Advances in Methods and Applications of Quantum Systems in Chemistry, Physics, and Biology (Progress in Theoretical Chemistry and Physics, 33) [1st ed. 2021] 3030683133, 9783030683139

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Progress in Theoretical Chemistry and Physics 33 Series Editors: Jean Maruani · Stephen Wilson

Alexander V. Glushkov Olga Yu. Khetselius Jean Maruani Erkki Brändas   Editors

Advances in Methods and Applications of Quantum Systems in Chemistry, Physics, and Biology

Advances in Methods and Applications of Quantum Systems in Chemistry, Physics, and Biology

Progress in Theoretical Chemistry and Physics VOLUME 33 Honorary Editors Rudolph A. Marcus (California Institute of Technology, Pasadena, CA, USA) Roy McWeeny (Università di Pisa, Pisa, Italy) Editors-in-Chief J. Maruani (formerly Laboratoire de Chimie Physique, Paris, France) S. Wilson (formerly Rutherford Appleton Laboratory, Oxfordshire, UK) Editorial Board E. Brändas (University of Uppsala, Uppsala, Sweden) L. Cederbaum (Physikalisch-Chemisches Institut, Heidelberg, Germany) A. Glushkov (Odessa State Environmental University, Odessa, Ukraine) E. K. U. Gross (Freie Universität, Berlin, Germany) K. Hirao (University of Tokyo, Tokyo, Japan) Chao-Ping Hsu (Institute of Chemistry, Academia Sinica, Taipei, Taiwan) R. Levine (Hebrew University of Jerusalem, Jerusalem, Israel) K. Lindenberg (University of California at San Diego, San Diego, CA, USA) A. Lund (University of Linköping, Linköping, Sweden) M. A. C. Nascimento (Instituto de Química, Rio de Janeiro, Brazil) P. Piecuch (Michigan State University, East Lansing, MI, USA) M. Quack (ETH Zürich, Zürich, Switzerland) S. D. Schwartz (Yeshiva University, Bronx, NY, USA) A. Tadjer (University Saint Kliment Ohridski, Sofia, Bulgaria) O. Vasyutinskii (Russian Academy of Sciences, St. Petersburg, Russia) Y. A. Wang (University of British Columbia, Vancouver, BC, Canada) Former Editors and Editorial Board Members I. Prigogine (†) J. Rychlewski (†) Y. G. Smeyers (†) R. Daudel (†) M. Mateev (†) W. N. Lipscomb (†) Y. Chauvin (†) H. W. Kroto (†) G. Delgado-Barrio (†) H. Ågren (*) V. Aquilanti (*) D. Avnir (*) J. Cioslowski (*)

W. F. van Gunsteren (*) H. Hubač (*) E. Kryachko (*) R. Lefebvre (*) M. P. Levy (*) G. L. Malli (*) P. G. Mezey (*) N. Rahman (*) S. Suhai (*) O. Tapia (*) P. R. Taylor (*) R. G. Woolley (*)

†: deceased; *: end of term More information about this series at http://www.springer.com/series/6464

Alexander V. Glushkov Olga Yu. Khetselius Jean Maruani Erkki Brändas •

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Editors

Advances in Methods and Applications of Quantum Systems in Chemistry, Physics, and Biology

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Editors Alexander V. Glushkov Department of Mathematics Odessa State Environmental University Odessa, Ukraine

Olga Yu. Khetselius Department of Mathematics Odessa State Environmental University Odessa, Ukraine

Jean Maruani LCP-MR CNRS and Sorbonne-Universités Paris, France

Erkki Brändas Department of Chemistry Uppsala University Uppsala, Sweden

ISSN 1567-7354 ISSN 2215-0129 (electronic) Progress in Theoretical Chemistry and Physics ISBN 978-3-030-68313-9 ISBN 978-3-030-68314-6 (eBook) https://doi.org/10.1007/978-3-030-68314-6 © Springer Nature Switzerland AG 2021 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, expressed 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

PTCP Aim and Scope

Progress in Theoretical Chemistry and Physics A series reporting advances in theoretical molecular and material sciences, including theoretical, mathematical and computational chemistry, physical chemistry and chemical physics, and biophysics.

Aim and Scope Science progresses by a symbiotic interaction between theory and experiment: Theory is used to interpret experimental results and may suggest new experiments; the experiment helps to test theoretical predictions and may lead to improved theories. Theoretical chemistry (including physical chemistry and chemical physics) provides the conceptual and technical background and apparatus for the rationalization of phenomena in the chemical sciences. It is, therefore, a wide-ranging subject, reflecting the diversity of molecular and related species and processes arising in chemical systems. The book series Progress in Theoretical Chemistry and Physics aims to report advances in methods and applications in this extended domain. It will comprise monographs as well as collections of papers on particular themes, which may arise from proceedings of symposia or invited papers on specific topics as well as from initiatives from authors or translations. The basic theories of physics—classical mechanics and electromagnetism, relativity theory, quantum mechanics, statistical mechanics, quantum electrodynamics— support the theoretical apparatus which is used in molecular sciences. Quantum mechanics plays a particular role in theoretical chemistry, providing the basis for the valence theories, which allow to interpret the structure of molecules, and for the spectroscopic models, employed in the determination of structural information from spectral patterns. Indeed, quantum chemistry often appears synonymous with theoretical chemistry; it will, therefore, constitute a major part of this book series. However, the scope of the series will also include other areas of theoretical chemistry, v

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such as mathematical chemistry (which involves the use of algebra and topology in the analysis of molecular structures and reactions); molecular mechanics, molecular dynamics, and chemical thermodynamics, which play an important role in rationalizing the geometric and electronic structures of molecular assemblies and polymers, clusters, and crystals; surface, interface, solvent, and solid-state effects; excited-state dynamics, reactive collisions, and chemical reactions. Recent decades have seen the emergence of a novel approach to scientific research, based on the exploitation of fast electronic digital computers. Computation provides a method of investigation which transcends the traditional division between theory and experiment. Computer-assisted simulation and design may afford a solution to complex problems which would otherwise be intractable to theoretical analysis and may also provide a viable alternative to difficult or costly laboratory experiments. Though stemming from theoretical chemistry, computational chemistry is a field of research in its own right, which can help to test theoretical predictions and may also suggest improved theories. The field of theoretical molecular sciences ranges from fundamental physical questions relevant to the molecular concept, through the statics and dynamics of isolated molecules, aggregates and materials, molecular properties and interactions to the role of molecules in the biological sciences. Therefore, it involves the physical basis for geometric and electronic structure, states of aggregation, physical and chemical transformations, thermodynamic and kinetic properties, as well as unusual properties such as extreme flexibility or strong relativistic or quantum field effects, extreme conditions such as intense radiation fields or interaction with the continuum, and the specificity of biochemical reactions. Theoretical chemistry has an applied branch (a part of molecular engineering), which involves the investigation of structure–property relationships aiming at the design, synthesis, and application of molecules and materials endowed with specific functions, now in demand in such areas as molecular electronics, drug design or genetic engineering. Relevant properties include conductivity (normal, semi-, and super-), magnetism (ferro- and ferri-), optoelectronic effects (involving nonlinear response), photochromism and photoreactivity, radiation and thermal resistance, molecular recognition and information processing, biological and pharmaceutical activities, as well as properties favoring self-assembling mechanisms, and combination properties needed in multifunctional systems. Progress in Theoretical Chemistry and Physics is made at different rates in these various research fields. The aim of this book series is to provide timely and in-depth coverage of selected topics and broad-ranging yet detailed analysis of contemporary theories and their applications. The series will be of primary interest to those whose research is directly concerned with the development and application of theoretical approaches in the chemical sciences. It will provide up-to-date reports on theoretical methods for the chemist, thermodynamician or spectroscopist, the atomic, molecular or cluster physicist, and the biochemist or molecular biologist who wish to employ techniques developed in theoretical, mathematical, and computational chemistry in their research programs. It is also intended to provide the graduate student with a readily accessible documentation on various branches of theoretical chemistry, physical chemistry, and chemical physics.

Preface

This volume collects 16 selected contributions from world-leading quantum scientists presented at the Twenty-Fourth International Workshop on Quantum Systems in Chemistry, Physics, and Biology (QSCP-XXIV), organized by Pr. Alexander Glushkov from Odessa State Environmental University (OSENU) at the Mozart Group Complex Odessa during August 18–24, 2019. Close to 100 scientists from 28 countries attended this meeting. The participants discussed the state of the art, new trends, and future evolution of concepts and methods, and their applications to a broad variety of problems in theoretical and computational chemistry, physics, and biology. The high-level attendance attained at this conference was particularly gratifying. It is the renowned interdisciplinary structure and friendly conviviality of QSCP meetings that make them so successful discussion forums. The workshop took place at the Mozart Group Complex Odessa, in connection with OSENU. Odessa is the largest city of Ukraine, a major tourist resort and a seaport and transport hub located on the northwestern shore of the Black Sea. Odessa was the site of a large Greek settlement as early as the middle of the sixth century BC. Nowadays, Odessa is often called the “Pearl of the Black Sea” or “Southern Palmyra”. Details of the Odessa QSCP meeting, including the scientific and social programs, can be found on the website: https://sites.google.com/view/qscp-2019. Altogether, there were 16 morning and afternoon sessions, where 36 plenary lectures and invited talks were given, and 3 evening poster sessions, with 72 posters being displayed. We are grateful to all the participants for making the QSCP-XXIV workshop a stimulating experience and a great success. QSCP-XXIV followed the traditions established at previous workshops: QSCP-I, organized by Roy McWeeny in 1996 at San Miniato (Pisa, Italy); QSCP-II, by Stephen Wilson in 1997 at Oxford (England); QSCP-III, by Alfonso Hernandez-Laguna in 1998 at Granada (Spain); QSCP-IV, by Jean Maruani in 1999 at Marly-le-Roi (Paris, France); QSCP-V, by Erkki Brändas in 2000 at Uppsala (Sweden); QSCP-VI, by Alia Tadjer and Yavor Delchev in 2001 at Sofia (Bulgaria); vii

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QSCP-VII, by Ivan Hubac in 2002 near Bratislava (Slovakia); QSCP-VIII, by Aristides Mavridis in 2003 at Spetses (Athens, Greece); QSCP-IX, by J.-P. Julien in 2004 at Les Houches (Grenoble, France); QSCP-X, by Souad Lahmar in 2005 at Carthage (Tunisia); QSCP-XI, by Oleg Vasyutinskii in 2006 at Pushkin (St. Petersburg, Russia); QSCP-XII, by Stephen Wilson in 2007 near Windsor (London, England); QSCP-XIII, by Piotr Piecuch in 2008 at East Lansing (Michigan, USA); QSCP-XIV, by G. Delgado-Barrio in 2009 at El Escorial (Madrid, Spain); QSCP-XV, by Philip Hoggan in 2010 at Cambridge (England); QSCP-XVI, by Kiyoshi Nishikawa in 2011 at Kanazawa (Japan); QSCP-XVII, by Matti Hotokka in 2012 at Turku (Finland); QSCP-XVIII, by M.A.C. Nascimento in 2013 at Paraty (Rio, Brazil); QSCP-XIX, by Cherri Hsu in 2014 at Taipei (Taiwan); QSCP-XX, by Alia Tadjer and Rossen Pavlov in 2015 at Varna (Bulgaria); QSCP-XXI, by Yan A. Wang in 2016 at Vancouver (BC, Canada); QSCP-XXII, by S. Jenkins and S. Kirk in 2017 at Changsha (China); QSCP-XXIII, by Liliana Mammino in 2018 at Kruger Park (South Africa). The lectures presented at QSCP-XXIV in the field of Quantum Systems in Chemistry, Physics, and Biology were grouped into eight areas, ranging from Concepts and Methods in Quantum Science to Computational Chemistry, Physics, and Biology through Relativistic Effects in Chemistry and Physics, Molecules and Atoms in Strong Electric and Magnetic Fields, Reactive Collisions and Chemical Reactions, Molecular Structure, Dynamics and Spectroscopy, and Molecular and Nano-materials. The width and depth of the topics discussed at QSCP-XXIV are reflected in the contents of this volume of the Springer book series Progress in Theoretical Chemistry and Physics, https://www.springer.com/series/6464, which includes four sections: I. II. III. IV.

Atomic Systems (5 papers); Molecular Systems (5 papers); Biochemistry and Biophysics (3 papers); Quantum Theory and Life Sciences (3 papers).

In addition to the scientific program, the workshop had its usual share of cultural events. There was a show at the world famous Odessa Opera and Ballet Theater, a visit to the Odessa Dolphinarium, and a Black Sea cruise. The award ceremony of the CMOA Prize and Medal took place during the traditional banquet. The CMOA Prize for junior scientists was shared between the two selected nominees: Joel Yuen-Zhou (University of California, San Diego, USA) and Eugeny Ternovsky (OSENU, Odessa, Ukraine). The prestigious CMOA Medal for senior scientists was awarded to Pr. Ria Broer (University of Groningen, The Netherlands).

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The venue of the following QSCP workshop was announced to be at Torun, Poland, in July 2020. But due to the Covid-19 pandemic, it was later postponed to July 2022. We are most grateful to the Local Patronage and Organization Committees for their help and dedication, which made the stay and work of participants both pleasant and fruitful. We would like to thank the personnel of the Mozart Group Complex Odessa as well as that of OSENU for their efficient logistics. Last but not least, we thank the International Scientific and Honorary Committees for their invaluable expertise and advice. We hope this book will provide timely and in-depth coverage of the latest advances in the theory and applications of quantum systems, and a basis for evaluating and generating new ideas in the widespread domains of chemistry, physics, and biology. It should be of interest primarily to those whose research is related to the development and applications of novel theoretical approaches in quantum chemistry, physics and biology. Odessa, Ukraine Odessa, Ukraine Paris, France Uppsala, Sweden

Alexander V. Glushkov Olga Yu. Khetselius Jean Maruani Erkki Brändas

Tribute and Recollections

Stephen Wilson (1950–2020) Stephen Wilson was a British scientist whose work focused on theoretical and computational molecular sciences and scientific computation. He pioneered the application of many-body theory to molecular systems, which became a widely used approach to electron correlation. His seminal book Electron Correlation in Molecules (Clarendon, Oxford, 1984) remains a classic in the field and a continuing inspiration to new generations of researchers. Stephen Wilson was born in Market Harborough, Leicestershire, UK, on February 2, 1950, to Hilda and Raymond Wilson. His marriage to Kate on April 5, 1980, proved to be a lifelong partnership and together they had two sons, James and Jonathan. Stephen was educated at King Edward-VII County Grammar School, where he was a pupil during 1961-1968. From his earliest years, he discovered not only the joys of chemistry, but also the pitfalls of experimentation. As an 8-year-old, he tried to make ginger beer, only to have the bottles explode in his mother’s conservatory, inviting the attention of swarms of wasps. By the age of ten, he was already well known at the local chemist’s shop. One day, when he appeared with a list for gunpowder, he was met by a firm “No!” from the kindly proprietor. It was no doubt that these formative experimental setbacks made him favor a career in Theoretical Science. Stephen gained his B.Sc. in Chemistry (1971) and his Ph.D. in Theoretical Chemistry (1975) under Joseph Gerratt. His thesis title was: A self-consistent group model for molecular wave functions. Pr. Roy McWeeny was Stephen’s Ph.D. examiner and there began an important mentorship and later friendship which was to stretch over Stephen’s lifetime. In 2018, Stephen finished editing Roy’s six volumes in the series Basic Books for Science, which Roy wanted to make available on the Internet free of charge. Roy’s wife, Virginia, decided not to tell her husband about Stephen’s death as she felt Roy, being 96, would not cope well with the sad memory of the young student whose talent he had seen come to fruition.

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In 1990, Stephen received a D.Sc. from Bristol University for his outstanding contribution to the advancement of Theoretical Chemistry. As acknowledged by Peter Grout, his long-term scientific colleague and friend, Stephen was not only a scholar with an encyclopedic knowledge in his field, he also belonged to that rare breed of truly innovative and creative thinkers. For Steve, his pursuit of science was not motivated by self-interest. He quietly avoided contention, focusing his energies on the business of furthering his field. This is not to say he was without ambition. He sought progression in the USA, where he worked on electron correlation as a postdoctoral fellow at Johns Hopkins University (1974–1976). He then became a NAS/NRS Research Associate at the NASA Goddard Institute for Space Studies in New York, where he remained until 1977. There he became interested in interstellar radicals and ions, successfully employing ab initio quantum chemical methods to identify three interstellar radicals and ions in the dust around a carbon star. This achievement fired Stephen's determination to investigate further new approaches to the electronic molecular structure. Steve was later to return to the UK as a Senior Research Associate on the Collaborative Project Electron Correlation in Molecules, at the Daresbury Laboratory in Cheshire. It was during this stay in England that he met his future wife and so the other strand of his life began, with new stars in his eyes. The recently married young couple then found themselves moving to Oxford, where Stephen’s second great success had brought them: the award of an SRC Advanced Fellowship (1980– 1985). During his time in the quaint Victorian House which was then the Department of Theoretical Chemistry in Oxford, Stephen was a prolific researcher and author. On its top floor, at the end of the rear staircase, he wrote his masterpiece: Electron Correlation in Molecules. During these 5 years in Oxford, he investigated the many-body perturbation theory of electron correlation, van der Waals interactions in inert gases and mixtures, relativistic electronic structure theory, and the fundamentals of the basis-set expansion technique. Stephen then moved on as Principal Scientific Officer to the Computational Science and Engineering Department of the Rutherford Appleton Laboratory, where he stayed from 1987 to 2007. There he worked on the many-body problem and relativistic quantum chemistry for molecules containing heavy atoms. He also worked on the development of parallel processing and high-accuracy approximation techniques in molecular electronic structure calculations. He brought methods of high-energy particle physics to theoretical chemistry by developing techniques inspired by Feynman diagrams, cutting through algebraic complexity in their implementation on high-performance computers. In 1990, with David Moncrieff of Florida State University, Stephen held a world record for writing a computer code based on this method which approached the theoretical maximum speed of the fastest computer in the world. It was an astounding result, for which he found himself awarded the title “Flop of the Year”. As time progressed, starting in the 1980s, the climate in Britain moved away from “blue-sky” research to commoditized science. In the words of Peter Grout, the most accurate description of Stephen Wilson’s professional activities is that he was a scholar, dedicated to the development of new ways of imagining the underlying

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machinery of the material world .... Steve’s scholarship led him to follow his own path and gain new insights into uncharted research areas. He [built] his own collaborative community in Britain and Europe and participated in a vibrant series of conferences and workshops over almost two decades. Stephen organized a number of Summer schools, notably at Jesus College in Oxford in 1997, 2001, 2002 and, in subsequent years, at Cosener’s House, the Rutherford Appleton conference center in Abingdon. For 10 years, he lectured at the Charles Coulson Summer Schools, with their multi-disciplinary approach across chemistry, physics, and biology. He also directed three NATO ASI’s and co-chaired ISTCP-IV in Paris in 2002. Together with J. Maruani (Paris), Stefan Christov (Sofia), Roy McWeeny (Pisa), Y. G. Smeyers (Spain), and E. Brändas (Uppsala), he founded the workshop series Quantum Systems in Chemistry and Physics (QSCP), and became a co-executive editor of the present Springer book series: Progress in Theoretical Chemistry and Physics (PTCP). The whereabouts of Stephen’s involvement in QSCP and PTCP are recalled in the Illustrated Overview of the Origins and Development of the QSCP Meetings which appeared in PTCP-B19 (Springer, 2009). Stephen published over 250 papers, mainly in Theoretical and Computational Chemistry but also in Computing Science and Numerical Analysis. He also produced many books, in particular Chemistry by Computer (Plenum, NY, 1986). After retirement in 2007, he co-authored a monograph with Ivan Hubac on the Brillouin-Wigner Methods for Many-body Systems (PTCP-A21). The collaboration with Ivan was to prove to be lasting and rich. Indeed, their final paper on Quantum Entanglement and Quantum Information in Biological Systems (published by Comenius University), using the Majorama quasi-particle formalism, conjectured that quantum mechanics may have a role in the information transfer across the hydrogen bonds between the base pairs of the DNA molecule. Aside from his own authored books and papers, Stephen edited 21 volumes. He was the Series Editor of Methods in Computational Chemistry and Physics and a Joint Editor-in-Chief of Progress in Theoretical Chemistry and Physics. A particular achievement of which he was very proud was his undertaking as an Editor-in-Chief of the Handbook of Molecular Physics and Quantum Chemistry, published in 3 volumes by Wiley. It was a huge work which took a great deal of his time and energy. In 2001, he wrote to Jacek Kobus, with whom he was to collaborate extensively in association with David Montcrieff over 15 years: “I was sorry I was not able to come (to the Kuusamo Symposium in Finland), but I seem to be doing nothing but ‘Handbook work’ at the moment.” In his eulogy at Steve’s funeral, Peter Grout spoke of his long-running collaboration with Ian Grant, FRS of Oxford University, as well as with Harry Quiney. He especially mentioned the legacy of that work at the interface of quantum chemistry and special relativity: “Harry … wishes to acknowledge his debt to Steve in the scientific training he received at Oxford … Steve was very generous of his time with both undergraduate and postgraduate students. He was always mindful of their difficulties often before they knew them themselves.”

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From the many letters of condolence that his wife received from his colleagues around the world, it is clear that Steve could do little else throughout his scientific life but reciprocate their innumerable ways of friendship. This he did wherever his advice or help could be of use. The scientific community to which he belonged was a vital and sustaining one. Without its energy and inspiration, his work would have been the less. Stephen Wilson died on September 2, 2020, in Oxford. He is survived by his wife, their two sons, and four small granddaughters. Stephen was, in the respective opinions of Peter Grout and Brian Sutcliffe, “a gentleman” and “a good man”. He will be remembered and missed by us all.

