmann–Schwinger equation of potential In .r/ D dr 0 dr .n/ K.r; r 0 /K.r; r 00 / K.r .n1/ ; r .n/ / : scattering. Further topics on scattering theory are covered (8.88) in other chapters (especially Chaps. 49 to 62) and in standard texts such as those by Joachain [38], Rodberg, and If the series converges, then the solution .r/ is approached Thaler [39], and Goldberger and Watson [40]. Owing esby the expansion. When the Schrödinger equation is cast as pecially to the wide variety of specialized techniques for an integral equation for scattering in a potential, this iteration solving integral equations, we briefly survey only a few of scheme leads to the Born series, the first term of which is the the most-frequently applied methods. incident, unperturbed wave, and the second term is usually referred to simply as the Born approximation. Integral Transforms Certain classes of integral equations may be solved using inSeparable Kernels tegral transforms such as the Fourier or Laplace transforms. If the kernel is separable, i.e., These integral transforms typically have the form n X Z 0 K.r; r / D fk .r/gk .r 0 / ; (8.89) (8.81) f .x/ D dx 0 K.x; x 0 /g.x 0 / ; kD1
where f .x/ is the integral transform of g.x 0 / by the ker- where n is finite, then substitution into the prototype integral nel K.x; x 0 /. Such a pair of functions is the solution of equation (8.82) yields the Schrödinger equation (spatial wave function) and its Z n X Fourier transform (momentum representation wave func.r/ D .r/ C fk .r/ dr 0 g.r 0 / .r 0/ : (8.90) tion). Arfken [10], Morse, and Feshbach [12], and Courant kD1
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Multiplying by fk .r/, integrating over r, and rearranging 8.3 Computational Linear Algebra yields the set of algebraic equations Previous sections of this chapter dealt with interpolation, n X differential equations, and related topics. Generally, discj D bj C aj k ck ; (8.91) cretization methodologies lead to classes of algebraic equakD1 tions. In recent decades, enormous progress has been made where in developing algorithms for solving linear-algebraic equaZ tions [41]. Many of the most widely adopted computational (8.92) linear algebra routines are available through Netlib [42], a ck D dr 0 gk .r 0 / .r 0/ ; Z portal developed to facilitate the distribution of such softbk D drfk .r/.r/ ; (8.93) ware for use in scientific computation. These routines include packages such as the Basic Linear Algebra Subprograms Z aj k D drgj .r/fk .r/ ; (8.94) (BLAS) [42, 43], which performs vector addition, dot products, matrix multiplications, etc., and the Linear Algebra or, if c and b denote vectors, and A denotes the matrix of Package (LAPACK, and ScaLAPACK its distributed memory implementation) [42, 44], which is used for solving linear constants aj k , 1 (8.95) systems of equations, eigenvalue problems, factorizations, c D .1 A/ b : and decompositions, etc. The eigenvalues are the roots of the determinantal equation. Here, we discuss methods for solving systems of equaSubstituting these into .1 A/c D 0 yields the constants tions such as ck that determine the solution of the original equation. This a11 x1 C a12 x2 C C a1n xn D b1 ; derivation may be found in the text by Arfken [10], along with an explicit example. Even if the kernel is not exactly a21 x1 C a22 x2 C C a2n xn D b2 ; separable, if it is approximately so, then this procedure can :: yield a result that can be substituted into the original equation : as a first step in an iterative solution. am1 x1 C am2 x2 C C amn xn D bm : (8.99)
Numerical Integration Perhaps the most straightforward method of solving an integral equation is to apply a numerical integration formula such as Gaussian quadrature. An equation of the form Z (8.96) .r/ D dr 0 K.r; r 0 /.r 0 / can be approximated as .rj / D
n X
wk K.rj ; rk0 /.rk / ;
(8.97)
kD1
where wk are quadrature weights, if the kernel is well behaved. However, such an approach is not without pitfalls. In light of the previous section, this approach is equivalent to replacing the integral equation by a set of algebraic equations. In this example, we have j
D
n X
Mj k k ;
(8.98)
kD1
so that the solution of the equation is found by inverting the matrix M . Since there is no guarantee that this matrix is not ill conditioned, the numerical procedure may not produce meaningful results. In particular, only certain classes of integral equations and kernels will lead to stable solutions.
In these equations, aij and bi form the set of known quantities, and xi must be determined. The solution to these equations can be found if they are linearly independent. Numerically, problems can arise due to truncation and round-off errors that lead to an approximate linear dependence [45]. In this case, the set of equations are approximately singular, and special methods must be invoked. Much of the complexity of modern algorithms comes from minimizing the effects of such errors. For relatively small sets of nonsingular equations, direct methods in which the solution is obtained after a definite number of operations can work well. However, for very large systems iterative techniques are preferable [46]. A great many algorithms are available for solving Eq. (8.99), depending on the structure of the coefficients. For example, if the matrix of coefficients A is dense, using Gaussian elimination takes 2n3 =3 operations; if A is also symmetric and positive definite, using the Cholesky algorithm takes a factor of 2 fewer operations. If A is triangular, that is, either zero above the diagonal or zero below the diagonal, we can solve the above system by simple substitution in only n2 operations. For example, if A arises from solving certain elliptic partial differential equations, such as Poisson’s equation, then Ax D b can be solved using multigrid methods in only n operations. We shall outline below how to solve Eq. (8.99) using elementary Gaussian elimination. More advanced methods,
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such as the conjugate gradient, generalized minimum residuals, and the Lanczos method are treated elsewhere [47]. To solve Ax D b, we first use Gaussian elimination to factor the matrix A as PA D LU , where L is lower triangular, U is upper triangular, and P is a matrix that permutes the rows of A. Then we solve the triangular system Ly D Pb and Ux D y. These last two operations are easily performed using standard linear algebra libraries. The factorization PA D LU takes most of the time. Reordering the rows of A with P is called pivoting and is necessary for numerical stability. In the standard partial pivoting scheme, L has 1s on its diagonal and other entries bounded in absolute value by 1. The simplest version of Gaussian elimination involves adding multiples of one row of A to others to zero out subdiagonal entries, and overwriting A with L and U . We first describe the decomposition of PA into a product of upper and lower triangular matrices,
The algorithm should now be clear. At the i-th row, ! j 1 X 1 0 Lij D Ujj Aij Li k Ukj ; j i 1 kD1
Uij D A0ij
i 1 X
Li k Ukj ;
j i :
(8.106)
kD1
We observe from the first line of these equations that the algorithm may run into numerical inaccuracies if any Ujj becomes very small. Now U11 D A011 , while in general Ui i D A0i i . Thus the absolute values of the Ui i are maximized if the rows are rearranged so that the absolutely largest elements of A 0 in each column lie on the diagonal. Note that the solutions are unchanged by permuting the rows (same equations, different order). The LU decomposition can now be used to solve the system. This relies on the fact that the inversion of a triangular (8.100) matrix is a simple process of back substitution. We replace A 0 D LU ; Eq. (8.99) by two systems of equations. Written out in full, where the matrix A 0 is defined by A 0 D PA. Writing out the the equations for a typical column of y look like indices, we obtain L11 y1 D b10 ; min.i;j / X L21 y1 C L22 y2 D b20 ; Li k Ukj : (8.101) A0ij D L31 y1 C L32 y2 C L33 y3 D b30 ; kD1 :: We shall make the choice : ; (8.107) (8.102) where the vector b0 is p 0 D Pb. Thus, from successive rows, we obtain y1 ; y2 ; y3 ; : : : in turn These equations have the remarkable property that the eleU11 x1 D y1 ; ments A0ij of each row can be scanned in turn, writing Lij 0 and Uij into the locations Aij as we go. At each position U12 x1 C U22 x2 D y2 ; .i; j /, only the current A0ij and already-calculated values of U13 x1 C U23 x2 C U33 x3 D y3 ; Li 0 j 0 and Ui 0 j 0 are required. To see how this works, consider :: the first few rows. If i D 1, : ; (8.108) Li i D 1 :
A01j D U1j ;
(8.103) and from successive rows of the latter, we obtain x1 ; x2 ; x3 ; : : : in turn. defining the first row of L and U . The U1j are written over Software libraries (Netlib [42], described above) also exthe A01j , which are no longer needed. If i D 2, ists for evaluating all the error bounds for dense and band matrices. Gaussian elimination with pivoting is almost alj D1 A021 D L21 U11 ; ways numerically stable, so the error bound one expects from A02j D L21 U1j C U2j ; j 2 : (8.104) solving these equations is of the order of n, where is related to the condition number of the matrix A. A good disThe first line gives L21 and the second U2j , in terms of exist- cussion of errors and conditioning is given in [3–5]. ing elements of L and U . The U2j and L21 are written over the A02j . (Remember that Li i D 1 by definition.) If i D 3, A031 D L31 U11 ;
j D1
A032 D L31 U12 C L32 U22 ;
j D2
8.4 Monte Carlo Methods
Owing to the continuing rapid development of computational D L31 U1j C L32 U2j C U3j ; j 3; (8.105) facilities and the ever-increasing desire to perform ab initio calculations, the use of Monte Carlo methods is becoming yielding in turn L31 ,L32 , and U3j , which are written over A03j . widespread as a means to evaluate previously intractable A03j
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multidimensional integrals and to enable complex modeling and simulation. For example, a wide range of applications broadly classified as quantum Monte Carlo have been used to compute, for example, the ground-state eigenfunctions of simple molecules. Also, guided random walks have found application in the computation of Green’s functions, and variables chosen randomly, subject to particular constraints, have been used to mimic the electronic distribution of atoms. The latter application, used in the classical trajectory Monte Carlo technique (CTMC) described in Chap. 62, allows the statistical quasiquantal representation of ion–atom collisions. CTMC is akin to another class of Monte Carlo simulations, classical molecular dynamics, which is used to describe systems ranging from molecules to solids as being composed of atoms whose movement is governed by classical mechanics subject to quantum mechanically derived potentials. Here, we summarize the basic tools needed in these methods and how they may be used to produce specific distributions and make tractable the evaluation of multidimensional integrals with complicated boundaries. Detailed descriptions of these methods can be found in [3–5, 18, 48].
8.4.1 Random Numbers An essential ingredient of any Monte Carlo procedure is the availability of a computer-generated sequence of random numbers that is not periodic and is free of other significant statistical correlations. Often, such numbers are termed pseudorandom or quasirandom, in distinction to truly random physical processes. While the quality of random number generators supplied with computers has greatly improved over time, it is important to be aware of the potential dangers that can be present. For example, many systems are supplied with a random number generator based on the linear congruential method. Typically, a sequence of integers n1 ; n2 ; n3 ; : : : is first produced between 0 and N 1 by using the recurrence relation ni C1 D .ani C b/ mod N ;
0 i < N 1 ; (8.109)
been developed, for example, the widely used tests published by L’Ecuyer and Simard [49]. The need for such tests has grown in the present era of parallel computing, relevant not only to use of parallel random number generators on supercomputers and clusters but also the use of personal computers and workstations employing multicore/multithread, computation. In fact, the scale of Monte Carlo calculations possible on contemporary platforms has exposed deficiencies of many commonly used methods of generating random numbers, leading to development of more robust techniques. These new methods seek to ensure the quality of random numbers, first of all for large sequences within a single stream, as well as for multiple streams (other threads on a single processing unit or across nodes within a cluster). A widely used set of parallel random number generators that pass robust tests was developed and described by Srinivasan, Mascagni, and Ceperley [50], for example.
8.4.2 Distributions of Random Numbers
8 Most distributions of random numbers begin with sequences generated uniformly between a lower and an upper limit, and are therefore called uniform deviates. However, it is often useful to draw the random numbers from other distributions, such as the Gaussian, Poisson, exponential, gamma, or binomial distributions. These are particularly useful in modeling data or supplying input for an event generator or simulator. In addition, as described below, choosing the random numbers according to some weighting function can significantly improve the efficiency of integration schemes based on Monte Carlo sampling. Perhaps the most direct way to produce the required distribution is the transformation method. If we have a sequence of uniform deviates x on .0; 1/ and wish to find a new sequence y that is distributed with probability given by some function f .y/, it can be shown that the required transformation is given by 31 2 y Z (8.110) y.x/ D 4 f .y/dy 5 : 0
where a; b; N , and the seed value n0 are positive integers. Real numbers between 0 and (strictly) 1 are then obtained by dividing by N . The period of this sequence is at most N and depends on the judicious choice of the constants, with N being limited by the word size of the computer. A user who is unsure whether the character of the random numbers generated on a particular computer platform is proper can perform additional randomizing shuffles or use a portable random number generator; both procedures are described in detail in the texts by Knuth [6] and Press et al. [3–5], for example. In addition, tests of random number sequences have
Evidently, the indefinite integral must be both known and invertible, either analytically or numerically. Since this is seldom the case for distributions of interest, other less direct methods are most often applied. However, even these other methods often rely on the transformation method as one stage of the procedure. The transformation method may also be generalized to more than one dimension [3–5]. A more widely applicable approach is the rejection method, also known as von Neumann rejection. In this case, if one wishes to find a sequence y distributed according to f .y/, one first chooses another function fQ.y/, called the
146
comparison function, which is everywhere greater than f .y/ on the desired interval. In addition, a way must exist to generate y according to the comparison function, such as use of the transformation method. Thus, the comparison function must be simpler or better known than the distribution to be found. One simple choice is a constant function that is larger than the maximum value of f .y/, but choices that are closer to f .y/ will be much more efficient. To proceed, y is generated uniformly according to fQ.y/, and another deviate x is chosen uniformly on .0; 1/. One then rejects or accepts y, depending on whether x is greater than or less than the ratio f .y/=fQ.y/, respectively. The fraction of trial numbers accepted depends on the ratio of the area under the desired function to that under the comparison function. Clearly, the efficiency of this scheme depends on how few of the numbers initially generated must be rejected, and, therefore, on how closely the comparison function approximates the desired distribution. The Lorentzian distribution, for which the inverse definite integral is known (the tangent function), is a good comparison function for a variety of bellshaped distributions such as the Gaussian (normal), Poisson, and gamma distributions. Especially for distributions that are functions of more than one variable and possess complicated boundaries, the rejection method is impractical, and the transformation method simply inapplicable. In the 1950s, a method to generate distributions for such situations was developed and applied in the study of statistical mechanics, where multidimensional integrals (e.g., the partition function) must often be solved numerically. It is known as the Metropolis algorithm. This procedure, or its variants, has more recently been adopted to aid in the computation of eigenfunctions of complicated Hamiltonians and scattering operators. In essence, the Metropolis method generates a random walk through the space of the dependent variables, and in the limit of a large number of steps in the walk, the points visited approximate the desired distribution. In its simplest form, the Metropolis method generates this distribution of points by stepping through this space, most frequently taking a step downhill but sometimes taking a step uphill. That is, given a set of coordinates q and a desired distribution function f .q/, a trial step is taken from the i-th configuration q i to the next, depending on whether the ratio f .q i C1/=f .q i / is greater or less than 1. If the ratio is greater than 1, the step is accepted, but if it is less than 1, the step is accepted with a probability given by the ratio.
D. Schultz and M. R. Strayer
gion is proportional to the volume, or definite integral, of the function defining that region. Although this idea is as true in one dimension as it is in n, unless there is a large number (large could be as little as three) of dimensions or the boundaries are quite complicated, the numerical quadrature schemes described previously are more accurate and efficient. However, since the Monte Carlo approach is based on just sampling the function at representative points rather than evaluating the function at a large number of finely spaced quadrature points, its advantage for very large problems is apparent. For simplicity, consider the Monte Carlo method for integrating a function of only one variable; the generalization to n dimensions being straightforward. If we generate N random points uniformly on .a; b/, then in the limit of large N , the integral is Zb
s hf 2 .x/i hf .x/i2 ; (8.111) N
1 f .x/dx hf .x/i ˙ N
a
where hf .x/i
N 1 X f .xi / N i D1
(8.112)
is the arithmetic mean. The probable error given is appropriately a statistical one rather than a rigorous error bound and is the one standard error limit. From this, one can see that the error decreases only as N 1=2 , more slowly than the rate of decrease for the quadrature schemes based on interpolation. Also, the accuracy is greater for relatively smooth functions, since the Monte Carlo generation of points is unlikely to sample narrowly peaked features of the integrand well. To estimate the integral of a multidimensional function with complicated boundaries, find an enclosing volume and generate points uniformly randomly within it. Keeping the enclosing volume as close as possible to the volume of interest minimizes the number of points that fall outside, and therefore increases the efficiency of the procedure. The Monte Carlo integral is related to the techniques for generating random numbers according to prescribed distributions described in Sect. 8.4.2. If we consider a normalized distribution w.x/, known as the weight function, then with the change of variables defined by Zx w.x 0 /dx 0 ;
y.x/ D
(8.113)
a
8.4.3 Monte Carlo Integration The basic idea of Monte Carlo integration is that if a large number of points is generated uniformly randomly in some n-dimensional space, the number falling inside a given re-
the Monte Carlo estimate of the integral becomes Zb a
1 f Œx.y/ f .x/dx ; N wŒx.y/
(8.114)
8 Computational Techniques
assuming that the transformation is invertible. Choosing w.x/ to behave approximately as f .x/ allows a more efficient generation of points within the boundaries of the integrand. This occurs since the uniform distribution of points y results in values of x distributed according to w and, therefore, close to f . This procedure, generally termed the reduction of variance of the Monte Carlo integration, improves the efficiency of the procedure to the extent that the transformed function f =w can be made smooth, and that the sampled region is as small as possible but still contains the volume to be estimated.
References 1. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (2002) 2. Burden, R.L., Faires, J.D.: Numerical Analysis. Thomson Brooks/Cole, Belmont (2005) 3. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes, the Art of Scientific Computing. Cambridge Univ. Press, Cambridge (2007) 4. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes in Fortran 77: The Art of Scientific Computing. Cambridge Univ. Press, Cambridge (1992) 5. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in Fortran 90: The Art of Parallel Scientific Computing. Cambridge Univ. Press, Cambridge (1996) 6. Knuth, D.E.: The Art of Computer Programming, Volume 2: Seminumerical Algorithms. Addison-Wesley, Boston (1998) 7. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Applied Mathematics Series, vol. 55. National Bureau of Standards/Dover, Washington/New York (1972). NIST Digital Library of Mathematical Functions: dlmf.nist.gov 8. Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products. Elsevier, Amsterdam (2007) 9. Zwillinger, D.: CRC Standard Mathematical Tables and Formulae. CRC, Boca Raton (2012) 10. Arfken, G.B., Weber, H.J., Harris, F.E.: Mathematical Methods for Physicists. Elsevier, Amsterdam (2013) 11. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge Univ. Press, Cambridge (2006) 12. Morse, P.M., Feshbach, H.: Methods of Theoretical Physics. McGraw-Hill, Boston (1999) 13. Courant, R., Hilbert, D.: Methods of Mathematical Physics. Interscience, New York (2009) 14. Margenau, H., Murphy, G.M.: The Mathematics of Physics and Chemistry. Van Nostrand, New York (1976) 15. Hamming, R.W.: Numerical Methods for Scientists and Engineers. McGraw-Hill, New York (1973) 16. Jeffreys, H., Jeffreys, B.S.: Methods of Mathematical Physics. Cambridge Univ. Press, Cambridge (1999) 17. Bender, C.M., Orszag, S.: Advanced Mathematical Methods for Scientists and Engineers. Springer, New York (1999) 18. Koonin, S.E., Meredith, D.C.: Computational Physics. AddisonWesley, Reading (1995) 19. Wikipedia: List of numerical-analysis software (2021). https://en. wikipedia.org/wiki/List_of_numerical_analysis_%software
147 20. Computer Physics Communications Program Library: www.cpc.cs.qub.ac.uk/ 21. Huber, P.J., Ronchetti, E.M.: Robust Statistics. Wiley, Hoboken (2009) 22. Young, H.D.: Statistical Treatment of Experimental Data: An Introduction to Statistical Methods. Waveland Press, Prospect Heights (1996) 23. Bevington, P.R., Robinson, D.K.: Data Reduction and Error Analysis for the Physical Sciences. McGraw-Hill, New York (2003) 24. Champeney, D.C.: Fourier Transforms and Their Physical Applications. Academic Press, London (1988) 25. Elliott, D.F., Rao, K.R.: Fast Transforms: Algorithms, Analyses, Applications. Academic Press, London (1982) 26. FFTW: https://en.wikipedia.org/wiki/FFTW; www.fftw.org 27. Lambert, J.D.: Numerical Methods for Ordinary Differential Equations: The Initial Value Problem. Chichester, New York (2000) 28. Shampine, L.F.: Numerical Solution of Ordinary Differential Equations. Chapman Hall, New York (1994) 29. Butcher, J.: The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods. Wiley, New York (1987) 30. Hall, G., Watt, J.M.: Modern Numerical Methods for Ordinary Differential Equations. Clarendon, Oxford (1976) 31. Gladwell, I., Wait, R. (eds.): A Survey of Numerical Methods for Partial Differential Equations. Clarendon, Oxford (1979) 32. Rektorys, K.: Variational Methods in Mathematics, Science, and Engineering. Springer Netherlands, Amsterdam (2012) 33. Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, R.J.: Concepts and Applications of Finite Element Analysis. Wiley, New York (2002) 34. Nürnberger, G.: Approximation by Spline Functions. Springer, Berlin, Heidelberg (1989) 35. DeBoor, C.: Practical Guide to Splines. Springer, Berlin (2013) 36. Light, J.C., Carrington, T.: Discrete variable representation and their utilization. Adv. Chem. Phys. 114, 263 (2000) 37. Light, J.C., Hamilton, I.P., Lill, J.V.: Generalized discrete variable approximation in quantum mechanics. J. Chem. Phys. 82, 1400 (1985) 38. Joachain, C.J.: Quantum Collision Theory. North Holland, Amsterdam (1987) 39. Rodberg, L.S., Thaler, R.M.: Introduction to the Quantum Theory of Scattering. Academic Press, New York (1970) 40. Goldberger, M.L., Watson, K.M.: Collision Theory. Dover, Mineola (2004) 41. Ciarlet, P.G.: Introduction to Numerical Linear Algebra and Optimisation. Cambridge Univ. Press, Cambridge (2001) 42. Netlib Repository at UTK and ORNL: www.netlib.org 43. en.wikipedia.org/wiki/Basic_Linear_Algebra_Subprogams 44. Wikipedia: LAPACK (2021). https://en.wikipedia.org/wiki/LAPACK 45. Golub, G., Van Loan, C.: Matrix Computations. Johns Hopkins Univ. Press, Baltimore (2013) 46. Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. Springer, Cham (2016) 47. George, A., Liu, J.: Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs (1981) 48. Kalos, M.H., Whitlock, P.A.: The Basics of Monte Carlo Methods. Wiley, Weinheim (2008) 49. L’Ecuyer, P., Simard, R.: TestU01: A C library for empirical testing of random number gen-erators. ACM Trans. Math. Softw. 33(4), 22 (2007) 50. Srinivasan, A., Mascagni, M., Ceperley, D.: Testing parallel random number generators. Parallel Comput. 29, 69 (2003)
8
148
D. Schultz and M. R. Strayer David Schultz David Schultz is Vice President for Research at Northern Arizona University and has held research and administrative positions at the University of North Texas, Oak Ridge National Laboratory, and the University of Tennessee. His interests are in computational atomic physics, using both molecular dynamics simulation and quantum mechanical, discrete variable representations, and in applications in plasma science and astrophysics.
9
Hydrogenic Wave Functions Robert N. Hill
Contents
Keywords
9.1 9.1.1 9.1.2 9.1.3
Schrödinger Equation Spherical Coordinates . Parabolic Coordinates . Momentum Space . . . .
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9.2
Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.3 9.3.1 9.3.2
The Coulomb Green’s Function . . . . . . . . . . . . . . 154 The Green’s Function for the Schrödinger Equation . . . 155 The Green’s Function for the Dirac Equation . . . . . . . 156
9.4 9.4.1 9.4.2 9.4.3 9.4.4
Special Functions . . . . . . . . . . . Confluent Hypergeometric Functions Laguerre Polynomials . . . . . . . . . Gegenbauer Polynomials . . . . . . . Legendre Functions . . . . . . . . . . .
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hydrogenic wave function Schrödinger equation Dirac equation Coulomb potential the Coulomb Green’s function
149 149 150 152
156 157 160 163 163
9.1 Schrödinger Equation The nonrelativistic Schrödinger equation for a hydrogenic ion of nuclear charge Z in atomic units is Z 1 r2 .r/ D E .r/ : (9.1) 2 r
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.1.1 Spherical Coordinates Abstract
This chapter summarizes the solutions of the one-electron nonrelativistic Schrödinger equation, and the one-electron relativistic Dirac equation, for the Coulomb potential. The standard notations and conventions used in the mathematics literature for special functions have been chosen in preference to the notations customarily used in the physics literature whenever there is a conflict. This has been done to facilitate the use of standard reference works such as Abramowitz and Stegun [1], the Bateman project [2, 3], Gradshteyn and Ryzhik [4], Jahnke and Emde [5], Luke [6, 7], Magnus, Oberhettinger, and Soni [8], Olver [9], Szegö [10], and the new NIST Digital Library of Mathematical Functions project, which has prepared a hardcover update [11] of Abramowitz and Stegun [1] and an online digital library of mathematical functions [12]. The section on special functions contains many of the formulas which are needed to check the results quoted in the previous sections, together with a number of other useful formulas. It includes a brief introduction to asymptotic methods. References to the numerical evaluation of special functions are given.
The separable solutions of Eq. (9.1) in spherical coordinates are (9.2) .r/ D Y`m .; /R` .r/ ; where Y`m .; / is a spherical harmonic as defined by Edmonds [13] and R` .r/ is a solution of the radial equation 2 d `.` C 1/ 1 d2 Z C R` .r/ D ER` .r/ : 2 dr 2 r dr r2 r (9.3) The general solution to Eq. (9.3) is
R` .r/ D r ` exp.ikr/ŒA1 F1 .aI cI z/ C BU.a; c; z/ ; (9.4) where 1 F1 and U are the regular and irregular solutions of the confluent hypergeometric equation defined in Eqs. (9.130) and (9.131) below, and p (9.5) k D 2E ; a D ` C 1 ik 1 Z ;
(9.6)
c D 2` C 2 ;
(9.7)
z D 2ikr :
(9.8)
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_9
149
150
R. N. Hill
.`/ A and B are arbitrary constants. The solution given in The coefficients ak;m in Eq. (9.15) are calculated recursively `1 singularity at r D 0 unless B D 0 from Eq. (9.4) has an r or a is a non-positive integer. The leading term for small r .2` C 2m C k C 1/ .`/ is proportional to r ` when B D 0 and/or a is a non-positive ak;m D 32.2k C m/.2` C m C 2k C 1/ integer. The large r behavior of the solution for Eq. (9.4) fol1 lows from Eqs. (9.134), (9.135), and (9.164) below. Bound .2` C 2m C k 1/ state solutions, with energy h .`/ .2` C 2m C k 1/ak1;m 1 i E D Z 2 n2 (9.9) .`/ C32.k m C 1/.2` C m k/ak;m1 ; (9.16) 2
are obtained when a D n C ` C 1 where n > ` is the prin- starting with the initial condition cipal quantum number. The properly normalized bound state .`/ a0;0 D 1 : solutions are s 2Z Rn;` .r/ D 2 n exp
Z.n ` 1/Š 2Zr ` .n C `/Š n
Zr .2`C1/ 2Zr Ln`1 ; n n
(9.10)
.2`C1/
where Ln`1 is the generalized Laguerre polynomial defined in Eq. (9.187). The relation in Eq. (9.188) shows that 1 F1 and U are linearly dependent in this case, so that Eq. (9.4) is no longer the general solution of Eq. (9.3). A linearly independent solution for this case can be obtained by .2`C1/ replacing the Ln`1 .2Zr=n/ in Eq. (9.10) by the second .2`C1/ (irregular) solution Mn`1 .2Zr=n/ of the Laguerre equation (see Eqs. (9.194), (9.196), and (9.197)). The first three Rn;` are R1;0 .r/ D 2Z 3=2 exp.Zr/ ; 3=2 1 1 .2 Zr/ exp Zr ; R2;0 .r/ D Z 2 2 3=2 1=2 1 1 1 Zr exp Zr : Z R2;1 .r/ D 2 3 2
(9.11) (9.12)
(9.17)
The expansion Eq. (9.14) converges uniformly in r for r in any bounded region of the complex r-plane. However, it converges fast enough so that a few terms give a good description of Rn;` only if r is small. The square root in Eq. (9.14) has not been expanded in inverse powers of n because it has a branch point at 1=n D 1=` which would reduce the radius of convergence of the expansion to 1=`. In some cases, large n expansions of matrix elements can be obtained by inserting Eq. (9.14) for Rn;` and integrating term by term; examples can be found in Drake and Hill [15]. An asymptotic expansion in powers of 1=n, which is valid from r equal to an arbitrary fixed positive number through the turning point at r D 2n2 =Z out to r D 1, can be assembled from Eqs. (9.133), (9.166)–(9.181), and (9.188) below. The Rn;` are not a complete set because the continuum has been left out. The Sturmian functions k;` , given by s ˇ 3 kŠ
k;` .ˇI r/ D .ˇr/` eˇr=2 .k C 2` C 3/ .2`C2/
Lk
.ˇr/ ;
(9.18)
(9.13) do form a complete orthonormal set. The positive constant ˇ, which is independent of k and `, sets the length scale for Additional explicit expressions, together with graphs of the basis set Eq. (9.18). some of them, can be found in Pauling and Wilson [14]. The Rn;` can be expanded in powers of 1=n [15] Rn;` .r/ D
2
9.1.2 Parabolic Coordinates
1=2
2Z .n C `/Š .n ` 1/Šn2`C4 1 h i X .`/ r 1=2 gk .8Zr/1=2 n2k ;
The Schrödinger equation (9.1) is separable in parabolic coordinates , , , which are related to spherical coordinates (9.14) r, , via
kD0 .`/
where the functions gk .z/ are finite linear combinations of Bessel functions .`/ gk .z/
Dz
3k
k X mD0
.`/ ak;m J2`C2mCkC1 .z/ :
D r C z D rŒ1 C cos./ ;
(9.19)
D r z D rŒ1 cos./ ;
(9.20)
D:
(9.21)
This separability in a second coordinate system is related to (9.15) the existence of a hidden O.4/ symmetry, which is also responsible for the degeneracy of the bound states [16, 17].
9
Hydrogenic Wave Functions
151
The solutions in parabolic coordinates are particularly con- where k, k , and k are the spherical coordinates of k. C venient for derivations of the Stark effect and the Rutherford can be split into an incoming plane wave and an outgoing scattering cross section. The separable solutions of Eq. (9.1) spherical wave with the aid of Eq. (9.134) below in parabolic coordinates are .r/ D exp.im/N./&./ ;
C .r/
(9.22)
D
in .r/
C
out .r/ ;
(9.31)
where
where 1 N./ D jmj=2 exp ik1 2 ŒA1 F1 .a1 I cI ik1 / CBU .a1 ; c; ik1 / ; 1 &./ D jmj=2 exp ik2 2 ŒC 1 F1 .a2 I cI ik2 /
(9.23)
CDU .a2 ; c; ik2 / ;
(9.24)
with 1 F1 and U defined in Eqs. (9.130), (9.131) below, and p k1 D ˙k2 D ˙ 2E ; 1 a1 D .jmj C 1/ ik11 ; 2 1 a2 D .jmj C 1/ ik21 .Z / ; 2 c D jmj C 1 :
(9.25) (9.26) (9.27) (9.28)
A, B, C , and D are arbitrary constants; is the separation constant. An important special case is the well-known Coulomb function
1 1 1 k Z C ik r C .r/ D 1 ik Z exp 2
1 1 F1 ik ZI 1I i.kr k r/ ; (9.29) which is obtained by orienting the z-axis in the k direction and taking m D 0, k1 D k2 D jkj, D Z C 12 ijkj. C is normalized to unit incoming flux [see Eq. (9.34) below]. In applications, Z is often replaced by Z1 Z2 , so that the Coulomb potential in Eq. (9.1) becomes CZ1 Z2 =r. Equation (9.232), the addition theorem for the spherical harmonics [13, p. 63 Eq. 4.6.6], and the D c D 1 special case of Eq. (9.163) below can be used to expand C into an infinite sum of solutions of the form Eq. (9.2)
1 ` X X ` C 1 ik 1 Z C .r/ D 4 .2` C 1/Š `D0 mD`
1 D exp ik r k 1 Z 2
1 U ik ZI 1I i.kr k r/ ; (9.32)
1 1 ik Z 1 exp ikr k 1 Z out .r/ D .ik 1 Z/ 2
1 (9.33) U 1 ik ZI 1I i.kr k r/ : in .r/
The functions in and out can be expanded for kr k r large with the aid of Eq. (9.164). The result is in .r/
exp ik r ik 1 Z ln.kr k r/ "
#2 1 X .i/n ik 1 Z C n nŠ .ik 1 Z/ nD0
.kr k r/n ;
i 1 ik 1 Z out .r/ .ik 1 Z/.kr k r/
exp ikr ik 1 Z ln.kr k r/ "
#2 1 n X i 1 ik 1 Z C n nŠ .1 ik 1 Z/ nD0 .kr k r/n :
9 (9.34)
(9.35)
Because Eq. (9.1) is an elliptic partial differential equation, its solutions must be analytic functions of the Cartesian coordinates (except at r D 0, where the solutions have cusps). The n D 0 special case of Eq. (9.138) shows that in and out are logarithmically singular at k r D kr. Thus in and out are not solutions to Eq. (9.1) at k r D kr. The logarithmic singularity cancels when in and out are added to form C , which is a solution to Eq. (9.1). Bound state solutions, with energy 1 E D Z 2 .n1 C n2 C jmj C 1/2 ; 2
(9.36)
are obtained when a1 D n1 and a2 D n2 where n1 and 1 .k ; k /Y`m .; / .2ik/` e k Z=2 Y`m n 2 are non-negative integers. The properly normalized bound
state solutions, which can be put into one–one correspon r ` eikr 1 F1 ` C 1 ik 1 ZI 2` C 2I 2ikr ; (9.30) dence with the bound state solutions in spherical coordinates,
152
R. N. Hill
are
s n1 ;n2 ;m .; ; /
where
D
ˇ 2jmjC4 n1 Šn2 Š 2 Z.n1 C jmj/Š.n2 C jmj/Š 1 exp im ˇ. C / 2
The volume element, which is easily obtained via the same series of transformations, is dV D r43 dr4 d˝4 ; 2
d˝4 D sin ./d sin./dd :
(9.47) (9.48)
The four-dimensional spherical harmonics [2, Vol. 2, .ˇ/L.jmj/ ./jmj=2 Ln.jmj/ n2 .ˇ/ ; 1 Chap. XI] are (9.37) s n.n ` 1/Š ` 1 (9.38) ˇ D Z.n1 C n2 C jmj C 1/ : Yn;`;m .; ; / D 2`C1 `Š sin ./ 2 .n C `/Š `C1 Œcos./Y`m .; / ; Cn`1
9.1.3 Momentum Space
(9.49)
`C1 is a Gegenbauer polynomial and n ` C 1 is where Cn`1 The nonrelativistic Schrödinger equation (9.1) becomes the an integer. They have the orthonormality property integral equation Z Z 0 .; ; /Yn0 ;`0 ;m0 .; ; /d˝4 Yn;`;m .p / 3 0 Z 1 2 d p D E.p/ (9.39) p .p/ 2 2 2 2 .p p 0 / D ın;n0 ı`;`0 ım;m0 : (9.50) in momentum space. Its solutions are related to the solutions Equations (9.229) and (9.230) with D 1, (9.231), and the in coordinate space via the Fourier transforms addition theorem for the three dimensional spherical harZ 3=2 3 monics Y `m can be used to show that (9.40) exp.ip r/.p/d p ; .r/ D .2 / ˚
1 Z 1 2 cos./ cos 0 sin./ sin 0 cos 0 t C t 2 3=2 3 .p/ D .2 / (9.41) exp.ip r/ .r/d r : ` 1 X n1 X X 2 2 n1 D t Yn;`;m .; ; / A trick of Fock’s [16, 18] can be used to expose the hidn nD1 `D0 mD` den O.4/ symmetry of hydrogen and construct the bound
0 0 0 ; ; Yn;`;m state solutions to Eq. (9.39). Let p, p , p and p 0 , p0 , p0 0 (9.51) be the spherical coordinates of p p and p . Change variables 0 0 0 0 0 from p, p to , via p D 2E tan.=2/ and p D holds for jtj < 1, where is the angle between p and p . p 0 0 0 0 Multiply both sides of Eq. (9.51) by Yn;`;m . ; ; /d˝40 2E tan. =2/. This brings Eq. (9.39) to the form (where d˝40 is d˝4 with , , replaced by 0 , 0 , 0 ) and p
2 2 Z 1 2E sec4 Œ=2.p/ use the orthogonality relation Eq. (9.50). The result can be 2 0
Z 4 0 0 0 rearranged to the form sec . =2/.p / sin . /d sin p dp dp D ; 2 2Œcos./ cos.0 / C sin./ sin.0 / cos. 0 / 2 1 n1 (9.42) 2 n t Yn;`;m .; ; / Z Yn;`;m .0 ; 0 ; 0 / sin2 .0 /d0 sin./dd where 0 is the angle between p and p 0 . Equation (9.42) D : is solved by introducing spherical coordinates and spheri1 2Œcos./ cos.0 / C sin./ sin.0 / cos. 0 /t C t 2 (9.52) cal harmonics in four dimensions via a natural extension of the procedure used in three dimensions. Going to polar co- Analytic continuation can be used to show that Eq. (9.52) ordinates on x and y yields the cylindrical coordinates r2 , , is valid for all complex t despite the fact that Eq. (9.51) is z; the further step of going to polar coordinates on r2 and restricted to jtj < 1. Comparing the t D 1 case of Eq. (9.52) 1 2 2 z yields spherical coordinates r3 , , . If there is a fourth with Eq. (9.42) shows that E D 2 Z n in agreement with coordinate w, spherical coordinates in four dimensions are Eq. (9.9), and that obtained via the additional step of going to polar coordinates on r3 and w. The result is .p / D normalizing cos4 .=2/ factor (9.43) x D r4 sin./ sin./ cos./ ; Y .; ; / : (9.53) n;`;m
y D r4 sin./ sin./ sin./ ;
(9.44)
z D r4 sin./ cos./ ;
(9.45) Transforming from back to p brings these to the form
(9.46) .p/ D Y`m p ; p Fn;` .p/ ; (9.54)
w D r4 cos./ :
9
Hydrogenic Wave Functions
153
The solutions to Eq. (9.60) in spherical coordinates have the form ! G.r/ m
.; / .r/ D ; (9.63) iF .r/ m .; /
where the properly normalized radial functions are s 2.n ` 1/Š np ` Fn;` .p/ D 22`C2 n2 `Š Z 3 .n C `/Š Z
Z 2`C4
.n2 p 2 C Z 2 /`C2 2 2 n p Z2 `C1 Cn`1 : n2 p 2 C Z 2 The first three Fn;` are
r
!1=2 1
C 12 m Yj C 1 j 1 ;m 1 C B 2 2 2C B j j 2 C 1 B C: D m !1=2
B C
C 12 C m @ A Yj C 1 j 1 ;mC 1 2 2 2 2 C 1 0
4
2 Z ; Z 3 .p 2 C Z 2 /2
32 Z 4 4p 2 Z 2 F2;0 .p/ D p ; 3 Z 3 .4p 2 C Z 2 / Z 5p 128 F2;1 .p/ D p : 3 3 Z 3 .4p 2 C Z 2 / F1;0 .p/ D 4
where, for positive energy states, G.r/ is the radial part of the large component and iF .r/ is the radial part of the small component. For negative energy states, G.r/ is the radial part (9.55) of the small component and iF .r/ is the radial part of the large component. is the two component spinor
(9.56) (9.57)
(9.64)
(9.58) The relativistic quantum number is related to the total angular momentum quantum number j by The Fn;` satisfy the integral equation 1
D ˙ j C : (9.65) 1 2 Z 2
0 0 0 p C p 02 1 2 Z Q` p Fn;` .p/ Fn;` p p dp 2 p 2pp 0 Because j takes on the values 12 , 32 , 52 , : : :, is restricted to 0 the values ˙1, ˙2, ˙3, : : : The spinor m
obeys the useful D EFn;` .p/ ; relations (9.59) which can be obtained by inserting Eq. (9.54) in Eq. (9.39). Here Q` is the Legendre function of the second kind, which is defined in Eq. (9.233) below.
m rO m
D ;
L m
D . C
(9.66) 1/m
;
(9.67)
where rO D r=r and L D r p with p D ir. Equations (9.66), (9.67), and the identity i L The relativistic Dirac equation for a hydrogenic ion of nu(9.68) p D . r/ O rO p C r clear charge Z can be reduced to dimensionless form by using the Compton wavelength „=.mc/ for the length scale can be used to derive the radial equations, which are and the rest mass energy mc 2 for the energy scale. The result is d 1C Z˛ Z˛ C G.r/ 1 C E C F .r/ D 0 ; .r/ D E .r/ ; (9.60) i˛ r C ˇ dr r r r (9.69) where ˛ D e 2 =.„c/ is the fine structure constant, and ˛, ˇ d 1 Z˛ are the usual Dirac matrices C F .r/ 1 E G.r/ D 0 : dr r r ! ! (9.70) I 0 0 : (9.61) ; ˇD ˛D 0 I 0 Equations (9.158), (9.159), (9.161), and (9.162) below can be used to show that the general solution to Eqs. (9.69) and Here is a vector whose components are the two by two Pauli matrices, and I is the two by two identity matrix given (9.70) is by ! ! G.r/ D r exp.r/.1 C E/1=2 fAŒf2 .r/ C f1 .r/ 0 1 0 i x D ; y D ; CBŒf4 .r/ C f3 .r/g ; (9.71) 1 0 i 0 ! ! F .r/ D r exp.r/.1 E/1=2 fAŒf2 .r/ f1 .r/ 1 0 1 0 : (9.62) ; I D z D CBŒf4 .r/ f3 .r/g ; (9.72) 0 1 0 1
9.2 Dirac Equation
9
154
R. N. Hill
where
f1 .r/ D Z˛1 1 F1 .aI cI 2r/ ;
(9.73)
f2 .r/ D a1 F1 .a C 1I cI 2r/ ;
(9.74)
f3 .r/ D U .a; c; 2r/ ;
f4 .r/ D Z˛1 C U .a C 1; c; 2r/ ;
(9.75)
1=2
1=2
D .1 C E/ .1 E/ ;
1=2 D 1 C 2 Z 2 ˛ 2 ; 1
(9.76) (9.77) (9.78)
a D 1 C EZ˛ ;
(9.79)
c D 3 C 2 :
(9.80)
The formulae for the 2 S1=2 excited state, with n D 1, j D 12 ,
D 1, and for the 2 P1=2 excited state, with n D 1, j D 12 ,
D 1, can be written together. They are 1=2 1 1p 2 2 1Z ˛ ; (9.90) C E1; D 2 2 s Z 3 ˛ 3 .2E1; /.1 C E1; /
2 G1; .r/ D 2 2E1; 4E1; C1 2 2 2E1; 1 =2
1
e
Œ.2E1; 1/.2E1; C / 1 ; s Z 3 ˛ 3 .2E1; /.1 E1; /
2 F1; .r/ D 2 2E1; 4E1; C1
(9.91)
A and B are arbitrary constants. Because is in general not an integer, the solutions have a branch point at r D 0, and be2E 2 2 come infinite at r D 0 when D ˙1, which makes negative. 1 1; e 1 =2 The solutions for E < 1 and E > C1 are in the continuum, Œ.2E1; C 1/.2E1; C / 1 ; (9.92) which implies that one of the factors .1 C E/1=2 , .1 E/1=2 is real with the other imaginary. Square integrable solutions, where D =E . For the 2 P excited state, with n D 0, 1 1; 3=2 with energy j D 12 , D 2, the formulae are 1=2 r Z Z 2˛2 1 ; (9.81) En; D 1C 2 D 1 Z 2˛2 ; (9.93) E jZj 0;2 .n C 1 C / 4 s are obtained when a D n where n is a non-negative integer. Z 3 ˛ 3 .1 C E0;2 / 2E0;2 1 =2 G0;2 .r/ D e ; (9.94)
The properly normalized square integrable solutions are 2 .1 C 4E0;2 / s Gn; .r/ D Cn; .2r/ exp.r/.1 C En; /1=2 Z 3 ˛ 3 .1 E0;2 / 2E0;2 1 =2
.2/ F .1/ .r/ D e : (9.95)
0;2 gn; .r/ C gn; .r/ ; (9.82) 2 .1 C 4E0;2 / Fn; .r/ D Cn; .2r/ exp.r/.1 En; /1=2
.2/ .1/ gn; .r/ gn; .r/ ; (9.83) 9.3 The Coulomb Green’s Function
1=2 .1/ gn; .r/ D Z˛1 L.2C2/ .2r/ ; (9.84) n
1=2 The abstract Green’s operator for a Hamiltonian H is the .2/ gn; .r/ D .n C 2 C 2/ Z˛1 inverse G.E/ D .H E/1 . It is used to write the solution .2C2/ Ln1 .2r/ ; (9.85) to .H E/ji D ji in the form ji D Gji. It has the spectral s representation 24 nŠ Cn; D : (9.86) Z X 1 Z˛ .n C 3 C 2/ (9.96) jej ihej j : G.E/ D E j E 1 When n D 0, jZ˛ j D j j, and the value of whose sign j is the same as the sign of Z˛1 is not permitted. Also, The sum over j in Eq. (9.96) runs over all of the spectrum .2C2/ .2/ L1 .2r/ is counted as zero, so that g0; .r/ D 0. of H , including the continuum. For the bound state part of The eigenvalues and eigenfunctions for the first four states the spectrum, the numbers Ej and vectors jej i are the eigenfor Z > 0 will now be written out explicitly in terms of the values and eigenvectors of H . For the continuous spectrum, variable D Z˛r. For the 1 S1=2 ground state, with n D 0, jej ihej j is a projection valued measure [19]. The representaj D 12 , D 1, the formulae are tion Eq. (9.96) shows that G.E/ has first order poles at the p E0;1 D 1 Z 2 ˛ 2 ; (9.87) eigenvalues. The reduced Green’s operator (also known as the s generalized Green’s operator), which is the ordinary Green’s 4Z 3 ˛ 3 .1 C E0;1 / E0;1 1
G0;1 .r/ D e ; (9.88) operator with the singular terms subtracted out, remains finite .2 / .1 C 2E0;1 / when E is at an eigenvalue. It can be calulated from s @ 4Z 3 ˛ 3 .1 E0;1 / E0;1 1
.red/ F0;1 .r/ D e : (9.89) .Ek / D lim (9.97) G .2 / Œ.E Ek /G.E/ : E!Ek @E .1 C 2E0;1 /
9
Hydrogenic Wave Functions
The coordinate and momentum space representatives of the abstract Green’s operator are the Green’s functions. The nonrelativistic Coulomb Green’s function has been discussed by Hostler and Schwinger [20–22]. A unified treatment of the Coulomb Green’s functions for the Schrödinger and Dirac equations has been given by Swainson and Drake [23–25]. Reduced Green’s functions are discussed in the third of the Swainson–Drake papers, and in the paper of Hill and Huxtable [26].
155
where # is the angle between r and r 0 . These expansions, and other integral representations, can be found in [20–25]. The partial wave expansion of G .S/ is
G .S/ r; r 0 I E X .S/
0 0 D g` r; r 0 I Y`m .; /Y`m ; : (9.105) `;m .S/
The radial Green’s function g` is a solution of the radial equation 2 1 d Z 2 d `.` C 1/ .S/
C E g` r; r 0 I 2 2 9.3.1 The Green’s Function for the Schrödinger 2 dr r dr r r Equation ı.r r 0 / D : (9.106) rr 0 .S/ The Green’s function G for the Schrödinger equation The standard method for calculating the Green’s function of (9.98) is a solution of a second order ordinary differential equation [27, pp. 354– 355] yields 1 2 Z r E G .S/ .r; r 0 I E/ D ı.r r 0 / : (9.98) 2 r .S/
g` r; r 0 I An explicit closed form expression for G .S/ is
.2Z/2`C2 .` C 1 / D exp 1 Z r C r 0 2`C1
.2` C 1/Š .1 / G .S/ r; r 0 I E D
0 `
2 jr r 0 j rr 1 F1 ` C 1 I 2` C 2I 2 1 Zr<
@ (9.107) U ` C 1 ; 2` C 2; 2 1 Zr> ; M; 1 .z1 / W; 1 .z2 / 2 2 @z1 where r< is the smaller of the pair r; r 0 and r> is the larger @ M; 1 .z1 / W; 1 .z2 / ; (9.99) of the pair r; r 0 . Matrix elements of g`.S/ can be calculated 2 2 @z2 with the aid of the formula for the double Laplace transform, where M;1=2 and W;1=2 are the Whittaker functions defined which is in Eqs. (9.132) and (9.133) below, and Z1 Z1
`C1 .S/
1=2 dr dr 0 rr 0 exp r 0 r 0 g` r; r 0 I D Z.2E/ ; (9.100)
0 0 (9.101) z1 D .2E/1=2 r C r 0 jr r 0 j ; 2`C2
4Z 2 2.2` C 1/Š 2`C3 (9.102) z2 D .2E/1=2 r C r 0 C jr r 0 j : D ` C 1 2Z . C Z/.0 C Z/ 1=2 The branch on which .2E/ is positive should be taken 2 F1 .2` C 2; ` C 1I ` C 2I 1 / ; when E < 0. When E > 0, the branch which corresponds to (9.108) incoming (or outgoing) waves at infinity can be selected with where 2Z. C 0 / the aid of the asymptotic approximation D ; (9.109) . C Z/.0 C Z/
.1 / (9.103) Matrix elements with respect to Slater orbitals can be calz exp 12 z2 ; G .S/ r; r 0 I E 2 jr r 0 j 2 culated from Eq. (9.108) by taking derivatives with respect which holds when z2 z1 . This approximation is ob- to and/or 0 to bring down powers of r and r 0 . Matrix tained by using Eqs. (9.130), (9.132), (9.133), and (9.164) elements with respect to Laguerre polynomials can be calin Eq. (9.99). A number of useful expansions for G .S/ can be culated by using Eq. (9.108) to evaluate integrals over the obtained from the integral representation generating function Eq. (9.199) for the Laguerre polyno
.S/ 0 mial [26]. Other methods of calculating matrix elements are G r; r I E discussed in Swainson and Drake [23–25]. The Green’s func 2 Z1 1 2Z tion GQ .S/ in momentum space is related to the coordinate D sinh. / coth
space Green’s function G .S/ via the Fourier transforms 2 0 Z n o
3 .S/ 0 1 0 1=2 G I E D .2 / r; r exp i p r p 0 r 0 I0 Z sinh. / 2rr Œ1 C cos.#/
exp 1 Z r C r 0 cosh. / d ; (9.110) (9.104) GQ .S/ p; p 0 I E d3 p d3 p 0 ;
9
156
R. N. Hill
Z
can be used to show that the radial functions gj k .r; r 0 I E/ GQ .S/ p; p 0 I E D .2 /3 exp i p r p0 r 0 satisfy the equation 3 3 0
.S/ 0 (9.111) G r; r I E d r d r : 1 0 d 1 Z˛ C 1E The Green’s function GQ .S/ is a solution of C B r dr r B Z C @
1 2 1 Z d 1C Z˛ A p E GQ .S/ p; p 0 I E C 1CE C 2 2 00 2 2 .p p / dr r r
00 0 3 00 ! ! .S/ 0 Q G p ;p IE d p D ı p p : (9.112) g11 .r; r 0 I E/ g12 .r; r 0 I E/ ı.r r 0 / 1 0 D : rr 0 An explicit closed form expression for GQ .S/ is 0 1 g21 .r; r 0 I E/ g22 .r; r 0 I E/
(9.122) GQ .S/ p; p 0 I E The solution to Eq. (9.122) is Z ı.p p 0 / i C D 1 !
1 h 2 2 2 jp p 0 j2 2 p 2 E 12 .p 0 /2 E 2p E g11 .r; r 0 I E/ g12 .r; r 0 I E/ .2/1C2 .a/ D .3 C 2/ g21 .r; r 0 I E/ g22 .r; r 0 I E/ 1q q 1q 1C " ! ! 2 F1 1; 1 I 2 I 1 1Cq 1Cq
0 G< .r/ 0 0 # r r G> .r / F> .r / 1Cq 1Cq F< .r/ ; 2 F1 1; 1 I 2 I 1q 1q !# !
(9.113) .r/ G > C# r r 0 G< .r 0 / F< .r 0 / ; (9.123) where F .r/ > s qD
2E jp p 0 j2
4E 2 4E p p 0 C .pp 0 /2
:
(9.114) where a is defined by Eq. (9.79), # is the Heaviside unit function, defined by 8 ˆ 0 ; #.x/ D 12 ; x D 0 ; ˆ : 0; x < 0 ;
9.3.2 The Green’s Function for the Dirac Equation
(9.124)
The Green’s function GD for the Dirac equation (9.60) is and the functions G< , F< , G> , and F> are special cases of a 4 4 matrix valued solution of the homogeneous solutions Eq. (9.71)–(9.80)
Z˛ i˛ r C ˇ E GD r; r 0 I E r
G< .r/ D r exp.r/.1 C E/1=2 Œf2 .r/ C f1 .r/ ; 0 (9.115) D ı r r I4 ; (9.125) where I4 is the 4 4 identity matrix. The partial wave expansion of GD is !
;m
;m X G11
iG12 0 ; (9.116) GD r; r I E D
;m
;m iG21 G22
;m
F< .r/ D r exp.r/.1 E/1=2 Œf2 .r/ f1 .r/ ; (9.126) G> .r/ D r exp.r/.1 C E/1=2 Œf4 .r/ C f3 .r/ ; (9.127)
1=2
F> .r/ D r exp.r/.1 E/
Œf4 .r/ f3 .r/ : (9.128)
where
0
;m m 0 0 D m G11
.; / ; g11 r; r I E ;
0
;m m 0 0 D m G12
.; / ; g12 r; r I E ;
0
;m m 0 0 D m G21 .; / ; g21 r; r I E ;
0
;m m 0 0 D m G22 .; / ; g22 r; r I E :
The functions G< .r/ and F< .r/ obey the boundary con(9.117) ditions at r D 0. The functions G .r/ and F .r/ obey > > (9.118) the boundary conditions at r D 1. Integral representa(9.119) tions and expansions for the Dirac Green’s function can be found in [23–25, 28]. Matrix element evaluation is discussed (9.120) in [23–25].
The identity
ı.r r 0 / ı r r 0 I4 D rr 0 ! 0 0 X m .; /m 0
. ; /
m 0 0 0 m .; / . ; /
;m (9.121)
9.4 Special Functions This section contains a brief list of formulae for the special functions which appear in the solutions discussed above. Derivations, and many additional formulae, can be found in
9
Hydrogenic Wave Functions
157
the standard reference works listed in the bibliography. For numerically useful approximations and available software packages, see Olver et al. [12], and Lozier and Olver [29].
9.4.1 Confluent Hypergeometric Functions The confluent hypergeometric differential equation is d d2 a w.z/ D 0 : z 2 C .c z/ dz dz
(9.129)
Equation (9.129) has a regular singular point at r D 0 with indices 0 and 1 c and an irregular singular point at 1. The regular solution to Eq. (9.129) is the confluent hypergeometric function, denoted by 1 F1 in generalized hypergeometric series notation. It can be defined by the series .c/ X .a C n/ z n : .a/ nD0 .c C n/ nŠ 1
1 F1 .aI cI z/
D
(9.130)
1 1 W ; .z/ D exp z z C 2 2 1 U C ; 2 C 1; z ; 2
(9.133)
are sometimes used instead of 1 F1 and U . For numerical evaluation and a program, see [30, 31]. The regular solution can be written as a linear combination of irregular solutions via 1 F1 .aI cI z/
D
.c/ .c/ ei a U .a; c; z/ C .c a/ .a/
ezCi .ac/ U .c a; c; z/ ; ) ( C1 ; Im z > 0 ; D 1 ; Im z < 0 :
(9.134) (9.135)
1 F1 can also be obtained from U as the discontinuity across a branch cut
z c1 exp.z/1 F1 .aI cI z/ .1 a/ .c/ D h 2 i c1
ze i U c a; c; ze i c1
i
U c a; c; ze i : ze i
The series Eq. (9.130) for 1 F1 converges for all finite z if c is not a negative integer or zero. It reduces to a polynomial of degree n in z if a D n where n is a positive (9.136) integer and c is not a negative integer or zero. The function 1 F1 .aI cI z/ is denoted by the symbol M.a; c; z/ in Abramowitz and Stegun [1], in Jahnke and Emde [5], and in The Wronskian of the two solutions is Olver [9], by 1 F1 .aI cI z/ in both of Luke’s books [6, 7] and d d U.a; c; z/ U.a; c; z/ 1 F1 .aI cI z/ 1 F1 .aI cI z/ in Magnus, Oberhettinger, and Soni [8], and by ˚.a; cI z/ in dz dz the Bateman project [2, 3] and Gradshteyn and Ryzhik [4]. exp.z/ : (9.137) D .c/z c The irregular solution to Eq. (9.129) is .a/ U .a; c; z/ D
.1 c/ 1 F1 .aI cI z/ .1 C a c/ .c 1/ 1c C z .a/ 1 F1 .1 C a cI 2 cI z/ :
A formula for U.a; c; z/ when c is the integer n C 1 can be obtained by taking the c ! n C 1 limit of the right-hand side of Eq. (9.131) to obtain (9.131)
U.a; n C 1; z/ D
.1/nC1 .a n/ " 1 ln.z/1 F1 .aI n C 1I z/ nŠ # 1 X .a C k/ak z k C ; .a/.k C n/Š
The function U.a; c; z/ is multiple-valued, with principal branch < arg z . It is denoted by the symbol U.a; c; z/ in Abramowitz and Stegun [1], in Magnus, Oberhettinger, and Soni [8], and in Olver [9], by .aI cI z/ in the first (9.138) of Luke’s books [6], by U.aI cI z/ in the second of Luke’s kDn books [7], and by .a; cI z/ in the Bateman project [2, 3] where and Gradshteyn and Ryzhik [4]. 8 The Whittaker functions M ; and W ; , which are related ˆ n k 1 ; 1 and 2 < N < 4l (d 3 –d 7 , f 3 – in LS Coupling f 11 , etc.). The recurring terms of a particular LS term type from d N and f N configurations are assigned sequential inThe centrality of LS coupling in the analysis and theoreti- dex numbers in the tables of Nielson and Koster [9]; the cal interpretation of atomic spectra has led to the acceptance index numbers stand for additional numbers having groupof notations and nomenclature well adapted to discussions theoretical significance that serve to differentiate the recurof particular structures and spectra [2]. The main elements ring terms. The first of those additional quantum numbers is
11
180
W. C. Martin et al.
Table 11.2 Allowed J -values for ljN equivalent electrons (jj ) coupling ljN
Allowed J -values
l1=2
1=2
2 l1=2
0
l3=2 and
3 l3=2
3=2
2 l3=2
0, 2
4 l3=2
0
5 l5=2 and l5=2
5=2
2 l5=2
0, 2, 4
and
4 l5=2
3 l5=2
3=2, 5=2, 9=2
6 l5=2
Q
R nllC1=2 nll1=2 , Q C R D N . The possible sets run from Q D N 2l (or zero if N < 2l) up to Q D N or Q D 2l C 2, whichever is smaller. The (degenerate) levels for a set with both Q and R nonzero have wave functions defined by the quantum numbers .˛J1 , ˇJ2 /J , with J1 and J2 deriving from the Q and R groups, respectively. The symbols ˛ and ˇ represent any additional quantum numbers required to identify levels. The J values of the allowed levels for each (˛J1 , ˇJ2 ) subset are obtained by combining J1 and J2 in the usual way.
11.8 Notations for Different Coupling Schemes
0
l7=2 and
7 l7=2
7=2
2 6 l7=2 and l7=2
0, 2, 4, 6
3 l7=2
3=2, 5=2, 7=2, 9=2, 11=2, 15=2
and
5 l7=2
4 l7=2
0, 2 2, 4 2, 2 4, 4 4, 5, 6, 8
8 l7=2
0
seniority [10]. It uniquely differentiates all recurring terms of d N and most of f N , except for a few terms of f 5 and f 9 , f 6 and f 8 , and f 7 . These remaining terms, which occur only in pairs, are further labeled A or B (in combination with seniority) to indicate Racah’s separation of the two terms. The index numbers of Nielson and Koster are in practice the most frequently used labels for the recurring terms of f N configurations. Use of their index numbers for the recurring terms of d N configurations perhaps has the disadvantage of substituting an arbitrary number for a quantum number (the seniority). The actual value of the seniority number is rarely needed, however, and a consistent notation for the d N and f N configurations is desirable. A table of the allowed LS terms of the l N electrons for l 3 is given in [10], with all recurring terms having the index numbers of Nielson and Koster as a following on-line integer. The group-theoretical labels are also listed. Thus, the d 3 2D term having seniority 3 is designated 2D2 , instead of 23 D, in this scheme; and the level having J D 3=2 is designated 2D2 3=2 .
11.7.2
jj Coupling
The Pauli exclusion principle restricts the set of possible J values for a group of N equivalent electrons having the same j -value, ljN . The allowed J -values for such groups are given in Table 11.2 for j D 1=2, 3=2, 5=2, and 7=2 (sufficient for 4 group has two allowed levels for each of l 3). The l7=2 the J -values 2 and 4. The subscripts distinguishing the two levels in each case are the seniority numbers [11]. The allowed levels of the configuration nl N may be obtained by dividing the electrons into sets of two groups
In this section, we give enough examples to make clear the meaning of the different coupling-scheme notations. Not all the configurations in the examples have been identified experimentally, and some of the examples of a particular coupling scheme given for heuristic purposes may be physically inappropriate. Cowan [3] describes the physical conditions for the different coupling schemes and gives experimental examples.
11.8.1 LS Coupling (Russell–Saunders Coupling) Some of the examples given below indicate notations bearing on the order of coupling of the electrons 1. 3d 7 4F 7=2
2. 3d 7 4F 4s4p 3P ı 6F ı9=2
3. 4f 7 8S ı 6s6p 2 4P 11P ı5
4. 3p 5 2P ı 3d 2 1G 2F ı7=2
5. 4f 10 3K2 6s6p 1P ı 3L ı6
6. 4f 7 8S ı 5d 7D ı 6p 8F 13=2
7. 4f 7 8S ı 5d 9D ı 6s 8D ı 7s 9D ı5
8. 4f 7 8S ı 5d 9D ı 6s 6p 3P ı 11F 8
9. 4f 7 8S ı 5d 2 1G 8G ı 6p 7F 0
10. 4f 2F ı 5d 2 1G 6s 2G 1P ı1 In the second example, the seven 3d electrons couple to give a 4F term, and the 4s and 4p electrons couple to form the 3P ı term; the final 6F ı term is one of nine possible terms obtained by coupling the 4F grandparent and 3P ı parent terms. The next three examples are similar to the second. The meaning of the index number 2 following the 3K symbol in the fifth example is explained in Sect. 11.7.1. The coupling in example 6 is appropriate if the interaction of the 5d and 4f electrons is sufficiently stronger than the 5d –6p interaction. The 7D ı parent term results from coupling the 5d electron to the 8S ı grandparent, and the 6p electron is then coupled to the 7D ı parent to form the final
11 8
Atomic Spectroscopy
F term. A space is inserted between the 5d electron and the 7 ı D parent emphasize that the latter is formed by coupling
to a term 8S ı listed to the left of the space. Example 7 illustrates a similar coupling order carried to a further stage; the 8 ı D parent term results from the coupling of the 6s electron to the 9D ı grandparent. Example 8 is similar to examples 25, but in 8 the first of the two terms that couple to form the final 11F term, i.e., the 9D ı term, is itself formed by the coupling of the 5d electron to the 8S ı core term. Example 9 shows an 8G ı parent term formed by coupling the 8S ı and 1G grandparent terms.
A space is again used to emphasize that the following 8G ı term is formed by the coupling of terms listed before the space. A different order of coupling is indicated in the final example, the 5d 2 1G term being coupled first to the external 6s electron instead of directly to the 4f core electron. The by a space to denote that it 4f 2F ı core term is isolated is coupled (to the 5d 2 1G 6s 2G term) only after the other electrons have been coupled. The notation in this particular case (with a single 4f electron) could be simplified by writing the 4f electron after the 2G term to which it is coupled. It appears more important, however, to retain the convention of giving the core portion of the configuration first. The notations in examples 15 are in the form recommended by Russell et al. [12] and used in both the Atomic Energy States [13] and Atomic Energy Levels [10, 14] compilations. The spacings used in the remaining examples allow different orders of coupling of the electrons to be indicated without the use of additional parentheses, brackets, etc. Some authors assign a short name to each (final) term, so that the configuration can be omitted in tables of classified lines, etc. The most common scheme distinguishes the low terms of a particular SL type by the prefixes a, b, c, : : : , and the high terms by z, y, x, : : : [14].
181
3 2 9=2; 2 11=2 4. 4d5=2 4d3=2
5. 4dC3 4d2 9=2; 2 11=2 The relatively large spin–orbit interaction of the 6p electrons produces jj -coupling structures for the 6p 2 , 6p 3 , and 6p 4 ground configurations of neutral Pb, Bi, and Po, respectively; the notations for the ground levels of these atoms are given as the first three examples above. The configuration in the first example shows the notation for equivalent electrons having the same j value ljN , in this case two 6p electrons each having j D 1=2. A convenient notation for a particular level .J D 0/ of such a group is also indicated. The second example extends this notation to the case of a 6p 3 configuration divided into two groups according to the two possible j values. A similar notation is shown for the 6p 4 level in the 2 /2 , third example; this level might also be designated .6p3=2 the negative superscript indicating the two 6p holes. The .J1 ; J2 /J term and level notation shown on the right in the fourth example is convenient because each of the two elec3 2 and 4d3=2 has more than one allowed total tron groups 4d5=2 Ji value. The assumed convention is that J1 applies to the 3 group) and J2 to group on the left (J1 D 9=2 for the 4d5=2 that on the right. Example 5 shows another possible representation of example 4 using a minus in the subscript to denote j D l 1=2 and a plus to denote j D l C 1=2.
11.8.3 J1 j or J1 J2 Coupling
ı 1. 3d 9 2D 5=2 4p3=2 5=2; 3=2 3
2. 4f 11 2H2 ı9=2 6s6p 3P ı1 9=2; 1 7=2
3. 4f 9 6H ı 5d 7H ı8 6s 6p 3P ı0 8; 0 8
ı 4. 4f 12 3H 6 5d 2D 6s 6p 3P ı 4F ı3=2 6; 3=2 13=2
ı 5. 5f 4 5I 4 6d3=2 4; 3=2 11=2 7s7p 1P ı1 11=2; 1 9=2
ı
4 5 6. 5f7=2 5f5=2 8; 5=2 21=2 7p3=2 21=2; 3=2 10 11.8.2 jj Coupling of Equivalent Electrons
ı 3 3 7. 5f7=2 5f5=2 9=2; 9=2 9 7s7p 3P ı2 9; 2 7
This scheme is used, for example, in relativistic calculations. 8. 2s 2p 2pC 0; 3=2 3=2 The lower-case j indicates the angular momentum of one 9. 2s 2p 2p 0 C 3=2 electron .j D l ˙ 1=2/ or of each electron in an ljN group. Various ways of indicating which of the two possible j-values The first five examples all have core electrons in LS couapplies to such a group without writing the j-value subscript pling, whereas jj coupling is indicated for the 5f core elechave been used by different authors; we give the j-values extrons in examples 6 and 7. Since the J1 and J2 values in the plicitly in the first four examples below. We use the symbols final (J1 ; J2 ) term have already been given as subscripts in Ji and j to represent total angular momenta. the configuration, the (J1 ; J2 ) term notations are redundant in all these examples. Unless separation of the configuration
2 1. 6p1=2 and final term designations is desired, as in some data tables,
2 0 ı one may obtain a more concise notation by simply enclos2. 6p1=2 6p3=2 3=2
2 ing the entire configuration in brackets and adding the final 2 3. 6p1=2 6p3=2 2 J value as a subscript. Thus, the level in the first example
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ı
can be designated as 3d 9 2D 5=2 4p3=2 3 . If the configuration and coupling order are assumed to be known, still shorter designations may be used; for example, fourth 3 ı 4the 3 ı F 3=2 13=2 level above might then be given as H 6 P
3 or H 6 ; 3P ı ; 4F ı3=2 13=2 . Similar economies of notation are of course possible, and often useful, in all coupling schemes. Example 8 shows a somewhat more compact representation of a case similar to example 6 with the j-values of the p electrons .j D l ˙ 1=2/ indicated by the plus and minus symbols in the subscripts. The s subshell does not need a subscript, since it has a unique possible j-value of 1=2. The parent term 0; 1=2 can be omitted before the last subshell 2pC , since the final term gives its J1 -value, J1 D 0, and the space before the last subshell makes the order of summation unambiguous. Another possible way to designate the same level is shown in example 9.
11.8.4 J1 l or J1 L2 Coupling (J1 K Coupling)
ı 1. 3p 5 2P ı1=2 5g 2 9=2 5
2. 4f 2 3H 4 5g 2 3 5=2
ı 3. 4f 13 2F ı7=2 5d 2 1D 1 7=2 7=2
ı 4. 4f 13 2F ı5=2 5d 6s 3D 3 9=2 11=2 The final terms in the first two examples result from coupling a parent-level J 1 to the orbital angular momentum of a 5g electron to obtain a resultant K , the K value being enclosed in brackets. The spin of the external electron is then coupled with the K angular momentum to obtain a pair of J values, J D K ˙ 1=2 (for K 6D 0). The multiplicity (2) of such pair terms is usually omitted from the term symbol, but other multiplicities occur in the more general J1 L2 coupling (examples 3 and 4). The last two examples are straightforward extensions of J1 l coupling, with the L2 and S 2 momenta of the external term (1D and 3D in examples 3 and 4, respectively) replacing the l and s momenta of a single external electron.
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the final term (in brackets). The multiplicity of the ŒK term arises from the spin of the external electron(s).
11.8.6 Coupling Schemes and Term Symbols The coupling schemes outlined above include those now most frequently used in calculations of atomic structure [3]. Any term symbol gives the values of two angular momenta that may be coupled to give the total electronic angular momentum of a level (indicated by the J-value). For configurations of more than one unfilled subshell, the angular momenta involved in the final coupling derive from two groups of electrons (either group may consist of only one electron). These are often an inner group of coupled electrons and an outer group of coupled electrons, respectively. In any case, the quantum numbers for the two groups can be distinguished by subscripts 1 and 2, so that quantum numbers represented by capital letters without subscripts are total quantum numbers for both groups. Thus, the quantum numbers for the two vectors that couple to give the final J are related to the term symbol as follows: Coupling scheme Quantum numbers for vectors Term symbol that couple to give J 2S C1 LS L; S L J1 J2 J1 ; J2 .J1 ; J2 / 2S2 C1 J1 L2 .! K/ K; S2 ŒK 2S2 C1 LS1 .! K/ K; S2 ŒK
11.9 Eigenvector Composition of Levels The wave functions of levels are often expressed as eigenvectors that are linear combinations of basis states in one of the standard coupling schemes. Thus, the wave function .˛J / for a level labeled ˛J might be expressed in terms of LS coupling basis states ˚.SLJ / .˛J / D
X
c.SLJ / ˚.SLJ / :
(11.2)
SL
11.8.5 LS1 Coupling (LK Coupling)
1. 3s 2 3p 2P ı 4f G 2 7=2 3
ı 2. 3d 7 4P 4s4p 3P ı D ı 3 5=2 7=2 The orbital angular momentum of the core is coupled with the orbital angular momentum of the external electron(s) to give the total orbital angular momentum L. The letter symbol for the final L value is listed with the configuration because this angular momentum is then coupled with the spin of the core (S 1 ) to obtain the resultant K angular momentum of
The c.SLJ / are expansion coefficients, and ˇ2 Xˇˇ ˇ ˇc.SLJ /ˇ D 1 :
(11.3)
SL
The squared expansion coefficients for the various SL terms in the composition of the ˛J level are conveniently expressed as percentages, whose sum is 100%. Thus, the percentage contributed by the pure Russell–Saunders state SLJ is equal to 100 jc.SLJ /j2 . The notation for
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Russell–Saunders basis states has been used only for concreteness; the eigenvectors may be expressed in any coupling scheme, and the coupling schemes may be different for different configurations included in a single calculation (with configuration interaction). Intermediate coupling conditions for a configuration are such that calculations in both LS and jj coupling yield some eigenvectors representing significant mixtures of basis states. Eigenvectors represented by the above equations are usually given in such a way that all terms in the summation use the same coupling scheme, and all eigenvectors for a given configuration use the same coupling scheme as well. The largest percentage in the composition of a level is called the purity of the level in that coupling scheme. The coupling scheme (or combination of coupling schemes if more than one configuration is involved) that results in the largest average purity for all the levels in a calculation is usually best for naming the levels. With regard to any particular calculation, one does well to remember that, as with other calculated quantities, the resulting eigenvectors depend on a specific theoretical model and are subject to the inaccuracies of whatever approximations the model involves. Theoretical calculations of experimental energy level structures have yielded many eigenvectors having significantly less than 50% purity in any coupling scheme. Since many of the corresponding levels have nevertheless been assigned names by spectroscopists, some caution is advisable in the acceptance of the level designations found in the literature. Moreover, the level designations, such as configuration and term labels, are not something permanent. Unlike the energy intervals, the wavefunctions and their composition are not directly observable. They are always a product of some theoretical interpretation. As theory evolves and improves with time, so does the interpretation of observed atomic properties, and, thus, some of the level designations are bound to change in the course of the development of the theory.
11.10 Ground Levels and Ionization Energies for Neutral Atoms Fortunately, most of the ground levels of the neutral atoms have reasonably meaningful LS-coupling names, the corresponding eigenvector percentages lying in the range from 55% to 100%. These names are listed in Table 11.3 except for Pb, Pa, U, and Np. The lowest few groundconfiguration levels of these four atoms comprise better [core].L1 S1 J1 /[valence]j .J1 j / terms than LS-coupling terms, the [core] and [valence] shells being 6s 2 and 6p 2 for Pb and 5f N and 6d 7s 2 for Pa, U, and Np. As noted in Sect. 11.8.2, the jj -coupling names given there for the ground levels of Bi and Po are more appropriate than the
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alternative LS-coupling designations in Table 11.3. For the heaviest elements, Sg, Bh, and Hs, only the J-values of the final terms are known. For brevity, the ionization energies in the table are given with no more than four digits after the decimal point. However, for all of the first 56 atoms, H through Ba, as well as for many of the heavier atoms, they are experimentally known with greater precision. The values in the table are from recent compilations [15, 16]. The National Institute of Standards and Technology (NIST) maintains an online Atomic Spectra Database (ASD) [15], which is updated with new and revised data every year. Each year, a few of the ionization energies tabulated here are revised. ASD provides full information about the uncertainties of the data and bibliographic references.
11.11 Zeeman Effect The Zeeman effect is of special interest because of the importance of Zeeman data in the analysis and theoretical interpretation of complex spectra. In a weak field, the J value remains a valid quantum number, although in general a level is split into magnetic sublevels [3, 17]. The Landé g-value of such a level may be defined by the expression for the energy shift of its magnetic sublevel having magnetic quantum number M , which has one of the 2J C 1-values, J , J C 1, : : :, J E D gMBB :
(11.4)
B is the magnetic flux density, and B is the Bohr magneton .B D e„=2me /. The wavenumber shift corresponding to this energy shift is D gM.0:46686B cm1 / ;
(11.5)
with B representing the numerical value of the magnetic flux density in teslas. The quantity in parentheses, the Lorentz unit, is about 1 or 2 cm1 for typical flux densities used to obtain Zeeman-effect data with classical spectroscopic methods. Accurate data can be obtained with much smaller fields, of course, by using higher-resolution techniques such as laser spectroscopy. Most of the g-values now available for atomic energy levels were derived by application of the above formula (for each of the two combining levels) to measurements of optical Zeeman patterns. A single transverse-Zeemaneffect pattern (two polarizations, resolved components, and sufficiently complete) can yield the J-value and the g value for each of the two levels involved. Neglecting a number of higher-order effects, we can evaluate the g value of a level ˇJ belonging to a pure LS-
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Table 11.3 Ground levels and ionization energies for neutral atoms Z 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 a
Element H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe
Ground configurationa 1s 1s 2 1s 2 1s 2 1s 2 1s 2 1s 2 1s 2 1s 2 1s 2 [Ne] [Ne] [Ne] [Ne] [Ne] [Ne] [Ne] [Ne] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Ar] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr] [Kr]
2s 2s 2 2s 2 2s 2 2s 2 2s 2 2s 2 2s 2 3s 3s 2 3s 2 3s 2 3s 2 3s 2 3s 2 3s 2
3d 3d 2 3d 3 3d 5 3d 5 3d 6 3d 7 3d 8 3d 10 3d 10 3d 10 3d 10 3d 10 3d 10 3d 10 3d 10
4d 4d 2 4d 4 4d 5 4d 5 4d 7 4d 8 4d 10 4d 10 4d 10 4d 10 4d 10 4d 10 4d 10 4d 10 4d 10
2p 2p 2 2p 3 2p 4 2p 5 2p 6
3p 3p 2 3p 3 3p 4 3p 5 3p 6 4s 4s 2 4s 2 4s 2 4s 2 4s 4s 2 4s 2 4s 2 4s 2 4s 4s 2 4s 2 4s 2 4s 2 4s 2 4s 2 4s 2 5s 5s 2 5s 2 5s 2 5s 5s 5s 2 5s 5s 5s 5s 2 5s 2 5s 2 5s 2 5s 2 5s 2 5s 2
4p 4p 2 4p 3 4p 4 4p 5 4p 6
5p 5p 2 5p 3 5p 4 5p 5 5p 6
Ground level 2 S 1=2 1 S0 2 S 1=2 1 S0 2 ı P 1=2 3 P0 4 ı S 3=2 3 P2 2 ı P 3=2 1 S0 2 S 1=2 1 S0 2 ı P 1=2 3 P0 4 ı S 3=2 3 P2 2 ı P 3=2 1 S0 2 S 1=2 1 S0 2 D 3=2 3 F2 4 F 3=2 7 S3 6 S 5=2 5 D4 4 F 9=2 3 F4 2 S 1=2 1 S0 2 ı P 1=2 3 P0 4 ı S 3=2 3 P2 2 ı P 3=2 1 S0 2 S 1=2 1 S0 2 D 3=2 3 F2 6 D 1=2 7 S3 6 S 5=2 5 F5 4 F 9=2 1 S0 2 S 1=2 1 S0 2 ı P 1=2 3 P0 4 ı S 3=2 3 P2 2 ı P 3=2 1 S0
Ionization energy (eV) 13:5984 24:5874 5:3917 9:3227 8:2980 11:2603 14:5341 13:6181 17:4228 21:5645 5:1391 7:6462 5:9858 8:1517 10:4867 10:3600 12:9676 15:7596 4:3407 6:1132 6:5615 6:8281 6:7462 6:7665 7:4340 7:9025 7:8810 7:6399 7:7264 9:3942 5:9993 7:8994 9:7886 9:7524 11:8138 13:9996 4:1771 5:6949 6:2173 6:6341 6:7589 7:0924 7:1194 7:3605 7:4589 8:3368 7:5762 8:9938 5:7864 7:3439 8:6084 9:0098 10:4513 12:1298
Z 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108
Element Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr Rf Db Sg Bh Hs
Ground configurationa [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Xe] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn] [Rn]
4f 4f 3 4f 4 4f 5 4f 6 4f 7 4f 7 4f 9 4f 10 4f 11 4f 12 4f 13 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14 4f 14
5f 2 5f 3 5f 4 5f 6 5f 7 5f 7 5f 9 5f 10 5f 11 5f 12 5f 13 5f 14 5f 14 5f 14 5f 14 5f 14 5f 14 5f 14
6s 6s 2 5d 6s 2 5d 6s 2 6s 2 6s 2 6s 2 6s 2 6s 2 5d 6s 2 6s 2 6s 2 6s 2 6s 2 6s 2 6s 2 5d 6s 2 2 5d 6s 2 3 5d 6s 2 4 5d 6s 2 5 5d 6s 2 6 5d 6s 2 7 5d 6s 2 9 5d 6s 5d 10 6s 5d 10 6s 2 5d 10 6s 2 5d 10 6s 2 5d 10 6s 2 5d 10 6s 2 5d 10 6s 2 5d 10 6s 2 7s 7s 2 6d 7s 2 2 6d 7s 2 3 . H 4 / 6d .4I ı9=2 / 6d .5I 4 / 6d 7s 2 7s 2 6d 7s 2 7s 2 7s 2 7s 2 7s 2 7s 2 7s 2 7s 2 2 6d 7s 2 3 6d 7s 2 4 6d 7s 2 6d 5 7s 2 6d 6 7s 2
An element symbol in brackets represents the electrons in the ground configuration of that element
6p 6p 2 6p 3 6p 4 6p 5 6p 6
7s 2 7s 2 7s 2
7p
Ground level 2 S 1=2 1 S0 2 D 3=2 1 ı G4 4 ı I 9=2 5 I4 6 ı H 5=2 7 F0 8 ı S 7=2 9 ı D2 6 ı H 15=2 5 I8 4 ı I 15=2 3 H6 2 ı F 7=2 1 S0 2 D 3=2 3 F2 4 F 3=2 5 D0 6 S 5=2 5 D4 4 F 9=2 3 D3 2 S 1=2 1 S0 2 ı P 1=2 .1=2; 1=2/0 4 ı S 3=2 3 P2 2 ı P 3=2 1 S0 2 S 1=2 1 S0 2 D 3=2 3 F2 .4; 3=2/11=2 .9=2; 3=2/ı6 .4; 3=2/11=2 7 F0 8 ı S 7=2 9 ı D2 6 ı H 15=2 5 I8 4 ı I 15=2 3 H6 2 ı F 7=2 1 S0 2 ı P 1=2 3 F2 4 F 3=2 J D0 J D 5=2 J D4
Ionization energy (eV) 3:8939 5:2117 5:5769 5:5386 5:4702 5:5250 5:5819 5:6437 5:6704 6:1498 5:8638 5:9391 6:0215 6:1077 6:1844 6:2542 5:4259 6:8251 7:5496 7:8640 7:8335 8:4382 8:9670 8:9588 9:2256 10:4375 6:1083 7:4167 7:2855 8:4181 9:3175 10:7485 4:0727 5:2784 5:3802 6:3067 5:89 6:1941 6:2655 6:0258 5:9738 5:9914 6:1979 6:2819 6:3676 6:50 6:58 6:6262 4:96 6:02 6:8 7:8 7:7 7:6
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a series of (core)nl levels from hydrogenic En -values may be due mainly to core penetration by the nl electron (low l-value series), or core polarization by the nl electron (high l-value series), or a combination of the two effects. In either case, it can be shown that these deviations can be approximately The independence of this expression from any other quantum represented by a constant quantum defect ıl in the Rydberg numbers (represented by ˇ) such as the configuration, etc., is formula, important. The expression is derived from vector coupling RZ Zc2 RZ Z 2 D 2c ; (11.9) Enl D 2 formulas by assuming a g value of unity for a pure orbital .n ıl / .n / angular momentum and writing the g value for a pure elecwhere Zc is the charge of the core and n D n ı is the tron spin as ge [18]. A value of 2 for ge yields the Landé effective principal quantum number. If the core includes only formula. If the anomaly of the magnetic dipole moment of closed subshells, the Enl -values are with respect to a value the electron, arising from small quantum electrodynamics efof zero for the (core)1 S0 level, i.e., the 1 S0 level is the limit fects [19], is taken into account, the value of ge is 2:0023193. of the (core)nl series. If the quantities in Eq. (11.9) are taken Schwinger g values obtained with this more accurate value as positive, they represent term values or ionization energies; for ge are given for levels of SL terms in [10]. the term value of the ground level of an atom or ion with The usefulness of gSLJ -values is enhanced by their relarespect to the ground level of the next higher ion is, thus, the tion to the g-values in intermediate coupling. In the notation principal ionization energy. used in Eq. (11.2) for the wave function of a level ˇJ in inIf the core has one or more open subshells, the series limit termediate coupling, the corresponding g-value is given by may be the barycenter of the entire core configuration, or X any appropriate sub-structure of the core, down to and ingSLJ jc.SLJ /j2 ; (11.6) gˇJ D cluding a single level. The Enl -values refer to the series of SL corresponding (core)nl structures built on the particular limit where the summation is over the same set of quantum structure. The value of the quantum defect depends to some numbers as for the wave function. The gˇJ -value is, thus, extent on which (core)nl structures are represented by the a weighted average of the Landé gSLJ -values, the weighting series formula. factors being just the corresponding component percentages. The quantum defect in general also has an energy depenFormulas for magnetic splitting factors in the dence that must be taken into account if lower members of J1 J2 and J1 L2 coupling schemes are given in [10, 18]. a series are to be accurately represented by Eq. (11.9). For an Some higher-order effects that must be included in more unperturbed series, this dependence can be expressed by the accurate Zeeman-effect calculations are treated by Bethe extended Ritz formula and Salpeter [5] and by Wybourne [18], for example. Highı D n n precision calculations for helium are given in [20]. See also a b Chaps. 14 and 15. D ı0 C C C::: ; (11.10) .n ı0 /2 .n ı0 /4 coupling term using the formula ( ) J.J C 1/ L.L C 1/ C S.S C 1/ gˇSLJ D 1 C .ge 1/ : 2J.J C 1/
11.12 Term Series, Quantum Defects, and Spectral-Line Series The Bohr energy levels for hydrogen or for a hydrogenic (one-electron) ion with nuclear charge Z are given by En D
RZ Z 2 ; n2
(11.7)
where RZ is the Rydberg constant for the appropriate nuclear mass; RZ can be found from the infinite-mass Rydberg constant R1 RZ D (11.8) me ; 1C M Z where me is the mass of the electron, and MZ is the mass of the nucleus. Equation (11.7) gives the energy relative to the ionization limit. For a multielectron atom, the deviations of
with ı0 , a, b : : : constants for the series (ı0 being the limit value of the quantum defect for high series members) [21]. The value of a is usually positive for core-penetration series and negative for core-polarization series. A discussion of the foundations of the Ritz expansion and application to high precision calculations in helium is given in [22] (see also Chap. 12). A spectral-line series results from either emission or absorption transitions involving a common lower level and a series of successive (core)nl upper levels differing only in their n values. The principal series of Na I, 3s 2S 1=2 np 2P ı1=2;3=2 (n 3) is an example. The regularity of successive upper term values with increasing n (Eqs. (11.9), (11.10)) is, of course, observed in line series; the intervals between successive lines decrease in a regular manner towards higher wavenumbers, and the series of increasing wavenumbers converges towards the term value of the lower level as a limit.
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11.13 Sequences
The above correspondence of names to ranges should not be taken as exact; the variation as to the extent of some of the Several types of sequences of elements and/or ionization named ranges found in the literature is considerable. For regions longer than 200 nm, wavelengths in stanstages are useful because of regularities in the progressive values of parameters relating to structure and other properties dard air are commonly tabulated. These wavelengths can be along the sequences. All sequence names may refer either to related to energy-level differences by conversion to the corresponding (vacuum) wavenumbers or frequencies [24, 25]. the atoms and/or ions of the sequence or to their spectra.
11.13.1 Isoelectronic Sequence
11.15 Wavelength Standards
A neutral atom and those ions of other elements having the same number of electrons as the atom comprise an isoelectronic sequence. (Note that a negative ion having this number of electrons is a member of the sequence.) An isoelectronic sequence is named according to its neutral member; for example, the Na I isolectronic sequence.
In 2001, the Comité International des Poids et Mesures recommended values for optical frequency standards from stabilized lasers using various absorbing atoms, atomic ions, and molecules [26]. These frequencies range from 29;054;057;446;579 Hz (10:318436884460 m; relative standard uncertainty 1:4 1013) for a transition in OsO4 to 1;267;402;452;889:92 kHz (236:54085354975nm; relative standard uncertainty 3:6 1013 ) for a transition in the 115 C In ion [26]. Extensive tables of wavenumbers for molecular transitions in the mid-IR range of 2:3 to 20:5 m are included in a calibration atlas published in 1998 [27]. Wavenumbers of Ar I [28] and Ar II [29] emission lines having uncertainties as small as 0:0003 cm1 are included in tables for these spectra covering a broad range from 222 nm to 5:865 m. Measurements of U and Th lines (575 to 692 nm) suitable for wavenumber calibration at uncertainty levels of 0:0003 or 0:0004 cm1 were reported in [30]. Comprehensive tables of lines for U [31, 32], Th [33], and I2 [34–37] are useful for calibration at uncertainty levels of 0:00003 to 0:003 cm1 , the line list of the Th spectrum extending down to 235 nm. A 1974 compilation gives reference wavelengths for some 5400 lines of 38 elements covering the range 1:5 nm to 2:5 m, with most uncertainties between 105 and 2 104 nm [38]. The wavelengths for some 1100 Fe lines selected from the Fe=Ne hollow-cathode spectrum have been recommended for reference standards over the range 183 nm to 4:2 m, with wavenumber uncertainties 0:0010:002 cm1 [39]. A comprehensive work on the Fe II spectrum [40] includes about 13;000 lines from VUV to far IR with wavenumber uncertainties between 0:0001 and 0:005 cm1 . Wavelengths for about 3000 VUV and UV lines (110400 nm) from a Pt=Ne hollow-cathode lamp have been determined with uncertainties of 0:0002 nm or less [15, 41]. More recent high-accuracy measurements of ultraviolet lines of Fe I, Ge I, Kr II, and Pt I, II include some wavelengths with uncertainties smaller than 105 nm [42]. The wavelengths tabulated for the Kr and Pt lines in [42] extend from 171 nm to 315 nm, and the accuracies of earlier measurements of a number of spectra useful for wavelength calibration are discussed.
11.13.2 Isoionic, Isonuclear, and Homologous Sequences An isoionic sequence comprises atoms or ions of different elements having the same charge. Such sequences have probably been the most useful along the d and f-shell rows of the periodic table. Isoionic analyses have also been carried out along p-shell rows, however, and a fine-structure regularity covering spectra of the p-shell atoms throughout the periodic table is known [23]. The atom and successive ions of a particular element comprise the isonuclear sequence for that element. The elements of a particular column and subgroup in the periodic table are homologous. Thus, the C, Si, Ge, Sn, and Pb atoms belong to a homologous sequence having np2 ground configurations (Table 11.3). The singly ionized atoms of these elements comprise another example of a homologous sequence.
11.14 Spectral Wavelength Ranges, Dispersion of Air The ranges of the most interest for optical atomic spectroscopy are: 220 m 7002000 nm 400700 nm 200400 nm 100200 nm 10100 nm < 10 nm
Mid-infrared (IR) Near IR Visible Near ultraviolet (UV) Vacuum UV (VUV) or far UV Extreme UV (EUV or XUV) Soft X-ray, X-ray
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11.16 Spectral Lines: Selection Rules, Intensities, Transition Probabilities, f Values, and Line Strengths
duced absorption W ./ D
ŒI./ I 0 ./ ; I./
(11.13)
The selection rules for discrete transitions are given in Tawhere I./ is the incident intensity at wavelength , e.g., ble 11.4. from a source providing a continuous background, and I 0 ./ the intensity after passage through the absorbing medium. The total reduced absorption of a homogeneous and optically 11.16.1 Emission Intensities thin absorbing medium of length l for a spectral line arising (Transition Probabilities) from a transition between levels i and k is obtained by integrating the above equation In the absence of self-absorption and stimulated emission, the total power radiated in a spectral line of frequency ZC1 per unit source volume and per unit solid angle is W ./ d Wi k D line D .4/1 h Aki Nk ;
0
(11.11) D
where Aki is the atomic transition probability and Nk the number per unit volume (number density) of excited atoms in the upper (initial) level k. For a homogeneous light source of length l and for the optically thin case, where all radiation escapes, the total emitted line intensity (SI quantity: radiance) is
Iline
ZC1 D line l D I./d D .4/1
0
hc 0
Aki Nk l ;
(11.12)
where I./ is the specific intensity at wavelength , and 0 the wavelength at line center.
e2 2 Ni fi k l ; 40 me c 2 0
(11.14)
where fi k is the atomic (absorption) oscillator strength (dimensionless), 0 is the central wavelength of the line, Ni is the density of atoms in state i, and the other quantities are fundamental constants. In astrophysics, the quantity given by the above equation is called the equivalent width of a spectral line [43]. Although it is not directly related to the width of the line profile, the term makes sense, as this quantity corresponds to the width of a rectangle having a unit height and the same area as the portion of the continuum absorbed by the spectral line.
11.16.3 Line Strengths Aki and fi k are the principal atomic quantities related to line intensities. In theoretical work, the line strength S is also widely used (Chap. 22)
11.16.2 Absorption f-values In absorption, the reduction of intensity of light passing through a medium is described by the quantity called the re-
S D S.i; k/ D S.k; i/ D jRi k j2 ; Ri k D h
k jP j
ii
;
(11.15) (11.16)
Table 11.4 Selection rules for discrete transitions
Rigorous rules
With negligible configuration interaction For LS coupling only
Electric dipole (E1) (allowed) 1. J D 0; ˙1 (except 0 6$ 0) 2. M D 0; ˙1 (except 0 6$ 0 when J D 0) 3. Parity change 4. One electron jumping, with l D ˙1, n arbitrary
Magnetic dipole (M1) (forbidden) J D 0; ˙1 (except 0 6$ 0) M D 0; ˙1 (except 0 6$ 0 when J D 0) No parity change No change in electron configuration; i.e., for all electrons, l D 0; n D 0
5. S D 0 6. L D 0; ˙1 (except 0 6$ 0)
S D 0 L D 0 J D ˙1
Electric quadrupole (E2) (forbidden) J D 0; ˙1; ˙2 (except 0 6$ 0; 1=2 6$ 1=2; 0 6$ 1) M D 0; ˙1; ˙2 No parity change No change in electron configuration; or one electron jumping with l D 0; ˙2; n arbitrary S D 0 L D 0; ˙1; ˙2 (except 0 6$ 0; 0 6$ 1)
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where i and k are the initial-state wavefunctions and the 11.16.5 Relationships Between Line final-state wavefunctions, and Ri k is the transition matrix eland Multiplet Values ement of the appropriate multipole operator P (computation of Ri k involves involves an integration over spatial and spin The relations between the total strength and f value of coordinates of all N electrons of the atom or ion). a multiplet (M) and the corresponding quantities for the lines of the multiplet (allowed transitions) are X Sline ; (11.21) SM D 11.16.4 Relationships Between A, f, and S
1 X fM D N gN i gi .Ji ; Jk / f .Ji ; Jk / I (11.22) The relationships between A, f , and S for electric dipole Jk ;Ji 1 (E1, or “allowed”) transitions in SI units (A in s , in m, S in m2 C2 ) are N is the weighted (multiplet) wavelength in vacuum Aki D
2 e 2 gi 16 3 f D S: i k me c0 2 gk 3h 0 3 gk
hc N D nN air D ; E
(11.17)
(11.23)
Numerically, in customary units (A in s1 , in nm, S in where atomic units, i.e., a02 e 2 ), E D Ek Ei X X 6:67025177 1013 gi 2:0261269 1015 Aki D fi k D S; D .gN k /1 gk Ek .gN i /1 gi Ei ; (11.24) 2 3 gk gk Jk Ji (11.18) and for S and E in atomic units, gN i and gN k are defined by Eq. (11.20), and n is the refractive index of air. 2 E SI (11.19) fi k D 3 gi gi and gk are the statistical weights, which are obtained from the appropriate angular momentum quantum numbers. Thus, for the lower (upper) level of a spectral line, gi.k/ D 2Ji.k/ C1 and for the lower and upper terms of a multiplet, the g-values are calculated from the corresponding quantum numbers X .2Ji.k/ C1/ D .2Li.k/ C1/ .2Si.k/ C1/ : (11.20) gN i.k/ D
11.16.6 Relative Strengths for Lines of Multiplets in LS Coupling
A general formula for the relative line strengths Srel of fine-structure components of LS multiplets is given by Sobel’man [44] ) ( i.k/ .2Ji C 1/.2Jk C 1/ Li Ji S ; Srel .i; k; S/ D Conversion relations between S and Aki for the most 2S C 1 Jk Lk 1 common forbidden transitions are given in Table 11.5. Os(11.25) cillator strengths f are rarely used for forbidden transitions, i.e., magnetic dipole (M1), electric quadrupole (E2), etc. where the indexes i and k correspond to the initial and final However, the left-hand parts of Eqs. (11.17) and (11.18) are levels, respectively, S is their common multiplicity, and the valid for any transition type and can be used to calculate fi k matrix in curly brackets is Wigner’s 6–j -symbol. The valfrom the normally tabulated Aki -values. ues of Srel in the above equation are normalized so that their [Numerical example: for the 1s2p 1P ı1 1s3d 1 D2 (al- sum over all multiplet components is unity. From this expreslowed) transition in He I at 667:815 nm (in air): gi D 3; gk D sion, it follows that the strongest, or principal, components of 5; Aki D 6:3705 107 s1 ; fi k D 0:71028; S D 46:860 a02 e 2 .] a multiplet are those for which the changes in J and L are Table 11.5 Conversion relations between S and Aki for forbidden transitions
SI unitsa
Numerically, in customary unitsb 5
Electric quadrupole
Aki D
16 S 15h 0 5 gk
Aki D
1:1199500 1013 S gk 5
Magnetic dipole
Aki D
16 3 0 S 3h 3 gk
Aki D
2:6973500 1010 S gk 3
A in s1 , in m. Electric quadrupole: S in m4 C2 . Magnetic dipole: S in J2 T2 A in s1 , in nm; S in atomic units: a04 e 2 D 2:012914421079 m4 C2 (electric quadrupole), e 2 h2 =16 2 m2e D 2B D 8:6007261 1047 J2 T2 (magnetic dipole). B is the Bohr magneton. a
b
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189
equal. The strongest of the principal lines correspond to the largest J-value of the initial level. The other, weaker components of the multiplet are called satellite lines. In order of decreasing intensity they are called the first-order satellites ( J D 0, L D ˙1) and second-order satellites ( J D C1, L D 1 or J D 1, L D C1). A discussion of normalization of Srel as well as extensive tables of their values are given in [45].
For large values of Z, roughly Z > 20, relativistic corrections become noticeable and must be taken into account.
f-value Trends f -values for high series members (large n0 values) of hydrogenic ions decrease according to
3
: (11.35) f n; l ! n0 ; l ˙ 1 / n0
11.18.2 Systematic Trends and Regularities in Atoms and Ions with Two or More Electrons The radiative lifetime k of an atomic level k is related to the
11.17 Atomic Lifetimes
sum of transition probabilities to all levels i lower in energy Atomic quantities for a given state or transition in an isoelecthan k !1 tronic sequence may be expressed as power series expansions X Aki : (11.26) in Z 1 k D i
The branching fraction of a particular transition, say to state i 0 , is defined as Aki 0 P D Aki 0 k : i Aki
Z 2 E D E0 C E1 Z 1 C E2 Z 2 C : : : ; 2
Z S D S0 C S1 Z f D f0 C f1 Z
1
1
C::: ;
(11.37)
2
C::: ;
(11.38)
C S2 Z C f2 Z
(11.36)
2
(11.27)
where E0 , f0 , and S0 are hydrogenic quantities. For transitions in which n does not change .ni D nk /, f0 D 0, since If only one branch .i 0 / exists (or if all other branches may be states i and k are degenerate. neglected), one obtains Aki 0 k D 1, and For equivalent transitions of homologous atoms, f values vary gradually. Transitions to be compared in the case of 0 (11.28) k D 1=Aki : alkalis are [46]
Precision lifetime measurement techniques are discussed in nl n0 l 0 Li ! .n C 1/l n0 C 1 l 0 Na Chaps. 18 and 19.
! .n C 2/l n0 C 2 l 0 Cu ! : : : :
11.18 Regularities and Scaling 11.18.1 Transitions in Hydrogenic (One-Electron) Species
Complex atomic structures, as well as cases involving strong cancellation in the integrand of the transition integral, generally do not adhere to this regular behavior.
11.19 Tabulations of Transition Probabilities
According to Eqs. (11.1) and (11.7), the nonrelativistic energy of a hydrogenic transition is The Aki -values for strong lines of selected elements are
2 2 2 .E/Z D .Ek Ei /Z D RZ Z 1=ni 1=nk : (11.29) given in Table 11.6. Data for some lines of the main spectral series of hydrogen are given in Table 11.7. Experimental and theoretical methods to determine A, Hydrogenic Z Scaling The spectroscopic quantities for a hydrogenic ion of nuclear f , or S-values, as well as atomic lifetimes are discussed in charge Z are related to the equivalent quantities in hydrogen Chaps. 12, 18, 19, and 22. Methods for critical evaluation of accuracy of exper.Z D 1/ as follows (neglecting small differences in the values imental and theoretical transition probabilities have been of RZ ) developed over several decades by the NIST Atomic Spec(11.30) troscopy Group [47, 48]. Examples of the work of this group .E/Z D Z 2 .E/H ; (11.31) can be found in [49, 50]. Comprehensive tables of A, f , .vac /Z D Z 2 .vac /H ; and S, including forbidden lines, are collected in the NIST 2 (11.32) SZ D Z SH ; ASD [15]. fZ D fH ; (11.33) For references to other sources of transition probability 4 (11.34) data, see Sect. 11.22. AZ D Z AH :
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Table 11.6 Wavelengths , upper energy levels Ek , statistical weights gi and gk of lower and upper levels, and transition probabilities Aki for persistent spectral lines of neutral atoms. Many tabulated lines are resonance lines (marked “g”), where the lower energy level belongs to the ground term Element Ag
Al
Ar
As
Au B
Ba
Be Bi
Br
C
Ca
Cd
a (nm) 328:07g 338:29g 520:91 546:55 308:22g 309:27g 394:40g 396:15g 104:82g 415:86 425:94 763:51 794:82 811:53 189:04g 193:76g 228:81 234:98 242:80g 267:60g 182:59g 182:64g 249:68g 249:77g 553:55g 649:88 705:99 728:03 234:86g 265:06 222:82g 289:80 298:90 306:77g 148:85g 154:07g 734:85 156:14g 165:70g 193:09 247:86 422:67g 430:25 558:88 616:22 643:91 228:80g 346:62 361:05 508:58
Ek (cm1 ) 30;473 29;552 48;744 48;764 32;435 32;437 25;348 25;348 95;400 117;184 118;871 106;238 107;132 105;463 52;898 51;610 54;605 53;136 41;175 37;359 54;768 54;768 40;040 40;040 18;060 24;980 23;757 22;947 42;565 59;696 44;865 45;916 44;865 32;588 67;184 64;907 78;512 64;087 60;393 61;982 61;982 23;652 38;552 38;259 31;539 35;897 43;692 59;498 59;516 51;484
gi 2 2 2 4 2 4 2 4 1 5 3 5 1 5 4 4 6 4 2 2 2 4 2 4 1 7 7 5 1 9 4 4 4 4 4 4 4 5 5 5 1 1 5 7 5 7 1 3 5 5
gk 4 2 4 6 4 6 2 2 3 5 1 5 3 7 6 4 4 2 4 2 4 6 2 2 3 7 9 7 3 9c 4 2 4 2 4 4 6 7 5 3 3 3 5 7 3 9 3 5 7 3
Aki (108 s1 ) 1:4 1:3 0:75 0:86 0:587 0:729 0:499 0:985 5:32 0:0140 0:0398 0:245 0:186 0:331 2:66 2:19 2:8 3:1 1:98 1:64 1:70 2:04 0:840 1:68 1:190 0:54 0:50 0:32 5:52 4:23 0:88 1:53 0:54 1:67 2:0 2:2 0:047 1:17 2:61 3:39 0:28 2:18 1:36 0:49 0:48 0:53 5:3 1:2 1:3 0:56
Accuracy b B B D D BC BC BC BC AA B BC B B B CC CC D D BC AC A A A A AC CC C CC A A CC B CC B DC DC C A A A CC BC CC D C D C D D C
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Table 11.6 (Continued) Element Cl
Co
Cr
Cs
Cu
F
Fe
Ga
Ge
He
Hg
a (nm) 134:72g 135:17g 452:62 725:66 340:51 345:35 350:23 356:94 357:87g 359:35g 360:53g 425:43g 427:48g 520:84 387:61g 455:53g 459:32g 852:11g 894:35g 217:89g 324:75g 327:40g 521:82 95:483g 685:60 739:87 775:47 358:12 371:99g 373:49 374:56g 385:99g 404:58 287:42g 294:36g 403:30g 417:20g 265:16g 270:96g 275:46g 303:91 53:703g 58:433g 388:86 402:62 447:15 587:57 253:65g 312:57 435:83 546:07
Ek (cm1 ) 74;226 74;866 96;313 85;735 32;842 32;431 32;028 35;451 27;935 27;820 27;729 23;499 23;386 26;787 25;792 21;946 21;765 11;732 11;178 45;879 30;784 30;535 49;942 104;731 116;987 115;918 117;623 34;844 26;875 33;695 27;395 25;900 36;686 34;782 34;788 24;789 24;789 37;702 37;452 37;702 40;021 186;209 171;135 185;565 193;917 191;444 186;102 39;412 71;396 62;350 62;350
gi
gk
4 2 4 6 10 10 10 8 7 7 7 7 7 5 2 2 2 2 2 2 2 2 4 4 6 6 4 11 9 11 5 9 9 2 4 2 4 1 3 5 5 1 1 3 9 9 9 1 3 3 5
4 2 4 4 10 12 8 8 9 7 5 9 7 7 4 4 2 4 2 4 4 2 6 4 8 6 6 13 11 11 7 9 9 4 6 2 2 3 1 3 3 3 3 9c 15 c 15 c 15 c 3 5 3 3
Aki (108 s1 ) 4:19 3:23 0:051 0:15 1:0 1:1 0:80 1:5 1:48 1:50 1:62 0:315 0:307 0:506 0:0039 0:01836 0:00794 0:3279 0:2863 0:91 1:395 1:376 0:75 5:98 0:487 0:353 0:323 1:02 0:162 0:901 0:115 0:0969 0:862 1:17 1:34 0:485 0:945 0:85 2:8 1:1 2:8 5.6634 17.989 0.09475 0.1160 0.2458 0.7070 0.0840 0.66 0.56 0.49
Accuracy b CC CC C D CC CC CC C B B B B B B BC AA AC AAA AAA CC AA AA CC BC CC C CC A A A A A A BC BC BC BC C C C C AAA AAA AAA AAA AAA AAA AC BC B B
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Table 11.6 (Continued) Element I In
K
Kr
Li
Mg
Mn
N
Na
Ne
Ni
a (nm) 178:28g 183:04g 303:93g 325:61g 410:17g 451:13g 404:41g 404:72g 766:49g 769:90g 557:03 587:09 760:15 811:29 323:27g 460:29 610:36 670:78g 202:58g 285:21g 470:30 518:36 279:48g 279:83g 280:11g 403:08g 403:31g 403:45g 119:96g 149:26 493:51 746:83 821:63 568:26 589:00g 589:59g 818:33 73:590g 74:372g 585:25 607:43 640:22 310:16 313:41 336:96g 341:48 352:45 361:94
Ek (cm1 ) 56;093 54;633 32;892 32;916 24;373 24;373 24;720 24;701 13;043 12;985 97;919 97;945 93;123 92;294 30;925 36;623 31;283 14;904 49;347 35;051 56;308 41;197 35;770 35;726 35;690 24;802 24;788 24;779 83;365 86;221 106;478 96;751 95;532 34;549 16;973 16;956 29;173 135;889 134;459 152;971 150;917 149;657 33;112 33;611 29;669 29;481 28;569 31;031
gi
gk
4 4 2 4 2 4 2 2 2 2 5 3 5 5 2 6 6 2 1 1 3 5 6 6 6 6 6 6 4 6 4 6 6 2 2 2 2
4 6 4 6 2 2 4 2 4 2 3 5 5 7 6c 10 c 10 c 6c 3 3 5 3 8 6 4 8 6 4 6 4 2 4 6 4 4 2 4 3 3 1 1 7 7 5 7 9 5 7
1 1 3 3 5 5 3 9 7 7 5
Aki (108 s1 ) 2.71 0.16 1.11 1.30 0.496 0.893 0.0115 0.0107 0.3779 0.3734 0.00980 0.0071 0.2732 0.3610 0.01002 0.2322 0.6856 0.3689 0.612 4.91 0.219 0.561 3.7 3.6 3.7 0.17 0.165 0.158 4.07 3.11 0.0176 0.196 0.226 0.101 0.616 0.614 0.429 5:88 0:440 0:615 0:603 0:5149 0:63 0:73 0:18 0:55 1:0 0:66
Accuracy b C D BC BC BC BC A BC AAA AAA BC C AA AAA AA AA AAA AAA BC A BC A C C C CC CC CC A BC B BC BC A AA AA AC A A AA BC AAA CC CC C C C C
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Atomic Spectroscopy
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Table 11.6 (Continued) Element O
P
Pb
Rb
S
Sc
Si
Sn
Sr Ti
Tl
U
a (nm) 130:22g 436:82 543:69 715:67 777:19 177:50g 178:28g 213:62 253:56 280:20 283:31g 368:35 405:78 420:18g 421:55g 780:03g 794:76g 147:40g 166:67 180:73g 469:41 390:75g 391:18g 402:04g 402:37g 250:69g 251:61g 288:16 500:61 594:85 284:00g 303:41g 317:50g 326:23 242:81g 460:73g 364:27g 365:35g 399:86g 521:04g 498:17 276:79g 351:92g 377:57g 535:05g 356:66g 357:16 358:49g
Ek (cm1 ) 76;795 99;681 105;019 116;631 86;631 56;340 56;091 58;174 58;174 46;329 35;287 34;960 35;287 23;793 23;715 12;817 12;579 67;843 69;238 55;331 73;921 25;585 25;725 24;866 25;014 39;955 39;955 40;992 60;962 57;798 38;629 34;641 34;914 39;257 41;172 21;698 27;615 27;750 25;388 19;574 26;911 36;118 36;200 26;478 26;478 28;650 38;338 27;887
gi
gk
5 3 7 5 5 4 4 6 4 5 1 3 5 2 2 2 2 5 5 5 5 4 6 4 6 3 5 5 3 3 5 3 5 5 1 1 7 9 9 9 11 2 4 2 4 11 17 13
3 9c 5 5 7 6 4 4 4 7 3 1 3 4 2 4 2 7 5 3 7 6 8 4 6 5 5 3 5 5 5 1 3 3 3 3 9 11 9 9 13 4 6 2 2 11 15 15
Aki (108 s1 ) 3:41 0:00758 0:0180 0:505 0:369 2:17 2:14 2:83 0:95 1:61 0:49 1:37 0:90 0:0177 0:0150 0:3812 0:3614 1:96 4:58 3:27 0:0084 1:66 1:79 1:63 1:65 0:547 1:68 2:17 0:028 0:0222 1:7 2:0 1:0 2:7 0:17 2:01 0:895 0:869 0:481 0:0389 0:660 1:26 1:24 0:625 0:705 0:24 0:13 0:18
Accuracy b A B CC B A C C C C CC CC B CC BC B AAA AAA CC CC CC D BC BC BC BC B B B D C D D D D CC AA A A A B CC C C B B B C B
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Table 11.6 (Continued) a (nm) 318:34g 411:18 437:92 438:47 119:20g 129:56g 146:96g 467:12 711:96 213:86g 330:26 334:50 636:23
Element V
Xe
Zn
Ek (cm1 ) 31;541 26;738 25;253 25;111 83;890 77;185 68;045 88;469 92;445 46;745 62;772 62;777 62;459
gi
gk
6 10 10 8 1 1 1 5 7 1 3 5 3
8 10 12 10 3 3 3 7 9 3 5 7 5
Accuracy b
Aki (108 s1 ) 2:52 1:00 1:15 0:92 5:93 2:53 2:73 0:0249 0:0839 7:14 1:20 1:70 0:47
BC BC BC BC BC AC AC B B A B B C
a
A “g” following the wavelength indicates that the lower level of the transition belongs to the ground term, i.e., the line is a resonance line. Wavelengths below 200 nm are in vacuum, and those above 200 nm are in air. b Accuracy estimates pertain to Aki -values: AAA, uncertainty within 0:3%; AA, within 1%; AC, within 2%; A, within 3%; BC, within 7%; B, within 10%; CC, within 18%; C, within 25%; DC, within 40%; D, within 50% c The Aki , , gi , and gk are multiplet values; see Eq. (11.20) and Sect. 11.16.5 Table 11.7 Some transitions of the main spectral series of hydrogen n–n0a 12 13 14 15 16 23 24 25
Customary nameb L˛ Lˇ L Lı L H˛ Hˇ H
c (nm) 121:56701 102:57220 97:25365 94:97429 93:78033 656:2819 486:1333 434:0471
gi d 2 2 2 2 2 8 8 8
gk 8 18 32 50 72 18 32 50
Aki e .108 s1 / 4:6986 5:5751.1/ 1:2785.1/ 4:1250.2/ 1:6440.2/ 4:4101.1/ 8:4193.2/ 2:5304.2/
n–n0a 26 27 34 35 36 37 38 45
Customary nameb Hı H P˛ Pˇ P Pı P Brackett-˛
c (nm) 410:17415 397:00788 1875:0976 1281:8070 1093:8086 1004:9369 954:5969 4052:269
gi d
gk
8 8 18 18 18 18 18 32
72 98 32 50 72 98 128 50
Aki e .108 s1 / 9:7320.3/ 4:3889.3/ 8:9860.2/ 2:2008.2/ 7:7829.3/ 3:3585.3/ 1:6506.3/ 2:6993.2/
a
Principal quantum numbers of the lower and upper levels L˛ is often called Lyman ˛, H˛ D Balmer ˛, P˛ D Paschen ˛ c Wavelengths below 200 nm are in vacuum; values above 200 nm are in air d For transitions in hydrogen, gi.k/ D 2.ni.k/ /2 , where ni.k/ , is the principal quantum number of the lower (upper) electron shell e The number in parentheses indicates the power of 10 by which the value has to be multiplied; uncertainties of these theoretical A-values are smaller than 0:3% b
11.20 Spectral Line Shapes, Widths, and Shifts
T is the temperature of the emitters in K, and M the atomic weight in atomic mass units (u).
Observed spectral lines are always broadened. The principal physical causes of spectral line broadening are Doppler and pressure broadening. The theoretical foundations of line 11.20.2 broadening are discussed in Chaps. 20 and 63.
Pressure Broadening
Pressure broadening is due to collisions of the emitters with neighboring particles (also Chaps. 20 and 63). Shapes 11.20.1 Doppler Broadening are often approximately Lorentzian, i.e., I./ / f1 C Œ. 0 /=1=2 2 g1 . In the following formulas, all FWHMs and Doppler broadening is due to the thermal motion of the emitwavelengths are expressed in nm, particle densities N in ting atoms. For a Maxwellian velocity distribution, the line cm3 , temperatures T in K, and energies E or I in cm1 . shape is Gaussian; the full width at half maximum intensity Resonance broadening (self-broadening) occurs only be(FWHM) is, in the same units as wavelength (), tween identical species and is confined to lines with the upper
7 1=2 D 7:16233 10 I (11.39) .T =M / or lower level having an electric dipole transition (resonance D 1=2
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Atomic Spectroscopy
195
line) to the ground state. The FWHM may be estimated as R1=2 ' 8:6 1028
gi gk
1=2 2 r fr Ni ;
(11.40)
where is the wavelength of the observed line; fr and r are the oscillator strength and wavelength of the resonance line, and gk and gi are the statistical weights of its upper and lower levels; Ni is the ground state number density. For the 1s2p 1 Pı1 1s3d 1 D2 transition in He I Œ D 667:815 nm; r .1s 2 1 S0 1s2p 1 Pı1 / D 58:4334 nm; gi D 1; gk D 3; fr D 0:2762 at Ni D 1 1018 cm3 : R1=2 D 0:0036 nm. Van der Waals broadening arises from the dipole interaction of an excited atom with the induced dipole of a ground state atom. (In the case of foreign gas broadening, both the perturber and the radiator may be in their respective ground states.) An approximate formula [51] for the FWHM, strictly applicable to hydrogen and similar atomic structures only, is
Table 11.8 Values of Stark-broadening parameter ˛1=2 of the Hˇ line of hydrogen (486:1 nm) for various temperatures and electron densities T (K) 5000 10;000 20;000 30;000
Ne (cm3 ) 1015 0:0787 0:0803 0:0815 0:0814
1016 0:0808 0:0840 0:0860 0:0860
1017 0:0765 0:0851 0:0902 0:0919
1018 ::: 0:0781 0:0896 0:0946
broadening in discharges and plasmas. The FWHM for hydrogen lines is
10 S,H ˛1=2 N e2=3 ; 1=2 D 2:50 10
(11.46)
where Ne is the electron density. The half-width parameter ˛1=2 for the Hˇ line at 486:1 nm, widely used for plasma diagnostics, is tabulated in Table 11.8 for some typical temperatures and electron densities [45]. This reference also contains ˛1=2 parameters for other hydrogen lines, as well as Stark width and shift data for numerous lines of other 11 2 2=5 3=10 elements, i.e., neutral atoms and singly charged ions (in ' 8:5 10 C .T =/ N ; (11.41) W 1=2 6 the latter, Stark widths and shifts depend linearly on Ne ). where is the atom-perturber reduced mass in atomic mass Other tabulations of complete hydrogen Stark profiles exist. units, N the perturber density, and C6 the interaction con- Griem [52] gives a theoretical foundation for treatment of Stark broadening of spectral lines in plasmas. stant; C6 may be roughly estimated as follows C6 D Ck Ci ;
(11.42)
11.21 Spectral Continuum Radiation
with
11.21.1 Hydrogenic Species 2 Œm6 =s ; Ci.k/ ' 9:1 1046 ˛d Ri.k/
(11.43)
Precise quantum-mechanical calculations exist only for hywhere ˛d is the mean dipole polarizability of the perturber drogenic species. The total power cont radiated (per unit source volume and per unit solid angle, and expressed in SI atom, 3IH 2 3 units) in the wavelength interval is the sum of radiation ˛d 4:5 a0 ; (11.44) due to the recombination of a free electron with a bare ion 4E (free–bound transitions) and bremsstrahlung (free–free tranIH being the ionization energy of hydrogen and E the ensitions) 2 ergy of the first excited level of the perturber atom, and Ri.k/ are the mean squared radii of the wavefunctions of the initial e6 Ne NZ Z 2 cont D and final levels of the transition, 2c 2 03 .6 me /3=2 2 1 hc IH 2 2 exp Ri.k/ 2:5 a0 ; (11.45) .kT /1=2 kT 2 .I Ei.k/ / 8 2 n0 < 2 Z2 I X Z IH fb H where I is the ionization energy of the radiator. Van der exp 2 3 : kT n n kT Waals broadened lines are redshifted by about one-third the n.Z 2 IH =hc/1=2 9 size of the FWHM. = 2 Z I 1 ı 1 H For the 1s2p P1 1s 3d D2 transition in He I, and with 1 C C Nfb exp ; (11.47) ff ; .n0 C 1/2 kT He as perturber: D 667:815 nm; I D 198;311 cm1 ; E D Ei D 171;135 cm1 ; Ek D 186;105 cm1 ; D 2. At T D 273 K and N D 1 1018 cm3 : W where Ne is the electron density, NZ the number density of 1=2 D 0:0012 nm. Stark broadening due to charged perturbers, i.e., ions and hydrogenic (bare) ions of nuclear charge Z, IH the ionizaelectrons, usually dominates resonance and van der Waals tion energy of hydrogen, n0 the principal quantum number of
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the lowest level for which adjacent levels are so close that they approach a continuum, and summation over n may be replaced by an integral (the choice of n0 is rather arbitrary; n0 as low as 6 can be found in the literature); fb and ff are the Gaunt factors, which are generally close to unity. (For the higher free-bound continua, starting with n0 C 1, an average Gaunt factor Nfb is used.) For neutral hydrogen, the recombination continuum forming H also becomes important [53]. In the equation above, the value of the constant factor before Ne is 6:065 1055 W m4 J1=2 sr1 . [Numerical example: for atomic hydrogen .Z D 1/, the quantity cont has the value 2:9 W m3 sr1 under the following conditions: D 3 107 m; D 1 1010 m; N e .D NZD1 / D 1 1021 m3 ; T D 12;000 K. The lower limit of the summation index n is 2; the upper limit n0 has been taken to be 10. All Gaunt factors fb , Nfb , and ff have been assumed to be unity.]
11.21.2
Many-Electron Systems
For many-electron systems, only approximate theoretical treatments exist, based on the quantum-defect method [54]. For results of calculations for noble gases, see, e.g., [55, 56]. Experimental work is centered on the noble gases [57, 58]. Modifications of the continuum by autoionization processes must also be considered. Near the ionization limit, the f values for bound-bound transitions of a spectral series .n0 ! 1/ make a smooth connection to the differential oscillator strength distribution df =d in the continuum [3].
11.22 Sources of Spectroscopic Data Access to most of the atomic spectroscopic databases currently online is given by links at the Plasma Gate server [59]. Extensive data from NIST compilations of atomic wavelengths, energy levels, and transition probabilities are available from the Atomic Spectra Database (ASD) at the NIST site [15]. Section 11.15 includes additional references for wavelength tables. The modern trend of online dissemination of digital data is exemplified by the Virtual Atomic and Molecular Data Centre (VAMDC) Consortium [60], which provides online access to many atomic and molecular spectroscopic databases through a unified portal.
References 1. Taylor, B.N., Thompson, A. (eds.): The International System of Units (SI) (2008) https://doi.org/10.6028/NIST.SP.330e2008. NIST Special Publication 330, available online at https://www. nist.gov/pml/special-publication-330
2. Condon, E.U., Shortley, G.H.: The Theory of Atomic Spectra. Cambridge Univ. Press, Cambridge (1935) 3. Cowan, R.D.: The Theory of Atomic Structure and Spectra. Univ. California Press, Berkeley (1981) 4. Demtröder, W.: Atoms, Molecules, and Photons. An Introduction to Atomic-, Molecular- and Quantum Physics, 2nd edn. Springer, New York (2006) 5. Bethe, H.A., Salpeter, E.E.: Quantum Mechanics of One- and Two-Electron Atoms. Plenum, New York (1977) 6. Karshenboim, S.G.: Phys. Rep. 422, 1 (2005). https://doi.org/10. 1016/j.physrep.2005.08.008 7. Edlén, B.: In: Flügge, S. (ed.) Encyclopedia of Physics, vol. 27, Springer, Berlin, Heidelberg (1964) 8. Russell, H.N., Saunders, F.A.: Astrophys. J. 61, 38 (1925). https:// doi.org/10.1086/142872 9. Nielson, C.W., Koster, G.F.: Spectroscopic Coefficients for the pn , dn , and fn Configurations. MIT Press, Cambridge (1963) 10. Martin, W.C., Zalubas, R., Hagan, L.: Atomic Energy Levels – The Rare-Earth Elements. U.S. Government Printing Office, Washington (1978). Nat. Stand. Ref. Data Ser., Nat. Bur. Stand. No. 60 11. de-Shalit, A., Talmi, I.: Nuclear Shell Theory. Academic Press, New York (1963) 12. Russell, H.N., Shenstone, A.G., Turner, L.A.: Phys. Rev. 33, 900 (1929). https://doi.org/10.1103/PhysRev.33.900 13. Bacher, R.F., Goudsmit, S.: Atomic Energy States. McGraw-Hill, New York (1932) 14. Moore, C.E.: Atomic Energy Levels. U.S. Government Printing Office, Washington (1971). Nat. Stand. Ref. Data Ser., Nat. Bur. Stand. No. 35 15. Kramida, A., Ralchenko, Y., Reader, J., NIST ASD Team: NIST Atomic Spectra Database, version 5.9 (2021) https://doi.org/10. 18434/T4W30F. Available online at https://physics.nist.gov/asd; This is one of several online NIST databases referred to in this chapter. The databases are accessible online at https://www.nist. gov/pml/productsservices/physical-reference-data 16. Sansonetti, J.E., Martin, W.C.: Handbook of Basic Atomic Spectroscopic Data, NIST (online database). https://physics.nist.gov/ PhysRefData/Handbook. These tables include selected data on wavelengths, energy levels, and transition probabilities for the neutral and singly-ionized atoms of all elements up through einsteinium (Z D 1–99); https://doi.org/10.18434/T4FW23 17. Bernath, P.F.: Spectra of Atoms and Molecules, 3rd edn. Oxford Univ. Press, New York (2015) 18. Wybourne, B.G.: Spectroscopic Properties of Rare Earths. Wiley, New York (1965) 19. Schwinger, J.: Phys. Rev. 73, 416 (1948). https://doi.org/10.1103/ PhysRev.73.416 20. Yan, Z.C., Drake, G.W.F.: Phys. Rev. A 50, R1980 (1994). https:// doi.org/10.1103/PhysRevA.50.R1980 21. Martin, W.C.: J. Opt. Soc. Am. 70, 784 (1980). https://doi.org/10. 1364/JOSA.70.000784 22. Drake, G.W.F.: Adv. At. Mol. Opt. Phys. 32, 93 (1994). https:// doi.org/10.1016/S1049-250X(08)60012-9 23. Fano, U., Martin, W.C.: In: Brittin, W.E., Odabasi, H. (eds.) Topics in Modern Physics, A Tribute to E.U. Condon, pp. 147–152. Colorado Associated Univ. Press, Colorado (1971) 24. Edlén, B.: Metrologia 2, 71 (1966). https://doi.org/10.1088/00261394/2/2/002 25. Peck, E.R., Reeder, K.: J. Opt. Soc. Am. 62, 958 (1972). https:// doi.org/10.1364/JOSA.62.000958 26. Quinn, T.J.: Metrologia 40, 103 (2003). https://doi.org/10.1088/ 0026-1394/40/2/316 27. Maki, A.G., Wells, J.S.: Wavenumber Calibration Tables from Heterodyne Frequency Measurements, version 1.3. NIST, Gaithersburg (1998) https://doi.org/10.18434/T49598. Originally
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28.
29.
30. 31. 32.
33. 34.
35.
36. 37. 38. 39.
40. 41.
42. 43. 44. 45. 46. 47. 48. 49. 50.
published as NIST Special Publication 821 (U.S. Government Printing Office, Washington 1987) Whaling, W., Anderson, W.H.C., Carle, M.T., Brault, J.W., Zarem, H.A.: J. Res. Natl. Inst. Stand. Technol. 107, 149 (2002). https:// doi.org/10.6028/jres.10714 Whaling, W., Anderson, W.H.C., Carle, M.T., Brault, J.W., Zarem, H.A.: J. Quant. Spectrosc. Radiat. Transf. 53, 1 (1995). https://doi. org/10.1016/0022-4073(94)00102-D Sansonetti, C.J., Weber, K.-H.: J. Opt. Soc. Am. B 1, 361 (1984). https://doi.org/10.1364/JOSAB.1.000361 Palmer, B.A., Keller, R.A., Engleman Jr., R.: Los Alamos National Laboratory Report LA-8251-MS, UC-34a (1980) Redman, S.L., Lawler, J.E., Nave, G., Ramsey, L.W., Mahadevan, S.: Astrophys. J. Suppl. Ser. 195, 24 (2011). https://doi.org/10. 1088/0067-0049/195/2/24 Redman, S.L., Nave, G., Sansonetti, C.J.: Astrophys. J. Suppl. Ser. 211, 4 (2014). https://doi.org/10.1088/0067-0049/211/1/4 Gerstenkorn, S., Luc, P.: Atlas du Spectre d’Absorption de la Molécule d’Iode entre 14 800–20 000 cm1 . Editions du CNRS, Paris (1978) Kato, H., Kasahara, S., Misono, M., Maasaki, B.: Doppler-Free High Resolution Spectral Atlas of Iodine Molecule 15 000 to 19 000 cm1 . Japan Society for the Promotion of Science, Tokyo (2000) Knöckel, H., Bodermann, B., Tiemann, E.: Eur. Phys. J. D 28, 199 (2004). https://doi.org/10.1140/epjd/e2003-00313-4 Salami, H., Ross, A.J.: J. Mol. Spectrosc. 233, 157 (2005). https:// doi.org/10.1016/j.jms.2005.06.002 Kaufman, V., Edlén, B.: J. Phys. Chem. Ref. Data 3, 825 (1974). https://doi.org/10.1063/1.3253149 Nave, G., Johansson, S., Learner, R.C.M., Thorne, A.P., Brault, J.W.: Astrophys. J. Suppl. Ser. 94, 221 (1994). and references therein https://doi.org/10.1086/192079 Nave, G., Johansson, S.: Astrophys. J. Suppl. Ser. 204, 1 (2013). https://doi.org/10.1088/0067-0049/204/1/1 Sansonetti, J.E., Reader, J., Sansonetti, C.J., Acquista, N.: Atlas of the Spectrum of a Platinum/Neon Hollow-Cathode Lamp in the Region 1130–4330 Å. J. Res. Natl. Inst. Stand. Technol. 97, 1–212 (1992). https://doi.org/10.6028/jres.097.002; online database available at https://www.nist.gov/pml/ultravioletspectrum-platinum-lamp Nave, G., Sansonetti, C.J.: J. Opt. Soc. Am. B 21, 442 (2004). https://doi.org/10.1364/JOSAB.21.000442 Stahler, S.W., Palla, F.: The Formation of Stars. Wiley-VCH, Weinheim (2004) Sobel’man, I.I.: Introduction to the Theory of Atomic Spectra. Pergamon Press, New York (1972) Cox, A.N. (ed.): Allen’s Astrophysical Quantities, 4th edn. AIP Press, Springer, New York (2000) Weiss, A.W.: J. Quant. Spectrosc. Radiat. Transf. 18, 481 (1977). https://doi.org/10.1016/0022-4073(77)90046-2 Wiese, W.L.: Phys. Scr. T65, 188 (1996). https://doi.org/10.1088/ 0031-8949/1996/T65/028 Kramida, A.: Fusion Sci. Technol. 63, 313 (2013). https://doi.org/ 10.13182/FST13-A16437 Wiese, W.L., Fuhr, J.R.: J. Phys. Chem. Ref. Data 38, 565 (2009). https://doi.org/10.1063/1.3077727 Fuhr, J.R., Wiese, W.L.: J. Phys. Chem. Ref. Data 39, 013101 (2010). https://doi.org/10.1063/1.3286088
197 51. Kunze, H.-J.: Introduction to Plasma Spectroscopy. Springer, New York (2009) 52. Griem, H.R.: Spectral Line Broadening by Plasmas. Academic Press, New York (1974) 53. Roberts, J.R., Voigt, P.A.: J. Res. Natl. Bur. Stand. 75A, 291 (1971). https://doi.org/10.6028/jres.075A.028 54. Sobelman, I.I.: Atomic Spectra and Radiative Transitions, 2nd edn. Springer, Berlin, Heidelberg (1992) 55. Schlüter, D.: Z. Phys. 210, 80 (1968). https://doi.org/10.1007/ BF01379872 56. Hofsaess, D.: J. Quant. Spectrosc. Radiat. Transf. 19, 339 (1978). https://doi.org/10.1016/0022-4073(78)90068-7 57. Wilbers, A.T.M., Kroesen, G.M.W., Timmermans, C.J., Schram, D.C.: J. Quant. Spectrosc. Radiat. Transf. 45, 1 (1991). https://doi. org/10.1016/0022-4073(91)90076-3 58. Sukhorukov, V.L., Petrov, I.D., Schäfer, M., Merkt, F., Ruf, M.-W., Hotop, H.: J. Phys. B 45, 092001 (2012). https://doi.org/10.1088/ 0953-4075/45/9/092001 59. Ralchenko, Y., Stambulchik, E.: Databases for Atomic and Plasma Physics (2018). Available at https://doi.org/10.34933/wis.000400 60. VAMDC Consortium: http://www.vamdc.org
William C. Martin William C. Martin’s research included the measurement and energy-level analysis of atomic spectra. He also published a number of critical compilations of atomic spectroscopic data, including a large volume for the rare-earth elements. As a Scientist Emeritus at NIST, Dr. Martin continued to work on internet-accessible atomic spectra databases. He passed away in 2013. Wolfgang Wiese Wolfgang Wiese is a physicist with extensive research background in atomic spectroscopy and in the critical tabulation of atomic reference data. He has worked at the National Institute of Standards and Technology for more than 40 years and has led the Atomic Physics Division from 1978 to 2004. He has authored 6 data volumes on Atomic Transition Probabilities, 15 book chapters and about 225 shorter research papers. Alexander Kramida Alexander Kramida received his PhD from the Moscow Institute of Physics and Technology in 1981. He worked at the Institute for Spectroscopy of the Russian Academy of Sciences. Since 2003, he works at the National Institute of Standards and Technology, USA on analysis of atomic spectra and development of databases and tools for research in atomic and plasma physics.
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High Precision Calculations for Helium
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Gordon W. F. Drake
Contents 12.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 199
asymptotic expansions to states of higher angular momentum L. Radiative transitions are discussed and oscillator strengths tabulated for a wide variety of transitions.
12.2 12.2.1
The Three-Body Schrödinger Equation . . . . . . . . 199 Formal Mathematical Properties . . . . . . . . . . . . . . 200
Keywords
12.3 12.3.1 12.3.2 12.3.3 12.3.4
Computational Methods . . . . Variational Methods . . . . . . . Construction of Basis Sets . . . Calculation of Matrix Elements Other Computational Methods .
helium variational methods Hylleraas coordinates energies relativistic and quantum electrodynamic corrections asymptotic expansions quantum defects radiative transitions oscillator strengths
12.4 12.4.1
Variational Eigenvalues . . . . . . . . . . . . . . . . . . 205 Expectation Values of Operators and Sum Rules . . . . 205
12.5 12.5.1 12.5.2
Total Energies . . . . . . . . . . . . . . . . . . . . . . . . 205 Quantum Defect Extrapolations . . . . . . . . . . . . . . 210 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . 210
12.6 12.6.1 12.6.2
Radiative Transitions . . . . . . . . . . . . . . . . . . . . 213 Basic Formulation . . . . . . . . . . . . . . . . . . . . . . 213 Oscillator Strength Table . . . . . . . . . . . . . . . . . . 215
12.7
Future Perspectives . . . . . . . . . . . . . . . . . . . . . 215
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200 200 201 202 204
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Abstract
Exact analytic solutions to the Schrödinger equation are known only for hydrogen and other equivalent two-body systems (Chap. 9). However, very high precision approximations are now available for helium and other three-body systems that are essentially exact for all practical purposes. This chapter summarizes the computational methods and tabulates numerical results for the ground state and several excited states up to n D 10 and L D 7. General formulas are presented for the evaluation of matrix elements in Hylleraas coordinates. Relativistic and quantum electrodynamic effects are included, along with quantum defect extrapolations to states of higher n and G. W. F. Drake () Dept. of Physics, University of Windsor N9B 3P4 Windsor, ON, Canada e-mail: [email protected]
12.1 Introduction Exact analytic solutions to the Schrödinger equation are known only for atomic hydrogen and other equivalent twobody systems (Chap. 9). However, very high precision approximations are now available for helium, which are essentially exact for all practical purposes. This chapter summarizes the computational methods and tabulates numerical results for the ground state and several singly excited states. Similar methods can be applied to other three-body problems.
12.2 The Three-Body Schrödinger Equation The Schrödinger equation for a three-body system consisting of a nucleus of charge Ze, mass M , and two electrons of charge e and mass me is "
# 2 X 1 1 P 2 C V .R N ; R i / D E ; (12.1) P2 C 2M N 2me i D1 i
where P i D .„=i/ri and V .R N ; R i / D
Ze 2 Ze 2 e2 C jR N R 1 j jR N R 2 j jR 1 R 2 j (12.2)
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_12
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depends only on the relative particle separations. Since the where ij D mi mj =.mi Cmj /, and N denotes the wave funccenter of mass (c.m.) is then an ignorable coordinate, it can tion averaged over a sphere centered at rij D 0. If vanishes be eliminated by defining the relative particle coordinates at rij D 0, then its leading dependence on rij is of the form rijl Ylm .r ij /, for some integer l > 0 [2]. Equation (12.7) apr i D Ri RN plies to any Coulombic system. The electron–nucleus cusp in the wave functions for hydrogen provides a simple example. to obtain " # Three-particle coalescences. Three-particle coalescences 2 1 X 2 1 p C p p C V .r 1 ; r 2 / D E ; (12.3) are described by the Fock expansion [3], as recently dis2 i D1 i M 1 2 cussed by Myers et al. [4]. For the S-states of He-like ions, the expansion has the form where D me M=.me C M / is the electron reduced mass, 1 Œj=2 X X and the term Hmp D p 1 p 2 =M is called the mass polar; r / D Rj .ln R/k j;k ; (12.8) .r 1 2 ization operator. For computational purposes, it is usual to j D0 kD0 measure distance in units of a D .me =/a0 and energies in units of e 2 =a D 2.=me /R1 , so that Eq. (12.3) assumes where [ ] denotes “greatest integer in”, and R D .r 2 C r 2 /1=2 1 2 the dimensionless form is the hyperradius. The leading coefficients are " # 2 1X 2 0;0 D 1 ; r r r 2 C V . 1 ; 2 / D " ; (12.4)
2 i D1 i M 1 1;0 D Zr1 C Zr2 12 r12 =R ; 2 r1 r2 2 D 2Z : (12.9) where i D r i =a , " D E=.e =a /, and 2;1 3 R2 Z Z 1 C :
1 2 j 1 2 j
(12.5) The next term 2;0 is known in terms of a lengthy expression [4, 5], but higher terms have not yet been obtained in closed form. The Fock expansion has been proved converThe limit =M ! 0 defines the infinite nuclear mass gent for all R < 12 [6] and extended to pointwise convergence problem with eigenvalue "0 and eigenfunction 0 . If the mass for all R [7, 8]. polarization term is treated as a small perturbation, then the total energy assumes the form Asymptotic form. The long-range behavior of many electron wave functions has been studied from several points e2 2 E D "0 C "2 C ; (12.6) of view [9–11]. The basic result of [12] is that at large dis"1 C M M me a 0 tances, the one-electron density behaves as V . 1 ; 2 / D
where "1 D h0 jr 1 r 2 j0 i determines the first-order spe(12.10)
1=2 .r/ r Z =t 1 e t r ; cific mass shift, and "2 is the second-order coefficient. The common .=me /"0 mass scaling of all eigenvalues deter- where t D .2I /1=2 , I is the first ionization potential (in a.u.), 1 1 mines the normal mass shift. Since =m D 1 =M , the and Z D Z N C 1 is the screened nuclear charge seen by shift is .=M /"0 . the outer most electron. For hydrogenic systems with principal quantum number n, I1 D .Z /2 =2n2 .
12.2.1
Formal Mathematical Properties
Two-particle coalescences. The exact nonrelativistic wave function for any many-body system contains discontinuities or cusps in the spherically averaged radial derivative with respect to rij as rij ! 0, where rij D jr i rj j is any interparticle coordinate. If the masses and charges are mi and qi , respectively, then the discontinuities are given by the Kato cusp condition [1] ! N 2 @ D ij qi qj .rij D 0/ ; (12.7) „ @rij rij D0
12.3 Computational Methods 12.3.1 Variational Methods Most high-precision calculations for the bound states of three-body systems such as helium are based on the Rayleigh–Ritz variational principle. For any normalizable trial function tr , the quantity Etr D
htr jH jtr i htr jtr i
(12.11)
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satisfies the inequality Etr E1 , where E1 is the true groundstate energy. Thus, Etr is an upper bound to E1 . The inequality is easily proved by expanding tr in the complete basis set of eigenfunctions 1 , 2 , 3 , of H with eigenvalues E1 < E2 < E3 < , so that tr D
1 X
(12.12)
where the ci are expansion coefficients. This can always be done in principle, even though the exact i are not actually known. If tr is normalized so that htr jtr i D 1, then P1 2 i D1 jci j D 1, and 2
2
Etr D jc1 j E1 C jc2 j E2 C jc3 j E3 C D E1 C jc2 j2 .E2 E1 / C jc3 j2 .E3 E1 / C E1 ;
λ3
E3 λ2
ci i ;
i D1
2
λ5
λ4
E E5 E4
E2
λ1 E1
1
2
3 N
4
5
Fig. 12.1 Diagram illustrating the Hylleraas–Undheim–MacDonald
(12.13) Theorem. The p , p D 1; : : : ; N are the variational eigenvalues for an
which proves the theorem. The basic idea of variational calculations, then, is to write tr in some arbitrarily chosen mathematical form with variational parameters (subject to normalizability and boundary conditions at the origin and infinity) and then adjust the parameters to obtain the minimum value of Etr . The minimization problem for the case of linear variational coefficients can be solved algebraically. For example, let j k ˛r1 ˇr2 e (12.14) p .˛; ˇ/ D r1i r2 r12
N -dimensional basis set, and the Ei are the exact eigenvalues of H . The highest p lie in the continuous spectrum of H
eigenvalue interleaving theorem, which states that as the dimensions of H and O are progressively increased by adding an extra row and column, the N old eigenvalues p fall between the N C 1 new ones. Consequently, as illustrated in Fig. 12.1, all eigenvalues numbered from the bottom up must move inexorably downward as N is increased. Since the exact spectrum of bound states is obtained in the limit N ! 1, no p can cross the corresponding exact Ep on its way down. Thus, p Ep for every finite N .
denote the members of a basis set, where p is an index labeling distinct triplets of nonnegative integer values for the powers fi; j; kg, and ˛; ˇ are (for the moment) fixed con12.3.2 Construction of Basis Sets stants determining the distance scale. If tr is expanded in the form Since the Schrödinger equation (12.4) is not separable in N the electron coordinates, basis sets which incorporate the X tr D cp p .˛; ˇ/ ; (12.15) r12 D jr 1 r 2 j interelectron coordinate are most efficient. pD1 The necessity for r12 terms also follows from the Fock expansion in Eq. (12.8). A basis set constructed from terms of then the solution to the system of equations @Etr =@cp D the form Eq. (12.14) is called a Hylleraas basis set [15]. (The 0, p D 1; : : : ; N , is exactly equivalent to solving the N basis set is often expressed in terms of the equivalent varidimensional generalized eigenvalue problem ables s D r1 C r2 , t D r1 r2 , u D r12 .) With p .˛; ˇ/ defined as in Eq. (12.14), the general form H c D Oc ; (12.16) for a state of total angular momentum L is where c is a column vector of coefficients cp , and H and O ŒL=2 XX have matrix elements Hpq D hp jH jq i and Opq D hp jq i. tr D Cp;l1 p .˛; ˇ/r1l1 r2l2 YlM .rO 1 ; rO 2 / 1 Ll1 L There are N eigenvalues 1 ; 2 ; : : : N , of which the lowest l1 D0 p is an upper bound to E1 . ˙ exchange ; (12.17) Extension to excited states. By the Hylleraas–Undheim– where MacDonald (HUM) theorem [13, 14], the remaining eigenX YlM .rO 1 ; rO 2 / D Yl1 m1 .rO 1 /Yl2 m2 .rO 2 / values 2 ; 3 ; : : : are also upper bounds to the exact energies 1 l2 L m1 ;m2 E2 ; E3 ; : : :, provided that the spectrum is bounded from be hl1 l2 m1 m2 jLM i low. The HUM theorem is a consequence of the matrix
(12.18)
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is the vector coupled product of angular momenta l1 , l2 for be transformed to the product of a three-dimensional angular the two electrons. The sum over p in Eq. (12.17) typically in- integral (ang) and a three-dimensional radial integral (rad) cludes all terms in Eq. (12.14) with i C j C k ˝, where ˝ is over r1 , r2 , and r12 . The transformation is an integer determining a so-called Pekeris shell of terms, and “ Z2 Z2 Z the exchange term denotes the interchange of r1 and r2 with dr 1 dr 2 D d d'1 sin 1 d1 .C/ for singlet states and ./ for triplet states. Convergence is studied by progressively increasing ˝. The number of terms is 0 0 0 rZ 1 1 1 1 Cr2 Z Z N D 6 .˝ C 1/.˝ C 2/.˝ C 3/ : r1 dr1 r2 dr2 r12 dr12 ; (12.20) Basis sets of this type were used by many authors, culmi0 0 jr1 r2 j nating in the extensive high-precision calculations of Pekeris and coworkers [16] for low-lying states, using as many as 1078 terms. Their accuracy is not easily surpassed because where 1 , '1 are the polar angles of r 1 , and is the angle of the rapid growth of N with ˝ and because of numeri- of rotation of the triangle formed by r 1 , r 2 , and r 12 about cal linear dependence in the basis set for large ˝. Recently, the r 1 direction. The polar angles 2 , '2 are then dependent their accuracy has been surpassed by two principal meth- variables. The basic angular integral is ods. The first explicitly includes powers of logarithmic and hYl1 m1 .1 ; '1 /Yl2 m2 .2 ; '2 /iang half-integral terms in p , as suggested by the Fock expanD 2ıl1 l2 ım1 m2 Pl1 .cos / ; (12.21) sion [17–19]. The most accurate calculation to date is that of Schwartz [20] who obtained 35 significant figures for the nonrelativistic ground-state energy of helium. This is partic- where cos rO 1 rO 2 denotes the radial function ularly effective for S-states. The second focuses directly on 2 r 2 C r22 r12 the multiple distance scales required for an accurate reprecos D 1 ; (12.22) 2r1 r2 sentation of the wave function by writing the trial function in terms of the double basis set [21] and Pl .cos / is a Legendre polynomial. The angular integral ŒL=2 i X Xh .1/ over vector-coupled spherical harmonics is [26] .2/ Cp;l1 p .˛1 ; ˇ1 / C Cp;l1 p .˛2 ; ˇ2 / tr D l1 D0 p
r1l1 r2l2 Yl1 l2 L .r 1 ; r 2 /
0
˙ exchange ;
.rO 1 ; rO 2 /iang hYlM0 l 0 L0 .rO 1 ; rO 2 / YlM 1 l2 L 1 2 X D ıL;L0 ıM;M 0 C P .cos / ;
(12.19)
where each p .˛; ˇ/ is of the form of Eq. (12.14) but with different values for the distance scales ˛1 ; ˇ1 and ˛2 ; ˇ2 in the two sets of terms. They are determined by a complete minimization of Etr with respect to all four parameters, producing a natural division of the basis set into an asymptotic sector and a close-range correlation sector. The method produces a dramatic improvement in accuracy for higher-lying Rydberg states (where variational methods typically deteriorate rapidly in accuracy) and is also effective for low-lying S-states [22, 23]. Nonrelativistic energies accurate to 1 part in 1016 are obtainable with modest computing resources. Another version of the variational method is the quasirandom (or stochastic) method in which nonlinear exponential parameters for all three of r1 , r2 , and r12 are chosen at random from certain specified intervals [24, 25]. The method is remarkably accurate and efficient for low-lying states but subject to severe roundoff error.
(12.23)
where C D 12 Œ.2l1 C 1/.2l10 C 1/.2l2 C 1/.2l20 C 1/1=2 .1/LC .2 C 1/ ! l10 l1 l20 l2 0 0 0 0 0
!( L 0
l1 l20
) l2 ; l10 (12.24)
and the sum over includes all nonvanishing terms. This can be extended to general matrix elements of tensor operators by further vector coupling [26].
Radial integrals. Table 12.1 lists formulas for the radial integrals arising from matrix elements of H , as well as those from the Breit interaction (Sect. 22.1.3). Although they can all be written in closed form, some have been expressed as 12.3.3 Calculation of Matrix Elements infinite series in order to achieve good numerical stability. The exceptions are formulas 5 and 10 in the table, which The three-body problem has the unique advantage that the became unstable as ˛ ! ˇ. More elaborate techniques for full six-dimensional volume element (in the c.m. frame) can these are discussed in [27]. Other cases can be derived by
12
High Precision Calculations for Helium
203 log
c ˛r1 ˇr2 c Table 12.1 Formulas for the radial integrals I0 .a; b; cI ˛; ˇ/ D hr1a r2b r12 e irad and I0 .a; b; cI ˛; ˇ/ D hr1a r2b r12 ln r12 e ˛r1 ˇr2 irad . .n/ D Pn1 1 C kD1 k is the digamma function, 2 F1 .a; bI cI z/ is the hypergeometric function, and s D a C b C c C 5. Except as noted, the formulas apply for a 1, b 1, c 1 ˛Cˇ ˛Cˇ 2 2 1. I0 .2; 2; 1I ˛; ˇ/ D ln C ln ˛ ˇ ˇ ˛ Œ.cC1/=2 X 2 cC2 2. I0 .a; b; cI ˛; ˇ/ D .c 1; s 0/ ŒFaC2iC2; bCc2iC2 .˛; ˇ/ C FbC2iC2; aCc2iC2 .ˇ; ˛/ c C 2 iD0 2i C 1 8 q j X .p C j /Š qŠ ˇ ˆ ˆ q 0; p 0 ˆ ˆ pC1 ˇ qC1 ˆ .˛ C ˇ/ j Š ˛ C ˇ ˆ j D0 < 1 j C1 X where Fp;q .˛; ˇ/ D pŠ ˛ jŠ ˆ q < 0; p 0 ˆ pCqC2 ˆ .j q/Š ˛ C ˇ ˆ ˆ˛ j DpCqC1 ˆ : a 0 pj 12 in a.u., and f0n is the 11 S n1 P oscillator strength where is Euler’s constant, and is the radius of an infinites(Sect. 12.6.1). imal sphere about r12 D 0 that is excluded from the range of integration (i.e. the principal value of the divergent integral). 12.5 Total Energies These two-electron QED corrections are known as the ArakiSucher terms [36, 37]. As discussed in Chaps. 22 and 28, relativistic and QED corFor a highly excited 1snl state, EL;2 ! 0, hı.r 1 / C rections must be added to the nonrelativistic eigenvalues of ı.r 2 /i ! Z 3 =, ln k0 ! ln k0 .1s/ D 2:984 128 555, and Sect. 12.4 before a meaningful comparison with measured EL;1 reduces to the Lamb shift of the 1s core state
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G. W. F. Drake
Table 12.2 Nonrelativistic eigenvalue coefficients "0 and "1 for helium State 1S 2S 2P 3S 3P 3D 4S 4P 4D 4F 5S 5P 5D 5F 5G 6S 6P 6D 6F 6G 6H 7S 7P 7D 7F 7G 7H 7I 8S 8P 8D 8F 8G 8H 8I 8K 9S 9P 9D 9F 9G 9H 9I 9K 10 S 10 P 10 D 10 F 10 G 10 H 10 I 10 K a
"0 .n 1L/ 2:903 724 377 034 119 60a 2:145 974 046 054 417 42a 2:123 843 086 498 101 36a 2:061 271 989 740 908 65a 2:055 146 362 091 943 54a 2:055 620 732 852 246 49a 2:033 586 717 030 725 44a 2:031 069 650 450 240 71a 2:031 279 846 178 685 00a 2:031 255 144 381 748 61a 2:021 176 851 574 363.5/ 2:019 905 989 900 83.2/ 2:020 015 836 159 984.4/ 2:020 002 937 158 742 7.5/ 2:020 000 710 898 584 71.1/ 2:014 563 098 446 60.1/ 2:013 833 979 671 73.2/ 2:013 898 227 424 286.5/ 2:013 890 683 815 549 7.3/ 2:013 889 345 387 313 22.3/ 2:013 889 034 754 279 72 2:010 625 776 210 87.2/ 2:010 169 314 529 35.2/ 2:010 210 028 457 98.1/ 2:010 205 248 074 013.1/ 2:010 204 386 224 772 55.7/ 2:010 204 182 806 482 04.2/ 2:010 204 120 606 191 32 2:008 093 622 105 61.4/ 2:007 789 127 133 22.2/ 2:007 816 512 563 811.7/ 2:007 813 297 115 014 1.6/ 2:007 812 711 494 024 1.1/ 2:007 812 571 828 655 81.1/ 2:007 812 528 549 584 59 2:007 812 512 570 229 31 2:006 369 553 107 85.3/ 2:006 156 384 652 86.5/ 2:006 175 671 437 641.6/ 2:006 173 406 897 324 6.8/ 2:006 172 991 627 586 3.2/ 2:006 172 891 903 619 14.2/ 2:006 172 860 732 382 57 2:006 172 849 096 329 78 2:005 142 991 748 00.8/ 2:004 987 983 802 22.4/ 2:005 002 071 654 250.6/ 2:005 000 417 564 668 2.9/ 2:005 000 112 764 318 0.3/ 2:005 000 039 214 394 52.2/ 2:005 000 016 086 516 19 2:005 000 007 388 375 88
"1 .n 1L/ 0:159 069 475 085 84 0:009 503 864 419 28 0:046 044 524 937.1/ 0:002 630 567 097 7.1/ 0:014 548 047 097.1/ 0:000 249 399 992 1.1/ 0:001 073 641 226 6.1/ 0:006 254 923 554 3.1/ 0:000 129 175 188 7.8/ 0:000 010 024 269 4.2/ 0:000 538 860 360 5.1/ 0:003 230 021 84.2/ 0:000 071 883 131.6/ 0:000 005 704 294 6.4/ 0:000 001 404 413 6 0:000 307 704 277.1/ 0:001 878 058 536.1/ 0:000 043 412 268 9.9/ 0:000 003 482 257.7/ 0:000 000 898 579 9.7/ 0:000 000 290 347 1 0:000 191 925 025.1/ 0:001 186 152 30.1/ 0:000 028 027 840.2/ 0:000 002 262 00.4/ 0:000 000 598 396 3.3/ 0:000 000 201 097 8 0:000 000 077 775 5 0:000 127 650 436.1/ 0:000 796 195 83.5/ 0:000 019 076 181.1/ 0:000 001 545 48.1/ 0:000 000 415 004 0.1/ 0:000 000 142 649 2.3/ 0:000 000 056 935 9 0:000 000 025 111 3 0:000 089 149 638 7.7/ 0:000 559 978 028.2/ 0:000 013 542 185.3/ 0:000 001 099 967 1.3/ 0:000 000 298 267 2.1/ 0:000 000 104 002 2 0:000 000 042 313 6 0:000 000 019 151 6 0:000 064 697 214.3/ 0:000 408 649 426 3 0:000 009 947 506 0.6/ 0:000 000 809 442.9/ 0:000 000 220 982.2/ 0:000 000 077 806 7 0:000 000 032 059 0.1/ 0:000 000 014 751 4
Aznabaev et al. [31] with a 22000-term basis set.
"0 .n 3L/
"1 .n 3L/
2:175 229 378 236 791 31 2:133 164 190 779 283 21a 2:068 689 067 472 457 19 2:058 081 084 274 275 33a 2:055 636 309 453 261 33a 2:036 512 083 098 236 30a 2:032 324 354 296 630 33a 2:031 288 847 501 795 54a 2:031 255 168 403 245 39a 2:022 618 872 302 312 27.1/ 2:020 551 187 256 25.1/ 2:020 021 027 446 911.5/ 2:020 002 957 377 369 4.4/ 2:020 000 710 925 343 92.1/ 2:015 377 452 992 862 19.3/ 2:014 207 958 773 74.1/ 2:013 901 415 453 792.7/ 2:013 890 698 348 532 0.2/ 2:013 889 345 416 952 96.3/ 2:013 889 034 754 301 55 2:011 129 919 527 626 21.4/ 2:010 404 960 007 94.2/ 2:010 212 105 955 595.2/ 2:010 205 258 374 865.1/ 2:010 204 386 250 217 93.6/ 2:010 204 182 806 512 04.1/ 2:010 204 120 606 191 340 2:008 427 122 064 721 42.6/ 2:007 947 013 771 12.1/ 2:007 817 934 711 706.3/ 2:007 813 304 535 090 8.5/ 2:007 812 711 514 424 82.9/ 2:007 812 571 828 685 73.1/ 2:007 812 528 549 584 61 2:007 812 512 570 229 306 2:006 601 516 715 010 67.3/ 2:006 267 267 366 41.4/ 2:006 176 684 884 697.2/ 2:006 173 412 365 043 0.7/ 2:006 172 991 643 665 0.3/ 2:006 172 891 903 645 88.2/ 2:006 172 860 732 382 60 2:006 172 849 096 329 780 2:005 310 794 915 611 3.2/ 2:005 068 805 497 8.1/ 2:005 002 818 080 232.8/ 2:005 000 421 686 603 6.7/ 2:005 000 112 777 003 1.4/ 2:005 000 039 214 417 41.2/ 2:005 000 016 086 516 22 2:005 000 007 388 375 88
0:007 442 130 706 04 0:064 572 425 024.4/ 0:001 896 211 617 81 0:018 369 001 636.2/ 0:000 025 322 839.1/ 0:000 742 661 516 18 0:007 555 178 98.1/ 0:000 029 442 651.2/ 0:000 009 669 639 6 0:000 363 697 136 49 0:003 810 911 035.1/ 0:000 019 568 85.1/ 0:000 005 406 490 0.5/ 0:000 001 404 001 3 0:000 204 329 479 10 0:002 184 346 463.1/ 0:000 012 742 22.3/ 0:000 003 268 458 6.8/ 0:000 000 898 123 7.7/ 0:000 000 290 346 7 0:000 125 981 736 89 0:001 366 500 8.3/ 0:000 008 563 121.3/ 0:000 002 110 58.3/ 0:000 000 598 005.1/ 0:000 000 201 097 3 0:000 000 077 775 5 0:000 083 070 552 34 0:000 911 053 5.3/ 0:000 005 971 123 4.3/ 0:000 001 436 452.2/ 0:000 000 414 690 4 0:000 000 142 648 7.2/ 0:000 000 056 935 9 0:000 000 025 111 3 0:000 057 628 311 52 0:000 637 531 359.6/ 0:000 004 306 538.6/ 0:000 001 019 651.2/ 0:000 000 298 019 8.1/ 0:000 000 104 001 9 0:000 000 042 313 6.1/ 0:000 000 019 151 6 0:000 041 598 811 52 0:000 463 433 718.8/ 0:000 003 198 298.8/ 0:000 000 748 926 4.2/ 0:000 000 220 785.3/ 0:000 000 077 806 2 0:000 000 032 058 9.2/ 0:000 000 014 751 4
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High Precision Calculations for Helium
207
Table 12.3 Eigenvalue coefficients "2 for helium State 1S 2S 2P 3S 3P 3D 4S 4P 4D 4F 5S 5P 5D 5F 5G 6S 6P 6D 6F 6G 6H 7S 7P 7D 7F 7G 7H 7I 8S 8P 8D 8F 8G 8H 8I 8K 9S 9P 9D 9F 9G 9H 9I 9K 10 S 10 P 10 D 10 F 10 G 10 H 10 I 10 K
1
"2 .n L/ 0:470 391 870.1/ 0:135 276 864.1/ 0:168 271 22.7/ 0:058 599 3124.4/ 0:066 047 859.3/ 0:057 201 299.9/ 0:032 522 293.2/ 0:035 159 71.6/ 0:032 150 91.2/ 0:031 274 336.4/ 0:020 647 26.9/ 0:021 847 6.3/ 0:020 510 1.2/ 0:020 013 498.6/ 0:020 003 560 8 0:014 261 796.4/ 0:014 902 86.9/ 0:014 199 4.2/ 0:013 896 984.2/ 0:013 891 179.6/ 0:013 889 619 1 0:010 438 2.2/ 0:010 818 6.2/ 0:010 405 09.3/ 0:010 209 2.3/ 0:010 205 61.5/ 0:010 204 590.2/ 0:010 204 276 7 0:007 968 944.3/ 0:008 211 7.5/ 0:007 950 7.4/ 0:007 815 9.3/ 0:007 813 563.1/ 0:007 812 855.4/ 0:007 812 642 9 0:007 812 563 0 0:006 282 5136.1/ 0:006 445 7.2/ 0:006 270 99.7/ 0:006 175 20.1/ 0:006 173 579 6.1/ 0:006 173 104.2/ 0:006 172 945 9.1/ 0:006 172 887 6 0:005 079 8362.8/ 0:005 197.1/ 0:005 072 4.4/ 0:005 001 76.2/ 0:005 000 55.2/ 0:005 000 193 5.2/ 0:005 000 080 3.4/ 0:005 000 036 9
3
"2 .n L/ 0:057 495 847 9.2/ 0:204 959 88.1/ 0:040 455 850 5.5/ 0:070 292 710.2/ 0:054 737 73.1/ 0:025 628 633 8.1/ 0:036 129 973.2/ 0:030 747 891.7/ 0:031 277 992 1.3/ 0:017 322 734 96 0:022 166 61.9/ 0:019 706 2.2/ 0:020 016 561.4/ 0:020 003 564 6 0:012 411 399 1.3/ 0:015 033 58.5/ 0:013 707 27.1/ 0:013 899 22.3/ 0:013 891 184.8/ 0:013 889 619 0 0:009 304 443 3.3/ 0:010 879.2/ 0:010 085 212.1/ 0:010 210 7.3/ 0:010 205 61.5/ 0:010 204 587.2/ 0:010 204 276 8 0:007 224 770 5.3/ 0:008 248 7.6/ 0:007 731 59.2/ 0:007 817 0.2/ 0:007 813 568.3/ 0:007 812 859.5/ 0:007 812 642 9 0:007 812 563 0 0:005 768 028 5.1/ 0:006 464 936 9.1/ 0:006 115 2.1/ 0:006 176 025 4.7/ 0:006 173 592.4/ 0:006 173 101.2/ 0:006 172 946 0.2/ 0:006 172 887 6 0:004 709 453 0.1/ 0:005 206 7.1/ 0:004 958 0.8/ 0:005 002 386.2/ 0:005 000 55.2/ 0:005 000 193 5.1/ 0:005 000 081.1/ 0:005 000 036 8
Table 12.4 Values of the reduced electron mass ratio =M , including binding energy corrections =M 104 5.443 205 7507(5) 2.723 695 0538(2) 1.819 212 0630(16) 1.370 745 634 66(5) 0.912 167 5131(2) 0.782 020 2727(6) 0.608 820 388(3)
Isotope 1 H 2 D 3 He 4 He 6 Li 7 Li 9 Be
(Sect. 29.7.1). Thus, EL;1 represents the electron–nucleus part of the QED shift with the factor of Z 3 = replaced by the correct electron density at the nucleus. Accurate values of ln k0 for two-electron atoms and ions are tabulated in [38]. For the low-lying S-states and P -states of helium [25, 39], ln k0 .11S/ D 2:983 865 861 ;
(12.42a)
1
(12.42b)
3
(12.42c)
1
(12.42d)
3
(12.42e)
ln k0 .2 S/ D 2:980 118 365 ; ln k0 .2 S/ D 2:977 742 459 ; ln k0 .2 P / D 2:983 803 377 ; ln k0 .2 P / D 2:983 690 995 :
For a 1snl state with large l, the asymptotic expansion [40, 41] 1 Z1 4 ln k0 .nl/ ln k0 .1snl/ ln k0 .1s/ C 3 n Z C 0:316 205.6/Z 6hr 4 inl C ˇ.1snl/
(12.43)
becomes essentially exact. Here, ln k0 .nl/ is the one-electron Bethe logarithm [42], and hr 4 inl D
16.Z 1/4 Œ3n2 l.l C 1/ : (12.44) .2l 1/2l.2l C 1/.2l C 2/.2l C 3/
The correction ˇ.1snl/ for higher order terms is ˇ.1snl 1 L/ D 95:8.8/hr 6i 845.19/hr 7i C 1406.50/hr 8i ;
(12.45)
ˇ.1snl 3 L/ D 95:1.9/hr 6i 841.23/hr 7i C 1584.60/hr 8i :
(12.46)
For example, for the 1s4f 1 F state, ˇ.4 1 F/ D 2:984 127 1493.3/. For higher Z, 1=Z, expansions of [38] should be used. 4. Relativistic finite mass corrections of O.˛2 =M /. Relativistic finite mass corrections come from two sources. First, a transformation to relative coordinates as in Eq. (12.3) is applied to the pairwise Breit interactions among the three
12
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G. W. F. Drake
Table 12.5 Nonrelativistic eigenvalues E D "0 C .=M /"1 C .=M /2 "2 for helium-like ions (in units of e 2 =a ) "0 .1 1S/ 0:527 751 016 544 377 2:903 724 377 034 119 5 7:279 913 412 669 305 9 13:655 566 238 423 586 7
Atom H He LiC BeCC
"1 .1 1S/ 0:032 879 781 852 30 0:159 069 475 085 84 0:288 975 786 393 99 0:420 520 303 439 44
"2 .1 1S/ 0:059 779 492 64.1/ 0:470 391 870.1/ 1:277 369 377 6.2/ 2:491 572 858 1.1/
Table 12.6 Expectation values of various operators for He-like ions for the case M D 1 (in a.u.) H 11:913 699 678 05.6/ 25:202 025 291 2.1/ 0:687 312 967 569 2:710 178 278 444.1/ 4:412 694 497 992.2/ 0:683 261 767 652 0:311 021 502 214 1:116 662 824 6.1/ 0:155 104 152 58.3/ 0:382 627 890 340 0:253 077 567 065 0:164 552 872 86.3/ 0:002 737 992 3.3/ 2:462 558 614.3/ 0:008 875 022 10.1/
Quantity hr12 i 2 hr12 i hr 1 r 2 i hr1 i hr12 i h1=r1 i h1=r12 i h1=r12 i 2 h1=r12 i h1=r1 r2 i h1=r1 r12 i hı.r1 /i hı.r12 /i hp 4 i hHoo i=˛ 2
LiC
He 1:193 482 995 019 2:516 439 312 833 0:064 736 661 398 0:929 472 294 874 1:422 070 255 566 1:688 316 800 717 0:945 818 448 800 6:017 408 867 0.1/ 1:464 770 923 350.1/ 2:708 655 474 480 1:920 943 921 900 1:810 429 318 49.3/ 0:106 345 371 2.2/ 54:088 067 230.2/ 0:139 094 690 556.1/
particles, generating the new terms [43] me D oo C so C 2 Hso ; M where
Z˛ 2 me X 1 pj p i C rO i rO i pj p i ; oo D 2M ri i;j (12.47) Z˛ me X 1 so D r i pj si : M r3 i ¤j i 2
(12.48)
Second, the mass polarization term Hmp in Eq. (12.3) generates second-order cross terms between Hmp and Hrel . If the wave functions are calculated by solving Eq. (12.4) in scaled atomic units, the Hmp correction is then automatically included to all orders, and the mass-corrected relativistic energy shift is (in units of e 2 =a0 ) 3 Erel D Hmass C HD C Hssc C Hoo me me 2me C oo C 1 C Hso C Hsoo C Hss C so ; M (12.49)
0:446 279 011 201 0:927 064 803 063 0:017 253 390 330 0:572 774 149 971 0:862 315 375 456 2:687 924 397 413 1:567 719 559 137 14:927 623 721 4.2/ 4:082 232 787 55.2/ 7:011 874 111 824.1/ 5:069 790 932 379 6:852 009 434 4.1/ 0:533 722 537 1.9/ 310:547 150 179.6/ 0:427 991 611 178.9/
BeCC 0:232 067 315 531 0:477 946 525 143 0:006 905 947 040 0:414 283 328 006 0:618 756 314 066 3:687 750 406 344 2:190 870 773 906 27:840 105 671 33.2/ 8:028 801 781 824.1/ 13:313 954 940 144.1/ 9:717 071 116 528 17:198 172 544 74.3/ 1:522 895 354 1.2/ 1047:278 491 476.2/ 0:878 768 694 709.1/
˛ 6 mc 2 / [44] and the spin-dependent parts up to order ˛ 5 Ry. The theory and a comparison with experiment are reviewed by Pachucki et al. [45]. The more accurate spin-dependent parts are particularly important in connection with the 2 3 PJ fine structure splittings [46] and determinations of the fine structure constant from atomic spectroscopy [47]. Recoil terms of order ˛ 4 m=M Ry are of importance in the interpretation of isotope shift measurements. The isotope shift has been extensively used to determine the nuclear charge radii of short-lived species such as the halo nuclei 6 He and 8 He [48]. For the energy terms of order ˛ 4 Ry, the dominant electron–nucleus part is known from the one-electron Lamb shift to be 427 ˛ 0 3 2 ln 2 C 0:538 931 EL;1 D Z˛ Z˛ 96 hı.r 1 / C ı.r 2 /i C O.˛ 5 / ; (12.50) and the electron–electron logarithmic part is [49] 0 EL;2 D ˛ 4 ln ˛ 1 hı.r12 /i ;
(12.51)
0 As an example, EL;1 contributes 50:336 MHz and 88:267 MHz to the 2 1P 2 1S and 2 3P J 2 3S 1 transition frequencies, respectively, while the differences between with =me D 1 =M . The difference Erel hHrel i1 cal- theory and experiment are 1 MHz and 7 MHz for the culated for infinite nuclear mass is the relativistic finite mass two cases [35]. Thus, two-electron corrections (for example, relativistic corrections of relative order Z˛ to EL;2 ) are evcorrection. idently small. Table 12.7 lists the calculated ionization energies for all 5. Higher-order corrections. All higher-order terms are now known in their entirety up to order ˛ 4 Ry (or equivalently states of helium up to n D 10 and L D 7. Some of the en-
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High Precision Calculations for Helium
209
Table 12.7 Total ionization energies for 4 He, calculated with RM D 3 289 391 006:600 MHz State 1S 2S 2P 3S 3P 3D 4S 4P 4D 4F 5S 5P 5D 5F 5G 6S 6P 6D 6F 6G 6H 7S 7P 7D 7F 7G 7H 7I 8S 8P 8D 8F 8G 8H 8I 8K 9S 9P 9D 9F 9G 9H 9I 9K 10 S 10 P 10 D 10 F 10 G 10 H 10 I 10 K a b c
E.n 1LL / 5 945 204 173.(36)a 960 332 038.0(1.9)a 814 709 146.46(40)a 403 096 132.(1) 362 787 968.23(5) 365 917 748.661(19)c 220 960 311.0(7) 204 397 210.75(4) 205 783 935.665(9)c 205 620 797.139 9(15) 139 318 258.4(3) 130 955 541.836(19) 131 680 211.860(5)c 131 595 041.497 5(9) 131 580 320.128 4(2) 95 807 681.97(18) 91 009 810.521(11) 91 433 655.795(3)c 91 383 852.028 7(5) 91 374 997.957 91(15) 91 372 940.609 15(5) 69 904 819.75(11) 66 901 127.504(7) 67 169 717.127 2(20)c 67 138 158.555 3(4) 67 132 455.945 34(10) 67 131 109.012 98(4) 67 130 692.477 700(15) 53 246 283.07(7) 51 242 587.369(5) 51 423 248.121 5(10)c 51 402 021.627 5(3) 51 398 146.236 30(8) 51 397 221.577 55(3) 51 396 931.941 336(10) 51 396 822.732 870(10) 41 903 979.20(5) 40 501 246.374(3) 40 628 480.253 2(9) 40 613 531.507 8(2) 40 610 783.197 32(7) 40 610 123.034 47(2) 40 609 914.515 382(10) 40 609 835.096 177(10) 33 834 679.62(4) 32 814 665.300(2) 32 907 601.904 8(6)c 32 896 683.095 55(13) 32 894 665.769 82(4) 32 894 178.908 49(2) 32 894 024.239 94(1) 32 893 964.925 90
E.n 3LL1 /
E.n 3LL /
E.n 3LLC1 /
876 078 646.9428(8)b 451 903 472.(2) 382 109 901.7(1) 366 018 892.691(23)c 240 210 377.3(7) 212 658 040.5(2) 205 842 547.795(11)c 205 621 029.597 1(19) 148 807 311.8(4) 135 203 442.99(11) 131 714 043.872(6)c 131 595 195.231 6(12) 131 580 370.942 0(3) 101 166 442.3(2) 93 472 041.50(6) 91 454 440.567(4)c 91 383 954.298 6(7) 91 375 027.414 6(2) 91 372 961.810 38(8) 73 222 269.28(12) 68 452 586.60(4) 67 183 264.565 5(20)c 67 138 228.556 6(5) 67 132 474.519 29(16) 67 131 122.364 48(6) 67 130 702.487 20(3) 55 440 834.08(8) 52 282 092.00(3) 51 432 523.230 4(20)c 51 402 071.108 6(3) 51 398 158.691 23(12) 51 397 230.522 15(5) 51 396 938.646 95(2) 51 396 827.927 806(10) 43 430 382.87(5) 41 231 283.22(2) 40 635 090.445 8(12) 40 613 567.554 0(2) 40 610 791.950 93(9) 40 610 129.316 61(4) 40 609 919.224 968(17) 40 609 838.744 754(10) 34 938 883.86(4) 33 346 784.339(14) 32 912 470.747 1(9)c 32 896 710.066 06(18) 32 894 672.154 58(6) 32 894 183.488 20(3) 32 894 027.673 24(1) 32 893 967.585 71(1)
876 108 263.8953(8)b 451 903 472.(2) 382 118 016.86(9) 366 020 217.716(23)c 240 210 377.3(7) 212 661 348.0(2) 205 843 102.990(11)c 205 621 502.013 5(19) 148 807 311.8(4) 135 205 105.24(11) 131 714 327.415(6)c 131 595 419.737 5(12) 131 580 529.514 3(3) 101 166 442.3(2) 93 472 992.51(6) 91 454 604.438(4)c 91 384 078.897 4(7) 91 375 119.133 0(2) 91 373 021.527 12(8) 73 222 269.28(12) 68 453 180.90(4) 67 183 367.678 1(20)c 67 138 305.063 7(5) 67 132 532.254 93(16) 67 131 159.970 01(6) 67 130 728.912 70(3) 55 440 834.08(8) 52 282 487.96(3) 51 432 592.271 0(20)c 51 402 121.532 8(3) 51 398 197.358 27(12) 51 397 255.714 73(5) 51 396 956.349 94(2) 51 396 841.051 166(10) 43 430 382.87(5) 41 231 560.17(2) 40 635 138.920 1(11) 40 613 602.576 6(2) 40 610 819.102 15(9) 40 610 147.010 06(4) 40 609 931.658 315(17) 40 609 847.961 702(10) 34 938 883.86(4) 33 346 985.598(13) 32 912 506.072 9(9)c 32 896 735.396 13(18) 32 894 691.944 56(6) 32 894 196.386 67(3) 32 894 036.737 13(1) 32 893 974.304 86(1)
1 152 842 741.4(1.3)a 876 110 555.0742(8)b 451 903 472.(2) 382 118 676.05(1) 366 020 292.981(23)c 240 210 377.3(7) 212 661 617.7(2) 205 843 138.986(11)c 205 621 287.969 4(15) 148 807 311.8(4) 135 205 240.76(11) 131 714 346.623(6)c 131 595 327.450 6(9) 131 580 446.456 2(2) 101 166 442.3(2) 93 473 070.00(6) 91 454 615.775(4)c 91 384 030.791 4(5) 91 375 071.110 59(15) 91 372 990.223 57(5) 73 222 269.28(12) 68 453 229.30(4) 67 183 374.898 6(20)c 67 138 276.717 8(4) 67 132 502.034 54(10) 67 131 140.256 92(4) 67 130 715.086 086(15) 55 440 834.08(8) 52 282 520.19(3) 51 432 597.142 1(20)c 51 402 103.368 7(3) 51 398 177.123 47(9) 51 397 242.508 48(3) 51 396 947.087 163(10) 51 396 834.201 597(10) 43 430 382.87(5) 41 231 582.71(2) 40 635 142.359 2(11) 40 613 590.209 2(2) 40 610 804.896 07(7) 40 610 137.734 90(2) 40 609 925.152 765(10) 40 609 843.151 026(10) 34 938 883.86(4) 33 347 001.972(13) 32 912 508.586 7(9)c 32 896 726.580 63(14) 32 894 681.591 36(4) 32 894 189.625 08(2) 32 894 031.994 58(1) 32 893 970.797 88
Pachucki, Patkóš and Yerokhin [45]. Fine structure relative to 2 3 P centroid at 876 106 246.0(7) MHz [45]. Yerokhin et al. [50].
12
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G. W. F. Drake
Table 12.8 QED corrections to the ionization energy included in Table 12.7 for the S and P -states of helium (in MHz) State 1S 2S 3S 4S 5S 6S 7S 8S 9S 10 S 2P 3P 4P 5P 6P 7P 8P 9P 10 P
0 EL;1 C EL;1 Singlet Triplet 41 286(36)a 2 810(2)a 4 059(1)a 858.34 1 030.29 349.09 402.29 174.93 196.80 99.807 110.505 62.221 68.113 41.369 44.904 28.885 31.147 20.959 22.482 48.9(4)a 1 254.3(7)a,b 35.13 344.96 15.15 142.33 7.816 71.911 4.540 41.256 2.866 25.824 1.923 17.223 1.352 12.055 0.987 8.764
0 EL;2 C EL;2 Singlet Triplet – – – 91.258 8.468 37.303 3.203 18.735 1.544 10.702 0.861 6.677 0.528 4.441 0.347 3.102 0.240 2.251 0.173 – – 19.559 12.376 8.413 5.035 4.348 2.529 2.529 1.446 1.598 0.904 1.073 0.602 0.755 0.421 0.551 0.306
where Z 1 is the screened nuclear charge, and n is the effective principal quantum number defined by an iterative solution to the equation n D n ı.n / ;
where ı.n / is the quantum defect defined by the Ritz expansion ı.n / D ı0 C
tries have been updated with more recent results by Pachucki et al. [45] for the low-lying S and P -states, and by Yerokhin et al. [50] for the D-states. As noted in the latter reference, there is evidently a systematic discrepancy between theory and experiment for all the nD-2S and nD-2P transitions. Except for using an updated value for the Rydberg states, the other entries are the same as previously tabulated in [51]. For the D-states and beyond, the uncertainties are sufficiently small that these states can be taken as known points of reference in the interpretation of experimental transition frequencies. However, long-range Casimir–Polder corrections [52] are not included since they still lack experimental confirmation [53]. The QED shifts are the largest for the S 0 and and P -states. The contributions from EL;1 C EL;1 0 EL;2 C EL;2 for these states are listed separately in Table 12.8. Applications to isotope shifts and measurements of the nuclear radius are discussed in Sects. 17.2 and 91.2.
ı2 ı4 C C ; 2 .n ı/ .n ı/4
(12.54)
with constant coefficients ıi . The absence of odd terms in this series is a special property of the eigenvalues of Hamiltonians of the form HC CV , where HC is a pure one-electron Coulomb Hamiltonian, and V is a local, short-range, spherically symmetric potential of arbitrary strength (see [54] for further discussion). For the Rydberg states of helium, odd terms must be included in the Ritz expansion in Eq. (12.54) due to relativistic and mass polarization corrections, but they can be removed again by first adjusting the energies according to Wn0 D Wn Wn ;
a
Total of EL;1 C EL;2 from [45]. b Centroid value without fine structure. See [45].
(12.53)
(12.55)
where, to sufficient accuracy [54] [see the discussion following Eq. (12.68)]
3˛ 2 .Z 1/4 2 .Z 1/2 Wn D RM C 4n4 M n2 5 1 C .˛Z/2 ; 6
(12.56)
with Z D 2 for helium. The quantum defect parameters listed in Table 12.9 provide accurate extrapolations to higher-lying Rydberg states, with Wn D RM =n2 C Wn :
12.5.2
(12.57)
Asymptotic Expansions
The asymptotic expansion method [35, 55] rapidly increases in accuracy with increasing angular momentum L of the Rydberg electron, and can be used to high precision for L 7. The method is based on a model in which:
1. The Rydberg electron, treated as a distinguishable particle, moves in the field of the core consisting of the He nucleus and a tightly bound 1s electron. As discussed in Sect. 15.1, the ionization energies of an iso2. The core, as characterized by its various multipole molated Rydberg series of states can be expressed in the form ments, is perturbed by the electric field of the Rydberg Wn D RM .Z 1/2 =n2 ; (12.52) electron.
12.5.1
Quantum Defect Extrapolations
12
High Precision Calculations for Helium
211
Table 12.9 Quantum defects for the total energies of helium with the Wn term subtracted [Eq. (12.56)] ıi ı0 ı2 ı4 ı6 ı8 ı0 ı2 ı4 ı6 ı8 ı0 ı2 ı4 ı6 ı8 ı0 ı2 ı4 ı6 ı0 ı2 ı4 ı6 ı0 ı2 ı4 ı0 ı2 ı4
Value 1 S0 0.139 718 064 86(21) 0.027 835 737(18) 0.016 792 29(41) 0.001 459 0(31) 0.002 922 7(65) 1 P1 0.012 141 803 603(64) 0.007 519 080 4(59) 0.013 977 80(15) 0.004 837 3(12) 0.001 228 3(29) 1 D2 0.002 113 378 464(49) 0.003 090 051 0(58) 0.000 008 27(22) 0.000 309 4(31) 0.000 401(14) 1 F3 0.000 440 294 26(62) 0.001 689 446(65) 0.000 118 3(20) 0.000 326(18) 1 G4 0.000 124 734 490(79) 0.000 796 230(12) 0.000 012 05(53) 0.000 013 6(69) 1 H5 0.000 047 100 899(61) 0.000 433 227 7(84) 0.000 008 14(26) 1 I6 0.000 021 868 881(17) 0.000 261 067 3(22) 0.000 004 048(67)
3
3
P0 0.068 328 002 51(27) 0.018 641 975(24) 0.012 331 65(57) 0.007 951 5(45) 0.005 448(10) 3 D1 0.002 885 580 281(22) 0.006 357 601 2(27) 0.000 336 67(11) 0.000 839 4(16) 0.000 379 8(72) 3 F2 0.000 444 869 89(22) 0.001 739 275(24) 0.000 104 76(76) 0.000 033 7(69) 3 G3 0.000 125 707 43(12) 0.000 796 498(19) 0.000 009 80(81) 0.000 019(11) 3 H4 0.000 047 797 067(43) 0.000 433 232 2(55) 0.000 008 07(16) 3 I5 0.000 022 390 759(20) 0.000 261 068 0(28) 0.000 004 042(87)
3
P1 0.068 357 857 65(27) 0.018 630 462(24) 0.012 330 40(57) 0.007 951 2(45) 0.005 450(10) 3 D2 0.002 890 941 493(25) 0.006 357 183 6(30) 0.000 337 77(11) 0.000 839 2(16) 0.000 432 3(75) 3 F3 0.000 448 594 83(28) 0.001 727 232(30) 0.000 152 4(9) 0.000 248 6(83) 3 G4 0.000 128 713 16(10) 0.000 796 246(15) 0.000 011 89(66) 0.000 014 1(85) 3 H5 0.000 049 757 614(51) 0.000 433 227 4(65) 0.000 008 13(19) 3 I6 0.000 023 768 483(14) 0.000 261 066 2(18) 0.000 004 076(58)
S1 0.296 656 487 71(75) 0.038 296 666(59) 0.007 513 1(12) 0.004 547 6(79) 0.002 180(14) 3 P2 0.068 360 283 79(23) 0.018 629 228(21) 0.012 332 75(51) 0.007 952 7(41) 0.005 451(9) 3 D3 0.002 891 328 825(26) 0.006 357 704 0(33) 0.000 336 70(13) 0.000 839 5(18) 0.000 381 1(83) 3 F4 0.000 447 379 27(21) 0.001 739 217(23) 0.000 104 78(71) 0.000 033 1(64) 3 G5 0.000 127 141 67(11) 0.000 796 484(17) 0.000 009 85(75) 0.000 019(10) 3 H6 0.000 048 729 846(45) 0.000 433 228 1(57) 0.000 008 10(16) 3 I7 0.000 023 047 609(26) 0.000 261 067 2(35) 0.000 004 04(11)
All terms are known up to A10 (see [35, 55] for detailed results, and [57] for derivations and corrections to earN lier work). The expansions for the terms "0 , "1 , and "2 in 2 2 X e .Z 1/ j D 2 C a A hr i ; (12.58) "nL Eq. (12.6) for helium are [57] 0 j nL 0 2n2 a0 j D4 1 9 4 69 6 3833 7 nL hr i C hr where the expectation value hr j inL is calculated with re- "0 D 2 2 hr i C 64 512 15 360 2n spect to the hydrogenic nL-electron wave function [56], and 55 923 908 185 9 957L.L C 1/ C hr 8 i hr i the series is truncated at the upper limit N , where the series 65 536 10 240 688 128 begins diverging. The leading coefficients Aj are 21 035 363 33 275L.L C 1/ 1 1 C C hr 10 i A4 D 2 ˛1 ; A5 D 0 ; A6 D 2 .˛2 6ˇ1 / ; 6 193 152 28 672 23 where ˛k is the 2k -pole polarizability of the hydrogenic core, C e .1;1/ e .1;2/ ; and ˇk is a nonadiabatic correction. The exact hydrogenic 20 (12.59) values are 9a03 15a05 43a05 ; ˛2 D ; ˇ1 D : ˛1 D 4 6 2Z Z 8Z 6 A systematic perturbation expansion yields an asymptotic series of the form
12
212
G. W. F. Drake
9 4 249 6 319 7 Table 12.10 Formulas for the hydrogenic expectation value hr j i hr i C hr i C hr i hnljr j jnli in terms of 256 3840 32 2p Z p .2l p C 2/Š .l C p/Š 34 659 14 419 9 957L.L C 1/ ; fpl D : Gpnl D pC1 8 C hr i hr i n .2l C p 1/Š .l p/Š 16 384 5120 3072 j hr j i .a0 / 155 027 773 24 155L.L C 1/ 10 2 12 G2nl C C hr i 6 193 152 8192 3 nG3nl 53 4 G4nl .3n2 f1l / C 4e .1;1/ e .1;2/ ; 5 nl 3 l (12.60) 5 2G5 Œ5n n.3f1 1/ nl 4 2 6 G6 Œ35n 5n .6f1l 5/ C 3f2l 1 45 4 165 6 2555 7 nL hr i C hr i "2 D 2 hr i C 7 2G7nl Œ63n5 35n3 .2f1l 3/ C n.15f2l 20f1l C 12/ 64 512 3072 2n 8 G8nl Œ462n6 210n4 .3f1l 7/ 268 485 957L.L C 1/ 598 909 9 8 C42n2 .5f2l 15f1l C 14/ 10f3l C hr i C hr i 32 768 2048 172 032 9 2G9nl Œ858n7 462n5 .3f1l 10/ C 42n3 .15f2l 75f1l C 101/ 1148 906 421 629 515L.L C 1/ 2n.35f3l 105f2l C 252f1l 180/ 10 C C hr i nl 49 545 216 114 688 10 G10 Œ6435n8 6006n6 .2f1l 9/ C 1155n4 .6f2l 44f1l C 81/ 2 6n .210f3l 1365f2l C 4648f1l 4566/ C 35f4l 251 .1;2/ C 14e .1;1/ : e 10 (12.61) The asymptotic formulas for the NFS relativistic corrections are [35, 58] The terms e .1;1/ and e .1;2/ are second-order dipole-dipole and 2 4 2 dipole-quadrupole perturbation corrections. Defining fpL D hHmass C HD i ! ˛ Z C h1 .nL/ C 1 .nL/ C .Z˛/ 2 .L C p/Š=.L p/Š, they are given by 8 14 4 5041 6 hr i hr i ; .2 ıj;k /22i C2j C1 .2L 2i/Š.2L 2j /Š 3Z 4 240Z 6 e .i;j / D n3 .2L C 2i C 1/Š.2L C 2j C 1/Š (12.63) 2i C2j .i;j / .i;j / L 2 2 .2L 2i 2j /ŠA B ˛ 3.Z 1/ 5 3.f1 C 8/ 6 4 C 2i C2j C1 ; hHoo i ! 2i C2j C2 i C hr i hr i ; hr n .2L C 2i C 2j C 1/Š n Z2 Z2 4Z 2 (12.64) (12.62) with where " # 2 4 .1;1/ 2 2 L L .Z 1/ ˛ 1 3 D 3n .3n 2f1 /.f1 2/.45 C 623f1 C 3640f2 A h1 .nL/ D (12.65) 2n3 4n L C 12 C 560f L / ; "nL 1 D
3
B .1;1/ D .9n2 7f1L /.3n2 f1L / ; A.1;2/ D 21n6 .94 500 C 122 850f1L 1126 125f2L 18 931 770f3L 11 171 160f4L 1029 600f5L 18 304f6L / 15n4 .94 500 444 150f1L C 7747 425f2L C 337 931 880f3L C 375 290 190f4L C 66 518 760f5L C 2880 416f6L C 29 568f7L / C 9n2 f1L .90 300 177 450f1L C 1738 450f2L C 133 125 575f3L C 160 040 870f4L C 29 322 216f5L C 1293 600f6L C 13 440f7L / C 2f1L f2L f3L .45 C 252f1L 1680f2L 2240f3L / ; B .1;2/ D 315n6 C 125n4 .3 5f1L / 7n2 f1L .43 39f1L / 27f1L f2L : The accuracy of the expansion for the "0 , "1 , and "2 can be reliably estimated to be one-half of the last hr j i term included in the sum. Formulas for the hr j i are given in Table 12.10.
is the leading one-electron Dirac energy, and ( Z 1 2 4 ˛ 2 ˛1 hr i .Z 1/hr 5 i 3 1 .nL/ D 2 n " Z1 6 4.2L 2/Š 9n2 5f1L 4 nC .2L C 3/Š n 2L C 1 L L 2 40f2 C 70f1 3 4 C.Z 1/ hr i 2L C 1 (12.66) is the correction due to the dipole perturbation of the Rydberg electron. The relativistic recoil terms due to mass polarization are hHmass C HD iRR 22.Z˛/2 .Z 1/ 4 ! hr i C 2.Z 1/ .nL/ 1 M 9Z 4 # " 2 5 ˛Z.Z 1/ 2 C C 4h1 .nL/ ; M 12 n (12.67)
12
High Precision Calculations for Helium
hHoo iRR C h oo i ˛ 2 4 .Z 1/4 1 3 ! Z C M n3 n 2L C 1 25Œ1 C 13f .Z/ 4 hr i ; 16Z 2
213
12.6 Radiative Transitions 12.6.1 Basic Formulation
(12.68) In a semiclassical picture, the interaction Hamiltonian with the radiation field is obtained by making the minimal coupling replacements with f .Z/ ' 1 C .Z 2/=6. The .5=12/Œ˛Z.Z 1/=n2 Ze term in Eq. (12.67) is the dominant contribution in helium for PN ! PN A.R N / ; L 4. It is included in Eq. (12.56) for Wn , along with the c e leading 1=n2 term from Eq. (12.61), and the 1=n4 term from P i ! P i C A i .R i / (12.73) c Eq. (12.65). The complete relativistic finite mass correction includes also the mass-scaling terms .=M /h4Hmass C in Eq. (12.1), where 3HD C 3Hoo i obtained by expanding =me in Eq. (12.49). The hı.r 1 /i term is [59] 2„ 1=2 i kR "O e (12.74) A.R/ D c !V 3 Z 1447 7 31 4 hr i C hr i hı.r 1 /i ! 2 4Z 3 32Z 5 is the time-independent part of the vector potential A.r; t/ D 31.Z 1/ A.r/e i !t C c:c for a photon of frequency !, wave vector k, 4 hr i C I (12.69) 2Z 3 M and polarization "O ? k normalized to unit photon energy „! in volume V . The linear coupling terms then yield hı.r 12 /i vanishes exponentially as 1=n2LC4 with increasing 2 L. The complete asymptotic expressions for the FS matrix Ze e X P i A.R i / ; (12.75) Hint D P N A.R N / C elements are summarized by the formulas Mc me c i D1 hnL3LJ jHFS jnL3LJ i ! TnL .J /fZ 3 C 2SL .J / C 2ae ŒZ 2
and from Fermi’s Golden Rule, the decay rate for spontaneous emission from state to 0 is
C .2 C ae /SL .J / C .=M /Œ2 4SL .J /g ; (12.70) 3
hnL LJ jHFS jnL1LJ i
w; 0 d˝ D
2 jhjHint j 0 ij2 f ; „
(12.76) p ! TnL .L/.Z C 1 C 2ae Z 2=M / L.L C 1/ ; 2 3 (12.71) where f D V ! d!=.2c/ „ is the number of photon states with polarization "O per unit energy and solid angle in the normalization volume V . In the long wavelength and elecwhere tric dipole approximations, the factor e i kR in Eq. (12.74) is 8 2 3 replaced by unity. After integrating over angles d˝ and sumˆ 2. To monitor the orientation n1 , the circular polarization must be measured p 5 n1 I C I D ; (13.32) 3 n0 2I C I since the coefficient D L0 vanishes for any linear polarization. Conversely, from n2 I I D ; 2n0 2I C I
(13.33)
alignment is given by linear polarization. However, the denominator in both cases is different from that common in polarization measurements, and indeed, the ratio .I C I /=.I C C I / is not proportional to the orientation but contains a contribution from the alignment, and .I I /=.I C I / is not linear in the alignment. The expressions for other total angular momenta j are the same except for the numerical coefficients on the LHS of Eqs. (13.32)and (13.33). Finally, if the desire is to measure the total excited-state population n0 without any polarization component (which might rotate in stray external fields), one can choose linear polarization at an angle to make D20 disappear, namely at the magic angle D arccos.31=2 / D 54:74ı . Of course, the same angle may be chosen to excite an unpolarized population.
2. Fano, U.: Rev. Mod. Phys. 29, 74 (1957) 3. Fano, U., Racah, G.: Irreducible Tensorial Sets. Academic Press, New York (1967) 4. Lewis, E.L.: Phys. Rep. 58, 1 (1980) 5. Baylis, W.E., Pascale, J., Rossi, F.: Phys. Rev. A 36, 4212 (1987) 6. Omont, A.: Prog. Quantum Electron. 5, 69 (1977) 7. Happer, W.: Rev. Mod. Phys. 44, 169 (1972) 8. Baylis, W.E., Bonenfant, J., Derbyshire, J., Huschilt, J.: Am. J. Phys. 61, 534 (1993) 9. Nikitin, E.E., Umanski, S.Y.: Theory of Slow Atomic Collisions. Springer, Berlin, Heidelberg (1984)
William E. Baylis Professor Baylis earned degrees in physics from Duke (B.Sc.), the University of Illinois (M.Sc.), and the Technical University of Munich (D.Sc.). He has authored two books, edited or co-edited four more, contributed 28 chapters to other volumes, and published over a hundred journal articles. His publications are in theoretical physics and emphasize atomic and molecular structure, atomic collisions, and interactions with radiation. His most recent work concerns relativistic dynamics, the photon position operator and wave function, and applications of Clifford algebra, especially to the quantum – classical interface. He is a fellow of the American Physical Society, past chair of the Divisions of Atomic and Molecular Physics and of Theoretical Physics of the Canadian Association of Physicists, a member of the international editorial boards of the Springer Series of Atomic, Optical, and Plasma Physics and of the journal Advances in Applied Clifford Algebras. He is currently a University Professor emeritus at the University of Windsor.
13
14
Atoms in Strong Fields S. Pedro Goldman and Mark M. Cassar
Contents
Keywords
14.1 14.1.1 14.1.2
Electron in a Uniform Magnetic Field . . . . . . . . . 223 Nonrelativistic Theory . . . . . . . . . . . . . . . . . . . . 223 Relativistic Theory . . . . . . . . . . . . . . . . . . . . . . 224
14.2 14.2.1 14.2.2 14.2.3
Atoms in Uniform Magnetic Fields Anomalous Zeeman Effect . . . . . . Normal Zeeman Effect . . . . . . . . . Paschen–Back Effect . . . . . . . . . .
14.3
Atoms in Very Strong Magnetic Fields . . . . . . . . . 226
14.4 14.4.1 14.4.2 14.4.3 14.4.4
Atoms in Electric Fields Stark Ionization . . . . . . Linear Stark Effect . . . . Quadratic Stark Effect . . Other Stark Corrections .
14.5
Recent Developments . . . . . . . . . . . . . . . . . . . . 229
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225 225 225 225
227 227 227 228 228
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Abstract
neutron star Landau level Zeeman effect Stark shift field atom
14.1 Electron in a Uniform Magnetic Field 14.1.1 Nonrelativistic Theory The nonrelativistic Hamiltonian (in Gaussian units) for an electron in an external field A is [1] e 2 1 p A B C eV ; (14.1) H D 2m c where A is the vector potential and V is the scalar potential. The second term in Eq. (14.1) must be included to account for the interaction of the electron magnetic moment with an external magnetic field. The potentials A and V are only defined to within a gauge transformation [2]:
Interest in the effect that electric and magnetic fields have on the internal structure of atoms is as old as quantum mechanics itself. In practical terms, an atom’s spectrum acts 1 @f : (14.2) A 0 D A C rf ; V 0 D V as its signature, and so it is important to understand how c @t electric and magnetic fields alter this characteristic. In this We choose the gauge r A D 0 in which the momentum chapter, a summary of the basic nonrelativistic and relaoperator p D i„r and vector potential A commute. B is tivistic theory of electrons and atoms in external magnetic a constant uniform magnetic field with vector potential fields is given. Extensions to the case of very strong fields are then introduced for both types of fields. 1 (14.3) A D B r : 2 S. P. Goldman () Dept. of Physics, Toronto Metropolitan University Toronto, Canada e-mail: [email protected] M. M. Cassar Zekelman School of Information Technology, St. Clair College Windsor, ON, Canada e-mail: [email protected]
is the magnetic moment of the electron: D
1 ge B ; 2
(14.4)
where the i are the Pauli spin matrices, B D e„=2me c is the Bohr magneton and ge is the electron g-factor which accounts for the anomalous magnetic moment of the electron (see Sect. 29.4).
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_14
223
224
S. P. Goldman and M. M. Cassar
Consider now the case of a free electron in a constant where ˛ and ˇ are the 4 4 matrices defined by Eq. (9.61). uniform field in the z-direction with no scalar potential, i.e., The eigenfunctions of the Dirac Hamiltonian Eq. (14.10) are V D 0 in Eq. (14.1). This case also describes an atom in the written as four-dimensional spinors: limit of strong magnetic fields such that the Coulomb inter0 1 0 1 .1/ 1 actions are negligible. In this case, which is different from ! B .2/ C B C Eq. (14.3), we have B C B1 C B C C B (14.11) D D B .3/ C B C ; ( @ A @ 1 A Ax D By .4/ 1 : (14.5) B D B zO with Ay D Ax D 0 where and are two-dimensional (Pauli) spinors and i For this field, the operators z , px and pz commute with and i are functions of the coordinates of the electron. the Hamiltonian and are therefore conserved. Calling their In the case V D 0, the lower component can be elimrespective eigenvalues z , px and pz , with 1 px ; py inated in the eigenvalue equation HD D E to obtain 1, the eigenstates are written as a Hamiltonian equation for only, similar to Eq. (14.1) [4]. For a field defined by Eq. (14.5) one writes in the form (as (14.6) in the nonrelativistic case) D ei.px xCpz z/=„ '.y/ : D ei.px xCpz z/=„ f .y/ :
Calling y0 D cpx =eB, ' satisfies
„2 d 2 1 ' C m!B2 .y y0 /2 ' 2 2m dy 2 ! pz2 D E C z B '; 2m
(14.12)
Here, f .y/ satisfies the equation
(14.7)
which is the Schrödinger equation for a one-dimensional harmonic oscillator with angular frequency !B D jejB=mc, the cyclotron frequency of the electron. The solutions to Eq. (14.7) give the eigenstates for an electron in an external homogeneous magnetic field. They are called Landau levels [1] with energies given by
„2 d 2 1 ' C m!B2 .y y0 /2 ' 2 2m dy 2 2 E m2 c 4 pz2 C z B D '; 2mc 2 2m
(14.13)
which reduces to the nonrelativistic form Eq. (14.7) in the limit E ! mc 2 . In Eq. (14.13), the term involving is not included arbitrarily as in Eq. (14.1) but is a consequence of HD , which predicts the value g D 2. The value ge used in Eq. (14.4) includes radiative corrections to the Dirac value. From Eq. (14.13) we obtain 1 p2 # " En D n C C ms „!B C z ; (14.8) 2 2 2m pz2 1 2 2 mc En D 2mc C n C C ms „!B C ; (14.14) 2 2 2m where „ms D ˙„=2 is the eigenvalue of the z-component of the spin operator s, and eigenfunctions given by and the eigenfunctions mDs are 'n .y/ D p
1=2 aB 2n nŠ y y0 .y y0 /2 exp Hn ; aB 2aB2
where aB D als [3].
14.1.2
0
1
(14.9)
D 1=2
p „=m!B and the Hn are Hermite polynomi-
Relativistic Theory
D 1=2
1 E C mc 2 n B C c B C B C 0 C; D NB B C pz n B C @ i„ p A 2n n1 aB 0 1 0 B E C mc 2 C B C B C n c B C; D N B i„ p C B C @ aB 2.n C 1/ nC1 A pz n
(14.15)
The relativistic analog of Eq. (14.1) is given by the Dirac Hamiltonian where n is defined as in Eqs. (14.6) and (14.9), and N is e (14.10) a normalization constant. HD D c˛ p A C ˇmc 2 C eV ; c
14 Atoms in Strong Fields
14.2 Atoms in Uniform Magnetic Fields 14.2.1
225
orbit interaction term is also neglected, then the energies relative to the field-free eigenvalues En are given by
Anomalous Zeeman Effect En .B/ D En C B B.ml C 2ms / :
(14.21)
Consider now the nonrelativistic Hamiltonian for a oneThe selection rules for transitions are ms D 0 and ml D electron atom in the presence of an external magnetic field 0; ˙1. The transition energy of a spectral line is split into B. The one-electron Hamiltonian can be written as [5] three components, the Lorentz triplet 1 Ze 2 1 2 p C .r /L S H D (14.22) En D En;0 C .ml /„!L ; 2m 40 r e2 B .B r/2 ; (14.16) where En;0 is the transition energy in the absence of a field, C .L C ge S / B C „ 8mc 2 and ! D jejB=2m is the Larmor frequency. Transitions with L
where
ml D 0 produce the line at the unshifted transition enZe 2 1 1 .r / D : (14.17) ergy; transitions with ml D ˙1 produce the shifted lines. 2m2 c 2 40 r 3 Lorentz triplets can be observed in many-electron atoms in The anomalous Zeeman effect corresponds to the case of which the total spin is zero. weak magnetic fields such that the magnetic interaction is small compared with the L S spin-orbit term. The energy shifts are obtained from a perturbation of Eq. (14.16) with 14.2.3 Paschen–Back Effect B D 0. The unperturbed wave functions are eigenfunctions of L2 , S 2 , J 2 and Jz , with J D L C S , but are not eigen- We add now to the results of the last section the first-order functions of Lz or Sz . The energy levels with given values of perturbation caused by the spin-orbit term. The calculation l, s and j split in the presence of a field defined by Eq. (14.5) can be performed in closed form for hydrogenic atoms, for according to which the contribution to the energy of the level n is [6] (14.18) Emj D gBBmj ; En D 0 ; for l D 0 where g is the Landé splitting factor given by ˛ 4 Z 4 c 2 En D ml ms j.j C 1/ s.s C 1/ C l.l C 1/ 3 g D gl 2n 2j.j C 1/ 1 l lC .l C 1/ ; for l ¤ 0 (14.23) j.j C 1/ C s.s C 1/ l.l C 1/ 2 C ge ; (14.19) 2j.j C 1/ where is the reduced mass, D mM=.m C M /. A general where ge is defined by Eq. (14.4), and gl D 1 me =M to expression of the relativistic solution for the Paschen–Back lowest order in me =M , where me is the electron mass and effect can be written for the one-electron case in the Pauli M is the nuclear mass. To a first approximation, it is often approximation in which the eigenstates are given in terms of sufficient to take ge D 2 (the Dirac value) and gl D 1. In the in Eq. (14.11). Call 0 the eigenfunctions with no external ˙ case of a many-electron atom, j , l and s are replaced by the field corresponding to j D l ˙ 1 with unperturbed energies total angular momenta J , L and S. In the one-electron case E 0 respectively. In terms of the 2dimensionless variables ˙ (neglecting corrections to gl or ge ) the g-value is simply s 4mj B B j C 12 ; D 1C C 2 ; D g ; (14.20) 1 EC E 2l C 1 lC2 4mj 1 1 1 C ; (14.24) D which shows that the splitting of the j D l C 2 levels is larger 2l C 1 1 than that of the j D l 2 levels. The selection rules for the splitting of spectral lines are ımj D 0 for components polar- the energies E˙ and wave functions ˙ of the states, in the ized parallel to the field ( components), and ımj D ˙1 for presence of the field B, are those perpendicular to the field ( components). E0 E0 E˙ D C .1 ˙ C 2mj / C .1 2mj / (14.25) 2 2
14.2.2
Normal Zeeman Effect
For moderately strong fields up to B 104 T, the quadratic .B r/2 term in Eq. (14.16) can be neglected. If the spin-
and r ˙ D ˙
1˙ 0 C C 2
r
1 0 : 2
(14.26)
14
226
S. P. Goldman and M. M. Cassar
14.3
Atoms in Very Strong Magnetic Fields
The case of very strong magnetic fields (i.e., B > 104 T), such as those encountered at the surface of neutron stars, is also called the quadratic Zeeman effect, as the last term in Eq. (14.16) is dominant. In this range, perturbation calculations fail to yield good results as the field is too large, and even at fields of the order B 107 T the Landau high B approximation of Eqs. (14.8) and (14.9) is not adequate. Very accurate calculations have been performed using variational finite basis set techniques for both the relativistic Dirac and nonrelativistic Schrödinger Hamiltonians. The calculations use the following relativistic basis set [7] that includes nuclear size effects (R is the nuclear size) and contains both asymptotic limits, the Coulomb limit for B D 0 and the Landau limit for B ! 1: ( .k/ 2 2 r qk 1C2n ean r ˇ ˝k r R .k;/ (14.27) D 2 .k/ nl bn r 1Cn erˇ ˝k r > R
Table 14.1 Relativistic ground state binding energy Egs =Z 2 and finite nuclear size correction ıEnuc =Z 2 (in a.u.) of hydrogenic atoms for various magnetic fields B (in units of 2:35 105 T). ıEnuc should be added to Egs Z 1 1 1 1 1 1 1 1 1 20 20 20 20 40 92 92
B 0 105 102 101 2 10 20 200 500 0 1 10 100 0 0 1
Egs =Z 2 0:50000665659748375 0:5000116564837 0:504981572360 0:5475324083429 1:0222180290 1:74780068 2:21540091 4:7271233 6:2570326 0:5026913084075098 0:50393086705 0:514950248 0:61237794 0:511129686143 0:5743381407377 0:574386987
ıEnuc =Z 2 1:55786 1010 1:5579 1010 1:5580 1010 1:5718 1010 3:23 1010 1:182 109 2:360 109 3:032 108 8:778 108 1:3372 106 1:34 106 1:3 106 1:4 106 1:1878 105 8:4155 104 8:4155 104
with ˝k D .cos /ljmk j .sin /jmk j eimk !k ;
Table 14.2 Relativistic binding energy E2S;1=2 for the 2S1=2 .mj D
(14.28) 12 / and E2P;1=2 for the 2P1=2 .mj D 12 / excited states of hydrogen (in a.u.) in an intense magnetic field B (in units of 2:35 105 T)
where n D 0; 1; : : : ; Nr ; q1 D q2 D k0 ; mk D k =2;
k D 1; 2; 3; 4;
q3 D q4 D
k00
D 1; 2; 3; 4 ;
;
1 D 3 D 1;
2 D 4 D 1 :
B 106 105 104 103 102 0.05 0.1 0.2 0.5 1 10 100
E2S;1=2 0:125002580164 0:12505204495 0:1296536428
E2P;1=2 0:125002283074 0:125006104950 0:12505096792 0:1254994694 0:1298513642
0:142018956 Here, k refers to the component .k/ in Eq. (14.11), and 0:1480917386 0:1624110524 and ˇ are variational parameters. The power of r at the origin 0:14898958 is given by 0:15081015 ( 0:16047107 0:26000934 j j if < 0 (14.29) k0 D 0:20895591 0:38266318 j j C 1 if > 0 ; 0:256191 0:4636641 ( j j C 1 if < 0 0 k0 D (14.30) where # is a two-component Pauli spinor: # D k # .For k z k i j j if > 0 ; even (odd) parity states, the value of l for the large comThe index refers to the two regular and two irregular solu- ponents (k D 1; 2) is an even (odd) number greater than or tions for r > R that match the corresponding powers at the equal to jmk j up to 2N (for even parity) or 2N C 1 (for odd parity), while for the small components (k D 3; 4) it is origin k0 and k00 . an odd (even) number greater than or equal to jmk j up to 2N C 1 (for even parity) or 2N (for odd parity), since the 1 D 0 ; 2 D 0 C 1; 3 D 0 ; 4 D 0 C 1 ; (14.31) small component has a different nonrelativistic parity than .k/ .k/ the large component. The coefficients an and bn are deterp 1 1 2 2 C ; (14.32) mined by the continuity condition of the basis functions and 0 D .˛Z/ ; D ˙ 2 2 ! ! their first derivatives at R. For a point nucleus, the section #1 #1 r R is omitted; for a nonrelativistic calculation, take ˛ D 0 !1 D ; !2 D ; 0 0 in the basis set. ! ! Table 14.1 presents relativistic (Dirac) energies for the 0 0 ; !4 D ; (14.33) ground state of one-electron atoms. Values for a point nu!3 D i#1 i#1 cleus and finite nuclear size corrections are given. Table 14.2
14 Atoms in Strong Fields
227
Table 14.3 Relativistic corrections ıE D .E ENR /=jER j to the nonrelativistic energies ENR for the ground state and n D 2 excited states of hydrogen in an intense magnetic field B (in units of 2:35 105 T). The numbers in brackets denote powers of 10 B 0:1 1 2 3 20 200 500 2000 5000
ıEgs 1:08Œ5 5:21Œ6 4:03Œ6 3:48Œ6 1:09Œ6 4.61Œ6 8.81Œ6 1.85Œ5 2.78Œ5
B
ıE2S;1=2 1:66Œ5 1:66Œ5
1Œ6 1Œ4 1Œ3 1Œ2 0.05 0.1 1 10 100
1:60Œ5 1:57Œ5 1:74Œ6 1:3Œ5 2:0Œ5 3:9Œ5
ıE2P;1=2 1:43Œ5 7:86Œ6 7:72Œ6 7:30Œ6 6:00Œ6 1:05Œ5 3:48Œ5 1:0Œ4
presents the relativistic energies for n D 2 excited states of hydrogen with the (negligible) finite nuclear size correction included. Table 14.3, which displays the relativistic corrections of the energies of the previous two tables, presents one of the most interesting relativistic results: the change in sign of the relativistic correction of the energy of the ground state at B 107 T.
14.4 Atoms in Electric Fields 14.4.1
there are no bound states any longer but resonances. The potential barrier is shallower the stronger the field; the well can contain a smaller number of bound states and ionization occurs.
14.4.2
Linear Stark Effect
The electric field Eq. (14.36) produces a dipole potential r 4 O ; (14.38) Y10 .r/ VF D eF z D eF r 3 which does not preserve parity. A first-order perturbation calculation for the energy En.1/ D hnjVF jni
(14.39)
yields null results unless the unperturbed states are degenerate with states of opposite parity. In the remainder of this chapter, atomic units will be used. Final results for energies can be multiplied by 2R1 hc to translate to SI or other units. The calculation can be carried out in detail for the case of hydrogenic atoms [8]. In this case it is convenient to work in parabolic coordinates: ' denotes the usual angle in the xy-plane, and Dr Cz ;
Stark Ionization
Dr z : (14.40) An external electric field F introduces the perturbing potenThe Hamiltonian for a hydrogenic atom with a field VF D tial 1 V D d F ; (14.34) 2 F . / from Eq. (14.38) is where dD
X
. C /H D hC ./ C h ./ : qi r i
(14.35)
i
The wave function is written in the form
1 is the dipole moment of the atom, and i runs over all elec .; ; '/ D p trons in the atom. In the case of strong external electric fields, 2Z bound states do not exist because the atom ionizes. Consider with the ˙ satisfying a hydrogenic atom in a static electric field F D F zO :
Vtot .r/ D
e2Z 1 C eF z : 40 r
h˙ .x/
(14.36)
(14.37)
C ./
˙ .x/
iml ' ./e
;
(14.42)
14 DE
˙ .x/
;
(14.43)
and x D for , and m2 2 d 1 d 2Z˙ h˙ .x/ D x C l2 F x ; (14.44) x dx dx x 2x 2
where x D for
The total potential acting on the electron is then
(14.41)
C
with Z D ZC C Z : Consider the z-dependence of this potential. Call
D p 2 2 x C y and v.z; / D V .x; y; z/. Unlike the Coulomb Using the notation p case in which vCoul .˙1; / D 0 resulting in an infinite num D 2E ; ber of bound states, now v.˙1; / D ˙1 and v has a local 1 maximum. On the z axis, this maximum occurs at zmax D p n˙ D Z˙ = .jml j C 1/ ; 2 Zjej=.40 F / for which v.zmax ; 0/ D 0. There is then n D n C n C jml j C 1 ; a potential barrier through which the electron can tunnel, i.e., C
(14.45)
228
S. P. Goldman and M. M. Cassar Table 14.4 Relativistic dipole polarizabilities for the ground state of hydrogenic atoms
nC ; N D 0; 1; : : : ; n jml j C 1 ; jml j D 0; 1; 2; : : : ; n 1 ; ın D nC n ;
Z 1 5 10 20 30 40 50 60 70 80 90 100
(14.46)
where n is the principal quantum number, the unperturbed eigenfunctions are 1
˙ .x/ D
n˙ Š 2 .n˙ C jml j/Š
1
1 2
1
l j/ .x/ ; e 2 x .x/ 2 jml j Ln.jm ˙
(14.47) .a/ where the Lb are generalized Laguerre polynomials (Sect. 9.4.2). The zero-order eigenvalues are ml C 1 .0/ : (14.48) Z˙ D n˙ C 2
For a field Eq. (14.36),
The first-order perturbation yields .1/ Z˙
1F D ˙ 2 6n˙.n˙ 4 C ml C 1/ C ml .ml C 3/ C 2 :
From these
Z D n
2 2F
n Z
1 En D ˛ n F 2 ; 2 (14.49)
;
(14.54)
where n D 2e 2 ˛ n ˛zz
3
nD1 4 ˛rel Z = a03 4:4997515 4:4937883 4:4751644 4:4008376 4:2775621 4:1062474 3:8881792 3:6250295 3:3188659 2:9721524 2:5877205 2:1686483
(14.50)
X jhnjzjmij2 : En Em m
(14.55)
m¤n
ın In terms of Eq. (14.46), a general nonrelativistic expression for the dipole polarizability of hydrogenic ions is [9]
and to first order in F , 1 E D 2 E .0/ C E .1/ ; 2 Z2 1 E .0/ D ; 2 n2 3 F E .1/ D n ın : 2 Z
˛n D
a03 n4
17n2 3 ın2 9m2l C 19 : 4 8Z
(14.56)
For the ground state of hydrogenic atoms, (14.51) ˛ nD1 D
9a03 : 2Z 4
(14.57)
14.4.3 Quadratic Stark Effect
Table 14.4 lists the relativistic values for the ground state ponD1 larizability ˛rel , obtained by calculating Eq. (14.55) using A perturbation linear in the field F yields no contribution relativistic variational basis sets [10]. The values are interpoto nondegenerate states (e.g., the ground state nC D n D lated by m D 0; n D 1). In this case, the lowest order contribution comes from the quadratic Stark effect, the contribution of a03 9 14 nD1 2 4 .0/ ˛ D C 0:53983.˛Z/ .˛Z/ : (14.58) 2 order F . The quadratic perturbation to a level En caused Z4 2 3 by a general electric field F can be written in terms of the symmetric tensor ˛ijn as 1 En.2/ D ˛ijn Fi Fj ; 2
(14.52)
Third Order Corrections For the energy correction cubic in the external field Eq. (14.36), one obtains [9]
with ˛ijn
˝ ˇ ˇ ˛ X hnjdi jmi mˇdj ˇn D 2 ; En Em m m¤n
where di is defined in Eq. (14.35).
14.4.4 Other Stark Corrections
(14.53) E .3/ D
3 3 n 7 F 32 Z ın 23n2 ın2 C 11m2l C 39 :
(14.59)
14 Atoms in Strong Fields
229
Relativistic Linear Stark-Shift of the Fine Structure of Hydrogen For a Stark effect small relative to the fine structure, the degenerate levels corresponding to the same value of j split according to s 3 1 2 nm 2 n jC ım Enj D F : (14.60) 4 2 j.j C 1/
presence of helium in the atmospheres of certain magnetic white dwarfs [19]. In recent years, the increased sophistication and resolution of observation techniques has not only increased the number of known astronomical objects, but also motivated the study of the effects of strong fields on heavier atoms [20]. Another interesting area of current research concerns the relationship between quantum mechanics and classically chaotic systems. For these studies, Rubidium Rydberg atoms are an ideal system since laboratory fields can easily push the Other Stark Corrections in Hydrogen atom to the strong-field limit [21–23]. The expectation value of the delta function, is, in a.u. [11], For a very useful review of various topics up to 1998 (14.61) see [24]; a more concise review, concerning the electronic 2h1sjı.r/j1si D 2 31F 2 : structure of atoms, molecules, and bulk matter, including For the Bethe logarithm ˇ defined by some properties of dense plasma, in strong fields, is given P 2 in [25]. jh1sjpjnij .En E1s / lnjEn E1s j ; (14.62) ˇ1s D n P 2 n jh1sjpjnij .En E1s /
References
the result is [12] ˇ1s D 2:290 981 375 205 552 301 2
C 0:316 205.6/ F :
(14.63)
These results are useful in calculating an asymptotic expansion for the two-electron Bethe logarithm [13].
14.5 Recent Developments The drastic change of an atom’s internal structure in the presence of external electric and magnetic fields is shown most clearly through the changes induced in its spectral features. Of these features, avoided crossings are a distinctive example. Recent work in this area by Férez and Dehesa [14] has suggested the use of Shannon’s information entropy [15], defined by Z S D
.r/ ln .r/dr ;
(14.64)
where .r/ D j .r/j2 , as an indicator or predictor of such irregular features of atomic spectra. By studying some excited states of hydrogen in parallel fields it was shown that, for the states involved, a marked confinement of the electron cloud and an information-theoretic exchange occurs when the magnetic field strength is adjusted adiabatically through the region of an avoided crossing. The field strengths studied are characteristic of compact astronomical objects, such as white dwarfs and neutron stars. Although the effects of strong magnetic fields on the structure and dynamics of hydrogen have been known for some time, knowledge of the helium atom in such fields has only recently become sufficient for comparison with astrophysical observations [16–18]. As one example of their importance, such studies have proven critical in showing the
1. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics (Course of Theoretical Physics) vol. 3. Pergamon, Oxford, p 456 (1977) 2. Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields (Course of Theoretical Physics) vol. 2. Pergamon, Oxford, p 49 (1975) 3. Messiah, A.: Quantum Mechanics. Wiley, New York, p 491 (1999) 4. Itzykson, C., Zuber, J.-B.: Quantum Field Theory. McGraw-Hill, New York, p 67 (1980) 5. Bethe, H.A., Salpeter, E.: Quantum Mechanics of One- and TwoElectron Atoms. Plenum, New York, p 208 (1977) 6. Bethe, H.A., Salpeter, E.: Quantum Mechanics of One- and TwoElectron Atoms. Plenum, New York, p 211 (1977) 7. Chen, Z., Goldman, S.P.: Phys. Rev. A 48, 1107 (1993) 8. Bethe, H.A., Salpeter, E.: Quantum Mechanics of One- and TwoElectron Atoms. Plenum, New York, p 229 (1977) 9. Bethe, H.A., Salpeter, E.: Quantum Mechanics of One- and TwoElectron Atoms. Plenum, New York, p 233 (1977) 10. Drake, G.W.F., Goldman, S.P.: Phys. Rev. A 23, 2093 (1981) 11. Drake, G.W.F.: Phys. Rev. A 45, 70 (1992) 12. Goldman, S.P.: Phys. Rev. A 50, 3039 (1994) 13. Goldman, S.P., Drake, G.W.F.: Phys. Rev. Lett. 68, 1683 (1992) 14. González-Férez, R., Dehesa, J.S.: Phys. Rev. Lett. 91, 113001 (2003) 15. Shannon, C.E.: Bell Syst. Tech. J. 27, 623 (1948) 16. Becken, W., Schmelcher, P., Diakonos, F.K.: J. Phys. B 32, 1557 (1999) 17. Becken, W., Schmelcher, P.: Phys. Rev. A 63, 053412 (2001) 18. Becken, W., Schmelcher, P.: Phys. Rev. A 65, 033416 (2002) 19. Jordan, S., Schmelcher, P., Becken, W., Schweizer, W.: Astron. Astrophys. 336, 33 (1998) 20. P. Schmelcher: private communication 21. von Milczewski, J., Uzer, T.: In: Schmelcher, P., Schweizer, W. (eds.) Atoms and Molecules in Strong External Fields, p. 199. Springer, Berlin, Heidelberg (1998) 22. Main, J., Wunner, G.: In: Schmelcher, P., Schweizer, W. (eds.) Atoms and Molecules in Strong External Fields, p. 223. Springer, Berlin, Heidelberg (1998) 23. Guest, J.R., Raithel, G.: Phys. Rev. A 68, 052502 (2003) 24. Schmelcher, P., Schweizer, W. (eds.): Atoms and Molecules in Strong External Fields. Springer, Berlin (1998) 25. Lai, D.: Rev. Mod. Phys. 73, 629 (2001)
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230
S. P. Goldman and M. M. Cassar Pedro Goldman Professor Pedro Goldman completed a Ph.D. in Relativistic Atomic Physics at the University of Windsor. His work in atomic physics includes pioneering work on relativistic variational basis sets, relativistic calculations for many-electron atoms and diatomic molecules, accurate calculations for atoms in strong magnetic fields and accurate calculations of QED energy corrections and of the energy levels of Helium. Presently his research is directed to the optimization of the radiation therapy of tumours. He has as well received numerous teaching awards.
Mark M. Cassar Mark M. Cassar received his Ph.D. from the University of Windsor, Canada in 2003. His past research focused on high-precision theoretical calculations for the energy level structure of three-body atomic and molecular systems. He is currently working on applications of artificial intelligence in medicine.
15
Rydberg Atoms Thomas F. Gallagher
Contents 15.1
Wave Functions and Quantum Defect Theory . . . . 231
15.2
Optical Excitation and Radiative Lifetimes . . . . . 233
15.3
Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . 234
15.4
Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . 236
15.5
Microwave Fields . . . . . . . . . . . . . . . . . . . . . . 237
15.6
Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
15.7
Autoionizing Rydberg States . . . . . . . . . . . . . . . 239
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Abstract
Rydberg atoms are those in which the valence electron is in a state of high principal quantum number n. They are of historical interest since the observation of Rydberg series helped in the initial unraveling of atomic spectroscopy [1]. Since the 1970s, these atoms have been studied mostly for two reasons. First, Rydberg states are at the border between bound states and the continuum, and any process that can result in either excited bound states or ions and free electrons usually leads to the production of Rydberg states. Second, the exaggerated properties of Rydberg atoms allow experiments to be done that would be difficult or impossible with normal atoms. Keywords
15.1 Wave Functions and Quantum Defect Theory Many of the properties of Rydberg atoms can be calculated accurately using quantum defect theory, which is easily understood by starting with the H atom [2]. We shall use atomic units, as discussed in Sect. 1.2. The Schrödinger equation for the motion of the electron in a H atom in spherical coordinates is 1 2 1 .r; ; / D E .r; ; / ; (15.1) r 2 r where E is the energy, r is the distance between the electron and the proton, and and are the polar and azimuthal angles of the electron’s position. Equation (15.1) can be separated, and its solution expressed as the product .r; ; / D R.r/Y`m .; / ;
where ` and m are the orbital and azimuthal-orbital angular momentum (i.e., magnetic) quantum numbers, and Y`m .; / is a normalized spherical harmonic; R.r/ satisfies the radial equation 2dR.r/ 2R.r/ `.` C 1/R d 2 R.r/ C ; C 2ER.r/ C D 2 dr rdr r r2 (15.3) which has the two physically interesting solutions f .`; E; r/ ; r g.`; E; r/ : R.r/ D r R.r/ D
microwave field Rydberg state radiative lifetime Rydberg atom photoionization cross section
T. F. Gallagher () Dept. of Physics, University of Virginia Charlottesville, VA, USA e-mail: [email protected], [email protected]
(15.2)
(15.4) (15.5)
The f and g functions are the regular and irregular Coulomb functions, which are the solutions to a variant of Eq. (15.3). As r ! 0, they have the forms [3] f .`; E; r/ / r `C1 ; g.`; E; r/ / r
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_15
`
;
(15.6) (15.7) 231
232
T. F. Gallagher
irrespective of whether E is positive or negative. As r ! 1, for E > 0 the f and g functions are sine and cosine waves, i.e., there is a phase shift of =2 between them. For E < 0, it is useful to introduce , defined by E D 1=2 2 , and for E < 0 as r ! 1 f D u.`; ; r/ sin v.`; ; r/ei ;
(15.8)
g D u.`; ; r/ cos C v.`; ; r/ei.C1=2/ ;
(15.9)
where u and v are exponentially increasing and decreasing functions of r. As r ! 1, u ! 1 and v ! 0. Requiring that the wave function be square integrable means that as r ! 0, only the f function is allowed. Equation (15.8) shows that the r ! 1 boundary condition requires that sin be zero or an integer n, leading to the hydrogenic Bohr formula for the energies ED
1 : 2n2
(15.10)
The classical turning point of an s wave occurs at r D 2n2 , and the expectation values of positive powers of r reflect the location of the outer turning point, i.e., ˝ k˛ r n2k : (15.11) The expectation values of negative powers of r are determined by the properties of the wave function at small r. The normalization constant of the radial wave function scales as n3=2 , so that R.r/ / n3=2 r `C1 for small r. Accordingly, the expectation values of negative powers of r, except r 1 , and any properties that depend on the small r part of the wave function, scale as n3 . Using the properties of the wave function and the energies, the n-scaling of the properties of Rydberg atoms can be determined. The primary reason for introducing the Coulomb waves instead of the more common Hermite polynominal solution for the radial function is to set the stage for single-channel quantum defect theory, which enables us to calculate the wave functions and properties of one-valence electron atoms such as Na. The simplest picture of an Na Rydberg atom is an electron orbiting a positively charged NaC core consisting of 10 electrons and a nucleus of charge C11. The ten electrons are assumed to be frozen in place with spherical symmetry about the nucleus, so their charge cloud is not polarized by the outer valence electron, although the valence electron can penetrate the ten-electron cloud. When the electron penetrates the charge cloud of the core electrons, it sees a potential well deeper than 1=r due to the decreased shielding of the C11 nuclear charge. For Na and other alkali atoms, we assume that there is a radius rc such that for r < rc the potential is deeper than 1=r, and for r > rc it is equal to 1=r. As a result of the deeper potential at r < rc , the radial wave function is pulled into the core in Na, relative to H, as shown in
Hydrogen r Sodium r
Fig. 15.1 Radial wave functions for H and Na showing that the Na wave function is pulled in toward the ionic core
Fig. 15.1. For r rc , the potential is a Coulomb 1=r potential, and R.r/ is a solution of Eq. (15.3), which can be expressed as ı
R.r/ D f .`; ; r/ cos ` g.`; ; r/ sin ` r ; (15.12) where ` is the radial phase shift. Near the ionization limit, E 0, and as a result, the kinetic energy of the Rydberg electron is greater than 1=rc ( 10 eV) when r < rc . As a result, changes in E of 0:10 eV, the n D 10 binding energy, do not appreciably alter the phase shift ` , and we can assume ` to be independent of E. The ` dependence of ` arises because the centrifugal `.` C 1/=r term in Eq. (15.3) excludes the Rydberg electron from the region of the core in states of high `. Applying the r ! 1 boundary condition to the wave function of Eq. (15.12) leads to the requirement that the coefficient of u vanish, i.e., cos ` sin./ C sin ` cos./ D 0 ;
(15.13)
which implies that sin. C` / D 0 or D n` =. Usually ` = is written as ı` and termed the quantum defect, and the energies of members of the n` series are written as ED
1 1 D 2 ; 2.n ı` /2 2n
(15.14)
where n D nı` is often termed the effective quantum number (Sect. 12.5.1). Knowledge of the quantum defect ı` of a series of ` states determines their energies, and it is a straightforward matter to calculate the Coulomb wave function specified in Eq. (15.12) using a Numerov algorithm [4, 5]. This procedure gives wave functions valid for r rc , which can be used to calculate many of the properties of Rydberg atoms with great accuracy. The effect of core penetration on the energies is easily seen in the energy level diagram of Fig. 15.2. The Na ` 2 states have the same energies as hydrogen, while the s and p states, with quantum defects of 1:35 and 0:85, respectively, lie far below the hydrogenic energies. Although it is impossible to discern in Fig. 15.2, the Na ` 2 states also lie below the hydrogenic energies. For these
15
Rydberg Atoms
E 0
233
1000 cm–1 8s
Sodium 8p
Hydrogen
8d
8f
4d
4f
8g
n=8
n=4
5s 4p
–10
3d
n=3
4s
uum begins. Nonetheless, below the limit the cross section is structured by the n spacing of 1=n3 between adjacent members of the Rydberg series. In any experiment, there is a finite resolution ! with which the Rydberg states can be excited. It can arise, for example, from the Doppler width or a laser linewidth. This resolution determines the cross section n for exciting the Rydberg state of principal quantum number n. Explicitly, n is given by n D
–20
PI 3 n !
:
(15.16)
3p
A typical value for PI is 1018 cm2 . For a resolution ! D 1 cm1 (6 106 a:u:) the cross section for exciting an n D 20 atom is 3 1017 cm2 . –30 From the wave functions of the Rydberg states, we can also derive the n3 dependence of the photoexcitation cross section. The dipole matrix element from the ground state to a Rydberg state only involves the part of the Rydberg state –40 3s wave function near the core. At small r, the Rydberg wave function only depends on n through the n3 normalization factor, and as a result, the squared dipole matrix element between the ground state and the Rydberg state and the cross section both have an n3 dependence. Fig. 15.2 Energy levels of Na and H Radiative decay, which is covered in Chap. 18, is, to some extent, the reverse of optical excitation. The general expresstates, it is not core penetration but core polarization that sion for the spontaneous transition rate from the n` state to 0 0 is responsible for the shift to lower energy. Contrary to our the n ` state is the Einstein A coefficient, given by [2] earlier assumption that the outer electron does not affect the ˛ 3 g> 4 3 inner electrons if r > rc , the outer electron polarizes the inner An`;n0 `0 D 2n`;n0 `0 !n`;n ; (15.17) 0 `0 3 2gn C 1 electron cloud even when r > rc , and the energies of even the high ` states fall below the hydrogenic energies. The leading where n`;n0 `0 and !n`;n0 ;`0 are the electric dipole matrix eleterm in the polarization energy is due to the dipole polariz- ments and frequencies of the n` ! n0 `0 transitions, gn and ability of the core, ˛d . For high ` states, it gives a quantum gn0 are the degeneracies of the n` and n0 `0 states, and g> is defect of [6] the greater of gn and gn0 . The lifetime n` of the n` state is 3˛d (15.15) obtained by summing the decay rates to all possible lower ı` D 5 : 4` energy states. Explicitly, Quantum defects due primarily to core polarization rarely X 1 exceed 102 , while those due to core penetration are often D An`;n0 `0 : (15.18) n` greater than 1. n0 `0 n=2
15.2 Optical Excitation and Radiative Lifetimes
Due to the ! 3 factor in Eq. (15.17), the highest frequency transition usually contributes most heavily to the total radiative decay rate, and the dominant decay is likely to be the lowest lying state possible. For low-` Rydberg states, the lowest lying `0 states are bound by orders of magnitude more than the Rydberg states, and the frequency of the decay is nearly independent of n. Only the squared dipole moment depends on n, as n3 , because of the normalization of the Rydberg wave function at the core. Consequently, for low-` states, (15.19) n` / n3 :
Optical excitation of the Rydberg states from the ground state or any other low-lying state is the continuation of the photoionization cross section PI below the ionization limit. The photoionization cross section, discussed more extensively in Chap. 25, is approximately constant at the limit. Above and below the limit the average photoabsorption cross section is the same, as evidenced by the fact that a discontinuity is not evident in an absorption spectrum, i.e., it is not possible to As a typical example, the 10f state in H has a lifetime of see where the unresolved Rydberg states end and the contin- 1.08 s [7].
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T. F. Gallagher
The highest ` states, with ` D n 1, have radiative lifetimes with a completely different n dependence. The only possible transitions are n ! n 1, with frequency 1=n3 . In this case, the dipole moments reflect the large size of both the n and n 1 states and have the n2 scaling of the orbital radius. Using Eq. (15.17) for ` D n 1 leads to
15.3 Electric Fields
As a starting point, consider the H atom in a static electric field E in the z-direction and focus on the states of principal quantum number n. The field couples ` and ` ˙ 1 states of the same m by the electric dipole matrix elements. Since the states all have a common zero field energy of 1=2n2 , n.n1/ / n5 : (15.20) and the off-diagonal Hamiltonian matrix elements are all proportional to E , the eigenstates are field-independent linear Another useful lifetime, n , is that corresponding to the av- combinations of the zero field ` states of the same m, and the erage decay rate of all `; m states of the same n. It scales as energy shifts from 1=2n2 are linear in E . In this first-order n4:5 [2, 8]. approximation, the energies are given by [2] Equation (15.17) describes spontaneous decay to lower 3 1 lying states driven by the vacuum. At room temperature, (15.25) E D 2 C .n1 n2 /nE ; 2n 2 300 K, there are many thermal photons at the frequencies of the n ! n ˙ 1 transitions of Rydberg states for n 10, and where n1 and n2 are parabolic quantum numbers these photons drive transitions to higher and lower states [9]. (Sect. 9.1.2), which satisfy A convenient way of describing blackbody radiation is in (15.26) n1 C n2 C jmj C 1 D n : terms of the photon occupation number n, N given by Consider the m D 0 states as an example. The n1 n2 D n 1 1 3 : (15.21) state is shifted up in energy by 2 n.n 1/E and is called nN D !=kT e 1 the extreme blue Stark state, and the n2 n1 D n 1 state 3 The stimulated emission or absorption rate Kn`;n0 `0 from state is shifted down in energy by 2 n.n 1/E and is called the extreme red Stark state. These two states have large perman` to state n0 `0 is given by nent dipole moments, and in the red (blue) state, the electron 3 spends most of its time on the downfield (upfield) side of the ˛ ng N 4 > 3 : (15.22) proton as shown in Fig. 15.3, a plot of the potential along Kn`;n0 `0 D n`;n0 `0 !n`;n 0 `0 3 2gn C 1
Summing these rates over n0 and `0 gives the total blackbody decay rate 1=nbb . Explicitly, X 1 D Kn`;n0 `0 : bb n` n0 `0
(15.23)
T at any given temperature is given The resulting lifetime n` by 1 bb D 1=n` C 1=n` : (15.24) T n`
For low-` states with 10 < n < 20, blackbody radiation produces a 10% decrease in the lifetimes, but for high-` states of the same n, it reduces the lifetimes by a factor of 10. Since bb / n2 , this term must dominate normal spontaneous 1=n` emission at high n. The above discussion of spontaneous and stimulated transitions is based on the implicit assumption that the atoms are in free space. If the atoms are in a cavity, which introduces structure into the blackbody and vacuum fields, the transition rates are significantly altered [10]. These alterations are described in Chap. 83. If the cavity is tuned to a resonance, it increases the transition rate by the finesse of the cavity (approximately the Q for low-order modes). On the other hand, if the cavity is tuned between resonances, the transition rate is suppressed by a similar factor.
E (arb. units) 0.001
0.0
–0.001
B R
–0.002
–0.003 –2800
–1600
0
1600
2800 z (a 0)
Fig. 15.3 Combined Coulomb–Stark potential along the z-axis when a field of 5 107 a:u: (2700 V=cm) is applied in the z-direction (solid lines). The extreme red state (R) is near the saddle point, and the extreme blue (B) state is held on the upfield side of the atom by an effective potential (dashed line) roughly analogous to a centrifugal potential
15
Rydberg Atoms
235
Energy (cm–1) –250 22s
–260 21p
–270 n = 20
–280
21s
–290 20p
–300 n = 19
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0 (kV /cm)
Fig. 15.4 Energies of Na m D 0 levels of n 20 as a function of electric field. The shaded region is above the classical ionization limit
the z-axis. We have here ignored the electric dipole couplings to other n states, which introduce small second-order Stark shifts to lower energy. As implied by Eq. (15.26), states of higher m have smaller shifts. In particular, the circular m D ` D n 1 state has no first-order shift since there are no degenerate states to which it is coupled by the field. The Stark effect in other atoms is similar, but not identical to that observed in H. This point is shown by Fig. 15.4, a plot of the energies of the Na m D 0 levels near n D 20. The energy levels are similar to those of H in that most of the levels exhibit apparently linear Stark shifts from the zero field energy of the high-` states. The differences, however, are twofold. First, the levels from s and p states with nonzero quantum defects join the manifold of Stark states at some nonzero field, given approximately by [4] ED
2ı`0 ; 3n5
(15.27)
If an atom has an energy E relative to the zero field limit, it can ionize classically if the energy E lies above the saddle point in the potential. The required field is given by ED
E2 : 4
(15.29)
Ignoring the Stark shifts and using E D 1=2n2 yields the expression 1 ED : (15.30) 16n4 The H atom ionizes classically as described above or by quantum mechanical tunneling, which occurs at slightly lower fields. Since the tunneling rates increase exponentially with field strength, typically an order of magnitude for a 3% change in the field, specifying the classical ionization field is a good approximation to the field, which gives an ionization rate of practical interest. The red and blue states of H ionize at very different fields, as shown by Fig. 15.5, a plot of the m D 0 Stark states out to the fields at which the ionization rates are 106 s1 [12]. First, note the crossing of the levels of different n mentioned earlier. Second, note that the red states ionize at lower fields than the blue states do, in spite of the fact that they are lower in energy. In the red states, the electron is close to the saddle point of the potential of Fig. 15.3, and it ionizes according to Eq. (15.29). If the Stark shift of the extreme red state to lower energy is taken into account, Eq. (15.30) becomes
where ı`0 is the magnitude of the difference between ı` and the nearest integer. Second, there are avoided crossings between the blue n D 20 and red n D 21 Stark states. In H, these states would cross, but in Na they do not because of the finite sized NaC core, which also leads to the nonzero quantum defects of the ns and np states. Field ionization is both intrinsically interesting and of great practical importance for the detection of Rydberg atoms [11]. The simplest picture of field ionization can be understood with the help of Fig. 15.4. The potential along 1 ED : (15.31) the z-axis of an atom in a field E in the z-direction is given 9n4 by In the blue state, the electron is held on the upfield side of the 1 (15.28) atom by an effective potential roughly analogous to a cenV D Ez : r
15
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T. F. Gallagher E ( 1000 cm–1) –500
–1000
–1500
0
50
100 (kV /cm)
Fig. 15.5 Energies of H m D 0 levels of n D 9; 10, and 11 as functions of electric field. The widths of the levels due to ionization broaden exponentially with fields, and the onset of the broadening indicated is at an ionization rate of 106 s1 . The broken line indicating the classical ionization limit, E D E 2 =4, passes near the points at which the extreme red states ionize
trifugal potential, as shown by Fig. 15.3. At the same field, the blue state’s energy is lower relative to the saddle point of its potential, shown by the broken line of Fig. 15.3, than is the energy of the red state relative to the saddle point of its potential, given by Eq. (15.28) and shown by the solid line of Fig. 15.3. As shown by the broken line of Fig. 15.5, the classical ionization limit of Eq. (15.29) is simply a line connecting the ionization fields of the extreme red Stark states. All other states are stable above the classical ionization limit. In the Na atom, ionization of m D 0 states occurs in a qualitatively different fashion [12]. Due to the finite size of the NaC core, crossings are avoided between the blue and red Stark states of different n, as is shown by Fig. 15.4. In the region above the classical ionization limit, shown by the shaded region of Fig. 15.4, the same coupling between hydrogenically stable blue states and the degenerate red continua leads to autoionization of the blue states [13]. As a result, all states above the classical ionization limit ionize at experimentally significant rates. In higher m states, the core coupling is smaller, and the behavior is more similar to H. Field ionization is commonly used to detect Rydberg atoms in a state selective manner. Experiments are most often conducted at or near zero field, and afterwards the field is increased in order to ionize the atoms. Exactly how the atoms pass from the low field to the high ionizing field is quite important. The passage can be adiabatic, diabatic, or anything in between. The selectivity is best if the passage is purely adiabatic or purely diabatic, for in these two cases, unique paths are followed.
In zero field, optical excitation from a ground s state leads only to final np states. In the presence of an electric field, all the Stark states are optically accessible, because they all have some p character. The fact that all the Stark states are optically accessible from the ground state allows the population of arbitrary ` states of nonhydrogenic atoms by a technique called Stark switching [6, 14]. In any atom other than H, the ` states are nondegenerate in zero field, and each of them is adiabatically connected to one, and only one, high field Stark state, as shown by Fig. 15.4. If one of the Stark states is excited with a laser and the field reduced to zero adiabatically, the atoms are left in a single zero field ` state. In zero field, the photoionization cross section is structureless. However, in an electric field, it exhibits obvious structure, sometimes termed strong field mixing resonances. Specifically, when ground state s atoms are exposed to light polarized parallel to the static field, an oscillatory structure is observed in the cross section, even above the zero field ionization limit [15]. The origin of the structure can be understood with the aid of a simple classical picture [16, 17]. The electrons ejected in the downfield direction can simply leave the atom, while the electrons ejected in the upfield direction are reflected back across the ionic core and also leave the atom in the downfield direction. The wave packets corresponding to these two classical trajectories are added, and they can interfere constructively or destructively at the ionic core, depending on the phase accumulation of the reflected wave packet. Since the phase depends on the energy, there is an oscillation in the photoexcitation spectrum. This model suggests that no oscillations should be observed for light polarized perpendicular to the static field, and none are. The oscillations can also be thought of as arising from the remnants of quasistable extreme blue Stark states that have been shifted above the ionization limit, and, using this approach or a WKB approach, one can show that the spacing between the oscillations at the zero field limit is E D E 3=4 [18, 19]. Further discussion can be found in Chap. 16. The initial photoexcitation experiments were done using narrow bandwidth lasers, so that the time dependence of the classical pictures was not explicitly observed. Using mode locked lasers it has been possible to create a variety of Rydberg wave packets [20, 21] and observe, in effect, the classical motion of an electron in an atom. Of particular interest is that it has been possible to directly observe the time delay of the ejection of electrons subsequent to excitation in an electric field [22].
15.4 Magnetic Fields To first order, the energy shift of a Rydberg atom due to a magnetic field B (the Zeeman effect) is proportional to the angular momentum of the atom. Since the states op-
15
Rydberg Atoms
tically accessible from the ground state have low angular momenta, the energy shifts are the same as those of low-lying atomic states. In contrast, the second-order diamagnetic energy shifts are proportional to the area of the Rydberg electron’s orbit and scale as B 2 n4 [23]. The diamagnetic interaction mixes the ` states, allowing all to be excited from the ground state, and produces large shifts to higher energies. The energy levels as a function of the magnetic field are reminiscent of the Stark energy levels shown in Fig. 15.5, differing in that the energy shifts are quadratic in the magnetic field. One of the most striking phenomena in magnetic fields is the existence of quasi-Landau resonances, spaced by E D 3„B=2, in the photoionization cross section above the ionization limit [24]. The origin of this structure is similar to the origin of the strong field mixing resonances observed in electric fields. An electron ejected in the plane perpendicular to the B fields is launched into a circular orbit and returns to the ionic core. The returning wave packet can be in or out of phase with the one leaving the ionic core, and, thus, can interfere constructively or destructively with it. While the electron motion in the xy-plane is bound, motion in the z-direction is unaffected by the magnetic field and is unbounded above the ionization limit, leading to resonances of substantial width. The Coulomb potential does provide some binding in the z-direction and allows the existence of quasistable three-dimensional orbits [25].
15.5 Microwave Fields
237
regime [27]. For ! > 1=n3 , the ionization field is more or less constant, and for ! > 1=2n2 , the process becomes photoionization. The ionization of nonhydrogenic atoms by linearly polarized fields has also been investigated at frequencies of up to 30 GHz, but the result is very different from the hydrogenic result. For ! 1=n3 and low m, ionization occurs at a field of E 1=3n5 [28]. This is the field at which the m D 0 extreme blue and red Stark states of principal quantum number n and n C 1 have their avoided crossing. For n D 20, this field is 500 V=cm, as shown by Fig. 15.4. How ionization occurs can be understood with a simple model based on a time-varying electric field. As the microwave field oscillates in time, atoms follow the Stark states of Na shown in Fig. 15.4. Even with very small field amplitudes, transitions between the Stark states of the same n are quite rapid because of the zero field avoided crossings. If the field reaches 1=3n5 , the avoided crossing between the extreme red n and blue n C 1 state is reached, and an atom in the blue n Stark state can make a Landau–Zener transition to the red n C 1 Stark state. Since the analogous red–blue avoided crossings between higher lying states occur at lower fields, once an atom has made the n ! n C 1 transition, it rapidly makes a succession of transitions through higher n states to a state that is itself ionized by the field. The Landau–Zener description given above is somewhat oversimplified in that we have ignored the coherence between field cycles. When it is included, we see that the transitions between levels are resonant multiphoton transitions. While the resonant character is obscured by the presence of many overlapping resonances, the coherence substantially increases the n ! n C 1 transition probability even when E < 1=3n5 . The fields required for ionization calculated using this model are lower than 1=3n5 , in agreement with the experimental observations. Nonhydrogenic Na states of high m behave like H, because no states with significant quantum defects are included, and the n ! nC1 avoided crossings are vanishingly small. Experiments on ionization of alkali atoms by circularly polarized fields of frequency ! show that for ! 1=n3 , a field amplitude of E D 1=16n4 is required for ionization [29]. This field is the same as the static field required. In a frame rotating with frequency !, the circularly polarized field is stationary and cannot induce transitions, so this result is not surprising. On the other hand, when the problem is transformed to the rotating frame, the potential of Eq. (15.28) is replaced by
Strong microwave fields have been used to drive multiphoton transitions between Rydberg states and to ionize them. Here, we restrict our attention to ionization. Ionization by both linearly and circularly polarized fields has been explored with both H and other atoms. Hydrogen atoms have been studied with linearly polarized fields of frequencies up to 36 GHz [26]. When the microwave frequency ! 1=n3 , ionization of m D 0 states occurs at a field of E D 1=9n4 .E 2 =4/, which is the field at which the extreme red Stark state is ionized by a static field. Due to the second-order Stark effect, the blue and red shifted states are not quite mirror images of each other, and when the microwave field reverses, transitions between Stark states occur. There is a rapid mixing of the Stark states of the same n and m by a microwave field, and all of them are ionized at the same microwave field amplitude, E D 1=9n4 . Important points are that no change in n occurs, and the ionization field ! 2 2 1 is the same as the static field required for ionization of the ex; (15.32) V D Ex r 2 treme red Stark state. As ! approaches 1=n3, the field falls below 1=9n4 due to n transitions to higher lying states, al- where 2 D x 2 C y 2 , and we have assumed the field to be lowing ionization at lower fields. This form of ionization can in the x-direction in the rotating frame. This potential has be well described as the transition to the classically chaotic a saddle point below E D 1=16n4 [30]. As n, or !, is raised
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T. F. Gallagher
so that ! ! 1=n3 , the experimentally observed field falls below 1=16n4 , but not so fast as implied by Eq. (15.32). Equation (15.32) is based solely on energy considerations, and ionization at the threshold field implied by Eq. (15.32) requires that the electron escape over the saddle point in the rotating frame at nearly zero velocity. For this to happen, when ! approaches 1=n3 , more than n units of angular momentum must be transferred to the electron, which is unlikely. Models based on a restriction of the angular momentum transferred from the field to the Rydberg electron are in better agreement with the experimental results. Small deviations of a few percent from circular polarization allow ionization at fields as low as E D 1=3n5 . This sensitivity can be understood as follows. In the rotating frame, a field with slightly elliptical polarization appears to be a large static field with a superimposed oscillating field at frequency 2!. The oscillating field drives transitions to states of higher energy, allowing ionization at fields less than E D 1=16n4. In the regime in which ! > 1=n3 , microwave ionization of nonhydrogenic atoms is essentially the same as it is in H [31]. In this regime, the microwave field couples states differing in n by more than 1, and the pressure or absence of quantum defects is not so important. Consequently, only for ! > 1=n3 is the microwave ionization of H and other atoms different.
(15.33)
in the M –Na collision, as first pointed out by Fermi [32]. First, consider the case where M is an atom. There are no energetically accessible states of atom M that can be excited by the low-energy electron, so the scattering must be elastic. The electron can transfer very little kinetic energy to M , but the direction of the electron’s motion can change. With this thought in mind, we can see that only the collisional mixing of nearly degenerate ` states of the same n has very large cross sections. The `-mixing cross sections are approximately geometric at low n [33]. If the M atom comes anywhere into the Rydberg orbit, scattering into a different ` state occurs. At high n, the cross section decreases, because the probability distribution of the Rydberg electron becomes too dilute, and it becomes increasingly likely that the M atom will pass through the Rydberg electron’s orbit without encountering the electron. The n at which the peak `-mixing cross section occurs increases with the electron scattering length of the atom. While `-mixing cross sections are large, n changing cross sections are small . 100 Å2 / since they cannot occur when the Rydberg electron is anywhere close to the outer turning point of its orbit [34]. If M is a molecule, there are likely to be energetically accessible vibrational and rotational transitions that can provide energy to or accept energy from the Rydberg electron, and this possibility increases the likelihood of n changing collisions with Rydberg atoms [11]. Electronic energy from the Rydberg atom must be resonantly transferred to rotation or vibration in the molecule. In heavy or complex molecules, the presence of many rotational-vibrational states tends to obscure the resonant character of the transfer, but in several light systems, the collisional resonances have been observed clearly [11]. Using the large Stark shifts of Rydberg atoms it is possible to tune the levels so that resonant energy transfer between two colliding atoms can occur [35] by the resonant dipole– dipole coupling, 1 2 : (15.34) Vd D R3
The long-range e – NaC interaction determines the energy levels of the Na atom. The short-range of the e – M and M – NaC interactions makes it likely that only one will be important at any given time. This approximation, termed the binary encounter approximation, is described in Chap. 60. The M – NaC interaction can only lead to cross sections of 10100 Å2 . On the other hand, since the electron can be anywhere in the cloud, the cross sections due to the e – M interaction can be as large as the geometric cross section of the Rydberg atom. Accordingly, we focus on the e – M interaction. Consider a thermal collision between M and an Na Rydberg atom. Typically, M passes through the electron cloud slowly compared with the velocity of the Rydberg electron, and it is the e – M scattering that determines what happens
Here, 1 and 2 are the dipole matrix elements of the upward and downward transitions in the two atoms, and R is their separation. At room temperature, this process leads to enormous cross sections, substantially in excess of the geometric cross sections. At the low temperatures (300 K) attainable using cold atoms, the atoms do not move, and, therefore, cannot collide. However, resonant dipole–dipole energy transfer is still observed due to the static dipole–dipole interactions of not two but many atoms [36, 37]. Since Rydberg atoms are easily perturbed by electric fields, it is hardly a surprise that collisions of charged particles with Rydberg atoms have large cross sections. In cold Rydberg atom samples, these large cross sections can lead to the spontaneous evolution to a plasma, since the macroscopic positive charge of the cold ions can trap any liberated
15.6 Collisions Since Rydberg atoms are large, with geometric cross sections proportional to n4 , one might expect the cross sections for collisions to be correspondingly large. In fact, such is often not the case. A useful way of understanding collisions of neutral atoms and molecules with Rydberg atoms is to imagine an atom or molecule M passing through the electron cloud of an Na Rydberg atom. There are three interactions e NaC ;
e M ;
M NaC :
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electrons, leading to impact ionization for a large part of the Fig. 15.6 SrC 5s and 5p states ( ); the Rydberg states of Sr Rydberg atom sample [38, 39].
15.7 Autoionizing Rydberg States The bound Rydberg atoms considered thus far are formed by adding the Rydberg electron to the ground state of the ionic core. It could equally well be added to an excited state of the core [40]. Figure 15.6 shows the energy levels of the ground 5s state of SrC and the excited 5p state. Adding an n` electron to the 5s state yields the bound Sr 5sn` state, and adding it to the excited 5p state gives the doubly excited 5pn` state, which is coupled by the Coulomb interaction to the degenerate 5s`0 continuum. The 5pn` state autoionizes at the rate n` given by [41] n` D 2jh5pn`jV j5s`0 ij2 ;
(15.35)
converging to these two ionic states are shown by (—), the continuum above the two ionic levels (///). The 5pn` states are coupled to the 5s`0 continua and autoionize
5p
5s
polarizes the SrC core, so that the outer electron is in a potential due to a charge and a dipole, and the resulting dipole states of the outer electron display a qualitatively different excitation spectrum than do states such as the 5pn` states, which are well described by an independent particle picture [44]. When both electrons are excited to very high-lying states, with the outer electron in a state of relatively low `, the classical orbits of the two electrons cross. Time domain measurements, made using wave packets, show that, in this case, autoionization is likely to occur in the first orbit of the outer electron [45].
where V denotes the Coulomb coupling between the nominally bound 5pn` state and the 5s`0 continuum. A more general description of autoionization can be found in Chap. 26. A simple picture, based on superelastic electron scattering from the SrC 5p state, gives the scaling of the autoionization rates of Eq. (15.35) with n and `. The n` Rydberg electron is in an elliptical orbit, and each time it comes near the core it References has an n-independent probability ` of scattering superelastically from the SrC 5p ion, leaving the core in the 5s state and 1. White, H.E.: Introduction to Atomic Spectra. McGraw-Hill, New gaining enough energy to escape from the Coulomb potential York (1934) 2. Bethe, H.A., Salpeter, E.A.: Quantum Mechanics of One and Two of the SrC core. The autoionization rate of the 5pn` state is Electron Atoms. Academic Press, New York (1975) obtained by multiplying ` by the orbital frequency of the n` 3. Fano, U.: Phys. Rev. A 2, 353 (1970) 3 state, 1=n , to obtain 4. Zimmerman, M.L., Littman, M.G., Kash, M.M., Kleppner, D.: n` D
` : n3
(15.36)
Equation (15.36) displays the n dependence of the autoionization rate explicitly and the ` dependence through ` . As ` increases, the closest approach of the Rydberg electron to the SrC is at a larger orbital radius, so that superelastic scattering becomes progressively less probable, and ` decreases rapidly with increasing `. The simple picture of autoionization given above implies a finite probability of autoionization each time the n` electron passes the ionic core, so the probability of an atom remaining in the autoionizing state should resemble stair steps [42], which can be directly observed using mode locked laser excitation and detection [43]. To a first approximation, the Sr 5pn` states can be described by the independent electron picture used above, but in states converging to higher lying states of SrC , the independent electron picture fails. Consider the SrC ` 4 states of n > 5. They are essentially degenerate, and the field due to an outer Rydberg electron converts the zero field ` states to superpositions much like Stark states. The outer electron
Phys. Rev. A 20, 2251 (1979) 5. Bhatti, S.A., Cromer, C.L., Cooke, W.E.: Phys. Rev. A 24, 161 (1981) 6. Freeman, R.R., Kleppner, D.: Phys. Rev. A 14, 1614 (1976) 7. Lindgard, A., Nielsen, S.E.: At. Data Nucl. Data Tables 19, 534 (1977) 8. Chang, E.S.: Phys. Rev. A 31, 495 (1985) 9. Cooke, W.E., Gallagher, T.F.: Phys. Rev. A 21, 588 (1980) 10. Haroche, S., Raimond, J.M.: Radiative properties of Rydberg states in resonant cavities. In: Bates, D., Bederson, B. (eds.) Advances in Atomic and Molecular Physics, vol. 20, Academic Press, New York (1985) 11. Dunning, F.B., Stebbings, R.F.: Experimental studies of thermalenergy collisions of Rydberg atoms with molecules. In: Stebbings, R.F., Dunning, F.B. (eds.) Rydberg States of Atoms and Molecules. Cambridge Univ. Press, Cambridge (1983) 12. Bailey, D.S., Hiskes, J.R., Riviere, A.C.: Nucl. Fusion 5, 41 (1965) 13. Littman, M.G., Kash, M.M., Kleppner, D.: Phys. Rev. Lett. 41, 103 (1978) 14. Jones, R.R., Gallagher, T.F.: Phys. Rev. A 38, 2946 (1988) 15. Freeman, R.R., Economou, N.P., Bjorklund, G.C., Lu, K.T.: Phys. Rev. Lett. 41, 1463 (1978) 16. Reinhardt, W.P.: J. Phys. B 16, 635 (1983) 17. Gao, J., Delos, J.B., Baruch, M.C.: Phys. Rev. A 46, 1449 (1992) 18. Gallagher, T.F.: Rydberg Atoms. Cambridge Univ. Press, Cambridge (1994)
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24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.
35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.
T. F. Gallagher Rau, A.R.P.: J. Phys. B 12, L193 (1979) Jones, R.R., Noordam, L.D.: Adv. At. Mol. Opt. Phys. 38, 1 (1997) Alber, G., Zoller, P.: Phys. Rep. 199, 231 (1991) Warntjes, J.B.M., Wesdorp, C., Robicheaux, F., Noordam, L.D.: Phys. Rev. Lett. 83, 512 (1999) Kleppner, D., Littman, M.G., Zimmerman, M.L.: Rydberg atoms in strong fields. In: Stebbings, R.F., Dunning, F.B. (eds.) Rydberg States of Atoms and Molecules. Cambridge Univ. Press, Cambridge (1983) Garton, W.R.S., Tomkins, F.S.: Astrophys. J. 158, 839 (1969) Holle, A., Main, J., Wiebusch, G., Rottke, H., Welge, K.H.: Phys. Rev. Lett. 61, 161 (1988) Sauer, B.E., Bellerman, M.R.W., Koch, P.M.: Phys. Rev. Lett. 68, 1633 (1992) Jensen, R.V., Susskind, S.M., Sanders, M.M.: Phys. Rep. 201, 1 (1991) Pillet, P., van Linden van der Heuvell, H.B., Smith, W.W., Kachru, R., Tran, N.H., Gallagher, T.F.: Phys. Rev. A 30, 280 (1984) Fu, P., Scholz, T.J., Hettema, J.M., Gallagher, T.F.: Phys. Rev. Lett. 64, 511 (1990) Nauenberg, M.: Phys. Rev. Lett. 64, 2731 (1990) Krug, A., Buchleitner, A.: Phys. Rev. A 66, 053416 (2002) Fermi, E.: Nuovo Cimento 11, 157 (1934) Gallagher, T.F., Edelstein, S.A., Hill, R.M.: Phys. Rev. A 15, 1945 (1977) Gounand, F., Berlande, J.: Experimental studies of the interaction of Rydberg atoms with atomic species at thermal energies. In: Stebbings, R.F., Dunning, F.B. (eds.) Rydberg States of Atoms and Molecules. Cambridge Univ. Press, Cambridge (1983) Gallagher, T.F.: Phys. Rep. 210, 319 (1992) Mourachko, I., Comparat, D., de Tomasi, F., Fioretti, A., Nosbaum, P., Akulin, V.M., Pillet, P.: Phys. Rev. Lett. 80, 253 (1998) Anderson, W.R., Veale, J.R., Gallagher, T.F.: Phys. Rev. Lett. 80, 249 (1998) Robinson, M.P., Laburthe-Tolra, B., Noel, M.W., Gallagher, T.F., Pillet, P.: Phys. Rev. Lett. 85, 4466 (2000) Dutta, S.K., Guest, J.R., Feldbaum, D., Walz-Flannigan, A., Raithel, G.: Phys. Rev. Lett. 86, 3993 (2001) Gallagher, T.F.: J. Opt. Soc. Am. B 4, 794 (1987) Fano, U.: Phys. Rev. 124, 1866 (1961) Wang, X., Cooke, W.E.: Phys. Rev. Lett. 67, 696 (1991) Pisharody, S.N., Jones, R.R.: Phys. Rev. A 65, 033418 (2002) Eichmann, U., Lange, V., Sandner, W.: Phys. Rev. Lett. 68, 21 (1992) Pisharody, S.N., Jones, R.R.: Science 303, 813 (2004)
Thomas F. Gallagher Thomas F. Gallagher received his PhD in Physics in 1971 from Harvard University and is now the Jesse W. Beams Professor of Physics at the University of Virginia. His research is focused on the use of Rydberg atoms to realize novel physical systems.
Rydberg Atoms in Strong Static Fields John B. Delos
16
, Thomas Bartsch, and Turgay Uzer
Contents 16.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 241
16.2
Semiclassical Approximations . . . . . . . . . . . . . . 242
16.3
Regular Trajectories and Regular Wave Functions
244
16.4 16.4.1 16.4.2 16.4.3
Chaotic Trajectories and Irregular Wave Functions Small-Scale Structure . . . . . . . . . . . . . . . . . . . . Large-Scale Structure of Energy Spectra . . . . . . . . . Chaotic Ionization . . . . . . . . . . . . . . . . . . . . . .
245 246 247 250
16.5
Nuclear-Mass Effects . . . . . . . . . . . . . . . . . . . . 250
16.6
Quantum Theories . . . . . . . . . . . . . . . . . . . . . 250
to the large-scale structure of the density of states and the absorption spectrum. This chapter is restricted to a discussion of Rydberg atoms in strong static fields. Related information on atoms in strong fields, on Rydberg atoms, and on the interaction of atoms with strong laser fields can be found elsewhere in the book. Keywords
semiclassical methods quantum chaos Rydberg states recurrence spectroscopy closed orbits external fields
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Abstract
Semiclassical approximations provide a connection between quantum mechanics and classical mechanics. That relationship is problematic when classical orbits are chaotic. Highly excited states of atoms have provided a laboratory for the study of this relationship. When classical trajectories are orderly, corresponding quantum spectra and wave-functions display the same type of order. When they are chaotic, the quantum spectrum is very complex, but short closed classical orbits are connected J. B. Delos () Butterfly Dynamics LLC Williamsburg, VA, USA e-mail: [email protected] T. Bartsch Mathematical Sciences, Loughborough University Leicestershire, UK e-mail: [email protected] T. Uzer School of Physics, Georgia Institute of Technology Atlanta, GA, USA e-mail: [email protected]
16.1 Introduction Confronting classical and quantum mechanics in systems whose classical motion is chaotic is one of the fundamental problems of physics, as evidenced by the enormous outpouring of research during the last three decades. Introductions to chaotic classical dynamics are given in [1, 2], and detailed discussions of quantum manifestations of classical chaos are given in [3–5]. Highly excited Rydberg atoms in external fields play a prominent role in this quest because they are among the best known examples of quantum systems whose classical counterparts are chaotic [6–11]. For a wide variety of field configurations and field strengths, their spectra can be measured to high precision. At the same time, since their Hamiltonians are known analytically, they are equally amenable to accurate theoretical investigations using either classical or quantum mechanics. This chapter is focused on semiclassical descriptions of Rydberg atoms in strong static fields. Related information on atoms in strong fields can be found in Chap. 14 of this Handbook, on Rydberg atoms in Chap. 15, and on the interaction of atoms with strong laser fields in Chaps. 76 and 78. Atomic units will be used throughout this chapter.
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_16
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Various configurations of external fields have been stud- the scaled Hamiltonian reads ied: 1. an electric field, which in hydrogen leads to separable and, therefore, integrable classical dynamics; 2. a magnetic field, which produces a transition from regular to chaotic classical dynamics and which sparked the interest in Rydberg atoms as prototype examples for the study of the quantum-classical correspondence [7, 12–14] and references therein; 3. parallel electric and magnetic fields [15]; 4. crossed electric and magnetic fields, which break all continuous symmetries of the unperturbed atom and, thus, allow one to study the transition from regularity to chaos in three coupled degrees of freedom [16–20]. The hydrogen atom is the prototype example for states with a single highly excited electron under the influence of strong external fields. For an electron in the hydrogen ground state, the influence of external electric or magnetic fields becomes comparable to that of the nuclear Coulomb field when the field strengths are in the order of the atomic units of electric field strength, F0 D e=.4 "0 a0 / D 5:142 206 42.44/ 1011 V=m, or magnetic field strength, B0 D ¯=.ea02 / D 2:350 517 42.20/ 105 T, which are far beyond experimental reach. However, the relative importance of the external fields scales with the principal quantum number n as n4 F and n3 B, so that for highly excited atoms, laboratory fields can easily be “strong”. In a nonhydrogenic atom, the influence of the inner-shell electrons can be summarized by means of a shortrange effective core potential or a set of quantum defects [21, 22]. For laboratory field strengths, the core is too small to be appreciably influenced by the external fields. For this reason, the field-free quantum defects can be used to model core effects even in the presence of external fields [23]. The Hamiltonian for a hydrogen atom in a zO -directed magnetic field and an electric field of arbitrary orientation is, in atomic units, H D
pQ 2 1 1 2 1 Q HQ D C L
Q C rQ FQ D EQ : z C 2 rQ 2 8
(16.4)
The scaled dynamics, thus, depends only on three parameters: the scaled energy EQ and the scaled electric field strength FQ and its polar angle. If the fields are parallel, then the scaled angular momentum LQ z is conserved, and it can be regarded as another fixed parameter, so, for example, if there is no electric field and LQ z D 0 the scaled dynamics depends only Q on the scaled energy E.
16.2 Semiclassical Approximations In Rydberg states, as in other highly excited quantum states, the deBroglie wavelength of the electron is short compared to the range over which the potential energy varies, so semiclassical approximations are appropriate means for insightful calculations. The semiclassical approximation in one dimension is a standard topic in textbooks on quantum mechanics, but satisfactory treatments in higher dimension are not common in textbooks. Therefore, below we give a brief description of semiclassical theory in two dimensions, which is adequate for many studies of Rydberg atoms in fields. Semiclassical approximations are typically constructed using numerical computations of classical trajectories. The semiclassical approximation in two dimensions is easiest to explain by thinking about elastic scattering of particles of mass m and energy E from a fixed potential energy V .r/ having long-range attraction and short-range repulsion [24]. In Fig. 16.1, a stream of classical particles comes in from the left and interacts with such a potential energy fixed at the ori-
1 1 p2 1 C BLz C B 2 2 C r F D E ; (16.1) 2 r 2 8
where 2 D x 2 C y 2 and Lz is the angular momentum component along the magnetic field axis. The dynamics depends on four parameters: the field strengths F and B, the energy E, and the angle between the fields. We can reduce the number of independent parameters by one if we exploit a scaling property of the Hamiltonian. In terms of the scaled quantities rQ D w 2 r ; pQ D wp ; EQ D w 2 E ; FQ D w 4 F ;
(16.2)
with the scaling parameter w D B 1=3 ;
Fig. 16.1 Two-dimensional scattering with long-range attraction and
(16.3) short-range repulsion. (Figure kindly provided by C. Schleif)
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gin. The radial .r/ and angular ./ motions are described by Œpx .t; b/; pz .t; b/ are single-valued vector functions of time and of the impact parameter. The classical action is likewise 1=2 2 dr l2 a single-valued function of .t; b/, D E V .r/ 2 dt m 2mr Zt d l @q.t; b/ (16.5) D dt : (16.6) p.t; b/ S.t; b/ D dt mr 2 @t (l D angular momentum). Particles having a large impact parameter b are only slightly deflected by the long-range attractive force. For smaller b, there is more deflection downward, which is said to be the negative scattering angle. The deflection reaches an extremum (the negative scattering angle has a minimum, called the “rainbow angle”), and then at smaller b, when the repulsive force dominates, particles are scattered upward, which is the positive scattering angle, until when b D 0, the scattering angle equals (backward scattering). The complete family of classical paths defines a dynamically allowed region of the plane, and two paths arrive at each point in this allowed region. The boundary between allowed and forbidden regions is called a “caustic”. (The dynamically forbidden region is much larger than the energetically forbidden region; the latter is the interior of a circle around the origin, where the potential energy exceeds the incident kinetic energy. The caustic touches the energetically forbidden region only on the trajectory having b D 0 [24]. Paths at larger b never touch the energetically forbidden region because of angular momentum repulsion, l 2 =2mr 2 .) Two properties of this family of classical trajectories are needed for the calculation of a semiclassical approximation to the quantum wave function, the classical actions and the classical densities. On each path, the position q.t; b/ D Œx.t; b/; z.t; b/ and the momentum p.t; b/ D Fig. 16.2 Potential energy, trajectories, and wave functions for two-dimensional scattering with long-range attraction and shortrange repulsion. (a) The potential energy, with a well and a barrier, and some trajectories from Fig. 16.1 shown in perspective. (b) Trajectories and the “incident” term in the wave function. (c) Trajectories and the “scattered” term in the wave function. (d) The complete wave function in the classically allowed region. (Figure kindly provided by C. Schleif)
a
t0
The classical density describes the fact that trajectories approaching the target represent a uniform flow, neighboring paths converge at the caustic, and scattered paths diverge from each other. This density is given by the function ˇ ˇ ˇ J.t0 ; b/ ˇ ˇ;
.t; b/ D ˇˇ J.t; b/ ˇ @q @.x; z/ where J .t; b/ D D ; (16.7) @.t; b/ @.t; b/ where t0 is the initial time on each trajectory. To construct a semiclassical wave function, we need to express the classical action and density as functions of position q. Since two paths arrive at each point in the allowed region, these quantities are two-valued functions of q. Each path that touches the caustic is divided into an “incident” (Inc) portion, prior to touching the caustic, and a “scattered” (Scat) portion, after touching the caustic. At large b, where paths never touch the caustic, the entire path is regarded as “Inc”, even the part that is going away from the target. (Quantum-mechanical descriptions of such collisions use an expansion in radial functions times spherical harmonics, and the radial function is divided into incoming and outgoing waves. The caustic does not coincide with the distance of
b
16 c
d
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closest approach of each trajectory, so our description of “incident” and “scattered” terms in the semiclassical wave function does not correspond directly to radial incoming and outgoing waves.) Numerical inversion and interpolation of the relationship between q and .t; b/ gives two-valued action and density functions, S˛ .q/, ˛ .q/, with ˛ D .Inc, Scat/. Then the semiclassical approximation to the wave function in the allowed region is given by a sum of an “incident” and a “scattered” wave function, S˛ .q/ ˛ 1=2 .q/ D Œ
.q/ exp i ; ˛ ˛ ¯ 2 ˛ D .Inc, Scat/ : (16.8) The absolute square of each term is the associated classical density, and the phase is the classical action with a correction ˛ =2, where ˛ is called the Maslov index. In this case, I D 0; S D 1. The incident and scattered and full wave functions are shown in Fig. 16.2b–d. One sees refraction of the Inc term following the classical trajectories, and reflection producing the nearly circular outgoing Scat term. In the full wave function, one sees interference between these two waves. Momentum-vector-fields are gradients of the two-valued action function, p˛ .q/ D rS˛ .q/, so r p˛ .q/ D 0. Any family of trajectories that forms a momentum-vector field with vanishing curl forms what in optics is called a “normal congruence” (in modern language, such a momentum-vector field defines a “Lagrangian manifold” in phase space). From any such vector field, a corresponding semiclassical wave function Eq. (16.8) can be constructed. Further details and mathematical theory are given in [24–28].
16.3 Regular Trajectories and Regular Wave Functions We return now to Rydberg states of atoms in fields. At energies below the ionization threshold, there are bound trajectories that stay in a finite region of space. Figure 16.3 shows a path in cylindrical coordinates . ; z/ followed by an electron in the presence of the attractive Coulomb force of a nucleus at the origin and forces from electric and magnetic fields directed along the z-axis. This is an example of a “regular” trajectory. The path is a perturbed Kepler ellipse; the eccentricity and orientation of the ellipse change continuously with time such that the ellipse does not close on itself. Also, the cylindrical coordinate cannot be negative, so in this picture, the ellipse is folded over across itself. For weak electric and magnetic fields, the principal action is approximately conserved, and its associated principal quantum number n is a good quantum number. If the fields
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are parallel, then the z-component of angular momentum Lz is conserved, and quantized as m¯. However, the total angular momentum is not conserved, and ` is not a good quantum number. It is replaced by an approximately conserved quantity, which can be written in various ways. Solov’ev [29] and Herrick [30] wrote it as D 4A 2 5A2z ;
(16.9)
where A is the Laplace–Runge–Lenz vector 1 1 r : ADp .p L L p/ r 2E 2
(16.10)
Waterland et al. [15] used hydrogenic action and angle variables and classical perturbation theory to write the conserved quantity in the form F .eccentricity, orientation/ ' constant f 0 :
(16.11)
Detailed analysis of either of these functions, Eq. (16.9) or Eq. (16.11), gives an understanding of all details of the regular classical trajectories. This regular path winds around in the . ; z/ plane such that it forms four well-defined curl-free momentum-vector fields p˛ .q/, with each having a well-defined density ˛ .q/. Accordingly, each of these four vector-fields “support” a semiclassical wave function of the form Eq. (16.8), and the full wave function is a linear combination of these four terms. However, it is necessary to combine the terms in such a way that the full semiclassical wave function corresponds to a quantum wave function that satisfies the boundary conditions, . ; z/ ! 0 as ! 1, z ! ˙1. This requires quantization conditions. In the 4-D phase space .q; p/, any surface having fixed energy E and satisfying Eq. (16.11) forms a 2-D torus labeled by .E; f 0 /, and the exact trajectory lies very close to
ρ 2000
1000
0 2000
1000
0
–1000
–2000 z
Fig. 16.3 A regular trajectory of an atomic electron in parallel electric and magnetic fields. This one has appropriately quantized action variables, so it is an “eigentrajectory”, from which a semiclassical eigenfunction can be calculated
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this surface. Any 2-D torus (the surface of a donut) has two distinct fundamental loops that cannot be shrunk to a point. Action variables are defined as Z
(16.12) Sj E; f 0 D p qI E; f 0 dq; j D 1; 2 j
around each of the two fundamental loops of each torus [25, 28, 31–33]. The tori form continuous families, since E and f 0 vary continuously. Out of those continuous families of tori, we select a discrete set having appropriately quantized values of action variables j 2 ¯; j D 1; 2 I Sj .E; f 0 / D kj C 4
(16.13)
kj is a positive integer, and j is a Maslov index (equal to 2 in this case). Those discrete tori are called “eigentori”, and the trajectories on them are called “eigentrajectories”. The momentum-vector fields associated with each eigentrajectory, with their associated densities and action functions, “support” a quantized semiclassical wave function. The local wavelength of the ˛ semiclassical term is ˛ D 2 ¯=jp˛ .q/j, and the quantization condition ensures that there will be a properly quantized number of wavelengths associated with each fundamental loop of each eigentorus. Thus, the corresponding exact quantum wave function will satisfy the necessary boundary conditions. In Fig. 16.4 is shown the complete quantum wave function (absolute squared) associated with the eigentrajectory that was shown in Fig. 16.3. (This eigenfunction was computed by accurate quantum methods, with no reference to the semiclassical wave function.) We see that the wave function primarily occupies the same region of space as the eigentrajectory, diffracting into the nearby classically forbidden region. The local wavelength and density of the quantum
wave function are consistent with the local momentum and density of the eigentrajectory [15]. This case illustrates the general principles for systems having regular trajectories [3, 4, 34, 35]. (1) Regular bound trajectories wind around a finite region of space such that they form curl-free momentum-vector fields. In the phase space of dimension 2n, bound regular trajectories lie on tori of dimension n. (2) Upon an appropriately quantized set of these regular trajectories, one can construct semiclassical wave functions such that the corresponding exact quantum wave function satisfies the necessary boundary conditions. In the simple cases (in the absence of tunneling), there is a one-to-one correspondence between eigentrajectories and eigenfunctions. (3) All trajectories are regular if the equations of motion are separable (i.e., the Hamilton–Jacobi equation is separable), or, more generally, if the system has n degrees of freedom and n conservation laws (technically, n “isolating integrals of the motion”). (4) Analysis of the conservation laws tells us everything we might want to know about the structure of the trajectories. (5) For weak perturbations of a system having regular trajectories, most of the trajectories are regular (Kolmogorov–Arnold–Moser (KAM) theorem [36–38]).
16.4 Chaotic Trajectories and Irregular Wave Functions
An example of a chaotic trajectory is shown in Fig. 16.5. As in Fig. 16.3, this is one continuous curve showing a path in . ; z/-space followed by a classical electron in the Coulomb field of a nucleus at the origin and a magnetic field along the z-axis (no applied electric field). In contrast to the curve shown in Fig. 16.3, which is similar to a machine-wound spool of twine, this path looks like a bowl of spaghetti that got dropped on the floor. Nevertheless, there is short-time order in the path: far from the nucleus, the electron’s path is close to a helix, circling around a magnetic field line, while close to the nucleus it is close to a Kepler ellipse. 2000 However, we can see no long-time order in this path. If we could draw a curve inside of which the magnetic field ρ were negligible and outside of which the Coulomb field were 2000 negligible, we could say that the magnetic field scatters the z 0 electron from one Kepler orbit to another, or we could say –2000 that the Coulomb field scatters the electron from one heρ 1000 lix to another; either way, the long-time behavior is chaotic. 0 Such trajectories arise when magnetic and Coulomb forces 2000 compete but neither dominates everywhere [7, 39]. This hap1000 0 pens as the energy approaches the ionization limit, or if the –1000 z –2000 magnetic field becomes extremely strong (in the kilotesla range). Fig. 16.4 Below is the eigentrajectory that was shown in Fig. 16.3, and There is a second way that chaos arises in Rydberg states above is an accurate eigenfunction for a Rydberg state of an electron in an atom in parallel electric and magnetic fields of atoms. The quantum or classical Hamiltonian for a hy-
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246 Fig. 16.5 A chaotic trajectory of an atomic electron in a strong magnetic field. (Courtesy of Donald Noid)
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drogen atom in an electric field is separable in parabolic coordinates, so all bound trajectories lie on tori. However for all other atoms, the small ion core scatters the electron from one torus to another, and the long-time behavior is chaotic. One may compare the motion to that of a billiard ball on a rectangular table containing a small round barrier [40–43]. Some properties of chaotic trajectories are the following. (1) They have a sensitive dependence on initial conditions: a small displacement of initial conditions grows exponentially, restricting predictability to short times. (2) Even in a chaotic system, there is order on short time scales. The trajectories obey Newton’s laws, and if the forces are smooth functions of position, then the paths are smooth functions of initial conditions (according to property (1), they become horrible and even uncomputable functions, but they are, in principle, smooth – as time increases, the derivative of final position with respect to initial position becomes gargantuan but finite). Thus, chaos is a long-time property of classical trajectories. (3) While regular bound trajectories are restricted to n-dimensional tori in the 2n-dimensional phase space, chaotic trajectories may fill the energy shell, in the sense of approaching arbitrarily closely to any point in it. Accordingly, at any spatial point, . ; z/, the magnitude of the momentum is determined by conservation of energy, but the direction of the momentum is not. Therefore, a path does not form a curl-free momentum-vector field, indeed, it does not form a momentum-vector field at all. Therefore, we cannot use chaotic trajectories to build quantum wave functions in the way that is done for regular trajectories. This is the “problem of quantum chaos”: if classical trajectories of a given system are chaotic, what can be said about the quantum states? For reviews of this problem see [3, 4, 34, 44]. Here, we discuss only Rydberg atoms in static fields. Two general principles apply. (A) Long-time chaos in classical mechanics carries over to irregular behavior of individual energy levels and individual eigenfunctions – these are most visible at high resolution of energy. (B) Short-time order in classical mechanics carries over to short-time order in quantum mechanics, and that in turn is connected to the
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large-scale structure of the energy spectrum and other quantum properties – these are most visible at low resolution of energy. Thus, different manifestations of chaos appear when we look at the spectrum on different scales [35].
16.4.1 Small-Scale Structure In Chap. 15, Fig. 15.5 shows energy levels of hydrogen in an electric field. It is seen that energy levels from different values of principal quantum number n cross each other. This is because the Schroedinger equation (as well as the Hamilton–Jacobi equation) is separable in parabolic coordinates. All bound motions are, therefore, regular, and states that cross have different parabolic quantum numbers. Figure 15.4 shows energy levels of sodium in an electric field. Now the states avoid crossing. The ion core scatters the electron from one parabolic state to another, and all eigenstates of the full Hamiltonian are superpositions of many different parabolic states. In this case, the avoided crossings are a quantum manifestation of the classical chaos that arises from core scattering. These figures suggest that regular behavior allows lots of degeneracies and near-degeneracies, but chaotic behavior produces avoided crossings, thus pushing eigenvalues apart. This repulsion of energy levels shows up in nearestneighbor statistics of the energies. For each energy eigenvalue, define s to be the absolute value of the energy gap to its nearest neighboring eigenvalue, and then for a large collection of eigenvalues find the probability distribution P .s/ of values of s. This is the nearest-neighbor spacing distribution. For Rydberg states of atoms, this nearest-neighbor spacing distribution was computed in [40] and is shown in Fig. 16.6, first for hydrogen, and then for lithium. The stepped line is the histogram, and the two dashed lines are an exponential, which permits lots of near-degeneracies, as is seen in regular spectra, and a Wigner distribution, P .s/ D . =2/s exp. s 2 =4/, which goes to zero as s ! 0, consistent with irregular spectra and their avoided crossings.
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Later, when the absorption spectrum of a hydrogen atom in a 6 T magnetic field was measured [14], it was found that the spectrum could be Fourier-decomposed into a sum of sinusoidal oscillations, X Tj E 0 Abs.E/ ' Cj sin C j ; (16.14) ¯ j and that these Tj are times of returning classical orbits of the electron. The shortest returning orbits produce oscillations of longest wavelength (on the energy axis), and successively longer returning orbits produce oscillations of successively shorter wavelength. This observation was the birth of recurrence spectroscopy, a different way of thinking about absorption spectra: instead of examining individual energy levels, the Fourier-transform of the absorption spectrum provides measures of closed classical orbits of the electron as it moves in the Coulomb field of the nucleus and the applied electric and magnetic fields [17, 48–53]. (In the case of atomic absorption spectra, the semiclassical connection is to orbits that are closed at the nucleus, not to periodic orbits.) The physical picture is shown in Fig. 16.7. Light shining on an atom (like a vibrator in a ripple tank, sending out a continuous stream of water waves) produces a continuous stream of electron waves moving out from the atom in all directions, with energy close to the ionization thresh0 0.5 1.0 1.5 2.0 2.5 3.0 old. That is a quantum process. However, when the waves s move away, we can follow the waves by following the corresponding “rays”, the outgoing classical trajectories. At large Fig. 16.6 Nearest-neighbor spacing distribution for Rydberg states of hydrogen (above) and for lithium (below). The stepped curve is the his- distances, the trajectories and waves are turned around by the togram computed from the spectrum; the light and heavy dashed lines applied fields. Some trajectories and waves go out in direcare, respectively, an exponential and a Wigner distribution [40] tions such that they return to the atom after a while. When they do, the waves refract and diffract around the atom, pro16.4.2 Large-Scale Structure of Energy Spectra ducing a steady stream of incoming waves, which interfere with the steady stream of outgoing waves, producing the osIf instead of looking at individual quantum states, we average cillations that are visible in the absorption spectrum. This the properties of these states over a finite band of energies, physical picture leads to a quantitative formula, then a relationship to classical periodic orbits appears. This X
connection was first recognized by Gutzwiller [4], and experCj .EI B; F / sin j .EI B; F / : Abs.EI B; F / D iments and theory on Rydberg states of atoms have shown the j connection most clearly [45–47]. (16.15) The first experimental evidence came from measurements Here, j labels the closed orbits, and j .EI B; F / is of the absorption spectrum of atoms in magnetic fields [12– 14]. At “low” energies, meaning absorption to Rydberg states a phase, in which the most important term is the classical around n 30 in laboratory magnetic fields ( 6 T), spectral action integrated around the closed orbit. The duration of the lines fall into orderly sequences, quantitatively described by closed orbit is Tj D @j =@E, which recovers the approxthe regular trajectories and wave functions discussed above. imate formula Eq. (16.14). The coefficient Cj .EI B; F / is With increasing energy, the spectrum becomes messier, slowly varying with energy and magnetic and electric field partly because of chaotic dynamics and partly because levels strengths; it contains factors representing the angular distriare not well resolved. Then at higher energies, orderly oscil- bution of outgoing waves, and the (square root of) the classilations appear. The wavelength of these oscillations on the cal density associated with the returning trajectories [45, 46]. energy axis E1 is related to the return time T1 of a closed This coefficient is called the “recurrence amplitude”, and its orbit perpendicular to the magnetic field, E1 D 2 ¯=T1 . absolute square is the “recurrence strength”.
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248 Fig. 16.7 Clockwise from lower left. Light shining on an atom (1) produces a continuous stream of outgoing electron waves (2), having energy close to the ionization threshold. When those waves get away from the atom, we can follow them by following the corresponding “rays” or classical trajectories (3). These trajectories travel away (4), pass through a caustic (5), a focus (6), and another caustic (7). We have drawn trajectories that return to the vicinity of the nucleus (8). When they do, they refract (9), and the waves diffract (10) around the nucleus, producing a steady stream of incoming waves that overlap the initial state (11) and interfere with the outgoing waves, producing interference oscillations in the absorption spectrum
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Scaled-variable methods provide important improvements to the experimental measurements [13]. The structure, size, and return times of orbits change with energy and field strengths, but it turns out that (for F D 0 and Lz D 0) the shapes of classical orbits depend only on the scaled energy, which is " D EB 2=3 (Eq. (16.2)). At constant scaled energy, the classical action is proportional to w D B 1=3 , and the coefficient Cj .EI B/ is nearly constant. It follows that if the absorption spectrum is measured varying E and B simultaneously to hold " fixed, and the absorption is recorded as a function of w, then the oscillations become almost perfectly sinusoidal, and the recurrence spectrum becomes a set of sharp peaks, one at the scaled classical action of each returning orbit. An example of a scaled-variable recurrence spectrum for hydrogen in a magnetic field is shown in Fig. 16.8 [54]. For atoms in an electric field F with no magnetic field present, the scaled energy is " D EF 1=2 . For this case, a family of recurrence spectra is drawn in a single graph in
Fig. 16.9 [55]. At all energies there is a closed orbit of the electron that goes up parallel to the electric field; its recurrence peak is barely visible in the upper left part of the figure. In certain ranges of energies, many other closed orbits are present. Some are shown at the bottom of Fig. 16.9, labeled (m=n), meaning m cycles of motion transverse to the field during n cycles of motion parallel to the field. The scaled actions of these orbits are close to the scaled action of n cycles of the parallel orbit. If we gradually lower the scaled energy, there is a point at which each of these closed orbits splits (bifurcates) out of the parallel orbit. At that point, orbits neighboring the parallel orbit return strongly focused on the residual ion, and the recurrence strength is large. Such points are marked in the figure. The detailed analysis of many such experiments has led to advances in classical and semiclassical mechanics. (1) In a magnetic field, one can see bifurcation and proliferation of closed orbits as order changes to chaos with increasing ".
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Mathematical theory shows that there are just five generic patterns of bifurcations of periodic orbits. In atomic systems, these generic patterns are modified by symmetries. CalculaFig. 16.9 Many recurrence spectra for an atom in an electric field are drawn in a single graph [55]. The curves in the horizontal plane represent the scaled action of the parallel orbit and its repetitions as a function of scaled energy. Locations of bifurcations are marked with small open circles. New orbits created in bifurcations have almost the same action as the corresponding return of the parallel orbit. Measured recurrence strengths are shown in the vertical direction. Recurrences are especially strong at scaled energies slightly lower than bifurcations. Orbits created by bifurcation of the parallel orbit are shown along the bottom. The 2=4, 4=6, and 6=8 orbits are repetitions of the 1=2, 2=3; and 3=4 orbits, respectively, so their shapes are identical
tions and theory show the shapes of bifurcated orbits. Also, sequences of bifurcations can be understood by reduction of the Hamiltonian to a local normal form. Measurements of recurrence spectra – intrinsically a quantum wave-interference phenomenon – have provided the most detailed and precise observations of bifurcations of classical periodic orbits and closed orbits [48, 56–63]. (2) Breaking cylindrical symmetry, by varying the angle between applied fields, weakens the recurrences in a systematic way. Breaking time symmetry, by applying a slowly oscillating field, has the same effect, and allows the electron orbit to be reconstructed from the recurrence spectrum [50, 51, 64]. (3) Semiclassical approximations diverge at bifurcations because there are boundaries between classically allowed and forbidden regions. The divergences are repaired by diffraction integrals such as Airy functions or Fresnel-type integrals. Diffraction into forbidden regions and also above-barrier reflection of waves produces “ghost recurrences” [54, 59, 61, 63, 65–71] (4) For nonhydrogenic atoms, the ion core scatters electrons from one returning orbit to another, producing “combination recurrences” [40, 72–75]. (5) Similar concepts give a theory of “width-weighted spectra”, related to energy-averaged decay rates of atoms [76]. Finally, we mention that recurrence spectroscopy associated with Rydberg states helped to stimulate the study of periodic orbits and their bifurcations in molecular systems [77–80].
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16.4.3 Chaotic Ionization At energies high enough that the electron can escape, we have a “chaotic scattering” system, or, for photoionization, a chaotic “half-collision”, or chaotic transport problem. The situation is closely related to the modern version of transition state theory of chemical reaction rates, and also has a connection with asteroid escape rates [81–86]. Consider the following experiment (which, as of this writing, has not yet been performed). An atom in a magnetic (and possibly electric) field is excited by a short laser pulse to an energy that permits escape of the electron. (The ideal experiment will excite to the minimum range of energies allowed by the uncertainty principle.) We might expect that the excited atom would decay by ionization exponentially with time. In fact, however, electrons come out in a chaotic sequence of short pulses. Classical calculations show that a graph of the time required for the electron to escape from the atom versus its initial direction has a fractal structure [87–91]. This fractal does not have the perfectly regular self-similar structure of mathematical fractals such as the Cantor set [92–95], nor is it a fractal with only statistical self-similarity. It is a distinct type, which has been called an “epistrophic” fractal. The definition and characterization of such fractals is given in [96–99], and application to the chaotic escape of electrons from atoms in fields is given in [100–106]. Quantum calculations [107] verify that the predictions obtained from this fractal description are correct, at least for short times. It appears that epistrophic fractals occur commonly (perhaps generically) in chaotic scattering because they have been identified both theoretically and experimentally in the escape of particles or light or sound from a vase-shaped cavity [108–110], and theoretically in the escape of atoms from a trap [111], ballistic atom pumps [112–114], fluid flow [115, 116], and perhaps also in the dynamics of the buckling of a nanobeam [117].
16.5 Nuclear-Mass Effects So far, only the relative motion of the electron with respect to the ionic core has been described. This is appropriate if the nucleus can be assumed to be infinitely heavy and, thus, not to take part in the motion. To include the effects of a finite nuclear mass, the description must start from the coupled two-body Hamiltonian and then work toward a separation of the internal dynamics from the center-of-mass (CM) motion. It turns out that in the presence of a magnetic field, unlike for the field-free two-body problem, a complete separation of the relative and CM motions is impossible. Instead, only a pseudo-separation can be achieved, where the relative and CM motions remain coupled through a new constant of motion called the pseudomomentum K [118]. This coupling
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introduces a number of novel effects into the dynamics [119– 123]. The influence of the CM motion on the internal dynamics is twofold. On the one hand, the motion of the atom in the magnetic field causes an induced electric field (motional Stark effect). On the other hand, the kinetic energy of the CM motion gives rise to an additional confining potential for the internal motion that could, in principle, locate the electron at a large distance from the nucleus, and produce atomic states with a huge dipole moment. Conversely, the motion of the CM is driven by the internal motion, the most strongly so in the case of vanishing pseudomomentum. It, thus, reflects the transition from regular to chaotic internal dynamics: regular internal motion leads to regular CM motion, whereas chaotic internal dynamics can give rise to chaotic behavior of the CM.
16.6 Quantum Theories This chapter has focused on semiclassical theories and the connections between classical trajectories and quantum wave functions in excited states of atoms in fields. For more information on quantum theory of atoms in fields, the reader is referred to [124–126].
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252 107. Topçu, T.Ü., Robicheaux, F.: Abstract 5005. In: APS 37th Meeting of the Division of Atomic, Molecular and Optical Physics 2006 (2006) 108. Hansen, P., Mitchell, K.A., Delos, J.B.: Phys. Rev. E 73, 66226 (2006) 109. Novick, J., Keeler, M.L., Giefer, J., Delos, J.B.: Phys. Rev. E 85, 016205 (2012) 110. Novick, J., Delos, J.B.: Phys. Rev. E 85, 016206 (2012) 111. Mitchell, K.A., Steck, D.A.: Phys. Rev. A 76, 31403 (2007) 112. Ivory, M.K., Byrd, T.A., Pyle, A.J., Das, K.K., Mitchell, K.A., Aubin, S., Delos, J.B.: Phys. Rev. A 90, 023602 (2014) 113. Byrd, T.A., Das, K.K., Mitchell, K.A., Aubin, S., Delos, J.B.: Phys. Rev. E 90, 052107 (2014) 114. Byrd, T.A., Delos, J.B.: Phys. Rev. E 89, 022907 (2014) 115. Budyansky, M.V., Uleysky, M.Y., Prants, S.V.: J. Exp. Theor. Phys. 99, 1018 (2004) 116. Budyansky, M., Prants, S.: In: Proceedings of the 2005 International Conference of Physics and Control, p. 556. (2005) 117. Collins, P., Ezra, G.S., Wiggins, S.: Phys. Rev. E 86, 056218 (2012) 118. Avron, J., Herbst, I., Simon, B.: Ann. Phys. 114, 431 (1978) 119. Schmelcher, P., Cederbaum, L.S.: arXiv preprint physics/9909045 (1999) 120. Schmelcher, P., Cederbaum, L.: Int. J. Quantum Chem. 40, 371 (1991) 121. Schmelcher, P.: Phys. Rev. A 64, 063412 (2001) 122. Melezhik, V., Schmelcher, P.: Phys. Rev. Lett. 84, 1870 (2000) 123. Schmelcher, P., Cederbaum, L.: Struct. Bonding 86, 27 (1997) 124. Ruder, H., Wunner, G., Herold, H., Geyer, F.: Atoms in Strong Magnetic Fields: Quantum Mechanical Treatment and Applications in Astrophysics and Quantum Chaos. Springer, Berlin (1994) 125. Friedrich, H., Eckhardt, B.: Classical, Semiclassical and Quantum Dynamics in Atoms. Springer, Berlin, Heidelberg (2014) 126. Friedrich, H.: Theoretical Atomic Physics, 4th edn. Springer, Berlin (2017)
J. B. Delos et al. John Delos John Delos received a PhD in Chemistry from MIT and spent most of his career in physics at William and Mary. His research interests have been in classical and semiclassical mechanics, atomic collisions, order and chaos, and excited states of atoms in fields. Currently, he studies heart rates and respiration of infants in neonatal intensive care units. Thomas Bartsch Thomas Bartsch received a PhD in Theoretical Physics from the University of Stuttgart, was a Postdoctoral Fellow at Georgia Tech, and is now Senior Lecturer in Mathematics at Loughboro University, UK. His research is centered on applications of nonlinear dynamics to atomic and molecular processes.
Turgay Uzer Professor Turgay Uzer obtained his doctorate at Harvard and was a Postdoctoral Fellow at CalTech. Currently he is Regent’s Professor in Physics at Georgia Institute of Technology. His research interests include: Rydberg atoms and molecules, semiclassical theories, nonlinear dynamics/chaos, intramolecular energy transfer, and chemical reactvity.
17
Hyperfine Structure Guy T. Emery
Contents 17.1 17.1.1 17.1.2 17.1.3
Splittings and Intensities . . . Angular Momentum Coupling Energy Splittings . . . . . . . . Intensities . . . . . . . . . . . . .
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17.2 17.2.1 17.2.2 17.2.3 17.2.4
Isotope Shifts . . . . . . . . . . . . . . . . Normal Mass Shift . . . . . . . . . . . . . Specific Mass Shift . . . . . . . . . . . . . Field Shift . . . . . . . . . . . . . . . . . . Separation of Mass Shift and Field Shift
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17.3 17.3.1 17.3.2 17.3.3
Hyperfine Structure . Electric Multipoles . . Magnetic Multipoles . Hyperfine Anomalies .
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Abstract
Hyperfine structure in atomic and molecular spectra is a result of the interaction between electronic degrees of freedom and nuclear properties other than the dominant one, the nuclear Coulomb field. It includes splittings of energy levels (and thus of spectral lines) from magnetic dipole and electric quadrupole interactions (and higher multipoles, on occasion). Isotope shifts are experimentally entangled with hyperfine structure, and the so-called field effect in the isotope shift can be naturally included as part of hyperfine structure. Studies of hyperfine structure can be used to probe nuclear properties, but they are an equally important probe of the structure of atomic systems, providing especially good tests of atomic wave functions near the nucleus. There are also isotope shifts owing to the mass differences between different nuclear species, and the study of these shifts provides useful atomic information, especially about correlations between electrons. G. T. Emery () Dept. of Physics, Bowdoin College Brunswick, ME, USA e-mail: [email protected]
Hyperfine effects are usually small and often, but not always, it is sufficient to consider only diagonal matrix elements for the atomic or molecular system and for the nuclear system. In some cases, however, matrix elements off-diagonal in the atomic space, even though small, can be of importance; one possible result is to cause admixtures sufficient to make normally forbidden transitions possible. Keywords
hyperfine structure isotope shift mass shift tensor operator diagonal matrix element In the diagonal case, one can picture each electron undergoing elastic scattering from the nucleus and returning to its initial bound state. As pointed out by Casimir [1, 2], however, the internal conversion of nuclear gamma-ray transitions involves the inelastic down-scattering from an excited nuclear state to a lower one as an electron goes from an initial bound state to the continuum. By further conversion of bound to continuum states, one sees the connection with electron scattering from the nucleus – elastic, inelastic, and break-up. Hyperfine structure of outer-shell electronic states is at the low momentum-transfer end of this chain of related processes. Some of the standard textbooks which discuss hyperfine structure are [3–10] and a few newer texts [11–14]. Especially relevant are [15–19] and the conference proceedings [20–22]. The study of hyperfine structure in free atoms, ions, and molecules is part of the more extensive research area of hyperfine interactions, which includes the study of atoms and molecules in matter, both at rest, for example as part of the structure of a solid, and moving, such as ions moving through condensed or gaseous matter. This more general subject also includes the ways in which atomic electrons shield the nucleus, or anti-shield it, from external or collective fields. Thus nuclear magnetic resonance, nuclear quadrupole reso-
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_17
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nance, electron-nuclear double resonance, recoilless nuclear absorption and emission, nuclear orientation, production of polarized beams, and many other widely used techniques, are intimately connected with hyperfine effects. Though hyperfine effects are ordinarily small in electronic systems, they can become much larger in exotic atoms: those with a heavier lepton or hadron as the light particle. Hyperfine effects are typically related to light particle density at the nucleus, or to expectation values of r 3 , and thus scale as the cube of the light particle mass. The study of muonic atoms has contributed importantly to knowledge of the nuclear charge distribution [23–26]. There has been considerable interest in pionic atoms, where the strong interaction also contributes to hyperfine structure (e.g., [27, 28]), and also in kaonic, antiprotonic, and other hadronic atoms [29, 30]. See especially [31] for recent work on antiprotonic helium. Some other examples of interaction between atomic and nuclear degrees of freedom are discussed in Chap. 91.
The shift operators are defined as before, and it is now F that is the generator of rotations of the (combined) system, or of the coordinate frame to which the system is referred. By the rules of combining angular momenta, the possible values of the quantum number F are separated by integer steps and run from an upper limit of J˛ C Iˇ to a lower limit of jJ˛ Iˇ j. The number of possible eigenvalues F is the smaller of 2J˛ C 1 and 2Iˇ C 1. Experimental values of the nuclear quantum number I may be found in a number of compilations [32–34].
17.1.2
Energy Splittings
Electromagnetic interactions between atomic electrons and the nucleus can be expanded in a multipole series HeN D
X
T k .N/ T k .e/ ;
k
X
k .1/i Tqk .N/Tq .e/
(17.4)
k;q
17.1 Splittings and Intensities 17.1.1
Angular Momentum Coupling
When the nuclear system is in an isotropic environment, each nuclear state ˇ has a definite value of nuclear angular momentum Iˇ „, where the possible values of Iˇ are related to the number of nucleons (protons plus neutrons) in the same way as those for J˛ are related to the number of electrons in electronic state j˛i. The nuclear operators, eigenstates, and eigenvalues are related to each other in the same way as for atomic angular momentum by I 2 jˇi D Iˇ .Iˇ C 1/jˇi ; Iz jˇi D Mˇ jˇi ;
(17.1)
where T k .N/ is an irreducible tensor operator of rank k operating in the nuclear space, and similarly T k .e/ operates in the space of the electrons. Since one is taking diagonal matrix elements (in the nuclear space, at least) in states that are to a very good approximation eigenstates of the parity operator, only even electric multipoles (E0, E2, etc.) and odd magnetic multipoles (M1, M3, etc.) contribute to the series. The effects of the parity nonconserving weak interaction are considered in Chap. 28. The term with k D 0 contributes directly to the structure (and fine structure) of atomic systems, and its dominant contributions come from the external r 1 electrostatic field of the nucleus. The hyperfine Hamiltonian is defined by subtracting that term to obtain X
T 0 .N/T 0 .i/ Ze 2 =ri Hhfs D
in units with „ D 1. Shift operators move the system from i X one M -value to another, as for the atomic system (see C T k .N/ T k .e/ ; (17.5) Chap. 2), and the operator I is the generator of rotations. kD1 When the combined atomic-nuclear system is considered, in an isotropic environment, it is the total angular momentum where Z is the nuclear charge number. The difference beof the combined system defined by tween the Ze 2 =r term(s) and the full monopole term is called the field effect or finite nuclear size effect in the isoF DJ CI ; (17.2) tope shift, and the remaining terms contribute dipole (k D 1), quadrupole (k D 2), and higher multipoles in hyperfine structhat has definite values. The state of the combined system ture. can be labeled by , so that Since the hyperfine Hamiltonian can be expressed as a multipole expansion, its contributions to the pattern of F 2 ji D F .F C 1/ji ; energy levels for the various F values in a given J˛ ; Iˇ Fz ji D M ji : (17.3) multiplet in first-order perturbation theory can be described
17
Hyperfine Structure
255
Electric quadrupole hfs is described by the quadrupole relatively simply in terms of 3–j and 6–j symbols. The contribution of the term which is the scalar product of electron hyperfine structure constant B. If we define the quantity K D ŒF .F C 1/ J.J C 1/ I.I C 1/, then and nuclear operators of multipole k is Ek .JIF; JIF / ( D ./J CI CF "
J J
k 0
J I
)
J k I F ! I k J I 0 J
I I
!#1 Ak ;
E2 .JIF; JIF /
1 B Œ3K.K C 1/=2 2J.J C 1/I.I C 1/ D 4 : J.2J 1/I.2I 1/ (17.12) The constant B is related to the tensor operators by (17.6) 1 B D ŒJ.2J 1/=.J C 1/.2J C 3/1=2 4 ˛ ˝ J T 2 .e/J
where for k 1, ˇ ˛ ˝ ˇ ˇ ˛ ˝ ˇ Ak D JJ ˇT k .e/ˇJJ II ˇT k .N/ˇII :
(17.7)
The commonly used hfs coefficients A, B, etc., are related to the Ak by A D A1 =IJ;
B D 4A2 ;
C D A3 ;
D D A4 : (17.8)
ŒI.2I 1/=.I C 1/.2I C 3/1=2 ˛ ˝ I T 2 .N/I :
(17.13)
For higher multipoles, see [36]. The multipole expansion is important because it is valid for relativistic as well as nonrelativistic situations, and for nuclear penetration effects (hyperfine anomalies discussed in Sect. 17.3.3) as well as for normal hyperfine structure. Its limitation comes from its nature as a first-order diagonal perturbation. Off-diagonal contributions, even when small, can perturb the pattern, but, more importantly, can lead to misleading values for the Ak coefficients, including the isotope shift.
The isotope shift A0 is the matrix element of the reduced monopole operator. The pattern of the splitting depends on the total angular momentum F wholly through the 6–j symbol. Since for k D 0 the value of the 6–j symbol is independent of F , the monopole term shifts all levels of the hyperfine multiplet equally, independent of the value of F . The F -dependence of the dipole contribution can be found from the fact that the same 6–j symbol would appear for any scalar product of k D 1 operators, for example J I. 17.1.3 Intensities But in this product space, with J , I , and F all good quantum numbers, the diagonal matrix elements of J I are just When hyperfine structure is observed as a splitting in an optical transition between different atomic levels, there are 1 hJ Ii D ŒF .F C 1/ J.J C 1/ I.I C 1/ ; (17.9) relations between the intensities of the components. The gen2 eral rule for reduced matrix elements of a tensor operator so that operating in the first part of a coupled space is [35, p. 152] E1 .JIF; JIF / 1 D AŒF .F C 1/ J.J C 1/ I.I C 1/ ; 2
˛ ˝ JIF Q .e/J 0 IF 0 (17.10)
where, in terms of reduced matrix elements according to the convention of Brink and Satchler [35, p. 152], (the first version given in Sect. 2.8.4)
CI CF 0 CJ
D .1/
( 0
1=2
.2F C 1/
F J0
˛ ˝ .2J C 1/1=2 J Q .e/J 0 :
F0 J
I
)
(17.14)
For a dipole transition ( D 1) connecting atomic states J ˛ ˝ A D ŒJ.J C 1/1=2 J T 1 .e/J ŒI.I C 1/1=2 and J 0 , with fixed nuclear spin I , the line strength SFF 0 of ˛ ˝ 1 0 I T .N/ I : (17.11) the hyperfine component connecting F and F is related to the line strength SJJ 0 by A is the magnetic dipole hyperfine structure constant for the ( )2 atomic level J and nuclear state I . M1 hfs shows the same F F0 1 0 SJJ 0 : (17.15) SFF 0 D .2F C 1/.2F C 1/ 0 pattern of splittings as spin-orbit fine structure, described J J I sometimes as the Landé interval rule.
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G. T. Emery
17.2 Isotope Shifts
where M is the nuclear mass and pj is the momentum of the j th electron. The matrix elements of these terms can be Two distinct mechanisms contribute to isotope shifts in strongly state-dependent, and the difference in their contriatomic energy levels and transition energies. First, there are butions for isotopes of different mass is called the specific shifts due to the different mass values of the isotopes; these mass shift (SMS). For a transition a ! b, the lowest-order mass shifts can again be separated into two kinds, the nor- SMS between isotopes A and A0 is mal mass shift and the specific mass shift. Second, there are shifts due to different nuclear charge distributions in differ.a; bI A; A0 / D K a;b .MA MA0 /=MA MA0 ; (17.19) ent isotopes. Shifts of this sort are called field shifts, and can be considered to be the monopole part of the hyperfine inter- where action. ˇ ˛ ˝ ˇP ˇ ˛ ˝ ˇP The usual convention is to describe an isotope shift in K a;b D aˇ j 5. As n increases, !1 moves further into the wing of the line, and when !1 reaches the Lorentzian tail of the Voigt line profile a transition to Eq. (20.19b) occurs, where k0 corresponds to a purely Lorentzian line. (That the core of the line does not have a Lorentzian shape does not matter, since the fraction of emission well into the Lorentzian wing is nearly the same as that of a pure Lorentzian line.) In the absence of a collision, a two level atom reradiates in its rest frame the same frequency it absorbed. Thermal motion redistributes this coherent scattering frequency within the Doppler envelope when the emission and absorption are in different directions, but it does not transfer it into the natural Lorentzian wing outside the Doppler envelope. This leads to the property that an atomic vapor will scatter frequencies in the natural wing, but will not emit in this wing unless it absorbed there or is excited by or during a collision. With line broadening collisions, a fraction =. C2c / of optical attenuation is coherently scattered and a fraction 2c =. C 2c / is redistributed into incoherent emission with a Lorentzian spectrum of width C 2c centered at !0 C ı in the reference frame of the moving atom. This redistributed emission can escape in the Lorentzian wing of the Voigt line. In this radiation transport problem, the consequence is that Eq. (20.19b) with k0 D n.2 =2 /. =c/.gu =gg /, corresponding to a Lorentzian with Total D 2c not C 2c , provides the best approximation to g1 in the density region where !1 is in the Lorentzian wing of the line. Since k0 / n=c and in the absence of a buffer gas c D kc n, where kc is the rate coefficient for self broadening collisions, g1 becomes independent of n. In fact, kc / as well, so g1 is also independent of . For the case of a J D 0 ground state and a J D 1 excited state, g1 D 0:21.=L/1=2 ; the broadening coefficient for other cases can be found in [60]. If the broadening is due to a buffer gas, c D kc nB in Eq. (20.19b) yields n 1=2 B I (20.20) g1 / n this has been studied in [61]. Post et al. have numerically evaluated g1 for all values of k0 L for slab and cylinder geometries, by integrating the radiative escape probability g.z/ over the fundamental mode distribution N.z/, where z is the position between the windows [62]. To obtain g.z/ they integrate over the angular
20 Line Shapes and Radiation Transfer
distribution of the emission, using T .x/ from the exact line shape. Thus all features of the calculation correspond to the Holstein–Biberman theory for an isolated line without approximation. As will now be discussed real atomic vapors are generally not that simple. Many atomic lines have multiple components due to hyperfine structure and isotope shifts; some components are isolated while others are separated by less than a Doppler line width and overlap. The absorption line shape then becomes a weighted sum over components, each with an equivalent Voigt shape. In a high density vapor or a plasma, collisions will usually distribute the excited state population between the isotopes and hyperfine states in proportion to their isotopic fraction and statistical weight. The emission line shape L.!/ is then a similarly weighted distribution over components. Since radiation only escapes in the wings of a line component at high k0 L, overlapping components act almost as a single component. If the line has M isolated components, the right-hand side of Eqs. (20.19a) and (20.19b) become sums over the fraction fj of the intensity in the j component times the escape probability for that component. The latter is obtained, for large k0 L, by replacing k0 with k0 fj in Eqs. (20.19a) and (20.19b). The net result, after summing over components, is an increase in gi by a factor of M in the Gaussian case and M 1=2 in the Lorentzian. This approximation was obtained by Holstein in the context of the Hg 254 nm radiation under conditions appropriate to the fluorescent lamp [63]. Walsh made a more detailed study of these overlapping components [64], and the dependence of g1 on the ratio of line separation to Doppler width is also given in [65].
20.2.3 Measurements The overall behavior of g1 versus n is shown in Fig. 20.8 for the Na(3P3=2 ) or D2 resonance line in pure Na vapor (i.e., the 3p 2 P3=2 3s 2 S1=2 transition) [65, 66]. In this type of experiment the fundamental mode decay rate is established by a combination of optimally exciting that spatial mode and of waiting until the fluorescence decay is exponential in time after termination of the excitation. A transition to approximately 1=n behavior, corresponding to Eq. (20.19a), is seen to occur at k0 L=2 5. At k0 L=2 100 the transition to n0 behavior, corresponding to a self-broadened Lorentzian line in Eq. (20.19b), can be seen. The behavior at k0 L < 5 fits the Milne diffusion theory [67] as well as the Post et al. theory shown as a solid line; this is also similar to T .L=2/, as seen in Fig. 20.7. For 5 < k0 L=2 < 100, the behavior is similar to Eq. (20.19a) (dashed line), but the Post et al. theory (solid line) is 20% higher due to the inclusion of the Na hyperfine structure (hfs splitting ' Doppler width). For k0 L=2 > 1000,
293
Sodium density (cm–3)
g1eff
10
100
11
10
12
13
1014
10
R = Гeff
1015 ГN = 2γc
10 –1
H(3P3/2)
10 –2
H(3P1/2)
10 –3 P(3P3/2) 10 –4
10 –1
100
101
10 2
10 3 10 4 Optical depth (k 0 L/2)
Fig. 20.8 Radiative escape probability g1 for Na vapor excited to the 3P3=2 state, for a slab geometry. The Holstein (H) approximation for the 3P3=2 –3S1=2 (D2) line and the 3P1=2 –3S1=2 (D1) line are indicated as dashed lines. The Post-type (P) calculation of [80] for the D2 line is indicated as a solid line. Solid squares are data from [66], and open circles are data from [65]. The effective escape probability corresponds to the D2 line rate at low densities but a combination of D1 and D2 at high densities
the Post theory converges to the Holstein–Lorentzian-line result with Total D 2c . The experiment is complicated in the 50 < k0 L=2 < 500 region by fine structure mixing [66]. The 3P3=2 state was excited, but at high densities, collisions populate the 3P1=2 state, which has a smaller g1 than the 3P3=2 state (Fig. 20.8). At low densities, g1eff D g1 .3P3=2 /, and at high densities these states are statistically populated and g1eff D 13 g1 .3P1=2 / C 2 g .3P3=2 /. The transition density where the fine structure 3 1 mixing rate R equals eff is indicated in Fig. 20.8. The theory is also complicated in this intermediate k0 L region by the necessity of including incomplete frequency redistribution [62]; this leads to the dip in g1 near k0 L 500. While the overall behavior of the data in Fig. 20.8 is consistent with the Post et al. theory, there is 30% systematic discrepancy at k0 L=2 D 10–100 and the dip near 500 is not seen. Part of this difference probably results from the experimental geometry, which was between a slab and a cylinder of radius R D L=2; g1 for the cylinder is 17% larger than the slab value used in Fig. 20.8. The fundamental mode decay rate has also been measured for the Hg 254 nm [68] and 185 nm [62] lines, for the Ne resonance line [69] and for the Ar resonance line [70]. The Hg measurements are complicated by multiple isotopes and hyperfine structure, producing a mixture of partially overlapping and isolated lines combined with density-dependent uncertainties in excited state populations of the various isotopes. Serious efforts to model and measure these effects
20
294
have been made [62, 64, 68, 71]. The Ne and Ar measurements have similar complications, as will now be discussed. In essence, g1 behaves like the Gaussian T .x D L=2/ in Fig. 20.7 until n is large enough for !1 to approach the collision induced Lorentzian wing of the Voigt line. g1 then decreases more slowly since the line wing does not fall off as rapidly as a Gaussian. With continued increase in n, !1 moves further into the Lorentzian wing, a broader spectral region escapes and g1 reaches a minimum. Finally, when the entire escaping spectral region is Lorentzian, g1 reaches the constant value described above. Independent and detailed treatments of this density region, including incomplete frequency redistribution, predict a dip in g1 as seen in Fig. 20.8 [62, 70, 72, 73]. However, this has not been clearly confirmed experimentally. In Fig. 20.8 this dip occurs where fine structure mixing also occurs, and in addition the data are higher than the calculations throughout this n region. Post et al. [62] did observe such a dip for the Hg (149 nm) resonance line, but the data do not fit the calculation at other densities; hyperfine and isotopic structure within the line cause major complications. This long-standing issue has finally been clarified by Menningen and Lawler [74], who measured the decay of the Hg (185 nm) resonance line following laser excitation. They observed a clear dip in g1 due to incomplete redistribution. They also carried out sophisticated Monte Carlo simulations, obtaining g1 values that compared favorably with the measurements. By extending the simulations over a large range of a parameter space, they constructed an analytic formula for g1 of a singlecomponent line in cylindrical geometry [75]. This formula includes effects of incomplete frequency redistribution and varying ratios of Doppler broadening, radiative broadening and collisional broadening, so that it can be applied to any resonance line. Payne et al. [70] did not observe the predicted dip for the Ar resonance line; again a minor isotope with an isolated line occurs and could be very important at these high optical depths. Phelps [69] reported such a dip for the Ne 74:3 nm resonance line, but with rather large uncertainties due to the necessity of correcting for other collisional effects. Again there are isotopes with isolated lines that may have effected the data. Thus, experiments have verified the essential aspects of the above theories, but quantitative agreement in all aspects has not yet been achieved. The fact that the escaping radiation is concentrated in the wings of the line, near the unity optical depth point !1 , is reflected in the emitted spectrum. Calculated examples are shown in [73]; the Gaussian case looks somewhat like the transmitted spectra in Fig. 20.6 for x L=4. These spectra, and all results described so far, are calculated assuming no motion of the atoms. This is appropriate in the central region of the vapor, because the distance moved in an excited state lifetime .Lv D v= / is much smaller than L. In fact, resonant collisions between excited and ground state
A. Gallagher
atoms further limits the distance an excited atom moves in one direction before transfer of excitation. However, near the window or wall of the container, atomic motion will cause wall collisions of excited atoms and loss of radiation. This loss will be primarily within the Doppler core of the line, since these frequencies can only escape if emitted near the vapor edge. This loss depends on the excited state density in the neighborhood of the wall, and can be significant if Lv > 1=k0 . The excited atom density near the wall must be self consistent with the radiation transport and wall quenching. This situation has been modeled and studied experimentally ([76] and references therein). Additional aspects of radiation trapping, such as higherorder spatial modes and non-uniform absorber distributions, can be significant in lighting plasmas (and trapped atom clouds). Propagator function techniques have been developed for modeling radiation transport when the excitation has unusual temporal or spatial character [77, 78]. Non-uniform absorber spatial distributions can be particularly important at high power densities, and have been considered in [79].
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
Lorentz, H.A.: Proc. Amst. Akad. 8, 591 (1906) Weisskopf, V.: Phys. Z. 34, 1 (1933) Holtsmark, J.: Ann. Phys. (Leipzig) 58, 577 (1919) Kuhn, H.G.: Philos. Mag. 18, 987 (1934) Margenau, H., Watson, W.W.: Rev. Mod. Phys. 8, 22 (1936) Jablonski, A.: Z. Phys. 70, 723 (1931) Jablonski, A.: Acta Phys. Polon. 27, 49 (1965) Lindholm, E.: Ark. Mat. Astr. Fys. 32, 1 (1945) Foley, H.M.: Phys. Rev. 69, 616 (1946) Foley, H.M.: Phys. Rev. 73, 259 (1948) Holstein, T.: Phys. Rev. 79, 744 (1950) Anderson, P.W.: Phys. Rev. 76, 647 (1949) Anderson, P.W.: Phys. Rev. 86, 809 (1952) Baranger, M.: Phys. Rev. 112, 855 (1958) Chen, S.-Y., Takeo, M.: Rev. Mod. Phys. 29, 20 (1957) Allard, N., Kielkopf, J.: Rev. Mod. Phys. 54, 1103 (1982) Griem, H.: Plasma Spectroscopy. McGraw-Hill, New York (1964) Griem, H.: Spectral Line Broadening by Plasmas. Academic Press, New York (1974) Jeffries, J.: Spectral Line Formation. Blaisdell, Waltham (1968) Mihalas, D.: Stellar Atmospheres. Freeman, San Francisco (1970) van Kranendonk, J.: Can. J. Phys. 46, 1173 (1968) Cooper, J.: Rep. Prog. Phys. 29, 35 (1966) Simons, J.: Energetic Principles of Chemical Reactions. Jones Bartlett, Boston (1983) Szudy, J., Baylis, W.E.: J. Quant. Spectrosc. Radiat. Trans. 17, 681 (1977) Walkup, R.E., Spielfiedel, A., Pritchard, D.E.: Phys. Rev. Lett. 45, 986 (1980) Ward, J., Cooper, J., Smith, E.W.: J. Quant. Spectrosc. Radiat. Trans. 14, 555 (1974) McCartan, D.G., Lwin, N.: J. Phys. B 10, 17 (1977) Stacey, D.N., Thompson, R.C.: Acta Phys. Polon. A 54, 833 (1978) Ottinger, C., Scheps, R., York, G.W., Gallagher, A.: Phys. Rev. A 11, 1815 (1975)
20 Line Shapes and Radiation Transfer 30. 31. 32. 33.
34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55.
O’Callaghan, M.J., Cooper, J.: Phys. Rev. A 39, 6206 (1989) Stoicheff, B.P., Weinberger, E.: Phys. Rev. Lett. 44, 733 (1980) Weber, K.H., Niemax, K.: Z. Phys. A 307, 13 (1982) Fuhr, J.R., Lesage, A.: Bibliography of Atomic Line Shapes and Shifts. NIST Special Publication 366. U.S. Government Printing Office, Washington (1993). Suppl. 4 Szudy, J., Baylis, W.E.: J. Quant. Spectrosc. Radiat. Trans. 15, 641 (1975) Stacey, D.N., Cooper, J.: J. Quant. Spectrosc. Radiat. Trans. 11, 1271 (1971) Pascale, J., Vandeplanque, J.: J. Chem. Phys. 60, 2278 (1974) Drummond, D., Gallagher, A.: J. Chem. Phys. 60, 3426 (1974) Sando, K.M., Wormhoudt, J.: Phys. Rev. A 7, 1889 (1973) Carrington, C.G., Gallagher, A.: Phys. Rev. A 10, 1464 (1974) Lam, L.K., Hessel, M.M., Gallagher, A.: J. Chem. Phys. 66, 3550 (1977) Tam, A., Moe, G., Park, W., Happer, W.: Phys. Rev. Lett. 35, 85 (1975) Holstein, T., Gallagher, A.: Phys. Rev. A 16, 2413 (1977) Carlston, J.L., Szoke, A.: J. Phys. B 9, L231 (1976) Omont, A.: Prog. Quantum Electron. 5, 69 (1977) Light, J., Szoke, A.: Phys. Rev. A 15, 1029 (1977) Burnett, K., Cooper, J., Ballagh, R.J., Smith, E.W.: Phys. Rev. A 22, 2005 (1980) Streater, A., Cooper, J., Sandle, W.J.: J. Quant. Spectrosc. Radiat. Trans. 37, 151 (1987) Tellinghausen, J.: J. Mol. Spectrosc. 103, 455 (1984) Mies, F.H.: J. Chem. Phys. 48, 482 (1968) Carrington, C.G., Drummond, D., Phelps, A.V., Gallagher, A.: Chem. Phys. Lett. 22, 511 (1973) Biberman, L.M.: J. Exp. Theor. Phys. U. S. S. R. 17, 416 (1947) Biberman, L.M.: J. Exp. Theor. Phys. U. S. S. R. 59, 659 (1948) Holstein, T.: Phys. Rev. 72, 1212 (1947) Holstein, T.: Phys. Rev. 83, 1159 (1951) Fioretti, A., Molisch, A.F., Müller, J.H., Verkerk, P., Allegrini, M.: Opt. Commun. 149, 415 (1998)
295 56. Molisch, A.F., Oehry, B.P.: Radiation Trapping in Atomic Vapours. Clarendon, Oxford (1998) 57. van Trigt, C.: Phys. Rev. 181, 97 (1969) 58. van Trigt, C.: Phys. Rev. A 4, 1303 (1971) 59. van Trigt, C.: Phys. Rev. 13, 726 (1976) 60. Carrington, C.G., Stacey, D.N., Cooper, J.: J. Phys. B 6, 417 (1973) 61. Huennekins, J., Park, H.J., Colbert, T., McClain, S.C.: Phys. Rev. 35, 2892 (1987) 62. Post, H.A., van de Weijer, P., Cremers, R.M.M.: Phys. Rev. A 33, 2003 (1986) 63. Holstein, T.: Phys. Rev. 83, 1159 (1951) 64. Walsh, P.J.: Phys. Rev. 116, 511 (1959) 65. Colbert, T., Huennekens, J.: Phys. Rev. 41, 6145 (1990) 66. Huennekins, J., Gallagher, A.: Phys. Rev. A 28, 238 (1983) 67. Milne, E.: J. Math. Soc. (London) 1, 40 (1926) 68. Holstein, T., Alpert, D., McCoubrey, A.O.: Phys. Rev. 85, 985 (1952) 69. Phelps, A.V.: Phys. Rev. 114, 1011 (1959) 70. Payne, M.G., Talmage, J.E., Hurst, G.S., Wagner, E.B.: Phys. Rev. A 9, 1050 (1974) 71. Anderson, J.B., Maya, J., Grossman, M.W., Lagushenko, R., Weymouth, J.F.: Phys. Rev. A 3, 2986 (1985) 72. Parker, G.J., Hitchon, W.N.G., Lawler, J.E.: J. Phys. B 26, 4643 (1993) 73. van Trigt, C.: Phys. Rev. A 1, 1298 (1970) 74. Menningen, K.L., Lawler, J.E.: J. Appl. Phys. 88, 3190 (2000) 75. Lawler, J.E., Curry, J.J., Lister, G.G.: J. Phys. D 33, 252 (2000) 76. Zajonc, A., Phelps, A.V.: Phys. Rev. A 23, 2479 (1981) 77. Lawler, J.E., Parker, G.J., Hitchon, W.N.G.: J. Quant. Spectrosc. Radiat. Trans. 49, 627 (1993) 78. Parker, G.J., Hitchon, W.N.G., Lawler, J.E.: J. Phys. B 26, 4643 (1993) 79. Curry, J.J., Lawler, J.E., Lister, G.G.: J. Appl. Phys. 86, 731 (1999) 80. Huennekins, J., Colbert, T.: J. Quant. Spectrosc. Radiat. Trans. 41, 439 (1989)
20
Thomas-Fermi and Other Density-Functional Theories
21
John D. Morgan III
Contents
Keywords
21.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 297
21.2 21.2.1 21.2.2 21.2.3 21.2.4 21.2.5
Thomas–Fermi Theory and Its Extensions . . . . . Thomas–Fermi Theory . . . . . . . . . . . . . . . . . . . Thomas–Fermi–von Weizsäcker Theory . . . . . . . . Thomas–Fermi–Dirac Theory . . . . . . . . . . . . . . Thomas–Fermi–von Weizsäcker–Dirac Theory . . . . Thomas–Fermi Theory with Different Spin Densities
21.3
Nonrelativistic Energies of Heavy Atoms . . . . . . . 302
21.4 21.4.1
General Density Functional Theory . . . . . . . . . . The Hohenberg–Kohn Theorem for the One-Electron Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Kohn–Sham Method for Including Exchange and Correlation Corrections . . . . . . . . . . . . . . . . . . . Density Functional Theory for Excited States . . . . . . Locality of Density Functional Theory . . . . . . . . . . Relativistic and Quantum Field Theoretic Density Functional Theory . . . . . . . . . . . . . . . . . . . . . .
21.4.2 21.4.3 21.4.4 21.4.5 21.5
. . . . . .
298 298 300 301 301 301
302 302 303 304 304 304
Recent Developments . . . . . . . . . . . . . . . . . . . . 304
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
Abstract
The key idea in Thomas–Fermi theory and its generalizations is the replacement of complicated terms in the kinetic energy and electron–electron repulsion energy contributions to the total energy by relatively simple functionals of the electron density . This chapter first describes Thomas–Fermi theory and then its various generalizations that attempt to correct, with varying success, some of its deficiencies. It concludes with an overview of the Hohenberg–Kohn and Kohn–Sham density functional theories.
J. D. Morgan III () Dept. of Physics & Astronomy, University of Delaware Newark, DE, USA e-mail: [email protected]
density functional theory neutral atom external potential electronic probability density Scott correction
21.1 Introduction In the early years of quantum physics, Thomas [1] and Fermi [2–5] independently invented a simplified theory, subsequently known as Thomas–Fermi theory, to describe nonrelativistically an atom or atomic ion with a large nuclear charge Z and a large number of electrons N . Many qualitative features of this model can be studied analytically, and the precise solution can be found by solving numerically a nonlinear ordinary differential equation. Lenz [6] demonstrated that this equation for the electrostatic potential could be derived from a variational expression for the energy as a functional of the electron density. Refinements to Thomas–Fermi theory include a term in the energy functional to account for electron exchange effects introduced by Dirac [7] and nonlocal gradient corrections to the kinetic energy introduced by von Weizsäcker [8]. Although the Hartree–Fock method or other more elaborate techniques for calculating electronic structure now provide much more accurate results (Chaps. 22–24), Thomas– Fermi theory provides quick estimates and global insight into the total energy and other properties of a heavy atom or ion. A rigorous analysis of Thomas–Fermi theory by Lieb and Simon [9, 10] showed that it is asymptotically exact in that it yields the correct leading asymptotic behavior, for both the total nonrelativistic energy and the electronic density, in the limit as both Z and N tend to infinity, with the ratio Z=N fixed. (In a real atom, of course, relativistic and other effects become increasingly important as Z increases.) However, Thomas–Fermi theory has the property that molecules do not bind, as first noted by Sheldon [11] and proved by
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_21
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Teller [12]. That the interatomic potential energy curve for a homonuclear diatomic molecule is purely repulsive was demonstrated by Balàzs [13]. This no binding property of clusters of atoms was used by Lieb and Thirring [14] to prove the stability of matter, in the sense that as the number of particles increases, the total nonrelativistic energy decreases only linearly rather than as a higher power of the number of particles, as it would if electrons were bosons rather than fermions. Lieb went on to explore the mathematical structure of the modifications of the Thomas–Fermi model when gradient terms (von Weizsäcker) and/or exchange (Dirac) terms are included [15, 16]. A review article by Spruch [17] explicates the linkage between long-developed physical intuition and the mathematically rigorous results obtained in the 1970s and 1980s. The older literature was reviewed by Gombás [18, 19] and by March [20]. An outgrowth of Thomas–Fermi theory is the general density functional theory initiated by Hohenberg and Kohn [21] and by Kohn and Sham [22], as discussed in Sect. 21.4.
21.2 Thomas–Fermi Theory and Its Extensions 21.2.1
Thomas–Fermi Theory
In a D-dimensional Euclidean space, the expectation value of the electronic kinetic energy operator in a quantum state can be approximated by Z D 2=D „2 2 D 2
.DC2/=D .r/dD r ; me D C 2 2˝D where ˝D D
D D=2
1 C D2
(21.1)
(21.2)
is the surface area of a unit hypersphere in D dimensions [17, p. 176]. These expressions can easily be derived by considering the energy levels of a system of a large number of noninteracting fermions confined to a D-dimensional box. Specialization to the physically interesting case of D D 3 yields the well-known expression Z 3 2=3 „2 2 3 2
5=3 .r/dr ; me 5 2˝3
(21.3)
where V .r/ is the Coulomb potential due toP a single nucleus [V .r/ D Z=r] or several nuclei [V .r/ D i Zi =jr R i j]. The electron–electron Coulomb repulsion energy in a threedimensional space is approximated by Z
.r/ .r 0 / 1 (21.6) drdr 0 ; U Œ D 2 jr r 0 j which tends to overestimate the actual repulsion energy because it includes the self-energy of the densities of individual electrons. This is, however, a higher-order effect for a system with a large number of electrons concentrated in a small region of space. As was suggested by Fermi and Amaldi [23], this overestimation can be approximately corrected for an atom with N electrons by multiplying this term by the ratio of the number of ordered pairs of different electrons to the total number of ordered pairs 1 N.N 1/ D1 : 2 N N
(21.7)
This is approximately correct for an atom, with many electrons concentrated close together, but it would still be an overestimate for a diffuse system, such as one composed of N electrons and N protons separated by large distances of O.R/, for which the ground-state electron–electron repulsion term should be proportional to 12 N.N 1/=R rather than to N times a constant of O.1/. For this reason, the Fermi–Amaldi correction, which complicates the mathematical analysis without eliminating the unphysical overestimation of the electron–electron repulsion term, is not usually included. It is evident that the treatment of both the electronic kinetic energy term and the electron–electron repulsion energy term depends on the assumption that the number N of electrons (actually, the number of electrons per atom) is large. Hence, the Thomas–Fermi model is sometimes called the statistical model of an atom. The three contributions to the total energy are now added together, and one seeks to minimize their sum, the Lenz functional [6] Z 3 2=3 „2 2 3 2
5=3 .r/dr EŒ D me 5 2˝3 Z Z
.r/ .r 0 / 1 C .r/V .r/dr C drdr 0 ; (21.8) 2 jr r 0 j
over all admissible densities . The mathematical question now arises: what is an admissible density? The answer was where provided by Lieb and Simon [9, 10]: a density for which both 3 3=2 Z ˝3 D
D 4 : (21.4) 1 C 32
.r/dr ; (21.9) The electron–nucleus attraction energy in a threedimensional space is given exactly by the total number of electrons, and Z Z (21.10)
.r/V .r/dr ; (21.5)
5=3 .r/dr ;
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Thomas-Fermi and Other Density-Functional Theories
299
which is proportional to the estimate of their kinetic energy, Another possibility is to do the constrained minimization are finite, automatically yields finite values of the other terms over all densities that obey in the expression for the energy. As Lieb and Simon proved, Z the minimization of this functional over all such densities
.r/dr D ; (21.18) yields a well-determined result. Carrying out the variation of EŒ with respect to yields where is the number of electrons, which for purposes of the Euler–Lagrange equation mathematical analysis is allowed to be nonintegral. One then 2=3 2 introduces a Lagrange multiplier , the chemical potential, 3 „ 2 2
2=3 .r/ 0D to correspond with this constraint, and thereby obtains the me 2˝3 Z Euler–Lagrange equation
.r 0 / 0 C V .r/ C : (21.11) dr jr r 0 j „2 2 3 2=3 2=3 2
.r/ 0D me 2˝3 The sum of the last two terms is, of course, the negaZ tive of the total electrostatic potential .r/, so one sees that
.r 0 / C V .r/ C (21.19) dr 0 C ; in Thomas–Fermi theory, the density is proportional to the jr r 0 j 3=2-power of the potential. To simplify subsequent manipuwhich holds wherever is positive. As was shown by Lieb lations, let and Simon [9, 10], this procedure, too, is well defined. The „2 2 3 2=3 „2 2 2=3 analogue of Eq. (21.13), the relationship between the density 2 D D p ; (21.12) 3 me 2˝3 2me and the electrostatic potential for the neutral atom, is now so that p 2=3 .r/ D .r/ : By Poisson’s theorem, " # X 2 .3/ r D 4 Zi ı .r R i / .r/ ;
p 2=3 .r/ D Œ.r/ C ;
(21.13)
where Œf C D max.f; 0/. The corresponding differential equation for the potential is "
(21.14) 2
r D 4
i
(21.20)
X
Zi ı .3/ .r R i /
i
3=2 # and from Eq. (21.13) one has D p 3=2 , so from the 3=2 3=2 p Œ.r/ C : integral equation for the electronic density , one obtains the (21.21) differential equation " # Lieb and Simon rigorously proved a large number of results X 2 .3/ 3=2 3=2 Zi ı .r R i / p r D 4 ; (21.15) concerning the solution of the Thomas–Fermi model. When i V .r/ is a sum of Coulomb potentials arising from a set of nuP for the potential . In the case of a single nucleus, the usual clei of positive charges Zi , with i Zi D Z, then the energy separation of variables in spherical polar coordinates yields E./ is a continuous, monotonically decreasing function of for 0 Z, and its derivative dE=d is the chemical pofor the ordinary differential equation tential ./, which vanishes at D Z. For in this range, there is a unique minimizing density , whereas for > Z 1 d2 .r/ D 4 p3=2 3=2 ; (21.16) there is no unique minimizing , since one can place arbi2 r dr trarily large clumps of charge with arbitrarily low positive whose similarity to Emden’s equation, which Eddington used energies arbitrarily far away from the nuclei. In the atomic to study the internal constitution of stars, was recognized by case, with a single nucleus, .r/ is a spherically symmetric Milne [24]. The numerical solution of this equation with the monotonically decreasing function of r. appropriate boundary conditions at r D 0 and r D 1 was Moreover, for an atom or atomic ion, the Thomas–Fermi discussed extensively by Baker [25], and accurate solutions density obeys the virial theorem tabulated by Tal and Levy [26]. The numerical solution determines that the total energy of a neutral atom is (21.22) 2hT i D hV i D 2E ;
E D 3:67874521: : : p1 Z 7=3 D 1:53749024: : : Z
7=3
Ry :
and for a neutral atom the electronic kinetic energy, electron– (21.17) nucleus attraction energy, and electron–electron repulsion
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J. D. Morgan III
energy terms in the expression for the total energy satisfy the ratios 3 W 7 W 1. It is straightforward to examine the behavior of the Thomas–Fermi density in the limit as either r ! 0 or r ! 1. For large r, the electron density vanishes identically outside a sphere of finite radius for a positive ion. For a neutral atom, the ordinary differential equation (21.16) for the potential can be analyzed to show that .r/ ' p
3p
2 r 4 ;
(21.23)
from which it follows that
.r/ '
3p
This singularity is integrable but unphysical, since it arises from the approximation of the local kinetic energy by 5=3 , which breaks down where is rapidly varying on a length scale proportional to 1=Z. In a real nonrelativistic heavy atom governed by the Schrödinger equation, the actual electron density at the nucleus is finite, being proportional to Z 3 . This unphysical singularity in the electron density in Thomas–Fermi theory can be eliminated by adding a gradient correction to the Thomas–Fermi kinetic energy term, as discussed in the following section.
21.2.2
3
Thomas–Fermi–von Weizsäcker Theory
r 6 ;
(21.24) The semiclassical approximation Eq. (21.3) for the quantum kinetic energy in terms of a power of the density is capable of independent of Z. This implies that as Z ! 1, a neutral improvement, particularly in regions of space where the denatom described by the Thomas–Fermi model has a finite size sity is rapidly varying. The incorporation of such corrections defined in terms of a radius within which all but a fixed leads to a gradient expansion for the kinetic energy [27]. The amount of electronic probability density is located. For ex- leading correction is of the form ample, if one defines the size of an atom as that value of ra for which Z 1
.r/dr D ; (21.25) 2 jrjra
one finds that in the large-Z limit ra D
8 3
1=3
3p :
(21.26)
In atomic units, p D 12 .3 2 /2=3 , and ra D .9 /2=3 a0 ' 9:3 a0 ;
(21.27)
which is about what one would expect for a real nonrelativistic atom with a large nuclear charge Z. On the other hand, the characteristic distance scale in Thomas–Fermi theory, defined as the average value of r, or in terms of a radius within which a fixed fraction of electronic probability density is located, is proportional to Z 1=3 , which shrinks to 0 as Z ! 1. The resolution of this paradox is that outside the typical core scale of distance set by Z 1=3 , within which most of the electron density is located, there resides in the mantle region a fraction of electrons proportional to Z 2=3 =Z D Z 1=3 , and almost all of these are concentrated within a sphere of radius of about 10 a0 . Moving deeper into the core and approaching the nucleus, the Z=r singularity in the electron–nucleus Coulomb potential dominates the smeared-out electron–electron potential, so one readily finds that
.r/ '
Z p r
3=2 :
(21.28)
„2 2m
Z
ˇ 1=2 ˇ2 ˇ r
.r/ˇ dr :
(21.29)
Addition of such a term to the Thomas–Fermi expression for the kinetic energy yields a theory that avoids many of the unphysical features of ordinary Thomas–Fermi theory at very short and moderately large distances. The more important points, as rigorously proved in Lieb’s review article [15, 16], are as follows. The leading features of the energy are unchanged; for large Z, the energy E.Z/ of a neutral atom or atomic ion is still proportional to Z 7=3 , but now there enter higher-order corrections arising from the gradient terms of order Z 7=3 Z 1=3 D Z 2 and higher powers of Z 1=3 . The maximum number of electrons that can be bound by an atom of nuclear charge Z is no longer exactly Z, but a slightly larger number; thus, Thomas–Fermi–von Weizsäcker theory allows for the formation of negatively charged atomic ions. It was further proven by Benguria and Lieb [28] that in the Thomas–Fermi–von Weizsäcker model a neutral atom can bind at most one extra electron, and that a neutral molecule can bind at most as many extra electrons as it has nuclei. The effect on the electronic density is more profound. While the general shape and properties of in the core and mantle regions is unchanged, the fact that r 1=2 need not, and in general does not, vanish when vanishes on some surface implies that for a positive ion no longer vanishes outside of a sphere, as it does in the case of Thomas–Fermi theory, but instead extends over all space. For positive ions, neutral atoms, and negative ions alike, differential inequality techniques [29] can be used to show that .r/ decays exponentially for large r, with the constant in the exponential proportional to ..//1=2 . For small r, the gradient terms
21
Thomas-Fermi and Other Density-Functional Theories
dominate the energy expression, so one finds that the electronic density no longer diverges as r ! 0, but instead tends to a finite limit, with a first derivative that obeys a relation analogous to the Kato cusp condition [30] The study of molecules within the Thomas–Fermi–von Weizsäcker model involves several subtleties and pitfalls, which can lead to physical absurdities. Since two neutral atoms with different nuclear charges will, in general, have different chemical potentials, a pair of such atoms placed a long distance apart will spontaneously ionize, with a small amount of electric charge being transferred from one to the other until the chemical potentials of the positively charged ion and the negatively charged ion become equal. The result is a long-range Coulomb attraction between them [31]. This phenomenon does not occur in the real world, since the amount of electric charge that can be transferred is quantized in units of e, and it is empirically true that the smallest atomic ionization potential exceeds the largest atomic electron affinity. For two neutral atoms with the same nuclear charges, the situation is more subtle. Nonetheless, a careful analysis shows that in this case, too, although no spontaneous ionization occurs, there is a long-range attractive interaction between them arising from the overlap of the exponentially small tails of the electron clouds. Since electron correlation is not included in this model, it could not be expected to describe attractive van der Waals forces. In summary, the Thomas–Fermi–von Weizsäcker model yields a more realistic picture of a single atom than does the Thomas–Fermi model. However, it does not provide a useful picture for understanding the interaction between atoms at large distances. These kinds of unphysical features provide a glimpse into the complicated nature of the universal density functional, which must include terms that rigorously suppress an unphysical feature like spontaneous ionization of a distant pair of heteronuclear atoms [32, 33]. It is evident from the mathematical properties of Thomas–Fermi–von Weizsäcker theory and related models that a density functional that fixes up the Thomas–Fermi expression simply by adding a few gradient terms and/or simple exchange terms and the like must still differ in important ways from the universal density functional, particularly for properties of extended systems.
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Minimization of the resulting Thomas–Fermi–Dirac energy functional over all admissible densities whose integral is yields a well-defined E./, which has the correct behavior for Z, and it has been shown that for an atom, the exchange correction to the energy is of order Z 5=3 . However, this model exhibits unphysical behavior for > Z, because one can obtain a completely artificial lowering of the energy by placing many small clumps of electrons R a large distance from the nucleus, for which the negative 4=3 dr term dominates the energy expression [15, 16]. At the conclusion of his original article, Dirac clearly stated that the correction he had derived, although giving a better approximation in the interior of an atom, gives a meaningless result for the outside of the atom [7]. It is, therefore, clear that any physically reasonable theory must somehow profoundly modify this correction in the region where the electronic density is very small.
21.2.4 Thomas–Fermi–von Weizsäcker–Dirac Theory One can also include the Dirac exchange correction in the Thomas–Fermi–von Weizsäcker energy functional. In this case, however, the mathematical foundations of the theory are still incomplete ([15, 16, pp. 638639]). Nonetheless, it is clear that this theory, too, suffers from the unphysical lowering of the energy by small clumps of electrons at large distances from the nucleus. In summary, one can say that the inclusion of Dirac’s exchange correction in its most straightforward form leads to an improvement of energies for positive ions or neutral atoms but to unphysical behavior for systems where the charge of the electrons exceeds the nuclear charge, in line with Dirac’s own observations on the limitations of his correction [7]. Here, we again see a manifestation of how complicated the behavior of the true universal density functional must be.
21.2.5 Thomas–Fermi Theory with Different Spin Densities
As was remarked by Lieb and Simon [10], it is possible to consider a variant of Thomas–Fermi theory with a pair of spin densities ˛ and ˇ for the spin-up and spin-down electrons, with the two adding together to produce the total 21.2.3 Thomas–Fermi–Dirac Theory electronic density . This theory was rigorously formulated and analyzed by Goldstein and Rieder [34]. Because the The effect of the exchange of electrons can be approximated, problem is nonlinear, the mathematical complications are following Dirac [7], by including in the Thomas–Fermi en- substantial, and the theory is not a trivial extension of orergy functional an expression of the form dinary Thomas–Fermi theory. Goldstein and Rieder first considered the case where the total number of electrons Z of each type of spin is specified in advance [35]. There 1 2 4=3 (21.30) is no mathematical obstacle to constructing such a spin3
4=3 .r/dr : 3 4
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polarized Thomas–Fermi theory, but it does not yield the kind of spontaneous spin-polarization that one observes in the ground states of many real quantum mechanical atoms and molecules, which is not surprising in view of the fact that such spin-polarization in accord with Hund’s first rule arises from exchange and correlation effects not included in this simple functional. However, in the case where the electronic spins (but not their currents) are coupled to an external magnetic field, the ground state is naturally spin-polarized [34].
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ergy agrees with the exact quantum energy through O.Z 5=3 /, with an error of smaller order [59]. Numerous auxiliary theorems and lemmas are published in [60–63]. Progress toward obtaining higher-order oscillatory terms is described in [64– 68]. The analytical evaluation of accurate approximations to the energy of a heavy atom, or at least of the contributions to that energy of all but the few outermost electronic orbitals, would be of particularly great value if it led to the construction of more accurate and better justified pseudopotentials [69–72] for describing the valence orbitals.
21.3 Nonrelativistic Energies of Heavy Atoms 21.4 General Density Functional Theory Thomas–Fermi theory suggests that Eq. (21.17) provides the leading term in a power series expansion for the nonrelativis- The literature on general density functional theory and its tic energy of a neutral atom of the form applications is enormous, so any bibliography must be selective. The reader interested in learning more could begin
E.Z/ D c7 Z 7=3 C c6 Z 6=3 C c5 Z 5=3 C : : : ; (21.31) by consulting a number of review articles [73–79], collections of articles [80–82], conference proceedings [83–93], with c7 D 1:53749024 : : : Ry, c6 D 1 Ry, and c5 ' and the textbooks by Parr and Yang [94] and by Dreizler and 0:5398 Ry. The c6 term was first calculated by Scott [36] Gross [95]. from the observation that it arises from the energy of the innermost electrons for which the electron–electron interaction can be neglected. The difference between the exact 21.4.1 The Hohenberg–Kohn Theorem and Thomas–Fermi energies for this case of noninteracting for the One-Electron Density electrons yields the correct c6 [17, 37]. A mathematically nonrigorous but physically insightful justification of the In 1964, Hohenberg and Kohn [21] argued that there exists Scott correction was provided in 1980 by Schwinger [38]. a universal density functional F Œ , independent of the exterThis result has now been rigorously proved, with upper and nal potential V .r/, such that minimization of the sum lower bounds coinciding [39–43]. Z The c5 term is much more subtle, since it arises from F Œ C .r/V .r/dr ; (21.32) a combination of effects from the exchange interaction and from the bulk motion of electrons in the Thomas– subject to the constraint Fermi potential. A general analytic procedure devised by Z Schwinger [44] yields the above value, in good agreement
.r/dr D N (a positive integer) ; (21.33) with a much earlier estimate by March and Plaskett [45]. A numerical check of these results, based on a fit to Hartree–Fock calculations for Z up to 290 with correlation corrections, yielded the values [46] c5 D 0:55 ˙ 0:02 Ry and c4 ' 0. It seems likely that, because of shell structure, the terms c4 and beyond have an oscillatory dependence on Z [47]. The oscillatory structure and other refinements of Thomas–Fermi theory are considered in a series of papers by Englert and Schwinger [48–54] and a book by Englert [55]. Iantchenko, Lieb, and Siedentop [56] proved Lieb’s strong Scott conjecture that for small r, the rescaled density for the exact quantum system converges to the sum of the densities of the bound noninteracting hydrogenic orbitals; the properties of this function were explored by Heilman and Lieb [57]. Fefferman and Seco [58] rigorously proved the correctness of Schwinger’s procedure for calculating not just the O.Z 6=3 / Scott correction but also the O.Z 5=3 / exchange term. Their full proof includes a demonstration that the Hartree–Fock en-
yields the ground state energy of a quantum-mechanical N -electron system moving in this external potential. However, Hohenberg and Kohn’s theorem is like a mathematical existence theorem; no procedure exists to calculate this unknown universal functional explicitly, which surely is extremely complicated if it can be written down at all in closed form. (E. Bright Wilson, however, defined it, implicitly and whimsically, as follows: “Take the ground-state density and integrate it to find the total number of electrons. Find the cusps in the density to locate all nuclei, and then use the cusp condition – that the radial derivative of the density at the cusp is minus twice the nuclear charge density at each cusp – to determine the charges on each nucleus. Finally solve Schrödinger’s equation for the ground-state density or any other property that is desired” (paraphrased by B. I. Dunlap, in [89, p. 3], from J. W. D. Connolly).)
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Moreover, Hohenberg and Kohn glossed over two problems: it is not clear a priori that every well-behaved
is derivable from a well-behaved properly antisymmetric many-electron wave function (the so-called n-representability problem, since n was used by Hohenberg and Kohn to represent the density of electrons), and it is also not clear a priori that every well-behaved density can be derived from a quantum-mechanical many-electron wave function , which is the properly antisymmetric ground-state wave function for a system of electrons moving in some external potential V .r/ (the so-called v-representability problem, since Hohenberg and Kohn used v in place of V ). The n-representability problem was solved by Gilbert [96] and by Harriman [97], who gave a prescription for starting from an arbitrary well-behaved and from it constructing a many-electron wave function that generated that [33, 98]. The v-representability problem is much more formidable, as demonstrated by the discovery that there are well-behaved densities that are not the ground-state densities for any fermionic system in an external potential V [98, 99]. Following Percus’ definition of a universal kinetic energy functional for independent fermion systems [100], Levy [101] proposed to circumvent this vrepresentability problem by modifying the definition of F Œ so that instead of being defined in terms of densities that might not be v-representable, it is defined as F Œ D minŒ ; .T C V / ;
(21.34)
with the minimum being taken over all properly antisymmetric normalized ’s that yield that . A great deal of effort has been devoted to trying to find approximate representations of the universal functional F Œ . One route is mathematical, and features a careful exploration of the abstract properties that F Œ must have. Another route is numerical, and can be characterized as involving the guessing of some ansatz with a general resemblance to Thomas– Fermi–von Weizsäcker–Dirac theory, with some flexible parameters determined by least-squares fitting of the energies resulting from insertion of Hartree–Fock densities into the trial functional to theoretical Hartree–Fock energies, or the like. If, however, the basic ansatz exhibits unphysical features in the case of negatively charged ions or heteronuclear molecules, it is not likely that the optimization of parameters in that ansatz will get one closer to the true universal density functional. In the opinion of this writer, significant progress in density functional theory based solely upon the one-electron density is likely to require a major revolution in our mathematical understanding of this field, with a useful procedure made explicit for constructing progressively better approximations to the universal density functional, which, like or other transcendental numbers, will probably never be written down exactly in closed form. Moreover, the numerical solution of the highly nonlinear Euler–Lagrange equations for
a very complicated density functional is likely to require large amounts of computer time, as well as problems with landing in local minima of the energy.
21 21.4.2 The Kohn–Sham Method for Including Exchange and Correlation Corrections Density functional theory posed solely in terms of the oneelectron density and based upon the Hohenberg–Kohn variational principle provides no general procedure for accurately calculating relatively small energy differences such as excitation energies, ionization potentials, electron affinities, or the binding energies of molecules. There is, however, a powerful method inspired by the Hohenberg–Kohn variational principle, which has been used with great success in the calculation of such quantities. This is the Kohn–Sham variational method [22]. The key idea in the Kohn–Sham variational method is to replace the nonlocal exchange term in the Hartree–Fock equations with an exchange-correlation potential, which, at least in principle, can be used to determine energies exactly. The oldest, simplest, and most common ansatz used for the exchange-correlation potential involves the local density approximation (LDA), in which one assumes that the exchangecorrelation potential for the actual system under study has the same functional form as does the exchange-correlation potential for a uniform interacting gas of electrons. If the density is not too small or not too rapidly varying, the exchange part of this potential can be approximated by 1=3 , which appears in Dirac’s first-order approximation for exchange energies, with a systematic procedure for deriving higher-order corrections in a gradient expansion. The correlation part of this potential is accurately known from Ceperley and Alder’s quantum Monte Carlo calculation of the properties of the uniform electron gas [102]. One, therefore, retains the important features of the quantum theory based on wave functions, with a determinantal approximation to , while approximately including exchange and correlation effects through a simply computable effective potential. Higher corrections, which are important for quantitative accuracy, can be incorporated by taking account of the variation of by means of a gradient expansion [27] involving r and higher derivatives [95, Chap. 7], thus yielding a generalized gradient approximation (GGA) for the exchange-correlation potential. The Kohn–Sham procedure has become the backbone for the vast majority of accurate calculations of the electronic structure of solids [78, 92]. In the 1990s, motivated by Becke’s work on constructing simple gradient-corrected exchange potentials [103–109], and incorporating the Lee, Yang, and Parr (LYP) expression for the correlation potential [110] derived from Colle and Salvetti’s correlationenergy formula [111–113], the Kohn–Sham method is find-
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ing increasing application in efficiently estimating relatively small energy differences of relevance to chemistry [114–116] (However, Becke’s gradient-corrected exchange potential does not have the correct 1=r behavior at large r, as was observed by several authors [117–119]). For definitive results, however, one must still resort to an ab initio theory that, at least in principle, converges toward the correct result. The generation of improved generalized gradient approximations has recently become a growth industry, with increasingly many proposals of increasingly greater complexity [103, 110, 119–131]. Inevitably, some expressions work better for some properties than for others. It is found that usually most of the errors in the long-range tails of the exchange and correlation potentials tend to cancel each other, thus leading to better overall energies than one could reasonably expect [132]. Under these circumstances, it is important to have benchmarks for testing the accuracy of the various approximations. Such comparisons have been carried out for two important sets of two-electron systems [133–135]: 1. A pair of electrons moving in harmonic potential wells and coupled by the Coulomb repulsion, which yields an exactly solvable system. 2. Helium-like ions of variable nuclear charge Z, for which extremely accurate energies and wave functions are available, which take account of the behavior of the exact but unknown wave function in the vicinity of all two-particle coalescences and the three-particle coalescence. The results indicate that the approximate exchange-correlation potentials differ quite considerably from the true exchange-correlation potentials, thus indicating the need for further analytical work in understanding how to design accurate exchange-correlation potentials and for devising tests of exchange-correlation potentials for larger atoms and for molecules. Another important way of testing the validity of various approximate exchange and correlation potentials is checking whether they obey inequalities imposed by such general properties as scaling and the Hellmann–Feynman theorem. Such general tests were devised by Levy and his coworkers [136–145], who found that many of the commonly used approximate potentials violate general inequalities that must be obeyed by the exact potential. These abstract results are helpful in designing potentials that should be better approximations to the true potential.
21.4.3 Density Functional Theory for Excited States
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the ground state of a given symmetry [146], but that leaves unresolved the issue of using density functional theory to calculate the energies of excited states for a given symmetry. Using the Rayleigh–Ritz principle for ensembles, general abstract procedures for generalizing density functional theory to excited state calculations were formulated by Theophilou [147] and by several other workers [148–158]. Unfortunately, the errors typically seem to be much larger than for ground-state density functional theory.
21.4.4 Locality of Density Functional Theory Although the locality of DFT was proved for a large class of functionals [159–161], this issue has come under dispute. The question that has been raised is whether there exists an exact Thomas–Fermi model for noninteracting electrons. If such an exact model does not exist, as it is a direct consequence of the Hohenberg-Kohn theorem, then DFT would be incomplete. Nesbet [162–166] argued that such a theory would be inconsistent with the Pauli exclusion principle for atoms of more than two electrons (or for a two-electron atom where both electrons are in the same spin state). The contention is that if only the total electron density were normalized (which corresponds to only one Lagrange multiplier), as in the TF model, then no shell structure can exist; hence such a system would violate the exclusion principle. A counterexample was constructed by Lindgren and Salomonson [167] showing that shell structure can, indeed, be generated through a single Lagrange multiplier. In addition, they verified numerically that a local Kohn–Sham potential can reproduce to high accuracy the many-body electron density and the 2s eigenvalue for the 1s2s 3 S state of neutral helium.
21.4.5 Relativistic and Quantum Field Theoretic Density Functional Theory At a formal level, one can discuss the development of density functional theory for a relativistic system of electrons. For an overview of this challenging subject, see the discussions by Dreizler and Gross [95, Chap. 8] and by Dreizler [168]. Much of the formalism carries over, but no good way of incorporating vacuum polarization corrections has yet been found.
21.5 Recent Developments
A recent article by J.P. Solovej [169] illustrates the good The Hohenberg–Kohn theorem and the Kohn–Sham method qualitative agreement of the Thomas–Fermi model’s predicwere originally formulated in terms of the ground elec- tions of the sizes of atoms with Z from 1 to 50 with the tronic state. These techniques can be extended to calculate empirical determinations of their radii in crystals by W.L.
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Bragg [170] and J.C. Slater [171], and presents some new conjectures about asymptotic formulas for the upper and lower limits of the radii and ionization energies of atoms as Z ! 1 (of course, ignoring relativistic effects). Many of the important articles by Elliott Lieb and his numerous coworkers on Thomas–Fermi theory and its extensions and their applications to understanding atomic structure and the stability of bulk matter were conveniently reprinted in a single large volume in 2001 [172]. A more readily comprehensible overview of the main results of this large body of work, and of the mathematical theorems used to obtain them, was published by Lieb and Seiringer in 2010 [173]. Ever since the mid-1990s there has continued to be exponentially growing interest in applications of density functional theory of the Kohn–Sham variety to atoms and molecules, especially those of chemical relevance that are too large for accurate ab initio electronic structure calculations. The awarding of the 1998 Nobel Prize in Chemistry to Walter Kohn and John A. Pople recognized their individual contributions to this increasingly important field. Their Nobel lectures were published the following year in the Reviews of Modern Physics [174, 175]. In a review article published in 2012, Kieron Burke illustrated the exponential growth in the numbers of articles on density functional theory published each year and noted they would soon exceed 10,000 articles per year [176]. Since a comprehensive summary of the wide-ranging developments in density functional theory during the past two decades is not feasible within the limited space available for this supplementary section, I will briefly cite some of the most extensive surveys of various aspects of this field that have appeared since 1995. Many aspects of density functional theory were reviewed in four consecutive volumes of Topics in Current Chemistry published in 1996 [177–180], and in 1999 an entire volume of Advances in Quantum Chemistry was devoted to density functional theory [181]. This has also been the subject of several textbooks [182–185] and conference proceedings [186–188]. The International Conferences on Density Functional Theory and Its Applications have been held in various European cities ever since 1995 in odd-numbered years. Recently, Burke [176] and Becke [189] and Yu, Li, and Truhlar [190] published review articles in the Journal of Chemical Physics. An article by Perdew and coworkers provides qualitative explanations of many of the features and paradoxes of Kohn– Sham density functional theory [191]. Tension persists between the proponents of computational methods who seek primarily to develop density functionals with relatively few parameters that obey mathematically derived constraints in the never-ending quest to approach ever more closely the unattainable “Holy Grail” of the exact density functional
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(e.g., [192]), and the developers and users of density functionals with few mathematical constraints and many adjustable parameters optimized for a class of molecules to generate predictions of the properties of similar molecules [193–195]. Within the context of Kohn–Sham density functional theory, Constantin, Snyder, Perdew, and Burke investigated the ionization potentials of neutral atoms and the behavior of the “ionization density”, defined as [4q r 2 . atom .r/ ion .r//], where “ion” refers to the singly charged atomic ion with one electron removed, for Z up to 2938, corresponding to a noble gas atom with a filled outer shell with row number n D 25 [196]. Their results indicate that as Z increases, the periodic properties of atoms persist and tend to distinct limits as Z ! 1, and that the averages of these periodic properties within a “shell” are close to the values given by Thomas– Fermi theory. These results are of mathematical interest, but of course, real atoms with such large nuclear charges do not exist, and even if they did, the effects of relativity and quantum electrodynamics would be profound not just on the orbitals in the “core” but also in the “mantle” and even in the outermost shell. Indeed, because the s-orbitals of atoms have appreciable probability density near the nucleus, there are significant relativistic effects on the physical and chemical properties of atoms already in the 6th period of the periodic table with atomic number Z running upward from 55 (cesium) [197, 198]. Both the unusual color of gold (Z D 79) and the unusually low melting point of mercury (Z D 80) are striking consequences of relativistic effects on the orbitals in the outermost shell of these atoms [199, 200]. See Chap. 23 for a review of the methodology for performing accurate relativistic calculations of the properties of atoms and atomic ions with large values of Z. Recent attention has been paid to the use in density functional theory of the Lieb–Oxford lower bound to the indirect part of the Coulomb energy [201] Z h Vee i U Œ CD d D r. .r//.DC1/=D ; (21.35) where h Vee i is the expectation value of the interelectronic interaction energy operator for an electronic wavefunction , and U Œ is the approximation to it with the corresponding one-electron density .r/ defined in Eq. (21.6), and CD is a dimension-dependent constant (D D 2 or 3) for which upper and lower bounds have been derived with various improvements. (As of 2016 it had been established that 1:4119 C3 1:6358.) Mirtschink, Seidl, and Gori-Giorgi demonstrated that this global lower bound obtained by integrating over all space cannot be converted into a local lower bound on the exchange-correlation potential xc .r/, which for large jrj decreases proportionally to 1=jrj, since such a lower bound would be violated in the outer regions of atoms or in the bond region of stretched molecules, where .r/ decreases exponentially with distance from the nuclei [202]. A further study
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of densities .r/, which challenge the Lieb–Oxford bound and thus provide lower bounds to C3 , was published by Seidl, Vuckovic, and Gori-Giorgi [203]. Generalizations of the original Lieb–Oxford bound that include integrals involving the absolute value of the gradient of and/or the gradient of fractional powers of were constructed by Benguria, Bley, and Loss [204] and by Lewin and Lieb [205]. Clearly, there is ample room for further theoretical improvements in this area, both in tightening the lower and upper bounds for C3 in the original Lieb–Oxford bound and in finding other bounds with more complicated functionals of . Developments in time-dependent density functional theory for chemical systems have been surveyed in several review articles and books [206–210]. A subtle point that was appreciated only recently is that the singularities in the electron–nucleus and electron–electron potential energies cause severe problems for the existence of Taylor series expansions in time [211, 212], which were used in the original Runge–Gross approach [213]. Acknowledgements I am grateful to Cyrus Umrigar and Michael P. Teter of Cornell University for generously providing sabbatical support in the spring of 1995. I should also like to thank them, as well as Elliott Lieb and Mel Levy, for helpful discussions. This work was supported by my National Science Foundation grant PHY-9215442.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
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J. D. Morgan III Solovej, J.P.: Mol. Phys. 114, 1036 (2016) Bragg, W.L.: Philos. Mag. 40, 169 (1920) Slater, J.C.: J. Chem. Phys. 41, 3199 (1964) Thirring, W. (ed.): The Stability of Matter: From Atoms to Stars – Selecta of Elliott H. Lieb, 3rd edn. Springer, Berlin, Heidelberg (2001) Lieb, E.H., Seiringer, R.: The Stability of Matter in Quantum Mechanics. Cambridge Univ. Press, Cambridge (2010) Kohn, W.: Rev. Mod. Phys. 71, 1253 (1999) Pople, J.A.: Rev. Mod. Phys. 71, 1267 (1999) Burke, K.: J. Chem. Phys. 136, 150901 (2012) Nalewajski, R.F. (ed.): Density Functional Theory I: Functionals and Effective Potentials. Topics in Current Chemistry, vol. 180. Springer, New York (1996) Nalewajski, R.F. (ed.): Density Functional Theory II: Relativistic and Time Dependent Extensions. Topics in Current Chemistry, vol. 181. Springer, New York (1996) Nalewajski, R.F. (ed.): Density Functional Theory III: Interpretation, Atoms, Molecules and Clusters. Topics in Current Chemistry, vol. 182. Springer, New York (1996) Nalewajski, R.F. (ed.): Density Functional Theory IV: Theory of Chemical Reactivity. Topics in Current Chemistry, vol. 183. Springer, New York (1996) Seminario, J.M. (ed.): Density Functional Theory. Advances in Quantum Chemistry, vol. 33. Academic Press, New York (1999) Koch, W., Holthausen, M.C.: A Chemist’s Guide to Density Functional Theory. Wiley, New York (2001) Fiolhais, C., Nogueira, F., Marques, M.: A Primer in Density Functional Theory. Lecture Notes in Physics, vol. 620. Springer, New York (2003) Sholl, D., Steckel, J.A.: Density Functional Theory: A Practical Introduction. Wiley, Hoboken (2009) Engel, E., Dreizler, R.M.: Density Functional Theory: An Advanced Course. Springer, Berlin, Heidelberg, New York (2011) Laird, B.B., Ross, R.B., Ziegler, T.: Chemical Applications of Density Functional Theory. ACS Symposium Series, vol. 629. American Chemical Society, Washington (1996) Dobson, J.F., Vignale, G., Das, M.P. (eds.): Electronic Density Functional Theory: Recent Progress and New Directions. Plenum, New York (1998) Joubert, D. (ed.): Density Functionals: Theory and Applications. Springer, New York (1998) Becke, A.D.: J. Chem. Phys. 140, 18A301 (2014) Yu, H.S., Li, S.L., Truhlar, D.G.: J. Chem. Phys. 145, 130901 (2016) Perdew, J.P., Ruzsinszky, A., Constantin, L.A., Sun, J., Csonka, G.I.: J. Chem. Theory Comput. 5, 902 (2009) Sun, J., Ruzsinszky, A., Perdew, J.P.: Phys. Rev. Lett. 115, 036402 (2015) Medvedev, M.G., Bushmarinov, I.S., Sun, J., Perdew, J.P., Lyssenko, K.A.: Science 355, 49 (2017) Medvedev, M.G., Bushmarinov, I.S., Sun, J., Perdew, J.P., Lyssenko, K.A.: Science 356, 496c (2017)
195. Kepp, K.P.: Science 356, 496 (2017) 196. Constantin, L.A., Snyder, J.C., Perdew, J.P., Burke, K.: J. Chem. Phys. 133, 241103 (2010) 197. Pyykkö, P., Desclaux, J.-P.: Acc. Chem. Res. 12, 276 (1979) 198. Thayer, J.S.: In: Barysz, M., Ishikawa, Y. (eds.) Relativistic Methods for Chemists Challenges and Advances in Computational Chemistry and Physics, vol. 10, pp. 63–97. Springer, Berlin, Heidelberg, New York (2010) 199. Norrby, L.J.: J. Chem. Educ. 68, 110 (1991) 200. Pyykkö, P.: Chem. Rev. 88, 563 (1988) 201. Lieb, E.H., Oxford, S.: Int. J. Quantum Chem. 19, 427 (1981) 202. Mirtschink, A., Seidl, M., Gori-Giorgi, P.: J. Chem. Theory Comput. 8, 3097 (2012) 203. Seidl, M., Vuckovic, S., Gori-Giorgi, P.: Mol. Phys. 114, 1076 (2016) 204. Benguria, R.D., Bley, G.A., Loss, M.: Int. J. Quantum Chem. 112, 1579 (2012) 205. Lewin, M., Lieb, E.H.: Phys. Rev. A 91, 022507 (2015) 206. Marques, M.A.L., Gross, E.K.U.: Annu. Rev. Phys. Chem. 55, 427 (2004) 207. Furche, F., Burke, K.: Ann. Rep. Comput. Chem. 1, 19 (2004) 208. Maitra, N.T.: J. Chem. Phys. 144, 220901 (2016) 209. Marques, M.A.L., Ullrich, C., Nogueira, F., Rubio, A., Burke, K., Gross, E.K.U. (eds.): Time-Dependent Density Functional Theory. Lecture Notes in Physics, vol. 706. (2006) 210. Marques, M.A.L., Maitra, N.T., Nogueira, F.M.S., Gross, E.K.U., Rubio, A. (eds.): Fundamentals of Time-Dependent Density Functional Theory. Lecture Notes in Physics, vol. 837. Springer, Berlin, Heidelberg, New York (2012) 211. Yang, Z.-H., Burke, K.: Phys. Rev. A 88, 042514 (2013) 212. Fournais, S., Lampart, J., Lewin, M., Sørenson, T.Ø.: Phys. Rev. A 93, 062510 (2016) 213. Runge, E., Gross, E.K.U.: Phys. Rev. Lett. 52, 997 (1984) 214. Lieb, E.H.: Int. J. Quantum Chem. 24, 243 (1983)
John D. Morgan III Dr Morgan, Associate Professor, obtained his BS from The George Washington University, his MSc in Theoretical Chemistry from Oxford University, and his PhD in Chemistry from Berkeley. He has served on the editorial boards of the Journal of Mathematical Physics and the International Journal of Quantum Chemistry. His interests include the application of sophisticated mathematical techniques to assist the accurate calculation of properties of atoms and molecules.
Atomic Structure: Variational Wave Functions and Properties Charlotte Froese Fischer
Contents
and Michel Godefroid
Abstract
22.1 22.1.1 22.1.2 22.1.3
Nonrelativistic and Relativistic Hamiltonians Schrödinger’s Hamiltonian . . . . . . . . . . . . . Dirac–Coulomb(–Breit) Hamiltonian . . . . . . . Breit-Pauli Hamiltonian . . . . . . . . . . . . . . .
22.2 22.2.1
Many-Electron Wave Functions . . . . . . . . . . . . . 311 Nonrelativistic Orbitals and LS Configuration State Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Relativistic Dirac Orbitals and jj Configuration State Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
22.2.2
22
. . . .
. . . .
. . . .
. . . .
309 310 310 310
22.3
Variational Principle . . . . . . . . . . . . . . . . . . . . 312
22.4 22.4.1 22.4.2 22.4.3 22.4.4 22.4.5 22.4.6
Hartree–Fock and Dirac–Hartree–Fock Methods . Diagonal Energy Parameters and Koopmans’ Theorem Hartree–Fock Operator . . . . . . . . . . . . . . . . . . . . Brillouin’s Theorem . . . . . . . . . . . . . . . . . . . . . The Dirac–Hartree–Fock Equations . . . . . . . . . . . . Numerical Solution of Variational Equations . . . . . . Properties of Hartree–Fock Solutions . . . . . . . . . . .
313 314 314 315 315 315 317
22.5 22.5.1 22.5.2 22.5.3 22.5.4 22.5.5
Multiconfiguration (Dirac)-Hartree–Fock Method Z-Dependent Theory . . . . . . . . . . . . . . . . . . . . The MC(D)HF Approximation . . . . . . . . . . . . . . Systematic Methods . . . . . . . . . . . . . . . . . . . . Excited States . . . . . . . . . . . . . . . . . . . . . . . . Autoionizing States . . . . . . . . . . . . . . . . . . . . .
318 318 319 320 321 321
22.6
Configuration Interaction Methods . . . . . . . . . . . 321
22.7 22.7.1 22.7.2 22.7.3 22.7.4
Atomic Properties . . . . . . . . . . . . . . . . . . Transition Data—Allowed and Forbidden Lines Electron Affinities . . . . . . . . . . . . . . . . . . Metastable States and Autoionization . . . . . . . Nuclear Effects . . . . . . . . . . . . . . . . . . . .
22.8
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
. . . . .
. . . . .
. . . . .
. . . . . .
. . . . .
324 324 324 325 326
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 C. Froese Fischer () Dept. of Computer Science, University of British Columbia Vancouver, Canada e-mail: [email protected] M. Godefroid Chemistry Department (SQUARES), Université libre de Bruxelles (ULB) Brussels, Belgium e-mail: [email protected]
This chapter describes variational methods for the determination of wave functions either in nonrelativistic (LS), quasi relativistic Breit–Pauli (LSJ ), or Dirac (jj ) theory. The emphasis is on Hartree–Fock and multiconfiguration Hartree–Fock theory with reference to similar Dirac theory. Although the underlying mathematics of the latter is technically different because of the properties of Dirac spinors, we draw parallels between the nonrelativistic and relativistic formulations that lay the foundation of, respectively, the Atomic Structure Package (ATSP) [1, 2] and the General Relativistic Atomic Structure Package (GRASP) [3, 4]. Some results from the application of these multiconfiguration methods are presented for a number of atomic properties. Although framed entirely in terms of nonrelativistic calculations, the present chapter reveals a pattern that can be, and actually is, explored in the review article of variational theory [5] as in the following Chap. 23 on Relativistic Atomic Structure. Keywords
complete active space configuration state functions Hamiltonians multiconfiguration wave functions orbital rotation quantum defect Rydberg series variational methods
22.1
Nonrelativistic and Relativistic Hamiltonians
A fully covariant relativistic many-body theory requires a field-theoretical approach, i.e., the use of quantum electrodynamics (QED) [6]. Although relativistic QED is the accepted theory describing the interaction of charged particles with the Maxwell field, it is too complicated to use for all but the simplest atomic structures. Almost all atomic structure calculations are, therefore, usually based on Hamil-
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_22
309
310
C. Froese Fischer and M. Godefroid
tonian models. For a time-independent Hamiltonian, we may where hD is the one-electron Dirac operator (shifted for the look for stationary solutions of the wave equation energy to coincide with nonrelativistic conventions), ˛ and ˇ are usual 4 4 Dirac matrices, c is the speed of light (c D .H E/ D 0 ; (22.1) 1=˛ D 137:035 999 139.31/ a: u:) according to the 2014 CODATA [10], and p D ir is the electron momentum operawhere H is the Hamiltonian operator for the system and E tor. For each electron, ˇ appearing in the relativistic singlethe total energy. The operator H depends on the sys- electron Dirac kinetic energy is replaced by ˇ 1 to align the tem (atomic, molecular, solid state, etc.) as well as the nonrelativistic rest energy of the electron, E D 0, and the relunderlying quantum mechanical formalism (nonrelativistic, ativistic one, E D me c 2 . The nuclear potential Vnuc .r/ can be Dirac–Coulomb, or Dirac–Coulomb–Breit, etc.). All these approximated as Z=r, as it is in Eq. (22.2), but this point nuHamiltonians are of the form T C V , where T refers to the clear charge model ignores the fact that real nuclei often have kinetic energy operator, and V approximates the electron– nonspherical charge density distributions that have physical nucleus and electron–electron interactions. effects (isotope shifts, hyperfine structures, etc.). A spherical average density distribution Vnuc .r/ is, therefore, frequently included when modeling heavier atoms. A more realistic nu22.1.1 Schrödinger’s Hamiltonian clear charge distribution can be a uniform nuclear charge distribution, a Fermi distribution, or other nuclear models deDeriving V from the (point) Coulomb interactions of the duced from nuclear scattering experiments (Chap. 23). classical potentials and describing the nucleus of charge Z as In Eq. (22.3), the electron–electron interaction is simply a point-charged nucleus of infinite mass, the nonrelativistic taken to be the static potential, while the full electron– Schrödinger Hamiltonian of a N -electron atom is, in atomic electron interaction should be included to approach QED. units (see Chap. 1 for their definition), The effective interaction of electrons depends of the choice of gauge potential. Their derivation in terms of the exchange X N of virtual photons leading to different gauge-dependent ver1 1 X 2 2Z ri C C ; (22.2) sions can be found in Chap. 23. In the Coulomb gauge, the Hnr D 2 i D1 ri r i F , while t F . According to Eq. (24.14), E D
Z X
el r;q>F It F
hptjV jqrihqrjV jp 0 ti .1/1C1 ; (24.18) " t "q "r C "p
The second correspondence rule is to identify sections and write down their energy denominators. After attributing to where the intermediate states are q; r > F and t F . It has each electron (vacancy) line a letter, denoting its state, the one vacancy and one electron–vacancy loop rt. An intermediate state in a diagram can be real or virtual. analytical expression for a diagram is given by It is real if the energy conservation law can be fulfilled, i.e., if for some values of the section energy the following relation Analytical Expression holds D (the product of all interaction matrix elements) X X "n "i : (24.19) ED 1 (all energy denominators) vac el
.1/L summed over all intermediate electron
If Eq. (24.19) can be fulfilled, a prescription for avoiding the (24.14) singularity in Eq. (24.13) is to substitute the expression "1 d by for Q, where where L is equal to the sum of the total number of vacancy !1 lines and closed vacancy or electron–vacancy loops. X X Although electrons are fermions, the summation in Q D lim E "n C "i C i !0 vac el Eq. (24.14) has no additional restrictions caused by the Pauli and vacancy states,
24 Many-Body Theory of Atomic Structure and Processes
DP E
X
"n C
!1 "i
vac
el
iı E
X
X el
"n C
367
X
! "i
;
(24.20)
vac
for "1 d . Here, P denotes that the principal value is to be taken on integration over intermediate state energies. The result of Eq. (24.20) can thus be complex. An intermediate state is virtual if the energy conservation law in Eq. (24.19) is violated for all values of the section energy.P In general, P the bigger the virtuality, i.e., the difference E el "n C vac "i , the smaller the contribution to the amplitude of the process.
The two interactions leading to Eq. (24.21) can be permuted, so that interaction 1 can be after interaction 2. This leads to extension of the sum to include states with q F . As a result, the summation in Eq. (24.22) must be performed over all states q. Interaction with the nucleus (nuclei in multiatomic objects) and the other electrons also affects the occupied (or vacancy) states i in Eq. (24.21), and, therefore, the latter must be modified by inserting the elements Eqs. (24.3b), (24.5), and (24.6) into them. Here again, the vacancy line in Eqs. (24.5) and (24.6) must be modified by including the corrections Eqs. (24.3b), (24.5), (24.6), and so on. Finally, the diagrammatic equation p
p =
p
q
p
+
q
+
p + i
i
24.1.4 Higher-Order Corrections and Summation of Sequences
i p
An important feature of the diagrammatic technique is the convenience in constructing higher-order corrections and in the summation of infinite sequences of diagrams. According to Eq. (24.13), each new interaction line leads to an additional interaction matrix element, an extra energy denominator, and summation over new intermediate states. An important example of infinite summation is that of determining the one-electron states. The interaction with the nucleus in atoms Eq. (24.3b) (or nuclei in multiatomic objects, e.g., molecules) and with atomic (or e.g., molecular) electrons Eqs. (24.5) and (24.6) is not small and must be taken into account nonperturbatively; i.e., these elements must be iterated infinitely. To simplify the drawing, only the element Eq. (24.5) is repeated, leading to the diagrammatic equation:
p
p
i +
+
+ q
q
q
q
(24.23)
is obtained. The doubled line for i emphasizes that the vacancy wave function is determined by an equation similar to Eq. (24.21). The corresponding analytical equation looks like Eq. (24.22), but includes also the Coulomb interaction with the nucleus and the exchange interaction with other atomic electrons. The summation over q in this equation is extended over all q, not only q > F . Multiplying the corresponding equation by .HO 0 "p / from the right (atomic units are used in this chapter: e D me D „ D 1/, whereRHO 0 D r 2 =2, and usP ing the completeness of the functions q jqihqj D ı.r r 0 /, results in the equation # " Z dr 0 r2 Z X 0 2 C ji .r /j "p p .r/ p p p q p q q' 2 r jr 0 rj = + + + ... i F 1 2 XZ dr 0 D .r 0 /p .r 0 /i .r/ ; 0 rj i jr i F i i i' (24.24) p p q + = for the electron wave function p .r/ in an atom; Z=r is the nucleus Z Coulomb potential. Here, i .r/ are wave functions determined by equations similar to Eq. (24.24). For i (24.21) multiatomic objects, instead of Z=r another potential U.r/ appears in these equations, called the Hartree–Fock (HF) equations. Indeed, everything in the infinite sum that is in front of the HF includes a part of interelectron interaction matrix eldashed line repeats the infinite sum itself, thus leading to ements, namely those given by Eqs. (24.5) and (24.6). The a closed equation of the form rest is called the residual interaction, and its inclusion leads Z beyond the HF frame, accounting for correlations. X 1 hpj Q D hpj C hpijV Q jqii hqj : (24.22) When a perturbative approach is used, it is essential to de"q C "p fine the zero-order approximation. In this chapter, and very q>F
24
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M. Y. Amusia
often in the literature, the Hartree–Fock approximation is used in this role. To simplify the drawing of diagrams, from now on single (rather than double) lines will represent electrons (vacancies), whose wave functions are determined in the HF approximation by Eq. (24.24). Obviously, in this case, elements Eqs. (24.3a), (24.5), and (24.6) should not be added to any other diagrams. The procedure used in deriving Eqs. (24.21) and (24.23) is, in fact, more general. Let us separate all diagrams describing elastic scattering that do not include a single one-electron or one-vacancy state as intermediate. Depicting their total contribution by a square, the precise one-particle state is determined by an infinite sequence of iterative diagrams that can be summed, similarly to Eq. (24.21), by p
p =
p +
q
q'
p
ˆª
+
ˆª
+…
ˆª
for one-electron and one-vacancy wave functions is used in what follows. All atomic characteristics and cross sections for atomic processes calculated with HF form the oneelectron approximation. Everything beyond the HF frame, i.e., caused by residual interaction, is called correlation corrections or correlations. They can be calculated using the many-body perturbation theory (MBPT) [4], random phase approximation (RPA) [19], and random phase approximation with exchange (RPAE) [15] or its generalized version GRPAE [15–17]. The simplest diagrammatic expression for the correlation energy is given by the two diagrams a
b
k i
q ˆª
k
j
p +…
+
j
i
n
n
(24.26)
q p =
q
p +
ˆª
.2/
p +
The analytical expression Ecorr for Eq. (24.26a) is
ˆª q
(24.25)
Here, the single line stands for an HF state. Using the correspondence rule Eq. (24.14), an analytical equation similar to the Schrödinger equation can be derived with the operator ˙O playing the role of an external potential. The essential difference is, however, that this potential in principle depends upon the energy and state of the particle. The same kind of iterative procedure leading to Eq. (24.21) or (24.23) will be used several times in this chapter. Other zero-order approximations can be chosen. But then diagrams with corrections of the type Eq. (24.3a) must be included, with the external static field potential equal to the difference between the HF and the chosen one. To calculate the numerical value of a given diagram or a sequence of diagrams one needs to know, according to the description given above, the matrix elements of external fields and interelectron interactions obtained with the help of one-electron HF wave functions. The required calculational procedures are described in [17] and [18].
