Transmission Electron Microscopy and Diffractometry of Materials [3rd ed.] 3540738851, 9783540738855

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Brent Fultz · James Howe Transmission Electron Microscopy and Diffractometry of Materials

High resolution transmission electron microscope (HRTEM) image of a lead crystal between two crystals of aluminum (i.e., a Pb precipitate at a grain boundary in Al). The two crystals of Al have different orientations, evident from their different patterns of atom columns. Note the commensurate atom matching of the Pb crystal with the Al crystal at right, and incommensurate atom matching at left. An isolated Pb precipitate is seen to the right. The HRTEM method is the topic of Chapter 10. Image courtesy of U. Dahmen, National Center for Electron Microscopy, Berkeley.

Brent Fultz · James Howe

Transmission Electron Microscopy and Diffractometry of Materials

Third Edition With 440 Figures and Numerous Exercises

123

Prof. Dr. Brent Fultz California Institute of Technology Materials Science and Applied Physics MC 138-78 Pasadena CA 91125 USA [email protected] http://www.its.caltech.edu/ matsci/btf/Fultz1.html

Prof. Dr. James M. Howe University of Virginia Department of Materials Science and Engineering P. O. Box 400745 Charlottesville VA 22904-4745 USA [email protected] http://www.virginia.edu/ms/faculty/howe.html

Library of Congress Control Number: 2007933070

ISSN: 1439-2674 ISBN 978-3-540-73885-5 3rd Edition Springer Berlin Heidelberg New York ISBN 978-3-540-43764-2 2nd Edition Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2001, 2002, 2008 The use of general descriptive names, registered names, trademarks, 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. Typesetting: supplied by the authors Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover Design: eStudioCalamar S.L., F. Steinen-Broo, Girona, Spain SPIN 12085156

57/3180/YL

543210

Printed on acid-free paper

Preface

Experimental methods for diffraction and microscopy are pushing the front edge of nanoscience and materials science, and important new developments are covered in this third edition. For transmission electron microscopy, a remarkable recent development has been a practical corrector for the spherical aberration of the objective lens. Image resolution below 1 ˚ A can be achieved regularly now, and the energy resolution of electron spectrometry has also improved dramatically. Locating and identifying individual atoms inside materials has been transformed from a dream of fifty years into experimental methods of today. The entire field of x-ray spectrometry and diffractometry has benefited from advances in semiconductor detector technology, and a large community of scientists are now regular users of synchrotron x-ray facilities. The development of powerful new sources of neutrons is elevating the field of neutron scattering research. Increasingly, the most modern instrumentation for materials research with beams of x-rays, neutrons, and electrons is becoming available through an international science infrastructure of user facilities that grant access on the basis of scientific merit. The fundamentals of scattering, diffractometry and microscopy remain as durable as ever. This third edition continues to emphasize the common theme of how waves and wavefunctions interact with matter, while highlighting the special features of x-rays, electrons, and neutrons. The third edition is not substantially longer than the second, but all chapters were updated and revised. The text was edited throughout for clarity, often minimizing sources of confusion that were found by classroom teaching. There are significant changes in Chapters 1, 3, 7, 8 and 9. Chapter 11 is new, so there are now 12 chapters in this third edition. Many chapter problems have been tuned to minimize ambiguity, and the on-line solutions manual has been updated. We thank Drs. P. Rez and A. Minor for their advice on the new content of this third edition. Both authors acknowledge support from the National Science Foundation for research and teaching of scattering, diffractometry, and microscopy. Brent Fultz and James Howe Pasadena and Charlottesville May, 2007

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Preface to the First Edition

Preface to the Second Edition We are delighted by the publication of this second edition by Springer–Verlag, now in its second printing. The first edition took over twelve years to complete, but its favorable acceptance and quick sales prompted us to prepare the second edition in about two years. The new edition features many re-writings of explanations to improve clarity, ranging from substantial re-structuring to subtle re-wording. Explanations of modern techniques such as Z-contrast imaging have been updated, and errors in text and figures have been corrected over the course of several critical re-readings. The on-line solutions manual has been updated too. The first edition arrived at a time of great international excitement in nanostructured materials and devices, and this excitement continues to grow. The second edition shows better how nanostructures offer new opportunities for transmission electron microscopy and diffractometry of materials. Nevertheless, the topics and structure of the first edition remain intact. The aims and scope of the book remain the same, as do our teaching suggestions. We thank our physics editors Drs. Claus Ascheron and Angela Lahee, and our production editor Petra Treiber of Springer–Verlag for their help with both editions. Finally, we thank the National Science Foundation for support of our research efforts in microscopy and diffraction. Brent Fultz and James Howe Pasadena and Charlottesville September, 2004

Preface to the First Edition Aims and Scope of the Book Materials are important to mankind because of their properties such as electrical conductivity, strength, magnetization, toughness, chemical reactivity, and numerous others. All these properties originate with the internal structures of materials. Structural features of materials include their types of atoms, the local configurations of the atoms, and arrangements of these configurations into microstructures. The characterization of structures on all these spatial scales is often best performed by transmission electron microscopy and diffractometry, which are growing in importance to materials engineering and technology. Likewise, the internal structures of materials are the foundation for the science of materials. Much of materials science has been built on results from transmission electron microscopy and diffractometry of materials. This textbook was written for advanced undergraduate students and beginning graduate students with backgrounds in physical science. Its goal is to acquaint them, as quickly as possible, with the central concepts and some details of transmission electron microscopy (TEM) and x-ray diffractometry

Preface to the First Edition

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(XRD) that are important for the characterization of materials. The topics in this book are developed to a level appropriate for most modern materials characterization research using TEM and XRD. The content of this book has also been chosen to provide a fundamental background for transitions to more specialized techniques of research, or to related techniques such as neutron diffractometry. The book includes many practical details and examples, but it does not cover some topics important for laboratory work such as specimen preparation methods for TEM. Beneath the details of principle and practice lies a larger goal of unifying the concepts common to both TEM and XRD. Coherence and wave interference are conceptually similar for both x-ray waves and electron wavefunctions. In probing the structure of materials, periodic waves and wavefunctions share concepts of the reciprocal lattice, crystallography, and effects of disorder. Xray generation by inelastic electron scattering is another theme common to both TEM and XRD. Besides efficiency in teaching, a further benefit of an integrated treatment is breadth – it builds strength to apply Fourier transforms and convolutions to examples from both TEM and XRD. The book follows a trend at research universities away from courses focused on one experimental technique, towards more general courses on materials characterization. The methods of TEM and XRD are based on how wave radiations interact with individual atoms and with groups of atoms. A textbook must elucidate these interactions, even if they have been known for many years. Figure 1.12, for example, presents Moseley’s data from 1914 because this figure is a handy reference today. On the other hand, high-resolution TEM (HRTEM), modern synchrotron sources, and spallation neutron sources offer new ways for wavematter interactions to probe the structures of materials. A textbook must integrate both these classical and modern topics. The content is a confluence of the old and the new, from both materials science and physics. Content The first two chapters provide a general description of diffraction, imaging, and instrumentation for XRD and TEM. This is followed in Chapters 3 and 4 by electron and x-ray interactions with atoms. The atomic form factor for elastic scattering, and especially the cross sections for inelastic electron scattering, are covered with more depth than needed to understand Chapters 5–7, which emphasize diffraction, crystallography, and diffraction contrast. In a course oriented towards diffraction and microscopy, it is possible to take an easier path through only Sects. 3.1, 3.2.1, 3.2.3, 3.3.2, and the subsection in 3.3.3 on Thomas–Fermi and Rutherford models. Similarly, much of Sect. 4.4 on core excitations could be deferred for advanced study. The core of the book develops kinematical diffraction theory in the Laue formulation to treat diffraction phenomena from crystalline materials with increasing amounts of disorder. The phase-amplitude diagram is used heavily in Chapter 7 for the analysis of diffraction contrast in TEM images of defects. After a treatment of diffraction lineshapes in Chapter 8, the Patterson function is used in Chapter 9 to treat short-range order phenomena,

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thermal diffuse scattering, and amorphous materials. High-resolution TEM imaging and image simulation follow in Chapter 10, and the essentials of the dynamical theory of electron diffraction are presented in Chapter 11 [now 12 in the third edition]. With a discussion of the effective extinction length and the effective deviation parameter from dynamical diffraction, we extend the kinematical theory as far as it can go for electron diffraction. We believe this approach is the right one for a textbook because kinematical theory provides a clean consistency between diffraction and the structure of materials. The phase-amplitude diagram, for example, is a practical device for interpreting defect contrast, and is a handy conceptual tool even when working in the laboratory or sketching on table napkins. Furthermore, expertise with Fourier transforms is valuable outside the fields of diffraction and microscopy. Although Fourier transforms are mentioned in Chapter 2 and used in Chapter 3, their manipulations become more serious in Chapters 4, 5 and 7. Chapter 8 presents convolutions, and the Patterson function is presented in Chapter 9. The student is advised to become comfortable with Fourier transforms at this level before reading Chapters 10 and 11 [now 10-12 in the third edition] on HRTEM and dynamical theory. The mathematical level is necessarily higher for HRTEM and dynamical theory, which are grounded in the quantum mechanics of the electron wavefunction. Teaching This textbook evolved from a set of notes for the one-quarter course MS/APh 122 Diffraction Theory and Applications, offered to graduate students and advanced undergraduates at the California Institute of Technology, and notes for the one-semester graduate courses MSE 703 Transmission Electron Microscopy and MSE 706 Advanced TEM, at the University of Virginia. Most of the students in these courses were specializing in materials science or applied physics, and had some background in elementary crystallography and wave mechanics. For a one-semester course (14 weeks) on introductory TEM, one of the authors covers the sections: 1.1, 2.1–2.8, 3.1, 3.3, 4.1-4.3, 4.6, 5.1–5.6, 6.1-6.3, 7.1–7.14. In a course for graduate students with a strong physics background, the other author has covered the full book in 10 weeks by deleting about half of the “specialized” topics. [He has never repeated this achievement, however, and typically manages to just touch section 10.3.] The choice of topics, depth, and speed of coverage are matters for the taste and discretion of the instructor, of course. To help with the selection of course content, the authors have indicated with an asterisk, “*,” those sections of a more specialized nature. The double dagger, “‡,” warns of sections containing a higher level of mathematics, physics, or crystallography. Each chapter includes several, sometimes many, problems to illustrate principles. The text for some of these problems includes explanations of phenomena that seemed too specialized for inclusion in the text itself. Hints are given for some of the

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IX

problems, and worked solutions are available to course instructors. Exercises for an introductory laboratory course are presented in an Appendix. When choosing the level of presentation for a concept, the authors faced the conflict of balancing rigor and thoroughness against clarity and conciseness. Our general guideline was to avoid direct citations of rules, but instead to provide explanations of the underlying physical concepts. The mathematical derivations are usually presented in steps of equal height, and we try to highlight the central tricks even if this means reviewing elementary concepts. The authors are indebted to our former students for identifying explanations and calculations that needed clarification or correction. Acknowledgements We are grateful for the advice and comments of Drs. C. C. Ahn, D. H. Pearson, H. Frase, U. Kriplani, N. R. Good, C. E. Krill, Profs. L. Anthony, L. Nagel, M. Sarikaya, and the help of P. S. Albertson with manuscript preparation. N. R. Good and J. Graetz performed much of the mathematical typesetting, and we are indebted to them for their careful work. Prof. P. Rez suggested an approach to treat dynamical diffraction in a unified manner. Both authors acknowledge the National Science Foundation for financial support over the years. Brent Fultz and James Howe Pasadena and Charlottesville October, 2000

Contents

1.

Diffraction and the X-Ray Powder Diffractometer . . . . . . . . 1.1 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Introduction to Diffraction . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Bragg’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Strain Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 A Symmetry Consideration . . . . . . . . . . . . . . . . . . . . . . . 1.1.6 Momentum and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.7 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Creation of X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Characteristic Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The X-Ray Powder Diffractometer . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Practice of X-Ray Generation . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Goniometer for Powder Diffraction . . . . . . . . . . . . . . . . . 1.3.3 Monochromators, Filters, Mirrors . . . . . . . . . . . . . . . . . . . 1.4 X-Ray Detectors for XRD and TEM . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Detector Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Position-Sensitive Detectors . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Charge Sensitive Preamplifier . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Other Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Experimental X-Ray Powder Diffraction Data . . . . . . . . . . . . . . 1.5.1 * Intensities of Powder Diffraction Peaks . . . . . . . . . . . . 1.5.2 Phase Fraction Measurement . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Lattice Parameter Measurement . . . . . . . . . . . . . . . . . . . . 1.5.4 * Refinement Methods for Powder Diffraction Data . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 6 7 9 10 10 13 14 16 20 23 23 25 28 30 30 34 36 37 38 38 45 49 52 54 55

2.

The TEM and its Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction to the Transmission Electron Microscope . . . . . . . 2.2 Working with Lenses and Ray Diagrams . . . . . . . . . . . . . . . . . . . 2.2.1 Single Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 66 66

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3.

Contents

2.2.2 Multi-Lens Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Modes of Operation of a TEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Dark-Field and Bright-Field Imaging . . . . . . . . . . . . . . . 2.3.2 Selected Area Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Convergent-Beam Electron Diffraction . . . . . . . . . . . . . . 2.3.4 High-Resolution Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Practical TEM Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Electron Guns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Illumination Lens Systems . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Imaging Lens Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Glass Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Lenses and Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Lenses and Phase Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Magnetic Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Image Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Pole Piece Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Lens Aberrations and Other Defects . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Spherical Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Chromatic Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.4 Astigmatism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.5 Gun Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 71 71 76 79 81 85 85 87 88 91 91 92 95 97 97 99 100 102 102 103 104 104 108 110 112 113

Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Waves and Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Coherent and Incoherent Scattering . . . . . . . . . . . . . . . . . 3.1.3 Elastic and Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . 3.1.4 Wave Amplitudes and Cross-Sections . . . . . . . . . . . . . . . 3.2 X-Ray Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Electrodynamics of X-Ray Scattering . . . . . . . . . . . . . . . 3.2.2 * Inelastic Compton Scattering . . . . . . . . . . . . . . . . . . . . . 3.2.3 X-Ray Mass Attenuation Coefficients . . . . . . . . . . . . . . . 3.3 Coherent Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 ‡ Born Approximation for Electrons . . . . . . . . . . . . . . . . 3.3.2 Atomic Form Factors – Physical Picture . . . . . . . . . . . . . 3.3.3 ‡ Scattering of Electrons by Model Potentials . . . . . . . . 3.3.4 ‡ * Atomic Form Factors – General Formulation . . . . . . 3.4 * Nuclear Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Properties of Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 119 122 123 124 128 128 132 134 136 136 141 144 148 153 153

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3.4.2 Time-Varying Potentials and Inelastic Neutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 * Coherent M¨ ossbauer Scattering . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 158 160 160

4.

Inelastic Electron Scattering and Spectroscopy . . . . . . . . . . . 4.1 Inelastic Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Electron Energy-Loss Spectrometry (EELS) . . . . . . . . . . . . . . . . 4.2.1 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 General Features of EELS Spectra . . . . . . . . . . . . . . . . . . 4.2.3 * Fine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Plasmon Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Plasmon Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 * Plasmons and Specimen Thickness . . . . . . . . . . . . . . . . 4.4 Core Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Scattering Angles and Energies – Qualitative . . . . . . . . 4.4.2 ‡ Inelastic Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 ‡ * Double-Differential Cross-Section, d2 σin /dφ dE . . . 4.4.4 * Scattering Angles and Energies – Quantitative . . . . . . 4.4.5 ‡ * Differential Cross-Section, dσin /dE . . . . . . . . . . . . . . 4.4.6 ‡ Partial and Total Cross-Sections, σin . . . . . . . . . . . . . . 4.4.7 Quantification of EELS Core Edges . . . . . . . . . . . . . . . . . 4.5 Energy-Filtered TEM Imaging (EFTEM) . . . . . . . . . . . . . . . . . . 4.5.1 Spectrum Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Energy Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Chemical Mapping with Energy-Filtered Images . . . . . . 4.5.4 Chemical Analysis with High Spatial Resolution . . . . . . 4.6 Energy Dispersive X-Ray Spectrometry (EDS) . . . . . . . . . . . . . 4.6.1 Electron Trajectories Through Materials . . . . . . . . . . . . 4.6.2 Fluorescence Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 EDS Instrumentation Considerations . . . . . . . . . . . . . . . . 4.6.4 Thin-Film Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.5 * ZAF Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.6 Artifacts in EDS Measurements . . . . . . . . . . . . . . . . . . . . 4.6.7 Limits of Microanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 165 165 167 169 173 173 175 177 177 180 184 186 187 189 191 193 193 193 196 197 200 200 203 205 208 211 213 215 217 217

5.

Diffraction from Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Sums of Wavelets from Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Electron Diffraction from a Material . . . . . . . . . . . . . . . . 5.1.2 Wave Diffraction from a Material . . . . . . . . . . . . . . . . . . . 5.2 The Reciprocal Lattice and the Laue Condition . . . . . . . . . . . . 5.2.1 Diffraction from a Simple Lattice . . . . . . . . . . . . . . . . . . .

223 223 224 226 230 230

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5.2.2 Reciprocal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Laue Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Equivalence of the Laue Condition and Bragg’s Law . . 5.2.5 Reciprocal Lattices of Cubic Crystals . . . . . . . . . . . . . . . 5.3 Diffraction from a Lattice with a Basis . . . . . . . . . . . . . . . . . . . . 5.3.1 Structure Factor and Shape Factor . . . . . . . . . . . . . . . . . 5.3.2 Structure Factor Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Symmetry Operations and Forbidden Diffractions . . . . 5.3.4 Superlattice Diffractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Crystal Shape Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Shape Factor of Rectangular Prism . . . . . . . . . . . . . . . . . 5.4.2 Other Shape Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Small Particles in a Large Matrix . . . . . . . . . . . . . . . . . . . 5.5 Deviation Vector (Deviation Parameter) . . . . . . . . . . . . . . . . . . . 5.6 Ewald Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Ewald Sphere Construction . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Ewald Sphere and Bragg’s Law . . . . . . . . . . . . . . . . . . . . . 5.6.3 Tilting Specimens and Tilting Electron Beams . . . . . . . 5.7 Laue Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 * Effects of Curvature of the Ewald Sphere . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231 233 233 234 235 235 237 242 243 247 247 252 252 256 257 257 259 259 262 262 266 266

Electron Diffraction and Crystallography . . . . . . . . . . . . . . . . . 6.1 Indexing Diffraction Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Issues in Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Method 1 – Start with Zone Axis . . . . . . . . . . . . . . . . . . . 6.1.3 Method 2 – Start with Diffraction Spots . . . . . . . . . . . . . 6.2 Stereographic Projections and Their Manipulation . . . . . . . . . . 6.2.1 Construction of a Stereographic Projection . . . . . . . . . . 6.2.2 Relationship Between Stereographic Projections and Electron Diffraction Patterns . . . . . . . . . . . . . . . . . . . 6.2.3 Manipulations of Stereographic Projections . . . . . . . . . . 6.3 Kikuchi Lines and Specimen Orientation . . . . . . . . . . . . . . . . . . 6.3.1 Origin of Kikuchi Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Indexing Kikuchi Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Specimen Orientation and Deviation Parameter . . . . . . 6.3.4 The Sign of s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Kikuchi Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Double Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Occurrence of Forbidden Diffractions . . . . . . . . . . . . . . . . 6.4.2 Interactions Between Crystallites . . . . . . . . . . . . . . . . . . . 6.5 * Convergent-Beam Electron Diffraction . . . . . . . . . . . . . . . . . . . 6.5.1 Convergence Angle of Incident Electron Beam . . . . . . . 6.5.2 Determination of Sample Thickness . . . . . . . . . . . . . . . . .

273 273 274 276 279 282 282 284 284 290 290 294 296 299 299 302 302 303 304 306 307

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6.5.3 Measurements of Unit Cell Parameters . . . . . . . . . . . . . . 6.5.4 ‡ Determination of Point Groups . . . . . . . . . . . . . . . . . . . 6.5.5 ‡ Determination of Space Groups . . . . . . . . . . . . . . . . . . . 6.6 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

309 314 325 330 330

Diffraction Contrast in TEM Images . . . . . . . . . . . . . . . . . . . . . . 7.1 Contrast in TEM Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Diffraction from Crystals with Defects . . . . . . . . . . . . . . . . . . . . 7.2.1 Review of the Deviation Parameter, s . . . . . . . . . . . . . . . 7.2.2 Atom Displacements, δr . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Shape Factor and t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Diffraction Contrast and {s, δr, t} . . . . . . . . . . . . . . . . . 7.3 Extinction Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The Phase-Amplitude Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Fringes from Sample Thickness Variations . . . . . . . . . . . . . . . . . 7.5.1 Thickness and Phase-Amplitude Diagrams . . . . . . . . . . . 7.5.2 Thickness Fringes in TEM Images . . . . . . . . . . . . . . . . . . 7.6 Bend Contours in TEM Images . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Diffraction Contrast from Strain Fields . . . . . . . . . . . . . . . . . . . . 7.8 Dislocations and Burgers Vector Determination . . . . . . . . . . . . 7.8.1 Diffraction Contrast from Dislocation Strain Fields . . . 7.8.2 The g·b Rule for Null Contrast . . . . . . . . . . . . . . . . . . . . 7.8.3 Image Position and Dislocation Pairs or Loops . . . . . . . 7.9 Semi-Quantitative Diffraction Contrast from Dislocations . . . . 7.10 Weak-Beam Dark-Field (WBDF) Imaging of Dislocations . . . . 7.10.1 Procedure to Make a WBDF Image . . . . . . . . . . . . . . . . . 7.10.2 Diffraction Condition for a WBDF Image . . . . . . . . . . . . 7.10.3 Analysis of WBDF Images . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Fringes at Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11.1 Phase Shifts of Electron Wavelets Across Interfaces . . . 7.11.2 Moir´e Fringes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12 Diffraction Contrast from Stacking Faults . . . . . . . . . . . . . . . . . 7.12.1 Kinematical Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12.2 Results from Dynamical Theory . . . . . . . . . . . . . . . . . . . . 7.12.3 Determination of the Intrinsic or Extrinsic Nature of Stacking Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12.4 Partial Dislocations Bounding the Fault . . . . . . . . . . . . . 7.12.5 An Example of a Stacking Fault Analysis . . . . . . . . . . . . 7.12.6 Sets of Stacking Faults in TEM Images . . . . . . . . . . . . . . 7.12.7 Related Fringe Contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13 Antiphase (π) Boundaries and δ Boundaries . . . . . . . . . . . . . . . 7.13.1 Antiphase Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13.2 δ Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14 Contrast from Precipitates and Other Defects . . . . . . . . . . . . . .

337 337 339 339 340 341 342 342 345 347 347 348 353 357 359 359 362 368 369 378 378 379 380 384 384 387 391 391 397 399 399 400 402 403 404 404 405 407

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7.14.1 Vacancies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14.2 Coherent Precipitates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.14.3 Semicoherent and Incoherent Particles . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

407 408 413 413 414

8.

Diffraction Lineshapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 8.1 Diffraction Line Broadening and Convolution . . . . . . . . . . . . . . 423 8.1.1 Crystallite Size Broadening . . . . . . . . . . . . . . . . . . . . . . . . 424 8.1.2 Strain Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 8.1.3 Instrumental Broadening – Convolution . . . . . . . . . . . . . 430 8.2 Fourier Transform Deconvolutions . . . . . . . . . . . . . . . . . . . . . . . . 433 8.2.1 Mathematical Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 8.2.2 * Effects of Noise on Fourier Transform Deconvolutions 436 8.3 Simultaneous Strain and Size Broadening . . . . . . . . . . . . . . . . . . 440 8.4 Diffraction Lineshapes from Columns of Crystals . . . . . . . . . . . 446 8.4.1 Wavelets from Pairs of Unit Cells in One Column . . . . 446 8.4.2 A Column Length Distribution . . . . . . . . . . . . . . . . . . . . . 448 8.4.3 ‡ Intensity from Column Length Distribution . . . . . . . . 450 8.5 Comments on Diffraction Lineshapes . . . . . . . . . . . . . . . . . . . . . . 451 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

9.

Patterson Functions and Diffuse Scattering . . . . . . . . . . . . . . . 9.1 The Patterson Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Atom Centers at Points in Space . . . . . . . . . . . . . . . . . . . 9.1.3 Definition of the Patterson Function . . . . . . . . . . . . . . . . 9.1.4 Properties of Patterson Functions . . . . . . . . . . . . . . . . . . 9.1.5 ‡ Perfect Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.6 Deviations from Periodicity and Diffuse Scattering . . . . 9.2 Diffuse Scattering from Atomic Displacements . . . . . . . . . . . . . . 9.2.1 Uncorrelated Displacements – Homogeneous Disorder . 9.2.2 ‡ Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 * Correlated Displacements – Atomic Size Effects . . . . . 9.3 Diffuse Scattering from Chemical Disorder . . . . . . . . . . . . . . . . . 9.3.1 Uncorrelated Chemical Disorder – Random Alloys . . . . 9.3.2 ‡ * SRO Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 ‡ * Patterson Function for Chemical SRO . . . . . . . . . . . 9.3.4 Short-Range Order Diffuse Intensity . . . . . . . . . . . . . . . . 9.3.5 ‡ * Isotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.6 * Polycrystalline Average and Single Crystal SRO . . . . 9.4 * Amorphous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 ‡ One-Dimensional Model . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 ‡ Radial Distribution Function . . . . . . . . . . . . . . . . . . . . .

457 457 457 458 459 461 463 467 469 469 472 477 481 481 485 487 488 488 490 491 491 495

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9.4.3 ‡ Partial Pair Correlation Functions . . . . . . . . . . . . . . . . 9.5 Small Angle Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Concept of Small Angle Scattering . . . . . . . . . . . . . . . . . . 9.5.2 * Guinier Approximation (small Δk) . . . . . . . . . . . . . . . . 9.5.3 * Porod Law (large Δk) . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 ‡ * Density-Density Correlations (all Δk) . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

500 502 502 504 508 510 512 513

10. High-Resolution TEM Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Huygens Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Wavelets from Points in a Continuum . . . . . . . . . . . . . . . 10.1.2 Huygens Principle for a Spherical Wavefront – Fresnel Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 ‡ Fresnel Diffraction Near an Edge . . . . . . . . . . . . . . . . . . 10.2 Physical Optics of High-Resolution Imaging . . . . . . . . . . . . . . . . 10.2.1 ‡ Wavefronts and Fresnel Propagator . . . . . . . . . . . . . . . 10.2.2 ‡ Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 ‡ Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Experimental High-Resolution Imaging . . . . . . . . . . . . . . . . . . . . 10.3.1 Defocus and Spherical Aberration . . . . . . . . . . . . . . . . . . 10.3.2 ‡ Lenses and Specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Lens Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 * Simulations of High-Resolution TEM Images . . . . . . . . . . . . . 10.4.1 Principles of Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Practice of Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Issues and Examples in High-Resolution TEM Imaging . . . . . . 10.5.1 Images of Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Examples of Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 * Specimen and Microscope Parameters . . . . . . . . . . . . . 10.5.4 * Some Practical Issues for HRTEM . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

517 518 518

11. High-Resolution STEM Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Characteristics of High-Angle Annular Dark-Field Imaging . . . 11.2 Electron Channeling Along Atomic Columns . . . . . . . . . . . . . . 11.2.1 Optical Fiber Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 ‡ Critical Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 * Tunneling Between Columns . . . . . . . . . . . . . . . . . . . . . 11.3 Scattering of Channeled Electrons . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Elastic Scattering of Channeled Electrons . . . . . . . . . . . 11.3.2 * Inelastic Scattering of Channeled Electrons . . . . . . . . 11.4 * Comparison of HAADF and HRTEM Imaging . . . . . . . . . . . . 11.5 HAADF Imaging with Atomic Resolution . . . . . . . . . . . . . . . . .