Picture taken by Jacek Karwowski near Windsor in 2007

Kate Wilson (December 2020) In consultation with Peter Grout, Harry Quiney, Ivan Hubac, Jacek Karwowski, and Jacek Kobus

Contents

Atomic Systems Auger Spectroscopy of Multielectron Atoms: Generalized Energy Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander V. Glushkov Advanced Relativistic Energy Approach in Electron-Collisional and Radiative Spectroscopy of Ions in Plasmas . . . . . . . . . . . . . . . . . . . Vasily V. Buyadzhi, Eugeny V. Ternovsky, Alexander V. Glushkov, and Anna A. Kuznetsova The Schrödinger Equation with Power Potentials: Exactly-Solvable Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jacek Karwowski and Henryk A. Witek Electron-b-Nuclear Spectroscopy of Atomic Systems and Many-Body Perturbation Theory Approach to Computing b-Decay Parameters . . . . Olga Yu. Khetselius, Valentin B. Ternovsky, Yulia V. Dubrovskaya, and Andrey A. Svinarenko Relativistic Quantum Chemistry and Spectroscopy of Some Kaonic Atoms: Hyperfine and Strong Interaction Effects . . . . . . . . . . . . . . . . . . Olga Yu. Khetselius, Valentin B. Ternovsky, Inga N. Serga, and Andrey A. Svinarenko

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Molecular Systems Atomic Electric Multipole and Polarizability Models for C6 X6 Molecules (X = F, Cl, Br) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Claude Millot

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A Quasiparticle Fermi-Liquid Density Functional Approach to Atomic and Diatomic Systems: Spectroscopic Factors . . . . . . . . . . . . . . 133 Alexander V. Glushkov, Anna V. Ignatenko, Andrey V. Tsudik, and Alexei L. Mykhailov Molecular Photoionization and Photodetachment Cross Sections Based on L2 Basis Sets: Theory and Selected Examples . . . . . . . . . . . . . 151 Bruno Nunes Cabral Tenorio, Sonia Coriani, Alexandre Braga Rocha, and Marco Antonio Chaer Nascimento Advanced Quantum Approach to Calculation of Probabilities of the Cooperative Electron-cc Vibrational-Nuclear Transitions in Spectra of Diatomics Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Alexander V. Glushkov, Eugeny V. Ternovsky, Valery F. Mansarliysky, and Pavel A. Zaichko Advanced Quantum-Kinetic Model of Energy Exchange in Atmospheric Molecules Mixtures and CO2 Laser-Molecule Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Olga Yu. Khetselius, Alexander V. Glushkov, Sergiy M. Stepanenko, Andrey A. Svinarenko, and Vasily V. Buyadzhi Biochemistry and Biophysics Roles of the Phenol OHs for the Reducing Ability of Antioxidant Acylphloroglucinols. A DFT Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Liliana Mammino Complexes in which Two Hyperjovinol-A Molecules Bind to a Cu2+ Ion. A DFT Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Liliana Mammino Adducts of Hydroxybenzenes with Explicit Acetonitrile Molecules. An ab initio and DFT Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Liliana Mammino Quantum Theory and Life Sciences Nonlinear Dynamics of Complex Neurophysiologic Systems Within a Quantum-Chaos Geometric Approach . . . . . . . . . . . . . . . . . . . 291 Alexander V. Glushkov and Olga Yu. Khetselius A Universe in Our Brain: Carnot’s Engine and Maxwell’s Demon . . . . 305 Erkki J. Brändas Structure Waves in Biopolymers and Biological Evolution Paths . . . . . . 331 Jean Maruani Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

Atomic Systems

Auger Spectroscopy of Multielectron Atoms: Generalized Energy Formalism Alexander V. Glushkov

Abstract An advanced version of relativistic energy formalism in the Auger spectroscopy of multielectron atomic systems is developed in order to calculate the fundamental energetic and spectroscopic parameters of the Auger decay process. The approach originally uses the Gell-Mann and Low adiabatic formulae in order to calculate an autoionization and Auger decays probabilities as well as the radiative oscillator strengths. The electron structure of a multielectron atom is calculated on the basis of the relativistic many-body perturbation theory (RMBPT) with ab initio model zeroth approximation and a correct accounting for the exchange-polarization corrections as the second and higher orders perturbation theory contributions. In order to provide gauge invariance performance, the RMBPT optimized zeroth approximation is generated on the basis of the relativistic criterion of minimization of the RMBPT second and higher orders exchange-polarization diagrams contributions into imaginary part of the atomic level energy shift. As an illustration, the results of computing the energy and spectral parameters of the resonant Auger decay for neon atomic system as well as some solids are listed. The results are compared with available experimental results as well as with the results, obtained within calculation on the basis of different semiempirical and ab initio methods. In whole there is a physically reasonable agreement between new theory results and experimental data. Keywords Auger spectroscopy · Multielectron atoms · Generalized relativistic energy formalism · Exchange-correlation effects

1 Introduction The accurate data on radiative, autoionization, Auger decay widths and probabilities, oscillator strengths and the corresponding energy parameters for different atomic, molecular systems as well as solids are of a great importance for different application in atomic, molecular and laser physics, quantum electronics, astrophysical analysis, A. V. Glushkov (B) Odessa State Environmental University, L’vovskaya str., bld. 15, Odessa 65016, Ukraine e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. V. Glushkov et al. (eds.), Advances in Methods and Applications of Quantum Systems in Chemistry, Physics, and Biology, Progress in Theoretical Chemistry and Physics 33, https://doi.org/10.1007/978-3-030-68314-6_1

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A. V. Glushkov

laboratory, thermonuclear plasma diagnostics, fusion research, etc. [1–159]. One could remind that the Auger electron spectroscopy is an effective method to study the electron structure of atomic and molecular systems, chemical composition of solid surfaces and near-surface layers etc. [1–8]. Study of the Auger spectra in atomic systems and solids gives the important data for the whole number of scientific and technological applications. In order to define the Auger decay process, one usually uses a terminology [1– 3] of a radiationless transition of an atomic system with an inner-shell vacancy in an initial state to a final state with inner vacancy, which is filled by an outer-shell electron. Simultaneously, a kinetic energy of the ejected Auger electron is determined within measured by Auger-electron spectroscopy. The key task is to calculate various Auger transition energies and autoionization, Auger decay widths and probabilities of atomic, molecular systems as well as solids. Usually one widely uses so called two-step model when calculating the Auger decay characteristics. Since the vacancy lifetime in an inner atomic shell is rather long (about 10−17 – −14 10 s), the atom ionization and the Auger emission are considered to be two independent processes. In the more correct dynamic theory of the Auger effect [1–4] the processes are not believed to be independent from one another. The fact is taken into account that the relaxation processes due to Coulomb interaction between electrons and resulting in the electron distribution in the vacancy field have no time to be over prior to the transition. In fact, a consistent Auger decay theory has to take into account correctly a number of correlation effects, including the energy dependence of the vacancy mass operator, the continuum pressure, spreading of the initial state over a set of configurations etc. [9–20]. The review of different theoretical approaches to computing the Auger decay characteristics can be found in Refs. [2–18]. In particular [18], a few theoretical schemes are ranging from semiempirical [4–6] to more sophisticated, ab initio methods, to calculate the Auger energy. As a rule, initially the correlation and relaxation effects arising from the response of the spectator orbitals to the creation of vacancies are partially or fully ignored. The Hartree-Fock (HF) approach has been applied to computing the Auger process parameters by Larkins and the nonrelativistic HartreeFock-Slater method with the use of local-density formalism is applied by McGuire (e.g. [3–6]). The most widespread theoretical studying the Auger spectra parameters is based on using the multi-configuration Dirac-Fock (MCDF) calculation [1–3]. The theoretical predictions based on MCDF calculations have been carried out within different approximations. One should remind using the multiconfiguration DiracFock (MCDF) method by Briancon and Desclaux, Bruneau [8] to calculate the KLL Auger transition energies of various atoms. The Dirac-Hartree-Slater (DHS) method by Chen et al. and the relativistic correlated local-density method by Vijayakumar and Gopinathan had been used to compute various Auger transition energies for processes like K-LL and L-MM transitions for a number of atomic systems. All calculation have demonstrated a great importance of accounting for the relativistic and correlation effects in order to predict the Auger energies with an adequate accuracy (e.g. [1–64]).

Auger Spectroscopy of Multielectron Atoms …

5

In this chapter we will present the fundamentals of a generalized relativistic energy formalism to Auger effect in atomic systems, solids and develop an advanced theoretical method to computing Auger electron energy and spectral parameters for complex atomic systems. It should be reminded that a model energy approach to autoionization and Auger processes in multielectron atoms and ions is developed by Ivanova-Ivanov et al. (see details, in Refs. [115–122]). A generalized gauge invariant version of the formalism been further developed in Refs. [118–122, 136–141]). Here, it is presented an effective approach to computing the radiative (imaginary part of energy shift) and Auger and autoionization decays (real part of energy shift) parameters for atomic (generally speaking any quantum) systems. The important point of computational scheme is provided by using the optimized relativistic wave functions basis set and correct treatment of the complex exchange-polarization corrections.

2 Relativistic Theoretical Method to Computing Auger Decay Energies and Widths in Multielectron Atoms 2.1 Auger Decay and an Energy Formalism Below we will present the key points of an effective relativistic method for computing the Auger process parameters of multielectron atoms. It should be noted that for obvious reasons, an energy-energy approach formulated below in Auger spectroscopy of atomic systems is practically analogous to the relativistic energy formalism of the decay of autoionization states in many-electron atoms (e.g. [2, 3] and Refs. therein). As a consequence, below one should consider fundamentals of a basic approach and only the new elements, related to the Auger spectroscopy of multielectron atoms. The key aspects of a method are in determination of an electron energy shift δE and construction of the corresponding secular matrix M (see details in refs. [115– 120]). An energy shift of any excited state of the multielectron atomic system can be represented as follows: δ E = Reδ E + iImδ E, Imδ E = −P/2

(1)

where Γ is an atomic level width. Generally speaking, it includes as the radiation as the autoionization widths simultaneously. It is worth to remind that it is further very suitable to expand the secular matrix elements in a perturbation theory (PT) series for the interelectron interaction [114, 115]: M = M (0) + M (1) + M (2) + M (3) , where M (0) is the energy contribution of the vacuum diagrams of all order of PT, and M (1) , M (2) , M (3) those of the one-, twoand three- quasiparticle diagrams respectively.

6

A. V. Glushkov

The complete description of matrix elements and connection with the corresponding series of Feynman diagrams is given in [114–120]. As usually, the secular matrix is computed between the whole groups of states with the same parity and total moment J. One could use two ways to refine the first order calculation. One is to enlarge the secular matrix dimension by adding new excited states; in this case each secular matrix element is calculated, the same as before, in the first order of PT. The second way is the inclusion of high order corrections by means of the special multi-particle polarization operator, not changing the matrix dimension. It is also worth to note an importance of the second order PT (atomic PT) diagrams contributing the imaginary and real energy part related to the investigated transitions, including the main direct and exchange polarization diagrams (fourth order of a quantum electrodynamical PT). The contribution of the direct and exchange polarization diagrams into the energy shift and the corresponding matrix elements can be consistently determined within the potential approach [13]. New effective twoquasiparticle polarizable operators had been earlier derived in the nonrelativistic approximation in Ref. [114, 115] and in the relativistic one in Ref. [112]. The technique of determination of the matrix elements of these polarization potentials as well as the PT first order matrix elements has been presented in detail in Refs. [114–135]. The Auger width is obtained from the adiabatic Gell-Mann and Low formula for the energy shift [3]. The direct contribution (accounting for a contribution of key direct Feynman diagrams) to the Auger level width with a vacancy nα l α jα mα is as follows:  λ

  2 Q λ (αkγβ)Q λ (βγ kα), (λ)( jα ) βγ ≤ f k> f

(2)

while the exchange diagram contribution is:   2    j j λ Q λ1 (αkγβ)Q λ2 (βγ kα) α γ 2 jk jβ λ1 ( jα ) λ λ βγ ≤ f k> f

(3)

1 2

The partial items of the

 βγ k

sum answer to contributions of α−1 → (βγ)−1 K

channels resulting in formation of two new vacancies βγ and one free electron k: ωk = ωα + ωβ – ωα . Within the frame of the RMBPT approach the Auger transition probability and the Auger line intensity are defined by the square of an electron interaction matrix element having the form [2–4]: ω V1234

1  = [( j1 )( j2 )( j3 )( j4 )[ / 2 (−1)μ λμ



j1 j3 λ m1 − m3 μ

 × ReQ λ (1234);

Br Q λ = Q Qul λ + Qλ .

(4)

Auger Spectroscopy of Multielectron Atoms …

7

The terms Q Qul and Q Br λ correspond to subdivision of the interparticle potenλ tial into the Coulomb part cos|ω|r 12 /r 12 and the Breit one, cos|ω|r 12 α 1 α 2 /r 12 . The Coulomb part Q Qul λ is expressed in terms of radial integrals Rλ, angular coefficients S λ according to Ref. [4]. The Breit interaction is known to change considerably the Auger decay dynamics in some cases. The concrete formulas for the Coulomb and Breit parts are presented in Refs. [4, 114–120, 128–130]. Expanding in more detail, the Auger decay width is expressed in terms of the sum of products of radial integrals and angular coefficients:  YaJ n 1 j1l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3l3 

j3 j4 J j4 − j2 +J +1 = (−1) j2 j1 a (2 j1 + 1)(2 j2 + 1)(2 j3 + 1)(2 j4 + 1)

  j1 j3 a j2 j4 a × 1/2 −1/2 0 1/2 −1/2 0   {al1l3 }{al2 l4 }Wa n 1 j1l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3l3    + (−1)l+a (2a + 1)Val n 1 j1l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3l3

(5)

l

    Here jl = j l and a1 l1 l3 denote the triangle condition for moments a1 ,l1 ,l3 . The quantities W and V are connected with the Coulomb and Breit parts of the interelectron interaction as follows:  Wa n 1 j1l1 n 2 l2 j2 ; n 4 l4 j4 n 3 j3l3 .  = Ra n 1 j1l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3l3   ∼ ∼ + Ra n 1 j1l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3l3  ∼  ∼ + Ra n 1 j1 l1 n 2 j2 l2 ; n 4 j4 l4 n 3 l3 j3  ∼  ∼ ∼ ∼ + Ra n 1 j1 l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3 l3 .  Val n 1 j1l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3l3    ∼ ∼ = Rl n 1 j1l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3 l3 X al j1l1 ;  ∼   ∼  + Rl n 1 j1 l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3l3 X al j1l1 ;  ∼   ∼  + Rl n 1 j1 l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3l3 X al j1l1 ;    ∼ ∼ + Rl n 1 j1l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3 l3 X al j1l1 ;

  j3l3 X al j2 l2 ;   j3l3 X al j2 l2 ;  j3l3 X al j2 l2 ;    j3l3 X al j2 l2 ;

(6)



j4 l4 j4 l4 

j4 l4 j4 l4



(7)

8

A. V. Glushkov

The definitions of the angular coefficients are presented in Refs. [114–120]. It is important to note that a probability of the Auger decay can be expressed through the matrix elements Re Qλ (12; 43), which differ from the imaginary matrix elements (the latter determines a probability of radiative decay) only by the definition of radial integrals: ∞ ReRλ (n 1 j1l1 n 2 j2 l2 ; n 4 j4 l4 ) =

∞ dr1r12

0

dr2 r22 0

rλ+1

Z λ(1) (ωr< )

Z λ(2) (ωr> )Rn 1 j1 l1 (r1 )Rn 2 j2 l2 (r2 )Rn 4 j4 l4 (r2 )Rn 3 j3 l3 (r1 )

(8)

and functions Z(1) and Z(2) are connected with the standard Bessel functions (see details, for example, in Ref. [115]). The calculation of all matrix elements, wave functions, Bessel functions etc. is reduced to solving the system of differential equations. The formulas for the autoionization (Auger) decay probability include the radial integrals where one of the functions describes electron in the continuum state. When calculating these integral, obviously, the correct normalization of the wave functions (see details in Refs. [115–120] is very important).  The integral Rl n 1 j1l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3l3 as well as the functions Z(1) and Z(2) can be effectively computed from the system of standard differential equations using the known universal and very effective Ivanova-Ivanov differential equations method [114–116]. Other details of calculational scheme as well as the PC atomic code “Superatom-ISAN” (the modified version 93) can be found in Refs. [4, 11, 12, 95, 114–130, 141–155].

2.2 The Elements of a Relativistic Many-Body Perturbation Theory and an Optimized One-Quasiparticle Representation As method of calculation of a multielectron atomic systems we use earlier developed formalism of the RMBPT with the Dirac-like zeroth approximation (see details, for example, in Refs. [2, 9–12, 91–96]). Below, the main emphasis will be placed on new elements of the method related to Auger spectroscopy. The relativistic Hamiltonian of multielectron atomic system can be wriritten as follows (in the atomic units): H=

 i

{αcpi − βc2 − Z /ri } +

 i> j

exp(i|ω|ri j )(1 − αi α j )/ri j ,

(9)

Auger Spectroscopy of Multielectron Atoms …

9

where α i β are the Dirac matrices, ωij is the transition frequency, Z is the charge of the atomic nucleus, and c is the speed of light. As a next step, one should construct the RMBPT Dirac-Kohn-Sham (DKS) zeroth approximation with the potential [4, 120, 148–150, 152, 155]: D (r ) + VX (r ) + VC (r |a)] V D K S (r ) = [VCoul

(10)

D where VCoul (r ) is the standard Coulomb potential, VX (r ) is the known Kohn-Sham (KS) exchange potential and VC (r |a) is the corresponding correlation functional; the latter has been taken in the form [4]. Naturally, the potential (10) is subtracted from the interelectron potential in (9) in a perturbation operator. The QED PT perturbation is then:

−V D K S (r ) − Jμ (x)Aμ (x),

(11)

where A is the vector of the electromagnetic field potential, J is the current operator. Next, the matrix M is constructed. The important aspect of the whole procedure is a correct treatment of the exchangepolarization effects such as the polarization and screening inter-quasi-particle interaction, a continuum pressure, an energy dependence of a mass self-energy operator etc. The entire description of the corresponding Feynman diagrams and the corresponding analytical expressions for matrix elements of the polarization and screening interaction potentials is given in Refs. [95, 114–135, 141–179]. In fact one should use the generalized relativistic Kohn-Sham density functional in the RMBPT zeroth approximation; naturally, the perturbation operator contents the interaction operator in (9) minus the zeroth approximation potential. Further the wave functions are corrected by accounting of the first order PT contribution. New effective twoquasiparticle polarizable operators had been earlier derived in the nonrelativistic approximation in Refs. [114, 115] and in the relativistic one in Ref. [120]. The technique of determination of the matrix elements of these polarization potentials as well as the PT first order matrix elements has been presented in detail in Refs. [114–135]. The important block of the calculational procedure is connected with using a procedure of optimization of the relativistic orbitals base. The main idea is reduced to minimization of the gauge dependent multielectron contribution Im E ninv of the lowest QED PT corrections (diagrams b, c in Fig. 1) to the radiation widths of atomic levels. According to Ref. [120], the standard exchange Hartree-Fock diagram contributes into imaginary part ImδE α as follows: Imδ E α (a)

 s

Imδ E(α − n; a).

(12)

10

A. V. Glushkov

In an energy approach the radiative decay probability is expressed through the imaginary part of electron energy of the system [114, 120] by means of the integrals  ImRλ n 1 j1l1 n 2 j2 l2 ; n 4 j4 l4 n 3 j3l3 . Further, according to the Ref. [118], one should determine the second order RMBPT (fourth order QED PT) direct and exchange polarization diagrams contribution into imaginary part ImδE and the corresponding result in the first case is as follows:     e2 dr1 dr2 dr3 dr4 I mδ E ninv (α − s|b) = −C 4π  1 1 ( + ) ω + ω ω − ωαs mn αs mn n> f,m≤ f × α+ (r1 )m+ (r2 )s+ (r4 )n+ (r3 ) · [(1 − α1 α2 )/r12 ] · {[α3 α4 − (α3 n 34 )(α4 n 34 )]/r34 × sin[ωαn (r12 + r34 )] + [1 + (α3 n 34 )(α4 n 34 )]ωαn cos[ωαn (r12 + r34 )]} × m (r3 )α (r4 )n (r2 )s (r1 ).

(13)

where the condition m ≤ f indicates the finite number of states in the core and the states of the negative continuum; C is the gauge constant. The optimized onequasiparticle representation is constructed by means of the minimization of the functional Im δE ninv , which leads to the integro-differential Dirac-Kohn-Sham-like equation for the ρ c . The description of the whole procedure can be found in Refs. [112, 118–122]. More consistent version is presented in Ref.[123]. In the Refs. [114, 118–123] it is also presented a scheme to take into account for continuum states within the generalized Sturm expansions method. Among examples of the effective using the Sturm expansions in various problems in atomic and molecular physics one should remind the schemes by Dalgarno et al., Buchachenko et al., Ivanova-Ivanov et al., Gruzdev et al., Glushkov et al. etc. (e.g., [1–4, 11, 12, 114]). Here we use the scheme [114]. Other details can be found in Refs. [4, 11, 12, 95, 114–130, 136–155]).

3 Auger Spectroscopy of Complex Atomic Systems and Solids: Illustrative Results 3.1 Auger Spectroscopy of Neon As illustration, below we list some results of computing the Auger transition energies in the neon and some solids. In Table 1 we present the data on the transition energies and angular anisotropy parameter β (for each parent state) for the resonant Auger

Auger Spectroscopy of Multielectron Atoms …

11

Table 1 Transition energies Ek (for each parent state for the resonant Auger decay to the 2s1 2p5 (1.3 P) np and 2s0 p6 2 S np (n = 3, 4) states of Ne+ (see text) Final state A = 2s1 2p5 B = 2s0 2p6

Exp.

Theory: (b)

Theory: (c1)

Theory: (c2)

Theory: This work

A(1 P)3p2 S

778.79

776.43

778.52

778.61

778.72

A(1 P)3p2 P

778.54

776.40

778.27

778.39

778.51

A(1 P)3p2 D

778.81

776.66

778.57

778.68

778.78

A(1 P)3p2 S

788.16

786.51

787.88

787.97

788.12

A(1 P)3p2 P

788.90

787.52

788.69

788.75

788.87

A(1 P)3p2 D

789.01

787.64

788.82

788.93

788.98

A(1 P)4p2 S

773.60

–

–

773.52

773.57

A(1 P)4p2 P

773.48

–

–

773.33

773.44

A(1 P)4p2 D

773.56

–

–

773.41

773.52

A(3 P)4p2 S

783.72

–

–

783.62

783.68

A(3 P)4p2 P

783.95

–

–

783.81

783.91

A(3 P)4p2 D

784.01

–

–

783.90

783.97

B(1 S)3p2 P

–

–

–

754.99

755.06

B(1 S)4p2 P

–

–

–

749.92

749.98

decay to the 2s1 2p5 (1.3 P) np and 2s0 p6 2 S np (n = 3, 4) states of Ne+ . There are listed the experimental data (Exp) [18] as well as the results by Pahler et al. (a) [15], theoretical ab initio Hartree-Fock results (b) [18], and data (c1 and c2) [7, 8] and our data, obtained within the relativistic many-body PT and a gauge-invariant QED PT method. In Table 2 we the data on the widths (meV) for the 2s1 2p5 (1.3 P)np and 2s0 p6 (1 S) np (n = 3, 4) slates of Ne+ . There are listed experimental data (Exp) [18], theoretical ab initio multi configuration Hartree-Fock results by Sinanis et al. (b) [16], singleconfiguration Hartree-Fock data by Armen-Larkins (a) [17], data [7, 8] (c1, c2) and our data, obtained within the relativistic many-body PT. Table 2 Widths (meV) for 2s1 2p5 (1.3 P)np and 2s0 p6 (1 S) np (n = 3, 4) states of Ne+ (see text) Final state

Exp.

Theory (a)

Theory (b)

Theory (c1, c2)

Theory: This work

A(1 P)3p2 S

530 ± 50 410 ± 50

687

510

524

526

A(1 P)3p2 P

42 ± 3

20.7

–

38

39

A(1 P)3p2 D

34 ± 4

40.2

–

32

35

A(3 P)3p2 S

120 ± 10 110 ± 40

18.8

122

118

120

A(3 P)3p2 P

19 ± 5

10.3

–

16

18

A(3 P)3p2 D

80 ± 10

62.3

–

72

74

12

A. V. Glushkov

The analysis of the presented results in Tables 1, 2 allows to conclude that the précise description of the Auger decay parameters requires the detailed accurate accounting for the exchange-correlation effects, including the particle-hole interaction, screening effects and iterations of the mass operator. The relativistic many-body PT approach provides more accurate results due to a considerable extent to more correct accounting for complex inter electron exchangecorrelation effects. It is important to note that using more correct gauge-invariant procedure of generating the relativistic orbital bases is directly linked with correctness of accounting for the exchange-polarization effects.

3.2 Auger Spectroscopy of Some Solids Below we present the advanced data for Auger electron energy in some solids. As mentioned above, the ejection probability of Auger electron from an atom via different channels associated with ionization from a core level is defined by the standard matrix element. In addition, the proportionality coefficient in the equation coincides with the electron impact ionization cross-section σ j of the level j. Of course, two aspects are to be considered when determining the exit probability of Auger electrons from an atom, namely, the radiative transition under neutralization of a hole at the level j and the possibility of a considerable change in the initial hole distribution at the core levels at the Auger decay via the radiative channel jkl associated as a rule with a considerable distinctions in the non-radiative transition probabilities [7–18]. For definiteness sake, let the ionization of L levels in a multi-electron atom be considered. The probability of the Auger electron emission from the atom via the channel L 3 Kl (taken as an example) is defined by the ionization cross-section of the level L 3 as well as by a certain effective cross-section depending on the ionization cross-sections of the levels L 1 , L 2 . The Auger line intensity is defined by three atomic constants: Ajkl = σ j f j ajkl , where ajkl is the non-radiative transition probability; fi is the Korster-Kronig coefficient; σ j , the ionization cross-section defined by the corresponding matrix element calculated for wave functions of bound state and continuum one. In the solids theory, to single out the important correlation and medium effects, the master definition of the state energy is usually rewritten as: E A ( jkl, 2S+1 L J ) = E( j) − E(k) − E(l) − (k, l; 2S+1 L J ),

(14)

where the item takes into account the dynamic correlation effects (relaxation due to hole screening with electrons etc.) To take these effects into account, the set of procedures elaborated in the atomic theory [114–120] is used. As illustration, in Table 3 we present the data on Auger electron energy for some solids calculated using the presented approach (column C), the semi-empirical method with using the Larkins’ equivalent core approximation [5] (column A), the

Auger Spectroscopy of Multielectron Atoms …

13

Table 3 Experimental data for the Auger electron energy for some solids and calculated values (see text) Ele-ment

Auger line

Na

KL 2,3 L 12,3 D2 KL 2,3 L 12,3 D2 L 3 M 4,5 M 14,5 G4 M 5 N 4,5 N 14,5 G4

Si Ge Ag

Experiment

Theory: A

Theory: B

Theory: C

994.2

993.3

994.7

994.1

1616.4

1614.0

1615.9

1616.2

1146.2

1147.2

1146.6

1146.1

353.4

358.8

354.1

353.2

perturbation theory approach (B) [7] as well as experimental data (e.g. [5, 7]). Other details can be found in Refs. [5–12]. The calculation accuracy of semiempirical and simple ab initio methods is quite low. The sophisticated approach can provide more accurate results that is connected with more correct accounting for complex electron interaction. Some improvement of the present data is connected with more correct accounting for complex inter electron exchange-correlation effects and using the optimized one-quasiparticle representation in the relativistic many-body PT. These factors significantly affect at agreement between theoretical and experimental data. It should be noted that using the Auger electron spectroscopy in analysis of the surface chemical composition and elements [1, 3, 30–34] requires consideration of Auger spectra and the corresponding characteristics of the Auger transitions, interpretation of effects like the shape transformations of the valence Auger spectra due to appearance of new lines, position and intensity changes of individual lines caused by the redistribution in the electron state density of the valence band. The correct theoretical estimations of the spectral characteristics are of critical importance for their full understanding.