24.2
.2/ D Ecorr
Z X
(24.27)
k;n>F Ii;j F
The analytical expression for Eq. (24.26b) differs from Eq. (24.27) by the sign and an exchange matrix element hk njV jj ii instead of a direct hk njV jij i one. As concrete calculations show, the contribution Eq. (24.26) overestimates the correlation energy by 10%. Diagrams Eq. (24.26) can also be used to describe the interaction potential of two atoms, designated A and B. Let the ki states belong to A and nj to B. At large distances R between the atoms, the contribution of Eq. (24.26b) is exponentially small. Because the vacancies i and j are located inside atoms A and B, respectively, the interelectron potential V D jr A r B C Rj1 at large distances R RA,B (RA,B are atomic radii) can be expanded as a series in powers of R1 . The first term giving a nonvanishing contribution to Eq. (24.27) is V ' R3 Œ.r A r B / 3.r A n/.r B n/, n being the unit vector in the direction of R. Substituted into Eq. (24.27), this potential leads to the expression
Calculation of Atomic Properties
24.2.1 Electron Correlations in Ground State Properties
hij jV jk nihk njV jij i : "i C "j "k "n
U.R/ D
C6 ; R6
(24.28)
for the interatomic potential [20], where Z X
jhijrjkij2 jhj jrjnij2 C6 : (24.29) A major advantage of the diagrammatic technique in many."i C "j "k "n / body theory is that it is usually unnecessary to know the total k;n>F Ii;j F wave function of the atom or other considered object. On the contrary, only actively participating electrons or vacan- Calculations [21] show that the inclusion of higher-order corcies appear in a diagram. The HF zero-order approximation rections is important for obtaining accurate values for Ecorr
24 Many-Body Theory of Atomic Structure and Processes
and C6 . However, to improve accuracy by taking into account the corrections to diagrams Eq. (24.26) requires considerable effort. Indeed, there are several types of corrections to Eq. (24.26) such as (i) screening of the Coulomb interelectron interaction by the electron–vacancy excitations, (ii) interaction between vacancies ij , (iii) interaction between electrons and vacancies ki.nj / .kj.ni//, and (iv) interaction between electrons k n. Corrections to the HF field itself, which acts upon electrons k; n and vacancies i; j , are discussed in Sect. 24.2.3. Screening of the Coulomb interelectron interaction is very important, and in many cases must be taken into account nonperturbatively. The simplest way to do this is to use RPA, which defines the effective interelectron interaction Q as a solution of an integral equation, shown diagrammatically by
369
RPAE corrections to ˛.!/ are discussed in Sect. 24.2.5 in connection with the photoionization process. Nondipole polarizabilities of other multipolarities can be obtained in the lowest order of interelectron interaction using Eq. (24.32) with a properly chosen interaction operator between the electromagnetic field and an electron, instead of P W D i F E r i .
24.2.2 Characteristics of One-Particle States
A single vacancy or electron can propagate from one instant of interaction to another, as described to zero order by elements Eq. (24.1b) [or Eq. (24.1a)] with dots Eq. (24.1f) at the ends. For an atom, this line represents a HF one-particle state with a given angular momentum, spin, and total momentum. k Accounting for virtual or real atomic excitations leads, for = + ~ ~ Г Г a vacancy, to a diagram similar to Eq. (24.25) but with opi (24.30) positely directed arrows. Because the interaction with these If V in Eq. (24.27) is replaced by Q , an expression for Ecorr excitations is usually much smaller than the energy distance between shells, in the sum over q only the term q D i, i bein RPA can be derived. Exchange is very important in atoms and molecules, so di- ing the considered vacancy state, need be taken into account. agram Eq. (24.30) can be modified to include this effect, thus Interaction with the vacuum leaves the angular momentum, spin, and total momentum unaltered. It can, however, change leading to the effective interaction in RPAE [15–17, 21]: the energy and lead to a finite lifetime for a vacancy state. k Analytically, the vacancy propagation in the HF approxik mation is described by the one-particle HF Green’s function = + + i Г Г i Г G HF : (24.31) (24.34) GiHF .E/ D 1=."i E/ : Replacing V in Eq. (24.27) by gives a rather accurate expression for Ecorr in RPAE. Taking into account screening Solving Eq. (24.25) for a vacancy i with only ˙i i terms inalso affects the long-range interatomic interaction consider- cluded gives ably by altering the constant C6 in Eq. (24.28). Gi .E/ D 1=Œ"i C ˙i i .E/ E : (24.35) The ground state energy of an atom or molecule is modified by an external field. For a not too intense electromagnetic field, the simplest correction to the ground state The pole in G.E/, which determines the vacancy energy, is energy is given by the diagrams shifted from E D "i to Ei D "i C ˙i i .Ei /. The quantity ˙i i .E/ is called the self-energy, and is, in general, a comk plex function of energy, its imaginary part determining the k i + lifetime of the vacancy i. Near Ei , Eq. (24.35) can be writi (24.32) ten in the form Considering a dipole external field, its interaction with the P atomic electrons is given by W D i F E r i , E being the Gi .E/ Fi =Œ"i C ˙i i .Ei / E D Fi =.Ei E/ ; (24.36) strength of the field. The ground state energy shift is given by E D ˛.!/E 2 =2, where ˛.!/ is the dynamical dipole where !1 polarizability, and ! is the frequency of the field. According ˇ ˇ .E/ @˙ i i to Eq. (24.32), ˛.!/ is determined by ˇ (24.37) Fi D 1 @E ˇEDEi Z X 2jhijzjkij2 ."k "i / ˛.!/ D ; (24.33) ."k "i /2 ! 2 is called the spectroscopic factor. It characterizes the probk>F Ii F ability for more complicated configurations to be admixed into a single vacancy state i [15, 22]. where z is a component of the vector r.
24
370
M. Y. Amusia
Along with Auger, radiative decay of vacancies takes An important problem is to calculate the self-energy part place. Its amplitude is presented by time reverse of Eq. (24.2b) ˙.E/. The first nonzero contributions are ./ diagram. The radiation width i in its HF approximation is a b given by equation i' j' i i' i +
j
j
n
k n
+ exchange terms.
./HF
i (24.38)
Specific calculations [16, 21] demonstrate that if the intermediate electron states n [in Eq. (24.38a)] and k n [in Eq. (24.38b)] are found in the field of vacancies jj 0 and i i 0 j , the diagrams Eq. (24.38) are able to reproduce the values of the correlation energy shift with about 5% accuracy. For outer subshell vacancies, the contributions Eqs. (24.38a) and (24.38b) are almost equally important, to a large extent canceling each other. For example, Eq. (24.38a) shifts the outer 3p vacancy in Ar to lower binding energies by 0:1 Ry, while the contribution of Eq. (24.38b) is 0:074 Ry. The total value 0:026 Ry is small and close to the experimental one, which is 0:01 Ry. For inner vacancies, Eq. (24.38a) is dominant because the intermediate states in Eq. (24.38b) have large virtualities and are, therefore, small. The main contribution to the sum over j 0 comes from the term j 0 D i 0 D i, which gives for the energy shift of level i Z X jhnjr 1jj ij2 .2/ : (24.39) "i D ˙i i ."i / D "n "j n>F Ij F
The value of Eq. (24.39) is positive. Most important higherorder corrections will be included if V in Eq. (24.27) is replaced by from Eq. (24.31). The physical meaning of diagram Eq. (24.38a) is transparent: it accounts for configuration mixing of one vacancy i and two vacancies jj 0 – one-electron n states in the lowest order in the interelectron interaction. Diagram Eq. (24.38b) is not as transparent, and for i D i 0 , its intermediate state appears to violate the Pauli principle. However, as noted in connection with Eq. (24.14), the Pauli principle should not be considered as a restriction in constructing intermediate states. Diagram Eq. (24.38a) and its exchange can have an imag.A/ inary part, which gives the probability of Auger decay i , calculated to the lowest order in the interelectron interaction. For Eq. (24.38a), one has .A/
i
D ImŒ˙i i .Ei / Z X jhi njV jjj 0ij2 ı."j C "j 0 "n "i / : D 2 j;j 0 F In>F
(24.40) is usually much smaller than ReŒ˙i i .Ei /, The width but there are several exceptional cases with abnormally large Auger widths, among which the most impressive is the 4pvacancy in Xe with its 4p 10 eV. .A/ i
D
1 X 3 ! jhijerjj ij2 ; c 3 i IPs , the polarizability in Eq. (24.33) is negative, while ˛.0/ > 0. This leads to a repulsive polarization inter-
action, rather than the usual attractive one. This difference affects the cross section qualitatively [25]. Another, more complicated, approximation substitutes k 0 .r 1 /k 00 .r 2 / by the product of the precise Ps wave function Ps .jr 1 r 2 j/ and the wave function of the free motion of the Ps center of gravity [26]. According to the diagrams Eq. (24.23), Eq. (24.44) describes the target with an additional electron. If the target is a neutral atom, the solution of Eq. (24.44) with discrete energy values describes negative ion states, both ground and excited ones. Again, as in Sect. 24.2.2, diagrams Eq. (24.47) with RPAE corrections form a reasonably good starting point for calculating the negative ion binding energies, even in cases when this binding is comparatively small, as in alkaline earth negative ions [27]. The inclusion of only the outer shell polarizability (j is a vacancy in the outer shell) leads to overbinding of the additional electron forming the negative ion. Only the inclusion of screening due to inner shell excitations yields good agreement with experiment. For instance, measurements of Ca affinity [28] give about 20 meV, while the calculations without inner shell excitations give about 50 meV [27]. Their inclusion must reduce the theoretical value considerably.
24.2.4 Two-Electron and Two-Vacancy States One can construct a diagrammatic equation for the wave function of a two-electron or a two-vacancy state by separating all diagrams describing two-electron (two-vacancy) scattering that do not include these states as intermediate, and denoting their total contribution by a circle. Then the exact two-electron (vacancy) state is determined by the infinite sequence of diagrams k
kk' =
k
k' k ˆ
k
k2 ˆ
+
k'
+ k'
k1 ˆ
+
k'1
k1 ˆ
+… k'1
k'2
k'
+…
i i'
k =
kk' +
k1 ˆ k'1
k'
kk'
ˆ
+ i
i'
(24.49)
The analytic equation for two electrons in an atom or any multiatomic system, interacting with each other can be written in the form HO 1HF C ˙O 1 C HO 2HF C ˙O 2 C QO ˘O 12 "
12 .r 1
r2/ D 0 :
(24.50)
24 Many-Body Theory of Atomic Structure and Processes
373
HF Here, HO 1.2/ is the HF part of the one-particle Hamiltonian in radiative Eq. (24.52b) decay rates. For inner shells, a speEq. (24.24), ˘O 12 is the effective interelectron interaction, and cific feature of such processes is that the released energy is about twice that for a single vacancy decay. Of course, in QO is the projection operator these cases, the decay probability is relatively small. It is not, however, necessarily much smaller than that of individual vaQO .1 n1 /.1 n2 / n1 n2 D 1 n1 n2 ; (24.51) cancies if the energies of the intermediate and initial states are close to each other. with n1.2/ being the Fermi step function, n1.2/ D 1 for 1.2/ The presence of residual two-body forces leads to effecF , and n1.2/ D 0 for 1.2/ > F . The function n1.2/ thus tive multiparticle interactions. The simplest diagram presenteliminates contributions of vacant states. The operator QO ing a three-electron interaction is given by takes into account the fact that propagation of two electrons (or, more precisely, quasi electrons due to the presence of n1 n'1 ˙O ) takes place in a system of other particles,which occupy n2 n'2 all levels up to the Fermi level. The presence of QO makes n3 n'3 (24.53) Eq. (24.50) essentially different from a simple two-electron O Schrödinger equation: Q requires that after each act of interaction described by ˘O 12 , both participants either remain The role of multiparticle interactions in the atomic structure is far from being clear. Diagrams similar to Eq. (24.53) are electrons or become vacancies. Diagrams Eq. (24.49) describe the states of two electrons important if it is of interest to calculate the energy levels of outside the closed shell core, or electron scattering by an atoms with three or more electrons (or vacancies) outside of atom with one electron outside the closed shells. In gen- closed shells. This is a very complicated calculation because eral, ˘O 12 is nonlocal, dependent upon ", and can have an even two electrons in the Coulomb field of the nucleus are imaginary part. The lowest-order approximation to ˘O 12 is a difficult three-body problem. V D jr 1 r 2 j1 . Equation (24.50) must be solved in order to obtain, for example, the excitation spectrum of two electrons in atoms with two electrons outside closed subshells, such as 24.2.5 Electron–Vacancy States Ca. Instead of V , the interaction from Eq. (24.31) can be used, which would account for screening due to virtual ex- The one-electron–one-vacancy state is the simplest excitacitation of inner shell electrons. For low-level two-electron tion of a closed shell system under the action of an external excitations, the energy dependence of ˘O 12 can be neglected. time-dependent field, represented by Eq. (24.2a). Beginning The same type of equation can be obtained for two- with the electrons and vacancies described in the HF approxivacancy states, describing their energies, decay widths, and mation, the result of residual interactions leads to excitations structure due to configuration mixing with more complex of more complex states, including those with two or more states. For inner and intermediate shell vacancies in atoms, electron–vacancy pairs. The interaction can also lead to a sinhowever, the corresponding corrections can be taken into ac- gle electron–vacancy pair. Let us concentrate on the latter count perturbatively, and the screening by outer shells is not case and separate all diagrams describing electron–vacancy essential. The admixture of outer shell excitation with inner interaction that do not include these states as intermediate vacancies is important, leading to so-called satellites of the and denote them by a circle. Then the exact electron–vacancy state is determined by the infinite sequence of diagrams main spectral lines. The interaction between vacancies leads to correlated k'' k' ki k k' k k two-vacancy decay processes in which the energy is carried = + +… + R R R away by a single electron or photon. Some diagrams exi i' i i' i i'' emplifying these processes in the lowest possible order of k interaction are
+
a
i1
b
j1
i
R k'
j1
i1
i'
i2
q
j3 ε
i2
j2
q
+
= i
φ
(24.52)
k' +
ki
k j2
+…
R i'
ki R k' i'
(24.54)
Contrary to the electron–electron case in Sect. 24.2.4, the Even in this order, there are several diagrams that together give the amplitude of the correlation Auger Eq. (24.52a) and analytical expression corresponding to Eq. (24.54) cannot be
24
374
M. Y. Amusia
in Eq. (24.19) must be multiplied by the square modulus of the ratio hkjD.!/jii=hkjd jii, with ! D "k C Ii . The RPAE corrections described by the term in round brackets in Eq. (24.57) are very large for outer and intermediate electron shells. Through the sum over i 0 , if only terms with the same energies "i 0 D "i are included, Eq. (24.57) accounts for intrashell correlations. By adding terms with "i 0 ¤ "i , the effect of intershell correlations is taken into account. ./RPAE k k' One can obtain the radiative width in RPAE i φ from Eq. (24.41), substituting hijd jj iby hijD.!j i /jj i. i' i (24.55) For atoms, Eq. (24.57) has to be solved numerically but can be presented in a symbolical operator form that creates The amplitude for elastic photon scattering is also expressed the possibility of qualitative analyses of its solutions via the exact electron–vacancy state, determined by (24.54) to be D.!/ D d C D.!/.!/U ; (24.58)
represented as a Schrödinger-type equation. Indeed, being symmetric under time reversal, Eq. (24.54) leads to an equation depending, unlike Eq. (24.50), upon the second power of the electron–vacancy energy !. Note that R in Eq. (24.54) and ˘O in Eq. (24.49) are different. It is necessary to solve Eq. (24.54) when calculating the photoabsorption amplitude, which can then be represented as
k'
k
where U is a combination of the direct Vd and exchange Ve Coulomb interelectron potentials, U D Vd Ve , .!/ D i' i (24.56) .!/ C .!/, .!/ D 1=.! ! 0 C i/, and .!/ D 1 2 1 2 1=.! C ! 0 /, with ! 0 being the excitation energy of the virThe other case where it is necessary to solve Eq. (24.54) tual electron–vacancy state. Thus, .!/ D 2! 0 =.! 2 ! 02 C is the scattering of electrons, both elastic and inelastic, by i/. Using from Eq. (24.31), one can present D.!/ as atoms with a vacancy in their outer shell, such as halogens. The simplest approximation to R is given by the Coulomb D.!/ D d C d.!/ .!/ : (24.59) interaction to lowest order Eqs. (24.4e) and (24.4f). With such R, the sequence of diagrams Eq. (24.54) is the same as Equation (24.58) allows a rather simple, also symbolic, soluEq. (24.31) and, thus, forms the RPAE, which is often used tion to describe photoionization and other atomic processes [15, D.!/ D d=Œ1 .!/U : (24.60) 16]. Both terms Eqs. (24.4e) and (24.4f) contribute to the electron–vacancy in Eq. (24.54) only if the external field is If the denominator in Eq. (24.60) has a solution ˝ deterspin independent, such as an ordinary photon. For a mag- mined by the equation netic interaction, which is proportional to spin, the term Eq. (24.4e) does not contribute. 1 .˝/U D 0 ; (24.61) φ
φ
24.2.6 Photoionization in RPAE and Beyond In RPAE, the photoionization amplitude hkjD.!/jii is determined by solving an integral equation obtained from Eqs. (24.54) and (24.55) using the correspondence rule Eq. (24.14) [15, 16] hkjD.!/jii D hkjd jii 0 Z X hk jD.!/ji 0 ihki 0 jV jik 0 k 0 ii C ! "k 0 C "i 0 C i k 0 >F Ii 0 F hi 0 jD.!/jk 0 ihkk 0 jV ji i 0 i 0 ii ; (24.57) ! C "k 0 "i 0 i where d is the dipole operator describing the photon– electron interaction, and V D jr 1 r 2 j1 . To obtain the RPAE photoionization cross section, the usual expression
at ˝ > I , where I is the atomic ionization potential, .˝/ > 0, then the cross section has a powerful maximum called a giant resonance with energy ˝. A giant resonance is of collective nature in the sense that it appears to be due to coherent virtual excitation of all electrons of at least one considered multielectron subshell. These intrashell correlations are most important for multielectron shells with large photoionization cross sections. Their inclusion leads to a quantitative description of the above-mentioned giant resonances – huge maxima in the photoionization cross sections. An example is the 4d10 photoionization cross section of Xe shown in Fig. 24.2 [16], where satisfactory agreement with experiment is demonstrated. In [16], one can find a number of examples of other Giant resonances. It appears that all RPAE intrashell time-forward diagrams, such as that on the first line in Eq. (24.54), and the first term in brackets in Eq. (24.57) with "i 0 D "i , may be taken Q jii. This is the into account by the matrix element hkjd
24 Many-Body Theory of Atomic Structure and Processes
375
ση (Mb)
σ (Mb)
30
1.0
20 0.5 10
0
0 4
5
6
7
8
9
10
11 φ (Ry)
2
4
6
8
10 φ (Ry)
Fig. 24.2 Photoabsorption in the vicinity of the 4d10 subshell threshold in Xe [8]. Solid line: RPAE; dashed line: experiment
Fig. 24.3 Photoionization of 5s2 electrons in Xe [8]. light brown solid line: RPAE with effects of 5p6 and 4d10 included; dotted line: 5s2 electrons only; grey solid line: with effect of 5p6 electrons; dark brown solid line: with effect of 4d10 electrons; black dashed line: experiment
one-electron approximation but with the function Qk .r/ calculated in the term-dependent HF approximation [15–18]. Term-dependency means that only the total angular momentum and spin and their projections for the electron–vacancy pair are conserved, being equal to that of the incoming photon. The individual values for the electron and vacancy angular momentum and spin are not considered to be good quantum numbers. Thus, the term-dependent HF includes a large fraction of RPAE correlations. RPAE allows the calculation of interference resonances. To describe them, let us consider a situation in which the direct HF amplitude ds is small, while there are other electrons with large photoionization amplitude Db ; Db .!/ ds . Then, from Eq. (24.58) one has
minimum is not seen in Fig. 24.3, since it lies at considerably higher !. RPAE is able to describe a number of other effects, such as giant autoionization resonance (decay of a powerful discrete excitation into a continuum, with which the excitation interacts strongly), continuous spectrum autoionization (modification of a broad continuous spectrum excitation due to its strong interaction with a narrow continuum that occurs in negative ions [27]), and quadrupole giant resonances [29]. Above we concentrated on dipole giant resonances. Quadrupole amplitudes in RPAE are determined by an equation similar to Eq. (24.58): Q.!/ D q C Q.!/.!/U ;
(24.63)
Ds .!/ ds C Db .!/.!/Ubs (24.62) where q is the quadrupole amplitude in HF approximation. Giant quadrupole resonance was found in excitations of if the intertransition interaction Ubs is not too small. The 4d6 electrons in Xe [30]. Its direct observation in photoabenhancement of the photoionization amplitude described by sorption is almost impossible, since the corresponding cross Eq. (24.62) manifests itself as a resonance in the partial cross section is very small due to the inclusion of the extra factor section of s electrons photoionization. Very often, the term ˛ 2 D 1=c 2 104 as compared to the dipole cross section. Db .!/.!/Ubs and ds are of opposite sign, so that the total However, the quadrupole amplitude leads to noticeable coramplitude acquires two minima, along with an extra max- rections to the angular distributions of photoelectrons where imum, thus forming a rather complicated structure in the their relative contribution is considerably bigger [16]. partial cross section, which was named interference or corNote that the amplitude of electron elastic scattering on relation resonance. an atom with a vacancy is expressed in RPAE via given by Usually, these resonances are manifestations of intershell Eq. (24.31) [31]. correlations. These are taken into account if the sum over i 0 For the inner or deep intermediate shells, RPAE proves to in Eq. (24.57) includes terms with "i 0 ¤ "i . An example is be insufficient. First, screening of the Coulomb interaction the 5s2 subshell in Xe, which is strongly affected by the outer between the outgoing or virtually excited electron and the va5p6 and inner 4d10 neighboring electrons. Due to this interac- cancy [see Eq. (24.4f)] must be taken into account. This can tion, the 5s2 cross section is completely altered, as illustrated be done by replacing V with from Eq. (24.31). The ionin Fig. 24.3 [15, 16]. The RPAE results predict a qualita- ization potential (or the energy of the vacancy i) must also tive feature of the experimental data, namely the formation of be corrected, which requires inclusion of at least the contria maximum and two minima in the cross section. The second bution from the first term of Eq. (24.38). It has been demon Db .!/.!/Ubs ds ;
24
376
a
M. Y. Amusia
Cross section, Mb 20
Cs3d
15
ns .!/ D 6ŒjQns .!/j=jDns.!/j cos. q d / ; (24.64)
10
5
0 720
b
730
740
750
760 770 Photon energy, eV
730
740
750
760 770 Photon energy, eV
β 2
1
0
–1
Figure 24.5 depicts the nondipole angular anisotropy parameter 5s for 5s electrons in Xe [34]. The parameter ns (in Fig. 24.5, n D 5) is given by the simple formula [21]
where Qns .Dns / are the RPAE (GRPAE) quadrupole (dipole) photoionization amplitudes, and q . d / are their phases. Thus, ns .!/ is sensitive to the presence of interference, dipole, and quadrupole resonances. The latter is presented by a small but noticeable maximum on the high energy slope of the huge maximum, caused by the presence of the giant dipole resonance. The variation of 5s near the 5s threshold is determined by the resonant behavior of cos. q d /, called phase resonance [29]. Close to inner shell thresholds, the Auger decay of a deep vacancy must be taken into account. Due to decay, the photoelectron instantly finds itself in the field of at least two vacancies instead of one, leading to considerable growth of the threshold cross section. Diagrammatically, the effect of decay may be described by k'
720
Fig. 24.4 Intradoublet resonance in 3d10 Cs. (a) Partial photoionization cross sections 5=2 (left peaks) and 3=2 (right peaks) [33]. Dashed curves represent the HF approximation. (b) Dipole angular anisotropy parameters ˇ5=2 and ˇ3=2 . Solid red line: data for 5=2 with account of 3=2; dashed red line: data for 5=2 without account of 3=2; solid black line: data for 3=2 with account of 5=2; dashed black line: data for 3=2 without account of 5=2
i
j1 j2 k2
(24.65)
Here, the double line emphasizes that starting from the moment of decay; the photoelectron moves in the field j1 j2 of a double instead of a single i vacancy. For inner vacancies, this is a strong effect, which can lead even to recapture of the photoelectron into some of the discrete levels in the field of the double vacancy j1 j2 . The instant change of the outgoing electron wave function due to rapid vacancy decay is called postcollision. This effect in photoionization can be taken into account by the diagram Eq. (24.65) when the Auger electron is much faster than the photoelectron [35]. If their speed is of the same order, their mutual Coulomb repulsion must be accounted for, leading to additional alteration of energy and angular redistributions. A photoelectron can excite or knock out another atomic electron. To lowest order in the residual interelectron interaction, this process can be represented by the diagrams Eqs. (24.66) and (24.67)
strated [32] that the screening of the electron–vacancy interaction can be taken into account by calculating the wave function of the virtually excited or outgoing electron in the selfconsistent HF field of an ion instead of that of a neutral atom. A method that uses only these one-particle wave functions in Eqs. (24.55) and (24.57) is called the generalized RPAE or GRPAE [16]. The use of this approximation improves the agreement with experiment near the intermediate shell thresholds considerably, decreasing the cross section value there and shifting its maximum to higher energies. GRPAE reveals intradoublet resonances that result from the interaction of electrons belonging to two components of the spin–orbit doublet, e.g., 3d3=2 and 3d5=2 in Xe, Cs, and Ba atoms [29, 33]. RPAE and GRPAE corrections do not only affect the cross sections but also the characteristics of the photoi2 a b k2 electron angular distribution, i.e., dipole and nondipole ani2 k2 gular anisotropy parameters [15, 21, 29]. As an example, k1 k k1 Fig. 24.4a presents the partial cross sections [33], while i Fig. 24.4b depicts the dipole anisotropy parameter ˇ [29] for i1 i1 (24.66) 3d5=2 and 3d3=2 electrons in Cs. The effect of intradoublet resonance – an additional maximum in the 3d5=2 cross secWhile the formation of an initial electron–vacancy ki1 tion under the action of 3d3=2 electrons – can be seen clearly. pair requires RPAE or GRPAE for its description, the sec-
24 Many-Body Theory of Atomic Structure and Processes
377
η
1
0.5
0 HF –0.5
24
RRPA RPAE –1 50
100
150
200 Photon energy, eV
Fig. 24.5 Nondipole anisotropy parameter 5s .!/ of 5s2 electrons in Xe [34]
ond step of Eq. (24.66a) can be reproduced reasonably well by the lowest-order term in V . It appears that process Eq. (24.66a) has a high probability of not too fast photoelectrons, changing the cross section [36] considerably for ionization by creating a vacancy i1 . Comparatively simple diagrams a
k i
i j n
b
k i j n
(24.67)
describe the photoionization process in which a more complicated state is created in the ion than a single vacancy, e.g., a state with two vacancies and one electron. Here, the double arrow indicates that the electron is in a discrete level n. If one considers cases when n also belongs to a continuous spectrum, the diagrams (Eqs. (24.65)–(24.67)) present in the lowest order the amplitude of two-electron–one-photon ionization. Usually, the contribution of this process to the total photon absorption cross section is small, but it contains very important information on the role of interelectron correlations in atoms. Diagrams (Eqs. (24.65)–(24.67)) present some corrections that mix electron–vacancy and two-electron–twovacancy configurations. Each additional interaction line increases the number of possible physical processes considerably. With the growth of the number of particles actively participating in a process, the calculational difficulties increase enormously. However, this is not a shortcoming of the diagrammatic approach but a specific feature of more and more complex physical processes.
Most prominent are multielectron effects in photoionization of outer and intermediate subshells and close to the inner thresholds of intermediate and heavy atoms. However, even at very high, although nonrelativistic, energies, the role of correlations is important. For example, the ionization probability for a dipole satellite to a l > 0 level decreases with the photon energy; ! grows as ! 7=2 , which is much slower than for the main line itself: ! 7=2l . Correlations alter the asymptotic behavior of the cross section of all subshells with l > 1 that acquire a universal behavior ! 9=2 instead of the HF value ! 7=2l [37]. Note that inelastic scattering of any fast charge particle can be studied and described as mentioned above, similarly to photoionization if one substitutes the dipole operator d in Eq. (24.57) for 4 exp.iqr/=q 2 , where q is the momentum transferred from the projectile to the target [20]. Fast particle scattering as a tool to study a target’s structure has some advantages as compared to photoionization [38].
24.2.7 Photon Emission and Bremsstrahlung The amplitude of photon emission in lowest order is given by the time-reversal of Eq. (24.2c): k
k' φ
(24.68)
This diagram represents ordinary Bremsstrahlung, i.e., a photon emission in the process of projectile deceleration in the field of the target. If the target has an internal structure like atoms, it can be really or virtually excited during the collision process. The simplest excitation means the creation of
378
M. Y. Amusia
an electron–vacancy pair. The annihilation of this pair results Eq. (24.69), for instance those that include simultaneous via the time-reversal of Eq. (24.2a) in photon emission. The photon emission and target excitation (ionization) [40, 41]. More general mechanisms of PR, other than those depicted process thus looks like in Eq. (24.69), are discussed in [42]. k
k'
k'' φ
i
(24.69) 24.2.8
To obtain the total Bremsstrahlung amplitude, the terms in Eqs. (24.68) and (24.69) must be summed. The polarization radiation (PR) created by the mechanism Eq. (24.69) has a number of features that are different from the ordinary Bremsstrahlung (OB) represented by Eq. (24.68). The intensity is proportional to 1=Mp2 for OB, where Mp is the projectile mass, and the spectrum, at least for high "k , is proportional to 1=!. On the other hand, the PR intensity is almost completely independent of Mp , and its frequency dependence is quite complex, being determined by the target polarizability ˛.!/ [39]. PR is most important for frequencies ! of the order of and higher than the target’s ionization potential. At sufficiently large distances and for neutral targets, PR starts to predominate over OB. Close to discrete excitations of the k 0 electron, the contribution Eq. (24.69) becomes resonantly enhanced. Higher-order corrections are important in the PR amplitude. First, the Coulomb interaction V in Eq. (24.69) must be replaced by from Eq. (24.31). The analytical expression for the total Bremsstrahlung amplitude, including RPAE corrections to Eq. (24.69), is given by the expression hkjA.!/jk 0 i 0
D hkjd jk i C
Z X
hkijV jk 0 k 00 i
k 00 >F Ii F
2."k 00 "i / hijD.!/jk 00 i : ! 2 ."k 00 "i /2 C i! (24.70) To derive the Bremsstrahlung spectrum, the usual general expression must be multiplied by the square modulus of the ratio hkjA.!/jk 0 i=hkjd jk 0 i. If the incoming electron is slow, corrections Eq. (24.47) also become important. The intermediate state in Eq. (24.47a) includes two electrons k 0 and k, and a vacancy i. The extent of the interaction between them could be considerable. An important feature of PR is that it is nonzero even if the projectile is neutral but is able to polarize the target. For example, it leads to emission of continuous spectrum radiation in atom–atom collisions, whose intensity for frequencies of the order of the ionization potentials is close to that in electron–atom collisions. The second and higher orders in the residual interaction involve processes that are more complicated than lim
!C0
RPAE in the Magnetic Channel
Above, we considered the absorption and emission of pure electric photons that do not interact with the spin of electrons. So, the total spin of excitation is S D 0. Of interest is RPAE in the magnetic channel with S D 1. The respective channels do not mix if the spin of an electron (vacancy) in the system under consideration is a good quantum number. The RPAES D1 is obtained from Eq. (24.57) by the contribution of direct terms in the V matrix element only Eq. (24.4f) and discarding Eq. (24.4e). Thus, the RPAES D1 amplitude hkjH .!/jii is determined by the following equation hkjH .!/jii D hkj jii Z 0 X hk jH .!/ji 0 ihki 0 jV jk 0 ii ! "k 0 C "i 0 C i k 0 >F Ii 0 F hi 0 jH .!/jk 0 ihkk 0 jV ji 0 ii ; ! "k 0 "i 0 i
(24.71)
where is the operator describing the interaction of an electron with the magnetic field. Just like Eq. (24.57), this equation can be presented in a symbolical operator form that creates the possibility of qualitative analyses of its solutions: H .!/ D ! C H .!/.!/Ve ;
(24.72)
where Ve is the exchange Coulomb interelectron potential Eq. (24.4f), .!/ D 1 .!/C2 .!/, 1 .!/ D 1=.! ! 0 Ci/, and 2 .!/ D 1=.! C ! 0 /, with ! 0 being the excitation energy of the virtual electron–vacancy state in the S D 1 channel. Equation (24.72) allows a rather simple, also symbolic, solution (24.73) H .!/ D ! =Œ1 C .!/V : e
Here, .!/ D 2! 0 =.! 2 ! 02 /. For ! < I1 , where I1 is the lowest possible excitation energy in the system under consideration, .!/ < 0, and according Eq. (24.73), the magnetic excitation enhances. If Œ1 C .0/Ve < 0, the system under consideration undergoes a phase transition, since S D 1 electron–vacancy pair creation proceeds without a supply of external energy. This is why open subshell systems have the maximum possible, compatible to the Pauli principle, spin, i.e., they obey the so-called Hund rule.
24 Many-Body Theory of Atomic Structure and Processes
24.2.9 Consideration of “Big Atoms” The theory presented can be applied to any many-body system. However, each object has some specific features. In the previous sections of this chapter, we concentrated on atoms. In addressing molecules, clusters, fullerenes, and endohedrals one can use two approaches. The first is “brutal force” or the first-principle approach. This approach employs the technique presented above where the action of all nuclei upon electrons is described by the potential U.r/ X (24.74) Zi =jr R i j ; U.r/ D
379 Cross section (Mb) 70 Xe free Xe@C60
60 50 40 30
4d Xe
20 10
24
All nuclei i
where R i is the location of the i-th nucleus. One-electron wave functions must be calculated using the Eq. (24.74) expression for U instead of Z=r, as in Eq. (24.57). Then all the techniques presented above can be applied directly. However, this is a difficult approach that requires solution of three-dimensional multicenter equations. For all but the simplest molecules, this approach is beyond the capabilities of modern computers. Simplifications are necessary to decrease the computational complexity for properties and processes in multiatomic formations. However, the so-called “big atom” [12] or endohedral, which is a fullerene CN stuffed by an atom A (denoted as A@CN ), permits essential simplification, making them only a little bit more complex than ordinary atoms. As one can read in a number of sources (e.g., [13]), a potential that acts upon an electron in an endohedral can be taken into account by adding to Eq. (24.57) UF .r/, which is, e.g., a square well of depth U0 , radius RF , and width b. This allows one to describe confinement resonances, which are manifestations of resonances in reflection of the photoelectron wave by UF .r/. This reflection strongly affects the encapsulated atom photoionization cross section. The Giant resonance of Xe 4d10 subshell, when Xe is caged in fullerene C60 , serves as an example. As a result the curve Fig. 24.2 is completely modified, as can be seen in Fig. 24.6 [43]. The fullerene electron shell is an object with a big polarizability [16]. This polarizability affects photon absorption in the polarizability maximum region. The treatment is essentially simplified in many cases, when, with good accuracy, one can consider RF b and RF ra , where ra is the atom’s A radius. The effects of the static fullerene shell potential and its polarizability lead to giant endohedral resonances, prominent maxima in the inner atom photoionization cross section. This effect is illustrated by the cross section of the 5p subshell of endohedral Xe@C60 depicted in Fig. 24.7 [44, 45]. The presence of a fullerene shell in A@C60 opens up a number of new channels in endohedral processes as com-
0
80
100
120
140 160 Photon energy (eV)
Fig. 24.6 Photoionization cross section of 4d-electrons for the Xe free atom and Xe@C60 in RPAE Cross section (Mb) 600
Xe free Xe@C60
500
Xe@C60 FRPAE
400 300 5p Xe 200 100 0
15
20
25 Photon energy (eV)
Fig. 24.7 Photoionization cross section of 5p-electrons for the Xe free atom in RPAE and Xe@C60 in RPAE and with account of fullerene shell polarization – FRPAE
pared to the atomic ones, e.g., in multielectron ionization and vacancy decay. In both of these channels, the interaction between atomic and fullerenes’ electrons is important. For example, the subvalent vacancies in noble gases decay only radiatively, since the decay energy is smaller than the outer electrons’ ionization potential. The presence of fullerenes’ electrons opens the Auger decay channel for subvalent vacancy [16, 38],
.A/A@CN if
D
./A
c 4 CN .!if / 3 ; 8 !if RF6
(24.75)
380
M. Y. Amusia
24.3
Phase (Rad) 9 8 7
p-wave
6 5 4 3 2 1 0
0
0.1
0.2
0.3
0.4
0.5 0.6 0.7 Electron energy (Ry)
Elastic cross section (a20) 1000 e/Ne, RPAE e/C60, Hartree
800
e/Ne@C60, RPAE 600
e/Ne@C60, RPAEF e/Ne@C60, RPAEFA
400 200 0 0
0.1
0.2
0.3
0.4
0.5 0.6 0.7 Electron energy (Ry)
Fig. 24.8 Phases and cross sections of the p-wave contribution in electron collisions with Ne@C60 . Index F marks the inclusion of fullerenes’ polarization potential, while FA denotes the mutual influence of Ne and C60 polarizations
Concluding Remarks
It is most convenient to apply diagrammatic techniques to closed shell objects whose ground state is nondegenerate. Degeneracy means that some of the energy denominators in Eq. (24.13) become zero with nonzero statistical weight. All such contributions must be summed to eliminate this degeneracy. This leads to strong mixing of some states. For example, the energy required for electron–vacancy transitions j n [see Eq. (24.38)] within an open shell is zero, thus leading to strong mixing of i and ij n states. If a pair with zero excitation energy has nonzero angular momentum, taking into account the mixing within such a pair destroys angular momentum as a characteristic of a one-vacancy state. This makes all calculations much more complicated, reflecting a specific feature of the degenerate physical system. In using the diagrams and formulas presented above, it is essential that the interelectron and electron–nucleus interactions be purely potential. Inclusion of retardation and spin-dependence in the interparticle interaction makes the calculations much more complicated. These parts of the interaction appear as relativistic corrections. They are comparatively small in all but the heaviest atoms and can be taken into account perturbatively. Beyond lowest order, these additional interactions are strongly altered when virtual excitations of electron–vacancy pairs and the Coulomb interaction between them is taken into account. An example is given by the sequence of diagrams k1
k'1
k2
k'2
k1
+
+
k1
k'1
k'2
+…
k'2
k2
.A/A@C
N where if is the width of the noble gas endohedral subvalent i vacancy decay via electron transition i ! j into the outer shell vacancy j , c is the speed of light, ./A is the pure atomic radiative decay probability, and CN .!if / is the endohedrals photoionization cross section at frequency !if . The width in Eq. (24.75) increases by four to six orders of magnitude, as compared to ./A . The presence of A in A@C60 strongly alters the lowenergy e C A@C60 cross section as compared to that of e C C60 , in spite of the fact that fullerene C60 is much bigger than the atom A. The quantum mechanical nature of this process is also reflected in a multitude of scattering resonances, maxima, and minima, including Ramsauer minimum in higher than l D 0 scattering waves, which is exemplified by Fig. 24.8 [46]. Of interest is the scattering eC C A@C60 that differs from e C A@C60 not only due to the lack of exchange but also due to effects of target polarization, which are different for e C A@C60 and eC C A@C60 processes [47].
k'1
k2
(24.76)
where the heavy dashed line stands for the spin-dependent interelectron interaction. Note that here there are no electron– vacancy loops as in Eq. (24.38) because the Coulomb interaction is unable to affect the electron spin and thus to transfer spin excitations. The same kind of diagrams describe the one-particle field acting upon an electron or vacancy due to the presence of spin–orbit interaction or weak interaction between electrons and the nucleus. For instance, the effective weak potential includes contributions from the sequence Wi
k1
Wi
k'
+ k
Wi k' +
k' + … k
(24.77)
This is another example demonstrating how flexible and convenient the many-body approach is for considering different processes and interactions.
24 Many-Body Theory of Atomic Structure and Processes
References 1. Feynman, R.P.: Quantum Electrodynamics. Benjamin, New York (1961) 2. Bethe, H.A., Goldstone, J.: Proc. R. Soc. Lond. A 238, 551 (1957) 3. Mattuck, R.D.: A Guide to Feynman Diagrams in the Many-Body Problem, 2nd edn. Dover Books on Physics. (2012) 4. Kelly, H.P.: In: Brueckner, K.A. (ed.) Advances in Theoretical Physics, vol. 2, pp. 75–169. Academic Press, New York (1968) 5. Amusia, M.Y.: Many-Body Effects in Electron Atomic Shells. A. F. Ioffe Physical-Technical Institute Publications, Leningrad, pp 1–144 (1968). in Russian 6. Amusia, M.Y.: X-Ray and inner-shell processes. In: Johnson, R.L., Schmidt-Böking, H., Sonntag, B.F. (eds.) AIP Conf. Proc., vol. 389, pp. 415–430. AIP Press, Woodbury (1997) 7. Amusia, M.Y., Connerade, J.-P.: Rep. Prog. Phys. 63, 41 (2000) 8. Amusia, M.Y.: Phys. Essays 13, 444 (2000) 9. Amusia, M.Y.: In: Konjevich, N., Petrovich, Z.L., Malovich, G. (eds.) The Physics of Ionized Gases, pp. 19–40. Institute of Physics, Belgrade (2001) 10. Cherepkov, N.A., Semenov, S.K., Hikosaka, Y., Ito, K., Motoki, S., Yagishita, A.: Phys. Rev. Lett. 84, 250 (2000) 11. Ipatov, A.N., Ivanov, V.K., Agap’ev, B.D., Ekardt, W.: J. Phys. B At. Mol. Opt. Phys. 31, 925 (1998) 12. Amusia, M.Y.: Chem. Phys. 414, 168 (2013) 13. Dolmatov, V.K.: Photoionization of atoms encaged in spherical fullerenes. Adv. Quantum Chem. 58, 13–68 (2009) 14. March, N.H., Young, W.H., Sampanthar, S.: The Many-Body Problem in Quantum Mechanics. Cambridge Univ. Press, Cambridge (1967) 15. Amusia, M.Y.: Atomic Photoeffect. Plenum Press, New York (1990) 16. Amusia, M.Y., Chernysheva, L.V., Yarzhemsky, V.G.: Handbook of Theoretical Atomic Physics, Data for Photon Absorption, Electron Scattering, Vacancies Decay. Springer, Berlin (2012) 17. Amusia, M.Y., Chernysheva, L.V.: Computation of Atomic Processes. Institute of Physics Publishing, Bristol-Philadelphia (1997) 18. Amusia, M.Y., Semenov, S.K., Chernysheva, L.V.: ATOM-M Algorithms and Programs for Investigation of Atomic and Molecular Processes. Nauka, Sankt-Peterburg (2016) 19. Pines, D.: The Many-Body Problem. Benjamin, New York (1961) 20. Landau, L.D., Lifshits, E.M.: Quantum Mechanics. Pergamon, Oxford, pp 319–322 (1965) 21. Amusia, M.Y., Cherepkov, N.A.: Case Stud. At. Phys. 5, 47 (1975) 22. Migdal, A.B.: Theory of Finite Fermi-Systems and Applications to Atomic Nuclei. Interscience, New York (1967) 23. Johnson, W.R., Guet, C.: Phys. Rev. A 49, 1041 (1994) 24. Amusia, M.Y., Cherepkov, N.A., Chernysheva, L.V.: JETP 124, 1 (2003) 25. Zhou, S., Parikh, S.P., Kauppila, W.E., Kwan, C.K., Lin, D., Surdutovich, A., Stein, T.S.: Phys. Rev. Lett. 73, 236 (1994) 26. Gribakin, G.F., King, W.A.: Can. J. Phys. 74, 449 (1996) 27. Ivanov, V.K.: J. Phys. B At. Mol. Opt. Phys. 32, R67 (1999)
381 28. 29. 30. 31. 32.
33. 34.
35. 36. 37. 38. 39. 40. 41.
42. 43. 44. 45. 46. 47.
Walter, C.W., Peterson, J.R.: Phys. Rev. Lett. 68, 2281 (1992) Amusia, M.Y.: Radiat. Phys. Chem. 70, 237 (2004) Johnson, W., Cheng, K.: Phys. Rev. A 63, 022504 (2001) Amusia, M.Y., Sosnivker, V.A., Cherepkov, N.A., Chernysheva, L.V., Sheftel, S.I.: J. Tech. Phys. 60, 1 (1990). in Russian Amusia, M.Y.: In: Becker, U., Shirley, D. (eds.) Photoionization in VUV and Soft X-Ray Frequency Regions, pp. 1–46. Plenum, New York (1996) Amusia, M.Y., Chernysheva, L.V., Manson, S.T., Msezane, A.Z., Radoevich, V.: Phys. Rev. Lett. 88, 093002 (2002) Hemmers, O., Guillemin, R., Kanter, E.P., Krassig, B., Lindle, D.W., Southworth, S.H., Wehlitz, R., Baker, J., Hudson, A., Lotrakul, M., Rolles, D., Stolte, W.C., Tran, I.C., Wolska, A., Yu, S.W., Amusia, M.Y., Cheng, K.T., Chernysheva, L.V., Johnson, W.R., Manson, S.T.: Phys. Rev. Lett. 91, 053002 (2003) Amusia, M.Y., Kuchiev, M.Y., Sheinerman, S.A.: J. Exp. Theor. Phys. 76(2), 470 (1979) Amusia, M.Y., Gribakin, G.F., Tsemekhaman, K.L., Tsemekhaman, V.L.: J. Phys. B 23, 393 (1990) Amusia, M.Y., Avdonina, N.B., Drukarev, E.G., Manson, S.T., Pratt, R.H.: Phys. Rev. Lett. 85(22), 4703–4706 (2000) Amusia, M.Y.: J. Electron Spectrosc. Relat. Phenomena 159, 81 (2007) Amusia, M.Y.: Phys. Rep. 162, 249 (1988) Tsitovich, V.N., Oiringel, I.M. (eds.): Polarizational Radiation of Particles and Atoms. Plenum, New York (1992) Astapenko, V.: Polarization Bremsstrahlung on Atoms, Plasmas, Nanostructures and Solids. Springer, Heidelberg, New York, Dordrecht, London (2013) Amusia, M.Y.: Radiat. Phys. Chem. 75, 1232 (2006) Amusia, M.Y., Baltenkov, A.S., Chernysheva, L.V., Felfli, Z., Msezane, A.Z.: J. Phys. B At. Mol. Opt. Phys. 38, L169 (2005) Amusia, M.Y., Baltenkov, A.S., Chernysheva, L.V.: JETP Lett. 87, 230 (2008) Amusia, M.Y., Baltenkov, A.S., Chernysheva, L.V.: JETP 134(2), 221 (2008) Dolmatov, V.K., Amusia, M.Y., Chernysheva, L.V.: Phys. Rev. A 95, 012709 (2017) Amusia, M.Y., Chernysheva, L.V.: JETP Lett. 106, 1 (2017)
Miron Ya. Amusia Dr Miron Amusia was Professor of Physics (Emeritus) at Hebrew University of Jerusalem, Israel, and Principal Scientist of the Ioffe Physical-Technical Institute, St. Petersburg, Russia. His research interests included the many-body theory of atoms, nuclei, endohedrals, and condensed matter. He received the Humboldt Research Award, Konstantinov and Frenkel medals and awards, Ioffe award, Kapitsa and Semenov medals, and was a member of the Russian Academy of Natural Sciences.
24
25
Photoionization of Atoms Anthony F. Starace
Contents 25.1 25.1.1 25.1.2 25.1.3 25.1.4 25.1.5
General Considerations . . . . . . . . . . . . . . . . . . The Interaction Hamiltonian . . . . . . . . . . . . . . . . Alternative Forms for the Transition Matrix Element . Selection Rules for Electric Dipole Transitions . . . . . Boundary Conditions on the Final State Wave Function Photoionization Cross Sections . . . . . . . . . . . . . .
383 383 384 385 385 386
25.2 25.2.1 25.2.2 25.2.3
An Independent Electron Model Central Potential Model . . . . . . . High Energy Behavior . . . . . . . . Near-Threshold Behavior . . . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
386 386 387 387
25.3 25.3.1 25.3.2 25.3.3 25.3.4
Particle–Hole Interaction Effects Intrachannel Interactions . . . . . . Virtual Double Excitations . . . . . Interchannel Interactions . . . . . . Photoionization of Ar . . . . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
388 388 389 389 389
25.4 25.4.1 25.4.2
Theoretical Methods for Photoionization . . . . . . . 390 Calculational Methods . . . . . . . . . . . . . . . . . . . . 390 Other Interaction Effects . . . . . . . . . . . . . . . . . . . 390
25.5
Related Photoionization Processes . . . . . . . . . . . 391
25.6 25.6.1 25.6.2
Applications to Other Processes . . . . . . . . . . . . . 391 Applications to Two and Three-Photon Ionization Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 Application to High-Order Harmonic Generation . . . 392
25.7
Future Directions . . . . . . . . . . . . . . . . . . . . . . 392
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Abstract
This chapter outlines the theory of atomic photoionization and the dynamics of the photon–atom collision process. Those kinds of electron correlation that are most important in photoionization are emphasized, although many qualitative features can be understood within a central field model. The particle–hole type of electron correlations are discussed, as they are by far the most important for describing the single photoionization of atoms near ionization thresholds. Detailed reviews of atomic pho-
toionization are presented in [1, 2] and [3]. Additional related aspects are well-described in two books [4, 5]. Also, reviews of experimental results for noble gas atom photoionization [6] and for metal atom photoionization [7] provide valuable information on the corresponding theoretical results. At the end of this chapter, some recent applications of the theory of atomic photoionization to few-photon ionization and to high-order harmonic generation are discussed. Keywords
electric dipole transition partial cross section cross section minimum resonance profile independent particle model
25.1
General Considerations
25.1.1
The Interaction Hamiltonian
Consider an N -electron atom with nuclear charge Z. In the nonrelativistic approximation, it is described by the Hamiltonian H D
N X pi 2 i D1
Ze 2 2m ri
C
N X
e2 : jr i rj j i >j D1
(25.1)
The one-electron terms in brackets describe the kinetic and potential energy of each electron in the Coulomb field of the nucleus; the second set of terms describes the repulsive electrostatic potential energy between electron pairs. The interaction of this atom with external electromagnetic radiation is described by the additional terms obtained upon replacing p i by p i C .jej=c/A.r i ; t/, where A.r i ; t/ is the vector po-
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_25
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A. F. Starace
tential for the radiation. The interaction Hamiltonian is, thus, the initial and final electronic states described by the atomic Hamiltonian in Eq. (25.1). Atomic units, in which jej D m D N X Cjej „ D 1, are used in what follows. Œpi A.r i ; t / C A.r i ; t/ p i Hint D 2mc i D1 e2 2 C jA.r ; t/j : (25.2) i 25.1.2 Alternative Forms for the Transition 2mc 2 Under the most common circumstance of single-photon ionization of an outer-subshell electron, the interaction Hamiltonian in Eq. (25.2) may be simplified considerably. First, the third term in Eq. (25.2) may be dropped, as it introduces two-photon processes (since it is of second order in A). In any case, it is small compared with single-photon processes since it is of second order in the coupling constant jej=c. Second, we choose the Coulomb gauge for A, which fixes the divergence of A as r A D 0; A thus describes a transverse radiation field. Furthermore, p and A now commute, and, hence, the first and second terms in Eq. (25.2) may be combined. Third, we introduce the following form for A
12
Matrix Element The matrix element of Eq. (25.4) is proportional to the maP trix element of the momentum operator i p i . Alternative expressions for this matrix element may be obtained from the following operator equations involving commutators of the exact atomic Hamiltonian in Eq. (25.1) "N # N X X p i D i ri ; H ; (25.5) "
i D1 N X i D1
pi ; H
i D1
# D i
N X Zr i i D1
ri3
:
(25.6)
Matrix elements of Eqs. (25.5) and (25.6) between eigen(25.3) states h 0 j and j f i of H having energies E0 and Ef , respectively, give This classical expression for A may be shown [8] to give N N X X photoabsorption transition rates that are in agreement with p i j f i D i!h 0 j ri j f i ; (25.7) h 0j those obtained using the quantum theory of radiation. Here, i D1 i D1 k and ! are the wave vector and angular frequency of the inN N X X Zr i i cident radiation, O is its polarization unit vector, and V is the h 0j pi j f i D j f i ; (25.8) h 0j ! ri3 spatial volume. Fourth, the electric dipole (E1) approximai D1 i D1 tion, in which expŒi.k r i / is replaced by unity, is usually P PN ! D Ef E0 . Matrix elements of N i D1 p i , i D1 r i , appropriate. The radii ri of the atomic electrons are usu- where PN 3 ally of order 1 Å. Thus, for 100 Å; jk r i j 1. Now and i D1 Zr i =ri are known as the velocity, length, and ac 100 Å corresponds to photon energies „! 124 eV. For celeration forms of the E1 matrix element. Equality of the outer atomic subshells, most of the photoabsorption occurs matrix elements in Eqs. (25.7) and (25.8) does not hold when for much smaller photon energies, thus validating the use of approximate eigenstates of H are used [9]. In such a case, the E1 approximation. (This approximation cannot be used qualitative considerations may help to determine which form uncritically, however. For example, photoionization of ex- is most reliable. For example, the length form tends to emcited atoms (which have large radii), photoionization of inner phasize the large r part of the approximate wave functions, subshells (which requires the use of short wavelength radi- the acceleration form tends to emphasize the small r part of ation), and calculation of differential cross sections or other the wave functions, and the velocity form tends to emphasize measurable quantities that are sensitive to the overlap of elec- intermediate values of r. If instead of employing approximate eigenstates of the tric dipole and higher multipole amplitudes all require that exact H , one employs exact eigenstates of an approximate the validity of the electric dipole approximation be checked.) N -electron Hamiltonian, then inequality of the matrix eleThe use of all of the above conventions and approximations ments in Eqs. (25.7) and (25.8) is a measure of the nonlocalallows the reduction of Hint in Eq. (25.2) to the simplified ity of the potential in the approximate Hamiltonian [10–12]. form The exchange part of the Hartree–Fock potential is an ex 1 N ample of such a nonlocal potential. Nonlocal potentials are Cjej 2c 2 „ 2 X O p i exp.i!t/ I (25.4) also implicitly introduced in configuration interaction calcuHint D mc !V i D1 lations employing a finite number of configurations [10–12]. Hint thus has the form of a harmonically time-dependent per- One may eliminate the ambiguity of which form of the E1 turbation. According to time-dependent perturbation theory, transition operator to use by requiring that the Schrödinger the photoionization cross section is proportional to the ab- equation be gauge invariant. Only the length form is consissolute square of the matrix element of Eq. (25.4) between tent with such gauge invariance [10–12]. A.r i ; t/ D
2c 2 „ !V
e O i.kr i !t / :
25 Photoionization of Atoms
However, equality of the alternative forms of the transition operator does not necessarily imply high accuracy. For example, they are exactly equal when one uses an approximate local potential to describe the N -electron atom, as in a central potential model, even though the accuracy is often poor. The length and velocity forms are also exactly equal in the random phase approximation with exchange (RPAE) [1, 3], which generally gives accurate cross sections for single photoionization of closed shell atoms. No general prescription exists,however, for ensuring that the length and velocity matrix elements are equal at each level of approximation to the N -electron Hamiltonian. However, if all corrections of a given order in interelectron interaction are taken into account, length and velocity matrix elements are equal [3].
385
N S; N AC ; `; L0 ; In Eq. (25.9), the quantum numbers ˛ L; S ; ML0 ; MS 0 (plus any other quantum numbers needed to specify the state of the ion AC uniquely) define a final state channel. All final states that differ only in the photoelectron energy " belong to the same channel. The quantum numbers L0 ; S 0 ; ML0 ; MS 0 , and tot D .1/` AC are the only good quantum numbers for the final states. Thus, the Hamiltonian Eq. (25.1) mixes final state channels having the same angular momentum and parity quantum numbers but differing quantum numbers for the ion and the photoelectron; i.e., difN S, N AC , and ` but the same L0 ; S 0 ; ML0 ; MS 0 and fering L; ` C .1/ A . 0
25.1.4 25.1.3
Selection Rules for Electric Dipole Transitions
Boundary Conditions on the Final State Wave Function
Photoionization calculations obtain final state wave functions satisfying the asymptotic boundary condition that the photoIf one ignores relativistic interactions, then a general atomic electron is ionized in channel ˛. This boundary condition is photoionization process may be described in LS-coupling as expressed as follows ˛E .r 1 s 1 ; : : : ; r N s N / A.L; S; ML ; MS ; A / C . ; ` ; m / 1 i ˛ 1 N AC /"`.L0 ; S 0 ; ML0 ; MS 0 / : (25.9) ! AC .LN S ! .r s ; : : : ; r O N sN / e 1 rN !1 ˛ 1 1 r i.2k˛ / 2 N X Here, the atom A is ionized by the photon to produce 1 i ˛0 1 ˛0 .r 1 s1 ; : : : ; rO N sN / e S˛ 0 ˛ ; 1 a photoelectron with kinetic energy " and orbital angular i.2k˛0 / 2 rN ˛0 C momentum `. The photoelectron is coupled to the ion A (25.15) with total orbital and spin angular momenta L0 and S 0 . In the where the phase appropriate for a Coulomb field is electric dipole approximation, the photon may be regarded as having odd parity, i.e., D 1, and unit angular momen1 1 tum, i.e., ` D 1. This is obvious from Eqs. (25.7) and (25.8), log 2k˛ rN C `˛ : (25.16) ˛ k˛ rN `˛ C 2 k˛ where the E1 operator is seen to be a vector operator. The component m of the photon in the E1 approximation is ˙1 The minus superscript on the wave function in Eq. (25.15) infor right or left circularly polarized light and 0 for linearly dicates an incoming wave normalization:, i.e., asymptotically polarized light. (The z axis is taken as kO in the case of circu ˛E has outgoing spherical Coulomb waves only in channel larly polarized light and as O in the case of linearly polarized ˛, while there are incoming spherical Coulomb waves in all light, where k and O are defined in Eq. (25.3).) Angular mochannels; S˛0 ˛ is the Hermitian conjugate of the S-matrix of mentum and parity selection rules for the E1 transition in scattering theory, ˛ indicates the coupled wave function of Eq. (25.9) imply the following relations between the initial the ion and the angular and spin parts of the photoelectron and final state quantum numbers wave function, k˛ is the photoelectron momentum in chan0 (25.10) nel ˛ and `˛ is its orbital angular momentum, and `˛ in L D L ˚ 1 D LN ˚ ` ; Eq. (25.16) is the Coulomb phase shift. ML0 D ML C m D MLN C m` ; (25.11) , experimenWhile one calculates channel functions ˛E 1 0 tally one measures photoelectrons that asymptotically have N (25.12) S DS DS˚ ; 2 well-defined linear momenta k˛ and well-defined spin states N LN MSN . The (25.13) m 1 , and ions in well-defined states ˛N LN SM MS 0 D MS D MSN C ms ; 2 (25.14) wave function appropriate for this experimental situation is A AC D .1/`C1 : related to the channel functions by uncoupling the ionic and Equation (25.14) follows from the parity .1/` of the photo- electronic orbital and spin angular momenta and projecting electron. The direct sum symbol ˚ denotes the vector addi- the photoelectron angular momentum states `˛ , m˛ onto the tion of A and B, i.e, A ˚B D ACB; ACB 1; : : : ; jABj. direction kO ˛ by means of the spherical harmonic Y`?˛ m˛ .k˛ /.
25
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A. F. Starace
Substitution of the final state wave function Eq. (25.17) in Eq. (25.20) permits one to carry out the numerous summations over magnetic quantum numbers and obtain the form
This relation is [2] ˛k N ˛ .r 1 s 1 ; : : : ; r N s N / X i`˛ exp.i` / ˛
D
1 2
d˛N ˛N D Œ1 C ˇP2 .cos / d˝ 4
k˛ X N LN `˛ m˛ jLML i hLM
`˛ m˛
Y`?˛ m˛ .kO ˛ /
LML SMS
N SN 1 m 1 jSMS i hSM 2 2
˛E .r 1 s1 ; : : : ; r N sN /
;
(25.17)
where the coefficients in brackets are Clebsch–Gordan coefficients. This wave function is normalized to a delta function in momentum space, i.e., Z
3 (25.18) ˛k N ˛ ˛N 0 k˛ 0 d r D ı˛N ˛N 0 ı.k˛ k˛0 / :
(25.21)
for the differential cross section [13]. Here, ˛N is the partial cross section for leaving the ion in the state ˛, N ˇ is the asymmetry parameter [23], P2 .cos / D 32 cos2 12 , and indicates the direction of the outgoing photoelectron with respect to the polarization vector O of the incident light. The form of Eq. (25.21) follows in the electric dipole approximation from general symmetry principles, provided that the target atom is unpolarized [15]. The partial cross section is given in terms of reduced E1 matrix elements [14] involving the channel functions in Eq. (25.15) by
1 ˇ2 ˇ N ˇ The factors i`˛ exp.i`˛ /k˛ 2 ensure that for large rN , Xˇˇ X 4 2 Œ1 1 ˇ ri k ˛E iˇ : (25.22) !ŒL ˛N D ˇh 0 k Eq. (25.17) represents a Coulomb wave (with momenˇ ˇ 3c i D1 `˛ L0 tum k˛ ) times the ionic wave function for the state ˛N plus a sum of terms representing incoming spherical waves. Thus, The ˇ parameter has a much more complicated expresonly the ionic term ˛N has an outgoing wave. One uses the sion involving interference between different reduced dipole wave function in Eq. (25.17) to calculate the angular distriamplitudes [16, 23]. Thus, measurement of ˇ provides inbution of photoelectrons. formation on the relative phases of the alternative final state channel wave functions, whereas the partial cross section in Eq. (25.22) does not. From the requirement that the differen25.1.5 Photoionization Cross Sections tial cross section in Eq. (25.21) be positive, one can see that If one writes H in Eq. (25.4) as H .t/ D H .0/ ei!t , then 1 ˇ C2. int
int
int
from first-order time-dependent perturbation theory, the transition rate for the transition from an initial state with energy 25.2 An Independent Electron Model E0 and wave function 0 to a final state with total energy Ef and wave function ˛k N ˛ is The many-body wave functions 0 and ˛E are usually ex 2 pressed in terms of a basis of independent electron wave dWk˛ D 2jh 0 jHint .0/j ˛k N ˛ ij functions. Key qualitative features of photoionization cross ı.Ef E0 !/k˛2 dk˛ d˝.kO ˛ / : (25.19) sections can often be interpreted in terms of the overlaps of initial and final state one-electron radial wave functions [2, The delta function expresses energy conservation, and the 14]. The simplest independent electron representation of the last factors on the right-hand side are the phase space facatom, the central potential model, proves useful for this purtors for the photoelectron. Dividing the transition rate by the pose. incident photon current density c=V , integrating over dk˛ , and inserting Hint .0/, the differential photoionization cross section is 25.2.1 Central Potential Model ˇ2 ˇ N ˇ ˇ X d˛N 4 2 k˛ ˇ ˇ p i j ˛k (25.20) In the central potential (CP) model, the exact H in Eq. (25.1) D ˇO h 0 j N ˛ iˇ : ˇ d˝ c ! ˇ i D1 is approximated by a sum of single-particle terms describing the independent motion of each electron in a central potential Implicit in Eqs. (25.19) and (25.20) is an average over initial V .r/ magnetic quantum numbers ML0 MS0 and a sum over final magnetic quantum numbers MLN MSN m 1 . The length form N X 2 pi 2 of Eq. (25.20) is obtained by replacing each p i by !r i (25.23) C V .ri / : HCP D 2m Eq. (25.7). i D1
25 Photoionization of Atoms
387
The potential V .r/ must describe the nuclear attraction and n` electrons’ binding energy due to repulsion between the the electron–electron repulsion as well as possible and must outer electrons and the photoelectron as the latter leaves the atom. The potential in Eq. (25.26) predicts hydrogen-like satisfy the boundary conditionse photoionization cross sections for inner-shell electrons with o (25.24) onsets determined by the outer-screening parameters Vn` . V .r/! Z=r and V .r/! 1=r r!1 r!0 The use of more accurate atomic central potentials in in the case of a neutral atom; HCP is separable in spherical place of the screened hydrogenic potential in Eq. (25.26) coordinates and its eigenstates can be written as Slater deter- generally enables one to obtain photoionization cross secminants of one-electron orbitals of the form r 1 Pn` Y`m .˝/ tions below the keV photon energy region to within 10% for bound orbitals and of the form r 1 P"` .r/Y`m .˝/ for of the experimental results [18]. For ` > 0, subshells and continuum orbitals. The one-electron radial wave functions photon energies in the keV region and above, the independent particle model becomes increasingly inadequate owing satisfy to coupling with nearby ns-subshells, which generally have larger partial cross sections at high photon energies [19]. For `.` C 1/ d 2 P"` .r/ C 2 " V .r/ P"` .r/ D 0 ; high, but still nonrelativistic, photon energies, i.e., ! mc 2 , dr 2 2r 2 (25.25) the energy dependence of the cross section for the n` subshell within the independent particle model is [20] subject to the boundary condition P"` .0/ D 0, and similarly 7 n` ! ` 2 : (25.27) for the discrete orbitals Pn` .r/. Hermann and Skillman [17] have tabulated a widely used central potential for each ele- However, when interchannel interactions are taken into acment in the periodic table, as well as radial wave functions count, the asymptotic energy dependence for subshells havfor each occupied orbital in the ground state of each element. ing ` > 0 becomes independent of ` [21] 9
25.2.2
High Energy Behavior
n` ! 2 .` > 0/ :
(25.28)
This result stems from coupling of the ` > 0 photoionization The hydrogen atom cross section, which is nonzero at thresh- channels with nearby s-subshell channels. old and decreases monotonically with increasing photon energy, serves as a model for inner-shell photoionization cross sections in the X-ray photon energy range. A sharp 25.2.3 Near-Threshold Behavior onset at threshold followed by a monotonic decrease above threshold is precisely the behavior seen in X-ray photoab- For photons in the vacuum ultra violet (VUV) energy region, sorption measurements. A simple hydrogenic approximation i.e., near the outer-subshell ionization thresholds, the phoat high energies may be justified theoretically as follows: toionization cross sections for subshells with ` 1 frequently (1) Since a free electron cannot absorb a photon (because have distinctly nonhydrogenic behavior [22, 24]. The cross of kinematical considerations), at high photon energies one section, instead of decreasing monotonically as for hydrogen, expects the more strongly bound inner electrons to be pref- rises above threshold to a maximum (the so-called delayed erentially ionized as compared with the outer electrons. maximum above threshold). Then it decreases to a minimum (2) Since the Pn` .r/ for an inner electron is concentrated in (the Cooper minimum [25, 26]) and rises to a second maxa very small range of r, one expects the integrand of the ra- imum. Finally, the cross section decreases monotonically at dial dipole matrix element to be negligible except for those high energies in accordance with hydrogenic behavior. Such values of r where Pn` .r/ is greatest. (3) Thus, it is only nec- nonhydrogenic behavior may be interpreted as due either essary to approximate the atomic potential locally, e.g., by to an effective potential barrier or to a zero in the radial dipole matrix element. We examine each of these effects in means of a screened Coulomb potential turn. The delayed maximum above outer subshell ionization thresholds of heavy atoms (i.e., Z & 18) is due to an effecZ sn` o (25.26) C Vn` Vn` .r/ D tive potential barrier seen by ` D 2 and ` D 3 photoelectrons r in the region of the outer edge of the atom Eq. (25.25). This appropriate for the n` orbital. Here, sn` is the inner-screening effective potential lowers the probability of photoelectron esparameter, which accounts for the screening of the nuclear cape until the photoelectrons have enough excess energy to o is the outer- surmount the barrier. Such behavior is nonhydrogenic. Furcharge by the other atomic electrons, and Vn` screening parameter, which accounts for the lowering of the thermore, in cases where an inner subshell with ` D 2 or 3
25
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A. F. Starace
is being filled as Z increases (as in the transition metals, the lanthanides and the actinides) there is a double well potential. This double well has profound effects on the 3p-subshell spectra of the transition metals, the 4d -subshell spectra of the lanthanides, and the 5d -subshell spectra of the actinides, as well as on atoms with Z just below those of these series of elements [27, 28]. The potential barrier effects on the photoionization cross sections of these elements are often referred to as “giant dipole resonances”. In fact, to describe the 4d subshell, interaction of at least all 4d electrons has to be taken into account. The many-electron nature of “giant dipole resonances” is presented in Chap. 24 (also [1]). Cross section minima arise due to a change in sign of the radial dipole transition matrix element in a particular channel [29, 30]. Rules for predicting their occurrence were developed in [25, 26]. Studies of their occurrence in photoionization from excited states [31], in high Z atoms [32], and in relativistic approximation [33] have been carried out. A proof has been given [34] that such minima do not occur in atomic hydrogen spectra. For other elements, there are further rules on when and how many minima may occur [35– 37]. Often within such minima, one can observe the effects of weak interactions that are otherwise obscured. Relativistic and weak correlation effects on the asymmetry parameter ˇ for s-subshells is a notable example [38]. Wang et al. [39] also emphasized that near such minima in the E1 amplitudes, one cannot ignore the effects of quadrupole and higher corrections to the differential cross section. Central potential model calculations [39] show that quadrupole corrections can be as large as 10% of the E1 cross section at such cross section minima, even for low photon energies. Particularly strong is the role of quadrupole effects in the so-called nondipole angular anisotropy parameters (e.g., [40]).
25.3
Particle–Hole Interaction Effects
The experimental photoionization cross sections for the outer subshells of the noble gases have played a prominent role in the development of the theory of photoionization for two reasons. These were among the first elements studied by experimentalists with synchrotron radiation beginning in the 1960s. Also, their closed-shell, spherically symmetric ground states simplified the theoretical analysis of their cross sections. The cross sections near the ionization thresholds can be understood in terms of interactions between the photoelectron, the residual ion, and the photon field that are called, in manybody theory language, particle–hole interactions (Chap. 24). These may be described as interactions in which two electrons either excite or de-excite each other out of or into their initial subshell locations in the unexcited atom. To analyze the effects of these interactions on the cross section, it is con-
a
nl
e–
l⬘ nl e–
⬘l⬘
nl
nl
b nl nl
e–
nl
l⬘
⬘l⬘
nl e–
c
n1l1
⬘⬘l⬘⬘
e– n1l1 e– n0 l0
n0 l 0
⬘l⬘
Fig. 25.1 MBPT diagrams (left) and scattering pictures (right) for three kinds of particle–hole interaction: (a) intrachannel scattering following photoabsorption; (b) photoabsorption by a virtual doubly excited state; (c) interchannel scattering following photoabsorption
venient to classify them into three categories: intrachannel, virtual double excitation, and interchannel. In many publications, however, another classification is used: intrasubshell and intersubshell correlations (Chap. 24). These alternative kinds of particle–hole interactions are illustrated in Fig. 25.1 using both many-body perturbation theory (MBPT) diagrams and more physical scattering pictures. We discuss each of these types of interaction in turn.
25.3.1 Intrachannel Interactions The MBPT diagram for this interaction is shown on the lefthand side of Fig. 25.1a; on the right-hand side, a slightly more pictorial description of this interaction is shown. The wiggly line indicates a photon, which is absorbed by the atom in such a way that an electron is excited out of the n` subshell. During the escape of this excited electron, it collides or interacts with another electron from the same subshell in such a way that the second electron absorbs all the energy imparted to the atom by the photon; the first electron is de-excited back to its original location in the n` subshell. For closed-shell atoms, the photoionization process leads to a 1 P 1 final state in which the intrachannel interaction is strongly repulsive. This interaction tends to broaden cross section maxima and push them to higher photon energies as compared with the results of central potential model calculations.
25 Photoionization of Atoms
389
the effect of weak channels on stronger channels. In addition, when the interacting channels have differing binding energies, their interchannel interactions lead to a resonance structure in the channel with lower binding energy (arising from its coupling to the Rydberg series in the channel with higher binding energy). At high photon energies, s-subshell partial cross sections dominate over ` > 0 subshell partial cross sections (Eqs. (25.27), (25.28)). Hence, interchannel interactions of 25.3.2 Virtual Double Excitations ` > 0 subshells with nearby s-subshells change independent particle model predictions significantly. In particular, as The MBPT diagram for this type of interaction is shown on noted in Sect. 25.3.2, such interactions can drastically change the left-hand side of Fig. 25.1b. Topologically, this diagram the magnitudes of the ` > 0 partial cross sections [19], as well is the same as that on the left-hand side in Fig. 25.1a. In fact, as their asymptotic energy behavior [21]. the radial parts of the two matrix elements are identical; only the angular factors differ. A more pictorial description of this interaction is shown on the right-hand side of Fig. 25.1b. The 25.3.4 Photoionization of Ar ground state of the atom before photoabsorption is shown to have two electrons virtually excited out of the n` subshell. An example of both the qualitative features exhibited by phoIn absorbing the photon, one of these electrons is de-excited toionization cross sections in the VUV energy region and of to its original location in the n` subshell, while the other the ability of the theory to calculate photoionization cross reelectron in ionized. These virtual double excitations imply actions accurately is provided by photoionization of the n D 3 a more diffuse atom than in central-potential or HF models, subshell of argon, i.e., with the effect that the overly repulsive intrachannel interacAr3s 2 3p 6 C ! Ar C 3s 2 3p 5 C e tions are weakened, leading to cross sections for noble gas atoms that are in very good agreement with experiment with ! Ar C 3s3p 6 C e : (25.29) the exception that resonance features are not predicted. As an impressive example of strong intrachannel coupling serves First calculations of photoionization cross section of Ar 3p6 the formation of so-called atomic giant resonance (Chap. 24, subshell with account of many-electron correlations was performed by Amusia et al. [41] in RPAE. Good agreement Fig. 24.2. with experiment was achieved. Figure 25.2 shows the MBPT calculation of Kelly and Simons [42], which includes both in25.3.3 Interchannel Interactions trachannel and interchannel interactions, as well as the effect of virtual double excitations. The cross section is in excellent The interchannel interaction shown in Fig. 25.1c is im- agreement with experiment [43, 44], even to the extent of deportant, particularly for s subshells. This interaction has scribing the resonance behavior due to discrete members of the same form as the intrachannel interaction shown in the 3s ! "p channel. Fig. 25.1a, except now when an electron is photoexcited out Figure 25.2 illustrates most of the features of photoionof the n0 `0 subshell, it collides or interacts with an electron ization cross sections described so far. First, the cross section in a different subshell – the n1 `1 subshell. This interaction rises to a delayed maximum just above the threshold because causes the second electron to be ionized, and the first elec- of the potential barrier seen by photoelectrons from the 3p tron to fall back into its original location in the n0 `0 subshell. subshell having ` D 2. For photon energies in the range of Interchannel interaction effects are usually very conspic- 4550 eV, the calculated cross section goes through a minuous features of photoionization cross sections. When the imum because of a change in sign of the 3p ! "d radial interacting channels have partial photoionization cross sec- dipole amplitude. The HFL and HFV calculations include the tions that differ greatly in magnitude, one finds that the strongly repulsive intrachannel interactions in the 1 P finalcalculated cross section for the weaker channel is completely state channels and calculate the transition amplitude using dominated by its interaction with the stronger channel. A the length (L) and velocity (V) form, respectively, for the good example of interchannel (or intershell) interaction is electric dipole transition operator Eq. (25.7). With respect to the photoionization cross section of 5s subshell in Xe that is the results of central potential model calculations, the HFL totally determined by 5s interaction with its many-electron and HFV results have lower and broader maxima at higher neighboring 5p and 4d subshells (Chap. 24, Fig. 24.3). At energies. They also disagree with each other by a factor of 2! the same time, it is often a safe approximation to ignore Inclusion of virtual double excitations results in length and Intrachannel interaction effects are taken into account automatically when the correct Hartree–Fock (HF) basis set is employed, in which the photoelectron sees a net Coulomb field due to the residual ion and is coupled to the ion to form the appropriate total orbital L and spin S angular momenta [3]. Any other basis set requires explicit treatment of intrachannel interactions.
25
390
A. F. Starace
σ (10 –18 cm2) 50 HFL 45 40 35 30 25 HFV
20 15 10 5 0 16
20
25
30
35
40
45 50 54 Photon energy (eV)
Fig. 25.2 Photoionization cross section for the 3p and 3s subshells of Ar. HFL and HFV indicate the length and velocity results obtained using HF orbitals calculated in a 1 P 1 potential. Dot-dash and dashed lines represent the length and velocity results of the MBPT calculation of Kelly and Simons [42]. Only the four lowest 3s ! np resonances are shown; the series converges to the 3s threshold at 29:24 eV. Experimental results are those of Samson [43] above 37 eV and of Madden et al. [44] below 37 eV (after [42])
velocity results that agree to within 10% with each other and with experiment, except that the resonance structures are not reproduced. Finally, taking into account the interchannel interactions, one obtains the length and velocity form results shown in Fig. 25.2 by dash-dot and dashed curves, respectively. The agreement with experiment is excellent, and the observed resonances are well reproduced.
25.4 25.4.1
Theoretical Methods for Photoionization Calculational Methods
Most of the ab initio methods for the calculation of photoionization cross sections (e.g., the MBPT method [45], the close-coupling (CC) method [46], the R-matrix method [47, 48], the random phase approximation with exchange (RPAE) method [1, 3, 49], the relativistic RPAE method [50], the transition matrix method [51, 52], the multiconfiguration Hartree–Fock (MCHF) method [53–55], time-dependent multiconfiguration methods [56], etc.) have successfully calculated outer p-subshell photoionization cross sections of the noble gases by treating in their alternative ways the key interactions described above, i.e., the particle–hole interactions. In general, these methods all treat both intrachannel and interchannel interactions to infinite order and differ only in their treatment of ground state correlations. (The exception is MBPT, which often treats interchannel interactions between weak and strong channels only to first or second order.) These methods, therefore, stand in contrast to central potential model calculations, which do not treat any
of the particle–hole interactions, and single-channel termdependent HF calculations, which treat only the intrachannel interactions. The key point is that selection of the interactions that are included in a particular calculation is more important than the method by which such interactions are handled. Treatment of the photoionization of atoms other than the noble gases presents additional challenges for theory. For example, elements such as the alkaline earths, which have s 2 outer subshells, require careful treatment of electron pair excitations in both initial and final states. Open-shell atoms have many more ionization thresholds than do the noble gases. Treatment of the resultant rich resonance structures typically relies heavily on quantum defect theory [55] (Chap. 32). All the methods listed above can be used to treat elements other than the noble gases, but a method that has come to prominence because of the excellent results it obtains for both alkaline earth and open-shell atoms is the eigenchannel R-matrix method [57].
25.4.2
Other Interaction Effects
A number of interactions, not of the particle–hole type, lead to conspicuous effects in localized energy regions. When treating photoionization in such energy regions, one must be careful to choose a theoretical method that is appropriate. Among the interactions that may be important are the following: Relativistic and spin-dependent interactions The fact that j D ` 12 electrons are contracted more than j D ` C 12 electrons at small distances has an enormous effect on the location of cross section minima in heavy elements [18, 58]. It may explain the large differences observed in the profiles of a resonance decaying to final states that differ only in their fine structure quantum numbers [59]. Impressive effects in the photoionization cross section appear due to interaction not only between inner but also intermediate sublevels ` C 12 and ` 12 [60]. Inner-shell vacancy rearrangement Inner-shell vacancies often result in a significant production of satellite structures in photoelectron spectra. Calculations for inner-subshell partial photoionization cross sections are often substantially larger than results of photoelectron measurements [61–63]. This difference is attributed to such satellite production that is often not treated in theoretical calculations, as well as to effect of inner-shell rearrangement, due to its ionization, upon the outgoing photoelectron [3, 49]. Polarization and relaxation effects Negative ion photodetachment cross sections often exhibit strong effects of core polarization near the threshold. These effects can be treated
25 Photoionization of Atoms
semiempirically, resulting in excellent agreement between theory and experiment [64]. Even for inner-shell photoionization cross sections of heavy elements, ab initio theories do not reproduce measurements near threshold without the inclusion of polarization and relaxation effects [65, 66]. An example The calculation of the energy dependence of the asymmetry parameter ˇ for the 5s subshell of xenon requires the theoretical treatment of all of the above effects. In the absence of relativistic interactions, ˇ for Xe 5s would have the energy-independent value of 2. Deviations of ˇ from 2 are, therefore, an indication of the presence of these relativistic interactions. The greatest deviation of ˇ from 2 occurs in the localized energy region where the partial photoionization cross section for the 5s subshell has a minimum. In this region, however, relativistic calculations show larger deviations from 2 than are observed experimentally. Innershell rearrangement and relaxation effects play an important role [67, 68] and must be included to achieve good agreement with experiment.
25.5
Related Photoionization Processes
One of the most intensively studied areas in atomic photoionization has been the double photoionization of the helium atom. Extensive sets of experimental measurements for the two electron angular distributions (i.e., the triply differential cross sections) have provided stringent tests for various theoretical models and their treatments of electron correlations. A number of excellent reviews of this field have been published recently [69–71]. Another intensively studied area has been the analysis and measurement of nondipole effects in photoionization, which were first observed in the X-ray region [72] but have been found to be significant even in the VUV photon energy region [73, 74]. In general, these effects stem from interference between electric quadrupole and the (usual) electric dipole transition amplitudes in differential cross sections (for a review, [75]). Besides asymmetries in the photoelectron angular distributions, nondipole effects also lead to new features for spin-resolved measurements [76, 77], and for the case of polarized atoms [78]. Nondipole effects have been predicted to be significant also in double photoionization of helium at relatively low photon energies [79]. In fact, the so-called “back-to-back” two-electron photoionization, predicted quite long ago [80] and experimentally observed recently [81], is entirely nondipole. Finally, both experimental and theoretical studies of ionic species have flourished since the 1990s. In particular, photodetachment of negative ions near excited atomic thresholds provides an opportunity to study correlated, three-body Coulomb states unencumbered by Rydberg series. The ad-
391
vent of powerful computer workstations in the 1990s enabled theorists to carry out numerical calculations for such high, doubly excited states with spectroscopic accuracy. Following experiments for photodetachment of H with excitations of atomic levels in H.n > 2/, theorists developed propensity rules for identifying and characterizing the dominant photodetachment channels [82–84]. Experimental and theoretical [85–88] interest also focused on the negative alkali ions (e.g., Li and Na ), which for low photon energies have outer electron detachment spectra grossly resembling that of H . However, the negative alkali spectra contain clear signatures of propensity-rule-forbidden states that become increasingly prominent as the atomic number increases (owing to the nonhydrogenic inner electron cores). A brief review of low-energy negative alkali photodetachment is given in [89]. Among the more general features brought to light by these studies is the mirroring of resonance profiles in alternative partial cross sections, which appears to be a very general phenomenon common to photodetachment and photoionization processes involving highly excited residual atoms or ions [90]. Recently, high-energy (K-shell) photodetachment of the negative ions Li and He (resulting in two-electron ionization) was studied both experimentally and theoretically [91, 92]. These studies represent the first results for inner-shell photodetachment. There is general agreement between theory and experiment well above the K edge, but the theoretical cross sections at the K edge are significantly higher than the experimental measurements. The latter discrepancy is now understood as arising from recapture of the low-energy detached electron following Auger decay of the inner-shell vacancy, which when taken into account theoretically has been shown to provide results that agree with experiment [93, 94]. Also, the first experimental data together with theoretical analyses were recently presented for photoionization of ground and metastable positive ions (OC and ScCC ) [95, 96]. With the advent of data for photoionization of positive ions it now becomes possible (using the principle of detailed balance) to make connections to data for electron–ion photo-recombination cross sections [96].
25.6
Applications to Other Processes
The detailed understanding of single-photon ionization processes in atoms described in this chapter have recently proved useful in interpreting more complex interactions of electromagnetic radiation with atoms. First, potential barrier effects, which are key features of single-photon ionization of the heavier rare gases, have been shown to also be key features of two and three-photon ionization processes in rare gases. Second, prominent features of the high-order harmonic generation (HHG) spectra of an atom have been
25
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A. F. Starace
shown to stem from features of that atom’s photoionization the HHG yield Y as the product of an electronic wave packet cross section. We discuss each of these two applications of W .E/ (corresponding to the first two steps) and the field-free the theory of atomic photoionization below. photorecombination cross section (PRCS) .E/ Y / W .E/ .E/;
25.6.1
Applications to Two and Three-Photon Ionization Processes
Potential barriers in the effective radial potential experienced by a photoexcited electron have been shown to result in dramatic, resonance-like effects in both two-photon ionization processes [97] and three-photon ionization processes [98] in rare gas atoms. In these two-photon and three-photon ionization processes, such potential barriers were found to affect not only the final state of the electron (as in ordinary photoionization), but also the intermediate-state electron wave packets corresponding to the absorption of one or two photons when the energy absorbed places the intermediate state wave packet in the energy vicinity of the effective potential barrier. These effects were illustrated for the generalized two and three-photon cross sections for ionization of Ar, Kr, and Xe within a single-active-electron, central-potential model [97, 98]. The results are relevant to experimental studies of nonlinear extreme ultraviolet and X-ray processes [99]. Numerous theoretical treatments of electron correlation effects relevant to few-photon ionization processes have been carried out (e.g., [100–103]). Interest in these theoretical treatments is likely to increase when experimental few-photon (nonlinear) ionization cross sections become available as a function of photon frequency.
Application to High-Order Harmonic Generation
The photoionization cross section of an atom has been found to have a direct connection to the high-order harmonic generation (HHG) spectrum of that atom, which is an unexpected relationship between linear and highly nonlinear interactions of an atom with electromagnetic radiation. The relation between the two processes may be understood by means of the quasi-classical interpretation of HHG as a three-step process [104–106] involving: (1) tunnel ionization of an electron in an atom by a strong low-frequency laser field; (2) propagation of the ionized electron in the (laser-dressed) continuum away from and back to the atomic ion by the oscillating laser field; and (3) recombination of the returning electron back to the atom ground state with emission of a harmonic photon. The approach that permits us to understand and describe atomic photoionization by a strong field, called “atomic antenna” was first suggested by Kuchiev [107]. It includes the above-mentioned steps (1) and (2). The three-step model suggests a phenomenological parametrization [108, 109] of
(25.30)
where E is the energy of the recombining electron, E˝ is the harmonic energy, and Ip is the target binding energy. This phenomenological parametrization has been confirmed by analysis of exactly solvable model systems [110–112]. The intensity of the harmonic radiation thus carries information on the PRCS of an electron whose momentum is directed along the linear polarization axis of the driving laser field. Since the photorecombination process is the inverse to that for photoionization, the PRCS is directly related (through the principle of detailed balance) to the photoionization cross section for photon energy E˝ . Thus, the form of the HHG spectrum replicates the shape of the photoionization cross section. Indeed, experimental and theoretical studies for the case of linearly polarized driving laser fields have shown that HHG spectra exhibit such key features of the corresponding photoionization cross sections as Cooper minima [110, 113] and potential barrier-induced (“giant dipole”) resonances [110, 114–116]. Extension of such studies to the case of two-dimensional driving laser waveforms has shown that this “HHG spectroscopy” can enable the extraction of not only photoionization cross sections but also the photoelectron angular distribution asymmetry parameters [117, 118].
25.7 25.6.2
E D E˝ Ip ;
Future Directions
A main new direction in photoionization is toward timedomain studies in the extreme ultraviolet (XUV) and X-ray regimes. First, advances in high-order harmonic generation have led to the production of isolated pulses of XUV light whose temporal width is measured in tens or hundreds of attoseconds [119]. The shortest such pulses have durations of only a few optical cycles so that (i) they have significant bandwidth, and (ii) the probability of photoionization depends on the phase of the electric field within the pulse envelope. Second, the construction of free-electron lasers (FELs) around the world has enabled studies of intense X-ray interactions with atoms (e.g., [120, 121]). Thus, inner-shell vacancy production and decay, satellite production, and multiple ionization phenomena are all being increasingly studied on timescales comparable to the durations of the FEL Xray pulses, which are estimated to be on the order of tens of femtoseconds. Of essential importance are also studies of photoionization of fullerenes and endohedrals that represent ordinary atoms “stuffed” inside a fullerene shell, such as, e.g., C60 [122].
25 Photoionization of Atoms
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Anthony F. Starace Dr Starace earned his PhD from the University of Chicago (1971) and became George Holmes University Professor of Physics at the University of Nebraska (2001). His primary research interests were the interaction of intense laser light with atoms, especially single and multiphoton detachment and ionization processes. He was a Fellow of the American Physical Society and the American Association for the Advancement of Science. He passed away in 2019.
26
Autoionization Anand K. Bhatia and Aaron Temkin
Contents 26.1 26.1.1 26.1.2 26.1.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Auger Effect . . . . . . . . . . . . . . . . . . . . . . . . . . Autoionization, Autodetachment, and Radiative Decay Formation, Scattering, and Resonances . . . . . . . . . .
26.2 26.2.1 26.2.2
Projection Operator Formalism . . . . . . . . . . . . . 396 Optical Potential . . . . . . . . . . . . . . . . . . . . . . . 396 Expansion of Vop : The QHQ Problem . . . . . . . . . . 396
26.3 26.3.1 26.3.2 26.3.3
Forms of P and Q . . . . . . . . . . . . . . . . . . . . . The Feshbach Form . . . . . . . . . . . . . . . . . . . . . Reduction for the N D 1 Target . . . . . . . . . . . . . Alternative Projection and Projection-Like Operators
. . . .
396 396 397 398
26.4 26.4.1 26.4.2 26.4.3
Width, Shift, and Shape Parameter . Width and Shift . . . . . . . . . . . . . . Shape Parameter . . . . . . . . . . . . . . Relation to Breit–Wigner Parameters .
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398 398 398 400
26.5 26.5.1 26.5.2 26.5.3
Other Calculational Methods . Complex Rotation Method . . . . Pseudopotential Method . . . . . . Exact Bound States in Continuum
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400 400 402 403
26.6
Related Topics . . . . . . . . . . . . . . . . . . . . . . . . 403
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395 395 395 396
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
tablished whereby the main element of the theory can be made into a bound state problem with the scattering elements built around it. The major constituent of both these features is accomplished with projection operators. A brief description of the above elements of the theory, centered around projection operators, is the aim of this chapter [2], although some additional methods are discussed in Sect. 26.5. Another class of resonances, shape resonances, is also briefly discussed, because one of the additional methods, complex rotation, reveals both Feshbach and shape resonances. Keywords
projection operator resonance parameter Feshbach resonance pseudopotential method autoionization state
26.1
Introduction
26.1.1 Auger Effect
Autoionization falls within the general class of phenomena known as the Auger effect. In the Auger effect, an atomic The phenomenon of autoionization, or more particularly system seemingly (Sect. 26.1.3) spontaneously decays into the autoionization state itself, is treated for the most part a partition of its constituent parts. in this chapter as a bound state. The process is rigorously a part of the scattering continuum, but, due mostly to the work of Feshbach [1], a rigorous formulation can be esAbstract
A. K. Bhatia () Laboratory for Astronomy & Solar Physics, NASA Goddard Space Flight Center Greenbelt, MD, USA e-mail: [email protected] A. Temkin () Laboratory for Solar and Space Physics, NASA Goddard Space Flight Center Greenbelt, MD, USA e-mail: [email protected]
26.1.2 Autoionization, Autodetachment, and Radiative Decay
If the initial, composite system is neutral, or positively charged, and its constituent decay particles are an electron and the residual ion, then the process is called autoionization. If the original system is a negative ion, so that the residual heavy particle system is neutral, then the process is technically called autodetachment; for the most part, the physics and the mathematical treatment are the same.
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_26
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A. K. Bhatia and A. Temkin
It is also possible, before electron emission takes place, 26.2.1 Optical Potential that the system will alternatively decay radiatively to an autoionization state of lower energy or a true bound state of the Straightforward manipulation of Eqs. (26.1) and (26.3) leads composite system. The latter process is called radiative sta- to an important relation between Q and P bilization (which is a basic part of dielectronic recombination (26.6) Q D .E QHQ/1 QHP : (Chap. 59)). From Eq. (26.6), a basic equation for P (which, by virtue of Eq. (26.5), contains all the scattering information) emerges (26.7) .PHP C Vop E/P D 0 : Autoionizing states are formed by scattering processes and photoabsorption. These are the inverses of the autoionization The most significant part of Eq. (26.7) is the optical potential and photon emission processes by which they can decay. In Vop given by the scattering process, formation of the autoionization state corresponds to a resonance in the scattering cross section (26.8) Vop D PHQ.E QHQ/1 QHP I (Chap. 49). Autoionization is the process that corresponds to the decay of the resonance. The decay of the resonance (auVop is a nonlocal potential; the most incisive way to give it toionization) is then seen to be the last half of the resonant meaning is to define the QHQ problem. scattering process. Strictly speaking, therefore, the resonant or autoionization state, although it may be long lived, is not completely stationary, and that is the reason that the word 26.2.2 Expansion of Vop : The QHQ Problem seemingly was used to describe the Auger process. The compound system can also be formed by absorption of photons The following eigenvalue problem constitutes the heart of impinging on a bound state (usually the ground state) of projection operator formalism the compound system, in which case the autionization state shows up as a line in the absorption spectrum. (26.9) QHQ˚ D E ˚ :
26.1.3 Formation, Scattering, and Resonances
n
26.2
Projection Operator Formalism
In the energy domain where the Schrödinger equation (SE) H D E
(26.1)
n
n
For calculational purposes, it is best to recast Eq. (26.9) in the variational form h˚QHQ˚i ı D0: (26.10) h˚Q˚i
describes scattering, the wave function does not vanish at in- This equation may yield a discrete plus a continuous spectrum in an energy domain where the SE has only a continufinity, i.e., (26.2) ous spectrum. Moreover, if the Q operator is appropriately lim ¤ 0 ; ri !1 chosen, then the discrete eigenvalues En are close to the where ri is the radial coordinate of the i-th electron. desired class of many-body resonances, called variously The basic idea of the projection operator formalism [1] is Feshbach resonances, core-excited [3], or doubly excited to define projection operators P and Q, which separate states. In terms of these solutions, Vop has the expansion into scattering-like (P ) and quadratically integrable (Q ) Z X parts PHQj˚ni.E En /1 h˚n jQHP : (26.11) Vop D D P C Q : (26.3) Implicit in Eq. (26.3) are completeness
P CQ D1;
idempotency
P2 D P ;
Q2 D Q ;
orthogonality PQ D 0 ; and the asymptotic properties 8
0, ju i by jai, for the N N discrete electron core states, and " by e N < 0, ju i by j`i, `
(27.29)
Z N
˝ ˇ Nˇ ˛ XX n ˇ ˇ x ˙DF n D hxjmi m a jV ja
and the corresponding hNR , used in the HF case, is p2 Z˛ : 2m r
(27.28)
where
Z˛ D ! hR .r/; hR .r/ D ˛ p C mˇ ; (27.25) r
hNR .r/ D
hnjn0 i D ınn0 :
The states jni (valence, core, and negative energy states) are orthonormal and complete. In the coordinate basis, Eq. (27.28) takes the more familiar form
(27.24)
N where ˙DF is the kernel defined in Eq. (27.27). For the R (DF) case,
g!1
in the DF (HF) approximation. The DF (HF) equation is the homogeneous equation corresponding to the inhomogeneous Eq. (27.24). Thus,
VDF .x; y/ d 3 yhyjni :
(27.30)
One can also generate equations corresponding to higher approximations than DF using the same approach. For example, one can obtain a Brueckner equation [20],
N ˙BN .en / jni D 0 ; en h ˙DF
(27.31)
and the states now satisfy the orthonormality condition Z lim
!!en
d 3 x d 3 yhnjxihyjn0 i ˚ ! ˚ .! en /ı 3 .x y/ hxj ˙BN .!/ ˙BN .en / jyi ! en
for the continuum of negative energy states (which do not (27.32) D ınn0 ; N contains appear in the HF approximation). The kernel ˙DF only core states jai, in both the DF and HF approximations, where the energy-dependent kernel ˙B .en / arises from irreand is given by the sum over states ducible Feynman diagrams involving two Coulomb photons. The kernel is given by N Z X 1 N jni D ˛ hmjxihajyi d 3x d 3 y hmj˙DF jx yj hmj˙BN .en /jni aD1 Z h ˇ ˇ ih ˇ ˇ i X 1=2 .hxjnihyjai hxjaihyjni/ m ˇ ˇi i ˇ ˇn D a ˇV ˇj j ˇV ˇa e C e e e N n a i j X
m a a;i;j n m Z a jV ja a jV jn h ˇ ˇ ih ˇ ˇ i X 1=2 m ˇ ˇa a ˇ ˇn aD1 C i ˇV ˇb b ˇV ˇi : (27.33) N e C e e e n i a b X n m a;b;i a jV ja R aD1 P involves summations over core states .a; b/, and summa(27.27) tion and integration over valence states .i; j /. In perturbation for arbitrary states m and n. The first term in brackets is the theory, the lowest order contribution of ˙B .en / is an ee corHartree term and the second is the electron exchange term relation term.
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Green’s Functions of Field Theory
27.3
409
The Four-Point Green’s Function
for the 2P and 2H cases, respectively. The antisymmetry under exchange follows from the definitions Eqs. (27.40) and The four-point Green’s function [4, 8, 11], or two-body prop- (27.41): agator, is defined as ./ .12/ D ./ .21/ ; ./ D '.ij / ; .ab/ ; (27.42) ˝ ˇ ˇ ˛ G N .12; 10 20 / N0 ˇT .1/ .10 / .2/ .20 / ˇN0 : (27.34) where it is understood that the suppressed spinor indices are interchanged as well as the coordinate and time variables. In order to avoid unnecessary complication, only simple The three amplitudes defined above satisfy Bethe– ladders of Vee V are considered. There are 4! possible Salpeter (BS) equations. The PH/HP case is the analog of time orderings. Of these, there are four groups of four with the positronium atom and the 2P case is analogous to He. t1 ; t10 > t2 ; t20 , t1 ; t10 < t2 ; t20 , t1 ; t2 > t10 ; t20 , and t1 ; t2 < t10 ; t20 , For the case of Coulomb ladder exchanges, to which we have corresponding to the particle–hole (PH/HP) and the two- restricted ourselves, these BS equations can be reduced to particle/two-hole (2P/2H) cases for the first eight and last simpler ones, with the relative time set equal to zero (the eight time orderings, respectively. For each of these cases, Salpeter equation [21]). The corresponding BS wave funcintroduce a total time and relative time variable defined by tions in the DF (HF) basis (rather than the coordinate basis), are for PH/HP, 1 1 0 T D .t1 C t10 / ; T D .t2 C t20 / ; Z 2 2 d 3 1 d 3 10 hmj1ih1j.i a/.0/j10 ih10 jni ; hmjN .i a/ jni 0 t D t1 t10 ; t D t2 t20 ; (27.35) (27.43) for the first eight cases (PH/HP) and with m D k and n D c, or m D c and n D k, and for 2P/2H, 1 1 Z T D .t1 C t2 / ; T 0 D .t10 C t20 / ; 2 2 hmnjN./ i d 3 1 d 3 2hmj1ihnj2ih12j./ .0/i ; (27.44) 0 t D t1 t2 ; t D t10 t20 ; (27.36) where ./ D '.ij / ; .ab/ . The states jii; jj i; jai; jbi; jmi, for the last eight cases (2P/2H). For a particular set of eight and jni label one-particle DF (HF) eigenkets and time orderings, a time translation with respect to the relevant ei ; ej ; ea ; eb ; em , and en , the corresponding eigenvalues. c.m. time t or t 0 , followed by a separate Fourier transformaIn the PH/HP case, the BS equation is tion for each case with respect to T T 0 (with integration .!.i a/ ek C ec /hijN .i a/ jai variable d˝), yields contributions with poles in the sepaZ nh ˇ ˇ i h ˇ ˇ i o X k ˇ ˇj k ˇ ˇb rately defined ˝-planes at hj j N hbj N jbi C jj i ; D ˇV ˇV ˇ ˇ .i a/ .i a/ b c j c !.i ˛/ D E.iN˛/ E0N ;
for PH
(27.37a)
E0N
for HP
(27.37b)
!.i ˛/ D
E.iN˛/
N C2 N !.ij / D E.ij / E0 ;
!.ab/ D
E0N
N 2 E.ab/
;
;
for 2P
(27.38a)
for 2H :
(27.38b)
Equations (27.37a), (27.37b) parallel Eqs. (27.19) and (27.20) as generalizations of Koopmans’ theorem. The spectral representation is of a form similar to Eq. (27.14), with wave functions corresponding to Eqs. (27.16) and (27.17) given by .i a/ .110 / h1j.i a/ .t/j10 i ei!.i a/ T ˝ ˇ ˇˇ E D N0 ˇT .1/ .10 / ˇ.iNa/
j;b
(27.45) In the 2P/2H case, the coupled pairs of BS equations are Z ˝ ˛ Xhc ˇˇ ˇˇm i˝ ˇ ˛ ˇN .!./ ec ed / cd jN./ D d ˇV ˇn mn ./ ; m>n
Z ˝ ˛ Xhk ˇˇ ˇˇm i˝ ˇ ˛ ˇN .!./ ek e` / k`jN./ D ` ˇV ˇn mn ./ ; m>n
(27.46) (27.39)
for the PH/HP case and '.ij / .12/ h12j'.ij / .t/i ei!.ij / T ˇ E D ˇ ˇ ˇ D N0 ˇT Œ .1/ .2/ˇN.ijC2 / ; ˛ 0 .ab/ .10 20 / h10 20 j.ab/ .t 0 / ei!.ab/ T ˇ D ˇˇ E 2 ˇ D N.ab/ ˇT .10 / .20 / ˇN0 ;
.!.i a/ ec C ek /hajN .i a/jii Z nh ˇ ˇ i h ˇ ˇ i o X c ˇ ˇj c ˇ ˇb hj j N hbj N jbi C jj i : D ˇ ˇ ˇV ˇV .i a/ .i a/ b j k k
(27.40)
with m and n in these equations labeling either both core, or both valence states. The wave functions also satisfy the additional conditions: hcjN .i a/ jd i D hkjN .i a/ j`i D 0 ; PH/HP (no 2P/2H terms)
(27.47)
hckj'N.ij / i D hckjN.ab/ i D 0 ; (27.41)
2P/2H (no PH/HP terms)
(27.48)
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G. Feldman and T. Fulton
The single indices a; b; c; d refer to core and i; j; k; ` to (with 3 replaced by r) valence DF (HF) states. The BS wave functions satisfy the N .rI !1 !2 / orthonormality conditions Z X juj iMjN` C1.r/hu` j X D .!1 "j C i/.!2 "` C i/ .hkjN .i a/ jcihcjN .i a/0 jki C hcjN .i a/ jkihkjN .i a/0 jci/ j`
k;c
D ı.i a/.i a/0 ;
(27.49)
C
Z X
N 1 jv iM .r/hv j
.!1 " i/.!2 " i/
C
(27.52)
for the PH/HP case, and X˝ ˇ ˛˝ ˇ ˛ X˝ ˇ ˛˝ ˇ ˛ N./ ˇcd cd ˇN. 0 / N./ ˇk` k`ˇN. 0 / ˙ c>d
!
k>`
D ı./. 0 / ; (27.50) where C./ corresponds to the 2H(2P) case and ./ D .ab/ or (ij ). The PH/HP case in the GFA, involving Coulomb ladders for the ee interaction, is just the R and NR RPA [2]. The labels c; d should also refer to antiparticles, but the contributions of the integrals from these terms to Eqs. (27.45) and (27.46) are negligible.
27.4
Radiative Transitions
For the majority of applications, one begins with the function N .12I 3/, the reducible three-point vertex: ˝ ˇ ˇ N .12I 3/ D N .1/j k .3/ 0 T
ˇ ˛ .2/ ˇN 0 :
(27.51)
The usual strategy is followed. The spectral representation serves to identify the functions of ultimate interest. (Energies are not relevant in this case.) One then generates Dyson equations in the chosen approximation by summing an infinite series of perturbation terms. There are 3! time orderings in Eq. (27.51). As with the four-point function, not all of them are subsequently useful. The two useful cases are t1 > t3 > t2 , and t2 > t3 > t1 . To obtain a spectral representation for these two cases, one first carries out a time translation of t3 , using the operator exp.iH t3 / of Eq. (27.3), so that t1 ! 1 D t1 t3 , t2 ! 2 D t2 t3 . One then introduces complete sets of intermediate states. The functions defined in connection with the two-point function in Eqs. (27.16) and (27.17) will now appear, as well as the radiative transition amplitude, defined in Eq. (27.10). If one next carries out a separate time translation of 1 and 2 and Fourier-transforms the resulting expressions, one obtains
We next define the three-point irreducible electron vertex N from the reducible vertex N (the electron legs in N are missing in N / as h1j N .rI !1 !2 /j2i Z Z d 3 10 d 3 20 h1jG!N1 j10 ih10 jN .rI !1 !2 /j20 i h20 jG!N2 j2i ; (27.53) and pick out the residues of the !1 and !2 poles at specific energies "m and "n . [N .rI !1 !2 / has no such poles.] With the equivalent of the DF(HF) approximation to the Dyson equations in Eq. (27.57) below, the corresponding kets form an orthonormal set, and scalar products are calculated with respect to hmj and jni. The second term on the right-hand side of Eq. (27.52) refers to hole states and is of less interest than the first term describing 1P–1P transitions. From the first term, the transition matrix element in terms of N is MfiN C1 .rI k0 / D hf jN .rI ef ei /jii :
(27.54)
Generation of a Dyson Equation The approximation of only Coulomb ladder ee interactions in the two-point Green’s function produces an infinite set of Feynman diagrams to which one end of a single transverse photon line is attached in all possible ways. A resummation N functions, which are of these diagrams generates the G!;Coul approximated by the DF (HF) propagators written in their spectral form. The scalar products leading to Eq. (27.54) can then be taken. A further simplification results because of the Coulomb ladder approximation, which is the same as in the four-point Green’s function case: the relative time can be set equal to zero, which corresponds to integrating over the relative frequency, ! D !1 !2 . With the total frequency defined as ˝ D 12 .!1 C !2 /, and the definitions Z 1 1 N N N .r; ˝/ D d! rI ! C ˝; ! 2 2 Z ikr ˝ ˇ ˇ ˛ ˇ ˛ ˝ ˇ N e d 3r p mˇN N .r; ˝/ˇn ; mˇN .˝/ˇn D 2k0 ˝mn D en em D k0 ;
1 ˝ 2
;
(27.55)
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411
with a relation similar to Eq. (27.55) for the quantity k .r/ defined in Eq. (27.11), i.e., Z eikr d 3r p hmjk .r/jni ; (27.56) hmj.k0 /jni D 2k0 the matrix elements in the DF (HF) basis are [14, 20] hmjN N .k0 /jni D hmj.k0 /jni Z h ˇ ˇ i X 1 m ˇ ˇn C ˇV ˇa hajN N .k0 /jj i ea ej C k0 j aj h ˇ ˇ i 1 m ˇ ˇn N N hj j C .k /jai : (27.57) ˇ ˇV 0 j ea ej k0 a As discussed in Sect. 27.2, the label a should include not just hole states but also negative energy states, which have been neglected in Eq. (27.57). Note also that only PH or HP matrix elements appear on the right-hand side of Eq. (27.57). Therefore, a closed set of inhomogeneous linear algebraic equations for hbjN N .k0 /j`i and h`jN N .k0 /jbi results from setting m D b, n D ` or m D `, n D b, respectively, in Eq. (27.57). These equations can be solved numerically, and the resulting hajN N .k0 /jj i and hj jN N .k0 /jai substituted in Eq. (27.57) to obtain the final result MfiN C1 .k0 / hf jN N .k0 /jii :
(27.58)
An integro-differential equation form, which provides the option of choosing alternative numerical techniques [14], is obtained from Eq. (27.57), after some rearrangement and passage to a coordinate basis. Defining mfi .k0 / from .k0 / in analogy with the definition of MfiN C1 .k0 / from N N .k0 / in Eq. (27.58), one has X MfiN C1 .k0 / D mfi .k0 / C .haj.k0 /jA i a
C hAC j.k0 /jai/ ; where Œh.r/ k0 ea hrjA˙ i C
(27.59)
Xh ˇˇ ˇˇ i r A˙ a ˇV ˇa a
Z X hrjj ihj jŒv˙ C V˙ jai ; D
(27.60)
j
h ˇ ˇ i h ˇ ˇ i j ˇ ˇa j ˇ ˇa ; hj jv jai D ˇV ˇ f i f ˇV ˇi ; Xnhj ˇˇ ˇˇ i hj ˇˇ ˇˇ io a a hj jV˙jai D : b ˇV ˇB˙ C B ˇV ˇb
hj jvC jai D
(27.61) (27.62)
b
The three-point Green’s function, as can be seen from this summary, describes transition amplitudes for radiative transitions between two valence states of atoms with closed shells (subshells) plus one electron (the 1P case). In the Coulomb
ladder approximation, closely related to the DF equation, the photon vertex h1jN .rI !1 !2 /j2i of Eq. (27.53), is nonlocal in space, rather than the local vertex h1jk .r/j2i of Eq. (27.11) [as follows from the factor of ı 3 .1r/ı 3 .2r/]. The presence of these additional nonlocal contributions to MfiN C1 .k0 / is made apparent in Eq. (27.59). The mfi .k0 / term is the contribution of the local vertex. Of course, if one knew the exact N C 1 electron wave functions and energies, only the local vertex would enter and a GI result would be obtained. However, the length and velocity versions of Eq. (27.12) are equal and GI even in the approximation just discussed [3, 7]. Gauge invariance is an essential constraint on radiative transition amplitudes. It has been proven for general gauges not only in the present approximation, but also in the ones discussed below [8, 10], all of them arising in the GFA. Since the effective potential in the DF (or HF) approximation is nonlocal, the effective current must also be nonlocal in order to maintain the GI. The somewhat more complicated Dyson equation satisfied by the nonlocal vertex corresponding to the Brueckner approximation has also been generated [22], and put in a numerically implementable form. An alternative approximation [3, 7] for transition matrix elements, proposed earlier [2] than the one just discussed, is based on the RPA [our PH/HP case, with wave function N .f a/ the solution of Eq. (27.45)]. Reference to Eqs. (27.6), (27.7), (27.10) and (27.11) gives immediately MfNa .rI !.f a/ / D k .r/hrjN .f a/ jri ;
(27.63)
where k0 D k D !.f a/ and N corresponds to atoms with a closed shell or subshell of electrons. Aside from being a different approximation to transition amplitudes than Eqs. (27.54), (27.63) covers a different set of cases than does Eq. (27.54), since the corresponding initial states i in Eq. (27.54) are restricted to core states a in Eq. (27.63). Thus, for example, the case N D 2 in Eq. (27.54) can describe transitions between any two valence states of Li. The corresponding case for Eq. (27.63) is N D 4, but only transitions to a higher level, originating in the 1s or 2s level of Li, can be described by the formalism. Finally, we shall discuss radiative transitions [8] for 2P atoms (closed shell/subshell plus two valence electrons). The general case involves a five-point nonlocal vertex. However, in the ladder approximation, this reduces to transition matrix elements which contain combinations of DF, 2P BS wave functions, and three-point functions. The final expressions are MfiN C2 .rI k0 /
! ˇ N N .r; k0 / N N .r; k0 / ˇˇ N ˛ N ˇ V V .i q/ : D .fp/ k0 H k0 H ˝
(27.64)
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G. Feldman and T. Fulton
The expressions appearing in this equation are, in more detail, in the DF basis: ˇ + * ˇ mˇˇ N N .r; k0 / ˇˇq hmjN N .r; k0 /jqiıns ; (27.65) ˇ ˇ k0 C eq em n ˇ k0 H ˇ s q
where m, n, q, s label DF states (P or H), js i jqsi jqijsi; Eq. (27.65) serves to define H , as a difference of two DF Hamiltonians, the argument r of N refers to the electron which emits the photon, and Z Z ˝m ˇ ˇ ˛ X m c ˝ ˇ ˛ X m ˇk ˝ ˇ ˛ ˇ ˇ N ˇN ˇV ˇN./ D n n jV jd cd ./ C n jV ` k` ./ : cd
k`
(27.66) In Eq. (27.66) we used the fact that, in the DF basis, jN./ i only has 2P or 2H components Eqs. (27.46), (27.48).
27.5
Radiative Corrections
This section summarizes radiative corrections for 1P/1H atoms, starting from the two-point Green’s function. The Dyson equations are generated and solved perturbatively for the energy to order ˛ 5 m (˛ 3 a.u.). The perturbation theory starts from the DF solution as the zero-order one. As done for the three-point function in Sect. 27.4, the Dyson equation for radiative corrections involving a single transverse photon is generated by expanding the two-point propagator to all orders in the Coulomb ladder approximation, inserting a transverse virtual photon in all possible ways, and then resumming. The resulting integral equation [23] is quite complicated, containing even the threepoint vertex in its inhomogeneous term, as well as a mass counter term to eliminate divergences. It is sufficient for the present purposes to expand these vertices through first order matrix elements of Vee V , but to exclude matrix eln ements of type Œm a jV ja (where a is the label for core states), since these are already included in the DF approximation [see Eq. (27.27)]. Other radiative corrections are also generated, which include Coulomb photons. Two of these corrections, involving only such photons, have already been referred to in the text between Eqs. (27.21) and (27.23). Finally, to order ˛ 5 m, corrections involving two transverse photons must be included. Systematic application of the pole-shifting process in Eq. (27.23) to the one- and two-transverse-photon and the Coulomb-photon expressions yields two types of terms: first, photon exchange (of one transverse photon or of two-photons of either kind, Coulomb or transverse) between core and valence states; and second, self-energy and vacuum polarization contributions. Among the photon exchange terms,
one can identify those contributions included in other approaches (at least for a few special cases, such as those arising from a single transverse photon interaction, and from two Coulomb interactions and Coulomb–Breit interactions with positive energy intermediate electron states). These are electron correlation terms which are included as part of a MBPT calculation [5], or one which involves consideration of an infinite subset of MBPT terms [6]. They need not be re-evaluated. The terms not covered by calculations of the type in [5, 6], which are of O.m˛ 5 c 2 /, involve retardation in Coulomb-transverse photon exchange, negative energy intermediate electron states for two Coulomb and Coulomb-transverse photon exchanges, two transverse photon exchange, self-energy and vacuum polarization terms, as well as anomalous magnetic moment corrections. After lengthy calculation, one obtains final results in finite analytic form, which are numerically implementable. The results given below are obtained after further approximations. First, the remnants of the original integral equation are solved iteratively. Second, the self-energy terms are calculated in a joint expansion [24–26] in ˛ and ˛Z, and are thus only valid for low Z in isoelectronic sequences. Finally, there are characteristic logarithmic terms generated by low virtual photon momenta in the self-energy contributions. A standard approach is to scale these with a factor of .˛Z/2 to obtain Bethe log (BL) terms as constants independent of ˛, together with constant log (CL) terms of the form ln.˛Z/2 . The BL terms are independent of Z for hydrogenic ions, and remain nearly so for other atoms. BL and CL terms are associated with both nuclear and ee Coulomb potentials. The ee CL terms arising from core and valence self-energy terms are canceled exactly by those coming from Coulomb and transverse photon exchange. Thus, the numerous and rather complicated corresponding ee BL terms, which also are individually small compared to their associated CL terms, should additionally almost cancel, and are therefore neglected. One finally obtains (with all state labels denoting principal and orbital quantum numbers as well as spin indices: n .n; `I ms /) E.n; `/ D Zjh0jnij2F .n; `/ ;
(27.67)
in a.u., where h0jni h0jn; 0i is the s-state HF wave function at the origin of coordinates, and serves to define an effective (shielded) nuclear charge Zn;eff : hrjn; `i n;` .r/; jh0jnij2
1 .Zn;eff ˛/3 : n3
(27.68)
(Just as in hydrogen, for which the final expressions for the radiative corrections require NR and not R wave functions and energies, so in this case we use HF (NR) and not DF (R) quantities.) F .n; `/ consists of a valence or hole contribution
27
Green’s Functions of Field Theory
413
Z i=2 X X .2ea ek eq / n ˇˇ ˇˇn Fv;h , a core term Fcore .n; `/, and one due to photon exchange U.2/ .n`/ D q V k between electrons Fee .n; `/: jh0jnij2 a eq ek k 5 contributions can be neglected, and Fig. 29.4 Fractional absolute values of the contributions to the theoret.2n/
Ce.2n/ D A1
ical g-factor anomaly of the free electron. QED contributions are due
.2n/ .2n/ .2n/ the mass-independent A1 (black markers), to the muon-dependent C A2 .xe / C A2 .xe£ / C : : : ; (29.11) to.2n/ .2n/
.2n/ A1
.2n/ A2 .x/
with coefficients and functions evaluated at x D xeX me =mX 1 for lepton X D or £. For n > 1, .2n/ the coefficients A1 include vacuum-polarization correc.2n/ tions with virtual electron/positron pairs, while A2 .x/ are vacuum-polarization corrections due to heavier leptons. For .4/ .6/ x ! 0, we have A2 .x/ D x 2 =45 C O.x 4 / and A2 .x/ D 2 4 x .b0 C b1 ln x/ C O.x / with b0 D 0:593274: : : and b1 D 23=135 [9, 10]. The O.x 4 / contributions are known and included in the calculations but not reproduced here. The .8/ .10/ functions A2 .x/ and A2 .x/ are also O.x 2 / for small x but are not reproduced here [11, 12]. Currently, vacuumpolarization corrections for the free-electron anomaly that depend on two lepton mass ratios can be neglected. Table 29.1 lists the relevant coefficients as used in the 2014 CODATA adjustment. Uncertainties in parenthesis here and elsewhere are combined one-standard-deviation statistical and systematic uncertainties. Additional references to the Table 29.1 Coefficients for the QED contributions to the electron .2n/ .2n/ anomaly. The coefficients A1 and functions A2 .x/, evaluated at x D xe me =m and xe£ me =m£ for the muon and tau leptons, respectively, are listed with two significant digits for ease of compar.2n/ ison; summed values Ce , based on the 2014 CODATA adjustment, are listed as accurately as needed for the tests described in this article. Missing values indicate that their contribution to the electron anomaly is negligible n 1 2 3 4 5
.2n/
A1 1=2 0:33 1:2 1:9 7:8
.2n/
A2 .xe / 0 5:2 107 7:4 106 9:2 104 3:8 103
.2n/
A2 .xe£ / 0 1:8 109 6:6 108 7:4 106
.2n/
Ce 0.5 0:32847844400: : : 1:181234017: : : 1:91206.84/ 7:79.34/
A2 .xe / (red markers), and to the tau-dependent A2 .xe£ / (blue markers) corrections, respectively. Contributions due to the weak and hadronic interactions are also shown. The horizontal orange line shows the relative uncertainty in the determination of ae .th/. The graph is based on the 2014 CODATA value for the fine-structure constant
original literature can be found in the paper on the adjust.8/ ment. It is worth noting that since 2014 the coefficient A1 has been evaluated virtually exactly in a tremendous effort .10/ described in [13], while A2 has been updated, shifting its value [14]. The electroweak contribution is ae .weak/ D 0:02973.23/ 1012
(29.12)
and is calculated as discussed in the 1998 CODATA adjustment but with the 2014 values of the Fermi coupling constant GF =.„c/3 and the weak mixing angle W [15]. The total hadronic contribution is ae .had/ D 1:734.15/ 1012
(29.13)
and is a sum of multiple vacuum-polarization contributions. Figure 29.4 shows a graphical representation of contributions to the electron anomaly. The QED corrections decrease roughly exponentially in size with order n for both mass-independent and dependent contributions. Contributions from virtual loops containing £ leptons are currently negligible. Weak and hadronic contributions are more important. The theoretical uncertainty of the anomaly (absent any uncertainty in the fine-structure constant) is dominated by two contributions: the mass-independent n D 5 QED correction and the hadronic contribution. In fact, it is given by uŒae .th/ D 0:037 1012 D 0:32 1010 ae
(29.14)
29
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E. Tiesinga and P. J. Mohr
and is significantly smaller than the uncertainty of the mostaccurate experimental value of 2:4 1010 ae [6]. Consequently, the relative uncertainty of the fine-structure constant is essentially the same as the experimental relative uncertainty. That is, equating the theoretical expression of ae .th/ and the experimental value yields ˛ 1 .ae / D 137:035999160.33/ Œ2:4 1010 : (29.15)
in kg in the revised SI to determine m.12 C/ with a relative uncertainty of 1:2 109 through error propagation. Similarly, relationships can be used for the mass of 28 Si.
29.6 Mass-Ratio Measurements Using the g-Factor of Hydrogen-Like Ions
Measurements of the spin-flip and cyclotron frequencies of The number in square brackets is the relative uncertainty of ground-state hydrogen-like atomic ions in a homogeneous the fine-structure constant, ur .˛/ D u.˛/=˛. magnetic field are currently the most accurate means to determine atom-to-electron mass ratios. This method relies on the ability of theorists to calculate the g-factor of the bound 29.5 Atom Recoil Experiments electron accurately. This approach is competitive as long as and Mass Spectrometry the trapped ion is relatively light. Accounting for the electron removal energies of ions becomes prohibitively hard for Atom recoil experiments are performed with the heavier heavy atoms. Similarly, computing the g-factor within the alkali-metal atoms, 87 Rb and 133 Cs. These atoms have easily framework of relativistic QED bound-state theory is more accessible atomic transitions in the optical frequency do- complex for heavier hydrogen-like ions as corrections with main, with excited states that only decay back to the ground higher powers in Z˛ are more important. For nuclei with state and can, thus, be brought to a near standstill by laser zero nuclear spin, the mass ratio follows from rearranging cooling techniques. For the 2014 CODATA adjustment, the Eq. (29.8) to most accurate measurement relied on the 87 Rb isotope [16] ˇ and gave 1 !s .Y qC / ˇˇ m.Y qC / D 2q (29.20) me gY .˛; : : :/ !c .Y qC / ˇexp m.87 Rb/ D 1;368;480;428:9.1:7/ m2 s Œ1:2 109 : „ and (29.16) q1 In the revised SI with a fixed reduced Planck constant, this X EI .Y mC / m.Y / m.Y qC / 87 D Cq ; (29.21) translates into an equally accurate measurement of m. Rb/ me me me c 2 mD0 in kg. A 2018 value for m.133 Cs/=„ for atomic cesium with a three times smaller relative uncertainty is reported in [17]. where EI .Y mC / is the positive ionization energy of ion Y mC , In mass spectrometry, the most accurate measurements and we recall that q D ZY 1. compare the mass of two or more low-charge-state ions. Over A broad program involving researchers from a numthe years, these measurements have been performed for most ber of European laboratories has measured spin-flip and stable and unstable atoms in the periodic table. After accyclotron frequency ratios and calculated the g-factor for counting for the mass of the electrons and binding energies, different ions, most notably 12 C5C and 28 Si13C . The measuremass data for neutral atoms were collated in [18, 19] in the ments themselves were performed at the Max-Planck Institut atomic mass unit mu . For 87 Rb and 28 Si, this 2012 database für Kernphysik (MPIK), Heidelberg, Germany. Recently regives ported values are [21, 22] 87
m. Rb/ !s 12 C5C D 86:9091805319.65/ Œ7:5 1011 ; Ar .87 Rb/ D D 4376:21050087.12/ Œ2:8 1011 ; (29.22) mu 12 C5C / ! . c (29.17) and
and [23]
Ar . Si/ D 27:97692653465.44/ Œ1:6 10 ; (29.18) !s 28 Si13C
D 3912:86606484.19/ Œ4:8 1011 : 28 Si13C ! c respectively, which have a much smaller relative uncertainty (29.23) than that for masses determined by atom recoil measureThese two frequency ratios are correlated with a correlation ments. The improved 2016 evaluation can be found in [7, 20]. coefficient r D 0:347 mainly due to image charge correcIn any case, from this recoil measurement, we calculate tions [24]. The authors of [24] also slightly reassessed the values given in the original references. The reassessed val12 m.12 C/ D m.87 Rb/ (29.19) 87 ues are quoted here. Ar . Rb/ 28
11
29 Tests of Fundamental Physics
a Ionization energy,
EI (units of 10 –3 mec2) 1
439
experiment. An evaluation of me =„ based on 28 Si data gives a consistent value with only a slightly larger uncertainty. The two values can be combined to find more accurate values for m.12 C/=me and me =„ as long as the strong correlations between the uncertainties of the g-factor, as well as those of the frequency ratio !s =!c , are taken into account. With the 2018 atom recoil experiments with 133 Cs [17], the uncertainty of me =„ has improved by a factor of 3.
b u(EI) C5+
(units of 10 –9 mec2) 2 C+
C 4+
0.8
1.5 0.6 1 0.4 0.5 0.2 C 0
0
C+ 1
C2+
C3+
C3+
C
2+
C4+ C5+
C 2
3 4 5 Charge state
0
29.6.1 Theory for the g-Factor of Hydrogen-Like Ions
0
1
2
3 4 5 Charge state
Fig. 29.5 Ionization energies (a) and their uncertainties (b) of the six mC charge states 12 C of carbon-12 expressed in units of me c 2 . Notice the large difference in scale between the two panels. In practice, ionization energies, EI , are reported in units of hc. The unit conversion of the energies and their uncertainties is based on the identity E=.me c 2 / D ˛ 2 =.2R1 / E=.hc/, where R1 is the Rydberg constant, and we observe that ˛ 2 =.2R1 /, with a relative uncertainty of 5 1010 , does not increase the uncertainty of the EI
The theoretical electron g-factors for 12 C5C and 28 Si13C have a relative uncertainty of 1:3 1011 and 8:5 1010 , respectively, as explained in the next section. Hence, the relative uncertainty for the mass ratio mY qC =me is 3:0 1011 and 8:5 1010 for these two ions, respectively. Next, we use Eq. (29.21) to determine the neutral atom to electron mass ratio. Figure 29.5 shows the ionization energies and their uncertainties for all charge states of 12 C in units of me c 2 . We observe that, as expected, the ionization energy increases with charge state, but that, possibly surprisingly, the ionization energy with the largest uncertainty by far is that for the singly charged carbon ion. The ionization energies are added to give the electron removal energy EB D 1:0569819.18/ 103 me c 2 from 12 C to 12 C5C . We combine the results of Sects. 29.5 and 29.6 to find the values m.12 C/ D 21;874:66183428.66/ Œ3:0 1011 me
We summarize the contributions to the theoretical description of the electron g-factor of ground S-state hydrogen-like ions. The main contributions to the g-factor can be categorized as gY D gD C grad C grec C gns C : : : ;
where gD is the Dirac (relativistic) value, and grad , grec , and gns are due to radiative, recoil, and nuclear size corrections, respectively. Other corrections, indicated by the dots, are negligible at this time. Numerical results for the various contributions are summarized in Table 29.2 for 12 C5C and Table 29.3 for 28 Si13C . The Dirac value is known exactly [25] from the Dirac equation for an electron in the field of a fixed point charge of magnitude Ze, where, for clarity, we omit subscript Y on Z for the remainder of this section. Its value is i p 2h (29.26) gD D 1 C 2 1 .Z˛/2 ; 3 with an uncertainty that is solely due to the uncertainty in ˛. Radiative corrections may be written as ˛ n X grad D 2 Ce.2n/ .Z˛/ ; (29.27) nD1 .2n/
where the functions Ce .x/ are evaluated at x D Z˛, correspond to contributions with n virtual photons, and are slowly varying functions of x. They are related to the coefficients .2n/ for the free electron Ce , defined in Sect. 29.4, by lim Ce.2n/ .x/ D Ce.2n/ :
x!0 .2/
and me me m.12 C/ m.87 Rb/ D „ m.12 C/ m.87 Rb/ „ D 8:637992726.11/ 103 m2 s
Œ1:2 109 : (29.24) We observe that the removal energy EB has a measurable effect on the mass of the carbon-12 atom but that the uncertainty of me =„ is currently limited by the atom recoil
(29.25)
(29.28)
The function Ce .x/ is computed as the sum of three contributions. The first contribution is a self-energy correction given by 32 1 247 1 .2/ 1 C x2 C x4 ln.x 2 / C Ce;SE .x/ D 2 6 9 216 8 8 ln k0 ln k3 C x 5 RSE .x/ ; 9 3 (29.29)
29
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E. Tiesinga and P. J. Mohr
where ln k0 D 2:984128556 and ln k3 D 3:272806545. Values for the remainder function RSE .x/ are based on extrapolations from numerical calculations at high Z. In particular, RSE .6˛/ D 22:160.10/ and RSE .14˛/ D 20:999.2/ based on the 2014 CODATA adjustment. In 2017, the values for RSE .x/ were significantly improved [26]. .2/ The second and third contributions to Ce .x/ are two lowest-order vacuum-polarization corrections [27]. In the second “wave function” correction, the vacuum polarization loop modifies the interaction between the bound electron and the Coulomb field of the nucleus, and in the third “potential” correction, the loop modifies the interaction between the bound electron and the external magnetic field. The sum of the one-photon vacuum polarization contributions are .2/
Ce;VP .6˛/ D 0:000001832142.12/ ;
for any n. This surprising result was derived in [33, 34], and for n D 1 and 2 it is evident from Eqs. (29.29) and (29.32). As the uncertainty due to uncalculated higher-order terms is .2n/ .2n/ negligible we use Ce .x/ ! .1 C x 2 =6/Ce for the three, four, and five (n D 3, 4, and 5) photon contributions with an .2n/ uncertainty solely determined by the uncertainty of Ce in Table 29.1. The corrections gD and grad are based on the assumption that nuclei have an infinite mass. The recoil correction to the g-factor associated with a finite mass is .0/ .2/ C grec ; grec D grec
(29.35)
corresponding to terms that are zero-order and first-order in .0/ ˛= , respectively. For grec , we have
(29.30)
(
) .Z˛/4 5 D .Z˛/ C .Z˛/ P .Z˛/ p and 3Œ1 C 1 .Z˛/2 2 .2/ 2 (29.31) Ce;VP .14˛/ D 0:0000505452.11/ ; me 2 me C .1 C Z/.Z˛/ ; mN mN for 12 C5C and 28 Si13C , respectively. (29.36) The n D 2 two-photon correction for the ground S-state where mN is the mass of the nucleus. Mass ratios, based is [28, 29] on the 2014 adjustment values of the constants, are 2 me =m.12 C6C / D 0:0000457275: : : and me =m.28 Si14C / D x .4/ .4/ Ce .x/ D 1 C Ce 0:0000196136: : : The authors of [35] evaluated the function 6 P .x/ numerically for a discrete set of x < 1, with the result 14 991;343 2 4 C x4 ln.x 2 / C ln k0 ln k3 P .6˛/ D 10:49395.1/ for hydrogenic carbon. For silicon, we 9 155;520 9 3 .2/ use the interpolated value P .14˛/ D 7:16223.1/. For grec , 2 2 679 1441 1441 C ln 2 C .3/ C O.x 5 / we have 12;960 720 480 ( ˛ .Z˛/2 me .2/ 0:3285778.23/ for x D 6˛ D : (29.37) grec D ; 3 mN 0:32917.15/ for x D 14˛ (29.32) The uncertainty in g .2/ is negligible compared to that of rec where .z/ is the Riemann zeta function. The quoted uncer- g .2/ . rad tainty is an estimate of uncalculated higher-order contribuFinally, the finite size of the nucleus leads to a small cortions given by [28] rection to the g-factor given by [36] ˇ 5 .4/ ˇ
.4/ 2 (29.33) u Ce .x/ D 2 ˇx Ce RSE .x/ˇ : 8 4 rN C::: ; (29.38) gns D .Z˛/ 3 C Since the remainder function differs only by about one percent for carbon and silicon, the main Z (or x) dependence of the uncertainty is given by x 5 , and we assume that the where rN is the nuclear rms charge radius, and C D „=.me c/ uncertainty of the two-photon correction is completely cor- is the reduced Compton wavelength of the electron. The related for the two ions. The authors of [30, 31] calculated authors of [37] calculated additional corrections but gave some of the two-loop vacuum polarization diagrams of or- numerical values for gns based on early 2rN values. We inder x 5 and found them to be on the order of the uncertainty corporate their corrections using an .rN / scaling and the rN D 2:4703.22/ fm and rN D 3:1223.24/ fm for in Eq. (29.32). The authors of [32] computed an additional updated 12 28 .4/ C and Si, respectively [38]. light-by-light contribution to Ce .x/, which shifts its value The theoretical value for the g-factor of the electron in within its uncertainty. It is not included here. .2n/ hydrogen-like carbon 12 or silicon 28 is the sum of the inThe leading binding correction to Ce .x/ is dividual contributions discussed above and summarized in Tables 29.2 and 29.3. For each contribution the tables also list x2 Ce.2n/ .x/ D 1 C (29.34) C : : : Ce.2n/ ; the uncertainty. For both ions the uncertainty is dominated by 6 .0/ grec
2
29 Tests of Fundamental Physics
441
Table 29.2 Theoretical contributions and total value for the g-factor of hydrogenic carbon 12 based on the 2014 CODATA recommended values of the constants and corresponding theory. The total g-factor has a relative uncertainty of 1:3 1011 Contribution Dirac gD .2/ gSE .2/ gVP g .4/ g .6/ g .8/ g .10/ grec gns g.12 C5C /
Value 1:9987213543921.6/ 0:002323672435.4/ 0:000000008511 0:000003545677.25/ 0:000000029618 0:000000000111 0:000000000001 0:000000087629 0:000000000408.1/ 2:001041590183.26/
Source Eq. (29.26) Eq. (29.29) Eq. (29.30) Eq. (29.32) Eq. (29.34) Eq. (29.34) Eq. (29.34) Eq. (29.35) Eq. (29.38)
Value 1:993023571557.3/ 0:00232891747.5/ 0:00000023481.1/ 0:0000035521.17/ 0:00000002966 0:00000000011 0:00000000000 0:00000020588 0:00000002053.3/ 1:9953489581.17/
Source Eq. (29.26) Eq. (29.29) Eq. (29.31) Eq. (29.32) Eq. (29.34) Eq. (29.34) Eq. (29.34) Eq. (29.35) Eq. (29.38)
that of the two-photon n D 2 correction g .4/ . The relative uncertainties of the 12 C5C and 28 Si13C values are 1:3 1011 and 8:5 1010 , respectively, sufficient for the purpose of determining competitive atom-to-electron mass ratios. Finally, the two g-factors have a correlation coefficient r D 0:79.
29.7
!H .1S1=2 –2S1=2 / D 2 2;466;061;413;187;035.10/ rad=s Œ4:2 1015 or H .1S1=2 –2S1=2 / D 2;466;061;413;187;035.10/ Hz in 2011 and !H .1S1=2 –2S1=2 / D 2 2;466;061;413;187;018.11/ rad=s Œ4:4 1015
Table 29.3 Theoretical contributions and total value for the g-factor of hydrogenic silicon 28 based on the 2014 CODATA recommended values of the constants and corresponding theory. The total g-factor has a relative uncertainty of 8:5 1010 Contribution Dirac gD .2/ gSE .2/ gVP g .4/ g .6/ g .8/ g .10/ grec gns g.28 Si13C /
The world’s best-known optical transition frequency is that for the 1S1=2 –2S1=2 Lyman-alpha line of hydrogen, which has been obtained by an experimental group in Garching, Germany [39, 40]. They quote
Hydrogen Atom Energy levels
Measurements of the hydrogen energy levels are currently the most precise way to determine the Hartree energy divided by the reduced Planck constant or, equivalently, the Rydberg constant. The measurements also help determine ˛ and the proton charge radius. The eigenstates are labeled by n`j , where n D 1; 2; : : : is the principal quantum number, ` D 0; 1; : : : ; n 1 is the nonrelativistic angular momentum quantum number, and j is the total angular momentum quantum number. Their energies are denoted by E.n`j / and, following the usual convention, we use S; P; D; : : : to denote ` D 0; 1; 2; : : : states. Our discussion will omit hyperfine effects from coupling of the electron to the magnetic and other moments of the proton, as at 2014 accuracy levels they can easily be accounted for.
or H .1S1=2 –2S1=2 / D 2;466;061;413;187;018.11/ Hz in 2013. The two values are correlated with a correlation coefficient r D 0:707 [41]. The frequency of this Lyman-alpha line is known to almost 15 significant digits. The analytical, theoretical determination of this transition frequency, which in its simplest description by Bohr equals 3Eh =.8h/, is equally impressive, although less accurate. It has 13 contributions ranging from the Dirac eigenvalue, QED corrections with one or more virtual photons and lepton-antilepton pairs, as well as nuclear size and recoil effects. Figure 29.6a gives a visual representation of the size and uncertainties of these contributions. We postpone the discussion of the name for and mathematical form of each of the corrections until Sect. 29.7.1. The sizes of the contributions vary by orders of magnitude, although size is not necessarily an indication of its uncertainty. The uncertainty of both the Dirac energy and the correction due to the proton charge radius are the largest at 10 kHz and reflect the uncertainty of constants Eh =„ and rp , respectively. The largest theoretical uncertainties are in the two-photon and three-photon corrections and reflect uncomputed terms. (The individual uncertainties of the Dirac energy and correction due to the proton charge radius are large compared to that of the experimental 1S1=2 –2S1=2 transition frequency. As a result the uncertainties are highly correlated with a correlation coefficient very close to one.) The 2P1=2 –2S1=2 transition, or Lamb shift, is equally important, because the Dirac or Bohr energies cancel, and the dominant contribution by far is the self-energy correction, a QED correction where the bound electron emits and absorbs a virtual photon. It is of order ˛ 3 Eh as seen in Sect. 29.7.1. The energy contributions to this transition are shown in Fig. 29.6b, while the two best measurements of this transition frequency are H .2P1=2 –2S1=2 / D 1;057;845:0.9:0/ kHz Œ8:5 106
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E. Tiesinga and P. J. Mohr
a H 1S1/2–2S1/2 Dirac energy Self energy Vacuum polarization Rel. recoil Proton charge radius Two photon Radiative recoil Muonic vacuum pol. Nucl. self energy Had. vacuum pol. Three photon Nucl. size on SE & VP Nucl. polarization 102
104
106
108
1010
1012
1014
1016 E/h (Hz)
in the proton charge radius, while uncertainties from uncomputed terms in the two-photon and three-photon corrections are much smaller than the experimental uncertainty. In a naive least-squares adjustment that includes only these two transitions, the Lyman-alpha line and Lamb shift determine (relatively inaccurate) values for ˛ and me . Data on other transitions and the measurements discussed in the previous sections significantly decrease their uncertainties, constrain the value for the proton charge radius, and check for consistency. Indeed, the focus has shifted to complementary transitions in H [44, 45] and measurements on and theory of muonic-hydrogen with the goal of resolving the discrepancies in the determination of the charge radius of the proton. In the 2014, CODATA adjustment muonic-hydrogen data was not included.
b H 2P1/2–2S1/2 Self energy Vacuum polarization Rel. recoil Proton charge radius Two photon Dirac energy Radiative recoil Muonic vacuum pol. Nucl. self energy Three photon Had. vacuum pol. Nucl. size on SE & VP Nucl. polarization
29.7.1
Theory for the Hydrogen Energy Levels
Theoretical values for hydrogen energy levels as for the bound-electron g-factor are determined by the Dirac eigenvalue, QED effects such as self-energy and vacuum polarization, and proton size and motion or recoil effects. Theoretical energies of different states are correlated. For example, for S-states, uncalculated contributions are primarily of the form of an unknown constant divided by n3 . This is taken into account by using covariances between levels in addition to the uncertainties of the individual levels. Hence, we distinguish 101 102 103 104 105 106 109 107 108 between components of uncertainty that are proportional to E/h (Hz) 1=n3 and those that are uncorrelated, where necessary. They are denoted by u0 and un , respectively. Fig. 29.6 Absolute value of the 13 contributions (gray bars) and their We now consider each of the contributions to the energy uncertainty (red and orange bars) to the hydrogen 1S1=2 –2S1=2 (a) and in turn, as well as explain how to combine the uncertainties 2P1=2 –2S1=2 (b) transition frequencies on a logarithmic scale. The valof the contributions. Table 29.4 gives a list of the contribuues are sorted by the size of the contribution. The label next to each contribution is defined in the text. The uncertainty of a contribution is tions, their size both in terms of fundamental constants, as either fully due to the uncertainty of constants within QED theory, here well as order-of-magnitude numerical values for the 1S – 1=2 the Hartree frequency, Eh =„, and proton charge radius (orange bars), transition. 2S 1=2 or by the estimated value of missing or uncomputed terms (red bars). The two transition frequencies have been measured to 10 and 9 kHz, respectively
and H .2P1=2 –2S1=2 / D 1;057;862.20/ kHz Œ1:9 105 ;
Dirac Eigenvalue The energy of an electron in a static Coulomb field with charge Ze with infinite mass is predominantly determined by the relativistic Dirac eigenvalue ED D f .n; /me c 2 ;
from research groups at Harvard University [42] and the University of Sussex [43], respectively. The theoretical value for where this transition from the 2014 CODATA least-squares adjustment is
(29.39)
1=2 .Z˛/2 ; (29.40) f .n; / D 1 C .n ı/2 6 H .2P1=2 –2S1=2 / D 1;057;843:7.2:1/ kHz Œ2:0 10 p with defect ı D j j 2 .Z˛/2 and is the angular and is in agreement with the experiments. The largest contri- momentum-parity quantum number ( D 1; 1; 2, 2, 3 for bution to the theoretical uncertainty is due to the uncertainty S1=2 , P1=2 , P3=2 , D3=2 , and D5=2 -states, respectively). States
29 Tests of Fundamental Physics
443
Table 29.4 List of contributions and their main dependence on fundamental constants to the hydrogen transition frequencies ordered by appearance in the text. The first two columns give the section in Sec. 29.7.1 in which the contribution is described in detail and the name of the contribution, respectively. The fundamental-constant dependence of a contribution in the third column is given in terms of the Hartree energy Eh D me c 2 ˛ 2 and four dimensionless variables with values small compared to one: the fine-structure constant ˛, the proton charge radius divided by the reduced Compton wavelength rp =C , and the mass ratios me =m and me =mp . The last column gives the order of magnitude of each of the contributions in Hz for the 1S1=2 –2S1=2 transition a b c d e e e f g h i j k
Contribution Dirac energy Relativistic recoil Nuclear polarizability Self-energy Vacuum polarization Muon vacuum polarization Hadronic vacuum pol. Two-photon Three-photon Nuclear size Nucl. size on SE & VP Radiative recoil Nuclear self-energy
Scaling Eh .me =mp /˛ 3 Eh – ˛ 3 Eh ˛ 3 Eh .me =m /2 ˛ 3 Eh – ˛ 4 Eh ˛ 5 Eh .rp =C /2 ˛ 2 Eh .rp =C /2 ˛ 4 Eh .me =mp /˛ 4 Eh .me =mp /2 ˛ 3 Eh
= h (Hz) 1015 106 102 1010 108 104 103 106 103 106 102 104 103
with the same n and j D j j 1=2 have degenerate eigenvalues. Finally, ` D j C 1=2j 1=2, and we retain the atomic number Z in the equations in order to classify the various contributions to the energies. Corrections to the Dirac eigenvalue that approximately take into account the finite mass of the proton mp are included in a more general expression for atomic energy levels. That is, we replace Eq. (29.39) by [46, 47] mr 1 EM D Mc 2 C f .n; / 1 Œf .n; / 12 2 M ) 4 2 1 1 ı`0 .Z˛/ mr C C : : : mr c 2 ; 2 .2` C 1/ n3 m2p (29.41)
Table 29.5 Values of the Bethe logarithms ln k0 .n; `/. Missing entries correspond to states for which no experimental measurements are available n 1 2 3 4 6 8 12
S 2:984128556 2:811769893 2:767663612 2:749811840 2:735664207 2:730267261
P
D
0:030016709 0:041954895
0:006740939 0:008147204 0:008785043 0:009342954
P where an D 2 ln.2=n/ 2 C 1=n 2 niD1 .1=i/ for ` D 0 and an D 1=Œ`.`C1/.2`C1/ otherwise. Values for the Bethe logarithms ln k0 .n; `/ are given in Table 29.5. Additional contributions to lowest order in the mass ratio and of higher order in Z˛ are
me .Z˛/6 me c 2 D60 C D72 Z˛ ln2 .Z˛/2 C : : : ; 3 mp n (29.43) D 4 ln 2 7=2 for ` D 0 and D where D 60 60 D
2 3 `.` C 1/=n2 =Œ.2` 1/.2` C 1/.2` C 3/ otherwise. Finally, D72 D 11=.60 / ı`0 . Recently, the coefficients D71 and D70 have been computed [49, 50], where, in particular, the D71 was found to be surprisingly large. In [49], the nuclear-size correction to ES C ER was also computed. The uncertainty in the relativistic recoil correction ES C ER is Œ0:1ı`0 C 0:01.1 ı`0 /ER : (29.44) ER D
Covariances follow from the 1=n3 scaling of the uncertainty.
Nuclear Polarizability For the energy correction due to the nuclear polarizability in hydrogen, we use EP .H/ D 0:070.13/h
ı`0 kHz : n3
(29.45)
where M D me C mp and mr D me mp =.me C mp / is the reduced mass. Note that in this equation, the energy of nS1=2 - The effect is neglected for states of higher `. states differs from that of nP1=2 -states. Self-Energy Relativistic Recoil The one-photon self-energy of an electron bound to a stationThe leading relativistic-recoil correction, to lowest order in ary point nucleus is Z˛ and all orders in me =mp , is [47, 48] ˛ .Z˛/4 .2/ m3r .Z˛/5 D F .Z˛/ me c 2 ; (29.46) E 2 SE 3 ES D 2 m c n e me mp n3 1 8 1 7 where the function ı`0 ln.Z˛/2 ln k0 .n; `/ ı`0 an 3 3 9 3 m m F .x/ D A41 ln.x 2 / C A40 C A50 x C A62 x 2 ln2 .x 2 / 2 p e 2 2 me ln 2 ı`0 mp ln ; C A61 x 2 ln.x 2 / C GSE .x/ x 2 ; mp m2e mr mr (29.47) (29.42)
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with A41 D 4=3 ı`0 , and A40 D .4=3/ ln k0 .n; `/ C 10=9 Two-Photon Corrections for ` D 0 and .4=3/ ln k0 .n; `/ 1=Œ2 .2` C 1/ otherwise. The two-photon correction is Next, A50 D .139=32 2 ln 2/ ı`0 , A62 D ı`0 , and ˛ 2 .Z˛/4 me c 2 F .4/ .Z˛/ ; (29.52) E .4/ D 1 1 28 n3 A61 D 4 1 C C C C ln 2 4 ln n 2 n 3 2 where n 1 2 77 1 601 C 1 C ı ı ı`1 ; `0 180 45n2 n2 15 3 j2
F .4/ .x/ D B40 C B50 x C B63 x 2 ln3 .x 2 / 96n2 32`.` C 1/ .1 ı`0 / : C 2 C B62 x 2 ln2 .x 2 / C B61 x 2 ln.x 2 / C B60 x 2 3n .2` 1/.2`/.2` C 1/.2` C 2/.2` C 3/ C B72 x 3 ln2 .x 2 / C B71 x 3 ln.x 2 / C : : : ; Values for GSE .˛/ in Eq. (29.47) are listed in Table 29.6. The uncertainty of the self-energy contribution is due to the uncertainty of GSE .˛/ listed in the table and is taken to be with type un . See [51] for details. 2 3 10 2 2179 9 Following convention, F .Z˛/ is multiplied by the factor B40 D ln 2 .3/ ı`0 2 27 648 4 .mr =me /3 , except the magnetic moment term 1=Œ2 .2` C 2 2 ln 2 197 3.3/ 1 ı`0 1/ in A40 , which is instead multiplied by the factor C ; .mr =me /2 , and the argument .Z˛/2 of the logarithms is re2 12 144 4
.2` C 1/ placed by .me =mr /.Z˛/2 . B50 D 21:55447.13/ ı`0, B63 D 8=27 ı`0 , and Vacuum Polarization 16 71 1 1 The stationary point nucleus second-order vacuum-polarB62 D ln 2 C .n/ C ln n C 2 ı`0 ization level shift is 9 60 n 4n 2 4 n 1 ˛ .Z˛/4 .2/ 2 C ı`1 ; EVP D H.Z˛/ m c ; (29.48) e 27 n2 n3 where H.x/ D H .1/ .x/ C H .R/ .x/ with .1/
H .1/ .x/ D V40 C V50 x C V61 x 2 ln.x 2 / C GVP .x/ x 2 ; with V40 D 4=15 ı`0 , V50 D 5 =48 ı`0 , and V61 D 2=15 ı`0 . .1/ Values of GVP .˛/ are given in Table 29.7. Moreover, .R/ H .R/ .x/ D GVP .x/ x 2 with .R/ GVP .x/
19 2 D C 45 27
1 31 2 x C : : : ; (29.49) 16 2880
for ` D 0 and zero otherwise. Higher-order terms are negligible. We multiply Eq. (29.48) by .mr =me /3 and include a factor of .me =mr / in the argument of the logarithm of the term proportional to V61 . Vacuum polarization from C pairs is
with Euler’s constant and Psi function .z/. Values and uncertainties for B61 and B60 are listed in Tables 29.8 and 29.9, respectively. For the S-state values, the first number in parentheses for B60 is the state-dependent uncertainty un .B60 /, while the second number in parentheses is the state-independent uncertainty u0 .B60 /. It is worth noting that recently [32] computed an additional light-by-light correction to B61 for S-states. It only shifts this coefficient within its current uncertainty. For S-states, the next term B72 is state independent, but its value is not known. The B71 coefficient is state dependent, although only the difference 427 16 ln 2 B71 .nS/ D B71 .nS/ B71 .1S/ D 36 3 3 1 1 C 2 C .n/ C ln n C : : : 4 n 4n
me 2 mr 3 ˛ .Z˛/4 4 me c 2 C : : : ; ı `0 n3 15 m me (29.50) is known with a relative uncertainty un .B71 / D 0:5 B71 . For our calculations, we use B71 .1S/ D 0. while the hadronic vacuum polarization is given by As with the one-photon correction, the two-photon cor.2/ .2/ rection is multiplied by the reduced-mass factor .mr =me /3 , (29.51) Ehad VP D 0:671.15/EVP : except the magnetic moment term proportional to 1=Œ .2` C Uncertainties are of type u0 . The muonic and hadronic 1/ in B40 , which is multiplied by the factor .mr =me /2 , vacuum-polarization contributions are negligible for higher- and the argument .Z˛/2 of the logarithms is replaced by .me =mr /.Z˛/2 . `-states. .2/
EVP D
29 Tests of Fundamental Physics Table 29.6 Values of the function GSE .˛/. Missing entries correspond to states for which no experimental measurements are available
Table 29.7 Values of the func.1/ tion GVP .˛/. No experimental data is available for missing entries. Zero values indicate that their contributions are negligibly small
Table 29.8 Values of B61 used in the 2014 CODATA adjustment. Zero values indicate that their contributions are negligibly small
Table 29.9 Values of B60 used in the 2014 CODATA adjustment. The uncertainties of B60 for S-states are explained in the text
445 n 1 2 3 4 6 8 12
S1=2 30:290240.20/ 31:185150.90/ 31:04770.90/ 30:9120.40/ 30:711.47/ 30:606.47/
P1=2
P3=2
0:97350.20/
0:48650.20/
1:1640.20/
0:6090.20/
n 1 2 3 4 6 8 12
S1=2 0:618724 0:808872 0:814530 0:806579 0:791450 0:781197
P1=2
P3=2
0:064006
0:014132
0:080007
0:017666
n 1 2 3 4 6 8 12
S1=2 48:95859024.1/ 41:06216431.1/ 38:904222.1/ 37:909514.1/ 36:963391.1/ 36:504940.1/
P1=2
P3=2
0:157775547.1/
0:092224453.1/
0:191192600.1/
0:121307400.1/
n 1 2 3 4 6 8 12
S1=2 81:3.0:3/.19:7/ 66:2.0:3/.19:7/ 63:0.0:6/.19:7/ 61:3.0:8/.19:7/ 59:3.0:8/.19:7/ 58:3.2:0/.19:7/
P1=2
P3=2
1:6.3/
1:7.3/
2:1.3/
2:2.3/
Three-Photon Corrections The three-photon contribution in powers of Z˛ is
E .6/ D
˛ 3 .Z˛/4
n3
me c 2 ŒC40 C C50 .Z˛/
C C63 .Z˛/2 ln3 .Z˛/2 C : : : : (29.53)
568 a4 85 .5/ 121 2 .3/ 84;071 .3/ C 9 24 72 2304
4787 2 ln 2 71 ln4 2 239 2 ln2 2 C 27 135 108 1591 4 252;251 2 679;441 C C ı`0 3240 9720 93;312
D5=2
0:007940.90/ 0:009130.90/
0:03163.22/ 0:03417.26/ 0:03484.22/ 0:03512.22/
D3=2
D5=2
0:000000 0:000000
0:000000 0:000000 0:000000 0:000000
D3=2
D5=2
0:0.0/ 0:0.0/
0:0.0/ 0:0.0/ 0:0.0/ 0:0.0/
D3=2
D5=2
0:015.5/ 0:014.7/
0:005.2/ 0:008.4/ 0:009.5/ 0:010.7/
100 a4 215 .5/ 83 2 .3/ 139 .3/ C C 3 24 72 18 25 2 ln2 2 298 2 ln 2 25 ln4 2 C C 18 18 9 239 4 17;101 2 28;259 1 ı`0 C ; 2160 810 5184 .2` C 1/
The leading term C40 is
C40 D
D3=2
P n 4 where a4 D 1 nD1 1=.2 n / D 0:517479061: : :. Only partial results for C50 have been obtained [52, 53]. We use C50 D 0 with uncertainty u0 .C50 / D 30ı`0 . Finally, we use C63 D 0 and un .C63 / D 1 for this unknown coefficient. Recently, Karshenboim and Ivanov [54] determined several of these coefficients. The dominant effect of the finite mass of the nucleus is taken into account by multiplying the term proportional to ı`0 by the reduced-mass factor .mr =me /3 and the term proportional to 1=Œ .2` C 1/, the magnetic mo-
29
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ment term, by the factor .mr =me /2 . The contribution from Nucleus Self-Energy four photons is expected to be negligible at the level of un- The nucleus self-energy correction for S-states of hydrogen is certainty of current interest.
Finite Nuclear Size For S-states, the leading and next-order correction to the level shift due to the finite size of the nucleus is given by ( mr rp Z˛ mr rp Z˛ ln C .n/ ENS D ENS 1 C me C me C n ) .5n C 9/.n 1/ 2 C .Z˛/ ; C 4n2 (29.54) where 2 2 mr 3 .Z˛/2 2 Z˛ rp ENS D me c : (29.55) 3 me n3 C
ESEN D
4Z 2 ˛.Z˛/4 m3r 2 c 3 n3 m2p mp ln ln k .n; `/ ; ı `0 0 mr .Z˛/2
(29.60)
with an uncertainty u0 given by Eq. (29.60), with the factor in the square brackets replaced by 0.5. For higher-`-states, the correction is negligible. Total energy and uncertainty: the energy E.nlj / of a level is the sum of the contributions listed above. Uncertainties in the energy due to the fundamental constants, i.e., ˛, me c 2 , etc., are taken into account through a least-squares adjustment. Uncertainties in the theory, i.e., from estimates of missing and uncomputed terms in the contributions, are taken The coefficients C and C are constants that depend on the into account with an energy correction ı.n`j /, with an uncharge distribution in the nucleus with values C D 1:7.1/ certainty that is the rms sum of the uncertainties of the and C D 0:47.4/ for hydrogen. individual contributions For the P1=2 -states in hydrogen, the leading term is X (29.61) u20i .n`j / C u2ni .n`j / ; u2 Œı.n`j / D .Z˛/2 .n2 1/ ENS D ENS : (29.56) i 2 4n For P3=2 -states and higher-`-states, the nuclear-size contribu- where u0i .n`j / and uni .n`j / are the components of uncertion is negligible. tainty u0 and un of contribution i. Covariances of the ıs are
X
Nuclear-Size Correction to Self-Energy u0i .n2 `j /u0i .n1 `j / : (29.62) u ı.n1 `j /; ı.n2 `j / D and Vacuum Polarization i For the lowest-order self-energy and vacuum polarization, the correction due to the finite size of the nucleus is The corrections ı.n`j /, their uncertainties, and covariances are input data in the least-squares adjustment. A value for 23 ENSE D 4 ln 2 (29.57) ı.n`j / returned by the adjustment that lies outside its un˛.Z˛/ENS ı`0 ; 4 certainty indicates either an underestimate of the value of and uncomputed terms in the contributions or a role for unexpected physics beyond QED. No such discrepancies have 3 ENVP D ˛.Z˛/ENS ı`0 ; (29.58) been found. 4 respectively.
Acknowledgments The authors gratefully acknowledge helpful conversations with David Newell.
Radiative-Recoil Corrections Corrections for radiative-recoil effects are
References m3r ˛.Z˛/5 2 m c ı e `0 m2e mp 2 n3 1. Sommerfeld, A.: Ann. Phys. 356, 1 (1916) 2. International Bureau of Weights and Measures (BIPM). https:// 35 2 448 2 6 .3/ 2 ln 2 C www.bipm.org/en/about-us/ 36 27 3. Newell, D.B.: Phys. Today 67, 35 (2014) 2 4. Eides, M.I., Grotch, H., Shelyuto, V.A.: Phys Rep 342, 63 (2001) C .Z˛/ ln2 .Z˛/2 C d61 .Z˛/ ln.Z˛/2 C : : : : 5. Mohr, P.J., Newell, D.B., Taylor, B.N.: Rev. Mod. Phys. 88, 3 035009 (2016) (29.59) 6. Hanneke, D., Fogwell, S., Gabrielse, G.: Phys. Rev. Lett. 100, The uncertainty is controlled by the unknown coefficient d61 120801 (2008) inside the square brackets. We use d61 D 0, u0 .d61 / D 10, and 7. Wang, M., Audi, G., Kondev, F.G., Huang, W.J., Naimi, S., Xu, X.: Chin. Phys. C(41), 030003 (2017) un .d61 / D 1. Corrections for higher-`-states are negligible. ERR D
29 Tests of Fundamental Physics 8. Feng, X.J., Zhang, J.T., Moldover, M.R., Yang, I., Plimmer, M.D., Lin, H.: Metrologia 54, 339 (2017) 9. Laporta, S., Remiddi, E.: Phys. Lett. B. 301, 440 (1993) 10. Laporta, S.: Nuovo Cimento 106, 675 (1993) 11. Kurz, A., Liu, T., Marquard, P., Steinhauser, M.: Nucl. Phys. B. 879, 1 (2014) 12. Aoyama, T., Hayakawa, M., Kinoshita, T., Nio, M.: Phys. Rev. D 91, 033006 (2015) 13. Laporta, S.: Phys. Lett. B. 772, 232 (2017) 14. Aoyama, T., Kinoshita, T., Nio, M.: Phys. Rev. D 97, 036001 (2018) 15. Particle Data Group: Chin. Phys. C 38, 090001 (2014) 16. Bouchendira, R., Cladé, P., Guellati-Khélifa, S., Nez, F., Biraben Phys, F.: Rev. Lett. 106, 080801 (2011) 17. Parker, R.H., Yu, C., Zhong, W., Estey, B., Müller, H.: Science 360, 191 (2018) 18. Audi, G., Wang, M., Wapstra, A.H., Kondev, F.G., MacCormick, M., Xu, X., Pfeiffer, B.: Chin. Phys. C 36, 1287 (2012) 19. Wang, M., Audi, G., Wapstra, A.H., Kondev, F.G., MacCormick, M., Xu, X., Pfeiffer, B.: Chin. Phys. C 36, 1603 (2012) 20. Huang, W.J., Audi, G., Wang, M., Kondev, F.G., Naimi, S., Xu, X.: Chin. Phys. C 41, 030002 (2017) 21. Sturm, S., Köhler, F., Zatorski, J., Wagner, A., Harman, Z., Werth, G., Quint, W., Keitel, C.H., Blaum, K.: Nature 506, 467 (2014) 22. Köhler, F., Sturm, S., Kracke, A., Werth, G., Quint, W., Blaum, K.J.: Phys. B 48, 144032 (2015) 23. Sturm, S., Wagner, A., Kretzschmar, M., Quint, W., Werth, G., Blaum, K.: Phys. Rev. A 87, 030501 (2013) 24. Sturm, S., Wagner, A., Kretzschmar, M., Quint, W., Werth, G., Blaum, K. (2015), private communication. 25. Breit, G.: Nature 122, 649 (1928) 26. Yerokhin, V.A., Harman, Z.: Phys. Rev. A 95, 060501 (2017) 27. Beier, T., Lindgren, I., Persson, H., Salomonson, S., Sunnergren, P., Häffner, H., Hermanspahn, N.: Phys. Rev. A 62, 032510 (2000) 28. Pachucki, K., Czarnecki, A., Jentschura, U.D., Yerokhin, V.A.: Phys. Rev. A 72, 022108 (2005) 29. Jentschura, U.D., Czarnecki, A., Pachucki, K., Yerokhin, V.A.: Int. J. Mass. Spectrom. 251, 102 (2006) 30. Jentschura, U.D.: Phys. Rev. A 79, 044501 (2009) 31. Yerokhin, V.A., Harman, Z.: Phys. Rev. A 88, 042502 (2013) 32. Czarnecki, A., Szafron, R.: Phys. Rev. A 94, 060501 (2016) 33. Eides, M.I., Grotch, H.: Ann. Phys. 260, 191 (1997) 34. Czarnecki, A., Melnikov, K., Yelkhovsky, A.: Phys. Rev. A 63, 012509 (2001) 35. Shabaev, V.M., Yerokhin, V.A.: Phys. Rev. Lett. 88, 091801 (2002) 36. Karshenboim, S.G.: Phys. Lett. A. 266, 380 (2000) 37. Glazov, D.A., Shabaev, V.M.: Phys. Lett. A. 297, 408 (2002) 38. Angeli, I.: Data Tables. At. Data. Nucl. 87, 185 (2004) 39. Parthey, C.G., Matveev, A., Alnis, J., Bernhardt, B., Beyer, A., Holzwarth, R., Maistrou, A., Pohl, R., Predehl, K., Udem, T.: Phys. Rev. Lett. 107, 203001 (2011) 40. Matveev, A., Parthey, C.G., Predehl, K., Alnis, J., Beyer, A., Holzwarth, R., Udem, T., Wilken, T., Kolachevsky, N., Abgrall, M.: Phys. Rev. Lett. 110, 230801 (2013) 41. T. Udem (2014), private communication.
447 42. Lundeen, S.R., Pipkin, F.M.: Metrologia 22, 9 (1986) 43. Newton, G., Andrews, D.A., Unsworth, P.J.: Philos. Trans. R. Soc. Lond. Ser. A 290, 373 (1979) 44. Beyer, A., Maisenbacher, L., Matveev, A., Pohl, R., Khabarova, K., Grinin, A., Lamour, T., Yost, D.C., Hänsch, T.W., Kolachevsky, N.: Science 358, 79 (2017) 45. Fleurbaey, H., Galtier, S., Thomas, S., Bonnaud, M., Julien, L., Biraben, F., Nez, F., Abgrall, M., Guéna, J.: arXiv, 1801:0881 (2018) 46. Barker, W.A., Glover, F.N.: Phys. Rev. 99, 317 (1955) 47. Sapirstein, J.R., Yennie, D.R.: chap. 12. In: Kinoshita, T. (ed.) in Quantum Electrodynamics, pp. 560–672. World Scientific, Singapore (1990) 48. Erickson, G.W.: J. Phys. Chem. Ref. Data 6, 831 (1977) 49. Yerokhin, V.A., Shabaev, V.M.: Phys. Rev. Lett. 115, 233002 (2015) 50. Yerokhin, V.A., Shabaev, V.: Phys. Rev. A 93, 062514 (2016) 51. Mohr, P.J., Taylor, B.N., Newell, D.B.: Rev. Mod. Phys. 84, 1527 (2012) 52. Eides, M.I., Shelyuto, V.A.: Phys. Rev. A 70, 022506 (2004) 53. Eides, M.I., Shelyuto, V.A.: Can. J. Phys. 85, 509 (2007) 54. Karshenboim, S.G., Ivanov, V.G.: Phys. Rev. A 98, 022522 (2018)
Eite Tiesinga Dr Eite Tiesinga received his PhD from Eindhoven University of Technology in The Netherlands in 1993. He works at the National Institute of Standards and Technology and the University of Maryland in the US. His research focuses on developing clocks and sensors using laser-cooled atoms. He maintains a web-based database of fundamental constants, an initiative by the Committee on Data of the International Science Council (CODATA). Peter Mohr Peter Mohr received his PhD from the University of California at Berkeley (1973) and spent some years at the Lawrence Berkeley Laboratory (1973–1978), Yale University (1978– 1985), the National Science Foundation (1985–1987), and the National Bureau of Standards/National Institute of Standards and Technology (from 1987). He held the Chair of the CODATA Task Group on Fundamental Constants (1999–2006) and was Chair of the Precision Measurement and Fundamental Constants Topical Group of the American Physical Society (2000–2001).
29
Atomic Clocks and Constraints on Variations of Fundamental Constants Savely G. Karshenboim, Victor Flambaum
Contents 30.1 30.1.1 30.1.2 30.1.3 30.1.4 30.1.5 30.1.6 30.2 30.2.1 30.2.2 30.2.3 30.3
Atomic Clocks and Frequency Standards . Caesium Atomic Fountain . . . . . . . . . . . . Trapped Ions . . . . . . . . . . . . . . . . . . . . Laser-Cooled Neutral Atoms . . . . . . . . . . Two-Photon Transitions and Doppler-Free Spectroscopy . . . . . . . . Optical Frequency Measurements . . . . . . . Limitations on Frequency Variations . . . . . Atomic Spectra and Their Dependence on the Fundamental Constants . . . . . . . The Spectrum of Hydrogen and Nonrelativistic Atoms . . . . . . . . . . . . Hyperfine Structure and the Schmidt Model . Atomic Spectra: Relativistic Corrections . . .
. . . .
. . . .
. . . .
. . . .
. . . .
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450 450 451 452
. . . . . . 452 . . . . . . 453 . . . . . . 453 . . . . . . 454 . . . . . . 454 . . . . . . 454 . . . . . . 454
30.3.4 30.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
30.3.2 30.3.3
. . 455 . . 456 . . 457
. . 457 . . 457
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
S. G. Karshenboim () Faculty of Physics, Ludwig-Maximilians-University Munich, Germany e-mail: [email protected] V. Flambaum () Dept. of Physics, University of New South Wales Sydney, Australia e-mail: [email protected] E. Peik Physikalisch-Technische Bundesanstalt Braunschweig, Germany e-mail: [email protected]
, and Ekkehard Peik
Abstract
Laboratory Constraints on Temporal Variations of Fundamental Constants . . . . . . . . . . . . . . Constraints from Absolute and Relative Optical Measurements . . . . . . . . . . Constraints from Microwave Clocks . . . . . . . . . A Model-Independent Constraint on the Variation of the Electron-to-Proton Mass Ratio from Molecular Spectroscopy . . . . . . . . . . . . . Other Model-Dependent Constraints . . . . . . . . .
30.3.1
30
Fundamental constants play an important role in modern physics, being landmarks that designate different areas. We call them constants, however, as long as we only consider minor variations with the cosmological time/space scale, their constancy is an experimental fact rather than a basic theoretical principle. Modern theories unifying gravity with electromagnetic, weak, and strong interactions, or even the developing quantum gravity itself, often suggest such variations. Many parameters that we call fundamental constants, such as the electron charge and mass (Chap. 1 and [1]), are actually not truly fundamental constants but effective parameters that are affected by renormalization or the presence of matter [2, 3]. Living in a changing universe we cannot expect that matter will affect these parameters the same way during any given cosmological epoch. An example is the inflationary model of the universe, which states that in a very early epoch, the universe experienced a phase transition, which, in particular, changed a vacuum average of the so-called Higgs field, which determines the electron mass. The latter was zero before this transition and reached a value close or equal to the present value after the transition. The problem of variations of constants has many facets, and here we discuss aspects related to atomic clocks and precision frequency measurements. Other related topics may be found in [4, 5]. Keywords
relativistic corrections frequency standards fundamental constants atomic clocks optical clocks
Laboratory searches for a possible time variation of fundamental physical constants currently consist of two important parts: (i) one has to measure a certain physical quantity at two different moments of time that are separated by at least a few
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_30
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450
years; (ii) one has to be able to interpret the result in terms of fundamental constants. The latter is a strong requirement for a cross comparison of different results. The measurements that may be performed the most accurately are frequency measurements, and thus, frequency standards or atomic clocks will be involved in most of the laboratory searches. With the introduction of laser cooling, ion trapping, frequency comb technology, atomic interferometry, and quantum entanglement, frequency metrology has shown great progress in the past two decades and will continue to do so for some time. The constraints on the variations of the fundamental constants obtained in this manner for the present epoch are now more stringent than those from other methods (astrophysics, geochemistry), when these are extrapolated under the assumption of a linear drift in time. Laboratory experiments on frequency comparisons allow for checks of reproducibility, a very clear interpretation of the final results, and a transparent evaluation procedure, making them less susceptible to systematic errors. The most advanced atomic clocks are discussed in Sect. 30.1. They are realized with many-electron atoms, and their frequency cannot be interpreted in terms of fundamental constants. However, a much simpler problem needs to be solved: to interpret their variation in terms of fundamental constants. This idea is discussed in Sect. 30.2. The current laboratory constraints on the variations of the fundamental constants are summarized in Sect. 30.3.
S. G. Karshenboim et al.
ual classical-physics flexibility, which allows their properties to change. In contrast, standards similar to the caesium clock have a frequency (or other property) that is determined by a certain natural constant, which is not flexible, being of pure quantum origin. It may change only if the fundamental constants are changing. In Sect. 30.3, the results obtained with the most advanced microwave and optical frequency standards are discussed. The former include the ones based on caesium and rubidium fountains, while the latter deal with precision spectroscopy with atomic beams using two-photon excitation, with lasercooled ions in traps and atoms in optical lattices.
30.1.1 Caesium Atomic Fountain Caesium clocks are the most accurate primary standards for time and frequency [7]. The hyperfine splitting frequency between the F D 3 and F D 4 levels of the 2S 1=2 ground state of the 133 Cs atom at 9:192 GHz has been used for the definition of the SI second since 1967. In a so-called caesium fountain [8] (Fig. 30.1), a dilute cloud of laser cooled caesium atoms at a temperature of about 1 K is launched upwards to initiate a free parabolic flight with an apogee at about 1 m above the cooling zone. A microwave cavity is mounted near the lower endpoints of the parabola and is traversed by the atoms twice – once during ascent and once during descent – so that Ramsey’s method of interrogation with separated oscillatory fields [6] can be realized. The total interrogation
30.1 Atomic Clocks and Frequency Standards 1 pair of vertical laser beams
Frequency standards are important tools for precision meaLaunching height surements and serve various purposes that, in turn, have different requirements that must be satisfied. In particular, it is not necessary for a practical frequency standard to reproClock duce a frequency that is related to a certain atomic transition, output although it may be expressed in terms of it. A well-known example is the hydrogen maser, where the frequency is affected = 9,192,631,770 Hz ± Ά Microwave by the wall shifts, which may vary with time [6]. For the cavity Microwave Quartz study of time variations of fundamental constants, it is necsynthesis oscillator essary to use standards similar to a primary caesium clock. Detection laser In this case, any deviation of its frequency from the unperElectronics for turbed atomic transition frequency should be known (within detection and control a known uncertainty) because this is a necessary requirement Photodetector for being a primary standard. 2 pairs of horizontal From the point of view of fundamental physics, the hylaser beams drogen maser is an artefact quite similar, in a sense, to the prototype of the kilogram held at the Bureau International Cold atoms: T = 1–2 μK des Poids et Mesures (BIPM) in Paris. Both artefacts are Launching velocity = 4 m/s somehow related to fundamental constants (e.g., the mass of the prototype can be expressed in terms of the nucleon masses and their number), but they also have a kind of resid- Fig. 30.1 Schematic of an atomic fountain clock
30 Atomic Clocks and Constraints on Variations of Fundamental Constants
time being on the order of 0:5 s, a resonance linewidth of 1 Hz is achieved, about a factor of 100 narrower than in devices using a thermal atomic beam from an oven. Selection and detection of the hyperfine state is performed via optical pumping and laser-induced resonance fluorescence. In a carefully controlled setup, a relative uncertainty in the low 1 1016 range can be reached in the realization of the resonance frequency of the unperturbed Cs atom. The averaging time that is required to reach this level of uncertainty is in the range of 104 to 105 s. One limiting effect that contributes to the systematic uncertainty of the caesium fountain is the frequency shift due to cold collisions between the atoms. In this respect, a fountain frequency standard based on the ground state hyperfine frequency of the 87 Rb atom at about 6:835 GHz is more favorable, since its collisional shift is lower by more than a factor of 50 for the same atomic density. While the caesium frequency is fixed by definition in the SI system, the 87 Rb frequency has been measured repeatedly and is one of the recommended secondary realizations of the second [9]. While a number of optical frequency standards have been evaluated with lower uncertainty, the realization of the caesium standards remains the base of international time and frequency measurements. A number of them, realized independently in numerous national metrological laboratories, are regularly compared via satellite time and frequency transfer to a fractional uncertainty of a few 1016 . At such a level of accuracy, the effects of general relativity should be carefully taken into account. From the gravitational potential of the Earth, a correction of 1 1016 will appear due to a displacement of the clocks by 1 m in the vertical direction.
30.1.2 Trapped Ions An alternative to interrogating atoms in free flight, and a possibility to obtain practically unlimited interaction time, is to store them in a trap. Ions are well suited because charged particles can be trapped in radio frequency ion traps (Paul traps [10], Chap. 79) that provide confinement around a fieldfree saddle point of an electric quadrupole potential. This ensures that the internal level structure is only minimally perturbed by the trap. Combined with laser cooling it is possible to reach the Lamb–Dicke regime, i.e., localization within a volume whose extension is smaller than the wavelength of the spectroscopic transition, leading to an elimination of the linear Doppler shift. A single ion, trapped in an ultrahigh vacuum is highly isolated from external perturbations, permitting excellent control of systematic frequency shifts [11]. The use of the much higher optical reference frequency allows one to obtain a stability that is superior to microwave frequency standards, although only a single ion is used to obtain the steering signal for the reference oscillator.
451
Metastable level Cooling transition (dipole allowed) “Forbidden” reference transition
Fig. 30.2 Double-resonance scheme applied in single-ion-trap frequency standards
A number of possible optical reference transitions with natural linewidths of the order of 1 Hz or below are available in different ions. Most ions of interest possess a level scheme where two transitions, a dipole-allowed transition and a forbidden reference transition of the optical clock, can be driven with two lasers from the ground state (Fig. 30.2). The dipole transition is used for laser cooling and for the optical detection of the ion via its resonance fluorescence. If the clock laser excites the ion to the metastable upper level of the reference transition, the fluorescence is interrupted, and every single excitation of the reference transition can thus be detected with unit efficiency as a dark period in the fluorescence signal. Different types of forbidden reference transitions are investigated [12]: ions with a single valence electron and s1=2 ground state possess metastable d levels that couple to the ground state via electric quadrupole (E2) radiation. This level scheme is found in CaC , SrC , YbC [13] and HgC [14, 15]. In YbC , the lowest excited level is an f state, coupled to the ground state only by an electric octupole (E3) transition [16]. In ions with two valence electrons, like AlC and InC , the total electronic angular momentum couples to 0 in the 1 S0 ground and 3 P0 lowest excited state. The vanishing angular momentum and high intrinsic symmetry of these states makes them less susceptible to level shifts induced by external fields. Hyperfine coupling with a nuclear spin mixes the P0 and P1 levels and induces a weak electric dipole transition between 3 P0 and 1 S0 . The absolute transition frequencies of several trapped ion optical frequency standards have been measured with an uncertainty that is essentially limited by the Cs reference (Table 30.1). Frequency standards based on 27 AlC and 171 YbC have been evaluated with relative systematic uncertainties in the 1018 range [12]. A distinctive feature of the AlC system is that it does not rely on direct laser cooling of the ion, because AlC does not possess a cooling transition at a technically convenient wavelength. Instead, the ion is cooled sympathetically via the Coulomb interaction by a laser-cooled ion of BeC or CaC in the same trap. The two ions form a quasi molecule with joint motional degrees of freedom. Laser excitation of motional sidebands is used to transfer information about the internal state of the AlC ion
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to a joint motional mode and onwards to an internal state of the coolant ion, where it can be read out via fluorescence detection. Such quantum logic protocols make it possible to perform precision spectroscopy on ions that cannot be directly laser cooled. The double resonance technique with the observation of dark periods in the fluorescence signal can also be employed if the reference transition is in the microwave domain, and a number of accurate measurements of hyperfine structure intervals in trapped ions have been performed. In particular, the HFS interval in 171 YbC has been measured several times [17–19] and can be used to obtain constraints on temporal variations.
S. G. Karshenboim et al.
certainty of these standards are the residual linear Doppler effect and phase front curvature of the laser beams that excite the ballistically expanding atom cloud. For an optical frequency standard of high precision, it is, therefore, necessary to confine the atoms in an optical lattice, i.e., in the array of interference maxima produced by several intersecting, reddetuned laser beams [23, 24]. The wavelength of the trapping laser can be chosen such that the light shift it produces in the ground and excited state of the reference transition are equal, and, therefore, it does not induce a shift of the reference frequency (Fig. 30.3). Such a so-called “magic” wavelength can be estimated with reasonable accuracy theoretically and later on verified experimentally. This approach is presently applied to the very narrow (mHz natural linewidth) 1S 0 ! 3P 0 transitions in neutral strontium, ytterbium, and mercury [12]. As in the case of trapped ions, absolute transition frequencies 30.1.3 Laser-Cooled Neutral Atoms of several so-called lattice clocks with neutral atoms have Optical frequency standards have been developed with laser- been measured with an uncertainty that is essentially limited cooled neutral atoms, most notably of the alkaline-earth by the Cs reference (Table 30.1), and systems based on Sr elements that possess narrow intercombination transitions and Yb have been evaluated with relative systematic uncerand the J D 0 ! 0 transitions discussed for ions above. The tainties in the 1018 range. atoms are collected and cooled in a magneto-optical trap. They may then be released and interrogated as free atoms by a sequence of laser pulses to realize a frequency-sensitive 30.1.4 Two-Photon Transitions and Doppler-Free Spectroscopy Ramsey–Bordé atom interferometer [20]. Of these systems, the one based on the 1S 0 ! 3P 1 intercombination line of 40 Ca at 657 nm has reached the lowest relative uncertainty The linear Doppler shift of an absorption resonance can also so far (about 2 1014 ) [21, 22]. Limiting factors in the un- be avoided if a two-photon excitation is induced by two counterpropagating laser beams. A prominent example that has been studied with high precision is the two-photon exU/h (MHz) 0.5 citation of the 1s ! 2s transition in atomic hydrogen. The 1 P = 0.5 W S0 precise measurement of this frequency is of importance for 3 w = 25 μm P0 the determination of the Rydberg constant and as a test of quantum electrodynamics (QED). Hydrogen atoms are 0 cooled by collisions in a cryogenic nozzle and interact with a standing laser wave of 243 nm wavelength inside a resonator. Since the atoms are not as cold as in laser-cooled Magic wavelength 813 nm –0.5 samples, a correction for the second-order Doppler effect is performed. The laser excitation is interrupted periodically, and the excited atoms are detected in a time-resolved manner so that their velocity can be examined. The experiments –1.0 on absolute frequency measurement were carried out at the Max-Planck-Institut für Quantenoptik, which is not a na500 1000 2000 λ (nm) tional metrological institute and does not possess its own primary caesium standard. The problem of the lack of such Fig. 30.3 Light shift of the 1 S0 ground state and 3 P0 first excited state a standard was resolved by use of a transportable caesium of Sr as a function of wavelength. At the ‘magic’ wavelength of 813 fountain [25] from the Observatoire de Paris and comparing nm both levels are shifted equally. Sr atoms in an optical lattice at this their hydrogen transition frequency with the caesium referwavelength do not show a frequency shift for the 1 S0 ! 3 P0 transition. Equivalent conditions can be found for reference transitions in other ence at PTB via a 920 km fiber link [26]. An accuracy of atoms about 4 1015 has been obtained.
30 Atomic Clocks and Constraints on Variations of Fundamental Constants
453
Table 30.1 Accurately measured atomic transition frequencies and their fractional uncertainty ([9, 12] and references therein). These transitions have been used in evaluations of temporal variation of fundamental constants Time
Ά t = 1/fr fn fr
f=0
fceo
Frequency
Fig. 30.4 Frequency comb generated from femtosecond laser pulses
30.1.5 Optical Frequency Measurements An essential tool for the measurement of optical frequencies with respect to a Cs clock in the microwave range is the so-called femtosecond laser frequency comb generator [27]. Briefly, a mode-locked femtosecond laser produces, in the frequency domain, a comb of equally spaced optical frequencies fn (Fig. 30.4) that can be written as fn D nfr C fceo (with fceo < fr ), where fr is the pulse repetition rate of the laser, the mode number n is a large integer (of order 105 ), and fceo (carrier-envelope-offset) is a shift of the whole comb that is produced by group velocity dispersion in the laser. The repetition rate fr can easily be measured with a fast photodiode. In order to determine fceo , the comb is broadened in a nonlinear medium, so that it covers at least one octave. Now, the second harmonic of mode n from the red wing of the spectrum, at frequency 2.nfr C fceo /, can be mixed with mode 2n from the blue wing, at frequency 2nfr C fceo , and fceo is obtained as a difference frequency. In this way, the precise relation between the two microwave frequencies fr and fceo , and the numerous optical frequencies fn , is known. The setup can now be used for an absolute optical frequency measurement by referencing fr and fceo to a microwave standard and recording the beat note between the optical frequency fo to be measured and the closest comb frequency fn . Vice versa, the setup may work as an optical clockwork, for example, by adjusting fceo to zero and by stabilizing one comb line fn to fo , so that fr is now an exact subharmonic to order n of fo . The precision of these transfer schemes has been investigated and was found to be so high that it will not limit the performance of optical clocks for the foreseeable future. Since the most precise optical frequency standards are now more stable and reproducible than primary caesium clocks, it is important to be able to measure the ratio of two
Atom, transition H, Opt AlC , Opt Sr, Opt Rb, HFS YbC , Opt E2 YbC , Opt E3 YbC , HFS HgC , Opt
f (GHz) 2;466;061 1;121;015 429;228 6:835 688;359 642;121 12:642 1;064;721
ur .1015 / 14 0.6 0.4 0.3 0.5 0.4 73 9
Ref. [25, 26]
[14, 15]
optical transition frequencies without the use of a microwave reference. A frequency comb generator can be configured for this task, for example by referencing the optical beat notes and fceo to fr . In this way, the dimensionless optical frequency ratio f1 =f2 D .n1 C fo1 =fr C fceo =fr /=.n2 C fo2 =fr C fceo =fr / can be measured to an accuracy that is only limited by the optical frequency standards. A number of such measurements have been performed. They will provide essential consistency checks for the developing system of optical frequency standards and build the basis for the most stringent laboratory tests for changes of the fine structure constant.
30.1.6 Limitations on Frequency Variations A variety of frequency standards described above have been successfully developed and their accuracy has been improved dramatically. This progress, as a consequence, has led to certain constraints on the possible variations of the fundamental constants. Considering frequency variations, one has to have in mind that not only the numerical value but also the unit may vary. For this reason, one needs to deal with dimensionless quantities, which are unit independent. During the last decade, a number of transition frequencies were measured in the corresponding SI unit, the hertz (Table 30.1). These-dimensional results are actually related to dimensionless quantities since a frequency measurement in SI is a measurement with respect to the caesium hyperfine interval ˚ f D 9;192;631;770
f ; fHFS .Cs/
(30.1)
˚ where f stands for the numerical value of the frequency f . (Most absolute frequency measurements have been realized as a direct comparison with a primary caesium standard.) In Sect. 30.3, in order to simplify notation, this symbol for the numerical value is dropped.
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30.2 Atomic Spectra and Their Dependence on the Fundamental Constants 30.2.1 The Spectrum of Hydrogen and Nonrelativistic Atoms
S. G. Karshenboim et al. Table 30.2 Magnetic moments and relativistic corrections for atoms involved in microwave standards. The relativistic sensitivity is defined in Sec. 30.2.3. Here, is an actual value of the nuclear magnetic moment, N is the nuclear magneton, and S stands for the Schmidt value of the nuclear magnetic moment; the nucleon g factors are gp =2 ' 2:79 and gn =2 ' 1:91 Z
Atom
=N
S =N
=S
87 Rb 2:75 gp =2 C 1 0:74 0:34 The hydrogen atom is the simplest atom, and one can easily 37 133 Cs 2:58 7=18 .10 gp / 1:50 0:83 calculate the leading contribution to different kinds of tran- 55 171 YbC 0:49 gn =6 0:77 1:5 sitions in its spectrum, such as the gross, fine, and hyperfine 70 structure. The scaling behavior of these contributions with the values of the Rydberg constant R1 , the fine structure 30.2.2 Hyperfine Structure and the Schmidt Model constant ˛, and the magnetic moments of proton and Bohr magneton is clear. The results for some typical hydrogenic The atomic hyperfine structure transitions are fNR .HFS/ D const. ˛ 2 cR1 (30.5) 3 B f .2p ! 1s/ ' cR1 ; 4
involves nuclear magnetic moments , which are different 1 2 f 2p3=2 2p1=2 ' ˛ cR1 ; for different nuclei; thus, a comparison of the constraints 16 on the variations of nuclear magnetic moments has a re16 2 p cR1 : (30.2) fHFS .1s/ ' ˛ duced value. To compare them, one may apply the Schmidt 3 B model [2, 3, 31], which predicts all the magnetic moments In the nonrelativistic approximation, the basic frequencies of nuclei with an odd number of nucleons (odd value of and the fine and hyperfine structure intervals of all atomic atomic number A) in terms of the proton and neutron gspectra have a similar dependence on the fundamental con- factors, gp , and gn , respectively, and the nuclear magneton only. Unfortunately, the uncertainty of the calculation within stants in Eq. (30.2) the Schmidt model is quite high (usually from 10 to 50%). The Schmidt model, being a kind of ab initio model, only fgross structure / cR1 ; allows for improvements that, unfortunately, involve more 2 ffine structure / ˛ cR1 ; effective phenomenological parameters. This would not re fHFS / ˛ 2 cR1 ; (30.3) ally improve the situation but return us to the case where B there are too many possibly varying independent parameters. A comparison of the Schmidt values to the actual data is prewhere stands for the nuclear magnetic moment. The pressented for caesium, rubidium, and ytterbium in Table 30.2. ence of a few electrons and a nuclear charge of Z ¤ 1 The measurements of the hyperfine intervals in those atoms makes theory more complicated and introduces certain mulsupply us with the constraints on the variation of the nuclear tiplicative numbers but involves no new parameters. The magnetic moments. importance of this scaling for a search for the variations of the atomic constants was first pointed out in [28] and was applied to astrophysical data. 30.2.3 Atomic Spectra: Relativistic Corrections Similar results may be presented for molecular transitions (electronic, vibrational, rotational, and hyperfine) [29] A theory based on the leading nonrelativistic approximation may not be accurate enough. Any atomic frequency can be felectronic / cR1 ; presented as fvibrational / .me =M /1=2 cR1 ; (30.6) f D fNR Frel .˛/ ; frotational / .me =M /cR1 : (30.4) where the first (nonrelativistic) factor is determined by a scalUp to now, no laboratory measurement with molecules ing similar to the hydrogenic transitions in Eq. (30.3). has been performed at a level of accuracy that is compet- The second factor stands for relativistic corrections that vanitive with atomic transitions (e.g., Sect. 30.3.3). However, ish at ˛ D 0; and thus, Frel .0/ D 1. they have widely been used in a search for variations of conThe importance of relativistic corrections for the hystants in astrophysical observations (e.g., [30]). perfine structure was first emphasized in [32]. Relativistic
30 Atomic Clocks and Constraints on Variations of Fundamental Constants
455
many-body calculations for various transitions were per- Table 30.3 The frequencies of different transitions in neutral atoms formed in [33–38]. A typical accuracy is about 10%. Some and ions, applied or tried for precision spectroscopy, and their sensitivity to variations in ˛ due to relativistic corrections results are summarized in Tables 30.2 and 30.3, where we
list the relative sensitivity of the relativistic factors Frel to Atom, transition H, 1s 2s 0:00 changes in ˛, @ ln Frel
D : @ ln ˛
(30.7)
Ca, 1S 0 3P 1
0:02
87
Sr, 1S 0 3P 0
0.06
88
Note that the relativistic corrections in heavy atoms are proportional to .Z˛/2 because of the singularity of relativistic operators. Due to this, the corrections rapidly increase with the nuclear charge Z. The signs and magnitudes of are explained by a simple estimate of the relativistic correction. For example, an approximate expression for the relativistic correction factor for the hyperfine structure (Casimir factor) of an s-wave electron in an alkali-like atom is [32] Frel .˛/ D p
40
1 : 1 .Z˛/2 1 .4=3/.Z˛/2 1
1
3
Sr, S 0 P 0
111
0.06
Cd, 1S 0 3P 0 Yb, S 0 P 0
0.31
174
Yb, 1S 0 3P 0
0.31
199
1
0.23
171
3
1
3
0.81
C 1
3
Hg, S 0 P 0
27
Al , S 0 P 0
0.01
40
CaC , S 1=2 D 5=2
0.15
88
C
Sr , S 1=2 D 5=2
0.43
InC , 1S 0 3P 0
0.18
115
C 2
171
Yb , S 1=2 D 3=2
171
YbC , S 1=2 F 7=2
6:0
199
HgC , 2S 1=2 2D 5=2
2:9
2
1:0
The related integral for the matrix element of the hyperfine interaction rapidly converges in a vicinity of the nucleus, on the distance about aB =Z, where aB is the Bohr radius. The wave function of a valence electron at such distance is proportional to the Coulomb wave function with zero energy since the energy of valence electron is negligible there, and nuclear charge is not screened. That allows one to perform an analytical calculation. A similar rough estimation for the energy levels may be performed for the gross structure
The first term here is a standard one, while the second term may be enhanced if the frequency f is very small. In other words, if a certain frequency is a difference between two transitions that have different nonrelativistic quantum numbers (which means that the value of fNR is not small) and has a [very] small value (f fNR ), it may be enhanced. A small eventual value of the frequency means that there is an accidental cancelation between the nonrelativistic contribution and the relativistic correction, which depends differently on ˛. An example of such a transition is 2 2 2 .Z˛/ 1 .Za ˛/ mc : (30.8) 1 C ED the one between two nearly degenerate states 4f10 5d 6s and 2n2 n j C 1=2 9 4f 5d 2 6s in atomic dysprosium. The presence of a strong en8 Here, j is the electron angular momentum, n is the effective hancement ' 5:7 10 there was pointed out in [38]. The value of the principle quantum number (which determines dysprosium transition was studied in [39, 40]. the nonrelativistic energy of the electron), and Za is the charge seen by the valence electron – it is 1 for neutral atoms, 2 for singly charged ions, etc. This equation tells us that , 30.3 Laboratory Constraints on Temporal for the excitation of the electron from the orbital j to the Variations of Fundamental Constants orbital j 0 , has a different sign for j > j 0 and j < j 0 . The difference of sign between the sensitivities of the ytterbium Logarithmic derivatives in Eq. (30.7) appear since we are and mercury transitions in Table 30.3 reflects the fact that in looking for a variation of the constants in relative units. YbC , a 6s-electron is excited to the empty 5d-shell, while in In other words, we are interested in a determination of, HgC , a hole is created in the filled 5d-shell if the electron is e.g., ˛=˛t, while the input data of interest are related to excited to the 6s-shell. f =f t. Their relation takes the form Usually, the sensitivity is roughly of order of unity (Ta@ ln f @ ln fNR @ ln ˛ ble 30.3). However, it may happen that it is greatly enhanced. D C : (30.10) One can rewrite the definition in Eq. (30.7) as @t @t @t
D
@ ln.f =fNR / @ ln fNR 1 @f D C : @ ln ˛ @ ln ˛ f @ ln ˛
(30.9)
If one compares transitions of the same type – gross structure, fine structure – the first term cancels.
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∂ ln f/∂t (10 –15 yr–1) 1
1/κ ∂ ln f/∂t (10 –15 yr–1) 0.8 Hg+
0
Yb+(E2)
Yb+(E3)
0.6
Sr 0.4
–1
Hg+
0.2 Al+/Hg+
–2 0
Yb+(E2)
Dy
Yb (E3)
H
–3
+
–0.2
–4
–0.4 –0.6 –6
–4
–2
0
2
κ
0
0.5
1.0
1/κ
Fig. 30.5 Frequency variations versus their sensitivity following Eq. (30.11)
Fig. 30.6 Frequency variations versus their sensitivity following Eq. (30.14): value of @ ln fopt =@t = as function of 1=
30.3.1 Constraints from Absolute and Relative Optical Measurements
Progress with relative optical frequency measurements [14, 15], comparing the transitions in a mercury ion and an ion of aluminum, delivers the constraints on ˛ variation
Absolute frequency measurements offer the possibility to compare a number of optical transitions with frequencies fNR , which scale as cR1 , with the caesium hyperfine structure. One can rewrite Eq. (30.10) as
@ ln ˛ D . 1:7 ˙ 2:5/ 1017 yr1 : @t
(30.13)
This relative-measurement result, which contributes the dominant constraint on the ˛ variation, cannot be presented (30.11) in such a plot as in Fig. 30.5 since the disappearance of the cR1 term makes the result equivalent to the result for where dimensional quantities, such as frequency and the an infinite value of the overall sensitivity in the terms of Rydberg constant, are stated in SI units in Eq. (30.1). This Eq. (30.11). equation may be used in different ways. For example, in There is an alternative equation, Fig. 30.5, we plot experimental data for @ ln fopt =@t as a func1 @ ln fopt tion of the sensitivity and derive a model-independent 1 @ ln cR1 @ ln ˛ D C ; (30.14) constraint on the variation of the fine structure constant and
@t
@t @t the numerical value of the Rydberg frequency cR1 in the SI unit of hertz. The latter is of great metrological impor- which can be applied for a graphic presentation. In Fig. 30.6, tance, being related to a common drift of optical clocks with the intercept at 1= D 0 is the ˛P constraint, while the slope respect to a caesium clock, i.e., to the definition of the SI reflects the variation of cR1 . To plot all the data we have to use a two-dimensional plot second. The SI definition of the meter is impractical and so, (see below). The constraints on the variations of ˛ and cR1 in practice, the optical wavelengths of reference lines caliare correlated, and the standard uncertainty ellipse, defined brated against the caesium standard are used to determine as the SI meter [9]. @ ln fopt @ ln cR1 @ ln ˛ D C ; @t @t @t
Equation (30.11) allows us a simple graphic interpretation (Fig. 30.5). The data around D 0 are those, whose frequencies are well described by nonrelativistic formulas and have a low sensitivity to ˛ variation. A direct comparison of two optical frequencies allows us a model-independent constraint on a variation of the fine structure constant ˛, which for relative optical measurements is given by . 1 2 /d ln ˛ d ln.f 1=f 2/ D : dt dt
X 1 @ ln fi @ ln ˛ @ ln R1 i 2 @t @t @t u i i
!2
D 1 C 2min ;
is presented in Fig. 30.7. Here, we sum over all available data: @.ln fi /=@t is the central value of the observed drift rate, ui its 1 uncertainty, and 2min the minimized 2 of the fit. The results of the fit are presented in Table 30.4. The progress in the constraint on the time variation of the fine structure constant and of the numerical value of the Ryd(30.12) berg constant is summarized in Table 30.5.
30 Atomic Clocks and Constraints on Variations of Fundamental Constants ∂ ln {R∞}/∂t (10 –15 yr–1) 0.3 2014 0.2
We have also to mention a ‘nonclock’ constraint on the ˛ variation obtained from the study of the dysprosium atom on a determination of the splitting between the 4f10 5d 6s and 4f9 5d 2 6s states. The most recent Dy constraint is [40]
Dy
+
Yb (E2)
@ ln.˛/ D .0:6 ˙ 0:7/ 1016 yr1 : @t
0.1 Hg+
0 –0.1
Sr
30.3.2 Constraints from Microwave Clocks Al+/Hg+
–0.2
Yb+(E3) –0.3 –0.3
–0.2
–0.1
0
0.1 0.2 ∂ ln α/∂t (10 –15 yr–1)
Fig. 30.7 Constraints on the time variations of the fine structure constant ˛ and the numerical value of the Rydberg constant (from [41])
Table 30.4 Model-independent laboratory constraints on the possible time variations of natural constants [41, 42] X ˛ fcR1 g Cs =B Rb =Cs Yb =Cs
@ ln X=@t 0:022.22/ 1015 yr1 0:01.14/ 1015 yr1 0:07.16/ 1015 yr1 0:15.9/ 1015 yr1 .3 ˙ 3/ 1014 yr1
Table 30.5 Progress in constraining a possible time variation of the fundamental constants: the results of 2004 (following [43, 44]), 2008 (following [45]), and of 2014 using data summarized in [42] but following [43, 44] (see [41] for details). Here, is the correlation coefficient
.@ ln ˛=@t; @ lnfR1 g=@t / Year 2004 2008 2014
457
@ ln ˛=@t .1015 yr1 / 0:4.2:0/ 0:018.23/ 0:022.22/
@ lnfR1 g=@t .1015 yr1 / 1:5.3:2/ 0:20.37/ 0:01.14/
0.13 0.17 0:08
A model-independent comparison of different HFS transitions is not simple because their nonrelativistic contributions fNR are not the same but involve different magnetic moments. Applying Eq. (30.15) to experimental data, one can obtain constraints on the relative variations of the magnetic moments of Rb, Cs, and Yb (Table 30.4).
30.3.3 A Model-Independent Constraint on the Variation of the Electron-to-Proton Mass Ratio from Molecular Spectroscopy Vibrational and rotational spectroscopy of molecules provide direct access to transition frequencies that depend on the electronic to nuclear mass ratio. Precision measurements have been performed with two-photon Ramsey spectroscopy of a rovibrational transition in a supersonic beam of SF6 . A relative uncertainty of 2:8 1014 has been obtained for the transition frequency at 28.4 THz, and from a sequence of measurements against a primary Cs clock spanning about 2 years, a limit on the temporal variation of the proton to electron mass ratio has been derived [46] @ ln.mp =me / D .3:8 ˙ 5:6/ 1014 yr1 : @t
This result makes use of the more stringent limit on a variation of the Cs magnetic moment, obtaining essentially The numerical value of the Rydberg constant, from the a model-independent constraint on a change of mp =me . point of view of fundamental physics, can be expressed in terms of the caesium hyperfine interval in atomic units, and its variation may be expressed in terms of the variations of 30.3.4 Other Model-Dependent Constraints ˛ and Cs =B . A constraint for the latter is presented in Table 30.4. Because of that, there is also an alternative pa- In order to gain information on constants more fundamental rameterization applied for the 2D constraint. It is based on than the nuclear magnetic moments, any further evaluation of the experimental data should consider the model. Here, the relation we apply the Schmidt model, which is far from perfect. It is @ ln.Cs =e / @ ln ˛ @ ln cR1 an ab initio model, but its accuracy for any individual case D C . C 2:83/ : (30.15) is unclear. The model-dependent constraints obtained by ex@t @t @t ploiting the Schmidt model are summarized in Table 30.6. To compare the results of two equivalent evaluations one has The nucleon g factors, in turn, depend on a dimensionto remember the correlations in the results for the fitting pa- less fundamental constant mq =QCD , where mq is the quark rameters (Table 30.5). The correlation is unimportant for the mass, and QCD is the quantum chromodynamic (QCD) scale. A study of this dependence may supply us with deep central values but affects the uncertainty of the outcome.
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Table 30.6 Model-dependent laboratory constraints on possible time variations of fundamental constants. The uncertainties here do not include uncertainties from the application of the Schmidt model
unclear. Because of a broad range of options there is a need for the development of as many different searches as possible, and the laboratory search for variations is an attractive X @ ln X=@t opportunity to open up a way that could lead to new physics. me =mp 0:02.16/ 1015 yr1 Precision frequency measurements with or without the p =e 0:08.16/ 1015 yr1 use of clocks are the most accurate among the other mea15 1 gp 0:07.5/ 10 yr surements. They are used not only to constrain the variation gn .3 ˙ 3/ 1014 yr1 of the values of the constants but also on some other possible effects of new physics, ranging from the violation of Lorentz insight into the possible variations of the more fundamental invariance to modified fundamental theories, opening a winproperties of Nature (see [33, 34] for details). This approach dow of opportunities for new physics [49]. is promising, but its accuracy needs to be better understood. Following the model suggested in [47, 48], the constraint on Acknowledgements The authors are very grateful to their colleagues the variation of the proton-to-electron mass ratio reads [42] and to participants of the ACFC-2003, 2007, 2011, and 2015 meet@ ln.mp =me / D .0:5 ˙ 1:6/ 1016 yr1 : @t
30.4 Summary The results collected in Tables 30.4 and 30.6 are competitive with or even more sensitive than data from other searches and have a more reliable interpretation. The results from astrophysical searches and the study of the samarium resonance from Oklo data (e.g., [4, 5]) possess sensitivity over extended time scales; however, they are more difficult to interpret. We have, for example, not assumed any hierarchy in variation rates or that some constants stay fixed while others vary, as it is done in the study of the position of the Oklo resonance. The evaluation presented here is transparent, and any particular calculation or measurement can be checked. In contrast, the astrophysical data show significant results only after an intensive statistical evaluation. The laboratory searches involving atomic clocks have definitely shown progress, and a further increase in the accuracy of these clocks can be expected, as well as an increase in the number of different kinds of frequency standards (e.g., optical Sr, SrC , Yb, YbC , Hg standards are being tried). Systems of enhanced sensitivity have been identified, for example in highly charged ions [49], and experiments for their study are under development. An optical clock based on a nuclear transition in Th-229 is also under consideration [50, 51]. Such a clock would offer different sensitivity to systematic effects, as well as to variations of different fundamental constants [52]. Laboratory searches are not necessarily limited to experiments with metrological accuracy. An example of a highsensitivity search with a relatively low accuracy is the study of the dysprosium atom for a determination of the splitting between the 4f10 5d 6s and 4f9 5d 2 6s states, which offers a great sensitivity value of ' 5:7 108 [38]. Variations of constants on the cosmological time scale can be expected, but the magnitude, as well as other details, are
ings for useful and stimulating discussions. The work of SGK was in part supported by DFG (under grant # KA 4645/1-1). The authors are grateful to V.A. Dzuba and, V.G. Ivanov for useful and stimulating discussions.
References 1. Mohr, P.J., Newell, D.B., Taylor, B.N.: Rev. Mod. Phys. 88, 035009 (2016) 2. Karshenboim, S.G.: Time and space variation of fundamental constants: motivation and laboratory search. In: Bigi, I.I., Faessler, M. (eds.) Time and Matter, p. 26. World Scientific, New Jersey, London, Singapore, Shanghai, Taipei, Chennai (2006) 3. Karshenboim, S.G.: Gen. Rel. Grav. 38, 159 (2006) 4. Karshenboim, S.G., Peik, E. (eds.): Astrophysics, Clocks and Fundamental Constants. Lect. Notes in Phys., vol. 648. Springer, Berlin, Heidelberg (2004) 5. S. G. Karshenboim, E. Peik (Eds.) Atomic Clocks and Fundamental Constants, Eur. Phys. J. Special Topics 163 (2008). Special issue 6. Ramsey, N.F.: Rev. Mod. Phys. 62, 541 (1990) 7. Bauch, A., Telle, H.R.: Rep. Prog. Phys. 65, 789 (2002) 8. Wynands, R., Weyers, S.: Metrologia 42, 64 (2005) 9. Riehle, F., Gill, P., Arias, F., Robertsson, L.: Metrologia 55, 188 (2018) 10. Paul, W.: Rev. Mod. Phys. 62, 531 (1990) 11. Dehmelt, H.: IEEE Trans. Instrum. Meas. 31, 83 (1982) 12. Ludlow, A.D., Ye, M.M.B.J., Peik, E., Schmidt, P.O.: Rev. Mod. Phys. 87, 637 (2015) 13. Stenger, J., Tamm, C., Haverkamp, N., Weyers, S., Telle, H.R.: Opt. Lett. 26, 1589 (2001) 14. Udem, T., Diddams, S.A., Vogel, K.R., Oates, C.W., Curtis, E.A., Lee, W.D., Itano, W.M., Drullinger, R.E., Bergquist, J.C., Hollberg, L.: Phys. Rev. Lett. 86, 4996 (2001) 15. Bize, S., Diddams, S.A., Tanaka, U., Tanner, C.E., Oskay, W.H., Drullinger, R.E., Parker, T.E., Heavner, T.P., Jefferts, S.R., Hollberg, L., Itano, W.M., Wineland, D.J., Bergquist, J.C.: Phys. Rev. Lett. 90, 150802 (2003) 16. Huntemann, N., Okhapkin, M.V., Lipphardt, B., Tamm, C., Peik, E.: Phys. Rev. Lett. 108, 090801 (2012) 17. Fisk, P.T.H., Sellars, M.J., Lawn, M.A., Coles, C.: IEEE Trans. UFFC 44, 344 (1997) 18. Fisk, P.T.: Rep. Prog. Phys. 60, 761 (1997) 19. Warrington, R.B., Fisk, P.T.H., Wouters, M.J., Lawn, M.A.: Proceedings of the 6th Symposium Frequency Standards and Metrology. World Scientific, Singapore, p 297 (2002) 20. Bordé, C.J.: Phys. Lett. A. 140, 10 (1989)
30 Atomic Clocks and Constraints on Variations of Fundamental Constants 21. Wilpers, G., Binnewies, T., Degenhardt, C., Sterr, U., Helmcke, J., Riehle, F.: Phys. Rev. Lett. 89, 230801 (2002) 22. Riehle, F., Degenhardt, C., Lisdat, C., Wilpers, G., Schnatz, H., Binnewies, T., Stoehr, H., Sterr, U.: An optical frequency standard with cold and ultra-cold calcium atoms. In: Karshenboim, S.G., Peik, E. (eds.) Astrophysics, Clocks and Fundamental Constants Lect. Notes in Phys., vol. 648, p. 229. Springer, Berlin, Heidelberg (2004) 23. Katori, H., Takamoto, M., Pal’chikov, V.G., Ovsiannikov, V.D.: Phys. Rev. Lett. 91, 173005 (2003) 24. Jun Ye, Kimble, H.J., Katori, H.: Science 320, 1734 (2008) 25. Parthey, C.G., Matveev, A., Alnis, J., Bernhardt, B., Beyer, A., Holzwarth, R., Maistrou, A., Pohl, R., Predehl, K., Udem, T., Wilken, T., Kolachevsky, N., Abgrall, M., Rovera, D., Salomon, C., Laurent, P., Hänsch, T.W.: Phys. Rev. Lett. 107, 203001 (2011) 26. Matveev, A., Parthey, C.G., Predehl, K., Alnis, J., Beyer, A., Holzwarth, R., Udem, T., Wilken, T., Kolachevsky, N., Abgrall, M., Rovera, D., Salomon, C., Laurent, P., Grosche, G., Terra, O., Legero, T., Schnatz, H., Weyers, S., Altschul, B., Hänsch, T.W.: Phys. Rev. Lett. 110, 230801 (2013) 27. Udem, T., Reichert, J., Holzwarth, R., Diddams, S., Jones, D., Ye, J., Cundiff, S.: T. W. H nsch, J. Hall: A new type of frequency chain and its application to fundamental frequency metrology. In: Karshenboim, S.G., Pavone, F.S., Bassani, F., Inguscio, M., Hänsch, T.W. (eds.) The Hydrogen Atom: Precision Physics of Simple Atomic Systems Lect. Notes in Phys., vol. 570, p. 229. Springer, Berlin, Heidelberg (2001) 28. Savedoff, M.P.: Nature 178, 688 (1956) 29. Thompson, R.I.: Astrophys. Lett. 16, 3 (1975) 30. Varshalovich, D.A., Ivanchik, A.V., Orlov, A.V., Potekhin, A.Y., Petitjean, P.: Current status of the problem of cosmological variability of fundamental physical constants. In: Karshenboim, S.G., Smirnov, V.B. (eds.) Precision Physics of Simple Atomic Systems Lect. Notes in Phys., vol. 627, p. 199. Springer, Berlin, Heidelberg (2003) 31. Karshenboim, S.G.: Can. J. Phys. 78, 639 (2000) 32. Prestage, J.D., Tjoelker, R.L., Maleki, L.: Phys. Rev. Lett. 74, 3511 (1995) 33. Flambaum, V.V.: Limits on temporal variation of fine structure constant, quark masses and strong interaction from atomic clock experiments. In: Hannaford, P., Sidorov, A., Bachor, H., Baldwin, K. (eds.) Laser Spectroscopy, p. 49. World Scientific, New Jersey, London, Singapore, Shanghai, Taipei, Chennai (2004) 34. Flambaum, V.V., Leinweber, L.B., Thomas, A.W., Young, R.D.: Phys. Rev. 69, 115006 (2004) 35. Dzuba, V.A., Flambaum, V.V., Webb, J.K.: Phys. Rev. Lett. 82, 888 (1999) 36. Dzuba, V.A., Flambaum, V.V., Webb, J.K.: Phys. Rev. A 59, 230 (1999) 37. Dzuba, V.A., Flambaum, V.V.: Phys. Rev. A 61, 034502 (2001) 38. Dzuba, V.A., Flambaum, V.V., Marchenko, M.V.: Phys. Rev. A 68, 022506 (2003) 39. Cingöz, A., Lapierre, A., Nguyen, A.-T., Leefer, N., Budker, D., Lamoreaux, S.K., Torgerson, J.R.: Phys. Rev. Lett. 98, 040801 (2007) 40. Leefer, N., Weber, C.T.M., Cing z, A., Torgerson, J.R., Budker, D.: Phys. Rev. Lett. 111, 060801 (2013) 41. Karshenboim, S.G., Ivanov, V.G.: Appl. Phys. B. 123, 18 (2017) 42. Huntemann, N., Lipphardt, B., Tamm, C., Gerginov, V., Weyers, S., Peik, E.: Phys. Rev. Lett. 113, 210802 (2014) 43. Peik, E., Lipphardt, B., Schnatz, H., Schneider, T., Tamm, Chr , Karshenboim, S.G.: Phys. Rev. Lett. 93, 170801 (2004) 44. Karshenboim, S.G., Flambaum, V.V., Peik, E.: In: Drake, G.W.F. (ed.) Springer Handbook of Atomic, Molecular and Optical
45. 46. 47. 48. 49. 50. 51. 52.
459 Physics, p. 455. Springer, NY (2005). (That is the same chapter but in the previous edition of the Handbook.) Karshenboim, S.G., Peik, E.: Eur. Phys. J. Special Top. 163, 1 (2008) Shelkovnikov, A., Butcher, R.J., Chardonnet, C., Amy-Klein, A.: Phys. Rev. Lett. 100, 150801 (2008) Flambaum, V.V., Tedesco, A.F.: Phys. Rev. C 73, 055501 (2006) Dinh, T.H., Dunning, A., Dzuba, V.A., Flambaum, V.V.: Phys. Rev. A 79, 054102 (2009) Safronova, M.S., Budker, D., DeMille, D., Kimball, D.F.J., Derevianko, A., Clark, C.W.: Rev. Mod. Phys. 90, 025008 (2018) Peik, E., Tamm, C.: Europhys. Lett. 61, 181 (2003) Peik, E., Okhapkin, M.: C. R. Phys. 16, 516 (2015) Berengut, J.C., Dzuba, V.A., Flambaum, V.V., Porsev, S.G.: Phys. Rev. Lett. 102, 210801 (2009)
Savely G. Karshenboim Dr Savely G. Karshenboim graduated in 1983 from St. Petersburg (then Leningrad) State University, from where he also received his PhD (1992) and DSc (1999). He was with D.I. Mendeleev Institute for Metrology, Max-PlanckInstitut für Quantenoptik (Garching), and Pulkovo Observatory. He is currently with Ludwig-Maximilians-Universität München. His scientific interests include precision physics of simple atoms, quantum electrodynamics, determination of fundamental constants, and the search for their variations. Victor Flambaum Dr Victor Flambaum is a Professor of Physics and holds a Chair of Theoretical Physics. PhD, DSc from the Institute of Nuclear Physics, Novosibirsk, Russia. He has about 400 publications in atomic, nuclear, particle, solid state, statistical physics, general relativity and astrophysics including works on violation of fundamental symmetries (parity, time reversal, Lorentz), test of unification theories, temporal and spatial variation of fundamental constants from Big Bang to present, many-body theory and high precision atomic calculations, as well as statistical theory of finite chaotic Fermi systems and enhancement of weak interactions. Ekkehard Peik Dr Ekkehard Peik is Head of the Time and Frequency Department at PTB in Braunschweig. His research interests are in the fields of optical clocks with laser-cooled and trapped ions, precision laser spectroscopy, the low-energy nuclear isomer of Th-229, the metrology of time and frequency, and tests of fundamental physics with atomic clocks.
30
Searches for New Particles Including Dark Matter with Atomic, Molecular, and Optical Systems Victor Flambaum
and Yevgeny Stadnik
Contents 31.1
Nongravitational Interactions of Spinless Bosons . . 462
31.2 31.2.1 31.2.2
New Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Macroscopic-Scale Experiments . . . . . . . . . . . . . . 463 Atomic-Scale Experiments . . . . . . . . . . . . . . . . . 463
31.3
Laboratory Sources . . . . . . . . . . . . . . . . . . . . . 464
31.4
Astrophysical Sources . . . . . . . . . . . . . . . . . . . 465
31.5 31.5.1 31.5.2 31.5.3
Cosmological Sources . . . . . . . Haloscope Experiments . . . . . . Spin-Precession Experiments . . . Time-Varying Physical Constants
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466 466 466 467
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Abstract
The “standard model” of physics has been successful in explaining most physical processes and phenomena that we see around us. However, despite the great success of the standard model, there remain a number of unresolved puzzles within the model, as well as questions about the self-consistency of the framework. Additionally, various independent astrophysical and cosmological observations contradicting the predictions of the standard model have been accumulating over the course of the past century. Many of these puzzles and unexpected observations can be elegantly explained by postulating the existence of at least one new particle or field outside of the present standard model. New particles can manifest their effects in many settings, ranging from effects on subatomic to galactic length V. Flambaum () Dept. of Physics, University of New South Wales Sydney, Australia e-mail: [email protected] Y. Stadnik School of Physics, The University of Sydney Sydney, Australia e-mail: [email protected]
31
scales. The nature of these effects depends on the specific particles and their nongravitational interactions. In this chapter, we give a brief overview of how atomic, molecular, and optical systems can be used to search for new particles. To illustrate the basic principles behind these methods, we focus on the simplest class of particles, namely new spinless bosons. Keywords
dark matter astrophysics axion fundamental constants electric dipole moment clock spectroscopy laser cavity interferometry magnetometry
The “standard model” of particle physics at present provides the most fundamental framework for understanding the basic building blocks of matter and describing various known interactions between these building blocks. The standard model does incredibly well in describing physical processes and phenomena that take place over a very broad range of energies and length scales, from explaining the binding of the constituents of atoms to understanding the formation and evolution of stars. However, numerous astrophysical and cosmological observations that cannot be explained by the standard model have been accumulating over the course of the past century. Observations of stellar orbits about the galactic center from as early as the 1930s indicate the presence of a nonbaryonic matter component that is traditionally termed “dark matter” (this nonbaryonic matter component does not appreciably emit or absorb electromagnetic radiation) [1]. Further astrophysical evidence for dark matter comes from measurements of angular fluctuations in the cosmic microwave background spectrum [2] and the need for nonbaryonic matter to explain the observed structure formation in our universe [1]. Additionally, distance and redshift measurements of supernovae show that the expansion of the universe is accelerating, indicating that the universe is being pushed apart by a repulsive
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_31
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force associated with a “dark energy” component [3]. These dark components (which are naturally explained by postulating the existence of at least one new particle or field) overwhelmingly dominate the observed matter-energy content of our universe, with ordinary baryonic matter making up only a small fraction of the total content [2]. Another profound mystery is the observed predominance of matter over antimatter in our universe—the problem of baryogenesis. The standard model contains the necessary ingredients to produce ever slightly more matter than antimatter; however, the observed predominance of matter over antimatter in our universe is much larger than can be accommodated within the standard model [4]. One of the key ingredients for baryogenesis is CP violation, which is the violation of the product of the charge parity (C D exchange of particles and antiparticles) and parity (P D inversion of spatial coordinates) symmetries. Additional sources of CP violation necessary to explain baryogenesis may come from new particles possessing CP-violating interactions with ordinary matter. Intriguingly, practically no CP violation has been observed in strong processes in the standard model (compared with the relatively large amount of CP violation in weak processes). This puzzling observation—termed the “strong CP problem”—is most elegantly explained by postulating the existence of a new low-mass feebly-interacting spinless boson called the axion [5]. In order to corroborate or refute models that claim to explain the above problems and observations via putative new particles, one needs experimental probes for such particles. New particles may arise in several different settings:
V. Flambaum and Y. Stadnik
be done, focusing mainly on new spinless bosons (which are the simplest possibility from the theoretical point of view) to help illustrate the basic principles behind the methods. We begin by presenting the simplest possible nongravitational interactions of spinless bosons with ordinary matter (Sect. 31.1). We then explain how atomic, molecular, and optical systems can be used to search for spinless bosons possessing nongravitational interactions in a broad variety of settings (Sects. 31.2–31.5). Unless explicitly stated otherwise, we adopt the natural units „ D c D 1 in this chapter.
31.1 Nongravitational Interactions of Spinless Bosons The possible nongravitational interactions of spinless bosons can be broadly distinguished on the basis of the parity symmetry (behavior under the inversion of spatial coordinates). The most relevant scalar-type (even-parity) interactions of a spinless boson with ordinary matter are: Llin. scalar D quad.
Lscalar D
gs
hs 4
4
F F
2 F F 2
X X
gs N
;
(31.1)
hs N
;
(31.2)
where the first term represents the interaction of the spinless boson with the electromagnetic field tensor F , and the second term represents the interaction of the spinless boson with a fermion field , with N D 0 the Dirac adjoint. Here, g;s and hs; are parameters that determine the relevant non(1) As mediators of new interactions between particles or gravitational interaction strengths. bodies (Sect. 31.2) The most relevant pseudoscalar-type (odd-parity) interac(2) Produced in laboratories or colliders (Sect. 31.3) tions of a spinless boson with ordinary matter are: (3) Produced in stars and astrophysical processes (Sect. 31.4) p p g gg (4) Constitute the observed dark matter or dark energy Lpseudoscalar D F FQ C G GQ (Sect. 31.5). 4 X 4 p i g N 5 ; (31.3) Atomic, molecular, and optical systems lie at the heart of some of the highest precision measurements currently available. Optical clocks, which measure transition frequencies in atoms and ions, have demonstrated a fractional precision at the level 1018 [6, 7]. Optical magnetometers, which measure magnetic fields using atoms, have demonstrated a magnetic field sensitivity at the level 1015 T Hz1=2 [8]. Laser interferometers (which have directly detected gravitational waves) have demonstrated an equivalent sensitivity to length fluctuations at the level 1023 Hz1=2 [9]. Can these extraordinary levels of precision and sensitivity be leveraged to search for new particles? The answer to this question is in the affirmative. Indeed, new particles arising in all types of settings described above can be sought for with experiments using atomic, molecular, and optical systems. In this chapter, we present a brief overview of how this can
where the first term represents the interaction of the spinless boson with the electromagnetic field tensor F , with FQ the dual field tensor, the second term represents the interaction of the spinless boson with the gluonic field tensor G, and the third term represents the interaction of the spinless bop son with a fermion field . Here, g;g; are parameters that determine the relevant nongravitational interaction strengths.
31.2 New Forces In the presence of the nongravitational interactions in Eqs. (31.1)–(31.3), new forces can be mediated between particles or bodies via the exchange of spinless boson(s). The
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simplest possibility involves the exchange of a single boson (1) Torsion pendula to search for anomalous torques [17–20] between two fermions in the presence of the linear-in- in- (2) Atom interferometers to search for anomalous accelerations [21, 24] teractions in the last terms of Eqs. (31.1) and (31.3). In this case, there are three distinct potentials that arise from the per- (3) Atomic clocks and other spectroscopy-based measurements to search for anomalous frequency shifts [12, 25, mutation of the two vertex types. In the nonrelativistic limit, 26]. these potentials take the following form [10, 11]: em r ; 4 r m em r 1 p s ; Vps .r/ D Cg1 g2 1 rO 2 C r r 8 m1 p p m 1 g1 g2 4 Vpp .r/ D 1 2 3 C 2 C ı.r/ 4 r r 3 " 3m 3 . 1 r/. O O C 2 C 2 r/ r3 r
The first two types of methods involve measurements of vector quantities, namely differences of torques and accelerations, respectively, of two different test bodies, while the (31.5) third method involves measuring a scalar quantity, namely the difference in the ratio of two transition frequencies at two different distances from a massive body. We mention that lunar laser ranging measurements can also be used to search #) m2 for spin-independent anomalous interactions [27]. r Various methods can be used to search for new spinm r dependent forces in macroscopic-scale experiments: e : (31.6) 4 m1 m2 (4) Torsion pendula to search for anomalous torques [28, 29] Here, m is the mass of the exchanged boson, 1 and 2 (5) Magnetometers to search for anomalous spin-precession effects [30–38]. denote the Pauli spin matrix vectors of the two fermions, rO is Vss .r/ D g1s g2s
(31.4)
the unit vector directed from fermion 2 to fermion 1, and r is the distance between the two fermions. In Eq. (31.5), the cross term (obtained by permuting the particle indices 1 $ 2) is implicit. Other relatively common potentials include the potential mediated by the exchange of a single boson between two bodies with nonzero electromagnetic energies in the presence of the linear-in- interaction in the first term of Eq. (31.1) [12], the potential mediated by the exchange of a pair of bosons between two fermions in the presence of the quadratic-in- interactions in the last term of Eq. (31.2) [13], and the potential mediated by the exchange of a pair of fermions (including neutrinos) between two particles [14]. In the limit when the mass of the exchanged particles is small, these potentials scale as V .r/ / 1=r, V .r/ / 1=r 3 and V .r/ / 1=r 5, respectively.
31.2.1 Macroscopic-Scale Experiments When the condition m 1=r is satisfied, the potentials in Eqs. (31.4)–(31.6) can be treated as long-range (since the exponential terms reduce to em r 1 when m r 1). Experiments performed on macroscopic length scales provide an excellent way of probing these new interactions. These types of experiments employ a massive body, such as the Sun, Earth, Moon, or a massive object in the laboratory, which functions as the source of new bosons. In order to detect effects associated with the anomalous interactions mediated by these bosons, a high-precision detector is required. Various methods can be used to search for new spin-independent forces in macroscopic-scale experiments:
Current limits from macroscopic-scale experiments on several types of nongravitational interactions of spinless bosons are shown in Figs. 31.1–31.3.
31.2.2 Atomic-Scale Experiments Compared with the macroscopic-scale experiments discussed in Sect. 31.2.1, the condition m 1=r holds up to much larger boson masses when a boson is exchanged between the constituents of an atom or molecule. This is because the interparticle separations between the constituents of atomic systems are much smaller than the length scales in macroscopic-scale experiments. Thus, phenomena originating on atomic and subatomic length scales are generally much more sensitive to bosons with larger masses. Various atomic-scale phenomena can be used to search for new forces: (1) Comparison of measured and predicted spectra of atoms, molecules and ions to search for new parity-conserving forces [14, 40, 44–49] (2) Comparison of measured and predicted parity-violating observables in atoms and molecules to search for new parity-violating forces [50] (3) Measurements of permanent electric dipole moments in atoms and molecules to search for new parity and timereversal-invariance-violating forces [39, 51]. There is an important difference between atomic-scale and macroscopic-scale experiments in the regime of a large boson mass, m 1=r. Macroscopic-scale experiments lose
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V. Flambaum and Y. Stadnik
log10 0
(
g γs –1
GeV
log10 (gNp) 0
)
New forces (atomic-scale) Light-shining-through-a-wall
–5
–2 New forces (macroscopic-scale) –4
–10
Magnetometry (DM)
New forces –15
–6
–20
–8
–25 –30
–10
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–20 –20
–15
–10
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–10 –15 –20 New forces (macroscopic-scale)
–30 –5
0
–5
0
( ) mϕ eV
Fig. 31.3 Limits on the linear pseudoscalar interaction of a spinless boson with nucleons, as defined in the last term of Eq. (31.3). The region in blue corresponds to constraints from a macroscopic-scale experiment that searches for new forces [31]. The region in magenta corresponds to constraints from molecular hydrogen spectroscopy measurements and comparison with theory [40]. The region in red corresponds to constraints from magnetometry measurements that search for the effects of a relic coherently oscillating field D 0 cos.m t /, which saturates the local cold dark matter (DM) content [41]. The region in light gray corresponds to astrophysical constraints pertaining to supernova energy-loss bounds [42, 43]
nomena, however, there is always a finite probability for two constituent particles to be located very close to each other, and so these types of experiments lose sensitivity to new forces much more slowly (at a power-law rate) when the boson mass becomes large. Current limits from atomic-scale experiments on several types of nongravitational interactions of spinless bosons are shown in Figs. 31.2 and 31.3.
New forces (atomic-scale)
–25
–10
log10
( )
Fig. 31.1 Limits on the linear scalar interaction of a spinless boson with the photon, as defined in the first term of Eq. (31.1). The region in green corresponds to constraints from light-shining-through-a-wall experiments [15, 16]. The region in blue corresponds to constraints from macroscopic-scale experiments that search for new forces [12, 17–21]. The region in red corresponds to constraints from atomic spectroscopy measurements that search for the effects of a relic coherently oscillating field D 0 cos.m t /, which saturates the local cold dark matter (DM) content [22, 23]
–5
–15
0 log10
–35
Astrophysical bounds (supernovae)
31.3 Laboratory Sources 5
10 log10
( ) mϕ eV
Fig. 31.2 Limits on the P; T -violating scalar-pseudoscalar nucleon– electron interaction mediated by a spinless boson , as given in Eq. (31.5) (which arises from the last terms of Eqs. (31.1) and (31.3)). The region in blue corresponds to constraints from macroscopic-scale experiments that search for new forces [28–30, 37, 38]. The region in magenta corresponds to constraints from atomic and molecular electric dipole moment experiments [39]
sensitivity to new forces exponentially quickly when the boson mass becomes large, because the interaction becomes contact-like, and so the very heavy boson cannot propagate between the source body and detector. In atomic-scale phe-
In the presence of the nongravitational interactions in the first terms of Eqs. (31.1) and (31.3), spinless bosons may interconvert with photons. Several different types of methods can be used to exploit this possible interconversion: (1) “Light-shining-through-a-wall” experiments [15, 16, 52, 53]. The basic idea here is to shine a powerful laser into a region of strong magnetic field. Some of the laser photons will convert into spinless bosons (provided that the energy of these photons is not less than the rest-mass energy of the spinless boson), which then pass through a wall that is impervious to photons (but not to the spinless bosons). A second strong magnetic field is applied on the other side of this wall in order to reconvert some of the transmitted spinless bosons back into photons for detection. In principle, it is not necessary for all of the incident laser photons to be blocked by the
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log10 0
(
g γp –1
GeV
465
31.4 Astrophysical Sources
)
In the presence of the nongravitational interactions in Eqs. (31.1)–(31.3), spinless bosons can be produced and subLight-shining-through-a-wall sequently emitted from the hot interiors of active stars (such –5 as the Sun) and dead stars (such as white dwarfs), as well as in supernovae explosions. Excessive emission of spinless Helioscopes bosons from astrophysical sources would contradict ob–10 servations and corresponding standard-model calculations, providing strong constraints on possible nongravitational inAstrophysical bounds Astrophysical bounds (stellar evolution) teractions of spinless bosons (Figs. 31.3–31.6) [42, 43]. If (supernovae) –15 spinless bosons are emitted from the nearest star (the Sun), Haloscopes (DM) then it also becomes feasible to search for these particles in –12 –10 –8 –6 –4 –2 0 2 terrestrial experiments. mϕ Spinless bosons emitted from the Sun can be detected log10 eV using helioscope experiments, which seek to exploit the interconversion of spinless bosons with photons in a strong apFig. 31.4 Limits on the linear pseudoscalar interaction of a spinless plied magnetic field [59, 60, 63–65]. Helioscope experiments boson with the photon, as defined in the first term of Eq. (31.3). The region in green corresponds to constraints from “light-shining- are somewhat similar to the “light-shining-through-a-wall” through-a-wall” experiments [15, 16]. The region in yellow corresponds experiments discussed in Sect. 31.3, except that the source to constraints from helioscope experiments that search for bosons emit- of spinless bosons in helioscope experiments is provided by ted from the Sun [59, 60]. The region in red corresponds to constraints nature. The nature of the spinless boson sources in these from haloscope experiments that search for the conversion of galactic dark matter (DM) bosons into photons [61, 62]. The regions in light two types of experiments is very different, however. In a gray correspond to astrophysical constraints pertaining to stellar evolu- “light-shining-through-a-wall” experiment, the energy of the tion and supernova energy-loss bounds [42, 43] spinless bosons depends on the frequency of the laser source used and can, thus, be altered. Additionally, since lasers wall. A tiny fraction of incident photons can be transmitted through the wall, so that an atomic or molecular transition g gp log10 can be resonantly induced involving the interference of phoGeV–1 0 ton and spinless-boson-induced amplitudes (assuming there exists a nongravitational interaction between the spinless boAstrophysical bounds (supernovae) son and electron) [54]. –5 (2) Experiments to search for vacuum birefringence and dichroism [55–58]. The basic idea in these types of experi–10 ments is to shine a polarized laser into a region of strong magnetic field. Vacuum birefringence involves different indices Astrophysical bounds –15 (BBN, DM) of refraction for light polarized parallel and perpendicular to an applied magnetic field and is caused by virtual spinless –20 bosons. Dichroism involves different absorptivities for light Magnetometry (DM) of different polarizations in an applied magnetic field and is –25 –20 –15 –10 –5 0 caused by the production of real spinless bosons. mϕ Although electric fields could also be used in these experlog10 eV iments, in practice it is much easier to generate a stronger magnetic field (in terms of the equivalent electromagnetic energy density) in the laboratory. An advantage of produc- Fig. 31.5 Limits on the linear pseudoscalar interaction of a spinless boson with gluons, as defined in the second term of Eq. (31.3). The ing spinless bosons in the laboratory is that the energies region in red corresponds to constraints from magnetometry measureof these bosons are fixed by energy conservation, so reso- ments that search for the effects of a relic coherently oscillating field nance techniques can be applied without having to scan over D 0 cos.m t /, which saturates the local cold dark matter (DM) an a priori unknown range of boson energies. Current lim- content [41]. The region in light gray corresponds to astrophysical constraints pertaining to supernova energy-loss bounds [77]. The reits from laboratory source experiments on several types of gion in dark gray corresponds to astrophysical constraints pertaining to nongravitational interactions of spinless bosons are shown in big bang nucleosynthesis (BBN) measurements, assuming that spinless bosons saturate the DM content [78–80] Figs. 31.1 and 31.4.
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Fig. 31.6 Limits on the quadratic scalar interaction of a spinless boson with the photon, as defined in the first term of Eq. (31.2). The region in red corresponds to constraints from atomic spectroscopy measurements that search for the effects of a relic coherently oscillating field D 0 cos.m t /, which saturates the local cold dark matter (DM) content [79, 90]. The region in light gray corresponds to astrophysical constraints pertaining to supernova energy-loss bounds [13]. The region in dark gray corresponds to astrophysical constraints pertaining to big bang nucleosynthesis (BBN) measurements, assuming that spinless bosons saturate the DM content [79]
are practically monochromatic sources of light, the resulting energy spectrum of spinless bosons is likewise sharply peaked in these types of laboratory experiments. On the other hand, in a helioscope experiment, the energies of the spinless bosons are determined by the core temperature of the Sun ( 1 keV), and the energy spectrum of the bosons is relatively broad. Current limits from solar source experiments on one type of nongravitational interaction of a spinless boson are shown in Fig. 31.4.
31.5 Cosmological Sources Low-mass (sub-eV) spinless bosons can be produced efficiently via nonthermal production mechanisms (which impart practically no kinetic energy to the bosons), such as “vacuum misalignment” in the early universe [66], and subsequently form a coherently oscillating classical field: D 0 cos.!t/, with the angular frequency of oscillation given by ! m c 2 =„, where m is the boson mass, c is the speed of light, and „ is the reduced Planck constant. The classical nature of this field arises due to the large number of low-mass bosons per reduced de Broglie volume. The oscillating bosonic field carries the energy density m2 02 =2, which may saturate the local cold dark matter (DM) energy local 0:4 GeV=cm3 [67]. If these bosons comdensity DM prise all of the DM, then the requirement that the boson de
Broglie wavelength does not exceed the DM halo size of the smallest dwarf galaxies gives the lower boson mass bound m ' 1022 eV. A variety of atomic, molecular, and optical experiments can be used to search for oscillating DM fields. The specific detection methods crucially depend on the particular nongravitational interactions between the bosonic DM and ordinary matter that are probed. In this section, we give a brief overview of the main types of detection methods for oscillating bosonic fields. We note that similar detection strategies can also be implemented to search for bosonic fields that form “clump-like” DM, except in this case, a network of detectors is required to unambiguously confirm the passage of such DM clumps [68–72]. Additionally, spinless bosonic fields with certain self-interactions are conjectured in “chameleonic” models of dark energy [73] and may be sought for with atom interferometry techniques [74, 75].
31.5.1 Haloscope Experiments In the presence of the nongravitational interaction in the first term of Eq. (31.3), spinless bosons may interconvert with photons. Spinless bosons that make up (part of) the galactic DM can be detected using haloscope experiments, which aim to convert galactic DM bosons into photons in the presence of a strong applied magnetic field inside a microwave cavity [61–63, 76]. Haloscope experiments are examples of “resonance-type” experiments, since the resonant frequency of a cavity mode must match the boson’s energy, hmode m c 2 . Although resonance-type experiments are sensitive to very feeble nongravitational interactions, the drawback of these types of experiments is that the boson mass (and hence energy) are not known a priori, meaning that these types of experiments have to scan over a very large range of frequencies in order to find a narrow resonance. Indeed, galactic bosonic DM in the vicinity of the Solar System is expected to have a root-mean-square velocity of vrms 300 km=s, giving the oscillating galactic bosonic field the finite coherence 2 2 106 =m , which is equivatime: coh 2 =m vrms lent to a relative width of !=! 106 . Current limits from haloscope experiments on one type of nongravitational interaction of a spinless boson are shown in Fig. 31.4.
31.5.2 Spin-Precession Experiments In the presence of the nongravitational interactions in the last two terms of Eq. (31.3), bosonic DM fields can induce a number of time-varying spin-dependent effects. In particular, the second term in Eq. (31.3) gives rise to timevarying electric dipole moments of nucleons [81] and atoms and molecules [82], with the angular frequency of oscilla-
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tion governed by the boson mass. The last term in Eq. (31.3) gives rise to anomalous time-varying spin-precession effects due to the motion of Earth through an apparent time-varying pseudo-magnetic field [77, 82, 83]. Various types of methods can be used to search for time-varying spin-dependent effects: (1) Atomic magnetometers and ultracold neutrons to search for time-varying anomalous spin-precession effects [41, 80, 82, 83] (2) Torsion pendula to search for time-varying anomalous torques [80, 82, 83] (3) Nuclear-magnetic-resonance techniques to search for the resonant build-up of transverse magnetization [77, 84, 85] (4) Resonant conversion of galactic DM bosons into photons in a magnetized material [86–89].
467
Instead of comparing two transition frequencies, it is also possible to compare a transition frequency against a reference frequency determined by the length of an optical cavity or an interferometer arm, or to compare two (different) cavities or arms [70, 93, 94]. In this case, the reference frequency scales roughly as !ref / 1=L / 1=aB if the solid length is allowed to vary freely, while the reference frequency is practically independent of the physical constants for a “fixed” length where fluctuations in the arm length are deliberately shielded. The sensitivity coefficients for laser/maser interferometry experiments depend on the specific configuration and mode of operation and were calculated in [70, 93, 94]. Like some of the methods discussed in Sect. 31.5.2 to search for time-varying spin-dependent effects, the observables in the types of experiments discussed in this section also have the attractive feature of scaling to the first power of the underlying interaction constant. Current limits on several types of nongravitational interactions of spinless bosons from experiments that search for apparent oscillations in the physical constants are shown in Figs. 31.1 and 31.6.
In the case of time-varying electric dipole moments, which can be sought for with methods (1) and (3), it is necessary to apply an electric field in the experiment. In the case of methods (1)–(3), the observables scale only to the first power of the underlying interaction constant. This is a particularly attractive feature of these types of methods, compared with the References methods discussed in Sects. 31.2–31.5.1, where the observ1. Bertone, G., Hooper, D., Silk, J.: Phys. Rep. 405, 279 (2005) ables scale either to the second or fourth power in a small 2. Ade, P.A.R., et al.: Astron. Astrophys. 594, A13 (2016). Planck interaction constant. Current limits from spin-precession exCollaboration periments on several types of nongravitational interactions of 3. Frieman, J.A., Turner, M.S., Huterer, D.: Ann. Rev. Astron. Astrophys. 46, 385 (2008) spinless bosons are shown in Figs. 31.3 and 31.5.
31.5.3 Time-Varying Physical Constants In the presence of the nongravitational interactions in Eqs. (31.1) and (31.2), bosonic DM fields can induce “apparent” variations in the physical constants [70, 79, 91]. One particularly powerful class of measurements to search for these apparent oscillations in the physical constants involve high-precision comparisons of atomic and molecular transition frequencies [22, 23, 70, 79, 90, 91], which were previously used to search for “slow temporal drifts” in the physical constants [92], Chap. 30. The basic idea of clock-comparison experiments is to compare two transition frequencies with different sensitivities to variations in the physical constants. For example, in the atomic units „ D e D me D 1, an atomic optical transition frequency scales opt as !opt / Frel .Z˛/, while an atomic hyperfine transition frehf .Z˛/.me =mN /, where Frel quency scales as !hf / Œ˛ 2 Frel are relativistic factors (which generally increase rapidly with the nuclear charge Z), and is the nuclear magnetic dipole moment. A summary of calculated sensitivity coefficients for various atomic, molecular, and nuclear transitions can be found in [92], Chap. 30.
4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
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Searches for New Particles Including Dark Matter with Atomic, Molecular, and Optical Systems Victor Flambaum Professor Victor Flambaum received his PhD and DSc from the Institute of Nuclear Physics, Novosibirsk, Russia. He has about 400 publications in atomic, nuclear, particle, solid state, statistical physics, general relativity and astrophysics, including works on the violation of fundamental symmetries, tests of unification theories, temporal and spatial variation of fundamental constants from the Big Bang to the present, many-body theory, and high-precision atomic calculations.
469 Yevgeny Stadnik Dr Yevgeny Stadnik completed his PhD at the University of New South Wales, Australia. He was a Humboldt Research Fellow at the Johannes Gutenberg University of Mainz, Germany, and a Kavli Fellow and Assistant Professor at Kavli IPMU within the University of Tokyo, Japan. Yevgeny is currently a DECRA Fellow at the University of Sydney, Australia. His research interests include the manifestations and phenomenology of dark matter in low-energy atomic and astrophysical phenomena.
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Contents 32.1 32.1.1 32.1.2 32.1.3 32.1.4
Parity Nonconserving Effects in Atoms . . Nuclear-Spin-Independent PNC Effects . . . Nuclear-Spin-Dependent PNC Effects . . . . PNC in Cesium and Implications for Particle and Hadronic Physics . . . . . . . . . . . . . . Current PNC Experiments . . . . . . . . . . . .
32.2 32.2.1 32.2.2 32.2.3
Electric Dipole Moments and Related Phenomena Experiments with Paramagnetic Systems . . . . . . . Experiments with Diamagnetic Systems . . . . . . . . Impact on Particle Physics . . . . . . . . . . . . . . . .
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Tests of the CPT Symmetry . . . . . . . . . . . . . . . . 477
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Lorentz Symmetry Tests . . . . . . . . . . . . . . . . . . 478 Electron–Photon Sector of SME . . . . . . . . . . . . . . 478 Proton and Neutron Sectors of SME . . . . . . . . . . . 480
32.5 32.5.1 32.5.2 32.5.3
AMO Tests of General Relativity . . . . . Tests of the Einstein Equivalence Principle Other AMO Tests of Gravity . . . . . . . . . Detection of Gravitational Waves . . . . . .
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Abstract
measurements, atomic magnetometers, laser and atom interferometers, and other precision AMO experiments. These advances bring forth a question: Do our current laws of fundamental physics hold at the level of experimental precision? Modern physics is based on a number of cornerstone principles such as Lorentz invariance, universality of free fall, local position invariance, constancy of fundamental constants, etc. However, in many theories beyond the Standard Model (BSM), these principles no longer hold, and discovery of their violation will be a first glimpse into the nature of BSM physics. AMO studies aimed to search for new physics beyond SM and to test the fundamental physics postulates are the subject of this chapter. This entire topic was recently reviewed in detail in [1]. In this chapter, we briefly discuss AMO studies in parity nonconservation, time-reversal violation including electric dipole moments, tests of the CP T theorem, tests of general relativity, and the search of the violation of local Lorentz invariance. The tests of quantum electrodynamics (QED), constraints on variations of fundamental constants, and searches for new particles, including dark matter, are discussed in other chapters.
While the Standard Model (SM) of particle physics has been very successful in describing as well as predicting numerous phenomena, it provides no insight into the naKeywords ture of dark matter and dark energy. Moreover, SM is not parity violation anapole moment electric dipole moconsistent with the present observable Universe, which ments CPT symmetry Lorentz symmetry tests of has a glaring imbalance of matter and antimatter. Despite general relativity gravitational waves Einstein equivdecades of effort, all attempts to unify gravity with other alence principle fundamental interactions remain unsuccessful. The advances in atomic, molecular, and optical (AMO) precision measurements, coupled with improved theoretical description, enabled new tests of fundamental physics. Revolutionary developments led to the extraordinary precision now reached by atomic clocks, other spectroscopy 32.1 Parity Nonconserving Effects in Atoms M. S. Safronova () Dept. of Physics & Astronomy, University of Delaware Newark, DE, USA e-mail: [email protected]
Under the parity transformation P , the position vector r transforms to r, which corresponds to mirror reflection and 180ı rotation. Until 1957, the invariance of the laws of
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_32
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physics under the process of parity inversion was assumed to hold. The concept of this invariance has its origin in atomic physics, where Laporte introduced it to explain certain aspects of the iron spectrum. The underlying theory of atomic structure, quantum electrodynamics, is an example of a theory that has this invariance. In 1956, it was suggested by Lee and Yang [2] that weak interaction processes are not invariant under parity inversion, and the parity violation in nuclear ˇ-decay was discovered by Wu et al. [3]. Zeldovich [4] discussed a possibility of observing parity nonconservation (PNC) in atoms but estimated the effect to be too small to be observed. The parity violation in atoms leads to a nonzero amplitude for atomic transitions otherwise forbidden by the parity selection rule, such as the 6s–7s transition in cesium. In 1974, the Bouchiats [5] showed that these effects in heavy atoms with a nuclear charge Z were enhanced by approximately a factor of Z 3 . In 1978, the first P -violating signal in an atom was observed in bismuth [6]. Parity violation was observed in other atoms, including lead [7], thallium [8, 9], cesium [10, 11], and ytterbium [12, 13]. Atomic parity violation has been discussed in detail in a number of reviews, most recently in [1, 14, 15]. The goals of high-precision atomic PNC studies are to search for new physics beyond the Standard Model by accurate determination of the weak charge and to probe hadronic parity violation. In the Standard Model, atomic PNC arises from weak interactions mediated by a Z boson. The contact contribution to the SM Hamiltonian density is given by [16] GF X .1/ Cq e N 5 e q N q C Cq.2/ e N e q N 5 q ; HPV D p 2 q
where u and d refer to up and down quarks, W is the weak mixing angle, and gA 1:26. We use a nucleon subscript N in expressions for the nucleon vector VN or axial-vector AN currents below. Substituting the value of the weak mixing angle, sin2 W D 0:23122.4/ [18] (MZ value) into the equation above shows that atomic PNC is dominated by the neutron contribution. Other contributions to atomic PNC are discussed in the next sections.
32.1.1 Nuclear-Spin-Independent PNC Effects Because the nucleon current VN Ae in the first term of Eq. (32.1) is of vector nature, one can introduce a conserved charge, the nuclear weak charge QW
QW D 2ZCp.1/ C 2NCn.1/ D Z 1 4 sin2 W N : (32.2) Here, Z is the number of protons and N the number of neutrons. This is the lowest-order QW N value, which is modified by SM radiative corrections at a level of a few percent [16, 18]. Assuming that the nucleon motion is nonrelativistic and averaging over nucleon distribution, the VN Ae contribution to HPV reduces to the effective electron-sector weak Hamiltonian, GF HW D p QW .r/5 ; 8
(32.3)
where .r/ is a nuclear density. The expression above ne(32.1) glects the difference between the proton and the neutron where q and e are field operators for quarks and electrons, distribution. It is accounted for by a so-called “neutron skin” respectively, sum q is over all quark flavors, are Dirac correction. The neutron skin is defined as the difference matrices, and 5 is the Dirac matrix associated with pseudo between the root-mean-square radii of neutron and proton scalars. The Fermi constant GF D 2:22 1014 a:u: quanti- distributions [19]. It is small, 0.2% for the Cs 6s–7s PNC fies the strength of the weak interactions. In Eq. (32.1), the amplitude, increasing for heavier atoms [20]. Due to the smallness of the GF , the matrix elements of first term describes a coupling of the electron axial-vector current to the quark vector current, Vq Ae , parameterized by the HW are exceptionally small. For example, PNC matrix 11 .1/ the constant Cq . The second term describes a coupling of element in Cs is 10 a:u:, while allowed electric-dipole the electron vector current to the quark axial-vector current, matrix elements are typically on the order of an atomic Aq Ve . The Vq Ae term dominates, because all the quarks con- unit (ea0 , where e is the electric charge, and a0 is the tribute coherently. The Aq Ve term contributes to the PNC Bohr radius). The PNC interaction due to the exchange of Z between two atomic electrons is much smaller than the effects that depend on nuclear spin. For the description of the atomic PNC, it is convenient electron-nucleon Z exchange. The correction due to weak .i / to combine Cq (i D 1; 2) quantities into the corresponding e–e processes in Cs is only 0.03% [20], significantly below both experimental and theoretical uncertainties. constants for protons and neutrons [17] 1
1 4 sin2 W ; 2 1 .1/ .1/ D Cu C 2Cd D ; 2 .2/ .1/ D Cn D gA Cp ; .1/
Cp.1/ D 2Cu.1/ C Cd D Cn.1/ Cp.2/
32.1.2 Nuclear-Spin-Dependent PNC Effects The nuclear spin-dependent PNC effects, contributing in open-shell atoms with nuclear spin I ¤ 0, are described by
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473
the effective interactions in the electron sector [1] GF HNSD D p .axial C NAM C hf / .˛ I/ .r/ ; 2
(32.4)
where ˛ is the velocity operator (˛i D 0 i ) for atomic electrons. The first term is associated with the AN Ve term of HPV given by Eq. (32.1). The second term arises from the anapole moment contribution, which generally dominates for heavy atoms. It is described in more detail below. The third term describes a combined effect of the hyperfine interactions and the Z-boson exchange interaction from the VN Ae term of HPV . The corresponding parameter hf was calculated for transitions of experimental interest by Johnson, Safronova, and Safronova [21]. Another nuclear spin-dependent PNC effect was pointed out and evaluated for cases of experimental interest in [22]: nuclei with spin I 1 have a weak quadrupole moment leading to a tensor weak interaction that mixes states of opposite parity for atomic levels with total angular momentum difference J 2. Nonzero nuclear multipolar electric (EJ ) and magnetic (MJ ) moments, such as charge (E0), magnetic dipole moment (M1), electric quadrupole moment (E2), etc., are all even under the parity transformation and T -reversal. Weak interactions inside the nucleus result in additional parity-odd nuclear moments. The nuclear anapole moment a is such a leading parity-odd, time-reversal even nuclear moment [23]. For a current density distribution J .r/, the anapole moment is given by Z (32.5) a D d 3 r r 2 J .r/ :
structure among all atoms for which the PNC effects were measured, allowing for accurate theoretical calculations of the PNC amplitude, required for the analysis of the experiment. In the text below, we omit the core electrons in the electronic level designations and only list the state of the valence electron. The PNC transition in cesium that has been studied is 6s ! 7s, which is forbidden by parity selection rules, i.e., the electric-dipole matrix element h7sjDj6si D 0. This transition becomes weakly allowed due to PNC effects, which mixes the 6s and the 7s states with the opposite parity np states. The 133 Cs nucleus has 78 neutrons and 55 protons, its nuclear spin is I D 7=2. The total angular momentum of atomic ns J D 1=2 states is then F D 3 or F D 4. Both of the transitions, 6s .F D 4/ ! 7s .F D 3/ and 6s .F D 3/ ! 7s .F D 4/ ;
have been measured, allowing the isolation of PNC effects that depend on the spin of the nucleus. The Boulder experiment [11] used the Stark interference method, applying the external electric field E to provide an additional admixture of the np states. This induces an electric-dipole transition with a transition amplitude proportional to E and a vector transition polarizability ˇ (which is determined separately). The 6s–7s transition is then excited with a 540 nm elliptically polarized laser light to get a nonzero interference between the Stark-induced amplitude and the parity-violating amplitude EPV . The ratio of the two amplitudes is measured, using The electromagnetic coupling of atomic electrons to the nureversals of the electric fields, changes in magnetic substates, clear anapole moment is given by ˛ A, where A D aı.r/ is and in laser polarization to further isolate the PNC effect. The the corresponding vector potential. Just as in the case of the final results for both 6s–7s transitions were reported to be nuclear magnetic moment, a is proportional to the nuclear ( spin. The anapole moment is related to the parameter NAM exp 1:6349.80/ F D 4 ! 3 E PV of Eq. (32.4) by (32.7) D Im ˇ 1:5576.77/ F D 3 ! 4 GF (32.6) a D p NAM I : in units of mV=cm. jej 2
The nonzero nuclear anapole moment was observed for the Nuclear spin-independent part first time in the Cs PNC experiment [11], providing insight The average of these values, combined with the value of ˇ, into hadronic weak interactions, as discussed in Sect. 32.1.3. gives spin-independent PNC amplitude, accurate to 0.35%. The value of the weak charge can be extracted from this averaged value using
32.1.3 PNC in Cesium and Implications for Particle and Hadronic Physics
exp
EPV D kPV QW ;
(32.8)
The most accurate PNC measurement was carried out in where kPV is the theoretical value of the PNC amplitude Cs [11]. Cs has a single electron outside a closed xenon- in terms of QW . Such a calculation in Cs with a subperlike core with 54 electrons. This is the simplest electronic cent accuracy was an extremely formidable task. The major
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theoretical effort involved a very accurate treatment of the many-body Coulomb correlations, requiring both implementation of the coupled-cluster method and the fourth-order many-body perturbation theory, accomplished in [20, 24]. Their kPV value, calculated with 0.27% uncertainty is the result of the most accurate and complicated PNC computation in heavy atoms accomplished to date. In addition, the small corrections due to the Breit interaction, radiative QED effects, and the neutron skin had to be calculated at this level of accuracy, which required the work of several groups. These calculations are reviewed by Derevianko and Porsev [14]. The resulting value of the weak charge
Fig. 32.2 Illustration of the nucleon–nucleon interaction, mediated by a meson exchange
N
N , ρ, ω
N
N
Atomic PNC studies are sensitive to extra Z bosons, setting limits on such particles with TeV masses, with limits only recently improved at the Large Hadron Collider [18]. APV is also uniquely sensitive to some “dark forces” [26], (32.9) with the limits on the 50 MeV dark boson shown in Fig. 32.1. QW .133 Cs/ D 73:16.29/Œ20 ; The AMO searches for new particles, including dark matter, where the first error is experimental and the second theoret- are further discussed in Chap. 31. ical [20, 24], in agreement with the SM value [18]. Recent reevaluation of small Coulomb correlation corrections, con- Nuclear spin-dependent part tributions of the core, and highly excited states [25], raised Next, we discuss the analysis of the Cs result in terms of the theoretical uncertainty to 0.5%, slightly shifting the value the spin-dependent PNC effects. The difference of the values of kPV but maintaining agreement with the SM. Theoretical for two transitions in Eq. (32.7), 0:077.11/ mV=cm, comeffort to resolve the issue of the accuracy of the highly ex- bined with the value of ˇ, gives the spin-dependent PNC amplitude, accurate to 15%. To extract the value of the cited terms is currently in progress. The Cs experiment, combined with theoretical calcula- anapole moment, the theoretical calculation of the nuclear NSD is needed, just as in the case tions, provides the most accurate low-energy test of the SM spin-dependent amplitude EPV electroweak sector. In combination with particle physics ex- of the spin-independent part, only using the HNSD Hamilperiments, atomic PNC demonstrated the 3% “running” (i.e., tonian (32.4), along with the calculations of axial and hf ; NSD was calculated in [27] using the linearized coupleddependence on the four-momentum transfer Q) of the elec- EPV troweak interactions across five orders of energy values. This cluster approach with 1% uncertainty, much exceeding the result is illustrated in Fig. 32.1 from [1] (adapted from [26]), experimental accuracy. The nuclear anapole moment arises due to nucleon– which shows the Cs atomic parity violation (APV) result tonucleon interaction, mediated by a meson (, !, ) exchange gether with values from particle physics experiments. illustrated in Fig. 32.2, where one of the nucleon–meson 2 2 vertexes is strong and another is weak and P -violating. sin (θ W)(Q ) Therefore, the experimental PNC result combined with the 0.242 Dark boson 50 MeV theoretical value of the spin-dependent PNC amplitude can 0.240 be used to probe weak hadronic interactions. Such an analyE158 Qweak v-DIS sis, carried out in [28], yielded the values of the weak meson– 0.238 nucleon couplings in disagreement with other constraints. 0.236 More recent limits on the values of the weak meson–nucleon couplings are given in [29], but the discrepancy remains un0.234 APV(Cs) solved, complicated by the difficultly in the required nuclear LEP Standard 0.232 Model calculation [28]. More experiments, including anapole moment measurements in other atoms or molecules are required SLAC 0.230 to solve this long-standing puzzle. –3 3 10
0.01
0.1
1
10 10 100 Momentum transfer Q, GeV/c
Fig. 32.1 Running of the weak mixing angle with momentum transfer Q. The solid red curve is the SM prediction. The Cs APV result is supplemented with data from particle physics experiments: E158, Møller [electron–electron] scattering; Q-weak, PNC electron–proton scattering, -DIS, deep inelastic scattering; LEP and SLAC results. The colored area comes from one of the “new physics” scenarios [26]: a dark boson of mass 50 MeV. From [1] (adapted from [26])
32.1.4 Current PNC Experiments The Yb PNC experiment (ongoing in Mainz, Germany) recently reported the measurements of atomic parity violation, made on a chain of ytterbium isotopes with mass numbers A D 170, 172, 174, and 176 [13]. The PNC effect in
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Yb is approximately 100 times larger than that of Cs. The advantage of the experiment with several isotopes is the potential of carrying out the analysis without the need for high-precision calculations, not currently feasible for the Yb PNC amplitude. Experiments with Dy, which has nearlydegenerate states of opposite parity, are being carried out by the same group. While PNC in Dy was not previously observed [30], it should be within the sensitivity of the new experiment. An experiment to remeasure PNC in the 6s–7s transition of Cs in underway in Purdue, using new two-pathway coherent control technique [31]. An Fr PNC experiment is being pursued at TRIUMF in Vancouver [32], which is complicated by the requirement that it be performed at a facility capable of producing needed radioactive isotopes. Measurements in a chain of Fr isotopes are planned, with the main goal of measuring nuclear anapole moments. The TRIUMF collaboration plans to measure PNC in both optical 7s–8s and ground state hyperfine transitions. Experiments to measure spin-dependent PNC effects in molecules are underway at Yale [33].
475
given by HEDM D d E :
(32.10)
While it would seem that EDMs of electrons and nucleons will simply lead to observable effects (i.e., shifts in energy levels) in atoms and molecules, this issue turns out to be more complicated. According to Schiff ’s theorem [37], there is no energy shift resulting from constituent particles’ EDMs when an E-field is applied to a neutral system, assuming nonrelativistic point particles in an electrostatic potential. This means that the other parts of the atomic system rearrange to completely screen the external E-field felt by the charged particle, otherwise it will experience a net acceleration. A number of effects (finite nuclear size, relativistic effects, magnetic interactions, and others) allow to evade the Schiff’s theorem leading to potentially observable experimental signals. In atoms and molecules, P; T -violating effects arise from EDMs of electrons, protons, and neutrons, P; T -violating electron–nucleon interactions and P; T -violating nucleon– nucleon interactions. Different atoms and molecules have various sensitivities to these effects, but one may divide them into two groups: paramagnetic (with unpaired electron spins) 32.2 Electric Dipole Moments and Related and diamagnetic (with closed electron shells but nonzero Phenomena nuclear spin). The first group is the most sensitive to the electron EDM (eEDM) and one type of electron–nucleus In this section, we continue with the discussion of the P interaction. The second is the most sensitive to EDMs of nuviolating effects but consider phenomena that are also odd cleons, purely hadronic (nucleon–nucleon) interactions, and under the time reversal .T /. Such searches probe the enother types of electron–nucleon interactions. ergy scales well above 1 TeV in particle theory models such as supersymmetry, considered to be natural extensions of the SM. Thus, the AMO studies of P; T -violating effects 32.2.1 Experiments with Paramagnetic Systems are of particular significance for particle physics. Numerous detailed reviews of this topic are available in the literature The relativistic motion of a bound electron in a heavy atom (e.g. [1, 34, 35] and references therein). Here, we discuss the can lead to energy shifts due to eEDM that are larger than most recent results. a shift for the free electron, i.e., EDM effects are enhanced The combined symmetry of charge conjugation C , parity in some atoms and molecules. For example, the EDM of a Tl transformation P , and time reversal T , i.e., CP T , is preatom is served in quantum field theory. Therefore, the violation of T implies the violation of CP . Tests of CP T invariance are d.205 Tl/ D Kde 573 de ; discussed in Sect. 32.3. The question of CP -violation is directly related to the problem of baryon asymmetry, observed where K is the enhancement factor calculated theoretically dominance of matter over the antimatter in our Universe. CP using a hybrid approach that combines configuration interviolation is required to generate baryon asymmetry [36], but action and a linearized coupled cluster method [38] with CP violation in the Standard Model is by far too small to a 3% accuracy. This enhancement makes measurement of account for it. Therefore, there must be other sources of CP EDM in heavy atoms and molecules containing heavy atoms violation leading to the effects already well within the preci- particularly attractive. Measurement of the EDM of 205 Tl sion of the experiments described in this chapter. As a result, atom [39] placed the bound on the electron EDM de < AMO searches for CP -violating effects have a particularly 1:6 1027 e cm. This limit was improved by the YbF molecular experiment to de < 1:05 1027 e cm [40]. The current high potential of discovering new physics. A permanent electric dipole moment (EDM) of a particle, best limit on eEDM comes from the ACME experiment with d, along its spin s violates both P and T . The Hamiltonian ThO molecules [41]; de < 8:7 1029 e cm with 90% confidescribing the EDM interaction with an electric field E is dence.
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The valence electrons in some molecules, such as ThO, can experience exceptionally large intramolecular effective electric field Eeff , on the order of 100 GV/cm. If an electron has an EDM, such enormous field (which can not be achieved in the lab) leads to an enhanced energy shift de Eeff , making these molecules particularly attractive for EDM searches. Unlike atoms, the ThO molecule can also be fully polarized applying an 100 V=cm external electric field. For the ˝doublet H3 1 electronic state in ThO, the effective field was calculated to be 80 GV=cm, with an uncertainty of about 10–20% ([42, 43] and references therein). The use of the ˝doublet states is particularly advantageous as they provide for “internal comagnetometry”, which allows for the reversal of the EDM interaction without changing any lab fields, minimizing the systematic effects. The H3 1 state has an additional advantage of the unusually small magnetic moment, which reduces the systematic effect due to spurious magnetic fields. To measure de , ACME performed a spin precession measurement on pulses of 232 Th16 O molecules from a cryogenic buffer gas beam source, which has the advantages of a low internal temperature, high flux, and low forward velocity in comparison with conventional molecular beam sources. Besides the electron EDM, ThO is also sensitive to the pseudoscalar electron–scalar nucleon T -violating interaction, quantified by the coupling CS . This effect would lead to an additional frequency shift, WS CS , where WS is calculated theoretically [42, 43]. The electron EDM limit above assumes CS D 0. Setting de D 0 yields CS < 6 109 [41]. In 2017, an electron EDM result from a new experiment at JILA with HfFC molecular ions was reported [44]. Cairncross et al. used a rotating electric field (E 20 V=cm) to trap and fully polarize ˝-doublet levels of a metastable 3 1 state. Combining the experimental result with the calculated value Eeff 23 GV=cm [45, 46], this experiment gives de < 13 1029 e cm at a 90% confidence level or CS < 14 109 . A combination of ThO and HfFC values may be used to improve the constraints further due to different relative sensitivities of the two experiments to de and CS . Improvements in ACME [47], HfFC , and YbF experiments are underway, and significant improvements in accuracy are expected. Kozyryev and Hutzler [48] recently proposed using polyatomic paramagnetic molecules, which can be laser cooled and have a level structure favorable for electron EDM measurements.
32.2.2
Experiments with Diamagnetic Systems
The Hg EDM search at the University of Washington [49, 50] is the most sensitive EDM experiment with a diamagnetic system. Because 199 Hg has a closed-shell 1 S0 electronic ground state and nuclear spin I D 1=2, the EDM must point
along the nuclear spin axis. The nuclear spins are polarized by optical pumping with a resonant laser beam. When spinpolarized 199 Hg atoms are immersed in electric and magnetic fields, the Larmor frequency is given by hL D 2jB ˙ dEj ;
(32.11)
where and d are magnetic and electric dipole moments, and ˙ sign denotes parallel (antiparallel) fields. The d ¤ 0 EDM experimental signature would be a Larmor frequency change correlated with the electric field when E is reversed relative to B. Hg atoms are contained in four nominally identical high-density vapor cells. The inner cells have strong, equal, and opposite E fields along the B-field axis, while the outer cells have E D 0 to cancel fluctuations not only in the average value of B but also in its first-order gradient. With 252 daily runs, the experiment was sensitive to the P T violating energy shift of E= h < 20 pHz. The final result for the Hg EDM was [49] dHg D .2:20 ˙ 2:75stat ˙ 1:48syst / 1030 e cm ; with a 95% confidence limit of dHg < 7:4 1030 e cm : Theoretical interpretation of this limit in terms of the Schiff moment S Hg , which is the leading-order P , T -violating nuclear moment not completely screened by the electron cloud, gives ˇ ˇ ˇSHg ˇ < 3 1013 e fm3 ; where results of several theoretical calculations were averaged to provide d Hg D 2:4 104 S Hg =fm2 : Further limits on neutron EDM and proton EDM were derived from the corresponding contributions to S Hg in a random-phase approximation calculation with core polarization [51] d Hg D .1:9 d n C 0:2 dp / fm2 : The resulting limits are dn < 1:6 1026 e cm and dp < 2:0 1025 e cm, where the other terms were assumed to be zero, and 30% uncertainly was assigned to the dp contribution. The 199 Hg experiment also set the best limits on the combined chromo-EDM of the up and down quarks, the observable QCD -parameter, NQCD < 1:5 1010 , and hadronic T -odd, P -odd couplings, pseudoscalar-scalar and tensor-tensor semileptonic couplings.
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Other experiments with diamagnetic systems were carried out with 129 Xe [52], 225 Ra atoms [53] and 205 TlF molecules [54]. The groups at Yale, Columbia, and the University of Massachusetts are constructing a new experiment (CeNTREX) to measure the Schiff moment of 205 Tl using a cryogenic beam of TlF molecules. A review of other new EDM experiments is given in [1].
32.2.3 Impact on Particle Physics Calculations of the size of the relevant T; P -violating parameters were made for a wide range of theoretical models, in particular those that include supersymmetry (SUSY) that is broken near the electroweak scale. Attractive features of such theories include predictions of superpartner particles with mass MSUSY MZ 0:1 TeV that provide potential dark matter candidates, a solution to the problem of the stabilization of the Higgs mass against radiative corrections, extra sources of CP -violation to resolve the matter-antimatter asymmetry, etc. These theories predict EDMs that are already well within the experimental reach and improving EDM sensitivity by 1–2 orders of magnitude will either yield a discovery or conclusively rule out many SUSY models. Even in the scenarios when primary contribution to EDMs comes from the two-loop diagrams, the electron EDM and quark chromo-EDM limits correspond to lower bounds of 2–4 TeV on the masses of the lighter SUSY particles, if ıSUSY 1 [1, 55]. For these types of SUSY particles, this is well beyond the direct reach of the Large Hadron Collider [18]. EDM experiments also set limits on models with extra scalar fields, analogous to the Higgs boson [34], also exceeding LHC bounds.
32.3
Tests of the CPT Symmetry
Current laws of physics are believed to be invariant under the CP T transformation. However, CP T symmetry breaking may arise in physics beyond the Standard Model, such as string theories [56]. The CP T invariance ensures the same masses and magnetic moments of particles and corresponding antiparticles [57, 58]. Therefore, comparisons of particle/antiparticle properties test CP T symmetry. Such experiments were recently reviewed in [1, 59–61]. We only provide a brief summary of the most recent results of the CP T N and antiprotests with antihydrogen (H), antiprotons p, tonic helium pHe N C . Antihydrogen experiments at CERN aimed at CP T and gravity tests include ATRAP, ALPHA, ASACUSA, AEgIS, and GBAR. Recently, the ALPHA experiment performed a laserspectroscopic measurement of the 1S–2S transition fre-
477
quency of antihydrogen using two-photon laser excitation with 243 nm light [62], which was a long-standing goal of the antihydrogen experiments. A comparison of this result with the 1S–2S frequency in hydrogen provided a test of with CP T invariance at a relative precision of 2 1010 . In 2017, an observation of the hyperfine spectrum of antihydrogen was reported [63], finding a ground-state hyperfine splitting in H to be 1420:4 ˙ 0:5 MHz. This result is consistent with a corresponding value in atomic hydrogen at the level of four parts in 104 . ALPHA collaboration reported an experimental limit on the charge Qe of antihydrogen, in which e is the elementary charge, to be jQj < 0:71 parts per billion (one standard deviation) [64]. This measurement constrained the relative difference between the positron and elementary charge at a level of about one part per billion (ppb), assuming charge superposition and using the best measured value of the antiproton charge [18]. Another focus of the antihydrogen experiments (GBAR, AEgIS, and ALPHA) is testing whether antimatter is affected by gravity in the same way as matter. Weakness of the gravitational interaction requires such direct laboratory tests to be conducted with neutral particles, making antihydrogen the best available system. These tests are complicated by the need for increased production of cold antihydrogen, requiring new cooling techniques [65, 66]. The goal of the Baryon Antibaryon Symmetry Experiment (BASE) is precise comparisons of the fundamental properties of antiprotons and protons for tests of CP T [67]. This experiment carries out measurements with single trapped particles. In 2017, BASE collaboration reported a parts-per-billion measurement of the antiproton magnetic moment [68] using an advanced cryogenic multi-Penning trap system. This experiment used a particle with an effective temperature of 300 K for magnetic field measurements and a cold particle at 0.12 K for spin transition spectroscopy, reporting the value of pN D 2:792 847 3441.42/N , where N is the nuclear magnetic moment (at the 68% confidence level). A measurement of the proton magnetic moment at the 0.3 ppb level was reported by the same team [69]. A comparison of the pN with the new proton value p D 2:792 847 344 62.82/N constrains CP T -violating effects. In the future, BASE collaboration proposes to use quantum-logic technologies to further advance the CP T tests. They plan to sympathetically cool and probe the (anti)proton using a coupling to an atomic “qubit” ion trapped in its vicinity via the Coulomb interaction. This technique has a potential to enable the proton and antiproton magnetic moment measurements at the parts per trillion level [69]. The BASE collaboration also compared the charge-tomass proton and antiproton ratios using high-precision cy-
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clotron frequency comparisons carried out in a Penning-trap In minimal SME, a general expression for the quadratic system, establishing a limit [70] Hermitian Lagrangian density in the presence of Lorentz violation describing a single spin-1=2 Dirac fermion of mass .q=m/p 1 D 1.69/ 1012 : m is given by [88] .q=m/p ! 1 Single-photon spectroscopy of antiprotonic helium, pHe N C, L D ic @ Mc 2 ; (32.12) 2 which consists of an ˛-particle, an electron, and an antiproton, was performed in [71]. Combining their experimental where is a four-component Dirac spinor, result with the high-precision calculations of the relevant transition frequencies [72], yielded a value of mpN =me D ! f @ g D f @ g g@ f ; 1836:152 673 4.15/ [71]. This value agrees with the CO1 DATA proton to electron mass ratio [73] at a level of 0.8 ppb. M D m C a C b 5 C H ; (32.13) 2 A recent review of “magnetometry” CP T tests, which do not require antimatter is given in [74]. and A number of very recent CP T tests described above mark a turning point from proof-of-principle experiments to high1 precision metrology and CP T comparisons, with prospects D C c C d 5 C e C i5 f C 2 g : for significant improvements in a near future. (32.14)
32.4 Lorentz Symmetry Tests Local Lorentz invariance (LLI) is the cornerstone of modern physics: the outcome of any local nongravitational experiment is independent of the orientation and the velocity of the (freely falling) apparatus. In the SM field theory, the Lorentz invariance and the CP T invariance, discussed in Sect. 32.3, are closely related. However, one may be violated but not the other in some BSM frameworks, which is a subject of recent debates (e.g. [75] and references therein). Current LLI tests span almost all fields of physics as separate violations of LLI are possible for all particles. The resulting limits and references are compiled in the Data Tables for Lorentz and CP T Violation [76]. A scaling argument is used to provide some guidance to the Lorentz symmetry violation (LV) searches: one may expect the LV effects to be suppressed by some power of the ratio R between the electroweak scale and the natural (Planck) energy scale for strings: R D mew =MPl D 2 1017 [56] or electron mass to Plank scale 4 1023 [77]. The LV tests in the electron sector are just at the R D 1 boundary, while some limits for other particles have already surpassed it. This presents a particular attraction of the LV tests: any verified positive signal will be an unambiguous sign of the new physics. Lorentz violation tests are analyzed in the context of an effective field theory known as the Standard Model extension (SME) [78]. The AMO LV tests include experiments with atomic clocks [79], other precision spectroscopy measurements [80], magnetometers [81, 82], rotating optical and microwave resonators [83–85], quantum-information trapped-ion technologies [86], and quartz oscillators [87]. In this section, we highlight the most recent AMO results; a detailed review is given in [1].
The first terms in the expressions for M and are from the SM Lagrangian. The coefficients in Eq. (32.13) have dimensions of mass, while the coefficients in Eq. (32.14) are dimensionless. Lorentz violation is parameterized by the coefficients b , c , d , e , f , g , and H . The field operators in Eqs. (32.13), (32.14) containing the coefficients c , d , and H are even under CP T , and the remaining ones are odd under CP T . Therefore, the results of the CP T experiments discussed in Sect. 32.3 can also be parameterized by SME. AMO experiments may be interpreted as Lorentz-invariance tests for the photon, electron, proton and neutron, and their combinations, with photon contributions appearing in all atomic experiments.
32.4.1 Electron–Photon Sector of SME Atomic LV experiments in the electron–photon sector [80, 86] use the different sensitivities of various energy levels to the hypothetic Lorentz violation. The quantization axis is set by the direction of the magnetic field, and the energy difference of two atomic levels with different LV sensitivities is monitored during the Earth’s rotation and motion around the Sun. No additional rotation of the experiment is required. The resulting data are combined with the theoretical calculations of the relevant LV-violating matrix elements and are analyzed to extract the SME coefficients. The SME coefficients are frame-dependent but have unique values in the specified frame. Generally, one uses the Sun centered celestial-equatorial frame (SCCEF) for the analysis of the experiments [89], indicated by the coordinate indexes T , X, Y , and Z. The experiment below set limits on the coefficient of the c tensor, which has nine components that need to be experimentally determined: parity-even cT T and cJK
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479
and parity-odd cTJ , where J; K D X; Y; Z. The elements cJK describe the rotational effects and have a leading order time-modulation period related to the sidereal day (12-h and 24-h modulation), while the cTJ and cT T describe the boost effects of the laboratory frame and have a leading order timemodulation period related to the sidereal year. Therefore, the terms cTJ are suppressed in comparison to cJK by a ratio of the Earth’s orbital velocity to the speed of light ˇ˚ 104 . 2 108 . The cT T term is suppressed by ˇ˚ The LV effect on the bound electronic states is described by the Hamiltonian [80, 88] 2 p 2U 1 .0/ .2/ .2/ ıH D C0 2 c00 C T ; (32.15) 3c 2me 6me 0 0 .2/
where p is the momentum of a bound electron, and T0 D p 2 3pz2 in nonrelativistic approximation. Interpretation of the LV experiments requires theoretical calculations of the expectation values of these operators for the states of interest. The second term in the parentheses gives the leading order gravitational redshift anomaly in terms of the Newtonian po.0/ .2/ tential U . The parameters C0 and C0 are elements of the c tensor in the laboratory frame defined by Eq. (32.14) .0/
C0 D c00 C .2=3/cjj ;
(32.16)
.2/ C0
(32.17)
D cjj C .2=3/c33 ; .2/
where j D 1; 2; 3. The other Ci components do not contribute to the energy shift of bound states. Fig. 32.3 Rotation of the quantization axis of the experiment with respect to the Sun as the Earth rotates. A magnetic field (B) is applied vertically in the laboratory frame to define the eigenstates of the system. As the Earth rotates with an angular frequency given by !˚ D 2 =.23:93 h/, the orientation of the magnetic field and, consequently, that of the electron wave packet (as shown in the inset in terms of probability envelopes) changes with respect to the Sun’s rest frame. The angle is the colatitude of the experiment. From [86]
The formalism above also applies to the LV violation in .2/ the nuclei, but the expectation values of the p 2 and T0 operators for the nuclear states determine the sensitivity. Here, we describe LV tests with Dy [80] and CaC [86], performed using very different techniques. Both experiments only involve isotopes with zero nuclear spin making them insensitive to nuclear LV effects. Dy has two opposite parity states with leading configurations [Xe]4f10 5d6s J D 10 (state A) and [Xe]4f9 5d2 6s J D 10 (state B) that are nearly degenerate. The E D EA EB energy difference is highly sensitive to the LV effects and can be measured directly by driving an electric–dipole transition between the states A and B with a radio-frequency (rf) field. The sign of the frequency shift is opposite for 162 Dy and 164 Dy allowing to reduce systematic effects. Repeated frequency measurements over the span of 2 years allowed to constrain eight of the nine elements of the c tensor, limiting Lorentz violation for electrons at the level of 1017 for the cJK components and improving bounds on gravitational redshift anomalies for electrons [90] by two orders of magnitude, to 108 . In the 40 CaC trapped ion experiment, the LV-dependent energy difference of the two Zeeman substates of the 2 D5=2 manifold was monitored for 23 h [86]. A pair of 40 CaC ions was trapped in a linear Paul trap, with an applied static magnetic field pointed up defining the quantization axis (i.e., a z axis of the lab frame). The direction of this magnetic field changes with respect to the Sun as the Earth rotates resulting in a rotation of the experiment as illustrated in Fig. 32.3.
Sun
32
χ ω 3:00 7:00 10:00
–1/2 +1/2
B
B
13:00 17:00
–5/2 |Ψ² =
20:00
+ |
Earth’s orbit Earth
+5/2 ²
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M. S. Safronova
Using the theoretical calculation of the matrix elements was monitored .2/ of the T0 operator for the 2 D5=2 atomic state, the energy jF D 3; mF D 3i difference E D EmJ D5=2 EmJ D1=2 of these two Zeeman jF D 3; mF D 3i levels induced by Lorentz violation is given by
E .2/ D 4:45.9/ 1015 Hz C0 : h .0/
Proton and Neutron Sectors of SME
Wolf et al. [79] used a cold Cs atomic clock, which is also the primary frequency standard defining the second, to test nucleon LLI. The Cs atomic clock operates on the jF D 3; mF D 0i $ jF D 4; mF D 0i
(32.20)
! jF D 4; mF D 3i :
(32.21)
To suppress the magnetic field fluctuations to the first order, (32.18) they used a combined observable
This experiment was not sensitive to the scalar C0 coefficient of Eq. (32.15). The main problem of such LV experiments is the effect of the magnetic field noise, which causes frequency shifts. The CaC experiment used quantum-information inspired techniques to create a decoherence-free two-ion product state, shown in Fig. 32.3, that is insensitive to magnetic field fluctuations to first order. Pruttivarasin et al. [86] noted that electron and photon sector Lorentz violation can not be separated in the LV experiments involving atomic energy levels as LV violation in the photon sector affects the Coulomb potential. Therefore, the experimental results may be interpreted in terms of either 0 D c C photon or electron LV violation described via c k =2, where two terms refer to the electron and photon LV, respectively. The CaC experiment improved the limits to the 0 coefficients of the LV-violation in the electron–photon cJK sector by a factor of 100. The LV tests with YbC octupole clock state, 4f13 6s2 2 F7=2 , and highly-charged ions proposed in [91, 92] promise another five orders of magnitude improvements in LV tests in the electron sector. Harabati et al. [93] proposed measuring transition energies of rare-earth ions doped in crystalline lattices, which can be highly sensitive to the electron SME parameters. LLI tests in the photon sector were carried out with rotating optical and microwave resonators [83–85].
32.4.2
! jF D 4; mF D 3i ;
(32.19)
hyperfine transition at 9.2 GHz, where F D J C I is the total angular momentum, and mF is a corresponding magnetic quantum number. This transition is insensitive to either Lorentz violation or first-order magnetic field effects. To test Lorentz symmetry, a combination of the clock transition above and two other transitions, which are LV sensitive,
c D C3 C 3 20 ;
(32.22)
where 0 , C3 , and 3 are frequencies of (32.19), (32.20), and (32.21) transitions. The combined observable is not sensitive to the Lorentzviolating tensor component from the electron sector but is sensitive to the nuclear (mostly proton) LV violation since the 133 Cs nucleus has one unpaired proton. The Cs clock experiment set the limits for SME parameters for the proton at the 1021 –1025 level. Recent reanalysis of the Cs experiment [94] placed improved bounds on the LV for the proton and neutron. Some of the most stringent clock-comparison tests of LLI [81, 95] were carried out using the self-compensating spin-exchange relaxation-free (SERF) comagnetometry scheme. The experiment of [95] was performed with K and 3 He coupled via spin-exchange collisions with the magnetic field along the z-direction tuned to the compensation point where the K-3 He SERF comagnetometer is insensitive to magnetic fields but highly sensitive to anomalous interactions that do not scale with the magnetic moments. The orientation of the apparatus was alternated between North–South or East– West every 22 s over the course of many days to test the LV effects. The results of this experiment probed neutron couplings to SME background fields (b ) at energy scales 1025 eV [95]. Recent work [96] analyzed the Coulomb interactions between the constituent particles of atoms and nuclei, and used comagnetometer experiments [81] to establish that the speed of light is isotropic to a part in 1028 . A related experiment [81] used a 21 Ne-Rb-K SERF comagnetometer with an order of magnitude better shotnoise-limited sensitivity to LLI violations. Moreover, this experiment probed tensor anisotropies in addition to the vector ones since the nuclear spin of 21 Ne is I D 3=2. To further control the gyro-compass effect, a version of this experiment is being carried out at the South Pole for further improvement of precision. A different scheme was used by [82] to test LV in the neutron sector with 3 He and 129 Xe atoms. We note a following discussion in the literature [97, 98] regarding this experiment. A novel approach to LLI tests in the matter sector was proposed and demonstrated by Goryachev et al. [99] using quartz bulk acoustic-wave oscillators.
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32.5 AMO Tests of General Relativity 32.5.1
Tests of the Einstein Equivalence Principle
According to the weak equivalence principle (WEP) the trajectory of a freely falling body is independent of its internal structure and composition. All bodies in a common gravitational field fall with the same acceleration, which is referred to as the universality of free fall (UFF). The equivalence principle implies that all forms of matter-energy respond to gravity in the same way, which may not be the case for most theories aimed at unifying all four fundamental interactions, such as string theories [100]. Therefore, WEP may be violated at small but measurable levels [101]. General tests of gravity including the EEP tests are reviewed in [100]. A review of the WEP tests and future proposals, including torsion-balance experiments, free-fall experiments, and measurement of relative motions of celestial bodies (for example, lunar laser ranging) is given in [102]. We briefly review best classical WEP limits before discussing quantum-based AMO tests of the WEP. If the WEP is violated, the gravitational mass of a body, mg , is not equal to its inertial mass, mI , and the acceleration a of a body in a gravitational field g is given by aD
mg g: mI
The violation of the WEP is generally quantified by the Eötvös ratio ˇ ˇ ˇ a1 a2 ˇ ˇ; ˇ D 2ˇ (32.23) a1 C a2 ˇ where a1 , a2 are the accelerations of the two falling bodies, which differ in their composition. The strictest WEP laboratory Earth-based constraints (at a 1013 level) is from the “Eöt-Wash” torsion-balance experiments, which compare differential accelerations of beryllium-aluminum and beryllium-titanium test-body pairs [103, 104]. The lunar laser-ranging experiments provided similar limits [105, 106]. A significant improvement in probing WEP is expected to come from space-based missions. MicroSCOPE is an ongoing Centre National d’Études Spatiales (CNES)/European Space Agency (ESA) gravity-research minisatellite mission [107]. Its first results have already yielded improved limits
phase shifts of two different types of matter waves without the use of classical gravimeters [108, 109]. The sensitivity of atom interferometers to WEP violations increases linearly with the momentum difference between the two matter waves and quadratically with the time of free fall. Therefore, increasing either of these parameters improves the sensitivity of the AI WEP tests, with an ultimate goal of the space-based tests. A proof-of-principle AI quantum tests of the universality of free fall were performed at the 108 –109 level [110, 111]. A scheme to compensate the effects of gravity gradients that pose a major challenge for high-precision tests of the WEP with atom interferometry was proposed recently [112]. Implementation of this scheme by Overstreet et al. [113] demonstrated that one can bring down the systematic effects due to the gravity gradients in WEP tests to one part in 1014 , making future AI tests eventually competitive with tests employing macroscopic masses. Hartwig et al. [114] proposed a long baseline (with over 10 m of free fall) AI test of WEP with Rb and Yb, estimated to reach 7 1013 accuracy in the Eötvös ratio. We note that using quantum matter for gravity tests provides the opportunity to perform qualitatively different (from the classical) WEP tests using “test masses” with a definite spin for a search of the spin-gravity coupling effects. An experiment to study the possible influence of entanglement between two test masses on the universality of free fall was proposed [115]. A number of quantum WEP tests in microgravity are being pursued with the promise of greatly increased precision of the current quantum WEP tests. In 2017, the QUANTUS collaboration [116] conducted the successful MAIUS 1 (matter-wave interferometry in microgravity) experiment aboard a sounding rocket at an altitude of up to 243 km above the Earth’s surface, conducting 100 discrete matter-wave experiments during the 6-min experimental phase of this flight.
32.5.2
Other AMO Tests of Gravity
Tests of gravity with antimatter were briefly discussed in Sect. 32.3. A number of AMO gravity tests are related to searches for exotic forces. Atomic spectroscopy was used to search for Yukawa-type fifth-forces [117] by studying the behavior of atomic transition frequencies at varying distances 15 D .1 ˙ 27/ 10 away from massive bodies. A search for subgravitational at a 2 confidence level. forces on atoms from a miniature source mass was carried Recent progress in control of the ultracold atoms started out in [118]. the field of the weak-equivalence tests with quantum matAn intriguing possibility was proposed in [119]: a reter using atom interferometers (AI) to directly compare the alization of the gravitational Aharonov–Bohm effect by
32
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measuring phase shifts with an atom interferometer due to a gravitational potential U in the absence of a gravitational force. Asenbaum et al. [120] measured a phase shift in an atom interferometer due to spacetime curvature across its wave function. In this experiment, interfering atomic wave packets were separated by a macroscopic (up to 16 cm) distance to study the interplay of recoil effects and gravitational curvature from a nearby Pb source mass.
M. S. Safronova
Mid-band Atomic Gravitational Wave Interferometric Sensor (MAGIS) collaboration assessed the technical feasibility of a satellite mission based on precision atomic sensors (involving both AI and atomic clocks) with laser-cooled atomic Sr [131]. MAGIS is expected to be sensitive to GWs of the 0.03 to 3 Hz observation band, intermediate between the LISA and advanced LIGO ranges, and also suited for performing dark matter searches.
References 32.5.3 Detection of Gravitational Waves In 2015, the Advanced LIGO detected the gravitational waves (GW) for the first time [121], starting the field of gravitationalwave astronomy. However, the detection capability of such terrestrial detectors is limited to GWs with frequencies above 10 Hz by seismic and Newtonian noise [122]. The detection of the gravitational waves at lower frequencies will significantly increase the number and mass range of potentially observed GW sources, allow for longer observation of the inspiralling binary stars before the merger, and may improve the chances of detecting relic GWs from the early evolution of the Universe [123]. This is the goal of space-based laser interferometry detectors such as the Laser Interferometer Space Antenna (LISA) [124] and TianQin [125]. Rapid progress of matter-wave interferometry and atomic clocks opens other approaches to gravitational wave detection, with a number of proposals for both terrestrial and space-based detectors ([126–129] and references therein). The Matter wave-laser based Interferometer Gravitation Antenna (MIGA) is a hybrid atom-laser detector in the 100 mHz–1 Hz frequency range [126]. MIGA will employ several atom interferometers simultaneously interrogated by the resonant mode of an optical cavity, with a cavity length of 200 m. It is the first very long baseline atom interferometry instrument, now under construction, to be located in dedicated tunnels, 500 m underground in France. Hogan and Kasevich [127] proposed a space-based GW detector with light-pulse AIs separated by a long baseline (over 100 000 km), capable of detecting GWs in the 0.1-mHz to 1-Hz frequency band, requiring only two satellites and potentially exceeding the sensitivity of proposed the LISA detector. AI GW detector that can operate in a resonant detection mode and can switch between the broadband and narrowband detection modes to increase sensitivity was proposed in [128]. Geiger [130] reviewed the perspective of using AI for GW detection in the mHz to about 10 Hz frequency band. Kolkowitz et al. [129] proposed to use atomic optical lattice clocks (rather than atom interferometers) for a sensitive, narrowband GW space-based detector in a twosatellite configuration.
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484 108. Hogan, J.M., Johnson, D.M.S., Kasevich, M.A.: In: Arimondo, E., Ertmer, W., Schleich, W.P., Rasel, E.M. (eds.) Atom Optics and Space Physics, p. 411. IOS, Amsterdam, Oxford, Tokyo, Washington D.C. (2009) 109. Kleinert, S., Kajari, E., Roura, A., Schleich, W.P.: Phys. Rep. 605, 1 (2015) 110. Zhou, L., et al.: Phys. Rev. Lett. 115, 013004 (2015) 111. Rosi, G., et al.: Nat. Commun. 8, 15529 (2017) 112. Roura, A.: Phys. Rev. Lett. 118, 160401 (2017) 113. Overstreet, C., Asenbaum, P., Kovachy, T., Notermans, R., Hogan, J.M., Kasevich, M.A.: Phys. Rev. Lett. 120, 183604 (2018) 114. Hartwig, J., et al.: New J. Phys. 17, 035011 (2015) 115. Geiger, R., Trupke, M.: Phys. Rev. Lett. 120, 043602 (2018) 116. Seidel, S.T., et al.: European Rocket and Baloon: Programmes and Related research. Esa Special Publ. 730, 309 (2015) 117. Leefer, N., Gerhardus, A., Budker, D., Flambaum, V., Stadnik, Y.: Phys. Rev. Lett. 117, 271601 (2016) 118. Jaffe, M., et al.: Nat. Phys. 13, 938–942 (2017) 119. Hohensee, M.A., Estey, B., Hamilton, P., Zeilinger, A., Müller, H.: Phys. Rev. Lett. 108, 230404 (2012) 120. Asenbaum, P., Overstreet, C., Kovachy, T., Brown, D.D., Hogan, J.M., Kasevich, M.A.: Phys. Rev. Lett. 118, 183602 (2017) 121. Abbott, B.P., et al.: Phys. Rev. Lett. 116, 061102 (2016). LIGO Scientific Collaboration, V Collaboration 122. Saulson, P.R.: Phys. Rev. D 30, 732 (1984) 123. Dimopoulos, S., Graham, P.W., Hogan, J.M., Kasevich, M.A., Rajendran, S.: Phys. Rev. D 78, 122002 (2008) 124. Danzmann, K., et al.: Assessment Study Report. ESA/SRE 3, 2 (2011) 125. Hu, X.-C., et al.: Class. Quantum Gravity 35, 095008 (2018) 126. Canuel, B., et al.: Sci. Rep. 8, 14064 (2018) 127. Hogan, J.M., Kasevich, M.A.: Phys. Rev. A 94, 033632 (2016) 128. Graham, P.W., Hogan, J.M., Kasevich, M.A., Rajendran, S.: Phys. Rev. D 94, 104022 (2016) 129. Kolkowitz, S., Pikovski, I., Langellier, N., Lukin, M.D., Walsworth, R.L., Ye, J.: Phys. Rev. D 94, 124043 (2016) 130. Geiger, R.: In: Augar, G., Plagnol, E. (eds.): An Overview of Gravitational Waves: Theory, Sources and Detection, pp. 285–313. World Scientific, New Jersey (2017) 131. Graham, P. W., Hogan, J. M., Kasevich, M. A., Rajendran, S., Romani, R. W.: arXiv:1711.02225 (2017)
M. S. Safronova Marianna Safronova Marianna Safronova is Professor of Physics at the University of Delaware. Her research interests include the study of fundamental symmetries and the search for physics beyond the standard model of elementary particles and fundamental interactions, the development of highprecision methodologies for calculating atomic properties and exploring their applications, atomic clocks, ultracold atoms and quantum information, longrange interactions, superheavy atoms, highly-charged ions, atomic anions, and other topics.
Part C Molecules
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Part C begins with a discussion of molecular structure from a theoretical/computational perspective using the Born–Oppenheimer approximation as the point of departure. The key role that symmetry considerations play in organizing and simplifying our knowledge of molecular dynamics and spectra is described. The theory of radiative transition probabilities, which determine the intensities of spectral lines, for the rotationally resolved spectra of certain model molecular systems is summarized. The ways in which molecular photodissociation is studied in the gas phase are outlined. The results presented are particularly relevant to the investigation of combustion and atmospheric reactions. Modern experimental techniques allow the detailed motions of the atomic constituents of a molecule to be resolved as
a function of time. A brief description of the basic ideas behind these techniques is given, with an emphasis on gas phase molecules in collision-free conditions. The semiclassical and quantal approaches to nonreactive scattering are outlined. Various quantitative approaches toward a description of the rates of gas phase chemical reactions are presented and then evaluated for their reliability and range of application. Ionic reactions in the gas phase are also considered. Clusters, which are important in many atmospheric and industrial processes, are arranged into six general categories, and then the physics and chemistry common to each category is described. The most important spectroscopic techniques used to study the properties of molecules are presented in detail.
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Contents 33.1 33.1.1
. 488
33.1.4
Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonadiabatic Ansatz: Born–Oppenheimer Approximation . . . . . . . . . . . Born–Oppenheimer Potential Energy Surfaces and Their Topology . . . . . . . . . . . . . . . . . . . . . Classification of Interstate Couplings: Adiabatic and Diabatic Bases . . . . . . . . . . . . . . . Surfaces of Intersection of Potential Energy Surfaces
33.2 33.2.1 33.2.2 33.2.3 33.2.4
Characterization of Potential Energy Surfaces . . The Self-Consistent Field (SCF) Method . . . . . . . Electron Correlation: Wave Function Based Methods Electron Correlation: Density Functional Theory . . . Weakly Interacting Systems . . . . . . . . . . . . . . . .
. . . . .
490 491 491 494 495
33.3 33.3.1 33.3.2 33.3.3
Intersurface Interactions: Perturbations Derivative Couplings . . . . . . . . . . . . . . Breit–Pauli Interactions . . . . . . . . . . . . Surfaces of Intersection . . . . . . . . . . . .
. . . .
. . . .
495 496 497 498
33.4 33.4.1 33.4.2 33.4.3
Nuclear Motion . . . . . . . . . . . . . . . . General Considerations . . . . . . . . . . . . Rotational-Vibrational Structure . . . . . . . Coupling of Electronic and Rotational Angular Momentum in Weakly Interacting Reaction Path . . . . . . . . . . . . . . . . . .
. . . . . . . 499 . . . . . . . 499 . . . . . . . 500
33.1.2 33.1.3
33.4.4
. . . .
. . . .
. . . .
. . . .
. . . .
. 488 . 489 . 489 . 490
. . . . . . . 500 . . . . . . . 501
33.5
Reaction Mechanisms: A Spin-Forbidden Chemical Reaction . . . . . . . . . 502
33.6
Recent Developments . . . . . . . . . . . . . . . . . . . . 504
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
Abstract
Molecular structure is a reflection of the Born–Oppenheimer separation of electronic and nuclear motion, which is in turn a consequence of the large difference between the electron and nuclear masses. One consequence of D. R. Yarkony () Dept. of Chemistry, The Johns Hopkins University Baltimore, MD, USA e-mail: [email protected]
this separation is the concept of a potential energy surface for nuclear motion created by the faster moving electrons. Corollaries include equilibrium structures, transition states, and reaction paths which are the foundation of the description of molecular structure and reactivity. However the Born–Oppenheimer approximation is not uniformly applicable and its breakdown results in perturbations in molecular spectra, radiationless decay, and nonadiabatic chemical reactions. There are many issues that can be addressed in a discussion of molecular structure, including the structure and bonding of individual classes of molecules, computational and/or experimental techniques used to determine or infer molecular structure, the accuracy of those methods, etc. In an effort to provide a broad view of the essential aspects of molecular structure, this Chapter considers issues in molecular structure from a theoretical/computational perspective using the Born–Oppenheimer approximation as the point of origin. Rather than providing a compendium of results, this chapter will explain how issues in molecular structure are investigated and how the questions that can be addressed reflect the available methodology. Even with these restrictions the scope of this topic remains enormous and precludes a detailed presentation of any one issue. Thus the abbreviated discussions in this work are supplemented by ample references to the literature. Several aspects of potential energy surfaces and their relation to molecular structure will be considered: (i) the electronic structure techniques used to determine a single point on a potential energy surface, and the interactions that couple the electronic states in question, (ii) the local properties of potential energy surfaces, in particular equilibrium structures and rovibrational levels that provide the link to experimental inferences concerning molecular structure, (iii) global chemistry deduced from potential energy surfaces including reaction mechanisms and reaction paths and (iv) phenomena resulting from the nonadiabatic interactions that couple potential energy surfaces.
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_33
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Equation (33.4) is valid for any complete set of electronic states depending parametrically on nuclear coordinates. The wave function potential energy surface reaction path parametrical dependence of ‰ e .rI R/ on R, denoted by the I configuration interaction conical intersection Born– semicolon, reflects the fact that the R are not dynamical variOppenheimer approximation ables in Eqs. (33.2) and (33.3). As a practical matter it is necessary to make a particular choice of ‰Ie .rI R/ in order to limit the size of the expansion in electronic states. An adiabatic electronic state is an eigenfunction of H e .rI R/ for 33.1 Concepts fixed R. A particularly useful choice of adiabatic state, denoted by ‰I0 .rI R/, employs H 0 .rI R/ rather than the full 33.1.1 Nonadiabatic Ansatz: H e .r; R/. These electronic wave functions satisfy Born–Oppenheimer Approximation Keywords
(33.5) H 0 .rI R/‰I0 .rI R/ D EI0 .R/‰I0 .rI R/ : Basic Quantities The total Hamiltonian for electronic and nuclear motion in The ‰I0 .rI R/ are determined up to a geometry dependent the space fixed coordinate frame, in atomic units, is phase. This phase is usually chosen such that the ‰I0 .rI R/ are real. This assumption is acceptable except in the situation N X 1 2 where there is a conical intersection on the potential energy H eN .r; R/ D r˛ C H e .rI R/ 2M˛ surface in question. In that case, the real-valued ‰I0 .rI R/ ˛D1 (33.1) changes sign when a closed loop surrounding the conical T nuc C H e .rI R/ ; intersection point is traversed, that is, ‰I0 .rI R/ is not singlewhere R denotes the 3N nuclear coordinates, with R D valued [3]. This geometric or Berry [4] phase condition .R 1 ; R 2 ; : : : ; R N /; R i D .Xi ; Yi ; Zi /, r denotes the 3M has consequences in such phenomena as the dynamic Jahn– electronic coordinates 2using similar conventions, T nuc is Teller effect [5, 6] but will not be addressed in detail in this the nuclear kinetic energy operator and H e .r; R/ is the total review. As a consequence of the parametric dependence of electronic Hamiltonian taken as ‰I0 .rI R/ on R, EI0 .R/ becomes a function of R and is (33.2) referred to as the nonrelativistic Born–Oppenheimer potenH e .rI R/ D H 0 .rI R/ C H rel .rI R/ : tial energy surface for reasons discussed below. Approaches Here H 0 .rI R/ is the nonrelativistic Born–Oppenheimer based on Eq. (33.5) are most appropriate for molecular systems with only light atoms. However with the use of pseuHamiltonian dopotential techniques [7], formally equivalent approaches M can be developed for heavier systems (Sect. 33.3.2). X X ZK 1 ri2 H 0 .rI R/ D Inserting Eq. (33.4) independent
into the time 2 i D1 jR k r i j K;i Schrödinger equation H eN .rI R/ E ‰ eN .rI R/ D 0 M N and taking the inner product with the electronic basis state 1 1 X ZK ZL 1X C ; C ‰I0 .rI R/ gives the following system of coupled equations 2 jr i rj j 2 jRK RL j i ¤j K¤L for the rovibronic functions ˇIL .R/ [8, 9]: (33.3)
nuc T C EI0 .R/ E ˇI .R/ and H rel .rI R/ is the relativistic contribution to the elecX BP KQ IJ .R/ C HIJ D .R/ tronic Hamiltonian, for light atoms conventionally treated J within the Breit–Pauli approximation [1] and discussed fur N n ther in Sect. 33.3.2. Note that in Eq. (33.3) the nuclear X o 1 IJ IJ .R/ r C r f .R/ f ˇJ .R/ ; ˛ ˛ kinetic energy is absent, so that only the r i are dynamical ˛ 2M˛ ˛ ˛D1 variables in Eqs. (33.2) and (33.3). Using the Born–Huang (33.6) approach [2] to the Born–Oppenheimer approximation, the total wave function ‰LeN .r; R/ is expanded in a basis of L electronic states, ‰Ie .rI R/. The total wave function for the where the state label L on ˇI has been suppressed, and system thus has the form
IJ IJ IJ (33.7) f IJ ˛ .R/ fX˛ .R/; fY˛ .R/; fZ˛ .R/ ; X X 1 eN e L JI ‰L .r; R/ D ‰I .rI R/ˇI .R/ : (33.4) KQ JI .R/ D kQW .R/ ; (33.8) ˛ W˛ 2M ˛ I W;˛
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Molecular Structure
ˇ ˇ @ 0 D ‰ .rI R/ ; @W˛ I r * + ˇ ˇ @ @ 0 0 D ‰ .rI R/ˇˇ ‰ .rI R/ ; @W˛ J @Wˇ0 I
fWJI˛ .R/ JI kQW 0 ˛W
ˇ
489
‰J0 .rI R/ˇˇ
(33.9)
Transition state Reaction path
(33.10)
r
with W D X; Y; Z; ˛ D 1 N . The subscript r on the matrix elements in Eqs. (33.9)–(33.10) denotes integration over all electronic coordinates and the final r˛ in Eq. (33.6) acts on Equilibrium Equilibrium structure structure .R/ and ˇ .R/. While this representation may not both f IJ J ˛ be optimal for a treatment of the nuclear dynamics, since it employs a space fixed frame representation, it is adequate for Fig. 33.1 Schematic representation of a transition state and reaction the present development as it clearly displays the origins of path connecting two minima the Born–Oppenheimer approximation and its breakdown. Reaction paths [17] connect the asymptotes, minima, and saddle points. They can best be defined as the steepest deModifications to the Nonrelativistic scent paths from a transition state structure down to local Born–Oppenheimer Potential Energy Surface minima [17], although methods for walking uphill (shallowWhen the interstate couplings on the right-hand side of est ascent path) [18] also exist. Unlike equilibrium structures Eq. (33.6) can be neglected, the nuclear motion is governed by and saddle points which are independent of the choice of coorthe effective potential given by V I .R/ EIe .R/C KQ II .R/ dinate system used to representR, the reaction path is coordiBP .R/ C KQ II .R/ and the Born–Oppenheimer nate system dependent. The intrinsic reaction coordinate [17], EI0 .R/ C HII approximation is valid. Here it is assumed that f II ˛ .R/ D the reaction coordinate in mass weighted cartesian coordi0, a sufficient condition for which is that the ‰I0 .rI R/ are nates R i ! R i Mi , is most frequently used (Sect. 33.4.4) chosen real-valued [see discussion following Eq. (33.5)]. These features of the potential energy surfaces and their EI0 .R/ is generally the principal R-dependent contribution determination will be discussed further below. Their deterBP .R/ is referred to as the relativistic contri- mination consists of two parts: (i) the level of treatment, i.e., to V I .R/. HII bution to the nonrelativistic Born–Oppenheimer potential en- the electronic structure technique used to determine E 0 .R/ I ergy surface and is discussed further in Sect. 33.3.2. The and (ii) the characterization of the features of E 0 .R/, i.e., I contribution KQ II .R/ is referred to as the adiabatic correc- minima, saddle points, and reaction paths [19, 20] at the level BP .R/, which are independent of of treatment chosen. In this chapter, only fully ab initio levels tion. Unlike EI0 .R/ and HIJ II Q mass, K .R/ is mass dependent Eqs. (33.8) and (33.10). It of treatment are considered [21, 22]. has been computed from first principles [10–12] and inferred from experiments [13, 14] principally for diatomic systems.
33.1.2 Born–Oppenheimer Potential Energy Surfaces and Their Topology EI0 .R/ is
the main focus of this chapter. There have been several recent reviews of ground state .I D 1/ potential energy surfaces [15, 16]. Equilibrium structures (stable or metastable species) I .R/ represent local minima on EI0 .R/, that is gW ˛ 0 e @EI .R/=@W˛ D 0 for all W˛ at R D R . There may be several equilibrium structures for a given set of nuclei; for example the atoms H, C and N form stable molecules HCN and HNC. Saddle points or transition states, extrema on EI0 .R/ with one negative eigenvalue of the Hessian matrix F I .R/, whose elements are defined by FWI ˛ Wˇ .R/ @2 EI0 .R/=@W˛ @Wˇ , represent mountain passes separating the various equilibrium structures and the asymptotes – values ofR corresponding to isolated molecular fragments. This situation is illustrated in Fig. 33.1.
33.1.3 Classification of Interstate Couplings: Adiabatic and Diabatic Bases Intersurface couplings, shown on the right-hand side of Eq. (33.6), result in the breakdown of the single potential energy surface Born–Oppenheimer approximation. Within the Born–Huang ansatz such a nonadiabatic process is interpreted in terms of motion on more than one Born– Oppenheimer potential energy surface [8]. From Eq. (33.6), it is seen that there are three types of matrix elements coupling the electronic states, KQ IJ .R/, H BPIJ .R/ and f IJ .R/. Two of these couplings, KQ IJ .R/ and f IJ .R/ arise from the nuclear kinetic energy operator T nuc , while the third coupling arises from the Breit–Pauli interaction H BP . This classification of the intersurface interactions is a consequence of the choice of the adiabatic electronic states through Eq. (33.5). Other choices of ‰I0 .rI R/, most notably the diabatic electronic states [23–27], are possible. In the adiabatic state basis, H 0 .rI R/ is diagonal and in the absence of relativistic effects, intersurface couplings
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originate exclusively from the derivative coupling terms in Eqs. (33.9) and (33.10). The diabatic basis seeks to transfer the intersurface coupling from these derivative operators to BP .R/. The diabatic a potential term analogous in form to HIJ 0d basis, ‰I .rI R/, is a unitary transformation of the adiabatic electronic basis, defined such that [24] ˇ ˇ @ 0d 0d ˇ ‰ .rI R/ ‰ .rI R/ D0; fRJI;d J ˇ @R I ˛ ˛ r
(33.11)
33.2 Characterization of Potential Energy Surfaces This section is concerned with the determination of the approximate solution of Eq. (33.5). For this purpose it is frequently convenient to re-express H 0 .rI R/ in second quantized [35] form as H 0 .rI R/ D
X i;j
hij ai aj C
1 X .iljj k/ai aj ak al ; 2 i;j;k;l
where R˛ is an internal coordinate. For polyatomic systems (33.16) .N > 2/, rigorous diabatic bases do not exist [27] and approximate diabatic bases are sought [27–30]. A discussion where hij and .iljj k/ are standard abbreviations [36] for the one electron (kinetic energy and nuclear–electron attracof this issue can be found in [31]. tion) and two electron (electron–electron repulsion) integrals respectively, and ai and ai are the fermion creation and destruction operators. The integrals and the creation and de33.1.4 Surfaces of Intersection of Potential struction operators are defined in terms of a basis of one Energy Surfaces electron functions, usually molecular spin-orbitals i i , i D ˛ or ˇ. Thus The intersurface couplings are most effective in promoting a nonadiabatic process when jV I .R/ V J .R/j is small for some J on the right-hand side of Eq. (33.6), so that
EIe .R/ C KQ II .R/ EJe .R/ C KQ JJ .R/ 0 BP .R/ C EIJ .R/ C KIJ .R/ EIJ
(33.12)
is small. Here 0 .R/ EI0 .R/ EJ0 .R/ ; EIJ KIJ .R/ KQ II .R/ KQ JJ .R/ ;
(33.13)
ˇ 1 ˝ hij .R/ D i .r k /i ˇ rk2 2 X ˇ ˛ ZK ˇj .r k /j rk jRK r k j K ˇ ˇ 1 .ij jkl/ D i .r m /i j .r m /j ˇˇ jr m r n j ˇ ˇ ˇ : ˇk .r n /k l .r n /l
(33.17)
(33.18)
r m ;r n
(33.14) Equation (33.18) is referred to as the Mulliken notation [37] : (33.15) for a two-electron integral. It differs from the frequently used [36] bra-ket representation hij jkli D .ikjj l/. The Since in general KIJ .R/ is quite small, and for the low molecular orbitals i .rj I R/ are in turn expanded in terms BP functions [38], .rj I R/, is itself small, of a basis set of atom-centered atomic number systems considered here HII PL we are led to consider regions of nuclear coordinate space with i .rj I R/ D P D1 P .rj I R/TP;i .R/. Note that here 0 .R/ is small, referred to as we have distinguished between a spin-orbital i .rj I R/i for which the magnitude of EIJ 0 avoided intersections when EIJ .R/ ¤ 0 and as conical and and a molecular orbital i .rj I R/, but we will follow the 0 .R/ D 0. The set usual convention of allowing the functions in Eqs. (33.17) Renner-type [32] intersections when EIJ 0 of R for which EIJ .R/ D 0, the noncrossing rule, was first and (33.18) to be molecular orbitals or spin-orbitals with the discussed in 1929 by von Neumann and Wigner [33]. The use being clear from the context. Since H 0 .rI R/ is independent of electron spin, it can rule states that for diatomic systems, crossings between two potential energy curves of the same symmetry are not pos- also be written in terms of the EQpq , spin-averaged excitation P sible (actually, are extremely rare), whereas for polyatomic operators [39], where EQpq D ap aq , p; q label orbitals systems, potential energy surface intersections are allowed. and D ˛; ˇ so that Mathematically the noncrossing rule is expressed in terms of X 1 X the dimensions of a surface, the surface of intersection, on H 0 .rI R/ D hij EQ ij C .ij jkl/EQ ij EQ kl ıj k EQ i l : 2 which a set of conditions can be satisfied [3, 34]. For eleci;j i;j;k;l tronic states of the same symmetry (neglecting electron spin (33.19) degeneracy), the dimension of the surface of conical intersection is K 2, where K is the number of internal nuclear The EQpq satisfy EQ ij ; EQ kl D EQ i l ıj k EQ kj ıi l as a consequence degrees of freedom. For states of different spin symmetry, of the commutation relations for the ai and ai and are generators of the unitary group [40], (Chap. 4). This observation the dimension of the surface of intersection is K 1. BP EIJ .R/
BP HII .R/
BP HJJ .R/
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allows the powerful machinery of the unitary group to be applied to the evaluation of EI0 .R/ [41] (Sect. 33.1.2). The determination of EI0 .R/ involves two steps; (i) the determination of the molecular orbitals i .rj I R/, and (ii) sometimes simultaneously, the determination of the approximate wave function ‰I0 .rI R/. Wave functions discussed in this section are constructed from functions that are antisymmetrized products of spinorbitals i .rj I R/i ‰I0 .rI R/ D A
M Y i D1
i .rj I R/i ;
with 2i .rj I R/ D 2i 1 .rj I R/; i D 1; : : : ; L and 2L D M , the SCF equations are "ri D hri C
X
2.irjjj / .ij jrj / D 0
(33.22)
j 2 fjocc g
for i , r occupied and virtual orbitals, respectively. Here, occupied orbitals jocc are those occurring in the orbital product in Eq. (33.21), and the remainder of the orbitals are referred to as virtual orbitals. The solution to Eq. (33.22) is referred to as the canonical SCF solution, provided the or(33.20) bitals satisfy the SCF equations
where i D ˛ or ˇ. The antisymmetrizer A is required to take into account the Fermi statistics of the electrons. The wave function in Eq. (33.20) is referred to as a Slater determinant and is an eigenfunction of Ms , the z-component of total electron spin. A linear combination of Slater determinants that is also an eigenfunction of S 2 is referred to as a configuration state function (CSF) [42]. Equation (33.20) corresponds to a particular distribution of electrons in the orbitals i .rj I R/. This distribution is referred to as an electron configuration. In principle, it is possible to learn everything about the topology of a potential energy surface from the pointwise evaluation of EI0 .R/. In practice however, the determination of topological features, including location of equilibrium structures, saddle points and reaction paths, characterization of quadratic and anharmonic force fields [43], and even the evaluation of derivative couplings [44], has benefitted immensely from techniques in which the energy gradient g I .R/ or higher derivatives are determined directly from knowledge of the wave function at the R in question. These analytic derivative techniques,, i.e., techniques that do not use divided difference differentiation, have been actively developed since their first introduction [45]. The techniques most commonly used to calculate EI0 .R/ and its derivatives are described below.
33.2.1 The Self-Consistent Field (SCF) Method The SCF Energy In this most basic treatment, the electronic wave function ‰I0SCF is taken as a single Slater determinant or a single CSF. The spin-orbitals are ˛ ˝ determinedˇto give an ˇextremum of the energy functional ‰I0SCF .rI R/ˇH 0 .rI R/ˇ‰I0SCF .rI R/ r . This gives rise to the self-consistent field conditions. For restricted closed shell wave functions [46], defined by the conditions
F i D i i ;
(33.23)
where the Fock operator F is defined by Fri D "ri and electronic energy is EI0 .R/ i D "i i . The corresponding P 0SCF .R/ D j 2 fjocc g ."jj C hjj /. EI
SCF Energy Derivatives The derivative of Ei0SCF .R/ can be expanded through third order using analytic gradient techniques [47, 48], and through fourth order using divided differences and more recently using analytic derivative techniques [43]. The ability to evaluate the energy derivatives or force field through fourth order is important in the determination of vibrational properties of molecules, as discussed in Sect. 33.4.2. Direct SCF The range of molecular systems accessible to treatment at the SCF level has been expanded considerably by the introduction of the direct SCF methods [49]. In this method, none of the two-electron integrals in the basis .pqjrs/ are stored during the iterative solution of Eqs. (33.22) and (33.23). Since the number of such integrals grows as L4 =8, direct SCF procedures avoid the L4 -storage bottleneck, enabling SCF wave functions to be determined for systems with more than 100 atoms and basis sets with more than 1000 functions. (If all the L4 =8 integrals were stored on disk this basis would require 1000 gigabytes).
33.2.2 Electron Correlation: Wave Function Based Methods
Wave functions more accurate than SCF wave functions are obtained by including the effects of electron correlation. The correlation energy is defined as EIcorr .R/ EI0 .R/ EI0SCF .R/. Methods for the determination of EIcorr .R/ are commonly classified as single reference or mulitreference L Y 0RSCF methods. In single reference methods, an SCF wave function .rI R/ D A 2i .r 2i I R/˛2i C1 .r 2i C1 I R/ˇ ‰I ‰I0SCF .rI R/ given by Eq. (33.20) is improved. In multirefi D1 (33.21) erence methods, the starting point is the space spanned by
33
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a set of terms like Eq. (33.20). This space is referred to as the reference space and the functions in the space as the reference configurations. Sometimes the molecular orbitals .r i I R/ for the reference space are the SCF orbitals of a single reference configuration calculation, but more often the .r i I R/ are chosen to satisfy multiconfigurational selfconsistent field (MCSCF) equations [50–53].
Single Reference Methods Second Order Møller–Plesset Perturbation Theory (MP2) In this approach [54], the solutions of the SCF equations are used to determine EI0 .R/ to second order as .abjjij /2 1 X EI0MP2 .R/ D EI0SCF .R/ ; 4 a C b i j
where ˆai , ˆab ij are single and double excitations from ˆ0 , and it is the amplitudes t that must be determined. Wave functions of the form in Eqs. (33.25)–(33.28) have the important property referred to as size consistency [60] or size extensivity [59]. Consider N noninteracting helium atoms. The electronic wave function for the ith helium atom in the natural orbital basis [61] can be written as ‰i D Œ1 C T2 .i/0 .i/, so that the N atom electronic wave function becomes Y Y ‰i D Œ1 C T2 .i/ˆ0 ‰D i
"
D exp
i
X
#
T2 .i/ ˆ0 D exp.T2 /ˆ0
(33.29)
i
Thus this exponential type of solution scales properly with (33.24) the number of particles. To obtain the amplitudes t, define HN D H 0 where i, j denote occupied orbitals, a, b denote virtual or- hˆ0 jH 0 jˆ0 i, E 0CC D E 0CC Chˆ0 jH 0 jˆ0 i E 0CC C I I I bitals, and EI00 , P D jˆ0 ihˆ0 j and Q D 1 P . Then, inserting Eqs. (33.25)–(33.28) into Eq. (33.5) gives the energy .abjjij / D .aijbj / .aj jbi/ D habjij i habjj ii : Eq. (33.30) and amplitude Eq. (33.31) equations, which deBoth gradients of EI0MP2 .R/ and direct implementations of fine the coupled cluster approach: MP2 [37, 55] are currently available. Higher order pertur(33.30) EI0CC D hˆ0 jHQ N jˆ0 i ; bation theories, also in common use, are discussed in the abc Q abc context of coupled cluster methods which are considered (33.31) hˆij k jHN jˆ0 i D 0 for all ˆij k ; next. where Coupled Cluster Method HQ N D exp.T /HN exp.T / Perhaps the most reliable method currently available for (33.32) D EI00 C exp.T /H 0 exp.T / : characterizing near equilibrium properties of moderately sized molecules is the coupled cluster approach [56–60] To appreciate the nature of these equations, consider the ap(Chap. 5). In this approach, the exact wave function is writproximation T D T2 constructed from SCF orbitals [36, 59], ten as a unitary transformation of a reference wave function
referred to as the coupled cluster doubles (CCD) level. At the SCF wave function ˆ0 .rI R/ usually, but not necessarily, this level the energy equation becomes ˆ0 .rI R/ D ‰I0SCF .rI R/ , a;b;i;j
‰I0CC D exp.T /ˆ0 : where T is an excitation operator defined as X Tp I T D
(33.25)
(33.26)
EI0CCD D hˆ0 j exp.T2 /H 0 exp.T2 /jˆ0 i D hˆ0 jH 0 T2 jˆ0 i C EI00 1 X D EI00 C ˆ0 H 0 ar as ai aj ˆ0 tijrs 4 i;j;r;s
p
D EI00 C
with Tp D
1 pŠ2
X
tijabc k aa ab ac ai aj ak ;
i;j;k;:::;a;b;c
1 X .rsjjij /tijrs 4 i;j;r;s
(33.33)
(33.27) for i; j occupied and r; s virtual orbitals. The amplitude equations become
where i; j; k denote occupied orbitals in ˆ0 and a; b; c denote virtual orbitals. Thus X 1 X ab ab ‰I0CC D ˆ0 C tia ˆai C tij ˆij C ; 4 i;a i;j;a;b
(33.28)
0 hˆrs ij j exp.T2 /H exp.T2 /jˆ0 i 1 2 1 2 rs 0 D hˆij j 1 T2 C T2 H 1 C T2 C T2 jˆ0 i 2 2 D0 (33.34)
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which reduces (after considerable commutator algebra) [36] function [50–53, 67]. This approach is capable of describing the internal or static correlation energy, the part of the to correlation energy that leads to sizeable separation of two .r C s i j /tijrs electrons in a pair, and near-degeneracy effects. A particX X pq ularly robust type of MCSCF wave function, the complete rs D .rsjjij / .rsjjpq/tij .kljjij /tkl active space (CAS) [68–70] wave function, includes in p>q k>l X Eq. (33.36) all CSFs arising from the distribution of the availrp sp C .ksjjjp/ti k .krjjjp/ti k able electrons among the active orbitals. Note that a CAS k;p wave function is size consistent. rp sp .ksjjip/tj k C .krjjip/tj k The remaining part of the correlation energy, the dynamic h X correlation, describes the two-electron cusp, i.e., the regions pq rs rp sq sq rp .kljjpq/ tij tkl 2 tij tkl C tij tkl of space for which two electrons experience the singularity in k>lIp>q the Coulomb potential. Empirically, it has been shown [71] i pq pq rp sq sq rp : (33.35) that wave functions capable of providing chemically accu 2 tirsk tj l C ti k tjrsl C 4 ti k tj l C ti k tj l rate descriptions of the dynamic correlation can be obtained The solution to Eq. (33.35) is obtained iteratively. The by augmenting the MCSCF or reference wave function with first iteration gives tijrs D .rsjjij /=.r C s i j /, all CSFs that differ by at most two molecular orbitals, douwhich when inserted into Eq. (33.33) gives Eq. (33.24), ble excitations Eq. (33.28) from those of the reference space. i.e., the MP2 result. If the quadratic terms in Eq. (33.35) Such wave functions are generally referred to as multireferare neglected, then the second iteration gives the third or- ence single and double excitation configuration interaction der Møller–Plesset (MP3) result [36]. The result of iterating (MR-SDCI) wave functions. First (second) order wave functions [72] include all single (single and double) excitations Eq. (33.35) to convergence gives the CCD result. Of the levels of coupled cluster treatments in current use, relative to a CAS reference space. those including T1 , T2 , T3 , and T4 in Eq. (33.25) provide the Multiconfigurational Self-Consistent Field (MCSCF) most reliable results [62–66]. Theory In the MCSCF Papproximation, a wave function Multireference Methods 0MCSCF .rI R/ D ˛ c˛I ˛ .rI R/ of the form in Eq. (33.36) Single reference methods provide probably the most power- ‰I ful tools for treating near-equilibrium properties of ground is to be determined. In this procedure, both the molecular I electronic state systems. In other instances, such as the study orbitals .rj I R/ andc .R/ are determined from the requireof electronically excited states, determination of global po- ment that tential energy surfaces and for systems with multiple open shells such as diradicals, multireference techniques are found to be extremely useful. In the multireference techniques discussed below, the wave function is written as ‰I0MRF .rI R/
D
X ˛
c˛I
˛ .rI R/
EI0MCSCF ˇ ˇ ˝ ˛ D ‰I0MCSCF .rI R/ˇH 0 ˇ‰I0MCSCF .rI R/ r ; (33.37)
be a minimum. In a popular variant, the state averaged MCSCF (SA-MCSCF) [50–53] procedure, the average energy (33.36) functional
where ˛ .rI R/ is a CSF. This expansion is usually referred to as a configuration interaction (CI) expansion, and the coefficients c I .R/ are referred to as CI coefficients [42]. Since Eq. (33.36) does not involve the exponential ansatz, it is not automatically size extensive. In the description of these wave functions, it is useful to generalize the notion of occupied and virtual orbitals to inactive, active, and virtual orbitals, where, referring to the CSF’s defining the reference space, inactive orbitals are fully occupied in all CSFs, virtual orbitals are not occupied in any CSF and the active orbitals are (partially) occupied in at least one CSF. Most multireference techniques begin with the determination of a multiconfigurational self-consistent field (MCSCF) wave function or state averaged MCSCF(SA-MCSCF) wave
E
0SAMCSCF
D
K X
ˇ ˇ ˝ wI ‰I0MCSCF .rI R/ˇH 0 ˇ
I D2
˛ ‰I0MCSCF .rI R/ r ;
(33.38)
where the weight vector w D .w1 ; w2 ; : : : ; wK / has only positive elements, is minimized. This procedure should be compared with the multireference CI approach described below in which a predetermined set of molecular orbitals is used. The optimum molecular orbitals and CI coeffients can be written as unitary transformations of an initial set of such quantities, i.e., ‰I0MCSCF D exp./ exp./ˆI0 ;
(33.39)
33
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D. R. Yarkony
where D
X
s;i as ai ;
i;s
D
X
ˇ ˛˝ ˇ l;n ˇˆ0l ˆ0n ˇ ;
Using unitary group techniques, this apparently intractable summation can be used to express the Aij kl as a simple finite (33.40) product [76].
n;l
and ; are general anti-Hermitian matrices. Since mi;j D mj;i mij for m D ; , the upper triangle of m forms a vector m E that enumerates the independent parameters of m. The MCSCF or SA-MCSCF equations can be succinctly formulated by inserting Eq. (33.39) into Eq. (33.37) or Eq. (33.38), expanding the commutators to second order and requiring @E=@mij D 0 [36]. The result provides a system of Newton–Raphson equations that can be solved iteratively for the ; [52, 53, 67, 73]. Multireference Configuration Interaction Theory: The MR-SDCI Method In multireference configuration interaction theory, the wave function is again of the form ‰I0MRCI .rI R/ D P I ˛ c˛ .R/ ˛ .rI R/, but now CSFs involving the large space of virtual orbitals are included. In this approach the CI coefficients are found for a predetermined set of molecular orbitals. The CSF expansions at the MR-SDCI level become quite large (110 million CSFs is routine), and even larger expansions are tractable using specialized methods. The c I satisfy the usual matrix equation
0 H EI0 c I D 0 ; (33.41) where 0 D h ˛ jH 0 j ˇ i H˛ˇ X ˛ˇ X ˛ˇ D Aij hij C Aij kl .ij jkl/ : i;j
i;j;k;l
(33.42)
Contracted CI and Complete Active Space Perturbation Theory (CASPT 2) The direct approach outlined above makes treatment of large MR-SDCI expansions possible. However, as the size of the reference space grows, the CSF space in the MR-SDCI expansion may become intractably large, particularly if the full second-order wave function is used. To avoid this bottleneck, the reference CSFs may be selected from the active space, and perturbation theory may be used to select CSFs involving orbitals in the virtual space [42]. The use of selection procedures complicates the implementation of direct techniques although recently progress in selected direct CI procedures has been reported [77–79]. Alternatively, new techniques have been developed that avoid this selection procedure. In these approaches, the MCSCF wave function itself is used as the reference wave function for CI or pertubation theory techniques. The use of a reference wave function rather than a reference space considerably reduces the size of the CSF space to be handled. In this approach, one of the principal computational complications is that the excited functions are not necessarily mutually orthogonal. Two computational procedures currently in wide use, known as contracted CI [80–83] and CASPT2 [84, 85], are based on this approach. The CASPT2 method is a computationally efficient variant of second order perturbation theory in which the reference wave function is a CAS-MCSCF wave function and thus may itself contain tens of thousands of CSFs. In this case the full multireference CI problem would be intractable owing to the large space of double excitations. A similar approach is adopted in the contracted CI method, in that the excitations are defined in terms of a general MCSCF reference wave function ‰I0MRF rather than the reference space as in the MR-SDCI methods described above.
It is the computationally elegant solution of Eq. (33.41) for large expansions that is the essence of modern MR-SDCI methods. Because of the large dimension of the CSF space, it is not possible (or even desirable) to find all the solutions of Eq. (33.41). The few lowest eigenstates and eigenvalues can be found [74] using an iterative direct CI procedure [75] in which a subspace is generated sequentially from the residual 33.2.3 Electron Correlation: defined at the kth iteration by Density Functional Theory X .k/ 0 .k1/ .k1/ H E ı c ; (33.43) D The approaches in Sect. 33.1.1 and 33.1.2 can be referred based approaches in the sense that dewhere for simplicity of notation, the state index I is sup- to as wave function 0 .R/ is accompanied by the determination termination of E I pressed. The computationally demanding step in this proce˛ˇ of the corresponding electronic wave function ‰I0 .rI R/. An Q Q dure is the efficient evaluation of the Aij kl D h ˛ jEij Ekl alternative approach is known as density functional theory ıj k EQ i l j ˇ i. Key to the efficiency of this evaluation is the fac(DFT) [86]. The ultimate goal of DFT is the determinatorization formally achieved by tion of total densities and energies without the determination X of wave functions, as in the Thomas–Fermi approximation. h ˛ jEQ ij j m ih m jEQ kl j ˇ i : h ˛ jEQ ij EQ kl j ˇ i D DFT is based on the Hohenberg–Kohn Theorem [87], which m (33.44) states that the total electronic density can be considered
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to be the independent variable in a multi-electron theory (Chap. 21). Computationally viable approaches exploit the Kohn and Sham formulation [88], which introduces molecular orbitals as an intermediate device. The essential features of the Kohn–Sham (KS) theory [86] are as follows [89]. Assume that the real N -electron system for a particular arrangement of the nuclei R has a total electron density .rI R/. Consider a system of N independent noninteracting electrons subject to a one-body potential V0 with total density 0 .rI R/ such that .rI R/ D 0 .rI R/. The corresponding independent particle orbitals, the Kohn– Sham orbitals iKS .rj I R/, satisfy a Hartree–Fock-like equation 1 (33.45) r 2 C V0 "i iKS D 0 2 with
0 .rI R/ D
X i
jiKS j2 :
(33.46)
outlined in Sect. 33.2.2 must be modified somewhat. These modifications arise from the finiteness, and hence incompleteness, of the basis (the set ) used to describe the molecular orbitals. Assume that the interaction of two molecules A and B is to be determined. Consider the description of molecule A as the distance between A and B decreases from infinity. The atom centered basis functions on molecule B augment those on molecule A, lowering its energy, independent of any physical interaction. This computational artifact serves to overestimate the interaction energy, and is known as the basis set superposition error [91, 92]. In chemically bonded systems where interaction energies are large, it is of negligible importance. However in weakly bonded systems for which the interaction energies may be on the order of 10 to 100 cm1 , the basis set superpostion error can be significant. The basis set superposition error can be reduced by the counterpoise correction [93, 94]. In this approach, the interaction energy is evaluated directly as
EI0int .R/ D EI0 .R/ EI0A .R/ C EI0B .R/ ; (33.48)
The relation between the energy of the ideal system and that 0A 0B of the true system E 0DFT .R/ is obtained from the adiabatic where EI .R/ and EI .R/, the energies of A and B respectively, are evaluated in the full basis. connection formula [89]. In order to determine E 0DFT .R/, functions iKS are required, which in turn means that V0 , the Kohn–Sham noninteracting one-body potential, must be determined. V0 is 33.3 Intersurface Interactions: Perturbations written as Eq. (33.3) The existence of interstate interactions can lead to unex(33.47) pected shifts in spectral lines as well as predissociation of V0 D VeN C VCoul C Vxc the states themselves [95]. These situations are illustrated in where VCoul is the Coulomb interaction corresponding to the Fig. 33.2a,b which present the 1; 2; 3 3 …g potential energy electron density, VeN is the electron-nuclear attraction inter- curves for Al2 and the corresponding derivative couplings action and Vxc is the exchange-correlation density. The effect f IJ .R/ respectively. The derivative couplings were evalof Eq. (33.47) is to isolate from V0 the straightforward con- uated using the method described in Sect. 33.3.1. In this tributions VCoul and VeN , and transfers our ignorance to the molecule, derivative couplings between the 2; 3 3 …g states remaining portion Vxc . Although, by the Hohenberg–Kohn are responsible for the perturbations in the vibrational levels theorem Vxc must exist, its determination remains the chal- of the bound 2; 3 3 …g states. Derivative couplings of these lenge of modern density functional theory, which currently states with the 1 3 …g state causes predissociation of all levuses approximate functional forms. These approximate treat- els in the 2; 3 3 …g manifold [96]. Sect. 33.1 describes two classes of interstate matrix elments of Vxc are quite useful in practice, and for large systems, DFT offers a promising alternative to wave func- ements that can lead to these nonadiabatic phenomena, the derivative coupling matrix elements in Eqs. (33.7)–(33.10) tion based methods. rel , which is usually treated for low Z systems within Through Eq. (33.45), the KS approach is formally similar and HIJ to SCF theory, although the Kohn–Sham theory is in princi- the Breit–Pauli approximation. An illustration of a nonadiaple exact. This formal similarity is exploited in the evaluation batic process induced by H rel is provided in Sect. 33.5. A key issue in the treatment of the electronic structure aspects of of the derivative E 0DFT .R/ [90]. these phenomena is the reliable evaluation of the interstate matrix elements. The interstate interactions are usually of 33.2.4 Weakly Interacting Systems most interest in regions of nuclear coordinate space far removed from the equlibrium nuclear configuration. Thus it When attempting to describe weak chemical interactions is desirable to evaluate these interactions using multirefersuch as van der Waals or dispersion forces, the techniques ence CI wave functions which (Sect. 33.1) are well suited
33
496
D. R. Yarkony
a E(cm–1) 40 000 35 000
33 ¦g
30 000 25 000 20 000
23 ¦g 13 ¦g
Al2
15 000 10 000 3.5
4.5
5.5
6.5 7.5 R(Al – Al) (a0)
b f IJ(a0–1) 4 3
f 12
Al2
2
relative to the earlier divided difference techniques [99]. The key ideas are given below. In this presentation, the standard, real-valued normalization is used. Additional contributions owing to the geometric phase, if required, must be evaluated separately as they do not follow from the electronic Schrödinger equation at a single point [100]. Differentiation of the ‰I0 .rI R/ defined in Eq. (33.36) gives: X @ @ 0 I ‰ .rI R/ D c .R/ .rI R/ @W˛ I @W˛ @ I C c .R/ : .rI R/ @W˛ (33.50) Thus fWJI˛ .R/, consists of two terms
f 23
fWJI˛ .R/ D
1 0 –1 3.5
f 13 4.5
5.5
6.5 7.5 R (Al – Al) (a0)
Fig. 33.2 (a) Adiabatic potential energy curves for the 1; 2; 3 3 …g states of Al2 from [96]. (b) Derivative couplings F Ij .R/ for .I; J / D 1; 3 3 …g ; 1; 2 3 …g and 2; 3 3 …g from [96]
CI
fWJI˛ .R/ C
CSF
fWJI˛ .R/ ;
where the CI contribution is given by X @ I CI JI J fW˛ .R/ D c .R/ c .R/ ; @W˛
(33.51)
(33.52)
and the CSF contribution has the form fWJI˛ .R/ X D cJ .R/
CSF
ˇ ˇ @ ˇ .rI R/ ˇ @W ˛
I .rI R/
cI .R/ : (33.53) for use in these regions of coordinate space. The evaluation r ; of nonadiabatic interactions, based on SA-MCSCF/CI wave Evaluation of CSFfWJI˛ .R/ is straightforward using analytfunctions, is discussed below. ical derivative techniques [101], so that the remainder of the discussion focusses on CIfWJI˛ .R/. From Eq. (33.52), 33.3.1 Derivative Couplings it would appear that the derivative of the CI coefficients @=@W˛ cI .R/ VWI ˛ ; .R/ would be required to evaluate From Eqs. (33.9)–(33.11), two classes of matrix elements CI fWJI .R/. This is quite costly and is in fact not necessary, ˛ JI .R/ and fWJI˛ .R/. In fact, techniques to since only the projection onto the state ‰J0 .rI R/ is required. are required, kQW ˛ Wˇ evaluate both of these exist [44]. However, it is common to Equation (33.52) for CIfWJI .R/ can be recast in a form similar ˛ JI .R/ by approximate kQW to that of g I .R/. This transformation of Eq. (33.52) enables ˛ Wˇ the explicit determination ofV IW˛ .R/ to be avoided, and is ˇ X @ ˇ the key to the efficient use of analytic gradient techniques in JI 0 kQW ‰J0 .rI R/ˇˇ‰M .rI R/ 0 .R/ D ˛ Wˇ the evaluation of f JI .R/. @W ˛ r M * + Differentiating Eq. (33.41) with respect to W˛ gives ˇ ˇ @ 0 0
0 ‰M .rI R/ˇˇ ‰ .rI R/ H EI0 .R/ V IW˛ .R/ @Wˇ0 I r X I @ 0 MI 0 D fWMJ f : (33.49) E .R/ c .R/ : (33.54) H D 0 Wˇ I ˛ @W˛ M J with the (in principle infinite) summation over states trun- Taking the inner product of Eq. (33.54) with c .R/ gives cated to reflect only the states explicitly treated in the CI JI fW˛ .R/ c J .R/ V IW˛ .R/ (33.55) nonadiabatic dynamics. Thus it is sufficient to discuss the @H 0 .R/ I 0 determination of fWJI˛ .R/. D EIJ .R/1 c J .R/ c .R/ (33.56) @W˛ The use of analytic derivative theory [97, 98] greatly im0 .R/1 hJI (33.57) proves the computational efficiency of evaluating fWJI˛ .R/ EIJ W˛ .R/ :
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Observe that Eq. (33.56) and Eq. (33.57) are not the Hellmann–Feynman theorem [101, 102] (Chap. 55), to which they bear a formal resemblance, since it is not the Hamiltonian operator H 0 .rI R/ but rather the Hamiltonian matrix H 0 .R/ that is being differentiated. Since the energy gradient has the form [44] @H 0 .R/ I @ EI0 .R/ D c I .R/ c .R/ ; @W˛ @W˛
(33.58)
H oo
0
1 r ij r ij pj p i ˛ 2 X p i pj A; D @ 4 jr ij j jr ij j3 j ¤i
(33.69) H ext D
˛2 X i˛ 2 X E .r i / p i si C E .r i / pi 2 i 4 i X C 2 H .r i / si ; (33.70) i
its relation to CIfWJI˛ .R/ is clear. This identification is the key E .r/ and H .r/ are electric and magnetic fields, and D step in the evaluation of CIfWJI˛ .R/ using analytic gradient e„=.2me / is the Bohr magneton. The physical significance of these terms is discussed is Chap. 22.1. One of the most techniques [97, 98]. important consequences of these relativistic effects is that total electron spin is no longer a good quantum number as it is 33.3.2 Breit–Pauli Interactions in H 0 .rI R/. The term H SD couple states corresponding to distinct eigenvalues of S 2 and lead to the nonadiabatic effects For light systems, it is possible to introduce relativistic ef- that are the subject of this section. fects using the Breit–Pauli approximation [1], in which the The Breit–Pauli approximation is most useful for light four component Dirac description of a single electron is re- atoms, but the approximation breaks down when Z beplaced by a two component (˛; ˇ) description. The H e .rI R/ comes large [104–106]. One of the principal effects omitted becomes [1, 103] in a treatment which includes only H SD is the relativistic contraction of the molecular orbitals [107] due to the (33.59) mass-velocity operator (H mass ), an effect whose importance H e .rI R/ D H 0 .rI R/ C H rel .rI R/ ; increases with Z. Several approaches exist which attempt to where, in parallel with Eqs. (33.4)–(33.13) for the atomic P correct this situation while retaining the spirit and simplificase, the relativistic correction H rel D 3kD1 H k .rI R/ can cations of the Breit–Pauli approximation. The first of these be divided into spin-dependent (SD) and spin-independent is the relativistic effective core potential (ECP) approxima(SI) parts, plus an external field interaction term H ext . These tion [108–111]. In this approach, the results of an atomic are given by Dirac–Fock calculation [112] are used to replace innermost (33.60) or core electrons of a given atom with (i) an effective one H 1 D H SD H so C H soo C H ss ; electron potential that modifies the electron-nuclear attracH 2 D H SI H mass C H D C H ssc C H oo ; (33.61) tion term in H 0 and (ii) an effective one electron spin-orbit 3 ext (33.62) operator, so that H DH ; H e .rI R/ ! H eECP .rI R/
where, in atomic units, H
so
˛ 2 X ZK .r i RK / p i si D ; 2 K;i jr i RK j3
D H 0ECP .rI R/ C H soECP .rI R/ : (33.63)
˛ 2 X r ij p i .si C 2sj / ; (33.64) ˇ ˇ3 ˇr ij ˇ 2 i ¤j X si sj 3.r ij si /.r ij sj / D ˛2 ; (33.65) jr ij j3 jr ij j5 i 4 / [29, 32, 33]. In either case, asymmetry correlation is done in Fig. 34.10 between Oh subgroup species fAg ; : : : ; T2u g and supergroup S6 species labeled by six-box spin-1=2 Young tableaus of 1 or 2 rows. Pauli–Fermi antisymmetry matches each spin tableau to its conjugate spatial tableau of 1 or 2 columns. Each matched pair belongs to a total nuclear spin of I D 0 (singlet state), I D 1 (triplet), I D 2 (quintet), or I D 3 (septet). This is correlated with one or more of the seven allowed Oh subgroup species fEu , T1g , T1u , T2g , A1g , A1u , A2u g, and three species Eg , T2u and A2g are forbidden. They would be allowed if the F-spin were greater than 1=2. If the F-spin is less, i.e., zero, then only A1g is allowed. Case 1 hyperfine SF6 spectra are sketched in the center and lower right-hand side boxes of Fig. 34.9d,e, as was done for CF4 in Fig. 34.7d,e. Case 1 has greater superfine cluster splitting between Oh species than spin multiplet splitting within them as seen by comparing these boxes to Fig. 34.10. The lowest A2 boxes hold an (I D 1) triplet and (I D 3) septet for spin weight 10 below A2u in Fig. 34.10. Above that lie T2g boxes with an (I D 0) singlet and (I D 2) quintet (weight 6), Eu with an (I D 1) triplet and (I D 2) quintet (weight 8), T1g or T1u triplet boxes, and finally A1g or A1u singlets. To form an SF6 molecule requires considerable spontaneous symmetry breaking to make six F atoms, free to undergo (at the very least) 6Š permutations and settle into O symmetry around a sulfur atom that has only 24 permutations. (A full Oh S6 correlation valid for any Fermi or Bose XY6 octahedral molecule is given in [32]) Thus, SF6 spectra in Fig. 34.9 involves a more severe symmetry downfall than that of CF4 in Fig. 34.7. This kind of downfall amounts to a breakdown of Herzberg’s rule concerning rovibronic spin species [2, 3]:
transitions between species are very strictly forbidden since the nuclear moments are so very slight. In other words, if subkilohertz interactions are negligible between states separated by MHz or GHz, then resonant mixing of different species is unlikely. However, it was only found later how symmetric molecules have tight level clusters that force subkilohertz near-degeneracy such as 6 1012 Hz for the A1 T1 E cluster n4 D 88 in Fig. 34.9c. Here, the exponential die-off of intracluster tunneling causes Case 2 superhyperfine mixing of species in the range from n4 D 88 down to about n4 D 78 where subkilohertz splitting begins. Also, on the right-hand side of Fig. 34.9c, the C3 clusters labeled by n3 D 88 and 87 may also fall into Case 2. When Oh species levels such as the A2 T2 E cluster n4 D 86 D 2 mod 4 in Fig. 34.9c are crushed together, the corresponding energies and states mix and reorganize as shown in Fig. 34.11. With O S6 symmetry of Case 1, all 6 O-axes fx xy N yz N zg N are equivalent. The three cluster levels are labeled by Oh symmetry species A2u (lowest J -tunneling energy: H 4S), Tg (middle: H ), Eu (highest: H C 2S), and the sixbox S6 tableaus correlated by Fig. 34.10 columns A2u , T2g , and Eu . The columns match each Oh to one or two spin multiplets. A2u is the winner with the highest hyperfine weight of 10: .I D 3/-septet C .I D 1/-triplet that gives a Bordé A2u spectra like the one shown in the lower right-hand side of Fig. 34.11. (Note that the Bordé spectral curve is a derivative of the seven peaks. Ideally, seven zeroes of that curve precisely locate septet peaks.) The middle Case 1 T2g has the lowest spin weight of 6: .I D 2/-quintetC.I C0/-singlet with an easily read Bordé spectrum of 5 peaks and central peak doubled. The top Eu has a spin weight of 8: .I D 2/-quintetC .I D 1/-triplet with a more confusing Bordé spectrum, possibly due to onset of Case 2 mixing that happens if ATE splitting S is approaching that of hyperfine multiplets. When the 6-axis J -tunneling rate S goes to zero, the SF6 becomes stuck to rotating on single-axis pair fz zg. N So four Fatoms rotate around the equator, while the other two take up a more sedate residence at North (z) and South (z) poles. Such restrictions of freedom or symmetry breaking is indicated symbolically by literally breaking the six-box tableaus of S6 into four-box and two-box pairs labeling subgroups of S4 ˝ S2 or D4 ˝ C2 . Each orbit tableau is Fermi-matched to a conjugate spin tableau (Fig. 34.10) as shown in Case 2 level side of Fig. 34.11. The whirling four-box nuclei enjoy greater spin-rotation coupling than their sedate two-box cohorts. To flip one of their four spins from # to " takes more energy than to flip a two-box spin. (The Bordé spectra in Fig. 34.11 indicate about four times more.) So raising #### to """" costs the entire width of superhyperfine pattern, while raising ## to "" costs only a triplet width. The scalar paired two-box spins l and four-box spins ll are invariants.
34
Molecular Symmetry and Dynamics
523
Eu B1u Eu B1u
(J + 2) –
(J +1) – (J +1) – J
–
E spin weight 3 + 5 = 8 E
J– (J – 1) –
One down
(J – 1) – (J – 2) –
Two down
(J + 2) +
β
T2 spin weight 1 + 5 = 6
T2g B2g (J +1) + J+ J+ T2g B2g (J – 1) +
β
T2
(J – 2) +
CASE 2 S→0 a = 1/6 s≈0≈t
CASE 1 S = 13 a = 1/6 s≈0≈t (J + 3) – A2u B1u (J + 2) –
24 cluster weight 8 + 6 + 10 = 24 3 (56) E8F12 2 A2
0
75.00 kHz
(J +1) – (J +1) – A2u B1u J– J– (J – 1) –
A 2 spin weight 3 + 7 = 10 A2 part of Q(28), n3 = 27 cluster at 28.464 691 25 THz
A2
(J – 1) – (J – 2) –
A2u1S6 =
(J – 3) –
–33
+ 0 33 Relative frequency (kHz)
Fig. 34.11 Case 1 S6 OfA1 T2 Eg levels mix to become Case 2 C4 ˝ C2h superhyperfine levels [9, 33]
34.5.4
Icosahedral Spherical Top Molecules: Extreme Spin Symmetry Effects
Yh . A semiclassical approach to rotational symmetry and dynamics is useful here since the rotational quantum constant is so small for the fullerenes (for C60 2B D 0:0056 cm1 or Until the recent discovery of fullerenes and the structure 168 MHz) [36–40]. of virus capsids, molecules with the highest icosahedral 2 HRot symmetry were thought to be rare or nonexistent in naSF6 D BSF6 J # "r r ture [34, 35]. With the discovery of Buckminsterfullerenes 7 .4/ 5 .4/ .4/ T C T4 C : : : C t044 T C (Buckyballs) C60 with molecular point symmetry Yh or Ih , 12 0 24 C4 one expects extreme dynamical symmetry effects. It helps (34.19a) to compare Yh -symmetric C60 to the Oh -symmetric SF6 first 2 by their symmetry-allowed rotation Hamiltonians of lowest HRot C60 D BC60 J # "r r rank [36–38] 11 .6/ 7 .6/ .6/ an extreme example of symmetry breaking effects, conT C T5 C : : : C t066 T C 25 0 25 C5 sider the Buckminsterfullerene or Buckyball molecule C60 , which has the highest possible molecular point symmetry
(34.19b)
34
524
Units of ¢J 6² 0.6
W. G. Harter and T. C. Reimer J = 100 K = 100 = 05
A 0.59750944939777 686 T1 . H . 580 T3 . 539
c C5 cluster
I=0 allowed
G
eigenlevels (15) A
0.6 K = 99 = 45
H G
H 0.25818069157638 5724525 T3 . G . 4991467
K = 96 = 15
K = 95 = 05
K = 94 = 45
0.6 K = 93 = 35 K = 92 = 25
–0.2 K = 91 = 15 K = 90 = 05
θ5cutoff
G 0.16832299606089 H . 80549983 39438180 T1 . I=0 A 0.088926413230 allowed 23871508 T1 . H . 20895138 19758132 T3 . G 0.0193551628 H . 433759 T1 . 179129 H –0.040906232 . 1044025 T3 G . 1073766 H –0.09217775 T3 . 77787 G . 90505 G –0.134590 H . 5469 T1 . 7470 A –0.16499
J = 0 mod 10 = 0, 10, ...
23
A
G
H
15
J = 4 mod 6 13 = 4, 10, 16, ...
J = 1 mod 10 = 1, 11, ...
T1
J = 5 mod 6 = 5, 11, 17, ...
13
35
H
J = 1 mod 6 = 1, 7, 13, ...
03 J = 0 mod 6
J = 8 mod 10 = 8, 18, ...
= 0, 6, 12, ...
45
T1
J = 9 mod 10 = 9, 19, ...
3-fold axis
J = 6 mod 10 = 6, 16, ...
K = 100 = 13
I=0 allowed
G
23
03
G G
15
T3
J = 2 mod 6 = 2, 8, 14, ...
J = 3 mod 6 = 3, 9, 15, ...
H
2-fold axis (saddle point)
–0.2
Superfine structure
T1
10 45 J == 44,mod 14, ...
T3
05
K = 99 = 03
Fine structure
–(2+ G+)S
05
H
H –0.17160 T3 –0.17325 G –0.1925 H –0.1954
C3 clusters
G
–(3 – G+)S –√5S
and C3 (inside) J = 5 mod 10 = 5, 15, ...
J = 7 mod 10
25 = 7, 17, ...
H
T3
25 J = 2 mod 10 = 2, 12, ...
35 J = 3 mod 10 = 3, 13, ...
d C3 cluster
I=0 T1 –0.16790 allowed
SEPARATRIX
(2 – G+)S
b T [6] species sequence ring C5 (outside)
= 26.6° 5-fold axis
K = 98 = 23
–0.2
T1
T3
(35)
√5S
(1+ G+)S
θ5 = 31.72°
θ 3 = 37.38°
θ 3cutoff = 10.8°
0.6
–0.2
(45)
a T [6] rotational energy surface K = 97 = 25
0.6
H
H 0.35917781105465 0903445 T3 . G . 0860308
–(3 – G+)S –√5S
G
–S –√5S
T3
(1+ G+)S
T3
(25)
0.6
0.6
H –(2+ G+)S
√5S
H
(05)
(2 – G+)S
T1
5S
T1
G 0.47203522166016 H . 62548 55067 T1 .
C5 clusters K = 98 = 35
√5S
H
eigenlevels
T1 H T3 G H
H T3
–0.210619 –0.219038 –0.222761 –0.225480 –0.230000 T3 G G H T1 A
–0.268730 –0.268956 –0.270745 –0.271551 –0.272490 –0.273042
T1 –0.33368009 H . 376797 T3 . 382899 G . 392051 H . 400540
T1
T3
(03)
G G H T1 A
√5S 2S
G
(13)
H
(1+ G+)S {(–1+ √13)/2}S (2 – G+)S –S –{(1+ √13)/2}S
0 –S
T1
–√5S –3S
H T3 (23)
G H
(1+ G+)S {(–1+ √13)/2}S (2 – G+)S –S –{(1+ √13)/2}S
Fig. 34.12 J D 100 Yh -symmetry sublevels. 12 C60 allows only the 4 A-species. 13 C60 allows all. (a) TŒ6 RES of 26:6ı C5 hills or 10:8ı C3 valleys hold (b, c) many C5 levels, and (d) few C3 levels
J D 30 eigenvalues of HRot SF6 in Fig. 34.4 are to be compared Rot to J D 100-values of HRot C60 in Fig. 34.12. Six C4 hills of HSF6 RES (rotational energy surface) compare to 12 C5 hills on Rot HRot C60 RES, and 8 C3 valleys of HSF6 RES compare to 20 C3 Rot valleys on HC60 RES. The C3 cut-off angle 3cut-off D 10:8 fl Rot ı (34.19c) for HRot C60 is half of 19:5 for HSF6 . So the huge J D 100 man-
Both have large rotor inertia I and a small rotor constant B D 2I1 . (For C60 : 2B D 0:0056 cm1 or 168 MHz. For SF6 : 2B D 0:091 cm1 ) [39, 40]. Both use Wigner–Eckart form for tensor matrix elements, ˝J ˇ .k/ ˇJ ˛ kJJ ˇ ˇ K 0 Tq K D CqKK 0 .J jjk jjJ / I
34
Molecular Symmetry and Dynamics
525
ifold has room for only two C3 clusters. A first comparison of SF6 and 13 C60 assumes fluorine, and 13 C nuclei causes no hyperfine splitting. (Here, we ignore Fig. 34.11 and assume that C60 rotates slowly with 13 C magnetic moments too weak to observe.) Just as order is maintained in a sequence of SF6 species between C3 valleys and C4 hills in Fig. 34.4, so also is the order of 13 C60 symmetry species maintained as C3 valley clusters morph into C5 hill clusters on either side of the separatrix in Fig. 34.12b. While SF6 or 13 C60 level sequence is maintained, their intra and intercluster level spacing is not. The C3 valley intracluster level spacing in Fig. 34.12c is quite different from the C5 hill intracluster level spacing in Fig. 34.12d, and similarly for SF6 cluster level eigenvalues in Table 34.5. Intracluster (superfine)eigenvalues have a quasi exponential tunneling “sneak” factor S, which was first defined for the asymmetric rotor by Eq. (34.13). It depends on local angular momentum mn of each Cn cluster and J -momentum cone polar angle J as diagrammed in Fig. 34.5. #m n J D acos p #m n
mn J.J C 1/
:
(34.20)
Asymmetric-rotor eigenvalues in Table 34.1 and CF4 or SF6 eigenvalues in Table 34.5 have numerical rationality and contrast withpthe rationality of C60 Golden ratios (GC D .1 C p 5/=2 and 13) of superfine eigenvalues in Fig. 34.12c,d. Rational frequencies give Poincaré periodicity, while irrational ones give quasi periodicity. However, the discussion of the similarity of SF6 and 13 C60 ends when the hyperfine structure of the latter overwhelms that of the former. This is revealed in the following section.
95, K D n5 D 90, and finally K D n5 D 99, in the second to lowest cluster in the C3 valley. The upper three are K D 0mod-5 D 05 waves orbiting C5 hills. K D n3 D 99 is the only 0-mod-3 wave in the tiny C3 valley of the RES and, thus, the valley’s only 12 C60 -allowed level in Fig. 34.12. Clearly, 12 C60 has a very sparse fine structure. This helps spectroscopists to assign lines and determine 12 C60 constants that will approximate those of 13 C60 . (If 12 C60 has molecular weight of 720 atomic units, then 13 C60 weights in around 780.) Recently (as this article was being written), the very first J-resolved 12 C60 spectra was observed in its 8.4 µm2 bands by Jun Ye’s group at JILA [44] that pioneered direct frequency comb spectroscopy. In this work, they used cavity-enhanced comb spectroscopy along with buffer gas cooling. [45, 46] This followed 27 years of failed attempts by other labs around the world. Now it suggests that a “Mt. Everest” of molecular spectroscopy, namely the 13 C60 spinmonster, might be attainable. There is both good and bad news concerning this. The good news: Fermi-symmetric 13 C60 has only ten times as many rotating spin- 12 nuclei as SF6 . The bad news is that the factor-of-10 lies in exponents! So, 13 C60 has about 260 hyperfine states or about 1:151018 spin states distributed among its ten species [32], as listed in Fig. 34.13. That is in contrast to Bosesymmetric 12 C60 , which has only a handful of A-states in each J -multiplet. Yet a single neutron entering 12 C60 to form 12 C59 13 causes many of the excluded levels to arise under lower Ch symmetry labels. SF6 spin weights f8; 3; 3; 6; 1; 1; 10g in Oh S6 correlation of Fig. 34.10 fit a sum identity of Oh irep dimension a
34.5.5
12
13
C60 versus C60 : a World of Difference in Spin- 21 Hyperfine Spectroscopy
b
Ag 9,607,679,885,269,312
= w(Ag)
0 1 2 3 4 5 6 7 8 9 10 1112 131415
Au 9,607,678,793,631,424
~ w(Ag)
A –30 –15
0
15
30
Total spin mz
SF6 has one stable F-isotope. C60 chooses from two staT1g 28,823,036,970,926,496 ~ 3w(Ag) T1 0 1 2 3 4 5 6 7 8 9 10 111213 141516 ble C isotopes: 12 C60 (spin 0) of 98.9% abundance or 13 C60 T1u 28,823,037,990,981,216 ~ 3w(Ag) 1 12 (spin- 2 ) at 1.1%. C spectroscopy is simpler than that of mz = –12 mz = +12 13 C. (The 14 C spin-0 isotope is even trickier, with its 5707T3g 28,823,036,970,926,496 ~ 3w(Ag) T3 year half-life ˇ-decay into 14 N.) It helps to compare purely T3u 28,823,037,990,981,216 ~ 3w(Ag) 12 13 bosonic C60 with purely fermionic C60 . In between are many mixed 12 Cn 13 C60n isotopomers. The first 12 C59 13 C is Gg 38,430,716,856,193,728 ~ 4w(Ag) G 33% likely and is an extreme example of isotopic symmetry G u 38,430,716,784,610,624 ~ 4w(Ag) breaking [41]. A single extra neutron reduces the very highest 3-D point symmetry Yh (120 operators) down to a lowly Hg 48,038,396,740,938,240 ~ 5w(Ag) H Ch (one planar reflection) [42, 43]. Hu 48,038,395,577,718,272 ~ 5w(Ag) 13 After comparing levels of SF6 and C60 above one should also compare SF6 (J D 30) levels in Fig. 34.4 to allowed 12 C60 (J D 100) levels in Fig. 34.12. Only four of its 201 Fig. 34.13 (a) Total spin weights and (b) mz -distributions for Yh S60 levels are Bose-allowed A-species: K D n5 D 100, K D n5 D symmetry species
34
526
W. G. Harter and T. C. Reimer
.` / times each of its spin weights. This equals its total spin dimension 26 D 64.
Œ
zero otherwise; U(2) fractions (m D 2) derive factors `S6 D f1; 5; 9; 5g in Eq. (34.21).
8`Eu C 3`T1u C 3`T1g C 6`T2g C 1`A1g C 1`A1u C 10`A2u D 16 C 9 C 9 C 18 C 1 C 1 C 10 D 64 D 26 :
Œ
`S6 D f1; 5; 9; 5g n T2g A2u T1g T1u Eu Eu 2u D `A Oh ; `Oh C `Oh ; `Oh C `Oh C `Oh C `Oh ; o A T2g A1u : (34.24) C ` C ` `O2g O O h h h
(34.21)
A similar identity uses products of U(2) spin- 12 dimensions (2S C 1) with ireps of group S6 whose tableaus match This is consistent with Eqs. (34.22), (34.23a), and U(2) tableaus that belong to each total spin S D 3; 2; 1; 0 (34.23b), and the Oh S6 correlation table in Fig. 34.10, if 1 made by six spin- 2 nuclei. we recall that each Fermi spin tableau has to be the transpose of its rotor tableau factor in order to make the FermiDirac-Pauli distribution a totally antisymmetric spin-rotor `S6 C `U.2/ `S6 26 D `U.2/ state. The hook-length formulae structure in Eqs. (34.23a) and (34.23b) shows that a tableau has the same dimensions as its transpose for both Sn and U(m). C `U.2/ `S6 C `U.2/ `S6 As powerful as tableau formulae are, they do not quite D .2S C 1 D 7/ 1 C .2S C 1 D 5/ 5 prepare us for the huge tableau dimensions and correlations C .2S C 1 D 3/ 9 C .2S C 1 D 1/ 5 : associated with 13 C60 and its permutation symmetry S60 con(34.22) taining its icosahedral symmetry subgroup Y S . The h 60 analogous spin weights in Fig. 34.13 for a Yh S60 correTableau hook-length formulas give the irep dimension for lation are truly enormous numbers when compared to the Sn and for U(m) ([15, p. 65]). analogous Oh S6 numbers in Fig. 34.10. Most C60 spin states have no Y orYh symmetry. Applying all 60 Y operators to a typical state generally gives 60 orthogonal spin 15Š 1 states spanning 60-by-60 regular representation and reduc`S15 D (34.23a) ing to 1A ˚ 3T1 ˚ 3T3 ˚ 4G ˚ 5H that obeys its group order hooklength 8 7 5 3 1 7 5 3 1 sum relation. product: 5 3 1
ı
2 1
integer product:
D 12 C 32 C 32 C 42 C 52 D 60 :
5 6 7 8 9
D
2 3
hooklength product:
(34.25)
4 5 6 7 3 4 5
`U5
Y D .`A /2 C .`T1 /2 C .`T3 /2 C .`G /2 C .`H /2
(34.23b)
8 7 5 3 1 7 5 3 1 5 3 1 2 1
The hook length h of a tableau box is the number nr of boxes to its right plus the number nb below it plus one for itself, that is, h D nr C nb C 1. The numerator for Sn formula in Eq. (34.23a) is nŠ (here n D 15). The denominator has exactly n-boxes with one hook length for each particle. The same denominator applies to U(m) irep dimension and Eq. (34.23b) for an n-particle-m-state system (here m D 5), but the numerator fills the same tableau frame with a matrix whose corner diagonal is (m; m; m; : : :), and ˙1st off-diagonals are ˙.m ˙ 1; m ˙ 1; m ˙ 1; : : :/, ˙2nd offdiagonals are ˙.m ˙ 2; m ˙ 2; m ˙ 2; : : :/, etc. The Eqs. (34.23a) and (34.23b) fractions always yield a positive integer if the desired representation exists and
So Yh species weights in Fig. 34.13 are (to one part in 106 ) proportional to the irep dimension. Thus, the approximate weight ratios are .A W T1 W T3 W G W H / D .1 W 3 W 3 W 4 W 5/ with relatively tiny ratio variations between odd (u) and even (g) parity species. However, as Fig. 34.13 shows, the arithmetic variations are in the billions! This is in contrast to SF6 , where the u and g ratios of Oh species differ markedly and even leave the three species A2g , Eg , and T2u with zero weight. Plots of species weight versus spin z-component .30 mz 30/ are shown in Fig. 34.13b. The height of each vertical line is proportional to the number of a given species A; T1 ; T3 ; G, or H that have the mz -value that line represents. Each species has its tallest line at mz D 0 and then falls quasi exponentially on either side. They become too small to plot for mz -values outside the interval .12 mz C12/ and thereafter are just lines of dots. The dots for scalar icosahedral A-species at extreme values of mz D C30 or mz D 30 are lines of height `Œ60;0 D 1 for the S60 tableaus below.
34
Molecular Symmetry and Dynamics
527
Fig. 34.14 Comparison of possible P(50) spectral structure for 12 C60 and 13 C60
Possible C60 rovibrational structure Q-branch
R-branch
P-branch
13
C60
80
70
60
50
40
30
10 N = 0 10
20
20
30
40
50
60
70
80
90
12
C60
AT1HT3 12 Relative
12 GHT1
cluster intensities
Fine structure
12 HT3G
12 HT3G
12 H G T1
13
C60
45
05
35
25
15
H 20 H T1 T3G A 05 or 03
Local symmetry clusters C5 {12-fold}
13
T1HT3GH 20
23
C5 {20-fold} 12
C60
Relative peak heights
Possible hyperfine splittings (may exceed superfine splittings)
5
Superfine structure
A
3
3 13
C60
1 H
T1
A
T3
Icoshedral symmetry species 12
C60
A
Dots for other species are zero at maximal spin momentum jmz j D 30. "
"
"
"
1 #
"
"
"
"
"
5 #
#
#
#
"
"
"
"
"
10 #
#
#
#
#
"
"
"
"
"
15 #
#
#
#
#
"
"
20 #
#
#
#
#
"
#
"
"
"
"
"
45 #
#
"
"
"
"
"
50 #
#
#
#
#
#
#
#
#
The opposite extreme is the spin tableau [30; 30] for totally paired-up zero-spin .S D 0 D mz /. "
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
#
(34.27)
"
"
"
"
55
(34.26)
"
"
mz D 0
#
"
mz D 30
60 #
#
#
#
#
mz D 30
34
528
W. G. Harter and T. C. Reimer
The S60 dimension `Œ30;30 for this zero-spin tableau is given zero. An asymmetric top with body-fixed spin is the following modified version of Eq. (34.1) by applying Eq. (34.23a) `Œ30;30 D
HRCS.Body-fixed/ D AR 2x C BRy2 C C R 2z C HrotorS
60 59 58 57 2 1 60Š D .31Š/.30Š/ 31 30 29 28 3 2 30 29 28 27 2 1
C . 0/ :
(34.30a)
D 3:8149865020923 1015 :
(34.28) The system total angular momentum is a conserved vector J D R C S in the lab frame and a conserved magnitude jJ j That number of zero-mz states is apportioned by ratios 1 W in the rotor-R body frame. So we use R D J S in place 3 W 3 W 4 W 5 .˙0:0001%/ to species A, T1 , T3 , G, and H , re- of R, spectively (the exact ratio calculation is tricky and required 2
HR;S.fixed/ D A.J x S x /2 C B J y S y an erratum [42, 43]). If ever the resulting panoply of 13 C60 superhyperfine spectra is observed, it will dwarf that of SF6 C C .J z S z /2 C HrotorS in Figs. 34.9 and 34.10. A plot comparing J D 100 superfine D AJ 2x C BJ y2 C C J 2z 2AJ x S x levels of 13 C60 to those of 12 C60 in Fig. 34.12 only begins to 0 2BJ y S y 2C J z S z C Hrotor : (34.30b) relate to SF6 levels in Fig. 34.4. An attempt to relate specS 13 12 tra of C60 to C60 is sketched by Fig. 34.14 as a highly speculative P(50) prognostication where only two A-singlet Gyro-spin components Sa are treated at first as constant clas12 C60 lines exist while 13 C60 has seven large clusters. The sical parameters Sa , .K D 50/05 .AT1 H T3 / cluster is a Case 1 guess for superfine HR;S.fixed/ D const 1 2ASx J x 2BSy J y 2CSz J z structure of astronomical numbers of hyperfine lines. The C AJ 2x C BJ y2 C C J 2z resolution assignment of such super-hyper-spectra will reX X quire a wealth of future technology and theory. D M0 T 0 C Dd T 1 C Qq T 2 : (34.30c) 0
d
d
q
q
This is a simple Hamiltonian multipole tensor operator expansion having here just a monopole T 00 term, three dipole 1 2 So far, the discussion has focused on Hamiltonians and in- T a terms, and two quadrupole T q terms shown in Fig. 34.15. volved functions of even multipolarity, i.e., constant .k D 0/, quadratic or quadrupole .k D 2/, quartic or hexadecapole a Monopole b Dipole c Quadrupole .k D 4/, while ignoring functions that are linear-dipole 1 T x .k D 1/ or cubic-octupole .k D 3/, for reasons of timeT 20 T 00 reversal symmetry. However, for composite “rotor-rotors” any multipolarity is possible, and the dipole is of primary utility. A composite rotor is one composed of two or more obT 1y jects with more or less independent angular momenta. This could be a molecule with attached methyl (CH3 ) “gyro” or 2z2 – x2 – y2 “pinwheel” subrotors, a system of considerable biological interest. It could be a molecule with a vibration or “phonon” excitation that couples strongly to rotation. Also, any nuclear T 1z or electronic spin with significant coupling may be regarded as an elementary subrotor. The classical analogy is a spaceT 2+2 – T 2–2 craft with gyro(s) on board. A rotor-rotor Hamiltonian has the following general interaction form
34.6 Composite Rotors and Multiple RES
Hrotor RCS D HrotorR C HrotorS C VRS :
(34.29)
√3 (x2 – y2)
A useful approximation assumes that the rotorS gyro is fastened to the frame of rotorR so the interaction VRS be- Fig. 34.15 The six lowest-order RES components needed to describe comes a constraint, does no work, and is thus assumed to be rigid gyro rotors
34
Molecular Symmetry and Dynamics
529
Each is a radial plot of a spherical harmonic function Yqk .'; #/ representing a tensor operator T kq T 00 D
J 2x
C
J y2
C
J 2z
3
T 1 CT 1 T 1x D J x D C1p 1 2 1 T1 T T y1 D J y D C1p 1 i 2 1 1 T z D Jz D T 0 T 2zz D T 2x 2 y 2
D
2J 2z
J 2x 2
J 2x
J y2
J y2
Precessing J vector
(34.31a)
S
Linear harmonic precession spectra
Lowest RE for gyro-rotor at North pole fixed point
(34.31b)
D T 20
2 T 22 T 22 : D p 6
(34.31c)
Highest RE for gyro-rotor at South pole fixed point
Fig. 34.16 The spherical gyro-rotor RES is a cardioid of revolution
The constant coefficients or moments indicate the strength of around gyro spin S each multipole symmetry, jRj D jJ S j is least, and rotor kinetic energy BR2 is least. 0 (34.32a) M0 D A C B C C C 3Hrotor S (Magnitudes jJ j and jSj are constant here.) The spherical Dx D 2ASx ; rotor-gyro RES in Fig. 34.16 is minimum along the bodyaxis CS and maximum along S , where BR2 is greatest. Dy D 2BSy ; As is the case for the rigid solid rotors in Figs. 34.1 Dz D 2CSz (34.32b) and 34.2, the RES energy topography lines determine the Qzz D .2C A B/=6 precession J -paths in the body frame, wherein gyro-S is Qx 2 y 2 D .A B/=2 ; (34.32c) fixed in Fig. 34.16. The left-hand rule gives J -precession sense in the body S -frame, i.e., all J precess anti-clockwise The scalar monopole RES in Eq. (34.32a) is a sphere, while relative to the “low” on the CS -axis, or clockwise relative to vector dipole RES in Eq. (34.32b) are bispheres pointing the “high” on the S -axis. In the lab, S precess clockwise along Cartesian axes, and the RES in Eq. (34.32c) resemaround a fixed J . ble quadrupole antenna patterns. Also, Fig. 34.15a–c plots Gyro-RES differ from solid rotor RES, which have two the six s, p, and d Bohr–Schrödinger orbitals that are analogs opposite “highs” and=or two opposite “lows” separated for the six octahedral J -tunneling states listed in Table 34.5. by saddle fixed points where the precessional flow direcThe asymmetric and symmetric rotor Hamiltonians in tion reverses, as can be seen in Fig. 34.2. The gyro-RES Eqs. (34.1) and (34.2) are combinations of a monopole in in Fig. 34.16 has no saddle fixed points and only one “high” Eq. (34.31a), which by itself makes a spherical rotor, and and one direction of flow with the same harmonic precesvarying amounts of the two quadrupole terms in Eq. (34.31c) sion frequency for all J -vectors between the high CS and to give the rigid rotor RES pictured previously in Figs. 34.1 low S -axes. This is because the spectrum of the gyro-rotor and 34.2. Both Q coefficients in Eq. (34.32c) are zero for Hamiltonian in Eq. (34.33) is harmonic or linear in the moa spherical top .A D B D C /, but only one is zero for a symmentum quanta K, metric top .A D B/. ˝J ˇ ˇJ ˛ ˇ ˇ (34.34) Combining monopole in Eq. (34.31a) with dipole terms in K H K D const: C BJ.J C 1/ 2BK : Eq. (34.31b) gives a gyro-rotor Hamiltonian in Eq. (34.30b) In contrast, even the symmetric rigid rotor spectrum in for a spherical rotor .A D B D C / that has the following form Eq. (34.4) is quadratic in K. Other rotors shown in Figs. 34.2 and 34.4 have levels that are quite nonlinear. H D const C BJ 2 gS J .where: g D 2A D 2B D 2C / I
(34.33)
H resembles a dipole potential m B for a magnetic moment m D gJ that precesses clockwise around a lab-fixed magnetic field B D S . (The PE is least for J along S .) Here, the Hamiltonian in Eq. (34.33) is a simple example of Coriolis rotational energy. It is least for J along S , where
34.6.1 3-D-Rotor and 2-D-Oscillator Analogy Linear levels belong to harmonic oscillators, not rotors, but the gyro-rotor’s linear spectrum highlights a 160-year-old analogy of 3-D rotor motion to 2-D vibration. Stokes [47]
34
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W. G. Harter and T. C. Reimer
described 2-D electric vibration or optical polarization, by a 3-D vector that became known as the Stokes vector S and labeled appropriately by the letter S . (Now we say S labels “spin”.) Stokes’ spin uses Hamilton quaternions [48, 49] redone 80 years later as Pauli spinor components [50] of a general 2-D Hermitian operator H , ! A B iC H D B C iC D D
ACD AD 0 C A C BB C C C ; 2 2
where 1 0
! 0 ; 1 ! 1 ; 0
!
1 0 ; 0 1 ! 0 i C D : i 0
(34.35)
This easily gives Schwinger’s 3-D angular momentum raising-lowering operators J C D J B C iJ C D a1 a2 and J D J B iJ C D a2 a1 , where two dimensions 1 and 2 are spin-up .C„=2/ and spin-down .„=2/ instead of the x-and y-polarized states envisioned by Stokes. The angular 3-D ladder operation is replaced by a simpler 2-D oscillator operation J C jn1 n2 i D a1 a2 jn1 n2 i p p D n1 C 1 n2 jn1 C 1; n2 1i ;
J jn1 n2 i D a2 a1 jn1 n2 i p p D n1 n2 C 1jn1 1; n2 C 1i :
(34.38)
2-D oscillator states are labeled by the total number N D .n1 C n2 / of quanta and the net quantum population N D J .n 1 n2 /. 3-D angular momentum states jK i are similarly 0 B D labeled by the total momentum J D N=2 D .n1 C n2 /=2 and 1 z-component K D N=2 D .n1 n2 /=2, just half (or „=2) Our labels A (“asymmetric-diagonal”), B (“bilateral- of N and N , n1 n2 balanced”), and C (“circular-Coriolis”) are mnemonic al a1 a2 ternatives to Pauli’s z, x, and y, respectively. The 2-D D ; n p j0; 0i D jn1 2 i Hamiltonian has the S J form of the Coriolis coupling in n1 Šn2 Š J CK J K Eq. (34.33). a1 a2 ˇJ ˛ ˇ Dp (34.39) j0; 0i ; K H D S0 1 C SA JA C SB JB C SC JC .J C K/Š.J K/Š D S0 J0 C SE J ; (34.36) where where n1 D J C K ; n2 D J K : A B C ; JB D ; JC D ; J0 D 1 ; JA D From this, Schwinger [52] rederived the Wigner matrices 2 2 2 J .˛ˇ/ appearing in Eqs. (34.5) and (34.6) and the DMK and kJJ 0 Clebsch–Gordan CqKK 0 coefficient or Wigner–Eckart tensor relation. This helps clarify RES approximation such as S0 D .A C D/=2 ; SA D .A D/ ; SB D 2B ; Eqs. (34.10) and (34.11) that use .J; K/-cone levels on RES, SC D 2C : * + ˇJ 0ˇ J ˇ ˇ k kJJ 0 0 The 2-D–3-D analogy is helped by using elementary oscillaD CqKK ˇT ˇ 0 hJ kkkJ i 0 q K K tor ladder a a operators in Eq. (34.37),
J kJJ J #K : C0KK hJ kkkJ i DJK 1 a1 a1 a2 a2 ; J 0 D N D a1 a1 C a2 a2 ; J A D 2 1 i JC D JB D a1 a2 C a2 a1 ; a1 a2 a2 a1 ; 34.6.2 Gyro-Rotors and 2-D-Local Mode Analogy 2 2 (34.37) The 2-D–3-D analogy provides insight into NMR spin [54], where laser quasi spin (Rabi-rotation) [55], rovibrational dynam! ! ics [56, 57], and the local mode formation [58, 59]. It also 1 0 0 1 a1 a1 D ; a1 a2 D ; has computational value. Part of this involves relating single 0 0 0 0 2-D-oscillator (Stokes model) to a model of two 1-D oscilla! ! tors with coordinates x1 D x and x2 D y. 0 0 0 0 ; a2 a2 D : a2 a1 D Two identical side-by-side oscillators have bilateral or B1 0 0 1 symmetry, and a HB Hamiltonian commutes with both B This is a Jordan–Schwinger map of 2-D oscillation to 3-D (a C45ı mirror reflection of axes ˙x ˙y) and with B (a 45ı mirror reflection of axes x ˙y) both of which rotation [51–53]. 0 D
A D
34
Molecular Symmetry and Dynamics
Fig. 34.17 A spherical gyrorotor becomes an asymmetric gyro-rotor by adding T 20
a
531
b
Spherical gyro-rotor or normal ±B-modes (1) (1) T (0) 0 + Dy T y
−B FIXED PT. Antisymmetric normal mode
Symmetric gyro-rotor or local ±A-mode normal –B-mode
Symmetric normal mode becomes UNSTABLE
C (or y)
s
A (or z)
c
Perturbed gyro-rotor or “soft” +B-mode
s
B (or x) +B FIXED PT. Symmetric normal mode
interchange oscillators. This means that to first order the Hamiltonian is HB D 2BB, i.e., a gyro-rotor T 1x with S along the B-axis, as shown in Fig. 34.17. (Added T 00 affects eigenvalues but not states.) Eigenvectors of HB are the symmetric and antisymmetric normal modes that belong to the fixed points on the S -vector or ˙B-axes of the Stokes space. If instead, the S -vector lies on the A-axis, the Hamiltonian is an asymmetric diagonal HA D 2AA matrix. From Eq. (34.35), we can see that the operator A reflects y into y but leaves x alone, so eigenvectors of HA are localized on the x-oscillator or the y-oscillator, but not both. Such motions are local modes but are not modes of HB , which does not commute with HA . Hamiltonian HB precesses a J vector from the CA-axis (local x-mode) around to the C -axis to the A (local ymode), then to the CC -axis, and then home to CA. The J -path is the largest equator of Fig. 34.17a. The ˙C -axes are what Stokes would label circular polarization chirality, right and left, respectively. In a B-beat, ˙C belong to resonant transitions where one vibrator’s phase is 90ı ahead and pumping up the other. Simple mode beat transfer dynamics is disrupted by adding anharmonic T 20 or T 2˙2 terms to existing B-symmetry terms T 1x and T 00 , as shown in Fig. 34.17b,c. In molecular rotation theory, the T 20 terms along with T 00 make the initial unperturbed Hamiltonian in Eq. (34.3) of a symmetric top, and gyro terms T 1q are viewed as perturbations. For vibration theory, the latter make up a normal-mode Hamiltonian, and the former T 20 term is viewed as an anharmonic perturbation. The effect of T 20 , seen in Fig. 34.17c, is to replace the stable fixed point CB (representing the (C)-normal mode) by a saddle point as B bifurcates (splits) into a pair of fixed points that head toward the ˙A-axes. So one normal mode dies and begets two stable local modes, wherein one mass
(1) (1) (2) (2) T (0) 0 + Dy T y + Q 0 T 0
+Α FIXED PT. −Α FIXED PT. Local mode-1 Local mode-2
may keep its energy without losing it to the other through the usual B-beating process. (The A-modes become anharmonically detuned.) Pairs of classical modes that each localize energy on one side of an RES in Fig. 34.17 are analogous to asymmetric top ˙K-precession pairs in Fig. 34.2. However, one must consider more than the classical aspects of RES pictures. Quantum-tunneling Hamiltonians such as Eq. (34.15) give a superfine doublet for each trajectory pair with (˙)combination eigenstates (Table 34.1), and they occupy both paths just as each gyro-spin doublet would have J both up and down the A-axis in a quantum picture.
34.6.3 Multiple Gyro-Rotor RES and Rotational Energy Eigensurfaces (REES) Just as quantum rotor theory allows J to occupy multiple paths, so also may J occupy multiple RES. In this way gyro-rotors differ from the analogous J precession around a lab-fixed B field or around the body-fixed S of a classical gyro. By allowing the S to be a quantum entity, the possibility arises for distribution over multiple RES [60, 61]. A simple quantum theory of S allows both CS and S at once. The RES for each is plotted one on top of the other, as in Fig. 34.18a, while component RES are shown in Fig. 34.18b for CS and in Fig. 34.18c for S . An energy sphere is shown intersecting an RES pair for an asymmetric gyro-rotor. If the spin S is set to zero, the pair of RES collapse to a rigid asymmetric top RES shown in Fig. 34.2 having angular inversion (time-reversal J ! J ) and reflection symmetry. The composite RES in Fig. 34.18a has inversion symmetry but lacks reflection symmetry. Its parts in Fig. 34.18b,c have neither symmetry due to their bodyfixed gyro-spins ˙S .