583 583 586 586 588 589 591 591 593 594 595

523 527 532 532 534 536 538 538 543 546 555 555 561 562 562 565 568 576 580 581

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11.5.1 * Effect of Defocus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Experimental Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 * Lens Aberrations and Their Corrections . . . . . . . . . . . . . . . . . 11.6.1 Cs Correction with Magnetic Hexapoles . . . . . . . . . . . . . 11.6.2 ‡ Higher-Order Aberrations and Instabilities . . . . . . . . . 11.7 Examples of Cs -Corrected Images . . . . . . . . . . . . . . . . . . . . . . . . 11.7.1 Three-Dimensional Imaging . . . . . . . . . . . . . . . . . . . . . . . . 11.7.2 High Resolution EELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

595 597 599 599 602 604 605 606 607 608

12. Dynamical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 ‡ * Mathematical Features of High-Energy Electrons in a Periodic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 ‡ * The Schr¨ odinger Equation . . . . . . . . . . . . . . . . . . . . . . 12.2.2 ‡ Kinematical and Dynamical Theory . . . . . . . . . . . . . . . 12.2.3 * The Crystal as a Phase Grating . . . . . . . . . . . . . . . . . . 12.3 First Approach to Dynamical Theory – Beam Propagation . . . 12.4 ‡ Second Approach to Dynamical Theory – Bloch Waves and Dispersion Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Diffracted Beams, {Φg }, are Beats of Bloch Waves, {Ψ (j) } . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Crystal Periodicity and Dispersion Surfaces . . . . . . . . . . 12.4.3 Energies of Bloch Waves in a Periodic Potential . . . . . . 12.4.4 General Two-Beam Dynamical Theory . . . . . . . . . . . . . . 12.5 Essential Difference Between Kinematical and Dynamical Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 ‡ Diffraction Error, sg , in Two-Beam Dynamical Theory . . . . 12.6.1 Bloch Wave Amplitudes and Diffraction Error . . . . . . . . 12.6.2 Dispersion Surface Construction . . . . . . . . . . . . . . . . . . . . 12.7 Dynamical Diffraction Contrast from Crystal Defects . . . . . . . 12.7.1 Dynamical Diffraction Contrast Without Absorption . . 12.7.2 ‡ * Two-Beam Dynamical Theory of Stacking Fault Contrast . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.3 Dynamical Diffraction Contrast with Absorption . . . . . 12.8 ‡ * Multi-Beam Dynamical Theories of Electron Diffraction . . Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

611 611 613 613 619 621 623 627 627 633 637 640 646 651 651 653 655 655 660 664 669 672 672

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677 References and Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682

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A. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Indexed Powder Diffraction Patterns . . . . . . . . . . . . . . . . . . . . . . A.2 Mass Attenuation Coefficients for Characteristic Kα X-Rays . A.3 Atomic Form Factors for X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . A.4 X-Ray Dispersion Corrections for Anomalous Scattering . . . . . A.5 Atomic Form Factors for 200 keV Electrons and Procedure for Conversion to Other Voltages . . . . . . . . . . . . A.6 Indexed Single Crystal Diffraction Patterns: fcc, bcc, dc, hcp . A.7 Stereographic Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.8 Examples of Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . A.9 Debye–Waller Factor from Wave Amplitude . . . . . . . . . . . . . . . A.10 Review of Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.11 TEM Laboratory Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.11.1 Preliminary – JEOL 2000FX Daily Operation . . . . . . . . A.11.2 Laboratory 1 – Microscope Procedures and Calibration with Au and MoO3 . . . . . . . . . . . . . . . . A.11.3 Laboratory 2 – Diffraction Analysis of θ′ Precipitates . A.11.4 Laboratory 3 – Chemical Analysis of θ′ Precipitates . . A.11.5 Laboratory 4 – Contrast Analysis of Defects . . . . . . . . . A.12 Fundamental and Derived Constants . . . . . . . . . . . . . . . . . . . . .

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691 691 692 693 697 698 703 713 717 720 721 728 728 732 735 739 740 742

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745

In section titles, the asterisk, “*,” denotes a more specialized topic. The double dagger, “‡,” warns of a higher level of mathematics, physics, or crystallography.

1. Diffraction and the X-Ray Powder Diffractometer

1.1 Diffraction 1.1.1 Introduction to Diffraction Materials are made of atoms. Knowledge of how atoms are arranged into crystal structures and microstructures is the foundation on which we build our understanding of the synthesis, structure and properties of materials. There are many techniques for measuring chemical compositions of materials, and methods based on inner-shell electron spectroscopies are covered in this book. The larger emphasis of the book is on measuring spatial arrangements of atoms in the range from 10−8 to 10−4 cm, bridging from the unit cell of the crystal to the microstructure of the material. There are many different methods for for measuring structure across this wide range of distances, but the more powerful experimental techniques involve diffraction. To date, most of our knowledge about the spatial arrangements of atoms in materials has been gained from diffraction experiments. In a diffraction experiment, an incident wave is directed into a material and a detector is typically moved about to record the directions and intensities of the outgoing diffracted waves.

2

1. Diffraction and the X-Ray Powder Diffractometer

“Coherent scattering” preserves the precision of wave periodicity. Constructive or destructive interference then occurs along different directions as scattered waves are emitted by atoms of different types and positions. There is a profound geometrical relationship between the directions of waves that interfere constructively, which comprise the “diffraction pattern,” and the crystal structure of the material. The diffraction pattern is a spectrum of real space periodicities in a material.1 Atomic periodicities with long repeat distances cause diffraction at small angles, while short repeat distances (as from small interplanar spacings) cause diffraction at high angles. It is not hard to appreciate that diffraction experiments are useful for determining the crystal structures of materials. Much more information about a material is contained in its diffraction pattern, however. Crystals with precise periodicities over long distances have sharp and clear diffraction peaks. Crystals with defects (such as impurities, dislocations, planar faults, internal strains, or small precipitates) are less precisely periodic in their atomic arrangements, but they still have distinct diffraction peaks. Their diffraction peaks are broadened, distorted, and weakened, however, and “diffraction lineshape analysis” is an important method for studying crystal defects. Diffraction experiments are also used to study the structure of amorphous materials, even though their diffraction patterns lack sharp diffraction peaks. In a diffraction experiment, the incident waves must have wavelengths comparable to the spacings between atoms. Three types of waves have proved useful for these experiments. X-ray diffraction (XRD), conceived by von Laue and the Braggs, was the first. The oscillating electric field of an incident x-ray moves the atomic electrons and their accelerations generate an outgoing wave. In electron diffraction, originating with Davisson and Germer, the charge of the incident electron interacts with the positively-charged core of the atom, generating an outgoing electron wavefunction. In neutron diffraction, pioneered by Shull, the incident neutron wavefunction interacts with nuclei or unpaired electron spins. These three diffraction processes involve very different physical mechanisms, so they often provide complementary information about atomic arrangements in materials. Nobel prizes in physics (1914, 1915, 1937, 1994) attest to their importance. As much as possible, we will emphasize the similarities of these three diffraction methods, with the first similarity being Bragg’s law. 1

Precisely and concisely, the diffraction pattern measures the Fourier transform of an autocorrelation function of the scattering factor distribution. The previous sentence is explained with care in Chap. 9. More qualitatively, the crystal can be likened to music, and the diffraction pattern to its frequency spectrum. This analogy illustrates another point. Given only the amplitudes of the different musical frequencies, it is impossible to reconstruct the music because the timing or “phase” information is lost. Likewise, the diffraction pattern alone may be insufficient to reconstruct all details of atom arrangements in a material.

1.1 Diffraction

3

1.1.2 Bragg’s Law Figure 1.1 is the construction needed to derive Bragg’s law. The angle of incidence of the two parallel rays is θ. You can prove that the small angle in the little triangle is equal to θ by showing that the triangles ABC and ACD are similar triangles. (Hint: look at the shared angle of φ = π2 − θ.) fronts of equal phase

D

e

e

C

e

ee

d B

q A

d sine

d sine

Fig. 1.1. Geometry for interference of a wave scattered from two planes separated by a spacing, d. The dashed lines are parallel to the crests or troughs of the incident and diffracted wavefronts. The total length difference for the two rays is the sum of the two dark segments.

The interplanar spacing, d, sets the difference in path length for the ray scattered from the top plane and the ray scattered from the bottom plane. Figure 1.1 shows that this difference in path lengths is 2d sinθ. Constructive wave interference (and hence strong diffraction) occurs when the difference in path length for the top and bottom rays is equal to one wavelength, λ: 2d sinθ = λ .

(1.1)

The right hand side is sometimes multiplied by an integer, n, since this condition also provides constructive interference. Our convention, however, sets n = 1. When we have a path length difference of nλ between adjacent planes, we change d (even though this new d may not correspond to a real interatomic distance). For example, when our diffracting planes are (100) cube faces, and 2d100 sinθ = 2λ ,

(1.2)

then we speak of a (200) diffraction from planes separated by d200 = (d100 )/2. A diffraction pattern from a material typically contains many distinct peaks, each corresponding to a different interplanar spacing, d. For cubic crystals with lattice parameter a0 , the interplanar spacings, dhkl , of planes labeled by Miller indices (hkl) are:

4

1. Diffraction and the X-Ray Powder Diffractometer

a0 , dhkl = √ h2 + k 2 + l 2

(1.3)

(as can be proved by the definition of Miller indices and the 3-D Pythagorean theorem). From Bragg’s law (1.1) we find that the (hkl) diffraction peak occurs at the measured angle 2θhkl :  √  λ h2 + k 2 + l 2 ) 2θhkl = 2 arcsin . (1.4) 2a0 There are often many individual crystals of random orientation in the sample, so all possible Bragg diffractions can be observed in the “powder pattern.” There is a convention for labeling, or “indexing,” the different Bragg peaks in a powder diffraction pattern2 using the numbers (hkl). An example of an indexed diffraction pattern is shown in Fig. 1.2. Notice that the intensities of the different diffraction peaks vary widely, and are zero for some combinations of h, k, and l. For this example of polycrystalline silicon, notice the absence of all combinations of h, k, and l that are mixtures of even and odd integers, and the absence of all even integer combinations whose sum is not divisible by 4. This is the “diamond cubic structure factor rule,” discussed in Sect. 5.3.2.

Fig. 1.2. Indexed powder diffraction pattern from polycrystalline silicon, obtained with Co Kα radiation.

One important use of x-ray powder diffractometry is for identifying unknown crystals in a sample of material. The idea is to match the positions and the intensities of the peaks in the observed diffraction pattern to a known pattern of peaks from a standard sample or from a calculation. There should be a one-to-one correspondence between the observed peaks and the indexed peaks in the candidate diffraction pattern. For a simple diffraction pattern as in Fig. 1.2, it is usually possible to guess the crystal structure with the 2

Procedures for indexing diffraction patterns from single crystals are deferred to Chap. 5.

1.1 Diffraction

5

help of the charts in Appendix A.1. This tentative indexing still needs to be checked. To do so, the θ-angles of the diffraction peaks are obtained, and used with (1.1) to obtain the interplanar spacing for each diffraction peak. For cubic crystals it is then possible to use (1.3) to convert each interplanar spacing into a lattice parameter, a0 . Non-cubic crystals may require an iterative refinement of lattice parameters and angles. The indexing is consistent if all peaks provide the same lattice parameter(s). For crystals of low symmetry and with more than several atoms per unit cell, it becomes increasingly impractical to try to index the diffraction pattern by hand. In practice, two approaches are used. The oldest and most reliable is a “fingerprinting” method. The International Centre for Diffraction Data (ICDD, formerly the Joint Committee on Powder Diffraction Standards, JCPDS) maintains a database of diffraction patterns from more than one hundred thousand inorganic and organic materials [1.1]. For each material the data fields include the observed interplanar spacings for all observed diffraction peaks, their relative intensities, and their hkl indexing. Software packages are available to identify peaks in the experimental diffraction pattern and then search the ICDD database to find candidate materials. Computerized fingerprint searches are particularly valuable when the sample contains a mixture of phases, and their chemical compositions are uncertain. When the chemical compositions of the crystallographic phases are known with some accuracy, however, the indexing of diffraction patterns is considerably easier. Phase determination is facilitated by finding candidate phases in handbooks of phase diagrams, and their diffraction patterns from the ICDD database. The problem is usually more difficult when multiple phases are present in the sample, but sometimes it is easy to distinguish individual diffraction patterns. The diffraction pattern in Fig. 1.3 was measured to determine if the surface of a glass-forming alloy had crystallized. The amorphous phase has two very broad peaks centered at 2θ = 38◦ and 74◦ . Sharp diffraction peaks from crystalline phases are distinguished easily from the amorphous peaks. Although this crystalline diffraction pattern has not been indexed, the measurement was useful for showing that the solidification conditions were inadequate for obtaining a fully amorphous solid. Beyond fingerprinting, another approach to structural determination by powder diffractometry is to calculate diffraction patterns from candidate crystal structures, and compare the calculated and observed diffraction patterns. Central to calculating a diffraction pattern are the structure factors of Sect. 5.3.2, which are characteristic of each crystal structure. Simple diffraction patterns (e.g., Fig. 1.2) can often be calculated readily, but structure factors for materials with more complicated unit cells require computer calculations. In its simplest form, the software takes an input file of atom positions, types, and x-ray wavelength, and calculates the positions and intensities of powder diffraction peaks. Such software is straightforward to use. In a more sophisticated variant of this approach, some features of the crystal structure, e.g.,

6

1. Diffraction and the X-Ray Powder Diffractometer

Fig. 1.3. Diffraction pattern from an as-cast Zr-Cu-Ni-Al alloy. The smooth intensity with broad peaks around 2θ = 38◦ and 74◦ , is the contribution from the amorphous phase. The sharp peaks show some crystallization at the surface of the sample that was in contact with the copper mold.

lattice parameters, are treated as adjustable parameters. These parameters are adjusted or “refined” as the software seeks the best fit between a calculated diffraction pattern and the measured one (Sect. 1.5.4). 1.1.3 Strain Effects Internal strains in a material can change the positions and shapes of x-ray diffraction peaks. The simplest type of strain is a uniform dilatation. If all parts of the specimen are strained equally in all directions (i.e., isotropically), the effect is a small change in lattice parameter. The diffraction peaks shift in position but remain sharp. The shift of each peak, ΔθB , caused by a strain, ε = Δd/d, can be calculated by differentiating Bragg’s law (1.1): d d 2d sinθB = λ, dd dd dθB 2 sinθB + 2d cosθB =0, dd ΔθB = −ε tanθB .

(1.5) (1.6) (1.7)

When θB is small, tanθB ≃ θB , so the strain is approximately equal to the fractional shift of the diffraction peak, although of opposite sign. For a uniform dilatation, the absolute shift of a diffraction peak in θ-angle increases strongly with the Bragg angle, θB . The diffraction peaks remain sharp when the strain is the same in all crystallites, but in general there is a distribution of strains in a polycrystalline specimen. For example, some crystallites could be under compression and others under tension. The crystallites then have slightly different lattice parameters, so each would have its diffraction peaks shifted slightly in angle as given by (1.7). A distribution of strains in a polycrystalline sample therefore causes a broadening in angle of the diffraction peaks, and the peaks at higher

1.1 Diffraction

7

Bragg angles are broadened more. This same argument applies when the interatomic separation depends on chemical composition – diffraction peaks are broadened when the chemical composition of a material is inhomogeneous. 1.1.4 Size Effects The width of a diffraction peak is affected by the number of crystallographic planes contributing to the diffraction. The purpose of this section is to show that the maximum allowed deviation from θB is smaller when more planes are diffracting. Diffraction peaks become sharper in θ-angle as crystallites become larger. To illustrate the principle, we consider diffraction peaks at small θB , so we set sinθ ≃ θ, and linearize (1.1) 3 : 2d θB ≃ λ .

(1.8)

If we had only two diffracting planes, as shown in Fig. 1.1, partiallyconstructive wave interference occurs even for large deviations of θ from the correct Bragg angle, θB . In fact, for two scattered waves, errors in phase within the range ±2π/3 still allow constructive interference, as depicted in Fig. 1.4. This phase shift corresponds to a path length error of ±λ/3 for the two rays in Fig. 1.1. The linearized Bragg’s law (1.8) provides a range of θ angle for which constructive interference occurs: λ−

λ λ < 2d(θB + Δθ) < λ + . 3 3

(1.9)

Wave Amplitude

unshifted shifted by //2 sum

0

2

4

6

8

Phase Angle (radians)

10

12

Fig. 1.4. The sum (bottom) of two waves out of phase by π/2. A full path length difference of λ corresponds to a phase angle of the wave that is 2π radians or 360◦ .

With the range of diffraction angles allowed by (1.9), and using (1.8) as an equality, we find Δθmax , which is approximately the largest angular deviation for which constructive interference occurs: λ . (1.10) Δθmax = ± 6d A situation for two diffracting planes with spacing a is shown in Fig. 1.5a. The allowable error in diffraction angle, Δθmax , becomes smaller with a larger 3

This approximation will be used frequently for high-energy electrons, with their short wavelengths (for 100 keV electrons, λ = 0.037 ˚ A), and hence small θB .

8

1. Diffraction and the X-Ray Powder Diffractometer

number of diffracting planes, however. Consider the situation with 4 diffracting planes as shown in Fig. 1.5b. The total distance between the top plane and bottom plane is now 3 times larger. For the same path length error as in Fig. 1.5a, the error in diffraction angle must be about 3 times smaller. For N diffracting planes (separated by a distance d = a(N − 1)) we have instead of (1.10), Δθmax ≃ ±

λ . 6(N − 1)a

(1.11)

Using (1.8) to provide the expression λ/(2a) ≃ θB for substitution into (1.11), we obtain: 1 Δθmax . (1.12) ≃ θB 3(N − 1)

A single plane of atoms diffracts only weakly, so it is typical to have hundreds of diffracting planes for high-energy electrons, and tens of thousands of planes for typical x-rays, so precise diffraction angles are possible for high-quality crystals.

Fig. 1.5. (a) Path length error, Δλ, caused by error in incident angle of Δθ. (b) Same path length error as in part (a), here caused by a smaller Δθ and a longer vertical distance.

It turns out that (1.12) predicts a Δθmax that is too small. Even if the very topmost and very bottommost planes are out of phase by more than λ/3, it is possible for most of the crystal planes to interfere constructively so that diffraction peaks still occur. For determining the sizes of crystals, a better approximation (replacing (1.12)) at small θ is: 0.9 Δθ , ≃ θB N

(1.13)

where Δθ is the half-width of the diffraction peak. The approximate (1.13) must be used with caution, but it has qualitative value. It states that the

1.1 Diffraction

9

number of diffracting planes is nearly equal to the ratio of the angle of the diffraction peak to the width of a diffraction peak. The widths of x-ray diffraction peaks are handy for determining crystallite sizes in the range of several nanometers (Sect. 8.1.1). 1.1.5 A Symmetry Consideration Diffraction is not permitted in the situation shown in Fig. 1.6 with waves incident at angle θ, but scattered into an angle θ′ not equal to θ. Between the two dashed lines (representing wavefronts), the path lengths of the two rays in Fig. 1.6 are unequal. When θ = θ′ , the difference in these two path lengths is proportional to the distance between the points O and P on the scattering plane. Along a continuous plane, there is a continuous range of separations between O and P, so there is as much destructive interference as constructive interference. Strong diffraction is therefore impossible. It will later prove convenient to formulate diffraction problems with the wavevectors, k0 and k, normal to the incident and diffracted wavefronts. The k0 and k have equal magnitudes, k = 2π/λ, because in diffraction the scattering is elastic. There is a special significance of the “diffraction vector,” Δk ≡ k−k0 , which is shown graphically as a vector sum in Fig. 1.6. A general principle is that the diffracting material must have translational invariance in the plane perpendicular to Δk. When this requirement is met, as in Fig. 1.1 but not in Fig. 1.6, diffraction experiments measure interplanar spacings 4 along Δk. – k0

6k

k0

k k n

e

e' O

P

wavefronts Fig. 1.6. Improper geometry for diffraction with θ = θ′ . The difference in path lengths is the difference in lengths of the two dark segments with ends at O and P. The vector Δk is the difference between the outgoing and incident wavevectors; n is the surface normal. For diffraction experiments, n  Δk. 4

b ≡ x/x, where x ≡ |x|. A hat over a vector denotes a unit vector: x

10

1. Diffraction and the X-Ray Powder Diffractometer

1.1.6 Momentum and Energy The diffraction vector Δk ≡ k − k0 , when multiplied by Planck’s constant, , is the change in momentum of the x-ray after diffraction:5 Δp =  Δk .

(1.14)

The crystal that does the diffraction must gain an equal but opposite momentum – momentum is always conserved. This momentum is eventually transferred to Earth, which undergoes a negligible change in its orbit. Any transfer of energy to the crystal means that the scattered x-ray will have somewhat less energy than the incident energy, which might impair diffraction experiments. Consider two types of energy transfers. First, a transfer of kinetic energy will follow the transfer of momentum of (1.14), meaning that a kinetic energy of recoil is taken up by motion of the crystal or Earth. The recoil energy is Erecoil = p2 /(2M ). If M is the mass of Earth, or even that a modest crystal, Erecoil is negligible (in that it cannot be detected today without heroic effort). When diffraction occurs, the kinetic energy is transferred to all atoms in the crystal, or at least those atoms within the spatial range described in Sect. 1.1.4. Second, energy may be transferred to a single atom, such as by moving the nucleus (causing atom vibrations), or by causing an electron of the atom to escape, ionizing the atom. A feature of quantum mechanics is that these events happen to some x-rays, but not to others. In general, the x-rays that undergo these “inelastic scattering” processes6 are “tagged” by one atom, and cannot participate in diffraction from a full crystal. 1.1.7 Experimental Methods The Bragg condition of (1.1) is unlikely to be satisfied for an arbitrary orientation of the crystallographic planes with respect to the incident x-ray beam, or with an arbitrary wavelength. There are three practical approaches for observing diffractions and making diffraction measurements (see Table 1.1). All are designed to ensure that Bragg’s law is satisfied. One approach, the “Debye–Scherrer” method, uses monochromatic radiation, but uses a distribution of crystallographic planes as provided by a polycrystalline sample. Another approach, the “Laue method,” uses the distribution of wavelengths in polychromatic or “white” radiation, and a single crystal sample. The combination of white radiation and polycrystalline samples produces too many diffractions, so this is not a useful technique. On the other hand, the study of single crystals with monochromatic radiation is an important technique, especially for determining the structures of minerals and large organic molecules in crystalline form. 5 6

b E/c = k b ω/c =  k, where This is consistent with a photon momentum of p = k c is the speed of light and E = ω is the photon energy. For x-rays, inelastic scattering is covered in Sect. 3.2, and parts of Chapter 4. For electrons, see Sect. 1.2 and Chapter 4, and for neutrons, see Sect. 3.4.2.

1.1 Diffraction

11

Table 1.1. Experimental methods for diffraction Radiation Sample

monochromatic

polychromatic

single crystal

single crystal methods

Laue

polycrystal

Debye–Scherrer

none

The “Laue Method” uses a broad range of x-ray wavelengths with specimens that are single crystals. It is commonly used for determining the orientations of single crystals. With the Laue method, the orientations and positions of both the crystal and the x-ray beam are stationary. Some of the incident x-rays have the correct wavelengths to satisfy Bragg’s law for some crystal planes. In the Laue diffraction pattern of Fig. 1.7, the different diffraction spots along a radial row originate from various combinations of x-ray wavelengths and crystal planes having a projected normal component along the row. It is not easy to evaluate these combinations (especially when there are many orientations of crystallites in the sample), and the Laue method will not be discussed further.

Fig. 1.7. Backscatter Laue diffraction pattern from Si in [110] zone orientation. Notice the high symmetry of the diffraction pattern.

The “Debye–Scherrer” method uses monochromatic x-rays, and equipment to control the 2θ angle for diffraction. The Debye–Scherrer method is most appropriate for polycrystalline samples. Even when θ is a Bragg angle, however, the incident x-rays are at the wrong angle for most of the crystallites in the sample (which may have their planes misoriented as in Fig. 1.6, for example). Nevertheless, when θ is a Bragg angle, in most powders there are some crystallites oriented adequately for diffraction. When enough crystallites are irradiated by the beam, the crystallites diffract the x-rays into a set

12

1. Diffraction and the X-Ray Powder Diffractometer

of diffraction cones as shown in Fig. 1.8. The apex angles of the diffraction cones are 4θB , where θB is the Bragg angle for the specific diffraction.

Fig. 1.8. Arrangement for Debye–Scherrer diffraction from a polycrystalline sample.

Debye–Scherrer diffraction patterns are also obtained by diffraction of monochromatic electrons from polycrystalline specimens. Two superimposed electron diffraction patterns are presented in Fig. 1.9. The sample was a crystalline Ni-Zr alloy deposited as a thin film on a single crystal of NaCl. The polycrystalline Ni-Zr gave a set of diffraction cones as in Fig. 1.8. These cones were oriented to intersect a sheet of film in the transmission electron microscope, thus forming an image of “diffraction rings.” In addition to the diffraction rings, a square array of diffraction spots is also seen in Fig. 1.9. These spots originate from some residual NaCl that remained on the sample, and the spots form a single crystal diffraction pattern.

Fig. 1.9. Superimposed electron diffraction patterns from polycrystalline Ni-Zr and single crystal NaCl.

Diffraction from polycrystalline materials, or “powder diffraction” with monochromatic radiation, requires the Debye–Scherrer diffractometer to pro-

1.2 The Creation of X-Rays

13

vide only one degree of freedom in changing the diffraction conditions, corresponding to changing the 2θ angle of Figs. 1.1–1.3. On the other hand, three additional degrees of freedom for specimen orientation are required for single crystal diffraction experiments with monochromatic radiation. Although diffractions from single crystals are more intense, these added parametric dimensions require a considerable increase in data measurement time. Such measurements are possible with equipment in a small laboratory, but bright synchrotron radiation sources have enabled many new types of single crystal diffraction experiments.