4 Conclusions To conclude, in this paper an advanced version of relativistic energy formalism in the Auger spectroscopy of multielectron atomic systems is developed in order to calculate the fundamental energetic and spectroscopic parameters of the Auger decay process. The approach originally uses the Gell-Mann and Low adiabatic formulae and the electron structure of a multielectron atom is calculated on the basis of the RMBPT with ab initio model zeroth approximation and a correct accounting for the exchange-polarization corrections. In order to provide gauge invariance performance, the RMBPT optimized zeroth approximation is generated on the basis of the relativistic criterion of minimization of the RMBPT second and higher orders exchange-polarization diagrams contributions into imaginary part of the atomic level energy shift. As an illustration, the results of computing the energy and spectral parameters of the resonant Auger decay for neon atomic system as well as some solids (Na, Si, Ge,

14

A. V. Glushkov

Ag; respectively the Auger lines are: KL 2,3 L 2,3 1 D2 , KL 2,3 L 2,3 1 D2, L 3 M 4,5 M 4,5 1 G4 , M 5 N 4,5 N 4,5 1 G4 ) are listed. The results are compared with available experimental results as well as with It should be underlined that the method presented is absolutely universal and can be used for quantum systems of any physical nature. It is important to note that earlier an universal energy approach had been applied to many problems of atomic, nuclear, molecular, laser spectroscopy, including a precise spectroscopy of complex multielectron atoms and multicharged ions, cooperative electron-muonhadron-aplha-beta-gamma-nuclear spectroscopy of atomic and molecular systems, spectroscopy of atoms and molecules in a laser field etc. [9, 95, 135–188]. Acknowledgements The author would like to thank Professors Olga Khetselius, Erkki Brändas, Jacek Karwowski, Ilya Kaplan, Jean Maruani, Boris Plakhutin and Andrey Svinarenko for useful discussions and comments and Dr Angeliki Athanasopoulou for the support. The author would like to thank the anonymous referees for useful comments too. The technical editorial assistance of Mr. Boopalan Renu and Mr. Muruga Prashanth are also very much acknowledged.

References 1. Aberg T, Hewat G (1979) Theory of Auger effect. Springer, Berlin 2. Glushkov AV (2020) Advanced relativistic energy approach in spectroscopy of autoionization states of multielectron atomic systems. In: Mammino L, Ceresoli D, Maruani J and Brändas E (eds) Advances in Quantum Systems in Chemistry, Physics, and Biology. Series: Progress in Theoretical Chemistry and Physics, vol 32. Springer, Cham, pp 3–31 3. Glushkov AV (2008) Relativistic quantum theory. Quantum mechanics of atomic systems. Astroprint, Odessa 4. Maruani J (2016) The dirac electron: from quantum chemistry to holistic cosmology. J Chin Chem Soc 63(1):33–48 5. Larkins FP (1976) Semi-empirical Auger electron energies. I. General method and K-LL line energies. J Phys B At Mol Opt Phys 9(1):47–58 6. Vijayakumar M, Gopinathan MS (1991) Theoretical Auger transition energies for atoms and ions through the relativistic and correlated local-density method. Phys Rev A 44(5):2850–2859 7. Glushkov AV, Ambrosov SV, Prepelitsa GP, Kozlovskaya VN (2003) Auger effect in atoms and solids: Calculation of characteristics of Auger decay in atoms, quasi-molecules and solids with application to surface composition analysis. Funct Mater. 10:206–210. Preprint OSENU NAMD-3 (2003) 8. Efimova EA, Chernyshev AS, Buyadzhi VV, Nikola LV (2019) Theoretical Auger spectroscopy of the neon: transition energies and widths. Photoelectronics. 28:24–31 9. Khetselius OYu (2011) Quantum structure of electroweak interaction in heavy finite Fermisystems. Astroprint, Odessa 10. Glushkov AV (2006) Relativistic and correlation effects in spectra of atomic systems. Astroprint, Odessa 11. Ambrosov SV, Glushkov AV, Nikola LV (2006) Sensing the Auger spectra for solids: New quantum approach. Sens Electr Microsyst Techn Issue 3:46–50 12. Glushkov AV, Buyadzhi VV, Chernyshev AS, Efimova EA, Tsudik AV (2020) Theoretical Auger spectroscopy of solids: sensing energy parameters. Sens Electr Microsyst Techn 17(1):21–28

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A. V. Glushkov theory of physical and chemical systems. Recent advances in the theory of chemical and physical systems. Series: progress in theoretical chemistry and physics, vol 15. Springer, Dordrecht, pp 285–299 Malinovskaya SV, Glushkov AV, Dubrovskaya YV, Vitavetskaya LA (2006) Quantum calculation of cooperative muon-nuclear processes: discharge of metastable nuclei during negative muon capture. In: Julien J-P, Maruani J, Mayou D, Wilson S, Delgado-Barion G (eds) Recent Advances in theory of physical and chemical systems. Recent advances in the theory of chemical and physical systems series: progress in theoretical chemistry and physics, vol 15. Springer, Dordrecht, pp 301–307 Malinovskaya SV, Glushkov AV, Khetselius OYu, Svinarenko AA, Mischenko EV, Florko TA (2009) Optimized perturbation theory scheme for calculating the interatomic potentials and hyperfine lines shift for heavy atoms in the buffer inert gas. Int J Quant Chem 109:3325–3329 Glushkov AV, Khetselius OYu, Lopatkin YM, Florko TA, Kovalenko OA, Mansarliysky VF (2014) Collisional shift of hyperfine line for rubidium in an atmosphere of the buffer inert gas. J Phys Conf Ser 548:012026 Khetselius OY, Lopatkin YM, Dubrovskaya YV, Svinarenko AA (2010) Sensing hyperfinestructure, electroweak interaction and parity non-conservation effect in heavy atoms and nuclei: new nuclear-QED approach. Sens Electr Microsyst Tech 7(2):11–19 Glushkov AV, Dan’kov SV, Prepelitsa G, Polischuk VN, Efimov AV (1997) Qed theory of nonlinear interaction of the complex atomic systems with laser field multi-photon resonances. J Techn Phys 38(2):219–222 Glushkov AV, Malinovskaya SV, Gurnitskaya EP, Khetselius OYu, Dubrovskaya Y (2006) Consistent quantum theory of recoil induced excitation and ionization in atoms during capture of neutron. J Phys Conf Ser 35:425–430 Glushkov AV, Malinovskay SV, Prepelitsa GP, Ignatenko V (2005) Manifestation of the new laser-electron nuclear spectral effects in the thermalized plasma: QED theory of co-operative laser-electron-nuclear processes. J Phys Conf Ser 11:199–206 Glushkov AV, Ambrosov SV, Loboda AV, Gurnitskaya EP, Prepelitsa GP (2005) Consistent QED approach to calculation of electron-collision excitation cross sections and strengths: Ne-like ions. Int J Quant Chem 104:562–569 Glushkov AV, Khetselius OYu, Loboda AV, Ignatenko AV, Svinarenko AA, Korchevsky DA, Lovett L (2008) QED approach to modeling spectra of the multicharged ions in a plasma: oscillator and electron-ion collision strengths. AIP Conf Proc 1058:175–177 Glushkov AV (1988) True effective molecular valency hamiltonian in a logical semiempricial theory. J Struct Chem 29(4):495–501 Glushkov AV (1990) Correction for exchange and correlation effects in multielectron system theory. J Struct Chem 31(4):529–532 Glushkov AV, Efimov VA, Gopchenko ED, Dan’kov SV, Polishchyuk VN, Goloshchak OP (1998) Calculation of spectroscopic characteristics 4 of alkali-metal dimers on the basis of a model perturbation theory. Opt Spectr 84(5):670–678 Svinarenko AA, Glushkov AV, Khetselius OY, Ternovsky VB, Dubrovskaya YuV, Kuznetsova AA, Buyadzhi VV (2017) Theoretical spectroscopy of rare-earth elements: spectra and autoionization resonance. In: Jose EA Orjuela (ed) Rare Earth Element. InTech, pp 83–104. https://doi.org/10.5772/intechopen.69314 Buyadzhi VV, Glushkov AV, Mansarliysky VF, Ignatenko AV, Svinarenko AA (2015) Spectroscopy of atoms in a strong laser field: New method to sensing AC Stark effect, multiphoton resonances parameters and ionization cross-sections. Sensor Electr Microsys Tech 12(4):27–36 Glushkov AV, Yu GM, Ignatenko AV, Smirnov AV, Serga IN, Svinarenko AA, Ternovsky EV (2017) Computational code in atomic and nuclear quantum optics: advanced computing multiphoton resonance parameters for atoms in a strong laser field. J Phys Conf Ser 905(1):012004

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180. Dubrovskaya YuV, Khetselius OYu, Vitavetskaya LA, Ternovsky VB, Serga IN (2019) Quantum chemistry and spectroscopy of pionic atomic systems with accounting for relativistic, radiative, and strong interaction effects. Adv Quant Chem, vol 78. Elsevier, pp 193–222. https://doi.org/10.1016/bs.aiq.2018.06.003 181. Malinovskaya SV, Dubrovskaya YV, Vitavetskaya LA (2005) Advanced quantum mechanical calculation of the beta decay probabilities In: Grzonka D, Czyzykiewicz R, Oelert W, Rozek T, Winter P (eds) Low energy antiproton physics. AIP: New York, AIP Conf Proc 796:201–205 182. Glushkov AV, Khetselius OYu, Svinarenko AA, Buyadzhi VV, Ternovsky VB, Kuznetsova AA, Bashkarev PG (2017) Relativistic perturbation theory formalism to computing spectra and radiation characteristics: application to heavy elements. In: Uzunov DI (ed) Recent studies in perturbation theory, InTech, pp 131–150. (https://doi.org/10.5772/intechopen.69102) 183. Glushkov AV (1992) Oscillator strengths of Cs and Rb-like ions. J Appl Spectrosc 56(1):5–9 184. Khetselius OY, Glushkov AV, Gurskaya MY, Kuznetsova AA, Dubrovskaya YV, Serga IN, Vitavetskaya LA (2017) Computational modelling parity nonconservation and electroweak interaction effects in heavy atomic systems within the nuclear-relativistic many-body perturbation theory. J Phys Conf Ser 905:012029 185. Buyadzhi VV, Zaichko PA, Gurskaya MY, Kuznetsova AA, Ponomarenko EL, Ternovsky VB (2017) Relativistic theory of excitation and ionization of Rydberg atomic systems in a black-body radiation field. J Phys Conf Ser 810:012047 186. Svinarenko AA, Khetselius OYu, Buyadzhi VV, Florko TA, Zaichko PA, Ponomarenko EL (2014) Spectroscopy of Rydberg atoms in a Black-body radiation field: relativistic theory of excitation and ionization. J Phys Conf Ser 548:012048 187. Glushkov AV, Khetselius OY, Svinarenko AA, Buyadzhi VV (2015) Methods of computational mathematics and mathematical physics. P.1. TES, Odessa 188. Glushkov AV (2012) Methods of a chaos theory. Astroprint, Odessa

Advanced Relativistic Energy Approach in Electron-Collisional and Radiative Spectroscopy of Ions in Plasmas Vasily V. Buyadzhi, Eugeny V. Ternovsky, Alexander V. Glushkov, and Anna A. Kuznetsova

Abstract An advanced relativistic approach to studying spectroscopic characteristics of the multicharged ions in plasmas is presented. The approach is based on the generalized relativistic energy approach combined with the optimized relativistic many-body perturbation theory with the Dirac-Debye shielding model as zeroth approximation, adapted for application to study of the spectral parameters of ions in plasmas. An electronic Hamiltonian for N-electron ion in plasmas is added by the Yukawa-type electron-electron and nuclear interaction potential. The transition probabilities and lifetimes for different excited states in spectrum of the Li-like calcium are computed within the consistent relativistic many-body approach for different values of the plasmas screening parameter (correspondingly, electron density and temperature) and compared with available alternative data. The results of relativistic calculation (taking into account the exchange and correlation corrections) of the electron collision cross-sections (strengths) of excitation of the transition between the fine-structure levels (2P3/2 − 2P1/2 ) of the ground state of F-like ions with Z = 19–26 and of the [2s2 1 S − (2s2p 1 P)] transition in the B-like O4+ are presented and analyzed. Keywords Electron-collisional processes · Multicharged ions · Relativistic energy approach · Debye plasmas

1 Introduction The properties of laboratory, thermonuclear (tokamak), laser-produced, astrophysical plasmas have drawn considerable attention over the last decades [1–76]. It is known that atomic ions play an important role in the diagnostics of a wide variety of plasmas [1–10]. Electron-ion collisions involving multiply charged ions, as well as various radiation and radiation-collisional processes, predetermine the quantitative characteristics of the energy balance of the plasmas [1–6, 15–20]. For this reason, the plasmas V. V. Buyadzhi (B) · E. V. Ternovsky · A. V. Glushkov · A. A. Kuznetsova Odessa State Environmental University, L’vovskaya str., 15, Odessa 65016, Ukraine e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. V. Glushkov et al. (eds.), Advances in Methods and Applications of Quantum Systems in Chemistry, Physics, and Biology, Progress in Theoretical Chemistry and Physics 33, https://doi.org/10.1007/978-3-030-68314-6_2

25

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V. V. Buyadzhi et al.

modelers and diagnosticians require absolute cross sections for these processes. The cross sections for electron-impact excitation of ions are needed to interpret spectroscopic measurements and for simulations of plasmas using collisional-radiative models. The electron-ion collisions play a major role in the energy balance of plasmas (e.g. [1–10]). Different theoretical methods were employed along with the Debye screening to study plasma medium. In the case of solving collision problems involving multi-electron atomic systems, as well as low-energy processes, etc., the structure of atomic systems should be described on the basis of rigorous methods of quantum theory. As a rule, the HartreeFock (HF) or Hartree-Fock-Slater (HFS) models implemented in the tight-binding approximation were used to describe the wave functions of the bound states of atoms and ions. Another direction is the models of the central potential (model potential, pseudopotential) implemented in the distorted wave approximation (DWA). It should be mentioned the currently widespread and widely used R-matrix method and its various promising modifications, as well as a generalization of the wellknown Dirac-Fock method for the case of taking into account multipolarity in the corresponding operators (see, e.g., [1–7]). It should be noted that, depending on the perturbation theory (PT) basis used, different versions of the R-matrix method received the corresponding names. For example, in specific calculations such versions as R-MATR-CI3-5R and R-MATR-41 R-matrix method were used using respectively wave functions in the multiconfiguration approximation, in particular, 5- and 41configuration wave functions. As numerous applications of the R-matrix method have shown, it has certain advantages in terms of accuracy and consistency over such popular approaches as the first-order PT method, as well as the distorted wave approximation taking into account configuration interaction (CI-DWBA), approximation of distorted waves using the HF basis (HF-DWBA), finally, the relativistic approximation of distorted waves with a 1-configuration and multi-configuration wave function of the ground state (SCGSRDWA, MCGS-RDWA, etc.). Improved models have also appeared in theories of the coupled-channel (VC) type VCDWA (Variational Continuum Distorted Wave), for example, a modification of the Vraun-Scroters type and others (see [1–5]). Various cluster methods have also been widely used (see in more detail Refs. [1–15]). Earlier we have developed a new version of a relativistic energy approach combined with the many-body perturbation theory (RMBPT) for multi-quasiparticle (QP) systems to study spectra of plasma of the multicharged ions, electron-ion collisional parameters [24–29]. The method is based on the Debye shielding model and energy approach. A new element of this paper is in using the effective optimized Dirac-Kohn-Sham method in general relativistic energy approach to collision processes in the Debye plasmas. In this paper, which goes on our work [8–11, 24–28], we present the results of computing the transition probabilities and lifetimes for different excited states in spectrum of the Li-like calcium for different values of the plasmas screening (Debye) parameter (respectively, electron density, temperature) and compared with available alternative spectroscopic data.

Advanced Relativistic Energy Approach in Electron-Collisional …

27

The computational approach used is based on the generalized relativistic energy approach combined with the optimized RMBPT with the Dirac-Debye shielding model as zeroth approximation, adapted for application to study the spectral parameters of ions in plasmas. An electronic Hamiltonian for N-electron ion in plasmas is added by the Yukawa-type electron-electron and nuclear interaction potential. The results of relativistic calculation (taking into account the exchange and correlation corrections) of the electron collision cross-sections (strengths) of excitation of the transition between the fine-structure levels (2P3/2 − 2P1/2 ) of the ground state of F-like ions with Z = 19–26 and of the [2s2 1 S − (2s2p 1 P)] transition in the B-like O4+ are presented and analyzed. It should be emphasized that an accurate treating the gauge dependent lowest perturbation theory multielectron contributions to radiation widths of atomic levels or radiation transitions probabilities is a fundamental requirement in order to construct the optimized one-electron representation in the many-body perturbation theory zeroth approximation. One could remember that the known relativistic many-body perturbation theory formalism is constructed with using the same ideas as the wellknown perturbation theory approach with the model potential zeroth approximation by Ivanov-Ivanova et al. [31–42]. But unlike the method by Ivanova et al. and similar method by Glushkov et al. [43–50], the PT zeroth approximation in our method is the Dirac-Debye-Hückel one. Computing the radiative and collisional characteristics of atoms and ions is performed within a gauge-invariant version of relativistic energy approach [43].

2 Advanced Relativistic Energy Approach in Electron-Collisional Spectroscopy The detailed description of our approach was earlier presented (see, for example, Refs. [24–29]). Therefore, below we are limited only by the key points. The generalized relativistic energy approach combined with the RMBPT has been in detail described in Refs. [7, 34–50]. It generalizes earlier developed energy approach. The key idea is in calculating the energy shifts E of degenerate states that is connected with the secular matrix M diagonalization. To construct M, one should use the Gell-Mann and Low adiabatic formula for E. The whole calculation is reduced to calculation and diagonalization of the complex IK matrix M and definition of matrix of the coefficients with eigen state vectors Bie,iv [7, 24, 25]. To calculate all necessary matrix elements one must use the bases of the 1QP relativistic functions. Within an energy approach the total energy shift of the state is usually presented as [31, 32]: E = ReE + iΓ /2,

(1)

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where Γ is interpreted as the level width and decay (transition) possibility P = Γ . The imaginary part of electron energy of the system, which is defined in the lowest PT order as [31, 32]: e2 4π

ImE(B) = −

|ω|

Vi jkl =

¨

dr1 dr2 Ψi∗ (r1 )Ψ j∗ (r2 )



|ωαn | Vαnαn

(2)

α>n> f [αn> f for electron and α j

(6) where α i ,α j are the Dirac matrices, c is the velocity of light and Z is a charge of an ionic nucleus. To generate the wave functions basis we use the optimized Dirac-Kohn-Sham potential with one parameter [43], determined on the basis of a relativistic energy formalism [31]. Modified PC numerical code ‘Superatom’ is used in all calculations. Other details can be found in Refs. [24–35, 91–100]. Certain ideas of a relativistic energy formalism in application to a quantum scattering topics have been presented in a literature (e.g., [7, 43–45]). The important quantity is a scattered part of energy shift Im ΔE. It can be presented in the form of integral over the scattered electron energy εsc:  dεsc G(εiv , εie , εin , εsc )/(εsc − εiv − εie − εin − i0) Im = π G(εiv , εie , εin , εsc )

(7) (8)

where εin and εsc are the energies of the incident and scattered electrons and G is a definite squired combination of the two-electron matrix elements.

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Further one could easily determine the collisional cross-section σ = −2 ImΔE (the details can be found in Refs. [6, 25, 26]). In particular, the collisional de-excitation cross section can be written in the following form [6, 25]: σ (I K → 0) = 2π



(2 jsc + 1)

jin , jsc

∗

⎧ ⎨ ⎩

IK < 0| jin , jsc | jie , jiv , Ji > Bie,iv

jie, jiv

2

⎫ ⎬ ⎭

(9a)

0| jin , jsc | jie , jiv , Ji    = (2 jie + 1)(2 jiv + 1)(−1) jie +1/2 × (−1)λ+Ji λ

× {δλ,Ji /(2Ji + 1)Q λ (sc, ie; iv, in)

jin . . . jsc . . . Ji + Q λ (ie; in; iv, sc)} jie . . . jiv . . . λ

,

(9b)

Here the quantity Qλ can be expressed thorough the known Coulomb-Yukawa (CY) and Breit (Br) matrix elements combinations [31–45]: Br Q λ = Q CY λ + Qλ

(10)

Br The Q CY λ , Q λ quantities contain the corresponding radial Wλ and angular Yλ integrals as follows:

     ˜ ˜ ˜ ˜ = W Q CY + W 124 3 Y 124 3 (1243)Y (1243) λ λ λ λ λ         ˜ Yλ 12˜ 43 ˜ + Wλ 1˜ 2˜ 4˜ 3˜ Yλ 1˜ 2˜ 4˜ 3˜ . +Wλ 12˜ 43

(11)

where the tilde designates that the large radial Dirac component f must be replaced by the small Dirac component g (other details can be found in Refs. [25, 26]). It should be noted that the Breit quantity can be analogically expressed thorough the same integrals. The effective collision strength Ω(I → F) is associated with a collisional cross section σ as follows (in the Coulomb units): σ (I → F) = Ω(I → F) · π /{(2Ji + 1)εin [(α Z )2 εin + 2]}, where α is the fine structure constant.

(12)

Advanced Relativistic Energy Approach in Electron-Collisional …

31

Other details can be found in Refs. [8–11, 14–20, 24–31, 91–124]. All computing is performed with using the modified PC atomic code “Superatom-ISAN” (the modified version 93).

3 Results and Conclusions Below we present the results of computing the energy, spectroscopic characteristics of some Li-like, B-like and F-like ions. The sought objects of research, firstly, belong to the class of complex relativistic many-electron atomic systems, in connection with which the approbation of the theory is extremely important and indicative just for such systems. Secondly, the sought multiply charged ions are of great interest for a number of applications in the field of laser physics and quantum electronics, in particular, the use of the plasma of the corresponding ions as an active medium for short-wavelength lasers, further in the field of diagnostics of astrophysical, laboratory plasma and plasma of a fusion reactor, tokamak and EBIT devices, as well as, of course, laser plasma. Firstly, we present our results on the transition probabilities and lifetimes for some excited states of the Li-like ion of calcium. The spectroscopic properties for plasma-isolated ion with μ = 0 have been considered. In Tables 1 and 2 there are listed probabilities values for transitions (E1, M1, and E2 channels) from the excited states to the low-lying states of Ca XVIII. Using these values, one could calculate the corresponding lifetimes of the excited states. The analysis shows that the presented data are in physically reasonable agreement with the NIST experimental data and theoretical results by relativistic coupled-cluster (RCC) method calculation (e.g. [3, 24, 25]). However, some difference between the corresponding results can be explained by using different relativistic orbital bases and by difference in the model of accounting for the screening effect as well as some numerical differences. In Tables 3 and 4 we list the numerical variations in the lifetimes of the 2p1/2 , 3s1/2 , 3p1/2 , 3d 3/2 , and 4s1/2 states in Ca XVIII for different μ values. Table 1 The transition probabilities (P) for some transitions in spectrum of Ca XVIII: RCC—relativistic coupled-cluster (RCC) method [3]; This—our work

Transition

Pf →i

Pf →i

f -i

RCC

This

2p1/2 -(E1)-2s1/2

1.31[9]

1.33[9]

2p3/2 -(E1)-2s1/2

2.00[9]

2.02[9]

2p3/2 -(M1)-2p1/2

7.00[2]

7.03[2]

2p3/2 -(E2)-2p1/2

2.54[−2]

2.57[−2]

3s1/2 -M1-2s1/2

2.04[4]

2.06[4]

3s1/2 -(E1)-2p1/2

3.01[11]

3.02[11]

3s1/2 -(E1)-2p3/2

6.22[11]

6.24[11]

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Table 2 The transition probabilities (P) for some transitions in spectrum of Ca XVIII (our data) Transition

Pf →i

f -i

This

3p1/2 -E1-2s1/2

2.37[12]

3p1/2 -M1-2p1/2

1.48[3]

3p1/2 -M1-2p3/2

6.78[4]

3p1/2 -E2-2p3/2

8.45[8]

3p1/2 -E1-3s1/2

1.72[8]

3p3/2 -E1-2s1/2

2.32[12]

3p3/2 -M1-2p1/2

1.24[4]

3p3/2 -E2-2p3/2

4.25[8]

3p3/2 -M1-2p3/2

2.78[4]

3p3/2 -E2-2p3/2

4.22[8]

3p3/2 -E1-3s1/2

2.66[8]

3p3/2 -M1-3p1/2

1.83[1]

3p3/2 -E2-3p1/2

2.13[−3]

Table 3 The dependence of the lifetimes (ps) of the 2p1/2 state in the Ca XVIII spectrum upon the screening parameter μ: RCC—relativistic coupled-cluster (RCC) method [3]; This—our work μ

2p1/2

2p1/2

RCC

This

0.133

741

738

0.667

494

492

1.000

334

332

1.250

242

241

1.429

192

190

0.60

140

138

Table 4 The dependence of the lifetimes (ps) of the 3lj, 4lj states in the Ca XVIII spectrum upon the parameter μ (this work) μ

3s1/2

3p1/2

3d3/2

0.133

1.07

0.428

0.143

1.62

0.667

1.26

0.518

0.688

2.54

1.000

1.53

0.658

0.206

4.81

1.250

1.85

0.849

0.262

12.48

1.429

2.20

1.072

0.336

82.77

4s1/2

Advanced Relativistic Energy Approach in Electron-Collisional … Table 5 The electron collision strengths of excitation the transition between the fine-structure levels (2P3/2 − 2P1/2 ) of the ground state of F-like ions with Z = 19–26

33

Ion

ICFT R-matrix

LS + JAJOM R-matrix

Ar X

0.582

0.420

Ca XII

0.162

0.160

Ti XIV

0.225

0.220

Cr XVI

0.112

0.100

Fe XVIII

0.132

0.110

Ion

Our data

Exp. [6]

Ar X

0.492

0.49

Ca XII

0.159

–

Ti XIV

0.252

–

Cr XVI

0.142

–

Fe XVIII

0.148

0.15

It is worth to note that our computing oscillator strengths within energy approach with different forms of transition operator (i.e. using the photon propagators in the form of Coulomb, Feynman or Babushkin) gives very close results. In Table 5 we present the results of our relativistic calculation (taking into account the exchange and correlation corrections) of the electron collision strengths of excitation the transition between the fine-structure levels (2P3/2 − 2P1/2 ) of the ground state of F-like ions with Z = 19–26. The energy of the incident electron is εin = 0.1294 · Z2 eV, T = z2 keV (z is the core charge), Ne = 1018 cm−3 . For comparison, in Table 5 there are also listed the calculation results based on the most advanced versions of the R-matrix method, nonrelativistic calculation data in the framework of the energy approach, and also the available experimental data (e.g. [1–3, 24, 25]). The analysis shows that the presented data are in physically reasonable agreement, however, some difference can be explained by using different relativistic orbital basises and different models for accounting for the plasma screening effect. This circumstance is mainly associated with the correct accounting of relativistic and exchange-correlation effects, using the optimized basis of relativistic orbitals (2s2 2p5 ; 2s 2p6 2s2 2p4 3l, l = 0–2) and, to a lesser extent, taking into account the effect of the plasma environment. The electron-ion collision characteristics for Be-like ions are of great interest for applications such as the diagnosis of astrophysical, laboratory, and thermonuclear plasmas, as well as EBIT plasmas (see, for example, [4, 5]). In the latter case, the characteristic values of electron density turn out to be significantly (several orders of magnitude) less than those considered above (1015 –1017 ). In particular, the socalled MEIBEL (the merged electron-ion beams energy-loss) experiment (1999), the results of which for a Be-like oxygen ion are presented in Fig. 1. In this figure there also listed the cross section (10−16 cm3 ) of the electron-collision excitation of the [2s2 1 S − (2s2p 1 P)] transition in the spectra of Be-like oxygen together with the

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V. V. Buyadzhi et al.