34
532
W. G. Harter and T. C. Reimer
a
Composite ±S rotational energy surface
b
Jz
J
following J -dependent body frame components
c
Forward gyro-spin +S = (1, 1, 1)
˝x D dx sin ˇ cos ;
˝y D dy sin ˇ sin ;
˝z D dz cos ˇ :
R
(34.41d)
The quantum REES shown in Fig. 34.19 results from substituting quantum spin S D 12 matrices 12 x , 12 y , or 12 z for S each algebraic Hamiltonian factor Sx , Sy , Sz of Eq. (34.40) Jx –S R J to make a matrix Hamiltonian in Eq. (34.41a)–(34.41d) that is then diagonalized. The resulting pair of eigenvalues are Time reversed gyro –S = (–1, –1, –1) plotted to make a pair of REES functions of polar body frame angles ˇ and . In comparing the classical composite RES in Fig. 34.18a or Fig. 34.20 with the quantum REES Fig. 34.18 Asymmetric gyro-rotor RES (classical body-fixed spin in Fig. 34.19 one may note that they have similar shapes in case); (a) composite ˙S ; (b) forward spin ˙S ; (c) reversed spin S regions where classical ˙J -surfaces of Fig. 34.20 are well S
Jy
Gyro-rotor Hamiltonians in Eq. (34.40) allow tunneling or mixing of multiple RES. A two-state spin1=2 gyro-spin model has a 2-by-2 Hamiltonian matrix and two-base RES. Hgyro D M0 J J C Dx Sx J x C Dy Sy J y C Dz Sz J z C Qxx J 2x C Qyy J y2 C Qzz J 2z ; (34.40) As in Eq. (34.7), J is approximated by classical vector components in the body frame. Jx D jJ j sin ˇ cos ;
Jy D jJ j sin ˇ sin ;
Jz D jJ j cos ˇ :
(34.41a)
However, the gyro-spin S uses its quantum representation p S D jSj =2 D 3 =2 from Eq. (34.34) ˝
Fig. 34.19 Quantum gyro-rotor REES views based on eigenvalues of Eqs. (34.41a)–(34.41d) with S D =2
˛ Hgyro D M0 J 2 C Qxx Jx2 C Qyy Jy2 C Qzz Jz2 C Dx jSkx Jx C Dy jSky Jy C Dz jSkz Jz
! jSk.Dx Jx iDy Jy / h.J / C Dz jSkJz D jSk.Dx Jx C iDy Jy / h.J / Dz jSkJz 1 0 .dx cos idy sin / h.J / C dz cos ˇ C B C B sin ˇ C; DB B.d cos C id sin / h.J / dz cos ˇ C y A @ x sin ˇ (34.41b) where h.J / D M0 J 2 C Qxx Jx2 C Qyy Jy2 C Qzz Jz2 ; and d D D jSjjJ j :
(34.41c)
The semiclassical Hgyro makes spin expectation in hS i to Fig. 34.20 Classical gyro-rotor c-RES views of Fig. 34.18a based on precess around an angular velocity crank vector ˝ with the Eq. (34.40)
34
Molecular Symmetry and Dynamics
separated. Wherever the classical RES cross the quantum REES differ most differently due to their intersurface resonance. Each REES resembles a wrapped symmetric top RES that is most perturbed at points where the two classical RES cross the RES pair intersection along the horizontal x-axis (in figure plane) becomes an REES avoided crossing. The RES pair intersection along the y-axis (out of figure plane) becomes an REES diabolical point so-named after a toy top called a diablo. A current term for such a dual-cone singularity is Dirac point. Near-crossing RES are the rotational equivalent of nearcrossing vibrational-potential (VES) employed in treatments of Jahn–Teller effects [62, 63]. The classical, semiclassical, and quantum theory for such loosely bound or fluxional systems is a rapidly growing subfield of atomic, molecular, and optical physics.
34.6.4 Multi-Quantum CF4 Rovibrational Polyads and REES Visualization Molecular quantum data comes first in the form of spectral line frequencies that are correlated with theory derived from Hamiltonian energy operator eigenvalues or differences thereof. Second comes a much larger dataset that includes spectral transition rates and intensities that are correlated with eigenvector matrices or density operators. As more AMOP phenomena become available researchers are trying to understand this deluge of data and find ways to visualize it. Section 34.4 provides a theory for level clustering of octahedral and tetrahedral molecules and shows a way to visualize underlying eigenstates using rotational energy surfaces (RES). A possibility for visualizing ever more complex spectral phenomena involves rovibronic energy eigenvalue surfaces (REES) introduced in Sect. 34.6.3. An example involves the CF4 3 =24 rovibrational polyad. It begins with a plot versus the total angular momentum J D 0–70 in Fig. 34.21 of level cluster bands found in this polyad as calculated by Boudon et al. [64, 65]. The plot uses squares ./ as boundary points containing C4 -level clusters such as are shown in Fig. 34.4 (right) and C4 in Fig. 34.6c. Triangular points .4/ mark bands of C3 -clusters shown in Fig. 34.4 (left) and C3 -correlated in Fig. 34.6a. Figsure 34.21 also has C2 -cluster regions marked by diamond .Þ/ points where C2 -correlated species in Fig. 34.6 (centre) might appear. These are rare in Oh or Td as the RES have saddle points at the C2 -axes as in Fig. 34.4 so they are unstable. The asymmetric rotor in Fig. 34.2 is a prime example of an RES with unstable C2 ˙y-saddle points.
533
Quantum levels C4 axes C3 axes C2 axes
1300
REES-9 J = 57 levels and Cn clusters
1290 REES-8 1280
REES-7
REES-6 1270
1260
REES-3 to 5
1250
REES-2
REES-1 J = 57
1240 0
10
20
30
40
50
60
70 J
Fig. 34.21 CF4 Molecular rovibrational level clusters for 3 =24 polyad J D 0 to J D 70. Adapted from Boudon [64]. Energy units are 1 kayser (1 cm1 )
However, overtone and combination resonances give rise to the larger clusters that have lower local symmetry of C2 and C1 (no symmetry). The 3 =24 REES for (J D 57)-states has five of its nine shells showing in Fig. 34.22 [66]. The innermost and lowest REES shell is nearly spherical at the scale of Fig. 34.22 and corresponds to the lowest cluster levels in Fig. 34.21 that come up if expanded, and would reveal usual ordering C3 -below-C4 separated by C2 saddle as in Fig. 34.4. The shell above that has a cubic (as opposed to octahedral) shape with inverted ordering C4 -below-C3 clearly shown by levels in the next-to-lowest J D 57 cluster bunch in Fig. 34.21. The three highest REES shells shown in Fig. 34.22 involve a level-cluster mashup in the middle of Fig. 34.21. A pair of shells form diabolic points on C2 -axes and avoidedcrossing pairs of orbits on both C3 - and C4 -axes. Above that
34
534
W. G. Harter and T. C. Reimer
ΘJJ = 7.55° = acos
J √(J(J + 1))
J = 57 minimum uncertainty J-cone
C1 J-orbits
C1 ⸦ Td correlation C1 01 A1 1 A2 1 E 2 T1 3 T2 3 C1 J-orbits
Fig. 34.22 CF4 Molecular (J D 57) REES-1-to-5 for 3 =24 polyad levels plotted in Fig. 34.21 (adapted from [66])
is an extraordinary REES shell with 24 C1 orbits indicated by arrows emerging from their groups. This fifth shell also supports many C3 orbits, a few C4 orbits, and a bizarre saddle shape on the C2 -axes. The 24-cluster of C1 orbits belongs to species correlation 01 " O D A1 ˚ A2 ˚ E ˚ T1 ˚ T2 of a regular representation of the isomorphic groups S4 Td O that now uses the identity subgroup C1 D f1g. The relevant C1 , C2 , C3 , and C4 correlation tables are matched to corresponding cluster orbits in Fig. 34.22, while each of the 24 C1 orbits lack rotational symmetry, they each do maintain a local reflection symmetry Cv D f1; v g. At much higher J one should expect to find Oh clusters of 48 species levels, indeed, a majority of them in analogy to the majority of regular S60 spin states of unit S1 symmetry in the correlation table of Fig. 34.13.
References 1. Herzberg, G.: Spectra of Diatomic Molecules. Molecular Spectra and Structure, vol. I. Van-Norstrand-Reinhold, New York (1950) 2. Herzberg, G.: Infrared and Raman Spectra of Polyatomic Molecules. Molecular Spectra and Structure, vol. II. VanNorstrand-Reinhold, New York (1945) 3. Herzberg, G.: Electronic Structure of Polyatomic Molecules. Molecular Spectra and Structure, vol. III. Van-NorstrandReinhold, New York (1966)
4. Wilson, F.B., Decius, V.C., Cross, P.C.: Molecular Vibrations. McGraw-Hill, New York (1955) 5. Papousek, D., Aliev, M.R.: Molecular Vibrational-Rotational Spectra, Studies Phys. Theor. Chem. 17. Elsevier, Amsterdam (1982) 6. Zare, R.N.: Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics. Wiley, New York (1988) 7. Harter, W.G.: Principles of Symmetry, Dynamics, and Spectroscopy. Wiley, New York (1993) 8. Harter, W.G., Patterson, C.W.: J. Math. Phys. 20, 1453 (1979) 9. Harter, W.G.: Phys. Rev. A 24, 192 (1981) 10. Harter, W.G., Patterson, C.W.: J. Chem. Phys. 80, 4241 (1984) 11. Harter, W.G.: Comp. Phys. Rep. 8, 319 (1988) 12. Sadovskii, D.A., Zhilinskii, B.I.: Mol. Phys. 65, 109 (1988) 13. Sadovskii, D.A., Zhilinskii, B.I.: Phys. Rev. A 47, 2653 (1993) 14. Pavlichenkov, I.M.: Phys. Rep. 226, 173 (1993) 15. Harter, W.G., Patterson, C.W., daPaixao, F.J.: Rev. Mod. Phys. 50, 37 (1978) 16. Harter, W.G., Patterson, C.W.: Phys. Rev. A 19, 2277 (1979) 17. Bunker, P.R.: Molecular Symmetry and Spectroscopy. Academic Press, New York (1979) 18. Stein, A., Rabinowitz, P., Kalder, A.: Exxon Labs (1977) 19. McDowell, R.S., Galbraith, H.W., Reisfield, M.J., Aldridge, J.P.: Los Alamos T-12. In: Hall, J., Carlsten, J. (eds.) Laser Spectroscopy, vol. III, p. 102. Springer, Berlin (1977) 20. Hecht, K.T.: J. Mol. Spectrosc. 5, 355 (1960) 21. Lea, K.R., Leask, M.J.M., Wolf, W.P.: J. Phys. Chem. Solids 23, 1381 (1962) 22. Dorney, A.J., Watson, J.K.G.: J. Mol. Spectrosc. 42, 1 (1972) 23. Fox, K., Galbraith, H.W., Krohn, B.J., Louck, J.D.: Phys. Rev. A 15, 1363 (1977) 24. Harter, W.G., Patterson, C.W.: J. Chem. Phys. 66, 4872 (1977) 25. Monostari, B., Weber, A.: J. Chem. Phys. 33, 1867 (1960) 26. Pfister, O.: Etude esperimentale et theorique des interactions hyperfines dans la bande de vibration 3 de la molecule 28 SiF4 . Univ., Paris-Nord, Paris (1993). Dissertation, 27. Borde, J., Borde, C., Salomon, C., Van Lerberghe, A., Ouhayoun, M., Cantrell, C.: In: Walther, H., Rothe, K. (eds.) Laser Spectroscopy, vol. IV, p. 142. Springer, Berlin (1979) 28. Borde, J., Borde, C.: J. Mol. Spectrosc. 78, 353 (1979) 29. Butcher, R.J., Chardonnet, C., Borde, C.: Phys. Rev. Lett. 70, 2698 (1993) 30. Borde, C., Ouhayoun, M., VanLerberghe, A., Salomon, C., Avrillier, S., Cantrell, C., Borde, J.: Phys. Rev. Lett. 45, 14 (1980) 31. Kim, K., Person, W., Seitz, D., Krohn, B.: J. Mol. Spectrosc. 76, 322 (1979) 32. Harter, W.G.: Phys. Rev. A 24, 192 (1981) 33. Borde, J., Borde, C.: Chem. Phys. 71, 417 (1982) 34. Kroto, H.W., Heath, J.R., O’Brian, S.C., Curl, R.F., Smalley, R.E.: Nature 318, 162 (1985) 35. Kratschmer, W., Lamb, W.D., Fostiropolous, K., Huffman, D.R.: Nature 347, 354 (1990) 36. Harter, W.G., Weeks, D.E.: Chem. Phys. Lett. 132, 387 (1986) 37. Weeks, D.E., Harter, W.G.: Chem. Phys. Lett. 144, 366 (1988) 38. Harter, W.G., Weeks, D.E.: J. Chem. Phys. 90, 4744 (1989) 39. Weeks, D.E., Harter, W.G.: Chem. Phys. Lett. 176, 209 (1991) 40. Harter, W.G., Reimer, T.C.: J. Chem. Phys. 94, 5426 (1991) 41. Reimer, T.C., Harter, W.G.: J. Chem. Phys. 106, 1326 (1997) 42. Harter, W.G., Reimer, T.C.: Chem. Phys. Lett. 194, 230 (1992) 43. Harter, W.G., Reimer, T.C.: Chem. Phys. Lett. 198, 429 (1992). erratum 44. Changala, P.B., Weichman, M.L., Lee, K.F., Fermann, M.E., Ye, J.: Science 363, 49–54 (2019) 45. Spaum, B., Changala, P., Patterson, D., Bjork, B., Heckl, O., Doyle, J., Ye, J.: Nature 533, 517 (2016)
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46. Bjork, B., Bui, T., Heckl, O., Changala, P., Spaum, B., Heu, P., Follman, D., Deutsch, C., Cole, G., Aspemeyer, M., Okumura, M., Ye, J.: Science 354, 444 (2016) 47. Stokes, G.: Proc. R. Soc. Lond. 11, 547 (1862) 48. Hamilton, W.R.: Proc. R. Ir. Acad. II, 424 (1844) 49. Hamilton, W.R.: Philos. Mag. 25, 489 (1844) 50. Pauli, W.: Z. Phys. 37, 601 (1927) 51. Jordan, P.: Z. Phys. 94, 531 (1935) 52. Schwinger, J.: In: Biedenharn, L.C., van Dam, H. (eds.) Quantum Theory of Angular Momentum, p. 229. Academic Press, New York (1965) 53. Biedenharn, L.C., Louck, J.D.: Angular Momentum in Quantum Physics. In: Rota, G.C. (ed.) Encyclopedia of Mathematics, vol. 8, p. 212. Addison Wesley, Reading (1981) 54. Rabi, I.I., Ramsey, N.F., Schwinger, J.: Rev. Mod. Phys. 26, 167 (1954) 55. Feynman, R.P., Vernon, F.I., Helwarth, R.W.: J. Appl. Phys. 28, 49 (1957) 56. Harter, W.G., dos Santos, N.: Am. J. Phys. 46, 251 (1978) 57. Lehmann, K.K.: J. Chem. Phys. 79, 1098 (1983) 58. Harter, W.G.: J. Chem. Phys. 85, 5560 (1986) 59. Li, Z., Xiao, L., Kellman, M.E.: J. Chem. Phys. 92, 2251 (1990) 60. Harter, W.G.: Phys. Rep. 8(319), 378–385 (1988) 61. Ortigoso, J., Kleiner, I., Hougen, J.T.: J. Chem. Phys. 110, 11688 (1999) 62. Jahn, H.A., Teller, E.: Proc. R. Soc. Lond. A161, 220 (1937) 63. Jahn, H.A., Teller, E.: Proc. R. Soc. Lond. A164, 117 (1938) 64. Boudon, V., Mitchell, J., Domanskaya, A., Maul, C., Georges, R., Benidar, A., Harter, W.G.: Mol. Phys. 58, 391–400 (2012) 65. Carlos, M., Gruson, O., Richard, C., Boudon, V., Rotger, M., Thomas, X., Maul, C., Sydow, C., Domanskaya, A., Georges, R., Soulard, P., Pirali, O., Goubet, M., Asselin, P., Huet, T.R.: J. Quant. Spectrosc. Radiat. Transf. 201, 2273 (2017) 66. Harter, W.G., Mitchell, J.C.: Int. J. Mol. Phys. 14, 796 (2013)
535 William G. Harter Professor Harter’s research centers on the theory of spectroscopy and what it reveals about quantum phenomena and symmetry principles of structure and dynamics. Current study focuses on how wave mechanics of light relates to matter waves and their relativistic symmetry ranging from intrinsic frames of floppy molecules to manifold dynamics of astrophysical objects. A strong educational effort is being developed to make modern theory more accessible.
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Contents
Abstract
35.1 35.1.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Intensity Versus Line-Position Spectroscopy . . . . . . 537
35.2 35.2.1 35.2.2 35.2.3
Molecular Wave Functions in the Rotating Frame . Symmetries of the Exact Wave Function . . . . . . . . . Rotation Matrices . . . . . . . . . . . . . . . . . . . . . . . Transformation of Ordinary Objects into the Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
538 538 539
35.3 35.3.1 35.3.2 35.3.3 35.3.4 35.3.5
The Energy–Intensity Model . . . . . States, Levels, and Components . . . . The Basis Set and Matrix Hamiltonian Fitting Experimental Energies . . . . . The Transition Moment Matrix . . . . Fitting Experimental Intensities . . . .
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539 539 540 541 541 542
35.4 35.4.1 35.4.2 35.4.3 35.4.4
Selection Rules . . . . . . . . . . . . . . . . . . . Symmetry Types . . . . . . . . . . . . . . . . . . Rotational Branches and Parity . . . . . . . . . . Nuclear Spin, Spatial Symmetry, and Statistics Electron Orbital and Spin Angular Momenta .
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542 542 542 543 544
35.5 35.5.1 35.5.2
Absorption Cross Sections and Radiative Lifetimes 545 Radiation Relations . . . . . . . . . . . . . . . . . . . . . . 545 Transition Moments . . . . . . . . . . . . . . . . . . . . . 545
35.6 35.6.1 35.6.2
Vibrational Band Strengths . . . . . . . . . . . . . . . . 546 Franck–Condon Factors . . . . . . . . . . . . . . . . . . . 546 Vibrational Transitions . . . . . . . . . . . . . . . . . . . . 547
35.7 35.7.1 35.7.2 35.7.3 35.7.4 35.7.5 35.7.6
Rotational Branch Strengths . . . . . Branch Structure and Transition Type Hönl–London Factors . . . . . . . . . . Sum Rules . . . . . . . . . . . . . . . . . Hund’s Cases . . . . . . . . . . . . . . . Symmetric Tops . . . . . . . . . . . . . . Asymmetric Tops . . . . . . . . . . . . .
35.8 35.8.1 35.8.2
Forbidden Transitions . . . . . . . . . . . . . . . . . . . 551 Spin-Changing Transitions . . . . . . . . . . . . . . . . . 551 Orbitally-Forbidden Transitions . . . . . . . . . . . . . . 551
35.9
Recent Developments . . . . . . . . . . . . . . . . . . . . 552
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 D. L. Huestis () Molecular Physics Laboratory, SRI International Menlo Park, CA, USA
This chapter summarizes the theory of radiative transition probabilities or intensities for rotationally-resolved (high-resolution) molecular spectra. A combined treatment of diatomic, linear, symmetric-top, and asymmetrictop molecules is based on angular momentum relations. Generality and symmetry relations are emphasized. The energy-intensity model is founded in a rotating-frame basis-set expansion of the wave functions, Hamiltonians, and transition operators. The intensities of the various rotational branches are calculated from a small number of transition-moment matrix elements, whose relative values can be assumed from the supposed nature of the transition, or inferred by fitting experimental intensities. Keywords
wave function angular momentum diatomic molecule transition moment transition matrix element
35.1 Overview 35.1.1 Intensity Versus Line-Position Spectroscopy The fact that atoms and molecules absorb and emit radiation with propensities that vary with wavelength is the origin of the field called spectroscopy. The relatively sharp intensity maxima are interpreted as corresponding to transitions between discrete states or energy levels. The frequencies or energies of these transitions are used as the primary source of information about the internal structure of the atom or molecule. Line positions can be measured with very high precision (1 ppm or better). Excellent calibration standards have been developed. The quality of these experimental data has attracted extensive analytical and theoretical effort.
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_35
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Sophisticated parametrized models have been developed in which the smallest shifts from the expected line positions can be used to identify perturbations or other subtle effects. For molecules, knowledge of the strengths of these transitions is far less well developed. One reason is that quantitative experimental data on rotationally-resolved absorption cross sections and emission intensities are much rarer and the experiments themselves are much more difficult to calibrate. Few measurements claim a precision better than 1% and agreement within 10% of measurements in different laboratories is typically viewed as good. This situation is undesirable because most applications of molecular spectroscopy are in fact measurements of intensity. In many cases the strengths of absorptions or emissions are used to infer gas composition, temperature, time evolution, or other environmental conditions. In other examples the actual absorption and emission is the primary interest. Among the most important of these are atmospheric absorption of solar radiation and the greenhouse effect.
35.2 Molecular Wave Functions in the Rotating Frame 35.2.1
Symmetries of the Exact Wave Function
The exact total wave function for any isolated molecule with well-defined energy and total angular momentum can be expressed in a basis-set expansion over configurations with well-defined internal quantum numbers, ˆexact D ˆtrans
X
ˇ
C˛ˇı ˆ˛rot ˆvib ˆelec ˆıespin ˆnspin :
˛ˇı
(35.1) In principle, the coefficients C˛ˇı can be found only by diagonalizing the exact Hamiltonian. In practice one attempts to find a sufficiently good approximation, containing only a few terms, with coefficients chosen by diagonalizing an approximate or model Hamiltonian. This is the basis of the energy–intensity model developed in Sect. 35.3. As discussed by Longuet-Higgins [1] and Bunker [2], there are only six true symmetries of the exact Hamiltonian of an isolated molecule: 1. 2. 3. 4. 5. 6.
translation of the center of mass; permutation of electrons; permutation of identical nuclei; time reversal or momentum reversal; inversion of all particles through the center of mass; rotation about space-fixed axes.
Of these, only the symmetries numbered 5 and 6 give quantum numbers (parity and the total angular momentum F ) that are both rigorous and useful spectroscopic labels of the states of the molecule. The other symmetries are convenient for simplifying the description of the molecular wave function, for the evaluation of relations between matrix elements, and for classification of molecular states according to approximate symmetries. The first symmetry, translation of the center of mass, allows the choice of a coordinate system referenced to the center of mass, and suppression of the portion of the wave function describing motion through space (as long as the molecule does not dissociate). Symmetry number 2, exchange of electrons, does not directly provide any labels or quantum numbers, since the Fermi–Dirac statistics of electrons require that all wave functions must be antisymmetric. However, it provides considerable information about the probable electronic states since it controls whether molecular orbitals can be doubly or only singly occupied. For most (low-Z) molecules, each state will have a nearly well-defined value of electron spin: singlet or triplet for example. Admixture of other spin values usually can be treated as a perturbation. These points will be elaborated in Sects. 35.4.4 and 35.7.4. Permutation of identical nuclei, symmetry number 3, also gives an identical quantum number to all the states of the molecule (˙1 depending on the character of the permutation and on whether nuclei with integral or half-integral spin are being permuted). It supplies little direct information about the energy separations between the states of the molecule. On the other hand, many molecules have identical nuclei in geometrically or dynamically equivalent positions. The existence of spatial symmetry, for nonplanar molecules, is really the same thing as permutational symmetry. Consequently, nuclear permutation, combined with inversion (symmetry number 5), is the basis for naming the states according to the approximate spatial symmetry group of the molecular frame and vibrational motion. These concepts will be explored in Sect. 35.4.3. Symmetry number 4, time reversal, is both subtle and simple. In the absence of external magnetic fields the Hamiltonian for a molecule will contain only even combinations of angular momentum operators, e.g., F˛ Fˇ , F˛ Lˇ , or F˛ Sˇ . Thus changing the signs of all the angular momenta should result in an equivalent wave function. This will require that matrix elements retain the same absolute value when the angular quantum numbers are reversed, leading in general to complex conjugation [3]. Spatial inversion, symmetry number 5, is always an allowed operation for any molecule, even if it appears to lack internal inversion symmetry. This operation can be considered as a symmetry of the spherically-symmetric laboratory
35 Radiative Transition Probabilities
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in which the molecule resides. If the molecule is linear, triatomic, rigid with a plane of symmetry, or is nonrigid with accessible vibrational or tunneling modes that correspond to plane reflections, inversion symmetry divides the states of the molecule into two classes, called parities. Perturbations can occur only between states of the same parity. For optical transitions, the change in parity of the states must match the parity of the operator. Otherwise, reflection of the molecule in a plane will interchange inconvertible optical isomers. Such optical isomers are energetically degenerate, so in all cases, inversion through the center of mass remains a valid symmetry of the rotating molecule. However, the separation of the states into two kinds does not provide any selection rules. The two parity classes are perfectly degenerate, thus there is always an allowed level with the correct parity either for perturbations or for optical transitions.
dom (thus the label (vesn)) can be thought of as the partial summation F/ ˆ.F;K .internal, spins/ vesn X ˇ C.FKF /ˇı ˆvib ˆelec ˆıespin ˆnspin ; D
(35.4)
ˇı
expressed in the internal or rotated coordinate system. Note that the FKF designation is only a parametric label. The rotational wave function has been absorbed into the rotation matrix.
35.2.3 Transformation of Ordinary Objects into the Rotating Frame
The assumption of rotational symmetry allows re-expression of matrix elements between total wave functions as a sum of matrix elements between internal wave functions. For 35.2.2 Rotation Matrices example, the tensor operator T .L/ belonging to the L representation of the rotation group, can be written in the rotating The final symmetry, rotation about the center of mass, re- frame as [5–8] X stricts the discussion to states with well-defined laboratory .L/ L T .lab/ D Dpq ./Tq.L/ .body/ ; (35.5) p angular momentum, and to re-expression of the exact wave q function by changing variables from laboratory coordinates to body-fixed or internal coordinates, and introducing the Eu- and can be used to evaluate matrix elements that might represent radiative transitions ler angles relating these two coordinate systems, ˇ 00 00 ˛ ˝ F 0 ;M 00 Cp ˇ .L/ ˇT F X F;M K .lab/ˇˆF ;MF p F;MF F F ˆrot .Euler angles/ ˆexact .lab/ D ˇ ˛ .2F 00 C 1/ 1=2 ˝ 00 00 KF F MF ; Lp ˇF 0 MF00 C p D 0 .2F C 1/ F/ ˆ.F;K .internal, spins/ : (35.2) vesn X˝ ˇ ˛ F 00 KF00 ; Lq ˇF 0 KF00 C q Here F is the total angular momentum of the molecule, qKF00 including vibrational, mechanical-rotation, electron-orbital, ˇ .F 00 ;K 00 / ˛ ˝ .F 0 ;KF00 Cq/ ˇ .L/ ˇT vesn .body/ˇˆvesn F ; (35.6) electron-spin, and nuclear-spin contributions. MF and KF q ˇ ˇ ˛ ˝ ˛ ˝ are the projections of F in the laboratory and body-fixed where F 00 MF00 ; Lp ˇF 0 MF00 C p and F 00 KF00 ; Lq ˇF 0 KF00 C q frames, respectively. In the majority of cases, the magnitude are Clebsch–Gordan coefficients that vanish if jF 0 F 00 j > L, of nuclear hyperfine interactions is sufficiently small that its jMF00 C pj > F 0 , or jKF00 C qj > F 0 . influence can be ignored when analyzing wave functions and This transformation forms the basis for the derivation of computing energies. Thus the quantum numbers J , MJ , and rotational branch strengths (Sect. 35.7.2) and for the deKJ , or just J , M , and K can be used. scription of electron motions that are weakly coupled to the Explanation is postponed of how the body-fixed frame is molecular frame (Sect. 35.7.4). to be selected, but for any choice, the wave function for rotation of the entire molecule can be expressed using a rotation matrix [4, 5] 35.3 The Energy–Intensity Model F KF .Euler angles/ ˆF;M rot .2F C 1/ 1=2 F DMF KF .; ; / : D 8 2
35.3.1 States, Levels, and Components (35.3)
The previous section introduced the concept of representing the wave function of a molecule as a product of five simpler For diatomics, Zare [5] suggests multiplying by .2 /1=2 and wave functions: setting D 0. The internal wave function for the vibrational, (35.7) electronic, electron-spin, and nuclear-spin degrees of free elec vib rot espin nspin :
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This construction yields a similar separation of the Hamiltonian, H Helec C Hvib C Hrot–fs C Hhf ;
(35.8)
and representations of the energies as sums of contributions, E Te C Gv C Fc .J / ;
(35.9)
and absorption or emission transition strengths as products, I Ielec Ivib Irot–fs Ihf :
(35.10)
Whatever theoretical arguments might favor such a separation, the real impetus is the empirical observation that most molecular absorption and emission spectra exhibit recognizable patterns arising from the dissimilar magnitudes of the energies associated with these five degrees of freedom. Separation of the wave function and the Hamiltonian into these four or five contributions facilitates the assignment of molecular spectra, in addition to suggesting models with parameters that can be adjusted to quantitatively represent the observed spectra. Most states of molecules are dominated by a single set of electronic and vibrational quantum numbers. Electronic states are often well separated. With each electronic state is associated a potential energy surface, the energy at the minimum being labeled Te . Motion of nuclei within this potential generates various levels corresponding to different vibrational quantum numbers, following regular patterns or progressions in energy, summarized by a small number of parameters called vibrational frequencies. The quantity Gv represents the energy of the the vibrational level above the potential minimum. For each vibrational level, a progression of rotational levels is expected. For linear molecules in electronic states without electronic angular momentum (i.e., 1 † states) the rotational energies are also reproduced by a few rotational constants. For more complicated molecules and electronic states, i.e., most cases, there are multiple energetically distinct levels with the same value of J (in addition to the 2J C 1 orientational degeneracy of each level). These multiple levels all share the same nominal quantum numbers (additional analysis may subdivide them into parity or permutational symmetry types). These sublevels are called components with energies expressed by the notation Fc .J /. The quantity Nc , the number of components expected, reflects the assignment of the nature of the vibronic state. For linear molecules there is a limited number of components corresponding to the various orientations of electron spin and orbital angular momentum. For example, a 2 … electronic state will have four components for each value of J (except for J D 1=2, where there are only two components). For nonlinear molecules the number of components increases with J , proportional to 2J C 1, corresponding to various possible projections of the total angular momentum onto the tumbling molecular frame.
The conclusion of this analysis is that a basis set be chosen, over which a model rotational and fine-structure Hamiltonian can be expressed. The wave functions then become vectors of numbers. A priori, only the form of the matrix elements and their dependence on J and body–frame projection (K or *) are known. Little is known in advance about how strong the interactions are in any given molecule. Thus one tends to write the Hamiltonian with parameters that are to be determined by fitting the observed energy levels. Similarly, the choice of the basis sets for the upper and lower states specifies the overall form of the matrix of transition moments between the basis functions. The transition can be chosen to be of a simple standard form, for example, parallel or perpendicular, with only one unknown parameter representing the overall strength of the transition. Alternatively, the transition matrix elements can be considered to be independently adjustable, within the symmetry restrictions that are required (time reversal) or assumed (spatial symmetry).
35.3.2 The Basis Set and Matrix Hamiltonian For linear molecules it is convenient to choose a basis set labeled by the projections of orbital and spin angular momenta in the body–frame coordinate system, represented symbolically by .2J C 1/ 1=2 J jƒ†I JM*i D DM* jƒ†i ; (35.11) 8 2 where * D ƒ C †. This is called the Hund’s case (a) basis set, which is an accurate representation in a single term if the body–frame angular momenta are nearly conserved. This is true if the spin-orbit interaction is larger than the separation between rotational levels. Under all circumstances, this basis set facilitates construction of the matrix Hamiltonian and representation of sources of transition probability [9]. One parametrization for the spin–rotation Hamiltonian is provided by Brown et al. [10–12]: H spin–rot D Te C Gv C Bv N 2 Dv N 4 1 C Av C ADv N 2 ; Lz Sz C
2 C v C Dv N 2 N S 1 C v C Dv N 2 ; 3Sz2 S 2 C 3 1
C v Lz Sz Sz2 3S 2 1 5 1 ov C oDv N 2 ; ƒ2C S2 C ƒ2 SC2 C 4 1 C pv C pDv N 2 ; ƒ2C S N C ƒ2 SC NC C 4 1 C qv C qDv N 2 ; ƒ2C N2 C ƒ2 NC2 C ; 4 (35.12)
35 Radiative Transition Probabilities
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where Œx; yC is the anticommutator .xy C yx/, and N D J S . Zare et al. [13] provide an alternative parametrization, with different interpretations of the spectroscopic constants .B; D; A; ; , etc.) because they multiply different symbolic operators. One significant difference is that Zare et al. use the mechanical angular momentum R D J L S as the expansion operator, rather than N , which might be called the spinless angular momentum. These differences mean that care must be taken in attempting to construct simulated spectra from published constants. In spite of much discussion in the literature, there is little theoretical foundation for preferring one parametrization over another, as long as the observed levels are accurately fit. In a number of cases naive assumptions about the origin of certain types of interactions have been overturned. For example, the spin–spin interaction, represented by the constant , is often dominated by level shifts due to off-diagonal spin-orbit perturbations [14]. For polyatomic molecules, a suitable basis set for expansion can be chosen to have a similar form [8, 15, 16] .2J C 1/ 1=2 J DMK jlƒ†i ; (35.13) jlƒ†I JMKi D 8 2
is to be accompanied by the appropriate hidden nuclear-spin basis function [8, 21]. For asymmetric top molecules, the Wang transformation divides the basis functions into four symmetry classes E˙ and O˙ according to the combining sign and whether K is even or odd. The eigenstates are often labeled by two projection quantum numbers called K1 and K1 . Assuming that A B C , the asymmetry parameter
D
.2B A C / .A C /
(35.17)
ranges from 1 for a prolate symmetric top .B D C / to 1 for an oblate symmetric top .B D A/. A, B, and C are the rotational constants, or reciprocals of the moments of inertia, about the three principal top axes. Each asymmetric top level can be correlated with specific symmetric top levels (i.e., K-values) in the two limits. The prolate limiting Kvalue is called K1 and the oblate limit is called K1 (i.e.,
D ˙1). Note that the symmetric-top principal axes rotate by 90ı during this correlation. The eigenstates are given additional symmetry names (ee, eo, oe, oo) according to whether K1 and K1 are even or odd. Papousek and Aliev [18] dis˙ ˙ where ƒ and † represent the projections of the electron- cuss the relations between the (E , O ) and (ee, eo, oe, oo) orbital (L) and spin (S) angular momenta, and l represents the labeling schemes. projection of the vibrational angular momentum (p for degenerate vibrational modes). This is the symmetric top basis set. Generalizing the work of Watson [17, 18], the parametrized 35.3.3 Fitting Experimental Energies Hamiltonian might be written in a form such as Having chosen a basis set and model Hamiltonian for both X ˛ˇı ˛ 2ˇ 2 the upper and lower levels, the observed transition energies 2 2 H rot D h l Jz JC C J J can be used to infer the numerical values of the constants ı that best fit the spectrum. The following quotation provides .J p/ .J L/ .J S / a good description of the process: .p L/ .p S / .L S /l ; The calculational procedure logically divides into three steps: (35.14)
where the fg indicates that an appropriately symmetric combination be constructed with anticommutators. For both linear and nonlinear molecules, it is convenient to use the Wang transformation [19] to combine basis functions with opposite sense of rotation: for diatomics
(1) The matrix elements of the upper and lower state Hamiltonians are calculated for each J value using initial values of the adjustable molecular constants; (2) both Hamiltonians are numerically diagonalized and the resulting sets of eigenvalues are used to construct a set of calculated line positions; and (3) from a least-squares fit of the calculated to the observed line positions, an improved set of molecular constants is generated. This nonlinear least-squares procedure is repeated until a satisfactory set of molecular constants is obtained.
1 This quotation is taken from the article by Zare et al. [13] p Œkƒj; †I J; M; *i ˙ j jƒj; †I J; M; *i I in which they describe the basis for the LINFIT computer 2 (35.15) program, one of the first to accomplish direct extraction of constants from diatomic spectral line positions based on nuand for polyatomics merically diagonalized Hamiltonians. 1 p Œjl; ƒ; †I J; M; Ki ˙ j l; ƒ; †I J; M; Ki : 2 (35.16) 35.3.4
The Transition Moment Matrix
For diatomic molecules, these combinations can be assigned Diagonalization of the model Hamiltonians for the upper and the parity ˙.1/J CS Cs , where s D 1 for † states, and 0 lower states yields vector wave functions that can be used for otherwise [13, 20]. For symmetric top molecules, each term calculating matrix elements, especially those needed to eval-
35
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D. L. Huestis
uate radiative transition probabilities. The wave functions for 35.4 Selection Rules diatomic molecules have the form X 00 0 J c 0 0 0 0 0 0 35.4.1 Symmetry Types bƒ 0 †0 jƒ † I J M * i ; J 0M 0 c0 D ƒ0 †0
X
(35.18) Selection rules are guidelines for identifying which transitions are expected to be strong and which are expected to ƒ00 †00 be weak. These rules are based on classifying rovibronic levSection 35.2.3 expresses matrix elements of laboratoryels into labeled symmetry types. Some symmetry distinctions frame operators in terms of matrix elements in the rotating are effectively exact: such as total angular momentum F , or body-fixed frame. Terms of the form laboratory-inversion parity. Others are approximate, derived ˇ ˝ 0 0 ˇ 00 00 ˛ K 0 K 00 D .J K / ˇTq.L/ .body/ˇˆ.J K / .q/ (35.19) from estimates that certain matrix elements are expected to be much larger than others. The most important of these are need to be evaluated. These terms are multiplied by zero if based on electron spin (for light molecules) and geometrical K 0 ¤ K 00 C q. In the diatomic basis set these become point-group symmetry (for relatively rigid polyatomics). In ˇ ˇ ˝ ˛00 ƒ0 †0 ƒ00 †00 D 0 ƒ0 †0 ˇTq.L/ .body/ˇƒ00 †00 .q/ ; (35.20) actual fact, no transition is completely forbidden. The multipole nature of electromagnetic radiation (electric-dipole, where .q/ is a phase factor described in Sect. 35.7.2. magnetic-dipole, electric-quadrupole, etc.) implies that any Only a few of these matrix elements are independent and change in angular momentum or parity is possible in princinonzero. For electric dipole transitions (L D 1), time-reversal ple. Practical interest emphasizes identification of the origin and inversion-symmetry can be used to establish the relation of the strongest source of transition probability, and estima0 00 *0 *00 D .1/* * *0 *00 : (35.21) tion of the strengths of the weak transitions relative to the The sign of D ˙1 is determined by the overall character stronger ones. The result is a collection of propensity rules of the electronic transition, and is related to the classification using selection rules as tools of estimation. Basis functions for expansion of the wave functions for of levels into e- and f -parity types and to the determination the upper and lower states were chosen in Sect. 35.3.2. The of which components are involved in the rotational branches first step in the symmetry classification of rovibronic levels (P , Q, and R). These concepts are elaborated in Sect. 35.4. consists of identifying various linear combinations of basis functions that block-diagonalize the exact or approximate Hamiltonians. Symmetry-type names are then assigned to 35.3.5 Fitting Experimental Intensities these linear combinations based on the value of F or J and For allowed transitions in linear molecules and symmetric knowledge of the symmetry properties of the underlying vitops, only one independent parameter is normally expected brational and electronic states. Thus each eigenfunction or in the transition moment matrix. Thus no additional infor- rovibronic level consists of an expansion over only one of mation is available from fitting the experimental rotational the kinds of linear combination, and the level can be assigned branch strengths (assuming the energy–intensity model is ad- a specific symmetry type. Similarly, the basis-set expansion leads to a matrix repequate). In diatomic molecules, the intensities of different resentation of the possible transitions. Spin- and spatialvibrational bands can be used to infer the internuclearsymmetry arguments establish relationships between these distance dependence of the electronic transition moment (for transition matrix elements, and provide estimates of which example, see Luque and Crosley [22]). For forbidden transitions and allowed transitions in asym- are much smaller than the others. Each combination of uppermetric tops, more than one independent parameter is ex- and lower-state symmetry types results in a specific pattern pected. The intensity of a single given rotational line can be of rotational branches. The most important patterns are J even (Q-branches) or odd (P - and R-branches), and intenexpressed in the form sity alternation for consecutive values of J (nuclear spin ˇ ˇ2 ˇX ˇ statistics). ˇ ˇ 0 00 Z 0 00 .line/ˇ ; (35.22) I line D ˇ ˇ 0 00 K K K K ˇ ˆ00J 00 M 00 c 00 D
00 00
J c 00 00 00 00 00 00 aƒ 00 †00 jƒ † I J M * i :
KK
where ZK 0 K 00 .line/ can be calculated in advance from the energies (wave functions) and quantum numbers alone, using the formulas in Sect. 35.7.2. Nonlinear least-squares fitting can be used to derive the best intensity parameters [23–26], analysis of which can help characterize the nature of the transition, and identify the sources of transition probability.
35.4.2 Rotational Branches and Parity The symmetry of time or momentum reversal implies that changing the signs of all the angular momenta should result in an equivalent wave function. For example, the phase con-
35 Radiative Transition Probabilities
543
and
vention .F;KF / ˆvesn .internal, spins/ F CKF
D .1/
F/ ˆ.F;K vesn
.internal; spins/
.F;KF / F/ xz ˆ.F;K D .1/KF ˆvesn ; vesn
(35.27)
(35.23)
where D 0 or 1=2 for integral or half-integral F , respectively, and is the parity label for the state, having values of ˙1. In linear molecules, levels with D C1 are called elevels while those with D 1 are called f -levels [28, 29]. Inversion symmetry can be combined with time reversal to establish that all matrix elements of the Hamiltonian can F F be taken to be real [27]. The wave function can also be exThe formula for matrix elements of optical transition opera- pressed in the form of the Wang transformation [19], uniting the ˙KF components, tors can also be reanalyzed, ˇ ˇ ˝ F 0 ;M 00 Cp .L/ 00 00 ˛ ˇT F .lab/ˇˆF ;MF .2F C 1/ 1=2 p F;MF .lab/ D ˆ exact 8 2 ˛ .2F 00 C 1/ 1=2 ˝ 00 00 0 00 X M ; LpjF M C p F D
F F F F F/ F/ .2F 0 C 1/ ˆ.F;K .1/KF C DM ˆ.F;K DM vesn vesn F KF F KF xz ˇ ˛ 1 X ˝ 00 00 KF 0 F KF ; Lq ˇF 0 KF00 C q 1=2
2 00 : 2.1 C ıKF 0 / qK ( (35.28) ˇ E D .F 0 ;K 00 Cq/ ˇ ˇ .L/ ˇ .F 00 ;KF00 / F ˇTq .body/ˇˆvesn vesn If the molecule is rigid and has a plane of symmetry, or is 0 00 nonrigid with accessible vibrational or tunneling modes that C .1/F CLF ) correspond to plane reflections, inversion symmetry divides ˇ ˇ E D .F 0 ;K 00 q/ ˇ .L/ ˇ .F 00 ;K 00 / vesn F ˇTq .body/ˇˆvesn F ; (35.25) the states of the molecule into two classes, according to the sign of . Perturbations can occur only for F D 0 and 0 00 D C1. For optical transitions, the change in parity of to establish that all contributions are purely real or purely the states must match the parity of the operator. Odd operimaginary [27]. Since such transition matrix elements will ators (e.g., electric-dipole) require 0 00 .1/F D 1. Even be used as absolute squares, they can be treated as if they operators (e.g., magnetic-dipole and electric-quadrupole) rewere purely real. quire 0 00 .1/F D C1. Parity classification of molecular states according to inversion through the center of mass is important for establishing which transitions are electric-dipole allowed and which 35.4.3 Nuclear Spin, Spatial Symmetry, states can perturb each other. As discussed by Larsson [4], and Statistics such classification is not without subtlety and opportunity for confusion. Inversion of the laboratory spatial coordinates For most molecules, the coupling of nuclear spin with the (isp , also called E [2]) is equivalent to a reflection of the electron-spin, electron-orbital, and frame-rotational angular molecule-fixed electronic and nuclear coordinates in an arbimomenta is sufficiently weak that treatment of the energetics trary plane followed by rotation of the molecular frame by of hyperfine interactions can be postponed. The first-order 180ı about an axis through the origin and perpendicular to effect of nuclear spin is that rovibronic wave functions for the reflection plane (if F is half-integral special care must be molecules containing identical nuclei must be combined with taken about the sense of rotation). It follows that appropriate nuclear spin wave functions in order to obtain F;MF the necessary Fermi–Dirac or Bose–Einstein nuclear permuisp ˆexact .lab/ tation symmetry. For many molecules, there exist combinaF;M D E ˆexactF .lab/ tions of nuclear permutations that correspond to combinaF D .1/F ˆF;M tions of frame rotations, laboratory inversions, and feasible exact .lab/ 1=2 vibrational motions (the rotational wave function makes .2F C 1/ D a contribution because renumbering the nuclei requires a re8 2 analysis of the Euler angles). For rigid molecules, these X F .F;KF / .1/F KF DM ˆ (35.26) xz permutations (possibly including inversion) can be used to vesn F KF KF generate the point symmetry group of the molecule. For can be chosen to establish that the relative phases of matrix elements of the Hamiltonian can be taken as ˇ ˇ E D .F;KF0 / ˇ ˇ F .F;K 00 / F DM 0 ˆvesn ˇH ˇDMF K 00 ˆvesn F F KF F ˇ ˇ E D .F;KF0 / ˇ ˇ F .F;K 00 / F : (35.24) D DMF K 0 ˆvesn ˇH ˇDMF K 00 ˆvesn F
35
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D. L. Huestis
fluxional molecules, with multiple energetically equivalent nuclear configurations, a rather large molecular symmetry group can result, one that may not correspond to any ordinary point group [1, 2]. In the discussion immediately following, consider the case of N occurrences of one kind of nucleus, the others being unique (e.g., PD3 ). The treatment can easily be extended to the case of multiple kinds of identical nuclei (e.g., C2 H6 ). The exact wave function can be rearranged into a sum over products of the form ˆexact D
X
.b/
ˆ.a/ rves ˆnspin ;
(35.29)
a;b .a/
where ˆrves is a rovibronic wave function belonging to the %.a/ representation of the symmetric group SN of permu.b/ tations over N objects, and ˆnspin is a nuclear spin wave function, belonging to the %.b/ representation of SN . In order to obtain the correct permutation symmetries for the overall wave function, the only terms that can appear in this sum are those for which the direct product %.a/ ˝ %.b/ contains the symmetric or antisymmetric representation, for bosons or fermions, respectively. Assumption of negligible hyperfine interactions allows evaluation of matrix elements of the form ˝ .c/ .d / ˇ ˇ .a/ .b/ ˛ ˆrves ˆnspin ˇH ˇˆrves ˆnspin ˇ ˇ .a/ ˛˝ .d / ˇ .b/ ˛ ˝ ˇ ˇ ˇ D ˆ.c/ (35.30) rves H ˆrves ˆnspin ˆnspin ; which vanishes unless %.a/ D %.c/ and %.b/ D %.d /. Thus the nearly-exact wave function can be written as a sum over products where the rovibronic and nuclear-spin factors correspond to basis functions from single known representations: ˆexact
X
.b/
ˆ.a/ rves ˆnspin ;
(35.31)
a;b
with %.a/ D %rves and %.b/ D %nspin . This divides the states of the molecule into a number of noninteracting symmetry classes, labeled by the representations of the symmetric group. In the absence of hyperfine interactions, optical transitions are possible only within a certain symmetry class. Thus the existence of spatial or dynamical symmetry implies that each rovibronic wave function transforms according to a particular representation of a subgroup of the permutation-inversion group (called CNPI by Bunker [2]). Each representation includes only specific values of nuclear spin, corresponding to the permutational properties of the nuclear spin wave functions. The most important effect of this analysis is to assign statistical weights or relative intensities to the different symmetry types. For example, the symmetry group for NH3 is D3h (including umbrella inversion), with representations A01 , A02 , A001 , A002 , E 0 , and E 00 . The A01 and A001
representations must be combined with the (nonexistent) antisymmetric spin function, yielding a statistical weight of 0. Similarly, A02 and A002 combine with the symmetric I D 3=2 spin function, with a statistical weight of 4 (i.e., 2I C 1). E 0 and E 00 combine with the nonsymmetric I D 1=2 spin functions, with a statistical weight of 2. This material is discussed from various viewpoints in numerous articles and text books, of which only a few can be cited here [1, 2, 8, 18, 21, 30–36]. See Chap. 34 for additional details and examples. Although this analysis appears rather complicated, the selection rules that result are actually the same, at least in simple cases, as the ones that are derivable from simpler ideas. For example, for a 1 †C g lower state, even J levels are permutation symmetric and have parity C1, while odd J levels are permutation antisymmetric and have parity 1. For a 1 †C u upper state, even J levels are permutation antisymmetric and have parity C1, while odd J levels are permutation symmetric and have parity 1. Both the parity selection rule and permutation-symmetry selection rule independently require that J D ˙1 for electric-dipole transitions. Similarly, that the permanent dipole moment of a symmetric-top molecule must lie along the body-fixed axis replicates the K D 0 selection rule for pure-rotation transitions provided by permutational symmetry arguments. This means that when simulating absorption and emission spectra, the nuclear-spin wave function can usually be ignored. The intensity alternation imposed by spin-statistics can be represented by multiplying each wave function by the appropriate factor, for example, 0 or Œ.2I C 1/=.2i C 1/N 1=2 , where I is the total nuclear spin, and i is the spin of one of the N equivalent nuclei. Group theory remains vital for understanding the relative strengths of vibrational transitions in polyatomics (see Cotton [37], for example, and Sect. 35.6.2) and becomes very interesting as interaction between vibration and rotation increases. For the purposes of this discussion, the most important issue is identification of which transition-moment matrix elements K 0 K 00 vanish and which are related by symmetry.
35.4.4 Electron Orbital and Spin Angular Momenta For all molecules, the strongest transitions tend to be those that conserve electron spin. The zero-order transitionmoment matrix is diagonal both in total spin and in spinprojection onto the body–frame axis. In the jƒ†i basis set for linear molecules this is expressed by ƒ0 †0 ƒ00 †00 D ƒ0 ƒ00 ı†0 †00 : .L/
(35.32)
The transition-moment tensor operator Tq .body/ can connect basis functions that differ in ƒ by at most L. Allowed
35 Radiative Transition Probabilities
545
Numerical values of the B coefficients can be derived from the optical absorption cross section, and thus (35.33) ƒ0 ƒ00 D 0; for jƒ0 ƒ00 j > 1 : Z 8 2 gl Aul D abs ./ d In the usual case that the upper and lower states each consist 2 c gu of only a single value of jƒj, there is only one independent, Z 8 2 nonzero, matrix element jƒ0 j;jƒ00 j with (35.40) D se ./ d : c2 0 00 (35.34) jƒ0 j;jƒ00 j D .1/jƒ jjƒ j jƒ0 j;jƒ00 j : Finally, the expression for the absorption oscillator strength See Sect. 35.8 for a discussion of spin-forbidden and is orbitally-forbidden transitions. Similar arguments and phase mc 3 gu relationships can be developed for polyatomic molecules fabs D .4 0 / Aul 2 2e2 8 gl with nonzero electron spin or degenerate vibrational or elecZ mc tronic states. D .4 0 / (35.41) abs ./ d : e 2 electric-dipole transitions thus satisfy
The emission oscillator strength is simply related to that for absorption: fem D .gl =gu /fabs . The oscillator strength offers considerable advantages as a means of reporting and comparing the strengths of radiative transitions. It is di35.5.1 Radiation Relations mensionless, obeys the simple sum rule (for electric-dipole Among the most important radiation relations is the con- transitions) X nection between the absorption cross section and the rate ful D number of electrons ; (35.42) of spontaneous emission. Einstein [38] introduced his A u and B coefficients to describe the rates of absorption and emission of radiation of a collection of two-level atoms or and is directly derivable from an experimental absorption molecules in equilibrium with a radiation field at the same cross section even before the assignment of the upper level temperature. The discussion here follows that of Condon and has been determined (i.e., before its degeneracy is known). Shortley [39], Penner [40], Thorne [41], and Steinfeld [42] (with corrections). Also see Chaps. 11, 18, 72 of this Handbook. The number of absorption events per unit volume per 35.5.2 Transition Moments unit time is written as In many cases, the intention is to construct model quantum (35.35) mechanical wave functions for the two states involved in Nl Blu ./ : the transition under study. In addition, ab initio electronic While the rate of emission is wave functions and matrix elements may be available (see (35.36) Chap. 33). Quantum mechanics suggests the following exNu Bul ./ C Nu Aul : pression for the Einstein A coefficient (see Sect. 12.6.1) In thermal equilibrium, the radiative energy density is given X ˇ˝ 0 ˇ ˇ 00 ˛ˇ2 64 4 3 1 1 by the Planck blackbody law ˇ 0 ˇerp ˇ 00 ˇ : Aul D u l 3 3hc 4 0 gu 0 00 3
u ;l ;p 1 8 h h=kT
./ D 1 ; (35.37) e (35.43) c3
35.5
Absorption Cross Sections and Radiative Lifetimes
and the ground and excited state densities satisfy a Boltz- The summation is over all three optical polarization directions p (i.e., rp runs over x, y, and z in the lab frame), all mann relationship, degenerate components l 00 of the lower state (i.e., gl of them), gu h=kT Nu 0 D ; (35.38) and all degenerate components u of the upper state (i.e., gu e Nl gl of them). This triple sum is also called the line strength Sul . where gu and gl are the degeneracies of the upper and lower Division by the upper-level degeneracy corrects for the fact states. The requirement that the rates of absorption and emis- that the transitions should be averaged rather than summed over the initial levels. sion must be equal leads to the relations In practice, choosing the appropriate degeneracy to divide 8 h 3 8 h 3 gl by is a question of some ambiguity. For atoms, it is suffiAul D Bul D Blu : (35.39) c3 c3 gu cient to understand how the individual matrix elements and
35
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D. L. Huestis
the line strength were calculated. For example, Bethe and Salpeter [43] use a degeneracy of (2L C 1) for Schrödinger wave functions for the hydrogen atom, and (2J C1) for Dirac wave functions. For molecules with internal angular momentum, i.e., everything other than 1 † states of linear molecules, the situation is much more complicated. For electric-dipole allowed transitions in light molecules, ab initio transition moments are calculated in a body-fixed coordinate system, ignoring spin, and not summed over anything. For diatomic molecules, following the work of Whiting et al. [44, 45], the transition probability from a single upperstate component (J 0 c 0 ) to a single lower-state component (J 00 c 00 ) is written as 64 4 3 1 AJ 0 c 0 J 00 c 00 D 3hc 3 4 0 1 0 00 (35.44) qv0 v00 jRe j2 SJc 0cJ 00 : 0 2J C 1 In this formula qv0 v00 jRe j2 represents the rotationless contribution to the transition moment, symbolically represented as a product of a vibrational overlap (qv0 v00 , i.e., a Franck– Condon factor) and an electronic-only component jRe j2 (Sect. 35.6.1). All of the rotational complexity is absorbed 0 00 into the rotational-branch strength factor SJc 0cJ 00 (Sects. 35.7.2 and 35.7.3). The issue to be addressed here is how to divide 0 00 numerical factors between jRe j2 and SJc 0cJ 00 . One approach is to construct an estimate for the rotationless transition probability X 1 AJ 0 c 0 J 00 c 00 ; (35.45) Av0 v00 D Nc0 c 0 J 00 c 00
and for spin-allowed transitions, the simple spin-free expressions for the electronic transition moments jRe j2 D jhƒjezjƒij2 for parallel transitions and ˇ2 ˇ ˇ ˇ 1 jRe j2 D ˇˇhƒ C 1je p .x C iy/jƒiˇˇ 2
(35.49)
(35.50)
for perpendicular transitions.
35.6 Vibrational Band Strengths 35.6.1 Franck–Condon Factors The Born–Oppenheimer separation of electron and nuclear motion suggests that during an optical transition between different electronic states the nuclei should change neither their position nor momentum. This concept was developed from semiclassical arguments by Franck [46] and justified quantum mechanically by Condon [47]. Following Herzberg [48] and Steinfeld [42] the vibronic transition moment can be written as ˇ ˇ ˝ ˛ v0 v00 D e0 v0 0 ˇˇ e00 v0000 ; Z D dR v00 .R/ v0000 .R/ Z dr e 0 .r; R/ e00 .r; R/.r; R/ ; Z D dR v00 .R/ v0000 .R/.R/ : (35.51)
If the R-dependence of .R/ is sufficiently weak, it can be where Nc0 is the number of internal spin-orbit components of factored out to obtain 0 00 the upper state. Whiting et al. suggest that SJc 0cJ 00 be normalZ ized such that for spin-allowed transitions 0 00 v v D Re dR v00 .R/ v0000 .R/ ; (35.52) X
0 c 0 c 00 SJ 0 J 00 D 2 ı0;ƒ0 ı0;ƒ00 2S C 1 2J C 1 : where Re is called the electronic transition moment. The c 0 J 00 c 00 (35.46) transition probability is proportional to the square of the above, which is usually written as The first factor is 1 for †–† transitions,
00 and 2 for all others.0 The final factor is replaced by 2J C 1 if the sum is over J I qv0 v00 Re2 ; (35.53) instead of J 00 . For spin-forbidden transitions the following is where a plausible extension of this sum rule, X ˇZ ˇ2
0 00 ˇ ˇ SJc 0cJ 00 D max Nc0 ; Nc00 2J 0 C 1 : (35.47) 0 00 ˇ ˇ ; 0 00 q D .R/ .R/ (35.54) dR 0 00 vv v v ˇ ˇ 0 00 00 cJ c
Section 35.7.3 provides a corresponding sum rule for poly- the square of the overlap between initial and final vibrational atomic molecules. This normalization yields wave functions, is called the Franck–Condon factor.
0 00 ! The Franck–Condon factors satisfy the sum rule 4 3 max Nc ; Nc 64 1 2 0 v 00 jRe j Av0 v00 D q v X X 3hc 3 4 0 Nc0 qv0 v00 D qv0 v00 D 1 (35.55) (35.48) v0 v 00
35 Radiative Transition Probabilities
547
provided the summations include the continuum vibrational wave functions above the dissociation limits. The Franck– Condon approach also can be used to calculate intensities and cross sections for bound–free [49] and free–free [50] emission and absorption. In some cases the variation of .R/ is significant. Calculation of the effect on the intensities can usually be handled by the r-centroid method in which the expression ˝ 0 ˇ ˇ 00 ˛ ˇ ˇ 00 0 R (35.56) rNv0 v00 D v 0 00 v h v0 j v00 i
Overtone bands, i.e., with jvj > 1, are observed, as dramatically illustrated by the v D 4; 5 emissions from the OH radical observed from the Earth’s night sky [54]. For polyatomic molecules, overtone and combination bands are often quite strong. The presence or absence of which is used to establish the symmetries of the vibrational modes. In general, it is difficult to construct quantitative vibrational intensity formulas with only a few parameters that can be inferred experimentally.
is used to calculate an effective internuclear distance for the transition. The transition strength is then proportional to qv0 v0 j.Nrv0 v00 /j2 . An advantage of this formulation is that the vibrational overlaps can be calculated from energy information only, before the transition moment function is known and before the transition strengths are investigated experimentally. The quantitative accuracy of the r-centroid method for transition moments that are not linear in the internuclear distance, has been addressed by a considerable literature, which has been summarized by McCallum [51]. A second complication arises from the fact that the vibrational wave functions themselves depend parametrically on the rotational angular momentum. Calculation of rotationally-dependent Franck–Condon factors is described by Dwivedi et al. [52] who also discuss the r-centroid method.
35.7 Rotational Branch Strengths
35.6.2
Vibrational Transitions
35.7.1
Branch Structure and Transition Type
The overall rotational structure of a molecular transition is determined by the relative values and phases of the body– frame transition-moment matrix elements, the relative values and phases of coefficients in the expansion of the upperstate and lower-state component wave functions over the angular-momentum-projection basis functions, the energy separations between the components, and the relative values and phases of the vector-coupling coefficients. In simple cases, each lower component might be connected to only a single upper component (Hund’s case (b) or symmetric tops), or J D ˙1 (P - and R-branches) may dominate over J D 0 (Q-branches). For diatomic molecules, symmetry arguments are used to divide the components into the two parity classes e and f . For electric dipole transitions, the selection rules from Sect. 35.4.2 imply that
Vibrational transitions derive their strength from the varia
0 00 tion of the permanent dipole moment of the molecule as Ne Ne C Nf0 Nf00 P - and R-branches
0 00 a function of geometry or internuclear coordinates. As de(35.60) Ne Nf C Nf0 Ne00 Q-branches scribed by several authors [31, 32, 34, 53] one can expand the dipole moment as a power series in the internal-Cartesian are expected, where Ne and Nf indicate the number of comor normal-mode coordinates ponents of each parity class (Ne and Nf differ by no more X @Mxyz 0 C (35.57) than one). Rotational branches are labeled with the notation Qi C Mxyz .Q/ D Mxyz R @Q i Jc 0 c 00 , using symbols P for 1, Q for 0, and R for C1. i R indicates the apparent change in mechanical rotational and calculate intensities from a formula like angular momentum (i.e., energy) (see Hund’s case (b) in ˇ ˇ ˇ˝ ˛ˇ2 (35.58) Sect. 35.7.4) and J indicates the actual change (i.e., quanI ˇ v10 v20 ˇM.Q/ˇ v100 v200 ˇ : tum mechanical). The labels c 0 c 00 indicate the components For homonuclear diatomic molecules, the dipole moment involved. Thus a P Q21 branch is expected to be red shaded vanishes identically, so there is no rovibrational spectrum. (R D 1) with J D 0 involving the second component of The dipole moment for heteronuclear diatomics is often close the upper state and the first (lowest) component of the lower to linear in the internuclear distance. The harmonic oscilla- state. The notation P Q21 .J / (or sometimes P Q21 .R D N / tor model suggests that transitions with v D ˙1 are the for † lower states) identifies an individual rotational line and strongest, with intensities approximated by specifies the rotational quantum number of the lower state involved in the transition. If the upper- and lower-state comˇ ˇ ˇ dM ˇ2 ˇ .v C 1/ : (35.59) ponent numbers are the same, one of them may be dropped. IvC1;v ˇˇ dR ˇ Thus P22 is sometimes written as P2 .
35
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D. L. Huestis
For symmetric top molecules, the rotational branches are labeled JK (e.g., P1 ). For asymmetric tops, the branches are labeled by JK1 ;K1 , where K1 and K1 are the prolate- and oblate-limit angular momenta projections (described in Sect. 35.3.2). The transition dipole moment commonly lies parallel or perpendicular to the body–frame axis. In the former case, K 0 K 00 vanishes for K 0 ¤ K 00 , and in the latter for K 0 D K 00 . Thus parallel bands correspond to K D 0 transitions, while perpendicular bands have K D ˙1. As enforced by the vector-coupling coefficients or Hönl–London factors described below, for low values of K (e.g., diatomics), K D 0 implies strong J D ˙1 (P and R) branches and weak J D 0 (Q) branches. On the other hand, K D ˙1 leads to Q-branches that are approximately twice as strong as either the P - or R-branches.
where ˇ ˛
˝ ˇ K 0 K 00 D K 0 ˇTq.L/ .body/ˇK 00 K 00 K 0
(35.65)
is the body–frame transition-moment matrix introduced in Sect. 35.3.4, with relative values that are hypothesized based on interpretation of the nature of the transition, calculated from ab initio wave functions, or inferred by fitting the observed rotational branch strengths. The Clebsch–Gordan expression ˇ 1=2 ˝ 00 00
˛
J K ; LK 0 K 00 ˇJ 0 K 0 J 0 ; K 0 ; J 00 ; K 00 D 2J 00 C 1
K 00 K 0 J 0 J 00 (35.66)
represents the transformation of the radiation field from the laboratory-frame to the body–frame, also related to the direction cosines used by many authors. The additional phase 35.7.2 Hönl–London Factors factors ( The matrix model Hamiltonians for the upper and lower C1 k 0 (35.67) .k/ D sgn.k/ D states have been diagonalized, yielding the wave functions 1 k < 0 !1=2
0 X 0 C 1 2J 0 0 J M J 0 have been included here to make the signs D bJK0 c 0 DM 0 K 0 .lab/ K 0 .body/ c0 and symmetry 2 8 0 0 00 00 0 0 00 K relations of K K and J ; K ; J ; K agree with those (35.61) already in use [9, 44, 55, 56]. They are related to the choice of the leading signs when TC and T are expressed as and ˙.Tx C iTy / and ˙.Tx iTy /. Their inclusion has no effect
00 !1=2 X 00 2J C1 00 00 for spin-allowed transitions with only one source of transiJ 00 ˆJc 00 M D aJK00 c 00 DM 00 K 00 .lab/ K 00 .body/ : 2 8 tion probability, e.g., purely parallel or perpendicular. K 00 In the usual case of electric-dipole (or magnetic-dipole) (35.62) radiation (i.e., L D 1), 2 isthe well-known Hönl–London In these expressions, the designations K 0 and K 00 are slightly factor [57]. J 0 ; K 0 ; J 00 ; K 00 is a real, signed quantity: negsymbolic. They represent the body–frame projection of the ative for JK > 0; or J D K D 0 and K < 0; otherwise total angular momentum and also a running index over basis positive [9]. functions. For complicated cases, more than one basis funcSetting L D 2 provides intensity formulas for electriction can have a given value of K. quadrupole [58], Raman [59], and two-photon [60, 61] Following Sect. 35.2.3, the rotational branch strength is transitions. Additional Rayleigh-like terms can appear for then written as K 0 D K 00 ¤ 0. Halpern et al. [61] also give formulas for X ˇ˝ 0 0 ˇ ˇ J 00 M 00 ˛ˇ2 c 0 c 00 J M .L/ ˇ ˇ 0 ˇT .lab/ˇˆc 00 SJ 0 J 00 D three-photon transitions in diatomics, expressed in terms of p c pM 0 M 00 Clebsch–Gordan coefficients with L D 3 and L D 1 (for ˇ ˇ X j*j 1). ˇ ˇ ˝ ˛ 0 K 00 ˇ ˇ .L/ .body/ˇK 00 D .2J 00 C 1/ˇ bJK 0 c 0 aJ 00 c 00 K 0 Tq ˇ 0 0 00 qKK ˇ2 ˇ 0 0 ˛ˇˇ 35.7.3 Sum Rules ˝ 00 00 ˇ J K ; Lq J K ˇ ˇ (35.63) The orthonormality relations for component eigenvectors or S
c 0 c 00 J 0 J 00
ˇ ˇ2 ˇX ˇ
0 00 ˇ K 0 0 00 00 ˇ Dˇ bJK ˇ ; 0 c 0 aJ 00 c 00 K 0 K 00 J ; K ; J ; K ˇ 0 00 ˇ
X
(35.68)
c0
X
KK
(35.64)
0
K bJK 0 c 0 bJ 0 c 0 D ıK 0 ;K ;
c 00
00
aJK00 c 00 aJK00 c 00 D ıK 00 ;K
(35.69)
35 Radiative Transition Probabilities
549
can be used to construct the sum rule Xˇ X 0 00 ˇ
ˇK 0 K 00 J 0 ; K 0 ; J 00 ; K 00 ˇ2 : SJc 0cJ 00 D c 0 c 00
(35.70)
K 0 K 00
Finally, the orthonormality relations of the Clebsch–Gordan coefficients result in X 0 00 ˇ
Xˇ ˇK 0 K 00 ˇ2 ; SJc 0cJ 00 D 2J 00 C 1 (35.71) J 0 c 0 c 00
X
S
J 00 c 0 c 00
c 0 c 00 J 0 J 00
K 0 K 00
ˇ Xˇ ˇK 0 K 00 ˇ2 : D 2J 0 C 1
(35.72)
K 0 K 00
As discussed in Sect. 35.5.2, it is convenient to have the K 0 K 00 matrix elements consist of numbers that represent the nature of the transition but not its strength, the latter being expressed by the vibrational (qv0 v00 ) and electronic (Re ) contributions. Following Sect. 35.5.2, for diatomic molecules, the orientational part K 0 K 00 is taken to have a fixed sum rule X
jK 0 K 00 j2 D max Nc0 ; Nc00 ;
(35.73)
K 0 K 00
where Nc0 is the number of components (K 0 values, or basis functions) for the upper state, and Nc00 is the number of components in the lower state. For polyatomic molecules, the sum rule can be written as
.2J C 1/2 2a C 2b C 2c max Nc0 ; Nc00 ; where 2a C 2b C 2c D jC j2 C j j2 C j0 j2 D 1
of energy levels. The advance of precision measurement of transition energies and the availability of sophisticated parametrical matrix models and fast computers on which to realize them, has reduced the importance of Hund’s cases for actual computations. In particular, the need to derive and implement numerous explicit energy and intensity formulas leads to unfortunate transcription errors. Nevertheless, they remain of value for qualitative and pedagogical understanding, especially for estimates of the relative intensities of rotational branches. Hund’s case (a) describes the situation in which ƒ and † are separately well-defined. This is a common case in which the separation between electronic states, i.e., different values of jƒj, is larger than the spin-orbit interaction, which in turn, is larger than the separation between rotational levels. At low J , there are .2S C 1/ pairs of nearly-degenerate energy levels separated from each other by the spin orbit constant: E Aƒ† C BJ.J C 1/. The wave functions are of the form JM *˙
.2J C 1/ 1=2 1 Dp 8 2 2
J J DMƒC† jƒ; †i ˙ DM ƒ† j ƒ; †i 1 D p Œjƒ; †I J; M; *i 2 ˙ j ƒ; †I J; M; *i ; (35.76)
K (35.74) with only two nonzero expansion coefficients aJ c . If both the upper and lower states are well described by Hund’s case (a), then each lower component is optically connected only to the upper components with the same value of j†j. Then
(35.75)
ˇ
ˇ˝ ˛ˇ2 0 00 SJ*0 J*00 D 2J 00 C 1 ˇ J 00 *00 ; 1*0 *ˇJ 0 *0 ˇ ;
(35.77) and Nc0 and Nc00 are the numbers of spin-electronic00 00 0 0 vibrational components in the upper and lower states, respec- with * D ƒ C † and * D ƒ C †. Hund’s case (b) indicates that ƒ is well-defined, but spintively. Also see Whiting et al. [44, 45] and Brown et al. [7]. orbit coupling is weak. The components correspond to welldefined values of N D J S , ranging from jJ Sj to .J C S/, with energies approximated by E BN.N C 1/, and 35.7.4 Hund’s Cases wave functions of the form X In diatomic molecules, several limiting cases are useful as JM .1/S C† hJ *; S†jN ƒi N˙ D short-hand or first-approximation concepts for classification † of energy levels and rotational branch strengths. These are 1 called the Hund’s cases [62–64]. They are distinguished p Œjƒ; †I J; M; *i 2 by the extent to which the electron orbital and spin angu˙ j ƒ; †I J; M; *i : (35.78) lar momenta are rigidly attached to the tumbling molecular frame, i.e., whether ƒ, †, and S are good quantum numbers. Hund’s cases are discussed in many journal articles and This equation is derivable from Mizushima’s equation (2-3in every textbook dealing with the rotational structure of di- 26) [14] and Zare’s equations (2.8), (2.26), and (3.105) [5], atomic spectra. An appealing recent description is provided using the lab-to-body transformation by Nikitin and Zare [65]. X S DM ./jS†i .body/ : (35.79) jSMS i .lab/ D In most works, the emphasis has been on finding a faS† † vorable zero-order approximation for perturbation expansion
35
550
It disagrees with Judd’s problem 9.1 [66] by a phase factor .1/J C2S C†N but agrees with Mizushima’s expansion of a 3 … state [14, p. 287] if the Clebsch–Gordan coefficients are taken from Condon and Shortley [39, p. 76]. If both the upper and lower states are well described by Hund’s case (b), these wave functions can be substituted into the general rotational-branch strength equations above. Following Edmonds [67, (6.2.8) and (6.2.13), and Table 5] yields the square of a product of Clebsch–Gordan and Racah coefficients
0 00 SJN0 JN00 D 2J 00 C 1 2J 0 C 1 2N 00 C 1 ˇ˝ ˇ ˛ ˇ N 00 ƒ00 ; 1ƒ0 ƒ00 ˇN 0 ƒ0
ˇ2 W N 0 ; J 00 ; N 00 ; J 0 I S; 1 ˇ : (35.80)
D. L. Huestis
cipal quantum number (n) increases. Spin-orbit coupling between these projections also diminishes. The eigenfunction components correspond to well-defined values of R D J L, ranging from jJ Lj to (J C L), with energies approximated by E BR.R C 1/, and wave functions of the form X JM .1/LCƒ hJ ƒ; LƒjR0ijLƒI JMƒi : RL D ƒ
(35.82)
Carroll [74] used intensity formulas provided by Kovacs [58]
1 to analyze, by spectral simulation, the 4p–15p †C u and 1 …u Rydberg states of N2 excited from the ground X 1 †C g. Three strong Q-form branches survive, corresponding to † and one from † † R0 D J 00 , two arising from … 0 00 D J 2) and case (a) branches. The remaining O-form (R The Clebsch–Gordan coefficient enforces the case (b) selec0 00 tion rule N D 0; ˙1, while the Racah coefficient provides S-form (R D J C 2) branches fade rapidly as n increases. the J D N propensity rule, which becomes more pre- With the phase conventions used here, this situation correcise as N increases. A similar propensity rule, F D J , is sponds to body–frame transition matrix elements satisfying common for transitions between hyperfine components (see 00 D 10 D 10 : (35.83) also Femenias [68]). Hund’s case (c) corresponds to the situation in which spin- In the opposite case, corresponding to a parity change of the orbit coupling is so strong that each level described by the parent-ion core [73], two of the Q-form branches are extinprojection * actually consists of multiple values of jƒj (e.g., guished, while one Q-form, one O-form, and one S-form mixing of † and … states) or multiple values of S (e.g., mix- branch remain. The transition matrix elements would satisfy ing of singlet and triplet spins). This limiting case is formally similar to Hund’s case (a), but no assumptions can be made 00 D 0 ; 10 D 10 : (35.84) about the relative magnitudes of transition-moment matrix elements *0 *00 . Any of which can be nonzero for j*j 1, Hund’s case (d) polyatomics are also known [8]. for example Hund’s case (e) would correspond to a situation in which L and S are strongly coupled to each other, but neither is
1ˇ *0 *00 strongly coupled to the internuclear axis. No examples are SJ 0CJ 00 ˙ D ˇ*0 *00 J 0 ; *0 ; J 00 ; *00 4
known for bound states of molecules. ˙ *0 *00 J 0 ; *0 ; J 00 ; *00
C *0 *00 J 0 ; *0 ; J 00 ; *00
ˇ2 ˙ *0 *00 J 0 ; *0 ; J 00 ; *00 ˇ : (35.81) 35.7.5 Symmetric Tops
The symmetry (sign) relations between *0 *00 J 0 ; *0 ; J 00 ; For transitions between nondegenerate vibronic states, the
*00 and *0 *00 J 0 ; *0 ; J 00 ; *00 determine whether transition moment must lie along the principal top axis, leadthis transition occurs only for J D ˙1 (P - and R-branches) ing to the selection rule K D 0. Otherwise, Hougen’s conor only for J D 0 (Q-branches). venient quantum number [33] G D ƒ C l K, provides the The interest and complexity of Hund’s case (c) were exem- selection rule G D 0; ˙n (for an n-fold major symmetry plified by a seminal work by Kopp and Hougen [69], who con- axis) (Sects. 35.3.2, 35.4.3, and Chap. 34). Transitions with sidered *0 D 1=2, *00 D 1=2 transitions, under the assumption G D ˙n are much weaker than those with G D 0 and are that both states could consist of arbitrary mixtures of 2 †1=2 not calculable from a simple formula. Branch intensities can and 2 …1=2 character. Each of the six rotational branches shows be calculated with the Hönl–London formulas of Sect. 35.7.2. constructive or destructive interference of parallel (* D 0) and perpendicular (* D ˙1) contributions. Hund’s case (c) also describes spin-orbit mixing collisions [70] or disso- 35.7.6 Asymmetric Tops ciation to specific spin-orbit limits [71–73]. Hund’s case (d) arises in the investigation of Rydberg se- In general, no assumptions can be made about the orienries [74], in which the separation between † and … from the tation of the transition moment. The vector representations same orbital configuration approach each other as the prin- (x ; y ; z ), (a ; b ; c ), and (0 ; C1 ; 1 ) can have any
35 Radiative Transition Probabilities
551
combination of independent nonzero values. It is common For example, first-order spin-orbit mixing would lead to that one of the (a ; b ; c ) values is significantly larger than terms of the form " ˇ ˇ ˝ 0 ˛ the others, especially for planar molecules with a two-fold X S ƒ†0 ˇHSO ˇS 00 ƒ00 †00 S 0S 0 symmetry axis. In this case one obtains a type A, B, or C band, S 00S 000 00 00 D ƒ0 †0 ƒ†0 ƒ†ƒ † E 00 if a , b , c dominates, respectively [31, 75]. The tradition ƒ # ˇ ˛! ˝ 0 0 0ˇ of analytic calculation of line strengths from explicit repreS ƒ † ˇHSO ˇS 00 ƒ†00 00 00 C Sƒ†S00 ƒ00 †00 : sentations of wave functions and transition moments leads to E 0 formulas of considerable complexity, with somewhat restric(35.89) tive assumptions [35, 36, 76]. In the more general notation of Sect. 35.7.2, the rotational line strength can be written as However, care must be taken in reducing these matrix eleˇ ments using the Wigner–Eckart theorem, for example, folˇX
ˇ 0 00 K 0 K 00 lowing Lefebvre-Brion and Field [55], in order to satisfy the b 0 0 a 00 00 0 ıK 0 K 00 C C ıK 0 K 00 C1 SJ 0 J 00 D ˇ ˇ 0 00 J J ˇ 2 * D 0 requirement for matrix elements of the rotationless KK 0 0 00 00 ˇˇ C ıK 0 K 00 1 J ; K ; J ; K ˇ ; Hamiltonian. ˇ (35.85)
35.8.2 Orbitally-Forbidden Transitions where 0 D c ;
jC j D j j ;
jC j2 C j j2 D 2a C 2b :
(35.86)
Papousek and Aliev [18] and Zare [5] follow the present formulation, but with somewhat less generality with respect to wave function expansion coefficients or transition moment components.
35.8
Forbidden Transitions
35.8.1 Spin-Changing Transitions
Even if the upper and lower states share the same value of electron spin, the transition may still be forbidden. The change in orbital angular momentum may be too large, jƒj > 1, or a change in reflection parity, † ! †C , may cause the zero-order transition matrix elements to vanish. Spin-orbit mixing with other 2S C1 ƒ states, as described above, is usually the largest source of transition probability. In addition, terms in the Hamiltonian of the form J L lead to contributions to the transition strength that increase with J , and that may mix-in higher values of * than were present in the zero-order ƒ† basis set for the upper and lower states. This situation can be represented by generalizing the formula from Sect. 35.7.2, following Huestis et al. [23], ˇ ˇ2 1 ˇX ˇ X
0 00 0 00 ˇ ˇ .i / K bJK K 0 K 00 .i / J 0 ; K 0 ; J 00 ; K 00 ˇ ; SJc 0cJ 00 D ˇ 0 c 0 aJ 00 c 00 ˇ 0 00 ˇ
The formalism presented above permits simulation of any i D1 KK allowed transition or forbidden transitions mediated by spin(35.90) orbit or spin-spin perturbations, or any perturbation that is diagonal in *. For spin-allowed transitions, the transition .0/ where K 0 K 00 is the rotationless contribution (K 0 K 00 from moment matrix is taken to be diagonal in and independent .˙1/ Sect. 35.7.2) and K 0 K 00 are the new rotation-assisted terms. of the spin projection, so that The new reflection-symmetry rule is ƒ0 †0 ƒ00 †00 D ƒ0 ƒ00 ı†0 †00 :
(35.87)
.i /
0
K 0 K 00 D .1/K K
00 Ci
.i /
K 0 K 00 :
The revised square-root Hönl–London factors are For forbidden transitions, or complicated Hund’s case
.0/ J 0 ; K 0 ; J 00 ; K 00 D J 0 ; K 0 ; J 00 ; K 00 (c) mixings, the transition moment matrix elements can be considered as independent variable parameters, limited only (from Sect. 35.7.2) and by the symmetry constraint
.˙1/ J 0 ; K 0 ; J 00 ; K 00 0 00 1=2
1 ˚ 0 0 (35.88) *0 *00 D .1/* * *0 *00 ; J J C 1 K0 K0 1 D 2
J 0 ; K 0 1; J 00 ; K 00 and the fact that terms with j*j > 1 will be multiplied by
1=2
zero. Alternatively, a specific set of candidate perturbers can C J 00 J 00 C 1 K 00 K 00 ˙ 1 be selected, and the ƒ- and †-dependence of their contri
J 0 ; K 0 ; J 00 ; K 00 ˙ 1 : butions to the transition-moment matrix evaluated explicitly.
(35.91)
35 (35.92)
(35.93)
552
D. L. Huestis
As in Sect. 35.7.2 the symbols K 0 and K 00 represent ƒ0 †0 References and ƒ00 †00 when used as labels, and *0 and *00 when used 1. Longuet-Higgins, H.C.: Mol. Phys. 6, 445 (1963) as numbers (a distinction that is relevant only when S jƒj 2. Bunker, P.R.: Molecular Symmetry and Spectroscopy. Academic and ƒ ¤ 0). This formulation is more symmetric than that Press, New York (1979) proposed by Huestis et al. [23], in that it explicitly allows for 3. Wigner, E.P.: Group Theory. Academic Press, New York (1959) either the upper or lower state to be mixed by rotation (of 4. Larsson, M.: Phys. Scr. 23, 835 (1981) 5. Zare, R.N.: Angular Momentum. Wiley, New York (1988) significance only for low J and ƒ > 1).
35.9 Recent Developments Added by Mark M. Cassar. Astronomical sky spectra are important for an understanding of processes both in Earth’s and other terrestrial environments. These spectra are the background spectra obtained through the slit of a spectrometer while excluding the object of primary interest to the astronomer—the star, galaxy, etc. The sky spectrum is subsequently subtracted from the object spectrum so that the final product contains no emissions from extraneous sources—nightglow, zodiacal light, and the light of other stellar objects. This operation then leaves the astronomer with purer astronomical spectra, which can then be compared to theoretical transition probability calculations to identify emission sources. This procedure has recently been used to identify the atomic oxygen green line in the Venus night airglow [77], which relied on an understanding of molecular oxygen emissions. In addition, interpretation of the intensities of molecular oxygen emissions also furthers the understanding of the elementary processes occurring in the Earth’s atmosphere [78]. Two recent studies have focused on the radiative properties of the CaN and 39 K85 Rb molecules. In the former study, the radiative transition probabilities and lifetimes for the A4 … X 4 † and B 4 † X 4 † band systems were calculated [79]. These results will in turn facilitate future spectroscopic studies of CaN showing the essential interplay between theory and experiment, which is required for a deeper understanding of these processes. (Radiative properties are sensitive to electronic coupling schemes and to configuration interaction, and thus present an important testing ground for theoretical models [80, 81].) The second study provides quantitative estimates for the radiative cooling of heteronuclear translationally ultracold molecules [82, 83]. By calculating the radiative transition probabilities for 39 K85 Rb, which lead to the radiative lifetime through the total Einstein A coefficient, it has been shown that under appropriate laboratory conditions such a cooling process is not relevant [84]. Acknowledgements This work was supported by the NSF Atmospheric Chemistry Program, the NASA Stratospheric Chemistry Section, and the NASA Space Physics Division.
6. Chang, E.S., Fano, U.: Phys. Rev. A 6, 173 (1972) 7. Brown, J.M., Howard, B.J., Kerr, C.M.L.: J. Mol. Spectrosc. 60, 433 (1976) 8. Helm, H., Lembo, L.J., Cosby, P.C., Huestis, D.L.: Photoionization and dissociation of the triatomic hydrogen molecule. In: Ehlotzky, F. (ed.) Fundamentals of Laser Interactions II. Springer, Berlin, Heidelberg (1989) 9. Hougen, J.T.: The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules. NBS Monograph, vol. 115. U.S. Government Printing Office, Washington, DC (1970) 10. Brown, J.M., Colbourn, E.A., Watson, J.K.G., Wayne, F.D.: J. Mol. Spectrosc. 74, 294 (1979) 11. Brown, J.M., Merer, A.J.: J. Mol. Spectrosc. 74, 488 (1979) 12. Brown, J.M., Milton, D.J., Watson, J.K.G., Zare, R.N., Albritton, D.L., Horani, M., Rostas, J.: J. Mol. Spectrosc. 90, 139 (1981) 13. Zare, R.N., Schmeltekopf, A.L., Harrop, W.J., Albritton, D.L.: J. Mol. Spectrosc. 46, 37 (1973) 14. Mizushima, M.: The Theory of Rotating Diatomic Molecules. Wiley, New York (1975) 15. Herzberg, G.: Molecular Spectra and Molecular Structure III. Electronic Spectra and Electronic Structure of Polyatomic Molecules. Van Nostrand, New York (1966) 16. Lembo, L.J., Helm, H., Huestis, D.L.: J. Chem. Phys. 90, 5299 (1989) 17. Watson, J.K.G.: Aspects of quartic and sextic centrifugal effects on rotational energy levels. In: Durig, J. (ed.) Vibrational Spectra and Structure, vol. 6, Elsevier, Amsterdam (1977) 18. Papousek, D., Aliev, M.R.: Molecular Vibrational-Rotational Spectra. Elsevier, Amsterdam (1982) 19. Wang, S.C.: Phys. Rev. 34, 243 (1929) 20. Helm, H., Cosby, P.C., Saxon, R.P., Huestis, D.L.: J. Chem. Phys. 76, 2516 (1982) 21. Townes, C.H., Schawlow, A.L.: Microwave Spectroscopy. McGraw-Hill, New York, p 1975 (1955). (reprinted, Dover, New York) 22. Luque, J., Crosley, D.R.: J. Quant. Spectrosc. Radiat. Transf. 53, 189 (1995) 23. Huestis, D.L., Copeland, R.A., Knutsen, K., Slanger, T.G., Jongma, R.T., Boogaarts, M.G.H., Meijer, G.: Can. J. Phys. 72, 1109 (1994) 24. Slanger, T.G., Huestis, D.L.: J. Chem. Phys. 78, 2274 (1983) 25. Kerr, C.M.L., Watson, J.K.G.: Can. J. Phys. 64, 36 (1986) 26. Dyer, M.J., Faris, G.W., Cosby, P.C., Huestis, D.L., Slanger, T.G.: Chem. Phys. 171, 237 (1993) 27. Di Lauro, C., Lattanzi, F., Graner, G.: Mol. Phys. 71, 1285 (1990) 28. Kronig, R.L.: Band Spectra and Molecular Structure. Cambridge Univ. Press, London (1930) 29. Brown, J.M., Hougen, J.T., Huber, K.-P., Johns, J.W.C., Kopp, I., Lefebvre-Brion, H., Merer, A.J., Ramsey, D.A., Rostas, J., Zare, R.N.: J. Mol. Spectrosc. 55, 500 (1975) 30. Wilson Jr., E.B.: J. Chem. Phys. 3, 276 (1935) 31. Herzberg, G.: Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic Molecules. Van Nostrand, New York (1945)
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553 63. 64. 65. 66. 67.
68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81.
82. 83. 84.
Hund, F.: Z. Phys. 40, 742 (1927) Hund, F.: Z. Phys. 42, 93 (1927) Nikitin, E.E., Zare, R.N.: Mol. Phys. 82, 85 (1994) Judd, B.R.: Angular Momentum Theory for Diatomic Molecules. Academic Press, New York (1975) Edmonds, A.R.: Angular Momentum in Quantum Mechanics, 2nd edn. Princeton Univ. Press, Princeton (1960). reprinted with corrections 1974 Femenias, J.L.: Phys. Rev. A 15, 1625 (1977) Kopp, I., Hougen, J.T.: Can. J. Phys. 45, 2581 (1967) Hickman, A.P., Huestis, D.L., Saxon, R.P.: J. Chem. Phys. 96, 2099 (1992) Helm, H., Cosby, P.C., Huestis, D.L.: J. Chem. Phys. 73, 2629 (1980) Helm, H., Cosby, P.C., Huestis, D.L.: Phys. Rev. A 30, 851 (1984) Cosby, P.C., Huestis, D.L.: J. Chem. Phys. 97, 6108 (1992) Carroll, P.K.: J. Chem. Phys. 58, 3597 (1973) Nielsen, H.H.: Phys. Rev. 38, 1432 (1931) Cross, P.C., Hainer, R.M., King, G.W.: J. Chem. Phys. 12, 210 (1944) Slanger, T.G., Crosby, P.C., Huestis, D.L., Bida, T.A.: Science 291, 463 (2001) Slanger, T.G., Copeland, R.A.: Chem. Rev. 103, 4731 (2003) Peligrini, M., Roberto-Neto, O., Machado, F.B.C.: Chem. Phys. Lett. 375, 9 (2003) Biémont, E., Garnir, H.P., Palmeri, P., Quinet, P., Li, Z.S., Zhang, Z.O., Svanberg, S.: Phys. Rev. A 64, 022503 (2001) Xu, H.L., Persson, A., Svanberg, S., Blagoev, K., Malcheva, G., Pentchev, V., Biémont, E., Campos, J., Ortiz, M., Mayo, R.: Phys. Rev. A 70, 042508 (1970) Stwalley, W.C., Wang, H.: J. Mol. Spectrosc. 195, 194 (1999) Bahns, J.T., Gould, P.L., Stwalley, W.C.: Adv. At. Mol. Opt. Phys. 42, 171 (2000) Zemke, W.T., Stwalley, W.C.: J. Chem. Phys. 120, 88 (2004)
David L. Huestis David L. Huestis received his PhD in Chemistry from the California Institute of Technology in 1973. He is a Fellow of the American Physical Society. His research activities include a wide range of experimental and theoretical investigations of fundamental kinetic and optical processes involving atoms, small molecules, liquids, and solids. Two major application areas have been chemical kinetics and optical physics of high-power visible and ultraviolet gas lasers and the optical emissions of terrestrial and planetary atmospheres.
35
Molecular Photodissociation
36
Abigail J. Dobbyn, David H. Mordaunt, and Reinhard Schinke
Contents
Introduction
36.1 36.1.1 36.1.2
Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 557 Scalar Properties . . . . . . . . . . . . . . . . . . . . . . . 557 Vector Correlations . . . . . . . . . . . . . . . . . . . . . . 557
36.2
Experimental Techniques . . . . . . . . . . . . . . . . . 558
36.3
Theoretical Techniques . . . . . . . . . . . . . . . . . . . 559
36.4 36.4.1 36.4.2 36.4.3
Concepts in Dissociation . Direct Dissociation . . . . . Vibrational Predissociation Electronic Predissociation .
36.5
Recent Developments . . . . . . . . . . . . . . . . . . . . 562
36.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
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Abstract
Molecular photodissociation is the photoinitiated fragmentation of a bound molecule [1]. The purpose of this chapter is to outline the ways in which molecular photodissociation is studied in the gas phase [2]. The results are particularly relevant to the investigation of the species involved in combustion and atmospheric reactions [3]. Keywords
potential energy surface absorption cross section exit channel partial cross section vector correlation wavepacket autocorrelation function
A. J. Dobbyn D. H. Mordaunt Göttingen, Germany R. Schinke () Max-Planck-Institute for Dynamics and Self-Organization Göttingen, Germany e-mail: [email protected]
Conceptually, the photodissociation process can be divided into three stages. During the first stage, the molecule absorbs a photon and is promoted to an excited state. This is generally an excited electronic state but can be a highly excited vibrational state in the ground electronic state. In the second stage, the transient complex evolves through a series of transition states, until finally, in the third stage, the molecule enters the exit channel and dissociates into the products. Schematically, this might be represented, for a triatomic molecule ABC (Fig. 36.1), as ABC C „! ! .ABC/+ ! AB.v; j / C C :
(36.1)
In the case of the triatomic molecule represented here, the dissociation involves the transformation of one of the vibrational modes to a translational, or dissociative, mode, another vibrational mode (the bending) to rotational motion of the products (j ), whilst the third vibrational mode is preserved (v). When the molecule is promoted to an electronic state that has a purely repulsive potential energy surface (PES), it undergoes very rapid dissociation, often in less than one vibrational period. This is called direct dissociation. However, the dissociation of the transient complex can be delayed, taking place over many vibrational periods. This is called indirect dissociation, or predissociation, and has been divided into three different categories [4], although as with the division between direct and indirect, this is sometimes somewhat arbitrary. Vibrational predissociation (Herzberg Type II) In this case, the transient complex is on a vibrationally adiabatic potential energy surface (this is an effective potential for the molecule when it is in a particular vibrational state v), which is not dissociative or which has a barrier to dissociation. Therefore, to dissociate it must either tunnel through the barrier, which is the only possibility for v D 0, or undergo
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_36
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˜) ABC(B ˜ B˜ – X
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ABC(A˜) σ (φ)
P[AB(X,α)] AB(X,α) + C
ABC(X˜) Dissociation coordinate R
Fig. 36.1 UV photodissociation of a triatomic molecule ABC into products AB(˛) and C, illustrating the total absorption cross section .!/, evolution of the molecular wave packet, and asymptotic product state distributions P .˛/, for direct and indirect dissociations on BQ and Q state PESs, respectively A
a nonadiabatic transition to a lower vibrational state, thereby transferring energy from a vibrational degree of freedom to the dissociative mode. This process of energy exchange is commonly called the intramolecular redistribution of vibrational energy (IVR). Rotational predissociation (Herzberg Type III) In this case, the transient complex is on a nondissociative rotationally adiabatic potential energy surface. Therefore, in a similar manner to vibrational predissociation, if it is to dissociate it must undergo a nonadiabatic transition to a lower rotational state, thereby transferring energy from rotation to the dissociative mode. Electronic predissociation (Herzberg Type I) In this case, the PES of the electronic state of the transient complex is not dissociative at the given energy, and in order to dissociate the molecule must undergo a nonadiabatic transition to a second dissociative electronic state. This involves the coupling of nuclear and electronic motion and, therefore, leads to a breakdown of the Born–Oppenheimer (BO) approximation. There are two main types of electronic predissociation. In the first case, there is only a very small coupling, and no actual crossing, between two different electronic states, and the transition between the two is driven by the very high density of vibrational states on
the second electronic state. This is called internal conversion for spin-allowed processes and intersystem crossing for spin-forbidden processes. In the second case, the transition between the electronic states is driven by strong coupling. This coupling can be vibronic (vibrational-electronic) in nature, e.g., for the Renner–Teller and Jahn–Teller effects, or purely electronic, as in the case of a conical intersection. Selection rules Two sets of selection rules apply to photodissociation. The first set governs the allowed states to which the molecule can be promoted by the photon. These selection rules are simply those for bound-state spectroscopy (Sect. 35.4). Note in particular the selection rule J D 0; ˙1. This has important practical implications since it means that a molecule that is initially rotationally cold remains so after absorption of a photon. Thus, those observables that are averaged over J will have a clear structure experimentally and will be easier to calculate theoretically. This is in contrast to scattering experiments, which in general involve a summation over many J states (Chap. 38). The second set of selection rules governs the dissociation process. The transient complex, or prepared .p/ state will undergo transitions to a final (f ) vibrational, rotational, or electronic state in order to dissociate; these transitions have their own set of selection rules. As for all selection rules, these are determined on the basis of symmetry. For a total wave function and a perturbation function W , which consists of the R coupling terms or neglected terms in the Hamiltonian, p W f d must be nonzero for a transition to take place. As W forms a part of the Hamiltonian, it is totally symmetric, and therefore the integral is nonzero only if the prepared and final state irreducible representations are equal, p D f . If there is a transition to an excited electronic state, the point groups of the initial and final states are often not the same, in which case the point group formed by the joint elements of symmetry is used, or, in the case where there is no stable geometry for one of the states, the symmetry of the potential is used. These selection rules are given in Table 52.1 for a diatomic molecule. If the motion can be separated into vibrational, rotational, and electronic parts, so that D v r e and W D W v C W r CW e , it is then possible to derive three separate selection rules: rp D rf for the rotational motion, i.e., conservation of internal angular momentum; ep D ef for the electronic motion; and vp D vf for the vibrational motion. Since the final vibrational state is in the continuum, in practice, all vibrational species ( vf ) are available at a given energy, so that the vibrational selection rule is not significantly restrictive. This separation is not possible in the case of electronic predissociation occurring through the Renner–Teller or Jahn–Teller effect, where it is necessary to consider the vibronic species of the initial and final states.
36 Molecular Photodissociation
36.1
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Observables
is given by the sum of the partial cross sections over all final product states Fundamental to any study of photodissociation is the meaX .!/ D .!; ˛/ : (36.5) surement or calculation of the characteristic properties, or ˛ observables, of the reaction, from which the underlying dynamics of the fragmentation process can be inferred. The rotational and vibrational product distributions P .!; ˛/
36.1.1 Scalar Properties The absorption cross section .!/ is a measure of the capacity of the molecule to absorb photons with frequency !. It is analogous to the line intensity in bound-state spectroscopy. Assuming that the light–matter interaction is weak (Chap. 72), and that the light pulse is on for a long time, the absorption cross section is given by O i ij2 ; .!/ / !fi jhf jE j
(36.2)
where i and f represent the initial and final states, whose energies differ by „!fi ; E is a unit vector in the direction of the polarization of the electric field, and O is the electric dipole operator of the molecule. Assuming the Born– Oppenheimer separation of electronic and nuclear motion, Eq. (36.2) can be rewritten as .!/ / !fi jhfrv jfi jirv ij2 ;
(36.3)
where the electronic transition dipole moment fi equals O ie i, and is, in general, dependent on the internal hfe jE j coordinates of the molecule. The superscripts (r, v) will be dropped from now on, and will refer to the wave function for the internal coordinates of the molecule. The absorption cross section reflects not only the nature of the transient complex but also its evolution through the transition states. For direct dissociation the absorption cross section is usually very broad and structureless. In contrast, the absorption cross section for predissociation is structured, containing lines that are normally Lorentzian in shape, and whose widths are related to the lifetime of the transient complex at that energy by D „= . The partial photodissociation cross sections .!; ˛/ are a measure of the capacity of the molecule to absorb photons with frequency ! and to yield products in quantum state ˛. They are defined by .!; ˛/ / !fi jhf˛ jfi ji ij2 ;
provide information about the amount of product formed by a photon with frequency ! in a particular rotational or vibrational state ˛. These are related to the partial cross sections by P .!; ˛/ D .!; ˛/= .!/. The rotational and vibrational product distributions reflect the nature of the transient complex as it enters the exit channel, as well as the dynamics in the exit channel. The branching ratios for different chemical species produced in photodissociation are defined as the fraction of the total number of parent molecules that produce the particular species of interest. In Eq. (36.1), the molecule ABC is dissociated into AB C C. It might equally well have dissociated to A C BC, or indeed A C B C C. It is clear then that there may be several different reaction schemes, or channels, for the photodissociation of one particular molecule. Thus, the branching ratio for forming AB is the yield of this first channel divided by the total dissociation yield into all possible channels. Further, it would sometimes be possible to produce AB, or any of the other chemical species, in various electronic, vibrational, or/and rotational quantum states. In this case, the branching ratio for forming AB(˛) is the yield of AB in the specific quantum state ˛ divided by the total yield of AB; however, this describes the branching into only this particular reaction channel. For the reaction scheme represented in Eq. (36.1), the quantum yields of the products AB and C are the same. However, for example, in the reaction ABC2 C „! ! .ABC2 /+ ! AB.v; j / C 2C ;
(36.6)
the quantum yield of C is twice that for AB. In general, the quantum yield of a particular product fragment for one reaction channel is the ratio of the number of fragments formed to the number of photons absorbed. However, it is again possible for a molecule to dissociate into various different reaction channels. In such a case, to obtain the overall quantum yield for a particular product, the quantum yield for each reaction channel must be summed over all the available reaction channels, taking into account the branching ratios for the channels.
(36.4)
where f˛ is the final wave function for the products in the quantum state ˛. The partial cross sections for direct dissociation are broad and featureless. For predissociation, similar structures are seen in the partial cross sections as in the absorption cross section. The absorption or total cross section
36.1.2 Vector Correlations Photodissociation is by its very nature an anisotropic process, as can be seen from Eq. (36.2). The operator O defines a specific axis in the molecular body-fixed frame of reference. At
36
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the instant of photoexcitation, O is preferentially aligned parallel to the polarization of the electric field E in the external laboratory space-fixed frame of reference. Hence, E defines a specific axis, and thus cylindrical symmetry in the bodyfixed frame. If fragmentation occurs on a timescale that is short compared with overall rotation of the excited complex, this correlation persists between the body-fixed frame and the space-fixed frame, and a wealth of information can be obtained. However, rotation of the transient complex prior to fragmentation serves to degrade this symmetry in the external body-fixed frame. Three vectors fully describe the photodissociation process for both the parent molecule and the products: (i) O in the body-fixed frame (and hence E , in the space-fixed frame, at the instant of photoexcitation); (ii) v, the recoil velocity of the products; and (iii) j , the rotational angular momenta of the fragments. Vector correlations can exist between all of these vectors [5]. The most commonly observed is the angular distribution of the photofragments I.; ˛/, i.e., the relation i 1 h 1 C ˇ.˛/P2 .cos / (36.7) I.; ˛/ / 4 between v and E ; P2 .x/ is the second-order Legendre polynomial, and is the angle between v and E . The anisotropy parameter ˇ.˛/ ranges between 1 for a perpendicular transition and C2 for a parallel transition. Thus, measuring the angular distribution of the fragments provides information about the type of electronic transition and, hence, the electronic symmetry of the excited state [6]. If the alignment between the body-fixed and space-fixed frames is destroyed, the angular distribution becomes isotropic and ˇ.˛/ D 0. The anisotropy parameter depends on the product channel ˛. A second vector correlation concerns the direction of j with respect to E . Fragmentation generates rotational motion in the nuclear plane; for a perpendicular transition, this is perpendicular to the plane containing the atoms, leading to the projection of j being preferentially aligned parallel to , O and thus E in the space-fixed frame. For a parallel transition, the opposite would be true, i.e., j would be aligned in the plane perpendicular to . O The alignment of j leads to polarized emission/absorption depending on whether molecules are created in an electronically excited/ground state. Therefore, the orientation of the product polarization with respect to the original photolysis polarization E also yields information about the symmetry of the electronic states involved in dissociation. The final association in this series is independent of the space-fixed frame, since v and j are both defined in the body-fixed frame. Unlike the two previous correlations, a long lifetime does not destroy the alignment, as it is not established until the bond breaks, and the two fragments recoil. For a tetratomic (or larger) molecule there are, in principle,
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a
b
Fig. 36.2 Spatial recoil anisotropies and Doppler line shape profiles for parallel and perpendicular transitions compared with an isotropic distribution
two possible sources of product rotational excitation: bending motion in a plane of the molecule producing fragments with v perpendicular to j or torsional motion leading to fragment rotation out of the plane. A prime example of this is the distinction between frisbee and propeller-type motion of the two OH fragments in the dissociation of hydrogen peroxide [7, 8]. Measurement of only the scalar properties cannot discriminate between these two possibilities, highlighting the additional information that can be gained about the bond rupture and the exit channel dynamics by the study of vector correlations (Fig. 36.2).
36.2 Experimental Techniques Early photochemical experiments used broad white-light continuum sources and large diffractometers [4]. However, it has been the development of lasers in combination with molecular-beam techniques that has dramatically increased the understanding of photodissociation processes. The everincreasing spectral and time resolutions of lasers, in addition to the power and range of wavelengths available, have made it possible to excite molecules selectively and with high efficiency. This has enabled state-specific preparation of the parent molecule, the study of time evolution, as well as the measurement of the scalar and vector properties of the asymptotic products [9]. Specification of the initial state A room-temperature sample of a gas will have a Boltzmann distribution over rotational states. Molecular beam
36 Molecular Photodissociation
techniques provide an improved specification of the initial angular momenta in the parent molecule by isentropically cooling its internal rotational energy [10]. Full quantum state specification can be achieved by various two-photon excitation schemes, e.g., stimulated emission pump (SEP) spectroscopy and vibrationally mediated dissociation. SEP is commonly used to study dissociation on the ground state PES; the molecules are excited to a stable upper electronic state, stimulated emission back to the ground state prepares a single quasi-bound state. Vibrationally mediated dissociation provides information about both ground and upper electronic states; the molecules are excited to a stable intermediate vibrational level on the ground state, and further excitation promotes this fully defined wave function to an upper dissociative electronic state [11]. Detailed measurement of the absorption cross section UV and VUV electronic spectroscopy has proven to be a very powerful tool for examining the interaction of a photon with a parent molecule. Absorption cross sections are typically measured by scanning the frequency domain and monitoring either the intensity of radiation absorbed or the flux of product molecules produced. State-specific detection of the product flux yields the partial cross section .!; ˛/. Explicit measurement of .!/ is a direct application of the Beer– Lambert law (Sect. 73.2) and, thus, depends on the length of the optical cavity; cavity ring-down spectroscopy with multiple passes through a cell provides an effective cell length of several tens of kilometers. Evolution of the transient complex The evolution of the molecular wave packet can be probed by time-resolved spectroscopy, as discussed in Chap. 37 and [12]. Real-time analysis of the molecular wave packet provides a direct insight into the forces acting during molecular photodissociation. This type of time-resolved spectroscopy and the energy-resolved spectroscopy described above are mutually exclusive due to the time-energy uncertainty principle (Sect. 84.3.1). Asymptotic properties The vast majority of photodissociation studies determine asymptotic properties of the dissociation process, measuring either internal energy, recoil velocity, or angular distributions of the dissociation products. The product state distributions are usually explicitly probed by laser-induced fluorescence (LIF), resonance-enhanced multiphoton ionization (REMPI) spectroscopy, or coherent Raman scattering with the relative populations of the products obtained via line intensities. The distribution and anisotropy of the recoil velocities are measured using Doppler spectroscopy or time-offlight (TOF) techniques [13]. Doppler spectroscopy uses the Doppler-broadening of lines in the LIF or REMPI ex-
559
citation spectra (Sect. 73.6); the profile of the line shape also depends on the recoil anisotropy of the probed species (Fig. 36.2). However, many important molecular fragments are not amenable to spectroscopic detection. Thus, though lacking the ultimate state specificity of spectroscopy, TOF techniques by virtue of their general applicability provide an appealing alternative route to determine the product state distributions. In TOF techniques, the time is recorded for photofragments to recoil a known distance from the interaction region to a detector. Due to total energy and momentum conservation, the translational energy distribution of a fragment state specifically detected directly implies the internal energy distribution of the other partner product. In less favorable cases where this is not possible, coincidence detection schemes are employed to define the partition between translational and internal energies. Rotation of the detection axis with respect to the polarization of the photolysis laser yields the recoil anisotropy. Advances in spatially resolved detection schemes are now providing an improved measure of vector correlations. Mass spectrometry can be used to measure branching ratios and quantum yields, which can also be obtained from the techniques described above.
36.3
Theoretical Techniques
The calculation of the observables of photodissociation can be carried out using quantum mechanics either in the timeindependent or the time-dependent frame, as well as using classical mechanics [1]. Theoretical studies have contributed greatly to the understanding of photodissociation processes, as they provide the ability not only to calculate the observables, but also, through the knowledge of the wave function, to view the dissociation dynamics directly. This has enabled the inference of the underlying dynamics from the observables of the reaction to be more precisely established. Due to computer limitations, the majority of quantum mechanical studies currently only treat three degrees of freedom fully, and, thus, have mainly concentrated on triatomic molecules. Jacobi coordinates are usually used, with the appropriate set for the dissociation of ABC into AB and C as follows: R, denoting the distance between the atom C and the center of mass of the AB fragment; r, denoting the internal vibration coordinate of AB; and , denoting the bending angle between R and r. The initial state i is generally taken to be a single bound state. It is obtained either by the solution of the Schrödinger equation at a particular energy (Sect. 33.4) or simply by taking a product of three Gaussians in the three coordinates, with the parameters of the Gaussians being determined from spectroscopic information on the ground state. Further, to calculate the observables it is also necessary to have information about fi . However, this is often assumed
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to be a constant, i.e., independent of the internal coordinates of the molecule. The Franck–Condon principle assumes that the nuclear geometry changes after the electronic transition and not during it. Therefore, a molecule, with a particular geometry, will, when promoted by the photon to the excited electronic state, be centered around the same geometry, which is thus referred to as the Franck–Condon region or point. To carry out any dynamical calculations it is necessary to have PESs for the electronic states involved. These are usually obtained from ab-initio calculations, which are described in Chap. 33. The accuracy of the PES surface largely determines the accuracy of the results obtained, as the PES essentially determines the dynamics of the fragmentation process. In the time-independent approach, a solution of the timeindependent Schrödinger equation O E/ ˛ D 0 ; .H
(36.8)
duced in order to take this into account. The most important of these use variational principles [14], such as that due to Kohn [15]. In the Kohn variational principle, the wave function in the inner or interaction region is expanded in a finite L2 basis. However, in the outer region, the wave function is expanded in an energy-dependent basis of outgoing and incoming waves, which are approximate solutions of the coupled channel equations. Other methods which can sometimes be used to indirectly extract information about the observables are stabilization [16] and complex scaling [17]. In the time-dependent approach [18], one solves the timedependent Schrödinger equation i„
@ O ˚.t/ ˚.t/ D H @t
(36.9)
for the wave packet ˚.t/ with initial condition ˚.0/ D i , i.e., it is assumed that the molecule is vertically promoted by an infinitely short pulse to the electronic state under consideration. The wave packet is a coherent superposition of stationary wave functions ˛ (Chap. 37), and since it comprises many of the stationary states, it contains all the information necessary to characterize the dissociation (Fig. 36.3). The total absorption cross section is given by
is sought for a specific total energy E subject to appropriate boundary conditions at infinite product separation. There are many different approaches to solving this problem, but they can be broadly separated into two groups: scattering methods ZC1 and L2 methods. The scattering methods involve the soludt S.t/ ei!t ; (36.10) .!/ / tion of the coupled channel equations described in Chap. 38. 1 These can be solved directly to yield the wave functions, which can then be used to calculate the observables, or they where the autocorrelation function S.t/ is defined as can be solved indirectly to provide similar information. The use of L2 methods, which attempt to expand in a finite baS.t/ D h˚.0/ j ˚.t/i : (36.11) sis set, is not directly applicable since the wave functions are in the continuum and spread out to infinite distances in the The autocorrelation function reflects the motion of the wave R coordinate. Thus, various modifications have been intro- packet, and is, therefore, a convenient means for visualizing
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Fig. 36.3 Time evolution of a wave packet in the dissociation of FNO in the S1 state
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36 Molecular Photodissociation
the molecular dynamics. The individual partial cross sections can be obtained in the limit t ! 1 by projection of the wave packet onto the stationary wave functions of the products, i.e., plane waves in the dissociation coordinate and vibrational-rotational wave functions for the free products.
36.4
Concepts in Dissociation
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Even for a purely repulsive PES, the potential may be very flat in the Franck–Condon region, so that the molecule may be able to undergo one internal vibration before it dissociates. In this case, there is a diffuse structure in .!/, associated with recurrences in the autocorrelation function. The spacing of the structures in .!/ are related to the period of the internal vibrations by E D 2 „=T . Diffuse structures in .!/ have also been linked to unstable periodic orbits.
There has been substantial experimental and theoretical work to elucidate the processes involved in photodissociation from 36.4.2 Vibrational Predissociation knowledge of the observables, and much progress has been made. In this section, an attempt is made to present some of Vibrational predissociation is dissociation delayed due to the trapping of the energy of the molecule in modes orthogothe simpler ideas which have emerged [1]. nal to the dissociation coordinate. It can be explained very clearly in the time-independent picture as resonances (sometimes known as Feshbach resonances), which are simply 36.4.1 Direct Dissociation extensions of the bound states into the continuum; .!/ in Direct dissociation is the very fast rupture of a bond after this case consists of a series of Lorentzian lines, whose width a molecule has been promoted to an electronic state that has is inversely proportional to the lifetime of the resonance. In a purely repulsive PES. A very clear picture of this process the case where the internal modes of the molecule are not can be obtained from wave packet calculations; the wave strongly coupled to each other or to the dissociation mode, packet that is placed on the repulsive surface moves directly these resonance states can often be assigned, with the numdown the PES and into the exit channel. The autocorrela- ber of quanta in each mode being specified. In this case, the tion function decays from one to zero in a short time and lines in .!/ form a series of progressions. The widths of does not show any recurrences, i.e., oscillations in the au- these lines, and thus the lifetimes of the resonance states, tocorrelation function. The absorption cross section .!/, often show trends relating the lifetimes to the assignment. which is the Fourier transform of the autocorrelation func- This is called mode specificity. In the case when the system tion in Eq. (36.10), is, therefore, a very broad Gaussian with is strongly mixed, it is not possible to make an assignment no structure. The breadth of .!/ is inversely proportional to of the resonances, and the lifetimes show strong fluctuations. the width of the autocorrelation function and, using simple This is called statistical state specificity [19]. The resonances can also be seen in the time-dependent classical pictures, can be taken to be approximately proportional to the steepness of the potential at the Franck–Condon picture, where the autocorrelation function shows many repoint. The partial cross sections have a similar structure to currences with periods T , depending on the fundamental the total cross section, although they have differing inten- frequencies of the internal modes ! D 2 =T (Chap. 37). The partial cross sections for vibrational predissociation sities and are shifted relative to each other on the energy also consist of Lorentzian lines, with positions and widths scale. exactly as for the total cross sections but with differing The product distributions can be predicted using simple classical pictures. These methods can be divided into intensities. The partial widths, which describe the rate of distwo groups, depending on the extent of the excitation/de- sociation into each product channel, are given by excitation, or coupling, in the exit channel. If there is very .!; ˛/ : (36.12) ˛ D little excitation/de-excitation in the exit channel, the rotatot .!/ tional and vibrational product distributions are best described using Franck–Condon mapping. Another model that gives In the weak coupling case, simple pictures can be used to good results for the rotational distributions is the impact pa- describe the product distributions. The rotational product rameter, or impulsive, model. If excitation/de-excitation in distributions can be explained using, again, the reflection the exit channel is not negligible, the product distributions principle; but in this case, instead of considering the distriare best described using the reflection principle. This relates bution of the initial wave function in , the distribution of the distribution of the initial wave packet in to the final the wave function at the transition state is used. The vibrarotational distributions through the classical excitation func- tional product distributions can often be well described by tion. Similarly, the distribution of the initial wave packet in examining vibrationally adiabatic curves. In the case when the modes are strongly coupled, the space is related to the final vibrational distributions through simple models break down. It is then sometimes possible another classical excitation function.
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to use statistical models to describe both the rates and the product distributions. One example of these unimolecularstatistical theories is the Ramsperger–Rice–Karplus–Marcus (RRKM) theory, which is widely used for the description of unimolecular dissociations [20]. Another example is phase space theory (PST) [21, 22], which is often used to calculate the product distributions for reactions that have no barrier. The quantum mechanical results fluctuate about these average values. These fluctuations, which can be considered as being independent of the system and can be described well by the predications of random matrix theory [23].
36.4.3 Electronic Predissociation Nonadiabatic transitions between two or more electronic states are a common phenomenon in photodissociation [24] as well as in other chemical reactions (Chap. 52). Such transitions can result in the production of both electronically excited and ground state fragments. Adiabatic molecular PESs can vary in complex fashions. Many of these contortions arise from avoided and real crossings of the surfaces, and in all such cases, the physical and chemical understanding is greatly facilitated by expressing the adiabats in terms of the diabatic states (Chap. 33). The electronic diabatic states are chosen to simplify the structure of the electronic wave functions by incorporating the off-diagonal or coupling elements as a pure potential energy term, rather than as a kinetic term, or a mixture of both. Under these conditions, the BO approximation is inadequate since there is coupling between the different adiabatic states, and the electronic and nuclear motion cannot be separated. Therefore, the solution of the time-dependent Schrödinger equation (Sect. 36.3) requires the set of coupled equations ! ! ! V11 C T11 1 V12 @ 1 D (36.13) i„ @t 2 V12 V22 C T22 2 to be propagated, where the coupling between the diabatic surfaces is in the potential (V ) and not the kinetic (T ) terms. The wave packet evolves on both diabatic (or adiabatic) surfaces and shows a complicated motion in moving between the two surfaces. The coupling between the nuclear and electronic motion can be thought of as resulting in the nuclear motion forcing the transfer of a valence electron to another molecular orbital. Since the efficiency of this transfer is greatest when the orbitals are degenerate, the crossings of the wave packet between the PESs are generally localized around their degeneracies. Finally, the wave packet moves out on the adiabatic surfaces towards the products, with which they are correlated.
A. J. Dobbyn et al.
36.5 Recent Developments In recent years, the field of photodissociation has seen a number of intriguing applications and comparisons between detailed experimental data and high-quality ab-initio calculations. These applications have become feasible mainly because of the possibility to construct accurate potential energy surfaces from first-principle electronic structure calculations. Cases in which the fragmentation proceeds via two or several electronic states have been especially concentrated on [25]. In these cases, the Born–Oppenheimer approximation is not valid, and the coupling between electronic and nuclear degrees of freedom is essential (Sect. 36.4.3). A nice example is the photodissociation of water in the second absorption band. Since water has only ten electrons, highly accurate potential energy surfaces have been calculated theoretically, and these have been used in extensive dynamics calculations – including motion on three potential energy surfaces [26]. The agreement between the calculated and the measured absorption cross section at room temperature is outstanding [27]. From the elaborate analysis of product state distributions (rotational, vibrational, and electronic), many details about the coupled motion on several potential energy surfaces have been learned [28, 29]. The electronic density of water is small, and therefore the photodissociation can be treated on a nearly exact level. For other triatomic molecules, with more electrons and a higher density of electronic states, this is generally not feasible. An important example is ozone, which plays a vital role in the atmosphere. The electronic structure of O3 is illustrated in Fig. 36.4, where many spinallowed as well as spin-forbidden fragmentation pathways are seen [30]. The photodissociation of ozone in the UV range has been the target of many experimental studies [31]. The interpretation of the many experimental results on the basis of realistic potential energy surfaces is a great challenge for theoretical chemistry. Photodissociation studies are particularly rewarding if the lifetime in the excited electronic state is long because then the absorption spectrum shows well-resolved lines (resonances), the widths of which are inversely related to the state-specific lifetimes [32]. A typical situation is the excitation of a particular vibrational-rotational state in a bound electronic state, which can decay only via coupling to a dissociative electronic state. The lifetime then reflects the coupling of this state to the continuum of the dissociative state (predissociation). An example is the photodissociation of HCO [33, 34]; in this case, the upper and the lower state are coupled by Renner– Teller coupling. A similar example is the photodissociation of HNO. For this molecule, the lower state has a deep potential well that supports long-lived states in its own continuum. The mixing between the quasi-bound states of the upper state with the resonance states of the lower state leads to interest-
36 Molecular Photodissociation
563
Potential energy (eV) B R
6
1
A
5
3
4 3
A' A'
A
1010 109
R
1
108 2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0 R1 (a0)
Potential energy (eV)
107 106
6
0
1000
1
A'' 3 A''
5
2000
3000
4000 E † (cm–1)
Fig. 36.5 Overview of the calculated dissociation rates of HOCl as a function of the excess energy. The solid line is the prediction of a statistical model. After [42]
4 3 2 1 0 2.0
1012 1011
B
2
0 2.0
k(E †) (s –1) 1013
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0 R1 (a0)
Fig. 36.4 Electronic structure of ozone. Shown are cuts through the potential energy surfaces for the singlet and triplet states. After [30]
ing behavior in the lifetime as a function of the rotational quantum number [35] (resonance between resonances). Resonances are also prominent features of ground state potential energy surfaces; they are the continuation of the true bound states into the continuum (Sect. 36.4.2). Since resonances determine the kinetics of chemical reactions, they are usually studied in the framework of unimolecular dissociations or unimolecular reactions [36]. On the other hand, these resonance states can be excited by photons, and therefore it is meaningful to discuss them also in the context of photodissociation. In the past few years, numerical methods have been developed to efficiently calculate the resonance parameters [37–39]; see [36] for a comprehensive overview. Several triatomic molecules with dramatically different intramolecular dynamics have been investigated. The main observation is a strong fluctuation in the resonance lifetimes over several orders of magnitude, even for molecules whose classical dynamics is chaotic, such as NO2 [40]. Figure 36.5 shows the results for HOCl ! HO C Cl [41, 42]. The large fluctuations of the lifetimes (or dissociation rates) are believed to affect the fall-off behavior of recombination rate coefficients [43].
The concept of first calculating a potential energy surface as function of all coordinates and then performing dynamics calculations is suitable only for triatomics. For molecules with more than four atoms, it is not applicable, simply because of the rapidly increasing number of degrees of freedom. For larger molecules, direct dynamics simulations, in which the methodology of classical trajectory simulations is coupled directly to electronic structure calculations, are the method of choice [44, 45]. In these simulations, the derivatives of the potential, which are required for the numerical integration of the equations of motion, are obtained directly from electronic structure theory without the need for an analytic potential energy surface. An important application of direct dynamics is the study of post-transition state intramolecular and unimolecular dynamics. When the dissociation proceeds through a transition state, it may be sufficient to start trajectories at the transition state and to follow them into the product channels [46, 47]
36.6 Summary Photodissociation of polyatomic molecules is an ideal field for studying the details of molecular dynamics. The primary goal of the experimental and theoretical approaches (Sects. 36.2 and 36.3) is to understand the connection between the observables (Sect. 36.1) and the underlying chemical dynamics (Sect. 36.4). Once this connection has been established, it is possible to have a detailed understanding of the dissociation dynamics, transition state geometries, and the PESs that ultimately govern the molecular chemical reactivity. The interplay between powerful experimental and
36
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theoretical techniques has enabled this goal to be realized for many photodissociation reactions.
References 1. Schinke, R.: Photodissociation Dynamics. Cambridge University Press, Cambridge (1993) 2. Okabe, H.: Photochemistry of Small Molecules. Wiley, New York (1978) 3. Wayne, R.P.: Chemistry of Atmospheres, 2nd edn. Oxford University Press, New York (1991) 4. Herzberg, G.: Molecular Spectra and Molecular Structure III, Electronic Spectra and Electronic Structure of Polyatomic Molecules. Van Nostrand, New York (1967) 5. Hall, G.E., Houston, P.L.: Ann. Rev. Phys. Chem. 40, 375 (1989) 6. Zare, R.N.: Angular Momentum. Wiley, New York (1988) 7. Grunewald, A.U., Gericke, K.-H., Comes, F.J.: J. Chem. Phys. 87, 5709 (1987) 8. Grunewald, A.U., Gericke, K.-H., Comes, F.J.: J. Chem. Phys. 89, 345 (1988) 9. Ashfold, M.N.R., Baggott, J.E. (eds.): Molecular Photodissociation Dynamics. Royal Society of Chemistry, London (1987) 10. Scoles, G. (ed.): Atomic and Molecular Beams Methods. Oxford University Press, New York (1988) 11. Crim, F.F.: Ann. Rev. Phys. Chem. 44, 397 (1993) 12. Manz, J., Wöste, L. (eds.): Femtosecond Chemistry. VCH, Weinheim (1995) 13. Ashfold, M.N.R., Lambert, I.R., Mordaunt, D.H., Morley, G.P., Western, C.M.: J. Phys. Chem. 96, 2938 (1992) 14. Nesbet, R.K.: Variational Methods in Electron–Atom Scattering Theory. Plenum, New York (1980) 15. Zhang, J.Z.H., Miller, W.H.: J. Phys. Chem. 92, 1811 (1990) 16. Lefebvre, R.: J. Phys. Chem. 89, 4201 (1985) 17. Reinhardt, W.P.: Ann. Rev. Phys. Chem. 33, 223 (1982) 18. Kosloff, R.: J. Phys. Chem. 92, 2087 (1988) 19. Hase, W.L., Cho, S., Lu, D., Swamy, K.N.: Chem Phys 139, 1 (1989) 20. Robinson, P.J., Holbrook, K.A.: Unimolecular Reactions. Wiley, London (1972) 21. Pechukas, P., Light, J.C.: J Chem Phys 42, 3285 (1965) 22. Pechukas, P., Light, J.C., Rankin, C.: J Chem Phys 44, 794 (1966) 23. Brody, T.A., Flores, J., French, J.B., Mello, P.A., Pandey, A., Wong, S.S.M.: Rev. Mod. Phys. 53, 385 (1981) 24. Dixon, R.N.: Chem. Soc. Rev. 23, 375 (1994) 25. Schinke, R.: Quantum mechanical studies of photodissociation dynamics using accurate global potential energy surfaces. In: Domcke, W., Yarkony, D.R., Köppel, H. (eds.) Conical Intersections. World Scientific, Singapore (2004) 26. van Harrevelt, R., van Hemert, M.C.: J Chem Phys 112, 5777 (2000) 27. Cheng, B.-M., Chung, C.-Y., Bahou, M., Lee, Y.-P., Lee, L.C., van Harrevelt, R., van Hemert, M.C.: J. Chem. Phys. 120, 224 (2004)
A. J. Dobbyn et al. 28. Fillion, J.H., van Harrevelt, R., Ruiz, J., Castillejo, M., Zanganeh, A.H., Lemaire, J.L., van Hemert, M.C., Rostas, F.: J. Phys. Chem. A. 105, 11414 (2001) 29. Harich, S.A., Yang, X.F., Yang, X., van Harrevelt, R., van Hemert, M.C.: Phys. Rev. Lett. 87, 263001 (2001) 30. Zhu, H., Qu, Z.-W., Tashiro, M., Schinke, R.: Chem Phys Lett 384, 45 (2004) 31. Matsumi, Y., Kawasaki, M.: Chem. Rev. 103, 4767 (2003) 32. Schinke, R., Keller, H.-M., Stumpf, M., Dobbyn, A.J.: J. Phys. B 28, 3081 (1995) 33. Neyer, D.W., Houston, P.L.: The HCO potential energy surface; probes using molecular scattering and photodissociation. In: Liu, K., Wagner, A. (eds.) The Chemical Dynamics and Kinetics of Small Radicals. World Scientific, Singapore (1994) 34. Weiß, J., Schinke, R., Mandelshtam, V.A.: J Chem Phys 113, 4588 (2000) 35. Weiß, J., Schinke, R.: J Chem Phys 115, 3173 (2001) 36. Grebenshchikov, S.Y., Schinke, R., Hase, W.L.: State-specific dynamics of unimolecular dissociation. In: Green, N. (ed.) Unimolecular Kinetics. Elsevier, Amsterdam (2003) 37. Moiseyev, N.: Phys Rep 302, 211 (1998) 38. Mandelshtam, V.A., Taylor, H.S.: J Chem Phys 102, 7390 (1995) 39. Mandelshtam, V.A., Taylor, H.S.: J Chem Phys 106, 5085 (1997) 40. Delon, A., Reiche, F., Abel, B., Grebenshchikov, S.Yu., Schinke, R.: J. Phys. Chem. A 104, 10374 (2000) 41. Skokov, S., Bowman, J.M.: J Chem Phys 110, 9789 (1999) 42. Hauschildt, J., Weiß, J., Beck, C., Grebenshchikov, S. Yu., Düren, R., Schinke, R., Koput, J.: Chem. Phys. Lett. 300, 569 (1999) 43. Hippler, H., Krasteva, N., Striebel, F.: Phys. Chem. Chem. Phys. 6, 3383 (2004) 44. Sun, L., Hase, W.L.: Born–Oppenheimer direct dynamics classical trajectory simulations. In: Lipkowitz, K.B., Larter, R., Cundari, T.R. (eds.) Review in Computational Chemistry, vol. 19, WileyVCH, Hoboken, NJ (2003) 45. Hase, W.L., Song, K., Gordon, M.S.: Comp. Sci. Eng. 5, 36 (2003) 46. Bolton, K., Schlegel, H.B., Hase, W.L., Song, K.: Phys. Chem. Chem. Phys. 1, 999 (1999) 47. Chen, W., Hase, W.L., Schlegel, H.B.: Chem Phys Lett 228, 436 (1994)
Reinhard Schinke Dr Reinhard Schinke received his PhD from the Physics department of the University of Kaiserslautern in 1976. His main areas of research are molecular dynamics, in particular energy transfer in atomic collisions, chemical reactions, photodissociation, and recombination processes. He retired in 2015.
Time Resolved Molecular Dynamics Volker Engel
and Patrick Nuernberger
Contents 37.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 565
37.2
The Principle of Time-Resolved Spectroscopy . . . . 566
37.3
Pump-Probe Scheme . . . . . . . . . . . . . . . . . . . . 568
37.4
Transient Absorption in the Liquid Phase . . . . . . 568
37.5 37.5.1 37.5.2
Further Implementations . . . . . . . . . Coherent Two-Dimensional Spectroscopy Ultrafast Dynamics Studied with X-Ray and Electron Pulses . . . . . . . . . . . . . . Dynamics and Control . . . . . . . . . . . .
37.5.3
37
. . . . . . . . 570 . . . . . . . . 570
developments in the ultrafast spectroscopy of molecular systems are outlined. These recent approaches comprise coherent two-dimensional spectroscopy, X-ray and electron diffraction, and also the possibility of quantum control with shaped laser pulses. Keywords
ultrashort pulse wave packet dynamics transient absorption coherent 2-D spectroscopy
. . . . . . . . 570 . . . . . . . . 571
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572
Abstract
The interaction of molecules with ultrashort laser pulses results in the preparation of nonstationary quantum states and may trigger photophysical and photochemical processes. As a result, measurements of molecular observables explicitly depend on time, thus reflecting the underlying dynamics. This chapter presents a theoretical description of a so-called pump-probe scheme that involves the preparation and detection of the molecular motion. As a particular example, the experimental realization of transient absorption spectroscopy is described. Utilizing this technique, a photochemical reaction proceeding differently in dependence on the solvent environment is discussed, illustrating the versatility of pump-probe measurements for investigating the dynamics of chemical reactions in real time. Further topical experimental V. Engel () Institute of Physical and Theoretical Chemistry, Universität Würzburg Würzburg, Germany e-mail: [email protected] P. Nuernberger Institute of Physical and Theoretical Chemistry, Universität Regensburg Regensburg, Germany e-mail: [email protected]
37.1 Introduction Time-resolved experiments have been performed on a multitude of molecular systems. Applications of spectroscopic techniques that work in the time domain range from the detection of simple vibrational motion of a diatomic molecule to the direct determination of relaxation times in polyatomic molecules in a liquid environment, or the recording of isomerization processes in biomolecules. The underlying principles of these experiments are more or less the same. Here, we summarize the basic ideas of transient spectroscopy. First, the quantum-mechanical description of a so-called “pump-probe” scheme is given, where two timedelayed interactions prepare and detect molecular dynamics (Sect. 37.2). This is followed in Sect. 37.3 by a summary of experimental realizations of the pump-probe scheme. To further illustrate the latter, an example of transient absorption measurements in the liquid phase is presented in Sect. 37.4. Finally, more recent developments and additional aspects are outlined in Sect. 37.5. There exists an extensive amount of literature on the topic of time-resolved molecular dynamics. To follow the developments over the years and keep track of the evolution of the exciting field and its latest innovations, it is instructive to consult the book series related to the biennial “International Conference on Ultrafast Phenomena” [1–11].
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_37
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37.2 The Principle of Time-Resolved Spectroscopy We exemplarily look at the phenomenon of wave packets and discuss how to coherently induce and subsequently observe them with ultrashort laser pulses. Regard a molecule with Hamiltonian HO 0 so that the time-independent Schrödinger equation reads HO 0 j
ni
D En j
ni
:
The obvious principle evolving from the above considerations is that a time-dependent experiment may reflect the induced dynamics, if we prepare the system in a superposition state. This is possible if it is subject to a time-dependent perturbation which, as we here are concerned with spectroscopic measurements, is a laser field. The time-dependent Schrödinger equation is then i„
(37.1)
@ j .t/i D HO 0 C WO 1 .t/ j .t/i ; @t
(37.8)
O Here, j n i are the eigenstates with eigenenergies En . The with the molecule–field interaction W1 .t/, which starts at the time-dependent Schrödinger equation for a general state reference time t D 0. Writing the state vector as X i j .t/i is j .t/i D bn .t/ e „ En t j n i ; (37.9) n @ (37.2) i„ j .t/i D HO 0 j .t/i : @t with time-dependent coefficients bn .t/, the Schrödinger If the state at the initial time t D 0 is an eigenstate, i.e., equation can, by projection, be transformed to a coupled system of linear differential equations for the coefficients, which j .0/i D j n i, the time-evolution simply reads reads j
n .t/i
i
D e „ En t j
ni
;
(37.3)
i„
X @ bn .t/ e i mn t h bm .t/ D @t n
O
m jW1 .t/j
ni
:
(37.10)
and the expectation value of an operator AO is timeindependent In order to find conditions for the preparation of wave packets, we regard the situation where the time-dependent O n .t/i D h n .0/jAj O n .0/i O hAi.t/ D h n .t/jAj interaction results in photon absorption. Inserting the explicit O D hAi.0/ : (37.4) form of the electric dipole interaction we have This means that if a measurement of an observable A with associated operator AO is performed on a molecule in an eigenstate, then the result of the measurement does not depend on time. Assume now that the state of the molecule at time t D 0 is a superposition of eigenstates so that j iD
X
cn j
ni
;
i„
with coefficients cn . Such superposition states are called wave packets, and their time dependence is given by X
i
cn e „ En t j
ni
1X @ h bm .t/ D @t 2 n
m
n
cm cn e i mn t h
O m jAj
ni
bn .t/ f1 .t/ e i.mn !1 /t :
The coefficients bm .t/ become time-independent when the (37.6) interaction is over (bm .t/ D b m ), and we have
:
Taking the expectation value of AO with respect to this state, we find that it now depends on time XX
O m jj
(37.12)
n
O hAi.t/ D
(37.11)
where O is the projection of the dipole operator on the polarization vector of the electric field, !1 the frequency of the laser pulse (commonly referred to as the pump pulse), and f1 .t/ is an envelope function. Then Eq. (37.10) takes the form
(37.5)
n
j .t/i D
1 WO 1 .t/ D O f1 .t/ e i !1 t ; 2
b m D bm .0/ C
1 X h i„ n
O m jj
ni
Imn .1/ ;
(37.13)
with the integrals ni ;
(37.7) 1 Imn .1/ D 2
Z1
dt bn .t/ f1 .t/ e i.mn!1 /t ; (37.14) with mn D .Em En /=„. The time-dependence is contained 0 in the phase factors that result in characteristic oscillations with frequencies determined by the energy differences be- where the argument .1/ refers to the properties of the pump pulse. In general, several of the coefficients will be nonzero tween the molecular eigenenergies.
37
Time Resolved Molecular Dynamics
567
so that, after the interaction, the state vector is a wave packet so that j .t/i D
X
bn e
„i En t
j
ni
:
(37.15)
d m D dm .0/ C
n
ˇ ˇ ˇ; ˇ
ˇ ˇ 2 D ˇˇ
mn
(37.16)
where the longest oscillation period Tmax is obtained for the smallest energy separation between two eigenstates. If the pulse duration Tp1 is longer than Tmax , and the excitation of the molecule is resonant with a single transition, i.e., mr nr !1 D 0, then, due to the oscillating phase factors, only one integral Imn .1/ is nonzero. This means that, for longer interaction times, a single eigenstate rather than a linear combination of states is excited. Thus, the condition for the preparation of wave packets is that the pulse duration is smaller than, or in the order of, the longest characteristic timescale for the molecular motion, i.e., Tp;1 < Tmax . Typical, oscillation periods for the vibrational motion of molecules are in the femtosecond range, so that time-resolved experiments need ultrashort pulses in the femtosecond regime. The same applies to the fastest photochemical reactions, which, for direct dissociations, take place on the sub-100fs timescale. On the other hand, rotational periods are in the order of picoseconds (and longer), so they can be resolved with ps laser pulses. Having prepared a wave packet with the pump interaction at the reference time t D 0, the question is how to detect its motion. This is realized in perturbing the system, at a defined delay time T , with another laser pulse (the probe pulse). Restricting the discussion to a second photon absorption, the interaction energy is as in Eq. (37.11), but the pulse envelope f2 .t/ starts at the delay time, and the frequency !2 may be different. The wave packet at time T is now written as j .t 0 /i D
X
i
0
dn .t 0 / e „ En .t CT / j
ni
;
(37.17)
n
where dn .0/ D b n , and the new time variable starts at t 0 D 0. Following the same procedure as above, one arrives at the coupled equations i„
1X @ dm .t 0 / D h 0 @t 2 n
O m jj
ni
Imn .2/ e i mn T ; (37.19)
The characteristic timescales for the molecular motion are where determined by the energy differences Tmn
1 X h i„ n
O m jj
ni /t 0
dn .t 0 /f2 .t 0 / e i.mn!2 e i mn T : (37.18) We may integrate this equation and regard times after the interaction when the coefficients become time-independent,
1 Imn .2/ D 2
Z1
0
dt 0 dn .t 0 / f2 .t 0 / e i.mn !2 /t :
(37.20)
0
Regarding the quantity Imn .2/ we see again that for a pulse being substantially longer than the oscillations times Tmn , the probe laser couples only levels that are in resonance with its frequency !2 . If we now regard the wave packet after the two laser interactions [Eq. (37.17) with dn .t 0 / D d n ], the expectation value of an operator AO depends not only on the propagation time t 0 but also on the delay time between the pump pulse and the probe pulse. This scheme is readily extended to experiments that involve more than two interactions. For example, in a third-order photon echo experiment, three pulses are delayed to each other. Then the expectation value of the dipole operator, which is the third-order polarization, depends on two delay times ; T and on the propagation (or detection) time t 0 . As described in the previous sections, the preparation of wave packets and the detection of their motion are essential to time-resolved spectroscopy [12–14]. If regarded in coordinate space, the probability density of a wave packet changes as a function of time. As an example, we regard a one-dimensional, coupled electron-nuclear motion. The employed model of this reduced dimensionality dynamics is such that the nuclear motion can be described with the Born–Oppenheimer approximation [15]. Then, the nuclear probability density moves in a bound-state potential, and the electronic density follows the nuclear dynamics. This is illustrated in the left-hand panels of Fig. 37.1. The densities move towards larger distances R, and upon reflection at the outer potential wall, the motion is reversed. The right-hand panels illustrate another case where the bound-state potential curve exhibits a potential barrier at R D 0, and the mean energy of the electron-nuclear wave packet is in the order of the barrier height. Then, a splitting occurs where the densities are partly reflected at the barrier. One-dimensional wave packet motion has been studied extensively using pump-probe fluorescence [16, 17]. Later, other techniques like pump-probe ionization [18, 19] or timeresolved CARS spectroscopy [20] were applied. A beautiful example is the detection of vibrational motion in the Na2 molecule via time-resolved photoelectron spectroscopy. Following theoretical studies [21, 22], it was documented that the time-dependent probability density of nuclear wave packets can be directly mapped onto the energy dependence of the photoelectron spectra [23–25].
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of ions or electrons generated by the probe pulse, e.g., from a molecular beam, a liquid microjet, a surface, or its adsor(R, t) (R, t) ρ ρ N N bate in a vacuum chamber system; optical detection may 4 comprise the emission intensity from a higher-lying state reached by the probe pulse (laser-induced fluorescence) or 2 the degree of attenuation of the probe pulse (transient absorption). 0 Generally, the signal detected reflects the temporal evolution of the system as induced by the pump pulse. In the case of wave packet dynamics, an oscillatory behavior can be de–2 tected, because the wave packet’s spatial distribution varies with time (Fig. 37.1), and properties like the ionization cross –4 section or the absorption coefficient are, thereby, transiently modulated. Beyond coherent effects, many processes can 20 20 40 40 60 60 80 t (fs) 80 t (fs) cause signal changes on an ultrafast timescale, among them r (Å) relaxation and solvation, or actual reactions originating from ρel (r, t) ρel (r, t) 4 the state reached by the pump pulse, e.g., photodissociation, energy or charge transfer, photoisomerization, rearrangement and other geometrical changes, intersystem crossing and 2 quenching, and very often a combination of several of these processes. 0 There are also ultrafast techniques that do not incorporate two laser pulses interacting with the sample. In fluorescence –2 upconversion as well as in fluorescence Kerr gating, the emitted light from a sample excited by the pump pulse is strobed by a second laser pulse (often called the gate pulse) in a non–4 linear process. Since the delay time between the pump (and 20 20 40 40 60 60 80 t (fs) 80 t (fs) thus the fluorescence) and the gate pulse is again adjustable in a well-defined fashion, the temporal emission profile is Fig. 37.1 Wave packet bound-state motion. From the solution of obtained. Great advances have also been achieved by an ulthe time-dependent Schrödinger equation for a simplified one- trashort pump pulse in combination with streak cameras or dimensional electron-nuclear motion, the nuclear ( N .R; t /) and electronic ( el .r; t /) probability densities are obtained. The motion pro- time-correlated single-photon counting, yet the best time resceeds adiabatically, so that the electronic density follows the nuclear olution can be achieved in the nonlinear approaches with geometry changes. Left-hand panels: the nuclear wave packet moves in very short laser pulses [27]. a bound-state potential outward until it is reflected at a potential wall. In the past decades, laser sources have been perpetually The electronic density adapts adiabatically to the nuclear motion. Rightimproved. In the visible spectral domain, pulse durations hand panels: the nuclear density is partly reflected at a potential barrier, and the same applies to the electronic density. The data was originally of 20 fs are routinely employed; soft X-rays even allow discussed in [15] the generation of attosecond pulses. The accessible spectral range has been continuously extended as well, so that pump-induced dynamics can be performed or probed with 37.3 Pump-Probe Scheme pulses from the far-infrared [29] or from the X-ray regime (Sect. 37.5.2), and basically all spectral regions in between. The pump-probe scheme outlined above can be realized in various ways. In the majority of experiments on ultrafast dynamics, the time resolution is obtained by employing two 37.4 Transient Absorption in the Liquid Phase laser pulses: the first one (termed pump pulse) interacts with the sample and starts the process to be studied, whereas af- Chemical reaction dynamics in solution can be elucidated ter a well-defined, adjustable delay time T , a second laser with ultrafast transient absorption spectroscopy in a unique pulse (called probe pulse) interacts and thereby monitors the fashion. A schematic experimental implementation is shown time evolution induced by the pump pulse [26–28]. In or- in Fig. 37.2a. The spectrally resolved intensity of the probe der to detect signals as a function of T , different detection beam is measured as a function of the pump-probe delay T . schemes have been realized and depend on the sample that To separate the transient from stationary absorption signals, is investigated. Nonoptical detection may record the amount the difference in optical density is determined from two R (Å)
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Fig. 37.2 Three ultrafast liquid-phase spectroscopy approaches (plotted with lenses and a minimal amount of optics for simplicity). a Transient absorption: a spectrometer determines the probe spectrum for different values of the pump-probe delay T . b Coherent 2-D spectroscopy in pumpprobe geometry. The spectrometer measures the probe spectrum for each combination of pump-probe delay T and the delay of a phase-stable pump-pulse pair, here generated by a liquid-crystal display (LCD) pulse shaper in a folded zero-dispersion compressor setup. Subsequent Fourier transformation with respect to yields 2-D spectra for each value of T . c Coherent 2-D spectroscopy in boxcars geometry: the phase-stable pump pulses and the probe pulse traverse the sample from different directions, a local oscillator heterodynes the signal field emitted. Thus, the spectrometer measures the interference of the signal field and the local oscillator for each combination of T and T (ps) 103
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measurements, one with and one without the pump pulse, respectively Ipumped .probe ; T / OD.probe ; T / D log10 I (37.21) Iunpumped .probe /
OD is positive if the probe encounters absorbing species that were not there in the absence of the pump pulse, i.e., excited reactants or products, whereas it is negative for stimulated emission from excited species or in case of the ground-state bleach caused by a reduced number of reactants in the ground state.
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axes (Fig. 37.4b), thereby allowing in an intuitive way the identification of couplings, transfer processes, or generally how a pumped species interacts with or turns into a probed one. This technique of coherent two-dimensional (2-D) spectroscopy [31–37] can be understood as an optical analogue of recording frequency-correlation spectra in 2-D nuclear magnetic resonance. Many different implementations of 2-D spectroscopy exist, all with pros and cons, the most versatile but possibly also experimentally most challenging being the boxcars geometry (Fig. 37.2c), where the three pulses all come from different directions, and the desired signal from the sample is emitted in yet another direction. With the help of an adequate local oscillator, that is a further pulse that together with the signal field impinges upon the detector, the signal field can be fully retrieved in both amplitude and phase, and thus the system’s response to the interacting fields is fully accessible. 2-D spectroscopy is successfully applied in many spectral ranges, from THz [38] to ultraviolet [39–43]. In 2-D infrared spectroscopy, vibrations are labeled by the pump event, and subsequent processes (coupling between vibrational modes, spectral diffusion, relaxation, chemical exchange, . . . ) are monitored, making it a versatile tool for assessing molec37.5 Further Implementations ular structure and associated dynamics [34, 35, 44]. 2-D electronic spectroscopy correlates electronic transitions and The arsenal of ultrafast experimental techniques is increas- has proven to be particularly powerful for unraveling elecing at an incredible speed. Besides extending to new spectral tronic couplings and exciton dynamics in light-harvesting regions and lower pulse durations, the nonlinearity of the systems [45, 46] and has opened the door to tackle the presslight–matter interaction and also the number of laser pulses ing question as to which extent coherent phenomena affect employed are adapted to unravel previously unexplored pho- functionality in chemical or biological systems [46]. tophysical and photochemical aspects. In the following, a few of these recent developments are outlined.
As an example, the photodynamics of Ph2 CN2 (diphenyldiazomethane) in different solvent environments are shown in Fig. 37.3. Upon excitation at T D 0 with a pump pulse centered at 285 nm, N2 is photolyzed off, and a reactive carbene 1 Ph2 C is formed, which is initially in a singlet configuration. The singlet absorption is observed around 355 nm (labeled S in Fig. 37.3). In the aprotic solvent MeCN (acetonitrile), 1 Ph2 C (S) will turn into a triplet 3 Ph2 C (T) within several hundred ps. In the presence of the protic solvent MeOH (methanol), a benzhydryl cation Ph2 CHC (B) or a hydrogen-bonded complex 1 Ph2 C HOMe (C) can be formed, depending on whether or not the close-by MeOH molecule is part of a hydrogen-bond network with other MeOH molecules, respectively. Both B and C react on to an ether that does not absorb in the covered spectral range. As can be seen in Fig. 37.3, the signal of the benzhydryl cation B is the most pronounced and decays the most rapidly in the absence of MeCN. Thus, these exemplary data sets emphasize that transient absorption is a powerful tool to reveal the different reaction pathways and the characteristic time scales of chemical reactions in solution.
37.5.2 37.5.1
Ultrafast Dynamics Studied with X-Ray and Electron Pulses
Coherent Two-Dimensional Spectroscopy
While in a transient absorption experiment, molecules are excited with a wavenumber Q pump and probed with Q probe (note that often frequency or wavenumber rather than wavelength is used for data presentation) after a delay time T , a transient absorption map as in Fig. 37.3 is a representation of the system response as a function of the latter two quantities only. To gain information on the role of the pump wavenumber, the spectral position of the pump pulse can be tuned, and each time a transient absorption map is recorded (Fig. 37.4a). However, the spectral bandwidth of the pump pulse limits the spectral resolution along Q pump . This can be overcome by splitting the pump pulse into a phase-stable pulse pair whose delay is scanned (e.g. by a pulse shaper as in Fig. 37.2b), a subsequent Fourier transform of the detected signal with respect to yields the dependence on Q pump . Hence, for each delay time T , which can be varied as well to capture the full dynamics, the information is spread along two frequency
In recent years, ultrashort pulses have been extended into the X-ray regime [47–52]. Laboratory-based sources often rely on a laser-induced plasma or a high-harmonic-generation (HHG) process. In the latter, photons at odd multiples of the initial laser frequency are generated in a strong-field nonlinear effect up to a spectral cutoff region, so that the radiation comprises coherent soft X-rays, which even allow the formation of attosecond pulses. Pump-probe schemes involving these soft X-rays opened a new way for unraveling nuclear and electron dynamics in atoms but also in molecules, both for gas-phase samples and in the condensed phase, providing insight into processes like charge transfer, ring opening, and photofragmentation [51–53]. Many of these experiments make use of a high photon-energy pump pulse in order to prepare core-excited states such that the following dynamics could be detected with a time-delayed optical probe pulse. In a second scenario, a visible or UV pump pulse initiates a chemical reaction that is followed by a time-delayed
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Fig. 37.4 Ultrafast spectroscopy data for the molecule 6-nitro-10 ; 30 ; 30 -trimethylspiro[2H-1-benzopyran-2; 20 -indoline] (6-nitro-BIPS). By trancentral sient absorption with different central pump wavenumbers Q pump (a), the dynamics can be investigated in a 3-D space with dimensions representing the central pump wavenumber, the pump-probe delay time T , and the probe wavenumber. 2-D spectroscopy (b) measurements can span the same 3-D space but without the disadvantage of a spectral blurring along the pump wavenumber axis as present in transient absorption as a consequence of the pump pulse’s spectral width. The graphs are adapted from [37]
X-ray pulse, which is then able to selectively excite core levels of the atoms involved in the process and thereby track the transient changes due to structural dynamics. This requires femtosecond hard X-ray pulses, as are available from storage rings with electron-bunch slicing and more recently from X-ray free-electron lasers (XFELs) delivering highintensity hard X-ray pulses with durations of a few fs. Hard X-ray pulses have helped to elucidate structural changes after photoexcitation, e.g., in iron complexes that exhibit rapid spin-crossover processes: for [FeII (bpy)3 ]2C , absorption changes near the K-edge of Fe disclose bond lengthening after spin-crossover [48], resonant inelastic X-ray scattering applied to Fe(CO)5 in an ethanol jet can separate triplet and ligation reaction channels on sub-ps time scales after photolysis of one CO [54]. These soft and hard X-ray approaches provide insight into structural dynamics not accessible in a similar fashion with ultrafast techniques in more conventional spectral regimes. A further example is the ring-opening reaction of 1,3-cyclohexadiene to 1,3,5-hexatriene, which has been extensively studied both with ultrafast spectroscopy and dynamics simulations (as e.g., summarized in [55]). The reaction dynamics could also be investigated by X-ray core ionization from an XFEL [56], the temporal evolution of the valence electronic structure was explored by soft X-rays near the carbon K-edge generated by HHG [57], while fs X-ray scattering in combination with calculated reaction trajectories provide real-space images of transient structures in this electrocyclic reaction [58].
The latter study is an example of a time-resolved X-ray diffraction experiment, an area that strongly benefits from recent advances in XFELs [49, 50]. Instead of using electromagnetic waves, a complementary approach with pulses of electrons (ultrafast electron diffraction) has been developed, with high sensitivity and spatiotemporal resolution so that structural dynamics are followed in real time [59, 60]. These powerful techniques provide an auspicious perspective for resolving structural dynamics with unprecedented possibilities, probing changes in molecular geometry rather than in energy.
37.5.3 Dynamics and Control As has been discussed throughout this chapter, molecular motion can be traced with the help of time-resolved spectroscopy. It is also possible to go one step ahead and use laser pulses to control molecular motion. For a long time, the field of “quantum control” has been dominated by theoretical work, see the textbooks [25, 61, 62]. However, the technical advances in laser technology have led to experimental realizations of proposed theoretical schemes. Employing sophisticated devices to modify the amplitude, phase, and/or polarization state of the laser pulses in a welldefined way [63], e.g., pulse shapers [64] as in Fig. 37.2b, the outcome of a chemical reaction can be influenced by specifically tailored light pulses. For the determination of the latter, feedback algorithms can be employed [65]. The range of photochemical reactions for which such an approach
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was demonstrated spans from molecular dimers in the gas phase via reactions on surfaces, organic molecules in solution, processes in solids, to large biological systems relevant in photosynthesis. It is worthwhile taking a look at one of the reviews [66–70] summarizing the accomplishments with quantum control approaches.
References 1. Conference proceedings Ultrafast Phenomena, vol. X: Springer Ser. Chem. Phys. 62 (1996) 2. Conference proceedings Ultrafast Phenomena, vol. XI: Springer Ser. Chem. Phys. 63 (1998) 3. Conference proceedings Ultrafast Phenomena, vol. XII: Springer Ser. Chem. Phys. 66 (2001) 4. Conference proceedings Ultrafast Phenomena, vol. XIII: Springer Ser. Chem. Phys. 71 (2003) 5. Conference proceedings Ultrafast Phenomena, vol. XIV: Springer Ser. Chem. Phys. 79 (2005) 6. Conference proceedings Ultrafast Phenomena, vol. XV: Springer Ser. Chem. Phys. 88 (2007) 7. Conference proceedings Ultrafast Phenomena, vol. XVI: Springer Ser. Chem. Phys. 92 (2009) 8. Conference proceedings Ultrafast Phenomena, vol. XVII: Oxford University Press, New York (2011) 9. Conference proceedings Ultrafast Phenomena, vol. XVIII: EPJ Web Conf. 41 (2013) 10. Conference proceedings Ultrafast Phenomena, vol. XIX: Springer Proc. Phys. 162 (2015) 11. Conference proceedings Ultrafast Phenomena, vol. XX: OSA Technical Digest (online) https://www.osapublishing.org/ conference.cfm?meetingid=29&yr=2016 12. Garraway, B.M., Suominen, K.-A.: Rep. Prog. Phys. 58, 365 (1995) 13. Yeazell, J., Uzer, T. (eds.): The Physics and Chemistry of Wave Packets. Wiley, New York (2000) 14. Manz, J.: Molecular wavepacket dynamics: theory for experiments 1926-1966. In: Sundström, V., Forsén, S., Eberg, U., Hjalmarsson, H. (eds.) Femtochemistry and Femtobiology: Ultrafast Reaction Dynamics at Atomic-Scale Resolution, pp. 80–319. Imperial College Press, London (1996) 15. Schaupp, T., Albert, J., Engel, V.: Eur. Phys. J. B 91, 97 (2018) 16. Rose, T.S., Rosker, M.J., Zewail, A.H.: J. Chem. Phys. 88, 6672 (1988) 17. Bowman, R.M., Dantus, M., Zewail, A.H.: Chem. Phys. Lett. 161, 297 (1989) 18. Baumert, T., Gerber, G.: Adv. Atom. Mol. Phys. 35, 163 (1995) 19. Fischer, I., Villeneuve, D.M., Vrakking, M.J.J., Stolow, A.: J. Chem. Phys. 102, 5566 (1995) 20. Materny, A., et al.: Appl. Phys. B 71, 299 (2000) 21. Seel, M., Domcke, W.: J. Chem. Phys. 95, 7806 (1991) 22. Meier, C., Engel, V.: Chem. Phys. Lett. 212, 691 (1993) 23. Assion, A., et al.: Phys. Rev. A 54, R4605 (1996) 24. Wollenhaupt, M., Engel, V., Baumert, T.: Annu. Rev. Phys. Chem. 56, 25 (2005) 25. Tannor, D.J.: Introduction to Quantum Mechanics: A TimeDependent Perspective. University Science Books, Sausalito (2007) 26. Fleming, G.R.: Chemical Applications of Ultrafast Spectroscopy. Oxford University Press, New York (1986)
V. Engel and P. Nuernberger 27. Simon, J.D.: Ultrafast Dynamics of Chemical Systems. Springer Science & Business Media, Dordrecht (1994) 28. Mukamel, S.: Principles of Nonlinear Optical Spectroscopy. Oxford University Press, New York (1995) 29. Schmuttenmaer, C.A.: Chem. Rev. 104, 1759 (2004) 30. Knorr, J., et al.: Nat. Comm. 7, 12968 (2016) 31. Mukamel, S.: Annu. Rev. Phys. Chem. 51, 691 (2000) 32. Jonas, D.M.: Annu. Rev. Phys. Chem. 54, 425 (2003) 33. Cho, M.: Two-Dimensional Optical Spectroscopy. CRC Press, Boca Raton (2009) 34. Fayer, M.: Annu. Rev. Phys. Chem. 60, 21 (2009) 35. Hamm, P., Zanni, M.: Concepts and Methods of 2D Infrared Spectroscopy. Cambridge University Press, New York (2011) 36. Fuller, F.D., Ogilvie, J.P.: Annu. Rev. Phys. Chem. 66, 667 (2015) 37. Nuernberger, P., Ruetzel, S., Brixner, T.: Angew. Chem. Int. Ed. 54, 11368 (2015) 38. Woerner, M., et al.: New J. Phys. 15, 025039 (2013) 39. Tseng, C., Matsika, S., Weinacht, T.C.: Opt. Express 17, 18788 (2009) 40. Selig, U., et al.: Opt. Lett. 35, 4178 (2010) 41. West, B.A., Moran, A.M.: J. Phys. Chem. Lett. 3, 2575 (2012) 42. Krebs, N., Pugliesi, I., Hauer, J., Riedle, E.: New J. Phys. 15, 085016 (2013) 43. Consani, C., Auböck, G., van Mourik, F., Chergui, M.: Science 339, 1586 (2013) 44. Kraack, J.P.: Top. Curr. Chem. 375, 86 (2017) 45. Brixner, T., et al.: Nature 434, 625 (2005) 46. Scholes, G.D., et al.: Nature 543, 647 (2017) 47. Pfeifer, T., Spielmann, C., Gerber, G.: Rep. Prog. Phys. 69, 443 (2006) 48. Bressler, C., Chergui, M.: Annu. Rev. Phys. Chem. 61, 263 (2010) 49. Schoenlein, R.W., Boutet, S., Minitti, M.P., Dunne, A.M.: Appl. Sci. 7, 850 (2017) 50. Barty, A., Küpper, J., Chapman, H.N.: Annu. Rev. Phys. Chem. 64, 415 (2013) 51. Krausz, F., Ivanov, M.: Rev. Mod. Phys. 81, 163 (2009) 52. Kraus, P.M., et al.: Nat. Rev. Chem. 2, 82 (2018) 53. Ramasesha, K., Leone, S.R., Neumark, D.M.: Annu. Rev. Phys. Chem. 67, 41 (2016) 54. Wernet, P., et al.: Nature 520, 78 (2015) 55. Deb, S., Weber, P.M.: Annu. Rev. Phys. Chem. 62, 19 (2011) 56. Petrovi´c, V.S., et al.: Phys. Rev. Lett. 108, 253006 (2012) 57. Attar, A.R., et al.: Science 356, 54 (2017) 58. Minitti, M.P., et al.: Phys. Rev. Lett. 114, 255501 (2015) 59. Zewail, A.H.: Annu. Rev. Phys. Chem. 57, 65 (2006) 60. Miller, R.J.D.: Annu. Rev. Phys. Chem. 65, 583 (2014) 61. Rice, S.A., Zhao, M.: Optical Control of Molecular Dynamics. Wiley, New York (2000) 62. Shapiro, M., Brumer, P.: Principles of the Quantum Control of Molecular Processes. Wiley, New York (2003) 63. Wollenhaupt, M., Assion, A., Baumert, T.: Femtosecond laser pulses: linear properties, manipulation, generation and measurement. In: Träger, F. (ed.) Springer Handbook of Lasers and Optics, pp. 937–983. Springer, New York (2007) 64. Weiner, A.M.: Opt. Commun. 284, 3669 (2011) 65. Judson, R.S., Rabitz, H.: Phys. Rev. Lett. 68, 1500 (1992) 66. Brixner, T., Gerber, G.: ChemPhysChem 4, 418 (2003) 67. Dantus, M., Lozovoy, V.V.: Chem. Rev. 104, 1813 (2004) 68. Nuernberger, P., Vogt, G., Brixner, T., Gerber, G.: Phys. Chem. Chem. Phys. 9, 2470 (2007) 69. Brif, C., Chakrabarti, R., Rabitz, H.: New J. Phys. 12, 075008 (2010) 70. Glaser, S.J., et al.: Eur. Phys. J. D 69, 279 (2015)
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Time Resolved Molecular Dynamics Volker Engel Volker Engel studied Physics at the University of Göttingen, where he also received his Doctorate in 1986. Afterwards he worked at UC Santa Barbara and the University of Freiburg. In 1994, he was appointed as Professor of Theoretical Chemistry at the University of Würzburg. The focus of his research is on time-dependent quantum mechanics and spectroscopy.
573 Patrick Nuernberger Patrick Nuernberger received his Doctorate in Physics from the University of Würzburg in 2007. He further worked at SUNY Stony Brook, Ecole Polytechnique, and RuhrUniversity Bochum. Since 2019, he has held the Chair of Physical Chemistry at the University of Regensburg. His research focuses on investigating femtosecond dynamics and photochemical reactions with advanced ultrafast spectroscopic methods and quantum control approaches.
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Nonreactive Scattering Alexandre Faure
, Francois Lique
Contents
, and David R. Flower
38.1
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
38.2
Quantal Method . . . . . . . . . . . . . . . . . . . . . . . 576
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Symmetries and Conservation Laws . . . . . . . . . . 577
the case of a diatom-diatom collision. Semiclassical and quasi-classical methods are also briefly introduced. The chapter ends with a comparison between theory and experiment for the benchmark CO–H2 system and an outline of future directions in the field.
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Coordinate Systems . . . . . . . . . . . . . . . . . . . . . 577
Keywords
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Scattering Equations . . . . . . . . . . . . . . . . . . . . 578
38.6 38.6.1 38.6.2
Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . 578 Interaction Potential . . . . . . . . . . . . . . . . . . . . . 578 Centrifugal Potential . . . . . . . . . . . . . . . . . . . . . 579
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Semi and Quasi-Classical Methods . . . . . . . . . . . 579
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Example: CO–H2 . . . . . . . . . . . . . . . . . . . . . . 579
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New Directions . . . . . . . . . . . . . . . . . . . . . . . . 580
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
Abstract
classical trajectory relative collision Coriolis coupling centrifugal potential Wigner coefficient quasi-classical and semi-classical scattering Feshbach resonances
38.1 Definitions The cross section for a transition from state i to f is defined classically as Z1
.i ! f / D 2 Pb .i ! f /bdb ; (38.1) The basic formulations of nonreactive scattering are presented in the sections to follow. The quantal approach 0 to this problem is outlined. Specific symmetries, and where Pb is the transition probability for impact parameter b their closely related conservation laws, which reduce the (Fig. 38.1). The impact parameter is related to the relative complexity of computation, are discussed, along with angular momentum quantum number, `, by the usual coordinate systems used to express the necessary scattering equations. The derivation of the matrix 2Eb 2 D `.` C 1/ elements needed for a given calculation is provided in (38.2) D k2 b2 ; A. Faure () Institute of Planetology and Astrophysics of Grenoble (IPAG), University of Grenoble Alpes (CNRS) Grenoble, France e-mail: [email protected]
where k is the wave number at relative collision energy E, and is the reduced mass; atomic units .e D me D „ D 1/ are used throughout. Differentiating Eq. (38.2) and setting d` D 1 in the quantal limit,
F. Lique CNRS, IPR (Institut de Physique de Rennes), Université de Rennes 1 Rennes, France e-mail: [email protected]
2` C 1 ; 2k 2 whence the quantum mechanical equivalent of Eq. (38.1) may be obtained X .2` C 1/P` .i ! f / ; (38.3) .i ! f / D 2 ki `
D. R. Flower Dept. of Physics, University of Durham Durham, UK e-mail: [email protected]
bdb D
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where xi D vi2 =2kB T , vi being the relative collision veloc1 ity; kB is Boltzmann’s constant, and .8kB T =/ 2 may be identified with the mean thermal velocity at temperature T . From Eqs. (38.8) and (38.9), h vii !f !i exp
"f "i D h vif !i !f exp ; kB T kB T
(38.10)
where "i ; "f denote the energies of the states i; f with respect to the reference level. Equation (38.10) relates the rate coefficients of forwards and reverse transitions to their relative degeneracies and excitation energies. Fig. 38.1 Classical scattering by a fixed scattering center. The trajectory is symmetric about the point of closest approach, which is the time origin
38.2 Quantal Method
The usual approach to nonreactive scattering [1] is based on the Born–Oppenheimer (BO) approximation. This approxki being the wave number in the initial state. If the initial imation postulates that electrons adapt adiabatically to the state is degenerate, and !i is the degeneracy, then motion of the much heavier nuclei. The BO approximation allows us to describe inelastic collisions as the motion of nu1 jT` .i; f /j2 ; (38.4) clei on the interaction potential V , which is a known function P` .i ! f / D !i of the nuclear coordinates. The total Hamiltonian may then where T` is an element of the transmission matrix T , which be written as contains all the information on the scattering event. The scat1 tering matrix S is related to the T matrix by H D rR2 C h1 .x 1 / C h2 .x 2 / 2 S D1T (38.5) (38.11) C V .R; x 1 ; x 2 / ; and thence to the reactance matrix K through
where R is the intermolecular distance, the first term represents the relative kinetic energy of the target and projectile, 1 C iK : (38.6) and h1 ; h2 are functions of the intramolecular nuclear coorS D 1 iK dinates x 1 ; x 2 . In the time-independent scheme, the eigenThe elements of the K matrix are real. Conservation of the functions 1 ; 2 of h1 ; h2 describe the internal motions of the colliding partners and form a basis for expanding the toincident flux of particles requires that tal wave function of the system X 2 jS.i; f /j D 1 : (38.7) X F .˛1 ˛2 `mjR/ f .R; x 1 ; x 2 / D R ˛1 ˛2 `m Microreversibility (time-reversal symmetry) implies that the Y`m .#; ˚/ 1 .˛1 jx 1 / 2 .˛2 jx 2 / : S matrix is symmetric, (38.12) S.i; f / D S.f; i/ ;
In Eq. (38.12), ˛1 ; ˛2 denote the sets of quantum numbers required to specify the states of the isolated molecules, and and hence Y`m .#; ˚/ is a spherical harmonic function of the angular 2 2 coordinates of the intermolecular vector R; F .˛1 ˛2 `mjR/ (38.8) .i ! f /ki !i D .f ! i/kf !f : are R-dependent expansion coefficients, which are solutions The thermally averaged (Maxwellian) rate coefficient for of the (time-independent) Schrödinger equation a transition from state i to f is .H E/ D 0 ; (38.13) 12 Z1 8kB T h vii !f D xi .i ! f / exi dxi ; (38.9) where E is the total energy. These solutions may be arranged as the columns of a square matrix, F .R/, in which each 0
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column is labeled by a different initial scattering state. The the original values of the coordinates, the corresponding radial functions must satisfy the physical boundary condi- eigenvalue p satisfies the equation p 2 D 1 or p D ˙1. For electromagnetic interactions, the commutation of P and H tions, implies conservation of the parity of the system. F .R/ ! 0 as R ! 0 If one takes advantage of the conservation laws associated with these symmetries of the system, substantial savings F .R/ ! J .R/A N .R/B as R ! 1 ; in computing time can be made: only one value of the total where J and N are diagonal matrices whose nonvanishing angular momentum and of the parity need to be considered elements are given by simultaneously, i.e., J and the parity are conserved during the collision. 1 2 Ji i D ki Rj` .ki R/ ; (38.14) 1
Ni i D ki2 Rn` .ki R/ ;
(38.15) 38.4
and j` ; n` are spherical Bessel functions of the first and second kinds; ki is the wave number in channel i (a given set of values of the quantum numbers ˛1 ˛2 `m/. The reactance matrix, K D BA 1 ;
(38.16)
yields the state-to-state cross sections. We note that the quantum scattering equations can also be solved in a timedependent scheme by propagation of a wave packet. In contrast to the time-independent approach, the wave-packet method is not adapted to cold and ultracold collisions due to difficulties in damping outgoing waves with long de Broglie wavelengths.
38.3
Coordinate Systems
The natural choice of coordinate system in which to express the interaction potential, V .R; x 1 ; x 2 /, is a body-fixed (BF) system in which the z-axis coincides with the direction of the intermolecular vector R D .R; #; ˚/. A rotation of the space-fixed (SF) coordinate system through the Euler angles .˚; #; 0/ generates such a BF frame. The intramolecular coordinates x 1 ; x 2 must then be expressed relative to the BF frame, as must the Laplacian operator rR2 , which appears in the expression for the total Hamiltonian, Eq. (38.11). The latter may be written as 1 @2 `2 R 2 2 R @R R 1 @2 .J j 12 /2 D R ; R @R2 R2
rR2 D
(38.17)
Symmetries and Conservation Laws
which are the forms suitable for calculations in SF and BF The expansion of the total wave function, Eq. (38.12), does coordinates, respectively. A unitary transformation relates the normalized eigennot explicitly incorporate the invariance of the Hamiltonian functions of a given parity in SF and BF coordinates. In Dirac under an arbitrary rotation in space or reflection in the co2 notation, ordinate origin. The Hamiltonian commutes with J , where J is the total angular momentum, and any component of J , jj12 `JM iSF Jx , Jy , or Jz , although these components do not commute X N N amongst themselves. Eigenfunctions of H may, therefore, be iBFhj12 ˝"JM jj12 `JM i; (38.18) D jj12 ˝"JM 2 chosen to be simultaneous eigenfunctions of J and (conven˝N tionally) Jz , with eigenvalues J.J C 1/ and M . If j 1 and j 2 are the angular momenta of the isolated collision partners, where ˝N D j˝j and ˝ is the projection of J on the BF z-axis. As the projection of ` on the intermolecular axis is then zero, ˝ is also the projection of j 12 on the BF z-axis. The j 12 D j 1 C j 2 absolute value of ˝ appears in the transformation because is their resultant, and jj12 ˙˝JM i are not eigenfunctions of the parity operator P , whereas the linear combinations J Dj C` ; 12
ˇ ˛ jj12 ˝JM i C "jj12 ˝JM i ˇj12 ˝"JM N where ` denotes the relative angular momentum of the two D 1 Œ2.1 C ı˝0 N / 2 molecules. The coupling of the angular momenta to the multipolar expansion of the electromagnetic field (i.e., the 1 interaction potential) gives rise to collisional selection rules. ." D ˙1/ are eigenfunctions of P . The factor Œ2.1 C ı˝0 N / 2 The parity operator, P , reflects the coordinates in the ensures the correct normalization of these functions. The eleorigin. Because two successive operations with P restore ments of the matrix that performs the unitary transformation
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in Eq. (38.18) are N jj12 `JM i hj12 ˝"JM 12 2.2` C 1/ j12 `J C˝0 D N ˝N ; .1 C ı˝0 N /.2J C 1/
D
ϕ θ2
(38.19) C B
j `J
12 where C˝0 N ˝N is a Clebsch–Gordan coefficient.
R
38.5 Scattering Equations Schrödinger’s equation for the scattering system may be reduced to a set of coupled, ordinary, second-order differential equations for the radial functions F .R/. Expressed in matrix form, these coupled-channel or close-coupling equations become d2 1 2 C W .R/ F .R/ D 0 : dR
θ1
A
Fig. 38.2 The body-fixed Jacobi coordinates system for the scattering
(38.20) of two rigid rotors
a least-squares fit to the original data points. If the collision There exists a set of equations in Eq. (38.20) for each value calculations are being done in the BF frame, the potential of the total angular momentum J and parity p (Sect. 38.3). matrix elements may be evaluated directly. However, if the The matrix W may be written as SF frame is to be used, the potential expansion must first be transformed into SF form. (38.21) W .R/ D k2 2V eff ; Consider the interaction between two rigid rotors. This example serves as a paradigm. The geometry of the colwhere k2 is a diagonal matrix whose nonvanishing elements lisional complex is characterized by three angles 1 , 2 , are and , and the distance R between the centers of mass of the two colliding partners (Fig. 38.2). The polar angles of
(38.22) the two rigid rotor with respect to R are denoted 1 and 2 , k˛21 ˛2 D 2 E "˛1 "˛2 I respectively, while denotes the dihedral angle. In the BF k˛1 ˛2 is the wave number at infinite separation .R ! 1/ frame, the potential can be expanded as when the collision partners are in eigenstates ˛1 ; ˛2 with X eigenenergies "˛1 ; "˛2 ; V eff is the matrix of the effective povl1 ;l2 I .R/sl1 ;l2 I .1 ; 2 ; / : V .R; 1 ; 2 ; / D tential, l1 ;l2 I (38.24) `2 Veff D V .R; x 1 ; x 2 / C ; (38.23) The basis functions s l1 ;l2 I .1 ; 2 ; / are products of associ2R2 ated Legendre functions Plm in which V is the interaction potential, and ` 2 =.2R2 / may be identified with the centrifugal potential. There exist standard computer codes for solving equations of the form in Eq. (38.20) [2–4].
38.6 Matrix Elements
sl1 ;l2 I .1 ; 2 ; / D
2 C 1 4 (
1=2
hl1 0l2 0 j l1 l2 0iPl10 .1 /Pl2 0 .2 / C
X
./m 2hl1 ml2 m j l1 l2 0i
m
38.6.1 Interaction Potential
)
Pl1 m .1 /Pl2 m .2 / cos.m/ ;
The interaction potential is usually expressed and computed in BF coordinates. For the purposes of the analysis, it is convenient to derive a multipolar expansion of the potential from where h: : : j : : :i is a Clebsch–Gordan coefficient.
(38.25)
38 Nonreactive Scattering
579
The Plm functions are related to spherical harmonics. Here, l1 ; l2 are associated to the rotational motion of the two collision partners. The matrix elements of the potential are 0 0 l JM i hj1 j2 j12 lJM jV .R; 1 ; 2 ; ; /jj10 j20 j12 X 0 D .4/3=2 vl1 ;l2 I .R/.1/j1 Cj2 Cj12 CJ l1 ;l2 ;
38.7 Semi and Quasi-Classical Methods
Œ.2j1 C 1/.2j2 C 1/.2j10 C 1/.2j20 C 1/ 0 C 1/.2l C 1/.2l 0 C 1/ .2j12 C 1/.2j12 0
1
.2l1 C 1/.2l2 C 1/ 2 .2 C 1/ j0 1 0 (
j1 0 0
l j12
l1 0 l 0 j12
!
j20 j2 0 0 8 )ˆj 0 < 12 j12 J ˆ :
l2 0
!
j20 j2 l2
l 0
9 j10 > = j1 ; > ; l1 ;
Eq. (38.27) are often replaced by their SF equivalent form `.` C 1/=.2R2 /. The matrix of the interaction potential, on the other hand, continues to be evaluated in BF coordinates. The net effect of these approximations is to ignore the rotation of the BF frame in the course of the collision, and the associated dynamical terms.
l 0 0
!
(38.26)
The above quantal method is the most accurate approach, but it is also computationally very expensive if the density of energy levels of both the target and the projectile is large. Besides decoupling approximations such as the coupled-states approach, semiclassical and quasi-classical approximations offer useful alternatives. The former consists in solving the time-dependent Schrödinger equation i„
@ D H ; @t
(38.29)
while in the latter, the classical Hamilton equations of motion are integrated where . a 6j -coefficient, / is a Wigner 3j -coefficient, g f n o a 9j -coefficient. and dq @H The close-coupling method has been applied to rigid symD ; (38.30) dt @p metric and asymmetric tops colliding with a spherical atom dp @H and/or a rigid rotor and to open-shell systems. The main difD ; (38.31) dt @q ference resides in the computation of the matrix elements of the potential, and we refer the reader to references [5–10]. where q are the coordinates and p the conjugate momenta. Their common feature is that at least one degree of freedom in the collision system is treated classically. Difficulties in the 38.6.2 Centrifugal Potential application of semiclassical theories have precluded their being employed routinely. In contrast, quasi-classical methods When evaluated in the SF frame, the matrix of the cen- have been widely employed in both reactive and nonreactrifugal potential is diagonal, with nonvanishing elements tive scattering studies. In the quasi-classical trajectory (QCT) `.` C 1/=.2R2 /. In the BF frame, the diagonal elements approach, batches of trajectories are sampled with random are (see Eq. (38.17)) (Monte Carlo) initial conditions and analyzed through statistical methods. State-resolved cross sections are extracted 2 2 N N j.J j 12 / =2R jj12 ˝"JM i hj12 ˝"JM by use of the correspondence principle combined with bin
D J.J C 1/ C j12 .j12 C 1/ 2˝N 2 = 2R2 ; histogram methods. QCT methods are very efficient, but, of (38.27) course, they ignore purely quantum effects such as interferences, tunneling, and resonances [13]. and, in addition, there are off-diagonal elements N j.J j 12 /2 =2R2 jj12 ˝N ˙ 1"JM i hj12 ˝"JM ˚
38.8 Example: CO–H2 D .1 C ı˝0 N / 1 C ı˝˙1;0 N N ˝N ˙ 1/ ŒJ.J C 1/ ˝. An important result of quantal theory for nonreactive scat1
N ˝N ˙ 1/ 2 = 2R2 : (38.28) tering is the prediction of a rich resonance structure at low Œj12 .j12 C 1/ ˝. collisional energy. Two different types of scattering resoThe matrix elements in Eq. (38.28), which are off-diagonal nances can occur, shape or Feshbach. A shape (or orbiting) N are associated with the rotation in space of the resonance corresponds to a quasi bound state that is confined in ˝, BF coordinate system (Coriolis coupling). In the coupled behind a centrifugal barrier and correlates with the initial or states approximation [11, 12], the off-diagonal elements in final monomer levels of the colliding partners. A Feshbach Eq. (38.28) are neglected, and the diagonal elements in resonance corresponds to a quasi bound state that corre-
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Cross section (Å 2) 30 25 20 15 10 5 0
0
5
10
15
20 25 Collision energy (cm –1)
Fig. 38.3 Cross section for the CO rotational excitation j D 0 ! 1 induced by normal-H2 as function of collision energy. The experimental data (circles) are compared to theory (full line)
lates with an energetically inaccessible or “closed” monomer level. Both types of resonances are predicted to occur at collision energies lower than the well depth of the interaction potential. Observing these resonances experimentally remained elusive until recent crossed molecular beam experiments [14, 15]. Such experimental data are presented in Fig. 38.3 for the CO excitation j D 0 ! 1 by normal-H2 . The measured resonance structures are in good quantitative agreement with the close-coupling calculations presented in [16], suggesting that theory and experiment have converged for this benchmark system.
38.9 New Directions An important application field of nonreactive scattering studies is astrophysics. Indeed, since collisions compete with radiative processes in altering the populations of energy levels, the estimation of molecular abundances in the interstellar medium from spectral line data requires collisional rate coefficients with the most abundant collision partners, i.e., He, H and H2 , and free electrons [17]. Hence, one of the major challenge for the coming years is the study of vibrational excitation of interstellar polyatomics since important molecules such as HCN, C2 H2 , or CH4 have been observed in vibrationally excited levels in various astrophysical environments, such as stellar envelopes and protoplanetary disks. Analysis of their emission spectra requires collisional data, and the development of new methods to deal with vibrational excitation will have to be considered.
Special attention will have to be given also to complex organic molecules with more than five atoms. There is only very little data for these species despite the fact that they are the object of many investigations using ground-based and space telescopes. For such heavy molecules with small rotational and vibrational constants, the main limitation is the computing resources, and approximate methods will have to be employed. Another important application field is cold chemistry. Novel experimental techniques allow the formation of molecules cooled down to very low temperatures (T < 1 K). These cold molecules allow new insights into molecular interaction dynamics. In particular, purely quantum effects are expected to dominate collisional processes at these low temperatures, and it will be essential to have accurate quantum theoretical results to support the new experiments.
References 1. Bernstein, R.B.: Atom–Molecule Collision Theory. Plenum, New York (1979) 2. Hutson, J.M., Le Sueur, C.R.: Comp. Phys. Commun. 241, 9 (2019) 3. Flower, D.R., Bourhis, G., Launay, J.-M.: Comput. Phys. Comm. 131, 187 (2000) 4. Alexander, M. H., Manolopoulos, D. E., Werner, H.-J., Follmeg, B., Ma, Q., Dagdigian, P. J., with contributions by Vohralik, P. F., Lemoine, D., Corey, G., Gordon, R., Johnson, B., Orlikowski, T., Berning, A., Degli-Esposti, A., Rist, C., Pouilly, B., van der Sanden, G., Yang, M., de Weerd, F., Gregurick, S., Klos, J., Lique, F.: HIBRIDON is a package of programs for the time-independent quantum treatment of inelastic collisions and photodissociation, More information and/or a copy of the code can be obtained from the website http://www2.chem.umd.edu/groups/alexander/ hibridon/hib43 5. Green, S.: J. Chem. Phys. 64, 3463 (1976) 6. Garrison, B.J., Lester, W.A., Miller, W.H.: J. Chem. Phys. 65, 2193 (1976) 7. Alexander, M.H.: J. Chem. Phys. 76, 3637 (1982) 8. Corey, G.C., Alexander, M.H.: J. Chem. Phys. 83, 5060 (1985) 9. Rist, C., Alexander, M.H., Valiron, P.: J. Chem. Phys. 98, 4662 (1993) 10. Phillips, T.R., Maluendes, S., Green, S.: J. Chem. Phys. 102, 6024 (1995) 11. McGuire, P., Kouri, D.J.: J. Chem. Phys. 60, 2488 (1974) 12. Pack, R.T.: J. Chem. Phys. 60, 633 (1974) 13. Faure, A., Lique, F., Wiesenfeld, L.: Mon. Not. Roy. Astron. Soc. 460, 2103 (2016) 14. Chefdeville, S., Stoecklin, T., Bergeat, A., Hickson, K.M., Naulin, C., Costes, M.: Phys. Rev. Lett. 109, 023201 (2012) 15. Chefdeville, S., Stoecklin, T., Naulin, C., Jankowski, P., Szalewicz, K., Faure, A., Costes, M., Bergeat, A.: Astrophys. J. 799, L9 (2015) 16. Faure, A., Jankowski, P., Stoecklin, T., Szalewicz, K.: Sci. Rep. 6, 28449 (2016) 17. Roueff, E., Lique, F.: Chem. Rev. 113, 8906 (2013)
38 Nonreactive Scattering
581 Alexandre Faure Alexandre Faure received his PhD from the University Joseph Fourier at Grenoble (France) in 1999. He worked at University College London in the UK and at the Laboratoire d’Astrophysique de Grenoble. His research focuses on computational molecular physics and is mainly concerned with processes of astrophysical relevance.
David R. Flower David R. Flower is Emeritus Professor of Physics at the University of Durham (UK). He was awarded his PhD by the University of London in 1969. After working at the Observatoire de Paris (Meudon, France) and at the ETH (Zuerich, Switzerland), he joined the Physics Department of the University of Durham in 1978. He has been Professor of Physics since 1994. His research interests are in atomic and molecular physics related to astrophysics.
François Lique François Lique received his PhD from Sorbonne University in Paris (France) in 2006. He worked at the University of Maryland in USA and at the University of Le Havre Normandy (France). He is now full professor at the University Rennes 1 (France). His research mainly focuses on the modeling of physical and chemical processes of astrophysical interest.
38
39
Gas Phase Reactions Eric Herbst
Contents
Keywords
39.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 583
39.2 39.2.1 39.2.2 39.2.3 39.2.4 39.2.5
Normal Bimolecular Reactions Capture Theories . . . . . . . . . . Phase Space Theories . . . . . . . Short-Range Barriers . . . . . . . . Complexes Followed by Barriers The Role of Tunneling . . . . . . .
. . . . . .
. . . . . .
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. . . . . .
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585 585 587 588 589 590
39.3 39.3.1 39.3.2 39.3.3
Association Reactions . . . . . . . . . Radiative Stabilization . . . . . . . . . . Complex Formation and Dissociation . Competition with Exoergic Channels .
. . . .
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591 591 592 593
39.4
Concluding Remarks . . . . . . . . . . . . . . . . . . . . 593
. . . . . .
. . . . . .
rate coefficient transition state theory microscopic reversibility phase space theory tunneling correction
39.1 Introduction As an example of a rate law, consider the rate of disappearance of reactant A in a gas mixture containing species A, B, and C . The rate law for this disappearance can be expressed by the equation
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
dŒA D kŒAa ŒBb ŒC c ; dt Abstract
The rates of gas phase chemical processes can generally be described by rate laws in which the rate of formation of products or disappearance of reactants is related to the product of the concentrations of reactants raised to various powers [1]. Rate laws are deterministic expressions that are usually accurate even though they are used to represent a stochastic reality. Rate equations may fail in the limit of small numbers of reacting particles, where both fluctuations and discrete aspects are important. Exceptions to the reliability of rate equations have been found recently in a variety of fields including surface chemistry on small particles [2]. Moreover, both Monte Carlo and master equation methods can be used in their place, given sufficient computing power [3–6].
where the symbols [ ] refer to concentration, and the rate coefficient k is dependent on temperature, and possibly other parameters such as the total gas density. The above law is not the most general that can be envisaged. For example, if species A reacts via more than one set of processes, more than one negative term on the right-hand side of the equation will be needed. In addition, if the reverse reaction to form A from products is appreciable, a positive term must also be included. At equilibrium, the rate of change of reactant must be zero. The relation above does not necessarily refer to one chemical reaction, but to a succession of elementary reactions known collectively as a mechanism. The most elementary reaction is a simple bimolecular process with a second order rate law of the type dŒA dŒC D D kŒAŒB ; dt dt
E. Herbst () Dept. of Chemistry and Dept. of Astronomy, University of Virginia Charlottsville, VA, USA e-mail: [email protected]
(39.1)
(39.2)
where two species A and B collide to form products, one of which can be labeled C . In this law, the rate coefficient k is related simply to the reaction cross section via the equation k D v ;
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_39
(39.3) 583
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where v is the relative velocity of reactants. The rate coefficient in a bimolecular process has units of volume per time, typically cm3 s1 . In a thermal system, the rate coefficient k.T / is averaged over all degrees of freedom of the reactants, both internal and translational. The most specific rate coefficient is termed a state-to-state coefficient and refers to a process in which reactant A in quantum state a reacts with reactant B in quantum state b at a specific translational energy T to form products in specific states separating with a specific translational energy [7]. Although normal bimolecular processes produce more than one product, it is also possible for the two reactants A and B to stick together if sufficient energy is released in the form of a photon A C B ! AB C h :
(39.4)
Such a process is called radiative association [8], and has mainly been studied for ion-molecule systems, i.e., reactions in which one of the two reactants is an ion. The process of radiative association is normally thought to occur in two steps. The first step produces a collision complex AB , which is a molecule existing in a transitory fashion above its dissociation limit
A C B ! AB :
(39.5)
Once formed, the complex, which is often thought of as an ergodic entity retaining memory only of its total energy and angular momentum, can either redissociate into reactants, or emit a photon of sufficient energy to stabilize itself
At low pressures, on the other hand, the rate law is second order. A simplified series of events, called the Lindemann mechanism, explains these limiting cases by invoking the activation of species A by strong inelastic collisions with bath gas M to form an activated complex A which can either be deactivated by inelastic collisions or spontaneously decompose, since it possesses sufficient energy to do so. The steps are written as A C M • A C M ;
A ! B C C ;
(39.8) (39.9)
where the spontaneous destruction of A can be thought of as a truly elementary unimolecular reaction, akin to spontaneous emission of radiation. Studies of spontaneous dissociation occupy an important place in gas phase reaction theory, and are discussed in Sects. 39.2.2 and 39.2.4 in the context of dissociation of intermediate complexes. If the rate coefficients for the forward and reverse processes in Eq. (39.8) are labeled k1 and k1 respectively, and the ratecoefficient for spontaneous dissociation of A into products is labeled k2 .s1 /, application of the steady-state principle to the concentration of the activated complex, namely, dŒA D k1 ŒAŒM k1 ŒA ŒM k2 ŒA 0 (39.10) dt leads to the rate law
dŒA D k 0 ŒAŒM ; dt
(39.11)
k1 k2 : k1 ŒM C k2
(39.12)
where AB ! AB C h :
(39.6)
Since redissociation of the complex is generally more rapid than radiative emission, radiative association rate coefficients are normally small. Emission of one infrared photon is sufficient to achieve stabilization, but stabilization of a complex via electronic emission may be a more rapid mechanism if suitable electronic states exist [8, 9]. The lifetime of the complex against redissociation into reactants is a strong direct function of the binding energy of the species and the number of atoms it possesses. In addition to bimolecular processes, two other types of reactions often referred to as elementary are unimolecular and termolecular reactions. Although complex reaction mechanisms are sometimes divided into unimolecular and termolecular steps, these are not strictly elementary because they can be subdivided into a series of bimolecular steps. In a unimolecular reaction [10, 11], a molecule A is destroyed at sufficiently high gas density by a process that has the seemingly simple first order rate law dŒA D kŒA : dt
(39.7)
k0 D
At low pressures, k1 ŒM k2 and k 0 k1 so that a second order rate law prevails. At high pressures, k1 ŒM k2 and k 0 k1 k2 =k1 ŒM so that the rate law becomes first order
dŒA k1 k2 ŒA : dt k1
(39.13)
Since both the activation and deactivation of complexes occur stepwise, rather than in single strong collisions, reality is far more complex than the Lindemann mechanism [12], and, at the highest degree of detail, master equation treatments are needed, especially for intermediate pressures [13]. In a termolecular reaction, which is actually the inverse of a unimolecular process [13], two species B and C collide to form a collision complex A , which can be regarded as the activated complex of stable species A. The collision complex can be stabilized by subsequent strong collisions with other species A C M ! A C M ;
(39.14)
39 Gas Phase Reactions
585
or can redissociate into reactants. If the complex formation step occurs with a bimolecular rate coefficient k2 , and k1 and k2 refer to complex stabilization and dissociation as in the unimolecular case, the rate law for the termolecular process can be written dŒA D k 0 ŒBŒC ŒM ; dt
(39.15)
where k0 D
k2 k1 : k1 ŒM C k2
(39.16)
At low pressures, k 0 k2 k1 =k2 and the rate law is third order. At high pressures, when every activated complex is deactivated, the rate law becomes second order since k 0 k2 =ŒM . Actually, at very low pressures radiative stabilization of the complex dominates and the rate law once again becomes second order. As in the unimolecular case, the complex does not really undergo strong inelastic collisions, so that reality is once again more complex than pictured here [13]. For detailed theories of thermal reactions, there is the additional problem in Eqs. (39.12) and (39.16) of deciding when thermalization of the partial rate coefficients should be undertaken. In reality, the partial rate coefficients refer to reactions with specific amounts of energy and angular momentum, and should not be thermally averaged before incorporation into the equations for k 0 .
39.2 Normal Bimolecular Reactions The rates of elementary (bimolecular) chemical reactions are governed by Born–Oppenheimer potential surfaces, which contain electronic energies and nuclear-nuclear repulsions. Reactions can be classified as exoergic or endoergic depending upon whether the 0 K energies of the products lie below or above those of the reactants, respectively. In the simplest types of exoergic chemical reactions, the potential energy flows downhill from reactants to products, or flows downhill from reactants to a global minimum (the reaction complex) after which it flows uphill to products. More commonly, the morphology of the potential surface is such that after some long-range attraction, the potential rises as old chemical bonds are broken before falling as new bonds are formed. Generally, there is a minimum energy pathway through the region of large potential referred to as the reaction coordinate. The system is said to traverse a transition state barrier, which refers to the configuration of atoms at which a potential saddle point occurs. The height of this transition state barrier is related to the activation energy barrier Ea in the classical Arrhenius rate law k.T / D A.T / exp
Ea ; kB T
for rate coefficients of bimolecular reactions which contain short-range barriers [1]. In the Arrhenius expression, kB is the Boltzmann constant, and the pre-exponential factor A.T / can be related to the form of the long-range potential, or to an equilibrium coefficient between the transition state and reactants (see the discussion of activated complex theory in Sect. 39.2.3). Although fits of experimental data over short temperature ranges often assume the pre-exponential factor to be totally independent of temperature, theories show that this is not strictly true in most instances. A more serious problem with the expression undoubtedly occurs at low temperatures since tunneling will clearly lead to deviations. Although, in principle, it is possible to calculate reaction cross sections and rate coefficients via the quantum theory of scattering, in practice few systems have been studied by this technique given the immense computational effort required [14, 15]. Another set of approaches, which has been used to study a large variety of systems, is known as classical molecular dynamics (see Chap. 62). In these approaches, the atoms move classically on the quantum mechanically generated Born–Oppenheimer potential surfaces. For many reactions, however, neither technique is applicable, and a variety of simpler approaches has been developed, using capture and statistical approximations; these approaches will be emphasized here. Indeed, the use of an ergodic complex in our preliminary discussion of association and unimolecular reactions above presages the use of statistical approximations. An excellent high-level review article on many of the topics covered here has been written by Troe [16].
39.2.1 Capture Theories For reactions that do not possess a potential energy barrier at short-range, it is tempting to apply long-range capture theories between structureless particles to calculate the reaction rate coefficient. Such theories assume that (a) all hard collisions lead to reaction, and (b) hard collisions occur for all partial waves up to a maximum impact parameter bmax or relative angular momentum quantum number Lmax . This is normally defined so that the reactant translational energy TAB is just sufficient to overcome a centrifugal barrier. The centrifugal barrier produces a long-range maximum in the effective potential energy function Veff given by the relation Veff .r; b/ D V .r/ C TAB b 2 =r 2 ;
(39.18)
where r is the separation between reactants and TAB b 2 =r 2 is the angular kinetic energy. If the reactants overcome the centrifugal barrier, they will spiral in towards each other in the absence of short-range repulsive forces. The approach has (39.17) had its most notable success for exoergic reactions between ions and nonpolar neutral molecules [17]. The long-range po-
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tential in this situation is simply (in cgs-esu units) V .r/ D e 2 ˛d =2r 4 ;
(39.19)
where ˛d is the polarizability of the neutral reactant. This potential leads to the Langevin rate coefficient
kD
2 vbmax
˛d D 2e
1=2 ;
(39.20)
where is the reduced mass of the reactants. The theory leads to a temperature-independent rate coefficient with magnitude 109 cm3 s1 . Numerous experiments from above room temperature to below 30 K confirm its validity for the majority of ion–molecule reactions, which appear rarely to possess potential energy surfaces with short-range barriers [18, 19]. An analogous central force potential for structureless neutral–neutral reactants is the van der Waals or Lennard– Jones attraction V .r/ D
C6 ; r6
(39.21)
where is the angle between the radius vector from the charged species to the center-of-mass of the dipolar species and the dipole vector. Adiabatic effective (centrifugal) potential energy curves can be defined at any given fixed intermolecular separation r by diagonalizing angular kinetic energy and potential energy matrices using a suitable basis set. If an atomic ion is reacting with a linear neutral, a suitable basis set would consist of spherical harmonics for the rotation of the linear molecule (angular momentum j ) as well as rotation matrices for the relative motion (angular momentum L) between the species. The eigenvalues of a matrix with fixed total angular momentum J then correspond to the effective radial potential energy functions Veff at each r. The L; j labeling of the potential curves can be ascertained by starting the calculation at sufficiently large r so that L and j are reasonably good quantum numbers. Additional approximations, such as the centrifugal sudden approximation, can be made to facilitate the calculation [25]. Making the adiabatic assumption that transitions between these potentials are not allowed at long range, one obtains an adiabatic capture rate coefficient for each initial value of j and translational (radial kinetic) energy TAB from the criterion
where C6 can be defined simply in terms of the ionization po(39.24) Veff .j; Lmax ; R/ D TAB ; tentials and polarizabilities of the reactants [20, 21]. The rate coefficient obtained using a capture theory with this longwhere R is the separation corresponding to the maximum range potential is given, after translational thermal averaging, effective potential. Unlike the Langevin approach, the rate by the equation [21] coefficients thermalized over translation for each j show an 1=3 1=2 .kB T /1=6 ; (39.22) inverse dependence on temperature, with the j D 0 state k.T / D 8:56 C6 showing the most severe dependence since the dipole is eswhere all quantities are in cgs units. Equation (39.22) leads sentially locked for this state. Thermal averaging over j to estimates for the rate coefficient at room temperature in typically leads to rate coefficients with an inverse depen1=2 and T 1 [26]. Rate the range 1010 109 cm3 s1 . Unlike the situation for dence on temperature between T 7 3 1 ion–molecule reactions, this estimate has not received much coefficients as large as 10 cm s can be obtained as attention mainly because most neutral–neutral reactions in- the temperature approaches 10 K. Although most ion–dipole volve activation energy. Even for those systems without reactions seem to obey capture theory models at low temperactivation energy, the approximation appears to lead to rate ature, there are exceptions. An alternative approach to ion–dipole reactions using coefficients that are too large by at least a factor of a few [21]. the classical concept of adiabatic invariants has been deIn place of the result of Eq. (39.22), kineticists often use the veloped [27]. In addition, variants to the capture theory simple hard-sphere model with atomic dimensions for the discussed here have been formulated. A statistical adiabatic reaction cross section. The hard-sphere model leads to a tem1=2 approach has been developed [23], in which adiabatic effecperature dependence of T , which is in disagreement with tive potential maxima are not used to define capture cross a whole series of new experiments on fast neutral–neutral resections, but to define rate coefficients via the activated actions down to temperatures near 10 K [22]. Both long-range potentials considered above are isotropic complex (transition state) theory discussed in Sect. 39.2.3, in nature. A variety of capture theories [23, 24] have been de- which posits thermal equilibrium between reactants and the veloped which take angular degrees of freedom into account. species existing at the potential maxima. One advantage For ion–molecule systems in which the neutral species has of this simplification is that it permits the adiabatic treata permanent dipole moment d , the long-range potential be- ment to be more easily extended to complex geometries, especially if perturbation approximations are utilized. Other comes approaches involve the variational principle; an upper bound e 2 ˛d ed cos to the capture rate coefficient can be determined within the V .r; / D ; (39.23) 4 2 2r r transition state, or bottleneck approach, which defines the
39 Gas Phase Reactions
transition state through minimization of the number of available vibrational/rotational states at the bottleneck [28] (see Sect. 39.2.3). Nonspherical capture theories have also been used [29, 30] to study rapid neutral–neutral reactions. The role of atomic fine structure at low temperatures is an especially interesting application; reactions involving atomic C and O with a variety of reactants have been studied. Use of an electrostatic potential shows that the reactivity of C or O atoms in their 3 P0 states with dipolar species is minimal. This is particularly important for atomic carbon since at low temperatures it lies primarily in its ground 3 P0 state. The choice of an electrostatic potential has been disputed [21] because experimental results for C–hydrocarbon reactions at room temperature are best understood if the long-range potential is dispersive (Lennard–Jones) in character rather than electrostatic. Reactions between radicals in 2 … states (e.g. OH) and 2 † states (e.g. CN) and stable molecules have also been considered, especially at low temperature, with long range potentials that contain both electrostatic and dispersion terms. The results can be compared with new low temperature experimental results on the CN–O2 reaction, but the temperature dependence is not matched by theory if the dispersion term is included [30, 31]. In general, even the most recent capture theories are not as reliable as those for ion–molecule systems [32]. Rapid neutral–neutral reactions can also be treated by transition state theories (see below) [33] or, rarely, by detailed quantum mechanical means [34]. The last five years have witnessed a burgeoning interest in rapid neutral–neutral reactions at low temperature studied with the so-called CRESU technique (an acronym for Cinétique de Réaction en Ecoulement Supersonique Uniforme) [22, 35–37]. The new data should provoke new attempts at theoretical understanding.
39.2.2 Phase Space Theories Capture theories tell us neither the products of reaction, if several sets of exoergic products are available, nor the distributions of quantum states of the products. The simplest approach to these questions for reactions with barrierless potentials is to make a statistical approximation – all detailed outcomes being equally probable as long as energy and angular momentum are conserved. Such a result requires strong coupling at short range. The most prominent treatment along these lines is referred to as phase space theory [38]. In this theory, the cross section for a reaction between two species A and B with angular momentum quantum numbers JA and JB and in specific vibrational-electronic states colliding with asymptotic translational kinetic energy TAB to form products C and D in specific vibrational-electronic states with angu-
587
lar momentum quantum numbers JC and JD is
39
.JA ; JB ! JC ; JD / „2 X D .2Li C 1/ 2TAB L ;J i
P .JA ; JB ; Li ! J /P 0 .J ! JC ; JD / ;
(39.25)
where J is the total angular momentum of the combined system, Li is the initial relative angular momentum of reactants, P is the probability that the angular momenta of the reactants add vectorially to form J , P 0 is the probability that the combined system with angular momentum J dissociates into the particular final state of C and D, is the reduced mass of reactants, and the summation is over the allowable ranges of initial relative angular momentum and total angular momentum quantum numbers. P 0 is equal to the sum over the final relative angular momentum Lf , of angular momentum allowed .J ! Lf ; JC ; JD / combinations leading to the specific product state divided by the sum of like combinations for all energetically accessible product and reactant states. The ranges of initial and final relative angular momenta are given by appropriate capture models (e.g. Langevin, ion–dipole, Lennard–Jones) as well as angular momentum triangular rules. This procedure involves the implicit assumption that strong coupling does not occur at long range; rather, adiabatic effective potentials can be assumed for initial and final states. The state-to-state rate coefficient is simply the cross section multiplied by the relative velocity of the two reactants. Summation over all product states, as well as thermal averaging over the reactant state distributions and the translational energy distribution, can all be undertaken. Strategies for summing/integrating over rotational/vibrational degrees of freedom have been given [39]. As can be seen by writing out the expression for P 0 , the phase space formula for the state-to-state cross section in Eq. (39.25) obeys microscopic reversibility 2TAB .JA ; JB ! JC ; JD / 0 D 20 TAB .JA ; JB
JC ; JD / ;
(39.26)
where the reduced mass and asymptotic translational energy of the products are denoted by primes. Thus, thermalization of the forward and backward rates leads to detailed balancing. As a detailed theory for prediction of reaction products and their state distributions, phase space theory and its variants have had mixed success. It is true that the theory correctly predicts that exoergic reactions occurring on barrierless potential surfaces proceed on every strong collision. It is also true, however, that the theory is generally not useful in predicting the branching ratios among several sets of exoergic products because potential surfaces do not
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E. Herbst
often show barrierless pathways for more than one set of products. When applied to product state distributions, the theory yields statistical results, so that population inversions in nondegenerate degrees of freedom (such as the vibrations of a diatomic molecule) are not replicated. With respect to more thermalized entities, such as total cross sections versus collision energy for endoergic reactions [40], the theory can be quite successful, especially when the potential surface involves a deep minimum or intermediate complex. In this instance, the strong coupling hypothesis comes closest to actuality. A deep well is also associated with a high density of vibrational states in the quasicontinuum above the dissociation limit of the molecular structure defined by the potential minimum [1, 41, 42]. Such a high density renders direct (specific) dynamical processess less likely. A useful variant of phase space theory if complex lifetimes are needed [43] (see Sects. 39.2.4 and 39.3) is based on a unimolecular decay theory of Klots [44]. The reaction is considered to proceed via a capture cross section to form the intermediate complex, which then can dissociate into all available reactant and product states consistent with conservation of energy and angular momentum. The complex dissociation rate kuni into a specific state can be obtained via the principle of microscopic reversibility in terms of the capture cross section from that state, obtained with P 0 D 1 in Eq. (39.25), to form the complex. In particular, if a complex with total angular momentum J can dissociate into one state of reactants A and B separating with translational energy TAB , then kuni
bined system, although typically only one surface leads to energetically accessible products. In such instances, it is normal to assume statistical partitioning, although fine structure effects can complicate this picture. A generalization of phase space theory to account for multiple potential surfaces has been proposed for the CC C H2 reaction [45]. Phase space theory has been used to predict product branching ratios for dissociative recombination reactions between polyatomic positive ions and electrons [46]; its success here has been limited at best.
39.2.3 Short-Range Barriers
Most reactions involving neutral molecules, as well as a minority of ion–molecule reactions, possess potential surfaces with short-range barriers. The experimental rate coefficients of these systems over selected temperature ranges are normally fit to the Arrhenius expression in Eq. (39.17). The simplest theoretical method of taking short-range potential barriers into account is the line-of-centers approach, which resembles the capture theories previously discussed [7]. In this crude approximation, it is assumed that structureless reactants colliding with impact parameter b along a repulsive potential must reach some minimum distance, d , for reaction to occur. If the potential energy in the absence of angular momentum at d is E0 , this condition implies that the asymptotic translational energy of reactants TAB must exceed the sum of E0 and the centrifugal energy TAB b 2 =d 2 , which in turn yields a maximum impact parame1 1 2 ter bmax . Thermal averaging of the rate coefficient k D vbmax D vib g.J / g.JA /g.JB /2TAB .JA ; JB ! J / ; (39.27) over a Maxwell–Boltzmann distribution then yields
where vib is the density of complex vibrational states obtained via a variety of prescriptions [1, 41, 42], and g is the rotational degeneracy. A cross section analogous to that in Eq. (39.25) can be formulated in terms of capture to form the complex multiplied by the complex dissociation rate into a particular state divided by the total (summed) dissociation rate. Note that the dissociation rate of the complex is 1 . Since vib is a strong function of the proportional to vib well depth, long-lived complexes are associated with large well depths (>1 eV). The Klots form for kuni is especially useful for ion–molecule systems, where the cross section for complex formation can be assumed to be Langevin or ion–dipole. The concept of a long-range potential is less useful for most neutral–neutral systems, so that unimolecular rate coefficients for unstable entities are normally obtained quite differently in terms of vib at the transition state (Sect. 39.2.4). For smaller reaction systems, especially those involving ions, it is quite common for the electronic states of reactants to correlate with more than one potential surface of the com-
k.T / D d 2
8kB T
1=2 exp
E0 ; kB T
(39.28)
which bears some resemblance to the Arrhenius expression if one equates the parameter E0 with the activation energy. Assuming T D 300 K, D 10 u, and d D 1 Å, the pre-exponential factor is 2:5 1011 cm3 s1 , which lies in a typical range. The simple line-of-centers approach reduces the problem to that of structureless particles. The standard method of including all degrees of freedom is to use canonical ensemble statistical mechanics and to imagine that the transition state is in equilibrium with reactants. In the transition state, one of the vibrational degrees of freedom of a normal polyatomic molecule is replaced by a degree of freedom (the reaction coordinate) along which the potential is a maximum (with a corresponding imaginary frequency of vibration). The reaction coordinate is treated as a separable translation, so that the reaction rate coefficient can be envisaged as the equilibrium coefficient between transition state (minus one
39 Gas Phase Reactions
589
coordinate) and reactants multiplied by the (average) speed of the transition state structure over the saddle point in the potential energy surface. The canonical result for k.T / is given by [1, 7] k.T / D
E0 kB T qAB exp h qA qB kB T
(39.29)
for reactants A and B, where E0 is the energy difference between the transition state and the reactants referred to zeropoint levels, the refers to the transition state, and the q are partition functions per unit volume. The partition functions can be factored into products representing electronic, vibrational, rotational, and translational degrees of freedom [47]. This formulation for k.T / is known as the activated complex theory (ACT); a more appropriate name would be transition state theory since the term activated complex is also used to refer to an unstable state of a molecule in a deep potential well. The rate coefficient can also be written in terms of the thermodynamic parameters H and S [1]. Both the size and temperature dependence of the preexponential factor depend critically on the characteristics of transition state and reactants. Originally, ACT theory was used mainly to fit transition state characteristics to rate data. Increasingly accurate ab initio calculations of potential surfaces now allow purely theoretical determinations of k.T / [48]. In addition to the assumption of a separable reaction coordinate that can be treated as a translation, several other assumptions have gone into the derivation of the ACT rate coefficient. First, molecules at the saddle point configuration (the transition state) have been arbitrarily chosen to be in equilibrium with reactants. More recently, the transition state assumption has been generalized to refer to that portion of the potential surface in which the reaction flux or number of states is a minimum [49], whether or not this occurs at a well-defined saddle point. Loose transition states can thus be defined even if there is no barrier along the reaction coordinate [50]. The procedure can be undertaken for transition states as a function of temperature, energy, or, in its most detailed version, energy and angular momentum. The variational theorem shows rate coefficients determined in this way to be upper bounds to the true rate coefficients, the more detailed procedures leading to the better bounds [23, 49, 50]. Secondly, the implicit assumption is made that translation along the reaction coordinate at the transition state structure invariably leads to products. Sometimes a transmission coefficient is used as an unknown factor in the expression for k.T / to account for the possibility that translation leads back to reactants instead. Thirdly, the assumption is made that two-body collisions can lead to canonical statistical equilibrium. This assumption becomes worse as temperature declines, because at low temperatures
the long-range centrifugal constraints on angular momentum become more significant [8]. The influence of the long-range potential on ACT is contained in the statistical adiabatic theory discussed earlier [23]. Here loose transition states are defined as the maxima of effective adiabatic potential curves, and Eq. (39.29) must be modified since, as in detailed variational transition state theory, the transition states are themselves dependent on energy and angular momentum. A related problem is how to treat angular momentum constraints at low temperatures for potential surfaces which also contain a large short-range barrier or tight transition state. One approach is discussed in Sect. 39.2.4. Rate coefficients derived from ACT do not indicate what the reaction products or sets of products might be. This information can be obtained from ab initio studies of the potential surface if different transition states lead to different products.
39.2.4 Complexes Followed by Barriers Since long-range forces are always attractive, it makes sense to consider theories in which attraction and repulsion occur sequentially. For ion–molecule systems, this is especially important because many potential surfaces are monotonically attractive from long range to formation of a deep minimum at short range, but possess transition state barriers in their exit channels which are not large enough to prevent reaction but which affect the reaction dynamics. In addition, there are ion–molecule systems with potential surfaces closer to the norm for neutral–neutral species; in these systems there is only a weak long-range minimum followed by a short-range transition state barrier with energy above that of reactants. For the former type and, more arguably, for the latter type of potential surface, one can assume that the reaction proceeds through initial formation of an ergodic complex, followed by dissociation of the complex back into reactants or over the transition state barrier. If the reactants are labeled A and B, the complex AB , and the transition state AB , the mechanism is A C B • AB ! AB ! Product ;
(39.30)
which leads to the steady-state rate law dŒA D kŒAŒB ; dt kcf kcd0 ; kD kcd C kcd0
(39.31) (39.32)
where kcf , kcd , and kcd0 are the rate coefficients for complex formation, redissociation into reactants, and dissociation into products over the transition state, respectively. One statistical approach to such systems is to use a capture theory for complex formation, to use the Klots formulation of complex redissociation into reactants, and to use a different theory for
39
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E. Herbst
the unimolecular dissociation of the complex into products. Transition state theories of unimolecular complex decay have been studied for many years [1]. The current version is called RRKM theory, after the four authors Rice, Ramsperger, Kassel, and Marcus [10–13]. This theory builds upon the earlier RRK approach, in which random intramolecular vibrational energy transfer leads to large amounts of energy in the bond to be broken. An alternative hypothesis, in which amplitudes of well-defined normal modes add up to extend the bond to be broken past a certain amount, is now discredited [51]. In the RRKM approach, which is perfectly analogous to ACT theory, an equilibrium is envisaged between the activated complex and transition state species. The main use of the theory is in thermal unimolecular decay where it explains complex activation, deactivation, and random unimolecular dissociation as a function of gas density. At high density in a thermal environment, an equilibrium among complex, activated complex, and transition state leads to a thermal unimolecular decay rate in Eq. (39.13) kRRKM .T / D
kB T qAB E0 exp ; h qAB kB T
ciation channels for which the reaction proceeds via a deep well and an exit channel barrier (see Sect. 39.3.3) [53–55]. Some authors prefer to use RRKM theory to describe redissociation of the complex into reactants [13, 56] by means of a loose transition state [57], which can be defined via a variety of variational and other methods [50]. Some recent work on unimolecular decay of polyatomic ions and neutrals following excitation indicates the superiority of the variational RRKM approach to unimolecular decay [58], especially if there is a significant amount of energy. Both the Klots and RRKM approaches to complex redissociation have problems associated with them: the Klots expression obeys microscopic reversibility but requires both a capture cross section and the assumption that the phase space formulation of the probability of individual quantum states is accurate; the RRKM expression does not obviously obey microscopic reversibility and, unless a variational calculation is performed, can create a somewhat artificial transition state. Reaction mechanisms with more than one potential barrier can also be treated via a combination of ACT, capture, (39.33) and unimolecular approaches [59, 60].
which is perfectly analogous to the ACT result. At lower pressures, the activated complex is not in thermal equilibrium with the stable complex and the details of complex activation and deactivation are important [12, 13]. In the microcanonical formulation of RRKM theory, the dissociation rate coefficient kRRKM as a function of (activated) complex total energy E and angular momentum J is given by
39.2.5 The Role of Tunneling
The role of tunneling in bimolecular and unimolecular reactions grows in importance as the temperature is lowered and hopping over potential barriers becomes more difficult. A simple one-dimensional correction for tunneling in the ACT expression for bimolecular rate coefficients, obtained by Wigner [1, 61], appears to be reasonable if the tunneling N ŒE E0 Erot .J / mechanism is not dominant. This correction > 1 is sim; (39.34) kRRKM .E; J / D ply the quantum correction to the partition function for the h ŒE Erot .J / reaction coordinate where N refers to the total number of vibrational states of the transition state from its minimum allowable saddle point .hi /2 C ; (39.35) 1 energy E0 through E, and refers to the density of vi24.kB T /2 brational states of the complex at energy E. For both the transition state and the complex, the available vibrational en- where i is the (imaginary) harmonic frequency at the saddle ergy is the total available energy minus the rotational energy point. Improved corrections have also been developed [62, Erot .J /, which is a function of the angular momentum. The 63] and, for the H C H2 reaction, tested versus accurate energy not used for vibration and rotation in the transition quantum calculations [64]. At very low temperatures, such as state is considered to belong to the separable reaction coor- those found in the interstellar medium, tunneling corrections dinate. The most common expressions for the number and are likely to be very large and to require proper multidimendensity of vibrational states are the semiclassical empirical sional treatments. values [1, 41, 42]; direct counting schemes also exist [52]. A one-dimensional tunneling correction to the RRKM Both empirical and direct counting refer to states represent- expression for the microcanonical unimolecular decay rate ing a bath of harmonic oscillators; anharmonic effects are coefficient also exists [65]. The effective potential representrarely treated. ing the reaction coordinate at the saddle point is assumed to A theory of reaction rates for the mechanism in be an Eckart barrier. The probability of tunneling for each Eq. (39.30) incorporating a capture theory, and the Klots and vibrational state of the transition state is computed, and this RRKM unimolecular decays has been applied successfully to probability takes the place of simply counting the state in the a variety of ion–molecule reactions in competition with asso- standard formula for kRRKM . In particular, N .E E0 Erot /
39 Gas Phase Reactions
591
in the equation for kRRKM .E; J / is replaced by X NQM .E E0 Erot / D P .E E0 Erot n / ; n
(39.36) where n is the vibrational energy of state n of the transition state with respect to its zero point energy, and the sum is over all states n for which the energy in the reaction coordinate (the energy in parentheses on the right-hand-side) is negative, but not so negative that the classical energy in the reaction coordinate lies below the minimum of the complex potential well. In general, only a few vibrational states of the transition state need be considered for the tunneling correction. This tunneling correction has been incorporated into statistical theories for the rate of reactions that proceed via complexes and transition state barriers [60]. It has been used recently in a calculation of the slow ion–molecule reaction C NHC 3 C H2 ! NH4 C H ;
(39.37)
which successfully reproduces the observation that, despite initially decreasing as the temperature is reduced below 300 K, the reaction rate coefficient begins to increase as the temperature is decreased below 100 K [66]. The parameters for the calculation were obtained from an ab initio calculation of the potential surface, which shows the system to proceed through a weakly-bound long-range complex before encountering a transition state with a rather small barrier of 0:25 eV. Although the dynamical theory is not quantitative (presumably because of the one-dimensional tunneling approximation), it does reproduce the isotope effects seen C when the reactants NHC 3 C D2 and ND3 C H2 are used, definitively showing that tunneling is the cause. The actual increase in rate at very low temperature comes from the fact that the tunneling rate is less dependent on temperature than the dissociation rate of the complex into reactants. Similar calculations have been performed for the analogous ion– molecule reaction C C2 HC 2 C H2 ! C2 H3 C H
(39.38)
to explain a similar observation, although in this latter instance there is still a controversy concerning whether or not the reaction is truly exoergic [67]. It is interesting to speculate on whether similar effects can be detected for analogous neutral radical–H2 reactants with moderate activation energy barriers, such as CCH C H2 ! C2 H2 C H :
(39.39)
39.3 Association Reactions Association reactions have been studied for both ion– molecule and neutral–neutral systems. For the ion–molecule case, association processes can be investigated in great detail at low densities near or in the radiative association regime because ions can be stored in low pressure traps. Review articles [69, 70] describe such experiments for small and large ions, respectively. The results of many higher pressure studies of termolecular ion–molecule association, especially those undertaken in the SIFT apparatus, are also available [71]. In general, below the high pressure limit, the rate of association reactions without activation energy is known to increase with (a) an increase in the size of the reactants, (b) an increase in the bond energy of the product species, and (c) a decrease in temperature. These trends are all reproduced by statistical theories in which association proceeds by the formation of a complex followed by radiative and/or collisional stabilization [8]. More limited data for termolecular neutral–neutral reactions, with and without short-range potential barriers, have also been reviewed [72]. Although radiative stabilization can proceed via a single photon, collisional stabilization proceeds most probably in a stepwise fashion rather than in a single strong collision. The most detailed statistical theories incorporating multistep collisional stabilization use master equation techniques; in general it is preferable to solve the inverse problem of unimolecular dissociation via detailed RRKM theory and then invoke detailed balance [13, 56]. Such theories are more successful than strong collision approaches, especially in considering the dependence of ternary association rates on pressure. This dependence can be especially difficult to treat when association competes with an exoergic channel which does not necessarily dominate because of a barrier in the exit channel.
39.3.1 Radiative Stabilization The problem of radiative association in the absence of competitive exoergic channels is in principle much more simple. In addition to the rates of complex formation and dissociation, one needs only the rate of stabilization via spontaneous emission. If the complex abundance is at steady state, the rate law for radiative association of species A and B is simply
dŒA (39.40) D kra ŒAŒB ; This system possesses a transition state of rather moderate dt energy [48], but the only long-range complex is presumably kcf krad ; (39.41) kra D the very weakly-bound van der Waals structure. Unlike the kcd C krad ion–molecule case, there are very few experimental studies of low temperature neutral–neutral reactions. Studies of pres- where kra represents the rate coefficient for radiative assosure broadening at very low temperatures do reveal, however, ciation and krad refers to the rate coefficient for radiative stabilization. If ternary association is considered in addition, the strong influence of the van der Waals bond [68].
39
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E. Herbst
estimates, as well as limited experimental data, show that krad varies between 10 and 1000 s1 for vibrational energies up to a few eV [69, 70, 73–75]. The radiative stabilization rate typically decreases slightly with increasing molecular size due to the inclusion of infrared inactive modes of vibration. There also appears to be a distinction between ions and neutral species, with rates for ions apparently faster. Since spontaneous emission rates between electronic states are often far more rapid than infrared vibrational rates, the possibility that radiative stabilization can proceed via electronic emission has been examined [9]. The well-studied association between CC and H2 appears to proceed via electronic relaxation, with a radiative stabilization rate one to two X X Pi Ai !j D PŒni AŒni !Œnj ; (39.42) orders of magnitude larger than would be the case for vikrad D i;j brational emission. Whether or not the association between Œni ;Œnj CHC 3 and H2 proceeds via an electronically excited complex where the sum is over initial states i and final states j , and has been debated for some time; the answer is apparently A is the Einstein coefficient for spontaneous emission. With negative [69]. some algebraic manipulation and the assumption that dipole selection rules apply, the result simplifies to
the rate coefficient for radiative association is the low pressure limit of a more complex rate expression. Radiative stabilization, which is normally considered to proceed via emission of a single infrared photon, has been studied in some detail [73–75]. If the unstable complex is imagined to be a collection of harmonic oscillators, each vibrational state i, which is best regarded as a Feshbach resonance, can be expressed as a set of occupation numbers ni1 ; ni2 ; : : : ; Œni for the assorted modes. If it is further assumed that upon formation, the complex has a probability Pi of being formed in state i, the rate of single-photon vibrational emission krad .s1 / is then given by
krad D
s X X
39.3.2 Complex Formation and Dissociation Pnkk nk Ak1!0
;
(39.43)
kD1 nk
where the index k refers to each of the s normal modes, the index nk to the occupation number of mode k, and the Einstein A coefficient, which here refers to the fundamental transition of mode k, can be expressed in terms of the absorption intensity (see Chap. 11).The probability Pnkk that mode k is excited to state nk is given by the statistical formula Pnkk D
0 .Evib nk hk /
vib ;
vib .Evib /
(39.44)
0 is the where vib is the vibrational density of states and vib reduced density of states when nk quanta are assigned to mode k. The role of overtone and combination contributions to the radiative emission rate has also been investigated [74], as have a canonical approximation to the microcanonical formulation discussed here [75] and the small additional effect of stabilization by sequential emission [73]. With the use of the classical approximation to the standard vibrational density of state functions [41, 42], Eq. (39.43) for krad reduces to a linear function of the vibrational energy
krad D
s Evib X Ak1!0 : s hk kD1
(39.45)
The formation and dissociation of the complex can be studied with a variety of statistical approximations other than what is discussed in this section. In particular, complex dissociation has been treated by a thermal RRKM approach [70, 75], an energy but not angular momentum specific RRKM approach [73], variational transition state theory, statistical adiabatic theory, and flexible transition state theory [72]. We first consider a simple thermal approximation for systems without activation energy incorporating microscopic reversibility [8]. In the limit that the complex redissociation into reactants is much more rapid than radiative stabilization, the rate coefficient kra for radiative association in the thermal model is kra D K.T /krad ;
(39.46)
where K.T / is the ratio between kcf and kcd in a canonical ensemble, and the radiative stabilization rate has been assumed to be independent of temperature. The equilibrium coefficient K.T / between complex and reactants can be rewritten in terms of partition functions in the standard way [47]. However, the partition function of the complex is best calculated via the equation Z1 qAB D
vr .E C D0 / exp 0
E dE ; kB T
The constant of proportionality between krad and Evib de(39.47) vr .D0 /kB T ; D0 kB T ; pends strongly on the fundamental intensities; many of these can be obtained from infrared absorption spectra, although where vr is the density of vibrational-rotational states [41, there is normally insufficient information for all of the active 42], E is the energy of reactants, and D0 is the bond energy modes in polyatomic molecules, especially ions. Reasonable of the complex. One immediate prediction of this approach
39 Gas Phase Reactions
593
(deriving mainly from the rotational partition functions of the of such competition is that it occurs via a sequential mechreactants) is that kra possesses a strong inverse dependence anism in which the reactants form a long-lived collision complex which either redissociates, dissociates into exoeron temperature since gic products, or is stabilized [53, 54]. The rate coefficient for K.T / / T .rA CrB C1/=2 ; (39.48) competitive radiative association is then given by the steadywhere r refers to the number of rotational degrees of freedom state expression (two for a linear molecule and three for a nonlinear one). It kcf also predicts a strong dependence on well depth and size of krad ; (39.49) kra D kcd C kcd0 C krad the complex since the density of complex states is a strong function of both these parameters. Although the thermal model agrees with the strong inverse temperature depen- where kcd0 refers to complex dissociation into products. The dence of the coefficients for both radiative and, more com- additional and normally large kcd0 term in the denominator monly, ternary ion–molecule association [8, 71] (in which means that the thermal and modified thermal theories cancase, the thermal model with strong collisions replaces krad not, in general, be used. The phase space treatment for this with a collisional rate coefficient), the absolute rate coeffi- mechanism [53, 54, 76, 77] shows that association cannot cients calculated tend to range up to an order of magnitude compete with exoergic channels unless there is a significant too high. Given the large range of values exhibited by radia- barrier in the exit channel which considerably slows the distive association rate coefficients .1020 109 cm3 s1 /, this sociation rate of the complex into products, especially for might not seem too large a problem. It has been shown, partial waves of high angular momentum. Such barriers tend however, that the thermal model is deficient, especially at to be large enough to slow dissociation down, but not large low temperatures because thermal equilibrium cannot be enough to require tunneling. Although the results of phase achieved by two-body collisions [8]. A revised approach, space calculations are in good agreement with experiment called the modified thermal theory, replaces the rovibra- for a variety of competitive systems [69, 76], they are once tional density of complex states in the thermal model with again inferior to RRKM calculations with master equation a vibrational density of states coupled with sums over the collisional deactivation when the competition involves colliallowable ranges of complex angular momentum achievable sional stabilization of the complex [13, 53, 54, 56]. Another mechanism exists for competition between assofrom the specific capture model assumed [8]. The result is ciation and normal exoergic channels [69]. Ab initio studies a somewhat lessened inverse temperature dependence and show that there is a parallel type of competition in which somewhat smaller rate coefficients, especially at low temthe product and association channels occur on different porperatures. Both modifications result in better agreement with tions of the potential surface, with a branching at long range experiment [8]. in the entrance channel. For example, the competing reacThe modified thermal approach considers structureless tions [69, 78] reactants; full consideration of the internal states of the reactants is achieved via the Klots version of phase space theory, in which kcf and kcd in Eq. (39.41) are the capture and unimolecular rates discussed in Sect. 39.2.2. The phase space treatment reduces to the modified thermal treatment for small reactant angular momentum if the possibility of saturation Œkcd .E; J / < krad may be ignored. Both conditions are normally met. The phase space approach has also been used for ion–molecule ternary association reactions, but here is coupled with the strong collision hypothesis and must be regarded as inferior to the more detailed RRKM calculations with master equation treatments for collisional stabilization [13, 56].
C C2 HC 2 C H2 ! C2 H3 C H ;
(39.50)
C C2 HC 2 C H2 ! C2 H4 ;
(39.51)
occur via distinct pathways. The former reaction is a possibly endoergic direct process in which the molecular hydrogen attacks perpendicularly, leading to the cyclic form of the C2 HC 3 ion, whereas the association reaction occurs via the deep well of the ethylene ion. The competition in the analogous C3 HC C H2 system is not currently well understood, with both parallel and series mechanisms suggested [69, 79].
39.4 Concluding Remarks 39.3.3 Competition with Exoergic Channels There have been numerous reactions reported (especially ion–molecule reactions) in which exoergic reaction channels compete with association channels, both radiative (at low pressure) and ternary (at higher pressures). One view
While the study of chemical reaction dynamics and kinetics is a relatively mature area of investigation, many challenges remain. A central one is the development of an efficient, fully quantum mechanical method for evaluating k.T / that can be applied to a range of systems [80]. A second set of
39
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challenges comes in the intrinsically non-adiabatic nature of chemical reactions. As a system moves from reactants to products, there is a dramatic change in the electronic wave function. In fact the very existence of a transition state reflects the avoided crossing of two potential energy surfaces. There is experimental and theoretical evidence that the fact that the dynamics takes place on multiple potential surfaces has a large effect on the gas-phase state-to-state dynamics, and eventually the rate coefficients. Finally, as the evaluation of the electronic structure of atoms and molecules becomes more easily accomplished through the use of sophisticated computer packages, researchers will be able to ask more detailed questions about the relationships between the topology of potential surfaces and the corresponding rate coefficients than was possible as recently as ten years ago [81]. Acknowledgements E. H. wishes to acknowledge funding from the National Science Foundation (US) for his research in astrochemistry via grant AST-1906489.
E. Herbst
23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.
35.
References 1. Johnston, H.S.: Gas Phase Reaction Rate Theory. Ronald, New York (1966) 2. Stantcheva, T., Shematovich, V.I., Herbst, E.: Astron. Astrophys. 391, 1069 (2002) 3. Gillespie, D.T.: J. Comp. Phys. 22, 403 (1976) 4. Charnley, S.B.: Astrophys. J. 509, L121 (1998) 5. Biham, O., Furman, I., Pirronello, V., Vidali, G.: Astrophys. J. 553, 595 (2001) 6. Green, N.J.B., Toniazzo, T., Pilling, M.J., Ruffle, D.P., Bell, N., Hartquist, T.W.: Astron. Astrophys. 375, 1111 (2001) 7. Levine, R.D., Bernstein, R.B.: Molecular Reaction Dynamics. Oxford University Press, New York (1974) 8. Bates, D.R., Herbst, E.: In: Millar, T.J., Williams, D.A. (eds.) Rate Coefficients in Astrochemistry, p. 17. Kluwer, Dordrecht (1988) 9. Herbst, E., Bates, D.R.: Astrophys. J. 329, 410 (1988) 10. Robinson, P.J., Holbrook, K.A.: Unimolecular Reactions. Wiley, New York (1972) 11. Forst, W.: Theory of Unimolecular Reactions. Academic Press, New York (1973) 12. Troe, J.: J. Phys. Chem. 83, 114 (1979) 13. Gilbert, R.G., Smith, S.C.: Theory of Unimolecular and Recombination Reactions. Blackwell, Oxford (1990) 14. Jakubetz, W., Sokolovski, D., Connor, J.N.L., Schatz, G.C.: J. Chem. Phys. 97, 6451 (1992) 15. Auerbach, S.M., Miller, W.H.: J. Chem. Phys. 98, 6917 (1993) 16. Troe, J.: Part. Adv. Chem. Phys. LXXXII(2), 485 (1992) 17. Su, T., Bowers, M.T.: In: Bowers, M.T. (ed.) Gas Phase Ion Chemistry, vol. 1, p. 83. Academic Press, New York (1979) 18. Rowe, B.R.: In: Millar, T.J., Williams, D.A. (eds.) Rate Coefficients in Astrochemistry, p. 135. Kluwer, Dordrecht (1988) 19. Anicich, V.G., Huntress, W.T.: Astrophys. J. Suppl. 62, 553 (1986) 20. Hirschfelder, J.O., Curtiss, C.F., Bird, R.B.: Molecular Theory of Gases and Liquids. Wiley, New York (1954) 21. Clary, D.C., Halder, N., Husain, D., Kabir, M.: Astrophys. J. 422, 416 (1994) 22. Herbst, E.: Cold chemistry and beyond: the astrochemical context. In: Rowe, B.R., Canosa, A., Heard, D.E. (eds.) Uniform Super-
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37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
51. 52. 53. 54. 55. 56.
57. 58. 59.
sonic Flows in Chemical Physics. World Scientific, London, UK, in press Smith, S.C., Troe, J.: J. Chem. Phys. 97, 8820 (1992) Clary, D.C.: In: Millar, T.J., Williams, D.A. (eds.) Rate Coefficients in Astrochemistry, p. 1. Kluwer, Dordrecht (1988) Clary, D.C.: J. Chem. Phys. 91, 1718 (1987) Adams, N.G., Smith, D., Clary, D.C.: Astrophys. J. 196, L31 (1985) Bates, D.R., Morgan, W.L.: J. Chem. Phys. 87, 2611 (1987) Smith, S.C., McEwan, M.J., Gilbert, R.G.: J. Phys. Chem. 93, 8142 (1989) Graff, M.M.: Astrophys. J. 339, 239 (1989) Clary, D.C., Stoecklin, T.S., Wickham, A.G.: J. Chem. Soc. Faraday Trans. 89, 2185 (1993) Rowe, B.R., Canosa, A., Sims, I.R.: J. Chem. Soc. Faraday Trans. 89, 2193 (1993) Dashevskaya, E.I., Maergoiz, A.I., Troe, J., Litvin, I., Nikitin, E.E.: J. Chem. Phys. 118, 7313 (2003) Georgievskii, Y., Klippenstein, S.J.: J. Chem. Phys. 118, 5442 (2003) Clary, D.C., Buonomo, E., Sims, I.R., Smith, I.W.M., Geppert, W.D., Naulin, C., Costes, M., Cartechini, L., Casavecchia, P.: J. Phys. Chem. A106, 5541 (2002) Chastaing, D., James, P.L., Sims, I.R., Smith, I.W.M.: J. Chem. Soc. Faraday Discuss. 109, 165 (1998) Rowe, B.R., Rebrion-Rowe, C., Canosa, A.: In: Minh, Y.C., van Dishoeck, E.F. (eds.) Astrochemistry: from Molecular Clouds to Planetary Systems, p. 237. Sheridan Books, Chelsea (2000) Chastaing, D., Le Picard, S.D., Sims, I.R., Smith, I.W.M.: Astron. Astrophys. 365, 241 (2001) Light, J.C.: Disc. Faraday Soc. 44, 14 (1967) Chesnavich, W.J., Bowers, M.T.: In: Bowers, M.T. (ed.) Gas Phase Ion Chemistry, p. 119. Academic Press, New York (1979) Gerlich, D.: J. Chem. Phys. 90, 3574 (1989) Whitten, G.Z., Rabinovitch, B.S.: J. Chem. Phys. 38, 2466 (1963) Whitten, G.Z., Rabinovitch, B.S.: J. Chem. Phys. 41, 1883 (1964) Bass, L.M., Chesnavich, W.J., Bowers, M.T.: J. Am. Chem. Soc. 101, 5493 (1979) Klots, C.E.: J. Phys. Chem. 75, 1526 (1971) Herbst, E., Knudson, S.K.: Chem. Phys. 55, 293 (1981) Galloway, E.T., Herbst, E.: Astrophys. J. 376, 531 (1991) Hill, T.L.: An Introduction to Statistical Thermodynamics. Addison-Wesley, Reading (1960) Harding, L.B., Schatz, G.C., Chiles, R.A.: J. Chem. Phys. 76, 5172 (1982) Truhlar, D.G., Garrett, B.C.: Acc. Chem. Res. 13, 440 (1980) Hase, W.L., Wardlaw, D.M.: In: Ashfold, M.N.R., Baggott, J.E. (eds.) Bimolecular Collisions, p. 171. Royal Society of Chemistry, London (1989) Slater, N.B.: Theory of Unimolecular Reactions. Cornell Univ. Press, Ithaca (1959) Stein, S.E., Rabinovitch, B.S.: J. Chem. Phys. 58, 2438 (1973) Bass, L.M., Cates, R.D., Jarrold, M.F., Kirchner, N.J., Bowers, M.T.: J. Am. Chem. Soc. 105, 7024 (1983) Herbst, E.: J. Chem. Phys. 82, 4017 (1985) Wlodek, S., Bohme, D.K., Herbst, E.: Mon. Not. R. Astron. Soc. 242, 674 (1990) ˙ R., ˙ Smith, S.C., Wilson, P.F., Sudkeaw, P., MacLagan, R.G.A. McEwan, M.J., Anicich, V.G., Huntress, W.T.: J. Chem. Phys. 98, 1944 (1993) Smith, S.C.: J. Chem. Phys. 97, 2406 (1993) Klippenstein, S.J., Faulk, J.D., Dunbar, R.C.: J. Chem. Phys. 98, 243 (1993) Chen, Y., Rauk, A., Tschuikow-Roux, E.: J. Phys. Chem. 95, 9900 (1991)
39 Gas Phase Reactions 60. Frost, M.J., Sharkey, P., Smith, I.W.M.: J. Phys. Chem. 97, 12254 (1993) 61. Wigner, E.P.: Z. Phys. Chem. B19, 203 (1932) 62. Truong, T.N., Truhlar, D.G.: J. Chem. Phys. 93, 1761 (1990) 63. Truong, T.N., Truhlar, D.G.: J. Chem. Phys. 97, 8820 (1993) 64. Takayanagi, T., Masaki, N., Nakamura, K., Okamota, M., Sato, S., Schatz, G.C.: J. Chem. Phys. 86, 6133 (1987) 65. Miller, W.H.: J. Am. Chem. Soc. 101, 6810 (1979) 66. Herbst, E., DeFrees, D.J., Talbi, D., Pauzat, F., Koch, W., McLean, A.D.: J. Chem. Phys. 94, 7842 (1991) 67. Smith, D., Glosik, J., Skalsky, V., Spanel, P., Lindinger, W.: Int. J. Mass Spectrom. Ion Proc. 129, 145 (1993) 68. Flatin, D.C., Holton, J.J., Beaky, M.M., Goyette, T.M., DeLucia, F.C.: J. Mol. Spectrosc. 164, 425 (1994) 69. Gerlich, D., Horning, S.: Chem. Rev. 92, 1509 (1992) 70. Dunbar, R.C.: In: Ng, C.Y., Baer, T., Powis, I. (eds.) Unimolecular and Bimolecular Ion-Molecule Reaction Dynamics, p. 270. Wiley, New York (1994) 71. Adams, N.G., Smith, D.: In: Fonijn, A., Clyne, M.A.A. (eds.) Reactions of Small Transient Species: Kinetics and Energetics, p. 311. Academic Press, New York (1983) 72. Davies, J.W., Pilling, M.J.: In: Ashfold, M.N.R., Baggott, J.E. (eds.) Bimolecular Collisions, p. 105. Royal Society of Chemistry, London (1989) 73. Barker, J.R.: J. Phys. Chem. 96, 7361 (1992) 74. Herbst, E.: Chem. Phys. 65, 185 (1982) 75. Dunbar, R.C.: Int. J. Mass Spectrom. Ion Proc. 100, 423 (1990) 76. Herbst, E., McEwan, M.J.: Astron. Astrophys. 229, 201 (1990) 77. Herbst, E., Dunbar, R.C.: Mon. Not. R. Astron. Soc. 253, 341 (1991) 78. Maluendes, S.A., McLean, A.D., Herbst, E.: Chem. Phys. Lett. 217, 571 (1994) 79. Maluendes, S.A., McLean, A.D., Yamashita, K., Herbst, E.: J. Chem. Phys. 99, 2812 (1993) 80. Miller, W.H.: Acc. Chem. Res. 26, 174 (1993) 81. Klippenstein, S.K., Harding, L.B.: J. Phys. Chem. A. 103, 9388 (1999)
595 Eric Herbst Dr. Eric Herbst is Commonwealth Professor of Chemistry and Astronomy at the University of Virginia. Herbst is a Fellow of the American Physical Society and the Royal Society of Chemistry (UK), where he won the Centenary Prize. His major work is in the field of astrochemistry, in which he studies how chemistry can deepen our understanding of the universe.
39
Gas Phase Ionic Reactions Abstract
40
Nigel G. Adams
Contents
40.1
40.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597
40.2
Reaction Energetics . . . . . . . . . . . . . . . . . . . . . 598
40.3
Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . 599
40.4 40.4.1 40.4.2
Reaction Processes . . . . . . . . . . . . . . . . . . . . . 600 Binary Ion–Neutral Reactions . . . . . . . . . . . . . . . 600 Ternary Ion–Molecule Reactions . . . . . . . . . . . . . 603
40.5
Electron Attachment . . . . . . . . . . . . . . . . . . . . 604
40.6 40.6.1 40.6.2
Recombination . . . . . . . . . . . . . . . . . . . . . . . . 604 Electron–Ion Recombination . . . . . . . . . . . . . . . . 604 Ion–Ion Recombination (Mutual Neutralization) . . . . 605
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
Abstract
Ionic reactions in the gas phase is a broad field encompassing a multitude of interactions between ions (both positively and negatively charged), electrons and neutrals. These can be as simple as the transfer of an electron between molecules, or can be complex with considerable bond breaking, reforming and rearrangement. Reactions of the general type AC; C B ;n ! products ;
(40.1)
illustrating this diversity, are given in Table 40.1. The reaction process can be considered as consisting of three parts: (a) the initial interaction, in which the colliding particles, AC; and B ;n are drawn together by an attractive potential, (b) the reaction intermediate and transition state, in which reactants are transformed into products and (c) the weakening interaction as the product particles separate. Interactions can also occur where only elastic scattering is involved but these are considered elsewhere. Keywords
rate coefficient proton affinity intermediate complex electron attachment dissociative recombination
Overview
Part (a) of the interaction is readily studied theoretically by classical mechanics in terms of reactant particle motions controlled by the interaction potential u.r/ between the particles, where r is the interparticle separation. For ion– neutral interactions, this can take the form (i) of an attractive ion-polarization or induced dipole potential [1] u.r/ D ˛d q 2 =2r 4, where ˛d is the polarizability of the neutral and q is the charge on the ion; anisotropy in the polarizability can also be taken into account [2], (ii) of an ion-permanent dipole potential [3] with u.r/ D qD cos =r 2 , where D is the permanent dipole of the neutral and is the angle that the dipole makes with r, and (iii) of an ion-quadrupole potential [4] with u.r/ D Qq.3 cos2 1/=2r 3, where Q is the quadrupole moment and is the angle the quadrupole axis makes with r. Such capture theories are considered in Chap. 39. Other interaction potentials can be considered, but these are of lesser significance. Coulombic interaction potentials u.r/ D q1 q2 =r, where q1 and q2 are the charges on the interacting particles, are appropriate for positive ion–negative ion recombination (more correctly termed mutual neutralization) and electron–ion recombination. For processes involving electrons, i.e., electron attachment and electron–ion recombination, the wave nature of the electron also has to be considered [5, 6]. In addition to providing a means of bringing the reactant particles together, the attractive interaction potential has another function. In part (b) of the reaction mechanism, where considerable rearrangement of the atoms in the colliding species may occur (i.e., in the intermediate complex and transition state), there may be barriers to reaction. Here, the kinetic energy gained from the interaction potential is available in the intermediate complex for overcoming such energy barriers (very much less interaction energy is available in neutral-neutral reactions and the effects of energy barriers are very much more evident [7]). Of course, this energy has to be reconverted to potential energy as the product particles separate, and is therefore not available to drive the overall
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_40
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Table 40.1 Examples illustrating the range of ionic reactions that can occur in the gas phase Reaction Process ArC C O2 ! OC 2 C Ar O C NO ! NO 2 2 2 C O2 C C He C O2 ! O C O C He 2C C3C C CC 60 C Corannulene ! Corannulene 60 C C O C N2 ! NO C N O C CH4 ! OH C CH3 H3 OC C HCN ! H2 CNC C H2 O OH C HCN ! CN C H2 O OH C H ! H2 O C e CC C C2 H2 ! C3 HC C H C CHC 3 C H2 C M ! CH5 C M Cl C BCl3 C M ! BCl 4 CM HeC 2 C e ! He2 .2He/ C h HCOC C e ! H C CO NOC C NO 2 ! NO C NO2 e C CCl4 ! Cl C CCl3 e C C7 F14 ! C7 F 14 e C O2 C M ! O 2 CM 13 C C C 12 CO • 12 CC C 13 CO OD C NH3 • OH C NH2 D
Reaction Type Nondissociative charge transfer/charge exchange Dissociative charge transfer Multiple charge transfer Ion/atom interchange or atom abstraction Atom abstraction Proton transfer Associative detachment Atom insertion Ternary collisional association Radiative electron–ion recombination Dissociative electron–ion recombination Ion–ion recombination (mutual neutralization) Dissociative electron attachment Nondissociative electron attachment Ternary collisionally stabilized attachment Isotope exchange
reaction. However, the amount of energy returned may differ from that initially converted to kinetic energy since the polarizability/dipole moment of the product neutral(s) may differ from those of the reactant neutrals. Generally, the final part (c) of the interaction has little influence on the magnitude of the reaction rate coefficient or on the product distribution once the products have significantly separated (i.e., beyond the range at which processes such as long range electron transfer can occur). Thus, to a major degree, part (b) of the mechanism and the reaction energetics determine the products of the reaction. Theories which address this part of the interaction are also discussed in Chap. 39 and these have met with mixed success. Thus, at present, experimental measurements are providing a more definitive understanding of reactions and their mechanisms. Also, the situation is not in general clear as to the form of the thermodynamic energy that governs whether a reaction will proceed spontaneously, i.e., whether it is controlled by the enthalpy change H in the reaction or by the Gibbs’ free energy change G D H TS, where S is the entropy change in the reaction [8]. This is discussed in more detail in Sect. 40.2. Rate coefficients and product distributions for these reaction processes are important in all ionized media where molecular species exist, such as interstellar gas clouds [9], planetary atmospheres [10] (including that of the Earth [11, 12]), comets (Chap. 87), the space shuttle environment [13], laser plasmas [14], plasmas used to etch semiconductors [15], hydrocarbon flames [16], etc.
40.2 Reaction Energetics The availability of sufficient energy is a primary consideration for determining whether a reaction can proceed spontaneously. Criteria for determining whether energy is given out or absorbed in a reaction are (i) exo- or endoergicity, E representing the internal energy change involved in a single interaction, (ii) exo- or endothermicity for an ensemble of particles in thermal equilibrium as defined by the enthalpy change per mole, H , in the reaction and (iii) exergonic or endergonic as defined by the Gibbs’ free energy change, G [17]. E is related to H at temperature T by HT NET ;
(40.2)
where N is Avogadro’s number. E is usually deduced from bond energies [18–20], ionization potentials [21], electron affinities [22, 23], proton affinities [22, 24, 25], gas phase basicities [22, 24–26], etc. HT0 is determined from the heats 0 by [27] of formation Hf;T HT0 D
X products
0 Hf;T
X
0 Hf;T ;
(40.3)
reactants
where the superscript 0 refers to the standard state [17] of the reactants and products [for example, see Eq. (40.1)]. GT0 is determined from HT0 and the entropy change ST0 by GT0 D HT0 TST0 ;
(40.4)
40 Gas Phase Ionic Reactions Abstract
599
where X
ST0 D
products
ST0
X
ST0 ;
(40.5)
reactants
and the ST0 are the standard entropies [17]. In cases where all 0 and ST0 are not available, they can often be deof the Hf;T duced by constructing other reactions involving the species of interest and other species for which the required thermodynamic information is known ([17] discusses the details of ways in which this can be achieved). Alternatively, the magnitude of a thermodynamic parameter can be calculated using equilibrium statistical thermodynamics, if the energies of all of the occupied molecular energy levels are known [28]. For studies where all reacting particles of a given type have the same energy, such as in beam/beam interactions where cross sections .E/ are measured, then E is most appropriate. Alternatively, for reactions involving an ensemble of particles in thermal equilibrium at a temperature T , such as those studied in high pressure mass spectrometer ion sources or afterglows where rate coefficients k.T / are measured, H and G are more appropriate. (The terms rate coefficient and rate constant are used interchangeably in the literature.) .E/ and k.T / are directly related via Z1 v.E/f .E/ .E/ dE ;
k.T / D 0
reacting species that are not involved in the interaction and which could be cooled as the reaction proceeds). Usually, the TS term is not sufficiently large that H and G have different signs so that confusion about the spontaneity of reactions often does not arise. Recently, however, reactions have been discovered which appear to proceed spontaneously even though H is positive, and there has been considerable discussion of this [8]. Eventually, all reaction processes will reach equilibrium as defined by G D RT ln K ;
(40.7)
where KD
d pCc pD a b pA pB
(40.8)
is the equilibrium constant for the reaction aA C bB • cC C dD :
(40.9)
The subscripts to the pressures p denote the components A to D, and the superscripts represent the stoichiometries [17, 28]. The equilibrium constant K also equals kf =kr , the ratio of the forward to reverse rate coefficients, obtained for thermalized particles at temperature T . Thus, from mea(40.6) surements of k or the equilibrium constant at a series of temperatures, H and S of the reaction can be separately obtained.
where v.E/ is the relative speed of the reactants and f .E/ is the Maxwell–Boltzmann energy distribution. Whether H or G is more important for determining reaction spontaneity depends on the degree of interaction of the reacting systems with the surroundings during the course of the whole reaction process. For example, if the reactions occur at low pressure such that the reaction time is much less than the collision time with the background gas, there will be no interactions with the surroundings, and the only energy available in the reactions will be H . Here, S can only determine the probability that the intermediate complex dissociates forward to products or back to reactants. At the other extreme, if the reactions are conducted at high pressure such that the reaction species are always in thermal equilibrium with the surroundings (the limiting case of this is reaction in solution), then the additional energy TS is available, and G determines whether the reactions are spontaneous. Obviously, for intermediate pressures, there is a varying degree of contact with the surroundings, and which of H and G is most applicable is more obscure. The definition of surrounding is also somewhat loose, since in the present context it represents anything that can provide a source of energy during the reaction (for example, vibrational modes of the
40.3
Chemical Kinetics
The rate coefficient k for the reaction process ACB !C CD
(40.10)
is defined by the rate equation [29] dŒA dŒB D D kŒAŒB ; dt dt
(40.11)
where the square parentheses represent the concentrations of the enclosed species, and the units of k are typically cm3 molec1 s1 (often molec1 is not written explicitly in the units). These k are deduced from the time variation of specific concentrations of thermalized reactants under a variety of conditions. For ion–neutral reactions, with A being the positive or negative ion species, the situation can usually be achieved in the laboratory where ŒB ŒA, i.e., a pseudo first-order reaction. Simple integration yields ŒA t D ŒA0 exp.k ŒB0 t / ;
(40.12)
40
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N. G. Adams
where the subscript 0 indicates that this concentration is time invariant. Similar circumstances apply to binary electron attachment reactions, where ŒA D Œe, the electron number density (the symbol ˇ is often used here in place of k). If the reaction proceeds by association that is stabilized by collision with a third body M then A C B C M ! AB C M ;
(40.13)
and the solution is ŒA t D ŒA0 exp.k ŒB0 ŒM 0 t / ;
(40.14)
with k now having units of cm6 molec2 s1 . Note that situations exist where binary and ternary reactions can occur simultaneously. If ŒB 6 ŒA, the solution of Eq. (40.11) is different. For the special case ŒB D ŒA, integration yields 1 1 D kt ; ŒA t ŒA0
(40.15)
and similarly for ŒB. This situation is usually achieved in the laboratory for the determination of k for electron–ion and ion–ion recombination (the symbols ˛e and ˛i respectively are often used to replace k here). In many applications where several processes contribute to changes in ŒA and ŒB, and where the simple limits are not applicable, the situation has to be analyzed numerically.
40.4
Reaction Processes
Recently, a nine volume series, concerned with all aspects of mass spectrometry, is being published of which Volume 1 deals with theoretical and experimental aspects of ionic reactions in great detail [30]. The reader is recommended to first consult the present chapter to get an overview of the reaction processes, and to go to this other text if more detail is required. The wide variety of possible reaction processes is listed in Table 40.1 by specific examples. However, the reaction mechanisms illustrated are completely general. Magnitudes of the rate coefficients for binary interactions between charged and neutral particles, vary from 107 cm3 s1 for electron attachment [31, 32] to 108 or 109 cm3 s1 for unit efficiency ion–molecule reactions [33–36]. The efficiency is defined as the ratio of the measured k to the theoretical collisional value, i.e., that determined using the appropriate interaction potential. Ion–molecule reactions involving molecules with large permanent dipoles (for example, HCN at 2:98 D and HCl at 1:08 D [37]), can have k > 107 cm3 s1 at low T [37–39] due to locking of the dipole along the
line joining the reactant species, thereby maximizing the strength of the interaction. Rate coefficients can be much smaller (by orders of magnitude) than these upper limits if the efficiency of part (b) of the mechanism is small. The part of the mechanism after collision occurs is treated by transition state theory in terms of the partition function of the transition state; where the transition state is, and its partition function, relates to the number of ways that the available energy can be distributed in the transition state [40]. More details of the theories are given in Chap. 39 and [41]. For dissociative electron–ion recombination, the upper limits on k are generally much larger as a consequence of the long range Coulombic interaction potential, varying from 107 cm3 s1 for diatomic ions to > 106 cm3 s1 for more polyatomic species [42–45]. For ion–ion recombination, the rate coefficients are about an order of magnitude smaller than these values because of the larger mass of the negative ion relative to the electron, and thus the smaller interaction velocity (the of the two processes are, in fact, similar) [46]. Electron–ion recombinations that are radiatively stabilized generally have a small k, as small as 1012 cm3 s1 [47], because of the large magnitude of the radiative lifetime of the reaction intermediate compared with the time for autoionization. For most reaction types, experimental rate coefficients are available at room temperature, and for some, the temperature and/or energy dependencies have also been determined [31, 32, 43, 44, 46, 48]. Generally, only the ion products are identified using mass spectrometry, and the product neutrals are not determined (unless the energetics allow only one neutral product), and the states of excitation of the products are not identified. For ternary processes involving association, k varies from totally saturated at the binary collision limit to very small ( 1032 cm6 s1 ) for ion–neutral reactions [33, 34] and similarly for electron attachment [49]. Little is known about collision stabilized recombination [50]. Each type of reaction process has its own characteristic behavior and dependence on temperature. Rate coefficients, temperature dependencies, and product distributions (where available) have been tabulated for ion–neutral reactions [33, 34, 51] and electron attachment [31, 32]. Less data are available for electron–ion and ion–ion recombination, so no attempts have been made to compile these. Data are available in [6, 42–44, 46].
40.4.1 Binary Ion–Neutral Reactions Over the years, these reaction processes have been reviewed several times, but from different perspectives. Most recently, studies of positive ion–molecules in flow tubes [52] and negative ion–molecule reactions from an organic mechanistic viewpoint [53] have been discussed.
40 Gas Phase Ionic Reactions Abstract
Charge Transfer and Charge Exchange These processes involve the exchange of an electron which can occur at relatively large interparticle separations (i.e., up to 6 Å [54]). Thus, in principle, k values can be larger than the collision limiting value. Such reactions are, in general, relatively fast, although there are some notable exceptions (for example, HeC C H2 , k D 1 1013 cm3 s1 ; NeC C H2 , k < 2 1014 and NeC C N2 , k D 1:1 1013 [33]). Attempts have been made to relate the efficiency of charge transfer to the Franck–Condon overlap between the neutral reactant and the product ion, with mixed success [55]. More energy is generally available in the positive ion reactions than the negative ion reactions since ionization potentials are much larger than electron affinities, and thus more dissociative products would be expected, as is usually observed. The reaction ( NC C N C He C 0:28 eV C He C N2 ! NC 2 C He C 9:00 eV ;
601
complex dissociates to products is similar to that available when it dissociates back to the reactants, and the k approximates to one-half the collisional value. For proton transfer reactions which are not highly exo- or endoergic, kf and kr can be measured and thus G determined (using Eq. (40.7)). If S can be determined in some way, or if kf =kr can be determined as a function of T , then H can be deduced and used to obtain the proton affinity difference (S can also be determined in the latter case). This has been used to construct proton affinity scales [24, 25]. Care is required in such studies to ensure that no vibrational excitation remains in the reactant ion due to its formation, and that the identities (i.e., isomeric forms) of the reactant and product ions are known (e.g. whether the ion is HCOC or the higher energy form COHC [62]). Often the individual reaction processes do not occur in isolation. For example, in the reactions of CHC 4 with COS, H2 S, NH3 , H2 CO and CH3 OH, both charge transfer and proton transfer are energetically possible, and both channels are observed. Here, the least exoergic channel is favored in all cases [63]. Proton transfer also occurs in negative ion reactions, for example those of OH and NH 2 giving H2 O and NH3 respectively (e.g. see Table 40.1), and these types of reactions are production sources for many negative ions [53, 64].
is particularly interesting and has been studied in considerable detail. The product distribution slightly favors NC C (60%) rather than NC 2 [56], with the N channel becoming even more important with increasing vibrational temperature of the N2 [57]. Spectroscopic emission studies [58] have shown that a significant fraction (5%) of the reactions pro2 C ceed by charge transfer into the NC 2 .C ˙g / state followed Ion–Atom Interchange and Atom Abstraction by the radiative decay In many simple cases, these two processes are the same, for
2 C C C 2 C N C ˙ !N X ˙ C h : (40.16) example 2
u
2
g
This channel competes with the predissociation
2 C C NC 2 C ˙u ! N C N :
ArC C H2 ! ArHC C H ;
(40.19)
(40.17) but not so for the more complicated molecules (e.g., O C CH4 ; see Table 40.1). Such reactions, when exoergic, usually A new mechanism has recently been observed in which occur close to the collisional rate although somewhat slower. charge transfer occurs in parallel with chemi-ionization (or, Series of such reactions with H2 occur in the interstellar medium and are responsible for producing the hydrogenaequivalently, electron detachment) tion in many of the species observed there, for example, in C C C and H3 OC CHC C e C He : (40.18) HeC C C60 ! C2C 3 production from CH , NH4 from NH 60 C from OH (see Chap. 86). The reaction, Such a process is, of course, only energetically possible for C (40.20) NHC high recombination energy ions like HeC and NeC [59]. 3 C H2 ! NH4 C H ; 3C Also, multiply charged C60 .C60 / has been seen to undergo two electron transfer with Corannulene and some Polycyclic is particularly interesting. At temperatures greater than Aromatic Hydrocarbons (PAH’s), generally in parallel with 300 K, the reaction shows an activation energy barrier of 2 kcal mol1 . The rate coefficient k decreases with decreasa whole series of other reaction channels [60]. ing T , reaching a minimum of 2 1013 cm3 s1 at 80 K Proton Transfer and then increases at lower T due to tunneling through the Where proton transfer is significantly exoergic, as deter- barrier [48]. This behavior occurs because, at the higher temmined by the difference in the proton affinities of the reactant peratures, the lifetime of the intermediate complex is not and product neutrals, reaction usually proceeds at the colli- sufficient for significant tunneling to occur, but there is suffisional rate [61]. If the reaction is close to thermoneutral, then cient energy in the reacting species to overcome the barrier. the amount of phase space available when the intermediate At lower temperatures, there is not sufficient energy available
40
602
N. G. Adams
to overcome the barrier, but the lifetime of the intermediate complex becomes long enough for significant tunneling to occur [65]. This explanation has been substantiated by the C isotopic studies of the reactions NHC 3 C D2 , and ND3 C H2 and D2 [66]. For some reactions, isotopic labeling has been used to identify the reaction mechanism. For example, the reaction OC C O2 ! OC 2 CO
(40.21)
In addition to charge and proton transfer, other channels requiring more rearrangement are evident.
Isotopically Labeled Reactants A great deal can be learned about reaction mechanisms in significantly exoergic ion–neutral reactions by isotopic labeling. Some such reactions have been mentioned above, but a particularly graphic example is the reaction
C CHC (40.24) has been shown to be predominantly charge transfer, rather 4 C CH4 ! CH5 C CH3 ; 18 C than ion–atom interchange, by labeling the ion as O [67, 68]. which was assumed to be either proton transfer and/or Hatom abstraction, depending on the interaction energy. By Associative Detachment studying the reaction with both the ion and the neutral reFor negative ions, an additional process is possible for which actants separately deuterated, the reaction was shown to be there is no equivalent positive ion analog (although there much more complex, with the product ions being CH4 DC C C is the related process of chemi-ionization, i.e., AB C C ! (10%), CH3 DC 2 (22%), CH2 D3 (43%) and CHD4 (25%) for C C ABC Ce ). In this process, the negative ion and the neutral the reaction of CH4 with CD4 [74]. The reaction clearly proassociate, and the association is stabilized by the ejection of ceeds via a long-lived intermediate in which there is a large the electron (see Table 40.1). Such reactions can only occur degree of isotope mixing before unimolecular decomposition when the electron detachment energy is less than the energy to products. of the bond that is produced. Therefore, these reactions usuA further class of isotopic reactions are those for which ally involve radical species which produce stable molecules. the exoergicity is provided only by the different zero point k values are usually an appreciable fraction of the collisional energies of the reactants and products. Many such reactions values [69, 70]. Infrared emissions have been detected from have been studied. Examples of the various types are: (i) The a series of these reactions [71], for example O with CO symmetrical charge transfer and F , Cl and CN with H, and show that the reactions 15 C 15 populate the highest vibrational levels that are energetically N2 C 14 N2 ! 14 NC (40.25) 2 C N2 ; accessible.
which proceeds in both directions at more than half the collisional rate. This implies that the reaction proceeds via long-range charge transfer, a conclusion substantiated by the fact that no mixed product 14 N 15 NC is produced [67]. (ii) Proton transfer reactions exemplified by 14 N2 HC C 15 N2 and H 12 COC C 13 CO, which are both exoergic by about 1 meV [48]. Some isotopic scrambling has been observed in the latter reaction by using the double isotopic substitu(40.22) tion [75], ( H13 C16 OC C 12 C18 O . 90%/ 12 18 C 13 16 H C O C C O! H13 C18 OC C 12 C16 O . 10%/ : (40.23)
Other Binary Ion–Molecule Reaction Channels As reactant species become more polyatomic, there is a greater variety of reaction processes that can occur, and these often occur in parallel. The processes that occur are too numerous to list so a few examples will have to suffice. Reactions that occur in isolation are insertion reactions of the type [72] CC C Cn Hm ! CnC1 HC mx C xH and SC C Cn Hm ! Hm1 Cn SC C H :
Multiple channels are very evident in reactions of ions produced from species with large ionization potentials and small proton affinities, for example, NHC [73], 8 H4 CNC C NH2 ˆ ˆ ˆ ˆ C ˆ ˆ 3=2kB T is required for ble, association reactions can still occur, both for positive and the complex (although still vibrationally excited) to be stable against unimolecular dissociation. However, there are cases negative ions. The reactions can be as simple as where it is believed that an electronically excited state is acC C He C He C He ! He2 C He ; (40.27) cessed [78, 79]. Examples are
40.4.2
Ternary Ion–Molecule Reactions
(40.33) CC C H2 ! CHC 2 C h to associations more complex than those listed in Table 40.1. Rate coefficients k3 at room temperature vary from 1 and 1031 cm6 s1 for reactions such as Eq. (40.27) to in excess (40.34) Cl C BCl3 ! BCl of 1 1025 cm6 s1 , this upper limit being due to experimen4 C h : tal constraints rather than the reaction itself. The mechanism As yet, radiative association has not been observed directly usually postulated is (i.e., by the detection of the emitted photon), however, the
kinetic evidence is strong for the existence of this process. A˙ C B • AB ˙ ; (40.28) For the more rapid collisional associations, competition
(40.29) with binary channels is possible, for example AB ˙ C M ! AB ˙ C M ; 8 C ˆ .70%/ ˆ where M stabilizes the excited intermediate, .AB ˙ / by re 1106 for polyatomic ions to 1 107 cm3 s1 for diatomic ions [42, 43]. For large k, there is little temperature dependence, temperature dependencies being more marked for the slower reactions.
40 Gas Phase Ionic Reactions Abstract
Two mechanisms were initially proposed: (i) the direct mechanism [86] where the neutralized ion undergoes a radiationless transition to a repulsive potential curve on which dissociation to products occurs, and (ii) the indirect mechanism [87] where the neutralized ion initially transfers to a Rydberg state and then undergoes a radiationless transition to the repulsive curve. For the former process, the theoretical temperature dependence is Te0:5 , while it is Tv1 TT1:5 for the latter, where the subscripts T and e refer to the thermal ion vibrational and electron temperatures respectively [6]. There is no reason why these two processes cannot occur in parallel, although it is not straightfoward to determine the relative contributions [87]. Fortunately, both processes are automatically included in multichannel quantum defect theory (MQDT) [88]. Experimental T -dependencies for reactions which are not close to the collisional limit have power law dependencies in the range 0:7 to 1:5 [43, 89, 90], i.e., between the theoretical predictions. In cases where there is detailed temperature data, e.g., for some hydroC C C C carbon ions (CHC 5 , C2 H3 , C2 H5 , C3 H7 , C6 H7 ) [89, 90], the dependence changes from the lower dependence at low temperature to the higher dependence at higher temperature. Less is known about the products. Detailed theory carC and NC ried out for the diatomic ions OC 2 , NO 2 , and to C a large degree HCO , are in general agreement with experiment [91, 92]. For the more polyatomic ions, the theory is not yet sufficiently quantitative [93, 94] and reliance is placed on experiment. H-atom contributions to the product distribuC C tions for ten ions (N2 HC , HCOC , HCOC 2 , N2 OH , OCSH , C C C C C H2 CN , H3 O , H3 S , NH4 and CH5 ) vary from 20% for OCSHC to 120% for CHC 5 [95]. OH is a substantial product (30 to 65%) in the above reactions where it is energetically possible [96] (i.e., excluding HCOC ), except in the case of OCSHC , perhaps indicating that the proton is exclusively on the S-atom rather than the O-atom (i.e., in the lowest energy form [97, 98]). More complete product distributions are now being obtained, initially using flowing afterglows (O2 HC , C C HCOC 2 , H2 O , H3 O ) [99], but more recently using storage C C C C C C rings (including H3 , CHC 2 , H2 O , NH2 , H3 O , CH3 , NH4 , C CH5 ) [100]. These studies are showing that fragmentation to give three products is a common, indeed dominant, mechanism [101]. Dissociative recombination has recently benn reviewed by [102]. Electron–ion recombination is an energetic process and electronically excited states can also be populated. Vibrational population distributions for these states can readily be determined if Einstein A coefficients are known for the observed transitions (I / AŒ, where I is the photon intensity, and Œ is the number density of the excited state). The N2 (B 3˘g ) state and the CO (a 3 ˘r ) state vibrational population produced in the recombinations of N2 HC , HCOC , C HCOC 2 and CO2 have been determined [103, 104]. Possible
605
mechanisms [105] have been suggested for this vibrational excitation: (i) the impulsive force on the molecular fragments as the neutralized ion rapidly dissociates, and (ii) the Franck– Condon overlap between the wave functions of the molecular products (in various vibrational states) and the wave function of this particular fragment in the neutralized ion before it dissociates. Theory underestimates the populations of the higher levels, but does predict the small observed oscillation in the occupancy of the various vibrational levels in the CO (a 3 ˘r ) state generated in the recombination of HCOC [106]. The theories of recombination discussed above assume favorable potential curve crossings, however, it has been shown both experimentally [107, 108] and theoretically [109] that such are not necessary when quantum tunneling can occur. Experimental evidence has been obtained in the recombinations of N2 HC and N2 DC where the populations of the 0 D 6 vibrational level of the N2 electronically excited (B 3˘g ) state is greatly enhanced ( 6 at 100 K) for N2 HC over N2 DC . This level is resonant with the D 0 vibrational level of the recombining ion, making tunneling more facile; and H atom tunneling is further enhanced because of the smaller mass [108].
40.6.2 Ion–Ion Recombination (Mutual Neutralization) Less information is available on this process than on electron–ion recombination; a detailed review has recently been published [110]. k values vary from about 4 108 to 1 107 cm3 s1 at room temperature [44, 46] with power law temperature dependencies of 0:4 for the only two systems that have been studied as a function of T (NOC C NO 2 and NHC 4 C Cl ) [46]. This is consistent with theory [111] in which the Landau–Zener approximation is used to determine the probability of a crossing between the Coulombic ion–ion attractive potential and the potentials of the neutral products. This transition occurs by an electron transfer (the optimum distance for such a transfer when a favorable crossing exists is about 10 Å). Theory gives that
k 2 v2 Q
2kB T
1=2
where Q is the cross section
Q D Rc2 1 C .Rc E/1 :
;
(40.43)
(40.44)
Rc is the crossing distance, E is the interaction energy and v is the relative velocity at the crossing point [111]. Once neutralized by electron transfer, the products continue undeflected with the velocity gained from the Coulombic field. Photon emission has been detected from a series of excited state products. Following the early detection [112] of
40
606
NO (A 2˙ C ) emissions from NOC C NO 2 , a variety of NO emissions have been detected from NOC recombinations with molecular (SF 6 , C6 F6 , and C6 F5 CH3 ) and atomic (Cl and I ) negative ions [113, 114] and He2 emissions from HeC 2 recombinations with C6 F6 and C6 F5 Cl [113]. These emissions were interpreted in terms of long-range electron transfer [112], however, in the recombination of KrC and XeC with SF 6 , KrF, and XeF, excimer emissions have been seen [113], suggesting an intimate encounter in these cases. Also, few data are available on the effects of pressure. For C the reactions SFC 3 C SF5 and NO C NO2 , k increases by about a factor of 4 between 1 and 8 Torr [46, 50].
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N. G. Adams 24. Lias, S.G., Liebman, J.F., Levin, R.D.: J. Phys. Chem. Ref. Data 13, 695 (1984) 25. Lias, S.G., Bartmess, J.E., Liebman, J.F., Holmes, J.L., Levin, R.D., Mallard, W.G.: J. Phys. Chem. Ref. Data 17, 1-861 (1988) 26. Bartmess, J.E., McIver, R.T.: The gas phase acidity scale. In: Bowers, M.T. (ed.) Gas Phase Ion Chemistry, vol. 2, p. 87. Academic Press, New York (1979) 27. Chase, M.W.: NIST-JANAF Thermochemical Tables, J. Phys. Chem. Ref. Data 25, 1069-1111 (1998) 28. McQuarrie, D.A.: Statistical Mechanics. Harper-Collins, New York (1976) 29. Steinfeld, J.I., Francisco, J.S., Hase, W.L.: Chemical Kinetics and Dynamics. Prentice-Hall, Englewood, p 1 (1989) 30. Armentrout, P.B.: The Encyclopedia of Mass Spectrometry vol. 1. Elsevier, Amsterdam (2003) 31. Christophorou, L.G., McCorkle, D.L., Christodoulides, A.A.: Electron attachment processes. In: Christophorou, L.G. (ed.) Electron-Molecule Interactions and their Applications, vol. 1, p. 477. Academic Press, Orlando (1984) 32. Smith, D., Adams, N.G.: Studies of plasma reaction processes using a flowing afterglow/langmuir probe apparatus. In: Lindinger, W., Mark, T.D., Howorka, F. (eds.) Swarms of Ions and Electrons in Gases, p. 284. Springer, Vienna (1984) 33. Ikezoe, Y., Matsuoka, S., Takebe, M., Viggiano, A.A.: Gas Phase Ion-Molecule Reaction Rate Constants through 1986. Maruzen, Tokyo (1987) 34. Anicich, V.G.: J. Phys. Chem. Ref. Data 22, 1469 (1993) 35. Anicich, V.: Astrophys. J. 84, 215 (1993) 36. Anicich, V.: An Index of the Literature for Bimolecular Gas Phase Cation-Molecule Reaction Kinetics. JPL Publication 03-19. JPL, Pasadena, California (2003) 37. Clary, D.C., Smith, D., Adams, N.G.: Chem. Phys. Lett. 119, 320 (1985) 38. Clary, D.C.: Mol. Phys. 54, 605 (1985) 39. Adams, N.G., Smith, D., Clary, D.C.: Ap. J. 296, L31 (1985) 40. McQuarrie, D.A.: Statistical Mechanics. Harper & Row, New York (1976) 41. Ridge, D.R.: Ion–molecule collision theory. In: Armentrout, P.B. (ed.) The Encyclopedia of Mass Spectrometry, vol. 1, p. 1. Elsevier, Amsterdam (2003) 42. Johnsen, R.: Int. J. Mass Specrom. Ion Proc. 81, 67 (1987) 43. Adams, N.G., Smith, D.: Laboratory studies of dissociative recombination and mutual neutralization and their relevance to interstellar chemistry. In: Millar, T.J., Williams, D.A. (eds.) Rate Coefficients in Astrochemistry, p. 173. Kluwer, Dordrecht (1988) 44. Mitchell, J.B.A.: Phys. Rep. 186, 216 (1990) 45. Mitchell, J.B.A., Rebrion-Rowe, C.: Int. Rev. Phys. Chem. 16, 201 (1997) 46. Smith, D., Adams, N.G.: Studies of ion–ion recombination using flowing afterglow plasmas. In: Brouillard, F., McGowan, J.W. (eds.) Physics of Ion-Ion and Electron-Ion Collisions, p. 501. Plenum, New York (1983) 47. Bates, D.R., Dalgarno, A.: Electronic recombination. In: Bates, D.R. (ed.) Atomic and Molecular Processes, p. 245. Academic Press, London (1962) 48. Adams, N.G., Smith, D.: Ion–molecule reactions at low temperatures. In: Fontijn, A. (ed.) Reactions of Small Transient Species, p. 311. Academic Press, London (1983) 49. Massey, H.W.S.: Negative Ions. Cambridge University Press, Cambridge (1976) 50. Smith, D., Adams, N.G.: Geophys. Res. Lett. 9, 1085 (1982) 51. Meot-Ner, M.: Temperature and pressure effects in the kinetics ion–molecule reactions. In: Bowers, M.T. (ed.) Gas Phase Ion Chemistry, vol. 1, p. 198. Academic Press, New York (1979) 52. Bohme, D.K.: Int. J. Mass. Spectrom. 200, 97 (2000)
40 Gas Phase Ionic Reactions Abstract 53. DePuy, C.H.: Int. J. Mass. Spectrom. 200, 79 (2000) 54. Smith, D., Adams, N.G., Alge, E., Villinger, H., Lindinger, W.: J. Phys. B 13, 2787 (1980) 55. Lauderslager, J.B., Huntress, W.T., Bowers, M.T.: J. Chem. Phys. 61, 4600 (1974) 56. Adams, N.G., Smith, D.: J. Phys. B 9, 1439 (1976) 57. Schmeltekopf, A.L., Ferguson, E.E., Fehsenfeld, F.C.: J. Chem. Phys. 48, 2966 (1968) 58. Inn, E.C.: Planet. Space. Sci. 15, 19 (1967) 59. Javahery, G., Petrie, S., Wang, J., Bohme, D.K.: Chem. Phys. Lett. 195, 7 (1992) 60. Javahery, G., et al.: Org. Mass Spectrom. 20, 1005 (1993) 61. Bohme, D.K.: The kinetics and energetics of proton transfer. In: Ausloos, P. (ed.) Interactions between Ions and Molecules, p. 489. Plenum, New York (1975) 62. McEwan, M.J.: Flow tube studies of small isomeric ions. In: Adams, N.G., Babcock, L.M. (eds.) Advances in Gas Phase Ion Chemistry, vol. 1, p. 1. JAI Press, Greenwich (1992) 63. Adams, N.G., Smith, D.: Chem. Phys. Lett. 54, 530 (1978) 64. DePuy, C.H.: SIFT-drift studies of anions. In: Almoster Ferreira, M.A. (ed.) Ionic Processes in the Gas Phase, p. 227. Reidel, Dortrecht (1984) 65. Barlow, S.E., Dunn, G.H.: Int. J. Mass Spectrom. Ion Proc. 80, 227 (1987) 66. Adams, N.G., Smith, D.: Int. J. Mass Spectrom. Ion Proc. 61, 133 (1984) 67. Fehsenfeld, F.C., Albritton, D.L., Bush, D.L., Fornier, P.G., Govers, T.R., Fourier, J.: J. Chem. Phys. 61, 2150 (1974) 68. Dotan, I.: Chem. Phys. Lett. 75, 509 (1980) 69. Fehsenfeld, F.C.: Associative detachment. In: Ausloos, P. (ed.) Interactions between Ions and Molecules, p. 387. Plenum, New York (1975) 70. Viggiano, A.A., Paulson, J.F.: Reactions of negative ions. In: Lindinger, W., Mark, T.D., Howorka, F. (eds.) Swarms of Ions and Electrons in Gases, p. 218. Springer, Vienna (1984) 71. Bierbaum, V.M., Ellison, G.B., Leone, S.R.: Flowing afterglow studies of ion reaction dynamics using infrared chemiluminescence and laser-induced fluorescence. In: Bowers, M.T. (ed.) Gas Phase Ion Chemistry, vol. 3, p. 1. Academic Press, New York (1984) 72. Smith, D., Adams, N.G., Ferguson, E.E.: Interstellar ion chemistry: Laboratory studies. In: Hartquist, T.W. (ed.) Molecular Astrophysics, p. 181. Cambridge University Press, Cambridge (1990) 73. Adams, N.G., Smith, D., Paulson, J.F.: J. Chem. Phys. 72, 288 (1980) 74. Smith, D., Adams, N.G.: Isotope exchange in ion–molecule reactions. In: Almoster Ferreira, M.A. (ed.) Ionic Processes in the Gas Phase, p. 41. Reidel, Dordrecht (1984) 75. Smith, D., Adams, N.G.: Ap. J. 242, 424 (1980) 76. Viggiano, A.A., Paulson, J.F.: J. Phys. Chem. 95, 10719 (1991) 77. Herbst, E., McEwan, M.J.: Astron. Astrophys. 229, 201 (1990) 78. Herbst, E., Schubert, J.G., Certain, P.R.: Ap. J. 213, 696 (1977) 79. Herd, C.R., Babcock, L.M.: J. Phys. Chem. 93, 245 (1989) 80. Smith, D., Adams, N.G.: Chem. Phys. Lett. 54, 535 (1978) 81. Warman, J.M., Sauer, M.C.: Int. J. Radiat. Chem. 3, 273 (1971) 82. Herd, C.R., Adams, N.G., Smith, D.: Int. J. Mass Spectrom. Ion Proc. 87, 331 (1989) 83. Naff, W.T., Compton, R.N., Cooper, C.D.: J. Chem. Phys. 54, 212 (1971)
607 84. Smith, D., Herd, C.R., Adams, N.G., Paulson, J.F.: Int. J. Mass Spectrom. Ion Proc. 96, 341 (1990) 85. Adams, N.G., Babcock, L.M., McLain, J.L.: Electron–ion recombination. In: Armentrout, P.B. (ed.) The Encyclopedia of Mass Spectrometry, vol. 1, p. 542. Elsevier, Amsterdam (2003) 86. Bates, D.R.: Phys. Rev. 78, 492 (1950) 87. Bardsley, J.N.: J. Phys. B 1, 349 (1968) 88. Giusti, A.: J. Phys. B 13, 3867 (1980) 89. McLain, J.L., Poterya, V., Molek, C.D., Adams, N.G., Babcock, L.M.: J. Phys. Chem. A. 108, 6704 (2004) 90. Ehlerding, A., et al.: J. Phys. Chem. A. 107, 2179 (2003) 91. Guberman, S.L.: Electron–ion continuum–continuum mixing in dissociative recombination. In: Rowe, B.R., Mitchell, J.B.A., Canosa, A. (eds.) Dissociative Recombination: Theory, Experiment and Applications, p. 47. Plenum, New York (1993) 92. Talbi, D., Ellinger, Y.: A theoretical study of the HCOC and HCSC electronic dissociative recombinations. In: Rowe, B.R., Mitchell, J.B.A., Canosa, A. (eds.) Dissociative Recombination: Theory, Experiment and Applications, p. 59. Plenum, New York (1993) 93. Bates, D.R.: Ap. J. 344, 531 (1989) 94. Galloway, E.T., Herbst, E.: Ap. J. 376, 531 (1991) 95. Adams, N.G., Herd, C.R., Geoghegan, M., Smith, D., Canosa, A., Rowe, B.R., Queffelec, J.L., Morlais, M.: J. Chem. Phys. 94, 4852 (1991) 96. Herd, C.R., Adams, N.G., Smith, D.: Ap. J. 349, 388 (1990) 97. Jasien, P.G., Stevens, W.J.: J. Chem. Phys. 83, 2984 (1985) 98. Scarlett, M., Taylor, P.R.: Chem. Phys. 101, 17 (1986) 99. Adams, N.G.: Adv. Gas-ph. Ion Chem. 1, 272 (1992) 100. Larsson, M.: Dissociative electron–ion recombination studies using ion synchrotrons. In: Ng, C.-Y. (ed.) Photoionization and Photodetachment, vol. II, p. 693. World Scientific, Singapore (2000) 101. Datz, S.: J. Phys. Chem. A. 105, 2369 (2001) 102. Adams, N.G., Poterya, V., Babcock, L.M.: Mass Spectrom. Revs (2005). (in press) 103. Adams, N.G., Babcock, L.M.: J. Phys. Chem. 98, 4564 (1994) 104. Skrzypkowski, M., Gougousi, T., Golde, M.F.: Spectroscopic emissions from the recombination. In: Larsson, M., Mitchell, J.B.A., Schneider, I.F. (eds.) Dissociative Recombination: Theory, Experiment and Applications, vol. IV, p. 200. World Scientific, Singapore (2000). of N2 OC , N2 OHC /HN2 OC , CO2 C , CO2 HC , HCOC /COHC , H2 OC , NO2 C , HNOC and LIF measurements of the H atom yield from H3 C 105. Bates, D.R.: Mon. Not. R. Astron. Soc. 263, 369 (1993) 106. Adams, N.G., Babcock, L.M.: Ap. J. 434, 184 (1994) 107. Butler, J.M., Babcock, L.M., Adams, N.G.: Mol. Phys. 91, 81 (1997) 108. Poterya, V., McLain, J.L., Adams, N.G., Babcock, L.M.: J. Phys. Chem. A 32, 7181-7186 (2005) 109. Bates, D.R.: Adv. Atom. Mol. Opt. Phys. 34, 427 (1994) 110. Adams, N.G., Babcock, L.M., Molek, C.D.: Ion–ion recombination. In: Armentrout, P.B. (ed.) The Encyclopedia of Mass Spectrometry, vol. 1, p. 555. Elsevier, Amsterdam (2003) 111. Olsen, R.E.: J. Chem. Phys. 56, 2979 (1972) 112. Smith, D., Adams, N.G., Church, M.J.: J. Phys. B 11, 4041 (1978) 113. Tsuji, M.: Adv. Gas-ph. Ion Chem. 4, 137 (2001) 114. Spanel, P., Smith, D.: Chem. Phys. Lett. 258, 477 (1996)
40
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Clusters Mary L. Mandich
Contents 41.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 609
41.2 41.2.1 41.2.2 41.2.3 41.2.4
Metal Clusters . . . . . . . . . . . . Geometric Structures . . . . . . . . . Electronic and Magnetic Properties Chemical Properties . . . . . . . . . Stable Metal Cluster Molecules and Metallocarbohedrenes . . . . .
41.3 41.3.1 41.3.2 41.3.3
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610 610 610 612
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Carbon Clusters . . . . . . . . . . . . . . . . . . . . . Small Carbon Clusters . . . . . . . . . . . . . . . . . . Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . Giant Carbon Clusters: Tubes, Capsules, Onions, Russian Dolls, Papier Mâché . . . . . . . . . . . . . .
. . 613 . . 613 . . 614 . . 615
41.4 41.4.1 41.4.2
Ionic Clusters . . . . . . . . . . . . . . . . . . . . . . . . . 615 Geometric Structures . . . . . . . . . . . . . . . . . . . . . 615 Electronic and Chemical Properties . . . . . . . . . . . . 616
41.5 41.5.1 41.5.2
Semiconductor Clusters . . . . . . . . . . . . . . . . . . 616 Silicon and Germanium Clusters . . . . . . . . . . . . . . 616 Group III–V and Group II–VI Semiconductor Clusters 617
41.6 41.6.1 41.6.2 41.6.3 41.6.4
Noble Gas Clusters . . . . Geometric Structures . . . . Electronic Properties . . . . Doped Noble Gas Clusters Helium Clusters . . . . . . .
41.7 41.7.1 41.7.2
Molecular Clusters . . . . . . . . . . . . . . . . . . . . . 620 Geometric Structures and Phase Dynamics . . . . . . . 620 Electronic Properties: Charge Solvation . . . . . . . . . 621
41.8
Recent Developments . . . . . . . . . . . . . . . . . . . . 621
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ters from, say, He2 to Na10000 , so do the properties of these clusters span a broad range. This chapter attempts to bring order to this diverse cluster kingdom by first sorting them into six general categories. Within each category, the physics and chemistry of the more or less similar cluster species are described. Particular emphasis is placed on the unique properties of clusters owing to their finite size and finite lattice. This chapter summarizes one of the youngest topics in this volume. Much of what is known is highly qualitative and has not yet been assembled into overarching tables or equations. Thus, this review is best regarded as a progress report on the current knowledge in this rapidly advancing field. Many of the concepts and the language used to discuss clusters are derived from condensed matter physics. The nature of these clusters impels such descriptions. Keywords
metal cluster excess electron carbon cluster molecular cluster fullerene cage
41.1 Introduction
Clusters discussed in this chapter are isolated species composed primarily of a single type of atom or molecule. Most References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 of these clusters are highly reactive and can only be made and studied under rarified conditions such as in a molecular Abstract beam. In keeping with the definition of clusters given above, Clusters are small aggregates of atoms or molecules this chapter will not attempt to cover the truly vast literawhich are transitional forms of matter between atoms or ture on atomic and molecular dimers and trimers. A great molecules and their corresponding bulk forms. Just as many of these species have been thoroughly characterized, this definition spans an incredibly broad range of clus- however they are better described as molecules rather than tiny clusters. Stabilized clusters in the form of Zintl ions, colloids, and nanoparticles have also been made. While it M. L. Mandich () is impossible to resist mentioning these clusters throughout NOKIA Corp. (retired) this chapter, they are more appropriate to condensed matter Martinsville, NJ, USA e-mail: [email protected] physics. Finally, experimental and theoretical techniques for © Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_41
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forming and studying clusters will not be covered; excellent the Mackay filled icosahedra, predominate the geometric reviews of these methods are available elsewhere [1–6]. structures [7, 10, 11] (The Mackay filled icosahedra contain concentric closed icosahedral shells of atoms plus one central atom. These structures are pentagonally symmetric and 41.2 Metal Clusters occur every n atoms by n D 13; 55; 1=3.2n C 1/.5n2 C 5n C 3/ [4]). These geometries have been deduced by a variClusters of a wide variety of refractory and nonrefractory ety of means. For example, abundance mass spectra of metal metals have been made and studied. These include clusters of clusters up to sizes containing thousands of atoms show inalkali metals, transition metals, coinage metals, main group tensity enhancements, termed magic numbers, at each cluster metals, and lanthanides. Clusters of alkali metals, particu- size which corresponds to a complete Mackay icosahedron. larly sodium, are the most well studied and understood. The Saturation coverages of transition metal clusters with varemerging picture of alkali clusters is that they behave much ious reagents can be explained in terms of covering the like quasifree electron metal spheres. Thus, approximations icosahedral faces of these clusters. The persistence of icosasuch as the jellium model used for describing bulk metals ex- hedral structures to large cluster sizes raises the question plain a large number of alkali metal properties quite well [2]. about where the crossover to the metallic packing occurs. Other metallic clusters, such as some main group metal and For example, a spherical piece of an fcc metal would have noble metal clusters, can also be understood within the jel- cuboctohedral symmetry rather than noncrystalline icosahelium model. Nonetheless, there are examples of metallic dral symmetry. Ultrafine metal particles containing 104 to clusters which deviate significantly from these simple mod- 105 atoms typically have bulk crystalline geometries [12]. els, particularly among the transition metal clusters. Little theoretical or experimental data currently exist to resolve the question of when the bulk crystalline structure emerges [10]. The point of crossover must involve kinetic 41.2.1 Geometric Structures as well as energetic factors. Although icosahedral packing is seen in beam experiments on metal clusters containing The ground state geometry is known accurately for only hundreds to thousands of atoms, high resolution electron mia handful of small metal clusters. Examples include lithium croscopy experiments on supported metal particles in this and sodium clusters containing up to nine atoms, where size range show fluctuating structures that can rapidly evolve electronic spectra are compared with accurate ab initio quan- between icosahedral, cuboctohedral, and other crystalline artum chemical calculations to deduce their structures [5–7]. rangements, depending on cluster temperature [13]. Such an approach is not a general one, either experimentally or theoretically. As the size and/or atomic valency of the metal cluster increases, the number of possible ground 41.2.2 Electronic and Magnetic Properties state structures grows enormously. Interpretation of experimental spectra requires theoretical guidance, which means A number of electronic properties have been measured that each one of these structures must be calculated using an for metallic clusters, particularly alkali metal and noble accurate electronic structure calculational method. Thorough metal clusters. These include ionization potentials, electronic theoretical investigation is only practical with current com- affinities, polarizabilities, and photoabsorption cross secputational tools for clusters containing up to about thirteen tions. The spherical jellium approximation is generally a good atoms. Methods such as molecular dynamics combined with density functional calculations have been used to speed up model for these properties in clusters of alkali and noble the process of finding and comparing various isomers [4, 8]. metals [2, 7]. This model treats the valence electrons of the Such approaches are still restricted by available computa- cluster as a delocalized sea of electrons smeared over a unitional power to clusters with relatively few atoms and valence form spherical background of ionic cores. The energy levels electrons per atom. Metal clusters of atoms with higher va- of such a jellium sphere can be calculated by confining the lency, e.g., mid-row transition metals, have proved to be jellium electrons to a three dimensional potential. A suitable quite difficult to treat accurately [9]. The large electron cor- form of this potential yields level spacings that are given relation problems inherent in these clusters must be treated by principal and angular momentum quantum numbers (n) semi-empirically, leading to large uncertainties in the relative and (l). There are no restrictions on l for a given n, and the degeneracy of a given l level is 2.2l C 1/. The levels order energies between isomers of different spin and geometry. Despite these difficulties, approximate geometries are as 1s, 1p, 1d, 2s, 1f, 2p, 1g, 2d, 3s, . The electron configuknown for many metal clusters. For cluster sizes greater than, ration of a cluster is given by filling successive energy levels roughly, several tens of atoms, there is strong experimental (termed shells) with the available valence electrons. Special evidence that spherical close packed geometries, particularly stability occurs for those clusters with a closed shell config-
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uration. This stabilization can be seen in the experimental measurements of the dependences of total binding energies, ionization potentials, electron affinities, and polarizabilities on cluster size [2, 7, 10, 11, 13, 14]. The potential used to calculate the energy levels within the jellium model has been formulated to include elliptical distortions of open shell clusters [7]. This refinement has been successful in describing the fine structures of these trends. Despite the usefulness of the jellium model, it is not applicable for very small or very large clusters. In many smaller clusters containing up to a few tens of atoms, electron-ionic lattice interactions cannot be neglected, and the spherical approximation is poor. For these clusters, accurate quantum chemical calculations must be used to determine the electronic structure. Alternatively, for clusters larger than roughly 500 to 3000 atoms (depending on which clusters and the cluster temperature), the level spacings become increasingly continuous, blurring the shell structure. When this occurs, other effects are seen to dominate the trends in cluster stability, particularly the stability arising from completing a geometric shell of atoms on the cluster [7, 10, 11, 15, 16]. The convergence of cluster electronic structure to bulk metal electronic structure has been seen in the evolution of metal cluster ionization potentials (IP ) and electron affinities (Ae ) as a function of cluster size. A good approximation for the overall trend in the IP as a function of cluster size (N is the number of atoms) is given by the electrostatic model for the work function of a classical conducting sphere [2] IP;N D WB C A
e2 ; RN
(41.1)
where WB is the work function of the bulk metal, e is the electron charge, and R is the cluster radius, which is often set proportional to N 1=3 . A is a constant which is found experimentally to have values of about 0:3 to 0:5; the theoretically derived value for A depends on the model used [2, 11]. A similar form of Eq. (41.1), where A is replaced by .A 1/, describes the Ae of a negatively charged cluster. Experimentally, the IP and Ae of alkali metal, noble metal, and some main group metal clusters behave as described by Eq. (41.1) with shell structure superimposed on the overall trend [2, 6, 14]. Equation (41.1) predicts a smooth convergence of the work function to the bulk value with increasing cluster size. This has been seen experimentally, such as in copper cluster valence and inner shell Ae which extrapolate smoothly to the corresponding bulk values [17]. In mercury clusters, the transition to bulk metallic behavior occurs more abruptly and clusters with less than 17 atoms appear to behave as nonmetals [18]. Transition metal clusters often deviate strongly from the trend in Eq. (41.1) [10]. Electronic spectra are available for a number of metal clusters including alkali, noble metal, transition metal, and aluminum clusters [7, 10, 11]. Small alkali clusters exhibit
rich spectra in the visible. Many of these spectra have been assigned using accurate electronic structure calculations as described above. Visible spectra of larger alkali metal clusters and other metal clusters are typified by giant resonances with cross sections reaching values as large as 2000 Å2 [6, 7]. In the absence of detailed electronic structure information, these spectra have been assigned using comparisons with bulk metal spectra. In particular, the giant resonances are assigned to collective excitations of the cluster valence electrons, in an analogy to bulk metal plasmon resonances. Theoretical treatments of these giant resonances for clusters have been derived from classical treatments of conducting spheres driven by an external electromagnetic field [7, 10]. While the blue shift of the resonance frequency with increasing cluster size is well predicted by the classical models, other details are less well described, such as the magnitude of the shift and the width of the resonance. Clusters which are known to be nonspherical from other measurements exhibit multiple resonance peaks; these have yet to be quantitatively described by theory. Transition metal clusters also show evidence of collective excitations; however, the magnitudes of these absorptions are 2 to 5 times larger than predicted by classical models [19]. An intriguing non sequitur occurs when the classical models are extended down to describe the resonances of small alkali clusters. This is illustrated by the example of Na8 , which exhibits a large single resonance in the photoabsorption spectrum. At first glance, the classical jellium model does fine: it predicts a spherical closed shell cluster which should exhibit a single resonance. Yet, a more thorough examination of the time dependence of this feature has revealed that it consists of four overlapping absorptions [10, 20]. These multiple absorptions clearly must arise from one-electron excitations, not a collective all-electron excitation [5]. Further work is needed to weave together the two disparate pictures of collective versus one electron excitations in metal clusters. Inner shell electrons of some metal clusters have been probed spectroscopically. Whereas excitations of the delocalized valence electrons primarily reflect the entire cluster environment, excitations of the localized inner shell electrons reflect the atomic environment. In mercury clusters, excitation of the 5d core electrons reveals a transition from insulating clusters at small sizes to more metal-like clusters with increasing s-p hybridization typical of bulk mercury [21]. Inner core electron spectra of copper and antimony clusters also reveal details regarding the evolution of the cluster lattice structure as a function of size [11, 17]. Such valuable information can be obtained from inner shell electron spectra of metal clusters that more experiments are warranted. Magnetic moments have been measured and described theoretically for a range of transition metal, rare earth, and Group III metal clusters [5, 7, 10, 22–24]. Examples of clusters which exhibit a positive magnetic moment include
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cobalt, iron, nickel, gadolinium, and terbium clusters. Clusters of metals such as vanadium, palladium, chromium, and aluminum are observed to be diamagnetic. The effective moment per atom, , in magnetic clusters is greater than in the bulk because of the lower average coordination number in clusters [24]. As the cluster size increases, the surface to volume atom ratio decreases and converges to the bulk value. The size of these magnetic clusters is smaller than the critical domain size for bulk ferromagnetism, thus they are best described as paramagnetic with a single moment given by N, where N is the number of atoms in the cluster. Stern–Gerlach deflection data have been obtained to measure . A critical parameter in interpreting these data is the so-called blocking temperature TB , the temperature at which the cluster moment unlocks from the cluster axes and orients thermally in an external field. For clusters with temperatures T > TB , the observed effective moment eff for paramagnetic clusters is given by [24] eff
NH kB T D coth ; kB T NH
(41.2)
where kB is the Boltzmann constant, and H is the magnetic field strength. Medium size iron and cobalt clusters behave well according to this classically derived expression. Deviations from this model are observed in internally cold clusters and rare earth clusters which can be explained by partial and complete locking of the magnetic moment to the cluster lattice. Also, in very large clusters, N becomes sufficiently large that alignment overwhelms thermal statistical behavior.
41.2.3 Chemical Properties The chemistry of both charged and neutral metal clusters has been studied, particularly for clusters of transition metals and Group III metals [6, 7, 25, 26]. Both chemisorption and physisorption are seen, depending on the type of metal cluster and reagent. In many cases, the observed chemistry is quite similar to that observed for the corresponding bulk metal. For example, platinum clusters dehydrogenate hydrocarbons, hydrogen chemisorbs on transition metal clusters with the exception of coinage metal clusters, and oxygen reacts readily with aluminum and iron clusters. Chemisorption reactions of metal clusters have been seen with a wide variety of reagents. Products, reaction rates, and activation barriers have been measured as a function of cluster size. With a few notable exceptions, such as the reactions of hydrogen with nickel and aluminum clusters, none of these reactions are understood at the microscopic level [10, 11]. Macroscopically, some correlations have been seen between reaction rates and other measured cluster properties. For example, some cluster reactions show evidence
that the shell structure affects reaction rates, with open shell clusters having much higher reaction rate and lower activation barriers than clusters with closed shells. In other cases, the pattern of cluster reaction rates with cluster size correlates with the cluster IP ’s. These correlations have been used to infer the essential cluster-reagent interaction which governs the reactivity for a given set of clusters with a given reagent. For example, open shell clusters and clusters with low IP ’s favor reactions which involve electron donation from the cluster to the reagent at a critical point along the reaction coordinate. Such generalities must be made with caution, however, since they do not hold up well over a broad range of cluster sizes, compositions, and reagents. Even a single cluster size can exhibit complex reactivity: isomers with different reactivity have been observed for niobium clusters and, in some cases, these isomers have been interconverted by annealing. Of course, complexity is to be expected, given the richness and diversity of chemistry that is known to occur for different metal systems in various electronic and geometric environments. Geometric structures of transition metal clusters have been studied using physisorption reactions. Weakly bound adsorbates such as hydrogen, water, and ammonia show strong saturation behavior in their uptake by metal clusters [10, 11, 27, 28]. Trends in the saturation coverage with cluster size yield information about the type of binding site and total number of binding sites for a given cluster size. Additional information regarding the nature of the adsorbate site is often available from studies of the corresponding physisorption on bulk metal surfaces. This knowledge is used to sort through possible cluster structures and deduce which geometries exhibit the correct number and type of adsorbate binding sites. For example, saturation coverages of iron, cobalt, and nickel clusters correlate with icosahedrally packed geometries over a wide range of sizes. Adsorbate binding energies have also been determined in some cases [27, 28].
41.2.4 Stable Metal Cluster Molecules and Metallocarbohedrenes Thus far, this section has focussed entirely on the properties of metal clusters isolated in the gas phase. This discussion would not be complete, however, without mentioning that a number of metal cluster molecules have been made which are sufficiently stable that they can be made in quantity and bottled [7, 10, 11, 29, 30]! Also, recently a new class of metal–carbon clusters has been discovered, termed metallocarbohedrenes, that are believed to be sufficiently stable and abundant that they can be made in bulk. The brief overview of these isolatable clusters given below is not meant to be complete but is intended to introduce the large and impressive body of work in this field. Interested readers
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are encouraged to consult the reviews cited and references therein. Metal cluster molecules consist of a metal cluster core surrounded by a stabilizing ligand shell. A large number of such metal cluster molecules has been made, including some with metal cores as large as 300 platinum or 561 palladium atoms. In many cases, crystals of single size clusters have been made with an exactly known number of metal atoms in the core. The availability of macroscopic samples of these clusters has made it possible to measure a number of their properties. Exact structures are known for many metal cluster molecules from X-ray crystallography. Electronic and magnetic properties have been determined which reveal the development of metallic behavior within the metal core. The ligand shell is found to interact strongly with the metal atoms on the surface of the metal core. This outer shell of metal atoms does not behave as a surface of metal atoms with free valence electrons such as is found on the surface of a bulk metal. Undoubtedly, such strong interactions between surface metal atoms and coordinating ligands are necessary to make a cluster which is sufficiently stable to be isolated and crystallized without coalescing. The core of atoms inside the outer metal shell of atoms does not appear to be greatly perturbed by the coordinating ligands. Studies of the electronic and magnetic properties of the core show the onset of metallic properties as a function of particle size and atomic packing. A distinct class of transition metal–carbon clusters has been recently been found which have been termed metallocarbohedrenes [31]. Within this class, the clusters M8 C12 (M D Ti, V, and Zr) are particularly abundant and stable. Extensive electronic structure calculations show that the structure of M8 C12 is metallic and should be viewed as a distorted M8 cube where each face is decorated with a C2 dimer [32]. This structure differs greatly from the corresponding bulk metal carbides, which have cubic rock salt crystalline forms. Furthermore, the metal carbide cluster formation conditions can be adjusted to yield cubic fragments of the bulk. The related metal nitride clusters are only observed to form cubic structures. Theoretical calculations on the metallocarbohedrene and cubic forms of these metal carbides and nitrides show that they are comparable in energy for the carbides, but that the cubic structures are much more stable in the metal nitride clusters [33]. Despite their apparent stability, none of the metallocarbohedrenes has yet been isolated and purified.
41.3 Carbon Clusters The discovery of the especially stable, spherical cluster of sixty carbon atoms, named buckminsterfullerene, has ignited an intense research effort in carbon clusters [7, 8, 10, 11, 34– 39]. The family of pure carbon clusters that have been made
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extends from the dimer all the way up to tubular shaped clusters containing thousands of atoms. Several of these clusters, most notably C60 and C70 , have been isolated as a single size in macroscopic quantities [40]. The fascinating properties of bulk materials made of pure and doped C60 and C70 are outside the scope of this review [8, 10, 11, 37, 39, 41]. Carbon clusters can be roughly grouped into three distinct classes. The first consists primarily of small carbon clusters which have linear or ring geometries. Hollow spheres (termed fullerenes) appear at about 28 carbon atoms and persist up to at least several hundred atoms. Very large carbon clusters containing hundreds to thousands of atoms assume various forms, including onion-like structures of concentric spheres and hollow tubes.
41.3.1 Small Carbon Clusters Theoretical and experimental studies have found that rings and linear chains are the most stable configurations for small carbon clusters. Below about ten atoms, linear cumulenes are the most stable structures for odd numbered carbon clusters and ionic carbon clusters [13, 26, 42, 43]. For the even numbered neutral carbon clusters, C4 , C6 , and C8 , however, cyclic and linear geometries are nearly isoenergetic [7, 11, 13, 42]. Various theoretical and experimental studies have yielded conflicting results as to which geometry is more stable for each of these three clusters [7, 42, 44]. Much of the controversy over the experimental evidence probably arises because the two structures are so close in energy that either isomer or both may be present, depending on the preparation conditions. By way of proof, direct experimental data have been found for coexisting linear and cyclic isomers of CC 79 and C11 [26, 44]. Also, some experimental probes are not equally sensitive to the signature from a linear versus a cyclic geometry. At ten carbon atoms, a distinct transition occurs from linear to cyclic structures [26, 43, 44]. Starting at this size, the additional bonding stabilization accrued by joining the two ends of the linear chain overcomes the strain energy resulting from ring closure. Stable monocyclic ring structures are observed for carbon clusters over a surprisingly wide range of sizes, even persisting into the size range where three dimensional fullerenes appear. Bicyclic rings, higher order polycyclic rings, and graphitic fragments also occur in this size range but many of these configurations appear to be metastable [43, 45]. Ae ’s and IP ’s have been measured for these small carbon clusters. Dramatic effects appear in the size dependence of the Ae ’s as a result of the changeover from linear to ring structures [44]. Ae ’s of the chain structures are noticeably higher than for ring structures for carbon clusters containing similar numbers of atoms. Distinct odd-even alternations in
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the Ae ’s are also seen, plus there is evidence for aromatic stabilization according to the 4n C 2 rule. No break in the IP ’s is seen across the structural transition, but the trend in the IP ’s also follows the 4n C 2 rule expected for aromaticity. Chemical reactions have been observed for both neutral and charged small carbon clusters, particularly for the cations [7, 26, 43]. In general, the linear cationic clusters, containing fewer than ten atoms, react readily with a variety of reagents and exhibit reactivity typical of carbenes. The ring shaped cationic clusters with ten or more atoms are much less reactive, and often show no detectable reaction with reagents that react efficiently with the smaller carbon clusters. It is this differential reactivity which has revealed the presence of coexisting linear and cyclic isomers. Evidence for polyacetylene versus cumulene structure in the small linear chains can also be seen in the cluster reaction patterns.
41.3.2 Fullerenes Fullerenes make up a class of carbon clusters with closed hollow carbon atom cage morphologies. The term fullerene was inspired by the geodesic domes of architect R. Buckminster Fuller, and has come into widespread usage despite its nonstandard nomenclature. The fullerene cage network is composed of interlocking rings of sp2 hybridized carbon atoms, where every carbon atom is bonded to three other carbon atoms. Five and six member rings predominate in the cages. Highly strained four and smaller membered rings are unfavorable. Seven and higher membered rings usually can facilely rearrange within the network to form five and six membered rings. Delocalization of the electrons over the cage contributes significantly to the stabilization of the fullerenes. Assuming at least five-membered rings, the smallest possible fullerene is C20 , consisting of twelve pentagons. However, the observed fullerenes all contain about 30 atoms or more. Larger fullerenes are formed by joining together pentagons, hexagons, and heptagons, with the pentagons providing the curvature necessary to close the cage. In fullerene structures composed entirely of pentagons, hexagons, and heptagons, it can be rigorously shown from Euler’s theorem that, N5 D N7 C 12, where N5 is the number of pentagons and N7 the number of heptagons. By far the most abundant fullerene that is seen is C60 . There are 1812 possible isomers for C60 but only one forms in great abundance [37], which has each of its 12 pentagons isolated and surrounded by its 20 hexagons. The resulting molecule, resembling a soccer ball, belongs to the highest point symmetry group Ih , where all the carbon atoms are equivalent [34–37]. The isolated pentagon rule arises out of minimizing strain and maximizing resonance energy and is a powerful tool for predicting the most stable fullerene isomers [37]. Besides C60 , a number of other fullerenes have been seen, most notably C70 which
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is the next larger fullerene that can be constructed using the isolated pentagon rule. The largest proven fullerene structure contains 84 carbon atoms. It is possible to construct increasingly larger fullerene cages containing hundreds of carbon atoms. Such giant fullerenes have yet to be conclusively verified in experiments, however evidence for their existence has been seen in bulk samples containing fullerene mixtures and in gas phase abundances of carbon clusters [36, 37, 39]. A number of the fullerenes has been made and purified as a single size in macroscopic quantities [37, 39, 40]. This has enabled their geometries to be determined quite accurately using a wide variety of spectroscopic and theoretical techniques. Even structures of fullerenes such as C76 , C78 , and C84 which occur as isomeric mixtures have been elucidated. The electronic structures of many of these fullerenes have been calculated and compared with experiment [37, 38]. Yet to be resolved is the issue of how these low entropy fullerene structures can form so efficiently in carbon vapor. One proposal is that cup-like prefullerenes form first which add new carbon moieties to close the cage. Other suggestions include coalescence of small carbon rings such as C6 or C10 , or stacking of intermediate size rings on a small seed ring such as C10 , or folding up of large defective graphitic fragments to form closed hollow spheroids [8, 10, 36, 37]. Some intriguing insights into this open question are provided by experiments on nonfullerene metastable forms of carbon clusters in the fullerene size regime [43]. These species can be made in the gas phase and are found to have bicyclic, graphitic, and other polycyclic ring shapes which are not in the form of partial fullerene cages. Upon annealing, some of these ring forms are seen to convert into the fullerenes, which suggests that they are important intermediates for forming the fullerene cage. Similar polycyclic ring shapes are observed during the initial melting of fullerenes in molecular dynamics simulations [46]. Since the fullerenes are hollow, much attention has focussed on putting something inside. A wide variety of noble gas atoms and metal atoms has been successfully loaded into fullerenes, forming a family of so-called endohedral complexes [36, 37, 39, 47–51]. Noble gas atoms encapsulated in fullerenes appear to assume central positions within the cage and do not perturb the overall electronic structure of the fullerene [11, 47, 52]. Endohedral complexes where the encapsulated species is a metal atom, called metallofullerenes, encompass a wide range of metals (M) and fullerene sizes from U@C28 to Sc3 @C82 (@ denotes that the metal atom is inside the fullerene). Separation and purification of a macroscopic amount has been achieved for several of these [48, 49, 51]. Theoretical and experimental data indicate that the endohedral metallofullerenes have considerable charge transfer from metal atom(s) to the fullerene cage [37, 47, 48, 51, 52]. This interaction affects the oxidation states of the metal atom(s) and the cage and, in some
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cases, causes the metal atom(s) to locate off-center. It is yet to be generally understood why, within the range of known fullerene sizes, only some form endohedral complexes, particularly C28 , C60 , C70 , C74 , C80 , C82 , and C84 . The M@C28 endohedral complex is especially intriguing because it appears that C28 itself does not form as an empty fullerene. The unsaturated surfaces of fullerenes undergo a broad range of chemical reactions. The availability of macroscopic samples of fullerenes, especially C60 and C70 , has enabled the preparation of numerous fullerene derivatives. With their high electron affinities, fullerenes readily form a wide variety of charge transfer compounds. These include the so-called exohedral complexes, where metal atoms are attached to the outside of the cage. Particularly stable exohedral metal-fullerene clusters have been observed which have one alkaline earth atom decorating each ring of the cage [53]. The unsaturated double bonds of the fullerenes can be functionalized with reagents such as halogens, aromatics, and alcohols [37, 39]. Some of the bulk forms of these complexes and derivatives exhibit amazing properties such as superconductivity at temperatures as high as 33 K seen in C60 films doped with alkali or alkaline earth metals [51]. References [37, 39, 51] contain recent reviews.
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Little experimental detail is available on the electronic or mechanical properties of these tubes. Theoretical calculations have been performed on the electronic properties of perfect tubes constructed in various ways [37, 51]. Numerous configurations of tubes are possible, depending on the tube diameter and the overall screw axis formed by the rows of carbon hexagons wound around the tube waist. The geometrical arrangement strongly affects the electronic properties of a tube. Appropriate choices of diameter and tilt angle of the screw axis yield tubes which are metallic or semiconducting. Tubes and capsules with perfect lattice arrangements are calculated to be extremely stiff, forming the strongest carbon fibers known. Almost no data exist on the chemical properties of carbon tubes, capsules, and onions. While they are stable enough to isolate in air, reaction occurs with O2 and CO2 at high temperatures which destroy the tubes [51]. Finally, as for the fullerenes, tubes, capsules, and onions have a hollow cavity which can be filled. These nanocapsules have been successfully loaded with lead and gold atoms, as well as crystalline metal carbide particles [51].
41.4 Ionic Clusters
A growing body of experimental and theoretical evidence on alkali halide and alkaline earth oxide clusters shows that these tiny clusters of ionic materials are ionically bound. Most of these clusters have highly ordered crystal structures Giant clusters of pure carbon have been found which have even at very small sizes. Thus, these clusters offer exceltubular, capsular, and spherical shapes [8, 10, 37, 51, 54]. lent systems for studying electron localization on finite-sized These clusters occur both as single entities and with multiple crystalline lattices. concentric layers. The basic structure consists of a spiralled or rolled graphitic sheet made up of hexagonal rings of sp2 hybridized carbon atoms. Just as in the fullerenes, pentag- 41.4.1 Geometric Structures onal rings provide the curvature required to form a ball or to cap the ends of tube shapes. Negative curvature has also Most ionic clusters of alkali halides and alkaline earth oxbeen seen which is believed to result from heptagonal rings. ides assume the cubic rock salt lattice typical of bulk sodium Carbon tubes and capsules are formed prolifically in the chloride [6, 7, 10, 13, 55]. This lattice is favored not only for same carbon arcs which produce fullerenes. Typical diam- clusters of rock salt cubic solids such as NaCl and NaF, but eters range from about 5 to 100 nm, depending on the inner also for other clusters such as Csx Iy where the bulk form has diameter and number of layers, and lengths of up to several the cesium chloride cubic crystal structure. Exceptions occur microns are seen. For concentric or spiralled structures, the when the cluster size is smaller than, roughly, a unit cube, or average interlayer spacing is 3:4 to 3:5 Å which is slightly when the cluster anions and cations differ greatly in ionic ralarger than in crystalline graphite. Although tubes and cap- dius, such as in lithium bromide clusters (even though solid sules are the most commonly seen morphologies, these are LiBr has the NaCl crystal structure) [10]. The bulk rock salt observed to convert to layered spherical onion structures un- lattice is retained even for clusters where the number of ander intense irradiation [8]. This suggests that collapsed onion ions and cations are unequal, as long as the deviation from structures are more stable. Currently, there exists some de- stoichiometry is not too great. In clusters with a large excess bate over whether the multilayer tubes or capsules consist of of alkali atoms, the extra metal atoms segregate to a face of layers of complete shells within shells, such as in a Russian the cluster and form a metallic overlayer [55]. Maximization of ionic interactions and minimization of doll, or whether these layers are so highly defective that the overall structures are best described as papier-mâché consist- surface energy leads to cuboid crystal morphologies for these clusters which have as many (1 0 0) faces as possible. Particing of numerous overlying graphitic fragments [54].
41.3.3 Giant Carbon Clusters: Tubes, Capsules, Onions, Russian Dolls, Papier Mâché
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ularly stable clusters occur for sizes where the cuboid lattice is completely filled and has nearly equal numbers of ions on all sides [55]. Excess electrons or holes also play a crucial role in the stability of ionic clusters. For example, while the ŒNa14 Cl13 C cluster forms a particularly stable cube (with 9 atoms on each square face and one atom in the center), its counterpart ŒNa13 Cl14 C does not because it has fewer holes than the available electrons; however, the anion ŒNa13 Cl14 does form a particularly stable cube [11, 55].
41.4.2
Electronic and Chemical Properties
The IP ’s, Ae ’s, photoabsorption spectra and photoabsorption cross sections have all been measured for ionic clusters, par
ticularly alkali halide Mj Xk clusters. Understanding the role of the coulombic interactions, particularly excess electrons, in these clusters is key to understanding the trends in their electronic properties with cluster size, composition, and overall charge. Excess electrons are known to localize in at least four distinct ways: at anion vacancies, in weakly bound surface states, on specific alkali metal ions, or in cation– anion pair dipole fields [11, 55, 56]. First, there is a large class of ionic clusters which contain equal numbers of electrons and holes, such as clusters
C in the series Mj C1 Xj . IP and Ae of clusters within this class reflect the overall stability associated with forming the perfect filled cuboid lattices described in Sect. 41.4.1. Such perfect clusters do not chemisorb polar molecules, whereas the imperfect clusters readily do [55]. The lowest energy absorptions for clusters in this class result from charge transfer excitations, just C as in perfect ionic solids. Spectra obtained
clusters show large cross sections for abfor Csj C1 Ij sorption, consistent with charge transfer, and features which converge towards the bulk for the perfectly cubic cluster, ŒCs14 I13 C [57]. Other ionic clusters do not contain equal numbers of electrons and holes; clusters with excess electrons have been the most studied. One such class consists of clusters with at least one anionic vacancy and one or more excess electrons. In this class, e.g., ŒNa14 Cl12 C , the excess electron localizes at the lattice site of the missing anion. This is conceptually similar to an F-center in an ionic bulk crystal. Enhanced stabilization is observed for those size clusters within this class where the excess electron sitting at an anionic site yields a filled cuboid lattice. Nonetheless, because the localized electron has a large zero-point energy, its binding energy is much less than an anion at the same site. This is reflected in both the first and second IP ’s, as well as the Ae ’s of this class of clusters [55]. Just as for F-centers in ionic crystals, clusters with an excess electron localized on an anionic vacancy have strong optical absorptions at energies well below the charge transfer bands [55].
The next class of ionic clusters with excess electrons consists of perfect cuboid ionic clusters which contain one or more excess electrons, but do not have a defect binding site such as an anionic vacancy or an excess metal atom. In these clusters, e.g., Na14 Cl13 , the electron is quite weakly bound in a surface state which primarily involves the surface metal cations. These clusters have particularly low electron binding energies; for example, the IP of 1:9 eV determined for Na14 F13 is the lowest IP measured for any compound [55]. In the other two known classes of ionic clusters with excess electrons, the extra electrons have been calculated to localize on a metal cation or in a dipole potential well [58]. Some evidence for these forms of localization has been seen in photoabsorption, IP ’s and Ae ’s [10, 11, 55]. For exam ple, the photoelectron spectra of some Naj C1 Clj clusters show that the two excess electrons are singlet-coupled, and localized either at an anion vacancy, or at a single Na site so that
they behave as a Na anion loosely bound to a neutral Naj Clj cluster. The spectral behavior of other clusters within this same series, however, suggests they have the two excess electrons in a triplet coupled state, forming the analog of a bipolaron in a solid [10].
41.5 Semiconductor Clusters Semiconductor clusters make up a class of clusters where, by analogy with bulk semiconductors, covalent forces are expected to dominate electronic and geometric structure. Silicon clusters are by far the most studied of the semiconductor clusters. Some information is available on germanium clusters and compound clusters made of Groups III and V atoms or Groups II and IV atoms. Also, there are the well known stable molecules of bulk semiconductors such as the P4 tetrahedron and various sulfur rings; these have been reviewed in detail elsewhere [3]. Clusters of other possible semiconductors have been made, but little data beyond their nascent distributions are available [3]. There is also a growing body of data on silicon, III–V, and, especially II–VI semiconductor clusters in the nanometer size regime where bulk samples of stabilized forms of these clusters have been made and isolated. The crystalline and electronic structure properties, particularly quantum confinement effects, are reviewed in [10, 11, 59–61].
41.5.1 Silicon and Germanium Clusters The geometries of small silicon clusters depart radically from microcrystalline fragments of the bulk silicon diamond lattice. The structures of silicon clusters containing up to about 13 atoms have been studied extensively both experimentally and theoretically [62]. These structures are more compact
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and, starting at Si7 which has a pentagonal bipyramidal structure, have higher coordination than silicon in the bulk lattice. Multiple isomers of similar energy also appear starting at about Si10 . For example, the tetracapped trigonal prism and the symmetric tetracapped octahedron structures of Si10 are nearly isoenergetic; microcrystalline fragments such as the adamantane form of Si10 are much higher in energy. The geometric structures of larger silicon clusters are less well known. The gross shapes of clusters containing up to about 60 silicon atoms have been found experimentally to undergo a transition between Si20 and Si30 from increasingly elongated structures to more spherical structures [26]. In the transition region, both prolate and oblate isomers are observed for a single cluster size. Multiple isomers over a wide range of silicon cluster sizes have been repeatedly observed in various experiments; it appears that somewhat different sets of isomers can be produced depending on the cluster formation conditions [26]. Elucidating the ground state structures of these larger clusters theoretically is, in general, an intractable problem and requires simplifying approaches. Use of semiempirical quantum mechanical techniques or silicon interaction potentials derived from bulk silicon, however, has led to unsatisfactory results, which suggests that the silicon atoms in silicon clusters are strongly reconstructed away from the usual sp3 silicon atom environment [63, 64]. Some consensus has recently emerged that larger silicon clusters consist of internal silicon atoms strongly interconnected to a surrounding cage that has been described as a buckled fullerene [63, 65, 66]. Such a silicon cage must not be construed as a true fullerene, however, since silicon does not form the strong double bonds which stabilize the interconnected carbon rings of the fullerenes. Within this scenario, the shape change is believed to occur at the point where the cage begins to contain one or more internal atoms which provide the additional bonding needed to stabilize a spherical geometry. IP ’s and Ae ’s, cohesive energies, and photoabsorption spectra have been measured for silicon clusters containing up to several hundred atoms. Silicon cluster IP ’s start near the Si atom IP , fall abruptly between 20 and 22 silicon atoms where the shape change has been observed, and then slowly converge towards the bulk work function. This convergence is apparently quite gradual since little change is seen between Si100 and Si200 [67]. Ae ’s are only known for silicon clusters up to 15 atoms and reflect the large structural changes which occur in this small size regime [3]. Indication of a structural shape change is not observed in either the trends in the cohesive energies or the electronic spectra for silicon clusters. The cohesive energies increase smoothly with increasing size and exceed the cohesive energy of bulk silicon by 1020% [68]. Silicon clusters exhibit strong sharp absorption spectra in the near UV. A common set of absorption features appears at Si15 and persists up to at least Si70 [69]. This suggests that the strong absorptions arise from
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localized Si–Si bond excitations, and that these obscure the delocalized excitations which are more sensitive to silicon cluster structure. The most unusual aspect of these spectra is that the common signature for these minute clusters which are strongly reconstructed from the bulk is, nonetheless, strikingly similar to the spectrum of bulk diamond-lattice crystalline silicon. Silicon cluster ions are observed to react with a variety of reagents [3, 26, 68]. For example, silicon clusters chemisorb small organic molecules such as ethylene, and inorganic molecules such as O2 , NH3 , and XeF2 . Some of these reactions are sufficiently exothermic that they cause cluster fragmentation, or loss of small neutral fragments. The chemistry of silicon clusters often bears a close relationship to that known for bulk silicon surfaces; however, the clusters are often much less reactive by as much as several orders of magnitude. Silicon cluster reactions reveal the presence of numerous isomers which differ in their reaction rate for a given reagent. Isomerization has been induced by adding sufficient thermal energy to some clusters; the resulting more stable cluster form may or may not be more reactive than the higher energy isomer. This is particularly well illustrated in reactions of oblate versus prolate isomers where the more reactive isomer varies depending on cluster size [26]. In general, differential chemistry is not a useful predictor of cluster geometry per se, and appears to correlate more readily with other factors such as the number and type of dangling bonds available for reaction. Much less is known about germanium clusters. The most stable structures that have been calculated for germanium clusters are quite similar to those for silicon clusters, although the overall binding energies are lower for germanium clusters [3, 7]. Available data such as Ae ’s and photoelectron spectra for germanium clusters also underscore their similarities to silicon clusters in general [3]. The chemistry of germanium clusters has not been reported.
41.5.2
Group III–V and Group II–VI Semiconductor Clusters
Geometric structures have been calculated for some of the smaller III–V and II–VI clusters, particularly aluminum phosphide, gallium arsenide, and magnesium sulfide clusters [3, 11, 70–72]. As for silicon clusters, these clusters differ significantly from microcrystalline fragments snipped from their corresponding bulk crystalline forms. Electronegativity differences between the two constituent atoms play a major role in determining and stabilizing these structures. This is manifested in several ways. The bonding arrangements in the most stable structures have, in general, alternating electropositive and electronegative atoms. In those clusters such as III–V clusters where electronegativity dif-
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Clusters have been made of all of the stable noble gases including helium. As a general class, these clusters are the most weakly bound of all clusters, and are held together only by van der Waals forces. These interactions are well understood theoretically, and thus noble gas clusters are excellent model systems for studying a variety of structural and electronic effects at finite sizes. Helium clusters behave much as quantum liquids, and are treated separately in the discussion below.
sizes, and different noble gas clusters exhibit somewhat different preferred arrangements. Starting at 100 to 250 atoms (depending on which noble gas), formation of closed-shell Mackay icosahedra dominates the structures [78]. Finally, at somewhere in the range 600 to 6000 atoms, the structures cross over to the close-packed fcc arrangement of the noble gas solids. The increased coordination characteristic of the Mackay icosahedra explains why this morphology is adopted at small cluster sizes. With the addition of each successive icosahedral shell however, this five-fold symmetric packing becomes increasingly strained as a result of both atom–atom radial compression and tangential dilation. At some size, icosahedral structures are no longer more stable than fcc structures, and the noble gas clusters undergo a phase change. Currently, the critical size required for fcc packing is a matter of some dispute. Experimental evidence for this transformation at 600 atoms comes from electron diffraction data [76]. The spectral signatures of molecules doped in noble gas clusters suggest however, that this transition occurs at 2000 atoms [77]. Theoretical studies find a range of critical sizes, depending on the treatment used, but typically favor an even larger size regime [4, 11, 13, 77]. Knowledge of accurate pair-wise interaction potentials for noble gases has enabled extensive simulations of the physical properties of noble gas clusters [4, 6, 7, 79–81]. Phenomena such as specific heat, bond length and coordination number fluctuations, phase equilibria, dynamical freezing and melting, isomerization, and solvation have been explored using various molecular dynamics simulations. Numerous effects peculiar to the finite sizes and proliferation of isomers in small noble gas clusters are observed. For example, the melting temperature decreases significantly with an overall decrease in cluster size. Internal diffusion rates depend strongly on cluster size and the stability of certain favored structures such as the complete Mackay icosahedra. A well-studied finite size effect is the dependence of the mean energy per particle in a cluster versus the particle temperature (the so-called caloric curve) in the region where the particle is observed to melt. Unlike in the bulk, calculations find hysteresis in the melting transition of small noble gas clusters where solid-like and liquid-like forms are observed to coexist. Such coexistence is quite size-dependent. It becomes more pronounced when the clusters are rapidly heated or cooled, and appears to be a finite time averaging effect [4, 6, 7].
41.6.1 Geometric Structures
41.6.2 Electronic Properties
The overall evolution of noble gas cluster structures as a function of size is well established [4, 6, 11, 76, 77]. Icosahedral packing dominates at small cluster sizes. This packing involves polyicosahedral structures for the small cluster
The fragility of noble gas clusters has hampered measurements of their IP ’s and Ae ’s. Extensive fragmentation is known to accompany ionization, making it difficult to establish the size of the ionized parent cluster. Nonetheless,
ferences are smaller, covalent interactions predominate ionic interactions and the energetics of the various geometric structures are similar to those of the covalent silicon clusters. In II–VI clusters, electronegativity differences are much larger, and these clusters have structures more comparable to those seen in ionic clusters where ionic interactions are maximized. Some data are available on the electronic structures of III– V clusters. A strong even–odd alternation occurs in both the IP ’s and Ae ’s of gallium arsenide clusters where those clusters having a total even number of atoms have higher IP ’s and lower Ae ’s than neighboring odd-numbered clusters [3, 73]. This has been explained by electronic structure calculations which find that, in general, the odd clusters are triplets while the even clusters of gallium arsenide are singlets [70– 72]. Electronic absorption spectra have also been recorded for indium phosphide clusters which exhibit strong differences with cluster size and stoichiometry [74]. These spectra are also consistent with odd-numbered Inx Py clusters having open-shell configurations, and even-numbered Inx Py clusters having closed-shell singlet ground states, even for clusters which are considerably off-stoichiometry. A strikingly similar strong continuum-like absorption appears in the blue end of spectra for the even-numbered Inx Py clusters. The onset of these bsorptions lies close to the bulk indium phosphide band gap. The overall spectral behavior, however, is quantitatively similar to the absorptions seen in semiconductor glasses. Little is known about the chemical properties of these clusters, with gallium arsenide clusters being the only ones studied. Hydrogen chloride is observed ubiquitously to etch ŒGax Asy ; however, multiple isomers are seen with differing degrees of reactivity [3]. Chemisorption of ammonia on ŒGax Asy C has also been seen with the highest rates of reactivity occurring for the stoichiometric (x D y) clusters [75].
41.6 Noble Gas Clusters
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IP ’s have been measured for cluster sizes containing up to several tens of atoms [82]. Measured IP ’s agree with theoretical predictions which are based on the assumption that the ionizing state within the cluster is a dimer, trimer, or higher n-mer cationic core [6, 82]. Absorption profiles of noble gas cations provide supporting evidence for the delocalization of the positive charge over a n-mer unit within the cluster, although consensus has yet to be reached on the size of this cationic cluster within the cluster [6, 11]. Several distinct types of electronic excitations are observed in absorption spectra of neutral noble gas clusters [6, 7, 11, 83]. Clusters containing less than 30 atoms exhibit broad absorptions near the atomic resonance lines. These absorptions are molecular-like, but have not been described in detail above the dimer. In addition, broad continuum absorptions assigned to Rydberg excitations are observed at these small sizes. Bulk-like excitations corresponding to surface and bulk excitons in solid noble gases emerge for clusters larger than 50 atoms. The profiles of these excitonic absorptions indicate that they arise from delocalized excitations analogous to Wannier excitons in bulk solids. These cluster exciton absorptions are blue-shifted relative to the bulk as a result of quantum confinement of the exciton within the small cluster. Another type of excitonic excitation has also been reported which appears to be a localized excitonic state whose character is sensitive to the cluster structure. Relaxation of cluster excited states is accompanied by extensive fragmentation [7]. Details of the relaxation processes differ according to noble gas, cluster size and cluster structure. Again, these differences can be traced to finite size effects in these noble gas clusters [4].
619
pulsive forces and collective dielectric effects as the cluster builds the solvent shell around the solute core. Both theoretical and experimental data illustrate this evolution and show how, once the first few solvent shells are established, the spectrum converges asymptotically to the matrix isolation value. For smaller clusters, the effects of too few or incomplete solvation shells appear as shifts and/or broadening of the spectral lines relative to the bulk [3, 4]. Differential solvation of the ground and excited states causes spectral shifts which are typically to lower energy since the excited state is usually more stabilized than is the ground state. Spectral broadening results from multiple isomers. Broadened electronic spectra have also been explained as a signature of cluster melting, although this interpretation has been questioned [3, 4, 84, 85]. Some infrared spectra of solute molecules have also yielded information on the nature of the solute binding site, revealing that the solvating noble gas cluster undergoes an icosahedral to fcc phase transition in a critical size regime [77]. Not all dopant species are found well-solvated, or wetted in the cluster interior. Both electronic and infrared dopant spectra indicate the existence of chromophores bound to the cluster surface. In some cases, the chromophore is ubiquitously nonwetting, such as for SF6 –Xen and SiF4 –Arn , and resides exclusively on the cluster surface [77]. In other systems, e.g., carbazole–Arn and CF3 Cl–Arn , both wetting and nonwetting are observed [7, 77, 84]. Systems such as these have enabled measurements of solute diffusion rates into or out of the cluster. Finally, wetting–nonwetting transitions have been observed which depend on cluster size and/or cluster temperature [7, 84]. Understanding this wide variety of behavior requires modeling cluster solute-solvent structures (in both ground and excited states in order to interpret elec41.6.3 Doped Noble Gas Clusters tronic spectra) as a function of size and temperature. To date, most of the theoretical simulations have focussed on smaller Various chromophores have been added to noble gas clus- clusters at a given temperature [4, 7, 77, 84, 85]. Nonetheless, ters as a microscopic variation of matrix isolation [3, 4, 7, theoretical models have provided qualitative and quantitative 10, 11, 77, 84]. A wide range of chromophores have been explanations for some of the observed spectral shifts, and studied including other noble gas atoms, metal atoms, small have shown that the propensity for wetting/nonwetting can polyatomic molecules such as SF6 , CH3 F, and HCl, and a va- be related to the degree to which maximum possible coordiriety of organic molecules such as benzene, carbazole, and nation within the solvent is attainable. naphthalene. Comparisons of the electronic and vibrational spectra of these guest molecules with their spectra in noble gas liquids or solid matrices have revealed information such 41.6.4 Helium Clusters as noble gas cluster structure, solvation effects, and solute Quantum effects play a dominant role in helium clusters diffusion. A crucial issue in understanding the spectral signatures since they consist of very weakly interacting small mass of doped noble gas clusters is the location of the solute in particles. Statistical effects are also expected since 3 He is or on the noble gas cluster. Chromophores well-embedded a fermion and 4 He is a boson. Overall, helium clusters are bein a noble gas cluster behave as molecules surrounded by lieved to be in fluid-like or superfluid-like states with highly a dielectric medium, which is most correctly viewed as both fluctional structures [86]. Neutral helium clusters have been made experimentally in imperfect and finite sized [4, 7, 84, 85]. The evolution of the spectral changes with cluster size reflect the interplay of re- sizes ranging from the dimer up to 106 atoms [4, 11, 77]. The
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diatomic He2 , has just been detected recently and is found to have a binding energy of only 107 eV with an average internuclear distance of 55 Å [87, 88]. Evidence for magic numbers in the helium cluster abundances is seen; however, the sizes of these especially stable clusters differ substantially from what is seen in the other noble gas clusters [4]. Helium cluster structures have been theoretically investigated using fully quantum mechanical treatments [4, 86]. These studies find that 4 Hen clusters should be bound at all sizes, but that a minimum number of atoms, 30, are required for 3 Hen clusters to be stable. The latter prediction has not yet been verified experimentally, owing to the expense of 3 He, and the difficulties of ascertaining the true size of helium clusters since they easily boil off atoms upon ionization. Calculations reveal that the packing of helium clusters is highly delocalized with no evidence of icosahedral morphology. The structures derived have been found to be extremely sensitive to temperature and total angular momentum. Cluster binding energies and densities increase smoothly and monotonically with increasing size, approaching bulk behavior at about 300 atoms. Thus, helium clusters do not exhibit any especially stable sizes (magic numbers) such as the heavier noble gases do. In short, helium clusters behave as liquid-like quantum fluids, and may even be superfluid-like at the temperatures required to stabilize them in beam experiments. Based on these calculations, it appears that the magic numbers observed experimentally in abundance spectra pertain to the more strongly bound ionized helium clusters, rather than the neutral clusters which are the subject of this discussion. Electronic absorption spectra have recently been recorded for clusters containing 50106 helium atoms [89]. Broad strong absorption bands are observed which do not behave like the Wannier exciton bands seen for the heavier noble gas clusters, nor are they well-described by the Frenkel excitonic model. Note that although the Wannier and Frenkel exciton models were originally developed for solids with translational symmetry, they are also good descriptions for excitations in liquid noble gases [4]. At this writing, the helium cluster electronic spectra cannot be compared with absorption spectra of liquid or solid helium because these latter spectra have not yet been measured! With this limitation and a current lack of sufficient theoretical guidance, the helium cluster spectra have not yet been thoroughly interpreted. Theoretical calculations are available which describe the collective excitations in helium clusters on the electronic ground state surface [86]. Spectra in this energy regime have not yet been recorded. Experimentally, it has been quite easy to dope helium clusters with various atomic and molecular species such as other noble gas atoms, oxygen, and SF6 [4, 11, 77]. Electrons, however, are not well-solvated and negatively charged helium clusters only appear for sizes containing more than 105 106 atoms [11]. Impurity species provide a spectro-
M. L. Mandich
scopic probe for studying the properties of the solvating helium cluster. Infrared spectra of helium clusters doped with SF6 show that the impurity molecule resides on the cluster surface, in contradiction of theoretical calculations which predict that it should be found inside a helium cluster. Understanding this discrepancy has stimulated further theoretical investigations that reveal the dramatic structural effects which can occur when angular momentum is added to these very fragile clusters during dopant pick-up [90].
41.7 Molecular Clusters Molecular clusters are in many ways similar to noble gas clusters. Both are weakly bound, and the interactions between the particles which make up the cluster can usually be described to a good approximation as the sum of pairwise interactions. The molecule constituency adds complexity, however, which is reflected in the diversity seen amongst these clusters. Molecular clusters provide model systems for studying solvation, including electron solvation, nucleation, and phase transitions at finite sizes. Furthermore, properties which are often difficult to study in the bulk, such as phase transition dynamics, are much more amenable to study in molecular clusters. Numerous molecular clusters have been made and studied. By far the most work has focussed on the smaller molecular clusters such as molecular dimers, trimers and tetramers. A number of outstanding reviews are available on this subset of molecular clusters [3, 4, 6, 76, 91, 92].
41.7.1 Geometric Structures and Phase Dynamics In contrast to noble gas clusters, molecular clusters typically assume bulk phase structures at relatively small sizes. Thus, the range of condensed phases of molecular solids is reflected in the variety of packing structures found in molecular clusters. At one extreme are clusters of small nearly-spherical molecules such CH4 and N2 which form clusters closely connected to noble gas clusters [76]. Small clusters of these molecules are packed in polyicosahedral arrangements at small sizes, and cross over to fcc structures at sizes containing several thousand atoms. At another extreme are clusters such as water clusters where stronger hydrogen bonding forces lead to the formation of crystalline networks such as in a diamond cubic phase [76]. Some molecular clusters appear to be liquid-like with thermal motion wiping out any persistent periodic order. The liquid-like structure observed in clusters of molecules such as benzene and various hydrocarbons may freeze into a crystalline or amorphous state at sufficiently low temperatures.
41
Clusters
Some molecular clusters have been found to exhibit a panoply of phases, depending on size and temperature. For example, not only do clusters of TeF6 assemble into the bulk body-centered cubic and orthorhombic phases, they can also be made in other phases including two trigonal forms, a rhombohedral phase, and two monoclinic phases [93, 94]. Similarly complex phase formation has been found in other clusters of other hexafluoride molecules such as SF6 , SeF6 , MoF6 , and WF6 . Phase transitions in molecular clusters have been examined both experimentally and theoretically using molecular dynamics. Transition temperatures are found to depend on cluster size, and span a much broader range than in the corresponding bulk phase transitions. Large rates for nucleation and phase changes are seen in molecular clusters. Once the critical nucleus, which initiates the phase change, forms in the interior of the cluster, the remainder of the cluster transforms to the new phase extremely rapidly. Critical nucleus sizes depend on the cluster size and are typically quite small, e.g., for TeF6 clusters, they contain only a few dozen molecules for clusters consisting of a few hundred molecules [10, 93, 94]. A key finding is that phase transitions in molecular clusters appear to violate Ostwald’s step rule, which states that equilibrium systems must pass through all intermediate-energy stable phases during a transition from a higher energy to a lower energy phase [10]. This noncompliance apparently occurs because of the speed at which molecular clusters undergo phase transitions: intermediate phases simply do not have time to form.
41.7.2
Electronic Properties: Charge Solvation
IP ’s, Ae ’s, and some spectral data are available for larger molecular clusters. Nearly all of the spectra measured pertain to clusters containing less than ten molecules [3, 4, 6, 76, 91, 92]. Precise molecular orientations within many of these clusters have been derived from detailed analyses of these spectra. Such information has not yet emerged from the few spectra that have been measured for larger molecular clusters [3, 6]. Alternatively, measurements of electronic properties of larger clusters have generally targeted a different issue: solvation of excess positive or negative charge in a restricted bulk-like system. Charged molecular clusters provide model systems for studying the mechanism of charge stabilization within the confines of a finite system. Excess positive charge appears to be highly localized in molecular clusters, residing on a small unit containing at most a few monomers. This positive core is surrounded and stabilized by overlying shells of molecules. For example, .CO2 /C n clusters behave as the dimer cation , surrounded by the remaining CO2 molecules [4]. .CO2 /C 2 Positively charged solutes such as alkali cations have also
621
been introduced to study solvation of positive charge within molecular clusters [10]. For those dopants which have much lower ionization potentials than the solvating molecules, the positive charge remains strongly localized on the impurity. The cluster molecules are observed to build up solvation shells around this central impurity. The first few solvation shells are strongly affected by the positive charge, which has a decreasing influence with each successive layer. The structure of such a cluster reflects the accommodation between the geometry of the molecules influenced by the central positive charge, and those in the outer solvation layers where intermolecular forces predominate [10, 95]. Evidence for intracluster reactions in positively charged clusters has also been seen, such as proton transfer in cationic water or ammonia clusters [3, 11]. Whereas virtually every molecular cluster containing as little as two molecules exists stably as a positive ion, the same cannot be said for the negative cluster ions. Measurements of the minimum number of molecules required within the cluster to support an additional electron show that, while the dimeric .HCl/ 2 , .SO2 /2 , and .H2 O/2 clusters are stable, 35 or 41 molecule-clusters of ND3 or NH3 , respectively, are required to stabilize an electron [4]. Spectroscopic studies show evidence for internal and external solvation of the excess electron in these anionic clusters. Charge solvation in water clusters has been examined in some detail. Of particular interest is the .H2 O/20 cluster which, from experimental and theoretical data, appears to form an especially stable, well-defined clathrate cage. The interior of this cage is large enough that it can and C does hold various cations such as NHC 4 , H3 O , and alkali ions [10]. This cage is not observed to surround negative ions or electrons, which instead, reside on the surface [10]. This is a general result for excess negative charge in water clusters in this size regime where excess charge can be external to the cluster. Excess electrons in water clusters are found on the cluster surface for small cluster sizes up to 60 to 70 water molecules. Above this size, the electron resides in the cluster interior, and behaves analogously to hydrated electrons in bulk water [4, 7].
41.8
Recent Developments
Added by Mark M. Cassar. Experimental and theoretical work on clusters has continued to be an active and rapidly growing area of research over the past decade. This section provides a non-exhaustive snapshot of some recent work in the vast field of cluster science. The original focus on the scalable properties of clusters (concerning a smooth transition from small particles to bulk matter) has now extended to include important non-scalable properties. These properties, particularly at the nanoscale
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level, have enormous potential for technological application [96–98]. Studies aimed at understanding the underlying atomic structure of noble-metal clusters and nanoparticles, which is the first step toward their controlled use in future nanotechnologies, e.g., catalysis, labeling, or photonics, have been carried out [99]. Ab initio all-electron molecular-orbital calculations for small (n D 711) and medium (n D 1220) silicon clusters Sin have been performed in order to study their structure and relative stability [100, 101]; such calculations are important in determining the scalability of present day semiconductor technology. Other interesting work has been done on the electron transfer properties of metal clusters that could act as conducting bridges between molecular wires [102]; and in the role that tetramanganese clusters play in one of the active photosynthesis sites (photosystem II) in green plants, and in certain bacteria and algae [103]. The reader is referred to various reviews that can be found in the literature: for time-resolved photoelectron spectroscopy (TRPES) of clusters, which allows the dynamics along the entire reaction coordinate to be followed, see [104]; for ultrafast dynamics in cluster systems and atomic clusters, see [105, 106]; for small carbon clusters, important in the chemistry of carbon stars, comets, interstellar molecular clouds, and hydrocarbon flames, see [107]; for the relation between electronic structure, atomic structure and magnetism of clusters of transition elements, see [97].
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46. Kim, S.G., Tomanek, D.: Phys. Rev. Lett. 72, 2418 (1994) 47. Bethune, D.S., Johnson, R.D., Salem, R.J., de Vries, M.S., Yannoni, C.S.: Nature 366, 123 (1993) 48. Wang, X.-D., Hazhizume, T., Xue, Q., Shinohara, H., Saito, Y., Nishina, Y., Sakurai, T.: Jpn. J. Appl. Phys. 32, L147 (1993) 49. Hino, S., Takahashi, H., Iwasaki, K., Matsumoto, K., Miyazaki, T., Hasegawa, S., Kikuchi, K., Achiba, Y.: Phys. Rev. Lett. 71, 4261 (1993) 50. Saunders, M., Jimenez-Vazquez, H.A., Cross, R.J., Mroczkowski, S., Gross, M.L., Giblin, D.E., Poreda, R.J.: J. Am. Chem. Soc. 116, 2193 (1994) 51. Ehrenreich, H., Spaepen, F. (eds.): Fullerenes in Solid State Physics vol. 48. Academic Press, New York (1994) 52. Cioslowski, J.: Ab initio electronic structure calculations on endohedral complexes of the C60 cluster. In: Davies, J.E.D. (ed.) Spectroscopic and Computational Studies of Supramolecular Systems, vol. 40, Kluwer, Netherlands (1992) 53. Zimmermann, U., Malinowski, N., Naher, U., Frank, S., Martin, T.P.: Phys. Rev. Lett. 72, 3542 (1994) 54. Zhou, O., Fleming, R.M., Murphy, D.W., Chen, C.H., Haddon, R.C., Ramirez, A.P., Glarum, S.H.: Science 263, 1744 (1994) 55. Whetten, R.L.: Acc. Chem. Res. 26, 49 (1993) 56. Barnett, R.N., Landman, U., Scharf, D., Jortner, J.: Acc. Chem. Res. 22, 350 (1989) 57. Li, X., Whetten, R.L.: J. Chem. Phys. 98, 6170 (1993) 58. Rajagopal, G., Barnett, R.N., Landman, U.: Phys. Rev. Lett. 67, 727 (1991) 59. Steigerwald, M.L., Brus, L.E.: Acc. Chem. Res. 23, 183 (1990) 60. Wang, Y.: Acc. Chem. Res. 24, 133 (1991) 61. Brus, L.: Adv. Mater. 5, 286 (1993) 62. Raghavachari, K., Curtiss, L.A.: Accurate theoretical studies of small elemental clusters. In: Langhoff, R.S. (ed.) Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy. Kluwer, Netherlands (1994) 63. Raghavachari, K.: Phase Transit. 24–26, 61 (1990) 64. Bingelli, N., Martins, J.L., Chelikowsky, J.R.: Phys. Rev. Lett. 68, 2956 (1992) 65. Kaxiras, E., Jackson, K.: Phys. Rev. Lett. 71, 727 (1993) 66. Röthlisberger, U., Andreoni, W., Parrinello, M.: Phys. Rev. Lett. 72, 665 (1994) 67. Fuke, K., Tsukamoto, K., Misaizu, F., Sanekata, M.: J. Chem. Phys. 99, 7807 (1993) 68. Jarrold, M.F.: Science 252, 1085 (1991) 69. Rinnen, K.-D., Mandich, M.L.: Phys. Rev. Lett. 69, 1823 (1992) 70. Al-Laham, M.A., Raghavachari, K.: J. Chem. Phys. 98, 8770 (1993) 71. Lou, L., Nordlander, P., Smalley, R.E.: J. Chem. Phys. 97, 1858 (1992) 72. Graves, R.M., Scuseria, G.E.: J. Chem. Phys. 95, 6602 (1991) 73. Jin, C., Taylor, K.J., Conceicao, J., Smalley, R.E.: Chem. Phys. Lett. 175, 17 (1990) 74. Rinnen, K.-D., Kolenbrander, K.D., DeSantolo, A.M., Mandich, M.L.: J. Chem. Phys. 96, 4088 (1992) 75. Wang, L., Chibante, L.P.F., Tittel, F.K., Curl, R.F., Smalley, R.E.: Chem. Phys. Lett. 172, 335 (1990) 76. Special issue on gas phase clusters, Chem. Rev. 86 (1986) 77. Goyal, S., Schutt, D.L., Scoles, G.: Acc. Chem. Res. 26, 123 (1993) 78. Miehle, W., Kandler, O., Leisner, T.: J. Chem. Phys. 91, 5940 (1989)
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Mary L. Mandich Mary Mandich is retired from NOKIA Corp. where she was a Technical Manager at Bell Laboratories and Distinguished Member ofTechnical Staff in the High Speed Optical Products Division. She obtained her Ph.D. degree in Physical Chemistry at Columbia University. She holds 9 US Patents and has authored 2 book chapters and 55+ scientific publications in chemistry, physics, and materials science.
41
Infrared Spectroscopy
42
Henry Buijs
Contents
Abstract
42.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 626
42.2
Historical Evolution of Infrared Spectroscopy Practice . . . . . . . . . . . . . . . . . . . 626
42.3
Quantitative Analysis by Infrared Spectroscopy . . 627
42.4
Molecular Spectroscopy . . . . . . . . . . . . . . . . . . 627
42.5
Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . 628
42.6
The Evolution of Fourier Transform Infrared Spectroscopy (FTIR) . . . . . . . . . . . . . . 628
42.7
Laser-Based Infrared Spectroscopy . . . . . . . . . . 629
42.8
Intensities of Infrared Radiation . . . . . . . . . . . . 629
42.9
Sources for IR Spectroscopy . . . . . . . . . . . . . . . 630
42.10
Relationship Between Source Spectrometer Sample and Detector . . . . . . . . . . . 630
42.11 42.11.1
42.11.3 42.11.4
Simplified Principle of FTIR Spectroscopy Interferogram Generation: The Michelson Interferometer . . . . . . . . . Description of Wavefront Interference with Time Delay . . . . . . . . . . . . . . . . . Operation of Spectrum Determination . . . . Optical Aspects of FTIR Technology . . . . .
42.12
The Scanning Michelson Interferometer . . . . . . . 633
42.13 42.13.1 42.13.2 42.13.3 42.13.4 42.13.5 42.13.6 42.13.7
Infrared Spectroscopy Application Activity 2020 Analytical Chemistry Laboratories . . . . . . . . . . Biomedical and Pharmaceutical Laboratories . . . . Forensic Investigation . . . . . . . . . . . . . . . . . . Infrared Spectroscopy in Quality Assurance Laboratories . . . . . . . . . . . . . . . . . Process Monitoring by Infrared Spectroscopy . . . . Environmental Monitoring . . . . . . . . . . . . . . . Remote Sensing . . . . . . . . . . . . . . . . . . . . . .
42.14
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 H. Buijs () FTS Consulting Québec, Québec, Canada e-mail: [email protected]
Coblentz determined the relation between infrared spectra and molecular identity. Infrared spectroscopy became a tool for chemical analysis widely used in analytical chemistry laboratories. Prism or grating “dispersive” spectrometers fulfill this function well. Molecular spectroscopy was first practiced in the visible and UV range using large high-resolution spectrographs to record the fine spectral lines of vapor phase molecules. Extension to the infrared region was impeded by the lack of sensitivity of single-element infrared detectors and the challenge of making ever larger spectrometers to achieve high resolution. Dispersive spectrometers were superseded by “interferometric” spectrometers that possessed a much larger optical throughput and, when combined with the Fourier technique, permitted multiplexing the entire spectrum efficiently on a single detector enabling extending highresolution molecular spectroscopy to the infrared region. As Fourier transform spectroscopy technology matured, measurement accuracy and reproducibility were also enabled. Quantitative infrared spectroscopy has opened a wide range of new applications in industry, environmental monitoring, and monitoring of atmospheric variations in relation to weather forecasting. Space probes use Fourier transform spectrometers to determine the chemical nature of planetary atmospheres and surfaces and even the measurement of the cosmic background radiation remnant. Low-cost Fourier transform spectrometers have also replaced dispersive spectrometers in analytical chemistry laboratories. Given its importance in a wide range of applications, a detailed description of the technique is provided. Keywords
infrared spectrometer FTIR chemical identification molecular structure determination quantitative analysis reproducibility remote sensing transmission absorption emission
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_42
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Introduction
Infrared spectroscopy consists of the measurement of interactions of waves of the infrared (IR) part of the electromagnetic spectrum with matter. The IR spectrum starts just beyond the red part of the visible spectrum at a wavelength of approximately D 700 nm extending to 2500 nm and is usually called near-IR, from D 2500 to 25,000 nm or D 2:5 to 25 µm called mid-IR, and from D 25 to 1000 µm called far-IR. For the mid-IR ( D 4000 to 400 cm1 ) and far-IR regions ( D 400 to 10 cm1 ), it is common practice to use frequency expressed in number of wavelengths per cm (wavenumbers, cm1 ) instead of wavelengths. The interactions observed in the mid-IR region principally involve energies associated with molecular structure change. Harmonic overtones, as well as combinations of mid-IR frequencies, are expressed in the near-IR region and can also be associated with the same molecular structure change as can be observed in the mid-IR but appear less obvious. Interactions observed in the far-IR involve molecular structure change having low energy. These are predominantly changes in molecular rotation energy. The most common IR analysis of a sample is by IR absorption spectroscopy. Samples may consist of solid or liquid samples, called condensed phase samples and gas phase samples. IR absorption spectroscopy involves transmitting a beam of intense IR radiation through the sample and observing the distribution of wavenumbers absorbed by molecules when compared to a spectrum measured without the sample. Condensed phase samples generally absorb strongly in the mid-IR region, requiring a path length through the sample of only a fraction of a mm. In the near-IR, these samples absorb at least an order of magnitude less, permitting path lengths through a sample of up to a cm. Gas phase samples absorb much less due to their lower density compared with condensed phase samples and may require a path length of several meters in the midIR. Molecules in a sample may also be studied by IR emission spectroscopy simply by observing specific wavenumbers being emitted by virtue of their nonzero absolute temperature. Finally, radiation reflected from a smooth surface of a solid sample also provides information about the molecular structure of the material by virtue of the anomalous dispersion associated with absorption bands. Historically, infrared spectroscopy started out as a qualitative method for identifying different molecular species present in a sample, as is also done by various titration methods used by chemists. Today, infrared spectroscopy has been extended to include accurate quantitative determination of molecular concentrations. This has enabled applications such as quality assurance and process control in chemical manufacturing and applications where multiple different spectrometers are required to provide quantitatively reproducible measurements. The evolution from qualitative to
increasingly accurate quantitative infrared analysis will be described. Molecular spectroscopy research covers spectral analysis at high spectral resolution covering the UV to the far-infrared. Infrared spectroscopy plays a key role in remote sensing. Satellite-based thermal infrared emission and near-infrared reflected solar spectroscopy is an essential part of weather forecasting, pollution monitoring, and climate change research related to Earth’s atmosphere. Infrared remote sensing is also used extensively in conjunction with interplanetary space probes, as well as astronomy.
42.2 Historical Evolution of Infrared Spectroscopy Practice After the discovery of infrared radiation by Herschel in 1800 [1, 2], it was not until the late nineteenth century that William Coblentz [3] obtained infrared spectra of many substances using both glass and salt prisms, reaching as far as D 15 µm in the mid-infrared. By studying the correlations amongst spectra of different substances Coblentz derived a number of important properties of infrared spectra, such as conservation of the unique spectral band positions and shapes of a given substance in mixtures of a sample. The unique correlation between the infrared spectral signatures and the molecular species being measured even in complex mixtures permitted Coblentz to establish infrared spectroscopy as a tool for identifying molecular structure for a variety of chemical species. These observations led to the concept of functional groups in infrared spectroscopy that permits us to elucidate a wide range of molecular structures from infrared and Raman spectra [4]. A great diversity of unique spectral signatures is found in the region from 400 to 2000 cm1 and is commonly known as the “fingerprint” region. It was not until the 1940s that infrared spectroscopy became a widely used tool for chemical analysis. This was driven by the need for chemical analysis in support of the development and manufacture of synthetic chemical products such as rubber and aviation fuels. Several thousand spectrometers were produced for this purpose. More widespread use of infrared spectrometers was enabled by the introduction of the Perkin Elmer model 137 “Infracord” [5] in 1956, a double-beam spectrometer that traced the sample transmittance directly on paper in real time. This quickly became a popular analytical tool for the chemistry laboratory displacing tedious titration methods for general chemical identification. Prisms made from various infrared transparent materials and diffraction gratings angularly disperse incident radiation as a function of the wavelength. Intensities at these wavelengths can be measured sequentially by scanning the dispersed radiation past a narrow slit. Gratings mostly re-
42 Infrared Spectroscopy
placed prisms for infrared spectroscopy starting in the 1950s, following improvements in the uniformity of groove spacing, as well as the technique of replicating gratings from masters, thereby greatly reducing cost. Today, the use of spectrometers consisting of a grating and a linear detector array covering the full dispersed spectrum simultaneously is common for the UV and visible regions, as well as the near-infrared down to about 1700 nm (5880 cm1 ). These are simple, easy to use spectrometers for which economical and performant detector arrays are available. Extension to the mid and far-IR of this combination has been impeded by the cost and complexity of linear detector arrays for these regions, which require operation at low temperature to achieve optimal sensitivity. With two-dimensional imaging detector arrays, a line with many fields of view along the length of a slit can be measured simultaneously, where the spectra are dispersed along the columns of the 2-D array perpendicular to the rows of fields of view. Scanning the slit across an extended field of view and repeatedly recording twodimensional data arrays builds up a hyperspectral image. Gratings have several significant advantages over prisms. Groove spacing provides flexibility in resolution and spectral region selection, gratings with blazed grooves improve optical efficiency for selected spectral regions, and concave gratings minimize the need for additional focusing optics. A limitation of gratings is that multiple different wavelengths are diffracted at the same angle, presenting multiple orders to a detector. Therefore, gratings require order selection filtering to avoid superposition of orders of different spectral regions. Starting in the 1950s infrared spectroscopy was being developed for use beyond the chemistry laboratory in three distinct areas. One area was the development of quantitative methods of chemical analysis, another was the development of high-resolution IR molecular spectroscopy, and, finally, a variety of remote-sensing applications.
42.3
Quantitative Analysis by Infrared Spectroscopy
Least squares comparison of a measured infrared spectrum with a reconstruction of spectral signatures for each of the components in a sample is impractical for condensed phase samples. Spectral features of a condensed phase sample compared with the spectral features of the same sample in the vapor phase shows a strong broadening due to intermolecular interactions of close neighbor molecules. Vapor phase spectral features typically have a width < 0:1 cm1 at atmospheric pressure, whereas in the condensed phase, these features often have widths of > 4 cm1 . In a mixture of different molecules, these intermolecular interactions vary somewhat from the homogeneous case, thus slightly distort-
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ing band shapes in a mixture with respect to pure component band shapes. Infrared spectroscopic determination of accurate molecular concentration for condensed phase samples, therefore, requires the acquisition of infrared spectra of prepared standard mixtures representative of the range of sample variations to be analyzed. These standards are used in a principal component analysis. Since the set of standard mixtures used for the principal component modeling may not permit exact reconstruction of the measured spectrum, a chemometric technique of “partial least squares” is employed [6, 7]. It permits controlling overrepresentation of a measured spectrum by the reconstructed spectrum. This technique was first employed in the determination of concentrations of protein and moisture in wheat via near-IR diffuse reflection to establish its commercial value [8]. It was soon realized that measurements of spectra employing different spectrometers presented significant errors. As a result, the prepared standards measured with different spectrometers expanded the number of standard spectra used as inputs for the principal component analysis. Improvement of grating spectrometers with respect to reproducibility such as to minimize the need for standard samples to be measured with multiple different spectrometers met with limited success. Small variations in slit geometry as well as scanning mechanisms prevented the achievement of adequate reproducible measurements. Improved reproducibility for spectrometers used at different sites was one of the driving forces for the development of Fourier transform spectroscopy.
42.4 Molecular Spectroscopy Molecular spectroscopy dates back to the 1920s, following the elegant description of molecular dynamics via quantum mechanics. It is carried out mainly using samples in the vapor phase where intermolecular interaction is minimized such that spectra consist of narrow spectral features. Resolving many different spectral features and determining precise wavelengths of these features is key to determining the detailed energy states of a molecule and, thereby, its structure [9]. An early study was carried out using large high-resolution monochromators and photographic recording of spectra. Extension to the infrared was enabled by the availability of sensitive infrared detectors operating at low temperatures starting in the 1950s. To achieve high resolution, large gratings are required to minimize dispersion by diffraction and projection of a greatly expanded spectral dispersion at a narrow slit requiring long focal length optics. Scanning all the wavelengths of a highly dispersed spectrum is time consuming using a single infrared detector and is much less efficient than photographic recording of the full spectrum. This was another driving force behind the development of Fourier transform spectroscopy [10].
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Remote Sensing
Measuring infrared spectra of remotely located targets is highly variable and is dependent on the temperature and emissivity of the targets. Spectrometers used for remote sensing require the best achievable sensitivity. Over time, Fourier transform spectrometers have become the spectrometers of choice for remote sensing not only for their favorable sensitivity but also for the flexibility of operating parameters, such as spectral resolution, spectral range, and measurement speed. Atmospheric remote sensing from satellites has become an essential tool for weather prediction by providing vertical distribution profiles of temperature and moisture, which are the most important thermodynamic drivers of weather. The technique is also used to determine the detailed composition of the atmosphere and its variations over time and geographic location.
42.6
The Evolution of Fourier Transform Infrared Spectroscopy (FTIR)
Infrared detectors represent an important aspect of infrared spectroscopy. Pyroelectric bolometric mid-infrared detectors operating at ambient temperature are widely used for chemistry laboratory infrared analysis. Using a pyroelectric detector with a scanning grating spectrometer, spectral measurements are time consuming, and the spectra are of modest quality. A Fourier transform spectrometer records an encoded signal for all wavelengths simultaneously on a single detector. The Fourier transform type spectrometer using a pyroelectric detector permits significant enhancement of measurement speed and quality. A semiconductor made from a small bandgap material can, in principle, provide broadband detection like a bolometer; however, thermal noise currents increase rapidly at smaller bandgaps, requiring operation at ever lower temperature. A proliferation of different bandgap infrared detectors are available, permitting optimization of sensitivity and degrees of cooling for targeted spectral regions. The demand for measurement speed, spectral resolution, reproducibility, and sensitivity are driven by a proliferation of applications of infrared spectroscopy. An alternative to the dispersive spectrometer was sought. In the 1960s, the Fourier transform spectrometer was seen as a strong candidate but required much technological development. Fourier transform spectroscopy was pioneered early in the twentieth century by Albert A. Michelson [11] as an alternative use of the two-beam interferometer he had invented for the purpose of determining the inhomogeneity of the velocity of light. However, due to a lack of tools to measure and process data he relied on visual observation of “visibility curves” that permitted him to characterize some aspect
of spectra. Lord Rayleigh [12] pointed out to Michelson that the relation between the observed visibility curves of the interferometer and their spectra is the Fourier transformation. Observing all wavelengths simultaneously with a Michelson interferometer, like recording the complete spectrum with photographic film, was not realized until 1949 by the work of P. Fellgett [13, 14]. Fellgett computed the Fourier transform of a limited set of measurements laboriously by hand. As digital computers started their exponential growth in capacity and computation speed, year over year Fourier transform spectroscopy became increasingly accessible. In the 1960s, Fourier transform spectroscopy became usable but required access to a large computing center. Early implementations of the scanning Michelson interferometer were sensitive to perturbations and required a skilled operator to maintain the alignment of the scanning mirror. The high cost and complexity of operating an early version of an FTIR limited its use to research centers. Nevertheless, the great increase in sensitivity and measurement speed provided access to measurement problems not available with dispersive spectrometers. As an example, Buijs and Gush [15] described the high-resolution measurement of the electric field induced absorption spectrum of hydrogen with a homemade Fourier transform spectrometer. In 1965, the interferometer scanning mirror positions were determined using a Hg198 lamp as monochromatic metrology source, since lasers were not yet available. Furthermore, the computation of the Fourier transform of the interferograms containing about 4000 numerical data points typically took up to 20 min on a large mainframe IBM 7094 computer. In the late 1960s, Cooly and Tukey [16, 17] presented a method of factoring the Fourier transform calculation making use of symmetries in the trigonometric matrix and, thereby, greatly reducing the number of calculations required; this is commonly called the fast Fourier transform or FFT. The same calculation as above on the same computer was realized in about 10 s. As well, in the late 1960s, lasers became available. The He-Ne laser permitted scan control of the moving mirror with much higher precision than the Hg198 lamp and over much longer mirror displacements. It is seen that over a period of 20 years Fourier transform spectroscopy progress depended greatly on new technological developments. By 1984 the introduction of the personal computer (PC) and advances in interferometer design permitted the realization of a low-cost versatile and easy to use Fourier transform spectrometer. The efficiency of measuring all wavelengths simultaneously called the multiplex advantage as described by Fellgett [13, 14] and the greater light throughput by elimination of the narrow slit as described by Jacquinot [18] allowed infrared spectra to be measured much more rapidly and with much greater signal-to-noise ratio. This development led to a rapid replacement of dispersive spectrometers in chemistry laboratories by FTIR, particularly for the mid-IR and the far-
42 Infrared Spectroscopy
IR. In the near-IR, the intensity of illumination sources, such the tungsten lamp, and the availability of high sensitivity of InGaAs, PbS, and other room-temperature semiconductor detectors allowed both near-IR grating spectrometers and FT-NIR to flourish. The availability of linear array detectors in the visible and UV that have intrinsic noise lower than photon statistics noise permit dispersive spectrometers to perform more favorably than FT spectrometers. Both techniques measure all spectral elements simultaneously; however, the array detector of a dispersive spectrometer localizes excess photon noise due to intense spectral features at certain elements of the detector array and avoids adding this noise to the lower photon noise of weaker spectral features at other elements of the detector array. The FT spectrometer, however, accumulates the photon noise of all spectral features on a single detector. This is particularly disadvantageous for FT spectrometers when measuring wide spectral intervals. UV and visible dispersive spectrometers with an array detector that has intrinsic noise lower than the source photon statistics noise can, therefore, measure weak spectral features with lower noise than strong features, effectively increasing the dynamic range of intensity measurement, which the FT spectrometer does not permit. With the maturing of FTIR technology, a great diversity of infrared spectroscopy applications has been enabled. With FTIR, spectral resolution and spectral range can be varied freely. Current scanning Michelson interferometers are constructed to maintain the precise alignment required either using “cube corner” retroreflectors instead of flat mirrors or servo-controlled dynamic alignment applied to flat mirrors. This advanced control of interferometer alignment has opened the possibility to operate FTIRs in hostile environments of vibration and temperature with minimal effect on performance. It has also permitted scanning the mirror distances as large as several meters to offer spectral resolution that far exceeds that of dispersive spectrometers. In the vapor phase at pressures well below atmospheric, spectral lines of specific energy transitions may be very narrow. Here the line width is limited by doppler shifts due to thermal motion of the molecules and is dependent on temperature and is linearly proportional to the molecular weight and frequency. As pressure is increased line width increases due to random collisions of molecules that are density dependent. At room temperature and low density in the mid-IR, spectral linewidths may be < 0:002 cm1 . FT spectrometers where the scanning mirror can be translated through several meters are also capable of resolution down to < 0:002 cm1 , which is far higher than any dispersive spectrometer, while still maintaining sufficient optical efficiency due to the Jacquinot advantage. These spectrometers are capable of covering the full infrared range and have enabled efficient extension of molecular spectroscopy to the infrared beyond photographic recording.
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42.7 Laser-Based Infrared Spectroscopy In the quest for the study of ever finer detail in infrared spectra, the use of tunable lasers has permitted still higher resolution and with significantly higher precision. Using cavity-enhanced absorption spectroscopy such as cavity ringdown or integrated cavity-enhanced spectroscopy, faint spectral lines can be observed with great clarity. Today, much research on molecular structure by infrared spectroscopy is performed using tunable laser spectroscopy. As well, cavity-enhanced absorption spectroscopy is used to measure ultralow concentrations of components in a gas mixture for the purpose of environmental monitoring and contamination assessments. Current tunable lasers have a limited spectral range capability requiring multiple lasers to cover a wide spectral range. Given its versatility for infrared spectroscopy applications it is useful to describe the common practice and elements of infrared Fourier transform (FTIR) spectroscopy.
42.8
Intensities of Infrared Radiation
For strong interactions of electromagnetic waves with matter, the emitted and absorbed intensities are governed by Planck’s radiation law in addition to the emissivity and absorptivity of the material. Planck’s radiation law for thermal radiation from an ideal black body is Pbb .T /d D
C1 3 d ; exp Khb T C 1
(42.1)
where is the frequency in cm1 , h is Planck’s constant, Kb is the Boltzmann constant, T is the temperature of the blackbody in K, and C1 is a proportionality constant. Depending on the definition of C1 , Pbb .T / may represent a radiation density per unit spectral interval cm1 in a cavity at temperature T in ergs=cm3 , or an energy flux emitted from a surface in W=.cm2 -steradians). At frequencies that are low compared with h=Kb T , the energy distribution increases with 2 and is approximately proportional to T at a given . At high frequency, the energy distribution falls off exponentially. In the near-IR, a high temperature is required to emit radiation. Room-temperature objects emit the most strongly in the 500 to 1800 cm1 region and emit negligible energy above 3000 cm1 . Materials cooled to liquid nitrogen temperature (77 K) only emit below 100 cm1 , while materials cooled to liquid He temperature (4.2 K) emit below 20 cm1 . In contrast to visible spectroscopy, IR absorption spectroscopy is complicated by emission of IR radiation from the sample and the surrounding environment.
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Sources for IR Spectroscopy
over a substantial thickness. Remote sensing of solid objects, therefore, relates to surface emission and (diffuse) reflection A silicon carbide element electrically heated to 1400 K pro- of IR radiation from the surrounding environment. vides a strong continuum of IR radiation over a major part of the IR spectrum. It is commonly used as a source of radiation for IR absorption spectroscopy. For near-IR 42.11 Simplified Principle of FTIR Spectroscopy spectroscopy, a tungsten filament lamp operated at 2800 K provides a strong continuum all the way up to the visible part of the spectrum; it is not useful below 3000 cm1 because of In FTIR spectroscopy, the spectrum of a beam of incident absorption by the glass or quartz envelope. Various electri- IR radiation is obtained by first generating and recording cally heated ceramic elements, such as the Nernst glower and an interferogram with a scanning Michelson interferometer. high-temperature carbon rods, have been devised to achieve Subsequently, the interferogram is inverted into the spectrum by means of a cosine Fourier transform. higher intensity.
42.10
Relationship Between Source Spectrometer Sample and Detector
Since a sample at room temperature emits IR radiation in the mid-IR, it is important to distinguish between the transmitted radiation used in the determination of its absorption spectrum and its emission spectrum. By employing an intense IR beam, the effect of emission is minimized. A further distinction is achieved by encoding the IR beam before it impinges on the sample. With classical grating or prism spectrometers, the source radiation is chopped by means of a mechanical chopper before it passes through the sample. The IR detector is provided with a means of synchronously decoding the chopped signal, thereby eliminating the emitted spectrum. Often, the chopper is arranged such that it alternately switches between an empty reference beam and the sample. The logarithm of the ratio of the demodulated sample and reference spectra provides the absorption spectrum directly. In Fourier transform infrared (FTIR) spectroscopy, a scanning Michelson interferometer provides the encoding function directly, and no chopper is required. The interferometer is commonly placed before the sample so that it does not encode the thermally emitted radiation of the sample. If it is not convenient to place the sample after the scanning Michelson interferometer, the absorption spectrum of a sample placed in front of the interferometer can be deduced by subtracting the separately recorded emission spectrum from the combined transmission plus emission spectrum. Infrared emission and reflection spectroscopy form the basis for remote sensing. Solid and gaseous (cloud) objects may be identified and quantified by direct observation of their IR spectra at a distance. Gaseous clouds reflect poorly, providing only transmitted or emitted IR radiation. Their emission spectrum is contrasted directly with the spectrum of the scene or object beyond the cloud. With a background at lower temperature than the gas cloud, the gas spectrum appears in emission, while with a warmer background, the gas spectrum appears in absorption. Only a few solid materials transmit IR radiation
42.11.1 Interferogram Generation: The Michelson Interferometer The scanning Michelson interferometer shown in Fig. 42.1 consists of a beamsplitter, which is a substrate with a dielectric coating such that 50% of an incident beam is reflected, and the remaining 50% is transmitted, and two plane mirrors (M1 and M2), one or both of which are translated along the direction of the beam. After splitting, the two equal amplitude wavefronts are propagated along different optical paths. The mirrors at the end of each path return the wavefronts to the beamsplitter, which then acts as a wavefront combiner. Because of their common coherent origin, the wavefronts interfere with one another when they combine. The state of interference is varied by scanning one or both of the mirrors such that there is a variable time delay between the two separated beams. The resulting intensity variation of the combined output beam as a function of relative time delay is the interferogram.
M2
Fixed mirror Compensator Beamsplitter M1 Moving mirror
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Fig. 42.1 The Michelson interferometer
42 Infrared Spectroscopy
42.11.2
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Description of Wavefront Interference with Time Delay
The constant term I0 provides no useful information about the spectrum. The inverse cosine Fourier transform of 2I0 .ı/ results in the spectrum S./ according to Z The intensity I0 .#/ at frequency # of a plane wave in space (42.10) S./ D 2 I0 .ı/ cos.2 ı/dı ; is given by the expectation value of its electric field vector i 2 # t Z E.#; t/ D E.#/e (42.11) S . / D 2 I0 .ı/ cos.2 ı/dı : I0 .#/ D hE.#; t/jE.#; t/i D E.#/2 : (42.2) Some early FTIR spectrometers were arranged with the The intensity at the output of the interferometer due to an insame dual beam for direct determination of the transmittance cident intensity I0 .#/ is given by I.#; ı/, where ı is the time of a sample as was done for dispersive spectrometers. It was delay between the two wavefronts, which have propagated quickly determined that the reproducibility of two spectra along different paths. Further, recorded sequentially using a single optical path was signifiI .#; ı/ D hE.#; t/ C E.#; t C ı/jE.#; t/ C E.#; t C ı/i cantly more precise than the reproducibility for two separate 1 (42.3) optical beams. The manipulation of the numerical represenD I0 .#/Œ1 C cos.2 #ı/ : 2 tations of the sequentially measured sample and reference As can be seen, the output intensity of a single frequency spectra became negligible using a digital computer. Contrary to classical spectrometers, where the spectrum source at the output of an ideal scanning Michelson interferometer fluctuates sinusoidally between zero and the input is sequentially scanned, there is no segregation of frequenintensity I0 ./ as the time delay ı between the separated cies of the input intensity. All frequencies in the source are wavefronts is varied by means of scanning one of the mirrors. modulated simultaneously by the scanning Michelson interThe quantity ı is related to mirror displacement x with ferometer into a single interferogram signal. It contributes respect to equal distance of the mirrors from the beamsplitter to a large advantage in sensitivity compared with dispersive spectrometers and is referred to as the Fellgett or multiplex by advantage. x (42.4) ı D 2 cos ; The reciprocal of the sampling interval x of optical c path difference determines the extent of the numerically where is the angle between of the wavefront and the optical computed spectrum. A higher density of sampling permits axis of the spectrometer, and the optical axis is the normal to a wider spectral range to be determined each mirror M1 and M2. From this, Eq. (42.3) becomes 1 n h x io 1 : (42.12) max D (42.5) I.; x.// D I0 ./ 1 C cos 2 2 cos 2x 2 c Beyond max , the spectrum repeats in reverse order, and be, or, using D c , yond 2max , the spectrum repeats as is. This is called spectral 1 aliasing and results from the incomplete knowledge of the I.; x.// D I0 . /f1 C cosŒ2 .2x cos /g : (42.6) full interferogram function between the discrete numeric rep2 Thus, the output intensity fluctuates at spatial frequency resentation. To ensure that the numeric representation of the interby 2x cos as a function of the mirror displacement x. The ferogram describes the continuous function uniquely, it is incident intensity generally consists of a distribution of intenimportant to band limit the interferogram information to the sities over many frequencies S./d with integrated intensity range 0 to max by means of optical and electrical filtering. Z Conversely, the higher the density of sampling in the spec(42.7) I0 D S./d : tral domain, the longer the interferogram needs to be. For The intensity fluctuations for all frequencies in the incident a wavenumber interval , the interferogram length is beam is given by 1 Z xmax D : (42.13) 2 I0 .ı/ D 1=2 S./Œ1 C cos.2 ı/d : (42.8) The second term on the right-hand side of Eq. (42.8) has the form of the cosine Fourier transform of the spectrum. By rearrangement of Eq. (42.8), it is given by Z 1 1 (42.9) dS./ cos 2 ıed D I0 .ı/ I0 : 2 2
42.11.3 Operation of Spectrum Determination The interferogram signal is detected by an IR detector that converts the intensity variations I0 .x/ as a function of different mirror positions x into an electrical signal. Continuous
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determination of the inverse cosine Fourier transform of the evolving interferogram requires continuous multiplication of the signal by cosine functions with all the different frequencies of the spectrum and integrating these products. It is, however, far more practical to capture the interferogram signal in numerical form, using an analog to digital converter, store the digitized data in computer memory, and compute the Fourier transform numerically after the mirror displacement range has been covered. A particularly convenient and accurate way to establish the sampling locations of the interferogram is the use of a single-frequency laser directed coaxially with the source radiation through the scanning Michelson interferometer. The intensity of the laser at the output of the interferometer is a pure cosine wave with one cycle per change in mirror movement of one half wavelength of the laser light, typically about 400 nm. It has long been the practice to digitize the interferogram signal at the precise mirror positions provided by the laser signal. However, as in digital audio recording, the currently highest fidelity of numerical representation of an analog signal is obtained with ˙ analog to digital (ADC) technology, which can represent a precisely linear signal dynamic range up to 224 to 1. However, these ADCs can only operate at a constant measurement rate. When applying a constant rate of digitizing to an interferogram signal the optical path different intervals measured depend on the velocity of scanning of the mirror of the interferometer. Since this is a mechanical displacement it is subject to velocity variations due to perturbations resulting from shocks and vibrations. The parallel monochromatic laser light passing through the interferometer (metrology signal) being a pure sinusoidal function of the optical path difference can provide path difference information at the same time that the infrared signal is digitized. By employing high-density “uniform time” sampling of an interferogram, intensities at a “uniform mirror displacement” can be accurately computed by interpolation based on the measured timing of the metrology signal information [19]. This approach has become a key feature in obtaining highly accurate and reproducible spectra, even in the presence of scan velocity perturbations. Errors in the spacing of measured interferogram data points along the path difference axis have the same effect as errors in the spacing of grooves of a diffraction grating; it leads to ghost features in the spectra. Residual sampling errors in modern FTIRs may be as low as subnanometers even in the presence of significant scan velocity variations. By the above-described approach, the numerical representation of the interferogram is determined at uniform intervals of mirror displacement x. The computed spectrum is then determined at uniform intervals of spatial frequency by the discrete cosine Fourier transform S.j / D
X n
2I0 .nx/ cos.2 j nx/ :
4
12 20 28 36 44 52 60 68 76 84 92 100 108 116
Fig. 42.2 The solid curve is the ILS (interpolated), and the dashed curve is ILS (interpolated) for an apodized interferogram. The vertical grid lines are locations where the output of the Fourier transform computation occur
From the orthogonality property of the discrete cosine Fourier transform, unique linearly independent information can occur only at spectral intervals equal to or greater than the sampling interval . Hence, the sampling interval in the spectrum is related to the achievable resolution. The full width at half maximum of the representation of a single frequency in the spectrum is 1:2 . This factor applies for the case where a single cosine wave interferogram has been abruptly truncated at the end of the mirror scan. The lineshape function for this case is quite oscillatory due to the abrupt termination of the interferogram signal at the end of the scan, known as the Gibbs phenomenon [20]. This is not always satisfactory, and frequently the interferogram is modified by a “windowing” or “apodization” function to make the lineshape more localized and monotonic. Apodization always results in an increase in the full width at half maximum. Figure 42.2 illustrates this.
42.11.4 Optical Aspects of FTIR Technology
The description of wavefront interference developed in Sect. 42.11.2 applies to interference of plane wave fronts only. A plane wave of IR radiation is obtained at the output of a collimator optics having an IR point source at its focus. In practice, a point source has insignificant intensity. A finite size source, which may be represented by a distribution of point sources in the focal plane of a collimator, provides a distribution of plane wavefronts with different angles of propagation through the scanning Michelson interferometer. As shown in Eq. (42.4), this distribution of angles results in a distribution of modulation frequencies as a function of mirror displacement x of the output intensity for a given IR wavenumber. This is illustrated in Fig. 42.3, which shows, for a single-frequency IR source, the distribution of output intensity modulation frequencies for an ideal point source a on the optical axis, a circularly symmetric distribution of uni(42.14) form intensity centered on the optical axis b, and, finally,
42 Infrared Spectroscopy
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a Point source on optical axis
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Fig. 42.3 left: distribution of interferogram modulation frequencies for a single optical frequency; right: the observed instrumental line shape functions (ILS) obtained from Fourier transformation of a finite data set representing the frequency distributions shown on the left
a circularly symmetric distribution positioned slightly off the axis of the collimator c [21]. As a result of illuminating the scanning Michelson interferometer with a finite size source, a single IR frequency source is observed having a distribution of modulation frequencies in its interferogram signal. This distribution limits the ability to resolve two closely spaced IR frequencies and determines the optical resolution limit of the FTIR; the larger the extent of the source, the more restricted the resolution becomes. The ratio = is the resolving power R of the scanning Michelson interferometer and, based on Eq. (42.4), is given by 1 ; (42.15) RD .1 cos m /
as shown with the centered circular illumination in Fig. 42.3. Any deviation from this, such as off-axis positioning of the circle, noncircular shapes, or a poorly focused centered symmetrical circle, will result in a gradual roll-off of the distribution on the low-frequency side only [21]. This results in an asymmetric spectral lineshape. For a collimator of given focal length and for a given resolving power, it is easily shown that the area of the source, or the stop that delineates it, is much larger than the slit area for classical grating spectrometers. This is particularly the case at high resolving powers. The stop that delineates the source extent for a scanning Michelson interferometer is often referred to as the Jacquinot stop or the field of view stop.
where m is the maximum off-axis angle of illumination. For small m , cos m 1 12 2 , so that
42.12 The Scanning Michelson Interferometer
Optical throughput is not only determined by the area of the (42.16) field of view stop but also the solid angle subtended by the rays traversing this area. The solid angle ˝ of rays traversing From Eq. (42.13), a resolution limit is also imposed by the a Jacquinot stop positioned in the focal plane of an input colmaximum length of the interferogram recorded. The interfer- limator is given by the ratio of the interferometer beam area ogram length-dependent resolution is constant for all spectral divided by the square of the focal length of the collimator. regions, while the optical resolution is proportional to the For a given collimator focal length, the area of the spectral frequency. Both resolution limits combine to give Jacquinot stop is inversely proportional to the resolving the overall resolution of an FTIR. At low resolution, the power. To maintain equal throughput, the area of the interavailable throughput is so high that the optical resolution is ferometer optics should be increased as the resolving power often negligible compared with the length resolution. At high is increased to offset the decrease in the Jacquinot stop area. resolution, the throughput is at a premium, and the optical It is common to construct interferometers with 2:5 cm resolution is often closely matched to the length resolution at diameter optics for resolving powers up to 5,000, 5:0 cm dithe frequency of interest. ameter optics for resolving powers up to 40,000, and 7:5 cm To ensure a symmetrical frequency distribution, the area diameter optics for resolving powers up to 1,000,000. integrated illumination must increase as sin , which is apIn order to obtain a uniform state of interference across the proximately linear for small angles, up to its maximum m , entire beam of the interferometer, the beamsplitter substrate R
2 : 2
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and the two mirrors must be flat to within a small fraction of the wavelength used. Also, these elements must be oriented correctly so that the optical path difference error across the beam is less than a small fraction of the wavelength. Figure 42.1 shows two substrates at the beamsplitter position. One of the substrates supports the beamsplitting coating, while the companion substrate of precisely the same thickness acts as a compensating element to insure identical optical paths through the two arms of the interferometer. To avoid secondary interference effects, both beamsplitter and compensator substrates are normally wedged. The direction of the wedges of the two substrates must be aligned again to insure symmetry in both arms of the interferometer. The maintenance of a close angular alignment tolerance of the two mirrors with respect to the beamsplitter in a stable manner over time and while scanning one of the mirrors has been a significant challenge of interferometer design and has also been the greatest weakness of FTIR. In early FTIR models, alignment was maintained by means of a stable mechanical structure and often using a highly precise linear air bearing for the scanning mirror. Satisfactory operation required a stable environment and frequent alignment tuning and could be achieved for mirror displacements of only several centimeters, thus limiting the maximum resolution. Different techniques have been developed to overcome this weakness of FTIR. The two most prominent are: (1) dynamic alignment of the interferometer, where optical alignment is servo-controlled using the reference laser not only for mirror displacement control but also for mirror orientation control, and (2) the use of cube corner mirrors in place of the flat mirrors in the interferometer. Dynamic alignment has the advantage of retaining simplicity in the optical design of the interferometer. On the other hand, it is more complex electronically. Cube corner mirrors have the property of always reflecting light 180ı to incident light independent of orientation. Cube corners always ensure wavefront parallelism at the point of recombination of the two beams in the interferometer. Cube corners lack a defined optical axis. In a cube corner interferometer, the optical axis is defined by the direction in which the wavefronts undergo zero shear. The scanning Michelson interferometer is normally provided with a drive mechanism to displace one of the mirrors precisely parallel to its initial position and at uniform velocity. The uniform velocity translates the mirror displacementdependent intensities into time-dependent intensities. This facilitates signal processing electronics. In some measurement scenarios, however, where the sample spectrum may vary with time, it is undesirable to deal with the multifrequency time-varying intensities of the interferogram signal. In this case, it is preferable to scan the moving mirror in a stepwise mode, where the mirror is momentarily stationary
H. Buijs
at the time of signal measurement and then advanced rapidly to the next position. The mirror scan velocity v can also be varied, so that electrical signals can have different frequencies for the same optical frequencies f D
: 2v
(42.17)
The scan velocity is normally selected to provide the most favorable frequency regime for the detector and electronics used and for the mechanical capabilities of the mirror drive. Typically, the velocity range is from 0.05 to 4 cm s1 , putting the frequencies in the audio range. At these velocities, the measurement scan may be completed in a shorter time than is desired for signal averaging purposes. In this case, it is common to repeat the scan a number of times and add the results together; this is called coadding of scans. As stated earlier, in FTIR there is no segregation of frequencies of the input intensity. All frequencies of the spectrum are transmitted through the interferometer simultaneously. In the commonly encountered case where the field of view is small, so that the interferogram length determines the resolution and lineshape, the lineshape closely matches the same analytical function for all spectral components. Along with the absence of dispersion in the interferometer providing a highly linear frequency scale that is calibrated at a single frequency FTIR provides a high degree of reproducibility. This is frequently used to reconstruct observed spectra based on predetermined spectral parameters for molecules present in the sample. By least squares fitting several attributes of the sample can be determined.