1.2 The Creation of X-Rays X-rays are created when energetic electrons lose energy. The same processes of x-ray creation are relevant for obtaining x-rays in an x-ray diffractometer, and for obtaining x-rays for chemical analysis in an analytical transmission electron microscope. Some relevant electron-atom interactions are summarized in Fig. 1.10. Figure 1.10a shows the process of elastic scattering where the electron is deflected, but no energy loss occurs. Elastic scattering is the basis for electron diffraction. Figure 1.10b is an inelastic scattering where the deflection of the electron results in radiation. The acceleration during the deflection of a classical electron would always produce radiation, and hence no elastic scattering. In quantum electrodynamics the radiation may or may not occur (compare Figs. 1.10a and 1.10b), but the average over many electron scatterings corresponds to the classical radiation field. Figure 1.10c illustrates two processes involving energy transfer between the incident electron and the electrons of the atom. Both processes of Fig. 1.10c involve a primary ionization where a core electron is ejected from the atom. An outer electron of more positive energy falls into this core hole, but there are two ways to dispose of its excess energy: 1) an x-ray can be emitted directly from the atom, or 2) this energy can be used to eject another outer electron from the atom, called an “Auger electron.” The “characteristic xray” of process 1 carries the full energy difference of the two electron states. The Auger electron was originally bound to the atom, however, so the kinetic energy of the emitted Auger electron is this energy difference minus its initial binding energy. After either decay process of Fig. 1.10c, there remains an empty electron state in an outer shell of the atom, and the process repeats itself at a lower energy until the electron hole migrates out of the atom. An x-ray for a diffraction experiment is characterized by its wavelength, λ, whereas for spectrometry or x-ray creation the energy, E, is typically more useful. The two are related inversely, and (1.16) is worthy of memorization: c (1.15) E = hν = h , λ 12.4 12.3984 ≃ . (1.16) E[keV] = λ[˚ A] λ[˚ A]

14

1. Diffraction and the X-Ray Powder Diffractometer

bremsstrahlung EDS background

elastic scattering

a

Imaging

b

inelastic scattering EELS background

characteristic x-ray

EDS, WDS decay channels

EELS

c

high-energy secondary

AES Auger electron

Fig. 1.10. a–c. Some processes of interaction between a highenergy electron and an atom: (a) is useful for diffraction, whereas the ejection of a core electron in (c) is the basis for chemical spectroscopies. Two decay channels for the core hole in c are indicated by the two thick, dashed arrows.

1.2.1 Bremsstrahlung Continuum radiation (somewhat improperly called bremsstrahlung, meaning “braking radiation”) can be emitted when an electron undergoes a strong deflection as depicted in Fig. 1.10b, because the deflection causes an acceleration. This acceleration can create an x-ray with an energy as high as the full kinetic energy of the incident electron, E0 (equal to its charge, e, times its accelerating voltage, V ). Substituting E0 = eV into (1.15), we obtain the “Duane–Hunt rule” for the shortest x-ray wavelength from the anode, λmin : 12.3984 hc = λmin [˚ . A] = eV E0 [keV]

(1.17)

The shape of the bremsstrahlung spectrum can be understood by using one fact from quantum electrodynamics. Although each x-ray photon has a distinct energy, the photon energy spectrum is obtained from the Fourier transform of the time dependence of the electron acceleration, a(t). The passage of each electron through an atom provides a brief, pulse-type acceleration. The average over many electron-atom interactions provides a broadband x-ray energy spectrum. Electrons that pass closer to the nucleus undergo stronger accelerations, and hence radiate with a higher probability. Their spectrum, however, is the same as the spectrum from electrons that traverse

1.2 The Creation of X-Rays

15

the outer part of an atom. In a thin specimen where only one sharp acceleration of the electron can take place, the bremsstrahlung spectrum has an energy distribution shown in Fig. 1.11a; a flat distribution with a cutoff of 40 keV for electrons of 40 keV. The general shape of the wavelength distribution can be understood as follows. The energy-wavelength relation for the x-ray is: c E = , (1.18) ν= h λ so an interval in wavelength is related to an interval in energy as: 1 dE = −ch 2 , (1.19) dλ λ ch (1.20) dE = − 2 dλ . λ The same number of photons must be counted in the interval of the wavelength distribution that corresponds to an interval in the energy distribution: I(λ) dλ = I(E) dE ,

(1.21)

so by using (1.19), the wavelength distribution is: ch dλ . (1.22) λ2 The negative sign in (1.22) appears because an increase in energy corresponds to a decrease in wavelength. The wavelength distribution is therefore related to the energy distribution as: I(λ)dλ = −I(E)

I(E) . (1.23) λ2 Figure 1.11b is the wavelength distribution (1.23) that corresponds to the energy distribution of Fig. 1.11a. Notice how the bremsstrahlung x-rays have wavelengths bunched towards the value of λmin of (1.17). The curve in Fig. 1.11b, or its equivalent energy spectrum in Fig. 1.11a, is a reasonable approximation to the bremsstrahlung background from a very thin specimen. The anode of an x-ray tube is rather thick, however. Most electrons do not lose all their energy at once, and propagate further into the anode. When an electron has lost some of its initial energy, it can reradiate again, but with a smaller Emax (or larger λmin ). Deeper within the anode, these multiply-scattered electrons emit more bremsstrahlung of longer wavelengths. The spectrum of bremsstrahlung from a thick sample can be understood by summing the individual spectra from electrons of various kinetic energies in the anode. A coarse sum is presented qualitatively in Fig. 1.11c, and a higher resolution sum is presented in Fig. 1.11d. The bremsstrahlung from an x-ray tube increases rapidly above λmin , reaching a peak at about 1.5 λmin . The intensity of the bremsstrahlung depends on the strength of the accelerations of the electrons. Atoms of larger atomic number, Z, have stronger I(λ) = ch

1. Diffraction and the X-Ray Powder Diffractometer

Intensity

16

a Energy

E0

Intensity

b

c

d 0

hc E0

Wavelength

Fig. 1.11. (a) Energy distribution for single bremsstrahlung process. (b) wavelength distribution for the energy distribution of part a. (c) coarse-grained sum of wavelength distributions expected from multiple bremsstrahlung processes in a thick target (d) sum of contributions from single bremsstrahlung processes of a continuous energy distribution.

potentials for electron scattering, and the intensity of the bremsstrahlung increases approximately as V 2 Z 2 . 1.2.2 Characteristic Radiation In addition to the bremsstrahlung emitted when a material is bombarded with high-energy electrons, x-rays are also emitted with discrete energies characteristic of the elements in the material, as depicted in Fig. 1.10c (top part). The energies of these “characteristic x-rays” are determined by the binding energies of the electrons of the atom, or more specifically the differences in these binding energies. It is not difficult to calculate these energies for atoms of atomic number, Z, if we make the major assumption that the atoms are “hydrogenic” and have only one electron. We seek solutions to the time-independent Schr¨ odinger equation for the electron wavefunction: Ze2 2 2 ∇ ψ(r, θ, φ) − ψ(r, θ, φ) = E ψ(r, θ, φ) . (1.24) 2m r To simplify the problem, we seek solutions that are spherically symmetric, so the derivatives of the electron wavefunction, ψ(r, θ, φ), are zero with respect to the angles θ and φ of our spherical coordinate system. In other words, we consider cases where the electron wavefunction is a function of r only: ψ(r). The Laplacian in the Schr¨ odinger equation then takes a relatively simple form:   Ze2 2 1 ∂ 2 ∂ ψ(r) − ψ(r) = E ψ(r) . (1.25) − r 2m r2 ∂r ∂r r −

Since E is a constant, acceptable expressions for ψ(r) must provide an E that is independent of r. Two such solutions are:

1.2 The Creation of X-Rays Zr

ψ1s (r) = e− a0 ,   Zr Zr − 2a ψ2s (r) = 2 − e 0 , a0

17

(1.26) (1.27)

where the Bohr radius, a0 , is defined as: 2 . (1.28) me2 By substituting (1.26) or (1.27) into (1.25), and taking the partial derivatives with respect to r, it is found that the r-dependent terms cancel out, leaving E independent of r (see Problem 1.7):   1 2 me4 1 (1.29) = − 2 Z 2 ER . En = − 2 Z 2 n 2 n a0 =

In (1.29) we have defined the energy unit, ER , the Rydberg, which is +13.6 eV. The integer, n, in (1.29) is sometimes called the “principal quantum number,” which is 1 for ψ1s , 2 for ψ2s , etc. It is well-known that there are other solutions for ψ that are not spherically-symmetric, for example, ψ2p , ψ3p , and ψ3d .7 Perhaps surprisingly, for ions having a single electron, (1.29) provides the correct energies for these other electron wavefunctions, where n = 2, 3, and 3 for these three examples. This is known as an “accidental degeneracy” of the Schr¨odinger equation for the hydrogen atom, but it is not true when there is more than one electron about the atom. Suppose a Li atom with Z = 3 has been stripped of both its inner 1s electrons, and suppose an electron in a 2p state undergoes an energetically downhill transition into one of these empty 1s states. The energy difference can appear as an x-ray of energy ΔE, and for this 1-electron atom it is:   1 3 1 − (1.30) Z 2 ER = Z 2 ER . ΔE = E2 − E1 = − 22 12 4 (The 1s state, closer to the nucleus than the 2p state, has the more negative energy. The x-ray has a positive energy.) A standard old notation groups electrons with the same n into “shells” designated by the letter series K, L, M. . . corresponding to n = 1, 2, 3 . . . . The electronic transition of (1.30) between shells L → K emits a “Kα x-ray.” A Kβ x-ray originates with the transition M → K. Other designations are given in Table 1.2 and Fig. 1.13. 7

The time-independent Schr¨ odinger equation (1.24) was obtained by the method of separation of variables, specifically the separation of t from r,θ,φ. The constant of separation was the energy, E. For the separation of θ and φ from r, the constant of separation provides l, and for the separation of θ from φ, the constant of separation provides m. The integers l and m involve the angular variables θ and φ, and are “angular momentum quantum numbers.” The quantum number l corresponds to the total angular momentum, and m corresponds to its orientation along a given direction. The full set of electron quantum numbers is {n, l, m, s}, where s is spin. Spin cannot be obtained from a constant of separation of the Schr¨ odinger equation, which offers only 3 separations for {r, θ, φ, t}. Spin is obtained from the relativistic Dirac equation, however.

18

1. Diffraction and the X-Ray Powder Diffractometer

Fig. 1.12. Characteristic x-ray energies of the elements. The x-axis of plot was originally the square root of frequency √ (from 6 to 24 ×108 Hz) [1.2].

Equation (1.30) works well for x-ray emission from atoms or ions having only one electron, but electron-electron interactions complicate the calculation of energy levels of most atoms.8 Figure 1.12 shows bands of data, which originate with electronic transitions between different shells. This plot of the relationship between the atomic number and the x-ray energy is the basis for Moseley’s laws. Moseley’s laws are modifications of (1.30). For Kα and Lα x-rays, they are:   1 1 (1.31) − = 10.204(Z − 1)2 , EKα = (Z − 1)2 ER 12 22   1 1 − = 1.890(Z − 7.4)2 . (1.32) ELα = (Z − 7.4)2 ER 22 32 8

Additional electron-electron potential energy terms are needed in (1.24), and these alter the energy levels.

1.2 The Creation of X-Rays

19

Equations (1.31) and (1.32) are good to about 1 % accuracy for x-rays with energies from 3–10 keV.9 Moseley correctly interpreted the offsets for Z (1 and 7.4 in (1.31) and (1.32)) as originating from shielding of the nuclear charge by other core electrons. For an electron in the K-shell, the shielding involves one electron – the other electron in the K-shell. For an electron in the L-shell, shielding involves both K electrons (1s) plus to some extent the other L electrons (2s and 2p), which is a total of 9. Perhaps Moseley’s law of (1.31) for the L → K transition could be rearranged with different effective nuclear charges for the K and L-shell electrons, rather than using Z–1 for both of them. This change would, however, require a constant different from ER in (1.31). The value of 7.4 for L-series x-rays, in particular, should be regarded as an empirical parameter. Table 1.2. Some x-ray spectroscopic notations label

transition

atomic notation

E for Cu [keV]

Kα1

L3 → K

2p3/2 → 1s

8.04778

M2,3 → K

3p → 1s

8.90529

M4,5 → L3

3d → 2p3/2

0.9297

Kα2 Kβ1,3 Kβ5 Lα1,2 Lβ1 Lβ3,4 Lη Ll

L2 → K

M4,5 → K

2p

1/2

→ 1s

3d → 1s

M4 → L2

3d → 2p1/2

M1 → L2

1/2

M2,3 → L1 M1 → L3

3p → 2s

3s → 2p

3s → 2p3/2

8.02783 8.99770 0.9498 1.0228 0.832 0.8111

Notice that Table 1.2 and Fig. 1.13 do not include the transition 2s → 1s. This transition is forbidden. The two wavefunctions, ψ1s (r) and ψ2s (r) of (1.26) and (1.27), have inversion symmetry about r = 0. A uniform electric field is antisymmetric in r, however, so the induced dipole moment of ψ2s (r) has zero net overlap with ψ1s (r). X-ray emission by electric dipole radiation is subject to a selection rule (see Problem 1.12), where the angular momentum of the initial and final states must differ by 1 (i.e., Δl = ±1). As shown in Table 1.2, there are two types of Kα x-rays. They differ slightly in energy (typically by parts per thousand), and this originates from the spin-orbit splitting of the L shell. Recall that the 2p state can have a total angular momentum of 3/2 or 1/2, depending on whether the electron spin of 9

This result was published in 1914. Henry Moseley died in 1915 at Gallipoli during World War I. The British response to this loss was to assign scientists to noncombatant duties during World War II.

20

1. Diffraction and the X-Ray Powder Diffractometer

Fig. 1.13. Some electron states and x-ray notation (in this case for U). After [1.3].

1/2 lies parallel or antiparallel to the orbital angular momentum of 1. The spin-orbit interaction causes the 1/2 state (L2 ) to lie at a lower energy than the 3/2 state (L3 ), so the Kα1 x-ray is slightly more energetic than the Kα2 x-ray. There is no spin-orbit splitting of the final K-states since their orbital angular momentum is zero, but spin-orbit splitting occurs for the final states of the M → L x-ray emissions. The Lα1 and Lβ1 x-rays are differentiated in this way, as shown in the Table 1.2. Subshell splittings may not be resolved in experimental energy spectra, and it may be possible to identify only a composite Kβ x-ray peak, for example. 1.2.3 Synchrotron Radiation Storage Rings. Synchrotron radiation is a practical source of x-rays for many experiments that are impractical with the conventional x-ray sources of Sect. 1.3.1. High flux and collimation, energy tunability, and timing capabilities are some special features of synchrotron radiation sources. Facilities for synchrotron radiation experiments are available at several national or international laboratories.10 These facilities are centered around an electron (or positron) storage ring with a circumference of about one kilometer. 10

Three premier facilities are the European Synchrotron Radiation Facility in Grenoble, France, the Advanced Photon Source at Argonne, Illinois, USA, and the Super Photon Ring 8-GeV, SPring-8 in Harima, Japan [1.4].

1.2 The Creation of X-Rays

21

The electrons in the storage ring have energies of typically 7 × 109 eV, and travel close to the speed of light. The electron current is perhaps 100 mA, but the electrons are grouped into tight bunches of centimeter length, each with a fraction of this total current. The bunches have vertical and horizontal spreads of tens or hundreds of microns. The electrons lose energy by generating synchrotron radiation as they are bent around the ring. The electrical power needed to replenish the energy of the electrons is provided by a radiofrequency electric field. This cyclic electric field accelerates the electron bunches by alternately attracting and repelling them as they move through a dedicated section of the storage ring. (Each bunch must be in phase with the radiofrequency field.) The ring is capable of holding a number of bunches equal to the radiofrequency times the cycle time around the ring. For example, with a 0.3 GHz radiofrequency, an electron speed of 3×105 km/s, and a ring circumference of 1 km, the number of “buckets” to hold the bunches is 1,000. As the bunches pass through bending magnets or magnetic “insertion devices,” their accelerations cause photon emission. X-ray emission therefore occurs in pulsed bursts, or “flashes.” The flash duration depends on the duration of the electron acceleration, but this is shortened by relativistic contraction. The flash duration depends primarily on the width of the electron bunch, and may be 0.1 ns. In a case where every fiftieth bucket is filled in our hypothetical ring, these flashes are separated in time by 167 ns. Some experiments based on fast timing are designed around this time structure of synchrotron radiation. Although the energy of the electrons in the ring is restored by the high power radiofrequency system, electrons are lost by occasional collisions with gas atoms in the vacuum. The characteristic decay of the beam current over several hours requires that new electrons are injected into the bunches. Undulators. Synchrotron radiation is generated by the dipole bending magnets used for controlling the electron orbit in the ring, but all modern “third generation” synchrotron radiation facilities derive their x-ray photons from “insertion devices,” which are magnet structures such as “wigglers” or “undulators.” Undulators comprise rows of magnets along the path of the electron beam. The fields of these magnets alternate up and down, perpendicular to the direction of the electron beam. Synchrotron radiation is produced when the electrons accelerate under the Lorentz forces of the row of magnets. The mechanism of x-ray emission by electron acceleration is essentially the same as that of bremsstrahlung radiation, which was described in Fig. 1.10 and Sect. 1.2.1. Because the electron accelerations lie in a plane, the synchrotron x-rays are polarized with E in this same plane and perpendicular to the direction of the x-ray (cf., Fig. 1.26). The important feature of an undulator is that its magnetic fields are positioned precisely so that the photon field is built by the constructive interference of radiation from a row of accelerations. The x-rays emerge from the

22

1. Diffraction and the X-Ray Powder Diffractometer

undulator in a tight pattern analogous to a Bragg diffraction from a crystal, where the intensity of the x-ray beam in the forward direction increases as the square of the number of coherent magnetic periods (typically tens). Again in analogy with Bragg diffraction, there is a corresponding decrease in the angular spread of the photon beam. The relativistic nature of the GeV electrons is also central to undulator operation. In the line-of-sight along the electron path, the electron oscillation frequency is enhanced by the relativistic factor 2(1 − (v/c)2 )−1 , where v is the electron velocity and c is the speed of light. This factor is about 108 for electron energies of several GeV. Typical spacings of the magnets are 3 cm, a distance traversed by light in 10−10 sec. The relativistic enhancement brings the frequency to 1018 Hz, which corresponds to an x-ray energy, hν, of several keV. The relativistic Lorentz contraction along the forward direction further sharpens the radiation pattern. The x-ray beam emerging from an undulator may have an angular spread of microradians, diverging by only a millimeter over distances of tens of meters. A small beam divergence and a small effective source area for x-ray emission makes an undulator beam an excellent source of x-rays for operating a monochromator. Brightness. Various figures of merit describe how x-ray sources provide useful photons. The figure of merit for operating a monochromator is proportional to the intensity (photons/s) per area of emitter (cm−2 ), but another factor also must be included. For a highly collimated x-ray beam, the monochromator crystal is small compared to the distance from the source. It is important that the x-ray beam be concentrated into a small solid angle so it can be utilized effectively. The full figure of merit for monochromator operation is “brightness” (often called “brilliance”), which is normalized by the solid angle of the beam. Brightness has units of [photons (s cm2 sr)−1 ]. The brightness of an undulator beam can be 109 times that of a conventional x-ray tube. Brightness is also a figure of merit for specialized beamlines that focus an x-ray beam into a narrow probe of micron dimensions. Finally, the x-ray intensity is not distributed uniformly over all energies. The term “spectral brilliance” is a figure of merit that specifies brightness per eV of energy in the x-ray spectrum. Undulators are tuneable to optimize their output within a broad energy range. Their power density is on the order of kW mm−2 ,and much of this energy is deposited as heat in the first crystal that is hit by the undulator beam. There are technical challenges in extracting heat from the first crystal of this “high heat load monochromator.” It may be constructed for example, of water-cooled diamond, which has excellent thermal conductivity. Beamlines and User Programs. The monochromators and goniometers needed for synchrotron radiation experiments are located in a “beamline,” which is along the forward direction from the insertion device. These components are typically mounted in lead-lined “hutches” that shield users from the lethal radiation levels produced by the undulator beam.

1.3 The X-Ray Powder Diffractometer

23

Synchrotron radiation user programs are typically organized around beamlines, each with its own capabilities and scientific staff. Although many beamlines are dedicated to x-ray diffraction experiments, many other types of x-ray experiments are possible. Work at a beamline requires success with a formal proposal for an experiment. This typically begins by making initial contact with the scientific staff at the beamline, who can often give a quick assessment of feasibility and originality. Successful beamtime proposals probably will not involve measurements that can be performed with conventional x-ray diffractometers. Radiation safety training,travel arrangements, operating schedules and scientific collaborations are issues for experiments at synchrotron facilities. The style of research differs considerably from that with a diffractometer in a small laboratory.

1.3 The X-Ray Powder Diffractometer This section describes the components of a typical x-ray diffractometer found in a materials analysis laboratory. The essential components are: • a source of x-rays, usually a sealed x-ray tube, • a goniometer, which provides precise mechanical motions of the tube, specimen, and detector, • an x-ray detector, • electronics for counting detector pulses in synchronization with the positions of the goniometer. Typical data comprise a list of detector counts versus 2θ angle, whose graph is the diffraction pattern. 1.3.1 Practice of X-Ray Generation Conventional x-ray tubes are vacuum tube diodes, with their filaments biased typically at –40 kV. Electrons are emitted thermionically from the filament, and accelerate into the anode, which is maintained at ground potential.11 Analogous components are used in an analytical TEM (Sect. 2.4.1), although the electron energies are higher, the electron beam can be shaped into a finelyfocused probe, and the electrons induce x-ray emission from the specimen. The operating voltage and current of an x-ray tube are typically selected to optimize the emission of characteristic radiation, since this is a source of monochromatic radiation. For a particular accelerating voltage, the intensity 11

The alternative arrangement of having the filament at ground and the anode at +40 kV is incompatible with water cooling of the anode. Cooling is required because a typical electron current of 25 mA demands the dissipation of 1 kW of heat from a piece of metal situated in a high vacuum. In a TEM, it is also convenient to keep the specimen and most components at ground potential.

24

1. Diffraction and the X-Ray Powder Diffractometer

of all radiations increases with the electron current in the tube. The effect of accelerating voltage on characteristic x-ray emission is more complicated, however, since the spectrum of x-rays is affected. Characteristic x-rays are excited more efficiently with higher accelerating voltage, V . In practice the intensity of characteristic radiation depends on V as: Ichar ∝ (V − Vc )1.5 ,

(1.33)

where Vc is the energy of the characteristic x-ray. On the other hand, the intensity of the bremsstrahlung increases approximately as: Ibrem ∝ V 2 Z 2 .

(1.34)

To maximize the characteristic x-ray intensity with respect to the continuum, we set: d Ichar d (V − Vc )1.5 = =0, dV Ibrem dV V2

(1.35)

which provides: V = 4Vc .

(1.36)

In practice, the optimal voltage for exciting the characteristic x-rays is about 3.5–4 times the energy of the characteristic x-ray. Combining the bremsstrahlung and characteristic x-ray intensities gives the wavelength distribution from an x-ray tube shown in Fig. 1.14. For this example of a tube with a silver anode, the characteristic Kα lines (22.1 keV, 0.56 ˚ A ) are not excited at tube voltages below 25.6 keV, which corresponds to the energy required to remove a K-shell electron from a silver atom. Maximizing the ratio of characteristic silver Kα intensity to bremsstrahlung intensity would require an accelerating voltage around 100 keV, which is impractically high. The most popular anode material for monochromatic radiation is copper, which also provides the benefit of high thermal conductivity. A modern sealed x-ray tube has a thin anode with cooling water flowing behind it. If the anode has good thermal conduction, as does copper, perhaps 2 kW of power (accelerating voltage times beam current) can be used before anode heating shortens excessively the tube life.12 An alternative type of xray tube has been developed to handle higher electron currents, and hence proportionately more x-ray emission. The trick is to construct the anode as a cylinder, and spin it at about 5,000 RPM during operation. Higher heat dissipations are possible with these rotating anode x-ray sources, perhaps 20 kW. Rotating anode x-ray sources are more expensive and complicated, however, because they require high mechanical precision in the rotating components, a leak-proof high vacuum rotating seal with provisions for water cooling, and continuous vacuum pumping. Both rotating anode and sealed tube x-ray 12

The efficiency of x-ray emission, the ratio of emitted x-ray power to electrical power dissipated in the tube, ǫ, is quite low. Empirically it is found that ǫ = 1.4 × 10−9 ZV , where Z is the atomic number and V is accelerating voltage.

1.3 The X-Ray Powder Diffractometer

25

Fig. 1.14. Intensity spectrum (in wavelength) of an x-ray tube with a silver anode [1.5].

sources require a regulated high voltage dc power supply for their operation. These high voltage generators include a feedback control circuit to adjust the thermionic emission from the filament to maintain a steady electron current in the tube. By using a direct beam slit (Fig. 1.15), a narrow x-ray beam can be obtained. By choosing this beam to be those x-rays that leave the anode surface at a shallow angle, geometrical foreshortening of the anode can be used to provide a line source. This shallow “take-off angle” of the x-ray tube is typically 3–6 degrees. 1.3.2 Goniometer for Powder Diffraction With the monochromatic radiation of the Debye–Scherrer method, we need equipment to control the angles between the x-ray source, specimen, and detector. Precise movements of the specimen and the detector with respect to the x-ray source are provided by a mechanical device called a “goniometer” (see Fig. 1.15). In practice, it is easiest to keep the bulky x-ray tube stationary, and rotate the specimen by the angle θ. To ensure that the scattered x-rays leave the specimen at angle θ, the detector must be rotated precisely by the angle 2θ.13 The goniometer may also provide for the rotation of the specimen in the plane of its surface by the angle φ, and in the plane of the goniometer by the angle ω. The angles φ and ω do not affect the diffraction pattern for a polycrystal with random orientations, but they are important for samples with crystallographic texture. 13

This “θ-2θ diffractometer” is less versatile than a “θ-θ diffractometer,” but the latter instrument requires precise movement of its x-ray tube.

26

tube

1. Diffraction and the X-Ray Powder Diffractometer

direct beam Soller slits

q t 2e = 0°

e direct beam slit

2e detector slit detector Soller Slits 2e = 90°

receiving slit

detector

Fig. 1.15. Schematic diagram of some typical components and angles of the goniometer for a θ-2θ x-ray diffractometer. The flat specimen is at the center of the goniometer circle, whose radius is typically 0.25–0.5 m.

To obtain good intensity, but well-defined diffraction angles, x-ray powder diffractometers usually employ a “line source,” which is narrow in the plane of the goniometer, but has a height of perhaps 1 cm perpendicular to this plane. Slits are used to collimate the incident and diffracted beams. The direct beam slit controls the “equatorial divergence” of the incident beam (the equatorial plane of the diffractometer is in the plane of the paper of Fig. 1.15). The divergence of the incident beam along the axis of the goniometer (perpendicular to the plane of the paper) must also be controlled to obtain well-defined diffraction angles. Control of “axial divergence” is achieved with Soller slits, which are stacked plates that slice the incident beam into a stack of beams, each with low axial divergence. Between the specimen and the detector is a detector slit to control equatorial divergence, and Soller slits to control axial divergence. The position of the detector is defined by the receiving slit. A divergent incident beam is a practical necessity for obtaining reasonable x-ray intensities at the detector. It would be unfortunate if the diffraction peaks were broadened in angle by the equatorial divergence of the incident beam, typically 1◦ . Fortunately, such broadening does not occur for the θ2θ goniometer of Fig. 1.15, which has “Bragg–Brentano” geometry. Bragg– Brentano geometry gives well-defined diffraction angles for finite slit widths and beam divergences, as shown with the aid of Figs. 1.16 and 1.17. In this goniometer, both detector and tube are on the circumference of a “goniometer circle” with the specimen in the center, as shown in Fig. 1.16. The beam divergence is indicated in Fig. 1.16 by the two ray paths from the tube to the detector. Although the two rays from the x-ray tube are incident at different angles on the specimen surface, if they pass through the receiving

1.3 The X-Ray Powder Diffractometer

27

slit they form the same angle, 180◦ −2θ, at the specimen. The Bragg–Brentano geometry illuminates a reasonable area of the specimen surface, and many ray paths have the same scattering angle. Good intensity and good instrument resolution are both achieved for powder samples. goniometer circle

focusing circle

specimen

tube

detector

180° < 2e

Fig. 1.16. Geometry of a Bragg– Brentano diffractometer. The two angles at the specimen are the same 180◦ – 2θ.