Fig. 1 Cross section for electron-collision excitation of the [2s2 1 S − (2s2p 1 P)] transition in the spectra of B-like O4+ : Experiment MEIBEL—points; Theory: R-matrix—solid line; our theory—dashed line

data from an alternative 3-configuration R-matrix calculation [4]. At energies below 20 eV there is a reasonable agreement between the theoretical and experimental, but, above 20 eV there is a discrepancy, which is due to different degrees of allowance for correlation effects (interaction of configurations) due to the difference in the bases used. To conclude, we have presented an effective relativistic approach to computing energy and spectroscopic characteristics of the multicharged ions in plasmas. It is consistently based on the generalized relativistic energy approach combined with the optimized relativistic many-body perturbation theory with the Dirac-Debye shielding model as zeroth approximation. The important theoretical aspect is connected with construction of an electronic Hamiltonian for N-electron ion in plasmas with addition of the Yukawa-type electron-electron and nuclear interaction potentials. As an illustration, the approach has been applied to computing probabilities and lifetimes for different excited states in spectrum of the Li-like calcium as well as the electron collision cross-sections (strengths) of excitation of the transition between the fine-structure levels (2P3/2 − 2P1/2 ) of the ground state of F-like ions with Z = 19–26 and the [2s2 1 S − (2s2p 1 P)] transition in the B-like O4+ The presented approach and obtained data can be used in different applications, namely, in atomic, molecular and laser physics, quantum electronics, astrophysical, laboratory, thermonuclear plasmas physics etc. (e.g. [1–21, 43]).

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The Schrödinger Equation with Power Potentials: Exactly-Solvable Problems Jacek Karwowski and Henryk A. Witek

Abstract Some aspects of quasi-exact and semi-exact solubility are discussed. In particular, quasi-exactly solvable potentials are obtained as solutions of inverse problems with preassumed wave functions. Then, quasi-exactly solvable equations are derived as polynomial reduction of semi-exactly solvable problems. The relations between the numerical accuracy of the eigenvalues and the radius of convergence of the power series expansion of the wave function are discussed in the last section. Keywords Schrödinger equation · Integrable potentials · Energy spectrum · Heun equation · Hessenberg determinant · Quasi-exact solubility · Semi-exact solubility · Harmonium

1 Introduction Exactly solvable Schrödinger eigenvalue problems, as for example harmonic oscillator or hydrogen atom, shape our way of thinking about fundamental properties of quantum systems. Their importance was always appreciated. Sophisticated theories of many-electron systems stem from exactly-solvable model of the one-electron atom, and quantum electrodynamics—from the harmonic oscillator. By the end of the last century, after very successful implementations of super-symmetry-based approaches and understanding their relation to the algebraic factorization method [1–3], it seemed that the family of exactly-solvable problems is nearly complete and that the criteria of solubility are well understood.

J. Karwowski (B) Institute of Physics, Nicolaus Copernicus University, 87-100 Toru´n, Poland e-mail: [email protected] H. A. Witek Department of Applied Chemistry, Center for Emergent Functional Matter Science, and Institute of Molecular Science, National Chiao Tung University, Hsinchu, Taiwan e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. V. Glushkov et al. (eds.), Advances in Methods and Applications of Quantum Systems in Chemistry, Physics, and Biology, Progress in Theoretical Chemistry and Physics 33, https://doi.org/10.1007/978-3-030-68314-6_3

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Nearly all exact solutions of the Schrödinger equation have been expressed in terms of polynomial reductions of hypergeometric or confluent hypergeometric functions, multiplied by asymptotic factors which define the behavior at the origin and at infinity. New directions of the development originated from the observation that the Schrödinger equation for a large class of potentials of physical importance may be transformed to an equation belonging to the Heun family of differential equations [4]. As a consequence, a new set of exactly solvable Schrödinger equations with eigenfunctions expressible in terms the Heun functions, have been discovered [5–8]. A vast majority of the one-particle eigenvalue problems have to be solved numerically. But, not always the entire spectrum of analytic solutions is needed. In many cases adequate information may be derived from the exact form of the wave functions for some particular states or for some specifically designed potentials. It is, in a way, surprising that these quasi-exact [9–12], conditionally exact [13–15], nextto-exact [16, 17] also known as semi-exact [18, 19] solutions attracted attention of researchers several decades after the formulation of quantum mechanics, after the set of unknown exactly solvable Schrödinger equations seemed to be nearly exhausted. Quasi-exactly solvable equations, where a finite number of solutions (in most cases just one) may be expressed analytically, have been intensively studied over last two decades. The foundations for this big branch of mathematical physics have been laid by Turbiner several decades ago [9–11]. Probably the best known example of a model leading to an equation belonging to this class is the Hooke atom, described by the biconfluent Heun equation [20]. This very interesting and apparently simple model has been studied in different contexts since several decades [21–23]. It has also been used as a textbook model explaining the behavior of the exact wave function at r12 = 0 singularity of the Coulomb interaction [24] and as a reference system used for checking the numerical accuracy of newly developed approaches [25]. The Schrödinger equation is referred to as conditionally-exactly-solvable if the complete set of analytic solutions may be obtained for specific values of the potential parameters [14, 15]. A large family of conditionally-solvable equations have recently been constructed by Ishkhanyan and coworkers [7, 8, 15]. Properly constructed conditionally-solvable problems offer new forms of exactly solvable potentials describing a variety of physical phenomena, as for example quark-antiquark interactions [26]. Finally, in the case of the next-to-exactly-solvable or semi-exactly-solvable equations the wave functions are expressed analytically, under condition that the eigenvalues are known [16, 19]. The Schrödinger equations with power potentials (polynomials in the radial variable r ) are semi-exactly-solvable and their solutions may be expressed in terms of r power series expansions with coefficients equal to the Hessenberg determinants. The eigenvalues have to be derived numerically except for the rare cases when a quasi-exact solution exists [19]. The finite accuracy of the numerical eigenvalues implies that the power series converges to the corresponding eigenfuntion for r < rmax . The value of rmax drammatically depends on the precision up to which the eigenvalue has been determined (c.f. Sect. 5). In this paper several aspects of quasi-exactly and semi-exactly solvable problems are briefly discussed. In particular, we discuss quasi-exact solubility as an inverse

The Schrödinger Equation with Power Potentials: Exactly-Solvable Problems

45

problem by constructing quasi-exactly solvable potentials corresponding to some preassumed forms of the wave functions. Next, quasi-exactly solvable equations are derived as polynomial reductions of semi-exactly solvable problems. In the last section the relation between the numerical accuracy of the eigenvalues and the radius of convergence of the power series expansion of the wave function is illustrated using as an example a quartic oscillator.

2 Separability It is commonly believed that separability is a prerequisite of exact solubility. This is not necessarily true. Probably the most spectacular are 70 years of efforts to solve exactly the helium atom problem, initiated by Vladimir Fock [27] and continued until now [28–31]. Several months ago Alexander Turbiner, in a private communication, stated To my understanding it is already time to abandon one-dimensional Schrödinger equation moving to higher dimensions. Still there are many new things lying of the surface even in 2D. Three body Coulomb problem is infinitely interesting subject.

and, in a very recent paper [32] reported on some essential progress in this direction, presenting an integrable and non-separable three-body system. The problem of two interacting particles described by Hamiltonian H(r1 , r2 ) = T(r1 ) + T(r2 ) + V (r12 ) . is separable to two one-particle spherically symmetric problems: the free motion of the center of mass and the relative motion described by the Schrödinger equation for a single particle confined in spherically-symmetric external potential V. But, separability is not possible in the relativistic case, when the interaction is transmitted with finite velocity. In general, problems of more than two particles are not separable. However, there are some exceptions. If interactions in a three-particle system are described by the following potential 2 2 + c23 r23 , V (r12 ) + c13 r13 then the system is separable to three spherically-symmetric one-particle equations: (1) Free motion of the center of mass of the three particles; (2) Spherical harmonic oscillator for the center of mass of two remaining quasi-particles; (3) Sphericallysymmetric equation for the third quasi-particle in an effective external potential Veff (r ) = V(r ) + a 2 r 2 , where r = r12 and a is a constant depending on c13 , c23 and on the particle masses [12].

46

J. Karwowski and H. A. Witek

It has been realized long ago that a system of N particles interacting with each other by a harmonic potential, known as the Moshinsky atom, is separable and exactly solvable [33]. Relatively recently it has been demonstrated that the Schrödinger equation for a system of particles in which disjoint pairs interact by arbitrary two-particle potentials while the remaining interactions are described by harmonic oscillator potentials is also, under certain conditions, separable [34, 35]. Special cases of this model are known as the Hookean atom or harmonium [36–38] and the Hookean molecules [39–41]. The problem reduces to a set of spherically-symmetric oneparticle equations. Their analysis, apart of the academic curiosity, supplies interesting data on the nature of various aspects of separability.

3 Quasi-exact Solutions as an Inverse Problem After the elimination of the angular part, the one-particle Schrödinger equation may be expressed as  2  d − W E (r ) Ψ (r ; E) = 0, (1) dr 2 where E is an eigenvalue if Ψ (r ; E) ∈ L 2 , and W E (r ) =

  l(l + 1) 1 d 2Ψ + 2m V(r ) − E = . r2 Ψ dr 2

(2)

The last equation gives a simple way to a plethora of non-trivial but easy to obtain, quasi-exact solutions. The formulation of this approach and a set of most interesting examples has been given by Turbiner in his recent review [11]. If we set an inverse problem: instead of looking for Ψ corresponding to a given V, we look for V corresponding to a given eigenfunction Ψ , then V(r ) = E +

1 2m



 1 d 2Ψ l(l + 1) . − Ψ dr 2 r2

It is convenient to define Φ(r ) = i.e. to take Ψ = C exp



(3)

1 dΨ , Ψ dr

 Φ(r )dr , where C is a normalization constant. Then   1 dΦ 2 V(r ) = E + Φ + , 2 dr

(4)

where, for simplicity, we set m = 1 and l = 0. Actually, the essential part of expression (4) defining the shape of the potential is

The Schrödinger Equation with Power Potentials: Exactly-Solvable Problems

V0 = Φ 2 +

dΦ = 2(V − E). dr

47

(5)

Thus, for an arbitrary nodeless Ψ ∈ L 2 one can easily find V(r ) for which Ψ describes the ground state. But, if we select a square-integrable function with n + 1 nodes, we get the potential for which this function describes the n-th excited state. A fascinating example of a non-trivial application of this procedure has been given in [11]. Taking Φ(x) = −a x 3 , with x ∈  −∞, ∞  and a > 0, we get a double well potential (6) V0 = a 2 x 6 − 3a x 2 with unusual spectral properties (for details see [11]). Another example, generating a family of interesting potentials, is associated with Ψ (r ) = r (b r + 1) e−α r i.e. Φ(r ) =

n+1

(7)

1 b + − α (n + 1) r n , r br + 1

with α > 0 and b ≥ 0 or b < 0 for, respectively, ground and first excited states. Then b α(n + 1) r n+1 − 1 V − E = Un (r ) − · (8) r br + 1 where Un (r ) =

 α (n + 1)  α (n + 1) r 2n − (n + 2) r n−1 . 2

From Eq. (8) one can derive several interesting special cases. For n = 0 we get V = −

b αr −1 α − · , r r br + 1

E =−

α2 . 2

(9)

If b = 0 we have the ground state of the hydrogen-like atom (α = Z and E 0 = −Z 2 /2). If b > 0 then the wave function (7) describes the ground state of a rather complicated potential (9). If b < 0 then setting α = −b = −Z /2 we get the first excited state of the hydrogen-like atom. If n = 1 then √ √ b ( 2α r − 1)( 2α r + 1) 2 2 , (10) V = 2α r − · r br + 1

E = 3 + 2 δ|b|,√2α α,

48

J. Karwowski and H. A. Witek

√ where δ is the Kronecker delta. By setting b = ± 2α we get either the ground state (b > 0) or the first excited state (b < 0) of a system described by combination of a parabolic a Coulomb potential √ 2α , V = 2α r ± r

E = 5α.

2 2

If b > 0 then the Coulomb potential is repulsive and if b < 0, it is attractive. If α = 1/2 then the lat equation may be recognized as one of quasi-exactly-solvable potentials in the the residual radial equation decribing the Hooke atom (c.f. Sect. 4). If b = 0, then for n > 0 and α > 0, we get ground states of an infinite set of oscillators, including the radial version of (6). For b = −[α (n + 1)]1/(n+1) we get the first excited states. Examples of the potentials for n = 2, 3, 4 and the corresponding ground-state wave functions are plotted in Fig. 1. A family of Lennard-Jones-like potentials may be derived from Φ(r ) =

√

D

 r 6 e

r

− a.

The resulting ground-state wave function reads

√ Ψ (r ) ∼ exp −

 D r  re 6 −ar . 5 r

It corresponds to the potential   r 6  3 e b− V(r ) = VLJ (r ) + 2 D r r √

and energy E =− where b =

√

√

D−b

2

(11)

,

D − a, m = 1/2, and VLJ (r ) = D

 r 12 e

r

−2

 r 6  e

r

is the Lennard-Jones potential. Potential V(r ) and the Lennard-Jones potential have the same asymptotic behavior. Additionally, if b = 3/re then V(re ) = VLJ (re ). A comparison of V(r ) with the Lennard-Jones potential for D = 2, re = 4 and b = 1/2, 3/4, 1, as well as plots of the corresponding ground-state wave functions, are given in Fig. 1. As one can see, if b = 3/4, then VLJ (r ) and V(r ) nearly overlap.

The Schrödinger Equation with Power Potentials: Exactly-Solvable Problems

49

Fig. 1 Left panel: Potentials (8) and ground-state wave functions for b = 0, α = 0.2 and n = 2 (green), n = 3 (blue), n = 4 (red); in all cases E = 0. Right panel: A comparison of V(r ) (11) for b = 1 (green), b = 3/4 (blue) and b = 1/2 (red) with the Lennard-Jones potential (black), the corresponding ground-state wave functions Ψ , and energies E

4 Quasi-exact Solutions as a Polynomial Reduction For a large class of potentials the bound-state wave function can be expressed as [19] Ψ (r ; E) ∼ φ0 (r ) F(r ) φ∞ (r ),

(12)

where φ0 (r ) = r λ and φ∞ (r ) = e−η(r ) are the asymptotic factors describing, respectively, the behavior at the origin and at infinity, η(r ) is a polynomial and F(r ) is an unknown function which does not influence the asymptotic behavior of Ψ . The independent variable r may stand for the original r but may also be obtained by a transformation of this variable as f (r ) → r . Since for our consideration it is irrelevant, the designation r is, for simplicity, retained in both cases. We assume that the substitution of (12) to the original Schrödinger equation leads to the following canonical equation for F(r ) 

 d d2 + T (r ) F(r ) = 0, P(r ) 2 + Q(r ) dr dr

where P(r ), Q(r ) and T (r ) are polynomials. In majority of exactly solvable equations known until the end of the 20-th century, the canonical equation is reducible to the hypergeometric differential equation   d2 d r (r − 1) 2 + (A r + B) + C F(r ) = 0, dr dr or to its confluent form [42]. Recently, several new exactly solvable potentials, leading to the Heun differential equation

50

J. Karwowski and H. A. Witek

  d2 d 2 r (r − 1)(r − a) 2 + (A r + B r + C) + D r + E F(r ) = 0, dr dr or one of its confluent forms have been constructed [7, 8]. If F(r ) has derivatives of all orders on its domain, i.e if it belongs to C ∞ class of functions, then it can be expanded to a power series of r F(r ) =

∞ 

an r n ,

(13)

n=0

where a0 = 0.1 The substitution of Eq. (13) to the canonical equation, leads to recurrence relations for the expansion coefficients. As one can easily see, the number of terms in the relations is equal to the number terms with different degrees of homogeneity in the canonical equations [19]. We get two-term recurrences, a j+1 = d j a j , for the hypergeometric equations, three term recurrences, a j+1 = d j a j + e j−1 a j−1 , for the Heun equations and, in general, we can have multi-term recurrence relations. The coefficients d j , e j , . . . are rational functions of parameters appearing in the canonical equation, i.e. are determined by the energy of the system and by the parameters of V(r ). If coefficients an are defined by an m-term recurrence then expansion (13) terminates at n = q if aq = 0 and the values of the pertinent recurrence relation coefficients are such that aq+1 = aq+2 = · · · = aq+m−1 = 0. In the case of two-term recurrences the termination condition, aq+1 = 0, implies dq = 0. The last condition defines the energy for which F(r ) is reduced to a polynomial. In other words, it defines the energy quantization condition for exactly solvable hypergeometric equations (as hydrogen atom, harmonic oscillator, quantum defect orbitals, Morse oscillator, and so on). A three-term recurrence (m = 3) terminates if two conditions are fulfilled. Then, the energy and one parameter in the potential are constrained. In general, the termination condition in the case of an m-term recurrence, determines the energy and m − 2 parameters in the potential. In effect we get quasi-exactly solvable equations. Usually, for one potential we have only one energy for which F(r ) is reduced to a polynomial. Obviously, for this very potential there exists also an infinite set of non-polynomial solutions. To this class of problems belong quasi-exactly solvable Heun equations. The best known examples are harmonium and shifted harmonic oscillator. A large set of other potentials has been discussed in [11]. Quasi-exact solutions, i.e., the ones for which expansion (13) is reduced to a polynomial, are essentially different from the remaining ones. One of these differences may be exemplified using two best known quasi-exactly solvable models the Hooke atom (harmonium) and the confined positronium. Hamiltonian describing two particles in a parabolic confinement

1 According

[43].

to the Borel’s theorem every power series expansion of this form is a Maclaurin series

The Schrödinger Equation with Power Potentials: Exactly-Solvable Problems

H(r1 , r2 ) =

51

p21 p2 ω2 m 1 r12 + m 2 r22 + V(r12 ) + 2 + 2m 1 2m 2 2

is separable [21]. The motion of the mass center is described by a spherical harmonic oscillator equation and the relative motion by a radial Schrödinger equation with Hamiltonian equal to h(r ) =

μ ω2 2 p2 (r ) + V(r ) + r , 2μ 2

(14)

where r12 = r . We assume hereafter that m 1 = m 2 = 1, i.e. μ= and

m1 m2 1 = , m1 + m2 2 s V(r ) = . r

If s = 1 we have harmonium and if s = −1 we have a confined positronium. In the Schrödinger equation  p2 (r ) + the transformation

 ω2 r 2 s + − E Ψs (r ; E) = 0, r 4

(15)

r → −r gives:   ω2 r 2 s p2 (r ) − + − E Ψs (−r ; E) = 0. r 4

(16)

By comparing Eqs. (15) and (16) one can see that E = E(s) = E(−s) and Ψ−s (r ; E) = Ψs (−r ; E),

(17)

i.e. the energies of harmonium and of confined positronium are the same and the wave function of the confined positronium may be obtained from the wave function of harmonium by the replacement r → −r . This subject was discussed by Turbiner [11, 23] and by Schulze-Halberg [44]. But, to our knowledge, it was never stressed that Eq. (17) is valid if, for a given s, Ψs (r ; E) ∈ L 2 and Ψ−s (r ; E) ∈ L 2 .

(18)

It was shown by a numerical check, that in the case of Eqs. (15) and (16), condition (18) is fulfilled for the polynomial (quasi-exact) solutions only [37, 45]. Therefore, we formulate the following conjecture:

52 Fig. 2 Solutions Ψ (r ; E) of Eq. (15)—right panels, and Eq. (16)—left panels, with s = 1 and ω = 0.5, for the ground state and for the first excited state. In the case of polynomial solutions corresponding to E = 1.25, both Ψ (r ; E) and Ψ (−r ; E) are square-integrable. In the remaining two cases r → −r transformation of the eigenfunction generates a non-square-integrable function

J. Karwowski and H. A. Witek

3

−r

−5

0

5

r

E=0.8545093 ground

2

1

0

−1

ground

2

1st excited 1 0 −1

E=1.25

E=1.25

2 1

1st excited

0 −1 −2

E=2.1901169

−3 −r

−5

0

5

r

If two particles in a parabolic confinement interact by a Coulomb potential s/r then E(s) = E(−s) and Ψ−s (r ; E) = Ψs (−r ; E)if and only if Ψ is a quasi-exact solution of the corresponding Schrödinger equation. This conjecture can be extended to a broader class of potentials. The behavior of the wave functions corresponding to two lowest states of harmonium and of confined positronium are shown in Fig. 2.