42.13 Infrared Spectroscopy Application Activity 2020 Infrared spectroscopy using FTIR has become a mature activity with many thousands of simple benchtop units produced. Due to innovative design and efficient manufacturing these FT spectrometers cost much less than the early pioneering versions. Below is a description of some of the areas of use of infrared spectroscopy using FTIR. The laboratory applications are routine, while applications outside the laboratory are more specialized, particularly some of the remote-sensing applications.
42.13.1 Analytical Chemistry Laboratories The traditional analytical chemistry laboratory is routinely equipped with an FTIR that covers the spectral range from about 400 to about 5000 cm1 . It is used mostly for the
42 Infrared Spectroscopy
analysis of liquid samples in transmittance and solids (in powder form) by diffuse reflectance. Condensed phase samples (liquids and solids) show spectral features that are mostly > 4 cm1 wide and are rarely narrower than 2 cm1 . Hence, the full width at half maximum intensity (FWHM), spectral resolution selected is mostly 4 cm1 . To avoid the need for spectral filtering, the numerically computed spectra typically cover the range from 0 to about 8000 cm1 or higher. The absorptivity of most condensed phase samples is high, particularly in the 400 to 2000 cm1 fingerprint region. A short pathlength is, therefore, selected to avoid regions of zero transmittance. An efficient sampling method is to place a drop of liquid sample or a small amount of power on the surface of an IR transparent crystal with a high index of refraction through which IR radiation is propagated via total internal reflection. The presence of the sample permits penetration of an evanescent wave several µm into the sample and, thereby, produces a transmittance spectrum corresponding to a path length of typically up to about 10 µm; this is called attenuated total internal reflection (ATR) spectroscopy. The measured spectra are typically used for the semiquantitative determination of molecular concentration. However, since the frequency axis in FTIR is calibrated via the scan control metrology laser, the precision (repeatability) is exceptionally high. In the numerical format, the method of spectral subtraction can bring to light subtle differences between comparable samples typically due to small amounts of contaminants mixed in the samples. Moreover, the scaled spectral subtraction is a powerful method of concentration determination.
42.13.2 Biomedical and Pharmaceutical Laboratories Medical formulations may consist of mixtures of complex molecular compounds. Infrared spectroscopy is one of the many analysis techniques employed for the determination of accuracy of formulation. It is found that the efficacy of a formulation may not only depend on a particular molecular formulation but also on the “chirality” of the formulation. In stereo chemistry, chirality provides the distinction between two enantiomers of otherwise identical molecules. Enantiomers are the right-hand and left-hand versions of otherwise identical molecules; a “spiral spring” may be coiled in two different ways, right-hand and left-hand, without affecting its spring property. Determination of the chirality of a sample by infrared spectroscopy consists of measuring the difference in absorption of right and left circular polarized radiation. Since this difference in absorption is typically five orders of magnitude weaker than the normal absorption
635
spectrum it is challenging to measure chirality. A Fourier transform spectrometer provided with a means of rapidly switching between right and left circular polarization, such as using a photoelastic modulator, permits demodulation of the difference absorption of the two circular polarizations [22].
42.13.3 Forensic Investigation Due to the capability of mid-IR spectroscopy to relate a large number of different substances to their unique infrared spectra, IR spectroscopy is an indispensable tool in forensic laboratories. The emphasis here is to analyze samples in many forms and frequently in exceedingly small amounts. The analysis of liquids by transmission is a small subset of the variety of samples that may occur. Many of the samples to be investigated may not be transparent or just have a surface contaminant that is of interest. Both specular and diffuse reflection are, therefore, often employed. In forensic investigation, sometimes extremely small amounts of sample may be of particular interest. Many microsampling accessories are available, amongst which the IR microscope is the most powerful. The IR microscope permits focusing the modulated IR beam onto spots as small as 10 µm and analysis of 50 to 100 µm areas is routine. To be sure the focusing down to an area of less than 100 µm utilizes only a small fraction of the available intensity of the FTIR. IR microscopes are normally provided with dedicated liquid nitrogen cooled Hg-Cd-Te semiconductor detectors. Increasingly available mid-IR, HgCd-Te imaging detector arrays with high image frame rates are being used in conjunction with IR microscopes. These provide the capability of examining microscopic samples with inhomogeneous composition distribution. A common property of infrared detectors is that their noise level is proportional to the square root of their area. Hence, a single pixel detector of a detector array with small area will have significantly lower noise than a larger single detector module. This permits recording a 2-D array of spectra of a microsample with low noise, where individual pixels may cover an area as small as a few µm. A complementary method used in forensic investigation and other applications is Raman spectroscopy. Spectra like mid-IR spectra of a sample are obtained via inelastic scattering of laser illumination from their surface. Raman spectroscopy is more tolerant to sample shape irregularity and texture and does not require contact. For some samples, fluorescence induced by the laser illumination can interfere with the Raman spectrum. Identifying the chemical makeup of samples from their measured mid-IR or Raman spectra is achieved using search algorithms applied to digital IR libraries of IR and Raman spectra. As many as 225,000 infrared spectra are available
42
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for searching. Because of small variations in sample accessories, instrument calibrations, and contaminants in the sample, the spectral identification is presented as a probable identification with a confidence index. Several probable identification candidates are returned with a decreasing confidence index.
42.13.4 Infrared Spectroscopy in Quality Assurance Laboratories
H. Buijs
42.13.5 Process Monitoring by Infrared Spectroscopy Continuous process monitoring of parameters such as temperature and pressure are common. In some processes, it is valuable to continuously monitor the product composition as well. Like methods used in quality assurance laboratories, process monitoring most frequently uses quantitative nearIR spectroscopy. An illustrative example is the blending of gasoline in refineries. There are several challenges in the blending of gasoline related to performance, as indicated by octane number, as well as insuring low viscosity in cold weather conditions and low evaporation in hot weather, and, finally, to conform to environmental restrictions. All these factors complicate the formulation, allowing only a limited number of different solvents to be used for blending. Figure 42.4 shows a nearinfrared spectrum of gasoline identifying the bands attributed to different types of hydrocarbons. In the range of permitted gasoline blends, the near-IR spectra such as shown in Fig. 42.4 vary by only a few percent in absorbance. To continuously monitor the blending process and report the required parameters requires highly accurate and stable spectroscopy; the overlay of spectra of the same sample measured with 14 different analyzers is shown in Fig. 42.4. The reporting of (nonspectroscopic) quality parameters is derived from chemometric models with many combinations of different known standard samples. Application of FT-NIR to this requirement has been shown to be exceptionally stable over time, as well as highly reproducible for different analyzers. The latter is significant because it avoids the costly and extensive recalibration of analyzers. Table 42.1 shows a typical performance summary achieved with a process FT-NIR that continuously monitors the blending process. It is used to control and adjust the blending process in real time.
A quality assurance laboratory evaluates conformity to the required specification of manufactured products or the concentration of essential components in products such as wheat, milk, medicinal formulations, and industrial chemical formulations. The demands of the QA laboratory are quite different from those of the analytical chemistry laboratory. The measurement accuracy (the agreement with a production standard) needs to be in line with the specifications of uniformity of the product. Increasingly, the manufacture of FTIR spectrometers includes a specification of reproducibility in addition to repeatability and low noise of the measured spectra. This means that the FTIR spectrometer, including the sampling arrangement, must conform to a limit of sameness of frequency scale calibration, sameness of instrumental lineshape function, FWHM, and sameness of the transmittance of a standard sample. Mid-IR sampling requires a short optical path through samples to avoid excessive absorption. Typically, the path length is below 50 µm and may be as low as a few µm. For quantitative concentration determinations, the path length is a required parameter. Therefore, the near-IR is preferred because, here, the harmonics and combinations of the mid-IR bands absorb much less strongly than the fundamental bands in the mid-IR and can be measured with longer path-length liquid cells that can have quite accurate path-length values. The determination of concentrations is done by means of chemometric models, such as partial least squares, which uses a set of standard samples to build a model. In order to 0.04 Primarily –CH3 insure accurate reproducibility of spectra, FTIR spectromePrimarily –CH 2– ters are tested with a “golden standard” sample for which 0.03 Mix of –CH 2– and –CH3 the spectrum can be rendered reproducible. Often, a sample, Aromatic such as toluene with 99.99% purity at constant temperature, 0.02 is used. This has led to the ability of deploying spectrometers at different measurement sites while maintaining identical Oxygenate 0.01 measurement results. In these applications, FTIR, due to its Pure aromatic laser-controlled scanning, provides significantly more accu0 rate measurements than a dispersive grating spectrometer. The above is an example of applying FTIR at short wave4000 4200 4400 4600 4800 lengths in the near-IR. Due to the brightness of near-IR light sources, the constraint of photon statistics noise in FTIR is Fig. 42.4 Near-IR absorbance spectra of gasoline identifying bands of different hydrocarbons not present.
42 Infrared Spectroscopy
637
Table 42.1 Summary of reproducibility, accuracy, and precision of measured components Gasoline property data Measurement Units
RON MON Total aromatics Total olefins RVP E200 E300 IBP 10% 30% 50% 70% 90% FBP
Octane number Octane number % % psi % % ı F ı F ı F ı F ı F ı F ı F
NIR Typical Std. dev. 0.30 0.30 1.50 1.00 0.15 1.50 1.00 3.70 6.00 2.90 4.80 4.20 4.90 6.70
ASTM
FTNIR transportability On-line to on-line North American refinery blender
Std. dev. 0.21 0.32 1.20 1.80 0.15 1.19 0.54 4.60 3.25 2.80 2.80 2.80 1.60 5.70
Absolute difference 0.05 0.05 0.11 0.05 0.09 0.03 0.10 0.41 0.21 0.95 0.06 0.51 0.75 1.05
Range 0.2–0.4 0.2–0.4
0.1–0.2
42.13.6 Environmental Monitoring Infrared spectroscopy is used extensively for the continuous monitoring of emission gases from a variety of industrial and utility smokestacks. The purpose is to comply with environmental regulations. Combustion processes produce large concentrations of carbon dioxide, as well as moisture. It is common to precipitate moisture from a gas sample prior to IR measurement. This permits the employment of simple, low-cost nondispersive IR analyzers (discrete narrowband filter-based analyzers). Particularly challenging is the monitoring of smokestacks that may emit water soluble gases such as HCl and HF. With a sample without precipitation of moisture, the infrared spectrum shows much interference by strong moisture bands covering large spectral regions. FTIR spectroscopy has been particularly effective in such applications because interference is minimized by choosing a spectral resolution that permits separating spectral features of the different components.
42.13.7 Remote Sensing In remote sensing, spectral information about remote objects is obtained by their thermal emission. At ambient temperature, the radiation emitted is the strongest in the 500 to 1800 cm1 region and is, therefore, widely used. Satellite-based remote sensing of Earth’s atmosphere is of particular interest to support weather prediction and to determine levels of pollution. Satellite-based thermal infrared measurements routinely provide input data for weather prediction and climate change models. Thermal emission from a gas depends on the opacity (absorbance) of the gas. Oxygen and nitrogen, which make up most of Earth’s atmosphere,
have virtually no opacity. Thermally emitted radiance by Earth’ atmosphere is dependent on small amounts of gases such as carbon dioxide, ozone, water vapor, and other minor constituents present in the atmosphere that have infrared absorption bands. The major thermodynamic drivers that affect Earth’s weather the most are temperature and water vapor variations. Meteorologists working with large computer simulations of local and extended weather patterns count on hourly updates of accurately measured three-dimensional maps (geographic and altitude resolved) atmospheric temperature, and water vapor distribution. In addition, the knowledge of detailed maps of surface emissivity and temperature, as well as albedo for solar reflection, are important drivers of weather. In Fig. 42.5 the reduced radiance near 650 cm1 is due to thermal emission to space by carbon dioxide in the upper atmosphere where it is much colder than 305 K. Likewise some reduced radiance near 1000 cm1 is due to colder ozone and beyond 1300 cm1 due to cold water vapor. It can be seen that if Earth had no atmosphere, it would radiate more radiance to space (smooth curve), keeping Earth at a lower temperature than in the presence of our atmosphere; this is a clear demonstration of the greenhouse effect. Temperature as a function of altitude can be derived from these spectra by realizing that the concentration of carbon dioxide is known and is uniformly mixed through the atmosphere. Having different band intensities provides the information to retrieve temperature as a function of altitude. The recent introduction of sophisticated new satellite-based Fourier transform spectrometer sounders has greatly improved the accuracy and detail in the vertical profiles of both temperature and moisture [23]. These new sounders have demanding requirements of spectral reproducibility, sensitivity, and speed of measurement. In order to map the entire globe
42
638 Fig. 42.5 A typical, clear sky, radiance in W=cm2 -sr-cm1 in the thermal infrared observed from space. The smooth line is a Planck curve for a highemissivity surface at 305 K
H. Buijs
1.80E–05 1.60E–05 1.40E–05 1.20E–05 1.00E–05 8.00E–06 6.00E–06 4.00E–06 2.00E–06 0.00E+00 500
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ter is a required tool for the chemistry laboratory. Expanding on this, today, the power of identifying the chemical nature of substances is incorporated into compact handheld FTIR and Raman spectrometers that permit noncontact “point and measure” for the determination of the molecular nature of chemicals. The sampling approach may be diffuse reflection and laser Raman scattering or on a small area point contact by means of attenuated total internal reflection (ATR). Due to continual improvement in operation stability, FTIR has become well known for its precise repeatability, including outstanding stability over time. The repeatability of FTIR is attributable to the highly precise measurement of the interferometer scan distance using a wavelength stabilized single-mode laser. The instrumental lineshape function (ILS) is determined by the precision of the scan distance measurement and not by any geometric factors in the instrument. This fundamental difference in the aspect of measurement precision has led to the development not only of high accuracy spectral measurements but additionally to excellent reproducibility amongst FTIRs. These developments have led to a reputation of FTIR as an accurate quantitative analysis tool. Increasingly, FTIR is being deployed for quality assurance in chemical manufacturing. An example is the determination of the hydroxyl value in a variety of surfactants. Here, easy-tooperate FTIR has replaced a laborious skill-demanding and costly titration method. These features have been quite useful in molecular spectroscopy research as well. This permits accurate reproducibility of the frequency scale from instrument to instrument. The instrumental lineshape function when defined by the field of view stop (sometimes called the Jacquinot stop) can be expressed by an analytical function and is invariant over a defined frequency range. This is a key factor in achiev42.14 Conclusion ing precisely reproducible spectral response with different instruments. Finally, a reason for the diversity of applications Infrared spectroscopy is widely used to identify the chemi- enabled with FTIR is its ease of adaptability to various needs cal nature of a range of different substances in a variety of for spectral resolution as well as scan velocities and spectral measurement settings. A compact low-cost FTIR spectrome- ranges.
more than once per day they typically produce over 400,000 spectra=day. Exploration of the Solar System by interplanetary probes was started by sending probes to the moon in order to determine the suitability of landing there. Mars has been extensively surveyed both on the surface and in its atmosphere. Besides visual images, infrared spectroscopy has been used to determine surface and atmospheric composition. One of the more recent probes is the Cassini–Huygens probe [24] to investigate Saturn and some of its moons. Saturn being almost 10 times further from the Sun as our Earth it receives much less heat from the Sun. Infrared spectroscopy of this probe is primarily centered on far-infrared thermal emission. A spectacular feat of infrared spectroscopy has been the measurement of the remnant of cosmic background radiation. Expected to be a Planck-type blackbody radiation at only an estimated 2.7 K, its spectrum occurs in the range 0 to about 20 cm1 and has a peak near 5 cm1 . Both the NASA COBE satellite mission [25] and the COBRA suborbital rocket experiment [26] used Fourier transform spectrometers. Since almost any part of the spectrometer might have a comparable temperature to the source of interest, the COBRA FTIR consisted of a precisely balanced dual-input and dual-output interferometer. The experiment measured the difference spectrum between the space view at one input against a calibrated blackbody at the second input. Adjusting the temperature of the blackbody to achieve a null interferogram provided a precise value for the temperature. Spectral analysis of the residual null interferogram confirmed that the structure of the spectrum closely matched the Planck blackbody curve at 2.736 K.
42 Infrared Spectroscopy
References 1. Herschel, W.: Experiments on the refrangibility of the invisible rays of the Sun. Phil. Trans. R. Soc. Lond. 90, 284–292 (1800) 2. Holden, E.S.: Sir William Herschel His Life and Works (2009). 3. Coblentz, W.W.: Investigations of Infra-Red Spectra. Carnegie Institution of Washington, Washington, DC (1905) 4. Lin-Vien, D., Colthup, N.B., Fateley, W.G., Grasselli, J.G.: The Handbook of Infrared and Raman Characteristic Frequencies of Organic Molecules. Academic Press, San Diego (1991). 5. PerkinElmer: 60 Years of Innovation in Infrared Spectroscopy. PerkinElmer, Shelton (2005). 6. Hirschfeld, T., Callis, J.B., Kowalski, B.R.: Chemical sensing in process analysis. Science 226(4672), 312–318 (1984) 7. Martens, H., Naes, T.: Multivariate Calibration. Wiley, New York (1989). 8. Williams, P.C.: Cereal Chem. 52, 561–576 (1975) 9. Herzberg, G.: Molecular Spectra and Molecular Structure: I. Spectra of Diatomic Molecules. Krieger, Malabar (1989) 10. Connes, J., Connes, P., Delouis, H., Guelachvili, G., Maillard, J.P., Michel, G.: Nouv. Rev. Opt. Appl. 1(1), 3 (1970) 11. Michelson, A.A.: Philos. Mag. 34, 280 (1892) 12. Rayleigh, J.W.S.: Philos. Mag. 34, 407 (1892) 13. Fellgett, P.B.: J. Phys. Radium 19(187), 237 (1958) 14. Fellgett, P.: Multichannel Spectroscopy (extended abstract). Ohio State University International Symposium on Molecular Spectroscopy, I1 (1952) 15. Buijs, H., Gush, H.: High resolution Fourier transform spectroscopy. J. Phys. Coll. 28(C2), 105–108 (1967). (archive, Journal de Physique) 16. Cooly, J.W., Tukey, J.W.: Math. Comp. 19, 296 (1965) 17. Forman, M.L., Steel, W.H., Vanasse, G.A.: J. Opt. Soc. Am. 56, 59 (1966) 18. Jacquinot, P.: J. Opt. Soc. Am. 44, 761 (1954) 19. Brault, J.W.: New approach to high-precision Fourier transform spectrometer design. Appl. Opt. 35(16), 2891–2896 (1996). https://doi.org/10.1364/AO.35.002891 20. Gibbs, J.W.: Fourier’s series. Nature 59(1522), 200 (1898) 21. Saarinen, P., Kauppinnen, J.: Appl. Opt. 31, 2353 (1992) 22. Nafie, L.: Vibrational Optical Activity: Principles and Applications. Wiley, Chichester (2011) 23. Strow, L., Motteler, H., Tobin, D., Revercomb, H., Hannon, S., Buijs, H., Predina, J., Suwinski, L., Glumb, R.: Spectral calibration and validation of the cross-track infrared sounder on the Suomi NPP satellite. J. Geophys. Res. Atmos. 118, 12486–12496 (2013). 24. Jennings, D.E., et al.: Composite infrared spectrometer (CIRS) on Cassini. Appl. Opt. 56(18), 5274–5294 (2017). 25. Boggess, N.W., et al.: The COBE mission: Its design and performance two years after launch. Astrophys. J. 397, 420–429 (1992). 26. Gush, H.P., Wishnow, E.H., Halpern, H.: Phys. Rev. Lett. 65(5), 537–540 (1990)
639 Henry Buijs Henry Buijs founded Bomem Inc. in 1973 to commercialize Fourier transform spectrometers for a wide range of applications. In 1962 he helped Dr Herbert Gush build and operate a Fourier transform spectrometer for the study of airglow from a balloon platform. He received his PhD from the University of British Columbia in 1970. Bomem Inc became a leader in high performance FT spectrometers for a very wide range of applications.
42
Laser Spectroscopy in the Submillimeter and Far-Infrared Regions
43
Kenneth Evenson and John M. Brown
Contents 43.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 641
43.2
43.2.2 43.2.3
Experimental Techniques Using Coherent SM-FIR Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . Tunable FIR Spectroscopy with CO2 Laser Difference Generation in a MIM Diode . . . . . . . . . . . . . . . . Laser Magnetic Resonance . . . . . . . . . . . . . . . . . TuFIR and LMR Detectors . . . . . . . . . . . . . . . . .
43.3
Submillimeter and FIR Astronomy . . . . . . . . . . . 645
43.4
Upper Atmospheric Studies . . . . . . . . . . . . . . . . 646
43.2.1
642 642 644 645
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
Abstract
Recent technical developments in sub-millimeter and farinfrared laser spectroscopy are described. This includes new laser sources, both side-band and difference-frequency generation. An experiment which uses fixed-frequency far-infrared lasers to study open-shell molecules (free radicals) is described; the technique is known as laser magnetic resonance (LMR). Sub-millimeter and far-infrared laser spectroscopies are finding expensive use in the detection and monitoring of molecules in astrophysical sources and in the earth’s atmosphere. Keywords
waveguide laser absorb water vapor polyatomic species fine structure transition laser magnetic resonance
43.1 Introduction Research in the submillimeter and far-infrared (SM-FIR) regions of the electromagnetic spectrum (1000 to 150 µm, 0:3 to 2:0 THz; and 150 to 20 µm, 2:0 to 15 THz, respectively) had been relatively inactive until about 30 years ago. Three
events were responsible for enhanced activity in this part of the electromagnetic spectrum: the discovery of far-infrared (FIR) lasers [1], the development of background-limited detectors [2], and the invention of the FIR Fourier transform (FT) spectrometer [3]. Following these developments major discoveries have taken place in laboratory spectroscopic studies [4, 5], in astronomical observations [6], and in spectroscopic studies of our upper atmosphere [7]. Rotational transition frequencies of light molecules (such as hydrides) lie in this region, and the associated electric dipole transitions are especially strong at these frequencies. In fact they are 10;000 times stronger than at microwave frequencies because they are 100 times typical microwave frequencies and their peak absorptivities depend approximately on the square of the frequency. Fine structure transitions of atoms and molecules also lie in this region; however, they are much weaker, magnetic dipole transitions. The observation of fine structure spectra is very important in determining atomic concentrations in astronomical and atmospheric sources and for determining the local physical conditions. Bending frequencies of larger molecules also lie in this region, but their transitions are not as strong as rotational transitions (typically a factor of 103 weaker). Spectral accuracy has been increased by several orders of magnitude with the extension of direct frequency measurement metrology into the SM-FIR region [8]. Transitions whose frequencies have been measured (including absorptions and laser emissions) are useful wavelength calibration sources (vac D c=) for FT spectrometers. FIR spectra of a series of rotational transitions have been measured in CO [9], HCl [10], HF [11], and CH3 OH [12] to be used for FT calibration standards. These lines are ten to a hundred times more accurately measured than can be realized in present state-of-the-art Fourier transform spectrometers; thus, they are excellent calibration standards. High-accuracy and highresolution spectroscopy has permitted the spectroscopic assignment of the SM-FIR lasing transitions themselves [13] resulting in a much better understanding of the lasing process.
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_43
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Astronomical spectroscopy in this region [6] may be in emission or absorption and is performed using either interferometric [14] (wavelength-based), or heterodyne (frequencybased) [15] techniques to resolve the individual spectral features. Most high resolution spectra of our upper atmosphere have been taken with FT spectrometers flown above the heavily absorbing water vapor region in the lower atmosphere. Emission lines are generally observed in these spectrometers [7].
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sideband technique [21, 24], and only the laser difference technique is described here.
43.2.1 Tunable FIR Spectroscopy with CO2 Laser Difference Generation in a MIM Diode
There are two different ways of generating FIR radiation using a pair of CO2 lasers and the MIM diode. One is by second-order generation, in which tunability is achieved by using a tunable waveguide CO2 laser as one of the CO2 lasers; it is operated at about 8 kPa (60 Torr) and is tunable by about ˙120 MHz. The second technique uses third-order 43.2 Experimental Techniques Using generation, in which tunable microwave sidebands are added Coherent SM-FIR Radiation to the difference frequency of the two CO2 lasers. The comThe earliest sources of coherent SM radiation came from plete spectrometer which can be operated in either second or harmonics of klystron radiation generated in point-contact third order is shown in Fig. 43.1. The FIR frequencies generated are semiconductor diodes [16]. Spectroscopically useful powers up to about one terahertz are produced. This technique (43.1a) second-order: fir D j1;CO2 W;CO2 j is being replaced by electronic oscillators which oscillate to over one terahertz [17]. The group at Cologne Univerthird-order: fir D j1;CO2 W;CO2 j ˙ mw (43.1b) sity, Germany [18] has been particularly successful with this approach. Spectroscopy above this frequency generally where fir is the tunable FIR radiation, 1;CO2 and W;CO2 are is performed with either lasers or FT spectrometers (see the CO2 laser frequencies, and mw is the microwave frequency. Chap. 42). Different MIM diodes are used for the two different orLaser techniques use either tunable radiation synthesized from the radiation of other lasers, or fixed-frequency SM-FIR ders: in second-order, a tungsten whisker contacts a nickel laser radiation and tuning of the transition frequency of the base and the normal oxide layer on nickel serves as the insuspecies by an electric or magnetic field. Spectroscopy with lating barrier; in third-order, a cobalt base with its natural tunable far-infrared radiation is called TuFIR spectroscopy. cobalt oxide layer is substituted for the nickel base. The Spectroscopy with fixed frequency lasers is called either third-order cobalt diodes produce about one third as much laser electric resonance (LER) or laser magnetic resonance FIR radiation in each sideband as there is in the second-order (LMR). LMR is applicable only to paramagnetic species and difference; hence third-order generation is not quite as senis noteworthy for its extreme sensitivity. LMR spectroscopy sitive as second-order. The much larger tunability, however, has been more widely applied than LER, and is discussed in makes it considerably easier to use. The common isotope of CO2 is used in both the wavethis chapter. Tunable SM-FIR radiation has been generated either by guide laser and in laser 2; in laser 1, one of four isotopic adding microwave sidebands to radiation from a SM-FIR species is used. Ninety percent of all frequencies from 0:3 to laser [19] or by using a pair of higher frequency lasers and 4:5 THz can be synthesized; the coverage decreases between generating the frequency difference [20]. The sideband tech- 4:5 and 6:3 THz. Ninety megahertz acoustooptic modulanique was first reported by Dymanus [19] and uses a Schottky tors (AOMs) are used in the output beams of the two CO2 diode as the mixing element. It has been used up to 4:25 THz lasers which irradiate the MIM diode; they increase the freand produces a few microwatts of radiation [21]. Groups at quency coverage by an additional 180 MHz and isolate the Berkley, California [22] and Cologne, Germany [23] have CO2 lasers from the MIM diode. This isolation decreases developed this technique to good effect. The CO2 laser amplitude noise in the generated FIR radiation, caused by the frequency-difference technique with difference generation in feedback to the CO2 laser from the MIM diode, by an order the metal-insulator-metal (MIM) diode covers the FIR region of magnitude; hence, the spectrometer sensitivity increases out to over 6 THz and produces about 0.1 µW. This technique by an order of magnitude. The radiations from laser 1 and the waveguide laser are uses fluorescence-stabilized CO2 lasers whose frequencies have been directly measured, and it is about two orders of focused on the MIM diode. Laser 2 serves as a frequency refmagnitude more accurate than the sideband technique. How- erence for the waveguide laser; the two lasers beat with each ever, it is somewhat less sensitive because of the decreased other in the HgCdTe detector and a servosystem offset-locks power available. There are several review articles on the laser the waveguide laser to laser 2. Lasers 1 and 2 are frequency
43
Laser Spectroscopy in the Submillimeterand Far-Infrared Regions
FIR detector
MIM diode Lens Microwave synthesizer (Third-order TuFIR only)
vFIR
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643
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vw
Iris
CO2 line identifier Iris Lens Lenses
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43 vw
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CO2 Reference cells
Fig. 43.1 Tunable far-infrared (TuFIR) spectrometer for second- or third-order operation using CO2 laser difference-frequency generation in the MIM diode
modulated using piezoelectric drivers on the end mirrors, and they are servoed to the line center of 4.3 µm saturated fluorescence signals obtained from the external low-pressure CO2 reference cells. In both second and third order, the CO2 reference lasers are stabilized to line center with an uncertainty of 10 kHz. The overall p uncertainty in the FIR frequency due to two lasers is thus 2 10 or 14 kHz. This number was determined experimentally in a measurement of the rotational frequencies of CO out to J D 38 (4:3 THz), with the analysis of the data set determining the molecular constants [9]. The CO2 radiation is focused by a 25 mm focal length lens onto the conically sharpened tip of the 25 µm diameter tungsten whisker. The FIR radiation is emitted from the 0:1 to 3 mm long whisker in a conical long-wire antenna pattern. Then it is collimated to a polarized beam by a corner reflector [25] and a 30 mm focal length off-axis segment of a parabolic mirror. The largest uncertainty in the measurement of a transition frequency comes from finding the centers of the Doppler broadened lines. This is about 0:05 of the line width for lines observed with a signal-to-noise ratio of 50 or better and corresponds to about 0:05 to 0:5 MHz. A linefitting program [26] improves the line center determinations by nearly
an order of magnitude. The FIR radiation is frequency modulated due to a 0:5 to 6 MHz frequency modulation of CO2 laser 1; this modulation is at a rate of 1 kHz. The FIR detector and lock-in amplifier detect at this modulation rate; hence the derivatives of the absorptions are recorded. Absorption cells from 0:5 to 3:5 m in length with diameters ranging from 19 to 30 mm have been used in the spectrometer. The cells have either glass, copper, or Teflon walls and have polyethylene or polypropylene windows at each end. These long absorption cells lend themselves naturally to electrical discharges for the study of molecular ions. A measurement of the HCOC line at 1 THz exhibited a signal-to-noise ratio of 100 W 1 using a 1 s time constant; this is the same signal-to-noise ratio that was obtained using the laser sideband technique. The instrumental resolution of the spectrometer is limited by the combined frequency fluctuation from each CO2 laser (about 15 kHz). This is less than any Doppler-limited line width and, therefore, does not limit the resolution. The entire data system is computerized to facilitate the data recording and optimize the data handling. Improvements in this TuFIR technique may come from either better diodes or detection schemes. The nonlinearity
644
K. Evenson and J. M. Brown
94.0 cm
Coupler
Pump beam
Micrometer drive
ZnSe window
Quartz spacer Moveable mirror
0
5
10
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20
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2T Magnet Quartz tube
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25 Centimeters
Fig. 43.2 40 µm to 1000 µm Far-infrared laser magnetic resonance (LMR) spectrometer using an optically pumped FIR laser
of the MIM diode is extremely small, and conversion efficiencies could be much larger if a more efficient diode is discovered. Differential detection (which requires more sensitive detectors) could also significantly improve the sensitivity and permit the detection of weaker lines.
43.2.2
Laser Magnetic Resonance
Laser magnetic resonance (LMR) is performed by magnetically tuning the transition frequencies of paramagnetic species into coincidence with the fixed frequency radiation from a laser. LMR is a type of Zeeman spectroscopy [27], and its chief forte is its extreme sensitivity. Approximately one hundred species have been observed in the SM-FIR region. These include atoms, diatomics (especially hydrides), polyatomics, ions, metastables, metastable ions [28], and many first observations of free radicals. These observations are summarized in several review articles on LMR [29–31]. The FIR LMR spectrometer at NIST, Boulder is shown in Fig. 43.2. This spectrometer has an intracavity paramagnetic sample in a variable magnetic field which is labeled sample volume in the figure. A regular 38 cm EPR magnet with a 7:5 cm air gap is used. The laser cavity is divided into two parts by a Brewster angle polypropylene beam splitter about 12 µm thick. A CO2 laser optically pumps the FIR laser gas. This pump beam makes many nearly perpendicular passes reflected by the walls of the gold-lined Pyrex tube. A 45ı coupler serves to couple out the FIR radiation from the cavity to the helium-cooled detector. Mirrors each with an 89 cm radius of curvature are used in the nearly confocal geometry of the 94 cm cavity. One of the gold-coated end
mirrors is moved with a micrometer to tune the cavity to resonance and also to determine the oscillating wavelength by moving the end mirror several half-wavelengths. The beam splitter is rotatable so that the polarization of the laser can be varied with respect to the magnetic field. Quartz spacers are used to minimize the thermal expansion of the cavity. This LMR spectrometer oscillates at wavelengths between 40 and 1000 µm. The technology of FIR lasers has been reviewed by Douglas [32]. The LMR technique requires a close coincidence (typically within 20 GHz) between the absorption line and the frequency of the laser. Zero field frequencies which are 100 times more accurate than those obtained from optical spectra are obtained from the analysis of the Zeeman spectra observed (i.e. using the laser frequency and the magnetic field values). The accuracy is within 1 or 2 MHz and has permitted the far infrared astronomical observation of many of these species. Atomic FIR LMR spectra are due to fine structure transitions and are magnetic dipole transitions; hence, they are several orders of magnitude weaker than electric dipole rotational spectra of molecules. The production of a sufficient atomic number density can be difficult; however, atomic flames have been very effective sources, and the high sensitivity of LMR has been the most successful spectroscopic technique for measuring these transitions. The atoms O, C, metastable Mg, S, Si, Fe, Al, NC , CC , PC , FeC , and FC have been measured by FIR LMR. Atomic Zeeman spectra are relatively simple to analyze, and fine structure frequencies accurate to within 1 MHz can be determined. In a FIR LMR spectrometer the sample is inside the laser cavity; hence, sub-Doppler line widths can be observed by
43
Laser Spectroscopy in the Submillimeterand Far-Infrared Regions
operating the sample at low pressure and observing saturation dips in the signals. This has permitted the resolution of proton hyperfine structure in a number of hydrides. The observation of the resolved hyperfine structure is useful in the identification of the species involved and also yields accurate values of the hyperfine splittings. Rotational spectra of most hydrides lie in this spectral region and many of the hydrides, such as OH, CH, SH, and NH have been observed. Others, such as CaH, MnH, TiH, and ZnH are excellent candidates for LMR studies. Ions are much more difficult because of their low concentrations; however, the use of a special microwave discharge operating in the magnetic field has proved to be an extremely productive source of ions for LMR studies [28]. The spectra of a number of polyatomic species has been observed by LMR: for example, NO2 , HO2 , HCO, PH2 , CH2 , NH2 , AsH2 , HO2 , HS2 , CH3 O, and CCH. The spectra of these polyatomic species are more difficult to analyze, resulting in a somewhat less accurate prediction of the zero-field frequencies. Spectra of the extremely elusive CH2 and CD2 radicals have finally yielded to analysis with the simultaneous observation of FIR LMR spectra [33] and IR LMR spectra [34]. The data yield rotational constants which predict the ground state rotational transition frequencies and permit the determination of the singlet–triplet splitting in that molecule [35]. In the last ten years, there has been an experimental push to higher frequencies in FIR spectroscopy. In the LMR spectrometer (shown in Fig. 43.2), this has been achieved principally by reducing the internal diameter of the goldlined tube in the pumping region to 19 mm so as to increase the overlap between the laser gain medium and the FIR radiation field in the laser cavity. As a result, many new laser lines have been discovered; the system operates down to below 40 µm. This has enabled molecules with larger fine-structure intervals to be studied (e.g., FO and HFC [36, 37]) and has also led to the first detection of vibration-rotation transitions in molecules with low-frequency bending vibrations (CCN, HCCN, FeH2 ) [38–40]. The sensitivity of TuFIR is only about 1% of that of laser magnetic resonance, but it is difficult to compare the two because the sample can be very large in the TuFIR spectrometer and is limited to about 2 cm3 in the LMR spectrometer.
43.2.3 TuFIR and LMR Detectors Four different detectors have been used in the TuFIR and LMR spectrometers: (1) an InSb 4 K, liquid 4 He-cooled, hot-electron bolometer, operating from 0:3 to 0:6 THz p with a noise equivalent power (NEP) of about 1013 W= Hz; (2) a gallium doped germanium bolometer, cooled to the lambda
645
point of liquid-helium, operating p from 0:6 to 8:5 THz with an NEP of about 1013 W= Hz; (3) a similar, but liquid 3 He-cooled bolometer, with a NEP two orders of magnitude smaller; and (4) an unstressed Ge:Ga p photoconductor, cooled 14 to 4 K, with an NEP of 10 W= Hz, operating from 2:5 to 8:5 THz. The optimization of detector systems for this new technique has been responsible for a significant improvement in sensitivity of these spectrometers.
43.3 Submillimeter and FIR Astronomy Submillimeter astronomy is reviewed in [6]. Many atomic species (both neutrals and ions) and more than 100 molecules have been detected in interstellar space; many were first observed in the submillimeter and FIR region. In the microwave and submillimeter region, radio (heterodyne) techniques are employed in the receivers. At higher frequencies, interferometric techniques are used [14]; however, heterodyne techniques recently have been employed at these higher frequencies and have resulted in the detection of CO [41] and OH [42] at 2:5 THz and a search for the IR band of methylene [43] at 30 THz. Submillimeter radio astronomy observatories (at high altitudes above much of the water vapor absorptions in our atmosphere) use laboratory determined frequencies. For example, the Kuiper Airborne Observatory and the submillimeter and FIR telescopes on Mauna Kea, Hawaii require frequencies accurate to about 1 MHz in their heterodyne receivers. For heterodyne detection, fixed frequency FIR gas lasers generally serve as local oscillators, and the frequencies of these lasers must be known. The discovery and direct frequency measurement of FIR laser lines has continued apace since the last major review of this topic was published in 1986 [44]. Almost 1000 lines were listed in that publication; the present number is at least twice this. Two main factors have fuelled this progress. First, the design of CO2 lasers has continued to develop so that now a single laser can produce 275 individual transitions from the regular, hot [45], and sequence [46] bands of CO2 with power levels from 2 to 30 W. Secondly, many new, short-wavelength laser lines have been discovered in a cavity specially designed to promote them. As a result, many high frequency lines have now been characterized, several in the range of 26 to 45 µm. The main lasing molecules used in this work are CH3 OH [47–50] and hydrazine, N2 H4 [51]. Recent observations of FIR transitions in molecules in the interstellar medium have been made from satellite-based platforms such as the Infrared Space Observatory (ISO) [52] using Fourier Transform methods. Attention has been paid to the measurement of less abundant isotopic forms, such as 18 OH and 17 OH, because of the information that they provide about star formation processes.
43
646
There are four main goals of laboratory SM-FIR spectroscopy which serve the needs of the SM-FIR radio astronomy field: 1. To provide accurate frequencies of SM-FIR species for their detection, 2. To find new far infrared active species, 3. To measure the frequencies of FIR species which can be used to calibrate Fourier transform spectrometers, and 4. To measure frequencies of far-infrared lasers for use as local oscillators in radio astronomy receivers (and to be used in the analysis of laboratory LMR data).
43.4 Upper Atmospheric Studies A very impressive set of SM-FIR spectra of our upper atmosphere has been observed using balloon-borne FT spectrometers flown at altitudes where the lines are narrow and the spectrometer is above the black, heavily absorbing water vapor transitions [7]. A number of very important species with strong SM-FIR spectra have been observed. They include: O2 , H2 O, NO, ClO, OH, HO2 , O3 , and O. Numerous lines have not yet been identified. It is difficult to calibrate these instruments absolutely because the spectra come from the species emitting at temperatures of about 200 K, and from an indeterminate path length. The high sensitivity of LMR might provide an alternate way of measuring the paramagnetic species in our upper atmosphere by flying an LMR spectrometer to high altitudes. A light-weight solenoid magnet could be used to increase the path length. Absolute concentrations of the paramagnetic species such as OH, HO2 , NO, and O could be obtained. Acknowledgements We have benefitted invaluably from the help and collaboration of I. G. Nolt of NASA, Langley for his assistance with the detector technology, and of Kelly Chance for his line shape fitting program and his assistance with our studies of upper atmospheric species.
References 1. Crocker, A., Gebbie, H.A., Kimmitt, M.F., Mathias, L.E.S.: Nature 201, 250 (1964) 2. Haller, E.E.: Infrared Phys. Technol. 35, 127–146 (1994) 3. Strong, J., Vanasse, G.: J. Opt. Soc. Amer. 49, 884 (1959) 4. Evenson, K.M., Broida, H.P., Wells, J.S., Mahler, R.J.: Phys. Rev. Lett. 21, 1038 (1968) 5. Zuidberg, B.F.J., Dymanus, A.: Appl. Phys. Lett. 32, 367 (1978) 6. Phillips, T.G.: Techniques of submillimeter astronomy. In: Burton, W.B. (ed.) Millimetre and Submillimetre Astronomy, p. 1. Kluwer, Doordrecht (1988) 7. Traub, W.A., Chance, K.V., Johnson, D.G., Jucks, K.W.: Proc. Soc. Photo-Opt. Instrum. Eng. 1491, 298 (1991) 8. Jennings, D.A., Evenson, K.M., Knight, D.J.E.: Proc. IEEE 74, 168 (1986)
K. Evenson and J. M. Brown 9. Varberg, T.D., Evenson, K.M.: IEEE Trans. Instrum. Meas. 42, 412 (1993) 10. Nolt, I.G., Radostitz, J.V., DiLonardo, G., Evenson, K.M., Jennings, D.A., Leopold, K.R., Vanek, M.D., Zink, L.R., Hinz, A., Chance, K.V.: J. Mol. Spectrosc. 125, 274–287 (1987) 11. Jennings, D.A., Evenson, K.M., Zink, L.R., Demuynck, C., Destombes, J.L., Lemoine, B.: J. Mol. Spectrosc. 122, 477 (1987) 12. Matsushima, F., Evenson, K.M., Zink, L.R.: J. Mol. Spectrosc. 164, 517 (1994) 13. Xu, L.-H., Lees, R.M., Evenson, K.M., Chou, C.-C., Shy, J.-T., Vasconcellos, E.C.C.: Can. J. Phys. 72, 1155 (1994) 14. Storey, J.W.V., Watson, D.M., Townes, C.H.: Int. J. Infrared Millimeter Waves 1, 15 (1980) 15. Röser, H.-P.: Infrared Phys. 32, 385 (1991) 16. Helminger, P., Messer, J.K., DeLucia, F.C.: Appl. Phys. Lett. 42, 309 (1983) 17. Belov, S.P., Gershstein, L.I., Krupnov, A.F., Maslovsky, A.V., Spirko, V., Papousek, D.: J. Mol. Spectrosc. 84, 288 (1980) 18. Winnewisser, G.: Vib. Spectrosc. 8, 241 (1995) 19. Zuidberg, B.F.J., Dymanus, A.: Appl. Phys. Lett. 32, 367 (1978) 20. Evenson, K.M., Jennings, D.A., Petersen, F.R.: Appl. Phys. Lett. 44, 576 (1984) 21. Verhoeve, P., Zwart, E., Versluis, M., Drabbels, M., ter Meulen, J.J., Meerts, W.L., Dymanus, A.: Rev. Sci. Instrum. 61, 1612 (1990) 22. Cohen, R.C., Busarow, K.L., Laughlin, K.B., Blake, G.A., Havenith, M., Lee, Y.T., Saykally, R.J.: J. Chem. Phys. 89, 4494 (1988) 23. Lewen, F., Michael, E., Gendriesch, R., Stutzki, J., Winnewisser, G.: J. Molec. Spectrosc. 183, 207 (1997) 24. Blake, G.A., Laughlin, K.B., Cohen, R.C., Busarow, K.L., Gwo, D.-H., Schmuttenmaer, C.A., Steyert, D.W., Saykally, R.J.: Rev. Sci. Instrum. 62, 1693–1701 (1991) 25. Grossman, E.N.: Infrared Phys. 29, 875 (1989) 26. Chance, K.V., Jennings, D.A., Evenson, K.M., Vanek, M.D., Nolt, I.G., Radostitz, J.V., Park, K.: J. Mol. Spectrosc. 146, 375 (1991) 27. Wells, J.S., Evenson, K.M.: Rev. Sci. Instrum. 41, 226 (1970) 28. Varberg, T.D., Evenson, K.M., Brown, J.M.: J. Chem. Phys. 100, 2487 (1994) 29. Evenson, K.M.: Faraday Discuss. 71, 7 (1981) 30. Russell, D.K.: Laser magnetic resonance spectroscopy. In: Symons, M. (ed.) Specialist Periodical Reports Electron Spin Resonance, vol. 16, p. 64. The Royal Sociey of Chemistry, London (1990) 31. Chichinin, A.I.: Magnetic Resonance: Laser Magnetic Resonance. In: Linton, J., Tranter, G., Holmes, J. (eds.) Encyclopedia of Spectroscopy and Spectrometry, pp. 1133–1140. Academic Press, London (2000) 32. Douglas, N.G.: Millimetre and Submillimetre Wavelength Lasers. Springer, Berlin, Heidelberg (1990) 33. Sears, T.J., Bunker, P.R., McKellar, A.R.W., Evenson, K.M., Jennings, D.A., Brown, J.M.: J. Chem. Phys. 77, 5348 (1982) 34. Sears, T.J., Bunker, P.R., McKellar, A.R.W.: J. Chem. Phys. 75, 4731 (1981) 35. McKellar, A.R.W., Bunker, P.R., Sears, T.J., Evenson, K.M., Saykally, R.J., Langhoff, S.R.: J. Chem. Phys. 79, 5251 (1983) 36. Tamassia, F., Brown, J.M., Evenson, K.M.: J. Chem. Phys. 110, 7273 (1999) 37. Allen, M.D., Evenson, K.M., Brown, J.M.: J. Mol. Spectrosc. 227, 13 (2004) 38. Allen, M.D., Evenson, K.M., Gillett, D.A., Brown, J.M.: J. Mol. Spectrosc. 201, 18 (2000) 39. Allen, M.D., Evenson, K.M., Brown, J.M.: J. Mol. Spectrosc. 209, 143 (2001)
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40. Körsgen, H., Evenson, K.M., Brown, J.M.: J. Chem. Phys. 107, 1025 (1997) 41. Boreiko, R.T., Betz, A.L.: Astrophys. J. 346, L97 (1989) 42. Betz, A.L., Boreiko, R.T.: Astrophys. J. 346, L101 (1989) 43. Goldhaber, D.M., Betz, A.L., Ottusch, J.J.: Astrophys. J. 314, 356 (1987) 44. Inguscio, M., Moruzzi, G., Evenson, K.M., Jennings, D.A.: J. Appl. Phys. 60, R161 (1986). Table 5 45. Maki, A.G., Chou, C.-C., Evenson, K.M., Zink, L.R., Shy, J.-T.: J. Mol. Spectrosc. 167, 211 (1994) 46. Chou, C.-C., Maki, A.G., Tochitsky, S.J., Shy, J.-T., Evenson, K.M., Zink, L.R.: J. Mol. Spectrosc. 172, 233 (1995) 47. Zerbetto, S.C., Vasconcellos, E.C.C.: Int. J. Infrared Millimeter Waves 15, 889 (1994) 48. Pereira, D., Telles, E.M., Strumia, F.: Int. J. Infrared Millimeter Waves 15, 1 (1994) 49. Zerbetto, S.C., Zink, L.R., Evenson, K.M., Vasconcellos, E.C.C.: Int. J. Infrared Millimeter Waves 17, 1049 (1996) 50. Telles, E.M., Odashima, H., Zink, L.R., Evenson, K.M.: J. Mol. Spectrosc. 195, 360 (1999) 51. Vascocellos, E.C.C., Zerbetto, S.C., Zink, L.R., Evenson, K.M.: J. Opt. Soc. Am. B. 15, 1839 (1998) 52. Ceccarelli, C., Baluteau, J.-P., Walmsley, M., Swinyard, B.M., Caux, E., Sidher, S.D., Cox, P., Gry, C., Kessler, M., Prusti, T.: Astron. Astrophys. 383, 603 (2002)
647 John M. Brown Professor Brown obtained his Ph.D. degree from the University of Cambridge in 1966. Before moving to Oxford in 1983, he was a Lecturer in the Department of Chemistry at Southampton University. He was a highresolution, gas-phase spectroscopist with a special interest in free radical species. In addition to experimental studies at all wavelengths, he was interested in the development of theoretical models to describe the experimental results. He was elected a Fellow of the Royal Society in 2003. He passed away in 2009 soon after his retirement.
43
Spectroscopic Techniques: Lasers
44
Paul Engelking
Contents
44.1
44.1 44.1.1 44.1.2 44.1.3 44.1.4
Laser Basics . . . . . Stimulated Emission Laser Configurations Gain . . . . . . . . . . Laser Light . . . . . .
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Interaction of Laser Light with Matter Linear Absorption . . . . . . . . . . . . . . . Multiphoton Absorption . . . . . . . . . . . Level Shifts . . . . . . . . . . . . . . . . . . . Hole Burning . . . . . . . . . . . . . . . . . Nonlinear Optics . . . . . . . . . . . . . . . Raman Scattering . . . . . . . . . . . . . . .
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Abstract
As a primary research tool, the laser plays a fundamental role in the spectroscopic study of atomic and molecular systems. This Chapter describes the basic operating principles, configurations, and characteristic parameters of lasers. Laser designs are discussed and then the details of the interaction of the laser light with matter delineated. The reader is also referred to other chapters in the book for further information on laser principles and types of lasers.
Laser Basics
44.1.1 Stimulated Emission A cross section 21 for absorption of radiation by a lower state 1 engenders a balancing cross section 12 for emission stimulated by radiation interacting with an upper state 2. Detailed balance relates these two cross sections according to g2 12 D g1 21 ;
(44.1)
where g1 and g2 are the statistical degeneracies of their respective states [1]. For a collection of emitting and absorbing states with densities n2 and n1 , amplification may occur when n2 12 > n1 21 , which leads to a requirement for an inversion of the state populations n2 =n1 > g2 =g1 :
(44.2)
The rate of spontaneous emission at frequency can be modeled itself by stimulated emission induced by a noise source of the magnitude of the density of states ./ 12 ./ D 12 ./ ./=c D 12 ./82 :
(44.3)
44.1.2 Laser Configurations
A practical laser combines a population inversion with a means for controlling the radiation. optical parametric oscillator gain medium integrated The basic laser source is the laser oscillator, an amplicross section frequency comb hole burning fier possessing positive feedback. The usual form is simply a piece of active gain medium placed inside a resonant optiP. Engelking () cal cavity (Fig. 44.1). Tunability is produced if the resonant Dept. of Chemistry and Chemical Physics Institute, University of cavity is frequency selective and adjustable (Fig. 44.2). Many Oregon Eugene, OR, USA laser sources use an amplifier after the oscillator. Keywords
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_44
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emission cross section is
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Fig. 44.1 Simple laser oscillator and beam parameters. Distances z are generally measured from the minimum beam waist w0 . The beam appears with a far-field divergence angle 0
a
(44.7)
For a Lorentzian lifetime-broadened line, the cross section for stimulated emission at the line center becomes
z Multipass cavity
100 % mirror
2 f12 F12 ./ : 82
12 D
2 f12 ; 4 2 12 2
(44.8)
where 12 is the full width at half maximum of the line.
b
44.1.4 Laser Light Lasers are inherently bright sources of radiation: the radiation field within a practical laser must be high enough for stimulated emission to compete with spontaneous emission. The effective source of spontaneous fluctuations approximates that of the density of states. In terms of the beam parameters photon flux J per solid angle ˝, and frequency , this is
z1
z2
d
c
Grating
Grating
Mirror
Beam expander
Fig. 44.2 Tunable laser oscillator geometries. a Fabry–Perot: tuning is usually done by changing the cavity length, although changing the index of refraction by changing the temperature or current is common with solid state laser diodes. b Littrow prism line selector: typical of atomic ion lasers capable of multiple line output. c Littrow grating tuning: common in pulsed dye lasers with high gain (> 10) per pass. Telescope increases resolution by filling, and reducing angular divergence at the grating. d Grazing incidence, mirror tuned, grating mount
2 2 d 2J : D d˝d r c 2
(44.9)
Beam quality is given by the product of the angular divergence times the beam width. Highest beam quality is associated with diffraction limited light emitted from a Gaussian spot. For circular laser beams traveling in the z-direction, this corresponds to a solution of the electromagnetic wave equation
44.1.3 Gain u.r; ; z/ D
.r; z/ exp.ikz/ ;
(44.10)
The fundamental gain per pass is given by G D J =J0 D expŒ. /L ;
where u is a polarization component of the field. For high (44.4) values of k D 2=, corresponding to short wavelength, the adiabatic radial solution is also Gaussian
where J =J0 is the ratio of light output to input, is the gain coefficient, is the nonradiative loss rate, and L is the path length. The gain coefficient
D n 12 depends upon the net inversion n , n D n2 .g1 =g2 /n1 :
˚ .r; z/ D exp i P C kr 2 =.2q/ ;
(44.11)
where the complex phase shift P , beam parameter q, and beam radius w are functions of z (44.5) z P .z/ D i ln 1 i w02 s z 2 z 1 (44.6) D i ln 1 C tan ; (44.12) w02 w02
If is the wavelength of radiation, F12 is the emission line shape function normalized over frequency , 2 is the lifetime of the transition, and f12 is the branching ratio for the upper state to undergo this transition, then the stimulated
q.z/ D iw02 = C z ; " # z 2 2 2 : w .z/ D w0 1 C w02
(44.13) (44.14)
44 Spectroscopic Techniques: Lasers
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Here, w0 is the beam waist parameter, the minimum width of the Gaussian beam at a focused spot. For Gaussian beams, the product of the minimum beam waist and beam divergence angle 0 is given by 0 w0 D = :
(44.15)
The beam waist and divergence follow optical imaging according to paraxial ray theory. Higher order circular modes with p radial nodes and l angular node planes are specified by multiplying Eqs. (44.11) and (44.12) by angular and radial factors to obtain p l
.r; ; z/ D 2r=w Lpl 2r 2=w 2 eil .r; z/ ; pl
y 6 Unstable 4 Plano r2 2
Unstable 0
(44.16) Ppl .z/ D .2p C l C 1/P .z/ :
(44.17)
Here, the functions Lpl .x/ are the Laguerre polynomials as defined in Sect. 9.4.2. The radial phase shifts produce a wave front curvature of effective radius
R D z C .2p C l C 1/=2 C w02 =2 w02 =z : (44.18) Modes with the same values of 2p C l have identical axial and radial phase shifts. The two polarization components of the electromagnetic field double the degeneracies of all modes considered here. Often these degeneracies are split in practice by optical inhomogeneities of the medium through which they pass. More details can be found in the summary of Kogelnick and Li [2], or in the texts by Verdeyen [3] or Svelto [4]. Some applications require knowledge of the electric field in additionto the flux density J . For purely sinusoidal single mode beams, the rms field is hJ 1=2 : (44.19) hEi D c0 Nonlinear effects are often expressed in terms of powers of the field by hE n i D 2.n1/=2 hEin
44
–2 Plano r1
Unstable –4
0.0
1.0
2.0
3.0
4.0 x
Fig. 44.3 Stability parameters for simple two-mirror laser cavities of length L and mirror radii of curvature R1 and R2 . Here, x D 1 .L=R1 L=R2 / is the mean curvature difference of the two mirrors; 2 y D 12 .L=R1 CL=R2 / is the mean curvature of the two mirrors. Cavities with parameters in the unshaded region are stable
Stability criteria are shown in Fig. 44.3. A cavity is stable when initial angles and displacements r of paraxial rays transform during a round trip into 0 and r 0 satisfying 2
160 nm. The scattering is a four-wave mixing process resulting in a series of output frequencies shifted from the pump laser frequency by multiples of the vibrational splitting of the ground electronic state of the gas [95]. Because
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G. Stark et al.
high anti-Stokes orders are not efficiently generated, H2 , with its large ground state vibrational splitting of 4155 cm1 , is often the gas of choice. The experimental requirements are relatively minimal; a tunable visible or UV source (e.g., a frequency-doubled dye laser) and a high-pressure gas cell (about 2–10 atm of H2 in a 1 m cell). Conversion efficiencies are a few percent for the first few anti-Stokes orders in H2 with pump laser energies of about 10–50 mJ [96]; efficiencies then drop continuously, reaching about 105 to 106 for the highest orders (n D 9 to 13) reported. Stokes seeding techniques [97] have been shown to increase efficiencies for the highest anti-Stokes orders by as much as a factor of 100. Third harmonic generation (! D 3!1 ), sum-frequency mixing (! D 2!1 C !2 ), and difference-frequency mixing (! D 2!1 !2 ) in appropriate noble gases and metal vapors are broadly applicable methods for producing tunable VUV light. These processes result from the presence of the third-order nonlinear term in the expansion of the induced macroscopic polarization of the gas as a power series in the electric field (Chap. 76). (The second-order term, which is responsible for frequency doubling in crystals, is zero for isotropic media such as gases.) The power generated in thirdorder effects is proportional to ˇ ˇ2 N 2 ˇ.3/ ˇ P12 P2 F ;
(46.7)
where N is the number density of the gas, .3/ is the thirdorder nonlinear susceptibility, P1 and P2 are the input laser powers at !1 and !2 , and the factor F describes the phase matching between the generated VUV light and the induced polarization [98, 99]. For third harmonic generation, P12 P2 is replaced by P13 . A critical constraint in third-order frequency conversion is the phase matching requirement; a comprehensive treatment is presented in [100]. The factor F is a function of the product bk, where b is the confocal beam parameter of the focused input radiation, and k is the phase mismatch between the generated VUV light and the input radiation k D k .2k1 C k2 / ;
(46.8)
where ki is the wave vector of the radiation with frequency !i . In the standard case of tight focusing (b L, where L is the linear dimension of the gas cell) for sum-frequency mixing and third harmonic generation processes, F is nonzero only for k < 0. Therefore, these techniques are applicable only in spectral regions where the gas exhibits negative dispersion. In contrast, for difference-frequency mixing, the factor F is nonzero in regions of both positive and negative dispersion, and wider tuning ranges are generally possible. Conversion efficiencies for sum-frequency and difference-frequency mixing schemes are usually in the range of 107 to 104 with input peak laser powers of 1–10 MW.
Resonant enhancement of .3/ , achieved by tuning the input radiation to transition frequencies in the gas, dramatically increases conversion efficiencies (by factors of about 104 ) and allows for much lower ( 1 kW) input laser powers. Most commonly, one of the incident frequencies is tuned to an allowed two-photon transition, with the second input frequency providing subsequent tunability in the VUV; resonant methods, therefore, require two tunable inputs. Metal vapors (e.g., Mg, Zn, Hg) are negatively dispersive over fairly broad regions of the VUV between 85 and 200 nm and are consequently used for sum-frequency mixing and third harmonic generation. Conversion efficiencies are further enhanced in these vapors by three-photon resonances (2!1 C !2 ) with the ionization continuum or broad autoionizing features [85]. One experimental drawback is the complexity associated with generating the metal vapors in ovens or heat pipes. Noble gases are generally less suited for sum frequency mixing because they exhibit negative dispersion over fairly limited spectral ranges in the VUV. However, they do provide an experimentally simple medium for difference-frequency mixing schemes. In particular, Xe, Kr, and Ar (as well as H2 ) have been used in resonant and nonresonant differencefrequency mixing schemes to produce tunable radiation over the 100–200 nm region [101–107]. The noble gases are also used for sum-frequency mixing and third harmonic generation. Kr, Ar, and Ne are used for the generation of tunable radiation down to 65 nm [101, 108–111]. Below the LiF transmission cutoff (105 nm), the gases are introduced as pulsed jets. In [112], a narrowband tunable extreme ultraviolet laser system based on resonance-enhanced four-wave mixing in Kr, is described, and in [113, 114], an ultranarrow bandwidth (0:008 cm1 ) system, utilizing resonanceenhanced sum-frequency mixing in rare gases, capable of producing coherent radiation between 62 and 124 nm is described. A representative summary of third-order frequency conversion schemes is presented in Table 46.1. Most third-order frequency conversion designs use nanosecond-pulsed lasers as primary sources. With the higher intensities (> 1013 W=cm2 ) that are associated with picosecond and femtosecond pulsed lasers, higher-order frequency conversion is readily achieved. These conversion techniques have been used to generate both fixed frequency and tunable coherent radiation shortward of 70 nm. Tunable radiation at 58 nm was produced through fifth harmonic generation in C2 H2 using the frequency-doubled output of a dye laser [124, 125]; a high repetition rate (up to 500 kHz) fiber-amplifier based system with harmonics down to 30 nm is reported by [90]. A picosecond tunable system with continuous tunability over the 40–100 nm region with sub-cm1 resolution has been developed using high-order harmonic generation in Ar and Kr [126]. Frequency combs, first up-converted into the VUV in 2005 [127, 128], allow for ultra-high precision
46 Spectroscopic Techniques: Ultraviolet
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Table 46.1 Representative third-order frequency conversion schemes for generation of tunable coherent VUV light Medium Sr Mg Zn Hg
Xe
Kr
Ar Ne
(nm) 165–200 178–196 121–174 106–140 142–182 117–122 104–108 85–125 160–206 140–147 113–119 117–150 110–116 120–124 127–180 72–83 102–124 97–105 72–74
Process res. diff. mixing res. sum mixing res. sum mixing res. sum mixing nonres. tripling res. sum mixing res. sum mixing res. sum mixing nonres. diff. mixing nonres. tripling nonres. sum mixing res. diff. mixing nonres. sum mixing nonres. tripling res. diff. mixing res. sum mixing res. diff. mixing nonres. tripling nonres. tripling
Reference [115] [116] [85] [117] [98] [118] [119] [102] [120] [121] [102] [122] [102] [123] [101] [101] [105] [108] [109]
spectroscopies and metrology in the VUV [129, 130]; the relevant techniques are reviewed in [92].
46.3
Spectrometers
46.3.1 Grating Spectrometers The design and characteristics of VUV grating spectrometers and monochromators are reviewed by many authors [131– 135]. Two basic types of spectrometer are used in the VUV: normal incidence instruments for 200 nm > > 30 nm and grazing incidence instruments for 50 nm > > 2 nm. The particulars of these two types are largely dictated by the low reflectivities of both metal and dielectric surfaces in the VUV. Concave gratings are used almost exclusively in VUV spectrometers. Such gratings provide both dispersion and focusing, thus eliminating the need for additional mirrors and their associated reflection losses. Most VUV spectrometers make use of the focusing properties of the Rowland circle, which is tangent to the grating at its center, lies in a plane perpendicular to the grating grooves, and has a diameter equal to the radius of the curvature of the grating [136]. A source on a horizontal Rowland circle is focused horizontally by the grating to a location also on the circle. The dispersion introduced by the grating results in a focused, diffracted spectrum lying on the Rowland circle. Almost all normal incidence and grazing incidence VUV instruments are designed with the entrance and exit slits lying on, or nearly on, the Rowland circle.
The image formed by a concave grating is not stigmatic, i.e., the vertical focus does not coincide with the horizontal focus. Hence, a point source is imaged into a vertical line on the Rowland circle [133, 137]. Astigmatism is particularly severe at grazing incidence angles, resulting in both loss of signal (the image of the entrance slit being larger in extent than the exit slit) and loss of resolution (the image of the entrance slit is curved in the dispersion direction). Aspherical concave gratings (e.g., toroidal gratings) reduce the astigmatism associated with conventional spherical gratings ([137] and references in [138]). The most important advances in concave grating production are the use of interference techniques to produce holographic gratings [138, 139] and the development of variable line spacing gratings [140, 141]. Interference techniques eliminate the periodic irregularities in conventionally ruled gratings that lead to spectroscopic “ghosts”, reduce the level of scattered light by significant amounts, allow for very high groove densities (e.g., 4800 mm1 ), and can be relatively easily applied to aspheric surfaces. Mechanically ruled variable line spacing gratings correct for spherical aberrations and allow for relatively simple focusing and scanning designs in EUV spectrometers and monochromators [141]. Normal incidence grating designs are appropriate for wavelengths greater than about 30–40 nm. The most common types include the Eagle, Wadsworth, and Seya– Namioka designs [131, 133, 134]. Eagle mounts, with entrance and exit slits on the Rowland circle and approximately equal angles of incidence and reflection (< 10ı ), can be either in-plane or off-plane. Photoelectric resolving powers =ı of 2 105 were achieved in the 100 nm region with 6.65 m Eagle mount instruments [142–144], although most of these large instruments have now been decommissioned. In the Wadsworth mount, the light source is at a large distance from the grating, and no entrance slit is required. This design is appropriate for collimated light sources such as synchrotron radiation. It has the advantages of high throughput and minimal astigmatism [132]. The Seya–Namioka mount is a Rowland circle instrument in which the angle subtended by the fixed entrance and exit slits relative to the center of the grating is approximately 70ı . The spectrometer remains in good focus for small grating rotations, resulting in a simple scanning mechanism and, thus, an inexpensive design. Seya–Namioka instruments provide high throughput at moderate spectral resolutions (typically 0.02–0.05 nm) [132]. The normal incidence reflectance of all standard metal coatings is no greater than a few percent for < 30 nm [133]; in this wavelength region, grazing incidence instruments take advantage of the total reflection of photons at extreme grazing angles. There is a sharp reflectance cutoff at photon energies above a characteristic energy that typically limits the use of grazing incidence optics to > 1 nm. Grazing
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incidence designs, tailored to the constraints of synchrotron radiation facilities, are described in [3, 132, 135]. Astigmatism becomes severe at grazing angles; interferometrically produced aspherical gratings and mechanically ruled variable line spacing gratings are used to reduce this aberration [139, 141]. The resolving power of a grazing incidence instrument is generally lower than that of a comparably sized normal incidence instrument, in large part because of the decreased effective width of the grating at grazing incidence. A number of nonstandard instrument designs are reported in the literature, often for use in VUV astronomy and aeronomy applications. These novel designs take advantage of the properties of conical diffraction [145–147] and dual-grating crossed dispersion or echelle mounts [148, 149] to reduce the effects of aberrations. Reference [131] reviews many specialized VUV spectrometer designs.
46.3.2 Fourier Transform Spectrometers Fourier transform spectroscopy (FTS) is a well-established technique for high-resolution emission and absorption spectroscopy at infrared and visible wavelengths (Chap. 42). An interferometric signal is recorded as a function of an optical path difference made possible, for example, by the displacement of a Michelson interferometer mirror. FTS can be understood as a transfer of optical frequencies into the audio range via the beating interferometric signal as a scan is recorded; the low-frequency interferogram is suitable for time-resolved photodetection. The technique has been extended into the near UV and VUV regions, where FTS is characterized by the large optical throughput, high spectral resolution, and accurate linear wavenumber scale available at longer wavelengths [150]. However, the multiplex advantage of FTS is not realized at UV wavelengths because the signal-to-noise ratio is photon-noise limited rather than detector limited. Moreover, the required stabilities, both for the light source and the interferometer optics, make the extension of this technique quite challenging at short wavelengths. A review of interferometric techniques in the VUV can be found in [151]. A scanning Michelson interferometer with a fused silica beamsplitter has achieved a resolving power of 1:8 106 at wavelengths down to 178 nm [150]. The same interferometer design, with a MgF2 beamsplitter, has achieved a resolving power of 8:5 105 at wavelengths down to 150 nm [152]. This spectral resolution is significantly better than that realized by the best VUV grating instruments [153]. At shorter wavelengths, FTS based on wave amplitude division interferometers is fundamentally limited by the transmission of VUV materials such as MgF2 ( 115 nm) or LiF ( 105 nm). The manufacture of VUV beamsplitters becomes extremely
G. Stark et al.
challenging due to both the strong absorption of these materials and the high optical quality required for a strong interferometric signal. Membrane-based beamsplitters have been proposed, but, up to now, these are mainly adapted to laser-based applications [154]. A Fourier transform spectrometer based on an all-reflecting wave-front-division interferometer has been developed to operate over a large spectral range, allowing the extension of FTS into the far VUV region [155]. Two roof-shaped reflectors split and reflect the incident light following a design similar to the standard Fresnel bimirror interferometer. The displacement of one reflector’s position is controlled and measured by a separate He-Ne stabilized laser interferometer. The instrument has been coupled to a UV-VUV undulatorbased synchrotron beamline, allowing Fourier transform absorption spectroscopy in the range 300–40 nm [156]. A raw line width of 0:13 cm1 was measured on the Rydberg series of Kr near 88 nm, corresponding to a resolving power greater than 106 [155]. In contrast to wave amplitude division interferometers, wavefront division interferometers require a certain degree of spatial coherence, and thus, this technique is not well adapted to low brilliance sources. Diffraction gratings can be used as division-of-amplitude beamsplitters to create all-reflecting interference spectrometers for VUV-XUV wavelengths; this idea has been applied to a Mach-Zender interferometer configuration [157, 158]. In parallel with the development of tunable laser-based laboratory sources in the UV-VUV range, original designs that can be related to FTS have been proposed for spectroscopy applications. When a VUV laser pulse created via high harmonic generation is split by a pair of mirrors, and one of the mirrors is translated to create a time delay between the resulting two pulses, the far field profile of the recombined pulses displays an interference pattern that can be exploited as a source diagnostic [159, 160]. Another method produces pulse pairs in the infrared that emerge as phase-locked harmonics in the VUV. Precise control of the pulse delay can be generated by a conventional Michelson interferometer [161, 162] or a movable wedge prism system [163]. This latter design has been applied to the measurement of the spatially resolved absorption of an inhomogeneous thin film in the VUV but with a rather limited spectral resolution [164]. Frequency comb sources operating in the infrared have been coupled to Fourier transform spectrometers, revealing a remarkable potential for unprecedented sensitivity and spectral resolution [165]. In recent years, an innovative approach combining two phase-locked frequency combs (dual comb spectroscopy) has led to high resolution FTS without any moving components [166–168]; a review presenting the advances and the application offered by frequency comb sources in the field of molecular and atomic spectroscopy can be found in [169]. This technique could possibly be implemented in the VUV by coupling a double comb to
46 Spectroscopic Techniques: Ultraviolet
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harmonic generation based on femtosecond enhancement surfaces upon exposure to air and humidity. Metal surfaces cavities [170]. such as tungsten and aluminum/Al2 O3 [172] must be used rather than the higher efficiency alkali halide coated surfaces. Sodium salicylate is the most commonly used VUV-visible 46.4 Detectors conversion phosphor; others include liumogen, terphenyl, and coronene [183, 184]. The conversion efficiency of sodium salThere is a wide variety of photon detectors with useful icylate (peak fluorescence about 430 nm) is relatively constant response and sensitivity at VUV wavelengths. With the ex- for VUV light from 30 to 200 nm. ception of those involving the photoionization of gases, VUV The review by Krizan [185] discusses recent advances in detectors are based on the same underlying principles as fast single-photon detection, including multianode PMTs and their counterparts in the visible and infrared regions – the hybrid PMT-microchannel-plate designs. common detection schemes are initiated by surface photoemission or electron–hole pair creation in semiconducting materials. The details of the design of VUV detectors, and 46.4.2 Microchannel Plates the constraints on their use, are often determined by the low levels of transmission of VUV light through suitable window The microchannel plate (MCP) detector is a photo-emissive and semiconducting materials. Nevertheless, VUV detectors array detector that combines the single-photon counting senwith single-photon counting sensitivities and/or imaging ca- sitivity of a PMT with high-resolution imaging capability. pabilities are widely available. Reviews of VUV detectors See [186, 187] for recent reviews; other general references can be found in [1, 171–175]. References [176–178] review include [171–173, 182, 188]. An MCP consists of an array of the operating characteristics and calibration of VUV detector semiconducting glass channels with diameters of about 10– standards. More general summaries of photon detector tech- 25 µm and length-to-diameter ratios of about 50:1. Electrons nologies are presented in [179, 180]. Below, we summarize ejected from the front surface of an MCP via the photoand compare the most common VUV detection schemes, in- electric effect are accelerated through the channels; repeated cluding the use of photomultiplier tubes, multichannel plate collisions with the channel walls result in an amplification detectors, silicon photodiodes, and charge-coupled devices. of the charge by about 106 . The exiting charge clouds are detected by a variety of position-sensing anode structures. Bare MCPs have quantum efficiencies of about 10% for 46.4.1 Photomultiplier Tubes 100 nm and a long wavelength cutoff of about 120 nm. Alkali halide coatings increase the quantum efficiency to Photomultiplier tubes (PMTs) [181] are used through the about 20% and extend MCP sensitivities toward longer waveVUV for single-photon-counting, nonimaging, applications. lengths. Feedback instabilities produced by positive ions creDark count rates are low (about 1 s1 ) for solar-blind tubes ated in the channels during the electron cloud amplification with 1 cm2 photocathodes; pulse rise times are about 1–10 ns are minimized through two common channel geometries: the ([182] for a general review of amplifying detectors in the chevron configuration, where two or more straight-channel VUV). In the 105–200 nm region, useful PMT window ma- MCPs oriented at different angles are cascaded [189], or terials are fused silica, MgF2 , and LiF; sapphire is also used a configuration with one set of curved channels [190]. in environments where ionizing radiation, which causes sapReadout schemes for determining the position of individphire to fluoresce, is not present. In the windowless region of ual detected photons rely on either direct detection of the the VUV wavelength range (2 nm 105 nm), two options resulting electron cloud or conversion of the electron cloud are available: the PMT may be operated bare, i.e., without into visible photons via a phosphor to produce an optical a window, or a fluorescent coating can be deposited on the win- image [182]. Direct detection schemes include centroiddow to down-convert the VUV light to longer wavelengths. detecting anodes, such as the wedge and strip design [191] Peak quantum efficiencies for VUV photomultiplier tubes and cross strip anodes [192]. Other designs are described are about 15–20%. Some of the most useful coatings for pho- in [193, 194]. tocathodes in the VUV are CsI, KBr, and other alkali halides. These materials combine high VUV quantum efficiencies with a solar-blind response – their relatively high photoelec- 46.4.3 Silicon Photodiodes tric work functions result in long wavelength cutoffs between 150 and 300 nm. Solar-blind PMTs have low dark count rates Broad use of silicon photodiodes at VUV wavelengths has traditionally been limited by the strong absorption of VUV and minimal response to stray ambient light. Bare PMTs, although used in some applications, are con- photons in the outer SiO2 passivation layer that covers the strained by the degradation of many photocathode and dynode p–n junction of these devices. Standard silicon detectors are
46
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sensitive throughout the infrared and visible regions and also in the soft X-ray ( < 2 nm) and X-ray regions. Significant improvements in silicon photodiode sensitivities in the VUV are realized by thinning the SiO2 passivation region to thicknesses of about 5–10 nm. Reference [195] reviews VUV semiconductor photodiode designs. There has been active development of non-silicon-based photodiodes as VUV detectors, e.g., wide bandgap materials such as GaN, e.g., [196, 197], and significant work on a variety of innovative semiconductor designs based on nanofabrication techniques – see [175, 198] for recent reviews. The development of photodiodes with appreciable VUV sensitivity and good temporal stability has led to their use as radiometric transfer standards [199]. Because silicon photodiodes respond strongly to radiation throughout the visible, IR, and X-ray regions, a method for rejection of stray light is essential for their effective use in the VUV. Reference [200] reports on the use of thin-film filters, deposited on the photodiode surface, to restrict the bandpass of the radiation impinging on the diode to selected VUV wavelengths. Wide bandgap semiconductors reject visible light with their natural solar-blind response [199].
46.4.4
Charge-Coupled Devices
CCDs are widely used for low light level imaging applications throughout the visible and near-IR regions. In the VUV, charge-coupled devices suffer from strong absorption both by surface gate structures and by the inactive passivation layer [182]. Two approaches are employed to overcome these limitations: the CCD front surface is overcoated with a photon down-converting phosphor [201] or a thinned and surface-treated CCD is back-illuminated with VUV light [202, 203]. Such techniques have resulted in CCD VUV quantum efficiencies that rival those of photoemissive devices such as microchannel plates. Recent CCD designs with multilayer antireflection coatings based on molecular beam epitaxial doping technologies [204, 205] achieve quantum efficiencies above 60% in the spectral region centered around 200 nm.
stants. Below, we briefly summarize the properties and uses of VUV optical materials. Comprehensive discussions can be found in [1, 206].
46.5.1 Windows Few bulk materials transmit light below 200 nm; none transmit light from about 2 to 105 nm. In the 105–200 nm wavelength region, the most common window materials are synthetic fused quartz or suprasil (with a short wavelength transmission cutoff of about 160 nm), sapphire (145 nm), CaF2 (125 nm), MgF2 (112 nm), and LiF (105 nm). Compilations of the transmission characteristics of these and other materials are presented in [133, 207, 208]. Single crystal windows with dimensions up to 10 cm can be obtained for most of these materials. Sapphire fluoresces when struck by ionizing radiation and is, therefore, not suitable for certain environments.
46.5.2
Capillaries
In the windowless region, 2–105 nm, it is standard practice to use differentially pumped slits or circular apertures to isolate different pressure regions, e.g., to isolate an absorption cell from a vacuum spectrometer. This technique has the drawback of requiring large pumping capacities and limiting optical throughputs. Thin films (see below) can be used below 105 nm as windows, but they provide limited transmission in narrow bandpasses and cannot support large pressure differentials. Differentially pumped MCP capillary arrays can serve as an alternative window in the VUV; see, for example, [209]. The capillaries are typically 3 mm in length and have diameters of 10–100 µm. Measured optical transmissions are in the 20–50% range throughout the VUV; tradeoffs must be considered between optical throughput and gas conductance. One significant limitation to an MCP window is its small angular aperture.
46.5.3 Thin Films
46.5
Optical Materials
The design of VUV instrumentation is dictated in large part by the constraints of VUV optical materials. Transmission through bulk materials is limited to > 105 nm, the short wavelength transmission limit of LiF. Normal incidence reflectance from metal surfaces and coatings decreases dramatically at short wavelengths; polarizers and narrowband interference filters are relatively difficult to produce because of the lack of materials with suitable optical con-
Below 105 nm, thin (about 10–200 nm) metallic films are used as transmission filters with bandpasses of approximately 10–50 nm. Details of filter properties are given in [208, 210]. Reference [211] describes composite metallic filters (e.g., Al=Ti=C, Ti=Sb=Al), developed for the Extreme Ultraviolet Explorer satellite, with transmission bandpasses of about 10–20 nm and high rejection at wavelengths of strong VUV geocoronal lines. The stability of metallic thin films under intense EUV irradiation is discussed in [212].
46 Spectroscopic Techniques: Ultraviolet
46.5.4
Coatings
Above 120 nm, the principal broadband reflector for VUV wavelengths is Al with a thin protective overcoat of MgF2 , having a normal incidence reflectance of 80 to 85% [213]. The normal incidence reflectivities of all materials drop dramatically below about 100 nm. A compilation of coating reflectivities at normal and grazing incidence is presented in [214]. Materials with the highest reflectivities include Os, Pt, Au, and Ir, with reflectivities of 15–30% from 30 to 110 nm [215]. Reference [216] reports grazing incidence reflection coefficients for Rh, Os, Pt, and Au from 5 to 30 nm; Rh has the highest reflectivity in this region.
46.5.5 Interference Filters and Multilayer Coatings The development of interference filters for VUV wavelengths has been limited by the availability of coating materials with both high transmission and appreciable range of refractive differentials [217]. Multilayer dielectric reflectors, thus, require many layers to compensate for the small reflectivities at the material interfaces. The theory of multilayer reflecting optics and optic designs are reviewed in [218]. A variety of multilayer coatings with high narrowband reflectivities have been developed, references include [219–226]. Metal-dielectric bandpass filters with reflectivities as high as 90% in the ultraviolet are described in [227]. Normal incidence optics with multilayer reflection coatings can also be used in the nominally grazing incidence spectral region below 30 nm [228–230]. References [231– 236] describe normal incidence gratings coated with Mo=Si multilayers. The performances of other multilayer coatings are described in [237–240].
46.5.6
Polarizers
The production and detection of linearly polarized light in the VUV is discussed in [133, 241]. Above about 112 nm, transmission polarizers, based on the birefringence of MgF2 , are often employed [242]. Reference [243] describes multilayer transmissive and reflective polarizing coatings for the hydrogen Lyman-alpha line. Reflection polarizers are usually used below 112 nm (e.g., [244]); [245] describe transmission multilayer polarizers for the 55–90 eV region. An analysis of double-reflection circular polarizers is given in [246], and multilayer transmission quarter wave plates for the generation of circular polarization are described in [247].
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G. Stark et al. Glenn Stark Professor Stark’s research interest is in the field of experimental molecular spectroscopy. His laboratory programs emphasize molecular transitions of interest to the astrophysics and aeronomy communities, primarily involving the measurement and interpretation of high-resolution absorption spectra of vacuum ultraviolet and extreme ultraviolet transitions. Related activities include Fourier transform spectroscopy of diatomic molecules and laser spectroscopies of diatomics. Nelson de Oliveira Nelson de Oliveira received a PhD in Optics and Photonics from the Université Pierre et Marie Curie (Paris) in 2001. His main scientific interest is the development of instrumentation and sample environments for spectroscopy applications, designed to operate in the VUV spectral range. He is currently working at the Synchrotron Soleil facility as a beamline scientist on the DESIRS beamline.
Peter Smith Peter Smith (PhD Physics Caltech 1972) retired in 2009 after 36 years at the Harvard-Smithsonian Center for Astrophysics, performing measurements of atomic and molecular parameters at ultraviolet wavelengths for analyses of astronomical spectra, designing/calibrating instruments for ultraviolet spectral radiometric measurements of the Sun from satellites, and helping manage instrument development for the SDO satellite. He failed at retirement and spent 2013–2016 helping manage development of an instrument for the Parker Solar Probe.
Part D Scattering Theory
684
Part D collects together the topics and approaches used in scattering theory. A handy compendium of equations, formulae, and expressions for the classical, quantal, and semiclassical approaches to elastic scattering is given; reactive systems and model potentials are also considered. The dependence of scattering processes on the angular orientation of the reactants and products is discussed through the analysis of scattering experiments, which probe atomic collision theories at a fundamental level. The detailed quantum mechanical techniques available to perform accurate calculations of scattering cross sections from first principles are presented. The theory of elastic, inelastic, and ionizing collisions of electrons with atoms and atomic ions is covered and then extended to include collisions with molecules. The powerful methods of quantum defect theory are discussed and applied to low-energy scattering processes, including multichannel processes. The standard scattering theory for electrons is extended to include positron collisions with atomic and molecular systems. Slow collisions of atoms or molecules within the adiabatic approximation are discussed; important deviations from this model are presented in some detail for the low energy case. Applications to collision processes involving Rydberg atoms in ultracold gases are particularly highlighted. The main methods in the theoretical treatment of ion–atom and atom–atom collisions are summarized with a focus on intermediate and high collision velocities. The molecular structure and collision dynamics
involved in ion–atom charge exchange reactions is studied. Both the perturbative and variational capture theories of the continuum distorted wave model are presented. The Wannier theory for threshold ionization is then developed. Studies of the energy and angular distribution of electrons ejected by the impact of high-velocity atomic or ionic projectiles on atomic targets are overviewed. A useful collection of formulae, expressions, and specific equations that cover the various approaches to electron–ion and ion–ion recombination processes is given. A basic theoretical formulation of dielectronic recombination is described, and its importance in the interpretation of plasma spectral emission is presented. Many of the equations used to study theoretically the collisional properties of both charged and neutral particles with atoms and molecules in Rydberg states are collected together; the primary approximations considered are the impulse approximation, the binary encounter approximation, and the Born approximation. The Thomas mass-transfer process is considered from both a classical and a quantal perspective. Additional features of this process are also discussed. The theoretical background, region of validity, and applications of the classical trajectory Monte Carlo method are then delineated. One-photon processes are discussed, and aspects of line broadening directly related to collisions between an emitting, or absorbing, atom and an electron, a neutral atom, or an atomic ion are considered.
Classical, Quantal, and Semiclassical Propagators and Applications to Elastic Scattering
47
Jan-Michael Rost
Contents 47.1
What Is Semiclassics? . . . . . . . . . . . . . . . . . . . 685
47.2 47.2.1 47.2.2 47.2.3
Quantum, Classical, and Semiclassical Propagators The Quantum Propagator as a Feynman Path Integral . The Classical Propagator . . . . . . . . . . . . . . . . . . The Semiclassical Propagator . . . . . . . . . . . . . . .
47.3
Advantages and Disadvantages of Semiclassics . . . 689
47.4 47.4.1 47.4.2
Applications to Elastic Scattering . . . . . . . . . . . . 689 Central Field . . . . . . . . . . . . . . . . . . . . . . . . . . 689 Center-of-Mass to Laboratory Coordinate Conversion 689
47.5 47.5.1 47.5.2 47.5.3 47.5.4 47.5.5 47.5.6 47.5.7
Quantal Elastic Scattering . . . . . . . . . Scattering Amplitude from the Propagator . Partial Wave Expansion . . . . . . . . . . . . Phase Shift and Cross Sections . . . . . . . . Identical Particles: Symmetry Oscillations . Low-Energy E ! 0 Scattering . . . . . . . . Nonspherical Potentials . . . . . . . . . . . . Reactive Systems . . . . . . . . . . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
690 690 690 691 692 693 695 695
47.6 47.6.1 47.6.2 47.6.3 47.6.4 47.6.5 47.6.6
Classical Elastic Scattering . . . . . . . . . . . . . Elastic Scattering Cross Section . . . . . . . . . . . Deflection Functions . . . . . . . . . . . . . . . . . . Glory and Rainbow Scattering . . . . . . . . . . . . Orbiting and Spiraling Collisions . . . . . . . . . . Deflection Function and Time Delay from Action Approximate Actions . . . . . . . . . . . . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
696 696 696 697 698 698 698
47.7 47.7.1 47.7.2 47.7.3 47.7.4 47.7.5 47.7.6 47.7.7 47.7.8
Semiclassical Elastic Scattering . . . . . . . . . . . . Semiclassical Amplitudes and Cross Sections . . . . . Diffraction and Glory Amplitudes . . . . . . . . . . . . Small-Angle (Diffraction) Scattering . . . . . . . . . . Small-Angle (Glory) Scattering . . . . . . . . . . . . . Oscillations in Elastic Scattering . . . . . . . . . . . . . Scattering Amplitude in Poisson Sum Representation Semiclassical Phase Shifts . . . . . . . . . . . . . . . . Semiclassical Amplitudes: Integral Representation .
. . . . . . . . .
698 698 700 701 701 702 702 702 704
. . . . . . . .
. . . . . . . .
. . . . . . . .
686 686 687 687
47.8 47.8.1 47.8.2 47.8.3 47.8.4
Coulomb Elastic Scattering . . . . . . . . . . . . . . . . Quantum Phase Shift, Rutherford and Mott Scattering Classical Coulomb Scattering . . . . . . . . . . . . . . . Semiclassical Coulomb Scattering . . . . . . . . . . . . . Born Approximation for Coulomb Scattering . . . . . .
47.9 47.9.1 47.9.2
Results for Model Potentials . . . . . . . . . . . . . . . 707 Classical Deflection Functions and Cross Sections . . . 707 Amplitudes and Cross Sections in Born Approximation 710
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
Abstract
Semiclassical methods predate even full quantum mechanics starting with Bohr’s atom model. Semiclassical propagators have been developed continuously in chemical physics ever since Miller’s seminal classical S-matrix. Semiclassics was revived with the success of periodic orbit quantization and chaotic dynamics entering the quantum domain. Today, semiclassical and even classical concepts are accepted to describe microscopic phenomena, from protein folding over plasma physics all the way to atoms in strong laser fields. Thereby, semiclassical approximations are used for computation as well as for physical insight. Explicit formulae are given for elastic scattering, always covering quantum, classical, and semiclassical expressions. Keywords
elastic scattering integral cross section transport cross section deflection function quasi-bound state semiclassical propagator quantum propagator
47.1 J.-M. Rost () Max Planck Institute for the Physics of Complex Systems Dresden, Germany e-mail: [email protected]
705 705 706 706 706
What Is Semiclassics?
Broadly speaking, semiclassics is asymptotic quantum theory taking „ as a small parameter and is, therefore, an approximation between full quantum mechanics and clas-
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_47
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686
sical mechanics. The latter has always played an important role for molecular dynamics to describe molecular reactions and spectra. More recently, it has been widely applied also in electronic problems, most prominently when bound electrons move under the influence of intense laser pulses and are eventually ionized. Hence, we will briefly provide the quantum (finite „) and the classical propagator („ D 0) before introducing the standard semiclassical propagator („ 1). What does a reader expect from a Handbook of AMO Physics chapter in the age of limitless internet access? If one is only vaguely familiar with a topic, e.g., semiclassics, one does not know how things are related and what may matter – my intention is to provide a guideline for this situation and some recipes. Newcomers may also not know which are the relevant methods, mechanisms and therefore key words, suitable to access additional resources from the internet. For this reason, we will put such phrases that are important in the context of semiclassics in italics in the following. Clearly, even for providing an overview (not a review!) the topics covered by semiclassics are by far too many. Hence, we do restrict ourselves to a subjective selection relevant for elastic scattering. The methods introduced are influenced by few-body physics, although most concepts are not restricted in the number of degrees of freedom (DoF). For the same reason, only nonrelativistic real-time propagation will be addressed, while imaginary time propagation describing thermal ensembles will not be covered, although it is treatable semiclassically as well. Furthermore, we will not touch semiclassical approximations to second quantization or many-body techniques even though there has been interesting progress. There are, of course, excellent monographs about semiclassics; a recent one being [1] with a time-dependent phase space perspective and intuitive arguments, written by Eric Heller, who has been deeply involved in chemical and atomic physics as well as in chaotic dynamics and billiards. A bit older and maybe easier to approach is the monograph [2], which is based on Matthias Brack’s research interest and provides an additional aspect related to nuclear and cluster physics. A real treat (and not so difficult to read) is Martin Gutzwiller’s book [3], and a more mathematical account but well approachable for physicists is provided by Alfredo Ozorio Alemeida in [4]. An original project lead by Predrag Cvitanovic is the online “chaos book”, with part V being relevant for semiclassics [5]. Out of the many reviews, we would like to point out two “classic” ones, namely that about elastic scattering by Berry and Mount [6], and arguably the foundation for the useful application of semiclassical propagators for quantum dynamics, Bill Miller’s review of his own work on the “classical S-matrix”, which is, in fact, semiclassical [7]. Directly relevant in particular for the first sections of this chapter is [8].
J.-M. Rost
47.2 Quantum, Classical, and Semiclassical Propagators 47.2.1
The Quantum Propagator as a Feynman Path Integral
The most general mathematical object describing evolution in the time domain is the propagator, iH„ t
0
K.q; q ; t/ hqj e
Z 0
jq i D
DŒq t e
iS Œq t „
;
(47.1)
where for simplicity we restrict ourselves to timeindependent Hamiltonians H . Semiclassically, any timedependent Hamiltonian can be turned by a formal extension of phase space into a time-independent one. Formulated as a Feynman path integral, the propagator contains all paths q t leading from q 0 to q in time t, where q is a state (often position or momentum) for an arbitrary number of degrees of freedom F , and the action S is the integral over the classical Lagrangian while the paths themselves are not classical in general [9]. Arguably, the path integral is not a very transparent expression. Moreover, it is difficult to compute. This is one motivation to resort to a semiclassical approximation to be introduced in Sect. 47.2.3. Analytical quantum propagators exist for a few problems only. The commonly known ones are those for the free particle of mass m and initial and final spatial coordinates x 0 and x, respectively, .F /
K0 .x; x 0 ; t/ D
m F2 m.x x 0 /2 (47.2a) exp 2 i„t 2i„t
and the harmonic oscillator with frequency ! F=2 m! .F / KHO .x; x 0 ; t/ D 2 i„ sin !t m!..x 2 C x 02 / cos !t 2xx 0 / exp : 2i„ sin !t (47.2b) The corresponding propagators in momentum space read .F / K0 .p; p 0 ; t/
itp 2 D ı.p p / exp 2m„ 0
(47.2c)
and for the harmonic oscillator
F=2 1 D 2 i„m! sin !t 2 .p C p 02 / cos !t 2pp0 exp ; 2i„m! sin !t
.F / KHO .p; p 0 ; t/
(47.2d)
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function. The quite transparent interpretation of the classical probability density equation (47.3) is with the help of Eqs. (47.4a) and (47.4b) a sum over individual probability densities for all classical trajectories that connect q and q 0 in time t. To obtain the normalization we ask “What is the probability of finding the system at time t D 0 with a moF Y mentum p 0 in the neighborhood p 0 , if it is located within K1 .xj ; xj0 ; t/ : (47.2e) q 0 of q 0 ?” We know that the system is in this neighborhood KF .x; x 0 ; t/ D j D1 for sure, and since the phase space integral in Eq. (47.3) is trivially fulfilled, we get This also implies, of course, that N particles in d dimensions 1 are simply described by F D 2Nd DoF. (47.5a) 1 D P .p 0 ; q 0 / p 0 q 0 D p 0 q 0 ; There are other scarce analytical results, mostly quite in volved and requiring an algebraic computer manipulation which means D p 0 q 0 , the phase space volume the tool to handle them, such as, e.g., for the Coulomb interac- system occupies initially (or more precisely at the time, tion [10]. the integration dp 0 dq 0 is performed). We can ask the same where the latter is easily obtained from Eq. (47.2b) through the equivalence x $ p=.m!/. Since propagation in each degree of freedom for both problems is independent, the propagator for F degrees of freedom is simply the product, e.g., in coordinate space
47.2.2
The Classical Propagator
Classically, densities (not amplitudes) and corresponding observables of a system are evolved in time. The quantum expression that answers the question “What is the probability of finding a system after time t at q given that it was initially at q 0 ” is answered by the probability density P .q; q 0 ; t/ D jK.q; q 0 ; t/j2 . Classically, it has the intuitive form of a correlation function over phase space, 1 Pcl .q; q ; t/ D 0
Z dq 0 dp 0 ı.q q t /ı.q 0 q0 / ;
(47.3)
where q t D q.t/ is a variable evolving under the system dynamics, and is a normalization constant. The phase space integral can, in principle, be taken at any time; here, we carry it out with the conjugate phase space variables at time t D 0, since then the integration over q 0 is trivial. For the integral over p 0 , one has to find those initial momenta pj0 that fulfil the remaining ı-function and the integral reduces to the sum over trajectories
question quantum mechanically: “What is the probability of finding momentum jp 0 i for a system in state jq 0 i?” Since there is nothing to propagate, the answer is ˇ ˇ ˇ eip0 q 0 =„ ˇ2 1 ˇ ˇ 0 0 2 0 0 2 jK.p ; q ; 0/j D jhp jq ij D ˇ D ; F ˇ ˇ .2 „/ 2 ˇ .2 „/F (47.5b) for F degrees of freedom. A comparison with Eq. (47.5a) reveals that the classical normalization compatible with the quantum result is F D .2 „/F , for F DoF. Obviously, this is a reasonable normalization, and if one wants to describe microscopic systems classically, it must be emphasized that this is a judicial choice. By no means does the classical propagator Eq. (47.3) contain an „-dependence per se.
47.2.3 The Semiclassical Propagator
Essentially an asymptotic expansion of the Feynman path integral Eq. (47.1) to first order in „, the semiclassical propagator is more subtle to formulate. The decisive step is the reduction of paths. It is justified by the argument of stationary phase: the phase SŒq t =„ varies substantially for X „ 1 for different paths connecting q and q 0 , rendering 0 Pcl .q; q ; t/ D Pj (47.4a) their added contribution marginal. The surviving paths have j an action that is stationary under variation of the paths, ˇ ˇ 1 ˇˇ @q t .p 0 ; q 0 / ˇˇ1 Pj D ; (47.4b) ıSŒq t =ıq t D 0. Since this stationarity condition defines the ˇ 0 0 ˇ @p 0 classical equations of motion, the surviving paths are exactly q t .pj ;q /Dq the trajectories following Newtonian dynamics. Hence, the where each trajectory contributes the weight equation stationary phase argument for small „ reduces the path inte(47.4b). Here, and in the following, ŒM D det M if M is gral to a sum over the trajectories. a matrix. This expression holds again in arbitrary dimensions, i.e., q and p can be vectors, and the absolute value The van Vleck–Gutzwiller (VG) propagator reads of the determinant from @q=@p 0 , the Jacobi matrix, measures X 1 ij iSj .q; q 0 ; t/ 0 2 the phase space volume around the trajectory j . The funcPj exp C ; Ksc .q; q ; t/ D 2 „ tion q D q t .p 0 ; q 0 /, and generally a set of final variables j as a function of the initial variables 0 , is called a deflection (47.6)
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where Pj is the classical probability density from Eq. (47.4b) with the normalization D .2 „/F for F degrees of freedom, as derived in Eqs. (47.5a) and (47.5b). At a first glance, the semiclassical propagator has the simple and intuitive form of a sum over classical trajectories whose weights are now complex amplitudes formed by the square root of the classical probability and multiplied by a phase factor given by the classical action of the trajectory, as originally suggested already in 1928 by van Vleck [11]. The truth is more complicated, as the semiclassical propagator retains a memory about the topology of the paths contributing between q and q 0 . At certain points in time, the trajectory bundle around traj. j may not span the full local phase space volume jŒ@q.t/=@pj0 j due to a focal point or a caustic. Propagating the trajectory across such a point where the determinant is singular may change the sign of Œ@q.t/=@pj0 . This is formally encoded in the Maslov phase j counting the number of sign changes of the eigenvalues of the Jacobi matrix in time along trajectory j , as pointed out by Gutzwiller only in 1967 [12]. Two features make the van Vleck–Gutzwiller propagator difficult to compute. To determine j is cumbersome and, also, to find the initial momenta p 0 that allow a trajectory to start at q 0 and end at q in time t requires numerically expensive root searches. An alternative formulation in phase space is particular suitable for numerical propagation.
The actual form of the probability density depends on a width parameter , which determines the admixture of the different blocks Mxy in the determinant R .p t ; q t / 12 1 1 2 Mqq C Mpp i „Mqp 2 Mpq D : 2 i „ (47.11)
Although this semiclassical propagator is not uniquely defined through its dependence on a suitable chosen parameter , it is exactly the feature that provides approximate global uniformization since at a caustic, R always remains finite, which is an important advantage over other forms. Secondly, as was already pointed out, one does not have to keep track of Maslov indices. Instead, one has to make R continuous as the radicant crosses the branch cut. Finally, the root search has been circumvented. All semiclassical propagators with this property use the initial value representation (IVR), i.e., the integral is over phase space coordinates at a single point in time (typically the initial time) only. By taking ! 1 or ! 0, one obtains the semiclassical IVR propagators in position or momentum space, respectively. Expressions take a particularly simple and appealing form if the propagator Eq. (47.7) is used in conjunction with waveThe Hermann Kluk propagator is given [13] as an integral functions that contain the Gaussians with the same width . over the initial conditions .p0 ; q0 / of the trajectories .q t ; p t /, The correlation function c.t/ D h j exp.iH=„/j i probing propagated under a classical Hamiltonian H , the spectrum of H is a general example. “ 1 K .q; q 0 ; t/ D d F q0 d F p0 R .p t ; q t / The semiclassical correlation function csc using .2 „/F Eq. (47.7) with a normalized Gaussian D . 2 = /3=4
i f .q; q 0 ; p 0 / reads exp S.p t ; q t / f .q; q t ; p t /f q 0 ; q0 ; p0 : „ (47.7) (47.12) csc .t/ D h jK .t/j i “ 1 Each trajectory is the center of a Gaussian D d F q0 d F p0 R.p t ; q t / F .2 „/ 2 i 0 i 0 0 0 2 0 f .q; q ; p / D exp .q q / C p .q q / : ; q / exp S.p t t 2 „ „
(47.8) g q0 ; p0 ; q 0 ; p 0 g q t ; p t ; q 0 ; p 0 ; (47.13) R The action S D .p qP H / dt accumulated along the trawhere jectory enters Eq. (47.7) as well as the probability density of 2 each trajectory R .p t ; q t /, which contains all four blocks
1 0 0 g q; p; q ; p D exp .q q 0 /2 2 2 .p p 0 /2 4 4 „ @x t ; (47.9) Mxy .t; 0/ D i 0 0 @y0 C .p C p /.q q / : (47.14) 2„ 2 RF RF of the monodromy matrix ! ! ! ıq t Mqq Mqp ıq0 D : ıp t Mpq Mpp ıp0
The integral in Eq. (47.12) is again over phase space representing the initial conditions .q0 ; p0 / of the trajectories (47.10) .q t ; p t / and is calculated in practice by Monte Carlo integration. The number of sampling points (trajectories to be run)
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to achieve convergence depends on the initial wave packet In particular, fragmentation into several particles is ideand the part of the spectrum of H covered by it through the ally suited for classical and semiclassical treatments since parameters .; p 0 ; q 0 /. one can follow the particle trajectories numerically to their asymptotic momenta and simply bin them into finite interAnalytical semiclassical propagators. For interactions up vals to arrive at a cross section. Analytical expressions that to quadratic order (constant, linear, and harmonic potentials) also give an idea about the singularities one encounters in hV 0 .x/i D V 0 .hxi/ holds, and, therefore, the quantum system classical and semiclassical scattering can be best exemplified observables can, indeed, be described with classical trajecto- with elastic scattering. A more exhaustive account of elastic ries. This is a consequence of the Ehrenfest theorem, which scattering is given in previous editions of the Handbook. casts quantum observables into a form similar to Hamilton’s equations dhxi D hpi ; m dt
dhpi D hV 0 .x/i : dt
47.4
Applications to Elastic Scattering
(47.15)
Elastic scattering is governed by the Hamiltonian H D 2 Hence, the quantum propagators for the free particle and p =.2m/ C V .r/ in the center-of-mass frame for the relative the harmonic oscillator in arbitrary dimensions Eqs. (47.2a)– motion of two collision partners with reduced mass m. (47.2e) are identical to the corresponding semiclassical propagators.
47.4.1
47.3
Advantages and Disadvantages of Semiclassics
Advantages of Semiclassics:
Central Field
For a conservative (real) potential V .r/ depending only on the distance r from the scattering center O, the total energy E > 0 and orbital the angular momentum L are conserved. Due to the different mathematical structures classical and quantum theory take advantage of this effect in quite different ways. Classically, the deflection function mapping the impact parameter b to the scattering angle in the scattering plane characterizes elastic scattering. Quantum mechanically, the phase shift for each angular momentum component is relevant, as it describes the change of the corresponding partial wave due to scattering. Alternatively, initial (vectorial) momentum p 0 is converted into final momentum p through scattering, and the ensuing scalar product p 0 p D 2mE cos encloses the scattering angle . This description can be applied in quantum and classical mechanics.
Relies on classical mechanics only but describes quantum interference
Reveals dominant structure of a quantum problem (via stationary phase)
Is exact for interactions up to quadratic order (Ehrenfest theorem)
Excellent approximation if dynamics evolves through many degrees of freedom
Formally offers a quantum formulation (complex amplitudes) and can, therefore, be seamlessly blended in approximation containing other quantum parts
Offers many possibilities for additional approximations
Is straightforward to compute. 47.4.2
Center-of-Mass to Laboratory Coordinate Conversion
Disadvantages of Semiclassics: Let 1 , 2 be the angles for scattering and recoil, respec “Hard” quantum properties, such as tunnelling, diffraction tively, of the projectile by a target initially at rest in the lab and entanglement, are not described. frame. Then,
While straightforward to code, the numerical effort may be exponential due to the nonlinearity of classical me1 . 1 /d˝1 D 2 . 2 /d˝2 chanics. .1;2/ D cm ./d˝cm ; 0 I
Instabilities and serious errors can occur at caustics, when .2/ .1/ .; / D cm . ; C / : (47.16) cm two stationary phase points (trajectories) coalesce. Uniformization (higher-order correction) is then necessary, which is tedious and often does not yield an improvement (A) Elastic scattering of two bodies with masses m1 and m2 without conversion of translational kinetic energy into worth the effort.
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in terms of the complex scattering amplitude f .E; / contained in the wavefunction
internal energy: .1/ C .2/ ! .1/ C .2/.
1 C 2x cos C x 2 1 . 1 / D cm ./ j1 C x cos j ˇ ˇ ˇ 1 ˇˇ ˇ 2 . 2 / D cm ./ˇ4 sin ˇ I 2 2
tan
1
3=2
1 .r/ ! exp.ikz/ C f ./ exp.ikr/ r as r ! 1.
I
1 1 . / ; 0 2 I 2 2 sin D ; x D m1 =m2 : .x C cos / D
m1 > m2 : As 0 c D cos1 .m2 =m1 /, 0
1
!
max 1
D sin1 .m2 =m1 /
r
v` .r/ D0 cos ` F` .kr/ C sin ` G` .kr/
! sin kr 12 ` C ` as r ! 1 :
(47.25) 47.5.3
Hence, v` .r/ is expressed in terms of the regular and irregular solution of the free radial Schrödinger equation (V D 0) containing the spherical Bessel j` and Neumann n` functions, respectively,
Phase Shift and Cross Sections
The scattering amplitude in terms of partial waves ` reads 1 1 X .2` C 1/Œexp.2i` / 1P` .cos / f ./ D 2ik `D0
D
F` .kr/ D .kr/j` .kr/ ( .kr/`C1 =.2` C 1/ŠŠ ; r ! 0
! sin kr 12 ` ; r !1;
1 X
f` ./ :
(47.33a)
`D0
(47.26) The phase shifts ` can be determined from the numerical solution of the radial Schrödinger equation Eq. (47.24). G` .kr/ D .kr/n` .kr/ The differential cross section (DCS) can be reexpanded in ( ` Legendre polynomials P` .cos /, .2` 1/ŠŠ=.kr/ ; r ! 0
(47.27) ! cos kr 12 ` ; r ! 1 : 1 d .E; / 1 X aL .E/PL .cos / ; (47.34a) D 2 d˝ k LD0 The standard asymptotic scattering solution Eq. (47.25) 1 `CL X X contains the effect of scattering through the phase shift ` a D .2` C 1/.2`0 C 1/ L in the regular standing wave. By changing the normalization `D0 `0 Dj`Lj of A` one can express this effect in many different ways. .``0 00 j ``0 L0/2 The most common ones are to add to the regular standing wave either an irregular standing wave of real amplitude K` sin ` sin `0 cos.` `0 / ; (47.34b) in Eq. (47.30) or, a spherical outgoing wave of amplitude T` in Eq. (47.31). Another possibility is to convert in Eq. (47.32) where .``0 mm0 j ``0 LM / are Clebsch–Gordan coefficients. the incoming spherical wave of unit amplitude to an outgoing spherical wave of amplitude S` . The different amplitudes The integral cross section simplifies considerably since are related as integration over the angle reveals the orthogonality of the partial waves, S` .k/ D exp.2i` / ;
(47.28a)
T` .k/ D sin ` exp.i` / ;
(47.28b)
K` .k/ D tan ` :
(47.28c)
Z 2
.E/ D 2
jf ./j d.cos / D 0
1 X `D0
` .E/ ;
(47.35a)
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ˇ2 ˇ1 ˇ 1 ˇˇX ˇ IA,S ./ D ! .2` C 1/Œexp 2i 1P .cos / ˇ ; ˇ ` ` ` ˇ 4k 2 ˇ
with the partial cross sections 4 X .2` C 1/ sin2 ` : k2 1
` .E/ D
`D0
(47.35b)
(47.41c)
`D0
A,S .E/ D
Optical theorem .E/ D .4 =k/ ImŒf .0/ :
!` .2` C 1/ sin2 ` ;
(47.41d)
`D0
(47.36) where A and S denote antisymmetric and symmetric wavefunctions (with respect to particle interchange) for collisions of identical particles with odd and even total spin St
S , P wave (` D 0; 1) net contribution
1˚ d D 2 sin2 0 C Œ6 sin 0 sin 1 cos.1 0 / d˝ k cos C 9 sin2 1 cos2 ; (47.37) .E/ D
4 k2
1 X
4 2 sin 0 C 3 sin2 1 : k2
(47.38)
A:
St odd
!` D 0 .` even/ I
!` D 2 .` odd/ I
S:
St even
!` D 2 .` even/ I
!` D 0 .` odd/ :
Spin states S t unresolved S/A combination
I./ D gA IA ./ C gS IS ./ ; (47.42a) For pure S-wave scattering, the DCS is isotropic. For pure .E/ D gA A .E/ C gS S .E/ ; (47.42b) P -wave scattering, the DCS is symmetric about D =2, where it vanishes; the DCS rises to equal maxima at D 0; . where gA and gS are the fractions of states with odd and even For combined S and P -wave scattering, the DCS is asym- total spins St D 0; 1; 2; : : : ; 2s. For fermions (F) with half metric with forward-backward asymmetry. integer spin s and bosons (B) with integral spin s, Collision integrals. Averages of .n/ .E/ a Maxwellian distribution at temperature T are
˝
.n;s/
sC2 1
over
exp.E=kT /E
dE :
gS D .s C 1/=.2s C 1/ ;
so that Eqs. (47.42a) and (47.42b) have the alternative forms
.n/ .E/ 0
sC1
gA D .s C 1/=.2s C 1/ ; gS D s=.2s C 1/ ;
B: gA D s=.2s C 1/ ;
Z1
.T / D .s C 1/Š.kT /
F:
(47.39)
Normalization factors are chosen so that the above expressions for .n/ and ˝ .n;s/ reduce to d 2 for classical rigid spheres of diameter d .
I.F/ D jf ./j2 C jf . /j2 I ; 1 1 .F/ D ŒS C A ŒS A =.2s C 1/ ; 2 2 I.B/ D jf ./j2 C jf . /j2 C I ; 1 1 .B/ D ŒS C A C ŒS A =.2s C 1/ ; 2 2
Mobility. The mobility K of ions of charge e in a gas of where the interference term is density N is given by the Chapman–Enskog formula 2 ID ReŒf ./f . / : 1 2s C 1 3e 1=2 .1;1/ .T / : (47.40) ˝ KD 8N 2M kT Example: for fermions with spin 1=2, " 1 2 X 47.5.4 Identical Particles: Symmetry Oscillations .2` C 1/ sin2 ` .E/ D 2 k `Deven # Colliding particles, each with spin s, in a total spin S t 1 X 2 resolved state in the range .0 ! 2s/ Particle interchange: C3 .2` C 1/ sin ` : .r/ D .1/St .r/ `Dodd 1 jf ./ f . /j2 ; 2 A,S .r/ ! Œexp.ikZ/ exp.ikZ/ 1 C Œf ./ f . / exp.ikr/ ; R IA,S ./ D
(47.43a) (47.43b) (47.43c) (47.43d)
(47.43e)
(47.44)
Symmetry oscillations originate from the interference between unscattered incident particles in the forward ( D 0) direction and backward scattered particles ( D ; ` D 0). Symmetry oscillations are sensitive to the repulsive wall of (47.41b) the interaction. (47.41a)
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If an ` D 0 bound level of energy En D „2 kn2 =2m is sufficiently close to the dissociation limit, the effective range 1
X and scattering lengths, re and as , respectively, are related by (47.45) .2` C 1/ sin2 s` t` ; ST .E/ D 2 k 1 1 `D0 (47.49) D kn C re kn2 C as 2 S,T where ` are the phase shifts for individual scattering by Wigner causality condition If ` provides the dominant the singlet and triplet potentials, respectively. contribution to f ./, then @` .k/ (47.50) as ; 47.5.5 Low-Energy E ! 0 Scattering @k where as is the scattering length (` D 0) and is a measure of Blatt–Jackson effective range formula For short-range the range of the interaction. potentials, Effective range formulae The Blatt–Jackson formula must
1 1 k cot 0 D C re k 2 C O k 4 (47.46a) be modified [14–16] for long-range interactions as follows. as 2 Singlet–triplet spin flip cross section
(1) Coulomb tail: V .r/ Z1 Z2 e 2 =r 1 2.K=a0 / D .1=as / C re k 2 Z1
2 2 (47.46b) u0 .r/ v02 .r/ dr ; re D 2 cot 0 K D 2 ˛ ln ˇ 0:5772 0 e 1 1 X
2 1 where u0 D sin.kr C 0 /= sin 0 is the k D 0 limit of the ; (47.51) n n C ˇ2 C ˇ2 potential-free ` D 0 radial wavefunction and normalized so nD1 that u0 .r/ goes to unity as k ! 0. The potential distorted where ˇ D Z1 Z2 e 2 =„v D Z1 Z2 =.ka0 / and ˛ D ` D 0 radial function v0 is normalized at large r to u0 .r/. Z1 Z2 e 2 =.2Ere /. The effective range is a measure of the distance over which v0 differs from u0 . The outside factor of 2 in Eq. (47.46b) is (2) Polarization: V .r/ D ˛ e 2 =2r 4 , d chosen such that re D a for a square well of range a. 4 tan 0 D as k C4 k 2 C4 as k 3 ln.ka0 / 3 3 The elastic cross section for k ! 0 can be expressed as C Dk 3 C F k 4 ; (47.52a) 4 2 .k ! 0/ D 2 sin 0 tan 1 D (47.52b) C4 k 2 as.1/ k 3 ; k 15
1 2 C4 k 2 D 4 as2 1 12 k 2 as re C k 2 as2 (47.47a) tan ` D .2` C 3/.2` C 1/.2` 1/
! 4 as2 1 C as k 2 .re as / ; (47.47b) C O k 2`C1 ; (47.52c) with effective range
with the
for ` > 1, where
˛d m 2m ˛d e 2 D : 2 „ 2 a0 me (47.47c) Example: e –Ar low energy collisions. The values D D 68:93a03 as D 1:459a0 I C4 D
scattering length as D lim f ./ : k!0
Relation with Bound Levels
as.1/ D 8:69a03 I
F D 97a04
Levinson’s theorem A radial potential V .r/ can support allow for an accurate fit to measurements [17]. n` bound states of angular momentum ` and energy En , such (3) Van der Waals: V .r/ D C =r 6 that ( 2mC 1 1 2 k3 k cot 0 D C re k n0 ; En < 0 as 2 15as2 „2 (47.48a) lim 0 .k/ D
k!0 n0 C 12 ; EnC1 D 0 ;
2mC 4 k 4 ln.ka0 / C O k 4 : (47.53) 2 (47.48b) lim ` .k/ D n` ; ` > 0 : 15as „ k!0
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e–Atom collisions with polarization attraction As k ! 0, Repulsion (C): with 2 D 2mC =„2 , the differential cross section is 2=.n2/ n 3 n 1 = : (47.60) as.C/ D d 8 C4 n2 n2 n2 2 2 k sin C C4 k ln.ka0 / C ; D as 1 C d˝ as 2 3 Attraction (): with n D =.n 2/, (47.54)
1 as./ D as.C/ 1 tan n tan SC 0 2 n cos n ; (47.61) and the elastic and diffusion cross sections are Z1 1n=2 2r0 SC 2 C4 k 0 D r n=2 dr D ; (47.62) 2 .k ! 0/ D 4 as 1 C n2 3as rm ./ .C/ 8 i D a cos n : (47.63) ha 2 s s C C4 k ln.ka0 / C ; (47.55) 3 Number of bound states 4 C4 k d .k ! 0/ D 4 as2 1 C 1 SC 1 5as N D int .n 1/ C1; (47.64) b n 0 2 8 2 C C4 k ln.ka0 / C : (47.56) 3 where int(x) denotes the largest integer of the real argument ./ x. For integer x, as is infinite, and a new bound state ap For e -noble gas collisions, the scattering lengths are pears at zero energy. He Ne Ar Kr Xe as .a0 / 1:19 0:24 1:459 3:7 6:5
Born approximation for phase shifts Z1
D k U.r/Œj` .kr/2 r 2 dr : For atoms with as < 0, a Ramsauer–Townsend minimum appears in both and d at low energies, provided that scat0 tering from higher partial waves can be neglected, because For D .` C 1=2/ ka, substitute from Eq. (47.52a), 0 ' 0 at k D 3as = C4 . E D 1=2 1
: k 2 r 2 Œj` .kr/2 D 1 2 =k 2 r 2 Semiclassical scattering lengths. In the limit E ! 0, the 2 semiclassical phase shift tends to Born S-wave phase shift 1=2 Z1 2m Z1 jV .r/j1=2 dr ; (47.57) SC 1 0 D B 2 „ tan 0 .k/ D U.r/ sin2 .kr/dr : k rm tan B` .k/
(47.65)
(47.66)
(47.67)
0
where rm is the turning point. Equation (47.57) can be used e˛r U0 , (ii) U D
to calculate some scattering lengths analytically and estimate Examples: (i) U D U0 ; 2 2 2 r C r r 0 the number of bound states in the potential [16]: U0
(i) tan B0 D ln 1 C 4k 2 =˛ 2 ; (a) Hard-core + well 4k 8 U0 B 2kr0 D / e 1 .1 C 2kr : (ii) tan ˆ 1; r < r 0 0 0 3 < 4kr0 V .r/ D V0 ; r0 r < r1 ˆ : Born phase shifts (large `) For ` ka, 0; r1 < r ;
Z1 (47.58a) as D 1 tan SC k 2`C1 0 =.k0 r1 / r1 ; B tan ` D U.r/r 2`C2 dr ; k02 D 2mV0 =„2 : (47.58b) SC Œ.2` C 1/ŠŠ2 0 D k0 .r1 rm / ;
(47.68) (47.69)
(47.70)
0
The phase-averaged scattering length is has i D r1 . (b) Hard-core + power law (n > 2) ( 1; r < r0 V .r/ D ˙C =r n ; r > r0 :
valid only for finite range interactions U.r > a/ D 0. Example: U D U0 , r a and U D 0, r > a. tan B` .` ka/ D U0 a2 (47.59)
.ka/2`C1 Œ.2` C 1/ŠŠ2 .2` C 3/
For ` ka, `C1 =` .ka=2`/2 .
:
(47.71)
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Classical, Quantal, and Semiclassical Propagators and Applications to Elastic Scattering
47.5.6
Nonspherical Potentials
The plane wave scattering states are k .r/ D exp.ik r/ D k .r/ ;
47.5.7 Reactive Systems All nonelastic processes (e.g., inelastic scattering and rearrangement collisions/chemical reactions) can be viewed as (47.72) a net absorption from the incident beam current vector J and modeled by a complex optical potential
and the full scattering solutions have the form .˙/
k .r/ k .r/ C
f .k; k0 / exp.˙ikr/ ; r
695
V .r/ D Vr .r/ C iVi .r/ :
(47.80)
(47.73a) The continuity equation is then 2 r J D Vi .r/j .r/j2 ; „
where the scattering amplitude is
(47.81)
E 1 D .C/ k0 .r/ j U.r/ j k .r/ : 4
(47.73b) so that particle absorption implies Vi > 0 and particle creation Vi < 0. Since particle conservation implies jS` j2 D 1, the phase shift Close to analytical solutions can typically only be obtained in the following. ı` .k/ D ` .k/ C i` .k/ (47.82a) f .k; k0 / D
Born approximation: Set kC D k in Eq. (47.73b). Then is also complex. Then fB .K/ D
1 4
Z U.r/ exp.iK r/dr ;
S` D A` .k/ exp.2i` / ;
(47.74)
(47.82b)
where the absorption (inelasticity) factor is with momentum transfer K D k k0 , and K D 2k sin 12 . For a symmetric potential, Z fB .K/ D
sin kr U.r/r 2 dr kr
(47.75)
A` D exp.2` / 1 :
The elastic, absorption, and total cross sections read fel ./ D
is consistent with Eq. (47.65) since
(47.82c)
1
1 X .2` C 1/ A` e2i` 1 P` .cos / ; 2ik `D0
X sin kr .2` C 1/Œj` .kr/2 P` .cos / : D kr
(47.83a)
1
(47.76) el .k/ D
`D0
The static e -atom scattering potential and Born scattering amplitude are Z j .r0 /j2 dr0 Ze 2 2 Ce ; V .r/ D r jr r0 j 2me 2 ŒZ F .K/ ; fB .K/ D „2 K2
(47.83b)
.2` C 1/ 1 A2` ;
(47.83c)
`D0
(47.83d)
(47.77) Upper limits to the partial cross sections are (47.78)
Z j .r/j2 exp.iK r/dr :
k2
`D0 1 X
.2` C 1/jA` e2i` 1j2 ;
tot .k/ D el .k/ C abs .k/ :
where the elastic form factor is F .K/ D
abs .k/ D
k2
1 X
(47.79)
4 .2` C 1/; `abs 2 .2` C 1/ ; 2 k k 4 4 el 2 .2` C 1/ D Im f` . D 0/ : k k
`el
(47.84a)
`tot
(47.84b)
For pure elastic scattering with no absorption, A` D 1. All nonelastic processes (0 A` < 1) are always accompanied For a pure Coulomb field, F .K/ D 0, and B .; E/ D by elastic scattering, even in the .A` D 0) limit of full abjfB .K/j2 reduces to Eq. (47.177e). sorption.
47
696
J.-M. Rost
which has a diffraction-shaped peak of width .ka/1 about the forward direction, and
Eikonal formulae for forward reactive scattering
fel ./ D ik
Z1
0 Z1
el .k/ D 2
1 e2iı 1 J0 .2kb sin /b db ; (47.85a) 2
0
abs .k/ D 2
Z1
1 e 4 b db ;
0
tot .k/ D 4
Z1
2
1 e
4 ImŒfel . D 0/ D 2 a2 k
(47.93)
is composed of a2 for classical absorption and a2 for edge (47.85b) diffraction or shadow (nonclassical) elastic scattering. This result also holds for the perfectly reflecting sphere ( a2 for classical elastic scattering and a2 for edge diffraction). (47.85c)
e j2 b db ;
2 2i
j 1 e
tot D
47.6 Classical Elastic Scattering
cos 2 b db ;
(47.85d)
Conservation of energy E > 0 and orbital angular momentum L for a central field V .r/ imply classically that the relative motion with reduced mass m takes place in the where the phase shift function ı D C i at impact parameplane perpendicular to the angular momentum vector L D ter b can be either the Jeffrey–Born phase 0 b p , where the impact parameter b and the initial momentum p0 with jp0 j D pa D .2mE/1=2 , normally taken Z1 V .r/ dr m ; (47.86) perpendicularly, define this plane. Hence, classically, elasıJB .b/ D 2 k„ .1 b 2 =r 2 /1=2 tic scattering reduces for central potentials to the analysis of b a one-dimensional deflection function #, mapping the initial impact parameter b into the final scattering angle in the where kb D D .` C 1=2/, or the eikonal phase scattering plane. This can be surprisingly complicated if traZ1 jectories exist that circle the scattering center several times m V .b; Z/ dZ : (47.87) before exiting with , corresponding to a nonmonotonic deıE .b/ D 2k„2 flection function. 1 0
Fraunhofer diffraction by a black sphere For a complex spherical well V 47.6.1 ( V D
Vr C iVi ; 0;
Ra:
The eikonal phase function Eq. (47.87) is ( ı.b/ D
1=2 mV =.k„2 / a2 b 2 ; 0
0ba b>a:
The absorption factor is h
1=2 i A.b/2 e4 D exp 2 a2 b 2 = ;
d The differential cross section d˝ asks which area of initial (47.88) conditions, typically parameterized by the impact parameter d D bdb, is scattered into the solid angle d˝ D d cos d. With the general formulation Eq. (47.4a) of classical propagators and Eqs. (47.96a) and (47.96b) the classical differential cross section for elastic scattering from a central (47.89) potential reads Z d D 2 db b ı.cos #.b; E/ cos / d˝ ˇ ˇ 2 X ˇˇ d# ˇˇ1 bj ˇ ; (47.94) D (47.90) j sin j j db ˇcos #.bj /Dcos
where D k„2 =.2mVi / is the mean free path towards absorption. For strong absorption, a, so that Za fel ./ D ik 0
1 J0 2kb sin b db ; 2
1 2
J1 2ka sin del D .ka/2 d˝ 2ka sin 12
!2 a2 ;
Elastic Scattering Cross Section
where the sum runs over all trajectories j connecting an impact parameter bi with a scattering angle . If cos #.b/ is d D bdb=d.cos /. monotonic, Eq. (47.94) simplifies to d˝
(47.91)
47.6.2
Deflection Functions
(47.92) The deflection function #.E; b/, (1 # < ) can be expressed through #.E; b/ D 2 .r ! 1I E; b/ with the
47
Classical, Quantal, and Semiclassical Propagators and Applications to Elastic Scattering
angle between initial (or final) momentum vector and the line that connects the scattering center with the point rm of closest approach of the scattering trajectory, defined through radial momentum p.rm / D 0. We define
Head-on collisions (b D 0): Overall repulsion: Overall attraction: Forward glory: Backward glory:
1
p.r/ D Œpa2 2mV .r/ L2 =r 2 2
Rainbow scattering: Deflection range: (47.95) Orbiting collisions: Diffraction scattering:
1
D Œpa2 2mV .r/ b 2 E=r 2 2 ; D Œpa2 2mVeff .r/1=2 ; with pa D
p
697
#.E; 0/ D . 0 < # . 1 # 0. # D 2n . # D .2n 1/ , n D 0;1; 2; : : : .d#=db/ D 0 at #r < 0. #r # . cf. Eq. (47.100). # ! 0 as b ! 1.
2mE. Then we can write The scattered particle may wind or spiral many times around (# ! 1) the scattering center. The experimentally observed quantity is the scattering angle (0 ), which (47.96a) is associated with various deflections
#.E; L/ D 2 .r ! 1I E; L/ Z1 @ p.r/dr ; D C2 @L rm
#i D C; ; 2 ˙ ; 4 ˙ ; : : :
#.E; b/ D 2 .r ! 1I E; b/ Z1 2 @ p.r/dr : D C pa @b
.i D 1; 2; : : : n/ (47.96b) resulting from n different impact parameters bi ; see Eq. (47.94).
rm
Small-angle diffraction scattering The small-angle scattering regime is defined by the conditions V .rm /=E 1, b rm , where D j#j. The main contribution to d=d˝ (47.97a) for small-angle scattering arises from the asymptotic branch of the deflection function # at large impact parameters b and is primarily determined by the long-range (attractive) part of the potential V .r/ (Sect. 47.7.3). (47.97b)
Small-angle scattering With V .rm /=E 1, b rm r Z1 ŒV .r / V .r/r dr m m #.E; b/ D
2 r 2 3=2 E r m r m
D
1 E
Z1
ŒV .rm / V .rm =x/dx .1 x 2 /3=2
0
;
Large-angle scattering The main contribution to d=d˝ for large-angle scattering arises from the positive branch of # at small b and is mainly determined by the repulsive part of the potential.
where x D rm =r. Other forms are 1 @ #.E; b/ D E @b
Z1 b
b D E
Z1
V .r/ dr .1 b 2 =r 2 /1=2
dV dr
b
1 @ D 2E @b
(47.97c)
47.6.3 Glory and Rainbow Scattering dr .r 2 b 2 /1=2
Z1 V .b; z/dz :
(47.97d) Glory The deflection function # passes through 2n (forward glory) or .2n C 1/ (backward glory) at finite impact parameters bg . Since sin D sin # D 0, the prefactor (47.97e) 1=j sin j in the cross section Eq. (47.94) diverges.
1
Rainbow The deflection function # passes through a minStraight-line paths With r 2 D b 2 C v 2 t 2 , Eq. (47.96b) imum at br , so that yields the impulse-momentum result (47.99a) #.b/ D #.br / C !r .b br /2 ; ˇ 1 Z 2 ˇ 1 d #ˇ p? 1 !r D >0: (47.99b) F? .t/ dt D ; (47.98) #.E; b/ D 2 db 2 ˇbr mv pa 1
where p? is the momentum transferred perpendicular to the incident direction, and F? D .@V =@r/.b=r/ is the impulsive force causing scattering. Special cases are:
The classical cross section diverges as 1 d br D Œ!r .r / 2 ; < r d˝ 2 sin
(47.99c)
47
698
J.-M. Rost
and is augmented by the contribution from the positive 47.6.6 Approximate Actions branch of #.b/. Jeffrey–Born phase function For small V =E and b rm , we get the action SJB , better known as the Jeffrey–Born 47.6.4 Orbiting and Spiraling Collisions phase, JB D SJB =„, Z1 V .r/dr m If strong enough, attractive interactions V .r/ can bind parti: (47.104) SJB .E; b/ D pa cles with E > 0 held inside the maximum of the effective po.1 b 2 =r 2 /1=2 b tential Veff .r/ D V .r/ C b 2 E=r 2 . The trajectory that touches this maximum at ro does not move radially and defines an Eikonal action For small V =E and a linear trajectory r 2 D orbiting resonance through the conditions b 2 C z 2 , the action reduces to ˇ dVeff ˇˇ Z1 D 0 and pb0 ;E .ro / D 0 (47.100) m ˇ dr rDro SEiko .E; b/ D V .b; z/ dz : (47.105) pa 1 which determine ro and bo . The corresponding total cross section is Applying Eq. (47.103a) to these actions one gets the corre(47.101) sponding deflection functions Eqs. (47.97a)–(47.97e). .E/ D b 2 : orb
o
For b < bo , the particle spirals inwards.
47.7 47.6.5 Deflection Function and Time Delay from Action Relevant is the change in action of a classical path from incoming momentum p 0 to final momentum p due to interaction with the potential V .r/ relative to the free path for a given L or b. In spherical momentum coordinates .p; L/, the action reduces to radial momenta integrals only, since L is conserved Zp S.E; L/ D .r r 0 /dp00 p0
Zpa D 2 .r.p/ r0 .p//dp
(47.102a)
Semiclassical Elastic Scattering
Semiclassical elastic scattering is conceptually quite involved since the scattering angle is defined over a finite range 2 Œ0; , while classically action and its conjugate, the deflection angle, are accumulated along a collision path without limits. Hence, several deflections differing by # D 2n contribute to differential scattering Eq. (47.94). Moreover, as a constitutive element for the construction of the scattering amplitude from the semiclassical propagator, we need the action in a “mixed representation”, that is, mapping .p 0 ; L0 / ! .p; #/ and not in full momentum representation .p 0 ; L0 / ! .p; L/ as defined in S.E; L/ of Eqs. (47.102a) and (47.2e). The change of variables is achieved with a Legendre transformation S C .#/ D S.E; L/ L#.E; L/ :
(47.106)
0
Decomposition into partial waves for constant angular momentum quantum number ` (Sect. 47.5.3) and the dep0 .r/dr : (47.102b) scription of elastic scattering with resulting phase shifts ` D 2 p.r/dr 2 rm b.E;L/ is a powerful technique. Its semiclassical formulation reThe boundary terms are the same with r.p/ and without quires quantizing the classical angular momentum such that potential r0 .p/, and, therefore, Eq. (47.102b) holds. The de- the classical L is replaced for semiclassical partial waves by 1 flection angle # is conjugate to L. Hence, the deflection L=„ ! D ` C ; (47.107) function is given by 2 which is a consequence of the Langer modification taking ˇ Z1 dr @S.E; L/ ˇˇ D 2L (47.103a) into account that the radial coordinate is only positive. #.E; L/ D @L ˇE p.r/r 2 Z1
Z1
rm
in Ragreement with Eqs. (47.96a), (47.96b) since 47.7.1 Semiclassical Amplitudes and Cross 1 2L b .p0 .r/r 2 /1 dr D . Similarly, the time delay , conSections jugate to E, is given by ˇ @S.E; L/ ˇˇ With the structure of the semiclassical propagator Eq. (47.6) .E; L/ D : (47.103b) ˇ and the classical cross section Eq. (47.94), the semiclassical @E L
47
Classical, Quantal, and Semiclassical Propagators and Applications to Elastic Scattering
699
scattering amplitude is a sum over contributions from tra- where the overall phases of each fj are jectories j starting with different impact parameters bj , or equivalently, angular momenta j , s1 D 2.1 / 1 =2 ; f ./ D
N X
fj ./ ;
(47.108)
(47.116a)
s2 D 2.2 / C 2 ;
(47.116b)
s3 D 2.3 / C 3 =2 ;
(47.116c)
j D1
where each classical path bj D bj ./ or SP-point j D j ./ contributes fj ./ to the amplitude h i 1=2 fj ./ D i˛j ˇj j ./ exp iSjC ./=„ #j0 D .d#=d/j ;
which are appropriate, respectively, to deflections #1 D at 1 , #2 D at 2 and #3 D at 3 , within the range # . The elastic differential cross section
(47.109) (47.110)
˛j D e˙i =4 I .C/ ; #j0 > 0 I
./ ; #j0 < 0 I (47.111a)
ˇj D e˙i =4 I .C/ ; #j > 0 I
./ ; #j < 0 : (47.111b)
./ D
3 X j D1
j ./ C 2
3 X
1=2 i ./j ./ cos.si sj / i 0 terms provide contributions from attractive collisions for which there are two points mn of stationary phase for each m in #m D .2m 1/ ˙ ˛. The backward glory amplitude at D ˛ is
fBG
2 1 (g) 2 1=2 1 XX D mn J0 .mn ˛/ eismn ; 0 k nD1 mD0 j#mn j
47.7.3
701
Small-Angle (Diffraction) Scattering
Diffraction originates from scattering in the forward direction by the long-range attractive tail of V .r/, where #, # 0 and ! 0. The cross section is formed from many contributions with small action S./ D „./ at a large angular momentum . A representation of the scattering amplitude as an integral over ./ avoids the classical singularity; see Sect. 47.7.8.
(47.134) 1 3 47.7.4 Small-Angle (Glory) Scattering (g) D 2.mn / C .2m 1/ mn : smn 2 4 (47.135) Amplitude and cross section The other contribution to forward scattering is the forward glory, which originates from In contrast to the Bessel amplitudes (below), these transi- the combined null effect of attraction and repulsion at a spectional formulae do not uniformly connect with the primitive ified glory impact parameter bg D g =k, where ./ attains semiclassical results for .f1 C f2 / away from the critical a maximum value of g . The m D 0 term of Eq. (47.131) yields glory angles. 1=2
Uniform Bessel amplitude for glory scattering The combined contributions from #1 D N C and #2 D N , where N D 2m, for forward and N D 2m 1 for backward scattering, yield [6] ˛j iN =2 . s21 /1=2 expŒiNs ./ e fG ./ D 2i 1 1=2 1=2 .1 C 2 /J0 s21 2 1 1=2 1=2 ; i.1 2 /J1 s21 2
i D 1; 2 ;
1 .s2 C s1 / ; 2
.C/; #j0 > 0I
(47.140c)
4 ImŒfD .0/ C fG .0/ k D D .E/ C G .E/ ;
.E/ D
(47.141a) (47.141b)
(47.137) where the diffraction cross section is Eq. (47.171), and where 4 2 G .E/ D 2 g k j#g0 j (47.138)
and ˛j D e˙i =4 I
(47.140b)
(47.136) Diffraction-glory oscillations
with mean sN21 ./ D
3 SG .E/ D 2g .E/ ; 4
(47.140a)
where J02 .x/ 1 14 x 2 C . The classical divergence is recovered Eq. (47.140b) over several oscillations ˛ ˝ by averaging with J02 .x/ D . x/1 .
where s21 ./ D s2 s1 is the difference of the collision phases Eq. (47.114a), si ./ D 2.i / i #i ;
fG ./ D G ./ expŒiSG .E/ ; ! 2g 2 J 2 .g / ; G ./ D 2 k j#g0 j 0
./; #j0 < 0 ; (47.139)
and the ordinary Bessel functions Jn .z/ satisfy the relationships J1 .z/ D J00 .z/, J1 .z/ D J1 .z/. This formula, valid for both forward ( 0) and backward ( ) glories, does uniformly connect the primitive result for (f1 C f2 ), valid when s21 1 and to the transitional results Eqs. (47.131) and (47.134), valid only in the vicinity of the glories.
!1=2
3 sin 2g .E/ 4 (47.142)
oscillates with E. For sufficiently deep attractive wells, the phase shift g successively decreases with increasing E through a series of multiples of =2. Writing g .E/ D .N 3=8/, maxima appear at N D 1; 2; : : : , and minima at N D 3=2; 5=2; 7=2; : : : . The glories are indexed by N in order of appearance, starting at high energies. g .E ! 0/ is related to the number n of bound states by Levinson’s theorem: 0 .E ! 0/ D .n C 1=2/ . Diffraction-glory oscillations also occur in the DCS at a frequency governed entirely by the energy variation of g .E/ and n of Eq. (47.175).
47
702
J.-M. Rost
When applied to Eq. (47.33a),
100 Glory number N 8
6
4
2
Symmetry oscillations
1
1 X
1
f ./ D .ik/
Z1 m
.1/
mD1
e2i./ 1
0
P 1 .cos / ei2m d ;
10
2
RT
Orbiting resonances
~v 0* –2/5
E*= 0.8
1
~v 0* –2/11 0.1 0.001
0.01
0.1
1
10
100
Fig. 47.1 All the various oscillatory effects for elastic scattering by a Lennard–Jones (12,6) potential of well depth and equilibrium distance Re . Ordinate D =.2 Re2 /, abscissa v D „v=.Re /
(47.144)
where ./ and P1=2 P .; / are now phase functions and Legendre functions of the continuous variable , being interpolated from discrete to continuous `. This infinitesum-of-integrals representation for f ./ is, in principle, exact. The particular merit of the Poisson sum in the present context is that the index m labels the classical paths that have encircled the (attractive) scattering center m times, and that the terms with m < 0 have no regions of stationary phase (SP). For deflections # in the range < # < , the only SP contribution is the m D 0 term.
47.7.7 Semiclassical Phase Shifts 47.7.5
Oscillations in Elastic Scattering
Semiclassical analysis ([6] and references therein) involves reducing Eq. (47.144) by the three approximations represented by cases A to C below. Since the integrands can oscillate very rapidly over large regions of , the main contributions to the integrals arise from points i of the stationary phase of each integrand. The amplitude can then be evaluated by the method of stationary phase, the basis of semiclassical analysis.
Figure 47.1 is an illustration [18, Chap. 1, p. 47] of all the various oscillatory structure and effects – Ramsauer–Townsend minimum (Sect. 47.5.5), orbiting resonances Eq. (47.204), diffraction-glory oscillations Eq. (47.141b), and symmetry oscillations Eq. (47.41d) – for elastic scattering by a Lennard–Jones .12; 6/ potential. Note the shift of velocity dependence from v 2=5 at low v to v 2=11 at high v; D 2 re2 A. Legendre Function Asymptotic Expansions is the averaged cross section 2 b02 in Eq. (47.168) at b0 D re . Main range: sin > 1 , not within 1 of zero or . The region > 1 probes the attractive part of the potential at low speeds, and < 1 probes the repulsive part at high P` .cos / D .2=. sin //1=2 cos. =4/ : (47.145) speeds. The four distinct types of structure originate from the Forward formula: within 1 of zero. nonrandom behavior of sin2 in Eq. (47.166). Orbiting tra(47.146a) P` .cos / D Œ= sin 1=2 J0 ./ ; jectories exist for E < 0:8 (Sect. 47.9). Z 1 J0 ./ ei cos d : (47.146b) 47.7.6 Scattering Amplitude in Poisson Sum 0
Representation
Backward formula: within 1 of . 1=2 With the help of the (exact) Poisson sum formula one P` .cos / D can convert a sum over discrete variables into an integral sin of a continuous variable. This allows one to directly de J0 Œ. / ei .1=2/ : (47.147) rive from the quantum partial wave expansion Eq. (47.33a) semiclassical expressions for the phase shift in terms of clas- Equations (47.145)–(47.147) are useful for analysis of caussical paths. The Poisson sum formula reads for D ` C 12 tics (rainbows), diffraction, and forward and backward glories, respectively. Also, a useful identity is [Eq. (47.107)] ( 1 X 1 4ı.1 cos /; > 0 Z 1 1 X
X .2` C 1/P` .cos / D F ` C 12 D .1/m F ./ ei2m d : (47.143) 0 D0; `D0 `D0
mD1
0
where ı.x/ is the Dirac delta function.
47
Classical, Quantal, and Semiclassical Propagators and Applications to Elastic Scattering
B. JWKB Phase Shift Functions Z1
Z1 ./ D
k .r/ dr rm
2 k 2 r
703
Uniform Airy result 1=2
2
AC ./ D a1 ./ ei.1 C =4/ F .21 / dr
(47.148a)
=k
3 21 Z 1 D lim 4 k` .r 0 / dr 0 kr 5 C ; r!1 2
(47.148b)
rm
with local wavenumber k2 .r/ D k 2 U.r/ 2 =r 2 :
C a2 ./ ei.2 =4/ F .21 / ;
1=2 ai ./ D 2 =ji00 j gi ./ ; i D 1; 2 ;
(47.154a) (47.154b)
21 ./ D 2 1 4 jz./j3=2 > 0 ; (47.154c) 3
1=4 p F Œ21 ./ D z Ai .z/ C iz 1=4 Ai0 .z/ ei.21 =2 =4/ ;
(47.149)
(47.154d)
where Ai and Ai0 are the Airy function and its derivative. The wavenumber is related to the impact parameter b and to This result holds for all separations .2 1 / in the loEqs. (47.102a) and (47.2e) through D S.E; „/=„ and the cation of stationary phases including a caustic (or rainbow), Langer modification Eq. (47.107) which is a point of inflection in , i.e. 1 D 2 , i0 D 0 D i00 . An equivalent expression is [6] p `.` C 1/ ` C 1=2
D D : (47.150) bD AC ./ D .a1 C a2 /z 1=4 Ai .z/ k k k p i.a1 a2 /z 1=4 Ai0 .z/ eiN ; (47.155) We note a useful identity. As r ! 1, 3 1 Zr 6
2 2 7 2 7 ! sin kr 1 ` : dr C sin6 k 2 4 r2 45 2
where the mean phase is N D 12 .1 C 2 /. The first form of Eq. (47.154a) is useful for analysis of widely separated regions of stationary phase when 21 0 and F ! 1. The equivalent second form Eq. (47.155) is valuable in the neighborhood of caustics or rainbows when the stationary phase (47.151) regions coalesce as a ! a . 1 2
k
JWKB phase functions are valid when variation of the po- Primitive result tential over the local wavenumber k1 .r/ is a small fraction For widely separated regions and , F ! 1 and 1 2 of the available kinetic energy E V .r/. Many wavelengths
can then be accommodated within a range r for a characA˙ ./ D a1 ./ ia2 ./ e˙i21 e˙i.1 C =4/ ; (47.156a) h teristic potential change V . The classical method is valid i A˙ ./ D a1 ./ exp ˙i 1 C when .1=k/.dV =dr/ .E V /. h 4 i : (47.156b) C a2 ./ exp ˙i 2 4 Phase–deflection function relation #./ D 2
@./ : @
Note that the minimum phase 1 is increased by =4, and (47.152) the maximum phase 2 is reduced by =4. Transitional Airy result In the neighborhood of a caustic or rainbow where 00 D 0, at the inflection point 1 D 2 D r , and then
C. Stationary Phase Approximations (SPA) to Generic Integrals Z1 ˙
A ./ D
g.I / expŒ˙i.I /d ; 1
(47.153)
ˇ ˇ ˇ 2 ˇ1=3 ˇ g.I r /Ai .z/ e˙i. Ir / ˇ A ./ D 2 ˇ 000 . / ˇ ˙
r
(47.157a) for parametric . In cases where the phase function has 2 : (47.157b) z D 000 two stationary points, a phase minimum 1 at 1 , and a phase j .r /j1=3 0 .I r / 0 00 00 0 maximum 2 at 2 , then i D 0, 1 > 0, 2 < 0, where i D .d=d/ at i and i00 D .d2 =d2 / for i D 1; 2. Since g is Only over a very small angular range does this result agree real, A D .AC / , gi ./ D g.; i /. in practice with the uniform result Eq. (47.154a), which
47
704
J.-M. Rost
uniformly connects Eqs. (47.156a) and (47.157a). These D r W 2 D 3 W SP’s coalesce. stationary-phase formulae are not only applicable to integrals > r W no classical attractive scattering. involving .; / but also to .t; E/ and .R; p/ combinations, C has no SP points. which occur in the method of variation of constants and in Franck–Condon overlaps of vibrational wavefunctions, reB. Forward Amplitude: sin < 1 spectively. 1=2 X Z1 1 1 im f ./ D e J0 ./ ik sin mD1 0 47.7.8 Semiclassical Amplitudes: Integral
(47.161a) e2i./ 1 e2im d : Representation
A. Off-Axis Scattering: sin > 1 Except in the forward and backward directions, Eq. (47.144) with Eq. (47.145) reduces to f ./ D
1 k.2 sin /1=2
1 X
.1/m
d 1=2 0
@ #.m / D 2 @
D 2m :
(47.158a) C. Backward Amplitude: O.1 / 1 1=2 ˙ .I m/ D 2./ C 2m ˙ ˙ =4 f ./ D k sin (47.158b) S C ˙ =4 ; Z1 1 X im C where S is the classical action Eq. (47.106) divided by „. e J0 Œ. / mD1 The stationary phase condition d ˙ =d D 0 yields the 0 deflection function # to the scattering angle relation eiŒ2./C.2m1/ d : C .;m/
ei
.Im/
ei
(47.161b)
Terms with m < 0, therefore, make no SP contribution to f ./ since # . The m D 0 term provides diffraction due to # ! 0, # 0 ! 0 at long range and a forward glory due to # ! 0 at a finite g and nonzero #g0 .
mD1
Z1
Stationary phase points: 0 .m / D 0.
;
#.i / D 2m ;
(47.162a)
(47.159)
Stationary phase points where i are points of stationary phase (SP). Since # @ 1, integrals with m < 0 have no SPs and vanish under the D .2m 1/ : (47.162b) #.m / D 2 SPA. For cases involving no orbiting (where # ! 1) and @ when > # > , integrals with m > 0 also vanish under SPA, so that the only remaining contribution from m D 0 to There are no SPs for m < 0. The m D 0 term provides a normal backward amplitude due to repulsive collisions (# D ), Eq. (47.158b) is and m > 0 terms are due to attractive half-windings. 1 f ./ D k.2 sin /1=2 D. Eikonal Amplitude Z1 h C i The m D 0 term of Eq. (47.161a) gives (47.160a) 1=2 ei ./ ei ./ d ; 0 ˙
./ D 2./ ˙ ˙ =4 :
(47.160b)
1 fE ./ D ik
Z1
e2i./ 1 J0 ./ d
(47.163a)
0 The attractive branch C contributes only negative deflecZ1 tions, and the repulsive branch contributes only positive
2i.b/ 1 J0 .kb/b db ; (47.163b) e D ik deflections and has one SP point at 1 where is maxi0 mum. Rainbow angle r : . C /00r D 0, so that # 0 .r / D 0, where and for potentials with cylindrical symmetry, kb can be re#.r / < 0 has reached its most negative value. placed by 2kb sin 12 D k b, and C < r W has two SP points 2;3 I Z ik 2i.b/ a maximum at 2 and fE ./ D 1 J0 .k b/ db : (47.164) e 2 a minimum at 3 :
47
Classical, Quantal, and Semiclassical Propagators and Applications to Elastic Scattering
From the optical theorem, Z1 sin2 .b; E/b db :
E .E/ D 8
(47.165)
0
705
Differential cross section d (47.172a) D fi2 ./ C fr2 ./ ; d˝ Z1 2 1 (47.172b) fi D sin2 ./ 1 2 2 d k 4 0 2 k D kD .E/ D 1 g1 .n/ 2 ; (47.172c) 4 16 Z1 1 1 2 2 (47.172d) sin 2./ 1 d fr D k 4 0 2 k D kD .E/ ; D 1 g2 .n/ 2 tan 4 16 n1 (47.172e)
E. Small-Angle Diffraction Corrected Scattering In small-angle scattering #, # 0 , and ! 0, where diffraction occurs due to scattering in the forward direction by the long-range attractive tail of V .r/. The main contributions to Eq. (47.163a) arise from a large number of small ./ at large . The Jeffrey–Born phase function Eq. (47.104) can, therefore, be used in Eq. (47.163b) for f ./ and in Eqs. (47.166) and (47.168), respectively, for .E/. A finite forward diffraction peak as ! 0 is obtained for f ./ in contrast to the classical infinite result. With Eq. (47.165) on gets the Landau–Lifshitz cross section where D is given by Eq. (47.171), and Z1 j f Œ2=.n 1/g2
2 : gj .n/ D 1 tan (47.166) sin JB .E; b/ b db : LL .E/ D 8 n1 Œ4=.n 1/ 0
The optical theorem Eq. (47.36) is satisfied, and
The Massey–Mohr cross section is
1=2
˛ 1 sin .E; b < b0 / D ; 2 Z1 MM .E/ D 2 b02 C 8 2JB .E; b/b db :
1 .E; b0 / D ; 2
(47.173)
˝
2
fD . 0/ D D .E/ eiSD .n/ ;
(47.174)
(47.167) where the (energy-independent) phase is SD .n/ D
(47.168)
.n 3/ : 2.n 1/
47 (47.175)
b0
An example For V .r/ D C =r n, the Landau–Lifshitz (LL) and Massey– Mohr (MM) cross sections are [Eq. (47.207a)] 2CF .n/ 2=.n1/ LL .E/ D .n 1/„v 1 2 sin ; (47.169) n1 n1 2CF .n/ 2=.n1/ 2n 3 ; (47.170) MM .E/ D .n 1/„v n2 p where F .n/ D . 12 n C 12 /= . 12 n/, and v is the relative speed. Both LL and MM have the general form 2=.n1/ C : (47.171) D .E/ D „v Ion–atom collisions 2=3
47.8
Coulomb Elastic Scattering
The Coulomb potential has the remarkable property that the quantum cross section (Rutherford cross section) agrees with the classical and semiclassical result, as well as with the Born approximation. Furthermore, the s-wave phase shift is semiclassically also correct, hence also the Mott cross section containing interferences due to identical particles is semiclassically correct. We will often use the abbreviations C D Z1 Z2 e 2 ; ˇ D C =„v D Z1 Z2 =.ka0 / ; rc .E/ D Z1 Z2 e 2 =.2E/ D ˇ=.2k/ :
47.8.1 Quantum Phase Shift, Rutherford and Mott Scattering , LL D 11:373,
For n D 4 attraction at low energy, D v and MM D 10:613. For n D 12 repulsion at high energy, D Direct solution of Eq. (47.24) yields v 2=11 , LL D 6:584, and MM D 6:296.
v` sin kr 12 ` C c` ˇ ln 2kr : Atom–atom collisions
2=5
(47.176)
(47.177a)
For n D 6 (attraction), D v , LL D 8:083, and MM D Coulomb phase shift 7:547 Fig. 47.1. Exact numerical calculations favor LL over c` D arg .` C 1 C iˇ/ D ImŒln .` C 1 C iˇ/ : (47.177b) MM ([19, pp. 1325] for details).
706
J.-M. Rost
For semiclassical Mott scattering, we also need the classical action in the scattering amplitude defined with the (47.177c) semiclassical propagator X dj 1=2 iS C .p;p0 /=„i =2 j f .p; p 0 / D e j (47.180) d˝ j
Coulomb S-matrix element
.` C 1 C iˇ/ S`c D exp 2ic` D : .` C 1 iˇ/ Coulomb scattering amplitude
ˇ exp 2ic` iˇ ln sin2 12
(47.177d) to determine d 1 D jf .p; p 0 / ˙ f .p; p 0 /j2 d˝ 2 Rutherford cross section 1 D jf ./ ˙ f . /j2 ; (47.181) d c 1 Z12 Z22 e 4 1 2 2 4 .E/ csc D r D ; (47.177e) c d˝ 16E 2 sin4 12 2 where the sum of the scattering amplitudes refers to bosons and the difference to fermions, while p ¤ p 0 denote final and which is the Coulomb differential cross section. initial momentum, respectively, and D arccosŒp p0 =.pp 0 /. The action between any two momenta p; p 0 that are conMott formula for elastic scattering of two indistinguish- nected by a classical trajectory of energy E under Coulomb able charged particles. From Eqs. (47.43a) and (47.43c), one interaction, is given by [3, p. 182], gets the positive (C, symmetric solutions, e.g., spin-zero p 1C C1 bosons as for 4 He–4 He) or negative (, antisymmetric so0 (47.182) ˚.p; p ; E/ D pa rc ln p lutions, e.g., spin- 12 fermions as for HC –HC , e ˙ –e ˙ ) 1C 1 c
f ./ D
2k sin2 12
:
d c 1 1 2 D rc .E/ csc4 C sec4 d˝ 2 2 1 1 2 2 ˙2 csc sec cos ; 2 2
D
.p 2 pa2 /.p 02 pa2 / : pa2 jp p 0 j2
We need the action only in the asymptotic regime outside (47.178) the range of the potential, p; p 0 ! pa , where for all nonzero scattering angles (p ¤ p 0 ), 1. Then [8]
where D 2ˇ ln tan 12 .
˚.p; p 0 ; E/ S C .p; p 0 / D lim 0 p;p !pa
47.8.2
with There is only a single trajectory, or in other words, the deflection function is monotonic. Hence, b2 rc .E/2
1=2
b.; E/ D rc .E/ cot 12 ;
;
(47.185)
2 .p pa2 /.p 02 pa2 / pa rc 0 D ln ; 2 pa4
(47.186)
where pa D .2mE/1=2 is the asymptotic momentum, and m the reduced mass of the two colliding particles. The constant (47.179b) 0 =„ D ı0 is the semiclassical phase shift for zero angular momentum, not dependent on the scattering angle . (47.179a)
and according to Eq. (47.94) and in agreement with Eq. (47.177e), we get d c 1 bdb rc .E/2 D D csc4 : d˝ d.cos / 4 2
(47.184)
D 2.0 pa rc lnŒsin #=2/;
Classical Coulomb Scattering
.b; E/ D j#j D 2 csc1 1 C
(47.183)
(47.179c)
47.8.4
Born Approximation for Coulomb Scattering
The scattering amplitude for Coulomb scattering reads accordingly [see Eq. (47.75)]
47.8.3
Semiclassical Coulomb Scattering fBc .p; p 0 / D
Since the monotony of the deflection function Eq. (47.179a) allows only for a single trajectory contributing to the cross section, the semiclassical Coulomb cross section between distinguishable particles reduces to the Rutherford cross section Eq. (47.179c).
rc .E/ 1 ; 2 2 sin .=2/
(47.187)
which gives the correct cross section for Rutherford scattering but not for Mott scattering since it lacks the relevant phase for the latter in contrast to the full quantum or semiclassical scattering amplitude.
47
Classical, Quantal, and Semiclassical Propagators and Applications to Elastic Scattering
47.9
707
Results for Model Potentials
Eqs. (47.188a)–(47.188d). For E > V0 and D #, define n2 D 1 V0 =E, b0 D na. Then ( Exact results for various quantities in classical, quantal, and 2 sin1 .b=na/ sin1 .b=a/ 0 b b0 semiclassical elastic scattering are obtained for the model po.b/ D 2 sin1 .b=a/ ; b0 b a ; tentials (a)–(s) in Table 47.1. (47.189a) and max D 2 cos1 n. For a given , the two impact parameters that contribute are
47.9.1
Classical Deflection Functions and Cross Sections
(a) Hard sphere
b1 ./ D
an sin 12 1 2n cos 12 C n2
1=2 ;
0 < b1 b0 (47.189b)
( 2 sin1 .b=a/; b a I (47.188a) .bI E/ D # D 0; b>a: 2
b./ D a cos ; 1 d D a2 I isotropic ; ./ D d˝ 4 D a2 (geometric cross section) I
1 b2 ./ D a cos ; 2
(47.189c)
(47.188b) For 0 , max (47.188c) (47.188d)
, ./, and are all independent of the energy E.
d 1 2 a2 n2 n cos 12 1 n cos 12 D a C
2 ; (47.189d) d˝ 4 4 cos 1 n2 C 1 2 cos 1 2
(b) Potential barrier For E < V0 , classical scattering is the same as for hard sphere reflection as given by
2
and d=d˝ D 0 for max . Finally, Zmax
Table 47.1 Model interaction potentials
b0 < b2 a :
D
d d˝ D a2 : d˝
D0
(a) (b) (c) (d) (e) (f)
Potential Hard sphere Barrier Well Coulomb .˙/ Finite-range Coulomb Pure dipole
(g)
Finite-range dipole
(h) (i)
Dipole + hard sphere Power law attractive
(j)
Fixed dipole + polarization
(k)
Fixed dipole + Coulomb
(l)
Lennard–Jones .n; 6/
(m)
Polarization .n; 4/
(n)
Multiple-term power law
(o) (p) (q)
Exponential Screened Coulomb Morse
V .r/ 1; r aI 0; r > a V0 ; r aI 0; r > a V0 ; r aI 0; r > a ˙C=r C=r C C=rs r rs I 0; r > rs ˙˛=r 2 1 1 ˙˛ 2 2 ; r aI 0; r > a r a ˙˛=r 2 ; r aI 0; r > a C=r n ; .n > 2/ De cos d ˛d e 2 r2 2r 4 De cos d e2 C 2 r r n 6 re n re 6 n6 n r r n 4 re n re 4 n4 n r r Cm Cn n D Vm .r/ Vn .r/ rm r V0 exp.˛r/ V0 exp.˛r/=r
e2ˇ.re r/ 2 eˇ.re r/
(r)
Gaussian
V0 exp.˛ 2 r 2 /
(s)
Polarization finite
V0 =.r 2 C r02 /2
47 (47.189e)
708
J.-M. Rost
(c) Potential well The results are similar to those for po- (g) Finite Range Dipole Scattering r02 D ˛=E, .r ˙ /2 D tential barrier case above, except that there is only a single b 2 ˙ r02 , .b0˙ /2 D a2 ˙ r02 . scattering trajectory with D #, and n D .1 C V0 =E/1=2 is the effective index of refraction for the equivalent problem in Repulsion (C): for b a, geometrical optics. Refraction occurs on entering and exiting
C the well. Then rmC b 2b 1 rm C C sin #.b/ D
rmC rm b0C .b/ D 2 sin1 .b=na/ sin1 .b=a/ ; (47.190a) b 2 sin1 ; (47.190b) .b D a/ D max D 2 cos1 .1=n/ ; a 1 an sin 2 #.0/ D ; #.b a/ D 0 ; D a2 : (47.194a) b./ D
(47.190c) 1=2 ; 1 1 2n cos 2 C n2
a2 n2 n cos 12 1 n cos 12 d Attraction (): for b > r0 , D
2 ; (47.190d) 1 1 2 d˝ 4 cos 2 n C 1 2n cos 2
rm b 2b 1 rm D a2 : (47.190e) C sin #.b/ D rm r b0 m (d) Rutherford or Coulomb The cross section for pure b ; (47.194b) 2 sin1 Coulomb scattering is discussed in Sect. 47.8.2. a #.r0 / ! 1 ;
(e) Finite-range Coulomb rc .E/ D
Z1 Z2 e 2 ; 2E
d r2 D c d˝ 4
˛.E/ D rc .E/=rs ; !2 1C˛ : ˛ 2 C .1 C 2˛/ sin2 12
(47.191) Repulsion (C): for 0 b b0 ,
rmC b
C 2b 1 rm C C sin #.b/ D a rmC rm b ; 2 sin1 a
Repulsion (C): # > 0, # D , 1 1 C 1 ; b .#/ D # 2 # 2 ˇ ˇ ˇ r02 ˇˇ 1 1 d ˇ: D ˇ 2 2 d˝ 4 sin .2 / ˇ
r02
Attraction (): # < 0. 1 1 b 2 .#/ D r02 C1 : j#j j#j C 2 2
(47.192a) (47.192b)
(47.193a)
There is an infinite number of (negative) deflections # D #n˙ associated with a given scattering angle
b0 b a I #.0/ D ;
D 2 n C ;
n D 0; 1; 2; : : : ;
(47.195a)
#.b/ D 2 sin1 .b=a/ ; #.b a/ D 0 ;
2
D a :
(47.195b) (47.195c)
Attraction (): for b > r0 , .rm b/ 2b 1 rm #.b/ D C sin rm r a m b ; 2 sin1 a #.b/ D #min at b D a ; #.0/ D ;
j#nC j j#n j
D a2 : (47.194c)
(h) Dipole plus hard sphere scattering r02 D ˛=E, .rm˙ /2 D b 2 ˙ r02 , .b0˙ /2 D a2 ˙ r02 .
(f) Pure dipole r02 .E/ D ˛=E.
2
#.b a/ D 0 ;
#.b a/ D 0 ;
D a2 :
(47.196)
(47.193b)
(47.193c) Orbiting or Spiraling Collisions From Sect. 47.6.5, the parameters are The infinite sum over contributions from bn˙ D b.#n˙ / for the attractive dipole yields Orbiting radius: r0 : ˇ ˇ 2 ˇ Focusing factor: F D Œ1 V .r0 /=E : ˇ d 1 r0 ˇ 1 ˇ: C D (47.193d) ˇ ˇ d˝ 4 sin 2 .2 /2 Orbiting cross section: orb D r02 F : D 2 n ;
n D 1; 2; 3; : : : :
47
Classical, Quantal, and Semiclassical Propagators and Applications to Elastic Scattering
709
(l) Lennard–Jones (n; 6) For the following two inter(i) Attractive power law potentials actions, there are two roots of E D Veff .r0 / D V .r0 / C 1 0 1 Veff .r0 / D 1 n V .r0 /; n > 2 ; 2 r0 V .r0 /. They correspond to stable and unstable circular 2 orbits [with different angular momenta associated with the 1=n minimum and maxima of the different Veff .R/]. Analytical .n 2/C n ; (47.197a) expressions can only be derived for the orbiting cross sec; F D r0 .E/ D 2E .n 2/ tion at the critical energy Emax , above which no orbiting can n .n 2/C 2=n orb .E/ D : (47.197b) occur. .n 2/ 2E For the Lennard–Jones .n; 6/ potential, orbiting occurs for E < Emax D 2Œ4=.n 2/6=.n6/ . The orbiting radius at Emax For the case n D 4 with V .r/ D ˛d e 2 =2r 4, this gives the is Langevin cross section
L .E/ D
2 r02
˛d e 2 D 2 2E
1=2
r0 .Emax / D re Œ.n 2/=41=.n6/ : (47.198) The orbiting cross section at Emax D 2.re =r0 /6 is
for orbiting collisions, and the Langevin rate
1=2 ; kL D vL .E/ D 2 ˛d e 2 =M
(47.199)
which is independent of E. The case n D 6 with V .r/ D C =r 6 is the van der Waals potential for which orb .E/ D
3 2 3 r0 D .2C =E/1=3 : 2 2
(47.200)
(j) Fixed dipole plus polarization potential
2 1=2
˛d e ; 2E 1=2 ˛d e 2 De cos d orb .E/ D 2 C : 2E E r02 .E/ D
(47.201a) (47.201b)
(47.204)
(m) Polarization (n; 4) As discussed for case (l),
47
4=.n4/ 2 ; Emax D n2 n 2 1=.n4/ r0 .Emax / D re ; 2 n orb .Emax / D 2 r02 ; .n 2/ p n D 12 W Emax D = 5 I
(47.205a) (47.205b) (47.205c)
Small-Angle Scattering For the power law potential V .r/ D C =r n, Eqs. (47.96a) and (47.96b) can be expanded in powers of V .r/=E to obtain analytic expressions for # and JB . The general form is #.b/ D
1 X V .b/ j j D1
Fj .n/ D
E 1=2
Fj .n/ D
1
.j C 1/
jn C
21 2jn
1 2
2 @ ; k @b
(47.206a)
:
(47.206b)
j C1
The leading j D 1 terms equivalent to Eq. (47.97d) are F1 .n/ F .n/, as defined following Eq. (47.169). Then to first order in V =E, (47.203a)
For all E and fixed rotations in the range 0 d max D cos1 e 2 =2De , orb .E/ D . De cos d /=E r02 .E/ :
orb D 2:4 re2 :
(47.202b)
(k) Fixed dipole plus Coulomb repulsion r02 .E/ D e 2 =2E :
n D 12W Emax
r0 D 1:22re I orb D 3:6 re2 :
For a locked-in dipole, the orientation angle is d D 0, and orb .E/ > 0, for all d , when E > Ec D .D 2 =2˛d /. all d from 0 to max D
over
1 Averaging 1 2 C sin =De , which satisfies orb .E/ > 0, for 2Er 0 2 all E, then " 1=2 1=2 # ˛d e 2 ˛d e 2 C horb .E/id D 2E 2Ec De E C 1 (47.202a) 4E Ec ! L .E/asE ! Ec :
3 2 n : r 2 0 n2 D 4=5 ; r0 D 1:165re ;
orb .Emax / D b02 .Emax / D
(47.203b)
k CF .n/ 1n ; b JB D 2E n1 CF .n/ 2=n d 1 D Ic ./ D : d˝ E n sin
(47.207a) (47.207b)
710
J.-M. Rost
From a log–log plot of sin .d=d˝/ versus E, C and n can (q) Morse potential both be determined. #.E; b/ D .2ˇb/ The integral cross sections for scattering by 0 is E
2ˇr e e K0 .2ˇb/ eˇre K0 .ˇb/ b Z Zmax h i p large b e2ˇ.re b/ 2 eˇ.re b/ ; ! . ˇb/1=2 .E/ D 2 Ic ./d.cos / D 2 b db E 0 0 br D re C .2ˇ/1 ln 2 ; CF .n/ 2=n #r D . ˇbr /1=2 .=2E/ ; D ; (47.208) E0 (47.213) !r D ˇ 2 j#r jre2 : where 0 is the smallest measured scattering angle corLarge-Angle Scattering responding to a trajectory with impact parameter bmax D For power law potentials V .r/ D C =r n, 1=n ŒCF .n/=E0 . A plot of ln .E/ versus ln E is a straight n line with slope .2=n/. X E .2j 1/=n #.b/ D Gj .n/ ; (47.214a) The Landau–Lifshitz cross section Eq. (47.169) and the V .b/ j D1 Massey–Mohr cross section Eq. (47.170) follow from use of 2j 1 .1/j 1 2 1=2 the JB phases Eq. (47.207a). ; (47.214b) Gj .n/ D .j / .k/ n n (n) Multiple-term power-law potentials with k D Œ.2j 1/=n j 1 . For the j D 1 term, 2
#.E; b/ D
1 ŒVm .b/F .m/ Vn .b/F .n/ : E
1=n
E G1 .n/b ; C 2=n C d Ic ./ D G12 .n/ ; D d˝ E
(47.209)
#.b/ D
For example, a Lennard–Jones .n; 6/ potential Table 47.1 has the following features
(47.214c) (47.214d)
which is isotropic. Including both j D 1 and 2 terms provides a good approximation to the entire repulsive branch of the deflection function #. Series Eq. (47.214a) for large anwhere ˛n D 6F .n/=ŒnF .6/ : gles and Eq. (47.206a) for small angles eventually diverge Rainbow: d#=db D 0 at br D .n˛n =6/1=.n6/ re : for impact parameters b < bc and b > bc , respectively, where (47.210a) #r D F .n/.E =E/.re =br /n ; C 1=n jn 2j1=n 1=2 b D n : (47.214e) c 1
3n 2E jn 2j1=2 (47.210b) !r D d2 #r =db 2 r D 2 j#.br /j : 2 br
Forward glory: # D 0 when bg D ˛n1=.n6/ re ;
47.9.2
(o) Exponential potential 1 V0 JB .E; b/ D kb K1 .˛b/ 2 E large b 1 V .b/ b 1=2 ! kb : 2 E 2˛
1 K D 2k sin ; 2 U0 D 2mV0 =„2 ; U0 =k 2 D V0 =E ; k 2 D 2mE=„2 ;
(47.211)
in the general expression Eq. (47.75) for the first Born approximation.
(p) Screened Coulomb potential #.E; b/ D ˛.V0 =E/K1 .˛b/ 1=2 large b 1 ! V .b/=E ; ˛b 2 k JB .E; b/ D V0 K0 .˛b/ 2E 1=2 large b b k ! : V .b/ 2E 2˛
Amplitudes and Cross Sections in Born Approximation
(a) Exponential V .r/ D V0 exp.˛r/ (47.212a)
(47.212b)
fB ./ D
2˛U0
; C K 2 /2 16 2 3˛ 4 C 12˛ 2 k 2 C 16k 4 ; U B .E/ D 3 0 ˛ 4 .˛ 2 C 4k 2 /3 V0 U0 E!1 4 ! : 3 E ˛4 .˛ 2
(47.215a)
(47.215b)
47
Classical, Quantal, and Semiclassical Propagators and Applications to Elastic Scattering
711
References (b) Gaussian V .r/ D V0 exp ˛ 2 r 2 1=2
U0 1. Heller, E.J.: The Semiclassical Way to Dynamics and Specexp K 2 =4˛ 2 ; fB ./ D (47.216a) 2 troscopy. Princeton University Press, Oxford (2018) 4˛ 2. Brack, M., Bhaduri, R.: Semiclassical Physics. Addison-Wesley, 2
U0 V0 Reading, Mass. (1997) 1 exp 2k 2 =˛ 2 : B .E/ D 4 3. Gutzwiller, M.C.: Chaos in classical and quantum mechanics. In8˛ E terdisciplinary Applied Mathematics. Springer, Berlin (1990) (47.216b) (c) Spherical well V .r/ D V0 for r < a, V .r/ D 0 otherwise U0 .sin Ka Ka cos Ka/ ; (47.217a) K3
V0 B .E/ D .U0 a4 / 1 .ka/2 C .ka/3 sin 2ka 2 E .ka/4 sin2 2ka : (47.217b) fB ./ D
(d) Screened Coulomb V .r/ D V0 exp.˛r/=r, V0 D Z1 Z2 e 2 , U0 D 2Z1 Z2 =a0 U0 ; C K2 V0 U0 4 U02 B .E/ D 2 2 ! : 2 ˛ .˛ C 4k / E ˛2 fB ./ D
˛2
(47.218a) (47.218b)
When ˛ ! 0, then fB ./ D U0 =K 2 . (e) e -atom V .r/ D Ne 2 ŒZ=a0 C 1=r exp.2Zr=a0 / ;
(47.219a)
H.1s/: N D 1, Z D 1; He.1s 2 /: N D 2; Z D 27=16. ! 2N 2˛ 2 C K 2 fB ./ D ; ˛ D 2Z=a0 ; (47.219b) a0 .˛ 2 C K 2 /2
a02 N 2 12Z 4 C 18Z 2 k 2 a02 C 7k 4 a04 : B .E/ D
3 3Z 2 Z 2 C k 2 a02 (47.219c) Also, fB decomposes Eq. (47.78) as fB .K/ D fBeZ .K/ C fBe e .K/F .K/ ;
4. Ozorio de Almeida, A.M.: Hamiltonian Systems: Chaos and Quantization. Cambridge University Press, Cambridge (1988) 5. Cvitanovi´c, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G.: Chaos: Classical and Quantum. ChaosBook.org, Niels Bohr Institute, Copenhagen (2016) 6. Berry, M.V., Mount, K.E.: Semiclassical approximations in wave mechanics. Rep. Prog. Phys. 35(1), 315 (1972) 7. Miller, W.H.: The Classical S-Matrix in Molecular Collisions vol. 30., pp 77–136 (1975) 8. Rost, J.M.: Semiclassical S-matrix theory for atomic fragmentation. Phys. Rep. 297, 271–344 (1998) 9. Kleinert, H.: Path Integrals in Quantum Mechanics Statistics and Polymer Physics. World Scientific, Singapore (1995) 10. Blinder, S.M.: Analytic form for the nonrelativistic Coulomb propagator. Phys. Rev. A 43, 13–16 (1991) 11. Van Vleck, J.H.: The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics. Proc. Natl. Acad. Sci. U.S.A. 14(2), 178–188 (1928) 12. Gutzwiller, M.C.: Phase-Integral Approximation in Momentum Space and the Bound States of an Atom. J. Math. Phys. 8(10), 1979–2000 (1967) 13. Herman, M.F., Kluk, E.: A semiclasical justification for the use of non-spreading wavepackets in dynamics calculations. Chem. Phys. 91(1), 27–34 (1984) 14. O’Malley, T.F., Spruch, L., Rosenberg, L.: Modification of Effective-Range Theory in the Presence of a Long-Range Potential. J. Math. Phys. 2(4), 491–498 (1961) 15. O’Malley, T.F.: Extrapolation of electron-rare gas atom cross sections to zero energy. Phys. Rev. 130, 1020–1029 (1963) 16. Gribakin, G.F., Flambaum, V.V.: Calculation of the scattering length in atomic collisions using the semiclassical approximation. Phys. Rev. A 48, 546–553 (1993) 17. Petovi´c, Z.L., O’Malley, T.F., Crompton, R.W.: J. Phys. B 28, 3309 (1995) 18. McDaniel, M.W., Mitchell, J.B.A., Rudd, M.A.: Atomic Collisions: Heavy Particle Projectiles. Wiley, New York (1993) 19. Massey, H.S.W., Burhop, E.H.S., Gilbody, H.B. (eds.): Electronic and Ionic Impact Phenomena. Clarendon, Oxford (1974)
(47.219d)
ij
where fB are two-body Coulomb amplitudes for .i; j / scattering, and F .K/ is the elastic form factor Eq. (47.79). (f) Dipole V .r/ D V0 =r 2 . fB ./ D U0 =2K :
(47.220)
(g) Polarization V .r/ D V0 .r 2 C r02 /2 U0 exp.Kr0 / ; (47.221a) 4 r0 3 U0 V0 B .E/ D Œ1 .1 C 4kr0 / exp.4kr0 / : 32r04 E (47.221b) fB ./ D
Jan-Michael Rost Jan-Michael Rost studied physics and philosophy in Munich and Freiburg. He received his PhD (1990) and spent postdoctoral years at the University of Washington, Seattle, and at Harvard University (1990–1993). He has been Director at the Max Planck Institute for the Physics of Complex Systems, Dresden, since 1999 and Professor of Quantum Dynamics at TU Dresden since 2000. His theoretical work includes ultrafast and ultracold highly excited quantum dynamics from atoms to nanosystems.
47
Orientation and Alignment in Atomic and Molecular Collisions
48
Nils Andersen
Contents 48.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 713
48.2 48.2.1 48.2.2
Collisions Involving Unpolarized Beams The Fully Coherent Case . . . . . . . . . . . The Incoherent Case with Conservation of Atomic Reflection Symmetry . . . . . . . The Incoherent Case Without Conservation of Atomic Reflection Symmetry . . . . . . .
48.2.3 48.3 48.3.1 48.3.2 48.3.3
. . . . . . . 714 . . . . . . . 714 . . . . . . . 717 . . . . . . . 717
Collisions Involving Spin-Polarized Beams The Fully Coherent Case . . . . . . . . . . . . The Incoherent Case with Conservation of Atomic Reflection Symmetry . . . . . . . . The Incoherent Case Without Conservation of Atomic Reflection Symmetry . . . . . . . .
. . . . . . 718 . . . . . . 718
Keywords . . . . . . 718 . . . . . . 720
48.4 48.4.1
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 The First Born Approximation . . . . . . . . . . . . . . . 721
48.5 48.5.1 48.5.2 48.5.3
Further Developments . . . . . . . . . . S ! D Excitation . . . . . . . . . . . . . . P ! P Excitation . . . . . . . . . . . . . . Relativistic Effects in S ! P Excitation
48.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
722 722 722 722
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722
Abstract
This chapter deals with the concepts of orientation and alignment in atomic and molecular physics. The terms refer to parameters related to the shape and dynamics of an excited atomic or molecular level, as it is manifested in a nonstatistical population of the magnetic sublevels. To take full advantages of the possibilities of this approach, one utilizes third generation experiments, i.e., scattering experiments that exploit the planar scattering symmetry, contrary to an angular differential cross section determination (a second generation experiment) having N. Andersen () The Niels Bohr Institute, Univerity of Copenhagen Copenhagen, Denmark e-mail: [email protected]
cylindrical symmetry, or a total cross section measurement (a first generation experiment) integrating over all scattering angles. In this way, one is often able to probe atomic collision theories at a more fundamental level, and in favorable cases approach a perfect scattering experiment in which the complex quantal scattering amplitudes are completely determined. This term was coined by Bederson [1] and has since served as an ideal towards which scattering experiments attempt to strive.
density matrix differential cross section coherence analysis reflection symmetry Stokes vector polarization orientation alignment
48.1
Introduction
The study of anisotropies has a long history in atomic physics, with light polarization, or Stokes parameter analysis, as a prominent example. A pioneering review by Fano and Macek [2] laid the mathematical and conceptual foundation for most of the later work. Advances in coincidence techniques, laser preparation methods, and the development of efficient sources for polarized electrons [3] have boosted the field further. Parallel developments of powerful computational codes have enabled a matching theoretical effort. In this way, detailed insights into the collision dynamics have been obtained, such as the locking radius model for low-energy atomic collisions, propensity rules for orientation in fast electronic and atomic collisions, and spin effects in polarized electron-heavy atom scattering. Presentations of these developments are contained in [4] and Chaps. 39, 53, 55, 67, 68, and 70. A comprehensive and critical review of the literature and the mathematical formalism was initiated by the National Institute of Standards and Technology (NIST) [5–7].
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_48
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Space limitations allow only presentation of the formalism for the simplest case of excitation. Recent developments in the description of processes involving polarized beams are included. The presentation is restricted to atomic outershell excitation. Other reviews describe excitation of inner shells [8] and molecular levels [9]. Related material on density matrices is contained in Chap. 7 [10].
48.2 48.2.1
d–2
d–1
d0
d1
d2
Collisions Involving Unpolarized Beams The Fully Coherent Case
p–1 p0 p1 Consider first the simplest nontrivial case of S ! P excitation. A general property of an atomic collision is that the total reflection symmetry with respect to the scattering plane of the total wave function describing the system is conserved. In simple cases, such as electron impact excitation of He singlet states, or atom excitation by fast, heavy particle ims0 pact, the projectile acts as a structureless particle, with only Reflection + + symmetry – – the target atom changing its quantum state. In this case, the reflection symmetry of the target wave function alone is conFig. 48.1 Reflection symmetry of the simplest spherical harmonics, served. corresponding to S, P, and D-states Figure 48.1 shows the angular parts of the simplest jlmistates with l D 0; 1; 2. The arrows indicate how the two families of states with positive or negative reflection symyc, zn, za xc, yn a b metry may couple internally. Thus, in an S ! P transition, + xa – an atom initially in an S-state may be excited to the (pC1 , xc, yn p1 ) subspace, while the p0 -state is not accessible. A chark in xa γ acteristic feature of the corresponding electron charge cloud γ kout zc, xn is that it has zero height, i.e., zero electron density in the dizc, xn rection perpendicular to the scattering plane. Furthermore, Θcoll the expectation value of the orbital angular momentum has a nonvanishing component in this direction only. c d The resulting electron charge cloud may, thus, have a shape as shown in Fig. 48.2a. We shall now discuss the parametrization of the wave function of this state and analyze the connection between the wave function and the corresponding photon radiation pattern emitted when the state decays back to the initial S-state, using the properties of electric dipole radiation.
Basic Definitions and Coordinate Frames The coordinate frames that are of particular use in describing the wave function are as follows. The collision frame, (x c ; y c ; z c ), is defined by zO c k kin and yO c k kin kout . The natural frame, (x n ; y n ; z n ), is defined by xO n k kin and zO n k kin kout . Finally, the atomic frame is identical to the natural frame, except that it is rotated by an angle around z n D z a such that x a is parallel to the major symmetry axis of the P-state charge cloud, as shown in Fig. 48.2. Most scattering calculations use the collision coordinate system as the reference frame, while mathematical analysis is of-
Fig. 48.2 a Shows the shape, or the angular part, of a P-state with positive reflection symmetry with respect to the scattering plane. Some relevant coordinate frames are also indicated; b shows a cut of this shape in the scattering plane. c is the angular distribution of photons emitted in the scattering plane; d is the polarization ellipse for light emitted in the direction perpendicular to the plane, as observed from above
ten most conveniently performed in the natural frame, where the algebra is particularly simple. Many expressions are even simpler in the atomic frame, but it has the disadvantage that the angle varies with collision parameters, such as impact velocity, impact parameter, etc.
48 Orientation and Alignment in Atomic and Molecular Collisions
The expansion of the P-state wave function in the three coordinate systems is ˇ ˛ c ˇ c c c pC1 C a0c jp0c i C a1 (48.1) j i D aC1 jp1 i ˇ ˛ n ˇ n n n D aC1 pC1 C a1 jp1 i (48.2) ˇ ˛ a ˇ a a a D aC1 (48.3) pC1 C a1 jp1 i:
715 Fig. 48.3 Fully coherent S ! P excitation may be described by two independent scattering amplitudes, characterized by their relative size and phase
f+1
δ
f–1
Conservation of reflection symmetry in the scattering plane implies c c D a1 ; aC1
a0n D 0 ; a0a D 0 :
optics (e.g., Born and Wolf [12]) the components of the (48.4) Stokes vector (P ; P ; P ) measured in the direction Cy c 1 2 3 (48.5) (Cz n ) perpendicular to the scattering plane and defined by (48.6) (48.15) IP1 D I.0ı / I.90ı / ;
The normalization condition implies ˇ c ˇ2 ˇ n ˇ2 ˇ D ˇa ˇ C jan j2 ja0c j2 C 2ˇaC1 C1 1 ˇ a ˇ2 a 2 ˇ ˇ D aC1 C ja1 j D1:
IP2 D I.45ı / I.135ı / ;
(48.16)
IP3 D I.RHC/ I.LHC/ ;
(48.17)
with I D I.0ı / C I.90ı /
(48.7)
D I.45ı / C I.135ı /
Finally, the am coefficients are related to the scattering amplitudes fm and the differential cross section ./ by 1
1
am D fm .kout =kin / 2 ./ 2 ˇ c ˇ2 ˇ ; etc. ./ D .kout =kin / jf0c j2 C 2ˇfC1
D I.RHC/ C I.LHC/ ;
(48.8) are given by (48.9)
P1 D
Except for an arbitrary phase factor, the pure state in Eq. (48.1) may thus be characterized by two dimensionless parameters. Traditionally, they have been chosen as (; ) defined as [11] or, alternatively, ˇ c ˇ2 ˇa ˇ 0 D ˇ ˇ2 (48.10) ˇ ˇ ; ˇac ˇ C 2ˇac ˇ2 0 C1
c c a0 : (48.11) D arg aC1 An alternative parametrization (L? ; ) is given by [5] ˇ n ˇ2 ˇ n ˇ2 ˇa ˇ ˇa ˇ L? D ˇ C1ˇ2 ˇ 1 ˇ2 ; (48.12) ˇan ˇ C ˇan ˇ C1 1 1 n n D arg a1 aC1 ˙ ; 2 2 1 (48.13) D .ı ˙ / ; 2 with the notation of Fig. 48.3. Referring to the natural coordinate frame, the expectation value of the electronic orbital angular momentum is, thus, h jLj i D .0; 0; L? / :
(48.18)
(48.14)
Coherence Analysis: Stokes Parameters We now discuss the information obtainable from a polarization analysis of the emitted light. In the notation of classical
2 1 ; p P2 D 2 .1 / cos ; p P3 D 2 .1 / sin ;
(48.20)
P1 D Pl cos 2 ;
(48.22)
P2 D Pl sin 2 ;
(48.23)
P3 D L? ;
(48.24)
(48.19)
(48.21)
with Pl D
q P12 C P22 :
(48.25)
Here, I./ is the intensity transmitted through an ideal linear polarizer with the transmission direction tilted by an angle with respect to the z c or x n -direction. Similarly, RHC (LHC) (right (left) hand circularly polarized light) refers to photons with negative (positive) helicity. Inspection of Eqs. (48.19) to (48.25) shows that determination of the Stokes vector in the direction perpendicular to the collision plane determines the wave function in Eq. (48.1) completely. A determination of the Stokes vector, thus, constitutes a perfect scattering experiment. The Stokes vector (P1 ; P2 ; P3 ) is a unit vector characterizing the state j i, and it may be represented by a point on the Poincaré sphere; see Fig. 48.4.
48
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According to the properties of electric dipole radiation, the pattern I./ in the scattering plane is
P3
I./ D
1 Œ1 Pl cos 2. / : 2
(48.31)
P2
A mapping of the radiation pattern in the collision plane determines and Pl , and thereby the absolute value of the angular momentum by
(P1, P2, P3)
Pl
q
2γ
jL? j D P1
Fig. 48.4 The Poincaré sphere. The Stokes vector for a pure state j i corresponds to a characteristic polarization ellipse, represented by a point on the unit sphere. Generally, the light is elliptically polarized, with the sense of rotation indicated by the arrows on the selected ellipses drawn. Special cases are the north and south poles, where the polarization is purely circular, and the equator, where it is purely linear, with the direction indicated by the orientation of the lines
Correlation Analysis Another experimental approach is to map the angular distribution of the photons emitted in the subsequent S ! P decay. For this purpose, we first note that the angular part - .; / of the electron probability density h j i corresponding to the wave function in Eq. (48.1) may be written as - .; / D
1 Œ1 C Pl cos 2. / sin2 : 2
(48.26)
Figure 48.2b shows a cut of this charge cloud in the scattering plane,
1 Pl2 :
(48.32)
However, the sign of L? cannot be determined by correlation analysis, and accordingly this approach does not qualify for classification as a perfect scattering experiment.
Density Matrix Representation Recalling that mn D am an , the density matrices in the various basis sets introduced above are 0
1 1 1 p .1 P1 / P2 C iP3 p .1 P1 / B 2 C 2 C p 1 B c B
D p B P2 iP3 2.1 P1 / P2 C iP3 C C; 8@ 1 A 1 p .1 P1 / P2 iP3 p .1 P1 / 2 2 (48.33) 0 1 2i 1 C L? 0 Pl e 1B C n (48.34)
D @ 0 0 0 A; 2 2i Pl e 0 1 L? 0 1 1 C L? 0 Pl 1B C (48.35)
a D @ 0 0 0 A: 2 Pl 0 1 L?
The density matrices illustrate the algebraic simplifications obtained using the natural coordinate frame. In the (48.27) following, this frame will be used unless otherwise stated, and we therefore suppress the superscript n below. The relative length, l, and width, w of the charge cloud are given by Postcollisional Depolarization Due to Fine Structure and Hyperfine Structure Effects 1 l D .1 C Pl / ; (48.28) After the collision, the isolated atom evolves under the influ2 ence of internal forces, such as fine structure and hyperfine 1 structure, until the optical decay. While typical collision w D .1 Pl / : (48.29) times are of the order 1015 s, the light emission occurs af2 ter a time interval of 109 s. This is long compared with the The degree of linear polarization Pl is, thus, a measure of the Larmor precession time of the electron spin ( 1012 s), and shape of the charge cloud, since one can, thus, assume that the atom has completely relaxed into, e.g., its 2 P1=2 and 2 P3=2 -states before the photon emisl w spin S D 12 , Pl D : (48.30) sion. For the simplest nontrivial case of electron l Cw the Stokes vector P.S/ of the subsequent 2 P ! 2 S transition
1 ; 2
1 D Œ1 C Pl cos 2. / : 2
48 Orientation and Alignment in Atomic and Molecular Collisions
is modified to become
3 1 D P1;2 .0/ ; P1;2 2 7 1 P3 D P3 .0/ ; 2
717
frame independent. A photon correlation experiment in the scattering plane may extract the (; Pl ) pair, while coher(48.36) ence analysis in the z-direction provides the complete set. An alternative set of (frame-dependent) parameters used in (48.37) particular for hydrogen is (; R; I ), with 1 .1 C P1 / ; 2 1 R D p P2 ; 8 1 I D p P3 : 8
and the Stokes vector is, thus, no longer a unit vector. In general, the Stokes vector components are reduced by depolarization factors ci (a table of ci coefficients for the most common values of electron and nuclear spin is given in Appendix B of [5]) Pi .S/ D ci Pi .0/ :
D
(48.38)
(48.41) (48.42) (48.43)
This suggests introduction of reduced polarizations P i D Pi .S/=ci :
(48.39) 48.2.3
The Incoherent Case Without Conservation of Atomic Reflection Symmetry
The reduced Stokes vector (P 1 ; P 2 ; P 3 ) is again a unit vector, and the formalism developed above for the spinless case can then be applied. We shall assume that this correction has In the general case, the assumption of positive reflection been made, and the bar will be dropped. symmetry with respect to the scattering plane for the wave function of the excited atom cannot be maintained. For example, for electron impact excitation of a P-state of a heavy 48.2.2 The Incoherent Case with Conservation atom, spin–orbit effects may be so strong that they flip the of Atomic Reflection Symmetry electron spin, thereby allowing population of the jp0n i-state. The picture outlined above has to be modified in cases where Thus, the total number of scattering amplitudes is now six. the experiment sums over several, in principle distinguish- Typical examples are mercury or the heavy noble gases. The able, channels, each of which, however, conserves reflection total angular momentum of the excited state is J D 1, the fine symmetry. Prototype examples are electron impact excitation structure being completely resolved in these cases. Strictly of hydrogen or light alkali atoms. Here, the excitation process speaking, the density matrix elements no longer describe the is described by singlet and triplet scattering amplitudes, re- electronic charge cloud, but rather the excited state (J D 1) spectively, and we have the possibility of direct and exchange distribution. Similarly, L? should be replaced by J? , but, for scattering. This doubles the number of scattering amplitudes simplicity, we keep the notation. This state radiates1 as a set from two to four. The experimental results are, thus, an inco- of classical oscillators, completely analogous to the P1 -state herent sum over these channels (singlet and triplet), the unrav- described above, so we shall maintain our previous notation, eling of which would require the application of spin-polarized simply replacing the term charge cloud density by oscillabeams; see Sect. 48.3. The atomic P-state can no longer be tor density. The main difference from the cases considered described as a single pure state, Eq. (48.1), but as a mixed above is that the density now displays a height; see Fig. 48.5. Blum, da Paixão, and collaborators, [13, 14], were the first state (Chap. 7). The expressions in Eqs. (48.33) to (48.35) for the density matrix are unchanged, but the matrix elements are to formulate a parametrization of the general case. Here, we now sums of the contributions from the individual channels shall use the expression for the density matrix in the natural (we shall discuss its decomposition below); Pl and L? are frame as the starting point and decompose it into the two terms now independent quantities, and the (reduced) Stokes vector with positive and negative reflection symmetry, respectively, P is generally no longer a unit vector. The degree of polarization P may thus be less than 1, i.e., P 2 D P12 C P22 C P32 D Pl2 C L2? 1 :
(48.40)
For electron impact excitation, deviation of the parameter P from unity may, thus, serve as a measure of the effect of electron exchange. Accordingly, we have three independent observables, e.g., (L? ; ; Pl ). This set of variables is
0
1 P3 1B
n D .1 h/ @ 0 2 C 2i Pl e 0 1 0 0 0 B C C h@0 1 0A ; 0 0 0
1 0 PlC e2i C 0 0 A 0 1 C P3 (48.44)
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N. Andersen
a
Table 48.1 Summary of cases of increasing complexity and the orientation and alignment parameters necessary for unpolarized beams; Np is the number of independent parameters, and Nd is the number of observation directions required
b
Variable Forces Representation Refl. symmetry Ang. mom. Align. angle Linear pol. Degree of pol. Height Np Nd
ya xa za
Sect. 48.2.1 Coulomb Wave func. C L? Pl P D1 hD0 2 1
Sect. 48.2.2 +exchange
mn C L? Pl P 1 hD0 3 1
Sect. 48.2.3 +spin–orbit
mn C; LC ? PlC PC 1 h0 4 2
xa
48.3
za
Collisions Involving Spin-Polarized Beams
ya
In this section, we discuss the additional information that can be gained by the application of a polarized beam compared Fig. 48.5 The top row shows classical oscillator densities for height with an unpolarized beam. Collisions with polarized electron parameters h D 0 and h D 1=3, respectively. The alignment angle is beams are discussed in particular, but most of the ideas pre D 35ı and PlC D 0:6 in both cases. Cuts along the symmetry axes are sented are easily generalized to beams of atoms in polarized, shown in the other rows or otherwise prepared, states. We shall keep the order of increasing complexity introduced in the previous section. where the linear polarization PlC is labeled with a C referring to the positive reflection symmetry. Similarly, we define LC ? P3 ; P
C2
P12
C
(48.45) 48.3.1 P22
C
P32
1:
The Fully Coherent Case
(48.46)
As can be seen in Sect. 48.2.2, a simple example of a fully The shape of the density is now characterized by the three coherent excitation process is He 1 1 S ! n 1 P excitation, for parameters which complete information can be obtained from experi1 ments using unpolarized beams. Consequently, in this case, (48.47) l D .1 h/ .1 C Pl / ; the application of polarized electrons will add nothing new. 2 1 Also, the polarization of the scattered electron can be trivially (48.48) w D .1 h/ .1 Pl / ; 2 predicted, since no change is possible in the scattering process. h D 00 ; (48.49) with l Cw Ch D 1. There are, thus, four independent observC 48.3.2 The Incoherent Case with Conservation ables, chosen as (LC ? ; ; Pl ; h). Again, this set of variables of Atomic Reflection Symmetry is frame independent. Determination of the height parameter evidently requires observation from a direction other than the hydrogen or light alkali atoms z n -direction. Traditionally, the Stokes parameter (P4 ; 0; 0) is The targets of interest here are 1 . Since we now have the possiwith an electron spin S D n 2 measured from the y -direction to obtain bility of triplet (t) and singlet (s) scattering, this doubles the .1 C P1 /.1 P4 / : (48.50) number of scattering amplitudes from two to four (recall that hD 4 .1 P1 /.1 P4 / f0 D 0); see Fig. 48.6. The amplitudes of interest are Thus, all four parameters may be obtained from analysis of the light coherence, but two directions of observation are necessary. Similarly, photon correlation analysis in two planes is required to extract (; PlC ; h); see, e.g., [5] for a discussion. The various cases with unpolarized beams discussed in this section are summarized in Table 48.1.
t D ˛C eiC ; fC1
(48.51)
t f1 s fC1 s f1
;
(48.52)
i
D ˛ e
D ˇC ei
C
;
(48.53)
i
:
(48.54)
D ˇ e
48 Orientation and Alignment in Atomic and Molecular Collisions Fig. 48.6 For 2 S ! 2 P electron impact excitation of hydrogen or light alkali atoms, four scattering amplitudes come into play
0
t1 1 C Lt? 0 Plt e2 i 1B C D 3w t u @ 0 0 0 A 2 t P tl e 2 i 0 1 Lt? 0 s1 1 C Ls? 0 Pls e2 i 1B C C w s u @ 0 0 0 A ; (48.63) 2 s Pls e 2 i 0 1 Ls?
t f +1
t f –1
δt Δ+1
s f +1
Δ–1 δ
s
Neglecting an overall phase, seven independent parameters are needed to characterize the amplitudes completely. Traditionally, one is chosen as the differential cross section u corresponding to unpolarized particles. Six additional dimensionless parameters may be defined: three to characterize the relative lengths of the four vectors, and three to define their relative phase angles. The density matrix is parametrized, in analogy to the unpolarized beam case, according to [15] 0 t1 1 C Lt? 0 Plt e2 i 1B C (48.55)
t D t @ 0 0 0 A; 2 t 2 i t t Pl e 0 1 L? 1B
s D s @ 2
1 C Ls? 0 s 2 i s Pl e
s1 0 Pls e2 i C 0 0 A; s 0 1 L?
where 2 C ˛2 ; t D ˛C 2 s D ˇC C ˇ2 ;
1 2 ˛C ˛2 ; t 1 2 Ls? D s ˇC ˇ2 ; 2˛C ˛ iıt t Plt e2i D P1t C i P2t D e ; t 2ˇC ˇ iıs s Pls e2i D P1s C i P2s D e : s Lt? D
where t t D ; s C 3 t 4u s s ws D s D D 1 3w t ; t C 3 4u 3 1 u D .3w t C w s /u D t C s : 4 4 wt D
s f –1
0
719
(48.64) (48.65) (48.66)
The six parameters u ; w t ; Lt? ; Ls? ; t , and s have now been introduced, leaving one parameter still to be chosen. Inspection of Fig. 48.6 suggests, for example, the angle C . The fourth angle, , is then fixed through the relation
(48.67) C D ı t ı s D 2 s t : The following set of six dimensionless parameters is thus complete w t ; Lt? ; Ls? ; t ; s ; C :
(48.68)
Detailed recipes for the extraction of the parameters from coherence experiments are somewhat complicated, and we refer to discussions in the literature [7, 15, 16]. Using spin(48.56) polarized electrons and spin-polarized targets, all parameters may be determined, except for information about singlettriplet phase differences, such as C . The reduced Stokes vector P of the unpolarized beam experiment is given by the singlet and triplet (unit) Stokes (48.57) vectors P s,t as (48.58) (48.69) P D 3 wt P t C ws P s ; (48.59) from which the set of parameters .L? ; ; Pl / for the unpolarized beam experiment may be evaluated from (48.60) (48.61)
L? D 3 w t Lt? C w s Ls? ; Pl e
2i
D 3w
t
Plt
e
2 i t
Cw
s
(48.70) Pls
e
2 i s
:
(48.71)
(48.62) Since, in general, Lt ¤ Ls and t ¤ s , this causes the (re? ? duced) degree of polarization P to be smaller than unity. In the case of an unpolarized beam, the total density matrix To summarize this section, Stokes parameter analysis may becomes the weighted sum of the two matrices s and t , i.e., provide five dimensionless parameters, the relative phase bet t 0 1 and f1 amplitudes and the relative phase tween the two fC1 1 C L? 0 Pl e2 i s s and f1 amplitudes, as well as the between the two f 1B C1 C
u D u @ 0 0 0 A relative sizes of all four amplitudes. However, none of the rel2 Pl e2 i 0 1 L? ative phases between any triplet and singlet amplitude can be
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Fig. 48.7 For J D 0 ! J D 1, electron impact excitation of heavy atoms six scattering amplitudes come into play since spin-flip may occur
1 1 # f 0; ; f0 D ˇ0 ei 0 ; 2 2 1 1 " i n f0 D ˛0e0 ; f 0; ; 2 2 n
f+1
δ
(48.76) (48.77)
where we have omitted Ji D Mi D 0 and Jf D 1. Equations (48.72)–(48.75) represent noflip amplitudes that leave Δ f+1 the projectile spin unchanged, while Eqs. (48.76) and (48.77) Δ– δ describe the cases where the electron spin is flipped. We first assume a polarization perpendicular to the scatterf–1 Δ0 ing plane, i.e., along the z-direction. In Eq. (48.44), the denf0 sity matrix for heavy atoms such as Xe or Hg was decomposed into a pair of matrices with one having positive reflection symf0 metry with respect to the scattering plane, and the other one having negative reflection symmetry, respectively. The extendetermined, and coherence analysis alone is, thus, not able to sion of this decomposition to the case of polarized electron provide a perfect scattering experiment. The missing phase beams is a pair of density matrices, one for spin-up electron may be extracted from the so-called STU parameters of the impact excitation and one for spin-down excitation, where up and down correspond to the initial spin component orientation scattered electron, ([16, 17] and Chap. 7). with respect to the scattering plane. Hence, δ
f–1
+
0
2
48.3.3 The Incoherent Case Without Conservation of Atomic Reflection Symmetry The results of coherence analysis of a J D 0e ! J D 1o transition will now be discussed. We only analyze the photon polarization in the exit channel, not the electron spin parameters. There are six independent scattering amplitudes for a J D 0 ! J D 1 transition (Fig. 48.7), thereby requiring the determination of one absolute differential cross section, five relative magnitudes, and another five relative phases of the scattering amplitudes. This large number of independent parameters leads to considerable complications. Nevertheless, the natural coordinate system enables disentangling of the scattering amplitudes and generalization of the parametrization of the density matrix for the case of unpolarized beams in a straightforward way [7, 18–20]. The nonvanishing amplitudes f n .Mf ; mf ; mi / in the natural frame (Fig. 48.7) for a J D 0 ! J D 1 transition are 1 1 " fC1 D ˛C eiC ; f n 1; ; 2 2 1 1 # f n 1; ; fC1 D ˇC ei C ; 2 2 1 1 " i f1 D ˛e f n 1; ; ; 2 2 1 1 # f n 1; ; f1 D ˇ ei ; 2 2
(48.72) (48.73) (48.74) (48.75)
1 C LC ? 1B 6
u D u 4.1 h/ @ 0 2 PlC e 2 i 0 13 0 0 0 B C7 C h @0 1 0A5 0 0 0
1 0 PlC e2 i C 0 0 A C 0 1 L?
D w " " C w # # 2 1 6 D w " u 4 1 h" 2 "1 C" C" 0 Pl e2 i 1 C L? C B @ 0 0 0 A " C" C" Pl e 2 i 0 1 L? 0 13 0 0 0 B C7 C h" @0 1 0A5 0 0 0 2 1 6 C w # u 4 1 h# 2
0
#1 C# C# 0 Pl e2 i 1 C L? C B @ 0 0 0 A # C# C# Pl e 2 i 0 1 L? 0 13 0 0 0 B C7 (48.78) C h# @0 1 0A5 :
0
0 0 0
48 Orientation and Alignment in Atomic and Molecular Collisions
Here, we have defined 2 ˛2 ˛C " D P3 ; 2 ˛C C ˛2
(48.79)
ˇ 2 ˇ2 # D 2C D P3 ; 2 ˇC C ˇ
(48.80)
C"
L? D C# L? "
"
"
#
#
#
2˛C ˛ ei. C / ; 2 ˛C C ˛2
(48.81)
2ˇC ˇ ei. 2 ˇC C ˇ2
(48.82)
C"
e2i D P1 C iP2 D
C#
e2i D P1 C iP2 D
Pl Pl
C/
2 C ˇ2 C ˇ02 ; # D ˇC
1 2 1 D 2
2 2 C ˇ2 C ˇ02 ˛C C ˛2 C ˛02 C ˇC " C # ;
C# 2
Pl
C L?
C# 2
C L?
ı "# C
C#
0
:
(48.97)
"
C" 2
D Pl
C# 2
D Pl
C C
"2 P3 #2 P3
C"
C#
.w " ; L? ; L? ; h" ; h# ; " ; # ; C ; 0 ; ı "# / :
(48.98)
(48.88)
#
Information about the remaining three phase angles may be (48.89) obtained in experiments with in-plane spin polarization. Further analysis shows that the generalized Stokes parameters in the y (or x)-direction with in-plane spin polarization Py or Px provides two additional phases. None of the relative " # phases C between fC1 and fC1 , etc., enter. Determination (48.90) of the final remaining angle requires determination of generalized STU parameters, describing the electron spin in the exit channel. Table 48.2 summarizes the various cases with polarized (48.91) beams discussed in this section. (48.92)
D1;
(48.93)
D1:
(48.94)
Extraction of these parameters is facilitated by the introduction of generalized Stokes parameters [18]. In this way, C" C# L? ; L? ; " ; # may be determined. If, in addition, h is known, e.g., by polarization analysis in the y-direction, the following set of seven dimensionless independent parameters can be derived from the generalized Stokes parameters in the z-direction C"
Sect. 48.3.3 +spin–orbit ";#
mn C; 10 9 2
(48.87)
h D w " h" C w # h#
D ˛02 C ˇ02 =.2u / ; Pl
Sect. 48.3.2 +exchange t,s
mn C 6 5 1
h# D ˇ02 = # ;
From these definitions it follows that C" " .1 h/ LC 1 h" L? ? Dw C# C w # 1 h# L? ; " C" .1 h/ PlC e2i D w " 1 h" Pl e2i # C# C w # 1 h# Pl e2i ;
C" 2
Sect. 48.3.1 Coulomb Wave func. C 2 2 1
h" D ˛02 = " ;
w D =.2 u / D 1 w :
C" 2
Variable Forces Representation Refl. symmetry Np NOA Nd
(48.85) A complete set of dimensionless independent parameters is then given by (48.86)
w " D " =.2 u / ; #
Table 48.2 Summary of cases of increasing complexity for spinpolarized beams. The number of independent dimensionless parameters Np is listed, along with NOA , the number determined from orientation and alignment only; Nd is the number of observation directions required
choice for the re(48.83) in analogy to Eq. (48.67). CA convenient 0 "# maining phase angles is . ; ; ı /, with (48.84)
2 " D ˛C C ˛2 C ˛02 ;
u D
;
721
L? ; L? ; h" ; h# ; w " I " ; # :
48.4 Example 48.4.1 The First Born Approximation As a simple, illustrative example, consider the predictions of the first Born approximation (FBA). Here, S ! P excitation by electron impact is described as creation of a pure p-orbital along the direction of the linear momentum transfer k D kin kout , along which there is axial symmetry. Evidently, LFBA D0; ?
(48.99)
(48.95) and, consequently,
This leaves three relative phases unknown. In the notation of PlFBA D 1 : (48.100) Fig. 48.7, we can see from inspection that The alignment angle is found from simple geometrical con(48.96) C D ı " ı # D 2 # " ; siderations; see Fig. 48.8. Denoting the incident energy by E
48
722
N. Andersen
48.5.3 Relativistic Effects in S ! P Excitation It has been discussed to what extent relativistic effects can be studied for excitation of the two fine structure components of the resonance transitions of heavy alkali atoms, such ∆k γ FBA as Rb or Cs. For electron-impact excitation, standard Stokes ΘFBA parameter analysis turns out to be extremely insensitive to k in ~ E1/2 the inclusion of relativistic effects in the numerical treatment, which explains the success of nonrelativistic theories. If spinFig. 48.8 Diagram for evaluation of the alignment angle in the first polarized electrons are used, either in the incident channel Born approximation through measurement of spin asymmetries or in the final channels by performing a time-reversed generalized Stokes and the energy loss by E, the relation between the projec- parameter experiment with a laser-prepared target and a spintile scattering angle #col and the alignment angle is directly polarized electron beam, distinct relativistic effects, typically read from the figure, at the 5% level, may be revealed [23]. kout ~ E – ∆ E
tan FBA D
sin #col ; cos #col x
1/2
(48.101)
48.6 Summary
where x D ŒE=.E E/1=2 . For E > 0, FBA is always A selection of fundamental formulas describing orientation negative, with its minimum value when k ? kout . Any theoand alignment in atomic collisions is given, with emphasis retical effort beyond the FBA involves serious computations. on the simplest case, S ! P excitation. A tutorial introduction to the field with a series of examples and applications may be found in a recent textbook [20]. Other examples and 48.5 Further Developments discussions of perfect scattering experiments may be found in [24].
48.5.1
S ! D Excitation
The generalization of the formalism of Sect. 48.3.2 to the References case of S ! D excitation involves the introduction of three scattering amplitudes, corresponding to a complete param1. Bederson, B.: Comments At. Mol. Phys. 1, 41 and 65 (1969) eter set of one cross section, two relative amplitude sizes, 2. Fano, U., Macek, J.H.: Rev. Mod. Phys. 45, 553 (1973) and two relative phases. Analysis shows that a full coherence 3. Kessler, J.: Polarized Electrons. Springer, Berlin, Heidelberg (1985) analysis of the light emitted in the subsequent D ! P optical 4. Hertel, I.V., Schmidt, H., Bähring, A., Meyer, E.: Rep. Prog. Phys. decay is not sufficient for a complete experiment, instead two 48, 375 (1985) solutions are obtained. A triple coincidence experiment may 5. Andersen, N., Gallagher, J.W., Hertel, I.V.: Phys. Rep. 165, 1 resolve the ambiguity [21]. (1988)
48.5.2
P ! P Excitation
By proper optical preparation of the atomic target, collision studies involving specific excited states may be performed as a function of the scattering angle. For collision-induced P ! P transitions, a systematic preparation of specific initial P-states, combined with Stokes parameter analysis of the radiation pattern from the final P-state, may lead to a complete scattering experiment. The corresponding complete set of nine parameters describes the process in terms of five independent scattering amplitudes. In addition to the charge cloud shape and orientation parameters, three Euler angles are needed to describe the atomic reference frame of the charge cloud with respect to the laboratory frame [22].
6. Andersen, N., Broad, J.T., Campbell, E.E.B., Gallagher, J.W., Hertel, I.V.: Phys. Rep. 278, 107 (1997) 7. Andersen, N., Bartschat, K., Broad, J.T., Hertel, I.V.: Phys. Rep. 279, 251 (1997) 8. Wille, U., Hippler, R.: Phys. Rep. 132, 129 (1986) 9. Greene, C.H., Zare, R.N.: Ann. Rev. Phys. Chem. 33, 119 (1982) 10. Blum, K.: Density Matrix Theory and Applications. Plenum, New York (1981) 11. Macek, J., Jaecks, D.H.: Phys. Rev. A 4, 2288 (1971) 12. Born, M., Wolf, E.: Principles of Optics. Pergamon, New York (1970) 13. Blum, K., da Paixão, F.T., Csanak, G.: J. Phys. B 13, L257 (1980) 14. da Paixão, F.T., Padial, N.T., Csanak, G., Blum, K.: Phys. Rev. Lett. 45, 1164 (1980) 15. Hertel, I.V., Kelley, M.H., McClelland, J.J.: Z. Phys. D 6, 163 (1987) 16. Andersen, N., Bartschat, K.: Comments At. Mol. Phys. 29, 157 (1993) 17. Bartschat, K.: Phys. Rep. 180, 1 (1989) 18. Andersen, N., Bartschat, K.: J. Phys. B 27, 3189 (1994)
48 Orientation and Alignment in Atomic and Molecular Collisions 19. Andersen, N., Bartschat, K.: corrigendum. J. Phys. B 29, 1149 (1996) 20. Andersen, N., Bartschat, K.: Polarization, Alignment, and Orientation in Atomic Collisions, 2nd edn. Springer, Cham (2017) 21. Andersen, N., Bartschat, K.: J. Phys. B 30, 5071 (1997) 22. Sidky, E.Y., Grego, S., Dowek, D., Andersen, N.: J. Phys. B 35, 2005 (2002) 23. Andersen, N., Bartschat, K.: J. Phys. 35, 4507 (2002) 24. Kleinpoppen, H., Lohmann, B., Grum-Grzhimailo, A.N.: Perfect/ Complete Scattering Experiments. Springer, Berlin, Heidelberg (2013) 25. Muktavat, K., Srivastava, R., Stauffer, A.D.: J. Phys. B 36, 2341 (2003)
723 Nils Andersen Nils Andersen is Professor of Physics at the Niels Bohr Institute of the University of Copenhagen. His main activities include experimental and theoretical studies of atomic collisions involving optically prepared states. Recent research interests include cold and ultracold collisions.
48
49
Electron–Atom, Electron–Ion, and Electron–Molecule Collisions Klaus Bartschat
, Jonathan Tennyson
Contents 49.1 49.1.1 49.1.2 49.1.3 49.1.4 49.1.5
for proper uncertainty quantification in such calculations has recently been emphasized [7]. 725 725 728 729 731
49.1.6
Electron–Atom and Electron–Ion Collisions . . . . . Low-Energy Elastic Scattering and Excitation . . . . . Relativistic Effects for Heavy Atoms and Ions . . . . . Multichannel Resonance Theory . . . . . . . . . . . . . . Solution of the Coupled Integro-Differential Equations Intermediate and High-Energy Elastic Scattering and Excitation . . . . . . . . . . . . . . . . . . . . . . . . . Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49.2 49.2.1 49.2.2 49.2.3 49.2.4
Electron–Molecule Collisions . . . . . . . . . . . Laboratory Frame Representation . . . . . . . . . Molecular Frame Representation . . . . . . . . . Inclusion of the Nuclear Motion . . . . . . . . . . Electron Collisions with Polyatomic Molecules
741 741 742 742 744
49.3 49.3.1 49.3.2
Electron–Atom Collisions in a Laser Field . . . . . . 744 Potential Scattering . . . . . . . . . . . . . . . . . . . . . . 744 Scattering by Complex Atoms and Ions . . . . . . . . . 746
. . . . .
. . . . .
. . . . .
. . . . .
735 737
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747
Abstract
This chapter summarizes the theory of electron collisions with atoms, ions, and molecules. Section 49.1 discusses elastic, inelastic, and ionizing collisions with atoms and atomic ions from close to threshold to high energies where the Born series becomes applicable. Section 49.2 extends the theory to treat electron collisions with molecules. Finally, the theory of electron–atom collisions in intense laser fields is discussed in Sect. 49.3. This chapter will not present detailed comparisons of theoretical predictions with experiment. Such comparisons are given in many books and review articles, such as [1–6]. The need K. Bartschat () Dept. of Physics & Astronomy, Drake University Des Moines, IA, USA e-mail: [email protected] J. Tennyson Dept. of Physics & Astronomy, University College London London, UK e-mail: [email protected]
, and Philip Burke
Keywords
electron collisions scattering amplitudes and cross sections potential scattering born series resonance theory collisions in laser fields optical-potential method coupled-channel methods close-coupling expansion
49.1
Electron–Atom and Electron–Ion Collisions
49.1.1 Low-Energy Elastic Scattering and Excitation In this section, we consider the process e C Ai ! e C Aj ;
(49.1)
where Ai and Aj are bound states of the target atom or ion and where the speed of the incident or scattered electron is of the same order or less than that of the target electrons actively involved in the collision. We assume initially that all relativistic effects can be neglected, which restricts the treatment to low-Z atoms and ions. The time-independent Schrödinger equation (TISE) describing the scattering of an electron by a target atom or ion containing N electrons and nuclear charge Z is HN C1 D E ;
(49.2)
where E is the total energy of the system. The .N C 1/electron nonrelativistic Hamiltonian HN C1 is given in atomic units by HN C1 D
N C1 X i D1
1 Z ri2 2 ri
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_49
C
N C1 X
1 ; r i >j D1 ij
(49.3) 725
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K. Bartschat et al.
where rij D jr i rj j, and r i and rj are the vector coordinates of electrons i and j relative to the origin of coordinates taken to be the target nucleus, which is assumed to have infinite mass. The target eigenstates ˚i and the corresponding eigenenergies wi satisfy the equation
where X N C1 x 1 ; x 2 ; : : : ; x N C1 represents the space and spin coordinates of all N C 1 electrons, x i r i i represents the space and spin coordinates of the i-th electron, and A is the operator that antisymmetrizes the first summation with respect to exchange of all pairs of electrons in accordance with the Pauli exclusion principle. The channel functions ˚ i , assumed to be n in number, are obtained by coupling (49.4) the orbital and spin angular momenta of the target states ˚i h˚i jHN j˚j i D wi ıij ; with those of the scattered electron to form eigenstates of the where HN is defined by Eq. (49.3) with N C 1 replaced total orbital and spin angular momenta L and S, their z comby N . The calculation of accurate target states is discussed ponents ML and MS , and the parity , where in Chap. 22. The solution of Eq. (49.2), corresponding to the process Eq. (49.1), has the asymptotic form (49.9) LM SM i ˚i 1 mi eiki z C r!1
X
2
L
˚j 1 mj fj i .; / eikj r : (49.5) 2
j
In Eq. (49.5), 1 mi and 1 mj are the spin eigenfunctions 2 2 of the incident and scattered electrons, where the direction of spin quantization is usually taken to be the incident beam direction, and fj i .; / is the scattering amplitude. The spherical polar coordinates of the scattered electron are denoted by r, , and . The wave numbers ki and kj are related to the total energy of the system by 1 1 E D wi C ki2 D wj C kj2 : 2 2
(49.6)
is conserved in the collision. The square-integrable correlation functions i allow for additional correlation effects not included in the first expansion in Eq. (49.8), which goes over a limited number of target eigenstates and possibly pseudostates. By substituting Eq. (49.8) into the Schrödinger equa tion (49.2), projecting onto the channel functions ˚ i and the square-integrable functions i , and eliminating the coefficients bij , we obtain n coupled integro-differential equations satisfied by the reduced radial functions Fij representing the motion of the scattered electron
The outgoing-wave term in Eq. (49.5) contains contributions from all target states that are energetically allowed; i.e., for which kj2 0. If the energy is above the ionization threshold, this includes target continuum states. For an atomic ion, a logarithmic phase factor is also needed (see Eq. (49.16)). The differential cross section for a transition from an initial state jii D jki ; ˚i ; 1 mi i to a final state jj i D 2 jkj ; ˚j ; 1 mj i is given by
S
`i .`i C 1/ 2.Z N / d2 2 C C ki Fij .r/ dr 2 r2 r Xn Vi` .r/F`j .r/ D2 `
Z1 C
o Ki` r; r 0 C Xi` r; r 0 F`j r 0 dr 0 : (49.10)
0
2
Here, `i is the orbital angular momentum of the scattered electron, while Vi` , Wi` , and Xi` are the local direct, (49.7) nonlocal exchange, and nonlocal correlation potentials, respectively. If the correlation potential, which arises from the and the total cross section is obtained by averaging over i terms in Eq. (49.8), is not included, Eqs. (49.10) are initial spin states, summing over final spin states, and inte- called the close-coupling equations. grating over all scattering angles. The direct potential can be written as In order to solve the Schrödinger equation to obtain the ˇ D scattering amplitude and cross section at low energies, we ˇ Vij .rN C1 / D ˚ i .x 1 ; : : : ; x N I rO N C1 N C1 /ˇ make a partial-wave expansion of the total wave function N X 1 N j .X N C1 / r rN C1 i D1 iN C1 n X ˇ E Fij .rN C1 / ˇ ˚ i .x 1 ; : : : ; x N I rO N C1 N C1 / DA .x ; : : : ; x I r O / ; (49.11) ˇ˚ 1 N N C1 N C1 j r kj dj i D jfj i .; /j2 ; d˝ ki
C
i D1 m X i D1
N C1
i .x 1 ; : : : ; x N C1 /bij ;
(49.8) where the integral is taken over all electron space and spin coordinates, except for the radial coordinate of the .N C1/-th
49 Electron–Atom, Electron–Ion, and Electron–Molecule Collisions
727
The dimensions of the matrices in these equations are na na , where na is the number of open channels at the energy unX max der consideration for the given . The hermiticity and timeVij .r/ D aij r 1 for r a ; (49.12) reversal invariance of the Hamiltonian ensures that K is real D1 and symmetric, while S is unitary and symmetric. The scattering amplitude defined by Eq. (49.5) can be where a is the range beyond which the orbitals in the target expressed in terms of the T -matrix elements. For a neutral states ˚i included in the first expansion in Eq. (49.8) are negtarget, ligible. The D 1 term in Eq. (49.12) gives rise, in second 12 X order, to the long-range attractive polarization potential 1 i `i `j .2`i C 1/ 2 fj i .; / D i k k i j 1˛ LS `i `j (49.13) V .r/ ! 4 r!1 2r .Li MLi `i 0jLML / seen by an electron incident on an atom. For an S-state atom 1 Si MSi mi jSMS in a state ˚0 , the dipole polarizability ˛ is given by 2
ˇ ˇ2 Lj MLj `j m`j jLML
4 1=2 PN ˇ ˇ X ˇh˚0 j 3 i D1 ri Y10 .rO i /j˚j iˇ 1 ˛D2 : (49.14) Sj MSj mj jSMS Tji Y`j m`j .; / wj w0 2 j (49.18) These long-range potentials have a profound influence on describes a transition from an initial state ˛i Li Si MLi MSi mi low-energy scattering. to a final state ˛j Lj Sj MLj MSj mj , where ˛i and ˛j repreThe exchange and correlation potentials, unlike the direct sent any additional quantum numbers required to completely potential, are both nonlocal, and the exchange potential vandefine the initial and final states, and .j1 m1 j2 m2 jJM / is ishes exponentially for large r. Explicit expressions for these a Clebsch–Gordan coefficient. The corresponding total cross potentials are too complicated to write down, except in the section, obtained by averaging over the initial magnetic case of e–H scattering, for which the direct and exchange quantum numbers, summing over the final magnetic quanpotentials were first given by Percival and Seaton [8]. In tum numbers, and integrating over all scattering angles, is practice, they are evaluated by general computer programs. X .2L C 1/.2S C 1/ ˇˇ ˇˇ2 The scattering amplitude and cross section can be obtot .i ! j / D 2 ˇT ˇ : tained by solving Eq. (49.10) for all relevant conserved ki LS 2.2Li C 1/.2Si C 1/ j i `i `j quantum numbers subject to the K -matrix asymptotic boundary conditions (49.19) electron. This potential has the asymptotic form
12
Fij ki r!1
sin i ıij C cos i Kij
for open channels .ki2 0/ I
It describes a transition from an initial target state ˛i Li Si to a final target state ˛j Lj Sj . In applications, it is also useful to define a collision strength by ˝.i; j / D ki2 .2Li C 1/.2Si C 1/tot .i ! j / ;
Fij 0
(49.20)
r!1
(49.15) which is dimensionless and symmetric with respect to interchange of the initial and final states denoted by i and j . For scattering by an ion, the above expression for fj i .; / is Here, modified by the inclusion of the Coulomb scattering amplitude when the initial and final states are identical. 1 z (49.16) i D ki r `i C ln.2ki r/ C i ; For incident electron energies insufficient to excite the 2 ki atom or ion, only elastic scattering is possible, and the above with z D Z N and i D arg .`i C 1 iz=ki /. The expressions simplify. Consider low-energy elastic electron S -matrix and T -matrix are related to the K -matrix defined scattering by a neutral atom in a 1 S ground state. Then the expression for the scattering amplitude Eq. (49.18) reduces by Eq. (49.15) through the matrix equations to I C iK 2iK 1
1 X ; T D S I D : S D f ./ D .2` C 1/ e2iı` 1 P` .cos / ; (49.21) I iK I iK 2ik (49.17) `D0 for closed channels .ki2 < 0/ :
49
728
K. Bartschat et al.
where D z=k, and .k 2 / is the analytic continuation of the quantum defects of the electron–ion bound states to positive energies. This quantum-defect theory enables spectroscopic observations of bound-state energies to be extrapolated to positive energies to yield electron–ion scat tan ı` D K11 : (49.22) tering phase shifts. For a negative ion, where the Coulomb potential is repulsive, the phase shift behaves as The corresponding expression for the total cross section is z then : (49.29) ı` ! exp 2 1 k!0 k 4 X tot D 2 .2` C 1/ sin2 ı` : (49.23) k This vanishes rapidly as k tends to zero, since z is now `D0 negative. A discussion of quantum-defect theory is given in The momentum-transfer cross section is defined as Chap. 50. Z D D 2 jf ./j2 .1 cos / sin d
where ` .D L D `i D `j / is the angular momentum of the scattered electron, k is its wave number, and the phase shift ı` can be expressed in terms of the K -matrix, which now has only one element, since na D 1, given by
49.1.2 Relativistic Effects for Heavy Atoms and Ions
0 1 4 X .` C 1/ sin2 .ı`C1 ı` / : D 2 k `D0
(49.24) As the nuclear charge Z of the target increases, relativistic effects become important even for low-energy scattering. There are two ways in which relativistic effects play a role. First, there is a direct effect corresponding to the relativistic distortion of the wave function describing the scattered electron by the strong nuclear Coulomb potential. Second, there is an indirect effect caused by the change in the charge distribution of the target due to the use of relativistic wave functions discussed in Chap. 23. We will concentrate on the direct effect in this section. For atoms and ions with small Z, the K -matrices can first be calculated in LS-coupling while neglecting relativistic ef(49.25) fects. The K -matrices are then recoupled to yield transitions between fine-structure levels. We introduce the pair-coupling scheme
This cross section is important when considering the diffusion of electrons through gases. At low incident electron energies, the behavior of the phase shift for an atom in an S-state is dominated by the long-range polarization potential Eq. (49.13). O’Malley et al. [9] showed that for s-wave scattering k cot ı0 satisfies the effective-range expansion 1 ˛ C 2k as 3as 2
˛k 2˛ 2 k ln C O k2 ; C 3as 16
k cot ı0 D
where as is the scattering length, while for ` 1
8 ` C 32 ` C 12 ` 12 C k 2 cot ı` D ˛
(49.26)
Li C S i D J i ;
J i C `i D K i ;
Ki C s D J ; (49.30)
Higher-order terms, which extend the energy range of usefulwhere J i is the total angular momentum of the target, ` i is ness of the expansion, were derived by Ali and Fraser [10]. Close to threshold, therefore, the total elastic cross section the orbital angular momentum of the scattered electron, s is its spin, and J is the total electronic angular momentum, has the form which together with the parity is conserved in the colli8 (49.27) sion. The transition from LS- to J `K-coupling involves the tot D 4as2 C 2 ˛as k C 3 recoupling coefficient When an electron is elastically scattered by a positive or negˇ ˇ 1 1 ative ion, these formulae for the low-energy behavior of the SI JMJ Œ.Li Si /Ji ; `i Ki ; I JMJ ˇˇ.Li `i /L; Si phase shift are modified. For scattering by a positive ion, 2 2 1 Seaton [11] showed that D Œ.2Ji C 1/.2L C 1/.2Ki C 1/.2S C 1/ 2
2 1 cot ı` .k/ D cot k ; (49.28) W .L`i Si Ji I Li Ki /W .LJSi I SKi / ; (49.31) 1 e2 2
49 Electron–Atom, Electron–Ion, and Electron–Molecule Collisions
where W .: : : / is a Wigner 6–j -symbol. The corresponding K -matrix elements transform as ˇ X ˇ 1 J Kij D Œ.Li Si /Ji ; `i Ki ; I JMJ ˇˇ 2 LS 1 SI JMJ Kij .Li `i /L; Si 2 ˇ ˇ
1 Lj `j L; Sj SI JMJ ˇˇ 2
1 Lj Sj Jj ; `j Kj ; I JMJ : 2 (49.32) This transformation was implemented in a computer program by Saraph [12, 13]. For intermediate-Z atoms and ions, relativistic effects can be included by adding terms from the Breit–Pauli Hamiltonian to the nonrelativistic Hamiltonian (Jones [14], Scott and Burke [15]). We write HNBPC1 D HNnrC1 C HNrelC1 ;
(49.33)
729
It describes a transition from an initial target state ˛i Ji to a final target state ˛j Jj . The corresponding collision strength is ˝.i; j / D ki2 .2Ji C 1/tot .i ! j / :
(49.35)
For high-Z atoms and ions, the Dirac Hamiltonian [16, 17] N C1 N C1 X X 1 Z (49.36) HNDC1 D c˛ p i C ˇ 0 c 2 C r r i i D1 i >j D1 ij must be used instead of Eq. (49.3), where ˇ 0 D ˇ 1 with ˛ and ˇ being the usual Dirac matrices. The expansion of the total wave functions for a particular JMJ takes the general form of Eq. (49.8). However, now the bound orbitals in the target and the correlation functions, as well as the orbitals representing the scattered electron, are represented by Dirac orbitals. These are defined in terms of large and small components P .r/ and Q.r/ by ! O / 1 Pa .r/ m .r; ; (49.37) .r; / D r Qa .r/ m .r; O /
where HNnrC1 is defined by Eq. (49.3) and HNrelC1 consists of one and two-body relativistic terms. The one-body terms are for the bound orbitals and ! (Sect. 22.1) O / 1 Pc .r/ m .r; ; (49.38) F .r; / D r Qc .r/ m .r; O / N C1 X 1 2 r4 (mass-correction term) I HNmass C1 D ˛ for the continuum orbitals. Here, a D n m, c D k m, and 8 i D1 i the spherical spinor is N C1 1 2 X 2 1 D1 X (Darwin term) I HN C1 D ˛ Z ri 1 8 r . r; O / D jj m Y`m` .; / 1 mi . / ; m `m i
m ` i i D1 2 2 m` mi N C1 X 1 @V 1 (49.39) HNsoC1 D ˛ 2 .` i si / (spin–orbit term) : 2 i D1 ri @ri where D j C 12 when ` D j C 12 , and D j 12 when 1 The two-body terms are less important and are usually not ` D j 2 . One can derive coupled integro-differential equations for included in collision calculations. The modified Schrödinger equation defined by Eq. (49.2), the functions Pc .r/ and Qc .r/ in a similar way to the with HN C1 replaced by HNBPC1 , is solved by adopting an ex- derivation of Eq. (49.10), except that these are now coupled pansion similar in form to Eq. (49.8) but now using the pair- first-order equations instead of coupled second-order equacoupling scheme in the definition of the channel functions tions. The K -matrix, and hence the S -matrix and T -matrix, and square-integrable functions. We then obtain coupled can be obtained from the asymptotic form of these equaintegro-differential equations similar in form to Eq. (49.10), tions. The total cross section in the jj -coupling scheme is from which the K -matrix, S -matrix, and T -matrix can be again given by Eq. (49.34), and the corresponding collision obtained. The corresponding total cross section in the pair- strength is given by Eq. (49.35). coupling scheme analogous to Eq. (49.19) is tot .i ! j / D
2ki2 .2Ji C 1/
X J
Ki Kj `i `j
ˇ2 ˇ ˇ ˇ .2J C 1/ˇTjJi ˇ :
49.1.3 Multichannel Resonance Theory
General resonance theories have been developed by (49.34) Fano [18, 19], Feshbach [20, 21], and Brenig and Haag [22].
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They are also discussed in Chap. 26. Here, we will limit our discussion to the effect that resonances have on electron collision cross sections. Following Feshbach, we introduce the projection operators P and Q, where P projects onto a finite set of low-energy channels in Eq. (49.8), and Q projects onto the orthogonal space. We limit our consideration to the space corresponding to a particular set of conserved quantum numbers . In this space, we have 2
P DP ;
2
Q DQ;
P CQ D1:
(49.40)
The Schrödinger equation (49.2) can then be written as P .H E/.P C Q/ D 0
(49.41)
side of Eq. (49.46). We find that the pole term on the righthand side of this equation gives rise to a Feshbach resonance, whose position is 1 1 Ei D "i C i ii D Ei;r ii ; 2 2
(49.47)
where the resonance shift is i D hi jQHP G0 PHQji i ;
(49.48)
and the resonance width is i D 2
Xˇ ˇhi jQHP j
0j i
ˇ2 ˇ :
(49.49)
j
The summation in this equation is taken over all continuum states corresponding to the operator on the left-hand side of Eq. (49.46), and these states are normalized to a ı-function Q.H E/.P C Q/ D 0 ; (49.42) in energy. In the neighborhood of the resonance energy Ei;r , the where we have omitted the subscript N C 1 on H and the S -matrix is rapidly varying with the form superscript on . After solving Eq. (49.42) for Q and ! substituting into Eq. (49.41), we find 1 1 i i 2 (49.50) S 02 ; S D S 0 I i 1 1 E Ei;r C 2 ii QHP E P D 0 ; P H PHQ Q.H E/Q (49.43) where S 0 is the slowly varying nonresonant or background S -matrix corresponding to 0 , and the partial widths i are where the term defined by and
Vopt D PHQ
1 QHP ; Q.H E/Q
(49.44)
1
hi jQHP j
0i
1
D i 2 i S 02 ;
(49.51)
called the optical potential, allows for propagation in the Q- where i i D 1. A corresponding resonant expression can be derived for the K -matrix (Burke [23]). space channels. Let us now diagonalize the S -matrix according to We now introduce the eigenfunctions i and eigenvalues "i of the operator QHQ by (49.52) S D A exp.2i/A | ; QHQi D "i i : (49.45) where A is an orthogonal matrix, while is a diagonal maIt follows that the discrete eigenvalues "i each give rise to trix whose diagonal elements, ıi , i D 1, . . . , na , are called poles in Vopt at "i . If the energy E is in the neighborhood of the eigenphases. If we define the eigenphase sum ısum by an isolated pole or bound state "i , we can rewrite Eq. (49.43) na X as ıi ; (49.53) ısum D 1 0 i D1 X jj ihj j @PHP PHQ QHP E AP "j E we can show from Eq. (49.50) that in the neighborhood of the j ¤i resonance the eigenphase sum follows a Breit–Wigner form ji ihi j QHP ; (49.46) D PHQ ! "i E 1 1 2 i ; (49.54) ısum .E/ D ı0;sum .E/ C tan Ei;r E where the rapidly varying part of the optical potential has been separated and put on the right-hand side of Eq. (49.46). This equation can be solved by introducing Green’s func- where ı0;sum is the slowly varying background eigenphase tion G0 and the solutions 0j of the operator on the left-hand sum obtained by replacing S by S 0 in Eqs. (49.52) and
49 Electron–Atom, Electron–Ion, and Electron–Molecule Collisions
731
Im(k)
Time delay (arb. u.) 1e + 06 1e + 05
Bound states 10 000 1000 Virtual state
100 Eigenphase sum (rad)
Re(k)
1 Resonances 0
Fig. 49.2 Structure of the S -matrix as a function of complex wavenumber, k. The features indicated correspond to poles of different physical meaning
–1
–2 0.8
0.9
1
1.1
1.2 Energy (eV)
Fig. 49.1 Comparison of the longest time delay (upper panel) with the eigenphase sum, given modulo (lower panel), at the same energy for a series of resonances found in e–NC 2 scattering. Note that with higher energy resolution, some of the time delay peaks can be split to reveal multiple resonances. Adapted from Little and Tennyson [25] with thanks to Sergei Yurchenko
(49.53). It follows from Eq. (49.54) that the eigenphase sum increases by radians in the neighborhood of the resonance energy. If there are m resonances, which may be overlapping, Eq. (49.54) generalizes to ! m 1 X 1 2 i tan ısum .E/ D ı0;sum .E/ C : (49.55) Ei;r E i D1
relative partial width associated with each channel. In this formulation, resonances have a Lorentzian shape as a function of energy. Figure 49.1 compares the structure of the eigenphase sum (plotted modulo ) with that of the longest time delay for a series of resonances; in both cases, the positions of the resonances are readily apparent. Analysis of the S -matrix as function of complex k also yields information of bound states, virtual states, and resonances. Figure 49.2 illustrates the location of these various features. Bound states are located as poles on the positive imaginary axis, as here 12 k 2 is negative. Virtual states, which can lead to pronounced features in low-energy scattering [26], form poles on the negative imaginary axis. Resonances give poles that are symmetrically placed on either side of the negative imaginary axis. Under these circumstances, the energy is complex; see Eq. (49.47). It is possible to characterize resonances by searching the S -matrix as a function of complex k [27], but in practice this is rarely done.
This result has proven to be useful in analyzing closely spaced resonances to obtain the individual resonance positions and widths. However, overlapping resonances can be difficult to fit using a Breit–Wigner form. An alternative approach, due 49.1.4 Solution of the Coupled Integro-Differential Equations originally to Smith [24], is based on the classical concept of time delay during a collision. Quantum-mechanically, the time delay in a collision can be represented in terms of the This section considers methods that have been developed for solving the coupled integro-differential equations (49.10). S -matrix and the time operator as These equations arise in low-energy electron–atom and S (49.56) electron–ion collisions. Section 49.2.2 shows that similar Q D i„S : E equations also arise in low-energy electron–molecule colliThe time delays are obtained by diagonalizing Q with reso- sions in the fixed-nuclei approximation. Thus, the methods nances manifesting themselves as the longest time delay or discussed in this section are also applicable to electron– largest eigenvalue. The corresponding eigenvector gives the molecule collisions.
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Fig. 49.3 The R-matrix method
Intermediate region: a < r < b; long-range multipole potentials, but no exchange between the projectile and the target electrons
Inner region: 0 r0 , ˇ D A
N X
˚i .˝/ fi .r/Iiˇ gi .r/Jiˇ :
(50.7)
i D1
Here, A represents an antisymmetrization operator, ˚i .˝/ is a channel eigenfunction of all spatial and spin degrees of freedom except the radial coordinate of the outermost electron. Note that the constant, energy-dependent matrices Iiˇ and Jiˇ determined by numerical solution of the Schrödinger equation inside r < r0 are not unique, because right-multiplication by any nonsingular matrix gives an equivalent set of linearly independent solutions. One quite common way
to standardize those solutions is to right-multiply by I 1 ˇi 0 , which gives the reaction matrix solutions, whose form at r > r0 is .K/
i 0
DA
N X i D1
˚i .˝/Œfi .r/ıi i 0 gi .r/Ki i 0 :
(50.8)
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For a time-reversal invariant system with a hermitian Hamil- where i D k1 ln 2ki `i2 C `i .ki /. This last expression i tonian, the reaction matrix K is real and symmetric, which shows the unphysical exponential divergence of the “energyleads to a unitary, symmetric scattering matrix, smooth” K-matrix solutions at r ! 1. But since the exponentially growing terms are explicitly displayed, it is simple 1 C iK SD : (50.9) to analytically find the N No -matrix of constants needed to 1 iK right-multiply the radial solution matrix to obtain the “physFor electric dipole photoabsorption processes from an initial ical” K-matrix or S-matrix solutions, state init to the final states described by the MQDT soluX .K/ .phys/ .K/ D i 0 Bi 0 ;j : j will depend on reduced dipole tions i 0 , the cross sections E D .K/ i0 K .1/ init matrix elements, di 0 D i 0 jjr jj , which are also norline or two of algebra, we can see that if a mally smooth functions of energy except for isolated poles. Specifically, with BPP (In cases where the poles are problematic or inconvenient we write B D BQP , then to have a physical K-matrix soluto deal with, an alternative formulation of the MQDT equa- tion in the open channels only, with exponential decay in the tions in terms of nonsingular eigenchannel quantum defects closed channels asymptotically, we require BPP D 1PP , and ˇ of the K-matrix, its orthonormal eigenvectors Uiˇ , and BQP i 0 ;j D .tan C KQQ /1 KQP , where D . `/ reduced eigenchannel dipole elements dˇ , whose physical is the diagonal matrix of negative energy phase parameters. interpretation was stressed by Ugo Fano, [15] can be con- Thus, we have algebraically enforced the required exponenvenient to work with instead of fK; d K g directly. See, for tial decay in the closed channels, giving the (smaller) No No instance, Eqs. 23–32 of [4].) Everything written above in this physical reaction matrix paragraph should look familiar to the reader conversant with K phys D KPP KPQ .tan C KQQ /1 KQP ; (50.11) ordinary scattering theory, but, in fact, there is a fundamental difference with quantum defect theory. In ordinary scattering one of the most fundamental equations of MQDT. In particutheory, the sum over channels i would only include the en- lar, the inverted matrix in the second term, which becomes ergetically open channels, but in quantum defect theory, this nearly singular at closed channel resonances, displays the sum is extended to include (weakly) closed channels as well. expected resonance structure explicitly from the infinite The inclusion of closed channels (iQ) has important number of Rydberg resonances that are attached to closedmathematical and physical implications. Most importantly, channel thresholds. Moreover, if we denote one recalls that .fi ; gi / are exponentially divergent at r ! d K;phys D dPK dQK .tan C KQQ /1 KQP (50.12) 1, which would seem initially undesirable. However, this exponential growth will be eliminated later in the calculation as the vector of reduced dipole matrix elements connecting of observables, as will be demonstrated shortly. Neverthe- an unpolarized initial state of total angular momentum J0 to less, one sometimes refers to this intermediate quantity, the the final states of angular momentum J described by this phys MQDT S-matrix, as the “unphysical” or “single-pass” S- K -matrix, then the total photoionization cross section at matrix. The reason for introducing this intermediate quantity frequency ! is given in atomic units by is because it normally has far less energy dependence than 4 2 !˛ tot d K;phys .1 C K phys K phys /1 d K;phys : the physical scattering matrix in the open channel subspace, D 3.2J C 1/ 0 which we denote as S phys . This will be demonstrated with an (50.13) example later, but first we show the rest of the steps needed The simplest nontrivial example of MQDT is a two-channel to calculate scattering or photoabsorption observables using system with two ionization thresholds at energies E1 ; E2 . this smooth information in K or S. The asymptotic forms For such a system, there are three energy ranges of inter.K/ of the MQDT solutions i 0 , at an energy E where the est, (i) E < E1 < E2 , where both channels are closed, and channel space includes both open ("i > 0; iQ) and closed only discrete bound states exist; (ii) E1 < E < E2 where one ("i < 0; iP ) channels, can be written in a partitioned matrix channel is open and one is closed, and all states have a mixed notation as follows continuum and bound state character (autoionization region, " .K/ # or Fano–Feshbach resonance region); and (iii) E1 < E2 < E i P ! ! r 1 where both channels are open, and there are only scattering .K/ i Q i 0 processes such as 1 ! 2. In the discrete region (i), there exists 2 q h 3 2m 1 only a set of discrete energy levels for which the wavefuncsin ki r C ki ln r C i ıi i 0 ki i 6 7 tion decays exponentially as r ! 1 in both channels. The 6 7 C cos ki r C k1i ln r C i Ki i 0 6 7 allowed energies are determined by a bound state quantizaq ; (50.10) 6 7 m 6 7 1 i i r 0 tion condition that enforces the exponential decay, and reads D r e Œsin . ` /ı i i i i 4 5 i i C cos .i `i /Ki i 0 C O.r i e i r /
det.tan C K/ D 0 :
(50.14)
50 Quantum Defect Theory
1.0
755
have been chosen arbitrarily equal 0 1:4 B K D @ 1:0 0:5
vs
0.8
0.6
0.4
0.2
0 1.7
1.9
2.1
2.3
2.5
2.7 vd
Fig. 50.1 Lu–Fano plot of the 1 P o bound levels of Sr, associated with the mixing of the 5snp and 4d np channels. Taken from Vaillant et al. [17]
for this illustrative example to 1 1:0 0:5 C 0:9 1:4 A: 1:4 3:8
(50.15)
The atomic photoionization spectrum is determined by this K-matrix, plus three independent electric dipole matrix elements, chosen for this model to equal fd1K ; d2K ; d3K g D f3:0; 4:0; 5:0g. Thus, in the notation of Eq. 50.13, we would say that in this energy range, where channel 1 is open and channels 2 and 3 are weakly closed, the lone channel belonging to the P subspace is channel 1, while channels 2 and 3 belong to the Q subspace. Hence, at energies just below the threshold E2 , there is only a single open channel (#1) with a complicated energy-dependent elastic scattering (physical) phase shift determined analytically, using Eq. 50.11 above, to be K13 K32 2 K 12 K33 CT3 K13 K31 phys ; tan ı K11 D K11 K32 K33 C T3 K22 C T2 KK23CT 33 3 (50.16) where Ti tan.i /. In the absence of channel interactions, a multichannel Rydberg spectrum would show a separate Rydberg series converging to every ionization threshold. But when shortrange channel interactions are taken into account, as they are via the MQDT K-matrix, a complicated series of interacting resonances that mutually perturb each other in their energy eigenvalues and autoionization linewidths. In the absence of channel 3, one would observe a simple regular Rydberg series of levels converging to the threshold E2 , with autoionization linewidths decreasing with the principal quantum number n in proportion to .n 2 /3 . But the presence of an interacting Rydberg series attached to threshold E3 produces qualitative modifications of the spectrum and phenomena such as Fano linewidth “q-reversal” and “nearzero-width” states. One of the latter is clearly apparent in the model photoabsorption spectrum shown in Fig. 50.2 that is blown up near the complex resonance that modifies the Rydberg series all the way from about 2 25 to 2 45.
In general, these roots must be found by a numerical search, and it must be remembered that there is a constraint relating the effective quantum numbers 1 and 2 , namely E D E1 212 D E2 212 . The roots of the determinantal 1 2 equation obey a double periodicity in 1 and 2 modulo an integer, which is strictly obeyed in the limit that the reaction matrix is energy independent. This double periodicity is emphasized in a popular representation of bound state energy levels for systems with two closed channel thresholds, which has come to be called the “Lu–Fano plot” [16]. The basic idea is to plot 1 or 1 mod 1/ in the most weakly closed channel as a function of 2 , and then one sees that all levels fall on a single universal curve in this diagram, defined by the roots of the determinantal Eq. (50.14). To find any specific level one must enforce also the constraint equation relating 1 and 2 through energy conservation. Fig. 50.1 shows an example of a Lu–Fano plot for bound 1 P o energy levels of atomic Sr lying below the 5s ionization threshold. In the autoionization region (ii), there is one channel (#2) that is energetically closed and one channel (#1) that is open. As an example of the type of spectrum you can expect in this region of a two-channel problem, consider the following Kmatrix (dimensionless) and reduced dipole matrix elements Semi-Empirical MQDT Versus Ab Initio MQDT (in a.u.). Historically, some of the early uses of MQDT were focused on developing models that could concisely characterize enThe following model problem illustrates the value of hav- ergy levels, autoionization widths, and bound and continuum ing the multichannel Rydberg structure “built-in” to the oscillator strengths. When the number of channels N is not theoretical description. Consider a system with three ion- too large, say, less than five or six, for a given overall symization thresholds at energies Ei D f0; 0:10; 0:105g a.u., and metry (total angular momentum, parity, etc.), then it is often an energy-independent 3 3 K-matrix. That symmetric K- possible to simply perform a least squares fit and determine matrix is defined by six independent real numbers, which an optimized N N K-matrix and a set of N electric dipole
50
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d K,phys (1 + K phys ) –1 d K,phys 70 60 50 40 30 20 10 0 20
25
30
35
40 45 50 Effective quantum number v 2
Fig. 50.2 Example photoabsorption spectrum for the three-channel model in the vicinity of a complex resonance, in the energy range below the second ionization threshold. This spectrum can be readily recomputed by the reader, for instance, by implementing Eqs. 50.12 and 50.13 with D i a 2 by 2 diagonal matrix in the closed-channel subspace i D 2; 3. Recall that the effective quantum number i is defined only in the closed channels, where it equals i D Œ2.Ei E/1=2
matrix elements connecting the K-matrix states to an initial state whose photoabsorption is of interest. This would involve fitting N.N C 1/=2 C N constants, if no theoretical input is used to reduce the number of parameters, and if their energy dependence is negligible over the energy range being fitted. Many early studies were carried out that determined these short-range MQDT parameters by fitting, as in [18], which gave fitted reaction matrices and dipole matrix elements that any experimentalist or theorist could use to accurately describe complex multichannel Rydberg spectra, e.g., for the rare gas atoms or for the alkaline earth metal atoms [19]. Other fits were carried out for the spectra of atomic silicon in a similar manner [20, 21] and, more recently, for several different symmetries of atomic strontium, by [17]. Armed with these fitted K-matrices and the experimentally known ionization thresholds, this allows immediate calculation of an infinite number of spectral levels, autoionizing resonance positions and widths, and spectral lineshapes. When the number of channels grows much larger than about five or six, however, the fitted K-matrices become far more difficult to determine from any manageable data set, and moreover, the fits tend to be nonunique. For multichannel atomic Rydberg spectra, it is then advantageous to directly compute the smooth MQDT information through nearly ab initio computations, followed by small empirical adjustments if extremely high accuracy is required in any particular application. For instance, to describe the photoionization of the atomic barium ground state at energies up to the 6p threshold, this involves determination of a 13 13 reaction matrix and 13 dipole amplitudes, which amounts to
a total of 104 parameters. Various methods for computing the MQDT parameters from first principles have been developed and applied over the years, including the Fano–Lee iterative eigenchannel R-matrix theory, reformulated as a noniterative variational R-matrix treatment by a few different groups, as was reviewed in [22] with applications to atoms in several different columns of the periodic table. Another successful treatment used the relativistic random phase approximation to compute the MQDT K-matrices needed to describe Rydberg spectra in each of the rare gas atoms from Ne to Xe; that approach was developed by Johnson and coworkers, who in [23] tabulated the MQDT information needed to describe J D 1 Rydberg bound and autoionizing states to nearspectroscopic accuracy.
50.2.3 Molecular Ionization Channels Treated by the Rovibrational Frame Transformation The starting point of any multichannel quantum defect calculation usually starts with a decision about which fragmentation channels are to be treated among the “open or weakly closed” set. Usually, all other channels are declared to be “strongly closed” in the energy range being studied, meaning that the enlarged or “unphysical” reaction matrix would have a dimension equal to the number of open or weakly closed channels for the specified symmetry being considered. For singly excited Rydberg states of a molecule such as H2 , the ionization channels jii (ignoring fine and hyperfine structure) would ˛be characterized by rovibrational ˇ eigenstates ˇvi .NiC `/JM of the molecular ion HC 2 coupled to the quantum numbers of the angular (`) and spin degrees of freedom of the outermost electron. We focus here on the spatial wavefunction of the singlet ungerade channels (omitting the singlet spinor common to all following expressions), which are predominantly p-waves (` D 1), and for a specific total angular momentum (J ) and projection .M /. The ionization threshold for channel jii occurs at an energy Ei . Owing to the small rotational and vibrational spacings, the set of ionization thresholds Ei in a molecule are comparatively dense compared to simple atoms. Consequently, most MQDT treatments in molecules have a rather large number of channels in the reaction matrix K discussed above, e.g., 20–100 channels is not uncommon for diatomic molecules, and for polyatomics such as H3 , the dimension of K is usually at least an order of magnitude larger. The structure of D the reaction matrix K can then E be represented C C O 0 as Ki i 0 D vi .Ni `/JM jKjvi .Ni 0 `/JM , and once Ki i 0 are known, and once the set of transition dipole matrix elements di 0 g connecting these final states to the molecular ground state are known, the standard formulas of MQDT presented
50 Quantum Defect Theory
757
above describe the ground state photoabsorption spectra over a wide range of energies. Determination of this rather large amount of information contained in Ki i 0 and di 0 g might seem a daunting challenge, however, as there are at least dozens of interacting channels, and those channel interactions are known to be quite strong for low `. In an exact theory, determination of fKi i 0 , di 0 g g would require solution of a large system of coupled equations, which has been formulated in principle as a “rovibrational close coupling theory” but rarely implemented in practice. Instead, the frame transformation approximation devised by Fano, Jungen, and their collaborators permits an effective and simple solution. This approximation is based on the Born–Oppenheimer separation of the electronic and nuclear motions. The idea is that the short-range electronion reaction operator K should be very nearly diagonal in a representation with a fixed angular momentum projection on the internuclear axis () and with a fixed value of the internuclear distance (R). In physical terms, the conservation of R in a short-range collision follows because the electron zips in and out of the short-range reaction volume so fast that the nuclei do not have time to move appreciably. This approximation identifies the structureD of the reaction maE O 0 ; `0 ' trix in the jR; `i representation as R; `jKjR ı.R R0 /ı0 tan ` .R/. Here, ` .R/ is the body-frame quantum defect function that can be extracted from accurate potential curves Un` .R/ of neutral Rydberg states computed in ab initio quantum chemistry, and from the ionic potential curve U C .R/, through the Rydberg formula at each R, i.e., in a.u., Un` .R/ U C .R/
1 : 2Œn ` .R/2
(50.17)
As written here, the weak n-dependence of the fixed-nuclei quantum defect function has been neglected, although in more sophisticated versions of this rovibrational frame transformation theory, it can also be incorporated into the theory. The main upshot is that the spectroscopy and scattering observables can be determined to an excellent approximation by now solving the MQDT equations on an arbitrarily fine energy mesh, using this reaction matrix computed by simple quadrature, XZ ˛ ˝ dR vi .NiC `/JM jR; ` Ki i 0 D
(A more robust singularity-free version of Eq. (50.18) that avoids divergences of the tangent functions can be formulated instead in terms of the sin ` .R/ and cos ` .R/ .N C / matrix elements.) Here, v .R/ is an ionic vibrational wavefunction, while the angular frame transformation coefficient relating Hund’s case d to b is known to be given in terms of a Clebsch–Gordan coefficient, e.g., in H2 , ˝
˛.`J / N C j D p ˛ (50.20) ˝ C 2 ı ;0 .`J /N C 0j` ; J : .1/J CN Examples of the power of this method, and the simplicity of the implementation, can be found in [24, 25]. In Fig. 50.3 shown here, the HD photoionization spectrum is plotted in a narrow energy range that is dominated by three-channel physics, and the qualitative resemblance to the three-channel model spectrum above is readily apparent. The ability of the rovibrational frame transformation theory, in combination with MQDT, to describe a complex experimental photoabsorption spectrum is also evident in Fig. 50.3. In addition to solving problems of molecular photoabsorption, in the presence of autoionization and predissociation, this basic formulation has been adopted to treat scattering processes, such as rotational and vibrational excitation of molecules, as well as dissociative recombination. Extensions of these methods to include the effects of hyperfine coupling have also been implemented by Jungen, Merkt, and coworkers, [26–28], although this involves more complexity in the angular momentum algebra, as one might expect. Length limitations do not permit a discussion of a key extension to frame transformation theory, namely the idea of the local frame transformation that is needed when a change of coordinates is required. This issue arises, for instance, for the theoretical description of electric field effects on photoabsorption spectra. A mostly analytical theory, originally formulated by Fano and Harmin, was later improved by Giannakeas et al. [30–33]. The theory involves a sphericalto-parabolic coordinate transformation of the short-range spherical reaction matrix of MQDT, and using related information it can obtain sub-MHz accuracy on Stark-shifted levels of multichannel atoms and molecules.
50.2.4 Ultracold Atom–Atom Collisions in MQDT ˝ ˛ C 0 tan ` .R/ R; `jvi .Ni 0 `/JM ; (50.18) One of the most exciting developments in ultracold physics using the well-known transformation between the two rep- in recent decades has been the demonstration of controllable atom–atom scattering lengths and many-body interactions resentations, namely the eigenchannel (jR; ˇ `i,C in Hund’s ˛ case b), and the fragmentation channel (ˇvi .Ni `/JM , in via magnetic (and other) Fano–Feshbach resonances. [34] Quantum defect theory has been generalized to effectively Hund’s case d) describe atom–atom van der Waals and many other differ˛ ˝ C ˛.`J / ˝ C .N C / : (50.19) ent types of long-range interactions [6–8, 10, 35] going v.N `/JM jR ; ` D v .R/ N j
50
758 Fig. 50.3 Photoionization spectrum of the HD molecule in its ground rovibrational state, in the energy range that exhibits a complex resonance. Shown are both the experimental measurement of [29] and the result of a rovibrational frame transformation theory calculation with no adjustable parameters
C. H. Greene
a Ion signal (arb. u.) 0.4 0.3 0.2 0.1 0 790.0
790.1
790.2
790.3
790.4
790.5
790.6 Wavelength (Å)
790.2
790.3
790.4
790.5
790.6 Wavelength (Å)
b Oscillator strength (arb. u.) 8 6 4 2 0 790.0
790.1
far beyond the original scope [1], which was limited to Coulombic long-range potentials. One of the most fruitful generalizations has been to treat ultracold collisions in the sub-mK range of temperatures, which can give a convenient parameterization of field-dependent S-wave scattering lengths a.B/ limk!0 tan kı`D0 and P -wave scattering volumes V .B/ limk!0 tankı3`D1 . In ultracold physics, e.g., relevant to Bose–Einstein condensation (BEC) and degenerate Fermi gases (DFG), it is particularly crucial to accurately describe the field-dependent Fano–Feshbach resonances, and this can be treated compactly using generalized MQDT [9, 10, 13, 35]. When quantum defect theory is generalized to handle a non-Coulombic long-range potential v LR .r/, a first key step is to identify a pair of independent solutions ff`0 ."; r/; g`0 ."; r/g to the radial Schrödinger equation of that potential, which are analytic functions of the energy " near the potential asymptote ." D 0/. These need not be fregular, irregularg, respectively. Analyticity in " is usually achieved by choosing them to be energy independent in the small r limit. These issues, and other details such as the modifications of the usual Wigner threshold laws in the presence of long-range power law potentials, are discussed in [10]. Figure 50.4 gives one simple example of the power of MQDT in the context of ultracold atom–atom collisions, specifically for 85 Rb 85 Rb s-wave collisions in the f D 2, mf D 2 state. In the range of energies and magnetic fields shown, the
Energy-dependent scattering length (arb. u.) 100 000
50 000
0
–50 000
–100 000
0
2
4
6
8 10 Energy (μK)
Fig. 50.4 The S-wave scattering length between two 85 Rb atoms, each in their f D 2, mf D 2 state, is shown as a function of energy for three different magnetic fields, close to the broad Fano–Feshbach resonance at Bres D 155G. These curves are all computed using multichannel quantum defect theory (MQDT), with a field-independent and energy-independent reaction matrix that was obtained using the methods described in [9]
physics can be described accurately using a 5 by 5 reaction matrix K that is independent of energy and field strength. The reaction matrix in this case was determined by solving the five coupled radial Schrödinger equations with a finiteelement R-matrix method. Note that over the energy range shown, there is only one energetically open collision chan-
50 Quantum Defect Theory
nel, and for this symmetry there are four closed channels, which are responsible for the Fano–Feshbach resonance apparent in Fig. 50.4.
References 1. Seaton, M.J.: Quantum defect theory. Rep. Prog. Phys. 46, 167– 257 (1983) 2. Fano, U.: Quantum defect theory of l uncoupling in H2 as an example of channel-interaction treatment. Phys. Rev. A 2, 353–365 (1970) 3. Jungen, C., Dill, D.: Calculation of rotational-vibrational preionization in H2 by multichannel quantum defect theory. J. Chem. Phys. 73, 3338–3345 (1980) 4. Greene, C.H., Jungen, C.: Molecular applications of quantum defect theory. Adv. At. Mol. Phys. 21, 51–121 (1985) 5. Gallagher, T.F.: Rydberg Atoms. Cambridge Monographs on Atomic. Cambridge University Press, Cambridge (2005) 6. Greene, C.H., Fano, U., Strinati, G.: General form of the quantumdefect theory. Phys. Rev. A 19, 1485–1509 (1979) 7. Greene, C.H., Rau, A.R.P., Fano, U.: General form of the quantum-defect theory. II. Phys. Rev. A 26, 2441 (1982) 8. Watanabe, S., Greene, C.H.: Atomic polarizability in negative-ion photodetachment. Phys. Rev. A 22, 158–169 (1980) 9. Burke, J.P., Greene, C.H., Bohn, J.L.: Multichannel cold collisions: simple dependences on energy and magnetic field. Phys. Rev. Lett. 81, 3355 (1998) 10. Ruzic, B.P., Greene, C.H., Bohn, J.L.: Quantum defect theory for high-partial-wave cold collisions. Phys. Rev. A 87, 032706 (2013) 11. Mies, F.H., Raoult, M.: Analysis of threshold effects in ultracold atomic collisions. Phys. Rev. A 62, 012708 (2000) 12. Gao, B.: Quantum-defect theory of atomic collisions and molecular vibration spectra. Phy. Rev. A 58, 4222–4225 (1998) 13. Gao, B.: Solutions of the Schrödinger equation for an attractive 1/r 6 potential. Phys. Rev. A 58, 1728–1734 (1998) 14. Li, W.H., Mourachko, I., Noel, M.W., Gallagher, T.F.: Millimeterwave spectroscopy of cold Rb Rydberg atoms in a magneto-optical trap: Quantum defects of the ns, np, and nd series. Phys. Rev. A 67, 052502 (2003) 15. Fano, U.: Unified treatment of perturbed series, continuous spectra, and collisions. J. Opt. Soc. Am. 65(9), 979–987 (1975) 16. Lu, K.T., Fano, U.: Graphic analysis of perturbed Rydberg series. Phys. Rev. A 2(1), 81 (1970) 17. Vaillant, C.L., Jones, M.P.A., Potvliege, R.M.: Multichannel quantum defect theory of strontium bound Rydberg states. J. Phys. B 47, 199601 (2014) 18. Aymar, M.: Rydberg series of alkaline-earth atoms Ca through Ba – the interplay of laser spectroscopy and multichannel quantum defect analysis. Phys. Rep. 110, 163–200 (1984) 19. Gounand, F., Gallagher, T.F., Sandner, W., Safinya, K.A., Kachru, R.: Interaction between two Rydberg series of autoionizing levels in barium. Phys. Rev. A 27, 1925–1938 (1983) 20. Ginter, D.S., Ginter, M.L.: Transitions to and 3-limit, 4-channel representations for the msmp3 , mpnd J=3o levels in Si-I and Sn-I. J. Chem. Phys. 85, 6536–6543 (1986)
759 21. Eiles, M.T., Greene, C.H.: Ultracold long-range Rydberg molecules with complex multichannel spectra. Phys. Rev. Lett. 115, 193201 (2015) 22. Aymar, M., Greene, C.H., Luc-Koenig, E.: Multichannel Rydberg spectroscopy of complex atoms. Rev. Mod. Phys. 68, 1015 (1996) 23. Johnson, W.R., Cheng, K.T., Huang, K.N., LeDourneuf, M.: Analysis of Beutler-Fano autoionizing resonances in the rare-gas atoms using the relativistic quantum-defect theory. Phys. Rev. A 22, 989– 997 (1980) 24. Greene, C.H., Jungen, C.: Molecular applications of quantum defect theory. Adv. At. Mol. Phys. 21, 51–121 (1985) 25. Child, M.S.: Theory of Molecular Rydberg States. Cambridge University Press, Cambridge (2011) 26. Osterwalder, A., Wuest, A., Merkt, F., Jungen, C.: High-resolution millimeter wave spectroscopy and multichannel quantum defect theory of the hyperfine structure in high Rydberg states of molecular hydrogen H2 . J. Chem. Phys. 121, 11810–11838 (2004) 27. Liu, J., Sprecher, D., Jungen, C., Ubachs, W., Merkt, F.: Determination of the ionization and dissociation energies of the deuterium molecule (D2 ). J. Chem. Phys. 132, 154301 (2010) 28. Cheng, C.-F., Hussels, J., Niu, M., Bethlem, H.L., Eikema, K.S.E., Salumbides, E.J., Ubachs, W., Beyer, M., Holsch, N., Agner, J.A., Merkt, F., Tao, L.-G., Hu, S.-M., Jungen, C.: Dissociation energy of the hydrogen molecule at 109 accuracy. Phys. Rev. Lett. 121, 013001 (2018) 29. Greetham, G.M., Hollenstein, U., Seiler, R., Ubachs, W., Merkt, F.: High-resolution VUV photoionization spectroscopy of HD between the X2 ˙gC C D 0 and C D 1 thresholds. Phys. Chem. Chem. Phys. 5(12), 2528–2534 (2003) 30. Fano, U.: Stark-effect of non-hydrogenic Rydberg spectra. Phys. Rev. A 24, 619–622 (1981) 31. Harmin, D.A.: Theory of the Stark-effect. Phys. Rev. A 26, 2656– 2681 (1982) 32. Robicheaux, F., Giannakeas, P., Greene, C.H.: Schwingervariational-principle theory of collisions in the presence of multiple potentials. Phys. Rev. A 92, 022711 (2015) 33. Giannakeas, P., Greene, C.H., Robicheaux, F.: Generalized localframe-transformation theory for excited species in external fields. Phys. Rev. A 94, 013419 (2016) 34. Chin, C., Grimm, R., Julienne, P., Tiesinga, E.: Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82(2), 1225–1286 (2010) 35. Gao, B., Tiesinga, E., Williams, C.J., Julienne, P.S.: Multichannel quantum-defect theory for slow atomic collisions. Phys. Rev. A 72, 042719 (2005)
Chris H. Greene Chris H. Greene earned his PhD in 1980 from the University of Chicago following undergraduate work in Nebraska. After postdoctoral work at Stanford, he held faculty positions in physics departments at Louisiana State University and the University of Colorado, and is currently at Purdue University. His research is in atomic and molecular and few-body theoretical physics.
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Positron Collisions Joshua R. Machacek
, Robert P. McEachran, and Allan D. Stauffer
Contents 51.1 51.1.1 51.1.2
Scattering Channels . . . . . . . . . . . . . . . . . . . . . 761 Positronium Formation . . . . . . . . . . . . . . . . . . . . 762 Annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . 762
51.2
Theoretical Methods . . . . . . . . . . . . . . . . . . . . 763
51.3 51.3.1 51.3.2 51.3.3 51.3.4 51.3.5 51.3.6
Particular Applications Atomic Hydrogen . . . . . Noble Gases . . . . . . . . Other Atoms . . . . . . . . Molecular Hydrogen . . . Water . . . . . . . . . . . . Other Molecules . . . . . .
51.4
Binding of Positrons to Atoms . . . . . . . . . . . . . . 768
51.5
Positronium Scattering . . . . . . . . . . . . . . . . . . . 768
51.6
Antihydrogen . . . . . . . . . . . . . . . . . . . . . . . . . 768
51.7
Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 768
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765 765 765 766 766 767 767
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Abstract
The positron is the antiparticle of the electron, having the same mass but opposite charge. Positrons undergo collisions with atomic and molecular systems in much the same way as electrons do. Thus, the standard scattering theory for electrons can also be applied to positron scatterJ. R. Machacek () Research School of Physics, Australian National University Canberra, ACT, Australia e-mail: [email protected] R. P. McEachran Research School of Physics, Australian National University Canberra, Australia e-mail: [email protected] A. D. Stauffer Dept. of Physics & Astronomy, York University Toronto, ON, Canada e-mail: [email protected]
ing. However, there are a number of important differences from electron scattering, which we outline below. Since the positron is a distinct particle from the atomic electrons, it cannot undergo an exchange process with the bound electrons during a collision, as is possible with electrons. Thus, the nonlocal exchange terms that arise in the description of electron scattering are not present for positrons. This leads to a simplification of the scattering equations from those for electrons. However, there are scattering channels available with positron scattering that do not exist with electrons. These are dealt with in Sect. 51.1. Historically, beams of low-energy positrons were difficult to obtain and, consequently, there is considerably less experimental data available for positrons than for electrons. This was particularly true for quantities that required large incident positron fluxes, such as differential scattering cross sections and coincidence parameters. The recent development of cold trap-based positron beams with high resolution and high brightness by the San Diego group [1] has enabled significant progress in experimental positron scattering, which is discussed in Sect. 51.3. Throughout this chapter we will employ atomic units unless otherwise noted. Keywords
positron differential elastic excitation ionization positronium formation total cross sections
51.1 Scattering Channels Positrons colliding with atomic and molecular systems have the same scattering channels available as for electrons, viz., elastic, excitation, ionization, and for molecules, dissociation. However, two channels exist for positrons that do not exist for electrons, viz., positronium formation and annihilation.
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_51
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Positronium Formation
target, then
Positronium, a bound state of an electron–positron pair (Chap. 28), can be formed during the collision of a positron with an atomic or molecular target. The positronium atom (Ps) can escape to infinity leaving the target in an ionized state with a positive charge of 1. Thus, this process can be difficult to distinguish experimentally from direct ionization, where both the incident positron and the ionized electron are asymptotically free particles. The positronium atom can exist in its ground state or in any one of an infinite number of excited states after the collision. These states form two series; a singlet series (S D 0) referred to as parapositronium (p-Ps) and a triplet series (S D 1) referred to as orthopositronium (o-Ps). The level structure of both these series is, to order ˛ 2 , where ˛ is the fine-structure constant, identical to that of hydrogen, but with each level having half the energy of the corresponding hydrogenic state. Positronium formation is a rearrangement channel, and thus, is a two-center problem. Because positronium is a light particle, having a reduced mass one-half that of an electron, the semiclassical type of approximations used in ion–atom collisions (Chap. 53) are not applicable here. We will discuss various theoretical approaches to this process in Sect. 51.2 and give references to experimental results in Sect. 51.3. Positronium formation in the ground state has a threshold that is 1/4 of a Hartree (6:80285 eV) below the ionization threshold of the target. This means that it is normally the lowest inelastic channel in positron scattering from neutral atoms. For atoms with a small ionization potential, such as the alkalis, this channel is always open. The energy range between the positronium threshold and the first excited state of the atom is known as the Ore gap. In this range, positronium formation is the only possible inelastic process.
51.1.2
Annihilation
Annihilation is a process in which an electron–positron pair is converted into two or more photons. It can occur either directly with a bound atomic electron or after positronium formation has taken place. The direct annihilation cross section for a positron of momentum k colliding with an atomic or molecular target can be written as [2] a D
˛ 3 Zeff 2 a0 ; k
(51.1)
Zeff D
N Z X
ˇ2 ˇ dr 1 dr 2 : : : dr N ˇ .r 1 ; r 2 ; : : : ; r N I r i /ˇ :
i D1
(51.2) While this formula can be naively derived by assuming that the positron can only annihilate with an electron if it is at the identical location, it actually follows from a quantum electrodynamical treatment of the process [3]. If the wave function is approximated by the product of the undistorted target wave function times a positron scattering function F .x/, then Z ˇ2 ˇ (51.3) Zeff D dr .r/ˇF .r/ˇ ; where is the electron number density of the target. Thus, in the Born approximation, where F is taken as a plane wave, Zeff simply becomes the total number of electrons Z in the target. However, a pronounced enhancement of the annihilation rate in the vicinity of the Ps formation threshold due to virtual Ps formation was predicted [4, 5]. Subsequently, the Born approximation was shown to be grossly inadequate by the San Diego group, who found annihilation rates (Zeff ) at room temperature that are an order of magnitude larger for some atoms and even up to five orders of magnitude larger in large hydrocarbon molecules. Furthermore, there is evidence that only the outer shell of electrons takes part in the annihilation process. Two mechanisms have been proposed in order to explain these large values for Zeff . One involves the enhancement of the direct annihilation process below the Ps formation threshold due to the attractive nature of the positron–electron interaction, which increases the overlap of positron and electron densities in the atom or molecule. The second mechanism is referred to as resonant annihilation, which occurs after the positron has been captured into a Feshbach resonance, where the positron is bound to a vibrationally excited molecule. A summary of the above results can be found in the review article by Gribakin et al. [6]. When a positron annihilates with an atomic electron, the most likely result is two 511 keV photons, if the positron– electron pair are in a singlet spin state (parapositronium). In the center-of-mass frame of the pair, the photons are emitted in opposite directions to conserve momentum. However, in the laboratory frame, the bound electron has a momentum distribution which is reflected in the photon directions not being exactly 180 degrees apart. This slight angular deviation, called the angular correlation, can be measured and gives information about the momentum distribution of the bound electrons. This quantity is given by [3]
N Z X where Zeff can be thought of as the effective number of elec- S.q/ D (51.4) dr 1 : : : dr i 1 dr i C1 : : : dr N trons in the target with which the positron can annihilate. If i D1 Z ˇ2 ˇ .r 1 ; r 2 ; : : : ; r N ; x/ is the wave function for the system of iqx ˇ ˇ ; dr dx e ; r ; : : : ; r I x/ ı.r x/ .r i 1 2 N i a positron, with coordinate x, colliding with an N -electron
51
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where q is the resultant momentum of the annihilating pair. In evaluating this quantity, the positron is assumed to be thermalized in the gas before undergoing annihilation. Experimentally, only one component of q is measured, so that S.q/ is integrated over the other two components of the momentum to obtain the measured quantity. The spin triplet component of an electron–positron pair (orthopositronium) can only decay with the emission of three or more photons that do not have well-defined energies. This is a much less probable process than the two-photon decay from the singlet component.
51.2 Theoretical Methods The basic theoretical approaches to the calculation of positron scattering from atoms and molecules were originally developed for electron scattering and later applied to the positron case. Thus, we emphasize here only the differences that arise between the electron and positron cases, both in the theoretical formulations, and in later sections, in the nature of the results. The lowest-order interaction between a free positron and an atomic or molecular target is the repulsive static potential of the target ˝ ˇ ˇ ˛ (51.5) Vs D 0 ˇV ˇ 0 ; where 0 is the unperturbed target wave function, and V is the electrostatic interaction potential between the positron and the target. Since this interaction has the opposite sign from that for electron scattering, the static potential also has the opposite sign in these two cases. On the other hand, the next higher-order of interaction is polarization, which arises from the distortion of the atom by the incident particle. If we represent this distortion of the target to first order by the wave function 1 , as in the polarized-orbital approximation, for example, then the polarization interaction can be represented by the potential ˝ ˇ ˇ ˛ Vp D 0 ˇV ˇ 1 : (51.6) This potential is attractive for both positron and electron scattering and has an asymptotic form with leading term ˛d =2r 4, where ˛d is the static dipole polarizability (Sect. 24.2.3) of the target. Thus, the static and polarization potentials for positron scattering from ground state systems are of opposite sign and tend to cancel one another. This leads to very different behavior from the electron case where they are of the same sign. In particular, the elastic scattering cross sections for positron scattering from an atom are much smaller than for electron scattering, and the phase shifts (Sect. 49.1.1) have very different magnitudes and dependencies on energy. This is illustrated for the case of scattering from helium in Fig. 51.1, where the results of the highly ac-
763 2
Total cross section (πa 0)
1
0.1
0.00
0.40
0.80 Momentum (a.u.)
Fig. 51.1 Total elastic cross sections for electron (dashed line) and positron (solid line) from helium atoms [8, 10, 11]. The higher-order phase shifts were calculated from effective range theory (Sect. 49.1.1)
curate variational calculations for scattering by electrons and positrons are shown. There is a corresponding difference in sign between the electron and positron s-wave phase shifts for very small values of the incident momentum. In fact, the positron phase shift goes through zero, which leads to the Ramsauer minimum in the positron total cross section, as shown in Fig. 51.1. The large difference in magnitudes between the electron and positron s and p-wave phase shifts leads to the large difference in the total elastic cross sections, as shown [7–9]. Higher-order terms in the interaction potential may also give important contributions to scattering cross sections. For a detailed discussion, see the article by Drachman and Temkin [12]. A simple potential scattering calculation using the sum of the static and polarization potentials, but without the exchange terms that are present for the electron case, can be applied to elastic scattering calculations for closed shell systems (Sects. 51.3.1, 51.3.2). The potentials defined above can also be used in a distorted-wave approximation (Chap. 49), which can be applied to excitation and ionization by positron impact. Once again, the complicated exchange terms that arise in electron scattering are absent here [13]. For positrons with high enough incident energies ( 1 keV), the first Born approximation will become valid (Chap. 49). Since the first Born approximation is independent of the sign of the charge of the incident particles, this indicates that as the incident energy increases, the corresponding cross sections for electron and positron scattering will eventually merge. From flux conservation arguments, this means that the positronium formation cross section will rapidly decrease as the incident energy increases. In
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fact, from experimental measurements, the total cross sections (summed over all possible channels) for electron and positron scattering appear to merge at a much lower energy than the cross sections for individual channels [14]. More elaborate calculations for high energy scattering have been carried out in the eikonal-Born series [15] (Chap. 49). These approximations allow for both polarization and absorption (i.e., inelastic processes) and yield good agreement with elastic experimental measurements of differential cross sections at energies above 100 eV. A detailed analysis [9, 15] of the various contributions to the scattering indicates that absorption effects due to the various open inelastic channels play a much more important role here than for electron scattering. A more elaborate treatment of positron scattering is based on the close-coupling approximation (Chap. 49), where the wave function for the total system of positron plus target is expanded using a basis set comprised of the wave functions of the target. Once again, there are no exchange terms involving the positron and, in principle, a complete expansion including the continuum states of the target would include the possibility of positronium formation. However, such an expansion is not practicable if one wants to calculate explicit cross sections for positronium formation. Even in cases where such cross sections are not required, the considerable effect that the positronium formation channels can have on the other scattering cross sections is best included by a close-coupling expansion that includes terms representing positronium states plus the residual target ion. There is a problem of double counting of states in such an expansion but, in practice, this does not appear to be a problem if the number of states in the expansion is not large. Also, in many cases, additional pseudo states have to be included in the expansion in order to correctly represent the long-range polarization interaction. A close-coupling expansion including positronium states is a two-center problem, i.e., it includes the centers of mass of both the target and the positronium states. This means that one is faced with a problem of considerable computational complexity [16–21]. Another way to take into account the effects of open inelastic channels without the complications of a full closecoupling approach is to use optical potentials. These are often based on a close-coupling formalism [9, 22] and lead to a complex potential, the real part of which represents distortions of the target (such as polarization), while the imaginary part allows for absorption (i.e., flux into open channels not explicitly represented). Such an approach was developed by Chen et al. [23] and applied to the noble gases [24]. Recently, a simple approach based upon an optical potential model, yielded good results for the positronium formation cross section in the noble gases [25]. Bray and Stelbovics [26] applied the convergent closecoupling (CCC) method to the scattering of positrons from
J. R. Machacek et al.
atoms. This method includes contributions from the continuum states of the target and sufficient terms in the expansion are included to ensure numerical convergence. Positronium states have been included via psuedostates for positron scattering with helium [27]. A single-center adiabatic-nuclei version has been used for positron scattering with molecular hydrogen [28]. The variational method (Chap. 49) uses an analytic form of trial wave function to represent the total system. The parameters of this analytic function are determined as part of the method. Given a trial wave function with sufficient flexibility and a large enough number of parameters, essentially exact results can be obtained in the elastic energy range and the Ore gap. Because the complexity of the trial function increases as the square of the number of electrons in the target, only positron scattering from hydrogen, helium and lithium, and the hydrogen molecule have been treated by this method to date [10, 29, 30]. Many-body theory has been applied to positron scattering from the noble gases below the positronium formation threshold. Of particular importance is the ability of manybody theory to account for positron–atom and electron– positron correlations, which have a large influence on both the scattering dynamics and the annihilation rate. Many-body theory has produced very good agreement with the most recent experimental results for elastic differential scattering [31]. In the case of direct ionization, there appears to be quite distinct threshold behavior of the cross sections for electron and positron collisions. For electrons, the Wannier threshold law (Sect. 56.3.1) has an exponent of 1:127, while a similar analysis for positrons [32] yields an exponent of 2:651. However, the existence of the positronium formation channel leaves in question whether this analysis will give the dominant term at the threshold. For a fuller discussion, see [33] and references therein. An experimental investigation of the threshold ionization of helium [34] obtained a value 2:2. This was followed by a theoretical investigation that considered anharmonic corrections to the Wannier threshold law to explain the experimental results [35]. A recent measurement within 2 eV of the ionization threshold obtained a value of 1:1 [36]. There has been an investigation [37] of the behavior of the elastic cross sections at the positronium formation threshold, which predicts the occurrence of a Wigner cusp for the noble gases using R-matrix methods. A comprehensive set of measurements using a high-resolution trap-based beams of the elastic scattering cross section about the positronium formation threshold revealed a family of Wigner cusps in the noble gases [38]. Subsequent measurements of the elastic scattering cross section about the positronium formation threshold in molecules that are isoelectronic with helium and neon, and did not show a Wigner cusp [39].
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51.3
Particular Applications
51.3.1 Atomic Hydrogen Because of the difficulty of taking measurements in atomic hydrogen, the available experimental data has so far been restricted to total cross sections, as well as to total ionization and positronium formation cross sections. Essentially exact variational calculations have been carried out in the elastic energy regime and the Ore gap [10]. These calculations are used to benchmark many-body theory [40]. Ionization cross sections have been measured by both the Bielefeld and London groups ([41, 42] and references therein) and have been calculated in a number of approximations [24, 26, 43–45]. However, disagreements between the experimental measurements mean that there is at present no reliable way of assessing the various theoretical approximations used. More elaborate calculations with asymptotically correct wave functions have been used to determine triple differential cross sections for ionization [46, 47]. However, the task of integrating these to produce total cross sections is a formidable one. The total positron–hydrogen cross section has also been measured by the Detroit group [48] and is in quite good agreement with calculations based upon the coupledpseudostate [45] and the convergent close-coupling [49] methods, except at very low energies where the experimental uncertainties are the greatest. In order to determine reliable positronium formation cross sections, the explicit positronium states have to be included. Several such calculations have been carried out [17–20]. These indicate the necessity of explicitly including positronium formation channels in the expansion of the total wave function in order to obtain accurate results, even for elastic scattering. The most recent calculations [45, 49, 50] are in quite good agreement with various experiments [41, 51] over the majority of the energy range. As is the case for electron scattering, positron cross sections exhibit resonances (Sect. 49.1.3). These have been extensively studied by Ho [52], who used variational and complex rotation methods.
51.3.2
Noble Gases
Because the noble gases are convenient experimental targets, a good deal of effort has gone into calculations for these targets, particularly for elastic scattering, ionization, and Ps formation. In the purely elastic energy range, i.e., for energies below the positronium formation threshold, the simple potential scattering approach using the static and polarization potentials defined above yields quite good results. Since the long-range behavior of the sum of the potentials
765
is attractive, the scattering phase shifts for positrons must be positive for sufficiently low energies. However, as its incident energy increases, the positron probes the repulsive inner part of the potential, and the phase shifts become negative. This behavior leads to the well-known Ramsauer minimum in the integral elastic cross sections (Chap. 49). The Ramsauer minimum has been observed in helium and neon ([53, 54] and references therein). In the case of argon, no minimum is observed given a sufficiently high-quality beam [55]. In general, this differs from electron scattering, where some low-energy phase shifts can be negative (modulo ) because of the existence of bound orbitals of the same symmetry. Another difference between positron and electron scattering is exhibited by the differential cross sections (Chap. 49). For electrons, the shape of the cross section is determined by a few dominant phase shifts, whereas for positrons, many phase shifts contribute to the final shape [56]. Because of this behavior, the differential cross sections for positron scattering have much less overall structure than for electron scattering. However, the differential cross sections for positrons for many of the noble gases have a single minimum at relatively small angles, both below and above the first inelastic channel [57, 58]. The earlier results have been reviewed by Kauppila et al. [59]. At intermediate energies, the simple potential scattering approximation is no longer sufficient, and the inelastic channels have to be taken into account via, for example, the use of an optical potential [9, 15, 23, 60] or convergent close-coupling method ([61] and references therein). Furthermore, in the inelastic scattering regime, the existence of open channels has a much more marked effect on the shape of the differential cross sections for positron scattering than that for electrons [57]. Absolute elastic differential cross sections were measured for argon, krypton, and xenon at low energies using a magnetized beam of cold positrons [58, 61, 62]. Magnetized beam measurements generally do not distinguish between forward and backward scattered positrons (or electrons). In this case, the reported elastic cross section is folded about 90 degrees. As with electrostatic beams, a portion of the angular scatter is indistinguishable from the incident beam, which plays an important role in the comparison of experiment and theory [55]. These results are in general agreement with a variety of different theoretical predictions [63–65]. A method that allows the scattering phase shifts to be extracted from the folded elastic differential scattering cross section was developed recently [66]. There is relatively little experimental data for the excitation of the noble gases. Some experimental work has been carried out for the lighter noble gases, helium, neon, and argon ([67] and references therein), and there is satisfactory agreement between these measurements and closecoupling [68], as well as distorted-wave [69] calculations.
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The first state-resolved absolute excitation cross sections for the 4s Œ1=2o1 and 4s Œ3=2o1 states of argon and the 6s Œ3=2o1 state of xenon have been measured by the San Diego group [6, 70]. Relativistic distorted-wave calculations are in satisfactory agreement with the experiment for argon [71] but less so for xenon [72]. Excitation of the n D 2 states of helium was measured by the ANU group [73]. Convergent close-coupling calculations are in good agreement for both the 2 1 S and 2 1 P states [27]. The total ionization and positronium cross sections have been measured extensively for all of the noble gases. In general, there is good agreement amongst the various experiments for the ionization cross section but much less so for the positronium formation cross section, with a recent exception [25]. A summary of the experimental work on the ionization and positronium cross sections for neon, argon, krypton, and xenon can be found in the article by Marler et al. ([74] and references therein). A summary of the corresponding theoretical work can be found in [24, 25]. Both double and triple-differential ionization cross sections have been measured for helium [75] and argon [76, 77]. Distorted wave Born (DWBA) and three-continuum approximation (3C model) have also been applied to these systems ([78, 79] and references therein). The formation of excited-state positronium has been measured for positron scattering from helium, argon, and xenon and has been compared to a number of theoretical results [80]. There exist some measurements of Zeff and angular correlation parameters for these gases [81], mainly at room temperature, and calculations for them have been made in the polarized-orbital approximation [82]. Measurements have been made at various temperatures and are in relatively good agreement with early calculations [83]. The combination of available gas-phase scattering cross sections for energies between 0 and 10 keV has been used to model a single track of a positron through argon [84].
51.3.3 Other Atoms In the case of positron scattering from the alkali atoms, the positronium formation channel is always open, and the simple potential scattering approach does not yield reliable results. The Detroit group has measured the total cross section, as well as the upper and lower bounds to the positronium formation cross section for sodium, potassium, and rubidium [48]. Although early close-coupling calculations of the elastic and excitation cross sections [85–87] were in surprisingly good agreement with the experimental total cross section, these calculations did not include the positronium formation channel. Subsequently, much more sophisticated calculations were carried out by the Belfast group using the
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coupled-pseudo-state method, which included both eigenstates of the target, as well as positronium [45, 88–90]. These calculations also showed the increasing importance of positronium formation in excited states for the alkalis: potassium, rubidium, and caesium. The overall agreement between experimental results and those from the coupledpseudo-state method are quite good for both potassium and rubidium; however, for sodium, the experimental positronium formation cross section is significantly above these theoretical calculations. A summary of the experimental work on the alkalis can be found in [48], while a corresponding summary of theoretical work is given in [45]. Substantial resonance features have been found in these positron–alkali atom cross sections using close-coupling calculations [91, 92]. More recently, various theoretical methods have been applied to the positron–lithium system ([93, 94] and references therein). The positronium formation threshold for magnesium is very low, only 0:844 eV, and hence, the elastic and positronium formation cross sections will dominate in the low-energy region. Upper and lower bounds to the Ps formation cross section in magnesium have been determined [48, 95] and are in agreement with both close-coupling calculations [96] and the results of many-body theory [97].
51.3.4 Molecular Hydrogen By its fundamental nature, molecular hydrogen has attracted considerable attention both experimentally and theoretically. The total elastic cross section has been measured by the Detroit group [51], the London group [98], and the ANU group [99]. There have been several theoretical calculations of this cross section by a variety of methods: Kohn variational [100], R-matrix [101], distributed positron model [102], a Schwinger multichannel method [101–103], and convergent close-coupling calculations by the Curtin group [104]. Calculations have shown the elastic cross section is strongly influenced by the positronium formation channel near the threshold. However, a recent measurement did not show any feature in the integral elastic cross section about the positronium formation threshold [39]. The vibrational .0 ! 1/ excitation cross section of molecular hydrogen has been measured between 0:55 and 4 eV and is in quite good agreement with theoretical calculations [105] (and reference therein). The San Diego group has also measured the electronic excitation of the B1 ˙ state from the threshold to 30 eV [70]. Their data are in reasonable agreement with the Schwinger multichannel calculation of Lino et al. [106]. The measured positron excitation cross section appears to be larger than that measured for electron excitation.
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The ionization cross section has been determined over a wide range of energies by a number of different groups ([107–111] and references therein). Since all of the above measurements are relative, they must be normalized to one another at particular energies. Theoretical calculations are in satisfactory agreement with the measurements ([112] and references therein). Electron capture to the continuum has been observed in triple-differential ionization measurements [113]. Reasonable agreement has been found between available experimental measurements and the distorted-wave Born approximation (with the Ward–Macek approximation; [79] and references therein). The positronium formation cross section has also been measured by a number of different groups and has been compared to the first Born approximation and a coupled-channels calculation, see [99] and reference therein. Differential elastic cross sections have been measured by the ANU group and compared to a number of calculations [99]. The shape of these measurements are in good agreement with the complex model potential calculation [114]. More recently, a single-center adiabatic-nuclei convergent close-coupling method [104] showed good agreement between experiment and theory for the differential elastic cross section below the positronium formation threshold. Rotational excitation cross sections have been calculated using the Schwinger multichannel method with fixed nuclei and the adiabatic rotational approximation ([115] and references therein). Recent progress in cryogenic positron traps by the San Diego group [116] will enable an experimental investigation of the rotational excitation of molecular hydrogen.
51.3.5 Water From a biological perspective, water is fundamental, and it was recently studied experimentally and theoretically. The total scattering cross has been measured by the ANU group and compared to previous measurements and calculations ([117] and references therein). The ionization cross section has been measured by the London group [118]. The total elastic cross section and total inelastic cross section less positronium formation have been measured by the ANU group [119]. These measurements were compared to R-matrix and the independent atom model (IAM) methods ([119] and references therein). Differential ionization cross sections have been measured by the London group [120]. Triple-differential ionization has been calculated using a second-order distorted-wave Born approximation [121].
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51.3.6 Other Molecules For diatomic and triatomic molecules, most of the experimental and theoretical work has been carried out for CO, CO2 , O2 , and N2 . Total cross sections for O2 , N2 , and CO2 have been measured from the threshold to several hundred eV ([98] and references therein). Recently, the independent atom model (IAM) was compared to measurements for C2 , N2 , and O2 [122] (and references therein). A spherical complex optical potential was used to determine the total scattering cross section for positron scattering from a number of simple molecules [123]. Relative differential cross sections have been measured for CO, CO2 , O2 , N2 , as well as N2 O, on both sides of the positronium formation threshold [124]. At low energies, the gases N2 , O2 , and CO exhibit a minimum in the DCS at small angles, as with the heavier noble gases. This minimum gradually disappears as the energy increases. Vibrational excitation cross sections for CO and CO2 have been measured [105] and are in excellent agreement with the theoretical calculations of [125] for CO, and in satisfactory agreement with theory [126] for CO2 . Electron excitation of the a1 ˘ and a01 ˙ states of N2 have been measured from threshold to 20 eV [70]. Interestingly, the positron cross section near the threshold is approximately double that for electrons. Impact ionization, positronium formation, and electronic excitation have been measured for N2 , CO, and O2 [127]. Additionally, positronium formation has been measured for CO2 [128]. For small polyatomic molecules, the majority of experimental and theoretical work was carried out for CH4 . This includes the total cross section and quasi-elastic (summed over vibration–rotational levels) differential cross sections [129]. At low energies, there is a minimum in these DCS at small angles, as for the heavier noble gases which, in turn, also disappears at higher energies [59]. The positronium formation cross section has also been measured [130]. Recently, a large number of total scattering cross section measurements of organic compounds were studied experimentally by the Trento group ([131] and references therein). This work has been extended to purine nucleobasis and other biologically relevant molecules [132] (and references therein). Additionally, a comparison of the total scattering cross section from chiral enantiomers was also made [133]. Positronium formation has been measured for methanol and ethanol [134]. Total and elastic differential scattering, and positronium formation cross sections for a number of biologically relevant molecules have been measured by the ANU group ([135, 136] and references therein). In these cases, the scattering cross sections are summed over rotational and vibrational states. Nucleobases that are solid at
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room temperature require a vacuum oven to produce vapor electron of the same velocity [148]. This has been explained of the molecules. In these cases, relative cross sections were in part by the polarization of the incident Ps [149]. The formation of the positronium negative ion, Ps , has determined experimentally, and IAM calculations set the abbeen observed [150], and resonance structure has also been solute scale at 150 eV. observed in the photodetachment of Ps [151, 152]. The formation of molecular positronium (Ps2 ) has been 51.4 Binding of Positrons to Atoms observed [153] using intense positron pulses at a metal surface. The scattering of positronium atoms by positronium There have been many recent investigations of the possible atoms has been investigated theoretically [154, 155] with an binding of positrons to a variety atoms. As was mentioned investigation of exciton-like nature of positronium [156]. in Sect. 48.1.2, such binding could greatly enhance the anniThe positronium atom is a boson, and, thus, there is fundahilation cross section and help explain the large measured mental interest in the creation of a Bose–Einstein condensate values of Zeff for both atoms and molecules. It has been of an antimatter–matter composite bose. A unique feature shown theoretically that a positron will bind to a large num- of a Ps-BEC is the possibility of stimulated annihilation or ber of one-electron and two-electron atomic systems ([137] the creation of a gamma-ray laser [157, 158]. More detailed and references therein). For one-electron systems, where theoretical treatments have appeared more recently [159– the ionization potential is less than 6:80285 eV, the dom- 161]. inant configuration is a polarized positronium (Ps) cluster moving in the field of the residual positive ion, while for twoelectron systems, with an ionization potential greater than 6:80285 eV, the dominant configuration involves a positron 51.6 Antihydrogen orbiting a polarized neutral atom [138]. So far, there is no experimental evidence for these positronic atoms [139]. Positron and positronium scattering play an important role in However, a significant amount of theoretical work has shown efforts to form antihydrogen, the bound state of an antiprothat many atoms can bind a positron ([140] and references ton and a positron. This antimatter atom is an ideal system to therein). test CPT. In the past decade, antihydrogen has been trapped Enhancement of the annihilation rate (Zeff ) is observed in [162] with the first spectroscopic measurements made repolyatomic molecules. The mechanism has been identified cently [163]. In addition, the formation of an antihydrogen as vibrational Feshbach resonances mediating positron bind- beam has been postulated [164]. Excited-state positronium ing to molecules ([6] and references therein). In the case of has been found to be extremely useful in the formation of linear alcohols, the binding energy depends linearly upon the antihydrogen using detailed CCC calculations [165]. molecular polarizability [141].
51.5 Positronium Scattering
51.7 Reviews
There has been interest in positronium (Ps) scattering from atomic and molecular targets for a considerable time; see the work of Fraser and coworkers [142] and reference therein, but only recently has this field seen much activity. This is a two-center problem with the Ps atom being a light particle, so semiclassical methods are not applicable. There was some theoretical work by Drachman and Houston [143, 144], and recently there has been close-coupling calculations for simple target systems [145]. This problem has also been studied by a pseudo-potential method ([146] and references therein). The long-range interactions in positron–hydrogen scattering have recently been investigated [147]. The London group has measured the total cross sections positronium scattering from the noble gases and small molecules (H2 , N2 , O2 , H2 O, CO2 , SF6 ). Unexpectedly, the total cross sections for positronium scattering have been found to be very similar to scattering by
For a number of years, a Positron Workshop has been held as a satellite of the International Conference on the Physics of Electronic and Atomic Collisions. The Proceedings [166– 182] give an excellent summary of the state of positron scattering research, both experimental and theoretical, and applications to astro and biophysics. There are several review articles on positron scattering in gases, including the early historical development [183], more comprehensive articles [2, 184–186], as well as more recent reviews [131, 187–191]. There are a number of reviews of the various antihydrogen efforts [192–195] with a special issue on antihydrogen and positronium [196]. A review on positronium laser physics [197] discusses the future for positron physics. The book [198] discusses various aspects of both experimental and theoretical positron physics.
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J. R. Machacek et al. Josh R. Machacek Josh Machacek received his PhD from the Australian National University. He was a postdoctoral scholar at JPL before returning to the ANU. He is a recipient of a DECRA Fellowship from the Australian Research Council and a US National Research Council Fellowship at the Air Force Institute of Technology. He is currently focused on quantum mechanically complete experiments involving positrons. Robert P. McEachran Professor McEachran received his PhD from the University of Western Ontario, Canada. He spent 2 years at the University College London (England) before joining York University in Toronto in 1964. In 1997, he accepted an Adjunct Professorship at the Australian National University. His current research interests are the theoretical treatment of electron/positron scattering from heavy atoms within a relativistic framework.
Allan D. Stauffer Allan Stauffer has published numerous papers in the field of electron and positron scattering from atoms and simple molecules. In collaboration with numerous colleagues, he has been involved with extensive scattering calculations and developed methods to carry out these investigations and has worked closely with groups involved in measuring these processes.
Adiabatic and Diabatic Collision Processes at Low Energies
52
Evgeny Nikitin and Alexander Kandratsenka
Contents 52.1 52.1.1 52.1.2 52.1.3
Basic Definitions . . . . . . . . . . . . . . . . . . . . Slow Quasi-Classical Collisions . . . . . . . . . . . Adiabatic and Diabatic Electronic States . . . . . . Nonadiabatic Transitions: The Massey Parameter
. . . .
52.2 52.2.1
Two-State Approximation . . . . . . . . . . . . . . Relation Between Adiabatic and Diabatic Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . Coupled Equations and Transition Probabilities in the Common Trajectory Approximation . . . . Selection Rules for Nonadiabatic Coupling . . . .
. . . 775
52.2.2 52.2.3 52.3 52.3.1 52.3.2 52.4 52.4.1
. . . .
. . . .
773 773 774 774
. . . 775 . . . 775 . . . 777
Single-Passage Transition Probabilities in Common Trajectory Approximation . . . . . . . . . . . . . . . . 778 Transitions Between Noncrossing Adiabatic Potential Energy Curves . . . . . . . . . . . . . . . . . . . . . . . . . 778 Transitions Between Crossing Adiabatic Potential Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779
52.4.2
Double-Passage Transition Probabilities . . . . . . . 780 Transition Probabilities in the Classically-Allowed and Classically-Forbidden WKB Regimes: Interference and Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . 780 Nonadiabatic Transitions near Turning Points . . . . . . 781
52.5 52.5.1 52.5.2
Multiple-Passage Transition Probabilities . . . . . . 782 Multiple Passage in Atomic Collisions . . . . . . . . . . 782 Surface Hopping . . . . . . . . . . . . . . . . . . . . . . . 782
of atoms is discussed within the two-state approximation. Analytical formulae for nonadiabatic transition probabilities are presented for particular models with reference to single and double-passage of coupling regions. The generalization for multiple passages is described. Keywords
potential energy surface potential energy curve Coriolis coupling nonadiabatic transition nonadiabatic coupling
52.1 52.1.1
Basic Definitions Slow Quasi-Classical Collisions
Slow collisions of atoms or molecules (neutral or charged) are defined as collisions for which the velocity of the relative motion of colliding particles v is substantially lower than the velocity of valence electrons ve v 1: ve
(52.1)
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783
If ve is estimated as ve 1 a.u. 108 cm=s, then Eq. (52.1) is fulfilled for medium-mass nuclei ( 10 amu) up to sevAbstract eral keV. Quasi-classical collisions are those for which the Adiabatic and diabatic electronic states of a system of atoms are defined and their properties are described. de Broglie wavelength dB for relative motion is substanNonadiabatic interaction for slow quasi-classical motion tially smaller than the range parameter a of the interaction potential (the WKB (Wentzel–Kramers–Brillouin) condition [1–4]) E. Nikitin () Dept. of Physics, Technion – Israel Institute of Technology Haifa, Israel e-mail: [email protected]
A. Kandratsenka () Dynamics at Surfaces, Max-Planck-Institute for Multidisciplinary Sciences Göttingen, Germany e-mail: [email protected]
dB a :
(52.2)
The two conditions in Eqs. (52.1) and (52.2) define the energy range within which collisions are slow and quasi classical. For medium-mass nuclei, this energy range covers collision energies above room temperature and below hundreds of eV. The parameter a should not be confused with
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_52
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another important parameter, L0 , which characterizes the extent of the interaction region. For instance, for the exchange interaction between two atoms, L0 corresponds to the distance of closest approach of the colliding particles, while a is the range of the exponential decrease of the interaction. Typically, L0 noticeably exceeds a.
52.1.2
Adiabatic and Diabatic Electronic States
Let r refer to a set of electronic coordinates in a body-fixed frame related to the nuclear framework of a colliding system and let R refer to a set of nuclear coordinates determining the relative position of nuclei in this system. A configuration of electrons and nuclei in a frame fixed in space is completely determined by r, R, and the set of Euler angles ˝, which relate the body-fixed frame to the space-fixed frame. With the total Hamiltonian of the system H .r; R; ˝/, the stationary state wave function satisfies the equation H .r; R; ˝/E .r; R; ˝/ D EE .r; R; ˝/ :
(52.3)
The electronic adiabatic Hamiltonian H.rI R/ is defined to be the part of H .r; R; ˝/ in which the kinetic energy of the nuclei and certain weak interactions are ignored. Typically, H.rI R/ includes the electron–electron and electron–nuclear electrostatic interaction and, possibly, the spin–orbit interaction. The adiabatic electronic functions n .rI R/ are defined as eigenfunctions of H.rI R/ at a fixed nuclear configuration R with set of electronic coordinates r H.rI R/
n .rI R/
D Un .R/
n .rI R/
:
(52.4)
Two basis sets
n
and n generate the matrices
hm jH jn i D Hmn ; hm jH jn i D Hmn C Dmn ; h
m jH j
ni
D Un .R/•mn C Dmn :
(52.5)
The eigenvalues of the matrix Hmn are Un ; Dmn and Dmn are the matrices of interactions ignored in the passage from the Hamiltonian H to H . All the above matrices are, in principle, of infinite order. For low-energy collisions, the use of finite matrices of moderate dimension will usually suffice. Diabatic PES are defined as the diagonal elements Hnn . The significance of the diabatic PES is that for velocities that are high [but still satisfy Eq. (52.1)] the system moves preferentially across diabatic PES, provided that the off-diagonal elements Hmn are small enough. An adiabatic function n .rI R/ can be constructed as a linear combination of the diabatic functions m .r/ as X Cnm.R/m .r/ : (52.6) n .rI R/ D m
52.1.3 Nonadiabatic Transitions: The Massey Parameter Deviations from the adiabatic approximation manifest themselves in transitions between different PES, which are induced by the dynamic coupling matrix D. At low energies, the transitions usually occur in localized regions of nonadiabatic coupling (NAR). In these regions, the motion of nuclei in different electronic states is coupled, and, in general, it cannot be interpreted as being driven by a single potential [3, 5–7]. An important simplifying feature of slow adiabatic collisions is that, typically, the distance between different NAR is substantially larger than the extents of each NAR. This makes it possible to formulate simple models for the coupling in isolated NAR, and, subsequently, to incorporate the solution for nonadiabatic coupling into the overall dynamics of the system. For a system of s nuclear degrees of freedom, there are the following possibilities for the behavior of PES within NAR:
The eigenvalues Un .R/ are called adiabatic potential energy surfaces (adiabatic PES). In the case of a diatom, the set R collapses into a single coordinate, the internuclear distance R, and the PES become potential energy curves PEC, Un .R/. The functions n .rI R/ depend explicitly on R and implicitly on the Euler angles ˝. The significance of the adiabatic PES is related to the fact that in the limit of very low velocities, a system of nuclei will move across a single PES. In this approximation, called the adiabatic approximation, the function Un .R/ plays the part of the potential energy, which drives the motion of the nuclei [1–4]. An electronic diabatic Hamiltonian is defined formally (i) If two s-dimensional PES correspond to electronic states as a part of H , i.e., H0 D H C H . The partitioning of of different symmetry, they can cross along an (s 1)H into H0 and H is dictated by the requirement that the dimensional line. For a system of two atoms, s D 1, and eigenfunctions of H0 , called diabatic electronic functions so two potential curves of different symmetry can cross n , depend weakly on the configuration R. The physical at a point. meaning of this weak dependence is different for different (ii) If two s-dimensional PES correspond to electronic states problems. A perfect diabatic basis set n .r/ is R indepenof the same symmetry, they can cross along an (s 2)dent; for practical purposes, one can use a diabatic set that dimensional line. For a system of two atoms, s D 1, and is considered R independent within a certain region of the so two potential curves of different symmetry cannot configuration space R. cross.
52 Adiabatic and Diabatic Collision Processes at Low Energies
775
(iii) If two s-dimensional PES correspond to electronic states one arrives at a set of coupled second-order equations for of the same symmetry in the presence of spin–orbit cou- the unknown functions nE .R/ (the scattering equations). In pling, they can cross along an (s 3)-dimensional line. the adiabatic approximation, the total wave function is represented by a single term in the first sum of Eq. (52.9) Statement (ii) applied to a two-atom system is known as the Wigner–Witmer noncrossing rule. In applications to a system (52.11) E .r; R/ D n .rI R/nE .R/ : with s > 1, it is frequently discussed in terms of the conical intersection [8]. The efficiency of the nonadiabatic coupling between two 52.2 Two-State Approximation adiabatic electronic states is determined according to the adiabatic principle of mechanics (both classical and quan52.2.1 Relation Between Adiabatic and Diabatic tum), by the value of the Massey (M) parameter M , which Basis Functions represents the product of the electronic transition frequency !el and the time nuc that characterizes the rate of change In the two-state approximation, two adiabatic functions of electronic function due to the nuclear motion. Putting k .rI R/ are assumed to be expressed as a linear combina!el U.R/=„ and nuc D L=v.R/, we get tion of two diabatic functions k .r/ via a rotation angle M .R/ D !el nuc D U.R/L=„v.R/ ;
(52.7)
1 .rI R/
D cos .R/1 .r/ C sin .R/2 .r/ ;
where U is the spacing between any two adiabatic PES, 2 .rI R/ D sin .R/1 .r/ C cos .R/2 .r/ : (52.12) and L is a certain range that depends on the type of coupling. The nonadiabatic coupling is inefficient at those The rotation angle, .R/, is expressed via the diagonal and configurations R where M .R/ 1. If M .R/ is less than or off-diagonal matrix elements of the adiabatic Hamiltonian H of the order of unity, the nonadiabatic coupling is efficient, in the diabatic basis 1 ; 2 and a change in adiabatic dynamics of nuclear motion is very 2H12 .R/ substantial. tan 2.R/ D : (52.13) The following relations usually hold for the parameters H11 .R/ H22 .R/ L; a; L0 for slow collisions The eigenvalues of H are L a L0 : (52.8) U1;2 D UN ˙ U ; When the nonadiabatic coupling is taken into account, the UN D .H11 C H22 /=2 D HN ; total (electronic and nuclear) wave function, E , can be repq 1 2 resented as a series expansion in n or n (the Euler angles .H /2 C 4H12 ; U D 2 ˝ are suppressed for brevity) H D H11 D H22 ; (52.14) X .rI R / . R / E .r; R/ D n nE implying n X D n .r/ nE .R/ : (52.9) H D U cos 2 ; n Here, nE .R/ and nE .R/ are the functions that have to be found as solutions to the coupled equations formulated in the adiabatic or diabatic electronic basis, respectively [3, 5– 7, 9]. In general, different contributions to the first sum in Eq. (52.9) can be associated with nonadiabatic transition probabilities between different electronic states. A practical means of calculating functions nE .R/ [or
nE .R/] consists of expanding them over certain basis functions &n .R0 /, where R0 denotes all coordinates R save a single coordinate R. Writing X &n .R0 /nE .R/ ; (52.10) nE .R/ D
H12 D .1=2/U sin 2 :
52.2.2
(52.15)
Coupled Equations and Transition Probabilities in the Common Trajectory Approximation
A two-state nonadiabatic wave function .r; R/ can be written as an expansion into either adiabatic or diabatic electronic wave functions .rI R/ D
1 .rI R/ ˛1 .R/
C
2 .rI R/ ˛2 .R/
.rI R/ D 1 .r/ˇ1 .R/ C 2 .r/ˇ2 .R/ ;
; (52.16)
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in which the nuclear wave functions satisfy two coupled sdimensional Schrödinger equations [1–3]. In the common trajectory (CT) approximation, the motion of the nuclei is described by the classical trajectory, i.e., by a one-dimensional manifold Q.t/ embedded in the s-dimensional manifold R. A section of PES along this onedimensional manifold determines a set of effective PEC. In the case of atomic collisions, Q coincides with the interatomic distance R, and the effective PEC are just ordinary PEC. A definition of a CT, or an effective potential that drives it, represents a challenging task which is not discussed here (e.g., references in the collection of notes [10, 11]). A CT counterpart of Eq. (52.16) is .r; t/ D
1 ŒrI Q.t/a1 .t/
C
2 ŒrI Q.t/a2 .t/
.rI t/ D 1 .r/b1 .t/ C 2 .r/b2 .t/ :
; (52.17)
The expansion coefficients ak .t/ satisfy the set of equations da1 D U1 .Q/a1 C iQP g.Q/a2 ; dt da2 D iQP g.Q/a1 C U2 .Q/a2 : i„ dt
i„
(52.18)
Here, the dynamical coupling function (defined in the adiabatic basis) g.Q/ D h
1j
@ j @Q
2i
D h
2j
@ j @Q
1i
D
d ; (52.19) dQ
arises from the action of the operator i„@=@t on the adiabatic functions and is expressed in terms of the angle .Q/ (defined in the diabatic basis by Eq. (52.13)). For low energies, function g.Q/ is localized near the NAR center, and Eqs. (52.18) decouple away from the NAR center. This property of coupled equations may be lost if they are formulated in the diabatic basis. The diabatic expansion coefficients bk .t/ satisfy the set of equations db1 D H11 .Q/b1 C H12 .Q/b2 ; dt db2 D H21 .Q/b1 C H22 .Q/b2 : i„ dt i„
The crossing conditions in the adiabatic and diabatic representations are
U.Qc / D 0 ; 2 2 .Qc / D 0 : H.Qc / C 4H12
(52.21)
Since Q represents a one-dimensional manifold, the crossing condition Eq. (52.21) is satisfied for complex values of Q D Qc , unless H12 D 0. Then, by definition, the location of the NAR center is identified with Qp D Re Qc . A characteristic width of an isolated NAR is determined by the width Qp of a peaked (at Q D Qp ) function g.Q/. Normally, Qp is about Im Qc [3]. A more general discussion of the crossing in the complex coordinate plane is given in the context of the hidden crossing [10]. For low energies, the Eq. (52.20) along a particular CT are partitioned into sets of adiabatic evolution and the sets of nonadiabatic transformation of the amplitudes. The spatial (or temporal) extension of the former is normally much larger compared to the latter, such that nonadiabatic transformation of the amplitudes can be described by Eq. (52.18) applied to an isolated NAR. Since the characteristic width of a NAR is small on the scale of the whole CT, the part of CT that is essential in describing the nonadiabatic transformation of ak .t/ can be simplified in order to make Eq. (52.18) more easily handled. This simplification (to be called the local common trajectory, LCT) is a crux of the analytical solution for model cases of nonadiabatic coupling (Sect. 52.3) within an isolated NAR. With the given functions U1 .Q/, U2 .Q/, g.Q/ (or H11 .Q/, H22 .Q/, and H12 .Q/), the Eq. (52.18) are completely defined by setting the LCT with Q D QLCT .t/. Equation (52.18) decouple on both sides of the NAR, say, at t < t ./ and t > t .C/ . At these values of t, the two-state function .r; t/ evolves adiabatically, and this behavior can be singled out by transformation (t ./ < tp < t .C/ ) 2 3 Zt 6 i 7 ak .t/ D aN k .t/ exp4 Uk ŒQLCT .t/dt 5 : (52.22) „ tp
The amplitudes aN k .t/, which become time-independent outside NAR, satisfy equations (52.20) 2 3 Zt daN 1 6i 7 Clearly, for a system of two atoms, Q R. U ŒQLCT .t/dt 5 ; D aN 2 QP LCT g.QLCT / exp4 dt „ Solutions to Eqs. (52.18) and (52.20) are equivalent, tp provided that the initial conditions are matched, and the trandaN 2 D aN 1 QP LCT g.QLCT / sition probability is properly defined with account taken for dt 2 3 the coupling of the diabatic states away from the NAR cenZt ter. It is customary to identify the center of the NAR with 6 i 7 exp4 U ŒQLCT .t/dt 5 : a value of Q D Qp , which corresponds to the real part of the „ tp complex-valued coordinate Qc at which two adiabatic PES cross and which possesses the smallest imaginary part [1]. (52.23)
52 Adiabatic and Diabatic Collision Processes at Low Energies
777
Thus, the amplitudes aN k .t/ are determined by two functions, U.Q/ and g.Q/, that evolve along a chosen LCT. These two functions are related to the two diabatic basis functions H12 .Q/ and H.Q/ by Eq. (52.14). It follows from Eq. (52.23) that aN 1 and aN 2 become time independent outside NAR. This property of the amplitudes in the adiabatic basis is lost for their counterparts in the diabatic basis. A solution of the equations of the nonadiabatic coupling across an isolated maximum of g.Q/ within the time interval ./ .C/ t ./ t t .C/ (LCT interval QLCT QLCT QLCT , accordingly) gives the single-passage (or one-way) matrix of C; , which connects the amthe nonadiabatic evolution, Nn;m ./ ./ .C/ .C/ plitudes fa1 ; a2 ! a1 ; a2 g on both sides of NAR. In particular, the probability P12 of the nonadiabatic transition 1 ! 2 and the probability 1P12 of the nonadiabatic survival 1 ! 1 are ˇ C; ˇ2 ˇ P12 D ˇN2;1
ˇ C; ˇ2 ˇ : and 1 P12 D ˇN1;1
(52.24)
The efficiency of nonadiabatic coupling in crossing an isolated NAR can be estimated by the Landau formula for transition probability in the near-adiabatic limit when P12 1 ˇ3 ˇ ˇ ˇ Ztc ˇ ˇ 2 ˇ 4 D exp ˇIm U ŒQLCT .t/dt ˇˇ5 ; „ˇ ˇ 2
P12
(52.25)
tr
where tc is a root of the equation QLCT .tc / D Qc , and tr is any real-valued time. The single-passage transition probabilities are discussed in Sect. 52.3. C; are known, they can be Once passage matrices Nn;m incorporated into the general scheme of nonadiabatic dynamics. In particular, the transition probabilities for the double passage of the same NAR are discussed in Sect. 52.4, and the nonadiabatic dynamics with multiple transitions is described in Sect. 52.5. Table 52.1 Selection rules for the coupling between diabatic and adiabatic states of a diatomic quasi molecule (w D g; u; D C; )
52.2.3
Selection Rules for Nonadiabatic Coupling
In the general case, the coupling between adiabatic states or diabatic states is controlled by certain selection rules. The most detailed selection rules exist for a system of two colliding atoms, since this system possesses a high symmetry (C1v or, for identical atoms, D1h point symmetry in the adiabatic approximation). In the adiabatic representation, the coupling is due to the elements of the matrix D. They fall into two different categories: those proportional to the radial nuclear velocity (coupling by radial motion or radial coupling) and those proportional to the angular velocity of rotation of the molecular axis (coupling by rotational motion or Coriolis coupling). Besides, D includes the spin–orbit and other weak interactions if they are not included in the adiabatic Hamiltonian. In a diabatic representation, the coupling is due to the parts of the interaction potential neglected in the definition of the diabatic Hamiltonian H0 . In typical cases, these parts (besides the nuclear motion) are: the electrostatic interaction between different electronic states constructed as certain electronic configurations (H0 corresponds to a selfconsistent field Hamiltonian); spin–orbit interaction (H0 corresponds to a nonrelativistic Hamiltonian); hyperfine interaction (H0 ignores the magnetic interaction of electronic and nuclear spins); as well as the electrostatic interaction between electrons and quadrupole moments of nuclei. Different definitions of the adiabatic electronic states (i.e., different Hund coupling cases [1]) of a system of two atoms are discussed in [11]; the respective selection rules are summarized in [12]. The selection rules for the above interactions in this case are listed in Table 52.1 for two conventional nomen. / clatures for molecular states: Hund’s case (a), 2S C1 w , and . / Hund’s case (c), ˝w . For molecular systems with more than two nuclei, the selection rules cannot be put in a detailed form since, in general, the symmetry of the system is quite low. For the important case of three atoms, a general configuration is planar (Cs symmetry); particular configurations correspond to an isosceles triangle if two atoms are identical (C2v symmetry),
Interaction Configuration interaction (electrostatic) Spin–orbit interaction Radial motion Rotational motion Hyperfine interaction
2S C1
./
w nomenclature D 0, S D 0 g 6• u, C 6• D 0; ˙1, S D 0; ˙1 g 6• u, C • D 0, S D 0 g 6• u, C 6• D ˙1, S D 0 g 6• u, C 6• D 0; ˙1, S D 0; ˙1 g • u, C •
./
˝w nomenclature ˝ D 0 g 6• u, C 6• ˝ D 0 g 6• u, C 6• ˝ D 0 g 6• u, C 6• ˝ D ˙1 g 6• u, C 6• ˝ D 0; ˙1 g • u, C •
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Table 52.2 Selection rules for dynamic coupling between adiabatic states of a system of three atoms Cs C2v
A0 00
A
A1 B1 A2 B2
A0 A1 z y; Rx Rz Ry
B1 y; Rx z Ry Rz
A00 A2 Rz Ry z y; Rx
B2 Ry Rz y; Rx z
to an equilateral triangle for three identical atoms (D3h symmetry), or to a linear configuration. For the last case, the selection rules are the same as for a system of two atoms. The selection rules for the dynamic coupling between adiabatic states classified according to the irreducible representations of the Cs and C2v groups are listed in Table 52.2. In Table 52.2, z and y refer to two modes of the relative nuclear motion in the system plane, Rz and Ry refer to two rotations about principal axes of inertia lying in the system plane, and Rx refers to a rotation about the principal axis of inertia perpendicular to the system plane. In-plane motion of nuclei couples the state of the same reflection symmetry; rotation of the system plane couples the state of the different reflection symmetry. The spin–orbit interaction, if included in matrices Dmn and Dmn, couples the states of different reflection symmetry. For a more detailed discussion see [12].
52.3
Single-Passage Transition Probabilities in Common Trajectory Approximation
when one passes to the adiabatic basis with the spacing U.Q/ and dynamical coupling g.Q/ U.Q/ D E g.Q/ D
q 1 2 cos 2# e˛.QQp / C e2˛.QQp / ;
˛ sin 2# e˛.QQp / : 2 1 2 cos 2# e˛.QQp / C e2˛.QQp / (52.27)
As # changes from very small values to =2, the spacing of adiabatic PEC changes from the narrow avoided crossing to the wide one and ultimately displays strong divergence. The adiabatic PEC cross at Qc D Qp ˙ 2i#=˛ :
This expression defines Qp as the center of NAR, Qp D Re Qc , while the characteristic range of NAR is Qp D Im Qc D 2#=˛. The adiabatic states become uncoupled outside the NAR, while diabatic states remain coupled on the asymptotic side of NAR. For this model, adiabatic wave functions coincide with diabatic functions before entering the coupling region (in the limit ˛.Q Qp / 1), but after exiting the coupling region (in the limit ˛.Q Qp / 1) they are #-dependent linear combinations of the diabatic functions. The definition of the exponential model within the LCT concept is completed once one specifies the LCT that crosses the NAR. It is taken to be a segment of a rectilinear trajectory, Q.t/ Qp D vp .t tp / ;
52.3.1
(52.28)
(52.29)
Transitions Between Noncrossing Adiabatic Potential Energy Curves
where vp D QP .t/j t Dtp is additional parameter of the model, and tp is the time corresponding to the center of NAR. The solution of coupled equations yields the transition Single-passage transition probabilities between noncrossing probability between adiabatic states adiabatic potential curves for different models can be classified by either diabatic or adiabatic Hamiltonians using the sinh. cos2 #/ sin2 # N common trajectory with the NAR region. The following disD ; (52.30) e P12 sinh cussion refers to the cases of one-dimensional motion, when the transition probability can be expressed analytically. Here, where D E=.„˛vp / is the asymptotic Massey paramea quite general model corresponds to exponential diabatic ter. Its value at the center of NAR is p D sin2 #. potentials and coupling (Nikitin (N) model [13]), with residSpecial cases of the exponential model correspond to the ual dynamical coupling neglected. It is formulated as linearly crossing diabatic potentials with the constant coupling (Landau–Zener (LZ) model [14–16]), parallel diabatic
H N .Q/ D E 1 cos 2# e˛.QQp / ; potentials with the exponential coupling (Demkov (D) model N .Q/ D .E=2/ sin 2# e˛.QQp / : (52.26) [17]), and the asymptotically degenerate diabatic potentials H12 (resonance (R) model [13]). The LZ model [14–16] is obtained from the exponential Here, # is a parameter that defines the interplay between the N off-diagonal element H12 .Q/ and the difference in the di- model by retaining Eq. (52.27) terms linear in # and ˛.Q agonal elements H N .Q/. In addition, Eq. (52.26) includes Qp / (i.e., for # 1, 1) asymptotic spacing E of diabatic PEC (for ˛.Q Qp / 1) H LZ .Q/ D E˛.Q Qp / F ˛.Q Qp / ; and characteristic scale 1=˛ of the exponential interaction. The meaning of the characteristic coordinate Qp is revealed,
LZ H12 D E# V :
(52.31)
52 Adiabatic and Diabatic Collision Processes at Low Energies
Here, the right-hand-side corresponds to the standard parameterization of the LZ model, where F D E˛ denotes the difference in slopes of the diabatic crossing PEC at Qp , and V is a constant. The pattern of the adiabatic PEC, with U.Qp / calculated from Eq. (52.31) as ˇ ˇq ˇ ˇ 2 2 2 ˇ U.Q/ D ˇ F .Q Qp / C 4V ˇˇ; (52.32) corresponds to an avoided crossing (or pseudo crossing). For this model, adiabatic wave functions coincide with diabatic functions on one side of the NAR, but on the other side, the former interchange with the latter accompanied by a sign reversal. When # 1 and 1, Eq. (52.30) yields the LZ transition probability LZ D exp.2# 2 / D exp.2V 2 =F vp /: P12
(52.33)
779
52.3.2
Transitions Between Crossing Adiabatic Potential Curves
The nomenclature of the adiabatic PEC within and outside a NAR differs for noncrossing and crossing adiabatic PES. For the former case, the assignment of the asymptotic states can be done according to their energies (e.g., U1 .Q/ > U2 .Q/), while for the latter case, the inequality sign is reversed as a system passes the NAR. When applying the model for the noncrossing adiabatic PEC to the crossing PEC, one should take into account this reversal of the nomenclature. Then the transition probability P12 between the crossing adiabatic PEC can be recovered from the survival probability for the noncrossing adiabatic PEC with the proper modification of parameters. A particular case of the linear crossing (LC) adiabatic potentials and the constant dynamical coupling corresponds to the Hamiltonian
Remarkable properties of the Landau–Zener model are: LZ (i) Expression Eq. (52.33) for the probability P12 is valid for arbitrary values of the exponent and not only for the large ones. In the latter case, the probability is very low, so that the system moves preferentially along a noncrossing adiabatic PEC. (ii) In the weak-coupling case (small value of the exponent in Eq. (52.33) termed narrow avoided crossing), the system moves preferentially along a crossing diabatic PES. The survival probability for a motion along the adiabatic PEC LZ D 2V 2 =.F vp /. is then very low, 1 P12
The D model [17] is obtained from Eq. (52.27) by setting # D =4 H D .Q/ D E ;
D H12 .Q/ D .E=2/ exp ˛.Q Qp / :
The transition probability in this case is D P12 D
exp./ : 1 C exp./
U LC .Q/ DF ˛.Q Qc / ; DLC 12 DD D const :
(52.37)
This Hamiltonian can be mapped onto the LZ Hamiltonian Eq. (52.31) by replacing the velocity vp at the NAR center with the velocity vc at the crossing point Qc . Taking into account the reversal of the LZ nomenclature for the adiabatic and diabatic functions in crossing the NAR, the nonadiabatic LC LZ is related to P12 as transition probability for the LC P12 ˇ LC LZ P12 D1 P12 .V; vp /ˇV !D;vp !vc
D1 exp 2D 2 =.„F vc / :
(52.38)
The matrix element D is often related to the Coriolis cou(52.34) pling between the adiabatic electronic states of different spatial symmetry. With D being proportional to the angular velocity of rotation of the molecular frame at the crossing point Qc , this coupling is normally weak, and in this limit (52.35) Eq. (52.38) reads
For this model, the adiabatic wave functions coincide with the diabatic functions on the one side of NAR, but on the other side, the former are expressed through the linear combinations of the latter with equal weights. The R model is defined by the condition E D 0, i.e., D 0. For arbitrary #, this case corresponds to the so-called accidental resonance (AR), and for # D =4, it corresponds to the symmetric resonance (SR). In these two cases, the transition probabilities read ˇ AR Nˇ D P12 D cos2 # ; P12 D0 ˇ SR Nˇ P12 D P12 D 1=2 : (52.36) D0;#D=4
LC 2D 2 =.„F vc / : P12
(52.39)
LC With P12 1, the system moves preferentially along the crossing PEC. However, sometimes the adiabatic PES are defined with the spin–orbit interaction neglected. In this case, the adiabatic electronic states are associated with different values of the electronic spin, and then the spin–orbit interaction is included into the coupling matrix element D. Then the approximation in Eq. (52.39) may not be valid, and one should use the general relation in Eq. (52.38) with the quantity D 2 in the exponent accounting for the contributions due to the dynamical (Coriolis) and static (spin–orbit) couplings.
52
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52.4
E. Nikitin and A. Kandratsenka
Double-Passage Transition Probabilities
where pk are the classical moments for motion across adiabatic potentials Uk .R/. Equation (52.41) was first derived by Stückelberg [18] for an avoided crossing situation under 52.4.1 Transition Probabilities assumption ˚12 D 0. in the Classically-Allowed If the single-passage transition probability is calculated in and Classically-Forbidden WKB Regimes: the LCT approximation, Eq. (52.41) represents a combinaInterference and Tunneling tion of LCT and WKB approximations. A further simplificaIf a system traverses several NAR, its dynamics can be char- tion of this expression corresponds to the replacement of the WKB acterized by successive matching of the different N matrices WKB phase, ˚12 , with its CT counterpart, by the intermediate matrices of the adiabatic evolution. The Ztt latter have a simple expression when the system motion sat
1 CT ˚12 D U R.t/ dt ; (52.43) isfies the standard WKB condition. In the case of an atomic „ tp collision, p the set R shrinks into a single coordinate, R, and vp D 2.E Up /= is expressed through the radial energy, E, of the relative motion of the atoms having the reduced where U.R/ D U1 .R/ U2 .R/, R.t/ is a CT, and, tp and tt mass, , and the potential energy, Up D UN .R/jRDRp . If there are the time moments corresponding to the NAR center and is only one NAR over the whole range of R, the colliding the turning points for the motion in the field of the CT poCT system traverses it twice, as the atoms approach and then tential, U .R/, respectively. An ambiguity in the definition CT recede, and there are two different paths between the cen- of U .R/ shows up in the approximation for P12 resulting WKB LC ter of the NAR, Rp , and the turning points, Rt1 and Rt2 , from the passage ˚12 ! ˚12 . When P12 in Eq. (52.41) is considered within the LZ or D on the adiabatic potential curves, U1 .R/ and U2 .R/. The models, the respective expressions are called Landau–Zener– double-passage nonadiabatic event 1 ! 2 consists of: (i) the Stückelberg (LZS) probability and Rosen–Zener–Demkov single-passage transition 1 ! 2, the adiabatic evolution Rp ! Rt2 ! Rp in the state j2i, the single-passage survival 2 ! 2; probability (although the interference phase was calculated (ii) the single-passage survival 1 ! 1, the adiabatic evolution in Eq. [19] for a special case of time-dependent coupling). Rp ! Rt1 ! Rp in state j1i, and the single-passage transition Analytical expressions for the dynamical phases for the mod1 ! 2. Accordingly, the double-passage nonadiabatic ampli- els discussed in Sect. 52.3 are also available [3, 6]. A particular example of the LZS probability corresponds for the transition 1 ! 2 reads tude AWKB 12 to the narrow avoided crossing between the adiabatic pop p tentials (or to the weak coupling between the crossing diAWKB D P12 exp.2i2 C 2i12 / P22 12 abatic curves). Here, one replaces the adiabatic potentials p p C P11 exp.2i1 C 2i21 / P12 ; (52.40) in Eq. (52.42) by their diabatic counterparts that define the WKB-D , with ˚12 D =4 [3]. Then Eq. (52.41), in phase, ˚12 where P22 D P11 D 1 P12 are the survival probabilities, the weak-coupling limit and with the LZS parameterization 21 and 22 are the WKB phases accumulated during the assumes the form derived by Landau (L) [14] adiabatic motion of a diatom from the center of NAR to 1=2 ˇ 8V 2 L the turning points and back, and 212 and 221 are the ˇ P12 .E/ E>U D p „F 2.E Up / so-called dynamical phases that originate from nonadiabatic
WKB-D dynamics in crossing the NAR. Then the double-passage sin2 ˚12 .E/ C =4 ; (52.44) ˇ ˇ WKB ˇ2 , assumes the form D ˇAWKB transition probability, P12 12 ˇ L which is valid for P12 .E/ˇE>Up 1. The unphysical diverˇ WKB gence in Eq. (52.44) at E ! Up is due to the use of the WKB P12 .E/ˇE>Up D approximation, when turning points are close to the crossing WKB 4P12 .1 P12 / sin2 .˚12 C ˚12 / : (52.41) point. For a correct expression in that case, see Sect. 52.4.2. In many applications, one can use the mean transition Here, probability hP12 i, which is obtained from P12 , by averaging over several oscillations, WKB .E/ D „.1 2 / „˚12 (52.45) hP12 i D 2P12 .1 P12 / ; ZRt1 ZRt2 p1 .R/ dR
D Rp
˚12 D 12 21 ;
p2 .R/dR ; Rp
such that the double-passage transition probability hP12 i is expressed only in terms of the single-passage transition prob(52.42) ability, P12 . Equation (52.45) is a simple example of the
52 Adiabatic and Diabatic Collision Processes at Low Energies
surface-hopping approximation [9], when one calculates the total transition probability as a results of two independent hop/survival events at the first and the second crossings of NAR: hop 1 ! 2, (survival 2 ! 2) C (survival 1 ! 1), hop 2 ! 1 (Sect. 52.5.2). The main condition of applicability of Eqs. (52.40)– WKB 1, i.e., the energy, E, is well above (52.45) is ˚12 the mean adiabatic potential at the NAR center, Up . In this case, the NAR region is classically accessible, and the singlepassage probabilities depend on the local kinetic energy E Up at the NAR center. With decreasing collision energy, WKB 1 for the WKB interference eventhe condition ˚12 tually breaks down, implying also the inapplicability of the LCT approximation for the single-passage transition probability. When the energy, E, drops well below Up , one can again use the WKB approximation for the description of tunneling nonadiabatic transitions [1, 3]. Then the system reaches NAR by motion in the classically forbidden range of R, and this motion is characterized by the imaginary-value ˇ WKB ˇ , which is obtained from the real-valued quantity ˚12 EUp
the complex energy plane from the classically accessible to the ˇclassically forbidden WKB motion. Under the conˇ WKB .E/ˇE 0 and Z2 < 0 refer to the nucleus and the antinucleus, respectively. The dependencies of the cross section Eq. (57.62) on Z1 , jZ2 j and turn out to be similar to those on Z1 , Z2 and for nonradiative electron transfer (see Eq. (57.42)) and have basically the same origin. In particular, the 1 -dependence of the cross section Eq. (57.62) reflects the decrease in the momentum overlap
57
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A. Voitkiv
Cross section (μb)
0.01
10 –3
10 –4
1
10
100 Collision energy (GeV/u)
Fig. 57.8 Formation of antihydrogen via bound–bound, p C U92C ! H.1s/ C U91C .1s/ (solid curve), and bound–free, p C U92C ! H.1s/ C U92C C e (dashed curve), pair production. For more information, the dotted curve displays the cross section for nonradiative capture, H.1s/C U92C ! pC C U91C .1s/, multiplied by 2
between the initial and final states with an increase in the impact energy.
57.4.4
More Advanced Theoretical Methods
In addition to the simplest theoretical methods, discussed in the previous sections, also more sophisticated models for pair production have been developed. In particular, they include one and two-center coupled-channel methods [19], direct numerical solutions of the time-dependent Dirac equation on a lattice [19], continuum-distorted-wave models [16], various eikonal approximations including the symmetric eikonal model [18], and the light-cone approximation [17].
References 1. McDowell, M.R.C., Coleman, J.P.: Introduction to the Theory of Ion-Atom Collisions. North-Holland, Amsterdam, London (1970) 2. Voitkiv, A.B., Ullrich, J.: Relativistic Collisions of Structured Atomic Particles. Springer, Berlin (2008) 3. Jackson, J.D.: Classical Electrodynamics, 3rd ed., Wiley, New York (1999) 4. Inokuti, M.: Rev. Mod. Phys. 43, 297 (1971) 5. McGuire, J.H.: Electron Correlation Dynamics in Atomic Collisions. Cambridge University Press, Cambridge (1997) 6. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics. Pergamon, Oxford (1977) 7. Bethe, H.A., Salpeter, E.E.: Quantum Mechanics of One- and Two-Electron Atoms. Academic Press, New York (1957) 8. Rose, M.E.: Relativistic Electron Theory. Wiley, New York (1975)
9. Berestetskii, V.B., Lifshitz, E.M., Pitaevskii, L.P.: Quantum Electrodynamics. Pergamon, Oxford (1982) 10. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1965) 11. Dedrick, K.G.: Rev. Mod. Phys. 34, 429 (1962) 12. Hillenbrand, P.-M., Hagmann, S., Jakubassa-Amundsen, D.H., Monti, J.M.: Phys. Rev. A 91, 022705 (2015) 13. In Chapter 53 of Springer Handbook of Atomic, Molecular and Optical Physics (2005). 14. Anholt, R.: Phys. Rev. A 19, 1004 (1979) 15. Landau, L.D., Pitaevskii, L.P., Lifshitz, E.M.: Electrodynamics of Continuous Media. Elsevier/Butterworth-Heinemann, Amsterdam, Boston (2008) 16. Crothers, D.S.F.: Relativistic Heavy-Particle Collision Theory. Kluwer Academic/Plenum, London (2000) 17. Baltz, A.J.: Phys. Rev. Lett. 78, 1231 (1997) 18. Voitkiv, A.B., Najjari, B., Shevelko, V.P.: Phys. Rev. A 82, 022707 (2010) 19. Eichler, J.: Lectures On Ion-Atom Collisions: From Nonrelativsitic To Relativistic Velocities. Elsevier, Amsterdam (2005) 20. Schulz, M.: Adv. At. Mol. Opt. Phys. 66, 508 (2017) 21. Moliere, G.: Naturforschung 2A, 133 (1947) 22. Salvat, F., Martinez, J.D., Mayol, R., Parellada, J.: Phys. Rev. A 36, 467 (1987) 23. Stolterfoht, N., DuBois, R.D., Rivarola, R.D.: Electron Emission in Heavy Ion-Atom Collisions. Springer, Berlin, Heidelberg (1997) 24. Voitkiv, A.B., Najjari, B.: Phys. Rev. A 85, 052712 (2012) 25. Lyaschenko, K.N., Andreev, O.Y., Voitkiv, A.B.: J. Phys. B 51, 055204 (2018) 26. Gumberidze, A., Thorn, D.B., Fontes, C.J., Najjari, B.: Phys. Rev. Lett. 110, 213201 (2013) 27. Greiner, W.: Relativistic Quantum Mechanics, 3rd edn. Springer, Berlin, Heidelberg, New York (2000) 28. Bates, D.R., McCarroll, R.: Prog. Roy. Soc. Lond. A 245, 175 (1958) 29. Najjari, B., Voitkiv, A.B., Artemyev, A., Surzhykov, A.: Phys. Rev. A 80, 012701 (2009) 30. Dewangan, D.P., Eichler, J.: Phys. Rep. 247, 59 (1994) 31. Dz. Belkic: Dz. Belkic. J. Comp. Meth. Sci. Tech. 1, 1 (2001) 32. Jakubassa-Amundsen, D.H., Amundsen, P.A.: Z. Phys. A 298, 13 (1980) 33. Eichler, J., Stöhlker, T.: Phys. Rep. 439, 1 (2007) 34. Baur, G., Hencken, K., Trautmann, D.: Phys. Rep. 453, 1 (2007) 35. Bertulani, C.A., Baur, G.: Phys. Rep. 163, 299 (1988) 36. Krause, H.F., Vane, C.R., Datz, S.: Phys. Rev. Lett. 80, 1190 (1998) 37. Krause, H.F., Vane, C.R., Datz, S.: Phys. Rev. A 63, 032711 (2001) 38. Vane, C.R., Krause, H.F.: Nim B 261, 244 (2007) 39. Voitkiv, A.B., Najjari, B., Di-Piazza, A.: New J. Phys. 12, 063011 (2010)
Alexander Voitkiv received his PhD from the State University (Tashkent, USSR, 1989) and the degree of Doctor of Physical-Mathematical Sciences from the Lebedev Physics Institute (Moscow, Russia, 2010). He worked at the Institute of Electronics (Tashkent) and is currently at the Heinrich-Heine-University (Düsseldorf, Germany). His research focuses on the interactions of ions, electrons, and photons with atoms and molecules.
58
Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes Edmund J. Mansky II and M. Raymond Flannery
Contents 58.1 58.1.1 58.1.2 58.1.3 58.1.4 58.1.5
Recombination Processes . . . . . . . . . . Electron–Ion Recombination . . . . . . . . . Positive–Ion Negative–Ion Recombination Balances . . . . . . . . . . . . . . . . . . . . . Neutral–Neutral Recombination . . . . . . . N-Body Recombination . . . . . . . . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
846 846 846 846 847 848
58.2 58.2.1 58.2.2 58.2.3 58.2.4 58.2.5
Collisional-Radiative Recombination . . Saha and Boltzmann Distributions . . . . . . Quasi-Steady State Distributions . . . . . . . Ionization and Recombination Coefficients Working Rate Formulae . . . . . . . . . . . . Computer Codes . . . . . . . . . . . . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
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. . . . . .
. . . . . .
849 849 850 850 850 850
58.3 58.3.1
Macroscopic Methods . . . . . . . . . . . . . . . . . Resonant Capture-Stabilization Model: Dissociative and Dielectronic Recombination . . . . Reactive Sphere Model: Three-Body Electron–Ion and Ion–Ion Recombination . . . . . . . . . . . . . . Working Formulae for Three-Body Collisional Recombination at Low Density . . . . . . . . . . . . Recombination Influenced by Diffusional Drift at High Gas Densities . . . . . . . . . . . . . . . . . .
58.3.2 58.3.3 58.3.4
. . 853
58.11.7
Dissociative Recombination . Curve-Crossing Mechanisms . Quantal Cross Section . . . . . Noncrossing Mechanism . . .
58.8 58.8.1 58.8.2
Mutual Neutralization . . . . . . . . . . . . . . . . . . . 860 Landau–Zener Probability for Single Crossing at RX . 860 Cross Section and Rate Coefficient for Mutual Neutralization . . . . . . . . . . . . . . . . . . 861
E. J. Mansky II () Eikonal Research Institute Bend, OR, USA e-mail: [email protected]
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
58.11 58.11.1
. . 853
58.7 58.7.1 58.7.2 58.7.3
. . . .
58.10.3 58.10.4 58.10.5
58.11.4 58.11.5 58.11.6
Field-Assisted Methods . . . . . . . . . . . . . . . . . . 857 . . . .
58.10.2
. . 851
58.6
. . . .
58.10.1
58.11.3
Hyperspherical Methods . . . . . . . . . . . . . . . . . 856 Three-Body Recombination Rate (Identical Particles) . . . . . . . . . . . . . . . . . . . . . . 856 N-Body Recombination Rate (Identical Particles) . . . . . . . . . . . . . . . . . . . . . . 857
. . . .
Microscopic Methods for Termolecular Ion–Ion Recombination . . . . . . . . . . . . . . . . . . . . . . Time-Dependent Method: Low Gas Density . . . . . . . . . . . . . . . . . . . . . Time-Independent Methods: Low Gas Density . . . . . . . . . . . . . . . . . . . . . Recombination at Higher Gas Densities . . . . . . . Master Equations . . . . . . . . . . . . . . . . . . . . . Recombination Rate . . . . . . . . . . . . . . . . . . .
. . 850
58.5 58.5.1
. . . .
58.10
58.11.2
Zero-Range Methods . . . . . . . . . . . . . . . . . . . . 855
. . . .
One-Way Microscopic Equilibrium Current, Flux, and Pair Distributions . . . . . . . . . . . . . . . . . . . 861
. . 850
58.4
58.5.2
58.9
. . . .
. . . .
857 857 858 860
58.12
. . 862 . . 862 . . . .
. . . .
Radiative Recombination . . . . . . . . . . . . . . . . . Detailed Balance and Recombination-Ionization Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kramers Cross Sections, Rates, Electron Energy-Loss Rates, and Radiated Power for Hydrogenic Systems . . Basic Formulae for Quantal Cross Sections . . . . . . . . . . . . . . . . . Bound–Free Oscillator Strengths . . . . . . . . . . . . . Radiative Recombination Rate . . . . . . . . . . . . . . . Gaunt Factor, Cross Sections, and Rates for Hydrogenic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exact Universal Rate Scaling Law and Results for Hydrogenic Systems . . . . . . . . . . . . . . . . . . .
863 864 865 865 866 867 867 869 871 871 872 873
Useful Quantities . . . . . . . . . . . . . . . . . . . . . . 873
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873
Abstract
This chapter collects together the formulae, expressions, and specific equations that cover various aspects, approximations, and approaches to electron–ion, ion–ion and neutral–neutral recombination processes. The primary focus is on recombination processes in the gas phase, both at thermal energies and in ultracold regimes. Recombination processes are ubiquitous in nature. These reactions occur in a wide variety of applications and are an important formation or loss mechanism of atoms and molecules. To illustrate the types of problems where recombination is important, we enumerate six broad areas in which recombination processes occur: (a) collisionalradiative recombination processes, involving hydrogen
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_58
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and helium, which are important in understanding the cosmic microwave background in cosmology [1–3]; (b) radio recombination lines involving electrons and ions, which are central to understanding the observed spectra from interstellar clouds and planetary nebulae [4, 5]; (c) recombination processes, involving electrons and holes are important in semiconductors [6, 7]; (d) electron–ion and ion–ion recombination processes, which are important in understanding the properties of plasmas, whether they are in the upper atmosphere, the solar corona, or industrial reactors on earth [8–10]; (e) atom–molecule recombination involving oxygen, which are important mechanism for forming ozone [11]; and, finally, (f) three-body recombination processes, involving neutral bosons, which are an important loss mechanism in ultracold Rydberg atom collisions, leading to the depletion of the Bose Einstein condensate (BEC) [12–14].
Field-assisted: Electron–ion recombination can also occur by application of a laser field, e C AC .i/ C h ! A.n`/ C h 0 ;
(58.5)
leading to high-order harmonic generation (HHG).
58.1.2 Positive–Ion Negative–Ion Recombination This proceeds via the following three processes: (e) Mutual neutralization AC C B ! A C B :
(58.6)
(f) Three-body (termolecular) recombination Keywords
AC C B C M ! AB C M : radiative recombination mutual neutralization dielectronic recombination dissociative recombination vibra- (g) Tidal recombination tional wave function ABC C C C M ! AC C B C M
(58.8a)
! BC C A C M ;
(58.8b)
58.1 58.1.1
(58.7)
where M is some third species (atomic, molecular, or ionic). Although (e) always occurs when no gas M is present, it is greatly enhanced by coupling to (f). The dependence of the rate ˛O on density N of background gas M is different for all three cases, (e)–(g).
Recombination Processes Electron–Ion Recombination
This proceeds via the following four processes:
Processes (a), (c), (d), and (e) are elementary processes in that microscopic detailed balance (proper balance) ex(a) Radiative recombination (RR) ists with their true inverses, i.e., with photoionization (both e C AC .i/ ! A.n`/ C h : (58.1) with and without autoionization) as in (c) and (a), associative ionization and ion-pair formation as in (d) and (e), (b) Three-body collisional-radiative recombination respectively. Processes (b), (f), and (g) in general involve e C AC C e ! A C e ; (58.2a) a complex sequence of elementary energy-changing meche C AC C M ! A C M ; (58.2b) anisms as collisional and radiative cascade and their overall rates are determined by an input–output continuity equation where the third body can be an electron or a neutral gas. involving microscopic continuum-bound and bound–bound (c) Dielectronic recombination (DLR) collisional and radiative rates.
e C AZC .i/ • AZC .k/ e n` .Z1/C
! An0 `0
.f / C h ;
(58.3)
58.1.3 Balances
(d) Dissociative recombination (DR) (58.4) Proper Balances Proper balances are detailed microscopic balances between Electron recombination with bare ions can proceed only forward and reverse mechanisms that are direct inverses of via (a) and (b), while (c) and (d) provide additional path- one another, as in ways for ions with at least one electron initially or for molecular ions ABC . Electron radiative capture denotes (a) Maxwellian: e .v1 / C e .v2 / • e .v10 / C e .v20 / ; the combined effect of RR and DLR. (58.9) e C ABC ! A C B :
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
847
Indirect pathways 8 ET (b) Saha: e C H.n`/ • e C HC C e (58.10) ˆ indirect < k! .A B/ C C ! AB C C A C B C C k Ex ; (58.14) between direct ionization from and direct recombination into ˆ indirect : ! .A C/ C B ! AB C C a given level n`; where the kinetic energy of the particles is redistributed;
Boltzmann: e C H.n`/ • e C H.n0 ; `0 /
(58.11) where .A B/ indicates an intermediate collision complex ET Ex formed between atoms A and B, and kindirect , kindirect are the between excitation and deexcitation among bound levels; reaction rates (cm3 =sec) for the transfer and exchange reac(58.12) tions in Eq. (58.14) (upper/lower), respectively. Prediction (d) Planck: e C HC • H.n`/ C h ; of rates k, in both the direct and indirect pathways, dewhich involves interaction between radiation and atoms in pends upon detailed knowledge of the molecular potential photoionization/recombination to a given level n`. energy surfaces, curve-crossings, and tunneling probabilities for specific reactants. Improper Balances Improper balances maintain constant densities via produc- LTE effective rate equation tion and destruction mechanisms that are not pure inverses of For systems in local thermodynamic equilibrium (LTE), the each other. They are associated with flux activity on a macro- effective rate equations are scopic level as in the transport of particles into the system for 1 d recombination and net production and transport of particles (58.15) ŒAB D kr ŒAŒB kd ŒAB ; ŒC dt (i.e. e ; AC ) for ionization. Improper balances can then exist between dissimilar elementary production–depletion pro- where the square brackets indicate the number density of the cesses as in (a) coronal balance between electron-excitation enclosed species, and k denotes the rate constant for threer into and radiative decay out of level n. (b) Radiative balance body recombination, while k the rate constant for collisiond between radiative capture into and radiative cascade out of induced dissociation level n. (c) Excitation saturation balance between upward X ŒAB.u/ collisional excitations n 1 ! n ! n C 1 between adjacent ku!b ; (58.16a) kr D ŒAŒB levels. (d) Deexcitation saturation balance between downb;u ward collisional de-excitations n C 1 ! n ! n 1 into and X ŒAB.b/ kb!u ; (58.16b) kd D out of level n. ŒAB (c)
b;u
58.1.4
with b and u representing bound and unbound states of the indicated collision complex, respectively. The state-to-state rate coefficients are defined by
Neutral–Neutral Recombination
Neutral–neutral reactions, leading to recombination or dissociation, generally proceed via resonances involving transi Z1 8kB T 1=2 2 tions among molecular electronic states. At thermal energies, .kB T / i !j .ET / ki !j D these reactions are studied via transition state theory, reaction 0 rate theory, and a wide range of semiclassical techniques. (58.17) exp.ET =kB T /ET dET ; See Chap. 37 of this Handbook and [15–19] for a general introduction to this vast literature in chemical physics and where is the reduced mass, E D E E is the translational T i chemistry. energy of the i-th state of the atom, and is the cross section for the indicated transition i ! j . Distinguishable Particles [3] When one or more reactants in the process are distinguish- NLTE effective rate equations able, the thermal reactions proceed via two broad pathways, For systems not in local thermodynamic equilibrium (NLTE) direct and indirect: the effective rate equations are Direct pathway
6 kdirect
A C B C C ! AB C C ;
(58.13)
where kdirect is the reaction rate (cm3 =s) for the formation of molecule AB and reactant C.
X d ŒAB.b/ D ŒC .ku!b ŒAB.u/ kb!u ŒAB.b// dt u X
C ŒC kb 0 !b AB.b 0 / kb!b 0 ŒAB.b/ ; b0
(58.18a)
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X d .kb!u ŒAB.b/ ku!b ŒAB.u// ŒAB.u/ D ŒC dt b X
C ŒC ku0 !u AB.u0 / ku!u0 ŒAB.u/ u0
C kfelastic !u ŒAŒB
ŒAB.u/ ; u
composed of heavy and light bosons, or bosons and fermions, or of a boson together with heavy and light fermions, and are analyzed using zero-range methods. See Table 1 of [25] for details on the various cases where the Efimov effect occurs in such systems and Sect. 58.4 for explicit expressions for ˛. (58.18b) For details on the Efimov effect, see Sects. 58.4 and 60.6 in this Handbook and [25, 26].
where u is the lifetime of the unbound state u, and kfelastic !u is the elastic two-body rate constant.
Identical Particles At thermal energies, reactions involving identical particles proceed as detailed above. In ultracold regimes, collisions involving identical bosons proceed via Fano–Feshbach or Föster resonances [20–22]. These resonances can be exploited by varying the applied magnetic or electric fields used in traps to create and control Bose–Einstein condensates (BEC). At thermal and higher energies, the rate-limiting step [23] is generally either collisional or radiative capture into highlying Rydberg states (n 1), followed by radiative decay into lower lying states via n and n; ` mixing collisions. In contrast, at ultracold regimes, the rate-limiting step is collisional ` mixing, wherein rapid collisional capture into very high-lying Rydberg states (n > 200) is followed by slow collisional-radiative decay. The result is a cloud of Rydberg atoms that has condensed into a long-lived macroscopic BEC, which can be controlled via the applied fields used in the trap. For three identical particles, this proceeds as A C A C A ! A2 C A.Ef / : Time-dependent equation ( d nA before onset of BEC 3˛ n3A ; D 1 3 dt 2 ˛ nA at BEC
Time-dependent equation d nA D ˛ nA n2B : dt
58.1.5 N-Body Recombination [27] This proceeds, for four identical particles, as 8 ˆ < A3 C A.Ef / A C A C A C A ! A2 C A2 ˆ : A4
trimer + atom ; dimer + dimer ; tetramer ; (58.23)
and has been observed [28] in an ultracold sample of Cs atoms at 30 nK. See Fig. 58.1 for an illustration of the allowed regimes for the various reactant branches in (58.23) as a function of energy and inverse scattering length. The general problem of N-body recombination is challenging both experimentally and theoretically. Samples involving alkali atoms at ultracold temperatures currently provide the best conditions for observing N-body recom(58.19) bination. At present, the hyperspherical method of solving the N-body Schrödinger equation has been used most extensively on the problem. See Sect. 58.5 for closed-form expressions for N-body recombination rates ˛.
(58.20) a0 A+A+A+A
1/a
D+A+A
T+A
D+D
Tetra 2 Tetra 1
for distinct atoms A and B. At thermal energies this is analyzed as above using either Fig. 58.1 Illustration of allowed regions for 4 bosons (A+A+A+A), LTE or NLTE formulations involving the relevant molecular Dimers + 2 bosons (D+A+A), 2 Dimers (D+D) and Trimers + bosons potential surfaces. At ultracold regimes [24] the mixtures are (T+A). See [28] for details
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
58.2
849
Collisional-Radiative Recombination
Production rates and processes The production rate for a level i is Radiative recombination X R Pi D n e nf Kfc i C n2e N C kci Process Eq. (58.1) involves a free-bound electronic transition f ¤i with radiation spread over the recombination continuum. It is X
the inverse of photoionization without autoionization and faC nf Af i C Bf i vors high-energy gaps with transitions to low n 1; 2; 3 and f >i
low angular momentum states ` 0; 1; 2 at higher electron C n e N C ˛O iRR C ˇi ; energies. Three-body electron–ion recombination Processes Eq. (58.2a,b) favor free-bound collisional transitions to high levels n, within a few kB T of the ionization limit of A.n/ and collisional transitions across small energy gaps. Recombination becomes stabilized by collisional-radiative cascade through the lower-bound levels of A. Collisions of the e AC pair with third bodies become more important for higher levels of n, and radiative emission is important down to and among the lower levels of n. In optically thin plasmas this radiation is lost, while in optically thick plasmas it may be reabsorbed. At low electron densities, radiative recombination dominates with predominant transitions taking place to the ground level. For process Eq. (58.2a) at high electron densities, three-body collisions into high Rydberg levels dominate, followed by cascade, which is collision dominated at low electron temperatures T e and radiation dominated at high T e . For process Eq. (58.2b) at low gas densities N , the recombination is collisionally radiatively controlled while, at high N , it eventually becomes controlled by the rate of diffusional drift Eq. (58.73) through the gas M.
58
(58.26)
where the terms in the above order represent (1) collisional excitation and deexcitation by e –A.f / collisions, (2) three-body e –AC collisional recombination into level i, (3) spontaneous and stimulated radiative cascade, and (4) spontaneous and stimulated radiative recombination. Destruction rates and processes The destruction rate for a level i is X ni Di D n e ni Kifc C n e ni Si f ¤i
C ni
X
Aif C Bif
f i
where the terms in the above order represent (1) collisional destruction, (2) collisional ionization, (3) spontaneous and stimulated emission, (4) photoexcitation, and (5) photoionization.
Collisional-radiative recombination [29] 58.2.1 Saha and Boltzmann Distributions Here, the cascade collisions and radiation are coupled via the continuity equation. The population ni of an individual exCollisions of A.n/ with third bodies such as e and M are cited level i of energy Ei is determined by the rate equations more rapid than radiative decay above a certain excited level d ni @ni n . Since each collision process is accompanied by its ex(58.24) D C r .ni vi / act inverse the principle of detailed balance determines the dt @t X (58.25) population of levels i > n . nf f i ni if D Pi ni Di ; D i ¤f
which involve temporal and spatial relaxation in Eq. (58.24) and collisional-radiative production rates Pi and destruction frequencies Di of the elementary processes included in Eq. (58.25). The total collisional and radiative transition frequency between levels i and f is if and the f -sum is taken over all discrete and continuous (c) states of the recombining species. The transition frequency if includes all contributing elementary processes that directly link states i and f , e.g., collisional excitation and deexcitation, ionization (i ! c) and recombination (c ! i) by electrons and heavy particles, radiative recombination (c ! i), radiative decay (i ! f ), possibly radiative absorption for optically thick plasmas, autoionization, and dielectronic recombination.
Saha Distribution This connects equilibrium densities nQ i , nQ e , and NQ C of bound levels i, of free electrons at temperature T e , and of ions by g.i/ h3 nQ i D exp.Ii =kB T e / ; C g e gA .2 m e kT e /3=2 nQ e NQ C (58.28) where the electronic statistical weights of the free electron, the ion of charge Z C 1, and the recombined e AC species of net charge Z and ionization potential Ii are g e D 2, gAC , and g.i/, respectively. Since ni nQ i for all i, then the Saha– Boltzmann distributions imply that n1 ni and n e ni for i ¤ 1; 2, where i D 1 is the ground state.
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the e AC or AC B continuum levels and the ground A.n D 1/ or the lowest vibrational levels of AB are, therefore, treated as two large particle reservoirs of reactants and products. These two reservoirs act as reactant and sink blocks C and S, which are, respectively, drained and filled at the same rate via a conduit of highly excited levels, which comprise an intermediate block of levels E . This C draining and S filling 58.2.2 Quasi-Steady State Distributions proceeds, via block E , on a timescale large compared with the short time for a small amount from the reservoirs to be The reciprocal lifetime of level i is the sum of radiative and redistributed within block E . This forms the basis of QSS. collisional components, and this lifetime is, therefore, shorter than the pure radiative lifetime R 107 Z 4 s. The lifetime 1 for the ground level is collisionally controlled and 58.2.4 Working Rate Formulae depends on n e , and, generally, is within the range of 102 and 104 s for most laboratory plasmas and the solar atmoFor electron–atomic–ion collisional-radiative recombination sphere. The excited level lifetimes i are then much shorter Eq. (58.2a), detailed QSS calculations can be fitted by the than 1 . The (spatial) diffusion or plasma decay (recombinarate [30] tion) time is then much longer than i , and the total number
˛O CR D 3:8 109 T e4:5 n e C 1:55 1010 T e0:63 of recombined species is much smaller than the ground-state 3 1 population n1 . The recombination proceeds on a timescale C6 109 T e2:18 n0:37 (58.31) cm s ; e much longer than the time for population/destruction of the ˛ optically excited levels. The condition for quasi-steady state, or QSS- which agrees with experiment for a Lyman 9 3 and T in the range 10 cm ne thick plasma with n e e condition, dni =dt D 0 for the bound levels i ¤ 1, therefore, 13 3 cm and 2:5 K T 4000 K. The first term is the 10 e holds. The QSS distributions ni , therefore, satisfy Pi D ni Di . pure collisional rate Eq. (58.61), the second term is the radiative cascade contribution, and the third term arises from collisional-radiative 58.2.3 Ionization and Recombination coupling.
C He , recombination in a high-pressure (5 – For e 2 Coefficients 100 Torr) helium afterglow, the rate for Eq. (58.2b) is [31]
Under QSS, the continuity Eq. (58.25) then reduces to a finite ˛O CR D .4 ˙ 0:5/ 1020 n e .T e =293/.4˙0:5/ i h set of simultaneous equations Pi D ni Di . This gives a matrix 27 10 C .5 ˙ 1/ 10 n.He/ C .2:5 ˙ 2:5/ 10 equation that is solved numerically for ni .i ¤ 1/ nQ i in terms of n1 and n e . The net ground-state population frequency per .T e =293/.1˙1/ cm3 =s : unit volume (cm3 s1 ) can then be expressed as (58.32)
Boltzmann Distribution This connects the equilibrium populations of bound levels i of energy Ei by
nQ i =nQj D Œg.i/=g.j / exp .Ei Ej /=kB T e : (58.29)
dn1 (58.30) D n e N C ˛O CR n e n1 SCR ; dt where ˛O CR and SCR , respectively, are the overall rate coefficients for recombination and ionization via the collisionalradiative sequence. The determined ˛O CR equals the direct .c ! 1/ recombination to the ground level supplemented by the net collisional-radiative cascade from that portion of bound-state population that originated from the continuum. The determined SCR equals direct depletion (excitation and ionization) of the ground state reduced by the deexcitation collisional radiative cascade from that portion of the bound levels accessed originally from the ground level. At low n e , ˛O CR and SCR reduce, respectively, to the radiative recombination coefficient summed over all levels and to the collisional ionization coefficient for the ground level.
The first two terms are in accord with the purely collisional rates Eqs. (58.61) and (58.64b), respectively.
58.2.5 Computer Codes A large number of computer codes for solving the collisional-radiative equations in astrophysical plasmas and fusion plasmas are available. See Table 58.1 for details.
58.3
Macroscopic Methods
58.3.1 Resonant Capture-Stabilization Model: Dissociative and Dielectronic Recombination
C , E , and S Blocks of Energy Levels
For the recombination processes in Eqs. (58.2a), (58.2b), and The electron is captured dielectronically, Eq. (58.53), into an (58.7), which involve a sequence of elementary reactions, energy-resonant long-lived intermediate collision complex of
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes Table 58.1 Recombination computer codes
Name ACQD AAID ADNT AATR AEMA ABUV RICO RECFAST HYREC LASER
851
Description Radiative recombination of hydrogenic ions COLLRAD CRModel COLRAD RATIP TRIP 1 Machine-learning recombination code for BBN BBN code Primordial hydrogen and helium recombination code Los Alamos suite of relativistic atomic physics codes
Reference/Link CPC 1 (1969) 31 CPC 12 (1976) 205 CPC 135 (2001) 135 CPC 44 (1987) 157 CPC 183 (2012) 1525 CPC 16 (1978) 129 ApJSS 181 (2009) 627 AA 475 (2007) 109 Phys. Rev. D 83 (2011) 043513 [61] & http://aphysics2.lanl.gov/tempweb
superexcited states d , which can autoionize or be stabilized Macroscopic Detailed Balance and Saha irreversibly into the final product channel f either by molec- Distribution ular fragmentation nQ d kc .d / K .T / D D D kc .d /a .d ! i/ d i kc C s Q .d ! i/ nQ e N a e C ABC .i/ • AB ! A C B ; (58.33) a (58.41a) as in direct dissociative recombination (DR), or by emission
!.d / h3 D exp Edi =kB T ; of radiation as in dielectronic recombination (DLR) 3=2 C .2 me kB T / 2! kc (58.41b) s .Z1/C .f / C h : e C AZC .i/ • AZC .k/ e n` ! An0 `0 a where Edi is the energy of superexcited neutral levels AB (58.34) above that for ion level ABC .i/, and ! are the corresponding statistical weights. Production Rate of Superexcited States d dnd / nd ŒA .d / C S .d / I D n e N C kc .d X dt a .d ! i 0 / ; A .d / D i0
S .d / D
X
s .d ! f 0 / :
f0
(58.35) Alternative Rate Formula X (58.36a) a .d ! i/s .d ! f / ˛Of D Kd i : A .d / C S .d / d (58.36b) Normalized Excited-State Distributions a .d ! i/ ; ŒA .d / C S .d / X X kc .d /P S .d / D Kd i d S .d / ˛O D
d D nd =nQ d D
Steady-State Distribution For a steady-state distribution, the capture volume is nd kc .d / D : n eN C A .d / C S .d /
(58.37)
d
D
X
(58.42)
(58.43) (58.44a)
d
kc .d /Œ d S .d /a .d ! i/ : (58.44b) Recombination Rate and Stabilization Probability d The recombination rate to channel f is Although equivalent, Eqs. (58.38a) and (58.42) are normally X kc .d /s .d ! f / invoked for Eqs. (58.33) and (58.34), respectively, since ; (58.38a) P S 1 for DR, so that ˛O ! k ; and for DLR ˛Of D DR c A S A .d / C S .d / d with n 50 so that ˛O ! Kd i s . For n 50, S A and and the rate to all product channels is ˛O ! kc . The above results Eqs. (58.38a) and (58.42) can also be derived from microscopic Breit–Wigner scattering theory X kc .d /S .d / : (58.38b) for isolated (nonoverlapping) resonances. ˛O D A .d / C S .d / d In the above, the quantities PfS .d / D s .d ! f /=ŒA .d / C S .d / ;
(58.39)
P S .d / D S .d /=ŒA .d / C S .d / ;
(58.40)
represent the corresponding stabilization probabilities.
58.3.2 Reactive Sphere Model: Three-Body Electron–Ion and Ion–Ion Recombination Since the Coulomb attraction cannot support quasi-bound levels, three body electron–ion and ion–ion recombination
58
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E. J. Mansky II and M. R. Flannery
do not, in general, proceed via time-delayed resonances but rather by reactive (energy-reducing) collisions with the third body M. This is particularly effective for A–B separations R R0 , as in the sequence kc
A C B • AB .R R0 / ; d s
AB .R R0 / C M • AB C M : s
(58.45a) (58.45b)
In contrast to Eqs. (58.33) and (58.34), where the stabilization is irreversible, the forward step in Eq. (58.45b) is reversible. The sequence Eqs. (58.45a) and (58.45b) represents a closed system where thermodynamic equilibrium is eventually established.
Steady-State Distribution of AB Complex kc s n D nA .t/nB .t/ C ns .t/ : (58.46) s C d s C d Saha and Boltzmann balances Saha: nQ A nQ B kc Boltzmann: nQ s s
D nQ d ; D nQ s I
Time-Independent Treatment The rate ˛O 3 given by the time-dependent treatment can also be deduced by viewing the recombination process as a source block C kept fully filled with dissociated species A and B maintained at equilibrium concentrations nQ A , nQ B (i.e., c D 1) and draining at the rate ˛O 3 nQ A nQ B through a steady-state intermediate block E of excited levels into a fully absorbing sink block S of fully associated species AB kept fully depleted with s D 0, so that there is no backward redissociation from block S. The frequency kd is deduced as if the reverse scenario, s D 1 and c D 0, holds. This picture uncouples ˛O and kd and allows each coefficient to be calculated independently. Both dissociation (or ionization) and association (recombination) occur within block E . If c D 1 and s D 0, then
D n =nQ D d =.s C d / ;
K D nQ =nQ A nQ B D kc =d D kc d ;
S
P D s =.s C d / D s s ;
(58.55a) (58.55b) (58.55c)
and the recombination coefficient is (58.47)
˛O D kc P S D kc . s d / D K s :
(58.56)
nQ is in Saha balance with reactant block C and in Boltzmann Microscopic Generalization balance with product block S. From Eq. (58.206), the microscopic generalizations of rate in Eq. (58.52) and probability in Eq. (58.55c) are, respectively, Normalized Distributions Z1 Zb0 n "
D D P D c .t/ C P S s .t/ ; (58.48a) (58.57a) ˛O D v " e d" 2b dbP S ."; bI R0 / ; nQ 0 0 nA .t/nB .t/ ns .t/ ; s .t/ D : (58.48b) c .t/ D R0 I nQ A nQ B nQ s P S ."; bI R0 / D i .R/ib .R/ dt h s id ; (58.57b) Stabilization and Dissociation Probabilities Ri s d S D .b/ P D ; P D : (58.49) where i .R/ D n."; bI R/=n."; Q bI R/; i is the frequency .s C d / .s C d / Eq. (58.203a) of .A–B/–M continuum-bound collisional Time-Dependent Equations transitions at fixed A–B separation R, Ri is the pericenter of the orbit, .i "; b/, and dnc (58.50a) D kc P S nQ A nQ B Œc .t/ s .t/ ; dt b02 D R02 Œ1 V .R0 /=E; " D E=kB T ; (58.57c) ˝ S˛ dns D (58.50b) D s P nQ s Œs .t/ c .t/ ; (58.57d) ˛O kc P ";b ; v D .8kB T =MAB /1=2 ; dt ˚ 2 dnc (58.57e) kc D R0 Œ1 V .R0 /=kB T v ; (58.51) D ˛O 3 nA .t/nB .t/ C kd ns .t/ ; dt where MAB is the reduced mass of A and B. where the recombination rate coefficient (cm3 =s) and dissociation frequency are, respectively, Low Gas Densities Here i .R/ D 1 for E > 0, kc s ˛O 3 D kc P S D ; (58.52) .s C d / IR0 IR0 S s d P ."; bI R0 / D .t/ dt D ds=i I (58.58) ; (58.53) kd D s P D D .s C d / Ri
which also satisfy the macroscopic detailed balance relation ˛O 3 nQ A nQ B D kd nQ s :
1
Ri
i D .N / is the microscopic path length towards the .A– (58.54) B/–M reactive collision with frequency D N v . For i
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
constant, the rate in Eq. (58.57a) reduces at low N to ZR0 ˛O D .v0 N / 0
V .R/ 4R2 dR ; 1 kB T
which is linear in the gas density N .
853
1026 300 2:5 N cm3 s1 ; M T
(58.64b)
(58.59) where the mass M of the gas atom is now in a.m.u. This result agrees with the energy diffusion result of Pitaevskii [33] when R0 is taken as the Thomson radius RT D 23 R e .
58.3.4 Recombination Influenced by Diffusional 58.3.3 Working Formulae for Three-Body Drift at High Gas Densities Collisional Recombination at Low Density For three-body ion–ion collisional recombination of the form Diffusional-Drift Current C C AC C B C M in a gas at low density N , set V .R/ D e 2 =R. The drift current of A towards B in a gas under an A –B attractive potential V .R/ is Then Eq. (58.59) yields K 8kB T 1=2 4 3 Re c 3 J .R/ D Drn.R/ rV .R/ n.R/ (58.65a) ˛O .T / D R 1 C .0 N / ; (58.60) e MAB 3 0 2 R0 @ O where R e D e 2 =kB T , and the trapping radius R0 , determined R: (58.65b) D D NQ A NQ B eV .R/=kB T @R by the classical variational method, is 0:41R e , in agreement with detailed calculation. The special cases are the following. Relative Diffusion and Mobility Coefficients
(a) e C AC C e D D DA C DB ; Here, 0 D 19 R2e for . e e / collisions for scattering anK D KA C KB ; De D K.kB T / ; (58.66) gles =2, so that where the Di and Ki are, respectively, the diffusion and mo300 4:5 c ˛O ee .T / D 2:7 1020 n e cm3 s1 (58.61) bility coefficients of species i in gas M. T in agreement with Mansbach and Keck [32].
Normalized Ion-Pair R-Distribution
n.R/ (b) AC C B C M : (58.67)
.R/ D NQ A NQ B expŒV .R/=kB T Here, 0 v 109 cm3 s1 , which is independent of T for polarization attraction. Then Continuity Equations for Currents and Rates 2:5 300 N cm3 s1 : (58.62) ˛O 3 .T / D 2 1025 @n T (58.68a) C r J D 0 ; R R0 ; @t ˛O RN .R0 / .R0 / D ˛ .1/ O : (58.68b) (c) e C AC C M Only a small fraction ı D 2m=M of the electron’s energy is lost upon . e M/ collision, so that Eq. (58.57a) for constant The rate of reaction for ion pairs with separations R R0 is ˛RN .R0 /. This is the recombination rate that would be ob is modified to tained for a thermodynamic equilibrium distribution of ion ZR0 ZEm pairs with R R0 , i.e., for .R R0 / D 1. 2 Q E/v dE (58.63a) ˛O eM D 0 N 4R dR n.R; 0
Steady-State Rate of Recombination
0
ZR0 D v e 0 N 0
Z"m V .R/ 2 4R dR " e" d" ; 1 E 0
(58.63b)
Z1 ˛O NQ A NQ B D
@n dR D 4R02 J.R0 / : @t
(58.69)
R0
where " D E=kB T , and Em D ıe 2 =R D "m kB T is the max- Steady-State Solution imum energy for collisional trapping. Hence, ˛O
.R/ D .1/ 1 ; ˛O TR .R/ 8kB T e 1=2 2 R e R0 Œ0 N (58.64a) ˛O eM .T e / D 4ı O ˛O RN .R0 / :
.R0 / D .1/Œ˛= me
R R0
(58.70a) (58.70b)
58
854
E. J. Mansky II and M. R. Flannery 1 1 With the binary decomposition 1 i D i A C i B ,
Recombination Rate ˛O RN .R0 /˛O TR .R0 / ˛O D ˛O RN .R0 / C ˛O TR .R0 / ( ˛O RN ; N ! 0 ! ˛O TR ; N ! 1 :
(58.71a)
Diffusional-Drift Transport Rate 01 11 Z V .R/=kB T e ˛O TR .R0 / D 4D @ dRA : R2
(58.78)
(58.71b) Exact b2 -Averaged Probability With Vc D e 2 =R for the AC –B interaction in Eq. (58.75b), and at low gas densities N ,
(58.72)
R0
With V .R/ D e 2 =R, ˛O TR .R0 / D 4KeŒ1 exp.R e =R0 /
P S D PA C PB PA PB :
4R0 3Vc .R0 / 1 3A;B 2Ei PA;B .E; R0 / D ; Œ1 Vc .R0 /=Ei
(58.79)
appropriate for constant mean free path i . 1
;
(58.73)
where R e D e 2 =kB T provides a natural unit of length.
Langevin Rate For R0 R e , the transport rate
(E,b2 )-Averaged Probability P S .R0 / in Eq. (58.75c) at low gas density is PA;B .R0 / D PA;B .E D kB T; R0 / :
(58.80)
(58.74) Thomson Trapping Distance When the kinetic energy gained from Coulomb attraction is tends to the Langevin rate which varies as N 1 . assumed lost upon collision with third bodies, then bound .A; B/ pairs are formed with R RT . Since E D 32 kB T Reaction Rate e 2 =R, then When R0 is large enough that R0 -pairs are in .E; L2 / equi librium Eq. (58.206), 2 e2 2 R D (58.81) D Re : T Z1 Zb0 3 kB T 3 ˛O RN .R0 / D v " e" d" 2b dbP S ."; bI R0 / (58.75a) Thomson Straight-Line Probability 0 0 1 The E ! 1 limit of Eq. (58.77) is Z
2 S " (58.75b) v " e d" b0 P ."I R0 /
T .bI RT / D 1 exp 2 RT2 b 2 =A;B : (58.82) PA;B ˛O TR ! ˛O L D 4Ke
0
2 P S .R0 / ; vbmax
where
V .R0 / 2 D R02 1 bmax ; kB T
(58.75c)
(58.76)
and b0 and " are given by Eq. (58.57c) and v by Eq. (58.57d). The probability P S and its averages over b and (b; E) for reaction between pairs with R R0 is determined in Eq. (58.75a–58.75c) from solutions of coupled master equations; P S increases linearly with N initially and tends to unity at high N . The recombination rate in Eq. (58.71a) with Eq. (58.75a) and Eq. (58.73), therefore, increases linearly with N initially, reaches a maximum when ˛O TR ˛O RN , and then decreases eventually as N 1 , in accord with Eq. (58.74).
The b 2 -average is the Thomson probability T .RT / D 1 PA;B
1 1 e2X .1 C 2X/ 2 2X
(58.83a)
for reaction of .AB/ pairs with R RT . As N ! 0 T .RT / ! PA;B
4 1 3 2 X 1 X C X2 X3 C 3 4 5 6 (58.83b)
and tends to unity at high N ; X D RT =A;B D N.0 RT /. These probabilities have been generalized [34] to include hyperbolic and general trajectories.
Thomson Reaction Rate Reaction Probability The classical absorption solution of Eq. (58.196) is
˛O T D RT2 v PAT C PBT PAT PBT 0 R 1 I0 (
4 dsi A 1 RT3 1 A C B ; N ! 0 P S .E; bI R0 / D 1 exp@ : (58.77) 3 ! i RT2 v; N !1: Ri
(58.84)
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
58.4
Zero-Range Methods
855
Shallow dimers When a > 0, there is a single bound state with binding energy ED D 1=.ma2 /, referred to as a shallow dimer state. The three-body recombination rate ˛s into a shallow dimer state is [26, 36] p 128 2 .4 3 3/ „a4 ˛s D ; 2 2 sinh .s0 / C cosh .s0 / tan2 Œs0 ln.a ? / C m (58.89)
Zero-range methods refer to scattering models that make use of the scattering length to characterize the collision. Zerorange methods are used in ultracold collisions where the de Broglie wavelength of the atoms is much larger than the range of their interactions, and hence the scattering is well described by the S-wave scattering length, a. The T-matrix element for S-wave scattering of two identical particles, mass m and energy E D k 2 =m, is where ? is the wave number of the shallow dimer state n? , is a constant, and s0 is the root of the transcendental equation 1 8 ; (58.85) T .k/ D s0 s0 8 m k cot ı0 .k/ {k D p sinh ; (58.90) s0 cosh 2 6 3 where ı0 .k/ is the S-wave phase shift. In general, one would with approximate numerical solution of s0 1:00624. The solve numerically a couple-channel problem to compute phase constant has been computed in [37, 38] and is of the phase shifts for all the states of interest. In ultracold order unity. collisions, the low-energy effective range expansion of the The three-body collision, at low energies in reaction S-wave phase shift can be utilized Eq. (58.19), that results in the formation of dimers A2 exhibits a universal property, wherein the spacing between 1 (58.86) adjacent energy levels of the dimer, E .n/ follows an expok cot ı0 .k/ D 1=a C rs k 2 C T 2 nential scaling independent of system to simplify the scattering calculation. The effective range „2 ?2 .n/ expansion serves to define the scattering length a and the ET ! . e2=s0 /nn? ; (58.91) m effective range rs . See Sect. 47.5.5 and Eq. (47.46b) in this Handbook, and [35] for details on the scattering length and as n ! 1, and where ? is the wave number associated with effective range. For the typical ultracold collision involving the dimer state labeled by the integer n? . The scaling relaalkali atoms in specific hyperfine states, the van der Waals tionship Eq. (58.91) was first described by Efimov [39, 40]. length, `vdW , and energy EvdW , provide natural length and Further details on the universal scaling property Eq. (58.91) called the Efimov effect can be found in Sect. 60.6 of this energy scales, respectively Handbook, and [20, 25, 26, 41]. 2 1=4 (58.87a) `vdW D .mC6 =„ / ; Deep dimers 3 6 1=2 ; (58.87b) EvdW D .m C6 =„ / Depending on the details of the short-range part of the dimer molecular potential, governed by the parameters a, ? , and for atom of mass m and van der Waals constant C6 . For alkali the inelasticity parameter ? , the dimer may support multiatom collisions, the scattering lengths are [35]: ple states below threshold, called deep dimer states. See [26], and references therein, for a complete discussion of these pa6 85 133 Li.a t / Rb.as / Cs.a t / rameters. a.a0 / 2160 2800 2400 The three-body recombination rate ˛d into a deep dimer state is [26] where the subscript s or t refer to singlet or triplet states, Cmax cosh.s0 / sinh.s0 / cosh ? sinh ? „a4 respectively. ˛d D ; sinh2 .s0 C ? / C sin2 Œs0 ln.a ? / C m Contrast these large values for alkali atom collisions with (58.92) the much smaller values for e C Rg atom collisions in Sect. 47.5.5 (Rg D He Xe). See Table 1 of [26] for a more where ? is the inelasticity parameter, and Cmax is defined complete tabulation. The elastic scattering cross section for p two bosonic atoms in the same spin state is 128 2 .4 3 3/ Cmax D ; (58.93) sinh2 .s0 / 2 8a ; (58.88) while s and retain their meaning as in Eq. (58.89). elastic .E/ D 0 1 C ma2 E Expressions for scattering lengths and effective ranges for shallow and deep dimers can be found in Sect. 60.6. where a is the scattering length.
58
856
E. J. Mansky II and M. R. Flannery
58.5
Hyperspherical Methods
Three-particle case
Hyperspherical methods are techniques of solving the Nbody Schrödinger equation where the coupled partial differential equations are reformulated in terms of hyperspherical coordinates. See Chap. 56 of this Handbook and [20] for details on hyperspherical methods.
q x12 C x22 C x32 C C xd2 :
(58.102a)
1 m3 .m1 C m2 / ; m1 C m2 C m3 m1 m2 m3 ; 2 D m1 C m2 C m3
xd 1 D R sin ˛d 1 cos ˛d 2 ; xd 2 D R sin ˛d 1 sin ˛d 2 cos ˛d 3 ;
(58.102b)
(58.103a) (58.103b)
for particles i with mass mi and position vector rEi , i D 13. The mass-scaled Jacobi coordinates, E1 and E2 determine the hyperangles and ' in the body frame x–y plane, yielding a rescaled Schrödinger equation for three identical particles
d 1 Y
E1 D .rE2 rE1 /= ; m1 rE1 C m2 rE2
E2 D rE3 ; m1 C m2
2 D
(58.94)
xd D R cos ˛d 1 ;
x2 D R
(58.101)
with,
d-dimensional coordinates [42] RD
R2 D 12 C 22 ; 0 R < 1 :
sin ˛j ;
j D1
15 2 1 @2 C C C V .R; ; '/ E D EE ; sin ˛j cos ˛1 ; (58.95a) x1 D R 2 @R2 8R2 2R2 (58.104) j D2 where the full three-particle interaction potential V is exwhere R is the hyperradius, xi the coordinates of the pressed in terms of the hyperradius R and hyperangles and particles, and ˛j the corresponding hyperangles. The set ', and the angular momentum operator is given in [20, Eq. (58.95a) is usually termed the canonical choice and have equations (22–25)]. the ranges d 1 Y
Coupled-channel expansion X ˚ .RI !/ Q ŒfE .R/ı 0 gE .R/K 0 ; E 0 D .d 1/=2 R The nonrelativistic kinetic energy operator is then separable (58.105) 2 2 where fE .R/ and gE .R/ are the regular and irregular radial „ TO D TR C ; (58.97a) functions, respectively; K 0 is the real symmetric reaction 2R matrix, and ! Q represents the set of hyperangles, while d is 2 „ 1 d 1 @ R ; (58.97b) TR D the dimension. 2 Rd 1 @R The usual technique is to solve numerically the set of coupled equations and matching to linear combinations of where is the N-body reduced mass regular and irregular functions at a large hyperradius R0 to (58.98) obtain the reaction matrix K. D .12 12;3 /1=.N 1/ ; 0 ˛1 2;
0 ˛i ;
i D 2; : : : ; d 1 : (58.96)
and, m1 m2 ; m1 C m2 .m1 C m2 /m3 D ; m1 C m2 C m3
12 D 12;3
58.5.1 (58.99a) (58.99b)
the usual reduced masses, while is the isotropic Casimir operator 2 D
X i >j
2ij ; ij D xi
@ @ xj : @xj @xi
Three-Body Recombination Rate (Identical Particles) ˛.E/ D
„k 192 2 X jS 0 j2 ; k5 0
(58.106)
p where k D 2E=„2 , and the S-matrix is determined from the reaction matrix using the standard expression Eq. (49.17). The sum includes all entrance channel three-body contin(58.100) uum states and exit channel 0 two-body bound states of A2 .