Further details of the focusing circle are shown in Fig. 1.17. It can be proved (see Problem 1.6) that the two ray paths from tube to detector make the same angle at the focusing circle (the angle 180◦ − 2θ of Fig. 1.16). It is also true that the dashed lines in Fig. 1.17, which bisect this angle, intersect at the bottom of the focusing circle, symmetrically between the tube and the detector. The dashed lines are normal to the diffracting planes. For strong diffraction, therefore, the optimal radius of curvature of the diffracting planes should be twice that of the focusing circle, and the sample surface should be curved along the focusing circle as shown in Fig. 1.17. Such crystals, known as “Johansson-cut” crystals, are specially prepared for x-ray optical devices, especially monochromators as discussed in Sects. 1.2.3 and 1.3.3. The geometry of Fig. 1.17 is the basis for the design of a high efficiency instrument known as a Seemann–Bohlin diffractometer. In this instrument a powder sample or thin film is spread over much of the circumference of the focusing circle. All divergent beams from the tube converge at the detector in Fig. 1.17 after diffraction by the 2θ angle. Different detector positions provide different 2θ angles. In the earliest days of the Debye–Scherrer technique, a stationary strip of film was placed around the goniometer circle, eliminating the need for precise mechanical movements. This concept has been extended to digital data acquisition with wide angle position-sensitive detectors (PSD), which intercept an arc of 120◦ or so (see chapter title image). Instead of detecting in sequence x-rays diffracted into angular intervals of about 0.1◦ , diffractions over the full

28

1. Diffraction and the X-Ray Powder Diffractometer symmetrically-cut sample surface radius = r crystal plane radius = 2r e e

2r

r asymmetrically-cut sample surface e

tube

e

detector

Fig. 1.17. Geometry of the focusing circle.

120◦ angle are detected simultaneously by the PSD. The obvious advantage of these PSD diffractometers is their high rate of data acquisition, which may be hundreds of times greater than conventional powder diffractometers with goniometer movements. 1.3.3 Monochromators, Filters, Mirrors Monochromatization of x-rays is best performed by Bragg diffraction from single crystals. A good monochromator can be built with a Johansson crystal (shown on its focusing circle in Fig. 1.17), together with slits located at the positions of the “tube” and the “detector.” This design makes efficient use of the divergent x-rays leaving the x-ray tube. The monochromated xrays form a non-parallel, convergent beam, however, and a non-parallel beam can be a disadvantage for some applications. A more parallel monochromatic beam can be produced with an “asymmetrically cut” curved single crystal. The asymmetrically cut crystal has its crystal planes aligned with those shown at the top of Fig. 1.17, but its surface is cut asymmetrically with respect to the diffracting planes, as shown on the right side of Fig. 1.17. The asymmetrically-cut crystal intercepts a broad range of incident angles. Its surface is foreshortened as seen from the detector, however, so its diffracted beam is less convergent. Compression of the beam divergence by a factor of 10 is possible with such an asymmetrically-cut crystal.

1.3 The X-Ray Powder Diffractometer

29

Installing a monochromator in the diffracted beam at the position of the detector14 in Fig. 1.15 can improve the signal-to-background ratio of the diffraction pattern. Diffractions from incident bremsstrahlung and other contamination radiations from the x-ray tube are no longer detected, because these radiations have the wrong wavelength to pass through the diffracted beam monochromator. Likewise, there is no detection of fluorescence x-rays emitted by the sample when excited by the incident beam. Sample fluorescence is usually emitted in all directions in front of the specimen, contributing a broad background to the measured diffraction pattern. Sample fluorescence can cause a serious background problem when there are elements in the specimen having atomic numbers, Z, that are less than the atomic number of the anode material by 2 to 5, or when the energetic bremsstrahlung from the x-ray tube is sufficiently intense (as can occur when the anode is a heavy element). Installing a monochromator in the incident beam, rather than the diffracted beam, can eliminate problems from diffracted bremsstrahlung and other contamination radiations, but an incident beam monochromator cannot prevent the detection of fluorescence from the specimen. Along the incident beam, it is sometimes useful to install a filter, typically a thin15 foil of absorbing material. This filter may be useful for suppressing the Kβ x-rays from the tube. If the foil is made from an element with an atomic number 1 less than that of the anode, the more energetic Kβ x-rays are attenuated strongly because they cause the foil to fluoresce. The desired Kα radiation does not induce fluorescence and is attenuated less. Finally, it should be noted that a detector with high energy resolution may not necessarily require a monochromator or filter, since discrimination of unwanted radiations can be accomplished electronically. Nevertheless, reducing the flux of unwanted radiations may improve the performance of the detector, especially at high count rates. The focusing of x-rays by curved mirrors was proposed by Kirkpatrick and Baez in 1948, but “K-B mirrors” have become important recently owing to improved fabrication methods and brighter x-ray sources. The essential idea is that the index of refraction of x-rays in most materials is slightly less than 1, typically about 0.99999. If the incident angle of an x-ray from vacuum to the material is less than a critical angle, total reflection will occur. These critical angles are small, of order 1◦ , so the x-ray beam makes only a glancing angle to the surface of the mirror. This sets stringent requirements over a substantial length of the surface of the mirror. For x-ray beams of narrow divergence and small diameters, as are typical of synchrotron undulator beams, curved K-B mirrors are practical for focusing the beam. Often two pairs of mirrors are 14

15

More precisely, the point labeled “tube” in Fig. 1.17 is located at the center of the “receiving slit” of Fig. 1.15 (and the drawing of Fig. 1.17 is rotated 90◦ clockwise). The thickness of a filter can be calculated with the method of Sect. 3.2.3.

30

1. Diffraction and the X-Ray Powder Diffractometer

used, one for focusing horizontally and the other vertically, producing a spot of a micron or so at the focal point.

1.4 X-Ray Detectors for XRD and TEM 1.4.1 Detector Principles An x-ray detector generates a pulse of current when it absorbs an x-ray. Several criteria are useful for characterizing its performance. First, the ideal detector should produce an output pulse for every incident x-ray. The fraction of photons that produce pulses is the “quantum efficiency” of the detector, QE. On the other hand, the detector and its electronics should not generate false pulses, or noise pulses. A “detective quantum efficiency” (DQE) combines the effects of quantum efficiency with signal-to-noise ratio (SN R) as a measure of how long different detectors (of the same geometry) must count to acquire data of the same statistical quality. The DQE is defined as square of the ratio of the SN R of the actual detector to the SN R of an ideal detector (where the SN R originates only with counting statistics): 2  SN Ractual , (1.37) DQE ≡ SN Rideal assuming the counting times for the actual and ideal detectors are equal.16 Second, the detector should produce a pulse of current having a net charge proportional to the energy of the x-ray photon. When detecting photons of the same energy, the voltage pulses from the electronics should all have the same height, or at least the distribution of pulse heights should be narrow. The width of this distribution for monochromatic x-rays is known as the detector energy resolution, usually expressed as a percentage of the x-ray energy. When acquiring a spectrum of characteristic x-rays, as in energy-dispersive spectrometry (EDS) in a TEM, energy resolution is a central concern. Energy resolution is less critical for x-ray diffractometry, but is still desirable because energy resolution allows the subsequent electronics to better discriminate against noise and unwanted radiations. Third, the amplitude of the detector pulses should remain steady with time, and should not vary with the incident x-ray flux. If the amplitude of the output pulses decreases at high count rates, the energy spectrum is blurred. There is also an undesirable “dead time” after the detection of a photon before the detector is able to detect a second one. This dead time 16

Suppose a detector does not generate noise of its own, but its QE = 1/2. For the same x-ray flux as an ideal detector, this detector would have half the signal and half the noise, but SN Ractual /SN R pideal is not 1.0. With half the countrate, counting statistics reduce this ratio to 1/2. The DQE of (1.37) would then be 1/2, so DQE = QE for detectors that do not generate false counts.

1.4 X-Ray Detectors for XRD and TEM

31

should be short. At high count rates, dead time can cause measured count rates to be sub-linear with the actual x-ray flux. (At extremely high fluxes, the count rate of some detectors can even fall to zero.) Finally, for EDS spectrometry in a TEM, it is important to maximize the solid angle subtended by the detector from the specimen. Table 1.3. Features of x-ray detectors Detector

Resolution at 10 keV Count Rate

Comments

gas-filled proportional scintillator Si[Li] intrinsic Ge silicon drift wavelength dispersive calorimetric avalanche photodiode

Fair (15 %)

< 30 kHz

robust

Poor (40 %) Good (2 %) Good (2 %) Good (2 %) Excellent (0.1 %)

Good ∼ 100 kHz Poor < 10 kHz < 30 kHz 200 kHz Good ∼ 100 kHz

Excellent (0.1 %) Fair (20 %)

robust liquid nitrogen liquid nitrogen –50◦ C mechanically delicate narrow acceptance Poor < 10 kHz research stage Excellent > 10 MHz electrically delicate

Some characteristics of x-ray detectors are summarized in Table 1.3. All can have high quantum efficiency, depending on the x-ray energy and the detector material or geometry. The gas-filled proportional counter is the oldest and simplest. The gas in this detector is ionized when it absorbs the x-ray energy. The electrons are attracted to the anode wire, which is biased at a high positive voltage. In the strong electric field near the anode wire, these electrons build up enough kinetic energy in a mean free path so they ionize additional gas atoms, and more electrons are created in this process of “gas gain.” The gas-filled proportional counter is inexpensive and has modest energy resolution, but its gas gain decreases with count rate. A scintillator is a piece of material, such as NaI made optically active by doping with Tl, that makes a brief flash of light when it absorbs an x-ray. The light is conducted to a photomultiplier tube, whose photocathode emits electrons when illuminated. The electron pulse is amplified further in the photomultiplier tube. Scintillation detectors are usable to very high count rates, but have poor energy resolution at typical x-ray energies. If energy resolution is not important, or if energy resolution is provided by a monochromator preceding the detector, a scintillation detector is often the best buy for a conventional x-ray diffractometer. The thickness of the scintillator should be sufficient to provide for strong absorption of the incident photon, and this thickness can be calculated from the mass-absorption coefficients discussed

32

1. Diffraction and the X-Ray Powder Diffractometer

in Sect. 3.2.3. The required thickness of the active region of a solid state detector can be obtained in a similar way. Solid state detectors have good energy resolution. They are silicon or germanium diodes, operated with reverse bias. Electrical contacts to the semiconductor surfaces are typically provided by thin layers of gold. Adjacent to the two contacts are p-type and n-type semiconductor, but most of the detector element is undoped, termed an “intrinsic” semiconductor. Intrinsic detectors often use pure Ge. Commercial silicon typically has a residual p-impurity content that requires compensation with an n-type impurity. Lithium is typically used for this purpose, and such an intrinsic detector is called a Si[Li] detector. An intrinsic semiconductor has no impurity levels in its band gap, so there is little thermally-activated current in reverse bias, especially when the detector is cooled with liquid nitrogen. An incident x-ray causes the excitation of electrons from the valence band into the conduction band, with an average energy per pair that is somewhat greater than the energy of the band gap. The high voltage of the reverse bias causes the electrons and holes to drift to their appropriate electrodes, providing a pulse of current through the diode. The total number of charge carriers is two times the energy of the x-ray photon divided by the average energy of the electron–hole pair. The net charge conducted across the diode is typically a few thousand electrons for typical x-ray energies. If the creation of each electron-hole pair required exactly the same energy, there would be a precise relationship between the x-ray energy and the current pulse, so the detector would have superb energy resolution. There is a statistical distribution of the electron–hole creation energies, however, causing differences in the number of electron–hole pairs generated by identical x-rays. When monochromatic x-rays each generate thousands of electron–hole pairs, the energy resolution is typically about 2 %. The energy resolution of a solid state detector remains good so long as the count rate is not excessive, so there is no interaction between charge carriers generated by different x-rays. Solid state detectors cause some spectral distortions and artifacts. When the primary ionization event occurs in the inactive “dead” layer near the contacts to the diode, not all the charge is collected. This causes the appearance of a low-energy tail on a spectrum from monochromatic radiation. Finally, the silicon itself can be ionized, with a threshold of 1.74 keV. If a silicon atom deep within the diode is ionized, most of this energy is eventually converted into electron-hole pairs and this presents no problem. However, if a silicon atom near the edge of the detector is ionized, this 1.74 keV energy may escape from the detector. Secondary “escape peaks” therefore appear in the energy spectrum from a Si[Li] detector. These escape peaks are located consistently at energies 1.74 keV below the energies of the main peaks in the spectrum. A typical experimental configuration for a solid state detector is presented in Fig. 1.18. To minimize thermal noise from the diode and from the pream-

1.4 X-Ray Detectors for XRD and TEM

33

plifier electronics, and to prevent damage to Si[Li] detectors by diffusion of Li during reverse bias, the detector is typically cooled with liquid nitrogen. The detector must then be kept in vacuum to prevent ice and hydrocarbon condensation on its surfaces. A beryllium window typically provides vacuum isolation for the detector, and this window must have sufficient thickness to withstand a pressure differential of 1 atmosphere. Unfortunately the beryllium window, the gold layer on the semiconductor, and the inactive (“dead”) layer of silicon near the gold contacts all attenuate the incident x-rays. This attenuation is particularly significant for x-rays with energies below 1 keV. The beryllium window confines energy-dispersive spectrometry (EDS) to the identification of elements of atomic number Z = 11 (sodium) or larger. Even for “ultrathin window EDS,” where polymeric films are employed, or “windowless EDS” where the detector and specimen share the same vacuum space, it is typically impractical to detect elements lighter than boron (Z = 5). As discussed in Sect. 4.6.2, the fluorescence yield of x-rays becomes very small for the lightest elements – excited states in these atoms usually decay by Auger electron emission.

Be window

Au

Si[Li] detector element

cryostat cold volume output

preamp input

X-ray

H.V.

Collimator

Fig. 1.18. Experimental configuration for a solid state detector. The cold volume of the cryostat is typically near the temperature of liquid nitrogen.

A silicon drift detector (SDD) is a new type of solid-state x-ray detector that is starting to see widespread service in energy dispersive spectroscopy. The detector is shaped as a thin disk of perhaps 300 μm thickness and 1 cm diameter, with its electron collector in the center of a flat surface. On the surface around the electron collector is a pattern of ring-shaped anodes that control the potential inside the disk, guiding the electron drift to the central current collector. The drift time is predictable, and more than one electron bunch at a time can be in transit to the electron collector. The field effect transistor at the preamplifier input can be integrated into the detector itself, further reducing capacitance. Some advantages of the SDD over a Si(Li) detector is its large area, high count rate owing to low capacitance (sub-pf),

34

1. Diffraction and the X-Ray Powder Diffractometer

and the requirement for only modest cooling, typically provided by a Peltier cooling system. An x-ray spectrometer is an integral part of an analytical transmission electron microscope. The vast majority of x-ray spectrometers in analytical TEM use solid state detectors, usually Si[Li], positioned with a direct view of the specimen. The energy resolution of an EDS spectrometer can lead to challenges when several elements are present in a sample. When characteristic energies are close together, peaks from individual energies may not be resolved. Such overlaps are common for the L and M lines of medium- and high-Z elements, respectively. It is a task of the spectrometer software to help untangle spectra with multiple peak overlaps, usually by fitting the measured spectrum to patterns of peaks from each element. A new type of x-ray detector is based on the calorimetric detection of x-ray energy. A superconducting wire held near its transition temperature is highly sensitive to small temperature excursions, and can be used to detect the heat energy deposited by an individual x-ray. The heat can be measured with sufficient accuracy to provide x-ray energy resolutions of 0.1 %, significantly better than solid state detectors. The limit on the energy resolution of such detectors is thermal noise, which can be suppressed by operating at temperatures below 0.1 K. Cryostats and cooling systems based on adiabatic demagnetization have been developed that allow operation of high performance calorimetric detectors for tens of hours per cooldown. At present the thermal response times of the detectors are somewhat long, limiting their maximum count rate. To shorten thermal response times, smaller detector geometries are under development. 1.4.2 Position-Sensitive Detectors High performance x-ray diffractometers have been built around positionsensitive detectors (PSD). Since a PSD detects x-rays at many angles simultaneously, it can minimize data acquisition times and improve counting statistics. There are many designs for PSDs, and all have unique features. Several types of PSDs are gas-filled counters. One design uses a resistive wire as an anode, and a preamplifier at each end of the anode wire. The position of the x-ray is determined by the difference in charge detected by the two preamplifiers. An x-ray that ionizes the gas at one end of the detector tube produces a larger pulse in the preamplifier connected to that end. Such detectors require that the resistivity of the anode wire be steady with time, and not affected by contamination from the detector gas, for example. A second type of gas-filled PSD makes use of time delays along electrical transmission lines. For example, the cathode surface may be subdivided into hundreds of independent plates, each connected to its neighbor by a small inductor and capacitor. Preamplifiers are located at each end of the cathode chain, and the time difference between their two signals is measured. The position of the x-ray is closer to the preamplifier that produces the earlier

1.4 X-Ray Detectors for XRD and TEM

35

pulse. The same time-delay concept is used in a design for a two-dimensional area detector. This detector uses crossed grids of anode wires, with wires running in the x-direction providing information on the y-coordinate of the event, and the wires running in the y-direction providing information on the x-coordinate. The individual wires in each anode grid are connected to their neighbors by an inductor and capacitor, providing a delay time along the grid. The electronics for these area detectors are complex and must be considered an integral part of the detector system. Gas-filled delay-line counters have low noise, but usually no energy resolution. Another type of area detector is based on a video camera system using charge-coupled-devices (CCDs). The CCD chips themselves serve as excellent, small x-ray detectors (assuming their active regions are sufficiently thick to stop the x-rays). They suffer radiation damage after a large number of detected x-rays, but are well-suited for low flux experiments. To reduce radiation damage, a thin scintillator can be used to stop the x-rays. The light from the scintillator enters the CCD by direct contact, or by focusing the light from a large scintillator onto the CCD through a lens or a tapered bundle of optical fibers. At low x-ray fluxes, thermal and readout noise may be a consideration for CCD area detectors, but a CCD area detector could have energy resolution at low x-ray fluxes when individual events are identified. Developments in semiconductor processing technologies have made possible a number of new types of PSDs based on silicon diodes or diodes of other semiconductor materials such as CdTe. Typically an array of square diode detectors is arranged over the surface of a large semiconductor chip. Each diode requires its own preamplifier and pulse processor electronics, and these are typically provided by a customized analog integrated circuit. Further electronic integration can include a multichannel analyzer (see next section), but this is typically shared (multiplexed) by a number of diodes, and can limit the peak rate of data acquisition. Pixellated diode PSD systems can provide full digital output, such as an alert to a detected event, followed by the pixel identification number and a number proportional to the energy of the event. The large and competitive marketplace for medical x-ray imaging equipment is spawning a number of developments in area detectors. Imaging plates, for example, are relatively inexpensive, and are handled in much the same way as photographic film. The plates include a layer of long-persistence phosphor, BaFI with Eu ions, for example. The x-ray excites the Eu2+ to Eu3+ , which persists for a day or so. The locations of the Eu3+ (the locations of the x-ray detections) are found by transporting the imaging plate to a readout unit, where a He-Ne laser beam is rastered over the entire plate. The Eu3+ is identified by its photostimulated blue light. The imaging plates themselves are later erased and reused. Unlike photographic film, the signal from imaging plates is linear over 6 or more decades, and the sensitivity to low exposures of x-rays (and electrons) is excellent. Energy resolution is usually impractical.

36

1. Diffraction and the X-Ray Powder Diffractometer

Although PSDs usually provide enormous improvements in data acquisition times (factors of 103 are possible), they have some limitations. Besides their higher cost, they have a reputation for requiring skill to operate and maintain. Several newer designs are robust and convenient, however. Most gas-filled PSDs do not provide energy resolution. This can be a problem in the presence of strong sample fluorescence as discussed in Sect. 1.3.3. 1.4.3 Charge Sensitive Preamplifier Typical charge sensitive preamplifiers have input circuits like the one in Fig. 1.19, shown with a gas-filled proportional counter. The capacitor, C, integrates the negative charge collected on the anode wire, causing a quick rise in resistance across the field effect transistor. A small value of C allows for a large rise in voltage and good sensitivity. On the other hand, small stray capacitances between the detector and the preamplifier can have a detrimental effect on the detector signal, so interconnections between detector and preamplifier are kept as short as possible. The resistor, R, bleeds away the voltage across C with a much longer time constant. Typically, RC = (107 Ω)(10−11 F) = 10−4 sec. A preamplifier with higher performance for solid state detectors is constructed with the detector output sent directly into an FET operational amplifier (cf., Fig. 1.18). This op-amp is configured as an integrator by using a capacitor in its feedback loop. The discharge of this capacitor is provided by a fixed resistance across it, or by an active circuit that discharges the capacitor when the integrated voltage exceeds a setpoint. + 1500 V gas-filled proportional counter

+

R C

JFET VOUT

VOUT RC tx1

time

t x2

tx3

Fig. 1.19. Input circuit for a simple charge sensitive preamplifier, here operating with a gasfilled proportional counter. The timedependent voltage across the field effect transistor (FET) is indicated schematically after detection of an x-ray at times tx1 , tx2 , and tx3 .

1.4 X-Ray Detectors for XRD and TEM

37

1.4.4 Other Electronics A full system for x-ray detection and spectroscopy is shown in Fig. 1.20. Following the preamplifier is a main amplifier. Its primary purpose is to shape the pulses into a convenient waveform, such as a Gaussian function with a width of a few microseconds, all the while ensuring that the height of the pulse remains proportional to the charge collected on the capacitor of the preamplifier. An important function of the main amplifier is to compensate for the slow decay set by RC of the preamplifier. This exponential decay is quite predictable. The main amplifier compensates for this decay in a process called “pole-zero” cancellation, which provides a flat voltage baseline following each sharp Gaussian pulse. The main amplifier may not separate two pulses from the preamplifier that arrive closely in time, causing an artifact where the two closely-spaced pulses are shaped into one large pulse. These large pulses appear in an x-ray spectrum at the sum of the energies of the real peaks, and this artifact is called a “sum peak.” The fraction of sum peaks becomes larger at high counting rates.

I

interface

buffer

MCA

A/D conv.

SCA

detector

P-Z amp.

high voltage

preamplifier

SCA out E computer

Fig. 1.20. Full x-ray spectroscopy system. The interface unit allows the computer to download the spectra, and allows the computer to control the electronic units. For an analytical TEM, the SCA output may be directed to a STEM unit for elemental mapping. An x-ray diffractometer may have the SCA output sent to a simple counter, and the subsequent electronic units such as the MCA may be needed only for calibration and diagnostic work.

For x-ray diffractometry, there are many small pulses coming out of the main amplifier that are low amplitude noise, or unwanted pulses from undesired types of radiations such as sample fluorescence. It is the job of the single channel analyzer (SCA) or window discriminator, to set upper and lower thresholds to define the pulses of interest. The counts from the SCA are accumulated in a counter or in a memory bin assigned to a particular 2θ angle of the goniometer. A computer system is typically used to synchronize

38

1. Diffraction and the X-Ray Powder Diffractometer

the stepper motors in the goniometer with the memory bin used for data acquisition. Besides its data acquisition and control functions, the computer also is often used for data display, storage, processing, and transmission to other computers. For analytical TEM, a solid state detector is used for the collection of the full spectrum of x-ray energies. For most work in analytical TEM, an energy spectrum is acquired by sending the shaped pulses from the main amplifier into a multichannel analyzer, or MCA. In the MCA, the pulse is first converted into a digital number by a fast analog-to-digital converter. A single count is added to the content of the MCA memory address corresponding to that number. With time, a histogram is collected in the memory of the MCA, displayed as the number of counts versus memory address. With an energy calibration provided by a source of known monochromatic photons,17 the (linear) correspondence between the memory address and the photon energy can be determined. This histogram can then be displayed as an x-ray energy spectrum. When performing elemental mapping, the SCA registers counts from a selected x-ray energy, and this SCA output is an input signal to the STEM raster display in Fig. 2.1.

1.5 Experimental X-Ray Powder Diffraction Data 1.5.1 * Intensities of Powder Diffraction Peaks18 Which crystals contribute to the Bragg peaks in a powder diffraction pattern? If nature demanded that diffracting crystallites were in exact Bragg orientations, then a powder containing a finite number of crystals would have zero crystallites that diffract. We observe diffractions with monochromatic radiation, so evidently the crystals need not be oriented perfectly. This is especially true if they are small and have broadened diffractions. In this section we consider the numbers of crystallites that are oriented “adequately” for diffraction. Three types of crystal orientations are shown in Fig. 1.21. Assume that the crystal at the left is oriented perfectly. It diffracts strongly, but there are very few such crystals. The one in the middle is misoriented a little. It does not diffract so strongly, but there are more such crystals. There are even more crystallites with the large misorientation of the right cube, but these crystallites contribute little to the diffracted intensity because they are far from the Bragg orientation. In powder diffraction 17 18

Such as a radioisotope source or a known atomic fluorescence. Throughout this book, an asterisk (*) in a section heading denotes a more specialized topic. For example, the results of the present section, (1.54) and (1.55), are important, but on a first reading the reader may choose to avoid the details of their derivation. Incidentally, a section heading with a double dagger (‡) indicates a higher level of mathematics.

1.5 Experimental X-Ray Powder Diffraction Data

39

we measure the number of crystallites that are within some small range of misorientations. The intensity of a powder pattern diffraction peak is controlled in part by geometrical aspects of the diffractometer and the sample. It is not our intent to calculate the absolute intensity of a powder diffraction peak because most x-ray diffractometry studies on materials are performed in a comparative way where absolute intensity is unimportant. It is instead important to know the systematic trend of how the intensities of the different (hkl) diffractions depend on the 2θ angle of the diffractometer. We consider individual effects as intensity correction factors, and present two examples of total correction factors in (1.54) and (1.55).

Fig. 1.21. Various orientations of crystallites with respect to the best orientation for diffraction.

Normals of Diffracting Planes Consider Fig. 1.22, where the incident and outgoing arrows make angles, θ, with the plane of the specimen. For a specific angle, θ, we want to know how many crystallites are oriented within an angular range suitable for diffracting into the diffraction cones of Fig. 1.8. The normals to these crystallites point to the ring drawn around the sphere in Fig. 1.22. Assuming the orientations of the crystallites are isotropic, we see that the number of these crystallites, and the diffracted intensity, is proportional to: I1 ∝ sin(90 − θ) = cosθ .