The Schrödinger Equation with Power Potentials: Exactly-Solvable Problems

53

5 Semi-exactly Solvable Problems The basis r n , n = 0, 1, 2, . . . in expansion (13) may be changed to an equivalent polynomial basis, gn (r ), by performing a non-singular linear transformation of the original basis. As an effect we get different recurrence relations for the expansion coefficients, in general more complicated. But, if derivatives of gn (r ) and products r gn (r ) are expressible in terms of gn  (r ) then the recurrence relations remain equally simple and the expansion itself may be easier to handle. Let us note, that if the conditions mentioned above are fulfilled, then instead of an integer index n we can take a real one, ν. Thus, instead of expansion (13) we set F(r ) =

∞ 

an gν+n (r ),

(19)

n=0

where

d gν (r ) = f (ν) gν−1 (r ). dr

The last equation implies that the basis functions gν (r ) form a generalized Appell sequence. Here are two examples of such sequences: 1. Monomials, gν (r ) ≡ r ν : 

 d2 ν−1 d gν (r ) = 0. − dr 2 r dr

d gν (r ) = ν gν−1 (r ), dr

r gν (r ) = gν+1 (r )

2. Hermite functions (polynomials if ν is integer), gν (r ) ≡ Hν (r ): 

 d2 d + 2ν gν = 0 − 2r dr 2 dr

d gν = 2 ν gν−1 (r ), dr

r gν (r ) = ν gν−1 (r ) +

1 gν+1 (r ). 2

If the recurrence terminates then F(r ) is reduced to a finite sum of the basis functions—a polynomial, a combination of the Hermite functions with non-integer indices, etc. Using different basis functions opens a way to finding new exactly solvable potentials for which the wave function is expressed in terms of a finite set of functions simpler than the Heun function [5, 7, 8, 15, 26]. A theorem, rediscovered several times (see e.g. [17, 19]) states that the m−diagonal Hessenberg determinants solve the m−term recurrence relations for an . In other words, expansion coefficients an which fulfill an m−term recurrence relation

54

J. Karwowski and H. A. Witek

Fig. 3 A contour map of Ψ (r ; E) for V(r ) = r 4 , Along the energy axis Ψ (r ; E) changes sign in the vicinity of each eigenvalue E n . The amplitudes of the oscillations rapidly increase with increasing r and approach infinity for r → ∞. For large r the eigenfunctions Ψ (r ; E n ) approach the nodal lines of Ψ (r ; E)

are equal to the nth–order, m−diagonal, Hessenberg determinants. For example, in the case of a three-term recurrence and n = 5 we have    d0 −1 0 0 0     e0 d1 −1 0 0    a5 =  0 e1 d2 −1 0  = d4 a4 + e3 a3 .  0 0 e2 d3 −1     0 0 0 e3 d4  Since the recurrence relation coefficients d j , e j , . . . are explicit functions of E, also an and, cosequently, F(r ) and Ψ (r ; E) [c.f. Eqs. (12) and (19)], are explicit function of E. Thus, for a given E, the analytic form of Ψ is known. If Ψ (r ; E) is square-integrable then E is an eigenvalue and Ψ (r ; E)—the corresponding eigenfunction. The form of the two-dimensional surface Ψ (r ; E) for V = r 4 is presented in Fig. 3. As one can see, for a fixed r , Ψ (r ; E) oscillates and the amplitude fast increases with increasing r . The eigenvalues are located in the vicinity of the nodal lines. Since E has to be determined numerically, its value is known up to a finite accuracy. Except for the rare cases of a polynomial reduction, numerically determined E is never equal to the exact eigenvalue. Therefore the expansion of F(r ) is always divergent (it is an asymptotic expansion) and has to be cut-off at a certain value of r . The dependence of the behaviour of Ψ (r ; E) on the accuracy of the eigenvalues is presented in Fig. 4 and in Fig. 5.2

2 In

expansion (13) we took 400 terms.

The Schrödinger Equation with Power Potentials: Exactly-Solvable Problems

55

Fig. 4 Semi-exact radial eigenfunction Ψ (r ; E) for V(r ) = r 4 . For large r the function blows up due to a finite accuracy of E

Fig. 5 Semi-exact radial eigenfunctions Ψ (r ; E n ), for V(r ) = r 4 with n = 0, 1, 2, 3, 4, 5

6 Final Remarks In this report we discuss some aspects of quasi-exact and semi-exact solubility of the radial Schrödinger equation. Quasi-exact solutions are presented from two perspectives. First, as results of inverse problems, where quasi-exactly solvable potentials are derived from preassumed wave functions. Second, as polynomial reductions of general, semi-exact solutions. The polynomial reductions may be obtained by a proper selection of the potential function. More precisely, by establishing certain relations between numerical constants in the potential. In the case of semi-exact solutions, the dependence of the formal solution Ψ (r ; E) of the Schrödinger equation on the energy parameter E has been elucidated on the example of a quartic oscillator. Also high degree of sensitivity of the convergence pattern of the power series expansion of Ψ (r ; E) on the accuracy of the eigenvalues has been demonstrated. Conflict of interest We (Jacek Karwowski and Henryk A. Witek) herewith declare that we have no conflict of interest.

56

J. Karwowski and H. A. Witek

Acknowledgements This work was financially supported by Ministry of Science and Technology of Taiwan (MOST108-2113-M-009-010-MY3) and the Center for Emergent Functional Matter Science of National Chiao Tung University from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE), Taiwan.

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Electron-β-Nuclear Spectroscopy of Atomic Systems and Many-Body Perturbation Theory Approach to Computing β-Decay Parameters Olga Yu. Khetselius, Valentin B. Ternovsky, Yulia V. Dubrovskaya, and Andrey A. Svinarenko Abstract The modern concepts of physical nature of a beta-decay are briefly presented as well as the main characteristics of a beta-decay, classification of the betatransitions, selection rules etc. It is presented a new relativistic approach to calculating the characteristics of the β-decay of atomic systems (nuclei), based on the combined relativistic nuclear model and relativistic many-body perturbation theory formalism with correct accounting for exchange-correlation, nuclear, radiation corrections. A relativistic many-body perturbation theory is applied to electron subsystem, and a nuclear relativistic middle-field model is used for nuclear subsystem. All correlation corrections of the second order and dominated classes of the higher orders diagrams are taken into account. Within the framework of the presented theory, the characteristics of a whole series of allowed (super-allowed) β-decays are calculated, namely, for 33 P → 33 S, 35 S → 35 Cl, 45 Ca → 45 Sc, 63 Ni → 63 Cu, 106 Ru → 106 Rh, 155 Eu → 155 Gd, 241 Pu → 241 Am decays. The effect of the chemical environment of an atom on the characteristics (integral Fermi function, half-life) of the beta-transitions is studied. The results of accurate calculation of the beta-decay parameters are presented and compared with alternative theoretical data. Results of computing the Fermi function of a β − -decay with different definitions of this function are presented. The effect of an atomic field type choice on the beta decay characteristics as well as the influence of accounting for the exchange-correlation effects in the wave functions of the discrete and continuous spectrum on the values of the Fermi and integral Fermi functions are calculated. The obtained data are analyzed and compared with available in literature. Keywords Beta-decay · Electron-beta-nuclear spectroscopy · Relativistic perturbation theory · Correlation, nuclear, radiative corrections · Integral fermi function

O. Yu. Khetselius (B) · V. B. Ternovsky · Y. V. Dubrovskaya · A. A. Svinarenko Odessa State Environmental University, L’vovskaya str., bld. 15, Odessa 65016, Ukraine e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. V. Glushkov et al. (eds.), Advances in Methods and Applications of Quantum Systems in Chemistry, Physics, and Biology, Progress in Theoretical Chemistry and Physics 33, https://doi.org/10.1007/978-3-030-68314-6_4

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1 Introduction 1.1 Nuclear Beta Decay: Modern Concepts Nuclear β-decay is a manifestation of the fundamental weak interaction of elementary particles (see, for example, [1–22]). According to modern concepts, beta decay is due to the transformations of quarks: in β − -decay, one d-quark of a nucleon turns into a u-quark, in β + -decay reverse transformation occurs. The main quanta of the weak interaction are the so-called intermediate bosons—particles of large mass: 81.8 (W ± ) i 91.2 (Z 0 ) GeV/s2 . It should be noted that nuclear β-decay is one of the three main types of radioactivity. The main elements of the modern theory of nuclear beta–decay are considered in many textbooks and monographies on nuclear physics (e.g. [1–10]). With electronic (β − )-decay, one of the neutrons of the nucleus turns into a proton with the emission of an electron and an electron antineutrino  νe [2, 3]: A Z XN

→

A Z +1 X N −1

+ e− +  νe

(1)

Here A—the mass number, Z—the charge of a nucleus, N—the number of neutrons. In positron (β + ) decay, one of the protons of the nucleus turns into a neutron with the emission of a positron and an electron neutrino νe [2, 3]: A Z XN

→

A Z −1 X N +1

+ e+ + νe

(2)

Beta decay is closely related to the so-called reverse β-processes: capture of an electron from the K-shell of an atom (K-capture) or less likely to be captured from L- and other shells (electronic capture): e−+ z A X N →

A z−1 X N +1

+ ve ,

(3)

νe ) +zA X N → ve (

A z±1 X N ∓1

  + e− e+

(4)

and also reverse β-decay:

In fact at the level of nucleons the processes described above (1)–(4) represent the following well-known fundamental transitions (at the quark level): n → p + e− + ν˜ e ,

(5)

p → n + e+ + νe ,

(6)

e− + p → n + νe .

(7)

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Naturally the energy conditions for β − -, β + -decays and electron capture differ. For example, the β − -decay energy Qβ -his defined by the following relationship (e.g. [2, 3]): Q β − = [m(Z , A) − Z m e ]c2 − [(m(Z + 1, A) − (Z + 1)m e + m e ]c2 = [m(Z , A) − m(Z + 1, A)]c2

(8)

where m(Z, A) is the mass of a neutral atom. The quantity Qβ -corresponds to the difference in the masses of the parent and daughter atoms. Naturally, the spectrum of emitted β-particles is continuous and their kinetic energy takes values from 0 to a certain boundary energy E 0 , determined by the relation (see, for example, [2, 3]): E 0 /c2 = M(A, Z ) − M(A, Z + 1) − m e − m v ,

(9)

where M—are the masses of the initial and final nuclei. The foundations of the theory of beta-decay were created by E. Fermi in 1934. Fermi proceeded from the four-fermionic interaction of nucleons and leptons by analogy with the effective electron-nucleon interaction in electrodynamics. In this case, it is important that, in contrast to the electromagnetic interaction, which is long-range, the four-fermionic Fermi interaction was contact (local). The Hamiltonian of the Fermi nucleon-lepton interaction is written in the form (see, for example, [1]):    Hβ = G β Ψ p γμ Ψn  e γ μ ν .

(10)

Here Gβ —is the coupling constant (Fermi constant); Ψ —four-component wave functions of interacting particles, satisfying the Dirac equation; Ψ e = Ψ + γ 0 — conjugate wave functions; γ μ —Dirac matrices, μ = 0, 1, 2, 3, 4; γ 0 = γ 0 ; γ i = − γ i (i = 1, 2, 3). In the original version of Fermi’s theory, the nucleon-lepton interaction had a purely vector form. Later it became clear that the weak interaction Hamiltonian can be a combination of relativistically invariant terms formed from a scalar (S), a pseudoscalar (P), a vector (V), an axial vector (A), and a tensor (T). The discovery of spatial parity nonconservation, the study of the correlations between the directions of emission of β-particles and neutrinos in β-decay of 35 Ar and 6 He nuclei, as well as the angular distributions of electrons and neutrinos in the decay of polarized neutrons showed that β-decay is mainly realized in the V-A-variant. The effective β-decay Hamiltonian used in modern calculations was proposed by R. F. Feynman and M. Gell-Man in 1958 and is written in the following form:  √  Hβ = Gβ / 2 J μ (x)L μ (x) + h.c.

(11)

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Here h.c.—these are Hermitian conjugate terms; J μ —nucleon current; L μ — lepton current; x—space-time coordinate; Gβ = Gμ · cosϑ c , where Gμ —universal constant of weak interaction; multiplier cosϑ c responds to processes without changing weirdness (ϑ c —so called Cabibbo angle); Constant Gβ = 1.4 · 10−49 erg cm3 is determined experimentally. The lepton current L μ is a combination of V–A terms with equal weights and is expressed through the wave functions of the electron and neutrino: L μ (x) = Ψ e (x)γ μ (1 + γ5 )Ψν ,

(12)

where γ 5 = iγ 0 γ 1 γ 2 γ 3 . The nucleon current J μ is also a combination of the vector and axial-vector terms J μ = V μ (x) − Aμ (x).

(13)

It cannot be written out explicitly in terms of the wave functions of nucleons; however, the matrix elements of V μ and Aμ between the nucleon states, which determine the characteristics of the nucleon-resonance, can be expressed through a small number of coupling constants gV , gM , gS , gp , gT (see, for example, [1–4]):

     2  2 gM q 2 v  σμν q + g S q qμ τ ± U N , (14) N V (0)± N = U N  gV q γμ + 2Mc

       2  2 gT q 2 v N  Aμ ± (0)± N = U N  g A q γμ + g P q qμ + σμν q γ5 τ ± U N , 2Mc (15) 





±

where N, N  —initial and final nucleons; U—Dirac bispinor (solution of the free Dirac equation); τ ± —increasing and decreasing isospin operators, converting neutron to proton and proton to neutron; σ νμ = 1/2 (γμ γν − γν γμ) ; N = 0, 1, 2, 3; qμ = (PN − PN )μ—transmitted fourth pulse; PN i PN —momenta of the initial and final states of the nucleon. From the hypothesis of conservation of the vector current it follows that: gV ≡ gV (0) = 1, gS (q2 ) = 0, gM (0) = μp − μn = 3.70, where μp , μn —abnormal magnetic moments of the proton and neutron in units nuclear magneton. Experimental studies of β-decay of nuclei have confirmed the hypothesis of conservation of the vector current and obtain a limitation on the constant gT , which characterizes the axial current of the second kind: |gT /gA |≤10−4 . The one-nucleon Hamiltonian H β is written in the form:  √  Hβ = Gβ / 2 gV (1L 0 − αL) − g A (γ5 L 0 − σ L 0 }τ ± .

(16)

Electron-β-Nuclear Spectroscopy of Atomic Systems …

63

Here gV and gA —vector and axial constants of nucleon-lepton interaction; 1 is the unit operator; α = γ0 γ are the Dirac matrices; σ = −γ0 γ γ5 are the Pauli spin matrices. As a result, the effective β-decay Hamiltonian is determined by two coupling constants—the vector gV and the axial-vector gA . Further development of the theory led to the creation of a unified theory of weak and electromagnetic interactions (and then the Standard Model), however, the existence of intermediate bosons has practically no effect on the theory of beta decay due to the smallness of the energy E ≤ 10 MeV in comparison with mW c2 . For this reason, in fact, the theory of electroweak interactions for β-decay is reduced to the theory of Feynman and Gell-Mann (see details in Refs. [1–4]).

1.2 Main Characteristics of β-Decay. Classification of β-Transitions As is known, the main characteristics of β-decay include the half-life T 1 /2 , the shape of β-spectra, β ± -γ-angle correlations, etc. An analysis of the fT 1 /2 values, together with the selection rules (see below), makes it possible to determine the unknown values of nuclear spins and parities, i.e. is one of the important methods of nuclear spectroscopy (see, for example, [1–9]). Since the fT 1 /2 values are directly related to the matrix elements of β-transitions, they also contain information about the nuclear structure. To determine the characteristics of beta decay, it is initially necessary to determine the amplitude of the process, which is determined by the known usual matrix element of the transition between the initial i and final f nuclear states:    M f i = f  Hβ i .

(17)

In the case of β-decay of a nucleon, the desired matrix element:

Mfi =

ψ +f (rl , . . . r A )Hβ (rl , . . . , r A )Ψi (rl , . . . , r A )d 3 rl , . . . , d 3 r A ,

(18)

where the effective Hamiltonian of the process H β is equal to the sum of the terms describing the β-decay of individual nucleons. It should be emphasized here that the theory describes not only one-nucleon transitions. In the wave functions of the initial and final states of nuclei, it is possible to take into account the effects of a multi-nucleon structure, including the possibility of collective excitations of the nucleus [2–20]. Naturally, in this approximation, the meson exchange currents (the corresponding terms describe the emission of the e  νe (e+ ν e ) pair by virtual mesons, which are exchanged by nucleons in the nucleus) are not taken into account. Also, the emission of a lepton pair by nucleons, which occurs due to the exchange of virtual mesons, is not taken into account.

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In fact, taking into account the meson exchange currents leads to the many-body operator H β . It should also be added that the contributions of the sought-for meson exchange currents to the β-spectra and half-lives can reach several percent. The βparticle spectrum is related to the matrix element M fi by the following expression [2]: N (E)d E =

G 2β 2π 2 c5 7

   M f i 2 pE(E 0 − E)2 d E,

(19)

where p and E—momentum and energy of the emitted β-particle; In deriving expression (19), it is usually assumed that mν = 0 and the recoil energy of the final nucleus is negligible compared to E 0 . If M fi does not depend on energy, then the shape of the β-spectrum is determined only by the “statistical” factor: N (E) ∼ pE(E 0 − E)2 .

(20)

When calculating the matrix elements M fi , a number of approximations are usually used, namely [3]: (i) being taken between nuclear states, some operators entering the formula for H β have matrix elements of the order 1, whereas others have matrix elements of order ν N /c, where ν N —characteristic velocity of a nucleon in a nucleus; (ii) the boundary energies E 0 are relatively small; therefore, the de Broglie wavelengths of the emitted leptons are large compared to the size R of a nucleus. When calculating M fi , an expansion in these small parameters is usually used. The neutrino wave function Ψ ν entering the lepton part of the matrix element Lμ(r) is described by a plane wave, i.e. [2, 3]: Ψv (r ) ∼ ex p(−iqr | ) ≈ 1 − iqr | − 1/2(qr | )2 + . . . .

(21)

Inside the nucleus (r < R) Ψ ν (r) ≈ const, and upon integration over the volume of the nucleus, the neutrino wave function does not lead to the dependence M fi from E. In the approximation of neglecting the interaction of the emitted β-particle with the Coulomb fields of the nucleus and the electron shell of the atom, its wave function can also be represented as an usual plane wave. Taking into account the Coulomb fields of the nucleus and the electron shell of the atom leads to a difference between the wave function and a plane wave; as a result, the wave function becomes dependent on the energy E even at pr|  1. Note that initially this circumstance was ignored and this often predetermined a significant error in calculating the characteristics of beta decay (for example, see details in Refs. [2–5]). To take into account the influence of the Coulomb interaction of the emitted β-particles on their energy spectrum, the so-called Coulomb correction factor is introduced, which is determined by the known Fermi function F(Z, E). When pr|  1 this factor is usually defined as the square of the ratio of the β-particle wave functions calculated with (Z = 0) and without (Z = 0) the Coulomb

Electron-β-Nuclear Spectroscopy of Atomic Systems …

65

field of the nucleus at the center (r = 0) or at the periphery (r = R) of the nucleus, i.e. [4, 12, 13]: F(Z , E) = |Ψe |2z /|Ψe |20 .

(22)

The approximation, in which only the leading nucleon contributions to the Hamiltonian H β are taken into account, and the lepton wave functions inside the nucleus are assumed to be independent of coordinates, is called allowed in the theory of beta decay. In this approximation, the spectrum of β-particles is described by the expression [2–4]: N (E)d E =



m 5e c4 G F(Z E) 2π 3 7 β √ ·E E 2 − 1(E

 2   2  n 2 V  1 + g 2 A  σ  .

− E)2 d E.    A

   (i)  1 ≡ f τ± i ,  

(23)

0

(24)

i=1

   A    (i) (i)  σ ≡ f σ τ± i .  

(25)

i=1

Here the energy is expressed in units of me c2 (me —is the electron mass); the first relation corresponds to the vector interaction C V and is called the Fermi matrix element, and the second relation corresponds to the axial-vector interaction C A and is called the Gamow-Teller matrix element. It is important to remind that the Coulomb field of a nucleus increases the probability of the emission of electrons and decreases the probability of the emission of positrons in the low-energy region. In addition, when the Fermi factor F(Z, E) is taken into account, the probability of electron emission during beta decay at the lower boundary of the β-spectrum does not vanish, but tends to a finite value. The influence of the Coulomb factor on the β-spectra and the probability of beta decay increase with increasing Z and decreasing E 0 . When calculating F(Z, E), it is also necessary to take into account the screening of the nuclear charge by atomic electrons (it is especially important in the case of β + -decay). It should be emphasized here that this effect has not yet found an adequate quantitative description in modern calculations (e.g. [1–13]). In many papers (e.g. [1–30]) the possibility of influencing the processes of nuclear decay with the participation of electrons of the atomic shell (K-capture and internal conversion) by ionizing the atom was considered. In the German research center GSI, it was experimentally shown that the effect of the presence or absence of electron shells in an atom can significantly change the entire decay scheme and, accordingly, the quantitative characteristics (e.g. [3]). The total probability w of beta decay per unit time can be represented as follows (see details, for example, in Refs. [1–4]):

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O. Yu. Khetselius et al.

   2  2      m 5e c4 2  2   w= G β g V  1 + g A σ  f, 3 7 2π 

E0 f =

 F(Z , E)E E 2 − 1(E 0 − E)2 d E.

(26)

(27)

1

The f value can be computed using the tabulated values of the Fermi function F(Z, E) [2]. A remarkable feature of the allowed transitions is the fact that all nuclear β-moments are concentrated in one factor, and the energy dependence is due only to a statistical factor and a function F(Z, E). In the expression for the normalized β- and ν˜ -spectra, the factor [C V2 < 1 >2 +C 2A < σ >2 ]

(28)

can be taken out from under the integral sign, and after abbreviations, the final expressions for calculating the spectra of allowed transitions can be obtained. By definition, the half-life T 1 /2 is related to the beta-decay probability w by the standard ratio: w = ln 2/T1/2 .

(29)

Then you can write: 

f T1/2

 2  2      2 = k/ g V  1 + g A σ  , 2

(30)

where k = 2π 3 ln 2 7 /m5e c4 G2β . The fT 1 /2 value is usually called the comparative half-decay period and plays an important role in the classification of β-transitions (see below). The function f takes into account the dependence of the beta decay probability on E 0 and Coulomb effects; therefore, fT 1 /2 , in contrast to the standard half-life T 1 /2 , depends only on Mfi . Next, we briefly consider the main classification of β-transitions. It should be noted right away that beta decay is characterized by a wide range of changes in the half-lives of T 1 /2 , usually from 10−2 s to 1016 years. Such a significant variation in the T1/2 values is explained by several reasons. First of all, this is due to the fact that the half-life strongly depends on E 0 (at E 0 me c2 , w ~ E 50 ), and the value of E 0 varies widely from 2.64 keV for the 187 Re → 187 Os transition to 13.43 MeV for 12 B → 12 C. On the other hand, depending on the spins and parities of the initial and final nuclear states, various terms in the effective beta decay Hamiltonian, whose matrix elements have different orders of magnitude, contribute to the process amplitude. Finally, the lepton pair emitted during beta decay can carry away different orbital angular momentum. With an increase in this moment, due to the centrifugal effect, the

Electron-β-Nuclear Spectroscopy of Atomic Systems …

67

values of the wave functions of leptons in the intranuclear region, and, consequently, the overlap integral of wave functions, which determines the matrix element Mfi. Accordingly, all β-transitions are divided into allowed and forbidden (e.g. [2]). Let’s consider the allowed transitions first. In the allowed approximation, the wave functions of leptons inside the nucleus are constant, and leptons do not carry away the orbital angular momentum. Moreover, if the spin of the nucleus does not change, then the total spin carried away by the lepton pair is also equal to 0. Such transitions are called Fermi transitions. In the case when the vector change in the nuclear spin (the total spin carried away by the lepton pair) is equal to 1, then, by definition, these transitions are called Gamow-Teller. The parity of nuclear states in allowed β-transitions does not change. As a result of the selection, the rules limiting the change in the total moment I and the parity π of the nucleus, in the case of allowed transitions of the Fermi type, are written in the form: ΔI = | I f − I i |= 0; Δπ ≡ π f π i = +1. For Gamow-Teller transitions, similar selection rules are: ΔI = 1, Δπ = +1. Further, in the modern classification, allowed transitions are subdivided into superallowed and hindered. The first include transitions between nuclear states with  similar  wave functions, as a result of which the integrals of their overlap are large ( ~1, σ ~1), and the values fT 1 /2 take minimum values. The super-allowed transitions include, in particular, transitions between states belonging to thesame isomultiplet (between analog states of nuclei). For supersolved β ± -transitions, 1 can be calculated exactly [2–4]. The fact is that A 

τ±i = T± ,

(31)

1 = [(T ∓ T3 ) · (T ± T3 + 1)]1/2 .

(32)

i=1

where T is the isotopic spin of the initial nucleus, T 3 is the isospin projection for the initial nucleus, numerically equal to ½ (Z − N). Here it is assumed that the βtransition occurs between pure isospin states; taking into account the meson exchange currents does not change this result, which is due to the conservation of isospin. In + + the case of super-allowed transitions √ 0 between neighboring terms of the   0 → isomultiplet: σ = 0 and, at T = 1: 1 = 2. For such super-resolved transitions, the f T 1 /2 values should be the same, which is in good agreement with available data (Table 1). The relationship (30) allows to determine the value of Gβ from the measured values of f T 1 /2 for 0+ → 0+ transitions: G β = (1.4057 ± 0.0016 ± 0.0070) · 10−49 erg. cm3 .