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
58.5.2
N-Body Recombination Rate (Identical Particles)
857
Detailed balance d 2 R d 2 I D ; ! 2 d˝n d˝k k 2 c 2 d˝k d˝n
58 (58.112)
„k 2 d 1 .d=2/ X jS 0 j2 ; (58.107) where the superscripts R and I refer to recombination and k 2 d=2 0 ionization, respectively. where .x/ is the Gamma function and d is the dimension. ˛N .E/ D N Š
58.6
58.7
Field-Assisted Methods
Field-assisted methods is a general term to encompass techniques used to compute recombination rates in cases where the reaction is assisted by the application of an external field. Currently, the most commonly applied field is laser-assisted recombination Eq. (58.5) in studying the spectrum in highorder harmonic generation (HHG). See Sects. 78.3, 78.4 and 80.6 in this Handbook for further details on HHG.
58.7.1
Dissociative Recombination Curve-Crossing Mechanisms
Direct Process Dissociative recombination (DR) for diatomic ions can occur via a crossing at RX between the bound and repulsive potential energy curves V C .R/ and Vd .R/ for ABC and AB , respectively. Here, DR involves the two-stage sequence kc a
R ˇ dlen 4 2 ! 3 ˇˇ sc D alen .k/ˇ2 ; 3 d˝k d˝n c k R ˇ dacn 4 2 ! ˇˇ sc D 3 aacn .k/ˇ2 ; d˝k d˝n c k
(58.108a) (58.108b)
where R is the differential cross section for Eq. (58.5), the subscripts len and acn refer to the length and acceleration form of the matrix elements, respectively; ! is the angular frequency of the released photon, k the momentum of the recombining electron, ˝n , ˝k the corresponding solid angles, and c the speed of light. He sc .k/ D a1 c1 e {.ı1 C1 / hug riusc alen k1 ; 1 sc .k/ D a1 c1 ZN e {.ı1 C1 / hug 2 iusc aacn k1 : r
(58.109a) (58.109b)
Rg (Ar–Xe) sc .k/ D a0 c0 e {.ı0 C0 / hug riusc alen k0
C a2 c2 e {.ı2 C2 / hug riusc k2 sc .k/ D a0 c0 ZN e {.ı0 C0 / hug aacn
a2 c2 ZN e {.ı2 C2 / hug
d
e C ABC .vi / • .AB /R ! A C B :
Differential cross section [43]
(58.110a) 1 sc iu r 2 k0
1 sc iu ; r 2 k2
where the constants a` and c` are defined r { ` 2` C 1 a` D ; 2k cos iY`mD0 ; c` D hY`mD0 g
(58.113)
The first stage is dielectronic capture whereby the free electron of energy D Vd .R/ V C .R/ excites an electron of the diatomic ion ABC with internal separation R and is then resonantly captured by the ion, at rate kc , to form a repulsive state d of the doubly excited molecule AB , which in turn can either autoionize at probability frequency a , or else in the second stage, predissociate into various channels at probability frequency d . This competition continues until the (electronically excited) neutral fragments accelerate past the crossing at RX . Beyond RX , the increasing energy of relative separation reduces the total electronic energy to such an extent that autoionization is essentially precluded, and the neutralization is then rendered permanent past the stabilization point RX . This interpretation [44] has remained intact and robust in the current light of ab initio quantum chemistry and quantal scattering calculations for the simple diatomics C C (OC 2 , N2 , Ne2 , etc.). Mechanism Eq. (58.113) is termed the direct process, which, in terms of the macroscopic frequencies in Eq. (58.113), proceeds at the rate ˛O D kc PS D kc Œd =.a C d / ;
(58.114)
where PS is probability for AB survival against autoionization from the initial capture at Rc to the crossing point RX . Configuration mixing theories of this direct process are (58.110b) available in the quantal [45] and semiclassical-classical path formulations [46].
Indirect Process (58.111a) In the three-stage sequence
(58.111b) e C ABC .viC / ! ABC .vf / e n ! .AB /d ;
with Y`m the spherical harmonics and ZN the atomic number.
! A C B
(58.115)
858
E. J. Mansky II and M. R. Flannery
the so-called indirect process [45] might contribute. Here, the accelerating electron loses energy by vibrational excita
tion viC ! vf of the ion and is then resonantly captured into a Rydberg orbital of the bound molecule AB in vibrational level vf , which then interacts one way (via configuration mixing) with the doubly excited repulsive molecule AB . The capture initially proceeds via a small effect – vibronic coupling (the matrix element of the nuclear kinetic energy) induced by the breakdown of the Born–Oppenheimer approximation – at certain resonance energies "n D E.vf /
E viC and, in the absence of the direct channel Eq. (58.113), would therefore be manifest by a series of characteristic very narrow Lorentz profiles in the cross section. Uncoupled from Eq. (58.113) the indirect process would augment the rate. Vibronic capture proceeds more easily when vf D viC C 1, so that Rydberg states with n 79 would be involved [for C HC 2 vi D 0 ], so that the resulting longer periods of the Rydberg electron would permit changes in nuclear motion to compete with the electronic dissociation. Recombination then proceeds as in the second stage of Eq. (58.113), i.e., by electronic coupling to the dissociative state d at the crossing point. A multichannel quantum defect theory [47] has combined the direct and indirect mechanisms.
of electrons of energy ", wavenumber k e , and spin statistical weight 2, for a molecular ion ABC .viC / of electronic statistiC in vibrational level viC is cal weight !AB ˇ ˇ2 !AB ˇ aQ ˇ 2 C k e 2! h2 !AB ˇˇ ˇˇ2 D aQ : 8 me " 2! C
DR ."/ D
(58.118)
is the electronic statistical weight of the dissociaHere, !AB tive neutral state of AB whose potential energy curve Vd crosses the corresponding potential energy curve V C of the ionic state. The transition T-matrix element for autoionization of AB embedded in the (moving) electronic continuum of ABC C e is the quantal probability amplitude
Z1 aQ .v/ D 2
Vd" .R/
C v .R/
d .R/
dR
(58.119)
0
for autoionization. Here, vC and d are the nuclear bound and continuum vibrational wave functions for ABC and AB , respectively, while Vd" .R/ D hd jHel .r; R.t//j" .r; R/ir;O"
Interrupted Recombination The process
D V"d .R/
(58.120)
are the bound-continuum electronic matrix elements coupling the diabatic electronic bound state wave functions a d .r; R/ for AB with the electronic continuum state wave nd á d n functions " .r; R/ for ABC C e . The matrix element is
C (58.116) an average over electronic coordinates r and all direcAB .v/ e n tions O of the continuum electron. Both continuum elecproceeds via the first (dielectronic capture) stage of tronic and vibrational wave functions are energy normalized Eq. (58.113) followed by a two-way electronic transition (Sect. 58.11.3), and with frequency d n and nd between the d and n states. All (58.121) .n; v/ Rydberg states can be populated, particularly those .R/ D 2jVd" .R/j2 in low n and high v since the electronic d n interaction varies as n1:5 with broad structure. Although the dissoci- is the energy width for autoionization at a given nuclear ation process proceeds here via a second-order effect (d n separation R. Given .R/ from quantum chemistry codes, and nd ), the electronic coupling may dominate the indirect the problem reduces to evaluation of continuum vibrational vibronic capture and interrupt the recombination, in contrast wave functions in the presence of autoionization. The rate to Eq. (58.115) which, as written in the one-way direction, associated with a Maxwellian distribution of electrons at feeds the recombination. Both dip and spike structure has temperature T is been observed [48]. Z kc
d
e C ABC .vi / • .AB /d ! A C B
˛O D v e
58.7.2
Quantal Cross Section
(58.122)
where v e is the mean speed (Sect. 58.12).
The cross section for direct dissociative recombination
e C ABC viC • AB r ! A C B
" DR ."/ e"=kB T d"=.kB T /2 ;
Maximum Cross Section and Rate Since the probability for recombination must remain less ˇ ˇ2 (58.117) than unity, ˇaQ ˇ 1, so that the maximum cross section and
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
rates are h2 !AB max .2` C 1/ ; (58.123) D DR ."/ D 2 k e 2! C 8 me " !AB
859
Approximate Capture Cross Section With the energy-normalized Winans–Stückelberg vibrational wave function .0/ d .R/
1=2
D jVd0 .R/j
ı.R Rc / ;
(58.129)
C
where has been replaced by 2.2` C 1/! under the as sumption that the captured electron is bound in a high-level where Rc is the classical turning point for .AB / relative motion, Eq. (58.126) reduces to Rydberg state of angular momentum `, and ( ) 2 C ! .R /j j max c v AB C ˛O max .T / D v e DR ." D kB T / (58.124a) c ."; v / D 2 ; Œ2 .Rc / k e 2! C jVd0 .Rc /j 1=2 300 .2` C 1/ cm3 =s : (58.124b) 5 107 (58.130) T where the term inside the braces in Eq. (58.130) is the effecCross section maxima of 5.2` C 1/.300=T / 1014 cm2 tive Franck–Condon factor. are therefore possible, being consistent with the rate Eq. (58.124b). Six Approximate Stabilization Probabilities (1) First-Order Quantal Approximation When the effect of autoionization on the continuum vi- A unitarized T -matrix is brational wave function d .R/ for AB is ignored, then TB a first-order undistorted approximation to the quantal ampliT D ˇ ˇ2 ; 1 C ˇ 12 TB ˇ tude Eq. (58.119) is so that PS D jT j2 =jTB j2 to give i dR ; (58.125) TB .v / D 2 PS .low "/ 0 2 1 2 D 1 C jTB j .0/ 4 where d is d in the absence of the back reaction of au8 ˇ1 toionization. Under this assumption, Eq. (58.118) reduces to ˇZ ˆ < h ˇ D 1 C 2 ˇˇ Vd" .R/ vC .R/ ˆ !AB ˇˇ C ˇˇ2 ˇ : ; (58.126) c ."; v C / D 2 TB v 0 C k e 2! Z1
C
h
Vd" .R/
C v .R/
(58.131)
.0/ d .R/
ˇ2 92 = i ˇˇ > .0/ ˇ .R/ dr d ˇ> ; ˇ; (58.132a)
which is then the cross section for initial electron capture C since autoionization has been precluded. Although the Born which is valid at low " when only one vibrational level v , T -matrix Eq. (58.125) violates unitarity, the capture cross i.e., the initial level of the ion is repopulated by autoionizasection Eq. (58.126) must remain less than the maximum tion. value (2) 2 At higher ", when population of many other ionic levels vfC !AB h !AB cmax D 2 D ; (58.127) occurs, then k 2! C 8 m " 2! C e
e
ˇ ˇ2 since ˇaQ ˇ 1. So as to acknowledge after the fact the effect of autoionization, assumed small, and neglected by Eq. (58.125), the DR cross section can be approximated as
2
32 Xˇˇ ˇˇ2 1 PS ."/ D 41 C ˇTB vfC ˇ 5 ; 4
(58.132b)
f
where the summation is over all the open vibrational levels DR "; v D c "; v PS ; (58.128) vfC of the ion. When no intermediate Rydberg AB .v/ states
are energy resonant with the initial e C ABC v C state, where PS is the probability of survival against autoioniza- i.e., coupling with the indirect mechanism is neglected, then tion on the Vd curve until stabilization takes place at some Eq. (58.128), with Eq. (58.132b), is the direct DR cross seccrossing point RX . tion normally calculated.
C
C
58
860
E. J. Mansky II and M. R. Flannery
(3) In the high-" limit, when an infinite number of vfC levels are populated following autoionization, the survival probability, with the aid of closure, is then
have spurred renewed theoretical interest because they both proceed at respective rates of .2 107 to 2 108 / cm3 s1 and 108 cm3 s1 at 300 K. Such rates are generally associated with the direct DR, which involves favorable curve crossings between the potential energy surfaces, V C .R/ 2 32 ZRX ˇ ˇ2 and Vd .R/ for the ion ABC and neutral dissociative AB ˇ 2 ˇ .0/ 2 4 5 : (58.133) states. The difficulty with Eqs. (58.137) and (58.138) is PS D 1 C jVd " .R/j ˇ d .R/ˇ dR that there are no such curve crossings, except at " 8 eV Rc for Eq. (58.137). In this instance, the previous standard (4) theories would support only extremely small rates when elecOn adopting in Eq. (58.133) the JWKB semiclassical wave tronic resonant conditions do not prevail at thermal energies. .0/ function for d , Theories [49–53] have been developed for application to processes such as Eq. (58.137). 2 32 ZRX .R/ 1 dR5 PS .high "/ D 41 C 2„ v.R/ Rc
2 D 41 C
1 2
ZtX
58.8
32
a .t/ dt 5
;
(58.134)
Mutual Neutralization
AC C B ! A C B :
(58.139)
tc
where v.R/ is the local radial speed of A B relative motion, Diabatic potentials .0/ .0/ and where the frequency a .t/ of autoionization is =„. Vi .R/ and Vf .R/ for initial (ionic) and final (covalent) states are diagonal elements of (5) Vif .R/ D hi .r; R/jHel .r; R/jf .r; R/ir ; (58.140) A classical path local approximation for PS yields 0 PS D exp@
ZtX
1 a .t/ dt A ;
where i;f are diabatic states, and Hel is the electronic (58.135) Hamiltonian at fixed internuclear distance R.
tc
which agrees to first order for small with the expansion of Eq. (58.134). (6) A partitioning of Eq. (58.113) yields PS D d =.a C d / D .1 C a d /1 ;
Adiabatic potentials for a two-state system h ˇ ˇ2 i1=2 ; (58.141a) V ˙ .R/ D V0 .R/ ˙ 2 .R/ C ˇVif .R/ˇ h i 1 .0/ .0/ (58.141b) V0 .R/ D Vi .R/ C Vf .R/ ; 2 h i .0/ .0/ .R/ D Vi .R/ Vf .R/ : (58.141c)
(58.136)
For a single crossing of diabatic potentials at RX , on adopting macroscopic averaged frequencies i and asso- Vi.0/ .RX / D Vf.0/ .RX / and the adiabatic potentials at RX are ciated lifetimes i D i1 . The six survival probabilities in .0/ Eqs. (58.132a), (58.132b), (58.132a), (58.133)–(58.136) are (58.142) V ˙ .RX / D Vi .RX / ˙ Vif .RX / ; all suitable for use in the DR cross section Eq. (58.128). with energy separation 2Vif .RX /.
58.7.3
Noncrossing Mechanism
58.8.1 Landau–Zener Probability for Single Crossing at RX
The dissociative recombination (DR) processes e C HC 3 ! H2 C H !HCHCH
On assuming .R/ D .R RX / 0 .RX /, where 0 .R/ D (58.137) d .R/=dR, the probability for single crossing is
at low electron energy ", and e C HeHC ! He C H
.n D 2/
(58.138)
Pif .RX / D expŒ.RX /=vX .b/ ; ˇ2 ˇˇ 2 Vif .RX /ˇ ; .RX / D „ 0 .RX /
(58.143a) (58.143b)
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
i1=2 h .0/ vX .b/ D 1 Vi .RX /=E b 2 =RX2 :
(58.143c)
Overall Charge-Transfer Probability From the incoming and outgoing legs of the trajectory, P X .E/ D 2Pif .1 Pif / :
861
R; E; L2 -distribution
n R; E; L2 dRdEdL2
2 8 =vR eE=kB T dR D dEdL2 : .2MAB kB T /3=2 4R2
58 (58.149)
(58.144) Case (b) .i R; E; L2 /-integrated quantities.
58.8.2
Cross Section and Rate Coefficient for Mutual Neutralization
1 ˙ 1 : vn .R; E/ vn˙ 2 2 i
(58.150)
4R2 ji˙ .R/dE D Œv" e" d"b02 ;
(58.151a)
b02
(58.151b)
Current:ji˙ .R/ D
ZbX M .E/ D 4 D bX2
Pif .1 Pif /b db
0 bX2 PM
Flux:
;
! .0/ Vi .RX / D 1 RX2 E ! 14:4 D 1C
RX2 I RX Å E.eV/
2
D R Œ1 V .R/=E :
.R; E/-Distribution: n.R; E/dRdE 2 E V .R/ 1=2 " Dp e d"dR kB T
(58.145a)
GMB .E; R/dR ;
(58.151c)
PM is the b 2 -averaged probability Eq. (58.144) for charge- which defines the Maxwell–Boltzmann velocity distribution transfer reaction within a sphere of radius RX . GMB in the presence of the field V .R/. The rate is Case (c) Z1
E; L2 -integrated quantities. 1=2 M ./ e d ; (58.146) ˛O M D .8kB T =MAB / 1 V .R/=kB T ve : 4 4R2 j ˙ .R/ D R2 v eV .R/=kB T :
0
j ˙ .R/ D
Current: where D E=kB T .
Flux: Distribution:
58.9
One-Way Microscopic Equilibrium Current, Flux, and Pair Distributions
All quantities on the RHS in the Cases (a)–(e) below are to be multiplied by NQ A NQ B Œ!AB =!A !B , where the !i denote the statistical weights of species i that are not included by the density of states associated with the E; L2 orbital degrees of freedom.
V .R/=kB T
n.R/ D e
4 2 eE=kB T dEdL2 : .2MAB kB T /3=2 (58.147)
1 vŒ1 V .R/=kB T ; (58.155) 4 V .R/ 2 v bmax v ; (58.156) 4R2 jd˙ .R/ D R2 1 kB T jd˙ .R/ D
(58.157)
n.E; L2 /dEdL2 D
4 2 R .E; L/ E=kB T e dEdL2 ; .2MAB kB T /3=2 (58.158)
H where R D dt D .@JR=@E/ is the period for bounded radialH motion of energy E and radial action JR .E; L/ D (58.148) MAB vR dR.
This flux is independent of R. For dissociated pairs E > 0, 4R2 ji˙ .R/dEdL2 D Œv" e" d"Œ2bdb :
(58.154)
Case (d)
E; L2 -distribution. For bound levels
Current:ji˙ .R/ D n˙ .R; E; L2 /vR n˙ i vR : Flux:4R2 ji˙ .R/dEdL2 D
(58.153)
When E-integration is only over dissociated states (E > 0), the above quantities are
n.R/ D Œ1 V .R/=kB T : Case (a) .i R; E; L2 /.
:
(58.152)
862
E. J. Mansky II and M. R. Flannery
Notation: MAB reduced mass MA MB =.MA C MB / R internal separation of AB ZRA E orbital energy 12 Mv 2 C V .R/ E V 1=2 2 e" dR ; (58.159) L n.E/dE D p d" orbital angular momentum kB T L2 2MEb 2 for E > 0 0 P vR radial speed jRj where RA is the turning point E D V .RA /. v mean relative speed .8kT =MAB /1=2 " normalized energy E=kB T pair distribution function nC n i Example i C ni ˙ For electron–ion bounded motion, V .R/ D Ze 2 =R, RA D ni component of ni with RP > 0 .C/ and RP < 0 ./. Ze 2 =jEj, R e D Ze 2 =kB T , " D E=kB T . Then R D 2.m=Ze 2 /1=2 .RA =2/3=2 , Case (e) E-distribution. For bound levels
58.10
ZRA
Re j"j R
1=2 dR D
2 5=2 1=2 R R ; 4 A e
(58.160)
0
Microscopic Methods for Termolecular Ion–Ion Recombination
At low gas density, the basic process AC C B C M ! AB C M
and
(58.167)
is characterized by nonequilibrium with respect to E. Dissociated and bound AC –B ion pairs are in equilibrium with (58.161) respect to their separation R, but bound pairs are not in Eequilibrium with each other; L2 -equilibrium can be assumed (58.162) for ion–ion recombination but not for ion–atom association reactions. At higher gas densities N , there is nonequilibrium in the For closely spaced levels in a hydrogenic e AZC system, ion-pair distributions with respect to R, E, and L2 . In the limit of high N , there is only nonequilibrium with respect to 2
dE dL ns .p; `/ D n E; L2 ; (58.163a) R. See [54] for full details. dp d` dE s n .p/ D n.E/ : (58.163b) 58.10.1 Time-Dependent Method: dp
2 2 e" 5=2 1=2 s R R n .E/dE D p d " 4 A e ! " 2e 2 R3e D p d" : 4j"j5=2
Low Gas Density
1 2 2 Using E D 2p 2 Z e =a0 and L2 D .` C 1=2/2 „2 for Energy levels Ei of AC –B pairs are so close that they form level .p; `/ then a quasi continuum with a nonequilibrium distribution over 2 2 Ei determined by the master equation dJR dL dE dL R .E; L/ D (58.164) Z1 dp d` dp d`
dni .t/ (58.168) ni if nf f i dEf ; D
dt (58.165) D h .2` C 1/„2 ; D
h3 2.2` C 1/ ns .p; `/ D eIp =kB T n eN C 2!AC .2 me kB T /3=2 ns .p/ h3 2p 2 D eIp =kB T ; C C n eN 2!A .2 me kB T /3=2
where ni dEi is the number density of pairs in the interval dEi about Ei , and if dEf is the frequency of i-pair collisions with M that change the i-pair orbital energy from Ei to ; (58.166a) between E and E C dE . The greatest binding energy of f f f the AC –B pair is D. (58.166b) Association Rate
in agreement with the Saha ionization formula Eq. (58.28), where N C is the equilibrium concentration of AZC ions in their ground electronic states. The spin statistical weights are !eA D ! e D 2.
Z1 A
PiS
R .t/ D
dni dEi dt
(58.169a)
D
D ˛N O A .t/NB .t/ k ns .t/ ;
(58.169b)
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
863
where PiS is the probability for collisional stabilization (re- The QSS condition (dni =dt D 0 in block E ) is then combination) of i-pairs via a sequence of energy changing Z1 ZE collisions with M. The coefficients for C ! S recombination D if dEf D if PfD dEf ; (58.177) Pi out of the C -block with ion concentrations NA .t/, NB .t/ (in 3 D D cm ) into the S block of total ion-pair concentrations ns .t/ and for S ! C dissociation are ˛O (cm3 s1 ) and k.s1 /, re- which involves only time-independent quantities. Under spectively. QSS, Eq. (58.176) reduces to the net downward current across bound level E, One-Way Equilibrium Collisional Rate and Detailed Z1 ZE Balance PiD PfD Cif dEf ; dEi (58.178) ˛O NQ A NQ B D Cif D nQ i if D nQf f i D Cf i ; (58.170) E D where the tilde denotes equilibrium (Saha) distributions.
Normalized Distribution Functions
which is independent of the energy level (E) in the range 0 E S of block E . The dissociation frequency k in Eq. (58.169b) is ZE
i .t/ D
ni .t/=nQ Si
;
s .t/ D c .t/ D NA .t/NB .t/=NQ A NQ B ;
ns .t/=nQ Bs .t/
;
(58.171)
k nQ s D
(58.172)
Z1 PiS PfS Cif dEf ; dEi
D
(58.179)
E
and macroscopic detailed balance ˛O NQ A NQ B D k nQ s is automatically satisfied; ˛O is the direct .C ! S/ collisional contribution (small) plus the (much larger) net collisional cascade downward contribution from that fraction of bound levels that originated in the continuum C ; kd is the direct dissociation frequency (small) plus the net collisional cascade upward (58.173) contribution from that fraction of bound levels that originated in block S.
where nQ Si and nQ B are the Saha and Boltzmann distributions.
Master Equation for i .t/ di .t/ D dt
Z1
i .t/ f .t/ if dEf :
D
Quasi-Steady State (QSS) Reduction Set t !1
i .t/ D PiD c .t/ C PiS s .t/ !1 ;
(58.174)
where PiD and PiS are the respective time-independent portions of the normalized distribution i , which originate, respectively, from blocks C and S. The energy separation between the C and S blocks is so large that PiS D 0 .Ei 0, C block), PiS 1 (0 > Ei S, E block), and PiS D 1 (S Ei D, S block). Since PiS C PiD D 1, then di .t/ D Œc .t/ s .t/ dt
Z1
58.10.2 Time-Independent Methods: Low Gas Density QSS rate Since recombination and dissociation (ionization) involve only that fraction of the bound state population that originated from the C and S blocks, respectively, recombination can be viewed as time independent with NA NB D NQ A NQ B ; ns .t/ D 0 ;
i D ni =nQ i
PiD PfD Cif dEf :
D
Z1 ˛O NQ A NQ B D
(58.175)
Recombination and Dissociation Coefficients Equation (58.174) in Eq. (58.169a) enables the recombination rate in Eq. (58.169b) to be written as Z1 ˛O NQ A NQ B D D
ZE
dEi E
PiD
(58.180a) (58.180b)
i f Cif dEf :
(58.180c)
D
QSS integral equation Z1
Z1 if dEf D
i D
f if dEf
(58.181)
S
Z1 is solved subject to the boundary condition PiD PfD Cif dEf : (58.176) PiD dEi D
i D 1.Ei 0/ ;
i D 0.S Ei D/ : (58.182)
58
864
E. J. Mansky II and M. R. Flannery
with the QSS analytical solution
Collisional energy-change moments
D
.m/
1 .Ei / D mŠ
Z1 .Ef Ei /m Cif dEf ;
0 (58.183)
D .m/ Di
i .Ei / D @
Z0
10 0 11 Z dE A@ dE A (58.188) D .2/ .E/ D .2/ .E/
Ei
1 d D h. E/m i : mŠ dt
S
(58.184)
of Pitaevskii [33] for ( e C AC C M) recombination where collisional energy changes are small. This distribution does Averaged energy-change frequency not satisfy the exact QSS condition Eq. (58.181). When For an equilibrium distribution nQ i of Ei -pairs per unit interinserted in the exact non-QSS rate Eq. (58.186b), highly acval dEi per second, curate ˛O for heavy-particle recombination are obtained. .1/
Di
d h Ei : dt
D
Bottleneck Method The one-way equilibrium rate (cm3 s1 ) across E, i.e., Eq. (58.180c) with i D 1 and f D 0, is
Averaged energy-change per collision .1/
.0/
h Ei D Di =Di
Z1
:
˛.E/ O NQ A NQ B D
Time-independent dissociation The time-independent picture corresponds to ns .t/ D nQ s ;
in analogy to the Eqs. (58.50a,58.50b).
E
macroscopic
reduction
ni dEi D
D
1 2
Z1
dEi D
i f if dEf
D Z1
Z1
(58.189)
D
of
Variational Principle The QSS condition Eq. (58.174) implies that the fraction PiD of bound levels i with precursor C are so distributed over i that Eq. (58.176) for ˛O is a minimum. Hence PiD or i are obtained either from the solution of Eq. (58.181) or from minimizing the variational functional Z1
Cif dEf :
This is an upper limit to Eq. (58.180c) and exhibits a mini
i D ni =nQ i PiS ; (58.185) mum at E , the bottleneck location. The least upper limit /. to ˛O is then ˛.E O
c .t/ D 0;
˛O NQ A NQ B D
ZE dEi
i f
2
Cif dEf ;
Trapping Radius Method Assume that pairs with internal separation R RT recombine with unit probability so that the one-way equilibrium rate across the dissociation limit at E D 0 for these pairs is ZRT ˛.R O T /NQ A NQ B D
Z0 dR
0
Cif .R/dEf ;
(58.190)
V .R/
where V .R/ D e 2 =R, and Cif .R/ D nQ i .R/if .R/ is the rate (58.186a) per unit interval .dRdEi /dEf for the Ei ! Ef collisional transitions at fixed R in (58.186b)
C A B Ei ;R C M ! AC B E
f
;R
C M : (58.191)
D
with respect to variational parameters contained in a trial analytic expression for i . Minimization of the quadratic functional Eq. (58.186b) has an analogy with the principle of least dissipation in the theory of electrical networks.
The concentration (cm3 ) of pairs with internal separation R and orbital energy Ei in the interval dRdEi about .R; Ei / is nQ i .R/dRdEi . Agreement with the exact treatment [54] is found by assigning RT D .0:48 0:55/.e 2 =kB T / for the recombination of equal mass ions in an equal mass gas for various ion–neutral interactions.
Diffusion-in-Energy-Space Method Integral Eq. (58.181) can be expanded in terms of energychange moments, via a Fokker–Planck analysis to yield the 58.10.3 Recombination at Higher Gas Densities differential equation As the density N of the gas M is raised, the recombination @ .2/ @ i Di D0; (58.187) rate ˛O increases initially as N to such an extent that there are @Ei @Ei increasingly more pairs n i .R; E/ in a state of contraction in
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
865
R than there are those nC Master Equations i .R; E/ in a state of expansion; i.e., 58.10.4 .R; E/ per unit interthe ion-pair distribution densities n˙ i val dEdR are not in equilibrium with respect to R in blocks Master Equation for n˙ .R/ n˙ R; Ei ; L2i [54] i C and E . Those in the highly excited block E , in addition, are not in equilibrium with respect to energy E. Basic sets 1 @ 2 ˙ ˙ n .R/jv R j R i Ei ;L2i of coupled master equations have been developed R2 @R
[54] for ˙ 2 2 the microscopic nonequilibrium distributions n R; E; L L Z1 Zf m h and n˙ .R; E/ of expanding .C/ and contracting ./ pairs D dEf dLf2 n˙ i .R/if .R/ with respect to A–B separation R, orbital energy
E, and or2 2 0 V .R/ bital angular momentum L . With n R; Ei ; Li ni .R/ and i ˙ using the notation defined at the beginning of Sect. 58.6, (58.195) nf .R/f i .R/ : the distinct regimes for the master equations discussed in Sect. 58.10.4 are: is coupled to the n The set of master equations [54] for nC i i
C set by the boundary conditions n D ni Ri at the Low N equilibrium in R, but not in E, L2 i Ri pericenter Ri for all Ei and apocenter RiC for Ei < 0 of the ! master equation for n E; L2 . 2 Ei ; L2i -orbit. Pure Coulomb equilibrium in L Attraction ! master equation for n.E/. Master Equations for Normalized Distributions [54] High N nonequilibrium in R, E, L2 ˙ ! master equation for ni .R/. L2 Highest N equilibrium in E; L2 but not in R Z1 Zf m ˙ @
! macroscopic transport equation ˙jvR j i D dEf dLf2 @R Eq. (58.68a) in n.R/. 0
V .R/
h i i˙ .R/ f˙ .R/ if .R/ :
Normalized Distributions For a state ii iE; L2 , ni .R/ n˙ .R/ ; ; i˙ .R/ D i˙ nQ i .R/ nQ i .R/ 1
i .R/ D iC C i : 2
Corresponding master equations for the L2 integrated distributions n˙ .R; E/ and ˙ .R; E/ have been derived [54].
i .R/ D
(58.192)
L2i D .2MAB Ei /b 2 ; Ei > 0 :
(58.193a) (58.193b)
2
D 2MAB R ŒEi V .R/ :
1 @ .R2 Ji / D R2 @R
0
@ iC .R/ i .R/ 1 jvR j 2 @R Z1 D V .R/
(58.198)
2
ZLf m
dEf dLf2 i .R/ f .R/ if .R/ : (58.199) 0
(58.194a)
(2) Bounded orbits of energy Ei < 0 can have L2i between 0 58.10.5 and L2ic .Ei / D 2MAB Rc2 ŒEi V .Rc / ;
dLf2
ni .R/if .R/ nf .R/f i .R/ ;
(58.193c)
(58.193d)
Z dEf
(58.197)
Lf2 m
Z1
V .R/
Maximum Orbital Angular Momenta (1) A specified separation R can be accessed by all orbits of energy Ei with L2i between 0 and L2im .Ei ; R/
Continuity Equations
C
Q˙ J i D nC i .R/ ni .R/ jvR j D i i ji ;
Orbital Energy and Angular Momentum 1 Ei D MAB v 2 C V .R/ ; 2 1 Ei D MAB vR2 C Vi .R/ ; 2 L2i Vi .R/ D V .R/ C ; 2MAB R2 Li D jR MAB vj ;
(58.196)
(58.194b)
where Rc is the radius of the circular orbit determined by @Vi =@R D 0, i.e., by Ei D V .Rc / C 12 Rc .@V =@R/Rc .
Recombination Rate
Flux Representation The R0 ! 1 limit of ˛O NQ A NQ B D 4R02 J.R0 /
(58.200)
58
866
E. J. Mansky II and M. R. Flannery
has the microscopic generalization 2
Z1
ZLi c
dEi dL2i 4R02 jQi˙ .R0 /
˛O NQ A NQ B D
0
V .R0 /
i .R0 /
iC .R0 / ;
Reaction Rate ˛RN .R0 / On solving Eq. (58.196) subject to .R0 / D 1, according to Eq. (58.68b), ˛O determined by Eq. (58.201) is the rate ˛O RN of recombination within the .A B/ sphere of radius R0 . The overall rate of recombination ˛O is then given by the full diffusional-drift reaction rate Eq. (58.71b) where the rate of (58.201) transport to R is determined uniquely by Eq. (58.72). 0
where L2ic is given by Eq. (58.194b) with Rc D R0 for bound states and is infinite for dissociated states, and where 58.11 IR0
i .R0 /
iC .R0 /
D
i .R/ ib .R/ C ic .R/ dt ;
Ri
Radiative Recombination
In the radiative recombination (RR) process e .E; `0 / C AZC .c/ ! A.Z1/C .c; n`/ C h ; (58.207)
(58.202)
the accelerating electron e with energy and angular momentum (E; `0 ) is captured, via coupling with the weak with quantum electrodynamic interaction .e=me c/A p associ2 ated with the electromagnetic field of the moving ion, into L VZ.R0 / Zf m
an excited state n` with binding energy In` about the parent
i .R/ib .R/ D dEf dLf2 i .R/ f .R/ ion AZC (initially in an electronic state c). The simultane0 V .R/ ously emitted photon carries away the excess energy h D E C In` and angular momentum difference between the ini if .R/ ; tial and final electronic states. The cross section Rn` .E/ for L2 Z1 Zf m RR is calculated (a) from the Einstein A coefficient for free–
dEf dLf2 i .R/ f .R/
i .R/ic .R/ D bound transitions or (b) from the cross section In` .h/ for 0 V .R0 / photoionization (PI) via the detailed balance (DB) relationship appropriate to Eq. (58.207). if .R/ : The rates hv e R i and averaged cross sections hR i for (58.203a) a Maxwellian distribution of electron speeds v e are then determined from either Collisional Representation L2ic
Z1 ˛O NQ A NQ B D
Z
dL2i
dEi V .R0 /
Z1
ZR0
0
i .R/ib .R/
˛O Rn` .T e / nQ i .R/ dR
0
˛ ˝ D v e Rn` .T e / ;
Ri
;
"Rn` ."/ exp."/d"
D ve
(58.208)
(58.204)
where " D E=kB T e , or from the Milne DB relation Eq. (58.281) between the forward and reverse macroscopic which is the microscopic generalization of the macroscopic rates of Eq. (58.207). Using the hydrogenic semiclassical In result ˛O D K s D ˛RN .R0 / .R0 /. of Kramers [34], together with an asymptotic expansion [55] The flux for dissociated pairs Ei > 0 is for the g-factor of Gaunt [56], the quantal/semiclassical cross section ratio in Eq. (58.287), Seaton [57] calculated 2 4R2 jvR jnQ ˙ i .R/ dE dL ˛O Rn` . D Œv" e" d"Œ2b dbNQ A NQ B ; (58.205) The rate of electron energy loss in RR is so the rate Eq. (58.204) as R0 ! 1 is Z1
Zb0 " e" d"
˛O D v 0
IR0
i .R/ib .R/ dt ; (58.206)
2b db 0
Ri
which is the microscopic generalization Eq. (58.57b) of the macroscopic result ˛O D kc P S of Eq. (58.56).
dE dt
Z1 "2 Rn` ."/ e" d" ;
D n e v e .kB T e / n`
(58.209)
0
and the radiated power produced in RR is
d.h/ dt
Z1 "hRn` ."/ e" d" :
D n eve n`
0
(58.210)
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
Standard conversions ED
p 2e =2m e
D
2
„2 k 2e =2m e
D Z 2 e 2 =2a0
D
k 2e a02
2 e =2a0
D " Z 2 e 2 =2a0 ;
E D h D „! D „k c D .In C E/
1 C n2 " Z 2 e 2 =2n2 a0 ;
h=In D 1 C n2 "; k 2e a02 D 2E= e 2 =a0 ;
k a0 D .h/˛= e 2 =a0 ;
k2 =k 2e D .h/2 = 2Eme c 2
D ˛ 2 .h/2 = 2E e 2 =a0 ;
(58.211a)
867
Cross sections are averaged over initial and summed over final degenerate states. For case (c), In
n1 1 X D 2 .2` C 1/In` I n
(58.211b) (58.211c)
(58.214a)
`D0
Rn
D
n1 X
2.2` C 1/Rn` :
(58.214b)
`D0
(58.211d) (58.211e)
58.11.2 Kramers Cross Sections, Rates, Electron Energy-Loss Rates, and Radiated Power (58.211g) for Hydrogenic Systems (58.211f)
(58.211h)
These are all calculated from application of the detailed balIH D e =2a0 ; ˛ D e =„c D 1=137:0359895 ; ance Eq. (58.212) to the original In .h/ of Kramers [34].
˛ 2 D me c 2 = e 2 =a0 ; In D Z 2 =n2 IH : (58.211i) 2
2
Semiclassical (Kramers) Cross Sections The electron and photon wavenumbers are k e and k , respec- For hydrogenic systems, tively. Z 2e2 In D 2 ; h D In C E : 2n a0
58.11.1
(58.215)
Detailed Balance and The results below are expressed in terms of the quantities Recombination-Ionization Cross Sections
Cross sections Rn` .E/ and In` .h/ for radiative recombination (RR) into and photoionization (PI) out of level n` of atom A are interrelated by the detailed balance relation g e gAC k 2e Rn` .E/ D g gA k2 In` .h/ ;
(58.212)
where g e D g D 2. Electronic statistical weights of A and AC are gA and gAC , respectively. Thus, using Eq. (58.211g) for k2 =k 2e , Rn` .E/ D
gA 2gAC
.h/2 n` .h/ : Eme c 2 I
The statistical factors are:
(a) For AC C e state c Sc ; Lc I "; `0 ; m0 :
gAC D .2Sc C 1/.2Lc C 1/ : (b) For A.n`/ state bŒSc ; Lc I n; `SL gA D .2S C 1/.2L C 1/ :
(58.213)
In ; kB T e 64a02 ˛ n p I0n D 3 3 Z2
D 7:907071 1018 n=Z 2 cm2 ;
8a02 ˛ 3 Z 2 e 2 =a0 p R0 .E/ D ; E 3 3
8a02 ˛ 3 Z 2 e 2 =a0 p : ˛O 0 .T e / D v e kB T e 3 3 bn D
(58.216)
(58.217) (58.218) (58.219)
PI and RR cross sections for level n In the Kramer (K) semiclassical approximation, n K I .h/
In h
3
I0n D K In` .h/ ; 2 In n K R .E/ D R0 .E/ n In C E D
(58.220) (58.221)
D 3:897 1020
1 2 n" 13:606 C n2 "2 cm ; where " is in units of eV and is given by
" D E=Z 2 2:585 102 =Z 2 T e =300 :
(58.222)
(c) For n` electron outside a closed shell gAC
D 1; gA D 2.2` C 1/ :
Equation (58.221) illustrates that RR into low n at low E is favored.
58
868
E. J. Mansky II and M. R. Flannery
Electron Energy Loss Rate
Cross section for RR into level n` n` K R
D .2` C 1/=n2 K Rn :
Rate for RR into level n ˛O Rn .T e / D ˛O 0 .T e /.2=n/bn ebn E1 .bn / ; which for large bn (i.e., kB T e In ) tends to
(58.223) Energy loss rate for RR into level n 1 bn ebn E1 .bn / dE D n e ˛O Rn .T e /kB T e ; (58.228a) dt n ebn E1 .bn / (58.224a) which for large bn (i.e. .kB T e / In ) tends to
n e ˛O Rn .T e /kB T e 1 bn1 C 3bn2 13bn3 C ; (58.228b)
˛O Rn .T e ! 0/ D ˛O 0 .T e /.2=n/
with Eq. (58.224a) for ˛O Rn . 1 bn1 C 2bn2 6bn3 C : (58.224b) Energy loss rate for RR into all levels n nf The Kramers cross section for photoionization at threshold is dE I0n , and dt
n n D n e kB T e ˛O 0 .T e / C ln bf C ebf E1 .bf /.1 bf / (58.225) R0 D 2R0 =nI ˛O 0 D 2˛O 0 =n (58.229a)
T b f provide the corresponding Kramers cross section and rate for D n e .kB T e / ˛O R .T e / ˛O 0 .T e /bf e E1 .bf / ; (58.229b) recombination as E ! 0 and T e ! 0, respectively. with Eqs. (58.226b) and (58.219) for ˛O RT and ˛O 0 . RR Cross sections and rates into all levels n nf Radiated Power Z1 Radiated power for RR into level n RT .E/ D Rn .E/ dn nf 1
d.h/ D n e ˛O Rn .T e /In bn ebn E1 .bn / ; (58.230a) D R0 .E/ ln.1 C If =E/ ; (58.226a) dt n
˛O RT .T e / D ˛O 0 .T e / C ln bf C ebf E1 .bf / : (58.226b) which for large b (i.e. .k T / I ) tends to n B e n
Useful integrals n e ˛O Rn .T e /In 1 C bn1 bn2 C 3bn3 C : (58.230b) Z1 x
e
ln x dx D ;
0 Z1
x 1 ex dx D E1 .b/ ; b
Zb ex E1 .x/ dx D C ln b C eb E1 .b/ ; 0
Zb Œ1 x ex E1 .x/dx
Radiated power for RR into all levels n nf (58.227a) d.h/ D n e ˛O 0 .T e /If : dt
(58.231)
(58.227b) To allow n-summation, rather than integration as in n n Eq. (58.226a), 1=2Rf , 1=2˛O Rf , 1=2hdE=dtinf , and 1=2hd.h/=dtinf , respectively, are added to each of the above expressions. The expressions valid for bare nuclei of (58.227c) charge Z are also fairly accurate for recombination to a core of charge Zc and atomic number ZA , provided that Z is identified as 1=2.ZA C Zc /. Differential cross sections for Coulomb elastic scattering
0 b
D C ln b C e .1 b/E1 .b/ ;
(58.227d) c .E; / D
b02 ; 4 sin4 12
b02 D .Ze 2 =2E/2 :
(58.232) where = 0:5772157 is Euler’s constant, and E1 .b/ is the first exponential integral such that The integral cross section for Coulomb scattering by =2 at energy E D .3=2/kB T is b eb E1 .b/ 1 b1 (58.233) c .E/ D b02 D R2e ; R e D e 2 =kB T : !1 b 1 C 2b 2 6b 3 C 24b 4 C : 9
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
Continuum-bound transition rate On summing over the two directions (g D 2) of polarization, (58.234a) the rate for transitions into all final photon states is Z
An`m E; kO e D Pif .E / dE dkO
Photon emission probability P D Rn .E/=c .E/ : This is small and increases with decreasing n as 3 In 8˛ 8 E : p P .E/ D 2 =a / h n .e 3 3 0
869
(58.234b)
D
4e 2 .h/3 jhn`m jrji .ke /ij2 : (58.241) 3„ .3„c/3
Transition frequency: alternative formula ˇ2 ˇ
An`m E; kO e D .2=„/ˇDf i ˇ ;
58.11.3 Basic Formulae for Quantal Cross Sections
(58.242)
Radiative Recombination and Photoionization where the dipole atom–radiation interaction coupling is Cross Sections 1=2 2! 3 The cross section Rn` for recombination follows from the Df i .k e / D hn`m je rji .k e /i : (58.243) continuum-bound transition probability Pif per unit time. It 3c 3 is also provided by the detailed balance relation Eq. (58.212) in terms of In` , which follows from Pf i . The number of ra- RR cross section into level .n; `; m/ Z diative transitions per second is 1 n`m R .E/ D Rn`m .k e /dkO e i h i h 4 C g e gA .E/ dE dkO e Pif .E / dE dkO Z
h3 .E/ D (58.244) An`m E; kO e dkO e : dp e dk C 8 m e E D g e gA v e R .k e / D g gA c I .k / ; 3 3 .2„/ .2/ (58.235) RR cross section into level .n; `/ .˛h/3 8 2 where the electron current (cm2 s1 ) is
.E/RIn` .E/ ; Rn` .E/ D 2 =a /E 3 2.e 0 Z 2mE v e dp e X Oe ; n` D (58.236) dE d k RI .E/ D dkO e (58.245) jhn`m jrji .ke /ij2 : .2„/3 h3 m
and the photon current (cm2 s1 ) is dk .h/2 c D c dE dkO : .2/3 .2„c/3
Transition T-matrix for RR (58.237)
Time-dependent quantum electrodynamical interaction 2h 1=2 e .O r/ ei.k r!t / A p D ie V .r; t/ D mc V i!t
V .r/ e
:
In the dipole approximation, eik r 1.
Number of photon states in volume V
E ; kO dE dkO D V .h/2 =.2„c/3 dE dkO
(58.247)
(58.248)
Continuum wave function expansion X 0
i .k e ; r/ D i` ei`0 RE`0 .r/Y`0 m0 kO e Y`0 m0 .r/ O :
Pif D Vf i
(58.246)
m
Photoionization cross section From the detailed balance in Eq. (58.235), In` is 2 C (58.238) 8 g n` I .h/ D ˛h A .E/RIn` .E/ : 3 gA
Continuum-bound state-to-state probability 2 ˇˇ ˇˇ2 Vf i ıŒE .E C In / „ D hn`m .r/V .r/ii .r; k e / :
a02 jTR j2 .E/ ; .ka0 /2 Z X ˇ ˇ ˇDf i ˇ2 dkO e : jTR j2 D 4 2
Rn` .E/ D
`0 m0
(58.239)
(58.249)
Energy normalization With .E/ D 1, Z
i .k e I r/i k0e I r dr D ı E E 0 ı kO e kO 0e : (58.240a)
D V ! 2 =.2c/3 d! dkO : (58.240b)
(58.250)
58
870
E. J. Mansky II and M. R. Flannery
Plane wave expansion eikr D 4
1 X
O k Y`m .r/ i` j` .kr/Y`m O ;
`D0
1 j` .kr/ sin kr ` .kr/ : 2 For bound states, O : n`m .r/ D Rn` .r/Y`m .r/
which is the generalization of the T -matrix Eq. (58.247) to include the effect of intermediate doubly excited autoion(58.251) izing states hj in energy resonance to within width j of the initial state i . The electrostatic interaction Pcontinuum N 2 1 initially produces dielectronic V De i D1 .r i r N C1 / (58.252) capture by coupling the initial state i with the resonant states j , which become stabilized by coupling via the dipole radia
1=2 PN C1 tion field interaction D D 2! 3 =3c 3 i D1 .er i / to the final stabilized state f . The above cross section for Eq. (58.3) (58.253) is valid for isolated, nonoverlapping resonances.
RR and PI cross sections and radial integrals Continuum Wave Normalization and Density of 2 3 States .˛h/ 8 Rn` .E/ D
.E/RI .EI n`/ : (58.254) 2 The basic formulae Eq. (58.245) for Rn` depends on the den3 2.e =a0 /E sity of states .E/, which in turn varies according to the For an electron outside a closed core, particular normalization constant N adopted for the continuum radial wave, gAC D 1; gA D 2.2` C 1/ ; . 4 2 ˛h .E/ 1 n` r; (58.258) I .h/ D (58.255a) RI .EI n`/ ; RE` .r/ N sin kr ` C ` 2 3.2` C 1/ Z1 ";`0 (58.255b) in Eq. (58.249) where the phase is Rn` D .R"`0 r Rn` /r 2 dr ; 0
ˇ ˇ ˇ ˇ ˇ ";`1 ˇ2 ˇ ";`C1 ˇ2 RI .EI n`/ D `ˇRn` ˇ C .` C 1/ˇRn` ˇ :
` D arg .` C 1 C iˇ/ ˇ ln 2kr C ı` :
(58.259)
(58.255c)
The phase corresponding to the Hartree–Fock short-range interaction For an electron outside an unfilled core (c) in the process Coulomb phase shift for electron motion
is ı` . The
C under Ze 2 =r is .` ı` / with ˇ D Z=.ka0 /. A C e ! A.n`/, the weights are For a plane wave k .r/ D N 0 exp.ik r/, C State i: ŒSc ; Lc I "; gA D .2Sc C 1/.2Lc C 1/. ˛
˝ 2 State f : Œ.Sc ; Lc I n`/S; L; gA D .2S C 1/.2L C 1/. k .r/jkE0 .r/ dk D .2/3 jN 0 j .k/ dk ı k k0 3
h 2 O dE dkO ı E E 0 ı kO kO 0 : jN 0 j .E; k/ .2L C 1/ mp RI .EI n`/ D .2Lc C 1/ (58.260) ( )2 X X
` L Lc 2L0 C 1 On integrating Eq. (58.260) over all E and kO for a single L0 `0 1 `0 D`˙1 L0 O states, N 0 and are then particle distributed over all jE; ki ˇ2 ˇ1 ˇ ˇZ interrelated by ˇ ˇ `max ˇˇ .R"`0 r Rn` /r 2 dr ˇˇ : (58.256) 2
ˇ ˇ (58.261) jN 0 j E; kO D mp= h3 : 0 This reduces to Eq. (58.255c) when the radial functions Ri;f do not depend on (Sc ; Lc ; S; L).
Cross Section for Dielectronic Recombination a02 jTDLR .E/j2 .E/ ; .ka0 /2 Z jTDLR .E/j2 D 4 2 dkO e ˇ2 ˇ Xˇˇ hf Dij hj V ii .k e / ˇˇ ˇ ; ˇ
ˇ ˇ E "j C ij =2 j n` DLR .E/ D
The incident current is O dEdkO e j dE dkO e D vjN 0 j2 .E; k/ (58.262a)
3 D 2mE= h dE dkO e D vdp e = h3 : (58.262b) Radial wave connection From Eqs. (58.249) and (58.251), N D .4N 0 =k/, so that the connection between N of Eq. (58.258) and .E/ is
(58.257)
O D jN j2 .E; k/
2m=„2 .2=/ : D k ka0 e 2
(58.263)
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
RR Cross Sections for Common Normalization Factors of Continuum Radial Functions (Four Choices) Choice a
2m=„2 .2=/ ; D N D 1I .E/ D k .ka0 /e 2 Z X ˇ ˇ 8 2 a02 ˇDf i ˇ2 dkO e ; Rn` .E/ D 3 .ka0 / m
871
2 k2 e dfn` ; 2 ke a0 dE 2 e dfn` : In` .h/ D 2 2 ˛a02 gAC a0 dE Rn` .E/ D 2 2 ˛a02 gA
(58.274a) (58.274b)
Semiclassical Hydrogenic Systems (58.264) gAC D 1 ; 2 n1 X k dFn Rn .E/ D Rn` .E/ D 2 2 ˛a02 2 ; k e dE gA D gn` D 2.2` C 1/ ;
(58.265)
(58.275)
`D0
where Df i of Eq. (58.243) is dimensionless.
dFn D dE
Choice b
n1 X
gn`
`D0
X dfn`m dfn` D2 : dE dE
(58.276)
`;m
(58.266) Bound–bound absorption oscillator strength N D k 1 I .E/ D 2m=„2 .k=/ ; s
For a transition n ! n0 , 3 2 2 e =a0 RI ˛h 16a XX 0 0 0 n`m Rn` .E/ D p 0 2 ; (58.267) 0 D 2 F fn`m (58.277a) 5 nn e =a E a0 3 2 0 `m `0 m0 " # where (58.255b) and (58.255c) for RI have dimen1 26 1 3 1 1
Eqs. D p ; (58.277b) sion L5 . n2 n02 n3 n03 3 3 Choice c 2m=„2 1=2 I .E/ D ; N Dk ! 8a02 ˛ 3 .h/3 RI Rn` .E/ D ;
2 3 a04 e 2 =a0 E
(58.268) (58.269)
where RI has dimensions of L4 . Choice d
4.a0 /2 3
(58.277c) (58.277d) (58.277e) (58.277f)
This semiclassical analysis yields exactly Kramers PI and associated RR cross sections in Sect. 58.8.2.
1=4
I N D 2m=„2 2 E Rn` .E/ D
dfn` I2 dFn 25 D p n n 3 D 2n2 ; dE dE 3 3 .h/ nIn2 25 ˛ 3 a02 ; Rn .E/ D p 3 3 E.h/ 26 ˛ n In 3 2 n` a0 ; I .h/ D p 3 3 Z 2 h n I 3 n Mb : D 7:907071 2 Z h
.E/ D 1 ; ! ˛ 3 .h/3 RI ;
2 e 2 a0 e 2 =a0 E
where RI has dimensions of L2 E
1
.
(58.270) (58.271) 58.11.5
Radiative Recombination Rate Z1 ˛O Rn` .T e /
" Rn` ."/ e" d"
D ve
(58.278a)
0
˛ v e Rn` .T e / ; ˝
58.11.4
Bound–Free Oscillator Strengths
For a transition n` ! E to E C dE, X Xˇˇ 0 0 ˇˇ2 dfn` 2 .h/ 1 D ˇr "` m ˇ ; dE 3 .e 2 =a0 / .2` C 1/ m 0 0 n`m `m Z X RI ."I n`/ D dkO e jhn`m jrji .ke /iEj2 m
X ˇˇ 0 0 ˇˇ2 m D ˇr "` n`m ˇ ; m;`0 ;m0
(58.278b)
where " is given in Eq. (58.57c), and hRn` .T e /i is the Maxwellian-averaged cross section for radiative recombination. In terms of the continuum-bound An` .E/, (58.272) Z1 dAn` " h3 n` e d" ; (58.279) ˛O R .T e / D .2 me kB T /3=2 d" 0 XZ dAn` (58.273) (58.280) D .E/ An`m .E; kO e / dkO e : dE m
58
872
E. J. Mansky II and M. R. Flannery
Milne Detailed Balance Relation In terms of In` .h/,
(a) Radiative Recombination Cross Section
˛O Rn` .T e / ˛ gA kB T e In 2 ˝ n` D ve .T / ; e I C mc 2 kB T e 2gA
Rn` .E/ (58.281)
where, in reduced units ! D h=In , T D kB T e =In D bn1 , the averaged PI cross section corresponding to Eq. (58.213) is ˝
˛ e1=T In` .T / D T
Z1 ! 2 In` .!/ e!=T d! : 1
When In` .!/ is expressed in Mb (1018 cm2 ), ˛O Rn` .T e /
D 1:508 10
13
300 Te
1=2
˛ ˝ In` .T / cm3 s1 :
In IH
2
D
gA gAC
˛ 2 .h/2 Gn` .!/K In .h/ 2E.e 2 =a0 / (58.288a)
D Gn` .!/K Rn` .E/ .2` C 1/ G .!/ Rn .E/ ; D n` n2 K Rn .E/ D Gn .!/K Rn .E/ ;
(58.288b) (58.288c) (58.288d)
(58.282) where the quantum mechanical correction, or Gaunt factor, to the semiclassical cross sections ( 1; !!1 (58.289) Gn` .!/ ! .`C1=2/ ; !!1 !
gA 2gAC
(58.283)
favors low n`-states. The `-averaged Gaunt factor is n1
X .2` C 1/Gn` .!/ : Gn .!/ D 1=n2
(58.290)
`D0
When I can be expressed in terms of the threshold cross Approximations for Gn : as " increases from zero, section 0n Eq. (58.217) as 3=4 4 28 In` .h/ D .In = h/p 0 .n/I .p D 0; 1; 2; 3/ ; (58.284) Gn ."/ D 1 C .an C bn / C an2 3 18 ˛ ˝ then In` .T / D Sp .T /0 .n/, where 7 7 ' 1 .an C bn / C an bn C bn2 3 6 S0 .T / D 1 C 2T C 2T 2 ; S1 .T / D 1 C T ; (58.285a)
2 2 2 S2 .T / D 1 ; (58.285b) where E D " Z e =2a0 , ! D 1 C n ", and
(58.285c) S3 .T / D e1=T =T E1 .1=T / an ."/ D 0:172825 1 n2 " cn ."/ ; T 1 4 2 1 T C 2T 2 6T 3 : (58.285d) 4 2 bn ."/ D 0:04959 1 C n " C n " cn2 ."/ ; 3
The case p D 3 corresponds to Kramers PI cross section 2=3 2 2=3 : 1Cn " cn ."/ D n Eq. (58.220) so that O Rn` .T e / K˛
.2` C 1/ 2 ˛O 0 .T e /S3 .T / n2 n K ˛O Rn` .T e ! 0/S3 .T / ;
D
(58.291a) (58.291b)
(58.292a) (58.292b) (58.292c)
Radiative Recombination Rate (58.286a) (58.286b)
1=2 such that K ˛O Rn` Z 2 = n3 T e as T D .kB T e =In / ! 0.
˛O Rn` .T e / D K ˛O Rn` .T e ! 0/Fn` .T / ; .2` C 1/ 2 n` ˛O R .T e ! 0/ D ˛O 0 .T e / ; n2 n
(58.293) (58.294)
in accordance with Eq. (58.224b).
58.11.6
Gaunt Factor, Cross Sections, and Rates for Hydrogenic Systems
e1=T Fn` .T / D T
Z1 1
Gn` .!/ !=T d! : e !
(58.295)
The Gaunt factor Gn` is the ratio of the quantal to Kramers (K) semiclassical PI cross section such that The multiplicative factors F and G convert the semiclassical val(Kramers) T e ! 0 rate and cross section
to their quantal 1=2 In` .h/ D K In .h/Gn` .!/ I for RR ues. Departures from the scaling rule Z 2 =n3 T e ! D h=In D 1 C E=In : (58.287) rates is measured by Fn` .T /.
58 Electron–Ion, Ion–Ion, and Neutral–Neutral Recombination Processes
58.11.7 Exact Universal Rate Scaling Law and Results for Hydrogenic Systems
n`
˛O Rn` .Z; T e / D Z ˛O R 1; T e =Z
2
(58.296)
873
58.12 Useful Quantities (a) Mean speed
8kB T 1=2 T 1=2 as exhibited by Eq. (58.281) with Eqs. (58.277e) and ve D D 1:076042 107 cm=s me 300 (58.282). 1=2 Recombination rates are greatest into low n levels and the D 6:69238 107 TeV cm=s ; `1=2 variation of Gn` preferentially populates states with ! 1=2 T 1=2
low ` 25. Highly accurate analytical fits for Gn` .!/ have v i D 2:51116 105 cm=s ; mp =mi 300 been obtained for n 20 so that Eq. (58.287) can be expressed in terms of known functions of fit parameters [58]. where .mp =m e /1=2 D 42:850352, and T D 11604:45 TeV reThis procedure (which does not violate the S2 sum rule) lates the temperature in K and in eV. has been extended to nonhydrogenic systems of neon-like Fe XVII, where In` .!/ is a monotonically decreasing function (b) Natural radius of !. The variation of the `-averaged values jV .R e /j D e 2 =R e D kB T : n1 X 300 14:4 e2 2 R D D 557 Å D Å: .2` C 1/Fn` .T / e n kB T T TeV `D0
is close in both shape and magnitude to the corresponding (c) Boltzmann average momentum semiclassical function S3 .T /, given by Eq. (58.295) with Z1 Gn` .!/ D 1. Hence the `-averaged recombination rate is 2 ep =2mkB T dp D .2 me kB T /1=2 : hpi D
˛O Rn .Z; T / D .300=T /1=2 Z 2 =n Fn .T / ; 1 1:1932 1012 cm3 s1 ;
(d) De Broglie wavelength
where Fn can be calculated directly from Eq. (58.295) or be approximated as Gn .1/S.T /. A computer program based on a three-term expansion of Gn is also available [59]. From a three-term expansion for G, the rate of radiative recombination into all levels of a hydrogenic system is ˛.Z; O T / D 5:2 1014 Z1=2 1 0:43 C ln C 0:47=1=2 ; 2
(58.297)
h h D .2 m e kB T /1=2 hpi 7:453818 106 D cm 1=2 Te 300 1=2 6:9194 D 43:035 ÅD Å: 1=2 Te TeV
dB D
O D cm3 =s. Tables [60] References where D 1:58 105 Z 2 =T and Œ˛ exist for the effective rate 0
˛O En` .T /
D
1 n 1 X X
0 0 ˛O Rn ` Cn0 `0 ;n`
(58.298)
n0 Dn `0 D0
of populating a given level n` of H via radiative recombination into all levels n0 n with subsequent radiative cascade (i ! f ) with probability Ci;f via all possible intermediate paths. Tables [60] also exist for the full rate ˛O FN .T / D
1 X n1 X
˛O Rn`
(58.299)
nDN `D0
of recombination, into all levels above N D 1; 2; 3; 4 of hydrogen. They are useful in deducing time scales of radiative recombination and rates for complex ions.
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Edmund J. Mansky II Edmund J. Mansky II received his PhD from Georgia Institute of Technology in 1985. His theoretical work involves statistical mechanics of dense gases. In addition, his interests are three-body recombination processes and collisions involving Rydberg states of atoms and molecules at thermal and ultracold temperatures. His day job is programming heliophysics data for the scientific community.
59
Dielectronic Recombination Michael Pindzola, Nigel Badnell, and Donald Griffin
Contents
59.1 Introduction
59.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 875
59.2
Theoretical Formulation . . . . . . . . . . . . . . . . . . 876
59.3 59.3.1 59.3.2
Comparisons with Experiment . . . . . . . . . . . . . . 877 Low-Z Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 High-Z Ions and Relativistic Effects . . . . . . . . . . . 878
59.4
Radiative-Dielectronic Recombination Interference
59.5
Dielectronic Recombination in Plasmas . . . . . . . . 879
878
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879
Abstract
Dielectronic recombination (DR) is a two-step process that greatly increases the efficiency for electrons and ions to recombine in a plasma. The process, therefore, plays an important role in the theoretical modeling of plasmas, whether in the laboratory or in astrophysical sources such as the solar corona. The purpose of this chapter is to present the theoretical formulation for DR and the principal methods for calculating rate coefficients. The results are compared with experiment over a broad range of lowZ ions and high-Z ions where relativistic effects become important. Keywords
dielectronic recombination plasmas Rydberg series distorted-wave approximation autoionization rate M. Pindzola () College of Sciences and Mathematics, Auburn University Auburn, AL, USA e-mail: [email protected] N. Badnell () Dept. of Physics, University of Strathclyde Glasgow Glasgow, UK e-mail: [email protected] D. Griffin Dept. of Physics, Rollins College Winter Park, USA
Electron–ion recombination into a particular final recombined state may be schematically represented as qC
e C Ai and qC
e C Ai
.q1/C
! Af
C „! ;
i h .q1/C .q1/C ! Af ! Aj C „! ;
(59.1)
(59.2)
where q is the charge on the atomic ion A, ! is the frequency of the emitted light, and the brackets in Eq. (59.2) indicate a doubly excited resonance state. The first process is called radiative recombination (RR), while the second is called dielectronic recombination (DR). Both recombination mechanisms are the inverse of photoionization. At sufficiently high electron density, three-body recombination becomes possible. The three-body mechanism is the inverse of electron impact ionization. The review article by Seaton and Storey [1] includes an interesting history of the theoretical work on dielectronic recombination. The process was first referred to as dielectronic recombination by Massey and Bates [2], after a suggestion of its possible importance in the ionosphere by Sayers in 1939. However, estimates of the rate coefficient for this process indicated that DR is not an important process in the ionosphere, where the temperatures are too low to excite anything but the lower energy resonance states. In 1961, Unsold, in a letter to Seaton, suggested that DR might account for a well-known temperature discrepancy in the solar corona. Seaton initially concluded that DR would not significantly increase recombination in the solar corona. However, he had only included the lower energy resonance states in his analysis; Burgess [3] showed that when one includes the higher members of the Rydberg series of resonance states that are populated at coronal temperatures, DR can indeed explain this discrepancy. Dielectronic recombination has since received much theoretical attention due, in part, to its importance in modeling
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_59
875
876
high-temperature plasmas. Various approaches to the theory are discussed in a review by Hahn [4] and in [5]. Recently, there have been various projects aimed at the generation of large quantities of DR data for use in astrophysical and fusion plasma modeling. One such project is based on the results of the AUTOSTRUCTURE code, with both the total and partial (i.e., resolved by recombined level) DR rate coefficients being archived. The methodology is outlined in Badnell et al. [6]. Data are calculated for all members of an isoelectronic sequence from H to Ar, along with various ions relevant to astrophysics or fusion, namely Ca, Ti, Cr, Fe, Ni, Zn, Kr, Mo, and Xe. Work has been completed for the hydrogen [7], helium [8], lithium [9], beryllium [10], boron [11], carbon [12], nitrogen [13], oxygen [14], fluorine [15], neon [16], sodium [17], magnesium [18], aluminum [19], and argon [20] isonuclear sequences. There has also been a large quantity of data generated using a fully relativistic Dirac–Fock code [21]. These data include calculations of Na-like ions [22] and H-like through to Ne-like [23] ions of certain astrophysically important elements. Interest in dielectronic recombination has increased dramatically in the last 20 years. Mitchell et al. [24] published the DR cross section for CC using a merged electron–ion beams apparatus, and Belic et al. [25] reported on a crossed beams measurement of the DR cross section for MgC . Also, Dittner et al. [26] published merged beams measurements of the DR cross section for the multiply charged ions B2C and C3C . Since that time, atomic physics experiments carried out using heavy-ion traps, accelerators, and storage-cooler rings have produced high-resolution mappings of the resonance structures associated with electron–ion recombination. The experiments have been carried out using a wide range of facilities and technologies, such as the test storage ring (TSR) at Heidelberg, the experimental storage ring (ESR) at Darmstadt, the accelerator-cooler ring facility at Aarhus, the electron beam ion trap (EBIT) at Livermore, and the electron beam ion source (EBIS) at Kansas State. A good review of the dramatic experimental progress in DR measurements is again found in the NATO proceedings [5].
M. Pindzola et al.
The set .˛1 J1 / represents the quantum numbers for the N electron target ion state, .˛0 J0 M0 / represents the quantum numbers for the (N C1)-electron recombined ion state, .k`j / represents the quantum numbers for the continuum electron state, .JM / represents the quantum numbers for the (N C1)electron system of target plus free electron state, and g1 is the statistical weight of a J1 level. The dipole radiation field operator is given by r DD
N C1 2! 3 X rs : 3 c 3 sD1
(59.4)
Continuum normalization is chosen as one times a sine function, and atomic units (e D „ D m D 1) are used. In the isolated-resonance approximation, the dielectronic recombination cross section for Eq. (59.2), in lowest-order perturbation theory, is given by 8 2 X X X 1 k3 2g1 `j JM M0 ˇ ˇ X ˇ h˛0 J0 M0 jDj˛i Ji Mi ih˛i Ji Mi jV j˛1 J1 `jJM i ˇ2 ˇ; ˇ ˇ ˇ E E C i =2 0 i i ˛ JM
DR D
i i
i
(59.5) where the set .˛i Ji Mi / represents the quantum numbers for a resonance state with energy Ei and total width i , and the electrostatic interaction between electrons is given by V D
N X
jr s r N C1 j1 :
(59.6)
sD1
By the principle of detailed balance, RR of Eq. (59.3) is proportional to the photoionization cross section from the bound state, while the energy-averaged DR may be written as hDR i D
2 2 X 1 Aa Ar ; k 2 ˛ J M 2g1 i i i
(59.7)
i
where the autoionization decay rate Aa is given by
59.2 Theoretical Formulation
Aa D
4 XX jh˛1 J1 `jJM jV j˛i Ji Mi ij2 ; k JM `j
(59.8)
In the independent-processes approximation, the two paths for recombination are summed incoherently. The radiative the radiative decay rate Ar is given by recombination cross section for Eq. (59.1), in lowest-order X of perturbation theory, is given by (59.9) Ar D 2 jh˛0 J0 M0 jDj˛i Ji Mi ij2 ; 2 M0 8 RR D 3 k and is the energy bin width. Each resonance level in X X X 1 ˇˇ˝ ˇ ˇ ˛ˇˇ2 Eq. (59.7) makes a contribution at a fixed continuum en ˇ ˛0 J0 M0 ˇD ˇ˛1 J1 `jJM ˇ : 2g1 ergy k 2 =2; thus hDR i plotted as a function of energy is `j JM M0 (59.3) a histogram.
59 Dielectronic Recombination
59.3
877
Comparisons with Experiment
59.3.1 Low-Z Ions For the most part, the agreement between the recent highresolution measurements and theoretical calculations based on the independent-processes and isolated-resonance (IPIR) approximations is quite good [27–45]. We illustrate the agreement with experiment obtained by calculations employing the IPIR approximations with three examples from the Li isoelectronic sequence. The pathways for dielectronic capture, in terms of specific levels, are given by
e C AqC .2s1=2 / ! A.q1/C .2p1=2 nlj /
(59.10) & A.q1/C .2p3=2 nlj / ; where a 1s2 core is assumed to be present. Both Rydberg series autoionize by the reverse of the paths in Eq. (59.10), while for sufficiently high n, the 2p3=2 nlj levels may autoionize to the 2p1=2 continuum. Both series radiatively stabilize by either a 2p ! 2s core orbital transition or by a nlj ! n0 l 0 j 0 valence orbital transition, where 2pj n0 l 0 j 0 with j D .1=2/; .3=2/ is a bound level. Dielectronic recombination cross section calculations [35] in the IPIR approximation for O5C are compared with experiment in Fig. 59.1. Fine-structure splitting of the two series of Eq. (59.10) is minimal for this light ion, so there appears only one Rydberg series. The 2p 6` resonances are located at 2:5 eV, the 2p 7` at 5:0 eV, and so on, accumulating at the series limit around 11:3 eV. Electric field effects on the highn resonances are strong in O5C , so that calculations were done for fields of 0, 3, 5, and 7 V=cm. Since the precise electric
field strength in the experiment is not known, the accuracy of an electric field-dependent theory, in this case, has yet to be determined. However, the effects of state mixing by extrinsic fields in the collision region for the dielectronic recombination of MgC have been investigated both experimentally [46] and theoretically [47]. There have been a significant number of recent experiments on low-Z ions. These experiments, in general, show good agreement with theory, see Fogle et al. [44] and Schnell et al. [45]. However, it is also clear that discrepancies remain between theory and experiment for certain low-energy resonances, due to the difficulty in calculating the energy positions of such resonances. As has been pointed out in Savin et al. [48] and Schippers et al. [49], this can lead to significant uncertainties in low-temperature DR rate coefficients. Calculating such low-energy resonances to sufficient accuracy for low-temperature DR rate coefficients remains a significant challenge for theory. Some success in this area has been achieved using relativistic many-body perturbation theory, obtaining very good agreement with low-energy resonance positions for a range of systems, see, for example, Lindroth et al. [50], Fogle et al. [51], and Tokman et al. [52].
¢σξ² (×10 –10 cm 3/s) 150 11
120
19 2p1/2 ∞
14 90
∞
2p 3/2
Experiment
60 30
¢σξ² (×10 –10 cm 3/s) 12.0
0 10.0
0
¢σξ² (×10 150
20 –10
40
60
3
cm /s) 10
8.0
80 Energy (eV) Theory
120 6.0 90
4.0
60
2.0 0.0 0.0
30 2.5
5.0
7.5
10.0 12.5 Energy (eV)
Fig. 59.1 Dielectronic recombination for O5C . Calculations were performed for fields of 0 V=cm (dotted curve), 3 V=cm (dashed curve), 5 V=cm (chain curve), and 7 V=cm (solid curve)
0
0
20
40
60
Fig. 59.2 Dielectronic recombination for Cu26C
80 Energy (eV)
59
878
M. Pindzola et al.
a Recombination rate (10–9 cm3/s) 2p1/2 nlj
5.6 n = 20
5.2
n = 21 2p3/2 6lj
j = 3/2
4.8
3/2
5/2
7/2
4.4 4.0 3.6 0
10
20 –9
30
40 Energy (eV)
30
40 Energy (eV)
3
b Recombination rate (10 cm /s) 4.8 4.4 4.0 3.6 0
10
20 –9
3
c Recombination rate (10 cm /s) 4.8
energy range of the experiment. Figure 59.3 shows that the perturbative relativistic, semirelativistic, and fully relativistic calculations for the dielectronic recombination cross section ride on top of a strong radiative recombination background. In principle, the fully relativistic theory contains the most physics, and thus it is comforting that on the whole it is in good agreement with the experiment. It is instructive, however, to see how well the computationally simpler perturbative relativistic and semirelativistic theories do for such a highly charged ion. There have been several recent experimental measurements on high-Z ions, in particular, for astrophysically abundant species. In general, there is good agreement between theory and experiment. Examples of high-Z element DR studies include those done on Fe21C and Fe22C by Savin et al. [48], and on Fe20C by Savin et al. [53]. There have also been recent comparisons between experiment and theory for heavy metal ions, for example W18C [54], W19C [55], W20C [56–59], and Au20C [60, 61]. Many calculations of electron–ion recombination rate coefficients and photoionization cross sections for astrophysically important species, and especially in connection with the iron project, have been carried out by Nahar et al. For recent work, see [62, 63].
4.4 4.0
59.4 Radiative-Dielectronic Recombination Interference
3.6 0
10
20
30
40 Energy (eV)
Fig. 59.3 Dielectronic recombination for Au76C . The curves show (a) perturbative relativistic, (b) semirelativistic, and (c) fully relativistic calculations for the dielectronic recombination cross section
59.3.2 High-Z Ions and Relativistic Effects Dielectronic recombination cross section calculations [38] in the IPIR approximation for Cu26C are compared with experiment in Fig. 59.2. The two fine-structure Rydberg series are now clearly resolved; the fine-structure splitting is about 27 eV for Cu26C . The 2p1=2 13` resonances are just above threshold, while the 2p3=2 11` resonances are found around 5:0 eV. Electric fields in the range 050 V=cm have little effect on the Cu26C spectrum. Overall, the agreement between theory and experiment is excellent. Electron–ion recombination cross section calculations [39] in the IPIR approximation for low-lying resonances in Au76C are compared with experiment in Fig. 59.3. The 2p1=2 nlj series limit is at 217 eV, while the 2p3=2 nlj series limit is at 2:24 keV, yielding a fine structure splitting of 2:03 keV. QED effects alone shift the 2p3=2 nlj series limit by 22:0 eV. Thus, accurate atomic structure calculations must be made to locate the 2p3=2 6lj resonances in the 050 eV
There has been a great deal of effort in recent years to develop a more general theory of electron–ion recombination that would go beyond the IPIR approximation to include radiative–dielectronic recombination interference and overlapping (and interacting) resonance structures [64–71]. In almost all cases, the interference between a dielectronic recombination resonance and the radiative recombination background is quite small and difficult to observe. The best possibility for observation of RR-DR interference appears to be in highly charged atomic ions. In the cases studied to date, the combination of electron and photon continuum coupling selection rules and the requirement of near energy degeneracy make the overlapping (and interacting) resonance effects small and difficult to observe. Heavy ions in relatively low stages of ionization are the best place to look, since there are resonance series attached to the large numbers of LS terms or fine structure levels. The distorted-wave approximation (Chap. 56) has been so successful in describing dielectronic recombination cross sections for most atomic ions because, for low charged ions, the DR cross section is proportional to the radiative rate, while for highly charged ions the DR cross section is proportional to the autoionization rate. Thus, the weakness of the distorted-wave method in calculating accurate autoionization rates for low charged ions is masked by a DR cross section
59 Dielectronic Recombination
that is highly dependent on radiative atomic structure. As one moves to more highly charged ions, the DR cross section becomes more sensitive to the autoionization rates, but at the same time, the distorted-wave method becomes increasingly more accurate.
59.5
Dielectronic Recombination in Plasmas
Dielectronic recombination is an important atomic process that is included in the theoretical modeling of the ionization state and emission of radiating ions, which is fundamental to the interpretation of spectral emission from both fusion and astrophysical plasmas (Chap. 86). The dielectronic recombination rate coefficient, into a particular final recombined state, is given by r ˛DR D
2 k 2 2 k hDR .i ! f /i exp ; (59.11) T 3 2T
where T is the electron temperature. Dielectronic recombination rate coefficients, from the ground and metastable states of a target ion into fully resolved low-lying states and bundled high-lying states of a recombined ion, are required for a generalized collisional radiative treatment [72, 73] of highly populated metastable states, the influence of finite plasma density on excited state populations, and of ionization in dynamic plasmas. We also note that contributions to recombination in plasmas may come from high Rydberg states just below the ionization threshold [74, 75].
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60
Rydberg Collision Theories Edmund J. Mansky II
Contents 60.1
Rydberg Collision Processes . . . . . . . . . . . . . . . 882
60.2 60.2.1 60.2.2 60.2.3 60.2.4 60.2.5 60.2.6 60.2.7
General Properties of Rydberg States . . . Dipole Moments . . . . . . . . . . . . . . . . . Interaction Potentials . . . . . . . . . . . . . . . Radial Integrals . . . . . . . . . . . . . . . . . . Line Strengths . . . . . . . . . . . . . . . . . . . Form Factors . . . . . . . . . . . . . . . . . . . . Impact Broadening . . . . . . . . . . . . . . . . Effective Lifetimes and Depopulation Rates .
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60.3 60.3.1 60.3.2 60.3.3 60.3.4 60.3.5
Correspondence Principles . . . . . . . . . Bohr–Sommerfeld Quantization . . . . . . . Bohr Correspondence Principle . . . . . . . Heisenberg Correspondence Principle . . . Strong Coupling Correspondence Principle Equivalent Oscillator Theorem . . . . . . . .
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60.4 60.4.1 60.4.2
Distribution Functions . . . . . . . . . . . . . . . . . . . 888 Spatial Distributions . . . . . . . . . . . . . . . . . . . . . 888 Momentum Distributions . . . . . . . . . . . . . . . . . . 889
60.5
Classical Theory . . . . . . . . . . . . . . . . . . . . . . . 889
60.6 60.6.1 60.6.2
Universality Properties . . . . . . . . . . . . . . . . . . . 890 Equal Mass Case . . . . . . . . . . . . . . . . . . . . . . . 891 Unequal Mass Case . . . . . . . . . . . . . . . . . . . . . . 891
60.7 60.7.1 60.7.2
Many-Body and Multiparticle Effects . . . . . . . . . 891 Many-Body Theory . . . . . . . . . . . . . . . . . . . . . . 891 Dilute Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . 891
60.8 60.8.1 60.8.2 60.8.3 60.8.4 60.8.5 60.8.6
Working Formulae for Rydberg Collisions Inelastic n,`-Changing Transitions . . . . . . Inelastic n ! n0 Transitions . . . . . . . . . . Quasi-Elastic `-Mixing Transitions . . . . . . Elastic n` ! n`0 Transitions . . . . . . . . . . Inelastic n` ! n`0 Transitions . . . . . . . . . Fine Structure n`J ! n`J 0 Transitions . . .
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892 892 893 894 894 894 895
60.9 60.9.1 60.9.2 60.9.3
Impulse Approximation . . . . . . . Quantal Impulse Approximation . . . Classical Impulse Approximation . . Semiquantal Impulse Approximation
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896 896 900 901
E. J. Mansky II () Eikonal Research Institute Bend, OR, USA e-mail: [email protected]
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60.10 60.10.1 60.10.2 60.10.3 60.10.4
Binary Encounter Approximation . . Differential Cross Sections . . . . . . . . Integral Cross Sections . . . . . . . . . . . Classical Ionization Cross Section . . . . Classical Charge Transfer Cross Section
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903 903 904 906 906
60.11 60.11.1 60.11.2 60.11.3 60.11.4 60.11.5
Born Approximation . . Form Factors . . . . . . . . Hydrogenic Form Factors Excitation Cross Sections Ionization Cross Sections Capture Cross Sections .
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Abstract
Collisions involving atoms or molecules in Rydberg orbitals are important in understanding a wide range of phenomena from the spectra of astrophysical objects, such as planetary nebulae and interstellar gas clouds, and industrial plasmas on Earth, to Bose-Einstein condensates and multi-qubit logic gates in quantum computing. This Chapter collects together many of the equations used to study theoretically the collisional properties of both charged and neutral particles with atoms and molecules in Rydberg states or orbitals, from thermal energies to ultracold temperatures, including the impulse approximation, binary encounter approximation and the Born approximation. Also covered are many new asymptotic methods and working formulae suitable for numerical computation, as well as models using scattering lengths and effective ranges. Readers interested in the basic quantum mechanical properties of Rydberg states may consult Chap. 15. The theoretical techniques used to study Rydberg collisions complement and supplement the eigenfunction expansion approximations used for collisions with target atoms and molecules in their ground (n D 1) or first few excited states (n > 1), as discussed in Chap. 49. Direct application of eigenfunction expansion techniques to Rydberg collisions, wherein the target particle can be in a Rydberg orbital with principal quantum number in the range n 100, is prohibitively difficult due to the need
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_60
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to compute numerically and store wave functions with n3 , or more, nodes. For n D 100 this amounts to 106 nodes for each of the wave functions represented in the eigenfunction expansion. Therefore, a variety of approximate scattering theories have been developed to deal specifically with the peculiarities of Rydberg collisions. Experiments in the ultracold regime, via the use of lasers and magnetic traps, has allowed access to aspects of many-body physics using collisions of Rydberg atoms in a gas that were previously inaccessible in condensed matter and nuclear physics. For the second edition of the Handbook we have changed the title of the chapter to reflect the broader array of theories now being used, in particular, in ultra-cold collisions of Rydberg atoms and the study of their universal properties.
(B) Ionizing collisions Direct and associative ionization ( AC C B.ˇ 0 / C e A ./ C B.ˇ/ ! (60.6) BAC C e : Penning ionization A ./ C B ! A C BC C e :
(60.7)
Ion-pair formation A ./ C B ! AC C B :
(60.8)
Dissociative attachment A ./ C BC ! AC B C C :
(60.9)
60.2 General Properties of Rydberg States
Keywords
Figure 60.1 illustrates the range of physical phenomena binary encounter Born approximation impulse approxi- where collisions of Rydberg atoms play a role from thermal mations line strengths correspondence principles form molecular beams down to the ultracold regime of the Bose– Einstein condensates (BEC) and degenerate Fermi gases. factors universality properties The types of theories required to understand Rydberg atom collisions range from Born, binary-encounter and impulse approximations at thermal energies to hyperspherical, zerorange, and many-body theories in the cold and ultracold regimes. A good general introduction to Rydberg atom colli60.1 Rydberg Collision Processes sion theories can be found in [1] and [2]. Table 60.1 displays the general n-dependence of a number (A) State-changing collisions Quasi-elastic `-mixing colof key properties of Rydberg states and some specific reprelisions sentative values for hydrogen. A .n`/ C B ! A .n`0 / C B :
(60.1)
Quasi-elastic J -mixing collisions: fine structure transitions with J D j` ˙ 1=2j ! J 0 D j` ˙ 1=2j are A .n`J / C B ! A .n`J 0 / C B :
(60.2)
Energy transfer n-changing collisions A .n/ C B.ˇ/ ! A .n0 / C B.ˇ 0 / ;
60.2.1 Dipole Moments Definition D i !f D eX i !f , where X X i !f D hf j eikrj rj ji i :
(60.10)
j
(60.3)
Hydrogenic dipole moments See Bethe and Salpeter [5] and the references by Khandelwal and coworkers [6–9] for details and tables.
where, if B is a molecule, the transition ˇ ! ˇ 0 represents an inelastic energy transfer to the rotational-vibrational degrees Exact expressions In the limit jkj ! 0, the dipole allowed of freedom of the molecule B from the Rydberg atom A . transitions summed over final states are Elastic scattering 8 2n5 ˇ ˇ ˇX1s!n ˇ2 D 2 n7 .n 1/ ; (60.11a) A ./ C B ! A ./ C B ; (60.4) 3 .n C 1/2nC5
1 1 2n7 5 ˇ ˇ 1 1 1 where the label denotes the set of quantum numbers n; ` or ˇX2s!n ˇ2 D 2 2 n 1 2 ; 3n3 1 C 1 2nC7 4 n2 n n; `; J used. 2 n Depolarization collisions (60.11b)
1 1 2n7 5 ˇ ˇ 12 A .n`m/ C B ! A .n`m0 / C B ; (60.5a) ˇX2p!n ˇ2 D 2 1 2 n 11 2 : (60.11c) 2nC7 3 0 144 n 1 C 1 n A .n`J M/ C B ! A .n`J M / C B : (60.5b) 2 n
60
Rydberg Collision Theories
Fig. 60.1 The world of ultracold atomic, molecular, and cluster systems [3]
883
T (K) 103
Effusive beams
102 1
10
1
Molecules and clusters in seeded supersonic jets
T = 2.7 K Black body radiation of universe
Matrix isolation
Molecule cooling in He supersonics
10 –1
Condensed phase & biomolecules
(4He) n (3He) n clusters neat and doped Superfluidity in 4He liquid & cluster
Fermion shell structure in 3He clusters
10 –2 10 –3
Super fluid 3He liquid clusters Photoassociated diatomic molecules
10 –4
Doppler cooling
Rydberg gas
10 –5 Optical molasses
Ultracold nucleation
Ultracold 10 –6 10 –7
BEC condensates
10 –8
6 Li degenerate fermi gas
10 –9
Table 60.1 General n-dependence of characteristic properties of Rydberg states. After [4] Property Radius (cm) Velocity (cm=s) Area (cm2 ) Ionization potential (eV) Radiative lifetime (s)a Period of classical motion (s) Transition frequency (s1 ) Wavelength (cm) p
a A0 D 8˛ 3 =.3 3/ .vB =a0 /
n-dependence n2 a0 =Z vB Z=n a02 n4 =Z 2
n D 10 5:3 107 2:18 107 8:8 1013
n D 100 5:3 105 2:18 106 8:8 109
n D 500 1:3 103 4:4 105 5:5 106
n D 1000 5:3 103 2:18 105 8:8 105
Z 2 R1 =n2 n5 Œ3 ln n .1=4/=.A0 Z 4 / 2=!n;n˙1 D hn3 =.2Z 2 R1 / !n;n˙1 D 2Z 2 R1 =.„n3 / n;n˙1 D 2c=!n;n˙1
1:36 101 8:4 105 1:5 1013 4:1 1013 4:6 103
1:36 103 17 1:5 1010 4:1 1010 4:6
5:44 105 7:3 104 1:9 108 3:3 108 570
1:36 106 7:22 hours 1:5 107 4:1 107 4:5609 103
Asymptotic expressions For n 1, ˇ ˇ2 5:731 13:163 24:295 C C n3 ˇX1s!n ˇ 1:563 C n2 n4 n6 39:426 58:808 C C ; (60.12a) n8 n10 ˇ ˇ2 180:785 1435:854 C n3 ˇX2s!n ˇ 14:658 C n2 n4 9341:634 54;208:306 C C n6 n8 292;202:232 C ; n10
(60.12b)
ˇ ˇ2 218:245 2172:891 n3 ˇX2p!n ˇ 13:437 C C n2 n4 17;118:786 117;251:682 C C n6 n8 731;427:003 C : (60.12c) n10
60
884
E. J. Mansky II
60.2.2
Interaction Potentials
˘ molecular states V .2/ .˘; ˇI / D
H(n)-H(n) long-range interactions [10]
X C2n .˘; ˇI / n3
(60.17a)
R2n
X 2 2 2 2 Dˇ .`; L; `0 ; L0 / ; (60.17b) C2n .˘; ˇI / D 4Q Q Q 3Q 1 3 1 2 3 5 R R `;L;`0 ;L0 1 `CLC`0 CL0 C2D2n 2 2 9 7 6Q1 Q5 15Q2 Q4 C 10Q3 O.R / ; R (60.13) where the Dˇ are defined
V!;! .R/ D
Dˇ .`; L; `0 ; L0 / D
for parallel-aligned dipoles; and, V
;! .R/
D
9˛ 2 X f1 .`; L; Ls I L t ; `0 ; L0 /G1 16 L ;L s t C f2 .`; L; Ls ; L t ; `0 ; L0 /G2 : (60.18)
2 2 6 Q C Q1 Q2 R3 1 R4 2 C 5 4Q1 Q3 C 3Q22 ŒQ1 Q4 C 2Q2 Q3 molecular states R 2 X D1 .`; L; `0 ; L0 / C 7 6Q1 Q5 C 15Q2 Q4 C 10Q32 O.R8 / ; R ; V .2/ D R`CLC`0 CL0 C2 (60.14) ``0 LL0
(60.19)
where the D1 are defined for antiparallel-aligned dipoles. Analytical expressions for the QL multipole moments for 1 X L D 0 5 are provided in Table 60.2. Full expressions for L G1 .`; L; `0 ; L0 ; Ls ; L t / D1 .`; L; `0 ; L0 / D 2 up to 10 are provided in [10]. Ls ;L t F1 .`; L; `0 ; L0 ; Ls ; L t / : 1;3
(60.20)
1;3
He(2 P)-He(2 P) Long-range dispersion coeffiSee [11] for detailed tabulations of the dispersion coefcients [11] Expressions for the second-order potential, .2/ ficients C and for the (lengthy) expressions for the various V , for the first three molecular states are: fi ; f1 ; f2 ; F1 ; G1 ; G2 functions appearing in Eqs. (60.16), (60.18), and (60.20). ˙ molecular states Cs-Cs and Rb-Rb pseudo-potentials [12] VLS .r/ for L D X C2n .˙; ˇI i / .2/ V .˙; ˇI i / D (60.15a) 0; 1 2n R n3 A ˛ 6 X (60.21a) V0S .r/ D er 4 1 e.r=rc / Di .`; L; `0 ; L0 / ; (60.15b) C2n .˙; ˇI i / D r 2r `;L;`0 ;L0 1 Zc ˛ 6 `CLC`0 CL0 C2D2n V1S .r/ D er Aer 4 1 e.r=rc / ; r 2r (60.21b) where the Di are defined where the fitted parameters for Cs and Rb are given in Table 60.3. For the case of Sr-Sr, see [13], while for e -Ne, see [14].
3˛ 2 X Di .`; L; ` ; L / D f .`; L; Ls I L t ; `0 ; L0 / 8 L ;L i 0
0
s
t
.G1 C .2 i/.2i / G2 / : Table 60.2 Multipoles QL for the Stark states j˙i of H(n)
(60.16)
Multipole Q0 .C/ D Q0 ./ Q1 .C/ D Q1 ./ Q2 .C/ D Q2 ./ Q3 .C/ D Q3 ./ Q4 .C/ D Q4 ./ Q5 .C/ D Q5 ./
Value 1 .3=2/n.n 1/ .2=22 /n2 .n 1/.5n 7/ .5=23 /n3 .n 1/.n 2/.7n 9/ .6=24 /n4 .n 1/.n 2/.21n2 77n C 62/ .42=25 /n5 .n 1/.n 2/.n 3/.11n2 39n C 30/
Asymptotic value .3=2/n2 .5=2/n4 .35=8/n6 .63=8/n8 .231=16/n10
60
Rydberg Collision Theories
885
Table 60.3 Fit parameters for Cs-Cs and Rb-Rb pseudo-potentials (from [12]) Cs
r0 0.01
˛ 402.2
7.2443
Rb
0.01
319.2
7.4975
State 1 S 3 S 1 P 3 P 1 S 3 S 1 P 3 P
60.2.3 Radial Integrals
4:5396 93:936 3:6681 4:1271 4:5642 68:576 4:2625 1:4523
nc D 2n n0 =.n C n0 / ;
Z1
0 0
rc 1.6848 2.6856 1.8031 2.1294 1.8883 2.3813 1.8869 1.8160
60
where
Definition n` Rn`
1.3304 7.5397 1.3195 2.2329 1.3438 9.9898 1.0055 4.8733
A
0
(60.26a)
Dn n ;
Rn` .r/rRn0 `0 .r/r 2 dr ;
(60.22)
(60.26b)
0
0
0
` D ` `; `> D max.`; ` / ; p x D e ; e D 1 .`> =nc /2 ;
(60.26c)
En` D Z 2 R1 =n2 ; n D n ı` ;
(60.27a)
(60.26d) where Rn` .r/ are solutions to the radial Schrödinger equation. See Chap. 9 for specific representations of Rn` for and J n .y/ is the Anger function. hydrogen. The energies of the states n` and n0 `0 are given in terms Exact results for hydrogen For `0 D ` 1 and n ¤ n0 [15], of the quantum defects ı by 0
n0 `1 Rn`
0
a0 .1/n ` .4nn0 /`C1 .n n0 /nCn 2`2 D Z 4.2` 1/Š.n C n0 /nCn0 .n C `/Š.n0 C ` 1/Š 1=2 .n0 C `0 /Š.n ` 1/Š 2 F1 .n C ` C 1; n0 C `I 2`I Y /
n n0 n C n0
2
0
0
En0 `0 D Z R1 =n ; n D n ı`0 :
(60.27b)
Sum rule For hydrogen Xˇˇ 0 ˇˇ2 Xˇˇ 0 ˇˇ2 n `C1 n `1 ˇRn` ˇ D ˇRn` ˇ
2
n0
2 F1 .n C ` 1; n0 C `I 2`I Y / ; (60.23) where Y D 4nn0 =.n n0 /2 . For n D n0 , ! a 3 p 2 0 n`1 D n n `2 : Rn` Z 2
02
D
n0 n2 a02 2 5n 2 Z2
C 1 3`.` C 1/ :
(60.28a) (60.28b)
See §61 of [5] for additional sum rules. (60.24)
Ze´ldovich effect [17]
aB 2 2 D cot ı ln ; (60.29) The hypergeometric functions 2 F1 in Gordon’s formulae 2as x0 above are increasingly difficult to compute numerically due to unavoidable roundoff errors. See Sect. 60.8 for working where a is the Bohr radius of the Coulomb field, ı is the B p formulae suitable for numerical computations. quantum defect, ln is Euler’s constant, x0 D 8r0 =aB , and r0 is the radius of the short-range field. Semiclassical quantum defect representation [16] ˇ 0 0 ˇ2 a 2 ˇ n`ˇ 0 ˇRn` ˇ D 60.2.4 Line Strengths Z ˇ 2 ˇ nc ` `> ˇˇ 1 J 1 .x/ Definition 2 nc ` `> 1C J C1 .x/ (60.30a) S.n0 `0 ; n`/ D e 2 .2` C 1/jr n0 `0 ;n` j2 nc ˇ 0 0 ˇ2 ˇ2 ˇ n`ˇ ˇ (60.30b) D e 2 max.`; `0 /ˇRn` ˇ ; 2 (60.25) C sin. /.1 e/ ˇˇ ;
886
E. J. Mansky II
where `0 D ` ˙ 1. For hydrogen, ea 2
0
.n n0 /2.nCn /3 Z .n C n0 /2.nCn0 /C4 ˚ 2 2 F1 .n0 ; n C 1I 1I Y /
2 ; 2 F1 .n0 C 1; nI 1I Y /
S.n0 ; n/ D 32
0
where D .E=R1 /.`C1=2/3=6, and K .x/ are Bessel functions of the third kind.
.nn0 /6
Line strength of line n X 1 S.n C k; n/ 3 : Sn S.n/ D k (60.31) k¤0
(60.39)
where Y D 4nn0 =.n n0 /2 .
Born approximation to line strength Sn [4] 2 2 X Semiclassical representation [18] R Z 1 1 1 41 B Sn D ln.1 C "e ="/ 1 E 2 4k k 4 32 ea0 2 .""0 /3=2 k¤0 0 S.n ; n/ D p G. n/ ; (60.32) 3 ." "0 /4 3 Z 4 "e X 0:60 1 5 C 1 where " D 1=n2 , "0 D 1=n02 , and the Gaunt factor G. n/ is 3 " C "e k k3 k¤0 given by "e 1:47"e Z 2 R1 p ˇ ˇ 0 C 0:82 ln 1 C ; D (60.33) G. n/ D 3ˇ nˇJ n . n/J n . n/ ; E " " C "e
(60.40)
where the prime on the Anger function denotes differentia- where " D jEn jZ 2 =R1 and "e D "=Z 2 R1 . tion with respect to the argument n. Equation (60.33) can be approximated to within 2% by the expression 1
1 : 4j nj
60.2.5 Form Factors (60.34) Fn0 n .Q/ D
Relation to oscillator strength X S.n0 `0 ; n`/ S.n0 ; n/ D `;`0
D 3e 2 a02
R1 X fn0 `0 ;n` : „! 0
X X ˇ˝ ˇ ˇ ˛ˇ ˇ n`mˇeiQr ˇn0 `0 m0 ˇ2 :
Connection with generalized oscillator strengths fn0 n .Q/ D
(60.35)
`;`
fn0 `0 ;n`
Z 2 E Fn0 n .Q/ : n2 Q2 a02
(60.42)
Semiclassical limit
Connection with radial integrals „! max.`; `0 / ˇˇ n0 `0 ˇˇ D : R 3R1 .2` C 1/ n`
(60.41)
`;m `0 ;m0
(60.36)
lim f
Q!0
n0 n
3 nn0 32 .Q/ D 2 3n n.n C n0 / 0 nJ n . n/J n . n/ ;
(60.43) Density of line strengths For bound–free n` ! E`0 transitions in a Coulomb field, the semiclassical representation [4] where Jm .y/ denotes the Bessel function. is Representation as microcanonical distribution R1 2 d Z S.n`; E/ D 2n.2` C 1/ 2Z 2 R1 dE „! dpjgn` .p/j2 Fn0 ;n` .Q/ D .2` C 1/ 03 n 1 J 0 .e /2 C 1 2 J .e /2 .p „Q/2 p 2 e ı En0 En` ; 2m 2m e 2 a02 ; (60.37) (60.44) R1 2 2 Z 2 2 q p dp dr 4Z R1 Ze
2 0 ı En Fn ;n .Q/ D where D „! n3 =2R1 and e D 1 ` C .1=2/ =n2 . The .nn0 /3 .2„/3 2m r asymptotic expression for 1 .p „Q/2 Ze 2 ı En0 ; (60.45)
4 2m r d 2.2` C 1/ R1 2 ` C 12 S.n`; E/ D
5 29 dE 3 2 „! n3 ; (60.46) D 2 3 2 h i e 2 a2 3.nn0/3 . 2 C C / . C 2 /3 0 2 2 K1=3 ./ C K2=3 ./ ; (60.38) R1 where D Qa0 =Z, and ˙ D j1=n ˙ 1=n0j.
60
Rydberg Collision Theories
887
60.2.6 Impact Broadening
Depopulation rates for trapped Rydberg states [22]
The total broadening cross section of a level n is
n D a02 =Z 4 n4 Sn :
nn`m 0 `0 m˙1 .B/ (60.47)
The width of a line n ! n C k is [19]
n2 n02 B B D 1 2 n n02 R1 hc
3 nn`m 0 `0 m˙1 .0/ ; (60.55)
where B is the external magnetic field, B is the Bohr magneton, and .0/ is the field-free depopulation rate. (60.48)
n;nCk D ne Œhvn i C hvnCk i ; where ne is the number density of electrons, and hvn i D
X
hvnCk;n i D
k¤0
D
n4 a02 vB Z 3 3=2
60.3
n4 Kn Z3
Z1 eE=kB T Sn 0
Correspondence Principles
(60.49a) Correspondence principles are used to connect quantum mechanical observables with the corresponding classical quan(60.49b) tities in the limit of large n. See [23] for details on the equations in this section.
E dE ; .Z 2 R1 /2
where D kB T =Z 2 R1 . See Chap. 63 for collisional line broadening. 60.3.1
Bohr–Sommerfeld Quantization I
60.2.7 Effective Lifetimes and Depopulation Rates Black-body radiation (BBR) depopulation rates [20] BBR D
2:14 1010 1 .s 1 / ; n5eff exp.315;780=n3effT / 1
n0
where ni D 0; 1; 2; : : : and ˛i D 0 if the generalized coordinate qi represents rotation, and ˛i D 1=2 if qi represents a libration.
EnCs En D hnCs;n s„!n ; s D 1; 2; : : : n ; (60.57)
where nCs;n is the line emission frequency, and !n is the angular frequency of classical orbital motion. The number of (60.51) states with quantum numbers in the range n is
2 : 3n2eff
N D
2:14 10 A .s 1 / : C nD exp.315;780B=n eff eff T / 1
(60.52)
(60.53)
Effective lifetimes [20] eff D
n D
D
D Y
D Y
.Ji wi /=.2„/D
i D1
.pi qi /=.2„/D ;
(60.58)
i D1
Semiempirical radiative lifetimes [21] 0 D s nıeff .ns:/
D Y i D1
10
21:4 1 A ı nD exp.315;780B=nC T / 1 s neff eff eff
(60.56)
60.3.2 Bohr Correspondence Principle
For nP and nD states, Eq. (60.50) overestimates the lifetimes for intermediate quantum numbers n 20. For those states a generalization of Eq. (60.50) using fitted parameters A; B; C , and D is appropriate, BBR D
pi dqi D 2„.ni C ˛i / ;
(60.50)
where neff represents an effective quantum number for n > 15. Effective sum rule X !nn0 f .nL ! n0 L0 / D
Ai D Ji wi
for systems with D degrees of freedom, and the mean value FN of a physical quantity F .q/ in the quantum state is X .q/ i!mn t FN D h jF .q/j i D am an Fmn e ; (60.59) n;m .q/
!1 .ns:/ (60.54)
See [20] for extensive tables of the effective lifetimes and fitted parameters A; B; C; D; s , and ı for the alkali atoms.
where the Fmn are the quantal matrix elements between timeindependent states. The first order S-matrix is Sf i
i! D 2„
Z1
2=! Z
V ŒR.t/; r.t1 /eis!.t1 t / dt1 ;
dt 1
0
(60.60)
60
888
E. J. Mansky II
where R denotes the classical path of the projectile and r the orbital of the Rydberg electron.
60.3.4 Strong Coupling Correspondence Principle The S-matrix is
60.3.3 Heisenberg Correspondence Principle Sf i
For one degree of freedom [23],
! D 2
m .r/F .r; R/n .r/ dr
8 9 Z1
m2
60.6 Universality Properties
Universality properties [32, 33] have been observed in all the E 0 I (60.87c) alkalis’ atoms in ultracold Rydberg collisions where the scatvl D v2 v1 ; vu D v10 C v20 ; E 0 : (60.87d) tering length, a, is very much greater than the effective range of the interaction, Reff . The result is that details of the two3. E "12 jQ"12 j 0, and 2m1 v1 m12 v2 body collision are characterized by a single parameter, a, the vl D v1 v2 ; vu D v1 C v2 ; E 0 I (60.87e) scattering length, that is universal and independent of the details of the short-range interaction at such low energies. vl D v10 v20 ; vu D v10 C v20 ; E 0 : (60.87f) In the case of three-body collisions, a universal property, eff first predicted in 1970 by Efimov [34], is that there is an infiIf 2m1 v1 < m12 v2 , then E .v1 ; v2 / D 0, nite number of shallow three-body bound states with binding 4. "12 C Q"12 E "12 Q"12 , and m1 < m2 .n/ energies ET that has an accumulation point at E D 0. The vl D v1 v2 ; vu D v10 C v20 ; E 0 I (60.87g) Efimov effect of the occurrence of an infinite number of vl D v10 v20 ; vu D v1 C v2 ; E 0 : (60.87h) three-body bound states is present for both the case of three identical bosons, and more generally for three bodies where eff If 2m1 v1 < m12 v2 , then E .v1 ; v2 / D 0. at least two of the three have large S-wave scattering lengths. In the general case, the bound states appear in the specSince v10 and v20 , given by Eqs. (60.85) and (60.86), respectrum at critical values of a that differ by multiplicative factors tively, must be real, E .v1 ; v2 / D 0 for E outside the of e=s0 , where s0 only depends on the masses and spin statisrange tics of the particles and is the solution of the transcendental 1 1 2 2 (60.88) equation, m2 v2 E m1 v1 ; 2 2 s0 s0 8 D p sinh : (60.92) s0 cosh which simply expresses the fact that the particle losing en2 6 3 ergy in the collision cannot lose more than its initial kinetic The solution of Eq. (60.92) yields s0 1:00624. energy. vl D v20 v10 ;
vu D v1 C v2 ;
60
Rydberg Collision Theories
891
Atom-dimer cross section (deep dimers)
Asymptotic number of bound states N !
s0 jaj ; ln r0
(60.93)
AD D 84:9
sin2 Œs0 ln.a=a? / 0:97 C sinh2 ? (60.100) sin2 Œs0 ln.a=a? / C sinh2 ?
for scattering length a and hard-core radius r0 . .nC1/
ET
.n/
=ET ! 1=515:03
at E D ED , where ED is the atom-dimer threshold Efimov (60.94) resonance in which S-wave scattering dominates.
as n ! C1 and with the scattering length a D ˙1.
60.6.2 Unequal Mass Case
Asymptotic spectrum: general case nn? „2 ?2
.n/ ET ! e 2=s0 : m
(60.95)
The two heavy masses are M , and the one light mass is m. Asymptotic spectrum: two heavy – one light [35]
60.6.1 Equal Mass Case
T
Asymptotic spectrum: three identical bosons nn? 2 2 1 „ ? .n/ ET ! ; 515:03 m
ET
.n/
.n/
p ! 103:0 exp 1:260 M=m 2
(60.96)
1 .n/ s0 ln.4Edeep =ET / ; 2
sin p s0 0:567142 M=m ;
(60.101a)
(60.101b) with n ! C1, the scattering length a D ˙1, and where ? .n/ is a three-body parameter that is the zeros of as n ! 1 with a D ˙1 and M m; T and Edeep are the width of the Efimov trimer resonance and the binding energy
1 sin s0 ln mET =.„2 ?2 / D 0 ; (60.97) of the dimer well, respectively. 2 with s0 being the solution of Eq. (60.92). See Table 1 of [32] for useful tabulations of scattering lengths for the alkali 60.7 Many-Body and Multiparticle Effects atoms. See §III.F of [33] for details on the ultracold experiments of the alkali atoms. 60.7.1 Many-Body Theory Atom-dimer scattering length (shallow dimers) aAD D .b1 b0 tanŒs0 ln.a ? / C ˇ/a ;
Gross–Pitaevskii Equation [36, 37] Z
(60.98) ŒT .x/ .x/ C
d3 x 0 V .x x 0 /j .x 0 /j2 .x/ D 0 ;
where b0 ; b1 , and ˇ are universal parameters, and ? is (60.102) the three-body parameter in Eq. (60.96). The parameters in Eq. (60.98) are universal in that they hold for all identical where is the Bose condensate wavefunction; T and V bosons, independent of the details of their short-range interare the kinetic and potential energy operators (respectively), actions. and is the chemical potential. See the standard monographs [38–41], on many-body theory for more general Atom-dimer scattering length a and effective range rs details. (deep dimers) aAD D .1:46 C 2:15 cotŒs0 ln.a=a? / C i? /a ; (60.99a)
60.7.2 Dilute Bose Gas rs;AD D .1:13 C 0:73 cotŒs0 ln.a=a? / C 0:98 C i? /2 a ; (60.99b) The Lee-Huang-Yang equation for the second correction to where a? is the scattering length when the Efimov reso- the thermodynamic ground state of a dilute Bose gas has nance is at the atom-dimer threshold, and is related to the been proven in a mathematically rigorous manner [42], three-body parameter, ? ; and ? is an inelasticity parameter, p 128 p 3 generally 0:1. See §6 and 7 of [32] for a full discussion. 2 3 E. / 4 a 1 C p
a C O. a / ; Atom and dimer here refer to one boson scattering with a pair 15 of bosons (dimer) in either a shallow or a deep well. (60.103)
60
892
E. J. Mansky II
where E. / is the energy density of the Bose gas with density
, and scattering length a. Further details on the use of Rydberg states in quantum information can be found in Chap. 85 of this handbook, and [43–46].
60.8
Working Formulae for Rydberg Collisions
Hydrogenic For the case of hydrogen, Dewangan [48] reformulated the hypergeometric functions appearing in Gor.˛;ˇ/ don’s formulae in terms of Jacobi polynomials Pi ./, a form more suitable for numerical computation .n ` C 1/.n `/ . 1;2`1/ n0 ;`1 .cos / P Rn;` D CJ .n C `/.n C ` 1/ n`C1 . C1;2`1/ 2 1 sin .cos / ; Pn`1 2 (60.110)
60.8.1 Inelastic n,`-Changing Transitions
where the constant CJ is, s ! A .n`/ C B ! A .n / C B C En0 ;n` ; (60.104) 1 .n C `/Š.n0 C ` 1/Š 2`C2 1 cot CJ D where En0 ;n` D En0 En` is the energy defect. The cross 4 .n ` 1/Š.n0 `/Š 2 section for Eq. (60.104) in the quasi-free electron model [47] .n ` 1/Š is C2` 1 sin ; 2 2 .n C ` 1/ 2as n0 ;n` .V / D
2 fn0 ;n` ./ ; ` n ; (60.105) and D n0 n 1. V =vB n03
0
where as is the scattering length for e C B scattering, D n a0 !n0 ;n` =V , !n0 ;n` D jEn0 ;n` j=„, En0 D R1 =n02 , and En` D R1 =n2 , with n D n ı` . Also, vB is the atomic unit of velocity (Chap. 1), and 2 2 4 fn0 ;n` ./ D tan1 ln 1 C 2 : (60.106) 2
Asymptotic Expressions [48]: Two Cases
(a) n n0 1 near continuum threshold The exact expression Eq. (60.110) has the following asymptotic form in terms of Airy functions (Ai.x/) s ` 0 `C1 0 .n / .n0 C ` 1/Š 4 n ;`1 0 ;n` ./ ! 1 as ! 0, and fn0 ;n` ./ ' Limiting cases: f R n n;`
n3=2 .n0 `/Š 8= 33 for 1. Then 4n0 .n0 C `/ 1 8 2 Ai 2as ˆ ˆ .2n0 C 1/2`C1=3 .2n0 C 1/1=3 ; !0; < 2 n03 .V =v / B (60.107) n0 ;n` .2` 1/ Ai.0/ 16as2 V n3 ˆ C 2`2=3 ˆ : ; 1: 2 .n0 /2`2=3 3vB jı` C nj3 1 1 Ai Rate coefficients .2n0 1/2`5=3 .2n0 1/1=3 (60.111) hn0 ;n` .V /i hVn0 ;n` .V /i=hV i (60.108a) suitable for practical computations, and where ` n0 . 2as2 D
(60.108b) 2 'n0 ;n` .T / ; VT =vB n03 (b) D n n0 1 nearby Rydberg states p " where VT D 2kB T =, T D n a0 !n0 ;n` =VT , n D n0 n, 0 ;`1 n 0 and is the reduced mass of A–B. The function 'n ;n` .T / R ' BJ J 1 .A1 / n;` in Eq. (60.108a) is given by # 1 1 A 2 1 'n0 ;n` .T / D eT =4 erfc T J C1 .A2 / ; .n ` C 1/.n `/ C1 2 A2 Z1 (60.112) du u T 4 p e ln 1 C 2 (60.109a) u T for small =n, J˛ .x/ is a Bessel function of order ˛, and the 0 constants are defined 8 T
2 n, where "e D E= Z 2 R1 , " D 1=n2 , "0 D 1=n02 , and " D " "0 .
60
894
E. J. Mansky II
(B) Rate coefficients [50] (three cases) With the definitions 0 3=2 ."" / 8a02 Z2 1 p D 4 2 ln.1 C "e ="/ 1 D v =v ; v D .8kB T /= ; (60.132) B B rms rms 4 Z n "e 4n ."/
f .y/ D y 1=2 1 .1 y/ey C y 3=2 Ei .y/ ; (60.133) "e ."0 /3=2 4 1 0:6 ı
C ; C 1 (60.134) y D .B as /2 4a02 n8 ; n " C "e ."/2 3n " (60.126) 1=4 (60.135) n D .ja j =4a / ;
(B) Heavy-particle–Rydberg atom collision n0 n
1
s
B
0
h 1=6 i1=3 5=6
; n2 D 0:7 jas jB = ˛d a03
where "e D m"=MZ 2R1 with heavy-particle mass and (60.136) charge denoted above by M and Z, respectively, and all other where ˛d is the dipole polarizability of A , then terms retain their meaning as in Eq. (60.125). 8 8a02 n4 ; n n1 ; ˆ ˆ ˆ ˆ ˆ ˆ4 1=2 a0 jas jB f .y/ ; n1 n n2 ; 60.8.3 Quasi-Elastic `-Mixing Transitions ˝ el ˛ < 2 2 ns 7.˛ /2=3 C 4as B X d B ˆ .`mixing/ ˆ n4 ˆ n` n0 `0 ;n` (60.127a) n n2 : ˆ 2 2 1=3 ˆ .˛ / 2:7a ˆ d B 0 s B : ` ¤` ; ( a0 n6 2 4 geo D 4a0 n ; n nmax ; (60.137) (60.127b) 2as2 vB2 =V 2 n3 ; n nmax : The two limits correspond to strong (close) coupling for 60.8.5 Inelastic n` ! n` 0 Transitions n nmax and weak coupling for n nmax , and expressions Eq. (60.127b) are valid when the quantum defect ı` of the ini- (A) Impact Parameter Theory with cutoffs [51, 52] tial Rydberg orbital n` is small; nmax is the principal quantum 1=2 D n` number, where the `-mixing cross section reaches a maxi- hn` i D 9:93 106 1=2 m T mum [49], e Te m 2=7 11:54 C log10 C 2 log10 Rc cm3 =sec : vB jas j Dn` : (60.128) nmax (60.138) Va0 For Rydberg atom–noble gas atom scattering, nmax D 8 to 20, where is the reduced mass of the system, m is the electron while for Rydberg atom–alkali atom scattering nmax D 15 to mass, Te is the temperature in K and, 2 30. Zp 6n2 .n2 `2 ` 1/ (60.139) Dn` D Zt
60.8.4 Elastic n` ! n` 0 Transitions A .n`/ C B ! A .n`0 / C B : (A) Cross sections elastic ns .V / D
2 Css as2 ; .V =vB /2 n4
valid for n ŒvB jas j=.4Va0 /1=4 with 8 Css D 2
p 1= Z 2
ŒK.k/2 dk ; 0
with Zp and Zt denoting the charge of the projectile and target, respectively. (60.129) Pengelly and Seaton’s expression Eq. (60.138) includes a sum over `0 D ` ˙ 1, and therefore only includes the contribution of dipole transitions. See [53] and [54] for further details on the contribution of multipole transitions to the (60.130) rate coefficients and line emissivities. See also [54] and [55] for tabulations of rate coefficients and line emissivities, and Eq. (14) of [54] for a modified version of Pengelly and Seaton’s expression which includes more accurately the contribution from small energies. The cutoff parameter, Rc is defined, ˚ (60.131) (60.140) Rc D min Rcd ; Rcnd ; RD
where K.k/ denotes the complete elliptic integral of the first kind.
with, Rcd D 0:72v
(60.141a)
60
Rydberg Collision Theories
895
Rcnd D 1:12„v=E kB Te 1=2 RD D 4 e2 ne
(60.141b) where, (60.141c)
A D sin2 sin2 .1 2 / 2
(60.147a)
2
(60.147b) B D sin .1 C 2 / sin .1 2 / where v is the projectile speed, , the lifetime of the initial state and E is the energy difference between non- and, degenerate initial and final states. cos 1 D `=n (60.148a) The individual cutoffs Rcd and Rcnd for degenerate and non0 degenerate states, respectively, are introduced to avoid the cos 2 D ` =n (60.148b) logarithmic divergence at large impact parameters. The third cutoff, RD is the Debye screening length, is used to model (D) Exact Quantal Theory for hydrogenic collisions [57, the effect of screening at large impact parameters. 59, 60] Vrinceanu and Flannery made L use of the isomorphism between SO(4) and SO.3/ SO.3/ to solve the (B) Exact Classical Theory for hydrogenic collisions [56, Schrödinger equation in the interaction picture, under as57] Exploiting the dynamical SO(4) symmetry of the hy- sumptions of an adiabatic, dipole and classical path, to yield drogen atom, Vrinceanu and Flannery solved exactly the the transition probability on the energy shell time-dependent classical equations for the transition proba#2 " n1 bility, `0 ` L 2`0 C 1 X .n/ 2 .2L C 1/ HjL ./ P`0 ` D 8 n j j j LDj`0 `j B < 0 0 ˆ ˆ 2`0 =n2 < KŒB=.BA/ (60.149) n p B > 0; A < 0 P``0 ./ D 2 BA ˆ „ sin .=2/ ˆ : KŒ.Bp A/=B where f g denotes the 6–j symbol, and, B > 0; A > 0 B s (60.142) .2j C 1/.2j L/Š L sin HjL ./ D .2L/ŠŠ where is the rotation angle, .2j C L C 1/Š 2
p .1CL/ 1 C ˛ 2 cos 1 C ˛ 2 ˚ C2j L cos (60.150) ; (60.143) cos D 2 2 2 .1 C ˛ / is a standard result from group theory [61, 62], and with, with K.k/ denoting the complete elliptic integral of the first p kind, ˛ D 3Zp an vn =2bv is the Stark parameter, ˚ is the 1 C ˛ 2 cos 1 C ˛ 2 ˚ (60.151) cos D polar angle swept out during the collision, and, 2 1 C ˛2 cos.u1 C u2 / cos A.`=n; `0 =nI / D (60.144a) where ˛ and ˚ have the same meanings as in (B) above. .˛/ 1 cos Cn .x/ is the ultraspherical, or Gegenbauer polynomials of cos.u1 u2 / cos See Table 1 of [60] for a list of B.`=n; `0=nI / D (60.144b) index n and order ˛ [63]. .n/ 1 cos exact expressions for P`0 ` for n D 15. where the angles u1 and u2 only depend on the initial and final states, respectively, cos u1 D 2`2 =n2 1 02
2
cos u2 D 2` =n 1
(60.145a) (60.145b)
P
A .n`J / C B ! A .n`J 0 / C B C EJ 0 J :
(60.152)
(A) Cross sections (two cases)
(C) Semiclassical Transition Probability [58] SC
60.8.6 Fine Structure n`J ! n`J 0 Transitions
2`0 .`=n; ` =nI / D „n2 sin 8 0 j sin j < j sin.1 2 /j ˆ ˆ < KŒB=A p j sin j > j sin.1 C 2 /j A ˆ ˆ : KŒA=B p j sin j < j sin.1 C 2 /j B 0
(60.146)
0
n`J n`J .V / D
2J 0 C 1 cnorm 4a02 n4 ; 2.2` C 1/
(60.153)
valid for n n0 .V /, and n`J 0 n`J .V
.`/ 2 CJ 0 J as2 vB2 .`/ n80 .V / /D 'J 0 J .J 0 J / 1 ; V 2 n4 2n8 (60.154)
60
896
E. J. Mansky II
valid for n n0 .V /, where the quantity n0 .V / is the 60.9 Impulse Approximation effective principal quantum number such that the impact parameter 0 of B (moving with relative velocity V ) equals the 60.9.1 Quantal Impulse Approximation radius 2n2 a0 of the Rydberg atom A ; n0 .V / is given by the Basic Formulation [66] solution to the following transcendental equation Consider a Rydberg collision between a projectile (1) of 2 .`/
.2` C 1/C charge Z1 and a target with a valence electron (3) in orbital v a 0 B s .`/ J J 'J 0 J J 0 J Œn0 .V / : n80 .V / D i bound to a core (2). The full three-body wave function for 2.2J 0 C 1/cnorm Va0 (60.155) the system of projectile + target is denoted by i . The relative distance between 1 and the center-of-mass of 23 is denoted The constant cnorm in Eq. (60.153) is equal to 5=8 if geo D by , while the separation of 2 from the center-of-mass of .`/ hr 2 in` , or 1 if geo D 4a02 n4 . The function 'J 0 J .J 0 J / in 13 is . Eq. (60.154) is given, in general, by [64, 65] Formal scattering theory .`/ .`/ .`/ (60.156a) 'J 0 J .J 0 J / D J 0 J .J 0 J /=J 0 J .0/ ; .C/ (60.161) i D ˝ .C/ i ; Z1 ` X .`/ .2s/ 2 2 J 0 J .J 0 J / D A`J 0 ;`J js .z/J s .z/zdz ; (60.156b) where the Möller scattering operator ˝ .C/ D 1 C G C Vi , and sD0 J 0 J Vi D V12 C V13 . vB Let m be a complete set of free-particle wave functions (60.156c) J 0 J D jı`J 0 ı`J j ; Vn satisfying where js .z/ is the spherical Bessel function, and the coef.`/ .2s/ (60.162) .H0 Em /m D 0 ficients CJ 0 J and A`J 0 ;`J in Eqs. (60.154) and (60.156b), respectively, are given in Table 5.1 of Beigman and LebeC dev [4]. The quantum defect of Rydberg state n`J is ı`J . For and define operators !ij .m/ by .`/
!ijC .m/m D 1 C
elastic scattering, JJ D 0, and 'JJ .0/ D 1. Symmetry relation .`/
JJ 0 .J 0 J / D
2J C 1 .`/ 0 .J 0 J / : 2J 0 C 1 J J
(60.157)
where Vij denotes the pairwise interaction potential between particles i and j (i; j D 1; 2; 3). Then the action of the full Green’s function G C on the two-body potential Vij is
(B) Rate coefficients n`J 0 i hn`J
1 Vij m ; Em H0 Vij C i (60.163)
!1=2 .`/ cnorm .2J 0 C 1/CJ 0 J D 2.2` C 1/ vB jas j 2 a0 F ./ ; VT a0
(60.158)
h i G C Vij D !ijC .m/ 1
C G C .Em E/ C V12 C V13 C V23 Vij h i !ijC .m/ 1 : (60.164)
where D n80 .VT /=n8 , and Projection operators
p 1 F ./ E2 ./ C .1 e / ;
(60.159)
where E2 .x/ is an exponential integral. Limiting cases D
n`J 0
n`J
E D
.`/
n
nmax
;
(60.160) where nmax D .3=2/1=8 n0 .V / if J 0 J 1.
(60.165a) !ijC
D
bijC
C1:
m
8 2J 0 C 1 2 4 ˆ ˆ < 2.2` C 1/ cnorm 4a0 n ; n nmax ; 2 CJ 0 J as2 vB2 ˆ ˆ : ; VT2 n4
bijC .m/ D !ijC .m/ 1 ; X bijC D bijC .m/jm ihm j ;
C
G Vij j
ii
D
X
(60.165b) C
G Vij jm ihm j
ii
(60.166a)
m
h i D bijC C G C V23 ; bijC CG
C
V12 C V13
j
Vij bijC .m/
ii
:
(60.166b)
60
Rydberg Collision Theories
897
references therein for details. With the definitions
Möller scattering operator
C
C
C C C !12 1 C G C V23 ; b13 C b12 ˝ C D !13
C C C G C V13 b12 : (60.167) C V12 b13
t 1 D K =a C v ; M1 ; aD M1 C m e k D ak2 .1 a/kf ;
Exact T -matrix Tif
4˛ı 2 ; .T 2ı/.T 2˛ı/ ı D iˇK p K ;
zD
ˇ ˇ C ˛ C D f ˇVf ˇ !13 C !12 1 i ˝ ˇ ˇ ˛
C C C f ˇVf ˇG C V23 ; b13 C b12 i ˝ ˇ ˇ
˛ C C C f ˇVf ˇG C V13 b12 C V12 b13 i : ˝
(60.168)
t D .K p/=a ;
T D ˇ2 C P 2 ; D aZ1 =K ; N./ D e=2 .1 i/ ; M2 bD ; M2 C m e K D ak1 .1 a/ki ; p D akf ki ;
ˇ D aZ1 =n ;
The impulse approximation to the exact T -matrix element and n is the principal quantum number, the impulse approxiEq. (60.168) is obtained by ignoring the second term involv- mation to the T -matrix becomes, in this case, ing the commutator of V23 , ˝ ˇ ˇ C ˛ imp (60.173) Tif f ˇV23 ˇ!13 i
C Z imp C (60.169) i ! i D !13 C !12 1 i : dK 1 D N./gi .t 1 /F .f; K ; p/ ; (60.174) 2 3 2 a t2 Impulse approximation: post form where, for a general final s-state, Z ˇ C ˛ imp C C !12 1 i : (60.170) Tif D h f jVf ˇ !13 F .f; K ; p/ D 'f .x/eipx
1 F1 i; 1I i Kx K x dx ; (60.175) The impulse approximation can also be expressed using incoming-wave boundary conditions by making use of the and gi .t 1 / denotes the Fourier transform of the initial state. prior operators The normalization of the Fourier transform is chosen such that momentum and coordinate spaceR hydrogenic wave func1 tions are related by 'n .r/ D .2/3 exp.it 1 r/gn .t 1 / dt 1 . Vij m ; !ij .m/m D 1 C Em H0 Vij i For the case f D 1s, (60.171a) X ˇ 3=2 @ !ij D !ij .m/jm ihm j : (60.171b) F .1s; K ; p/ D p I .; 0; ˇ; K ; p/ @ˇ m p .1 i/ˇ D 8 ˇ 3=2 T2 The impulse approximation Eq. (60.170) is exact if V23 is i # a constant since the commutator of V23 vanishes in that case. T i.ˇ iK/ C T .T 2ı/ T 2ı (60.176)
Applications [66] evaluated at ˇ D aZ1 . For the case f D 2s, C
C
(1) Electron capture X C H.i/ ! X.f / C H . imp
Tif
D
˝ f
ˇ
ˇV12 C V23 ! C C ! C 1 12 13
˛ i
ˇ 3=2
F .2s; K ; p/ D p
:
(60.172)
@ @2 C ˇ 2 I ; 0; ˇ; K ; p @ˇ @ˇ (60.177)
Wave functions: i D eiki 'i .r/, f D e ikf 'f .x/, m D evaluated at ˇ D aZ1 =2. For a general final ns-state f , .2/3 expŒi.K x C k /, where the 'n are hydrogenic p 4 T 2˛ı i wave functions. I .; ˛; ˇ; K ; p/ D T T 2ı If X above is a heavy particle, the V12 term in Eq. (60.172) may be omitted due to the difference in mass between the .U cosh ˙ iV sinh / ; projectile 1 and the bound Rydberg electron 3. See [66] and (60.178)
60
898
E. J. Mansky II
where the complex quantity U C iV is
where
12 C i i U C iV D .4z/ .1 C i/ 2 F1 .i; iI 1 2iI 1=z/ :
(60.179)
(2) Electron impact excitation e C H.i/ ! e C H.f / :
(60.180)
Neglecting V12 and exchange yields the approximate T matrix element imp
Tif
ˇ ˇ C ˛ ˇV13 ˇ! i 13 Z Z Z1 1 D dx dr eiq 'f .r/ .2a/3 x Z dK N./gi .t 1 /eit 1 r 1 F1 .i; 1I iKx iK x/
˝
f
(60.181)
Z Z1 D dK N./gi .t 1 /gf .t 2 / .2a/3
I ; 0; 0; K ; q ;
8 3b 2 D ˛D D 2b 2 2 e A.cos / D 4 C D .˛ 2 ˇ 2 /3=2 .˛ 2 ˇ 2 /1=2
16D D .3 C 4˛/b 2 48 2 D 2 b 2 C . 2 ı 2 /5=2 . 2 ı 2 /3=2 !
32 D 3b 2 C A.cos / ; (60.188) . 2 ı 2 /1=2 with A.cos / given by Eq. (60.185), and K2 K q2 ˛ Db Cv C 2 C C E cos ; (60.189a) a aq " 2 2 #1=2 q K 2 2 ; (60.189b) sin 4v q C E ˇD aq 2
2
ı D 4ˇ; D 4˛ C D ; 4bq .q C 2K cos / ; DD a
(60.189c) (60.189d)
while and N./ retain their meaning from Eq. (60.173). (60.182) (4) Ionization e C H.i/ ! e C HC C e .
where imp
Tif
I .; 0; 0; K ; q/
i q2 4 N./g .k bq/ ; i q2 q 2 2q K (60.190)
i ˇ2 C q 2 4 D lim 2 ˇ!0 ˇ C q 2 ˇ 2 C q 2 C 2q K 2iˇK (60.183) where K D a.k bq v/ and q D ki kf and exchange ˇ ˇi and V12 are neglected. ˇ 2K 4 ˇ (60.184) D 2 ˇˇ1 C cos ˇˇ A.cos / ; q q (5) Rydberg atom collisions [23, 68]
with ( A.cos / D
A C B.n/ ! A C B.n0 / 1; e ;
cos > q=2K ; cos < q=2K ;
C
(60.185)
and cos D KO q, O t 2 D t 1 C bq and q D ki kf . (3) Heavy-particle excitation [67] HC C H.1s/ ! HC C H.2s/ : imp
Tif
D
Z1 215=2 b 5 a3 q 2
Z1
(60.186) Z1
dK N.K/K 2 0
d.cos /
(60.191)
!ACB Ce :
(60.192)
Consider a Rydberg collision between a projectile (3) and a target with an electron (1) bound in a Rydberg orbital to a core (2) [23, 68]. Full T -matrix element ˇ .C/ ˇ ˝ ˛ 0 Tf i .k3 ; k03 / D f .r 1 /eik3 r 3 ˇV .r 1 ; r 3 /ˇi .r 1 ; r 3 I k3 / ; (60.193) with primes denoting quantities after the collision, and where the potential V is
1
ˇ ˇi ˇ ˇ 2K A.cos / ; ˇˇ1 C cos ˇˇ e q
(60.187)
V .r 1 ; r 3 / D V13 .r/ C V3C .r 3 C ar 1 / ;
r D r1 r3 ; (60.194)
60
Rydberg Collision Theories
899
with a D M1 =.M1 C M2 /, while the subscript C labels the (2) Plane-wave final state core. The impulse approximation to the full, outgoing wave .C/ f .r 1 / D .2/3=2 exp.i01 r 1 / ; (60.204) function i is written as gf .k01 / D ı.k01 01 / ; Z (60.205) imp 3=2 0 0 gi .k1 /˚.k1 ; k3 I r 1 ; r 3 /dk1 ; i D .2/ Tf i .k3 ; k3 / D gi .k1 /T13 .k; k / ; k1 D 1 P ; (60.206) (60.195) Z ˇ2 ˇ ˇ2 dif MAB 2 k30 ˇˇ 1 ik1 r 1 D gi .k1 /ˇ ˇf13 .k; k0 /ˇ : .r /e dr : (60.196) gi .k1 / D 0 i 1 1 M13 k3 .2/3=2 dkO 3 dk01 (60.207) Impulse approximation Z Z (3) Closure imp 0 dk1 dk01 gf .k01 /gi .k1 /T13 .k; k0 / Tf i .k3 ; k3 / D Z X
gf .k01 /gf .k001 / D ı.k01 k001 / ; (60.208) (60.197) ı P .k01 k1 / ; f
where T13 is the exact off-shell T -matrix for 13 scattering, ˝ ˛ (60.198) T13 .k; k0 / D exp.ik0 r/jV13 .r/j .k; r/ :
diT 0 dkO 3
D
Z ˇ2 ˇ ˇ2 kN30 MAB 2 ˇˇ gi .k1 /ˇ ˇf13 .k; k0 /ˇ dk1 ; k3 M13 (60.209)
The delta function in Eq. (60.197) ensures linear momentum, K D k1 C k3 D k01 C k03 , is conserved in 13 collisions, with where k01 D .M3 =M /.k1 C k3 / k03 . Conditions for validity of Eq. (60.209): (a) k3 is high (60.199a) k01 D k1 C .k3 k03 / k1 C P ; enough to excite all atomic bound and continuum states, and 02 2 M3 be approximated by k3 , or by .k1 C k3 / k03 D k C P : (60.199b) (b) k3 D .k3 2"f i =MAB / can k0 D M1 C M3 an averaged wavenumber kN30 D .k32 2N"f i =MAB /1=2 , where the averaged excitation energy is Elastic scattering Z Ti i .k3 ; k3 / D
gf .k1 /gi .k1 /T13 .k; k/dk1 ;
(60.200)
"Nf i D lnh"f i i D
X
fij ln "ij
X
j
!1
fij
where k D .M3 =M /k1 C .M1 =M /k3 , and M D M1 C M3 . and the fij are the oscillator strengths. Integral cross section: for 3–(1,2) scattering, Z ˇ ˇ MAB 2 k30 (4) Peaking approximation ˇhgf .k1 C P/ˇ if .k3 / D M13 k3 peak Tf i .k3 ; k03 / D Ff i .P/T13 .k; k0 / ; ˇ2 0 0 ˇ O f13 .k; k /gi .k1 /i dk3 ; (60.201) where Ff i is the inelastic form factor where MAB is the reduced mass of the 3–(1,2) system, and Z M13 the reduced mass of 1–3. The 13 scattering amplitude Ff i .P/ D gf .k1 C P/gi .k1 /dk1 f13 is given by D h f .r/ exp.iP r/j i .r/i : 1 2M13 0 f13 .k; k0 / D .k; k / : (60.202) T 13 4 „2 (5) T13 D T13 .P/
Six Approximations to Eq. (60.197)
Tf i .k3 ; k03 / D T13 .P/Ff i .P/ :
(1) Optical theorem 1 2MAB Ti i .k3 ; k03 / tot .k3 / D k3 „2 Z ˇ ˇ 1 ˇgi .k1 /ˇ2 v13 T .v13 / dk1 ; D 13 v3
;
(60.210)
j
(60.211)
(60.212) (60.213)
(60.214)
(6) T13 D constant f13 as D constant scattering length. 0
(60.203)
T where 13 is the total cross section for 13 scattering at relative speed v13 . The resultant cross section Eq. (60.203) is an upper limit and contains no interference terms.
kZ3 Ck3 ˇ ˇ 2as2 MAB 2 ˇFf i .P/ˇ2 P dP ; if .k3 / D 2 M13 k3 k3 k30
( tot .k3 / D
(60.215) 4as2 ; hv1 i4as2 =v3 ;
v3 v1 v3 v1 :
(60.216)
60
900
E. J. Mansky II
60.9.2 Classical Impulse Approximation
Validity Criteria
(A) Ionization For electron impact on heavy particles [70], the cross section for ionization of a particle moving with velocity t by a projectile with velocity s is
(A) Intuitive formulation [68]
Particle 3 scatters separately from 1 and 2, i.e., r12 A1;2 ; the relative separation of (1,2) the scattering Zs 2 2s dz A.z/ 1 lengths of 1 and 2. Q.s; t/ D 2 C B.z/ ; u m2 z2 z (ii) 13 r12 , i.e., the reduced wavelength for 13 relative 1 motion r12 . Interference effects of 1 and 2 can be ignored and 1, 2 treated as independent scattering centers. where (iii) 23 collisions do not contribute to inelastic 13 scat" tering. 1 3=2 1 3=2 x x A.z/ D 01 (iv) Momentum impulsively transferred to 1 during collision 2st 3 3 02 (time coll ) with 3 momentum transferred to 1 due to # V12 , i.e., 1=2 1=2 2 2 x02 x01 2 s Ct
(i)
P h
n` j
rV12 j
n` icoll
:
(60.217)
For a precise formulation of validity criteria based upon the two-potential formula, see the Appendix of [68].
(60.221)
2 2 s t 2 1=2 1=2 x ; (60.222a) x 02 01 2ts 3 h
2 1=2 1 1=2 2 .m B.z/ D C m / s t x x 1 2 02 01 2m2 st 3 i 1=2 1=2 : (60.222b) .m2 m1 / x02 x01
Two classes of interaction in A–B.n/ Rydberg collisions justify use of the impulse approximation for the T -matrix for For electron impact, Eq. (60.222b) is evaluated at m1 D 1. 13 collisions: (i) quasi-classical binding with Vcore D const:, The remaining terms above are given by and (ii) weak binding with E3 Ec h
n .r/jV1C .r/j
„2 h n .r 1 /j r2j 2M12 1
n .r 1 /i
n .r/i
;
(60.218a)
j"n j ;
(60.218b)
where E3 is the kinetic energy of relative motion of 3, and Ec is the energy shift in the core. The fractional error is [69] f13 Ec „ C delay 1 ; „ E3
(60.219)
s 2 D v22 =v02 ;
t 2 D v12 =v02 D E1 =u ;
E2 D m2 v22 ; u D v02 D Ionization potential of target ; x0i D .s 2 C t 2 2st cos i / ; i D 1; 2 ; 8 ˆ < 0 ˙ 1 ; j 0 ˙ 1 j 1 cos i D 1 ;
0 ˙ 1 > 1 ; ˆ : 1 ;
0 ˙ 1 < 1 r z z 1 m1 z 1C 2 1 2 :
0 ˙ 1 D 1 ˙ 2 m2 st t s
where k31 is the reduced wavelength of 3, f13 is the Equal mass case: (m1 D m2 ) scattering amplitude for 13 collisions, and delay is the time 8 4 1 2.s 2 1/3=2 ˆ delay associated with 13 collisions. ˆ ; 1 s2 t 2 C 1 ; < 2 2 3s u t Special case: for nonresonant scattering with delay D 0, Q.s; t/ D ˆ 4 1 3 ˆ 2 : C 3 2t ; s2 t 2 C 1 : 3s 2 u2 s2 t 2 f13 j"n j 1; (60.220) (60.223) E3 which follows from Eq. (60.219) upon identifying the shift Integrating over the speed distribution (Sect. 60.4), in the core energy Ec with the binding energy j"j. Z1 Condition Eq. (60.220) is less restrictive than Q.s; t/t 2 dt 32 1 ; (60.224) Q.s/ D Eq. (60.218a) or Eq. (60.218b) since f13 can be either less u2 .t 2 C 1/4 0 than or greater than .
60
Rydberg Collision Theories
901
which is then numerically evaluated. For electrons, the inte- where C denotes that the integration range Œ1; C1 is regral can be done analytically with the result stricted such p that the integrand is real and positive and that j1 "j < y 2 a2 . The dimensionless variables a and y 8 above are defined as Q.y/ D 3y 2 .y C 1/4 " 2 2
jV .R/jV 2 ; a2 D Ii V 2 ; (60.229) y2 D 5y 4 C 15y 3 3y 2 7y C 6 .y 1/1=2 me me
C 5y 5 C 17y 4 C 15y 3 25y 2 C 20y tan1 .y 1/1=2 p ˇ# ˇp ˇ y C y 1ˇ 3=2 ˇ ; 24y ln ˇˇ p p y y 1ˇ with y D s 2 . Thomson’s result QT .y/ D
4 1 1 : 1 y u2 y
(B) Electron-loss cross section [30]
and with Pi .r/=r representing the Hartree–Fock–Slater radial wave function for shell i, with normalization Zri ŒPi .r/2 dr D 1 :
(60.230)
0
The ionization potential and number of electrons in shell i are denoted above, respectively, by Ii and Ni . (60.225) (D) Total capture cross section [30] X total i capture .V / D capture :
(60.231)
i
A.V / C B.u/ ! AC C e C B.f / ;
(60.226)
(E) Universal capture cross section [30] A universal curve where B.f / denotes that the target B is left in any state (ei- independent of projectile mass M and charge Z is obtained ther bound or free) after the collision with the projectile A. from the above expressions for the capture cross section by The initial velocity of the projectile is V , while the veloc- plotting the scaled cross section ity of the Rydberg electron relative to the core is u, and the E 11=4 ionization potential of the target B is I . D total (60.232) e total capture M 11=4 Z 7=2 3=4 capture Z1 versus the scaled energy 1 dx T .x u/ N loss D 2 3 =4 e D me E ; E (60.233) M I 8x 1 . x/2 2 C where me is the mass of the particle captured, which is usuŒ1 C . x/2 3 ally taken to be a single electron, and I is the ionization ! 1 potential of the target. The term in Eq. (60.232) is the couC ; (60.227) 2 2 pling constant in the target potential, V .R/ D me =R2 , which Œ1 C . C x/ the electron being captured experiences during the collision. p where D V =u, N D I =.1=2/me uN 2 , uN D 2I =me , and T See [30, Fig. 11] for details. is the total electron scattering cross section at speed x u. N The cross section Eq. (60.227) is valid only for particles being stripped (or lost from the projectile), which are not strongly 60.9.3 Semiquantal Impulse Approximation bound. See [30, 71, 72] for details and numerous results. Basic expression [68, 73, 74] (C) Capture cross section from shell i [30] ˇ2 k102 k30 M3 2 ˇˇ d i D gi .k1 /ˇ capture .V / d"dP dk1 dkd1 J55 k3 M13 ˇ ˇ2 CZ.C1/ ri Z ˇf13 .k; k0 /ˇ : (60.234) 25=2 Ni 2 0 D dr d.cos /ŒP .r/ i 3V 7 J55 is the five-dimensional Jacobian of the transformation 0 C .1/ p
2 2
0 1 C y 2 4" " y 2 1 C "2 C a2 y 2 (60.235a) .P; "; k1 ; k; 1 / ! kO 3 ; k01 ; p ; 3 3=2 9=2 2 2 2 r " .1 C a / y a @.P; "; k1 ; k; 1 / : (60.235b) J55 D
(60.228) @ cos 30 30 ; k10 ; cos 10 ; 10
60
902
E. J. Mansky II
Expression for elemental cross section [68] .P; "; k1 ; k; 1 / representation, d"dP jgi .k1 /j2 k12 dk1 d1 d D 2 2 v1 M13 v3 jf13 .k; k0 /j2 dg 2 q
; 2 g 2 g 2 g2 gC
In the
2 where v10 ."/ D maxŒ0; .2"=M /, and the limits to the P integral are
P C D P C ."; v1 I v3 /
D min M.v10 C v1 /; MAB .v30 C v3 / ; (60.239a) (60.236)
p 2 D .1=2/B ˙ .1=4/B 2 C , and where g˙ B D B."; P; v1 I v3 /
a P2 2 3 2 02 2 02 D C v1 C v1 C v3 C v3 C 2 .1 C a/2 M13 M13 4"." C 3 / ; P2 C D C."; P; v1 I v3 /
v 2 C av32 P 2 D 1 C v12 v32 v102 v302 2 1 C a M13 4 3 2 3 2 C v1 C v32 C 2 v12 ." C 3 / "v32 ; M13 P M2 M3 aD ; M1 .M1 C M2 C M3 / f M 1 D M1 .1 C M1 =M2 / ; 2" 2." C 3 / ; v302 D v32 ; v102 D v12 C f MAB M1
P D P ."; v1 I v3 / ˇ ˇ ˇ
ˇ D max M ˇv10 v1 ˇ; MAB ˇv30 v3 ˇ ;
and unless P C > P , the P integral is zero. (ii) f13 D f13 .g; / is a function of relative speed and scattering angle. Then 1 .v3 / D 2 v3
d D
(m-averaged
and
"1
v10
Wn` .v1 / dv1 v1
ZgC g
g dg S.v1 ; gI v3 /
C
p
jf13 .g; /j2 d.cos / C
.cos
cos /.cos
cos
/
;
(60.240) where S.v1 ; gI v3 / D
1 -
jf13 .P; g/j2 dg 2 d"dP Wn` .v1 /dv1 q
; 2 2 2v1 M13 v3 g2 g2 g2 g2 C
Z1 d"
Z
(60.237) where the speed distribution Wn` is given by Eq. (60.73). Two representations for 1–3 scattering amplitude [68]
1=2
M13 : .1 C a/ v12 C av32 ag 2 1Ca (60.241)
Scattering angle cos
Hydrogenic systems gi .k1 / D gn` .k1 /Y`m .1 ; 1 / The gn` are the hydrogenic wave functions in momentum space. See Chap. 9 for details on hydrogenic wave functions. sections
Z"2
and 3 is the change in internal energy of particle 3, while " denotes the energy change in the target 12.
Elemental cross integrated)
(60.239b)
˙
-limits,
D cos ˙ ."; v1 ; gI v3 / (60.242) n
1 g 2 2 2 ˛.˛ C " Q / ˙ ˇ ! C ˇ ˛ D 0 2 g ˛ C ˇ2 1=2 o ; (60.243) .˛ C "/ Q2
where ˛ D ˛.v1 ; gI v3 / 1a 2 1 2 2 D M13 v1 v3 C g ; 2 1Ca ˇ D ˇ.v1 ; gI v3 / h
2 i1=2 1 ; D M13 2v12 C 2v32 g 2 g 2 v12 v32 2 a ! D g 0 =g ; "Q D " C 3 : 1Ca
(i) f13 D f13 .P; g/ is a function of momentum transferred and relative speed. Then Special case: f13 D f13 .P/ .v3 / D
Z"2
1 2 2 M13 v3
d" "1
ZgC q
g
Z1 v10
Wn` .v1 /dv1 v1
ZP C dP
.v3 / D
P
jf13 .P; g/j2 dg 2
; 2 g 2 g 2 g2 gC
Z"2
2 2 M13 v3
Z1 d"
"1
v10
Wn` .v1 /dv1 v1
Z
PC
(60.238)
jf13 .P /j2 dP :
P
(60.244)
60
Rydberg Collision Theories
60.10
903
Binary Encounter Approximation
Cross Section per Unit Momentum Transferred per Unit Steradian The basic assumption of the binary encounter approximation Differential relationships is that an excitation or ionization process is caused solely d' d2 ˛ d2 ˛ d' by the interaction of the incoming charged or neutral projec˛E;P D D D ˛P;' : (60.252) dP dE dP d' dE dE tile with the Rydberg electron bound to its parent ion. If, for example, the cross section depends only on the momentum transfer P to the Rydberg electron (as in the Born approxi- For distinguishable particles, mation), then the total cross section is obtained by integrating (60.253a) ˛P D 2v1 P;' D 2 ˛P;' ; P over the momentum distribution of the Rydberg electron. ˇ ˇ 2 The basic cross sections required are given in Sect. 60.10.1. 4Z12 Z22 e 4 P ˇˇ eiP ˇˇ (60.253b) ˛P;' D ˇ P2 ˇ ; For further details see [75] and references therein. v ˇ2 ˇ 8Z12 Z22 e 4 ˇˇ eiP ˇˇ ˛E;P D v1 E;P D p ˇ 2ˇ : (60.253c) v1 v2 X P 60.10.1 Differential Cross Sections Cross Section per Unit Momentum Transfer Let the masses, velocities, and charges of the particles be (m1 ; v1 ; Z1 ; e) and (m2 ; v2 ; Z2 ; e), with v D jv1 v2 j denoting the relative velocity, and quantities after the collision are denoted by primes. Then for distinguishable particles, P D
8Z12Z22 e 4 P v2
ˇ ˇ ˇ exp.iP / ˇ2 ˇ ˇ ˇ P2 ˇ ;
(60.245)
where the phase shift P is P D 2 ln.P =2v/ C 20 C ;
(60.246)
For identical particles, ˙ ˛P;'
˙ ˛E;P
ˇ ˇ2 4Z12 Z22 e 4 P ˇˇ eiP eiS ˇˇ D ˇ P 2 ˙ S2 ˇ ; v ˇ2 ˇ 8Z12 Z22 e 4 ˇˇ eiP eiS ˇˇ ˙ D v1 E;P D p ˇ ˙ 2ˇ ; S v1 v2 X P 2
(60.254b)
where
X D cos2 C 2 vO 1 PO vO 2 PO cos C 1
2
2 vO 1 PO vO 2 PO D .cos min cos /.cos cos max / 2 v .Emax E/.E Emin / ; D v1 v2 P
and with
(60.254a)
(60.255a) (60.255b)
(60.255c) m1 m2 Z1 Z2 e 2 .1 C i/ D ; D ; e2i0 D : m1 C m2 „v .1 i/ with being the angle between the velocity vectors v1 and (60.247) v . 2 For the special case of electron impact, M2 D me , Z2 D For identical particles, 1, and ˇ ˇ2 8Z12 Z22 e 4 P ˇˇ eiP eiS ˇˇ 8Z12 e 4 P˙ D ˙ ; (60.248) ˇ ˇ (60.256) ./ D p ; 2 2 2 E;P v P S v12 v2 P 4 X p ˙ where P is given by Eq. (60.246), and S is ./ D 8e 4 v12 v2 X E;P
! 1 2 cos P S 1 S D 2 ln.S=2v/ C 20 C ; (60.249) C 4 ; (60.257) P4 S P 2S 2 while 0 is given by Eq. (60.247). The momenta P and S
transferred by direct and exchange collisions, respectively, where P S D 2 ln.P =S/ D 2e 2 =„v ln.S=P /, and X is given by Eq. (60.255a). are given by P D m1 .v1 v01 / D m2 .v1 v02 / ; S D m1 .v1 v02 / D m2 .v01 v2 / :
(60.250a) Integrated Cross Sections (60.250b) For incident heavy particles Z
The collision rates (in cm3 =s) are ˛O P D v1 P ;
˛O P˙
D
E;P D v1 P˙
:
(60.251)
0
1 4Z 2 e 4 E;P ./ sin d D 2 1 4 : 2 v1 v2 P
(60.258)
60
904
E. J. Mansky II
60.10.2 Integral Cross Sections
For incident electrons Z ˙ qE;P
D
1 ˙ E;P ./
2
e .T / C A.E2 / ! e .E/ C AC C e ; sin d
(60.266)
(60.259a)
where T is the initial kinetic energy of the projectile electron, while the Rydberg electron, initially bound in potential Ui to v12 C v22 P 2 =2m2e 2E 2 =P 2 4e 4 1 the core AC , has kinetic energy E2 . The cross section per C D 2 ˇ 2 ˇ3 2 4 ˇ ˇ v1 v2 P 4 me v1 v2 2E=me unit energy E is denoted below by E . See the review by ! Vriens [75] for details. 2˚ ˇ ˙ 2 2ˇ 2 ; For electron impact, there are two collision models: the me P ˇv1 v22 2E=me ˇ unsymmetrical collision model of Thomson and Gryzinski (60.259b) assumes that the incident electron has zero potential energy, and the symmetrical collision model of Thomas and Burgess where ˚ can be approximated [76] by assumes that the incident electron is accelerated initially by ˇ! ˇ ˇ ˇ the target (and thereby gains kinetic energy) while losing an ˇ ˇ R1 ˇ1=2 ˇ E ˇ ; (60.260) equal amount of potential energy. ˇ ˇ ˇ ln ˚ cos ˇ ˇE E E ˇ E3 E2 ˇ 3 2 Unsymmetrical model (two cases) and E3 is defined in [76]. e 4 1 4E2 1 4E2 ˚ C C 2C ; E D T E2 3E 3 D 3D 3 ED Cross Sections per Unit Energy (60.267) For incident heavy particles (three cases) which is valid for D D T E2 E 0, or 2Z12 e 4 1 2me v22 C E D ; (60.261) e 4 1 4T 0 1 4T 0 ˚ 2 2 2 E 3E C C 2C E D me v1 T E2 3E 3 D 3jDj3 EjDj 0 1=2 which is valid for 2v1 v2 C v20 , E 2me v1 .v1 v2 /, or T ; (60.268) E 2 2 4 Z e 1 E D 2 1 3 4v13 .v20 v2 /2 ; (60.262) which is valid for D 0 and T E; and where T 0 T E. 2 3v1 v2 E Symmetrical model (two cases) which is valid for v20 v2 2v1 v20 Cv2 , 2me v1 .v1 v2 / e 4 1 4E2 1 4E2 ˚ E 2me v1 .v1 Cv2 /, or otherwise, E D 0 for E me v1 .v20 C E D C C 2C ; Ti E 2 3E 3 Xi 3Xi3 EXi v2 /. (60.269) For incident electrons (two cases) 0
which is valid for Xi T C Ui E 0, with Ti T C 2e 4 1 2me v22 1 2me v22 2˚ C C C ˙ ; Ui C E2 , and 3E 3 D2 3D 3 ED me v12 E 2 e 4 1 4Ti0 1 4Ti0 ˚ (60.263) C C C E D Ti E 2 3E 3 3jXi j3 EjXi j Xi2 which is valid for me .v2 v20 / me .v10 v1 /, me .v20 C v2 / 0 1=2 Ti me .v1 C v10 /, D 0, or ; (60.270) E2 2me v102 2e 4 1 1 which is valid for 0 Ti0 E2 , T 0, with Ti0 Ti E, E D C C 2 2 E2 3 3E D me v1 and where ˚ is given by Eq. (60.260). ! For incident heavy particles, the unsymmetrical model in v10 2me v102 2˚ C ˙ ; (60.264) Eq. (60.267) should be used. 3jDj3 EjDj v1 E˙ D
Single-Particle Ionization which is valid for me .v20 v2 / m.v1 v10 /, me .v1 C v10 / The total ionization cross section per atomic electron for in me .v20 Cv2 /, D 0. In the expressions above, the exchange cident heavy particles is energy D transferred during the collision is ! 2Z12e 4 1 me v22 1 1 1 1 1
; C Qi D D D me v12 me v202 D me v12 me v22 E : Ui me v12 3Ui2 2me v12 v22 2 2 2 2 (60.271) (60.265)
60
Rydberg Collision Theories
905
Ionization rate coefficients For heavy-particle imwhich is valid for Ui 2me v1 .v1 v2 /, or pact [70], 1 Z12 e 4 1 Qi D C Ui me v12 2me v2 .v1 C v2 / a02 128 3 3 D
b b 3=2 hQi h i 2
me
9 3=2 2v13 C v23 2Ui =me C v22 C ; 3v2 Ui2 1 58 8 C b 35 b b 2 (60.272) 3 3 3
3 2 which is valid for 2me v1 .v1 v2 / Ui 2me v1 .v1 C v2 /, C ab 5 4 2 3a2 C ab C b 2 3 2 or otherwise Qi D 0 for Ui 2me v1 .v1 C v2 /.
15 For electron impact, C 9a C 5b 16 a4 ln 4 2 1 2 1 dir ex int 35 5 Qi D Qi C Qi C Qi : (60.273) 2 2 3 C a C 3a C 4a C 8a ; 2 6 2 (60.281) In the unsymmetrical model, Qiex diverges, hence the exchange and interference terms above are omitted in the where unsymmetrical model for electrons to obtain Qidir
e 4 1 2E2 1 D C ; T Ui 3Ui2 T E2
(60.274)
D .4 /1 ; D C 2 tan1 ;
which is valid for T E2 C Ui , or Qidir
2e 4 .T Ui /3=2 D p ; 3T Ui2 E2
a D .1 C 2 /1 ; (60.275)
which is valid for Ui T E2 C Ui . In the symmetrical model, Qidir D Qiex
Qiint
e 4 1 1 E2 2 E2 D C ; Ti U i T 3 Ui2 T 2 2˚ 0 e 4 T D ln ; Ti T C U i U i
where ˚ 0 can be approximated by [76] " # 1=2 R1 E1 0 ˚ D cos ln ; E1 C Ui Ui and E1 is defined in [76]. The sum of Eqs. (60.276) and (60.277) yields e 4 1 1 2 E2 E2 C 2 Qi D Ti U i T 3 Ui T T ˚0 ln ; T C Ui Ui
D v1 =v0 ;
b D .1 C 2 /1 : ˇ ˇ !3 64e 4 v05 ˇˇ E P ˇˇ2 2 hE;P dP dEi D v dP dE ; 2me ˇ 0 3v12 P 4 ˇ P where .1=2/me v02 is the ionization energy of H.1s/.
(60.276)
Scaling laws Given the binary encounter cross section for ionization by protons of energy E1 of an atom with binding (60.277) energy ua , the cross section for ionization of an atom with different binding energy ub and scaled proton energy E10 can be determined to be [30]
(60.278)
ion E10 ; ub D
u2a ion .E1 ; ua / ; u2b
E10 D .ub =ua /E1 ;
(60.282) (60.283)
where ion .E; u/ is the ionization cross section for removal of a single electron from an atom with binding energy u by impact with a proton with initial energy E.
(60.279) Double ionization See [77] for binary encounter cross section formulae for the direct double ionization of two-electron which is also obtained by integrating the expression in atoms by electron impact. Eq. (60.270) for E , Excitation Excitation is generally less violent than ioniza1 2 .T Z CUi / tion, and hence binary encounter theory is less applicable. E dE : (60.280) Binary encounter theory can be applied to exchange excitaQi D tion transitions, e.g., e C He.n1 L/ ! e + He.n03 L/, with Ui
60
906
E. J. Mansky II
The limits E`;u to the E integration in each of the four the restriction of large incident electron velocities. The cross section is cases is indicated in the appropriate validity conditions. The constants a and b above are given by U ZnC1 Qeex D E;ex dE 4m1 m2 1 E aD E C v .m m / ; v 1 2 1 2 1 2 Un .m1 C m2 /2 2 " !# 4m1 m2 1 e 4 1 2 E2 E2 1 bD E1 E2 v1 v2 .m1 m2 / : D C ; 2 .m1 C m2 / 2 Ti TnC1 Tn 3 T2 T2 n
nC1
(60.284) valid for T UnC1 , with Tn T C Ui Un and TnC1 T C Ui C UnC1 , or
The expressions above for ion .v1 ; v2 / must be averaged over the speed distribution of v2 before comparison with experiment. See [29] for explicit formulae for the case of a delta function speed distribution.
ZT Qeex
D
E;ex dE Un
e Ti
4
1 1 2 E2 E2 C 2 2 Ui Tn 3 Ui Tn
60.10.4 Classical Charge Transfer Cross Section
(60.285) Applying the classical energy-change cross section result in Eq. (60.84) of Gerjuoy [28] to the case of charge-transfer yields the four cases [29] valid for Un T UnC1 ; Un and UnC1 denote the excitation energies for levels n and n C 1, respectively. E Z u .eff/ E .v1 ; v2 /dE CX .v1 ; v2 / D
;
60.10.3 Classical Ionization Cross Section
E`
D
6v2 =m2 e 4 2v23 ; .E/2 E 3v12 v2
Applying the classical energy-change cross section result in Eq. (60.84) of Gerjuoy [28] to the case of electron-impact ionization yields the four cases [29] which is valid for 0 < E < b, or Eu Z eff E .v1 ; v2 I m1 =m2 / d.E/ ion .v1 ; v2 /
CX .v1 ; v2 / D
E`
2 Z1 Z2 e 2 2v23 6v2 D ; .E/2 m2 E 3v12 v2
e 4 v1 =m1 v2 =m2 3 2 E 3v1 v2
03
! v2 v23 v103 C v13 C ; .E/2
(60.289)
(60.290)
(60.286) which is valid for b < E < a, or 2v102 e 4 CX .v1 ; v2 / D 2 ; .E/2 3v1 v2
which is valid for 0 < E < b, or ion .v1 ; v2 / D
2
Z1 Z2 e 2 3v12 v2 !
4 v1 2v10 4 v2 2v20
2 C 2
2 ; m21 v1 v10 m2 v2 v20 (60.287)
which is valid for b < E < a, or .Z1 Z2 e 2 /2 2v103 ion .v1 ; v2 / D ; .E/2 3v12 v2
(60.288)
which is valid for E > a, 2m2 v2 > jm1 m2 jv1 , or otherwise is zero for E > a, 2m2 v2 < jm1 m2 jv1 .
(60.291)
which is valid for E > a, me v2 > .m1 me /v1 , or otherwise CX D 0 when E > a, me v2 < .m1 me /v1 . The above expressions for CX .v1 ; v2 / must be averaged over the speed distribution W .v2 / before comparison with experiment. See [29] for details. The constants a and b above are as defined in Sect. 60.10.3, and the limits E`;u are given by 1 me v12 C UA UB ; 2 1 Eu D me v12 C UA C UB ; 2 p v2 D 2UA =me ; E` D
(60.292a) (60.292b) (60.292c)
60
Rydberg Collision Theories
907
where UA;B are the binding energies of atoms A and B. 60.11.2 Hydrogenic Form Factors The expressions above for CX diverge for some v1 > 0 if UA < UB . If UA D UB , then CX diverges at v1 D 0. Bound–bound transitions In terms of D t=Z, To avoid the divergence, employ Gerjuoy’s modification, 16 jF1s;1s j D
(60.297a) E` D .1=2/me v12 C UA . 2 ; 4 C 2
60.11 Born Approximation See the reviews in [78, 79], as well as any standard textbook on scattering theory, for background details on the Born approximation, and [80–85] for extensive tables of Born cross sections.
60.11.1 Form Factors
2 jF1s;2s j D 217=2
3 ; 4 2 C 9 3 jF1s;2p j D 215=2
3 ; 4 2 C 9
2 2 4 7=2 27 C 16 jF1s;3s j D 2 3 4 ;
9 2 C 16
2 11=2 3 27 C 16 jF1s;3p j D 2 3
4 ; 9 2 C 16
(60.297b) (60.297c) (60.297d) (60.297e)
2 (60.297f) jF1s;3d j D 217=2 37=2
The basic formulation of the first Born approximation 4 : 9 2 C 16 for high-energy heavy-particle scattering is discussed in Sect. 57.1. For the general atom–atom or ion–atom scatterBound–continuum transitions In terms of the scaled wave ing process vector D ka0 =Z for the ejected electron,
0 0 (60.293) A.i/ C B.i / ! A.f / C B.f / ; 28 2 1 C 3 2 C 2 exp.2= / 2 jF1s; j D 3 3
; 3 1 C . /2 1 C . C /2 1 e2= with nuclear charges ZA and ZB , respectively, let „K i and (60.298) „Kf be the initial and final momenta of the projectile A, and „q D „Kf „K i be the momentum transferred to the target. where D tan1 Œ2 =.1 C 2 2 /. Expressions for the Then Eq. (57.4) can be written in the generalized form bound–continuum form factors for the L-shell (2` ! ) and tC M-shell (3` ! ) transitions can be found in [86] and [87], Z ˇ2 dt ˇˇ 8a02 ˇ i 0f 0 A respectively. See also §4 of [88] for further details. if D ˇZA ıif Fif .t/ˇ s2 t3 t General Expressions and Trends ˇ ˇ2 ˇ ˇ ˇZB ıi 0 f 0 FiB0 f 0 .t/ˇ ; (60.294) For final ns states
n1 24 n .n 1/2 C n2 2 where the momentum transfer is t D qa0 , and s D v=vB is 2 jF1s;ns j D the initial relative velocity in units of vB . The form factors 2 Œ.n C 1/2 C n2 2 nC1 are sin2 .n tan1 x C tan1 y/ ; (60.299) N A X ˇ ˇ ˝ ˛ FifA .t/ D ˚fA ˇ exp.it r a =a0 / ˇ ˚iA ; (60.295) where kD1
2 xD ; where NA is the number of electrons associated with atom A 2 C 1 n2 / n. and similarly for FifB .t/. The limits of integration are t˙ D For final n` states [89] jKf ˙ Ki ja0 . For heavy-particle collisions, tC 1 and t D .Ki Kf /a0 Eif me Eif ' 1C ; 2s 4M s 2
yD
2 : 2 1 C n2
(60.300)
F1s;n` ./ (60.296)
where M D MA MB =.MA C MB /. Limiting cases As discussed in Sect. 57.1, for the case i D f , FifA .t/ ! NA as t ! 0, so that ZA FifA .t/ ! ZA NA . For the case i ¤ f , FifA .t/ ! 0 as t ! 0 and t ! 1.
p .n ` 1/Š 1=2 2` C 1.` C 1/Š D .i/ 2 .n C `/Š
2 2 2 .n`3/=2 .n 1/ C n n`C1 Œ.n C 1/2 C n2 2 .nC`C3/=2 h i .`C2/ .`C2/ .`C2/ an` Cn`1 .x/ bn` Cn`2 .x/ C cn` Cn`3 .x/ ; ` 3.`C1/
(60.301)
60
908
E. J. Mansky II
with coefficients an` , bn` and cn` given by
where
an` D .n C 1/ .n 1/2 C n2 2 ; p bn` D 2n Œ.n 1/2 C n2 2 Œ.n C 1/2 C n2 2 ;
cn` D .n 1/ .n C 1/2 C n2 2 ; and the argument n2 1 C n2 2
xDp : Œ.n C 1/2 C n2 2 Œ.n 1/2 C n2 2
X
jF1s;n` j2
`
1 2 n 1 C n2 2 3 n3
.n 1/2 C n2 2
For analytical expressions for A.n/, B.n/, and C.n/ for final np and nd states, see [90, 91]. General trends in hydrogenic form factors [92] The inelastic form factor jFn`!n0 `0 j oscillates with `0 on an increasing background until the value ! 1=2 2.n C 3/ 1 (60.307) `0max D min .n0 1/; n .n C 1/ 2
Summation over final ` states jF1s;n j2 D
28 n9 .n 1/2n6 ; 32 .n C 1/2nC6
29 n11 n2 C 11 .n 1/2n8 ; B.n/ D 32 5.n C 1/2nC8
28 n13 313n4 1654n2 2067 .n 1/2n10 : C.n/ D 32 52 7.n C 1/2nC10 A.n/ D
8 7 2
D2 n
(60.302)
Œ.n C 1/2 C n2 2 nC3
60.11.3 Excitation Cross Sections
which for large n becomes 2
jF1s;n j
28 2 3 2 C 1 3n3 . 2 C 1/6
For initial 2s and 2p states,
is reached, after which a rapid decline for ` > `0max occurs. See [92] for illustrative graphs.
exp
4 2 . C 1/
Atom–Atom Collisions [93, 94]
:
(60.303)
Single excitation For the process A.i/ C B ! A.f / C B ;
(60.308)
Eq. (60.294) reduces to 3 4 1 6 1 2 1 jF2s;n j D 2 n C n n C n Z1 ˇ2 3 2 16 48 8a02 dt ˇˇ A ˇˇ2 ˇˇ (60.309) Fif ZB FiB0 i 0 ˇ : if D 2 3 1 19 2 s t C n2 2 n2 C n4 t 3 3 48 5 7 Double excitation For the process 4 4 2 6 6 n Cn Cn 3 6 (60.310) H.1s/ C H.1s/ ! H.n`/ C H.n0 `0 / ; 2
1 2 2 n4 n 1 C n 1 2 Z (60.304)
2 nC4 ; ˇ2 ˇ ˇ 8a2 dt ˇˇ 1 1s;n0 l 0 ˇ ˇF1s;n0 `0 ˇ2 : n C 1 C n2 2 1s;n` D 20 (60.311) F 1s;n` 2 s t3 4 9 2 1 n 2 11 7 t jF2p;n j2 D n2 C n4 3 4 24 192 Special cases are [95] 23 2 1 4 4 2 2 5
n n C n 230 a02 880t4 C 396t2 C 81 6 24 4 1s;2s ; (60.312a) 1s;2s D
2
1 2 4t 2 C 9 11 2 2 n4 495s n 1 C n (60.305) 2 2 nC4 : 1 230 34 a02 1s;2p n C 1 C n2 2 2 (60.312b) 1s;2p D 11 ;
11s 2 4t2 C 9
Power series expansion 2 1 [7] 229 32 44t2 C 9 1s;2p (60.312c) 1s;2s D 11 ;
55s 2 4t2 C 9 2 4 6 8 jF1s;ns ./j D A.n/ C B.n/ C C.n/ C ;
(60.306) with t2 D 9= 16s 2 1 C 3me = 4M s 2 C .
2
4 7 2
60
Rydberg Collision Theories
909
Ion–atom collisions For the proton impact process HC C H.1s/ ! HC C H.n`/ ;
(60.313)
4E nE ln Jn
Eq. (60.294) reduces to
1s;n`
Validity criterion The Born approximation is valid provided that [98]
8a02 D s2
Z1
ˇ2 dt ˇˇ F1s;n` .t/ˇ ; 3 t
t
(60.322)
for transitions n ! n0 when n, n0 1 and jn n0 j 1. The (60.314) constant Jn is undetermined (see [98] for details) but is generally taken to be the ionization potential of level n.
with t D 1 n2 =.2s/.
60.11.4 Ionization Cross Sections Asymptotic Expansions e .k/ C H ! e .k 0 / C HC C e . / : (60.323)
2 2 2 4a0 n 1 jX1s!ns j 1s;ns D 24s 2 n2 The general expression for the Born differential ionization 1 n2 C 11 cross section can be evaluated in closed form using screened Cs .n/ 2 C hydrogenic wave functions. The differential cross section per s 20n2 s 4 4 2 incident electron scattered into solid angle d˝k 0 , integrated 313n 1654n 2067 ; (60.315) over directions for the ejected electron (treated as distinC 4 6 8400n s guishable) is [99–101] 10 2 7 2 a0 n .n 1/2n5 1s;np D 3s 2 .n C 1/2nC5 ˇ2 4k 0 ˇˇ Fn`; 0 ˇ .q/d˝k 0 d 0 ; I.; / d˝k 0 d 0 D 2 4 C 11 n 4 Q kq a0 ZB Cp .n/ C ln s 2 C 10n2 s 2 (60.324) 313n4 1654n2 2067 ; (60.316) where the form factor is given by Eq. (60.299) for the case C 5600n4 s 4
2
n` D 1s, with the ejected electron wavenumber and mo211 a02 n2 4 n5 n2 1 .n 1/2n7 mentum transferred q in the collision, 0 D a0 =ZQ B , q D 1s;nd D 2 2 2nC7 0 3 5s .n C 1/ .k k/a0 =ZQ B , being scaled by the screened nuclear charge 1 11n2 C 13 ZQ B appropriate to the n`-shell from which the electron is Cd .n/ 2 C ; (60.317) 2 4 ejected. The total Born ionization cross section per electron s 28n s is where Cs .2/ D 16=5, Cs .3/ D 117=32, Cs .4/ 3:386, and kCk Z max Z 0 B 1:3026 1:7433 16:918 D d 0 I.q; 0 / dq ; (60.325) ion C C ; (60.318) n Cp .n/ D n3 n5 n7 0 kk 0 2:0502 7:6125 C ; (60.319) n Cd .n/ D n3 n5 which is generally evaluated numerically. with n
28 n7 .n 1/.2n5/ : 3.n C 1/.2nC5/
(60.320)
60.11.5 Capture Cross Sections
Electron capture Further asymptotic expansion results can be found in [6–9]. A general expression for Born excitation and ionization (60.326) AC C B.n`/ ! A.n0 ` 0 / C B C : cross sections for hydrogenic systems in terms of a parabolic coordinate representation (Chap. 9) is given in [96, 97]. In the Oppenheimer–Brinkman–Kramers (OBK) approximation [102], Number of independent transitions Ni between levels n Z1 and n0 [96, 97] ˇ ˇ M 2 vf ˇFn`!n0 `0 ˇ2 ; 0 0 D d.cos / (60.327) n`;n ` h n i n 2„3 vi 2 0 C : (60.321) Ni D n n 1 3 3
60
910
E. J. Mansky II
where vi D vi nO i , vf D vf nOf , is the angle between nO i and Table 60.4 Coefficients C.ni `i ! nf `f / in the Born capture cross section formula in Eq. (60.330) nOf , M D MA MB =.MA C MB /, and “ jFn`;n0 `0 j D
ZA e 2 dr ds 'i .r/'f .s/ r
ei.˛rCˇs/ ;
(60.328)
with MA ˛ D kf nOf C ki nO i ; MA C me MB ˇ D ki nO i kf nOf ; MB C m e vf MB .MA C me / ki D ; „ .MA C MB C me / vf .MB C me /MA kf D : „ .MA C MB C me /
JS D
(60.329)
and the capture cross section is JS a02 C.ni `i ; nf `f / .ni `i ; nf `f / D p2 Z1 F .ni `i ; nf `f I x/ dx ;
28 ZA5 ZB5
2s
25 ZA5 ZB5
3s
28 ZA5 ZB5 =33
2p
25 ZA5 ZB7
3p
213 ZA5 ZB7 =36
3d
215 ZA5 ZB9 =39
4s
22 ZA5 ZB5
4p
5ZA5 ZB7
4d
ZA5 ZB9
4f
ZA5 ZB11 =20
Table 60.5 Functions F .ni `i ! nf `f I x/ in the Born capture cross section formula in Eq. (60.330)
1 32 56 127 C 2 C 4 192 p p tan1 12 p 32 60 83 C 2 C 4 48p p p
1 1 2 tan 2 p 16 32 C C 31 C ; 24p 2 p2 p4
C.1s ! nf `f /
1s
C.2s ! nf `f / D C.1s ! nf `f /=8 C.2p ! nf `f / D C.1s ! nf `f /=24
The Jackson–Schiff correction factor [103] is
nf `f
nf `f
F .ni `i ; nf `f I x/
1s 2s 2p 3s 3p 3d 4s 4p 4d 4f
x 6 .x 2b 2 /2 x 8 .x b 2 /x 8 Œx 2 .16=3/b 2 x C .16=3/b 4 2 x 10 .x b 2 /.x 2b 2 /2 x 10 .x b 2 /2 x 10 .x 2b 2 /2 .x 2 8b 2 x C 8b 4 /2 x 12 .x b 2 /Œx 2 .24=5/b 2 x C .24=5/b 4 2 x 12 .x b 2 /2 .x 2b 2 /2 x 12 .x b 2 /3 x 12
F .2s; nf `f I x/ D .x 2a2 /2 x 2 F .1s; nf `f I x/ F .2p; nf `f I x/ D .x a2 /x 2 F .1s; nf `f I x/
(60.330) References
x
with mvi ZB ZA ; bD ; ; aD „ ni nf .
x D p 2 C .a C b/2 p 2 C .a b/2 4p 2 :
pD
The coefficients C in Eq. (60.330) are given in Table 60.4, while the functions F are given in Table 60.5 [102]. In Table 60.5, the appropriate values of a and b are indicated by the quantum numbers ni , `i and nf , `f .
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E. J. Mansky II Edmund J. Mansky II Edmund J. Mansky II received his PhD from Georgia Institute of Technology in 1985. His theoretical work involves statistical mechanics of dense gases. In addition, his interests are three-body recombination processes and collisions involving Rydberg states of atoms and molecules at thermal and ultracold temperatures. His day job is programming heliophysics data for the scientific community.
Mass Transfer at High Energies: Thomas Peak Jack C. Straton
and James H. McGuire
Contents 61.1
The Classical Thomas Process . . . . . . . . . . . . . . 913
61.2 61.2.1 61.2.2 61.2.3 61.2.4
Quantum Description . . . . . . . . . . . . . . . Uncertainty Effects . . . . . . . . . . . . . . . . . . Conservation of Overall Energy and Momentum Conservation of Intermediate Energy . . . . . . . Example: Proton–Helium Scattering . . . . . . .
61.3
Off-Energy-Shell Effects . . . . . . . . . . . . . . . . . . 916
61.4
Dispersion Relations . . . . . . . . . . . . . . . . . . . . 916
61.5
Destructive Interference of Amplitudes . . . . . . . . 916
61.6
Recent Developments . . . . . . . . . . . . . . . . . . . . 917
. . . . .
. . . . .
. . . . .
. . . . .
914 914 914 914 915
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917
Abstract
Thomas peaks correspond to singular second-order quantum effects whose location may be determined by classical two-step kinematics. The widths of these peaks (or ridges) may be estimated using the uncertainty principle. A second-order quantum calculation is required to obtain the magnitude of these peaks. Thomas peaks and ridges have been observed in various reactions in atomic and molecular collisions involving mass transfer and also ionization. Transfer of mass is a quasi-forbidden process. Simple transfer of a stationary mass M2 to a moving mass M1 is forbidden by conservation of energy and momentum. If M1 < M2 , then M1 rebounds; if M1 D M2 , then M1 stops, and M2 continues on; and if M1 > M2 , then M2 leaves faster than M1 . In none of these cases do M1 and M2 leave J. C. Straton () Dept. of Physics, Portland State University Portland, OR, USA e-mail: [email protected] J. H. McGuire Dept. of Physics, Tulane University New Orleans, LA, USA e-mail: [email protected]
61
together. Thomas [1] understood this in 1927 and further realized that transfer of mass occurs only when a third mass is present, and all three masses interact. The simplest allowable process is a two-step process now called a Thomas process [2, 3]. Quantum mechanically [4], the second Born term at high energies is the largest Born term and corresponds to the simplest allowed classical process, namely the Thomas process. While the classically forbidden first Born term is not zero (saved by the uncertainty principle), the first Born cross section varies at high v as v 12 , in contrast to the second Born cross section, which varies as v 11 , thus dominating. Higher Born terms correspond to multistep processes that are unlikely in fast collisions where there is not enough time for complicated processes. The higher Born terms (n > 2) are also smaller than the second Born term. Keywords
dispersion relation intermediate state uncertainty principle transfer ionization destructive interference
61.1 The Classical Thomas Process The basic Thomas process is shown in Fig. 61.1. Here, the entire collision is coplanar since particles 1 and 2 go off together (that is what is meant by mass transfer). We assume that all the masses and the incident velocities v are known. Thus, there are six unknowns, v0 , vf , and v3 , each with two components. Conservation of momentum gives two equations of constraint for each collision. Conservation of overall energy gives a fifth constraint, and conservation of energy in the intermediate state (which only holds classically) gives a sixth constraint. With six equations of constraint, all six unknowns may be completely determined. The allowed values of v0 , vf , and v3 depend on the masses M1 , M2 , and M3 . For example, in the case of the transfer of an electron from
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_61
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1 + (2, 3)
61.2 Quantum Description
(1, 2) + 3 Zf
Mf v f M1 v
61.2.1 Uncertainty Effects α
Z1
Zf mf vf
mf v'
α
Zf γ M3 v3 Z3
In quantum mechanics, energy conservation in the intermediate states may be violated within the limits of the uncertainty principle, E „=t, where t is the uncertainty in time of mass transfer. It is not possible to determine if mass transfer actually occurs at the beginning, in the middle, or at the end of the collision. Thus, we choose t D r= N v, N where rN is the size of the collision region, and vN is the mean collision velocity. Taking rN a0 =Ztarget and vN v, the projectile velocity, we have
Fig. 61.1 Diagram for mass transfer via a Thomas two-step process C
C
atomic hydrogen to a proton, i.e., p C H ! H C p , it is easily verified that the angles are ˛ D .M2 =M1 / sin 60ı , ˇ D 60ı and D 120ı , where m0 D M2 D mf , and M1 D M3 D Mp D 1836 me . For the case eC C H ! Ps C pC , it may be verified that ˛ D 45ı , ˇ D 45ı and D 90ı . In general, the intermediate mass m0 may be equal to M1 ; M2 , or M3 . We shall regard these as different Thomas processes and label them B, A, and C, respectively. The standard Thomas process (the one actually considered by Thomas in 1927) is case A and corresponds to the first example given above. Lieber diagrams for the Thomas processes A, B, and C [5] are illustrated in Fig. 61.2. In the Lieber diagram, mass regions in which solutions exist for processes A, B, and C are shown. (An equivalent diagram was given earlier by Detmann and Liebfried [3].) There are some regions in which two-step processes are forbidden. In these regions, the theory of mass transfer is not fully understood at present.
A
1
Two-step forbidden
2
M1
M2
M3
M1
C only
Here, a0 is the Bohr radius, and Ztarget is the nuclear charge of the target in units of the electron charge. Within this range of energy E, the constraint of energy conservation in the intermediate state does not apply.
61.2.2 Conservation of Overall Energy and Momentum Conservation of overall energy and momentum then gives three equations of constraint on the four unknowns vf and v3 [6], namely,
M1 v D Mf C MQf vf C M3 v3 ;
M1 v 2 D Mf C MQf vf2 C M3 v32 ;
2v3 vO D 2 cos D
M3 C
(61.1)
(61.2) (61.3)
M2
B A and C allowed
„Ztarget „ „vN „v D : D D t rN a0 =Ztarget a0
where Mf MQ f is the mass of the upper (lower) particle in the final state of the bound system shown in Fig. 61.1, in which m0 is the mass of the intermediate particle M1 ; M2 or M3 . From Eqs. (61.2) and (61.3) it may be shown that the velocity of the recoil particle is constrained by the condition
M1 /M2
3
E
M1
M3
M1 C M2 C M3 v3 M2 v : (61.4) M1 v M3 v3
Thus, the magnitude and the direction of v3 are not independent. Specifying either v3 or vO 3 is sufficient, together with the equations of constraint, to determine the energies and directions of all particles in the final state. Similarly, one may express the equations of constraint in terms of vf and vOf .
M2
A and B allowed B only
61.2.3 Conservation of Intermediate Energy
Two-step forbidden 1
2
Fig. 61.2 Lieber diagram for mass transfer
3
M3 / M2
In a classical two-step process, the projectile hits a particle in the target, and the intermediate mass m0 then propagates and subsequently undergoes a second collision. Quantum
61
Mass Transfer at High Energies: Thomas Peak 10–23 d σ (arb. unit) dv3
p + He
915
61.2.4 Example: Proton–Helium Scattering
H + He+++ e
2
The effect of the constraints of conservation of overall energy and momentum may be seen in Fig. 61.3, where a sharp ridge is evident in the reaction
10–28
10–33
pC C He ! H C HeCC C e ;
Rec oil spe ed, v
3
v–10v0
v
(61.8)
and M1 D Mp , M2 D M3 D me . Here, v3 is the speed of the recoiling ionized target electron, and the target nucleus is not directly involved in the reaction. The width of the sharp ridge v+ 10v0 is due to the momentum spread of the electrons in helium and may be regarded as being caused by the uncertainty princiπ 3π π ple since this momentum (or velocity) spread corresponds to 4 2 2 Recoil angle, γ p D „=r, where r is taken as the radius of the helium atom. The locus of the sharp ridge in Fig. 61.3, corresponding to Fig. 61.3 Counting rate (or cross section) on the vertical axis versus recoil speed v3 and recoil angle for a Thomas process in which a pro- conservation of overall energy and momentum, is given by ton (projectile) picks up an electron from helium (target). The captured Eq. (61.8). The locus of the weaker ridge, corresponding to electron bounces off the second target electron the conservation of energy in the intermediate state, is given mechanically, this corresponds to a second Born term rep- by Eq. (61.7). The intersection of these two loci gives the resented by V1 G0 V2 , where V represents an interaction and unique classical result suggested by Thomas. The width of these ridges may be estimated from the uncertainty principle G0 is the propagator of the intermediate state, namely, as described above. 1 Experimental evidence for the double ridge structure has G0 D .E H0 C i/ been reported by Palinkas et al. [7], corresponding to the 1 D iı.E H0 / C } ; (61.5) calculations given in Fig. 61.3, but at a collision energy of E H0 1 MeV, as shown below. The data in Fig. 61.4 corresponds to a slice across the where } is the Cauchy principal value of G0 , which excludes sharp ridge of Fig. 61.3 at v D v3 . The solid line is a second the singularity at E D H0 . This singularity corresponds to conservation of energy in the intermediate state. It is this d2σ/dEd⍀ (cm2/sr eV) singularity that gives rise to the weaker secondary ridge in Fig. 61.3 at v3 D v. The width of the secondary ridge is given approximately by E D „=t, as discussed above. The intersection of the ridges is the Thomas peak. At very 10–25 high collision velocities, the Thomas peak dominates the total cross section for mass transfer. The constraint imposed by conservation of intermediate energy may be expressed by replacing the speed of the recoil 10–26 particle v3 by the scaled variable K D M3 v3 =m0v. Then it may be shown that the conservation of intermediate energy may be expressed in the form [6]:
M3 v3 m0 v
2 K2 D 1 :
(61.6)
The constraints of conservation of overall energy and momentum, i.e., Eq. (61.4), may be easily written in terms of K as M2 1 m0 K 0 ; (61.7) 2 cos D r M2 m K where r D .M1 C M2 C M3 /M2 =.M1 M3 /.
10–27
10–28 60
90
120 Emission angle (deg)
Fig. 61.4 Observation of a slice of the Thomas ridge structure in pC C He ! H C HeCC C e at 1 MeV by Palinkas et al. [7]
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sequently, it obeys the dispersion relation
Scattering amplitude (10–6 a. u.) p+H 2.0
H+p
50 MeV
1 ReŒf2 ./ D }
Thomas peak
ZC1
ImŒf2 .0 / 0 d ; 0
1
1.0
ReT2 (off-shell)
ImT2 (on-shell)
ImŒf2 ./ D
1 }
ZC1
ReŒf2 .0 / 0 d ; 0
(61.9)
1
0.0
where Re f2 and Im f2 denote the real and imaginary parts of f2 . Thus, the energy-nonconserving part of f2 is related to an integral over the energy-conserving part and vice versa. –1.0 In the case of the dielectric constant it is well known that the 1 2 3 4 5 λ = (4M sin ϑ–2 )2 real and imaginary parts of are also related by a dispersion relation, namely the Kramers–Kronig relation [12]. Resonances are usually a function of energy E. The width Fig. 61.5 Energy-conserving (on-shell) and energy-nonconserving contributions to the second Born scattering amplitude of a resonance gives the lifetime of the resonance. Classically, is how long the projectile orbits the target before it leaves, corresponding to a delay or shift in time of the proBorn calculation [8, 9] at 1 MeV. The bump of data above jectile during the interaction. If the width of the resonance is a smooth background is the indication of the ridge structure. E, then the lifetime is D „=E; E and are conjugate variables. The Thomas peak is an overdamped resonance in scattering angle, corresponding to a shift in the impact parameter of the scattering event [13]. However, unlike energy 61.3 Off-Energy-Shell Effects resonances, our Thomas resonance in the scattering angle seems to have no classical analog [14]. In Eq. (61.6), the Green’s function G0 contains an energyconserving term iı.E H0 /, which is imaginary, and a real energy-nonconserving term }Œ1=.E H0 /. The latter 61.5 Destructive Interference of Amplitudes does not occur classically; it is permitted by the uncertainty principle and represents the contribution of virtual (off-theIt has already been noted that the location of the Thomas energy-shell or energy-nonconserving) states within ˙E D peaks depends on the mass of the collision partners. For the „=t about the classical value E D H0 . This quantum term process also represents the effect of time-ordering in the second Born (61.10) pC C Atom ! Atom C H ; amplitude [10]. In plane wave second-Born calculations, the off-energy-shell term gives the real part of the scattering am- there are two separate Thomas peaks [2, 15] corresponding plitude f2 , while the on-shell (energy conserving) term gives to cases A and B in the Lieber diagram (Fig. 61.2). Experithe imaginary part of f2 . These two contributions are shown mental evidence exists for both peaks. The standard Thomas in Fig. 61.5. peak occurs at small forward angles [11], while the second Half of the Thomas peak comes from energy- peak [16] occurs at about 60ı . If the mass of the projectile is nonconserving contributions that are not included in reduced, the positions of these Thomas peaks move toward a classical description. Also, the energy-nonconserving con- one another [17], as illustrated in Fig. 61.6. tribution plays a significant role in determining the shape of When M1 D M2 , then both Thomas peaks occur at 45ı . the standard Thomas peak, which has been observed [11]. This occurs in positronium formation where M D M D m , 1
2
e
i.e., eC C He ! Ps C HeCC C e :
61.4 Dispersion Relations
(61.11)
In cases A and B of Fig. 61.2, the two V1 G0 V2 second Born terms are of opposite sign because V2 is of opposite sign in Because of the form of the Green’s function of Eq. (61.6), diagrams A and B. This leads to destructive interference for the second Born contribution f2 to the scattering amplitude 1s 1s electron capture (which is dominant at high velocihas a single pole in the lower half of the complex plane. Con- ties), as was first discussed by Shakeshaft and Wadehra [17].
61
Mass Transfer at High Energies: Thomas Peak
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References
dσ d⍀
Thomas peak
1. 2. 3. 4.
p+
5. 6.
60 Peak 60
ϑT
ϑ
dσ d⍀
7. 8.
e+
9. 10. 11.
Destructive interference
12.
45
ϑ
Fig. 61.6 Change of position and nature of the Thomas peaks with decreasing projectile mass
Consequently, the observed Thomas peak structure is expected [18, 19] to be quite different for eC impact than for impact of pC or other projectiles heavier than an electron. The double ridge structure for transfer ionization of helium by eC is expected to differ significantlyfrom the structure shown in Fig. 61.3. Understanding such destructive interference between resonant amplitudes may give deeper insight into the physical nature of the intermediate states in this special few-body collision system.
61.6 Recent Developments In the late 1990s, observations [20] of the Thomas peak in the case of transfer ionization (where one electron is ionized, and another is transferred) differential in the momentum of the ejected electron provided new specific detail on the kinematics of the two-step process [21]. In 2001, the Thomas peak was discussed [22] in the context of quantum time ordering. In this case, time ordering, surprisingly, is not significant at the center of the peak, in contradiction to the classical picture that there is a definite order in the two-step process for transfer ionization. However, time ordering does contribute to the shape of the Thomas peak. In 2002, the Stockholm group [23] reported that at very high velocities, the ratio of transfer ionization to total transfer approaches the same asymptotic limit as in double to single ionization in (non-Compton) photoionization, namely 1:66%. This was interpreted in terms of a common shake process occurring when the wavefunction collapses after a sudden collision.
13. 14. 15. 16. 17. 18. 19. 20.
21. 22.
23.
Thomas, L.H.: Proc. Soc. A 114, 561 (1927) Shakeshaft, R., Spruch, L.: Rev. Mod. Phys. 51, 369 (1979) Detmann, K., Liebfried, G.: Z. Phys. 218, 1 (1968) Simony, P.R.: A second order calculation for charge transfer. Ph.D. Thesis. Kansas State University, Manhattan, KS, USA (1981) Lieber, M.: private communication (1987) McGuire, J.H., Straton, J.C., Ishihara, T.: In: Pindzola, M.S., Boyle, J.J. (eds.) The Application of Many-Body Theory to Atomic Physics. Cambridge University Press, Cambridge (1994) Palinkas, J., Schuch, R., Cederquist, H., Gustafsson, O.: Phys. Rev. Lett. 22, 2464 (1989) McGuire, J.H., Straton, J.C., Axmann, W.C., Ishihara, T., Horsdal, E.: Phys. Rev. Lett. 62, 2933 (1989) Briggs, J.S., Taulbjerg, K.: J. Phys. B 12, 2565 (1979) McGuire, J.H.: Adv. At. Mol. Opt. Phys. 29, 217 (1991) Horsdal-Pederson, E., Cocke, C.L., Stöckli, M.: Phys. Rev. Lett. 57, 2256 (1986) Jackson, J.D.: Classical Electrodynamics. Wiley, New York, p 286 (1975) Weaver, O.L., McGuire, J.H.: Phys. Rev. A 32, 1435 (1985) McGuire, J.H., Weaver, O.L.: J. Phys. B 17, L583 (1984) McGuire, J.H.: Indian J. Phys. 62B, 261 (1988) Horsdal-Pederson, E., Loftager, P., Rasmussen, J.L.: J. Phys. B 15, 7461 (1982) Shakeshaft, R., Wadehra, J.: Phys. Rev. A 22, 968 (1980) Igarashi, A., Toshima, N.: Phys. Rev. A 46, R1159 (1992) McGuire, J.H., Sil, N.C., Deb, N.C.: Phys. Rev. A 34, 685 (1986) Mergel, V., Dörner, R., Achler, M., Khayyat, K., Lencinas, S., Euler, J., Jagutski, O., Nüttgens, S., Unverzagt, M., Spielberger, L., Ru, W., Ali, R., Ullrich, J., Cederquist, H., Salin, A., Wood, C.J., Olson, R.E., Belkic, D., Cocke, C.L., Schmidt-Böcking, H.: Phys. Rev. Lett. 79, 387 (1997) Tolmanov, S.G., McGuire, J.H.: Phys. Rev. A 62, 032771 (2000) Godunov, A.L., McGuire, J.H., Ivanov, P.B., Shipakov, V.A., Merabet, H., Bruch, R., Hanni, J., Shakov, K.: J. Phys. B 34, 5055 (2001) Schmidt, H.T., Cederquist, H., Fardi, A., Schuch, R., Zettergren, H., Bagge, L., Kallberg, A., Jensen, J., Resfelt, K.G., Mergel, V., Schmidt, L., Schmidt-Boecking, H., Cocke, C.L.: In: Burgdoerfer, J., Cohen, J.S., Datz, S., Vane, C.R. (eds.) Photonic, Electronic and Atomic Collisions, p. 720. Rinton, Princeton, NJ (2002) Jack C. Straton Dr Straton is an Associate Professor at Portland State University, where his teaching focuses on astronomy, diversity, art, and social responsibility. His research ranges from antiracist pedagogy to nanometrology to quantum scattering theory, with a present focus on calculating the effects of laser stimulation to the attachment of a second positron to the antihydrogen atom. James H. McGuire Dr McGuire is Murchison Mallory Professor Emeritus and past Department Chair at Tulane University. He is a past Chair of the Division of Atomic, Molecular, and Optical Physics (DAMOP) of the American Physical Society. His research interests are in electron correlation dynamics, entanglement, complexity and correlation, and quantum time.
61
Classical Trajectory and Monte Carlo Techniques Marcelo Ciappina Ronald E. Olson
, Raul O. Barrachina
Contents 62.1 62.1.1 62.1.2 62.1.3
Theoretical Background . . . . . . . . Hydrogenic Targets . . . . . . . . . . . . Nonhydrogenic One-Electron Models Multiply Charged Projectiles and Many-Electron Targets . . . . . . .
. . . . . . . . . . 919 . . . . . . . . . . 919 . . . . . . . . . . 920 . . . . . . . . . . 920
62.2
Region of Validity . . . . . . . . . . . . . . . . . . . . . . 921
62.3 62.3.1 62.3.2 62.3.3 62.3.4 62.3.5 62.3.6 62.3.7
Applications . . . . . . . . . . . . Hydrogenic Atom Targets . . . . Pseudo-One-Electron Targets . . State-Selective Electron Capture Exotic Projectiles . . . . . . . . . Heavy-Particle Dynamics . . . . Ion-Molecule Collisions . . . . . Strong Laser Field Ionization . .
62.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 925
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921 921 922 922 923 923 923 924
, Francisco Navarrete
62
, and
tively applied to a large number of chemical systems to determine reaction rates and final state vibrational and rotational populations (e.g., Karplus et al. [2]). For atomic physics problems, a major step was introduced by Abrines and Percival [3], who employed Kepler’s equations and the Bohr–Sommerfield model for atomic hydrogen to investigate electron capture and ionization for intermediate velocity collisions of HC C H. An excellent description is given by Percival and Richards [4]. The CTMC method has a wide range of applicability to strongly coupled systems, such as collisions by multiply charged ions [5]. In such systems, perturbation methods fail, and basis set limitations of coupled-channel molecular-orbital and atomic-orbital techniques have difficulty in representing the multitude of active excitation, electron capture, and ionization channels.
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925
Keywords Abstract
The classical trajectory Monte Carlo (CTMC) method originated with Hirschfelder, who studied the H C D2 exchange reaction using a mechanical calculator [1]. With the availability of computers, the CTMC method was acM. Ciappina () Physics Program, Guangdong Technion – Israel Institute of Technology Shantou, Guangdong, China e-mail: [email protected] R. O. Barrachina Bariloche Atomic Centre, National Atomic Energy Commission Bariloche, Argentina e-mail: [email protected] F. Navarrete Institute of Physics, University of Rostock Rostock, Germany e-mail: [email protected] R. E. Olson Physics Dept., University of Missouri-Rolla Rolla, MO, USA e-mail: [email protected]
electron capture differential cross section target nucleus angular scattering classical trajectory Monte Carlo
62.1
Theoretical Background
62.1.1 Hydrogenic Targets Let us consider a simple three-body collision system comprised of a fully-stripped projectile (a), a bare target nucleus (b), and an active electron (c), where each pair of particles of charge Zi and Zj , and positions ˇ ˇ ri and rj , interact via a Coulomb potential Zi Zj =ˇri rj ˇ. The classical Hamiltonian for the system is H D
X jpi j2 X Zi Zj ˇ ˇ; C ˇri rj ˇ 2mi
i Da; b; c
(62.1)
i ¤j
where pi and mi are the momenta and masses of the particles, respectively.
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_62
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case. However, the boundary conditions for the long and short-range interactions are poorly satisfied. To improve the electronic representation of the target, model potentials derived from quantum mechanical calculations are now routinely used. Here, the simple solution of Kepler’s equation cannot be applied. However, Peach et al. [7] and Reinhold and Falcón [8] have provided the appropriate methods that yield a target representation that is correct under the microcanonical distribution. The method of Reinhold and Falcón is popular because of its ease of use (62.2) and generalizability. For the effective interaction potential, dri =dt D @H=@pi dpi =dt D @H=@ri : (62.3) Garvey et al. [9] have performed a large set of Hartree–Fock calculations and have parametrized their results by a potenA fourth-order Runge–Kutta integration method is suitable tial V .R/, which depends on the distance to the target R, as to solve them because of its ease of use and its ability to vary V .R/ D ŒZ NS.R/=R ; (62.6) the time step size. This latter requirement is essential since the time step can typically vary by three orders of magnitude where Z and N denote the nuclear charge and number of during a single trajectory. In essence, the CTMC method is a computer experiment. nonactive electrons in the target core, respectively, and S.R/ Total cross sections for a particular process are determined is the screening of the core given by by (62.7) S.R/ D 1 f.=/Œexp.R/ 1 C 1g1=2 ; 2 (62.4) R D .NR =N /bmax ; where and are screening parameters. Screening paramewhere N is the total number of trajectories run within a given ters are given in [9] for all ions and atoms with Z 54. This maximum impact parameter bmax , and NR is the number potential can also be used for the representation of partially of positive tests for a reaction, such as electron capture or stripped projectile ions. Even though it seems natural to think ionization. Angle and energy differential cross sections are that the inclusion of more active electrons in the description easily generalized from the above. As in an experiment, the of the problem would give rise to more accurate cross seccross section given by Eq. (62.4) has a standard deviation of tions, it is worth noticing that it could also lead to unphysical autoionization, as was pointed out in [10], where a possible (62.5) remedy to this problem is shown as well. R D R Œ.N NR /=N NR 1=2 ; The initial state of the target is completely determined by five random numbers given by Kepler’s equation that constrain the plane and eccentricity of the electron’s orbit, and another one is used to determine the impact parameter within the range of interaction [4, 5]. From Eq. (62.1), one obtains a set of 18 coupled (this number can be reduced by energy-momentum conservation considerations) first-order differential equations that determine the time evolution of the Cartesian coordinates and momenta of each particle,
1=2
which for large N is proportional to 1=NR . Here lies one of the major difficulties associated with the CTMC method: 62.1.3 Multiply Charged Projectiles and Many-Electron Targets it takes considerable computation time to determine minor or highly differential cross sections because, to decrease the statistical error of a cross by a factor of 2, four times as many Multiple ionization and electron capture mechanisms in energetic collisions between multiply charged ions and manytrajectories must be evaluated. electron atoms is poorly understood because major approximations must be made to solve the many-electron problem 62.1.2 Nonhydrogenic One-Electron Models associated with transitions between two centers. For a representative collision system such as For many-electron target atoms, it is sometimes adequate to (62.8) treat the problem within a one-electron model and employ AqC C B ! A.qj /C C B i C C .i j /e ; the independent electron approximation to approximate the atomic shell structure [6]. For an accurate calculation, it is it is essential to use a theoretical method able to predict sinecessary to use an interaction potential that simulates the multaneously the various charge states of the projectile and screening of the target nucleus by the passive electrons, i.e., recoil ions, and also the energy and angular spectra of the the electrons that do not participate in the reaction. One can ejected electrons. Theoretical methods based on the indesimply apply a Coulomb potential with an effective charge pendent electron model fail because the varying ionization Zeff obtained from, for example, Slater’s rules. Then, the energies of the electrons are not well represented by concomputational procedure is the same as for the hydrogenic stant values, especially for outer shells. At present, the most
62
Classical Trajectory and Monte Carlo Techniques
successful implementation is the nCTMC method, which is a direct extension of the hydrogenic CTMC method to an n-electron system and has been able to make reasonable predictions of the cross sections and scattering dynamics of such strongly coupled systems [11]. As such, the number of coupled equations rises to 6n C 12, where n is the number of electrons included in the calculation. In the nCTMC technique, all interactions of the projectile and target nuclei with each other and the electrons are implicitly included in the calculations. The inclusion of all the particles then allows a direct determination of their angular scattering, along with an estimate of the energy deposition to electrons and heavy particles. Postcollision interactions are included between projectile and recoil ions with the electrons; however, electron–electron interactions are introduced only in the bound initial state via a screening factor in a central-field approximation. This theoretical model has been very successful in predicting the single and double differential cross sections for the ionized electron spectra. Moreover, since a fixed target nucleus approximation is not used, this method has been the only one available to help understand and predict the results in the field of recoil-ion momentum spectroscopy [12].
62.2
921
Zp/Zt 100 CTMC CDW 10 MO
BORN
1 AO 0.1 0.1
1
10
100 v/ve
Fig. 62.1 Approximate regions of validity of various theoretical methods; ZP =ZT is the ratio of the projectile charge to the target charge, and v=ve is the ratio of the collision velocity to the velocity of the active target electron. Theoretical methods: molecular orbital (MO), atomic orbital (AO), classical trajectory Monte Carlo (CTMC), continuum distorted wave (CDW), and first-order perturbation theory (BORN)
and is thus restricted to strongly coupled systems. The continuum distorted wave (CDW) method (Chap. 56) greatly extends the region of applicability of first-order perturbation methods and has demonstrated validity in high-charge state ionization collisions.
Region of Validity
The CTMC method has a demonstrated region of applicability for ion–atom collisions in the intermediate velocity regime, particularly in the elucidation of both heavy-particle and electron collision dynamics. The method can be termed a semiclassical method in that the initial conditions for the electron orbits are determined by quantum mechanically based interaction potentials with the parent nucleus. Since the method is most suitable for strongly coupled systems, it has been applied successfully to a variety of intermediate energy multiply charged ion collisions. Figure 62.1 describes pictorially and the regions of validity of various theoretical models. Both the atomic orbital (AO) and molecular orbital (MO) basis set expansion methods (Chap. 53) work well until ionization strongly mediates the collision, since the theoretical description of the ionization continuum is not well founded and relies on pseudo states to span all ejected electron energies and angles. We have arbitrarily limited these methods to a projectile charge to target charge ratio of, ZP =ZT , ' 8, since above this value the number of terms in the basis set becomes prohibitively large. The CTMC method does not include molecular effects, and thus it is restricted to low velocities, except in the case of highcharge-state projectiles that capture electrons into high-lying Rydberg states, which are well described classically. Likewise, at high velocities, the method is inapplicable in the perturbation regime, where quantum tunneling is important,
62.3
Applications
62.3.1 Hydrogenic Atom Targets The original application of the CTMC method to atomic physics collisions was done on the HC C H system [3]. Here, the electron capture and ionization total cross sections were found to be in very good accord with experiment. The Abrines and Percival procedure casts the coupled equations into the center-of-mass (c.m.) coordinate system to reduce the three-body problem to 12 coupled equations. However, this reduction complicates extensions of the code to laser processes and collisions in the presence of electric fields or with many electrons. An ideal application for the CTMC method is for collisions involving excited targets. Such processes are well described classically, while basis set expansion methods show limited applicability due to computer memory constraints. Considerable early work was done on Rydberg atom collisions, including state-selective electron capture, ionization, and electric fields [13–15]. Presently, there is a resurgence of work on Rydberg atom collisions because new crossed-field experimental techniques allow the production of these atoms with specific spatial orientations and eccentricities [16, 17], and even Rydberg positronium atoms have been achieved experimentally [18].
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For hydrogenic ion–ion collision processes, one must nucleus (excitation) as be careful to apply the CTMC method only for projectile
charges ZP ZT because after the initialization of the active E D Z 2 = 2n2c : (62.9) electron’s orbit and energy, there is no classical constraint on the orbital energy of a captured electron. For a low-charge- Then, nc is related to the principal quantum number n of the state ion colliding with a ground state highly charged ion, final state by the condition [24] one will obtain unphysical results because a captured elec 1=3 1=3 tron will tend to preserve its original binding energy. Thus, 1 1 < nc n n C : .n 1/ .n C 1/ n n excess probability will be calculated for electron orbits that 2 2 lead to unrealistic deeply bound states of the projectile. (62.10)
62.3.2 Pseudo-One-Electron Targets Collisions involving alkali atoms are of interest because of their relevance to applied programs, such as plasma diagnostics in tokamak nuclear fusion reactors. They are also a testing ground for theoretical methods since experimental benchmarks are difficult to realize with hydrogenic targets but are amenable for such cases as alkali atoms. In such collisions, it is essential to use a theoretical formalism that correctly simulates the screening of the nucleus by the core electrons (Sect. 62.1.2), since a simple Zeff =R Coulomb potential is inadequate for both large and small R. One can also apply the methods of Sect. 62.1.2 to partially or completely filled atomic shells [19]. This works reasonably well for collisions with a low charge state ion such as a proton, but it gets less accurate as the charge of the projectile becomes greater than that of the target nucleus. The reason is that the independent electron approximation [6] must be applied to the calculated transition probabilities in order to simulate the shell structure. This latter method can only maintain its validity if the transition probability is small. Otherwise, it will greatly overestimate the multiple electron removal processes since the first ionization potential is inherently assumed for each subsequent electron that is pulled out from the shell, leading to an underestimation of the energy deposition.
62.3.3 State-Selective Electron Capture One of the powers of the CTMC method is that it can be applied to electron capture and excitation of high-lying states that are not accessible with basis set expansion techniques [20]. For the C4C colliding on Li, and C6C and N7C colliding on H, where AO calculations and experimental data exist, the CTMC method agrees quite favorably with both [21–23]. The procedure consists in defining first a classical number nc related to the calculated binding energy E of the active electron to either the projectile (electron capture) or target
The electron’s normalized classical angular momentum lc D .n=nc /.r k/, is related to the orbital quantum number l of the final state by l < lc l C 1 :
(62.11)
The magnetic quantum number ml is then obtained from 2ml 1 2ml C 1 lz < ; 2l C 1 lc 2l C 1
(62.12)
where lz is the z-projection of the angular momentum obtained from calculations [25]. In principle, it is also possible to analyze the final-state distributions from the effective quantum number n D n ıl ;
(62.13)
where ıl is the quantum defect. In this latter case, it is necessary to sort the angular momentum quantum numbers first and then sort the principal quantum numbers. One of the main applications of the method explained in this section can be found in the description of the charge transfer of one electron from a neutral atom or molecule to an excited state of a highly charged ion, from which it deexcites by releasing electromagnetic radiation. It was uncovered in the 1970s that charge exchange (CX) reactions can be used as a diagnostic tool in nuclear fusion reactors, because the CX between impurities – constituted by multiply charged ions – and a H or D neutral beam provides a method for determining the concentration of fully stripped ions and for measuring ion temperature and plasma rotation [26]. This technique, known as charge exchange spectroscopy (CXS), has become widespread as a diagnostic in high-temperature – mostly tokamak – plasmas [27, 28]. Furthermore, the discovery in 1996 [29, 30] that comets undergo a highly efficient generation of X-rays [31–33], originated by the CX between cometary tail neutral gas and the highly charged ions from the solar wind (SWCX), opened up a vast new field of Xray astrophysics, leading to subsequent studies that revealed many other astrophysical locations where a similar process
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occurs, ranging from the heliosphere, planetary atmospheres, the interstellar medium, galaxies, and could even be linked to dark matter radiation [34]. Due to the high relevance of these applications, numerous experimental studies have been performed at the energy intervals of interest. For instance, experiments to study SWCX and CXS in the range from of 1 to 3 keV/amu and of 1 to 40 keV/amu, respectively, have been modeled using the state-selective CTMC method to successfully calculate the nl-resolved CX cross sections [35–37].
62.3.4 Exotic Projectiles The study of collisions involving antimatter projectiles, such as positrons and antiprotons, is a field that has grown rapidly over the past two decades, spurred on by experimental advances [38, 39]. Such scattering processes are of basic interest, and they also contribute to a better understanding of normal matter–atom collisions. Antimatter–atom studies highlight the underlying differences in the dynamics of the collision, as well as on the partitioning of the overall scattering. In the Born approximation, ionization cross sections depend on the square of the projectile’s charge and are independent of its mass. Thus, the comparison of the cross sections for electron, positron, proton, and antiproton scattering from a specific target gives a direct indication of higher-order corrections to scattering theories [40, 41]. Early work using the CTMC method concentrated on the spectra of ionized electrons for antimatter projectiles [42]. Later work focused on the angular scattering of the projectiles during electron removal collisions, such as positronium formation, on ratios of the electron removal cross sections, and on ejected electron “cusp” and “anticusp” formation. Two complete reviews on antiproton and positron impact collisions, which compare various theoretical results and available experiments, are given in [43, 44].
62.3.5 Heavy-Particle Dynamics A major attribute of the CTMC method is that it inherently includes the motion of the heavy particles after the collision. A straight-line trajectory for the projectile is not assumed, nor is the target nucleus constrained to be fixed. This allows one to compute easily the differential cross sections for projectile scattering or the recoil momenta of the target nucleus. As a computational note, the angular scattering of the projectile should be computed from the momentum components, not the position coordinates after the collision, since faster convergence of the cross sections using the projectile momenta is obtained. For recoil ion momentum transfer studies, one must initialize the target atom such that the c.m. of the nucleus plus its electrons has zero momentum so that there is
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no initial momentum associated with the target. A common error is to initialize only the target nucleus momenta to zero. Then the target atom after a collision has an artificial residual momentum that is associated with the Compton profile of the electrons because target–electron interactions are included in the calculations. Examples of recoil and projectile scattering cross sections are given in [11, 12]. The field of recoil ion momentum spectroscopy rapidly expanded in the early 2000s (e.g., [45, 46]), and the CTMC theoretical method impacted on the interpretation and understanding of experimental results because it inherently provides a kinematically complete description of the collision products. For the studied systems, primarily He targets because of experimental constraints, it is necessary for the theoretical method to follow all ejected electrons and the heavy particles after a collision. As an example, it has been possible to observe the backward recoil of the target nucleus in electron capture reactions, which is due to conservation of momentum when the active electron is transferred from the target’s to the projectile’s frame [47]. Theoretical methods were tested further with magneto-optical traps (MOT) that provide frozen alkali metal atomic targets (T 1 mK) from which to perform recoil ion studies [48]. For three and four-body systems, it is now possible to measure the momenta of all collision products. These observations provide a severe test of theory, since all projectile and target interactions must be included in the calculations. The CTMC method addresses all projectile interactions with the target nucleus and electrons. Thus, it is possible to calculate fully differential cross sections (FDCS), the most detailed accessible experimental observable. It is of interest that triply differential cross sections (TDCS) calculated using the CTMC method compare very favorably with sophisticated continuum distorted wave methods [49]. The CTMC technique allows one to incorporate electrons on both the projectile and the target nuclear centers. Furthermore, all interactions between centers are included. The only interaction that needs to be approximated is the electron– electron interaction on a given center. Here, simple screening parameters derived from Hartree–Fock calculations are employed to eliminate nonphysical autoionization. Within this many-electron model, the signatures of the electron–electron and electron–nuclear interactions on the dynamics of the collisions have been observed [50, 51]. Further, projectile ionization studies can be undertaken [52–54].
62.3.6 Ion-Molecule Collisions Ion-molecule ionization collisions share many aspects with the processes described in 62.1.3, but its study is much more complex because, apart from being multielectronic, the target is composed of several nuclei. Due to its simplicity, the
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first of these systems to be analyzed by a CTMC method was the H2 molecule [55], for which a fixed internuclear axis (Franck–Condon approximation) randomly orientated was assumed, and the electrons initialized in terms of two one-electron microcanonical distributions constructed from the quantum mechanical ground state of the molecule. For more complex molecules, apart from the Franck–Condon approximation, which is a typical assumption, many other approximations must be performed, which give rise to different calculation methods. Single ionization and electron capture in ion-H2 O collisions (for different projectile charges) have been successfully described by the use of one-center screened Coulomb potentials [56–58], in a similar fashion as was described for many-electron atoms in 62.3.1. In all the cases, a single active electron approximation is also invoked. We can also mention other methods that introduce approximations on the projectile-target interaction and are based on the classical over-barrier charge transfer process [59, 60], called COB-CTMC. In COB-CTMC, electrons bound to the target molecule are included in the calculation through the course of the collision. At each time step, the internuclear ion-molecule potential is evaluated, and the most weakly bound electrons are released if its maximum is lower than the binding target energy. After this, the electrons change from being virtual to being an active part of the dynamics of the process, and a further condition is usually established to account for the spatial electron density. This method has been successfully applied to the study of ionization of DNA and RNA uracil by highly charged ion impact [61]. Finally, as was already mentioned in the first paragraph of this section, other methods describe the multicenter nature of the target by fixing the position of the nuclei but rotating the internuclear axis randomly at each trajectory calculation and have been applied to the study of the ionization by ion impact of not only H2 [55] but also H2 O molecules [62].
62.3.7 Strong Laser Field Ionization The study of the nonlinear electron dynamics in the ionization of atoms by intense few-cycle laser pulses has gained attention over the last years (e.g., [63]). The limitations of a quantum description for this type of processes come from the fact that perturbation theory is, in most of the cases, not applicable, and nonperturbative calculations are sometimes prohibitively expensive, from a computational viewpoint. The straightforward extension of the CTMC method to photoionization problems is a promising solution to this problem, and it consists simply of adding the external laser field to the particles Hamiltonian, while the rest of the method remains essentially the same as that derived by Abrines and Percival [3]. As a remarkable example of the power of
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this method, we can mention the work of Ho et al. [64], who performed a fully classical self contained study of the atomic nonsequential double ionization (NSDI) [65]. Their procedure discards all quantum effects, even tunneling, but describes the dynamics in a complete form, which is a very important feature because electron–electron correlation is supposed to play an instrumental role in this process. The electrons are initialized with an energy corresponding to the two-electron quantum ground state, and then the dynamics are governed by the Coulomb interaction among the particles and their interaction with the laser electric field. Even though the CTMC method cannot include tunneling by default, features on the cross sections found experimentally for the double ionization of He and Ne [66, 67], and single ionization of He, Ne, and Ar [68] by laser fields at high intensities, on the range of 1015 W/cm2 , have been described by nonrelativistic semiclassical CTMC methods that include an ad hoc tunneling step [69]. These theoretical works are based on a method developed by Cohen [70], which treats both the description of the initial state and the dynamics of the particles in the standard way. The improvement of this method on the description of the process is that, using the WKB approximation, tunneling is allowed in the direction of the field starting from classical turning points. At points rc for which dz=dt D 0 and E.t/ rc > 0, being zO the polarization direction of the electric field E.t/, the tunneling probability is calculated as follows: 8 9 Zsout < = Ptun D exp 2.2/1=2 ŒV .rc C s zO / V .rc /ds ; : ; 0
(62.14) where is the reduced mass. If this probability is greater than a random number between 0 and 1, the particle then O at which V .rc C sout zO / D tunnels out at the point rc C sout z, V .rc /. On the contrary, the trajectory remains unaltered. For an extension of this model, which includes references to a broad variety of applications of this method, see [71]. Finally, it is also worth giving a brief description of the extension of CTMC method to treat relativistic problems [72]. For laser intensities above 1018 W/cm2 , the motions of electrons become highly relativistic because the mass of the electron starts becoming comparable to its rest mass. Here, we can mention two methods, which are similar in that the equations of motion must be solved relativistically but treat the initial state in a different way depending on the nucleus charge. If the charge of the nucleus is lower than 10, it is possible to treat the electron nonrelativistically in its initial state [73], whereas if it is higher than that value, it needs to be treated relativistically [74]. Additionally, it is possible to study both the ion and electron dynamics driven by an ultrastrong laser field (1018 1026 W/cm2 ) by applying a fully relativistic extension of the CTMC approach [75].
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62.4
Conclusions
In many ways it is surprising that a pure classical model can be successful in a quantum mechanical world, especially since the classical radial distribution for the hydrogen atom is described so poorly. However, hydrogen’s classical momentum distribution is exactly equivalent to the quantum one, and since collision processes are primarily determined by velocity matching between projectile and electron, reasonable results can be expected. Moreover, the CTMC method preserves conservation of flux, energy, and momentum, and Coulomb scattering is the same in both quantal and classical frameworks. Of significant importance is that the CTMC method is not restricted to one-electron systems and can be easily extended to more complicated systems involving electrons on both projectile and target. For these latter cases, multiple electron capture and ionization reactions can be investigated. On the projectiles side, the success of the CTMC process driven by both charged particles and photons is worth mentioning.
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Marcelo Ciappina Dr Ciappina received his PhD from the Balseiro Institute at Argentina in 2005 and a Research Professor degree from the Czech Academy of Sciences in 2019. His academic training includes postdoctoral stays at renowned institutions in Germany, Singapore, USA, Spain and Czechia. At present, he is Associate Professor at the Guangdong Technion – Israel Institute of Technology (GTIIT), Shantou, China.. His areas of interest include attosecond science and atomic collisions.
Raul O. Barrachina Dr Barrachina received his PhD from the Balseiro Institute at Argentina in 1985. His academic training also includes postgraduate studies in philosophy and the history of science, and postdoctoral stays in USA, France, Italy, and Spain. At present, he is Fellow of the Institute of Physics (IoP) and researcher of the Argentine Atomic Energy Commission (CNEA) and the Council of Scientific and Technical Research (CONICET). Francisco Navarrete Francisco Navarrete received his PhD from Balseiro Institute, National University of Cuyo at Bariloche, Argentina in 2016. He has worked at the National Atomic Energy Commission in Argentina, at the Department of Physics of Kansas State University in the US, and at the group Quantum Theory and Many-Particle Systems at the Institute of Physics, University of Rostock in Germany, where he currently conducts his research. He has made contributions to the fields of atomic collisions and strong-field physics. Ronald E. Olson Ronald E. Olson, Curators’ Professor of Physics earned his PhD from Purdue University in 1967. He is a Fellow of the American Physics Society and a Fulbright Fellow in France. His research interests concentrate on theory of elastic and inelastic total and differential scattering cross sections and studies of multiply charged ion–atom collisions, Rydberg atom collisions, negative ion detachment mechanisms, and Penning and associative ionization.
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Isolated Lines . . . . . . . . . . . . Semiclassical Theory . . . . . . . . Simple Formulae . . . . . . . . . . Perturbation Theory . . . . . . . . Broadening by Charged Particles Empirical Formulae . . . . . . . . .
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Abstract
One-photon processes only are discussed and aspects of line broadening directly related to collisions between the emitting (or absorbing) atom and one perturber are considered. Pointers to other aspects are included and a comprehensive bibliography of work on atomic line shapes, widths, and shifts is regularly updated [1]. The perturber may be an electron, a neutral atom or an atomic ion and can interact weakly or strongly with the emitter. The emitter is either a hydrogenic or a nonhydrogenic atom that is either neutral or ionized. In general, transiG. Peach () Dept. of Physics and Astronomy, University College London London, UK e-mail: [email protected]
tions in nonhydrogenic atoms can be treated as isolated, that is the separation between neighboring lines is much greater than the width of an individual line. When the emitter is hydrogen or a hydrogenic ion, the additional degeneracy of the energy levels with respect to orbital angular momentum quantum number means that lines overlap and are coupled. Pressure broadening is a general term that describes any broadening and shift of a spectral line produced by fields generated by a background gas or plasma. The term Stark broadening implies that the perturbers are atomic ions and/or electrons, and collisional broadening implies that the ‘collision’ model is appropriate; this term is often used to describe an isolated line perturbed by electrons. Neutral atom broadening indicates neutral atomic perturbers; this implies short-range emitter–perturber interactions, which in turn influence the approximations made. The broadening of molecular lines and bands is not considered here, however Hartmann, Boulet, and Robert [2] have published a comprehensive review of collisional effects on molecular spectra. This includes a discussion of laboratory experiments and models with their consequences for applications. General reviews of the theory of pressure broadening have been given [3–5], and in Chap. 2, Chap. 11, Chap. 15, Chap. 20, and Chap. 49 topics relevant to the theory of collisional broadening of spectral lines are discussed. A workshop on the topic of spectral line shapes in plasmas (SLSP) has been established to compare codes and different computational and analytic methods, see [6]. The International Conference on Spectral Line Shapes (ICSLS) and the Serbian Conference on Spectral Line Shapes in Astrophysics (SCSLSA) are devoted exclusively to this subject. Keywords
line profile orbital angular momentum impact approximation semiclassical theory radio line
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_63
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G. Peach
63.1 Impact Approximation
in terms of reduced matrix elements that are independent of magnetic quantum numbers. They are defined by
If the perturbers are rapidly moving, the broadening and shift (63.7) hhif jji 0 f 0 ii D Dif i 0 f 0 hhif kki 0 f 0 ii ; of the line arise from a series of binary collisions between the atom and one of the perturbers. The theory assumes that where although weak collisions may occur simultaneously, strong X collisions are relatively rare and only occur one at a time. .1/Ji CJi 0 Mi Mi 0 Dif i 0 f 0 D The impact approximation is valid if ! ! 1 Jf 1 Jf 0 Ji Ji 0 w 1 ; VN =„ 1 ; (63.1) ; Mi Mf Mi 0 Mf 0 where w is the half-width at half-maximum, is the average (63.8) time of collision and VN is the average emitter–perturber inters ˛˛ ˝˝ 0 0 action. It is not only widely applicable to electron and neutral i f w C id ı.! !if /I if X atom broadening, but also, for certain plasma conditions, to Dif i 0 f 0 D broadening by atomic ions. The power radiated per unit time Mi ;Mi 0 ;Mf ;Mf 0 and per unit interval in circular frequency !, in terms of the ˇ ˝˝ s ˇ ˛˛ i 0 f 0 ˇ w C id i.! !if /I ˇif ; (63.9) line profile I .!/, is P .!/ D
4 !4 I .!/ : 3 c3
(63.2)
For an isolated line produced by a transition from an upper energy level i to a lower level f , the line profile is Lorentzian with a shift d :
with s D 1; 1. For the line profile Eq. (63.3), the width and shift each have a single element, D 2w D hhif jj2wjjif ii;
d D hhif jjdjjif ii ; (63.10)
where is the full width at half-maximum. 1 w ; (63.3) hhif jjif ii
2 Throughout the rest of this article, it will be assumed that ! !if d C w 2 collisions only connect the set of upper levels i; i 0 or the set 0 and if the profile is for a transition between an upper set of of lower levels f; f , which is valid when w !; pressure levels i; i 0 and a lower set f; f 0 with quantum numbers Ji Mi , broadening of spectral lines is also assumed to be independent of Doppler broadening. However, for microwave spectra Ji 0 Mi 0 , Jf Mf , and Jf 0 Mf 0 , then of molecules, w can be of the order of !, and collisions conX 1 necting the upper to the lower levels become important; for 0 0 I .!/ D Re hhif jji f ii further details, see Ben-Reuven [7, 8]. Also for microwave i i 0 ff 0 ! spectra, pressure broadening and Doppler broadening cannot ˇ DD 1 ˇˇ EE
be considered to be independent effects, and a generalized 0 0 ˇ i f ˇ w C id i ! !if I ˇif ; theory has been developed by Ciuryło and Pine [9]. (63.4) I .!/ D
where I is the unit operator, and w and d are width and shift 63.2 Isolated Lines operators. In Eq. (63.4), is an operator corresponding to the dipole line strength defined by 63.2.1 Semiclassical Theory hhif jji 0 f 0 ii q 2 hijrjf i hi 0 jrjf 0 i ;
(63.5)
The motion of the perturber relative to the emitter is treated where r represents the internal emitter coordinates, q 2 D classically and is assumed to be independent of the internal e 2 =.40 / is the square of the electronic charge, and 0 is states of the emitter and perturber. This common trajectory is the permittivity of vacuum in SI units of F m1 . On taking specified by an emitter–perturber separation the average over all degenerate magnetic sublevels, R R.b; v; t/; b v D 0 ; (63.11) X 1 0 0 I .!/ D Re hhif kki f ii where v is the relative velocity, and b is the impact parameter. i i 0 ff 0 The time-dependent wave equation for the emitter–perturber ! EE DD system is
1 i 0 f 0 w C id i ! !if I if d D H ; (63.12) i„ (63.6) dt
63
Collisional Broadening of Spectral Lines
and the eigenfunctions H0
i
D Ei
i
i
;
929
for the unperturbed emitter obey i D 0; 1; 2; : : : :
(63.13) Simple formulae are useful for making quick estimates but in individual cases may give results in error by a factor of 2 or more. If it is assumed in Eq. (63.17) that
If .r; R/ is expanded in the form .r; R/ D
X
aj i .t/
j .r/ exp.iEj t=„/
63.2.2 Simple Formulae
;
(63.14)
Vij .R/ D Vjj .R/ıij ;
(63.22)
j
where Vjj .R/ is a simple central potential
where initially at time t D 1 aj i .1/ D ıj i
(63.15)
j D i; f ;
(63.23)
and Cj depends only on the state j of the emitter, and if the relative motion is along the straight line
and Eqs. (63.12)–(63.14) give i„
Vjj .R/ D „Cj Rp ;
X
daj i aki Vj k exp i!j k t ; D dt
R D b C vt ;
(63.24)
k
i; j; k D 0; 1; 2; : : : ; where
av
Z Vj k .R/ D
(63.16) then i h
Si 0 i .b; v/Sf0 f .b; v/ D exp 2i i f ;
j .r/V
.r; R/
k .r/ dr
(63.17) where the phase shifts are
in which V .r; R/ is the emitter–perturber interaction, and „!j k D Ej Ek :
(63.25)
(63.18)
1 j .b; v/ D 2„
Z1 Vjj .R/dt ;
j D i; f :
(63.26)
1
Integration of equations Eq. (63.16) for 1 t 1 gives Equations (63.20)–(63.26) give the unitary scattering matrix S , with elements ˇ ˇ !2=.p1/ ˇp ˇCp ˇ w C id D N vN (63.19) Sj i .b; v/ aj i .1/ ; i; j D 0; 1; 2; : : : : vN p3 i Then % (63.27) exp ˙ ˛p ; p1 p1 Z1 where w C id D 2N vf .v/ dv 0 2p 3 1=.p1/ D % ; ˛ Z1h p i p1 4
ıi 0 i ıf 0 f Si 0 i .b; v/Sf 0 f .b; v/ bdb ; av p % 12 .p 1/ 0
; ˇp D % 12 p (63.20) Cp D Ci Cf ; where Ji 0 D Ji and Jf 0 D Jf , N is the perturber density and Z1 8kB T 1=2 [ ]av denotes an average over all orientations of the collivN D vf .v/ dv D : (63.28) sion and over the magnetic sublevels [Eq. (63.9)]. In (63.20), 0 f .v/ is the Maxwell velocity distribution at temperature T for an emitter–perturber system of reduced mass : In (63.27), the ˙ sign indicates the sign of Cp , ˛p ' 1 for p 3, and %. / is the gamma function. 3=2 v 2 The cases p D 3; 4 and 6 correspond to resonance, 2 exp ; f .v/ D 4v 2kB T 2kB T quadratic Stark and van der Waals broadening, respectively. Z1 This approximation is invalid for the dipole case (p D 2) for f .v/ dv D 1 : (63.21) which Eq. (63.27) is not finite. The dipole–dipole interaction (p D 3) occurs when emitter and perturber are identical 0
63
930
G. Peach
atoms (apart from isotopic differences). If the level i is con- 63.2.3 Perturbation Theory nected to the ground state by a strong allowed transition with absorption oscillator strength fg i , and the perturbation of the An approximate solution of Eq. (63.16) is given by [3–5, 11, level f can be neglected by comparison, then 12] q 2 fg i ˇ ˇ ; 2me ˇ!g i ˇ p 1 cd D 1 C p ln 2 C 3 D 1:380173 : 2 3 C3 D cd
4 gg gi
cd D
Z
1 X 2 Vj k .t/ exp i!j k t dt „ 1
k
Zt
1=2
;
Vj i .t/ exp i!j i t dt
1
(63.29)
Also, if gj is the statistical weight of level j , the constant cd may be replaced by an empirical value
Z1
i Sj i .b; v/ D ıj i „
(63.30)
1
0
0 0 Vki t exp i!ki t dt :
(63.35)
1
and this gives a width correct to about 10% [10]. Equation This gives a cross section for the collisional transition i ! j : (63.27) does not predict a finite shift. Quadratic Stark broadZ1 ening occurs when a nonhydrogenic emitter is polarized by
ij .v/ D 2 Pij .b; v/ av b db ; electron perturbers. Then q2
˛i ˛f ; C4 D 2„
0
i; j D 0; 1; 2; : : : ;
(63.31)
(63.36)
where ˛i and ˛f are the dipole polarizabilities of states i where and f , respectively. Van der Waals broadening occurs when ˇ ˇ2 Pij .b; v/ D ˇıj i Sj i .b; v/ˇ ; the emitter and perturber are nonidentical neutral atoms. If X energy level separations of importance in the perturbing atom 2 ReŒ1 Si i .b; v/ D Pij .b; v/ ; are much greater than those of the emitter (e.g., alkali spectra j broadened by noble gases), C6 is given by C6 D
q2 2 ˛d ri rf2 ; „
(63.38)
correct to second-order in V .r; R/ on both sides. Using (63.32) Eqs. (63.10), (63.20), and (63.36)–(63.38), the full width is
where ˛d is the dipole polarizability of the perturber. The mean square radii can be calculated from the normalized radial wave functions 1r Pnj lj .r/ or estimated from Z1
Z1 vf .v/ dv
DN 0
2 4
Pn2 lj .r/r 2 dr
rj2 D
(63.37)
X
ij .v/ C
j ¤i
j
X
3 fj .v/ C e if .v/5 ;
(63.39)
j ¤f
0
'
nj2 a02 h 2 i
5nj C 1 3lj lj C 1 ; 2 2z j D i; f ;
where the sums are taken over all energy-changing transitions, the tilde indicates an interference term, and (63.33) e if .v/ D 2
in which the effective principal quantum numbers nj are given by Ej
z2 Ih ; nj2
z D Ze C 1 ;
(63.34)
where Ih D hcR1 is the Rydberg energy, Ze is the charge on the emitter, and z D 1 in this case.
Z1
e if .b; v/ b db ; P av
(63.40)
0
in which ˇ 1 ˇ2 ˇ Z ˇ ˇ1 ˇ
ˇ e P if .b; v/ D ˇ Vi i .t/ Vff .t/ dt ˇˇ : ˇ„ ˇ 1
(63.41)
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Collisional Broadening of Spectral Lines
931
63.2.4 Broadening by Charged Particles The total emitter–perturber interaction is zZp q 2 Zp q 2 ' V0 .R/ C V .r; R/ ; R jR rj
(63.42)
where Zp is the charge on the perturber, and V0 .R/ D
Ze Zp q 2 ; R
V .r; R/ D Zp q 2
r R : (63.43) R3
If Ze D 0, the relative motion is described by Eq. (63.24), but if Ze ¤ 0, the trajectory is hyperbolic and is given by
Ze Zp q 2 d 2R D rV D R; 0 dt 2 R3
(63.44)
Eq. (63.51), } indicates the Cauchy principal value. If Ze D 0, then ˇ ˇ ˇ!ij ˇ D0; ˇDb ; ıD1 (63.52) v in Eqs. (63.50) and (63.51). Approximation Eq. (63.48) breaks down at small values of b because of assumption Eq. (63.43) and the lack of unitarity of S as given by Eq. (63.35). This problem is discussed elsewhere [3, 11, 12], and all methods used involve choosing a cutoff at b D b0 , where b02 & rf2 , and using Eq. (63.48) only for b > b0 . For b b0 , an effective constant probability is introduced, and the method works well as long as the contribution from b b0 is small. For b > b0 (or ˇ > ˇ0 ), the contribution to ij .v/ in Eq. (63.39) is evaluated using Eqs. (63.47)–(63.50) and (63.52), where Z1 db A.ˇ; / (63.53) D e ˇ0 Ki0 .ˇ0 /Ki .ˇ0 / : b
with the resulting hyperbola characterized by a semimajor axis a and an eccentricity , where ˇ ˇ ˇZe Zp ˇq 2 b0
2 2 2 : (63.45) b Da 1 ; a D A similar treatment exists for the quadrupole contribution to v 2 V .r; R/ in Eq. (63.43) [3, 11, 12]. On using Eqs. (63.17), (63.20), (63.36)–(63.41), and (63.43), Vi i .t/ D 0 ;
e if .b; v/ D 0 ; P
e if .v/ D 0 ;
(63.46)
63.2.5 Empirical Formulae
and An empirical formula based on the theory of Sect. 63.2.4 for the width of an atomic line Stark broadened by elec(63.47) w C id D 2N vf .v/ dv trons has been developed [13]. Konjevi´c [14] has reviewed 0 the data available for nonhydrogenic lines and has provided Z1 X X simple analytical representations of the experimental results Qij .b; v/ C Qfj .b; v/ b db ; for widths and shifts. j ¤f 0 j ¤i The full half-width is given by Eq. (63.39), where e if D 0, and where Z1 X 8 2 „ 2 1 4Zp2 Ih2 a02 fij vfe .v/ j k .v/ dv D p v ˇ ˇ Qij .v/ D ŒA.ˇ; / C iB.ˇ; / ; 3 3 me k¤j „me ˇ!ij ˇ b 2 v 2 0 2 3
2 ReŒQij .v/ D Pij .b; v/ av : (63.48) X ejj 0e T 4Tj g.xj / C g .xjj 0 /5 ; j D i; f ; If lj 0 Dlj ˙1 ˇ ˇ (63.54) aˇ!ij ˇ 2 1 ; (63.49) ; ˇ ; ı 2 where v Z1 the functions A.ˇ; / and B.ˇ; / in Eq. (63.48) are given by 2me 1=2 1 v D v 1 fe .v/ dv D ; (63.55) kB T A.ˇ; / D ı exp. /ˇ 2 0 ˇ ˇ2 ˇ ˇ2 ˇ 0 ˇ .v/ D f .v/ with D me . In Eq. (63.54), and f e ˇKi .ˇ/ˇ C ı ˇKi .ˇ/ˇ ; (63.50) 2 h i 3nj 1 2
Z1 0 0 nj C 3lj lj C 1 C 11 ; (63.56) Tj D A.ˇ ; / dˇ 2ˇ 2z 9 } ; (63.51) B.ˇ; / D 2 02 .ˇ ˇ /
l> 2 0 e Rjj T jj 0 D
(63.57) 0 n l ; n l ; l> ; > < 2lj C 1 (63.50), where Ki .ˇ/ is a modified Bessel function.ˇ In Eq.
ˇ l< D min lj ; lj 0 ; l> D max lj ; lj 0 ; the sign corresponds to Ze Zp D ˙ˇZe Zp ˇ, and in Z1
63
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G. Peach
63.3 Overlapping Lines
with
Rjj 0
nl> ; nl< ; l>
Z1
a01
Pnl
>
l> .r/rPnl
; nl< ; l>
ejj 0 nl ; l> nl ; nl ; l> DR Z1 > > < 0 0 3nl> 2 hhi f kw C id kif ii D 2N vf .v/ dv 1=2 nl> ; nl< ; l> ; (63.59) nl> l>2 0 2z Z1h i
and nl> ; nl< ; l> is tabulated elsewhere [15]. The effective ıi 0 i ıf 0 f Si 0 i .b; v/Sf0 f .b; v/ b db : av principal quantum numbers nl> and nl< of the states j and j 0 0 (63.63) in Eqs. (63.57)–(63.59) both correspond to principal quantum number nj and ' 1 for nl> nl< 1. The effective The superscripts and suffixes e and i will be used to denote Gaunt factors g.x/ and e g .x/ are given by electronic and ionic quantities respectively, and em indicates that averaging over magnetic quantum numbers has not been 3kB T e g.x/ D 0:7 1:1=z C g.x/ ; x D ; (63.60) carried out. In the impact approximation, electron and ion 2E contributions evaluated using (63.63) are additive, and the matrix to be inverted is of order ni nf . However, under typical where conditions in a laboratory plasma, e.g., a hydrogen plasma with Ne D Ni D 1022 m3 , perturbing atomic ions cannot be x 2 3 5 10 30 100 treated using the impact approximation. The ions collectively g.x/ 0:20 0:24 0:33 0:56 0:98 1:33 generate a static field at the emitter that produces first-order Stark splitting of the upper and lower levels. The ions are randomly distributed around the emitter, and the field distriis used for x < 50, and for x > 50 bution W .F / used assumes that each ion is Debye screened p by electrons; allowance is made for these heavy composite 4 jEj 3 1 x ; (63.61) perturbers interacting with each other as well as with the C ln e g.x/ D g.x/ D 2 3z Ih emitter. If the ion field has a slow time variation, ion dynamic effects can produce significant changes in the line shape, see with Eqs. (63.60) and (63.61) joined smoothly near x D 50. [3, 4, 6, 18]. The energy E D Ej is given by Eq. (63.34), and xj and xjj 0 The shift produced by electron perturbers is very small, in Eq. (63.54) are evaluated using and the usual model adopted is to assume that the ions split the line into its Stark components, and that each compoˇ ˇ 2z 2 ˇ ˇ (63.62) nent is broadened by electron impact. In both cases, only Ej D 3 Ih ; Ejj 0 D Ej Ej 0 : nj the dipole interactions in Eq. (63.43) are included, and the profile is symmetric. Then (63.4) takes the form For Ze D 2 and 3, Eq. (63.54) is generally accurate to within Z X 1 ˙30% and ˙50%. For Ze 4, Eq. (63.54) is less accurate, as I .!/ D Re W .F / dF hhif jji 0 f 0 ii relativistic effects and resonances become more important. 0 0 i i ff ˇ DD Accuracy increases for transitions to higher Rydberg levels
1 ˇˇ EE 0 0 ˇ i ; w C id i ! ! f ˇ ˇif if I as long as the line remains isolated. Tables in Appendices IV and V of [3] give widths (63.64) for atoms with Ze D 0; 1, and other semiempirical formulas based on detailed calculations have been developed by and the destruction of the degeneracy by the ion field means to be inverted in Eq. (63.64) is of orSeaton [16, 17] for use in the Opacity Project where simple that the matrix 2 der ni nf . Inclusion of higher multipoles in V .r; R/ in estimates of many thousands of line widths are required.
63
Collisional Broadening of Spectral Lines
933
Eq. (63.43) introduces small asymmetries. The Stark rep- and in Eq. (63.69) resentation for the hydrogenic wave functions is often used Z1h i because it diagonalizes the shift matrix in Eq. (63.64). The em Q i 0 f 0 if .v/ D 2 PQie0 f 0 if .b; v/ b db transformation is given by (see Chap. 9.1.2 and Chap. 14.4.2) av
(63.71)
0
0
nj 1 X ˇ ˛
1=2 ˇnj Kj mj D .1/K 2lj C 1 lj Djmj j ! ˇ ˛ N N lj ˇnj lj mj ; M1 M2 mj
j D i; i 0 ; f; f 0 ;
by analogy with Eqs. (63.36) and (63.40), where av0 indicates an average over all orientations of the collision only. On using Eqs. (63.48)–(63.50), (63.59), and (63.8) with ji D li , ji 0 D li 0 , jf D lf , and jf 0 D lf 0 , h i PQie0 f 0 if .b; v/
(63.65)
av0
8Ih a02 D Dif i 0 f 0 RQ i 0 f 0 if A.0; 0/ ; 3me b 2 v 2
where quantum number Kj replaces lj , and ˇ ˇ 1
nj D Kj C Kj0 C ˇmj ˇ C 1 ; N D nj 1 ; 2 1 0 ˇˇ ˇˇ 2Kj C mj C mj C 1 ; KD 2
0 Kj nj 1 ; 1 mj C Kj0 Kj ; M1 D 2 1 M2 D mj C Kj Kj0 : 2
(63.72) where RQ i 0 f 0 if 2 4
X
RQ ij2 .ni ; li > / C
lj Dli ˙1
X
3
2 RQfj nf ; lf > 5
lj Dlf ˙1
ıli 0 li ılf 0 lf
(63.66)
2RQ i 0 i .ni ; li > /RQf 0 f nf ; lf > ıli 0 li ˙1 ılf 0 lf ˙1 : (63.73)
For the electron impact broadening, it is convenient to separate the energy-changing and the zero energy-change transitions, so that Eq. (63.39) is generalized to give From Eqs. (63.45) and (63.49)–(63.53) 0 C Qem hhi 0 f 0 j2we jif ii ; em D em
Zb1
where 0 em
(63.67)
ˇ ˇ ˛˛ ˝˝ i 0 f 0 ˇ2w0e ˇif D Ne 2
Z1 vfe .v/ dv 0
b0
A.0; 0/ D ı ; ( db ln.1 =0 / ; Ze ¤ 0 ; A.0; 0/ D b ln.b1 =b0 / ; Ze D 0 :
(63.74)
3
The impact approximation neglects electron–electron correlations and the finite duration of collisions, so the long-range em ijem .v/ C fj .v/5ıi i 0 ıff 0 ; (63.68) 4 dipole interaction leads to a logarithmic divergence at large j ¤i j ¤f impact parameters in Eq. (63.74). Therefore, a second cutoff we jif ii Qem hhi 0 f 0 j2e parameter is introduced, which is chosen to be the smaller of Z1 the Debye length bD and v: D Ne vfe .v/ dv Q iem (63.69) 0 f 0 if .v/ : " # 1=2 kB T 0 ; v ; (63.75) b1 D min bD ˇ ˇ ˇ ˇ 4q 2 Ne ˇ ˇ ˇ ˇ In Eq. (63.68), ni nj ¤ 0 and nf nj ¤ 0 in the first and respectively, and in Eq. (63.69) .ni 0 ni / D
second terms, but estimating in this case is not straightforward; it depends nf 0 nf D 0. The matrix element Eq. (63.68) can be evalon the splitting of the Stark components [3]. uated using Eqs. (63.36), (63.48), (63.50), and (63.52) as before. In Eq. (63.68), X
X
ijem .v/
Z1h i Pije .b; v/ D 2
63.3.2 Infrared and Radio Lines av0
bdb ;
0
i; j D 0; 1; 2; : : : ;
If the density Ni is low enough, the impact approximation be(63.70) comes valid for the perturbing atomic ions, and since impact
63
934
G. Peach
shifts are unimportant, Eq. (63.6) gives X 1 I .!/ D Re hhif kki 0 f 0 ii 0 0 i i ff EE DD
1 0 0 i f we C wi i ! !if I if ;
63.4 Quantum-Mechanical Theory 63.4.1 Impact Approximation
The scattering amplitude for a collisional transition i ! j is given in terms of elements of the transition matrix T D 1 S (63.76) by
where in Eq. (63.76) 0
0
(63.77) C Qe,i hhi f k2we,i kif ii ; Z1 ˛˛ ˝˝ i 0 f 0 2w0e,i if D Ne,i vfe,i .v/ dv
e,i D 0 e,i
0 e,i
0
3 2 X e,i X e,i ij .v/ C fj .v/5ıi 0 i ıf 0 f ; 4
f kj ; ki f j Mj kj ; i Mi ki X 2i 0 O ki Yl 0 m0 kOj ill Ylm D
1=2 ki kj lml 0 m0
T j Mj l 0 m0 I i Mi lm ; (63.81)
(63.78)
where the quantities ki lm and kj l 0 m0 refer to the motion of the perturber relative to the emitter before and after the Qe,i hhi 0 f 0 k2wQ e,i kif ii collision, and i and j represent all nonmagnetic quantum Z1 numbers associated with the unperturbed states i and j of D Ne,i vfe,i .v/ dv Q ie,i0 f 0 if .v/ : (63.79) the emitter. The total energy of the emitter–perturber system 0 is given by In general, cross sections for electron and heavy-particle „2 2 impact are roughly comparable for the same velocity EJ D Ej C "j ; "j D k ; and, hence, different impact energies. Therefore, using 2 j Eqs. (63.78) and (63.79), .J; j / D .I; i/; .F; f / ; (63.82) 0 0 (63.80) e i ; Qe Qi ; and for an isolated line, is given by Eq. (63.39), where and this result is consistent with Eq. (63.64)
approximation for high-density plasmas. If ni nf D 1; 2, say, as nf kj 1 ij .v/ D increases, the relative contributions from Eqs. (63.78) and ki 4gi (63.79) decrease because there is increasing coherence, and, X Z ˇ
ˇ e i ˇf j Mj kj ; i Mi ki ˇ2 dkO i dkOj ; hence, cancelation in Q i 0 f 0 if .v/ and Q i 0 f 0 if .v/ between the Mi Mj effects of levels i; i 0 and f; f 0 . Radio lines of hydrogen are observed in galactic HII reki D v=„ ; gions where principal quantum numbers are of the order of (63.83) nf ' 100, temperatures are Te D Ti ' 104 K, and densities are Ne D Ni ' 109 m3 . For a comprehensive review of the the- and the interference term Q if .v/ Q if if .v/ is given by ory and observation of radio recombination lines, see Gordon Z and Sorochenko [19]. 1 X Q if .v/ D Dif i 0 f 0 jf .i Mi 0 k0 ; i Mi k/ If is the full half-width and Q is the full half-width when 4 M M 0 i i only contributions Eq. (63.79) are retained, the effect of canMf Mf 0 ˇ2 celation is illustrated by 0 f .f Mf 0 k0 ; f Mf k/ˇ dkO dkO ; (63.84) ni nf D 1 Electrons Protons C Electrons j ¤i
nf 5 10 15 20 25 50 100
Q = 0.82 0.40 0.17 0.08 0.05 0.00 0.00
j ¤f
= Q 0.99 0.95 0.86 0.73 0.59 0.17 0.01
with k D k 0 D v=„. From Eqs. (63.8) and (63.20), 2 Z1 „ 1 N w C id D f .v/ dv v 0
1 X
h i .2l C 1/ 1 Si i .l; v/Sff .l; v/ ;
lD0
Further details are given by Peach [20].
(63.85)
63
Collisional Broadening of Spectral Lines
935
in Eq. (63.81), where j0 denotes all other quantum numbers required to describe state j that do not change during the (63.86) collision. Then .vb/2 H) „2 l.l C 1/ ; ˇ and the integral over b has been replaced by a summation X ˇ Lj lLjT S 12 S T ˇLj Mj SMS lm 1 ms D CM mM T CM m M T over l. In Eq. (63.85)), Si i .l; v/Sff .l; v/ is given by ˇ j S s S S 2 T T T where
Lj Mj S MS
ˇ ˇ 1 ˇˇLj Sl LjT MjT S T MST ; 2 (63.92)
Si i .l; v/Sff .l; v/ X
1 Dif i 0 f 0 SI i 0 Mi 0 lm0 I i Mi lm D .2l C 1/ M M 0 M M 0 i
i f mm0
f
SF f 0 Mf 0 lm0 I f Mf lm ;
j j j
3 where Cm11 m2 2 m 3 is a vector coupling coefficient, the super(63.87) script T denotes quantum numbers of the emitter–perturber system, and 12 , ms are the spin quantum numbers of the scatand subscripts I and F are introduced to emphasize that the tered electron. On using Eq. (63.92), Eq. (63.90) is replaced S-matrix elements correspond to different total energies EI by and EF defined by Eq. (63.82). If scattering by the emitter in a state j is treated using a central potential, the amplitude hhi 0 f 0 kw C id kif ii D .„=me /2 N for elastic scattering is X 0
.1/Li CLi 0 ClCl 2LTi C 1 2LfT C 1 1 0
i X LTi LfT S T l l 0 .2l C 1/Tjj .l; v/Pl kO kO ; (63.88) f k0; k D )( ) 2k ( T
T lD0 2S C 1 LfT LTi 1 Lf LTi 1 where 2.2S C 1/ Li Lf l Li 0 Lf 0 l 0
Z1 1 Tjj .l; v/ D Sjj .l; v/ D exp 2ij .l; k/ ; 1 fe .v/ dv ıl 0 l ıLi 0 Li ıLf 0 Lf j D i; f ; (63.89) v 0 [Eqs. (63.23), (63.25), and (63.26)]. For the case of overlap1 T T 01 T T 0 S Sl S I L Sl S L L L I i i ping lines, Eq. (63.63) becomes 2 i 2 i 2 Z1 1 T T 01 T T „ 1 0 Sl S I L Sl S L L S L ; f f F N f .v/ dv hhi 0 f 0 kw C idkif ii D 2 f 2 f v (63.93) 0 1 h i X where, for an isolated line, the width and shift are given by .2l C 1/ ıi 0 i ıf 0 f Si 0 i .l; v/Sf0 f .l; v/ Eq. (63.93) with Li 0 D Li and Lf 0 D Lf . For hydrogenic lD0 (63.90) systems, where states i; i 0 and f; f 0 with different angular momenta are degenerate, a logarithmic divergence occurs for on generalizing Eq. (63.85) and using Eq. (63.87). Formularge values of l and l 0 [Eq. (63.74)], and must be removed lae Eqs. (63.85) and (63.90) have been obtained by assuming by using Eqs. (63.75) and (63.86). that a collision produces no change in the angular momentum If a jj coupling scheme is used j Mj ) j0 Jj Mj , j D of the relative emitter–perturber motion. This corresponds i; i 0 ; f; f 0 in Eq. (63.81), and to the assumption of a common trajectory in semiclassical ˇ theory and means that the total angular momentum of the ˇ ˇJj Mj lm 1 ms emitter–perturber system is not conserved. This assumption ˇ 2 ˇ is removed in the derivation of the more general expressions E X Jj jJ T l 12 j ˇ T T ; (63.94) CM m0jM T Cmm D 0 ˇJj ljJj Mj given in the following sections. sm j
JjT MjT j m0
then Eq. (63.93) becomes
63.4.2 Broadening by Electrons Different coupling schemes can be used to describe the emitter–perturber collision. For LS coupling, j Mj ) j0 Lj Mj SMS ;
j D i; i 0 ; f; f 0 ;
j
(63.91)
hhi 0 f 0 kw C idkif ii D .„=me /2 N X J CJ C2J T Cj Cj 0 1 .1/ i i 0 f .2JiT C 1/.2JfT C 1/ 2 T T 0 0 Ji Jf jj l l
63
936
G. Peach
(
JfT
JiT
1
)(
JfT
JiT
)
1
Z1
1 f .v/ dv ıl 0 l ıJi 0 Ji ıJf 0 Jf v 0
SI Ji 0 l 0 JiT I Ji lJiT SF Jf 0 l 0 JfT I Jf lJfT : (63.98)
Ji 0 Jf 0 j 0 j Z1 1 fe .v/ dv ıl 0 l ıj 0 j ıJi 0 Ji ıJf 0 Jf v Ji
Jf
0
SI .Ji 0 l 0 j 0 JiT I Ji ljJiT / SF .Jf 0 l 0 j 0 JfT I Jf ljJfT / ; (63.95) where Ji 0 D Ji and Jf 0 D Jf for an isolated line. If the spectrum of the emitter is classified using LS coupling, it is often sufficient to use energies defined by ELj S D
X Jj
2Jj C 1
ELj SJj
2Lj C 1 2Sj C 1
(63.96)
For many cases of practical interest, transitions of type Ji ! Jf are isolated and so have line profiles given by Eqs. (63.3), (63.10), and (63.98), where Ji D Ji 0 and Jf D Jf 0 [10]. In order to obtain the S -matrix elements in Eq. (63.98), it is usually sufficient to use adiabatic potentials for the emitter– perturber system that have been calculated neglecting fine structure. Since T is typically a few hundred degrees, only coupling between adiabatic states that tend to the appropriate separated-atom limit are retained in the scattering problem. The coupled scattering equations are then solved with fine structure introduced by applying an algebraic transformation to the adiabatic potentials, and using the observed splittings of the energy levels. The Born–Oppenheimer approximation is valid, and details are given in [21].
and obtain the S -matrix elements in an LS coupling scheme. They are then transformed to jj coupling by using the algebraic transformation S Jj 0 l 0 j 0 JjT I Jj ljJjT 63.5 One-Perturber Approximation
0 1=2
D 2Jj C 1 2Jj 0 C 1 .2j C 1/ 2j C 1 63.5.1 General Approach and Utility X .2LjT C 1/.2S T C 1/ If only one perturber is effective in producing broadening, LjT S T 8 98 9 I .!/ can be obtained by considering a dipole transition beT 0 T ˆ < Lj l Lj > =ˆ < Lj 0 l Lj > = tween initial and final states I and F of the emitter–perturber 1 S 12 S T S ST system. Then P .!/ is given by Eq. (63.2), where 2 ˆ > > : ;ˆ : ; Jj j JjT Jj 0 j 0 JjT I .!/ D Œı.! !IF /hhIF jjIF iiav ; 1 T T 01 T T S Lj 0 Sl Lj S I Lj Sl Lj S ; (63.97) „!IF D EI EF ; (63.99) 2 2 and introducing the splitting of the fine structure compo- and av denotes an average over states I and a sum over states nents in Eq. (63.4) or (63.6). If in LS coupling the line is F [4]. Wave functions J are given by X isolated, but nevertheless the broadened fine structure com
(63.100) J .r; R/ D O j .r/ kj ; kj0 I R ; ponents overlap significantly, then the interference terms in j Eq. (63.6) must be included. where J D I; F , and O is an operator that takes account of any symmetry properties of the emitter–perturber system. 63.4.3 Broadening by Atoms The energies EI and EF are given by Eq. (63.82). The perturber wave functions for initial state j0 and final state j are The formal result is very similar to Eq. (63.95), but in this expanded in the form case, the relative motion only gives rise to orbital angular X
momentum. Thus, 1=2 ilj0 kj0 Ylj mj kOj0 kj ; kj0 I R D 2i 0
0
0
2
hhi f kw C id kif ii D .„=/ N X T 0 .1/Ji CJi 0 C2Jf ClCl .2JiT C 1/.2JfT C 1/ JiT JfT l l 0
(
JfT
JiT
1
Ji
Jf
l
)(
JfT
JiT
1
Ji 0
Jf 0
l0
)
0
lj0 mj0 lj mj
1
O F j ; j0 I R ; (63.101) Ylj mj R R where j denotes a channel characterized by j D j Mj lj mj ;
j D 0; 1; 2; : : : ;
(63.102)
63
Collisional Broadening of Spectral Lines
937
[Eq. (63.81)]. In Eq. (63.101), the radial perturber wave func- 63.5.2 Broadening by Electrons tion has the limiting forms If LS coupling is used, definition of channel j in
lj C1 R F j ; j0 I R R!0 Eq. (63.104) is replaced by h
1=2 R!1k ıj j0 exp ij j 1 j D Lj Slj LjT S T ; j D i0 ; i; i 0 ; f0 ; f; f 0 ; (63.110)
i 2 SJ j I j0 exp ij ; (63.103) [cf. Eqs. (63.91) and (63.92)]. Then assuming that the weights ui0 of all the levels i0 that effectively contribute to with the line are the same, Eq. (63.106) becomes 1 z
j D kj R lj ln 2kj R X ˝˝ 2 ki
˛˛ 1 Li S Lf S kkLi 0 S Lf 0 S I .!/ D Ne z 2 C arg % lj C 1 C i ; i i i 0 0 kj f f f 0 0 zD
q 2 Ze Zp : „2
ıli lf ıli 0 lf 0 ıLT LT ıLT LT ıLT LT0 ıLT LT 0 i0 i i f f f f i
0 T 0 0 C 1 2S .1/Li CLi 0 Cli Cli 0 2.2S C 1/ T
T 2Li C 1 2Lf C 1 )( ( ) 1 LfT LTi LfT LTi 1 Li Lf l Li 0 Lf 0 li 0 Z1 1 (63.111) fe .v/ dv F .; v/ ; v
(63.104)
The coupled equations obtained by using Eqs. (63.12), (63.13), (63.100), and (63.101), where (63.105) .r; R; t/ D J .r; R/ exp.iEJ t=„/
are integrated to give functions F j ; j0 I R . Using Eqs. (63.100) and (63.101), Eq. (63.99) becomes I .!/ D
1 N 2
X
˝˝ ˛˛ ui0 i f jji 0 f 0
0
i0 i i 0 f0 f f 0
Z1
where F .; v/ is Eqs. (63.107) and (63.111). If
defined by (63.106) the functions F j 0 ; j0 I R in Eq. (63.107) are replaced by their asymptotic forms Eq. (63.103), then
1 f .v/ dv F .; v/ ; v
0
where
2
F .; v/ ' ! 2 .„=me /
Z1
h
i ıi0 i ıf0 f SI i I i0 SF f I f0 h
i ıi0 i 0 ıf0 f 0 SI i 0 I i0 SF f 0 I f0 : (63.112)
F i ; i0 I R F f ; f0 I R dR
F .; v/ D 0
Z1
F i 0 ; i0 I R F f 0 ; f0 I R dR ;
On substituting Eq. (63.112) into Eq. (63.111), summing over i0 and f0 and using the unitary property of the S(63.107) matrix,
0
u i0 D g i0
X
gi00 ;
i00
vD
„ki0 :
(63.108)
D
The one-perturber approximation is valid when ! j! !if j w I
I .!/
V VN ;
(63.109)
where V is the effective interaction potential required to produce a shift !. In the center of the line, many-body effects are always important, and the one-perturber approximation diverges as ! ! 0. In many cases, there is a region of overlap where criteria Eqs. (63.1) and (63.109) are all valid, but when ! 1, Eq. (63.99) is a static approximation, since the average time between collisions is ! 1 .
1 ! 2
X
˝˝
˛˛ Li S Lf S kkLi 0 S Lf 0 S
LTi LfT S T l l 0
hhi 0 f 0 kwkif ii ; (63.113) where hhi 0 f 0 kwkif ii is given by Eq. (63.93). Line shape Eq. (63.113) is identical to that obtained from Eq. (63.6) when ! w. If the jj coupling scheme specified by Eq. (63.94) is used, and channel j is defined by %j D Jj lj jj JjT ;
j D i0 ; i; i 0 ; f0 ; f; f 0 ;
(63.114)
63
938
G. Peach
0 lj . 400, whereas j . 2. Therefore, transitions take place between channels defined by
Equation (63.106) becomes [cf. Eq. (63.95)] I.!/ X
1 Ne 2
D
˝˝
˛˛ Ji Jf kkJi 0 Jf 0
i0 i i 0 f0 f f 0
ıli lf ıli 0 lf 0 ıji jf ıji 0 jf 0 ıJiT JiT ıJ T J T ıJiT J T0 ıJ T J T0 f0 f
0
f0 f
0 i
Ji CJi 0 C2JfT Cji Cji 0
.1/ 1
2JiT C 1 2JfT C 1 (2 )( JfT JiT 1 JfT Ji Jf ji Ji 0 1 Z 1 fe .v/ dv F .; v/ ; v
JiT
1
Jf 0
ji 0
0
%j D j Lj SLj ;
j D i; f ;
where the unperturbed emitter in state j has quantum numbers Lj S, the quantum number j represents the projection of the orbital angular momentum on the internuclear axis, and Eq. (63.100) is replaced by J .r; R/ D O
)
X
j .rI R/
kj ; kj0 I R ;
where kj D kj0 , and j .rI R/ is the wave function for molecular state j . In Eq. (63.119), the only molecular states (63.115) retained are those that correlate with emitter states i and f . The scattering is described by
63.5.3 Broadening by Atoms and Molecules
"
#
lj lj C 1 2z 2 d2 2 2 Vj .R/ C kj dR2 R2 R „ Fj .R/ D 0 ;
ˇ ˇ where kj D kj0 and In the wings of a line where „j!j ' ˇEJi EJi 0 ˇ ; j D 0 0 i0 ; i; i ; f0 ; f; f , coupling between the fine structure levels Fj .R/ D Pvj lj .R/ or is important. If channel j is defined by
Fj .R/ F j ; j I R D Fki li .R/ %j D
;
0
0
j D i0 ; i; i ; f0 ; f; f ;
(63.116)
Equation (63.106) becomes [cf. Eq. (63.115)] I .!/ D
1 N 2
X i0 i i 0 f0 f f 0
(63.119)
j
where F .; v/ is given by Eqs. (63.107) and (63.114).
Jj lj JjT
(63.118)
˝˝
˛˛ Ji Jf kkJi 0 Jf 0
(63.120)
(63.121)
for vibrational or free states, respectively, and Vj .R/ is the potential energy of state j . Free–free transitions always contribute to the line profile, but bound–free and free–bound transitions only contribute on the red and blue wings, respectively. On using Eqs. (63.82), (63.99), and (63.109), „! D "i "f , and "j becomes the energy of bound state j with vibrational quantum number vj when "j < 0. If
ıli lf ıli 0 lf 0 ıJ T J T ıJ T J T ıJ T J T0 ıJ T J T0 i0 i i0 i f0 f f0 f
ˇ2 ˇ1 T Ji CJi 0 C2Jf Cli Cli 0 T ˇ ˇZ .1/ 2Ji C 1 2JfT C 1 ˇ
ˇ ˇ ˇ N ( )( ) G ; "i ; "f ˇ Fi .R/ .R/F f .R/ dRˇ T T T T 1 1 Jf Ji Jf Ji ˇ ˇ 0 Ji Jf li Ji 0 Jf 0 li 0 [cf. Eq. (63.107)], where 1 Z 1 Z (63.117) f .v/ dv F .; v/ ; v N .R/ D q 0 i .rI R/r f .rI R/ dr
(63.122)
(63.123)
and (63.116). where F .; v/ is given by Eqs. (63.107) is the dipole moment, then using Eq. (63.119)), the free–free ˇ ˇ In the far wings, where „j!j ˇEJi EJi 0 ˇ ; j D contribution is given by i0 ; i; i 0 ; f0 ; f; f 0 , an adiabatic approximation is valid. AdiaN X batic states of the diatomic molecule formed by the emitter– ui ıli lf .2li C 1/ I0 .!/ D perturber system are considered in which the total spin is 2 i f assumed to be decoupled from the total orbital angular moZ1 mentum of the electrons. The coupling between rotational f .v/ (63.124) dv G .; "i ; "f / ; and electronic angular momentum can also be neglected, bev 0 cause, typically, contributions to the line profile come from
63
Collisional Broadening of Spectral Lines
939
where "i D 12 v 2 , and ui is the relative weight of state i [cf. Eqs. (63.106)–(63.108)]. The bound–free contribution is
been made over the last 30 years in developing theories that take into account many of the key features of the intermediate problem, and they are often successful in predicting line proN X files for practical applications [3–5, 22–24]. More recently, I1 .!/ D ui ıli lf .2li C 1/ 2 time-dependent many-body problems have been tackled usi f X
ing computer-oriented approaches that invoke Monte Carlo g."i /G ; "i ; "f ; and other simulation methods to study line broadening in i (63.125) dense, high-temperature plasmas, see, for example, [25]. In this chapter, the emphasis has been on aspects of the subject and the free–bound contribution is that relate directly to electron–atom and low-energy atom– 1 X „! atom scattering. Many experts in the fields of electron–atom ui ıli lf .2li C 1/ exp I2 .!/ D N 2 kB T and atom–atom collisions are still not exploiting the direct i f applicability of their work to line broadening. It is hoped X
g "f G ; "i ; "f ; (63.126) that this contribution will encourage more research workf ers to study these fascinating problems that not only provide links with plasma physics and, in particular, with the physics where of fusion plasmas, but also with a quite distinct body of 2 laboratory-based experimental data. „ f .v/ g."/ D 2 2 v 3=2 2 " 2 „ D 8 exp ; (63.127) References 2kB T kB T on using Eq. (63.21) with " D 12 v 2 . The full line profile is then given by Eqs. (63.124)–(63.126), so that I .!/ D
2 X
Ij .!/ :
(63.128)
j D0
The satellite features that are often seen in line wings arise because turning points in the difference potential
Vi .R/ Vf .R/ produce a phenomenon analogous to the formation of rainbows in scattering theory. The JWKB approximation is often used for the functions Fkj lj .R/ and can be shown to lead to the correct static limit in which transitions take place at fixed values of R called “Condon points”, i.e., the Franck–Condon principle is valid. Further details are given in [4, 5, 22–24].
63.6 Unified Theories and Conclusions The pressure broadening of spectral lines is, in general, a time-dependent many-body problem and as such cannot be solved exactly. After all, even the problem of two free electrons scattered by a proton is still a subject of active research. There is no practical theory that leads to the full static profile in the limit of high density (or low temperature) and to the full impact profile in the limit of low density (or high temperature). As with so many problems in physics, it is the intermediate problem that is intractable because no particular feature can be singled out as providing a weak perturbation on a known physical situation. However, much progress has
1. Kramida, A., Fuhr, J.R.: Atomic Spectral Line Broadening Bibliographic Database, Version 3.0. National Institute of Standards and Technology, Washington, D.C. (2017) 2. Hartmann, J.-M., Boulet, C., Robert, D.: Collisional Effects on Molecular Spectra. Elsevier, Amsterdam (2008) 3. Griem, H.R.: Spectral Line Broadening in Plasmas. Academic Press, New York (1974) 4. Peach, G.: Adv. Phys. 30, 367 (1981) 5. Allard, N., Kielkopf, J.R.: Rev. Mod. Phys. 54, 1103 (1982) 6. Stambulchik, E., Calisti, A., Chung, H.-K., González, M.Á. (eds.): Spectral Line Shapes in Plasmas. MDPI, Basel, Switzerland (2015) 7. Ben-Reuven, A.: Phys. Rev. 145, 7 (1966) 8. Ben-Reuven, A.: Adv. Atom. Molec. Phys. 5, 201 (1969) 9. Ciuryło, R., Pine, A.S.: J. Quant. Spectrosc. Radiat. Transf. 67, 375 (2000) 10. Lewis, E.L.: Phys. Rep. 58, 1 (1980) 11. Sahal-Bréchot, S.: Astron. Astrophys. 1, 91 (1969) 12. Sahal-Bréchot, S.: Astron. Astrophys. 2, 322 (1969) 13. Dimitrijevi´c, M.S., Konjevi´c, N.: J. Quant. Spectrosc. Radiat. Transf. 24, 451 (1980) 14. Konjevi´c, N.: Phys. Rep. 316, 339 (1999) 15. Oertel, G.K., Shomo, L.P.: Astrophys. J. Suppl. 16, 175 (1969) 16. Seaton, M.J.: J. Phys. B 21, 3033 (1988) 17. Seaton, M.J.: J. Phys. B 22, 3603 (1989) 18. Kogan, V.I., Lisitsa, V.S., Sholin, G.V.: Rev. Plasma Phys. 13, 261 (1987) 19. Gordon, M.A., Sorochenko, R.L.: Radio Recombination Lines. Springer, New York (2009) 20. Peach, G.: J. Astrophys. Astr. 36, 555 (2015) 21. Leo, P.J., Peach, G., Whittingham, I.B.: J. Phys. B 28, 591 (1995) 22. Szudy, J., Baylis, W.E.: J. Quant. Spectrosc. Radiat. Transf. 15, 641 (1975) 23. Szudy, J., Baylis, W.E.: J. Quant. Spectrosc. Radiat. Transf. 17, 269 (1977) 24. Szudy, J., Baylis, W.E.: Phys. Rep. 266, 127 (1996) 25. Calisti, A., Godbert, L., Stamm, R., Talin, B.: J. Quant. Spectrosc. Radiat. Transf. 51, 59 (1994)
63
Part E Scattering Experiment
942
Part E focuses on the experimental aspects of scattering processes. Recent developments in the field of photodetachment are reviewed, with an emphasis on accelerator-based investigations of the photodetachment of atomic negative ions. The theoretical concepts and experimental methods for the scattering of low-energy photons, proceeding primarily through the photoelectric effect, are given. The main photon–atom interaction processes in the intermediate energy range are outlined. The atomic response to inelastic photon scattering is discussed; essential aspects of radiative and radiationless transitions are described in
the two-step approximation. Advances such as cold-target recoil-ion momentum spectroscopy are also touched upon. Electron–atom and electron–molecule collision processes, which play a prominent role in a variety of systems, are presented. The discussion is limited to electron collisions with gaseous targets, where single collision conditions prevail, and to low-energy impact processes. The involving neutral molecules at chemical energies is presented. Applications of single-collision scattering methods to the study of reactive collision dynamics of ionic species with neutral partners are discussed.
64
Photodetachment David Pegg and Dag Hanstorp
Contents 64.1
Negative Ions . . . . . . . . . . . . . . . . . . . . . . . . . 943
64.2 64.2.1 64.2.2 64.2.3 64.2.4 64.2.5
Photodetachment . . . . . . . . . . . Photodetachment Cross Sections . . . Threshold Behavior . . . . . . . . . . . Structure in Continuum . . . . . . . . Photoelectron Angular Distributions Higher-Order Processes . . . . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
944 944 945 946 946 946
64.3 64.3.1 64.3.2 64.3.3 64.3.4
Experimental Procedures . Production of Negative Ions Interacting Beams . . . . . . . Light Sources . . . . . . . . . Detection Schemes . . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
947 947 947 948 948
64.4 64.4.1 64.4.2 64.4.3
Measuring Properties of Negative Ions Electron Affinities . . . . . . . . . . . . . . Bound States . . . . . . . . . . . . . . . . . . Continuum Processes . . . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
948 948 949 949
64.5 64.5.1 64.5.2
Investigation of Fundamental Processes . . . . . . . . 950 Threshold Studies . . . . . . . . . . . . . . . . . . . . . . . 950 Photodetachment Using Short Laser Pulses . . . . . . . 950
64.6 64.6.1 64.6.2
Observations and Applications of Negative Ions . . 950 Natural Occurrence of Negative Ions . . . . . . . . . . . 950 Applications of Negative Ions . . . . . . . . . . . . . . . 951
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effect. In the gaseous phase, the photoelectric effect is referred to as either photoionization (atoms and positive ions) or photodetachment (negative ions). This chapter reviews the basics of negative ions, photodetachment, and the experimental methods used when the photodetachment process is applied to the study of negative ions. Negative ions of molecules and cluster exist, but this article covers only atomic negative ions. The monographs of Massey [1] and Smirnov [2] and a review by Bates [3] offer good introductions to the subject of negative ions. The focus will be on accelerator-based measurements. Reviews of experimental methods used to investigate the properties of negative ions have been published by, for example, Andersen [4], Pegg [5], and Rienstra-Kiracofe [6]. Keywords
photodetachment laser photodetachment threshold spectroscopy Wigner law doubly excited states Fano resonance profile photoelectron angular distributions resonance ionization spectroscopy anomalous threshold behavior collinear laser-ion beam geometry neutral atom detection
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 951
Abstract
64.1
Negative Ions
Investigations of photon–ion interactions have grown rapidly over the past few decades, primarily due to the inInterest in negative ions stems from the fact that their struccreased availability of laser and synchrotron light sources. ture and dynamics are qualitatively different from those of At photon energies below about 1 keV, the dominant raisoelectronic atoms and positive ions. This can be traced diative process is the electric dipole induced photoelectric to the nature of the force that binds the outermost electron. In the case of atoms and positive ions, the outermost D. Pegg () electron moves asymptotically in the long-range Coulomb Dept. of Physics, University of Tennessee field of the positively charged core. The relatively strong Knoxville, TN, USA 1=r potential is able to support an infinite spectrum of e-mail: [email protected] bound states that converge on the ionization limit. In conD. Hanstorp () trast, the outermost electron in a negative ion experiences Dept. of Physics, University of Gothenburg the short-range induced-dipole field of the atomic core. The Gothenburg, Sweden e-mail: [email protected] relatively weak 1=r 4 polarization potential can typically sup© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_64
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port only a single bound state. The weakness of the attractive force is reflected in the binding energies of the outermost electron, i.e., the electron affinities, which are typically an order of magnitude smaller than the ionization energies of atoms. Weakly bound systems such as negative ions are ideally suited for investigations of the effects of electron correlation, which plays an important role in determining the structure and dynamics of many-electron systems [7]. As a result of the more efficient shielding of the nucleus by the atomic core, the electron-–electron interactions become relatively more important than the electron–nucleus interaction. A major goal of photodetachment experiments is to measure correlation-sensitive quantities such as electron affinities, the energies and widths of resonant states, and asymmetry parameters describing photoelectron angular distributions. These quantities provide sensitive tests of the ability to incorporate electron correlation into theoretical models. The stimulating interplay between experiment and theory continues to help elucidate the role of many-electron effects in the structure and dynamics of atomic systems. There exist a number of bound excited states of negative ions, but most of them have the same parity as the ground state [8]. As such, they are long lived and radiate via electric dipole-forbidden transitions. Dipole-allowed transitions have been observed in a few ions. In these cases, excited states connect to the ground state or other excited states of opposite parity [9–12]. Multiple excitations in a negative ion leads to the production of states embedded in continua lying above the detachment limits. Such quasi-discrete states are unbound and decay rapidly via the allowed process of Coulomb autodetachment. Most negative ions exhibit a rich spectrum of these short-lived states [5]. They are manifested as resonance structures at well-defined energies in photodetachment spectra. Most, but not all, elements form stable negative ions. Exceptions are, for instance, the noble gases, nitrogen and magnesium. Here, the degree of correlation is insufficient to attach an extra electron. Negative ions of other atoms such as Ca have ground states that are very weakly bound and, therefore, easily destroyed [13, 14]. Some negative ions, such as Be [15], are formed in states that are embedded in continua but are metastable due to violations of the selection rules for Coulomb autodetachment. A prominent example is the 1s2s2p4 P metastable state of He [16, 17], which has a binding energy of 77 meV relative to the 1s2s3 S metastable state of He. Its decay via autodetachment is strongly suppressed by the selection rule on spin.
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coworkers [21, 22] followed soon after. Essentially all information on the structure and dynamics of negative ions comes from controlled experiments in which electrons are detached from the ions when they interact with photons or other particles. Photodetachment is the preferred method since the energy resolution associated with such measurements is typically much higher than that attainable in any particle-induced detachment process. Generally, one or more electrons can be detached from a negative ion following the absorption of one or more photons in the photodetachment process. To date most measurements of the structure of negative ions involve the simplest process of single-electron detachment following single-photon absorption. In this process, the photon transfers energy and angular momentum to the negative ion, which subsequently breaks up into a neutral atom and an electron. The initial energy and angular momentum are conserved and shared in the final state by the residual atom and the free electron.
64.2.1 Photodetachment Cross Sections
The total cross section for photodetachment is proportional to the probability of detaching one electron while leaving the residual atom in its ground state or any of its excited states. The partial cross section describes the detachment probability for leaving the atom in a specific state, which could be either the ground or an excited state. The differential cross section, finally, represents the angular distribution of the detached electrons. Examples of the photon energy dependence of these three types of cross sections are shown in Fig. 64.1. Fermi’s golden rule predicts the photodetachment cross section to be proportional to the product of the square of the dipole matrix element connecting the initial and final states and the density of final states. The general behavior of a photodetachment cross section is that it starts at zero at threshold, rises to a peak that occurs at an energy of a few, to as much as tens of, eV above threshold, and then asymptotically approaches zero at higher energies. In Fig. 64.1a, the total cross section for the photodetachment of Na is given as an example. The increase in the probability of detachment just above threshold depends on the density of final states and the ability of the outermost electron to overcome a possible centrifugal barrier. The decrease in the cross section at higher energies is due to the declining overlap between the initial bound state wave function and the rapidly oscillating wave function representing the emitted electron. On top of the otherwise 64.2 Photodetachment smooth behavior of the photodetachment cross sections different structures occur. One prominent feature that can be Pioneering photodetachment experiments using lasers began observed in certain special cases is called a Wigner cusp (see in the mid 1960s with the work of Hall and coworkers [18– the sharp peak around 2.7 eV in Fig. 64.1a). This structure 20]. The use of a tunable dye laser by Lineberger and arises at the opening of an excited state channel when there
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a Cross section (Mb) 200 8 4s
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Fig. 64.1 Typical photodetachment cross sections. a The solid line shows the calculated total cross section for photodetachment of Na by Liu and Starace [23], whereas the opened (filled) circles indicate the dipole length (velocity) results of Moores and Norcross [24]. The inset shows a magnification of the region from 3.5–4.9 eV where resonances can be observed. The figure is reprinted with permission from [23]. Copyright 1999 by American Physical Society. b Measured partial cross section for 4p3=2 photodetachment channel of K . The solid line is a fit to the Wigner law (Eq. (64.1)), which is used to extract the threshold position from the data. Reprinted with permission from [25]. Copyright 2000 by American Physical Society. c Theoretical differential cross section for F expressed as the asymmetry parameter ˇ [26]. The figure is reprinted with permission from [26]. Copyright 1987 by American Physical Society J
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6 5 Photon energy (eV) b Partial photodetachment cross section (arb. u.) 60 0
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Threshold Behavior
Theoretical aspects of the general subject of threshold laws has been reviewed by Rau [27]. Cross sections for photodetachment are zero at threshold, in contrast to the finite value characteristic of photoionization cross sections. The threshold behavior for systems without Coulomb interaction is determined by the dynamics of the residual atom and the emitted electron. In general terms, Wigner demonstrated that for a two-body final state, the near-threshold cross section depends only on the dominant long-range interaction between the two product particles. The Wigner law [28] determines the energy dependence of the near-threshold cross section for the photodetachment of a single electron from an atomic negative ion. It can be written as D Ak 2lC1 D B.E Eth /lC1=2 ;
0
(64.1)
where A and B are constants, and k and l represent the wavenumber and angular momentum quantum number of the c β-parameter detached electron, respectively; E is the photon energy, and 2.0 Eth is the threshold energy for the photodetachment process. As a result of the electric dipole selection rules, the detached 1.5 electron is represented, in general, by two partial waves l D 1.0 l0 C 1 and l0 1, where l0 is the angular momentum of the bound electron in the negative ion prior to detachment. In 0.5 the case of photodetachment involving electrons with l > 0, 0 the angular momentum dependence arises from the centrifugal force, l.l C 1/=r 2. The 1=2 exponent is associated with –0.5 the density of continuum states. Shorter-range final state in–1.0 teractions, such as those due to the electric dipole polarizability of the residual atom, will not change the form of the –1.5 0 20 40 60 80 100 120 threshold behavior, as long as the potential falls off faster than Photon energy (eV) 1=r. They will, however, limit the range of validity of the Wigner law. The polarizability of the atom increases rapidly with the degree of excitation, making the range of validity is a strong configuration interaction between the ground state correspondingly smaller. There is no a priori way of deterand excited state final continuum channels [23]. Other, more mining the range of validity of the Wigner law in any parcommon, structures called resonances are the result of multi- ticular experiment. Measured threshold data is usually fit to ple excitation in the negative ion (see the inset of Fig. 64.1a). the Wigner law in order to determine the threshold energy, Cusps and resonances will be discussed in more detail in as is shown in Fig. 64.1b in the case of K . It is possible to extend the range of the fit beyond that of the Wigner law. Sects. 64.2.2 and 64.2.3, respectively. 2.1180
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2.1182 Photon energy (eV)
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O’Malley [29], for example, considered the effects of atomic multipole forces on threshold behavior. O’Malley’s formalism does not treat polarization explicitly. This is, however, accounted for in the modified effective range theory of Watanabe and Greene [30]. Anomalous threshold behavior can be observed in cases where there is a coupling of two final state channels as a result of configurational mixing. For example, a sharp feature has been observed in the partial cross section for the photodetachment of Na via the p-wave channel as a result of the strong coupling between this channel and the s-wave channel. The sharp peak at threshold in the p-wave channel is called a Wigner cusp, as can be seen in Fig. 64.1a.
64.2.3 Structure in Continuum Photodetachment cross sections are often modulated by resonance structure at certain photon energies. The structure arises when more than one electron, or a core electron, is excited. These unbound quasi-discrete states are embedded in the continua above the first (or higher) detachment limit and are, therefore, subject to decay via the spontaneous process of autodetachment. The allowed autodetachment process is induced by the relatively strong electrostatic interaction between the outermost electrons. This process causes quasidiscrete continuum states to be very short lived. The decay of a quasi-discrete state in the continuum by autodetachment is manifested as a resonance structure in the detachment cross section, as can be seen in the inset of Fig. 64.1a. The shape of a resonance is determined by the interference between the two pathways for reaching the same final continuum state: direct detachment and detachment via the quasi-discrete state embedded in the continuum. A resonance can be parametrized by fitting it to a Fano profile [31] .q C "/2 ; 1 C "2 „! Er ; "D =2
.!/ D 0
(64.2) (64.3)
section can, within the dipole approximation, be written in the form d (64.4) D Œ1 C ˇP2 .cos / : d˝ 4 Here, represents the total cross section, is the angle between the directions of the linear polarization vector of the photons and the momentum of the outgoing electrons, and P2 .cos / is the second-order Legendre polynomial. The asymmetry parameter, ˇ, characterizes the angular distribution or anisotropy of the photoelectrons. The angular distribution of the free electrons in the photodetachment process has been treated theoretically by Cooper and Zare [32]. They showed that the ˇ parameter, in a central-potential model, can be described by 2 2 C .l C 1/.l C 2/lC1 l.l 1/l1
ˇD
6l.l C 1/l1 lC1 cos.ılC1 ıl1 / 2 2 .2l C 1/ll1 C .l C 1/lC1
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(64.5)
where l is the angular momentum of the electron prior to photodetachment, and ı are the dipole radial matrix elements and the residual phase shift of the interfering waves, respectively. A bound electron in an s-orbital .l D 0/ yields an outgoing electron represented by a pure p-wave. In this case, ˇ D 2 for all photon energies and the angular distribution has a pure cos2 distribution. If, however, the bound state electron is in a p-orbital .l D 1/ prior to detachment, the outgoing electron will be represented by both s-waves and d-waves. Close to threshold the d-wave is suppressed due to the centrifugal barrier, and the s-wave dominates. In this case, ˇ D 0, and the angular distribution is isotropic. At higher energies, the value of ˇ will be determined by the relative amplitudes of the swaves and the d-waves, as well as their relative phases. This behavior can be seen in Fig. 64.1c, in the case of F . Most angular distribution measurements have been performed with a crossed laser and ion beams geometry [33, 34]. However, Hanstorp et al. [35] investigated the interference between swaves and d-waves arising from the photodetachment of O using a collinear beams geometry. They introduced an approximation in the analysis of a measurement of the energy dependence of the ˇ parameter. They showed that the phase difference between the s-waves and d-waves can be determined from the value of ˇ at the minimum. A ˇ-value of 1 means that the two waves are completely out of phase.
where denotes the cross section, 0 the nonresonant cross section, „! the photon energy, Er the resonance energy, the resonance width, and q a shape parameter. The resonance parameters (Er , ) can be extracted from a fit to Eq. (64.2). If the selection rules on the allowed Coulombinduced process are violated, the state may live much longer. Such metastable states eventually decay via autodetachment 64.2.5 induced by weaker higher-order interactions.
Higher-Order Processes
With the advent of high-power pulsed lasers, it has become possible to observe multiphoton detachment processes. Here, a single electron is ejected following the absorption of two or For an unpolarized target, such as an ion beam, the angular more photons. Hall et al. [18] performed the first such meadistribution of the photoelectrons, i.e., the differential cross surement on I . Early work in this area has been reviewed by
64.2.4
Photoelectron Angular Distributions
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Blondel [36]. More recently, Bergues et al. [37] used a shortpulse, high-intensity laser to study photodetachment in the strong field regime. In contrast, multielectron detachment involves the detachment of two or more electrons following the absorption of a single photon. This process, which appears to be initiated by the detachment of an inner shell electron, requires photons with energies higher than can be generated by conventional lasers. Such measurements can, however, be performed at synchrotron radiation [38–40] and free-electron laser sites [41].
64.3
Experimental Procedures
Experimental investigations of negative ions have been performed using ion traps [16], linear accelerators [42], and storage rings [43]. The focus of this article is on the use of photodetachment in accelerated-based measurements, which is by far the most commonly used method.
64.3.1 Production of Negative Ions Negative ions are created in exoergic attachment processes when an electron is captured by an atom or molecule. These quantum systems are weakly bound with diffuse outer orbitals. As a consequence, they are easily destroyed in collisions with other particles. The production of negative ions with a density sufficiently high for spectroscopic studies poses a challenge to the experimentalist since processes involved in their creation must compete with more probable destruction processes. The most versatile source for the production of negative ions for accelerator-based experiments is the Cs sputter ion source [44]. This source has been used to generate a wide variety of atomic, molecular, and cluster negative ions. Alternatively, negative ions can be produced in a plasma source, either by direct extraction [45] or by extraction of positive ions that are converted to negative ions through a sequential double-charge transfer in an alkali vapor [15]. Fig. 64.2 Schematic showing both the crossed and collinear beam geometries. Negative ions (solid black line) are bent twice using two electrostatic deflectors. The trajectories of positive ions (dotted black line), neutral atoms (dashed black line), photoelectrons (blue lines), and the laser beam (red lines) are also shown. CEM represents “channel electron multiplier” detectors
64.3.2 Interacting Beams The most common method used for spectroscopic studies of negative ions is to form a unidirectional beam in an accelerator. Here, the ions are extracted from the ion source and focused to form a collimated beam that is accelerated to a desired energy of typically 1–30 keV. A bending magnet is used to select a single mass beam that is essentially monoenergetic. The ions then drift to the interaction region through a beamline that is maintained at low pressure to minimize destructive collisions with the residual gas. Negative ions have also been injected into storage rings, where repeated passes through the interaction region take place. The well-defined spatial dimensions of an ion beam readily permit an efficient overlap with a beam of photons. Figure 64.2 schematically shows two different geometries that have been used in photodetachment measurements. If the ion and laser beam are merged collinearly, the product particle that is usually detected is the residual atom [46]. In this geometry, the laser beam can be either copropagating or counterpropagating with respect to the ion beam. If, however, the two beams are crossed with respect to each other, either the residual atom or the photoelectrons are detected. The crossed beam arrangement is best suited for spectroscopic and angular distribution studies of the detached photoelectrons since they are, in this case, collected from a spatially well-defined interaction region. Further, crossed beams are advantageously used for absolute cross section measurements using the so-called animated beams method [47]. Here, the overlap of the ion and photon beam is determined by sweeping the laser beam through the ion beam. The collinear beam arrangement, on the other hand, yields the best energy resolution. In almost all collinear beam experiments, the residual atoms have constituted the signal. The sensitivity of a measurement is significantly enhanced by the use of the large interaction volume defined by the collinear laser and ion beams. The energy resolution is determined by the convolution of the laser bandwidth and the Doppler broadening. Doppler broadening is the result of the divergence of both the ion and laser beams and the longitudinal velocity spread of the ions that originates from the motion
Electron spectrometer X– e–
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of the ions in the ion source. The broadening due to the divergences of the two beams is greatly reduced by the use of a collinear interaction geometry. In this case, the derivative of the Doppler angle is zero since the two beams are superimposed at a crossing angle of zero. Furthermore, the longitudinal velocity spread is compressed in the collinear geometry [48]. In collinear beam experiments, the energy resolution will essentially be determined by the bandwidth of the light source. There will remain, however, a Doppler shift of the center-of-mass energy arising from the fast-moving ions. This energy shift can be determined experimentally by performing separate experiments, one with the ion and laser beams parallel and another with them antiparallel. The corrected energy can then be determined, to all orders, by taking the geometric mean of the two results [49].
64.3.3 Light Sources Most photodetachment experiments to date have been performed using lasers. CW lasers have been used for highresolution studies but, nowadays, pulsed lasers are most often preferred. The large peak powers characteristic of pulsed lasers are a necessity in, for example, multiphoton experiments. Lasers, in combination with nonlinear techniques such as frequency doubling or optical parametric oscillators, are used in photodetachment experiments to span a wavelength range from the ultraviolet to the infrared. Optical parametric oscillators have also become available commercially. In order to investigate inner shell excitation and detachment processes it is necessary to access the VUV or X-ray region. These regions are still outside the limits of conventional lasers and can only be accessed at synchrotron radiation or free-electron laser facilities.
64.3.4 Detection Schemes Photodetachment events are usually monitored by detecting the particles (electrons, atoms, or positive ions) produced in the breakup of the negative ion. The heavy residual particles, atoms or positive ions, can be detected in both crossed and collinear beam experiments. The neutral atoms are efficiently detected by allowing them to impact on a solid target [50]. The secondary electrons thus produced can then be detected and used as the signal. The major source of background arises from collisions of the negative ions with the atoms and molecules of the residual gas in the vacuum chamber. By maintaining a high vacuum, typically 109 mbar or better, the background contribution can be kept at a tolerable level. The signal-to-noise ratio can be further enhanced by using timegated detection, which becomes possible if a pulsed laser is used. The selectivity and sensitivity of a measurement is im-
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proved significantly if the residual atoms are state-selectively detected, i.e., by performing a partial cross section measurement. This can be achieved using the method of resonance ionization spectroscopy [42]. Here, a second laser is used to excite the residual atom to a state near the ionization limit. This resonance step is followed by electric field ionization of the high-lying, Rydberg state. In this case, the resulting positive ions constitute the signal. An alternative scheme for monitoring photodetachment events involves the detection of photoelectrons. This method is the most efficient if the laser and ion beams are perpendicularly crossed. The total cross section is measured by detecting the yield of photoelectrons, whereas the partial and differential cross sections are obtained by measuring their energy and angular distributions, respectively. Initially, electrostatic analyzers [51] and time-of-flight (TOF) spectrometers [52] were used in such studies. However, more recent measurements have favored a velocity mapping imaging (VMI) technique [53]. In a VMI spectrometer, an electric field is applied to the interaction region to guide all of the emitted electrons onto a position-sensitive detector. Velocity and angular information can be determined from the position on the detector, allowing both energy-resolved and angle-resolved measurements with a detection efficiency approaching 100%.
64.4 Measuring Properties of Negative Ions 64.4.1 Electron Affinities Initially, most electron affinities (EA) were determined using photoelectron spectroscopy since such experiments could be conducted with a fixed-frequency laser, as long as the photon energy exceeded the EA. More recent EA measurements, however, involve the laser photodetachment threshold (LPT) method. Here, the normalized yield of residual atoms arising from photodetachment is recorded as a function of the photon energy in the near-threshold region of the cross section. The threshold position is obtained by fitting the Wigner law (Eq. (64.1)) to the data. The threshold can be accurately determined if a p-orbital electron is detached into a pure s-wave continuum. Then, according to the Wigner law, a sharp E 1=2 energy dependence at threshold is predicted. For many elements, however, an s-wave threshold is either not accessible within the energy range of the available laser, or the lowest energy process involves the detachment of an s-orbital electron. In the latter case, the cross section near threshold will exhibit a slowly rising onset, characteristic of a p-wave continuum. Under such circumstances, a high-resolution study of the EA would require access to an excited state threshold in order to detach into an s-wave continuum. The residual atom will then be left in a specific excited state, which can be detected using the RIS method, as was the case of K , shown
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Photodetachment
in Fig. 64.1b. This technique has been applied, for example, to negative ions of the alkalis [25], alkaline earths [42], and helium [17]. Until recently, all LPT measurements involved stable negative ions. However, the method has now been used at the CERN facility to make the first measurements of the electron affinity of a radioactive atom, in this case the radioactive isotope 128 I [54] and 211 At [55]. The LPT method has been used to accurately determine the electron affinities of many atoms. However, the most accurate affinity measurements to date have been performed using the photodetachment microscope method, pioneered by Blondel et al. [56]. Here, the photodetachment takes place in the presence of a weak electric field that provides two paths for the detached electron to travel from the interaction region to a position sensitive detector. In addition to yielding the highest resolution, the method also provides a beautiful demonstration of quantum interference. The electron affinities of the atoms of the elements was last reviewed in 1999 by Andersen, Haugen, and Hotop [8]. Blondel is currently maintaining an updated list [57].
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ion was made on Os by Bilodeau and Haugen [9]. La was first studied using photoelectron spectroscopic methods to determine the energies of several of these bound states [61]. In a more recent photodetachment experiment, Walter et al. [11] used a crossed beams geometry to identify excited bound states in La . A two-step process was employed in which the first photon was used to excite the ion to an intermediate bound state, and a second photon was used to detach an electron from the ion in this state. One of the observed resonances was identified with the intermediate 3 D1 odd parity state that was excited from the 3 F2 even parity ground state. More recently, Kellerbauer et al. [62] showed, by performing a high resolution experiment, that this transition is a promising candidate for laser cooling. In 2019 Tang et al. [12] showed that also in Th there exist an electric dipole transition. Laser cooling of negative ions is of considerable interest to the antimatter physics community since, potentially, it could be used to sympathetically cool antiprotons.
64.4.3 Continuum Processes 64.4.2
Bound States
Long-lived excited states of negative ions with the same parity as the ground state are known to exist. For example, the simplest, associated with the ground state configuration, would be excited fine structure levels and those arising from term spitting. More highly excited metastable states, with different electron configurations but the same parities as the ground state, are also present in a number of negative ions. The energy of such states has been determined using electron spectroscopy [58]. These states are long lived since they can only radiate via forbidden radiative transitions driven by M1, E2, or even higher-order processes. The principal method of measuring the lifetimes of these long-lived states is to use an ion storage ring. The time-varying population of the excited state due to radiative decay can be sampled by laser photodetachment on each trajectory around the ring [59]. The latest measurements have been performed at DESIREE in Stockholm, which is cryogenically cooled to an operating temperature of 13 K, resulting in an extremely low background pressure of 1014 mbar. Such an environment greatly reduces the destruction of the negative ions by particle collisions. The longest lifetime observed so far, obtained in a study of S , is 503 ˙ 54 s [43]. More recently such experiments are now also conducted at the CSR storage ring in Heidelberg [60]. Until relatively recently, it was thought that bound excited states of negative ions that were able to radiate via allowed electric dipole transitions, were nonexistent. Now, however, it has been shown that such states are present in at least three ions, Os , [9] Ce [10], La [11] and Th [12]. The first clear observation of a bound–bound transition in a negative
Detachment continua contain a wealth of structure, and many Feshbach resonances in negative ions have been studied. The simplest negative ion is the two-electron H ion. This three-body Coulomb system is fundamentally important in our understanding of the role played by electron correlation in the atomic structure. The pioneering measurements of the photodetachment of one and two electrons from the H ion were performed by Bryant and coworkers [63, 64]. In this work, the ion-frame photon energy was scanned by Doppler tuning the angle between a relativistic H beam and a laser beam. ASTRID (Aarhus Storage Ring Denmark) was later used in two measurements of resonance structure in H with much higher resolution than the previous measurements [65, 66]. Subsequent, collinear beam experiments by Hanstorp and coworkers revealed resonance structure in the photodetachment continua of the metastable He ion [67] and the alkali-metal negative ions [68, 69]. Synchrotron radiation has been used over the past few years in order to study how negative ions respond to the absorption of high-energy photons. Photons in the VUV and X-ray regions will excite and/or detach inner shell electrons. Multiple electron detachment appears to be initiated by the excitation of a core electron. This process triggers the ejection of one or more valence electrons, either by shake-off or by interactions of the detached core electron with the valence electrons as it leaves the atom. For example, measurements of the absolute cross sections for the detachment of two electrons from the closed shell ions Cl [38] and Li [39] have been reported. The Advanced Light Source (ALS) has been used to investigate resonance structure and multielectron detachment processes, where, for instance, resonances
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associated with the photodetachment of inner-shell electrons have been observed in K [70] and He [71]. Currently, the most intense light source in the UV regime is the freeelectron laser (FEL). Pedersen and coworkers used a FEL to study the branching of single-to-double detachment in the photodetachment of O [41]. The intense light pulse from the FEL permitted the photoelectron spectrum to be studied using a magnetic bottle spectrometer. More recently, multiphoton detachment of negative ions are also conducted at the PETRA III synchotron radiation source [40].
64.5
Investigation of Fundamental Processes
Signal (arb. u.) 3 2 1 0 0 0.4
270 0.8
1.2 El. momentum (arb. u.)
180 90 0
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–90
A weakly bound quantum system such as a negative ion alFig. 64.3 Momentum and angular distributions observed in the strong lows one to study fundamental photon–ion interactions in laser field-induced photodetachment of Br . Reprinted with permission atomic physics. from [76]. Copyright 2008 by American Physical Society
64.5.1
Threshold Studies
are particularly suited for such experiments since the initial state is well defined. In addition, long-range 1/r potentials are absent in the final state arising from photodetachment. Kiyan et al. [53] conducted a number of experiments on multiphoton detachment using a velocity-mapping imaging technique. For example, they used intense femtosecond laser pulses to study excess photon detachment (EPD) in the F ion, in which many more photons were absorbed than were necessary to reach the detachment threshold [37]. It was also observed that the angular distribution of the ejected photoelectrons became highly directional along the laser polarization vector, as can been seen in the case of EPD of Br shown in Fig. 64.3 [76]. Kiyan et al. also used the photodetachment process to create a ground state atom in carbon [77] and silicon [78] that was in a superposition of its fine structure levels. In this superposition state, the atom exhibited spatial oscillations. The oscillations were monitored by applying a second laser, in a pump-probe arrangement, to photoionize the atom.
The photodetachment threshold, as described by Wigner (Eq. (64.1)), is unique in the sense that an atomic process can be described with a very simple analytical expression. Small deviations from this law can be used to observe various physical processes. For example, Larson and coworkers applied this approach to negative ions in order to study the fundamental problem of particle breakup in magnetic [72] and electrical [73] fields. Small oscillations in the threshold cross sections were observed superimposed on the normal Wigner behavior. Atomic dipole polarizabilities are normally positive (as would be expected from a classical model of a dipole in an electric field). However, in certain special cases, an atomic polarizability can be negative, which can only be explained in a quantum context. In recent experiments involving the photodetachment of K [74] and Na [75], anomalous threshold behavior was observed in which the onset of the photodetachment process occurred at a photon energy well above the predicted threshold energy. This was attributed to 64.6 Observations and Applications of Negative Ions the fact that the highest members of a manifold of excited states in alkali atoms have a negative polarizability. In such Negative ions occur naturally, in both astrophysical environcases, the Wigner law is no longer valid. ments (plasma state) and terrestrial environments (matter in the liquid and gaseous phases). They also play a role in nu64.5.2 Photodetachment Using merous applications.
Short Laser Pulses The combination of a weakly bound quantum system such as a negative ion and the strong electric fields associated with intense laser radiation is suitable for fundamental investigations of multiphoton detachment processes. The internal structures of the negative ions of hydrogen and the halogens
64.6.1 Natural Occurrence of Negative Ions Negative ions are difficult to detect in natural environments due to the rarity of bound excited states and, consequently, the lack of emission/absorption spectra. One of the first iden-
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tifications of negative ions in nature was made by Wildt in References 1939 [79]. He suggested that absorption seen in the blackbody radiation spectrum from the sun was the result of 1. Massey, H.S.: Negatives Ions. Cambridge University Press, Cambridge (1976) photodetachment of H in the photosphere. This prediction 2. Smirnov, B.M.: Negative Ions. McGraw-Hill, New York (1982) was confirmed in subsequent calculations and experiments 3. Bates, D.R.: Negative Ions: Structure and Spectra. Adv. At. Mol. by Branscomb and Smith [80]. In 2006, the CH6 ion was deOpt. Phys. 27, 1–80 (1991) tected in the interstellar medium [81]. This first identification 4. Andersen, T.: Atomic negative ions: Structure, dynamics and collisions. Phys. Rep. 394(4-5), 157–313 (2004) was followed by the observation of other ions in the follow5. Pegg, D.J.: Structure and dynamics of negative ions. Rep. Prog. ing years [82]. In addition, negative ions occur in abundance Phys. 67(6), 857–905 (2004) both in liquids, where Cl ions in the sea is the most obvious 6. Rienstra-Kiracofe, J.C., Tschumper, G.S., Schaefer, H.F., Nandi, example, and in gases, such as in the Earth’s atmosphere. S., Ellison, G.B.: Atomic and molecular electron affinities: pho7.
64.6.2
Applications of Negative Ions
Negative ions have a role in numerous applications. Several applications are based on the fact that the outermost electron in a negative ion is weakly bound and, therefore, easy to remove. For example, in the field of thermonuclear fusion, a commonly used method of heating the plasma in a tokamak reactor is to inject an intense, high-energy beam of neutralized D ions. The removal of the outermost electron has so far been achieved by the use of particle collisions; however, the process could be made more efficient if photodetachment were used [83]. The same neutralization principle has also been used to produce fast beams of atoms for particle collision experiments [84]. Currently, one of the most important applications of negative ions is in the field of accelerator mass spectrometry (AMS) [85], which is the most sensitive method for trace element analysis. AMS is based on a tandem accelerator, in which a beam of negative ions produced at the low-energy end of the accelerator is converted into positive ions at the high-voltage terminal. In many cases, however, a major limitation on sensitivity is the presence of interfering atomic isobars in the beam leaving the ion source. The photodetachment process has the potential to overcome such problems by overlapping the negative ions beam with a laser beam prior to injection into the tandem accelerator. The interfering isobar can then be removed from the ion beam by photodetachment provided that the EA of the isobar of interest is larger than the interfering isobar. However, photodetachment cross sections are, unfortunately, typically small due to the nonresonant nature of the bound– free process. As a consequence, only a small fraction of the interfering isobar will normally be detached, even when a powerful laser is employed in a collinear geometry. One viable solution to this problem is to slow down the ion beam in an RF cooler in order to increase the interaction time. It has been shown that using photodetachment in combination with an RF cooler can reduce the presence of an interfering isobar by many orders of magnitude [86]. A successful proof-ofprinicple of this method has been performed at the VERA facility in Vienna, [87].
8.
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David Pegg holds a BSc (Manchester, 1963) and a PhD (New Hampshire, 1970). He has worked at the University of Tennessee, Knoxville (1970–2006) with a research base at Oak Ridge National Laboratory (1971–2000). His research interests lie in the structure and dynamics of positive and negative ions. He has taken part in collaborative research at Lund, Lyon, Brookhaven, and Lawrence Berkeley National Laboratories and is in long-term collaboration with the University of Gothenburg. Dag Hanstorp received his PhD at the University of Gothenburg in 1992 and now holds a professorship at the same university. He was Chairman of the Scientific Council for Natural and Engineering Sciences at the Swedish Research Council 2019–2021. His research is focused on laser spectroscopy of negative ions and optical levitation.
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Contents 65.1 65.1.1 65.1.2
Theoretical Concepts . . . . . . . . . . . . . . . . . . . . 955 Differential Analysis . . . . . . . . . . . . . . . . . . . . . 956 Electron Correlation Effects . . . . . . . . . . . . . . . . 958
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Experimental Methods . . . . Synchrotron Radiation Source Photoelectron Spectrometry . . Resolution and Natural Width
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energy h, angular momentum j D 1, and parity D 1 interacts with a free atom or molecule A, having total energy Ei , angular momentum Ji , and parity i to produce an electron of energy ", spin s D 1=2, orbital angular momentum `, total angular momentum j , and parity e D .1/` and an ion AC with final total energy Ef , angular momentum Jf , and parity f . This process can be written as the reaction .h; j D 1; D 1/ C A.Ei ; Ji ; i / ! AC .Ef ; Jf ; f / C e Œ"; `sj; e D .1/` : (65.1) Conservation laws require that
Abstract
h C Ei D " C Ef ; Theoretical and experimental aspects of the atomic phoJ i C j D Jf C s C ` ; toelectric effect at photon energies up to about 1 keV are presented. Relevant formulae and interpretations are (65.2) i D f e D .1/` f : given for the various excitation and decay processes. Techniques and results of photoelectron spectrometry in Since Ef Ei becomes quite large for inner shells or deep core levels, scattering of low-energy photons involves the reconjunction with synchrotron radiation are emphasized. moval of an electron from a valence or shallow core level. In the low-energy regime, from the first ionization threshold to Keywords h 1 keV, the photoelectric effect accounts for more than photoelectric effect partial cross section double ioniza- 99:6% of the photon interactions in the elements, with elastion natural width Auger decay tic scattering contributing the remainder [1, 2]. Ionization by inelastic scattering, the Compton effect, assumes increasing importance with the higher photon energies and the lower Z elements. Above the first ionization potential, the total 65.1 Theoretical Concepts photoabsorption cross section and the photoionization cross section are essentially equivalent at the lower photon enerScattering of low-energy photons proceeds predominantly gies. through the photoelectric effect. In this process a photon of The cross section if for producing a given final ionic state in the photoionization process is given by D. Caldwell () Physics Division, National Science Foundation Alexandria, VA, USA e-mail: [email protected] M. Krause Oak Ridge National Laboratory Oak Ridge, TN, USA e-mail: [email protected]
if D
ˇ2 4 2 ˛ 2 X ˇˇ hf jTO ji iˇ ; k
(65.3)
where k is the photon momentum, TO is the transition operator, i and f are the wave functions of the initial and final states, and the summation includes an average over all initial
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_65
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states and a summation over all the unobserved variables in expressions derived from a single-particle model [1, 4]. For the final state. A detailed derivation of this expression, in- the central field potential, cluding the different forms for TO , is given in the articles by 4 2 ˛ 2 Fano and Cooper [3] and Starace [4] and in Chap. 25. The if D ao Nn` h 3 total cross section is given by the sum of all these different `C1 ` partial cross sections, if . 2 2 R C RC ; (65.7) 2` C 1 2` C 1
65.1.1
where Nn` is the occupation number of the subshell, and
Differential Analysis
Detailed information about the photoionization process can be obtained most directly in emission measurements, especially those involving the photoelectron. The resulting photoelectron spectrum yields the energy and intensity for a given interaction. Further differentiation is obtained by varying the angle of observation and by a spin analysis of the photoelectron. Hence, electron emission analysis can reveal all energetically allowed photoprocesses connecting an initial atomic state i to a final ionic state f and yield their dynamic properties. When averaged over the spin, the differential cross section dif =d˝ is given in terms of the partial cross section if and an expression involving an expansion in Legendre polynomials of order n with the coefficients Bn X dif if Bn Pn .cos / ; D d˝ 4 n
(65.4)
where the angle is measured between the direction of the emitted electron and the unpolarized incoming photon beam. In the dipole approximation, which describes the dominant process at low energy, only the terms containing P0 and P2 contribute. Then Eq. (65.4) reduces, for a photon beam with linear polarization p, to dif ˇif if D 1C .1 C 3p cos 2/ ; d˝ 4 4
ˇif D
2 `.` 1/R2 C .` C 1/.` C 2/RC
2 .2` C 1/ `R2 C .` C 1/RC 6`.` C 1/RC R cos
: 2 .2` C 1/ `R2 C .` C 1/RC
(65.8)
The subscripts C and refer to the .` C 1/ and .` 1/ channels respectively, and D ıC ı is the difference in phase shift between these two allowed outgoing waves. The parameter R˙ is the radial dipole matrix element connecting the electron in the bound orbital with orbital angular momentum ` with the outgoing wave having orbital angular momentum ` ˙ 1. Effects of the electron correlation on the direct photoionization process can result in values for ˇ which are not reproduced by the Cooper–Zare expression in Eq. (65.8) [4]. The contribution of the different partial waves to the outgoing wave function can, however, be ascertained through the angular momentum transfer formalism developed by Fano and Dill [6]. In this approach, one defines the angular momentum transferred from the photon to the unobserved variables jt as j t D j ` D Jf C s J i ;
(65.9)
where the second portion of the equality results from the con(65.5) servation of angular momentum. For each allowed value of jt the associated transfer can be defined as either parity favored or parity unfavored according to whether the product i f where the angle lies in the plane perpendicular to the diis equal to C.1/jt or .1/jt respectively. (All symbols rection of propagation and is measured with respect to the have the same definition as in Eq. (65.1).) Calculation of the major axis of the polarization ellipse [5]. Then, the differpartial cross section for the production of a given final state ential cross section, or photoelectron angular distribution, is characterized by the values Jf and s is then determined from characterized by the single angular distribution or anisotropy the cross section corresponding to each angular momentum parameter ˇif for a particular process i ! f . For observation transfer according to at the so-called pseudomagic angle m , defined as X .jt / : (65.10) if D 1 1 jt m D cos1 ; (65.6) 2 3p The associated anisotropy parameter ˇif is derived from the differential cross section dif =d˝ becomes proportional a similar sum to the angle-integrated, or partial, cross section if . X X In the absence of correlation effects, the partial cross secif ˇif D .jt /fav ˇ.j t /fav .jt /unfav : tion if for the production of an individual final state and the jt Dfav jt Dunfav corresponding anisotropy parameter ˇif are given by simple (65.11)
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Photon–Atom Interactions: Low Energy
957
The second equation derives from the fact that ˇ.jt / for each parity-favored value must be calculated separately, whereas for the parity-unfavored case ˇ.jt / D 1 always. The physical effect described by the angular momentum transfer approach is the interaction between an electron and the anisotropic distribution of the other electrons in the atom. Thus, it becomes most useful in the case of ionization from an open-shell atom having an extra electron or a hole in a shell with ` 6D 0. An illustrative example is the 3s ionization of chlorine. Here ˇif D 2 identically in Eq. (65.8) because only the single value ` D 1 is allowed in the single-particle model. However, the three possible values, jt D 0; 1; 2 are allowed, of which only the first corresponds to the Cooper– Zare or single-particle, central field result [4]. That ˇ 6D 2 for 3s ionization of atomic chlorine has been demonstrated experimentally [7]. It is generally the case for ionization of elements which are found naturally in the atomic state that there is an equal population in all the fine-structure components of the initial state. This is because of the relatively small energies associated with the fine-structure splitting. (This does not necessarily apply to atomic species generated through a process of molecular dissociation or high-temperature metal vaporization.) Thus, the determination of all cross sections and angular distributions involves an average over these fine-structure components. However, it is possible to generate atoms in which one of the fine-structure components is preferentially populated. In this case there can also be a preferential ionization to a particular J -component of the final ionic state even in the limit in which the electron correlation is neglected, i.e., the geometrical limit. The partial intensities for the production of a given ionic state characterized by the angular momenta Lf Sf Jf by removal of an electron from an orbital ` of a state characterized by Li Si Ji are given by L S J R` f f f
ŒJf ŒLf ŒSf D g.`; Li ; Si ; Lf ; Sf / Œ1=2Œ` 92 8 > ˆ ` 1=2 j j D`C1=2 = < X Œj Li Si Ji : > ˆ ; : j D`1=2 Lf Sf Jf
detection is carried out at a specific angle, and the ionization is from a given fine-structure component of the initial state to a given fine-structure component of the final state. In the latter case, the transverse spin polarization is given by P D
2 sin cos ; 1 C ˇP2 .cos /
(65.13)
for linearly polarized radiation, and by P D
2 sin cos ; 2 ˇP2 .cos /
(65.14)
for unpolarized radiation [11, 12]. The angle is the same as in the angular distribution measurement; the parameter is the spin parameter analogous to ˇ; and P2 .cos / is the Legendre polynomial of order 2. Yet another parameter which describes the differentiation inherent in the photoionization process is the alignment A, which reflects an anisotropy in the quadrupole distribution of the angular momentum Jf of the ion [12]. For the cylindrically symmetric coordinate system appropriate to dipole photon excitation, only one moment A0 of the distribution is nonzero. This is defined by P 2 mj 3mj Jf .Jf C 1/ .mj / P A0 D ; (65.15) Jf .Jf C 1/ mj .mj / where .mj / is the partial cross section for production of a given mj component of Jf . A very useful approach to the interpretation of the alignment can be obtained through the angular-momentum-transfer formalism [13]. In this approach the angular momentum transfer jt is defined as j t D j Jf :
(65.16)
In contrast to the case of the electron angular distribution, it is possible to derive an alignment for each value of jt as a function of the angular momentum Jf of the ion. The net alignment is then the incoherent sum of the contributions corresponding to each (65.12) X X A0 .jt / .jt / .jt / : (65.17) A0 D jt
Here the term in curly brackets is a 9–j symbol, and the notation ŒJ D 2J C 1 is used. The quantities g.`; Li ; Si ; Lf ; Sf / are weighting factors determined solely by the initial-state wave function. For the case in which ` represents a closed shell, these factors are equal to unity [8]. In situations in which the target atoms possess an initial orientation, i.e., have an average value h Jz i 6D 0, or if the ionization is performed with circularly polarized radiation, the electrons which are produced have a net spin [9, 10]. It is also possible that unpolarized atoms which are ionized by unpolarized photons can have a net spin, provided that the
jt
If the photoionization produces an ion in an excited state which decays by photoemission, the parameter A0 is reflected either in the angular distribution of the fluorescence photons I./ or the linear polarization P measured at one angle, typically 90ı , according to 1 I./ D I0 1 h.2/ A0 P2 .cos / 2 3 .2/ C h A0 sin2 cos.2/ cos.2/ (65.18) 4
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or, for D =2 and D 0, P D
3h.2/ A0 I. D 0/ I. D =2/ ; D I. D 0/ C I. D =2/ 4 C h.2/ A0
(65.19)
respectively. The angle is the angle at which the fluorescence is determined, and the angle is measured between the axis of the polarization selected by the detector and the quantization axis. The polarization of the fluorescence is given by D .cos ; i sin ; 0/. The quantity h.2/ is a ratio of 6–j symbols depending on the angular momenta Jf of the intermediate ion and the final state Jf0 J J 0
h.2/ D .1/ f f ( Jf Jf 1 1
2 Jf0
) ( Jf 1
Jf 1
2 Jf
) :
(65.20)
When it is energetically allowed, a hole in a shallow innershell will preferentially undergo Auger decay, emitting an electron with an energy "A determined by the energy difference between the energy Ef of the ion and Ef0 of the state of the doubly-charged ion to which the decay occurs. Angular analysis of the Auger electrons reflects the alignment of the intermediate ionic state, which is different from, and does not bear a one-to-one relationship to, the angular distribution parameter ˇ of the photoelectrons. Normally, Auger decay is regarded as a two-step process in which the first step is the production of the hole and the release of the primary photoelectron, followed by the decay and the release of the second electron. Within this approximation [14], the angular distribution of the Auger electrons takes on the simple form I0 (65.21) Œ1 C ˛2 A0 P2 .cos / : I./ D 4 Here P2 .cos / is the second-order Legendre polynomial, and ˛2 is the matrix element corresponding to the Auger decay. For the specific case in which the Auger decay is to a final ionic state of 1 S0 symmetry, ˛2 is purely geometric, and a measurement of the angular distribution leads directly to a determination of the alignment. Correspondingly, if the alignment of a specific state can be determined through such a decay, then analysis of the angular distribution of the decay to other states provides a value for ˛2 .
65.1.2
known, theory can focus on the many-body aspects of the process (Chap. 24). Electron correlation manifests itself in many ways. Most prominent are the appearance of autoionization structure due to the excitation of one or two electrons, the production of correlation satellites due primarily to the ionization of one electron accompanied by the excitation of another, and the creation of two continuum electrons in a double ionization process. Autoionization resonances are perhaps the oldest known features associated with electron correlation (Chap. 26). These features arise when the absorption of a photon creates a localized state which lies in energy above at least one ionization limit. This state is then degenerate in energy with a state of an electron in the continuum, and the interaction between these states results in the decay of the quasi-localized state into the continuum. Such resonance states appear in an absorption spectrum in the form of strong, localized variations over an energy range characteristic of the width of the feature, which is in turn related to the lifetime of the state by D „= :
(65.22)
In contrast to absorption features between bound states, autoionization resonances are characterized by having an asymmetric line shape. When only one localized state and one continuum are involved, these line shapes can be derived analytically, as first shown by Fano [15] and later by Shore [16], resulting in simple parametrized forms which are suitable for numerical calculation of overlap integrals for determining widths. For the Fano profile ./ D a
. C q/2 C b ; 2 C 1
(65.23)
E Er ; . =2/
(65.24)
with D
the parameter q describes the asymmetry of the line, Er is the resonance position, and is the width of the line. The parameters a and b reflect the relative contributions to states in the continuum which do and do not interact with the autoionizing state respectively. The energy Er does not correspond to the peak energy Em of the resonance feature but is related to the maximum through
Electron Correlation Effects Em D Er C
: 2q
The primary focus of advanced studies in photoionization is to determine the role played by electron correlation in The Shore profile, the structure and dynamics of electron motion above the A C B lowest ionization threshold. Because the form of the inter; ./ D C./ C 2 C1 action potential for the Coulomb interaction is very well
(65.25)
(65.26)
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Photon–Atom Interactions: Low Energy
959
describes the same phenomenon except that the interpretation of the parameters A and B is different. In this case, they represent products of dipole and Coulomb matrix elements. C.) is the continuum contribution. From an experimental point of view, the parametrized forms for the Fano and Shore profiles are very useful as a basis for fitting autoionization spectra. However, they both have the limitation that they only describe the interaction of an isolated state with the continuum. While they can be extended to include several continuua [4], they do not allow for an interaction among two or more localized states [17, 18]. Nevertheless, it is possible to use these functions to achieve often good fits of states which do interact with each other, as these functions are mathematical representations of localized resonances in a continuous spectral distribution. If this is done, the parameters no longer have the physical meaning which they have for the noninteracting case. Mixing of discrete ionization channels with competing continuum channels adds complexity to the photoionization process, not just in the classical autoioinization regime but also in the vicinity of inner shells [19–21]. In a rigorous application of the Mies formalism [17], feasible with modern computer power, even complex experimental spectra can now be satisfactorily interpreted and reproduced. A case in point is the excitation spectrum from the 2p level of the openshell chlorine atom [19]. The process of autoionization is discussed in more detail in Chap. 26. In Fig. 65.1 an example is shown of the set of 2s2 p3 .4 S/ ! 2s2p3 np; n 3 autoionization resonances which decay into the 3 P ground state of the NC ion [22]. The energies En of these resonances are related to the ionization limit E1 of the series by the Rydberg formula En D E1 R1 =.n s /2 ;
(65.27)
where n is the principal quantum number and s is the quantum defect characteristic of a given series and reflecting the short-range electrostatic interactions of the electron with the ion core. Values of s for s, p, d, and f electrons have been calculated for atoms and ions up to Z D 50 [23]. For high precision work, the reduced Rydberg constant RM D
R1 1 C 5:485 799 104 =.MA me /
(65.28)
should be used instead of the value R1 for infinite nuclear mass (Chap. 1). The atomic mass MA and the electron mass me are in a.u. A process closely related to the autoionization phenomenon is resonant Auger decay. This process differs from the ordinary Auger process [24] in that an electron from an inner shell is not ionized but excited to either a partially filled or an empty subshell. It may be viewed either as an Auger process or as autoionization. Such an inner-shell excited state
Cross section (Mb) 30
N+(3P)
25 20 15 10 5 n=3 0 17.5
18.0
4 18.5
19.0
5
6 78
19.5 20.0 20.5 Photon energy (eV)
Fig. 65.1 Autoionization resonances 2s2 2p3 .4 S/ ! 2s2p3 np in atomic nitrogen
of a neutral atom (molecule) lies above one or more of the ionization limits of the singly ionized species and consequently must decay by electron emission unless the decay is forbidden by selection rules. As a result, resonance structure will appear superimposed on the continua of direct photoionization from the various subshells. From a most general point of view, the resonant Auger process can be considered as resonances in the continua of single photoionization, while the ordinary Auger process can be regarded as resonances in the continua of double photoionization. If excitation proceeds to a partially filled subshell within a principal shell, as, for example, Mn 3p ! 3d [20], interference between the direct photoionization channels and the indirect resonance channel may be strong, and the lineshapes are given by Eq. (65.23) with arbitrary q values and a =b ratios. If, however, the excitation proceeds to an empty shell, as, for example, Mg 2p ! ns or nd, n 4, interference with the direct channels is likely to be negligible, and the resulting resonances are distinguished by essentially Lorentzian line shapes (as for normal Auger lines) with q 1 and a =b 1 in Eq. (65.23). For a given excitation state a number of resonance peaks may arise because more than one ionization channel is usually available and, in addition, the excited electron can change its orbital from n to n0 D n ˙ 1; 2; : : : in a shakeup or shakedown process [25]. As a consequence of the electron–electron interactions which occur simultaneously with the electron–photon interaction, ions are produced in states which do not correspond to those which would be expected based on an interpretation using an independent particle model, which allows for only a single-electron transition (Chap. 25). Evidence for these states appears as correlation satellites in the photoelectron spectrum, the Auger electron spectrum or the X-ray spectrum [26]. Figure 65.2 [27] presents as an example the photoelectron spectrum of argon produced by photons with h D 60:6 eV. In addition to the 3s main line of single electron
65
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D. Caldwell and M. Krause
103 counts /channel 55° 4
3s 13
3
22
10 2
27
4
16
1
1
29
/ E .21/=4 ;
6
0 28
cesses, the threshold cross section is finite, as it is for single electron photoionization, in accord with Wigner’s theorem [30] (Sect. 64.2.2). In the case of double photoionization, the cross section is zero at threshold and then rises according to Wannier’s law [31] (Chap. 56). For the motion of two electrons with essentially zero kinetic energies in the field of the ionic core,
31
34
37
40 43 46 Binding energy (eV)
Fig. 65.2 Photoelectron spectrum (PES) of 3s; 3p satellites in argon at a photon energy of 60:6 eV. Note the reduced intensity of the satellites compared to the 3s main line
photoionization (and the 3p main lines not shown) numerous satellite lines are seen as the manifestation of two-electron transitions involving ionization-with-excitation correlations. It is convenient to categorize the satellites in a photoelectron spectrum according to various electron correlations, as, for example, initial state interactions which mix different configurations into the initial state, and final state interactions, which include core relaxation and electron–electron interactions in the final ionic state, electron–continuum, and continuum–continuum interactions. While initial-state correlations are essentially independent of the photon energy, final-state correlations depend on the energy of the photon through the interactions with the continuum channels. However, the heuristic value of placing correlation effects into a strict classification scheme is limited by the fact that their relative strengths depend on the basis set used in a particular theoretical model and its expansion into a fully correlated system within a given gauge [3, 4, 28, 29] (see also Chap. 25). Another manifestation of double-electron processes is the simultaneous excitation of two electrons to bound states. These states may decay by electron or photon emission and are seen as resonance structures above the thresholds of inner-shell ionization or near autoionizing members of Rydberg series. As single or double ionization continua are usually strong in the spectral range of the double excitations, interference occurs, and the lineshapes can display dispersion forms. Typically, the cross section for the sum of all correlation processes is between 10 and 30% of that for single photoionization, but may exceed this range considerably in special cases. Photon scattering near thresholds is complex because of the possibility of strong interactions between the various particles created and the different modes of deexcitation (Chap. 66). In the case of ionization-with-excitation pro-
(65.29)
where depends on the value of the nuclear charge Z through D
1 1 .100Z 9/ 2 : 2 .4Z 1/
(65.30)
For Z D 1 the Wannier exponent has the value 1:127. In the case of Auger decay following ionization at threshold, interaction between the two electrons results in a shift in the energy of the Auger electron and a corresponding shift in that of the photoelectron (to conserve energy), as well as an asymmetry in the shape of the Auger electron peak and a corresponding asymmetry in the photoelectron peak shape. In this so-called post-collision interaction (Chap. 66), the lineshape, averaged over angles, has the form [32] K."/ D
. =2/ f ."/ ." "A /2 C . =2/2
(65.31)
with 1 " "A f ."/ D exp 2 tan : sinh. / . =2/
(65.32)
In the above equations, " is the energy of the Auger line, "A is the nominal Auger energy, p is thepinitial hole-state width, and the parameter D 1= 2"e 1= 2"A , with "e being the energy of the photoelectron, and "A "e .
65.2
Experimental Methods
An overview of the experimental approaches to the study of photon interactions at low energies is given in Fig. 65.3. The sketch emphasizes the interaction of a polarized photon beam with a small static or particle-beam target of atoms, ions, molecules, or clusters, and the detection of the reaction products at various angles in a plane perpendicular to the direction of propagation of the photon beam, where the general equation in Eq. (65.5) is valid. Emission products, such as electrons, ions, or photons, may be studied by way of the total yields, which can be related to the total photoionization cross section, or by differential analysis in a spectrometer according to energy, intensity, emission angle, and polarization.
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Photon–Atom Interactions: Low Energy
Dispersive element and detector ˆz
961
Photon monitor (flux, polarization)
yˆ
Total yield detector (ions, electrons, photons) E
Photon source (laser, lamp, plasma, synchrotron)
θ
xˆ
Dispersive element and detector (electron, fluorescence, or ion spectrometer, spin or polarization analyzer) Target source (oven, gas discharge, plasma, static gas, particle beam)
Fig. 65.3 Generic arrangement for detection of particles in an emission measurement. The incoming radiation is assumed to be linearly polarized along the z-axis
at a critical photon energy given by hc D Ee2 , where Ee is the energy of the electrons in the ring and is a constant characteristic of the ring. Synchrotron radiation can also be generated by introducing additional magnetic field structures [41] into the ring, such as undulators or wigglers, which produce a deviation of the electron motion from a straight path in a well-defined manner. Wiggler radiation has the same spectrum as a bending magnet, except that the critical energy is generally much higher because the effective magnetic fields can be larger than those of the bending magnet. Undulator radiation is very different in that it consists of a sharp spiked profile of about a 1% bandwidth at energies determined by the magnetic field within the undulator and by the electron beam energy in the storage ring. Synchrotron radiation, no matter what the magnetic field structure of the source, requires monochromatization before it can be used for experiments. For the wavelengths of interest in low-energy photon scattering, this can be achieved by using grating instruments with a metallic coating on the grating surface. The highest resolution possible is obtained through the use of a normal incidence monochromator (NIM) with a plane grating set at normal incidence. However, because the reflectivity of the metallic coating at normal beam incidence decreases drastically as the photon energy increases, use of a NIM has an upper limit of about 40 eV. At energies above this, up to about 1 keV, gratings can still be used but must be mounted at grazing incidence. There is a number of functional designs for these grazing-incidence instruments which vary in the shape of the grating – spherical, toroidal, or plane surfaces (SGM, TGM, PGM) – and the associated optics. Above 1 keV, gratings are no longer suitable, and crystal diffraction must be used. While the radiation emerging from a beamline which couples the monochromator to a bending magnet, wiggler, or undulator has a high degree of linear polarization, varying from 80 to 99% in the plane of the electron orbit, a useful flux of circularly polarized radiation can be derived from out-of-plane radiation [33], by the use of multiple reflection optics, or from a helical undulator.
The various particles may be measured independently, simultaneously, or in coincidence. The photon monitor provides the information for normalization of the data with regard to flux and polarization. The photon monitor can also be used for a measurement of the total photoionization cross section, equivalent to the photoabsorption or photoattenuation coefficient at low photon energies. For this purpose, the size of the target source is advantageously increased in the direction of the photon beam. While experimental apparatus differs, sometimes drastically, for the photon sources as well as for the spectrometry of electrons, ions, and fluorescence photons, many features are common, and the relationships of the measured quantities to basic properties of the atoms and the photon–atom interaction are similar. Thus, the following will place emphasis only on the roles of the synchrotron radiation source and photoelectron spectrometry, whereas specific references to other methods can be found elsewhere [5, 33–38]. 65.2.2
65.2.1
Synchrotron Radiation Source
The primary source of photons over a broad energy range for experiments in the VUV and soft X-ray region of the spectrum is the synchrotron radiation source [39, 40]. In a synchrotron or electron storage ring, radiation is produced as the electrons are bent to maintain the closed orbit. Such bending magnet radiation is emitted in a broad continuous spectrum which begins in the infra-red and ends sharply
Photoelectron Spectrometry
The primary particle emitted in photoionization is the photoelectron. Hence, a photoelectron spectrum provides a detailed view of the photon interaction by (a) specifying the individual processes from an initial state i to a final state f by way of the electron energy, (b) determining their differential and partial cross sections by recording the number of electrons as a function of emission angle, and (c) measuring the polarization of the electrons by a spin analysis (spin polarimetry). The experimental approach is governed largely by the relations in Eqs. (65.3), (65.5), and (65.6). The num-
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D. Caldwell and M. Krause
ber of electrons Nif (e) detected per unit time at an angle within an energy interval d" and within a solid angle d˝ is given by Nif .e/ D GN.h/N.A/f .h/f ."/
dif d" d˝
(65.33)
where G is a geometry factor, which includes the source dimensions, N.h) the number of photons, N.A/ the number of atoms in the source, f .h) and f .") efficiency factors depending, respectively, on photon and electron kinetic energies, and dif =d˝ is the differential cross section for a particular transition i ! f . Equation (65.33) assumes that d˝ and d" are sufficiently small that integration over the pertinent parameters is not needed. Since Nif / dif =d˝, a measurement at two angles, e.g., D 0ı and 90ı , yields the electron angular distribution parameter ˇif according to Eq. (65.5), and a measurement at m in Eq. (65.6) yields the partial cross section if . In the case of closed-shell atoms, n`j notation is sufficient to designate single ionization to an "` continuum, e.g., 3p1=2;3=2 ! "s or "d in argon, but for open-shell atoms LSJ notation is required, e.g., 3P5 .2 P03=2 / ! 3P4 .3 Pe2;1;0 ; 1 De2 ; 1 Se0 /"`. 2 De ; 2 Pe ; 2 Se / in chlorine. Similarly, for ionization-with-excitation transitions, the final state requires an open-shell designation. The sum of the partial cross sections is equal to the total photoionization or absorption cross section X if (65.34) tot D i;f
where the if encompass (a) single ionization events in all energetically accessible subshells n`j , or the LSJ multiplet components, (b) ionization-with-excitation events (shakeup or shakedown), and (c) double ionization events (shakeoff). All if can be determined from a photoelectron spectrum from its discrete peaks (cf. Fig. 65.2) and from the continuum distribution of multiple ionization. However, the latter process is measured more readily by observing the multiply charged ions in a mass spectrometer. The differentiation afforded by measuring the various partial cross sections, and the associated ˇ parameters, can be augmented by differentiating the continuum channels according to the spin using a spin polarimeter [33]. In closed-shell atoms, this allows for an experimental determination of the relevant matrix elements and phase shifts, and hence for a direct comparison with theory at the most basic level [34, 35]. In a more global measurement, the cross section tot is obtained by ion or mass spectrometry from tot D .AC / C .A2C / C .A3C / C :
(65.35)
Generally, the charge states can be correlated with the various initial photoionization processes if allowance is made for
Auger transitions and the fluorescence yield upon exceeding the binding energies of core levels. If the charge states are not distinguished, as in a total ion yield measurement, tot is obtained directly. Similarly, a direct measurement of the global quantity tot is obtained by the total electron yield, although care must be exercised to avoid discrimination by angular distribution effects. At photon energies below about 1 keV, ionization, absorption, and attenuation are virtually equivalent, and tot can also be determined in an ion chamber setup [36] or in a photoabsorption measurement in which the number of photons N absorbed in a source of length d and having an atom density n is given by N D N.ph/Œ1 exp.tot nd /
(65.36)
with N (ph) being the flux of incident photons. As a rule, in all experiments employing the relation in Eq. (65.5), the total, partial, or differential cross sections are determined on a relative rather than an absolute scale because it is very difficult to know accurately such factors as the geometry of the source volume and the number density. However, once a single absolute value of tot or any if is available, all relative values of the other quantities can be converted to absolute values. An electron spectrometric experiment can be carried out in three different operational modes, as defined in Fig. 65.4. In the most conventional mode, PES, the photon energy is fixed, and a scan of the electron kinetic energy reveals all the electron-emission processes possible and yields their properties. The CIS (constant ionic state) mode is especially suited to follow continuously a selected process as a function of photon energy by locking onto a given state Ef Ei (denoted by EB ) which requires a strict synchronization of the photon energy (h) and electron kinetic energy (") during a scan. This mode is particularly advantageous to elucidate resonance features, such as autoionization resonances. Finally, a CKE (constant kinetic energy) scan allows one to access various processes sequentially or, most importantly, follow a process of fixed energy, such as an Auger transition, as a function of photon energy. This description also includes the technique of zero-kinetic-energy measurements. Most frequently, the PES and CIS modes are employed, and Fig. 65.5 gives a self-explanatory example of an actual experiment directed at the characterization of the argon 3s ! np autoionizing resonances. It should be stressed that the cross section tot can be partitioned into its components by CIS scans that differentiate between the 3p1=2 and 3p3=2 doublet states. Energy analysis can be performed either by electrostatic energy analyzers or by time-of-flight techniques. The latter is well-suited to those electrons which have very low ki-
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Photon–Atom Interactions: Low Energy
963
65.2.3 Resolution and Natural Width
Photon energy, hv
CIS (EB = hv – = const.)
CKE ( = hv – EB = const.) EB
PES ( hv = EB + = const.)
Electron kinetic energy
Fig. 65.4 Energy relationship among the three different operational modes of the technique of photoelectron spectrometry. EB is the binding energy of the level
PES
3s3p6np(1p10) n = 100
E
3s23p5 l 3p1 2
3p3 2
2
1
n=6 n=5
n=4 3s np 26.62 eV
CIS –σtot
3p
l
3s23p6(fs0)
Fig. 65.5 Connection between the PES and CIS techniques as illustrated by the 3s ! np autoionization resonances in argon
netic energy, including threshold electrons with " 0 eV. Of the electrostatic energy analyzers now in use, two designs are prevalent, the cylindrical mirror analyzer (CMA) and the hemispherical analyzer, where the latter readily lends itself to the application of multichannel detectors.
The details that can be gleaned from an experiment using photons depend on the resolution achievable with the particular photon source and spectrometry used, the particular excitation or analysis modes and the target conditions chosen, and, ultimately, on the natural width of either the levels or transitions examined as well as any fine structure present. Generally, the instrumental and operational resolution should approach, but need not exceed by much, the natural width inherent in the photoprocess under scrutiny. The demands are most severe for processes involving outer levels because of their typically very narrow widths, and are relatively mild for processes involving inner levels [5, 35]. It is desirable that in the former case the resolving power (the inverse of the resolution) of the instrument exceed 105 , while in the latter case 104 may suffice. If the target atoms move randomly, a resolution limit is set by the thermal motion, namely " D 0:723 ."T =M /1=2 .meV/ ;
(65.37)
65 where, in an experiment involving photoelectron spectrometry, " is the kinetic energy of the photoelectron in eV , T the temperature in K, and M the mass in a.u. of the target atom. This contribution can be limited in first order by employing a suitably directed atomic beam. The experimental peak-width and shape generally contain the natural width; the extent to which instrumental factors enter depends on the specific experiment. In a measurement of the total or partial photoionization cross section in which either the fluorescence photons, the electrons, or the ions are monitored, the resolution of the photon source (often called the bandpass) is the only instrumental contributor. This applies specifically to the CIS mode of electron spectrometry, in which features are scanned in photon energy and the electron serves solely as a monitor. However, in such a CIS study, or a corresponding fluorescence study, the resolution of the electron or fluorescence spectrometer must be adequate to be able to distinguish adjacent processes. For the example of Fig. 65.5, the 3p3=2 and 3p1=2 levels of Ar need to be separated if the partial photoionization cross sections are to be determined across the resonances. In such a case of more than one open ionization channel, the natural widths of the features will be identical in all channels [4], but the shapes may be different. In emission processes subsequent to initial photoionization, namely electron (Auger) decay or photon (X-ray) emission, the resolution of only the spectrometer performing the detection counts on the instrumental side. In the PES mode (Fig. 65.4) the observed lines contain contributions from all sources, the photon source or photon monochromator, the electron analyzer, thermal broadening, and the natural level width. In the special case of photo-
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processes near inner thresholds, the post-collision interaction influences the position and shape of photoelectron and Auger lines (Chap. 66). Excluding threshold regions and the resonant Raman effect, the line profile observed in the various experiments is given by the Voigt function Z (65.38) V .!; !0 / D L.! 0 !/G.! 0 !0 /d! 0 : In this the Lorentzian function L represents the natural level or transition profile, and the Gaussian function G is representative of the window functions of the dispersive apparatus. Although the integral representing the Voigt profile has no analytic form, it can be represented for practical purposes by the analytic Pearson-7 function [42] C ." "0 /2 ; P7 ."/ D A 1 C B 2C
(65.39)
where A is the peak height, "0 the peak position, B the nominal half-width-half-maximum of the peak, and C the shape of the peak. In the limit in which C D 1, this function is identically a Lorentzian; in the limit C ! 1, the function is essentially Gaussian. Use of this function allows one to fit the resulting photoelectron spectrum using standard numerical techniques. If the width of the feature is the only quantity of interest, the simple approximate expression L D1 V
G V
2
Additional Considerations
Although low-energy photon interactions are well described nonrelativistically in the dipole approximation, relativistic and higher multipole effects which become increasingly important at higher energies cannot be ignored even below 1 keV. Spin-orbit effects [44] and relativistic effects [45] are of special significance even at low energies in Cooper minima, where one of the transition matrix elements becomes zero. Moreover, the use of intermediate coupling, which includes both the spin-orbit and electrostatic interactions [46], is required in open-shell systems, as exemplified for the halogen atoms and atomic oxygen [47]. Level energies of heavy elements also require a relativistic treatment [48] (Chap. 23), and it is natural to employ relativistic formulations for calculating the spin parameters appearing in photoionization [49]. Although low energy photon scattering is dominated by the dipole contribution, experiments and theory have shown higher multipole effects to be present at h 0, and this energy appears as excess kinetic energy of the products after the collision. For an endoergic process, Q < 0 and must be provided by the initial kinetic energy of the reactants, so that the corresponding cross section is usually small at low collision energies. If Q is zero or very small, it is a resonant or near-resonant electron capture process, and the resulting cross section is appreciable even at very low energies. Electron capture by multiply charged ions from atoms is predominantly an exoergic process, for which cross sections
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Fig. 68.1 Typical cross section variation with collision energy for electron capture by a multiply charged ion from hydrogen. The low-energy behavior depends on the structure of the quasi molecule formed during the collision
may also be large at low energies. In this case, electrons are preferentially captured into excited levels of A.qk/C . As shown in Fig. 68.1, the typical cross section behavior for single-electron capture (k D 1) by a multiply charged ion from atomic hydrogen is described by the q 1015 cm2 scaling law. At intermediate and high collision energies, the initial ionic charge is no longer the major determinant of the cross section, whereas at low energies, the cross section strongly depends on the structure of the transient quasi molecule formed during the collision. For the thermal energy region (not shown in Fig. 68.1), the cross section shows a 1/v dependence (v is the collision velocity) due to the attractive ion-induced dipole [1]. Several techniques like gas target, crossed ion beam, merged beam methods, etc., have been developed for total cross section measurements. For a multiply charged ion collision with a neutral atom, as shown in Fig. 68.2, electron capture reaction channels that are moderately exoergic produce curve crossings at internuclear separations where there is sufficient overlap of the electron clouds for electron capture to be a likely process. The Landau–Zener curve-crossing model [2], the classical over-barrier model for single capture [3], and the extended classical over-barrier model for multiple capture [4] are useful in predicting the important final product states, and in providing a semiquantitative interpretation. In the case of single-electron capture by a bare multiply charged ion (of charge q D Z) from a hydrogen atom, the principal quantum number np of the most probable final ionic state is given in
1
2
A+q–1 + B+
40 10 20 Internuclear separation (a. u.)
4
Fig. 68.2 Potential energy curves for the collision of a multiply charged ion AqC with a multielectron atom B
this model [3] by s np D
2Z 1=2 C 1 : Z C 2Z 1=2
(68.2)
The internuclear separation Rp at which the potential curves cross in this case is given by Rp D
2.Z 1/ : Z 2 =n2p 1
(68.3)
Multiple electron capture (k > 1) from multielectron atoms occurs predominantly into multiply excited levels, which stabilize either radiatively, leading to stabilized or true double capture, or via autoionization. The latter process is usually referred to as transfer ionization, AqC C B ! A.qk/C C B mC C .m k/e C Q : (68.4) At low energies, transfer ionization is particularly important in collisions of highly charged ions with multielectron atoms. There has been much discussion of the role of electron correlation in the multiple-electron capture process. An issue is the relative importance of the mechanism whereby several electrons are transferred (in a correlated manner) at a single curve crossing compared with electrons transferred sequentially at different curve crossings. Measurements of the distribution of final product electronic states as well as its angular scattering can provide insight into electron capture collision mechanisms [5]. This motivated the development of experimental detection techniques beyond the total cross section measurement for measuring the nl-resolved differential cross sections (n and l
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997
are the principal quantum number and angular momentum quantum number, respectively) and angular differential cross sections. To this end, traditional electron spectroscopy, energy gain spectroscopy, photon emission spectroscopy, as well as recoil ion momentum spectroscopy, etc., have been developed [6]. A lot of progress has been achieved by measuring nl-resolved differential cross sections. There have been some measurements of the electron dipole and current density following electron transfer [7, 8]. The next generation of electron capture experiment would be expected to measure nlm-resolved differential cross sections (m is the magnetic substate quantum number), i.e., the captured electron is initially prepared in a specific nlm state. Other inelastic ion–atom collision processes, such as direct electronic excitation and ionization, are endothermic, with relatively small cross sections that fall off with decreasing energy below a few tens of keV=u. Exceptions are collisions involving Rydberg atoms and collisional excitation of fine structure transitions, for which the required energy transfer is relatively small. There is a only a small amount of experimental data available for direct excitation and ionization processes at low collision energies.
68.2
In designing the gas cell for measurements of total cross sections, d2 and the beam detector must be large enough that elastic scattering may be eliminated as a contributor to the measured attenuation. Usually d2 is made larger than d1 . Measurement of the gas pressure in a target cell is usually made using a capacitance manometer connected to the cell via a tube whose conductance is much larger than that of the gas cell apertures so that, to a good approximation, the pressure will be the same in both the manometer and the gas cell. For gas jet targets, the effective target thickness NL is usually determined by in-situ normalization to some well-known cross section. The quantity in Eq. (68.5) refers to an effective cross section for removing projectile ions from the incident beam, which is the sum of cross sections for all such processes. In many cases, a single process (e.g., electron capture) is dominant, and primarily describes that process. Whether or not a collision process removes a projectile ion from the reactant beam depends on the configuration of the experiment. For example, the projectile particle may remain physically in the beam after passing through the target, but with a changed charge due to a collision. This would be registered as an attenuation of the primary ion beam if the beam is charge analyzed after the reaction.
Experimental Methods for Total Cross Section Measurements 68.2.2 Gas-Target Product Growth Method
In the present context, a total cross section measurement refers to an integration or summation over scattering angles, product kinetic energies and (frequently) electronic states. The total cross section is usually measured as a function of relative collision energy or velocity.
68.2.1 Gas-Target Beam Attenuation Method The attenuation of a collimated ion beam of incident intensity I0 in a differentially pumped gas target cell or gas jet is related to the collision cross section by I D I0 eNL ;
(68.5)
where I is the intensity after traversing an effective length L of the target gas, and N is the number density of target atoms. For a gas target cell with entrance and exit apertures of diameter d1 and d2 , which are much less than the physical length z of the gas cell, L is given to a good approximation by z C .d1 C d2 /=2. This is valid under molecular flow conditions, for which the mean free path between collisions of target atoms is much larger than the dimensions of the gas cell.
The product growth method is similar to the beam attenuation method; the major difference is that the growth of reaction products is measured rather than the loss of reactant projectiles. The products may be derived from either the projectile beam or the gas target, or both. The main advantage of this method is its higher degree of selectivity of a specific collision process. In addition, the reactants and products can usually be registered simultaneously, or in some cases in coincidence, eliminating the sensitivity of the measurement to temporal variation of ion beam intensity. An important criterion is that the density of the target gas be low enough that single collision conditions prevail (i.e., that the likelihood of an ion passing through the gas target and interacting with more than one target atom is negligibly small). This must, in general, be satisfied in order for Eq. (68.5) to correctly relate the measured attenuation to the collision cross section of interest and is critical to the product growth method [9]. In this case, under single-collision conditions, one may set the number of products Ip D I0 I , and Eq. (68.5) may then be written as Ip D 1 eNL NL : I0
(68.6)
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The approximate expression is useful for NL 1, which is a requirement for single-collision conditions. It is also important that the products not be lost in a subsequent collision in the gas cell, so the magnitudes of cross sections for the loss of products in the target gas must also be considered. The products in such an experiment may be derived from either the projectile beam or the target gas (e.g., collection of slow product ions in a gas cell), or from both in coincidence to enhance the specificity of the method. The method may, in principle, be used to determine either total or differential cross sections, depending on the degree of selectivity of collision products.
xq+
Neutral beam detector Collision region (50 cm)
H
H H+
x(q–1)+
xq+ Faraday cup Channel electron multiplier
Fig. 68.3 The merged beams arrangement used at the multicharged ion research facility at Oak Ridge National Laboratory to study low-energy electron capture collisions of multicharged ions with H [10]
68.2.3 Crossed Ion and Thermal Beams Method
and is given by ”
Replacement of the gas target cell by an effusive thermal beam is advantageous for studying collisions of ions with reactive species such as atomic hydrogen, as well as for collecting slow ion products. The use of accelerating electrodes or grids for slow charged products allows coincident detection of fast and slow products, permitting measurement of ionization as well as electron-capture cross sections. The use of an effusive source or gas jet precludes accurate measurement of the effective target thickness, and in-situ normalization to the cross section for some well-known process is usually employed. A comprehensive discussion of such methods as applied to collisions of multiply charged ions with atomic hydrogen is given by Gilbody [9].
68.2.4
Fast Merged Beams Method
Cross sections for ion–atom collision processes at very low energies have been measured by merging fast beams of ions and neutral atoms [10], as in Fig. 68.3. In this case, is determined from experimental parameters by D
vC v0 1 R qe ; IC I0 jvC v0 j L˝
(68.7)
where R is the number of products detected per second, is the product detection efficiency, q is the charge state of projectile ion, e is the electronic charge, IC is the ion current, I0 is the neutral “current” (number/second) of atoms, vC and v0 the laboratory velocities of the ion and atom beams, vrel their relative velocity, and ˝ is the overlap integral and is a measure of the spatial overlap of the two beams along the merged path L. Rigorously, the overlap of the beams should be determined by integrating the product of the current density jC and j0 of the two beams at every point along the merged path
˝D”
jC .x; y; z/j0 .x; y; z/dxdydz ” : jC .x; y; z/dxdydz j0 .x; y; z/dxdydz (68.8)
A variety of three-dimensional and approximate schemes have been developed to measure the overlap of the two beams (Chap. 4 in [6] and references therein). The relative collision energy Erel in eV=u is given by s Erel E0 EC C 2 D mC m0
EC E0 cos ; mC m0
(68.9)
where EC , mC , and E0 , m0 are the energies and masses of the ion and atom, respectively, and is their reduced mass. For collinear merged beams, D 0 and Erel can be reduced to zero by making the two beam velocities the same. In practice, the finite divergences of the beams place a lower limit on the energy and the energy resolution. The fast neutral atom beam can be created by neutralizing an accelerated positive ion in a gas via electron capture, or by electron detachment of a negative ion beam either in a gas, or by using a laser. In gas collisions, a small fraction of the neutral beam is produced in excited n-levels (with an n3 distribution), which may influence the measurements. With fast colliding beams, the maximum effective beam densities are invariably much smaller than the background gas density, even under ultrahigh vacuum conditions. For example, a typical 10 keV proton beam with a circular cross section of diameter 3 mm and a current I D 10 A has an average effective density n D I =.eav/ D 1:6 106 cm3 (a is the beam cross sectional area, and v is its velocity). It is, therefore, necessary either to modulate the beams or to use coincidence techniques to separate signal events due to beam–beam collisions from background events produced by collisions of either beam with residual gas. A typical twobeam modulation scheme is described in [10]. To eliminate
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the production of spurious signals, the signal detector is gated “on” a short time after the beams are switched, and the beam modulation period is made much shorter than the pressure time constant of the vacuum system. Absolute electron-capture cross sections have been measured for Si4C CD collisions to energies below 1 eV=u, where the attractive ion-induced–dipole (polarization) interaction is expected to play a role [1]. The inverse velocity dependence of the cross section in this region is suggestive of the classical Langevin orbiting model for ion–neutral collisions [11].
68.2.5 Trapped Ion Method
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recoil ion momentum spectroscopy (COLTRIMS) in collisions of ion beams with gas targets.
68.3.1 Photon Emission Spectroscopy At low energies, electron capture from an atom by a charged ion usually populates excited levels. Since the timescale of photon emission deexcitation is several orders of magnitude larger than the collision time, and photon emissions usually show discrete spectroscopic structure, the measured photon spectra contain state-selective information. For example, photon emission spectra can be used to measure emission cross sections and electron capture state-selectivity involving angular momentum quantum states [16, 17]. A calibrated grating or crystal spectrometer can be used to record spectra. By summing the signal under a peak area in the spectra, the emission cross sections can be calculated via the relation
The trapped ion method is used to determine rate coefficients and effective cross sections for ion–atom collisions at near thermal energies. The technique involves storing ions in an electrostatic or electromagnetic trap and measuring the rate of loss of ions from the trap after a small quantity of neuS./ 4 q tral gas is admitted [12]. Like the beam attenuation method, ; (68.10) em D the trap technique cannot distinguish different processes that ! K./Qc NL cause ions to be lost from the trap. The mean collision energy is estimated from an analysis of the ion dynamics in the trap. where em is the emission cross section, ! is the solid anCollisions at temperatures of a few K and below have been gle of observation, q is the charge of the projectile ions, Qc is the accumulated ion charge, K./ is the wavelength disachieved in, e.g, a magneto-optical (MOT) trap. persive quantum yield of the detection system, S.) is the measured signal, N is the effective target density, and L is the observation length. 68.2.6 Swarm Method Similar measurements have also been performed with an x-ray quantum microcalorimeter (XQC) detector based on The swarm method, using the flowing afterglow, drift tube, or cryogenic techniques [18], with which the photon spectral selected-ion flow tube, has been used successfully to study range is extended beyond the visible and vacuum ultraviolet ion–atom collisions at very low energies [13]. Ions are injected into a homogeneous electric field and drift through to the X-ray region. The measurements are able to cover all a suitable low-density buffer gas such as helium. The ions direct and many possible cascade emissions. Complimentamove as a swarm whose mean energy depends on the applied rily, with an electron beam ion trap, those X-ray emissions electric field and on collisions with the buffer gas and can be due to charge exchange arising from forbidden transitions varied from the near-thermal region to tens of eV. The method can be observed [19–21]. These long-lived transitions are not involves measuring the additional attenuation of the directed accessible with the traditional ion extraction mode. ion swarm by a known quantity of added reactant gas and is the major technique that has been used for the study of ion–atom collisions at near-thermal energies [14]. As with ion beam at- 68.3.2 Translational Energy Spectroscopy tenuation and ion trap methods, the drift tube is not selective of the process that leads to attenuation of the ion swarm. Translational energy loss or energy gain spectroscopy provides a convenient method to determine the distribution of final states in low-energy ion–atom collisions that are either 68.3 Methods for State-Selective endoergic or exoergic (Q ¤ 0). For example, this method has Measurements been used extensively by the Belfast group [22] to study the predominantly exoergic process of electron capture by mulFour principal methods have been developed and applied to tiply charged ions from hydrogen atoms. An ion beam with the measurement of partial cross sections for the population a well-defined energy is directed through a gas target, and of specific product electronic states in ion–atom collisions the energy of the product ion beam is energy analyzed after involving electron capture [15]. These are photon emission, the collision. Since the energy gain or loss to be measured is translational energy gain, electron emission, and cold target only a very small fraction of the initial kinetic energy, it is
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usually necessary to reduce the initial energy spread to a few tenths of an eV by decelerating the reactant ion beam prior to energy selection by an electrostatic analyzer. If the scattering angle of the ion is very small, its energy change is approximately equal to Q. Since the ion beam is attenuated by deceleration and energy analysis, cross sections for collisionally populating specific states are determined by normalizing the measured product-state distributions to total cross section data. The attainable state resolution is not as good as for photon emission spectroscopy.
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68.3.3 Electron Emission Spectroscopy As noted in Sect. 68.1, multiple-electron capture by multiply charged ions from atoms at low energies occurs primarily into multiply excited states, which decay either radiatively or via autoionization [5]. The latter decay pathway (transfer ionization) provides an experimental method to determine the product ionic states by ejected-electron spectroscopy. Analysis of electrons emitted into the forward (ion beam) direction (zero degree spectroscopy) offers significant advantages for analysis of low-energy electrons with high resolution [23]. Since a gas jet is often employed, and absolute electron collection and spectrometer efficiencies are difficult to determine, some normalization procedure is usually employed to determine state-selective cross sections by this method. Electrons and product projectile or recoil ions have also been detected in coincidence to increase the specificity of the method.
68.3.4 Recoil-Ion Momentum Spectroscopy
Detector
Fig. 68.4 Schmatic of the COLTRIMS apparatus used for ion–atom electron capture
the primary beam by the electrostatic deflector downstream of the collision zone and directed to a projectile PSD, while the primary beam is collected by a Faraday cup. The PSD detectors can record time and position information of the recoil ions and projectile ions. By making a time coincidence between the recoil ions and scattered projectile ions, the desired electron capture process can be identified, and background events are suppressed. Three-dimensional momentum vectors of the recoil ions can be constructed using kinematical expressions. The binding energy difference of the active electrons in the state-selective electron capture is reflected in the measured longitudinal momentum distributions of recoil ions. Angular differential information is obtained by the measured transversal momentum of the recoil ions, which is equivalent to that of the scattered projectile ions according to momentum conservation. More recently, COLTRIMS was updated by replacing the supersonic gas jet with a MOT trap where more targets are available by laser cooling. This updated COLTRIMS is the so-called MOTRIMS [26], with which electron capture measurements have achieved an unprecedented resolution in lowenergy ion–atom collisions. In addition, it is promising to combine x-ray detecting techniques with COLTRIMS, with which the spectroscopy of charged ions as well as collision dynamics can be studied in electron capture processes [27].
Over the past 20 years, a significant experimental technique development in ion–atom collisions is cold-target recoil ion momentum spectroscopy [24, 25], also called a reaction microscope. Its powerful multifold differential measurements have advanced ion–atom electron capture to the point where the details of state-selectivity become available. It has also been used to identify molecular or cluster geometry configurations and dissociation dynamics. As shown in Fig. 68.4, the core components of a COLTRIMS apparatus are a time-of-flight (TOF) spectrometer, an ultracold supersonic gas jet, and two-dimensional position sensitive detectors (PSD). For an ion–atom electron capture collision, an energetic ion beam interacts with a precooled and localized supersonic gas jet at a right angle in the center References of the TOF spectrometer, which consists of a homogenous 1. Pieksma, M., Gargaud, M., McCarroll, R., Havener, C.C.: Phys. electric field acceleration-region and field-free drift region. Rev. A 54, R(13) (1996) The slow recoil ions produced are accelerated by the elec2. Salop, A., Olson, R.E.: Phys. Rev. A 13, 1312 (1976) tric field away from the collision zone towards a recoil-ion 3. Ryufuku, H., Sasaki, K., Watanabe, T.: Phys. Rev. A 23, 745 (1980) PSD. The charge changed projectile ions are separated from
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4. Niehaus, A.: J. Phys. B 19, 2925 (1986) 5. Barat, M., Roncin, P.: J. Phys. B 25, 2205 (1992) 6. Shafroth, S.M., Austin, J.C.: Accelerator-based Atomic Physics Techniques and Applications. AIP Press, Woodbury (1997) 7. Havener, C.C., Westerveld, W.B., Risley, J.S., Tolk, N.H., Tully, J.C.: Phys. Rev. Lett. 48, 926 (1982) 8. Havener, C.C., Rouze, N., Westerveld, W.B., Tully, J.C.: Phys. Rev. Lett. 53, 1049 (1984) 9. Gilbody, H.B.: Adv. At. Mol. Phys. 22, 143 (1986) 10. Havener, C.C., Huq, M.S., Krause, H.F., Schulz, P.A., Phaneuf, R.A.: Phys. Rev. A 39, 1725 (1989) 11. Gioumoussis, G., Stevenson, D.P.: J. Chem. Phys. 29, 294 (1958) 12. Church, D.A.: Phys. Rep. 228, 254 (1993) 13. Kaneko, Y.: Comm. At. Mol. Phys. 10, 145 (1981) 14. Lindinger, W.: Phys. Scr. T 3, 115 (1983) 15. Janev, R.K., Winter, H.: Phys. Rep. 117, 266–387 (1985) 16. Van Zyl, B., Dunn, G.H., Chamberlain, G., Heddle, D.W.O.: Phys. Rev. A 22, 1916 (1980) ´ c, D., Brazuk, A., Dijkkamp, D., de Heer, F.J., Winter, H.: J. 17. Ciri´ Phys. B 18, 3639 (1985) 18. McCammon, D., Almy, R., Apodaca, E., Bergmann Tiest, W., Cui, W., Deiker, S., Galeazzi, M., Juda, M., Lesser, A., Mihara, T., Morgenthaler, J.P., Sanders, W.T., Zhang, J.: Astrophys. J. 576, 188 (2002) 19. Beiersdorfer, P., Olson, R.E., Brown, G.V., Chen, H., Harris, C.L., Neill, P.A., Schweikhard, L., Utter, S.B., Widmann, K.: Phys. Rev. Lett. 85, 5090 (2000) 20. Allen, F.I., Biedermann, C., Radtke, R., Fussmann, G., Fritzsche, S.: Phys. Rev. A 78, 032705 (2008) 21. Tawara, H., Takács, E., Suta, T., Makónyi, K., Ratliff, L.P., Gillaspy, J.D.: Phys. Rev. A 73, 012704 (2006) 22. Gilbody, H.B.: Adv. At. Mol. Opt. Phys. 32, 149 (1994) 23. Stolterfoht, N.: Phys. Rep. 146, 316 (1987) 24. Dörner, R., Mergel, V., Jagutzki, O., Spielberger, L., Ullrich, J., Moshammer, R., Schmidt-Böcking, H.: Phys. Rep. 330, 95 (2000) 25. Ullrich, J., Moshammer, R., Dorn, A., Dörner, R., Schmidt, L.P.H., Schmidt-Böcking, H.: Rep. Prog. Phys. 66, 1463 (2003) 26. DePaola, B., Morgenstern, R., Andersen, N.: Adv. At. Mol. Opt. Phys. 55, 139 (2008) 27. Ali, R.: AIP Conf. Proc. 926, 216 (2007)
1001 Charles Havener Charles Havener received his PhD in Physics from North Carolina State University in 1983. This was followed by a postdoc and research staff position at Oak Ridge National Laboratory. The merged-beam apparatus he developed measures charge exchange cross sections from meV to keV collision energies for ions on atomic H and is currently used to measure high-resolution x-ray spectra. Ruitian Zhang Ruitian Zhang received his PhD from the Lanzhou University in 2013. He worked at Helmholtz Centre for Heavy Ion Research in Germany, at Oak Ridge National Laboratory and Columbia University in the City of New York, and at the Institute of Modern Physics, Chinese Academy of Sciences in China. He is currently working on experimental atomic collision physics, focusing on heavy ion– atom collision dynamics. Ronald Phaneuf Phaneuf earned a PhD in Physics from the University of Windsor (1973) and has since been engaged in experimental research on interactions of ions with electrons, photons, atoms, and molecules using mergedbeams and crossed-beams techniques. He was a postdoctoral fellow at JILA and researcher and program manager for atomic physics and plasma diagnostics for fusion at Oak Ridge National Laboratory. He is Professor Emeritus (University of Nevada).
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Contents 69.1 69.1.1 69.1.2
Basic One-Electron Processes . . . . . . . . . . . . . . 1003 Perturbative Processes . . . . . . . . . . . . . . . . . . . . 1003 Nonperturbative Processes . . . . . . . . . . . . . . . . . 1009
69.2
Multielectron Processes . . . . . . . . . . . . . . . . . . 1009
69.3 69.3.1 69.3.2
Electron Spectra in Ion–Atom Collisions . . . . . . . 1012 General Characteristics . . . . . . . . . . . . . . . . . . . 1012 High-Resolution Measurements . . . . . . . . . . . . . . 1013
69.4 69.4.1 69.4.2 69.4.3
Quasi-Free Electron Processes in Ion–Atom Collisions . . . . . . . . . . . . . . . . . . . . . . . Radiative Electron Capture . . . . . . . . . . . . Resonant Transfer and Excitation . . . . . . . . Excitation and Ionization . . . . . . . . . . . . .
69.5 69.5.1 69.5.2
Some Exotic Processes . . . . . . . . . . . . . . . . . . . 1014 Molecular Orbital X-Rays . . . . . . . . . . . . . . . . . . 1014 Positron Production from Atomic Processes . . . . . . . 1015
. . . .
. . . .
. . . .
. . . .
. . . .
1013 1014 1014 1014
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015
Abstract
collisions involving outer-shell processes can be studied using relatively small accelerators, while those involving inner-shell processes require larger van de Graaffs, LINACs (linear accelerators), etc. Because the motion of the inner-shell electrons is dominated by the nuclear Coulomb field of the target, and because transitions involving these electrons take place rather independently of what transpires with the outer-shell electrons, it has proven to be somewhat easier to understand one-electron processes involving inner-shell electrons. Thus, for a long time, a great deal of the work on fast ion–atom collisions concentrated on inner-shell processes involving heavytarget atoms. However, more recently, new experimental techniques have led to a shift of this focus to inelastic processes involving light-target atoms. Furthermore, present investigations go beyond the one-electron picture to include the influence of the electron–electron interaction. The present chapter outlines some of the developments in this area over a very active past few decades. The literature is vast, and only a small sampling of references is given. Emphasis is on experimental results (for the theory, see Part D of this book).
This chapter deals with inelastic processes that occur in collisions between fast, often highly charged, ions and atoms. Fast collisions are here defined to be those for Keywords which V =ve 1, where V is the projectile velocity and ve the orbital velocity of this electron. For processes involvdouble ionization Compton profile recoil peak plane ing outer-shell target electrons, this implies V & 1 a.u., wave bear approximation single-ionization cross section or the projectile energy & 25 keV=a:m:u. For inner-shell electrons, typically, V & Z2 =n a.u., where Z2 is the target nuclear charge and n the principal quantum number p of the active electron. A useful relationship is V D 6:35 E=M , where V is in a.u., E is in MeV, and M is in a.m.u. Fast 69.1 Basic One-Electron Processes M. Schulz () Dept. of Physics, Missouri University of Science & Technology Rolla, MO, USA e-mail: [email protected] L. Cocke Dept. of Physics, Kansas State University Manhattan, KS, USA e-mail: [email protected]
69.1.1 Perturbative Processes Inner-Shell Ionization of Heavy Targets For ion–atom collisions involving projectile and target nuclear charges Z1 and Z2 , respectively, the parameters 1 D Œ„V = Z1 e 2 /2 and 2 D Œ„ve =.Z2 e 2 /2 are useful in characterizing the strength of the interaction between Z1 , Z2 , and
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_69
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the target electron. If 2 1 , (i.e., Z1 =Z2 V =ve ), the effect of the projectile on the target wave function can be treated perturbatively. Perturbation treatments of inner-shell ionization by lighter projectiles have been extensively studied and reviewed [1–8]. Two well-known formulations have been used: the plane wave Born approximation (PWBA) [1– 4] and the semiclassical approximation (SCA) [9, 10]. The former represents the nuclear motion with plane waves, while the latter is formulated in terms of the impact parameter b with the nuclear motion treated classically. For straight line motion of the nuclei, the results are equivalent [11]. The total cross section for ionizing the K-shell of a target of charge Z2 by a projectile of charge Z1 is given within the PWBA by
i D 8Z12=Z24 2 f .2 ; 2 / a02 ;
U 2k σk /Z12 (keV2 cm2) 10–19
10–20
10–21
10–22
10–23
(69.1)
where 2 D 2uk n2 =Z22 , and uk is the target binding energy. The function f rises rapidly for V < ve , reaching a value near unity near V D ve and falling very slowly thereafter. Tables of f for K and L-shell ionization are given in [3, 4]. Figure 69.1 shows a comparison of experimental data for K vacancy production by protons with PWBA calculations, and with a classical binary encounter approximation [12] for a large range of proton data [6]. For larger Z1 , corrections to the PWBA and SCA must be made for the effective increase of uk due to the presence of the projectile during the ionization, for nuclear projectile deflection, for relativistic corrections, and for the polarization of the electron cloud, as reviewed in [13–17]. Total cross section measurements for inner-shell vacancy production in the perturbative region are reviewed in [15, 16]. In the SCA treatment, the heavy-particle motion is taken to be classical, and the evolution of the electronic wave function under the influence of the projectile field is calculated by time-dependent perturbation theory. The assumption of classical motion is valid if the Bohr parameter K D 2Z1 Z2 e 2 =.„V / is much larger than unity [18]. If this condition is satisfied, the projectile scattering angle can be associated with a particular b through a classical deflection function. For K-shell ionization, the action occurs typically at sufficiently small b that a screened Coulomb potential is sufficient for calculating the deflection. In the absence of screening, D r0 =b, where r0 D Z1 Z2 e 2 =E with and E expressed in either the laboratory or center of mass system. Calculations for K and L-shell ionization have been carried out [10]. The typical ionization probability P .b/ for V ve and b D 0 is P .0/ .Z1 =Z2 /2 . For V < ve , P .b/ decreases with increasing b with a characteristic scale length of rad D V =!, the adiabatic radius, where ! is the transition energy. For V > ve , P .b/ cuts off near the K-shell radius of the target. A more sophisticated relativistic SCA program has been written [19] and is widely used for calculating P .b/,
10–24
10–25 10–3
10–2
10–1
100
101 E/μUk
Fig. 69.1 Comparison of experimental cross sections for K-shell vacancy production with PWBA (dashed line) and binary encounter (solid line) theories; Uk is the target binding energy in keV and the projectile=electron mass ratio [6]
cross sections, and probabilities differential in the final electron energy and angle. Experimentally, the probability P .b/ for inner-shell ionization can be determined from P .b/ D
Y 1 ; !˝ N Œ.b/
(69.2)
where Y is the coincidence yield for the scattering of N./ ions into a well-defined angle .b/ accompanied by X-ray (or Auger electron) emission with fluorescence yield ! into a detector of efficiency and solid angle ˝ [20]. The necessary ! can be obtained from calculations for neutral targets [21] (Chap. 66). However, they must be corrected for changes due to extensive outer-shell ionization during the collision. Such corrections are particularly important for targets with low fluorescence yields, for Z2 below 30, and for collisions in which the L-shell is nearly depleted in the collision [15, 16]. Values of P .b/ have been measured for many systems and generally show good agreement (better than 10%) with the SCA for fast light projectiles such as protons, with increasing deviation as higher Z1 or slower V are used [22]. Examples of P .b/ for K vacancy production for several systems are shown in Fig. 69.2, showing the evolution away from the SCA as the collision becomes less perturbative.
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P(b) /Z 21 Projectile /Target 10–3
Z2 /Z1
F(5+)/Ar
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O(6+)/Cu
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0.35
10–3
–3
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10–3
p/Cu
29
0.35
SCA 1.0
2.0
3.0
b/rad
Fig. 69.2 P .b/ for K-shell vacancy production versus b=rad for several systems (see text). The ratio V =ve is designated as V here. For protons p, agreement with the SCA theory is found [10], while for higher Z1 =Z2 , P .b/ moves to larger impact parameters as one leaves the perturbative region [22]
Ionization of light-target Atoms Ionization of light-target atoms by bare ion impact is a particularly suitable process to study the atomic few-body problem. In the case of an atomic hydrogen target the collision represents a three-body system, i.e., the simplest system for which the Schrödinger equation is not analytically solvable. However, because of the experimental difficulties associated with atomic hydrogen, measurements with this target species are rare [23, 24] and experimental studies have focused on helium targets. Here, the collision still constitutes a relatively simple four-body system. With regard to the few-body problem, studies of ionization processes have the important advantage that, in contrast to pure excitation and capture processes, the final state involves at least three independently moving particles. Detailed information about the few-body dynamics in a collision can be extracted from multiply differential measurements. This can be accomplished by measuring the kinematic properties (e.g., energy, momentum, ejection angle) of one or more of the collision fragments. The first experimental multiply differential single-ionization cross sections were obtained by studying the ionized electron spectra as a function of energy and ejection angle. Such studies were reviewed by Rudd et al. [25] and are discussed in more detail in
Sect. 69.3. More recently, complementary multiply differential data were obtained by measuring projectile energy-loss spectra as a function of the scattering angle in p C He and p C H collisions [24, 26, 27]. A comprehensive picture of ionizing collisions can be obtained from kinematically complete experiments. In such a study, the momentum vectors of all collision fragments need to be determined. However, in the case of single ionization, it is sufficient to directly measure the momentum vectors of any two particles in the final state; the third one is then readily determined by momentum conservation. For ionization by electron impact, this has been accomplished by momentum-analyzing the scattered and the ejected electrons ([28] for a review). For ion impact, this approach is difficult because of the very small scattering angles and energy losses (relative to the initial collision energy) resulting from the large projectile mass. Consequently, the only kinematically complete experiments involving a direct projectile-momentum analysis were reported for light ions at relatively low projectile energies [29]. For heavy-ion impact at high projectile energies, in contrast, the complete determination of the final space state is only possible through a direct measurement of the ejected electron and recoil-ion momenta [30]. The technology to measure recoil-ion momenta with sufficient resolution, and therefore to perform kinematically complete experiments for heavy-ion impact, has only become available over the last 2 decades ([31–33] for reviews). Figure 69.3 shows measured (top) and calculated (bottom) three-dimensional angular distributions of electrons ionized in 100 MeV=a:m:u: C6C C He collisions for fully determined kinematic conditions [34]. The arrows labeled po and q indicate the direction of the initial projectile momentum and the momentum transfer defined as the difference between p0 and the final projectile momentum pf . This plot is rich in information about the dynamics of the ionization process. The main feature is a pronounced peak in the direction of q. It can be explained in terms of a binary interaction between the projectile and the electron, i.e., a first-order process, and is thus dubbed the binary peak. A second, significantly smaller, structure is a contribution centered on the direction of q (called the recoil peak). This has been interpreted as a twostep mechanism where the electron is initially kicked by the projectile in the direction of q and then backscattered by the residual target ion by 180ı . Although this process involves two interactions of the electron, it is nevertheless a first-order process in the projectile–target atom interaction. Therefore, as expected for this very large value of 1 D 100 (in a.u.), the ionization cross sections are dominated by first-order contributions. The basic features of the data in Fig. 69.3 are well reproduced even by the relatively simple first Born approximation (FBA). Furthermore, the calculation shown at the bottom
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x
y
q
Fig. 69.3 Three-dimensional angular distribution for fully determined kinematic conditions of electrons ionized in 100 MeV=a:m:u: C6C C He collisions. Top, experimental data; bottom, CDW calculation (see text)
of Fig. 69.3, which is based on the more sophisticated continuum distorted wave approach (CDW) ([35–37], also Chap. 56), yields practically identical results to the FBA. In the CDW method, higher-order contributions are accounted for in the final-state scattering wavefunction. Apart from this good overall agreement, a closer inspection of the comparison between experiment and theory also reveals some significant discrepancies. While in the calculation the binary and recoil peaks are sharply separated by a minimum near the origin, in the data, this minimum is almost completely filled up, giving rise to a ring-like shape of the recoil peak. This was explained by a higher-order ionization mechanism involving an interaction between the projectile and the residual target ion [34, 37, 38]. Although the contribution of this process to the total cross section is negligible, it is a very surprising result that for selected kinematic conditions higher-order processes can be important even at large projectile energies. A sobering conclusion of recent research on ionization of light-target atoms is that even well inside the perturbative regime the atomic few-body problem is not nearly as well understood as was previously assumed based on studies for restricted collision geometries. At large perturbation, the lack of understanding is dramatic [39]. Part of these discrepancies may be related to a realization that was recently made in studies of ionization of H2 by proton impact at a much smaller projectile energy. There it was found that interference structures are present in the scattering angle-dependent cross sections when the incoming
projectiles are coherent relative to the dimension of the target molecule but absent if the projectiles are incoherent [40]. The coherence properties of the projectiles, in turn, are basically determined by the width of the projectile wave packet in momentum space. If the momentum spread is too large, the phase relation between the overlapping amplitudes is not well defined, and no interference structure is observable. In contrast, in scattering theory, it is usually assumed that the incoming projectiles can be described in terms of a plane wave or a distorted wave, which corresponds to a sharp momentum, i.e., to fully coherent projectiles. Therefore, theory may predict interference structures that are not observable in experiments if the projectiles are incoherent. In the ionization of atomic targets interference can arise from the superposition of different (nonobservable) impact parameters leading to the same (observable) scattering angle. Formally, this type of interference is reflected in the Fourier transform, which converts an impact parameter transition amplitude to a scattering angle-dependent amplitude. If the projectile is represented by a broad wave packet in momentum space, rather than, e.g., a plane wave, the Fourier transform can be altered considerably. Even in the coherent case, the interference resulting from the Fourier transform is not necessarily reflected by an oscillating pattern in the cross sections. Nevertheless, the effect of the momentum wave packet on the Fourier transform could explain differences between cross sections for coherent and incoherent projectiles and, thereby, the aforementioned discrepancies between theory and experiment on ionization of atomic targets. This interpretation seemed to be confirmed by a study of ionization for a similar collision system as 100 MeV=a:m:u C6C C He, but for projectiles with a much higher degree of transverse coherence [41]. There, the discrepancies to theory were drastically reduced. Furthermore, this interpretation is supported by theoretical analyses [42, 43]. Another experimental study, also performed for a collision system with a perturbation parameter similar to 100 MeV=a:m:u C6C C He, was consistent with calculations assuming coherent projectiles [44]. However, there the smallest coherence length realized in the experiment was several orders of magnitude larger than for 100 MeV=a:m:u C6C and larger than the size of the target atom. A theoretical study showed that under these circumstances, much smaller decoherence effects are to be expected than for 100 MeV=a:m:u C6C collisions [43]. Nevertheless, further studies on the role of the projectile coherence properties are needed for collisions with small perturbation parameters. While for much slower collisions coherence effects have been reported for several cases ([45] for a recent review), the question of the extent to which the discrepancies between experiment and theory for the 100 MeV=a:m:u C6C C He collision system can be explained by such effects has not yet been settled. Initially, recoil-ion and electron-momentum spectrometers were operated in conjunction with supersonic gas jets
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η = 1.0
q = 1.0 au Eel = 1.5 eV
Fig. 69.4 Fully differential, three-dimensional angular distribution of electrons ejected in 7 MeVLi2C C Li collisions
1007
tum space. In order to achieve good momentum resolution this field needs to be as uniform as possible, which seems to be in conflict with the requirement of a field gradient in order to trap the target atoms magnetically. This problem was overcome with a sophisticated switching device so that the trapping magnetic field can be turned off during a brief electron detection period [48]. With this apparatus, the only fully differential cross sections for ion impact ionization of targets other than He or H2 were measured, and the resolution was improved by about a factor of 3. In Fig. 69.4, a measured fully differential three-dimensional angular distribution of electrons ejected in 7 MeV Li2C C Li collisions is shown [49]. In the region of the binary peak, the data reveal a considerably richer structure than for 100 MeV=a:m:u C6C C He. This was explained in terms of a combination of the nodal structure of the initial state wave function and the interaction between the nuclei of the collision partners.
Excitation Inner-shell excitation can be treated within the same perturbative framework, which leads to a cross section given in terms of the generalized oscillator strength for the transition [50–52]. For inner-shell vacancy production by light projectiles, excitation is generally much smaller than ionization, since the oscillator strengths are strongest for transitions to low-lying occupied orbitals, as reviewed by Inokuti [51]. Excitation cross sections can be deduced from photon production cross sections and from inelastic energy-loss experiments. An example of the cross section for excitation of the n D 2 level of H by protons, measured by the latter technique, is shown in Fig. 69.5 [53].
delivering target beams with a temperature around 1 K. While these setups made kinematically complete experiments possible, which, in turn, led to major advances in our understanding of the few-body dynamics, they also entail some drawbacks compared to conventional approaches. These include: a) the recoil-ion momentum resolution is limited by the achievable temperature that can be reached in supersonic gas jets. b) The energies of electrons detected simultaneously with the recoil ions is limited by the extraction voltage on the spectrometer. For the large voltages required to detect large-energy electrons with sufficiently large solid angle, the recoil-ion momentum resolution is compromised. c) The method is restricted to molecular and noble gas Contributions of individual terms in dσ/dκ targets. However, the recoil-ion momentum resolution gets worse with increasing target mass. As a result, many applications are actually restricted to H2 and He targets. While the second problem is inherent to the spectrometer design, the first and third problems can be addressed by a different target design. A successful approach was to replace the supersonic jet with a gas target cooled in a magneto-optical trap (MOT). Initially, this method was only used in conjunction with momentum spectroscopy of the recoil ions, but no σ (n = 3) ejected electrons were momentum analyzed [46, 47]. Tarσ (n = 2) Continuum terms get temperatures reduced by about two orders of magnitude compared to gas jets were achieved, resulting in drastically improved recoil-ion momentum resolution. Furthermore, in σ (n = 4) principle, recoil ions can be momentum analyzed with much improved resolution for any target that can be optically 0 10 20 30 40 pumped, independent of the target mass. Energy loss (eV) One enormous challenge posed by a MOT target is the momentum analysis of ejected electrons. Conventional setFig. 69.5 Energy loss spectrum for 50 keV protons in atomic hydrogen, ups operating with supersonic gas jets use a magnetic field showing excitation to discrete states in H proceeding smoothly into ionin order to detect electrons for a sufficiently large momen- ization at the continuum limit [53]
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Capture As Z1 =Z2 is raised, the probability for direct transfer of inner-shell electrons from the projectile to the target becomes competitive with, and can even exceed, that for ionization of the target electron into the continuum. The first-order perturbation treatment for electron capture, given by Oppenheimer [54] and by Brinkman and Kramers [55] (OBK) [11, p. 379] results in a cross section per atom OBK D 29 .Z1 Z2 /5 =5 V 2 5 n3 ˇ 5 a02 ;
(69.3)
from a filled shell to all final states n, where
1 h ˇ D V 2 V 4 C 2V 2 Z22 = 2 C Z12 =n2 4
2 i C Z22 = 2 Z12 =n2 :
(69.4)
Both the PWBA=SCA and the OBK cross sections maximize near the matching velocity, but the OBK falls off much more strongly with increasing V beyond this, eventually falling as V 12 , while the ionization cross section only falls as V 2 ln V . The OBK amplitude for capture is simply the momentum space overlap of the initial wave function with the final state wave function, where the latter is simply a bound state on the projectile but moving at a velocity V relative to the initial bound state. The integral is done only over the transverse momentum, since the longitudinal momentum transfer is fixed by energy conservation [11]. This capture amplitude thus depends heavily on there being enough momentum present in the initial and=or final wave function to enable the transfer, and the loss of this match is what leads to the steep decrease in the OBK cross section above velocity matching. Cross sections for K-shell capture have been measured by detection of K Auger electrons and K X-rays in coincidence with charge capture by the projectiles [22, 56, 57]. On the basis of these and much other data on electron capture, the OBK is a factor of approximately 3 too large [56–59]. This factor comes from a fundamental failure of first-order perturbation theory for electron capture. As was pointed out already in 1927 by Thomas [60], who proposed a classical two-collision mechanism for capture, it is essential that the electron interacts with both nuclei during the collision in order to be captured (Chap. 61). In quantum theories, this corresponds to the fundamental need to include second-order terms (and higher) in the capture amplitude. In the limit of large V , the second-order cross section decreases more slowly than the OBK term, as V 10 , and is, thus, asymptotically larger than the first-order term [61]. At large V , the coefficient of the V 12 term, the dominant one at most experimentally reachable V , is 0:29 times the OBK cross section, when the theory is carried out to second order in the projectile potential [61, 62]. Roughly speaking, this provides an explanation for the factor of 3. Much more sophisticated treatments of high-velocity capture are now available [63–
71]. The underlying role of the second-order scattering process was confirmed experimentally by the detection of the Thomas peak in the angular distribution of protons capturing electrons from He and H [72, 73] (Chap. 61). In spite of the basic importance of second-order amplitudes in perturbative capture, the OBK gives an excellent account of the relative contributions from and to different final shells over a large range of V above ve , and is thus, when appropriately reduced, still useful as an estimate for perturbative capture cross sections between well-defined and n for large V . For electron capture, as in the case of ionization (see Sect. 69.1.1, Inner-Shell Ionization of Heavy Targets), the development of recoil-ion momentum spectroscopy (RIMS) has enabled much more detailed studies of the collision dynamics. The transverse (perpendicular to the beam direction) recoil-ion momentum component p? reflects the closeness of the collision both relative to the target nucleus and the electrons. The longitudinal (parallel to the beam direction) component pz , on the other hand, is related to the internal energy transfer Q in the collision by (in a.u.) pz D Q=V nV =2 ;
(69.5)
where n is the number of captured electrons. A measurement of pz is, therefore, equivalent to a measurement of Q. The advantage over measuring Q from the projectile energy loss is that at large collision energies, a much better energy resolution is achievable. A sample Q measurement with RIMS is shown in Fig. 69.6 [74]. Very recently, RIMS was applied to study capture processes in collisions with an atomic hydrogen target [75]. This could be an important breakthrough in advancing our understanding of the atomic
Counts (arb. units) He2++ He– – He1+ (n) + He1+(n )
0.25 MeV
(1, 2) & (2, 1) (n, n ) = (1, 1)
(1, 3) & ( 3, 1)
0.26 a.u. FWHM
–3
–2
( 2, –1
0 P
rec
2) 1 (a. u.)
Fig. 69.6 Longitudinal momentum spectrum of recoil ions from 0:25 MeV He2C capturing a single electron from a cold He target, showing clear resolution of capture to n D 1 from that which leaves target or projectile excited [74]
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few-body problem as it opens the possibility to perform kinematically complete experiments on the true three-body system X ZC C H, where X can be any bare projectile.