(1.38)

Slit Width Not all of the x-rays diffracted into the ring in Fig. 1.22 are seen by the detector. The detector has a receiving slit with a limited horizontal width, as shown in Fig. 1.23. Owing to the horizontal width of its receiving slit, the detector collects a larger fraction of x-rays from diffraction cones of smaller 2θ. The fraction detected is proportional to: I2 ∝

1 . sin(2θ)

(1.39)

40

1. Diffraction and the X-Ray Powder Diffractometer

90 1). The frequency of the light remains constant, however, so the wavelengths of light in the two media must be related as: n2 λ1 = , λ2 n1

(2.14)

and the wavelength is shorter in glass. The electromagnetic field of the light wave in the air drives the fields in the adjacent glass, so the spacing between wave crests must be the same on both sides of the air/glass interface. The matching of wave crests is shown in Fig. 2.29 – note that to accommodate a shorter wavelength in the glass, 2 , bends towards the surface normal. The the direction of the light ray, k separations, l, between wave crests along both sides of the air/glass interface are equal, so the angles, θ1 and θ2 , are related to the wavelengths as: λ1 = l sinθ1 , λ2 = l sinθ2 .

(2.15) (2.16)

By substituting (2.15) and (2.16) into (2.14), we obtain Snell’s law: n1 sinθ1 = n2 sinθ2 .

(2.17)

Fig. 2.29. Matching of wave crests at the interface between two media, with n1 < n2 . The wave crests are drawn as sets of parallel lines that match at the vertical interface.

2.5.2 Lenses and Rays Glass lenses focus light by means of curved surfaces. To make a lens that functions as shown in Fig. 2.6 or 2.7, each light ray reaching the lens must bend by an angle that depends on its distance from the optic axis. Consider the symmetrical arrangement in Fig. 2.30 where the small object and image are at equal distances from the center of the lens. By the symmetry of this arrangement, when the rays are inside the lens they must be traveling parallel

2.5 Glass Lenses

93

to the optic axis. For one particular off-axis ray, we can achieve the correct bend with the flat interface of Fig. 2.29 if we tilt the interface with respect to the optic axis. A flat interface provides only one angle of tilt, however, so it cannot provide the correct bend for rays at all angles. A ray that is inclined further from the optic axis requires a larger angle of bend, so it must reach a part of the lens where the surface normal is tilted more steeply from the optic axis. We need a curved lens so that the angle of tilt of the lens surface is larger as we move away from the center of the lens. For focusing, a glass lens will have a convex curvature, and by the symmetry of our problem the back side of the lens must have the opposite curvature. The ray traveling along the optic axis should not be bent, so it should encounter surfaces with normals parallel to the optic axis. normals to lens surface

Fig. 2.30. Ray paths through a symmetrical double convex lens with symmetrically-positioned object and image.

To analyze the bending of light at curved surfaces, we can work with either the rays, or the phases of the wave crests. We first calculate the shape of a lens surface using “ray tracing” and Snell’s law. For simplicity, we consider the symmetrical case of a double convex lens with object and image planes equidistant from the lens as in Fig. 2.30. This case is simple because we know that by symmetry, off-axis rays travel parallel to the optic axis when they are passing through the lens. In Fig. 2.31, the shape of the lens surface is given by the unknown function x(R). For reference we set x(0) = 0 (at the center of the lens), and for a convex lens we know that x(R) must increase with R, where R is the radial position on the lens. At R, the normal of the lens surface makes the angle φ with respect to the optic axis. By the symmetry of our problem, upon entering the lens the ray must bend by the angle θ so that it becomes parallel to the optic axis. Looking at the enlargement in Fig. 2.31, we apply Snell’s law: n1 sin(θ + φ) = n2 sinφ ,

(2.18)

where the index of refraction inside the lens is n2 , and outside it is n1 . Approximately, for small angles, θ, and thin lenses (small φ), (2.18) is: n1 (θ + φ) ≃ n2 φ .

Now from Fig. 2.31 we see that the angles θ and φ are

(2.19)

94

2. The TEM and its Optics lens surface dx

lens normal

lens surface

e+q ray

e ray q e nlar gem e

x=0

dR q

nt

q

x

e ray

ray

R

e 2f 2

Re

2

(2f) +Re

o Fig. 2.31. Geometry for the design of a lens surface, x(R), using ray optics.

R , (2.20) 2f dx φ= . (2.21) dR We substitute (2.20) and (2.21) into the approximate form of Snell’s law of (2.19):   R dx dx + , (2.22) n1 ≃ n2 2f dR dR n1 dx R= (n2 − n1 ) , (2.23) 2f dR

x

R n1 ′ ′ (2.24) R dR = (n2 − n1 ) dx′ , 2f θ=

0

0

n1 R 2 . x= n2 − n1 4f

(2.25)

Since x ∝ R2 , (2.25) predicts a parabolic shape for our thin lens. A parabolic shape is indistinguishable from a spherical shape when the lens is as thin as we have assumed. Equation (2.25) also shows that the thickness of the lens is inversely proportional to its focal length, so strongly-focusing lenses are thicker. Our lens has a smaller thickness when the glass has a large index of refraction, so the difference n2 − n1 is large. Assuming that n1 = 1, as is approximately the case for air, our convex lens must be made from material

2.5 Glass Lenses

95

with an index of refraction greater than 1. It is interesting, however, that for n2 < n1 , we would focus with a concave lens. 2.5.3 Lenses and Phase Shifts Instead of the ray-tracing approach of the previous Sect. 2.5.2, the analysis of phase shifts of wave crests provides an alternative way to design a lens. The light traveling from object to image is redrawn in Fig. 2.32 in terms of its wave crests. The lens turns the diverging wave from the left into a converging wave on the right by altering its phase. Relative to the wave traveling straight down the optic axis, those rays more inclined to the optic axis can be advanced in phase by the lens, as indicated by the arrows in the top part of Fig. 2.32. Alternatively, converging wave crests on the right can be achieved by retarding the phase of the waves closer to the optic axis (this is the actual case for glass lenses in air). The precision of the lens in providing these phase shifts determines its accuracy in focusing.

Fig. 2.32. To perform focusing, a lens must provide differential phase shifts (arrows) for rays at different distances from the optic axis. Top: focusing by advancing the phase of offaxis rays. Bottom: focusing by delaying the phase of on-axis rays.

With Fig. 2.32 we can see almost immediately that a focusing glass lens must have spherical surfaces. The bottom construction in Fig. 2.32 shows that the required phase delay through the glass must be larger in the center of the lens. Since the phase delay is proportional to the thickness of the glass, the transformation of an outgoing spherical wavefront into a converging spherical wavefront requires a spherical lens. We now use the method of phase shifts to calculate the required thickness, 2τ , at the center of our spherical lens in the symmetrical case of Fig. 2.31. We seek a vertically-flat wavefront at the center of the lens, i.e., for all ray

96

2. The TEM and its Optics

paths to the center of the lens (the vertical dashed line in Fig. 2.32), the total number of wave periods must be the same. Since there are more wave periods per unit length in the glass, by thickening the glass we add wave periods to the on-axis ray path, compensating for its shorter path compared to off-axis rays. The on-axis ray path has two segments to the center of the lens, a long segment outside the lens of length 2f − τ , and a short segment of length τ inside the lens, giving a total number of wave periods, φon : n2 n1 +τ . (2.26) φon = (2f − τ ) λ λ Since we seek only the total thickness of our lens, which we know from Fig. 2.32 to have spherical surfaces, we consider only one off-axis ray that reaches the lens at Re , the outermost edge of the lens where it is infinitesimally thin. The length of this ray path is obtained from a triangle in Fig. 2.31 having edge lengths 2f and Re , with a right angle at the exact center in the lens. The number of wave periods along this off-axis path to the periphery of the lens is φoff : n1 . (2.27) φoff = (2f )2 + Re2 λ For focusing as in Fig. 2.32 we demand that the number of wave periods to the center of the lens is the same both along the optic axis and at the edge of the lens at Re , so φon = φoff , (2f − τ ) n1 + τ n2 = (2f )2 + Re2 n1 .

(2.28) (2.29)

It is expedient to  approximate 2f > Re , so the radical in (2.29) becomes 2f 1 + Re2 /(8f 2 ) , giving   Re2 (2.30) τ (n2 − n1 ) ≃ n1 2f 1 + 2 − 2f n1 , 8f n1 Re2 . (2.31) τ = n2 − n1 4f

This is the half-thickness of the lens at its center, and not the entire function x(R) of (2.25). Nevertheless, (2.31) and (2.25) predict the same lens shape because at the edge of the lens, τ = x(Re ), and we had already used Fig. 2.32 to show that the surface of the lens is spherical. The phase shift approach to lens design is also consistent with Fermat’s principle. This minimum principle states that between two points, a ray takes the path requiring the least time. From object point to image point in the idealized Fig. 2.32, the wave fronts make an instantaneous jump along the arrows, so all paths from point to point require the same time. More realistically, in our design of a real lens using phase shifts, we ensured an equal number of wave periods for all rays through the lens. Since the wave frequency is a constant, this ensures equal transit times for all rays. From object point to image point, all ray paths through the lens require the same time.

2.6 Magnetic Lenses

97

2.6 Magnetic Lenses 2.6.1 Focusing

Br (r & 0) –z +z

r

e r

0

Br(z > 0)

0

z coordinates

Bz (any r in solenoid)

–z

0

+z

Fig. 2.33. Center: Magnetic field in and around a short solenoid. Also shown are the Bz and Br components. The cylindrical coordinate system is on the right.

Magnetic lenses in transmission electron microscopes are short solenoids. Some features of the magnetic field in and near the solenoid are indicated in Fig. 2.33. The components of the magnetic field along the coordinate di z , denoted Br , Bθ , Bz , are all quite different. By cylindrical , θ, rections: r symmetry, Bθ = 0, and need not be considered further, but Br (r, z) = 0 and Bz (r, z) = 0. The exact shape of the magnetic field is difficult to calculate because magnetic lenses have ferromagnetic pole pieces, whose properties are nonlinear with lens current, and cannot be modeled well. Nevertheless, the most important features of the magnetic field can be deduced from the symmetry of the solenoid: • Br vanishes in the plane z = 0 (in Fig. 2.33, z = 0 at the center of the solenoid).

98

2. The TEM and its Optics

• Br is antisymmetric under reflection across the plane z = 0. • Br reaches its peak value at some distance away from the solenoid. • For a given value of z, near the optic axis Br increases with r, since Br = 0 at r = 0. • Bz is largest at z = 0, and decreases monotonically with increasing |z| in a manner that is sometimes assumed to be a Gaussian function. • At large values of |z| and moderate r, Br > Bz . The focusing action of a magnetic lens is understood by analyzing the Lorentz force on the moving electron: F = −e v×B .

(2.32)

Fz = +evθ Br ,

(2.33)

Fθ = −e (vz Br − Bz vr ) ,

(2.34)

 + vz z , and In terms of the components of the electron velocity, v = vr  r + vθ θ  , the components of the force on the  + Bθ θ + Bz z magnetic field, B = Br r electron, F , are (given that Bθ = 0): Fr = −evθ Bz .

(2.35)

To understand the electron trajectory under these forces, we employ Fig. 2.34, which must be understood as a 3-dimensional image with the following perspective. As the optic axis runs from left to right in the plane of the paper, it also rises slightly above the plane of the paper. Four planes perpendicular to the optic axis are drawn obliquely. The magnetic lens in the figure is assumed to be symmetric about the center of the optic axis (analogous to Fig. 2.30).

Fig. 2.34. Electron trajectory through a magnetic lens. See text for details.

2.6 Magnetic Lenses

99

We trace the path of an electron that leaves a point on the leftmost plane (object plane), and comes to a focus at a point in the center of the rightmost plane (image plane). Initially the electron travels at an angle to the optic axis, but travels directly above it. Before the electron reaches the solenoid, it senses the pre-field, which is almost entirely radial, Br ≫ Bz . Because our Bz points to the right, on the left of the lens center Br points towards the optic axis (cf., Fig. 2.33). The cross product of the velocity (actually vz ) and Br gives a force, Fθ (Fig. 2.34), that points out of the plane of the paper. Such a force gives a new velocity component vθ , which causes the electron trajectory to spiral upwards out of the plane of the paper. The electron is still moving away from the optic axis, however, so no focusing has yet taken place. The new vθ component of the velocity enables focusing. As the electron rotates about the optic axis with velocity vθ , it enters the region where Bz  and Bz z  provides the force towards the is strong. The cross product of vθ θ optic axis, Fr , that is needed for focusing. In our assumed symmetrical optical arrangement, the velocity of the electron away from the optic axis, vr , becomes zero exactly at the center of the magnetic lens. At this point the electron is not traveling parallel to the optic axis, but is moving in a helix with velocity components vθ and vz . As the electron passes to the right of center of the magnetic lens, the velocity vθ causes further focusing. At the same time, Br has changed its sign, and vθ , which has reached its maximum at the center of the lens, begins to decrease. By symmetry, our lens will reduce vθ to zero by the time the electron exits the post-field of the lens, so the spiraling motion stops. The electron now moves directly towards the optic axis, and comes to a focus at the rightmost plane of Fig. 2.34. The focal length of the lens decreases as the lens current increases because vθ is greater, and so is Bz . 2.6.2 Image Rotation Although the electron comes to a final focus by traveling in a straight line, this line is no longer directly above the optic axis, as was the initial trajectory. The path has been rotated out of the plane of the paper by the angle θ, which may be large. An important consequence of this rotation is that the image itself is rotated by the angle θ. This rotation increases with the magnetic field in the lens, which increases with the current through the lens coil (not necessarily proportionally, however, owing to the characteristics of the ferromagnetic pole piece material). The following approximate formula (with units E[eV], θ[radians, R]), Bz [G]) is useful for estimating the angle of rotation of the image:

0.15 Bz dz . (2.36) θ[R] = √ E axis

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2. The TEM and its Optics

For a typical case with 100 keV electrons passing through a lens with a 10 kG field and a length of 0.5 cm: θ≃

0.15 × 104 × 0.5 ≃ 2.5 R . 3 × 102

(2.37)

The square root of E in (2.36) is interesting. Larger Lorentz forces are exerted on higher velocity electrons, but they spend proportionately less time in the lens. The angle that an electron is displaced depends quadratically on the time that it is subjected to the force. Consequently, an electron with twice the velocity is bent half as much by a magnetic lens. High voltage electron microscopes require powerful magnetic lenses and long columns. The current in the intermediate lens is decreased when the microscope is switched from image mode to diffraction mode. Consequently there is a difference in the rotation of the observed diffraction pattern and the image. Knowing this “image rotation” at different magnifications is crucial when using SAD patterns to relate crystallographic directions to directions in the image, as in contrast analyses of defects. It is a traditional laboratory exercise to measure the image rotation of the microscope by examining MoO3 crystals, which are orthorhombic and elongated along their 001 directions. One such crystal is shown in Fig. 2.35. To show the image rotation with respect to the SAD pattern, the images in Fig. 2.35 are double exposures of a diffraction pattern plus an image. The diffraction pattern is in the same orientation in the six images, but as the magnification is increased, we see that the image of the elongated MoO3 particle rotates clockwise. The rectilinear SAD pattern does not line up with the particle image because of the difference in image rotation. With measurements like these, the image rotation can be calibrated (see Appendix A.11.2.C). There is one more consideration in relating a SAD pattern to an image, best seen by reference to Figs. 2.11 and 2.16. There is an extra cross-over (a point where the rays cross the optic axis) when making an image than when making a diffraction pattern. Consequently, the diffraction pattern is inverted (through its center) with respect to the image. When relating photographic negatives of the SAD pattern to their corresponding images, one of the negatives should be rotated by an angle of 180◦ before correcting for the image rotation. In some magnification ranges, modern microscopes may provide an even number of additional crossovers in image mode, so this rotation of 180◦ may be unnecessary. Other microscopes may use electron optical designs where one magnetic lens may compensate for the image rotations of other lenses. Be sure you know the characteristics of your microscope before relating images to SAD patterns. 2.6.3 Pole Piece Gap To minimize the spherical aberration of the objective lens, there is only a small gap between the pole pieces of this lens. This leaves little space for the

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Fig. 2.35. Double exposures of SAD and BF images of a MoO3 crystal (made by burning a Mo wire in an oxy-acetylene torch and collecting some soot on a holey carbon TEM grid). The diffraction patterns are unchanged between the six images. The rotation angle, φ, of the image with respect to the diffraction pattern was measured to be: φ = 15◦ at 10 kX, φ = 27◦ at 20 kX, φ = 38◦ at 30 kX, φ = 48◦ at 40 kX, φ = 58◦ at 50 kX, φ = 69◦ at 60 kX. The microscope was an older Siemens 1A, which had no changes in lens modes for different ranges of magnifications. Unlike most modern instruments, it therefore had no abrupt inversions of the image with increasing magnification.

specimen, which is located between the pole pieces of the objective lens, and it is challenge to design a good specimen stage (the specimen holder and the mechanism to move it). With a “side-entry stage” as shown in Fig. 2.4, the specimen is held at the end of a long, non-magnetic rod that enters into the gap between two pole pieces of the objective lens. At the end of the rod is a hard jewel bearing that makes contact with a matched surface in the stage assembly, giving a point of rotation for the sample rod. A side-entry stage allows adjustment of the center of tilt of the specimen to be “eucentric,” so no horizontal translations occur during tilting. For a side-entry stage, the gap in the pole pieces is usually large, making it easier to place EDS detectors or electron detectors near the specimen for analytical work. Specimen drift has been a challenge for side-entry stage design, but the stability of side-entry holders is now so good that they are used in most modern microscopes.

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2.7 Lens Aberrations and Other Defects Important performance criteria for a TEM are the smallest spatial feature that can be resolved in a specimen, or the smallest focused electron beam that can be formed on a specimen. It turns out that these performance criteria are determined largely by the performance of the objective lens of the microscope. The objective lens of a TEM, like all magnetic lenses, has aberrations that impair its performance. To understand microscope resolution we must first understand lens aberrations. Section 2.8 then shows how lens aberrations and other defects determine the performance of a TEM. 2.7.1 Spherical Aberration Spherical aberration changes the focus of off-axis rays. The further the ray deviates from the optic axis, the greater its error in focal length. All magnetic lenses have a spherical aberration coefficient that is positive; those rays furthest from the optic axis are focused most strongly. For reference we define the true image plane (sometimes called the “Gaussian image plane”) as the image plane for paraxial8 imaging conditions. The angle of illumination into the lens is defined as the aperture angle, α, (cf., Fig. 2.26), and in paraxial imaging conditions α is very small. Spherical aberration causes an enlargement of the image of a point P to a distance QQ′ in the Gaussian image plane in Fig. 2.36. The minimum enlargement of point P occurs in front of QQ′ and is termed the “disk of least confusion.” For a magnetic lens, the diameter, d′s , of the disk of least confusion caused by spherical aberration is: 3

d′s = 0.5M CS (αOA ) ,

(2.38)

where Cs is the spherical aberration coefficient (usually 1–2 mm), αOA is the aperture angle of the objective lens (see Fig. 2.36), and M is magnification.9 At the specimen itself, the corresponding diameter of uncertainty, ds , is: 3

ds = 0.5CS (αOA ) .

(2.39)

The positive value of Cs is the key problem in phase-contrast (highresolution) transmission electron microscopy.10 This is discussed at length in Chapter 10, but it is now possible to see with Fig. 2.36 why the objective lens is defocused slightly when making HRTEM images. Defocus moves the point O closer to the lens, so the ray crossings near point P move to the right in Fig. 2.36. The blur on the image plane becomes smaller than QQ′ . This 8 9

10

In a paraxial imaging condition, the rays are near the optic axis, making only small angles with respect to the optic axis. For this third-order spherical aberration, the disk of least confusion is smaller than QQ′ at the image plane by a factor of 4. Do not confuse the radius at the image plane, M CS (αOA )3 , with the circle of least confusion. Correctors for spherical aberration, discussed in Section 11.6, have changed this situation, but at a price.

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103

defocus is a compromise for achieving the best performance at the largest practical diffraction vector, Δk (corresponding to a large αOA ). In HRTEM, the spherical aberration of the objective lens is typically analyzed in terms of errors in phase shifts. The top drawing of Fig. 2.32 shows the phase shifts needed for perfect focusing. Compared to the phase shifts in this drawing, a lens with positive spherical aberration advances excessively the phases of the off-axis electron waves.

Fig. 2.36. Lens with positive spherical aberration, showing a closer focus for offaxis rays.

2.7.2 Chromatic Aberration The index of refraction of glass is somewhat dependent on the wavelength of the light. The focal length of a simple lens therefore depends on wavelength, so a sharp image cannot be made with white light and simple lenses. Magnetic lenses also suffer from chromatic aberration. Electrons with different energies, when entering a lens along the same path, come to different focal points. The spread in focal lengths is proportional to the spread in energy of the electrons. There are two main sources of this energy distribution. First, the electron gun does not produce monochromatic electrons. Typically, less than ±1 eV of energy spread can be attributed to irregularities of the high voltage supply. Electrons emitted thermionically from a hot filament have a Maxwellian distribution of velocities that provides an energy distribution with a broad tail extending to about 1 eV. With high beam currents, the electron-electron interactions at the condenser crossover cause an energy spread of ±1 eV through a phenomenon known as the Boersch effect. The specimen itself is the other important cause of an energy spread of electrons. Inelastic scatterings of the high-energy electrons by plasmon excitations are a common way for electrons to lose 10–20 eV. Thin specimens minimize the blurring of TEM images caused by chromatic aberration. The disk of least confusion for chromatic aberration corresponds to a diameter at the specimen, dc : dc =

ΔE Cc αOA , E

(2.40)

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where ΔE/E is the fractional variation in electron beam voltage, Cc is the chromatic aberration coefficient (approximately 1 mm), and αOA is the aperture angle of the objective lens. 2.7.3 Diffraction An aperture truncates the k-space components of an image, discussed in the context of HRTEM with (2.11). In optics this effect is explained as “diffraction” from the edge of an aperture. It contributes a disk of confusion of diameter corresponding to a distance at the specimen, dd : 0.61λ , (2.41) αOA where λ is the electron wavelength and αOA is the aperture angle of the objective lens. Equation (2.41) is the classic Rayleigh criterion for resolution in light optics. In essence, (2.41) states that when the intensity between two point (Gaussian) sources of light reaches 0.81 of the maximum intensity of the sources, they can no longer be resolved. This effect is demonstrated in the series of images in Fig. 2.37. In the top image (a), the aperture opening of an imaging lens is so small that the two point sources on the right are not resolved by the Rayleigh criterion. Increasing the size of the aperture opening as in (b) and (c) reduces the diffraction effect, improving the resolution. dd =

Fig. 2.37. Effect of small (a), medium (b) and large (c) objective apertures on the resolution of point sources of light. After [2.11].

2.7.4 Astigmatism Astigmatism occurs when a lens does not have perfect cylindrical symmetry. The focusing strength of the lens then varies with angle θ (see Fig. 2.38),

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105

again leading to a spread of focus and a disk of least confusion. Two lenses of the TEM require routine corrections for astigmatism. The first condenser lens, C1, must be “stigmated” to produce a circular incident beam on the specimen. Similarly, the astigmatism of the objective lens blurs the image and degrades resolution, so it is necessary to adjust the objective lens stigmators when making high-resolution images. In Figs. 2.38–2.40 referring to astigmatism, the right hand side of the optic axis tilts slightly downwards and up from the plane of the paper. The lens is drawn as a flat disk that is seen obliquely, and so appears as an ellipse. A cylindrical coordinate system with coordinates {r, θ, z} is used, with z along the optic axis. The key point about the astigmatic lens in Fig. 2.38 is that its focal length varies with the angle θ. For paraxial rays entering the lens at the same distance from the optic axis (same r, but different θ), this lens has a weaker focus for the top and bottom rays than for the rays above and below the plane of the paper. The type of astigmatism important for magnetic lenses can be described with a simple model. In Fig. 2.39, the astigmatic lens of Fig. 2.38 is modeled as a perfect lens or radial symmetry, plus a second lens with curvature in only one direction.

Fig. 2.38. Different ray paths through an astigmatic lens.

Fig. 2.39. Model of the astigmatic lens of Fig. 2.38.

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Unlike the spherical aberration of a conventional TEM, it is possible to correct accurately the astigmatism of the objective lens in a TEM with “stigmator” adjustments. This correction can in fact be performed so well that astigmatism has a negligible effect on image resolution. The correction for astigmatism, or “stigmation,” is specified by an angle and a strength. In Fig. 2.40, a stigmator lens has corrected the astigmatism of the lenses in Figs. 2.38 and 2.39. The axis of the stigmator is perpendicular to that of the first non-cylindrical lens, and its strength is approximately the same. Figure 2.40 shows that when we have corrected the astigmatism, however, we have also changed the focus of the lens. All rays come to the same focal point, but this point is now a bit closer to the lens.

Fig. 2.40. Astigmatism correction for the lens of Figs. 2.38 and 2.39.

A stigmator in a modern TEM is a pair of magnetic quadrupole lenses arranged one above the other.11 For electrons coming from above into the plane of the paper, the focusing action of a quadrupole lens is as shown in Fig. 2.41. The Lorentz forces can squeeze and elongate the beam to form a circle from, say, an oval. If the strength of one of the N-S pairs in the quadrupole is stronger than the other pair, beam deflection occurs. The quadrupole lenses used for stigmation can be used simultaneously for beam deflection. Correction of objective lens astigmatism is one of the more difficult skills to learn in electron microscopy.12 This correction is particularly critical in high-resolution TEM, where the image detail depends on the phases of the beams, and hence on the cylindrical symmetry of the magnetic field of the objective lens. The astigmatism correction is tricky because three interdependent adjustments are needed: 1) main focus, 2) adjustment (focus) of xstigmator, and 3) adjustment of y-stigmator. These three adjustments must 11 12

The pair are rotated 45◦ with respect to each other to allow different orientations for the perpendicular x and y axes. The other is getting the beam exactly on the optic axis of the objective lens by performing a voltage or current center adjustment (Sect. 10.5.3).

2.7 Lens Aberrations and Other Defects

N

N S

107

S

S

S

N

N

Fields

Forces

Fig. 2.41. Magnetic fields and forces on electrons traveling down through a magnetic quadrupole lens.

be performed in an iterative manner, using features in the image as a guide. The stigmation procedure is a bit of an art, and a matter of personal preference. A holey carbon film is an ideal specimen for practicing this correction, as illustrated in Fig. 2.42 with the faint Fresnel rings caused by diffraction from the edge of the hole (see Sect. 10.3). Figures 2.42a-c below show overfocused, focused, and underfocused images of a holey carbon film when the astigmatism is small. When the objective lens is overfocused (strong current) or underfocused (weak current) with respect to the Gaussian image plane, dark and bright Fresnel fringes, respectively, appear around the inside of the hole. When the astigmatism is corrected properly, the Fresnel ring is uniform in thickness around the periphery of the hole. Figure 2.42d shows a fringe of uneven thickness caused by poor astigmatism correction in an overfocused image. Adjusting Fresnel fringes around a hole are helpful for learning about stigmation, but these adjustments are inadequate for HRTEM work, which is typically performed at the highest magnification of the instrument. For HRTEM work, astigmatism corrections can be performed with the “sandy” or “salt and pepper” contrast of the amorphous carbon film that forms on the surface and edge of the specimen. When the astigmatism is small, one can adjust the focus control to give the minimum contrast in the image. At this focus, if the x- and y-stigmators are adjusted independently, the “salt and pepper” contrast in the amorphous film will increase and streak out in the perpendicular x or y directions. To completely eliminate astigmatism, the stigmators are adjusted to minimize the contrast of the amorphous film. This “minimum contrast condition” (which is near “exact” or “Gaussian” focus) is achieved by: 1 finding the focus where contrast is minimized and the image appears flat and featureless, 2 adjusting the x-stigmator to further reduce the contrast, 3 adjusting the y-stigmator to further reduce the contrast, 4 repeating steps 1–3 iteratively until minimum contrast is obtained. When the astigmatism is eliminated, slight overfocusing or underfocusing from the minimum contrast condition gives sharp, radially-symmetric detail in the amorphous film. This detail turns from black to white as the focus is changed from above to below the specimen. If the detail tends to smear out in perpendicular directions as one rocks the objective lens focus back and forth through minimum contrast, further astigmatism correction is probably re-

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quired. This minimum contrast focus condition, incidentally, is an important reference point needed in HRTEM work.