(33)

+ + Further, it is worth to remind that the  Gamow-Teller transitions 0 → 1 are characterized by a single matrix element σ = 0 and can be used to obtain information

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Table 1 Characteristics of some super-allowed β-transitions πf

Transition

Iiπi → I f

T 1/2

E 0 (keV)

fT 1/2

n→p

½+ → ½+

11.7 + 0.3 min

782 ± 1

1187 ± 35

½+ → ½+

3.87 · 108 s

18.65 ± 0

1132 ± 40 808 ± 32

3H

→

→

6 He 17 F

3 He 6 Li

→ 17 O

0.813 ± 0.7 s

3500 ± 2.0

5/2+ → 5/2+

66.0 ± 0.5 s

1748 ± 6

2380 ± 40

0+

→

1+

35 Cl

→ 35 Ar

3/2+ → 3/2+

1.804 ± 0.21 s

4948 ± 30

5680 ± 400

14 O

→ 14 N

0+ → 0+

71.36 ± 0.09 s

1012.6 ± 1.4

3066 ± 10

34 Cl

→

0+ → 0+

1.565 ± 0.07 s

4460 ± 4.5

3055 ± 20

42 Sc

→ 42 Ca

0+ → 0+

0.6830 ± 0.0015 s

5409 ± 2.3

3077 ± 9

→ 46 Ti

0+ → 0+

0.4259 ± 0.0008 s

6032.1 ± 2.2

3088 ± 8

→

0+ → 0+

0.2857 ± 0.0006 s

6609.0 ± 2.6

3082 ± 9

46 V

50 Mn

34 S

50 Cr

on the value of the axial-vector coupling constant gA . The most accurate value gA = −1.254 ± 0.007 is obtained from the data on β-decay of the neutron. The so-called hindered transitions differ from the super-allowed transitions by a relatively weak overlap of the wave functions of the initial and final nuclear states, as a result of which the matrix elements turn out to be small compared to the matrix elements of the super-allowed transitions [2]. Another type of β-transitions is called forbidden transitions. Selection rules for matrix elements of forbidden transitions are derived similarly to the case of allowed transitions, while the expression for the matrix element after applying the procedure for separating the reduced matrix elements (β-moments) has a rather complicated and inconvenient form for practical use. To simplify it, the so-called normal approximation is used, based on the fact that nuclear β-moments have different orders of magnitude. The small parameters by which these quantities are estimated are: nucleon velocity V N , nucleus radius R, Coulomb smallness parameter aZ. The order of smallness of the β-moments included in the expansion of the matrix element determines the degree of inhibition of β-transitions. Forbidden transitions include transitions in which a lepton pair carries away the orbital angular momentum and (or) the main contribution to the process amplitude is made by small matrix elements from the operators γ5 , α in the effective Hamiltonian Hβ . Forbidden transitions are classified according to the degree of smallness of the matrix element. Transitions of the first   include  transitions  order   of exclusion described by the matrix elements such as α, r, γ5 , [σr], (σr), Bij , where

  A     α a τ±a i , α = f  

(34)

a=1

  A     r = f r a τ±a i ,   a=1

(35)

Electron-β-Nuclear Spectroscopy of Atomic Systems …

Bij ≡ σi xj + σj xi 2/3(σ r )σij ;

69

(36)

etc. Here i, j = 1, 2, 3; xi —vector component r. The first two matrix elements are due to the vector current, the rest—to the axial one. Matrix elements containing the value r arise when a lepton pair carries away the orbital angular momentum 1. Selection for matrix elements γ5 , (σr) are as follows:  I = 0, π =  rules  −1. For α, r and [σr], the selection rules are: I π = 1− , 0− (transitions 0 ↔ 0 are prohibited). In the transitions described by matrix transitions of the first forbidden, the lepton pair carries away the total moment 2, and the selection rules are as follows: Iπ = 2−, 1− , 0− (forbidden transitions 0↔ 0, 0 ↔ 1, ½ ↔ ½). The matrix elements γ5 and α are of order of smallness (νN /c). For matrix elements containing r, it is natural to expect that the order pR|è ≤ E 0 R|èc. However, this is only true for unique transitions. With an increase in the order of exclusion, the number of the corresponding matrix elements that determine the transition probability increases, and the difficulty of analyzing the experimental data increases; in this case, the matrix elements themselves decrease in order of magnitude [2]. Selection rules for β-transitions of the nth order of prohibition: π = (−1)n , I ≤ n for ordinary transitions and ≤ n +1 for unique transitions. With an increase in n and a decrease in matrix elements, the value of fT 1 /2 increases. Although the range of its variation is narrower than for T 1 /2 , it turns out to be quite large, so here it is convenient to characterize β-transitions by the value lg f T 1 /2 (see Table 2). Further, before proceeding to a detailed analysis of the current state of calculations of the characteristics of beta decay, we note some experimental aspects of the problem, following to Refs. [2, 4]. Usually β-spectra are experimentally investigated, as a rule, using beta-spectrometry. To conclude this subsection, let us note that in many cases, beta decay occurs not into one state of the daughter nucleus, but into two or more states. In this case, the experimentally observed β-spectrum is composed of two or more partial spectra with different values of the boundary energies. Such β-spectra are usually called complex. Investigation of β-spectra near E 0 allows obtaining information on the neutrino mass mν . The β-spectrum study of the 3 H (E 0 = 18.61 keV) gave mν < 35 eV/s2 . Results obtained with emission of the β-spectrum 3 H: 14 eV < mν < 46 eV need further confirmation (see details in Refs. [1, 7–11, 25]).

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Table 2 Selection rules for β-transitions of various types Transition type Allowed over-authorized hindered

Selection rules  I = 0, 1 π = +1

Forbidden first ban

I = 1.0

Unique first ban

π = −1

Second ban

I = 2 π = −1

Unique second prohibition

I = 3 π =+1

Third ban

I = 3 π =+1

Unique third prohibition

I = 3 π = −1

Fourth ban

I = 4 π = −1 I = 4 π =+1

lg fT 1/2

lg fnT 1/2

3.5 ± 0.2 5.7 ± 1.1 7.5 ± 1.5

8.5 ± 0.7

12.1 ± 1.0

11.7 ± 0.9

18.2 ± 0.6

15.2 (40 K)

227 (115 ln)

2 Theoretical Method. Relativistic Many-Body Perturbation Theory 2.1 Determination of the Probability of Beta Decay As is known [4], the perturbation theory method is usually used in calculating the probability of β-decay, since the corresponding interaction constant g is characterized by significant smallness. For this well-known reason, in practice, the calculations are limited to taking into account only first-order terms corresponding to direct transitions from the initial state to the final state. The probability of a transition from an initial state |i > with energy E i to a certain final state < f | with energy E f per unit time under the condition E 0 = E f − E i is determined by the well-known Fermi golden rule expression: dwξ f =

2π    2 d N f Hβ i · | E=E0 ,  dE

(37)

where, naturally, the matrix element is determined by the form of the interaction Hamiltonian Hβ and the wave functions of the initial ψ i and final ψ f states of the nucleus:

   f  Hβ ξ = ψ ∗f Hβ ψi d 3r1 . . .d 3r A (38)

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71

The determination of the square of the matrix element is reduced to integration over the volume of the nucleus and averaging over all unobservable variables. The quantity dd NE | E=E0 determines the density of the final states of the system per unit of energy. The detailed expressions for the probability (number of β − , ν˜ -particles with energies in the range from E to E + dE) of transitions can be found in Refs. [2–4, 12–15]. In what follows, we restrict ourselves to considering allowed and over-allowed transitions. It is generally known that allowed transitions make the most significant contribution to the total spectrum of β-decay of a nucleus, while the contribution of forbidden transitions usually amounts to only a few percent of the total intensity. The specific contribution of these and other transitions to the β-decay probability is usually described using in the expression for the Hamiltonian of the interaction and the β-decay probability of the expansion of the lepton current in a series in terms of small parameters characteristic of β-decay (see [2, 3]). Where the zero term of such an expansion describes the most intense allowed β-transitions, and the next terms of the expansion correspond to forbidden transitions of various degrees of forbiddenness. Let us consider further the allowed transitions in more detail. The energy distribution of β-particles in this case has a standard form: dwβ (E)/d E =

1 2 G · F(E, Z ) · E · p · (E 0 − E)2 · |M|2 . 2π 3 F

(39)

Here GF —is the weak interaction constant; E, p = (E 2 − 1)½ —total energy and momentum of a β-particle; |M|—energy-independent matrix element for allowed β-transitions. The Fermi function F can be represented as follows: F(E, Z ) =

1 2 (g 2 + f +1 ), 2 p 2 −1

(40)

where are the icons ±1 = κ, κ = (l − j)/(2j + 1). In Eq. (40) the functions f +1 and g-1 are relativistic electron radial wave functions, which are calculated at the boundary of a spherical nucleus with radius R0 (see, for example, [2]) or the values of these functions at zero (amplitudes of the expansion of functions in a series at zero), as it is shown in Refs. [3, 12, 13]. In our calculations, we use the latter option everywhere. The corresponding integral Fermi function f is given by the definition:

E0 f (E 0 , Z ) =

F(E, Z ) · E · p · (E 0 − E)2 d E.

(41)

1

The half-life of beta decay in this notation is:   T1/2 = 2π 3 ln 2/ G 2 |M|2 f (E 0 , Z ) .

(42)

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O. Yu. Khetselius et al.

An important point of the theory is the correct normalization of the relativistic electron radial functions f κ and gκ , at which, for large values of the radial variable gi (r ) → r −1 [(E + 1)/E]1/2 sin( pr + δi ),

(43)

f i (r ) → r r −1 (i/|i|)[(E − 1)/E]1/2 cos( pr + δi ).

(44)

A detailed description of the methodology for calculating the electronic and nucleon wave functions within the framework of the formalism of the relativistic nuclear and relativistic (QED) many-body perturbation theory is given in Refs. [4, 6, 11, 18–46] (see also [47–61]). Here, we note that the numerical solution of all equations, as well as the entire calculation of the characteristics of β-decay and atomic corrections were performed on the basis of a modified numerical complex “Superatom-M”. The functions of the continuous spectrum were found iteratively in the field of the daughter atom. The condition for the self-consistency of the functions of the continuous spectrum is reduced to the fact that the normalized functions at two adjacent iterations differ by less than 10−5 in relation to their values at the maximum point of the function. For different energies, to achieve the required accuracy, it was required from 3 (at higher energy) to 11 (at low energy) iterations. When calculating the normalizing factor, the procedure of averaging over the oscillation period of the continuous spectrum function was used (the matching condition included the difference between the values of the averaged normalizing factors at two adjacent periods of no more than 0.025%). To achieve the required accuracy, the Dirac equations were integrated (on a semilogarithmic scale) to the distances from the core, at which the continuum function passes 6–8 periods. As usual, when calculating the integrals of strongly oscillating functions, the damping factor exp (−dr) was introduced, the value of the parameter d in which was chosen based on the accuracy requirement at a level of ~0.005%.

2.2 Relativistic Many-Body Perturbation Theory Here we present a brief description of the key moments of our approach (more details can be found in Refs. [21–60]). Fundamental spacts of accounting for the QED radiative corrections and physical nature of these ones is described in Refs. [61–85]. Within our approach, the electron wave functions zeroth basis is found from the generalized Dirac-Kohn-Sham equation solution with a mean-field (MF) selfconsistent potential: D (r ) + VX (r ) + VC (r |b)] VM F = V D K S (r ) = [VCoul

(45)

Electron-β-Nuclear Spectroscopy of Atomic Systems …

73

D Here VCoul (r ) is the standard Coulomb-like potential, VC (r |b) is a correlation potential (the known Lundqvist-Gunnarsson-like definition for VC (r |b) with ab intio optimization parameter b is used; for details, see below and Refs. [10, 48–52] and VX (r ) is an exchange potential [48, 49]. The known Kohn-Sham definition for VX (r ) is as follows (in atomic units):

 3 [β + (β 2 + 1)1/2 ] 1 ln , · − 2 β(β 2 + 1)1/2 2 

VX [ρ(r ), r ] =

VXK S (r )

(46)

where β = [3π 2 ρ(ri )]1/3 /c

(47)

In order to describe a nuclear subsystem we use the known relativistic mean-field model [2, 3]. In concrete calculation the most preferable version of this model is so called NL3-NLC version (c.g., Refs. [2, 3, 16, 62]). The total relativistic Dirac Hamiltonian for a multielectron system has the following form [2, 10]: H=

 i

{αcpi − βc2 − Z /ri } +



exp(i|ω|ri j )(1 − αi α j )/ri j ,

(48)

i> j

where α i ,α j are the Dirac matrices, ωij is the transition frequency. It should be noted that the magnetic interaction in the lowest order on parameter of the fine structure constant α 2 (α is the fine structure constant) as well as the retarding effect are taken into account in the relativistic interelectron interaction potential. As it is indicated earlier, all correlation corrections of the second order and dominated classes of the higher orders diagrams are taken into account within a formalism of many-body perturbation theory [11, 21–46]. The principal important point of a total approach is in using a generalized relativistic energy approach to construction of an optimized basis set of electron wave functions. According to Glushkov-Ivanov-Ivanova method [48, 49, 52, 53] optimization of electron wave function set and gauge invariance performance can be reached by means of the minimization of contribution into imaginary part of radiation width Im δE for the multi-electron system due to the QED perturbation theory fourth order Feynman diagrams ones. The details of a whole procedure can be found in Refs. [2, 3, 10, 48, 49, 52, 53]. The next very important aspect of a whole procedure is an accurate consideration of the QED or radiation corrections. There are developed a few accurate methods of accounting for the QED corrections. In our approach we use the generalized procedures, described in detail in Refs. [2, 10, 50, 63, 67]. In order to account for a vacuum polarization effect, the generalized UehlingSerber potential approach is used and modified to account for the high-order radiative

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O. Yu. Khetselius et al.

corrections according to the procedure [2, 10]. It can be represented in the following form: U (r ) = −

∞

2α 3πr

√      t2 − 1 2α dt exp −2r t α Z 1 + 1 2t 2 ≡− C(g), (49) t2 3πr

1

where g = r/(αZ). A more correct and consistent approach is presented in Refs. [4, 11]. Taking into account the nuclear finite size effect modifies the potential (49) as follows:

U

FS

2α 2 (r ) = − 3πr

3 

∞

d r

1

√      2  t 2 − 1 ρ(r  )    dt exp −2t r − r α Z · 1 + 1 2t , |r − r  | t2 (50)

The details of some alternative approaches can be found in Refs. [62–84]. Other details of the used method and PC code are described in Refs. [2, 10, 21–45]. All calculations are performed with using the numeral codes SuperAtom (Nucleus) (modified versions 93) [21–61, 85–113].

3 Results 3.1 Characterization of a Number of Allowed Beta Transitions and the Results of Calculating the Characteristics of Beta Decay The following beta decays were selected as objects of study, the results of which will be presented below (their characteristics are given in Table 3): 33 P → 33 S, 35 S → Table 3 Characteristics of a number of allowed β-transitions lg ft πf

Z mat → Z daught

Iiπi → I f

Type

E 0 (keV)

T 1 /2

lg ft

15 → 16

1/2+ →3/2+

Allowed

249

25.3 days

5.0

16 → 17

3/2+ →3/2+

«»

167.4

87.4 days

5.0

20 → 21

7/2− → 7/2−

Above

257

165 days

6.0

63 Cu

28 → 29

1/2− →3/2−

«»

65.8

100 days

6.6

106 Ru

→ 106 Rh

44 → 45

0+ → 1+

«»

39.4

367 days

4.3

155 Eu

→ 155 Gd

63 → 64

5/2+ →3/2+

«»

140.7

4.9 years

7.4

241 Pu

→

94 → 95

5/2+ →3/2−

First ban

20.8

14.4 years

5.8

Decay 33 P

→

35 S

→ 35 Cl

33 S

45 Ca

→ 45 Sc

63 Ni

→

241 Am

Electron-β-Nuclear Spectroscopy of Atomic Systems …

75

Cl, 45 Ca → 45 Sc, 63 Ni → 63 Cu, 106 Ru → 106 Rh, 155 Eu → 155 Gd, 241 Pu → 241 Am. Most of the considered beta decays belong to the number of transitions with a low boundary energy and correspond to different ranges of values of the atomic nucleus charge Z (see Table 1). Almost all transitions, the characteristics of which are given in Table 1, are allowed (as well as super-allowed). The choice of such transitions, naturally, is determined by the important circumstance that for such transitions the formulas for the decay probability are exact. Of course, for forbidden beta transitions, the theory naturally becomes more complicated. The corresponding formulas are more complicated than in the case of allowed transitions and should generally contain six nuclear matrix elements. A detailed presentation of the theoretical aspects of their calculation is given, for example, in [2, 4]. In the so-called ξ-approximation known in the theory of beta decay, where the parameter ξ is introduced, determined by the expression: ξ = αZ/2R0 1, (Z—the nuclear charge, R0 —the radius) usually neglect small terms, and the remaining sum of nuclear matrix elements does not give an additional dependence on the lepton energy and turns out to be analogous to the allowed case. Moreover, it is written as a constant factor |M|2 . Recall that if the condition ξ-approximation ξ 1 is satisfied, the Coulomb effects lead to an increase in the wave function of the electron inside the atomic nucleus, as a result of which these matrix elements are of the order of smallness ~Z/137, and no pR|è. At the same time, it is well known that the sought condition ξ > > 1 turns out to be satisfied for most β-transitions. Returning to the transition 241 Pu → 241 Am, it should be noted that this transition is not a unique one of the first ban. The parameter ξ for the decay of plutonium is ξ = 18 (i.e. ξ 1). It is well known that for the overwhelming majority of such first-forbidden transitions the formulas for the decay probability are applicable with a sufficiently high degree of accuracy. In Ref. [2] the results of the test calculation of the probabilities and half-lives 34 Cl → 34 S, 42 Sc T 1 /2 of a number of super-resolved beta transitions, in particular,  42 ± → Caare listed. Recall that for superallowed β -transitions 1 can be calculated exactly 1 = [(T ± T 3 ) · (T ± T 3 + 1)]1 /2 , where T 3 —isospin projection for the initial nucleus, numerically equal to ½ (Z − N). If the β-transition occurs between pure isospin states, then taking into account the meson exchange currents (as a rule, contributing to a few percent) does not change this result, which is due to the conservation of isospin. + → 0+ between neighboring terms of In the case of super-allowed transitions 0√  the isomultiplet σ = 0 and, at T = 1, 1 = 2. For such super-allowed transitions, the f T 1/2 values are almost the same. The performed calculation (the well-known Gauss model was used to determine the charge distribution in the nucleus) gave the following values of the half-lives for the transitions 34 Cl → 34 S (1.55 s), 42 Sc → 42 Ca (0.67 s). The sought data are in good agreement with the experimental values (respectively: 1.565 ± 0.007 s; 0.683 ± 0.002 s). For comparison, we present similar calculation data in the framework of 35

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O. Yu. Khetselius et al.

the standard model of the Dirac-Fock atom (1.52; 0.64), as well as in the framework of the Hartree-Fock-Slater method (1.4; 0.6) [12, 13]. Thus, in the approach proposed by us, it is more correct to take into account exchange-correlation and other effects. It is easily to find that the accuracy of calculations within the framework of the standard and optimized DF method is quite acceptable. It seems important to study in more detail the influence of the choice of the atomic field on the values of the Fermi function.

3.2 Results of Computing an Effect of Atomic Field Type Choice on the Beta Decay Characteristics In this subsection, we present the results of evaluating the influence on the Fermi function of the choice of the type of atomic field. Note that in a number of papers (see, for example, Refs. [2–13]) various methods were used to calculate the characteristics of beta decays, in particular, the method of the self-consistent nonrelativistic atomic field of the Hartree-Fock-Slater (HFS), the method of the relativistic self-consistent field of the HFS (taking into account relativistic corrections in the Breit-Pauli approximation), the classical and improved versions of the Dirac-Fock (DF) method etc. In order to compare different approaches, the calculation of the Fermi function F(E, Z) is carried out under conditions similar to [12, 13], namely, in all cases the values of the functions on the boundary of the nucleus R0 = 1.202×A1/3 fm with the same A. The corresponding numerical results of the influence of the choice of the atomic field (HFSrel , HFSnonrel ; Dirac-Kohn-Sham (DKS) approach) on the Fermi function F(E, Z) for different beta decays are given in Tables 4 and 5. The parameters are calculated as the test quantities: 1 = {[FHFSrel (E, Z )/FHFSnonrel (E, Z )] − 1} · 100%,

(51)

Table 4 Effect on the Fermi function F(E, Z) of a choice of the atomic field (HFSrel , HFSnonrel , DKS): value 1 (%) E kin (keV)

Z = 20

Z = 80

Z = 95

10

−0.05

−0.34

−0.56

50

−0.03

−0.34

−0.55

100

+0.01

−0.34

−0.45

500

+0.08

−0.30

−0.40

Note Here 1 = {[F HFSrel (E, Z)/F HFSnonrel (E, Z) ] − 1} · 100%, where F HFS (E, Z)—Fermi function in atomic field HFSrel; F HFSnonrel (E, Z)—Fermi function in atomic field HFSnonrel [2, 12, 13]

Electron-β-Nuclear Spectroscopy of Atomic Systems …

77

Table 5 Effect on the Fermi function F(E, Z) of a choice of the atomic field (HFSrel , HFSnonrel , DKS): value 2 (%) E kin (keV)

Z = 20

Z = 44

Z = 63

Z = 80

Z = 95

10

−0.08

−0.10

−0.24

−0.56

−0.79

50

−0.06

−0.08

−0.23

−0.55

−0.77

100

+0.04

−0.07

−0.18

−0.54

−0.68

500

+0.13

−0.06

−0.14

−0.51

−0.61

Note Here 2 = {[F DKS (E, Z)/F HFSnonrel (E, Z)] − 1} · 100%, where FHFS (E, Z)—Fermi function in atomic field HFS; F DKS (E, Z)—Fermi function in atomic field DKS [11–13]

2 = {[FDKS (E, Z )/FHFSnonrel (E, Z )] − 1} · 100%,

(52)

where F HFSrel (E, Z)—Fermi function in atomic field of the HFSrel ; F HFSnonrel (E, Z)—Fermi function in atomic field of the HFSnonrel ; F DKS (E, Z)—Fermi function in atomic field of the DKS. Note also that in all three calculations, the effect of the finite size of the nucleus was taken into account within the framework of the model of a uniformly charged ball. As the calculation has shown, the use of the alternative Gauss model has practically no effect on the results obtained, although it is more convenient computationally. Analysis of the data obtained shows that for small and medium values of the nucleus (for example, Z ~20), the difference between the data obtained on the basis of the relativistic HFS and DKS methods turns out to be insignificant, amounting to hundredths of a percent. At large values of Z (up to Z = 95, calculations in the HFSrel field gave a 0.5% lower value for F(E, Z), and in the DKS field by 0.8%, in comparison with the nonrelativistic HFSnonrel values. The reason for this difference is obviously related to the well-known effect of relativistic compression of orbitals. The wave function of the continuous spectrum is more screened from the charge of the atomic nucleus by the relativistic field of atomic electrons than by the nonrelativistic one, and the more accurately relativistic effects are taken into account, the greater the effect.

3.3 Results of Computing the Fermi Function of β − -Decay with Different Definitions of This Function In this subsection, the difference in the values of the Fermi function F(E, Z) for the β − -decay is numerically estimated when choosing different definitions for the desired quantity. As indicated above, the Fermi function F(E, Z) was calculated by us both at the nuclear boundary and near zero. In the first case, the Fermi function 2 (R0 ) F(E, Z) was calculated using the values of the radial electron wave functions f +1 2 (R0 )—at the boundary of the nucleus (uniformly charged spherical nucleus), + g−1 in the second, the Fermi function was calculated using the squared amplitudes of

78

O. Yu. Khetselius et al.

2 2 2 2 the expansion (Nκ=+1 + Nκ=−1 ) radial electron wave functions f +1 (0) +g−1 (0) at r → 0 [2, 12, 13]. A convenient value characterizing the desired difference is the parameter:

3 = {[F(E, Z , R = 0))/F(E, Z , R = R0 ] − 1} · 100%,

(53)

where F(E, Z, R = R0 )—Fermi function value calculated with values of radial electron wave functions at the nucleus boundary; F(E, Z, R = 0)—the value of the Fermi function calculated using the amplitudes of the expansion of the radial wave functions near zero. The results of calculating the differences in the values of the Fermi function F(E, Z) for β-decay when choosing two different definitions of this quantity are given in the Table 6. The results of our calculation within the framework of the ODF method are presented, as well as for comparison for a number of values of the kinetic energy the data of estimates within the framework of the relativistic HFS (e.g. [12, 13]). Analysis of the results shows that with an increase in the atomic number Z, the difference in the values of the Fermi function determined by different methods sharply increases. The change in the integral Fermi function f(E 0, Z) turns out to be similar. In particular, the calculation showed that the function f increases for decays 33 P → 33 S (E 0 = 249 keV), 35 S → 35 Cl (E 0 = 167 keV) by 2–4%, 63 Ni → 63 Cu (E 0 = 65.8 keV)—5%, 155 Eu → 155 Gd (E 0 = 140.7 keV)—12%, 241 Pu → 241 Am (E 0 = 20.8 keV)—32% (when passing from the definition of F(E, Z) by functions at the boundary of the nucleus to the definition of F(E, Z), calculated from the amplitudes at zero). Note that in the literature there have been various points of view on the correctness and acceptability of one or another approach to the definition of the Fermi function. In our opinion (see also [2, 3, 5, 12]), the determination of the Fermi function using the amplitudes of the expansion of wave functions near zero is more justified and rational. As indicated in [2, 11–13, 18], an additional factor in favor of this statement Table 6 The difference in the Fermi function F(E, Z) for β-decay when choosing different definitions for this quantity: 3 = {[F(E, Z, R = 0))/F(E, Z, R = R0 ] − 1} · 100%, where F(E, Z, R = R0 ) calculated with the values of radial electron wave functions at the nucleus boundary, and F(E, Z, R = 0)—using the amplitudes of the expansion of the radial wave functions near zero (R0 = 1.2 A1/3 fm); HFS—work data [12, 13]; DKS—calculational data within the framework of the DKS PT (e.g. [2, 11–13, 18]) E kin (keV)

3 (%) Z = 20

Z = 44

Z = 63

HFS

DKS

0.1

1.35

1.39

5.44

12.72

1.0

1.37

1.42

5.53

12.84

50

1.38

1.45

5.58

500

1.50

1.58

5.84

Z = 80

Z = 95 HFS

DKS

23.25

33.9

36.8

23.36

34.1

37.2

12.95

23.58

34.2

37.6

13.10

24.61

35.5

39.88

Electron-β-Nuclear Spectroscopy of Atomic Systems …

79

Table 7 Formation region of the integral Fermi function f(E 0 , Z) for β-decay (our data) E 0 (keV)

β-decay

x/E 0 = 0.3

0.5

0.7

0.9

20.8

241 Pu

→

241 Am

67

89

99

100

39.4

106 Ru

→ 106 Rh

66

88

98

100

65.8

63 Ni

65

87

97

100

140.7

155 Eu

63

84

96

100

167.4

35 S

→ 35 Cl

58

81

95

100

249

33 P

→ 33 S

53

78

93

100

257

45 Ca

→

52

77

91

100

y (%)

→ 63 Cu →

155 Gd

45 Sc

is the fact that, based on the amplitudes of the expansion of the electronic wave functions at zero, one usually calculates, for example, the electronic factor of the EO conversion (EO), corrections to the internal conversion coefficients to take into account anomalies etc. Let us now consider the question of the region of formation of the integral Fermi function f(E 0 , Z). A convenient parameter for this estimate is the quantity used in a number of papers (see, for example, [2, 12]):

x y=

E0 F(E, Z )E p(E 0 − E) d E/

F(E, Z )E p(E 0 − E)2 d E.