1009 σ π δ
4fσ
3d
69.1.2
3p Cl 3s Cl
Nonperturbative Processes
Fano–Lichten Model When the collision becomes increasingly perturbative, either due to a decreased V or increased Z1 =Z2 , higher-order effects become generally more important. One approach to account for such contributions is the continuum distorted wave–eikonal initial state (CDW-EIS) model ([35, 36, 76], see also Chap. 56 and Sect. 69.1.1). The range of validity of CDW-EIS is roughly given by Z1 =V 2 1 [35]. Therefore, if the perturbation is large due to the projectile charge, the collision may still be treated perturbatively provided that the collision energy is sufficiently large. Otherwise, the perturbation treatment is replaced by a molecular orbital treatment. Fano and Lichten [77] pointed out that the ratio V =ve can be small for inner orbitals even for V of several a.u., and thus an adiabatic picture of the collision holds. K vacancy production cross sections become much larger than the perturbation treatments above predict and extend to much larger b. In the molecular orbital picture, the collision system is described in terms of time-dependent molecular orbitals (MO) formed when the inner shells of the systems overlap. Vacancy production occurs due to rotational, radial, and potential coupling terms between these orbitals during the collision. The independent electron model is used, but the results in any specific collision are quite sensitive to the occupation numbers (or vacancies) in the initial orbitals. These are very difficult to control in ion–atom collisions in solids and even problematic in gases, since outer-shell couplings can produce vacancies at large internuclear distances, which then enable transfers at smaller distances. Numerous reviews of the subject are available, including [78–82]. The most famous MO ionization mechanism involves the promotion of the 4f orbital in a symmetric collision (Fig. 69.7), which promotes both target and projectile L electrons to higher energies where they are easily lost to the continuum during the collision. There are now many treatments of inner-shell vacancy production mechanisms based on MO expansions (Chaps. 53, 55). For the case of K vacancy production in quasi-symmetric collisions, an important MO mechanism is the transfer of L vacancies in the projectile to the K-shell of the target through the rotational coupling between 2p and 2p orbitals, which correlate to the L and K-shells, respectively, of the separated systems (Fig. 69.7) [83, 84]. The process can be dynamically altered by the sharing at large b between L vacancies of target and projectile through a radial coupling mechanism [85, 86]. This sharing mechanism can also give rise to the direct transfer of K vacancies from
3p 3s 2p Ar 2s Ar 2p Cl 2s Cl
2pπ 2p
Rot coup 2pσ
2s
Radial coup
ls Cl ls Ar
lsσ
ls R=O
R=
Fig. 69.7 Schematic correlation diagram for the Cl–Ar system, indicating the rotational coupling and radial couplings important for K vacancy production and the 4f orbital whose promotion leads to L vacancy production [89]
projectile to target (KK sharing). In symmetric systems, the KK sharing results in an oscillation of the K vacancy back and forth between target and projectile during the collision and leads to an oscillatory behavior of the transfer probability with V and b [87, 88]. Both of the above vacancy production mechanisms are electron transfer processes rather than direct ionization processes in that no inner-shell electron need be liberated into the continuum. Between the perturbation region and the full MO region the importance of transfer increases relative to ionization. While the MO correlation diagrams and mechanisms are qualitatively useful, actual close-coupling calculations for both inner and outer-shell processes are often carried out using atomic orbitals instead of molecular orbitals, as well as other basis sets (Chaps. 53, 55).
69.2 Multielectron Processes In a single collision between multielectron partners, two or more electrons may be simultaneously excited or ionized. The electric fields created during a violent ion–atom collision are so large that at small impact parameters the probability of such multielectron processes can be of order unity. While there are many similarities between ion–atom collisions and the interaction of atoms with photons (X-rays or short laser pulses) or electrons, the dominance of multielectron processes is very much less common in the photon and
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vp = v
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103
Kβ(2p)3 Kβ(2p)2
Kβ(2p)1
102 101
2.80
2.70
2.60
2.50
2.40 Wavelength (Å)
Fig. 69.8 K X-ray spectrum of Ti for various projectiles showing dominance of multielectron transitions when K vacancies are collisionally produced by heavily ionizing projectiles (16 O beam) [90]
electron cases. As an example, when a K-shell electron is removed from a target atom (which is selective on small impact parameters) by the passage of a fast highly charged ion through its heart, the probability that L-shell electrons will be removed at the same time can be large. This gives rise to target X-ray and Auger-electron spectra that are dominated by satellite structure [20]. For example, the spectrum of X-rays from Ti bombarded by 30 MeV oxygen shows that the production of the K vacancy is accompanied by multiple L vacancy production and that the dominant K X-rays are those of systems that are missing several L electrons [90] (Fig. 69.8). When a gas target is used, the recoil target ion is heavily ionized and=or excited electronically without receiving much translational kinetic energy. In the impulse approximation the transverse momentum p? received by the target from a projectile passing at impact parameter b is given in a.u. by p? D 2Z1 Z2 =bV . This expression ignores the exchange of electronic translational momentum but gives a good estimate. The resulting recoil energies are typically quite small, ranging from thermal to a few eV. This subject was reviewed in [74, 91]. These slow moving recoils have been used to provide information about the primary collision dynamics, and as secondary highly charged ions from a fast-beam-pumped ion source. Such an ion source has, for moderately charged ions, a high brightness and has been used extensively for energy-gain measurements. The primary recoil production process is difficult to treat without the independent electron model, and even in this model, the nonperturbative nature of the collision makes the theory difficult. The most successful treatments have been the CTMC (Chap. 62) and a solution of the Vlasov equation [75].
Studies of many-electron transitions in collisions of bare projectiles with a He target are particularly suitable to investigate the role of electron–electron correlation effects because such collisions represent the simplest systems where the electron–electron interaction is present. Such studies have been performed extensively for a variety of processes, such as double ionization, transfer-ionization, double excitation, transfer-excitation, or double capture ([92–94] for reviews). It is common to distinguish (somewhat artificially) between such correlations in the initial state, the final state, and during the transition (dynamic correlation). From a theoretical point of view, the biggest challenge is to describe electron–electron correlation effects and the dynamics of the two-center potential generated by the projectile and the target nucleus simultaneously with sufficient accuracy. In the case of double ionization, an experimental method, based on the so-called correlation function [95], was developed to analyze electron–electron correlations independently of the collision dynamics. Here, a measured two-electron spectrum (for example, the momentum difference spectrum of both ejected electrons) is normalized to the corresponding spectrum one would obtain for two independent electrons. An example of such a correlation function R is shown in Fig. 69.9 for three very different collision systems (1 ranging from 0:05 to 100 and 2 from 0:01 to 0:5 in a.u.). The similarity in these three data sets illustrates that R is remarkably insensitive to the collision dynamics. Rather, the shape of R is determined predominantly by correlations in the final state [95, 97]. However, for selected kinematic conditions, R can also be sensitive to initial-state correlations [98]. Clear signatures of initial-state correlations
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1011
R 100 MeV/amu C6+ + He 3.6 MeV/AMU Au53++ He 3.6 MeV/AMU Au53++ Ne
0.8
0.4
0.0
– 0.4
– 0.8 0
2
4
6
8 | p1–p2 | (a. u.)
Fig. 69.9 Correlation function R for double ionization in the collisions indicated in the legend as a function of the momentum difference between the two electrons [95]; R is defined as R D Iexp =IIEM 1, where Iexp is the directly measured momentum spectrum and IIEM the one obtained for independent electrons
were found in the recoil-ion momentum spectra for transferionization [99]. Early attempts to identify dynamic correlations were based on measurements of the ratio of double to singleionization cross sections [100, 101]. From such studies, it was found that at small V , double ionization is dominated by an uncorrelated mechanism involving two independent interactions of the projectile with both electrons. In contrast, at large V , the double to single ionization ratio asymptotically approaches a common value for all collision systems [102]. This is indicative of the dominance of first-order doubleionization mechanisms, where the projectile interacts with only one electron, and the second electron is ionized through an electron–electron correlation effect. This may either be a rearrangement process of the target atom adjusting to a new Hamiltonian (shake-off, an initial-state correlation) or a direct interaction with the first electron (i.e., dynamic correlation). However, a recent, nearly kinematically complete experiment on double ionization in p C He collisions revealed that even at large V , higher-order contributions are not negligible [96]. In Fig. 69.10, the ejection angles of both electrons are plotted against each other for almost completely determined kinematics. For comparison, the bottom part of Fig. 69.10 shows the corresponding spectra for elec-
a 2
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270
0 2 4 6 8 10 12 14 16 18
0 6 12 18 24 30 36 42 48 54 60
Fig. 69.10 Differential double-ionization cross sections in 6 MeV p C He (a–c) and 2 keV e C He (d–f) collisions as a function of the polar emission angle of both electrons, which are emitted into the scattering plane. The electrons have equal energy, and data are shown for small (a,d), medium (b,e), and large momentum transfers (c,f) [96]
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tron impact at the same V [103]. For both projectiles, the basic features of these spectra are determined by the electric dipole selection rules, which again is indicative of dominating first-order contributions. However, a closer inspection of the comparison between the proton and electron impact data shows some nonnegligible differences. Since in a firstorder treatment, the cross sections should be identical for both projectile species, this demonstrates that higher-order contributions cannot be ignored. Another efficient method to distinguish different doubleionization mechanisms, like e.g., first from higher-order processes, is to analyze four-particle Dalitz (4-D) plots [104]. Here, the data are presented in a tetrahedral coordinate system, where each tetrahedron plane represents one final-state particle (i.e., the scattered projectile, the two ejected electrons, and the residual target ion). The quantities that are plotted are the relative squared momenta i D pi2 =˙pj2 so that ˙ i D 1. For a given data point, each i for the four particles is given by the distance of that data point from the corresponding tetrahedron plane. An example of such a 4-D plot for double ionization in 6 MeV p C He collisions is shown in Fig. 69.11 [105]. The spectrum is dominated by a strong peak structure at the intersection line between the planes representing the two ejected electrons. This intersection line corresponds to events for which both ejected electrons have a small momentum compared to the heavy particles, i.e., momentum exchange occurs mostly between the nuclei. This shows that a higherorder mechanism involving the nucleus–nucleus interaction plays an important role even at this relatively large projectile energy. Further information about the underlying doubleionization mechanism was obtained from a comparison with various theoretical models [105]. In fact, a new mechanism,
Projectile (right side)
Recoil ion (back side) Electron 2 (bottom side)
Electron 1 (front side)
Fig. 69.11 Four-particle Dalitz plot for double ionization of helium by 6 MeV proton impact
which can be viewed as a hybrid between a correlated and an uncorrelated process, could be identified. Here, the projectile only ejects one electron directly; the second electron is ejected by an interaction with the first electron (like in the correlated mechanism). Nevertheless, the projectile does interact with the second electron as well (like in the uncorrelated mechanism), but only after the latter has already been promoted to the continuum.
69.3
Electron Spectra in Ion–Atom Collisions
69.3.1 General Characteristics An ionizing collision between a single ion and a neutral atom ejects electrons into the continuum via two major processes. Electrons ejected during the collision form broad features or continua and are traditionally referred to as delta rays; electrons ejected after the collisions from the Auger decay of vacancies created during the collision form sharp lines in the spectra. The distributions of energy and the angle of all electrons determine the electronic stopping power and characteristics of track formation of ions in matter, and the study of these distributions in the binary encounter of one ion with one atom forms the basis of any detailed understanding of these averaged quantities. Figure 69.12 shows a typical electron spectrum from the collision of a fast O ion with O2 [106, 107]. Electrons from the projectile can be identified in the cusp peak (electron loss, P) or ELC, and the O-K-Auger (P) peak. Electrons from the target include the soft (large b) collision electrons (T), which are ejected directly by Coulomb ionization by the projectile, the binary collision (or encounter) electrons coming from hard collisions between projectile and quasi-free target electrons, and the target O-K-Auger (T) electrons. The electron loss peak is widely called the cusp peak because the doubly differential cross section in the laboratory d 2 =dEd˝ becomes infinite, in principle, if it is finite in the projectile frame. In general, this peak may also contain capture to the continuum. All of these features have been heavily studied; some reviews can be found in [108, 109]. Capture to the continuum [109] is an extension of normal capture into the continuum of the projectile and is not a weak process. Both it and ELC produce a heavy density of events in the electron-momentum space centered on the projectile velocity vector and, thus, appear strongly only at or near zero degrees in the laboratory and at ve ' V . The binary encounter electrons at forward angles occur at ve Š 2V . For relatively slow collisions, it was found that the ELC and binary peaks are just part of a more general and complex structure of the electron spectra [110]. Additional peaks in the forward direction were found for ve Š nV , where
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Ion–Atom Collisions – High Energy
1013
Cross section Electron energy (10 –17cm2/sr) 6 30 MeV 05+on O2 Soft collisions (T) 5 O – K Auger (T) 4 Binary Electron collisions (T) loss (P) 3 O–K Auger (P) 2 25°
1
d2σ/d d 6
10 –20 cm2/eV sr) 3.91 MeV B4+ + H2
2l2l 5 4
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2lnl series limit
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0 1
30°
0
0 dσ/d 10 –18 cm2/sr)
e– + B4+(1s)
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Fig. 69.12 Electron spectrum from 30 MeV O on molecular oxygen. See text for an explanation of features [106, 107]
n can, in principle, be any integer number. These structures reflect a bouncing back and forth (known as the Fermi shuttle) between the projectile and the target core before the electron eventually gets ejected from the collision system. In electron spectra for molecular targets, additional structures were found that were not observed for atomic targets [111]. These were initially interpreted as an interference effect. The electronic wavefunction has maxima at the atomic centers of the molecule. Since in the experiment, it cannot be distinguished from which center the electron is ionized, both possibilities have to be treated coherently. However, more recent studies showed that at small electron energies the structures in the electron spectra reflect vibrational excitation of the molecule [112].
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Fig. 69.13 High-resolution Auger electron spectrum from H-like B on H2 , showing resolved lines from doubly excited projectile states lying on top of a continuum due to electron elastic scattering [115]. The bottom part shows an R-matrix calculation, which does not account for the experimental resolution. The smooth line in the upper figure is the R-matrix calculation convoluted with the experimental spectrometer resolution
the multiple outer-shell ionization that attends the inner-shell vacancy producing event, in close analogy to X-ray spectra (see Sect. 69.2). Projectile Auger electron spectra suffer from kinematic broadening due to the finite solid angle of the spectrometer and velocity of the emitter, but at 0ı to the beam this problem vanishes, and the resolution in the emitter frame is actually enhanced by the projectile motion, such that for electrons with eV energies in the projectile frame, resolutions in the meV region are possible [113, 114]. The highest resolution Auger lines from ion–atom collisions have been recorded for electrons ejected from projectiles. A sample spectrum is shown in Fig. 69.13. From such high resolu69.3.2 High-Resolution Measurements tion spectra, one-electron processes in which one electron is excited, captured, or ionized can be distinguished from the The Auger electron spectra provide detailed information configuration of the emitting state. about inner-shell vacancy production mechanisms. When coupled with fluorescence yields, Auger electron production probabilities, and cross sections can be converted into the 69.4 Quasi-Free Electron Processes in Ion–Atom Collisions corresponding quantities for vacancy production [15, 108]. This is best done when sufficient resolution can be obtained to isolate individual Auger lines. The Auger spectra in ion– At sufficiently low V , those electrons not actively involved atom collisions are often completely different from those in a transition play only a passive role in screening the obtained from electron or photon bombardment because of Coulomb potential between the nuclei and, thereby, create
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a coherent effective potential for their motion. However, at high V , the colliding electrons begin behaving as incoherent quasi-free particles capable of inducing transitions directly via the electron–electron interaction. Such processes signal their presence through their free-particle kinematics, as if the parent nucleus were not present. For example, a projectile ionization process requiring energy U has a threshold at .1=2/me V 2 ' U , in collisions with light targets where the quasi-free picture is meaningful. The threshold is not sharp, due to the momentum distribution or Compton profile of the target electrons. Within the impulse approximation, the cross section for any free electron process can be related to the corresponding cross section for the ion–atom process by folding the free electron cross section into the Compton profile [115– 117].
69.4.1
Radiative Electron Capture
The first quasi-free electron process to be observed was radiative electron capture (REC), the radiative capture of a free electron by an ion. Conservation of energy and momentum is achieved by the emission of a photon that carries away the binding energy. The cross section exceeds that for bound-state capture at high V . Radiative electron capture was observed through the X-ray spectra from fast heavy projectiles for which the electrons of light targets appear to be quasi free. The corresponding free electron process was seen [118] and has recently been heavily studied in EBIT [119], cooler [120], and storage rings [121]. Total cross sections for REC have also been deduced from measured total capture cross sections at large V , where REC dominates the bound-state capture [122]. At high velocities, the cross section for radiative capture to the K-shell of a bare projectile is given approximately by
n D n= 2 C 4 2:1 1022 cm2 ;
(69.6)
into a bound state on the projectile (Chap. 59). If the doubly excited state so populated decays radiatively, resonant radiative recombination is achieved (DR); if it Auger decays, resonant elastic scattering has occurred. The process has long been known to be important as a recombination process in hot plasmas [124] but was not observed in the laboratory until 1983 [125–127]. The corresponding ion–atom process, known as resonant transfer and excitation (RTE), was seen a bit earlier by Tanis et al. [128]; see [129, 130] for reviews of both DR and RTE.
69.4.3 Excitation and Ionization Excitation and ionization of inner shells of fast projectile ions by the quasi-free electrons of light targets (usually He or H2 ) have been identified and studied. This process competes with excitation and ionization by the target nucleus [131–133], and special signatures must be sought to distinguish the processes. In the case of excitation, the e–e excitation populates states through the exchange part of the interaction, which is excluded for the nuclear excitation, and this has been used to separate this mechanism [134]. For ionization, enhancements of the ionization cross section above the Born result for nuclear ionization [135], coincident charge exchange measurements [136], and projectile [137] and recoil-ion momentum spectroscopy [138, 139] have been used to distinguish the two processes. Rapid development in the production of good sources of beams of highly charged ions (EBIS=T, ECR; [140]) have made these studies possible. Continued study of this field in heavy-ion storage rings is now achieving resolutions of meV and is opening broad new opportunities for data of unprecedented high quality for electron–ion collisions [141]. Recently, the first kinematically complete experiment on projectile ionization by quasi-free electrons was reported [142]. The observed features are qualitatively similar to those found for ionization of neutral target atoms by free electron impact.
p where [123] D EB =E0 , EB is the binding energy of the captured electron, E0 the energy of the initial electron in the ion frame, and n the principal quantum number of the cap- 69.5 Some Exotic Processes tured electron. The theory seems to be in good agreement with experiment for capture to all shells of fast bare pro- 69.5.1 Molecular Orbital X-Rays jectiles, although a small unexplained discrepancy between theory and experiment exists for capture to the K-shell [121]. A typical time duration for an ion–atom encounter is 1017 s, which is much shorter than Auger and X-ray lifetimes, so that hard characteristic radiation is emitted by 69.4.2 Resonant Transfer and Excitation the products long after the collision. There remains, however, a small but finite probability that X-rays or Auger electrons Dielectronic recombination in electron–ion collisions is the can be emitted during the collision, in which case the radiaprocess whereby an incident electron excites one target elec- tion proceeds between the time-dependent molecular orbitals tron and, having suffered a corresponding energy loss, drops formed in the collision and reflects the time evolution of the
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Ion–Atom Collisions – High Energy
energies and transition strengths between the orbitals. Such molecular orbital X-radiation (MOX) has now been observed in many collision systems [143] and is reviewed in [144– 146]. MOX spectra have been studied in total cross sections as well as a function of impact parameter. In the latter case, oscillating structures in the MOX spectra are seen [145, 147], due to the interference between amplitudes for the emission of X-rays with the same energy on the incoming and outgoing parts of the trajectory. The formation of transient molecular orbitals in close collisions between highly charged ions provides opportunities for studying the electrodymamics of very highly charged systems [148]. For example, two uranium nuclei passing within one K-shell radius of each other form a transient molecule whose energy levels resemble those of an atom of charge 184.
69.5.2
Positron Production from Atomic Processes
Investigating the MOX interference patterns in such exotic systems offers the interesting prospect of performing spectroscopy on superheavy ions. Reinhart et al. [149] predicted that, for such highly charged species, the binding energy of the united atom Kshell exceeds twice the rest mass energy of the electron, and that if a K vacancy is either brought into the collision or created during it, spontaneous electron–positron pair production occurs (the decay of the charged vacuum) with the electron filling the K-hole. However, further analysis showed that the dominant mechanism for positron production (other than those resulting from the decay of nuclear excitations) likely results from the dynamic time dependence of the fields during the collision [150]. Experiments showed evidence for such positron production in collisions at 6 MeV=u [151], but reported sharp lines in the positron spectra were later attributed to an error in the data analysis. Present theories for the production of lepton pairs in the close collision of two highly charged systems predict that the cross section grows rapidly with the collision energy. Electrons produced in such a process may end up in bound states on either collision partner, and thus represent a new chargechanging mechanism. At highly relativistic velocities, the cross section for this process exceeds that for any other charge-changing process. In a heavy-ion collider, such as RHIC, this process could limit the ultimate storage time for the counterpropagating beams, since charge-exchanged ions are lost. The cross section has been measured recently by Vane et al. [152], for 6:4 TeV S on several targets (the highest energy ion–atom collision experiment performed to date) and are in good agreement with theory [153]. The bound-
1015
state capture was measured at lower energy by Belkacem et al. [154], with similarly good agreement. The extension of ion–atom collisions to such extreme velocities has opened the field for the study of processes that not even imagined a short time ago.
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M. Schulz and L. Cocke Michael Schulz Michael Schulz received his PhD from Heidelberg University in 1987. After positions at ORNL and Kansas State University, he joined Missouri S&T in 1990. He is Curators’ Distinguished Professor and Director of the Laboratory for Atomic, Molecular, and Optical Research. His research focuses on the atomic few-body problem. He is a Fellow of the APS and received a Distinguished Scientist Fellowship from the Chinese Academy of Sciences.
70
Reactive Scattering Hongwei Li
, Arthur G. Suits
Contents
, and Yuan T. Lee
Abstract
70.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1019
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Experimental Methods . . . . Molecular Beam Sources . . . Reagent Preparation . . . . . . Detection of Neutral Products A Typical Signal Calculation .
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Experimental Configurations . . . . . . . . . . Crossed-Beam Rotatable Detector . . . . . . . . Laboratory to Center-of-Mass Transformation Product Imaging . . . . . . . . . . . . . . . . . . .
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Elastic and Inelastic Scattering The Differential Cross Section . . Rotationally Inelastic Scattering . Vibrationally Inelastic Scattering Electronically Inelastic Scattering
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Reactive Scattering . . . . . . . . Harpoon and Stripping Reactions Rebound Reactions . . . . . . . . . Long-Lived Complexes . . . . . .
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Recent Developments . . . . . . . . . . . . . . . . . . . . 1031
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H. Li () State Key Laboratory of Molecular Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences Dalian, Liaoning, China e-mail: [email protected] A. G. Suits Dept. of Chemistry, University of Missouri Columbia, MO, USA e-mail: [email protected] Y. T. Lee Institute of Atomic and Molecular Science, Academia Sinica Taipai, Taiwan e-mail: [email protected]
This chapter presents the experimental aspects of reactive scattering from neutral–neutral chemical reactions under single-collision conditions. It begins with a discussion of the principle of molecular beam sources, developments of reagent preparation, various detection methods of neutral products, and a typical signal calculation. Two main crossed-beam configurations are also summarized here: one with a rotatable ionizer mass spectrometer detector and the other one with velocity-map ion-imaging detection. The differential cross section is introduced and discussed in application to elastic, inelastic, and reactive scattering. Various mechanisms are illustrated in reactive scattering, such as harpoon, stripping, rebound, and long-lived complexes mechanisms. In the last part, recent developments and advances are summarized to show the power of crossed-beam techniques to study the chemical reaction dynamics. Keywords
differential cross section Newton diagram velocity-map imaging relative velocity vector supersonic beam
70.1
Introduction
This chapter presents a résumé of the methods commonly employed in scattering experiments involving neutral molecules at chemical energies, i.e., less than about 10 eV. These experiments include the study of intermolecular potentials, the transfer of energy in molecular collisions, and elementary chemical reaction dynamics. Closely related material is presented in Chaps. 37, 39, and 40, as well as in other chapters on quantum optics.
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_70
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H. Li et al.
Experimental Methods Molecular Beam Sources
The development of molecular beam methods has transformed the study of chemical physics [1]. Supersonic molecular beam sources allow one to prepare reagents possessing a very narrow velocity distribution with very low internal energies, ideal for use in detailed studies of intermolecular interactions. Early experiments generally employed continuous beam sources, but since the 1980s more intense pulsed beam sources have come into common usage [2]. The advantages of pulsed beams primarily arise from the lower gas loads associated with their use, hence reduced demands on the pumping system. If any component of the experiment is pulsed (pulsed laser detection, for example), then considerable advantages may be obtained by also pulsing the beam. Although the theoretical descriptions of pulsed and continuous expansions are essentially equivalent, in practice, some care is required in employing pulsed beams because the temperature and velocity distributions may change dramatically through the course of the pulse. Free jet expansions are supersonic because the dramatic drop in the local temperature in the beam is associated with a drop in the local speed of sound. A detailed description of the supersonic expansion may be found in [3–5]. In practice, many of the detailed features associated with a supersonic expansion may be ignored, and one may assume an isentropic expansion into the vacuum. For an isentropic nozzle expansion of an ideal gas, the maximum terminal velocity is given by q (70.1) vmax D 2COp T0 ;
Table 70.1 Collision numbers for coupling between different modes. V, R, and T refer to vibrational, rotational, and translational energy, respectively. Each entry is the typical range of ZAB V R
V 10:53
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By seeding heavy species in light gases one may accelerate them to superthermal energies. Supersonic beams are characterized by the speed ratio, i.e., the mean velocity divided by the velocity spread: v ; Sp 2kT =m
(70.6)
where T is the local translational temperature, or by the Mach number v M p : (70.7) kT =m For the purpose of order-of-magnitude calculations, the number density on axis far from the nozzle may be estimated as 2 d n n0 ; (70.8) x where n0 is the number density in the stagnation region, d is the nozzle diameter, and x is the distance from the nozzle. The number density versus speed distribution of a nozzle beam is well described as a Gaussian characterized by the speed ratio S and a parameter ˛ D v0 =S, where v0 is the most probable velocity: 2 v S n.v/ D v 2 exp : (70.9) ˛
where, for an ideal gas, the heat capacity is R O Cp D ; 1 m
Cooling efficiencies for the various internal degrees of free(70.2) dom correlate with the efficiency of coupling of these modes with translation, hence they vary widely. Coupling of modes R is the gas constant, m is the molar molecular mass, T0 is A and B are expressed by the collision number ZAB : the temperature in the stagnation region, and is the heat ZAB ZAB ; (70.10) capacity ratio. For ideal gas mixtures, and assuming Cp independent of temperature for the range encountered in the where AB is the bulk relaxation time and Z the collision frequency. This represents the number of collisions between expansion, one may use effective inelastic events. Typical values are summarized in X X i Xi Cpi D Xi CNp D R; (70.3) Table 70.1. R-T coupling is relatively efficient, while V-T i 1 i i coupling is quite inefficient, so that vibrational excitation may not be effectively cooled in the expansion. and the average molar mass X m N D Xi m i ; (70.4) i
where Xi is the mole fraction of component i, to obtain an estimate of the maximum velocity for a mixture: s 2CNp T0 : (70.5) vmax D m N
70.2.2
Reagent Preparation
Molecular beam methods may be used in conjunction with a variety of other techniques to prepare atoms or molecules in excited or polarized initial states (Chap. 48), to generate unstable molecules or radicals [6, 7], or to produce
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beams of refractory materials such as transition metals or carbon [8, 9]. Some of the common techniques are outlined below. Optical pumping of atoms to excited electronic states is a useful means of reagent preparation, and this topic is presented in detail in Chap. 11. This technique further allows one, using polarized lasers, to explore the influence of angular momentum polarization in the reagents on the collision dynamics [10]. Most of these studies have been performed using alkali and alkaline earth metals since there exist strong electronic transitions and convenient narrowband visible lasers suitable for use with these systems. Laser excitation may also be used to generate vibrationally excited molecules in their ground electronic states. The techniques employed include direct IR excitation using an HF chemical laser [11], population depletion methods [12], and various Raman techniques [13]. Adiabatic passage techniques, such as stimulated Raman adiabatic passage (STIRAP), Starkinduced Raman adiabatic passage (SARP), and rapid adiabatic passage (RAP), opened new opportunities for coherent laser control of atomic and molecular processes and were applied to prepare molecules in a single excited rovibrational level in their ground electronic states with near 100% efficiency. They are applied to use for different molecules with differing advantages. STIRAP allows efficient population transfer through a third upper electronic intermediate state [14]. SARP can be used for molecules that have no vibrational transition moment and no conveniently accessible excited states, like H2 [15]. RAP only requires a single laser but is only applicable for molecules that possess a strongly allowed vibrational transition [16]. Stimulated emission pumping (SEP) [17] is an alternative technique for preparing very highly excited vibrational states, which does not require a coherent laser but needs convenient electronically excited states, as is the case for NO. Metastable atoms may also be prepared by laser photolysis of a suitable precursor. O(1D) is readily prepared by photolysis of ozone or N2 O, for example [18]. Alternatively, RF or microwave discharges may be used to produce metastable species or reactive atoms or radicals [19]. These techniques may also be used to prepare ground-state atoms; for example, hot H atom beams are frequently produced by photolysis of HI or H2 S [20]. Such atomic or molecular radical beams may also be generated by pyrolysis in the nozzle. In this case, care must be taken to minimize recombination through careful choice of the temperature, nozzle geometry, and transit time through the heated region. Beams of refractory materials are now commonly generated using laser ablation sources [8, 9]. Typically, these employ a rod or disk of the substrate of interest, which is simultaneously rotated and translated to provide a fresh surface for ablation at each laser pulse. A laser beam is focused on the substrate and timed to fire just as a carrier gas pulse passes over. Laser power and wavelength must be optimized for a given sub-
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strate. Lasers operating in the IR, visible, and UV have all been employed. Aligned or oriented molecules have been prepared using multipole focusing [21, 22], and more recently using strong electric fields (“brute force”) [23]. In the former case, specific quantum states are focused by the field. In the latter case, so-called pendular states are prepared from the low rotational levels of molecules possessing large dipole moments and small rotational constants. The ability to orient these molecules can be estimated on the basis of the Stark parameter ! D E=B, where is the dipole moment, E the electric field strength, and B the rotational constant. Orientation is feasible for low rotational levels of molecules when the Stark parameter is on the order of 10 or higher [23]. Manipulating the motion or control over the velocity of molecules in a molecular beam is achieved by concepts from charged-particle accelerator physics. Stark [24, 25] and Zeeman [26, 27] decelerators have been developed to control the motion of molecules that possess an electric or magnetic dipole moment using time-varying electric and magnetic fields, respectively. These decelerator techniques can not only slow the molecular beam to a narrow distribution of velocities but also select molecules into a single rotational quantum state.
70.2.3 Detection of Neutral Products Broadly speaking, detection of neutral molecules is accomplished either by optical (spectroscopic) or nonoptical techniques. Nonoptical methods usually involve nonspecific ionization of neutral particles, most commonly by electron impact, followed by mass selection and ion counting. Thermal detectors such as cryogenic bolometers are also finding widespread application in molecular beam experiments owing to their remarkable sensitivity [28]. In general, optical methods may rely on resonant or nonresonant processes, hence they may or may not enjoy quantum-state selectivity. Both photoionization and laser-induced fluorescence methods are now in common usage, usually in applications where quantum-state resolved information is desired. The advantage of nonoptical methods is primarily one of generality: all neutral molecules may be detected and branching into different channels readily measured. Quantum-state resolution is more difficult to achieve using nonoptical detection methods, but both vibrationally and rotationally resolved measurements have been obtained by these means [29, 30]. The primary advantage of spectroscopic detection is the aforementioned possibility of quantum-state specificity. Another unique opportunity afforded by spectroscopic probes is the measurement of product alignment and orientation. In addition, in some cases, background interference may be reduced or eliminated using state-specific probes, thereby affording enhanced signal-to-noise ratios.
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Nonoptical Techniques Detectors based on nonspecific ionization remain the most commonly used in molecular beam experiments, owing to the ease of subsequent mass selection, and the convenience and sensitivity of ion detection. Surface ionization is a sensitive means of detecting alkali atoms and other species exhibiting low ionization potentials [31]. Surface ionization occurs when a neutral atom or molecule with a low ionization potential sticks on a surface with a high work function and is subsequently desorbed. Typically, these detectors employ a hot platinum or oxidized tungsten wire or ribbon for formation and subsequent desorption of the ions, which is surrounded by an ion collector. They are very efficient for the detection of alkali atoms and molecules whose ionization potentials are . 6 eV. All neutral gas molecules may be ionized by collision with energetic electrons, and electron beam ionizers may be produced that couple conveniently to quadrupole mass spectrometers [32]. Collision of a molecule with a 100–200 eV electron leads predominantly to formation of the positive ion and a secondary electron. Other processes also occur and can be very significant: doubly or triply charged ions may be formed and, importantly, molecules can fragment yielding many daughter ions in addition to the parent ion. These fragmentation patterns vary with different molecules and may further show a strong dependence on molecular internal energy, so particular care must be taken to determine the role of these phenomena in each particular application. It is often necessary to record data for the parent ion and daughter ions for a given product channel and compare them to eliminate contributions arising from cracking of the parent molecule or other species [33]. Electron impact ionization probabilities for most species exhibit a similar dependence on electron energy, rising rapidly from the ionization potential to a peak at 80–100 eV, then falling more slowly with increasing collision energy. The ionization cross section for different species scales with molecular polarizability according to a well-established empirical relation [34]: p ion D 36 ˛ 18 ;
(70.11)
where ion is in Å2 and ˛, the molecular polarizability, is in Å3 . This relation can be used to estimate branching ratios in the absence of any other means of calibrating the relative contributions of two different channels. The ionization rate is given by dŒMC D Ie ŒM ; dt
(70.12)
where Ie is the electron beam intensity, typically 10 mA=cm2 or 6 1016 electrons=(cm2 s), and ŒM is the number density of molecules M in the ionizer. If one assumes an ionization
cross section ion of 1016 cm2 for collision with 150 eV electrons (a typical value for a small molecule), the ionization probability for molecules residing in the ionizer is then dŒMC 1 D Ie D 6 1016 1016 D 6 s1 : (70.13) dt ŒM However, product molecules arriving in the detector are not stationary. Typically, product velocities are on the order of 500 m=s. If the ionization region has a length of 1 cm, the residence time of a product molecule is on the order of 2 105 s. Consequently, the ionization probability of product molecules passing through the ionizer is dŒMC D 2 105 6 D 1:2 104 : dt ŒM
(70.14)
Although this does not appear very efficient (indeed, it is four orders of magnitude less so than surface ionization), nevertheless, if the background count rate is sufficiently low, then good statistics may be obtained with signal levels as low as 1 Hz. Thus, for detection based on electron impact ionization, a key factor determining the sensitivity of the experiment is the background count rate at the masses of interest.
Spectroscopic Detection Spectroscopic detection methods usually involve either laserinduced fluorescence (LIF) or resonant photoionization (REMPI) (Chap. 46). Alternative techniques such as laserinduced grating methods and nonresonant vacuum ultraviolet (VUV) photoionization are also being applied to scattering experiments. VUV photoionization can detect molecules and radicals with ionization energy lower than the photon energy that provides universal detection ability but does not resolve any vibrational or rotational information. Essential to the use of spectroscopic methods for reactive scattering studies is an understanding of the spectrum of the species of interest. This may be challenging for many reactive systems because the products may be produced in highly excited vibrational or electronic states that may not be well characterized. Additional spectroscopic data may be required. Franck–Condon factors are necessary to compare the intensities of different product vibrational states, while a calibration of the relative intensities of different electronic bands requires a measure of the electronic transition moments. In some cases, one must include the specific dependence of the electronic transition moment on the internuclear distance by integrating over the vibrational wave function. Populations corresponding to different rotational lines may be compared after the appropriate correction, which is represented by the Hönl– London factors, only for isotropic irradiation and detection. This is certainly not the case for most laser-based experiments. Generally, the detailed dependence of the excitation and detection on the relevant magnetic sublevels must be
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considered [35–37]. Caution is required in using any spectroscopic method involving a level that is predissociated. This may lead to a dramatic decrease in the associated fluorescence or photoionization yield if the predissociation rate approaches or exceeds the rate of fluorescence or subsequent photoionization. An important question in any experiment based on spectroscopic detection is whether product flux or number density is probed. This question is considered in detail in several articles [13, 38]. It depends on the lifetime of the state that is probed, the relative time that the molecule is exposed to the probe laser field, and its residence time in the interaction region. Saturation phenomena are also important, yet not necessarily easily anticipated. Complete saturation does not readily occur because excitation in the wings of the laser beam profile becomes more significant as the region in the center of the beam becomes saturated [39]. LIF was a widely used spectroscopic technique in inelastic and reactive scattering experiments [35, 40, 41] in the 1980s to 1990s. It has been used to measure state-resolved total cross sections [42] and differential cross sections in electronic [43], vibrational, and rotationally inelastic scattering [12], as well as reactive scattering [44]. With the development of high-power tunable lasers and the broad use of imaging methods, REMPI is becoming a more general technique [45, 46]. REMPI has the advantages associated with ion detection, namely considerable convenience in mass selection and efficient detection, in addition to the capability of quantum-state selectivity. Disadvantages associated with REMPI arise primarily from higher laser power employed compared with LIF. Caution is required in attempting to extract quantitative information from REMPI spectra if one or several of the steps involved in the ionization process are saturated. This is of particular concern at the high laser powers necessary for multiple photon transitions. Another disadvantage, similar to LIF measurement, is the lack of available spectroscopic information for larger molecules, which limits its versatility in terms of universal detection. An alternative to direct photoionization involves excitation of products to metastable Rydberg states, followed by field ionization some distance from the interaction region. This technique has the advantage of very low background and is capable of extraordinary time-of-flight resolution. One example is the H atom Rydberg time-of-flight (HRTOF) method [47, 48] pioneered by the late Karl Welge and coworkers [49] for the hydrogen exchange reaction, D C H2 . This approach employs a conventional scattering geometry and is suitable only for experiments yielding product H or D atoms. Despite this narrow focus, owing to the general importance of hydrogen elimination reactions and the remarkable resolution of the technique, this has been an important development. The H or D atom products are excited to long-lived high-n Rydberg states in a 1 C 10 excitation scheme at the interaction region. They fly through a field-free region and impinge upon a rotat-
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able field-ionization detector. The result is very high velocity resolution, largely because the spreads in the beam velocities make a negligible contribution to the product velocity spread since the H atoms are moving so fast. In addition, the dimensions of the scattering volume and ionization region may easily be made small relative to the flight length. Using this technique, Welge and coworkers achieved fully rotationally resolved differential cross sections for the hydrogen exchange reaction. Now, this technique has also been widely applied by the Yang group in studying the benchmark systems such as F C H2 and Cl C H2 [50–52]. They were able to observe quantum effects from the reactive resonances trapped in the transition-state region of those reactions.
70.2.4
A Typical Signal Calculation
For a crossed-beam system in which a beam of atoms A collides with a beam of molecules B yielding products C and D, the rate of formation of C is given by dNC D nA nB r gV ; dt
(70.15)
where nA and nB are the number densities of the respective reagents at the interaction region, r is the reaction cross section, g the magnitude of the relative velocity between the reactants, and V the volume of intersection of the beams. For a typical experiment employing continuous supersonic beams, the number densities of the atomic and molecular reactants are 1011 –1012 cm3 and the scattering volume 102 cm3 . For g D 105 cm=s and r D 1015 cm2 , the rate of product formation dNC =dt D 1011 molecules=s. The kinematics and energetics of the reaction then determines the range of laboratory angles into which the products scatter and the magnitude of the scattered signal. If the products scatter into 1 sr of solid angle, and the detector aperture is 3 103 sr (roughly 1ı in both directions perpendicular to the detector axis), then the detector receives 3 107 product molecules=s. Given the detection probability obtained above, 3600 product ions=s are detected. This is adequate to obtain very good statistics in a short time, as long as the background count rate is not considerably higher. For a nonspecific detection technique, such as electron bombardment ionization coupled with mass filtering, it is necessary to use ultrahigh vacuum ( 1010 Torr) in the detector region to minimize interference from background gases. The residual gases are then primarily H2 and CO, with number densities on the order of 106 cm3 . Differential pumping stages, each of which may reduce the background by two orders of magnitude, are generally used to lower the background from gases whose partial pressures are lower than the ultrahigh vacuum limit of the detector chamber. However, this differential pumping helps only for those molecules that
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do not follow a straight trajectory through the detector. The pumping before reaching the ionizer. Ions formed by eleccontribution from the latter is given by tron impact on the neutral products are then extracted into a quadrupole mass spectrometer with associated ion counter. nA 0 ; (70.16) A chopper wheel is generally used at the entrance to the den D 4 x 2 tector to provide a time origin for recording time-of-flight where n is the number density of molecules effusing from an spectra. orifice of area A, and n0 is their number density at a disPseudo-random sequence chopper disks provide optitance x on the axis downstream. For a distance of 30 cm mal counting statistics while maintaining a high-duty cycle and a main chamber pressure of 3 107 Torr, this corre- (50%) [53]. The detector may be rotated about the interacsponds to a steady-state density of 105 molecules=cm3 at tion region, typically through a range of 120ı or so, allowing the ionizer, a reduction of six orders of magnitude. Three one to examine products scattered at a range of laboratory anstages of differential pumping are, thus, the most useful un- gles. In addition to time-of-flight detection, one of the beams der these conditions, since the primary source of background may be gated on and off for background subtraction, and the is then molecules following a straight trajectory from the detector moved to record the integrated signal at each labomain chamber. A liquid helium-cooled surface opposite the ratory angle. Two kinds of measurements are typically made in these detector entrance may then be useful to minimize scattering experiments: time-of-flight spectra and angular distributions. of background molecules into the ionizer. Usually, one is interested in obtaining the complete productflux versus velocity contour map, since this contains full 70.3 Experimental Configurations details of the scattering process. This is obtained by measuring a full angular distribution as well as time-of-flight data at many laboratory angles. The results are then simu70.3.1 Crossed-Beam Rotatable Detector lated using a forward convolution fitting procedure to obtain The configuration illustrated in Fig. 70.1 represents a stan- the underlying contour map [54–56]. Because scattering of dard, now widely used [32], usually with two continuous isotropic reagents exhibits cylindrical symmetry about the beams fixed at 90ı . The molecular beam sources are differen- relative velocity vector, it is sufficient to measure products tially pumped and collimated to yield an angular divergence scattered in any plane containing this vector to determine the of about 2ı . The beams cross as closely as possible to the product distribution. This is not true for structured particles nozzles, with a typical interaction volume of 3 mm3 . Scat- (e.g., involving atoms in P states); however, this azimuthal tered products pass through an aperture on the front of anisotropy has been used to explore the impact parameter dethe detector, thence through several stages of differential pendence of the reaction dynamics [57]. Fig. 70.1 Experimental arrangement for F C D2 ! DF C F reactive scattering. Pressure (in Torr) indicated in each region
Velocity sector
Cold trap D2 beam source Liquid nitrogen feedline
10
–4
10–4 Heater –6
10 Effusive F atom source
Synchronous motor Cross-correlation chopper
Skimmer Turning fork chopper
10–7 10
Ultrahigh vacuum differentially pumped mass spectrometer detector
–9
10–10 10–11
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In a typical reactive scattering experiment, A C BC ! AB C C, either of the two products may be detected. Conservation of linear momentum requires that the center-ofmomentum (CM) frame momenta of the two products must sum to zero. It is, thus, only necessary to obtain the contour map for one of the products. The choice of detected product is usually dictated by kinematic considerations, although one may choose to detect a product that is kinematically disfavored if its partner happens to have a mass with a large natural background in the detector. Kinematic considerations can be critical in assessing the suitability of a given system for study. It is very important that one of the products be scattered entirely within the viewing range of the detector in order to obtain a complete picture of the reaction dynamics. The advantages of crossed beams employed in conjunction with an electron impact ionizer-mass spectrometer detector derive primarily from the universality of the detector. No spectroscopic information is required, and there are no invisible channels, such as may occur with spectroscopic detection methods. In addition, the resolution of these machines may be increased almost arbitrarily; indeed, even rotationally inelastic scattering has been studied [58]. The disadvantages are complementary to the advantages: the universal detector implies that quantum-state resolution is not achieved directly, although in favorable cases, the product vibrational states may be resolved in the translational energy distributions [29, 59]. In addition, if the product of interest represents a mass that receives interference from one of the beam masses, background interference may be problematic. The kinematic considerations mentioned above may also preclude study of certain systems. However, the kinematic requirements for the imaging approach discussed below are complementary to those of the rotatable-detector configuration.
v=1 v=2 0
v=3 vF
vH2
Fig. 70.2 Newton diagram for collision of F with H2 with superimposed CM flux versus velocity contour map
V CM divides the g into two segments corresponding to the CM velocities of the two reagents. The magnitude of these vectors, uF and uD2 , are inversely proportional to the respective masses. If scattered DF products are formed with a laboratory scattering angle # and a laboratory velocity vDF , as shown in Fig. 70.2, this corresponds to DF backscattered with respect to the incident F atom, in the CM system. It is common to refer the scattering frame direction to the atomic reagent in an A C BC ! AB C C reaction, for example, to make clear the dynamics of the process. In this case, the backscattered DF arises as a result of a direct rebound collision. Some useful kinematic quantities are summarized here. For beams A and BC intersecting at 90ı , the angle of the CM velocity vector with respect to A is given by #CM D arctan
MBC vBC : MA vA
For an arbitrary Newton diagram with angle ˛ between the two beams, the magnitude of the relative velocity is 2 2vA vBC cos ˛ ; g 2 D vA2 C vBC
70.3.2
(70.18)
(70.19)
the relative velocity vector is
Laboratory to Center-of-Mass Transformation
g D vA vBC ; Angular and velocity distributions measured in the laboratory frame must be transformed to the CM frame for theoretical interpretation. Accounts of this transformation and details concerning the material presented below may be found in [60–63], among others. The Newton diagram is useful to aid in visualizing the transformation and in understanding the kinematics of a given collision system. For the scattering of F C D2 , for example, shown in Fig. 70.2, a beam of fluorine atoms with a velocity vF is crossed by a beam of D2 , velocity vD2 , at 90ı . The relative velocity between the two reactants is g D vF vD2 , and the velocity of the CM of the entire system is
(70.20)
and the collision energy is Ecoll D 12 i g 2 ;
(70.21)
where i is the reduced mass of the initial collision system. The magnitude of the CM frame velocity of particle A before collision is uA D
mBC g: mA C mBC
(70.22)
The final relative velocity is V CM
MF vF C MD2 vD2 D I MF C MD2
(70.17)
g 0 D vAB vC ;
(70.23)
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For the case of continuous final velocities, the are velocity dependent and are related by
with magnitude s 0
g D
2Eavail ; F
(70.24)
where the available energy Eavail is Eavail D Ecoll C Eint;reac C Eexo Eint;prod ;
d3 d3 D J ; d2 ˝ dv d2 ! du so that here the Jacobian is given by
(70.25) J D
in which Eint;reac is the internal energy of the reactants, Eexo is the exoergicity of the reaction, and Eint;prod is the internal energy of the products. One must transform the laboratory intensity I.˝/ d2 =d2 ˝ into I.!/ . d2 =d2 !/, the corresponding CM quantity. For the crossed-beam configuration described in Sect. 70.3.1, the laboratory distributions are distorted by a transformation Jacobian that arises because the laboratory detector views different CM frame solid angles, depending on the scattering angle and recoil velocity. For the spectroscopic experiments described in Sect. 70.3.3, the Jacobian is unity (the CM velocity represents a simple frequency offset of the Doppler profiles, for example); however, the transformation of the scattering distributions from the recorded quantities (two-dimensional projections or intensity versus wavelength) to recoil velocity distributions may be complex. Two cases must be considered for the configuration discussed in Sect. 70.3.1: one in which discrete velocities result (such as elastic or state-resolved scattering experiments), and one in which continuous final velocities are measured. For the first case, the laboratory and CM differential cross sections are independent of the respective product velocities v and u, and these quantities are related by d2 d2 D J ; d2 ˝ d2 !
(70.26)
so that the transformation Jacobian is given by d2 ! J D 2 : d˝
(70.31)
d2 ! du : d2 ˝ dv
In this case, we consider a recoil volume element d (in velocity space), which must be the same in both coordinate frames: dCM D u2 du d2 ! D dlab D v 2 dv d2 ˝ ;
J D
v2 : u2
(70.34)
The laboratory intensity is then related to that in the CM frame by Ilab .v; #/ D
v2 u2
ICM .; u/ :
Nlab .v; #/ D
Ilab .v; #/ v D 2 ICM .u; / : v u
d2 ˝ D
cos.u; v/ dA ; v2
so that the Jacobian for the first case is given by J D
v2 : u2 cos.u; v/
(70.36)
The usual flux versus velocity contour map is a polar plot of the quantity ICM .u; /. The product velocity distributions are then I.u/ D
Z I.u; / sin d ;
I.; u/ sin d d D 2 0
(70.37)
and the translational energy distributions are ˇ ˇ ˇ du ˇ ˇ ˇ / D I.u/ I.E T where dA is a surface element of the product Newton sphere. ˇ dE ˇ : T The laboratory solid angle corresponding to this quantity is dA ; u2
(70.35)
For a mass spectrometer detector with electron bombardment ionizer, one measures the number density of particles rather than the flux, so that the recorded signal is given by
For discrete recoil velocities, the CM solid angle is d2 ! D
(70.33)
so that the Jacobian is
“ (70.27)
(70.32)
(70.28)
(70.29) 70.3.3
(70.38)
Product Imaging
An important approach now is based on direct imaging of the scattered product distribution. The technique was first used to record state-resolved angular distributions of methyl rad(70.30) icals from the photodissociation of methyl iodide [64] and
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has since been widely employed to study unimolecular photodissociation dynamics [65]. In 1993, Houston and coworkers first applied imaging detection in a crossed-beam experiment to record state-resolved inelastic scattering of NO with argon [66]. Since then, velocity map imaging combined with crossed-beam apparatus has been widely used to measure inelastic and reactive scattering of bimolecular reactions [67]. The crossed-beam configuration coupled with velocity map imaging detection, which was first used by Houston and coworkers and now widely adopted to use by many research groups, is shown schematically in Fig. 70.3. The two skimmed supersonic beams cross at right angles, and scattered products are state-selectively ionized on the axis of a time-of-flight mass spectrometer using resonant photoionization (or ionized by a VUV nonresonant photoionization without state selectivity). The ion cloud thus formed continues to expand with its nascent recoil velocity as it drifts through the flight tube. The ions then strike a microchannel plate coupled to a phosphor screen. The latter is viewed by a video camera gated to record the signal at the mass of interest. The images are, thus, two-dimensional projections of the nascent three-dimensional product distributions. From this projection, it is necessary to generate the three-dimensional distribution. This is a tomographic reconstruction, achieved using an inverse Abel transform or related method, as widely used in photodissociation studies [68–70]. It is a direct inversion procedure feasible for cases in which the image is the projection of a cylindrically symmetric object, with its axis of symmetry parallel to the image plane. This analysis yields a unique product contour map directly from the image, but it is difficult to incorporate apparatus functions, and it is sensitive to noise in the data. The problem of the noise or uncertainty induced from the reconstruction process was overcome by an important innovation in ion imaging, time slicing, in the early twenty-first century. Time slicing was first reported by Kitsopoulos and coworkers [71] using delayed pulsed extraction in 2001, followed by two closely related velocity mapping versions in 2003: “DC slicing” [72] and “time-sliced imaging” [73]. Ideally, DC slice-imaging techniques do not require any reconstruction procedure in the analysis; however, in practice, most slice-imaging experiments achieve a velocity slice width of 10–25% or more around the center of the distribution due to the limited experimental conditions. This will carry significant out-of-plane elements that can blur the spectrum, lose fine resolution, and underestimate the contribution from slow recoiling products. To overcome these limitations, a finite-slice analysis (FINA) slice reconstruction method was recently developed [74, 75] to remove these outof-plane elements from a sliced image. This method allows reconstruction procedure at any velocity slice width and can even be applied to nonsliced or conventional images. The advantages of the imaging method again derive from its reliance on a spectroscopic probe, so that quantum-state
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CCD camera
Phosphor screen Back plate Front plate
MCP detector
TOF axis
Ion optics
Molecular beam 1
Molecular beam 2 Skimmer
Skimmer Pulsed valve
Ionization probe laser
Pulsed valve
Fig. 70.3 Crossed molecular beam apparatus with product imaging detection
resolution is possible, and background interference may be avoided. In addition, it possesses a multiplexing advantage since the velocity distribution is recorded for all angles simultaneously. Imaging relies exclusively on photoionization, unlike the Doppler methods, which may use either photoionization or LIF. This is somewhat disadvantageous since the available photoionization schemes are limited, and often high laser power is necessary to achieve adequate signal intensity. As a result, background ions can be a problem. In general, resonantly enhanced two-photon ionization, i.e., Œ1 C 1, detection schemes are thus preferable. As we mentioned earlier in the spectroscopic detection in Sect. 70.2.3, nonresonant VUV photoionization is an alternative ionization method for the velocity map imaging technique when a REMPI scheme is not available for probed products. VUV photoionization does not have quantum-state resolution but provides universal detection that can ionize all molecules and radicals with their ionization energy lower than the VUV photon energy. VUV photoionization from an excimer laser serves as a convenient and powerful probe with crossed beams for a variety of systems [76]. This combination of crossed molecular beam experiment with a high-resolution velocity map imaging technique directly measures the product velocity-flux contour map that reflects the underlying dynamics of the bimolecular reaction. High-resolution product images provide coupling information between scattering angles, i.e., the differential cross section and the energy partitioning into translational and internal degrees of freedom
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of the products [77–79]. The velocity map imaging tech- differential cross section d gives the rate of all collisions nique has been a powerful probe tool and been used in many leading to deflection angles in the solid angle element d!: crossed-beam studies, as discussed in Sect. 70.6. dN./ / I./ d! D I./2 sin./ d : (70.39) dt
70.4
Elastic and Inelastic Scattering
The incremental cross section is d D I./ d! D 2 b db, When particles collide, they may exchange energy or recou- so ple it into different modes, they may change their direction b : (70.40) I./ D of motion, and they may even change their identity. The sin .d =db/ study of these processes reveals a great deal of information about the forces acting between the particles and their in- For classical particles, the relation between the deflection ternal structure. It is useful to begin with a summary of the function and the potential is dominant features of elastic and inelastic scattering. 1=2 Z1 dR b2 V .R/ ; (70.41) 1 D 2b R2 ET R2 70.4.1 The Differential Cross Section R0
Figure 70.4 illustrates the relation between the deflection function and the impact parameter b for a realistic potential containing an attractive well and a repulsive core. For large b there is no interaction, hence no deflection. At smaller values of b, the attractive part of the potential is experienced, and some positive deflection results. At a smaller value of b, br , the influence of the attractive component of the potential reaches a maximum, giving the greatest positive deflection: this is the rainbow angle by analogy with the optical phenomenon. There is another value of the b for which point the attractive and repulsive parts of the potential balance, yielding no net deflection. This is the glory impact parameter bg . For yet smaller values of b, the interaction is dominated by the repulsive core, and rebound scattering gives a negative deflection function. The important expressions related to the differential cross section are summarized here [80]. For scattering involving an isotropic potential, the deflection angle is # D jj. The
where V .R/ is the potential as a function of interparticle distance R, R0 is the turning point of the collision, and ET the collision energy. In the high energy limit, for large b R0 , .b; ET / /
V .b/ : ET
(70.42)
For a long-range potential V .R/ proportional to Rs , 2=s
ET 2.1C1=s/ I./ D const.
(70.43)
For a potential exhibiting a minimum, the rainbow angle r is proportional to the collision energy and clearly resolved when the collision energy is three to five times the well depth. In addition, supernumary rainbows and quantummechanical “fast oscillations” occur in the d , and these provide a sensitive probe of the interaction. Accurate interatomic potentials are routinely obtained from elastic scattering experiments [62, 81].
2
70.4.2
br*
Rotationally Inelastic Scattering
* 1 bg
0
– 0 + ψ
+
Fig. 70.4 Schematic diagram showing the relation between impact parameter b and deflection function
Classical scattering involving an anisotropic potential results in another rainbow phenomenon, distinct from that seen in pure elastic scattering and notable in that it does not require an attractive component in the potential. These rotational rainbows are equivalently seen in a plot of integral cross section against change in rotational angular momentum j or in the differential cross section for a particular value of j . The rotational rainbow peaks arise from the range of possible orientation angles in a collision involving an anisotropic potential. When there is a minimum in d=d for a given j , the differential cross section reaches
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larger cross sections than V-T processes [89, 90]. In addition, spectroscopic methods may be used to overcome some of the background problems that hamper the study of the latter. Often, quenching of electronically excited states involves curve crossing mechanisms, so that very effective coupling of electronic to vibrational energy may occur. Spin–orbit changing collisions of Ba(1P ) with O2 or NO, for example, occur by a near-resonant process and result in almost complete conversion of electronic energy to vibrational excitation of the product [91]. The analogous collisions with N2 and H2 , however, reveal very repulsive energy release with little concomitant vibrational excitation. Both processes likely occur 1 r;cl j via curve crossings of the relevant electronic states, but the ; (70.44) 2 sin AB D near-resonant mechanism occurs by way of an ionic intermep0 2 diate. where j is the rotational angular momentum, p0 is the initial linear momentum, r;cl is the classical rainbow position, and A and B are the semimajor and semiminor axes 70.5 Reactive Scattering of a hard ellipse potential. The classical rainbow positions occur somewhat behind the quantum-mechanical and ex- Reactive differential cross sections reveal several distinct asperimental rainbow positions, so the classical rainbow may pects of the chemical encounter. The angular distributions be estimated as the point at which the peak has fallen to themselves may be used to infer the lifetime of the col44% of the experimental value. Real molecular potentials lision intermediate: long-lived complexes exhibit forward– may be far from ellipsoids, however, such detailed quan- backward symmetry along the relative velocity vector. In titative insight into the potential requires a comparison of this case, “long-lived” means on the order of several rotascattering data with quantum-mechanical calculations. The tional periods. The rotational period of the complex may open-shell, close-coupled quantum-mechanical (CC QM) thus be used as a clock to study the energy dependence scattering calculations have been proved successfully in the of the intermediate’s lifetime. The angular distributions furrotationally inelastic scattering of NO with rare gases. Inter- ther reveal the relation of initial and final orbital angular esting phenomena, like scattering resonances [84], diffrac- momentum. Sharply peaked angular distributions generally tion oscillations [85], and stereodynamic effects [86], have indicate strongly correlated initial and final orbital angular been observed in the joint work of high-resolution imaging momentum vectors. Finally, the product translational energy detection with crossed-beam experiments and CC QM calcu- release contains the details of the energy disposal and reveals a wealth of information about the thermodynamics of the lations. process, the existence of barriers, and sometimes even the geometry of the transition state. Together, the angular and translational energy distributions reveal many of the details 70.4.3 Vibrationally Inelastic Scattering of the potential energy surface. The dynamics of reactive collisions fall broadly into three There has been no direct observation of the differential cross section of T-V or V-T energy transfer involving neu- main categories characterized by distinct angular and entral molecules owing to the small cross sections for these ergy distributions. The three categories are harpoon/stripping processes. Integral cross section data are available, however. reactions, rebound reactions, and long-lived complex formaAbove threshold, the latter has shown a linear dependence of tion. Some reactions may exhibit more than one of these on collision energy for D 1, quadratic for D 2 and mechanisms at once, or the dynamics may change from one cubic for D 3 [87]. In addition, a great deal of informa- to another as the collision energy is varied. tion on vibrational relaxation processes has been obtained in cell experiments [88]. a maximum [82]. The rotational rainbow peak occurs at the most forward classically allowed value of the scattering angle, and d drops rapidly at smaller angles. The rainbow moves to more backward angles with increasing j because the larger j -changing collisions require greater momentum transfer and, hence, must arise from lower impact parameter collisions. For heteronuclear molecules, two rainbow peaks may be observed, corresponding to scattering off either side of the molecule. One can relate the location of the rainbow peak to the shape of the potential using a classical hard ellipsoid model [83]:
70.5.1 70.4.4
Electronically Inelastic Scattering
Harpoon and Stripping Reactions
It was known in the 1930s that collisions of alkali atoms with halogen molecules exhibit very large cross sections and yield A wealth of information is available on electronically inelas- highly excited alkali halide products. These observations tic scattering systems, since these in general exhibit much were accounted for by the harpoon mechanism proposed by
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M. Polanyi. Because alkali atoms have low ionization poten- probability for remaining on the diabatic curve) is given by tials and halogen molecules large electron affinities, as the p D 1 eı ; (70.48) alkali atom approaches the molecule, electron transfer may occur at long range. These processes are considered in detail in Chap. 52 [92, 93]. The harpooning distance Rc at which where this curve crossing takes place may be estimated simply as 1 2 2 2 H12 Rc b2 2 the distance at which the Coulomb attraction of the ion pair ıD ; (70.49) 1 2 g Rc is sufficient to compensate for the endoergicity of the electron transfer: and H12 is the coupling matrix element between the two 2 curves; H12 may be estimated from an empirical relation that e ; (70.45) is accurate within a factor of three over a range of ten orders Rc D IP Ae of magnitude [94]. In atomic units, where IP and Ae represent the ionization potential and elecp (70.50) H12 D I1 I2 Rc e0:86Rc ; tron affinity of the electron donor and recipient, respectively. For R in Å and E in eV, this relation is where 14:4 p p : (70.46) Rc D R D . I C I2 / (70.51) 1 c IP Ae Due to the large Coulombic attraction between the ion pair, reaction proceeds immediately following electron transfer. The crossing distance may then be used to estimate the effective reaction cross section. The vertical Ae is not necessarily the appropriate value to use in estimating these crossing distances; stretching of the halogen bond may occur during approach, so the effective Ae is generally somewhere between the vertical and adiabatic values. Often, there exists some repulsion between the atoms in the resulting halogen molecular ion, so that electron transfer is accompanied by dissociation of the molecule in the strong field of the ion pair. The alkali ion, having sent out the electron as the “harpoon”, then reels in the negative ion, leaving the neutral halogen atom nearly undisturbed as a spectator. Because these events occur at long range, there is no momentum transfer to the spectator atom, and it is a simple matter to estimate the anticipated angular and translational energy distributions in this spectator-stripping limit. The product molecule is scattered forward (relative to the direction of the incident atom), and for the reaction A C BC ! AB C C, the final CM velocity for the product AB is given by
is the reduced crossing distance, and I1 and I2 are the initial and final ionization potentials of the transferred electron. One finds electron transfer probabilities near unity for curve crossing distances below about 5 Å, dropping to zero for crossing at distances greater than about 8 Å. These estimates are based on electron transfer in atom–atom collisions, and it is important to remember that atom–molecule collisions occur on surfaces rather than curves, so the crossing seam may cover a broad range of internuclear distances.
70.5.2
Rebound Reactions
Another common direct reaction mechanism is the rebound reaction exemplified by F C D2 ! DF C D [29]. The CM product flux versus velocity contour map obtained for this reaction is shown in Fig. 70.2. Due to the favorable kinematics and energetics in this case, the FD product vibrational distribution is clearly resolved and peaks at v D 2. The dominant v D 2 product peaks at a CM angle of 180ı (referred to the direction of the incident F atom). This rebound scattering is characteristic of reactions exhibiting a barrier in the entrance channel. Rebound scattering implies small b collisions, and MC uBC ; (70.47) this serves to couple the translational energy efficiently into u0AB D MAB overcoming the barrier. Small b collisions necessarily have where uBC is the initial CM velocity of the BC molecule. smaller cross sections however, since the cross section scales This spectator-stripping mechanism may occur in systems quadratically with b. other than harpoon reactions, and it is useful to remember it as a limiting case. The likelihood of electron transfer at these crossings 70.5.3 Long-Lived Complexes may be estimated using a simple Landau–Zener model [93] (Chap. 52). For relative velocity g, impact parameter b, A third, important reaction mechanism involves the formaand crossing distance Rc , the probability for undergoing tion of an intermediate that persists for some time before disa transition from one adiabatic curve to another (that is, the sociating to give products. If the collision complex survives
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for many rotational periods ( 1011 s), then the CM angular distribution exhibits a characteristic forward–backward symmetry, usually with peaking along the poles. The latter occurs because the initial and final orbital angular momenta tend to be parallel (and perpendicular to the initial relative velocity vector). When there exist dynamical constraints enforcing some other relation, as in the case F C C2 H4 , then sideways scattering may be observed, despite a lifetime of several rotational periods [95–97]. For some systems exhibiting this long-lived behavior, the rotational period may be used as a molecular clock to monitor the lifetime of the complex. By increasing the collision energy until the distribution begins to lose its forward–backward symmetry, one can investigate the internal energy of the system just when its lifetime is on the order of a rotational period. Systems that have an inherent symmetry may exhibit this forward–backward symmetry in the scattering distributions despite lifetimes that are considerably shorter than a rotational period. This is the case for O(1D) reacting with H2 , for example [98]. This reaction involves insertion of the O atom into the H2 bond, resulting in an intermediate that accesses the deep H2 O well and contains considerable vibrational excitation. Trajectory calculations show that the complex dissociates after a few vibrational periods, but the distribution exhibits forward–backward symmetry because the O atom is equally likely to depart with either H atom.
70.6
Recent Developments
Astonishing progress in reactive scattering methods have continued in the past decade, and a few highlights are summarized here. These advances have taken the form of improvements in detection methods, or in some cases, entirely new experimental approaches and new achievements that people have made from crossed-beam experiments. One of the most important of these is the high-resolution H=D-atom Rydberg tagging time-of-flight (HRTOF) technique in the crossed molecular beam experiment, which is suitable for the study of bimolecular reaction yielding product H or D atoms. Yang and coworkers applied this method to systematically study the benchmark reaction of FCH2 and look for reactive resonances in this elementary bimolecular reaction. The lowest barrier height for the F C H2 reaction is 1.63 kcal mol1 , [99]; however, H products were probed when collision energies were lower than the barrier height that indicates some tunneling effects here [100]. Based on the conservation of the linear momentum, such high-resolution H-atom TOF spectra of the H atom products resolve the rovibrational states of the HF (v 0 D 1; 2; 3; j 0 ) coproducts. Pronounced forward-scattered HF products in the v 0 D 2 vibrational state were clearly observed at the collision energy of 0.52 kcal mol1 . Full quantum scattering calculations
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based on an accurately constructed potential energy surface indicate that this is attributed to the Feshbach resonances trapped in the peculiar HF.v 0 D 3/ H0 vibrationally adiabatic potential (VAP) and decay to the HF.v 0 D 2/ product channel via strong coupling between HF.v 0 D 3/ H0 and HF.v 0 D 2/ H0 adiabatic curves [100–102]. The advantage of HRTOF is its very high resolution, but it can only detect H=D products at one angle each time and requires rotating the detector in order to get full differential cross section (DCS) information. This could be improved by combining it with velocity map imaging (VMI) detection, since VMI can measure the product angular distribution (proportional to DCS distribution) at all angles together and can measure the translational energy distribution simultaneously. The problem for imaging the H=D-atom is that the H=D-atom is light, and electron recoil will blur the image and result in spectrum resolution that is not good. Very recently, Wang and coworkers applied the 1 C 10 REMPI threshold ionization technique to eliminate the electron recoil for H(D)-atom detection that gives rise to a high-resolution time-sliced velocity map ion imaging apparatus [103]. Such a high-resolution experiment allowed them to observe the geometric phase effect in the H C HD ! H2 C D reaction for the first time. Compelling applications of time-sliced VMI with crossed beams can be found in the work of Liu and coworkers on state-resolved detection on methyl radicals following reactions of methane with X (X: F, Cl, O(3P ), and OH) [104–106]. Unlike an atom detected in HRTOF, polyatomic molecules/radicals have vibrational and rotational excitation when they are first produced. In general, their REMPI schemes are not available for all vibrational and rotational states, i.e., the methyl radical only has the origin bands and the umbrella-mode excited bands known in its first REMPI spectrum [107]. Nevertheless, the methyl ion images provide quantum-state correlated differential cross sections for this type of reactions directly, allowing comparison to theory at an unprecedented level of detail [108–111]. Another significant new direction in detection strategies is the use of near-threshold VUV product ionization, especially for larger molecules whose REMPI probe schemes are not available. This is a universal approach, in that little advance spectroscopic information is required, but it is selective in that dissociative ionization is minimized, and sometimes isomer-selective detection may be achieved. This approach has been used in synchrotron-based studies of Cl atom reactions [112], radical–radical reactions [113] and in transition metal reactions [114, 115], and in product imaging studies of oxygen and chlorine atom reactions using the F2 excimer at 157 nm [77, 79, 116]. Inspired by the threshold VUV detection methods, Casavecchia and coworkers recently advanced the use of near-threshold electron impact ionization in a conventional universal crossed-beam config-
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uration [117]. Their recent results show the great promise of this technique to deliver higher signal-to-noise ratio and to minimize fragmentation processes in the detection step that may obscure the underlying dynamics [118]. A final note concerns developments in the study of chemical dynamics in the cold and ultracold regime with crossed beams. Quantum phenomena such as resonances dominate at low temperatures where few partial waves contribute to the collision. One can reduce the collision energy by slowing the beams with a Stark decelerator [119], merged beams [120, 121], or by making the small intersection angles of the two beams as shown in Eq. (70.19) [122–124]. These approaches are ushering in a new era in scattering studies, and reactive applications of these methods hold promise for interesting new effects.
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H. Li et al. Hongwei Li Hongwei Li received his PhD in 2016 from the University of Pennsylvania. He then joined the Suits group at the University of Missouri as a postdoctoral fellow. He is now a professor at Dalian Institute of Chemical Physics, CAS, China. His research focuses on generating different radical sources to investigate their reaction dynamics and kinetics with other molecules.
Arthur G. Suits Arthur G. Suits obtained his PhD in 1991 with Yuan T. Lee at Berkeley. He has held positions at Lawrence Berkeley and Brookhaven National Laboratories and faculty positions at Stony Brook and Wayne State. He is now Curators’ Distinguished Professor of Chemistry at the University of Missouri. He has devoted his career to studying the detailed dynamics of elementary chemical reactions.
Ion–Molecule Reactions
71
James M. Farrar
Contents
Keywords
71.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1035
71.2
Specification of Cross Sections . . . . . . . . . . . . . . 1036
71.3 71.3.1 71.3.2 71.3.3 71.3.4 71.3.5
Instrumentation . . . . . . . . . . . . Reactant Ion Preparation . . . . . . . Reactant Mass Selection . . . . . . . . The Collision Region . . . . . . . . . . Product Detection . . . . . . . . . . . . Imaging Methods in Velocity Space .
71.4
Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 1039
71.5 71.5.1
Recent Examples of State-Resolved Measurements 1040 Velocity-Angle Differential Cross Sections 0 .Vrel ; jVrel / . . . . . . . . . . . . . . . . . . . . . . . . . 1040 State-Resolved Cross Sections .n0 jn; Vrel / and Rate Constants k.n0 jn; T / . . . . . . . . . . . . . . . 1040
71.5.2 71.6
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1037 1037 1037 1038 1038 1039
The Future of the Field . . . . . . . . . . . . . . . . . . . 1042
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042
Abstract
Ionized matter is ubiquitous in Nature, occurring over a large range of length scales and energy densities, in environments ranging from plasmas to condensed phases. In the gas phase, collisions that occur between ions and neutral atoms and molecules provide important microscopic routes for the formation of species that influence the macroscopic behavior of chemical and physical systems such as flames and planetary atmospheres. The term “ion– molecule” reactions generally applies to an entire class of gas phase collision phenomena in which electrons and atoms can be transferred among the approaching reactants to form products with new chemical bonds and distributions of charge. Such reactions are generally regarded as low energy phenomena, occurring over a collision energy range from thermal up to several eV. J. M. Farrar () Dept. of Chemistry, University of Rochester Rochester, NY, USA e-mail: [email protected]
state-resolved products mass spectrometry cross section reaction dynamics total cross section flux distribution vibrational state total reaction cross section collision complex
71.1 Introduction Mass spectrometry has played a crucial role in the discovery and the study of ion–molecule reactions. At the beginning of the twentieth century, J.J. Thomson discovered that operating his positive ray parabola apparatus in a hydrogen atmosphere produced signals at a mass to charge ratio of 3, which he correctly attributed to the species H3 [1]. Later experiments by Hogness and Lunn [2] employing mass spectrometry with a high-pressure ion source demonstrated unequivocally that this species was produced by a very fast reaction between the primary ionization product HC 2 and molecular hydrogen. The pioneering work of Tal’rose in Moscow on ion formation in methane gas irradiated by electrons [3, 4] initiated the study of ion–molecule reactions as a distinct discipline within the field of chemical kinetics and established the central role of such reactions in radiation chemistry. The same reactions were also observed by Stevenson and coworkers [5–7] in the ion source of a laboratory mass spectrometer. Their analysis [8] via an orbiting collision model [9] predicted that ion–molecule reactions proceed with encountercontrolled rates in excess of 109 cm3 molecule1 s1 , corresponding to thermal energy cross sections of magnitude 1016 –1015 cm2 . This recognition and its confirmation by experiment has made ion–molecule reactions critical contributors to the modeling of planetary atmospheres, interstellar chemistry, and reactive plasmas. Studies of chemical dynamics that focus on how reactant energy is consumed and how available energy appears in products have highlighted the
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_71
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importance of scattering processes and the critical role that features of multidimensional potential surfaces play to control reaction dynamics. This chapter builds on the relationship between ion– molecule reactions and advances in mass spectrometry. Under single-collision conditions, which may be achieved when a collimated ion beam interacts with neutral reactants also formed into a beam, the experimental data for product formation take the form of one of a number of quantities called “cross sections”. In its simplest, phenomenological form, the total cross section parameterizes the attenuation of an ion beam with a collision gas of number density n and interaction length L according to Eq. (71.1) D .nL/1 ln
Io : I
(71.1)
In this expression, Io denotes the initial ion beam intensity, and I is the intensity measured after attenuation. Although Eq. (71.1) defines a cross section including all processes that scatter reactant ions from the beam, it is useful to think of the more specific cross section for chemical reaction as a major contributor to , providing a means to estimate the total rate of product formation dN˙ D Vrel n1 n2 V : dt
(71.2)
In this expression, is the total reaction cross section, Vrel is the relative speed of the collision partners, n1 is the number density of the primary (ion) beam, n2 is the number density of the collision gas, and V is the collision volume defined by the overlap of the ion beam with the target gas. This expression can be converted to a particularly useful form for computing signal levels from experimental parameters such as ion beam current I˙ , cross section , neutral number density n2 , and attenuation length L as follows dN˙ D 6:25 107 I˙ n2 L ; dt
(71.3)
where is expressed in Å2 , I˙ in units of 109 A, n2 in cm3 , and L, the attenuation length, in cm. An ion beam of current 1 109 A, intersecting a target of length 1 cm at a pressure of 103 Torr, corresponding to a number density of 3:5 1013 cm3 , and reacting with a cross section of 1 Å2 yields a total rate of product formation of 2 107 s1 . The cross section used to estimate total product formation rates is phenomenological, and in order to understand measurements, we need to define a hierarchy of cross sections based on the incident velocities and quantum states of reactants, as well as the quantum states, relative velocities, and scattering angles of products. Experimental conditions
may also require us to consider quantities that are averages over initial states, sums over final states, or are averaged over initial and final velocity distributions. The next section provides assistance in interpreting cross sections for these less well-specified processes.
71.2 Specification of Cross Sections An “ideal” experiment is one in which reactants with wellspecified quantum numbers, collectively denoted n, collide at a precisely defined relative velocity V , resolving products in quantum states n0 scattered through center of mass scattering angle . Technological advances, particularly in laser preparation of quantum state-selected reactants and in statespecific product detection, have made this goal a near-reality in favorable circumstances, particularly in neutral–neutral interactions, e.g., in experiments on H C H2 and its isotopic variants. Most experiments, however, require the experimenter to settle for a less precisely defined system in which one carries out partial averages over initial conditions or partial summations over final states. Figure 71.1 shows that the detailed differential cross section described above and denoted .n0 ; jn; Vrel / can be averaged over scattering angle to yield a state-to state cross section at a fixed collision velocity Vrel denoted as .n0 jn; Vrel /. Averaging this cross section over a Maxwell–Boltzmann distribution of molecular speeds at a specified temperature T yields the detailed state-to-state rate constant k.n0 jn; T /, while summation over the final states n0 and averaging over the initial states n yields the thermal rate constant k.T /. The rate constant is a thermally averaged, multiple-collision property and not the subject of this chapter, although k.T / plays an important role in modeling applications. Less detailed differential cross sections .n0 ; jn; Vrel / arise from averaging over initial states n and summing over final states n0 to yield cross sections 0 and scattering angle at dependent on product velocity Vrel 0 ; jVrel /. An averfixed collision velocity, denoted by .Vrel age over scattering angles and product velocities yields the velocity dependent total cross section .Vrel /, and its average over the Maxwell–Boltzmann reactant speed distribution produces the thermal rate constant k.T / once again. As one moves from more highly averaged quantities to the detailed cross sections, more sophisticated reactant preparation and product detection schemes are required to extract the desired information content. More highly specified cross sections lead to lower signal levels as defined in Eq. (71.3). The subject of this chapter will be a discussion of the various cross sections shown in Fig. 71.1, emphasizing new experimental methods and results that focus on the determination of cross sections for state-selected reactants and for state-tostate processes.
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σ(n ,θ n,V ) n, n
σ(V ,θ V ) ¢¢²²θ, v
¢²θ σ(n n,V )
n, n
¢²v k(n n,T)
σ(V )
¢²v
tials and probed optically have also led to major advances in the quest to measure state-to-state cross sections.
n, n k(T )
Fig. 71.1 Relationships among differential and total cross sections, and rate constants. Brackets denote averages over indicated variables or averages over initial states and summations over final states
71.3 Instrumentation Instrumentation to study ion–molecule reactions is quite diverse, and numerous literature sources are available for further discussion [10, 11]. A typical instrument has an ionization source that may allow for some degree of reactant state specification, a primary mass selector, a collision region, and a detector, consisting of a mass spectrometer or employing a spectroscopic technique allowing product quantum state and molecular identification. The “traditional” method for determining high-resolution/highinformation content cross sections employs continuous ion and neutral beams in a crossed beam configuration. Product detection with an energy/mass analyzer that rotates about the collision center allows pointwise construction of the velocity space distribution of reaction products. In principle, with adequate resolution, product vibrational state-resolved angular distributions may be measured. Experiments conducted at Rochester on the O C D2 reactive system [12, 13] and at Freiburg on the H C D2 system [14, 15] have approached this limit. However, except for optical detection of reaction products, rotational state resolution has yet to be achieved. These limitations reflect the low signal levels associated with increasing resolution. Significant progress in extending the scope of highresolution experiments has come from the application of pulsed methods of reactant preparation and pulsed techniques for product detection. Matching the time structures for reactant formation and product detection has yielded promising results for measuring cross sections with enhanced information content and for extending the range of chemical complexity of reactants and products. Elegant methods employing ions trapped by electrodynamic poten-
71.3.1 Reactant Ion Preparation Ion–molecule reactions were the first chemical reactions for which initial quantum state selection was achieved. Photoionization has been the method of choice for vibrational and vibrational–rotational state selection of molecular ions. Historically, single-photon VUV ionization methods played a special role in illustrating fundamental concepts of reactivity. The selective role of reactant vibrational excitation in overcoming potential energy barriers to endothermic chemical reactions was first demonstrated in the pioneering work of Chupka and coworkers on the HC 2 CHe system [16]. Many more studies on systems of comparable complexity have followed; the recent development of pulsed-field ionization – photoelectron secondary ion coincidence (PFI-PESICO) on this same system and others [17, 18] provides a picture of the high resolution that is now accessible to simple ion–molecule systems. In systems of greater complexity, Anderson and colleagues [19–21] applied guided beam methods to reactions of polyatomic ions prepared by photoionization to demonstrate “mode-selective” chemistry. Ng and coworkers demonstrated recently [22] that single rotational states in H2 OC can be prepared and that significant dynamical effects can be observed in the reactions with D2 . Laser-based multiphoton ionization methods have also been applied to the production of state-selected ions. Measurements of photoelectron spectra of emitted electrons are often necessary to assess the vibrational state purity of the photoions, as demonstrated by Zare and coworkers [23] on NHC 3 prepared in the 2 umbrella bending mode. Although such methods are, in principle, capable of producing usable populations of state-selected ions, examples are relatively few in number [20, 24]. State selection of NC 2 , for example, to produce rovibronic ground-state ions has been accomplished multiphoton ionization of N2 via the P by0 (2 C 1) 0 v D 0, J D 2 intermediate state [25]. a001 C g Less selective ionization methods forming a range of internally excited states have been teamed with ion trapping methods that contain excited ions in a small volume with a cold collision gas to cool them to the lowest vibrational state. Examples will be discussed below.
71.3.2 Reactant Mass Selection “Traditional” mass selection methods using deflecting fields are continuous in nature. To complement the quantum-state selection methods described above, which are intrinsically pulsed, time-of-flight methods are ideally suited to add mass
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CCD camera Photomultiplier MCP + phosphor screen Velocity map imaging spectrometer
Electron gun
8-pole rf ion trap 100–400 K Ion lens
Wiley–McLaren mass spectrometer
Piezoelectric pulsed valve
Piezoelectric pulsed valve with plasma electrode
Fig. 71.2 Schematic view of the Innsbruck ion–molecule crossed-beam imaging spectrometer. (Reprinted from [27] with permission from the PCCP Owner Societies)
selection to reactant ion preparation schemes. The standard Wiley–McLaren time-of-flight (TOF) method [26] produces a train of spatially compressed, time-separated ions with the proper time structure to mate with pulsed methods for neutral reactant preparation and time-dependent methods for product detection. Wester and coworkers have combined these methods in instrumentation in the last few years [27]. Figure 71.2 shows a schematic of this instrument, illustrating the formation of a pulsed beam of mass selected ions by a Wiley–McLaren TOF mass spectrometer, cooling of the ions in a multipole trap, and injection of those ions into the collision region along with pulsed neutral reactants. The collision region, the dimensions of which are controlled by the spatial extent of the overlap of the pulsed reactant beams, is monitored by a velocity map imaging (VMI) detector, which is also intrinsically pulsed. The operation of such a detection scheme will be discussed in a later section.
71.3.3 The Collision Region The potential difference between the volume where ions are created and where they undergo reaction generally determines the incident collision energy of an ion–molecule system. In a “traditional” experiment with continuous beams, the collision volume is typically held at a constant potential, most often ground potential. With pulsed beams, the collision volume can be held at ground potential when the pulsed reactant beams intersect but can be pulsed to a nonzero potential to extract product ions into an imaging detector, for example. Constraints on such extraction pulses will be discussed in a later section.
Fig. 71.3 Collision- and state-controlled ion–molecule reactions. Schematic of the experimental setup of the Basel instrument. NC 2 ions were prepared in their rotational–vibrational ground state using a threshold-photoionization scheme and were sympathetically cooled into a Coulomb crystal of laser-cooled CaC ions (inset). The state-selected NC 2 ions were exposed to rotationally cold neutral N2 molecules introduced into the ion trap via a doubly skimmed supersonic molecular beam. (Reproduced from [11] by permission of John Wiley & Sons Ltd)
Significant progress in the study of ion–molecule collisions has also been achieved by allowing reaction to occur within the volume of a multipole ion trap such that all product ions are collected and analyzed. Willitsch and collaborators [11, 25, 28] have made noteworthy use of the idea of “sympathetic cooling”, as shown in Fig. 71.3, to produce state-selected ions whose reactions can be initiated by pulsing a reactant gas into the trap. Further details are discussed below.
71.3.4 Product Detection Mass spectrometry and angle-resolved product energy and mass analysis have opened the door to cross section measurements of product-state resolved ion–molecule reactions. The ability to determine product state-resolved cross sections via product kinetic energy analysis is determined by the kinetic energy resolution of the detection system. The majority of extant crossed-beam experimental studies construct three-dimensional velocity space distributions with a series of one-dimensional sections through the full distribution such that, in any interval of time, only a single detection element in velocity space can be measured. Typically, kinetic energy distributions at N energy points are collected at M different scattering angles, yielding a dataset of dimension N M . Acquisition of the product signal occurs in a pointwise manner, in which any given measurement interval only collects signals in a single element of the velocity space. The total signal level calculated by Eq. (71.2) must be distributed among all N M elements of velocity space,
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providing a practical limit on this method’s abilities. As de- ciple. Products with different masses describe spheres with scribed below, imaging methods allow all N M velocity different delay times, providing mass-selective detection of space elements to be measured in a single time window. products via temporal gating of the MCP detector. Electrons from the MCP are accelerated into a phosphor screen, the light from which is detected by a CCD camera. The sig71.3.5 Imaging Methods in Velocity Space nal detected on the phosphor screen is a two-dimensional representation of the three-dimensional distribution. One Originally proposed in the late 1980s by Paul Houston and method for recovering the true three-dimensional distribuDavid Chandler [29] to study the velocity distributions of tion employs inverse Abel transformation of the data [31]. the products of pulsed laser photolysis of molecules such Time-sliced methods have also been developed that gate the as CH3 I, methods for reconstructing velocity space flux imaging detector such that slices of the product spheres can distributions have undergone significant development since be analyzed [32]. The azimuthal symmetry of the flux distrithat time. The velocity focusing method of Parker and Ep- bution about the relative velocity vector is sufficient to ensure pink [30] is a particularly important improvement: the use of that equatorial slices represent the full distribution. an additional electrostatic lens after the extraction region alOne limitation of the simplest version of VMI is its inlows all points in the plane of the intersecting beams with the herent two-dimensional focusing capability. The thickness same Vx and Vy components to be imaged to a single point. of the collision volume along the focusing direction can deSuch a geometry is particularly useful in crossed beam ap- grade resolution. Three-dimensional imaging methods that plications. In this geometry, two pulsed beams of reactants correlate arrival position and time have been applied sucintersect in a small volume, and the reactive products sepa- cessfully to reduce this “chromatic aberration” contribution rate with their characteristic velocities. The locus of points in to this degradation of resolution [27, 33]. the center of the mass coordinates for each product quantum state is distributed, perhaps nonuniformly, on the surface of a sphere. After the beams intersect, a repeller plate pushes the 71.4 Kinematics ions up a drift tube. During this drift time, a given productstate sphere grows in coordinate space, until the sphere is The sliced or inverse Abel-transformed images represent flattened on the MCP detector. Spheres associated with in- product ion flux in laboratory Cartesian velocity coordidividual quantum states of a single product of unique mass nates [34, 35] (vx , vy ), and a simple velocity shift to the are “nested”, having a common origin, and produce concen- center of mass tric rings of intensity on the MCP detector. Figure 71.4 shows uDvC (71.4) a schematic of an instrument that illustrates the imaging prinPhosphor screen
MCP detector
yields barycentric distributions in Cartesian coordinates, symbolized by P .ux ; uy /. Barycentric recoil speed and scattering angle are given by the following expressions u D .u2x C uy2 /1=2 ; uy D tan1 : ux
Drift tube
(71.5) (71.6)
The relationship between Cartesian and polar flux intensity is given by the following expression Ic:m: .u; / D u2 P .ux ; uy / :
Velocity focusing lens
Crossed beams Repeller plate
Fig. 71.4 Schematic of crossed beam imaging instrument. Product flux extracted from the collision volume with a single center-of-mass speed describes a sphere in velocity space. This flux is projected onto a multichannel plate/phosphor screen detector and recorded with a CCD camera
(71.7)
Product angular and kinetic energy distributions, computed by averaging the flux distribution over recoil speed and scattering angle, respectively, and denoted by the bracket notation below, require integration over the Cartesian flux. The angle-averaged relative translational energy distribution of products, P .ET0 /, is given by the following expression Z hP .ET0 /i D
d sin u P .ux ; uy / : 0
(71.8)
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Integration over specific angular regions provides a means to assess how energy disposal may depend on scattering angle. For example, the angular distribution for a particular product may arise from a superposition of long-lived complex decay with flux distributed symmetrically about the relative velocity vector, and direct reaction, with an asymmetric flux distribution. The speed-averaged angular distribution is computed as follows Z1 du u2 P .ux ; uy / :
hg./iu D
(71.9)
0
which more than a single reaction product has been detected by VMI. Figure 71.5 shows images for the NHC 3 product and the HCNHC product at a collision energy of 1.5 eV. The NHC 3 charge transfer product appears in velocity space in the vicinity of the neutral NH3 beam, consistent with long-range electron transfer that results in negligible momentum transfer to the ionic product. The formation of HCNHC around the centroid indicates that this product arises from a long-lived collision complex. Product images are readily separated in time since the extraction process acts like a TOF mass spectrometer, with the lower mass NHC 3 products arriving at the detection plane before the heavier HCNHC products. The detector is gated to allow only a single product to be imaged at a time. This instrument has also been used with free-radical reactants, yielding the first crossed beam VMI studies of ionradical reactions [37, 38]. Wester and coworkers [27] have used the instrument shown in Fig. 71.2 to study ion–molecule interactions with an emphasis on increased reactant complexity. A longstanding interest of the group is the second-order nucleophilic SN 2 reaction of the form
The kinetic energy and angular distributions described in Eqs. (71.8) and (71.9) are the primary quantities that are subject to comparison with theory. Depending on the resolution of a given experiment, the limits of integration in Eq. (71.8), for example, can be controlled to allow relative vibrational state populations to be extracted from experimental data. Most often, dynamical calculations in the form of quasi-classical trajectories or an appropriate approximation in quantum theory can be undertaken with a parametrized (71.10) X C CH3 Y ! CH3 X C Y : potential energy surface, in order to probe a particular region of the surface or to test a particular dynamical approxima- Among the recent examples from the group is a series of tion. studies in which X D OH and Y D I, with the incoming nucleophile OH solvated by a variable number of H2 O molecules [39]. Such studies are critical in making the con71.5 Recent Examples of State-Resolved nection between the gas phase and the solution phase in Measurements which nucleophilic substitution is a benchmark reaction in organic chemistry. Figure 71.6 shows product images for the The study of ion–molecule reactions in the context of scat- I (H2 O)1;2 products of the reactions tering or collision phenomena is a technologically dependent activity. As single-collision methods have advanced from OH .H2 O/1;2 C CH3 I ! CH3 OH C I .H2 O/1;2 (71.11) crossed-beam geometries to configurations in which reactant states may be controlled and/or selected, and products at collision energies of 0.5 and 1.5 eV [27]. The images show can be detected state-selectively, either kinematically or op- significant dynamical signatures that depend both on collitically, information content has increased, and tests of theory sion energy and degree of solvation. Most important, the data have become more stringent. In the final section of this chap- provide evidence of the degree of chemical complexity that ter, examples of a few of the more recent and advanced may be probed in contemporary experiments. experiments will be presented. They should be viewed as a representative sampling rather than an exhaustive compilation of recent advances in the field of ion–molecule reactions. 71.5.2 State-Resolved Cross Sections .n0 jn; Vrel /
and Rate Constants k.n0 jn; T /
71.5.1
Velocity-Angle Differential Cross Sections 0 ; jVrel / .Vrel
Implementing VMI detection in a crossed beam experiment with a continuous ion beam and a pulsed neutral beam has led to a number of new results. The CC C NH3 reaction, important in circumstellar shells, has been studied in the author’s lab, [36] providing one of the first examples of a system in
Measurements of state-to-state total cross sections as a function of initial relative velocity have advanced significantly in the past decade. Improved resolution in photoionization, advances in methods for confinement of ions in multipole traps, and the use of sympathetic cooling techniques have been responsible for these advances. Until recently, photoionization methods were limited to elucidation of the role of reactant vibrational excitation in
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Fig. 71.5 Product images for charge transfer and C–N bond formation from the CC C NH3 reaction, superimposed on the most probable Newton diagrams at a collision energy of 1.5 eV. The left-hand image corresponds to the formation of NHC 3 by charge transfer, and the righthand image corresponds to the formation of HCNHC products by C–N
σ (Å 2)
vr (m/s) 1000
bond formation. The circles denote the loci of center of mass speeds that correspond to the maximum product values allowed by energy conservation. (Reprinted with permission from [36]. Copyright 2012, American Institute of Physics)
–
H 2O+ (X 2B1; v+1v +2v+3 = 000; N +Ka+Kc+) + D2 → H 2DO+ + D
OH (H 2O) 60
N +Ka+Kc+ = 211
0
N +Ka+Kc+ = 111
40 –1000 1000
0.5 eV
1.5 eV
OH–(H 2O)2
20
0
–1000
0
0.5 eV –1000
N +Ka+Kc+ = 0 00
0.1
1
10 Ecm (eV)
1.5 eV 0
1000 –1000 vx (m/s)
0
1000 vx (m/s)
Fig. 71.6 Crossed-beam reactive scattering images of the I reaction product for the reaction of OH (H2 O) and OH (H2 O)2 with CH3 I. The translational energy difference between adjacent rings is 1 eV, the outer rings again mark the kinematic cutoff. (Reprinted from [27] with permission from the PCCP Owner Societies)
promoting reactivity. A recent example from the laboratory of Ng at UC Davis [22] has provided convincing evidence for the role of reactant rotational excitation in promoting atom transfer in the system H2 OC C D2 ! H2 DOC C D :
(71.12)
The reactant H2 OC species was prepared in the 211 , 111 , and 000 asymmetric rotor states by vacuum ultraviolet (VUV) photoionization, and was transported to the interaction re-
Fig. 71.7 Comparison of the values for the reaction C H2 OC (X2 B1 I v1C v2C v3C D 000; NKa C KcC D 000 ; 111 ; and 211 / C D2 ! C H2 DO C D in the Ecm range of 0.05–10.00 eV. (Reprinted with permission from [22]. Copyright 2012, American Institute of Physics)
gion with a quadrupole mass filter. The collision region consists of a gas cell where ions are confined by an octupole trap, and products are mass-analyzed by another quadrupole mass filter. The geometry allows absolute total cross sections to be measured. Figure 71.7 shows plots of energy-dependent total cross sections from 0.05 to 10 eV, indicating a strong rotational state dependence at the lowest collision energies. Relative to ground-state ions, rotational excitation enhances the cross section by as much as 30%. Quasi-classical trajectory studies [40] have shown that rotational excitation reorients the H2 OC ion to allow the oxygen atom to achieve the necessary orientation with respect to H2 , thereby accessing the transition state that leads to the H3 OC H complex and subsequent product formation.
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71.6 The Future of the Field
Compared to the examples covered in the last edition of the Handbook, many experimental methods such as ion imag40 ing have reached maturity, and the potential of new methods, 20 particularly ion trapping and cooling, is just beginning to be realized. New methods have allowed collision studies to be carried out with single ions and at very low collision ener10 gies, approaching 1 K. In a field as technologically dependent as collision physics and chemistry, there is every expectation N + = 0, F1 5 that new phenomena in the quantum regime will be discovN + = 1, F1 ered and that new chemistry with ultracold reagents will be N + = 1, F2 elucidated. In the same way that single-molecule methods N + = 2, F1 0 N + = 2, F2 have revolutionized the study of condensed matter, we also expect that the ability to “pick apart the ensemble” in gas 0 1 2 3 4 5 6 phase collisions will yield new insights into reactivity and Reaction time (s) provide exciting theoretical challenges for many years to Fig. 71.8 Dynamics of the populations in different spin-rotational come. 60
states NC , F1;2 of sympathetically cooled NC 2 ions as a function of the time of exposure to the molecular beam of neutral N2 molecules. The populations in the different states have been probed by laser-induced charge transfer (LICT) with Ar atoms. (Reproduced from [11] by permission of John Wiley & Sons Ltd)
The high degree of control that can be achieved in ion– molecule reactions studied in ion trapping configurations is illustrated beautifully in recent work by Willitsch and coworkers. By preloading the multipole trap with CaC , cooling those ions, and establishing a “Coulomb crystal” [28] in which the individual trapped ions are arranged in a regular spatial pattern, reactions of CaC ions with CH3 F, CH2 F2 , and CH3 Cl have been studied [41, 42]. Ground-state NC 2 ions produced by multiphoton ionization may also be injected into the trap and cooled by the process of sympathetic cooling to form a bicomponent Coulomb crystal, as shown in Fig. 71.3. The individual trapped NC 2 ions may then be reacted with rotationally cooled N2 molecules to produce rotationally excited product NC 2 ions. Rotational excitation cross sections may be detected by employing a laser-induced charge transfer (LICT) method. Photoexcitation of the reaction products with a 787 nm laser pumps rotationally excited NC 2 , via the first electronically excited state, into vibrationally excited NC 2 , which selectively undergoes LICT with Ar. This detection scheme results in time-dependent depletion of the ground rotational state of NC 2 , and growth of excited rotational state populations, as shown in Fig. 71.8. The data yield cross sections .n0 jn; T / and rate constants k.n0 jn; T / for rotational excitation in NC 2 –N2 collisions. Equally impressive is the fact that these cross sections may be determined for the reactions of single ions.
References 1. Thomson, J.J.: Rays of Positive Electricity. Longman, Green, and Company, Essex, p 116 (1913) 2. Hogness, T.R., Lunn, E.G.: Phys. Rev. 26, 0044 (1925) 3. Frankevich, E.L., Tal’rose, V.L.: Dokl. Akad. Nauk. SSSR 119, 1174 (1958) 4. Tal’rose, V.L., Frankevich, E.L.: Dokl. Akad. Nauk. SSSR 111, 376 (1956) 5. Schissler, D.O., Stevenson, D.P.: J. Chem. Phys. 24, 926 (1956) 6. Stevenson, D.P.: J. Phys. Chem. 61, 1453 (1957) 7. Stevenson, D.P., Schissler, D.O.: J. Chem. Phys. 23, 1353 (1955) 8. Gioumousis, G., Stevenson, D.P.: J. Chem. Phys. 29, 294 (1958) 9. Langevin, P.: Ann. Chim. Phys. 5, 245 (1905) 10. Farrar, J.M., Saunders, J.W.H.: Technique for the Study of Ion– Molecule Reactions. John Wiley & Sons, New York (1988) 11. Willitsch, S.: In: Rice, S.A., Dinner, A.R. (eds.) Advances in Chemical Physics, vol. 162, p. 307. (2017) 12. Carpenter, M.A., Farrar, J.M.: J. Phys. Chem. A. 101, 6475 (1997) 13. Carpenter, M.A., Farrar, J.M.: J. Phys. Chem. A. 101, 6870 (1997) 14. Zimmer, M., Linder, F.: Chem. Phys. Lett. 195, 153 (1992) 15. Zimmer, M., Linder, F.: J. Phys. B At. Mol. Opt. Phys. 28, 2671 (1995) 16. Chupka, W.A., Russell, M.E.: J. Chem. Phys. 49, 5426 (1968) 17. Qian, X.M., Zhang, T., Chang, C., Wang, P., Ng, C.Y., Chiu, Y.H., Levandier, D.J., Miller, J.S., Dressler, R.A., Baer, T., Peterka, D.S.: Rev. Sci. Instrum. 74, 4096 (2003) 18. Zhang, T., Qian, X.M., Tang, X.N., Ng, C.Y., Chiu, Y., Levandier, D.J., Miller, J.S., Dressler, R.A.: J. Chem. Phys. 119, 10175 (2003) 19. Chiu, Y.H., Fu, H.S., Huang, J.T., Anderson, S.L.: J. Chem. Phys. 101, 5410 (1994) 20. Anderson, S.L.: Accounts Chem. Res. 30, 28 (1997) 21. Qian, J., Green, R.J., Anderson, S.L.: J. Chem. Phys. 108, 7173 (1998) 22. Xu, Y.T., Xiong, B., Chang, Y.C., Ng, C.Y.: J. Chem. Phys. 137, 241101 (2012)
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23. Morrison, R.J.S., Conaway, W.E., Ebata, T., Zare, R.N.: J. Chem. Phys. 84, 5527 (1986) 24. Green, R.J., Anderson, S.L.: Int. Rev. Phys. Chem. 20, 165 (2001) 25. Tong, X., Nagy, T., Reyes, J.Y., Germann, M., Meuwly, M., Willitsch, S.: Chem. Phys. Lett. 547, 1 (2012) 26. Wiley, W.C., McLaren, I.H.: Rev. Sci. Instrum. 26, 1150 (1955) 27. Wester, R.: Phys. Chem. Chem. Phys. 16, 396 (2014) 28. Willitsch, S.: Int. Rev. Phys. Chem. 31, 175 (2012) 29. Chandler, D.W., Houston, P.L.: J. Chem. Phys. 87, 1445 (1987) 30. Eppink, A., Parker, D.H.: Rev. Sci. Instrum. 68, 3477 (1997) 31. Dribinski, V., Ossadtchi, A., Mandelshtam, V.A., Reisler, H.: Rev. Sci. Instrum. 73, 2634 (2002) 32. Townsend, D., Minitti, M.P., Suits, A.G.: Rev. Sci. Instrum. 74, 2530 (2003) 33. Trippel, S., Stei, M., Cox, J.A., Wester, R.: Phys. Rev. Lett. 110, 163201 (2013) 34. Wolfgang, R., Cross, R.J.: J. Phys. Chem. 73, 743 (1969) 35. Friedrich, B., Herman, Z.: Collect. Czech. Chem. Commun. 49, 570 (1984) 36. Pei, L., Farrar, J.M.: J. Chem. Phys. 136, 204305 (2012) 37. Pei, L.S., Carrascosa, E., Yang, N., Falcinelli, S., Farrar, J.M.: J. Phys. Chem. Lett. 6, 1684 (2015) 38. Pei, L.S., Farrar, J.M.: J. Phys. Chem. A. 120, 6122 (2016) 39. Otto, R., Brox, J., Trippel, S., Stei, M., Best, T., Wester, R.: J. Phys. Chem. A. 117, 8139 (2013) 40. Song, H.W., Li, A.Y., Guo, H., Xu, Y.T., Xiong, B., Chang, Y.C., Ng, C.Y.: Phys. Chem. Chem. Phys. 18, 22509 (2016) 41. Willitsch, S., Bell, M.T., Gingell, A.D., Procter, S.R., Softley, T.P.: Phys. Rev. Lett. 100, 043203 (2008) 42. Gingell, A.D., Bell, M.T., Oldham, J.M., Softley, T.P., Harvey, J.N.: J. Chem. Phys. 133, 194302 (2010)
1043 James Farrar James Farrar received his PhD degree from The University of Chicago in 1974. He is currently Professor Emeritus at the University of Rochester, where he served as Professor of Chemistry from 1976 to 2018. His research has been concerned with the application of mass spectrometry, molecular beams, and computational chemistry to the study of ion–molecule interactions.
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Part F Quantum Optics
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Part F presents a coherent collection of the main topics and issues found in quantum optics. Optical physics, which is concerned with the dynamical interactions of atoms and molecules with electromagnetic fields, is first discussed within the context of semiclassical theories, and then extended to a fully quantized version. The theoretical techniques used to describe absorption and emission spectra using density matrix methods are developed. Applications of the dark state in laser physics is briefly mentioned. The basic concepts common to all lasers, such as gain, threshold, and electromagnetic modes of oscillation are described. Recent developments in laser physics, including single-atom lasers, two-photon lasers, and the generation of attosecond pulses are also introduced. The current status of the development of different types of lasers – including nanocavity, quantumcascade, and free-electron lasers – are summarized. The important operational characteristics, such as frequency range and output power, are given for each of the types of lasers described. Nonlinear processes arising from the modifications of the optical properties of a medium due to the passage of intense light beams are discussed. Additional processes that are enabled by the use of ultrashort or ultra-intense laser pulses are presented. The concept of coherent optical transients in atomic and molecular systems is reviewed; homogeneous and inhomogeneous relaxation in the theory are properly distinguished. Multiphoton and strong-field pro-
cesses are given a theoretical description. A discussion of the generation of sub-femtosecond pulses is also included. General and specific theories for the control of atomic motion by light are presented. Various traps used for the cooling and trapping of charged and neutral particles and their applications are discussed. The fundamental physics of dilute quantum degenerate gases is outlined, especially in connection with Bose–Einstein condensation. de Broglie optics, which concerns the propagation of matter waves, is presented with a concentration on the underlying principles and the illustration of these principles. The fundamentals of the quantized electromagnetic field and applications to the broad area of quantum optics are discussed. A detailed description of the changes in the atom–field interaction that take place when the radiation field is modified by the presence of a cavity is given. The basic concepts needed to understand current research, such as the EPR experiment, Bell’s inequalities, squeezed states of light, the properties of electromagnetic waves in cavities, and other topics depending on the nonlocality of light are reviewed. Applications to cryptography, tunneling times, and gravity wave detectors are included, along with recent work on “fast light” and “slow light”. Correlations and quantum superpositions, which can be exploited in quantum information processing and secure communication, are delineated. Their link to quantum computing and quantum cryptography is given explicitly.
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Contents 72.1 72.1.1 72.1.2 72.1.3
Multipole Expansion . . . . . . . . . Electric Dipole (E1) Interaction . . . Electric Quadrupole (E2) Interaction Magnetic Dipole (M1) Interaction . .
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72.2 72.2.1 72.2.2 72.2.3 72.2.4
Lorentz Atom . . . . . . . . . . . . . . . . . Complex Notation . . . . . . . . . . . . . . Index of Refraction . . . . . . . . . . . . . . Beer’s Law . . . . . . . . . . . . . . . . . . . Slowly Varying Envelope Approximation
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72.3 72.3.1 72.3.2 72.3.3 72.3.4 72.3.5
Two-Level Atoms . . . . . . . Hamiltonian . . . . . . . . . . . Rotating Wave Approximation Rabi Frequency . . . . . . . . . Dressed States . . . . . . . . . . Optical Bloch Equations . . . .
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Relaxation Mechanisms . . . . . . . . Relaxation Toward Unobserved Levels Relaxation Toward Levels of Interest . Optical Bloch Equations with Decay . Density Matrix Equations . . . . . . . .
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72.5 72.5.1 72.5.2 72.5.3
Rate Equation Approximation Steady State . . . . . . . . . . . . Saturation . . . . . . . . . . . . . . Einstein A and B Coefficients .
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72.6 72.6.1 72.6.2 72.6.3
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cal physics, while quantum optics treats the interaction of atoms or molecules with quantized electromagnetic fields. A significant part of optical physics and quantum optics is the study of near-resonant atom–field interactions and concentrates on nonperturbative dynamics, where the effects of the optical fields have to be kept to all orders. The atomic properties themselves are assumed to be known. The vast majority of problems in light–matter interactions can be treated quite accurately within semiclassical theories. However, an important class of problems where this is not the case is presented in Chaps. 82–84. While much of optical physics and quantum optics ignores the effects of the electromagnetic fields on the centerof-mass motion of the atoms, important topics such as atomic trapping and cooling (Chap. 79) and de Broglie optics (Chap. 81) rely in an essential way on such mechanical effects of light. The present chapter deals with more traditional aspects of optical physics, where these effects are ignored. Keywords
Rabi frequency multipole expansion rotating wave approximation atomic coherence optical Bloch equations dressed states Einstein A and B coefficients Lorentz atom two-level atom
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056
Abstract
Optical physics is concerned with the dynamical interactions of atoms and molecules with electromagnetic fields. Semiclassical theories, which study the interaction of atoms with classical fields, are often said to comprise optiP. Meystre () University of Arizona Tucson, AZ, USA e-mail: [email protected]
72.1 Multipole Expansion Conside a test charge q of mass m localized within an atom and acted upon by an external electromagnetic field with electric field E .r; t/ and magnetic field B.r; t/. In the multipole expansion formalism [1, 2], the electric and magnetic interaction energies between the charge and the electromagnetic field are Zr (72.1) Ve D VE0 .t/ q ds E .R C s; t/ ; 0
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_72
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Zr ds v B.R C s; t/ ;
Vm D q 0
and the magnetic energy becomes (72.2) Vm .t/ D
3 X
Bi .R; t/ i D1 respectively. (The material of this chapter is discussed in deX e„ tail in a number of texts and review articles. We cite such Œg` .˛/`i .˛/ C gs .˛/si .˛/ references rather than the original sources whenever pos2m˛ ˛ sible.) These energies correspond to the work done by the 3 X @Bi .R; t/ X e„ 2 electric and magnetic components of the Lorentz force in g` .˛/`i .˛/rj .˛/ @Rj 2m˛ 3 first moving the charge to a stationary origin of coordinates ˛ i;j D1 at a point R and then to a location r relative to R. Here, C gs .˛/si .˛/ C VE0 .t/ represents the energy of the charge when located at the reference point R. It may be expressed in terms of the (72.7) VM1 .t/ C VM2 .t/ C : electrostatic potential .R; t/ as VE0 .t/ D Cq.R; t/. A Taylor series expansion of Ve .t/ and Vm .t/ about r D 0 yields 1 72.1.1 Electric Dipole (E1) Interaction X 1 @ n1 r E .R; t/ ; r Ve .t/ D VE0 .t/ q nŠ @R nD1 For optical fields whose wavelength is large compared with (72.3) the interacting atom, only the first few terms in the Taylor 1 expansions of Ve .t/ and Vm .t/ need to be retained. The first n q„ X @ n1 Vm .t/ D ` B.R; t/ ; r term, VE0 .t/, of Ve .t/ is the net charge of the atom, which m nD1 .n C 1/Š @R vanishes for neutral atoms. The second term, VE1 .t/, is the (72.4) electric dipole interaction energy. Introducing the electric where „` D r p is the angular momentum of the test charge dipole (E1) moment, relative to the coordinate origin R. Here, use of the mechaniX dD q˛ r.˛/ ; (72.8) cal momentum p D mr, P instead of the canonical momentum, ˛ neglects the electromagnetic component of the momentum responsible for diamagnetic effects. or Z In addition to the electromagnetic interaction, electrons and nuclei are characterized by a spin magnetic moment d D d3 r .r/r ; (72.9) ms D .q„=2m/gs s, where s is the spin of the test charge and gs its gyromagnetic factor, equal to 2:002 : : : for elec- for a charge distribution, this contribution to the interaction trons. The factor q„=2m is the particle’s magneton. The spin energy may be re-expressed as magnetic moment yields an additional term to the magnetic (72.10) VE1 .t/ D d E .R; t/ : energy Vm , which becomes 1 e„ X 1 @ n1 The E1 term dominates most optical phenomena. Vm .t/ D r 2m nD1 nŠ @R 2 (72.5) 72.1.2 Electric Quadrupole (E2) Interaction g` ` C gs s B.R; t/ ; nC1 where the orbital g-factor is g` D q=e (g` D 1 for an elec- The V .t/ contribution to v .t/ describes electric E2 e tron). For an ensemble f˛g of charged particles q˛ in an atom, quadrupole (E2) interactions. In terms of the quadrupole these expressions are to be summed over all particles. Thus, tensor the electric energy becomes Z 3 (72.11) Q D 3 d3 r .r/ri rj ; X XX q˛ .R; t/ q˛ ri .˛/Ei .R; t/ Ve .t/ D ˛ i D1 ˛ VE2 .t/ becomes # " 3 1 X X 3 q˛ ri .˛/rj .˛/ @ 1 X 2 i;j D1 ˛ VE2 .t/ D Qij Ej .R; t/ : (72.12) 6 i;j D1 @Ri @ Ei .R; t/ C @Rj Alternatively, E2 interactions can be expressed in terms of R 3 .2/ VE0 .t/ C VE1 .t/ C VE2 .t/ C ; (72.6) the traceless quadrupole tensor Qij D d r .r/ .3ri rj
72 Light-Matter Interaction
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ıij r 2 /. E2 interactions are typically weaker than E1 interactions by a factor a0 =, where a0 is the Bohr radius, and is the wavelength of the transition. Since a0 = is very small for optical transitions, E2 interactions are typically neglected in quantum optics.
72.2.1 Complex Notation The study of light–matter interactions is simplified by the introduction of complex variables [8–11]. For example, an electric field E .R; t/ D
72.1.3 Magnetic Dipole (M1) Interaction
X
En cos.!n t/ ;
(72.18)
n;
The first term in the multipole expansion of the magnetic in- where is the polarization vector of the field Fourier comteraction (Sect. 72.6) is the magnetic dipole M1 interaction, ponent at frequency !n , is expressed as VM1 D m B.R; t/ ;
(72.13)
of a magnetic moment m in a magnetic field, where X q˛ „ mD Œg` .˛/`.˛/ C gs .˛/s.˛/ 2m˛ ˛ D B .L C 2S / ;
E .R; t/ D E C .R; t/ C E .R; t/ ;
(72.19)
where the positive frequency part of the field is E C .R; t/ D (72.14)
1X En expŒi.kn R !n t/ : 2 n;
(72.20)
and we have used the fact that for electrons g` D 1 and Due to the linearity of Eq. (72.16), it is sufficient to study gs ' 2. The Bohr magneton B is the response of the Lorentz atom to a plane monochromatic electric field of frequency !, complex amplitude E and poe„ ˛ea0 B D D ; (72.15) larization . Introducing the complex dipole moment 2mc 2 where ˛ is the fine structure constant. Thus, M1 interactions d D er D P expŒi.k R !t/ C c:c: ; (72.21) tend to be smaller than E1 interactions by a factor of order ˛=2. The connection between m and angular momentum J where P is in general complex for E real, and the complex is m D J , where is the gyromagnetic ratio. polarizability ˛.!/ via P D ˛.!/E , then
72.2 Lorentz Atom
˛.!/ D
!02
e 2 =m : ! 2 i0 !
(72.22)
The Lorentz atom consists of a classical electron harmonically bound to a proton. It provides a framework for understanding a number of elementary aspects of the electric 72.2.2 Index of Refraction dipole interaction between a single atom and light [3–7]. Assuming that the center of mass (CM) motion of the atom is From the Maxwell wave equation unaffected by the field, and neglecting magnetic effects, the equation of motion of the electron is 1 @2 1 @2 P.R; t/ 2 ; (72.23) r E .R; t/ D 2 c 2 @t 2 0 c 2 @t 2 0 d e d 2 C C !0 r D E .R; t/ ; (72.16) dt 2 2 dt m where P.R; t/ is the electric polarization, given by the elecwhere !0 is the electron’s natural oscillation frequency, and tric dipole density of the medium P D N d, N being the 0 represents a frictional decay rate that accounts for the ef- atomic density, the plane wave dispersion relation is fects of radiative damping. For the classical Lorentz atom !2 (72.24) k 2 D 2 n2 .!/ ; (72.17) 0 D 2!02 r0 =3c ; c where r0 D e 2 =40 mc 2 is the classical electron radius. This where the index of refraction n.!/ is damping arises physically from the radiation reaction of the s field radiated by the atom on itself. In the E1 approximation, N˛.!/ the electric field is evaluated at the location R of the atomic : n.!/ D 1 C c.m. 0
(72.25)
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Beer’s Law
the z-axis) can be expressed in the form
1 Since the polarizability ˛.!/ is normally complex, so is the E .R; t/ D E C .R; t/ ei.kz!t / C c:c: ; (72.30) 2 index of refraction. Its real part leads to dispersive effects, while its imaginary part leads to absorption. Specifically, such that ReŒn.!/ 1 has the form of a standard dispersion curve, ˇ Cˇ ˇ Cˇ ˇ ˇ ˇ @E ˇ ˇ ˇ ˇ ˇ positive for ! !0 < 0 and negative for ! !0 > 0, while ˇ ! ˇE C ˇ ; ˇ @E ˇ k ˇE C ˇ : ˇ (72.31) ˇ ˇ ˇ ˇ ImŒn.!/ is a Lorentzian curve peaked at ! D !0 . The inten@t @z sity absorption coefficient a.!/ is It is further consistent within this approximation to assume that the polarization takes the form a.!/ D 2 ImŒn.!/!=c " #1=2
2 2 i! C !0 ! 2 Ne 2! 1 : Im 1 C D P.R; t/ D P C .R; t/ ei.kz!t / C c:c: ; (72.32)
2 c m0 ! ! 2 2 C 2 ! 2 2 0
(72.26) with ˇ Cˇ For atomic vapors, the corrections to the vacuum index of ˇ Cˇ ˇ @P ˇ ˇ ˇ ˇ ˇ (72.33) ˇ @t ˇ ! P : refraction are normally small, so that the square root in Eq. (72.26) can be expanded to first order, giving Under these conditions, known as the slowly varying enve ! 2 Ne 2 : (72.27) lope approximation [5], Maxwell’s wave equation reduces to a.!/ D
0 mc ! 2 ! 2 2 C 2 ! 2 0 @ k C 1 @ E C .z; t/ D P .z; t/ : (72.34) C The intensity of a monochromatic field propagating along the @z c @t 2i0 z-direction through a gas of Lorentz atoms is, therefore, atHence, in the slowly varying envelope approximation, we igtenuated according to Beer’s law given by nore the backward propagation of the field [12]. The slowly I.!; z/ D I.!; 0/ ea.!/z : (72.28) varying amplitude and phase approximation is essentially the same, except that it expresses the electric field envelope in If the index of refraction at a given frequency becomes purely terms of a real amplitude and phase. imaginary, no electromagnetic wave can propagate inside the medium. In a plasma this is the case for field frequencies smaller than the plasma frequency 72.3 Two-Level Atoms s A large number of optical phenomena can be understood by Ne 2 : (72.29) !p D considering the interaction between a quasi-monochromatic m0 field of central frequency ! and a two-level atom, which simWhile the Lorentz atom model gives an adequate description ulates a (dipole-allowed) atomic transition [5, 7, 10, 11, 13– of absorption and dispersion in a weakly excited absorb- 17]. This approximation is well justified for near-resonant ing medium, it fails to predict the occurrence of important interactions; i.e., ! ' !0 . The next three sections discuss phenomena such as saturation and light amplification. This the model Hamiltonian for this system in the semiclassical is because, in this model, the phase of the induced atomic approximation where the electromagnetic field can be dedipoles with respect to the incident field is always such that scribed classically. The formal results are then extended to the polarization field adds destructively to the incident field. the case of a quantized field, where the electric field is treated The description of light amplification requires a quantum as an operator. treatment of the medium, which gives a greater flexibility to the possible relative phases between the incident and polarization fields. 72.3.1 Hamiltonian
72.2.4
Slowly Varying Envelope Approximation
Light–matter interactions often involve quasi-monochromatic fields for which the electric field (taken to propagate along
In the absence of dissipation mechanisms, the dipole interaction between a quasi-monochromatic classical field and a two-level atom is H D „!e jeihej C „!g jgihgj d E .R; t/ ;
(72.35)
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where jei and jgi label the upper and lower atomic levels, to perform the RWA. In the RWA, the atomic system is deof frequencies !e and !g , respectively, with !e !g D !0 , scribed by the Hamiltonian and R is the location of the center of mass of the atom. The
H D „!0 sz d sC E C .R; t/ C s E .R; t/ ; (72.41) electric dipole operator Eq. (72.8) couples the excited and ground levels and may be expressed as or, in a frame rotating at the frequency ! of the field, d D d d .jeihgj C jgihej/ ;
1
H D „ sz d sC E eikR C h:c: ; 2
(72.36)
where d is a unit vector in the direction of the dipole and d the matrix element of the electric dipole operator between the ground and excited state, which we take to be real for simplicity. We also neglect the vector character of d and E .R; t/ in the following, assuming, for example, that both d and are parallel to the x-axis. The Hamiltonian Eq. (72.35) may then be expressed as
(72.42)
where D !0 ! is the atom–light detuning. (Note that the alternate definition ı D ! !0 is frequently used in the literature.) In the rest of this chapter, we consider atoms placed at R D 0.
72.3.3 Rabi Frequency
H D „!e jeihej C „!g jgihgj d .jeihgj C jgihej/ The dynamics of the two-level atom is conveniently ex E C .R; t/ C E .R; t/ ; (72.37) pressed in terms of its density operator , whose evolution is given by the Schrödinger equation where we have generalized the notation of Eq. (72.20) in an d
i obvious way. One can introduce the pseudo-spin operators D ŒH; ; (72.43) dt „ sz D .jeihej jgihgj/=2 ;
where ee D hej jei and gg D hgj jgi are the upper and lower state populations Pe and Pg , respectively, while the D jeihgj ; (72.38) sC D ? are called off-diagonal matrix elements eg D hej jgi D ge the atomic coherences, or simply coherences, between levels and redefine the zero of atomic energy to introduce the comjei and jgi. These coherences play an essential role in optical monly used form physics and quantum optics, since they are proportional to
C the expectation value of the electric dipole operator. H D „!0 sz d .sC C s / E .R; t/ C E .R; t/ : The evolution of Pg .t/ and Pe .t/ D 1 Pg .t/, is char(72.39) acterized by oscillations at the generalized Rabi frequency s
72.3.2
1=2
; ˝ D ˝12 C 2
Rotating Wave Approximation
Under the influence of a monochromatic electromagnetic field of frequency !, atoms undergo transitions between their lower and upper states by interacting with either the positive or the negative frequency part of the field. The corresponding contributions to the atomic dynamics oscillate at frequencies !0 ! and !0 C !, respectively, and their contributions to the probability amplitudes involve denominators containing this same frequency dependence. For near-resonant atom– field interactions, the rapidly oscillating contributions lead to small corrections, the first-order one being the Bloch–Siegert shift, whose value near resonance is, for ! ' !0 , [17]. ı!eg
ˇ C ˇ 2 d ˇE ˇ=„ D 4!
(72.44)
where the Rabi frequency ˝1 is ˝1 D d E =„, (or ˝1 D d E .d /=„ when the vector character of the electric field and dipole moment are included). Specifically, assuming that the atom is initially in its ground state jgi, the probability that it is in the excited state jei at a subsequent time t is given by Rabi’s formula Pe .t/ D .˝1 =˝/2 sin2 .˝t=2/ :
(72.45)
At resonance ( D 0), the generalized Rabi frequency ˝ reduces to the Rabi frequency ˝1 . (In addition to the texts on quantum optics already cited, see also [18].)
(72.40) 72.3.4
Dressed States
to lowest order in d E =„!. The neglect of these terms is the Semiclassical Case rotating wave approximation (RWA). Note that it is normally The atomic dynamics can alternatively be described in terms inconsistent to regard an atom as a two-level system and not of a dressed states basis instead of the bare states jei and
72
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P. Meystre
E1 |g²
Δ
where the creation and annihilation operators a and a obey the boson commutation relation Œa; a D 1 (Chap. 6), and the coupling constant r ! gDd (72.49) 20 „V
|e² E2
Fig. 72.1 Dressed levels of a two-level atom driven by a classical monochromatic field as a function of the detuning D !0 !
jgi [17]. The dressed states j1i and j2i are eigenstates of the Hamiltonian Eq. (72.42), and, by convention, the state j1i is the one with the greatest energy. They are conveniently expressed in terms of the bare states via the Stückelberg angle =2 as j1i D sin jgi C cos jei ; j2i D cos jgi sin jei ;
(72.46)
where sin.2/ D ˝1 =˝, cos.2/ D =˝. The corresponding eigenenergies are 1 E1 D C „˝ ; 2 1 E2 D „˝ : 2
included. The Hamiltonian of the total atom–field system becomes
1 H D „!0 sz C „! a a C C „g sC a C a s ; 2 (72.48)
is the vacuum Rabi frequency, with V being a photon normalization volume. This Hamiltonian defines the Jaynes– Cummings model, [7, 19] which is discussed in more detail in Chap. 83. The dressed states of the atom–field system are the eigenstates of the Jaynes–Cummings model. Since, in the RWA, the dipole interaction only couples states of same excitation number, e.g., je; ni and jg; n C 1i, where jni is an eigenstate of the photon number operator, a ajni D njni, with n an integer, the diagonalization of the Jaynes–Cummings model reduces to that of the semiclassical driven two-level atom in each of these manifolds. Hence, the dressed states are j1; ni D sin n jg; n C 1i C cos n je; ni ; j2; ni D cos n jg; n C 1i sin n je; ni ;
(72.50)
with p tan.2n / D 2g n C 1= :
(72.47)
These energies are illustrated in Fig. 72.1 as a function of the field frequency !. The dressed levels repel each other and form an anticrossing at resonance ! D !0 . As the detuning varies from positive to negative values, state j1i passes continuously from the excited state jei to the bare ground state jgi, with both bare states having equal weights at resonance. The distances between the perturbed levels and their asymptotes for j j ˝1 represent the AC Stark shifts, or light shifts, of the atomic states when coupled to the laser. From Fig. 72.1, the AC Stark shift of jgi is positive for < 0 and negative for > 0, while the jei state shift is negative for < 0 and positive for > 0.
Quantized Field The concept of dressed states can readily be generalized to a two-level atom interacting with a single-mode quantized field in the dipole and rotating wave approximations. The atom and its dipole interaction with the field are still described by the Hamiltonian Eq. (72.41), except that the positive and negative frequency components of the field are now operators, and the free field Hamiltonian must be
(72.51)
(The factor of 2 difference between this and the semiclassical case is due to the use of a running waves quantization scheme, while the semiclassical discussion was for standing waves.) The corresponding eigenenergies are E1n D „.n C 1/! „Rn ; E2n D „.n C 1/! C „Rn ;
(72.52)
where Rn D
1p 2 C 4g 2 .n C 1/ : 2
(72.53)
Chapter 83 shows that by including the effects of spontaneous emission, this picture yields a straightforward interpretation of a number of effects, including the Burshtein– Mollow resonance fluorescence spectrum. Dressed states also help to elucidate the interaction between two-level atoms and quantized single-mode fields, as occur, for example, in cavity QED ([19] and Chap. 83). Their generalization to the case of moving atoms offers simple physical interpretations of several aspects of laser cooling, see Chap. 79 and [20].
72 Light-Matter Interaction
1053
72.3.5 Optical Bloch Equations
theory. One advantage of describing the atomic state in terms of a density operator is that the physical interpretation of Introducing the density operator matrix elements ab D its elements allows us phenomenologically to add various rehaj jbi, where a; b can be either e or g, as well as the real laxation terms directly to its elements. quantities U D eg ei!t C c:c: ; i!t
V D i eg e
72.4.1 Relaxation Toward Unobserved Levels
C c:c: ;
If the relaxation mechanisms transfer populations or atomic (72.54) coherences toward uninteresting or unobserved levels, their description can normally be given in terms of a Schrödinger the equations of motion for the density matrix elements ij D equation, but with a complex Hamiltonian. In contrast, if all hij jj i may be expressed, with Eq. (72.43), as levels involved in the relaxation mechanism are observed, a more careful description, e.g., in terms of a master equadU tion, is required. Specifically, in the case of relaxation to D V ; dt unobserved levels, the evolution of the atomic density opdV erator, restricted to the levels of interest, is of the general D U C ˝1 W ; dt form [16] dW (72.55) D ˝1 V : dt i d
(72.57) D Heff Heff ; dt „ These are the optical Bloch equations, as discussed extensively in [5, 17]. Physically, U describes the component of where the atomic coherence in phase with the driving field, V the (72.58) Heff D H C O ; component in quadrature with the field, and W the atomic inversion. The optical Bloch equations have a simple geometrical interpretation offered by thinking of U , V , and W H being the atom–field Hamiltonian and O the nonas the three components of a vector called the Bloch vector Hermitian relaxation operator, defined by its matrix elements U , whose equation of motion is „ (72.59) hnjO jmi D n ınm : 2i dU D˝ U ; (72.56) dt Both inelastic collisions and spontaneous emission to unobserved levels can be described by this form of evolution. In where ˝ D .˝1 ; 0; /. Thus, U precesses about ˝, of the framework of this chapter, inelastic, or strong, collisions length ˝, while conserving its length. The evolution of are defined as collisions that can induce atomic transitions a two-level atom driven by a monochromatic field is, thus, into other energy levels. mathematically equivalent to that of a spin- 21 system in two magnetic fields B 0 and 2B 1 cos !t, which are parallel to the z- and x-axis, respectively, and whose amplitudes are such 72.4.2 Relaxation Toward Levels of Interest that the Larmor spin precession frequencies around them are ! and 2˝1 cos !t, respectively. In optics, this vectorial picA master equation description is necessary when all levels inture is often referred to as the Feynman–Vernon–Hellwarth volved are observed [7, 17]. This master equation can rapidly picture [21]. It is very useful in discussing the coherent trantake a complicated form if more than two levels are involved. sient phenomena discussed in Chap. 77. We give results only for the case of a two-level atom and upper to lower-level spontaneous decay and elastic or soft collisions; i.e., collisions that change the separation of energy 72.4 Relaxation Mechanisms levels during the collision but leave the level populations unchanged. In that case, the atomic master equation takes the In addition to their coherent interaction with light fields, form atoms suffer incoherent relaxation mechanisms, whose orii d
gin can be as diverse as elastic and inelastic collisions and D ŒH; .sC s C sC s 2s sC / dt „ 2 spontaneous emission. Collisional broadening is discussed 1 in Chap. 63, while a QED microscopic discussion of sponta(72.60) ph C 2ph sz sz ; 2 neous emission is described in Chap. 83 in terms of reservoir W D ee gg ;
72
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P. Meystre
where the free-space spontaneous decay rate is found from 72.4.4 Density Matrix Equations QED to be In the general case, it is necessary to consider the density 1 4d 2 !03 ; (72.61) operator equation (72.43) instead of the optical Bloch equa D 40 3„c 3 tions. The equations of motion for the components of
become, for the general case of complex ˝1 , and ph is the decay rate due to elastic collisions. It is possible to express the classical decay rate d ee 1
D e ee i˝1 Qeg C c:c: ; Eq. (72.17) in terms of the quantum spontaneous emission dt 2 rate Eq. (72.61) as d gg 1
D g gg C i˝1 Qeg C c:c: ; dt 2 (72.62) D cl fge ; d Qeg ˝1
(72.66) D . C i / Qeg i
ee gg ; dt 2 where fge is the oscillator strength of the transition. The various oscillator strengths characterizing the dipole-allowed where D .e C g /=2 C ph , and Qeg D eg ei!t . In the case transitions from a ground state jei to excited levels jei obey of spontaneous decay from the upper to the lower level, these the Thomas–Reiche–Kuhn sum rule equations become X fge D 1 ; (72.63) 1
d ee D ee i˝1 Qeg C c:c: ; e dt 2 d gg 1
where the sum is on all levels dipole-coupled to jgi. AssumD C ee C i˝1 Qeg C c:c: ; dt 2 ing that d and the polarization of the field are both parallel d Qeg ˝1
to the x-axis, this gives (72.67) D . C i / Qeg i
ee gg ; dt 2 2m!0 (72.64) where D =2 C ph . Equations (72.67) are completely fge D jhgjxjeij2 : „ equivalent to the optical Bloch equations (72.65).
72.4.3 Optical Bloch Equations with Decay
72.5 Rate Equation Approximation
In general, the optical Bloch equations cannot be generalized to cases where relaxation mechanisms are present. There are, If the coherence decay rate is dominated by elastic collihowever, two notable exceptions corresponding to the fol- sions, and hence is much larger than the population decay rates e and g , Qeg can be adiabatically eliminated from the lowing situations. equations of motion Eqs. (72.66) and (72.67) to obtain the 1. The upper level spontaneously decays to the lower level rate equations (Sect. 72.2) only, while the atom undergoes only elastic collisions.
d ee 2. Spontaneous emission between the upper and lower levels D e ee R ee gg ; dt can be ignored in comparison with decay to unobserved
d gg levels, which occur at equal rates e D g D 1=T1 . (72.68) D g gg C R ee gg ; dt Under these conditions, Eq. (72.55) generalizes to dU D U=T2 V ; dt dV D V =T2 C U C ˝1 W ; dt dW D .W Weq /=T1 ˝1 V ; dt
and
d ee D ee R ee gg ; dt
d gg D C gg C R ee gg ; dt
(72.69)
(72.65) respectively, where the transition rate is
(72.70) R D j˝1 j2 L. /=.2/ ; where we have introduced the longitudinal and transverse relaxation times T1 and T2 , with T1 D 1= and T2 D .1=2T1 C ph /1 in the first case, and T2 D .1=T1 C ph /1 in the sec- and we have introduced the dimensionless Lorentzian ond case. The equilibrium inversion Weq is equal to zero in
L. / D 2 = 2 C 2 : (72.71) the second case since the decay is to unobserved levels.
72 Light-Matter Interaction
1055
The transitions between the upper and lower state are, thus, described in terms of simple rate equations. Adding phenomenological pumping rates e and g on the right-hand side of Eq. (72.68) provides a description of the excitation of the upper and lower levels from some distant levels, as would be the case in a laser. The equations then form the basis of conventional, single-mode laser theory. In the absence of such mechanisms, the atomic populations eventually decay away.
72.5.1
Steady State
In the case of upper to lower-level decay, the state populations reach a steady state with inversion [5, 17]
72.5.3 Einstein A and B Coefficients When atoms interact with broadband radiation instead of the monochromatic fields considered so far, Eq. (72.69) still apply, but the rate R becomes R ! Beg %.!/ ;
(72.76)
where %.!/ is the spectral energy density of the inducing radiation. Einstein’s A and B coefficients apply to an atom in thermal equilibrium with the field, which is described by Planck’s blackbody radiation law %.!/ D
1 „! 3 ; 2 3 „!=k BT 1 c e
(72.77)
where T is the temperature of the source, and kB is Boltz(72.72) mann’s constant. Invoking the principle of detailed balance, which states that at thermal equilibrium, the average numwhere s is the saturation parameter. In the case of pure radia- ber of transitions between arbitrary states jii and jki must be tive decay, ph D 0, s is given by equal to the number of transitions between jki and jii, one finds ˝12 =2 : (72.73) sD 2 „! 3 Aki =4 C 2 D 2 3 ; (72.78) Bki c In steady state, the other two components of the Bloch vector U are given by where Aki is the rate of spontaneous emission from jki to P jii, and k D i Aki is the level width. s 2 (72.74) Ust D ˝1 1 C s Wst D
1 D ; C 2R 1Cs
72.6 Light Scattering
and
s ˝1 1 C s
(72.75) Far from resonance, the approximation of a two-level or fewlevel atom is no longer adequate. Two limiting cases, which Ust varies as a dispersion curve as a function of the detuning are always far from resonance, are Rayleigh scattering for , while Vst is a Lorentzian of power-broadened half-width low frequencies and Thomson scattering for high frequencies. at half-maximum . 2 =4 C ˝ 2 /1=2 . Vst D
I
1
72.5.2
Saturation
As the intensity of the driving field, or ˝12 , increases, Ust and Vst first increase linearly with ˝1 , reach a maximum, and finally tend to zero as ˝1 ! 1. The inversion Wst , which equals 1 for ˝1 D 0, first increases quadratically and asymptotically approaches Wst D 0 as ˝1 ! 1. At this point, where the upper and lower state populations are equal, the transition is said to be saturated, and the medium becomes effectively transparent, or bleached. (This should not be confused with self-induced transparency,which is discussed in Chap. 77.) The inversion is always negative, which, in particular, means that no steady-state light amplification can be achieved in this system. This is one reason why external pump mechanisms are required in lasers.
72.6.1 Rayleigh Scattering Rayleigh scattering is the elastic scattering of a monochromatic electromagnetic field of frequency !, wave vector k, and polarization by an atomic system in the limit where ! is very small compared with its excitation energies [4, 17]. To second order in perturbation theory, the Rayleigh scattering differential cross section into the solid angle ˝ 0 about the wave vector k0 with k 0 D k, and polarization 0 is X fge d 2 4 0 2 D r ! . / 0 2 d˝ 0 !eg e
!2 ;
(72.79)
where r0 is the classical electron radius, the sum is over all states jei, fge is the E1 oscillator strength Eq. (72.64), and
72
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P. Meystre
!ge is the transition frequency. The corresponding total cross References section is 1. Shore, B.W.: The Theory of Coherent Atomic Excitation. Wiley, !2 New York (1990) 8 r02 ! 4 X fge 2. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Photons and D : (72.80) 2 3 !eg Atoms: Introduction to Quantum Electrodynamics. Wiley, New e
72.6.2
Thomson Scattering
Thomson scattering is the corresponding elastic photon scattering by an atom in the limit where ! is very large compared with the atomic ionization energy, yet small enough compared to ˛mc 2 =„, that the dipole approximation can be applied. The differential cross section for this process is [4, 17]
2 d D r02 0 ; 0 d˝
(72.81)
and the total cross section is D
8 2 r : 3 0
(72.82)
This is a completely classical result, which exhibits no frequency dependence.
72.6.3 Resonant Scattering We finally consider elastic scattering in the limit where ! is close to the transition frequency !0 between jgi and jei. Provided that no other level is near resonant with the ground state, the resonant scattering differential cross section is [6, 17]
. =2/2 9 d 2 0 2 D ; 0 d˝ 0 16 2 2 C . =2/2
(72.83)
where 0 D 2c=!0 is the wavelength of the transition, and is the spontaneous decay rate Eq. (72.61). The total elastic scattering cross section is D
3 2 : 2 0
York (1989) 3. Sommerfeld, A.: Optics. Academic Press, New York (1967) 4. Jackson, J.D.: Classical Electrodynamics. Wiley, New York (1975) 5. Allen, L., Eberly, J.H.: Optical Resonance and Two-Level Atoms. Dover, New York (1987) 6. Milonni, P.W., Eberly, J.H.: Laser Physics. Wiley, New York (1988) 7. Meystre, P., Sargent III, M.: Elements of Quantum Optics, 4th edn. Springer, Berlin, Heidelberg (2007) 8. Born, M., Wolf, E.: Principles of Optics, 4th edn. Pergamon, Oxford (1970) 9. Glauber, R.J.: Optical coherence and photon statistics. In: DeWitt, C., Blandin, A., Cohen-Tannoudji, C. (eds.) Quantum Optics and Electronics. Gordon and Breach, New York (1965) 10. Mandel, L., Wolf, E.: Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge (1995) 11. Zubairy, M.S., Scully, M.O.: Quantum Optics. Cambridge University Press, Cambridge (1997) 12. Shen, Y.R.: The Principles of Nonlinear Optics. Wiley, New York (1984) 13. Orszag, M.: Quantum Optics, 3rd edn. Springer, Berlin, Heidelberg (2016) 14. Schleich, W.P.: Quantum Optics in Phase Space. Wiley-VCH, Weinheim (2001) 15. Sargent III, M., Scully Jr., M.O., Lamb, W.E.: Laser Physics. Wiley, Reading (1977) 16. Stenholm, S.: Foundations of Laser Spectroscopy. Wiley, New York (1984) 17. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Atom– Photon Interactions: Basic Processes and Applications. Wiley, New York (1992) 18. Knight, P.L., Milonni, P.W.: Phys. Rep. 66, 21 (1980) 19. Berman, P.P. (ed.): Cavity QED. Academic Press, Boston (1994) 20. Cohen-Tannoudji, C.: Atomic motion in laser light. In: Dalibard, J., Raimond, J.M., Zinn-Justin, J. (eds.) Fundamental Systems in Quantum Optics. North-Holland, Amsterdam (1992) 21. Feynman, R.P., Vernon, F.L., Hellwarth, R.W.: J. Appl. Phys. 28, 49 (1957)
(72.84)
In contrast to the nonresonant Rayleigh and Thomson scattering cross sections, which scale as the square of the classical electron radius, the resonant scattering cross section scales as the square of the wavelength. For optical to frequencies, 0 =r0 ' 104 , giving a resonant enhancement of about eight orders of magnitude. This illustrates why near-resonant phenomena, which form the bulk of the following chapters, are so important in optical physics and quantum optics.
Pierre Meystre’s research ranges from theoretical quantum optics and ultracold science to quantum optomechanics. He is a recipient of the Senior Scientist Research Prize of the Humboldt Society, the R.W. Wood Prize, and the Willis E. Lamb Award. He is an Emeritus Regents Professor at the University of Arizona and a Honorary Professor at East China Normal University.
73
Absortion and Gain Spectra Stig Stenholm
Contents 73.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1057
73.2
Index of Refraction . . . . . . . . . . . . . . . . . . . . . 1057
73.3
Density Matrix Treatment of the Two-Level Atom . 1058
73.4
Line Broadening . . . . . . . . . . . . . . . . . . . . . . . 1059
73.5
The Rate Equation Limit . . . . . . . . . . . . . . . . . 1060
73.6
Two-Level Doppler-Free Spectroscopy . . . . . . . . . 1062
73.7
Three-Level Spectroscopy . . . . . . . . . . . . . . . . . 1064
73.8
Special Effects in Three-Level Systems . . . . . . . . 1065
73.9
Summary of the Literature . . . . . . . . . . . . . . . . 1067
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067
73.1 Introduction In many cases, departures from the thin sample limit, such as beam attenuation, light scattering and radiation trapping (Sect. 73.3) may be important. However, the properties of laser devices themselves depend in an essential way on these effects, making a self-consistent treatment of their properties necessary. At the other extreme, the spectroscopy of dilute gases is well characterized by ensemble averages over the properties of the individual particles, interrupted by occasional brief collisions. Ensemble averages, however, may no longer apply to recent experiments probing a single atomic particle in a trap, as discussed in Chap. 79.
Abstract
This chapter develops theoretical techniques to describe absorption and emission spectra, using concepts introduced in Chap. 72, and density matrix methods from Chap. 7. The simplest cases are treated, compatible with the physics involved, and more realistic applications are referred to in other chapters. Vector notation is not used, but it can be inserted as required. Only steady-state spectroscopy is covered; for time-resolved transient techniques see Chap. 77. Laser technology has greatly expanded the potential of atomic and molecular spectroscopy, but the same techniques for describing the interaction of light with matter also apply to the traditional arc lamps and flash discharges, and the more recent synchrotron radiation sources. Keywords
density matrix rotate wave approximation hole burning density matrix element radiation trapping
73.2 Index of Refraction As discussed in Sect. 72.2.2, the complex index of refraction for a medium containing harmonically bound charges (electrons) [1] with natural frequency !0 is s N˛.!/ N˛.!/ 1C "0 2"0 !
2 i! C !0 ! 2 Ne 2 D1C
2m"0 ! 2 ! 2 2 C 2 ! 2
n.!/ D
1C
0
D n0 C i n00 :
(73.1)
The expansion is valid when the density of atoms N is low. A plane wave can be written in the form 0
E / eikz D ei!nz=c D ei!n z=c e!n
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_73
00 z=c
:
(73.2)
1057
1058
S. Stenholm
The absorption of light through the medium then shows a res- into its equations of motion, and theoretical derivations often provide master equations for the density matrix (Chap. 7). onant behavior near ! !0 determined by The two-level Hamiltonian Eq. (72.35) can be written as ! 2 " # Ne ! n00 D d E .R; t/ „!0 =2
2 2 2m"0 ! ! 2 C 2 ! 2 H D : (73.6) 0 d E .R; t/ „!0 =2 2 =2 Ne : (73.3) The equation of motion for the density matrix Eq. (72.67) is 4m"0 !0 .! !0 /2 C 2 =4 then
This is called an absorptive lineshape. When the single elecid E d
ee D ee C cos !t ge eg ; tron is harmonically bound, its interaction with radiation dt „
is found in this response. For a real atom, the response d id E
gg D ee cos !t ge eg ; of the electron is divided among the various transitions to dt „
other states. The fraction assigned to one single transition d id E
eg D . C i!0 / eg C cos !t gg ee : is characterized by the oscillator strength fn as discussed in dt „ (73.7) Sect. 72.4.2. In ordinary linear spectroscopy, the laser is tuned through Here is the spontaneous decay rate given by Eq. (72.61) the resonance ! !0 , and the value of !0 is determined from and the lineshape Eq. (73.3). Several closely spaced resonances (73.8) D =2 C ph ; can be resolved if their spacing is larger than their widths ˇ ˇ where ph derives from all processes that tend to randomˇ .1/ .2/ ˇ (73.4) ize the phase between the quantum states jei and jgi, such as ˇ!0 !0 ˇ > : collisions (Chaps. 7 and 20), noise in the laser fields and therThis defines the spectral resolution (Chap. 11). mal excitation of the environment in solid state spectroscopy. The velocity of light in the medium is seen to be given by The Greek letters and correspond to the longitudinal rethe expression laxation rate (T1 -process) and the transverse relaxation rate " !# (T2 -process), respectively. !02 ! 2 c Ne 2 In the rotating wave approximation (RWA) the density : ceff D 0 c 1
n 2m"0 ! 2 ! 2 2 C 2 ! 2 matrix equations Eq. (73.7) become identical to Eq. (72.67) 0 with (73.5) (73.9)
eg D Qeg ei!t : This expression shows a dispersive behavior around the position ! D !0 , where the modification of the velocity dis- Using the condition of conservation of probability appears. Below resonance ! < !0 , the velocity of light is (73.10)
ee C gg D 1 ; lower than in vacuum. This derives from the fact that the poto Eq. (72.67) are [2] larization is in phase with the driving field. Thus, by storing the steady state solutions 2 1 ˝1 the incoming energy, the driving field retards the propagation ; (73.11)
ee D of the radiation. 2 2 C 2 C ˝12 =
For a harmonically bound charge, the refractive index i˝1 gg ee
Qeg D Eq. (73.1) always stays absorptive and it is independent of 2 C i the intensity of the laser radiation. This no longer holds for i i˝ 1 discrete level atomic systems. In order to see this, we con; (73.12) D 2 2 C 2 C ˝12 = sider the two-level atom in Sect. 73.3. where ˝1 is the Rabi frequency from Eq. (72.44), and D !0 ! is the detuning. The induced polarization is then ! " !# 73.3 Density Matrix Treatment
0 d
ee eg of the Two-Level Atom P D N Tr dO D N Tr d 0 ge gg
The response of atoms to light is conveniently expressed in D Nd ei!t Qeg C ei!t Qge
terms of the density matrix . In addition to the direct phys(73.13) D N ˛ E .C/ C ˛ E ./ ; ical meaning of the density matrix elements discussed in Sect. 72.3.3, the density matrix formalism is advantageous where N is the density of active two-level atoms. Setting because the various relaxation mechanisms effecting the 1 E .C/ D E ei!t ; (73.14) atomic resonances can be introduced phenomenologically 2
73
Absortion and Gain Spectra
the complex polarization is ˛.!/ D
i C d2 ; „ 2 C 2 C ˝12 =
1059
additional levels occurs, these must be included (Sect. 73.5). The term ph contains all perturbing effects effecting each sin(73.15) gle atom. For low enough pressures, collisional perturbations are proportional to the density of perturbing atoms so that
and from Eq. (73.1), the complex index of refraction is
ph D coll p ;
(73.19)
N˛.!/ where p is the pressure of the perturbing gas and coll is 2"0 a constant of proportionality. This is called collision or pres i C Ne 2 f0 sure broadening (Chap. 63), whose order of magnitude can D1C ; (73.16) 4"0 m!0 2 C 2 C ˝12 = be estimated to be the inverse of the average free time between collisions. For high pressure (usually of the order of where f0 D 2d 2 m!0 =„e 2 is the oscillator strength. Summing torrs), the linearity in Eq. (73.19) breaks down. When identiover all possible transitions yields the f-sum rule Eq. (72.63), cal atoms collide, resonant exchange of energy may also take (Chap. 22). place. The third term in Eq. (73.18) is the power broadening The imaginary part of Eq. (73.16) shows exactly the term. It derives from the effect of the laser field on each indisame absorptive behavior as in the harmonic oscillator model vidual atom. All such relaxation processes that are active on of Sect. 73.2 [see Eq. (73.3)]. However, the additional each and every individual atom separately are called homofactor of ˝12 = in the denominator makes the line ap- geneous broadening processes. For a detailed discussion of pear broader than in the harmonic case; the line is power line broadening, consult Chap. 20. broadened. Physically, this derives from a saturation of the In contrast to homogeneous broadening, Doppler broadtwo-level system in which the population of the upper level ening is characterized by an atomic velocity parameter v becomes an appreciable fraction of that of the lower level. which varies over the observed assembly. In a thermal asIn the limit ˝1 ! 1, Eq. (73.16) shows that n.!/ ! 1 and sembly at temperature T (e.g., a gas cell), the velocity the atom-field interaction effectively vanishes. In this limit, distribution is
ee ! 12 Eq. (73.11), and the field induces as many upward v2 1 transitions as downward transitions. exp 2 ; (73.20) P .v/ D p When radiation at the frequency ! propagates in a medium 2u 2u2 of two-level atoms, the energy density is where u2 D kB T =M . A particular atom with velocity v in 2 Nd ! z the direction of the optical beam with wave vector k then : I.z/ / E E / exp 2 2 experiences the Doppler-shifted frequency ! kv relative to „"0 c .! !0 / C (73.17) a stationary atom, and the effective detuning becomes n.!/ D 1 C
Far from resonance (j! !0 j ), the medium is transpar(73.21) 0 D C kv ; ent; but near resonance, damping is observed. The impinging radiation energy is deposited in the medium and propagation replacing . The population in the lower level from is impeded. This is called radiation trapping. In spectroscopy, Eq. (73.11) is then the phenomenon is seen as a prolongation of the radiative de 1 ˝ 2 cay time; the spontaneously emitted energy is seen to emerge : (73.22)
gg D 1 1 2 . C kv/2 C 2 C ˝12 = from the sample more slowly than the single atom lifetime implies. The atoms in the lower level, originally distributed according to Eq. (73.20), are now depleted from the velocity group around 73.4 Line Broadening (73.23) v D .! !0 /=k : The effective width of a spectral line from Eq. (73.16) is r The width of the depleted region is given by eff of
˝2 1 eff D 2 C ˝12 ' C ph C 1 C O ˝14 : Eq. (73.18). This region is called a Bennett hole. When the 2 2 laser frequency ! is tuned, the hole sweeps over the velocity (73.18) distribution of the atoms. The atomic response is saturated at The various contributions are as follows [3]. The term con- the velocity group of the hole, indicating that spectral hole tains all the transverse relaxation mechanisms. If decay to burning has occurred.
73
1060
S. Stenholm
The observed spectrum is obtained by averaging the sin- in solids. The resonant light selectively excites atoms at gle atom response Eq. (73.15) over the velocity distribution. those particular positions which make the atoms resonant. Thus only these spatial locations are saturated, and the pheFrom the imaginary part, the absorption response is nomenon of spatial hole burning occurs. This has been C1 Z 2 2 investigated as a method for storing information, signal pro ev =2u d2 ˛ 00 .!/ D p dv : cessing, and volume holography. 2 2 2 . C kv/ C C ˝1 = „ 2u2 It is, however, possible that the effects of collisions can 1 (73.24) counteract the inhomogeneous broadening. In order to see this we observe that the induced atomic dipole is proportional In the limits ˝1 ! 0 (no saturation), and ku (the to the induced density matrix element, from Eq. (73.9), Doppler limit), the Lorentzian line shape sweeps over the entire velocity profile, finding a resonant velocity group ac(73.28)
eg D Qeg ei.kz.t /!t / / eikvt : cording to Eq. (73.23) as long as v u. Thus, the linear spectroscopy sees a Doppler broadened line of width ku. If the atoms now experience collisions characterized by an average free time of flight , the phase kvt cannot build up This is called inhomogeneous broadening. In the unsaturated regime, the atomic response function coherently for times longer than this duration, the atomic Eq. (73.24) is proportional to the imaginary part of the func- velocity is quenched on the average, and the full Doppler profile cannot be observed. To see how this comes about, tion consider a time t . During this period the atom experi ZC1 2 2 ences on the average nN D t= collisions. Assuming a Poisson exp x =2 1 dx ; (73.25) distribution, the probability of n collisions in time t is V .z/ D p zx 2 2 1
pn D
enN n nN : nŠ
(73.29) at z D i and D ku. This is the Hilbert transform of the Gaussian, and its shape is called a Voigt profile [4]. For ku, it traces over the Gaussian, but for large detunings Taking the average of Eq. (73.28) over the time t D n with it always goes to zero as slowly as the Lorentzian, i.e., as the distribution pn yields nN 1 . The profile has been widely used to interpret the data 1 X ikv n e of linear spectroscopy. The function is tabulated [5] and its
Neg / e nN n nŠ expansion is nD0
D enN exp nN eikv p
r n 1 X iz= 2 1 2 2 V .z/ D 2 ; (73.26) ikvt (73.30) e exp tk v : 2 nD0 .1 C n=2/ 2 and it has the continued fraction representation, 1
V .z/ D
:
2
z
2 2
z
3 2
z z
4 2 z
This heuristic derivation suggests that for long enough interaction times (t /, the large velocity components are suppressed. This tends to prevent the tails of the velocity dis(73.27) tribution from contributing to the observed spectral profile. The effect is called collisional narrowing or Dicke narrowing. It is an observable effect, but the narrowing cannot be very large. To overcome the Doppler broadening one has to turn to nonlinear laser methods (Sect. 73.6).
In addition to velocity, any other parameter shifting the individual atomic resonance frequencies !0 by different amounts for the different individuals leads to inhomogeneous broadening. The detuning is then different for different members of the observed assembly, and a line shape similar to Eq. (73.24) applies. The distribution function must be replaced by the one relevant for the problem. In practice a Gaussian is almost always assumed. An example of inhomogeneous broadening is the influence of the lattice environment on impurity spectroscopy
73.5 The Rate Equation Limit Consider now a generalized theory for the case of several incoming electromagnetic fields of the form E .R; t/ D
X1 i
2
Ei .R/ ei!i t Ci'i C c:c: :
(73.31)
The index i may range over several laser sources, the output of a multimode laser or the multitude of components of
73
Absortion and Gain Spectra
1061
the spontaneous emission which forces the entire population to the lower level. The terms proportional to W describe induced emission, with upward transitions proportional to W gg and downward transitions proportional to W ee . In the absence of spontaneous emission, they strive to equalize the population of the two levels. Using Eq. (73.10), the steadystate solution is where the detuning is W
ee D C 2W (73.33) i D !0 !i : 1 1 dE 2 : (73.39) D The response of the atom now separates into individual con2 2 2 „ C C .d E =„/2 = tributions oscillating at the various frequencies !i according This is clearly seen to agree with the solution Eq. (73.11) as to is expected in steady state. X .i / i!i t Ci'i Although the rate equations were derived in the limit
eg e : (73.34)
eg D Eq. (73.36), the rate coefficient Eq. (73.38) has a special sigi nificance in the limit ! 0. In this limit, the factor This resolution is of key importance to the theory. = 1 1 With the multimode field, Eq. (72.67) for the level occulim D ı. / 2 2 „ !0 C „ pation probabilities become (73.40) D ı.„! „!0 / ; d d
ee D gg enforces energy conservation in the transition. Using the field dt dt X in Eq. (73.14), and the interaction from Eq. (73.6), the offi D ee C diagonal matrix element is 2 i 1 d Ei i!i t Ci'i jhejH jgij D d E : (73.41)
ge c:c: : (73.35) e 2 „ With these results, Eq. (73.38) can be written in the form of Insertion of the steady state result Eq. (73.32) into this equa- Fermi’s Golden Rule tion yields a closed set of equations for the level occupation 2 (73.42) jhejH jgij2ı.„! „!0 / ; W D probabilities. These are called rate equations. „ To justify the above steps, consider the single frequency usually derived from time dependent perturbation theory. case again. The off-diagonal time derivatives can be neReturning to the multimode rate equations, an incoherent glected when broad band light source has many components that conˇ ˇ tribute to the sum over field frequencies. In the case of flash ˇ ˇd ˇ eg ˇ j C i jj eg j : (73.36) pulses, thermal light sources, or free-running multimode ˇ ˇ dt lasers the spectral components are uncorrelated. Inserting This can be surmised to hold when the phase relaxation con- Eq. (73.32) into Eq. (73.35) yields tributions to are large Eq. (73.8) or the detuning j j is d d
ee D gg large. Insertion of Eq. (73.32) into Eq. (73.35) for the single dt dt mode case gives d2
D
C
gg ee ee 2
d 2„ (73.37)
ee D ee W ee gg ; X dt Ei Ej ei.!j !i /t ei.'i 'j / 2 : (73.43) C 2 j i;j where the rate coefficient is given by The contributions from the different terms i ¤ j average to d E 2 = zero either by beating at the frequencies j!i !j j or by in: (73.38) W D 2 2„ 2 C 2 coherent effects from the random phases 'j . Thus, only the Multiplying Eq. (73.37) by the density of active atoms N coherent sum survives to give d d then produces the conventional rate equations for the popu ee D gg dt dt lations Nee D N ee and Ngg D N gg . X
D ee W .i / gg ee ; (73.44) Two physical effects can be discerned in Eq. (73.37): ini duced and spontaneous emission. The term with gives a flashlight or a thermal source. Each component carries its own amplitude Ei . In steady state, the generalization of Eq. (73.12) becomes
i X d Ei gg ee i!i t Ci'i e ; (73.32)
eg D 2 i „ C i i
73
1062
S. Stenholm
where the rate coefficients W .i / are given by Eq. (73.38) with the appropriate detunings j D !0 !j . This is a rate equation in the limit of many uncorrelated components of light, i.e., for a broad band light source. In this case the incoherence between the different components justifies the use of a rate approach, and no assumption like Eq. (73.36) is needed. Thus, the limit ! 0 Eq. (73.40) is legitimate, and the W .i / can be calculated in time dependent perturbation theory from Fermi’s Golden Rule. In the limit of an incoherent broad band light source, the sum in Eq. (73.44) can be replaced by an integral. In particular, this is allowed for incandescent light sources as used in optical pumping experiments [6]. Pumping of lasers by strong lamps or flashes are also describable by the same rate equations. In amplifiers and lasers, the atoms must be brought into states far from equilibrium by incoherent optical excitation or resonant transfer of excitation energy in collisions (Chap. 74). In the two-level description, the atomic levels are constantly replenished. The normalization condition Eq. (73.10) is then no longer appropriate; often the density matrix is normalized so that Tr. / directly gives the density of active atoms. With pumping into the levels, one must allow for decay out of the two-level system in order to prevent the atomic density from growing in an unlimited way. This decay takes the atom to unobserved levels. In the rate equation approximation, the pumping and decay processes can be described by terms added to the equations of the form Eq. (72.68) d
ee D e e ee ; dt d
gg D g g gg : dt
d2 „
i C .0/ .0/
:
gg ee 2 C 2
For weakly excited atoms, gg 1, and Eq. (73.47) agrees with the unsaturated limit of Eq. (73.16). The dispersion of a light signal behaving according to Eq. (73.48) is called normal, i.e., according to the harmonic model in Sect. 73.2. Below resonance (! < !0 ), n0 is larger than unity, implying a reduction of the velocity of light. As a function of !, the curve Eq. (73.48) starts above unity, and passes below unity for ! > !0 . This is normal dispersion. .0/ .0/ However, for an inverted medium ( gg < ee ), n0 is less than unity for low frequencies and goes through unity with a positive slope. This is called anomalous dispersion and signifies the presence of a gain profile. In such a medium, ˛ 00 D ImŒ˛.!/ is of opposite sign, as seen from Eq. (73.47); in the inverted medium, ˛ 00 becomes negative near D 0. From Eq. (73.2), this indicates a growing electromagnetic field, i.e., an amplifying medium. The amplitude grows, and the assumption of a small signal becomes invalid. Then saturation has to be included, either at the rate equation level or by performing a full density matrix calculation. This regime describes a laser with saturated gain. In steady state, the two levels become nearly equally populated ( ee gg ), and the operation is stable. The theory of the laser is discussed in detail in Chap. 74.
73.6 Two-Level Doppler-Free Spectroscopy
(73.46)
A population inversion exists when this is negative. The population difference Eq. (73.46) is modified when the effects of spontaneous and induced processes are added, as in Eq. (73.37). Then the transitions saturate because of the induced processes. Using Eq. (73.46) in Eq. (73.12), the calculated polarizability without saturation is
˛.!/ D
N˛ 0 .!/ 2"0 Nd 2 .! !0 / .0/ .0/
: D1
ee 2"0 „ .! !0 /2 C 2 gg (73.48)
n0 .!/ D 1 C
(73.45)
In a laser, the level jgi is usually not the ground state of the system. From Eq. (73.45), the steady state population is .0/ .0/ ee D g =g e =e :
gg
According to Eq. (73.1), the index of refraction is
(73.47)
The linear absorption of a scanned laser signal defines linear spectroscopy and gives information characterizing the sample. However, Sect. 73.4 shows that inhomogeneous broadening masks the desired information by dominating the line shape. The availability of laser sources has made it possible to overcome this limitation, and to use the saturation properties of the medium to perform nonlinear spectroscopy. This section discusses how Doppler broadening can be eliminated to achieve Doppler-free spectroscopy. Similar techniques may be used to overcome other types of inhomogeneous line broadening; a general name is then hole-burning spectroscopy (see the discussion in Sect. 73.4). Other aspects of nonlinear matter-light interaction are found in Chap. 76. Equation (73.24) shows that a single laser cannot resolve beyond the Doppler width. However, if a strong laser is used to pump the transition, a weak probe signal can see the hole burned into the spectral profile by the pump. This technique is called pump-probe spectroscopy. Because the probe
73
Absortion and Gain Spectra
1063
is taken to be weak, perturbation theory may be used to calculate the induced polarization to lowest order in the probe amplitude only. In the field expansion Eq. (73.31), define the strong pump amplitude to be E1 and the weak probe E2 at frequency !2 . .2/ From the resolution Eq. (73.34), the component eg carries the information about the linear response at frequency !2 . If the field E2 propagates in a direction opposite to that of E1 , its detuning is
This denotes the energy absorbed from the field E1 , as induced nonlinearly by the intensity E12 . The resonance is still at ! D !0 , but with a homogeneous atomic line shape. In the Doppler limit, the Doppler broadening is only seen in the prefactor 2 ˝1 ˝12 D ; (73.53) ku ku
which shows that only the fraction .=ku/ of all atoms can contribute to the resonant response. The first factor on the (73.49) right-hand side of Eq. (73.53) is the dimensionless saturation 02 D 2 kv factor. The Doppler-free character of this spectroscopy derives (as compared with 01 D 1 C kv for E1 ). Since !1 ' !2 , from the fact that the two fields burn their two separate Benthe two k-vectors are nearly equal in magnitude. nett holes at the velocities The linear response now becomes .2/ D
eg
id E2 1
gg ee : 2„ C i. 2 kv/
kv1 D 1 ;
With only the signal E1 present, the population difference follows directly from Eq. (73.22). The linear response at frequency !2 is then
.2/ D
eg
and kv2 D 2 :
d E2 i C . 2 kv/ 2„ 2 C . 2 kv/2 1 ˝12 1 : . 1 C kv/2 C 2 C ˝12 = (73.51)
When these two groups coincide, i.e., when v1 D v2 , the probe E2 sees the absorption saturated by the pump field E1 and a decreased absorption is observed. For !1 D !2 , the two holes meet in the middle at zero velocity. With two different frequencies, one can make the holes meet at a nonzero velocity to one side of the Doppler profile. The decreased absorption seen in these experiments is called an inverted Lamb dip Chap. 74. The results derived here are based on a simplified view of the pump-probe response which in turn is based on the rate equation approach. Certain coherent effects are neglected, which would considerably complicate the treatment [2]. Section 73.7 discusses these effects in the three-level system where they are more important. In addition to measuring the probe absorption induced by the pump field, it is also possible to observe the dispersion of the probe signal caused by the saturation induced by the pump. Assuming that the pump introduces the velocity dependent population difference
This is the linear response of atoms moving with velocity v. To obtain the polarization of the whole sample, we must average over the velocity distribution using the Gaussian weight Eq. (73.20). The first term in Eq. (73.51) gives the linear response in the form of a Voigt profile, as discussed in Sect. 73.4. This part of the response carries no Dopplerfree information. The second terms contain the nonlinear response. This shows the details of the homogeneous features
ee gg D .v/ ; under the Doppler line shape. For simplicity we assume the Doppler limit, ku, and neglect the variation of the Gaus- then from Eq. (73.50), Re.n/ is sian over the atomic line shape. We also neglect the power Z 2 kv Nd 2 broadening due to the field E1 and obtain, using Eqs. (73.13), 0 .v/dv : n D1C 2 2 2„" . (73.20), and (73.51), 0 2 kv/ C 2 d ˝ 2 2 p 1 ˛ .!/ D „ 2 u ZC1 dv h ih i 2 2 2 2 . . kv/ C C kv/ C 2 1 1 2 p 2 d 2˝1 : D „ 4 ku .! !0 /2 C 2 (73.52) 00
(73.54)
(73.50)
(73.55)
(73.56)
From Eq. (73.2), the phase of the electromagnetic signal feels the value of n0 through the factor E / exp.i! n0 z=c/ :
(73.57)
Thus, by modifying .v/, a pump laser can control the phase acquired by light traversing the sample, and thereby control the optical length of the sample. A real atom has magnetic sublevels, which are coupled to light in accordance with dipole selection rules (see the
73
1064
S. Stenholm
discussion in Chap. 35). If a pump laser is used to affect in Eq. (73.59) is neglected; in that case, the solution is populations in the various sub-levels differently, the opti
ee d2 E2 cal paths experienced by the differently circularly polarized
Qf e D 2„ 2 C k2 v if e components of a linearly polarized probe signal are differ d1 E1 d2 E2 ent. Thus, its plane of polarization will turn, corresponding to the Faraday effect. By tuning the pump and the probe over 2„ 2„ the spectral lines of the sample, the turning of the probe po Qeg larization provides a signal to investigate the atomic level . 2 C k2 v if e / structure. This method of polarization spectroscopy can be 1 : (73.61) used both with two-level and three-level systems [7]. . 1 C 2 / C .k1 C k2 /v ifg
73.7
Three-Level Spectroscopy
New nonlinear phenomena appear when one of the levels in the two-state configuration, jei say, is coupled to a final level jf i by a weak probe. This may be above the level jei (the cascade configuration denoted by &) or below jei (the lambda configuration ) (if the third level were coupled to jgi we would talk about the inverted lambda or V configuration) [8]. For simplicity, only the & configuration is discussed here. Assume that the level pair jgi $ jei is pumped by the field E1 and its effect probed by the field E2 coupling jei $ jf i. The dipole matrix element is d2 D hf jH jei. The RWA is now achieved by introducing slowly varying quantities through the definitions
f e D ei.k2 z!2 t / Qf e ;
fg D eiŒ.k1 Ck2 /z.!1 C!2 /t Qfg ;
(73.58)
d1 E1 „
2
eg 1 ; 2 2ee . 1 C k1 v/2 C eg
d1 E1 1 2„ 1 C k1 v ieg
(73.62)
From Eq. (73.61), these matrix elements give A
Qf e D 1 C k1 v ieg " 2eg 1
ee 1 C k1 v C ieg 2 C k2 v if e 1
2 C k2 v if e 1 . 1 C 2 / C .k1 C k2 /v ifg
(73.59)
# ;
(73.63)
where
where the second step detuning is now 2 D !f e !2 :
ee D
Qeg D
and omitting all components oscillating at multiples of the optical frequencies. From the equation of motion for the density matrix, the steady state equations are [2] . 2 C k2 v if e / Qf e d2 E2 d1 E1 D
ee
Qfg ; 2„ 2„
. 1 C 2 / C .k1 C k2 /v ifg Qfg d2 EŒ 2 d1 E1 D
Qeg
Qf e ; 2„ 2„
The result to lowest order in E2 follows by replacing the density matrix elements for the two-level system jei $ jgi by their results calculated without this field. This shows that the polarization induced at the frequency !2 consists of two parts: one induced by the population excited to level jei by the field E1 proportional to ee , and the other one is induced by the coherence Qeg created by the pump. Retaining only the former produces a rate equation approximation, called a two-step process. This misses important physical features which are included in the second term called a two-photon or coherent process. In order to see the effects of the two terms most clearly, consider the two-level matrix elements from Eqs. (73.11) and (73.12) in lowest perturbative order with respect to the pump field E1 , i.e.,
(73.60)
d2 E2 d1 E1 2 AD : 2„ 2„
(73.64)
The imaginary part of this yields the absorptive part of the These equations contain the lowest order response propor- polarization at the frequency !2 . The first term becomes the tional to E2 , including some coherence effects. The coher- product of two Lorentzians, and is an incoherent rate contrience effects remain to lowest order, even when the last term bution. It dominates when the induced population ee decays
73
Absortion and Gain Spectra
1065
much more slowly than the induced coherence Qeg , i.e., when Eq. (73.63) can be written in the form ee eg . 1 A The significance of the second term in Eq. (73.63) is
Qf e D 2 : (73.70) 1 . 1 C 2 / C .k1 C k2 /v ifg evident in the limit when no phase perturbing processes intervene, so that If the two fields E1 and E2 have the same frequency but are counterpropagating, then k1 C k2 D 0, and no velocity 1 eg D ; ee D ; dependence occurs in Eq. (73.70) for the two-photon reso2 nance. All atoms in the sample contribute to the strength of 1 1 (73.65) the resonance, and then polarization is obtained directly by fg D ff ; f e D .ff C / : 2 2 multiplication with the total atomic density. Equation (73.13) For v D 0, Qf e becomes then gives ! fg d22 d1 E1 2 1 A 00 : (73.71) ˛2 D
2
Qf e D 2 : (73.66) 1 2 „ 2„ 1 C . =2/ 1 C 2 i ff 2! !fg C 2 1
fg
2
Neglecting the decay of the final level jf i, ff ! 0, the ab- This is a sharp Doppler-free resonance on the two-photon transition !fg . The advantage is that all atoms contribute sorption becomes proportional to with the sharp line width fg , which is not easily affected
jAj by phase perturbations because of the two-photon nature ı. 1 C 2 / : (73.67) Im Qf e D 2 of the transition. The disadvantage is the weakness of the 1 C . =2/2 transition, which is caused by the large detuning, making In this limit, strict energy conservation between the ground d1 E1 =j 1 j 1 in most cases. With tunable lasers, however, state and the final state must prevail; the final state must be this Doppler-free spectroscopy method has been used sucreached by the absorption of exactly two quanta. The delta cessfully in many cases. function in Eq. (73.67) indicates precisely this: 1 C 2 D !f e C !eg .!1 C !2 /
73.8
Special Effects in Three-Level Systems
D !fg !1 !2 (73.68) We continue our considerations of a three-level system but this time in the V -configuration. Thus, we have a ground The detuning and the width of the intermediate state affect state jgi coupled to a doublet of excited states fjei; jf ig the total transition rate, but not the condition of energy con- through the interaction servation. The presence of the second term in Eq. (73.63) (73.72) V D ˝1 jgihej C ˝2 jgihf j C h.c. ; makes the resonance contributions at D0:
2 D !fg !2 D 0
(73.69) where the couplings are due to radiation fields and given by
cancel approximately. Only a two-photon transition remains; in the configuration this would be a Raman process (Chaps. 66 and 76). If the velocity dependence in Eq. (73.63) is retained, the nonlinear response of an atomic sample must be averaged over the velocity distribution given by the Gaussian Eq. (73.20). The computations become involved, but the results show more or less well resolved resonances around the two positions Eqs. (73.68) and (73.69), i.e., the coherent twophoton process and the single step rate process jei ! jf i appearing due to a previous single step process jgi ! jei; this is the two-step process. A special situation arises when the intermediate step is detuned so much that no velocity group is in resonance, i.e., j 1 j j 2 j kv for all velocities contributing significantly to the spectrum. Then the second coherent term of
˝i D
di Ei : 2„
(73.73)
We use the rotating wave approximation and set the detunings to !e D !e !1
and !f D !f !2 ;
(73.74)
where the frequency !i derives from the coupling field Ei . The energy Eg D 0. The time-dependent Schrödinger equation for this system is then written as icPg D ˝1 ce C ˝2 cf ;
(73.75)
icPe D !e ce C ˝1 cg ;
(73.76)
icPf D !f cf C ˝2 cg :
(73.77)
73
1066
S. Stenholm
We now introduce the two new variables
cC D ˝N 1 ˝1 ce C ˝2 cf
cNC D ˝N 1 ˝2 ce ˝1 cf ;
In this case, we may let ˝1 come on later than ˝2 . Then we may have (73.78) 0 1 ˝1 B C lim @ q AD0; 2 2 ˝2 C ˝1 0 1
(73.79)
t !1
where ˝N 2 D ˝22 C ˝12 :
˝2 B C lim @ q AD1 ˝22 C ˝12
(73.80)
t !1
The equations of motion follow: and
˝2 ˝1 !e ce !f cf icPNC D ˝N ˝N
1 h ˝1 ˝2 !e !f cC : D 2 N ˝
i C !e ˝22 C !f ˝12 cNC ; ˝1 ˝2 N g !e ce C !f cf C ˝c icPC D ˝N ˝N 1 h
D 2 !e ˝12 C !f ˝22 cC ˝N
i N g: C ˝1 ˝2 !e !f cNC C ˝c
(73.81)
0
(73.86)
1
˝1 B C lim @ q AD1; t !1 2 2 ˝2 C ˝1 0 1 ˝2 B C lim @ q AD0: ˝22 C ˝12
t !1
(73.87)
Both couplings are thus pulses, but they occur with a slight time delay. If we now start the system in the state jei, we find (73.82) from Eq. (73.84) that
lim jNCi D jei ; (73.88) t !1 We notice that cNC is not coupled to the ground state, but, in general, its coupling to the state cC provides an indirect and coupling. This indirect coupling, however, can be made to (73.89) lim jNCi D jf i : t !C1 disappear, if we set !e D !f !. Then we find the equations Thus, by keeping the system in this uncoupled state we can adiabatically transfer its population beteen the states withicPNC D !cNC ; out involving any population of the intermediate state jgi. N Especially if this is an upper state, which may decay and icPC D !cC C ˝cg ; N (73.83) dephase rapidly, the proposed population transfer may be icPg D ˝cC : greatly advantageous. Because it is usually applied in the This describes a pair of coupled quantum levels and a single -configuration, it is termed Stimulated Raman Adiabatic uncoupled level. Thus, if we start in the ground state, this lat- Passage (STIRAP). The dark state has found a wide range of applications in ter level can never be populated. It remains unpopulated and laser physics. As we may use the method to pass radiation is called a dark state. This state corresponds to the superpothrough a medium without any absorption, it has led to the sition phenomenon of light-induced transparency. It can also be 1 used to affect the index of refraction without having the ac(73.84) jNCi D .˝2 jei ˝1 jf i/ : companying absorption. The absorptive part of a resonance ˝N normally manifests itself as a quantum noise; utilizing the Using the coupling operator Eq. (73.72), we find the matrix dark state idea one may reduce the noise in quantum devices. element The dressing of the levels due to the special features of the interaction has also made it possible to achieve lasing withhgjV jNCi D 0 : (73.85) out an inversion of the bare levels. These topics, however, will not be treated here. We also notice that the dark state can be found if the states jei A special application of the method to affect the refractive and jf i are the lower ones, i.e., we have the -configuration. index deserves a more detailed consideration. We look at the Alternatively, if we start the system off in the dark state, relationship between the electric field vectors in the medium: it will never be able to get out of this state. This is also taken D.!/ D "0 E C N˛.!/E.!/ D .!/"0 E.!/ ; (73.90) to hold true if we let the couplings depend slowly on time.
73
Absortion and Gain Spectra
1067
where .!/ is the susceptibility of the medium. From an exceedingly slow propagation of light pulses. Such slow light has been shown to travel at only a few kilometers per Maxwell’s equations we find the relation Eq. (72.24) hour, which is a most remarkable result. The drawback is, !2 !2 2 2 k D n .!/ 2 D 2 Œ1 C .!/ : (73.91) however, that this can only occur over a very narrow frec c quency range, as follows from the assumption of a strong If we take the real parts of the quantities, this describes the dependence on frequency. propagation of waves in the medium. Now we may use the relations Eq. (73.16) to estimate the function ˛.!/ in the case when we have two weak fields 73.9 Summary of the Literature exciting the three-level system in the V -configuration. We assume that both couplings have the same frequency, !1 D Much of the material needed to formulate the basic theory of interaction between light and matter can be found in the text !2 D !. We write book [2]. A comparison between the harmonic model and !
! !f .! !e / the two-level model is given by Feld [1]. The density maC
.!/ I 2 trix formulation is presented in detail in [2]. The influence of .! !e /2 C 2 ! !f C 2 (73.92) various line broadening mechanisms on laser spectroscopy is discussed in the book [3]. The Voigt profile is related to the in the weak field limit we may assume the two processes to error function, which is treated in the compilation [5]. The add independently. The parameter numerical evaluation of the Voigt profile is discussed in [4]. Rate equations are commonly used in laser theory and they Nd 2 (73.93) are derived for optical pumping and laser-induced processes D 2„"0 in the lectures [6]. The Doppler-free spectroscopy was dehas the dimension of a frequency and indicates the strength veloped in the 1960s and 1970s by many authors following of the interaction. It is clear that tuning the frequency be- the initial discovery of the Benett hole by Bill R. Bennett Jr. tween the levels may give rise to a value of zero for the and the Lamb dip by Willis E. Lamb Jr. Much of the piosusceptibility. For a large enough , we write, in the neigh- neering work can be found in the book [3]. The three-level work has been reviewed by Chebotaev [8]. Various applicabourhood of the zero, tions of lasers in spectroscopy are treated by Levenson and 2 N ; (73.94) Kano [7]. For references to other topics, we refer to the spe.!/ 2 .! !/ cialized chapters of the present book. Many features of the quantum dynamics of a few-level 1 where !N D 2 .!e C !f /. system are found in [9, 10]. The theoretical methods to We now have the relation Eq. (73.91) to determine the treat such systems are presented in detail in [11]. The endispersion relation, and assuming the effect of the medium suing physical processes are presented in [12] with much to be substantial, we may derive an expression for the group additional material on quantum optics phenomena. The bavelocity in the medium sic theory of the dark state and many of its applications in @k spectroscopy and laser physics are found in [13]. A rather : (73.95) vg1 D @! complete review of adiabatic processes induced by delayed pulses is the article [14]. The slowing down and stopping of We find light is reviewed in [15]. A very recent article with earlier 2k 2! ! 2 2 D 2 .1 C / C 2 : (73.96) references is [16]. vg c c 2 Even though D 0 near the point !, N we still have ! ck, References so that 2 1. Feld, M.S.: In: Balian, R., Haroche, S., Liberman, S. (eds.) Fron c tiers in Laser Spectroscopy, p. 203. North-Holland, Amsterdam c c: (73.97) vg D ! ! 1 C 2 (1977) The last inequality follows from the fact that in all cases !. We have thus found that utilizing the interference of two near quantum levels, the refractive index may acquire a very strong dependence on frequency. This may manifest itself in
2. Stenholm, S.: Foundations of Laser Spectroscopy. Wiley, New York (1984) 3. Letokhov, V.S., Chebotaev, V.P.: Nonlinear Laser Spectroscopy. Springer, Berlin, Heidelberg (1977) 4. Thompson, W.J.: Comp. Phys. 7, 627 (1993) 5. Abramowitz, M., Stegun, I.E.: Handbook of Mathematical Functions. Dover, New York (1970)
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1068 6. Cohen-Tanoudji, C.: In: Balian, R., Haroche, S., Liberman, S. (eds.) Frontiers in Laser Spectroscopy, p. 1. North-Holland, Amsterdam (1977) 7. Levenson, M.D., Kano, S.S.: Introduction to Nonlinear Laser Spectroscopy. Academic Press, New York (1988) 8. Chebotaev, V.P.: In: Shimoda, K. (ed.) High-Resolution Spectroscopy. Springer, Berlin, Heidelberg (1976) 9. Shore, B.W.: Simple Atoms and Fields. The Theory of Coherent Atomic Excitation, vol. 1. Wiley, New York (1990) 10. Shore, B.W.: Multilevel Atoms and Incoherence. The Theory of Coherent Atomic Excitation, vol. 2. Wiley, New York (1990) 11. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: AtomPhoton Interactions, Basic Processes and Applications. Wiley, New York (1992) 12. Mandel, L., Wolf, E.: Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge (1995) 13. Scully, M.O., Zubairy, M.S.: Quantum Optics. Cambridge University Press, Cambridge (1997) 14. Vitanov, N.V., Fleischauer, M., Shove, B.W., Bergmann, K.: In: Bederson, B., Walther, H. (eds.) Advances of Atomic, Molecular and Optical Physics, vol. 46, p. 55. Academic Press, New York (2001) 15. Matsko, A.B., Kocharovskaya, O., Restoutsev, Y., Welch, G.R., Zibrov, A.S., Scully, M.O.: In: Bederson, B., Walther, H. (eds.) Advances in Atomic, molecular, and Optical Physics, vol. 46, p. 191. Academic Press, New York (2001) 16. Bajcsy, M., Zibrov, A.S., Lukin, M.D.: Nature 26, 368 (2003)
S. Stenholm Stig Stenholm Stig Stenholm was Professor of Laser Physics and Quantum Optics at the Royal Institute of Technology, Stockholm. He studied Technical Physics at the Helsinki Institute of Technology and Mathematics. He worked at the Research Institute for Theoretical Physics in Helsinki until 1997, when moving to Stockholm. His research included spectroscopy, quantum optics, and informatics. He passed away in 2017.
74
Laser Principles Ralf Menzel and Peter W. Milonni
Contents 74.1
Gain, Threshold, and Matter–Field Coupling . . . . 1069
74.2
Continuous Wave, Single-Mode Operation . . . . . . 1071
74.3
Laser Resonators and Transverse Modes . . . . . . . 1074
74.4
Photon Statistics . . . . . . . . . . . . . . . . . . . . . . . 1076
74.5
Multimode and Pulsed Operation . . . . . . . . . . . . 1077
74.6
Instabilities and Chaos . . . . . . . . . . . . . . . . . . . 1079
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079
Abstract
Despite their great variety and range of size, power, wavelength, and temporal, spatial, and polarization characteristics, all lasers involve certain basic concepts, such as gain, threshold, and electromagnetic modes of oscillation [1–6]. In addition to these universal characteristics are features, such as Gaussian beam modes, that are important to such a wide class of devices and must be included in any reasonable compendium of important laser concepts and formulas. We have, therefore, included here both generally applicable results as well as some more specific but widely applicable ones. Keywords
gain coefficient laser resonators Gaussian beams laser linewidth frequency stabilization ultrashort laser pulses
R. Menzel () Dept. of Physics, University of Potsdam Potsdam, Germany e-mail: [email protected] P. W. Milonni Theoretical Division, Los Alamos National Laboratory Los Alamos, NM, USA
74.1 Gain, Threshold, and Matter–Field Coupling All lasers involve some medium that amplifies an electromagnetic field within some band of frequencies. At the simplest level of description, the amplifying medium changes the intensity I of a field according to the equation dI D gI ; dz
(74.1)
where z is the coordinate along the direction of propagation, and g is the gain coefficient, typically expressed in cm1 . Amplification occurs as a consequence of stimulated emission of radiation from the upper state (or band of states) of a transition for which a population inversion exists, i.e., for which an upper state has greater likelihood of occupation than a lower state. Different types of lasers may be classified by the pump mechanisms used to achieve population inversion (Chap. 75). In the case that the amplifying transition involves two discrete energy levels, E1 and E2 > E1 , the gain coefficient at the frequency is given by g2 2 A g./ D N2 N1 S./ : 8 n2 g1
(74.2)
Here, D c= is the transition wavelength, A (s1 ) is the Einstein A coefficient for spontaneous emission for the transition, and g2 , g1 are the degeneracies of the upper and lower energy levels. In nearly every case, these quantities are fixed characteristics of the medium, independent of the laser intensity or the pump mechanism; N2 and N1 are the population densities (cm3 ) of the upper and lower levels, respectively, and S./ is the normalized transition lineshape function (Chap. 20); n is the refractive index at frequency and has contributions from all nonlasing transitions and from the host medium. Equations (74.1) and (74.2) describe either amplification or absorption, depending on whether the population inversion N2 .g2 =g1 /N1 is positive or negative.
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_74
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Equation (74.2) is a standard expression for the gain coefficient [2], which can also be written in terms of the stimulated emission cross section [4] or the oscillator strength of the laser transition [5]. By far the most common configuration is that in which the gain medium is contained in a cavity bounded on two sides by reflecting surfaces. The mirrors, defining what is called the laser resonator or cavity, allow feedback, i.e., the redirection of the field back into the gain medium for multipass amplification and sustained laser action. The two mirrors allow the field to build up along the directions parallel to the optical axis and to form a pencil-like beam of light. The design of the resonator determines the spectral, spatial, temporal, and polarization properties of the laser radiation within the limitations set by the properties of the chosen laser medium. In order to sustain laser action, the gain in intensity due to stimulated emission must equal or exceed the loss due to transmission through the mirrors, scattering, absorption in the host medium, and diffraction. Transmission through at least one of the mirrors is, of course, required to obtain a usable laser beam, and it is typically the dominant field loss mechanism. If the mirror reflectivities are r1 and r2 , then the intensity I is reduced by the factor r1 r2 in a round-trip pass through the cavity, while according to Eq. (74.1) the gain medium causes the intensity to increase by a factor exp.2g`/ in the two passes through the gain medium of length `. Equating of the gain and loss factors leads to the threshold condition for laser oscillation: g gt , where the threshold gain is
R. Menzel and P. W. Milonni
This is a typical result: the population inversion required for laser oscillation tends to be relatively small compared with the total population. Calculations of population inversions and other properties of the gain medium are based on rate equations, or more generally, the density matrix . This density matrix and, thus, the gain coefficient g can be functions of time, space, frequency, and polarization of the laser light. However, in many instances, the laser process in the medium is fairly well described in terms of two energy eigenstates, other states appearing only indirectly through pumping and decay channels. In this case, is a 2 2 matrix whose elements satisfy (Chap. 72) 1
P22 D .2 C / 22 i ˝ 21 ˝ 12 ; 2 1
P11 D 1 11 C 22 C i ˝ 21 ˝ 12 ; 2 1
P21 D . C i / 21 i˝. 22 11 / ; 2
(74.6)
. Here, 2 and 1 are, respectively, the rates of with 12 D 21 decay of the upper and lower states due to all processes other than the spontaneous decay from state 2 to state 1 described by the rate D A. , which is 2 times the homogeneous linewidth (HWHM) of the transition, is the rate of decay of off-diagonal coherence due to both elastic and inelastic processes; in general, .1 C 2 C /=2. ˝ D d21 E =„ is the Rabi frequency (Chaps. 72 and 77). (The Rabi frequency is often defined as 2d 21 E =„.) E is the complex amplitude of the electric field; i.e., the electric field is 1 (74.3) gt D ln.r1 r2 / C a ;
2` E .r; t/ D Re E .r; t/ ei.kr!t / Š Re E .r; t/ eikz ei!t : a being an attenuation coefficient associated with any loss (74.7) mechanisms that may exist in addition to reflection losses at It is assumed that E is slowly varying in time compared the mirrors. Suppose, for example, that a laser has a 50 cm gain cell with the oscillations at frequency ! .D 2 /, and that the and mirrors with reflectivities 0.99 and 0.97, and that absorp- wave vector k is approximately k zO D .n!=c/Oz, where zO is tion within the host medium of the gain cell is negligible. a unit vector pointing in the direction of propagation. Finally, Then the threshold gain is gt D 4 104 cm1 . If the las- D !0 ! in Eq. (74.6) is the detuning of ! from the central ing transition is the 6328 Å Ne transition of the He-Ne laser, transition frequency !0 D .E2 E1 /=„ of the lasing transiA Š 1:4 106 s1 , n Š 1:0 and, assuming a pure Doppler tion. Rapidly oscillating terms involving !0 C ! are ignored in the rotating-wave approximation that pervades nearly all lineshape, of laser theory (Chap. 72). 4 ln 2 1=2 1 In most lasers, is so large compared with the diagonal (74.4) S./ D decay rates that the off-diagonal elements of may be as ıD sumed to relax quickly to the quasi-steady values obtained at line center, where ıD is the width (FWHM) of the by setting P12 D 0 in Eq. (74.6). Then the diagonal density Doppler lineshape (Sect. 73.4). For T D 400 K and the Ne matrix elements satisfy the rate equations atomic weight, ıD Š 1500 MHz and S./ Š 6:3 1010 s. Then the threshold population difference required for laser j˝j2 =2 D . C /
. 22 11 / ;
P 22 2 22 oscillation is 2 C 2 g2 8 n2 gt j˝j2 =2 Š 2:8 109 cm3 : (74.5) N2 N1 D 2 . 22 11 / : (74.8)
P11 D 1 11 C 22 C 2 g1 AS./ C 2 t
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Such rate equations, usually expressed equivalently in terms of population densities N2 , N1 rather than occupation probabilities 22 ; 11 , are the basis of most practical models of laser oscillation. These equations, or, more generally, the density matrix equations, must also include terms accounting for the pump mechanism. In the simplest model of pumping, one adds a constant pump rate 2 to the right-hand side of the equation for 22 to obtain
P22 D 2 .2 C A/ 22
j˝j2 =2 74.2 . 22 11 / : (74.9) 2 C 2
In the case of an inhomogeneously broadened laser transition, equations of the type Eqs. (74.6) and (74.8) apply separately to each detuning arising from the distribution of atomic or molecular transition frequencies. In writing these equations, we have assumed a nondegenerate electric dipole transition. The generalization to magnetic or multiphoton transitions, or to a case where the amplification is due, for instance, to a Raman process, is straightforward but of less general interest. A more realistic treatment of the electromagnetic field than that based on Eq. (74.1) proceeds from the Maxwell equations, which, for a homogeneous and nonmagnetic medium, lead to the equation 1 2 r EC 2ik T
equations are the basis of semiclassical laser theory, wherein the particles constituting the gain medium are treated quantum mechanically whereas the field is treated according to classical electromagnetic theory. Aside from fundamental linewidth considerations and photon statistics (Sects. 74.2 and 74.4), very few aspects of lasers require the quantum theory of radiation.
@ 4 i! 1 @ ED C N 21 : @z c @t nc
(74.10)
Here, N is the density of active atoms, .d 12 /, and rT2 @2 =@x 2 C @2 =@y 2 . This result assumes the validity of the rotating-wave approximation as well as the assumption that E is slowly varying compared with exp.ikz/ and exp.i!t/. In the plane-wave approximation, Eq. (74.10) becomes 4 i! @ 1 @ ED (74.11) C N 21 : @z c @t nc
Continuous Wave, Single-Mode Operation
In the case of steady-state, continuous-wave (CW) operation, the appropriate matter–field equations are those obtained by setting all time derivatives equal to zero. Equation (74.11), for instance, becomes dE 2 !N jd j2 1 D . 22 11 /E ; dz 3n„c C i or, in terms of the intensity I , dI 4 !N jd j2 . 22 11 /I D dz 3n„c 2 C 2 2 A .N2 N1 /S./I D g./I ; D 8 n2
(74.13)
for the nondegenerate case under consideration. Here, jd j2 D 3jd 12 j2 , Nj D N jj , S./ D 2 .= /=. 2 C 2 / is the Lorentzian lineshape function for homogeneous broadening, and A D 4! 3 jd j2 n=3„c 3 is the spontaneous emission rate in the host medium of (real) refractive index n. Local (Lorentz–Lorenz) field corrections will, in general, modify these results, but such corrections are ignored here [7]. The steady-state solution of the density matrix or rate equations gives, similarly, g./ D
(More generally, the velocity c on the left-hand sides of Eqs. (74.10) and (74.11) should be replaced by the group velocity vg associated with nonresonant transitions. If there is substantial group velocity dispersion, it is sometimes necessary to include a term involving the second derivative of E with respect to t.) Equations (74.6) or (74.8) and (74.10) or (74.11) are coupled matter–field equations whose selfconsistent solutions determine the operating characteristics of the laser. The density matrix or rate equations must be modified to include pumping, as in Eq. (74.9), and the field equations must be supplemented by boundary conditions and loss terms. Laser media based on more complicated energy-level schemes (e.g., “three-level” and “four-level” lasers [2–6]) are described similarly with more general rate or density-matrix equations. With these modifications, the
(74.12)
g0 ./ ; 1 C I =Isat
(74.14)
where the saturation intensity Isat , like the small-signal gain coefficient g0 ./, depends on decay rates and other characteristics of the lasing species. Thus, in the plane-wave approximation, the growth of intensity in a homogeneously broadened laser medium is typically described by the equation g0 ./I dI : D dz 1 C I =Isat
(74.15)
This equation, supplemented by boundary conditions at the mirrors and possibly other terms on the right-hand side to account for any distributed losses within the medium, determines the intensity in CW, single-mode operation.
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The simplest model for calculating output intensity as- r2 IC .L/, IC .0/ D r1 I .0/ for mirrors at z D 0 and z D L sumes that the intensity is uniform throughout the laser gives, for Iout D t1 I .0/ C t2 IC .L/, the formula [5, 8] cavity. In steady state, the gain exactly compensates for the r p r1 r2 loss; i.e., g./ D gt , the “gain clamping” condition for CW Iout D Isat t2 C t1 p p
p r1 r1 C r2 1 r1 r2 lasing. Equation (74.14) then implies that the steady-state in
p tracavity intensity is (74.24) g0 ./` C ln r1 r2 : I D Isat Œg0 ./=gt 1 :
(74.16) Analysis of this result gives an optimal output coupling that reduces to Eq. (74.19) in the limit t C s 1. Curves for opIf I is assumed to be the sum of the intensities of waves timal output coupling and I as a function of g and s are out 0 propagating in the Cz and z-directions, i.e., I D IC C I , given by Rigrod [8]. These results are based on several asthen the output intensity from the laser is sumptions and approximations: the gain medium is assumed to be homogeneously broadened and to saturate according to (74.17) Iout D t2 IC C t1 I ; the formula Eq. (74.14); g0 and Isat are taken to be constant throughout the medium; the field is approximated as a plane where t2 , t1 are the mirror transmissivities at the right and left wave; field loss processes occur only at the mirrors; and inmirrors, respectively. The uniform intensity approximation terference between the left and right-going waves is ignored. implies that IC D I D I =2, and Interference of the counterpropagating waves in 1 a standing-wave, single-mode laser modifies the gain satIout D .t2 C t1 /Isat Œg0 ./=gt 1 : (74.18) 2 uration formula Eq. (74.14) as follows: Suppose one of the mirrors is perfectly reflecting, so that t1 D 0 and t2 D t. If the reflectivity r of the transmitting mirror is close to unity, then gt Š .1=2`/.1r/ D .1=2`/.t Cs/, where s is the fraction of the incident beam power that is scattered or absorbed at the output mirror. Then 2g0 ./` 1 Iout Š Isat t 1 ; (74.19) 2 t Cs and it follows that the optimal output coupling, i.e., the transmissivity that maximizes the output intensity, is p (74.20) topt D 2g0 ./`s s : This output coupling gives the output intensity hp i2 p max D Isat g0 ./` s=2 I Iout
(74.21)
g.; z/ D
g0 ./ 1 C .2IC =Isat / sin2 kz
(74.25)
in the case of small output coupling, where IC Š I as assumed in Eq. (74.18). (The general case of arbitrary output coupling with spatial interference of counterpropagating waves is somewhat complicated and is not considered here.) The sin2 kz term is responsible for spatial hole burning: holes are burned in the curve of g.; z/ versus z at points where sin2 kz is largest. This spatially dependent saturation of the gain acts to reduce the output intensity, typically by as much as about 30%, compared with the case where interference of counterpropagating waves is absent or ignored. Spatial hole burning tends to be washed out by atomic motion in gas lasers and is absent entirely in pure traveling-wave ring lasers. If complete spatial hole burning based on Eq. (74.25) is assumed, the output intensity is s ! g0 ./ t 1 g0 ./ 1 C (74.26) Iout D Isat 2 gt 4 2gt 16
g0 ./Isat is the largest possible power per unit volume extractable as output laser radiation at the frequency . More generally, when mirror reflectivities are not necessarily close to unity, IC ¤ I and both IC and I vary with the axial coordinate z. In this more general case, Eqs. (74.14) in the case where one mirror is perfectly reflecting. In inhomogeneously broadened media, the gain coeffiand (74.15) are replaced by cient is obtained by integrating the contributions from all g0 ./ g.; z/ D (74.22) possible values of 0 . The different contributions saturate 1 C ŒIC .z/ C I .z/=Isat differently, depending on the detuning of 0 from the cavity mode frequency . If spatial hole burning and power broadand ening are ignored, then, to a good approximation, the gain dIC dI (74.23) saturates as D g.; z/IC ; D g.; z/I ; dz dz g0 ./ (74.27) g./ D p where it is assumed that g0 ./ is independent of z, and that 1 C I =Isat all cavity loss processes occur at the mirrors. The solution of these equations with the boundary conditions I .L/ D in typical inhomogeneously broadened media.
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Oscillation on a single longitudinal mode (Sect. 74.3) may be realized simply by making the cavity length L small enough that the mode spacing c=2nL (Eq. (74.28)) exceeds the spectral width of the gain curve. This is possible in many gas lasers where the spectral width is small, and in semiconductor lasers, where L is very small. More generally, the gain clamping condition g./ D gt implies that the cavity mode frequency having the largest small-signal gain g0 ./ saturates the gain g./ down to the threshold value gt , while the gain at all other mode frequencies then lies below gt . In other words, the gain clamping condition implies single-mode oscillation. However, this conclusion assumes homogeneous broadening and also that spatial hole burning is negligible, so that the gain saturates uniformly throughout the cavity. High-pressure gas lasers, where the line broadening is due primarily to collisions and is, therefore, homogeneous, tend to oscillate on a single mode because spatial hole burning is largely washed out by atomic motion. On the other hand, homogeneously broadened solid-state lasers can be multimode as a consequence of spatial hole burning. Single-mode oscillation in inhomogeneously broadened media is generally more difficult to achieve because of spectral hole burning, which makes the simple gain clamping argument inapplicable. However, single-mode oscillation can be enforced in any case by introducing, in effect, an additional loss mechanism for all mode frequencies except one. This is commonly done with a Fabry–Pérot etalon inside the cavity having a free spectral range that is large compared with the spectral width of the gain curve. By choosing the tilt angle appropriately, a particular resonant frequency of the etalon can be brought close to the center of the gain curve, while all other resonance frequencies lie outside the gain bandwidth. Laser oscillation at a fixed polarization can likewise be achieved by discriminating against the orthogonal polarization, as is done when Brewster angles are employed. Laser oscillation, in general, does not occur precisely at one of the allowed cavity mode frequencies. Associated with the sin kz dependence of the intracavity field is the condition kL D N , or DN
c 2nL
.N an integer/ ;
(74.28)
where L is the geometrical distance between the resonator mirrors, ` ( L) is the length of the gain medium, Lopt D n` C .L `/ is the optical length of the cavity, and N D Nc=2L is an empty-cavity mode frequency. The laser oscillation frequency will, therefore, be different, in general, from any of the allowed empty-cavity mode frequencies. If the refractive index n./ here is attributable primarily to the lasing transition, as opposed to the host material, or other nonlasing transitions, then the following relation between n./ and g./ may be used in the case of homogeneous broadening (for instance [5]): n./ 1 D
0 0 g./ ; 4 ı0
(74.31)
where 0 , 0 , and ı0 are the wavelength, frequency, and homogeneous linewidth (HWHM), respectively, of the lasing transition. This implies that D
0 ıc C N ı0 ; ıc C ı0
(74.32)
for the laser oscillation frequency , where ıc
cg./` 4 L
(74.33)
is the cavity bandwidth. Thus, the actual lasing frequency is not simply one of the allowed empty-cavity frequencies N , but is rather pulled away from N toward the center of the gain profile. This is called frequency pulling. If spatial hole burning is absent, then g./ D gt in steadystate oscillation, and the cavity bandwidth ıc D cgt `=4 L is the largest for lossy cavities. Most lasers fall into the good cavity category, that is, ıc ı0 , so that Eq. (74.32) may be approximated by Š N C .0 N /ıc =ı0 :
(74.34)
Similar results apply to inhomogeneously broadened lasers. For a Doppler broadened medium, for instance, the frequency pulling formula is q Š N C .0 N /.ıc =ıD / 4 ln 2 ;
(74.35) for the cavity mode frequencies in the plane-wave approximation. If the gain medium of refractive index n does not fill for good cavities. These results show that frequency pulling the entire length L between the mirrors, then Eq. (74.28) is is most pronounced in lasers with large peak gain coefficients replaced by and narrow gain profiles, such as the 3.51 µm He-Xe laser, as observed experimentally [9]. Nc=2 ; (74.29) D Spectral hole burning leads to especially interesting conLopt sequences in Doppler-broadened gas lasers. Since two travor eling waves propagating in opposite (˙z) directions will ` strongly saturate spectral packets of atoms with oppositely (74.30) Œn./ 1 D N ; L Doppler-shifted frequencies, a standing-wave field will burn
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two holes on opposite sides of the peak of the Doppler profile. When the mode frequency is exactly at the center of the Doppler profile, however, the two holes merge, the field now being able to saturate strongly only those atoms with zero velocity along the z-direction. In this case, since the field feeds off a single spectral packet of more strongly saturated atoms, there is a dip in the output power compared with the case when the mode frequency is detuned from the line center. This dip in the output power at the line center is called the Lamb dip. It can be used to determine whether a gas laser is Doppler broadened, and more importantly, to stabilize the laser frequency to the center of the dip. Highprecision measurements often require that a laser operate at a very stable frequency, in spite of mirror vibrations and other noise processes. Lamb-dip frequency stabilization is based on an “error signal” that is the difference between the measured output intensity and the intensity at the center of the Doppler gain profile. A feedback circuit controls the bias voltage across a piezoelectric element used to vary the cavity length so as to minimize the error signal and thereby to stabilize the laser frequency. Fractional frequency stabilities of about 1 part in 1010 can by obtained by Lamb-dip stabilization. The widely used Pound–Drever–Hall frequency stabilization technique employs a Fabry-Pérot cavity to provide an error signal proportional to the intensity reflected when the field frequency differs from a cavity resonance frequency. This method has the advantage that it is not restricted to stabilization of laser frequencies near that of a laser material resonance [10–13]. With high-finesse etalons it has been possible to obtain short-time fractional frequency stabilities 1016 and laser linewidths < 40 mHz [13]. Semiclassical laser theory suggests that CW laser radiation could ideally be perfectly monochromatic, since the amplitude and phase of the field given by Eq. (74.7) are time-independent. However, when quantum electrodynamic considerations are built into laser theory, it is found that spontaneous emission, which adds to the number of photons put in the lasing mode by stimulated emission, causes a phase diffusion that results in a Lorentzian linewidth (FWHM) D
i2 ıc N2 8 h h ; N2 N1 Pout ng C 1 C ıc =ı0
N2 8 h .ı0 /2 : N2 N1 Pout
D
N2 8 h .ıc /2 : N2 N1 Pout
(74.38)
The “Schawlow–Townes linewidth” is a fundamentally quantum-mechanical, finite linewidth that persists no matter how small various sources of “technical noise”, such as mirror jitter, are made. Although it has been observed in highly stabilized gas lasers, it is negligible compared with technical noise in conventional lasers. However, in free-running semiconductor lasers, L is very small and, consequently, ıc is large, and the fundamental lower limit to the laser linewidth can be the dominant contribution to the laser linewidth. However, linewidths 10100 MHz observed in semiconductor lasers are too large to be explained by these expressions for , and two modifications are necessary, each of them involving a multiplication of the Schawlow–Townes linewidth by a certain factor:
! 0 D 1 C ˛ 2 K I
(74.39)
˛ is called the Henry ˛ parameter and is associated with a coupling between phase and intensity fluctuations above the laser threshold [15, 16]. Values of ˛ between about 4 and 6 are typical in semiconductor injection lasers, and consequently, the correction to the Schawlow–Townes linewidth due to the Henry ˛ parameter is substantial. The K factor [17–19] arises as a consequence of the deviation from the spatially uniform intracavity intensity assumed in the derivation of the Schawlow–Townes formula [20, 21]. (Intracavity intensities along the optical axis are approximately uniform only in the case of low output couplings.) The fundamental quantum-mechanical linewidth under consideration can be associated with vacuum field fluctuations, which, according to general fluctuation–dissipation theory, will increase as the cavity loss increases. This explains why the Petermann K factor deviates increasingly from unity as the output coupling (cavity loss) increases. Values of K between 1 and 2 appear to be typical for lossy, stable resonators [22, 23], but (74.36) much larger values are possible for unstable resonators [24].
where Pout is the output laser power and ng D d.n!/=d! is the group index associated with nonlasing transitions in the laser medium. In the bad cavity limit in which ıc ı0 , this reduces to the expression originally derived by Schawlow and Townes [14] when it is assumed that the contribution (ng ) from nonlasing transitions is negligible: D
In the more common case of a good cavity,
(74.37)
74.3
Laser Resonators and Transverse Modes
The assumption that the complex field amplitude E .r; t/ is slowly varying in z compared with exp.ikz/ leads to the paraxial wave equation Eq. (74.10), rT2 E C 2ik
@E D0; @z
(74.40)
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for a monochromatic field in vacuum. If E .x; y; z/ is specified in the plane .x; y; z D 0/, it follows that “ i E .x; y; z/ D dx 0 dy 0 z 0 2 C.yy 0 /2 =2z
E .x 0 ; y 0 ; 0/ eikŒ.xx /
: (74.41)
(x, y)
(x, y) z0 θ
w0
w0
w(z)
z=0 Beam waist
Thus, in the case of a laser resonator, the field E .x; y; L/ at Intensity Intensity the mirror at z D L is related to the field E .x; y; 0/ at the mirror at z D 0 by Fig. 74.1 Variation of the beam radius w.z/ of a Gaussian beam with “ the propagation distance z from the beam waist i 0 0 E .x; y; L/ D dx dy L for stable resonators, where the g parameters are defined in 0 2 0 2 E .x 0 ; y 0 ; 0/ eikŒ.xx / C.yy / =2L terms of the (spherical) mirror curvatures Ri and the mirror “ separation L by gi 1 L=Ri ; Ri is defined as positive or dx 0 dy 0 K.x; yI x 0 ; y 0 /E .x 0 ; y 0 ; 0/ : negative depending on whether the mirror i is concave or con(74.42) vex, respectively, with respect to the interior of the resonator. Plane–parallel mirrors (Ri ! 1) can be used, but they are Similarly, the field at z D 0 after one round-trip pass through difficult to keep aligned and have large diffractive losses. the resonator is Among stable configurations, the symmetric confocal res“ onator with R1 D R2 D L has the smallest beam radii at the E .x; y; 0/ D dx 0 dy 0 K.x; yI x 0 ; y 0 /E .x 0 ; y 0 ; L/ mirrors, while the concentric resonator with R1 D R2 slightly “ greater than L=2, has the smallest beam waist (Fig. 74.1). The hemispherical resonator (R1 D 1, R2 slightly greater D dx 0 dy 0 K.x; yI x 0 ; y 0 / than L) is relatively easy to keep aligned and allows the beam “ radius w at mirror 2 to be adjusted by slight changes in L. 00 00 0 0 00 00 00 00 dx dy K.x ; y I x ; y /E .x ; y ; 0/ The fundamental Gaussian beam modes of stable res“ onators may be constructed from the free-space solutions Q yI x 00 ; y 00 /E .x 00 ; y 00 ; 0/ : dx 00 dy 00 K.x; of the paraxial wave equation. The most important (lowest(74.43) order) solution for this purpose is By definition, a mode of the resonator is a field distribution that does not change on successive round-trip passes through the resonator. More precisely, since an empty cavity is assumed, a mode will be such that the field spatial pattern remains the same except for a change in amplitude per pass. A longitudinal mode is defined by the value of k in exp.ikz/, whereas a transverse mode is defined by the corresponding .x; y/ dependence and satisfies the integral equation Z Q yI x 0 ; y 0 /E .x 0 ; y 0 ; z/ ; E .x; y; z/ D dx 0 dy 0 K.x;
2
2
A ei.z/ eik.x Cy /=2R.z/ e.x E .x; y; z/ D q 1 C z 2 =z02
2 Cy 2 /=w 2 .z/
; (74.46)
where A is a constant, .z/ D tan1 .z=z0 /, and R.z/, w.z/, and z0 are the radius of curvature of surfaces of constant phase, the beam radius, and the Rayleigh range, respectively. The confocal parameter, 2z0 , is also used to characterize Gaussian beams. Here, z0 w02 =, where w0 is the beam radius at the waist at z D 0 (Fig. 74.1), and R.z/ and w.z/ (74.44) vary with the distance z as follows:
where z defines any plane between the mirrors, and accounts for the change per pass of the complex field amplitude. Iterative numerical solutions of this equation for laser resonator modes were first presented by Fox and Li [25]. Laser resonators may be classified as stable or unstable according to whether a paraxial ray traced back and forth through the resonator remains confined in the resonator or escapes. This leads to the condition 0 g1 g2 1 ;
R.z/ D z C z02 =z ;
q w.z/ D w0 1 C z 2 =z02 :
(74.47)
The divergence angle of a Gaussian beam (Fig. 74.1) is D = w0 . The ABCD law for Gaussian beams [2–6] allows the effects of various optical elements on Gaussian beam propagation to be calculated in a relatively simple fashion. For instance, a Gaussian beam incident on a lens of focal length (74.45) f at its waist is focused to a new waist at a distance d D
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R. Menzel and P. W. Milonni
f =.1 C f 2 =z02 / behind the lens, and the beam radius at the Secondly, the resonator structure itself may introduce new waist is complications that preclude closed-form solutions even for the empty cavity. This is generally true, for instance, of 1 f q ; (74.48) unstable resonators, where iterative numerical solutions of w00 D w0 1 C f 2 =z 2 the integral equation (Eq. (74.44)) are necessary for accu0 rate predictions of modes. In such computer simulations, the which is approximately f = w0 D f for tight focus- mode structure must be determined self-consistently with ing. This shows that a Gaussian beam can be focused to the numerical solutions of the density matrix or, more coma very small spot, on the order of a wavelength. On the monly, rate equations for the gain medium. other hand, beam expanders consisting of two appropriately The diffraction-limited divergence angle of the ideal spaced lenses may be used to increase the beam radius by lowest-order (TEM00 ) Gaussian mode, D = w0 , may be the ratio of the focal lengths in order to realize very small difficult to realize in practice. The deviation from the ideal divergence angles. diffraction-limited field is characterized by a beam quality Gaussian beam modes of laser resonators have radii of M 2 defined by writing the actual divergence angle of the outcurvature R.z/ that match in magnitude those at the positions put field as of mirrors. The beam radii at the mirrors and the location of the beam waist with respect to the mirrors may be expressed (74.50) M D M 2 = w0 : in terms of , L, and g1 g2 . The empty-cavity mode frequencies are given by For the lowest-order Hermite–Gaussian mode M 2 D 1 by definition, whereas for the first-order mode M 2 D 3 and c 1 p N D N C cos1 g1 g2 ; (74.49) increases to 13 for the sixth-order mode, for example. Com2L parable values are obtained for the circularly symmetric where N is an integer. For a host medium of refractive index Gauss–Laguerre modes [4]. For a low-power gas laser, M 2 n, c=2L is replaced by c=2nL, and Eq. (74.49) then gener- might be around 1.1, whereas for a high-power laser, it might alizes the plane wave result Eq. (74.28) to account for both be 10 or more. The brightness L of a laser beam, defined longitudinal and transverse effects in the determination of the as the power per unit area per unit solid angle, is, therefore, cavity mode frequencies. inversely proportional to M 2 and may be defined as [4] The assumption of Gaussian modes presupposes that the P resonator mirrors are large enough to intercept the entire ; (74.51) LD 2 beam without any spillover; i.e., that a w1 ; w2 , where .M 2 /2 w1 ; w2 are the beam radii at the mirrors, and a is an effective mirror cross sectional radius; this is very easy to realize. This where P is the beam power. Unstable resonators, although inherently lossy, offer some implies that the Fresnel number NF a2 =L 1. Diffraction important advantages for high-gain and high-power lasers. losses generally increase with decreasing Fresnel numbers. Higher-order Gaussian modes, where the Gaussian func- Thus, whereas the Gaussian modes of stable resonators typtions of x and y in Eq. (74.46) are replaced by higher-order ically involve small beam radii, the distinctly non-Gaussian Hermite (or Laguerre) polynomials, or combinations of them modes of unstable resonators often have large mode volumes (hybrid modes) [4], can be produced via suitable resonator and make more efficient use of the available gain volume. Phase conjugation mirrors can be used to improve the beam designs and may be used in special applications. In general, it is not possible to write closed-form expres- quality of high-power lasers [4]. sions for laser modes. There are at least two reasons for this, the first being that the gain medium cannot, in general, be regarded as a simple amplifying element that preserves 74.4 Photon Statistics the basic empty-cavity mode structure. In general, the gain coefficient and the refractive index of the laser material, es- Optical fields may be characterized and distinguished by pecially in the high-power regime, are functions of space, their photon statistical properties (Chap. 82 and [26–30]). In time, frequency, and light intensity. In low-power gas lasers, a photon counting experiment, the number of photons registhe spatial variations of the gain and refractive index are tered at a photodetector during a time interval T is measured sufficiently mild that the lasing modes can be accurately de- and used to infer the probability Pn .T / that n photons are scribed as Gaussians. However, there can be strong gain and counted in a time interval T . If the probability of counting index variations that themselves play an important role in a photon at the time t in the interval dt is denoted I.t/dt, determining the modes of the laser, as in index-guided and where I.t/ is the intensity of the field, and is a factor depending on the microscopic details of the photon absorption gain-guided semiconductor lasers or in fiber lasers.
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process and on the photodetector geometry, then it may be linewidth ı0 , the gain tends to saturate homogeneously, and shown from largely classical considerations that [5, 28, 29] the total output power on all modes is well described by the analysis outlined in Sect. 74.2. 3n * 2 ZT + Pulsed laser operation adds the further complication of R 14 0 0 5 0T dt 0 I.t 0 / ; (74.52) temporal variations to the CW theory outlined in the precedPn .T / D dt I.t / e nŠ ing sections. It is possible, nevertheless, to understand some 0 of the most important types of multimode and pulsed operawhere the average h i is over the intensity variations dur- tion using relatively simple models. ing the counting interval. (The fully quantum-mechanical One method of obtaining short, high-power laser pulses treatment involves normally ordered products of electric field is Q switching [31]. In a very lossy cavity, the gain can operators [28, 29].) In the simplest case of constant intensity, be pumped to large values before the threshold condition is Pn follows the Poisson distribution: met and gain saturation occurs. If the cavity loss is suddenly decreased via an optical switch, there will be a rapid buildup (74.53) of intensity because the gain is far above the (now reduced) Pn D n n en =nŠ ; threshold value. The switching of the cavity loss is called where n D I T . Since a single-mode laser field can be Q switching, the quality factor Q being defined as =2ıc . thought of approximately as a classical stable wave [29], it is The buildup time for a Q-switched pulse is typically on the not surprising that its photocount distribution is found both order of the cavity round-trip time [4], resulting in pulses theoretically and experimentally to satisfy Eq. (74.53), which of duration on the order of 10 ns for solid-state lasers, for is characteristic of a coherent state of the field (Chap. 82). example. Q switching requires that the gain medium be caBecause of spontaneous emission a laser cannot produce ex- pable of retaining a population inversion over a time much actly a coherent state of the field but, if the phase diffusion larger than the Q-switched pulse duration, and, in particular, due to spontaneous emission is sufficiently slow, laser radia- that the spontaneous emission lifetime be relatively long. tion on sufficiently short time scales can be treated in effect The pulse duration can be reduced to approximately a single as a coherent-state field to a very good approximation. cavity round-trip time by Q switching from low reflectivA thermal light source, by contrast, follows the Bose– ity mirrors to 100% reflectivities, and then switching the Einstein distribution, reflectivity of the outcoupling mirror from 100% to 0% at the peak of the amplified pulse. In addition to this pulsed (74.54) transmission mode is cavity dumping, where both mirrors Pn D nn =.1 C n/nC1 ; have nominally 100% reflectivity and the intracavity power if the time interval T is short compared with the coherence is “dumped” by an acousto-optic or electro-optic intracavity time of the light; i.e., if T 1, where is the band- element that deflects the light out of the cavity. The pulse width. If T 1, Pn is again Poissonian.Thus, if a quasi- duration achieved in this way is again roughly a round-trip monochromatic beam of light is made from a natural source time. Cavity dumping can be employed with CW lasers and by spatial and spectral filtering, it has measurably different does not require the long energy storage times necessary for photon counting statistics from a single-mode laser beam ordinary Q switching. of exactly the same bandwidth and average intensity. These Shorter pulses can be realized by mode locking, where the differences are exhibited, for example, in measurements of phases of N longitudinal cavity modes are locked together. intensity correlations of the Brown–Twiss type [28–30]. In (The evolution of the mode-locking technique is reviewed such experiments, thermal photons (or more generally pho- in [32].) In the simplest model, assuming equal amplitudes tons from a “chaotic” source [29]) have a statistical tendency and phases of the individual modes, the net field amplitude to arrive in pairs (“photon bunching”), whereas the photons is proportional to from a laser arrive independently. .N 1/=2 X X0 sinŒ.!0 C n /t C 0 X.t/ D
74.5
Multimode and Pulsed Operation
Multimode laser theory is generally much more complicated than single-mode theory, particularly in the case of inhomogeneous broadening with both spectral and spatial hole burning. In certain situations, however, considerable simplification is possible. For instance, when the cavity mode frequency spacing is small compared with the homogeneous
nD.N 1/=2
D X0 sin.!0 t C 0 /
sin.N t=2/ : sin. t=2/
(74.55)
The temporal variation described by this function for large N is a train of spikes of amplitude NX0 at times tm D 2 m= , m D 0, ˙1, ˙2, , the width of each spike being 2 =.N /. In the case of a mode-locked laser, D 2 .c=2L/, and the
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R. Menzel and P. W. Milonni
output field consists of a train of pulses separated in time by T D 2 = D 2L=c. The peak amplitude of the spikes is proportional to N , and the duration of each spike is approximately 2L=cN . The spectrum of such a field may be described as a “comb” of frequencies with “teeth” separated by the angular frequency . The regular, controllable comb of frequencies in the output of a mode-locked laser serves as a “ruler” for measurements of absolute optical frequencies (optical frequency metrology) [33–35]. The maximum number Nmax of modes that can actually be phase-locked is limited by the spectral width g of the gain curve: Nmax D
g 2L D g : c=2L c 2L 1 D : cNmax g
(74.57)
Mode locking thus requires a gain bandwidth large compared with the cavity mode spacing, and the shortest and most intense mode-locked pulse trains are obtained in gain media having the largest gain bandwidths. Trains of picosecond pulses are routinely obtained with liquid dye and solid gain media having gain bandwidths g 1012 s1 or more. Various techniques, employing acoustic or electro-optic modulation or saturable absorbers, are used to achieve mode locking. The different methods all basically rely on the fact that a modulation of the gain or loss at the mode separation frequency c=2L tends to cause the different modes to oscillate in phase. Such a modulation is achieved passively when a saturable absorber is placed in the laser cavity; because of the intensity-dependent loss caused by the saturable absorber, the multimode intensity oscillates with a beat frequency that is impressed on the saturated loss coefficient. Kerr lens mode locking (KLM) [36–38] is based on the self-focusing of a field in a medium with a nonlinear refractive index (Chap. 76) n D n0 C n2 I :
Chirped pulse Glass fiber
b
Grating Chirped input pulse
Blue
Red
Output pulse
Grating
(74.56) Fig. 74.2 Nonlinear pulse compression by frequency chirping. In (a),
Similarly, the shortest pulse duration is min D
a
(74.58)
With such a nonlinear index the field can be strongly selffocused (assuming n2 > 0) where the intensity I is largest. The basic idea in KLM is that there is an intensity-dependent loss, as in a saturable absorber. However, this intensitydependent loss now occurs within the laser medium; the extent to which the field is susceptible to diffractive losses depends on the degree of self-focusing and, therefore, the intracavity intensity. Megawatt mode-locked pulses with durations and repetition rates on the order of femtoseconds and 100 MHz, respectively, are routinely produced with KLM Ti:sapphire lasers (Chap. 75). Ultrashort pulse generation is possible with other nonlinear techniques. The colliding-pulse laser [39] is a threemirror ring laser in which two mode-locked pulse trains
the nonlinear refractive index of a glass fiber results in a time-dependent frequency of the transmitted pulse, and in (b), a pair of diffraction gratings is used to produce frequency-dependent path delays such as to temporally compress the pulse
propagate in opposite senses and overlap in a very thin ( 10 µm) saturable absorber placed in the ring in addition to the gain cell. The cavity loss is least when the two pulses synchronize to produce the highest intensity, and, therefore, the lowest loss coefficient in the saturable absorber. The short length of the absorbing cell forces the pulses to overlap within a very small distance and, therefore, to produce very short pulses (10 µm=c 30 fs pulse duration with a c=L 100 MHz repetition rate). Another method of ultrashort pulse generation relies on , i.e., a time dependent shift of the frequency of an optical pulse [40–43]. In a medium (e.g., a glass fiber) with a nonlinear refractive index (Eq. (74.58)), a field experiences an instantaneous phase shift .t/ that depends on the instantaneous intensity. Therefore, as I.t/ increases toward the peak intensity of the pulse, .t/ increases (assuming n2 > 0), whereas .t/ decreases as I.t/ decreases from its P peak value. The frequency shift .t/ is such that the instantaneous frequency of the pulse is smaller at the leading edge and larger at the trailing edge of the pulse (Fig. 74.2), resulting in a stretching of the pulse bandwidth. Following this spectral broadening by the nonlinear medium, the pulse can be compressed in time by means of a frequency-dependent delay line such that the smaller frequencies, say, are delayed more than the higher frequencies. The trailing edge of the pulse can, therefore, catch up to the leading edge, resulting in a shorter pulse whose duration is given by the inverse of the chirp bandwidth. The delay line can be realized with a pair of diffraction gratings (Fig. 74.2). More than two decades ago this pulse compression technique was used to compress 40 fs amplified pulses from a colliding pulse laser to 8 fs, corresponding to about four optical cycles [44]. One of the most important techniques in ultrashort, highpower laser pulse generation is chirped pulse amplification (CPA) [45, 46] in which a laser pulse is chirped, temporally
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stretched, and then passed through one or more amplifiers. The lengthening in time of the pulse by factors typically 103 –104 prior to amplification allows greater energy extraction from the amplifier and avoids optical damage and nonlinear effects causing spatiotemporal pulse distortion. After amplification, pulse compression is performed with a grating pair. The Ti:sapphire (TiWAl2 O3 ) amplifier is particularly attractive for femtosecond CPA because of its very large spectral width and high saturation fluence. The wavelength dependence of the linear refractive index n0 in Eq. (74.58) results in a group velocity dn0 1 vg D c n0 d
(74.59)
74.6 Instabilities and Chaos Mode-locked pulses and solitons exemplify an ordered dynamics, as opposed to the erratic and seemingly random intensity fluctuations that are sometimes observed in the output of a laser. In fact, it is possible, under certain circumstances, for laser oscillation to exhibit deterministic chaos; i.e., an effectively random behavior that can nevertheless be described by purely deterministic equations of motion. Extensive experimental and theoretical work on unstable and chaotic behavior in a wide variety of laser systems has been reported [58–61]. Instabilities of single transverse mode dynamics have also been studied, especially in connection with spontaneous spatial pattern formation [61]. The high gain and strong nonlinearities of diode lasers can result in chaotic behavior, which is of interest for possible applications including secure communications and random number generation [62].
that, if dn0 =d < 0, is such that higher frequencies propagate more rapidly than lower frequencies. Assuming n2 > 0, on the other hand, the nonlinear part of the index causes a delay of higher frequencies with respect to lower ones, as discussed above. This leads to soliton solutions of the wave equation, such that the opposing effects of the linear and nonlinear References dispersion are balanced, and the pulse propagates without distortion. Soliton lasers, with pulse durations ranging from 1. Sargent III, M., Scully, M.O., Lamb Jr., W.E.: Laser Physics. picoseconds down to 100 fs, depending on the fiber length, Addison-Wesley, Reading, Massachusetts (1974) 2. Yariv, A.: Quantum Electronics, 3rd edn. Wiley, New York (1989) have been made with solid-state lasers and intracavity optical 3. Siegman, A.E.: Lasers. University Science Books, Mill Valley fibers [47]. (1986) The generation of coherent, extreme ultraviolet and x4. Menzel, R.: Photonics. Linear and Nonlinear Interactions of Laser ray radiation by creating a population inversion and gain Light and Matter, 2nd edn. Springer, Berlin (2007) 5. Milonni, P.W., Eberly, J.H.: Laser Physics. Wiley, Hoboken (2010) (Eq. (74.2)) on atomic transitions is impractical due to the 6. Svelto, O.: Principles of Lasers, 5th edn. Springer, New York very short lifetimes and large energies required to populate (2010) excited levels of such transitions; there is the additional prob7. DiBartolo, B.: Optical Interactions in Solids. Wiley, New York lem that conventional optical components like mirrors and (1968) 8. Rigrod, W.W.: J. Appl. Phys. 36, 27 (1965) lenses are unavailable at x-ray wavelengths. Such radiation is 9. Casperson, L.W., Yariv, A.: Appl. Phys. Lett. 17, 259 (1970) generated in free electron “lasers” (FELs) by undulator radia10. Drever, R.W.P., Hall, J.L., Kowalski, F.V., Hough, J., Ford, G.M., tion from relativistic electron beams in a “wiggler” magnetic Munley, A.J., Ward, H.: Appl. Phys. B. 31, 97 (1983) field [48, 49]. (For reviews of FEL developments and x-ray 11. Black, E.D.: Am. J. Phys. 69, 79 (2001) 12. Alnis, J., Matveev, A., Kolachevsky, N., Udem, Th , Hänsch, T.W.: FELs: [50, 51].) Phys. Rev. A 77, 053809 (2008) The high intensities produced with mode locking and 13. Kessler, T., Hagemann, C., Grebing, C., Legero, T., Sterr, U., chirped pulse amplification allow highly nonlinear atom– Riehle, F., Martin, M.J., Chen, L., Ye, J.: Nat. Photon. 6, 687 field interactions, such as the generation of high harmonics (2012) of the central frequency of an incident pulse. This has made 14. Schawlow, A.L., Townes, C.H.: Phys. Rev. 112, 1940 (1958) possible the generation of attosecond pulses [52, 53]. High- 15. Lax, M.: Phys. Rev. 160, 290 (1967) 16. Henry, C.H.: IEEE J. Quantum Electron. 18, 259 (1982) harmonic generation (HHG) has been achieved by focusing 17. Petermann, K.: IEEE J. Quantum Electron. 19, 1391 (1979) high-intensity, femtosecond infrared pulses onto atoms with 18. Siegman, A.E.: Phys. Rev. A 39, 1253 (1989) high ionization potentials (inert gases), resulting in the gen- 19. Siegman, A.E.: Phys. Rev. A 39, 1264 (1989) eration of coherent extreme ultraviolet pulses and tabletop 20. Goldberg, P., Milonni, P.W., Sundaram, B.: Phys. Rev. A 44, 1969 (1991) x-ray “lasers” [54]. HHG has been explained by a classi- 21. Goldberg, P., Milonni, P.W., Sundaram, B.: Phys. Rev. A 44, 4556 cal “three-step” model [55, 56] involving (i) ionization of (1991) the atom by the field, (ii) the field drives the freed electron 22. Hamel, W.A., Woerdman, J.P.: Phys. Rev. Lett. 64, 1506 (1990) back to the ion, and (iii) the emission of high-harmonic ra- 23. Eijkelenborg, M.A., Lindberg, A.M., Thijssen, M.S., Woerdman, J.P.: Phys. Rev. A 55, 4556 (1997) diation by the accelerated electron. General features of the 24. Cheng, K.-J., Mussche, P., Siegman, A.E.: IEEE J. Quantum Elecclassical model are recovered in fully quantum-mechanical tron. 30, 1498 (1994) 25. Fox, A.G., Li, T.: Bell Syst. Tech. J. 40, 453 (1961) theory [57].
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1080 26. Glauber, R.J.: Phys. Rev. 130, 2529 (1963) 27. Glauber, R.J.: Phys. Rev. 131, 2766 (1963) 28. Mandel, L., Wolf, E.: Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge (1995) 29. Loudon, R.: The Quantum Theory of Light, 3rd edn. Clarendon, Oxford (2000) 30. Meystre, P., Sargent III, M.: Elements of Quantum Optics, 2nd edn. Springer, Berlin, Heidelberg (1991) 31. McClung, F.J., Hellwarth, R.W.: J. Appl. Phys. 33, 828 (1962) 32. Haus, H.A.: IEEE J. Sel. Top. Quantum. Electron. 6, 1173 (2000) 33. Hall, J.L.: Rev. Mod. Phys. 78, 1279 (2006) 34. Hänsch, T.W.: Rev. Mod. Phys. 78, 1297 (2006) 35. Cundiff, S.T., Ye, J.: Rev. Mod. Phys. 75, 325 (2003) 36. Spence, D.E., Kean, P.N., Sibbett, W.: Opt. Lett. 16, 42 (1991) 37. Brabec, T., Spielmann, Ch , Curley, P.F., Krausz, F.: Opt. Lett. 17, 1292 (1992) 38. Yefet, S., Pe’er, A.: Appl. Sci. 3, 694 (2013) 39. Fork, R.L., Greene, B.I., Shank, C.V.: Appl. Phys. Lett. 38, 671 (1981) 40. Treacy, E.B.: IEEE J. Quantum Electron. 5, 454 (1969) 41. Grischkowsky, D., Balant, A.C.: Appl. Phys. Lett. 41, 1 (1982) 42. Mollenauer, L.F., Stolen, R.H., Gordon, J.P., Tomlinson, W.J.: Opt. Lett. 8, 289 (1983) 43. Fujimoto, J.G., Weiner, A.M., Ippen, E.P.: Appl. Phys. Lett. 44, 832 (1984) 44. Shank, C.V.: Science 233, 1276 (1986). and references therein 45. Strickland, D., Mourou, G.: Opt. Commun. 56, 219 (1985) 46. Maine, P., Strickland, D., Bado, P., Pessot, M., Mourou, G.: IEEE J. Quantum Electron. 24, 398 (1988) 47. Mollenauer, L.F., Stolen, R.H.: Opt. Lett. 9, 13 (1984) 48. Madey, J.: J. Appl. Phys. 42, 1906 (1971) 49. Deacon, D.A.G., Elias, L.R., Madey, J.M.J., Ramian, G.J., Schwettman, H.A., Smith, T.I.: Phys. Rev. Lett. 38, 892 (1977) 50. Pellegrini, C.: Eur. Phys. J. H 37, 659 (2012) 51. Huang, Z., Kim, K.-J.: Phys. Rev. Special Top. Accel. Beams 10, 034801 (2007) 52. Krausz, F., Ivanov, M.: Rev. Mod. Phys. 81, 163 (2009) 53. Gallmann, L., Cirelli, C., Keller, U.: Annu. Rev. Phys. Chem. 63, 447 (2012) 54. Popmintchev, T., et al.: Science 336, 1287 (2012) 55. Kulander, K.C., Schafer, K.J., Krause, J.L.: eds. In: Piraux, B., L’Huillier, A., Rza¸z˙ ewski, K.K. (eds.) Super-Intense Laser-Atom Physics. Plenum, New York (1993)
R. Menzel and P. W. Milonni 56. Corkum, P.B.: Phys. Rev. Lett. 71, 1994 (1993) 57. Lewenstein, M., Balcou, P., Ivanov, M.Y., L’Huillier, A., Corkum, P.B.: Phys. Rev. A 49, 2117 (1994) 58. Casperson, L.W.: Spontaneous pulsations in lasers. In: Harvey, J.D., Walls, D.F. (eds.) Laser Physics. Springer, Berlin, Heidelberg (1983) 59. Milonni, P.W., Shih, M.-L., Ackerhalt, J.R.: Chaos in Laser– Matter Interactions. World Scientific, Singapore (1987) 60. Abraham, N.B., Arecchi, F.T., Lugiato, L.A. (eds.): Instabilities and Chaos in Quantum Optics II. Plenum, New York (1988) 61. Lugiato, L., Prati, F., Brambilla, M.: Nonlinear Optical Systems. Cambridge University Press, Cambridge (2015) 62. Sciamanna, M., Shore, K.A.: Nat. Photon. 9, 151 (2015)
Ralf Menzel Ralf Menzel studied physics, received his PhD from the Humboldt University, Berlin, in 1978, and finished his Habilitation in Physical Chemistry at the Technical University, Berlin, in 1989. In 1994 he became a full Professor for Experimental Physics in Photonics at the University of Potsdam. His research interests are in the fields of lasers, nonlinear optics, measurement techniques for biological and medical applications, and quantum optics. Peter Milonni Peter Milonni received his PhD from the University of Rochester in 1974. He is currently a Laboratory Fellow and Guest Scientist at Los Alamos National Laboratory and a Research Professor at the University of Rochester. He previously had positions with the US Air Force, the Perkin-Elmer Corporation, and the University of Arkansas. His research interests include quantum optics and electrodynamics.
75
Types of Lasers Richard S. Quimby and Richard C. Powell
Contents 75.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
75.2 75.2.1 75.2.2 75.2.3 75.2.4 75.2.5
Single-Atom Transitions . Neutral Atom Gas Lasers . Ion Lasers . . . . . . . . . . Metal Vapor Lasers . . . . . Rare-Earth Ion Lasers . . . Transition Metal Ion Lasers
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75.3 75.3.1 75.3.2 75.3.3
Molecular Transitions . . . Molecular Vibrational Lasers Dye Lasers . . . . . . . . . . . Excimer Lasers . . . . . . . .
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75.4 75.4.1 75.4.2
Solid-State Transitions . . . . . . . . . . . . . . . . . . . 1090 Semiconductor Lasers . . . . . . . . . . . . . . . . . . . . 1090 Quantum Cascade Lasers . . . . . . . . . . . . . . . . . . 1091
75.5
Free Electron Lasers . . . . . . . . . . . . . . . . . . . . 1092
75.6 75.6.1 75.6.2 75.6.3 75.6.4
Nonlinear Optical Processes Raman Lasers . . . . . . . . . . Optical Parametric Oscillators Frequency Conversion . . . . . High Harmonic Generation . .
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has become an essential component of industries such as telecommunications, manufacturing, medicine, and photonics. The various types of lasers can be categorized in a number of different ways; for example, in terms of spectral range, temporal characteristics, pumping mechanism, or by the phase (solid, liquid, gas, plasma) of the laser gain medium. In keeping with a handbook on atomic, molecular and optical physics, the types of lasers will be organized here more fundamentally, according to the type of quantum states involved in the lasing transition. Lasing can arise, for example, from transitions between quantum states that are primarily associated with either a single atom, a single molecule, or an extended solid. But it can also arise from stimulated scattering and nonlinear parametric processes that do not require population inversion between the energy levels of a medium. Keywords
fiber laser solid state laser color center laser emission pulsed lasers
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094
Abstract
The availability of coherent light sources has revolutionized atomic, molecular, and optical science. Since its invention in 1960, the laser has become the basic tool for atomic and molecular spectroscopy and for elucidating fundamental properties of optics and optical interactions with matter. The unique properties of laser light have found many practical applications as well, and the laser R. S. Quimby () Dept. of Physics, Worcester Polytechnic Institute Worcester, MA, USA e-mail: [email protected] R. C. Powell Optical Sciences Center, University of Arizona Tuscon, AZ, USA e-mail: [email protected]
75.1
Introduction
Various types of lasers are listed in Table 75.1. The list is not comprehensive but illustrates the variety and properties of the most important lasers that are currently of interest for applications. Included in the table are examples of specific lasers of a given type, the type of lasing medium, and the typical spectral range. Also indicated is the nature of the transition: between electronic states, between vibrational states, a combination of electronic and vibrations, or not involving transitions between discrete energy levels of the medium (nonlinear processes). In this chapter, the emphasis will be on understanding the basic operational principles of and differences between the laser types, with details of a few representative lasers as space allows. For a more comprehensive discussion of different laser types, the reader is
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_75
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Table 75.1 Overview of laser types, organized by the nature of the lasing transition Laser type Single-atom transitions Neutral atom gas lasers Ion gas lasers Metal vapor lasers Rare-earth ion lasers
Transition metal ion lasers Molecular transitions Molecular vibrational lasers (IR) Dye lasers (visible) Excimer lasers (UV) Solid-state transitions Semiconductor lasers
Quantum cascade lasers Free electron lasers Nonlinear optical processes Raman lasers Optical parametric oscillators Frequency conversion High harmonic generation
Typical lasers
Host
Transition type
Spectral range
He-Ne ArC Cu vapor Er:YAG Yb:silica glass fiber Yb thin disk lasers Ti:sapphire Cr:ZnSe/S
Low-pressure gas Low-pressure gas High-temp metal Dielectric crystal glass fiber thin disk crystal Dielectric crystal semiconductor
Elec Elec Elec Elec
Vis/IR Vis/IR UV/vis UV/vis/IR
Elec-vib
IR tunable
CO2 Rhodamine 6G ArF
Gas mixture Liquid (ethanol) Gas discharge
Vib Elec-vib Elec-vib
IR Vis UV
Double heterostructure quantum well VCSEL QCL
III/V semiconductor II/VI semiconductor
E-h interband transition
Near IR-vis
Synchrotron facility linac facility Phosphosilicate fiber Yb fiber laser pump OPA & OPO Frequency doubling frequency difference generation Table-top XUV soft X-ray source
III/V semiconductor Intraband transition II/VI semiconductor High-energy electron Radiation from beam in vacuum electron oscillations
IR
Glass optical fiber
Stimulated scattering
Near IR-vis
Transparent dielectric crystal Transparent dielectric crystal Noble gas
Parametric process
Near-mid IR
IR-vis-UV-Xray
Parametric process
Mid IR THz High field $ atom interaction Extreme UV soft X-ray
referred to textbooks on lasers [1–5], general handbooks on suppressed relative to emission. This makes “lasing without lasers [6–11], books on specialized laser types [12–17], and inversion” [19] possible (see also Sect. 78.2.7). authoritative websites [18]. For our purpose, transitions will be considered singleatom type even if the atoms are not completely isolated or in the gaseous or plasma phase. Certain atoms, such as the rare-earth or transition metals, can occur as ionic impurities 75.2 Single-Atom Transitions in an insulating dielectric solid, and they have optical transitions that are between energy levels of one atom. The host The simplest type of laser transition, and the one usually con- solid does interact with the impurity atom, perturbing its ensidered when explaining or learning about laser theory, is ergy levels to various degrees, but nonetheless the transition a transition between energy levels associated with a single can be described as between these perturbed levels of a single atom. As discussed in the previous chapter, it is generally atom. necessary to have population inversion (more atoms in the upper level than in the lower level) in order to have amplification of light in the medium, and thus lasing. There are 75.2.1 Neutral Atom Gas Lasers exceptions to this general rule, however. For example, atomic transitions can occur between two energy level manifolds, The signature laser of this type is the He-Ne laser, one of with a quasi-thermal distribution of population within each the first lasers to be developed in the early 1960s, and still manifold. This can lead to unequal absorption and emission in use today. Although the lasing medium is a mixture of cross sections at the lasing wavelength, relaxing the re- two gases, only Ne is involved in the lasing transition, He quirement of population inversion. Another exception occurs serving only to populate the 3s levels of Ne by collisional when two closely spaced lower levels are put into a coherent energy transfer. Lasing occurs on Ne transitions from the exsuperposition state, such that absorption to a higher level is cited 3s levels to the lower 3p levels in the infrared (3.39 µm),
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Types of Lasers
and also to the lower 2p levels in the visible (543.3, 594.1, and 632.8 nm) and infrared (1.15, 1.529 µm). For operation at the visible wavelengths, it is necessary to suppress lasing on the much stronger 3.39 µm transition, for example by using end mirrors with less than 0.5% reflectivity at that wavelength. The gas mixture ( 10% Ne) is excited in a low pressure (4–7 mbar) discharge tube of small diameter (0.5– 2 mm), with 1–5 kV and 4–11 mA. The tube diameter is limited because of the need for wall collisions to return the Ne to the ground state, and this limits the practical output power to tens of mW. Although not scalable to high power, He-Ne lasers are still relevant for applications such as interferometry, where long coherence lengths are needed. For typical tube lengths of 20–50 cm, there is simultaneous lasing of two to three longitudinal modes, and this limits the coherence length to 20 cm. However, operation on a single longitudinal mode and frequency stabilization can result in coherence lengths greater than 1 km. Although diode lasers have replaced He-Ne lasers for many applications, these long coherence lengths along with the nearly diffraction-limited output of the He-Ne laser remain distinct advantages, and it continues to be relevant for interferometry and educational instruction.
75.2.2
Ion Lasers
Ion lasers operate on transitions between energy levels of ionized noble gas atoms such as Ar or Kr. The noble gas (argon, for example) is excited in a high voltage (100–400 V) and high current (5–30 A) electrical discharge. The upper laser level is excited by a two-step process: first, the neutral argon atom is ionized by electron collisions ( 16 eV), and then the ArC ions are raised to an excited electronic state ( 20 eV) by further electron collisions. Lasing occurs on the ArC 4p ! 4s transition ( 2 eV). The gas pressure is carefully adjusted to give the greatest possible density of argon atoms, while at the same time allowing the electron mean free path to be great enough to give the electrons sufficient kinetic energy when they collide with Ar atoms. Even with optimum tube pressure, the gain coefficients are small, and this requires long cavity lengths, very high mirror reflectivities, and low cavity losses. To minimize reflection losses at the plasma tube windows, they are tilted at Brewster’s angle (Brewster windows) and must be kept very clean. The spectral output of argon lasers is a series of green and blue lines, most prominently at 514.5 and 488 nm, which can be selected by an intracavity prism. Krypton lasers (5p ! 5s transition) emit in the green and red, most prominently at 521, 568, and 647 nm. The wallplug efficiency is quite low ( 0:1%), and for output powers greater than 1 W, water cooling is needed to extract the waste heat. Ion lasers were at one time the most important source of high-power laser light in the visible but have decreased in
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popularity due to competition from frequency-doubled solidstate lasers such as Nd:YAG.
75.2.3
Metal Vapor Lasers
These lasers are similar in operation to the neutral atom gas or ion lasers discussed above, except that the excitation process begins with vaporizing a solid or liquid to produce the gas for lasing, then followed by normal electrical discharge pumping. A typical ion laser of this class is the helium– cadmium laser. For the excitation processes, metallic Cd is evaporated and mixed with He. Then a DC electric discharge excites the He atoms and ionizes the Cd. The excited He atoms transfer their energy to the Cd ions, and lasing occurs between electronic levels of the Cd ion. The main emission lines are 441.6 and 325 nm, with CW output powers in the range 10–150 mW. The wall-plug efficiency is low, between 0.002% and 0.02%. A typical neutral atom laser of this type is the copper vapor laser, with emission lines at 510.5 and 578.2 nm. These lasers operate only in the pulsed mode since the lower levels of the laser transitions are metastable and the transition is self-terminating. These lasers have high gain (10–30%/cm), and high wall-plug efficiency ( 1:0%), but require high temperatures to create the vapor.
75.2.4
Rare-Earth Ion Lasers
This section and the next (Sect. 75.2.5) discuss lasers in which the gain medium is a dielectric solid (crystal or glass), doped with metal impurity ions to form an optically active center. The solid host itself is not usually optically active, and this type of laser has come to be referred to as a “solidstate laser”. This is in contrast to the case of semiconductor lasers (Sect. 75.4), in which the solid itself is the optically active material. Although semiconductor lasers might be considered a kind of “solid-state laser” (they are, after all, a laser that is in the solid state), conventional terminology is to reserve the term “solid-state laser” for those based on impurity-doped dielectric solids. When elements of the lanthanide series (Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb) are doped as impurities into an dielectric crystal or glass, they typically become triply ionized, e.g., Er3C . These rare-earth ions have atomic-like transitions between levels of the same .4f/n configuration, where n ranges from 1 for Ce to 13 for Yb [20]. The host matrix perturbs these levels only weakly, due to the shielding of the optically active inner 4f shell by the filled outer 5s and 5p shells. The intermediate coupling scheme applies here, where spin–orbit interactions are small but not negligible compared to electrostatic interactions [21]. In this
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case, the LS Russell–Saunders states with the same J are mixed, forming states that are linear combinations of different LS with a common value of J . The notation used to identify a given level is that of the principal LSJ component (i.e., 2SC1 LJ ). The local electric field from surrounding atoms in the host splits the (2J C 1) MJ levels of each LSJ state through the Stark effect, resulting in a manifold of closely spaced sublevels [22]. For even values of n, there are 2J C 1 Stark levels, but for odd n, there are .2J C 1/=2 levels, due to Kramer’s degeneracy. The three types of splittings of the 4f states differ from each other by roughly an order of magnitude: electrostatic ( 104 cm1 ), spin–orbit ( 103 cm1 ), and Stark effect ( 102 cm1 ). Radiative .4f/n ! .4f/n transitions are not electric dipole allowed by parity, and although magnetic dipole and quadruple transitions are allowed, they are very weak. However, the local electric field mixes some of the higher 5d levels into the nominal 4f states, allowing so-called “forced electric dipole” transitions with an oscillator strength of 106 [23]. Typical absorption and emission cross sections are on the order of 1020 cm2 , and with rare-earth ion concentrations of 1020 cm2 ( 1% doping), the gain coefficient can be as high as 1 cm1 . This is intermediate between the very low gain typical of gas lasers and the very high gain of semiconductor lasers. There are many varieties of rare-earth-doped lasers, and they have become one of the most important lasers for a variety of applications. The main characteristics that distinguish the different types are the nature of the optical pump, the type of solid host, the geometry of the host, and, of course, the selection of a particular rare-earth ion. In what follows, each of these aspects will be examined to illustrate the large and still growing variety of rare-earth-doped lasers. Rare-earth doped lasers are excited optically and were originally pumped by lamps. However, the brightness of lamps is limited, and their spectral output does not couple efficiently with the narrow absorption lines of the rare earth. These issues were overcome with the advent of efficient and powerful diode lasers for pumping (Sect. 75.4.1), and today diode laser pumping of solid-state lasers has largely replaced lamp pumping [24]. The first host materials used in solid-state lasers were oxide crystals such as Al2 O3 (sapphire) and Y3 Al5 O12 (YAG). Fluoride crystals such as LiYF3 (YLF) were found to have desirable properties, such as lower nonradiative relaxation between levels, which allows transitions further into the infrared. Glass hosts were tried early on [25], but the low thermal conductivity presented thermal management problems. It was not until the advent of fiber lasers, with a very small diameter that allows efficient cooling, that glass hosts began to compete seriously with crystal hosts. Just as with crystals, it was found that fluoride-based glasses such as ZBLAN [26] exhibit lower nonradiative relaxation, and hence the potential
R. S. Quimby and R. C. Powell
for lasing further into the infrared. A major difference between crystal and glass hosts is the smoothness of the optical spectra. Crystals provide similar crystal field sites for every dopant ion, while the disorder associated with glass structure gives many different types of local crystal field sites for the dopant ions, leading to inhomogeneous broadening of the spectral lines. As a result, absorption and emission spectra for rare earths in crystals show a great deal of structure [22], while in glasses, the spectra are significantly smoothed out, which is an advantage for tunable lasers. More recently, there has been a developing interest in glass ceramics [27], where active ions embedded in nanocrystals see essentially a crystalline environment, while on a macroscopic scale, the material is amorphous and can be molded and drawn into shapes like a glass. This topic is under active development. The classic geometry of the gain medium is a cylindrical rod, typically 5–15 cm long and a few mm in diameter. This is still a common geometry, although for higher powers, rectangular slabs with Brewster’s angle ends are sometimes used [14]. At high powers where thermal lensing limits the output power of traditional rod or slab geometries, there is increasing interest in thin-disk lasers [14], which have the active medium in the form of a thin disk mounted directly on a reflecting element and heat sink. Both pumping and lasing occur perpendicularly to the plane of the disk, which is efficiently cooled with the temperature gradient in the direction of lasing, avoiding thermal lensing issues. Fiber lasers take the opposite geometric approach – instead of being short with large diameter like a disk, they are long and have a small diameter [28, 29]. In both cases, the surface-to-volume ratio is large, leading to efficient cooling. Comparing the two approaches, fiber lasers are better suited for high CW power, whereas thin-disk lasers are better for pulses with high peak power. In a fiber laser, the fiber core is doped with the active ion, while the cladding is undoped. It is usually end-pumped with another laser, most often with a diode laser but sometimes with another fiber laser. For very high power operation, the diode laser sources that are available do not provide a Gaussian beam that can be coupled into the core of a single-mode fiber. In this case, a doubleclad fiber is used, which has a second inner cladding into which the pump light is launched and guided. The fraction of pump light absorbed per unit length is smaller than in core pumping, but for a sufficiently long fiber, most of the pump light can be absorbed. This amounts to a brightness converter, in which low brightness pump light is converted into high brightness lasing light. The output power in a fiber laser is ultimately limited by nonlinear optical effects, due to the confinement of significant power in the very small mode area of the core. This led to the development of large-mode-area (LMA) fibers [30], which reduce the light intensity in the core. Another approach is hollow core fibers, a special class of photonic
75
Types of Lasers
Fig. 75.1 Low-lying energy levels of selected rare-earth 3+ ions, showing the wavelength of important lasing transitions in µm
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1/λ (103 cm –1) 3
P2 1
3
P0
20
G4
4
F7/2
2
H11/2 S3/2
5
S2
4
1
D2 2
H11/2
5
F5
4
F9/2
3
F2 F3
3 5
I4
4
F5/2
I6
1.06 0.94
F4 F3
H9/2+6F11/2
6
4
3
I15/2
F2 3 H6
2.9
H11/2
H5
4
I13/2
3
F4
5
I7
1.0 μm
4
I13/2
3
H5
4
I11/2
H4
H13/2
crystal fibers in which the light propagates mostly in air, avoiding nonlinear effects in the glass [30]. The guiding of light in such fibers is due to the photonic bandgap effect, rather than the index guiding characteristic of traditional optical fibers. Although optical fibers are typically made of glass, there is increasing interest in single-crystal fibers (SCF), especially for very high power ultrashort pulse amplification [31]. These can have diameters from 100 µm to as large as 1 mm, approaching the dimensions of a traditional rod but with guiding of the pump light much as in a traditional fiber amplifier. There are many possible laser transitions for the 12 optically active rare-earth ions, but only a few important examples can be mentioned here. A more comprehensive listing can be found elsewhere [11, 20, 23]. Figure 75.1 shows the more important lasing transitions in the visible (Pr), near-IR (Nd, Yb, Er), and mid-IR (Dy, Ho, Er, Tm). The development of efficient GaN diode lasers [32–34] at 440 nm has made practical the direct pumping of the Pr 3 P levels to give visible lasers in the blue (490 nm), green (540 nm), and red (600, 640 nm). Typical hosts for these lasers are fluorozirconate glass (ZBLAN) and the crystal LiYF4 (LYF). In the near-IR, the Nd:YAG laser has been and continues to be one of the most widely used solid-state lasers for scientific, industrial, and medical applications, operating most efficiently at 1064 nm, but also at 946, 1123, and 1319 nm. Originally pumped with lamps, it is now mostly pumped with high power diode lasers. When the 1064 beam is frequency
1.9
2.1 3.0 6
Dy
4
5
H15/2
I9/2
Nd
1.5
6
4
3
Pr
F5/2
3
2.7
6
3
0
2
5
3
0.490 0.540
2.3
I11/2
G4
0.600
4
H4
I5
1
0.640
3
5
F3/2
10
4
I9/2
4
Ho
Er
2
3
I15/2
I8
H6
Tm
F7/2
Yb
doubled (Sect. 75.6.3), the 532 nm output has replaced the 514.5 nm line of the more bulky and less efficient argon ion laser for many applications. The Nd:YAG laser is now, in turn, gradually being replaced by the more compact and efficient Yb fiber laser, which operates in the same 1 µm wavelength region. These Yb lasers have been scaled to the kW range and higher in recent years, pumped by increasingly efficient and powerful diode laser stacks [35]. Also important in the near-IR is the Er3C 4 I13=2 ! 4 I15=2 transition at 1.5–1.6 µm, which falls right at the wavelength of minimum attenuation in silica glass fibers. With Er doped in silica fiber, this transition forms the basis for the erbiumdoped fiber amplifier (EDFA), which is largely responsible for the explosive growth in telecommunications bandwidths in the past two decades [28, 36]. The EDFA can be pumped with diode lasers at 980 or 1480 nm, the latter having a higher potential efficiency due to a smaller quantum defect (difference between pump and lasing photon energy). In the case of 980 nm pumping, the fiber is sometimes codoped with Yb, which absorbs the pump more strongly and transfers the energy to Er nonradiatively. Output powers can be several to 100 W CW [11]. Er-doped crystals also operate with high power on this transition, for example, Er:YAG at 1645 nm. For operation in the 2 µm region, both Tm (1.9 µm) and Ho (2.1 µm) have transitions that terminate on the groundstate manifold (three-level systems). Tm has an especially large tuning range of up to 250 nm in a fiber and 300 nm in a crystal. Another unique advantage of Tm is the cross-
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relaxation that occurs at high (> 2%) Tm concentrations when the 3 H4 level is pumped at 790 nm [37]. In this process, the excited Tm3C exchanges energy nonradiatively with a neighboring Tm3C in the ground state, leaving both ions in the 3 F4 . The result is two ions in the upper laser level for a single pump photon absorbed – a quantum efficiency of 200%. Another mid-IR transition in Tm is the 3 H4 ! 3 H5 transition at 2.3 µm, which can also be pumped at 790 nm. In this case, however, the Tm concentration is kept below 2% because cross-relaxation would have the undesirable effect of depleting the upper laser level. The first excited state of Ho3C is a bit lower in energy than that of Tm3C , and its ground state transition is somewhat longer in wavelength, 2:1 µm. Unfortunately, Ho3C does not have a convenient wavelength for high power diode laser pumping, so Tm is often codoped to efficiently absorb the pump light at 790 nm. The excited Tm3C cross relaxes as described above, and then the Tm3 F4 level transfers its energy to a nearby Ho3C in the ground state, exciting it to the 5 I7 . This process can be efficient with proper adjustment of the Tm and Ho concentrations [38]. In the 3 µm spectral region, there are transitions in Dy, Ho, and Er that have proven successful in fiber and crystalline solid-state lasers. For these lasers, fluoride hosts are preferred due to decreased multiphonon quenching of the laser transition and increased host transparency. The Dy (6 H13=2 ! 6 H15=2 ) transition is three-level, whereas the Ho (5 I6 ! 5 I7 ) and Er (4 I11=2 ! 4 I13=2 ) transitions are four-level. The four-level transitions have the disadvantage of a lower laser level that is metastable with a long (ms) lifetime, and this limits the output power due to accumulation of population in the excited states. Three techniques that have been used to reduce this problem are codoping with a quencher ion, energy transfer upconversion (ETU), and cascade lasing. An example of the first method is codoping Pr into a Hodoped host, which results in energy transfer from the lower laser level of Ho (5 I7 ) to a nearby Pr ion, recycling the Ho ion to the ground state [39]. Similarly, codoping of Pr into an Erdoped host depopulates the lower laser level of Er (4 I13=2 ). An important example of the second method (ETU) occurs in highly doped Er materials [40], whereby two neighboring Er ions both in the lower laser level (4 I13=2 ) exchange energy, leaving one in the ground state (4 I15=2 ) and the other in the excited state (4 I9=2 ). The (4 I9=2 ) then relaxes nonradiatively to the (4 I11=2 ). This has a double benefit: the lower laser level is quenched while the upper level population is simultaneously increased. A similar ETU process can occur in highly doped Ho systems [41]. In the third method, simultaneous lasing (cascade lasing) is established on, for example, both the Ho (5 I6 ! 5 I7 ) transition at 2.9 µm, and the (5 I7 ! 5 I8 ) transition at 2.1 µm [42]. For Er, cascade lasing can be established on the (4 I11=2 ! 4 I13=2 ) and (4 I13=2 ! 4 I15=2 ) transitions [43]. This method is especially advantageous for high-power op-
R. S. Quimby and R. C. Powell
eration, because it not only solves the population bottleneck problem, but also reduces the heat load and cooling requirements. In contrast to Ho and Er, a Dy 3 µm laser does not suffer from population bottlenecking. As a three-level system, however, it requires high pump intensity to reach lasing threshold, and, therefore, the fiber geometry with fluoride glass is most efficient.
75.2.5
Transition Metal Ion Lasers
When transition metal elements of the fourth row of the periodic table (Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) are doped as impurities into an dielectric crystal or glass, they can take on various oxidation states, such as 2+, 3+, or 4+. In the oxidation process, they first lose the two outer 4s electrons and then the 3d electrons according to the oxidation state. Example electron configurations for the optically active electrons are (3d)1 for Ti3C , and (3d)3 for Cr3C . Unlike the case of the rare-earth ions, there is little shielding of these optically active 3d states from external electric fields, and the energy level positions are strongly influenced by the host. The quantum states become characterized by the symmetry of the ion’s crystallographic site, rather than by the atomic states of an isolated ion [44, 45]. This strong ion–host interaction leads to broadly tunable optical transitions, best understood in terms of a “configurational coordinate diagram” [21]. In this picture, the collective motion of all nearby host atoms is represented by a single coordinate Q, and the Born–Oppenheimer approximation is used to separate the degrees of freedom into electronic (fast) and nuclear (slow) parts. The energies of the ion’s different electronic levels are calculated assuming a stationary Q and then plotted as a function of Q as shown in Fig. 75.2. For each electronic state, there is a minimum energy at some value of Q, about which the system oscillates with a frequency given by the curvature of E.Q/. This gives rise to vibrational sublevels superimposed on the electronic states, with the total quantum state now characterized by the electronic state and an integer number of vibrational quanta. Optical transitions are nearly vertical in this diagram, because the electron motion is much faster than the nuclear motion during the transition. During the optical transition, the atoms of the host can be thought of as “frozen in position” (the Franck–Condon principle). When the minima for the two electronic states occur at different values of Q, there is a shift to lower energy (longer wavelength) between absorption and emission, known as the “Stokes shift”. Both the absorption and emission spectra are then broadened and shifted as shown in Fig. 75.3. When the two minima occur at nearly the same Q, however, as is the case for rare-earth 4f transitions and certain pairs of transition metal states, the op-
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Types of Lasers
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Electronic energy
4
T1b (t2 e2)
E/B 80 Excited state 2
D 60
Absorption Ground state
2
F 40
A1 (t2 e 2)
4
T1a (t2 e 2)
4
T2 (t2 e 2) T2 (t2 3)
2
2 D H,2P 2 G 20 4 P
2
2
Emission
2
2
T1 (t2 3) E (t2 3)
Configurational coordinate Q 4
F
Fig. 75.2 Configuration coordinate diagram for strong electron-lattice interaction
4
0
1
2
3
4 Dq/B
A2 (t2 3)
Fig. 75.4 Energy E=B of transition metal ion states versus crystal field strength Dq=B [11]
Cross section
Stokes shift
Absorption
0
Emission
Wavelength
Fig. 75.3 Stokes shift between absorption and emission
tical transition is mostly purely electronic (“zero-phonon”), with a very weak phonon sideband. Taking Cr3C as an example, Fig. 75.4 shows how the energy E=B of different atomic orbitals in a site having octahedral symmetry vary with the crystal field strength Dq=B, relative to the ground state. This is known as a Tanabe–Sugano diagram [21]. For certain states (2 E, 2 T), the energy difference with the ground state changes little as the field strength changes, indicating weak ion–lattice coupling. At sufficiently high field strength, these are the lowest-lying excited states, leading to sharp spectral absorption and emission for transitions to and from the ground state. This explains why the ruby laser (Cr3C :Al2 O3 ), the first laser to be demonstrated in 1960, operates at a single wavelength of 694 nm, with no tunability. Other crystalline hosts for Cr3C provide a “low field” envi-
ronment, such that the 4 T2 level lies lower in energy than the 2 E. The first tunable laser based on Cr3C was Cr3C :BeAl2 O4 (alexandrite), which is tunable from 700–820 nm with good efficiency and thermal properties [46]. Other hosts for Cr3C that are similarly tunable have been demonstrated, such as LiCaAlF6 and LiSrAlF6 , but these have poorer crystal properties and lower power output than alexandrite [47]. The really significant advance in tunable lasers after alexandrite was the introduction of the Ti-sapphire laser (Ti3C :Al2 O3 ) in 1982 [10, 48]. The (3d)1 configuration for Ti3C leads to a very simple energy level diagram, with a single absorption band from 450–550 nm and tuning range from 670–1100 nm. This wide tunability, along with the lack of excited-state absorption, high slope efficiency ( 60%), high crystal thermal conductivity, scalability to high power, and good mechanical and chemical hardness, have made the Tisapphire laser one of the most important and commercially successful of all solid-state lasers. The same broad emission bandwidth that gives wide tunability also allows ultrashort mode-locked pulses, and pulse widths of 5 fs have been achieved. Pumping was originally with an Ar ion laser in the blue/green, and then with frequency-doubled Nd:YAG lasers at 532 nm. Recent progress in high-power blue diode lasers based on GaN (Sect. 75.4.1) now allows direct diode laser pumping [34], which permits a much more compact and efficient Ti-sapphire laser. Other valence states of Cr are possible for tunable solidstate lasers, in particular Cr4C and Cr2C . Cr4C has the (3d)2 configuration, and the best hosts are found to be YAG (Y3 Al5 O12 ) with 1350 < < 1550 nm, and Mg2 SiO4 , with 1200 < < 1350 nm. The main drawbacks of Cr4C systems
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are excited-state absorption (ESA) at the lasing wavelength and nonradiative relaxation that reduces the quantum efficiency. Furthermore, careful material preparation is needed to make sure that only the Cr4C oxidation state is present and not Cr3C [49]. More promising is the divalent state Cr2C, most notably Cr2C :ZnSe, which has a lasing transition tunable from 2.0–3.1 µm [50]. The many advantages of this system include high absorption and emission cross sections ( 1018 cm2 ) due to the tetrahedral site symmetry, near unity quantum efficiency due to the low phonon energy of the chalcogenide host, and no significant ESA because all relevant ESA transitions are spin-forbidden. For all these reasons, the Cr2C :ZnSe laser has come to be called the “Ti-sapphire laser of the mid-IR”, and has been commercialized [35]. A rather new type of impurity-doped solid-state laser is the bismuth-doped fiber laser. Although bismuth is not actually a transition metal ion, it is discussed here because its optical properties in the near-IR, like those of transition metal ions, depend strongly on the host. The actual nature of the Bi-related near-IR emitting centers is not yet understood, despite extensive research. The emitting center may be not a single ion, but rather clusters of ions of some valence (Bi5C , Bi3C , Bi2C , and BiC ), perhaps in association with an oxygen vacancy. Despite this current lack of understanding, such fiber lasers were first demonstrated in the range 1140–1215 nm using Bi-doped aluminosilicate fibers, and subsequent work has extended the range to include 1270–1800 nm using newly developed phosphogermanosilicate, germanosilicate, and silica fibers [51, 52]. Although these are not currently as efficient as rare-earth-doped fiber lasers, a better understanding of the nature of the optical centers may allow improved efficiency.
75.3
Molecular Transitions
This section considers lasers involving transitions between the quantum states associated with a molecule. Vibrational transitions in a gas (e.g., the carbon dioxide laser) are discussed in Sect. 75.3.1, electronic-vibrational transitions in a liquid (e.g., dye lasers) in Sect. 75.3.2, and electronicvibrational transitions in a gas (e.g., excimer lasers) in Sect. 75.3.3.
75.3.1
Molecular Vibrational Lasers
The flagship laser of this type is the carbon dioxide (CO2 ) laser, operating on transitions between different vibrational states of the CO2 molecule [14]. It was developed at Bell Laboratories by Patel in 1964, and while the basic operational principles have changed little over the years, the applications have evolved as newer types of lasers have been developed. Once the go-to laser for cutting, welding,
a
O
C
O
1 Symmetric stretch mode
b
2 Bending mode
Lase 10.6 μm
Rotational sublevels
(100)
(001) Lase 9.6 μm
3 Asymmetric stretch mode 1
Collisional energy transfer
(020) (010)
(000)
0 1
2 CO2
3 N2
Fig. 75.5 Vibrational modes (a) and laser transitions (b) in a CO2 laser [4]
marking, surface treatments, as well as spectroscopy in the 9.2–11 µm region, it now faces competition with high-power Yb fiber lasers in the 1 µm region, and quantum cascade lasers (Sect. 75.4.1) in the 2–12 µm region. However, the CO2 laser still remains relevant for certain applications [53] because of its unique combination of longer wavelengths ( 10 µm), high power (kW CW), good beam quality (often diffraction limited), and spectral purity (linewidth < 1 kHz). CO2 is a linear molecule and has three vibrational modes, as indicated in Fig. 75.5. Transitions from the asymmetric stretch mode to the symmetric stretch or bending modes give rise to the two principle laser transitions around 10.6 and 9.6 µm, each having a number of closely spaced ( 10 GHz) rotational sidebands. Rotational states with quantum number J have energies / J.J C 1/, and vibrational-rotational transitions obey the selection rule Jfinal D Jinitial ˙ 1, with the result that the rotational sideband energies are evenly spaced in frequency. The so-called P branch has Jfinal D Jinitial C 1, and the R branch has Jfinal D Jinitial 1 [1, 54]. Tuning the laser to discrete lines in the range 9.2–11 µm region is possible using a diffraction grating in place of the end mirror. With no wavelength selection, lasing occurs on the 10.6 µm line on the 10P branch with J 20. Since the thermal population of the different rotational states depends on gas temperature, cooling of the gas is needed for stable high-power operation. Excitation of the CO2 vibrational levels occurs in a highvoltage ( 15 kV/m) gas discharge tube, either sealed or with flowing gases. The gas mixture contains CO2 , N2 , and He, with approximate ratios 1, 2, and 8, respectively. The N2 serves to selectively enhance excitation of the CO2 upper laser level by collisional energy transfer, while the He helps depopulate the lower CO2 laser level and also cool the gas due to its high thermal conductivity. Other gases that are
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sometimes added include Xe (few %), which absorbs fast electrons in the discharge that would otherwise disassociate the CO2 molecules. The wall-plug efficiency is rather high ( 10–15%) for a gas laser, and output power scales with increasing tube length ( 80 W/m). Tube diameters are limited to 5–10 mm due to limits on heat extraction. Other vibrational transition lasers include the carbon monoxide (CO) laser operating at 5–6.5 µm, and the nitrous oxide (NO2 ) laser operating at 10–11 µm, although these are much less commonly used. Some vibrational lasers utilize a chemical reaction to achieve an excited vibrational state, and are termed chemical lasers. Examples are the hydrogen fluoride (HF) laser operating at 2.6–3.3 µm, and the deuterium fluoride (DF) laser operating at 3.5–4.2 µm.
75.3.2
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not a problem, because the triplet states can relax back to the ground state between pulses. For CW operation, however, it is necessary to circulate the dye and move it rapidly through a small nozzle in a thin jet of flowing liquid across the optically pumped area. This limits the ability to make the laser compact and robust. Eventually, the dye degrades and needs to be replaced for both continuous wave (CW) and pulsed operation. Most significantly, perhaps, are the safety and convenience issues dealing with messy and, in many cases, carcinogenic dyes. Because of these drawbacks, newer options such as frequency-doubled solid-state lasers or fiber lasers are now often seen as more desirable for applications once served by the dye laser. Dye lasers do, however, continue to serve in niche applications such as laser isotope separation [59] and laser guide stars at large terrestrial telescopes [60].
Dye Lasers 75.3.3 Excimer Lasers
Dye molecules consist of a hydrocarbon backbone chain terminated by a more complicated structure on each end. The electrons in the conjugate double bonds give broad and efficient absorption and emission spectra in the visible, making them useful for lasers as well as decorative coloration. Developed by Sorokin and Lankard in 1966, dye lasers became very popular during the next three decades as one of the few sources of widely tunable visible and near-IR laser light [55]. The optically active electrons in the conjugate double bonds are strongly coupled to vibrational modes of the molecule, and the configurational coordinate diagram of Fig. 75.2 can be used to understand the large Stokes shift between absorption and emission, as previously discussed for transition metal ions (Sect. 75.2.5). A typical tuning range for one particular dye is 50–60 nm, and by sequentially choosing different dyes it is possible to cover the range 300–1200 nm [56]. The wide emission bandwidth makes dye lasers attractive for mode-locking, and they have played an important role in developing ultrashort pulse technology, first in the ps and then the fs region [57]. Dye lasers have desirable properties in the frequency as well as time domain. With CW pumping (Ar ion laser or frequency-doubled Nd:YAG, for example), spectral linewidths in the tens of kHz can be obtained on a single longitudinal cavity mode [58]. To avoid spatial hole burning, which would otherwise cause multimode output, an etalon or a ring laser configuration can be used. In pulsed operation (e.g. flashlamp, N2 laser, excimer laser) the linewidth is longer, typically hundreds of MHz. Despite the dye laser’s desirable properties, it does have some drawbacks that have decreased its popularity in the past two decades. One issue is the need to avoid triplet quenching, in which an excited-state population accumulates in the lowest triplet spin level, followed by excited-state absorption, which quenches laser action [55]. In pulsed operation, this is
The excimer laser was developed in the 1970s and operates in the UV region in a pulsed mode [61]. The lasing transition is between electronic energy states of an “excited dimer” molecule, or excimer, which has the interesting property that bound states only exist when the molecule is in the excited state. When the molecule returns to the ground state, it dissociates, breaking up into two separate atoms. Taking KrF as a typical example, the lasing process is illustrated in Fig. 75.6, which shows the electronic energy of a KrF “molecule” as a function of the separation of the two nuclei. This is similar in concept to the configurational coordinate diagram of Fig. 75.2, where Q is now simply the internuclear separation R. When an atom of Kr and F are brought together, each in their ground state, the energy of the system increases monotonically due to Coulomb repulsion, and the “molecule” is unbound. This is characteristic of the noble gases (He, Ne, Ar, Kr, and Xe), which are generMolecule energy (eV)
Bound excited state
(KF)*
5 Lasing Unbound ground state 0
Kr + F 2.3
Nuclear separation (10 –10 m)
Fig. 75.6 An excimer laser operates on a bound ! unbound transition [4]
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ally unreactive and not inclined to form stable molecules. If the KrF molecule is promoted to the next-highest electronic state, however (by electron bombardment in an electrical discharge, for example), then the energy curve has a minimum at R D 0:23 nm. This gives rise to a bound state at this value of R, which remains stable as long as the molecule remains in the excited state ( 10 ns for KrF). During this excitedstate lifetime, lasing can occur on the transition from the bound excited state to the unbound ground state. Since there is no stable population in the lower laser level, this is an example of a perfect four-level system. In KrF, lasing occurs at a wavelength of 248 nm. Other noble-gas–halide pairs operate in the same fashion, for example XeCl at 309 nm, XeF at 351 nm, and ArF at 193 nm. Excimer lasers have moderately high efficiencies ( 1– 4%), short pulse duration ( 10 ns), high pulse energies (0.1–1 J), and can be scaled up to high average power (100 W). They have numerous applications, including photolithography [62], materials processing, and ophthalmology (LASIK – laser-assisted in situ keratomileusis). The short wavelengths are strongly absorbed by most materials, a distinct advantage for precision cutting. The principal disadvantage of these lasers is the need to work with highly reactive gases such as fluorine, which requires proper ventilation and safety precautions. Typically, the gases are circulated through the electrical discharge chamber to maintain the purity of the reactants. A high concentration of helium (about 2 atm pressure) is added as a buffer gas to facilitate the reactions between noble and halide gases.
75.4
Solid-State Transitions
This section considers lasers based on transitions between extended, band-like states in a solid. These can be divided into two categories: semiconductor lasers, in which the transitions occur between conduction and valence band, and quantum cascade lasers (QCL), in which transitions occur between electronic sublevels of the conduction band. Semiconductor lasers are typically pumped by an electric current, in which case, they are also referred to as diode lasers. When the semiconductor is pumped optically (the optically pumped semiconductor laser, OPSL), it is often configured in the thin-disk geometry for high-power operation, similar to that of the thin-disk solid-state laser [63]. Semiconductor lasers have high gain coefficients ( 103 cm1 ), which allows for compact photonic devices.
75.4.1
Semiconductor Lasers
The fundamental light emission process in semiconductor lasers arises from radiative recombination of electrons
and holes. In the simplest case, this occurs at the junction of an n-type material having excess electrons (e.g., n-Alx Ga1x As) and a p-type material having excess holes (e.g., p-Alx Ga1x As). At this p–n junction, there is an electric field that keeps the charges separated until they are “injected” by an applied voltage into the depletion region, where light emission occurs. A double-heterostructure (DH) laser [64, 65] improves the performance by introducing a middle layer of a different composition (e.g., GaAs) sandwiched between the original p and n-type materials. This provides several advantages over the single homojunction laser: (1) a well-defined recombination region (given by the thickness of the middle layer) that confines the carriers, reducing lasing threshold and improving temperature stability, (2) index guiding of the emitted light due to the index difference between middle and outer layers, and (3) reduced reabsorption of emitted light in the outer layers, due to the higher bandgap there. In choosing materials for these heterostructures, lattice matching at the junctions is important to minimize strain. Common III-V compound families that exhibit good lattice matching and have wavelengths of interest for photonic applications include Alx Ga1x Iny P1y (635– 670 nm), Alx Ga1x As (720–850 nm), Inx Ga1x As (900– 1100 nm), and Inx Ga1x Asy P1y (1000–1650 nm). Most recently there has been significant progress in developing high-power blue diode lasers, based on the Inx Ga1x N family [32], with operation at 380, 405, 450, and 470 nm. This permits efficient diode laser pumping of certain solid-state lasers, as discussed in Sects. 75.2.4 and 75.2.5. Quantum well (QW) structures are often used to improve the diode laser performance [66]. These are thin ( 10 nm) regions within the active gain region that have a smaller bandgap than the surrounding material, forming a potential well for charge carriers. The quantum states become localized in the well, resulting in a ladder of discrete energy levels that is similar to the familiar “particle in a box” of elementary quantum mechanics. The laser wavelength then depends not only on the host material but also on the width of the wells. This flexibility, along with improved device performance, has made QW structures common in commercially available laser diodes. There are a number of configurations and geometries for the semiconductor laser cavity [14, 65]. In an edge emitter, laser light propagates along the plane of the thin semiconductor layers and is emitted from the edge of the semiconductor chip. The simplest of these is the Fabry–Perot (FP) laser, in which Fresnel reflections from the cleaved semiconductor end facets suffice for reaching the lasing threshold, due to the very high gain coefficient. Another option is a feedback mirror external to the semiconductor, which allows insertion of an intracavity element for modulation or control of polarization. Single-frequency operation can be obtained in a distributed feedback (DFB) laser, which has an undulat-
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ing element of period incorporated throughout the gain region of the waveguide. The laser frequency is then given by the Bragg condition D c=.2n/, where n is the effective waveguide index of refraction. A variation of this is the distributed Bragg reflector (DBR) laser, where the Bragg reflection occurs only at the cavity ends, separate from the gain region. The gain region in an edge emitter is usually confined in the lateral dimension to a width 10 µm, known as the stripe geometry. This improves stability at high powers by confining both light and charge carriers to a smaller, well-defined area. For high-power applications, multiple stripes are incorporated in a single layer of a diode laser, and for even higher powers, a number of multistripe lasers are packaged in a linear array, in a diode laser bar. These bars can be stacked to provide a planar array with very high ( kW) power suitable for pumping high-power solid-state lasers [14]. In contrast to the edge emitters, there is an entirely different class of semiconductor laser in which the emission is perpendicular to the plane of the semiconductor chip [67]. These “vertical cavity surface emitting lasers” (VCSELs) consist of a small circular active region of thickness 1 µm and diameter 10 µm, with Bragg reflecting mirrors above and below formed by a series of dielectric layers with alternating refractive index. Since the fractional gain in one pass through the active region is only 1%, the Bragg reflectors must have a reflectivity R > 99:5%. The small cavity length L has the advantage of naturally promoting single-frequency operation, since the mode spacing c=.2nL/ is greater than the width of the gain spectrum. VCSELs have a number of other advantages, including a symmetrical emission pattern with diffraction-limited divergence, suitable for efficient coupling to single-mode optical fibers. They can be modulated up to 25 GHz due to the small surface area (and small capacitance) and formed into arrays that are individually modulable. CW lasers can have linewidths of 200 MHz. A practical advantage of VCSELs is that thousands of individual devices can be processed on a single wafer and tested at various stages of the manufacturing process, increasing reliability and yield. Fig. 75.7 Electron energy levels in a quantum cascade laser [11]
2.1
Closely related to the VCSEL is the vertical externalcavity surface-emitting laser (VECSEL), in which one of the Bragg reflecting elements is replaced by an external mirror in an extended cavity. This allows intracavity elements such as saturable absorbers for mode-locking or nonlinear crystals for second harmonic generation (SHG). VECSELs have potential as a compact high-power fs laser source [68].
75.4.2
In the quantum cascade Laser (QCL), light generation arises not from electron–hole recombination but rather from electron transitions from one quantum-well (QW) state to another, all within the conduction band [69]. The bandgap and valence band states play no role, in contrast to a conventional semiconductor laser. The gain medium consists of alternate layers of higher and lower bandgap material, forming a series of closely spaced QWs. A voltage is applied across the device, which causes the electron’s potential energy to vary linearly with position, as shown in Fig. 75.7. There is no p–n junction, so in a circuit, the device behaves much like a resistor. It is termed a unipolar circuit element, because its operation depends on only one type of charge carrier (i.e., the electron). Conventional laser diodes, in contrast, are bipolar circuit elements. As electrons flow through the QCL opposite to the electric field direction, they lose potential energy, not continuously, but in steps, as they jump from one QW to the next. Electrons in the lowest energy level of each QW see a potential barrier on both sides but can tunnel through with a probability that depends on the barrier height and width. In the lasing process, an electron first tunnels through a number of QWs in the “electron injection” region, as shown in Fig. 75.7, and then tunnels into a higher state of the laser active QW. Light is emitted as the electron relaxes to the lower level, followed by tunneling into the next injection region. This process is repeated many times (typically 20–25) during the transit of a single electron across the device, so that a single electron gives rise to many individual photons.
1 1 1
4.7
Electron
3
6.1
Laseractive
Quantum Cascade Lasers
4.5 4 3.8 3.4
Electron injection
Conduction band 1.5 nm hω
2 1
3 8.2nm
Laser active
AlInAs barriers hω
2 1 Electron injection
Photon emission Electron
Laseractive
GaInAs QWs
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The most important application of the QCL is as a light source in the mid-IR wavelength range (4–12 µm) [69]. For a conventional diode laser, this would require working with materials with a very small bandgap energy, which is problematic due to the high levels of thermally generated electrons and holes. Since the photon energy in the QCL is independent of the band gap, wide bandgap materials can be used in the layers for better performance. The lasing wavelength can be selected simply by choosing the proper QW width and spacing. In fact, it is possible to operate the QCL laser at multiple wavelengths simultaneously, by varying the QW energy levels across the device. QCLs have also been operated in the THz region [70]. A variation of the QCL concept is the interband cascade laser (ICL), which differs from the QCL in that the lasing transition is from QW states in the conduction band to QW states in the valence band. However, unlike a traditional interband transition, the transition in the ICL is between QW states in adjacent layers of the material. There must be sufficient overlap between the wavefunctions of the QW states in the two adjacent layers for this method to work. These lasers operate in the 3–6 µm range and can have advantages in terms of power dissipation [71].
75.5
Free Electron Lasers
The free electron laser (FEL) is very different from the other lasers discussed so far; the electrons emitting the radiation are not bound to any atom. A beam of high-energy (relativistic) electrons in a vacuum is sent through an undulator (or “wiggler”), where they are deflected by a transverse magnetic field that periodically reverses direction. The resulting oscillatory motion of the electrons creates a dipole radiation pattern in the electron rest frame, which becomes a beam of light colinear with the electron beam in the laboratory frame. The light beam has wavelength Š =.2 2 /, where is the undulation period, and is the relativistic factor. Varying the electron beam kinetic energy . 1/me c 2 changes the emission wavelength, and this leads to a very wide tunability, a hallmark of the FEL. By using different combinations of beam energy and undulation period, FELs have been demonstrated with wavelengths from the microwave region ( mm) to hard X-rays ( 0:1 nm) [72, 73]. Free electron lasers can be operated as either an amplifier or an oscillator, depending on the availability of reflecting optics at the laser wavelength. In the oscillator configuration, reflecting elements are placed at each end of the undulation region, and laser action proceeds much in the same way as in a traditional laser. Spontaneous emission provides the seed light for subsequent amplification, and very high gain is not required since the light makes multiple passes through the cavity. In the absence of mirrors, the device acts as a single-
pass amplifier, which amplifies either a seed pulse that is injected from one end or spontaneous emission. High gain is desirable in this case. With seed pulse injection, the system is referred to as a master oscillator power amplifier, or MOPA, which has the advantage of preserving the spectral and noise properties of the seed, while increasing its power in the undulator region. With no seed pulse, the output is actually superradiance rather than lasing, termed amplified spontaneous emission (ASE) in general laser theory or selfamplified spontaneous emission (SASE) in connection with FELs. This has less temporal coherence than laser oscillation but still has high spatial (transverse) coherence, and therefore high brightness (brilliance). The main drawback of the FEL is the need for a high-energy electron accelerator, found only at a few large facilities around the world [73]. Both linear accelerator (linac-based) and storage ring facilities are available with FEL sources, and users must schedule time at the facility that meets their requirements for wavelength, power, temporal structure of output, etc. There is increasing interest in an X-ray FEL (XFEL), a spectral region where few other laser sources are available [74]. Due to a lack of suitable mirrors, the XFEL operates in the single-pass amplifier mode. An important figure-of-merit is the single-pass gain, which can be as high as 107 for < 1 nm. Preliminary success has led to governmental support of a number of linac facilities, including the Linac Coherent Light Source (LCLS) in the US, the SPring– 8 Compact SASE Source (SACLA) in Japan, the European X-FEL in Germany, and the SwissFEL XFEL in Switzerland. All of these use a room-temperature LINAC except for the Swiss facility, which uses superconducting accelerator technology that enables orders of magnitude higher brilliance. The LCLS operates in the hard X-ray wavelength range 0.12–1 nm, with pulse energies of 1–3 mJ and pulse duration from 40–500 fs. The SACLA produces 0.06 nm hard X-ray pulses with pulse energies 0:5 mJ and pulse duration 2–10 fs.
75.6
Nonlinear Optical Processes
The final section of this chapter in concerned with various types of nonlinear optical processes that can be used to extend the wavelength range of available lasers. These processes arise from the optical behavior of materials at high light intensity and are discussed from a theoretical viewpoint in the next chapter. In these processes, the medium plays a different role than in a conventional laser, where the medium stores energy in electronic or vibrational energy states, with lasing occurring due to transitions between those states. Instead, nonlinear processes such as stimulated Raman scattering, optical parametric amplification and oscillation, second harmonic generation (SHG), and difference
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frequency generation (DFG) use the medium as an intermediary, by which two or more optical waves can interact with each other. High-harmonic generation is a special case of extreme nonlinearity, in which electrons are forced to emit radiation by an incident pump beam but in a manner quite different from that of a conventional laser.
75.6.1
Raman Lasers
In Raman scattering, a pump photon of frequency p is incident on a molecule or solid with vibrational modes of frequency fv . After scattering, the photon has frequency s D p fv , which is lower than that of the incident photon because some of the photon’s energy was converted into vibrational energy. This process occurs spontaneously with low efficiency, but at sufficiently high incident intensity, stimulated emission of the scattered light can occur, making the process more efficient. The gain (fractional increase per unit length) of the scattered “signal” light is proportional to the intensity of the incident “pump” light and is described by a third-order nonlinear process (Sect. 76.5.1). To make a Raman laser, the gain medium is placed between mirrors with high reflectivity at the signal frequency s and high transmission at the pump frequency p . The pump beam comes in along the cavity axis from a separate laser. Optical fiber is an ideal medium for a CW Raman laser because of its good light confinement (high intensity) over a long path length [75]. Typical frequency shifts for fluoride glass fiber are 550– 650 cm1 (16.5–19.5 THz), while for phosphosilicate glass, the shift is as high as 1330 cm1 ( 40 THz). The conversion efficiency in a fiber can be so high that cascaded Raman shifts are possible, where the first-order Raman shift becomes the pump for the second-order Raman shift, and so on, up to six to seven orders [76]. Raman conversion can also be accomplished in bulk crystals, especially diamond, which has a high Raman gain and excellent thermal properties. CW Raman lasing in diamond has been achieved with a pump threshold of 11 W at 1064 nm [77].
75.6.2
Optical Parametric Oscillators
In an optical parametric oscillator (OPO), a high-intensity pump beam of frequency p incident on a crystal inside an optical resonator creates two additional optical beams, a signal beam of frequency s and an idler beam of frequency i [78, 79]. There is no physical distinction between signal and idler, but by convention the signal is the higher of the two frequencies. This three-wave mixing process is mediated by the second-order nonlinear susceptibility .2/ of the crystal, which is nonzero only for crystals that lack inversion sym-
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metry (Sect. 76.3). The signal light originates from quantum fluctuations (optical parametric fluorescence), with no need to seed the oscillator with an external signal. The optical cavity can be resonant with only s (singly resonant oscillator – SRO) or with both s and i (doubly resonant oscillator – DRO), the latter having a lower pump threshold for lasing. An important requirement for efficient parametric amplification is that both energy and crystal momentum be conserved, a condition known as phase matching (Sect. 76.2.2). In a single crystal, this can be accomplished by varying the angle of a birefringent crystal (critical phase matching) or varying the temperature of the crystal at fixed angle (noncritical phase matching). An alternative to these perfect phase matching schemes is quasi-phase-matching (QPM), in which the nonlinear crystal is periodically modified in the direction of light propagation. In this scheme, the accumulated phase mismatch in one section is reversed in the next section, giving an average conversion efficiency somewhat smaller than for perfect phase matching, but still good. This approach has the advantage of allowing more flexibility in the choice of crystal axis and polarization, so a higher value of .2/ may be used. A commonly used QPM material with high .2/ is periodically poled lithium niobate (LiNbO3 ) or PPLN [80].
75.6.3
Frequency Conversion
Two other parametric processes that depend on .2/ are second harmonic generation (SHG), outlined in Sect. 76.3.2, and difference frequency generation (DFG), outlined in Sect. 76.3.4. Phase matching is required for both of these. In SHG (also known as frequency doubling), high-intensity light at frequency p is incident on a crystal, resulting in a newly generated lightwave at frequency 2p . Efficient SHG requires high incident intensity, which tends to be naturally available in pulsed operation. For CW operation, the nonlinear crystal can be located inside a laser cavity, to take advantage of the high intracavity intensity there [81], or inside a separate resonator tuned to the harmonic wavelength [82]. An important example is frequency doubling of the Nd:YAG laser ( D 1064 nm) to green light at 532 nm [83], which has played a large role in advancing the technology of Ti:sapphire lasers (Sect. 75.2.5). The difference-frequency generation process is similar to that of the OPO, except that two strong beams at frequencies p and s are incident on the crystal, rather than just one at p . A third beam at i D p s is generated, which, in spite of being labeled the “idler”, is actually the desired output. DFG is often used to generate mid-IR light from nearIR lasers. For example, tunable coherent light in the range 6.4–7.5 µm has been generated by mixing a Tm fiber laser at 2010 nm with a Yb-fiber-pumped OPO in orientationpatterned GaAs [84]. DFG can also be used to generate THz
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radiation, using two lasers with closely spaced frequencies, incident on a semiconductor.
75.6.4
High Harmonic Generation
High harmonic generation (HHG) is of interest as a coherent source of extreme ultraviolet (XUV, 30–100 nm) and soft X-ray (0.2–30 nm) radiation and is generated by focusing intense ( 1014 W=cm2 ) femtosecond laser pulses on a gaseous target. This process involves an extreme (nonperturbative) atom–laser interaction, in which the electric field of the pump light is comparable to the Coulomb field of the atom. The light generation process can be understood as occurring in three steps. First, an electron in the atom’s bound state tunnels through the potential barrier that has been lowered by the strong laser field, freeing the electron from the atom. Second, the electron is accelerated by the laser field, first away from the parent ion and then back to it when the laser field reverses its direction in the second half-cycle. Third, the electron recombines with the ion, releasing the kinetic energy it has acquired and the binding energy as an optical pulse. Since this process repeats every half-cycle of the fundamental field, it gives rise to a series of attosecond pulses that create a harmonic spectrum going up to very high order [85]. The photon energy of the maximum harmonic component scales with pump intensity I and wavelength as hmax / I 2 . This allows table-top generation of X-ray pulses in the few nm region and is a more accessible technology for many researchers than the use of large free electron facilities. Although HHG naturally results in a pulse train, single attosecond pulses can be obtained in a number of ways by careful control of the spectral phase [86]. Thus, HHG not only allows table-top access to these short wavelengths but also opens the door to the optical investigation of phenomena on an attosecond timescale.
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Richard Quimby Richard Quimby received his PhD from the University of Wisconsin, Madison in 1979. After a stay at the Center for Laser Studies at the University of Southern California, he joined the Physics Department at Worcester Polytechnic Institute in 1982. His research interests include radiative and nonradiative processes in rare-earth doped glasses and crystals, for laser and optical amplifier applications. Richard Powell Richard Powell was educated in physics at the United States Naval Academy and Arizona State University. He has been a research scientist and professor at Air Force Cambridge Research Laboratories, Sandia National Laboratory, and Lawrence Livermore National Laboratory, Oklahoma State University, and the University of Arizona. He has authored two textbooks and over 260 scientific papers on laser spectroscopy and solid-state laser development.
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, Alexander L. Gaeta, and Enno Giese
Contents 76.1 76.1.1 76.1.2 76.1.3 76.1.4
Nonlinear Susceptibility . . . . . . . . . . Tensor Properties . . . . . . . . . . . . . . . Nonlinear Refractive Index . . . . . . . . . Quantum Mechanical Expression for .n/ Hyperpolarizability . . . . . . . . . . . . . .
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76.2 76.2.1 76.2.2 76.2.3 76.2.4
Wave Equation in Nonlinear Optics Coupled-Amplitude Equations . . . . . Phase Matching . . . . . . . . . . . . . . Manley–Rowe Relations . . . . . . . . . Pulse Propagation . . . . . . . . . . . . .
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76.3 76.3.1 76.3.2 76.3.3 76.3.4 76.3.5 76.3.6 76.3.7
Second-Order Processes . . . . . . . . . . . Sum-Frequency Generation . . . . . . . . . . Second Harmonic Generation . . . . . . . . Parametric Amplification and Oscillation . Difference-Frequency Generation . . . . . . Two-Mode Squeezing . . . . . . . . . . . . . Spontaneous Parametric Down-Conversion Focused Beams . . . . . . . . . . . . . . . . .
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76.4 76.4.1 76.4.2 76.4.3 76.4.4 76.4.5 76.4.6 76.4.7
Third-Order Processes . . . . . . . . . . Third-Harmonic Generation . . . . . . . Self-Phase and Cross-Phase Modulation Four-Wave Mixing . . . . . . . . . . . . . Self-Focusing and Self-Trapping . . . . . Saturable Absorption . . . . . . . . . . . . Two-Photon Absorption . . . . . . . . . . Nonlinear Ellipse Rotation . . . . . . . .
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R. W. Boyd () Dept. of Physics, University of Ottawa Ottawa, ON, Canada Institute of Optics, University of Rochester 14627 Rochester, NY, USA e-mail: [email protected] A. L. Gaeta Applied Physics and Applied Mathematics, Columbia University New York, NY, USA e-mail: [email protected] E. Giese Institut für Angewandte Physik, Technische Universität Darmstadt Darmstadt, Germany e-mail: [email protected]
76.5 76.5.1 76.5.2
Stimulated Light Scattering . . . . . . . . . . . . . . . 1106 Stimulated Raman Scattering . . . . . . . . . . . . . . . . 1106 Stimulated Brillouin Scattering . . . . . . . . . . . . . . 1106
76.6 76.6.1 76.6.2 76.6.3
Other Nonlinear Optical Processes High-Order Harmonic Generation . . Electro-Optic Effect . . . . . . . . . . Photorefractive Effect . . . . . . . . .
76.7 76.7.1 76.7.2
New Regimes of Nonlinear Optics . . . . . . . . . . . . 1108 Ultrafast and Intense-Field Nonlinear Optics . . . . . . 1108 Nonlinear Plasmonics and Epsilon-Near-Zero Effects . 1109
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Abstract
Nonlinear optics is concerned with the propagation of intense beams of light through a material system. The optical properties of the medium can be modified by the intense light beam, leading to new processes not present in a material that responded linearly to an applied optical field. These processes can lead to the modification of the spectral, spatial, or polarization properties of the light beam or the creation of new frequency components. More complete accounts of nonlinear optics including the origin of optical nonlinearities can be found in [1–4]. Both the Gaussian and MKS system of units are commonly used in nonlinear optics. Thus, we have chosen to express the equations in this chapter in both the Gaussian and MKS systems. Each equation can be interpreted in the MKS system as written or in the Gaussian system by omitting the prefactors (e.g., 1=4 "0 ) that appear in square brackets at the beginning of the expression on the right-hand-side of the equation. Keywords
nonlinear refractive index stimulate Brillouin scattering nonlinear susceptibility difference-frequency generation nonlinear polarization
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_76
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Nonlinear Susceptibility
[i.e., .i; ! /] may be permuted along with the pairs associated with the applied field components. For example, for the In linear optics it is customary to describe the response of second-order susceptibility, this condition implies that a material in terms of a macroscopic polarization PQ (i.e., .2/ .2/ (76.5) ij k .! I !m ; !n / D kj i .!n I !m ; ! / : dipole moment per unit volume) that is linearly related to the .1/ Q applied electric field E through the linear susceptibility . If full permutation symmetry holds, and in addition all the In order to extend the relationship between PQ and EQ into the frequencies of interest are well below any of the transition nonlinear regime, the polarization is expanded in a power sefrequencies of the medium, the .n/ are invariant upon free ries of the electric field strength. We express this relationship permutation of all the Cartesian indices. This condition is mathematically by first decomposing the field and the polarknown as the Kleinman symmetry condition. ization into their frequency components such that EQ .r; t/ D
X
E .r; !l / ei!l t ;
(76.1) 76.1.1
Tensor Properties
l
Q P.r; t/ D
X
P.r; !l / ei!l t ;
(76.2) The spatial symmetry properties of a material can be used to predict the tensor nature of the nonlinear susceptibility. For example, for a material that possesses inversion symmewhere the summations are performed over both positive and try, all the elements of the even-ordered susceptibilities must negative frequencies. The reality of EQ and PQ is then assured vanish (i.e., .n/ D 0 for n even). The number of independent by requiring that E .r; !l / D E .r; !l / and P.r; !l / D elements of the nonlinear susceptibility for many materials P .r; !l /. In this case, the general expression for the can be substantially fewer than the total number of elements. Cartesian component i of the polarization at frequency ! For example, in general, .3/ consists of 81 elements, but is given by for the case of isotropic media such as gases, liquids, and hX glasses, only 21 elements are nonvanishing, and only three .1/ Pi .! / D Œ"0 ij .! /Ej .! / of these are independent. The nonvanishing elements consist .3/ .3/ .3/ j of the following types: i ijj , ij ij , and ijj i , where i ¤ j . X X .2/ In addition, it can be shown that .! I !m ; !n / C l
ij k
j k .mn/
.3/
j kl .mno/
(76.3)
where ij kl refer to field components, and the notation .mn/, for example, indicates that the summation over n and m should be performed such that ! D !m C!n is held constant. Inspection of Eq. (76.3) shows that the .n/ can be required to satisfy intrinsic permutation symmetry, i.e., the Cartesian components and the corresponding frequency components [e.g., .j; !j / but not .i; ! /] associated with the applied fields may be permuted without changing the value of the susceptibility. For example, for the second-order susceptibility, .2/
.2/
ij k .! I !m ; !n / D i kj .! I !n ; !m / :
.3/
.3/
(76.6)
76.1.2 Nonlinear Refractive Index
i
Ej .!m /Ek .!n /El .!o / C ;
.3/
i i i i D i ijj C ij ij C ijj i :
Ej .!m /Ek .!n / X X .3/ ij kl .! I !m ; !n ; !o / C
(76.4)
For many materials, the refractive index n is intensitydependent such that n D n0 C n2 I ;
(76.7)
where n0 is the linear refractive index, n2 is the nonlinear refractive index coefficient, and I D Œ4 "0 n0 cjEj2 =2 is the intensity of the optical field. For the case of a single, linearly polarized light beam traveling in an isotropic medium or along a crystal axis of a cubic material, n2 is related to .3/ by 1 12 2 .3/ n2 D (76.8) .!I !; !; !; !/ : 2 16 "0 n20 c i i i i
For the common situation in which n2 is measured in units of If the medium is lossless at all the field frequencies taking cm2 =W, and .3/ is measured in Gaussian units, the relation part in the nonlinear interaction, then the condition of full becomes permutation symmetry is necessarily valid. This condition 2 cm 12 2 107 .3/ states that the pair of indices associated with the Cartesian i i i i .!I !; !; !; !/ : (76.9) D n2 W n20 c component and the frequency of the nonlinear polarization
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There are various physical mechanisms that can give rise to a nonlinear refractive index. For the case of induced molecular orientation in CS2 , n2 D 3 1014 cm2 =W. If the contribution to the nonlinear refractive index is electronic in nature (e.g., glass), then n2 2 1016 cm2 =W.
76.1.3 Quantum Mechanical Expression for .n/ The general quantum mechanical perturbation expression for the .n/ in the nonresonant limit is
76.1.4 Hyperpolarizability The nonlinear susceptibility relates the macroscopic polarization P to the electric field strength E . A related microscopic quantity is the hyperpolarizability, which relates the dipole moment p induced in a given atom or molecule to the electric field E loc (the Lorentz local field) that acts on that atom or molecule. The relationship between p and E loc is hX pi .! / D Œ"0 ˛ij .! /Ejloc .! / j
C
C
(76.12)
0 i1 iga ian1 in 1 a1 a2 n1 an an g
.!a2 g !2 !n / .!an g !n /
; (76.10)
where ! D !1 C C !n , N is the density of atoms or molecules that compose the material, 0 .g/ is the probability that the atomic or molecular population is initially in the state g in thermal equilibrium, ia11 a2 is the i1 -th Cartesian component of the .a1 a2 / dipole matrix element, !a1 g is the transition frequency between the states a1 and g, and PF is the full permutation operator that is defined such that the expression that follows it is to be summed over all permutations of the pairs .i0 ; ! /, .i1 ; !1 / .in ; !n / and divided by the number of permutations of the input frequencies. Thus, the full expression for .2/ consists of six terms and that for .3/ consists of 24 terms. (Under conditions of resonant excitation, relaxation phenomena must be included in the treatment, and the density matrix formalism must be used [4]. The resulting equation for the nonlinear susceptibility is then more complicated.) In the limit in which the frequencies of all the fields are much smaller than any resonance frequency of the medium, the value of .n/ can be estimated to be .n/ Ñ
1 "0
2 „!0
ij kl .! I !m ; !n ; !o /
i Ejloc .!m /Ekloc .!n /Elloc .!o / C ;
n
X X j kl .mno/
1 .!a1 g !1 !n /
ˇij k .! I !m ; !n /Ejloc .!m /Ekloc .!n /
j k .mn/
.n/
i0 in .! I !1 ; : : : ; !n / X 1 N D PF
0 .g/ n "0 „ ga a 1
XX
n
where ˛ij is the linear polarizability, ˇij k is the first hyperpolarizability, and ij kl is the second hyperpolarizability. The nonlinear susceptibilities and hyperpolarizabilities are related by the number density of molecules N and by localfield factors, which account for the fact that the field E loc that acts on a typical molecule is not, in general, equal to the macroscopic field E . Under many circumstances, it is adequate to relate E loc to E through use of the Lorentz approximation 1 4 P.!/ : (76.13) E loc .!/ D E .!/ C 4 "0 3 To a good approximation, one often needs to include only the linear contribution to P.!/, and thus the local electric field becomes E loc .!/ D L.!/E .!/ ;
(76.14)
where L.!/ D fŒ"1 0 ".!/ C 2g=3 is the local field correction factor, and ".!/ is the linear dielectric constant. Since P.!/ D N p.!/, Eqs. (76.3) and (76.12) through Eq. (76.14) relate the .n/ to the hyperpolarizabilities through .1/
ij .! / D L.! /N˛ij .! / ;
(76.15)
.2/
N ;
(76.11)
ij k .! I !m ; !n / D L.! /L.!m /L.!n / Nˇij k .! I !m ; !n / ;
(76.16)
.3/ where is a typical value for the dipole transition moment, ij k .! I !m ; !n ; !o / D L.! /L.!m /L.!n /L.!o / and !0 is a typical value of the transition frequency be N ij kl .! I !m ; !n ; !o / : tween the ground state and the lowest-lying excited state. (76.17) For the case of .3/ in Gaussian units, the predicted value is .3/ D 3 1014 , which is consistent with the measured val- For simplicity, the analysis above ignores the vector characues of many materials (e.g., glass) in which the nonresonant ter of the interacting fields in calculating L.!/. A generalization that does include these effects is given in [5]. electronic nonlinearity is the dominant contribution.
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76.2 Wave Equation in Nonlinear Optics 76.2.1
!1 and !2 are combined through second-order nonlinear interaction to create a third wave at frequency !3 D !1 C !2 . Assuming full permutation symmetry, the amplitudes of the nonlinear polarization for each of the waves are
Coupled-Amplitude Equations
The propagation of light waves through a nonlinear medium is described by the wave equation 1 4 @2 Q 1 @2 P: r EQ 2 2 EQ D c @t 4 "0 c 2 @t 2 2
.2/
P NL .z; !1 / D Œ"0 2eff A.z; !3 /A .z; !2 / eikz ; (76.22) .2/
(76.18)
P NL .z; !2 / D Œ"0 2eff A.z; !3 /A .z; !1 / eikz ; (76.23) .2/
P NL .z; !3 / D Œ"0 2eff A.z; !1 /A.z; !2 / eikz ;
(76.24)
For the case in which EQ and PQ are given by Eq. (76.1), the where k D k1 C k2 k3 is the wave-vector mismatch field amplitudes associated with each frequency component (Sect. 76.2.2), and .2/ is given by eff can be decomposed into their plane wave components such X .2/ that .2/ eff D ij k .uO 1 /i .uO 2 /j .uO 3 /k ; (76.25) X ij k ikn r E .r; !l / D A n .r; !l / e ; n
X
where .uO l /i is the i-th Cartesian component of uO l . For sim(76.19) plicity, the subscripts on each of the field amplitudes have n been dropped, since only one spatial mode at each frequency where kn D n.!l /!l =c is the magnitude of the wave-vector contributed. The resulting coupled-amplitude equations are kn . The amplitudes A n and P n are next decomposed into .2/ dA.!1 / 1 i4 !1 eff vector components whose linear optical properties are such D A.!3 /A .!2 / eikz ; (76.26) that the polarization associated with them does not change as dz 4 n.!1 /c the field propagates through the material. For example, for .2/ 1 i4 !2 eff dA.!2 / a uniaxial crystal these eigenpolarizations could correspond D A.!3 /A .!1 / eikz ; (76.27) dz 4 n.!2 /c to the ordinary and extraordinary components. In order to .2/ 1 i4 !3 eff dA.!3 / describe the propagation and the nonlinear coupling of these (76.28) D A.!1 /A.!2 / eikz : eigenpolarizations, the vector field amplitudes are expressed dz 4 n.!3 /c as P.r; !l / D
P n .r; !l / eikn r ;
A n .r; !l / D uO ln An .r; !l / ; P n .r; !l / D uO ln Pn .r; !l / ;
76.2.2
Phase Matching
(76.20)
where uO ln is the unit vector associated with the eigenpolarization of the spatial mode n at frequency !l . If the fields are assumed to travel along the z-direction, and the slowlyvarying amplitude approximation @2 An =@z 2 2kn @An =@z is made, the change in the amplitude of the field as it propagates through the nonlinear medium with no linear absorption is described by the differential equation 1 i2 !l NL dAn .!l / P .!l / ; (76.21) D˙ dz 4 "0 n.!l /c n where PnNL is the nonlinear contribution to the polarization amplitude Pn , n.!l / is the linear refractive index at frequency !l , and the plus (minus) sign indicates propagation in the positive (negative) z-direction. Sections 76.3 and 76.4 give expressions for the PnNL for various second- and thirdorder nonlinear optical processes. Equation (76.21) is used to determine the set of coupled-amplitude equations describing a particular nonlinear process. For example, for the case of sum-frequency generation, the two fields of frequency
For many nonlinear optical processes (e.g., harmonic generation) it is important to minimize the wave-vector mismatch in order to maximize the efficiency. For example, if the field .2/ amplitudes A.!1 / and A.!2 / are constant and eff does not depend upon z, the solution to Eq. (76.28) yields for the output intensity I.L; !3 / D
1 64 3 "0 h i2 .2/ 32 3 eff !32 I.!1 /I.!2 /L2
n.!1 /n.!2 /n.!3 /c 3 sinc .kL=2/ ; 2
(76.29)
in terms of sinc x D .sin x/=x, where I.L; !3 / D .4 "0 /n.!3 /cjA.L; !3 /j2 =2 , and I.!1 / and I.!2 / are the corresponding input intensities. Clearly, the effect of the wave-vector mismatch is to reduce the efficiency of the generation of the sum-frequency wave. The maximum propagation distance over which efficient nonlinear coupling can
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occur is given by the coherence length Lc D
2 : k
(76.30)
As a result of the dispersion in the linear refractive index that occurs in all materials, achieving phase matching over typical interaction lengths (e.g., 5 mm) is nontrivial. For the case in which the nonlinear material is birefringent, it is sometimes possible to achieve phase matching by insuring that the interacting waves possess some suitable combination of ordinary and extraordinary polarization. Other techniques for achieving phase matching include quasi-phase matching [5], which .2/ relies on a eff that varies in the propagation direction z, and the use of the mode dispersion in waveguides [6]. However, the phase-matching condition is automatically satisfied for certain nonlinear optical processes, such as twophoton absorption (Sect. 76.4.6) and Stokes amplification in stimulated Raman scattering (Sect. 76.5.1). One can tell when the phase-matching condition is automatically satisfied by examining the frequencies that appear in the expression for the nonlinear susceptibility. For a nonlinear susceptibility of the sort .3/ .!1 I !2 ; !3 ; !4 /, the wave-vector mismatch in the corresponding set of coupled differential equations is given, in general, by k D k2 C k3 C k4 k1 . Thus, for the example of Stokes amplification in stimulated Raman scattering, the nonlinear susceptibility is given by .3/ .!1 I !1 ; !0 ; !0 /, where !0 .!1 / is the frequency of the pump (Stokes) wave, and, consequently, the wave-vector mismatch vanishes identically.
of light fields that have esoteric quantum statistical properties (Chaps. 82 and 84). A nonlinear optical process that satisfies the Manley– Rowe relations is called a parametric process. Conversely, a process for which field energy is not conserved, and thus Manley–Rowe relations cannot be formulated, is said to be nonparametric. Thus, parametric processes are described by purely real .n/ , whereas nonparametric processes are described by complex .n/ .
76.2.4 Pulse Propagation If the optical field consists of ultrashort (< 100 ps) pulses, it is more convenient to work with the temporally varying amplitude, rather than with the individual frequency components. Thus, for a linearly polarized plane wave pulse propagating along the z-axis, the field is decomposed into the product of a slowly varying amplitude A.z; t/ and a rapidly varying oscillatory term such that Q E.r; t/ D A.z; t/ ei.k0 z!0 t / C c:c: ;
(76.32)
where k0 D n0 !0 =c. For a pulse propagating in a material with an intensity-dependent refractive index, the propagation can be described by the nonlinear Schrödinger equation @A iˇ2 @2 A D ijAj2 A ; C @z 2 @ 2
(76.33)
where ˇ2 D .d2 k=d! 2 /j!D!0 is the group velocity dispersion parameter, D t z=vg is the local time for the 76.2.3 Manley–Rowe Relations pulse, vg D Œ.dk=d!/j!D!0 1 is the group velocity, and D Œ4 "0 n2 n0 !0 =2 is the nonlinear refractive index paUnder conditions of full permutation symmetry, there is no rameter. flow of power from the electromagnetic fields to the medium, and thus the total power flow of the fields is conserved. The flow of energy among the fields can be described by the 76.3 Second-Order Processes Manley–Rowe relations. For example, for the case of sumfrequency generation, one can deduce from Eqs. (76.26)– Second-order nonlinear optical processes occur as a consequence of the second term in expression Eq. (76.3), i.e., (76.28) that processes whose strength is described by .2/ .! I !m ; !n /. These processes entail the generation of a field at frequency d I.!1 / d I.!2 / d I.!3 / D D : (76.31) ! D !m C!n in response to applied fields at (positive and/or dz !1 dz !2 dz !3 negative) frequencies !m and !n . Several examples of such The expressions in square brackets are proportional to the processes are described in this section. flux of photons per unit area per unit time and imply that the creation of a photon at !3 must be accompanied by the annihilation of photons at both !1 and !2 . Similar relations can 76.3.1 Sum-Frequency Generation be formulated for other nonlinear optical processes that are governed by a nonlinear susceptibility that satisfies full per- Sum-frequency generation produces an output field at fremutation symmetry. Since this behavior occurs at the photon quency !3 D !1 C !2 for !1 and !2 both positive. It is useful, level, nonlinear optical processes can lead to the generation for example, for the generation of tunable radiation in the UV
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if !1 and/or !2 are obtained from tunable lasers in the vis- relations, as described above in Sect. 76.2.3. Because of the ible range. Sum-frequency generation is described in detail relation !3 D !1 C !2 , the annihilation of an !3 photon must be accompanied by the simultaneous creation of photons !1 in Sects. 76.2.1–76.2.3. and !2 . From Eq. (76.36) it is apparent that the intensity of the generated fields depends on the phase relations among 76.3.2 Second Harmonic Generation the three incoming fields, and the amplification is, therefore, a phase sensitive process. Second harmonic generation is routinely used to convert the An optical parametric oscillator can be constructed by output of a laser to a higher frequency. It is described by placing the nonlinear optical material inside an optical res.2/ .2!I !; !/. Let be the power conversion efficiency onator that provides feedback at !1 and/or !2 . When such from frequency ! to 2!. Assuming that phase matching is a device is excited by a wave at !3 , it can produce output perfect, and the pump wave has the fundamental frequency frequencies !1 and !2 that satisfy !1 C !2 D !3 . Optical parametric oscillators are of considerable interest as sources !, a derivation analogous to that for Eq. (76.29) yields of broadly tunable radiation [7].
(76.34) D tanh2 z= l ; where the characteristic conversion length l is given by p c n.!/n.2!/ : l D 4 4 !.2/jA1 .0/j
76.3.4 Difference-Frequency Generation
Difference-frequency generation can be used to create light in the infrared and far infrared by generating the difference frequency !2 D !3 !1 (where !3 and !1 are positive, and Note that the conversion efficiency asymptotically ap- !3 > !1 ) of two incident lasers. Consider the case in which proaches unity. In practice, conversion efficiencies exceeding a strong (undepleted) pump wave at frequency !3 and a weak 80% can be achieved. (signal) wave at !1 are incident on a nonlinear medium described by .2/ .!2 I !3 ; !1 / D .2/ .!1 I !3 ; !2 /. The amplitude A3 of the strong wave can be taken as a constant, and 76.3.3 Parametric Amplification and Oscillation Eq. (76.36) describes the solution to a system of coupled differential equations. Since there is no incident idler field, we Under the assumption that the pump at !3 in a second-order have A2 .0/ D 0, and the solution reduces to process is undepleted and strong, the system of coupled dif ik ferential Eqs. (76.26) and (76.27) can be solved analytically. sinh gz ; (76.39) A1 .z/ D A1 .0/ cosh gz C The solution for A.!j / D Aj and up to a global phase factor 2g r is [4] n1 !2 A3
A .0; !1 / sinh gz : (76.40) A2 .z/ D i n2 !1 jA3 j g k sinh gz A1;2 .z/ D A1;2 .0/ cosh gz C i 2g Equation (76.40) describes the spatial growth of the r n2;1 !1;2 A3 difference-frequency signal. i A .0/ sinh gz ; (76.36) n1;2 !2;1 g jA3 j 2;1 (76.35)
where the gain for conditions of perfect phase matching is given by
2 D
1 16 2
2 16 2 .2/ !12 !22 jA3 j2 ; k1 k2 c 4
(76.37)
and the generalized gain for arbitrary phase matching is g 2 D 2 .k=2/2 :
(76.38)
Equation (76.36) shows simultaneous (and exponential) amplification of the two fields A1 and A2 , often called signal and idler. The process is referred to as parametric amplification. It is a direct consequence of the Manley–Rowe
76.3.5 Two-Mode Squeezing To describe the quantum mechanical properties of light– matter interactions, one has to quantize the electromagnetic fields participating in the nonlinear interaction. However, the Heisenberg equations of motion of the respective field operators are very similar to their classical counterparts from Eqs. (76.26)–(76.28), and one can just replace Aj ! const: p !j =nj aOj and Aj by the Hermitian conjugate of this operator. Note that the photon annihilation operators aOj are defined per field mode, and the proportionality factor depends on the quantization volume. Note further that the commutation re lation ŒaOj ; aO l D ıj;l is preserved upon propagation of the quantum field. For an undepleted and strong pump, the so-
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lution to the Heisenberg equations of motion is in analogy to Unlike for the case of its classical counterpart, the output photon number n1;2 .z/ D jvj2 is nonvanishing even for a vacEq. (76.36) the Bogoliubov transformation [8] uum input n1;2 .0/ D 0 because of the noncommutativity of (76.41) the photon creation and annihilation operators. This means aO 1;2 .z/ D uaO 1;2 .0/ C v aO 2;1 .0/ ; that one can not only squeeze input light, but also the vacwith the coefficients uum itself. Moreover, a weak nonlinear interaction or a weak k sinh gz (76.42) pump field leads to spontaneous emission. In this case, u D cosh gz C i 2g k=2 and Eq. (76.45) reduces to i3 sinh gz ; (76.43) v D i e g 2 2 2 (76.46) nSPDC 1;2 .L/ D L sinc .kL=2/ ; where 3 is the phase of the pump wave, and g as well as are defined as in Eqs. (76.38) and (76.37), respectively. Note that juj2 jvj2 D 1, which ensures unitary of the amplifica- for a bulk crystal of length L. A more sophisticated treatment tion process. This transformation leads to a squeezing of the shows that there are not only photons spontaneously emitquadratures, as demonstrated in the following. ted, but they are strongly correlated. The photons are always The two-mode quadratures of the electromagnetic field generated in pairs and share common properties, a feature p are defined by xO ˙ D ŒaO 1 ˙ aO 1 C aO 2 ˙ aO 2 = ˙8. They corre- that is called entanglement. There have been numerous exspond to the quantum analog of the real and imaginary parts perimental verifications of polarization entanglement, but of the electric fields. From Eq. (76.41) with 3 D =2 and also entanglement of continuous variables such as energy k D 0 (for phase matching), we find that the two-mode and time or transverse position and momentum, discussed in quadratures are amplified or deamplified according to [9] Chap. 84. Position-momentum entanglement is a direct consequence of the shape of the crystal and the pump beam in (76.44) SPDC. The two-photon wave function .q ; q / of signal xO ˙ .z/ D e˙ z xO ˙ .0/ : 1 2 and idler photons with a transverse momentum q 1 and q 2 , 2 2 Hence, the variances x˙ D h.xO ˙ hxO ˙ i/ i of these respectively, is proportional to [10] 2 quadrature operators can be written as x .z/ D ˙
e˙2 z 2 x˙ .0/ and are amplified and suppressed, respectively. Since the variances correspond to the intrinsic noise of .q 1 ; q 2 / / A3 .q 1 C q 2 / eikL=2 sinc.kL=2/ ; (76.47) the electric fields, the squeezing process can lead to its suppression. Therefore, squeezed light is used to increase the sensitivity of interferometers and gives a sensitivity below with k D k3 .q 1 C q 2 / k1 .q 1 / k2 .q 2 /. Here, kj correthe shot-noise level. Single-mode squeezing and the associ- sponds to the longitudinal component of the respective wave ated squeezing operator are discussed in Chaps. 82 and 84. vector. A3 .q 3 / is the angular distribution of the pump field. For degenerate down-conversion (!1 D !2 ), we find in the paraxial regime that k D 3 .q 1 q 2 /2 =.4 /. For a plane 76.3.6 Spontaneous Parametric wave pump, A3 .q 3 / becomes very narrow and the transverse Down-Conversion wave vectors become anticorrelated, which is implied by momentum conservation. The squared modulus of Eq. (76.47) is With the results from the preceding section, it is possible to proportional to the coincidence count rate of signal and idler look at the squeezing of vacuum input states and the limit of photons with the respective momenta. a small coupling z, which leads to spontaneous parametric With a Fourier transform, the two-photon wave function down-conversion (SPDC). In this process, a pump photon is from Eq. (76.47) is transformed into position space, and it spontaneously converted into one signal and one idler pho- may reveal that the transverse positions are correlated. The ton, without the process being seeded by input fields. The combination of position correlation and momentum anticortwo generated photons always appear pairwise and are, there- relation can lead to the apparent violation of a Heisenberg fore, often used as a source of single photons. The detection uncertainty relation for a linear combination of the two posiof an idler photon implies that there is a single photon in the tions and momenta, which is a clear witness of continuoussignal field. variable entanglement. In fact, a photon pair that is not For any uncorrelated input state, it is easy to show with entangled cannot have correlated positions of the signal and Eq. (76.41) and the commutation relations that the expecta- idler photons and simultaneously anticorrelated momenta, as tion value nj D haOj aOj i of the photon number is these are two conjugate variables. However, for an entangled photon pair, quantum correlations allow for simultaneous pon1;2 .z/ D juj2 n1;2 .0/ C jvj2 Œn2;1 .0/ C 1 : (76.45) sition correlation and momentum anticorrelation [11, 12].
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76.3.7 Focused Beams For conceptual clarity, much of the discussion so far has assumed that the interacting beams are plane waves. In practice, the incident laser beams are often focused into the nonlinear material to increase the field strength within the interaction region and, consequently, to increase the nonlinear response. However, it is undesirable to focus too tightly, because doing so leads to a decrease in the effective length of the interaction region. In particular, if w0 is the radius of the laser beam at the beam waist, the beam remains focused only over a distance of the order b D 2 w02 =, where is the laser wavelength measured in the nonlinear material. For many types of nonlinear optical processes, the optimal nonlinear response occurs if the degree of focusing is adjusted so that b is several times smaller than the length L of the nonlinear optical material.
76.4 Third-Order Processes A wide variety of nonlinear optical processes are possible as a result of the nonlinear contributions to the polarization that are third-order in the applied field. These processes are described by .3/ .! I !m ; !n ; !o / as defined in Eq. (76.3) and can lead not only to the generation of new field components (e.g., third-harmonic generation) but can also result in a field affecting itself as it propagates (e.g., self-phase modulation). Several examples are described in this section.
76.4.1
Third-Harmonic Generation
where I.!/ is the input intensity of the fundamental field. .3/ As a result of the typically small value of eff in crystals, it is generally more efficient to generate the third harmonic by using two .2/ crystals in which the first crystal produces second harmonic light, and the second crystal combines the second harmonic and the fundamental beams via sumfrequency generation. It is also possible to use resonant enhancement of j.3/ j in gases to increase the efficiency of third-harmonic generation [13].
76.4.2 Self-Phase and Cross-Phase Modulation The nonlinear refractive index leads to an intensitydependent change in the phase of the beam as it propagates through the material. If the medium is lossless, the amplitude of a single beam at frequency ! propagating in the positive z-direction can be expressed as A.z; !/ D A.0; !/ ei
NL .z/
;
(76.50)
where the nonlinear phase shift NL .z/ is given by NL .z/ D
! n2 I z ; c
(76.51)
and I D Œ4 "0 n0 cjA.0; !/j2 =2 is the intensity of the laser beam. If two fields at different frequencies !1 and !2 are traveling along the z-axis, the two fields can affect each other’s phase; this effect is known as cross-phase modulaNL .z/ for each of the waves tion. The nonlinear phase shift 1;2 is given by NL 1;2 .z/ D
!1;2 n2 .I1;2 C 2I2;1 /z : c
(76.52) Assuming full-permutation symmetry, the nonlinear polarization amplitudes for the fundamental and third-harmonic For the case of a light pulse, the change in the phase of beams are the pulse inside the medium becomes a function of time. In .3/ P NL .z; !/ D Œ"0 3eff A.z; 3!/ŒA .z; !/2 eikz ; this case, the solution to Eq. (76.33) shows that in the ab.3/ NL 3 ikz sence of group-velocity dispersion (GVD) (i.e., ˇ2 D 0), the P .z; 3!/ D Œ"0 eff ŒA.z; !/ e ; time-varying amplitude A.z; / is of the form of Eq. (76.50), (76.48) except that the temporal intensity profile I./ replaces the .3/ where k D 3k.!/ k.3!/ and eff is the effective third- steady-state intensity I in Eq. (76.51). As the pulse proporder susceptibility for third-harmonic generation and is agates through the medium, its frequency becomes time .2/ defined in a manner analogous to the eff in Eq. (76.25). dependent, and the instantaneous frequency shift from the If the intensity of the fundamental wave is not depleted by central frequency !0 is given by the nonlinear interaction, the solution for the output intensity !0 n2 z @I @ NL ./ I.L; 3!/ of the third-harmonic field for a crystal of length L D : (76.53) ı!./ D is @ c @t h i2 48 2 ! 2 .3/ This time-dependent self-phase modulation leads to a broadeff 1 I.L; 3!/ D ening of the pulse spectrum and to a frequency chirp across 256 4 "20 n.3!/n.!/3c 4 the pulse. 3 2 2 kL If the group velocity dispersion parameter ˇ2 and the non I.!/ L sinc ; (76.49) 2 linear refractive index coefficient n2 are of opposite sign, the
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nonlinear frequency chirp can be compensated by the chirp due to group velocity dispersion, and Eq. (76.33) admits soliton solutions. For example, the fundamental soliton solution is s 1 (76.54) A.z; t/ D sech eiz=2LD ; jLD j p
1105
76.4.4 Self-Focusing and Self-Trapping
Typically, a laser beam has a transverse intensity profile that is approximately Gaussian. In a medium with an intensitydependent refractive index, the index change at the center of the beam is different from the index change at the edges of the beam. The gradient in the refractive index created by the beam can allow it to self-focus for n2 > 0. For this condition where p is the pulse duration, and LD D p2 =ˇ2 is the dis- to be met, the total input power of the beam must exceed the persion length. As a result of their ability to propagate in critical power Pcr for self-focusing, which is given by dispersive media without changing shape, optical solitons .0:61/2 ; (76.57) Pcr D show a great deal of promise in applications such as optical 8n0 n2 communications and optical switching. For further discuswhere is the vacuum wavelength of the beam. For powsion of optical solitons, see [14]. ers much greater than the critical power, the beam can break up into various filaments, each with a power approximately equal to the critical power. For a more extensive discussion 76.4.3 Four-Wave Mixing of self-focusing and self-trapping, see [15, 16]. Various types of four-wave mixing processes can occur among different beams. One of the most common geometries is backward four-wave mixing used in nonlinear spec- 76.4.5 Saturable Absorption troscopy and optical phase conjugation. In this interaction, two strong counterpropagating pump waves with amplitudes When the frequency ! of an applied laser field is sufficiently A1 and A2 and with equal frequencies !1;2 D ! are injected close to a resonance frequency !0 of the medium, an appreinto a nonlinear medium. A weak wave, termed the probe ciable fraction of the atomic population can be placed in the wave, (with frequency !3 and amplitude A3 ) is also inci- excited state. This loss of population from the ground state dent on the medium. As a result of the nonlinear interaction leads to an intensity-dependent saturation of the absorption among the three waves, a fourth wave with an amplitude A4 and the refractive index of the medium (for a more detailed is generated, which is counterpropagating with respect to the discussion: Chap. 73.3) [4]. The third-order susceptibility as probe wave and with frequency !4 D 2! !3 . For this case, a result of this saturation is given by 2 the third-order nonlinear susceptibilities for the probe and ıT2 i 1 jj T1 T2 ˛0 c (76.58) .3/ D conjugate waves are given by .3/ .!3;4 I !; !; !4;3 /. For
2 ; 2 "0 3 !0 „ 1 C .ıT2 /2 constant pump wave intensities and full permutation symmetry, the amplitudes of the nonlinear polarization for the probe where is the transition dipole moment, T1 and T2 are and conjugate waves are given by the longitudinal and transverse relaxation times, respectively (Chap. 72.4.3), ˛0 is the line-center weak-field intensity abP NL .z; !3;4 / sorption coefficient, and ı D ! !0 is the detuning. For the
D ˙Œ"0 6.3/ jA1 j2 C jA2 j2 A3;4 C A1 A2 A4;3 eikz ; 3s $ 3p transition in atomic sodium vapor at 300 °C, the (76.55) nonlinear refractive index n2 107 cm2 =W for a detuning ıT2 D 300. where k D k1 C k2 k3 k4 is the phase mismatch, which is nonvanishing when !3 ¤ !4 . For the case of optical phase conjugation by degenerate four-wave mixing (i.e., !3 D !4 D 76.4.6 Two-Photon Absorption !, and A4 .L/ D 0), the phase conjugate reflectivity RPC is When the frequency ! of a laser field is such that 2! is close jA4 .0/j2 2 D tan . L/ ; (76.56) to a transition frequency of the material, it is possible for RPC D jA3 .0/j2 two-photon absorption (TPA) to occur. This process leads to a contribution to the imaginary part of .3/ .!I !; !; !/. In
p where D 1=16 2 "0 24 2 !.3/ =.n0 c/2 I1 I2 and I1;2 the presence of TPA, the intensity I.z/ of a single, linearly are the intensities of the pump waves. Phase-conjugate re- polarized beam as a function of propagation distance is flectivities greater than unity can be routinely achieved by I.0/ performing four-wave mixing in atomic vapors or photore; (76.59) I.z/ D 1 C ˇI.0/z fractive media.
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where ˇ D 1=16 2 "0 24 2 ! Im .3/ =.n0 c/2 is the TPA where R .!0;1 / .3/ .!0;1 I !0;1 ; !1;0 ; !1;0 /, the Raman coefficient. For wide-gap semiconductors such as ZnSe at susceptibility, actually depends only on the frequency differ800 nm, ˇ 108 cm=W. ence * D !0 !1 and is given by
76.4.7 Nonlinear Ellipse Rotation The polarization ellipse of an elliptically polarized laser beam rotates but retains its ellipticity as the beam propagates through an isotropic nonlinear medium. Ellipse rotation occurs as a result of the difference in the nonlinear index changes experienced by the left-circular and right-circular components of the beam, and the angle of rotation is 1 D n!z=c 2 12 2 .3/ 1 D .!I !; !; !/.IC I /z ; 2 16 "0 n20 c xyyx (76.60)
R .!0;1 / D
1 1 N.@˛=@q/20 ; (76.62) 2 2 "0 6M !v * 2i *
where the minus (plus) sign is taken for the !0 (!1 ) susceptibility, M is the reduced nuclear mass, and .@˛=@q/0 is a measure of the change of the polarizability of the molecule with respect to a change in the intermolecular distance q at equilibrium. If the intensity of the pump field is undepleted by the interaction with the !1 field and is assumed to be constant, the solution for the intensity of the !1 field at z D L is given by I.L; !1 / D I.0; !1 / eGR ;
(76.63)
where the SRS gain parameter GR is where I˙ are the intensities of the circularly polarized comp 1 O 2. !1 ponents of the beam with unit vectors O ˙ D .xO ˙ iy/= ImŒR .!1 /I0 L GR D 48 2 Nonlinear ellipse rotation is a sensitive technique for de16 2 "0 .n1 c/2 .3/ termining the nonlinear susceptibility element xyyx for (76.64) D gR I0 L ; isotropic media and can be used in applications such as optical switching. gR is the SRS gain factor, and I0 is the input intensity of the pump field. For !1 < !0 (!1 > !0 ), the !1 field is termed the Stokes (anti-Stokes) field, and it experiences exponen76.5 Stimulated Light Scattering tial amplification (attenuation). For sufficiently large gains (typically GR & 25), the Stokes wave can be seeded by sponStimulated light scattering occurs as a result of changes in taneous Raman scattering and can grow to an appreciable the optical properties of the material that are induced by the fraction of the pump field. For a complete discussion of the optical field. The resulting nonlinear coupling between dif- spontaneous initiation of SRS, see [17]. For the case of CS , 2 ferent field components is mediated by some excitation (e.g., g D 0:024 cm=MW. R acoustic phonon) of the material that results in changes in Four-wave mixing processes that couple a Stokes wave its optical properties. The nonlinearity can be described by having ! < ! and an anti-Stokes wave having ! > ! , 1 0 2 0 a complex susceptibility and a nonlinear polarization that where ! C ! D 2! , can also occur [4]. In this case, 1 2 0 is of third order in the interacting fields. Various types of additional contributions to the nonlinear polarization are stimulated scattering can occur. Discussed below are the two present and are characterized by a Raman susceptibility of processes that are most commonly observed. the form .3/ .!1;2 I !0 ; !0 ; !2;1 /. The technique of coherent anti-Stokes Raman spectroscopy is based on this four-wave mixing process [18].
76.5.1
Stimulated Raman Scattering
In stimulated Raman scattering (SRS), the light field interacts with a vibrational mode of a molecule. The coupling between the two optical waves can become strong if the frequency difference between them is close to the frequency !v of the molecular vibrational mode. If the pump field at !0 and another field component at !1 are propagating in the same direction along the z-axis, the steady-state nonlinear polarization amplitudes for the two field components are given by
76.5.2
Stimulated Brillouin Scattering
In stimulated Brillouin scattering (SBS), the light field induces and interacts with an acoustic wave inside the medium. The resulting interaction can lead to extremely high amplification for certain field components (i.e., Stokes wave). For many optical media, SBS is the dominant nonlinear optical process for laser pulses of duration > 1 ns. The primary apP NL .z; !0;1 / D Œ"0 6R .!0;1 /jA.z; !1;0 /j2 A.z; !0;1 / ; plications for SBS are self-pumped phase conjugation and (76.61) pulse compression of high-energy laser pulses.
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If an incident light wave with wave vector k0 and frequency !0 is scattered from an acoustic wave with wave vector q and frequency *, the wave vector and frequency of the scattered wave are determined by conservation of momentum and energy to be k1 D k0 ˙ q and !1 D !0 ˙ *, where the (C) sign applies if k0 q > 0 and the () applies if k0 q < 0. Here, * and q are related by the dispersion relation * D vjqj where v is the velocity of sound in the material. These Bragg scattering conditions lead to the result that the Brillouin frequency shift *B D !1 !0 is zero for scattering in the forward direction (i.e., in the k0 direction) and reaches its maximum for scattering in the backward direction given by
1107
D g0
**B %B2 I0 L 2 *2B *2 C .*%B /2
D gB I0 L ;
(76.69)
gB is the SBS gain factor, I0 is the input intensity of the pump field, and
1 !02 e2 g0 D 2 "0 n0 c 3 0 v%B
(76.70)
is the line-center (i.e., * D ˙*B ) SBS gain factor. For * > 0 (* < 0), the !1 field is termed the Stokes (anti-Stokes) field, and it experiences exponential amplification (attenuation). *B D 2!0 vn0 =c ; (76.65) For sufficiently large gains (typically GB & 25), the Stokes wave can be seeded by spontaneous Brillouin scattering and where n0 is the refractive index of the material. can grow to an appreciable fraction of the pump field. For The interaction between the incident wave and the scata complete discussion of the spontaneous initiation of SBS, tered wave in the Brillouin-active medium can become see [19]. For CS2 , g0 D 0:15 cm=MW. nonlinear if the interference between the two optical fields can coherently drive an acoustic wave, either through electrostriction or through local density fluctuations resulting from the absorption of light and consequent temperature 76.6 Other Nonlinear Optical Processes changes. The following discussion treats the more common electrostriction mechanism. 76.6.1 High-Order Harmonic Generation Typically, SBS occurs in the backward direction (i.e., O since the spatial overlap between k0 D k0 zO and k1 D k1 z), If full permutation symmetry applies, and the fundamental the Stokes beam and the laser beam is maximized under field ! is not depleted by nonlinear interactions, then the inthese conditions and, as mentioned above, no SBS occurs tensity of the q-th harmonic is given by in the forward direction. The steady-state nonlinear polarization amplitudes for backward SBS are 1 I.z; q!/ D 4 .4 "0 /.q1/=2 P NL .z; !0;1 / D Œ"0 6B .!0;1 /jA.z; !1;0 /j2 A.z; !0;1 / ; (76.66) 2 q 2 ! 2 2 I.!/ q 2 n .q!/c n.!/c where B .!0;1 / .3/ .!0;1 I !0;1 ; !1;0 ; !1;0 /, the Brillouin ˇ .q/ ˇ2 ˇ .q!I !; : : : ; !/Jq .k; z0 ; z/ˇ ; susceptibility, depends only on * D !0 !1 and is given by (76.71) 1 !02 e2 1 B .!0;1 / D ; (76.67) "0 24 2 c 2 0 *2B *2 i%B * where k D Œn.!/ n.q!/!=c,
where the minus (plus) sign is taken for the !0 (!1 ) susZz 0 0 eikz dz ceptibility, e is the electrostrictive constant, 0 is the mean Jq .k; z0 ; z/ D ; (76.72) density of the material, and %B is the Brillouin linewidth .1 C 2iz 0 =b/q1 z0 given by the inverse of the phonon lifetime. If the pump field is undepleted by the interaction with the !1 field and is assumed to be constant, the solution for the output intensity of z D z0 at the input face of the nonlinear medium, and b is the confocal parameter (Sect. 76.3.7) of the fundamental beam. the !1 field at z D 0 is given by Defining L D z z0 , the integral Jq can be easily evaluated I.0; !1 / D I.L; !1 / eGB ; (76.68) in the limits L b and L b. The limit L b corresponds to the plane-wave limit in which case where the Brillouin gain coefficient GB is given by 2 2 2 kL 1 2 !1 .k; z ; z/j D L sinc : (76.73) jJ q 0 GB D ImŒB .!1 /I0 L 48 2 16 2 "0 .n0 c/2
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The limit L b corresponds to the tight-focusing configu- this effect mimics that of the nonlinear refractive index described in Sect. 76.1.2, but it differs from the nonlinear ration in which case refractive index in that the change in refractive index is inJq .k; z0 ; z/ dependent of the overall intensity of the incident light field 8 0 : .q 2/Š 2 torefractive effect can occur only in materials that exhibit Note that in this limit, the q-th harmonic light is only gen- a linear electro-optic effect and contain an appreciable denerated for positive phase mismatch. Reintjes et al. [20, 21] sity of trapped electrons and/or holes that can be liberated by observed both the fifth and seventh harmonics in helium gas, the application of a light field. Typical photorefractive matewhich exhibited a dependence on I.!/, which is consistent rials include lithium niobate, barium titanate, and strontium with the I q .!/ dependence predicted by Eq. (76.71). How- barium niobate. A typical photorefractive configuration might be as folever, more recent experiments in gas jets have demonstrated the generation of extremely high-order harmonics, which do lows: two beams interfere within a photorefractive crystal to not depend on the intensity in this simple manner (Chap. 78 produce a spatially modulated intensity distribution. Bound charges are ionized with greater probability at the maxima for further discussion of this nonperturbative behavior). than at the minima of the distribution and, as a result of the diffusion process, carriers tend to migrate away from regions of large light intensity. The resulting modulation of 76.6.2 Electro-Optic Effect the charge distribution leads to the creation of a spatially The electro-optic effect corresponds to the limit in which the modulated electric field that produces a spatially modufrequency of one of the applied fields approaches zero. The lated change in refractive index as a consequence of the linear electro-optic effect (or Pockels effect) can be described linear electro-optic effect. For a more extensive discussion, by a second-order susceptibility of the form .2/ .!I !; 0/. see [23]. This effect produces a change in the refractive index for light of certain polarizations, which depends linearly on the strength of the applied low-frequency field. More gener- 76.7 New Regimes of Nonlinear Optics ally, the linear electro-optic effect induces a change in the amount of birefringence present in an optical material. This 76.7.1 Ultrafast and Intense-Field Nonlinear Optics electrically controllable change in birefringence can be used to construct amplitude modulators, frequency shifters, optical shutters, and other optoelectronic devices. Materials Additional nonlinear optical processes are enabled by the commonly used in such devices include KDP and lithium use of ultrashort (< 1 ps) or ultraintense laser pulses. For niobate [22]. If the laser beam is propagating along the optic reasons of basic laser physics, ultraintense pulses are necaxis (i.e., z-axis) of the material of length L, and the low- essarily of short duration, and, thus, these effects normally frequency field Ez is also applied along the optic axis, the occur together. Ultrashort laser pulses possess a broad frenonlinear index change n D ny nx between the compo- quency spectrum, and, therefore, the dispersive properties of nents of the electric field polarized along the principal axes the optical medium play a key role in the propagation of such pulses. The three-dimensional nonlinear Schrödinger of the crystal is given by equation must be modified when treating the propagation 1 (76.75) of these ultrashort pulses by including contributions that n30 r63 Ez ; n D 4 can be ignored under other circumstances [24, 25]. These additional terms lead to processes such as space–time couwhere r63 is one of the electro-optic coefficients. The quadratic electro-optic effect produces a change in pling, self-steepening, and shock-wave formation [26, 27]. the refractive index that scales quadratically with the applied The process of self-focusing is significantly modified under DC electric field. This effect can be described by a third- short-pulse (pulse duration shorter than approximately 1 ps) excitation. For example, temporal splitting of a pulse into order susceptibility of the form .3/ .!I !; 0; 0/. two components can occur; this pulse splitting lowers the peak intensity and can lead to the arrest of the usual col76.6.3 Photorefractive Effect lapse of a pulse undergoing self-focusing [28]. Moreover, optical shock formation, the creation of a discontinuity in The photorefractive effect leads to an optically induced the intensity evolution of a propagating pulse, leads to suchange in the refractive index of a material. In certain ways, percontinuum generation, the creation of a light pulse with
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an extremely broad frequency spectrum [29]. Shock effects and the generation of supercontinuum light can also occur in one-dimensional systems, such as a microstructure optical fiber. The relatively high peak power of the ultrashort pulses from a mode-locked laser oscillator and the tight confinement of the optical field in the small ( 2 m) core of the fiber yield high intensities and strong self-phase modulation, which results in a spectral bandwidth that spans more than an octave of the central frequency of the pulse [30]. Such a coherent octave-spanning spectrum allows for the stabilization of the underlying frequency comb of the mode-locked oscillator and has led to a revolution in the field of frequency metrology [31]. Multiphoton absorption [32] constitutes an important loss process that becomes important for intensities in excess of 1013 W=cm2 . In addition to introducing loss, the electrons released by this process can produce additional nonlinear effects associated with their relativistic motion in the resulting plasma [33, 34]. For very large laser intensities (greater than approximately 1016 W=cm2 ), the electric field strength of the laser pulse can exceed the strength of the Coulomb field that binds the electron to the atomic core, and nonperturbative effects can occur. A dramatic example is that of high-harmonic generation [35–37]. Harmonic orders as large as the 341-st have been observed, and simple conceptual models have been developed to explain this effect [38]. Under suitable conditions, the harmonic orders can be suitably phased so that attosecond pulses are generated [39].
76.7.2
Nonlinear Plasmonics and Epsilon-Near-Zero Effects
Despite their intrinsic loss, plasmonic materials (i.e., metals) have shown stronger nonlinear effects than insulators or dielectrics [40]. The optical properties of such materials are determined by the dynamics of quasi-free electrons in the conduction band and can be understood in terms of the Drude model for sufficiently low frequencies. In this model, the permittivity
(76.76) .!/ D 1 !p2 = ! 2 C i! shows a strong dependence on the frequency ! and is determined by the high-frequency permittivity 1 , a damping constant , and the plasma frequency
1109
ementary charge. Note that for most plasmonic materials the optical response deviates significantly from the Drude model due to interband transitions at optical frequencies. However, the model can be improved by including additional contributions to the permittivity that arise in a Lorentzoscillator description of bound electrons. Even though such models give a better quantitative description of the response, Eq. (76.76) describes the qualitative behavior of plasmonic materials very well [41]. The permittivity from Eq. (76.76) has some remarkable features. For example, the real part of can take negative values. Moreover, it may even vanish for ! 2 D !p2 =1 (neglecting terms of the order of 2 ). This resonance gives rise to a new class of epsilon-near-zero (ENZ) materials, which possess unique nonlinear optical properties. For a vanishing real part of , the real part of the refractive index becomes small and, following Eq. (76.8), the nonlinear refractive index n2 is enhanced [42, 43]. This heuristic explanation captures some features of the outstanding optical response of some plasmonic materials. Other nonlinear mechanisms that dominate in such materials [44–47] are (i) saturable absorption due to the filling of the conduction band, (ii) quantum-size effects for small particles where the electrons in the conduction band are confined to a small volume and, thus, not completely free, (iii) ponderomotive effects in Fermi–Dirac metals where the electron density is depleted by high intensities, and (vi) hot-electron effects where the distribution of electrons in the conduction band is modified by the absorption of heat from high intensities. The latter effect leads to a shift of the frequency where the real part of the permittivity vanishes, which can be easily understood in terms of Eq. (76.77), since a modified distribution changes the effective mass m and by that !p . Note that such thermal effects are not instantaneous and depend relatively strongly on the laser pulse parameters. A field of magnitude jE0 j incident on such an ENZ medium is also significantly enhanced inside the material. Since the perpendicular component of the electric displacement is continuous at the interface, the electric field inside a material depends on the inverse of the permittivity. For an angle of incidence , the magnitude of the field jEmat j within the material is
1=2 jEmat j D jE0 j cos2 C 1 sin2
(76.78)
!p2 D Œ1=.4 0 / 4 Ne 2 =m :
(76.77) and, therefore, enhanced for an oblique angle of incidence and small values of . In fact, the field enhancement leads Here, N is the electron density, and m the effective mass to an angle-dependence for the nonlinear optical response of of the electrons in the conduction band, e denotes the el- ENZ materials [42].
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References 1. Bloembergen, N.: Nonlinear Optics. Benjamin, New York (1964) 2. Shen, Y.R.: Nonlinear Optics. Wiley, New York (1984) 3. Butcher, P.N., Cotter, D.: The Elements of Nonlinear Optics. Cambridge University Press, Cambridge (1990) 4. Boyd, R.W.: Nonlinear Optics, 3rd edn. Academic Press, Boston (2008) 5. Armstrong, J.A., Bloembergen, N., Ducuing, J., Pershan, P.S.: Phys. Rev. 127, 1918 (1962) 6. Stegeman, G.I.: In: Agrawal, G.P., Boyd, R.W. (eds.) Contemporary Nonlinear Optics. Academic Press, Boston (1992). Chap. 1 7. See for example the Special Issue on: Optical Parametric Oscillation and Amplification. J. Opt. Soc. Am. B. 10 (1993) 8. Klyshko, D.N.: Photons and Nonlinear Optics. Gordon and Breach, New York (1989) 9. Loudon, R., Knight, P.L.: J. Mod. Opt. 34, 709 (1987) 10. Walborn, S.P., Monken, C.H., Pádua, S., Souto Ribeiro, P.H.: Phys. Rep. 495, 87 (2010) 11. Howell, J.C., Bennink, R.S., Bentley, S.J., Boyd, R.W.: Phys. Rev. Lett. 92, 210403 (2004) 12. D’Angelo, A., Kim, Y.-H., Kulik, S.P., Shih, Y.: Phys. Rev. Lett. 92, 233601 (2004) 13. Miles, R.B., Harris, S.E.: IEEE J. Quant. Electron. 9, 470 (1973) 14. Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, Boston (1989) 15. Akhmanov, S.A., Khokhlov, R.V., Sukhorukov, A.P.: In: Arecchi, F.T., Schulz-Dubois, E.O. (eds.) Laser Handbook. North-Holland, Amsterdam (1972) 16. Marburger, J.H.: Prog. Quant. Electr. 4, 35 (1975) 17. Raymer, M.G., Walmsley, I.A.: In: Wolf, E. (ed.) Progress in Optics, vol. 28, North-Holland, Amsterdam (1990) 18. Levenson, M.D., Kano, S.: Introduction to Nonlinear Spectroscopy. Academic Press, Boston (1988) 19. Boyd, R.W., Rzazewski, K., Narum, P.: Phys. Rev. A 42, 5514 (1990) 20. Reintjes, J., She, C.Y., Eckardt, R.C., Karangelen, N.E., Elton, R.C., Andrews, R.A.: Phys. Rev. Lett. 37, 1540 (1976) 21. Reintjes, J., She, C.Y., Eckardt, R.C., Karangelen, N.E., Elton, R.C., Andrews, R.A.: Appl. Phys. Lett. 30, 480 (1977) 22. Kaminow, I.P.: An Introduction to Electro-Optic Devices. Academic Press, New York (1974) 23. Günter, P., Huignard, J.-P. (eds.): Photorefractive Materials and Their Applications. Springer, Berlin, Heidelberg (1989). Part I (1988), Part II (1989) 24. Brabec, T., Krausz, F.: Phys. Rev. Lett. 78, 3283 (1997) 25. Ranka, J.K., Gaeta, A.L.: Opt. Lett. 23, 534 (1998) 26. Rothenberg, J.E.: Opt. Lett. 17, 1340 (1992) 27. Yang, G., Shen, Y.R.: Opt. Lett. 9, 510 (1984) 28. Ranka, J.K., Schirmer, R., Gaeta, A.L.: Phys. Rev. Lett. 77, 3783 (1996) 29. Gaeta, A.L.: Phys. Rev. Lett. 84, 3582 (2000) 30. Ranka, J.K., Windeler, R.S., Stentz, A.J.: Opt. Lett. 25, 25 (2000) 31. Jones, D.J., Diddams, S.A., Ranka, J.K., Stentz, A., Windeler, R.S., Hall, J.L., Cundiff, S.T.: Science 288, 635 (2000) 32. Kaiser, W., Garrett, C.G.B.: Phys. Rev. Lett. 7, 229 (1961) 33. Sprangle, P., Tang, C.-M., Esarez, E.: IEEE Trans. Plasma Sci. 15, 145 (1987) 34. Wagner, R., Chen, S.-Y., Maksemchak, A., Umstadter, D.: Phys. Rev. Lett. 78, 3125 (1997) 35. Agostini, P., Fabre, F., Mainfray, G., Petite, G., Rahman, N.K.: Phys. Rev. Lett. 42, 1127 (1979) 36. Chang, Z.: Phys. Rev. Lett. 79, 2967 (1997) 37. Chang, Z.: Phys. Rev. Lett. 82, 2006 (1999)
R. W. Boyd et al. 38. Corkum, P.B.: Phys. Rev. Lett. 71, 1994 (1993) 39. Kienberger, H.R., Spielmann, C., Reider, G.A., Milosevic, N., Brabec, T., Corkum, P., Heinzmann, U., Drescher, M., Krausz, F.: Nature 414, 509 (2001) 40. Maier, S.A.: Plasmonics: Fundamentals and Applications. Springer, New York (2007) 41. Vial, A., Grimault, A.S., Macías, D., Barchiesi, D., Lamy De La Chapelle, M.: Phys. Rev. B 71, 085416 (2005) 42. Alam, M.Z., De Leon, I., Boyd, R.W.: Science 352, 795 (2016) 43. Reshef, O., Giese, E., Alam, M.Z., De Leon, I., Upham, J., Boyd, R.W.: Opt. Lett. 42, 3225 (2017) 44. Hache, F., Ricard, D., Flytzanis, C.: J. Opt. Soc. Am. B 3, 1647 (1986) 45. Hache, F., Ricard, D., Flytzanis, C., Kreibig, U.: Appl. Phys. A Mater. Sci. Process. 47, 347 (1988) 46. Boyd, R.W., Shi, Z., De Leon, I.: Opt. Commun. 326, 74 (2014) 47. Clerici, M., et al.: Nat. Commun. 8, 15829 (2017)
Robert Boyd Robert Boyd received the BS degree in Physics from MIT and the PhD degree from UC Berkeley under the supervision of Charles Townes. Dr Boyd holds faculty appointments at the University of Rochester and University of Ottawa. His research interests include optical physics and quantum effects in nonlinear optics. He has written two textbooks. Alexander L. Gaeta Alexander Gaeta received his BS, MS, and PhD in Optics from the University of Rochester in Rochester, NY in 1983, 1985, and 1991, respectively. He remained there as a postdoctoral associate from 1991 to 1992. Gaeta joined Columbia Engineering as the David M. Rickey Professor of Applied Physics and Materials Science in 2015. Prior to that, Gaeta was the Samuel B. Eckert Professor of Engineering at Cornell University and was Chair of the School of Applied and Engineering Physics from 2011 to 2014. He is a Fellow of the American Physical Society, the Institute of Electrical and Electronics Engineers, and of Optica (formerly the Optical Society of America) and is also the founding Editor-in-Chief of the journal Optica. Enno Giese Enno Giese received his Dr rer nat from Ulm University in 2015 and subsequently worked on quantum nonlinear optics, quantum imaging, and nonlinear interferometry in the Department of Physics, University of Ottawa. His current position is at the Technical University of Darmstadt, where he focuses on matter wave optics and interferometry in microgravity and in curved space–time for tests of relativity.
77
Coherent Transients Joseph H. Eberly and Carlos R. Stroud Jr.
Contents 77.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1112
77.2
Origin of Relaxation . . . . . . . . . . . . . . . . . . . . 1112
77.3
State Evolution . . . . . . . . . . . . . . . . . . . . . . . . 1113
77.4
Numerical Estimates of Parameters . . . . . . . . . . 1114
77.5 77.5.1 77.5.2 77.5.3 77.5.4
Homogeneous Relaxation . . . Rabi Oscillations . . . . . . . . . Bloch Vector and Bloch Sphere Pi Pulses and Pulse Area . . . . Adiabatic Following . . . . . . .
77.6 77.6.1 77.6.2
Inhomogeneous Relaxation . . . . . . . . . . . . . . . . 1116 Free Induction Decay . . . . . . . . . . . . . . . . . . . . 1116 Photon Echoes . . . . . . . . . . . . . . . . . . . . . . . . . 1117
77.7 77.7.1 77.7.2 77.7.3
Resonant Pulse Propagation . . . . . . . . . . . . . Maxwell–Bloch Equations . . . . . . . . . . . . . . . Index of Refraction and Beer’s Law . . . . . . . . . . The Area Theorem and Self-Induced Transparency
. . . .
1117 1117 1117 1118
77.8 77.8.1 77.8.2
Multilevel Generalizations . . . . . . . . . . . . . . . . Rydberg Packets and Intrinsic Relaxation . . . . . . . . Multiphoton Resonance and Two-Photon Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pump-Probe Resonance and Dark States . . . . . . . . . Induced Transparency . . . . . . . . . . . . . . . . . . . .
1118 1118
77.8.3 77.8.4 77.9
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1114 1114 1114 1115 1115
1119 1120 1121
Disentanglement and “Sudden Death” of Coherent Transients . . . . . . . . . . . . . . . . . . . 1121
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123
Abstract
Coherent transients are excited in atomic and molecular systems when a phase relation is able to persist at least J. H. Eberly () Dept. of Physics and Astronomy, University of Rochester Rochester, NY, USA e-mail: [email protected] C. R. Stroud Jr. The Institute of Optics, University of Rochester Rochester, NY, USA e-mail: [email protected]
semistably between an exciting light field and the system’s response. A steady state can occur whenever the multiplicity of “reservoir” effects in the background has reached a stable average. The theory of coherent transients typically makes an important distinction between two types of average, labeled either homogeneous or inhomogeneous. The terms refer to the type of randomness existing over the experimental “system” and originating in the system’s environment. An example of inhomogeneous randomness and relaxation to a steady state is found in the distribution of transition frequencies of otherwise identical atoms or molecules trapped as impurities in a host crystal and arising from the host crystal’s random strain fields, which induce a permanent energy-level shift in each of the impurities under study. These unchanging level shifts allow, in principle, the experimenter to assign permanent energy-level information in a one-by-one way to system members. Examples of homogeneous randomness are also common. For example, because of continuing collisions, energy-level shifts throughout a vapor cell are random and are constantly changing. Their effect is individually transient, and any consequent relaxation effects are observable only homogeneously over the entire sample, allowing no information to be retained about individual atoms and molecules. We remark at the end of this chapter that the existence of vector-space non factorization, called “quantum entanglement” by Schrödinger in 1935, leads to nonintuitive effects in transients that are currently still under some discussion as not being completely explained. One example is the predicted, and multiply observed, phenomenon labelled ESD, which stands for entanglement sudden death.
Keywords
wave packet Rabi frequency free induction decay Rabi oscillation Bloch sphere photon echo Area theorem qubit
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_77
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J. H. Eberly and C. R. Stroud Jr.
Introduction
cuit, the bias voltage can be adjusted to form such a two-level quantum system [8]. The coherent oscillations of the induced dipole are subject to the influence of the environment. One way that the dipole interacts with the environment is through the field that the dipole itself generates. This field interacts back on the dipole, causing damping and a frequency shift. In classical circuits, we refer to this as the impedance of free space and the frequency shift as self-inductance [9]. The impendance of free space is given exactly by
When an electromagnetic field is incident on matter it polarizes the matter by causing a displacement between the electrons and the nuclei. As the electromagnetic field oscillates the induced polarization also oscillates. The nature of this induced oscillation is determined by both the incident field and the internal dynamics of the matter. This same physics occurs whether the matter is made of isolated atoms or molecules, dopant atoms, quantum dots, or color centers in a host medium, or bulk matter: an insulator, semiconductor p (77.3) or metal. The bulk matter might even be an electronic circuit Z D 0 =0 D 120 (Ohms) ; in the form of a radio receiver or a superconducting quantum interference device (SQUID). and the self-inductance depends on the exact shape of the For all of these systems, the energy is exchanged between dipole radiator. In the case of a circuit, this resistance and inthe field and the matter according to ductance will be a part of the impedance in the circuit. In the case of an atomic or molecular dipole, the impedance of free Z dWm 3 (77.1) space will lead directly to the Einstein A-coefficient damping D d rJE; dt and the Lamb shift of the quantum transition. If the radiator is placed in a cavity, the self-field of the dipole radiator will where Wm is the energy of the matter, J is the induced curbe modified leading to the Purcell-effect-enhanced damping rent, and E is the electric field. This relation holds both rate [10] and modification of the Lamb shift. for classical variables and the corresponding quantum operators. For most cases of interest, the dipole approximation is appropriate. The atoms, molecules, and quantum dots are 77.2 Origin of Relaxation small compared with the wavelength of optical fields, and the circuits are small compared with the wavelength of the We will examine the transient effects that are present as a remicrowave fields with which they interact. The electric field sult of interactions that lead to relaxation. The first step is is then constant over the integration region and can be taken to mention the two types of relaxation, homogeneous and outside the integral. The integral over the current density can inhomogeneous, and to distinguish between coherent and then be carried out reducing the energy absorption integral to incoherent evolution. Relaxation occurs whenever the environment of a physical system fluctuates randomly, that is, it dp dWm D E; (77.2) is too irregular and too complex to be treated fundamentally. dt dt Typically, such fluctuation leads to degradation of some dewhere p is the dipole moment associated with the absorber. gree of regularity in the evolution of a particular subsystem If both the field and the induced dipole are oscillatory, of interest. The inability to track short-term irregularities rethen whether the matter is absorbing or emitting energy quires signal-smoothing of some kind, and this means a loss depends on the phase relation between the two. Coherent op- of microscopic coherence. The time scales of environmentical transients are excited in a quantum system when a stable tal fluctuations determine the division between homogeneous phase relation persists between (typically) an exciting elec- and inhomogeneous decoherences, as follows. tric field and the system’s electronic response. The extreme When environmental fluctuations are sufficiently rapid sensitivity of phase-dependent effects is responsible for the that all dynamical systems in a macroscopic sample exmany applications of optical transient techniques in atomic, perience the whole range of fluctuations in a time that molecular and optical (AMO) physics [1–7]. While all of our is short compared with the time scale of interest, the rediscussion thus far applies equally to many classical or quan- sultant relaxation is called homogeneous. If environmental tum systems, we must specialize somewhat to discuss the differences exist randomly over a macroscopic sample, but dynamic response of the induced polarization. We will as- change relatively slowly in time, then the relaxation is called sume that our dipole is associated with a single resonance in inhomogeneous. As examples, weak distant collisions are exa quantum system. A quantum resonance is associated with perienced constantly by all atoms at thermal equilibrium in two energy levels. We can usually ignore all of the other a vapor cell and give rise to homogeneous relaxation. If the levels of quantum system and reduce our consideration to vapor is sufficiently dilute, the same atoms may neverthea “two-level atom”, equivalently in modern context inch an less retain their own individual velocities and, thus, transition atomic “qubit”. Even in the case of a superconducting cir- frequencies for long times. These transition frequencies are
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relatively fixed in time, as are the frequency shifts experienced by impurity atoms in crystal lattices. However, they are random over a distribution of shifts and give rise to inhomogeneous relaxation. Fundamentally, the distinction between homogeneous and inhomogeneous relaxation is artificial, depending on a separation of timescales that may not always exist. Nevertheless, when it exists, the distinction provides an extremely useful way to classify coherent transients. It is one of the foundations of the subject.
77.3
State Evolution
1113
with the understanding that E ˙ .t/ is a field amplitude that is very slowly varying in comparison to the applied field’s carrier frequency oscillations e˙i!t . Equations (77.6) and (77.7) for the level amplitudes can be used to obtain the equations of motion for the level probabilities jcg .t/j2 and jce .t/j2 , with an important conservation equation as a consequence: d jcg .t/j2 C jce .t/j2 D 0 ; dt
(77.9)
indicating that all of the initial probability remains confined to those two levels. Under these approximations, pure state evolution is fully conservative. However, as indicated in Chap. 72, it is usually impossible to describe fully an interacting two-level system by a state vector such as Eq. (77.4) simply because of environmental fluctuations. There is no way, consistent with the two-level model, to add damping terms such as i„ce to a state amplitude equation such as Eq. (77.6) and (77.7) to account for the relaxation due to those fluctuations. The density matrix approach is required. The pure state Eq. (77.4) is equivalent to the fully coherent density matrix coh :
A very weakly excited dipole transition in a ground-state quantum system responds linearly to an applied timedependent electric field. This is the case when transitions to upper quantum levels are negligibly excited, and this is the basis for classical and linear Lorentzian dielectric theory in Chap. 72. However, the generally unequal spacing of dipoleallowed levels introduces the possibility of partial or even complete transfer of occupation probability from the ground level to just one of the upper levels. If the transition frequency !0 between upper and lower levels is nearly resonant
coh j‰.t/ih‰.t/ with !, the frequency of the exciting electric field, one can, to first approximation ignore other levels and deal with a muchD cg .t/jgi C ce .t/jei hgjcg .t/ C hejce .t/ ; simplified “two-level” quantum system. Its wave function is (77.10) then simply written in terms of lower and upper (ground and excited) states jgi and jei and two complex time-dependent with matrix elements . / D c c , etc. The actual density coh ge g e coefficients: matrix is the same as the one in Eq. (77.10), after it has been averaged over environmental randomness: (77.4) j‰.t/i D cg .t/jgi C ce .t/jei : !
.t/gg .t/ge
.t/ D ; (77.11) We will adopt the Hamiltonian described for such a two .t/eg .t/ee state system in Eqs. (72.38)–(72.41): where .t/ge D Œcg .t/ce .t/avg. , etc. „!0 We note that restriction to two levels implies full conserjeihej jgihgj H D 2h vation of level probability: gg C ee D 1, and the Hermitian (77.5) i C , so that three real varicharacter of means that ge D eg d jeihgjE .t/ C jgihejE .t/ ; ables are sufficient to exhibit the entire temporal evolution of the two-level system’s density matrix. As in Chap. 72, we and apply the Schrödinger equation i„ @t@ j‰.t/i D H j‰.t/i to will denote by U and V the amplitudes of the components of obtain equations of motion for the quantum state amplitudes. the time-dependent dipole expectation value that are, respecSome minor algebra leads to tively, in-phase and in-quadrature with the field. That is, U and V are given by „!0 (77.6) i„cPg D cg dE .t/ce ; 2 hex.t/i D h‰.t/jexj‰.t/i D Refd.U iV / ei!t g ; „!0 C (77.7) ce dE .t/cg ; i„cPe D (77.12) 2 where the positive and negative frequency radiation fields are where d is the matrix element of the transition dipole. Still following Chap. 72, for simplicity both dipole moment madefined in Eq. (72.30): trix element and field amplitude have been taken to be real 1 scalars. The complications of vector notation add little to (77.8) E ˙ .t/ D E ˙ .t/ e i!t ; 2 a discussion of the principles of coherent transients.
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J. H. Eberly and C. R. Stroud Jr.
Chapter 72.3.5 shows that U and V are dynamically coupled to each other and to the inversion W through three equations introduced by Felix Bloch initially for spin resonance theory and now called “optical Bloch equations” and commonly abbreviated as OBE: dU D V ; dt dV D U C *1 W ; dt dW D *1 V : dt
(77.13)
Furthermore, Bloch showed that weak Markovian interactions with environmental oscillators had the effect of decohering the evolution of U , V , and W in specific ways. These decohering influences can be accounted for by adding damping terms into the U , V , and W (i.e., density matrix) equations, rather than into the state vector equations. Chap. 72 shows that when Bloch’s damping terms are included, the OBEs are given by Eq. (72.55): dU D V U=T2 ; dt dV D U C *1 W V =T2 ; dt dW D *1 V .W Weq /=T1 ; dt
a bandwidth that engages some or many other “resonances”. For perspective, in the atomic case, one can note that with p 1 ns and with an intensity less than about 1 GW=cm2 , a laser pulse can be tuned to an isolated atomic resonance, and the interaction can be described in terms of a simple two-level theory. However, laser pulses as short as a few fs in duration, or with intensities greater than 1020 W=cm2 , are available, and such pulses can quasi-resonantly excite many more than just one upper level. For example, if a 1 ps pulse (bandwidth approximately 20 cm1 ) were tuned so that it resonantly excited the n D 95 Rydberg state of an atom, it would also simultaneously and coherently excite all the levels from n D 67 to the continuum limit, while a 10 fs pulse would excite all the levels from n D 4 to the continuum. In other extended cases, in which multiphoton transitions are possible, or simultaneous excitation by a number of near-resonant fields occurs, the appropriately generalized versions of these same parameters apply.
77.5
Homogeneous Relaxation
Homogeneous relaxation is ubiquitous and can be dominant when the wave function phase is rapidly and irregularly interrupted, as from quantum acoustic or other collisional in(77.14) teractions. In the absence of an exciting field (i.e., *1 D 0), the solutions of the OBEs are where D !0 ! is the detuning, and the so-called Rabi .U iV / D .U0 iV0 / e.1=T2 Ci/t ; frequency *1 is given by „*1 D d E . These are the key equa(77.15) W D 1 C .W0 C 1/ et =T1 ; tions of the theory of optical transients [1, 2]. Clearly, *1 is, in frequency units, the dipole interaction energy between the electric field and the quantum system, where the subscripts denote values at t D 0. Then the roles but it has a significance beyond this, as is discussed in of T1 and T2 as relaxation times are clear. They are homogeneous because they apply to each atom individually. Sect. 77.5.2.
77.4
Numerical Estimates of Parameters
The nature of the coherent interaction between a quantum system and a time-dependent electric field is controlled by the relative size of a number of frequencies or rates. In the case of single-photon transitions, they include and *1 , the detuning and Rabi frequency defined above, 1=T2 the transverse, and 1=T1 the longitudinal damping rates, given in Eq. (77.14) and 1=T the inhomogeneous linewidth, and 2 =p the transform bandwidth of the exciting field or pulse. All of these frequencies are defined in Chap. 2, with the exception of the last, so we should mention its effect. A wide excitation bandwidth can overcome the two-level assumption by leading to excitation of several or many levels beyond the single one we have been calling “upper”. This can occur with any exciting pulse, from radio and microwave to X-ray frequencies, if the pulse is short enough to provide
77.5.1
Rabi Oscillations
The OBEs predict damped, although well-defined (coherent, many-period) oscillations of the inversion W with the angular frequency *1 even in the presence of T1 and T2 if *1 is large enough: *1 T1 1 and *1 T2 1. These are usually called Rabi oscillations and were originally called optical nutations following the terminology of nuclear magnetic resonance. Figure 77.1 shows the behavior of the Bloch variables undergoing Rabi oscillations in a representative case.
77.5.2
Bloch Vector and Bloch Sphere
Coherent dynamical behavior is simplest for times much shorter than the relaxation times T1 and T2 . In this case, the
77
Coherent Transients
1115
W
1.0
77
W V
14
0.5
12
U
–V
0.0
–U 1
–0.5 2
–1.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0 t/ T2
Fig. 77.1 Damped Rabi oscillations of the atomic variables after sudden turn-on of the field. In this example, T1 D T2 , T2 D 1, and *1 T2 D 15
Fig. 77.2 Orbits of the Bloch vector on the unit sphere for various ratios of the detuning to the Rabi frequency *1
77.5.3
Pi Pulses and Pulse Area
damping terms can be dropped from the OBEs and the reThe exactly resonant . D 0/ undamped OBEs can be sulting equations can be written in the form (Chap. 72.3.5) solved analytically even for arbitrarily time-dependent exciting pulse envelopes. The solutions are dU (77.16) D U with D .*1 ; 0; / ; dt U.t; 0/ D 0 ; V .t; 0/ D sin .t/ ; where U D .U; V; W / is the Bloch vector, and acts as a torque vector defining the axis and rate of precession. Conservation of probability is reflected in the unit length of the undamped Bloch vector: U U D 1. All possible states of an undamped two-level quantum system can be mapped onto a unit sphere in U –V –W space. Conventionally, W defines the polar axis with the ground state at the south pole, and the excited state at the north pole. Points on the sphere surface between the poles represent coherent superpositions of the two states. The azimuthal angle represents the phase between the expectation value of the quantum dipole moment and the incident field. In Fig. 77.2, the solutions to Eq. (77.16) are shown for the case of a square pulse applied to the system in its ground state at t D 0. The solutions in this case are *1 sin *t ; * *1 V .t; / D 2 .1 cos *t/ ; * *2 W .t; / D 1 C 12 .1 cos *t/ : *
W .t; 0/ D cos .t/ ;
(77.18)
a rotation of the Bloch vector in the V –W plane. The rotation angle is called the pulse area, defined by the integrale Zt
d dt *1 .t / D „ 0
.t/ 1
Zt
0
dt 0 E .t 0 / :
(77.19)
1
The area under the envelope of the Rabi frequency is the same as the angle through which an exactly-resonant Bloch vector turns due to the pulse. If D , the system is driven from the ground state exactly to the excited state. This is “ pulse” inversion. A “2 -pulse” takes the system from the lower state through the excited state and back to the lower state. The Bloch vector rotation angle does not depend upon the shape of the field pulse, only on the area of the pulse.
U.t; / D
77.5.4
Adiabatic Following
The Bloch vector picture for fixed and *1 can be used for semiquantitative predictions even if and *1 are time dependent, if they change slowly. For example, if is moved slowly, the Bloch vector follows closely. It is possible to For any , the solution orbit is a circle on the surface of the achieve complete inversion smoothly in this way. If the field sphere with the orbit passing through the south pole. The rate is initially tuned far below resonance so is large and negat which the system precesses about q the circle is given by the ative, then points approximately toward the south pole of generalized Rabi frequency * 2 C *21 . the Bloch sphere. As shown in Fig. 77.3, the Bloch vector (77.17)
1116
J. H. Eberly and C. R. Stroud Jr. W
Free induction decay – dipoles dephase B A
A –V
t=t –
t=0
B
After π pulse – dipoles rephase –U
U
B A
⍀
Fig. 77.3 In adiabatic inversion, the Bloch vector precesses in a small cone about the torque vector as the torque vector goes from straight down to straight up
t=t +
A
B
t = 2t
Fig. 77.4 The ensemble of dipole moments spreads due to the distribution of resonance frequencies. The distribution of Bloch vectors in the U –V plane is shown at various times after the initial short-pulse excitation. By the time t 0 , the dipoles have spread uniformly around the unit circle. A -pulse then flips the relative orientation of the dipoles, so that they subsequently rephase
then precesses about the torque vector in a very small circle near the south pole. If the field frequency is now slowly changed (chirped) so that goes from a large negative value to a large positive value, then every atomic Bloch vector will More specifically, if all systems are first exposed to a 0 continue to precess rapidly around the torque vector and follow it as it proceeds from pointing nearly straight down to pulse, so that at t D 0 pointing nearly straight up. In this way, the population can U.0; / D 0 ; be transferred from lower to upper levels. V .0; / D sin 0 ;
77.6
Inhomogeneous Relaxation
W .0; / D cos 0 ;
(77.21)
then if E D 0 for t > 0, an individual system with detuning The fact that individual otherwise identical quantum systems evolves according to Eq. (77.15): in a sample may have slightly different resonance frequencies produces a number of notable phenomena. Given a dis(77.22) U iV D i sin 0 e.1=T2 i/t : tribution g./ of detunings in a dilute sample of density N , the macroscopic collective polarization can be written as The macroscopic polarization is found by summing the individual .U iV / values over the detuning distribution g./. P .t/ D N hex.t/i If we temporarily ignore competition from homogeneous Z
i!t d : (77.20) decay (take 1=T2 0) and assume the most common inhoD Nd g./ Re .U iV / e mogeneous lineshape (i.e., Doppler–Maxwellian):
77.6.1
Free Induction Decay
Free induction refers to evolution of the polarization in the absence of an exciting field. For *1 D 0, the Bloch vector of a system with < 0 precesses counterclockwise in the “transverse” U –V plane. In a macroscopic sample, there are many values of , and about as many are positive as negative. Thus, a collection of Bloch vectors, all pointing in the V direction at t D 0, will rapidly fan out in the U –V plane due to differing positive and negative precession rates, and after a short time, the net U and V values will be zero. This is free induction decay (FID) of polarization and is indicated in the top row of Fig. 77.4.
T 2 2 g./ D p e.0 / T =2 ; 2
(77.23)
where 1=T is defined as the width (standard deviation) of the Doppler distribution, and 0 is the detuning of the zerovelocity atoms, then the collective result is
P .t/ D Nd sin 0 sin !t ei0 t exp t 2 =2T 2 :
(77.24)
The detuning “inhomogeneity” in the sample leads to dephasing of the collective dipole moment at the rate 1=T , so the inhomogeneous relaxation time is obviously T . This is illustrated by the decrease of collective alignment of Bloch vectors in the top row of Fig. 77.4. The Doppler effect for
77
Coherent Transients
1117
atoms makes a common example, and in a typical room tem- of mass of the atom rather than the internal electron coordiperature gas, a visible transition has a Doppler width given nate. The field is generalized from Eq. (77.2) to by 1=.2 T / 1:5 GHz so that T 1010 s. 1 E.t; z/ D ŒE .t; z/ ei.Kz!t / C c:c: ; (77.26) 2
77.6.2
Photon Echoes
and the macroscopic polarization is the correspondingly generalized form of Eq. (77.20): A photon echo is generated by pulse-induced recovery of Z ˚ a nonzero P .t/ after free induction decay (FID) and P .t/ ! P .t; z/ D Nd g./ Re .U iV / ei.Kz!t / d : 0. This analog of the spin echo effect is possible when each (77.27) near-resonant quantum system retains its own detuning for a relatively long time, i.e., up to an average collision time T2 . During FID, the U –V projection of every atom’s Bloch vector precesses steadily clockwise or counterclockwise, depending on the sign of its . Thus, the Bloch vectors could be rephased, if they could all be forced at the same moment to reverse their relative sense of precession. The prototypical echo scenario begins with FID at t D 0, as an established P .t/ approaches zero for t T . This is followed by injection of a -pulse at the time t 0 , where t 0 T . The effect of the pulse is to reverse the sign of V and W (recall Eq. (77.21)), in effect flipping the equatorial plane of the Bloch sphere upsidedown. Thus, for t t 0 , we have Bloch vectors fanning back together. The macroscopic polarization obeys
The difference between K and k D !=c indicates that the refractive index is nonzero. The traveling-pulse rotating frame is also obtained by replacing !t by !t Kz. If the field envelope E is slowly varying, its second derivatives can be dropped when E.t; z/ is substituted into Maxwell’s wave equation. The resulting dispersive and absorptive reduced wave equations are Z
2 2 2 K k E D 4 k Nd g./U.t; z; / d ; Z @ @ E D 2 k 2 Nd g./V .t; z; / d : Ck K @z @.ct/ (77.28)
P .t/ D Nd sin 0 sin !t exp .t 2t 0 /2 =2T 2 et =T2 ; The Bloch equations along with these reduced wave equa(77.25) tions form the self-consistent Maxwell–Bloch equations that are used to treat resonant propagation problems in quantum where the last factor reinserts the effect of homogeneous optics and laser theory [1, 2, 5, 11]. dipole damping. We require T2 t 0 T for a strong echo signal in the neighborhood of t D 2t 0 . The result is illustrated in Fig. 77.4. An “echo” of the ini- 77.7.2 Index of Refraction and Beer’s Law tial excitation at t D 0 appears at the time t D 2t 0 . After this, FID occurs again. This second decay can also be reversed by If a weak field propagates in a ground-state medium (W applying another -pulse, and so on, until t T2 , at which 1), the Bloch equations have simple quasi-steady-state sotime the inevitable and irreversible homogeneous relaxation lutions cannot be avoided. The scenario employing -pulse reversal *1 is only the most ideal, leading to the most complete echo, and ; (77.29) U D 2 C 1=T22 other pulse areas will also lead to echoes. A more important *1 =T2 factor is that the reversing pulse must be short enough that : (77.30) V D 2 negligible dephasing takes place during its application. C 1=T22
77.7 77.7.1
Resonant Pulse Propagation Maxwell–Bloch Equations
When U and V are substituted back into the reduced wave equations Eq. (77.28), the dispersive equation gives the index of refraction n D K=k: Z g./ 4 Nd 2 2 d ; (77.31) n 1D 2 „ C 1=T22
Time-dependent atomic dipole moments created by applied fields are themselves a source of fields, another form of and the absorptive equation predicts steady-state field attencoherent transient. We limit discussion to plane-wave propa- uation during propagation: gation in the z-direction. Note that we use z rather than Z for 1 @ convenience, although in the dipole approximation, the coorE D ˛B E : (77.32) @z 2 dinate entering our equations is the coordinate of the center
77
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J. H. Eberly and C. R. Stroud Jr.
The constant ˛B given by 2
˛B D
4 Nd ! „cT2
δυ/δz
Z 2
g./ d C 1=T22
(77.33)
is called the extinction coefficient or the reciprocal Beer’s length. Since field intensity I.z/ is proportional to E 2 , the solution to the absorptive equation is I.z/ D I.0/ e˛B z :
0
π
2π
3π
υ
(77.34)
This relation is called Beers law. Both the dispersive and absorptive results are familiar from classical physics [12], with the important distinction that here the quantum „dependence is apparent in the oscillator strength, which is only an empirical parameter in Lorentzian dielectric theory [11, 12].
4π
Fig. 77.5 McCall–Hahn area theorem for an absorbing medium. On propagation the area of the pulse will follow the arrows toward one of the stable values, 0; 2 ; 4 ; : : :
77.8
Multilevel Generalizations
77.8.1 Rydberg Packets and Intrinsic Relaxation 77.7.3
The Area Theorem and Self-Induced Transparency
A form of pulse propagation with no classical analog arises in the short-pulse limit. By integration over the entire pulse, the absorptive Maxwell equation becomes an equation for @=@z, where is the pulse area defined in Eq. (77.19). In the short-pulse limit, the relaxation terms in the OBEs can be ignored and when substituting from the OBEs we obtain the McCall–Hahn area theorem [1]: 1 @ .z/ D ˛B sin .z/ : @z 2
(77.35)
When sin , this predicts the same exponential attenuation as Eq. (77.32) for the case of a small area pulse, .z/ , but in the case of larger area pulses, the behavior dictated by Eq. (77.35) is quite different. The area decreases during propagation for areas in the range 0 < .z/ < , but it increases for areas < .z/ < 2 . As seen from Fig. 77.5, this change of area with propagation causes the pulse area to evolve to one of the stable values 0; 2 ; 4 : : : There is one special pulse, a soliton solution with area exactly 2 , which propagates without shape change in the short-pulse limit, given by t z=v 2„ E .t; z/ D sech ; (77.36) d where is the pulse duration, which is arbitrary but must satisfy the short-pulse inequality T1 ; T2 . The soliton’s group velocity is determined by the dispersive wave equation to be c : (77.37) vD 1 C 12 ˛B c
A short laser pulse can populate a band of excited states, and their probability amplitudes will exhibit mutual coherence [13]. This single-atom coherence is transient, even without collisions or other external perturbations to disrupt it, and its decay can be called intrinsic relaxation. The decay is basically a dephasing. The dipole moments associated with the excited band interfere due to the wide variety of resonance frequencies of the states in the superposition. Because of the discreteness of the energy levels of any bound quantum system, this relaxation has its own unique characteristics, including similarities with both homogeneous and inhomogeneous decay. The wave function for a coherently excited system can be expressed in the interaction picture in the form [13] X bn .t/ ei!n t n .r/ ; (77.38) ‰.r; t/ D a.t/ g .r/ C n
where g .r/ is the ground-state wave function, and n labels the states in an upper band with excitation frequencies !n !. If j!n !j !n , the transition frequency !n can be expanded about the principal quantum number n of the resonant excited state En D „! to obtain @2 !n @!n 1 C C .n n/2 @n 2Š @n2 2 2 C .n n/2 C : (77.39) D !n C .n n/ TK TR
!n D !n C .n n/
Thus, 2 =TK is the mean frequency separating neighboring levels, i.e., TK =2 D „ .E/, where .E/ is the density of excited states, and 2 =TR is the mean change in this frequency separation; TK is the same as the Kepler period of a classical orbit, and TR is the revival time. Substituting the The group velocity can be slower than the speed of light by Bohr frequencies into the definitions for TK and TR yields TR D nTK =3. orders of magnitude if ˛B c 1.
77
Coherent Transients
a
1119
a
|e² φ φ
b φa
|j²
77
|j² φb |e²
|g²
|g²
b Fig. 77.7 Model two-photon resonances. Two photons couple the ground state jgi with an excited state jei. Many intermediate nonresonant levels jj i are present. In (a) we have the cascade system and in (b) the ƒ pump-probe system
The coherent quantum wave packet behaves like a classical particle for many Kepler periods, gradually spreading out as the second exponential in the sum Eq. (77.40) begins to contribute. This spreading of the wave packet produces the intrinsic relaxation of the collective dipole moments from the various transitions. However, because the levels are discrete, this decay is not permanent but is reversed and leads d to a spontaneous “revival” of the original wave packet [6], without the need for a -pulse to produce an “echo”. In its evolution toward the revival, the wave packet passes through a number of fractional revivals in which miniature replicas of the original wave packet are equally spaced around the orbit, each traveling at the velocity of a particle traveling in a corresponding classical Kepler orbit. This comFig. 77.6 The free evolution of a Rydberg wave packet made up of plex time evolution arises from the spreading of the wave a superposition of circular-orbit states centered about n D 360. (a) Ini- packet all the way around the orbit, so that the head and tail of tially, the wave packet is to good approximation a minimum uncertainty the packet interfere with each other, producing interference wave packet in all three dimensions, but after 12 orbits, (b) the packet has spread all the way around the orbit so that the head and tail of the fringes. The further evolution of this fringe pattern produces wave packet overlap, producing interference fringes. (c) After 40 orbits, the various revival phenomena shown in Fig. 77.6. c
t D TR =3, the fringes have produced the one-third fractional revival in which three miniature replicas of the original wave packet are equally spaced around the orbit. (d) After 120 orbits, t D TR , the complete wave packet revival occurs
77.8.2
Multiphoton Resonance and Two-Photon Bloch Equations
For times t TR , the expansion can be truncated after the third term. Then the wave function is given approximately by Multiphoton transitions (Fig. 77.7a) introduce new coherent X transient phenomena. If levels jgi and jei have the same par‰.r; t/ a.t/ g .r/ C ei!n t bnCm .t/ ei2 mt =TK ity, two photons from the same laser field are sufficient to m excite level jei. For simplicity, we regard jei as a single state, 2 ei2 m t =TR nCm .r/ ; (77.40) but any number of intermediate levels jj i of opposite parity where m D n n. For high Rydberg states, n 1, the may be present. Substituting the state vector timescales associated with the two exponentials inside the X sum are quite different. For times t TK , the individual .t/jgi C bj .t/ ei!t jj i C ae .t/ ei2!t jei j‰.t/i D a g levels are not resolved, thus the laser excites what is effecj tively a continuum with a density of states .E/. In that (77.41) case, the ground-state population simply decays exponentially at the rate given by first-order perturbation theory, % D into the Schrödinger equation yields .2 =„/d 2 E 2 .E/. At longer times t TK , the first expodag 1X nential contributes, but the second does not, giving a simple *gj bj ; (77.42) i D dt 2 j Fourier series time dependence. In this regime, the evolution of the wave function is just periodic motion of a wave packet dbj 1 (77.43) i D jg bj Œ*jg ag C *je ae ; around a Kepler orbit, as is illustrated in Fig. 77.6a. dt 2
1120
J. H. Eberly and C. R. Stroud Jr.
i
dae 1X *ej bj ; D eg ae dt 2 j
(77.44)
where the s are the detunings, and the *s are the Rabi frequencies for the dipole-allowed transitions. For example, *gj D dgj E =„, and jg D .Ej Eg /=„ ! and eg D .Ee Eg /=„ 2!. If the states jj i are not too close to resonance, the bj oscillate rapidly and to a first approximation average to zero. A better approximation is to retain the small nonzero solution for bj obtained by setting dbj =dt D 0 in Eq. (77.43) to obtain bj D
Œ*jg ag C *je ae ; 2jg
jei have the same parity. Thus, the quantity U .2/ iV .2/ is the two-photon analog of U iV in the original OBEs; however, it cannot serve as a source term in the Maxwell equation. In the case of continuous wave (CW) applied fields, the solutions to the two-photon OBEs are formally identical to those for a two-level atom. In the case of pulsed fields, however, the detuning 2 .t/ is automatically “chirped” in frequency by the Stark shifts. This chirping may significantly modify the dynamics. Multiphoton generalizations of the Bloch equations can be made for other arrangements and numbers of levels. The two-photon version applies to threelevel ƒ and V configurations as well as to the cascade system shown in Fig. 77.7a, for which they were derived.