Fig. 2.42. Images of a small hole, showing Fresnel rings that change with focus and stigmation. The images are: (a) overfocused, (b) focused, (c) underfocused, and (d) astigmatic. After [2.12].

2.7.5 Gun Brightness Many TEM measurements require a small-diameter beam on the sample. The smallest diameter of a focused electron beam is determined by 1) the quality of the lens used in focusing, and 2) the performance of the electron gun. The important gun parameter is brightness, β, which is depicted with the three sources located at the top of Fig. 2.43. All three sources in Fig. 2.43 emit the same current, and they send the same current density into the lens, which focuses the rays on the sample below. The sources to the left have the higher brightness, however, and sources with higher brightness are better for making the smallest electron beams on the sample. The reason is that the rays from the brighter sources have higher accuracies in the angles formed with respect

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to the optic axis – note the untidy ray paths from the source of Fig. 2.43c caused by its large size. If the rays entering a lens originate from a point source, each ray enters at the correct angle to be focused into a point image. For a source of lower brightness, errors in this angle of arrival at the lens surface lead to a blurred point. The focused spot on the specimen is, in fact, an image of the source itself, so it should be easiest to form a small spot when the source itself has a small size. The source of Fig. 2.43c has the lowest brightness. Nevertheless, the focused beams in Figs. 2.43b and 2.43c are the same size. To make a small spot on the specimen with the low brightness source of Fig. 2.43c, however, the lens in Fig. 2.43c must provide stronger focusing, i.e., a larger angle of convergence. Good focusing with a large angle of convergence requires a lens with low spherical aberration. In other words, focusing an electron beam to a small point requires both a bright source and a high quality lens.

Fig. 2.43. a–c. Formation of focused electron beams with sources of differing brightness. For all 3 sources (at top) the currents (number of lines) are the same, and the current densities at the white disks are the same. The brightness of the sources decreases from left to right, owing to a larger area (or smaller current density) at the source.

More quantitatively, the electron gun  brightness, β, is defined as the cur rent density per solid angle A/ cm2 sr , measured at the source of the electrons. Brightness is a conserved quantity when lenses are ideal. For example, after a lens focuses the electrons as in Fig. 2.43c, the radius of the focused electron beam is reduced by a factor of two compared to the source, but the angle of convergence is increased by a factor of 2. In other words, the current density has increased by a factor of 4, and the solid angle has increased by a factor of 4, leaving the current density per solid angle unchanged. Where the focused beam hits the specimen:

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β=

j0 . πα2p

(2.42)

Here j0 is the current density (A/cm2 ) in the beam on the specimen, and αp is the semi-angle of beam convergence. We can relate the beam size to the brightness of the electron gun and the convergence angle of the lens (assuming perfect lenses). The beam diameter, d0 , is related to the total beam current, Ip , by the relationship between current and current density:  2 d0 j0 . (2.43) Ip = π 2 Substituting (2.42) into (2.43), solving for d0 , and defining C0 : 4Ip C0 β ≡ . d0 = παp αp

(2.44)

For a given beam current, Ip , small values of the beam diameter, d0 , are obtained by increasing the brightness, β, or by increasing the semi-angle of convergence, αp . Because of lens aberrations, however, αp has a maximum value, and β is limited by the design of the electron gun. Equation 2.44 shows that the beam √ diameter d0 improves (becomes smaller) in proportion to the product αp β, as suggested by the previous discussion of Fig. 2.43.

2.8 Resolution We now collect the results of Sect. 2.7 and obtain a general expression for the resolution of the electron microscope for its two important modes of operation. In STEM (or nanobeam TEM) mode we are concerned with the smallest diameter of an electron probe that can be formed on a specimen. In high-resolution imaging, we are concerned with the smallest feature that can be resolved. A general expression for the beam size, dp , and image resolution can be obtained by summing in quadrature13 all diameters of the disks of least confusion from the previous sections, ds , dc , dd and d0 : d2p = d2s + d2c + d2d + d20 .

(2.45)

Substituting the diameters of these disks of least confusion from (2.39), (2.40), (2.41) and (2.44):  2 ΔE C02 + (0.61λ)2 2 6 2 + 0.25Cs αp + αp Cc . (2.46) dp = α2p E 13

This is strictly valid only when all broadenings are of Gaussian shape, so that convolutions of these different beam broadenings have a Gaussian form (see Sect. 8.1.3).

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111

Fig. 2.44. Example of a parametric plot of minimum beam size versus beam aperture. Lower curve can be used for spatial resolution of a TEM. E0 = 100 keV, Cs = Cc = 2 mm, ΔE = 1 eV, β = 105 A cm−2 sR−1 . After [2.10].

For a thermionic gun of low brightness, C0 ≫ λ, and the contributions of dd and dc can be neglected. Figure 2.44 shows how the diameters d0 and ds superpose to produce a minimum beam diameter, dmin , at an optimum aperture angle, αopt , for a constant Ip . The optimum aperture angle is found by setting ddp /dαp = 0, giving:   81   41 4 C0 , (2.47) αopt = 3 Cs and substitution in (2.46) yields:   18 1 3 1 3 3 dmin = C04 Cs4 ≃ 0.96 C04 Cs4 . 4

(2.48)

For a field emission gun, C0 ≪ λ, and the contributions of d0 and dc can be neglected. This is also true for the important case of image resolution in a TEM. Superposition of the remaining terms again yields a minimum, as shown in Fig. 2.44. In this case, αopt and dmin are given by:

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αopt = 0.9



λ Cs

3

 14 1

dmin = 0.8 λ 4 Cs4 .

,

(2.49) (2.50)

These expressions can be used to estimate the optimum aperture angle and the resolution limit of a high-resolution TEM. Equation (2.50) is especially important for evaluating the capabilities of different TEM instruments. Notice that the resolution depends more strongly on λ than Cs . This encourages the use of high accelerating voltages (small λ). Small gaps in objective pole pieces are also used to minimize Cs . Issues of resolution in HRTEM are developed in greater detail in Chapter 10.

Further Reading The contents of the following are described in the Bibliography. M. De Graef: Introduction to Conventional Transmission Electron Microscopy (University Press, Cambridge 2003). J. W. Edington: Practical Electron Microscopy in Materials Science, 1. The Operation and Calibration of the Electron Microscope (Philips Technical Library, Eindhoven 1974). P. J. Goodhews and F. J. Humphreys: Electron Microscopy and Microanalysis (Taylor & Francis Ltd., London 1988). P. Grivet: Electron Optics, revised by A. Septier, translated by P. W. Hawkes (Pergamon, Oxford 1965). P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley, and M. J. Whelan: Electron Microscopy of Thin Crystals (R. E. Krieger, Malabar, Florida 1977). D. C. Joy, A. D. Romig, Jr. and J. I. Goldstein, Eds.: Principles of Analytical Electron Microscopy (Plenum Press, New York 1986). R. J. Keyse, A. J. Garratt-Reed, P. J. Goodhew and G. W. Lorimer: Introduction to Scanning Transmission Electron Microscopy (Springer BIOS Scientific Publishers Ltd., New York 1998). M. H. Lorretto: Electron Beam Analysis of Materials (Chapman and Hall, London 1984). L. Reimer: Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, 4th Ed. (Springer–Verlag, New York 1997). F. G. Smith and J. H. Thomson: Optics, 2nd Ed. (John Wiley & Sons, New York 1988). G. Thomas and M. J. Goringe: Transmission Electron Microscopy of Materials (Wiley–Interscience, New York 1979). D. B. Williams and C. B. Carter: Transmission Electron Microscopy: A Textbook for Materials Science (Plenum Press, New York 1996).

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Problems 2.1 Conservation of gun brightness on the optic axis implies that the current density of the beam focused on the specimen j0 is given by: j0 = πβα2p ,

(2.51)

where β is the electron gun brightness and αp is the semi-angle of convergence of the focused electron beam. The maximum value of β can be estimated as:   jc E (1 + E/2E0 ) , (2.52) βmax = 1+ π kTc where jc is the current density at the cathode (filament), jc = ATc2 e(−φ/kTc ) , A is Richardson’s constant, A = 30 A·cm−2 ·K−2 , Tc is the cathode temperature, φ is the work function, k is Boltzmann’s constant, E = eU (kinetic energy of electron), e is the electron charge, U is the accelerating potential and E0 is the rest energy of an electron, E0 = m0 c2 , where m0 is the rest mass of the electron and c is the speed of light. (a) Using (2.52), plot βmax versus E for U = 100, 300 and 500 kV and Tc = 1500, 1750 and 2000 K, assuming a LaB6 cathode. (b) Using (2.51), plot the maximum current density jmax versus semi-angle of convergence αp over the range of 10−2 –10−4 rad, using βmax obtained for the conditions in part a. (c) Based on these results, what is the best way to maximize the current density j0 = jmax at the specimen? Why? 2.2 The force of a homogeneous magnetic field B and the velocity of an electron normal to this field v⊥ results in a circular motion of the electron about the optic axis with the radius of the trajectory r given as: r = mv⊥ /eB = [2m0 E(1 + E/2E0 )]1/2 /eB ,

(2.53)

where m = relativistic mass of electron and the other terms have the same meaning as in Problem 2.1. (a) Assuming that v⊥ is approximately equal to the velocity of the incoming electron, calculate the radius that an electron travelling at 100 and 400 kV would make about the optic axis with a magnetic field of 2.5 Wb/m2 (Tesla). See Appendix A.12 for the velocity of an electron as a function of kV. (b) A field of 2.5 Wb/m2 is about the maximum value that can be obtained due to the saturation magnetization of the pole-piece material. What consequence does this result have on the focussing of electrons at higher accelerating voltages?

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2.3 (a) Derive the equations for the optimum aperture angle αopt and the minimum size of the focused beam dmin for a thermionic gun, starting with (2.45), which considers the final size of the focused beam as equal to the square root of the sum of the squares of the beam diameters due to effects of current, diffraction, spherical aberration and chromatic aberration. (b) Starting with (2.45), also derive the expressions for dmin and αopt when dd and ds are the important terms, as in Sect. 2.8. (c) Using the results obtained in part b, plot αopt (rad) and dmin (nm) as functions of the accelerating voltage at 100, 200 and 400 kV, for Cs = 1 and 3 mm. What can you conclude from these graphs? 2.4 Ray diagrams for a TEM in the bright-field and diffraction modes are shown in Fig. 2.45. If a crystallographic direction in the specimen points to the right, how is this direction oriented in the final image screen? How will the diffraction spot for these crystallographic planes be oriented on the final screen? Neglect any rotation of the beam other than 180◦ crossovers.

Fig. 2.45. Ray diagrams for Problem 2.4. After [2.10].

2.5 Using the simplified representation of an electron microscope in Fig. 2.46, draw ray diagrams to show the difference between:

Problems

115

(a) bright-field imaging, (b) dark-field imaging, and (c) selected-area diffraction. (You must use a ruler for drawing straight lines.)

specimen objective lens objective aperture intermediate aperture imaging lens

viewing screen

Fig. 2.46. Framework for ray diagrams for Problem 2.5. You may enlarge it for convenience.

2.6 (a) Determine the aperture angles for the three objective apertures in Fig. 2.26, assuming that the ring pattern is from an Au specimen at 100 kV. (b) Suppose that you wanted to install an aperture with the largest size in part a. What actual radius should the aperture have (in microns) if the focal length of the lens is 2.0 mm? 2.7 (a) Use the lens formula (2.1) to prove that the depth of focus is M 2 times as large as the depth of field. (b) Use the lens formula to prove that the error in focus, d, equals αM D1 , as given on the right side of Fig. 2.27. (Hint: For electron lenses it is acceptable to assume small angles, so the lens formula can be expanded as, for example, 1/(l1 + δ) = (1/l1 ) (1 − δ/l1 ).) 2.8 Electrons accelerated through a potential of 100 kV pass through a crystal with a mean inner potential V of 30 volts. Calculate the mean refractive index, n, of the crystal, where: n = λ0 /λv ,

(2.54)

and λ0 is the wavelength of electrons in vacuum and λv is the wavelength of the electrons in the specimen. (Hint: The electron energy, kinetic plus potential, is conserved.) 2.9 (a) What is phase-contrast (HRTEM) microscopy and what limits the resolution in this mode? (b) Amplitude-contrast microscopy uses diffraction contrast with conventional modes of the TEM. What limits resolution in this imaging mode?

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2. The TEM and its Optics

2.10 (a) Although the wavelength of electrons used in TEM is very small (λ ≪ 0.1 nm), the best resolution in the TEM is not better than about 0.15 nm. Why? (b) The wavelength of light in the visible range is 400 < λ < 800 nm and the resolution of the light microscope is about 0.5 μm. Why? 2.11 (a) Plot the electron wavelength, λ, versus accelerating voltage, U , for electrons from 50 kV to 1 MV using non-relativistic values. (b) On the same plot, show the relativistic correction: h λ=   2me eU 1 +

eU 2me c2

 12 .

(2.55)

(c) For a cubic crystal with a lattice parameter of 0.415 nm, how large is the relativistic correction for the Bragg angle of a (111) diffraction, given electron energies of 100 keV, 300 keV and 1 MeV? Express your answer as a fraction of the nonrelativistic Bragg angles. 2.12 Use ray diagrams to sketch the filament crossover onto the specimen for a double condenser lens system in the following conditions: (a) overfocus (b) in focus (c) underfocus. 2.13 This questions refers to the three selected area diffraction patterns from a polycrystalline element in Problem 1.3. Using these diffraction patterns, determine Planck’s constant. Useful Data: lattice parameter 4.078x10−10 m, camera-length 0.345 m, high voltages (a) 60 keV (b) 80 keV (c) 100 keV, rest mass of electron 9.1×10−28 g, 1 eV = 1.6×10−12 erg. 2.14 Prove the lens formula (2.1): 1 1 1 = + f d1 d2

(2.56)

(Hint: In Fig. 2.47, f is fixed by the lens, and with a constant h1 , α is also fixed.) lens h1

`

_ h2 f

d1

d2

Fig. 2.47. Framework for derivation of lens formula in Problem 2.14.

Problems

117

2.15 Figure 2.48 shows a specimen consisting mostly of transmitting material with a disk of a diffracting crystal in its center. Using a ruler, draw ray diagrams illustrating the STEM mode of operation (rastered beam on specimen) to show why the central disk of material is bright in dark-field STEM mode (detector off axis in position to detect diffraction) and the surrounding transmitting material appears dark.

Fig. 2.48. Schematic of STEM operation in Problem 2.15.

2.16 In practice, the SAD technique is limited to obtaining diffraction patterns from regions no smaller than 0.5 μm in size. The source of this problem is the positive, third-order spherical aberration of the objective lens. The ray diagram in Fig. 2.49 is given for reference. (M is the magnification.) The solid rays to the right of the lens are drawn for a perfect objective lens. The dashed rays show the effect of positive spherical aberration. Notice that with spherical aberration the diffracted rays (but not the transmitted rays) are bent too strongly by the objective lens. This error depends on the cube of the angle, α (units are [radians]). If an intermediate aperture were placed as the dark vertical line at the right, this aperture would pass diffracted rays and the transmitted rays that do not originate from quite the same region of the specimen. The diffracted rays would originate from an area on the specimen that is displaced above the dark line on the left of the drawing. With small apertures, this error can get so bad that the selected area for a high order diffraction spot does not overlap at all the selected area for the transmitted beam. You are given a microscope with an objective lens having Cs = 3 mm, and 100 keV electrons. (a) How large is the displacement between the selected area on the specimen for the transmitted beam, and the selected area from a (600) diffraction of Fe (lattice constant = 2.86 ˚ A)? (b) Same question for an (800) diffraction.

118

2. The TEM and its Optics

(c) Using your answers to parts a and b as examples, explain why it is impractical to obtain full diffraction patterns in SAD mode from very small particles. transmitted ray

diffracted ray

intermediate image plane

_

specimen objective lens

M Cs _3

Fig. 2.49. Effect of spherical aberration on accuracy of area selection in SAD mode for Problem 2.16.

2.17 This question refers to Fig. 2.50, not drawn to scale. A lens in the center of the optic axis focuses rays traveling from left to right by providing aphase  shift as indicated. We define the dimensionless phase shift as: φ = 2π 2δλ λ . (a) If the phase shift of the lens is:  r 2 φ (r) = 4π , (2.57) 100λ what is the focal length, f , of the lens for small r (in units of λ)? (b) If the phase shift of the lens is:  r 2  r 4 φ (r) = 4π + 4π , (2.58) 100λ 100λ what is the focal length for a ray reaching the lens at r = 100λ? (c) For the phase shift of part b, sketch qualitatively a few wave crests to the right of the lens in Fig. 2.50. (d) What is the qualitative effect of the quartic term in part b on the performance of the lens? (e) Could φ (r) have a cubic term if the lens has cylindrical symmetry?

h

2 bh r

Fig. 2.50. Parameters for phase shifts of lens in Problem 2.17.

3. Scattering

3.1 Waves and Scattering This chapter explains how waves (and particles) are scattered by individual atoms. The emphasis is on elastic scattering – inelastic scattering is the topic of Chap. 4. Diffraction, as shown with Bragg’s Law in Fig. 1.1 and developed further in Chap. 5, is a type of cooperative elastic scattering by a group of atoms. Diffraction requires “coherent scattering,” characterized by a precise relationship between the phases of the incident and scattered waves. The scattered wave is the sum of component waves, “wavelets” as we call them, emanating from the different atoms in the sample. In diffraction, phase differences between these outgoing wavelets cause constructive or destructive interferences at different angles around the sample, e.g., the appearance of Bragg diffraction peaks. 3.1.1 Wavefunctions Phase. A wavefunction ψ(x, t) describes the structure of a wave (its crests and troughs) along position x, at any time t. The mathematical form

120

3. Scattering

ψ(kx − ωt) accounts for how the wave amplitude shifts in position with increasing time, t. The argument of the wavefunction, kx − ωt, is called the “phase” of the wave. It includes two constants: k (the wavevector), and ω (the angular frequency). The phase kx − ωt is dimensionless, so it can be used as the argument of a sine function or a complex exponential, for example. Our mathematical form causes the entire structure of the wave ψ(kx − ωt) to move to more positive x with increasing t. This is clear if we recognize that a particular wavecrest in ψ exists at a particular value of phase, so for larger t, the wave amplitude moves to larger x for the same value of kx − ωt.1 One-Dimensional Wave. One-dimensional waves are simple because they have no vector character. Suppose the wave is confined a region of length L. The wavefunction and its intensity are: 1 ψ1D (x, t) = √ e+i(kx−ωt) , L

(3.1)

1 1 ∗ I1D = ψ1D (x, t) ψ1D (x, t) = √ e+i(kx−ωt) √ e−i(kx−ωt) , (3.2) L L 1 (3.3) I1D = . L If ψ1D (x, t) were an electron wavefunction, the intensity, I1D , would be a probability density. The prefactor in (3.1) ensures proper normalization in the interval L, with a probability of 1 for finding the electron in the interval:

L

L 1 dx = 1 . (3.4) I1D dx = P = L 0 0 Plane Wave. In three dimensions, a plane wave is: 1 ψ3Dpl (r, t) = √ e+i(k·r−ωt) , V

(3.5)

which has an intensity and a normalization analogous to those for the onedimensional wavefunction. The spatial part of the phase, k·r, is illustrated for a snapshot in time in Fig. 3.1 for two orientations of r: with k·r = 0, and with k  r. Along the direction of r √ in Fig. 3.1a there is no change in the phase of 3.1b the phase the wave (here ψ3Dpl (r, t) = 1/ V e+i(0−ωt) ), whereas in √Fig.+i(kr−ωt) changes most rapidly along r (here ψ3DPlan (r, t) = 1/ V e ). The dot product k · r for the phase in (3.5) gives the plane wave its anisotropy in space. Spherical Wave. By placing the origin of a spherical coordinate system at the center of the spherical wave, the spherical wave has its simplest form: 1 e+i(kr−ωt) . ψ3Dsph (r, t) = √ r V 1

(3.6)

We say ψ(kx − ωt) travels to the right with a “phase velocity” of ω/k. The wave ψ(kx + ωt) travels to the left.

3.1 Waves and Scattering

121

Fig. 3.1. Plane wave with k oriented to the right, with orientations of r being (a) along the wave crests, perpendicular to k, (b) parallel to k.

If the center of the spherical wave is the distance r0 away from the origin of the coordinate system: 1 e+i(k|r−r0 |−ωt) . ψ3Dsph (r, t) = √ |r − r 0 | V

(3.7)

Figure 3.2 shows a vector construction for r − r0 , which can be obtained by connecting the tail of −r0 to the arrow of r. At distances far from the scattering center, where the curvature of the spherical wave is not important, it is often useful to approximate the spherical wave as a plane wave with r − r 0 pointing along the direction of k.2 Fig. 3.2. Spherical wave with k oriented away from the center of wave emission. (a) with coordinate system for r having its origin at center of wave emission. (b) with coordinate system for r having an arbitrary origin.

Phase Factor. A phase factor, e−i∆k·R or e−i(∆k·R+ωt) , has the mathematical form of a plane wave (3.5), and is associated with a particular wavelet, but beware. A phase factor is not a wave. A phase factor proves handy when two or more wavelets are scattered from different points in space at {Rj }, typically separated by some atomic distances. What is important after the long path to the detector is how the wavelets interfere with each other – constructively or destructively – and this is accounted for by sums of phase factors like this:  ψphf (Δk) = (3.8) e−i∆k·Rj . {Rj }

2

This is often useful because real scatterers typically emit spherical waves, but Fourier transforms require plane waves.

122

3. Scattering

The definition Δk ≡ k−k0 (illustrated in the chapter title image) is repeated a number of times in this book. This Δk is a difference in the wavevectors of two actual waves. Dot products like Δk · Rj give phase differences between wavelets, but Δk is not an actual wavevector. Chapter 5 develops these concepts, but the reader is now forewarned that exponentials containing Δk are not waves, but phase factors. 3.1.2 Coherent and Incoherent Scattering Coherent scattering preserves the relative phases of the wavelets, {ψrj }, scattered from different locations, {rj }, in a material. For coherent scattering, the total scattered wave, Ψcoh , is constructed by adding the amplitudes of the scattered wavelets:  Ψcoh = ψrj . (3.9) rj

The total coherent wave therefore depends on the constructive and destructive interferences of the wavelet amplitudes. Diffraction experiments measure the total coherent intensity, Icoh : 2    ∗ (3.10) Icoh = Ψcoh Ψcoh =  ψr j  . rj

On the other hand, “incoherent scattering” does not preserve a phase relationship between the incident wave and the scattered wavelets. For incoherent scattering it is incorrect to add the amplitudes of the scattered wavelets, {ψrj }. Incoherently-scattered wavelets do not maintain phase relationships, so they cannot interfere constructively or destructively. The total intensity of incoherent scattering, Iinc , is the sum of individual scattered intensities:    ψr 2 . Ir j = Iinc = (3.11) j rj

rj

Because measurable intensities are added in incoherent scattering, the angular distribution of incoherent scattering from a group of N identical atoms is the same as for a single atom, irrespective of how these N atoms are positioned in space. The total intensity is simply N times larger. Some types of incoherent scattering occur with a transfer of energy from the wave to the material, and these processes can be useful for spectroscopic analysis of the atom species in a material. It is important to emphasize the difference between the right-hand sides of (3.10) and (3.11). Because the intensity of coherent scattering in (3.10) first involves the addition of wave amplitudes, coherent scattering depends on the relative phases of the scattered wavelets and the relative positions of the N atoms in the group. Coherent scattering is useful for diffraction experiments. Incoherent scattering is not. This chapter describes in sequence the

3.1 Waves and Scattering

123

four types of scattering having coherent components that allow for diffraction experiments on materials: • x-rays, which are scattered when they cause the atomic electrons to oscillate and re-radiate, • electrons, which are scattered by Coulomb interactions when they penetrate the positively-charged atomic core, • neutrons, which are scattered by nuclei (or unpaired electron spins), and • γ-rays, which are scattered when they resonantly excite a nucleus, which later re-radiates. 3.1.3 Elastic and Inelastic Scattering Besides being “coherent” or “incoherent,” scattering processes are “elastic” or “inelastic” when there is, or is not, a change in energy of the wave after scattering. We can therefore construct four word pairs: (coherent elastic) (coherent inelastic) (incoherent elastic) (incoherent inelastic) Diffraction experiments need coherent elastic scattering, whereas spectroscopies that measure intensity versus energy often use incoherent inelastic scattering. The case of incoherent elastic scattering is also common, and occurs, for example, when phase relationships between scattered wavelets are disrupted by disorder in the material. Incoherent elastic intensity does not show the sharp diffractions associated with crystalline periodicities, but has a broad angular dependence. Finally, coherent inelastic scattering is used in neutron scattering studies of excitations in materials, such as such as phonons (vibrational waves) or magnons (spin waves), that have precise energy-wavevector relationships. In some phonon studies, a neutron loses energy when creating a phonon (so it is inelastic), but the scattering amplitude depends on the phases of the atom movements in the phonon with respect to the neutron wavevectors (so it is coherent). A deeper and more rigorous distinction between coherent and incoherent scattering involves our knowledge about the internal coordinates of the scatterer: • Consider a simple oscillator (a bound electron, for example) that is driven by an incident wave and then re-radiates. There is a transfer of energy from the incident wave to the oscillator, and then to the outgoing wave. Suppose we know in full detail how the coordinates of the oscillator respond to the incident wave. Since the scattering process is determined fully, the phases of all outgoing wavelets have a precise and known relationship to the phase of the incident wave. The scattering is coherent. • On the other hand, suppose the coordinates of this oscillator were coupled to another system within the material (a different electron, for example), and furthermore suppose there is freedom in how the oscillator can interact

124

3. Scattering

with this other system. (Often differing amounts of energy can be transferred from the oscillator to the other system because the transfer occurs by a quantum mechanical process that is not deterministic.) If this energy transfer is different for different scatterings, we cannot predict reliably the phase of the scattered wavelet. The scattering is incoherent. It is therefore not surprising that incoherence is often associated with inelastic scattering, since inelastic scattering involves the transfer of energy from the scatterer to another component of the material. Incoherence does not imply inelastic scattering, however, and inelastic scattering is not necessarily incoherent. 3.1.4 Wave Amplitudes and Cross-Sections Cross-Sections. X-rays, electrons, neutrons, and γ-rays are detected oneat-a-time in scattering experiments. For example, the energy of an x-ray is not sensed over many positions, as are ripples that spread to all edges of a pond of water. Either the entire x-ray is detected or not within the small volume of a detector. For x-ray scattering by an individual atomic electron as described in the next section, the scattering may or may not occur, depending on a probability for the x-ray–electron interaction. An important quantity for scattering problems is the “cross-section,” σ, which is the effective “target area” presented by each scatterer. With crosssections it is handy to think of a number, N , of scatterers in a sample of area A as in Fig. 3.3. The probability of scattering is equal to the fraction of sample area “blocked” by all N scatterers. For thin samples when the scatterers do not overlap, the N scatterers block an area equal to N σ. The fraction of rays removed from the incident beam is the blocked area divided by the total area: σx σ =N = ρ σx . (3.12) N A Ax Here the density of scatterers, ρ ≡ N/(Ax) has units [scatterers cm−3 ].

x

m A

Fig. 3.3. These 7 scatterers occupy the fraction 0.2 of the sample area, A, and therefore remove the fraction 0.2 of the rays from the incident beam. From (3.12): σ = (0.2/7)A. In the thin sample limit, the number of scatterers and the amount of scattering increase in proportion to thickness, x, but σ remains constant.