2

0

(54)

0

Table 7 shows the calculational data [11, 18, 20] on the formation region of the integral Fermi function f(E 0 , Z) for a series of β-decays, in particular, decays: 241 Pu → 241 Am, 106 Ru → 106 Rh, 63 Ni → 63 Cu, 155 Eu → 155 Gd, 35 S → 35 Cl, 33 P → 33 S, 45 Ca → 45 Sc. Analysis of the data obtained (Table 7) shows that for energy values from x = 0.7E 0 and further to x = 0.9E 0 , 100% of the integral for the function f(E 0 , Z). At an energy value x = 0.5E 0 , about ~80% of the integral for the function f(E 0 , Z). As a result, it turns out that the corrections, which are significant for small values of the energy of the emitted β-particle, affect the integral Fermi function. Next, we will study the question of the quantitative characteristic of taking into account the exchange-correlation effects in the wave functions of the discrete and continuous parameters of the Fermi functions.

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3.4 An Effect of Accounting for Exchange-Correlation Effects in Wave Functions on the Values of the Integral Fermi Function The many-body DKSPT allows for an effective accounting for exchange effects, as well as correlation effects. In this subsection, we quantitatively study the influence of taking into account the sought effects in the electronic wave functions on the values of the Fermi function and the integral Fermi function. It should be noted that the issue of accounting for exchange was considered earlier in the literature (e.g., [2, 5, 11–13, 18, 20]). Table 8 shows the results of calculating the contribution of the value of complete accounting for exchange in the electronic wave functions of discrete and continuous spectra to the values of the integral Fermi function f(E 0 , Z); on the basis of various approaches [2, 3, 5, 12, 13, 18], transitions are considered: 35 S → 35 Cl, 63 Ni → 63 Cu, 33 P → 33 S, 106 Ru → 106 Rh, 155 Eu → 155 Gd, 241 Pu → 241 Am. As a convenient parameter that determines the desired contribution, we took the quantity: 4 = {[ f (E 0 , Z )DKS/ f (E 0 , Z ]HFSr el )] − 1} · 100%,

(55)

5 = {[ f (E 0 , Z )DF/ f (E 0 , Z ]HFSr el )] − 1} · 100%,

(56)

where f(E 0 , Z)DKS—integral Fermi function calculated in the DKS PT approximation with an effective accouting for the exchange-correlation effects; f(E 0 , Z)DF— integral Fermi function calculated in the DF approximation with accouting for the exchange effects; f(E 0 , Z]HFSrel )—integral Fermi function calculated in the HFSrel approximation with incomplete accounting for exchange effects. The values of the Table 8 Contribution of the value of the complete account of exchange in the electronic wave functions of discrete and continuous spectra to the values of the integral Fermi function f(E 0 , Z) for some transitions Decay 35 S

→ 35 Cl

E 0 (keV)

f (E 0 , Z) ODF

f (E 0 , Z) DF

f (E 0 , Z) HFSrel

4 (%)

5, (%)

167.4

1.3461 · 10−2

1.3556 · 10−2

1.3682 · 10−2

-1.6

0.9

106 Ru

→ 106 Rh

39.4

6.2375 · 10−4

6.4304 · 10−4

6.6304 · 10−4

-5.9

3.0

155 Eu

→ 155 Gd

140.7

8.6124 · 10−2

8.7025 · 10−2

8.8817 · 10−2

-3.0

2.0

241 Pu

→ 241 Am

20.8

1.5896 · 10−3

1.6424 · 10−3

1.7208 · 10−3

−7.6

-4.6

Note Here Δ4 = {[f(E 0 , Z)DKS/f(E 0 , Z] HFS rel )] − 1} · 100%, where f(E 0 , Z]HFSrel )—integral Fermi function calculated in the HFS rel approximation with incomplete account of exchange effects; Δ5 = {[ f(E 0 , Z)DF/f(E 0 , Z] HFS rel )] − 1} · 100%, where f(E 0 , Z)DF—integral Fermi function calculated in the DF approximation with full account of exchange (exchange-correlation) effects

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f (E 0 , Z)DF are obtained on the basis of the classical Dirac-Fock (DF) calculation [12, 13]. As can be seen from the data obtained, with an increase in the completeness of accounting for exchange (and further exchange-correlation) effects in the wave functions of the discrete and continuous spectrum, the correction to the energy increases with a decrease in the boundary energy. The relative change in the integral Fermi function, for example, for the 241 Pu → 241 Am transition is 7.6%.

3.5 An Effect of Accounting the Exchange-Correlation Effects in Wave Functions on the Values of the Fermi Function Next, we will consider the topic of the influence of accounting for the exchangecorrelation effects in the wave functions of the discrete and continuous spectrum on the values of the Fermi function. Tables 9, 10 and 11 shows the data of the DKS calculation of the values of the Fermi function F(E, Z) for decays: 106 Ru → 106 Rh, 63 Ni → 63 Cu, 241 Pu → 241 Am. For comparison, the same table also shows some values of the function F(E, Z), calculated by the HFSrel method, by the DF method, as well as in the Coulomb field approximation, taking into account the finite dimensions of the nucleus (data taken from [2, 5, 11–13, 18, 20]). As characteristic parameters determining the contribution of the sought effects, it is convenient to operate with the quantities: Δ6 = {[F(E, Z )DKS/F(E, Z ]HFSr el )] − 1} · 100%,

(57)

Δ7 = {[F(E, Z )DFexc /F(E, Z ]HFSr el )] − 1} · 100%,

(58)

Table 9 The functions F(E, Z) and the quantitative estimate of accounting for exchange (correlation) effects in the wave functions of the discrete and continuous spectra on F (transition: 106 Ru → 106 Rh) Z = 45

F(E, Z) 106 Ru → 106 Rh; E 0 = 39.4 (keV)

DKS

DFexc

HFSrel

Coulomb

0.5140

84.0896

86.3579

93.6620

2.6582

38.7468

39.6767

41.2162

6.3456

25.6138

26.1625

16.767

16.0979

28.233

12.6722

39.314

10.8742

E –β kin (keV)

−Δ6 (%)

−Δ7 (%)

−Δ8 (%)

95.3163

10.2

7.8

1.7

42.0030

6.0

3.7

1.9

26.8605

27.3434

4.6

2.6

1.8

16.3667

16.6530

16.9466

3.3

1.7

1.7

12.7921

12.9745

13.2067

2.3

1.4

1.8

10.9863

11.1218

11.3237

2.2

1.2

1.8

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Table 10 The functions F(E, Z) and the quantitative estimate of accounting for exchange (correlation) effects in the wave functions of the discrete and continuous spectra on F (transition: 63 Ni → 63 Cu) E− β kin (keV)

F(E, Z) 63 Ni → 63 Cu; Z = 29; E 0 = 65.8 (keV)

Δ6 (%)

Δ8 (%)

DKS

HFSrel

Coulomb

0.85858

29.3482

31.5491

31.8710

−7.0

−1.0

4.4394

13.4120

13.9167

14.0385

−3.6

−0.9

10.547

8.8125

9.0867

9.1751

−3.0

−1.0

28.002

5.6139

5.7411

5.8094

−2.2

−1.2

47.159

4.5391

4.6076

4.6644

−1.5

−1.2

65.657

4.0197

4.0652

4.1132

−1.1

−1.2

Table 11 The functions F(E, Z) and the quantitative estimate of accounting for exchange (correlation) effects in the wave functions of the discrete and continuous spectra on F (transition: 241 Pu → 241 Am) E− β kin (keV)

0.27137

Z = 95

F(E, Z) 241 Pu → 241 Am; E 0 = 20.8 keV

Δ6 (%)

Δ7 (%)

Δ8 (%)

DKS

DFexc

HFSrel

Coulomb

2014.27

2075.86

2316.49 1018.29

2431.60

−13.0

−10.4

−4.7

1069.57

−7.3

−5.6

−4.8

1.4033

944.400

961.517

3.3341

621.735

634.238

661.040

694.165

−5.9

−4.1

−4.0

8.8517

391.342

394.909

406.591

426.528

−3.8

−2.9

−4.7

14.907

303.169

306.220

313.858

329.084

−3.4

−2.4

−4.6

20.755

259.003

260.587

266.528

279.230

−2.8

−2.2

−4.5

Δ8 = {[F(E, Z )HFSr el )/F(E, Z )Coulomb ) − 1) · 100%,

(59)

where F(E 0 , Z)DKS—Fermi function, calculated in the optimized DKS approximation with an effective accounting for exchange-correlation effects; F(E 0 , Z)DFexc — Fermi function, calculated in the DF approximation with accounting for exchange effects; F(E 0 , Z]HFSrel )—Fermi function, calculated in the HFSrel approximation with incomplete accounting for exchange effects; F(E, Z)Coulomb —Fermi function, calculated in the Coulomb approximation. As can be seen from the data obtained (see Tables 9, 10 and 11), the correction associated with taking into account the exchange-correlation effects in the electronic wave functions of the discrete and continuous spectra at low energies significantly exceeds the correction for screening (with respect to the Coulomb field), which is found using the HFSrel method, however, with increasing energy, the screening correction is gradually compared with the exchange contribution.

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It is easy to understand that the construction of the well known Curie plot according to our calculated data F(E, Z), as well as according to the data of the standard DF calculation (e.g. [2, 5, 11–13, 18, 20]), in comparison with similar data based on the HFSrel method, in the region of low energy values will have excess over a straight line drawn through points with higher energy. Then, such an excess can simulate a massive neutrino with a nonzero mass in the amount of 1.8% of the number of decays.

4 Conclusions We have briefly presented the modern concepts of physical nature of a beta-decay phenomenon and considered the key fundamental parameters of a nuclear betadecay, classification of the beta-transitions, selection rules etc. An effective relativistic approach to calculating the characteristics of the β-decay for different of atomic systems (nuclei) is presented and based on the combined relativistic nuclear model and relativistic many-body perturbation theory formalism with the DiracKohn-Sham zeroth approximation and correct accounting for exchange-correlation, nuclear, radiation corrections. A relativistic many-body perturbation theory is applied to electron subsystem, and a nuclear relativistic middle-field model is used for nuclear subsystem. The results of computing the characteristics of a whole series of allowed (superallowed) β-decays are presented, namely, for the 33 P → 33 S, 35 S → 35 Cl, 45 Ca → 45 Sc, 63 Ni → 63 Cu, 106 Ru → 106 Rh, 155 Eu → 155 Gd, 241 Pu → 241 Am decays. The effect of the chemical environment of an atom on the characteristics (integral Fermi function, half-life) of β-transitions is studied. We presented the results of accurate calculation of the beta-decay parameters and compared with alternative theoretical data. Results of computing the Fermi function of a β − -decay with different definitions of this function are presented too. The effect of an atomic field type choice on the beta decay characteristics as well as the influence of accounting for the exchangecorrelation effects in the wave functions of the discrete and continuous spectrum on the values of the Fermi and integral Fermi functions are estimated.

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81. Glushkov AV, Khetselius OYu, Malinovskaya SV (2008) Optics and spectroscopy of cooperative laser-electron nuclear processes in atomic and molecular systems—new trend in quantum optics. Eur Phys J ST 160:195–204 82. Glushkov AV, Ivanov LN, Letokhov VS (1991) Nuclear quantum optics. Preprint of Institute of Spectroscopy, USSR Academy of Sciences, Troitsk N4 83. Chernyakova YuG, Vitavetskaya LA, Bashkaryov PG, Serga IN, Berestenko AG (2015) The radiative vacuum polarization contribution to the energy shift of some levels of the pionic hydrogen. Photoelectronics 24:108–111 84. Kuznetsova AA, Vitavetskaya LA, Chernyakova YG, Korchevsky DA (2013) Calculating the radiative vacuum polarization contribution to the energy shift of 2p-2s transition in pionic deuterium. Photoelectronics 22:122–127 85. Glushkov AV, Ambrosov SV, Loboda AV, Chernyakova Yu, Svinarenko AA, Khetselius OYu (2004) QED calculation of the superheavy elements ions: energy levels, radiative corrections, and HFS for different nuclear models. Nucl Phys A Nucl Hadr Phys 734:21 86. Glushkov AV, Khetselius OYu, Svinarenko AA (2013) Theoretical spectroscopy of autoionization resonances in spectra of lanthanide atoms. Phys Scr T 153:014029 87. Glushkov AV, Khetselius OYu, Svinarenko AA, Buyadzhi VV, Ternovsky VB, Kuznetsova AA, Bashkarev PG (2017) Relativistic perturbation theory formalism to computing spectra and radiation characteristics: application to heavy element. In: Dimo I (ed) Recent studies in perturbation theory. InTech, Uzunov, pp 131–150. https://doi.org/10.5772/intechopen.69102 88. Svinarenko AA (2014) Study of spectra for lanthanides atoms with relativistic many-body perturbation theory: Rydberg resonances. J Phys Conf Ser 548:012039 89. Svinarenko AA, Glushkov AV, Khetselius OYu, Ternovsky VB, Dubrovskaya YuV, Kuznetsova AA, Buyadzhi VV (2017) Theoretical spectroscopy of rare-earth elements: spectra and autoionization resonance. In: Jose EA (ed) Rare earth element. InTech, Orjuela, pp 83–104. https://doi.org/10.5772/intechopen.69314 90. Glushkov AV, Butenko YuV, Serbov NG, Ambrosov SV, Orlova VE, Orlov SV, Balan AK, Dormostuchenko GM (1996) Calculation and extrapolation of oscillator strengths in Rb-like, multiply charged ions. Russ Phys J 39(1):81–83 91. Glushkov AV (1992) Oscillator strengths of Cs and Rb-like ions. J Appl Spectr 56(1):5–9 92. Glushkov AV, Khetselius OYu, Lopatkin YM, Florko TA, Kovalenko OA, Mansarliysky VF (2014) Collisional shift of hyperfine line for rubidium in an atmosphere of the buffer inert gas. J Phys Conf Ser 548:012026 93. Glushkov AV, Malinovskaya SV, Chernyakova YuG, Svinarenko AA (2004) Cooperative laserelectron-nuclear processes: QED calculation of electron satellites spectra for multi-charged ion in laser field. Int J Quant Chem 99(5):889–893 94. Glushkov AV, Malinovskaya SV, Filatov VV (1989) S-matrix formalism calculation of atomic transition probabilities with inclusion of polarization effects. Sov Phys J 32(12):1010–1014 95. Glushkov AV, Svinarenko AA, Ignatenko AV (2011) Spectroscopy of autoionization resonances in spectra of the lanthanides atoms. Photoelectronics 20:90–94 96. Glushkov AV (1990) Relativistic polarization potential of a many-electron atom. Sov Phys J 33(1):1–4 97. Khetselius OYu (2015) Optimized perturbation theory for calculating the hyperfine line shift and broadening of heavy atoms in a buffer gas. In: Nascimento M, Maruani J, Brändas E, Delgado-Barrio G (eds) Frontiers in quantum methods and applications in chemistry and physics. Progress in theoretical chemistry and physics, vol. 29. Springer, Cham, pp 55–76 98. Khetselius OYu, Zaichko PA, Smirnov AV, Buyadzhi VV, Ternovsky VB, Florko TA, Mansarliysky VF (2017) Relativistic many-body perturbation theory calculations of the hyperfine structure and oscillator strength parameters for some heavy element atoms and ions. In: Tadjer A, Pavlov R, Maruani J, Brändas E, Delgado-Barrio G (eds) Quantum systems in physics, chemistry, and biology. Progress in theoretical chemistry and physics, vol 30. Springer, Cham, pp 271–281 99. Malinovskaya SV, Glushkov AV, Khetselius OYu, Svinarenko AA, Mischenko EV, Florko TA (2009) Optimized perturbation theory scheme for calculating the interatomic potentials and hyperfine lines shift for heavy atoms in the buffer inert gas. Int J Quant Chem 109:3325–3329

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100. Glushkov AV, Khetselius OY, Malinovskaya SV (2008) Spectroscopy of cooperative laser– electron nuclear effects in multiatomic molecules. Mol Phys 106:1257–1260. https://doi.org/ 10.1080/00268970802158262 101. Glushkov AV, Malinovskaya SV, Khetselius OY, Loboda AV (2009) Monte-Carlo quantum chemistry of biogene amines. Laser and neutron capture effects. AIP Conf Proc 1102:175–177. https://doi.org/10.1063/1.3108371 102. Glushkov AV, Khetselius OY, Loboda AV, Ignatenko A, Svinarenko A, Korchevsky D, Lovett L (2008) QED approach to modeling spectra of the multicharged ions in a plasma: oscillator and electron-ion collision strengths. AIP Conf Proc 1058:175–177. https://doi.org/10.1063/ 1.3026437 103. Glushkov AV, Khetselius OYu, Kruglyak YuA, Ternovsky VB (2014) Calculational methods in quantum geometry and chaos theory. P.3. OSENU (TEC), Odessa 104. Glushkov AV, Malinovskaya SV, Loboda AV, Shpinareva IM, Prepelitsa GP (2006) Consistent quantum approach to new laser-electron-nuclear effects in diatomic molecules. J Phys Conf 35:420–424 105. Glushkov AV, Malinovskaya SV, Loboda AV, Shpinareva IM, Gurnitskaya EP, Korchevsky DA (2005) Diagnostics of the collisionally pumped plasma and search of the optimal plasma parameters of x-ray lasing: Calculation of electron-collision strengths and rate coefficients for Ne-like plasma. J Phys Conf Ser 11:188–198 106. Malinovskaya SV, Glushkov AV, Khetselius OYu (2008) New laser-electron nuclear effects in the nuclear γ transition spectra in atomic and molecular systems. In: Wilson S, Grout PJ, Maruani J, Delgado-Barrio G, Piecuch P (eds) Frontiers in quantum systems in chemistry and physics. Progress in theoretical chemistry and physics, vol 18. Springer, Dordrecht, pp 525–541. https://doi.org/10.1007/978-1-4020-8707-3_24 107. Glushkov AV (1994) Calculation of parameters of the interaction potential between excited alkali atoms and mercury atoms-the Cs-, Fr-Hg interaction. Opt Spectr 77(1):5–10 108. Glushkov AV (1994) New form of effective potential to calculate polarization effects of the π-electronic states of organic molecules. J Struct Chem 34(5):659–665. https://doi.org/10. 1007/BF00753565 109. Khetselius OYu (2013) Forecasting evolutionary dynamics of chaotic systems using advanced non-linear prediction method. In: Awrejcewicz J, Kazmierczak M, Olejnik P, Mrozowski J (eds) Dynamical systems applications (Lodz Univ.), vol T2, pp 145–152 110. Khetselius OYu (2008) Hyperfine structure of atomic spectra. Astroprint, Odessa 111. Glushkov AV, Khetselius OYu, Svinarenko AA, Buyadzhi VV (2015) Spectroscopy of autoionization states of heavy atoms and multiply charged ions. TEC, Odessa 112. Glushkov AV, Khetselius OYu, Svinarenko AA, Buyadzhi VV (2015) Methods of computational mathematics and mathematical physics. P.1. TES, Odessa 113. Glushkov AV (2012) Methods of a chaos theory. Astroprint, Odessa

Relativistic Quantum Chemistry and Spectroscopy of Some Kaonic Atoms: Hyperfine and Strong Interaction Effects Olga Yu. Khetselius, Valentin B. Ternovsky, Inga N. Serga, and Andrey A. Svinarenko

Abstract We present a consistent relativistic approach to calculation of energy and spectral parameters of the kaonic exotic atomic systems with accounting for the nuclear radiative (quantum electrodynamics), hyperfine and strong interactions. The approach is naturally based on using the relativistic Klein-Gordon-Fock equation with introduction of electromagnetic and strong interactions potentials. To take a strong kaon-nuclear interaction into account, the generalized optical potential method is applied. In order to take the nuclear (the finite nuclear size effect) and radiative (quantum electrodynamics) corrections into account, the generalized Uehling-Serber approach is applied. The elements of the hyperfine structure theory of the kaonic atoms (KA) are presented. As an illustration, there are results of calculating the binding energies of various atomic levels in a hydrogen KA obtained within the Hlike model of Iwasaki, the method of Indelicato et al. and our approach (here the Fermi model of the charge distribution in the nucleus is used). Using our calculated “electromagnetic” values of the transition energy and a set of available latest experimental values, it is calculated a shift of the 1s level in kaonic hydrogen, due to the strong kaon-nucleon interaction; the calculated “electromagnetic” value of the transition energy and further comparison with the experimental value of the transition allowed to obtain a theoretical estimate of the “strong” shift in kaonic hydrogen, which is in excellent agreement with the DEAR experimental data. In addition, the results of calculating the energy (electromagnetic) contributions (the main Coulomb correction, correction for vacuum polarization, relativistic correction for the recoil effect, a hyperfine shift) to the energy of the 8k-7i, 8i-7h transitions in the spectrum of kaonic nitrogen are presented and compared with the alternative theoretical data by Indelicato et al. Keywords Quantum mechanics and spectroscopy · Kaonic atoms · Relativistic many-body perturbation theory · Klein-Gordon-Fock equation · Strong kaon-nuclear optical potential · Hyperfine structure O. Yu. Khetselius (B) · V. B. Ternovsky · I. N. Serga · A. A. Svinarenko Odessa State Environmental University, L’vovskaya str., 15, Odessa 65016, Ukraine e-mail: [email protected] © Springer Nature Switzerland AG 2021 A. V. Glushkov et al. (eds.), Advances in Methods and Applications of Quantum Systems in Chemistry, Physics, and Biology, Progress in Theoretical Chemistry and Physics 33, https://doi.org/10.1007/978-3-030-68314-6_5