(77.45)
77.8.3 Pump-Probe Resonance and Dark States which can be used to eliminate bj from the equations for ag and ae . This is called adiabatic elimination of dipole coherence. In this approximation, levels jgi and jei are directly coupled to each other, and two-photon coherence arises. The coupling of levels jgi and jei is similar to the two-level coupling described in Sects. 77.3–77.6, and two-photon Bloch equations analogous to Eq. (77.16) are the result: dU .2/ D .2/ V .2/ ; dt dV .2/ D .2/ U .2/ C *.2/ W .2/ ; dt dW .2/ D *.2/ V .2/ : dt
Dark states or trapping states occur whenever a fielddependent linear combination of active levels is dynamically disconnected from the other levels. This occurs, e.g., in a pump-probe interaction, which fits the scenario of Fig. 77.7b, if two lasers instead of one are used to excite level jei from the ground level via two-photon resonance. A strong steady laser a is applied for the jgi ! jj i transitions and a weak tunable “probe” laser b for the jj i ! jei transitions. In the simplest format, eg D 0, and all the jj i levels can be combined into a single level labeled j2i. The three-level state vector can be written in terms of field-free states
(77.46)
j‰.t/i D ag jgi C b2 j2i C ae jei ; Here the superscript (2) indicates that the variables are idenor, in terms of field-dependent dressed states, tified with the two-photon jgi ! jei transition. The various coefficients are similarly generalized [14]. j‰.t/i D AT jT i C b2 j2i C AS jSi ; For example, the two-photon Rabi frequency is given by *.2/
1 X dgj dje 2 E ; 2 j „2 jg
(77.49)
(77.50)
where *jT i *a jei *b jgi, *jSi *a jgi C *b jei and (77.47) AS .t/ *1 Œ*a ag C *b ae ; 1
and the two-photon detuning .2/ incorporates the laserinduced level shifts .2/ ej C jg 1 X jdej j2 E 2 .t/ 1 X jdjg j2 E 2 .t/ C C : 4 j „2 ej 4 j „2 jg (77.48) The last two terms give the difference in the AC Stark shifts of the upper and lower levels produced by the laser field. As before, W .2/ is the inversion, but U .2/ and V .2/ are somewhat different. They cannot be directly tied to the expectation value of a dipole moment because here levels jgi and
AT .t/ * Œ*a ae *b ag :
(77.51) (77.52)
p The normalizing factor is * *a 2 C *b 2 . The state jT i is an eigenvector of the three-level rotating wave approximation (RWA) Hamiltonian, with eigenvalue zero, and the amplitude AT .t/ is a constant of motion. Thus, jT i is termed a trapping state, and the population in jT i is inaccessible to the (possibly very strong) laser fields. At twophoton resonance, this conclusion is robust, not depending strongly on the idealized conditions assumed here. In fact, AT , the trapping state amplitude, is an adiabatic invariant, remaining constant to first order even under slow changes in *a and *b . In a pump-probe experiment, this trapping state is observed as an abrupt drop in probe absorption, as
77
Coherent Transients
the probe frequency is tuned through two-photon resonance. Since only two-photon resonance is required (both transitions can be equally detuned) this coherent transient effect has no analog in two-level physics. The ideal method for exciting the trapping state from the ground state uses another coherent transient process called counterintuitive pulse sequencing in which pulse b is turned on first. The trapping state jT i is essentially the ground state jgi if *a D 0. Thus, if *b is turned on first, and then *a is turned on later, the ground state adiabatically becomes the trapping state, and all initial probability flows smoothly with it. An essential point is the ease with which pump-probe adiabaticity is maintained, particularly for strong fields on resonance, in contrast to one-photon adiabaticity, which is never achieved at strong-field resonance. In the pump-probe case, one must only satisfy the inequality ˇ ˇ ˇ d*a
2 d*b ˇˇ 2 3=2 ˇ ; (77.53) * * a ˇ *a C *b ˇ dt b dt which is automatically accomplished by counterintuitive pulse sequencing. The inequality allows one to tolerate a rapid change of *b while *a D 0. Then after *b has reached a high value, *a can also be turned on very rapidly because the right-hand side of Eq. (77.53) is already very large. This is “counterintuitive” excitation because if the population is in level jgi, it is “natural” to turn on pulse *a first, not *b . It can be dramatically beneficial to use counterintuitive excitation when it is important to avoid relaxation associated with level j2i.
77.8.4
Induced Transparency
1121
depend only on a single variable Z uT , where u is the pulse’s constant velocity in the moving frame. Soliton solutions are given (for a D b ) by *a D A sech KZ ;
*b D B tanh KZ ;
(77.57)
and ag D tanh KZ ; 2iKu sech KZ ; a2 D A B ae D sech KZ ; A
(77.58)
where the parameters A; B; Ku are nonlinearly related to the pulse length : Ku 1=
and .2=/2 D A2 B 2 :
(77.59)
The moving frame velocity is given by 1=u D 2=A2 , and the expression for the lab frame velocity V is 1=V D 1=c C 2=A2 . If B ! 0, then A ! 2=, which is the exact McCall–Hahn formula for the two-level one-pulse soliton amplitude [1]. No adiabatic condition was invoked in obtaining the soliton solutions. The physical measure of adiabaticity comes from the pulse duration . If is short, an appreciable population appears transiently in level j2i, but if is long (an adiabatic pulse), the population skips level j2i and goes directly from jgi to jei and back again during the pulse. Note that the sech and tanh functions are ideally counterintuitive, with pulse b starting infinitely far ahead of pulse a. In practice, the infinite leading edge of the tanh function plays no role and can be truncated to several times without appreciable change in the character of the pulse pair.
The foregoing results for three-level excitation can be extended to resonant pulse propagation in three-level media. 77.9 The equations governing simultaneous two-pulse evolution in the local-time coordinates cT ct z and z are
Disentanglement and “Sudden Death” of Coherent Transients
The existence of entanglement (nonseparability) of states is (77.54) the most prominent evidence that quantum mechanics allows possibilities not known in classical physics. This has con(77.55) sequences for coherent transients, allowing them to exhibit nonintuitive effects unlike any discussed up to this point. An example that directly illustrates this point shows that two where, as before, entangled atoms whose inversion and coherences decay ex2 ponentially can have an entanglement that does not decay 4 da N !a ; etc: (77.56) similarly smoothly and reaches zero long before the atoms a D „c reach their final states. This shows that transients of an entanNote that the bilinear combination 2ag a2 corresponds to U gled pair of atoms can be very different from the transients iV in Eq. (77.27). affecting the atoms separately. Soliton-like pulses can propagate in three-level media. We imagine the situation sketched in Fig. 77.8, two twoBoth pulses must compete for interaction with level j2i. They level atoms A and B of the type discussed at length already, @*a D ia ag a2 ; @ @*b D ib ae a2 ; @
77
1122
J. H. Eberly and C. R. Stroud Jr.
Atom A
Atom B
Cavity A
Cavity B
Fig. 77.8 Illustration of a setup in which two partially excited atoms A and B are located inside two spatially separated cavities that are possibly very remote from each other. The two atoms are assumed to initially be entangled, but they have no interaction
prepared by an earlier interaction in a partially excited entangled mixed state and subsequently located remotely from each other, without further interaction. They each must eventually, because of spontaneous emission, come to their ground states, creating the final joint state jgiA ˝ jgiB , which is clearly not entangled, a factored simple product. The question is, what is the manner of evolution by which the entanglement of the pair evolves toward zero? The standard master equation methods [15] for investigating spontaneous emission [16] can be applied to each atom separately since they are not interacting with each other. For any initial state AB .0/, the density operator at t can be expressed as
AB .t/ D
4 X
K .t/ AB .0/K .t/ ;
(77.60)
and the upper level occupation of atom A can be found to A t =0 decay exactly as expected: ee .t/ D aC1 , where 0 is 3 e the usual spontaneous emission lifetime. For nonlocal transients, we need a time-dependent measure of entanglement, and there are several options [19], all related to the joint entropy of the two-atom system. We will use the concurrence C.t/ of Wootters [20], which has the convenient normalization 1 C.t/ 0, where C D 1 represents completely entangled atoms (such as in a pure Bell state, for example), and C D 0 denotes the complete absence of entanglement. For the specific case a D 1, one finds for the state Eq. (77.61) the initial concurrence C.0/ D 23 , indicating a state with partial two-party coherence (incomplete entanglement). At time t, one then finds: C AB .t/ D
2 maxf0; et =0 f .t/g ; 3
(77.63)
p where f .t/ D 1 2! 2 C ! 4 , and ! 1 et =0 . The strikingly nonintuitive consequence of this transient is that C.t/ reaches 0 abruptly after a finite time (disentanglement suffers a “sudden death”), if 2! 2 C ! 4 1. In fact, only a minor algebraic rearrangement shows that this strange condition must occur, and that the finite sudden death time t0 is given by " p # t0 2C 2 ln 0:53 : (77.64) 0 2
D1
The nonlocal coherent transient behavior of entanglement for the entire range of allowed a values [18] is shown in where the so-called Kraus operators [17] K .t/ are available Fig. 77.9. This shows that the concurrence undergoes familin closed form in this case [18]. iar smooth and infinitely long decay only for a values in the For illustration purposes, we will choose a partially coherlimited range 13 > a 0. Otherwise, sudden (finite-time) ent initial state, expressed by a two-atom density matrix with death occurs sooner or later. In our example, nonlocal cohera single free parameter a: ence becomes zero most abruptly after a finite time for a D 1, 0 1 the case that was calculated in Eq. (77.63). It appears that a 0 0 0 B C these results are not exceptional. Sudden termination of en1 B0 1 1 0 C C: tanglement has also been predicted for two-party continuum
AB .0/ D B (77.61) 3B 0 C @0 1 1 A 0 0 0 1a Here, the convention is to label the rows and columns in the “inverted” order ee; eg; ge; gg. That is, the probabilities are a=3 that both atoms are in state jei at t D 0 and .1 a/=3 in state jgi, and with probability 1=3, they are in either one of the opposite pair states, with the sum equaling 1, of course. Local transient decay, e.g., of either atom’s inversion W , can be evaluated separately from their reduced density matrices,
A TraceB f AB g, etc. For example,
A D B hej AB jeiB C B hgj AB jgiB ! 1 1Ca 0 D ; 3 0 2a
C (ρ) 1 2 1
3 3 0 0.5
0
1 0.5
1
2 2.5 t 3
2 a
3
3
Fig. 77.9 Effect of spontaneous emission on concurrence of two two-
(77.62) level atoms, given the initially entangled mixed state Eq. (77.61) depending on the single parameter 1 a 0
77
Coherent Transients
states, as well as for qubit pairs experiencing only T2 decay, in contrast to the combined T1 and T2 decay appropriate to spontaneous emission as treated here.
References 1. Allen, L., Eberly, J.H.: Optical Resonance and Two-Level Atoms. Dover, New York (1987) 2. Shore, B.W.: Theory of Coherent Atomic Excitation vol. 1 & 2. Wiley, New York (1990) 3. Meystre, P., Sargent III, M.: Elements of Quantum Optics. Springer, New York (1990) 4. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: AtomPhoton Interactions. Wiley, New York (1992) 5. Scully, M.O., Zubairy, M.S.: Quantum Optics. Singapore, Cambridge (1997) 6. Schleich, W.P.: Quantum Optics in Phase Space. Wiley-VCH, New York (2001) 7. Gerry, C.C., Knight, P.L.: Introductory Quantum Optics. Singapore, Cambridge (2005) 8. M. H. Devoret, A. Wallraff, and J. M. Martinis: arXiv:condmat/0411174 (2004). 9. Stratton, J.: Electromagnetic Theory. McGraw-Hill, New York (1941) 10. Purcell, E.M.: Phys. Rev. 69, 681 (1946) 11. Milonni, P.W., Eberly, J.H.: Lasers. Wiley, New York (1988) 12. Jackson, J.D.: Classical Electrodynamics, 2nd edn. Wiley, New York (1975) 13. Fedorov, M.V.: Atomic and Free Electrons in a Strong Light Field. World Scientific, New York (1997) 14. Faisal, F.H.M.: Theory of Multiphoton Processes. Plenum, New York (1987) 15. This citation is intended for cross-reference to the location in the Handbook where the Master Equation approach to density matrix evolution is explained.
1123 16. This citation is intended for cross-reference to the location in the Handbook where spontaneous emission is treated systematically. 17. Daffer, S., Wódkiewicz, K., McIver, J.K.: A complete discussion of Kraus operators appropriate to Bloch vector evolution is given in. J. Mod. Opt. 51, 1843 (2004) 18. Yu, T., Eberly, J.H.: Phys. Rev. Lett. 93, 140404 (2004) 19. This citation is intended to refer to the chapter on Quantum Information. 20. Wootters, W.K.: Phys. Rev. Lett. 80, 2245 (1998)
Joseph H. Eberly Joseph H. Eberly is Andrew Carnegie Professor of Physics and Professor of Optics at the University of Rochester. He has published more than 400 research and review papers as well as several books dealing with a variety of topics in quantum and classical optics, including spontaneous revivals in cavity QED, photon–atom interactions, nondiffracting beams, entanglement versus duality, high-field atomic attophysics, and nonlinear propagation of short optical pulses. Carlos R. Stroud Jr. Carlos R. Stroud Jr. is Professor of Optics, Professor of Physics, and Director of the Center for Quantum Information at the University of Rochester where he works in a variety of areas of experimental and theoretical quantum optics and atomic physics. His group pioneered the area of Rydberg electron wave-packet physics observing localization, decays, revivals, and interferometry with a single electron.
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Multiphoton and Strong-Field Processes Marcelo Ciappina
, Alexis A. Chacon S.
Contents
78.5
78.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1125
78.2 78.2.1 78.2.2 78.2.3 78.2.4 78.2.5 78.2.6 78.2.7 78.2.8
Weak-Field Multiphoton Processes . . . . . Perturbation Theory . . . . . . . . . . . . . . . Resonant-Enhanced Multiphoton Ionization . Multielectron Effects . . . . . . . . . . . . . . . Autoionization . . . . . . . . . . . . . . . . . . . Coherence and Statistics . . . . . . . . . . . . . Effects of Field Fluctuations . . . . . . . . . . Excitation with Multiple Laser Fields . . . . . Waveforms . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
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1126 1126 1127 1127 1127 1127 1128 1128 1128
78.3 78.3.1 78.3.2 78.3.3 78.3.4 78.3.5 78.3.6 78.3.7 78.3.8
Strong-Field Multiphoton Processes . . . . . Nonperturbative Multiphoton Ionization . . . . Tunneling Ionization . . . . . . . . . . . . . . . . Multiple Ionization . . . . . . . . . . . . . . . . . Above Threshold Ionization . . . . . . . . . . . High-Order Harmonic Generation . . . . . . . . Stabilization of Atoms in Intense Laser Fields Molecules in Intense Laser Fields . . . . . . . . Microwave Ionization of Rydberg Atoms . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
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1129 1129 1129 1130 1130 1131 1133 1133 1134
78.4 78.4.1 78.4.2 78.4.3 78.4.4 78.4.5
Strong-Field Calculational Techniques Floquet Theory . . . . . . . . . . . . . . . . Direct Integration of the TDSE . . . . . . . Volkov States . . . . . . . . . . . . . . . . . Strong-Field Approximations . . . . . . . . phase-space Averaging Method . . . . . .
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1135 1135 1135 1136 1136 1136
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M. Ciappina () Physics Program, Guangdong Technion – Israel Institute of Technology Shantou, Guangdong, China e-mail: [email protected] A. A. Chacon S. Dept. of Physics, Pohang University of Science and Technology and Max Planck Research Initiative/Korea Pohang, Korea, Republic of e-mail: [email protected] M. Lewenstein The Barcelona Institute of Science and Technology, Quantum Optics Theory, ICFO and ICREA Av. Carl Friedrich Gauss 3, 08860 Castelldefels, Spain e-mail: [email protected]
78
, and Maciej Lewenstein
Atto-Nano Physics . . . . . . . . . . . . . . . . . . . . . . 1137
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138
Abstract
The excitation of atoms by intense laser pulses can be divided into two broad regimes: the first regime involves relatively weak optical laser fields of long duration, and the second involves strong fields of short duration. In the first case, the intensity is presumed to be high enough for multiphoton transitions to occur. The resulting spectroscopy is not limited by the single-photon selection rules for radiative transitions. However, the intensity is still low enough for a theoretical description based on perturbations of field-free atomic states to be valid, and the time dependence of the field amplitude does not play an essential role. In the second case, the field intensities are too large to be treated by perturbation theory, and the time dependence of the pulse must be taken into account. In addition to a detailed picture of the two subjects described above, we include a discussion on the generation of attosecond pulses and applications of both high-order harmonic generation (HHG) and above-threshold ionization (ATI). Furthermore, we incorporate a brief summary about the incipient field of atto-nano physics. Keywords
strong laser fields Rydberg atoms multiphoton ionization multiphoton processes attosecond physics
78.1
Introduction
The excitation of atoms by intense laser pulses can be divided into two broad regimes determined by the characteristics of the laser pulse relative to the atomic response. The first regime involves relatively weak optical laser fields of long
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_78
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duration (> 1 ns), and the second involves strong fields of short duration (< 10 ps). These will be referred to as the weak-long (WL) and strong-short (SS) cases, respectively. In the case of atomic excitation by WL pulses, the intensity is presumed to be high enough for multiphoton transitions to occur. The resulting spectroscopy of absorption to excited states is potentially much richer than single-photon excitation because it is not limited by the single-photon selection rules for radiative transitions. However, the intensity is still low enough for a theoretical description based on perturbations of field-free atomic states to be valid, and the time dependence of the field amplitude does not play an essential role. The SS case is fundamentally different in that the atomic electrons are strongly driven by fields too large to be treated by perturbation theory, and the time dependence of the pulse as it switches on and off must be taken into account. Atoms may absorb hundreds of photons, leading to the emission of one or more electrons, as well as photons of both lower and higher energy. Because the flux of incident photons is high, a classical description of the laser field is adequate, but the time-dependent Schrödinger equation (TDSE) must be solved directly to obtain an accurate representation of the atom–field interaction. For SS pulses of optical wavelength, it is sufficient in most cases to consider only the electric dipole (E1) interaction term defined in Chap. 72. The atom–field interaction can then be expressed in either the length gauge or the velocity gauge [1] (Chap. 22). In the length gauge, the TDSE is i„
@‰.r; t/ D H0 C er E .t/ ‰.r; t/ ; @t
Since lasers must usually be focused to reach the strong-field regime, measured electron and ion yields include contributions from a distribution of field strengths. The photoemission spectrum, on the other hand, contains a coherent component due to the macroscopic polarization of all the atoms and is, therefore, sensitive also to the laser phase variations within the focal volume. In this chapter, methods for solving Eqs. (78.1) and (78.3) are discussed along with details of the atomic emission processes. Three relevant books provide excellent introductions to this subject [1–3]. Further developments are well described in the proceedings of the International Conferences on Multiphoton Physics [4–7] and the NATO workshop on SuperIntense Laser-Atom Physics [8, 9].
78.2 Weak-Field Multiphoton Processes 78.2.1
Perturbation Theory
Since atomic ionization energies are generally ' 10 eV, while optical photons have energies of only a few eV, several photons must be absorbed to produce ionization, or even electronic excitation in the case of the noble gases. For WL pulses, the electronic states are only weakly perturbed by the electromagnetic field. The rate of an n-photon transition can then be calculated using the n-th order perturbation theory for the atom–field interaction. For an incident photon number flux of frequency !, the rate is
(78.1)
.n/
Wi !f D 2
where H0 is the field-free atomic Hamiltonian, r the collective coordinate of the electrons, and E .t/ the electric field of the laser given by where
E .t/ D E .t/ xO cos.!t C '/ C yO sin.!t C '/ : (78.2)
2 ˛! e2
n ˇ ˇ ˇ .n/ ˇ2 ˇTi !f ˇ ı.!i C n! !f / ; (78.5)
˝ ˇ .n/ Ti !f D f ˇd GŒ!i C .n 1/!
d GŒ!i C .n 2/! Here, E .t/ defines the envelope, ' is the so-called carrier ˛ envelope phase (CEP), and characterizes the polarization: d GŒ!i C ! d ji ; (78.6) linear if D 0 and circular if jj D 1. In the velocity gauge, the TDSE is jii is the i-th eigenstate of the field-free atomic Hamiltonian, d D e O r, with O the polarization direction, and 2 ie„ e @ .r; t/ 2 A .t/ .r; t/ : D H0 A.t/ r C i„ @t mc 2mc 2 X jj ihj j (78.3) :
G.!/ D (78.7) ! !j C i%j =2 j Rt Here, A.t/ D c E t 0 dt 0 is the vector potential of the laser field. The solutions of Eqs. (78.1) and (78.3) are related The sum over j includes an integration over the continuum by the phase transformation for all sequences of E1 transitions allowed by angular momentum and parity selection rules. Methods for calculating ie r A.t/ .r; t/ : (78.4) cross sections and rates in the weak-field regime are de‰.r; t/ D exp „c scribed in [1, 10] and in Chap. 25.
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Multiphoton and Strong-Field Processes
78.2.2
Resonant-Enhanced Multiphoton Ionization
For multiphoton ionization, ! can be continuously varied because the final state in Eq. (78.5) lies in the continuum. If ! is tuned so that !i C m! Ñ !j for some contributing intermediate state jj i in Eq. (78.7), then in that state lies an integer m photons above the initial state, and the corresponding denominator vanishes (to within the level width %j ), producing a strongly peaked resonance. Since it takes k D n m additional photons for ionization, the process is called m; k resonant-enhanced multiphoton ionization (REMPI). Measurements of the photoelectron angular distribution are useful in characterizing the resonant intermediate state. Calculations using the semi-empirical multichannel quantum defect theory to provide the needed matrix elements have been very successful in describing experimental results. This technique is discussed in more detail in Chap. 49. The perturbation equation Eq. (78.5) indicates that the rate for nonresonant multiphoton ionization scales as n for an n-photon process [11]. However, this is not the case for REMPI since the resonant transition saturates, and Eq. (78.5) no longer applies. Then the rate can be controlled either by the m-photon resonant excitation step or by the number of photons k needed for the ionization step.
78.2.3
Multielectron Effects
Multiply excited states can play a role in multiphoton excitation dynamics. These states are particularly important if their energies are below or not too far above the first ionization potential. Configuration expansions including these states have been used successfully in studies, for example, of the alkaline earth atoms, which have many low-lying doubly excited states. The presence of these states can also enhance the direct double ionization of an atom [12].
1127
78.2.5 Coherence and Statistics Real laser fields exhibit various kinds of fluctuations and so are never perfectly coherent. The effects of such fluctuations on the complex electric field amplitude
E.t/ D E .t/ exp i!t C '.t
can be modeled by a variety of stochastic processes [16], depending on the conditions [10, 17–20], as follows. For continuum wave (CW) lasers, a phase diffusion model (PDM) is often used, for which E .t/ D E D const: and '.t/ P D
p
Autoionization
The configuration interaction between a bound state and an adjacent continuum leads to an absorption profile in the single photon ionization spectrum with a Fano lineshape. The actual lineshape reflects the interference between the two pathways to the continuum. Autoionizing states can also be probed via multiphoton excitation [13, 14]. Because, in the strong-field regime, coupling strengths and phases change with intensity, the lineshapes can be strongly distorted by changing the incident intensity. At particular intensities, the phases of the excited levels can be manipulated to prevent autoionization completely. Then a trapped population with energy above the ionization limit can be created [15].
2bF .t/ ;
(78.9)
where F .t/ describes white noise by a real Gaussian function [16] characterized by the averaged values hF .t/i D 0, hF .t/F .t 0 /i D 2bı.t t 0 /. The stochastic electric field then has an exponential autocorrelation function ˝
ˇ ˛
ˇ E.t/E .t 0 / D E 2 exp b ˇt t 0 ˇ i!.t t 0 / ; (78.10)
and a Lorentzian spectrum of width b. Far-off resonance, such a Lorentzian spectrum often gives unrealistic results, and the model Eq. (78.9) is then replaced by an Ornstein– Uhlenbeck process, '.t/ R D ˇ '.t/ P C
p 2bˇF .t/ ;
(78.11)
where the parameter ˇ for ˇ b plays the role of a cutoff of the Lorentzian spectrum. A multimode laser with a large number P M of independent modes has a field of the form E.t/ D jMD1 Ej expŒi!j t C i'j .t/, and according to the central limit theorem [16], can be described for large M as a complex Gaussian process defined to be a chaotic field, P E.t/ D .b C i!/E.t/ C
78.2.4
(78.8)
q ˝ ˛ 2b jE.t/j2 F .t/ ;
(78.12)
where F .t/ is now a complex white noise, and ! is the central frequency of the field. The field, Eq. (78.12) has an exponential autocorrelation function and a Lorentzian spectrum of width b. Various other stochastic models have been discussed in the literature. These include Gaussian fluctuations of the real amplitude of the field E .t/; Gaussian chaotic fields with non-Lorentzian spectra; non-Gaussian, nonlinear diffusion processes (that describe for instance a laser close to threshold [16]); multiplicative stochastic processes (that describe a laser with pump fluctuations [18]) and jumplike Markov processes [21–23]. Statistical properties of laser fields can sometimes be controlled experimentally to a great extent [19, 20].
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M. Ciappina et al.
Effects of Field Fluctuations
Since the response of systems undergoing multiphoton processes is, in general, a nonlinear function of the field intensity (and, in particular, of the field amplitude), it depends in a complex manner on the statistics of the field. The enhancement of the nonresonant multiphoton ionization rate illustrates the point. According to the perturbation equation Eq. (78.5), the rate of an n-photon process is proportional to n ; i.e., to I n , where I is the field intensity. For fluctuating fields, the average response is thus Dˇ D E ˇ2n E .n/ Wi !f / I n / ˇE.t/ˇ :
(78.13)
Phase fluctuations (as described by PDM) do not affect the average. On the other hand, for complex chaotic fields, the average is
inversion [28] in which the dressed atom can have an inverted population, allowing gain, even though in terms of the undressed states, the lower level has the largest population. A laser can produce dramatic changes in the index of refraction of an atomic medium [29], creating, at specific frequencies, laser-induced transparency for a second, probe laser field. Multistep ionization, where each step is driven by a laser at its resonant frequency, has resulted in two useful applications. These are: efficient atomic isotope separation [23]; and the detection of small numbers of atoms in a sample, called single-atom detection. This technique is extremely sensitive because the use of exact resonance for each step yields very large cross sections for ionization, and the efficiency of collecting ions is high [30].
78.2.8 hI i Ñ nŠhI i ; n
n
Waveforms
(78.14)
i.e., there exists a significant enhancement of the rate for n > 1. Field fluctuations lead to more complex effects in resonant processes. Two well-studied examples are the enhancement of the AC Stark shift in resonant multiphoton ionization [24], and the spectrum of double optical resonance – a process in which the AC Stark splitting of the resonant line is probed by a slightly detuned fluctuating laser field [18]. Double optical resonance is very sensitive not only to the bandwidth of the probing field but also to the shape of its frequency spectrum.
78.2.7 Excitation with Multiple Laser Fields The simultaneous application of more than one laser field produces interesting and novel effects. If a laser and its second (2!) or third (3!) harmonic are combined and the relative phase between the fields controlled, product state distributions and yields can be altered dramatically. The effects include reducing the excitation or ionization rates in the ! 3! case [25] or altering the photoelectron angular distributions and the harmonic emission parity selection rules using a ! 2! laser source [26]. A laser field can dress or strongly mix the field-free excited states, including the continuum, of an atom. This can produce a number of effects depending on how the dressed system is probed. By coupling a bound, excited state with the continuum, ionization strengths and dynamics are altered, resulting in new resonance-like structures where none existed before. This effect is called laser-induced continuum structure, or LICS [15, 27]. This general idea has been exploited to design schemes for lasers without
One step beyond would be to drive the atomic or molecular system with a so-called multicolor laser field, i.e., with a laser source created starting from different laser frequencies, not necessarily a multiple of each other (these sources are also known as waveforms or light-field transients). Waveform-controlled light transients with a bandwidth spanning almost two octaves have been demonstrated at microjoule energy and gigawatt peak power levels. These particular sources allow temporal confinement of optical radiation to less than 1 fs in subcycle waveforms. With their power substantially enhanced, these extreme waveforms may open up a new stage in nonlinear optics and attoscience. This is mainly due to, among other things, the feasibility of suppressing ionization up to unprecedented peak intensities and instantaneous ionization rates approaching optical frequencies, respectively. A prototypical three-color few-cycle system is already available, and it offers a conceptually simple route to scaling multioctave optical waveform synthesis to the multiterawatt regime (e.g., [31]). To this end, with the three channels delivering few-cycle pulses in the visible (VIS), 0:45 0:65 m, near-infrared (NIR), 0:7 1:3 m, and mid-infrared (MIR), 1:6 2:7 m, spectral ranges are recombined using a set of dichroic chirped mirrors to yield one single beam. Additionally, the feasibility of superoctave optical waveform synthesis was recently demonstrated in the NIR-VIS-ultraviolet (UV) spectral range by seeding a three-channel and, more recently, a four-channel, synthesizer consisting of broadband chirped mirrors with a continuum originating from a Ti:sapphire-laser-driven hollowfiber/chirped-mirror compressor. Further degrees of freedom for waveform sculpting can be introduced by shaping the amplitude and phase of the spectra of the individual channels, e.g., via an acousto-optic pulse shaper and/or a spatial light modulator (for a recent review see, e.g., [32]).
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Multiphoton and Strong-Field Processes
78.3
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Strong-Field Multiphoton Processes
78.3.1 Nonperturbative Multiphoton Ionization
Recently developed laser systems can produce very short pulses, some as short as a few to tens of femtoseconds, while at the same time maintaining the pulse energy so that the peak power becomes very high. In this regard, e.g., focused intensities up to 1019 W=cm2 have been achieved. Because the pulses are short, atoms survive to much higher intensities before ionizing, making possible studies of laser–atom interactions in an entirely new regime [33]. A discussion of the status of short-pulse laser development is given in Chap. 75 and in [34]. With increasing intensity, higher-order corrections to Eq. (78.6) contribute to the transition rate. The next order correction comes from transitions involving two additional photons, one absorbed and one emitted, leading to the same final state. One effect of these terms is to shift the energies of the excited states in response to the oscillating field. This is called the dynamic or AC Stark shift. The AC Stark shift of the ground state tends to be small because of the large detuning from the excited states for long wavelength photons. On the other hand, in strong fields, the shift of the higher states and the continuum can become appreciable. Electrons in highly excited states respond to the oscillating field in the same manner as a free electron. Their energies shift with the continuum by the amount
The breakdown of perturbation theory for nth-order multiphoton processes occurs when the higher-order correction terms become comparable to the n-th-order term. Assuming that the dipole strength is / ea0 , where a0 is the Bohr radius, and the detuning is ı / !, the ratio of an (n C 2)-order contribution to the n-th-order term from Eq. (78.6) is [1]
Up D
.1 C /e 2 E 2 ; 4m! 2
RnC2;n Ñ
2 ˛! ea0 2 I !a 2 D ; (78.16) 2 e ! I 2!
where „!a Ñ 27:2114 eV is the atomic unit of energy e 2 =a0 , and I Ñ 3:50945 1016 W=cm2 is the intensity corresponding to an atomic unit of field strength, given by Ea D
1=2 ˛c m=a03 Ñ 5:1422 109 V=cm. The atomic unit of intensity itself is defined by Ia D a „!a D ˛cEa2 ;
(78.17)
which is 6:436414.4/ 101 5 W=cm2 . Thus, I D Ia =.8 ˛/. For photon energies of 1 eV, RnC2;n becomes unity for I 1014 W=cm2 . Because of the large number of (n C 2)-order terms, perturbation theory actually fails for I > 1013 W=cm2 . Above this critical intensity, nonresonant n-photon ionization ceases to scale with the n dependence predicted by perturbation theory.
(78.15)
where Up is the cycle-averaged kinetic energy of a free electron in the field, and defines the polarization of the field Eq. (78.2); Up is called the ponderomotive or quiver energy of the electron. For strong laser fields, Up can be several eV or more, meaning that during a pulse, many states shift through resonance as their energies change by an amount larger than the incident photon energy. The resulting intensity-induced resonances can dominate the ionization dynamics. Electrons promoted into the continuum acquire the ponderomotive energy, oscillating in phase with the field. In a linearly polarized field, the amplitude of the quiver motion of a free electron, given by e E =m! 2 , can become many times larger than the bound-state orbitals. If the initial velocity of an electron is small after ionization, it can be accelerated by the laser electric field and back into the vicinity of the ion core, when the field reverses its direction. The subsequent rescattering process, amongst other effects, changes the photoelectron energy and angular distributions, and allows the emission of high-energy photons [35, 36]. This simple dynamical picture forms the basis of the current understanding of many strong-field multiphoton processes.
78.3.2 Tunneling Ionization At sufficiently high intensity and low frequency, a tunneling mechanism changes the character of the ionization process. For lasers in the infrared (IR) or optical range, a strongly bound electron can respond to the instantaneous laser field since the oscillating electric field varies slowly on the timescale of the electron. The Coulomb attraction of the ion core combines with the laser electric field to form an oscillating barrier through which the electron can escape by tunneling, if the amplitude of the laser p field is large enough. The DC rate for this process is e E = 2mIP , where IP is the ionization potential of the electron. When this rate is comparable to the laser frequency, tunneling becomes the most probable ionization mechanism [37–39]. The ratio of the incident laser frequency to the tunneling rate is called the Keldysh parameter and is given by q D
IP =2Up ;
(78.18)
which is less than unity ( 1) when tunneling dominates and larger than unity ( 1) when multiphoton ionization
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governs the laser–matter process. Interestingly, many experiments are carried out in an intermediate or crossover region, defined by 1. Another way to interpret is to note that we can write p D T =L , where T is the Keldysh time (defined as T D E2IP ) and L D 2 =! is the laser period. Hence, serves as a measure of nonadiabaticity by comparing the response time of the electron wavefunction to the period of the laser field.
78.3.3 Multiple Ionization Excitation and ionization dynamics are dominated by singleelectron transitions in the strong-field regime. Although atoms can lose several electrons during a single pulse, the electrons are released sequentially. There is no convincing evidence of significant collective excitation in atoms in strong fields, even though it has been extensively sought. Once one electron is excited in an atom, the remaining electrons have much higher binding energies. As a result, the laser field is unable to affect them significantly until it reaches a much higher intensity. By that time, the first electron has already been emitted. Simultaneous ejection of two electrons occurs as a minor channel (< 1%) in strong-field multiple ionization. Although it is possible that doubly excited states of atoms could assist in the double ionization, in the helium and neon cases studied, these states are unlikely to be contributors [40]. Correlated multielectron ionization became a very hot subject over the past 20 years. Various mechanisms were proposed and demonstrated theoretically and experimentally. Perhaps the most important is the recollision mechanism, in which one electron tunnels out and is driven back to the nucleus by the laser field, where it recollides and releases a second atomic or molecular electron [41].
78.3.4
Above Threshold Ionization
In strong optical and IR laser fields, electrons can gain more than the minimum amount of energy required for ionization. Rather than forming a single peak, the emitted electron energy spectrum contains a series of peaks separated by the photon energy. This is called above-threshold-ionization, or ATI [42–45]. The peaks appear at the energies Es D .n C s/„! IP ;
(78.19)
where n is the minimum number of photons needed to exceed IP , and s D 0; 1; : : : is called the number of excess photons or above threshold photons carried by the electron. Calculations in the perturbative regime for ATI are given for hydrogen in [11].
Peak Shifting As the intensity approaches the nonperturbative regime, the AC Stark shift of the atomic states begins to play a significant role in the structure of the ATI spectrum. The first effect is a shift of the ionization potential, given roughly by the ponderomotive energy Up . Additional photons may then be required in order to free the electron from the atom, i.e., enough to exceed IP C Up . If the emitted electron escapes from the focal volume while the laser is still on, it is accelerated by the gradient of the field. The quiver motion is converted into radial motion, increasing the kinetic energy by Up and exactly canceling the shift of the continuum. In this situation, the electron energies are still given by Eq. (78.19). However, when Up exceeds the photon energy, the lowest ATI peaks disappear from the spectrum. In this long pulse limit, no electron is observed with energy less than Up . This is called peak shifting and names the situation where the dominant peak in the ATI spectrum moves to higher order as the intensity increases. ATI Resonance Substructure If the laser pulse is short enough (< 1 ps for the typical laser focus), the field turns off before the electron can escape from the focal volume. Then the quiver energy is returned to the field, and the ATI spectrum becomes much more complicated. The observed electron energy can now be computed by Es .short pulse/ D .n C s/„! .IP C Up / ;
(78.20)
which results shifted an amount Up with respect to the long pulses case Eq. (78.19). Electrons from different regions of the focal volume are thus emitted with different ponderomotive shifts, introducing a substructure in the spectrum, which can be directly associated with AC Stark-shifted resonances [42, 43].
ATI in Circular Polarization The above discussion is particularly appropriate for the case of linear polarization, where the excited states of the atom can play a significant role in the excitation. On the contrary, in a circularly polarized field, the orbital angular momentum L must increase one unit with each photon absorbed, so that multiphoton ionization is allowed only to states that have high L, and, hence, a large centrifugal barrier. The lower energy scattering states thus cannot penetrate into the vicinity of the initial state. Thus, the ATI spectrum in circular polarization peaks at high energy and is very small near threshold. ATI Applications In the few-cycle regime, a precise knowledge of the CEP is instrumental to adequately characterize the subsequent nonlinear laser–matter phenomena. Then, the CEP plays
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Multiphoton and Strong-Field Processes
a major role in, for instance, the production and control of single attosecond pulses and the investigation of nonsequential double-electron ionization [46–48], amongst others. Recently, experimental observations of ATI driven by few-cycle IR-laser pulses have demonstrated that using a socalled “stereo” technique measurement of electrons emitted on the left and on the right with respect to the linear laser polarization, it is possible to extract the CEP of the driving laser [49]. In addition, another important application is to use energetic rescattered ATI electrons to estimate the bond distance in molecules. For instance, recent measurements in acetylene (C2 H2 ) [50] have proved that is entirely possible to extract structural information of the chemical internuclear distance between the carbon atoms in the CC pair and the carbon and hydrogen atoms in the CH one, both with unprecedented high accuracy. This singular technique is named laser-induced electron diffraction (LIED) and promises to lead the time-resolved studies in complex molecules [51]. The two above-mentioned examples, namely the fewcycle CEP characterization and the LIED technique, show the capabilities of ATI to interrogate nature and answer fundamental questions about both the electron dynamics and molecular structure, just to name a few. Hence, ATI provides a powerful tool to not only give light about time-dependent atomic and molecular processes but also to configure an invaluable ally for extracting information of the driving laser itself [49].
1131
Plateau and Cutoff The well-known three-step model [35, 36], which in 2018 turned 25 years old and combines quantum and classical aspects of laser-atom physics, accounts for many strong-field phenomena. In this model, the electron first tunnels out [55] from the ground state of the atom through the barrier formed by the Coulomb potential and the laser field. Its subsequent motion can be treated classically, and primarily consists of oscillatory motion in phase with the laser field. If the electron returns to the vicinity of the nucleus with kinetic energy T , it may recombine into the ground state with the emission of a photon of energy .2n C 1/„! T C IP , where n is an integer. (The relation of this recombination process to atomic photoionization is discussed briefly in Chap. 25.) The maximum kinetic energy of the returning electron turns out to be T Ñ 3:2 Up , resulting in a cutoff in the harmonic spectrum at the harmonic of order Nmax Ñ .IP C 3:2Up /=„! :
(78.21)
Theoretical Methods Calculation of harmonic strengths requires the evaluation of the time-dependent dipole moment of the atom, d.t/ D h‰.t/jerj‰.t/i :
(78.22)
The strength of harmonics emitted by a single atom are then related to the Fourier components of d.t/, or more precisely, its second time derivative, dR .t/. The induced dipole moment d.t/ can be directly evalu78.3.5 High-Order Harmonic Generation ated from the numerical [56] or Floquet [26] solutions of the TDSE. Good agreement with numerical and experimenHigh-order harmonic generation (HHG) in noble gases is tal data is also obtained using the strong-field approximation a rapidly developing field of laser physics [33, 52–54]. discussed below and a Landau–Dyhne formula. This latter When an SS pulse interacts with an atomic gas, the atoms approach can be considered to be a quantum mechanical imrespond in a nonlinear way, emitting coherent radiation plementation of the three-step model [57]. at frequencies that are integral multiples of the laser frequency. Due to the inversion symmetry of the atom, only odd harmonics of the fundamental are emitted. In the high- Propagation and Phase-Matching Effects intensity .> 1013 W=cm2 /, low-frequency regime, the har- A single atom response is not sufficient to determine the monic strengths fall off for the first few orders, followed by macroscopic response of the atomic medium. Because difa broad plateau of nearly constant conversion efficiency and ferent atoms interact with different parts of the focused laser then a rather sharp cutoff [52, 53]. The plateau extends to beam, they feel different peak field intensities and phases well beyond the hundredth order of the 800–1000 nm inci- (which actually undergo a rapid shift close to the focus, dent wavelengths, using the light noble gases as the active due to geometrical effects). The total harmonic signal results medium. Moreover, there has also been experimental ev- from coherently adding contributions from single atoms, acidence of HHG from ions. Considering the laser-induced counting for propagation and interference effects. The latter electron dynamics behind HHG natively develops on a sub-fs effects can wipe out the signal completely if a constructive timescale, HHG provides a source of very bright, short- phase matching does not take place. pulse, coherent XUV radiation, which can have several The propagation and phase-matching effects in the strongadvantages over the other known sources, such as the syn- field regime [58] can be studied by solving Maxwell’s chrotron. equations for a given harmonic component of the electric
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field EM .r/ Eq. (72.23), r 2 EM .r/ C nM .r/EM .r/ D
1 M! 2 PM .r/ ; 0 c (78.23)
where nM .r/ is the refractive index of the medium (which depends on atomic, electronic, and ionic dipole polarizabilities), while PM .r/ is the polarization induced by the fundamental field only. The latter can be expressed as
PM .r/ / N.r/dM M.r/ exp iM.r/ ;
(78.24)
where N.r/ is the atomic density, dM .r/ is the M-th Fourier component of the induced dipole moment, and .r/ is a phase shift coming from the phase dependence of the fundamental beam due to focusing. All of these quantities may have a slow time dependence, reflecting the temporal envelope of the laser pulse. Typically, phase matching is the most efficient in the forward direction. In general, the strength and spatial properties of a harmonic depend in a very complex way on the focal parameters, the medium length, and the coherence length of a given harmonic. Propagation and phase-matching effects can lead to a shift of the location of the cutoff in the harmonic spectrum [59].
Harmonic Generation by Elliptically Polarized Fields The three-step model implies that for harmonic emission it is necessary that the tunneling electrons return to the nucleus and recombine into their initial state. According to classical mechanics, there are many trajectories in a linearly polarized field that involve one or more returns to the origin. However, there are practically no such trajectories in elliptically polarized fields. As a result, the three-step model predicts a strong decrease of the harmonic strengths as a function of the laser ellipticity. This prediction has been already confirmed experimentally [60]. The Generation of Attosecond XUV Pulses Manipulation of generated harmonics by allowing the temporal beating of superposed high-order harmonics can produce a train of intense and very short spikes, on the order of 100 as and shorter, where 1 as D 1018 s [61–63]. The structural characteristics of the generated pulse trains depend on the relative phases of the harmonics combined. Employing driving pulses that were themselves only a few femtoseconds long, experimental groups in Vienna [64] and Paris [65] reported the first observations and measurements of such subfemtosecond UV/XUV light pulse trains. Very recently, a new world record was set in Florida, where a single 53-attosecond X-ray pulse, with a photon energy enough to reach the carbon K-edge, was measured [66]. The scientific
importance of breaking the femtosecond barrier is obvious: the timescale necessary for probing the motion of an electron in a typical bound, the valence state, is measured in attoseconds (atomic unit of time 24 as). Attosecond pulses will allow the study of the time-dependent dynamics of correlated electron systems by freezing the electronic motion, in essence exploring the structure with ultrafast snapshots. A crucial aspect for all attosecond pulse generation is the control of spectral phases. Measurements of the timing of the attosecond peaks relative to the absolute phase of the IR driving field have been accomplished [67]. This provides insight into the recollision, the key step in the harmonic generation process. Also, the control of the group velocity phase relative to the envelope of the few-cycle driving pulses allows the production of reproducible pulse trains [68]. Thus, the highly nonperturbative, nonlinear multiphoton interactions of very short IR or visible light pulses with atoms or molecules is becoming a novel, powerful, and unique source for studies of the fastest quantum electronic processes known to date.
HHG Applications One of the singular aspects of HHG is the possibility to produce and control both trains of and single attosecond pulses. These ultrashort light sources are extremely important for the investigation of the ultrafast process in atomic and molecular systems. The Auger decay, the delay in the photoemission absorption via an XUV source and doubleelectron ionization are examples of phenomena that occur on the subfemtosecond time domain. HHG allows not only to get knowledge about the ultrafast electron dynamics inside atoms and molecules, but it also has also been demonstrated that it configures an indispensable tool to extract atomic and molecular structural information [69, 70]. The fascinating technological advances of pump-probe experiments, where attosecond pulses and delayed IR lasers are being employed to firstly trigger the system and later interrogate its dynamical time evolution, have allowed us to address unexplored questions on the above-mentioned processes. The extension of these techniques to more complex systems, such as solid structures and biological materials, is already in the pipeline. HHG in Bulk Matter So far, the cornerstone of attosecond science applications has been the HHG phenomenon in gases. However, there still exists a clear disadvantage of HHG due to its poor conversion efficiency—the ratio between the outgoing XUV and incoming IR photon fluxes. Recent advances in light sources within the mid-infrared (MIR) domain, 1:7 7 m, have opened a new avenue in the investigation of HHG using condensed matter materials [71]. In particular, the control of such MIR sources allows us to overcome the low-efficiency problem and substantially increase the photon flux per produced at-
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Multiphoton and Strong-Field Processes
tosecond pulse. This characteristic could lead to unexplored physical processes in that condensed matter phase. As is known, in an insulator, there exists the concept of band energy dispersion for electrons in its valence or conduction bands. For instance, the HHG in semiconductors (dielectrics) is governed by electron or holes bound wavefunctions that are not spatially (or energetically) localized. Holes, in addition, also have the possibility to move on their own valance band while, on the other hand, the electron dynamics takes place entirely in the conduction band. This leads to two main contributions to the HHG process, namely, one dictated by an intraband oscillation and other by an interband electron–hole current [72, 73]. This emerging field is attracting the attention of both the attosecond science and condensed matter communities. Join efforts are being carried out to address fundamental questions such as how particle collisions can be mimicked in a solid through HHG processes [74] or how Bloch oscillations might be observed in real time [75]. Additionally, questions related to topological insulators and how their features might affect the harmonic emissions were recently brought up [76, 77].
78.3.6 Stabilization of Atoms in Intense Laser Fields It has been argued [78] that in very intense laser fields of high frequency, atoms undergo dynamical stabilization and do not ionize. The stabilization effect can be explained by gauge transforming the TDSE Eq. (78.1) to the Kramers– Henneberger (K–H) frame; i.e., a noninertial oscillating frame that follows the motion of the free electron in the laser field. The K–H transformation consists of replacing r ! r C ˛.t/, where, for the linearly polarized monochro
matic laser field, ˛.t/ D x˛ O 0 cos.!t '/; ˛0 D e E = m! 2 is the excursion amplitude of a free electron, and xO is the polarization direction. The TDSE in the K–H frame is 2 2
@‰.r; t/ „ r i„ D C V r C ˛.t/ ‰.r; t/ ; (78.25) @t 2m
1133
near these minima, thus exhibiting a dichotomy. The ionization rates from the K–H bound states are induced by the higher Fourier components of V .r C ˛.t//. For large enough ˛0 , the rates decrease, if either the laser intensity increases, or the frequency decreases. Numerical solutions of the TDSE [79, 80] show that stabilization, indeed, occurs for laser field strengths and frequencies of the order of one atomic unit. More importantly, stabilization is possible even when the laser excitation is not monochromatic but, rather, is produced by a short laser pulse. Physically, free electrons in a monochromatic laser field cannot absorb photons due to the constraints imposed by energy and momentum conservation. Absorption is possible only in the vicinity of a potential, such as the Coulombic attraction of the nucleus. In the case of a strong excitation, i.e., when ˛0 is much larger than the Bohr radius, the electron spends most of the time very far from the nucleus, and, therefore, does not absorb energy from the laser beam. Therefore, stabilization, as viewed from the K–H frame, has a classical analog. Other mechanisms of stabilization based on the quantum mechanical effects of destructive interference between various ionization paths have also been proposed [81]. Due to classical scaling (Sect. 78.3.8) stabilization is predicted to occur for much lower laser frequencies, if the atoms are initially prepared in highly excited states. If additionally, the initial state has a large orbital angular momentum corresponding to classical trajectories that do not approach the nucleus, stabilization is even more easily accomplished. For instance, the stabilization of a 5g Rydberg state of neon has been reported [82]. Although recently interest in stabilization in intense laser fields became very limited, the K–H transformation and the stabilization effect have been discussed in a completely different situation. In [83, 84], the stabilization of atoms in a shaken trap Bose–Einstein condensate was proposed. In fact, recently, a K–H trick was used to develop an experimental simulator of ultrafast processes in strong laser fields using ultracold trapped atoms [85].
78.3.7 Molecules in Intense Laser Fields i.e., it describes the motion of the electron in an oscillatory potential. In the high-frequency limit, this potential may be Molecular systems are much more complex than atoms bereplaced by its time average cause of the additional degrees of freedom resulting from nuclear motion. Even in the presence of a laser field, the 2 =! Z electron and nuclear degrees of freedom can be separated
! dt V r C ˛.t/ ; (78.26) by the Born–Oppenheimer approximation, and the dynamVK–H .r/ D 2 ics of the system can be described in terms of motions 0 on potential energy surfaces. In strong fields, the Born– and the remaining Fourier components of V Œr C˛.t/ treated Oppenheimer states become dressed, or mixed by the field, as a perturbation. When ˛0 is large, the effective potential creating new molecular potentials. Because of avoided crossEq. (78.26) has two minima close to r D ˙x˛ O 0 . The cor- ings between the dressed molecular states, the field induces responding wave functions of the bound states are centered new potential wells in which the molecules become trapped.
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These states, known as laser-induced bound states, are stable against dissociation, but exist only while the laser field is present [86]. Their existence affects the spectra of photoelectrons, photons, and the fragmentation dynamics. If the field is strong enough, many electrons can be ejected from a molecule before dissociation, producing highly charged, energetic fragments [87]. Such experiments are similar to beam–foil Coulomb explosion studies of molecular structure. However, because of changes from the field-free equilibrium geometries in laser dissociation, the energies of the fragments lie systematically below the corresponding values from Coulomb explosion studies. Additionally to the latter, a similar phenomenon was intensively studied with atomic clusters [88].
a decrease in the effective „ toward the classical limit. In view of this classical scaling, experimental and theoretical results are usually analyzed in terms of the scaled variables. Since the classical dynamics generated by the Hamiltonian Eq. (78.28) exhibits chaotic behavior in some regimes, the dynamics of the corresponding quantum system is frequently referred to as an example of quantum chaos [91–93].
Regimes of Response By varying the initial n0 , several regimes of the scaled parameters can be covered. The experimentally measured response of Rydberg atoms in microwave fields can be divided into six categories:
The tunneling regime For !Q 0:07, the response of the system is accurately represented as tunneling through the slowly oscillating potential Similar phenomena appear in the ionization of highly excited barrier composed of the Coulomb and microwave potentials. hydrogen-like (Rydberg) atoms by microwave fields [89, 90], but the dynamical range of the parameters involved is differThe low-frequency regime ent from the case of tightly bound electrons. Recent developFor 0:05 !Q 0:3, the ionization probability exhibits disments have greatly extended techniques for the preparation tinct structures (bumps, steps, changes of slope) as a function and detection of Rydberg states. Since, according to the of the field strength. The quantum probability curves might equivalence principle, highly excited Rydberg states exhibit be lower or higher than the corresponding classical countermany classical properties, a classical perspective of ionizaparts, calculated with the aid of the phase averaging method tion yields useful insights (Sect. 78.4.5). (Sect. 78.4.5).
78.3.8 Microwave Ionization of Rydberg Atoms
Classical Scaling The classical equations of motion for an electron in both a Coulomb field and a monochromatic laser field polarized along the z-axis are invariant with respect to the following scaling transformations: Q; p / n1 0 p
r / n20 rQ ;
t / n30 tQ ;
' / 'Q ;
!/
Q n3 0 !
Q ; E / n4 0 E :
The semiclassical regime For 0:1 !Q 1:2, the ionization probabilities agree well for most frequencies with the results obtained from the classical theory. In particular, the onset of ionization and appearance intensities (i.e., the intensities at which a given degree of ionization is achieved) coincide with the onset of chaos in the classical dynamics. In the ionization probabilities, resonances appear that correspond to the classical trapping (78.27) resonances [91–94].
In the scaled units, the Hamiltonian HQ D n2 0 H of the system becomes
The transition region For 1 !Q 2, the differences between the quantum and clas
pQ 2 1 (78.28) sical results are visible. Quantum ionization probabilities are HQ D C zQ e EQ cos !Q tQ C 'Q ; 2m rQ frequently lower and appearance intensities higher than their classical counterparts. i.e., it depends only on !Q and EQ . In experiments, the principal quantum number n0 of the prepared initial state typically ranges from 1 to 100. The high-frequency regime Classical scaling extends to the fields of other polariza- For !Q 2, quantum results for ionization probabilities are tions and to pulsed excitations, provided that the number of systematically lower and appearance intensities higher than cycles in the rise, top, and fall of the pulse is kept fixed. their classical counterparts. This apparent stability of the This scaling does not hold for a quantum Hamiltonian, unless quantum system has been attributed to three kinds of effects: one also rescales Planck’s constant, „Q D „=n0 . In practice, quantum localization [94], quantum scars [95], and perhaps increasing n0 , keeping EQ and !Q constant, corresponds to the stabilization of atoms in intense laser fields (Sect. 78.3.6).
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1135
into a time-independent eigenvalue problem [26, 97]. From The photoeffect regime When the scaled frequency becomes greater than the single- Floquet’s theorem, the eigenfunctions for a perfectly periodic photon ionization threshold, the system undergoes single- Hamiltonian of the form photon ionization (the photoeffect). X H D H0 C HN eiN !t (78.29) N ¤0 Quantum Localization The classical dynamics changes as the field increases. Chaotic trajectories start to fill the phase space and, as the can be expressed as KAM tori (describing periodic orbits) [91–93] break down, X eiN !t N : (78.30) ‰.t/ D eiX t =„ the motion becomes stochastic, resembling a random walk. N This process, in which the mean energy grows linearly in time, is termed diffusive ionization. In quantum theory, difPutting this into the time-dependent Schrödinger equation refusion corresponds to a random walk over a ladder of suitably sults in an infinite set of coupled Floquet equations for the chosen quantum levels. However, both diagonal and offharmonic components N . In the velocity gauge, the Floquet diagonal elements of the evolution operator, which describe equations are quantum mechanical amplitudes for transitions between the levels, depend in a quasi-periodic manner on the quantum .X C N „! H0 / N D VC N 1 C V N C1 ; (78.31) numbers of the levels involved. Such quasiperiodic behavior is quite analogous to a random one. Electronic wave pack- where, for a vector potential of amplitude A, ets, which initially spread in accordance with the classical e laws, tend to remain localized for longer times due to deVC D Ap ; (78.32) structive quantum-interference effects. Quantum localization 2mc is an analog of the Anderson localization of electronic wave and V D VC . The Eq. (78.31) has been solved, after trunfunctions propagating in random media [94]. cation to a manageable number of terms, using many techniques to provide what are called the quasi-energy states of Quantum Scars Even in the fully chaotic regime, classical phase space the laser-atom system. The eigenvalues X of these equations contains periodic, although unstable, orbits. Nevertheless, are complex, with Im.X/ giving the decay or ionization rate quantum mechanical wave-function amplitudes can become for the system. The generated rates are found to be very aclocalized around these unstable orbits, resulting in what are curate as long as the pulse length of the laser field is not too called quantum scars. The increased stability of the hydrogen short, at least hundreds of cycles. The eigenfunctions provide atom at !Q Ñ 1:3 has, in fact, been attributed [95] to the ef- the amplitudes for the photoelectron energy spectra, and the fects of quantum scars. These effects are very sensitive to time-dependent dipole of the state can be related to the phofluctuations in the driving laser field. Control of the laser toemission spectrum of the system. Yields for slowly varying noise, therefore, provides a powerful spectroscopic tool to pulses can be constructed by combining the results from the study such quantal phenomena [96]. Using this tool, it has individual, fixed-intensity calculations [26]. The Floquet method can be applied for any periodic become possible to demonstrate the effects of quantum scars in the intermediate regime of scaled frequencies, i.e., for val- Hamiltonian. In strong enough fields of high frequency, the Floquet equations can be truncated to a very small set in the ues less than but close to 1. K–H frame [78].
78.4
Strong-Field Calculational Techniques 78.4.2
The SS pulse regime requires a nonperturbative solution of the TDSE. We describe here two of the most used approaches: the explicit numerical solution of the TDSE and the Floquet expansion technique. In addition to these, several approximate methods have been proposed.
Direct Integration of the TDSE
Methods for the direct solution of the time-dependent Schrödinger equation (TDSE) are described in general in Chap. 8 and in [98, 99] for multiphoton processes. The wave functions are defined on spatial grids or in terms of an expansion in basis functions. The time evolution is obtained by either explicit or implicit time propagators. All these meth78.4.1 Floquet Theory ods are capable of generating numerically exact results for an atom with a single electron in a short pulsed field for a wide The excitation and ionization dynamics of an atom in range of pulse shapes, wavelengths, and intensities. The soa strong laser field can be determined by turning the problem lutions are time-dependent wave functions for the electrons,
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which can be analyzed to obtain excitation and ionization rates, photoelectron energies, angular distributions, and photoemission yields. The ability to generate an explicit solution of the TDSE allows the study of arbitrary pulse shapes and provides insight into the excitation dynamics. For multielectron atoms, one generally has to limit the calculations to that for a single electron in effective potentials, which represent, as well as possible, the influences of the remaining atomic electrons. This approach is called the single-active electron (SAE) approximation, and it gives generally accurate results for systems with no low-lying doubly excited states, for example, the noble gases [98]. In these cases, the excitation dynamics is dominated by the sequential promotion of a single electron at a time.
electronic wave function corresponds to a bare bound state. On the other hand, the final, continuum states of the electron are described by dressed wave functions that account for the free motion of the electron in the laser field. In the simplest case, such dressed states are Volkov states Eq. (78.33). AlterO natively, the time-reversed S-matrix is obtained by dressing the initial state and using field-free scattering states for the final state. Yet another method consists of expanding the electronic continuum–continuum dipole matrix elements in terms corresponding to matrix elements for free electrons plus corrections due to the potential [57]. In the latter version of the SFA, the amplitude of the electronic wave function b.p/, corresponding to an outgoing momentum p, is given by ZtF
78.4.3
Volkov States
b.p/ D i
dt d p eA.t/=c E .t/ exp iS.tF ; t/=„ :
0
A laser interacting with a free electron superimposes an os(78.34) cillatory motion on its drift motion in response to the field. The wave function for an electron with drift velocity v D Here, dŒp eA.t/=c denotes the dipole matrix element for „k=m is given by the transition from the initial bound state to the continuum state in which the electron has the kinetic momentum p ‰V .r; t/ eA.t/=c, tF is the switch-off time of the laser pulse, and 1 0 Zt h i2 e i " # 2 ZtF „k A.t 0 / dt 0 A eiŒkeA.t /=„cr ; D exp@ p eA.t 0 /=c 2m„ c 0 C IP (78.35) S.tF ; t/ D dt 2m (78.33) t Rt where A.t/ D c E .t 0 /dt 0 is the vector potential of the field; ‰V is called a Volkov state. In a linearly polarized field, the electron oscillates along the direction of polarization with an amplitude ˛0 D e„A=.mc!/. In the strong-field regime, this amplitude can greatly exceed the size of a bound-state orbital. Volkov states provide a useful tool that can be applied in various strong-field approximations discussed in Sect. 78.4.4.
is a quasi-classical action for an electron that is born in the continuum at t and propagates freely in the laser field. The form of the expression Eq. (78.34) is generic to the SFA.
78.4.5 phase-space Averaging Method
The methods of classical mechanics are particularly useful in describing the microwave excitation of highly excited (Rydberg) atoms [89, 90] (Sect. 78.3.8), but have also been 78.4.4 Strong-Field Approximations applied to describe HHG, stabilization of atoms in super intense fields, and two-electron ionization [100–102]. There have been several attempts to solve the TDSE in the The classical phase-space averaging method [103] solves strong-field limit using approximate but analytic methods. Newton’s equations of motion Such strong-field approximations (SFA) typically neglect all p the bound states of the atom except for the initial state. In the ; (78.36) rP D m tunneling regime ( < 1), and a quasi-static limit (! ! 0), pP D r V .r/ eE .t/ ; (78.37) one can use a theory [55] in which the ionization occurs due to the tunneling through the Coulomb barrier distorted by the electric field of the laser. The wave function is constructed as for an electron interacting with an ion core and an external a combination of a bare initial wave function of the electron laser field. A distribution of initial conditions in phase space (close to the nucleus) and a wave function describing a mo- is chosen to mimic the initial quantum mechanical state of tion of the electron in a quasi-static electric field (far from the the system, and a sample of classical trajectories is generated. nucleus). In a second approach [37–39], the elements of the Quantum mechanical averages of physical observables are scattering matrix SO are calculated assuming that initially the then identified with ensemble averages of those observables
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over the initial distribution. Since the dynamics of multiphoton processes is very complex, the neglected phases in this approach generally cause negligible errors, and the results can be in quite good agreement with quantum calculations. Additionally, an examination of the trajectories provides details of the excitation dynamics, which are often difficult to extract from a complex time-dependent wave function.
78.5
Atto-Nano Physics
The interaction of ultrashort strong laser pulses with larger systems has recently received much attention; it has led to a consequent advance in our understanding of the attosecond to few-femtosecond electronic and nuclear dynamics. For instance, the interaction of atomic clusters with strong ultrafast laser fields has long been known to lead to the formation of nanoplasmas in which there is, on the one hand, a high degree of charge localization and ultrafast dynamics and, on the other hand, the emission of both very energetic (multiple keV) electrons and highly charged ions (in the MeV energy range). Interestingly, the most recent utilization of ultrashort few-cycle pulses ( 10 fs) to trigger the laser–matter processes has succeeded in isolating the electron dynamics from the longer ion dynamics timescale (which can essentially be considered as frozen) revealing a higher degree of fragmentation anisotropy in both electrons and ions compared to the isotropic distributions found from much longer (multicycle) pulses (100 fs). Additionally, interactions of intense lasers with nanoparticles, such as micron and submicron scale liquid droplets, lead to hot plasma formation. An important role is found for enhanced local fields on the surface of these droplets, the so-called hot spots. Furthermore, studies of driving bound and free charges in larger molecules, e.g., collective electron dynamics in fullerenes and in graphene-like structures, proton migration in hydrocarbon molecules, charge migration in proteins and biomolecules, amongst others, could be included in this category. In addition, biological applications of atto-nano physics could be envisaged as well, e.g., to explain the DNA-protein interactions in solutions of living cells, study the induced covalent crosslink between aromatic amino acids and peptides, and characterize the protein–protein interactions in living cells. In turn, laserdriven broadband electron wavepackets have been used for static and dynamic diffraction imaging of molecules, obtaining structural information with subnanometer spatial and subfemtosecond temporal resolution (Sect. 78.3.4). Tailored ultrashort and intense fields have also been used to drive electron dynamics and electron or photon emission from (nanostructured) solids (for a recent compilation: [104]). The extraordinary progress seen recently has been largely driven by advances in both experimental, e.g., laser technology, and engineering, e.g., nanofabrication, tech-
1137
niques. Amongst the remarkable achievements in just the most recent years are the demonstration of driving electron currents and switching the conductivity of dielectrics with ultrashort pulses, controlling the light-induced electron emission from nanoparticles and nanotips, and the subcycledriven photon emission from solids. Furthermore, the intrinsic electron propagation and photoemission processes have been investigated on their natural, attosecond timescales. A key feature of light-nanostructure interactions is the enhancement, by several orders of magnitude, of the electric near-field and its local confinement on a subwavelength scale. From a theoretical viewpoint, this field localization presents a unique challenge: we have at our disposal strong fields that change on a comparable spatial scale of the oscillatory electron dynamics that are initiated by those same fields. This peculiar property entails profound consequences in the underlying physics of the conventional strong-field phenomena. In particular, it violates one of the main assumptions that the modeling of strong-field interactions is based upon: the spatial homogeneity of laser fields in the volume of the electronic dynamics under scrutiny (this hypothesis was per se considered in all the previous sections of this chapter). Interestingly, an exponential growing attraction in strongfield phenomena induced by plasmonic-enhanced fields was sparked by the controversial work of Kim et al. [105]. These authors claimed having observed efficient HHG from noblegas atoms driven by the plasmonic-enhanced field generated by bow-tie metallic nanostructures. Although later on the interpretation of the outcomes was demonstrated to be incorrect [106], Kim’s paper definitively stimulated a steadily constant interest in the plasmonic-enhanced HHG and ATI. We should mention, however, a very recent result of the same group, which clearly seems to be well justified and, as such, opens new perspectives and ways toward efficient HHG using nanostructures [107]. In this recent contribution, the authors demonstrate experimentally plasmonic-driven HHG by devising a metal-sapphire nanostructure that provides a solid tip as the HHG emitter instead of gaseous atoms. Measured EUV spectra show odd-order harmonics up to 60nm wavelengths, without the plasma atomic lines typically seen when using gaseous atoms as the HHG-driven media. This experimental data confirm that the plasmonic HHG approach is a promising way to make real coherent EUV sources for nanoscale near-field applications in spectroscopy, microscopy, lithography, and attosecond physics (for another recent related experiment, see, e.g., [108]). Within the conventional assumption, both the laser electric field, E .r; t/, and the corresponding vector potential, A.r; t/, are spatially homogeneous in the region where the electron moves and only their time dependence is considered, i.e., E .r; t/ D E .t/ in Eq. (78.2) and A.r; t/ D A.t/ in Eq. (78.4). This is a genuine assumption considering the usual electron excursion ˛0 is bounded roughly by a few nanometers in the NIR, for typical laser intensities, and sev-
78
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eral tens of nanometers for MIR sources (note that ˛0 / Ep2 , where is the wavelength of the driving laser and E D I , where I is the laser intensity). Hence, electron excursions are very small relative to the spatial variation of the field in the absence of local (or nanoplasmonic) field enhancement. On the contrary, the fields generated using surface plasmons are spatially dependent on a nanometric region. As a consequence, all the standard theoretical tools in the strong-field ionization toolbox described previously, ranging from purely classical to frequently used semiclassical and complete quantum mechanical descriptions, have to be reexamined. For a comprehensive review about atto-nano physics see, e.g., [109] Acknowledgments ML acknowledges support from: ERC AdG NOQIA; Agencia Estatal de Investigación (R&D project CEX2019000910-S, funded by MCIN/AEI/10.13039/501100011033, Plan National FIDEUA PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2, Proyectos de I+D+I “Retos Colaboración” QUSPIN RTC2019-007196-7), MCIN via European Union NextGenerationEU (PRTR-C17.I1); Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya through the European Social Fund FEDER and CERCA program (AGAUR Grant No. 2017 SGR 134, QuantumCAT\U16-011424, co-funded by ERDF Operational Program of Catalonia 2014–2020); EU Horizon 2020 FET-OPEN OPTOlogic (Grant No 899794); National Science Centre, Poland (Symfonia Grant No. 2016/20/W/ST4/00314); European Union’s Horizon 2020 research and innovation programme under the Marie-Skłodowska-Curie grant agreement No 101029393 (STREDCH) and No 847648 (“La Caixa” Junior Leaders fellowships ID100010434: LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012, LCF/BQ/PR21/11840013). MFC acknowledges financial support from the Guangdong Province Science and Technology Major Project (Future functional materials under extreme conditions – 2021B0301030005).
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M. Ciappina et al. Marcelo Ciappina Dr Ciappina received his PhD from the Balseiro Institute at Argentina in 2005 and a Research Professor degree from the Czech Academy of Sciences in 2019. His academic training includes postdoctoral stays at renowned institutions in Germany, Singapore, USA, and Spain. At present, he is Senior Researcher at ELIBeamlines. His areas of interest include attosecond science and atomic collisions. Alexis A. Chacon S. Alexis A. Chacon S. obtained his PhD in Quantum Optics-Physics at Universidad de Salamanca-Spain (2015). He then moved to QIT group led by Lewenstein at ICFO to expand his background to electron and photon emissions from atoms and solids. Alexis joined LANL-USA in 2017 and is currently Group Leader at MPI-Korea pursuing ultrafast physics in quantum materials. Maciej Lewenstein Maciej Lewenstein received his PhD in Essen, Germany, in 1983. He has worked at the Center for Theoretical Physics, Harvard University, CEA Saclay, and Leibniz University, Hannover. In 2005 he became ICREA Professor and Group Leader at ICFO in Castelldefels. His interests include quantum optics, quantum physics, quantum information, many-body theory, attophysics, and statistical physics. He is also an acclaimed jazz writer and critic.
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Cooling and Trapping Juha Javanainen
Contents 79.1
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1141
79.2 79.2.1 79.2.2 79.2.3
Control of Atomic Motion by Light . General Theory . . . . . . . . . . . . . . Two-State Atoms . . . . . . . . . . . . . Multistate Atoms . . . . . . . . . . . . .
79.3 79.3.1
Magnetic Trap for Atoms . . . . . . . . . . . . . . . . . 1148 Evaporative Cooling . . . . . . . . . . . . . . . . . . . . . 1148
79.4 79.4.1 79.4.2 79.4.3
Trapping and Cooling of Charged Particles Paul Trap . . . . . . . . . . . . . . . . . . . . . . . Penning Trap . . . . . . . . . . . . . . . . . . . . . Collective Effects in Ion Clouds . . . . . . . . .
79.5 79.5.1 79.5.2
Experimental . . . . . . . . . . . . . . . . . . . . . . . . . 1152 Free Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 1152 Trapped Particles . . . . . . . . . . . . . . . . . . . . . . . 1152
79.6 79.6.1 79.6.2
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 1153 Cold Molecules . . . . . . . . . . . . . . . . . . . . . . . . 1153 Quantum Systems of Internal and CM States . . . . . . 1154
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1142 1142 1144 1147
1149 1149 1150 1151
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154
Abstract
At the time of writing, the book [1] appears to be the standard reference on cooling and trapping of atoms. Reviews of various vintages with a substantial component on trapped particles include [2–5]. Optical lattices binding atoms, discussed in a tutorial manner in [6], is presently a prominent frontier. Additional references ranging from pioneering works to representative recent examples are given as leads into specific topics. No assignment of credit or priority is implied. Keywords
spontaneous emission Wigner function trap particle laser cool Paul trap
79.1
Notation
In this chapter, the lower and upper states of an optical transition are denoted by the respective labels g and e, for ground and excited. The notation Jg ! Je stands for a transition in which the lower and upper levels have the angular momentum degeneracies 2Jg C 1 and 2Je C 1. The resonance frequency of the transition is !0 . The detuning of the driving monochromatic light of frequency ! from the atomic resonance is defined as
Interactions of light with an atomic particle are accompanied by exchange of momentum between the electromagnetic field and the atom. Narrowband resonance radiation from tunable lasers enhances the ensuing mechanical effects of light to the extent that it is possible to stop atoms emanating from a thermal gas and to trap atoms with light. We will also discuss charged ions trapped by types of electromagnetic fields other than light, and possibly D ! !0 : (79.1) cooled with light. The emphasis is on basic theoretical concepts and experimental procedures. Cooling and trapSpontaneous emission is taken to be the sole mechanism of ping of atomic particles is now a basic tool and large line broadening, so that the rate of spontaneous emission % swaths of modern AMO physics depend on it, so the disand the half-width half-maximum (HWHM) linewidth of the cussion of applications is necessarily cursory. transition are related by D %=2. The Rabi frequency is * D DE =„, where D is the reduced dipole moment matrix J. Javanainen () element that would apply to a transition with unit Clebsch– Department of Physics, University of Connecticut Gordan coefficient, and E is the electric field amplitude of Storrs, CT, USA e-mail: [email protected] the laser. The corresponding intensity scale is the saturation © Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_79
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Table 79.1 Laser cooling parameters for the lowest S1=2 –P3=2 transition of hydrogen and most alkalis (the D2 line). Also shown are the nuclear spin I and the ground state hyperfine splitting hfs Parameter m vr % TD "r Tr Is I hfs
1
6
H 1:67 121:6 326 99:7 2390 13;400 643 14;500 1=2 1420
7
Li 9:99 671:0 9:89 5:87 141 73:7 3:54 5:08 1 228:2
Li 11:7 671:0 8:48 5:87 141 63:2 3:03 5:08 3=2 803:5
23
39
Na 38:2 589:2 2:95 9:79 235 25:0 1:20 12:5 3=2 1772
K 64:7 766:7 1:34 6:05 145 8:71 0:418 3:51 3=2 461:7
40
K 66:4 766:7 1:30 6:05 145 8:49 0:408 3:51 4 1286
85
Rb 141 780:2 0:602 6:07 146 3:86 0:185 3:34 5=2 3036
87
Rb 144 780:2 0:588 6:07 146 3:77 0:181 3:34 3=2 6835
133
Cs 221 852:3 0:352 5:22 125 2:07 0:0992 2:20 7=2 9193
Units 1027 kg nm cm=s 2 MHz K 2 kHz K mW=cm2 2 MHz
79.2 Control of Atomic Motion by Light
intensity Is D
4 2 „c% ; 33
(79.2)
defined in such a way that the light intensity I satisfies * D % ) I D Is :
(79.3)
If multiple laser beams are explicitly mentioned, laser intensity and Rabi frequency are quoted for one of the equally intense beams. The momentum of a photon with the wave vector k is „k. The recoil velocity vr D
„k m
(79.4)
equals the change of the velocity of an atom of mass m when it absorbs a photon with wave number k D 2 =. The kinetic energy of an atom with velocity vr and the corresponding frequency, Er D
1 2 mv ; 2 r
"r D
Er ; „
(79.5)
79.2.1
General Theory
The Hamiltonian The mechanical effects of light may be derived from the Hamiltonian HO D HO A C HO CM C HO F dO EO .r/ O ;
(79.7)
where HO A , HO CM , and HO F are the Hamiltonians for the internal degrees of freedom of the atom, center-of-mass (CM) motion of the atom, and free electromagnetic field. The quantized electric field is EO .r/, O where rO is the CM position operator. The dipole operator dO acts on the internal degrees of freedom of the atom. Since the dO EO .r/ O term couples all degrees of freedom, the possibility of influencing CM motion by light immediately follows. The inclusion of the quantized CM motion is the essential ingredient not contained in traditional theories of light–matter interactions. For an atom with mass m trapped in a possibly anisotropic harmonic oscillator potential with frequencies i .i D x; y; z/, the CM Hamiltonian is X m 2 rO 2 pO 2 i i HCM D C ; (79.8) 2m i Dx;y;z 2
are referred to as recoil energy and recoil frequency. Two where pO is the CM momentum operator. For a free atom, temperatures, the Doppler limit TD and the recoil limit Tr , i D 0. are often cited in laser cooling. They are Master Equation (79.6) With the aid of Markov and Born approximations, the vacuum modes of the electromagnetic field may be eliminated as described in Chap. 7.1.6. This gives a master equation for where kB is the Boltzmann constant. the reduced density operator O that contains the internal and Table 79.1 lists numerical values of pertinent parameters CM degrees of freedom of the atom. Relaxation terms profor laser cooling and trapping using the D2 line for most staportional to % and are all that is left of the quantized fields. ble and long-lived alkali isotopes and hydrogen. Except for Consider as an example a two-state atom in a traveling the spontaneous decay rates % (and the derived quantities TD , wave of light with the electric field strength Is ), data greatly exceeding the displayed precision are widely 1 available. The spontaneous decay rate is taken either from [7] (79.9) E.r; t/ D E ei.kr!t / C c.c. 2 or [8]. TD D
„ ; kB
Tr D
Er ; kB
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Cooling and Trapping
1143
O are Master equations are conveniently written using Wigner sphere. Representative expressions for W .n/ functions to represent the CM motion. Given the internal1 state labels i and j D g or e, and the three-dimensional W .n/ O D ; 4 variables r; p, the Wigner functions are defined as 3 3 Z 1 .eO n/ O 2 ; 1 C .eO n/ O 2 : 1 8 16 d3 u eiup=„
ij .r; p/ D (79.16) .2 „/3 1 1 r ujhij jj O ijr C u : (79.10) These apply, respectively, for isotropic spontaneous emis2 2 sion, for spontaneous emission in a m D 0 transition, and in The Wigner function is one of the quantum mechanical m D ˙1 transitions; eO stands for the unit vector in the diquasi-probability distributions, Chap. 82.5, with the special rection of the quantization axis for angular momentum. Only property that the marginal distribution of r obtained by the p dependence has been denoted explicitly in the Wigner integrating over p coincides with the correct quantum prob- functions, as the recoil effects displayed on the right-hand ability density function for position, and vice versa with r sides of Eqs. (79.11)–(79.14) take place at a fixed position r. and p interchanged. In the rotating wave approximation, Semiclassical Theory Chap. 72.3.2, the master equations are Suppose that vCM vr and CM , where and CM are the * ikr 1 d timescales for light-driven changes of the internal state and
ee .p/ D % ee .p/ C i e Oge p „k CM motion of the atom. Then the internal degrees of freedom dt 2 2 may be eliminated adiabatically from the master equations 1 eikr Oeg p „k ; in favor of the position-momentum distribution for the CM 2 motion, (79.11) X Z
i i .r; p; t/ ; (79.17) f .r; p; t/ D d i* O ee .p C „k n/ O
gg .p/ D % d2 n W .n/
i dt 2 1 where the sum runs over the internal states of the atom. Tech eikr Oge p C „k 2 nically, the recoil velocity vr is treated as an asymptotically 1 small expansion parameter. The result is the Fokker–Planck eikr Oeg p C „k ; 2 (79.12) equation for the CM motion d i*
Oge .p/ D . C i/ Oge .p/ dt 2 1 eikr gg p „k 2 1 ikr e ee p C „k ; 2 d i*
Oeg .p/ D . i/ Oeg .p/ C dt 2 1 eikr gg p „k 2 1 ikr
ee p C „k : e 2
X @2 d @ .Dij f / : f D .F f / C dt @p @pi @pj i;j
(79.13)
(79.14)
Here, the convective derivative that describes the motion of the atom in the absence of light is X @ @ p @ d mi ri I D C dt @t m @r @pi i
(79.15)
O is the angular distribution of cf. HCM in Eq. (79.8); W .n/ spontaneous photons, and the integral runs over the unit
(79.18)
In this semiclassical theory, the CM motion of the atom is regarded as classical. The atom moves under the optical force F .r; p; t/, which models the coarse-grained flow of momentum between the electromagnetic field and the atom; Dij .r; p; t/, with i; j D x; y; z, is the diffusion tensor. Diffusion is an attempt to model quantum mechanics with a classical stochastic process, including discreteness of recoil kicks, random directions of spontaneous photons, and random timing of optical absorption and emission processes. A general prescription exists for calculating the force and the diffusion tensor for an arbitrary atomic level scheme and light field [9]. However, the formally correct analysis of diffusion has not proven to be particularly useful, and here only the force is considered at length. Let VO .r/ be the dipole interaction operator coupling the driving field and the internal state for an atom at position r. By assumption, VO .r/ has been rendered slowly varying in time with the aid of a suitable rotating wave approximation. To compute the force, one takes
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J. Javanainen
an atom that travels along a hypothetical trajectory unper- For the force in Eq. (79.20) with constant ˇ D ˇ0 and turbed by light in such a way that at time t it arrives at the D.z; p/ D D0 , the steady state of the Fokker–Planck equaphase space point (r; p), whereupon the density operator of tion is a thermal distribution of the form the internal degrees of freedom is %. O The force is then ! m 2 x 2 ˇ0 m p 2 ; (79.23) C f .x; p/ D K exp @VO D0 2m 2 Fi .r; p; t/ D Tr %O : (79.19) @ri where K is a normalization constant. Quantum Theory Since Wigner functions give correct quantum mechaniWhen either vCM / vr or ' CM , the full quantum theory cal marginal distributions for r and p, expectation values of of cooling and trapping is needed. Master equations such kinetic and potential energy may be calculated from the disas Eqs. (79.11)–(79.14) must then be solved without the as- tribution function Eq. (79.23) as if it were a classical phase sumption that vr is small. For instance, a truncated basis of space density. For a free atom with D 0, the temperature is plane waves may be used to expand the CM state. Density directly proportional to the kinetic energy, matrix equations are then solved numerically either directly D0 or by resorting to quantum Monte-Carlo simulations [10]. : (79.24) T D Such solutions for free (i D 0) particles are very few. Our ˇ0 mkB approach and terminology are, therefore, semiclassical by default. However, for a trapped particle with ¤ 0 the Fokker–Planck equation may be valid all the way to the quantum mechaniQualitative origin of laser cooling cal zero-point energy. Then, temperature and energy are no Velocity-dependent dissipative forces are needed for cool- longer directly proportional to one another. For a trapped paring. They arise because the evolution of the internal state of ticle, the safe interpretation of Eq. (79.23) is that the total CM a moving atom has a finite response time . The atom con- energy of the particle is veys the memory of the field it has sampled over the length D0 ` D v on its past trajectory. If ` , a nonequilibrium com: (79.25) ED ponent proportional to ` is present in the density operator of ˇ0 m the internal state of the atom. Further interactions with light convert this component into a velocity-dependent force of the form 79.2.2 Two-State Atoms F D mˇv ;
ˇ / I :
(79.20)
A two-state or two-level atom, discussed in detail in Chap. 72.3, stands for a closed (cycling) transition with one lower state and one excited state. In practice, a two-state system is often realized by driving a J ! J C 1 transition with circularly polarized light. This leads to optical pumping to the states with the maximal (or minimal) component of angular momentum along the quantization axis, say, to the transition m D J ! m0 D J C 1. (79.21) vc : Two types of force are generally distinguished: light pres sure, or scattering force, or spontaneous force, and dipole, or One-dimensional considerations gradient, or induced force. However, the distinction is neiMost specific results cited here are one-dimensional. By de- ther exclusive, nor exhaustive. Here, the two types of force fault, the propagation direction of light and the direction of are approached by way of examples. vector quantities other than light polarization is eO x . The relevant components of position, velocity, and momentum are Traveling Waves denoted by x, v, and p. The general one-dimensional Fokker–Planck equation for Light pressure a particle trapped in a harmonic oscillator potential with Consider a cycle of absorption and spontaneous emission. In a CM oscillation frequency is the absorption, the atom receives a photon recoil kick in the 2 @ @ p @ @ @ direction of the laser beam, while in the spontaneous emisC m 2 x f D .Ff / C 2 .Df / : sion the recoil kick has a random direction and zero average. @t m @x @p @p @p (79.22) The atom is, on the average, left with a velocity change equal
If the damping constant ˇ is positive, Eq. (79.20) describes exponential damping of the velocity on the timescale ˇ 1 . In the contrary case ` , when the atom travels many wavelengths during the memory time, linear dependence of force on velocity breaks down. The watershed is the critical velocity or velocity capture range
79
Cooling and Trapping
1145
to vr . The rate of such processes equals % ee , which gives the Phenomenology in Multimode Fields force along k and equal to Doppler cooling in standing waves *2 =2 (79.26) Next, take an atom in two counterpropagating plane waves of F D Fm 2 C 2 .v/ C *2 =2 light. At low intensity, * %, forces of the form Eq. (79.26) for the two beams may be added when averaged over a wavein magnitude. Here, the maximum of light pressure force, length. For velocities well below the critical velocity a convenient scale for optical forces, is % 1 ; (79.31) vc,D D (79.27) Fm D mvr % ; k 2 the wavelength-averaged force is of the form of Eq. (79.20), and .v/ D kv
(79.28)
N ; ˇN D F D mˇv
4*2 .2 C 2 /2
"r :
(79.32)
is the effective detuning, which includes the Doppler shift When light is tuned below the atomic resonance (red detunexperienced by the moving atom. ing with < 0), exponential damping of the atomic velocity with the time constant ˇN 1 ensues. No matter which way the Diffusion For a traveling wave, the diffusion coefficient accompanying atom moves, it is always Doppler tuned toward resonance with the light wave that propagates opposite to its velocity, light pressure is and away from resonance with the light wave that propagates along its velocity. Net momentum transfer, therefore, D .1 C ˛/*2 D 2 2 2 2 2 opposes the motion of the atom. This is known as Doppler „k % 4Œ .v/ C C * =2
2 4 cooling. 2 .v/ 3 * ; (79.29) 4Œ2 .v/ C 2 C *2 =23 Optical molasses For three pairs of counterpropagating waves in three orthogwhere onal directions, Eq. (79.32) is valid in all coordinate direcZ tions, and, hence, as a vector equation between the force F O eO x n/ O 2 (79.30) ˛ D d2 n W .n/. and velocity v. For < 0, an atom experiences an isotropic viscous damping force, as if it were moving in a thick liquid. depends on W .n/, O see Eq. (79.16). Representative values are Such a field configuration is dubbed optical molasses. Two ˛ D 1=3 for isotropic spontaneous emission, and ˛ D 2=5 counterpropagating beams make a one-dimensional optical (˛ D 3=10) for spontaneous emission with m D 0 (m D molasses. ˙1) with respect to a quantization axis that is perpendicular (parallel) to the direction eO x . Limit of Doppler cooling The expression Eq. (79.29) is the formally complete re- Under the conditions of Eq. (79.32), the diffusion coeffisult from the expansion in recoil velocity. The first term on cients for the two counterpropagating beams averaged over the right is easily obtained by computing the variance of the a wavelength may be added, and the v D 0 form suffices for atomic momentum in a cycle of absorption and spontaneous slow atoms. This yields emission and multiplying it by the rate of such processes % ee . The second term, rarely seen in theoretical analyses N D 0/ .1 C ˛/ *2 D.v D : (79.33) and henceforth ignored, is related to photon antibunching: „2 k 2 % 2.2 C 2 / The subsequent spontaneous emissions do not happen independently of one another at the rate % ee , but after an The random diffusive motion of the atom corresponds to emission, there is some delay before the atom gets excited diffusive heating that competes with Doppler cooling. In and can emit again. equilibrium, the temperature is Spontaneously emitted photons cover all of the 4 solid N D 0/ angle, and so do the directions of photon recoil kicks on jj D.v „ D .1 C ˛/ C : (79.34) T D the atom. Absorption from a light wave traveling in a parN B 4kB jj mˇk ticular direction leads to transverse diffusion also in the orthogonal directions, which is not accounted for by the one- Equation (79.34) also applies to three-dimensional Doppler cooled molasses, provided one uses ˛ D 1 corresponding dimensional Eq. (79.29).
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the beams may be deliberately offset to average out extraneous standing wave patterns. Very intense lasers with a large detuning, including CO2 and Nd-YAG, can been used. As the % optical force and the rate of spontaneous emission scale, rejj D D : (79.35) 2 spectively, as I = and I =2 , the heating from spontaneous emission may be negligible. For three-dimensional molasses, the Doppler limit TD of A standing wave of light makes a periodic array of optical Eq. (79.6) is obtained. traps called an optical lattice. Optical lattices may be set up For * > %, the performance of Doppler cooling deterioin 1-D, 2-D, and 3-D configurations. rates. Qualitatively, power broadening increases the effective linewidth . Induced diffusion Random motion of atoms in velocity space owing to abDipole forces sorptions and induced emissions of photons with different Dipole forces are the resonant analog of the ponderomotive momenta leads to induced diffusion. Contrary to diffusion in forces discussed in Chap. 78.3. They arise from successions a traveling wave as in Eq. (79.29), induced diffusion does of absorption and induced emission driven by photons with not saturate at high intensity. Instead, the diffusion coeffidifferent momenta. Such processes occur only if there is cient continues to grow linearly with I . Induced diffusion is more than one wave vector present in the field, i.e., if there is another reason why the lowest Doppler cooling temperatures an intensity gradient. For a zero-velocity, two-state atom, the are generally reached at low (I < Is ) light intensities. gradient force in a monochromatic field with the local total intensity I.r/ is Sisyphus effect In a standing wave at high intensity and large detuning, an2 rI.r/ 4„ other kind of optical force becomes important that cannot be : (79.36) F g .r/ D 2 C 2 Œ1 C 2I.r/=Is Is categorized either as light pressure or gradient force. As explained in Chap. 72.3.4, one may diagonalize the The dipole force may be derived from the potential energy Hamiltonian to obtain the dressed atom-field states. Because the light field depends on position, so do the energies of 2 2 I.r/=Is : (79.37) Vg .r/ D 2„ ln 1 C the dressed states and their decompositions into plain atomic 2 C 2 states. In Fig. 79.1 drawn for > 0, the dressed state with The atoms are strong-field seekers for red detuning ( < 0) a minimum (maximum) at the field nodes coincides with the bare ground state (excited state) at the nodes. At the and weak-field seekers for blue detuning ( > 0). Suppose the amplitude of the dipole moment induced on antinodes, the admixtures of ground and excited states are the atom equals d D ˛ E .r/, where E .r/ is a slowly vary- evened out to some extent. The energy of a dressed state acts as potential energy for ing (or stationary) local electric field amplitude, and ˛ is the the CM motion of an atom residing in that particular state. polarizability. This is true in the limit of a large detuning or In fact, the gradient force is the force derived from these poa low light intensity, in the case when the response of the atom is isotropic. In the limit of large detunings, the dipole tential energies, averaged over the occupation probabilities moment is also either in phase ( ! 1) or out of phase of the dressed states. The occupation probability is larger for with the electric field, so the polarizability is real. Then, there the dressed state with a larger ground-state admixture. From Fig. 79.1 one, therefore, sees that the atom predominantly reis a general expression for the dipole force, sides in the dressed state that has a minimum of energy at the antinodes. The atom is a weak-field seeker, as it should for 1 (79.38) > 0. F g .r/ D ˛ rjE .r/j2 ; 4 Spontaneous emission remains to be considered. It gives with the obvious potential energy. For a CO2 or Nd-YAG rise to transitions between the dressed states. These transilaser, the static polarizability of the atom serves as a prac- tions may go both ways between the dressed states because tical approximation for ˛. the states are, in general, superpositions of the bare ground state and the excited state. The rate of spontaneous transiOptical trap and optical lattice tions from one dressed state to another increases (decreases) Dipole forces are utilized in an optical trap for atoms, and with the excited (ground) state admixture of the initial state. even molecules. A common configuration is the crossedIn reference to Fig. 79.1, suppose that the atom is combeam dipole force trap, two crossing laser beams tuned below ing from the left in the upper dressed state. The probability resonance, and focused to the same spot. The frequencies of that the atom makes a transition to the lower dressed state, to added transverse diffusion. The minimum temperature is reached at
79
Cooling and Trapping
Fig. 79.1 Qualitative origin of the Sisyphus effect. The hatched pattern represents a standing light wave. The energies of the two dressed states are drawn as a function of position, along with a few filled circles representing the admixture of the ground state in each dressed state at selected field positions. Larger circles correspond to larger ground-state admixtures, and hence, larger equilibrium populations of the dressed state. This figure applies for tuning of the laser above the atomic resonance (blue detuning)
as marked by the downward vertical arrow, is largest at the antinode. If this transition takes place, near the next node the most probable transition is as shown by the upward vertical arrow. In this manner, the atom spends most of its time at an uphill climb against the potential, and is, therefore, slowed down. In reference to Greek mythology, this is called the Sisyphus effect. In the two-state model atom, cooling takes place when the laser frequency is higher than the atomic resonance frequency.
1147
Polarization Gradient Cooling As explained in connection with Eq. (79.20), a finite memory time of the internal atomic state may lead to damping of the CM motion. For a two-state atom, internal equilibration arises from spontaneous emission. The timescale is D % 1 , and Doppler cooling ensues. However, an atom whose ground state has angular momentum degeneracy is also subject to optical pumping. If the polarization of light varies as a function of position, optical pumping is needed to reach local equilibrium. The pumping timescale p / I 1 then becomes relevant for a moving atom. The associated cooling is known as polarization gradient cooling. Its hallmark is that, for low I , the damping coefficient ˇ / I p is independent of intensity. Two detailed mechanisms of polarization gradient cooling have been described [11], although in three-dimensional light fields they are intertwined. The Sisyphus effect works like the Sisyphus effect for a two-state atom, except that it relies on light shifts and optical pumping within the ground state manifold. Induced orientation cooling is analogous to Doppler cooling. Velocity dependence of optical pumping in counterpropagating waves leads to pumping to a state for which the force due to the wave propagating opposite to the atom exceeds the force due to the wave propagating along with the atom.
C – molasses As an example consider one-dimensional C – molasses, which consists of two counterpropagating waves with opposite circular polarizations. The net polarization is linear everywhere, but the direction of polarization rotates as the point of observation is displaced along the propagation axis; Semiclassical Versus Quantum Theory hence the name corkscrew molasses. At low intensity and When "r , the root-mean-square (RMS) velocity of low velocity, the force on an atom with a 1 ! 2 transition is a cooled two-state atom is always vr , and semiclassical theory is valid. Under the same condition "r , the 60 „k 2 v : (79.39) F D Doppler-limit RMS velocity is also less than the critical ve17 5 2 C 2 locity vc,D from Eq. (79.31). Velocity expansions such as in Eqs. (79.32) and (79.33) are then justified. In this configuration, only induced orientation cooling conIn the contrary case, / "r , the full quantum theory of tributes. Cooling again takes place for < 0, and the trapping and cooling must be employed. The cooled velocity resulting temperature is distribution cannot be expected to be thermal and tempera ture is ill defined, but on dimensional grounds, one expects 292 C 1045 2 2I % (79.40) T D TD : that the minimum kinetic energy for cooled free atoms is 300.2 C 2 / Is jj comparable to the recoil energy Er . Experimental molasses For I %=Is jj < 1 and jj %, the temperature Eq. (79.40) reduces to the form
79.2.3
Multistate Atoms
T DC
„*2 2I % DC TD : kB jj Is jj
(79.41) Energy levels of atomic systems usually have angular momentum degeneracy. In addition, the polarization of light, in general, depends on position. A combination of these aspects Under these conditions, the same I = scaling is approxileads to phenomena beyond the two-state atomic model. mately observed also in three-dimensional, six-beam optical
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molasses operating with atoms that have a degenerate ground state. The constant C depends on the degeneracy of the transitions and on the polarizations of the molasses beams. On dimensional-analysis grounds, one expects C 1; values in the range 0:25 < C < 0:5 have been reported [12].
states take the place of velocity-dependent Doppler shifts. The restoring force and the damping coefficient of Doppler cooling are closely related. For the coordinate directions u D x; y; z, the relation is
Limit of cooling While the expressions in Eqs. (79.40) and (79.41) suggest that T goes all the way to zero as I ! 0 or ! 1, in practice there is a lower limit of T reached in polarization gradient cooling. This is because the velocity capture range falls below the speeds of the would-be cooled atoms. The temperature eventually starts to rise abruptly when jj is increased, or I is decreased. The empirical rule of thumb is that T 10 Tr is the lowest temperature one can expect.
(79.43)
Magneto-Optical Trap Since spontaneous forces may be strong already at modest light intensities, I Is , the use of light pressure to trap neutral atoms appears desirable. However, the optical Earnshaw theorem states that (in the limit of low I ) the spontaneous force on a two-state atom is sourceless. While confinement may be possible in some directions, escape routes for atoms remain open in others. Three-dimensional trapping of a twostate atom with light pressure is not possible. A magneto-optical trap (MOT) defeats the Earnshaw theorem by relying on angular momentum degeneracy. Consider an atom with a 0 ! 1 transition in a magnetic field B that depends linearly on position around the zero at x D 0. Suppose that the gradient of B is chosen in such a way that the m D 1 (m D 1) magnetic substate of the excited state has the higher (lower) energy for x > 0, and that the atom is illuminated by ˙ polarized beams propagating in the ˙x-directions, tuned below resonance. When the atom is displaced from x D 0 in either direction, it is closer to resonance with the beam that pushes it back toward x D 0. This makes the restoring force responsible for trapping. A magneto-optical trap can be set up also in three dimensions. A quadrupole magnetic field of the form ˇ 1 1 @Bz ˇˇ z eO z x eO x y eO y (79.42) B.r/ Ñ ˇ @z ˇ 2 2
ˇ ˇ mge B ˇˇ @Bu ˇˇ Fu D u u; u D ˇ ; „k ˇ @u ˇ
where ge is the Landé factor of the excited state. A magneto-optical trap may similarly be based on the induced-orientation mechanism of polarization gradient cooling. This might actually be the mechanism of many magneto-optical traps. Atoms in a well-aligned magneto-optical trap reside near the zero of B, so that the magnetic field has little effect on polarization gradient cooling. Trapping and cooling are achieved simultaneously.
79.3
Magnetic Trap for Atoms
The magnitude B.r/ of a magnetic field may have a minimum in free space, as in Eq. (79.42). A particle with a magnetic dipole moment then experiences a trapping potential U.r/ D B.r/ if and B are antiparallel; remains locked antiparallel to B if the field seen by the moving dipole satisfies the adiabatic condition ˇ ˇ B 1 ˇˇ dB ˇˇ B ˇ dt ˇ „
(79.44)
(cf. Sect. 77.5.4). However, if B.r/ D 0 at the minimum, the adiabatic condition is violated, and the dipole may flip (Majorana transition). The particle may end up in a repulsive potential and get expelled from the trap. This could become a problem at low temperatures, when the particles accumulate near the minimum of the potential. Trap configurations are, therefore, designed in which B.r/ ¤ 0 at the minimum.
79.3.1 Evaporative Cooling
rD0
is produced by reversing the direction of current in one of the two Helmholtz coils. Three orthogonal pairs of light beams, each in the C – configuration, complete the trap. The magnetic field is sourceless. To compensate for the ensuing signs of the field gradients, one of the C – corkscrews has the opposite handedness from the other two (with the Cx, Cy and Cz-directions as the reference for handedness). The mechanism of the magneto-optical trap for the 0 ! 1 configuration is the same as the mechanism for Doppler cooling, except that position-dependent level shifts of the excited
A magnetic trap is often combined with evaporative cooling. The most energetic atoms from the tail of the thermal distribution escape from the trap, whereupon the average energy of the remaining atoms decreases. The atoms then thermalize to a lower temperature. Successful operation of evaporative cooling requires a high enough rate of elastic collisions so that the atoms thermalize in a time short compared with the lifetime of the sample. In order to sustain the evaporation, the effective depth of the trap is typically lowered as the atoms cool.
79
Cooling and Trapping
79.4
Trapping and Cooling of Charged Particles
1149
of the ion is stable in all directions. A Paul trap normally operates in the first stability region of Eq. (79.47). Stable motion may be qualitatively divided into forced micromotion at the frequency !Q of the external drive, and slower secular motion of the center of the micromotion. If U D 0, the secular motion takes place in an effective ponderomotive potential UP , Sect. 78.3, equal to the cycleaveraged kinetic energy in the micromotion. Explicitly,
q 2 V 2 x 2 C y 2 C 4z 2 q 2 E 2 .r/ D ; (79.50) UP .r/ D 4m!Q 2 4m!Q 2 %40
Since the potential ˆ.r/ of a static electric field satisfies Laplace’s equation, ˆ.r/ cannot have an extremum in free space. A static electric field, therefore, cannot serve as an ion trap (Earnshaw’s theorem). Paul and Penning traps circumvent this limitation by making use of an alternating electric field and a magnetic field, respectively. Cooling is often employed to assist trapping. Before going any further, it should be noted that neutral atoms trapped in sufficiently deep far-off resonance optical where E .r/ is the AC field amplitude. This is an anisotropic dipole traps and optical lattices may behave essentially like harmonic oscillator potential characterized by the oscillation ions in a Paul trap. They are, therefore, subject to similar frequencies cooling and detection methods. p 2qV : (79.51) z D 2x;y D m!% Q 20
79.4.1
Paul Trap
Quantization of CM motion The separation of micromotion and secular motion is excelQ Ignoring the lent, and the trap is stable, when x;y;z !. micromotion, the CM motion of the ions in the potential Configuration Consider an ideal trap whose surfaces are hyperboloids of UP .r/ may be quantized readily. The energy of a state with revolution; see Fig. 79.2. The two endcaps and the interven- nx;y;z quanta in the coordinate directions x, y, z is ing ring are equipotential surfaces of the quasi-static electric X 1 potential ED „i ni C : (79.52) 2 i Dx;y;z
2 ˆ0 .t/ 2z x 2 y 2 ; (79.45) Variations of the Paul trap ˆ.x; y; z/ D 2%20 Usually, little practical advantage arises from realization of the ideal shape. Even a single electrode with an applied AC is the distance from the center to the ring, z D where % 0 p 0 %0 = 2 is the distance to the endcaps, and ˆ0 .t/ is a voltage voltage may work as a Paul trap. A linear trap is basically a two-dimensional Paul trap with an added static longitudinal applied between the endcaps and the ring, potential to prevent escape of the ions from the ends of the Q : (79.46) trap. A closed race track Paul trap is obtained by bending ˆ0 .t/ D U V cos !t a linear trap into a ring. Motion of an ion In the ideal three-dimensional Paul trap, Newton’s equations Cooling of motion for the coordinates u D x, y, or z may be recast as Origin of laser cooling Mathieu’s equations, The Wigner function formalism in Eqs. (79.11)–(79.14) is d2 u exact as long as the trapping forces are harmonic, but occaC .au 2qu cos 2/u D 0 ; (79.47) d 2 sionally it is more convenient to think in terms of harmonic oscillator eigenstates of a trapped particle instead of posiwhere D !t=2 Q is a dimensionless quantity proportional to tion and momentum. In this view, a plane wave of light can time, the parameters are effect transitions between motional states because the position of the ion is regarded as a quantum mechanical operator. 8qU az D 2ax;y D ; (79.48) The coupling matrix element between two harmonic oscillam!Q 2 %20 tor eigenstates n1 and n2 due to light is proportional to 4qV ; (79.49) qz D 2qx;y D m!Q 2 %20 Kn1 n2 D hn2 jeikrO jn1 i I
Trapping
and m and q are the mass and charge of the particle. Stable jKn1 n2 j2 may be viewed as the Franck–Condon factor for the trapping ensues when au and qu are such that the motion transition n1 ! n2 .
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J. Javanainen
Laser cooling in one dimension Consider the motion of the ion in one of the principal-axis directions x; y; z, with denoting the corresponding CM frequency. In the common case where , Doppler cooling basically works as with a free atom. In the contrary case, , cooling may be achieved by tuning the laser to ! D !0 . Resonant photoabsorption starting with n quanta in the CM motion decreases the quantum number from n to n 1, and the subsequent spontaneous emission, on the average, leaves the CM energy nearly untouched. The net effect is reduction of the CM energy by „ in such a Raman process. Since the oscillating ion sees a frequency-modulated laser with sidebands, this method of cooling is called sideband cooling. For one-dimensional motion of a two-state ion in a traveling light wave at low I , the velocity damping rate for tuning below resonance, < 0, is ˇD
2*2 jj "r ; Œ. C /2 C 2 Œ. /2 C 2
in a given two-state system, but there is a workaround. After Doppler cooling on one transition, the experiment may switch to sideband cooling using another, narrower, transition. A narrow resonance may also be achieved by using the two-photon resonance in a three-state ƒ configuration as an effective two-state system, see Chap. 77.8.2. Laser cooling in three dimensions Either by design or chance, no two of the i are precisely degenerate. If the damping rate ˇ and the trap frequencies i satisfy ˇ / ji j j ;
i ¤j ;
(79.57)
the motion of the ion in each principal axis direction of the trap is cooled independently of the other directions. For x;y;z , a single beam propagating approximately
plaser (79.53) in the direction 1= 3 .eO x C eO y C eO z / suffices to cool all components of the secular motion.
and the expectation value of the CM energy is Energy in micromotion Possibly with the aid of compensating static electric fields, one cooled ion may be confined near the zero of the trap!
ping AC electric fields. The energy in the micromotion is ./2 C 2 .C/2 C 2 2 C 2 C 2 C˛ ; then comparable to the energy in the secular motion. 2 C 2
„ ED 4jj
(79.54) where ˛ characterizes the angular distribution of spontaneous emission, see Eq. (79.30). The result in Eq. (79.54) is useful when either "r or "r . The case with "r ; is for Doppler cooling; the temperature from Eq. (79.54) coincides with Eq. (79.34) for a free atom. The case with "r ; corresponds to sideband cooling in the Lamb– Dicke regime, in which the cooled ion is confined to a region much smaller than . In connection with sideband cooling, it is convenient to cite the expectation number of harmonic oscillator quanta hni instead of energy or temperature; the latter are 1 ; E D „ hni C 2
79.4.2
Penning Trap
Trapping Configuration A Penning trap might use the same hyperboloid-shape electrodes as an ideal Paul trap. However, a DC voltage U is applied between the endcaps and the ring, and a constant magnetic field B in the direction of the trap axis z is added. The magnetic field forces an ion escaping toward the ring to turn back.
1 „
Motion of an ion : ln 1 C hni1 kB The motion of an ion is a superposition of three periodic (79.55) components. For the hyperboloid-shape trap, Eq. (79.45) and Fig. 79.2, the three components are completely decoupled. For optimal sideband cooling, and D , the result Firstly, in the axial direction, the ion executes oscillations at is the axial frequency 4 1 2 hni D .1 C 4˛/ CO : (79.56) 2qU 1=2 4 : (79.58) z D m%20 In principle, by decreasing the linewidth , the ion may be put arbitrarily close to the ground state of the CM har- Secondly, the ion undergoes cyclotron motion in the plane monic oscillator. Such a tuning of linewidth is not practical perpendicular to the trap axis. As a result of the electric field, T D
79
Cooling and Trapping
1151
essentially as for a free atom. However, energy should be added to the magnetron motion in order to reduce the magnetron radius and velocity. The solution is to aim a finite-size laser beam off the center of the trap in such a way that an ion experiences a higher (lower) intensity over the part of its magnetron orbit in which it travels in the direction of (opposite to) the laser beam. With a proper choice of the parameters, the ensuing addition of energy overcomes Doppler cooling of the magnetron motion.
U – Vcos ω˜ t
z0
~
ϱ0
Fig. 79.2 Electrode configuration and voltages of an ideal hyperboloid Paul trap
the frequency of the cyclotron motion c0 D
1 c C 2
1 2 1 2 4 c 2 z
1=2 (79.59)
is displaced from the cyclotron frequency c D qB=m of a free ion. Thirdly, the guiding center of cyclotron motion rotates about the trap axis at the magnetron frequency m D
1 c 2
1 2 1 2 4 c 2 z
1=2 :
(79.60)
The frequencies typically satisfy m z c0 :
(79.61)
Magnetron motion has unusual properties. It takes up the majority of the electrostatic energy in the transverse directions, which in the absence of the magnetic field would lead to expulsion of the ion. Relative to a stationary ion at the trap center, the energy of the magnetron motion is bounded from above by zero. The radius, as well as velocity and kinetic energy of magnetron motion, decreases with increasing total energy. The energy for a state with nc , nz , and nm quanta in the cyclotron, axial, and magnetron motions is, therefore, 1 1 C „z nz C E D „c0 nc C 2 2 1 „m nm C : 2
Other cooling methods Precision measurements are carried out in Penning traps with objects that do not have an internal level structure suitable for laser cooling, and, thus, other cooling methods are used. For light particles such as electrons, characteristic times of radiative damping of the cyclotron motion are in the subsecond regime, and, hence, so are the equilibration times with blackbody radiation. Cooling is accomplished by enclosing the trap in a low-temperature (e.g., liquid helium) environment. For protons and heavier particles, the equilibration times of the cyclotron motion with the environment are impracticably long, and the same applies to the axial and magnetron motions even for electrons. A workable cooling scheme for the axial motion is based on the charges that the oscillating particle induces on the endcaps. The charges generate currents in an external circuit connecting the endcaps. The endcaps are coupled to a cooled resonant circuit tuned to the axial frequency, and axial motion relaxes to thermal equilibrium with the resonant circuit. A variant of this resistive cooling, in which the ring is split into electrically insulated segments, is used to cool the cyclotron motion of protons and heavier ions. Magnetron motion of an electron or proton is cooled by sideband cooling. An electric field with components in both the z-direction and xy-plane, and tuned to ! D z C m , drives transitions that may either increase or decrease the number of quanta in each mode. However, the matrix elements favor transitions with nz D 1 and nm D 1. Pumping of the axial motion is canceled by axial cooling, while an equilibrium with low kinetic energy ensues for the magnetron motion. Ideally, the ratio of kinetic energies becomes Tkin;m
(79.62)
Tkin;z
D
m : z
(79.63)
Cooling
79.4.3 Collective Effects in Ion Clouds Laser cooling For ions in a practical Penning trap, the frequencies c0 , z As soon as there is more than one ion in the trap, and m are < . If k is not orthogonal to either the cy- Coulomb interactions between the ions profoundly shape the clotron motion or the axial motion, Doppler cooling proceeds physics [13].
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J. Javanainen
Ion Crystal In the standard Paul trap, radio frequency heating [14] due to transfer of energy from micromotion to secular motion limits the number of ions that can be cooled efficiently by a laser. Nevertheless, if a low enough temperature is reached, the ions settle to equilibrium positions corresponding to a minimum of the joint trapping and Coulomb potentials and form a crystal. Strongly Coupled Plasma Cooling of a large number of ions is possible in a Penning trap. However, magnetron motion becomes uniform rotation of the entire cloud, and Coulomb interactions set a lower limit on the attainable radius of the cloud. This leads to a lower limit on the kinetic energy and second-order Doppler shift. In a corotating frame, the ions behave like a onecomponent plasma on a neutralizing background. The characteristic parameter for a one-component plasma with charge per particle q and density n is %P D
4 n 3
1=3
q2 ; 4 0 kB T
(79.64)
essentially the ratio of the Coulomb energy between two nearest-neighbor ions divided by the thermal kinetic energy. %P > 1 indicates a strongly coupled plasma; for %P > 2 and %P > 170 liquid and solid phases are expected in an infinite plasma. Experiments with a Penning trap have produced %P ' 300. Concentric shells of ions or various more or less crystalline arrangements are seen, depending on the experimental conditions.
Sympathetic Cooling In a trap that holds two or more species of charged particles, cooling of the motion of one species is transferred by the Coulomb interactions to the other species. This sympathetic cooling broadens the selection of species accessible to ion cooling methods.
79.5
return such atoms to circulation. Other species such as alkaline earths (Sr) and lanthanides (Yb) are occasionally used. They offer very narrow optical transitions and isotopes with varying hyperfine structures (including none). Originally the experiments often started with a longitudinal deceleration and cooling of an atomic beam by a counterpropagating laser beam. To compensate for the change of the Doppler shift of the atoms while they slowed down, the position-dependent magnetic field of a tapered solenoid shifted the transition frequency of the atoms to keep them near resonance while they moved down the solenoid. A magneto-optical trap then scooped up some atoms, cooled them further, and held them. Nowadays, this Zeeman slower is mostly supplanted by various schemes in which a MOT directly captures atoms from the low-velocity wing of the thermal distribution. Depths of neutral-atom traps are far below 1 K. Storage times in a MOT are typically on the order of seconds, limited at high densities by exothermic binary collisions and at low densities by collisions with the atoms of the background gas. Absorption imaging of the atom cloud is the standard diagnostic tool, giving both (the projection to a plane of) the spatial distribution of the atoms, and after proper calibration, the number of the atoms. If cooling and trapping is suddenly turned off, the atom cloud flies apart ballistically, and the velocity distribution gets converted into a position distribution. Analysis of time-delayed absorption images, thus, gives the momentum distribution. The evolution of the experimental techniques during the era of laser cooling and trapping has been nothing short of spectacular. Acousto-optic and electro-optic modulators enable control of temporal variations and frequencies of the light; the spatial shape of the light field may be tailored using spatial light modulators such as digital micromirror devices; by moving an optical trap of light around quickly, one can “paint” a nearly arbitrary time-averaged optical potential energy for the atoms [15]; it is possible to assemble an arbitrary defect-free 2-D array of atoms one atom at a time [16], and so on. Digital control of the apparatus, data acquisition, and data analysis is an indispensable element in the experiments.
Experimental 79.5.2
79.5.1
Trapped Particles
Free Particles
Historically, the focus has been on alkali metal vapors, to a large extent because the required laser frequencies can be generated using inexpensive diode lasers. Hyperfine structure of the ground state of the alkalis complicates experiments because the atoms may end up in an inert hyperfine level outside the active cooling/trapping transitions. To counteract this, appropriately tuned repumper light is added to
Both Paul and Penning traps behave like a conservative potential and scatter rather than confine a particle coming from the outside. One method to load a trap is to generate the ions in situ, e.g., by letting a beam of atoms and electrons collide inside the trap. Time-dependent electric potentials are another loading method. The trapped species is injected through a hole in the endcap, and the opposing endcap is raised to an electric potential that makes the entering parti-
79
Cooling and Trapping
cles stop. The potential is then lowered before it ejects the particles. A single electron, positron, proton, antiproton, or ion may be loaded. Typical depths of ion traps are 1 eV or 104 K. With the aid of cooling, the storage time may be made infinite for all practical purposes. Trap frequencies are measured by observing the resonances excited by added AC fields. For instance, an electric field near the axial resonance frequency may be coupled between the ring and one endcap. A resonance circuit coupled between the ring and the other endcap is used to detect the resonance. Alternatively, ejection of the driven ions is monitored. For an ion with a dipole-allowed resonance transition, fluorescence of a single ion is readily detected. Even absorption of a single ion may be measurable. Various methods of finding the temperature have been devised. At temperatures of 1 K and higher, Doppler broadening of a dipole-allowed optical transition is observable. The size of the single-ion cloud is a measure of temperature. Finally, motional sidebands in the absorption of a narrow transition ( ), not necessarily the same transition as the one used for cooling, may be measured to find hni. In the Lamb–Dicke regime, only the carrier absorption at D 0 and sidebands at D ˙ are significant, and the ratios of the peak absorptions are
1153
79.6 Applications
Trapping and cooling offer increased interaction times between the atoms/ions and the light. This leads to reduced transit time broadening, and, indeed, to macroscopic (> 1 s) interaction times. Laser cooling in a magneto-optical trap routinely gives temperatures so low that the Doppler width is below the natural linewidth of the cooling transition. A homogeneously broadened atomic sample is thus prepared. Cooling also enables reduction of the second-order Doppler effect. Spectroscopy and various frequency measurements have traditionally been the primary beneficiary. Potential applications range from detection of the change of natural constants over time (Chap. 30) to such feats of technology as the Global Positioning System. Another broad area drawing from low temperatures is experiments, and in the future possibly technologies, that rely on quantum mechanical behavior of either light or atoms. Nowadays, nearly all cutting-edge experiments in quantum optics start with laser cooling. Trapping and manipulation of individual atomic particles is also an established practice. In this chapter we, by necessity, limit the discussion of applications to very few topics in which the interplay between the internal degrees of freedom and the CM motion is the central theme. At the moment, the most prominent broad "r "r (79.65) application area of laser cooling and trapping is probably ˛ W ˛0 W ˛C D hni W 1 W .1 C hni/ : quantum gases under conditions when the particle statisFluorescence carries similar information. tics matters, such as the use of optical lattices as quantum In an ion crystal, the ions have collective vibration modes simulators of condensed matter systems. These are taken up akin to phonons, instead of the three vibration modes along separately in Chap. 80. the principal axes of the trap of a single ion. Doppler cooling and sideband cooling work for such collective modes much 79.6.1 Cold Molecules like they work for the vibration modes of a single ion. Ion traps and optical dipole traps make it possible to isolate an individual atom scale particle for studies for a practi- Ultralow-temperature molecules may be produced by startcally infinite time, which enables clean experiments on vari- ing with an ultralow-temperature atomic gas or gases, and ous fundamental aspects of quantum mechanics and quantum using photoassociation or Feshbach resonances to convert electrodynamics. Quantum jumps are a case in point. Sup- pairs of atoms into diatomic molecules [18]. Direct laser pose that, in addition to an optically driven two-level system, cooling of molecules was long thought impossible because of a single ion has a third “shelving” state. Suppose further that the proliferation of vibrational and rotational states. In genthe ion infrequently makes a transition to the shelving state, eral, there is no closed transition that allows enough cycles stays there for a long time compared with the timescale of of absorption and spontaneous emission so that the net exspontaneous emission of the active system, and then returns change of momentum between light and the molecules would to the two-level system. When the ion makes a transition be comparable to the momentum of a room-temperature to the shelving state, fluorescence from the two-level sys- molecule. However, in certain molecules, such as CaF and tem suddenly ceases, and the fluorescence reappears equally SrF, there are diagonal transitions between vibrational states abruptly when the ion returns to the two-level system. These with the Franck–Condon factor very close to unity, giving jumps in light scattering are the quantum jumps [17]. They a simulacrum of a closed two-level system. One still has are a method to detect a weak transition with an enormous to compensate for the leakage from the almost-closed tranamplification; a single transition to or from the shelving state sition because the Franck–Condon factors of the unwanted may mean the difference between the presence or absence of transitions are not exactly zero, branching between hyperfine levels is possible as in atoms, and so on. A number of billions of fluorescence photons.
79
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J. Javanainen
laser frequencies may be needed to repump the molecule ergy in the rotating frame lies halfway between the states, the into the desired transition. Nonetheless, the basic program Hamiltonian reads " 2 # of laser cooling and trapping has been demonstrated for pOx „ „* ikx e diatomic molecules [19]. One of the foreseeable goals is 2 H D 2m„* 2 : (79.66) pO 2 a Bose–Einstein condensate of polar molecules, a novel sys 2 eikx 2mx C „ 2 tem because of the strong anisotropic long-range electric After a unitary transformation generated by dipole interactions between the molecules. # " 0 eikx=2 ; (79.67) U D 0 eikx=2 79.6.2 Quantum Systems of Internal
and CM States
the Hamiltonian may be written in terms of Pauli spin matrices as A vibrational mode of a trapped ion and an effective two 2 state system for the internal degrees of freedom make a real„k „* „ 1 pOx C z x z : (79.68) H D ization of the Jaynes–Cummings model (discussed in detail 2m 2 2 2 in Chap. 83.4.1) possible. Moreover, sideband cooling enables the experimenter to put the mode cleanly in its lowest The term / pOx z that arises from the expansion of the square quantum state. These observations have inspired quantum- is the spin–orbit coupling, seemingly a direct coupling bestate engineering with the objective of generating an arbitrary tween motional and internal states of the atom. A Raman state of the vibrational motion of the ion [20]. In many-ion scheme with two beams propagating in opposite directions crystals, the collective vibration modes may be used to cou- and the intermediate state far-off resonance offers a potenple and entangle the internal degrees of freedom of two or tial realization. A main goal here, again, is clean AMO more ions. As discussed in Chap. 85, quantum information physics simulations of prominent systems in condensed matter physics. A similar topic, synthetic gauge fields that arise processing has been demonstrated in this manner. Fully coherent manipulation of joint internal and CM from coupling between internal and motional states of an states of free atoms is also possible. For instance, two mo- atom, is discussed at some length in Chap. 80. tional states of a ground-state atom with the velocities v and v C 2vr can be coupled by two light pulses with opposite propagation directions. If both pulses are off resonance from References a transition between the ground and excited state of the atom, 1. Metcalf, H., van der Straten, P.: Laser Cooling and Trapping. and their frequency difference matches the difference of the Springer, New York (1999) kinetic energies 12 mv 2 and 12 m.v C 2vr /2 , a Raman scheme 2. Wineland, D., Itano, W.M., Van Dyck Jr, R.: High-resolution spectroscopy of stored ions. Adv. At. Mol. Phys. 19, 135–186 (1983) is set up that may transfer the atom between the two mo3. Brown, L.S., Gabrielse, G.: Geonium theory: Physics of a single tional states. Variations of this Bragg diffraction technique electron or ion in a Penning trap. Rev. Mod. Phys. 58, 233–311 have been used to coherently split an atomic wave packet (1986). https://doi.org/10.1103/RevModPhys.58.233 to components with the velocities differing by tens of recoil 4. Stenholm, S.: The semiclassical theory of laser cooling. Rev. Mod. Phys. 58, 699–739 (1986). https://doi.org/10.1103/RevModPhys. velocities [21]. Recombining the wave packet makes an in58.699 terferometer that is sensitive to the forces, real and inertial, 5. Knoop, M., Madsen, N., Thompson, R.C. (eds.): Physics with that acted on the atom while the wave packet was split. AnalTrapped Charged Particles. Imperial College Press, London ogous atom interferometers are being developed for purposes (2014) 6. Morsch, O., Oberthaler, M.: Dynamics of Bose–Einstein condensuch as gravimetry. The hope is improved sensitivity; in the sates in optical lattices. Rev. Mod. Phys. 78, 179–215 (2006). literature one sees claims of fundamental scaling of the sensihttps://doi.org/10.1103/RevModPhys.78.179 tivity between atom interferometers and light interferometers 7. (2018). http://steck.us/alkalidata with mc 2 =„!0 . 8. (2018). https://physics.nist.gov/PhysRefData/ASD/lines_form. html More generally in the way of coupling between internal 9. Javanainen, J.: Density-matrix equations and photon recoil for and CM degrees of freedom, a field called (in the atomic conmultistate atoms. Phys. Rev. A 44, 5857–5880 (1991). https://doi. text maybe confusingly) spin–orbit coupling [22] emerged org/10.1103/PhysRevA.44.5857 during the past decade and has inspired much theoretical 10. Castin, Y., Mølmer, K.: Monte Carlo wave-function analysis of 3D optical molasses. Phys. Rev. Lett. 74, 3772–3775 (1995). https:// and experimental work. In an elementary example, consider doi.org/10.1103/PhysRevLett.74.3772 a two-level atom without damping moving in the x-direction 11. Dalibard, J., Cohen-Tannoudji, C.: Laser cooling below the in the plane wave of light with the amplitude proportional to Doppler limit by polarization gradients: Simple theoretical modeikx . Using the vector-matrix representation for the internal els. J. Opt. Soc. Am. B 6, 2023–2045 (1989). https://doi.org/10. 1364/JOSAB.6.002023 states fjei; jgig and in the convention that the zero of the en-
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12. Gerz, C., Hodapp, T.W., Jessen, P., Jones, K.M., Phillips, W.D., Westbrook, C.I., Molmer, K.: The temperature of optical molasses for two different atomic angular momenta. EPL 21, 661 (1993) 13. Bollinger, J.J., Wineland, D.J., Dubin, D.H.E.: Non-neutral ion plasmas and crystals, laser cooling, and atomic clocks. Phys. Plasmas 1, 1403–1414 (1994). https://doi.org/10.1063/1.870690 14. Nam, Y., Weiss, D., Blümel, R.: Explicit, analytical radiofrequency heating formulas for spherically symmetric nonneutral plasmas in a Paul trap. Phys. Lett. A 381(40), 3477–3481 (2017). https://doi.org/10.1016/j.physleta.2017.09.001 15. Henderson, K., Ryu, C., MacCormick, C., Boshier, M.G.: Experimental demonstration of painting arbitrary and dynamic potentials for Bose–Einstein condensates. New J. Phys. 11, 043030 (2009) 16. Barredo, D., de Léséleuc, S., Lienhard, V., Lahaye, T., Browaeys, A.: An atom-by-atom assembler of defect-free arbitrary twodimensional atomic arrays. Science 354, 1021–1023 (2016). https://doi.org/10.1126/science.aah3778 17. Blatt, R., Zoller, P.: Quantum jumps in atomic systems. Eur. J. Phys. 9, 250 (1988) 18. Danzl, J.G., Mark, M.J., Haller, E., Gustavsson, M., Hart, R., Aldegunde, J., Hutson, J.M., Nägerl, H.-C.: An ultracold highdensity sample of rovibronic ground-state molecules in an optical lattice. Nat. Phys. 6, 265 (2010) 19. Barry, J.F., McCarron, D.J., Norrgard, E.B., Steinecker, M.H., DeMille, D.: Magneto-optical trapping of a diatomic molecule. Nature 512, 286 (2014) 20. Leibfried, D., Blatt, R., Monroe, C., Wineland, D.: Quantum dynamics of single trapped ions. Rev. Mod. Phys. 75, 281–324 (2003). https://doi.org/10.1103/RevModPhys.75.281 21. Müller, H., Chiow, S.-W., Long, Q., Herrmann, S., Chu, S.: Atom interferometry with up to 24-photon-momentum-transfer beam splitters. Phys. Rev. Lett. 100, 180405 (2008). https://doi.org/10. 1103/PhysRevLett.100.180405 22. Zhai, H.: Degenerate quantum gases with spin–orbit coupling: A review. Rep. Prog. Phys. 78, 026001 (2015)
1155 Juha Javanainen Juha Javanainen is Professor of Physics at the University of Connecticut. He has worked on a number of topics in theoretical quantum optics.
79
Quantum Degenerate Gases
80
Juha Javanainen
Contents 80.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1157
80.2 80.2.1 80.2.2 80.2.3
Elements of Quantum Field Theory Bosons . . . . . . . . . . . . . . . . . . . Fermions . . . . . . . . . . . . . . . . . . Bosons Versus Fermions . . . . . . . . .
. . . .
. . . .
. . . .
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. . . .
. . . .
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1158 1158 1159 1159
80.3 80.3.1 80.3.2 80.3.3 80.3.4 80.3.5 80.3.6
Basic Properties of Degenerate Gases . . . Atoms Are Trapped . . . . . . . . . . . . . . . . Atom–Atom Interactions . . . . . . . . . . . . Model Hamiltonian . . . . . . . . . . . . . . . . Bosons . . . . . . . . . . . . . . . . . . . . . . . Meaning of the Macroscopic Wave Function Fermions . . . . . . . . . . . . . . . . . . . . . .
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1160 1160 1160 1160 1160 1164 1165
80.4 80.4.1 80.4.2 80.4.3
Experimental . . . . . . . . . . . . . Preparing a BEC . . . . . . . . . . . Preparing a Degenerate Fermi Gas Monitoring Degenerate Gases . . .
. . . .
. . . .
. . . .
. . . .
. . . .
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1165 1165 1166 1166
80.5 80.5.1 80.5.2
BEC Superfluid . . . . . . . . . . . . . . . . . . . . . . . 1167 Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . 1167
80.6 80.6.1 80.6.2 80.6.3
Optical Lattice as Quantum Simulator Basics of the Optical Lattice . . . . . . . . Strongly Correlated Systems . . . . . . . . Topological Phases of Matter . . . . . . . .
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strongly correlated systems. Inside AMO physics the approach is in the vein of quantum optics, as opposed to atomic/molecular structure and collisions. For the most part, the coverage is on elementary concepts and basic material. The exception is Sect. 80.6, where a few topical issues are addressed. The article [1] has become the standard reference on the mean-field physics of a Bose–Einstein condensate (BEC), [2] is particularly explicit about the excitations of a BEC, and [3] reviews the basics of the concurrent condensed-matter approach to quantum degenerate gases. The monograph [4] is a recent broad overview of BEC physics. Here, references are usually not given for topics that are discussed in these sources or where a full discussion can traced from them. Otherwise, references are meant to be entries to the literature. Assignment of credit or priority is never implied. Keywords
magnetic trap optical trap Feshbach resonance phase space density condensate wave function
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172
Abstract
The purpose of this chapter is to summarize the physics of dilute quantum degenerate gases. Given the broad activity in the field, many choices have to be made regarding the topics to include and the style of the discussion. Emphasis is on AMO physics, as opposed to condensed-matter physics. Two related choices are that virtually nothing is said about temperature dependence, and that the focus is on one-particle and mean-field theories as opposed to J. Javanainen () Department of Physics, University of Connecticut Storrs, CT, USA e-mail: [email protected]
80.1 Introduction Bose–Einstein condensation in dilute alkali metal vapors has realized a source of atoms with properties analogous to the properties of laser light, and ultralow-temperature Fermi gases have also come under study. Quantum degenerate gases have become a main theme in AMO physics. Dilute-vapor systems are weakly interacting and subject to a degree of experimental control not seen before in traditional low-temperature condensed-matter systems. Ultralowtemperature gases thereby reinvigorated, say, investigations of superfluids. More recently, the main thrust has been to use AMO techniques to simulate experimentally condensed-
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_80
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matter systems, some of which have been subject to much Its Hermitian conjugate, the creation operator, behaves as theoretical study in the past but have lacked clean implep mentations. This applies, in particularly, to various lattice ak j : : : ; nk ; : : :i D nk C 1 j : : : ; nk C 1; : : :i : (80.2) systems [5, 6]. It follows that
80.2 Elements of Quantum Field Theory
ak ak j : : : ; nk ; : : :i D nk j : : : ; nk ; : : :i ;
A Bose–Einstein condensate and a degenerate Fermi gas are both consequences of particle statistics, exchange symmetries of the many-particle wave function. It is possible, in principle, to deal directly with the wave functions, but in practice, analyses of many-body systems are usually carried out using the methods of second quantization and field theory. In first quantization, the particles are labeled as if each one had a unique tag on it, and the wave function for more than one indistinguishable particle must be symmetrized explicitly. In second quantization, the question is how many particles are in a given one-particle state without any distinction between identical particles. The exchange symmetries are then taken care of automatically. Here, we briefly summarize [7] elementary features of quantum field theories for both bosons and fermions.
(80.3)
and so nO k D ak ak is called the number operator for the state k. Correspondingly, NO D
X
ak ak
(80.4)
k
is the operator for the total number of particles in the system. The annihilation and the creation operators have the usual boson commutators,
Œak ; ak 0 D ak ; ak 0 D 0 ;
ak ; ak 0 D ıkk 0 :
(80.5)
The boson field operator is defined as O .x/ D
X
uk .x/ak :
(80.6)
k
80.2.1 Bosons
The commutator for the field operator,
Particles with an integer value of the angular momentum O .x/; O .x 0 / D ı.x x 0 / ; (80.7) obey the Bose–Einstein statistics. The characteristic property is that a one-particle state can accommodate an arbitrary follows from boson commutators and the completeness of number of bosons. the wave functions fuk .x/gk . The orthonormality of the wave functions gives the expression State Space for Bosons Specifically, first consider one particle whose states are completely specified by a set of quantum numbers k. As a notational device for the purposes of the present chapter, all of the quantum numbers are assumedly mapped in a one-to-one fashion to nonnegative integers, and, correspondingly, the quantum numbers are written k D 0; 1; 2; : : :. The quantum numbers incorporate a description of the state of the centerof-mass (CM) motion of the particle. We, therefore, have an orthonormal basis of wave functions to represent any state of a particle, fuk .x/gk , where x stands for the CM coordinate. Given the one-particle states, the postulate is that the Fock states jn0 ; n1 ; : : : ; n1 i with nk D 0; 1; 2; : : : particles in the states k D 0; 1; 2; : : : form an orthonormal basis for the many-body system.
Second-Quantized Operators for Bosons The annihilation operator for the state k, ak , is defined by ak jn0 ; n1 ; : : : ; nk ; : : : ; n1 i p D nk jn0 ; n1 ; : : : ; nk 1; : : : ; n1 i :
Z NO D
(80.8)
for the particle number operator. The positive operator n.x/ O D O .x/ O .x/
(80.9)
evidently represents the density of the particles at the position x. The second-quantized operators introduced thus far can be used to express all observables acting on indistinguishable bosons. The most relevant here are the one and two-particle operators. One-particle operators, such as kinetic energy, act on one particle at a time, while two-particle operators, such as atom–atom interactions, refer to two particles. In first quantization, these are of the form O1 D
(80.1)
d 3 x O .x/ O .x/ ;
X n
V .x n / ;
O2 D
1X u.x n ; x n0 / ; 2 0 n¤n
(80.10)
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where the sums run over the labels of the particles. The corresponding second-quantized operators are Z
regarded as a constant, Eq. (80.14) may be regarded as an equation for the critical temperature Tc or the critical density nc for Bose–Einstein condensation.
OO 1 D
d 3 x O .x/V .x/ O .x/ ; (80.11) Z 1 80.2.2 Fermions d 3 x d 3 x 0 O .x/ O .x 0 /u.x; x 0 / O .x 0 / O .x/ : OO 2 D 2 (80.12) Particles with a half-integer angular momentum obey the Fermi–Dirac statistics. Each Fock state may then only have When the particles have internal degrees of freedom in addi- the occupation numbers nk D 0; 1. The conventional definition to the CM motion, such as hyperfine and Zeeman states, tion of the annihilation operator contains a phase factor, it is convenient for the present purposes to regard particles in each internal state as a separate species. Thus, if the quantum ak jn0 ; n1 ; : : : ; nk ; : : : ; n1 i Pk1 number breaks up into k fp; ˛g, where p stands for quanD nk .1/ pD0 np jn0 ; n1 ; : : : ; nk 1; : : : ; n1 i ; (80.15) tum numbers of the center of the mass and ˛ for the quantum numbers of the internal state, it is expedient to define a quanand fermion operators are governed by the anticommutator tum field for each species ˛ as O ˛ .x/ D
X
up˛ .x/ ap˛ :
ŒA; BC AB C BA
(80.13)
p
Mechanisms that cause transitions between the internal states couple the fields O ˛ .x/ with different ˛.
States of Bosons To complete the transformation from wave function quantum mechanics to second quantization, the state of the system must be specified in second quantization. For instance, take the Hamiltonian HO and the particle number operator NO . According to statistical mechanics a system characterized by the temperature T and chemical potential is in the state with O O the density operator O D e.H N /=kB T =Z, where the grand O O partition function is Z D Tr e.H N /=kB T . Bose–Einstein Condensate The state of a boson system of particular interest here is the BEC. In an ideal gas, condensation entails a macroscopic fraction of the particles occupying the ground state of the CM motion. Condensation is a phase transition that occurs when either the density of the gas is increased, or the temperature is lowered. In a homogeneous ideal Bose gas, the governing parameter is the phase space density , defined in three spatial dimensions as
(80.16)
rather than the commutator. For instance, Œak ; ak 0 C D 0 ;
ak ; ak 0
C
D ıkk 0 :
(80.17)
Except for the use of anticommutators in lieu of commutators, all formal expressions for field operators and one and two-particle operators written down for bosons in Sect. 80.2.1 remain valid as stated.
Degenerate Fermi Gas A degenerate Fermi gas realized in a dilute atom vapor is the fermion counterpart of a BEC. The basic parameter of a free noninteracting Fermi gas is the Fermi energy, the chemical potential at zero temperature. It is given by F D
„2 kF2 I 2m
1=3 ; kF D 6 2 n
(80.18)
where n once more is the density for the relevant fermion species. In the limit of zero temperature, the Fermi gas makes a Fermi sea; the states below the Fermi energy are filled with one particle each, the states above the Fermi energy are empty. The gas begins to show substantial deviations from the Maxwell–Boltzmann statistics and may be regarded as degenerate when the temperature is below the Fermi tem 2 3=2 perature, T / TF D F =kB . Except for a numerical factor, in 2 „ n; (80.14) D terms of density and temperature, the condition T D TF is the mkB T same as the condition for Bose–Einstein condensation. where m is the mass of the condensing atoms, T is the temperature, and n is the density of the condensing species. For the purposes of quantum degeneracy, each internal state of 80.2.3 Bosons Versus Fermions an atom behaves as a separate species. Bose–Einstein con
densation takes place when the phase space density satisfies Isotopes of alkali metals with an odd mass number 7 Li, D 2:612. Depending on whether density or temperature is 23 Na, 39 K, 85 Rb, 87 Rb, 133 Cs make Bose–Einstein gases,
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while isotopes with an even mass number 6 Li, 40 K make Fermi–Dirac gases. Atoms are composite particles consisting of fermions, and how they may act as bosons is a legitimate question. Whether a satisfactory formal answer to this question exists may be debated, but in practice, atoms seem to obey the correct statistics in processes that do not expose their individual constituents. When two bosonic atoms with integer angular momenta combine into a molecule, the molecule has an integer angular momentum and behaves as a boson. On the other hand, two fermionic atoms also make a bosonic molecule. Models for the latter type of system are basically ad hoc since, at this point in time, no microscopic theory for such a rearrangement of the statistics appears to exist. Nonetheless, empirically, diatomic molecules formed by combining two fermionic atoms indeed appear to be bosons.
where a is the s-wave scattering length. Qualitatively, the scattering length is positive (negative) if the interaction is repulsive (attractive).
80.3.3 Model Hamiltonian Quantum field theory for a single-component Bose gas usually starts with the Hamiltonian Z HO D
Basic Properties of Degenerate Gases C
80.3.1 Atoms Are Trapped Quantum degenerate alkali vapors are typically prepared in an atom trap. Close to the bottom almost every trap is a three-dimensional harmonic oscillator potential completely characterized by the principal-axis directions and the corresponding trap frequencies !i , the (angular) frequencies at which a single atom would oscillate back and forth in the given principal-axis direction. In the principal-axis coordinate system the trapping potential reads 1X m!i2 xi2 : 2 i D1
(80.22)
where the Hamiltonian density is O .x/ D O .x/ H
80.3
d 3 x HO .x/ ;
„2 2 r C V .x/ O .x/ 2m
2 „2 a O .x/ O .x/ O .x/ O .x/ : m
(80.23)
As in Sect. 80.2.1, O .r/ is the boson field operator, the kinetic energy „2 r 2 =2m and trapping potential V .x/ are one-particle operators, and atom–atom interactions are governed by the two-particle operator u of Eq. (80.21). Analogous models can be written down for multicomponent boson and fermion fields, for conversion between atoms and molecules, and so on.
3
V .x/ D
(80.19) 80.3.4
Bosons
Gross–Pitaevskii Equation It is convenient to introduce the characteristic harmonicoscillator frequency scale and the corresponding harmonicoscillator length scale as Mean-field approximation Conventionally, the next step for bosons is to go over to the r „ : (80.20) corresponding classical field theory. The result is referred to !N D .!1 !2 !3 /1=3 ; ` D m!N as mean-field theory or semiclassical theory. Formally, one first writes down explicitly the Heisenberg equation of motion for the boson field O ,
80.3.2 Atom–Atom Interactions
@ O At low temperatures/energies, only s-wave collisions are sig.x; t/ D O .x; t/; HO ; (80.24) i„ @t nificant. In the theory of quantum degenerate gases, these are frequently represented by a pseudopotential tailored to give the correct s-wave phase shift. Except for a mathematical and then declares that in the equations of motion, O ! is subtlety that is usually ignored, for two atoms the pseudopo- a classical field and no longer a quantum field. We call the macroscopic wave function of the condensate. This approxitential is mation is precisely analogous to using the classical instead of 4 „2 a the quantum description for the electric and magnetic fields (80.21) ı.x 1 x 2 / ; u.x 1 ; x 2 / D m of the light coming out of a laser.
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Time-dependent Gross–Pitaevskii equation condensate must then be presumed lost. In the absence of The ensuing time-dependent Gross–Pitaevskii equation an external potential, a condensate with a negative scatter(GPE) is ing length is unconditionally unstable against collapse. For a bounded condensate, the increase in the kinetic energy „2 2 @ coming with the decreasing size may hold off the collapse, .r; t/ D r C V .r/ .r; t/ i„ @t 2m provided the number of atoms in the condensate is sufficiently small. Simple dimensional-analysis arguments give 4 „2 a (80.25) the condition of stability in a harmonic trap as N jaj / `. C j .r; t/j2 .r; t/ : m Behavior attributed to a collapse has been observed in This equation is nonlinear, and normalization of the macro- 7 Li with a negative scattering length. By using a Feshbach scopic wave function is important. Quantum mechanically, resonance it is also possible to adjust the scattering length, the particle number operator is given by Eq. (80.8), so the see Eq. (80.66), which has led to further demonstrations of normalization for a system with N particles naturally reads collapse-like physics. Z (80.26) Healing length d 3 x j .x; t/j2 D N : Consider the time-independent GPE Eq. (80.28) without an Time evolution under Eq. (80.25) preserves the normaliza- external potential and scale the various quantities: tion. Obviously, and in accordance with Eq. (80.9), D n.x/ D j .x/j2
(80.27)
is the local density of the gas. Time-independent Gross–Pitaevskii equation Solutions to the time-dependent GPE of the form .x; t/ D .x/ eit =„ are stationary states with no time evolution in the physics. The analog of the energy of a stationary state in ordinary quantum mechanics is called chemical potential. The corresponding wave function satisfies the time-independent GPE
p
N D N n ;
4 „2 na ; m
1 : x D xI N Dp 8 na
(80.29)
Here, n is the density scale for the gas, and the length scale is . In terms of these new variables, the time-independent GPE reads ˇ ˇ2 N N D rN 2 N C ˇN ˇ N :
(80.30)
There is a solution in all of space with N D 1, N D 1. If for some reason, such as an edge of the sample, the condensate wave function must vanish, the length scale over which 4 „2 a „2 2 2 (80.28) the wave function grows back to one (in the scaled units) is D r CV C jj : 2m m on the order of unity. p In fact, Eq. (80.30) has the solution N .x/ N D tanh zN = 2 in the half-space zN 0. The quantity Both GPEs are nonlinear variants of the Schrödinger equais the minimum length scale over which a condensate wave tion, and in other contexts they are often referred to as function can build up to the density n. It is called the healing nonlinear Schrödinger equations. The nonlinear term aplength. proximates the interaction energy of an atom with the other atoms in an averaged way based on the local density of the Thomas–Fermi approximation atoms, hence the term mean-field theory. Without atom–atom interactions, the ground state of the trapping potential V .x/ would be the lowest-energy (lowest ) Sign of scattering length solution to Eq. (80.28). However, experience has shown that The qualitative properties of a condensate, as per the GPE, even modest repulsive atom–atom interactions (a > 0) spread depend on the sign of the scattering length. For repulsive out the macroscopic wave function of the condensate a great atom–atom interactions or no atom–atom interactions, a 0, deal. With increasing size comes decreasing kinetic energy, both the time-dependent and the time-independent forms are according to the Heisenberg uncertainty principle. This sugmathematically well behaved. Unless otherwise noted, below gests the Thomas–Fermi approximation, in which the kinetic the scattering length is always assumed nonnegative. energy term in Eq. (80.28) is simply ignored. The density of In the case of a negative scattering length, a BEC may, the gas is then easily solved to be in principle, decrease its energy without a bound by col( lapsing to a point. Mechanisms such as three-body recomm ŒV .x/ ; > V .x/ 4 „2 a (80.31) n.x/ D bination and molecule formation would eventually set in as 0; otherwise ; the density increases, and the collapse would stop, but the
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an inverted image of the trapping potential. The normalization Eq. (80.26) can be used to find the relation between chemical potential and particle number, and all of the unknown quantities may, in principle, be found. For a harmonic potential the Thomas–Fermi approximation can be worked out explicitly with the results 15Na 1=5 15Na 2=5 1 ; RD` ; D „!N 2 ` ` 1 ` 15Na 2=5 : n.0/ D 8 `3 a ` (80.32)
where „q and „ are the momentum and energy associated with the excitation relative to the momentum and energy of the original flow. The GPE mixes the field and its complex conjugate , so that two small amplitudes u and v are needed for the excitations. The ansatz Eq. (80.36) is a solution to the time-dependent GPE to the lowest nontrivial order in u and v if these amplitudes and the frequency of the excitation satisfy the eigenvalue equation ! ! ! 0 u u q C 0 C q v D : 0 q C 0 q v v v (80.37)
The quantity R represents the size of the condensate. In particular, it equals the radius of the spherical condensate if the trap is isotropic with !1 D !2 D !3 . Finally, n.0/ is the central, maximum, density of the atoms. The relevant dimensionless parameter is Na=`, which can easily be much larger than unity in the experiments. When the Thomas–Fermi approximation is accurate, the chemical potential exceeds the typical level spacing of the harmonic-oscillator trap, and the condensate is larger than the ground-state wave function of the harmonic oscillator would be. Ordinarily, the condensate is also much larger than the healing length.
The remaining problem is that the eigenvalue equation has two solutions for each q, which gives twice as many smallexcitation modes as there are degrees of freedom. The extra modes are the penalty one pays for the linearization of the GPE. The criterion juj2 jvj2 > 0 picks out the correct smallexcitation modes. The corresponding dispersion relation for the excitations is q (80.38) .q/ D q v C q .q C 20 / :
Small Excitations in a BEC Linearizing the GPE The time-dependent GPE is nonlinear but may be linearized around a stationary solution. Consider the special case without a trapping potential, V .x/ 0. The stationary solutions are plane waves, p (80.33) .x/ D n eipx : This corresponds to a flow of the gas at the velocity v D „p=m and with a momentum „p per atom. The chemical potential of such a mode is 1 D p C 0 ; „ 2 where
(80.34)
For a stationary BEC with v D 0 and in the limit q ! 0, Eq. (80.38) gives Ñ cq. This confirms the identification of c as the speed of sound. In the BEC experiments the condensates are usually trapped, but in principle the same analysis of small-excitation modes may be carried out both numerically and in a myriad of analytical approximations. The generic result is that the trap frequencies lend their frequency scale to small excitations. At low enough temperatures, excitation frequencies calculated in this way agree well with the experiments. Within the mean-field approximation, small excitations may be analyzed similarly in all boson systems, for instance, in a multicomponent Bose–Einstein condensate or a joint atom–molecule condensate. The evolution frequencies may be complex, which signals a dynamical instability of the stationary configuration; there are small-excitation modes that grow exponentially. The instability of a free gas with a negative scattering length, which is apparent in Eq. (80.38) for 0 < 0, is a simple example.
p „ 8 na „ Bogoliubov theory cD D m m Bogoliubov theory is the many-body quantum version of the (80.35) analysis of small excitations. The idea is to treat the condenare the dispersion relation of free atoms, a peculiar analog of sate mode 0 , containing n0 atoms, separately in the field operator, the rest energy, and the speed of sound in the BEC. The ansatz for small deviations from the stationary solup O D n0 0 C ı O ; (80.39) tion is written as p .x; t/ D n ei.px „ t / 1 C u ei.qx t / C v ei.qx t / ; expand the Hamiltonian in the lowest nontrivial (second) (80.36) order in the remnant quantum field ı O , and diagonalize. „p 2 p D ; 2m
mc 2 0 D ; „
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Split-step Fourier method The superposition principle does not hold for the solutions of the time-dependent GPE, and the excited states are usually not orthogonal to one another in any useful sense. Methods based on eigenstate expansions for solving the time-dependent GPE are cumbersome at best. Instead, one often simply integrates the GPE as a partial differential equation in time. A number of different methods are used, but here we only discuss an elementary split-step Fourier method [8]. This is an exceedingly popular algorithm for parabolic equations, which is easy to implement and with minor modifications also solves the time-independent GPE in any number of dimensions. Thus, consider integration of Eq. (80.25) forward in time over a step from t to t C t. For this purpose, assume first 2 When the so-called gas parameter na3 is much smaller than that j j in the GPE were a constant equal to its value at time unity, at low enough temperatures most of the atoms are in t, then the evolution over the time step t would be given by the condensate. Mean-field theory and the GPE are expected „r 2 to apply, and empirically, they do. .x; t C t/ D exp it C U.x/ .x; t/ ; 2m (80.41) Numerical Methods for GPE
The result is small-excitation modes with the annihilation operators Ak , where k stands for the appropriate quantum numbers. It turns out that the core mathematics of Bogoliubov theory is the same as the mathematics of small excitations, but two features are added. First, Bogoliubov theory explicitly shows that the coefficients u and v in the analog of Eq. (80.37) need to satisfy juj2 jvj2 D 1 to ensure boson commutators for the operators Ak . Second, with quantum fluctuations, atom–atom interactions force atoms out of the condensate even at zero temperature. In a homogeneous (untrapped) condensate, in the limit na3 1, at T D 0, the fraction of noncondensate atoms is r N n0 8 na3 D : (80.40) N 3
Mathematical properties of the GPE Let us momentarily assume that, by separation of variables or by some fiat, the problem of solving the time-independent GPE has been rendered one-dimensional. The ordinary Schrödinger equation is linear, and any constant multiple of a solution is also a solution. One parameter, e.g., the logarithmic derivative of the wave function at a given point in space, determines a stationary state completely. This does not hold for the corresponding GPE, for which the values of the wave function and its derivative can be specified independently at (almost) every fixed point in space. As a result of the added flexibility, and unlike the Schrödinger equation, the GPE has bounded solutions for continuous ranges of the values of the chemical potential . However, the time-independent GPE Eq. (80.28) comes with the added normalization condition Eq. (80.26). Normalization quantizes the values of for the bound states. In practice one might, for instance, find a solution that satisfies the boundary conditions for a given with the shooting method, then adjust the value of until the normalization holds. Techniques used in the first numerical analyses of the time-independent GPE in the context of atom vapor condensates were variations of this theme. Such schemes are not feasible in spatial dimensions greater than one. In general, there is one solution to the time-independent GPE that can be chosen to be positive everywhere, the ground state with the lowest chemical potential. Excited steady states exist, but only a few, such as the flowing states of Eq. (80.33) and vortices discussed in Sect. 80.5.1, have obvious physical meanings. As the GPE is nonlinear, excited states are not the same as small excitations.
where U.x/ is a known function of position. In the algorithm the exponential is first split approximately, for instance, as „r 2 C U.x/ exp i t 2m
t „r 2 t „r 2 Ñ exp i exp i t U.x/ exp i 2 2m 2 2m TQ UQ TQ : (80.42) The exponential of the kinetic-energy operator is diagonal in the Fourier representation. Consequently, carrying out the Fourier transformation F and its inverse gives the split-step algorithm .t C t/ D F 1 TQ F UQ F 1 TQ F
.t/ ;
(80.43)
with an efficient implementation using the fast Fourier transformation. For the GPE, U.x/ includes the mean-field contribution that depends on the wave function itself, so the question arises as to which approximation of to use when one operates with UQ . The maybe surprising answer is that the very that emerges from the first step F 1 TQ F .t/ leads to the lowest order of error in t [9]. The split-step algorithm preserves the normalization of the macroscopic wave function, and features an accurate compact approximation of the exponential function of the kinetic-energy operator. Integration in imaginary time The split-step algorithm also provides a global method to find the ground state. To this end, the time-dependent GPE is integrated in imaginary time, i.e., replacing t ! it, starting
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from a more or less random initial wave function and normalizing after every step. For the Schrödinger equation, this process would increase the amplitude of the lowest-energy component in the wave function with respect to the other components exponentially in the imaginary time, and eventually leave only the ground state. It is not clear that the same should hold for the nonlinear GPE, but often it does. However, in nonlinear problems split-operator methods often exhibit spurious behavior. Successful use of these techniques requires experience and skill in the art of numerical methods.
J. Javanainen
A precise formal meaning of the macroscopic wave function, and of Bose–Einstein condensation for interacting systems, is found by considering the one-particle density matrix ˝ ˛
.x; x 0 / D O .x/ O .x 0 / ;
(80.47)
which is sufficient to determine the expectation value of any one-particle operator. This is the position representation of a positive operator with the trace equal to particle number (or technically, its expectation value) N . Therefore, an orthonormal set of eigenfunctions f k .x/gk and the corresponding nonnegative eigenvalues nk with
Local-Density Approximation X While an experimental BEC is usually trapped, it is often nk D N (80.48) much easier to study the theory for a formally infinite hok mogeneous condensate. As long as the phenomena under investigation involve length scales much smaller than the size exist such that the one-particle density matrix may be put in of the condensate, trapping cannot affect the behavior of the the form gas locally. Under such conditions, one may analyze the gas X at each position x as if it were homogeneous, and at the end nk k .x/ k .x 0 / : (80.49)
.x; x 0 / D of the calculations average over the density distribution. The k unit-normalized distribution of the density of the gas used in The system is a BEC if in the thermodynamic limit at least the averaging is one eigenvalue nk is on the order of the number of partiR cles (if the thermodynamic limit exists and is sensible). The 3 ı % n.x/ n.x/ d x R : (80.44) P .%/ D usual case is that only one state, call it k D 0, has such n.x/ d 3 x a large eigenvalue. The macroscopic wave function is essenp tially the corresponding eigenfunction, n0 0 , and n0 For instance, gives the number of condensate atoms. If there is more than p one macroscopic eigenvalue, the condensate is called frag15 n.0/ % % mented. H.%/H.n.0/ %/ (80.45) P .%/ D 4Œn.0/5=2 Another interpretation of the macroscopic wave function comes from statistical mechanics. In a continuous (secondholds for the Thomas–Fermi approximation with the maxi- order) phase transition typically a symmetry of the system is mum density n.0/. The Heaviside step functions H restrict spontaneously broken. For example, below the Curie temthe density to the correct range 0 % n.0/. As an example, perature, a single-domain ferromagnet magnetizes in some the average density in the Thomas–Fermi model is specific direction, and the state has a lower symmetry than the rotationally invariant Hamiltonian of an isotropic ferroZ 4 (80.46) magnet. Any quantity that appears in a continuous phase %P .%/ d% D n.0/ : 7 transition and characterizes the breaking of the symmetry may be called an order parameter. The macroscopic wave function can be viewed as the order parameter associated 80.3.5 Meaning of the Macroscopic Wave with spontaneous breaking of the global phase or gauge Function symmetry of quantum mechanics. Specifically, in quantum mechanics, the state of the system is unchanged if the wave Here, the macroscopic wave function has been introduced function is multiplied by an arbitrary complex phase factor by replacing a boson field theory with a classical field theory. e i' . However, to write the wave function as .x/ already The intuitive interpretation is that, for interacting par- implies a preferred phase, and likewise even if the wave functicles, the atoms condense not to the ground state of the tion is adorned with random but, for any given condensate, confining potential but to the one-body state whose wave fixed phase, as in ei' .x/. For one condensate the random phase is inconsequential. function is the macroscopic wave function. This notion may be criticized on various grounds, but in practice it makes Suppose, however, that two BECs with the wave functions ei'1 1 .x/ and ei'2 2 .x/ are combined. If the macroscopic a useful picture.
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wave functions behave as wave functions should, the combination of two condensates displays the density n.x/ D j ei'1 1 .x/ C ei'2 2 .x/j2 . There should be an interference pattern between the condensates. Two BECs indeed produce an interference pattern when they are combined. From the quantum optics viewpoint, a condensate is a given number of atoms in a given one-particle state, a number state, and cannot possess any phase at all. This seems to contradict the observations of an interference pattern. The resolution is that the process of measuring the phase difference in itself creates a phase difference, even if there initially was none [10].
The quantities !, N `, R, and n.0/ have the same meaning as in the BEC case. The Thomas–Fermi approximation for fermions should be applicable whenever N 1. In a one-component Fermi gas at low temperature, atom– atom interactions are typically negligible for a multitude of reasons. There is no s-wave scattering, and the presence of the Fermi sea tends to suppress repulsive interactions. However, in the case of attractive interactions between two species, the Fermi sea may be thermodynamically unstable; the energy may be lowered by pairing fermions into Cooper pairs. This is the mechanism behind the BCS theory of superconductivity. BCS theory is a mean-field theory and also allows one to describe a superconductor or a fermion superfluid with a wave function.
80.3.6 Fermions Excitations in a Fermi Gas If the interactions do not render a fermion system into a BCS superfluid, the elementary excitations of a degenerate weakly interacting Fermi gas with short-range interactions are basically atom–hole pairs. Both the static Fermi gas and its excitations have been major themes throughout the history of quantum physics, and we will not attempt a more detailed discussion.
Static Fermi Gas
Thomas–Fermi approximation Consider an ideal single-species Fermi gas of trapped atoms. The original Thomas–Fermi approximation (Chap. 21) was formulated for fermions, namely, electrons, and in the present case, it is modified as follows. For the atom density n.x/ at position x, at a low temperature, the corresponding local chemical potential is approximated according 80.4 to Eq. (80.18) as 2=3
„2 6 2 n.x/ : F .x/ D 2m
(80.50)
Experimental
80.4.1 Preparing a BEC
In a trapped gas, the density n.x/ is self-determined from Given the trapping potential V .x/, the density of the gas ad- the atom number N , and the condition for a BEC in an ideal justs in such a way that the sum of the external potential gas is most readily expressed in terms of the total number of energy and the local chemical potential, the Fermi energy, atoms as is a constant across the gas, N 1=3 : (80.54) kB Tc D 0:94 „!N 2=3
„2 6 2 n.x/ D; (80.51) V .x/ C In practice, at the bottom of the trap, the conditions on tem2m perature and density for a BEC are similar to the conditions the global chemical potential. One may solve the density for for a BEC in a free gas. In the thermodynamic limit, such that a given chemical potential as !N ! 0, N ! 1 with !N N 1=3 held constant, below the critical 8
3 temperature Tc , the fraction of condensate atoms behaves as < p2 m 32 V .x/ 2 a function of temperature T as ; V .x/ < I (80.52) n.x/ D 3 2 „3 : 3 0; otherwise : T n0 : (80.55) D1 N Tc Finally, the integral of the density over all space should equal the atom number, which gives an equation to determine the The experimental realizations of alkali vapor condensates are chemical potential . based on techniques of laser cooling and trapping of atoms. For a harmonic trap, this program can be carried out exThe following discussion overlaps with, and is supplemented actly, analytically, with the result that by, the material in Chap. 79. 1=3 1=3 2=3 1=3 1=6 A BEC in a dilute atomic gas is typically prepared using D 6 N „!N ; R D 2 3 N ` ; p a two-stage process. First, a magneto-optical trap is used to 2 N : (80.53) capture a sample of cold atoms and to cool it to a temperan.0/ D p 3 2 `3 ture of the order of a few tens of microkelvin. The atoms are
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then transferred to a magnetic or optical trap for evaporative cooling that leads to condensation. A magnetic trap is based on a combination of two ideas. First, if an atom that starts out with its magnetic moment antiparallel to the magnetic field moves slowly enough in a position-dependent magnetic field, its magnetic moment remains adiabatically locked antiparallel to the magnetic field. The energy of the atom is then a minimum where the magnetic field is a minimum. Second, the absolute value of the magnetic field may have a minimum in free space. The minimum is then a trap for atoms whose magnetic moments are suitably oriented. The downside is that only atoms in the right magnetic (Zeeman) states are trapped. While the atoms cool down, they accumulate at the center of the trap. The center should not be a zero of the magnetic field, because at zero field an atom would lose the lock between the directions of the magnetic moment and the magnetic field. In a time-orbiting potential (TOP) trap, a time-dependent magnetic field is added in such a way that the zero of the magnetic field orbits around the center of the trap. If the frequency at which the zero moves is high enough, the atoms see a time-averaged effective potential with a minimum at the center of the trap, and do not dwell around the zero of the magnetic field. Alternatively, it is possible to wind a coil in such a way that it makes a magnetic field whose absolute value has a minimum that is not zero. In this kind of Ioffe–Pritchard trap, the winding of the wire resembles the seams on a US baseball. It is also possible to condense atoms trapped in a far-off resonant optical trap based on the dipole forces of light, instead of the magnetic trap. For tuning below the resonance, the atoms are strong-field seekers. An arrangement with two crossed laser beams focused to the same spot is typical. With extreme off-resonant light from, say, a CO2 or a Nd:YAG laser, absorption of photons, and the associated photon recoil kicks and heating, may be negligible. The frequencies of the two beams may be made slightly different by using, for instance, an acousto-optic modulator, whereby a potentially harmful standing-wave pattern averages out. The advantage of a far-off resonant optical trap is that it is not particularly sensitive to the Zeeman state, so that it can hold atoms in many states. The idea of evaporative cooling is that the most energetic atoms escape from the trap, and then the remaining atoms thermalize to a lower temperature. Some atoms are lost in the process, but with the decreasing temperature, the density at the trap center nonetheless tends to increase, and the phase space density increases even more due to the cooling. Evaporative cooling is forced by actually or effectively lowering the depth of the trap. For instance, a radio frequency field is applied to the atoms. The transition frequency between the Zeeman states depends on the magnetic field and increases toward the edges of the trap. Atoms are re-
J. Javanainen
moved where the radio frequency (RF) is on resonance and drives transitions to untrapped Zeeman states. Thus, while the atoms cool and concentrate at the center, the radio frequency is swept down in such a way that the “RF knife” removing the atoms slides in from the edge of the trap. At some point along the evaporation, a condensate abruptly emerges. The temperature can be further lowered by continuing evaporative cooling, albeit at the expense of loss of atoms. A typical time needed to prepare a condensate is on the order of a second, and a condensate may live for tens of seconds in a good vacuum.
80.4.2 Preparing a Degenerate Fermi Gas A single-species, very low-temperature Fermi gas is usually an uninteresting system, as the Fermi–Dirac statistics forbids s-wave interactions between the atoms, and the gas is nearly ideal. In the experiments the gas usually has two or more species, different states of the same atom. The interactions between the species are then comparable in strength to the interactions between bosonic atoms. Evaporative cooling works in a two-species gas and can be used to prepare a degenerate Fermi gas either in an optical or a magnetic trap. One can also use a gas of bosons, and, indeed, a BEC, as a refrigerator. The lowest attainable temperatures with direct cooling are on the order of 0:1 TF . In the quest toward lower temperatures, methods have been devised to reduce the entropy of the atoms rather than the energy.
80.4.3 Monitoring Degenerate Gases Orders of Magnitude As a rule of thumb, the trapping frequencies in a magnetic trap are !N 2 10 Hz, while the frequencies in an optical trap may reach into the kHz regime. A typical oscillator length is ` 1 m. A usual number of atoms is N 107 . Scattering lengths are of the order of a 10 nm. The size of a degenerate gas is in the neighborhood of R 0:1 mm, the maximum density is about n.0/ 1015 cm3 , and the BEC transition temperature and the Fermi temperature are of the order of Tc TF 1 K. However, much lower temperatures are readily reached in a BEC. Time-of-Flight Imaging Usually the observation of a degenerate gas in an experiment is by time-of-flight imaging. The trap is suddenly removed, whereupon the gas expands freely. After the atom cloud has grown to a size large enough compared to the wavelength of the resonant light used to monitor the gas, an absorption image of the cloud is taken. This gives the projection of the density of the gas onto a plane perpendicular to the direction
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Vortices As an example, in a fluid rotating as a whole at the angular velocity *, the line integral around a loop at the distance r from the axis of rotation would be 2 r 2 *, which is not permitted for an arbitrary r. Instead, upon an attempt to make a BEC rotate, the angular velocity will be carried by vortex lines. These are lines through the condensate, entering and exiting at the surface, such that each vortex carries a quantum of circulation. At the core of a vortex, the flow velocity should be infinite to sustain a finite circulation, which is physically impossible. Nature solves this problem by making the vortex core normal (not BEC), so that the 80.5 BEC Superfluid macroscopic wave function does not apply. The diameter of the vortex core is of the order of the healing length , given 80.5.1 Vortices by Eq. (80.29). When the trapping potential on the atoms is rotated, it is convenient to study the physics in the corotating frame. Flow Velocity in a Superfluid By manipulating the Heisenberg equations of motion for Given a frame rotating at the angular velocity and the a Bose field under the Hamiltonian Eq. (80.22) it is easy to angular momentum operator L D x p per particle, transderive the equation of continuity for the atoms, formation to the rotating frame adds the one-particle term of propagation of the imaging light. Except for the effects of atom–atom interactions, after a sufficiently long time of free flight the density reflects the initial momentum distribution of the atoms. Time-of-flight images bear the signs of both condensation in a Bose gas and quantum degeneracy in a Fermi gas. Nontrivially, other features of interest such as vortex cores are also preserved and can be detected after the free expansion. The downside is that the time-of-flight method is destructive. After each snapshot, the sample will have to be prepared again.
@ nO C r jO D 0 ; @t
(80.56)
Hr D L
(80.61)
„ O O O O jO D i : r r 2m
(80.57) to the Hamiltonian. Any particular configuration of vortices is a thermodynamically stable equilibrium if it is the minimum of energy in the corotating frame. For a trapped This identifies nO and jO as the operators for atom density and condensate, at zero rotation velocity the state without voratom current density. The corresponding mean-field quanti- tices is the energy minimum, and increasing the rotation ties are again obtained when the boson fields are replaced speed makes states with an increasing number of vortices the with the corresponding classical fields. Writing the classical stable configuration. However, a vortex configuration may field in terms of the density n.x; t/ and phase '.x; t/ in the be metastable and live for a long time even if it is not the form minimum of energy. Conversely, even the energy-minimum configuration of vortices must first be nucleated. Since the p i' (80.58) circulation can only have quantized values, it cannot change D ne ; in a continuous process. It takes a zero condensate density the local flow velocity v D j =n becomes somewhere to create or destroy a vortex. These alternatives provide a large number of experimental „ (80.59) scenarios involving rotation of the trap or stirring of the conv D r' : m densate, condensation of a rotating normal gas by taking it across the transition temperature, and so forth. For instance, The velocity field is irrotational. when a trap containing a BEC is rotated, vortices are generated at the surface where they start their lives as dynamical Quantization of Circulation Integration of the flow velocity around an arbitrary loop gives instabilities. The vortices then drift in and form a regular vortex array. When the rotation is halted, the vortices drift out to I 2 „ „ the surface and disappear. ; dl v D ' D p m m p D 0; ˙1; ˙2; : : : ; (80.60) nO D O O ;
80.5.2 Superfluidity since the change of the phase ' around a closed loop must be an integer multiple of 2 . Equation (80.60) expresses the quantization of circulation in a superfluid. A medium described by a macroscopic wave function, such as a BEC, cannot sustain arbitrary flow velocities.
A BEC also has the remarkable property that it may sustain persistent currents that are completely immune to viscosity. The qualitative reason may be seen from the dispersion relation of small excitations Eq. (80.38). As long as the flow
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speed jvj is less than the speed of sound c, all excitation energies are positive, so that the flowing state is the state of lowest energy and is thermodynamically stable. On the other hand, when the flow velocity exceeds the speed of sound, the system has excitations that lower the energy, .q/ < 0 for some q. The flowing state is then not a minimum of energy. The flow is not thermodynamically stable, and it decays when it interacts with an environment by sending off small excitations. The speed of sound gives the Landau critical velocity for superfluidity. The critical velocity c tends to zero when the atom–atom interactions vanish with a ! 0. While the condensate wave function may be written down whether or not the atoms interact, superfluidity and persistent flows rely on the interactions. The same applies to vortices, as in the limit of a noninteracting gas the healing length and the radius of the vortex core tend to infinity. The conventional picture is that superfluid flow in an inhomogeneous medium is thermodynamically unstable if the local flow velocity somewhere exceeds the local (densitydependent) speed of sound. In practice, in liquid He experiments and numerical simulations of dilute condensates, the supercurrent often dissipates by shedding vortices when it flows past an obstacle. Persistent flow in a toroidal geometry would be an ideal test case for superfluidity, and such experiments have, in fact, been carried out with dilute atomic condensates [11]. Nonetheless, no clear picture of the quantitative limits to the flow velocity of a superfluid has emerged from these experiments.
J. Javanainen
adiabatically in a cold gas and captures atoms without undue heating of the CM motion, but with a random number of atoms at random lattice sites. Nevertheless, loading of the desired initial configuration atom by atom has also been demonstrated [13]. Such developments have lent credence to the idea of using an optical lattice system as a quantum simulator [5, 6, 14]—basically as an analog quantum computer.
80.6.1 Basics of the Optical Lattice Dipole forces of standing-wave fields of light generate a periodic potential, an optical lattice, on the atoms [15]. If the light is detuned far enough from atomic resonances, absorption and spontaneous emission are negligible, and the potential is conservative. In one dimension, the prototypical potential energy for the motion of the atoms is of the form V .x/ D V0 sin2 kx ;
(80.62)
where k is the wave number of the lattice light, and the depth of the lattice V0 can be inferred from the known parameters of the atoms and the light as explained in Chap. 79.2.2. With multiple laser beams propagating in different directions and having different polarizations, a wide variety of 2-D and 3-D optical lattices may be set up analogously. As is usual with periodic potentials, the one-particle energy eigenstates are characterized by a quasimomentum or lattice momentum q, and group into bands typically separated by bandgaps. In an experimental technique called band mapping, the lattice potential is adiabatically removed, whereupon such Bloch states get converted into eigenstates 80.6 Optical Lattice as Quantum Simulator of ordinary momentum. Upon further free evolution, moLattice models have been, and are, widely studied in menta get mapped into observable positions of the atoms. condensed-matter physics. Even though they may look de- In this way, the distribution of the atoms among the quasiceptively simple, quantum systems with interactions between momenta and bands may be determined. the particles usually cannot be solved exactly even with massive numerical computations. Certain static properties Bose–Hubbard Model of bosonic systems are accessible to quantum Monte Carlo Consider as an example a 1-D optical lattice that holds spin-0 methods even in fairly large models, but less so for fermions atoms, with the sites labeled by an integer n. The simplified because of the notorious “sign problem” [12]; and exact nu- Hamiltonian, the Bose–Hubbard model, reads merical solutions of time dependence in, say, boson lattice i HO 1 Xh models are restricted to the scale of about ten sites with ten J anC1 an C an1 an C Uan an an an : D „ 2 n particles. The thermodynamic limit of a strongly interacting system is typically addressed with in-principle uncontrolled approximations, such as a judicious mean-field theory. However, progress in AMO physics has given us materially flawless optical lattices to hold the atoms, lattice parameters may be controlled by varying the parameters of the lattice lasers, and even the strength of the atom-atom interactions may be tuned using Feshbach resonances. In addition, it is possible to monitor the atoms with single-site resolution. In most experiments, the optical lattice is raised
(80.63)
The operator an annihilates and the operator an creates an atom at the site n, so the terms / J describe tunneling of the bosons from a site to a neighboring site. The term / U models the interactions of the atoms that are in the same site. Boundary conditions ordinarily matter little for a long lattice; for convenience, we assume periodic boundary conditions for a lattice with L sites, so that the sites n and n C L are
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regarded as the same. This case could be physically realized by making a 1-D ring lattice. The key consideration in the physics of the Bose–Hubbard model is the competition between tunneling / J and atom– atom interactions / U .
Tuning of the Parameters Without the interactions, the model Eq. (80.63) corresponds to the tight-binding approximation to the band structure of a 1-D solid, with the energies in the single Bloch band ranging from „J to „J . The wave functions of the states annihilated by an are the corresponding Wannier states n .x/. We use vector notation for the position since in reality the lattice is confined in the transverse directions, making a wave function in 3-D, and the atom–atom interactions depend on the volume density of the atoms anyway. For the standard contact interaction Eq. (80.21) characterized by the scattering length a, we have 4 „a U D m
Z d 3 x jn .x/j4 :
(80.64)
On the other hand, the exact band structure for the potential energy Eq. (80.62) of an optical lattice comes from the Mathieu equation. A comparison of the width of the first band with the tight-binding approximation gives the estimate 8 Er J Dp „
V0 Er
s
3=4 exp 2
V0 Er
! ;
(80.65)
where Er is the photon recoil energy, and we assume that V0 ' Er . Since the depth of the optical lattice V0 is proportional to the intensity of the lattice light and the tunneling amplitude J depends strongly on V0 , J may be varied over a wide range by changing the intensity of the lattice light. Near a Feshbach resonance at the magnetic field B0 , the scattering length varies with the magnetic field applied to the atoms, B, as a D a0 1
B B B0
;
(80.66)
where a0 , B0 , and B are constants depending on the atomic species and the Feshbach resonance in question. Under favorable conditions, it may be possible to vary the scattering length over a wide range, and even to change its sign, by varying the magnetic field applied to the atoms.
Discrete Nonlinear Schrödinger Equation The mean-field counterpart of the Bose–Hubbard model is found in the usual way, by positing that in the Heisenberg equations of motion the operators an are no longer boson operators but instead complex numbers ˛n . Furthermore, we
scale these numbers so that the average of j˛n j2 over the lattice equals 1, which brings in the average site occupation nN D N=L for a total of N bosons in a lattice of length L. The result, an analog of the GPE, d N j˛n j2 ˛n ; ˛n D 12 J.˛nC1 C ˛n1 / C nU dt
(80.67)
is often called the discrete nonlinear Schrödinger equation (DNLSE). DNLSE has interesting properties of its own. For instance, for attractive (a < 0) atom–atom interactions, it may have solutions called gap solitons, clusters of atoms that propagate along the lattice (or stand still) while retaining their shape. They are “gap solitons” because formally their energies are outside of the energies of the Bloch bands, i.e., in the bandgaps.
80.6.2 Strongly Correlated Systems Superfluid–Mott Insulator Transition In generic terms, a many-body system that cannot be adequately described by a mean-field theory is called strongly correlated. In particular, the mean-field version of the Bose– Hubbard model Eq. (80.67) misses an essential piece of physics that emerges when the interactions become sufficiently strong. When tunneling dominates, the system is in what is referred to as the superfluid phase. Fluctuations of the atom number between the sites are relatively large. On the other hand, if atom–atom interactions dominate, fluctuations in atom number become costly in energy; a configuration with zero atoms in one site and two in the next site has a much higher energy than a configuration with one atom in both sites. The same number of atoms in each site corresponds to the Mott insulator phase. According to calculations carried out using the so-called Gutzwiller ansatz, the ground state of an optical lattice inserted in an atom trap then consists of regions with the same integer number of atoms at the lattice sites within each region [16]. When the parameters of the system are varied, in what is known as a quantum phase transition [17] the system should abruptly switch between the phases. This superfluid-Mott insulator transition has been observed experimentally [18]. The observation of the transition is by means of phase coherence. In the superfluid phase, the system is characterized by a global macroscopic wave function. When the atoms are abruptly released from the lattice, atoms originating from different sites are capable of interference, and the interference pattern reflects the lattice structure. On the other hand, in the Mott insulator phase, the lattice sites are in number states with little phase coherence between them, and there is no interference pattern.
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Quantum Gas Microscope The superfluid–Mott insulator transition has established optical lattices as viable simulators of lattice models and has inspired much current following [5, 6, 14]. As an example of the ongoing technical advances, we describe cursorily the quantum gas microscope [19]. This is literally a microscope, one with an objective lens that has a very large (effectively 0.8) numerical aperture. This entails resolution close to a wavelength. The microscope views a 2-D lattice of the atoms. However, in this particular experiment, the lattice is not produced by added standing waves but by imaging a judiciously designed mask backwards through the objective to the plane of the atoms. The lattice is turned on slowly and collects cold atoms from a suitable trap. The physics experiment then commences. After the experiment, the lattice is switched much deeper. It then confines the atoms to its individual potential wells, and facilitates a measurement of the number of the atoms at each lattice site; or at least the parity of the atom numbers, since the trapped atoms experience pairwise laser-assisted exothermic collisions and are lost from the trap until only one or zero atoms per site remains. Laser beams forming an optical molasses illuminate the atoms, and the atoms fluoresce. The deep confinement and a tuning of the laser conducive to cooling eliminate heating from photon recoil kicks, so that the atoms stay in place and send out a number of fluorescence photons. This improves the signal-to-noise ratio to the extent that an individual atom can be detected at the resolution of a single lattice site. Not only atom density, but also position correlations of the atoms can be recorded for comparison with theory. By making use of resonance spectroscopy, the spins of the atoms and the correlations of the spins between the sites may also be determined. Phase Diagrams in Lattice Systems There is, for all practical purposes, an infinity of lattice systems to study: different lattice structures; bosons versus fermions, various spins, real, or simulated with internal states of the atoms; short-range on-site interactions versus long-range coupling of the spins via magnetic or (for polar molecules) electric dipole interactions; and so on. A number of implementations relying on AMO techniques have been considered and are being considered. In the tradition of condensed-matter physics, the goal is to find the phases and phase diagrams; either to compare with existing theoretical arguments, or to discover new phases experimentally. We will say nothing further about this kind of work.
J. Javanainen
periments, in which a steady-state system is suddenly altered and then let evolve, have the potential to deliver reliable results not available in any other way. The broad experience in physics is that, when left alone, a large system thermalizes and can then be described by statistical mechanics, at least to the extent permitted by the conserved quantities. The issue is, does this always happen, and how; especially in isolated systems when it is not possible to appeal to the interactions with the environment as the cause of the thermalization. Time evolution in wellcontrolled lattice systems [20] is an enterprise with potential for breakthroughs in our understanding of the foundations of statistical mechanics.
80.6.3 Topological Phases of Matter As another current line of development, also a spin-off from condensed-matter physics, we discuss topological band structures and topological phases of matter [14, 21].
Phase of Tunneling in a Lattice Consider as a preparatory example a static magnetic field acting on a charged free particle. In the minimum coupling substitution, the Hamiltonian reads H D
1 ŒpO qA.x/2 : 2m
(80.68)
If there actually is no magnetic field so that the vector potential satisfies B D r A D 0, the vector potential may be “gauged away” with the canonical transformation 3 2 Zx iq dl A 5 .x/ ; (80.69) .x/ ! Q .x/ D exp4 „ x0
where x 0 is an arbitrary reference point. With r A D 0, the integral is independent of the path. In particular, the integral vanishes over any closed path. In contrast, if a magnetic field is present, the integral depends on the path, and the canonical transformation as written in Eq. (80.69) is ill defined. The integral becomes a phase factor eiˆ , where the phase ˆ.P; x 0 ; x/ depends on the path P . If a wave packet is split at x 0 and the components are sent along two different paths to recombine at x, they will have a phase difference modifying the interference Z q d B (80.70) ˆ D „
Time Dependence S As was mentioned already, time dependence in large, strongly interacting many-body quantum systems usually proportional to the magnetic flux through an arbitrary suroverwhelms the theoretical tools that we have available face S with the paths of the two wave packets as its edge. today, including exact numerical computations. Quench ex- This is how the Aharonov–Bohm effect comes about.
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By making use of the interplay between internal and CM degrees of freedom of an atom, it is possible to realize a Hamiltonian of the form Eq. (80.68) also for a neutral atom whose CM motion does not directly couple to electromagnetic fields, and even with r A ¤ 0. This is the idea of synthetic gauge fields [22]. Suppose now that in a 1-D lattice, there is a change of phase ' of the wave function when an atom tunnels from site n to the site n C 1. The counterpart of the Bose–Hubbard model would be i 1 Xh i' HO J e anC1 an C ei' an1 an C Uan an an an : D „ 2 n
a four-site plaquette. This means that the translation that leaves the lattice unchanged is not a, but 2a. Since the unit cell has become two times as big, the Brillouin zone is half the size, and 1/4 in area. The result is that the original lowest band splits into (in principle) four energy bands. Moreover, as a result of the accumulated phases, the topology of these bands may be nontrivial. The topology is characterized by an invariant called the Chern number. Suppose one first solves for the one-body Bloch states in an energy band, u.qx ; qy /, and forms the Berry curvature * D i.h@qx uj@qy ui h@qy uj@qx ui/ ;
(80.72)
then the Chern number is computed as an integral over the first Brillouin zone, Z A priori such a phase change is inconsequential, as one can 1 CD (80.73) * d2 q : remove it by redefining the inconsequential phases of the 2 states corresponding to an atom at different sites n. However, if it is possible to have a closed loop among the lattice sites The value would be zero for a topologically trivial band, and the phase changes do not add up to an integer multiple while in the experiments actually three bands are found (one of 2 along the way, the phases ' cannot be gauged away in with twice as many states as the other two) with the Chern this way. This could happen in a 1-D ring lattice. Among the numbers 1, 2, and 1. consequences is a nontrivial ground state in which the atoms It is not possible to modify the Hamiltonian continuously flow around the ring. in such a way that the Chern number suddenly changes Here, the emphasis is on 2-D lattice systems. It could cor- between integer values unless “something drastic” happens respondingly happen, for instance, that in a square lattice the along the way. In this case, the drastic event is that the gap bephase changes in a loop around a plaquette of the four atoms tween bands closes. In the experiments, there is an auxiliary at the corners of a unit cell add up to something other than 0 control variable to take the system across such a topological or another integer multiple of 2 . phase transition. A phase change in tunneling can be engineered by havOne way of detecting the Chern number is to apply a force ing external light drive a transition between the sites, instead that moves the atoms. As one might expect from the analogy of the ordinary tunneling. This is because the phase of the between the engineered phases and a magnetic field, in an light gets imprinted on the transition. In practice, compli- analog of the Hall effect, the atoms get translated also in the cated arrangements are needed [23, 24] to effect a nontrivial direction perpendicular to the applied force. This translation accumulation of phase around a closed loop, for one thing may be measured and is proportional to the Chern number because the phase changes of a laser field itself add up to (under the particular experimental conditions), which gives zero around a closed loop; but it has been achieved experi- a measurement of the latter. mentally. Modulation of the lattice potential, e.g., by shaking the lattice, is another approach to the same effect [25, 26]. We Significance of Topology do not go into the details of either type of idea, but stipulate The circulation of a superfluid around the core of a vortex that an effective magnetic field with a nontrivial phase accu- is a topological property, characterized by discrete values. mulation can be achieved. The effective magnetic field can A vortex is, therefore, resilient; it vanishes if the density be enormous compared to the real magnetic fields that can of the superfluid vanishes, but small perturbations will not change its circulation. The same applies to the Chern numbe reached in a laboratory. ber, which will not change unless the perturbations of the Topological Band Structure lattice are so drastic that a bandgap closes. The integer quanAs an example, we briefly consider a recent experiment [24]. tum Hall effect with a precisely determined quantized value The starting point here is (conceptually) a square lattice with of the Hall conductance occurs because the Hall conductance a lattice constant a in the xy plane. The reciprocal lattice is equal to the product of a Chern number and a combination is also square, with the lattice constant 2 =a, and so is the of natural constants [21]. first Brillouin zone that contains the values of the lattice moIn practice, there is no such thing as an infinite lattice. menta q for the lowest energy band. By using optical fields, A finite 2-D lattice with a topologically nontrivial band strucan accumulated phase ˆ D =2 is put in each loop around ture is believed to have chiral edge modes. They are localized (80.71)
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at the edges of the lattice, are chiral in that the atoms propagate in one direction around the lattice only (in analogy with cyclotron motion), and are topologically protected, i.e., their character does not change in small perturbations of the lattice or of the edge. From the AMO perspective, the long-term goal is to use topologically protected variables, hopefully insensitive to decoherence, for quantum computations.
References 1. Dalfovo, F., Giorgini, S., Pitaevskii, L.P., Stringari, S.: Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999). https://doi.org/10.1103/RevModPhys.71.463 2. Fetter, A.L.: Bose–Einstein condensates in dilute trapped atomic gases. J. Low Temp. Phys. 129, 263–321 (2002). https://doi.org/ 10.1023/A:1021408428662 3. Bloch, I., Dalibard, J., Zwerger, W.: Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008). https://doi. org/10.1103/RevModPhys.80.885 4. Pitaevskii, L., Stringari, S.: Bose-Einstein Condensation and Superfluidity. Intenational Series of Monographs in Physics. Oxford University Press, Oxford (2016) 5. Lewenstein, M., Sanpera, A., Ahufinger, V.: Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems. Oxford University Press, Oxford (2012) 6. Gross, C., Bloch, I.: Quantum simulations with ultracold atoms in optical lattices. Science 357, 995–1001 (2017). https://doi.org/10. 1126/science.aal3837 7. Fetter, A.L., Walecka, J.D.: Quantum Theory of Many-Particle Systems. Dover, Mineola (2003) 8. Feit, M.D., Fleck Jr., J.A., Steiger, A.: J. Comp. Phys. 47, 412 (1982) 9. Javanainen, J., Ruostekoski, J.: Symbolic calculation in development of algorithms: Split-step methods for the Gross–Pitaevskii equation. J. Phys. A 39, L179 (2006) 10. Javanainen, J., Yoo, S.M.: Quantum phase of a Bose–Einstein condensate with an arbitrary number of atoms. Phys. Rev. Lett. 76, 161–164 (1996). https://doi.org/10.1103/PhysRevLett.76.161 11. Kumar, A., Eckel, S., Jendrzejewski, F., Campbell, G.K.: Temperature-induced decay of persistent currents in a superfluid ultracold gas. Phys. Rev. A 95, 021602 (2017). https://doi.org/10. 1103/PhysRevA.95.021602 12. Wikipedia contributors: Numerical sign problem – Wikipedia (2019) 13. Barredo, D., de Léséleuc, S., Lienhard, V., Lahaye, T., Browaeys, A.: An atom-by-atom assembler of defect-free arbitrary twodimensional atomic arrays. Science 354, 1021–1023 (2016). https://doi.org/10.1126/science.aah3778 14. Goldman, N., Budich, J.C., Zoller, P.: Topological quantum matter with ultracold gases in optical lattices. Nat. Phys. 12, 639 (2016)
J. Javanainen 15. Morsch, O., Oberthaler, M.: Dynamics of Bose–Einstein condensates in optical lattices. Rev. Mod. Phys. 78, 179–215 (2006). https://doi.org/10.1103/RevModPhys.78.179 16. Jaksch, D., Bruder, C., Cirac, J.I., Gardiner, C.W., Zoller, P.: Cold bosonic atoms in optical lattices. Phys. Rev. Lett. 81, 3108–3111 (1998). https://doi.org/10.1103/PhysRevLett.81.3108 17. Sachdev, S.: Quantum Phase Transitions, 2nd edn. Cambride University Press, New York (2011) 18. Greiner, M., Mandel, O., Esslinger, T., Hänsch, T.W., Bloch, I.: Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39 (2002) 19. Bakr, W.S., Gillen, J.I., Peng, A., Fölling, S., Greiner, M.: A quantum gas microscope for detecting single atoms in a Hubbardregime optical lattice. Nature 462, 74 (2009) 20. Trotzky, S., Chen, Y.-A., Flesch, A., McCulloch, I.P., Schollwöck, U., Eisert, J., Bloch, I.: Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas. Nat. Phys. 8, 325–330 (2012). https://doi.org/10.1038/nphys2232 21. Xiao, D., Chang, M.-C., Niu, Q.: Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010). https://doi.org/ 10.1103/RevModPhys.82.1959 22. Dalibard, J., Gerbier, F., Juzeli¯unas, G., Öhberg, P.: Colloquium: Artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523–1543 (2011). https://doi.org/10.1103/RevModPhys.83.1523 23. Jaksch, D., Zoller, P.: Creation of effective magnetic fields in optical lattices: The Hofstadter butterfly for cold neutral atoms. New J. Phys. 5, 56–56 (2003). https://doi.org/10.1088/1367-2630/5/1/ 356 24. Aidelsburger, M., Lohse, M., Schweizer, C., Atala, M., Barreiro, J.T., Nascimbène, S., Cooper, N.R., Bloch, I., Goldman, N.: Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162 (2014) 25. Struck, J., Ölschläger, C., Weinberg, M., Hauke, P., Simonet, J., Eckardt, A., Lewenstein, M., Sengstock, K., Windpassinger, P.: Tunable gauge potential for neutral and spinless particles in driven optical lattices. Phys. Rev. Lett. 108, 225304 (2012). https://doi. org/10.1103/PhysRevLett.108.225304 26. Goldman, N., Dalibard, J.: Periodically driven quantum systems: Effective Hamiltonians and engineered gauge fields. Phys. Rev. X 4, 031027 (2014). https://doi.org/10.1103/PhysRevX.4.031027
Juha Javanainen Juha Javanainen is Professor of Physics at the University of Connecticut. He has worked on a number of topics in theoretical quantum optics.
81
De Broglie Optics Carsten Henkel and Martin Wilkens
Contents
Abstract
81.1
Wave-Particle Duality . . . . . . . . . . . . . . . . . . . 1173
81.2 81.2.1 81.2.2 81.2.3 81.2.4
The Hamiltonian of de Broglie Optics Gravitation and Rotation . . . . . . . . . . Charged Particles . . . . . . . . . . . . . . Magnetic Moments . . . . . . . . . . . . . Atoms . . . . . . . . . . . . . . . . . . . . .
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81.3 81.3.1 81.3.2 81.3.3 81.3.4 81.3.5
Evolution of De Broglie Waves Light Optics Analogy . . . . . . WKB Approximation . . . . . . Phase and Group Velocity . . . . Paraxial Approximation . . . . . Raman–Nath Approximation . .
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81.4 81.4.1 81.4.2 81.4.3 81.4.4
Refraction and Reflection Atomic Mirrors . . . . . . . Atomic Cavities . . . . . . . Atomic Lenses . . . . . . . . Atomic Waveguides . . . .
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81.5 81.5.1 81.5.2 81.5.3 81.5.4
Diffraction . . . . . . . . . . . . . . . . Fraunhofer Diffraction . . . . . . . . . Fresnel Diffraction . . . . . . . . . . . Near-Resonant Kapitza–Dirac Effect Atom Beam Splitters . . . . . . . . . .
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81.6 81.6.1 81.6.2 81.6.3
Interference . . . . . . . . . . . . . . . . . . . . . . . . . . Interference Phase Shift . . . . . . . . . . . . . . . . . . . Internal-State Interferometry . . . . . . . . . . . . . . . . Manipulation of Cavity Fields by Atom Interferometry
1182 1182 1184 1184
81.7 81.7.1 81.7.2
Coherence of Scalar Matter Waves . . . . . . . . . . . 1184 Atomic Sources . . . . . . . . . . . . . . . . . . . . . . . . 1185 Atom Decoherence . . . . . . . . . . . . . . . . . . . . . . 1185
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C. Henkel () M. Wilkens () Institute of Physics and Astronomy, University of Potsdam Potsdam, Germany e-mail: [email protected] M. Wilkens e-mail: [email protected]
De Broglie optics concerns the propagation of quantum mechanical particle waves, their reflection, refraction, diffraction, and interference. The basic principles of De Broglie optics, which are quite similar to the principles of ordinary light optics, do not depend much on the specific nature of the sort of particles under consideration – electron, neutron, atom, ion, or molecule, but the focus in this chapter is on the De Broglie optics of atoms and molecules. This sort of particle comes with a variety of internal degrees of freedom, which are easily addressed and manipulated with electromagnetic fields, in particular laser fields, and thereby allow for quite a broad spectrum of different applications in lithography, imaging, and precision measurement. Keywords
atomic beam matter wave Fresnel zone plate Sagnac effect atom optics
81.1 Wave-Particle Duality The wave-particle duality of quantum mechanics states that freely traveling material particles of mass M and momentum p are properly described by a plane wave of wavelength dB D 2 „=p, named in honor of Prince Victor Louis De Broglie, and frequency ! D „k 2 =.2M /, with k the wavenumber k D 2 =dB . Originally meant to account for the orbitals of a Coulombbound electron, the ultimate wave character of particles has meanwhile been confirmed for all fundamental particles, but also for the center-of-mass (CM) motion of composite particles like ions, atoms, and even large molecules. These waves have been dubbed matter waves, although they do not encode
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_81
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a density variation or the like, but rather a probability amplitude for the center-of-mass degree of freedom of the particle under consideration. The optics of matter waves comprises, among others, electron optics and electron microscopy [1], electron holography [2], neutron optics and neutron imaging [3], in particular neutron interferometry [4], and atom optics [5]. Being strongly driven by the progress of laser cooling – see Chap. 79 on cooling and trapping – but also nanofabrication, the subject of atom optics is relatively new, and applications are currently being developed in atom interferometry [6], atom lithography, [7] and atom imaging [8]. These days, the wave character of particles has been confirmed for molecules with more than 430 atoms [9], and atom optics is promoted as a competitive method for the determination of the gravitational redshift [10], a particle’s proper time [11], Newton gravitational constant [12], gravimetry [13], inertial sensing [14], and even the exploration of possible manifestations of dark energy [15]. Interference experiments with atoms and molecules have been suggested to explore the boundaries between quantum and classical mechanics [16, 17]. In this domain, processes called decoherence or localization are expected to change the behavior of particles; quantal features like superpositions and particlewave duality are fading out, and the particles start to behave as expected from classical kinetic theory [18]. In terms of the Bohr radius a0 , the fine-structure constant ˛, and the electron mass me , Eh me 1=2 2 ˛a0
D 2 dB D
a0 ; 2T M v=c M=me
(81.1)
where v D p=M is the velocity, T D .1=2/M v 2 is the kinetic energy, and Eh is the atomic unit of energy (Table 1.4 of Sect. 1.2). For electrons, dB 1:226426 nm=.T =eV/1=2 . For thermal samples, the mean kinetic energy is of the porder of kB T , and, thus, the mean momentum scales / T , and Eq. (81.1) is replaced by the thermal De Broglie wavelength, which is defined in Eq. (81.76). In terms of the atomic mass unit u: th D 1:74.5/ nm .M=u/1=2 .T =K/1=2 . At room temperature, the thermal De Broglie wavelength of atomic samples is sub-Å, but for temperatures on the micro or nano-Kelvin scale, even in dilute samples the wavelength may well exceed the mean interparticle spacing, in which case, the quantum statistical effects of indistinguishability become manifest – see Chap. 80 on quantum degenerate gases. However, even in the nondegenerate regime, the physics of de Broglie waves differs from that of light waves in a number of regards. First, the dispersion relation of free particles is quadratic in the wavenumber, giving rise to spatial spreading of wave packets even in one-dimensional configurations. Second, particles may be brought to rest, which is impos-
sible for photons. Finally, particles in beams are subject to pair interactions, giving rise to nonlinear optical phenomena. Electrons and ions, for example, experience a particle density-dependent Coulomb broadening in the focus of a lens (Boersch effect [2]). Similarly, atoms or molecules interact via short range interparticle van der Waals forces, giving rise to density-dependent nonlinear effects, see Sect. 81.2.4 “Atom Optical Nonlinearity”, below.
81.2 The Hamiltonian of de Broglie Optics A typical de Broglie optical experiment involves a source, a beam of particles produced by that source, an array of optical elements, possibly a probe, other optical elements placed behind the probe, and, finally, a detector. Optical elements are collimators, apertures, lenses, mirrors, and beam splitters. A probe could be a crystal, a biological sample, or just another unknown optical element whose properties are to be investigated. To avoid proliferation of notation, we refer to any object placed in the beam path simply as an optical element. Here, we shall assume that the particles are far from quantum degeneracy, and the effects of particle interactions can safely be ignored. A large class of optical elements is then well described by a Hamiltonian model in which the ensemble of particles is described by a wave function .x; t/ D hxj .t/i whose time evolution is governed by the Schrödinger equation i„
@ j .t/i D H.t/j .t/i ; @t
(81.2)
with a one-particle Hamiltonian of the generic form H.t/ D
ŒpO A.x; O t/2 O t/ : C U.x; 2M
(81.3)
Here, pO and xO denote the canonically conjugate operators of the center of mass momentum and position, respectively. The Cartesian components of pO and xO obey the fundamental commutation relation
(81.4) xO i ; pOj D i„ıij : The vector potential A.x; t/ and scalar potential energy U.x; t/ account for the interaction of the particle with all the optical elements, probes, and samples that are placed between the source and the detector, including the effects of gravitation and rotation.
81.2.1 Gravitation and Rotation All particles are subject to the influence of gravitation and rotation, which may both be viewed as being special cases
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of an accelerated frame of reference. The effects of uniform 81.2.4 Atoms gravitation are described by Many optical elements in atom optics are based on the meU.x/ D M g x ; A.x/ D 0 ; (81.5) chanical effects of the radiation interaction. In the electric dipole approximation, the interaction of a single atom with where g describes the direction and magnitude of gravthe electromagnetic field is described by itational acceleration. The effects of uniform rotation are described by O .x; t/ D dO E .x; t/ ; U M O .x; t/ D B.x; t/ dO ; A (81.11) U.x/ D .˝ x/2 ; A.x/ D M .˝ x/ ; (81.6) 2 O where the direction and magnitude of ˝ refer to the ori- where d is the operator of electric dipole transition. The O entation of the axis of rotation and the angular velocity, vector potential A is due to the motional correction of the respectively. Here, U.x/ is the potential of the centrifugal electric dipole interaction. Usually it is neglected; see however, the paragraph “Electric dipole phase” in Sect. 81.6.1. force, while A.x/ is the potential of the Coriolis force. For a monochromatic field of frequency !
81.2.2
Charged Particles
E .x; t/ D E .C/ .x/ ei!t C h:c: ;
(81.12)
.C/ The interaction of charged particles (electrons, ions) with the where E .x/ defines both polarization and spatial characteristics of the field. A standing-wave laser field with optical electromagnetic field is described by axis in the x-direction and linear polarization , for example, is described by U.x; t/ D q˚.x; t/ ; A.x; t/ D qA.x; t/ ; (81.7)
where q is the particle charge (q D e for electrons), and (81.13) E .C/ .x/ D E0 f .x; y; z/ cos.kx/ ; A and ˚ denote the gauge potentials of the electric field where E0 is the electric field amplitude, k D !=c is the strength E and magnetic induction B, respectively, wavenumber, and the slowly varying function f .x; y; z/ ac@ (81.8) counts for the transverse profile of the laser field. E .x; t/ D A.x; t/ r˚.x; t/ ; @t B.x; t/ D r A.x; t/ : (81.9) Essential state approximation The electric dipole operator dO acts in the Hilbert space of The quantum mechanics of charged particles is invariant unelectronic states of the atom. In the particular case that the der a gauge transformation A 7! A 0 D A C r, ˚ 7! ˚ 0 D polarization of the laser field is spatially uniform and that ˚ P combined with a local phase transformation of the spontaneous emission does not play a role, two states are wave function, 7! 0 D eiq=„ . In particular, all measurgenerally sufficient, and the atom may be modeled by a twoable quantities in the De Broglie optics of charged particles, level atom with electronic levels jei and jgi of energy Ee and including the phase shift in the Aharonov–Bohm effect, are Eg , respectively (Ee > Eg ). With the pseudo-spin represenindependent of the gauge – see Sect. 81.6.1 “Aharonov– tation Bohm Effect” below. ! ! 1 0 jei 7! ; jgi 7! ; (81.14) 0 1
81.2.3 Magnetic Moments
The interaction of a magnetic dipole moment O with the electromagnetic field is described by O .x; t/ D U O B.x; t/ ; O .x; t/ D A
O E .x; t/ : c2
the electric dipole operator assumes the form ! 0 1 O ; d 7! d d 1 0
(81.15)
(81.10) where d is the matrix element of the dipole transition, and d denotes its polarization. The prefix “pseudo” of a pseudo Here, the vector potential is due to the motional correction spinor indicates that it does not transform in the spin-.1=2/ of the magnetic dipole interaction. Usually, it is neglected. representation of the physical rotation group. However, it does play an important role for the Aharonov– The laser field is assumed to be near resonant with the Casher effect (Sect. 81.6.1). e $ g transition at !0 D .Ee Eg /=„, and we denote by
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!0 ! the atom-laser detuning. Using the rotating- with eigenvalues determined by the dressed Rabi frequency wave approximation (Sect. 72.3.2) in an interaction picture p with respect to the laser frequency, the Hamiltonian describ˝.x/ D j˝1 .x/j2 C 2 ; (81.23) ing the atomic dynamics – both internal and center-of-mass – is given by and coefficients (we suppress the x-dependence for notational clarity) ! O „ ˝1 .x/ pO 2 ! ! ; (81.16) H D cos #2 ˛C ˛ ei' sin #2 2M 2 ˝1 .x/ O D SO : (81.24) ˇC ˇ ei' sin #2 cos #2 where Here, the Stückelberg angle # #.x/ and phase angle ' 2d d E .C/ .x/ (81.17) '.x/ are defined in terms of the polar representation of the ˝1 .x/ D „ fictitious magnetic field is the spatially dependent bare Rabi frequency. In the field ˝ D ˝.cos ' sin #; sin ' sin #; cos #/ : (81.25) Eq. (81.13), ˝1 .x/ is cosinusoidal with peak value ˝0 D 2d E0 =„. In the dressed state basis, the transformed Hamiltonian asThe Hamiltonian Eq. (81.16) may be written in the form sumes the form H D
pO 2 „ C ˝.x/ O O ; 2M 2
(81.18)
where O is the vector of Pauli matrices and ˝.x/ D ReŒ˝1 .x/; ImŒ˝1 .x/; :
2 O .x/ pO A O „ Q O O O O C ˝.x/ O 3; H S HS D 2M 2
(81.26)
with matrix-valued vector potential (81.19)
O .x/ D i„S .x/ŒrS.x/ : A
(81.27)
As it stands, the Hamiltonian Eq. (81.18) describes the precession and center-of-mass motion of the atomic pseudo spin in an external fictitious magnetic field ˝.x/ [Eq. (81.10)]. Spatial variations of this field give rise to the atom-optical Stern–Gerlach effect, i.e., the splitting of the atomic center of mass wave function (Sect. 81.5.3).
This matrix is not diagonal; its off-diagonal elements describe nonadiabatic transitions between the dressed states. If the detuning is sufficiently large, and the atom moves sufficiently slowly, these nonadiabatic transitions may be neglected in a first approximation. In such an approximation, which is akin to the Born–Oppenheimer approximation in molecular physics, the dynamics of the atom is described by Adiabatic approximation two decoupled Hamiltonians of the generic form Eq. (81.3), In the position representation, the Hamiltonian Eq. (81.16) with scalar potentials U given by acts on state vectors of the form of a pseudo spinor .x; t/ D
e .x; t/ g .x; t/
!
„ 2
U.x/ D ˙ ˝.x/ ;
:
(81.20)
(81.28)
and vector potentials A.x/ given by the diagonal elements of Eq. (81.27). The vector potential is usually neglected. If Alternatively, this state vector may be expanded in terms of included, it describes the Berry phase of the mechanical efthe local eigenvectors ˛˙ .x/, ˇ˙ .x/ of the interaction mafects of the radiation interaction of a two-level atom. trix, also called dressed states, The idea behind the adiabatic approximation is that the ! ! internal state of the atom, which is described by any of the ˛C .x/ ˛ .x/ C .x; t/ ; .x; t/ D C .x; t/ locally varying dressed states, has enough time to adjust ˇC .x/ ˇ .x/ smoothly to the motion of the atom. For the important case (81.21) of strong detuning j j j˝ .x/j, this assumption is usu1 ally well justified. In this case, an atom in the dressed ground where state experiences a potential that is approximately given by ! ! O .x/ ˛˙ .x/ D ˙ „˝.x/ ˛˙ .x/ ; „j˝1 .x/j2 (81.22) U U.x/ D : (81.29) 2 ˇ˙ .x/ ˇ˙ .x/ 4
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For red detuning, we have > 0, in which case the atom is attracted towards regions of high intensity (high-field seeker). For blue detuning, the atom is repelled from such regions (low-field seeker). A potential similar to Eq. (81.29) also applies for complex particles like molecules whose transitions are detuned far from the light frequency. The potential is proportional to the dynamic polarizability ˛.!/ and the field intensity.
If U D 0 at the entrance to the interaction region, E is the kinetic energy of the freely traveling de Broglie wave, and k0 is the related wavenumber. Comparing Eq. (81.31) with the scalar Helmholtz equation of electromagnetic theory and identifying
Atom optical nonlinearity In atom optics, nonlinear phenomena occur due to atom– atom interactions in the ensemble, the nature of which may be significantly influenced by laser light [19]. In the presence of laser light, atom–atom interactions mainly result from photon exchange, which, in most cases, leads to a repulsive interaction. Details of the microscopic basis of atom–atom interactions, in particular those concerning cold collisions, are presented in Chap. 54. Characteristic effects of nonlinear atom optics like four-wave mixing [20] and parametric amplification [21] have been observed with the highly dense samples provided by Bose–Einstein degenerate gases (Chap. 80).
as an index of refraction for matter waves, one observes the complete analogy of scalar optics of stationary matter waves and monochromatic light waves. This analogy can be generalized for spinor-valued wave functions, which would correspond to vector wave optics in anisotropic index media. However, in contrast to light, spinor-valued wave functions do not obey a transversality condition. In Eqs. (81.31)–(81.33), the parameter E describes the kinetic energy of the incoming beam. Thus, E is positive, and, therefore, nE < 1 for positive values of the potential, while nE > 1 for negative values of the potential. For neutrons, one generally has nE < 1. This contrasts to the index of refraction for light waves, which is generally larger than 1. For electrons, ions, and atoms both nE < 1 and nE > 1 may be realized.
E U.E/ nE .x/ E
1=2 (81.33)
81.3 Evolution of De Broglie Waves Since the Schrödinger equation is a linear partial differential 81.3.2 WKB Approximation equation, de Broglie optics shares most of its principles with principles of other wave phenomena, and in particular with Waves are described by amplitude and phase. Particles are described by position and momentum. The link between the optical principles of electromagnetic waves. these concepts is provided by Hamilton’s ray optics. For scalar matter waves, a ray follows a classical trajectory. The optical signature of the ray is the phase associated 81.3.1 Light Optics Analogy with it. The quantum mechanical version of Hamilton’s ray The analogy of de Broglie optics and light optics becomes optics is obtained in the WKB (Wenzel, Kramers, and Brilparticularly transparent for monoenergetic beams of scalar louin) approximation of the stationary Schrödinger equation particles. Such beams are described by a time harmonic wave (Eq. (81.30)). Any solution of Eq. (81.30) may be written in the form function .x; t/ D eiEt =„ E .x/, where E .x/ obeys the stationary Schrödinger equation iW .x/ ; (81.34) E .x/ D A.x/ e (81.30) H EDE E: with real-valued A.x/ and W .x/. In the WKB approximation, A.x/ is slowly varying, and Setting A.x/ D 0 in Eq. (81.3) for simplicity, this equation assumes the form U.x/ r 2 C k02 1 E
ZP E .x/
D0;
(81.31)
Zx A.x 0 / dx 0 ;
1
nE .s/ds C „
W .x/ D k0 Pi
(81.35)
xi
where the wavenumber k0 is related to the energy E via the which is called eikonal in Hamilton’s ray optics. In this exdispersion relation pression, nE .s/ nE .x.s//, where x.s/ denotes the classical trajectory of energy E connecting the point Pi with the „2 k02 point P , x i and x are the coordinates of Pi and P , respecE : (81.32) 2M tively, and ds jdxj is the element of arc length measured
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Let the z-axis be the central optical axis of symmetry along the classical trajectory. Note that the second contribution in Eq. (81.35) is generally gauge dependent. However, along which the optical elements are aligned. Using the for closed loops, which are frequently encountered in in- ansatz E .x/ D eik0 z .x; yI z/ and dropping @2 =@z 2 in terferometry, the gauge-dependence disappears by virtue of a slowly varying envelope approximation one obtains Stokes theorem, which transforms the path integral into an @ area integral over the rotor of A. i„v0 .x; yI z/ The eikonal Eq. (81.35) may also be written in the form @z of a reduced action „2 2 (81.39) r? C U.x; yI z/ .x; yI z/ ; D 2M Zx Zx 1 p.x 0 / dx 0 D k.x 0 / dx 0 ; (81.36) W .x/ D „ where r 2 D @2 =@x 2 C @2 =@y 2 , and v D „k =M is the lonxi
xi
where p.x/ is the local value of the canonical momentum of the particle, k.x/ p.x/=„ is the corresponding wave vector, and the integral is evaluated along the classical trajectory of the particle. Note that in the presence of a vector potential, p.x/ and dx are no longer parallel as a result of the difference between canonical momentum p and kinetic momentum M dx=dt p A. The WKB approximation becomes invalid in the vicinity of caustics where neighboring rays intersect, and the amplitude A.x/ becomes large. There, connection formulae are used to find the proper phase factors picked up by the ray in traversing the caustics. Depending on the topology of the intersecting rays, different classes of diffraction integrals provide uniform approximations for the wave amplitude near caustics. For further details, see [22].
81.3.3 Phase and Group Velocity The velocity of a particle that traverses a region of negative potential increases so that p.x/ > p0 , and the phase R advances: ıW D Œp.x/ p 0 dx > 0. In quantum mechanics, the classical velocity corresponds to the group velocity, while the evolution of the phase is determined by the phase velocity. The phase and group velocities of de Broglie waves are given by r E E 1 D ; (81.37) vp .x/ p.x/ nE .x/ 2M r 2E @E D nE .x/ ; (81.38) vg .x/ @p.x/ M
0
?
0
gitudinal velocity of incoming particles. This equation has exactly the form of a time-dependent Schrödinger equation in two dimensions, with z=v0 playing the role of a fictitious time t. With this interpretation, the spatial evolution of phase fronts along z can be analyzed in dynamical terms of particles moving in the xy-plane.
81.3.5 Raman–Nath Approximation In the Raman–Nath approximation (also called the shorttime, thin-hologram, or thin-lens approximation), the r?2 term in Eq. (81.39) is neglected. The potential U.x; yI z/ then acts as a pure phase structure, and the solution of Eq. (81.39) becomes .x; yI z/ D 2 3 Zz i exp4 dz 0 U.x; yI z 0 /5.x; yI zi / : „v0
(81.40)
zi
In terms of a classical particle moving under the influence 2 of U, the approximation loses validity for .1=2/M v? & U, which is just a quarter cycle for a harmonic oscillator.
81.4 Refraction and Reflection
Consider a particle beam of energy E incident on a medium with constant index of refraction nE . The boundary plane at z D 0 in Fig. 81.1 divides the vacuum from the medium. At the boundary, the beam is partially reflected and partially respectively. Note that the product vp vg D E=M is indepentransmitted, with the angles determined by Snell’s law of redent of nE .x/. fraction
81.3.4 Paraxial Approximation
sin ˛ D nE sin ˇ ;
The paraxial approximation is useful in describing the evolu- and the law of reflection tion of wave-like properties and/or distortion of wavefronts ˛ D ˛0 : in the immediate neighborhood of an optical ray.
(81.41)
(81.42)
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De Broglie Optics
1179
α'
β z
Here, n is the light index of refraction of the dielectric, k is the wave number of the light beam in vacuo, and i is its angle of incidence. If the light is blue-detuned from the atomic resonance, an incident beam of ground state atoms experiences the repulsive potential
α
U.x/ D U>0
Fig. 81.1 Reflection geometry
The coefficients of reflectivity R, and transmittivity T D 1 R are given by the Fresnel formula ˇ ˇ ˇ cos ˛ nE cos ˇ ˇ2 ˇ : R D ˇˇ cos ˛ C nE cos ˇ ˇ
(81.44)
If E < U, then nE is imaginary, and total reflection occurs for all ˛. In neutron optics, this total mirror reflection requires ultracold neutrons (T 0:5 mK). It has important applications for the storage of ultracold neutrons in material cavities, and neutron microscopy using spherical mirrors. For details, see [3].
81.4.1 Atomic Mirrors
(81.46)
For ˛ > ˛Q c , the evanescent field acts as a nearly perfect mirror, the imperfections being due to nonadiabatic transitions into the wrong dressed state, possible spontaneous emission, and scattering off an inhomogeneous potential, inherited from the roughness of the dielectric-vacuum interface. Reflection of atoms by evanescent laser light was demonstrated by Balykin et al. [23] at grazing incidence and by Kasevich et al. [24] at normal incidence.
(81.43)
For nE > 1, the interface is attractive, and R 1, with R ! 1 only for glancing incidence ˛ ! =2. For nE < 1, the interface is repulsive, and total reflection (R D 1) occurs for ˛ ˛Q c , where ˛Q c D sin1 nE is the critical angle. For ˛ > ˛Q c , the de Broglie wave becomes evanescent, with z inside the medium (z > 0), where E .z > 0/ e
1=2
D k0 sin2 ˛ sin2 ˛Q c :
i 1=2 „ h 2 2 z C 2 j j : ˝0 e 2
Reflection of atoms by magnetic near fields Magnetic near fields are produced above substrates with a spatially modulated permanent magnetization or close to arrays of stationary currents. In the vacuum above the substrate, the field decays approximately exponentially over a length comparable to the scale of the magnetic modulation. The motion of atoms that cross such inhomogeneous magnetic fields sufficiently slowly is governed by the analog of the adiabatic potential described in Sect. 81.2.4 U.x/ D
ˇ ms ˇˇ B.x/ˇ ; s
(81.47)
where is the magnetic moment, and ms is the (conserved) projection of the atomic spin s onto the local magnetic field direction. A repulsive mirror potential is achieved for spin states with ms < 0; these weak field seekers are repelled from the strong fields close to the substrate. Experiments have used magnetic recording media like magnetic tapes or hard disks, arrays of current-carrying wires, or amorphous magnetic substrates, see [25].
Inelastic processes, such as diffuse scattering and adsorption, inhibit coherent reflection of atoms from bare surfaces. The 81.4.2 Atomic Cavities surface must, therefore, be coated either with material of low adsorptivity (noble gas), or electromagnetic fields (evanes- Atomic reflections are used in the two kinds of cavities procent light or magnetic fields, see below). posed so far: the trampoline cavity and bicolored traps. The trampoline cavity, also called the gravito-optical cavReflection of atoms by evanescent laser light ity, consists of a single evanescent mirror facing upwards, the Evanescent light fields are produced by total internal reflec- second mirror being provided by gravitation [26]. A stable tion of a light beam at a dielectric–vacuum interface. In the cavity is realized with the evanescent laser field of a parabolivacuum (z 0), the field decays exponentially away from cally shaped dielectric–vacuum interface, see the experiment the interface on a characteristic length 1 , where by Aminoff et al. [27]. A cavity with transverse confinement provided by a hollow blue-detuned laser beam was
2 2 1=2
D k n sin i 1 : (81.45) demonstrated by Hammes et al. [28]. By combining red and
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blue-detuned beams guided by an optical fiber, one can form that has seen a rapid evolution recently [33]. A planar waveguide is provided by the one-dimensional confinement in an array of traps, as shown by Vetsch et al. [29]. an optical standing wave or an atomic mirror combined with gravity. The discrete nature of the waveguide modes in that case was demonstrated by lowering an absorbing ob81.4.3 Atomic Lenses ject [34]. Linear waveguides can be modeled by the parabolic De Broglie waves may be focused by refraction from transverse potential Eq. (81.48) that now extends along the a parabolic potential or by diffraction, e.g., by a Fresnel zone waveguide axis (the z-axis). Physical realizations include plate (Sect. 81.5.2). Consider focusing by the parabolic po- hollow, blue-detuned laser beams, hollow fibers whose inner wall is coated with blue-detuned evanescent light [35], tential interference patterns from three or four far-detuned beams ( 1 2 2 2 (also known as optical lattices) [36], and magnetic field minM!f .x C y / ; w z 0 2 U.x; yI z/ D ima along current-carrying wires, possibly combined with 0; otherwise : homogeneous bias fields [37]. With typical thermal atomic (81.48) ensembles, these waveguides operate in a multimode regime, and coherent operation has been demonstrated only with For ground-state atoms, such a potential is realized in the Bose–Einstein condensates. A strong transverse confinevicinity of the node of a blue-detuned standing-wave laser ment that facilitates monomode operation, can be achieved field of transverse width w. In this case, with miniaturized wire networks deposited on a solid substrate. This approach has led to the fabrication of atom !rec 1=2 ! f D ˝0 ; (81.49) chips [38, 39], and integrated atomic interferometers were j j demonstrated in [40, 41]. where !rec D „k 2 =2M is the recoil frequency. The comparison of the Raman–Nath approximation 81.5 Diffraction Eq. (81.40) at z D 0 with the phase fronts of a spherical wave converging towards a point xf D .0; 0; f /, shows that U de- The diffraction of matter waves is described by the soluscribes a lens of focal length tion of the Schrödinger equation (Eq. (81.30)) subject to the boundary conditions imposed by the diffracting object. For v02 a plane screen ˙ made of opaque portions and apertures, : (81.50) f D w!f2 the solution in the source-free region behind the screen is given by the Rayleigh–Sommerfeld formulation of the HuyThe Raman–Nath approximation is only valid for a thin lens gens principle w f and breaks down for w > wRN D v0 =2!f . In the latZ ter case, oscillations of the particles in the harmonic potential dd eik0 R k0 .x/ D E become relevant, a phenomenon sometimes called channeli 2 R ing. Channeling may be used to realize thick lenses with ˙ focal length f D wRN corresponding to a quarter oscillation nR i 1C ./ ; (81.51) period. k0 R R Focusing of a metastable helium beam using the antinode of a large period standing-wave laser field has been demon- where D .; ; / denotes coordinates of points on ˙ , n is strated [30]. Such a field is produced by reflecting a laser an inwardly directed normal to ˙ at a point , and R D x . beam from a mirror under glancing incidence. The standing A diffraction pattern only becomes manifest in the diffracwave forms normal to the mirror surface. Similar interfer- tion limit r d , where r is the distance to the observation ence patterns provide arrays of thick lenses that have been point, and d is the length scale of the diffracting system. exploited in atom lithography to focus an atomic beam onto The two diffraction regimes are then the Fraunhofer limit a substrate [7]. r d 2 =dB and the Fresnel regime r d 2 =dB , also called near-field optics.
81.4.4 Atomic Waveguides 81.5.1 Fraunhofer Diffraction Atomic waveguides can be realized with potentials that confine atoms in one or two dimensions [26, 31, 32]. These In the Fraunhofer limit, the field at position .x; y/ on a screen devices are key elements for integrated atom optics, a field at a distance L downstream from the diffracting object is
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De Broglie Optics
1181
given by
w
Z
dd eik0 L iL 2 i.kx Cky / e .; I 0/ ;
.x; yI L/ D k0
k0
(81.52)
where kx D k0 x=L, ky D k0 y=L. The field at the observation screen is thus given by the Fourier transform of the field in the object plane; i.e., the momentum representation of the x diffracted state. Since most diffraction experiments in atom optics are performed in the Fraunhofer limit, most calculaz tions are done in the momentum representation. Atomic diffractions from microfabricated transmission Fig. 81.2 Geometry of the Kapitza–Dirac effect gratings [42] and double slits [43] have been observed. Recent experiments have extended de Broglie wave diffraction Due to the periodicity of the standing-wave light field, the to heavier, complex particles like fullerenes (C60 ) [44] and transverse momenta of the transmitted atom waves differ by even larger molecules [9]. multiples of „k from the transverse momentum of the undiffracted wave. For the important case of strong detuning, and assuming that the incoming atoms are in their electronic 81.5.2 Fresnel Diffraction ground state moving with transverse momentum px D 0, the Typical applications of Fresnel diffraction are Fresnel zone px -distribution of the outgoing wave is given in the Raman– plates and the effects of Talbot and Lau. Fresnel zone Nath approximation (Sect. 81.3.5) by plates are microfabricated concentric amplitude structures ˇ !ˇ2 ˇ ˇ that act like lenses. They are frequently employed in op˝0 2 ˇ ˇ (81.53) Prob.px D 2n„k/ / ˇJn ˇ ; tics of ˛-particles and neutrons. In atom optics, focusing ˇ 8j j ˇ with a Fresnel zone plate was first demonstrated by Carnal where Jn is a Bessel function of order n, D w=v0 is an efet al. [45]. The Talbot effect and the related Lau effect refer to the fective interaction time, w being the width of the laser field, self-imaging of a grating of period d , which appears down- and v0 the longitudinal velocity of the atoms. The distribustream at distances that are integral multiples of the Talbot tion Eq. (81.53) was observed by Gould et al. [47]. For & RN , where RN D .!rec ˝0 2 =j j/1=2 , the Raman– length L D 2d 2 =dB . For a discrete set of distances smaller than the Talbot length, images of the grating appear with Nath approximation becomes invalid. As a result of Doppler smaller periods d=n, n D 2; 3; : : : For applications in matter- related phase-mismatch, the momentum spread saturates and shows a sequence of collapse and revival as a function of . wave interferometry, see [46]. If the detuning is too small to allow for a scalar description, the two-level character of the atoms must be taken into account. For the particular case of D 0, the ground state 81.5.3 Near-Resonant Kapitza–Dirac Effect evolves p into an equal superposition of the two diabatic states The near-resonant Kapitza–Dirac effect refers to the diffrac- 1= 2.jei ˙ jgi/ while entering the interaction region. Intion of two-level atoms from a standing-wave laser field side the interaction region, these states experience potentials with a spatially uniform polarization. The dynamics of the that differ only by their sign. For atomic beams with a small effect is described by the Hamiltonian Eq. (81.16) with spatial spread ıx 2 =k, the diabatic states experience opthe mode function of the laser field given by Eq. (81.13). posite forces, leading to a splitting of the atomic beam called Consider atoms traveling predominantly in the z-direction, the atom-optical Stern–Gerlach effect [48]. Diffraction similar to the Kapitza–Dirac effect is also i.e., orthogonal to the axis of the laser field, with energy .1=2/M v02 „˝0 (Fig. 81.2). Kapitza–Dirac diffraction is possible with other polarizable particles. Experiments with then observed in transmission, which in the theory of diffrac- molecules have been demonstrated by Gerlich et al. [49]. tion is called Laue geometry. In the paraxial approximation (Sect. 81.3.4) for motion in the z-direction, and assuming that the laser profile is homogeneous in the y-direction, the 81.5.4 Atom Beam Splitters description reduces to an effectively one-dimensional model for the quantum mechanical motion along the x-axis of the Beam splitters are optical devices that divide an incoming beam into two outgoing beams traveling in different direclaser field.
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tions. For thermal neutrons, beam splitters may be realized by diffraction from perfect crystals in Laue geometry. For atoms, they can be realized using diffraction from crystalline surfaces, microfabricated structures (Sect. 81.5.1), or by using diffraction from an optical standing wave. The Kapitza–Dirac effect, for example, may be exploited to split an atomic beam coherently using Bragg reflection at the lattice planes provided by the periodic intensity variations of a standing-wave laser field [50]. This process is resonant for an incoming atomic beam traveling with transverse momentum px D „k because it is energetically degenerate with the diffracted beam traveling with transverse momentum pNx D „k. This level degeneracy is lifted while the atoms enter the interaction region. In the Bragg regime, the lifting happens slowly enough that only the momentum states j ˙ „ki participate in the diffraction (two-beam resonance), and their populations show Pendellösung-type oscillations as a function of the transit time. The frequency of the oscillations is given by ıE=„, where ıE is the energy splitting of the two beams inside the interaction region. For a transit time given by a quarter period of the Pendellösung, a 50% beam splitting is observed [50]. In principle, Bragg resonances may also be realized for higher diffraction order px D n„k $ pNx D n„k. However, in this case, intermediate momentum states become populated (multibeam resonance), which makes the higher-order Bragg resonances less suitable for beam splitting purposes. More promising for the realization of an atomic beam splitter is the magneto-optical diffraction which refers to the diffraction of three-level atoms from a laser field with a periodic polarization gradient (lin ? lin configuration) (Chap. 79), and a magnetic field aligned parallel to the optical axis of the laser field. This configuration realizes an interaction potential in the form of a blazed grating, i.e., a phase grating with an approximately triangular variation of phase. In an experiment by Pfau et al., transverse splitting of a beam of metastable helium by 42„k was observed [51]. Diffraction from an evanescent standing wave involves Bragg reflection of atoms under glancing incidence from the periodic grating of a blue detuned evanescent standing-wave laser field [52]. Diffraction at normal incidence has been demonstrated with sufficiently slow atoms [53, 54] and can be described by a generalization of the Raman–Nath approximation (Sect. 81.3.5).
81.6 Interference
C. Henkel and M. Wilkens
vision of wavefront, and the Mach–Zehnder interferometer as a paradigm for interferometers based on division of amplitude. In de Broglie optics, the latter type is realized in the form of a three-grating interferometer, since division of amplitude is achieved by diffraction at gratings rather than by semitransparent mirrors. Experiments with this geometry have been reported for atoms [55] and recently for more massive, complex molecules [56].
81.6.1 Interference Phase Shift From a fundamental point of view, any interferometer is a ring. At the entrance port of a three-grating interferometer displayed in Fig. 81.3, for example, the wave function is split into two coherent parts that evolve spatially along different paths and, subsequently, come together at the exit port where they are superimposed to produce two outgoing waves ˙
1 Dp . 2
1
˙
2/
;
(81.54)
where the components from path 1 and 2 are given by 1
D A1 exp.iW1 / ;
D A2 exp.iW2 / : (81.55) p For simplicity, assume A1 D A2 D A0 = 2. The relative flux of the outgoing waves is then 2
I˙ D
1
1 ˙ cos ; 2
(81.56)
where D W1 W2 is the relative phase of the two components. The sinusoidal variations of I˙ with varying are called interference fringes, and =2 is called the fringe order number. In the WKB approximation, the phases W1 and W2 are given by Z (81.57) Wi D W0 C k.x/ dx; i D 1; 2 ; i
1
+
2
–
ϑ
While the overall phase of a wave function is not observable, interferometry makes detectable the relative phases of D D two components 1 , 2 in a superposition D 1 C 2 . Two types of interferometers are most common: the Young double-slit as a paradigm for interferometers based on di- Fig. 81.3 Geometry of the three-grating interferometer
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De Broglie Optics
1183
where the integrals extend over the classical paths 1 and 2, Sagnac effect. The Sagnac effect refers to fAg in a rotatrespectively, and dx is an element of displacement along the ing interferometer. Inserting Eq. (81.6) into Eq. (81.64) and paths. Using Eq. (81.57), the relative phase is assuming that the axis of rotation is oriented perpendicular to the plane of the interferometer, the Sagnac phase shift is Z (81.58) given by D k.x/ dx ; C
Sa D where C is the closed interferometer loop. Note that on path 2, the path element dx in Eq. (81.58) and k are antiparallel. Usually, the absolute value of is not measured, but only variations, called phase shifts, which result from displacements of the diffraction gratings or placement of optical elements into the beam path. Phase shifts come in two categories: dispersive and geometric. If a phase shift depends on v0 , it is called dispersive, otherwise it is called geometric. Geometric phases depend only on the geometry of the interferometer loop. The Sagnac effect (see below), for example, may be geometric. A phase that depends neither on v0 nor on the geometry of the interferometer loop is called topological. The Aharonov–Bohm effect (see below) is topological. Using Eq. (81.35), becomes D 0 C fUg C fAg ; where for weak potentials U and A,
4M˝A ; „
(81.65)
where A is the geometric area enclosed by the interferometer loop; Sa may be dispersive or geometric depending on the type of interferometer. In a Young double-slit, A is independent of energy, and Sa is geometric. In a three-grating interferometer, the area is A #D 2 , where # is the splitting angle; see Fig. 81.3. In this case, Sa is dispersive because of the velocity dependence of # 2„k=M v0 . The Sagnac effect for de Broglie waves was first observed by Werner et al. [57] using a neutron interferometer.
Aharonov–Bohm effect. The Aharonov–Bohm effect refers to the fAg of charged particles encircling a magnetic flux line [58]. Inserting A from Eq. (81.7) into Eq. (81.64), and assuming the particles of charge q encircle a line of flux (81.59) ˚ once, one finds AB D
q˚ : „
(81.66)
M v0 .L1 L2 / ; „ I 1 UŒx.s/ds ; fUg D „v0 I 1 fAg D A.x/ dx ; „ 0 D
(81.60) A characteristic feature of the Aharonov–Bohm effect is that the particles actually never see the magnetic field of the flux (81.61) line, which is confined to some region inaccessible to the particles; AB is strictly topological and only depends on the (81.62) linking number of the interferometer loop and the flux line. Its appearance is characteristic for all gauge theories. For furand Li is the geometric length of the path i. For a constant ther details and a summary on its experimental verification, potential U0 intersecting the interferometer on a length w, see [2]. fUg is given by
Aharonov–Casher effect. The Aharonov–Casher effect refers to fAg of a magnetic spin encircling an electric line : (81.63) fUg D charge [59]. Inserting A from Eq. (81.10) into Eq. (81.64), „v0 one obtains for proper alignment of and E , Using Stokes’ theorem, fAg may be written in the manijj r0 festly gauge-invariant form ; (81.67) AC D 2 B Z 1 (81.64) where r is the classical electron radius, is the Bohr magŒr A.x/ da ; fAg D 0 B „ neton, D e= el , el being the electric line charge density; where the integral extends over the area enclosed by the in- AC is topological only if the spin is aligned parallel to the terferometer, and da is an infinitesimal area element. electric line charge, and both are oriented perpendicular to the plane of the interferometer; AC for atoms has been obDispersive phase shifts. Atom interferometers have been served by Sangster et al. [60]. able to measure phase shifts of the form Eq. (81.61) due to, for example, the atomic level shift in an electric field (Stark Electric dipole phase. Electric dipole phase refers to effect) or to coherent forward scattering by background gas the fAg of an electric dipole moment encircling a magnetic line charge [61]. Inserting A from Eq. (81.11) into atoms, see [6]. U0 w
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C. Henkel and M. Wilkens
Eq. (81.64), one obtains for proper alignment of d and B dE D 2
jd j a0 ; ea0
momentum via homodyne detection of the cavity field. For further details, see [67] and Chap. 82.
(81.68)
where a0 is the Bohr radius, and D ˚0 =%mg , ˚0 being the 81.7 Coherence of Scalar Matter Waves magnetic flux unit, and %mg being the magnetic line charge density. In analogy to the Aharonov–Casher effect, dE is The general solution of the free Schrödinger equation topological, provided that d is aligned parallel to the mag- (Eq. (81.2)) may be written in the form netic line charge, and both are oriented perpendicular to the Z interferometer plane. The electric dipole phase has been ex.x; t/ D d3 k a.k/ ei.kx!.k/t / ; (81.70) perimentally measured using atom interferometry [62]. where !.k/ E=„ D „k2 =2M . If the coefficients a.k/ are known, the state represented by .x; t/ is called a pure state. Otherwise it is called a mixed state, and physical quantities Manipulation of the internal state of atoms by means of elec- are obtained by an ensemble average over the possible realtromagnetic fields makes it possible to realize interferometric izations of a.k/. setups that involve separation of paths in internal space rather The degree of coherence of matter waves is described by than in real space. Examples of such interferometers are the the autocorrelation function of .x; t/ optical Ramsey interferometer [63], the stimulated Raman interferometer [64], and interferometers using static electric (81.71) .x; tI x 0 ; t 0 / .x 0 ; t 0 / .x; t/ ; and magnetic fields [65, 66].
81.6.2 Internal-State Interferometry
81.6.3 Manipulation of Cavity Fields by Atom Interferometry The entanglement of atomic states and quantized field states opens novel possibilities for manipulating and/or measuring nonclassical field states in a cavity. In the adiabatic limit, for example, and assuming sufficient detuning between the atom and the cavity field, the interaction and CM motion of an atom traversing a cavity is well described by the potential Eq. (81.29) U.x/ D „
g2 f .x/2 a a ;
(81.69)
where g is the vacuum Rabi frequency, f .x/ is a cavity mode function, and a, a denote cavity photon annihilation and creation operators, respectively (see Chap. 82). Because of the presence of the photon-number operator a a in Eq. (81.69), the deflection and phase shift of an atom traversing the cavity is quantized, displaying essentially the photon number statistics in the cavity. The quantized deflection is sometimes called the inverse Stern–Gerlach effect. Due to the entanglement of atom and cavity states, and the position dependence of the interaction strength, the phase shift induced by U.x/ in a standing-wave cavity may be used to measure either the atomic position via homodyne detection of the cavity field or the photon statistics via atom interferometry. In a ring cavity, the entanglement of CM motion and cavity field may be used to measure the atomic
where the overline . / denotes the ensemble average over the possible realizations of a.k/. In light optics, .x; tI x 0 ; t 0 / is called the mutual coherence function. In particular, for equal times, .x; tI x 0 ; t/ describes the spatial coherence, and for equal positions, .x; tI x; t 0 / describes the temporal coherence. For a beam of particles, coherence may be either longitudinal (measured along the beam) or transverse (measured across the beam). In contrast to light optics, there is no simple relation between longitudinal coherence and temporal coherence because the dispersion relation of matter waves is quadratic in the wavenumber. The spatial coherence function is intimately related to the quantum mechanical density operator of the particles (Chap. 7)
.t/ D j .t/ih .t/j :
(81.72)
In the position representation, one has hxj .t/jx 0 i .x; x 0 I t/ D .x; tI x 0 ; t/ :
(81.73)
Longitudinal and temporal coherence of a particle beam is determined mainly by the source of the beam. The thermal fission reactors used in neutron optics and the ovens used in atom optics are analogous to black-body sources in light optics. In contrast, the transverse coherence is mainly determined by the way the particles are extracted from the oven to form a beam.
81
De Broglie Optics
81.7.1
1185
Atomic Sources
Supersonic beams. Supersonic beams are produced by supersonic expansion of a high-pressure gas that is forced To describe thermal sources, consider a single particle in an through an appropriately designed nozzle. The expansion oven of temperature T and volume V , assuming that a.k/ produces a velocity distribution in the longitudinal direction, which is approximately Gaussian with a velocity ratio and a.k0 / are statistically independent v=ıv 1020. a.k/ a.k0 / D .k/ı.k k0 / ; (81.74) Pulsed beams. Pulsed beams are produced by chopping where any of the beams described above. Important applications for pulsed beams are the resolution of temporal coherence 3 „2 k 2 (81.75) and the mapping of the relative phases of the a.k/ in matter .k/ D th exp V 2M kB T wave interferometry. accounts for the thermal distribution of wavenumbers, and
Laser-like source of atoms. In these sources, many atoms 1=2 with integral spin (Bosons) occupy one and the same quan2 „2 th D (81.76) tum state of motion. Their operational principle is rooted in M kB T the quantum statistical effects of indistinguishability. It may be viewed in close analogy to the operational principle of denotes the thermal de Broglie wavelength. Using Eqs. (81.74)–(81.76) in Eq. (81.71), the mutual co- an ordinary laser (Chap. 74) and the mechanism underlying Bose–Einstein condensation (Chap. 80). Laser-like sources herence function becomes have, indeed, been achieved by letting a small current of 1 1 0 0 atoms leak out of a trapped Bose–Einstein condensate [69]. .x; tI x ; t / D V f1 C i.t t 0 /=th g3=2 .x x 0 /2 =2th exp ; (81.77) 81.7.2 Atom Decoherence 1 C i.t t 0 /=th where th D
„ ; kB T
In any interferometer, the contrast of the interference fringes quantitatively measures the coherence of the wave involved. (81.78) Partially coherent beams show an output flux given by
is the thermal coherence time. According to Eq. (81.77), the spatial coherence of a thermal state falls off in a Gaussian manner on a scale given by th . The temporal coherence, in contrast, falls off algebraically on a timescale given by th . Expressed in physical units, one has 1:74.5/ nm ; th D p .M=u/.T =K/ 7:63 ps th D ; .T =K/
(81.79)
where u is the atomic mass unit.
Atomic Beams Effusive beams. Effusive beams are produced from thermal sources by a suitable set of collimators placed in front of the opening of the oven. This produces a Maxwell– Boltzmann distribution of atomic velocities in the longitudinal direction. The coherence properties in the transverse direction are described by the van Cittert–Zernike theorem [68]; for details see any textbook on classical optics.
I˙ D
1 1 ˙ Re C ei ; 2
(81.80)
instead of Eq. (81.56), with a complex number C . One has jC j 1, with the maximum achieved for a pure state; the phase of C is measured by scanning the interferometer phase shift . In de Broglie interferometry, coherence can be lost when the interfering matter wave gets entangled with other systems. This happens for atoms, for example, due to the emission or scattering of photons, as soon as the detection of these photons permits, in principle, the resolution of spatially separated paths in the interferometer. In fact, the width of .x; tI x 0 ; t/ as a function of x x 0 , also called the spatial coherence length, is reduced to the photon wavelength after a single scattering event; see [70]. Interference can be restored when the emitted photons are detected and correlated with the atom output [71, 72]. Collisions with background gas atoms between the optical elements of a three-grating interferometer also reduce coherence, as has been shown with fullerene molecules [73]. Furthermore, coherence is lost when atoms interact with random electromagnetic fields. This has become relevant for atom reflection from evanescent light because of the roughness of the dielectric surface
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used [74]. The coherent operation of integrated atom optics near metallic surfaces is limited by thermally excited electromagnetic near fields [75], as shown in experiments by Harber et al. [76]. Finally, there are suggestions that decoherence may be ever more efficient for particles with larger masses, due to the interaction with the gravitational field or internal degrees of freedom [17, 18]. Interferometer experiments with large molecules [16] and nanoparticles [77] have been proposed to resolve this issue.
References 1. Pozzi, G.: Particles and Waves in Electron Optics and Microscopy. Academic Press, Cambridge (2016) 2. Tonomura, A.: Electron Holography. In: Shimoda, K. (ed.) Springer Series in Optical Sciences, vol. 70, Springer, Berlin, Heidelberg (1993) 3. Utsuro, M., Ignatovich, V.K. (eds.): Handbook of Neutron Optics. John Wiley & Sons, Hoboken (2010) 4. Rauch, H., Werner, S.A.: Neutron Interferometry. Clarendon Press, Oxford (2000) 5. Meystre, P.: Atom Optics. Springer, Berlin, Heidelberg (2001) 6. Cronin, A.D., Schmiedmayer, J., Pritchard, D.E.: Optics and interferometry with atoms and molecules. Rev. Mod. Phys. 81, 1051 (2009) 7. Rohwedder, B.: Resource Letter AON-1: Atom optics, a tool for nanofabrication. Am. J. Phys. 75, 394 (2007) 8. Witham, P.W., Sanchez, E.: A simple approach to neutral atom microscopy. Rev. Sci. Instr. 82, 103705 (2011) 9. Gerlich, S., Eibenberger, S., Tomandl, M., Nimmrichter, S., Hornberger, K., Fagan, P.J., Tüxen, J., Mayor, M., Arndt, M.: Quantum Interference of Large Organic Molecules. Nat. Commun. 2, 263 (2011) 10. Müller, H., Peters, A., Chu, S.: A precision measurement of the gravitational redshift by the interference of matter waves. Nature 463, 926 (2010) 11. Lan, S.-Y., Kuan, P.-C., Estey, B., English, D., Brown, J.N., Hohensee, M.A., Müller, H.: A clock directly linking time to a particle’s mass. Science 339, 554 (2013) 12. Rosi, G.: Precision measurement of the Newton gravitational constant using cold atoms. Nature 510, 518 (2014) 13. dos Santos, F.P., Bonvalot, S.: Cold-Atom Absolute Gravimetry. In: Grafarend, E. (ed.) Encyclopedia of Geodesy. Springer, Cham (2016) 14. Barrett, B., Antoni-Micollier, L., Chichet, L., Battellier, B., Lévèque, T., Landragin, A., Bouyer, P.: Dual matter-wave inertial sensors in weightlessness. Nat. Commun. 7, 13786 (2016) 15. Hamilton, P., et al.: Atom-interferometry constraints on dark energy. Science 349, 849 (2015) 16. Arndt, M., Hornberger, K.: Testing the limits of quantum mechanical superpositions. Nat. Phys. 10, 271 (2014) 17. Pikovski, I., Zych, M., Costa, F., Brukner, C.: Universal decoherence due to gravitational time dilation. Nat. Phys. 11, 668 (2015) 18. Joos, E., Zeh, H.D., Kiefer, C., Giulini, D., Kupsch, J., Stamatescu, I.-O.: Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd edn. Springer, Berlin, Heidelberg (2003) 19. Anderson, B.P., Meystre, P.: Nonlinear atom optics. Contemp. Phys. 44, 473 (2003) 20. Deng, L., et al.: Four-wave mixing with matter waves. Nature 398, 218 (1999) 21. Inouye, S., et al.: Phase-coherent amplification of atomic matter waves. Nature 402, 641 (1999)
C. Henkel and M. Wilkens 22. Berry, M.V., Upstill, C.: Catastrophe optics: morphology of caustics and their diffraction patterns. In: Wolf, E. (ed.) Progress in Optics, vol. XVIII, pp. 259–346. North-Holland, Amsterdam (1980) 23. Balykin, V.I., Letokhov, V.S., Ovchinnikov, Y.B., Sidorov, A.I.: Quantum-state–selective mirror reflection of atoms by laser light. Phys. Rev. Lett. 60, 2137 (1988) 24. Kasevich, M.A., Weiss, D.S., Chu, S.: Normal-incidence reflection of slow atoms from an optical evanescent wave. Opt. Lett. 15, 607 (1990) 25. Wang, Y., et al.: Magnetic lattices for ultracold atoms and degenerate quantum gases. Sci. Bull. 61(14), 1097 (2016) 26. Dowling, J.P., Gea-Banacloche, J.: Evanescent light-wave atom mirrors, resonators, waveguides, and traps. In: Berman, P.R. (ed.) Adv. At. Mol. Opt. Phys, vol. 37, pp. 1–94. Academic Press, New York (1997) 27. Aminoff, C.G., et al.: Cesium atoms bouncing in a stable gravitational cavity. Phys. Rev. Lett. 71, 3083 (1993) 28. Hammes, M., et al.: Optical and evaporative cooling of caesium atoms in the gravito-optical surface trap. J. Mod. Opt. 47, 2755 (2000) 29. Vetsch, E., et al.: Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber. Phys. Rev. Lett. 104, 203603 (2010) 30. Sleator, T., Pfau, T., Balykin, V., Mlynek, J.: Imaging and focusing of an atomic beam with a large period standing light wave. Appl. Phys. B 54, 375 (1992) 31. Balykin, V.I.: Atom waveguides. In: Bederson, B., Walther, H. (eds.) Adv. At. Mol. Opt. Phys, vol. 41, pp. 181–260. Academic Press, San Diego (1999) 32. Hinds, E.A., Hughes, I.G.: Magnetic atom optics: mirrors, guides, traps, and chips for atoms. J. Phys. D Appl. Phys. 32, R199 (1999) 33. Amico, L., et al.: Roadmap on Atomtronics, AVS Quantum. Sci. 3, 039201 (2021) 34. Nesvizhevsky, V.V., et al.: Quantum states of neutrons in the Earth’s gravitational field. Nature 415, 297 (2002) 35. Ito, H., et al.: Laser spectroscopy of atoms guided by evanescent waves in micron-sized hollow optical fibers. Phys. Rev. Lett. 76, 4500 (1996) 36. Bloch, I.: Ultracold quantum gases in optical lattices. Nat. Phys. 1, 23 (2005) 37. Schmiedmayer, J.: Guiding and trapping a neutral atom on a wire. Phys. Rev. A 52, R13 (1995) 38. Reichel, J., Vuleti´c, V. (eds.): Atom Chips. Wiley-VCH, Weinheim (2011) 39. Keil, M., Amit, O., Zhou, S., Groswasser, D., Japha, Y., Folman, R.: Fifteen years of cold matter on the atom chip: promise, realizations, and prospects. J. Mod. Opt. 63, 1840 (2016) 40. Wang, Y.-J., et al.: Atom Michelson interferometer on a chip using a Bose–Einstein condensate. Phys. Rev. Lett. 94, 090405 (2005) 41. Berrada, T., van Frank, S., Bücker, R., Schumm, T., Schaff, J.F., Schmiedmayer, J.: Integrated Mach–Zehnder interferometer for Bose–Einstein condensates. Nat. Commun. 4, 2077 (2013) 42. Keith, D.W., Schattenburg, M.L., Smith, H.I., Pritchard, D.E.: Diffraction of atoms by a transmission grating. Phys. Rev. Lett. 61, 1580 (1988) 43. Carnal, O., Mlynek, J.: Young’s double-slit experiment with atoms: a simple atom interferometer. Phys. Rev. Lett. 66, 2689 (1991) 44. Arndt, M., et al.: Wave-particle duality of C60 molecules. Nature 401, 680 (1999) 45. Carnal, O., et al.: Imaging and focusing of atoms by a Fresnel zone plate. Phys. Rev. Lett. 67, 3231 (1991)
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46. Clauser, J.F., Reinsch, M.: New theoretical and experimental results in Fresnel optics with applications to matter-wave and X-ray interferometry. Appl. Phys. B 54, 380 (1992) 47. Gould, P.L., Ruff, G.A., Pritchard, D.E.: Diffraction of atoms by light: the near-resonant Kapitza–Dirac effect. Phys. Rev. Lett. 56, 827 (1986) 48. Sleator, T., et al.: Experimental demonstration of the optical Stern– Gerlach effect. Phys. Rev. Lett. 68, 1996 (1992) 49. Gerlich, S., et al.: A Kapitza–Dirac–Talbot–Lau interferometer for highly polarizable molecules. Nat. Phys. 3, 711 (2007) 50. Martin, P.J., Oldaker, B.G., Miklich, A.H., Pritchard, D.E.: Bragg scattering of atoms from a standing light wave. Phys. Rev. Lett. 60, 515 (1988) 51. Pfau, T., et al.: Magneto-optical beam splitter for atoms. Phys. Rev. Lett. 71, 3427 (1993) 52. Henkel, C., et al.: Theory of atomic diffraction from evanescent waves. Appl. Phys. B 69, 277 (1999) 53. Landragin, A., et al.: A reflection grating for atoms at normal incidence. Europhys. Lett. 39, 485 (1997) 54. Stehle, C., Bender, H., Zimmermann, C., Kern, D., Fleischer, M., Slama, S.: Plasmonically tailored micropotentials for ultracold atoms. Nat. Photon. 5, 494 (2011) 55. Keith, D.W., Ekstrom, C.R., Turchette, Q.A., Pritchard, D.E.: An interferometer for atoms. Phys. Rev. Lett. 66, 2693 (1991) 56. Juffmann, T., Ulbricht, H., Arndt, M.: Experimental methods of molecular matter-wave optics. Rep. Progr. Phys. 76, 086402 (2013) 57. Werner, S.A., Staudenmann, J., Colella, R.: Effect of Earth’s rotation on the quantum mechanical phase of the neutron. Phys. Rev. Lett. 42, 1103 (1979) 58. Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485 (1959) 59. Aharonov, Y., Casher, A.: Topological quantum effects for neutral particles. Phys. Rev. Lett. 53, 319 (1984) 60. Sangster, K., Hinds, E.A., Barnett, S.M., Riis, E.: Measurement of the Aharonov–Casher phase in an atomic system. Phys. Rev. Lett. 71, 3641 (1993) 61. Wilkens, M.: Quantum phase of a moving dipole. Phys. Rev. Lett. 72, 5 (1994) 62. Lepoutre, S., Gauguet, A., Trénec, G., Büchner, M., Vigué, J.: He–McKellar–Wilkens topological phase in atom interferometry. Phys. Rev. Lett. 109, 120404 (2012) 63. Bordé, C.J.: Atomic interferometry with internal state labelling. Phys. Lett. A. 140, 10 (1989) 64. Kasevich, M., Chu, S.: Atomic interferometry using stimulated Raman transitions. Phys. Rev. Lett. 67, 181 (1991) 65. Sokolov, Y.L., Yakovlev, V.P.: Measurement of the Lamb shift in the hydrogen atom (n = 2). Sov. Phys. JETP 56, 7 (1982) 66. Robert, J., et al.: Atomic interferometry with metastable hydrogen atoms. Europhys. Lett. 16, 29 (1991) 67. Haroche, S., Raimond, J.M.: Manipulation of non classical field states by atom interferometry. In: Berman, P.R. (ed.) Cavity Quantum Electrodynamics, Adv. At. Mol. Opt. Phys., suppl. 2, pp. 123–170. Academic Press, Boston (1994)
1187 68. Taylor, B., Schernthanner, K.J., Lenz, G., Meystre, P.: The van Cittert–Zernike theorem in atom optics. Opt Commun 110, 569 (1994) 69. Köhl, M., Hänsch, T.W., Esslinger, T.: Measuring the temporal coherence of an atom laser beam. Phys. Rev. Lett. 87, 160404 (2001) 70. Pfau, T., et al.: Loss of spatial coherence by a single spontaneous emission. Phys. Rev. Lett. 73, 1223 (1994) 71. Chapman, M.S., et al.: Photon scattering from atoms in an atom interferometer: coherence lost and regained. Phys. Rev. Lett. 75, 3783 (1995) 72. Kurtsiefer, C., et al.: Observation of correlated atom-photon pairs on the single particle level. Phys. Rev. A 55, R2539 (1997) 73. Hackermüller, L., et al.: Decoherence in a Talbot–Lau interferometer: the influence of molecular scattering. Appl. Phys. B 77, 781 (2003) 74. Landragin, A., et al.: Specular versus diffuse reflection of atoms from an evanescent-wave mirror. Opt. Lett. 21, 1591 (1996) 75. Henkel, C., Wilkens, M.: Heating of trapped atoms near thermal surfaces. Europhys. Lett. 47, 414 (1999) 76. Harber, D.M., McGuirk, J.M., Obrecht, J.M., Cornell, E.A.: Thermally induced losses in ultra-cold atoms magnetically trapped near room-temperature surfaces. J. Low. Temp. Phys. 133, 229 (2003) 77. Romero-Isart, O., et al.: Large quantum superpositions and interference of massive nanometer-sized objects. Phys. Rev. Lett. 107, 020405 (2011)
Carsten Henkel Carsten Henkel is Docteur en Sciences at the Université Paris-Sud Orsay. In 1997, he went to the University of Potsdam where he was appointed Professor (apl.) of Quantum Optics in 2015. His research interests are in nano-optics and ultracold atoms.
Martin Wilkens Martin Wilkens received his PhD in Physics from Essen University. He spent his postdoctoral years in Warsaw, Tucson, and Konstanz. He has been Professor of Quantum Theory at the University of Potsdam since 1997. His research interests are in degenerate quantum gases, quantum information processing, and general relativity.
81
Quantum Properties of Light Da-Wei Wang
82
and Girish S. Agarwal
Contents 82.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1189
82.2
Quantization of the Electromagnetic Field . . . . . . 1190
82.3
Quantum States . . . . . . . . . . . . . . . . . . . . . . . 1190
82.4
Field Observables: Quadratures . . . . . . . . . . . . . 1192
82.5
Phase-Space Representations of the Light: P, Q, and Wigner Functions . . . . . . . . . . . . . . . 1192
82.6
Squeezed State . . . . . . . . . . . . . . . . . . . . . . . . 1194
82.7
Detection of Quantum Light by Array Detectors . . 1196
82.8
Two-Mode Squeezed States . . . . . . . . . . . . . . . . 1197
82.9
Quantum Entanglement . . . . . . . . . . . . . . . . . . 1198
82.10
Non-Gaussian Nonclassical States . . . . . . . . . . . . 1199
82.11
Beam Splitter, Interferometer, and Measurement Sensitivity . . . . . . . . . . . . . . . 1200
many of these ideas are applicable to spin systems, twolevel systems, and many other bosonic systems, like ionic motion and mechanical (phonon) degrees of freedom in optomechanical systems. An extensive discussion of the quantumness and its applications in quantum metrology can be found in the book [1]; the summary in this Handbook chapter is tailored along the lines of this book. We start with a historical introduction to the quantum theory of radiation. Keywords
quantum entanglement coherent state squeezed state CAT states Nonclassical light Quantum sensing Standard quantum noise limit Heisenberg limit Detection of quantum fields
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203
82.1 Introduction Abstract
This article focuses on the understanding of issues where the quantumness of the radiation field is important. To this end, we introduce the experimental signatures of the strict quantum character of the radiation field. Besides the fundamental interest in quantumness, we discuss the important consequences of it, especially in beating the standard quantum noise limit. The quantum character of the field can be used for achieving unprecedented sensitivity. This article is focused on the radiation field, but D.-W. Wang () Department of Physics, Zhejiang University Hangzhou, China e-mail: [email protected] G. S. Agarwal Institute for Quantum Science and Engineering, Texas A&M University College Station, TX, USA e-mail: [email protected]
The quantum theory of light has its beginnings in the work of Max Planck in 1900. After a tedious struggle in explaining the black-body radiation, Planck found that in order to fit the thermal radiation curve, he had to assume the exchange of energy between a black body and the radiation fields with an undividable unit of energy, „!, where „ D 1:06 1034 J s is the Planck constant and ! is the circular frequency of the light. However, Planck did not regard light itself as being composed of particles with this unit energy. The idea of light quanta was proposed by Einstein in 1905. In studying the thermal dynamic properties of light, Einstein realized that the thermal dynamic equations of light allow a corpuscular theory of light, i.e., the energy of light is carried in discrete quantized packets. Let us first see how the quantum nature of light affects its thermal statistical distribution. If light behaves like classical waves, the intensity distribution at a certain temperature is 1 I=hI i ; (82.1) e P .I / D hI i
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_82
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where hI i is the average intensity. Quantum theory requires decomposed into plane waves with wave vector k and pothat the probability that the light field contains n photons is, larization s, given by the Bose–Einstein distribution, X Aks A D p "ks eikri!k t C c.c. ; (82.5) n V hni 1 k;s ; (82.2) pn D 1 C hni 1 C hni where the mode functions Rof the plane wave satisfy the or0 thogonal condition, .1=V / V d3 reikrik r D ı.k k0 /. where Planck’s law gives the average photon number hni D The R energy of the electromagnetic fields, U D 1=Œexp.„!=kB T / 1 at the frequency ! with kB the Boltz.1=8 / V ŒE 2 .r; t/ C B 2 .r; t/d3 r can be written in terms mann constant, and T is the temperature. In the large photon of the vector potential, number limit, hni 1, 1 X !k2 (82.6) U D jAks j2 : 1 n 2 c2 ; (82.3) pn exp k;s hni hni which is identical to Eq. (82.1) since hI i / hni. However, in the low-temperature limit, when hni 1 or smaller than 1, pn drops to zero much faster than Eq. (82.1). Another drastic difference is in the fluctuations of the fields. From Eq. (82.1), the variance of the intensity I 2 D hI i2 , while the fluctuation in the photon numbers according to Eq. (82.2) is n2 D hni2 C hni. Einstein called hni2 and hni the particle and wave parts of the fluctuations, respectively. At the low-temperature limit, the fluctuation of the photon numbers is larger than that of the classical statistics. In a remarkable experiment [2], the moments of the Bose–Einstein distribution were measured down to the mean number of the order of unity with deviation in the second moment less than 10%. The nature of the quantization of the electromagnetic field was not fully understood until Dirac found the second quantization after quantum mechanics was formulated by Heisenberg and Schrödinger in 1925–1926. The wave nature of the light is governed by the Maxwell equations, which give the mode function for the electromagnetic modes. In each of the modes, the amplitude of the field is quantized using the fact that photons obey Bose statistics.
What was revealed in Planck’s black-body radiation is that P the above energy should be quantized, U D k;s nks „!k , where nks is the number of quanta in the mode ks. By making comparison between these two expressions of the energy, we realize the correspondence, !k jAks j2 $ nks : 2 „c 2
(82.7)
This is the quantization of the field amplitudes. This quantization is done by introducing the annihilation and creation operators for the electromagnetic field, aks and aks , with the commutation relations h i aks ; ak0 s 0 D ık;k0 ıss 0 ; Œaks ; ak0 s 0 D 0 :
(82.8)
The number operator can be written as,
nks D aks aks :
(82.9)
From Eqs. (82.7) and (82.9), we can relate Aks D p 2 „c 2 =!k aks , p and the vector potential can be rewritten P as A D k;s "ks 2 „c 2 =.!k V /aks eikri!k t C H.c., where H.c. stands for Hermitian conjugate. With A, it is straightforward to write E and B in the form of a and a . Most importantly, the total energy of the electromagnetic fields, 82.2 Quantization of i.e., the Hamiltonian can be written as the Electromagnetic Field X 1 H D „! C : (82.10) n k ks According to the Maxwell equations, which characterize the 2 k;s wave nature of the electromagnetic fields, the electromagnetic fields in free space can be written with the vector potential A,
82.3
1 @A E D ; c @t
B Dr A ;
Quantum States
(82.4) After the field is quantized, it is natural to ask what kind of quantum states photons can possess. In the following, we where E and B are the electric field and the magnetic first introduce several photonic states that people frequently field, respectively. We employ CGS units because the physics encounter, i.e., the Fock states, the coherent states and the community uses these widely. In a volume V , A can be thermal states.
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The Fock state. The quantization of the electromagnetic field is most obviously represented by the quantized value of the number operator a a, which is directly related to the Hamiltonian of the electromagnetic fields through Eq. (82.10). Fock states jni are the eigenstates of the number operators, a ajni D njni :
is Poissonian with the mean value hni D j˛j2 . The variance is equal to its mean value, hn2 i D hn2 i hni2 D hni :
(82.18)
One important property of the coherent state is that they form a complete set, Z 1 (82.11) (82.19) d2 ˛j˛ih˛j D 1 ;
For a single mode, the commutation relation in Eq. (82.8) where ˛ D x C iy and d2 ˛ D dxdy with x and y being real reduces to Œa; a D 1. It is easy to find that ajni is also an numbers. However, different from the complete set formed eigenstate of the number operator, by the number states, the basis states of the coherent states are not orthogonal,
(82.12) a aajni D aa 1 ajni D .n 1/ajni : j˛j2 jˇj2 h˛jˇi D exp ˛ ˇ : (82.20) 2 2 Therefore, ajni D cjn 1i with c being a constant, which can be determined from the inner product hnjaajni D hn The nonorthogonality of the coherent states leads to an p 1jc cjn 1i that c D n. This then leads to important property of the coherent state, namely, we can represent a coherent state in the form of other coherent states, Z an 1 (82.13) jni D p j0i : (82.21) hˇj˛ijˇid2 ˇ : j˛i D nŠ Obviously, there is a lower limit of the number in the number Any coherent state can be generated by the displacement of the vacuum state, states, aj0i D 0 :
j˛i D D.˛/j0i ;
(82.14)
(82.22)
where D.˛/ is the displacement operator Here, j0i is also called the vacuum state, which is also an (82.23) D.˛/ D exp.˛a ˛ a/ : eigenstate of the annihilation operator. Note that h0jH j0i D „!=2 for a single mode. This is called the zero-point energy The displacement operator has the following important propof the mode. It diverges for infinitely many-mode fields. This erties, cannot be measured, and hence no physical consequence exD 1 .˛/ D D .˛/ D D.˛/ ; ists. However, the differences of the zero-point energy are finite and are related to the Casimir force and the van der D .˛/G.a; a /D.˛/ D G.a C ˛; a C ˛ / ; Waals force. D.˛/D.ˇ/ D D.˛ C ˇ/ expŒ.˛ˇ ˛ ˇ/=2 : In the following, we are going to introduce a more general (82.24) form of the eigenstate of the annihilation operator. Here, G is any function of annihilation and creation operaThe coherent state. A coherent state is an eigenstate of the tors. annihilation operator a, aj˛i D ˛j˛i ;
The thermal state. The Fock states and the coherent states are both pure states. In contrast, thermal states are mixed (82.15) states and can only be represented by the density matrices,
and in the basis of the number states, j˛i D
X
ej˛j
2 =2
n
T D n
˛ p jni : nŠ
T D
1 X nD0
hnin ; nŠ
(82.25)
(82.16) where ˇ D 1=kB T . In terms of the Fock states,
The photon number distribution pn D ehni
exp.ˇ„! a a/ ; TrŒexp.ˇ„! a a/
(82.17)
hnin jnihnj ; .1 C hni/nC1
(82.26)
where the average photon number hni D 1=Œexp.ˇ„!/ 1, which is Planck’s law.
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82.4
Field Observables: Quadratures
The above three kinds of states are different in their quantum statistical properties, which can be measured by physical observables of the fields. However, the annihilation and creation operators are not Hermitian or directly measurable. Instead, the measurable quantities in experiments are, besides the photon number distributions, the amplitudes and phases of the fields, e.g., the real and imaginary parts of the electric fields. These are called in-phase and out-of-phase quadratures, which can be measured by homodyne detection. We can define two Hermitian quadrature operators, X and Y, XD
a a a C a : p ;Y D p 2 2i
The commutation relation between X and Y is ŒX; Y D i ;
(82.27)
light can be represented by harmonic oscillations of the fields. Classically, we can describe the center of mass of a particle with its position and momentum. For an ensemble of particles, we can define a probability distribution P .x; p/ (phase-space distribution) to study their statistics. However, in quantum mechanics, x and p do not commute, and thus cannot have definite values at the same time. E. Wigner tried to incorporate quantum mechanics into the classical statistics. By doing this, the noncommutative nature in quantum mechanics is demonstrated by negative values of the Wigner quasi probability. The same problem exists in a probability description of the light field with respect to its amplitude and phase. In this section, we are going to introduce several representations of light in phase space.
P-representation. Glauber and Sudarshan proposed to use the coherent states as a basis to describe the light field [3, 4]. The coherent state basis, even though composed by states (82.28) that are nonorthogonal, can represent any radiation field
analogous to the position-momentum commutation relation. Therefore, we can represent the operator Y with Y D i.@=@X/. The Hamiltonian H D „!.a a C 1=2/ can be rewritten as H D .X 2 C Y 2 /=2, which is similar to the one of the harmonic oscillators. The number state can be represented with variable X in a similar way as in the harmonic oscillator, p 2 n .X/ D hXjni D .2n nŠ /1=2 Hn .X/eX =2 ; (82.29) where Hn .X/ is the Hermite polynomial of degree n. The variance of the quadratures reflects the fluctuations in the amplitudes and phases of the fields,
Z P .˛/j˛ih˛jd2˛ ;
D
(82.31)
R with the normalization condition P .˛/d2 ˛ D 1. This representation is called Glauber–Sudarshan P -representation. The P .˛/ is not necessarily positive, and thus cannot be simply regarded as a probability. For the Fock states, D jnihnj, the P -function is highly singular, Pn .˛/ D ej˛j
2
@2n 1 ı 2 .˛/ : nŠ @˛ n @˛ n
(82.32)
For a coherent state j˛0 i, the P -function is completely classical,
ha2 i C ha2 i 1 ha C a i2 C hni C : (82.33) Pc .˛/ D ı .2/ .˛ ˛0 / : 2 2 2 ha2 i C ha2 i 1 ha a i2 Likewise for a thermal state T D exp.ˇ„! a a/= hY 2 i D C hni C C : 2 2 2 (82.30) TrŒexp.ˇ„! a a/, the P -function is classical [3], 1 j˛j2 For Fock states or thermal states, hX 2 i D hY 2 i D hni C PT .˛/ D exp : (82.34) hni hni 1=2. For coherent states, hX 2 i D hY 2 i D 1=2. The coherent states are the minimum uncertainty states as the product The P -representation is convenient to evaluate normally orhXY i takes the minimum value 1=2. dered quantities, such as hX 2 i D
Z
82.5
Phase-Space Representations of the Light: P, Q, and Wigner Functions
ham an i D
P .˛/˛ m ˛ n d2 ˛ :
(82.35)
We will use the singularity or negativity of the P -function to The quantum nature of light is evidently represented by characterize the strict quantum nature of the field. This is bethe number states. On the other hand, in classical elec- cause the experimentally measured observables correspond m tromagnetic theory, according to the Maxwell equations, to the normally ordered moments like ha an i.
82
a
Quantum Properties of Light
1193
b W1(α)
WT(α) 0.25
0.3 0.2
0.2
0.1 0
0.15
−0.1 −0.2
0.1
−0.3 −0.4
0.05
−0.5 0 −2
−1
0 Re(α)
1
2
−2
−1
0 Im(α)
1
2
−2
−1
0 Re(α)
1
2
−2
−1
0 Im(α)
1
2
Fig. 82.1 The Wigner function of a thermal state (a) and single-photon state (b)
Q-representation. Different from the P -functions, the Q- obtained by using different characteristic functions, such as function has the properties of a classical probability distribu- the Wigner function in the following. tion, The Wigner representation. We next discuss the Wigner 1 (82.36) function. This function has great practical utility as it alQ.˛/ D h˛j j˛i : ways exists and can be experimentally measured, and, thus, Q-functions are convenient to evaluate antinormally ordered nonclassical aspects are easily accessible via the Wigner function. The Wigner function is defined as moments of a and a , Z Z 1 1 han am i D Tr an j˛ih˛jam d2 ˛ W .˛/ D 2 d2 ˇTrŒ D.ˇ/eˇ˛ Cˇ ˛ ; (82.41) Z 1 D ˛ n ˛ m h˛j j˛id2 ˛ where D.ˇ/ is the displacement operator. The characteris Z tic function of the Wigner function is CW D TrŒ D.ˇ/. For D ˛ n ˛ m Q.˛/d2 ˛ : (82.37) a coherent state ji, the Wigner function is 2 By substituting Eq. (82.31) in Eq. (82.36), we obtain the (82.42) Wc .˛/ D exp.2j˛ j2 / ; relation between the P and Q-functions, Z which is a Gaussian distribution with a width 1=2. The 1 Q.˛/ D (82.38) Wigner function is related to the P -function, P .ˇ/ exp.j˛ ˇj2 /d2 ˇ : Z The Q-function of a thermal field is 2 (82.43) W .˛/ D P ./ exp.2j˛ j2 /d2 : 2 1 j˛j QT .˛/ D exp ; (82.39) .1 C hni/ 1 C hni The Wigner function of a thermal state is which is Gaussian. 1 j˛j2 WT .˛/ D exp ; (82.44) The P and Q-functions can be written as integrals of their .hni C 1=2/ hni C 1=2 characteristic functions, Z which is Gaussian as shown in Fig. 82.1a. 1 P .˛/ D 2 d2 ˇeˇ˛ Cˇ ˛ CP .ˇ/ ; The Wigner function of a Fock state is more complicated, Z 1 2 2 Q.˛/ D 2 d2 ˇeˇ˛ Cˇ ˛ CQ .ˇ/ ; (82.40) (82.45) Wn .˛/ D .1/n e2j˛j Ln .4j˛j2 / ;
and CQ .ˇ/ D where Ln .x/ is the n-th order Laguerre polynomial. Obviwhere CP .ˇ/ D Tr.eˇa eˇ a / ˇ a ˇa e /. Other phase-space representations can be ously, Wigner functions can be negative. Shown in Fig. 82.1b Tr.e
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D.-W. Wang and G. S. Agarwal
is the Wigner function of the single-photon number state W1 .˛/ D
2 2j˛j2 .4j˛j2 1/ ; e
The expansion of the squeezed state in the Fock-state basis is given by
(82.46) ji D p
1 X
1 cosh r
p ein .tanh r/n
nD0
.2n/Š j2ni ; nŠ2n
(82.51)
which is clearly maximally negative at j˛j D 0. For an arbitrary state, the Wigner function at the zero where all the Fock states with odd numbers are missing. point is used in many tests of nonclassicality, The action of the squeezing operator to the annihilation operator yields 1 2 2X n
nn .1/ D hexp.i a a/i ; (82.47) W .0/ D nD0 a./ D S ./aS./ D a cosh r C a ei sinh r : (82.52) which is related to the average value of the parity operator or Note that Eq. (82.52) also represents the Bogoliubov transexp.i a a/. For a single-photon Fock state W .0/ D 2= . formation. The squeezed state is the vacuum state of the Another important property is that the Wigner function is transformed annihilation operator square-integrable, a./ji D 0 : (82.53) Z Tr 2 D
d2 ˛W 2 .˛/ 1 :
(82.48)
To see the effect of squeezing, let us define a general quadrature X ,
The negativity of the Wigner function at the origin has been used to test the nonclassicality. Haroche and collaboraaei C a ei X D ; (82.54) p tors [5] measured the negativity of the Wigner function for 2 the Fock state j1i using cavity QED. More recently, this was measured for ionic motion (boson) [6]. Further, the negativ- which is conjugate to X C =2 and satisfies the commutation ity of the Wigner function for a set of states equivalent to relation ŒX ; X C =2 D i. The variance of X=2 of a squeezed non-Gaussian states including Fock states is measured [7]. state ji is
82.6
.X=2 /2 D
Squeezed State
One of the most important nonclassical fields is the squeezed state, which was discovered by Yuen [8]. It only gained popularity after Caves proposed its great advantage in measurements beyond the standard quantum noise limit (SQL) [9]. The effectiveness of the squeezed states was first demonstrated by Kimble’s group by a 3-dB sensitivity enhancement [10]. We will discuss this issue later. Here, we first discuss the measurable properties of the squeezed states.
1 2r e ; 2
(82.55)
while for the conjugate quadrature, .X=2C =2 /2 D .1=2/e2r . Therefore, the squeezed state has a squeezed quadrature X=2C =2 and a stretched quadrature X=2 . To see the nonclassicality of the squeezed state, we introduce the squeezing parameter [8] S D hW.X /2 Wi hX i2 ;
(82.56)
where W W stands for the normal ordering of the operators between them; S is related to the variance of the quadrature by Squeezing operator. We have shown that the coherent state S D .X /2 1=2. The negative value of S is a sufficient can be generated by applying a displacement operator on the condition for nonclassicality. Obviously, for the squeezed vacuum state. The squeezed state ji can also be generated state, S=2 =2 D .1 e2r /=2 < 0, which is a signature by applying a squeezing operator S./ on the vacuum state, of the nonclassicality of the squeezed vacuum. The representation of the squeezed state in the quadratureji D S./j0i ; (82.49) X space can be obtained from Eq. (82.53),
where .X/ D 2
2
S./ D exp.a =2 a =2/ ; i
and D re is a complex number.
(82.50)
.0/
1 2 cosh r ei sinh r exp X ; (82.57) 2 cosh r C ei sinh r
which is a Gaussian wavefunction with chirped width (due to the complex parameter in the exponent).
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Quantum Properties of Light
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a Wξ (α)
b Wξ (α)
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 −5
5 0 Re(α)
5
−5
0 −5
5 0 Re(α)
0 Im(α)
5
−5
0 Im(α)
82
Fig. 82.2 The Wigner function of a squeezed state with r D 1, D 0 (a), and D (b)
Phase-space representations. While the P -function of the The squeezed thermal state. The squeezed thermal state squeezed states does not exist, the Q-function and Wigner can be obtained by applying the squeezing operator to the thermal state density matrix, function are tanh r i 2 1 i 2 2 exp .e ˛ C e ˛ / j˛j ; Q .˛/ D cosh r 2 (82.58)
D S./ T S ./ :
(82.63)
The Wigner function of the squeezed thermal state is and 2 W .˛/ D exp.2j˛ cosh r ˛ ei sinh rj2 / :
W .˛/ D (82.59)
2 .2hni C 1/ 2 i 2 exp j˛ cosh r ˛ e sinh rj : 2hni C 1 (82.64)
The Wigner functions of a squeezed state with the same squeezing amplitude but different squeezing angles are plotted in Fig. 82.2. The Wigner function for the squeezed states is Gaussian and does not exhibit a nonclassical nature, which Nonclassicality. To characterize the nonclassicality of the squeezed states, we need to introduce a physical quantity that is truly reflected by the P -function. can tell us its quantum feature. Mandel observed that the photon number distribution of the coherent state is Poissonian The squeezed coherent state. The squeezed coherent state and realized that a state with narrower variance is the feature is defined by first squeezing the vacuum state and then disof a nonclassical field. He introduced the parameter [11], placing it, j; ˇi D D.ˇ/S./j0i :
(82.60)
QM D
h.a a/2 i ha ai2 ha ai ha ai
(82.65)
The annihilation operator transforms under these two operato quantify the nonclassicality of the field [12–15]. The P tions, function of a coherent field is a delta function, which is in accord with the classical probability distribution. The QM of a.; ˇ/ D S ./D .ˇ/aD.ˇ/S./ a classical field is zero. In contrast, there are singularities in i (82.61) D a cosh r C a e sinh r C ˇ : the P -function of the number states. The QM of a number state is 1. The negativity of QM is a sufficient condition for The Q-function and Wigner function of j; ˇi are a nonclassical field. While the Mandel QM factor is super-Poissonian for Qˇ .˛/ D Q .˛ ˇ/ ; a squeezed state, QM ./ D 1 C 2 sinh2 r D 1 C 2ha ai, the Wˇ .˛/ D W .˛ ˇ/ : (82.62) QM factor for a squeezed coherent state, e.g., with phase
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D.-W. Wang and G. S. Agarwal
yields statistics resembling a binomial distribution [17],
γ 1.0 3.5
0.9
3.0
0.8 0.7
2.5
0.6
2.0
0.5
1.5
0.4
1.0
0.3
0.5
0.2
NŠ hW N k .1 /k Wi ; kŠ.N k/Š
–0.5 0
1
2
3
4
5 β
(82.67)
where W W denotes the normal ordering, k is the number of recorded clicks, and O describes the measurement operator for a click of a single on–off detector including its imperfections. In fact, coherent light, describing a classical radiation field, yields a true binomial distribution. Consequently, nonclassical light can be identified via the binomial parameter as follows [18],
0
0.1 0
ck D
QB D N
.k/2 N N k.N k/
1 < 0 ;
(82.68)
where kN and .k/2 are the mean value and the variance of the click-counting distribution ck , respectively. The binomial Fig. 82.3 The Mandel QM -factor as a function of r and ˇ for the parameter generalizes the Mandel Q-parameter [11] to the squeezed coherent state j; ˇi, D case of measurements with on–off detectors. Moreover, in the limit of an infinite number of detectors, the binomial click statistics converges to the actual Poisson distribution of the D , photons. 4 2 2 2 As an example, let us consider the click-counting statissinh r C 2ˇ sinh r C sinh r 2 2 tics of an n-photon state measured with N on–off detectors cosh r 2ˇ sinh r cosh r ; (82.66) QM .; ˇ/ D with a quantum efficiency . This scenario yields QB D ˇ 2 C sinh2 r .N 1/.Œ1 2=N n Œ1 =N 2n/.1 Œ1 =N n/1 Œ1 =N n, which is negative for all efficiencies > 0 (N 2 can be negative, as shown in Fig. 82.3. and n 1). This means that an n-photon state represents a sub-binomial, i.e., nonclassical, state of light [18]. For a large number of detectors (N 1), we approach the wellknown Mandel parameter for photons, QB .1 1=N /, 82.7 Detection of Quantum Light as was mentioned previously. This example demonstrates the by Array Detectors direct application of the method for verifying quantum propIn this section, we discuss only the array detectors that erties of quantized radiation fields, as shown in Fig. 82.4a. have recently become in vogue in the study of the statis- Furthermore, the click-counting theory can be directly emtics of quantum fields although photon counting devices have ployed to analyze data and characterize experiments [19–22]. been extensively used in past. The characterization of quantum light requires the detection of photons. However, ideal Efficiency η photon-number-resolving measurement devices are typically 1.0 0.8 0.6 0.4 0.2 0 0 not available. For example, avalanche photodiodes in the n=5 n=4 Geiger mode can only distinguish between the presence n=3 –0.2 (“click” or “on”) and the absence (“no click” or “off”) of photons. To overcome this deficiency, different measurement n=2 –0.4 schemes have been proposed and implemented. One of the most successful approaches is the multiplexed detection of –0.6 photons [16]. In such a measurement layout, a beam of light is split into N beams with equal and reduced intensities. Sub–0.8 sequently, experimentally accessible on–off detectors, such as avalanche photodiodes, can be used to measure the result–1.0 n=1 ing quantum light fields. Binomial parameter Q B Typically, the photon-number distribution takes the form of a quantum version of a Poisson distribution. However, Fig. 82.4 QB < 0 certifies the nonclassicality of n-photon states as the theory of multiplexing layouts with N on–off detectors a function of for only two on–off detectors (N D 2)
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Besides the applications to certify quantum properties we obtain the two-mode squeezing operator based on the photon-number statistics, the click-counting 2 2 theory additionally allows for more advanced techniques to exp.a b ab/ D exp d d 2 2 characterize and manipulate quantum light. For example, 2 employing multiplexing detectors in interferometers [23] un2 exp c C c ; (82.73) 2 2 covers the click-counting counterpart of squeezing [24]. In addition, click-counting detectors can be used to implement which is a product of two single-mode squeezing operators. advanced multiphoton heralding, addition, and subtraction The Q-function of the two-mode squeezed states is protocols to generate nonclassical quantum states [25]. For instance, the heralding is based on the conditional click1 2 counting statistics of a two-mode light field, which has been Q .˛; ˇ/ D jh˛; ˇjij experimentally compared with the joint click-counting statis1 D 2 tics [26]. cosh
ri exp .e tanh r˛ ˇ C c.c./ j˛j2 jˇj2 ; (82.74)
82.8
Two-Mode Squeezed States
which is a Gaussian distribution of Re˛, Reˇ, Im˛, and Imˇ. The Wigner function of the two-mode squeezed vacuum The squeezing operator can be generated in parametric is downconversion (PDC) into the same optical modes. More
4 generally, if the generated modes are different, the PDC proW .˛; ˇ/ D 2 exp 2j˛ cosh r ˇ sinh rei j2 cess results in the two-mode squeezing. 2j˛ sinh rei C ˇ cosh rj2 : (82.75) The two-mode squeezing operator. The two-mode squeezed state can be generated from the operator The quadrature representation is S./ D exp.a b ab/ ;
(82.69)
where D rei is a complex number. We apply S./ to the two-mode vacuum state j0; 0i, 1 exp.ei tanh ra b /j0; 0i cosh r 1 1 X in e .tanh r/n jn; ni : D cosh r nD0
ji D S./j0; 0i D
(82.70)
.Xa ; Xb / D hXa ; Xb ji D
1 1 1 X in e tanhn r n p cosh r nD0 2 nŠ 2
2
Hn .Xa /Hn .Xb /e.Xa CXb /=2 1 Dp .1 2 / cosh2 r 2Xa Xb .Xa2 C Xb2 /2 exp 1 2 1 2 2 .Xa C Xb / ; 2
(82.76)
The interesting property of the two-mode squeezed states where D ei tanh r. This distribution of the quadratures is is that the two modes must contain the same number of plotted in Fig. 82.5. photons. The operators a and b transform under two-mode squeezing Cauchy–Schwartz inequality. The nonclassicality of the two-mode squeezed states can be measured by the Cauchy– Schwarz inequalities. For two well-defined functions f a./ D S ./aS./ D a cosh r C b ei sinh r ; b./ D S ./bS./ D b cosh r C a ei sinh r ; (82.71) and g, 2the Cauchy–Schwarz inequality is hf f ihg gi jhf gij if the average is made with positive probability distributions. ConsideringR the P -representation of the which is a two-mode Bogoliubov transformation. By intro- two-mode density matrix D P .˛; ˇ/j˛; ˇih˛; ˇjd2 ˛d2 ˇ, ducing two linearly combinational modes of a and b, letting f D ˛ ˛ and g D ˇ ˇ, the Cauchy–Schwarz inequality in phase-space distribution is ab aCb (82.72) dD p ; cD p ; (82.77) ha2 a2 ihb 2 b 2 i jha ab bij2 : 2 2
82
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D.-W. Wang and G. S. Agarwal
|Ψ(Xa, Xb)|
Entanglement has been widely used in quantum information [42–44]. We briefly introduce the basic properties of entanglement in the following. Using the Peres–Horodecki criterion [45, 46] on the negativity of the partial transpose of the density matrix, we can measure the entanglement by the logarithmic negativity,
2
0.4 0.3 0.2
EN . / D log2 Œ1 C 2N. / :
0.1
(82.80)
Here, N. / is the absolute value of the sum of the negative eigenvalues of the partial transpose density matrix PT , 0 0 which is obtained by taking the transpose of the operators Xa Xb associated with the mode b. In the following, we calculate 5 −5 the parameter EN . / for the state in Eq. (82.70). The density Fig. 82.5 Quadrature distribution of a two-mode squeezed state matrix is 0 −5
5
j .Xa ; Xb /j2 for r D 1 and D 0
A violation of this inequality can be regarded as a sufficient condition of nonclassicality. We define a parameter [27–29] p ha2 a2 ihb 2 b 2 i 1 I D ha ab bi
D
X
cn cm jnnihmmj :
(82.81)
nm
Its partial transpose is (82.78)
PT D
X
cn cm jnmihmnj :
(82.82)
nm
to measure this deviation. The quantity I has been used to measure the nonclassicality of both photons [30–32] and matter waves [33]. More recently, such a violation of non- This matrix can be block-diagonalized within the basis states classicality using Eq. (82.78) was studied in phonon–photon jmni and jnmi for any combination n ¤ m, and the eigeninteractions [34]; obviously, for a coherent state I D 0. For values are ˙jcn cm j. The negativity is the two-mode squeezed states, 0 1 !2 X X EN . / D log2 @1 C jcn cm jA D log2 jcn j : ha ai (82.79) I D 2 D sech2r < 0 : n n¤m h.a a/ i (82.83) Therefore, the two-mode squeezed state is essentially nonFor the two-mode squeezed classical. .tanh r/n = cosh r, we obtain
82.9
Quantum Entanglement
The two-mode states that cannot be written as a direct product of two quantum states, such as the two-mode squeezed states ji ¤ ja ijb i, are called entangled states. Historically, entanglement was proposed by Einstein, Podolsky, and Rosen to challenge the completeness of quantum theory [35]. The entangled nonseparable states indicate that the measurement of one particle will project the other particle to a certain quantum state, even if the other particle is far away. They suggested the existence of hidden variables to explain the “spooky action at a distance” between entangled particles. Bell proposed an inequality to distinguish the predictions between the theories of hidden variables and quantum entanglement [36]. Many experiments have been done, and the results overwhelmingly support the quantum theory [37–41].
vacuum state
EN . / D log2 .e2r / :
jcn j D
(82.84)
The entanglement is measured by the squeezing parameter. The reduced density matrix of mode a after tracing over the mode b is
.a/ D Trb jihj D
1 1 X tanh2n r jnihnj ; cosh2 r nD0
(82.85)
which is a thermal state. Therefore, by measuring one mode, we lose the nonclassicality of the other mode. In particular, if we measure b and obtain a Fock state jni, we immediately know that the a-mode is also in the state jni. This is extensively used in experiments of generating single photons, which is called the heralded source of single photons.
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Quantum Properties of Light
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Parametric downconversion. We can use parametric down- Cat states. In 1935, Schrödinger proposed a thought experconversion to generate two-mode squeezed states by consid- iment, which contains a radioactive atom, a vial of cyanide, and a cat. The electron emitted during the decay of the ering a three-wave mixing process with the Hamiltonian atom can trigger some mechanical device, which finally (82.86) results in the release of the cyanide and kills the cat. HowH D „.Gf a b C H.c./ ; ever, a quantum mechanical description of the atom allows where the frequency of the mode f is !f D !a C !b , G is a superposition of the decayed and undecayed states. Conserelated to .2/ , and the geometric factors of the crystal. If quently, the cat is in a superposition of dead and alive states. !a D !b , the Hamiltonian is reduced to one containing the This later became known as Schrödinger’s cat and led to a2 -term, which describes single-mode squeezing. a lively debate over the completeness of quantum mechanThe squeezing parameter r D gt is proportional to the ics. interaction time t, and g is an effective coupling constant. When the excitation number is large, a classical state of The evolution of the two mode operators is described by a harmonic oscillator is naturally described by a coherent Eq. (82.71). The evolution of the quadratures of the comstate. A superposition of two different coherent states is, binational mode operators c and d in Eq. (82.72) obeys, therefore, thought of as a superposition of two classical states and traditionally called the cat states, d ˙d d.t/ ˙ d .t/ D e˙r p ; p
2 2 (82.91) j˛0 ic D N 1 j˛0 i C ei j˛0 i ; c ˙ c c.t/ ˙ c .t/ D e r p : (82.87) where N D .2 C 2e2j˛0 j2 cos /1=2 is a normalization factor. p 2 2 Typically, a nonlinear medium like the Kerr medium can proAfter having passed through the crystal, the operators are ob- duce such cat states provided that the nonlinearity is strong. tained by scaling the ones input into the crystal. No quantum The Kerr medium can also produce a superposition of many noise terms are added. This property is useful in phase- coherent states lying on a circle [49]. The strong Kerr intersensitive amplifiers, since one quadrature is amplified, while action can be realized in cavity QED and related setups. Such the other quadrature is attenuated. The signal-to-noise ratio cat states have been created in trapped ions [6], and cavof the quadrature, ity quantum electrodynamic systems [50] with a dispersive cavity [51]. Multicomponent cat states as a superposition of hXd .t/i hXd i q Dq ; (82.88) several coherent states have also been realized in superconhXd2 .t/i hXd .t/i2 hXd2 i hXd i2 ducting qubit systems [52, 53]. The coherent superposition of the two states results in inp where Xd D .d C d /= 2 does not change with time. terference of the wavefunction in the quadrature space, Type II parametric downconversion is widely used to genjc .x/j2 D jhxj˛0 ic j2 erate entangled photons. The Hamiltonian can be written as D N 2 1=2 i1 i2 h (82.89) H D i„g.e aH bV C e aV bH H.c./ ; 2 2 e.xx0 / C e.xCx0 / i where the photons with orthogonal polarizations travel in dif2 2 C2e.x Cx0 / cos. 2y0 x/ ; (82.92) ferent directions. To first order in the coupling, the photons generated are entangled, p where we have written ˛0 D .x0 C iy0 /= 2. The oscillation jH1 ; V2 i C ei jV1 ; H2 i of the probability appears in the complementary way of x ; (82.90) j i D and y. When y0 D 0, c .x/ does not oscillate, but the wave2 function with the conjugate variable c .y/ oscillates. where is a phase factor. Such entangled states have been The nonclassicality of the cat states can be shown by its realized in experiments for up to 10 photons [47, 48]. CurWigner function, rently, the entanglement is widely used in the fields of quantum information science and quantum technology. 2N 2 Wc .˛/ D h 2
82.10
Non-Gaussian Nonclassical States
e2j˛˛0 j C e2j˛C˛0 j
2
i 2 C2e2j˛j cos. C 4Im˛0 ˛/ ;
(82.93) In this section, we introduce the quantum properties of two important non-Gaussian nonclassical states: the cat state and where the oscillation of the cosine function results in negathe single-photon added state. tivity of the Wigner function, as is shown in Fig. 82.6.
82
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D.-W. Wang and G. S. Agarwal
For a photon-added coherent state, the result is
Wc(α) 0.6
Wc.a/ .˛/ D
0.4 0.2
2 .1 C j˛0 j2 /
4j˛j2 2.˛ ˛0 C ˛˛0 / 1 C j˛0 j2 2
e2j˛˛0 j ;
(82.98)
0 −0.2 −0.4 −5
0 Re(α)
5
0 Im(α)
−5
5
Fig. 82.6 The Wigner function of a cat state with ˛0 D 2 and D 0
Single-photon-added states. If we add a photon to a classical state, we can generate a nonclassical state [54]. A simple example is the vacuum state. By adding a photon, it becomes a single-photon state, which is highly nonclassical. The density matrix of a single-photon-added state is related to the original density matrix by
.a/ D
a a : Tr.a a/
which can be negative. The quadrature representation of the single-photon-added coherent classical state is p hxja j˛i 1 2x 2˛ .a/ Dp p c .x/ ; (82.99) c .x/ D p 2 1 C j˛j2 1 C j˛j2 p which has a p zero at x D ˛= 2. Here we have used a D .x @=@x/= 2. The single-photon-added coherent states [56–58] and thermal states [59] have been realized in experiments, and several properties, as originally predicted [54, 55], have been measured. The negativity of the Wigner function mentioned above has been tested using the photon addition or subtraction on Gaussian states [60].
(82.94)
82.11 Beam Splitter, Interferometer, and Measurement Sensitivity The Q-function and P -function are related to the original one by
Beam splitter. At a beam splitter as shown in Fig. 82.7, the input modes a and b are transformed to the output modes c and d by
2
Q.a/ .˛/ D
h˛ja aj˛i j˛j D Q.˛/ ; haai haa i
P .a/ .˛/ D
ej˛j @2 2 P .˛/ej˛j : haa i @˛@˛
2
(82.95)
It is clear that the Q-function can become zero (for the vacuum state) and the P -function can become negative, even if the original functions are classical. For example, it is easy to calculate the P -function of the single-photon-added thermal state [55], 1 C hni hni .a/ 2 (82.96) PT .˛/ : j˛j PT .˛/ D hni2 1 C hni
c d
!
t1 D r1
r2 t2
!
! ! a a ; DU b b
(82.100)
where r1;2 and t1;2 are the two reflection and transmission coefficients at the two sides of the beam splitter. The lossless beam splitter has the requirement that U is unitary, which leads to, jt1 j2 C jr1 j2 D jt2 j2 C jr2 j2 D 1; t1 r2 C r1 t2 D 0 : (82.101)
.a/
For j˛j2 < hni=.1 C hni/, PT .˛/ < 0. Fig. 82.7 Beam splitter The P -function for a single-photon-added coherent state is singular. It is more convenient to use the Wigner function to characterize its nonclassical behavior. The general relation between the Wigner functions after adding the single photon is, 1 1 @ 1 @ .a/ ˛ ˛ W .˛/ : W .˛/ D haa i 2 @˛ 2 @˛ (82.97)
d
r 2, t 2
a
c
r1, t1
b
82
Quantum Properties of Light
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D1
No. of coincidence counts in 10 min 1000
aout
800 bout
C 600
D2
B
C'
φ
M2
BS2
400 200 0
260
280
300
320 340 360 Position of beam splitter (μm)
B'
A
a
A'
Fig. 82.8 The second-order correlation function as a function of the time delay (controlled by the position of the beam splitter) between the two single photons. Adopted from [61]
For a symmetric beam splitter, r1 D r2 , t1 D t2 , we can rewrite U as ! cos i sin : (82.102) U D i sin cos
M1
BS1
82
b
Fig. 82.9 Mach–Zehnder interferometer. Adopted from [1]
as shown in Fig. 82.9. In one of the two split light paths, we have a tunable phase . The output fields at detectors D1 and D2 are, a C ib 1 i b C ia aout .D1 / D p p ie p When D =2, it is a 50–50 beam splitter. 2 2 2 A remarkable feature of the beam splitter is the Hong– i=2 Ou–Mandel effect [61]. Two photons incident at a 50–50 / a sin C b cos ; D .ie 2 2 beam splitter simultaneously bunch together to come out b C ia a C ib 1 from either one of the output modes. This can be easily seen bout .D2 / D p ei p i p by 2 2 2 i=2 / a cos b sin : (82.104) D .ie 1 2 2 j1; 1iout D .c C id /.d C ic /j0i 2 i If one of the input modes, e.g., b, is the vacuum state, the (82.103) D p .j2; 0i C j0; 2i/ : output photon numbers are 2 Two single photons get entangled to become a two-photon NOON state [62] at a beam splitter, i.e., two detectors cannot click at the same time, if the two single photons arrive at the beam splitter simultaneously. Note that a NOON state is one in which N photons can exist either in one mode orpthe other mode, i.e., it has the structure .jN; 0i C j0; N i/= 2. The second-order correlation function g .2/ ./ as a function of the time delay between the two single-photon pulses was measured in the original Hong–Ou–Mandel experiment, as shown in Fig. 82.8. However, if n > 1, we cannot obtain an n-photon NOON state, although the output state is still entangled. Mach–Zehnder interferometer. The Mach–Zehnder interferometer is composed of two successive beam splitters. The two output modes of the first beam splitter are directed to the two input modes of the second beam splitter by two mirrors,
ha ai ; 2 hbout bout i D cos2 ha ai : 2
haout aout i D sin2
(82.105)
Therefore, the phase can be measured by the detectors. Sensitivity of the interferometer. Now we calculate the sensitivity of the Mach–Zehnder interferometer with respect to the phase . We take the signal to be the difference between the intensities of the aout and bout modes. We assume that mode a is in a coherent state j˛i, and the mode b is either in vacuum or squeezed vacuum state, then
S D hbout bout i haout aout i D .hb bi j˛j2 / cos ı.j˛j2 hb bi/ ;
(82.106)
1202
D.-W. Wang and G. S. Agarwal
for small ı around D =2, i.e., D =2 C ı. We next calculate the noise in the measured signal .S/2 D hS 2 i hSi2 :
(82.107)
|β² b0 Pump
a1
χ
OPA
bf
a2
Db
OPA
b1 a0 b2 The calculation based on the properties of the squeezed vacaf |α² Da uum and coherent states shows that
.S/2 D j˛j2 h.bei C b ei /2 i C sinh2 r Fig. 82.10 Mach–Zehnder interferometer with two OPAs as the beam
2 2r splitter 2 i ˛ D j˛je : D j˛j e C sinh r ; (82.108) Instead of half-reflection mirrors, we use two optical paraWe chose the phases such that the quadrature bei C b ei metric amplifiers (OPAs) (Fig. 82.10) to mix the fields in the is squeezed. Thus, the signal-to-noise ratio becomes two interfering paths. The same pumping field pumps the two OPAs. The idler fields and signal fields after the first OPA is, S hb b a ai Dp N h.b b a a/2 i hb b a ai2 a1 D a0 C b0 ;
D
.j˛j2 sinh2 r/ı ; .j˛j2 e2r C sinh2 r/1=2
b1 D b0 C a0 ;
(82.109)
(82.112)
with D cosh r and D ei sinh r. The object in one of the which for r D 0 is .S=N / D j˛jı leading to the shot noise interfering paths brings a phase factor, a2 D ei a1 and b2 D limit or the standard quantum noise limit (SQNL) of the b1 . The second OPA transforms a2 and b2 to phase sensitivity ı D 1=.j˛j/. Clearly, the above equation, af D a2 b2 ; Eq. (82.109), shows that with squeezed light applied to one of the ports, the phase sensitivity improves considerably (82.113) b D b a : f
1 ı j˛j er
2
2
if j˛j2 sinh2 r :
(82.110) There is a relative phase between the two OPA transformations. The relation between the input fields and output fields is We say that we have sensitivity better than the SQNL. 2r D 1=2, the SQNL is For 3-dB squeezing p of the field e beaten by a factor 2. This was the proposal of Caves [9], first experimentally implemented by Kimble [10]. The great utility of using squeezed light in advanced LIGO has been extensively considered [63]. Schnabel’s group reported a direct measurement of 15-dB squeezed vacuum states of light at 1064 nm from a doubly resonant, type I optical parametric amplifier operated below threshold. They managed to push down the noise levels from various contributing sources [64].
af D ei .Aa0 C Bb0 / ;
bf D Ab0 C Ba0 ;
(82.114)
where A D 2 jj2 ei and B D .1 ei /. Since jAj2 jBj2 D 1, the input fields and the output fields are related by a Bogoliubov transformation, which is dependent on . If the input field is a vacuum state, we obtain,
haf af i D hbf bf i D jBj2 :
(82.115)
Heisenberg limited sensitivity. Thus, the discussion above We define the signal as shows that the SQNL can be beaten by illuminating the vacuum port in conventional interferometer by squeezed vacS haf af C bf bf i D 8jj2 jj2 sin2 : (82.116) 2 uum. In addition, we can have the interferometer where both input ports are illuminated with two-mode squeezed The variance of the signal is vacuum. This leads to several advantages, as shown both theq p oretically [65] and experimentally [66]. Besides beating the S D h.af af C bf bf /2 i S 2 D 2 jBj2 C jBj4 : SQNL it is also possible to achieve Heisenberg limited sen(82.117) sitivity, i.e., ı
1 j˛j2
(82.111)
At the point D 0, we obtain the sensitivity D
by using the SU(1,1) interferometers [67–69].
S 1 D : j@S=@j 2jj
(82.118)
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Quantum Properties of Light
Since the average photon number in each OPA arm is around jj, this result yields the Heisenberg limited sensitivity. This interferometer was realized by Jing et al. [70]. The SU(1,1) interferometer and its variants are currently gaining momentum in the ultrasensitive measurements of phase [71, 72]. Besides being of great utility, the two-mode squeezed states in metrology may also turn out to be of special importance in bioimaging. Two-photon excitation microscopy (TPEM) is the most widely used technique for optically imaging tissues at depths up to several cell layers below the surface. However, because two-photon absorption is such a weak effect, and scattering in biological tissues is so strong, TPEM requires excitation with high-power lasers that can damage or disrupt biological function. It is known that the use of entangled light leads to two-photon absorption that is proportional to intensity [73–75]. Goodson’s group reported that two-photon absorption in some molecules can be significantly enhanced when the two incident photons are energy-time entangled with one another [76]. Clearly, the real advantages of entangled photon excitation are just beginning to appear now [77, 78] and should have far-reaching consequences for bioimaging. We mention that the nonclassical states for atoms like squeezed states and cat states have been introduced [79–81]. The review by Pezzè et al. [82] discusses in detail the metrological applications of the nonclassical states of atoms. We also note that many of the ideas of nonclassicality from the radiation field are being applied to general bosonic degrees of freedom. An important example of mesoscopic systems is the optomechanical interactions involving mechanical mirrors in high-quality cavities. Squeezing of mechanical mirrors and Fock states of mirrors have been realized [83, 84]. The entanglement of mirrors has also been reported [85, 86]. The field of optomechanics is likely to remain very active in the nonclassical properties of mechanical mirrors. Acknowledgments GSA thanks Barnabas Kim and Jan Sperling for help in preparing this article. He also thanks his collaborators who worked on nonclassical aspects of the radiation fields, two-level systems, and general bosonic systems. DWW thanks Wei Feng for technical help and support. The work of GSA is supported by the Air Force Office of Scientific Research [Award Nos. FA 9550-18-1-0141, FA 9550-20-1-0366] and The Welch Grant award No. [A-1943-20180324]. DWW is supported by National Natural Science Foundation of China [No. 11874322] and the Basic Science Research Funding of Zhejiang University.
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Da-Wei Wang Da-Wei Wang obtained his PhD from the Chinese University of Hong Kong in 2012. He worked at Texas A&M University as a postdoc, Research Assistant Professor, and Research Associate Professor from 2012 to 2017. He joined Zhejiang University, China, in 2017 and works on quantum optics and quantum information with atoms and superconducting circuits. Girish S. Agarwal Girish S. Agarwal received his PhD in 1969 from Rochester. He is Fellow of the Royal Society UK and author of a book and 700 research papers. His work has been recognized by receiving the Max-Born Prize of the Optical Society of America, the Physics Prize of the World Academy, a Humboldt Research Award, and honorary doctorates.
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83
Entangled Atoms and Fields: Cavity QED Qiongyi He
, Wei Zhang
Contents 83.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1208
83.2 83.2.1 83.2.2 83.2.3
Atoms and Fields . . . . . . . . . . . . Two-Level Atoms . . . . . . . . . . . . . Electromagnetic Fields . . . . . . . . . . Dipole Coupling of Fields and Atoms
. . . .
. . . .
. . . .
1208 1208 1208 1209
83.3 83.3.1 83.3.2 83.3.3 83.3.4 83.3.5 83.3.6
Weak Coupling in Cavity QED . . . . . . . . . . . Radiating Atoms in Waveguides . . . . . . . . . . . . Trapped Radiating Atoms and Their Mirror Images Radiating Atoms in Resonators . . . . . . . . . . . . Radiative Shifts and Forces . . . . . . . . . . . . . . . Experiments on Weak Coupling . . . . . . . . . . . . Cavity QED and Dielectrics . . . . . . . . . . . . . .
. . . . . . .
. . . . . . .
1210 1210 1210 1211 1212 1213 1213
83.4 83.4.1 83.4.2 83.4.3 83.4.4
Strong Coupling in Cavity QED . . . . . . . . . . The Jaynes-Cummings Model . . . . . . . . . . . . Fock States, Coherent States, and Thermal States Vacuum Splitting . . . . . . . . . . . . . . . . . . . . Strong Coupling in Experiments . . . . . . . . . . .
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1213 1214 1215 1216 1216
83.5 83.5.1 83.5.2 83.5.3 83.5.4
Micromasers . . . . . . . . . . . . Maser Threshold . . . . . . . . . . Nonclassical Features of the Field Trapping States . . . . . . . . . . . Atom Counting Statistics . . . . .
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1217 1218 1218 1219 1219
83.6 83.6.1 83.6.2
Cavity Cooling . . . . . . . . . . . . . . . . . . . . . . . . 1220 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . 1220 Cavity Cooling Experiments . . . . . . . . . . . . . . . . 1221
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Q. He () School of Physics, Peking University Beijing, China e-mail: [email protected] W. Zhang Dept. of Physics, Renmin University of China Beijing, China e-mail: [email protected] D. Meschede Inst. f. Applied Physics, Rheinische Friedrich-Wilhelm Universität Bonn Bonn, Germany e-mail: [email protected]
, Dieter Meschede, and Axel Schenzle
83.7 83.7.1 83.7.2 83.7.3
Cavity QED for Cold Atomic Gases . . Atomic Ensembles in a Cavity . . . . . . . Bose–Einstein Condensate in a Cavity . . Cavity Optomechanics with Cold Atoms .
. . . .
1222 1222 1223 1225
83.8 83.8.1 83.8.2
Applications of Cavity QED . . . . . . . . . . . . . . . Quantum Nondemolition (QND) Counting of Photons Detecting and Trapping Atoms Through Strong Coupling . . . . . . . . . . . . . . . . . . . . . . . . Single-Photon Sources . . . . . . . . . . . . . . . . . . . . Generation of Entanglement . . . . . . . . . . . . . . . .
1225 1225
83.8.3 83.8.4
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1226 1227 1227
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228
Abstract
Although the concept of a free atom is of use as a first approximation, a full quantum description of the interaction of atoms with an omnipresent electromagnetic radiation field is necessary for a proper account of spontaneous emission and radiative level shifts such as the Lamb shift. This chapter is concerned with the changes in the atom– field interaction that take place when the radiation field is modified by the presence of a cavity. An atom in the vicinity of a plane perfect mirror serves as an example of cavity quantum electrodynamics [1–5]. The interaction between atom and intracavity light field can significantly change the spontaneous and stimulated emission of light and induce transitions of the atom between different quantum states. In the case of strong coupling, it also becomes possible for a single atom to control the transmission of light through the cavity, and for a single photon to deterministically change the state of the atom.
Keywords
cavity mode photon number Rydberg state Casimir force Rydberg atom
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_83
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Q. He et al.
a Pe
b
spin operators are
Pe
1
1
0.5
0.5
x D C C ; y D i.C / ;
0
0
1
2
0
3 t/τ
c
z D C C D C ; ;
0
1
2
3 ⍀ /t
d ⍀
(83.1)
with C D jeihgj and D jgihej. The quadratures (out-ofphase components) of the atomic polarization are given by x and y , while z is the occupation number difference. The free atom Hamiltonian is Hatom D
1 „!0 z ; 2
(83.2)
where „!0 D Ee Eg is the transition energy. φ0
(φ – φ0)
φ0
(φ – φ0)
Fig. 83.1 Upper row: excitation probability of an excited atom. (a) Exponential decay in free space or bad cavities in the weak coupling limit. (b) Oscillatory evolution in good cavities or in the strong coupling case. Lower row: the spectral signature of exponential decay is a Lorentzian line shape (c), while the so-called vacuum Rabi splitting (d) is observed in the strong coupling case
83.2.2 Electromagnetic Fields Classical Fields Classical electromagnetic fields have longitudinal and transverse components E .r; t/ D E l .r; t/ C E t .r; t/ :
(83.3)
83.1 Introduction In the Coulomb gauge, the longitudinal part is the instantaIn the weak coupling regime, the coupling of an excited atom neous electric field. The transverse part is the radiation field to a broad continuum of radiation modes leads to exponen- that obeys the wave equation tial decay (Fig. 83.1a), as first described by Weisskopf and 1 @2 1 @ 2 Wigner [6]. Spontaneous emission may be enhanced or supr 2 2 E t .r; t/ D j .r; t/ : (83.4) c @t 0 c 2 @t pressed in structures such as waveguides or bad cavities. Cavities also introduce van der Waals forces and the subtle In empty space, the driving current density j .r; t/ vanishes, Casimir level shifts [7]. In the strong coupling regime, the excited atom is strongly and the field may be expanded in a set of orthogonal modes as coupled to an isolated resonant cavity mode. In the absence X of damping, an oscillatory exchange of energy between the E .t/ ei! r u .r/ C c.c. ; (83.5) E t .r; t/ D atom and the field replaces exponential decay (Fig. 83.1b) with a coherent evolution in time. Experimental investigations of these effects began [8] with the development of with slowly varying amplitudes E .t/. The spatial distribusuitable resonators and techniques for producing atoms with tions u .r/ obey the vector Helmholtz equation long-lived excited states and strong dipole transition mo ! 2 ments. 2 (83.6) r C u .r; t/ D 0 ; c depending on geometric boundary conditions as imposed by conductive or dielectric mirrors, waveguides, and resonators. In free space, plane wave solutions u .r; t/ D u eikr have 83.2.1 Two-Level Atoms a continuous index D .k; / with wave vector k and an index for the two independent polarizations. The orthogoMost of the essential features of cavity QED can be elucinality relation dated by the two-level model atom discussed in Chaps. 72, Z 74, and 81 (also [9]). A ground state jgi and an excited state 1 u u d3 r D ı (83.7) jei are coupled to the radiation field by a dipole interaction. V V Using the formal equivalence to a spin-1/2 system, the Pauli
83.2 Atoms and Fields
83
Entangled Atoms and Fields: Cavity QED
applies. For a closed cavity, V is the resonator volume. In waveguides and free space, an artificial boundary is introduced and then increased to infinity at the end of a calculation, such that the final results do not depend on V . Many of the properties of cavity QED can be described by a classical theory [10] where the classical fields interact with electric dipoles via the so-called dipole approximation.
1209
This interaction causes the atom to exchange energy with the radiation field. In the dipole approximation, the interaction can be reduced to the commonly used form of p E [11], with the coupling strength proportional to the component of the atomic dipole moment d eg D qhejr eg jgi along the electric field. The coupling constant g .r/ D jd eg u .r/E j=„ :
(83.13)
Quantum Fields In the rotating-wave approximation (RWA) (Chaps. 72 and The quantum analog of the classical transverse field in 74), Eq. (83.4) is obtained through a quantization of its harmonic X modes leading to a number state expansion. Field operators (83.14) HRWA D „ g C a C g a ; obey standard commutation relations Œa ; a D ı; , and for a single mode with index , the amplitude E in Eq. (83.5) where we have used the atomic operators of Eq. (83.1). is replaced by the corresponding operator For the simplest scenario of a two-level atom interacting (83.8) with a monomode cavity, the Hamiltonian under RWA reE .t/ D E a ei!t ; E .t/ D E a ei!t : duces to the Jaynes–Cummings (JC) model The normalization factor E is chosen such that the energy
1 difference between number states jni and jn C 1i in the HJC D „!0 z C „!C a a C „ gC a C g a ; 2 volume V is „! , giving (83.15) E E D
„! : 20 V
(83.9) where !C denotes the cavity mode frequency, and the zeropoint energy of the field is neglected. Note that the JC model is valid only if the RWA can be employed to describe the The Hamiltonian of the free field is coherent dipole interaction between the atom and the field X mode. This usually requires that the coupling constant g is 1 HField D „! a a C : (83.10) relatively much smaller than the energy scales of !0 and !C . 2 Research beyond the RWA has been conducted in the regime In the Coulomb gauge, the vector potential A.r/ is related to where the coupling constant g is comparable with the atomic and field frequencies. the electric field E D @A=@t by In a continuous electromagnetic spectrum, the atom inter E acts with a large number of modes having quantum numbers a ei!t C a ei!t u .r/ : (83.11) A .r; t/ D , yielding exponential decay of an excited atomic level at ! the rate [6] The ground state j0i is called the vacuum state. While 2 X Xˇˇ ˇˇ2
the expectation value hnjEjni D 0 for a number state, the (83.16) g ı ! !0 : eg D 2 „ variance is not zero, since hnjE E jni > 0, giving rise to nonQ k vanishing fluctuations of the free electromagnetic field. Here, we have separated the discrete () Q and the continuous part (wave vector k) of the mode index . If g Eq. (83.13) does not vary much across a narrow resonance, then 83.2.3 Dipole Coupling of Fields and Atoms ˇ ˇ2 X eg ' 2 ˇg .!0 /ˇ
Q .!0 / : (83.17) The combined system of atoms and fields can be described Q by the product quantum states ja; ni of atom states jai and field states jni. The interaction Hamiltonian HI of the atom The density of states corresponding to the continuous mode and the radiation field is given by (The A 2 -term plays an index k of dimension can be evaluated on a -dimensional important role in energy shifts and can only be neglected fictitious volume V ./ as when radiative processes involving energy exchange are conZ1 X sidered.) V ./
Q D ı.!;k d kı.!;k Q !/ ! Q !/ ; .2 / 2 k 0 q q HI D p A.r/ C (83.12) A 2 .r/ : (83.18) m 2m
83
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Q. He et al.
provided that !.k/ is known, and by converting the sum over plane wave vectors k into an integral. (This is formally accomplished by taking the limit of k D 2 = l for large l, where l is a linear dimension of an artificial resonator, and the resonator volume is V D l 3 . If the relation between mode spacing k and geometric dimension is nonlinear in a more complex geometry, this analysis can be very complicated.)
The Rate of Spontaneous Emission In free space [!.k/2 D .ck/2 ], the sum in Eq. (83.17) contributes a factor of 2, due to polarization, to the total density of states in free space, free .!/ D V ! 2 = 2 c 3 . When the vector coupling of atom and field Eq. (83.13) is replaced by its average in isotropic free space, that is, by 1/3, the result eg D Aeg D
2 e 2 reg !3
3 0 „c 3
(83.19)
/ free(φ0) 15
free
10
cav 5
0
0
1
2
3
4 (φ – φ0)
Fig. 83.2 Modification of the average vacuum spectral density ( TE C
TM ) in a parallel plate cavity (thick line) compared with free space (thin line)
is obtained for the decay rate Aeg as measured by the natural containing the area of the plates, V D Ad , giving linewidth eg . !c Œ!
free .!c / ;
TE .!/ D 2!c2 83.3 Weak Coupling in Cavity QED !c Œ! C 1
TM .!/ D
free .!c / ; (83.21) 2!c2 The regime of weak cavity QED generally applies when an atom is coupled to a continuum of radiation modes. This is where Œx is the largest integer in x, and !c D c=d gives the always the case with mirrors, waveguides, or bad cavities. waveguide cutoff frequency. Below !c , the TE-mode density The signatures of weak cavity QED are modifications of the clearly vanishes and, with the pictorial notion of turning off rate of spontaneous emission, as well as the existence of van the vacuum introduced by Kleppner [12], inhibition of rader Waals and Casimir forces. Formally, this regime is well diative decay is obvious. Figure 83.2 shows the calculated described by perturbation theory. mode density for a parallel plate waveguide. The decay rate can be calculated from Eq. (83.17), with the spatial variation of g included. This configuration was used for the 83.3.1 Radiating Atoms in Waveguides first experiments, which showed the suppression of spontaneous emission in both the microwave and the near optical Within the continuous spectrum of a waveguide, the radia- frequency domain [13, 14] with atomic beams. tive decay of an excited atomic level remains exponential, and eg may be determined as in the preceding section. We now consider the modifications of spontaneous decay 83.3.2 Trapped Radiating Atoms in a parallel plate waveguide. According to Eq. (83.17), the and Their Mirror Images theoretical problem is reduced to a geometric evaluation of mode densities. Between a pair of mirrors it is convenient to Boundary conditions imposed by conductive surfaces may distinguish TEnk and TMnk modes, where n is the number also be simulated by appropriately positioned image charges. of half-waves across the gap of width d . The dispersion re- Inspired by classical electrodynamics, this image charge lation !.k/ reflects the discrete standing-wave part (n =d ) model can be successfully used to determine the modifiand a running-wave part as in free space, cations of radiative properties in confined spaces. In the simplest case, an atom is interacting with its image produced n D 0; 1; 2; : : : TM by a plane mirror. Trapped atoms and ions allow one to conD c jkj C n =d trol their relative position with respect to a mirror to distances n D 1; 2; : : : TE : (83.20) below the wavelength of light. Hence they are ideal objects for studying the spatial dependence of the mirror-induced The average mode density [du D 1, Eq. (83.13)] is evaluated modifications of their radiative properties. In an experiment [Eq. (83.18), D 2] with an appropriate quantization volume with a single trapped ion (Fig. 83.3), its radiation field was 2 !n;k
2
2
2
83
Entangled Atoms and Fields: Cavity QED
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493-nm photon counts in 0.2 s
a Open resonator
1800 1600 1400 1200 1000 800 600
b Closed resonator
400 200 0
–200
Movable mirror
–100
0
Ion trap
100 200 Mirror shift (nm)
Detector
Fig. 83.3 Sinusoidal variation of the D 493 nm spontaneous emission rate of a single trapped Ba ion caused by self-interference from a retroreflecting mirror. The experimental arrangement is sketched at the bottom [15]
Fig. 83.4 Two frequently used resonator types for cavity QED: (a) open Fabry–Perot (FP) optical cavity; (b) closed pillbox microwave cavity
for by an effective mode density of Lorentzian width ! for a single isolated mode,
.!/ D
! =2Q 1 : .! ! /2 C .! =2Q /2
(83.23) superposed onto its mirror image [15, 16], yielding a sinusoidal variation of both the spontaneous decay rate and the Bad and Good Cavities mirror-induced level shift with excellent contrast. The modification of spontaneous decay is again calculated from Eq. (83.17). For an atomic dipole aligned parallel to the mode polarization, and right at resonance, ! D !0 , the 83.3.3 Radiating Atoms in Resonators enhancement of spontaneous emission is found to be proportional to the Q-value of a selected resonator mode Resonators In a resonator, the electromagnetic spectrum is no longer conegcav
ju.r/j 3Q3 3Q3 2 tinuous, and the discrete mode structure can also be resolved D D D ; (83.24) ju.r/j egfree
free 4 2 V 4 2 Veff experimentally. While a resonator is only weakly coupled to external electromagnetic fields, it still interacts with a large thermal reservoir through currents induced in its walls. The total damping rate is due to resistive losses in the walls ( wall ) and also due to transmission at the radiation ports, 1= D
D wall C out . An empty resonator stores energy for times D
Q ; !
(83.22)
and the power transmission spectrum is a Lorentzian with width ! D ! =Q . The index , for instance, represents the TElm and TMlm modes of a pillbox microwave cavity, or the TEMklm modes of a Fabry–Perot (FP) interferometer (Fig. 83.4). When cavity damping remains strong, eg , the atomic radiation field is immediately absorbed, and Weisskopf–Wigner perturbation theory remains valid. In this socalled bad cavity limit, resonator damping can be accounted
where the effective mode volume is Veff D V =ju.r/j2 . The lowest possible value Veff ' 3 is obtained for ground modes of a closed resonator. For an atom located at the waist of an open FP cavity with length L, it is much larger. Special limconc D iting cases for concentric and confocal cavities are Veff conf D L2 =2 , respectively, where (R=D) 2 L.R=D/ and Veff gives the ratio of mirror radius to cavity diameter. At resonance, the atomic decay rate grows with Q , whereas the resonator damping time constant is reduced. Eventually, the energy of the atomic radiation field is stored for such a long time that reabsorption becomes possible. Perturbative Weisskopf–Wigner theory is no longer valid in this good cavity limit, which is separated from the regime of bad cavities by the more formal condition egcav > :
(83.25)
83
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The strong coupling case is considered explicitly in Sect. 83.4.
Antenna Patterns Since the reflected radiation field of an atomic radiator is perfectly coherent with the source field, the combined radiation pattern modifies the usual dipole distribution of a radiating atom. The new radiation pattern can be understood in terms of antenna arrays [17]. For a single atomic dipole in front of a reflecting mirror, for example, one finds a quadrupole-type pattern due to the superposition of a second, coherent image antenna. In some of the earliest experimental investigations on radiating molecules in cavities, modifications of the radiation pattern were observed [18].
83.3.4 Radiative Shifts and Forces
Q. He et al.
state. In perturbation theory, the van der Waals energy shift of an atomic level jai is vdW
hajq 2 .d 2 xO t /2 C 2.d 2 zO /2 jai D : 64 0 z 3
(83.26)
Since the van der Waals force is anisotropic for electronic components parallel (Oz) and perpendicular (xO t ) to the mirror normal, the degeneracy of magnetic sublevels in an atom is lifted near a surface. The total energy shift is 1 kHz for a ground-state atom at 1 m separation and is very difficult to detect. However, the energy shifts grow as n4 since the transition dipole moment scales as n2 . With Rydberg atoms, van der Waals energy shifts have been successfully observed in spectroscopic experiments [19].
The Retarded Limit: Casimir Forces At large separation, retardation becomes relevant, since the contributions of individual atomic oscillation frequencies in Eq. (83.26) cancel by dephasing, thus reducing the vdW . A residual Casimir–Polder [20] shift may be interpreted as the polarization energy of a slowly fluctuating field with squared amplitude hE 2 i D 3„c=640 z 4 originating from the vacuum field noise
When the radiation field of an atom is reflected back onto its source, an energy or radiative shift is caused by the corresponding self-polarization energy. An atom in the vicinity of a plane mirror (Fig. 83.5) again makes a simple model system. Since the energy shift depends on the atom wall separation z, it is equivalent to a dipole force F dip whose 1 3„c˛st details depend on the role of retardation. Here, we distin CP D ; (83.27) 4 4 0 8 z guish between the two cases where no radiation energy is exchanged between the atom and the field (van der Waals, where ˛st is the static electric polarizability. The vacuum Casimir forces), and where the atomic radiation causes forces field noise CP replaces vdW at distances larger than charby self-interference. acteristic wavelengths and is even smaller. Only indirect observations have been possible to date, relying on a deThe Unretarded Limit: van der Waals Forces flection of polarizable atoms by this force [21, 22]. The When the radiative round-trip time tr D 2z=c is short Casimir–Polder force can also be regarded as an ultimate, compared with the characteristic atomic revolution period cavity-induced consequence of the mechanical action of light 2 =!eg , retardation is not important. In this quasi-static on atoms [23]. It is an example of the conservative and limit, van der Waals energy shifts for decaying atomic dispersive dipole force, which is even capable of binding dipoles vary as z 3 with the atom–wall separation. Such a polarizable atom to a cavity [24]. a shift is also present for a nonradiating atom in its ground Radiative Self-Interference Forces Spontaneous emission of atoms in the vicinity of a reflecting a 2 b σ wall also provides an example of cavity-induced modifiπ π π σ 0 cation of the dissipative type of light forces, or radiation pressure. If the returning field is reabsorbed, the spontaneous 1 emission rate is reduced, and a recoil force directed away –0.5 from the mirror is exerted. If the returning radiation field causes enhanced decay, a recoil towards the mirror occurs σ Гcav /Гfree Δcav /Гfree 0 due to stimulated emission. 1 0 1 2 2z/λ 0 1 2 2z/λ If the photon is detected at some angle with respect to the normal vector connecting the atom with the mirror surface, Fig. 83.5 (a) Normalized rate of modified spontaneous emission in the two paths for the photon are possible: it can reach a detector vicinity of a perfectly reflecting wall for and orientation of the directly or following a reflection off the wall. At small atom– radiating dipole. (b) Corresponding energy shift of the resonance frequency. The shaded area indicates contribution of static van der Waals mirror separation, these paths are indistinguishable, the atom interaction is thus left in a superposition of two recoil momentum states.
83
Entangled Atoms and Fields: Cavity QED
83.3.5 Experiments on Weak Coupling Perhaps the most dramatic experiment in a weak coupling cavity QED is the total suppression of spontaneous emission. For the experiments that were carried out with Rydberg atoms and for a low-lying near-infrared atomic transition [13, 14], it is essential to prepare atoms in a single decay channel. In addition, the atoms must be oriented in such a way that they are only coupled to a single decay mode (see the model waveguide in Fig. 83.4). This may be interpreted as an anisotropy of the electromagnetic vacuum or as a specific antenna pattern. An important problem in detecting the modification of radiative properties – changes in emission rates as well as radiative shifts – arises from their inhomogeneity due to the dependence on atom–wall separation. This difficulty has been overcome by controlling the atom–wall separation at microscopic distances through light forces [19], or by using well localized trapped ions [15, 16]. Furthermore, spectroscopic techniques that are only sensitive to a thin layer of surface atoms [25] have been used to clearly detect van der Waals shifts. An atom emitting a radiation field in the vicinity of a reflecting wall will experience an additional dipole optical force caused by its radiation field. This force has been observed as a modification of the trapping force holding an ion at a fixed position with respect to the reflector [26]. Conceptually most attractive and experimentally most difficult to detect is the elusive Casimir interaction. Only for atomic ground states is this effect observable, free from other much larger shifts. The influence of the corresponding Casimir force on atomic motion was observed in a variant of a scattering experiment, confirming the existence of this force in neutral atoms [21, 22]. The success of this experiment shows that spectroscopic techniques involving the exchange of photons are not suitable for the Casimir problem. A notable exception could be Raman spectroscopy of the magnetic substructure in the vicinity of a surface. In general, scattering or atomic interferometry experiments are more promising methods. The experiment by Brune et al. [27] may be interpreted in this way.
83.3.6 Cavity QED and Dielectrics There are two variants of dielectric materials employed to study light–matter interaction in confined space: conventional materials, such as glass or sapphire, and artificial materials called photonic materials or metamaterials. While dielectric materials are theoretically more difficult to treat than perfect mirrors, since the radiation at least partially enters the medium, they have a similar influence on radiative decay processes. One new aspect is, however, the
1213
coupling of atomic excitations to excitations of the medium, which was observed for the case of a surface-polariton in [28]. Cavities with dimensions comparable to the wavelength promise the most dramatic modification of radiative atomic properties, but micrometer-sized cavities for optical frequencies with highly reflecting walls are difficult to manufacture. So-called whispering gallery modes of spherical microcavities [29] have been intensely studied, but no simple way of coupling atoms to these resonator modes has yet been found. On the other hand, dielectric materials with a periodic modulation of the index of refraction may exhibit photonic bandgaps in analogy with electronic bandgaps in periodic crystals [30, 31]. Electronic phenomena of solid-state physics can then be transferred to photons. For example, excited states of a crystal dopant or a quantum dot cannot radiate into a photonic bandgap, the radiation field cannot propagate, and the excitation energy remains localized. The bandgap behaves like an empty resonator, and if a resonator structure is integrated into the device, the regime of strong coupling [32, 33] can be achieved with such photonic structures. An overview of suitable systems can be found in [34].
83.4 Strong Coupling in Cavity QED Strong coupling of atoms and fields is realized in a good cavity when < eg Eq. (83.25). The Hilbert space of the combined system is then the product space of a single twolevel atom and the countable set of Fock states of the field, H D Hatom ˝ Hfield ;
(83.28)
which is spanned by the states jnI ai D jnijai :
(83.29)
The interaction of a single-cavity mode with an isolated atomic resonance is now characterized by the Rabi nutation frequency, which gives the exchange frequency of the energy between atom and field. For an amplitude E corresponding to n photons, p ˝.n/ D g n C 1 : (83.30) This is the simplest possible situation of a strongly coupled atom–field system. The new energy eigenvectors are conveniently expressed in the dressed-atom model [35] jC; ni D cos jg; n C 1i C sin je; ni ; j; ni D sin jg; n C 1i C cos je; ni ;
(83.31)
p with tan 2 D 2g n C 1=.!0 ! /. The separate energy structures of free atom and empty resonator are now replaced
83
1214
Q. He et al.
not degenerate with respect to the energy H . The initial state problem corresponding to Eq. (83.33) is solved by elementary methods in terms of the expansion
j .t/i D
l+, 1>
1 X 2 X
Cnj .t/jn; j i ;
(83.35)
nD0 j D1
le, 1>, lg, 2> l–, 1>
where the expansion coefficients are l+, 0>
le, 0>, lg, 1>
Cn1 .t/
l–, 0>
Fig. 83.6 Level diagram for the combined states of noninteracting atoms and fields (left) which are degenerate at resonance. Degeneracy is lifted by strong coupling of atoms and fields (right) yielding new dressed eigenstates
ı 1 D Cn .0/ cosŒ˝.n/t i sinŒ˝.n/t 2˝.n/ p ng 2 i C .0/ sinŒ˝.n/t ˝.n/ n1 1 exp i! n t (83.36) 2
and by the combined system of Fig. 83.6. At resonance, the new eigenstates are separated by 2„˝R , where ˝R D g is the vacuum Rabi frequency.
83.4.1 The Jaynes-Cummings Model In a microscopic laser, simple atoms are strongly coupled to a single mode of a resonant or near-resonant radiation field. Collecting atomic and field operators from Eqs. (83.2), (83.10), and (83.14), this situation is described by the Jaynes–Cummings model Hamiltonian Eq. (83.15) [36, 37]
( Cn2 .t/
Cn2 .0/
D
cosŒ˝.n C 1/t
) ı sinŒ˝.n C 1/t Ci 2˝.n C 1/ ! p g n C 1 1 i C .0/ sinŒ˝.n C 1/t ˝.n C 1/ nC1 1 exp i! n C t ; (83.37) 2
with ı D ! !0 the detuning between the atom and cavity, and ˝.n/ D .1=2/.ı 2 C 4g2 n/1=2 is the generalized Rabi fre j 1 quency. The coefficients Cn .0/ are determined by the initial D „!0 z C „! a a C „ g C a C g a : 2 preparation of atom and cavity mode. The result simplifies (83.32) considerably for ı D 0 to
HJC D Hatom C Hfield C HRWA
The JC model Eq. (83.32) represents the most basic and, at the same time, the most informative model of strong coupling in quantum optics. It consists of a single two-level atom interacting with a single mode of the quantized cavity field. The time evolution of the system is determined by i„
@ DH @t
:
(83.33)
This model can be solved exactly due to the existence of the additional constant of motion
N D a a C z C 1 ;
(83.34)
i.e., conservation of the number of excitations. Its eigenvalues are the integers N , which are twofold degenerate except for N D 0. The simultaneous eigenstates of H and N are the pairs of dressed states defined in Eq. (83.31), which are
j .t/i D
1 n X mD0
Cm1 .0/ ei! .m1=2/t
p cos g mt jmI 1i
p i sin g m t jm 1I 2i C Cm2 .0/ ei!m u.mC1=2/t
p cos g m C 1 t jmI 2i
p o i sin g m C 1t jm C 1I 1i : (83.38) j
The coefficients Cn .0/ represent any initial state of the system, from uncorrelated product states to entangled states of atom and field. There exist numerous generalizations of this model that include more atomic levels and several coherent fields.
83
Entangled Atoms and Fields: Cavity QED
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83.4.2 Fock States, Coherent States, and Thermal States
The Coherent State Consider the case where the field is initially prepared in a coherent state
We now illustrate the properties of the Jaynes–Cummings model by specifying the initial state. Assume that the atom and field are brought into contact at time t D 0 and that all correlations that might exist due to previous interactions are suppressed.
Rabi Oscillations If the atom is initially in the excited state, and the field contains precisely m quanta, then Cnj .t
D 0/ D ın;m ıj;2 :
1 X
˛n 2 j˛i D exp ˛a ˛ a j0i D ej˛j =2 p jni ; nŠ nD0 (83.47)
while the atom starts from the excited state Cnj .0/ D ej˛j
j .t/i D ei! .mC1=2/t
p cos g m C 1 t jmI 2i
p i sin g m C 1 t jm C 1I 1i :
j˛jn p ıj;2 : nŠ
(83.48)
In this case, the general solution specializes to
(83.39) j .t/i D
The solution of Eq. (83.33) assumes the form
2 =2
1 X ˛n 2 p ei!.nC1=2/t ej˛j =2 nŠ nD0
p cos g n C 1 t jnI 2i
p i sin g n C 1 t jn C 1I 1i ;
(83.49)
(83.40) and the occupation probability of the excited state is # " 1 2n X p
j˛j 1 2 The occupation probabilities of the atomic states evolve in 1C ej˛j cos 2g n C 1 t : n2 .t/ D 2 nŠ time according to nD0 (83.50) n2 .t/ D h .t/j2ih2j .t/i
p From here, detailed quantitative results can only be obtained (83.41) D cos2 g m C 1 t ; by numerical methods [38]. However, if the coherent state n1 .t/ D h .t/j1ih1j .t/i contains a large number of photons j˛j2 1, the essential dy
p 2 (83.42) namics can be determined by elementary methods. Initially, D sin g m C 1 t : the population oscillates with the Rabi frequency ˝1 g j˛j, which is proportional to the average amplitude of the field, as The photon number and its variance are expected from its classical counterpart. With increasing time, ˝ ˛ hn.t/i D .t/a a .t/ the coherent oscillations tend to cancel due to the destructive
p 2 (83.43) interference of the different Rabi frequencies in the sum D m C sin g m C 1 t ; ˝
˝ ˛2 ˛ i 1h .gt /2 =2 h2 ni D .t/ a a a a .t/ : (83.51) .t/ D j˛j t/ e 1 C cos.2g n 2
p 2 sin2 2g m C 1 t D : (83.44) However, strictly aperiodic relaxation of n .t/ is impossible 2 4 since the exact expressions, Eqs. (83.35) and (83.36), repp In the limit of large m, g m C 1 is proportional to the field resent a quasi-periodic function which, given enough time, amplitude, and the classical Rabi oscillations in a resonant approaches its initial value with arbitrary accuracy. field are recovered. The nonclassical features of the states For short times, the oscillating terms in the sum cancel are characterized by Mandel’s parameter each other due to the slow evolution of their frequency with n. However, consecutive terms interfere constructively for ˝ 2 ˛ n hni 1 : (83.45) larger times tr , such that the phases satisfy QM D hni (83.52) nC1 .tr / n .tr / D 2 : For the present example, For j˛j2 1, the increment of the arguments is
p 2 1 sin 2g m C 1 t g tr I
p QM D 1 C (83.46) nC1 n D ; (83.53) 4 m C sin2 g m C 1 t j˛j QM 0 indicates the classical regime, while Q 0 can only and, therefore, the first revival of the Rabi oscillations occurs be reached by a quantum process. approximately at tr D j˛j=g . A clear distinction of Rabi
83
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Q. He et al.
of vanishing temperature [42]. Since energy is exchanged between the nondecaying atom and the decaying cavity mode, cavity damping is modified in a characteristic way due to the presence of the atom. The technical details can be found in [43].
Population n2(t), α = 4 1
1.75
0.5
83.4.3 Vacuum Splitting 0.25
0
0
20
40
60 gt
Fig. 83.7 Rabi oscillations, dephasing, and quantum revival
oscillation, collapses, and revivals requires a clear separation of the three timescales t1 t2 t3 ;
(83.54)
In the classical case, the eigenvalues of the interaction-free Hamiltonian are degenerate at resonance. The atom–field interaction splits the eigenvalues and determines the Rabi frequency of oscillation between the two states. One consequence is the existence of sidebands in the resonance fluorescence spectrum [44]. In the quantum case, the field itself is treated as a quantized dynamical variable determined from a self-consistent solution for the complete system of atom plus field. The vacuum Rabi frequency ˝vac D g remains finite and accounts for the spontaneous emission of radiation from an excited atom placed in a vacuum. In the limiting case of a single atom interacting with the quantized field, the photon number n can only change by ˙1, and the population oscillates with the frequency ˝.n/ given by Eq. (83.30). For an ensemble of N atoms, n can, in principle, change by up to ˙N . However, if the field and atoms are only weakly excited, the collective frequency of the ensemble is determined by the linearized Maxwell–Bloch equations. The eigenfrequencies are given by q i 1h ˙ D i.? C / ˙ 4g2 N .? /2 ; (83.57) 2
where t1 .g j˛j/1 for Rabi oscillation, t2 g1 for collapse, and t3 j˛j=g for revival. The discovery of the collapse and revival of Rabi oscillations is one of the key demonstrations of photons [38–40]. The typical features of the transient evolution starting from a coherent state are shown in Fig. 83.7. With time increasing even further, revivals of higher order occur, which spread in time and finally can no longer be separated order by order. It is also proposed that by carefully manipulating the initial states and atom–light coupling during evolution, arbitrary Fock states can be created with prechosen evolution time [41]. where ?1 is the phase relaxation time of the atom and 1 the decay time of the resonator. This is the polariton disperThe Thermal State sion relation in the neighborhood of the polariton gap. The Consider a microwave resonator brought into thermal contact spectral transmission with a reservoir, inducing loss on a timescale 1 and thermal ˇ ˇ ˇ Œ C i.!0 !/ ˇ2 excitation. The dissipative time evolution is described by the ˇ ˇ (83.58) T .!/ D T0 ˇ master equation .! C /.! / ˇ
P D .L0 C L/
˚ iŒH; C .nth C 1/ a; a C a ; a „ ˚ C nth a ; a C a ; a ;
(83.55)
where nth D Œexp.ˇ„!/ 11 , at T D kB ˇ 1 , is the equilibrium population of the cavity mode, L0 symbolizes the unitary evolution according to the Jaynes–Cummings dynamics, and L is a dissipation term. The solution of this model can be expressed in terms of an eigenoperator expansion of the equation L D :
of an optical cavity containing a resonant atomic ensemble of N atoms reveals the internal dynamics of the coupled system and a splitting of the resonance line occurs; T0 is the peak transmission of the empty cavity. The splittingpincreases either with the number of photons, approaching n C 1 in the presence ofpa single atom or with the number of atoms, approaching N in the resonator when the field is weak. The latter case is demonstrated in Fig. 83.8 [45] for an optical resonator with 110 atoms interacting with a field that contains, on average, much less than a single photon.
(83.56) 83.4.4
The eigenvalues that determine the relaxation rates, as well as the eigenoperators, are known in closed form for the case
Strong Coupling in Experiments
In order to achieve strong coupling experimentally, it is necessary to use a high-Q resonator in combination with a small
83
Entangled Atoms and Fields: Cavity QED
0.05
n N = 10.7 atoms
0 –20
–10
0
n
10 20 Frequency Ω (MHz) N = 1.0 atoms
0.10 (i)
the single-atom level. The interaction or transit time T is usually much shorter than the lifetime Ry of the Rydberg states involved. For this reason, circular Rydberg states with quantum numbers l D m D n 1 are particularly suitable. Rydberg atoms [49] are prepared in an atomic beam, selectively excited to an upper level, and then sent through a microwave cavity where the upper and lower levels are coupled by the electromagnetic field. If the atom is detected in the lower of the coupled levels as it leaves the resonator, the excitation energy has been stored in the resonator field. Thus, the evolution of the resonator field is recorded as a function of the atomic interaction. A microwave cavity in interaction with a single or a few Rydberg atoms is called a micromaser (formerly a one-atom maser) [8]. The experimental conditions may be summarized as g >
0.05 (i) (ii) 0 –20
1217
–10
(ii)
0
10 20 Frequency Ω (MHz)
Fig. 83.8 Intracavity photon number (measured from a transmission experiment, [45]) as a function of probe frequency detuning, and for two values of N , the average number of atoms in the mode. Thin lines give theoretical fits to the data, including atomic number and position fluctuations. Curve (ii) in the lower graph is for a single intracavity atom with optimal coupling g
effective mode volume. This condition was first realized for ground modes of a closed microwave cavity [8] and later also for open-cavity optical resonators (Fig. 83.6) [45]. It is interesting to control the interaction time of the atoms with the cavity field. In earlier experiments, this was typically achieved by selecting the passage time for an atom transiting the cavity. The advancement of atom-trapping methods has also led to the observation of a truly one-atom laser at optical frequencies [46]. More recently, this situation has also been realized for artificial atoms including superconducting systems [47, 48] and quantum dots [32, 33].
Rydberg Atoms and Microwave Cavities At microwave frequencies, very low-loss superconducting niobium cavities are available with Q 1010 . Resonator frequencies are typically several tens of GHz and can be matched by atomic dipole transitions between two highly excited Rydberg states. By selective field ionization, the excitation level of Rydberg atoms can be detected, and hence it is possible to measure whether a transition between the levels involved has occurred. The efficiency of this method approaches unity, so that experiments can be performed at
1 1 > : > T Ry
(83.59)
Strong Coupling in Open Optical Cavities At optical wavelengths, a cavity with small Veff in Eq. (83.24) is clearly more difficult to construct than at centimeter wavelengths. However, dielectric coatings that allow very low damping rates ! =Q for optical cavities are now available. Very high finesse F ' 107 (which is a more convenient measure for the damping rate of an optical FP interferometer) has been achieved. By reducing the volume of such a highQ cavity mode, strong coupling of atoms and fields at optical frequencies has been demonstrated [45]. In open structures, the atoms can still decay into the continuum states with a rate . Therefore, the condition for strong coupling in such systems is usually given as g2
>1:
(83.60)
83.5 Micromasers Sustained oscillations of a cavity mode in a microwave resonator can be achieved by a weak beam of Rydberg atoms excited to the upper level of a resonant transition. For a cavity with a Q 1010 , much less than a single atom at a time, on average, suffices to balance the cavity losses. Operation of a single atom maser has been demonstrated [8]. The atoms enter the cavity at random times, according to the Poisson statistics of a thermal beam, and interact with the field only for a limited time. In order to restrict the fluctuations of the atomic transit time, the velocity spread is reduced. This is achieved either by Fizeau chopping techniques or by making use of Doppler velocity selection in the initial laser excitation process. Since most of the time no atom is present, it is natural to separate the dynamics into two parts [50]:
83
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Q. He et al.
1. For the short time while an atom is present, the state evolves according to the Jaynes–Cummings dynamics, where H is defined in Eq. (83.32), iŒH;
.t/ P D ; „
0.8
nth = 0 nth = 0.1
(83.61) 0.6
and damping can safely be neglected. The formal solution is abbreviated by .t/ D F .t t0 / .t0 /. 2. During the time interval between successive atoms, the cavity field relaxes freely toward the thermal equilibrium according to Eq. (83.55) with L0 D 0
.t/ P D L ;
Average photon number ¢n² 1
(83.62)
with the formal solution .t/ D expŒL.t t0 / .t0 /.
0.4 0.2 0
0
5
10 15 Normalized transit time τ
Fig. 83.9 Average photon number as a function of the normalized transit time defined by Eq. (83.69)
The time development of the micromaser, therefore, consists of an alternating sequence of unitary F .t/ and dissipative 83.5.1 Maser Threshold e.Lt / evolutions. Atoms enter the cavity one by one at random times ti . Until the next atom enters at time ti C1 , the evolution The steady-state distribution determines the mean photon ti is given by number of the resonator as a function of the operating conditions
.ti C1 / D exp.Ltp /F ./ .ti / ; (83.63) 1 X where tp D ti C1 ti , and is the transit time. If ti C1 nPn : (83.68) hni D ti on average, then tp ti C1 ti . After averaging Eq. (83.63) nD0 over the Poisson distribution P .t/ D R exp.Rtp / for tp , where R is the injection rate, the mean propagator from atom A suitable dimensionless control parameter is to atom is r R 1 R # D g : (83.69) (83.64) F ./h .ti /i : h .ti C1 /i D 2
RL After excitation, the reduced density matrix of the field alone For # 1, the energy input is insufficient to counterbalance the loss of the cavity, effectively resulting in a negligible becomes diagonal after several relaxation times 1 photon number. With increasing pump rate R, a threshold is hnjTratom . /jmi D Pn ın;m : (83.65) reached at # ' 1, where hni increases rapidly with R. In contrast to the behavior of the usual laser, the single-atom maser Due to the continuous injection of atoms, the field never be- displays multiple thresholds with a sequence of minima and comes time independent but may relax toward a stroboscopic maxima of hni as a function of # [51]. This can be related state defined by to the rotation of the atomic Bloch vector. When the atom undergoes a rotation of about during the transit time , h .ti C1 /i D h .ti /i : (83.66) a maximum of energy is transferred to the cavity, and hni is maximized. The converse applies if the average rotation is The state of the cavity field can be determined in closed form a multiple of 2 . This behavior is shown in Fig. 83.9. The by iteration minima in hni are at # ' 2n . n Y nth C Ak Pn D N ; (83.67) .nth C 1/ kD1
83.5.2 Nonclassical Features of the Field
where N guarantees normalization of the trace, and Ak D p .R=n/ sin2 .g n/, and exact resonance between cavity Fluctuations can be of classical or of quantum origin. The mode and atom is assumed. Since all off-diagonal elements variance of the photon number vanish in steady state, Eq. (83.67) provides a complete de
2 D hn2 i hni2 (83.70) scription for the photon statistics of the field.
83
Entangled Atoms and Fields: Cavity QED
1219
Variance of photon number distribution 4 nth = 0 nth = 0.1 3
2
1
0
0
5
at zero temperature, nq still represents an upper barrier that cannot be overcome, since damping only causes downward transitions. Even in the presence of dissipation, generalized trapping states exist with a photon distribution that vanishes for n > nq and has a tail towards smaller photon numbers n nq . However, thermal fluctuations at finite temperatures destabilize the trapping states since they can momentarily increase the photon number and allow the distribution to jump over the barrier n D nq . Nevertheless, even for nth < 107 , remnants of the trapping behavior persist and can be seen in the transient response of the micromaser (Sect. 83.5.4).
10 15 Normalized transit time τ
83.5.4 Atom Counting Statistics Fig. 83.10 Variance normalized on the average photon number 2hni =h i. Values below unity indicate regions of nonclassical behavior
Direct measurements of the field in a single-atom maser resonator are not possible because detector absorption would drastically degrade its quality. However, the field can be deis a measure of the randomness of the field intensity. Classi- duced from the statistical signature of the atoms leaving the cal Poisson statistics require that 2 hni. A value below resonator. The probability P .n/ of finding n atoms in a beam during unity indicates quantum behavior, which has no classical analog. In Fig. 83.10, the variance is plotted as a function of an observation interval t is given by the classical Poisson #. Regions of enhanced fluctuations 2 > hni alternate with distribution regions with sub-Poissonian character 2 < hni [52]. When .Rt/n eRt : (83.72) P .n/ D hni approaches a local maximum, it is accompanied by large nŠ fluctuations, while at points of minimum field strength, the fluctuations are reduced below the classical limit. This fea- Information on the field inside is then revealed by the conture is repeated with a period of # ' 2 but finally washes ditional probability W .n; jgi; m; jei; T / of finding n atoms in the ground state and m atoms in the excited state during out at large values of #. The large variance of n is caused by a splitting of the a time t. Since there are only two states, it is sufficient to photon distribution Pn into two peaks, which gives rise to determine the probability bistability in the transient response [53]. The sub-Poissonian 1 X behavior of the field is reflected in an increased regularity of W .n; jgi; m; jei; t/ (83.73) W .n; jgi; t/ D the atoms leaving the cavity in the ground state. mD0
83.5.3 Trapping States
for being in the ground state [55]. For n D 0, the probability of observing no atom in the ground state during the period t is
W .0; jgi; t/ If cavity losses are neglected, operating conditions exist D Tr. stst / expfL C RŒOjgi C .1 /Ojei 1tg ; which lead directly to nonclassical, i.e., Fock states. If the (83.74) cavity contains precisely nq photons, an atom that enters the resonator in the excited state leaves it again in the same state where Ojj i D hj jF ./jj i Eq. (83.61) and stst is the steady provided the condition [54] state of the maser field. This probability is closely related to the waiting time statistic P2 .0; jgi; t/ between two succesp (83.71) sive ground-state atoms, a property that is easily determined g nq C 1 D 2q in a start–stop experiment. For an atom detector with finite is satisfied, i.e., the Bloch vector of the atom undergoes q com- quantum efficiency for state selective detection, the waitplete rotations. Such a photon Fock state jnq i is referred to as ing time probability is a trapping state in the literature of maser. Note that the same term is also used in other contexts with very different defi- P2 .0; jgi; t/ Tr. stst /Ojgi expŒL C RŒOjgi C .1 /Ojei 1T Ojgi nitions. If the maser happens to reach a trapping state jnq i, : D ŒTr. stst /Ojgi 2 the photon number nq can no longer increase irrespective of (83.75) the flux of pump atoms. With the inclusion of cavity damping
83
1220
Q. He et al.
P2(0, |b², η; T) 105 104 103 102 101 100 10–1 10–2 10–3 10–4 10–5 –2 –1
0
1
2
3
4
5 6 log(T/Tcav)
Fig. 83.11 Waiting time probability for atoms in the ground state while the cavity is operated at the vacuum trapping-state condition
trapped at a specific position. Thus, the external degree of freedom associated with the kinetic motion of atoms needs to be taken into account to achieve a complete description of the system. As an example, the cavity modes can exert light forces on moving atoms and induce a significant effect in transmission spectroscopy measurements [56–58]. Another important example and one of the most promising applications of the cavity QED dynamics involving atomic motion is the realization of cavity cooling, i.e., the dissipation of kinetic energy through the cavity photon-loss channel in a controlled manner. The realization of cavity cooling has been proved to be an essential step for the experimental achievement of strongly coupled cavity QED systems with sufficiently long interaction times and precise control of atomic motion. In the weak coupling regime, the atom can be cooled by coupling with cavity photons if the pumping is red-detuned with !p !C < 0, in which case the average frequency of emitted photons is higher than that of the pumping laser due to the increase of mode density around the cavity frequency. The energy, hence, has to be compensated by the loss in kinetic energy of the atomic center-of-mass (CM) motion. In the strong-coupling regime, however, the atom–light scattering process becomes complicated with the nonnegligible photon reabsorption. For a high-finesse FP cavity, the dynamic cavity cooling effect in this case can be interpreted in the frequency domain with a Sisyphus-type mechanism using the dressed-state picture [59]. For the case of a ring cavity, the intuitive photon scattering picture can still be used but with a full velocity dependence on the radiation pressure [60].
How a specific field state is reflected in the atom counting statistics will be illustrated for two situations: the region of sub-Poisson statistics and the region where the trapping condition is satisfied. Increased regularity of the cavity field QM 0 manifests itself in increased regularity of groundstate atoms in the beam. The statistical behavior exhibits antibunching, i.e., P2 .0; jgi; t/ has a maximum at finite t, indicating repulsion between successive atoms in comparison with a Poissonian beam. If the transit time is chosen in such a way that g ' 2 , the chance of observing an initially excited atom in the ground state is negligible. At some point, however, an unlikely thermal fluctuation occurs, adding a photon. The p rotation angle of the Bloch vector suddenly increases to 2 2 ' 3 , and the atoms tend to leave the cavity in the ground state. After a typical cavity lifetime, the field decays, and the trapping condition is restored again. Under this operation condition, the statistics of ground-state 83.6.1 Master Equation atoms is governed by two time constants: As a simplest example, we consider a system of a single two1. A short interval, in which successive atoms leave the cav- level atom trapped in a single-mode optical cavity, which is ity in the ground state after a thermal fluctuation driven by a monochromatic pumping laser. Under the dipole 2. A long time interval, in which the trapping condition is approximation and the RWA, the Hamiltonian of the system maintained, and all atoms leave the resonator in their ex- can be written as cited state until the next fluctuation occurs. H D HGJC C HCoM C Hpump : (83.76) The probability P2 .0; jgi; t/ is plotted in Fig. 83.11. The plot clearly shows the two time regimes that govern the imperfect The generalized Jaynes–Cummings (GJC) term is similar to trapping situation. the JC model in Eq. (83.15)
83.6 Cavity Cooling In previous sections, we presented some important aspects of cavity QED by examining the effect of coupling between a single-cavity photon and the atomic (pseudo-)spin degree of freedom. However, an experimental realization and a perfect control of such coupling requires very cold atoms
HGJC D
1 „!0 .r/z C „!C a a 2 C „ gf .r/C a C g f .r/a ;
(83.77)
where the atomic frequency !0 .r/ acquires explicit dependence on the CM coordinate r of the atom owing to a differential AC-Stark shift induced by a far-detuned external trapping potential exerted on the atom. Besides, the
83
Entangled Atoms and Fields: Cavity QED
1221
atom–photon coupling strength is spatially modulated with a function f .r/ according to the electric field strength of cavity mode. For a standing-wave mode in an FP cavity, the coupling modulates in magnitude with f .r/ D cos.kx/, while for a propagating-wave mode in a ring cavity, the coupling modulates in phase with f .r/ D ei kx . In both cases, the intracavity mode is assumed without loss of generality to be along the x-direction. The motion of the atomic CM degree of freedom is described by the term HCoM D
p2 C Vtrap .r/ ; 2m
(83.78)
where m is the atomic mass, and Vtrap .r/ denotes the trapping potential. If pumped by a driving laser at frequency !p , the pumping Hamiltonian takes the form Hpump D i„.a a/ei!p t
C i„˝p .r/ei!p t .C / ;
(83.79)
can significantly simplify the Hamiltonian Eq. (83.76) by adiabatically eliminating the atomic excited state. This approximation is valid provided that the atomic transition between the two internal spin states is far detuned from the pumping laser or has a large spontaneous decay rate. In either case, the atomic operator can be approximated as
gf .r/a C ˝p .r/ : i.!p !0 / C
(83.83)
By substituting the expressions of and C D back into Eq. (83.76), we obtain an effective Hamiltonian in which the CM motion of the atom is coupled to the cavity mode Heff D
p2 C Vtrap .r/ 2m „ .!p !C / U0 jf .r/j2 a a
C „eff .r/ f .r/a C f .r/a ;
(83.84)
where the first term describes a pumping laser applied along where the position-dependent effective pumping strength is the cavity direction to couple the cavity mode with amplitude given by , and the second corresponds to a driving directly on the .!p !0 /g˝p .r/ atomic spin degree of freedom at a position-dependent Rabi : (83.85) eff .r/ D .!p !0 /2 C 2 frequency ˝p .r/. The dynamics of the system can be described by the denIn this effective model, one can see clearly that the cavity sity operator , which satisfies the master equation photons provide a trapping potential „U0 jf .r/j2 a a to the atomic CM motion with i d
(83.80) D ŒH ; C Lcav C Latom ; dt „ g 2 .!p !0 / : (83.86) U0 D .!p !0 /2 C 2 where the Liouville operators are introduced to describe the dissipation of cavity photons and atoms. If the environment The gradient of this effective potential acts as a force exerted is Markovian, this photon decay can be approximated by on the atom. This force acquires an explicit dependence on
Lcav D a a C a a 2a a ; (83.81) the amplitude of the cavity mode, which depends not only on the momentary position of the atom but has a memory with the cavity decay rate. Another decay channel of the effect owing to the finite decay rate . The force, hence, besystem is the spontaneous decay of atom from the excited to comes velocity dependent, which, in a semiclassical theory the ground state, accompanied by the emission of a photon of atomic motion, leads to an effective viscous friction that can cool the atom. into the environment. This process takes the form Latom D ŒC C C
Z
2
d2 rO ? h.rO ? / eik0 r? eik0 r? C ; (83.82)
where is the spontaneous decay rate, rO ? is the directional unit vector in the transversal plane, r? D r rO ? is the projection, and k0 D !0 =c is the recoil wave vector of the atom. The function h.rO ? / is present to describe the directional distribution of spontaneous decay of specific atomic transition. In general, it is not possible to solve the master equation (83.80) analytically, even for a single atom. If the population of the atomic excited state is negligible, one
83.6.2 Cavity Cooling Experiments Cavity cooling of single atoms was first demonstrated with 85 Rb atoms trapped in an FP cavity [61]. In this experiment, the trapping field was red-detuned with respect to the atom, while the cooling laser was a blue-detuned probing field with frequency !p > !0 . From Eqs. (83.84) and (83.86), the interaction between the probing laser and the atom induces to a blueshift of the cavity frequency by an amount of U0 jf .r/j2 , which then leads to an increase of the energy stored in the cavity field with a cost of kinetic energy
83
1222
of the atoms. The cooling effect was demonstrated via the observation of extended storage times and improved localization of 85 Rb atoms from time-resolved measurement of the cavity transmission signal. As a result, a cooling rate of ˇ=m D 21 kHz is achieved, which exceeds the estimated value of 4 kHz for blue-detuned Sisyphus cooling of a twolevel atom in free space, or with the Doppler cooling rate of 1.5 kHz at equivalent atomic saturation. With the advanced progresses of free-space laser cooling and trapping, other achievements have been obtained in the exploration and applications of single-atom cavity QED systems. These include the high-precision measurements demonstrated the basic cavity QED model in the optical domain [62, 63], the quantum anharmonic domain of the Jaynes–Cummings spectrum [64, 65], the generation of squeezed light [66], the development of a deterministic singlephoton source [67, 68], the realization of the long-time sought atom–photon quantum interface [69, 70] and single-atom quantum memory [71], and the realization of electromagnetically induced transparency with a single atom [72].
83.7 Cavity QED for Cold Atomic Gases Another new research direction in cavity QED is about the hybrid system of cold and ultracold atomic ensembles and high-finesse optical resonators. With the common coupling of atoms to the cavity field, there exits a long-range interatomic interaction mediated by coherent scattering of cavity photons, an ingredient which is usually absent in free-space cold atom experiments. As a result, this strongly interacting many-body system can present novel quantum phenomena with strong correlation. Besides, the coupling between the cavity mode and the atomic motion can induce a frequency shift of the cavity frequency and its backaction on mechanical motion, which may lead to a self-organization of atoms and an implementation of cavity optomechanics.
83.7.1 Atomic Ensembles in a Cavity
Q. He et al.
tion energy can still be enhanced coherently within an atomic ensemble. Next, we discuss the character of the cavitymediated interatomic interaction in two different pumping geometries, namely, pumping the cavity either directly or indirectly via light scattering off the laser-driven atoms. In the case of cavity pumping, the detuning between the driving laser and the dispersively shifted cavity resonance depends on the position of all atoms, which, in turn, experience the optical dipole force of the intracavity field. Under the adiabatic approximation with small atomic velocities and low saturation, the interaction potential among the ensemble of N atoms takes the form [73] V .r 1 ; : : : ; r N / D
„.!p !0 /jj2 .!p !0 / C .!p !C / tan1
2 .!p !0 /.!p !C C geff / : (83.87) .!p !0 / C .!p !C /
The collective coupling strength geff depends on the spatial distribution of the N atoms. This cavity-mediated long-range interatomic interaction gives rise to an asymmetric deformation of the normal-mode splitting, which has been observed experimentally [74]. The situation drastically changes if the system is atom pumped by a driving laser from a direction perpendicular to the cavity axis. In this case, photons are Rayleigh scattered by atoms into the cavity mode. Photons scattered from different atoms will interfere either destructively or constructively, depending on the relative positions of the atoms. Consider as a simple example, two atoms trapped in a cavity. If the two atoms are separated by odd integer multiples of the halfwavelength, the photons scattered into the cavity have the same magnitude but opposite signs, resulting in destructive interference and a vanishing cavity field intensity. On the other hand, if the two atoms are separated by even integer multiples of the half-wavelength, the photons scattered off the two atoms interfere constructively to yield a fourfold enhancement of the field intensity. This collective enhancement of scattering photons from multiple atoms is referred to as superradiance [75, 76]. Approximately, in the limit of U0 ! 0, the collective potential of the atomic ensemble reads
The interatomic interaction in a cavity is mediated by the cavity mode by its direct coupling to the atomic electricdipole moments. However, the nature is inherently different „.!p !C /jeff j2 from the induced dipole–dipole interaction, i.e., the van der V .r 1 ; : : : ; r N / D .!p !C /2 C 2 Walls interaction in free space. In a cavity, the interaction 2 32 strength does not decay with the interatomic separation and N X depends only on the local coupling of the atoms to the cavity 4 cos.kxj / cos.kzj /5 : (83.88) field. More importantly, the interaction is a global coupling j D1 with the ensemble of atoms collectively coupling to the cavity field and experiencing the resulting backaction. Thus, Here, we assume that the cavity mode and the pumping laser even in cases where the interaction between a single atom are both standing waves along the x and z-directions, respecand cavity mode is not strong enough, the collective interac- tively. Notice that if one moves along the cavity direction
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Entangled Atoms and Fields: Cavity QED
over a wavelength, the potential varies from its maximal value to zero, leading to a contrast of unity regardless of the atom–cavity coupling constant g. This is clearly distinct from the case of cavity pumping, where the atoms cause only a small modulation of the cavity intensity as g ! 0. This observation suggests that significant many-body effects can be encountered even in the weak-coupling regime in the atom pumping geometry. The long-range interaction between atoms is the origin of various collective dynamical effects. As an example, a thermal cloud of cold atoms interacting with a single-mode FP cavity shows a phase transition upon tuning the atomic pumping power P from a direction perpendicular to the cavity axis [77, 78]. For pumping power less than a critical value Pcr , the atoms distribute uniformly to minimize its kinetic energy. The light scattered from atoms in different positions then interferes destructively, and the average intensity of the cavity field is zero. When the pumping power P > Pcr , however, the atoms arrange themselves to form a checkerboard pattern to compensate the long-range interaction energy, so that the light interferes constructively to achieve a macroscopic cavity field. The self-organization of atoms in a transversally driven FP cavity was first demonstrated in 2003 [79]. In that experiment, a total of N 107 Cs atoms prepared at a temperature of 6 µK were pumped by a strong enough red-detuned laser to achieve collective emission of light into the cavity. As a result, the emission rate exceeded the free-space single-atom Rayleigh scattering rate by a factor of up to 103 , and the spatial configuration of atoms featured a spontaneous symmetry breaking into either the odd or even sites of a checkerboard pattern, which was revealed by measuring jumps in the phase of emitted cavity field relative to the pumping laser. In a ring-cavity with two propagating-wave modes, the selforganization of atoms was also observed with transverse pumping [80]. However, the system spontaneously breaks a continuous translational symmetry rather than a discrete Z2 symmetry as in the FP cavity.
83.7.2
Bose–Einstein Condensate in a Cavity
As compared to a thermal cloud of cold atoms, the system of a Bose–Einstein condensate (BEC) coupling with an optical cavity is of particular importance as it corresponds to some conceptually fundamental models of atoms coupling to a single-mode light field. When the atoms are Bose condensed into a single motional quantum state, the number of degrees of freedom required to describe the system can be substantially reduced. Therefore, the experimental platform can be used to mimic some model Hamiltonians of matter– light interaction, including the Tavis–Cummings or the Dicke
1223
model, as well as the generic model for cavity optomechanics. Under the mean-field approximation, which is valid in the presence of a cavity-mediated global coupling among all atoms, the condensate wave function and the cavity field are assumed by their amplitudes of expectation .r; t/ !
p Nc .r; t/ ;
a.t/ ! ˛.t/ ;
(83.89)
where Nc is the number of condensed atoms, and .r/ is the normalized wave function. These mean fields for atoms and cavity field satisfy the Gross–Pitaevskii-like equations i„@ t .r; t/ „2 r 2 D C Vtrap .r/ 2m C Nc gatom j.r; t/j2 C „U0 j˛.t/j2 cos2 .kz/ C „eff Œ˛.t/ C ˛ .t/ cos.kx/ cos.kz/ .r; t/ ; i@ t ˛.t/
D !C !p C Nc U0 hj cos2 .kz/ji i ˛.t/ C i C Nc eff hj cos.kx/ cos.kz/ji ;
(83.90)
where the interatomic contact interaction gatom D 4 „2 as =m with as the s-wave scattering length. On the one hand, the dynamics of the cavity field involves spatial averages over the condensate density distribution. On the other hand, the atoms are also affected by the backaction from the cavity field according to the potential-like terms, which depend on the amplitude ˛ and intensity j˛j2 of the cavity field in the Gross–Pitaevskii equation for atoms. The mean-field equations are usually solved numerically to obtain the timeevolution of the system and the steady-state solutions for both atoms and cavity field [81]. As in a thermal atomic ensemble, the BEC trapped inside a cavity can also self-organize to emit an intracavity field collectively. The key difference in the case of BEC is that the atomic motion is also quantized, and the transition is now between a uniform distribution and a periodic array of atoms in the cavity-induced lattice potential. As a consequence, the transition point is determined by the competition of kinetic energy and potential energy associated with the spatial modulation of atomic density. Under the mean-field approximation, the steady state of the system can be determined by the numerical solutions of the Gross–Pitaevskii-like equations for both the cavity field ˛ and the atomic mode .r/. By assuming for simplicity that the atomic motion is only along the cavity axis and the cavity pumping D 0, one can define an order parameter # D hj cos.kx/ji, which describes the nonuniform spatial
83
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which contains a sum over all atoms with index j that couple to the field mode. It is instructive to introduce the collective spin operators
Order parameter Θ 1.0
0.8
1 X .j / Sz D ; 2 j D1 z N
0.6
S˙ D
0.4
N X
.j /
˙ ;
(83.92)
j D1
0.2
0 0
65
130 195 √N η (in units of ωR)
together with the total spin operator S D .Sx ; Sy ; Sz /, where N is the number of atoms, S˙ D Sx ˙ iSy , and .j / are the Pauli spin operators for the j -th atoms. In this notation, the Dicke Hamiltonian becomes
HDicke D „!0 Sz C „!C a a C „g SC a C a S : Fig. 83.12 The steady-state order parameter # plotted as a function of the effective cavity pumping strength eff , indicating the selforganization of a Bose–Einstein condensate in an FP cavity [81]
modulation of the atoms. As can be seen from a typical result of the numerical solution shown in Fig. 83.12, the order parameter takes a nonzero value beyond a critical pumping strength eff , indicating self-organization of the density of atoms. Self-organization of a BEC was experimentally achieved in 2010 [82]. In that experiment, a BEC of about 105 atoms was trapped inside a high-finesse FP cavity and was illuminated from a transversal direction by a far red-detuned standing-wave laser beam. By gradually increasing the power of the transverse laser beam beyond a critical value, a sharp rise of the intracavity field intensity and a macroscopic populations in the atomic momentum states .px; pz/ D .˙„k; ˙„k/ were observed, indicating a transition to the self-organization phase. Above this critical pump power, the relative phase between pump field and cavity field was observed to stay constant, which demonstrates that the system reached a steady state. Self-organization of a laser-driven BEC in an optical cavity can also be considered as a realization of the Dicke quantum-phase transition in an open system, where the quantized atomic motion acts as a macroscopic spin, which strongly couples to the cavity field. The Dicke model, or equivalently, the Tavis–Cummings model, is proposed to study the collective interaction between a set of two-level atoms and a single light mode
HDicke
N X 1 D „!0 z.j / C „!C a a 2 j D1 N X .j / C a C a .j / ; C „g j D1
(83.93) Here, we assumed that all atoms have the same frequency !0 and that they are coupled with the same strength g. Note that the eigenstates of the Dicke Hamiltonian are highly degenerate, reflecting the fact that there are many possible ways to accommodate a fixed number of excitations. The eigenstates of the Dicke Hamiltonian, usually referred to as Dicke states, can be labeled by the quantum numbers of the spin operators S 2 jJ; M i D J.J C 1/jJ; M i and Sz jJ; M i D M jJ; M i, where J D 0; 1; : : : ; N=2, and M D J; J C 1; : : : ; J . In the weak excitation limit of J C M N , the Dicke states behave like a harmonic oscillator, as can be seen from jhJ; M C 1jSC jJ; M ij2 D .J C M C 1/.J M / .J C M C 1/N :
(83.94)
Thus, in this regime, the Dicke model can be regarded as p a pair of harmonic oscillators coupled at a rate of N g. If there is only one excitation in the system, the eigenstate is equivalent to that of the JC model, except that the p coupling strength is collectively enhanced by a factor of N . However, for the case of more than one excitation, the energy spectrum acquires a significant difference of anharmonicity. For the case of large number of excitations, the Dicke Hamiltonian has been investigated extensively in the context of superradiance, during which process a macroscopic number of the atomic ensemble decays collectively and emits a large pulse of radiation. This can be seen from SC S jJ; M i D .J C M /.J M C 1/jJ; M i ; (83.95)
which is proportional to the spontaneous emission rate of the ensemble. For the case of M D J , the emission rate is pro(83.91) portional to N as J D N=2. For the case of M D 0, however, it is proportional to N 2 . This increase can be understood as
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Entangled Atoms and Fields: Cavity QED
1225
a result of strong correlations between the atoms in the ensemble during the emission process. The self-organization of a BEC-cavity system can be regarded as the Dicke model by considering a pair of motional states as two pseudo-spin states [82, 83]. The two motional states are given by the uniform condensate mode jpx ; pz i D j0; 0i and the coherent superposition of the four momentum states j˙„k; ˙„ki, where x and z denote the cavity and pump directions, respectively. Coherent light scattering between the transverse pump beam and the cavity mode couples these two momentum states via two distinguishable Raman channels, resulting in a tunable interaction between the cavity mode and the corresponding collective spin degree of freedom. Experimentally, the phase boundary was mapped out as a function of pump-cavity detuning !p !C as shown in Fig. 83.13 [82].
detuned from the cavity field and hence contribute as a heat bath. Another route to realize the cavity optomechanics Hamiltonian is reported in 2008 [86]. In that experiment, a BEC is prepared in an external harmonic trapping potential, extending over several periods of the cavity standing-wave mode structure. Initially, all condensate atoms are prepared in the zero-momentum state jp D 0i. The dispersive interaction with the cavity field diffracts atoms into the symmetric superposition of momentum states j˙2„ki along the cavity axis. If the diffraction into higher-order momentum modes can be neglected, the dynamics of the coupled system can be well described by the optomechanics Hamiltonian Eq. (83.96).
83.8 83.7.3 Cavity Optomechanics with Cold Atoms The hybrid atom–cavity system can also be used to study an important model of cavity optomechanics, where a harmonically suspended mechanical element interacts with an intracavity single-mode field. In certain limiting cases, where the motional degrees of freedom of atoms can be reduced to a single-mode harmonic oscillator [84], the effective Hamiltonian of Eq. (83.84) can be significantly simplified, leading to a generic form of the cavity optomechanics Hamiltonian HOM
G D „!m c c „ .!p !C / .c C c/ a a 2
C i„.a a/ ;
(83.96)
where c and c denote creation and annihilation operators of the mechanical oscillator at frequency !m , G is the dispersive shift of the cavity frequency induced by the atomic density modulation, and is the pumping strength along the cavity axis. The key ingredient of the experimental realization of the cavity optomechanics Hamiltonian Eq. (83.96) is that the cavity field must affect and sense predominantly a single collective motional mode of the atomic ensemble, which hence can be regarded as the harmonically suspended mechanical element. One possible way to realize this condition is to trap ultracold atoms in a far-detuned intracavity lattice potential such that the atoms form a stack of tightly confined atom clouds [85]. Each atom cloud is harmonically suspended with oscillation frequency !m and extends along the cavity axis by only a fraction of the optical wavelength. A cavity mode, whose periodicity differs from that of the trapping lattice potential, couples strongly to a single collective center-of-mass mode of the atomic stack. All remaining collective modes are
Applications of Cavity QED
83
83.8.1 Quantum Nondemolition (QND) Counting of Photons When the object of interest consists of only a few atoms and a few photons, the puzzling consequences of quantum mechanical measurement become visible. In the case of the micromaser, the information on the state of the field is imprinted in a subtle way on the atomic beam. While photon counting is normally a destructive operation, the dispersive part of the photon–atom interaction may be used to determine the photon number inside a resonator without altering it on average. Dispersive effects shift the phase of an oscillating atomic dipole without changing its state. The phase shift due to the field in the resonator can be measured in a Ramsey-type experiment [87]. Consider an atom with two transitions jgi ! jei and jei ! jii. The first is far from resonance with the cavity, and the second is close to resonance but with a detuning ıi e D ! !i e large enough so as not to change the cavity photon number as the atom passes through. The dynamic Stark effect of the jgi ! jei transition frequency due to state jii is then !eg
2
p gi e n C 1 D : ıi e
(83.97)
If the resonator is now placed between the two Ramsey cavities, which are tuned to !R !eg , such that the polarization of the jei ! jgi transition is rotated by =2, then the additional phase shift !eg and, hence, the photon number n can be measured. Here, is the transit time through the optical resonator. Since Rydberg states have a large coupling constant g , the phase shift due to a single atom is detectable [87]. This proposal was demonstrated experimentally in 2007 at ENS [88]. With the ability of developing super high-
1226 Fig. 83.13 Dicke-model phase diagram. The mean-intracavity photon number is finite in the superradiant phase with large pumping power and small pumpcavity detuning [82]
Q. He et al.
Pump-cavity detuning (MHz) 0 5 0
10
15
Pump lattice depth (Er) 25 30
20
a
n–
–10
45 30 15 0 0
–20
225 150 75 0
c
b
3
6
9 0 Time (ms)
3
2
3 Time (ms)
–30
–40
Mean photon number n– 10 0 101 102 103
–50 0
200
quality microwave cavity, the experimental group successfully performs a QND measurement on single photons via dispersive interaction of single atoms in circular Rydberg states. The atoms have a high principal quantum number n D 50, and the highest possible angular and magnetic quantum numbers .l D n 1; jml j D n 1/, hence acquire a life time of the order D 30 ms. The experiment can witness the birth, life, and death of a photon nondestructively. A complete measurement of n requires a sequence of N atoms because a single Ramsey measurement only determines whether the atom is in state jei or jgi, and hence !eg to within ˙ =2. Since each measurement provides one binary bit of information, a sequence of N measurements can, in principle, distinguish 2N possible Fock states for the photon field. However, with a monoenergetic beam, integral multiples of 2 remain undetermined. A distribution of velocities, and hence transit times, is, therefore, desirable. An entropy reduction strategy for selecting an optimal velocity distribution, based on the outcome of previous measurements, is described in [89]. The experimental demonstration was also reported in 2007 by the same group at ENS [90], where the experimental setup was refined to distinguish states with photon number n 7. As a consequence of the uncertainty principle, a measurement of the photon number destroys all information about the phase of the field. In the present case, the noise in the conjugate variable (the phase) is prevented from coupling back on the measured one, and hence the measurement is called a quantum nondemolition experiment. Many other aspects of phase diffusion, entangled states, and quantum measurements in the micromaser are discussed in [91].
400
600
800
1000
1200 Pump power (μW)
83.8.2 Detecting and Trapping Atoms Through Strong Coupling One of the key ingredients of cavity QED experiments is to deterministically localize atoms at desired positions in a cavity, where the atom–photon coupling can be well calculated and tuned. From Fig. 83.8 it is obvious that an atom traveling through the cavity will modify the transmission properties of this cavity. The ability of tuning strong coupling, thus, enables the experimenter to detect the presence of a single atom dispersively by monitoring cavity transmission or reflection. Laser-cooled atoms have low velocities and spend sufficient time in the cavity even in free flight to generate the transmission signal shown in Fig. 83.14. The signals correspond to individual atom transits, and the shape depends on the detuning of the probe laser from the resonantly interacting cavity–atom system [56, 92]. Thus, the strongly coupled cavity QED system can work as a sensitive single-atom detector, and can help us to extract the temperature and the statistical properties of cold atoms, which have great potential in time-resolved atom–cavity microscopy and in tracking single-atom trajectories. The same scheme can also be applied to monitor a specific collective motion in an atomic ensemble [85, 93, 94]. In these experiments, the cavity mode is shifted in frequency by the strong interaction with the center-of-mass motion of the atomic ensemble. This provides the ability to realize continuous nondestructive measurement of a quantum many-body system. The strong coupling between atom and cavity mode can also be used to trap atoms at specified positions. If an atom
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a T(∆) 1
0.1
0.01
0
0.2
0.4
0.6
0.8 1 Time (ms)
0.2
0.4
0.6
0.8 1 Time (ms)
b T(∆) 2 1.5 1 0.5 0
0
c T(∆) 4 3
Cavity-QED systems offer an attractive light–matter process for the generation of such photon-bit-streams, or singlephoton sources [99]. In such devices, a single-photon state can, for instance, be created by Raman processes involving a classical field, which serves as the control parameter for the process, and the vacuum field of the optical resonator. The Raman process leaves a single photon in the cavity, which only weakly interacts with the atom. If the resonator has suitable transmission properties, this photon will then escape with predetermined frequency, shape, and propagation direction. Deterministic single-photon sources have been realized with quantum dots [100, 101], single molecules [102], and also with slow [103, 104] or trapped [105, 106] cold atoms and ions [97, 107] inside optical cavities. The high efficiency of photon sources also paves the route towards quantum memory and quantum network, which is essential for providing cluster states in one-way quantum computing [92] and for the quantum simulation of complex solid-state systems [93]. Successive operations of photon generation, photon storage, and photon retrieval were successfully demonstrated in a hybrid atom–cavity system [71].
2 1 0
83.8.4 Generation of Entanglement 0
0.2
0.4
0.6
0.8 1 Time (ms)
Fig. 83.14 Transmission of a strongly coupled cavity for individual atom transits. Cesium atoms and cavity are in perfect resonance at D 852 nm, while the probe laser is increasingly detuned to the red side of the resonance from top to bottom [125]
absorbs a photon inside the cavity, a strong dipole force can be exerted due to the inhomogeneous field distribution of the cavity mode. Trapping of atoms with a single photon was achieved [92], and from the time variation of the cavity transmission, a reconstruction of atomic trajectories became possible. A similar mechanism is also implemented in a hybrid system of ion trap and optical cavity to localize single ions [95, 96]. This has allowed for very long trapping times of more than 90 min, as well as excellent control over the position of trapped ions within a subwavelength precision [97]. Recently, position control was also demonstrated individually for two ions trapped in the same cavity [98].
83.8.3 Single-Photon Sources
In the middle of the 1990s, it was realized that fully controlled quantum systems can be used to implement a revolutionary type of information processing, now called quantum computing [108]. From the beginning, cavity QED has conceptually played an important role for experimental realizations, since it offers a route to manipulate, in principle, all physical parameters of a coherently interacting system. With the well-established microwave-cavity–Rydberg–atom system, it was proven that the generation of correlated and nonlocal, so-called entangled quantum states, is possible [109]. The first application of cavity QED was the transfer of the strong coupling idea to the combined internal and motional quantum states of trapped ions [110]. Here, the harmonic oscillation of the ion replaces the electric field of the conventional cavity-QED system. This quantum gate was realized with a system of two trapped ions coupled to each other by Coulomb forces [111]. Ideas about how to use the strong coupling of atoms and photons [112–114] for the generation of atom–photon, photon–photon, or atom–atom systems (by insertion of more than one atom) abound and become possible with the aid of experimental capabilities in the preparation and control of atoms in high-finesse cavities.
Coherent laser fields are considered the ultimate source of classical radiation fields, and they are characterized by the random arrival time of photons. Nonclassical light sources Atom–Photon Entanglement with, for instance, a regularized stream of photons offer in- A key advantage of the hybrid atom–cavity QED system is teresting properties for low-noise measurement applications. the potential to work as a quantum interface through which
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the quantum state can be faithfully transferred from one References medium to another. By driving a vacuum-stimulated Raman 1. Haroche, S.: Cavity QED is reviewed in detail. In: Dalibard, J. adiabatic passage into a superposition of two atomic ground (ed.) Fundamental Systems in Quantum Optics. Elsevier, Amsterstates, for example with different orientations of the atomic dam (1992) 87 spin of Rb atoms [70], the internal state of a single atom 2. Meschede, D.: Phys. Rep. 211(5), 201–250 (1992) can be entangled with the polarization state of a single pho3. Berman, P. (ed.): Cavity Quantum Electrodynamics. Academic Press, Amsterdam (1994) ton.
Photon–Photon Entanglement Starting from the atom–photon entangled state, one can drive another vacuum-simulated Raman adiabatic passage into a single atomic ground state with a well-defined spin orientation to map the quantum state of the atom into a second single photon. This process disentangles the atom and the light and creates an entangled photon pair. The two photons are emitted one after the other into the same spatial mode, hence have never overlapped with each other [115]. Atom–Atom Entanglement By reversing the role of field and atom, the atom–photon entanglement can also be transferred to a second atom to create atom–atom entanglement. Multiparticle entanglement is considered as a crucial resource of quantum simulation, quantum computation, and quantum-enhanced metrology. The largest number of atoms ever to be entangled in an FP cavity is about 3000 [116] and is more than 40 in an optical fiber cavity [117]. The entanglement between two ions and two atomic ensembles via coupling to cavity mode were also demonstrated in experiments [118, 119]. Quantum Network Using the aforementioned atom–photon entangled state as building blocks, an elementary quantum network can be implemented with two fiber-linked optical cavities, each containing a single trapped atom as a stationary quantum node. This scheme offers a clear perspective of both addressability and scalability because the atoms are trapped in their corresponding cavities, and more constituent cavities can be added to an existing network without much apparent complication. Besides, the component cavities can, in principle, be arranged in any geometry and two-party links can be established at will, both in time and space. Elementary quantum network links implementing teleportation protocols between remote trapped atoms and atom– photon quantum-gate operations have been demonstrated in experiments [115, 120–122]. Besides, quantum networking between two cavities has also been experimentally demonstrated with atomic ensembles [123, 124]. These achievements represent a big step towards the goal of realizing an elementary quantum network and a feasible quantum computing system.
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Q. He et al. Qiongyi He Qiongyi He received her PhD from Jilin University in Changchun, China in 2007 and then worked at the University of Queensland in Brisbane, at Swinburne University of Technology in Melbourne, Australia, and at Peking University in China. She now works on quantum optics and information, with a focus on quantum entanglement, nonlocality, and quantum information applications. Wei Zhang Wei Zhang received his PhD from Georgia Institute of Technology in 2006 and then worked at the University of Michigan at Ann Arbor and at Renmin University of China. He now works on quantum simulation and computation in artificial quantum systems, including cold quantum gases, optical microcavities, and trapped ions.
Dieter Meschede Professor Dieter Meschede teaches at the Institute for Applied Physics in Bonn. After his studies in Hanover and Cologne and having been awarded his Dr rer nat in Munich in 1984, he first worked at Yale University. Then he became Senior Scientist at the MPI for Quantum Optics, Garching. He has been Professor of Physics since 1990, first in Hanover and since 1994 in Bonn. Axel Schenzle Professor Schenzle worked on various aspects of theoretical quantum optics, the description of classical and quantum-mechanical noise in microscopic and mesoscopic systems, Bose–Einstein condensation, quantum information theory, and quantum computing and decoherence. He was Deputy Rector of the University of Munich and Dean for many years. He passed away in 2016.
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Quantum Optical Tests of the Foundations of Physics L. Krister Shalm, Aephraim M. Steinberg, Paul G. Kwiat, and Raymond Y. Chiao
Contents
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84.6.2 84.6.3 84.6.4 84.6.5 84.6.6 84.6.7 84.6.8 84.6.9
Bell’s Inequalities as a Game . . . . . . . . . . Loopholes in Bell Tests . . . . . . . . . . . . . Closing the Loopholes in Bell Tests . . . . . . Polarization-Based Entangled Sources . . . . Other Entanglement Sources for Bell Tests . Advanced Experimental Tests of Nonlocality Nonlocality Without Inequalities . . . . . . . . Connection to Quantum Information . . . . .
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Single-Photon Tunneling Time . . . . . . . . . . . An Application of EPR Correlations to Time Measurements . . . . . . . . . . . . . . . . . Superluminal Tunneling Times . . . . . . . . . . . . Tunneling Delay in a Multilayer Dielectric Mirror Interpretation of Tunneling Time . . . . . . . . . . Other Fast and Slow Light Schemes . . . . . . . . .
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Gravity and Quantum Optics . . . . . . . . . . . . . . 1250 Gravitational Wave Detection . . . . . . . . . . . . . . . 1250 Gravity and Quantum Information . . . . . . . . . . . . . 1251
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Introduction: The Photon Hypothesis . . . . . . . . . 1232
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Quantum Properties of Light . . . Vacuum Fluctuations: Cavity QED Two-Photon Light Sources . . . . . Squeezed States of Light . . . . . .
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Nonclassical Interference . . . . . . . . . . . Single-Photon and Matter-Wave Interference “Nonlocal” Interference Effects and Energy-Time Uncertainty . . . . . . . . . . . . Two-Photon Interference . . . . . . . . . . . .
84.3.3 84.4 84.4.1 84.4.2 84.4.3 84.4.4
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Complementarity and Coherence . . . Wave-Particle Duality . . . . . . . . . . . Quantum Eraser . . . . . . . . . . . . . . . Vacuum-Induced Coherence . . . . . . . Suppression of Spontaneous Parametric Downconversion . . . . . . . . . . . . . .