3.1 Waves and Scattering

125

For x-ray scattering, the total cross-section depends on the total number of electrons in the material. As mentioned after (3.11), for incoherent scattering the picture is then complete – the spatial distribution of the scattered intensity is obtained by adding the intensities from independent scattering events. This simple approach is appropriate, for example, for x-ray Compton scattering and absorption processes, as described in Sects. 3.2.2 and 3.2.3. Coherent scattering, as in (3.9), requires further consideration of the wave amplitudes and interferences before the cross-section is calculated. The spatial redistribution of scattered intensity can be spectacularly large (cf., Fig. 1.2), but the total coherent cross-section remains constant. By rearranging the atom positions in a material, the constructive and destructive interferences of coherent scattering are altered and the angles of scattering are redistributed, but for the same incident flux the scattered energy is conserved (for x-rays or γ-rays), or the total number of scattered particles remains the same (electrons and neutrons). The flux of scattered x-rays, electrons, neutrons, or γ-rays at the distance r from the scatterer decreases as 1/r2 along rˆ . A scattered photon carries energy, so the radiated energy flux also decreases as 1/r2 from the scatterer. The energy of a photon is proportional to E ∗ E, so the electric field, E, has an amplitude that must decrease as 1/r from the center of scattering. For scattered x-rays, we relate the electric field along  r to the incident electric field at the scatterer, E0 :

E0 , (3.13) r where the constant of proportionality would include any angular dependence. The electric fields E(r) and E0 in (3.13) have the same units, of course, so the constant of proportionality has units of length. The square of this “scattering length” is the cross section per steradian, as we next show for electron scattering (but the argument pertains to all waves). E(r) ∝

Cross-Section for Wave Scattering. Here we find the cross-section for wave scattering. Consider the total flux, Jsc (R), scattered through a unit area of surface of a sphere at radius R around the scatterer. The incident beam has a flux Jin over an area A. The ratio of all scattered electrons to incident electrons, Nsc /Nin , is: Jsc (R) 4πR2 v |ψsc (R)|2 4πR2 Nsc = . = Nin Jin A v |ψin |2 A

(3.14)

We consider elastic scattering for which the incident and scattered electrons have the same velocity, v, but for inelastic scattering these factors do not cancel. We use the spherical wave (3.6) for ψsc (R) and the plane wave (3.5) for ψin . For both waves, the exponential phase factors, multiplied by their complex conjugates, give the factor 1. The normalization factors also cancel, so (3.14) becomes:

126

3. Scattering

Nsc |fel |2 4πR2 , = Nin R2 A

(3.15)

where fel /R is the fraction of the incident electron amplitude that is scattered into a unit area of the sphere at radius R. Figure 3.3 helps demonstrate the fact that the ratio of the cross-section σ to the area A of the incident beam equals the ratio of scattered to incident electrons, Nsc /Nin : Nsc σ 4π|fel |2 = , = A Nin A σ = 4π|fel |2 .

(3.16) (3.17)

The scattering of an x-ray by a single atomic electron can be treated in the same way, but we need to account for the electric dipolar pattern of x-ray radiation with a factor of 2/3 in the cross-section, 2 8π  fx1e  , σx1e = (3.18) 3 where fx1e is the scattering length. This fx1e is the actual constant of proportionality to convert (3.13) into an equality (cf., (3.30) below). Anisotropic scattering is the rule rather than the exception, however, so simple cross-sections like those of (3.17) are usually inadequate, even if altered by factors like the 2/3 used in (3.18). The “differential scattering cross-section,” written as dσ/dΩ, contains the angular detail missing from the total cross-section, σ. The differential scattering cross-section, dσ/dΩ, is the piece of area offered by the scatterer, dσ, for scattering an incident x-ray (or electron or neutron) into a particular increment in solid angle, dΩ. The concept of dσ/dΩ is depicted Fig. 3.4. Note that dσ/dΩ relates an increment in area (on the left) to an increment in solid angle (on the right).

solid angle d11

dm1 dm2 dm3 6cm2

m area

4/ [sR] d12 d13 6sR

For the simple case of isotropic scattering,  2 dσ = f  , dΩ

Fig. 3.4. The differential scattering cross-section, dσ/dΩ, for three paths past a scatterer. The third path, dσ3 /dΩ3 , misses the scatterer and contributes only to the forward beam. The paths with areas dσ1 and dσ2 make contributions to the total crosssection for scattering, σ, and these contributions are included when the intensity is integrated over the differential solid angles dΩ1 and dΩ2 .

(3.19)

3.1 Waves and Scattering

127

which is a constant. For anisotropic scattering, (3.19) is generalized with a scattering length, f (k0 , k), that depends on the directions of the incident and outgoing wavevectors, k0 and k, respectively:  2 dσ = f (k0 , k) . (3.20) dΩ We expect to recover the total cross-section, σ, by integrating dσ/dΩ over all solid angle, σ=

4π

dσ dΩ . dΩ

(3.21)

sphere

Substituting (3.19) into (3.21) and integrating gives (3.17), as expected. Special Characteristics of Coherent Scattering. Compare the differential scattering cross-sections for coherent x-ray scattering by a single electron at rj , dσx1e,rj /dΩ, and an atom having Z electrons, dσatom /dΩ:  2 dσx1e,rj (k0 , k) = fx1e,rj (k0 , k) , (3.22) dΩ 2  dσatom (3.23) (k0 , k) = fatom (k0 , k) . dΩ In coherent scattering we sum wave amplitudes (cf., (3.9)), so for coherent scattering we sum the scattering lengths of all Z electrons to obtain the scattering length of an atom: fatom (k0 , k) =

Z 

fx1e,rj (k0 , k) .

(3.24)

rj

Note that (3.24) is a sum of the fx1e,rj , but (3.23) is the square of this sum. Equation (3.23) can predict that the coherent x-ray scattering from an atom with Z electrons is Z 2 times stronger than for a single electron, and this proves to be true in the forward direction. The total cross-section for coherent scattering must increase linearly with the number of scatterers (here the number of electrons, Z). Consequently the coherent scattering is suppressed in other directions if a scaling with Z 2 is allowed in special directions. The angular distribution of coherent scattering must be different for the atom and for the single electron. That is, fx1e (k0 , k) and fatom (k0 , k) must have different shapes (they must depend differently on k0 and k). The following is an inequality for coherent scattering (although its analog for incoherent scattering is an equality): Z

 dσx1e,r j ,coh dσatom,coh (k0 , k) = (k0 , k) . dΩ dΩ r

(3.25)

j

Integrating (3.25) gives an equality for coherent (and incoherent) scattering:

128

3. Scattering

4π

dσatom,coh (k0 , k)dΩ = dΩ

4π  Z dσx1e,r j ,coh (k0 , k)dΩ , dΩ r

sphere

sphere

(3.26)

j

because with (3.21) we see that (3.26) equates the individual electron crosssections to the total cross-section of the atom: σatom,coh = Zσx1e,coh .

(3.27)

The process of actually performing the sum in (3.24) evidently requires delicacy in accounting for the phase relationships between the x-ray wavelets scattered into different angles, and knowledge about the electron density of the atom. This is the subject of Sects. 3.3.2–3.3.4, which focus on the dependence of the scattered intensity, I, on the incident and outgoing wavevectors, k0 and k.

3.2 X-Ray Scattering 3.2.1 Electrodynamics of X-Ray Scattering Classical electrodynamics can help explain how electric dipole radiation from an atom depends on the x-ray frequency, ω. We seek the ω-dependence of the x-ray scattering length of one electron, fx1e , when the one electron is driven by the electric field of an incident wave. The electron is bound to an atom so that its displacement provides a harmonic restoring force, and therefore a resonant frequency, ωr . The equation of motion for our electron is: d2 x dx eE0 iωt + ωr2 x = e , +β (3.28) dt2 dt m where x is along the direction of the incident electric field, E0 . The variable β is an internal damping constant divided by the electron mass, m, and ω is the frequency of the incident wave. The following solution for x(t) can be verified by substitution into (3.28): x(t) =

eiωt eE0 . 2 m ωr − ω 2 + iβω

(3.29)

The product of x (t) and the charge of the electron, e, is an oscillating dipole moment. Classical electrodynamics predicts the radiation intensity from this dipole oscillator – the radiated electric field, E, is proportional to the acceleration of the dipole. The acceleration is (3.29) times −ω 2 . The complete expression for E in the equatorial plane of the dipole is:  2  ω2 E0 e E0 = fx1e , (3.30) E(r, t) = 2 2 2 mc ωr − ω + iβω r r  2  e ω2 . (3.31) fx1e ≡ 2 2 mc ωr − ω 2 + iβω

3.2 X-Ray Scattering

129

The factor fx1e defined in (3.31) is the “x-ray scattering factor,” of one atomic electron. In (3.30) we neglected the time dependence, eiω(t−r/c) (r is the distance from the electron), but we can later multiply our field by this factor if we so choose. Since x-rays have energies comparable to the energies of interatomic electronic transitions, ω may or may not be close to the resonant frequency ωr of a particular atom in our specimen. We now consider in sequence all three possibilities: ω > ωr , ω < ωr , ω ≃ ωr . ω ≫ ωr . First consider the case where the frequency of the incident radiation is very high. The weak intraatomic forces are not so important for our high energy x-ray, so the mass of the electron limits its acceleration in the same way as for a free electron. The term ω 2 dominates the denominator of (3.30) (the intraatomic damping for the electron, β, is also neglected), and (3.30) becomes  2  e E0 E0 = −2.82 × 10−13 , (3.32) E(r, t) = − mc2 r r

where r is in units of cm and r0 = e2 /(mc2 ) = 2.82 × 10−13 cm is the “classical electron radius.” The negative sign tells us that the electric field of the scattered wave is out-of-phase with the electric field of the incident wave. The intensity of the scattered wave is:  2 2 2 e E0 e4 I0 I0 = 2 4 2 = 7.94 × 10−26 2 . (3.33) I(r, t) = E ∗ E = 2 2 mc r m c r r

Equation (3.33) gives the strength of “Thompson scattering.” This result can be converted to a total cross-section by multiplying by 4π(2/3) to account for polarization and all solid angles. With such a small cross-section, of order 10−24 cm2 ≡ 1 barn, a single free electron is a rather weak elastic scatterer of x-rays. A mole of electrons provides significant scattering, however. ω ≪ ωr . Now consider (3.30) for the case where the frequency of the incident radiation is very low. The intraatomic forces are important for the scattering of a low-energy x-ray, so the behavior is dominated by the stiffness of the restoring force binding the electron to the atom: E(r, t) = +

e2 E0 ω 2 . mc2 r ωr2

(3.34)

The large displacements required for low frequency radiations are difficult to achieve because of the harmonic restoring force, and we see that the intensity of the scattered wave goes as ω 4 .3 The electric field of the scattered wave is in phase with that of the incident wave. 3

This is why the sky is blue. Visible light is of low energy compared to excitations of electrons in the molecules of the atmosphere.

130

3. Scattering

ω  ωr . Finally, consider the case near resonance. We are forced to use (3.30) in full detail, and we break it up into real and imaginary parts: E0 , where : r ω2 e2 = 2 ≡ fx1e . ωr − ω 2 + iβω mc2

′ ′′ E(r, t) = (fx1e + ifx1e ) ′ ′′ fx1e + ifx1e

(3.35) (3.36)

′ ′′ We separate the real part, fx1e , and imaginary part, fx1e , for the scattering from our single electron (by multiplying the numerator and denominator by the complex conjugate of the denominator):   ω 2 ωr2 − ω 2 e2 ′ , (3.37) fx1e ≡ 2 (ωr2 − ω 2 ) + β 2 ω 2 mc2 ′′ fx1e ≡

−βω 3

e2 . 2 (ωr2 − ω 2 ) + β 2 ω 2 mc2

(3.38)

′ As we have already seen, the real part, fx1e , dominates the scattering at ′′ , driving frequencies far from resonance |ω −ωr | ≫ 0. The imaginary part, fx1e is nearly zero for very low and very high ω. It corresponds to a secondary component of the scattered wave that is shifted in phase by π/2 with respect ∗ fx1e = to the primary scattered wave. To get an intensity, we multiply: fx1e ′ ′′ ′ ′′ ′2 ′′2 (fx1e − ifx1e ) (fx1e + ifx1e ) = fx1e + fx1e , so the intensity of this secondary scattered wave adds to that of the primary scattered wave. (This also shows ′′ does not affect intensities.) Near resonance (ω ≃ ωr ), the that the sign of fx1e ′′ imaginary part of the scattering factor, fx1e , approaches −ω/β[e2 /(mc2 )]. On ′ , vanishes at the the other hand, the real part of the scattering factor, fx1e resonance frequency of our one electron. It turns out that the overall intensity of scattering decreases at the resonance of a K electron, for example, but to demonstrate this we must know more about β, and we must consider wave emission from all electrons of the atom, most of which are not near resonance. A more rigorous way to calculate x-ray scattering intensities from atomic electrons is to use quantum mechanical perturbation theory. A calculation employing the Schr¨ odinger equation is not too difficult, but it would take us a bit afield, and is not included here. The essential steps are:

• start with the atomic electrons in their stationary states (atomic wavefunctions), • set up a perturbation hamiltonian (proportional to A · grad, where A is the vector potential), • calculate the probability current density of the moving electrons, from which a dipole strength is obtained, and • calculate from classical electrodynamics the scattered wave field. This more rigorous approach (performed with Hartree-Fock wavefunctions, ′ that are similar to those of the classical for example) provides results for fx1e

3.2 X-Ray Scattering

131

′′ ′′ approach, but a distinct difference occurs for fx1e . The magnitude of fx1e is zero for frequencies less than:

Eαβ , (3.39)  where Eαβ is the energy difference between two electron states with labels ′′ α and β. In other words, fx1e describes how x-ray energy is absorbed by exciting the atomic electron from α into a higher energy state β. This occurs occurs only when ω ≥ ωr . The “atomic form factor for x-ray scattering,” fx , is the x-ray scattering amplitude (e.g., (3.30)) for a particular species of atom. This fx is the sum of the amplitudes from the individual electrons at the atom, the fx1e considered above. We can now understand how the scattering of a particular type of xray, a Cu Kα x-ray of 8.05 keV energy, depends on the atomic number of the atom. The Appendix includes a table of atomic form factors for high-energy x-rays and a graph of “dispersion corrections.” These are useful resources, so the reader is encouraged refer to Appendices A.3 and A.4 for the following discussion. The convention in Appendix A.4 is to write the x-ray scattering factor as: ωr =

fx = Z + f ′ + if ′′ ,

(3.40)

where Z is the atomic number. The terms f ′ and f ′′ are the “H¨ onl dispersion corrections,” and are used to make corrections to the x-ray scattering factors for heavy elements and for elements for where ω ≃ ωr . Equation (3.40) is in “electron units.” For actual scattering intensities, we need to multiply fx∗ fx by the Thompson cross-section ((3.33), et seq.), less a small amount of Compton scattering described in the next section. As we move up the periodic table of elements, there are more electrons about the atom (equal to Z). For all elements, most of their electrons are bound at energies less than 8.05 keV. To a first approximation, the x-ray scattering factor increases as Z, and the scattered intensity increases as Z 2 . (To approximate an absolute intensity, we could multiply (3.33) by Z 2 .) In Appendix A.3, this trend is most evident in the far-left column for s = 0, representing coherent scattering in the forward direction. Notice how neutral atoms have larger form factors than positive ions, which have fewer electrons. With increasing Z, the electron energy levels of the atom become more and more negative. For the light elements, a Cu Kα x-ray has a high frequency compared to the characteristic ωr of all electrons, and our analysis of case 1 ′ , and f ′′ is not large. As Z increases (ω > ωr ) is reasonable, since fx ≃ Zfx1e to 28 (Ni), however, we approach the case ω ≃ ωr for the K-shell electrons. For elements below Ni there is x-ray absorption by “photoelectric” emission of K-electrons, since the Cu Kα x-ray has sufficient energy to remove a K-shell electron. This K-shell ionization is rather strong for the element Co (Z = 27), which fluoresces intensely in a beam of Cu Kα x-rays. For Ni and heavier elements there is no absorption by K-shell ionization. The graph of x-ray

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3. Scattering

dispersion corrections in Appendix A.4 shows that f ′′ drops abruptly between Co and Ni. Additionally, the real part of the scattering factor is changed because the K electrons are near resonance (ω ≃ ωr ). In passing through resonance, the scattering from the K electrons of the atom changes phase from oscillating against the incident wave to oscillating with the incident wave. Around Ni, the scattering from K electrons starts to become out-ofphase with the scattering from the rest of the atom. Equation (3.37) shows that near resonance there is a large reduction of the real part of the scattering factor from the K-electron of Ni (and its neighbors on the periodic table), termed “anomalous scattering.” (This is seen as a dip in the real part, f ′ , of the dispersion correction curve in Appendix A.4.) For elements with Z > 28, (3.34) shows that the K-shell electrons scatter out-of-phase with the rest of the atom, but increasingly weakly as ωr becomes larger. As we move further up in the periodic table, this whole process repeats for the L-shell electrons around the element Sm (Z = 62). The x-ray atomic form factor has another important feature described in Sect. 3.3.2. The fx is a function of Δk (where Δk ≡ k − k0 , and is shown in Fig. 3.9). Notice how the atomic form factors for high-energy x-rays decrease from left to right across the table in Appendix A.3 (where the variable s = Δk/(4π)). For both electron and x-ray scattering, the Δk-dependence of f originates with the finite size of the atom. Were the atom infinitely small, f would depend weakly on Δk, and (3.40) would be valid for all Δk. Because the x-ray wavelengths are comparable to atomic sizes, however, the Δk-dependence of f (Δk) must be considered explicitly. Typically this is done by taking f from (3.40), determining Δk from the diffraction angle, θ, and then multiplying by a tabulated function for the atom of interest.4 For x-ray and electron scattering, the functions fx (Δk) and fel (Δk) are tabulated in Appendices A.3 and A.5. 3.2.2 * Inelastic Compton Scattering In addition to x-ray fluorescence following the excitation of a core electron, another inelastic x-ray scattering process is important for x-ray experiments. Compton scattering, discovered in 1923, was helpful in elucidating the particle nature of light, but tends to be a nuisance in diffraction work.5 Compton scattering is a relativistic scattering of a photon by a free electron. Here we perform an adequate analysis nonrelativistically for the usual case where the change in photon energy after the collision is not too large. The incident photon, traveling along the x-direction in Fig. 3.5, has an initial energy x . (Recall that the photon momentum Ephoton = hν0 and momentum (hν0 /c) is its energy divided by the speed of light.) The electron is at rest initially, with zero momentum and zero kinetic energy. After the collision, the photon is deflected by the angle, 2θ. 4 5

With k ≡ 2π/λ, Δk = (4π sinθ)/λ, and s = sin θ/λ. Compton scattering is incoherent and inelastic.

3.2 X-Ray Scattering y

133

photon out x

photon in

2e

Fig. 3.5. Geometry for Compton scattering of a photon by an electron.

After the collision the photon has the energy hν ′ , since it lost an amount of energy hΔν ≡ hν0 − hν ′ to the electron. The electron now has an xand y-component of momentum, and we have two equations for momentum conservation. Along the y-direction the initial momentum is zero, so the momenta of the electron and photon are equal and opposite after the collision.  , the electron momentum is the change in the x-component of the Along x photon momentum. The electron momentum has these x- and y-components: hν ′ sin2θ , c h (ν0 − ν ′ cos2θ) . pel x = c Now we impose the nonrelativistic conservation of energy. All energy the photon goes into the kinetic energy of the electron: 1  el 2  el 2  hΔν ≡ hν0 − hν ′ = , px + py 2me  h2  2 2 ′ ′ hΔν = − ν cos2θ) + (ν sin2θ) , (ν 0 2me c2  h2  2 hΔν = ν + ν ′2 − 2ν0 ν ′ cos2θ . 2me c2 0 pel y =

(3.41)

(3.42) lost by (3.43) (3.44) (3.45)

Approximately, when Δν is small, so ν0 = ν ′ and: h2 ν02 (1 − cos2θ) , m e c2 Ephoton Δν = (1 − cos2θ) , ν0 Ereste−

hΔν =

(3.46) (3.47)

where Ereste− = me c2 is the rest mass energy equivalence of the electron: 511 keV. Typical x-ray energies are much smaller than this energy – a Cu Kα photon has an energy of about 8 keV, for example. The relative energy loss of the photon predicted by (3.47) is therefore small. The Compton scattering of an x-ray is incoherent because there are degrees of freedom in each scattering event associated with the atomic electron. Compton scattering provides a background intensity in x-ray diffraction patterns that can be understood as follows. The outer electrons of an atom are the ones that can participate in Compton scattering because they can become unbound from the atom and carry momentum when they acquire the energy of hΔν. Compton scattering by outer electrons is more likely at higher diffraction angles 2θ, where hΔν is as large as 125 eV for Cu Kα radiation,

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3. Scattering

for example. The Compton background therefore rises with 2θ angle. The core electrons of heavier atoms do not participate in Compton scattering, since they are bound too tightly. The relative amount of Compton scattering versus coherent scattering therefore decreases with the atomic number of the element. It turns out that the total inelastic Compton scattering intensity plus the total elastic intensity are exactly equal to the Thompson scattering. 3.2.3 X-Ray Mass Attenuation Coefficients As an x-ray beam passes through a material, the energy of each x-ray remains constant, but there is a decrease in the number of x-rays in the beam. At the depth x, the increment of thickness of a material, dx, scatters a number of xrays, dI, removing them from the beam. The number of lost x-rays, −dI(x), equals the product of 1) the increment of thickness, dx, 2) the number of x-rays present at x, I(x), and 3) a material coefficient, μ: −dI(x) = μI(x)dx , dI(x) = −μI(x) , dx I(x) = I0 e−μx .

(3.48) (3.49) (3.50)

The product in the exponent, μx, must be dimensionless, so μ has dimensions of [cm−1 ]. When μx is small, it equals the fraction of x-rays removed from the incident beam. From Fig. 3.3 we know that this fraction also equals N σ/A, so: Nσ N μ= = σ, (3.51) Ax V where N/V has units [atoms cm−3 ] and σ is the scattering cross-section with units [cm2 ]. Since density varies with the type of material, tabulations such as the one in Appendix A.2 provide “mass attenuation coefficients,” which are ratios μ/ρ. Here the density, ρ, has units [g cm−3 ], so the coefficients μ/ρ have units [cm−1 ]/[g cm−3 ]=[cm2 g−1 ]. Exponents in (3.50) are products (μ/ρ) × ρ × x, and are, of course, dimensionless. As a typical application of mass attenuation coefficients tabulated in Appendix A.2, consider the characteristic depth of penetration for Cu Kα x-rays in a sample of iron metal. This is obtained readily: the mass attenuation coefficient is 302 g−1 cm2 , the density of iron is 7.86 g cm−3, and the inverse of the product of these numbers gives 4.2 μm. For comparison, the table also shows that higher energy Mo Kα x-rays are more penetrating in iron, having an e−1 reduction in intensity (e−1 = 0.368) over a distance of 34 μm. It is straightforward to calculate the composite mass attenuation coefficient for a compound or an alloy. (We obtain a different expression from (1.67), however, which involved multiple phases.) In all absorption problems, the point to remember is that the net x-ray scattering depends on the number

3.2 X-Ray Scattering

135

and types of atoms in the path of the beam. The composite mass attenuation coefficient is obtained from the mass attenuation coefficients, μi , for the different elements, i, weighted by their atomic fractions in the material, fi :  fi μi . (3.52) < μ >= i

For use with tabulated values of μ/ρ, however, we must use mass fractions. For example, consider the attenuation of Cu Kα radiation in an Fe-25at.%Al alloy, which has a density of 6.8 g cm−3 . We attribute 13.9% of the density to Al and 86.1% to the Fe because the alloy composition is Fe-13.9 wt.% Al. For Cu Kα radiation the product, < μρ >FeAl , is:

 < μρ >FeAl = 0.139·49.6 + 0.861·302 6.8 = 1815 cm−1 . (3.53)

This gives a characteristic length of 5.5 μm. Interestingly, if we assume that the scattering is due entirely to iron, we obtain a characteristic length of 5.7 μm. In this example the mass attenuation is dominated by the iron in the material, primarily because iron is the stronger x-ray attenuator (and secondarily because iron is the majority species). Figure 3.6 is an x-ray penetration image of an important work of art, “Blue Boy,” by Thomas Gainsborough. Many minerals are used in paint pigments, but in Gainsborough’s day the mineral lead carbonate was used for the color white. The lead dominates the x-ray absorption, and in this (negative) image the light regions correspond to a high lead density.6

Fig. 3.6. Left: Negative image of x-ray penetration through the canvas “Blue Boy,” by Thomas Gainsborough. Right: The portrait surface photographed with reflected light. After [3.1].

The material coefficient, μ, originates with both inelastic and elastic scattering. For x-rays with energies from 1 to 20 keV, however, the mass attenu6

Notice the dog in the lower right, which Gainsborough evidently decided was inappropriate for the portrait. The top of the x-ray image also shows the collar of another person, indicating the canvas itself was used for a previous portrait.

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3. Scattering

ation coefficient is dominated by photoelectric absorption, where an incident x-ray loses energy by exciting an electron out of the atom. Photoelectric absorption requires the energy of the incident x-ray to be greater than the binding energy of an atomic electron. The mass absorption coefficients are larger for elements where the x-ray energy exceeds a binding energy of an atomic electron. For Cu Kα x-rays, for example, this causes a 7-fold increase in mass absorption coefficient for Co over that of Ni. The energy of a Cu Kα x-ray is 8.05 keV, whereas the energy required for exciting a K-electron from Co is 7.71 keV, and from Ni it is 8.33 keV.