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1 Introduction At present, an exotic atom is usually understood as a bound or quasi-stationary complex, which is obtainedas a result of the landing  of a heavy  negatively charged particle (hadron, lepton) X X = μ− , π − , K − , p − , − , ... on an ordinary atom [1–50]. Hence the name of various types of exotic pionic, hyper in particular,    atoms, onic, and KA of interest to us. Antihydrogen p − e− , muonium μ− e− , and other systems are sometimes referred to such systems. The progress observed in the last decade in the theoretical and experimental study of hadronic atoms has been noted in a number of rather interesting reviews both on the physics of KA and on the physics of other hadronic atoms (see, for example, [51–60]). Despite the more than 70-year period of the development of the physics of hadronic atoms, until the early 2000s, the situation with the data on the energy parameters of most kaonic, pionic, and other atoms was rather confusing [1–50]. Moreover, in recent years, the situation in the physics of kaonic atoms (KA) continues to change rapidly, the most striking example of which is the recent solution to the problems of kaonic hydrogen and helium (see [1–4] and the text below), due, among other things, to huge experimental errors. The study of kaonic atoms has become especially relevant in the light of the well-known progress of experimental studies (at meson factories in the laboratories of LAMPF (USA), PSI (Switzerland), TRIUMF (Canada), IFF (Russia), RIKEN (KEK, Japan), RAL (United Kingdom), DEAR at the DAPNE (Italy) and further substantial development of modern nuclear theory, quantum mechanics of atoms etc. At present, it is customary to consider (see, e.g., [1–10]) that the main tasks of modern physics of the nucleus, elementary particles and high energies are to check the consequences and search for violations of the Standard Model of electroweak interactions with the aim of generalizing it, determining the neutrino masses, elucidating cosmological consequences from the physics of the microworld, etc. Experimental studies here, as a rule, are developing in two complementary directions, in particular, the basis of the first is the construction of high-energy accelerators and unique detectors in order to detect new particles and interactions and verification of theoretical models [1–56]. The second direction is precisely the physics of hadronic atoms or, as is often indicated, the physics of intermediate energies, including the determination of the energy and spectral parameters of systems, as well as the search for rare decays and reactions with already known particles, the detection of violations of the fundamental properties of symmetry, the study of atomic and molecular processes with the participation of hadronic, including kaonic atoms. It should be noted that the most correct approach to description of the kaonic atomic systems should be based on the principles of a modern consistent quantum chromodynamics with some elements of a quantum electrodynamics in a case of multielectron kaonic atoms. One could remind that consistent quantum chromodynamics represents a fundamental gauge theory of strong interactions with the interacting coloured quarks and gluons. Due to the strong interaction effect, there is a shift of energies of the low-lying levels from the purely electromagnetic values and the finite lifetime of the state corresponds to an increase in the observed level width. A

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few serious measurements are performed for different light and heavy kaonic atoms (e.g. [1–5]). The most spread theoretical methods to study energy and spectral characteristics of kaonic atomic systems are described in Refs. [1–56]. In Refs. [43–56] ab initio schemes to the Klein-Gordon-Fock equation solution and further determination of the X–ray spectra for multi-electron kaonic atoms are presented with the different procedure for accounting for the nuclear, quantum electrodynamics, interelectron and kaon-electron interaction, exchange–correlation effects. Another extremely fundamental aspect of the theory of hadronic atoms, in particular, KA, is associated with taking into account the radiation, QED corrections to the energy of the atom (transition energies, etc.) [57–74]. First of all, we are talking about taking into account the effect of vacuum polarization, as well as the less significant contribution for KA, due to the self-energy part of the Lamb shift. In [1, 4, 5, 42, 43, 57–78], a review of the current state of calculating radiative corrections to the energies of levels in kaonic atoms is given, and existing problems are analyzed, in particular, the difficulties of calculating the required radiative corrections in the case of heavy systems. For a point nucleus in the first order of PT in Z α (Z is the nuclear charge; α is the fine structure constant), the vacuum polarization addition to the nucleus potential is the known Uehling-Serber potential U˜ (r ). Taking into account the finiteness of the size of the nucleus modifies this potential. In the well-known works of WichmannKroll (see [1, 42, 43, 57–74]), a method was developed that makes it possible to calculate the vacuum polarization in all orders by α (Z α)n . In principle, the polarization shift could be calculated as a first-order correction to the potential U˜ (r ). Convenient techniques with application to the description of the spectra of ordinary heavy atoms and multiply charged ions have been proposed and implemented in the papers by Declaux -Indelicato, Mohr, Saperstein, Johnson et al. (see, for example, [1–4, 50–98]). However, it should be noted that the main techniques based on the expansion of the contribution of radiative corrections by Z α, naturally, work only for systems with low Z, i.e., light atoms. In the case of heavy systems, in particular, KA, these approaches will obviously not give correct results. In Refs. [2, 81, 82, 91, 92] we have presented an effective relativistic approach to calculation of spectra and the spectroscopic properties of the heavy kaonic (pionic) multielectron atomic systems. The approach is based on the Klein-Gordon-Fock equation solution with simultaneous accounting for electromagnetic and strong kaonnuclear interactions. The modified method of optical potential is used to take a strong kaon-nuclear interaction into consideration. The consistent procedures, in particular, such as an advanced Uehling-Serber model and model potential approach are applied to take the main nuclear and quantum electrodynamics corrections into account. The results of calculation of the energy and spectral parameters for the kaonic atoms of He, 184 W, 207 Pb, 238 U, with taking the radiation (vacuum polarization), nuclear (finite size of a nucleus) and the strong kaon-nuclear interaction corrections into account have been presented. In this chapter we present the generalization of theory in order to determine the hyperfine and strong interaction effects as well as to calculate the

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probabilities of the radiative transitions in spectra of the hydrogen (including the kaonic hydrogen puzzle) and nitrogen kaonic atoms.

2 Relativistic Theory of Kaonic Atoms with Accounting for the Nuclear, Hyperfine and Strong Interaction Effects 2.1 The Klein-Gordon-Fock Equation and Electromagnetic Interactions in Kaonic System New version of a relativistic theory of kaonic atomic systems with accounting for the nuclear, radiation, hyperfine and strong interaction effects has been in detail presented in Refs. [2, 81, 82, 91, 92]. So, here it is worth to consider only some new model elements. However, at once, we present shortly the summary of modern kaon data [1–10]. There are 4 types of kaons: negatively charged K− (composition: s-quark + uantiquark), mK − = 493.667 ± 0.013 meV; lifetime (1.2384 ± 0.0024)10−8 s and radius: r K- = 0.560 ± 0,031 fm. Its antiparticle is a positively charged kaon K+ (composition: u-quark + s-antiquark). Naturally, due to the CPT symmetry, mK + = mK − , t = tK − should take place (modern data: m = 0.032 ± 0.090 meV; t = ˜ 0 have the following composition: d-quark (0.11 ± 0.09)10−8 s). Neutral kaons K0 K + s-antiquark and s-quark + d-antiquark, mK o = 497.648 ± 0.022 meV. Since K0 ˜ 0 appear as a result of strong interaction, they decay due to weak and the antiparticle K interaction and represent a composition of 2 weak eigenstates: short-lived neutral K = KS (“K-short”; decays into 2 pions and tK o = 8.958 × 10−11 s) and long-lived neutral K = KL (“K-long”; decays into 3 pions and tK o = 5.18 × 10−8 s). The kaonic wave functions are determined from solution of the known relativistic Klein-Gordon-Fock equation: {

1 [E + eV0 (r )]2 + 2 ∇ 2 − m 2 c2 }φ(x) = 0 c2

(1)

Here c is a velocity of light, E is the total energy of atomic system, V 0 is a sum of electric potential of a nucleus and strong interaction potential and the Uehling-Serber potential. To determine the electric potential of the nucleus, we used the Fermi model with charge distribution ρ(r ) [1]: ρ(r ) = ρ0 /{1 + exp[(r − c)/a)]}

(2)

where parameter a = 0.523 fm, and parameter c is chosen so that the root-meansquare radius is determined by the expression: 1/2 = (0.836 · A1/3 + 0.5700) fm. As an alternative, as usual in atomic calculations, the empirically determined

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Z-dependence for the effective radius is used [1]. Other versions of the nuclear electric potential are presented by the Gauss and a homogeneously charged sphere models [92, 93]. The standard point of our theoretical approach is connected with determination of the electric and radiative potentials within the effective algorithm based on the differential equations method. This is the method originally developed by Ivanova and Ivanov [86] and further has been often used in solving many problems of atomic, molecular, nuclear and laser spectroscopy (e.g. [1, 38, 39, 86, 87, 99–140]). It is important to underline too that in order to determine an electrical interaction between a nucleus of finite size (radius of R1 ) and kaon (radius R2 ), one should use, for example, the potential, introduced by Indelicato et al. [42, 43, 50] (e.g. Refs. [2, 81, 82, 91, 92] too). The next principally important block of our approach includes an accurate treatment of the radiative (quantum electrodynamics) effects (e.g., [37–41, 57–68, 86–90]). We have used an effective the generalized Uehling-Serber approach to accounting the radiative corrections, in particular, vacuum-polarization one. The standard Uehling-Serber potential can be written as follows: 2α U (r ) = − 3πr

∞

√     2 t 2 − 1 2α dt exp −2r t α Z 1 + 1 2t ≡− C(g), 2 t 3πr

1

g = r/α Z

(3)

where α-constant of fine structure, which in fact (even taking into account the finite size of the nucleus) takes into account the main contributions of the order [α(Z α)]n , but does not take into account the known contributions of Källen-Sabry, WichmannKroll and others. A more correct form of the Uehling-Serber potential is U (r ) = −(2a/3 pr )C(g), where C is the so-called Uehling-Serber integral, but as C(g) the generalized function e.g. [1, 37–40, 86–90]) is used and then performed the transition from the potential U for the point core to the potential for the finite core. To take into account the effect of electron shielding (in the case of the nitrogen atom), the usual potential of a self-consistent electron field is used. The whole procedure of accounting for the QED corrections is in detail described in Refs. [1, 85–87, 91–93] as well as the fine corrections, provided by relativistic recoil, reduced mass and other effects. Further in order to calculate the radiation transition probabilities or a radiation width in spectra of the kaonic atom we apply our traditional relativistic energy formalism in the version [37–39, 66, 67, 85–87]. A total energy level shift δ E can be presented in the following form:

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δ E i = Reδ E i + iImδ E i = Reδ E i − (i/2)ir ,

(4)

where ir is a level radiation width. It should be noted that an oscillator strength (or transition probability) is directly connected with ir (Pi ~ ir ) and further is determined by combination of amplitudes (ωij is a frequency of the i-j transition). The detailed procedure for computing the radiative transition matrix elements as well as the hyperfine structure characteristics is presented in Refs. [1, 48–50, 66, 67, 72–77, 88, 90]. All computing is carried out with using the PC code Superatom (version 98).

2.2 Model Approach to Study of the Strong and Hyperfine Interactions in Kaonic Atoms As it is indicated, the most correct approach to description of the kaonic atomic systems should be based on the principles of a modern consistent quantum chromodynamics with some elements of a quantum electrodynamics in a case of multielectron kaonic atoms. Indeed a quantum chromodynamics represents a fundamental gauge theory of strong interactions with the interacting coloured quarks and gluons. From the other side, since we are interested by relatively low energy physics of the kaonic atomic systems, one should use different model potential methods to determine the strong kaon-nuclear interaction in these systems (e.g. [1–56]). In this case the total Klein-Gordon-Fock equation taking into account the strong kaonnuclear interaction VN can be written as follows:   2 2  ∇ + c−2 (E − VF S )2 − μ2 c2 ψ = 2μVN ψ.

(5)

where the standard phenomenological optical potential with the proton ρ p and neutron ρn densities is written taken as follows [12]: VN = −

MK 2π [1 + ][A K p ρ p (r ) + A K n ρn (r )], μ MN

(6)

All the parameters of the potential (6) are described in Refs. [12, 45, 51–53]. As it is noted in Ref. [1, 12], the key disadvantage of the used potential (5) approach is connected with inaccurate determination of its parameters, including the proton and neutron densities, the effective K-nucleon scattering lengths (e.g., Refs. [1, 45, 51–53]). It should be noted that if the experimental value of energy E exp is known, then one could easily calculate a strong kaon-nucleus interaction shift of the energy levels: E N = E exp − (E K G F + E F S + E Q E D + E),

(7)

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In Eq. (7) in the brackets different purely electromagnetic contributions (respectively, an energy of kaon in a case of point nucleus, the nuclear finite size and QED effects terms). The detailed procedure for computing the strong interaction corrections is presented in Refs. [1, 48–50, 66, 67, 72–77, 88, 90]. All computing is carried out with using the PC code Superatom (version 98).

3 Some Results and Conclusions 3.1 Spectrum of Kaonic Hydrogen and “Kaonic Hydrogen Puzzle”. The Strong Interaction Effects Below we present some results of calculation of the energy and spectroscopic characteristics for kaonic atoms of the hydrogen and nitrogen. The kaonic hydrogen atom is of a considerable interest as a meson atom that has no electron subsystem. The results of experimental study of the hydrogen KA has been presented, for example, in Refs. [3, 4]. In particular, the X-ray 2-1 transition in the kaon H spectrum is studied obtained in experiment by SIDDHARTA Collaboration (see details in Refs. [3, 4]). In Table 1 there are presented the results of calculating the binding energy of different atomic levels in a hydrogen KA (in keV) obtained in the H-like model of Iwasaki, the method by Indelicato et al. and our approach. The Fermi model of a charge distribution in a nucleus is used in our computing [53–56]. In principle, all approaches naturally give fairly close results. Note that the contribution of radiation corrections here is extremely small, in contrast to KA with a large value of the nuclear charge. In Table 2 there are presented the experimental and theoretical values of the energy (keV) of the 2-1 transition to the hydrogen KA [1–4, 7, 43, 44, 53–56]. Experimental data ware listed in Refs. [3, 4]. Theoretical results are obtained on the Table 1 Calculated binding energies of different atomic levels in hydrogen KA (in keV)

Table 2 The experimental Eexp and theoretical Ec values of energy (keV) of the 2-1 transition in the hydrogen KA spectrum (see text)

Level

Indelicato et al.

Iwasaki

Our data

1s

8.63360

8.634

8.63380

2p

2.15400

2.154

2.15390

3p

0.95720

0.957

0.95710

Ec , This work

Ec [44, 50]

Eexp [1–4]

6.481

6.480 6.482

6.44 ± 0.4 6.675 ± 0.15 6.96 ± 0.33

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basis of calculations within the method by Indelicato et al. [42, 43] and our relativistic approach. As in the case of the energies of atomic levels, there is a fairly good agreement between the theoretical results (in fact, the electromagnetic contributions to the transition energy!), which is explained by the insignificant role of radiative corrections (in the absence of electrons). We have performed the calculation of the transition energy using charge distribution models in the form of a uniformly charged ball, the Gauss distribution and the Fermi model. The difference in the corresponding values of energies averaged a few eV (compared to keV), thus, in this case, the choice of the charge distribution model is not critical. On the other hand, for radiative transitions between low-lying energy levels (n ~ 1), as in our case, the contribution due to the strong interaction is very significant. The strong kaon-nucleon interaction induces a shift and broadening of the 1s level in the spectrum of kaonic hydrogen. The corresponding shift in the presence of an experimental value of the transition energy, say, Eexp (2p-1s) and a “precisely” defined “electromagnetic” correction EEM is defined as: E(1s) = Eexp (2p − 1s) − EEM (2p − 1s). According to the experimental data by M. Iwasaki et al. (KEK 1997), as well as T. Ito, R. Hayano, S. Nakamura et al. (2007), the measured shift is as follows: E1s = −323 ± 63(stat.) ± 11(syst.) eV It is appropriate to present the results of earlier experiments (see, for example, [41]), in particular, according to the measurements by Davies et al. (1979): E1s = +40 − 50 eV, Bird et al. (1983) E1s = +180 − 190 , Izycki et al. (1980) E1s = +260 − 270 eV. Finally, the most recent DEAR (DAFNE Exotic Atom Research) experiment, performed on the DAFNE facility at the Frascatti laboratory (Frascatti, Italy, 2005), allowed to get the following result: E1s = −194 ± 37(stat.) ± 6(syst.)eV; Now using the “electromagnetic” values of the transition energy, obtained in theoretical calculation and the latest experimental values available, it is not difficult to estimate the shift of the 1s level in spectrum of the kaonic hydrogen due to the strong kaon-nucleon interaction. For different values Eexp (2p-1s) then one could obtained:

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E1s = −6440 + 6481 = 41эВ, E1s = −6675 + 6481 = −194 eV, E1s = −6960 + 6481 = −479 eV. Note that the exact coincidence of the theoretically estimated (−194 eV) and the experimental and measured “strong” shift here is, obviously, fortunately random. In contrast to the considered below kaonic helium, the “strong” shift of the 1s level in the hydrogen atom is rather large in absolute value. In any case, the calculated “electromagnetic” value of the transition energy and further comparison with the experimental value of the transition allowed us to obtain a theoretical estimate of the “strong” shift in kaonic hydrogen, which is in excellent agreement with the DEAR experimental shift.

3.2 Spectrum of Kaonic Nitrogen. Hyperfine Structure and Radiative Transitions Probabilities The kaonic nitrogen atom (14 N), like the previous case of the kaonic hydrogen, belongs to the light KA. Its study is of a great interest, first of all, from the point of view of developing new X-ray standards. As a model of the charge distribution in the nucleus, we applied the model of a uniformly charged ball, the Gauss model, and the Fermi model. The influence of the choice of the potential describing the effect of vacuum polarization on the energy parameters of KA has been in details studied too. To take into account the effect of vacuum polarization, we used the standard Uehling-Serber potential and the generalized potential of the form [1] taking into account the finite size of KA nucleus. The relativistic QED corrections of higher orders are also taken into account, including the relativistic recoil correction. In Table 3 there are presented the results of calculating the energy (electromagnetic) contributions (the main Coulomb correction, the correction for the vacuum polarization, the relativistic correction for the recoil effect and the hyperfine shift) to the 8k-7i transition energy in the spectrum of kaonic nitrogen: the data of calculations on the basis of theory by Indelicato et al. [43, 50] and our theoretical approach. Table 3 Energy contributions (in eV) to the 8k-7i transition energy in the spectrum of kaonic nitrogen

Contributions

8k-7i, Theory [43, 50]

8k-7i Present work

Coulomb contribution

2968.4565

2968.4492

1.1789

1.1778

Vacuum polarization Relativistic recoil effect Hyperfine shift Full energy Error

0.0025

0.0025

−0.0006

−0.0007

2969.6373

2969.6288

0.096

0.096

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Table 3 also shows the error caused by the inaccuracy in determining the mass of the K− kaon. Our values, given in Table 3, correspond to the Gaussian model of charge distribution in the nitrogen nucleus. Calculation using other models showed the difference, which is for the model of a uniformly charged ball (0.8 eV) and the Fermi model (0.5 eV). The value corresponding to the correction for vacuum polarization, obtained in the approximation of the standard Uehling-Serber potential (i.e., without taking into account the contribution of the vacuum of polarization corrections of higher orders in the parameter Z α, namely Wichmann-Kroll, Calen-Sabri, etc.) is 1.1665 eV, while the corresponding value with accounting for the indicated corrections is 1.1778 eV. According to estimates by Indelicato et al. [43, 50], who performed a complete calculation of the vacuum of polarization corrections of higher orders in the parameter, the sought contribution is 0.01 eV. Thus, the use of the generalized UehlingSerber potential turns out to be more efficient in comparison with the standard Uehling-Serber approximation, which is usually used in calculating the spectra of both usual (purely multielectron) and exotic atomic systems. On the other hand, for atoms with a low nuclear charge, the sought contribution to the vacuum of polarization corrections remains insignificant. Naturally, with an increase in the nuclear charge, in the transition to heavy KA, this contribution will increase significantly. The PT formalism for evaluating the vacuum of polarization corrections of higher orders in terms of Z α, naturally, ceases to be correct, and a nonperturbative approach is required here. In Table 4 we present the results of calculating the energies (in eV) of transitions between the components of the hyperfine structure 8k-7i in a spectrum of the kaonic nitrogen: (1) the theoretical data, obtained within the theory by Inelicato et al. [43, 50] and our theoretical approach. Similarly, in Table 5 we present the results of calculating a probabilities A (in 1013 s−1 ) of the transitions between the components of the hyperfine structure 8k-7i in the spectrum of kaonic nitrogen: (1) the theoretical data, obtained on the basis of calculation within the theory by Indelicato et al. [43, 50] and (2) data, obtained on the basis of our theoretical approach. Analysis of the data presented shows, in principle, a reasonable agreement between the results of both theories. It should be Table 4 Energies (in eV) of transitions between the components of the hyperfine structure 8k-7i in the spectrum of kaonic nitrogen

F-F

E, Theory [43, 50]

E, Present work

8-7

2969.6365

2969.6289

7-6

2969.6383

2969.6298

7-7

2969.6347

2969.6264

6-5

2969.6398

2969.6345

6-6

2969.6367

2969.6284

6-7

2969.6332

2969.6248

Relativistic Quantum Chemistry and Spectroscopy … Table 5 Probabilities A (1013 c−1 ) of transitions between the components of the hyperfine structure 8k-7i in the spectrum of kaonic nitrogen

101

F-F

A Theory [43, 50]

A This work

8-7

1.54 × 1013

1.51 × 1013

7-6

1.33 ×

1013

1.32 × 1013

7-7

1.31 ×

1013

1.29 × 1013

6-5

1.15 × 1013

1.12 × 1013

6-6

0.03 ×

0.02 × 1013

6-7

–

1013

0.004 × 1013

noted that the radiative corrections are taken into account in our theory within the combined generalized Uehling-Serber approach and method [2]. This, on the one hand, explains the difference in the results, on the other hand, the data we obtained should be considered as the most accurate at the moment. The same applies to the analysis of the obtained values of the probabilities of transitions between the components of the hyperfine structure 8k-7i in the spectrum of kaonic nitrogen. The considered transitions in the spectrum of kaonic nitrogen actually belong to the so-called Rydberg transitions, which largely demonstrate hydrogenlike properties; therefore, as a rule, the results of various theories for such transitions are, as a rule, in good agreement with each other. In Table 6 we present the results of our calculation of the energy (electromagnetic) contributions (the main Coulomb correction, the correction for vacuum polarization, the relativistic correction for the recoil effect and the hyperfine shift) to the 8i7h transition energy in the spectrum of kaonic nitrogen. The data on the energy contributions presented here correspond to the use of the Gaussian model of the charge distribution in the nucleus. Similarly to the previous case, an error is also indicated due to the in accuracy in determining the mass of the K− kaon. The value corresponding to the correction for the polarization of the core, obtained in the generalized Uehling-Serber approximation (i.e., taking into account the contribution of the vacuum of polarization corrections of higher orders in the parameter Z α) is 1.8758 eV. Further, in Table 7 we present the results of our calculation of the energies (in Table 6 Energy contributions (in eV) to the 8i-7h transition energy in the spectrum of kaonic N

Contributions

8i-7h Present work

Coulomb contribution

2968.5344

Vacuum polarization

1.8758

Relativistic recoil

0.0025

Hyperfine shift Full energy Error

−0.0009 2970.4118 0.096

102 Table 7 Energies (in eV) of transitions between the components of the hyperfine structure 8i-7h in the spectrum of kaonic nitrogen

O. Yu. Khetselius et al. F-F

E, This work

7-6

2970.4107

6-5

2970.4135

6-6

2970.4086

5-4

2970.4193

5-5

2970.4114

5-6

2970.4073

eV) of transitions between the components of the hyperfine structure 8i-7h in the spectrum of kaonic nitrogen. Correspondingly, in Tables 8 and 9 we present the results of our calculation of probabilities A (in 1013 s−1 ) of transitions between the hyperfine structure components 8i-7h and 7h-6g in the spectrum of kaonic nitrogen. Note that, for the first time in a theory of the kaonic atomic systems, a consistent relativistic energy approach has been generalized and applied to calculating the probabilities of radiative transitions between the components of the hyperfine structure. To conclude, let us note that a consistent relativistic approach to calculation of energy and spectral parameters of the kaonic exotic atomic systems with accounting for the nuclear radiative (quantum electrodynamics), hyperfine and strong interactions is presented. The approach is naturally based on using the relativistic KleinGordon-Fock equation with introduction of electromagnetic and strong interactions Table 8 Probabilities A (1013 , s −1 ) of transitions between components of the hyperfine structure 8i-7h in the spectrum of kaonic nitrogen

Table 9 Probabilities A (1013 , s−1 ) of transitions between the components of the hyperfine structure 7h-6g in the spectrum of kaonic nitrogen

F-F

P, This work

7-6

1.16 × 1013

6-5

0.99 × 1013

6-6

0.96 × 1013

5-4

0.81 × 1013

5-5

0.02 × 1013

5-6

0.005 × 1013

F-F

P, This work

6-5

0.82 × 1013

6-6

0.76 × 1013

5-4

0.42 × 1013

5-5

0.01 × 1013

5-6

0.001 × 1013

Relativistic Quantum Chemistry and Spectroscopy …

103

potentials. To take a strong kaon-nuclear interaction into account, the generalized optical potential method is applied. As an illustration, different results of computing the energy and spectral characteristics for some kaonic atoms are presented. In particular, the results of calculating the binding energies of various atomic levels in the hydrogen and nitrogen kaonic atoms are listed and obtained within the H-like model of Iwasaki, the method of Indelicato et al. and our approach (here the Fermi model of the charge distribution in the nucleus is used). In addition, the results of calculating the energy (electromagnetic) contributions (the main Coulomb correction, correction for vacuum polarization, relativistic correction for the recoil effect, a hyperfine shift) to the energy of the 8k-7i, 8i-7h transitions in the spectrum of kaonic nitrogen are listed too. In principle, one should keep in mind that a physically reasonable agreement between experimental and theoretical data for the kaonic atomic systems can be achieved only with simultaneous accurate and correct consideration of relativistic, radiation, and nuclear effects. The further improvement and refinement of the theoretical approach and increasing the calculational data accuracy should include the corresponding development of model of the strong kaon-nuclear interaction such as receiving more exact data about the kaon-nuclear potential parameters.

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