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Measurements in Quantum Mechanics Quantum (Anti-)Zeno Effect . . . . . . . . Quantum Nondemolition . . . . . . . . . . Quantum Interrogation . . . . . . . . . . . . Weak Measurements . . . . . . . . . . . . . Direct Measurements of a Wave Function
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84.6 84.6.1
The EPR Paradox and Bell’s Inequalities . . . . . . . 1240 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . 1240
L. K. Shalm () Dept. of Physics, University of Colorado at Boulder Boulder, CO, USA e-mail: [email protected] A. M. Steinberg Dept. of Physics, University of Toronto Toronto, ON, Canada e-mail: [email protected] P. G. Kwiat Dept. of Physics, University of Illinois at Urbana-Champaign Urbana, IL, USA e-mail: [email protected] R. Y. Chiao Departments of Physics and Engineering, University of California Merced Merced, CA, USA e-mail: [email protected]
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Abstract
Quantum mechanics began with the solution of the problem of blackbody radiation by Planck’s quantum hypothesis: in the interaction of light with matter, energy can only be exchanged between the light in a cavity, and the atoms in the walls of the cavity by the discrete amount E D h, where h is Planck’s constant, and is the frequency of the light. Einstein, in his treatment of the photoelectric effect, reinterpreted this equation to mean that a beam of light consists of particles (“light quanta”) with energy h. The Compton effect supported this particle viewpoint of light by demonstrating that photons carried momentum, as well as energy. In this way, the wave-particle duality of quanta made its first appearance in connection with the properties of light. It might seem that the introduction of the concept of the photon as a particle would necessarily also introduce the concept of locality into the quantum world. However, in view of observed violations of Bell’s inequalities, exactly the opposite seems to be true. Here, we review some
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_84
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recent results in quantum optics, which elucidate nonlo- where N123 is the rate of triple-coincidences between decality and other fundamental issues in physics. tectors D1 , D2 , and D3 ; N1 is the singles rate at D1 ; and N12 and N13 are double-coincidence rates. Then from Schwarz’s inequality [6, 9, 10], ˛ 1 for any classical wave. Keywords In essence, since the wave divides smoothly, the coinciwave packet entangle state beamsplitter Bell inequality dence rate between D and D is never smaller than the 2 3 weak measurement “accidental” coincidence rate, even when measurements are conditioned on an event at D1 . (The Hanbury-Brown and Twiss experiment [11] can be explained classically, because 84.1 Introduction: The Photon Hypothesis thermal fluctuations lead to “bunching”, or a mean coincidence rate, which is greater than the mean accidental rate; In spite of the successes of quantum electrodynamics, and cf. Sect. 74.4.) By contrast, the indivisibility of the photon of the standard model in particle physics, there was for leads to strong anticorrelations between D2 and D3 , making a time considerable resistance to the concept of the photon as ˛ arbitrarily small. In agreement with this quantum mechania particle. Many papers have been written trying to explain cal picture, Grangier et al. observed a 13-standard-deviation all optical phenomena semiclassically, i.e., with the light violation of the inequality [9], corroborating the notion of viewed as a classical wave, and the atoms treated quantum the “collapse of the wave packet” as proposed by Heisenmechanically [1–4]. We first present some quantum optics berg [12]. phenomena that exclude this semiclassical viewpoint. In an early experiment, Taylor reduced the intensity of a thermal light source in Young’s two-slit experiment, until, on the average, there was only a single photon passing 84.2 Quantum Properties of Light through the two slits at a time. He then observed a twoslit interference pattern, which was identical to that for 84.2.1 Vacuum Fluctuations: Cavity QED a more intense classical beam of light. In Dirac’s words, the apparent conclusion is that “each photon then interferes The above considerations necessitate the quantization of the only with itself” [5]. However, a coherent state, no matter electromagnetic field, which in turn leads to the concept of how strongly attenuated, always remains a coherent state vacuum fluctuations [4] (Sect. 82.2). Difficulties with this (Sect. 83.4.2); since a thermal light source can be mod- idea, such as the implied infinite zero-point energy of the unieled as a statistical ensemble of coherent states, a stochastic verse, have led some researchers to attempt to dispense with classical wave model yields complete agreement with Tay- this concept altogether, along with that of the photon, in evlor’s observations. The one-by-one darkening of grains of ery explanation of electromagnetic interactions with matter. film can be explained by treating the matter alone quantum Of course, it is impossible to explain all phenomena, such as mechanically [2]; consequently, the concept of the photon spontaneous emission and the Lamb shift, without some kind need not be invoked, and the claim that this experiment of fluctuating electromagnetic fields (Chap. 70), but one can demonstrates quantum interference of individual photons is go a long way with an ad hoc ambient classical electromagnetic noise-field filling all of space, in conjunction with the unwarranted [6]. This weakness in Taylor’s experiment can be removed radiation reaction [1, 4]. In particular, even the Casimir attraction between two by the use of nonclassical light sources; as discussed by conducting plates can be explained semiclassically in terms Glauber [7], classical predictions diverge from quantum ones of dipole forces between electrons in each plate as they only when one considers counting statistics or photon correlaundergo zero-point motion and induce image charges in tions. In particular, two-photon light sources, combined with the other plate. Nevertheless, the effects of cavity QED coincidence detection, allow the production of single-photon (n D 1 Fock) states with near certainty. In the first such exper- (Chap. 71) [13, 14], including the influence of cavity-induced iment [8], two photons, produced in an atomic cascade within nanoseconds of each other, impinged on two beamsplitters 2 and were then detected in coincidence by means of four photomultipliers placed at all possible exit ports. In a simplified version of this experiment [9], one of the beamsplitters and its two detectors are replaced with a single 1 3 detector D1 (Fig. 84.1). We define the anticorrelation parameter ˛ N123 N1 =N12 N13 ;
(84.1) Fig. 84.1 Triple-coincidence setup of Grangier et al. [9]
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boundary conditions on energy levels and spontaneous emission rates, are most easily unified via quantization of the electromagnetic field. By coupling highly excited Rydberg atoms to photons in a high-finesse superconducting microwave cavity, Brune et al. observed single-photon-driven Rabi oscillations [15] and used these to study decoherence effects [16], atom– photon entanglement [17], and quantum nondemolition measurements [18] (Sect. 84.5.2). Kimble et al. [19] and Rempe et al. [20] performed similar experiments, coupling atoms to small optical cavities, and even trapping the atoms with light fields at the single-photon level [21, 22]. By monitoring the amplitude of the light transmitted through the cavity (which depends on the precise location of the atom inside the cavity volume), the trajectory of the atom can be determined with ultrahigh resolution, much smaller than an optical wavelength [23, 24].
84.2.2
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Fig. 84.2 Conical emissions of downconversion from a nonlinear crystal. Photon energy depends on the cone opening angle, and conjugate photons lie on opposite sides of the axis, e.g., the inner “circle” orange photon is conjugate to the outer “circle” deep-red photon, etc.
is a weaker effect. Using this third-order nonlinearity, it is possible to create a correlated signal and idler photon pair through the annihilation of two pump photons in a process known as four-wave mixing.
Two-Photon Light Sources
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84.2.3 Squeezed States of Light The quantum aspects of electromagnetism are made more striking with a two-photon light source, for example, when two highly correlated photons are produced in spontaneous parametric downconversion, or parametric fluorescence [25– 27], or four-wave mixing. In this process, a “pump” photon produced in a laser spontaneously decays inside a crystal with a .2/ nonlinearity into two highly correlated lower energy photons, conventionally called the “signal” and the “idler” (Sect. 76.3.6). (The quantum state of the light is more correctly written as j i / jvacuumi C j1is j1ii C 2 j2is j2ii , but since the amplitude of the downconversion process itself is very weak ( is of order 104 to 106 ), one often neglects the terms containing two or more pairs. However, recent experiments have begun to exploit these higher-order terms, e.g., to investigate multiphoton quantum entangled states of up to 10 and 12 photons [28].). As shown in Fig. 84.2, a rainbow of colored cones is produced around an axis defined by the direction of the pump beam (for the case of type-I phase-matching), with the correlated downconversion photons always emitted on opposite sides of the pump (Sect. 76.2.2). Their emission times are within femtoseconds of each other, so that detection of one photon implies with near certainty that there is exactly one quantum present in the conjugate mode [29]. In type-I phase-matching, the correlated photons share the same polarization, while in type-II phase-matching, they have orthogonal polarizations (Sect. 76.2.2). We will see below (Sect. 84.6.5) how both of these can enable the production of photons that are entangled in polarization, as well as in other degrees of freedom. Downconversion can only take place in a crystal having inversion symmetry; otherwise .2/ D 0. However, the next order nonlinearity (.3/ ) can occur in any optical material, although it
The creation of correlated photon pairs is closely related to the process of quadrature-squeezed light production (Sect. 82.6 and the review in [30]). For example, when the gain arising from parametric amplification in a downconversion crystal becomes large, there is a transition from spontaneous to stimulated emission of pairs. This gain is dependent on the phase of amplified light relative to the phase of the pump light. As a result, the vacuum fluctuations are reduced (“squeezed”) below the standard quantum limit in one quadrature but increased in the other, in such a way as to preserve the minimum uncertainty-principle product [31]. This periodicity of the fluctuations at 2! is a direct consequence of the fact that the light is a superposition of states differing in energy by 2„! – the quadrature-squeezed vacuum state ji D exp
1 1 aa a a j0i 2 2
(84.2)
represents a vacuum state transformed by the creation (a a ) and destruction (aa) of photons two at a time. Essentially any optical processes operating on photon pairs (e.g., four-wave mixing [32, 33]) can also produce such squeezing. Amplitude squeezing involves preparation of states with a well-defined photon number, i.e., states lacking the Poisson fluctuations of the coherent state. The possibility of producing such states (e.g., via a constant-current-driven semiconductor laser [34], or more recently a quantum dotbased source [35–38]) demonstrates that “shot noise” in photodetection should not be thought of as merely the result of the probabilistic (à la Fermi’s Golden Rule; Sect. 73.5) excitation of quantum mechanical atoms in a classical field but
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as representing real properties of the electromagnetic field, accessible to experimental control. This longstanding field of fundamental research has at long last begun to be applied to gravitational-wave detection (Sect. 84.8) and to quantum metrology [39].
84.3
Nonclassical Interference
84.3.1 Single-Photon and Matter-Wave Interference The first truly one-photon interference experiment [9] used the cascade source discussed in Sect. 84.1. One of the photons was directed to a “trigger” detector, while the other, thus prepared in an n D 1 Fock state, was sent through a Mach–Zehnder interferometer. The output photon, detected in coincidence with the trigger photon, showed fringes with a visibility > 98%. Dirac’s statement that a single photon interferes with itself is thus verified. The pioneering atomic matter-wave interferometry experiments of Pritchard used standing-light gratings and nanofabricated diffraction gratings to construct Mach–Zehnder-type interferometers for sodium atoms [40] and molecules [41] from a supersonic source. Chu and Kasevich significantly advanced the field by the use of STIRAP (stimulated Raman pulses) to produce coherent beamsplitters for cold atomic beams, also in a Mach–Zehnder-type interferometer, but whose source were cesium atoms cooled in and launched from a magneto-optical trap (MOT) [42]. Matter-wave interferometry has since been applied to precision measurements of the acceleration g due to Earth’s gravity [43], gravity gradiometry [44], and Sagnac matter-wave gyroscopes [45, 46]. To date some of the largest molecular systems to display quantum interference of this sort are large molecules like carbon 60 (“buckyball”) and carbon 70 [47]. These are significant in that the average de Broglie wavelength of the molecules, emitted from an oven, was 2.8 pm, actually about 350 times smaller than the molecule itself. Multislit diffraction has also been demonstrated with the biological molecule porphyrin and with fluorofullerenes (C60 F48 ) [48]. With a mass of 1632 amu, the latter are currently the largest single objects to display interference. Marshall, Simon, Penrose, and Bouwmeester separately suggested that it may be possible to put a micron-scale mirror (with 1014 atoms) into a superposition of resolvable spatial locations [49] – the mirror, part of a high-finesse optical cavity forming one arm of a Michelson interferometer, could be mounted on a highquality mechanical oscillator, whereby the interaction with a single photon would change the frequency of the oscillator. During the last decade, progress along these lines has been dramatic. The quantum optical control of nano and micromechanical devices through motional coupling to optical cavity
fields or superconducting circuits, i.e., quantum optomechanics, has now been firmly established [50–52]. Recent examples include ground-state laser cooling of the center of mass motion of a micromechanical device comprising up to 1011 atoms [53], as well as the generation of quantumsqueezed states [54–56] and of non-Gaussian states [57, 58] of motion. Regarding interference phenomena, Ramsey interference has been observed with a micromechanical device coupled to a superconducting qubit [57], and quantum entanglement comprising one [59] and two [60–62] mechanical resonators has been demonstrated, including the violation of a Bell inequality [63]. Still, while masses of these solidstate resonators comprise the collective motion of up to 1015 atoms, the displacement involved in the demonstrated interference experiments is minute (specifically, on the order of one or a few mechanical quanta, which translates to nuclear-scale spatial displacements). Larger spatial displacements may be possible using levitated solid-state systems coupled to cavities [64], which is now becoming a focus of intense research and which holds even the promise of matterwave interference in this extreme large-mass regime [65, 66]. Two other systems, demonstrating Bose–Einstein condensation (BEC), have also produced evidence of macroscopic quantum coherence. In the experiments of Andrews et al., atoms from two different atomic vapor BEC clouds were allowed to fall onto the same detection region, and displayed interference fringes [67] (more recently interference from an array of 30 independent BECs was observed [68]). In some ways, this is the matter-wave equivalent of the famous Pfleegor–Mandel experiment [69], in which light from two separate lasers displays interference, even when attenuated to the single-photon level. The explanation in terms of the indistinguishability of the underlying processes is that one cannot ascertain from which laser source a given photon originated. However, this explanation must be applied carefully to the situation of the two atomic BECs; unlike the lasers, the BEC clouds can – at least in principle – be prepared with a definite number of atoms, and it would therefore seem that one could, in principle, determine which cloud emitted a given detected atom. However, this determinacy is rapidly lost after a few atoms are detected [70]. Once the number becomes uncertain, a well-defined relative phase of the two BECs is established, according to the number-phase uncertainty relation .N2 N1 /.2 1 / 1=2. Finally, quantum coherence (although not explicitly spatial interference as in the previous examples) has been detected in the operation of a Josephson-junction linked superconducting loop. Specifically, the group of Mooij was able to prepare a superposition of clockwise and counter-clockwise circulating electrical currents [71]. The 0:5 A currents corresponded to the motion of millions of Cooper pairs. This superconducting system has become a major focus of quantum computing efforts (Chap. 85), with many groups and
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companies around the world making rapid progress with superconducting quantum bits [72–83].
F1 UV laser
D1
Idler KDP
84.3.2 “Nonlocal” Interference Effects and Energy-Time Uncertainty
Beam dump Signal
The energy-time uncertainty principle, Et „=2 has been tested in a downconversion interference experiment [84]. The downconversion process conserves energy and momentum:
F2 D2 Fixed QWP
„!0 D „!1 C „!2 ;
(84.3)
„k0 „k1 C „k2 ;
(84.4)
where „!0 („k0 ) is the energy (momentum) of the parent photon, and „!1 („k1 ) and „!2 („k2 ) are the energies (momenta) of the daughter photons; k1 and k2 sum to k0 to within an uncertainty given by the reciprocal of the crystal length [85]. Since there are many ways of partitioning the parent photon’s energy, each daughter photon may have a broad spectrum, and hence a wave packet narrow in time; indeed, detecting one of the photons with a fast detector will localize the other to a narrow pulse. However, !1 C !2 D !0 is extremely well defined, so that the difference in the daughter photons’ arrival times and the sum of their energies can be simultaneously known to high precision. Thus, the daughter photons of a parent photon of sharp energy E0 are in an energy-entangled state, a nonfactorizable sum of product states [86]: ZE0 dE A.E/jEijE0 Ei ;
j i D
(84.5)
0
where A.E/ is the probability amplitude for the production of two photons of energies E and E0 E. Measuring the energy of one of the photons to be E1 “collapses” the system to the state jE1 ijE0 E1 i, implying an instantaneous increase of the width of the other photon’s wave packet. In the experiment, one photon was used as a trigger, while the other was sent into an adjustable Michelson interferometer (Fig. 84.3) used to measure its coherence length. If the trigger photon was detected after passing through an interference filter F1 of narrow width E, then the conjugate photon occupied a broad wave packet of duration t „=E and displayed interference. When there was no trigger, no fringes were observed, implying a much shorter wave packet. This is a nonlocal effect in that the photons can be arbitrarily far away from each other when the collapse occurs.
84.3.3 Two-Photon Interference In the above experiments, interference occurs between two paths taken by a single photon. An early experiment to
F3
Rotatable QWP D3
Fig. 84.3 The energy–time uncertainty relation and wave-function collapse were studied by investigating the effect of various filters before the detectors in a single-photon interference experiment [9, 84]
demonstrate two-photon interference using the downconversion light source was performed by Ghosh and Mandel [88]. They looked at the counting rate of a detector illuminated by both of the twin beams. No interference was observed at the detector, because although the sum of the phases of the two beams emitted in parametric fluorescence is well defined (by the phase of the pump), their difference is not, due to the number-phase uncertainty principle. However, when Ghosh and Mandel looked at the rate of coincidence detections between two such detectors whose separation was varied, they observed high-visibility interference fringes. Whereas in the standard two-slit experiment, interference occurs between the two paths a single photon could have taken to reach a given point on a screen, in this case, it occurs between the possibility that the signal photon reached detector 1 and the idler photon detector 2, and the possibility that the reverse happened. This experiment provides a manifestation of quantum nonlocality; interference occurs between alternate global histories of a system, not between local fields. At a null of the coincidence fringes, the detection of one photon at detector 1 excludes the possibility of finding the conjugate photon at detector 2. Such interference becomes clearer in the related interferometer of Hong, Ou, and Mandel [89] (Fig. 84.4). The identically polarized conjugate photons from a downconversion crystal are directed to opposite sides of a 50–50 beamsplitter, such that the transmitted and reflected modes overlap. If the difference in the path lengths L prior to the beamsplitter is larger than the two-photon correlation length (of the order of the coherence length of the downconverted light), the photons behave independently at the beamsplitter, and coincidence counts between detectors in the two output ports are observed half of the time – the other half of the time, both photons travel to the same detector. However, when
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84.4 Complementarity and Coherence
A
84.4.1 Wave-Particle Duality
UV
Nonlinear crystal
D2
Fig. 84.4 Simplified setup for a Hong–Ou–Mandel (HOM) interferometer [89]. Coincidences may result from both photons being reflected or both being transmitted. When the path lengths to the beamsplitter are equal, these processes destructively interfere, causing a null in the coincidence rate. In a modified scheme, a half-waveplate in one arm of the interferometer (at “A”) serves to distinguish these otherwise interfering processes, so that no null in coincidences is observed. Using polarizers before the detectors one can “erase” the distinguishability, thereby restoring interference [108]. (Sections 84.4.2 and 84.6.5)
L 0, such that the photon wave packets overlap at the beamsplitter, the probability of coincidences is reduced, in principle to zero if L D 0. One can explain the coincidence null at zero path-length difference using the Feynman rules for calculating probabilities: add the probability amplitudes of indistinguishable processes that lead to the same final outcome, and then take the absolute square. The two indistinguishable processes here are both photons being reflected at the beamsplitter (with Feynman amplitude r r) and both photons being transmitted (with Feynman amplitude tt). The probability of a coincidence detection is then ˇ ˇ ˇ i i 1 1 ˇ2 Pc D jr r C t tj2 D ˇˇ p p C p p ˇˇ D 0 ; 2 2 2 2 (84.6) assuming a real transmission amplitude, and where the factors of i come from the phase shift upon reflection at a beamsplitter [90, 91]. The possibility of a perfect null at the center of the dip is indicative of a nonclassical effect. Indeed, classical field predictions allow a maximum coincidence-fringe visibility of only 50% [92]. The tendency of the photons to travel off together at the beamsplitter can be thought of as a manifestation of the Bose–Einstein statistics for the photons [93]. In practice, the bandwidth of the photons, and hence the width of the null, is determined by filters and/or irises before the detectors [85]. Widths as small as 5 µm have been observed, corresponding to time delays of only 15 fs [94]. Consequently, one application is the determination of single-photon propagation times with extremely high time resolution (Sect. 84.7). The dependence of the dip visibility on the indistinguishability of the photons is now commonly used to characterize quantum light sources [95–98].
The complementary nature of wave-like and particle-like behavior is frequently interpreted as follows: due to the uncertainty principle, any attempt to measure the position (particle aspect) of a quantum leads to an uncontrollable, irreversible disturbance in its momentum, thereby washing out any interference pattern (wave aspect) [99, 100]. This picture is incomplete though; no “state reduction”, or “collapse”, is necessary to destroy interference, and measurements that do not involve reduction can be reversible. One must view the loss of coherence as arising from an entanglement of the system wave function with that of the measuring apparatus (MA) [101]. Previously interfering paths can thereby become distinguishable, such that no interference is observed. Consider the simplest experiment, a Mach–Zehnder interferometer with a 90ı polarization rotator in arm 1. If horizontally polarized light is input, the state p before the recombining beamsplitter is .j1ijV i C j2ijH i/= 2, where j1i and j2i label the path of the photon. Because the polarization – playing the role of the MA – labels the path, no interference is observed at the output. Englert [102] has introduced a generalized relation quantifying the interplay between the wave-like attributes of a system (as measured by the fringe visibility V ) and the particle-like character (as measured by the distinguishability D of the underlying quantum processes): V 2 C D2 1 :
(84.7)
The equality holds for pure input states. This relation has now been well verified in optical systems like that described above [103], as well as in atom interferometry (Sect. 81.6) [104]. In the latter, the role of the polarization was played by internal energy states of an atom diffracted off a standing light wave.
84.4.2 Quantum Eraser The interference lost to entanglement may be regained if one manages to “erase” the distinguishing information. This is the physical content of quantum erasure [105, 106]. The primary lesson is that one must consider the total physical state, including any MA with which the interfering quantum has become entangled, even if that MA does not allow accessible which-path information [107]. If the coherence of the MA is maintained, then interference may be recovered.
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Quantum Optical Tests of the Foundations of Physics
Coincidence rate (s–1) 450 45, 45 Theory 400 45, 22 350 Theory 45, –45 300 Theory 250
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Ds
s1 BS
NL1 i1
s2
A NL2
UV
i1, i2
B
Di
200 150
Fig. 84.6 Setup used in [117]. The idler photons from the two crystals are indistinguishable; consequently, interference fringes may be observed in the signal singles rate at detector Ds . Additional elements at A and B can be used to make a quantum eraser
100 50 0 –1110
–1090
–1070
–1050
–1030 –1010 –990 Trombone prism position (μm)
Fig. 84.5 Experimental data and scaled theoretical curves (adjusted to fit observed visibility of 91%) with polarizer 1 at 45ı and polarizer 2 at various angles. Far from the dip, there is no interference, and the angle is irrelevant [108]
The first demonstration of a quantum eraser was based on the interferometer in Fig. 84.4 [108]. A half-waveplate inserted into one of the paths before the beamsplitter serves to rotate the polarization of light in that path. In the extreme case, the polarization is made orthogonal to that in the other arm, and the r r and t t processes become distinguishable; hence, the destructive interference that led to a coincidence null does not occur. The distinguishability can be erased, however, by using polarizers just before the detectors. In particular, if the initial polarization of the photons is horizontal, and the waveplate rotates one of the photon polarizations to vertical, then polarizers at 45ı before both detectors restore the original interference dip. If one polarizer is at 45ı and the other at 45ı , interference is once again seen but now in the form of a peak instead of a dip (Fig. 84.5). There are four basic measurements possible on the MA (here the polarization) – two of which yield which-path information, one of which recovers the initial interference fringes (here the coincidence dip), and one of which yields interference antifringes (the peak instead of the dip). In some implementations, the decision to measure wave-like or particle-like behavior may even be delayed until after detection of the original quantum, an irreversible process [114, 115]. However, in all cases, one must correlate the results of measurements on the MA with the detection of the originally interfering system. This requirement precludes any possibility of superluminal signaling. This is an extension of the original delayed-choice discussion by Wheeler [109] and the experiments by Hellmuth et al., Alley et al., Jacques et al., and Kim et al. [110–113], in which the decision to display wave-like or particle-like aspects in a light beam may be delayed until after the beam has been split by the appropriate optics.
84.4.3 Vacuum-Induced Coherence A somewhat different demonstration [116, 117] of complementarity involves two downconversion crystals, NL1 and NL2, aligned such that the trajectories of the idler photons from each crystal overlap (Fig. 84.6). A beamsplitter acts to mix the signal modes. If the path lengths are adjusted correctly, and the idler beams overlap precisely, there is no way to tell, even in principle, from which crystal a photon detected at Ds originated. Interference appears in the signal singles rate at Ds , as any of the path lengths is varied. If the idler beam from crystal NL1 is prevented from entering crystal NL2, then the interference vanishes, because the presence or absence of an idler photon at Di then “labels” the parent crystal. Thus, one explanation for the effect of blocking this path is that coherence is established by the idler-mode vacuum field seen by both crystals. Experiments have also been performed in which a timedependent gate is introduced in the idler arm between the two crystals [118]. As one expects, the presence or absence of interference depends on the earlier state of the gate, at the time when the idler photon amplitude was passing through it.
84.4.4 Suppression of Spontaneous Parametric Downconversion A modification [119] of this two-crystal experiment uses only a single nonlinear crystal (Fig. 84.7). A given pump photon may downconvert in its initial rightward passage through the crystal or in its left-going return trip (or not at all, the most likely outcome). As in the previous experiment, the idler modes from these two processes are made to overlap; moreover, the signal modes are also aligned to overlap. Thus, the left-going and right-going production processes are indistinguishable and interfere. The result is that fringes are observed in all of the counting rates (i.e., the coincidence rate and both singles rates) as any of the mirrors is translated. A different interpretation is as a change in the spontaneous emission of the downconverted photons, akin to the suppression of spon-
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Signal
Pump LiIO3 Idler
Fig. 84.7 Experiment to demonstrate enhancement and suppression of spontaneous downconversion [119]
taneous emission in cavity QED demonstrations, discussed in Sects. 84.2.1 and 83.3. Subsequent work has shown that, in a sense, there are always photons between the downconversion crystal and the mirrors, even in the case of complete suppression of the spontaneous emission process [120]. This same conclusion should also apply in the atom case, although in contrast to that system, for the downconversion experiment, the distances to the mirrors are much longer than the coherence lengths of the spontaneously emitted photons. One recent application of this phenomenon is to study effective nonlinearities at the single-photon level [121]. Finally, with the inclusion of waveplates to label the photons’ paths and polarizers to erase this information, an improved quantum eraser experiment was also completed, in which the which-path information for one photon was carried by the other photon [115].
84.5 84.5.1
Measurements in Quantum Mechanics Quantum (Anti-)Zeno Effect
A strong measurement of a quantum system will project it into one of its eigenstates [101]. If the system p evolves slowly out of its initial state: j .t D 0/i D j0 i ! 1 t 2 = 2 j0 i C t=j1 i, then repeated measurements with an interval much less than can inhibit this evolution. If there are N total measurements within , then the probability for the system to still be in the initial state is P ./ D .1 .=N /2 = 2 /N ! 1 as N ! 1. This phenomenon, known as the quantum Zeno effect [122], has been experimentally observed using three levels in a 9 BeC ion [123]. The ion was prepared in state jii and weakly coupled to state jf i via RF radiation that induced a slow Rabi oscillation between the two states. Thus, in the absence of any intervening measurements, the ion evolved sinusoidally into state jf i. When rapid measurements were made (by a laser strongly coupling state jf i to readout state jri, hence leading to strong fluorescence only if the atom was in state jf i), the effect was to inhibit the jii ! jf i transition.
Note that here it was the absence of fluorescent photons that projected the state at each measurement back into the state jii. Kofman and Kurizki pointed out that the above inhibition phenomenon depends on there being a bounded number of final states (the ion example had only one). If, instead, the measurement process actually increases the number of accessible final states jf i, then one obtains the “anti-Zeno” effect, in which the jii ! jf i rate is enhanced rather than suppressed by frequent measurements [124]. For example, in the ion example, this would be the case if the jii ! jf i transition were spontaneous (allowing all frequencies) instead of driven (proceeding only at the driving Rabi frequency). The anti-Zeno effect has been observed by monitoring the survival time (against tunneling escape) of atoms trapped in an accelerating far-detuned standing wave of light [125].
84.5.2
Quantum Nondemolition
The uncertainty principle between the number of quanta N and phase of a beam of light, N 1=2 ;
(84.8)
implies that to know the number of photons exactly, one must give up all knowledge of the phase of the wave. In theory, a quantum nondemolition (QND) process is possible [126]: without annihilating any of the light quanta, one can count them. It might seem that this would make possible successive measurements on noncommuting observables of a single photon, in violation of the uncertainty principle; it is the unavoidable introduction of phase uncertainty by any number measurement that prevents this. QND schemes [127] often employ the intensity-dependent index of refraction arising from the optical Kerr effect (Sect. 76.4.3) – the change in the index due to the intensity of the “signal” beam changes the optical phase shift on a “probe” beam [128–131]. Other proposals include using the Aharonov–Bohm effect to sense photons via the phase shift their fields induce in passing electrons [132, 133]. Two classic experimental realizations [18] of a QND measurement – of the photon number in a microwave cavity – were performed by passing Rydberg atoms [134, 135] in a superposition of ground and excited state through the cavity. The interaction with the cavity photon is equivalent to a 2 -pulse (Sect. 83.8). The result is that, in the absence of any photon, the quantum state of the atoms after the cavity was unchanged; with a photon in the cavity, the ground state acquired an extra relative phase of , which was then detected by measuring the atom’s quantum state. Using similar schemes, successful nondemolition measurements have now been performed on microwave photons in microwave circuits [136, 137].
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84.5.3 Quantum Interrogation The previous section described techniques to measure the presence of a photon without absorbing it. Now we discuss a method – quantum interrogation – to optically detect the presence of an object without absorbing or scattering a photon. The possibility that the absence of a detection event – a “negative-result” measurement – can lead to wave-packet reduction was first discussed by Renninger [138] and later by Dicke [139]. Here, we consider the gedanken experiment proposed by Elitzur and Vaidman, a simple single-particle interferometer, with particles injected one at a time [140]. The path lengths are adjusted so that all the particles leave a given output port (A) and never the other (B). Now suppose that a nontransmitting object is inserted into one of the interferometer’s two arms – to emphasize the result, we consider an infinitely sensitive “bomb”, such that interaction with even a single photon will cause it to explode. By classical intuition, any attempt to check for the presence of the bomb involves interacting with it in some way, and by hypothesis this will inevitably set it off. Quantum mechanics, however, allows one to be certain some fraction of the time that the bomb is in place, without setting it off. After the first beamsplitter of the interferometer, a photon has a 50% chance of heading towards the bomb and, thus, exploding it. On the other hand, if the photon takes the path without the bomb, there is no more interference, since the nonexplosion of the bomb provides welcher Weg (“which way”) information (Sect. 84.4.1). Thus, the photon reaches the final beamsplitter and chooses randomly between the two exit ports. Some of the time (25%), it leaves by output port B, something that never happened in the absence of the bomb. This immediately implies that the bomb (or some object) is in place – even though (since the bomb is unexploded) it has not interacted with any photon; Elitzur and Vaidman termed this an “interaction-free measurement”. (We prefer the more general description “quantum interrogation”, which then includes cases – e.g., detecting a semitransparent or quantum object – where it may not be possible to logically exclude the possibility of an interaction.) It is the possibility that the bomb could have interacted with a photon that destroys interference. An initial experimental implementation of these ideas [141] used downconversion to prepare the single photon states (Sect. 84.1) and a single-photon detector as the “bomb”. Subsequently, the technique was implemented incorporating focusing lenses, which would enable the image (more correctly, the silhouette) of an object to be determined with less than one photon per “pixel” being absorbed [142]. By adjusting the beamsplitter reflectivities in the above example, one can achieve at most a 50% fraction of measurements that are interaction free. An improved method, relying on the quantum Zeno effect [122] (Sect. 84.5.1), was discovered with which one can, in principle, make this fraction
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arbitrarily close to 1 [141]. For example, consider a photon initially in cavity #1 of two identical cavities coupled by a lossless beamsplitter whose reflectivity R D cos2 . =2N /. If the photon’s coherence length is shorter than the cavity length, after N cycles, the photon will with certainty be located in cavity #2, due to an interference effect. However, if cavity #2 instead contains an absorbing object (e.g., the ultrasensitive bomb), at each cycle there is only a small chance (D 1 R) that the photon will be absorbed; otherwise, the nonabsorption projects the photon wave packet entirely back onto cavity #1. After all N cycles, the total probability for the photon to be absorbed by the object is 1 RN , which goes to 0 as N becomes large. In practice, unavoidable losses in the system limit maximum number of cycles and, hence, the achievable performance [143]. The photon effectively becomes trapped in cavity #1, thus indicating unambiguously the presence of the object in cavity #2. This quantum-Zeno version of interrogation was first implemented using the inhibited rotation of a photon’s polarization (the object to be detected blocked one arm of a polarizing interferometer through which the photon was repeatedly cycled), achieving an efficiency of 75% [143]. A cavity-based implementation, in which the presence of the absorbing object inside a high-finesse cavity vastly increased the reflection off the cavity [144], detected the presence of the object with only 0.15 photons on average being absorbed or scattered [145].
84.5.4 Weak Measurements Aharonov, Albert, and Vaidman extended quantum measurement theory by introducing “weak measurement”, a procedure that determines a physical property of a quantum system belonging to an ensemble that is both preselected and postselected [146, 147]. In the standard theory, a quantum system is measured by entangling its eigenstates with distinguishable pointer states of a measurement device, completely resolving the observable spectrum. The measurement can be weakened by increasing the overlap of the pointer states, consequently reducing the resolution of the eigenstates. When performed between two measurements this can give surprising results – in contrast to ordinary expectation values, the pointer can lie outside the range of the eigenvalue spectrum of the measured observable O. The “weak value” Ow D hini jOjfin i=hfin jini i between preselected jini i and postselected jfin i states completely characterizes the outcome of the weak measurement. Weak measurements do not disturb each other, so that the weak values of noncommuting observables can be measured simultaneously [148]. Weak measurements have played an important role in addressing many fundamental issues in quantum mechanics, from the tunneling time [149, 150] (Sect. 84.7) to Hardy’s
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paradox [151, 152] (Sect. 84.6.8) to the role of momentum disturbance in complementarity experiments [153] to verification of Ozawa’s precision-disturbance relationship [154], which shows that a measurement need not disturb a system as much as predicted by a naive reading of the Heisenberg relation [155]. They have also permitted the study of negative kinetic energies [156], by postselecting a particle in a classically forbidden region, the direct reconstruction of the Bohmian trajectories of particles in a two-slit experiment [157], and a demonstration that a single postselected “shutter” can block two slits at once [158]. One other potentially powerful application of weak measurements is the amplification of weak signals, which was first demonstrated by amplifying the birefringence-induced small displacement of optical fields [159]. More recently it was used to detect very small deflections of an optical beam ( 2 is observed, this rules out any local hidden variable description of the experiment. Strictly speaking, in a Bell test, we are never able to prove that quantum mechanics is correct, but rather we are only trying to falsify the notion that local hidden variables are fundamental to nature. The stronger correlations in quantum systems not only have important philosophical implications about the structure of reality but are part of the reason why emerging technologies like quantum computers should have
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a massive performance advantage over classical computers in certain situations. It should also be noted that, in principle, the maximum achievable score in this Bell game is 4 for a perfectly correlated system, which is even greater than what quantum mechanics allows for. While we do not have any evidence that the universe permits correlations stronger than quantum mechanics, models of such systems, known as PR boxes [189], can serve as a useful theoretical tool.
84.6.3 Loopholes in Bell Tests To unambiguously violate a Bell inequality one must experimentally satisfy several conditions. The first experiments to violate Bell’s inequalities were carried out by John Clauser [190] and later by Aspect and colleagues [191], but due to technological limitations needed to make additional assumptions. These extra assumptions open up “loopholes” that could allow a local hidden variable theory to replicate the stronger correlations predicted by quantum mechanics. Strictly speaking, these experiments could only rule out a subset of local hidden variable models. If Alice and Bob are close together, then there is the possibility that hidden variables controlling Alice and Bob’s photons could signal one another to coordinate their outcomes. To prevent this, Alice and Bob must be far enough apart so that their measurements can be individually completed before any such signals traveling at the speed of light could influence the outcomes. If this condition is not satisfied, it opens the so-called “locality loophole”. There is a subset of the locality loophole, sometimes referred to as “freedom of choice”, where Alice and Bob must assume that their measurement choices are independent of one another and the source of photons. This is usually addressed through the use of independent random number generators at Alice and Bob, but strictly speaking they can never guarantee that something in their shared past light cones did not correlate their settings choices. The extreme example of this is “superdeterminism”, the idea that the universe is deterministic with its entire history set in stone at beginning of time. Such theories are not testable, and therefore exist outside the realm of experimental physics. Because a Bell test is statistical in nature, if there is too much loss, it is possible for a local hidden variable to appear to violate a Bell inequality. A local hidden variable could bias the observed correlations by preferentially discarding, through nondetection, certain outcomes. In this way, the detected events would appear to violate Bell’s inequalities, but the total ensemble of detected and undetected events would not. To prevent this, at least 2/3 of the photons sent to Alice and Bob must be detected. If this efficiency threshold cannot be met, then one must assume that the detected events are fairly sampled from the total ensemble leading to the “fair-sampling” loophole (also known as the “detector loop-
L. K. Shalm et al.
hole”). All of the experiments discussed above that closed the locality loophole were unable to simultaneously close the fair-sampling loophole. In order to understand the detection loophole, it is convenient to consider the Clauser–Horne (CH) form of the Bell inequality [183]. This form of the inequality incorporates loss directly and is convenient because it only requires Alice and Bob to use one detector each. Now Alice and Bob must report either the detection of a photon (C) or no detection (0) every trial (any outcomes are now just mapped to 0). The CH inequality is: NCC .a; b/ C NCC .a; b 0 / C NCC .a0 ; b/ NCC .a0 ; b 0 / NC .a/ C NC .b/ :
(84.12)
The left-hand side of this equation is the number of coincidence events where in a given trial Alice and Bob both detect a photon. The term NC .a/ (NC .b/) on the right-hand side represent the total number of photons Alice (Bob) detects while in setting a (b) irregardless of what the other party measures. For certain choices of a, a0 , b, and b 0 , quantum mechanics predicts that the left-hand side of the CH inequality can exceed the right-hand side. However, in practice, this is very difficult to observe, since the probability of detecting a coincidence falls as 2 ( is the detection efficiency), compared to the singles counts on the right-hand side, which fall only as . For a maximally entangled state, such as the one described in Eq. (84.9), a detection efficiency 83% is required to violate the CH inequality [192]. If one uses a nonmaximally entangled state of the form: j i D cos./jH1 ; V2 i sin./jV1 ; H2 i ;
(84.13)
then the required detection efficiency may be reduced to 67% as goes to 0 for the appropriate analysis settings [193, 194]. Finally, the CH inequality in Eq. (84.12) can be converted into a “hypothesis test”. Doing so can prevent additional forms of potential statistical manipulation. For example, if Alice and Bob normalize their correlations by the total number of counts, they are implicitly making the assumption that every trial was independent and identical. Instead, by using an appropriate hypothesis-testing framework it is possible to avoid these subtle analysis assumptions. In the hypothesis-testing framework, a p-value is reported which is the probability that, if local realism is true, it would be able to produce a violation of the Bell inequality greater than or equal to the observed violation.
84.6.4 Closing the Loopholes in Bell Tests In 1982, Aspect et al. used entangled photons from an atomic cascade to perform the first Bell test where Alice and Bob were separated far enough apart (12 m) that their measurement outcomes satisfied the locality constraints [191]. This
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was followed up by a first attempt to have Alice and Bob vary their measurement settings while the photons were in flight towards the measurement stations (early experiments used passive components that could be interpreted as allowing the photons to make the decisions as to how they would be measured). In this experiment, function generators at Alice and Bob produced a periodic voltage to drive home-built acoustic optic modulators that switched the incoming photons between two different measurement setups. However, because the generators used to drive the measurement switching were periodic, the locality loophole could not be fully addressed. In 1998, Weihs et al. were able to fully close the locality loophole by using entangled photons generated by the process of spontaneous parametric downconversion along with fast random number generators that drove electronic polarization analyzers. The photons were sent to measurement stations 400 m apart, and while they were in flight, the measurement basis was randomly and independently set to ensure space-like separated observers [195]. However, in that experiment, the detection efficiency was less than 5%. Currently, the locality loophole has been closed on a continental scale–in 2018, Yin et al. performed a Bell test with Alice and Bob separated by 1203 km using an entangled source of photons on board a satellite in low Earth orbit [196]. The first experiment to close the detection loophole was the entangled-ion experiment of Rowe et al. [197] in 2001. In this experiment the entangled variables were the hyperfine energy levels of 9 BeC ions. By employing a cycling transition that leads to the emission of many photons if the atom is in one of the states, a detection efficiency in excess of 98% was achieved, allowing an 8- Bell inequality violation. However, because the ions were separated by only 3 µm in the same linear Paul trap, and in fact were measured using the same laser pulse, there was no possibility of closing the locality loophole. A decade later the detection loophole was finally closed using photons in two experiments [198, 199] that utilized superconducting transition edge detectors with efficiencies > 98%. These detectors use a superconducting material operating near the transition point to measure the energy deposited by one or more photons. However, in these experiments the detectors used for Alice and Bob were cooled in the same cryostat, preventing the locality loophole from being closed. It wasn’t until 2015, 51 years after Bell’s original paper, that three groups were able to independently close these loopholes at the same time in the same experiment. At TU Delft Hensen et al. used entangled spins in Nitrogenvacancies in diamonds separated by 1.3 km to violate a Bell inequality with a value of S D 2:42 ˙ 0:20 [200]. This corresponds to a p-value of 0.039 for refuting local realism. The other two experiments were performed using entangled photons. At the National Institute of Standards and Technologies in Boulder, Shalm et al. were able to close
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all the loopholes with > 5 confidence, and violate a Bell inequality with a p-value of 5:3 109 [201], while in Vienna a loophole-free Bell test was conducted at the Hofburg Imperial Palace by Giustina et al., measuring a p-value of 3:74 1031 [202]. In 2016 a fourth loophole-free Bell test used entangled Rb87 atoms separated by 398 m by Rosenfeld et al. in Munich to obtain a violation of S D 2:221 ˙ 0:033, corresponding to a p-value of 2:57 109 [203]. While in principle it is not possible to close the freedomof-choice loophole due to super determinism, with some additional assumptions one is able to push back the earliest time that a local hidden variable could have correlated Alice’s and Bob’s measurement choices by using light from distant stars or quasars to choose the measurement settings used. This requires the assumption that the detectors, telescopes, and other electronic equipment used by Alice and Bob are not correlated with one another, and that the mechanisms that produce the light in the stellar objects are short-lived. For example, it might take a photon 100,000 years to escape the deep plasma core of a star due to the short mean-free path [204, 205], but an unimpeded neutrino could escape at a much earlier time to carry the information to the Earth. Recently, in 2018 Li et al. were able to perform a loopholefree Bell test using polarization entangled photons that used the random arrival times of light from nearby stars to make Alice and Bob’s measurement choices. This pushes the earliest point at which Alice’s and Bob’s measurement choices could be correlated to 11 years in the past [206]. A similar experiment, albeit not loophole-free, used high-redshifted light from distant quasars to chose Alice’s and Bob’s settings, and pushing back the earliest time a local hidden variable model could have exerted an influence to approximately 7.8 billion years ago (corresponding to 96% of the space-time volume in the past light cone of the experiment) [207]. Taken together these experiments provide convincing proof that local realism is not a viable theory of nature.
84.6.5 Polarization-Based Entangled Sources The process of downconversion (see Sect. 76.3.6) provides a convenient way to generate entangled pairs of polarizationentangled photons. To create entangled photons, two different processes need to interfere with one another. For example, consider two identical type-I phase-matched crystals placed back-to-back, but with the second crystal rotated by 90ı . The first crystal will downconvert a H -polarized pump into V -polarized pairs while the second crystal downconverts a V -polarized pump into H -polarized pairs. By coherently pumping p the two crystals with light polarized at j45i D .jV i C jH i/= 2, there is an equal probability of producing a pair of photons in either the first or second crystal. Since there is no knowledge of which process took place,
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p one obtains the entangled state .jHH i C ei' jV V i= 2 [208]. More generally, pumping ˛jV i C ei' ˇjH i produces arbitrary nonmaximally entangled states of the form ˛jHH i C ei' ˇjV V i [209]. Using type-II phase-matching one also can produce polarization entanglement from a single crystal [210]. One member of each downconversion pair is emitted along an ordinary polarized cone while the other is emitted along an extraordinary polarized cone. If the photons happen to be emitted along the intersection of the two cones, neither photon will have a definite p polarization – they will be in the state .jH V i C jVH i/= 2. This entanglement source has now been used in a variety of quantum investigations, including Bell inequality tests [195, 210], quantum cryptography [211, 212] (Sect. 85.2.1) and teleportation [213, 214], and as a resource for studying entanglement of more than two photons [215–218].
84.6.6 Other Entanglement Sources for Bell Tests The advent of parametric downconversion has also led to the appearance of several nonpolarization-based BI tests, using, for example, an entanglement of the photon momenta (Fig. 84.8) [219]. By the use of small irises (labeled ‘A’ in Fig. 84.8), only four downconversion modes are examined: 1s, 1i, 2s, and 2i. Beams 1s and 1i correspond to one pair of conjugate photons; beams 2s and 2i correspond to a different pair. Photons in beams 1s and 2s have the same wavelength, as do photons in beams 1i and 2i. With proper alignment, after the beamsplitters there is no way to tell whether a pair of photons came from the 1s-1i or the 2s-2i paths. Consequently, the coincidence rates display interference, although the singles rates at the four detectors indicated in Fig. 84.8 remain constant. This interference depends on the difference of phase shifts induced by rotatable glass plates Pi and Ps in paths 1i and 2s, respectively, and is formally equivalent to the polarization case considered above, in which it is the difference of polarization-analyzer angles that is rele-
A KD*P
1s
Ds Di
2i
UV Ps 1i A
2s
Di⬘ Pi
Ds⬘
Fig. 84.8 Outline of apparatus used to demonstrate a violation of a Bell’s inequality based on momentum entanglement [219]
vant. By measuring the coincidence rates for two values for each of the phase shifters – a total of four combinations – the experimenters were able to violate an appropriate BI. One interpretation is that the paths taken by a given pair of photons are not elements of reality. Momentum conservation in the downconversion process (Eq. (84.4)) also leads to entanglement directly in the spatial modes in the correlated photons. For example, Zeilinger et al. [220, 221] and Langford et al. [222] demonstrated entanglement between the orbital angular momentum of the photons, of the form .jC1; 1i C j0; 0i C j1; C1i, where 0 and ˙1, respectively, denote modes with no orbital angular momentum (Gaussian spatial profiles) and ˙„ (Laguerre– Gauss vortex modes). Note that this enables one to investigate correlations for degrees of freedom that reside in larger Hilbert spaces than do the two-level systems (e.g., polarization) discussed above. The nonlocal spatial correlations of the downconversion photons have also given rise to many interesting experiments in the area of quantum imaging [223–226], where one is able top obtain spatial resolution beyond that predicted by the usual N shot-noise limitations. Several groups [227, 228] have implemented a different BI proposal [229], based on energy-time entanglement of the photons. One member of each downconverted pair is directed into an unbalanced Mach–Zehnder-like interferometer, allowing both a short and a long path to the final beamsplitter; the other photon is directed into a separate but similar interferometer. There arises interference between the indistinguishable processes (“short-short” and “long-long”), which could lead to coincidence detection. Using fast detectors to select out only these processes, the reduced state (Eq. (84.9)) is 1
j i D p jS1 ; S2 i ei jL1 ; L2 i ; 2
(84.14)
where the letters indicate the short or the long path, and the phase is the sum of the relative phases in each interferometer. Although no fringes are seen in any of the singles count rates, the high-visibility coincidence fringes (Fig. 84.9) lead to a violation of an appropriate BI. One conclusion is that it is incorrect to ascribe to the photons a definite time of emission from the crystal, or even a definite energy, unless these observables are explicitly measured. A more general interpretation, applicable to all violations of Bell’s inequalities, is that the predictions of QM cannot be reproduced by any completely local theory. It must be that the results of measurements on one of the photons depend on the results for the other, and these correlations are not merely due to a common cause at their creation [230, 231]. This same sort of arrangement, modified to work with a pulsed pump, has been used to demonstrate a violation of local realism with photons separated by 10.9 km [232].
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Coincidence counts N12 (per 24 s) 700 600
Φ2 = 0 Visib. = 92.8
0.9%
Φ2 = 90 Visib. = 90.3 0.7%
500 400 300 200 100 0
Interferometer 1 phase shift, Φ1
Fig. 84.9 High-visibility coincidence fringes in a Franson dualinterferometer experiment [229] for two values of the phase in interferometer 2 as the phase in interferometer 1 is slowly varied. The curves are sinusoidal fits
In a related experiment, a similar system was used to place limits on the “speed of collapse” of the two-photon wave function, i.e., how fast a nonlocal “signal” would need to propagate from one side of the experiment to the other to account for the measured nonclassical correlations. Depending on some assumptions about the detection process and which inertial frame of reference is considered, the nonlocal influence speed was constrained to be at least 104 c to 107 c [233]. In one interesting variant, the researchers arranged to have moving detectors, such that in the local reference frame of each detector, it was the other detector that initiated the collapse. Due to the experimental difficulty of accelerating actual detectors to high velocities, a rapidly rotating absorbing disk was placed close to one output port of a polarizing beamsplitter; following ideas discussed in Sect. 84.5.3, the nonabsorbance of the photon by the absorber was deemed sufficient to cause a reduction of the wave function. As expected, the measured correlations were in no way reduced, but this experiment did rule out one alternative theory of nonlocal collapse [234]. Finally, by using more than two possible creation times, one can also demonstrate entanglement in higher dimensional Hilbert spaces. For instance, using a mode-locked pulsed laser, Gisin’s group demonstrated entanglement for a two-photon state at least of dimension 11 [235].
84.6.7 Advanced Experimental Tests of Nonlocality The previously described tests of Bell’s inequality actually have little to do with quantum mechanics per se, other than that it is quantum correlations that enable a violation; the conclusions of such a violation only say that no local realistic description is sufficient to explain the results. Just as the
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CHSH inequality [183] bounds the observable correlations of a local realistic description to jSj 2, Tsirelson [236] showed that, according to quantum theory, jSj has an upp per bound of 2 2 2:82843. One immediate question is how close experimental realizations have come to reaching this value. The loophole-free experiments discussed in Sect. 84.6.4 have values of S that, while larger than the local realistic bound of 2, are still far from the maximum possible value permitted by quantum mechanics. A recent experiment [237] that did not attempt to close the detection and locality loopholes provides the best test of the upper bound to date, using photon pairs in a maximally entangled state of polarization to obtain S D 2:82759 ˙ 0:00051. Other generalizations to the Bell inequality have been studied and have obtained some counterintuitive results. For example, one modified Bell inequality has the property that the maximum quantum violation can only be obtained using partially entangled states [238], and another cannot be violated at all by a maximally entangled state. Experimental violations [239] of these modified inequalities have been observed, with results matching the (quantum) theoretical predictions extremely well, thus illustrating the phenomenon of “more nonlocality with less entanglement” [238, 240, 241]. In fact, the four correlation functions that make up S (Eq. (84.11)) are each bounded by 1, and Popescu and Rohrlich showed that the no-signaling principle that prevents superluminal communication is compatible with values up to S D 4 [242]; a theoretical ultra-nonlocal construct that enables one to achieve this maximal value is called a “PR box” after its originators. A 2.828 maximal quantum violation of the CHSH inequality could, thus, be interpreted as arising from a 41% contribution ofpa PR box and a 59% contribution from a local theory: 2 2 D .0:59 2/ C .0:41 4/, where the 2 and 4 represent the maximum values for S attainable with a local theory or nonsignaling nonlocal theory, respectively. One can instead consider a “chained Bell inequality” [243, 244] in which the usual two measurement settings on either side are replaced by a value n > 2; in this case, as n increases, the quantum violation approaches the bound imposed by the no-signaling principle. Consequently, one can place a stronger bound on the “nonlocal” content that would be required to achieve the observed results. In the experiment reported in [239], nonlocal resources (such as PR boxes) would be needed in 87:4 ˙ 0:1% of the experimental rounds in order to match the observed chained Bell inequality violation; this represents the most nonlocal correlations ever produced experimentally (this experiment did not attempt to close the detector or locality loophole). A subsequent experiment using 9 BeC ions closed the detection loophole in a chained-Bell inequality while certifying that the nonlocal content consistent with their results was at least 67.3% [245]. A remaining challenge is performing a fully loophole-free chained Bell inequality.
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α
α⬘ a
a⬘
Φa
γ
Φc
c
c⬘
Φb
b
⬘
b⬘
γ⬘
Fig. 84.10 A three-particle gedankenexperiment to demonstrate the inconsistency of quantum mechanics and any local realistic theory. All beamsplitters are 50–50 [116]
84.6.8 Nonlocality Without Inequalities
are 0, it will occasionally happen (1=8-th of the time) that there will be a triple coincidence of all primed detectors. Using a “contrafactual” approach, we ask what would have happened if a had been =2 instead. By the locality assumption, this would not change the state from the source, nor the fact that detectors ˇ 0 and 0 went off. However, from Eq. (84.16), the probability of a triple coincidence for primed detectors is zero in this case; therefore, we can conclude that detector ˛ would have “clicked” if a had been =2. Similarly, if b or c had been =2, then detectors ˇ or would have clicked. Consequently, if all the phases had been equal to =2, we would have seen a triple coincidence between unprimed detectors. However, according to Eq. (84.16), this is impossible: the probability of triple coincidences between unprimed detectors when all three phases are equal to =2 is strictly zero! Hence, if one believes the quantum mechanical predictions for these cases of perfect correlations, it is not possible to have a consistent local realistic model. Downconversion experiments have enabled the production of three and four-photon GHZ states, with results in good agreement with theory (the all-or-nothing arguments given above become inequalities in any real experiment) [215, 216, 248]. In 2014, Erven et al. were able to close the locality loophole while violating the Mermin inequality by 9- , although they were unable to close the detector loophole simultaneously. By similar arguments, Hardy has shown the inconsistency of quantum mechanics and local realism in a gedankenexperiment with just two particles [249, 250]. When the arguments are suitably modified to deal with real experiments, inequalities once again result; these have also been experimentally violated, using nonmaximally entangled states from downconversion [251, 252], underscoring the vast disagreement between quantum theory and locality.
In the above experiments for testing nonlocality, the disagreement between quantum predictions and Bell’s constraints on local realistic theories are only statistical. Greenberger, Horne, and Zeilinger (GHZ) pointed out that in some systems involving three or more entangled particles [246, 247], a contradiction could arise even at the level of perfect correlations. A schematic of one version of the GHZ gedankenexperiment is shown in Fig. 84.10. The source at the center is posited to emit trios of correlated particles. Just as the Rarity-Tapster experiment selected two pairs of photons (Fig. 84.8), the GHZ source selects two trios of photons; these are denoted by abc and a0 b 0 c 0 . Hence, the state coming from the source may be written as
p j i D jabci C ja0 b 0 c 0 i = 2 : (84.15) 84.6.9 After passing through a variable phase shifter (e.g., a ), each primed beam is recombined with the corresponding unprimed beam at a 50–50 beamsplitter. Detectors (denoted by Greek letters) at the output ports signal the occurrence of triple coincidences. The following simplified argument conveys the spirit of the GHZ result. Given the state Eq. (84.15), one can calculate from standard QM the probability of a triple coincidence as a function of the three phase shifts:
Connection to Quantum Information
The properties of entangled photons and the nonlocal correlations they share play a critical role in many quantum information applications. The information carried by photons cannot be cloned perfectly, which has led to protocols for securely exchanging one-time pads for use in cryptography [211, 212, 253–260]. The stronger correlations present between entangled photon-pairs can also be exploited to transfer two bits of information using a single photon [261– 263], a process known as superdense coding. Since it is not possible to perfectly clone a quantum state, techniques have 1 (84.16) been devised to use an entangled pair of photons to instead P .1 ; 2 ; 3 / D Œ1 ˙ sin.a C b C c / ; 8 “teleport” or move quantum information from one location where the plus sign applies for coincidences between all to another through the use of Bell state measurements and unprimed detectors, and the minus sign for coincidences be- classical communications [213, 264–267]. This process of tween all primed detectors. For the case in which all phases teleportation may form a fundamental building block in fu-
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ture quantum networks, allowing quantum information to be transmitted over long distances [268]. Such a network would enable networked quantum computing, connecting quantum circuits based on solid-state or matter quantum bits using light. Finally, loophole-free Bell tests are a fundamental component in new types of security protocols. A hacker or attacker who wishes to control or predict the outcome of a Bell test has at their disposal a set of tools equivalent to those of a hidden-variable theory. They are hoping to use their added knowledge, perhaps hidden from the experimenter, to manipulate the results. However, if a loophole-free Bell violation can be obtained, one can rule out the influence of such an attacker even if they control part of the equipment used. For this reason, protocols based on loophole-free Bell tests are referred to as being “device-independent” [269–273]. In 2017, the first fully device-independent protocol was carried out to generate certifiably secure random numbers using the loophole-free Bell test developed at NIST in Boulder, CO [274]. As the technology behind loophole-free Bell tests improves, a wider-range of device-independent applications promises to become more prevalent. As an indicator of the importance of these fundamental tests of (non)locality, the 2022 Nobel prize in physics was awarded to John F. Clauser, Alain Aspect, and Anton Zeilinger “for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science,” i.e., based on the research described above [182, 183, 190, 191, 195, 213, 268].
84.7 84.7.1
Single-Photon Tunneling Time An Application of EPR Correlations to Time Measurements
In this section, we discuss experiments involving the quantum propagation of light in matter. Due to the sharp time correlations of the paired photons from spontaneous downconversion, one can use the HOM interferometer (Sect. 84.3.3, Fig. 84.4) to measure very short relative propagation delay times for the signal and idler photons. One early application was, therefore, to confirm that single photons in glass travel at the group velocity [94]. At least until recently, the only quantum theory of light in dispersive media was an ad hoc one [275]. The shift of the interference dip resulting from a medium introduced into one of the interferometer arms can be accurately measured by determining how much the path lengths must be changed to compensate the shift and recover the dip. This result suggests that when looking for a microscopic description of dielectrics, it is unnecessary to consider the medium as being polarized by an essentially classical electric field due to the collective action of all photons present, and reradiating accordingly. Linear di-
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electric response is not a collective effect in this sense – each photon interferes only with itself (as per Dirac’s dictum) as it is partially scattered from the atoms in the medium. The single-photon group velocity thus demonstrates “waveparticle unity”. The standard limitation for measurements of short-time phenomena is that to have high time-resolution, one needs short pulses (or at least short coherence lengths), but these, in turn, require broad bandwidths and are, therefore, very susceptible to dispersive broadening. It is a remarkable consequence of the EPR energy correlations of the downconversion photons (Sect. 84.3.2) that time measurements made with the HOM interferometer are essentially immune to such broadening [94, 276, 277]. In effect, the measurement is sensitive to the difference in emission times while the broadening is sensitive to the sum of the frequencies. While frequency and time cannot both be specified for a given pulse, the crucial feature of EPR correlations such as those exhibited by downconverted photons is that this difference and this sum correspond to commuting observables and both may be arbitrarily well defined. The photon that reaches detector 1 could either have traversed the dispersive medium and been transmitted or traversed the empty path and been reflected, leaving its twin to traverse the medium. The medium, thus, samples both of the (anticorrelated) frequencies, leading to an automatic cancelation of any first-order (and in fact, all odd-order) dispersive broadening. Measurements can be more than five orders of magnitude more precise than would be possible via electronic timing of direct detection events, and in principle, better than those performed with nonlinear autocorrelators (which rely on the same fundamental physics as downconversion but do not benefit from a cancelation of dispersive broadening).
84.7.2
Superluminal Tunneling Times
Another well-known problem in the theory of quantum propagation is the delay experienced by a particle as it tunnels. There are difficulties associated with calculating the “duration” of the tunneling process, since evanescent waves do not accumulate any phase [278–280]. First, the kinetic energy in the barrier region is negative, so the momentum is imaginary. Second, the transit time of a wave-packet peak through the barrier, defined in the stationary phase approximation by ./ @Œarg t.!/eikd =@! ;
(84.17)
tends to a constant as the barrier thickness diverges, in seeming violation of relativistic causality. It is shown in [91] that such saturation of the delay time is a natural consequence of time-reversal symmetry, and in [281] that one can deduce
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from the principle of causality itself that every system possesses a superluminal group delay, at least at the frequency where its transmission is a minimum. For example, the transmission function for a rectangular barrier, t.k; / D
eikd 2
2
k cosh d C i 2k sinh d
;
ment against interpreting superluminal tunneling predictions as a mere artifact of the nonrelativistic Schrödinger equation. Also, one is denied the recourse suggested by some workers [291] of interpreting the superluminal appearance of transmitted peaks to mean that only the high-energy components (which, for matter waves, traveled faster even before (84.18) reaching the barrier [292]) were transmitted.
leads in the opaque limit ( d 1) to a traversal time of 2m=„k , independent of the barrier width d . The same result applies to photons undergoing frustrated total internal reflection [91], when the mass m is replaced by n2 „!=c 2 , and similar results apply to other forms of tunneling. Some researchers have, therefore, searched for some more meaningful “interaction time” for tunneling, which might accord better with relativistic intuitions and perhaps have implications for the ultimate speed of devices relying on tunneling [282]. The “semiclassical time” corresponds to treating the magnitude of the (imaginary) momentum as a real momentum. This time is of interest mainly because it also arises in Büttiker and Landauer’s calculation of the critical timescale in problems involving oscillating barriers, which they take to imply that it is a better measure of the duration of the interaction than is the group delay [278]. The Larmor time [283] is one of the early efforts to attach a “clock” to a tunneling particle, in the form of a spin aligned perpendicular to a small magnetic field confined to the barrier region. The basic idea is that the amount of Larmor precession experienced by a transmitted particle is a measure of the time spent by that particle in the barrier. This clock turns out to contain components corresponding both to the distance-independent “dwell time” and the linear-indistance semiclassical time. Experiments are being done with cold atoms, which have begun to resolve these two contributions [284]. Curiously, the most common theories for tunneling times become superluminal in certain cases anyway, whether or not they deal with the motion of wave packets. Pioneering tests of tunneling times included both optical and solid-state approaches [285]. In recent years, new experimental results have begun to appear both in attosecond strong-field ionization measurements and in cold-atoms experiments [286–289], but there is still disagreement about their interpretation. Below, in Sect. 84.7.4, we discuss theoretical status of different tunneling-time proposals. Here, we will restrict ourselves to discussing the time of appearance of a peak of a single-photon wave packet. Optical tests offer certain unique advantages [290], including the ease of construction of a barrier with no dissipation, very little energy dependence, and a superluminal group delay. The transmitted wave packets suffer little distortion and are essentially indistinguishable from the incident wave packets. At a theoretical level, the fact that photons are described by Maxwell’s (fully relativistic) equations is an important argu-
84.7.3 Tunneling Delay in a Multilayer Dielectric Mirror A suitable optical tunnel barrier can be a standard multilayer dielectric mirror. The alternating layers of low and high index material, each one quarter-wave thick at the design frequency of the mirror, lead to a photonic bandgap [293] analogous to that in the Kronig–Penney model of solid state physics (Sect. 83.3.6). The gap represents a forbidden range of energies, in which the multiple reflections will interfere constructively so as to exponentially damp any incident wave. The analogy with tunneling in nonrelativistic quantum mechanics arises because of the exponential decay of the field envelope within the periodic structure, i.e., the imaginary value of the quasi momentum. The same qualitative features arise for the transmission time: for thick barriers, it should saturate at a constant value, as was verified in a recent experiment employing short classical pulses [300]. A more direct analogy, that of waveguides beyond cutoff, yielded similar results in a classical microwave experiment [294], while another paper reported superluminal effects related to the penetration of diffracted or “leaky” microwaves into a shadow region [295]. All these experiments involve very small detection probabilities, just as in Chu and Wong’s pioneering experiment on propagation within an absorption line [296]. However, it has been predicted that superluminal propagation could occur without high loss or reflection [297– 299] by operating outside the resonance line of an inverted medium (Sect. 74.1). One can understand the effect as offresonance “virtual amplification” of the leading edge of a pulse. The phenomenon was investigated at the single-photon level by using the high time-resolution techniques discussed in Sect. 84.7.1 to measure the relative delay experienced by downconversion photons [301] when such a tunnel barrier (consisting of 11 layers) was introduced into one arm of a HOM interferometer. The transmissivity of the barrier was relatively flat throughout the bandgap (extending from 600 to 800 nm; Fig. 84.11), with a value of 1% at the gap center (700 nm), where the experiment was performed. The HOM coincidence dip was measured both with the barrier (and its substrate) and with the substrate alone (Fig. 84.12). Each dip was subsequently fitted to a Gaussian, and the difference between their centers was calculated. When several such runs
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Transmission probability (%) 100
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Fig. 84.11 Transmission probability for the tunnel barrier used in [301] (light solid curve); heavy, dashed, and dotted curves show group delay, Larmor time, and semiclassical time; d=c D 3:6 fs is shown for comparison
Coincidence counts (normalized)
Without barrier
With barrier
Δt ≈ –2 fs Time
Fig. 84.12 Coincidence profiles with and without the tunnel barrier map out the single-photon wave packets. The lower profile shows the coincidences with the barrier; this profile is shifted by 2 fs to negative times relative to the one with no barrier (upper curve): the average particle which tunnels arrives earlier than the one which travels the same distance in air
were combined, it was found that the tunneling peak arrived 1:47 ˙ 0:21 fs earlier than the one traveling through air, in reasonable agreement with theoretical prediction of approximately 1.9 fs. Taking into account the 1.1 µm thickness of the barrier, this implies an effective photon tunneling velocity of 1:7c. The results exclude the “semiclassical” time but are consistent with the group delay. A recent investigation of the energy dependence of the tunneling time was performed by angling the dielectric mirror, thus shifting its bandgap [150]. The data confirm the group delay in this limit as well and rule out identification of Büttiker’s Larmor time with a peak propagation time.
84.7.4
Interpretation of Tunneling Time
Even though a wave-packet peak may appear on the far side of a barrier sooner than it would under allowed propagation, it is important to stress that no information is transmitted faster than c, nor, on average, is any energy. These effects occur in the limit of low transmission, where the transmitted wave packet can be considered as a “reshaped” version of the leading edge of the incident pulse [285, 296, 302, 303]. At a physical level, the reflection from a multilayer dielectric is due to destructive interference among coherent multiple reflections between the different layers. At times before the field inside the structure reaches its steady-state value, there is little interference, and a nonnegligible fraction of the wave is transmitted. This preferential treatment of the leading edge engenders a sort of “optical illusion”, shifting the transmitted peak earlier in time. A signal, such as a sharp onset, relies on high-frequency components, which would not benefit from this illusion but instead travel more slowly than c. Even for a smooth wave packet, no energy travels faster than light; most is simply reflected by the barrier. Only if one considers the Copenhagen interpretation of quantum mechanics, with its instantaneous collapse, does one find superluminal propagation of those particles that happen to be transmitted. This leads to the question of whether it is possible to ask which part of a wave packet a given particle comes from. In one paper it was argued that transmitted particles do, in fact, stem only from the leading edge of the wave packet [304]. While it is true that the transmission only depends on causally connected portions of the incident wave packet, further analysis revealed that simultaneous discussion of such particle-like questions and the wave nature of tunneling ran afoul of the complementarity principle [305]. In essence, labeling the initial positions of a tunneling particle destroys the careful interference by which the reshaping occurs (as in the quantum eraser). However, one picture in which the transmitted particles really do originate earlier is the Bohm–de Broglie model of quantum mechanics [306, 307]. This theory considers to be a real field (residing, however, in configuration space, thus incorporating nonlocality) that guides point-like particles in a deterministic manner. It reproduces all the predictions of quantum mechanics without incorporating any randomness; the probabilistic predictions of QM arise from a range of initial conditions. Bohm’s equation of motion has the form of a fluid-flow equation, v.x/ D „r arg .x/=m , implying that particle trajectories may never cross, as velocity is a single-valued function of position. Consequently, all transmitted particles originate earlier in the ensemble than all reflected particles [305, 308]. This approach yields trajectories with welldefined (and generally subluminal) dwell times in the barrier region. However, the fact that the mean tunneling delay of Bohm particles diverges as the incident bandwidth becomes
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small, along with other interpretational issues [309, 310], leaves open the question of whether time scales as defined by the Bohm model have any physical meaning. The “weak measurement” approach of Aharonov et al. [146, 147] or equivalently, complex conditional probabilities obeying Bayes’s theorem [149, 311], can be used to address the question of tunneling interaction times in an experimentally unambiguous way. The real part of the resulting complex times determines the effect a tunneling particle would have on a “clock” to which it coupled, while the imaginary part indicates the clock’s backaction on the particle. They unify various approaches such as the Larmor and Büttiker–Landauer times, as well as Feynman path methods. In addition, they allow one to discuss separately the histories of particles that have been transmitted or reflected by a barrier, rather than discussing only the wave function as a whole. Interestingly, these calculations do not support the assertion that transmitted particles originate in the leading edge of a wave packet.
L. K. Shalm et al.
Building on the concepts of electromagnetically induced transparency [323], two groups succeeded in that year in utilizing the steep dispersion curves that can be generated by extremely narrow holes in absorption lines to slow light by a remarkable factor, to speeds as low as 17 m=s [324, 325]. Two experiments succeeded in bringing light to a standstill [326, 327] 2 years later. This can be understood in terms of a “dark-state polariton” model, in which the propagating photon is adiabatically converted into a (stationary) metastable atomic excitation. In addition to the obvious possibilities for storage of optical pulses [328, 329], there have been several proposals and demonstrations of generating extremely strong optical nonlinearities at the single-photon level that may be directly applicable to quantum information processing [330–337]. For a review on anomalous optical propagation velocities, see [338].
84.8
Gravity and Quantum Optics
84.8.1 Gravitational Wave Detection 84.7.5 Other Fast and Slow Light Schemes In addition to the case of tunneling described already, apparently superluminal propagation was observed in a Bessel beam geometry [312], and in the case of an inverted medium [313], as described in [297–299]. While the former case may be explained geometrically, the latter – in which superluminal group velocities occur without significant gain, loss, or distortion – raises difficult questions about the speeds of propagation of both energy and information. For two contrasting perspectives, see [314] and [315]. Much more work has followed [313], including theoretical treatments of the role of quantum noise in preventing superluminal information transfer [316] and attempts to experimentally compare the velocity of information transfer with the group velocity [317]. The latter work seemed to verify the claim, previously tested only in an electronic analog [318], that even in the regime of superluminal propagation, new information was limited to causal speeds. Some dispute has persisted [319], and it seems clear that a more rigorous definition of information velocity is required; somewhere between the idealized extremes of infinitely sharp signal fronts and strictly finite signal bandwidths lies the real world, and neither the front velocity nor the group velocity should be expected to completely describe the behavior of actual information-carrying pulses. At the same time, the definition of the energy velocity in active media cannot be resolved without reviving the long-standing conundrum about how to apportion energy between the propagating field and energy stored in the medium [320, 321]. Since 1999 there has actually been much more excitement over so-called “slow light” than over “fast light” [322].
According to general relativity, gravitational waves can be produced and detected by moving mass distributions [339– 341]. However, gravitational waves are coupled only to time-varying mass quadrupole moments in lowest order, P since the mass dipole moment is j mj rj D M Rcm and Rcm , for a closed system can only exhibit uniform rectilinear motion. Current efforts focus on detecting gravitational waves (typically at 100 Hz to 1 kHz) from astrophysical sources, such as supernovae or collapsing binary stellar systems. For example, it is expected that in the nearby Virgo cluster of galaxies, several such events should occur per year, each yielding a fractional strain (L=L) of 1021 on Earth. The first major effort to detect gravitational waves began in the 1960s and utilized a large cylindrical mass (sometimes known as a “Weber bar”, after its inventor) whose fundamental mode of acoustical oscillation is resonantly excited by time-varying tidal forces produced by the passage of a gravitational wave [341]. However, these devices were not sensitive enough to detect gravitational waves. More recently, a large amount of research has been devoted to using optical interferometry to detect gravitational waves. A passing gravitational wave alters the relative path length in the arms of a Michelson interferometer, thereby slightly shifting the output fringes. Although the effective gravitational mass of the light is much smaller than that of the Weber bar, very long interferometer arms (2–4 km, with a Fabry–Perot cavity in each arm to increase the effective length) more than make up for this disadvantage. The signal-to-noise ratio for the detection of a fringe shift depends on the power of the light. The US initiative, called Advanced LIGO (Laser Interferometer Gravitational-Wave Observatory [342]) uses 20 W
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from a Nd:YAG laser and an additional external mirror to recirculate the unmeasured light, thus increasing the stored light power up to 100 kW. There are three LIGO interferometers, located in Hanford, Washington (with both a 2-km and a 4-km version), and Livingston, Louisiana (4-km version), respectively. The registration of coincident events at the separated interferometers allows one to rule out terrestrial artifacts, but many problems involving seismic and thermal isolation, absorption and heating, intrinsic thermal noise, and optical quality had to be addressed. In September 2015, both the Louisiana and Hanford LIGO interferometers registered the first direct terrestrial observation of gravitational waves. The signal, which was coincident at both detection site, swept up from 35 Hz to 250 Hz, and corresponded to a peak gravitational strain of 1:0 1021 [343]. The gravitational waves were most likely produced by the merger of two black holes that converted 3 solar masses worth of energy into gravitational waves. This discovery was awarded the 2017 Nobel prize. Since then several other binary black hole mergers have been detected, and in August 2017 gravitational waves were detected from the inspiral of two binary neutron stars [344]. Currently, a world-wide network of gravitational wave detectors is coming online with operational locations in Italy (VIRGO) and Germany (GEO600), along with planned detectors in Japan (KAGRA) and India (IndIGO). Because the standard quantum noise limit of these detectors is ultimately determined by the vacuum fluctuations incident on the unused input ports of the interferometers, it is, in principle, possible to achieve reduced noise levels by using squeezed vacuum instead [341, 345] (Sect. 84.2.3). Currently, at lower frequencies, it is the radiation pressure of the circulating light in the interferometers at LIGO that set the noise floor–adding more power to the laser will not lead to a simple gain in sensitivity. Using squeezed light instead could circumvent this problem by yielding increased sensitivity at current power levels. At the GEO600 gravitational detector in Hanover, Germany, squeezed light has been used to improve the observable volume of the universe by approximately a factor of 2 [346], and plans are underway to incorporate this technology in future versions of LIGO [347]. A more ambitious project aims to establish an interferometric gravitational wave detector in space using a network of satellites, where the fundamental baseline can be several million kilometers long, allowing the device to detect gravity waves with much lower frequencies than ground-based systems [348]. It has also been suggested that matter waves that interact with gravity waves inside a matter-wave interferometer (Sect. 84.3.1) could lead to a sensitive method to detect gravitational waves [349]. Such a “Matter-Wave Interferometric Gravitational-Wave Observatory” (MIGO) may allow the detection of primordial gigahertz gravity waves arising from the Big Bang [350]. Moreover, quantum mechanical detec-
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tors based on the use of macroscopically coherent entangled states may enable quantum transducers that can interconvert between electromagnetic and gravitational waves, based on time-reversal symmetry [351].
84.8.2 Gravity and Quantum Information A fundamental connection between black holes and quantum information was suggested recently [352–354]. The basic idea is that every physical object, including a black hole, can be thought of as a computer that unitarily transforms input states to output states; i.e., “in” quantum bits (“qubits”) can always be reversibly interconverted into “out” qubits, thus obeying time-reversal symmetry. Hawking blackbody radiation from a black hole, thus, contains as much information in an evaporating black hole as originally fell into it, when this process is viewed as a reversible quantum-scattering problem. To resolve the paradox of the apparent loss of information of matter falling into a black hole, Lloyd and Ng propose that pairs of entangled photons can materialize at the event horizon of a black hole. One member of the photon pair flies outward to become the Hawking radiation; the other falls into the black hole and hits the singularity together with the matter that formed the hole. The annihilation of the infalling photon acts a measurement on the infalling matter in a quantum teleportation-like process, transporting the information contained in the infalling matter to the outgoing Hawking radiation, using the Horowitz–Maldacena mechanism [355].
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L. Krister Shalm Krister is a Sr. Research Associate at the University of Colorado, Boulder and an Associate of the National Institute of Standards and Technologies. He uses photons to study quantum networking as well as perform fundamental tests of quantum mechanics. When he isn’t in the lab he can be found swing dancing.
Paul G. Kwiat Paul G. Kwiat is the Bardeen Chair in Physics, at the University of Illinois, in Urbana-Champaign. A Fellow of the American Physical Society and the Optical Society of America, he studies the phenomena of entanglement, quantum interrogation, quantum erasure, and optical implementations of quantum information protocols. He can’t resist a good swing dance.
Aephraim M. Steinberg Aephraim Steinberg works on experimental quantum optics and laser cooling, with specific emphasis on foundational questions in quantum mechanics (especially quantum measurement) and on quantum information. His obsession is with tunneling times; in 1994, and in 2005, he demonstrated (with Kwiat and Chiao) the superluminal tunneling of photons; he is starting an experiment to probe tunneling times for Bose-condensed atoms through optical barriers.
Raymond Y. Chiao Professor Chiao was awarded his PhD by MIT in 1965. He has been Professor of Physics at Berkeley since 1977. His research interests are nonlinear and quantum optics, low-temperature physics as applied to astrophysics, and the relationship between general relativity and macroscopic quantum matter. He is writing a book with J.C. Garrison on quantum optics.
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Quantum Information Daniel F. V. James, Peter L. Knight
Contents
Abstract
85.1 85.1.1
Entanglement and Information . . . . . . . . . . . . . 1260 Testing for and Quantifying Entanglement . . . . . . . . 1260
85.2 85.2.1 85.2.2
Simple Quantum Protocols . . . . . . . . . . . . . . . . 1262 Quantum Key Distribution . . . . . . . . . . . . . . . . . 1262 Quantum Teleportation . . . . . . . . . . . . . . . . . . . . 1262
85.3 85.3.1 85.3.2 85.3.3 85.3.4
Quantum Logic . . . . . . . . . . . Single-Qubit Operations . . . . . . . Two-Qubit Operations . . . . . . . . Multiqubit Gates and Networks . . Cluster-State Quantum Computing
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85.4 85.4.1 85.4.2 85.4.3 85.4.4
Quantum Algorithms . . . . . . . Deutsch–Jozsa Algorithm . . . . . Grover’s Search Algorithm . . . . Shor’s Factor-Finding Algorithm Other Algorithms . . . . . . . . . .
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85.6 85.6.1 85.6.2 85.6.3 85.6.4 85.6.5
The DiVincenzo Checklist . . . . . Qubit Characterization, Scalability Initialization . . . . . . . . . . . . . . Long Decoherence Times . . . . . . Universal Set of Quantum Gates . . Qubit-Specific Measurement . . . .
85.7 85.7.1 85.7.2
Physical Implementations . . . . . . . . . . . . . . . . . 1268 Linear Optics . . . . . . . . . . . . . . . . . . . . . . . . . 1268 Trapped Ions . . . . . . . . . . . . . . . . . . . . . . . . . . 1269
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1270 D. F. V. James () Dept. of Physics, University of Toronto Toronto, ON, Canada e-mail: [email protected] P. L. Knight Dept. of Physics, Imperial College London London, UK e-mail: [email protected] S. Scheel Institute of Physics, University of Rostock Rostock, Germany e-mail: [email protected]
, and Stefan Scheel
Since its inception over a century ago, quantum mechanics has been an indispensable tool for understanding atomic structure and has been used very successfully in the calculation energy levels, natural lifetimes, and cross sections; but for the most part, the philosophical interpretation of quantum mechanics has been left to others. However, after the work of Bell in the 1960s showed that the peculiarly nonlocal nature of quantum correlations could be tested in the lab, a number of atomic physicists turned to the experimental study of entanglement and quantum measurement. By the mid 1990s, it was becoming increasingly apparent that the peculiar quantum correlations and quantum superpositions might be exploited in quantum information processing, and that practical devices to exploit this potential might well be within the grasp of current experimental techniques. This led to an explosive growth of the subject over the past 25 years, fueled by the long-term prospects of quantum computing, quantum-enhanced metrology, and quantum cryptography. Today, quantum information is a vast area, embracing physics, chemistry, electrical engineering, material science, mathematics, and computer science. In this chapter, we introduce some of the basics of the subject, with an emphasis on atomic, molecular, and optical physics. Keywords
entangled state quantum teleportation quantum information processing quantum gate dense coding
The fundamental resource in quantum information processing is generally considered to be entanglement between spatially separated subsystems. Entanglement is a quantummechanical effect and has led to numerous speculations about the validity of quantum mechanics itself for its apparent paradoxical implications. Most, if not all, of these difficulties have been resolved and can be mostly attributed
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to the simple fact that paradoxical behavior is incompatible with common sense or everyday experience. These quantummechanical correlations have numerous applications in quantum key distribution (QKD), quantum communication, dense coding, and act as the main resource in quantum computing. We will briefly touch upon some mathematical issues concerning separability and quantification of entanglement before describing how quantum key distribution and teleportation work. After that, a brief discussion of single-qubit and two-qubit quantum gates follows, before we describe some simple quantum algorithms. The issues of error correction and fault tolerant computation as well as DiVincenzo’s checklist (which any realization should satisfy) provide the background for the discussion of some physical implementations, of which we will pay most attention to those involving atomic, molecular, and atomic physics. We are acutely aware of the fact that we can give only a brief introduction into what has become a major field of investigation over the last quarter-century. There are a number of textbooks available that cover the vast literature on this subject. Perhaps the most well-known is the widely cited quantum information textbook by Nielsen and Chuang [1]; a number of more recent works are also recommended as starting points for more in-depth study [2–7].
85.1
Entanglement and Information
As was already noted, entanglement comes about if a quantum-mechanical system can be divided into two or more distinguishable parts. (Distinguishability of the subsystems is important for applications of entanglement and is implicit in Schrödinger’s original definition of the term [8]; systems such as the two-electron ground state of helium form a spin-singlet system, but the two electrons are not distinguishable, thus strictly speaking are not entangled.) As an example, consider a two-photon emission process from a spin-zero particle by which two photons escape in opposite directions. Given that the photons are spin-one particles, their spin projections onto some axis must be mutually opposite. As there is no prior information about the actual orientation of the spin, the part of the photon wave function associated with the spin degree is, therefore, 1 j i D p .j"#i j#"i/ : 2
(85.1)
origin is called entanglement. (Classical entanglement, i.e., an electromagnetic field, which is nonfactorizable in some representation, has received some attention in the last few years ([9] and references therein); although there are interesting parallels with quantum entanglement, the key difference is the significance of the quantum wavefunction and the collapse of multipartite states due to measurement, which is fundamentally quantum mechanical.) An important and inviolable principle of special relativity is that no information can be transmitted faster than the speed of light. Because the (classical) information concerning the measurement result on one particle needs to be transmitted via a causal classical channel, this experiment respects the letter of the law; however, the instantaneous collapse of the distant photon’s state can be said to violate the spirit of the law. This issue of nonlocality has been seen as a vital part in understanding the foundations of quantum mechanics itself (Chap. 84). In 1935, Einstein, Podolsky, and Rosen argued on the basis of entangled states that quantum mechanics is incomplete [10]. They were most concerned about the existence of elements of reality within strongly correlated quantum systems and initiated the debate on quantum nonlocality. The nonexistence of so-called local hidden variable theories for the description of states like Eq. (85.1) was demonstrated by John Stewart Bell [11]. He showed that maximally entangled states violate certain inequalities (now called Bell’s inequalities), which local hidden variable models would have obeyed. Later experiments showed the correctness of Bell’s demonstration [12–16]. In classical information theory, the unit of information is called a bit, which can be defined as the amount of information contained in a yes–no question. As a matter of fact, “bit” is the abbreviation for “binary digit” and refers to Boolean algebra in which the allowed states of a system are the logical 0 and the logical 1. Therefore, by abuse of language, one bit (as a unit) is the information carried by one bit (as a binary digit) [18]. In quantum mechanics, however, due to its inherent linearity, two “quantum bits” (or “qubits” – this word, which has now been included the Oxford English Dictionary, seems to have been first used in [17] can be in superpositions of the logical states j01i and j10i (or j"#i and j#"i, as in the example above). This typical example of an entangled state shows that quantifying the amount of information contained in a quantum state is different from what is known in classical information theory because of the superposition property.
The striking feature of this type of quantum state is that it describes correlations of two spatially separated particles. If 85.1.1 Testing for and Quantifying the polarization state of one photon is measured in any basis Entanglement (e.g., linear polarization, circular polarization, etc.), the state of the other photon, which can be far away, is then instantly From the above it is clear that entangled states play a mapredetermined. This nonlocal correlation of purely quantum jor role in defining the differences between classical and
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quantum information. Thus, it is clearly useful to be able to determine whether or not a particular state is entangled. Consider a quantum state of two distinct subsystems A and B. Note that these subsystems themselves might consist of ensembles of particles, in which case we are looking at a bipartite cut through the whole system. Suppose subsystem A is described by an orthonormal basis set fj˛i i; i D 1; 2; : : : NA g and subsystem B by an orthonormal basis set fjˇk i; k D 1; 2; : : : NB g, then the most general state of the bipartite system can be written as NA X NB X
where, to account for the uncertainty of the state of subsystem B, we have averaged out all its possible states; mathematically this is described by taking a partial trace over subsystem B. A simple and useful measure of how mixed a state is given by the trace of the square of its density matrix, called the purity, defined as ˚ P . / O D Tr O2 :
(85.5)
(85.2) The purity has the maximum value of 1, corresponding to pure states; its smallest value is 1=N , (where N is the dimeni D1 kD1 sionality of ), O corresponding to maximally mixed states. As where ci k are probability amplitudes, which form the elewe have seen, a pure reduced density matrix implies a separaments of a NA NB matrix. Using the singular value deble state, for which the entanglement is zero; for a maximally composition on this matrix [19], the state may be rewritten in entangled state, we would expect the reduced density matrix a simpler manner known as the Schmidt decomposition, i.e., to be maximally mixed, corresponding to complete ranN X domness. In this case, the entanglement is so strong and p pn jun ijvn i ; (85.3) comprehensive that discarding subsystem B has resulted in jAB i D nD1 a completely random state for subsystem A. Hence, it is natwhere fpn ; n D 1; 2; : : : N g are real, positive valued con- ural to define a measure of entanglement for the pure state P stants, such that N nD1 pn D 1; fjun i; n D 1; 2; : : : N g and jAB i as: fjvn i; v D 1; 2; : : : N g are sets of orthonormal vectors in subsystems A and B, respectively. The constants pn and r the vectors fjun ig are, respectively, the eigenvalues and the N C .AB / D .1 P. OA // eigenvectorsP of the NA NA Hermitian, positive-definite maN 1 NB v trix Aij D kD1 ci k cj k , and N is the number of nonzero ! u N u N X eigenvalues; similarly, P the NB NB Hermitian, positivet pn2 : 1 (85.6) D A definite matrix Bkl D N N 1 i D1 ci k ci l has eigenvectors fjvn ig; nD1 the eigenvalues of Aij and Bkl are identical, hence if NB < NA , then some of the eigenvalues of Bkl will be zero, and vice versa). The Schmidt number must be greater than or This quantity is called the concurrence [20, 21]. By defequal to 1, and at most, equal to the smaller of the two sub- inition, it takes values between 0 (separable states) and 1 system dimensions NA and NB . If (and only if) the Schmidt (maximally entangled states). For the simplest nontrivial number of the state is 1 the state of the combined system can case, that of two qubits, the concurrence of the state jAB i D be written as a simple product of a state of subsystem A and ˛j##i C ˇj#"i C j"#i C ıj""i is given by C D 2j˛ı ˇj; a state of subsystem B; such states are called separable. If for the state Eq. (85.1), C D 1, implying that it is maximally the Schmidt number is greater than 1, no such factorization entangled. exists, and the state is then entangled. We have concentrated here on bipartite entanglement In order to find some notion of the extent of the entan- of pure states. This is a major simplification and ignores glement of a state, let us now suppose we were to discard a great body of work on the entanglement both of mixed subsystem B and concentrate our attention on subsystem A. states and of multipartite states. The fundamental issue when How could we describe the state of this system? If we assume dealing with mixed states is that one must somehow sepathat the Schmidt number is greater than 1, then there can be rate the inherent mixture of the state as a whole from the no pure state description, and one must consider the reduced mixture induced by discarding one of the subsystems; for density operator for subsystem A, viz. pure states, the former is zero. Since quantum information jAB i D
OA D TrB fjAB ihAB jg D
ci k j˛i ijˇk i ;
N X
Aij j˛i ih˛j j
i;j D1
D
N X nD1
pn jun ihun j ;
usually assumes devices that are prepared in pure states, and their performance rapidly degrades as their systems undergo decoherence, the taxonomy of mixed-state entanglement measures, while fascinating, is at best tangential to the (85.4) main thrust of this article; the interested reader should consult some of the recent review articles on the topic [22, 23].
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Simple Quantum Protocols
Table 85.1 BB84 protocol for secret key distribution. “Alice” sends information encoded in either of two basis sets. “Bob” randomly chooses a measurement basis that is publicly communicated. For those cases when sender and receiver chose the same basis, the receiver’s measurement yields a secure bit
In this section, we describe the historically first and simplest quantum protocols – quantum key distribution and quantum teleportation – that make use of inherently “quantum” Sender (Alice) Receiver (Bob) Generated key properties of quantum-mechanical systems. These are ei- D fH; V g None ther entanglement or, in the case of the simplest version V fH; V g 0 of quantum key distribution, properties of the quantum- A fH; V g None mechanical measurement process. We should mention here H fH; V g 1 fD; Ag 0 the pioneering work of Holevo [24], who showed that there D fD; Ag None are fundamental limits on the amount of information that V fD; Ag 1 can be extracted by measurements. The application of his A H fD; Ag None ideas to channel capacity and communication [25, 26] are well described in [1], and space limitations prevent us from is the quantum-mechanical measurement process itself that elaborating on it in this chapter. provides security of the protocol. Nor can any eavesdropper (generically known as Eve) hope to intercept the signal with85.2.1 Quantum Key Distribution out being detected: individual photons can neither be split in a beam splitter (the photon will either go to Bob, so Eve Historically, the earliest protocol that used quantum- cannot see it, or to Eve, in which case Bob does not receive mechanical features in order to realize some specific task it), nor can it be cloned by some device [29] without inducthat could not have been performed classically was a pro- ing a detectable increase in the error rate. The first quantum tocol for secure distribution of a key in cryptography, known key distribution experiments were reported in [30]. However, as the BB84 protocol after its inventors Bennett and Bras- imperfections in the generation and detection of photons, sard and the year of its invention [27]. Although it is transmission losses, and polarization drift cause an actual commonly referred to as the first example of quantum in- experimental realization to be far from ideal. In practice, formation processing, it does not make use of entanglement; encodings other than polarization may be used (for examan entanglement-based method for quantum key distribution ple, a time-binned interferometric basis). Despite these error was introduced some years later by Ekert [28]. The BB84 sources, unconditionally secure quantum key distribution has protocol works in the following way. The sender, generically been demonstrated and implemented over distances of up to called Alice, prepares a random sequence (or string) of single 100 km in both fibers and free space. For reviews of theophotons in a polarization state, which is chosen out of a set of retical and experimental aspects of quantum cryptography, four basis states, horizontally and vertically (H and V ) po- see [31, 32]. larized, and diagonal (45ı ) and antidiagonal (135ı ) (D and A) polarized. In each of the two basis sets fH; V g, fD; Ag, one of the states is used to encode the logical value 0 (say 85.2.2 Quantum Teleportation in H and D), and the other states encode the logical value 1 (V and A). The random sequence is sent to the receiver, An important utilization of entanglement as a necessary reBob, who performs measurements on the sequence of sig- source can be found in quantum teleportation. The task of nals by randomly choosing analogous basis states. The result teleportation is to transmit the complete information of an will be another string of 0’s and 1’s that generically does arbitrary unknown quantum state to a spatially different lonot coincide with the original string. To rectify this prob- cation with the aim of recreating it. The simplest and most lem, sender and receiver communicate over a classical public obvious way to perform this task would be to take the quanchannel, where the sender announces the sequence of basis tum object that is prepared in the original state and physically sets in which the photon states were prepared. The receiver transport it to a different location. However, sometimes this is compares its sequence of randomly chosen basis states with not possible because, for example, an ion needs to be stored the announced string and keeps all measurement results for in a trap and cannot be moved. The next obvious thing to do which the choice of basis had been the same. In that way, would be to measure the quantum state and to recreate it at a different position using the classical information obtained a common secret key is established (Table 85.1). The security against eavesdropping of this simple protocol during the measurement. However, single measurements on comes from the fact that even by knowing the measure- a quantum system yield only partial information and multiple ment basis (say fH; V g) no information has been revealed measurements on many identically prepared copies would about the choice of the actual bit value (H or V ). Hence, it have to be performed.
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Output state 2-bit classical information
Bell measurement
Pauli rotation
Entangled resource
Input state
Fig. 85.1 Outline of an ideal teleportation protocol
The protocol (Fig. 85.1), which was originally proposed in [33], makes use of the existence of maximally entangled states. Let the unknown quantum state that is to be teleported be a qubit superposition state of the form: j i D ˛j0i C ˇj1i ;
j˛j2 C jˇj2 D 1 :
(85.7)
C Then p one prepares a maximally entangled state j i D .1= 2/.j01i C j10i/, which is one of the four orthonormal Bell states, defined by
1 j i D p .j01i ˙ j10i/ ; 2 1 ˙ j˚ i D p .j00i ˙ j11i/ : 2 ˙
We then form the tensor product state j j
C A ijBC i
C A ijBC i
(85.8)
85.3
as
1
D p ˛j0A 0B 1C i C ˛j0A 1B 0C i 2 C ˇj1A 0B 1C i C ˇj1A 1B 0C i ; 1 C i.˛j0C i C ˇj1C i/ D jAB 2 C jAB i.˛j0C i ˇj1C i/ C C j˚AB i.˛j1C i C ˇj0C i/ C j˚AB i.˛j1C i ˇj0C i/ ;
C , that flips the sign of the state j1i, whereas on obtaining C j˚AB i or j˚AB i the operations to be applied have to be O x or O z O x D i O y , respectively. Note that this quantum teleportation protocol works with perfect fidelity only if a maximally entangled state has been used. In the course of the Bell measurement, the quantum information is used up, and two classical bits of information (the measurement result) have to be communicated to C in order to restore the original quantum state. In this sense, entanglement can be regarded as a resource or “fuel” for certain tasks in quantum information processing. The first experimental demonstrations of teleportation of qubits were performed using photons as qubits by the groups of Zeilinger in Vienna and De Martini in Rome [34, 35], although these could not implement a complete Bell-state measurement (i.e., they could only distinguish one or two of the four Bell states). The first experiments to demonstrate full teleportation with highly entangled states, strong projective measurement, and complete state reconstruction were performed by the groups of Blatt at Innsbruck, Austria and, independently, Wineland at Boulder in Colorado USA, both using trapped ions [36, 37]. Since then, it has been implemented on other systems and over longer distances; in 2017 a collaboration headed by Pan demonstrated the teleportation of photons over distances of up to 1400 km from a ground station to a satellite in low Earth orbit [38].
Quantum Logic
As we have seen, quantum information relies on devices consisting of qubits. In order to perform any quantum information processing task, one must implement unitary operations on them collectively, in order to create the desired quantum state. For a large number of qubits, this would seem a task daunting in its complexity; fortunately, it has been demonstrated that all unitary operations on a large qubit register can be broken down into single-qubit operations and operations (85.9) involving pairs of qubits.
where we have explicitly indexed the relevant subsystems. After performing a joint measurement on subsystems A and B in the Bell basis (this is called a Bell-state measurement), one obtains one of four possible results. If the measurement C i, then the subsystem C is, indeed, prepared result was jAB in the original unknown quantum state j i, and hence the state has been “teleported” from subsystem A to C . For all other measurement results the outcome is not exactly the same quantum state as intended, but the difference is just a unitary transformation, which is uniquely determined by the outcome of the Bell measurement. For example, measur i means one has to perform a O z -operation on qubit ing jAB
85.3.1 Single-Qubit Operations It is instructive to give an example of how to classify all possible unitary operations that can act on a single qubit. A unitary operation acting upon the basis states fj0i; j1ig can be represented by a unitary .2 2/-matrix, and hence is a matrix that represents an element of the unitary group U.2/. This group has four generators, the identity matrix and the three Pauli matrices. All unitary .2 2/-matrices are linear combinations of those four matrices. Given the way they
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controlled-NOT gate is to interchange the states j10i $ j11i, whereas the controlled-phase gate changes the phase of the IO D j0ih0j C j1ih1j basis state j11i by and leaves all other states unchanged. The controlled-NOT gate has an interpretation as a sum gate O x D j0ih1j C j1ih0j O z D j0ih0j j1ih1j (85.10) in that it performs a mapping jx; yi ! jx; x ˚ yi, where the addition has to be taken modulo 2. Moreover, it acts as an and, by virtue of the commutation rules for U.2/-generators, entangling gate when acting on tensor products of superpoO y D i O x O z . Sometimes, the short-hand notation XO D O x , etc., sitions. is used. A particularly useful single-qubit gate that is not just one 85.3.3 Multiqubit Gates and Networks O of the Pauli operators is the Hadamard gate p H . In terms of O Its Pauli operators, it is defined as HO D .1= 2/.XO C Z/. purpose is to transform each basis state p into equal superpo- To realize a unitary operation on many qubits for a particular algorithm one would need a network of single-particle sitions of basis p states, i.e., j0i ! .1= 2/.j0i C j1i/ and and multiparticle quantum gates. Quantum networks enable j1i ! .1= 2/.j0i j1i/. The Hadamard gate is used to initialize an equal superposition of all possible N -qubit ba- a prepared input state to be transformed by the appropriate sis states from the state j0i˝N (i.e., the tensor product of N unitary operator to a final state, which is then measured. Deutsch’s model of quantum networks enables us to dequbits in state j0i). Hence, compose the network into component gates in diagrammatic N form [39]. The task is then to optimize the sequence of gates. O 1 X HO i j0i˝N D p jxk i ; (85.11) One can treat quantum gates acting on N qubits as being el2N k i D1 ements of the group, which has generators. This, however, is not a particularly transparent or useful way of looking where the jxk i are all 2N possible N -digit binary numbers. at these gates. Much more useful, and of immense practical importance, is a result essentially from linear algebra, which states that every N -qubit gate can be generated by 85.3.2 Two-Qubit Operations a network that consists only of very few elementary building blocks consisting of one or two-qubit operations, the Similarly to the single-qubit case, one can write down all so-called universal set of quantum gates [40]. This set conpossible unitary operations on two qubits by noting that they tains all possible single-qubit rotations and one nontrivial constitute a representation of U.4/; the 16 generators of this two-qubit gate, such as the controlled-NOT or controlledgroup can be found by forming the tensor products of pairs phase gate mentioned above. the four generators of U.2/, listed above; all possible twoqubit operators can be written as a linear combination of these generators. 85.3.4 Cluster-State Quantum Computing Particular examples of nontrivial two-qubit gates are the controlled-NOT and the controlled-phase gate, defined in The original paradigm for a quantum computer was a register terms of Pauli operators as of qubits, on which unitary operations would be performed, act upon basis states, they can be written as
CO 12 D j01 ih01 j ˝ IO2 C j11 ih11 j ˝ XO2 ; ˘O 12 D j01 ih01 j ˝ IO2 C j11 ih11 j ˝ ZO 2 ;
(85.12)
where in both cases, qubit 1 acts as the “control” and qubit 2 acts as the “target” (Fig. 85.2). The net effect of the
Control Target
00²
00²
01²
01²
10²
11²
11²
10²
Fig. 85.2 Symbol and truth table of the controlled-NOT gate. The target qubit is flipped depending on the state of the control qubit
and eventually the answer would be read out by performing a measurement on some or all of the qubits. A system of N qubits in a pure entangled state would be a quantum system of 2N dimensions; however the readout could at most procure N bits of information. This indicates both the potential and the drawbacks of the scheme: a quantum computer has an enormous memory on which parallel operations might be performed most efficiently; however, only a small amount of the information present in that memory can ever be accessed. An important new paradigm for quantum computing was introduced by Raussendorf and Briegel in 2001 [41]. Cluster-state quantum computing (also called one-way quantum computing, or measurement-based quantum computing) envisions the initial creation of an entangled state of a very
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large number of qubits, called a cluster state. If the algorithm N N N H H 0 x x one wishes to implement requires N register qubits to store information and M quantum gates to process it, then the Uf size of the cluster state needs to be N M physical qubits. The quantum algorithm may then be implemented by mea1 H y y f (x) surements of a subset of the qubits, followed by a series of single-qubit operations conditioned on the outcome of those measurements, followed by another series of measurements Fig. 85.3 Gate network for implementing the Deutsch–Jozsa algorithm and conditional single-qubit operations, and so on. The resultant final state can be shown to be equivalent to the output is the result: state of a traditional, circuit-based quantum computer.
85.4
Quantum Algorithms
The search for algorithms that would run faster on a quantum computer than on any classical computer is a formidable task. By “faster”, we actually mean that the number of elementary tasks required to perform a given computation should be drastically reduced. The hope is that eventually one will find algorithms that provably run exponentially faster on a quantum computer compared to a classical computer. Here, we briefly discuss three well-known algorithms; investigation of other useful quantum algorithms remains an area of intense on-going research (for reviews: [42–44]).
85.4.1
Deutsch–Jozsa Algorithm
Let us give a particularly instructive example due to Deutsch and Jozsa [45]. Suppose one is given a function f .x/, which is Boolean (i.e., f .x/ has only two possible values: 0 or 1), and whose argument is a N -bit number, (i.e., x 2 f0; 1; : : : ; 2N 1g). From the start it is assumed f .x/ is either constant (i.e., it always has the value 0 or 1) or that it is balanced, (i.e., it returns the value 0 and 1 with equal probability). The task is to find out whether f .x/ is constant or balanced. Although this seems a contrived problem, it serves to illustrate the potential power of quantum computers. Classically, to be certain of getting the correct answer, one needs to evaluate the function at least 2N 1 C 1 times to find the answer (i.e., half of the possible values of x, plus 1). With a quantum computer, only a single evaluation of the function is required. First, N qubits are prepared in a superposition of all possible input states, using the Hadamard gate from Eq.p (85.11); and a single qubit is prepared in the state jyi D .1= 2/.j0i j1i/; then a unitary operator that evaluates the function f .x/ is applied, i.e., UOf jxijyi D jxijy ˚ f .x/i. Since the N qubits containing the value of the argument x are in a superposition of all possible values, a single evaluation of the function suffices (Fig. 85.3). The following state
X 1 p jxk i.jf .xk /i j1 ˚ f .xk /i/ 2N C1 k 1 X D p .1/f .xk / jxk i jyi ; 2N k
(85.13)
where we have used the fact that f .x/ can only take the value 0 or 1 in order to simplify the result. The final step is to repeat the Hadamard single-qubit unitaries of the argument register qubits and perform a measurement on all of these qubits; the probability of allpthe output P qubits being in state j0i can be shown to be j.1= 2N / k .1/f .xk / j2 , which will be 1 if the function is constant, and 0 if it is balanced. Thus, a measurement outcome other than 0 on any of the N argument qubits then tells that the function f .x/ is balanced. This quantum parallelism is at the heart of the increase in speed that occurs in quantum computation. The Deutsch– Jozsa algorithm involving a few qubits has been implemented on various platforms, most compellingly using ion traps [46].
85.4.2
Grover’s Search Algorithm
In contrast to the preceding example, which always gives the desired answer after exactly one trial, the quantum search algorithm by Grover [47] uses a procedure that amplifies the sought after result by a method called “inversion about average”. The goal of Grover’s algorithm is to search an unsorted database with 2N entries out of which only one entry fulfills a given criterion; for example, searching through an alphabetic telephone directory to find the person who has some specific telephone number. As in the Deutsch–Jozsa algorithm described above, the query is simultaneously run on all 2N possible N -qubit basis states, being prepared in an equal superposition. It is assumed that the state that satisfies the search criterion will acquire a phase shift of . After this step, the inversion about average is carried out. It is represented by a diffusion operator DO D 2PO IO, where IO is the identity operator, and PO is a projection operator that averages each input vector with respect to its components. Compared to the previous average value of probability amplitudes, after each of
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these steps, the magnitude of the desired state increases by O.2N=2 /. This procedure is repeated, and after only O.2N=2 / steps, a projective measurement yields the desired result with probability of O.1/, or more precisely, of more than a half. This is a quadratic increase in speed compared to classical search algorithms, which need O.2N / steps.
85.4.3 Shor’s Factor-Finding Algorithm Suppose one were given an integer n known to be the product of two primes, a and b. A systematic way to determine a and b would be to attempt to divide n by all the prime nump bers less than n. If n, written in binary, has L digits (i.e., L D dlog2 .n/e), then the prime number theorem [48, p. 47] implies that one must perform this division O.2.L=2/C1 =L/ times; in other words, the complexity of finding factors is exponential in the size of the number to be factored. While more efficient algorithms exist, they all have this unfortunate exponential scaling. This mathematical fact is central to the security of public-key encryption systems [48, pp. 212–217], and thus an algorithm that can find factors more efficiently will have far-reaching implications. Shor’s algorithm for quantum computers provides just such an algorithm [49]. The basic idea is to exploit conventional number theory to convert the factor finding to finding the period of particular modular exponential function. While a conventional computer would require multiple evaluations to determine the period of a function, a quantum computer, using the sort of parallelism we have already seen in the Deutsch–Jozsa algorithm, would require a single evaluation. A quantum Fourier transform (analogous to the fast Fourier transform executed on quantum registers) followed by measurement allows the period of the function to be determined with high probability. Once the period of this modular exponential function is determined, the factors of n may be deduced straightforwardly. The total number of operations required is a polynomial function of the number of bits L in the number to be factored. Implementation of simplified versions of Shor’s algorithm, using various tricks and short-cuts, have been reported using various platforms [50–53]; the first potentially scalable experimental demonstration (albeit on a small scale, i.e., factoring 15, using 5 qubits) was reported by Blatt and coworkers in 2016 using trapped ions [54].
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microscopic interactions to study the emergence of macroscopic behavior ([57, 58], for some recent experiments); the efficient calculations of molecular energy spectra in Quantum Chemistry [59–61]; Boson Sampling, the interference of harmonic oscillator modes in nonclassical states (typically single-photon states) in a large interferometer, the output of which can be shown to be related to computational intractable problems such as the calculation of permanents of matrices [62–64]; and Quantum Machine Learning [65], plus many others. A significant milestone in the development of viable quantum computing was passed in 2019 with the demonstration of quantum supremacy using a 53 qubit quantum processor developed by the team at Google led by John Martinis [66]. They carried out a calculation in about 200 seconds which, had a conventional computer been used, would have required 10 000 years.
85.5
Error Correction
As we have discussed, the essence of quantum information processing is the use of quantum superpositions, interference, and entanglement. However, quantum interference is fragile. It appears in practice that it is very difficult to maintain a superposition of states of many particles in which each particle is physically separated from all the others. Entanglement turns out to be incredibly delicate. The reason for this is that all systems, quantum or classical, are not isolated; they interact with everything around them: local fluctuating electromagnetic fields, the presence of impurity ions, coupling to unobserved degrees of freedom of the system containing the qubit, etc. These fluctuations destroy quantum interference. A simple analogy is the interference of optical waves in Young’s double-slit experiment. In that apparatus, waves from two spatially separated portions of a beam are brought together. If the two parts of the beam have the same phase, then the fringe pattern remains stable. However, if the phase of one part of the beam is drifting with respect to the other, then the fringe pattern will be washed out. And the more slits there are in the screen, the lower the visibility for the same amount of phase randomization per pair of slits. The sensitivity of an N -qubit register to decoherence is even worse, as a maximally entangled N -particle state decoheres at a rate N times faster than a single particle [67], one of the reasons why the world around us appears so classical. A single bit of 85.4.4 Other Algorithms information lost to an unobserved degree of freedom will result in the reduction of the quantum superposition to a mixed Other algorithms that are attracting considerable attention state. Yet, correcting errors due to environmental interactions and are being pursued both theoretically and experimentally is essential if a quantum computer is to be constructed; to include quantum simulations [55, 56], i.e., the use of quan- do “fault tolerant” computing we need to be able to execute tum devices to emulate physical systems with prescribed many gate operations coherently within the decoherence time
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if we are to have a chance of building a scalable quantum register [68]. It might appear that the problem of stabilizing a register of qubits is hopeless, like trying to balance several pencils on their tips on the deck of a ship in a storm. Yet, amazingly, quantum mechanics provides a way to solve this problem, through the use of even higher levels of entanglement. In 1996, Steane and, independently, Shor and coworkers, showed that encoding information in entangled sets of qubits offered the opportunity to execute quantum error correction [69, 70]. That one can do this is a remarkable consequence of entanglement. In classical information processing, inevitable environmental noise is dealt with by error correction. In its simplest form, this involves repeating the message transmission or calculation until a majority result is obtained. However, there are more efficient ways, for example, the use of a parity check on a block of bits. It turns out that a similar notion can be applied to a quantum register. However, the application is not straightforward because the contents of a register cannot be measured without destroying the superposition state encoded in it. The problem then is to determine what errors might be present in a quantum register without looking at the qubits. The elegant solution is to entangle the qubits in question with those in an ancillary register and measure the ancillary register. Because the two registers are correlated, the results of the measurement of the ancilla reveal any errors present in the processing register without destroying any coherent superpositions in the processing register itself. The first experimental demonstration of quantum error correction used liquid-state nuclear magnetic resonance (NMR) techniques [71]; but given the highly mixed nature of the “pseudo-pure” quantum states in NMR, which precludes the existence of any entanglement [72], this work had limited implications for quantum information processing. In 2004, Wineland’s group in Boulder succeeded in implementing quantum error correction in an entangled system using trapped ions [73, 74]; since then, multiple, repetitive error correction has been demonstrated by Blatt’s group. Another way to prevent the register coherence from falling apart is to know a little about the sort of noise that is acting on it. If the noise has some very slow components (or those with very long wavelengths), then it is sometimes possible to find certain combinations of qubit states for which the noise on one qubit exactly cancels the noise on another. These qubit states live in a “decoherence-free subspace” (DFS) [75–77]. A computer will then be immune to environmental perturbations if all the computational states lie in this DFS. Decoherence-free subspaces were first realized experimentally using photonic qubits by Kwiat and coworkers in 2000 [78]. A remarkable result, which has important implications for the long-term viability of quantum information processing, is the quantum threshold theorem, which states in essence that
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if a qubit can be realized with a high enough degree of protection against environmental errors, then through the process of concatenation (i.e., encoding one logical qubit using noisy qubits; each of which might be encoded on another set of qubits, and so on), quantum computations can be performed with arbitrary accuracy [79]. As in the field of quantum algorithms, fault-tolerant quantum computing remains a highly active area of research ([80], for a recent overview).
85.6 The DiVincenzo Checklist So much for the theory of what we would like to with quantum computers; there remains the vexatious topic of how they might be realized in practice. The short answer is that no one knows for certain how (or, indeed, if) practical quantum computers might be realized. DiVincenzo has given a list of requirements that a physical implementation must fulfill in order to qualify as a sensible candidate for an implementation of quantum information processing [81]; we will review these here and then use them to assess how various technologies are developing. We will concentrate on those implementations that currently seem the most promising, rather than discuss every proposal that initially seem promising but has over the years fallen by the wayside.
85.6.1 Qubit Characterization, Scalability Each physical implementation must be tested upon how qubits should be encoded. For a qubit being essentially a twolevel system, this task is generally not too difficult. Several candidates, such as electronic or nuclear spin, photon polarization, and choice of path in an interferometer, degenerate ground states of an atom or ion, charge or flux states in superconducting quantum interference devices (SQUIDs), or exciton population, have all been explored. Much more challenging will be the question as to whether there are fundamental or technological limitations of having many of those qubits being operated upon separately, hence whether the system can be scaled up to contain potentially many qubits.
85.6.2
Initialization
Once the qubits have been specified, each quantum information processing or quantum computation task needs to be able to start from a well-defined state. This can be basically any quantum state of the many-qubit system as long as it is a product state and can be prepared error-free. Commonly, this state is then called the ground state and is denoted by j0i˝N .
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85.6.3 Long Decoherence Times
85.7
In order to ensure error-free computation without loss of purity of quantum superpositions, the decoherence times that are relevant for the quantum operation should be much longer than the gate operation time itself. In most situations, decoherence limits the number of qubits that can be worked on simultaneously, thus affecting the scalability of the system. Typical examples of decoherence processes are heating mechanisms in ultracold systems, such as ion-trap or atom-chip experiments, spin relaxation in NMR-type experiments, or absorption in linear optical elements. Generally, decoherence is unavoidable due to the basic principles upon which quantum information processing is supposed to work. Avoiding decoherence means isolating the system from the outside world, the environment. However, controlling the interaction between subsystems always has the negative effect of bringing the system in contact with the environment and, therefore, necessarily introduces decoherence. Once one has accepted that decoherence is unavoidable, ways have to be found to guard against it. Several error-correction schemes have been proposed that can correct for certain small amounts of decoherence as described in Sect. 85.5.
Over the past quarter of a century, a number of architectures capable of attempting an assault on the rigors of the DiVincenzo criteria have been proposed and tested; many have fallen by the wayside, a few have persevered. In general, atomic and optical systems based on nonclassical light (e.g., single photons) [82] or cold trapped atoms [83, 84] have led the way, since these had been studied for many years for their intrinsic scientific interest, and had already the proven capability to produce entangled quantum states and/or the high degree of protection against environmental decoherence necessary for quantum computation. In what follows, we will concentrate our attention on these. However, they possess a decided disadvantage as a scalable architecture for a full-scale quantum computer with say 105 qubits and capable of performing useful tasks: in order that computation technology be robust and durable, it is taken for granted that a solid-state-based architecture will ultimately be necessary. Such architectures, based on quantized magnetic flux in superconducting circuits [85, 86] or on individual spin impurities in a semiconductor host [87, 88], have also shown considerable promise. In such a rapidly developing field, the relative merits of the various technological implementations may well reverse themselves within the space of a few years.
85.6.4
Physical Implementations
Universal Set of Quantum Gates
A necessary prerequisite for quantum information processing and quantum computing is the ability to generate a set of quantum gates that can be considered universal. With such a set, it will then be possible to generate all other quantum gates by concatenating them to form suitably arranged networks. The choice of which set out of the many possible is taken depends on the physical implementation itself. Basically, it is determined by the operations that are intrinsically simple for the given interaction Hamiltonian. In some applications, such as the ion-trap experiments, the controlled-NOT gate is preferred as the nontrivial two-qubit gate, whereas in linear optical networks, one rather works with the controlled-phase gate.
85.6.5 Qubit-Specific Measurement The last requirement is to be able to read out the result of the computation. That is, there has to be a way of providing a selective projective measurement. This proves to be a major challenge in most proposals for implementing quantum computing. Examples of the challenges involved are the necessity to provide photon-number resolving photodetectors, single-electron charge measurement devices, or single-spin measurements.
85.7.1
Linear Optics
The use of photons as carriers of quantum information seems to be a straightforward matter: they are easy to produce in numbers; much of the technology to store and to manipulate photons in mature due to its dominance in conventional telecommunications; they show both spin-like behavior (i.e., polarization) and boson-like behavior, making them versatile quantum systems. The first serious proposal for a quantum logic gate was made using photons as qubits in 1989 [89]. The major drawback for photons as qubits is the lack of strong nonlinearities at the single-photon level needed to implement two-qubit logic gates. However, a different approach, based on a generalization of the well-known quantum-optical Hong–Ou–Mandel interference effect [90], was proposed in 2001 [91, 92]. In this scheme, information is encoded in number states (also called Fock states): one photon can be in one of two distinct modes of free space or of a photonic waveguide. Since photons are bosons, there is a natural tendency to bunch together in interference experiments, which in a sense is a nonlinearity on the level of single or few photons. The trick is to use conditional measurements or measurement-induced nonlinearities. The idea was first realized experimentally in [93], where a polarization encoding was used. Fig. 85.4 shows the schematic
85
Quantum Information
Ancilla photon
1269
PBS
Control qubit
Target qubit PBS 45
Fig. 85.4 Setup of an all-optical controlled-NOT gate. Control and target qubits are encoded in the polarization of single photons. These are fed into polarizing beam splitters (PBS), one of which is rotated by 45ı
setup of a simplified version of a controlled-NOT gate with one single ancilla photon (after [94]). Measurement-induced nonlinearities make use of the fact that unitary transformations in a larger Hilbert space, e.g., with added auxiliary photon modes combined with photodetection, can yield effective nonlinearities [95]. The drawback is, however, that the wanted nonlinearity is conditioned on the appearance of a certain measurement pattern, which means that these schemes work only with a certain probability [96]. The set of quantum gates that can be considered fundamental differs slightly from most other physical implementations. Within the qubit encoding in photon number states, the gates that can actually be implemented efficiently are those that act within Fock layers (subspaces of fixed total photon number), such as the controlled-phase gate, or the swap gate. Other gates that do not fall into this class require excessively more resources unless other types of qubit encodings are used simultaneously. Gate operation times can be very fast and are only limited by the gating times of the photodetectors. Other attractive features of this approach are that devices can work at room temperature and can readily be integrated into photonic waveguides. However, a major experimental challenge is mode matching in larger networks and interferometric setups, as well as scalability [96]. The cluster state approach discussed above (Sect. 85.3.4) is actively being pursued using linear optical implementations.
85.7.2
Trapped Ions
The most advanced method of quantum information processing in terms of the number of qubits and the number of gates
that have been unequivocally demonstrated is the ultracoldions scheme proposed by Cirac and Zoller in 1995 [97]. The ions are stored in a linear Paul trap, in which radio frequency electric quadrupole fields are used to generate confining potentials [98, 99]. The ions are cooled into their motional ground states by Doppler cooling [100] and further cooled to the quantum ground state by resolved sideband cooling [101]. Since the ions are strongly coupled by their mutual Coulomb repulsion, their oscillations are resolved into a number of normal modes, rather like pendulums connected by springs [102]; each of these normal modes are quantum harmonic oscillators that interact with all of the ions in the trap, enabling a quantum data bus to connect the ion qubits. The qubits are encoded into two long-lived electronic states of individual ions, either transitions to metastable excited states (i.e., with lifetimes of a second or more, since decays are dipole forbidden) or Raman coupling between sublevels of the ground state, thereby avoiding the decoherence that an upper-state lifetime, or laser phase drift can generate. Lasers focused on individual ions are then used to control both the internal states of individual qubits and, by detuning, to couple the qubits to the quantum oscillation modes of the ions, thereby enabling two-qubit quantum logic gates. Readout is performed by observation of fluorescence from individual ions. To date 20 qubits have been trapped, cooled, coherently manipulated, and shown to be entangled [103].
85.8
Outlook
We have discussed the basic ideas behind quantum information and described a few possible applications. However, the immense wealth of ideas and possible routes have barely been touched upon. In the past few years, the field has generated tremendous excitement, with many new strands of investigation evolving rapidly. Quantum key distribution is becoming a commercial technology. For quantum computing, only one or two dozen qubits have been entangled and manipulated experimentally so far, and a deployable quantum computer remains decades away, although new algorithm research suggests that demonstrations of the quantum advantage in computation may be a reality in a much shorter time. Acknowledgements We like to thank A. Al-Quasimi, A. Beige, R. Blatt., J. Eisert, E. A. Hinds, V. Kendon, P. G. Kwiat, W. J. Munro, M. B. Plenio, A. G. White and many others for numerous stimulating discussions on this exciting subject. Funding by the Natural Science Engineering Research Council (Canada), UK Engineering and Physical Sciences Research Council (EPSRC), the Royal Society, the European Commission, and the Alexander von Humboldt foundation are gratefully acknowledged.
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1271 Daniel James Daniel James was born in Manchester, UK. He received his PhD from the University of Rochester in 1992, supervised by Professor Emil Wolf. He was a staff member in the Theoretical Division, Los Alamos National Laboratory (1994–2005), after which he joined the faculty of the University of Toronto. He has been engaged in quantum information research since 1995. Peter Knight Peter Knight is Emeritus Professor at Imperial, a past President of the Institute of Physics, 2004 President of the Optical Society of America, Chair of the UK National Quantum Technology Programme Strategy Advisory Board and chairs the Quantum Metrology Institute at the National Physical Laboratory. His research centres on quantum technology.
Stefan Scheel Stefan Scheel received his PhD (Dr rer nat) from the Friedrich Schiller University, Jena, Germany, in 2001. He worked in the Quantum Optics and Laser Science Group at Imperial College London until 2012, when he moved to Rostock, Germany as a Full Professor. His research interests include QED in dielectrics, fluctuation-induced phenomena, integrated waveguide quantum optics, and Rydberg physics with atoms and semiconductor excitons.
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Part G Applications
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Part G is concerned with the various applications of atomic, molecular, and optical physics. A summary of the processes that take place in photoionized gases, collisionally ionized gases, the diffuse interstellar medium, molecular clouds, circumstellar shells, supernova ejecta, shocked regions, and the early universe are presented. The principal atomic and molecular processes that lead to the observed cometary spectra, as well as the needs for basic atomic and molecular data in the interpretation of these spectra, are focused on. The basic methods used to understand planetary atmospheres are given. The structure of atmospheres and their interac-
tion with solar radiation are detailed, with an emphasis on ionospheres. Atmospheric global change is then studied in terms of the applicable atomic and molecular processes responsible for these changes. Various applications of atomic and molecular physics to phenomena that occur at surfaces are reviewed; particular attention is placed on the application of electron–atom and photon–atom scattering processes to obtain surface specific structural and spectroscopic information. The effect of finite nuclear size on the electronic energy levels of atoms is also detailed; and conversely, the electronic structure effects in nuclear physics are discussed.
Applications of Atomic and Molecular Physics to Astrophysics
86
Stephen Lepp, Phillip C. Stancil, and Alexander Dalgarno
Contents
lar shells, supernova ejecta, shocked regions, and the early universe.
86.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1275
86.2
Photoionized Gas . . . . . . . . . . . . . . . . . . . . . . 1276
Keywords
86.3
Collisionally Ionized Gas . . . . . . . . . . . . . . . . . 1278
86.4
Diffuse Molecular Clouds . . . . . . . . . . . . . . . . . 1278
86.5
Dark Molecular Clouds . . . . . . . . . . . . . . . . . . 1280
86.6
Circumstellar Shells and Stellar Atmospheres . . . . 1281
charge exchange dielectronic recombination early universe molecular cloud photoionization planetary nebula radiative association recombination supernova remnant three-body recombination Wilkinson microwave anisotropy probe
86.7
Supernova Ejecta . . . . . . . . . . . . . . . . . . . . . . 1282
86.8
Shocked Gas . . . . . . . . . . . . . . . . . . . . . . . . . 1283
86.9
The Early Universe . . . . . . . . . . . . . . . . . . . . . 1283
86.10
Atacama Large Millimeter/Submillimeter Array . . 1284
86.11
Recent Developments . . . . . . . . . . . . . . . . . . . . 1285
86.12
Other Reading . . . . . . . . . . . . . . . . . . . . . . . . 1285
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286
Abstract
The range of physical conditions of density, temperature, and radiation fields encountered in astrophysical environments is extreme and can rarely be reproduced in a laboratory setting. It is not only reliable data on known processes that are needed but also a deep understanding so that the relevant processes can be identified and the influence of the conditions in which they occur fully taken into account. We present here a summary of the processes that take place in photoionized gas, collisionally ionized gas, the diffuse interstellar medium, molecular clouds, circumstelS. Lepp () Dept. of Physics & Astronomy, University of Nevada, Las Vegas Las Vegas, NV, USA e-mail: [email protected] P. C. Stancil Dept. of Physics & Astronomy, The University of Georgia Athens, GA, USA e-mail: [email protected]
86.1 Introduction Almost all our information about the universe reaches us in the form of photons, though neutrinos, cosmic rays, and, recently, gravitational waves provide additional insight. Traditional observational astronomy is based on measurements of the distribution in frequency and intensity of the photons that are emitted by astronomical objects and detected by instrumentation on ground-based and space-borne telescopes. Information about the earliest stages in the evolution of the universe before galaxies and stars had formed is carried to us by blackbody background photons that attended the beginning of the universe. The photons that are the signatures of astronomical phenomena are the result of a plethora of physical and chemical processes reaching into a host of disciplines, including nuclear physics; plasma physics; solidstate physics or surface sciences; and atomic, molecular, and optical physics. The processes that modify the photons on their journey from distant origins through intergalactic and interstellar space to the Earth belong mostly to the domain of atomic, molecular, and optical physics, as do the instruments that detect and measure the arriving photons and their spectral distribution. The spectra are used to classify galaxies and stars and to identify other astronomical entities and phenomena such as quasars, active galactic nuclei, gravitational lensing, jets and outflows, pulsars, supernovae, novae, supernova remnants, nebulae, masers, protostars, shocks,
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_86
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molecular clouds, circumstellar shells, accretion disks, protoplanetary disks (PPDs), galaxy clusters, and black holes. Quantitative analyses of the spectra of astronomical sources of photons and of the atomic, molecular, and optical processes that populate the atomic and molecular energy levels and give rise to the observed absorption and emission require accurate data on transition frequencies and wavelengths, oscillator strengths, cross sections for electron impact, rate coefficients for radiative, dielectronic, and dissociative recombination, and cross sections for heavyparticle collisions involving charge transfer, excitation, ionization, dissociation, fine structure and hyperfine transitions, collision-induced absorption, and line broadening. Data on ion–molecule and neutral particle reaction rate coefficients are central to the interpretation of measurements of chemical composition in molecular clouds, circumstellar shells, PPDs, and supernova ejecta. The range of physical conditions of density, temperature, and radiation fields encountered in astrophysical environments is extreme and can rarely be reproduced in a laboratory setting. It is not only reliable data on known processes that are needed but also a deep understanding of the microphysics so that the relevant processes can be identified and their influence on the environments in which they occur fully taken into account. We present here a summary of the processes that take place in photoionized gas, collisionally ionized gas, the diffuse interstellar medium, molecular clouds, circumstellar shells, PPDs, supernova ejecta, kilonovae, shocked regions, and the early universe.
minosity is low, but the ionized regions can still be detected by radio observations. Planetary nebulae are smaller in extent and denser. They have a passing similarity in appearance to planets. Planetary nebulae are produced by the photoionization of shells of gas that have been ejected from the parent star as it evolved to its final white dwarf stage. Because the core of the parent star is very hot, the irradiated gas is more highly ionized than are emission nebulae and has a distinctive spectrum. Photoionized gas is also found around novae. Novae are stars that have undergone spasmodic outbursts, and they are surrounded by faint shells of ejected gas, photoionized and excited by the stellar radiation. Some supernova remnants, which are what remains after a massive star has exploded, have spectra that also appear to be emanating from photoionized gas. The source of ionization may be synchrotron radiation. Figure 86.1 shows the X-ray emission spectrum of a supernova remnant. The nuclei of starburst galaxies have spectra like those of emission nebulae. They result from gas photoionized by radiation from hot stars created in a period of rapid star formation. Active galactic nuclei, such as quasars, have a different spectrum characterized by broad lines indicating a large range of velocities. Photoionized gas is the most likely interpretation, although the ionizing source may be an accretion disk around a compact object, such as a black hole. The ionization structure in a photoionized gas is determined by a balance of photoionization X.m1/C C h ! XmC C e
(86.1)
XmC C e ! X.m1/C C h
(86.2)
and radiative
86.2
Photoionized Gas
The universe contains copious sources of energetic photons, most often due to radiation from hot stars. As a consequence, much of the baryonic material of the universe exists as photoionized gas. Photoionized gas produces the visible emission from emission nebulae, planetary nebulae, nova shells, starburst galaxies, and probably active galactic nuclei [1]. Emission nebulae are extended regions of luminosity in the sky. They arise from the absorption of stellar radiation by the gas surrounding one or more hot stars. The gas is ionized by the photons and excited and heated by the electrons released in the photoionizing events. A succession of ionization zones is created in which highly ionized regions give way to less ionized gas with increasing distance from the central star as the photon flux is diminished by geometrical dilution and by absorption. The outer edge of a nebula is a front of ionization pushing out into the neutral interstellar gas. The densities are typically between 100 and 10,000 cm3 , with temperatures between 5000 and 15,000 K. Nebulae are also called HII regions. At low densities, the lu-
and dielectronic
XmC C e ! X.m1/C ! X.m1/C C h
(86.3)
recombination, while in plasmas with a significant population of neutral hydrogen and helium, charge transfer recombination XmC C H ! X.m1/C C HC ; mC
X
.m1/C
C He ! X
C
C He
(86.4) (86.5)
may also play a role. Many detailed calculations of the ionization structure of photoionized regions have been carried out [1]. The ionizing source spectra of hot stars can be obtained from calculations of stellar atmospheres; see Sect. 86.6. Approximate values of cross sections for photoionization for a wide range of atomic and ionic systems in many stages of ionization are available [3–6]. Calculations of higher precision and reliability that incorporate the contributions
86
Applications of Atomic and Molecular Physicsto Astrophysics
Counts / 320 s 17
16
15
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14 Ne IX 1s2 – 1s 2p, s R F I
20 Fe XVII 2p6–2p53d 15
10
O Fe XVII 2p6–2p53s
12
Wavelength (Å)
Fe XVII 2p6–2p5 4d Ne X Ly α
VIII Ly γ
Ly β
13
Ne IX 1s2 –1s 3p
Ly δ
5
0 700
800
900
1000
1100 Energy (eV)
Fig. 86.1 X-ray spectrum of the supernova remnant Puppis A as observed by the Einstein satellite. Note the high level of ionization with hydrogenlike ions of oxygen and neon, suggesting a high temperature. After [2]
from autoionizing resonance structures exist for specific systems [7]. They are undergoing continual improvements as increasingly more powerful computational techniques are brought to bear on the calculations. The cross sections for radiative recombination are obtained by summing the cross sections for capture into the ground and excited states of the recombining system, typically by applying detailed balance to the state-resolved photoionization cross sections. Because of the contribution from highly excited states that are nearly hydrogenic, the rate coefficients are similar for different ions of the same excess nuclear charge. They vary slowly with temperature. In contrast, dielectronic recombination is a specific process whose efficiency depends on the energy level positions of the resonant states. For nebular temperatures, the rate coefficients vary exponentially with temperature. Explicit calculations have been carried out for many ionic systems [8–10]. Because the photoionization cross sections of the major cosmic gases hydrogen and helium diminish rapidly at high frequencies, multiply charged ions and neutral gas coexist in cosmic plasmas produced by energetic photons and charge-transfer recombination may control the ionization structure. For multiply charged ions with excess charge, greater than 2, charge transfer is rapid. For doubly charged and singly charged ions, the cross sections are sensitive to the details of the potential energy curves of the quasi molecule formed in the approach of the ion and the neutral particle and the interaction mechanism. Few reliable data exist. Some recent calculations may be found in the papers [11–13]. Photoionized gas is heated by collisions of the energetic photoelectrons and cooled by electron impact excitation of metastable levels, principally of OC and O2C , NC and N2C ,
and SC and S2C , followed by emission of photons that escape from the nebula. Considerable attention has been given to the determination of the rate coefficients [14]. The resulting cooling rates increase exponentially with temperature and keep the temperature of the gas between narrow limits. Some contribution to cooling occurs from recombination and from free-free emission by electrons moving in the field of the positive ions. The luminosity of the photoionized gas comes from the photons emitted in the cooling processes and from radiative and dielectronic recombination. The radiative recombination spectrum of hydrogen extends from the Ly ˛ line at 121:6 nm to radio lines at meter wavelengths. The recombination spectrum can be predicted to high accuracy, and calculations for a wide range of temperature, density, and radiation environments have been carried out for diagnostic purposes [15–18]. Electron impact and proton impact-induced transitions are important in determining the energy-level populations and the resulting spectrum. Stimulated emission often affects the intensities of the radio lines, especially those from extragalactic sources. Comparisons of the predicted intensities in the visible and infrared with theoretical predictions yield information on interstellar extinction in the nebula and along the line of sight. The relative intensities of the lines emitted by different metastable levels depend exponentially on the temperature. The relative intensities of the lines at 500:7 and 436:3 nm originating in the 1 D2 and 1 S0 levels of O2C vary as exp.33;170=T /, and are commonly used to derive the temperature T . The electron density can be inferred from the lines emitted from neighboring levels with different radiative lifetimes
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for which there occurs a competition between spontaneous emission and quenching by impact with thermal electrons. There are many possible combinations of lines. The lines at 372:89 and 372:62 nm emitted by the 2 D3=2 and 2 D5=2 levels of NC are readily observable, and their relative intensity can be used to estimate the electron density. Radiative and dielectronic recombination lines are often seen in the spectra, as are a few lines due to charge transfer recombination. Fluorescence of starlight and resonance fluorescence of lines emitted in the nebula, called Bowen fluorescence by astronomers, also contribute to the spectra of photoionized gases. A large variety of data are needed to adequately interpret the observations. Two X-ray satellites, Chandra and XMM-Newton, both launched in 1999, have greatly increased our ability to detect hot ionized gas in stars, supernova remnants, active galaxies, and other regions [19]. In particular, Chandra has allowed us for the first time to directly observe the hot gas between galaxies [20]. The state of modeling photoionized clouds has been reviewed by Ferland [21]. He also highlights the great need for atomic and molecular data for analyzing these clouds.
86.3
Collisionally Ionized Gas
Hot gas is found in the coronae of stars and, particularly, the Sun, and in young supernova remnants, in the hot phase of the interstellar medium, and in intergalactic space. In a hot gas, the ionization is produced by impact ionization due to the fast thermal electrons and removed by recombination, both radiative and dielectronic [22]. The rate coefficients for electron impact ionization and for recombination for any given ionization stage are functions only of temperature, and hence so is the resulting ionization distribution. When ionization and recombination balance, coronal equilibrium is attained, in which the ionization structure is specified by the temperature. Recombination at high temperatures is dominated by dielectronic recombination. At high temperatures, dielectronic recombination is stabilized by transitions in which the core electrons are the active electrons. The associated emission lines lie close in frequency to that of the resonant transition of the parent ion, resulting in so-called satellite lines. Together with lines generated by electron impact excitation, they provide a powerful diagnostic probe of density and temperature. In many circumstances, such as in supernova remnants, coronal equilibrium does not hold, and the ionization and recombination must be followed as functions of time, a situation referred to as nonequilibrium ionization. The temperature also evolves as the hot plasma is cooled by electron impact excitation and ionization. The recombining gas produces X-rays and extreme UV radiation, which modify the ionization structure. There is a par-
–log P/ ne nH (erg 21
H
cm3/s)
C
O
He
22
N
Fe
Si Mg
23
Ne
(OII) 4.0
Si
5.0
6.0
7.0 log T (K)
Fig. 86.2 Total emissivity and emissivity by element as a function of temperature in coronal equilibrium. The heavy solid curve is total emissivity, and the lighter lines are contributions from individual elements. After [23]
ticular need for more reliable data on high-energy photoionization cross sections, on collision cross sections for electrons and positive ions, and on the energy levels and transition probabilities of highly stripped complex ions. In particular, for M-shell ions, dielectronic recombination becomes important at low temperatures, but the reliability of theoretical rate coefficients are uncertain due to a large number of near threshold resonances. Currently, available atomic data have been used to update coronal ionization equilibrium calculations [24]. Figure 86.2 shows the emissivity of coronal gas.
86.4 Diffuse Molecular Clouds Diffuse molecular clouds are intermediate between the hot phase of the galaxy and the giant molecular clouds where much of the gas resides. They are called diffuse because they have optical depths of order unity, or less, allowing photons to penetrate from outside the cloud and affect the chemical composition. The atoms and molecules are observed in absorption against background stars. Translucent clouds, defined to have optical depths between about 2 and 5 are intermediate between diffuse and dark clouds where photons from the outside still affect the chemistry. They can be observed both in absorption against a background source or in emission in the radio. The gas temperature is 100200 K at the edges of a diffuse cloud with a density of about 100 cm3 . In a typical diffuse cloud, the temperature decreases to about 30 K at the center, while the density increases to about 300800 cm3 . The chemistry is driven by ionization from interstellar UV photons and from cosmic rays, composed mostly of highenergy protons.
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Interstellar UV photons ionize species that have ionization potentials less than that of atomic hydrogen. Atomic hydrogen is so pervasive in the galaxy that UV photons with energies higher than 13:6 eV are absorbed near the source. The UV flux is a very important parameter in determining the composition of a diffuse cloud. Photodissociation provides destruction that limits the buildup of more complex species, and so diffuse clouds are typically dominated by simpler diatomic species. However, a number of polyatomic molecules, including SO2 , H2 CO, and C3 H2 , have been observed in diffuse clouds, while the so-called diffuse interstellar bands suggest the presence of large organic molecules. Understanding the origin of these large molecules in diffuse clouds remains an important area of research [25]. Species with ionization potentials greater than hydrogen are mainly ionized by cosmic rays. Cosmic rays are highenergy nuclei that stream through the galaxy. The cosmic-ray ionization rate, the number of cosmic-ray ionizations per second per particle, is an important parameter in interstellar chemistry. A lower limit to the cosmic-ray ionization rate may be set by measured high-energy cosmic rays reaching Earth, giving an ionization rate of 1017 s1 . More realistic estimates of the cosmic-ray ionization rate from looking at recombination lines suggest values of a few 1017 s1 . Earlier work by studying recombination lines suggests values of a few 1017 s1 . However, more recent studies based on observations of molecular ions in the galactic disk indicate a much higher rate of 5:3 1016 s1 , although with only marginal evidence for a decrease in the rate with extinction into diffuse clouds [26]. On the other hand, the hydroxyl radical OH is produced in a manner similar to that discussed below in Sect. 86.6 and removed by photodissociation. Thus, in diffuse clouds, OH may be used to measure the cosmic-ray ionization rate, subject to uncertainties in the OH photodissociation rate and the HC 3 dissociative recombination rate. The OH abundances give rates of several 1017 s1 for many diffuse clouds. The carbon chemistry begins with the ionization of C by UV photons: C C h ! CC C e :
(86.6)
The carbon ion cannot react directly with H2 by CC C H2 ! CHC C H ;
(86.7)
as this reaction is endothermic by 0:4 eV. Instead, the chemistry proceeds by the slow radiative association process CC C H2 ! CHC 2 C h :
(86.8)
The CHC 2 ion may either dissociatively recombine CHC 2 C e ! CH C H ;
(86.9)
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or react with molecular hydrogen C CHC 2 C H2 ! CH3 C H :
(86.10)
The CHC 3 then undergoes dissociative recombination CHC 3 C e ! C C H2 C H
(86.11)
! CH C H C H
(86.12)
! CH2 C H
(86.13)
! CH C H2 ;
(86.14)
where the products are listed in order of decreasing likelihood. The CH is removed by photodissociation CH C h ! C C H
(86.15)
and by photoionization CH C h ! CHC C e :
(86.16)
CH may also be removed by reactions with oxygen or nitrogen atoms to form CO and CN, respectively. One of the outstanding problems in diffuse clouds is to understand the large abundance of CHC relative to CH. The difficulty is in finding a mechanism that can produce sufficient CHC without producing additional CH. Since most reaction paths go through reaction Eq. (86.7), this is the most likely candidate. What is needed is some extra energy to overcome the endothermicity. This energy must come from either hot CC or from hot or vibrationally excited H2 . The most popular model is gas heated by a shock, possibly a magnetic shock, in which ions stream relative to the neutrals, giving a high effective energy. Unfortunately, although these shock models can reproduce the CHC abundances, they also predict relative velocities between the CHC and CH, which are not often observed. Recently, there have been suggestions that turbulence in the cloud could account for the CHC abundance. The most comprehensive models of diffuse clouds are by van Dishoeck and Black [27]. A collection of photodissociation rates and photoionization rates is given by Roberge et al. [28], and a more recent compilation has been made by Heays et al. [29]. The UV flux is predominantly from stars and may be as much as 105 times larger near an HII region (Sect. 86.2) than it is in the general interstellar medium. Regions in which the chemistry is dominated by photons are referred to as photondominated regions or photodissociation regions (PDRs) [30]. In the presence of high UV flux, the cloud is much warmer than in a typical diffuse cloud. Temperatures may reach 1000 K near the edge of the cloud and 100 K far into the cloud. The chemistry differs from traditional diffuse cloud chemistry in that the high temperatures allow endothermic reactions to proceed. Sternberg and Dalgarno [31] have published a comprehensive model of photondominated regions.
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Dark Molecular Clouds
Table 86.1 Molecules observed in interstellar clouds
Much of the mass of the galaxy is in the form of dark molecular clouds. The molecular clouds are sites of new star formation. They are composed primarily of hydrogen, with about 10% helium and trace amounts of heavier elements. They have densities of approximately 103 or 104 cm3 and temperatures between 10 and 20 K, and often contain denser clumps. The clouds are optically thick, and so photons from the outside are absorbed on the surface of the clouds. The interiors are heated and ionized by cosmic rays that penetrate deep into the cloud. The temperatures are too low to sustain much neutral chemical activity in the clouds, and cosmic-ray ionization is important in driving the chemistry. In dense clouds, the cosmic rays both initiate the chemistry and limit it through the production of HeC and through cosmic-ray induced photons. Currently, about 200 molecules have been detected in the interstellar medium, many of which have also been found in other galaxies. Table 86.1 gives a partial list, ranging from simple diatomics to the largest being C70 . Most molecules, except H2 , are formed in the gas phase by ion–molecule reaction sequences initiated by a cosmic-ray ionization. The fact that isomers, such as HCN and HNC, are seen in approximately equal abundances suggests a low density gas phase environment. Reactions on surfaces and the formation of grains are not well understood but are surely important. It does appear that many complex organic molecules, such as dimethyl ether, are primarily formed on ice mantles [32]. The Infrared Space Observatory (ISO) has provided a tremendous amount of data on cold regions in our own galaxy and allowed us to directly observe the icy mantles of dust grains [33]. The chemistry of molecular clouds is dominated by ion– molecule reactions driven by cosmic-ray ionization. The cosmic rays primarily ionize H2 : H2 ! HC 2 C e ;
(86.17)
producing both HC 2 and fast electrons. The fast electrons produce additional ionizations. The HC 2 quickly reacts with H2 C to form H3 , C HC 2 C H2 ! H3 C H :
H2 CHC C2 CO NH CS SO NS PN SiN H2 O HCN HCO N2 HC HNO SO2 C2 O N2 O H2 CO NH3 HNCO C3 H C3 S C2 H2 HCNHC CH4 HCOOH HC3 N HCCNC NH2 CH CH2 NH H2 CCCC C5 H CH3 OH NH2 CHO CH2 CHCN C6 H CH3 CHO CH3 C3 N .CH3 /2 O HC7 N HC9 N
Hydrogen Methylidyne ion Carbon Carbon monoxide Amidogen Carbon monosulfide Sulphur monoxide Nitrogen sulfide Phosphorus nitride Silicon nitride Water Hydrogen cyanide Formyl Protonated nitrogen Nitroxyl Sulfur dioxide Carbon suboxide Nitrous oxide Formaldehyde Ammonia Isocyanic acid Propynylidyne Tricarbon sulfide Acetylene Protonated hydrogen cyanide Methane Formic acid Cyanoacetylene Ethynyl isocyanide Cyanamide Methanimine Butatrienylidene Pentynylidyne Methyl alcohol Formamide Vinyl cyanide Hexatrinyl Acetaldehyde Methyl cyanoacetylene Dimethyl ether Cyanohexatriyne Cyano-octatetra-yne
which quickly reacts with H2 to form H3 OC in an abstraction (86.18) sequence
The HC (86.20) OHC C H2 ! H2 OC C H ; 3 reacts with other species by proton transfer, which then drives much of the interstellar chemistry. C C H2 O C H2 ! H3 O C H : (86.21) As an example of the production of more complex molecules in interstellar chemistry, we examine the reaction The H3 OC then undergoes dissociative recombination to networks leading to the production of water H2 O and the form water and OH hydroxl radical OH. The HC 3 ions formed by cosmic-ray ionization react with atomic oxygen to form OHC (86.22) H3 OC C e ! H2 O C H ; C O C HC 3 ! OH C H2 ;
(86.19)
! OH C H2 :
(86.23)
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Table 86.1 (Continued) CH OH CN COC NO SiO SOC SiS HCl NH2 C2 H HNC HCOC H2 S OCS HCSC C2 S H2 CN H2 CS HCNS HOCOC C3 N C3 O H3 OC C3 H2 H2 CCC CH2 CO HNCCC C4 H CH2 CN CH3 CH CH3 SH HCC2 HO HC3 NHC CH3 C2 H HC5 N CH3 NH2 HCOOCH3 CH3 C4 H CH3 CH2 CN CH3 CH2 OH HC11 N
Methylidyne Hydroxyl Cyanogen Carbon monoxide ion Nitric oxide Silicon monoxide Sulfur monoxide ion Silicon sulfide Hydrogen chloride Amino radical Ethynyl Hydrogen isocyanide Formyl ion Hydrogen sulfide Carbonyl sulfide Thioformyl ion Dicarbon sulfide Methylene amidogen Thioformaldehyde Isothiocyanic acid Protonated carbon dioxide Cyanoethynyl Tricarbon monoxide Hydronium ion Cyclopropenylidene Propadienylidene Ketene Cyanoacetylene isomer Butadinyl Cyanomethyl radical Methyl cyanide Methyl mercaptan Propynal Protonated cyanoacetylene Methyl acetylene Cyanodiacetylene Methylamine Methyl formate Methyl diacetylene Ethyl cyanide Ethyl alcohol Cyano-decapenta-yne
The water is removed by reactions with neutral or ionized carbon, which eventually lead to the production of CO. OH is primarily removed by reactions with atomic oxygen, leading to O2 . The CO and O2 are removed by reactions with HeC . The HeC , generated by cosmic-ray ionization of helium, does not react with H2 and is available to remove species by reactions such as CO C HeC ! CC C O C He :
(86.24)
Water and OH are also removed by UV photons generated within the cloud. The clouds are too thick for external UV
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photons to penetrate, but cosmic rays excite H2 into electronically excited states that decay through emission of UV photons. These internally generated photons play an important role in determining the composition of the cloud. Gredel et al. [34] compiled a list of the photodissociation and photoionization rates for cosmic-ray induced photons. Modern chemical networks for molecular clouds include several hundred species and several thousand reactions. Standard sets of reaction rates are provided by the UMIST (University of Manchester Institute of Science and Technology) dataset [35, 36] and the KInetic Database for Astrochemistry (KIDA) [37]. The datasets may be obtained from the UMIST Astrochemistry (udfa.ajmarkwick.net) and KIDA (kida.obs. u-bordeaux1.fr) home pages. The clouds also contain dust particles as evidenced by the extinction curves with the observed depletions of heavier elements. The importance of surface chemistry on these dust particles to interstellar clouds is still uncertain. Dust particles are the best candidates for the sites of molecular hydrogen formation because known gas phase reactions fail to produce H2 in the quantities observed.
86.6 Circumstellar Shells and Stellar Atmospheres The continuum emission from a star is very nearly that of a blackbody at the photosphere. This emission is then absorbed and redistributed by the atmosphere of the star. The spectrum of the star is, thus, determined by its atmosphere. In the hottest stars, most of the material is ionized, and the absorption lines are predominantly those of ions, while in the coldest stars, molecular lines are prominent. Kurucz has calculated models with continuum spectra and the inclusion of a large number of absorption lines [38]. There are two major projects for calculating the required atomic data. In 1984, an international collaboration named the Opacity Project was set up to calculate accurate atomic data needed for opacity calculations [39, 40]. The other earlier project is called OPAL. The two sources of data are compared in [41, 42]. The Opacity Project data was updated by Badnell et al. [43], while another set of opacity data is available from Los Alamos [44]. Low and intermediate mass stars eject circumstellar envelopes in their red giant phase near the end of their evolution. Circumstellar envelopes are an important part of astronomy, and they are a likely location for dust formation. They provide an interesting environment for studying molecules because they represent a transition between very high-density stellar atmosphere environments to low-density interstellar environments. These objects evolve to become planetary nebulae (Sect. 86.2). We are fortunate in having one example, IRC +10216, which is very close to the Sun. The brightest 10 µm source
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beyond the solar system, IRC +10216 is a carbon-rich star surrounded by the dust and gas it ejected in a strong stellar wind. The central star is so shielded that it is almost undetectable at optical wavelengths and was not discovered until the 2 µm survey. IRC +10216 is where most of the circumstellar molecules are detected and has greatly increased our understanding of circumstellar envelopes. The envelopes are ejected by the red giant in its final phase of evolution, the asymptotic giant branch (AGB). The mass loss rates increase to 104 solar masses per year, and temperatures in the envelope are of order 1000 K. Close to the star, the density is high, and the chemistry is characteristic of thermal equilibrium. The situation is quite unlike any interstellar environments. For example, in IRC +10216, the HNC is over 100 times less abundant than HCN, whereas in molecular clouds, they have about the same abundance. If the star is oxygen-rich, such as the AGB star IK Tau, large amounts of H2 O are formed, and if it is carbonrich, C2 H2 and other hydrocarbons are abundant. This high-temperature environment forms both molecules and grains. The high obscuration of the central source indicates that grains are formed in these envelopes. Polycyclic aromatic hydrocarbons (PAHs) or similar carbonaceous species, are observed in carbon-rich planetary nebulae. These large molecules must have been produced when the object was in the AGB phase. As the material flows out from the star, the density and temperature decrease. As the density becomes lower, threebody reactions become less important, and at some point the products of these reactions are frozen out in a similar manner to the evolution of molecules in the early universe (Sect. 86.12). In the outermost portions of the circumstellar envelope, molecules are dissociated by interstellar UV photons. The penetrating UV radiation is shielded by dust, H2 , and CO. The relative abundances can vary rapidly with radius, and observations provide abundances and radial distributions, a wealth of data for modelers. Circumstellar chemistry was reviewed by Omont [45], and recent chemical models are given in [46–48].
86.7
Supernova Ejecta
A supernova, the explosion of a massive star following core collapse, is one of the most spectacular displays in the universe. The explosion occurs when the iron core of a massive star collapses to form a neutron star, and the rebound shock and neutrino flux eject the outer portion of the star. The ejected portions of the star are rich in heavy elements produced in the interior of the progenitor star. We are fortunate to have had in our lifetime a supernova that was close and in an unobscured line of sight. Supernova 1987A, the first supernova observed in 1987, went off in the
Large Magellanic Cloud, a small satellite galaxy close to our own. It was the first supernova visible to the naked eye in nearly 400 years (since the Kepler supernova in 1604). Using the full range of modern astronomical instruments has allowed us to get detailed spectra of the evolving ejecta, which has greatly increased our knowledge of supernovae. We will use SN1987A as an example of supernova ejecta. Initially the temperature of the ejecta of SN1987A was high, 106 K, but it quickly cooled through adiabatic expansion and radiation from the photosphere. The temperature leveled off at several thousand degrees because of heating by radioactive nuclei, first 56 Ni and then 56 Co, formed in the explosion. The dynamics is homologous free expansion: the velocity scales linearly with the radius r.t/ D vt, where v is the velocity and t the time since the explosion. The ejecta were at first optically thick, and the spectrum resembled that of a hot-star continuum with absorption lines from the surface. After a few days, the temperature dropped, but the ejecta remained optically thick and continued to show strong continuum emission. As the ejecta expanded, the temperature dropped, the ejecta became optically thin, and the resulting spectrum was dominated by strong emission lines, superficially resembling an emission nebula (Sect. 86.3). The emission is dominated by neutral atoms and singly ionized species. The gas is heated and ionized by gamma rays from radioactive decay. The gamma rays Compton scatter, producing X-rays and fast electrons. The X-rays further ionize the gas and produce multiply charged ions through the Auger process. These multiply charged ions recombine through charge transfer with neutral atoms. Further charge transfer determines the relative ionization of different species, with the lowest ionization potential species more ionized than the higher ionization potential species. The development of the infrared and optical spectrum of Supernova 1987A was reviewed by McCray [49]. One of the great surprises in the spectrum of Supernova 1987A was the discovery of molecules in the infrared region. CO, SiO, and possibly HC 3 have been identified. In the absence of grains, molecules must be formed through either three-body or radiative processes. In the supernova ejecta, the densities are too low for three-body processes to be effective, and molecules are formed through radiative association reactions. The molecules are removed by reactions with HeC , and the molecular abundances put a constraint on how much helium can be mixed back into the region with carbon and oxygen [50]. Since the detection of molecules in SN 1987A, there have been many more observations of CO molecules in Type II supernova, and they may even occur in every Type II [51]. CO has also been observed in a Type Ic supernova [52]. It remains a puzzle as to why the molecules are not rapidly removed by helium ions. A calculation of the O C HeC
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system [53] found that radiative charge transfer is much faster than direct charge transfer for temperatures below 106 K but still too slow to significantly reduce the helium ion abundance in supernova ejecta. The most likely explanation remains that mixing is not complete in supernova ejecta, and the molecules survive in regions of relatively low helium abundance. More recent observations have detected H2 [54], and the Atacama Large Millimeter/submillimeter Array (ALMA) has also detected HCOC and SO, which are discussed in Sect. 86.10.
86.8
Shocked Gas
Shock waves occur in compressible fluids when the pressure gradients are large enough to generate supersonic motion, or when a disturbance is propagating through the fluid at supersonic velocities. Because information about the disturbance cannot propagate upstream in the fluid faster than the speed of sound, the fluid cannot respond dynamically until the shock arrives. The shock then compresses, heats, and accelerates the fluid. The boundary separating the hot compressed gas and the upstream gas is the shock front in which the energy of directed motion of the shock is converted to random thermal energy. Shocks are ubiquitous in the interstellar medium where they are driven by the ionization fronts of expanding HII regions or nebulae, by outflowing gas accompanying stellar birth and evolution, and by supernova explosions. If the shock velocity is above 50 km=s, the shock gas is excited, dissociated, and ionized. The subsequent recombination and cooling radiation produces photons that may ionize and dissociate the gas components ahead of and behind the shock. This precursor radiation modifies the effects of the shock and influences its dynamical and thermal evolution. Fast shocks destroy all molecules by dissociating H2 by collisions with H, H2 and He and with electrons. Exchange reactions with H atoms destroy the other molecular species. At low densities, radiative stabilization occurs, and dissociation is less efficient. Molecules reform in the cooling postshock gas. Slower shocks do not cause ionization or dissociation, but the chemical composition and the ion composition are modified by reactions taking place in the warm gas. The response of the interstellar gas to slow shocks is significantly affected by the presence of a magnetic field. In some ionization conditions, a magnetic precursor may occur in which a magnetosonic wave carries information about the shock, and the ionized and neutral components of the gas react differently to the shock. Many different kinds of shocks have been identified [55]. A very fast shock with a velocity of hundreds of km=s, such as are driven by supernova explosions, creates a hot dilute cavity in the interstellar medium with a temperature of millions of degrees. The density is low, and the gas cools and
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recombines slowly. Overlapping supernova-induced cavities may be responsible for the hot gas that occupies a considerable volume of the interstellar medium in the Galaxy and in some external galaxies. The conditions are far from coronal equilibrium as the gas cools more rapidly than it recombines. The cooling radiation appears as soft X-rays and UV emission lines with a characteristic spectrum. As the gas cools below 10;000 K, molecular formation occurs. Molecular hydrogen is formed on the surfaces of grains as in molecular clouds and by the negative ion sequence that is effective in the early universe (Sect. 86.9). With the formation of H2 in a still warm gas, the chemistry is driven by exothermic and endothermic reactions with H2 . Thus, OH is produced by the reaction of O atoms, and H2 O by the further reaction of OH with H2 . Enhanced abundances of other neutral and ionic molecules are the products of subsequent reactions with OH. The reactions of SC and S with OH lead to SOC and SO, and their simultaneous presence may be an indicator of a dissociative shock. There are, in addition, physical indicators of shocks, such as asymmetric line profiles indicating high velocities. In a nondissociative shock in a molecular gas, reactions with warm H2 dominate the chemistry as it does in the cooling zone of a dissociative shock. The composition is controlled by the post shock temperature and the H/H2 ratio. The warm H2 changes the ionic composition by converting CC into CHC . Evidence for a nondissociative shock is the infrared emission from H2 . The thermal emission from collisionally excited vibrational levels in shock-heated gas is readily distinguished from that discussed in Sect. 86.4 arising from UV pumping in a PDR. Emission from H2 has been detected in numerous objects in the Galaxy and in many distant external galaxies. In external galaxies, X-rays may contribute to the H2 infrared spectrum through heating the gas and through excitation by photoelectron pumping to excited states followed by a downward cascade [56, 57].
86.9 The Early Universe Molecules appeared first in the universe after the adiabatic expansion had reduced the matter and radiation temperature to a few thousand degrees, and recombination occurred, creating a nearly neutral universe. The small fractional ionization that remained was essential to the formation of molecules. Molecular hydrogen formed through the sequences HC C H ! HC 2 C h ;
(86.25)
HC 2
C H ! H2 C H ;
(86.26)
H C e ! H C h ;
(86.27)
C
and
H C H ! H2 C e ;
(86.28)
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which separated out of the cosmic flow. Three-body recombination reactions such as
H +, ν
e
–
H
H
ν
H e
H(n = 2) ν H(n = 2)
e
e
H2
H C H C H ! H2 C H
(86.38)
Li C H C H ! LiH C H
(86.39)
[59] and
ν
H–
H(n = 2)
e, ν H
e
were likely major sources of molecules as the density increased. H+ H+2 H+3 The Wilkinson Microwave Anisotropy Probe (WMAP) H ν H2 has given us the best map of the universe at the time of recombination and given us general confirmation of the big Fig. 86.3 Diagram showing the important reactions in the production bang model. Perhaps the most surprising result is that stars of hydrogen molecules in the early universe seem to have formed much sooner than would have been expected, about 180 million years after the big bang [60]. This the protons and electrons acting as catalysts. Many other makes it even more difficult to understand how the universe atomic and molecular processes occurred (Fig. 86.3), some evolves from the relative uniformity at the time of recombination to the collapse and formation of the first objects so involving excited hydrogen atoms. Thus, quickly, a problem that is certainly controlled by atomic and (86.29) molecular processes. The relevant atomic and molecular proH C H2 ! HC 3 C e cesses have been reviewed by the authors [61]. was a source of HC . 3
The universe contained trace amounts of deuterium and 7 Li nuclei with which heteronuclear molecules could be made. Molecules with dipole moments may leave an imprint 86.10 Atacama Large Millimeter/Submillimeter Array on the cosmic blackbody background radiation that permeates the universe. The deuterated molecule HD forms from Breakthroughs in astronomy are often driven by major im(86.30) provements in instrumentation [62]. The Atacama Large DC C H2 ! HD C HC ; Millimeter/Submillimeter Array (ALMA) provides angular and H2 DC from resolution and sensitivity improvements of over an order of magnitude leading to quicker and better resolved specC C (86.31) D C H3 ! H2 D C H ; tra for many astrophysical environments. To fully interpret HDC C H2 ! H2 DC C H ; (86.32) these spectra will require additional work in laboratory astrophysics so that breakthroughs in understanding may occur. It and is expected that ALMA will revolutionize our understanding C of astrophysical environments. Some early ALMA results are ! H D C H : (86.33) HD C HC 2 2 highlighted in this section. Lithium hydride is formed through By using the improved spatial resolution of ALMA, observations were able to resolve the dust production and show Li C H ! LiH C h ; (86.34) that Supernova 1987A produced about 0.2 solar masses of (86.35) dust [63]. These observations demonstrated that the dust was Li C H ! LiH C e ; Li C H ! LiH C e : (86.36) formed in the inner ejecta and has not yet been processed through the shock front. If the dust survives the processing There are many destruction processes, of which of the shock, then supernova are an important cosmological (86.37) source of dust. In addition, a molecular line survey for SuLiH C H ! Li C H2 pernova 1987A detected both HCOC and SO in the ejecta of may be the most severe, although its rate coefficient is un- a supernova for the first time [64]. The spectral resolution of certain. The chemistry of the early universe is summarized the molecular survey is also helping to understand the mixing that occurs in the expanding ejecta. in [58]. Using ALMA astronomers were able to resolve the milThe formation of molecules was a crucial step in the fragmentation of the first gravitationally collapsing objects, limeter size dust grains around a young A-type star Oph IRS
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48 [65]. By comparing this strongly asymmetric distribution of millimeter grains to that of smaller dust particles and molecules, both of which are symmetric about the orbit, the authors infer that the larger grains may be trapped. In their model, the trapping is due to a larger preexisting planet in the system. This trap could allow the dust grains to keep growing to sizes that allow for additional planet formation. With improved sensitivity, ALMA has detected additional building-blocks-of-life molecules in the interstellar medium, including iso-propyl cyanide [66] in the star-forming region of Sagittarius B2 and methyl isocyanate in the solar-type protostar IRAS16293-2422 B [67]. The detection of these potential precursors to biological molecules is interesting and hints at the proposed idea that life and self-replicating molecules formed first in the interstellar medium and then rained down on the surfaces of planets. With its increased sensitivity, ALMA may give insight into this idea for the appearance of life.
86.11 Recent Developments In 2015, a new era opened up when gravitational waves from a binary black hole merger were detected for the first time [68]. More recently, a merger between two neutron stars was observed [69]. The neutron star merger, known as a kilonova, was also observed as a gamma-ray burst, and transient emission was observed in several electromagnetic bands [70]. The observations confirmed a long standing prediction that the violent collision between neutron stars could lead to gamma-ray bursts and the production of heavy elements [71]. This start to multimessenger astronomy, combining the electromagnetic and gravitational waves along with recent theoretical studies [72], suggests that a binary neutron star merger is a major source for production of gold and other heavy elements. The Herschel Space Observatory, a far infrared and submillimeter telescope active 2009 through 2013, greatly increased our knowledge of molecular astrochemistry. A special issue of Astronomy and Astrophysics was released in 2010 with Herschel’s first results with papers on observations of the solar system, star formation, interstellar medium, galaxy evolution, and cosmology [73]. Herschel had four main objectives: to study the formation of galaxies in the early universe, the creation of stars and their interaction with the interstellar medium, the observation of chemical composition of solar system objects, and the molecular chemistry in the universe. The observatory made the first multiline detection of molecular oxygen in space [74, 75]. Another rapidly expanding field involves planets outside our solar system. The detection and characterization of molecules in exoplanets will be critical to understanding how they formed, and what the planets are like. It will be an im-
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portant factor in the search for life on these planets. So far, CO and H2 O have been detected [76], as well as TiO [77]. However, further detections should be forthcoming, particularly with the launch of the James Webb Space Telescope. A database of molecular line lists for species that might be detected in exoplanet atmospheres is currently being maintained and updated [78]. In summary, satellite data along with continued ground observations continually raise new astrophysical puzzles, puzzles that are controlled and probed by atomic and molecular processes. The astrophysical community owes a great debt to both atomic and molecular laboratory measurements and theoretical models of energy levels, reaction rates, and transition probabilities. In order to continue to progress in our understanding of the universe we will need to continue to fund the work needed for the understanding of the atomic and molecular processes that control it.
86.12 Other Reading Astronomy is one of the oldest sciences and one of the fastest evolving. Advances in technology are rapidly increasing the sensitivity and resolution of our instruments, and so new observations and more sophisticated models lead to an ever greater understanding of the universe. This means that books will often be somewhat dated when they appear. However, the series Annual Review of Astronomy and Astrophysics is a good source of recent review articles. In addition, the following books are recommended as good introductions or overviews of a particular field [1, 79–86]:
Astrophysical Quantities by Allen Astrophysical Formula by Lang Interstellar Chemistry by Duley and Williams Physical Processes in the Interstellar Medium by Spitzer Astrophysics of Gaseous Nebula and Active Galactic Nuclei by Osterbrock and Ferland Spectroscopy of Astrophysical Environments edited by Dalgarno and Layzer Molecular Astrophysics edited by Hartquist The Physics and Chemistry of the Interstellar Medium by Tielens The Physics of the Interstellar and Intergalactic Medium by Draine.
Many sources of atomic and molecular data are listed and discussed in a special issue of Revista Mexicana de Astronomia y Astrofisica, 23 March 1992 [7]. For details on atomic spectroscopy, the books [87, 88]:
The Theory of Atomic Structure and Spectra by Cowan
Atomic Spectra and Radiative Transitions by Sobelman.
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For details on molecular spectroscopy, the books [89, 90]:
Molecular Spectra and Molecular Structure by Herzberg
Spectra of Atoms and Molecules by Bernath.
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78. Tennyson, J., Yurchenko, S.N.: Atoms 6, 26 (2018) 79. Allen, C.W.: Astrophysical Quantities. Athlone, London (1973) 80. Lang, K.R.: Astrophysical Formula. Springer, Berlin, Heidelberg (1980) 81. Duley, W.W., Williams, D.A.: Interstellar Chemistry. Academic Press, London (1984) 82. Spitzer, L.: Physical Processes in the Interstellar Medium. Wiley, New York (1978) 83. Dalgarno, A., Layzer, D.R.: Spectroscopy of Astrophysical Plasmas. Cambridge University Press, Cambridge (1987) 84. Hartquist, T.: Molecular Astrophysics. Cambridge University Press, Cambridge (1990)
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Contents 87.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1289
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87.3 87.3.1 87.3.2 87.3.3 87.3.4 87.3.5 87.3.6 87.3.7
Excitation Mechanisms . . . Basic Phenomenology . . . . . Fluorescence Equilibrium . . . Swings and Greenstein Effects Bowen Fluorescence . . . . . . Electron Impact Excitation . . Prompt Emission . . . . . . . . OH Level Inversion . . . . . . .
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less and low-density coma, with no gravity or magnetic field, is a unique spectroscopic laboratory, and in recent decades, observations of comets have expanded over the entire electromagnetic spectrum, from X-ray to the submillimeter and millimeter. Although Rosetta has elucidated many key questions, issues concerning the basic molecular composition and the elemental abundances of both the volatile and refractory components of the cometary nucleus, as well as the comet-to-comet variation (particularly between new and evolved periodic comets) of these quantities still remain to be solved. A recent (although slightly out-of-date) compendium covering all aspects of cometary science is the Comets II book [1], while a review of coma chemistry is given by Mumma and Charnley [2]. Keywords
comets cometary coma spectroscopy ultraviolet fluorescence
Abstract
Prior to spacecraft missions to comets, beginning with the Giotto and Vega spacecraft to comet 1P/Halley (the P/ signifies a periodic comet) in March 1986, determinations of the volatile composition of a cometary coma were derived from spectroscopic analyses. Detailed modeling was then used to infer the volatile composition of the cometary nucleus. Since then there have been additional missions, including Deep Impact (2005), Stardust (2006), EPOXI (2010), and most recently, Rosetta, that flew along with comet 67P/Churyumov-Gerasimenko for over 2 years, that have greatly advanced our knowledge of the physics, chemistry, and geology of comets. Only a few comets have been studied this way, but the results provide a framework, or “ground truth”, for remote observations from Earth and spaceborne observatories, both past and future. This chapter focuses on the principal atomic and molecular processes that lead to the observed spectrum as well as the needs for basic atomic and molecular data in the interpretation of these spectra. The largely collision-
87.1
Introduction
Comets are small bodies of the solar system believed to be remnants of the primordial solar disk. As long-period comets are believed to have formed near the orbits of Uranus and Neptune and were subsequently ejected into an Oort cloud of some 40;000 AU in extent, these objects likely preserve a record of the volatile composition of the early outer solar system and so are of great interest for the physical and chemical modeling of solar system formation. The comets arrive in the inner solar system as a result of galactic perturbations. The cometary volatiles are vaporized as their orbits bring them closer to the Sun, and it is solar radiation that initiates all of the processes that lead to the extended coma. Gas vaporization also leads to the release of dust into the coma, and the scattering of sunlight by dust is the major source of the visible coma and dust tail of comets. Somewhat fainter, and much more extended, is the plasma tail, resulting from
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_87
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photoionization by solar extreme UV radiation of the neutral duced directly in ionic form and will not be counted using volatiles and their subsequent interaction with the solar wind. this approach. In addition, another fraction exists in the coma in the solid grains, and this component will not be included either, except for a small amount volatilized by evaporation 87.2 Observations or sputtering by energetic particles. The composition of the grains, although not the absolute abundance, has been deIn a review in 1965, Arpigny [3] summarized the known termined from in situ measurements made by the Halley molecular and atomic emissions detected in the visible re- encounter spacecraft [12] and can be inferred, although not gion of the spectrum (here defined as 3000 to 11;000 Å) as unambiguously, from reflection spectroscopy of cometary follows: dust in the 35 µm range. The advent of spaceborne platforms for observations in radicals: OH; NH; CN; CH; C3 ; C2 ; and NH2 ; the VUV has produced a wealth of new information about the C C ions: OHC ; CHC ; COC volatile constituents of the coma. The A 2 ˙ C X 2 ˘ .0; 0/ 2 ; CO ; and N2 ; band of the OH radical at 3085 Å was well known metals: Na; Fe : from ground-based spectroscopic observations, but as this The only known atomic feature was the O I forbidden red wavelength lies very close to the edge of the atmospheric doublet at 6300 and 6364 Å. From the radicals and ions, one transparency window, the strength of this feature (relative to could infer the presence of their progenitor parent molecules that of other species) was not appreciated until 1970, when such as H2 O, NH3 , HCN, CO, and CO2 , directly vaporizing comet Bennett (C/1969 Y1) was observed from space by from the comet’s nucleus. The metals, seen only in comets the Orbiting Astronomical Observatory (OAO-2). The OAOpassing close to the Sun, were assumed to come from the va- 2 spectrum also showed a very strong, broadened H I Ly-˛ porization of refractory grains. The inventory of metals was emission from H, the other principal dissociation product of soon expanded to include K, CaC , Ca, V, Cr, Mn, Ni, and H2 O. The broad shape of Ly-˛ seen in the OAO-2 spectrum Cu, from observations of the Sun-grazing comet Ikeya-Seki is due to the large spatial extent of the atomic H envelope, (C/1965 S1) [4, 5], and H2 OC was identified in comet Ko- the result of a high velocity acquired in the photodissociahoutek (C/1973 E1). This latter comet was also the first to be tion process and a long lifetime against ionization. Later, at extensively studied at wavelengths both shortward and long- the apparition of comet Kohoutek (C/1973 E1), atomic O and ward of the visible spectral range. C were identified in the spectra, and direct UV images of the The first parent molecule to be directly identified was CO, H coma, as well as of the O I and C I emissions, were obwhich fluoresces in the Fourth Positive system .A 1 ˘u tained from sounding rocket experiments. X 1 ˙ C / in the VUV [6]. In principle, the molecular species Between 1978 and 1996, over 50 comets were observed should be detectable through their radio and sub-mm rota- spectroscopically over the wavelength range 12003400 Å tional transitions or through the detection of vibrational bands by the International Ultraviolet Explorer (IUE) satellite or individual rovibrational lines in the near IR. Water was first observatory [13, 14]. Most of the spectra were obtained directly detected through rovibrational lines near 2.7 µm in at moderate resolution ( D 610 Å), although highcomet Halley and in many subsequent comets [7]. The detec- dispersion echelle spectra ( D 0:20:3 Å) are useful for tion of species such as H2 CO, H2 S and CH3 OH was made some studies, particularly those of fluorescence equilibrium possible by the development of more sensitive instrumental (Sect. 87.3.2). The launch of the Hubble Space Telescope techniques. More than two dozen parent molecules have been (HST) in 1990, together with subsequent enhancements to identified spectroscopically [8, 9]. Many more parent species, the spectroscopic instrumentation that were made on-orbit, including O2 and N2 , were detected by the ROSINA mass marked another advance in sensitivity as well as the abilspectrometer on Rosetta [10]. Isotope ratios, particularly the ity to observe in a small field-of-view very close to the D/H ratio, in molecules such as HDO have been determined nucleus. This yielded the first detection of CO Cameron from IR and submillimeter observations [11]. band emission, a direct measure of CO2 being vaporized The ultimate result of solar photolysis (and to a lesser de- from the nucleus [15]. Both the Space Telescope Imaging gree, the interaction with the solar wind) is the reduction of Spectrograph, installed in 1997, and the Cosmic Origins all of the cometary volatiles to their atomic constituents. The Spectrograph, installed in 2009, enabled high-resolution obatomic inventory is somewhat easier to derive as the reso- servations of the CO Fourth Positive System [16, 17]. An nance transitions of the cosmically abundant elements H, O, example of a STIS (Space Telescope Imaging Spectrograph) C, N, and S all lie in the VUV and, in principle, the total spectral image of the region of the CO Fourth Positive syscontent of these species in the coma can be determined by tem is shown in Fig. 87.1 [16]. The geocentric distance of an instrument with a suitably large field of view. Of course, comet 153P/Ikeya-Zhang was 0.43 AU so that the image, top a fraction of the atomic species of each element will be pro- to bottom, spans 2800 km on either side of the nucleus.
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eral comets was made by the Submillimeter Wave Astronomy Satellite (SWAS) [22], the Odin satellite [23], and the Herschel observatory [24]. This line cannot be observed from ground-based telescopes because of the strong absorption by water vapor in the terrestrial atmosphere. 5 Prior to 1996, X-rays had not been detected in comets, and the conventional wisdom was that they were unlikely to be produced in the cold, rather thin cometary atmosphere. The discovery of soft X-ray emission (E < 2 keV) from comet Hyakutake (C/1996 B2) by the Röntgen Satellite (ROSAT) 0 thus came as a surprise [25]. Since then, X-ray emission has been detected from over a dozen comets using ROSAT and four other space observatories, the Extreme Ultraviolet Explorer, BeppoSAX, the Chandra X-ray Observatory (CXO), and Newton-XMM [26]. The earliest observations 5 were at very low spectral resolution, making it difficult to select amongst the possible excitation mechanisms: charge exchange, scattering of solar photons by attogram dust particles, energetic electron impact and bremsstrahlung, collisions between cometary and interplanetary dust, and solar 1400 1500 1600 1700 X-ray scattering and fluorescence. The more recent CXO Wavelength (Å) observations, at higher spectral resolution, favor the charge exchange of energetic minor solar wind ions such as O6C , O7C , C5C , C6C , and others, with cometary gas, principally –16.30 –15.68 –15.44 –15.28 –15.17 –15.07 –15.00 –1 –2 –1 –2 Log surface brightness (erg s cm Å arcsec ) H2 O, CO, and CO2 , as the primary mechanism. This mechanism would explain why the X-ray intensity appears to be Fig. 87.1 STIS spectral image of comet 153P/Ikeya-Zhang, showing independent of the gas production rate of the comet, and that the brightness variations along the slit. The spectral region containing the peak emission is offset from the location of the comet’s the strong geocoronal H I 1216 and O I 1302 has been excluded. The carbon multiplets at 1561 Å and 1657 Å are the strongest features and nucleus. This conclusion is also supported by recent laboappear relatively constant, as they are dissociation products extending ratory work on the charge transfer of highly ionized species far into the coma. The zero point in the vertical direction marks the with cometary molecules [27] and by theoretical calculations location of the nucleus. From [16] of state-specific cascades [28]. The X-ray emission thus tells us more about the solar wind than about the gaseous compoThe launch of the Far Ultraviolet Spectroscopic Explorer sition of comets. (FUSE) in 1999 provided access to the spectral region between 900 and 1200 Å at very high spectral resolution and has led to the detection of H2 (Sect. 87.3.4), upper lim- 87.3 Excitation Mechanisms its on Ar and N2 , and some three dozen emission lines identified as solar Lyman-˛ pumped H2 fluorescence from 87.3.1 Basic Phenomenology vibrationally excited H2 resulting from the photodissociation Coma abundances may be derived from spectrophotometric of H2 CO. [18, 19]. Several other satellite observatories have contributed measurements of either the total flux or the surface brightness unique cometary observations in the UV and sub-mm spec- in a given spectral feature. The uncertainty in the derived tral windows. The Solar and Heliospheric Observatory abundances includes not only the measurement uncertainty, (SOHO) has two valuable instruments: the SWAN (Solar but also uncertainties in the atomic and molecular paramWind Anisotropies) instrument provides sky maps in H I Ly- eters and, in the case of surface brightness measurements, ˛ at 1ı resolution and has observed over 20 comets since uncertainties in the model parameters used. Thus, relative 1996 [20]. The UVCS (Ultraviolet Coronograph Spectrom- abundances, derived from observations of different comets eter) provides far-UV spectra and images of comets close with the same instrument and under similar geometrical conto the Sun, where HST and FUSE are prohibited from ob- ditions, are often more reliable. serving, and has recently detected C2C in the tail of comet Atoms and ions in the cometary coma emit radiation priC/2002 X5 (Kudo-Fujikawa) [21]. The direct detection of marily by means of resonance re-radiation of solar photons. H2 O in the fundamental rotational line at 557 GHz in sev- For the cosmically abundant elements H, C, N, O, and S, their Offset (arcsec)
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strongest resonance transitions are in the VUV. The few ex- where ˝ is the solid angle subtended by the aperture. The ceptions are noted below. Assume that the coma is optically brightness, in turn, is related to N i , the average column denthin in these transitions. The total number of species i in the sity of species i within the field of view by coma is (87.7) Bi D 106 gi .r/N i : (87.1) Mi D Qi i .r/ ; The evaluation of Qi from N i requires the use of a model of the density distribution of the species i (Sect. 87.4.2). A similar treatment can be applied to the excitation of the near-infrared vibrational transitions of cometary parent molecules since the direct pumping by solar IR radiation far exceeds the indirect pumping of ground state vibrational levels through electronic transitions excited by the solar UV flux [8]. However, this does not apply to the rotational (87.2) transitions, which are controlled by collisional excitation, Li D Mi gi .r/ ; primarily collisions with H2 O. In this case, the observed rowhere the fluorescence efficiency, or g-factor, gi .r/ D tational temperature may be regarded as a reliable measure gi .1 AU/r 2 , is of the kinetic temperature of the coma gas. e 2 2 f Fˇ !Q photons s1 atom1 ; gi .1 AU/ D mc 2 (87.3) 87.3.2 Fluorescence Equilibrium
where Qi is the production rate (atoms or molecules s1 ) of species i, and i .r/ is its lifetime at heliocentric distance r, i .r/ D i .1 AU/r 2 . The r dependence arises from photolytic destruction processes induced by solar UV radiation and to a lesser degree by the solar wind, as described in Sect. 87.4.1. The luminosity, in photons s1 , in a given transition at wavelength , is then
where f is the absorption oscillator strength, Fˇ is the In the case of low-resolution spectroscopy, where an atomic solar flux per unit wavelength interval at 1 AU, and !Q is the multiplet or molecular band is unresolved, the evaluation of albedo for single scattering, defined for a line in an atomic the g-factor Eq. (87.3) does not require knowledge of the popmultiplet as ulation of either atomic fine structure levels or molecular rotaAj tional levels in the ground state of the transition. Furthermore, ; (87.4) !Q D P the assumption that the solar flux does not vary over the mulj Aj tiplet or band allows us to use the total transition oscillator where Aj is the decay rate. If a given multiplet is not re- strength. This assumption is more often than not invalid besolved, then !Q D 1. For diatomic molecules, fluorescence to cause of the Fraunhofer structure of the solar spectrum in the other vibrational levels becomes important, and the evalua- near-UV and visible region and the emission line nature of the tion of !Q depends on the physical conditions in the coma, as spectrum below 2000 Å. For high-resolution spectra, the gdiscussed in Sect. 87.3.2. Thus, for a comet at a geocentric factors for each individual line must be calculated separately, distance , the total flux from the coma for the transition is and the relative populations of the ground state levels must be included. There are three cases to be considered: Li Qi gi .r/i .r/ D ; (87.5) Fi D 2 2 4 4 1. The g-factor, or probability of absorption of a solar phoand the product gi .r/i .r/ is independent of r. ton, is less than the probability that the species will be Unfortunately, the scale lengths (the product of lifetime dissociated or ionized, i.e., gi < .i /1 . In this case, the and outflow velocity) of almost all of the species of interground-state population is not affected by fluorescence, est in the UV are 105 106 km at 1 AU. Thus, total flux and a Boltzmann distribution at a suitable temperature measurements require fields of view ranging from several (typically 200 K at 1 AU) corresponding to the producarc-minutes to a few degrees. This has been done only in tion of the species may be used. This is often the case in the the case of a few isolated sounding rocket experiments. Most far-UV, where the solar flux is low, such as for the Fourth information about the UV spectra comes from observations Positive band system of CO. For atomic transitions from made by orbiting satellite observatories whose spectrographs triplet ground states, such as is found with O, C and S, 00 00 have small apertures (e.g., 10 20 for IUE) and, thus, samdownward fine-structure transitions are fast enough to efple only a very small part of the total coma. In this case, again fectively depopulate all but the lowest fine structure level. 0 assuming an optically thin coma, the measured flux Fi in the 2. The species undergoes many photon absorption and emisaperture can be converted to an average surface brightness sion cycles in its lifetime, and the ground-state population Bi (in units of Rayleighs): is determined (usually after five or six cycles) by the (87.6) Bi D 4 106 Fi0 ˝ 1 ; fluorescence branching ratios. This is the condition of
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fluorescence equilibrium, which applies for almost all radicals observed in the visible and near-UV regions. The general procedure is to solve a set of coupled equations of the form X X dna pab C nb pba ; D na dt N
bD1
N
L/N 8
–15
rH2
erg/(s molecule))
1.00 AU; unquenched 1.00 AU; quenched 0.25 AU; unquenched 0.25 AU; quenched
6
(87.8)
bD1
4
where na is the relative population of level a, and pab and pba are transition rates out of and into this level, respectively. The na are normalized to unity. The steady state, obtained after many cycles, is given by dna =dt D 0. Since the downward transition rates are determined only by quantum mechanics, while the absorption rates depend on the magnitude of the solar flux, the steady-state population varies with distance from the Sun with higher rotational levels (as for the case of CN [3]) populated closer to the Sun. In some cases, where only a few cycles occur, the equations are integrated numerically. As the gfactor varies with time, it also effectively varies with the position of a species in the coma. Care must also be taken when spectra taken with small apertures are analyzed, as the transit time for an atom or molecule to cross the aperture may be . gi .r/1 . In practice, these considerations are often not important. 3. The same as 2., except that the density is sufficiently high that collisional transitions must be included in addition to the radiative transitions between levels. However, as the collisional rates are poorly known, in practice a collision sphere is defined such that a molecule traveling radially outward from this sphere suffers only one collision with other molecules or atoms. Outside this sphere, fluorescence equilibrium is assumed to hold, while inside, a thermal distribution of ground-state levels is used. A rough estimate of the radius Rc of the collision sphere, based on the radial outflow model of Sect. 87.4.2, is given by [29, 30] Rc D 103
Q km ; 1029
(87.9)
where Q is the total production rate in molecules s1 .
87.3.3 Swings and Greenstein Effects Swings [31] first pointed out that because of the Fraunhofer absorption lines in the visible region of the solar spectrum, the absorption of solar photons in a molecular band would vary with the comet’s heliocentric velocity rP , leading to differences in the structure of a band at different values of rP when observed at high resolution. In Eq. (87.8), this corresponds to evaluating the pba D pba .Pr /. For typical comets observed near 1 AU, rP can range from 30 to
2
0 –60
–40
–20
0
20 40 60 Heliocentric velocity (km/s)
Fig. 87.2 OH (0,0) band g-factor as a function of heliocentric velocity. After [32]
C30 km=s, while in certain cases of comets with small perihelia, the range can be twice as large. This effect of the Doppler shift between the Sun and the comet is commonly referred to as the Swings effect. Even for observations at low resolution, the Swings effect must be taken into account in the calculation of the total band g-factor, and this has been done for a number of important species such as OH, CN, and NH. A particularly important case, that of the OH A 2 ˙ C X 2 ˘ (0,0) band at 3085 Å, which is often used to derive the water production rate of a comet, is illustrated in Fig. 87.2, which also shows the dependence of fluorescence equilibrium on heliocentric distance [32]. While this effect was first recognized in the spectra of radicals in the visible range, a similar phenomenon occurs in the excitation of atomic multiplets below 2000 Å, where the solar spectrum makes a transition to an emission line spectrum. For example, the three lines of the O I 1302 multiplet have widths of 0:1 Å, corresponding to a velocity of 25 km=s, so that knowledge of exact solar line shapes is essential for a reliable evaluation of the g-factor for this transition [33]. A similar effect occurs for the S I multiplets at 1429 and 1479 Å [17]. A differential Swings effect occurs in the coma since atoms and molecules on the sunward side of the coma, flowing outward towards the Sun, have a net velocity that is different from those on the tailward side, and so, if the absorption of solar photons takes place on the edge of a line, the g-factors will be different in the two directions. Differences of this type appear in long-slit spectra in which the slit is placed along the Sun–comet line (the Greenstein effect [34]). Again, an analogue in the far-UV has been observed in the case of O I 1302 [35]. Although it is also possible to explain the observation by nonuniform outgassing, this was considered unlikely, as all of the other observed emissions had symmetric spatial distributions. The measurement of the
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Greenstein effect immediately leads to a determination of the emissions. Since the photoionization rate of water (and of the important minor species such as CO and CO2 ) is 106 s1 mean outflow velocity of the given species. at 1 AU, and the efficiency for converting the excess electron energy into excitation of a single emission is of the order of 87.3.4 Bowen Fluorescence a few percent, the effective excitation rate for any emission will be 108 s1 or less at 1 AU [40]. Since the efficienFor heliocentric velocities > 30 km=s, the Doppler shift re- cies for resonance scattering or fluorescence for almost all the duces the solar flux at the center of the absorption line to known cometary emissions are much larger, electron impact a very small value, so that the O I 1302 line is expected may be safely neglected except in a few specific cases. to appear weakly, if at all, in the observed spectrum. Thus, The cases of interest are those of forbidden transitions, it was a surprise that this line appeared fairly strongly in where the oscillator strength, and consequently the g-factor, two comets, Kohoutek (C/1973 E1) and West (C/1975 V1), is very small. The primary example is the O I 5 S2 3 P2;1 whose values of rP were both > 45 km=s at the times of obser- doublet at 1356 Å, which was observed faintly in several vation. The explanation invoked the accidental coincidence bright comets such as Hyakutake and Hale-Bopp. This emisof the solar H I Ly-ˇ line at 1025:72 Å with the O I 3 D3 P sion appeared very conspicuously in measurements of the transition at 1025:76 Å, cascading through the intermediate coma of 67P/Churyumov-Gerasimenko made by the Alice 3 P state to the 3 S upper level of the 1302 Å multiplet [33]. far-ultraviolet spectrograph on Rosetta and was found conThis mechanism, well known in the study of planetary neb- centrated within a few km of the nucleus, a region usually not ulae, is referred to as Bowen fluorescence [36]. The g-factor resolvable from Earth-based observations [41]; O I 1356 due to Ly-ˇ pumping is an order of magnitude smaller than is produced by dissociative electron impact of the principal that for resonance scattering but is sufficient to explain the molecular species H2 O, CO2 , and O2 , with the brightness of observations and to confirm that H2 O is the dominant source this multiplet relative to O I 1304, C I 1657, and H I Ly-ˇ, of the observed oxygen in the coma. diagnostic of the composition of the near-nucleus coma [42]. Ly-ˇ is also coincident with the P1 line of the (6,0) However, calculations of excitation rates require electron imband of the H2 Lyman system .B 1 ˙uC X 1 ˙gC / leading pact cross sections at energies where the electron flux is high, to fluorescence in the same line of several (6,v 00 ) bands, the usually near threshold [39], and these are poorly known. strongest being that of the (6,13) band at 1608 Å [37]. This line is, however, difficult to observe because of the nearby strong CO Fourth Positive bands. Recently, the shorter wave- 87.3.6 Prompt Emission length (6,1), (6,2), and (6,3) bands have been detected in three comets using FUSE, and the derived H2 column In cases where the dissociation or ionization of a molecule abundance was found to be consistent with a water photodis- leaves the product atom or molecule in an excited state, the sociation source [18]. Another interesting example occurs for decay of this state with the prompt emission of a photon proNe I, where the second resonance transition at 629:74 Å co- vides a useful means for tracing the spatial distribution of the incides with the strong solar O V line at 629:73 Å. This line parent molecule in the inner coma. The products of interest was used to set a sensitive upper limit on the Ne abundance for the water molecule are described in Sect. 87.4.1. Prompt in the coma of comet Hale-Bopp (C/1995 O1) [38]. emission includes both allowed radiative decays (such as from the A 2 ˙ C state of OH) as well as those from metastable states such as O(1 D), since the latter will move 150 km (for 87.3.5 Electron Impact Excitation a comet at a geocentric distance of 1 AU, 1 arc-second corresponds to a projected distance of 725 km) in its lifetime. The The photoionization of parent molecules and their dissocia- O I 1 D 3 P transition at 6300 and 6364 Å has been used extion products leads to the formation of a cometary ionosphere tensively as a ground-based monitor of the water production whose characteristics are only now known as a result of mea- rate with the caveat that other species such as OH, CO and surements by the Rosetta Plasma Consortium [39]. In princi- CO2 may also contribute to the observed red line emission. ple, electron impact excitation, which is often the dominant In addition, when the density of H2 O is sufficient to produce source of airglow in the atmospheres of the terrestrial planets, observable red line emission, it is also sufficient to produce also contributes to the observed emissions, particularly in the collisional quenching of the 1 D state, and this must also be UV, and so must be accounted for in deriving column densi- considered in the interpretation of the observations. The analties from the observed emission brightnesses. However, one ogous 1 D 3 P transitions in carbon occur at 9823 and 9849 Å can use a very simple argument, based on the known energy and provide similar information about the production rate of distribution of solar UV photons, to demonstrate that elec- CO. Carbon atoms in the 1 D state, whose lifetime is 4000 s, tron impact excitation is only a minor source for the principal are known to be present from the observation of the resonantly
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scattered 1 Po 1 D transition at 1931 Å [43], and the 9849 Å line has been detected in comet Hale-Bopp [44]. The OH A 2 ˙ C X 2 ˘ prompt emission competes with that produced by the resonance fluorescence of OH and is difficult to detect, except close to the nucleus (inside 100 km) where the density of water molecules exceeds that of OH by a few orders of magnitude [45]. Again, the reason is that at the wavelengths below the threshold for simultaneous dissociation and excitation, the Sun has much less flux than at the resonance wavelength. On the other hand, only a few rotational lines are excited in fluorescence equilibrium [32], while the prompt emission is characterized by a very hot rotational distribution, so in principle the two components may be separated although observations at very high spatial and spectral resolution are required. OH prompt emission has also recently been detected in the infrared at 3.28 µm [46].
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a spherical surface is conserved), which assume isotropic gas production but appears to hold equally well for the case of Halley, which clearly was not outgassing uniformly over its surface [49]. The photolysis of H2 O can proceed by: H2 O C h ! OH C H
a 0
a
C
! OH.A ˙ / C H 1
b
! H2 C O. D/
b0
! H2 C O.1 S/
b
00
! H2 O C e
d
C
e
1450 Å
! H C H C O. P/ C
! H2 C O C e C
1304 Å 984 Å
! H C OH C e C
1357:1 Å 1770 Å
3
c
f
87.3.7 OH Level Inversion
2424:6 Å
2
! OH C H C e
684:4 Å 664:4 Å 662:3 Å
The right-hand column gives the energy threshold for each reaction, in wavelength units. The products are subsequently removed by:
An important consequence of fluorescence equilibrium in the OH radical is the UV pumping of the hyperfine and g OH C h ! O C H 2823:0 Å -doublet levels of the X 2 ˘3=2 .J D 3=2/ ground state, C which results in a deviation of the population from statisti928 Å h ! OH C e cal equilibrium [32]. Depending on the heliocentric velocity, 844:79 Å i H2 C h ! H C H this departure may be either inverted or anti-inverted, givC 803:67 Å j ! H2 C e ing rise to either stimulated emission or absorption against C the galactic background at 18-cm wavelength. This technique 685:8 Å k !HCH Ce has been used extensively since 1974 to monitor the OH proC 910:44 Å l O C h ! O C e duction rate in comets, even of those that appear close to the 911:75 Å m H C h ! HC C e Sun [47]. The resulting radio emissions are easily quenched by collisions with molecules and ions, the latter giving rise to a fairly large Rc that must be accounted for in interpreting Reactions l and m can also occur by resonant0 charge ex-0 the derived OH column density. Nevertheless, the radio and change with solar wind protons. Reactions a , b, and b correspond to the production of prompt emission, as disUV measurements give reasonably consistent results [47]. cussed in Sect. 87.3.6. The determination of column densities of H, O, and OH simultaneously was convincing evidence that the dominant volatile species in the cometary nucleus 87.4 Cometary Models was H2 O, long before the direct infrared detection of this species in the coma. 87.4.1 Photolytic Processes Detailed cross sections for the absorption of UV photons by each of the reactants, including proper identification of As an example of the photolytic destruction processes occurthe final product states, is necessary for the evaluation of the ring in the coma, consider the dominant molecular species, water. Water vapor is assumed to leave the surface of the photodestruction rate in the solar radiation field of each of nucleus with some initial velocity v0 and flow radially out- the above reactions. These rates are evaluated at 1 AU using ward, expanding into the vacuum, and increasing its velocity whole disk measurements of the solar flux by integrating the according to thermodynamics [48]. Even though collisions cross section are important at distances typically up to 104 km (depending Zth on the density and, consequently, on the total gas production (87.10) Jd D Fˇ d d : rate), the net flow of H2 O molecules is radially outward, such 0 that the density varies as R2 near the nucleus, where R is the cometocentric distance. This is the basis for spherically sym- Another quantity of interest in coma modeling is the exmetric coma models (the number of particles flowing through cess velocity (or energy) of the dissociation or ionization
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P. D. Feldman
products, and this requires knowledge of the partitioning of energy between internal and translational modes for each reaction [50]. Qualitatively, the photodissociation and photoionization rates can be estimated from the threshold energies given in the table above, since the solar flux is decreasing very rapidly to shorter wavelengths. It is customary to specify the lifetime against photodestruction i , of species i, which is just X .Jd /j ; (87.11) .i /1 D j
where the sum is over all possible reaction channels, as well as the lifetimes into specific channels. Processes with thresholds near 3000 Å have lifetimes 104 s, those with thresholds near 2000 Å an order of magnitude longer, while those with thresholds below Ly-˛, such as most photoionization channels, have lifetimes 106 s, all at 1 AU. In addition to uncertainties in the details of the absorption cross sections, further uncertainty is introduced into the calculation of Jd by the lack of knowledge of the solar flux at the time of a given observation due to the variability of the solar radiation below 2000 Å, and most importantly, below Ly-˛, where there have not been continuous space observations for more than a decade. The solar UV flux is known to vary considerably both with the 27-day solar rotation period and with the 11-year solar activity cycle. Also, at any given point in its orbit, a comet sees a different hemisphere of the Sun than what is seen from Earth. Huebner et al. [51] have compiled an extensive list of useful photodestruction rates using mean solar fluxes to represent the extreme conditions of solar minimum and solar maximum. They also include the excess energies of the dissociation products. A detailed analysis of the rates for H2 O and OH, using surrogate solar indices such as the 10:7 cm solar radio flux, or the equivalent width of the He I line at 1.083 µm, is in good agreement with observations [52]. Similar analyses still remain to be carried out for other important species, such as CO and NH3 .
87.4.2
Density Models
although detailed analysis revealed that the velocity of the water molecules increased from 0:8 km=s at about 1000 km from the nucleus to 1:1 km=s at a radial distance of 10;000 km. The dependence of outflow velocity on heliocentric distance remains uncertain, although Delsemme [53] has suggested an r 1=2 -dependence based on thermodynamic arguments. Submillimeter observations of H2 O have sufficient spectral resolution to permit the mapping of outflow velocities along various lines-of-sight to the comet [23]. For the dissociation products, the modeling is more complex. The simplest model assumes continued radial outflow, although at a different velocity, such as to maintain a constant flux of the initial particle across an arbitrary spherical surface surrounding the nucleus [54]. This model, which is valid only at distances equal to a few ˇi , is widely used, as the densities can be easily expressed in analytical form. However, as surface brightness measurements are often made with small fields of view close to the nucleus, this model can lead to a factor of 2 error from the neglect of the dissociation kinematics. Since the solar photodissociation often leaves the product fragments with 12 eV of kinetic energy [50, 51], the resultant motion (which is assumed to be isotropic in the parent molecule’s rest frame), will contain a large nonradial component. Several approaches have been developed to account properly for the kinematics; notably, the vectorial model of Festou [55] and the average random walk model (a Monte Carlo method) of Combi and Delsemme [56]. The latter model has been extended to include time-dependent gas kinetics so as to properly account for regions of the coma where the gas is not in local thermodynamic equilibrium [57]. In addition to the photodestruction chains, chemical reactions, particularly ion–molecule reactions, can alter the composition within the collision zone defined in Sect. 87.3.2. While such reactions may produce numerous minor species, they do not erase the signatures of the original parent molecules. In fact, detailed chemical models have clearly demonstrated the need for complex molecules to serve as the parents of the observed C2 and C3 radicals in the coma, strengthening the connection between comet formation and molecular cloud abundances. Thus, the photochemical chains provide a valid means of relating the coma composition to that of the nucleus.
For parent molecules produced directly by sublimation from the surface of the comet, a spherically symmetric radial outflow model is often adopted. Such a model assumes a steady-state gas production rate Qi and a constant outflow velocity v, and gives rise to a density distribution as a func- 87.4.3 Radiative Transfer Effects tion of cometocentric distance R given by The results of a model calculation for the density of a species Qi R=ˇi must be integrated over the line of sight to obtain the cole ; (87.12) ni .R/ D 4 vR2 umn density at a given projected distance from the nucleus where ˇi D vi is the scale length of species i. The basic and then integrated over the instrumental field of view for validity of this model was demonstrated by the Giotto neu- comparison with the observed average surface brightness or tral mass spectrometer measurements of H2 O and CO2 [49], derived average column density Ni . This assumes that the
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coma is optically thin, and that all atoms or molecules have an equal probability of absorbing a solar photon. In practice, this is true for all molecular emissions except perhaps within 1000 km of the nucleus (i.e., for observations made at better than 100 resolution). Since the cross sections at line center can be very large for an atomic resonance transition, the optical depth for the abundant species can exceed unity, and radiative transfer along both the line of sight to the Sun and that to the Earth must be considered. This is not a trivial problem, as the velocity distribution of the atoms, particularly the component due to the excess energy of the photodestruction process, must be well known, as must be the shape of the exciting solar line. The most thoroughly studied case to date is that of H I Ly-˛, whose angular extent, in direct images, can exceed several degrees on the sky [58]. An interesting case arises for resonance transitions between an excited 3 S1 state and the ground 3 P2;1;0 state, as for O and S, particularly the latter, as its concentration near the nucleus can be quite large due to the rapid decay of one of its parents, CS2 . For S I the three lines at 1807, 1820, and 1826 Å are not observed to have their statistical intensity ratio of 5 W 3 W 1, except at large distances from the nucleus. This is explained by noting that fine structure transitions will lead to all of the S atoms reaching the J D 2 ground state in a time short compared with that for absorbing a solar photon, and that the emitted 1807 Å photons will be reabsorbed and can then branch into the other two lines. The detailed solution to this problem has led to the conclusion that H2 S was the primary source of sulfur rather than CS2 , whose other product, CS, was simultaneously observed in the UV [59]. Millimeter and submillimeter observations of comet Hale-Bopp (C/1995 O1) subsequently showed that SO, SO2 , and OCS were also minor sources of atomic sulfur, comparable in abundance to CS2 [60]. Another minor source is S2 , initially observed in only one comet, IRAS-Araki-Alcock (C/1983 H1) [61], but recently seen in three additional comets by HST. The origin of S2 in the cometary nucleus has been discussed in the context of Rosetta mass spectrometer measurements of this species [62].
87.5
Summary
This brief chapter can only hint at the wealth of observational data spanning the entire electromagnetic spectrum now routinely acquired at almost every comet apparition allowing for a statistically significant assessment of comet diversity and formation scenarios. The Rosetta mission has revolutionized our knowledge of comets [63] and will serve as a benchmark for inferring the properties of comets from remote observations. Nevertheless, since space missions to comets are rare, Earth-based observations of comets will continue to play an important role in understanding the physical and chemical
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environments of these objects left over from the formation of the solar system, as well as the diversity of comets, as spacecraft missions are not able to visit long-period comets.
References 1. Festou, M.C., Weaver, H.A., Keller, H.U. (eds.): Comets II. Univ. Arizona Press, Tucson (2004) 2. Mumma, M.J., Charnley, S.B.: Ann. Rev. Astron. Astrophys. 49, 471 (2011) 3. Arpigny, C.: Ann. Rev. Astron. Astrophys. 3, 351 (1965) 4. Preston, G.W.: Astrophys. J. 147, 718 (1967) 5. Slaughter, C.D.: Astron. J. 74, 929 (1969) 6. Feldman, P.D., Brune, W.H.: Astrophys. J. Lett. 209, L45 (1976) 7. Mumma, M.J., Weaver, H.A., Larson, H.P., Davis, D.S., Williams, M.: Science 232, 1523 (1986) 8. Bockelée-Morvan, D., Crovisier, J., Mumma, M.J., Weaver, H.A.: In: Festou, M.C., Weaver, H.A., Keller, H.U. (eds.) Comets II. Univ. Arizona Press, Tucson (2004) 9. Crovisier, J., Bockelée-Morvan, D., Colom, P., Biver, N.: C. R. Phys. 17, 985 (2016) 10. Le Roy, L., Altwegg, K., Balsiger, H.: Astron. Astrophys. 583, A1 (2015) 11. Bockelée-Morvan, D., Calmonte, U., Charnley, S.: Space Sci. Rev. 197, 47 (2015) 12. Jessberger, E.K., Kissel, J.: In: Newburn, R.L., Neugebauer Jr., M., Rahe, J. (eds.) Comets in the Post-Halley Era, p. 1075. Kluwer, Dordrecht (1989) 13. Festou, M.C., Feldman, P.D.: In: Kondo, Y. (ed.) Exploring the Universe with the IUE Satellite, p. 101. Reidel, Dordrecht (1987) 14. Festou, M.C.: International Ultraviolet Explorer – Uniform Low Dispersion Data Archive Access Guide: Comets. ESA SP-1134. ESA, Noordwijk (1990) 15. Weaver, H.A., Feldman, P.D., McPhate, J.B., A’Hearn, M.F., Arpigny, C., Smith, T.E.: Astrophys. J. 422, 374 (1994) 16. Lupu, R.E., Feldman, P.D., Weaver, H.A., Tozzi, G.-P.: Astrophys. J. 670, 1473 (2007) 17. Feldman, P.D., Weaver, H.A., A’Hearn, M.F., Combi, M.R., Dello Russo, N.: Astron. J. 155, 193 (2018) 18. Feldman, P.D., Weaver, H.A., Burgh, E.B.: Astrophys. J. Lett. 576, L91 (2002) 19. Feldman, P.D.: Astrophys. J. 812, 115 (2015) 20. Mäkinen, J.T.T., Bertaux, J.-L., Pulkkinen, T.I., Schmidt, W., Kyrölä, E., Summanen, T., Quémerais, E., Lallement, R.: Astron. Astrophys. 368, 292 (2001) 21. Povich, M.S., Raymond, J.C., Jones, G.H., Uzzo, M., Ko, Y.-K., Feldman, P.D., Smith, P.L., Marsden, B.G., Woods, T.N.: Science 302, 1949 (2003) 22. Neufeld, D.A., et al.: Astrophys. J. Lett. 539, L151 (2000) 23. Lecacheux, A., et al.: Astron. Astrophys. 402, L55 (2003) 24. Hartogh, P., Crovisier, J., de Val-Borro, M.: Astron. Astrophys. 518, L150 (2010) 25. Lisse, C.M., et al.: Science 274, 205 (1996) 26. Cravens, T.E.: Science 296, 1042 (2002) 27. Greenwood, J.B., Williams, I.D., Smith, S.J., Chutjian, A.: Phys. Rev. A 63, 62707 (2001) 28. Kharchenko, V., Dalgarno, A.: Astrophys. J. Lett. 554, L99 (2001) 29. Festou, M.C., Rickman, H., West, R.M.: Astron. Astrophys. Rev. 4, 363 (1993) 30. Festou, M.C., Rickman, H., West, R.M.: Astron. Astrophys. Rev. 5, 37 (1993) 31. Swings, P.: Lick. Obs. Bull. 131, xix (1941) 32. Schleicher, D.G., A’Hearn, M.F.: Astrophys. J. 331, 1058 (1988)
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1298 33. Feldman, P.D., Opal, C.B., Meier, R.R., Nicolas, K.R.: The Study of Comets. In: Donn, B., Mumma, M., Jackson, W., A’Hearn, M., Harrington, R. (eds.) NASA SP-393, p. 773. NASA, Washington (1976) 34. Greenstein, J.L.: Astrophys. J. 128, 106 (1958) 35. Dymond, K.F., Feldman, P.D., Woods, T.N.: Astrophys. J. 338, 1115 (1989) 36. Bowen, I.S.: Publ. Astron. Soc. Pac. 59, 196 (1947) 37. Feldman, P.D., Fastie, W.G.: Astrophys. J. Lett. 185, L101 (1973) 38. Krasnopolsky, V.A., Mumma, M.J., Abbott, M., Flynn, B.C., Meech, K.J., Yeomans, D.K., Feldman, P.D., Cosmovici, C.B.: Science 277, 1488 (1997) 39. Galand, M., Héritier, K.L., Odelstad, E.: Mon. Not. R. Astron. Soc. 462, 331 (2016) 40. Cravens, T.E., Green, A.E.S.: Icarus 33, 612 (1978) 41. Feldman, P.D., A’Hearn, M.F., Bertaux, J.-L.: Astron. Astrophys. 583, A8 (2015) 42. Feldman, P.D., A’Hearn, M.F., Bertaux, J.-L.: Astron. J. 155, 9 (2018) 43. Tozzi, G.P., Feldman, P.D., Festou, M.C.: Astron. Astrophys. 330, 753 (1998) 44. Oliversen, R.J., Doane, N., Scherb, F., Harris, W.M., Morgenthaler, J.P.: Astrophys. J. 581, 770 (2002) 45. Bertaux, J.-L.: Astron. Astrophys. 160, L7 (1986) 46. Mumma, M.J., et al.: Astrophys. J. 546, 1183 (2001) 47. Gérard, E.: Astron. Astrophys. 230, 489 (1990) 48. Mendis, D.A., Houpis, H.L.F., Marconi, M.L.: Fund. Cosm. Phys. 10, 1 (1985) 49. Krankowsky, D.: In: Newburn, R.L., Neugebauer Jr., M., Rahe, J. (eds.) Comets in the Post-Halley Era, p. 855. Kluwer, Dordrecht (1989) 50. Okabe, H.: The Photochemistry of Small Molecules. Wiley, New York (1978) 51. Huebner, W.F., Keady, J.J., Lyon, S.P.: Solar Photo Rates for Planetary Atmospheres and Atmospheric Pollutants. Springer, New York (1992)
P. D. Feldman 52. Budzien, S.A., Festou, M.C., Feldman, P.D.: Icarus 107, 164 (1994) 53. Delsemme, A.H.: In: Wilkening, L.L. (ed.) Comets, p. 85. Univ. Arizona Press, Tucson (1982) 54. Haser, L.: Bull. Acad. R. Sci. Liège 43, 740 (1957) 55. Festou, M.C.: Astron. Astrophys. 95, 69 (1981) 56. Combi, M.R., Delsemme, A.H.: Astrophys. J. 237, 633 (1980) 57. Combi, M.R.: Icarus 123, 207 (1996) 58. Richter, K., Combi, M.R., Keller, H.U., Meier, R.R.: Astrophys. J. 531, 599 (2000) 59. Meier, R., A’Hearn, M.F.: Icarus 125, 164 (1997) 60. Bockelée-Morvan, D., Lis, D.C., Wink, J.E., Despois, D., Crovisier, J., Bachiller, R., Benford, D.J., Biver, N., Colom, P., Davies, J.K., Gérard, E., Germain, B., Houde, M., Mehringer, D., Moreno, R., Paubert, G., Phillips, T.G., Rauer, H.: Astron. Astrophys. 353, 1101 (2000) 61. A’Hearn, M.F., Feldman, P.D., Schleicher, D.G.: Astrophys. J. Lett. 274, L99 (1983) 62. Mousis, O., Ozgurel, O., Lunine, J.I.: Astrophys. J. 835, 134 (2017) 63. Taylor, M.G.G.T., Altobelli, N., Buratti, B.J., Choukroun, M.: Philos. Trans. R. Soc. A 375, 20160262 (2017)
Paul Feldman Dr Feldman was Professor Emeritus of Physics and Astronomy at the Johns Hopkins University, where he has been since 1967. He received his PhD in Physics from Columbia University in 1964. His last work was in space ultraviolet astronomy and spectroscopy with a focus on the study of the atmospheres of comets, planets, and the Galilean satellites.
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Contents 88.1 88.1.1 88.1.2
Basic Structure of Atmospheres . . . . . . . . . . . . . 1299 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1299 Atmospheric Regions . . . . . . . . . . . . . . . . . . . . 1300
88.2 88.2.1 88.2.2
Density Distributions of Neutral Species . . . . . . . 1304 The Continuity Equation . . . . . . . . . . . . . . . . . . 1304 Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . 1305
88.3 88.3.1 88.3.2
Interaction of Solar Radiation with the Atmosphere 1305 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1305 The Interaction of Solar Photons with Atmospheric Gases . . . . . . . . . . . . . . . . . . . 1306 Interaction of Energetic Electrons with Atmospheric Gases . . . . . . . . . . . . . . . . . . . 1309
88.3.3 88.4 88.4.1 88.4.2 88.4.3 88.4.4 88.4.5
Ionospheres . . . . . . . . . . Ionospheric Regions . . . . . Sources of Ionization . . . . . Nightside Ionospheres . . . . Ionospheric Density Profiles Ion Diffusion . . . . . . . . . .
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1311 1311 1312 1317 1320 1322
88.5
Neutral, Ion, and Electron Temperatures . . . . . . . 1323
88.6
Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . 1325
88.7
Planetary Escape . . . . . . . . . . . . . . . . . . . . . . 1332
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334
Abstract
We describe here the neutral and ionic structures of atmospheres, including the processes that determine the atmospheric layers, the distributions of the species, and the temperature profiles. We focus on the upper atmosphere, which comprises the thermosphere and the ionosphere, two regions that overlay and interact with each other. We describe the interaction of near and extreme ultraviolet solar photons and energetic electrons with the atmosphere and their role in ionization and dissociation of J. L. Fox () Dept. of Physics, Wright State University Dayton, OH, USA e-mail: [email protected]
atmospheric species. We also review the production and loss processes that are important in the formation of the different layers of the dayside and nightside ionospheres, including ion and neutral diffusion. The processes that determine the neutral, ion, and electron temperatures are discussed. We review the processes that are important in the production of the luminosity of the upper atmospheres, including dayglow, nightglow, and auroras. Finally, we describe atmospheric escape processes, including thermal and nonthermal mechanisms. Keywords
solar zenith angle terrestrial planet outer planet dissociative recombination suprathermal electron
88.1 Basic Structure of Atmospheres 88.1.1 Introduction In a stationary atmosphere, the force of gravity is balanced by the plasma pressure gradient force in the vertical direction, and the variation of pressure P .z/ with altitude above the surface z is governed by the hydrostatic relation dP .z/ D .z/g.z/ ; dz
(88.1)
where .z/ D n.z/m.z/ is the mass density, n.z/ is the number density, and m.z/ is the weighted average mass of the atmospheric constituents. In general, variables such as P , T ,
, g, n, and even m are functions of altitude, although it will often not be shown explicitly in the equations that follow for the sake of compactness. The acceleration of gravity g is usually taken to be the vector sum of the gravitational attraction per unit mass and the centrifugal acceleration due to the rotation of the planet g.r/ D
GM ! 2 r cos2 ; r2
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_88
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where r D r0 C z is the distance from the center of the planet, r0 is the planetary radius, M is the planetary mass, G D 6:670 108 dyn cm2 g2 is the gravitational constant, is the latitude, and ! is the angular velocity of the planet. When the hydrostatic relation Eq. (88.1) is combined with the ideal gas law in the form
sion frequencies) having lower homopause altitudes. It must also be borne in mind that the homopause is not a sharp boundary, and that the transition from mixing to diffusion occurs gradually. In the terrestrial atmosphere, the homopause is near 100 km at n.z/ 1013 cm3 . Below the homopause, the mixing ratios (or fractions by number) of the constituent gases, apart from those minor or trace species whose density (88.3) profiles are determined by photochemistry or physical loss P D nkB T ; processes, are fairly constant with altitude. Characteristics of where kB is Boltzmann’s constant, and T is the temperature, the homopauses of the planets are presented in Table 88.1. and integrated, the barometric formula 0 P .z/ D P0 exp@
Zz
1 1 dz 0 A ; H.z 0 /
(88.4)
z0
for the pressure P .z/ above a reference level (denoted by the subscript 0) as a function of altitude results. The pressure scale height H.z/ is defined as H.z/ D
kB T : mg
(88.5)
In the lower and middle atmosphere, the mass m in Eq. (88.5) is the weighted average mass of the atmospheric constituents. When the ideal gas law Eq. (88.3) is substituted into the barometric formula Eq. (88.4), the altitude distribution 1 0 z Z T0 1 exp@ dz 0 A ; n.z/ D n0 T .z/ H.z 0 /
(88.6)
z0
for the number density n.z/ above a reference altitude is obtained. Integration of Eq. (88.1) or Eq. (88.6) from z to infinity shows that the column density above that altitude is approximately N.z/ D n.z/H.z/. Thus, the scale height can be thought of as the effective thickness of the atmosphere. Throughout the atmosphere, gravity exerts a force on each particle that is proportional to its mass. Below the homopause, however, the tendency of the species to separate out under the force of gravity is overpowered by large scale mixing processes, such as turbulence and/or convection. Thus in its lower and middle regions, the homosphere, the atmosphere is well-mixed. The upper boundary of this region is called the homopause (or turbopause). Above this level, the major transport process is diffusion, and each species is distributed according to its own scale height. The homopause is defined as the level at which the time constants for mixing and diffusion are equal, and usually occurs at n.z/ 1011 1013 cm3 , depending on the strength of vertical mixing for a given planet. Since molecular diffusion coefficients vary from one species to another, the exact altitude of the homopause is species-dependent, with smaller species (or those that have smaller momentum transfer colli-
88.1.2 Atmospheric Regions The division of atmospheres into regions is based on the temperature structure of the terrestrial atmosphere, which is shown in Fig. 88.1. In the troposphere of a planet, above the boundary layer, T decreases at close to the adiabatic lapse rate ( ) for the constituent gases from the surface to the tropopause. For an atmosphere that is a mixture of ideal gases, D g=cp , where cp is the specific heat of the gas mixture at constant pressure. The presence of a condensible constituent, such as water vapor in the terrestrial troposphere and that of other planets, such as Mars and the outer planets, and ammonia, C2 - or higher hydrocarbons in the atmospheres of the outer planets and satellites, decreases because upward motion leads to cooling and condensation, which releases latent heat. On Earth, the dry adiabatic lapse rate is about 10 K km1 , and the moist adiabatic lapse rate is about 46 K km1 in the lower to middle troposphere. The average lapse rate is about 6:5 K km1 , and the altitude of the tropopause varies from about 9 to 16 km from the poles to the equator. The composition of the lower atmosphere of the Earth is given in Table 88.2. Above the terrestrial tropopause lies the stratosphere, a region of increasing T that is terminated at the stratopause, near 50 km. This increase in T is caused by absorption of solar near-UV radiation by ozone in the Hartley bands and continuum (200310 nm). In the terrestrial mesosphere, which lies above the stratosphere, T decreases again to an absolute minimum at the mesopause, where T 180 K and n.z/ 1014 cm3 . Above the mesopause, in the thermosphere, T increases rapidly to a constant value, the exospheric temperature, T1 . The value of T1 in the terrestrial atmosphere depends on solar activity and is usually between about 700 and 1500 K. Figure 88.1a also shows altitude profiles of the noon and midnight thermospheric temperature for four values of the F10:7 index (the 2800 MHz flux in units of 1022 W m2 Hz1 at 1 AU), which represent different levels of solar activity. Analogous temperature profiles of other planetary bodies as a function of pressure are shown in Fig. 88.1b. It can be seen that the temperature profiles above some of the tropopauses do not reach a maximum value that signals the stratopause. In
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Table 88.1 Homopause characteristics of planets and other solar system bodies Planet Venusb
Altitude (km) 135
K (cm2 s1 ) 4(7)a
T (K) 199
Earthc
100
1(6)
185
Marsd
130–140
2.3(8)
163
Jupitere
500f
2(6)
600
3(7) (3–7)(7) 5(3)
325
5(6)
210
Saturng Uranusg Neptuneg Titanh Tritoni Plutoj
1010f (900–1400) 350f 700f 1010 (1000–1100) 30–40 0–12
137
4(8) 150–180 (1.2(8)–1.8(9)) (2–8)(3) 50 (1–4)(3) 40–50
nt (cm3 ) 1.4(11)
P (µbar) 4.7(3)
Composition (fraction by number) CO2 (76%), N2 (7:6%), O(9:3%), CO(6:7%), N(0:16%), C(0:01%) 1.3(13) 0.3 N2 (77%), O2 (18%), O(34%), Ar(0:7%), He(9:5 ppm), H(1:3 ppm) 2.8(10) 6:2.4/ CO2 (86%), N2 (4:3%), Ar(2:7%), O(3:2%), CO(3:1%), NO(40 ppm), H2 (22 ppm) 1.4(13) 0.4 H2 (95%), He(4:1%), H(0:055%), CH4 (200 ppb), C2 H2 (1:2 ppb), C2 H4 (2:5 ppb), C2 H6 (0:12 ppb) 6(11) 2:7.2/ H2 (9699%), He(1:23:1%), CH4 (16 ppm), H(0:173:7%) 1.93(15) 36 H2 (93%), He(6:7%), H(17 ppm), CH4 (0:8%), C2 H2 (0:83 ppb), C2 H6 (0:46 ppb), 2.8(12) 0:83 H2 (96:6%), He(3:42%), CH4 (0:48 ppm), H(30 ppm), C2 H6 (0:4 ppb) 3:2.9/ 6:4.5/ N2 (9496%), H2 (3:4%), CH4 (2:1%), C2 H2 (0:034%), C2 H4 (0:04%), HCN(0:024%), C2 H6 (60:5 ppm) 3.7(14) 2.7 N2 (99:9%), CH4 (3 ppm), H2 (70 ppm) 1.85(15) 11.5 N2 (99:7%), CH4 (0:3%), C2 H2 (274 ppb), C2 H6 (230 ppm), (5(14)–2(15)) C2 H4 (700 ppb)
Read as 4 107 K from von Zahn et al. [1] and model atmosphere from Hedin et al. [2], for 1500 h, 15ı N latitude, F10:7 D 150 c From The US Standard Atmosphere [3] d Fit to MAVEN model for F10:7 D 120, near subsolar point e From photochemical model of Kim [4] f Altitude above the 1 bar level g From Vervack, Jr. and Moses [5] h From Magee et al. [6] i From Strobel and Zhu [7] j From Young et al. [8] a
b
Table 88.2 Molecular weights and fractional composition of dry air in the terrestrial atmospherea Species N2 O2 Ar CO2 Ne He Kr Xe CH4 H2
Molecular weight (g mol1 ) 28:0134 31:9988 39:948 44:00995 20:183 4:0026 83:80 131:30 16:04303 2:01594
Fraction by volume 0:78084 0:209476 0:00934 0:00042099b 0:00001818 0:00000524 0:00000114 0:000000087 0:000002 0:0000005
a
Taken from The US Standard Atmosphere [3], except as noted 2022 June global averaged data value. The CO2 mixing ratio is increasing at an average annual rate of 0.489%. Value from Earth System Research Laboratory Global Monitoring Division of the National Oceanic and Atmospheric Administration (NOAA) (website http://gml. noaa.gov/ccgg/trends)
b
these atmospheres, the region between the tropopause and the mesopause is usually called the mesosphere for the inner planets and the stratosphere for the outer planets. The exosphere is assumed to be a nearly collisionless region of the thermosphere that is bounded from below by
a level called the “exobase”. The assumption involved in deriving the altitude of the exobase is that a particle traveling upward at or above the exobase will, with high probability, escape from the gravitational field of the planet. The exobase on Earth is located at about 450500 km, depending upon solar activity. The properties of the classical exobase will be discussed in the context of escape of species from atmospheres in Sect. 88.7. The surface P and T on Mars are about 6 mbar and 230 K, respectively. Due to the effect of dust storms, the extent of the Martian troposphere is highly variable, with a lapse rate that is 23 K km1 compared with the adiabatic lapse rate of 4:5 K km1 and a variable thickness of 2050 km [11]. The atmosphere of Mars, like many planetary atmospheres, does not have a stratopause. A roughly isothermal mesosphere extends from the tropopause to the base of the thermosphere at about 90 km. Sometimes a peak in the temperature between the tropopause and the mesosphere is caused by the absorption of solar radiation by dust that is lifted to high altitudes by dust storms, which are most common at perihelion. The thermospheric T is sensitive to solar activity and, since Mars has a very eccentric orbit, to heliocentric distance; T1 varies from about 180 to 350 K. Near the surface of Venus, T & 700 K, and P 95 bar; T decreases with a mean lapse rate of 7:7 K km1 ,
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a
Table 88.3 Composition of the lower atmospheres of Mars and Venus
Altitude (km) 600
Species
F10.7 = 75
250
140
200
CO2 N2 40 Ar O2 CO H2 O He Ne Kr Xe H2 CH4 SO2 H2 S
500
400
300
200
Mixing ratio Mars 0.949a 0.0279a 0.0208a 0.0013a 0.00174a 0.0003c (10˙6) ppme 2.5 ppmf 0.3 ppmf 0.08 ppmf (15˙5) ppmg 0.69 ppmi – –
Venusj 0.965 0.035 20–50 ppmb 0.3 ppm 20–30 ppm (30˙10) ppmd (9˙6) ppme 5–13 ppm 0.02–0.4 ppm – > 0.1 ppmh – 150 ppm 1–3 ppm
a
From Franz et al. [12] Includes all isotopes of Ar c Highly variable. From Jakosky and Haberle [13] d Value applies to altitudes below the cloud deck. From [14] and Taylor et al. [15] e From Krasnopolsky and Gladstone [16] f Owen [17] g At base of thermosphere; Krasnopolsky and Feldman [18] h From the lower atmosphere model of Yung and DeMore [19] at 100 km; value is almost certainly higher i Variable. Background value measured at Gales Crater by Webster et al. [20] j From Lodders and Fegley, Jr. [21] except as noted b
100
Mesopause Stratopause Tropopause
0
0
500
1000
1500 Temperature (K)
b Log p (μbar) –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8
Pluto
Titan
Jupiter
Saturn
Mars Triton Venus
Neptune
Earth Thermosphere
Uranus
Mesopause Mesosphere
Earth
Stratopause Stratosphere Tropopause Troposphere
Saturn 0
200
Jupiter 400
Venus 600
800 1000 Temperature (K)
Fig. 88.1 (a) Vertical distribution of temperature in the terrestrial atmosphere. The altitudes of the tropopause, stratopause, and mesopause are indicated. The thermospheric temperatures depend on solar activity, and profiles are shown for four values of the F10:7 index, from 75 (low solar activity) to 250 (high solar activity). The solid and dashed curves are for noon and midnight, respectively. After the MSIS model of Hedin [9]. (b) Vertical distribution of neutral temperatures of planets and satellites plotted on a common pressure scale. The regions of the atmospheres that are analogous to the terrestrial atmosphere are shown. From Strobel [10]
compared with the adiabatic lapse rate of 8:9 K km1 , from the surface to about 50 km. The region from 50 to 60 km contains the major cloud layer, and the tropopause is usually considered to be at about 60 km.In the mesosphere, between about 60 and 85 km, T decreases slowly from about 250 to 180 K
and is nearly constant from 85 km to the mesopause at 100 km. The daytime exospheric temperature is only weakly dependent on solar activity, varying from about 230 to 300 K from low to high solar activity. The slow retrograde rotation of the planet, which results in a period of darkness that lasts 58 Earth days, leads to the relative isolation of the nightside thermosphere, where T is found to decrease above the mesopause to a constant value of T1 100 K. Because of this, the nightside Venus thermosphere has been called the cryosphere. The compositions of the lower atmospheres of Mars and Venus are given in Table 88.3. The giant planets, Jupiter, Saturn, Uranus, and Neptune do not have solid surfaces, so their atmospheric regions are defined either in terms of pressure, altitude above the cloud tops, or more commonly, altitude above the 1 bar level. The temperature structures of all but Uranus are influenced by internal heat sources that the terrestrial planets do not possess. The temperature structures near the tropopause can be determined from IR observations and radio occultation data, and at thermospheric altitudes from ultraviolet solar and stellar occultations performed by various spacecraft. Below 300 mbar on Jupiter, the lapse rate is close to adiabatic (1:9 K km1 ); T at 1 bar is about 165 K, and the tropopause occurs near 140 mbar, where T 110 K. At 1 mbar, T again reaches 160170 K. Temperature inversions
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Table 88.4 Composition of the lower atmospheres of Jupiter and Saturn Species H2 He CH4 NH3 H2 O C2 H6 PH3 C2 H2 C2 H4 20 Ne 36 Ar 84 Kr 132 Xe
Mixing ratio Jupiter 0:864b 0:136b 0:00181b 0:00061b 520 ppmb 5:8 ppma 1:1 ppma 0:11 ppma 7 ppbb 21 ppmc 16 ppmc 8 ppbc 8 ppbc
Table 88.5 Composition of the lower atmospheres of Uranus and Neptunea Species
Saturn 0:881b 0:119b 0:0047d 0:00016c 220 ppbb 7:0 ppmb 4:5 ppmb 0:3 ppmb 0:2 ppbb
H2 He CH4 HD CH3 D C2 H6 C2 H2 CO NH3 a
Neptune 0:80 0:19 0:010:02 192 ppm 12 ppm 1:5 ppm 60 ppb 0:62:2 ppm < 600 ppb
After Lodders and Fegley, Jr. [21]
Table 88.6 Composition of the lower atmosphere of Titana Species N2 CH4 H2 CO 40 Ar C2 H6 C3 H8 C2 H2 C2 H4 C4 H2 HCN H2 O
a
After Strobel [23] After Lodders and Fegley, Jr. [21] c After Niemann et al. [24] d After Fletcher et al. [25] b
have been reported in the stratosphere and are probably due to absorption of solar radiation by dust or aerosols. Temperatures derived from the Voyager UV stellar and solar occultations show that T increases from about 200 K near 1 µbar to an exospheric value of about 1000 K [22]. For P > 500 mbar on Saturn, the lapse rate approaches the adiabatic value of 0:9 K km1 , and the tropopause, near the 100 mbar level, is characterized by T 80 K. Above the tropopause, the temperature increases to about 140 K near a P 1 mbar, and above this altitude, the temperature is essentially constant. Above 1000 km, T increases to a T1 400500 K, e.g. [22]; T1 appears to increase with latitude (Vervack, Jr. and Moses [5]). The mixing ratios of the species in the lower atmospheres of Jupiter and Saturn are given in Table 88.4. The tropopauses on both Uranus and Neptune occur near 100 mbar, where T 50 K. The lapse rates in the troposphere are 0:7 and 0:85 K km1 for Uranus and Neptune, respectively. The temperatures in the Uranus thermosphere range from 500 K near 107 bar (about 1000 km above the 1 bar level) to an exospheric value of about 700800 K. In the range 300 to 600 km on Neptune, T is characterized by a nearly constant value of 160180 K. Above 600 km, where P 1 µbar, T increases again to a value that is probably about 700 K [26, 27]. The compositions of the lower atmospheres of Uranus and Neptune are given in Table 88.5. Titan, which is a satellite of Saturn, has an N2 =CH4 atmosphere of intermediate oxidation state. The mixing ratios of components of the lower atmosphere are given in Table 88.6. The surface P and T are 1:5 bar and 94 K, respectively; T decreases above the surface to about 70 K at the tropopause, which occurs at an altitude of 42 km and
Mixing ratio Uranus 0:825 0:152 0:023 148 ppm 8:3 ppm .10 ˙ 1/ ppb 10 ppb < 27 ppb < 100 ppb
Mixing ratiob 0:9510:984 0:049 9:6 104 47 ppm 43 ppm 13 ppm 0:6 ppm 3:7 ppm 0:110:22 ppm 1:184:2 ppb 0:15 ppm 0:4 ppbc
a
From measurements obtained by the Cassini spacecraft, including the Huygens probe, except as noted b From Lodders and Fegley, Jr. [21], except as noted c Value pertains to an altitude of 400 km. From the Infrared Space Observatory (ISO) measurements of Coustenis et al. [28]
a pressure of 128 mbar. A reanalysis of the Voyager 1 solar occultation experiment showed that, above the tropopause, the temperature increases to a peak value of about 176 K at an altitude of about 300 km. The temperature then decreases to a T1 of 153158 K [29]. The Huygens Atmospheric Structure Instrument (HASI) on the Cassini spacecraft measured temperature profiles that exhibited wave-like structure in the range 2001400 km (Fulchignoni et al. [30]). A review of the thermal structure and dynamics of the upper atmosphere of Titan has been presented by Yelle et al. [31]. Triton is a satellite of Neptune. It also has an N2 atmosphere with small amounts of methane, CO, H2 , H, N, C, and HCN. The mixing ratios at 10 km are given in Table 88.7. The surface P is about 19 µbar, and the surface temperature is about 37 K. Methane in the troposphere is in equilibrium with a surface methane frost at about 3850 K. The temperature decreases with a lapse rate of about 0:1 K km1 from the surface to the tropopause. The tropopause temperature is about 36 K, and occurs in the 812 km region. The middle
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Table 88.7 Composition of the atmosphere of Tritona
a b
Species N2 CO
Mixing ratio 0:99 ˙ 0:01 0.0016
CH4 H2 N N H
600 ppm 75 ppm 3:8 105 ppm 290 ppm 1 ppm
Comments Below 200 km At surface; uncertain by a factor of 3b At surfaceb From model From model 100 km; from model 30 km; from model
From [32]. Values are at 10 km, except as noted From Lellouch et al. [33]
Table 88.8 Number densities of neutral and ion species at the surface of Mercurya Species Nab Mgb Cab Hb Alb Fe Mn CaC K a b
Number density (cm3 ) (1 103 1 105 ) (550) (6:64) 70140 0:20:6 Detected Detected Detected Detected
Table 88.9 Composition of the lower atmosphere of Plutoa
Scale height (km) 75–110 300–2000 1000–2500 600–800 515–675 Scale height not measured Not measured Not measured Not measured
For references, see Killen et al. [35]; McClintock et al. [36] Variable spatially and temporally
atmosphere is isothermal with a temperature of about 52 K from 25 to 50 km, increasing to 78 K near 150 km [34]; T then rises to a constant maximum T1 in the range 80100 K above 450 km. Mercury does not have a troposphere, mesosphere, or stratosphere; the surface of the planet is the exobase. Nevertheless, several atomic species and ions have been identified previous to and during the recent MESSENGER mission. They are listed in Table 88.8. Among the possible sources of atmospheric species are interactions with the incoming solar wind, dust, and meteoroids. There are also a number of sources that act to release of atoms from the surface and include radiation pressure, thermal desorption, impact vaporization, photon-stimulated desorption, ion sputtering, and chemical sputtering (Killen et al. [35]; McClintock et al. [36]). Ions produced by photoionization of neutrals may be picked up by the solar wind and lost from the atmosphere. The dwarf planet Pluto forms a binary system with its satellite Charon. The radius of Pluto is 1190 km, and that of Charon is about 617 km. The atmosphere of Pluto is mostly N2 , with small amounts of methane, the C-2 hydrocarbons C2 H2 , C2 H4 , and C2 H6 , and haze (Table 88.9), which were detected by the solar occultations carried out by the New
Species N2 CH4 C2 H6 C2 H2 C2 H4 H2 O HCN CO
Mixing ratio 0.997 0.0031 20 ppm 5 ppm 6 ppm 100 ppbb 10–100 ppbc (515˙40) ppmd
a
From Young et al. [8] and Hinson et al. [37] at 100 km From the model of Strobel and Zhu [7] c Below 100 km, increases at higher altitudes. From Lellouch et al. [38] d At surface. From Lellouch et al. [38] b
Horizons spacecraft (e.g., Young et al. [8]). The appearance of haze is common to atmospheres in which methane is present, even in small abundance; it signals the presence of higher hydrocarbons, including benzene, and for Pluto, Titan, and Triton, large nitrogen-containing organic compounds. The mixing ratio of CO was measured as 515 ppm and is evenly mixed in the atmosphere of Pluto; HCN was detected by the interferometer on the ALMA (Atacama Large Millimeter Array). Below 100 km, a mixing ratio for HCN of 0:010:1 ppm was found, rising to 40 ppm at 800 km (Lellouch et al. [38]). H2 O with an abundance of 0:1 ppm at the surface was inferred in order to explain the observed temperature profile (Strobel and Zhu [7]). The surface pressure and temperature on Pluto are 11:5 µbar and 45 K, respectively. The thermal structure of the atmosphere is influenced by the large thermal escape flux at the top of the atmosphere, by adiabatic cooling, and by H2 O rotational cooling. The temperature profile shows a shallow (if any) troposphere, above which T increases to a maximum (stratopause) temperature of about 110 K at an altitude of 30 km, due to absorption of solar UV radiation. Above that altitude, T decreases to form a broad minimum of about 63 K, before rising to an exospheric temperature of 70 K. Charon has no apparent atmosphere.
88.2 Density Distributions of Neutral Species 88.2.1 The Continuity Equation The density distribution of a minor neutral species j in an atmosphere is determined by the continuity equation @nj C r ˚j D Pj Lj ; @t
(88.7)
where ˚j is the flux of species j , and Pj and Lj are the chemical production and loss rates, respectively. If only the
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vertical direction is considered, the divergence of the flux where fk is the mixing ratio of species k. The binary diffubecomes @˚j =@z, and ˚j D nj wj , where wj is the vertical sion coefficient can be expressed as velocity of the species, and nj is its number density. In one3kB T dimensional models, transport due to turbulence and other Dj k D ; (88.13) 16nt j k ˝j k macroscopic motions of air masses is often parametrized like molecular diffusion, using an eddy diffusion coefficient K where j k is the reduced mass in place of the molecular diffusion coefficient Dj . The total mj mk transport velocity wj is then the sum of the diffusion velocity ; (88.14) j k D wjD and the eddy diffusion velocity wjK mj C mk wj D wjD C wjK :
(88.8)
If there are no net flows of major constituents, wjD and wjK satisfy the equations .1 C ˛jT / dT 1 1 dnj D Dj C C nj dz Hj T dz 1 dnj 1 dT 1 wjK D K C C : nj dz Havg T dz wjD
and nt D nj C nk is the total number density. The collision integral ˝j k is given by ˝j k D
! ;
(88.9)
5=2 1 2 1=2 2kB T Z1 v 2 D 5 Q .v/v exp dv ; 2kB T
(88.15)
0
(88.10) where v is the relative velocity of the particles, QD .v/ is the diffusion or momentum transfer cross section
In these expressions, ˛jT is the thermal diffusion factor (the ratio of the thermal diffusion coefficient to the molecular diffusion coefficient), and the pressure scale height Havg for a mixed atmosphere is given by Eq. (88.5) with m D mavg , the average molecular mass. For a stationary atmosphere, if molecular diffusion greatly exceeds eddy diffusion, and if photochemistry can be neglected, then wjD D 0. The resulting number density distribution is called diffusive equilibrium, and is given by
Z D
jelk .; v/.1 cos / sin d ;
Q .v/ D 2
(88.16)
0
and jelk .; v/ is the differential cross section for elastic scattering of species j and k through angle in the centerof-mass frame. In practice, Dj k is often expressed as bj k =nt , where nt is the total number density, and bj k is the binary collision parameter, which is usually given in tabulations in the semiempirical form b D AT s . Here, A and s (0:5 . s . 1:0) 1C˛jT Zz 0 are parameters that are fitted to the data. The binary collision dz T0 : (88.11) parameter appears, for example, in the expression for the difexp nj .z/ D nj .z0 / T Hj fusion limited flux of a light species to the “exosphere” of z0 a planet (e.g., Hunten [40]) (Sect. 88.7). K When mixing processes dominate, and wj D 0, the distribution is given by Eq. (88.6), with m D mavg in the definition 88.3 Interaction of Solar Radiation of the scale height H (Eq. (88.5)).
with the Atmosphere 88.2.2
88.3.1 Introduction
Diffusion Coefficients
The source for all atmospheric processes is ultimately the interaction of solar radiation, either photons or particles, with atmospheric gases. Since visible photons arise from the photosphere of the sun, which is characterized by T 6000 K, the solar spectrum in the visible and IR is similar to that of a black body at 6000 K. At longer (radio) and shorter (UV and X-ray) wavelengths, the photons arise from parts of the chromosphere and corona where the temperatures are higher (104 106 K). Thus, the photon fluxes differ substan(88.12) tially from those that would be predicted for a 6000 K black body. Photons in the extreme and far UV regions of the spec-
In the thermosphere of a planet, above the homopause, the major transport mechanism is diffusion or transport by random molecular motions. The characteristic time D for molecular diffusion is approximately Hj2 =Dj . The diffusion coefficient for a species j in a multicomponent mixture is usually taken as a weighted mean of inverse binary diffusion coefficients Dj k X fk 1 D ; Dj Dj k k¤j
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auroras are usually found in an oval around the magnetic poles, where the dipole field lines enter the atmosphere. On 107 Venus and Mars, diffuse and variable auroras are observed 106 on the nightside. Since Mars has a remanent crustal mag5 netic field, discrete auroras are also seen on the nightside. On 10 Earth, low-latitude auroras, which arise from heavy particle 104 precipitation, have also been observed. The primary particles 1000 that are responsible for the Jovian aurora may be electrons, protons, or heavy ions originating from its satellite Io. The 100 latter source produces a well-defined “footprint” near the au10 roral oval. Due to charge transfer, heavy particles spend part 1 of their lifetime as neutral species, and their paths may then 0 1000 2000 3000 diverge from magnetic field lines. In any case, a large fracWavelength (Å) tion of the effects of auroral precipitation is due to secondary electrons, regardless of the identity of the primary particles. Fig. 88.2 Solar fluxes in units of 106 photons cm2 s1 at 1 AU for 13000 Å at a resolution of 1 Å. From Eparvier [41]. See also Cham- In addition to producing emissions of atmospheric species in the visible UV and IR portions of the spectrum, auroral partiberlin et al. [39] cles ionize and dissociate atmospheric species and contribute trum are absorbed in the terrestrial thermosphere and X-rays to heating the neutrals, ions, and electrons. in the lower thermosphere and mesosphere. The solar Lyman ˛ line at 1216 Å penetrates through a window in the O2 absorption cross sections to about 75 km. Near-UV photons are 88.3.2 The Interaction of Solar Photons absorbed by ozone in the stratosphere, and visible radiation with Atmospheric Gases is not appreciably attenuated by the atmosphere. The wavelength ranges that are most important for aeron- The number flux of solar photons in a small wavelength inomy are the UV and X-ray regions. A solar spectrum in the terval around at an altitude z can, for the most part, be UV and X-ray regions at moderate solar activity (F10 D 120) computed from the Beer–Lambert absorption law is presented in Fig. 88.2. The ratio of high solar activity fluxes to low solar activity fluxes is near unity at wavelengths (88.17) F .z/ D F1 expŒ.; z/ ; longward of 2000 Å but increases to factors that range between 2 and 3 over much of the extreme UV. At wavelengths where F1 is the solar photon flux outside the atmosphere, between about 100 and 550 Å, the ratio of high to low soand .; z/ is the optical depth which, in the plane parallel lar activity fluxes reaches values as high as 100. The fluxes approximation, is given by at X-ray wavelengths arise principally from solar flares and can increase by orders of magnitude from low to high solar 1 XZ activity. nj .z 0 /ja ./ sec dz 0 : (88.18) .; z/ D The sun also emits a stream of charged particles, the soj z lar wind, which flows radially outward in all directions, and consists mostly of protons, electrons, and alpha particles. Here, a ./ is the absorption cross section of species j at j The average number density of solar wind protons is about wavelength , and the solar zenith angle is the angle of the 3 1 5 cm , and the average speed is about 400450 km s at sun with respect to the local vertical. Earth orbit (1 AU). The interaction of these particles with the For greater than about 75ı , the variation of the solar magnetic field (either induced or intrinsic) of a planet, and zenith angle (SZA) along the path of the radiation cannot be ultimately with the atmosphere, is the source of auroral ac- neglected; the optical depth must be computed by numerical tivity. Terrestrial auroras arise mostly from precipitation of integration along this path in spherical geometry. For electrons with energies in the kilovolt range, although mea- 90ı , the optical depth is sured spectra vary widely. An example of a primary electron auroral spectrum is shown in Fig. 88.3. Terrestrial auroral 1 XZ emissions maximize in the midnight sector, but dayside cusp nj .z 0 /ja ./ .; z/ D auroras are produced by lower energy electrons, and diffuse j z proton auroras are also observed. ro C z 2 2 0:5 0 Since charged particles are constrained to move along sin dz : (88.19) 1 ro C z 0 magnetic field lines, for planets with intrinsic dipole fields, (106 photons cm –2 s –1 Å–1)
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Fig. 88.3 Sample of energy spectrum of precipitating auroral electrons and ions (protons) as measured by the Defense Meteorology Satellite Program (DMSP) satellite. The abscissa is log (base-10) of the energy in electron-volts, and the ordinates are log (base 10) of the energy fluxes of the electrons (top) and protons (bottom). Also shown are the universal time (UT) geographic latitude (LAT) and longitude (LON), magnetic latitude (MLAT) and longitude (MLT), the integrated energy flux (Electron JE) and ion flux (Ion JE), and average energy (Avg E) of the electrons and ions. From Paxton [42]
Differential directional energy flux spectrum UTC = 1996-243T01:48:30.000 Electron log energy flux (cm –2 sr–1 s –1) 8
7
6 Ion log 6 energy flux (cm –2 sr–1 s –1)
5
2 LAT: –80.37 LON: 1.31
For larger than 90ı , the optical depth is given by 1 X Z 2 nj .z 0 /ja ./ .; z/ D j
zs
ro C zs 2 2 ı 0:5 0 sin 90 dz 1 ro C z 0 Z1 nj .z 0 /ja ./ z
1
ro C z ro C z 0
0:5
2 2
sin
dz
0
;
(88.20) where zs is the tangent altitude, the point at which the solar zenith angle is 90ı for the path of solar radiation through the atmosphere. Of course, we need not write sin 90ı explicitly in Eq. (88.20), but its inclusion helps to elucidate the origin of the equation. In a one-species atmosphere, the rate of absorption of solar photons of wavelength is q a ./ D F a ./n :
(88.21)
For an isothermal atmosphere in which H.z/ is approximately constant, the absorption maximizes where .; z/ D 1. This is a fairly good approximation even for regions of the atmosphere where the H.z/ is not constant. The altitude of unit optical depth is shown for wavelengths from X-rays to
3 MLAT: –69.78 MLT: 23:29
4 Log channel energy (eV)
Electron JE: 7.59e+11 Avg E: 3.76e+03
Ion JE: 2.12e+09 Avg E: 1.94e+03
the near UV for overhead sun in the terrestrial atmosphere in Fig. 88.4a. Note the accidental window in the O2 photoabsorption cross sections at Lyman alpha (1216 Å), which was mentioned above. Similar plots for Venus, Mars, Jupiter, and Saturn are shown in Fig. 88.4b–e, respectively. N2 does not absorb longward of about 105 nm, so in the terrestrial atmosphere, O2 and O3 are the primary absorbers between 100 and 220 nm, while ozone dominates the absorption for wavelengths in the range 220320 nm. On Venus and Mars, CO2 is the main absorber of FUV and EUV radiation, although at wavelengths less than about 100 nm, N2 , CO, and O also contribute. On Jupiter and Saturn, H2 absorbs radiation in the 8421116 Å range in discrete transitions to rovibrational levels of excited states, which decay either to rovibrational levels or to the continuum of the ground state. Since the cross sections vary by several orders of magnitude, this leads to oscillations in the depths of penetration, especially in the wings of the lines and gaps between groups of lines. The photons are abruptly stopped near the methane homopause, when the hydrocarbon densities are large enough to absorb the photons. Similarly, in the thermospheres of Titan and Pluto, absorption by N2 in the range 8501020 Å is like that of H2 ; it undergoes discrete line absorption to predissociating states. Methane and hydrocarbon hazes are the primary absorbers of UV radiation between 1600 Å and the effective absorption threshold of N2 of about 12:14 eV (Strobel and Zhu [7]; Zhang et al. [45]). The threshold for photoabsorption of CH4 is effectively near 1600 Å, above which the cross sections are very small (e.g., Huebner et al. [46]). Higher hydrocarbons,
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b
Altitude (km) 300
Altitude (km) 600
110° 250
500 Venus τ=1
200
400
150
105°
300 100°
100
200
95° 90°
0
500
1000
1500 2000 Wavelength (Å)
60° 0
c
Altitude (km)
0
500
1000
1500 2000 Wavelength (Å)
d Altitude (km)
160
1000
Mars 7.5° SZA
800
140
600 120 400 100 200
80
0
500
1000
0
1500 2000 Wavelength (Å)
0
e Altitude for Tau = 1 (km)
500
1000 Log p (μbar)
4000 –7 –6
3000
–5 –4
2000
–3 –2 1000
0
–1 0 1 2 0
200
400
600
800
1000
1400 1200 Wavelength (Å)
1500 2000 Wavelength (Å)
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Fig. 88.4 The altitude where D 1 versus wavelength: (a) Earth, (b) Venus. The optical depths for various solar zenith angles (SZA) from 60ı to 110ı . The solar zenith angle is the angle that the line of sight to the sun makes with the vertical. (c) Mars, for near-subsolar conditions (SZA D 5ı –10ı ) and moderate solar activity, (d) Jupiter [43], (e) Saturn, wavelength range 11500 Å, high-resolution values for SZA D 27ı . The oscillations in the 8421116 Å range are characteristic of the high-resolution photoabsorption cross sections of H2 . In this region, the photons can be seen to penetrate to the region of hydrocarbon absorption, where they are abruptly absorbed. Soft X-rays with wavelengths in the range 40150 Å are absorbed in the altitude range of 7001000 km above the 1-bar level, from Kim et al. [44] J
however, such as C2 H2 , C2 H4 , and C2 H6 , which are formed order to model this effect, the differential (with respect to anchemically subsequent to photoabsorption of methane, ab- gle) cross sections for photoionization ji .; / are necessary. The differential cross section is sometimes expressed as sorb longward of 1600 Å. The interaction of UV photons with atmospheric gases i ji ./ h 1 produces ions and photoelectrons through photoionization, 1 ˇ./P2 .cos / ; ji .; / D (88.28) which may be represented as 4 2 X C h ! XC C e ; and photodissociative ionization AB C h ! AC C B C e :
(88.22) where is the angle between the incident photon beam and the ejected electron, P2 is a Legendre polynomial, and ˇ is an asymmetry parameter. (88.23)
88.3.3 Interaction of Energetic Electrons
In these equations, X represents any atmospheric species; with Atmospheric Gases A is either an atom or a molecular fragment, and AB is a molecule. The energy of the photoelectron in reaction Suprathermal electrons, which are here denoted e and Eq. (88.22) is given by include both energetic photoelectrons and auroral primary electrons, can also ionize species through electron-impact Epe D h IX Eex ; (88.24) ionization and in reaction Eq. (88.23) it is Epe D h Ed IA Eex ;
X C e ! XC C e C e0 (88.25)
(88.29)
and electron-impact dissociative ionization
where Ij is the ionization potential of species j , Ed is the AB C e ! AC C B C e C e0 : (88.30) dissociation energy of molecule AB, and Eex is the sum of the internal energy states of the products. Neutral fragments, which may be reactive radicals, are also produced in pho- In these reactions, e represents the energy degraded photoelectron or primary electron, and e0 the secondary electron. todissociation The energy of the secondary electron E e0 in an electronAB C h ! A C B : (88.26) impact ionization process Eq. (88.29) is given by The rate of ionization of a species j by a photon of wavelength at an altitude z is given by
E e0 D Ee IX Eex Ee ;
(88.31)
where Ee is the energy of the primary or photoelectron, (88.27) Ee is the energy of the degraded primary or photoelectron, and Eex is the internal excitation energy of the product ion where ji ./ is the photoionization cross section. The rate for or neutral fragments. For the dissociative ionization process photodissociation is given by a similar expression, with the Eq. (88.30) the dissociation energy of the molecule must also photoionization cross section replaced by the photodissocia- be subtracted as well. tion cross section. The expression above must be integrated Energetic electrons can also dissociate atmospheric over the solar spectrum to give the total rate. In addition, it is species. In this process, often necessary to take into account ionization and/or dissociation to different final internal states of the products, so the (88.32) AB C e ! A C B C e ; partial cross sections or yields are needed. In the atmospheres of magnetic planets, photoelectrons the energy of the degraded electron is may travel upward along the magnetic field lines to the con(88.33) Ee D Ee DAB Eex ; jugate point, where the field line re-enters the atmosphere. In qji .; z/ D F .z/ji ./nj .z/ ;
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where DAB is the dissociation energy of molecule AB. Col- .E Ij /=2, is not serious, although for high-energy auroral lisions with suprathermal electrons can also promote species electrons, a larger upper limit may be required. to excited electronic, vibrational, or rotational states An estimate of the number of ionizations in a gas produced by a primary electron with energy Ep is Ep =Wip , (88.34) where Wip is the energy loss per ion pair produced, which AB C e ! AB C e ; approaches a constant value as the energy of the electron where the dagger denotes internal excitation. The energy lost increases. This constant is also found to increase as a funcby the electron is, thus, the excitation energy of the species. tion of fractional ionization because of the larger amount of In determining the rate of ionization, dissociation, and energy lost to thermal electrons as the fractional ionization excitation by photoelectrons, the local energy loss approxi- increases. Empirical values are available for Wip for many mation, that is, the assumption that the electrons lose their gases, and usually fall in the range 3040 eV [47]. The total loss function or stopping cross section for an energy at the same altitude where they are produced, is fairly good near the altitude of peak photoelectron production. The electron with incident energy E in a gas j is given by the mean free path of an electron near 150 km on Earth is about expression 30 m. Substantially above the altitude of peak production X jk .E/Wjk Lj .E/ D of photoelectrons, transport of electrons from below is imk portant, and use of the local energy loss assumption causes .EI the excitation, ionization, and dissociation rates to be unZ j /=2 dji .E/ derestimated. For keV auroral electrons, the computation of C .Ij C Ws / dWs ; (88.37) dWs the energy deposition of the electrons must consider their 0 transport through the atmosphere. Thus, the elastic total and differential cross sections for electrons colliding with neu- where W k is the energy loss associated with excitation of j tral species must be employed, as well as the inelastic cross species j to excited state k. The differential cross section is sections, and the angles through which the electrons scatter usually adopted from an empirical formula that is normalized must be taken into account. so that In general, the excitation rate qjk .z/ of a species j to an .EI excited level k with a threshold energy Ek at an altitude z by Z j /=2 i dj .E/ electron impact is given by dWs ; (88.38) ji .E/ D dWs 0 Z1 dF .z; E/ k dE ; (88.35) where i .E/ is the total ionization cross section at primary qj .z/ D nj .z/ jk .E/ j dE Ek electron energy E. One formula in common use is that employed by Opal et al. [48] to fit to their data where jk .E/ is the excitation cross section at electron energy E, and dF .z; E/=dE is the differential flux of electrons (bedji .E/ A.E/ D (88.39) i
2:1 ; tween energies E and E C dE). The ionization rate qj .z/ of dWs 1 C Ws =W a species with ionization potential Ij due to electron impact is given by where A.E/ is a normalization factor, and W is an empirically determined constant, which has been found to be equal Z1 .EI Z j /=2 i d .E/ to within a factor of about 50% to the ionization potential for dF .z; E/ j qji .z/ D nj .z/ dWs dE ; a number of species. dWs dE Ij 0 For energy loss due to elastic scattering by thermal elec(88.36) trons, an analytic form of the loss function such as that proposed by Swartz et al. [49] may be used i where dj .E/=dWs is the differential cross section for pro 2:36 duction of a secondary electron with energy Ws by a primary E kB Te 3:37 1012 ; (88.40) Le .E/ D electron with energy E. The integral over secondary enE 0:94 n0:03 E 0:53kB Te e ergies Ws terminates at .E Ij /=2 because the secondary electron is by convention considered to be the one with the where Te is the electron temperature and ne is the number smaller energy. Since the average energy of photoelectrons density of ambient thermal electrons. For high-energy auroral electrons, the rate of energy loss is less than 20 eV, the error incurred in cutting off the integrals in Eqs. (88.35) and (88.36) at 200 eV or so, rather than per electron per unit distance over the path s of the electrons
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Since the energy bins should be smaller than the typical in the atmosphere can be estimated using the continuous energy loss in order to obtain accurate rates for the excitation slowing down approximation (CSDA) as processes, it is often convenient to treat rotational excitaX dE tion also as a continuous process, with a pseudo-collision nj .z/Lj .E/ sec C ne .z/Le .E/ sec ; D ds j frequency similar to that for elastic scattering from ambient (88.41) electrons Eq. (88.43) with where is the angle between the path of the primary electron s and the local vertical. In the CSDA, all the electrons of a given energy are assumed to lose their energy continuously and at the same rate. The rate of energy loss (dE=ds) is integrated numerically over the path of the electron, which degrades in energy until it is thermalized. In this approximation, inelastic processes are assumed to always scatter the electrons forward, so cross sections that are differential in angle are not required. Because electrons actually lose energy at different rates, however, and because elastic and inelastic scattering processes do change the direction of the electrons, the CSDA gives an estimate for the rates of electron energy loss processes that is increasingly inaccurate as the energy of the electron decreases. In practice, discrete energy loss of electrons can be easily treated numerically if the local energy loss approximation is valid. The spectrum of electrons must be divided into energy bins that are smaller than the energy losses for the processes, and the integrals in Eqs. (88.35) and (88.36) are replaced by sums over energy bins. Since elastic scattering of electrons by neutrals changes mostly the direction of the incident electron, and not its energy, only inelastic processes need be considered. In order to compute excitation and dissociation rates, only integral cross sections are required; the scattering angle is unimportant. For ionization, of course, the energy distribution of the secondary electrons must be considered but not the scattering angles of either the primary or secondary electrons. Below the lowest thresholds for excitations, energetic electrons lose their energy in elastic collisions with thermal electrons. The process of energy loss to thermal electrons is often approximated as continuous, rather than discrete. The collision frequency jk for a discrete electron-impact excitation process k of a species j is given by
dE D ve .E/nj Ljrot .E/ ; dt
(88.45)
where the loss function for rotational excitation is given by X X J;J 0 0 jJ j .E/WjJ;J : (88.46) Ljrot .E/ D J
J0
In this expression, jJ is the fraction of molecules j in the 0 rotational level J , and jJ;J is the measured or computed cross section for electron-impact excitation of species j from 0 rotational state J to rotational state J 0 , and WjJ;J is the associated energy loss. The slowing down of high energy auroral primary or the concomitant high energy secondary electrons arises from both elastic and inelastic scattering processes and cannot be treated using the local energy loss approximation. In solving the equations for electron transport, the angle through which the primary electron is scattered, as well as the change in energy of the primary electron and the production of any secondaries, must be taken into account. Thus, differential cross sections for the elastic and inelastic scattering of electrons by neutral species are required. The detailed equations for electron transport have been presented by, for example, Rees [50]. Several methods for approximating the energy deposition of auroral electrons are currently in use. The CSDA has already been discussed, but it provides only a rough approximation to the depth of penetration of the electrons, and the rates of excitation, ion production, and other energy loss processes. In the two-stream approximation, the electrons are assumed to be scattered in either the forward or the backward direction [51]. Implementation of this method requires only the backscattering probabilities, rather than complete differential cross sections. The method has been generalized to multistream models, in which the solid angle range of the k k (88.42) electrons is divided into 20 or more intervals, so more or j .E/ D nj .z/ve .E/j .E/ : less complete differential cross sections are required [52, 53]. For energy loss due to elastic scattering from thermal elec- Monte Carlo methods have also been used to model auroral trons, a pseudo-collision frequency e may be defined as electron precipitation [54]. 1 dE ; (88.43) e .E/ D E dt 88.4 Ionospheres where E is the grid spacing in the calculation, and the energy loss rate is
dE D ve .E/ne Le .E/ ; dt
where Le is taken from, for example, Eq. (88.40).
88.4.1 Ionospheric Regions
(88.44) The division of the dayside ionosphere into regions is based on the structure of the terrestrial ionosphere, which consists of overlapping layers of ions. These layers are the result of
88
1312
J. L. Fox Altitude (km)
log electron flux (cm –2 s –1 eV–1 sr–1) 10
Topside ionosphere
600
Earth
172km 8
500 6 100 km X+
400
4
300
F2 XY
200
100
102
+
Cosmic rays X-rays < 10 Å
EUV
Ly α 1216 Å
E
XY –
D
103
2
F1
Bottomside ionosphere
UV Ly β 1026 Å X-rays (10–100 Å) 104
105 106 Electron density (cm–3)
Fig. 88.5 Ionospheric regions and primary ionization sources. After Bauer [55]
changes both in the composition of the thermosphere and in the sources of the ionization, and are shown schematically in Fig. 88.5. The major molecular ion layer is the F1 layer, which is produced by absorption of EUV (1001000 Å) photons by the major thermospheric species, and occurs where the ion production maximizes. The E layer is below the F1 layer and is produced by shorter and longer wavelength photons that are absorbed deeper in the atmosphere: soft X-rays and Lyman ˇ, which can ionize O2 and NO (Fig. 88.4a). In the D region, the densities of negative ions become appreciable, and large densities of positive cluster ions appear. These ions are produced by harder X-rays, with . 10 Å, and Lyman ˛, which penetrates to about 75 km, where it ionizes NO. The highest altitude peak in the terrestrial ionosphere is the F2 peak, which occurs near or slightly below 300 km, where the major ion is OC . In the ionospheres of the outer planets, an F2 peak is formed from HC . The F1 region is formed from HC 3 , and near or below that layer by hydrocarbon ions, as shown for Saturn in Fig. 88.10e. The peak density in the F2 region occurs where the chemical lifetime of the (atomic) ion is comparable to the characteristic time for transport by diffusion ( H 2 =D).
88.4.2 Sources of Ionization As discussed in Sect. 88.3, ionization can be produced either by solar photons and photoelectrons during the daytime or
0
20
40
60
80
100 120 140 160 180 200 Energy (eV)
Fig. 88.6 Primary photoelectron spectrum for the terrestrial atmosphere at 172 km (near the F1 peak) and at 100 km. The spectrum at 100 km is significantly harder than that at 172 km
by energetic particles and secondary electrons during auroral events. Photoelectrons have sufficient energy to carry out further ionization if they are produced by photons with . 500 Å. These photons penetrate further and exhibit larger solar activity variations than longer-wavelength ionizing photons. Thus, the ionization rate due to photoelectrons peaks below the main photoionization peak. Primary flux spectra of photoelectrons produced near the F1 peak (172 km) and below the ion peak (100 km) are shown in Fig. 88.6. The primary spectrum at the ion peak consists mostly of lower-energy electrons, whereas at 100 km, the lower-energy electrons are depleted, and there are relatively larger fluxes of electrons with E & 50 eV. Figure 88.7 shows the primary and steady-state photoelectron spectra near the ion peaks on Venus and Titan. The major ions produced in the ionospheres of the Earth and planets are usually those from the major thermospheric C C C C C C species: NC 2 , O2 , and O on Earth; CO2 , O , N2 , and CO C C C C on Venus and Mars; H2 , H , CH4 , and He on the outer planC C ets; NC 2 , N , and CH4 (and higher hydrocarbon ions) in the C C C ionospheres of Titan and Pluto; and NC 2 , N , CH4 , and C in the ionosphere of Triton. In the presence of sufficient neutral densities, however, ion–molecule reactions transform ions whose parent neutrals have high ionization potentials to ions whose parent neutrals have low ionization potentials. This is a rigorous rule only for charge transfer reactions, but it applies more often than not in other ion–molecule reactions as well. Because of transformations by ion–molecule reactions, the major ions in the F1 regions of the dayside ionospheres C of Earth, Venus, and Mars are OC 2 and NO , despite the large differences in composition between the thermosphere of the Earth and the thermospheres of Venus and Mars. A diagram illustrating the ion chemistry in the ionospheres of the terrestrial planets is shown in Fig. 88.8. The vertical positions of the ions in this figure represent the relative ionization po-
88
Aeronomy
1313 Table 88.10 Ionization potentials (IP ) of common atmospheric species (computed with data taken from [56], except as noted)
a log electron flux (cm–2 s –1 eV–1 sr–1) Venus 140 km
10
High IP Species He Ne N2 H2 N CO CO2 O H HCN OH
Initial
8
6 Steady-state 4
2
0
50
100
150
200 Energy (eV)
b log electron flux (cm–2 s –1 eV–1 sr–1) 8 Titan 1000 km 6
4
0
Steady-state
0
50
Medium IP Species IP (eV) H2 O 12:612 CH4 12:51 O2 12:07 C2 H6 11:52 C2 H2 11:40 C 11:26 C3 H8 10:95 CH 10:64 C2 H4 10:51 CH2 10:40 S 10:35
Ionized by Ly ˛ Species IP (eV) C4 H2 10:18 CH3 9:73 NO 9:264 C2 H5 8:13 HCO 8:10 C3 H7 8:09 Mg 7:65 trans-HCNH 7:0a cis-HCNH 6:8a Ca 6:11 Na 5:14
From [57]
Many ion–molecule reactions proceed at or near gas kinetic (or collision) rates. The interaction of an ion with a nonpolar molecule is dominated by the ion–induced dipole interaction, for which the interaction potential is 12 ˛d q 2 =r 4 , where ˛d is the polarizability of the neutral, q is the charge on the ion, and r is the distance between the particles. The Langevin–Gioumousis–Stevenson (LGS) or merely the Langevin rate coefficient, kL , is then given by
Initial
2
a
IP (eV) 24:59 21:56 15:58 15:43 14:53 14:01 13:76 13:62 13:60 13:60 13:00
100
150
200 Energy (eV)
Fig. 88.7 Computed primary and steady-state spectra for photoelectrons near the F1 peak on Venus at 1 eV resolution (a), and Titan at 0:5 eV resolution (b). For Titan, no precipitation of electrons from Saturn’s magnetosphere is assumed. The steady-state spectra are averaged over three intervals in both plots
kL D 2 q
˛d
1=2 ;
(88.47)
where is the reduced mass of the two species. For a singly charged ion, with ˛d in Å3 and in atomic mass units, this formula reduces to 2:34 109 .˛d =/1=2 . The rate coefficient for an ion with a polar molecule is " 1=2 # tentials of the parent neutrals. In regions where there are 2 2 q 1=2 ; (88.48) kd D 1=2 ˛d C cd sufficient neutral densities the ionization flows downward. kB T Table 88.10 shows ionization potentials (IP ) for several major and minor species present in planetary thermospheres. where d is the dipole moment, and c is a constant that is Major atmospheric species generally have IP & 1216 eV unity in the locked dipole approximation and is about 0:1 in ( < 7751000 Å). Only a few species can be ionized by the average dipole orientation (ADO) theory. The value of the strong solar Lyman alpha line (1216 Å, 10:2 eV), includ- c has been shown to be a function of D =˛ 1=2 and can be ing NO, and a few hydrocarbon radicals and stable species, determined either experimentally or graphically, as proposed such as CH3 , C2 H5 , C4 H2 , and benzene. Metal atoms, which by Su and Bowers [58]. Proton transfer reactions in particare produced in the lower thermospheres and mesospheres ular have been shown to proceed at rates that are close to of planets from ablation of meteors, have very low ionization the Langevin (or capture) rates. Theories for ion–quadrupole potentials, and some can be ionized by photons with wave- interactions have also been developed, and the resulting forlengths longer than 2000 Å. mulas can be found in, for example, the review by Su and In ionospheres where hydrogen and sufficient neutral den- Bowers [58]. Measured rate coefficients for ion–molecule resities are present, ionization flows from protonated species actions have been compiled by Anicich et al. [59], Ikezoe whose parent neutrals have small proton affinities to species et al. [60], and Anicich [61]. formed by protonation of neutrals that have large proton Loss of ionization in planetary atmospheres proceeds affinities. There are no in situ measurements of the ion com- mainly by dissociative recombination of molecular ions, position of the outer planets, but models predict that HC 3 and which may be represented by hydrocarbon ions dominate the lower ionospheres. In regions (88.49) where meteor ablation occurs, metal ions may also be found. ABC C e ! A C B :
88
1314
J. L. Fox
He
N2
e,hv
e,hv
e,hv
e,hv
e,hv e,hv e,hv
O2 N2
e,hv
N2 NO
N +2 15.6
N
CO
CO, CO2
CO2
CO+ 14.01
O
O
N(2D)
CO2
O2, NO
CO2
CO2
e,hv
e,hv
e,hv
e,hv
e,hv
O2,CO2
O+(2D) 16.0
CO2
N O C O +2 13.76
O
H O+(4S) 13.62
O
H+ 13.60
O2
O,O2
O2 O2
CO,C O2
NO
O +2 12.1
CO2,O2 C
C+ 11.26
+
O2
NO
O,O2
N+ 14.54
N2, N, NO
CO2
e,hv
H
O
CO2
CO2,O2
He+ 24.5
CO
C NO
NO 9.76
NO
N2, NO
NO
N(2D) N, NO, N(2D)
Fig. 88.8 Ion chemistry in the ionospheres of the terrestrial planets. The numbers under the names of the ions indicate the ionization potentials of the parent neutral. In the presence of sufficient neutral densities, the ionization flows downward, and the importance of dissociative recombination for the molecular ions increases as the ionization potentials of the parent neutrals decrease
Dissociative recombination coefficients are characteristically large, about 107 cm3 s1 , at the electron temperatures Te typical of planetary ionospheres, which are usually within a factor of a few of the neutral T 150800 K near the molecular ion density peak. Daytime peak electron densities tend to be in the range 104 106 cm3 ; fractional ionizations are small, about 105 , near the F1 peak. The relative importance of ion–molecule reactions and dissociative recombination in the destruction of a particular ion is determined by the relative densities of electrons and neutrals with which the ions can react. In general, molecular ions whose parent neutrals have high IP are transformed by ion–molecule reactions preferentially to loss by dissociative recombination, and their peak densities occur higher in the atmosphere. Ions for which dissociative recombination is an important loss mechanism near the ion peaks of the terrestrial planets include NOC and OC 2 , and in the atmospheres of and hydrocarbon ions. For ions whose the outer planets, HC 3 C parent neutrals have very high IP , such as NC 2 and H2 , dissociative recombination is rarely important as a loss process, except at very high altitudes. They may, however, be im-
portant as a source of vibrationally or electronically excited fragments or hot atoms. Atomic ions may be destroyed by radiative recombination XC C e ! X C h ;
(88.50)
but the rate coefficients are small, less than 1012 cm3 s1 at the typical Te of planetary ionospheric peaks [62], and decrease with increasing electron temperature. Atomic ions may dominate at high altitudes, where neutral densities are low, but in such regions, loss by diffusion may be more important than chemical recombination. The ions diffuse downward to altitudes where the neutral densities are higher and are then destroyed in ion–molecule reactions. The major ions in the topside ionospheres of the planets tend to be atomic ions: OC in the ionospheres of Earth and Venus and HC in the ionospheres of the outer planets. On Mars, however, the OC peak density appears not to exceed that of OC 2 even at high altitudes. Model thermospheres for Earth and selected planets and satellites are shown in Fig. 88.9a–j. Measured or computed
88
Aeronomy
1315
a Altitude (km)
b Altitude (km)
1000
400 F10.7 = 240
Earth F10.7 = 75
800 N
He
He
H
O2
600 O2 H
O
O N
H
200
300
O
Ar
400 Ar
N2
N2 0
0
Venus
He
5
10 15 20 0 Log densities (cm –3)
5
CO
10 15 20 200 Log densities (cm –3)
NO
C O2
H2
c Altitude (km)
N
CO2
Mars
H
Ar
H2 600
103
104
105
106
107
108
109 1010 1011 1012 1013 Density (cm –3)
d Altitude (km) He
200
3000
400
400
2500
600
800
H
2000
T O
CO NO
1500
Ar N
C
100 1000 10
10
5
10
6
He
1000 CO2
O2 4
Temperature (K) 1000
H2
N2
200
N2
100
10
7
500 10
8
10
9
10
10
11
10
12
13
10 10 10 Density (cm –3)
e Altitude (km)
88
CH4
14
0 10 –15
Log p (μbar)
10 –5
10 –10
10 0
105
1015 1010 Density (cm –3)
f Altitude (km)
Temperature (K) 0 200 400 600 800 1000
–6
3000
Uranus 2500
–5
2500 Tn
H
–4
H
He
2000
2000
Te
H2
1500 H2
–3
He
1500 H 2O
–2 –1
500
C 2 H4
C2 H 2
CH3 104
1000
CH4
1000
106
108
1010
C2 H 6 1012
0 1 2 1014 1016 Density (cm –3)
500 C2H 2
C2 H 6 CH4
C3H8 5
10
15 Log densities (cm –3)
1316
J. L. Fox
h Geopotential height (km)
g Altitude (km)
Altitude (km) 1600 Ingress Egress 1400 N2 N2 1200 CH4 CH4 1000 C2 H 2 C2 H 2 C 2 H4 C 2 H4 800 C2 H 6 C2 H 6 Haze Haze 600
2500 600 2000
Neptune
H
500 H2
1500
400
He 400
300 1000 200 CH4
500
100
C3H8 C2 H 2
200
C2 H 6 0
5
10
i Radial distance (km) 4200
0
20
40 60 ϕ(CH4)
4000 3800
3400
100 120 140 160 180 200
1014 Density (cm –3)
N2
1100
700 600
Ar
500 400
ϕ(H 2)
300
Counts per second 104
200 100 0 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 Number density (m –3) Altitude (km) 500
103 2
Pressure (μbar)
C C(kCNN = 0)
400
10 –4
10 101
N
300
10 –1 104
10 –3 H
200
103
10 –2
102 CH4
100
10
H2
1
10
10 –1 60
0
800 Temperature
2800 l(CH4)/10 ϕ(CH4) p(H 2)/10 2600 105 106 107 108 109 1010 1011 1012 1013 1014 1015 1016 1017 1018 10191020 Number density (cm –3), flux (cm –2 s –1)
10
1012
900
3200 3000
1010
1000
CH4
3600
108
j Altitude (km) 80
ϕ(H 2) H2
106
15 Log densities (cm ) –3
70
80
90 100 Mass-to-charge ratio
0
0
2
4
6
10 –1
10 0 101 8 10 12 Density log10 (cm –3)
88
Aeronomy
1317
Fig. 88.9 Model thermospheres for the Earth, planets, and satellites. The curves are number density profiles and are labeled by the species they represent. (a) Earth, based on the NRLMSIS model of [9] for a latitude of 0ı and a local time of noon for low (left) and high (right) solar activities. (b) High solar activity model of Venus, based on the model of [2] for 15ı N latitude 15:00 LT. (c) Moderate (F10:7 D 120) solar activity model of Mars, based on MAVEN NGIMS (Neutral Gas and Ion Mass Spectrometer) measurements that are averaged over 5 km bins from March, April, and May 2015 for the SZA range of 5ı –10ı . The CO2 , O, N2 , CO, Ar, and He profiles (solid curves) are fitted to data. The N, C, and NO profiles (dashed curves) are computed self-consistently from a model, as are the H and H2 profiles. (d) Jupiter, basic JGITM (Jupiter Global Ionosphere-Thermosphere Model) after Egert et al. [63]. (e) Saturn, after Kim et al. [44] from the model from Vervack and Moses [5]. The neutral and electron temperatures are also shown, as well as C-1 and C-2 hydrocarbons and H2 O. (f) Uranus, plotted with model data from Moses and Poppe [64]. (g) Neptune, plotted with model data from Moses and Poppe [64]. (h) Pluto. Densities of species derived from line-of-sight integrated densities obtained from solar occultations of the New Horizons spacecraft as a function of geopotential height and altitude. The species are labeled by the color-coded names in the upper right corner. The circles are from data, and the dashed curves, as well as the dotted curves for N2 and CH4 are from a model. The N2 densities at altitudes below about 110 km are from the radio occultation profiles carried out by the New Horizons REX instrument. Also shown (in purple) are the extinction coefficients multiplied by 1015 of the hazes. Both ingress and egress values are shown. From Young et al. [8]. (i) (top) Titan model, after Strobel [65]. The measured densities of N2 , Ar, CH4 , and H2 are shown with dashed curves for radial distances above about 3500 km. (The radius of the surface is 2575 km.) The solid curves of these species’ densities are from a model. Also shown are the upward fluxes of CH4 and upward and downward fluxes of H2 , denoted by (CH4 ), and (H2 ), respectively, and the adopted temperature profile. The loss rates of methane and production rates of H2 are also shown in units of cm3 s1 divided by 10. (bottom) The diversity of neutral species with mass/charge ratios of 60–99, in counts s1 found in the atmosphere of Titan by the Ion and Neutral Mass Spectrometer (INMS) instrument on the Cassini spacecraft. The range of mass/charge is adapted for investigations of benzene and its derivatives. Masses less than 60 amu are not shown but are also present due to the presence of higher hydrocarbons that are ultimately formed from photolysis of methane. The points are the averaged values over the 960980 km range, and the solid line is from a model. From Vuitton et al. [66]. (j) (top) Triton total neutral densities (largely N2 ) as a function of altitude, after Strobel and Zhu [7]. The three curves represent models with different neutral temperature profiles. The red (solid), black (long dashes), and brown (short dashes) curves are those for exospheric temperatures of 97, 90, and 75 K, respectively. Note that the units of number density are SI units of m3 rather than the traditional cm3 . (bottom) Density profiles of minor species on Triton, including H2 , N, H, and CH4 . The two profiles that represent C densities are for slight differences in the chemistry J
ion density profiles for the Earth and selected planets and ionosphere is sunlit to about 105ı SZA. The nightside ionosatellites are shown in Fig. 88.10a–j. sphere has been probed by radio occultation experiments on the Mariner 9 orbiter, the Viking spacecraft [76], and by the Mars Global Surveyor (MGS) spacecraft radio sci88.4.3 Nightside Ionospheres ences (RS) experiment [77]. Until recently, there was no composition information about the nightside ionosphere of Nightside ionospheres can result from several sources, in- Mars. cluding remnant ionization from the dayside, like OC in the More recently, however, the Neutral Gas and Ion Mass terrestrial ionosphere. While the lower molecular ion layers Spectrometer (NGIMS) on the MAVEN spacecraft has mearecombine, the F2 peak persists through the night, although it sured ion densities in situ on Mars from periapsis of about rises, and the peak density is reduced by a factor of 10. Elec- 150 km (with occasional “deep dips” down to 125130 km) tron density profiles for day and night at high and low solar to 500 km at all SZAs, including the nightside. On both the activities are shown in Fig. 88.11. In the auroral regions of dayside and the nightside of Mars, in addition to the major the Earth, the precipitating electrons may also produce sig- ions found on the dayside, populations of protonated species, nificant ionization, which maximizes in the midnight sector including water-group ions, that is, those that contain only H of the auroral oval. and O were measured (e.g., Benna et al. [78]; Fox [79]; Fox The nightside ionosphere of Venus is highly variable but et al. [80]). has been shown to contain the same ions as the dayside The nightside ion densities for SZAs from 110ı to 120ı ionosphere. The ion and electron densities are, however, were shown to be smaller by an order of magnitude or more smaller by factors of 10 or more than those of the day- than those of the near-terminator dayside 75ı –80ı SZA, and side ionosphere, and the average peak in the electron density the composition in the lower ionosphere changed from OC 2 C C C profile is about .12/ 104 cm3 . It is produced by a combi- and COC 2 to O2 , NO , and HCO , the latter three of which nation of precipitation of suprathermal electrons that have have longer chemical lifetimes than COC 2 . For long-lived been observed at high altitudes in the umbra, and trans- molecular ions, the major loss process is usually dissociative port of atomic ions (mostly OC ) at high altitudes from the recombination, but the dissociative recombination coefficient dayside. For Mars, geometry limits the solar zenith angle of HCOC is particularly uncertain. As for all the known probed by radio occultation (RO) measurements beyond the nightside ionospheres, the nightside ion densities have been terminator of the nightside to values of about 130ı . The shown to be quite variable (Girazian et al. [81]).
88
1318
J. L. Fox
a Altitude (km)
b Altitude (km) 400
1000
N+
Ne –
Venus +
H O 500
+
He+
He+
CO+
300 H+
N+ CO+2
300 NO
250 200
N+
N+2
100
N+2 e
200
O+
NO+ + O2
Ne – O+2
NO+
O+2
O+
+
O+2
150
C
+
100
102
103
104
105
106 Number (cm 3)
c Altitude (km)
1
10
100
1000
104
105 106 Density (cm –3)
Altitude (km) 400
400 +
Ar
Mars
HOC+
O+
O++
O+2
300 He
300
ArH+
HNO+
+
e
CH+
N+2 NH+
N+ 200
N2H+
200
HCO+
C+ NO+
OCOH+
CO+2
CO+ 100 0.1
100 1
10
100
104 105 Density (cm –3)
1000
0.001
0.01
0.1
d Pressure (mbar)
10 –10
Electron (model) Electron (observed)
10 –9 H+3 10
–8
H+
10 –7 10 –6 10
CH+5
–5
1×101
1×102
1×103
1×104
1×106 1×105 Density (cm –3)
1
10
100 1000 Density (cm –3)
88
Aeronomy
1319
e Altitude (km)
Altitude (km) 2000
Log p (μbar)
4000
Log p (μbar)
–7 H+
H +2
3000
–3
H3O+
–6
H+3
1500
CH+5
–5
C2H+3 –2
–4
2000 He+
C3H+5
1000
–3
–1
–2 1000
C5H +n
–1 0 1
10 –4
10 –2
10 0
102
H3O+
500 10 –1
104 106 Density (cm –3)
10 0
C 4H+9 101
0
C6H +n
102
1 104 105 Density (cm –3)
103
f Altitude (km) 10 000 Uranus
g Altitude (km)
8000
3000 Neptune 2500 6000
H+3
2000
H+
4000
1500
1000
2000 Ingress Egress 0 102
103
Mg+
500 –1
104 105 106 Electron number density (cm –3)
0
Density (cm –3) 1000
h Altitude (km) 3000 Dusk Avg Dawn Avg Zero Voyager 1 RPWS Ta in RPWS Ta out RPWS Tb in RPWS Tb out Michigan MHD model
2500
2000
CH4+ CH +5 NH4+ CH +3 + + N ,CH 2
10
C2H 5+
C3H+4
CH 2NH +2
10
CH +3 +
C3H +5 CH 3CNH+
CHCN C3H +3
C3H7+
C2H+7
HC3NH+
C 4H +5 + C2H 3CNH+ C 4H 3NH
C 4H +3
10
30
20
C5H +5
C 4H+7
C2H 5CNH+
500
40
C6H +5 C5H 3NH+ HC5NH+ C6H+7 C 4H 5NH+
50
C5H+7
+ C5H +9 C6H 3
C5H 5NH+
C7H+7 + C6H 3NH+ C6H 5NH C7H +9
C7H +5
0
1000
2000 4000 3000 Electron density (cm –3)
0.1
50
60
70
80
C8H +3
C6H7NH+
1 0
88
3 4 5 Log ion densities (cm –3)
1
100
1000
2
HCNH+
100
0.1 10 1000
1500
1
90
100 m/z
1320
J. L. Fox
i Altitude (km)
Pressure (bar)
800
3000
Ne N2H+
H+
j Radius (km)
10 –6
N+2
600
2500 10
400
N+
C+
–5
10 –4
2000
10 –3 200 10 –2 Egress Ingress 0
0
1
2
3
4 5 log10 density (cm –3)
10 –1 10 0 101
1500
102
103 Electron density (cm –3)
Fig. 88.10 Ionospheric density profiles. (a) Measured ion density profiles for Earth [67]. (b) Computed ion density profiles for Venus, for high solar activity and a solar zenith angle of 45ı . (c) Ion density profiles for Mars, for moderate solar activity (F10:7 D 120) and a SZA range of 5ı –10ı . C C (Left) major ions and (right) protonated species. The model densities of the major ions OC 2 , O , and CO2 are compared to the MAVEN NGIMS C C C C measurements. Not shown are the density profiles of water ions (HC , HC , H , OH , OH , H O ). The model has no ambient water other than 3 2 3 2 that produced through ion chemistry. (d) Jupiter, precomet impact model without hydrocarbon ion chemistry. After Maurellis and Cravens [68]. (e) Saturn, after Kim et al. [44], who adopted the model of Vervack, Jr. and Moses [5]. The total electron density profiles are shown with a thick curve. The SZA is 27ı , and the magnetic dip angle is 45ı . (Left) model density profiles of the H-containing ions and HeC . (Right) model density profiles of hydrocarbon ions and H3 OC . (f) Uranus. Voyager 2 radio occultation electron density profiles for ingress and egress. From Lindal et al. [69]. (g) Neptune. From a model of Lyons [70], also [71]. (h) (Left) Titan. Electron density profiles measured by various instruments including the Voyager 1 radio occultation and in situ measurements of electron densities by the Cassini RPWS (Radio and Plasma Wave Science) LP (Langmuir probe) on the first two flybys, Ta and Tb, along with the dawn and dusk averages from the Cassini radio occultations, and one model. From Kliore et al. [72]. (Right) Titan. Cassini INMS (Ion and Neutral Mass Spectrometer) ion density measurements compared to a model at 1100 km. The abscissa is the mass/charge ratios from 0 to 55 is shown on the upper panel, and those from 45 to 100 are shown on the lower panel. From the review by Vuitton et al. [73]. (i) Triton. From the model of Strobel and Summers [74]. (j) Pluto. Upper limits to electron density profiles from a model (Hinson et al. [37]). That for the subsolar point (SZA D 0ı ) is shown in red, and that for the terminator region is in orange. No ionosphere was detected, however, by the New Horizons’ radio science experiment
On satellites whose trajectories take them into the magnetospheres of the planet that they orbit, such as Titan around Saturn, or Triton around Neptune, ionization on the dayside or nightside may be produced by the impact of energetic electrons from these magnetospheres. The nightside ion densities of Titan have been shown to be a factor of 4 smaller than those of the dayside (Ågren et al. [82]). At this time, there is no information available about the nightside ionospheres of the outer planets. Radio occultations can only probe the near-terminator ionospheres because of geometrical considerations. Just as on the dayside, only modeling can predict the identity of the major ions.
lost locally by dissociative recombination. The ionization rate q i in a one-species Chapman layer for monochromatic radiation is given by qi D F in ;
where i is the ionization cross section, and F D F 1 expŒ.z/ is the local solar flux. For an isothermal atmosphere, the scale height is approximately constant and, therefore, n D n0 exp.z=H /. Sometimes an ionization efficiency i is defined such that i D i a :
88.4.4 Ionospheric Density Profiles
(88.51)
(88.52)
Near threshold, the ionization efficiency for molecules is usually about 0:30:7, but it increases rapidly to 1.0 at Density profiles of molecular ions have often been approxi- shorter wavelengths, usually less than 600 Å. mated as idealized Chapman layers. A Chapman layer of ions Since the maximum ionization rate in an isothermal atis one in which the ions are produced by photoionization and mosphere occurs where the optical depth ( D nH a sec )
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1321
a
Altitude (km) 800
Reduced altitude (Z/H) 8 6
700 Solar min
600
4
Solar max
2 75
60
0
500
30
0
–2 400 –4 F2
300
Day
b
Night 200 F1
108
109
1010
1011
0.4
0.6 0.8 1 Production rate (Q/Qmax)
Reduced altitude (Z/H) 8
4
D 0
0.2
6
E
100
0
1012 1013 Electrons (m–3)
Fig. 88.11 Typical midlatitude ionospheric electron density profiles for sunspot maximum and minimum, day and night. After Richmond [75]
2 75 0
60
30 0
–2 –4 –3
–2
–1 0 log [production rate–1(Q/Qmax)]
is unity and, therefore, n D 1=. a H sec /, the maximum ionization rate in a Chapman layer is Fig. 88.12 Chapman production profile as a function of altitude on lin1
i D qmax;
i
i qmax;0
F D : e a H sec sec
(88.53)
If the altitude of maximum ionization for overhead sun is defined as z D 0, then n0 D . a H /1 , and, expressing F 1 i in terms of qmax;0 , the ionization rate is z i q i .z/ D qmax;0 exp 1 sec ez=H : H
ear (a) and semilog plots (b). The production rate is divided by the maximum production rate and the altitude by the scale height H , which is assumed to be constant. The origin on the altitude scale corresponds to the point of maximum absorption for overhead sun
density of an ion in a Chapman layer (in the photochemical equilibrium region) is given by ni .z/ D
(88.54)
It is apparent that at high altitudes (z ! 1), the ionization profile follows that of the neutral density, and below the peak (z ! 1), the ionization rate rapidly approaches zero. As the solar zenith angle increases, the peak rises, and the magnitude of the density maximum decreases. Figure 88.12 shows a production profile for an idealized Chapman layer on both linear and semilog plots. The asymmetry with respect to the maximum is more obvious for the semilog plot. If photochemical equilibrium prevails, the production rate of the major molecular ion is equal to the loss rate due to dissociative recombination
D
q i .z/ 1=2 ˛dr i qmax;0 1=2 ˛dr
1 z 1 z=H exp sec e : 2 2H 2 (88.56)
Actual ion density profiles differ from the idealized Chapman profile for several reasons. First, ionization is produced by photons over a range of wavelengths, which do not all reach unit optical depth at the same altitude. Second, thermospheres are usually not isothermal near the altitude of peak ion production. Third, photoionization is supplemented by photoelectron-impact ionization, which peaks lower in the atmosphere. Fourth, the major ion produced is often transformed by ion–molecule reactions before it can recombine dissociatively; finally, the underlying neutral atmosphere (88.55) undergoes changes with SZA. Nonetheless, the idealized q i .z/ D ˛dr ni ne D ˛dr n2i ; concept of the Chapman profile is useful in attaining a genwhere ˛dr is the dissociative recombination coefficient, ni is eral understanding of the shape of ion profiles and their the ion density, and ne is the electron density. Therefore, the behavior as the solar zenith angle changes. In addition, ion
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J. L. Fox
layers produced by auroral precipitation may take on a sim- is a reduced temperature. Numerically, ln is about 15, and ilar appearance to a Chapman-type layer, although energetic the collision frequency is approximately [84] electrons are not extinguished, as are photons, in ion pro1=2 duction. Although attempts have been made to express real Zi2Zs2 is ns D 1:27 ; (88.61) i s ionospheric profiles in terms of Chapman layer theory, ion 3=2 mi Tis layers rarely, if at all, fit true, or even slightly modified Chapman layers. where Z is the species charge number, and m are in amu, and the number density is in units of cm3 . The ion densities can be computed by solving the ion con88.4.5 Ion Diffusion tinuity equation, which is similar to Eq. (88.7) for neutral species, and in one dimension is Above the photochemical equilibrium layer of the ionosphere, upward and downward transport of ions must be @˚i @ni (88.62) C D Pi Li ; considered. The motions of ions, neutrals, and electrons are @t @z coupled, and the momentum equation, which determines the fluxes or velocities of the ions must take into account these where the ion flux is given by ˚i D ni wi . In general, it is interactions. The interaction of an ion, denoted by a sub- impossible to solve the momentum equation for the ion difscript i, with a neutral species denoted by a subscript n, is fusion velocity wi in closed form, except for the special cases through the ion–induced dipole attraction or, for the diffu- of a single major ion and of a minor ion moving through sion of an ion through its parent neutral, by resonant charge a dominant ion species. If only motion of the ions parallel transfer. For the former process, the ion–neutral diffusion co- to magnetic field lines is considered, the vertical velocity of efficient is given by a dominant ion (for which ni ne ) moving through a stationary neutral atmosphere is kB T ; (88.57) Din D mi in wi D Da sin2 I 1 dni d.Te C Ti / mi g 1 where mi is the mass of the ion, and the ion–neutral momen C C ; ni dz k.Te C Ti / Te C Ti dz tum transfer collision frequency (88.63) 2 1=2 ˛n e n n mn in D 2:21 ; (88.58) where I is the magnetic dip angle, and the ambipolar diffumi C mn in sion coefficient is defined as where ˛n is the polarizability of the neutral species (Dalgarno et al. [83], Schunk and Nagy [84]). For the resonant charge transfer interaction, the diffusion coefficient is Dinct
1 3. =2/1=2 kB Ti 1=2 D ; 1=2 ct m .1 C T i 8nn Qin n =Ti /
(88.59)
Da D
kB .Te C Ti / : mi in
(88.64)
For a minor ion i diffusing through a major ion species j , its velocity is given by kB Ti =mi ij C in 1 dni 1 d.Te C Ti / Te =Ti dne mi g C C C ni ds ne ds kB Ti Ti ds ij wj : (88.65) ij C in
wi D
ct is the average charge transfer cross section [85]. where Qin It should be noted that the forgoing effect does not apply to accidental resonances, since it arises from the indistinguishability of the nuclei. The momentum transfer collision frequency for Coulomb interactions between an ion i and another ion or electron denoted by the subscript s is For regions in which there are large gradients in the ion or electron temperatures, thermal diffusion may also be impor 3=2 2 2 ei es is 16 1=2 ns ms tant in determining the ion density profiles, especially those is D ln ; (88.60) of light ions such as HC and HeC . Equations for ion dis3 mi C ms 2kB Tis 2is tributions in which thermal diffusion is included have been where es is the charge on species s, ln is related to the presented by, for example, Schunk and Nagy [84]; see also Debye shielding length, and Tis D .ms Ti C mi Ts /=.mi C ms / references therein.
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88.5
Neutral, Ion, and Electron Temperatures
The temperature distribution in planetary thermospheres/ ionospheres can be modeled by solving the equation for conservation of energy, which, in a simplified form in the vertical direction is N @Tm @Tm @ (88.66)
m D Qm L m nm kB 2 @t @z @z where N is the number of degrees of freedom (three for an atom and five for a diatomic molecule), the subscript m refers to the neutrals, ions or electrons, m is the thermal conductivity, Qm is the volume heating rate, and Lm is the volume cooling rate. If horizontal variations are considered, the model becomes multidimensional, and advective terms must be added to the equations. The values of Tm are also affected by compression or expansion due to subsidence or upwelling, respectively. Viscous heating may be a factor where there are local regions of intense energy input, such as in auroral arcs. These terms are not shown in the energy equation above but may be found in standard aeronomy texts, such as Banks and Kockarts [85], Whitten and Popoff [86], Rees [50], and Schunk and Nagy [87]. For planets with intrinsic magnetic fields, the electrons and ions are constrained to move along magnetic field lines, and the second term on the left-hand side of Eq. (88.66) must be multiplied by a factor sin2 I . The neutral thermospheres of planets are mostly heated by absorption of solar radiation in the 102000 Å range, although on planets with powerful auroras, electron precipitation may be an important source of heat. Absorption of EUV radiation (1001000 Å) largely results in ionization of the major thermospheric species, in which most of the excess energy is carried away by the photoelectron. The photoelectron, however, may produce further dissociation or excitation of neutral species along the path to thermalization, and these processes may result in neutral heating. Photons near and longward of ionization thresholds in the FUV may lose their energy in photodissociation, in which the excess energy of the photon appears as kinetic or internal energy of the fragments. Chemical reactions that follow ionization or dissociation release much of the absorbed solar energy as heat. Although the partitioning of kinetic energy released between the product species can easily be determined by conservation of energy and momentum, the fraction of energy that appears as internal or kinetic energy must be determined by measurements or theoretical calculations. If vibrationally or electronically excited states are produced in these interactions, however, the energy may be radiated to space, thus producing cooling. This may occur promptly if the radiative lifetime is short, or subsequent to an energy transfer process from a long-lived metastable species to a species for which radiation to a lower state is allowed. If the metastable species is quenched, however, its energy can also appear as
1323
heat. Thus, the energy partitioning in chemical reactions and in the interactions of photons and photoelectrons with atmospheric species is important in understanding the temperature structure of thermospheres. A heating efficiency is often defined as the fraction of energy absorbed at a given altitude that appears locally as heat. The heating efficiencies are in the range 30%–40% in the terrestrial lower thermosphere. Above 200 km, the heating efficiency decreases because the energy of the important metastable species O(1 D) is lost as radiation rather than by quenching [88]. The heating efficiencies in the thermospheres of Venus and Mars are about 20% from 100 to 200 km [89, 90], and on Titan, they range from 20% to 30% from 800 to 2000 km. A column averaged heating efficiency for the Jovian thermosphere has been computed as 53% [91]. On Venus and Mars, CO2 is the major absorber of far UV radiation, whereas on the Earth, O2 plays that role. In the F1 regions of the ionospheres of the terrestrial planets, dissociative recombination of molecular ions tends to be the major source of heating. Below the F1 peak, photodissociation and neutral–neutral reactions, including quenching of metastable species, dominate. Since CH4 is a very strong absorber, the major heating mechanisms in the thermosphere of Titan are photodissociation and neutral–neutral reactions, both above and below the F1 peak. The few data that exist suggest that electron-impact dissociation is unimportant as a source of neutral heating, although further measurements would certainly be of benefit. Profiles of the heat sources in the terrestrial thermosphere and those of Mars and Titan are also shown in Fig. 88.13. Important cooling processes in planetary thermospheres include downward transport of heat by molecular and eddy conduction and infrared cooling from rotational and vibrational excitation of IR active species such as NO, CO, and CO2 . Excitation of the fine structure levels of atomic oxygen and subsequent emission at 63 and 147 µm also plays a role in cooling the neutral species in the thermospheres of the terrestrial planets. In the outer planets and their satellites, hydrocarbon molecules such as CH4 and C2 H2 are the primary thermospheric IR radiators. The global circulation may play a role in redistributing the heat that is deposited in the dayside or auroral thermosphere [95, 96]. In order to model heating rates, cross sections for processes in which solar photons or photoelectrons interact with neutral species, and rate coefficients and product yields for chemical reactions of ions and neutral atmospheric species are necessary. In addition, it is necessary to know, for example, how much of the energy released appears as internal energy of the products in chemical reactions and how much appears as kinetic energy of the products. Knowledge of energy transfer processes, including vibration–vibration (V– V) and translation–vibration (T–V) transfer between atmospheric species is also important. For example, a particularly
88
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J. L. Fox
a Altitude (km) 250 Chemical reactions
O2+ DR
200
Quenching Photodissociation
Electron impact 150
100
0
1
2
Fig. 88.13 Heating rates for the thermospheres of (a) Mars, (b) Earth, and (c) Titan. In (b), the curve labeled “Thermal-e” is the heating rate due to collisions between the neutrals and electrons, “OC ” is the heating rate due to reactions of OC with neutrals; “ion-neutral” is the heating rates due to other exothermic neutral–ion reactions. “Neutral– neutral” is the heating rate due to neutral–neutral chemical reactions; “e-recombination” is the heating rate due to dissociative recombination reactions; N2 -dis is the heating rate due to dissociation of N2 ; SR-dis is the heating rate due to absorption by O2 in the Schuman–Runge bands and continuum. From Richards [92]. In (c), the heating rates in the atmosphere of Titan due to five sources are shown. EUV represents solar heating, CDN represents heating due to thermal conduction, AER represents heating by aerosols, MAG is the heating due to magnetospheric ion precipitation, NIR is the heating due to absorption of the near infrared bands of CH4 . After Snowden [93]; see also Snowden and Yelle [94] J
3 4 5 log10 heating rate (eV cm –3 s –1)
peratures. Near the F1 peak, Te is usually larger than Ti , but T i begins to diverge from Te at higher altitudes. The source e-recombiTotal heating rates nation 450 for the energetic electrons on the dayside is largely photoionization, which, as discussed above, produces photoelectrons Thermal-e 400 Ionwith average energies in the 1520 eV range. In slowing 350 neutral down, these electrons lose their energy in inelastic processes 300 with neutrals until E 12 eV. At this point, elastic scatter+ O ing by the thermal electron population becomes the dominant Neutral250 N2-dis neutral SR-dis energy loss process for the suprathermal electrons and the 200 major source of heat for the thermal electrons. Other heat150 ing mechanisms for thermal electrons include deactivation of electronically or vibrationally excited species, and, for the 100 1 10 102 103 104 105 terrestrial planets, quenching of the fine structure levels of O. Heating rate (eV cm –3 s –1) As for the neutrals, heat in the electron gas is redistributed c Altitude (km) by conduction at a rate that depends on the electron ther1400 mal conductivity. This quantity is inversely proportional to EUV 1300 the sum of the momentum transfer collision frequencies of 1200 electrons with ions, neutrals, and ambient electrons. Cooling mechanisms for thermal electrons include Coulomb colli1100 sions with ions, rotational excitation of molecules, and, for 1000 the terrestrial planets, excitation of the fine structure levels NIR of O. Because of the large mass difference, elastic collisions CDN 900 between neutrals and electrons are not effective in transferAER 800 ring kinetic energy. MAG In the ionosphere, Ti is elevated above Tn at high altitudes 700 principally because of Coulomb collisions of ions with ener600 getic electrons. Another potentially important source of heat 100 101 102 103 104 Heating rate (eV cm –3 s –1) input to the ions near the ion peak is exothermic ion–neutral reactions, including quenching of metastable ions, such as OC .2 D/, by neutrals. In the presence of electric fields, Joule important cooling process for the thermospheres of the terres- heating may be important and can cause Ti to exceed Te . trial planets is excitation of the CO2 15 µm 2 bending mode The ions cool in elastic collisions and resonant charge in collisions with energetic O, and subsequent radiation [97]. transfer with neutral species, which are characterized by The deexcitation rate coefficient is several percent of gas ki- lower temperatures than the ions. The cooling rate for elastic netic, which is anomalously large for a V–T process [98]. collisions is In the lower ionosphere, the electrons and ions are in thermi 3 mal equilibrium with the neutral species, but at higher alti in kB .Ti Tn / : (88.67) Lin D 2ni m C m 2 tudes the plasma temperatures deviate from the neutral temi n b Altitude (km)
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Collisions between ions and neutrals (other than their parents) are dominated by the ion–induced dipole interaction. The momentum transfer collision frequency in is, thus, given by Eq. (88.58). Resonant charge transfer between an ion and its parent neutral, such as OC C O ! O C OC ;
ues measured by the Cassini Plasma Spectrometer (CAPS) and the Radio and Plasma Wave Science/Langmuir Probe (RPWS/LP) on the Cassini spacecraft (Ågren et al. [82]; Crary et al. [102]).
(88.68) 88.6
leads to very effective ion cooling, which dominates at sufficiently high temperatures. Examples of Ti and Te profiles are shown in Fig. 88.14a–f for various bodies, including Earth, Venus, Mars, Titan, and Saturn. The International Reference Ionosphere (IRI) temperature profiles for the terrestrial thermosphere show the close coupling between the electrons, ions, and neutrals at low altitudes and the ions and electrons at high altitudes; Ti increases with increasing solar activity at low altitudes and approaches Te at high altitudes. The values of Te and Ti are about 20003000 K at 1000 km on the dayside and slightly smaller on the nightside [105]. Electron and ion temperatures in the terrestrial ionosphere have been discussed by Rees [50], Banks and Kockarts [85], Whitten and Popoff [86], and Schunk and Nagy [106]. In the Venus atmosphere, Te and Ti were measured by instruments on the PVO, including the Langmuir Probe (LP) (e.g., Brace et al. [107]) and the Retarding Potential Analyzer (ORPA) (e.g., Miller et al. [99]). The values of Ti were found to be in the range 20002500 K at high altitudes and to be insensitive to the solar zenith angle, except beyond about 150ı SZA, where Ti increases to values of 50006000 K [99]. Measured values of T e do not vary appreciably with the solar zenith angle; the high altitude values are in the range 40006000 K [108]. The STATIC (Superthermal and Thermal Ion Composition) experiment on the MAVEN spacecraft found that Ti on Mars decouples from the Tn near 180200 km and approaches values of about 10001200 K at high altitudes (McFadden [100]). The electron temperatures on Mars have been measured by the LPW (Langmuir Probe and Waves) instrument on the MAVEN spacecraft (e.g. Ergun et al. [101]); Te is predicted to diverge from the Tn in the lower ionosphere (120130 km) and has been found to approach values of 30004000 K at high altitudes. The plasma temperatures on Titan have been measured and predicted by models. Electrons from Saturn’s magnetosphere may interact with the Titan ionosphere during the part of its orbit that is within Saturn’s magnetosphere. In this case, Te is predicted to increase to values that are about 5000 K near 2000 km. The models of Richard et al. [103] showed that the predicted altitude profiles of the ion and electron temperatures depend greatly on the magnetic field orientation near a given location. Their computed results are shown in Fig. 88.14d and e. They deviate significantly from the val-
Luminosity
The luminosity that originates in the atmospheres of the planets is generally classified as dayglow, nightglow, or aurora. Dayglow is the luminosity of the dayside atmosphere that occurs as a more or less direct result of the interaction of solar radiation with atmospheric gases. Among the sources of dayglow are photodissociative excitation and simultaneous photoionization and excitation. Dayglow may also include scattering of solar radiation by processes that are selective, such as resonance scattering by atoms and fluorescent scattering by molecules, but the term generally excludes nonselective scattering processes, such as Rayleigh scattering. In resonance scattering, the absorption of a photon by an atom in the ground state causes a (usually dipole allowed) transition to a higher electronic state A C h ! A ;
(88.69)
followed by the emission of a photon as the state decays back to the ground state A ! A C h :
(88.70)
The wavelength of the emitted radiation is very nearly the same as that of the radiation absorbed. The cross section for absorption of a line in the solar spectrum is a ./ D 12
$2 c 2 A21 ./ ; $1 8 2
(88.71)
where the subscript 1 indicates the lower state and 2 the upper state, $ is the statistical weight of the state, and D c= is the frequency of the transition; A21 is the Einstein A coefficient for the transition, and ./ is the lineshape function, which in this equation is normalized so that the integral over all frequencies is unity. If the linewidth is determined by the spread of velocities of the species, the lineshape ./ is a Doppler (Gaussian) profile 0 2 c 2 c p exp ; D ./ D 0 u2 u0
(88.72)
where 0 is the frequency at the line center. The variable u D .2kB T =m/1=2 is the modal velocity of a gas in thermal
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a Altitude (km)
b Altitude (km)
2000
800 Earth 700 F10.7 = 150
1500
600
Nightside
Dayside
500 1000
Tn
Tn
Ti
400
Ti Te
Te
300
500 200
1000
100
104 100 Temperature (K)
c Altitude (km)
1000
100 102
104 Temperature (K)
103
104 105 Temperature (K)
d Altitude (km) 2200 2000
600
Radial (D1)
Surface anchor (D2)
Mars
With T8 magnetospheric electrons (D6) With T5 magnetospheric electrons (D6)
Triple neutral densities (D3)
1800 1600 400 1400
Tn Ti
Te
Nested field lines for SZA = 90.93 (D4) and SZA = Cassini (D5)
1200
200 1000
100
104 Temperature (K)
1000
Crary et al. (2009) ion temperature
800 100
150
e Altitude (km)
200
250
300 Ion temperature (K)
f Altitude (km)
1800
3000 Solar sources with T8 (lobe like) magnetospheric electron fluxes (Rymer et al., 2009) (D6)
1700 1600
RPWS (Agren et al., 2009)
2500
1500 2000
1400 1300
SZA = 90.93 nested field lines (D4)
1200
1500
1100 1000
SZA = Cassini nested field lines (D5)
Solar sources with T5 (plasma sheet) magnetospheric electron fluxes (D7)
1000
900 800
0
500
1000
1500 2000 Electron temperature (K)
500 100
200
300
400 500 Temperature (K)
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Fig. 88.14 Neutral (Tn ), ion (Ti ), and electron (Te ) temperature profiles for selected planets. (a) For the dayside terrestrial ionosphere from the International Reference Ionosphere (IRI) for equinox, noon, and moderate solar activity (F10:7 D 150). (Left) dayside; (right) nightside. The Te is found to not vary substantially with SZA. Downloaded from the IRI website irimodel.org. (b) Smoothed median ion (dashed curve) and electron (solid curve) temperatures in the Venus ionosphere as measured by the Pioneer Venus (PV) retarding potential analyzer and the Langmuir probe, respectively. The electron temperature profile is essentially constant with the solar zenith angle. The ion temperature profile applies to solar zenith angles between 0 and 90ı . After [99]. (c) Mars. Temperatures fit to the MAVEN model for F10:7 D 120; Tn is fit to the CO2 density profile; Ti is fit to the ion temperatures at 200 km and at high altitudes McFadden [100]. It is assumed to deviate smoothly from the neutral temperature profile at 180 km. The Te profile is based on that of Ergun et al. [101] at 200 km and above and adjusted smoothly to converge with the value of Tn at 125 km. (d) Titan Ti dayside models for various magnetic field configurations compared to measurements of the Cassini spacecraft (Crary et al. [102]). From Richard et al. [103]. (e) Titan (Te ) dayside models for various magnetic field configurations compared to measurements of the Cassini RPWS/LP (Ågren et al. [82]). From Richard et al. [103]. (f) Saturn, Tn , Ti , and Te for 07:00 LT (local time). The dash-dot curve is the Tn profile. The solid curve is the Te profile, and the dotted and dashed curves are for HC and HC 3 , respectively. From Moore et al. [104] J
equilibrium at temperature T . The width of the line at half where F ./ is the solar flux in units of photons cm2 s1 Hz1 , and F ./ is the flux in units of photons maximum, D is cm2 s1 per unit wavelength interval. It should be noted that 20 u 1=2 D D (88.73) for radiative transfer purposes, the photon flux that we have .ln 2/ : c called F ./ is sometimes denoted F ./. It is customary in If the linewidth is determined by the natural lifetime, the pro- aeronomy to define a g-factor, which is the probability per atom that a photon will be resonantly scattered in a particular file is a Lorentzian transition L =2 ; (88.74) L ./ D q2 A21 . 0 /2 C .L =2/2 ; (88.80) g21 D P i A2i where L D R =2 is the linewidth at half-maximum, and where the sum in the denominator is over all the lower states R D 2 C 1 ; (88.75) i that are accessible from the upper state 2. The g-factor for unattenuated solar radiation is often quoted at the mean sun– where 2 and 1 are the inverse radiative lifetimes of the Earth distance or at a particular planet. The volume emission levels 2 and 1, respectively. Collisional broadening also rate "21 .z/ for resonance scattering of a solar photon is then results in a Lorentzian lineshape. If both Doppler and nat- given by ural broadening mechanisms are important, the lineshape is "21 .z/ D g21 n1 .z/ ; (88.81) a convolution of the two profiles, called a Voigt profile Z1
where n1 .z/ is the number density of atoms in level 1. In fluorescent scattering, a photon is absorbed by D . 0 /L . 0 /d 0 : (88.76) V ./ D a molecule in a vibrational state v producing an excited elec1 tronic state with a vibrational quantum number v 0 a The absorption cross section 12 integrated over all frequencies is proportional to the absorption oscillator strength f12 AB.v/ C h ! AB .v 0 / : (88.82) Z1 a 12 ./d D 0
e 2 f12 ; me c
This is followed by emission, at wavelengths that are usually (88.77) the same as or longer than that of the absorbed photon, to a range of vibrational levels v 00 of a lower state
where me is the mass of the electron; A21 is related to the oscillator strength through A21
$1 8 2 e 2 2 $1 8 2 e 2 D f D f12 : 12 $ 2 me c 3 $2 me c2
e 2 $2 4 f12 D F ./ A21 ; me c $1 8 c
(88.83)
The volume emission rate of a transition from a level v 0 of 00 (88.78) the upper electronic state to a vibrational level v of a lower electronic state at an altitude z is given by
The excitation rate q2 of an upper level 2 by resonance scattering is given by q2 D F ./
AB .v 0 / ! AB.v 00 / C h 0 :
Av0 v00 ; "v0 v00 .z/ D n.z/gv0 v00 D n.z/qv0 P v Av 0 v
(88.84)
(88.79) where Av0 v00 is the transition probability, n.z/ is the number density of the molecular species at altitude z, and qv0 is the
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J. L. Fox
excitation rate of vibrational level v 0 of the upper electronic state from a range of lower states v. The latter quantity is qv0 D
X
Normalized NO spectrum 0.030 NO δ (C2Π → X2Π), v' = 0
2
v F ./
v
e 2 fvv0 ; me c 2
(88.85)
0.025
0
1 2 3
0
0.020
where v is the fraction of molecules in the v vibrational level. A pedagogical review of these quantities and the relationships between them can be found in Hilborn [109] (erratum in Hilborn [110]). Dayglow also includes emissions that are the result of the interaction of atmospheric species with the photoelectrons produced in solar photoionization, either by direct excitation
AC e
!A Ce ;
(88.86)
or by simultaneous dissociation and excitation AB C e ! A C B C e ;
(88.87)
or ionization and excitation X C e ! XC C e :
5 NO γ (A2Σ → X2Π), v' = 0
4
1
2
3
4
5
260
300 280 Wavelength (nm)
0.015 0.010 0.005 0 180
200
220
240
Fig. 88.15 Far ultraviolet NO nightglow spectrum of Venus obtained with the SPICAV (Spectroscopy for the Investigation of the Characteristics of the Atmosphere of Venus) instrument on the Venus Express spacecraft. Show is an average of 771 spectra obtained during orbit 516 when the tangent altitude was between 90 and 120 km. The sum of all the intensities is normalized to 1. The NO ı (C 2 ˘ to X2 ˘ ) and (A2 ˙ to X2 ˘ ) bands are shown. From Royer et al. [118]
(88.88)
a tracer of day-to-night transport and is found near 2 am just south of the equator. The normalized NO nightglow spectrum as measured by the SPICAV (Spectroscopy for the Investigation of the Characteristics of the Atmosphere of Venus) instrument on the Venus Express is shown in Fig. 88.15 (Royer et al. [118]). Similar phenomena have recently been observed by the ultraviolet spectrometer on the Mars Express spacecraft [120], and by the MAVEN UVIS, but the circulation seems to be mostly pole to pole, or equator to pole, depending upon the season. In any case, the day-to-night circulation seems to be much more complicated than that of Venus. Auroral emissions are defined here as those produced by the impact of particles other than photoelectrons. Although auroras are usually thought of as confined to the polar regions of the Earth and the outer planets, Venus, which does not have an intrinsic magnetic field, exhibits UV emissions on the nightside that are highly variable and cannot be explained as nightglow. It has been proposed that the emissions are produced by precipitation of soft electrons into the nightside thermosphere, and the auroras are seen as diffuse and variable emissions on the nightside of the planet [121]. Mars, which has only a remanent crustal magnetic field, mostly in the southern hemisphere, exhibits discrete emissions that are concentrated over magnetic field anomalies, and more diffuse auroral emissions, which are seen over more extended N C O ! NO C h ; (88.89) regions. The SPICAM instrument on the Mars Express (e.g., Bertaux et al. [122], Gérard et al. [123]), and the IUVS inproducing emission in the ı and bands of NO (e.g., Stewart strument on the MAVEN spacecraft (Schneider et al. [124]) et al. [119]). The point on the disk of maximum intensity is have observed both diffuse and discrete aurorae. Electron impact processes are particularly important in producing excited states that are connected to the ground state by dipole forbidden transitions, whereas resonance and fluorescent scattering are largely limited to transitions that are dipole allowed. Dayglow emissions may also result from prompt chemiluminescent reactions, which occur when fragments or ions produced by dissociation or ionization recombine with the emission of a photon. As an example, the dayglow spectrum of the Earth from 1200 to 9000 Å is shown in Fig. 88.16. An ultraviolet spectrum of Mars as measured by the MAVEN Imaging Ultraviolet Spectrometer (IUVS) is shown in Fig. 88.17a; that of Saturn as measured by the Ultraviolet Imaging Spectrograph (UVIS) on Cassini (Fig. 88.17b); the UV dayglow of Saturn and Uranus, which were measured by the Voyager spacecraft, are compared in Fig. 88.17c; and the dayglow of N2 , along with NII and NI lines on Titan in Fig. 88.17d. Nightglow arises from chemiluminescent reactions of species whose origin can be traced to species produced during the daytime or that have been transported from the dayside. For example, on Venus, O and N produced on the dayside are transported by the subsolar to antisolar circulation to the nightside, where they subside and radiatively associate
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a Rayleighs/Å N2, Vegard–Kaplan Dayglow
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.10 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 3.4 3.5 3.6 +
NOη
500 400
1.0
N2 , First negative
0.0 1.1
0.1
0.2 0.3 0.4 0.5 1.3 1.4 1.5 1.6
1.0 2.1 3.2
2.0 3.1
N2, Second positive 0.0
Lα(1216)
2.0 3.1
3.0
N2, LBH
Nll(2143)
Oll(2470) Mgll(2795)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
300
0.6 1.7
1.0 2.1 3.2
1.1 2.2 3.3
0.11
0.12 1.12
0.13 1.13
2.13
0.14 1.14
2.14
0.0 1.1 2.2 3.3
0.1 1.2 2.3 3.4
1.15 2.15
0.2
0.3
1.3
1.4
2.4
2.5
3.5
3.6
0.1 0.2 0.3 0.4 0.5 1.2 1.3 1.4 1.5 1.6 2.3 2.4 2.5 2.6 2.7 3.4 3.5 3.6 3.7 3.8 3.9
1.7 2.8
Call(3934)
Ol(2972)
1.0 1.11.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Call(3970)
200
[Nl](5200)
Lα(1216)
Ol (1304) Ol (1356)
Oll(4351) Oll(4368) Oll(4415)
Nl(3466)
100
1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 5000 5200 Wavelength (Å)
Rayleighs/Å
N2, First positive 8.4 7.3 6.2
1.5
2.5
6.3 5.2 4.1 8.5 7.4
N2, Atmospheric
0.4
0.3 1.4
500
5.1 4.0
2.0
2.6
3.6 4.0
5.0
3.0 8.6
5.3
6.4
7.5
2.0 7.6
8.7 0.0 1.1
1.0 2.1
N+2 Meinel
3.1
4.2
3.0
4.1
2.1
3.2
0.1
2.2 2.0
5.1
4.3
5.4
6.5
3.1
4.2
5.2
1.0
1.2
5.3
[Ol](6300)
400
[Ol](6364)
[Oll](7320)
Ol(8446)
300 200
[Nl](5200)
Ol(7774)
[Ol](5577)
[Nl](5755)
100
5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000 7200 7400 7600 7800 8000 8200 8400 8600 8800 9000 Wavelength (Å)
b U
Intensity (kR) 0.30
B
V
R
I
[OI] 557.7
0.25 0.20
OH (Meinel) [OI] 630.0, 636.4
6-2
0.15
9-4 5-1
0.10 0.05
88
Na I D 589.3
NO2
O2 (Herzberg)
6-1
8-3
7-2
0 N+2 1N
N2 2P 315.8 337.0 353.5, 357.6 375.4, 380,4
100
50
N+2 1N 391.4 427.8 470.8
O+2 1N [OI] 525.3, 527.5 557.7 630.0 N2 1P 636.4 N+2 M O2 ATM
N2 2P [OI] O+2 1N
0 300
400
500
600
700
800 Wavelength (nm)
Fig. 88.16 (a) Terrestrial dayglow spectrum measured in a single 32-s exposure by the Arizona Imager/Spectrograph on board the Space Shuttle [111]. (b) Terrestrial near ultraviolet, visible, and near-IR nightglow (top) and aurora (bottom) at Dome A, Antarctica. The major features are labeled by their sources. From Sims et al. [112]
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a Intensity (DN/nm)
Intensity (DN/nm)
200 H Ly
O 297.2
50
200
CO 4PG
N2 VK(0,6)
OI
CO (B−X)
100
N2 VK(0,5)
C II
CO+2 UVD
CO Cameron
300
CI
150
100
0 120
140
0
160 Wavelength (nm)
200
250
CO+2 FDB
300 Wavelength (nm)
c Counts / 3840S
b R/Å
20
25
40
60
80
Channel 120
100
UVIS spectrum at 300 km MRH Best fit with airglow H 2 spectrum
Raman Lyα Solar reflection CI
20
450 H2Ly H2Wr
15 300 10
H2 a–b
5 150
L(6.12) P(1) at 1581.11 Å
L(6.11) P(1) at 1546.72 Å
L(6.9) P(1) at 1461.97 Å
0.5
L(6.7) P(1) at 1365.66 Å
1.0
L(11.9) R(1) at 1356.07 Å
L(6.5) P(1) at 1265.68 Å
20 eV electron impact H 2 spectrum Airglow H 2 spectrum
L(11.14) R(1) + P(1) at 1493.83 &1495.56 Å
0 Arbitary units 1.5
L(6,13) P(1) + L(5.12) R(1) at 1607.50 &1607.90 Å
HI Rydberg
0 600
1.0 0.8
1300
1400
1500
1600 1700 Wavelength (Å)
1200
1400 1600 Wavelength (Å)
N I, N II and H Ly–β N II 1085: 3.9 ± 0.2 R
0.6 0.4 0.2 0 –0.0 1.0 0.6
1200
1000
d Radiance (R/Å)
0.8 0
800
CY(3,4,6): 6.5 ± 0.4 R BH1: 1.9 ± 0.3 R CY(v' = 3) 1 CY(v' = 4) 1 2 CY(v' = 6) 1
0.4
2
4
BH1(v' = 1) 1 2 3 5
2
3
4
5
6
4
5
6
7
8
9
7
3
4
0.2 0 –0.0
800
850
900
950
1000
1050
1100 1150 Wavelength (Å)
Fig. 88.17 Dayglow spectra of selected planets. (a) Martian dayglow spectrum (intensity as a function of wavelength) in the FUV to NUV, as measured by the MAVEN IUVS. The major features are labeled on the plot. From Jain [113, 114]. (b) (Top) Saturn airglow spectrum, as measured by the Cassini UVIS at a minimum ray height (MRH) of 300 km, compared to a fitted airglow H2 spectrum. (Bottom) spectrum produced by the impact of 20 eV electrons compared to the airglow spectrum. From Gustin et al. [115]. (c) Comparison of dayglow spectra from Uranus (heavy line) and Saturn (thin line) recorded by Voyager 2 [116]. (d) Titan. Ultraviolet airglow of Titan in the wavelength range from about 780 to 1150 Å. (Top) Some atomic lines of NI, NII, and Lyman-beta are identified. (Bottom) Some bands of the N2 Carroll-Yoshino (CY) and Birge-Hopfield 1 (BH1) systems. From Stevens et al. [117]
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a kR/Å Atomic hydrogen Lyman series beta alpha
1000
Solar lines CIV
Molecular hydrogen
HeII CI
SiII
Lyman and Werner bands
0.100
0.010
0.001
Heavy: northern aurora Light: equatorial air glow 1000
b Signal (c 100 s –1 ch–1)
1200
1400
1600
Signal (c 100 s –1 ch –1)
Saturn high-altitude aurora
1800 Wavelength (Å) Saturn high-altitude aurora
H 2 (B, B', B'', C, D, D') Rydberg
102
103 H Lyα 101
102
10 0
10 –1 500
H 2 B Rydberg
H 2 C Rydberg
101
600
700
800
900
1000
1100
10 0 1200 1100 λ (A)
1200
1300
1400
1500
1600
1700 λ (A)
Fig. 88.18 (a) Spectra of the northern aurora (heavy line) and the equatorial airglow of Jupiter (light line) taken by the UVIS spectrometer the Cassini spacecraft on closest approach on 30 December 2000. In both spectra, the Lyman and Werner bands of molecular hydrogen are prominent. They are excited by photoelectron impact in the airglow and by primary and secondary auroral electrons. Locations of some of the brighter bands are indicated by dotted lines. Also shown are the Lyman-alpha and beta lines of atomic hydrogen. At wavelengths longer than 1550 Å, the airglow spectrum is swamped by reflected UV sunlight; some solar lines are indicated. At shorter wavelengths, UV sunlight is absorbed by methane and acetylene, rather than reflected. The solar signal is much weaker in the auroral spectrum than in the equatorial, due to the higher angles of incidence and emission in the former [125]. (b) Auroral synthetic spectrum of Saturn as predicted for the Cassini Ultraviolet Imaging Spectrograph in the EUV (left) and FUV (right); H2 bands and H emissions are shown. From Esposito et al. [126]
The intensities of airglow and aurora are usually measured in units of brightness called Rayleighs. One Rayleigh is an apparent column emission rate at the source of 106 photons cm2 s1 integrated over all angles or 106 =4 photons cm2 s1 sr1 . A comparison of auroral and dayglow emissions as measured by the Cassini UVIS as the spacecraft flew by Jupiter is shown in Fig. 88.18a (Stewart [125]). A discussion of terrestrial airglow and auroral emissions can be found in Rees [50]. Meier [127] has reviewed
spectroscopy and remote sensing of the terrestrial ultraviolet emissions. The airglows of Mars and Venus have been reviewed by Barth [11], Fox [128], and Paxton and Anderson [129]. Airglow and auroral emissions on the outer planets have been reviewed by Atreya et al. [130], Atreya [22], and Strobel [116]. Airglow in the atmospheres of the planets has been reviewed by Slanger and Wolven [131]. Airglow and aurorae of solar system bodies have been reviewed by Slanger et al. [132].
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Planetary Escape
Equation (88.90) reduces to
The escape of species from atmospheres can occur by thermal and nonthermal mechanisms. Thermal processes include Jeans escape and hydrodynamic escape. Jeans escape is essentially evaporation of the energetic tail of the Maxwell–Boltzmann distribution, while hydrodynamic escape is a large-scale expansion of the atmosphere that occurs when the average molecular velocity is near or above the escape velocity. Although Jeans escape still occurs for light species in the thermospheres of small solar system bodies, until recently, hydrodynamic escape was thought to have been operative only in the early history of the smaller solar system bodies when the solar flux in the UV was higher. Now, hydrodynamic escape has been postulated to occur for light species in the atmospheres of comets, or, for example, for H2 escape from Titan (Strobel [133]). There is an intermediate region between Jeans and hydrodynamic escape called “slow hydrodynamic escape”. The transition from hydrodynamic to Jeans escape is discussed by Volkov et al. [134]. Nonthermal escape mechanisms, which dominate for heavy species on smaller bodies such as Mars and Titan and for all species on Venus and Earth, include both photochemical and mechanical processes. A pedagogical discussion of escape processes can be found in Chamberlain and Hunten [135]. Because of the exponential rate of change of density with altitude in thermospheres, escape is sometimes assumed to occur only in the exosphere, that is, at or above an altitude historically called the “exobase” or “critical level”. The classically defined exobase is the altitude where the mean free path of a species, l D .n c /1 (where c is the collision cross section), is equal to the atmospheric scale height. The probability that a particle, moving upward from the exobase with sufficient velocity will actually escape without suffering another collision is then 1= e. The condition l D H , therefore, reduces to nH c D 1 or, equivalently, to N D . c /1 , where N is the column density. A typical collision cross section has been assumed to be about 3 1015 cm2 , which places the “exobase” at the altitude above which the column density is about 3:3 1014 cm3 . Thus, the “exosphere” defined in this way is not a truly collisionless region. Whether the trajectory of a particle moving upward at a given altitude is ballistic (bound) or escaping (free) in a collision-free region is determined by its total energy E, which is the sum of its kinetic and potential energies 1 E D mvc2 C 2
Zrc
mGM dr ; r2
ED
1 2 mv mgc rc ; 2 c
(88.91)
where gc is the gravitational acceleration at the critical level. The escape velocity at the exobase, vesc , is then defined by the condition vesc D .2gc rc /1=2 . In the “exobase” approximation, particles with velocities greater than the escape velocity are assumed to escape if their velocity vectors are oriented in the upward hemisphere and if they undergo no further collisions. In this approximation, the angular distribution of the products of collisions is assumed to be isotropic, and therefore half of the sufficiently energetic particles meet this criterion. The exobase approximation has been shown not to be valid for photochemical escape mechanisms by, for example, Fox and Ha´c [136]. In reality, the elastic and inelastic cross sections for collisions of different pairs of species vary over nearly an order of magnitude and are functions of energy in the relevant hyperthermal energy region ( 1–10 eV), however, and thus the location of the exobase should vary from species to species. In addition, for similar mass species pairs, the angular differential cross sections in the center-of-mass frame for the hyperthermal energy region are also strongly forward peaked, and escape of atoms, such as O from the Martian thermosphere, originate over a large range of altitudes that extends from well below to above the theoretical exobase. Measurements of angular differential cross sections, however, show that the collisions of comparable mass particles are strongly forward-peaked, so the nonthermal escape of a given species occurs over a range of altitudes, which varies from several scale heights above to well below the “exobase”. Use of the exobase approximation has been shown to be in error for photochemical escape of O, C, and N on Mars (e.g., Fox and Ha´c [136]) but may have some relevance for thermal escape mechanisms. It also useful as a reference altitude that shows the escape region to within several scale heights above and below it. The radius, gravitational acceleration, escape velocity, and scale heights at the equatorial “exobases” of the planets are given in Table 88.11. In the thermal Jeans process, escape occurs when particles in the high-energy tail of the Maxwellian distribution attain the escape velocity. The escape flux, ˚J is given by nc u ˚J D p .1 C c / exp.c / ; 2
(88.92)
where u D .2kB T =m/1=2 is the modal velocity, and is the (88.90) gravitational potential energy in units of kB T
1
D
mgr r GM m D D : rkB T kB T H
(88.93) where the symbols have the same meaning as in Eq. (88.2), and the subscript c refers to the critical level, the base of the Sometimes, a correction factor is applied to the expression assumed collision-free region. If E < 0, the particle is bound. for the escape flux to account for the suppression of the tail
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Table 88.11 Exobase properties of solar system bodies Planet Mercuryc Venusd Earthe Marsf Jupiterg Saturnh Uranusi Neptunei Titanj Tritonk Plutok
a
rc (km) 2440 6250 6878 3605 73 000 62 968 31 800 27 300 4000 2203 2900
a
gc (cm s2 ) 371 831 841 331 2236 821 561 919 56:1 29:5 10:3
a
vesc;c (km s1 ) 4:25 10:2 10:8 4:74 57:2 32:2 18:9 22:4 2:12 1:14 0:774
a,b
Havg (km) – 17 71 21:8 250 192 66 250 85 90 253
a Values given are those at the equatorial exobase and assume that the thermosphere corotates with the planet b Average value computed from Havg D kB T =mavg g but does not represent the local pressure scale height, except for cases where there is one major constituent c Exobase is at the surface d Model from Hedin et al. [2] for F10:7 D 150, 45ı solar zenith angle. e MSIS model for F10:7 D 150, equator, 45ı solar zenith angle [9] f Model fit to MAVEN data for F10:7 D 120 conditions; April 2015, 7:5ı solar zenith angle g Model from [4, 22] h From Koskinen et al. [137] and from Vervack, Jr. and Moses [5] i Model from Moses and Poppe [64] j Model from Strobel [133], Strobel [65] k Model from Strobel and Zhu [7]
of the distribution due to the escape of the energetic particles [138, 139]. Photochemical processes that produce energetic fragments include photodissociation and photodissociative ionization, photoelectron impact dissociation and dissociative ionization, as well as exothermic chemical reactions. The most important example of the latter are dissociative recombination reactions, which are very exothermic and tend to produce neutral fragments with large kinetic energies. Charge transfer processes such as OC C O ! OC C O
(88.94)
HC C O ! OC C H
(88.95)
or
can produce fast neutrals if the ion temperature is larger than the neutral temperature, as is usually the case at altitudes well above the exobase of a planet, or if solar wind heavy ions, including protons, directly impact the atmosphere. In modeling these processes, the kinetic energy distribution (in the laboratory frame) of the product species is important, as well as the cross sections or reaction rates. Physical or collisional escape mechanisms include sputtering and knock-on. Sputtering can occur when a heavy ion
picked up by the solar wind (the projectile) collides with an atmospheric species (the target) and in the process produces a back-splash in which the accelerated neutral may be ejected from the atmosphere. Since for comparably sized species, there is no backscattering in the laboratory frame, either the collision must be glancing or the process must involve multiple collisions. The sputtering phenomenon has been described in detail in the monograph by Johnson [140]. In knock-on, hot atmospheric neutral species, such as O atoms produced in exothermic chemical reactions at sufficiently high altitudes can collide with a (usually) lighter species, such as H, imparting sufficient kinetic energy to allow it to escape. Modeling these processes requires knowledge of the ion–neutral and neutral–neutral elastic, inelastic, and angular differential collision cross sections. The escape rate of a light species from a planetary atmosphere may be limited by diffusion of the species from the lower atmosphere to the exosphere, rather than by the escape process itself [141]. The limiting upward flux, i of a species i with mixing ratio fi can be estimated as i
bi fi ; Ha
(88.96)
where Ha is the scale height that is based on the average mass of the constituents of the thermosphere at the homopause, and bi is the binary collision parameter introduced in Sect. 88.2.2. Equation (88.96) above is usually evaluated at the homopause, with the mixing ratio taken from a suitable altitude in the middle atmosphere, but above the cold trap (where the species condenses), if one exists. The limiting flux obtains, if and only if the mixing ratio is constant with altitude. The effect of photochemistry can be accounted for if all chemical forms of the species are considered in the calculation of fi (Hunten [40]). For example, on Titan, H2 sometimes escapes hydrodynamically and sometimes by Jeans escape at the diffusion limit, but CH4 escapes at 75% of its diffusion limiting rate. On Triton, the mixing ratio of CH4 near the surface is much smaller than that in the lower atmosphere of Titan. It is chemically destroyed in the lower atmosphere before it can escape (e.g., Strobel [65]). On Pluto, CH4 escapes with significant rates, exceeding 1025 s1 ; H and H2 escape at their diffusion limit (Strobel [10]). Even if energetic particles released near the exobase of a planet do not have enough energy to escape, they may travel to great heights along ballistic orbits before falling back to the atmosphere. These particles are said to form hot atom coronas. Hot H and O coronas surround Earth, Venus, and Mars (Fig. 88.19). In addition, hot C and hot N coronas have been detected or predicted to surround Venus and Mars. Since the escape velocities of the giant planets are so large (tens of km s1 ), escape of even H is not possible. Nonthermal processes at high altitudes may, however, lead to the formation of hot H coronas. Reviews of the H and O coronas
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Relative intensity 5000 4000 3000
Venus dark – Venus bright HIRES UT 20 Nov 99 15:59 – 16:08 500s Airmass = 1.7 Velocity = 12.45 km/s
Venus A = 4191 Г = 0.103 Å T = 5577.574 Å
Terrestrial A = 3094 Г = 0.103 Å T = 5577.344 Å
2000 1000 0
5576
5576.5
5577
5577.6
5578 5577.5 Wavelength (Å)
Fig. 88.19 Spectrum of the oxygen green line taken on the nightside of Venus by the Keck/HIRES on 20 November 1999. The individual terrestrial and Doppler-shifted Venusian components are shown by the dashed lines (from Slanger et al. [142])
of Venus have been presented by Fox and Bougher [96] and by Nagy et al. [143]. The exospheres of various solar system bodies and the coronas of Mars, Venus, and Titan have been reviewed by Futaana et al. [144]. See Chamberlain and Hunten [135] for a detailed discussion of planetary coronal population processes.
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Jane Fox Jane Fox received her PhD in Chemical Physics from Harvard University and has held positions at the State University of New York at Stony Brook and the Harvard/Smithsonian Astrophysical Observatory. She is a Fellow of the American Geophysical Union. Her research has focuses on the chemistry, luminosity, heating of the thermospheres/ionospheres of the planets, and their evolution.
Applications of Atomic and Molecular Physics to Global Change Gonzalo González Abad
Contents 89.1 89.1.1 89.1.2
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337 Global Change Issues . . . . . . . . . . . . . . . . . . . . 1337 Structure of the Earth’s Atmosphere . . . . . . . . . . . 1338
89.2 89.2.1 89.2.2 89.2.3
Atmospheric Models and Data Needs . . . Modeling the Thermosphere and Ionosphere Heating and Cooling Processes . . . . . . . . . Atomic and Molecular Data Needs . . . . . .
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89.3 89.3.1 89.3.2 89.3.3
Tropospheric Warming/Upper Atmosphere Cooling Incoming and Outgoing Energy Fluxes . . . . . . . . . . Tropospheric “Global” Warming . . . . . . . . . . . . . . Upper Atmosphere Cooling . . . . . . . . . . . . . . . . .
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Stratospheric Ozone . . . Production and Destruction The Antarctic Ozone Hole Arctic Ozone Loss . . . . . Global Ozone Depletion . .
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details, and there will continue to be rapid advances in the future. This means that any article or book discussing this topic becomes out of date as soon as it is written. Nevertheless, several recent references on these topics are recommended [1–3]. Atomic and molecular structure and spectroscopy, as well as collision processes involving atoms, molecules, ions and electrons, are important to the study of all planetary atmospheres. Keywords
polar vortex sulfate aerosol ozone layer total ozone column ozone monitoring instrument air quality global warming greenhouse effect stratospheric ozone depletion
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Abstract
While there has been a general understanding and appreciation of the science involved in both global warming and stratospheric ozone depletion by atmospheric scientists for some time, a detailed understanding and rigorous proof have often been lacking. Over the last 20 years, there have been many advances made in filling in the G. González Abad () Harvard-Smithsonian Center for Astrophysics Cambridge, MA, USA e-mail: [email protected] K. Chance Harvard-Smithsonian Center for Astrophysics Cambridge, MA, USA e-mail: [email protected] K. P. Kirby American Physical Society College Park, MD, USA e-mail: [email protected]
89.1 Overview 89.1.1 Global Change Issues Over the last several decades there has been increasing concern about the global environment and the effect of human perturbations on it. This whole area, which involves a wide range of scientific disciplines, has become known as global change. Knowledge of processes taking place in the atmosphere, oceans, land masses, and plant and animal populations, as well as the interactions between these various Earth–system components is essential to an overall understanding of global change – both natural and human induced. The processes of atomic and molecular physics find greatest application in the area of atmospheric global change. The three major issues that have received significant attention in both the media and the scientific literature are: (1) global warming, due to the buildup of infrared-active gases; (2) stratospheric ozone depletion due to an enhancement of de-
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_89
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structive catalytic cycles; and (3) air quality and tropospheric pollution. These three problems are thought to be largely caused by atmospheric pollutants due to industrialized human society. However, the general problem of air pollution and its direct effects on plant and animal populations will not be addressed here.
89.1.2 Structure of the Earth’s Atmosphere The vertical temperature structure of the Earth’s atmosphere shown in Fig. 89.1 provides an important nomenclature that is widely used [4]. The atmosphere is divided into regions called “-spheres”, in which the sign of the temperature gradient with respect to altitude, dT =dz, is constant. The regions in which the temperature gradient changes sign are called “-pauses”. The precise altitude pertaining to each of these regions can vary depending upon latitude and the time of year. In the troposphere, covering the range from 0 to 15 km above the Earth surface. The temperature steadily decreases with altitude. This is the most complex region of the atmoFig. 89.1 Vertical temperature profile of the atmosphere
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sphere, as it interacts directly with plant and animal life, land masses, and the oceans. It is the region in which weather occurs. The change in sign of the temperature gradient at the tropopause to a positive dT =dz in the stratosphere is due to heating by absorption of solar ultraviolet radiation, which photodissociates O2 and O3 . The stratosphere, extending from 15 to 50 km above the Earth, contains the ozone layer, which shields the Earth’s surface form harmful ultraviolet radiation in the range 280–320 nm. At the stratopause, the heating processes have become too weak to compete with the cooling processes, and throughout the mesosphere, approximately 50–85 km, the temperature again decreases with increasing altitude. Cooling processes, which will be discussed in Sect. 89.2.2, involve collisional excitation of molecular vibrational modes, which decay by radiating to space. The coldest temperatures in the atmosphere are found at the mesopause, where the temperature gradient once again becomes positive. From approximately 70 km upward, a very diffuse plasma called the ionosphere exists due to photoionization of atoms and molecules by short wavelength (UV and EUV) solar radiation.
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Throughout the thermosphere, which extends from approximately 90 km upward, heating occurs because the atmosphere has become so thin that there are very few collisions and, thus, inefficient equilibration of the highly translationally excited atoms and ions with the molecular species, which can radiate in the infrared. This “bottleneck” for energy loss causes increased heating. In the thermosphere and ionosphere, the thermal inertia is very small, and there are huge temperature variations, both diurnally and with respect to solar activity. The densities are low enough in the thermosphere, ionosphere, and mesosphere that the primary processes determining the chemical and physical characteristics of these regions are two-body processes and “half-collision events” discussed elsewhere in this volume: dissociative recombination, photoionization, photodissociation, charge transfer, and collisional excitation of molecular rotation and vibration. As the altitude decreases, the density increases. Then, threebody interactions, interactions on surfaces (of aerosols and ices), and complex chemical cycles together with dynamical effects such as winds determine the chemical and physical characteristics of the stratosphere and troposphere.
such a model include: a solar spectrum of photon fluxes as a function of wavelength, concentrations of neutral species, and photoabsorption, photodissociation, and photoionization cross sections as functions of wavelength. The computer code brings all these elements together, calculating opacity as the solar radiation propagates downward through the atmosphere, keeping track of ion production and electron production. Additional steps are required to calculate the abundances of trace species such as NOC , necessitating the inclusion of all relevant ion–neutral reactions. Electron energy degradation can be tracked by including inelastic collisions of electrons with ions, atoms, and molecules. The primary neutral species are N2 , O2 , O, and He. Below about 100 km, the atmosphere is fully mixed by turbulence in the ratio 78% N2 , 21% O2 , and 1% trace species such as O3 and CO2 . Above 100 km, turbulence dies out, and the atmospheric species are in diffusive equilibrium, distributed by their molecular weight, with atomic oxygen dominating above 150 km and He dominating much higher. The major C C C ions are NC 2 , O2 , O , and NO .
89.2.2 Heating and Cooling Processes
89.2 Atmospheric Models and Data Needs While models are absolutely essential to the study of any system as complex as the Earth’s atmosphere, they play a particularly fundamental role in exploring global change issues. Models not only provide predictions of future changes but also allow exploration of sensitivities to particular parameters. Comparing the results of a model with observations ultimately tests and challenges scientific understanding. Of critical importance is the atomic and molecular data that goes into the models. Generally, atmospheric models become increasingly complex as altitude decreases. General circulation models (GCMs), incorporating thousands of chemical reactions, global wind patterns, and abundances of large numbers of trace species, require supercomputers in order to model aspects of the troposphere and stratosphere [5]. Tropospheric chemistry and transport models, such as GEOS-CHEM [6], and CAM-Chem [7] model the sources, evolution, transport, and sinks of pollution, as well as the oxidative capacity of the troposphere.
In the upper atmosphere, heating occurs through absorption of short wavelength solar radiation to produce ionization and dissociation and is mediated by collisions between electrons, ions, and neutrals. Ions and electrons are created during the daytime and to a great extent disappear during the night with the absence of solar radiation. Processes such as dissociative recombination, the primary electron loss mechanism, heat the gas: 3 3 e C OC 2 ! O. P/ C O. P/ C 7 eV ;
eC
NC 2
4
4
! N. S/ C N. S/ C 6 eV :
(89.1) (89.2)
Cooling takes place when the kinetic energy of the gas is transformed through collisions into internal energy, which can then be radiated away. The primary coolant above 200 km is the fine structure transition of atomic oxygen, h
O.3 P1 / ! O.3 P2 /, which is excited by thermal collisions and radiates at 63 µm. From approximately 120 to 200 km, the fundamental band of NO, v D 1 ! v D 0, which is excited by collisions with atomic oxygen and radiates at 5.3 µm, dominates the cooling. Below 120 km and throughout the mesosphere and stratosphere, the primary coolant is the 2 89.2.1 Modeling the Thermosphere band of CO2 radiating at 15 µm. This transition is excited by collisions of CO2 and atomic oxygen. Cooling throughand Ionosphere out most of the atmosphere is accomplished through trace The data necessary for modeling of the thermosphere and species because the major molecular species, N2 and O2 , are ionosphere are described here. The primary components of not infrared active.
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89.2.3 Atomic and Molecular Data Needs Knowledge of rate coefficients for ion–neutral and neutral– neutral reactions as functions of vibrational and rotational excitation of the reactants is becoming increasingly important, as there is recent evidence of more internal excitation of molecular species than had previously been thought [8]. Accurate photoabsorption, photodissociation, and photoionization cross sections as functions of wavelength for all the relevant species are important parameters determining the reliability and ultimate accuracy of an atmospheric model. Compilations of data, such as that by Conway [9] and Kirby et al. [10] are very useful but can rapidly become outdated. The Smithsonian Astrophysical Observatory maintains the world standard database, HITRAN, for molecular line parameters and absorption cross sections from the microwave through the ultraviolet for analysis of atmospheric spectra [11]. Discussions of the needs for atomic and molecular data in the context of space astronomy, but including applications to atmospheric physics, can be found in a book edited by Smith and Wiese [12]. Atomic and molecular data are available at the Virtual Atomic and Molecular Data Center Consortium [13].
89.3
Tropospheric Warming/Upper Atmosphere Cooling
89.3.1 Incoming and Outgoing Energy Fluxes The overall temperature of a planet is determined by a balance between incoming and outgoing energy fluxes. In a steady state, the planet must radiate as much energy as it absorbs from the sun. The Earth, radiating as a black body at an effective temperature TE , obeys the Stefan–Boltzmann law in which the energy emitted is expressed as TE4 4 RE2 , with the Stefan–Boltzmann constant, and RE the radius of the Earth. An equation expressing the equality of energy absorbed and energy emitted can be written as [14] Fs RE2 .1 A/ D TE4 4RE2 ;
(89.3)
where A is the albedo of the Earth (the fraction of solar radiation reflected from, rather than absorbed by, the Earth), Fs is the solar flux at the edge of the Earth’s atmosphere, and RE2 is the Earth’s area normal to the solar flux. Solving this equation for TE , one obtains TE 255 K (18 ı C). The sun, which has a surface temperature of approximately 6000 K, emits most of its radiation in the 0.2–4.0 µm region of the spectrum (200–4000 nm). The upper atmosphere of the Earth (thermosphere, ionosphere, mesosphere, and stratosphere) absorbs all the solar radiation shortward of 320 nm. The atmosphere of the Earth absorbs only weakly in the visible region of the spectrum where the solar flux peaks.
The Earth, with an effective radiating temperature of 255 K, emits mainly long-wavelength radiation in the 4– 100 µm region. Molecules naturally present in the atmosphere in trace amounts, such as carbon dioxide, water, and methane, absorb strongly in this wavelength region [15]. Radiation coming from the Earth is thus absorbed, reradiated back to the surface and thermalized through collisions with the ambient gas. This trapping of the radiation produces an additional warming of 33 K. Thus, the mean surface temperature of the Earth is 288 K, not 255 K as found for TE above. This effect of the Earth’s atmosphere is known as the greenhouse effect. The greenhouse effect is what makes Earth habitable for life as we know it. Gases, both natural and manmade, which absorb strongly in the 4–100 µm region, are known collectively as greenhouse gases.
89.3.2 Tropospheric “Global” Warming According to the Intergovernmental Panel on Climate Change (IPCC) Fifth Assessment Report (AR5) the “globally averaged combined land and ocean surface temperature data as calculated by a linear trend, show a warming of 0.85 [0.65 to 1.06] ı C, over the period 1880 to 2012.” The IPCC has also concluded that “Human influence on the climate system is clear. This is evident from the increasing greenhouse gas concentrations in the atmosphere, positive radiative forcing, observed warming, and understanding of the climate system” [16]. The Arctic region has warmed by an estimated 1 ı C in the past two decades, leading to substantial changes in the cryosphere. Antarctic sea ice was stable from 1840 to 1950 but has since declined sharply. The extent of sea ice shows a 20% decline since about 1950 [17, 18]. From air bubbles trapped at different depths in polar ice, it is possible to determine carbon dioxide and methane concentrations several thousand years ago. In 2011, CO2 levels reached 390.5 ppm, 40% greater than in 1750 [19]. In 2013, CO2 levels passed the 400 ppm threshold for the first time. Over the next century, the total amount of CO2 in the atmosphere since 1900 is expected to double to as much as 600 ppm [20]. This increase is due primarily to the burning of fossil fuels. Although methane is present at levels several orders of magnitude less than CO2 , it is increasing much more rapidly. Methane concentrations have more than doubled over the last 200 years due to industrial processes, fuels, and agriculture [19]. The manmade chlorofluorocarbons (CFCs), which have been widely used as refrigerants and in industry, increased in the atmosphere at a rate of over 5% per year between the 1970s and the 1990s. Due to the Montreal Protocol there has been a reduction or inversion of this trend [21]. Ozone, which is a primary component of chemical smog, is a pollutant when it occurs in the troposphere and an ef-
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in the stratosphere, mesosphere, and thermosphere, but as a greenhouse gas it is involved in heating the troposphere. The explanation for this revolves around the collision physics issue of quenching versus radiating. In the troposphere, CO2 absorbs infrared radiation coming from the Earth, exciting the 2 vibrational bending mode at 15 µm. The excited molecule can either reradiate or collisionally de-excite. In the lower atmosphere, where densities are large, the lifetime against collisions is very short, and the excited molecule is rapidly quenched. This transfer of energy from radiation through collisions into the kinetic energies of the colliding partners results in a net heating. In the stratosphere and above, atomic oxygen collisions with CO2 excite this same bending mode, but at these higher altitudes, densities are lower and quenching is greatly reduced. The excited molecule radiates, and the radiation escapes into space. A net cooling results because the opacity is low at these altitudes. The cooling of the stratosphere has also been linked with ozone trends [25]. Roble and coworkers [26, 27] investigated the doubling of CO2 and CH4 concentrations (as predicted for the twentyfirst century) in the mesosphere and at the lower boundary of the thermosphere. Using sophisticated atmospheric general circulation models, they predict that the stratosphere, mesosphere, and thermosphere will show significant cooling – the largest cooling of 40–50 ı C occurring in the thermosphere. The extent of this cooling very much depends on the rate coefficient for the O C CO2 excitation of the 2 bending mode. Rishbeth and Roble [26] assumed a value for this rate coefficient of 1 1012 cm3 =s, intermediate between the value of Sharma and Wintersteiner [28] (61012 cm3 =s) and an earlier value of 2 1013 cm3 =s used by Dickinson [29]. The Sharma and Wintersteiner value, based on observations of 15 µm emission in the atmosphere around 100–150 km, was recently confirmed by Rodgers et al. [30], but Pollock et al. [31] obtained a value of 1:2 1012 cm3 =s in laboratory experiments. Using the larger rate coefficient would result in even greater cooling [32]. The overall consequences of such a large temperature decrease in the upper atmosphere have not been fully explored – particularly the question as to how the dynamics of the atmosphere will be affected. Since many chemical reactions depend on temperature, there may be considerable readjustments in the vertical distribution of minor species in the atmosphere. Cooler temperatures cause the atmosphere to contract, reducing densities and, consequently, satellite drag. Cooler temperatures may also increase the occurrence of polar stratospheric clouds, thereby affecting ozone depletion 89.3.3 Upper Atmosphere Cooling (Sect. 89.4). Most significantly, tropospheric warming and upper atmoThe buildup of CO2 has an even greater effect on the sphere cooling both result from a buildup of CO2 . The size temperature in the upper atmosphere than on that in the tro- of the predicted cooling is greater by an order of magnitude posphere [24]. As discussed in Sect. 89.2.2, CO2 is a coolant than the amount of the predicted heating. Thus, it may be fective greenhouse gas. It has been increasing worldwide as well. This buildup of CO2 , CH4 , CFCs, and tropospheric O3 causes a problem. In much of the spectral region from 5–100 µm, there is 100% absorption of radiation by the atmosphere – due mainly to naturally occurring water vapor. There is, however, a region of rather weak absorption, from 7–15 µm, known as the “atmospheric window”. Increased concentrations of the greenhouse gases strengthen the absorption in this region, tending to “close” this window, thus increasing the infrared opacity of the atmosphere. The increased opacity causes an immediate decrease in the thermal radiation from the planet–atmosphere system, forcing the temperature to rise until the energy balance is restored [22]. It is difficult to prove that the buildup of greenhouse gases is the cause of the observed temperature rise. Other possible causes include slight changes in solar activity and irradiance, and changes in ocean currents, which may have a profound effect on global temperature and climate. These are areas of active research. Given the increase in concentrations of greenhouse gases that has occurred and is predicted to continue, the change in radiative heating of the troposphere can be calculated. Models generally predict an increase in tropospheric temperatures ranging from 1.5 to 5 ı C, considering different emission scenarios over the next century [23]. The 3.5 ı C range in temperature is due to the ways that different models incorporate climate feedbacks. Climate feedbacks include water vapor, snow and sea ice, and aerosols and clouds. Rising temperatures increase the concentration of water vapor, which is itself a greenhouse gas, producing further warming. Rising temperatures reduce the extent of reflective snow and ice, thus reducing the Earth’s albedo. This leads to increased absorption of solar radiation, further increasing temperatures. Clouds and aerosols both contribute to the albedo, thereby reducing the solar flux reaching the Earth, and absorb infrared radiation causing temperatures to rise. The modeling of clouds and aerosols and their radiative properties is very difficult and is one of the largest sources of uncertainty in the climate models. Understanding the role that the ocean, with its giant heat capacity, plays in global warming and identifying and quantifying the various interactions occurring at the ocean–atmosphere interface are vital areas of research, which will affect the size of the predicted temperature increase. At present, there are few obvious opportunities for traditional atomic and molecular physics to play a significant part in global-warming research.
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possible to monitor the global warming trend by observing in each of these cases is to convert ozone and atomic oxygen (otherwise known as odd-oxygen) into molecular oxygen: the predicted cooling in the upper atmosphere. There is evidence of this cooling in the stratosphere and NO C O3 ! NO2 C O2 ; mesosphere. Keckhut and coworkers [33] found cooling rates of 1:1 ˙ 0:6 K at 25 km, 1:7 ˙ 0:7 K between 35–50 km and NO2 C O ! NO C O2 ; 3:3 ˙ 0:9 K at 60 km per decade, while Santer and coworkers (89.7) NET W O3 C O ! 2O2 ; linked these changes in the thermal structure of the atmosphere with human activities [34]. Gadsden [35] also found and that the frequency of occurrence of noctilucent clouds, the highest-lying clouds in the atmosphere, has more than douNO C O3 ! NO2 C O2 ; bled over the last 25 years. He calculated that this change NO2 C O3 ! NO3 C O2 ; could result from a decrease in the mean temperature at the NO3 C h ! NO C O2 ; mesopause of 6.4 ı C during this time period. However, inNET W 2O3 ! 3O2 : (89.8) creased concentrations of water produced by oxidation of increased amounts of methane may be responsible for the OH C O3 ! HO2 C O2 ; more frequent appearance of these clouds. This is an ongoHO2 C O3 ! OH C 2O2 ; ing area of research. NET W 2O3 ! 3O2 ; (89.9) and the halogen cycle, in which Z D Cl or Br:
89.4 Stratospheric Ozone
Z C O3 ! ZO C O2 ;
89.4.1 Production and Destruction
ZO C O ! Z C O2 ; NET W
O3 C O ! 2O2 : (89.10) Ozone production takes place continually in the stratosphere during daylight hours, as molecular oxygen is photodissociated, and the resulting oxygen atoms undergo three-body The following series of reactions couples the HOx and halogen cycles: recombination with O2 : O2 C h ! 2O ;
(89.4)
HO2 C ZO ! HOZ C O2 ;
O C O2 C M ! O3 C M :
(89.5)
HOZ C h ! OH C Z ; Z C O3 ! ZO C O2 ;
Ozone can be destroyed through photodissociation: O3 C h ! O2 C O ;
OH C O3 ! HO2 C O2 ; (89.6)
but because an oxygen atom is produced, which immediately recombines with another O2 to form O3 , no net loss of O3 results. The photodissociations of O2 and O3 are important heating processes in the stratosphere. The amount of ozone in the stratosphere is quite variable, changing significantly with the seasons and with latitude. In the lower stratosphere, much of the ozone is created over the equatorial regions and then transported toward the poles. In addition to being photodissociated, O3 is destroyed by reactions with radicals that are involved in catalytic cycles. The shorthand notation for the major cycles, NOx , HOx , and ClOx (BrOx ) refers to the catalytically active forms involved in the cycles. Our knowledge about the relative importance of these catalytic cycles in ozone destruction has increased dramatically over the last decades. A number of these cycles are given below, with the ozone-destroying step listed first, and the rate-limiting step closing the catalytic cycle and regenerating the ozone-destroying radical listed last. The net effect
NET W
2O3 ! 3O2 :
(89.11)
Finally, the reaction set BrO C ClO ! Br C Cl C O2 ; Br C O3 ! BrO C O2 ; Cl C O3 ! ClO C O2 ; NET W
2O3 ! 3O2
(89.12)
is also important in the halogen destruction cycle. The coupling between these different cycles by reactions such as HO2 C NO ! OH C NO2
(89.13)
turns out to be very important in understanding the details of ozone destruction, such as how much each mechanism contributes to the destruction as a function of altitude and in the presence of aerosols. Wennberg et al. [36] recently showed that catalytic destruction by NO2 , which for two decades was considered to be the predominant loss process, accounted for
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less than 20% of the O3 removal in the lower stratosphere during May 1993. They further showed that the cycle involving the hydroxyl radical accounted for nearly 50% of the total O3 removal, and the halogen-radical chemistry was responsible for the remaining 33%. The NOx and HOx cycles occur naturally, whereas the ClOx and BrOx cycles are due mainly to manmade chemicals – CFCs and halons. The amplification that takes place through a catalytic cycle is the reason that these chemicals, which are only present at the level of parts per trillion, can have such a destructive effect. It is useful to think in terms of a total chemical budget for a radical, such as Cl, that enters into a cycle. Chlorine is put into the stratosphere when chemicals such as CF2 Cl2 are released into the atmosphere. Such compounds are chemically inert and insoluble in water, and therefore are not easily cleansed out of the lower atmosphere. In the stratosphere, however, the CF2 Cl2 is subjected to solar UV radiation and is photodissociated, producing the Cl radical. Chlorine exists in the upper atmosphere in catalytically active forms, Cl and ClO, as well as in stable, reservoir species, HCl and ClONO2 . The total chlorine budget consists of both the catalytically active plus reservoir species. Reactions that reduce the formation of reservoir species, or convert reservoir species to catalytically active forms, contribute to the ozone destruction. Photolysis of stable reservoir species, such as ClONO2 , can produce catalytically active forms. Bromine has an identical cycle to that of chlorine but is 50 to 100 times more destructive than Cl because it does not react readily to go into its reservoir form, HBr. A knowledge of the photodestruction rates of all such species is important for an understanding of the overall ozone photochemical depletion problem. Studies of the Antarctic ozone hole show that gas phase photochemical cycles, as given above, are not the whole story with respect to ozone depletion. Heterogeneous chemistries taking place on the surfaces of ice crystals and sulfate aerosols also play an important role. These are discussed briefly in Sects. 89.4.2 and 89.4.4.
ing the 1990s [38]. Recent data shows that this phenomenon continues, but due to the implementation of the Montreal protocol there are signs of decline in the Antarctic ozone depletion [39, 40]. The link between the release and buildup of manmade CFCs and the ozone hole over Antarctica was convincingly established by Anderson et al. [41] through in situ observations from high altitude aircraft flights into the polar vortex during the end of the polar night and the beginning of Antarctic spring in 1987. The polar vortex is a stream of air circling Antarctica in the winter, creating an isolated region that becomes very cold during the polar night. Flights into the vortex were able to document a heightened, increasing level of ClO and a monotonically decreasing O3 concentration over a 3–4 week time period during late September and early October. The mechanism repartitioning the chlorine from its reservoir form into its catalytically active form is a heterogeneous process occurring on the surfaces of polar stratospheric clouds. At the cold temperatures during the polar night, polar stratospheric clouds form, consisting of ice and nitric acid trihydrate. Gaseous ClONO2 collides with HCl that has been adsorbed onto the surface of the cloud crystals. Chlorine gas is liberated, and the nitric acid formed in the reaction remains in the ice [41]: HCl C ClONO2 ! Cl2 .g/ C HNO3 :
(89.14)
As solar radiation starts to penetrate the region at the beginning of spring, the Cl2 molecules are rapidly photodissociated, producing Cl atoms, which initiate the catalytic destruction of O3 . As there are few oxygen atoms available to complete the catalytic cycles, several mechanisms for regenerating the Cl and Br radicals that involve only the ClO and BrO molecules themselves have been proposed. Mechanism I [42] ClO C ClO C M ! .ClO/2 C M ; .ClO/2 C h ! Cl C ClOO ; ClOO C M ! Cl C O2 C M ;
89.4.2 The Antarctic Ozone Hole
2 .Cl C O3 ! ClO C O2 / ;
NET W 2O3 ! 3O2 : (89.15) The ozone depletion problem was largely theoretical until the Mechanism II [43] discovery of the ozone hole over Antarctica. Following the 1985 announcement by Farman et al. [37] of ground-based ClO C BrO ! Cl C Br C O2 ; observations of significant decline in O3 concentrations durCl C O3 ! ClO C O2 ; ing springtime in the Southern Hemisphere, it was possible to map this event using archived satellite data beginning in Br C O3 ! BrO C O2 ; 1979. The data depict a worsening event throughout the early NET W 2O3 ! 3O2 : (89.16) 1980s. In 1987, 70% of the total O3 column over Antarctica was lost during September and early October, and the areal While Mechanism I accounts for 75% of the observed ozone extent of the hole was 10% of the Southern Hemisphere. loss, the sum of I and II yields a destruction rate in harmony The ozone hole continued to grow in depth and width dur- with the observed O3 loss rates [41].
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89.4.3 Arctic Ozone Loss The region around the North Pole does not appear to exhibit an ozone hole as severe as that found in the Antarctic. Several factors lessen the probability of a significant ozone hole developing in the Arctic. First, a stable polar vortex does not get well established due to increased atmospheric turbulence from the greater land surface area in the Northern Hemisphere. Second, temperatures during the Arctic winter do not get as cold as during the Antarctic winter, so that polar stratospheric clouds (PSCs) do not form as easily. As seen in the preceding section, the surfaces of PSCs play an essential role in the O3 destruction mechanisms in the Antarctic. However, ozone levels are showing 20–25% reductions during February and March [44] over a much larger area around the North Pole than in the South. Thus, ozone destruction is taking place during the transition from polar winter to spring in the Arctic, but the phenomenon is more widespread, diffuse, and not as well contained as in the Antarctic.
89.4.4 Global Ozone Depletion Over the last four decades, satellite instruments have measured the total ozone column in the atmosphere. During this time ozone levels decreased globally. Only recently have they stabilized as a consequence of the policies introduced in Montreal protocol agreements. A recent analysis indicates the first evidence of recovery of stratospheric ozone levels, with diminished rates of ozone loss at altitudes of 35–45 km, coupled with a slowdown in the increase in stratospheric loading of chlorine [38]. Heterogeneous reactions on aerosol surfaces, as well as the homogeneous gas phase chemical cycles mentioned earlier, must be invoked to explain the global decline in ozone levels. A particularly important reaction appears to be the hydrolysis of N2 O5 on sulfate aerosols. This occurs very rapidly, converting reactive nitrogen, NO2 , into its reservoir species HNO3 : sulfate aerosol
N2 O5 C H2 O ! 2HNO3 :
(89.17)
The N2 O5 is formed at night by reaction of NO2 and NO3 . Following the hydrolysis of N2 O5 , there is less reactive NO2 available to convert ClO into its reservoir species, ClONO2 , and less NO2 around to convert OH into the reservoir species, HNO3 . A heightened sensitivity of the ozone to increasing levels of CFCs develops. It has been shown that certain regions of significantly depleted ozone also show high concentrations of sulfate aerosols [45]. In addition, measurements of the ratio of catalytically active nitrogen to
total nitrogen can be reproduced using the above heterogeneous reaction, and not by using gas phase processes alone. The study of further mechanisms at varying altitudes and latitudes is an active area of research. Record low global ozone measurements, 2 to 3% lower than any previous year, were reported beginning in 1992 [46] and continuing well into 1993. The increase in naturally occurring aerosols due to the eruption of Mount Pinatubo in June 1991 appears to explain this decline. During the winter of 1993–1994, the total ozone levels returned to levels slightly above normal [47], presumably because the excess aerosols had been removed from the stratosphere by natural sedimentation processes. The continuing buildup of CO2 is predicted to contribute to increase cooling of the stratosphere. Declining temperatures in the stratosphere may increase the frequency of formation of polar stratospheric clouds that drive the destructive heterogeneous chemistry creating the Antarctic ozone hole. An increased occurrence of these clouds outside of the polar regions could affect ozone levels globally. There are also indications that certain ozone depletion chemistries taking place on the surface of sulfate aerosols may also be enhanced by lower temperatures [45]. Ozone itself is the dominant heat source in the lower stratosphere. Decreasing the amount of ozone drives temperatures still lower [48]. It is unfortunate that the two most significant atmospheric global change effects – the buildup of CO2 and the enhanced ozone destruction due to manmade CFCs – both cause decreasing temperatures in the stratosphere, which may further enhance the destructiveness of the ozone’s photochemical cycles.
89.5 Atmospheric Measurements Ground-based observations, as well as measurements made by instruments carried aloft in satellites, balloons, rockets, and high-flying aircraft, allow to explore the atmosphere. Measurements may be made either in situ or by remote sensing techniques. The region of the atmosphere from 60 to 120 km, encompassing the mesosphere and lower thermosphere and ionosphere, is studied by in situ measurements using sounding rockets as it is too high for balloons and aircraft and too low (i.e., too much drag) for satellites. A comprehensive description of rocket measurements is provided by Larsen [49]. Remote sensing experiments are essential to observe the upper atmosphere. A comprehensive book on the subject is recommended [50]. Most of the instruments used to make atmospheric measurements were developed in molecular physics and spectroscopy laboratories. Even experiments utilizing sophisticated techniques, such as laser-induced fluorescence, and instruments, such as Fourier transform spectrometers, are be-
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ing flown on payloads. An excellent compendium of satellite instruments for stratospheric research has been assembled by Grant [51]. A detailed description of past satellite instruments can be found in the Encyclopedia of Atmospheric Science [52–60]. A combination of good laboratory experiments, theoretical calculations, and ingenuity is necessary to extract accurate information from measurements made in the atmosphere. For instance, in order to understand the complicated interactions of the different photochemical cycles involved in ozone chemistry, spectroscopic emissions and absorptions of the many trace species are used to measure concentration profiles. An accurate knowledge of the emission spectroscopy of species such as OH, HO2 , H2 O2 , H2 O, O3 , HNO3 , NO2 , N2 O, N2 O5 , HNO3 , ClNO3 , BrO, HCl, HOCl, and ClO is essential. Such measurements provide a rigorous test of atmospheric models. The NASA EOS Aura satellite, launched in 2004, carries instruments making global measurements of a number of these species [61–63]. One year before, in August 2003, the Canadian Space Agency launched the ACE-FTS instrument [64]. Taking advantage of the high signal-to-noise measurements inherent to the solar occultation technique it was able to measure vertical profile distributions of more than 60 molecules and isotopologues globally. In order to analyze the data and deconvolve some of the line profiles to give information on concentrations as a function of altitude, molecular data such as line strengths and pressure broadening coefficients are needed [11]. Until recently, it was impossible for remote sensing experiments to distinguish between ozone occurring in the stratosphere (where it is formed naturally) and ozone occurring in the troposphere (where it is a pollutant). Satellite instruments such as the ESA Global Ozone Monitoring Experiment (GOME), the Scanning Imaging Absorption Spectrometer for Atmospheric Chartography (SCIAMACHY), and the Ozone Monitoring Instrument (OMI) have a broad enough spectral coverage and a high enough resolution that the temperature dependence of the ozone absorption features from 300–340 nm, known as the Huggins bands, can be used to separate out the ozone concentrations in the middle and lower atmospheres [65, 66]. The future of atmospheric remote sensing from space looks very promising. Projects around the world are increasing the number and quality of observations, allowing scientists to investigate the time and spatial scales previously out of reach. The Sentinel-5 Precursor satellite launched by the European Space Agency in 2017 carrying the TROPOMI instrument [67] is starting to show what future geostationary instruments, like the US Tropospheric Emissions: Monitoring of Pollutants (TEMPO) [68], the Korean Geostationary Environment Spectrometer (GEMS), and the European Space Agency’s Sentinel-4 instrument, each having even higher spatial resolution, will be capable of. To keep track
of past and current satellite missions, as well as instrumental techniques for atmospheric observation, the NASA Earth Science site (https://www.nasa.gov/topics/earth/index.html) and the ESA Earth Observation site (https://www.esa.int/ Our_Activities/Observing_the_Earth) are extremely useful.
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40. Strahan, S.E., Douglass, A.R.: Decline in Antarctic Ozone Depletion and Lower Stratospheric Chlorine Determined From Aura Microwave Limb Sounder Observations. Geophys. Res. Lett. 45(1), 382–390 (2018) 41. Anderson, J.G., Toohey, D.W., Brune, W.H.: Free Radicals Within the Antarctic Vortex: The Role of CFCs in Antarctic Ozone Loss. Science 251(4989), 39–46 (1991) 42. Molina, L.T., Molina, M.J.: Production of chlorine oxide (Cl2O2) from the self-reaction of the chlorine oxide (ClO) radical. J. Phys. Chem. 91(2), 433–436 (1987) 43. Prather, M.J., Watson, R.T.: Stratospheric ozone depletion and future levels of atmospheric chlorine and bromine. Nature 344(6268), 729–734 (1990) 44. Manney, G.L., et al.: Chemical depletion of ozone in the Arctic lower stratosphere during winter 1992–93. Nature 370(6489), 429–434 (1994) 45. Fahey, D.W., et al.: In situ measurements constraining the role of sulphate aerosols in mid-latitude ozone depletion. Nature 363(6429), 509–514 (1993) 46. Gleason, J.F., et al.: Record Low Global Ozone in 1992. Science 260(5107), 523–526 (1993) 47. Hofmann, D.J., et al.: Recovery of stratospheric ozone over the United States in the winter of 1993–1994. Geophys. Res. Lett. 21(17), 1779–1782 (1994) 48. Ramaswamy, V., et al.: Stratospheric temperature trends: observations and model simulations. Rev. Geophys. 39(1), 71–122 (2001) 49. Larsen, M.F.: OBSERVATIONS PLATFORMS – Rockets. In: North, G.R., Pyle, J., Zhang, F. (eds.) Encyclopedia of Atmospheric Sciences, 2nd edn., pp. 285–289. Academic Press, Oxford (2015) 50. Houghton, J.T., Taylor, F.W., Rodgers, C.D.: Remote Sounding of Atmospheres. Cambridge University Press, Cambridge (1984) 51. Grant, W.B.: Ozone Measuring Instruments for the Stratosphere. Optical Society of America, Washington D.C. (1989) 52. Dudhia, A.: SATELLITE REMOTE SENSING – Temperature Soundings. In: Holton, J.R. (ed.) Encyclopedia of Atmospheric Sciences, pp. 1985–1998. Academic Press, Oxford (2003) 53. Fischer, H., Hase, F.: OBSERVATIONS FOR CHEMISTRY (REMOTE SENSING) – IR/FIR. In: Holton, J.R. (ed.) Encyclopedia of Atmospheric Sciences, pp. 1499–1508. Academic Press, Oxford (2003) 54. Hays, P.B., Skinner, W.R.: SATELLITE REMOTE SENSING – Wind, Middle Atmosphere. In: Holton, J.R. (ed.) Encyclopedia of Atmospheric Sciences, pp. 2012–2024. Academic Press, Oxford (2003) 55. Healy, S.B.: SATELLITE REMOTE SENSING – GPS Meteorology. In: Holton, J.R. (ed.) Encyclopedia of Atmospheric Sciences, pp. 1965–1972. Academic Press, Oxford (2003) 56. King, M.D., Herring, D.D.: SATELLITES – Research (Atmospheric Science). In: Holton, J.R. (ed.) Encyclopedia of Atmospheric Sciences, pp. 2038–2047. Academic Press, Oxford (2003) 57. Shepherd, G.G.: OPTICS, ATMOSPHERIC – Optical Remote Sensing Instruments. In: Holton, J.R. (ed.) Encyclopedia of Atmospheric Sciences, pp. 1595–1601. Academic Press, Oxford (2003) 58. Stolarski, R.S., McPeters, R.D.: SATELLITE REMOTE SENSING – TOMS Ozone. In: Holton, J.R. (ed.) Encyclopedia of Atmospheric Sciences, pp. 1999–2005. Academic Press, Oxford (2003) 59. Vaughan, G.: OBSERVATIONS FOR CHEMISTRY (REMOTE SENSING) – Lidar. In: Holton, J.R. (ed.) Encyclopedia of Atmospheric Sciences, pp. 1509–1516. Academic Press, Oxford (2003) 60. Waters, J.: OBSERVATIONS FOR CHEMISTRY (REMOTE SENSING) – Microwave. In: Holton, J.R. (ed.) Encyclopedia of Atmospheric Sciences, pp. 1516–1528. Academic Press, Oxford (2003)
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61. Levelt, P.F., et al.: The Ozone Monitoring Instrument: overview of 14 years in space. Atmos. Chem. Phys. 18(8), 5699–5745 (2018) 62. Beer, R.: TES on the aura mission: scientific objectives, measurements, and analysis overview. IEEE Trans. Geosci. Remote. Sens. 44(5), 1102–1105 (2006) 63. Waters, J.W., et al.: The Earth observing system microwave limb sounder (EOS MLS) on the aura satellite. IEEE Trans. Geosci. Remote. Sens. 44(5), 1075–1092 (2006) 64. Bernath, P.F.: The Atmospheric Chemistry Experiment (ACE). J. Quant. Spectrosc. Radiat. Transf. 186, 3–16 (2017) 65. Liu, X., et al.: Ozone profile and tropospheric ozone retrievals from the Global Ozone Monitoring Experiment: Algorithm description and validation. J. Geophys. Res. 110(D20), D20307 (2005) 66. Liu, X., Bhartia, P.K., Chance, K., Spurr, R.J.D., Kurosu, T.P.: Ozone profile retrievals from the Ozone Monitoring Instrument. Atmos. Chem. Phys. 10(5), 2521–2537 (2010) 67. Veefkind, J.P., et al.: TROPOMI on the ESA Sentinel-5 Precursor: A GMES mission for global observations of the atmospheric composition for climate, air quality and ozone layer applications. Remote. Sens. Environ. 120, 70–83 (2012) 68. Zoogman, P., et al.: Tropospheric emissions: Monitoring of pollution (TEMPO). J. Quant. Spectrosc. Radiat. Transf. 186, 17–39 (2017)
Gonzalo Gonzalez Abad Gonzalo Gonzalez Abad works as a physicist at the Harvard-Smithsonian Center for Astrophysics, where he is member of the Tropospheric Emissions: Monitoring of Pollution satellite instrument science team for measuring North American air pollution hourly from geostationary orbit (tempo.si.edu). He measures Earth’s atmosphere composition from aircraft, the ground, and satellites, with a special interest in pollution and air quality. Kelly Chance Kelly Chance heads the Tropospheric Emissions: Monitoring of Pollution satellite instrument for measuring North American air pollution hourly from geostationary orbit (tempo.si.edu). He measures Earth’s atmosphere from balloons, aircraft, the ground, and satellites, including the stratospheric ozone layer, greenhouse gases, and atmospheric pollution. He is an author a textbook on spectroscopy and radiative transfer of planetary atmospheres. Kate Kirby Kate Kirby has a PhD in Chemical Physics from the University of Chicago and is currently Director of the Institute for Theoretical Atomic, Molecular, and Optical Physics. Her research interests center on theoretical studies of ultracold molecule formation and atomic and molecular structure and processes that are of interest to astronomy and atmospheric physics. Such processes include photoionization, photodissociation, radiative association, charge transfer, and line-broadening.
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Surface Physics Erik T. Jensen
Contents 90.1
Low Energy Electrons and Surface Science . . . . . 1349
90.2 90.2.1
90.2.3
Electron–Atom Interactions . . . . . . . . . . . . . . Elastic Scattering: Low Energy Electron Diffraction (LEED) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inelastic Scattering: Electron Energy Loss Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . Auger Electron Spectroscopy . . . . . . . . . . . . . . .
90.3 90.3.1 90.3.2 90.3.3 90.3.4
Photon–Atom Interactions . . . . . . . . . . . Ultraviolet Photoelectron Spectroscopy (UPS) Inverse Photoemission Spectroscopy (IPES) . X-Ray Photoelectron Spectroscopy (XPS) . . . X-Ray Absorption Methods . . . . . . . . . . . .
90.4 90.4.1 90.4.2
Atom–Surface Interactions . . . . . . . . . . . . . . . . 1357 Physisorption . . . . . . . . . . . . . . . . . . . . . . . . . 1357 Chemisorption . . . . . . . . . . . . . . . . . . . . . . . . . 1357
90.5
Recent Developments . . . . . . . . . . . . . . . . . . . . 1357
90.2.2
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of book series that deal with areas of particular interest are published at regular intervals, such as surface chemistry [9], surface vibrations [10] and stimulated desorption processes [11]. Keywords
Auger electron spectroscopy electron energy loss spectroscopy resonant scattering surface sensitivity core hole electron diffraction electron-molecule collisions photoelectron spectroscopy X-ray absorption atom scattering
90.1 Low Energy Electrons and Surface Science
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358
Abstract
This chapter describes various applications of atomic and molecular physics to phenomena that occur at surfaces. Particular attention is placed on the application of electron- and photon-atom scattering processes to obtain surface specific structural and spectroscopic information. The study of surfaces and interfaces touches on many fields of pure and applied science. In particular there are applications in the fields of semiconductor processing, thin film growth, catalysis, corrosion and fundamental physics in two-dimensions. A number of recent texts cover surface physics in general [1–4] as well as specific areas such as experimental techniques [5], surface electron spectroscopies [6, 7], and the application of synchrotron radiation to surface science [8]. Also a number E. T. Jensen () Dept. of Physics, University of Northern British Columbia Prince George, BC, Canada e-mail: [email protected]
To obtain information specific to the first few layers of atoms at the surface of a solid, techniques must be devised that discriminate between signals from the surface and from the bulk of the material. If a solid has dimensions of 1 cm3 , then only approximately one atom in 107 is located at the surface. This makes the application of bulk techniques to the study of surfaces (e.g. X-ray diffraction [12]) problematic. There are several experimental approaches to achieving surface sensitivity. If the probe used is an atom or molecule that can be scattered from or desorbed from the surface, then these species can be analyzed by techniques such as mass spectrometry or resonant ionization. A more recent innovation has been the use of scanning probe microscopies [13] (e.g., the scanning tunneling microscope and its variants) which exploit a surface sensitivity such as the surface valence electron density. However the most common surface analytical techniques use the intrinsic surface sensitivity of low energy (102000 eV) electrons. The surface specificity of low energy electrons arises from the very short inelastic mean free path (MFP) for these electrons in a solid. This property can be seen in Fig. 90.1 which
© Springer Nature Switzerland AG 2023 G.W.F. Drake (Ed.), Springer Handbook of Atomic, Molecular, and Optical Physics, Springer Handbooks, https://doi.org/10.1007/978-3-030-73893-8_90
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Mean free path λ (Å) 100 7 6 5 4 3 2
10 7 6 5 4 3 3 4 5 6
10
2
3 4 5 6
100
2 3 4 5 6 1000 2 Electron energy (eV)
Fig. 90.1 The variation of the inelastic mean free path with electron kinetic energy (universal curve). Based on Briggs and Seah [14]
plots measured values of the inelastic MFP for a number of materials and electron energies between 1 and 2000 eV. The curve is called universal because the same general trend of short inelastic MFP is observed for nearly all materials [14]. The dominant energy loss mechanisms are valence band excitations (plasmons and electronic excitations), and since most materials have similar valence electron densities, the resultant inelastic scattering MFP is to a good approximation, material independent. The very short inelastic MFPs for low energy electrons has the result that any that escape from the solid without having undergone inelastic scattering can only have originated very close to the surface, usually within a few atomic layer spacings. Many of the surface analysis techniques described in this chapter use this surface sensitivity to obtain surface structural and spectroscopic information. The same techniques are easily applicable to conducting and semiconducting materials, and can also be applied to insulating materials, if the charging effects are dealt with in some manner.
90.2
Electron–Atom Interactions
90.2.1 Elastic Scattering: Low Energy Electron Diffraction (LEED) The elastic scattering of low energy (20500 eV) electrons at surfaces is historically important in physics as the experiments of Davisson and Germer provided early experimental evidence of the wave nature of electrons. After these initial experiments, the technique was largely unused until the advent of cleaner ultra-high vacuum systems and surface preparation techniques in the 1960’s. LEED is one of the most important and widely used surface characterization techniques due to its surface sensitivity and wide utility [5]. Diffraction from a two-dimensional net of scatterers results in a two-dimensional array of reciprocal lattice rods oriented normal to the surface. All kinematically allowed
rods intersect the Ewald sphere at all energies—unlike the case in bulk X-ray diffraction. The short inelastic MFP for scattering low energy electrons yields the surface sensitivity of LEED for a crystal surface. The kinematic theory used in X-ray diffraction is not directly applicable to LEED since the low energy electrons can undergo several elastic collisions in the surface region. The elastic MFPs for low energy electrons are comparable in magnitude to the inelastic mean free paths shown in Fig. 90.1. This multiple scattering does not affect the positions of the diffraction beams, but does alter their intensities due to interference effects. Kinematic analysis is sufficient to determine the diffraction beam positions for a proposed structure, and this is the most common use for LEED. A diffraction pattern is often sufficient to determine the surface periodicity and unit mesh size. However determining the surface crystal basis from the intensities of the diffracted LEED beams requires moderately sophisticated calculations to be performed for proposed structures. Quantitative LEED generally compares measured I(V) curves (the intensity I of a particular diffraction beam as a function of electron energy measured in volts) with a calculated I.V / profile for a proposed surface crystal structure. These calculations determine the propagating electron wave functions LEED that take into account electron-ion core scattering cross sections and phase shifts, available multiple scattering pathways, inelastic scattering cross sections and Debye-Waller effects. The structural parameters are varied systematically until agreement can be reached with the experimental I.V / curves [15, 16]. These calculations can yield surface atom positions to better than 0:1 Å vertically and 0:2 Å horizontally. Over 1000 surface structures have been determined using LEED and associated techniques [17]. Clean surfaces often have a different crystallography than simple termination of the bulk crystal structure. Due to the absence of neighbors above the surface, the surface atoms can undergo both relaxation (change in interlayer spacing) and reconstruction (changes in periodicity and bonding) [2]. While relaxation and reconstruction do occur on all types of surfaces, the reconstructions found on semiconductors are most striking. The strong covalent bonds that are broken when a semiconductor is cleaved give rise to high energy dangling bonds. The surface energy is minimized by having the surface reconstruct to reduce the total number of these bonds. The resultant structure is formed through a balance between eliminating as many dangling bonds as possible and the resultant stress caused in other bonds due to the displacements of the surface atoms. One example of this is the clean Si(111) surface, in which 49 surface atoms form a new periodic arrangement to make up the reconstructed Si(111)– (77) unit cell [2]. The number of dangling bonds is thereby reduced from 49 to 19.
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Inelastic Scattering: Electron Energy Loss Spectroscopy
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the surface region) can be expanded in terms of the normal displacements i from equilibrium according to
V r; ŒR D V r; ŒR 0 Low energy surface excitations such as phonons, plasmons N and electron-hole pair excitations can be studied using inX
ˇˇ C ri V r; ŒR ˇ i