3.3 Coherent Elastic Scattering 3.3.1 ‡ Born Approximation for Electrons Almost without a second thought, we treat electron scattering as a wave phenomenon with the electron wavefunction satisfying the Schr¨ odinger wave equation. An electron diffraction pattern, with its series of spots or rings as in Fig. 1.9, is certainly evidence of wave behavior. The interpretation of the electron wavefunction is different from that of a simple wave, however. Suppose we were to turn on an electron beam and watch the formation of the diffraction pattern of Fig. 1.9, using a detector capable of recording impacts of individual electrons. When the electron beam is turned on, bright flashes are seen at points on the detector screen. Each individual event occurs at a particular point on the detector, and does not appear as a continuous ring. With time, an obvious bias appears, where the points of detection are most frequently at the positions of the rings and spots of the diffraction pattern. This behavior motivates the interpretation of the electron wavefunction in terms of probabilities – specifically, the electron probability is the electron wavefunction times its complex conjugate (which gives a real number). Usually this probabilistic interpretation can be ignored when we consider a diffraction pattern from many electrons, and we can consider electron diffraction as the diffraction of any other type of wave. When individual electron events are considered, however, we may have to recall the probabilistic interpretation of the electron wavefunction because individual electron detections look like particles rather than waves. Another point to remember is that wave behavior is a characteristic of an individual electron. When considering a diffraction pattern involving multiple electrons, we do not add the amplitudes of multiple wavefunctions. At the viewing screen, we add the intensities of individual electrons. The interactions between different high-energy electrons are not coherent. Our picture of scattering begins with one electron as a wave incident on an atom. This wave looks like a plane wave because it comes from a distant source. The wave interacts with the nucleus and electron cloud of the atom, and an outgoing wave is generated. This outgoing wave is something

3.3 Coherent Elastic Scattering

137

like a spherical wave originating at the atom, although its intensity is not isotropic. Figure 3.7 shows the geometry, wavevectors and position vectors for our electron scattering problem. Here both r and r ′ are large compared to the size of the scatterer. Because we consider elastic scattering, the magnitudes of the incident and scattered wavevectors are equal, i.e., k = k0 . Our plane wave incident from the left, Ψinc , is of the standard form: 

Ψinc = ei(k0 ·r −ωt) .

(3.54)

In what follows we neglect the time dependence to emphasize the manipulations of the spatial coordinates. We can always recover the time-dependence by multiplying our results by e−iωt . A spherical wave, Ψscatt , travels outwards from the center of scattering. The scattered wave has the form: eik|r−r | , |r − r′ | 

Ψscatt = f (k0 , k)

(3.55)

where the scattering length f (k0 , k) of Sect. 3.1.4 varies with the orientation of k0 and k, r ′ is now used to locate the center of the scatterer, and the difference, r − r′ , is the distance from the scatterer to the detector. The intensity of Ψscatt falls off with distance as 1/r2 , as we expect:    2 e−ik|r−r | eik|r−r | ∗ Ψscatt = f (k0 , k) Iscatt = Ψscatt , |r − r ′ | |r − r ′ |  2 1 Iscatt = f (k0 , k) 2 . |r − r ′ |

(3.56) (3.57)

Fig. 3.7. Wavevectors and position vectors for electron scattering.

To obtain the scattering length f (k0 , k), we must solve the Schr¨odinger equation for the incident electron inside the scattering atom (the mass of the electron is m, and its coordinates in the atom are r ′ ): −

2 2 ∇ Ψ (r ′ ) + V(r ′ ) Ψ (r ′ ) = E Ψ (r ′ ) , 2m 2 2 ∇ Ψ (r′ ) + E Ψ (r ′ ) = V(r ′ ) Ψ (r ′ ) , 2m

(3.58) (3.59)

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3. Scattering

which we write as:   2 ∇ + k02 Ψ (r ′ ) = U (r′ ) Ψ (r ′ ) ,

(3.60)

after having made the two definitions:

2mE , (3.61) 2 2mV(r ′ ) . (3.62) U (r′ ) ≡ 2 The formal approach7 to finding the solution of the Schr¨ odinger equation in this problem makes use of Green’s functions. A Green’s function, G(r, r′ ), provides the response at r for a point scatterer at r ′ :  2  (3.64) ∇ + k02 G(r, r′ ) = δ(r ′ ) . k02 ≡

We find the Green’s function in a quick way by starting with an identity:

1 eikr eikr = eikr ∇2 − k 2 , (3.65) r r r  2  eikr 1 ∇ + k2 = eikr ∇2 , (3.66) r r Recall that: 1 ∇2 = −4πδ(r) , so (3.67) r  eikr  2 ∇ + k2 = −eikr 4πδ(r) . (3.68) r The right-hand side simplifies because it equals zero everywhere except at r = 0, due to the nature of the δ-function. At r = 0, however, eikr = 1. From our identity (3.65) we therefore obtain: ∇2

 2  eikr ∇ + k2 = −4πδ(r) . (3.69) r We make a shift of the origin: r → r − r′ (so we can see more easily how the outgoing wave originates at the scatterer – see Fig. 3.7). After doing so, we can identify our Green’s function by comparing (3.64) and (3.69): 7

An intuitive shortcut from (3.64) to (3.71) is to regard (∇2 + k20 ) as a scattering operator that generates a scattered wavelet proportional to U (r ′ )Ψ (r ′ ). The scattered wavelet must also have the properties of (3.55) for its amplitude and phase versus distance. The scattered wavelet amplitude from a small volume, d3 r ′ , about r ′ is:  eik|r−r | 3 ′ d r , dΨscatt (r, r ′ ) = U (r ′ )Ψ (r ′ ) (3.63) |r − r ′ |

which is a spherical wave at r originating at r ′ . This approach is even more intuitive for x-ray scattering, which is proportional to the number of electrons about the atom. For x-rays, U (r ′ ) becomes ρ(r ′ ), the electron density. The result is the same as (3.83) below, but with a different prefactor and ρ(r ′ ) instead of V(r ′ ).

3.3 Coherent Elastic Scattering  1 eik|r−r | . G(r, r ) = − 4π |r − r′ |

′

139

(3.70)

With our Green’s function in hand, we construct Ψscatt (r) by integrating. The idea is that to obtain the total wave amplitude at r, we need to add up the spherical wavelet amplitudes emanating from all r ′ (each of form (3.70)), weighted by their strengths. This weight is the right-hand side of (3.60):

Ψscatt (r) = U (r ′ ) Ψ (r ′ ) G(r, r′ ) d3 r′ . (3.71)

Formally, the limits of integration cover all of space, but in fact it is only important to extend them over the r′ where U (r′ ) is non-zero (approximately the volume of the atom). The total wave at r, Ψ (r), has both incident and scattered components: Ψ = Ψinc + Ψscatt ,

2m ik0 ·r Ψ (r) = e + 2 V(r′ ) Ψ (r ′ ) G(r, r ′ ) d3 r ′ . 

(3.72) (3.73)

Up to here our solution is exact. It is, in fact the Schr¨ odinger equation itself, merely transformed from a differential equation to a integral equation appropriate for scattering problems. The problem with this integral equation (3.71) is that Ψ appears both inside and outside the integration, so an approximation is generally required to proceed further. The approximation that we use is the “first Born approximation.” It amounts to using a plane wave, the incident plane wave, for Ψ in the integral: 

Ψ (r ′ ) ≃ eik0 ·r .

(3.74)

The first Born approximation assumes that the wave is undiminished and scattered only once by the material. This assumption is valid when the scattering is weak.8 8

Extending the Born approximation to higher orders is not difficult in principle. Instead of using an undiminished plane wave for Ψ (r ′ ), we could use a Ψ (r ′ ) that has been scattered once already. Equation (3.73) gives the second Born approximation if we use do not use the plane wave of (3.74) for Ψ (r ′ ), but rather: Z  2m Ψ (r ′ ) = eik 0 ·r + 2 V(r ′′ ) Ψ (r ′′ ) G(r ′ , r ′′ ) d3 r ′′ , (3.75)  where we now use a plane wave for Ψ (r ′′ ): 

Ψ (r ′′ ) ≃ eik 0 ·r .

(3.76)

The second Born approximation involves two centers of scattering. The first is at r ′′ and the second is at r ′ . The second Born approximation is sometimes used when calculating the scattering of electrons with energies below 30 keV from heavier atoms such as Xe. For solids, however, the second and higher Born approximations are not used very frequently. If the scatterer is strong enough to violate the condition of weak scattering used in the first Born approximation, the scattering will also violate the assumptions of the second Born approximation.

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3. Scattering

We simplify (3.70) by making the approximation that the detector is far from the scatterer. This allows us to work with plane waves at the detector, rather than outgoing spherical waves. To do so we align the outgoing wavevector k along (r − r′ ) as shown in Fig. 3.7. The product of scalars, k |r − r ′ |, in the exponential of a spherical wave emitted from r′ , is then equal to k · (r − r′ ) of a plane wave, 

G(r, r′ ) ≃ −

1 eik·(r−r ) . 4π |r|

(3.77)

In (3.77) we also assumed that the origin is near the scatterer, so |r| ≫ |r ′ |, simplifying the denominator of our Green’s function.9 Returning to our exact integral equation (3.73), we obtain the approximate scattered wave (the first Born approximation for the scattered wave) by using (3.74) and (3.77) in (3.73):

ik·(r−r  ) m ′ ik0 ·r  e ik0 ·r d3 r ′ , (3.78) V(r ) e Ψ (r) ≃ e − 2π2 |r|

 m eik·r Ψ (r) = eik0 ·r − (3.79) V(r ′ ) ei(k0 −k)·r d3 r ′ . 2π2 |r| If we define:

Δk ≡ k − k0 , Ψ (r) = eik0 ·r −

ik·r

m e 2π2 |r|

(3.80) 

V(r ′ ) e−i∆k·r d3 r ′ .

(3.81)

The scattered part of the wave is: eik·r f (Δk) , where : |r|

 m f (Δk) ≡ − V(r ′ ) e−i∆k·r d3 r′ . 2 2π

Ψscatt (Δk, r) =

(3.82) (3.83)

The factor f (Δk) is the scattering factor of (3.55), which we have found to depend on the incident and outgoing wavevectors only through their difference, Δk ≡ k − k0 . We recognize the integral of (3.83) as the Fourier transform of the potential seen by the incident electron as it goes through the scatterer. In the first Born approximation: The scattered wave is proportional to the Fourier transform of the scattering potential. The factor f (Δk) of (3.83) is given various names, depending on the potential V (r) (we changed notation: r ′ → r). When V (r) is the potential of a single atom, Vat (r), we define fel (Δk) as the “atomic form factor”: 9

If we neglect a constant prefactor, this assumption of |r − r ′ | = |r| is equivalent to assuming that the scatterer is small compared to the distance to the detector.

3.3 Coherent Elastic Scattering

fel (Δk) ≡ −

m 2π2

Vat (r) e−i∆k·r d3 r .

141

(3.84)

Alternatively, we can use the potential for the entire crystal for V(r) in (3.83) (this is developed in Chap. 5). When V(r) refers to the entire crystal, however, the first Born approximation of 3.81 is generally not reliable because multiple scattering will invalidate the assumption of (3.74). This assumption is, nevertheless, the basis for the “kinematical theory of diffraction,” which we develop for its clarity and its qualitative successes. It is possible to transcend formally the single scattering approximation, and develop a “dynamical theory” of electron diffraction by considering higher order Born approximations, but this has not proved a particularly fruitful direction. Modern dynamical theories take a completely different path. 3.3.2 Atomic Form Factors – Physical Picture For coherent elastic scattering, which provides the basis for diffraction measurements, it turns out that the scattered wave is strongest in the forward direction. The “atomic form factor” describes the decrease of the scattered wave amplitude at angles away from the forward direction. It is the Fourier transform of the shape of the scattering potential (3.84). For electron and x-ray scattering, which involve the atomic electrons, the shape of the scattering potential is comparable to the “shape of the atom.” A consequence for x-ray and electron diffraction experiments is that Bragg diffractions at higher angles are attenuated significantly, and this angular dependence is important for any quantitative understanding of diffraction intensities. The present section discusses the origin and characteristics of the atomic form factor for both x-ray and electron scattering.10 A physical interpretation of the dependence on Δk of the atomic form factor for electron scattering, fel (Δk), can be provided with Fig. 3.8 and a rewritten (3.84):  fel (Δk) = (3.85) Δfel,j e−i∆k·rj . j

Equation (3.85) describes an atom as being composed of many small subvolumes, {ΔVj } at positions {rj }. Each sub-volume is able to emit a scattered wavelet with a phase factor e−i∆k·rj . The wavelet amplitude from the j th sub-volume is Δfel,j , where: m Δfel,j ≡ − Vat (rj ) ΔVj . (3.86) 2π2 The same approach can be used to understand the x-ray atomic form factor – we substitute Δfx,j for Δfel,j in (3.85). Here the amplitude of the scattered 10

For neutron or M¨ ossbauer γ-ray scattering, however, the scattering potential originates with the tiny volume of the nucleus. Nuclear form factors have no dependence on Bragg angle in the energy ranges of materials science.

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3. Scattering

high-energy x-ray wavelet, Δfx,j , depends on the electron density of the atom, ρ (r j ), (cf., (3.32)) as: e2 ρ (rj ) ΔVj . (3.87) mc2 The amplitude of each emitted wavelet in (3.85) depends on the Coulombic potential at that sub-volume (for electron scattering), or the electron density at that sub-volume (for high-energy x-ray scattering). Away from the atom, along the direction of k = k0 + Δk, the amplitude of the scattered wave is determined by the constructive and destructive interferences between wavelets emitted from the different sub-volumes. Equation (3.85) shows that this interference is set by the sums of the phase factors, e−i∆k·rj , weighted by the appropriate Δfel,j (3.86), or Δfx,j (3.87). In the forward direction where k = k0 and Δk = 0, the exponential in (3.85) is e−i0r = 1 for all values of rj . The wavelets emitted from all volume elements add constructively. In other directions where Δk = 0, however, this exponential may vary from +1 to +i to –1 to –i, depending on rj . The consequence is that when Δk = 0, there are cancelling contributions in (3.85), suppressing fel (Δk) and fx (Δk). Figure 3.8 illustrates how an incident plane wave, moving through an atom from left to right, is scattered coherently in the forward direction. This figure shows a set of wavelets emanating from small square volume elements of different density (the r j in (3.85)) At the left of Fig. 3.8 are two isolated points. The picture on the right of Fig. 3.8 is a geometrical construction showing how the interference between the waves scattered from all elements in the row interfere constructively in the forward direction, but at larger angles the interference is increasingly destructive. Only a row of volume elements is shown in Fig. 3.8, but similar results will be found for adjacent rows. (It is not possible for the wavelets from adjacent rows to suppress the destructive interference from the row shown in the figure.) Figure 3.9 is another illustration that shows how at intermediate scattering angles, or at intermediate Δk, the wavelets from the atom have a destructive interference that suppresses the intensity of coherent scattering. It also shows that the phase error, averaged over all scattering sub-volumes of the atom, becomes larger with the ratio of atomic size to wavelength. We expect more destructive interference for the set of wavelets emanating from larger atoms. As noted in the previous paragraph, however, an exception occurs for scattering at small angles (θ ∼ 0), where there is minimal phase difference between the scattered waves. The Δk-dependence of the atomic form factors f (Δk) is therefore different for large and small atoms – large atoms have a more rapid decrease in f (Δk) with Δk than do small atoms. This can also be understood as a wider atom having a narrower Fourier transform. Atomic form factors for electrons and x-rays are provided as tables in Appendices A.3 and A.5. Note that these tables present the form factor as a function a scalar variable, s ≡ Δk/(4π), rather than a vector, Δk. Most of the electrons about an atom form closed shells of spherical symmetry, so their Δfx,j ≡ −

3.3 Coherent Elastic Scattering

143

Fig. 3.8. Illustration of how scattering from different parts of an atom lead to coherence in the forward direction, but destructive interference at larger angles. The incident wave is from the left. Left drawings show isolated scattering subvolumes with their outgoing wavelets.

coherent elastic scattering is isotropic. It is difficult to detect anisotropies in the atomic form factor, so we usually substitute fat (Δk) → fat (Δk). (Figure 3.11 is a spectacular exception, however.)

Fig. 3.9. For large angles of scattering, phases are better preserved for waves scattered from small atoms than from large ones.

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3. Scattering

3.3.3 ‡ Scattering of Electrons by Model Potentials The potential that causes electron scattering, V (r) in (3.81) or (3.83), is Coulombic in origin. Coulomb interactions are potent, and electrons used in TEM are scattered much more strongly than x-rays used in x-ray diffraction. The positive nucleus provides a negative (attractive) contribution, but the nucleus is screened by the atomic electrons that provide a positive (repulsive) contribution. Since the atom is electrically neutral, outside the atom there is a cancellation of the electric fields from the nucleus and the atomic electrons. The incident electron is therefore unaffected by this neutral atom until it gets quite close. In fact, the high-energy electron must actually penetrate the electron cloud of the atom for scattering to occur. Inside the atom, the high-energy electron senses a net positive charge because the screening of the nuclear charge is not complete. Detailed calculations of this scattering require accurate densities of the atomic electrons. Section 3.3.4 shows how an accurate calculation of fel (Δk) can be performed if the atomic electron density is known. ‡ Screened Coulomb Potential. In this subsection we use a simple “screened Coulomb” potential to obtain an approximate analytical result. This screened Coulomb potential, V(r), is: Ze2 −r/r0 . (3.88) e r The exponential factor accounts for the screening of the nuclear charge by the atomic electrons, and r0 is an effective Bohr radius for the atom. Interestingly, the exponential decay also facilitates the mathematics of working with a potential that is otherwise strong at very large distances. We now use the first Born approximation, (3.83), to calculate the atomic scattering factor, f (Δk), as the Fourier transform of V (r) :

m e−i∆k·r V(r) d3 r . (3.89) fel (Δk) = − 2π2 V (r) = −

all space

Substituting the potential (3.88) into (3.89):

e−r/r0 3 mZe2 d r. e−i∆k·r fel (Δk) = 2 2π r

(3.90)

all space

The integral, I (Δk, r0 ), in (3.90) occurs in other contexts, so we pause to solve it. Some readers may prefer to skip ahead to the result in (3.101), or go directly to the next subsection on Thomas–Fermi and Rutherford models.

e−r/r0 3 d r, e−i∆k·r (3.91) I (Δk, r0 ) ≡ r all space

which is the 3-dimensional Fourier transform of the screened Coulomb potential (3.88). It is natural to use spherical coordinates:

3.3 Coherent Elastic Scattering

I (Δk, r0 ) =

∞ π 2π

e−i∆k·r

e−r/r0 2 r sinθ dθ dφ dr . r

145

(3.92)

r=0 θ=0 φ=0

The trick for working with the exponential in (3.92), e−i∆k·r , is to align the vector Δk along the z-axis so that Δk · r = Δkz. Also, since z = r cosθ: dz = −r sinθ dθ .

(3.93)

The limits of integration are changed as: θ = 0 =⇒ z = r ,

(3.94)

θ = π =⇒ z = −r .

(3.95)

With the substitution of (3.93)–(3.95) into (3.92): I (Δk, r0 ) =

∞ −r 2π

e−iΔkz e−r/r0 dφ(−dz)dr ,

(3.96)

e−iΔkz e−r/r0 dz dr .

(3.97)

r=0 z=r φ=0

∞ r

I (Δk, r0 ) = 2π

r=0 z=−r

Writing the exponential as e−iΔkz = cos(Δkz) − i sin(Δkz), the z-integration of the sine function vanishes by symmetry in the interval −r to +r, and the cosine integral is:

r +2 sin(Δkr) , (3.98) cos(Δkz) dz = Δk z=−r

 Using (3.98) for the z-integration which does not depend on the direction Δk. in (3.97), we obtain: 4π I (Δk, r0 ) = Δk

∞ sin(Δkr) e−r/r0 dr .

(3.99)

r=0

Equation (3.99) is the Fourier transform of a decaying exponential. This integral can be solved by twice integrating by parts.11 The result is a Lorentzian function:

∞ Δk sin (Δkr) e−r/r0 dr = . (3.100) Δk 2 + r12 r=0

11

0

Defining U ≡ Re−r/r0 and dV ≡ sin(Δkr) dr, we integrate by parts: R U dV = U V − V dU . The integral on the right hand side is evaluated as: R∞ (Δkr0 )−1 r=0 cos(Δkr) e−r/r0 dr, which we integrate by parts again to obtain: R∞ R − (Δkr0 )−2 r=0 sin(Δkr) e−r/r0 dr. This result can be added to the U dV on the left hand side to obtain (3.100).

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3. Scattering

We substitute the result (3.100) into (3.99), completing the evaluation of (3.91):

4π e−r/r0 3 d r= . (3.101) e−i∆k·r I (Δk, r0 ) = 2 r Δk + r12 0

all space

For later convenience, we now obtain a related result. The use of an exponential screening factor to perform a Fourier transform of the Coulomb potential is a useful mathematical trick. By letting r0 → ∞, we suppress the screening of the Coulomb potential, so e−r/r0 = 1 in (3.88). The Fourier transform of this bare Coulomb potential, with its mathematical form of 1/r, is obtained easily from (3.101):

4π 1 . (3.102) e−i∆k·r d3 r = r Δk 2 all space

Thomas–Fermi and Rutherford Models. With the result (3.101) for a screened Coulomb potential, we can continue with the calculation of fel (Δk) in (3.90):   2Ze2 m 1 . (3.103) fel (Δk) = 2 Δk 2 + r12 0

We need an expression for the effective Bohr radius of a multi-electron atom, r0 . Specifically, we need the fact that r0 decreases with Z. Using a result from the Thomas–Fermi model of the atom, we approximate r0 as the Bohr 1 radius of hydrogen times Z − 3 : r0 =

1 1 2 Z − 3 = a0 Z − 3 . 2 e me

(3.104)

We substitute this result for the effective Bohr radius of our Thomas-Fermi atom, r0 , in (3.103): fel (Δk) =

2Za0 Δk 2 a20

2

+Z3

.

(3.105)

It is interesting to compare the dependence on atomic number, Z, of fel (Δk) to the Z-dependence of its counterpart for x-ray scattering, fx (Δk). X-ray scattering from an atom involves the atomic electrons only (the nucleus is too massive to accelerate). The magnitude of fx (Δk) increases approximately in proportion to Z 1 because there are Z electrons about the atom. Equation (3.105) shows a different trend for electron scattering by an atom. For the usual case in TEM imaging where Δka0 is of order unity, the electron scattering factor of the atom, (3.105), increases with atomic number somewhat slower than Z 1 .12 The electron scattering factor would have 12

The cross-section from all contributions |fel (Δk)|2 therefore decreases somewhat slower than Z 2 .

3.3 Coherent Elastic Scattering

147

increased linearly with Z if the effective Bohr radius of the atom, r0 , were independent of Z (cf., (3.103)). The effective Bohr radius, r0 , decreases with Z because nuclei of heavier atoms attract more closely their core electrons. For heavier atoms, the incident high-energy electron does not sense a significant fraction of the nuclear positive charge until it gets rather close to the nucleus. These “close trajectories” are less probable, so the potent nuclear potential of high-Z elements is encountered less frequently than the nuclear potential of lighter elements. When the high-energy electron does pass close to the nucleus, the electron is deflected by a large angle. In this event the screening by the atomic electrons is less important, since the high-energy electron sees the nucleus more directly. For high-angle scattering, fel (Δk) is approximately proportional to 2 Z 1 (e.g., assume Δk is large in (3.105), so Δk 2 a20 ≫ Z 3 ). The conventional approach to high energy Coulomb scattering arises in a rather different but classic example – the scattering of energetic α-particles (He nuclei) by atoms. Rutherford and his students, Geiger and Marsden, were surprised by their observation of high angle scatterings of the α-particles. Rutherford correctly interpreted this phenomenon as the discovery of the atomic nucleus, which causes high angle deflections of the α-particles when they pass near it. His analysis of this high angle scattering assumed an unscreened Coulomb potential from a stationary nucleus, and the atomic electrons were neglected. We obtain the differential cross-section for this Rutherford scattering of electrons, dσR /dΩ, from (3.103) and (3.20) in the limit of no screening (i.e., r0 → ∞):  2 4Z 2 4Z 2 e4 m2 dσR . = = fel (Δk) = 4 dΩ  Δk 4 a20 Δk 4

(3.106)

Rutherford calculated his result with classical mechanics. The familiar form of the “Rutherford scattering cross-section” is obtained by straightforward substitutions into (3.106) of: Δk = 4π sinθ/λ, p = h/λ (which removes the quantum mechanics), and E = p2 /(2m): Z 2 e4 dσR = . (3.107) dΩ 16E 2 sin4 θ Equation (3.107) is useful for understanding some features of electron scattering at high angles. The probability that an incident electron is scattered at a high angle increases quadratically with the charge of the nucleus, Z 2 , and decreases quadratically with the kinetic energy of the incident electron, E 2 . High angle scattering is not nearly so likely as scattering at smaller angles, owing to the factor sin4 θ in the denominator.13 High-angle scattering contributes to mass-thickness contrast, but not to the diffraction contrast used in imaging studies of materials (Chap. 7). 13

In the present usage the angle θ is defined as half the total angle of scattering, consistent with our definition of the Bragg angle.

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3.3.4 ‡ * Atomic Form Factors – General Formulation In Sect. 3.3.3 we calculated the atomic form factor for electron scattering using a specific model of the atom. The model had the virtue of providing an analytical result, plus other results needed later in the book, but this “screened Coulomb model” is not a very accurate picture of an atom. Here we develop a rigorous but less specific expression for the atomic form factor for elastic electron scattering. The important input to the form factor expression will be the electron density of the atom, ρ(r), but this must be obtained independently. The resultant “Mott formula” also provides an important link between the atomic factors for electron scattering and x-ray scattering, fel (Δk) and fx (Δk). As in Sect. 3.3.3, we start with (3.84) for electron scattering:

m Vat (r) e−i∆k·r d3 r . (3.108) fel (Δk) = − 2π2 Instead of (3.88), we use a general form of Vat (r) comprising an attractive term from the nucleus (of atomic number Z) and a repulsive term from the atomic electrons (with electron density ρ(r)): Ze2 + Vat (r) = − |r|

+∞

e2 ρ(r ′ ) 3 ′ d r , |r − r′ |

(3.109)

−∞

which we substitute into (3.108): ⎛ ⎞ +∞ +∞

2 2 ′ Ze m e ρ(r ) 3 ′ ⎝− + d r ⎠ e−i∆k·r d3 r . (3.110) fel (Δk) = − 2π2 |r| |r − r ′ | −∞

−∞

We define a new variable, R ≡ r − r ′ , so r = R + r ′ , and rearrange the second term in (3.110): mZe2 fel (Δk) = 2π2

+∞

−∞

me2 − 2π2

1 −i∆k·r 3 e d r |r|

+∞

−∞

1 −i∆k·R 3 e d R |R|

+∞

 ρ(r ′ )e−i∆k·r d3 r ′ .

(3.111)

−∞

Two of the integrals of (3.111) are Fourier transforms of 1/r, for which we use (3.102): +∞

−∞

4π 1 −i∆k·r 3 e . d r= |r| Δk 2

(3.112)

Using the result of (3.112) in (3.111), we obtain a general expression for the electron form factor of an atom:

3.3 Coherent Elastic Scattering

⎛ ⎞ +∞

2me2 ⎝ fel (Δk) = 2 2 Z − ρ(r)e−i∆k·r d3 r⎠ .  Δk

149

(3.113)

−∞

The nuclear and electronic contributions to the electron scattering have provided the two terms in (3.113), and the 1/r character of the Coulomb potential provides the factor of 1/Δk 2 , which is multiplied by 2/a0 = 2m(e/)2 . Equation (3.113) is an important expression for the atomic form factor for electron scattering, given the electron density of the atom, ρ(r). It is called the “Mott formula.” The simple first term in (3.113), the electron wave amplitude scattered by the nucleus, when multiplied by its complex conjugate, gives the Rutherford cross-section (3.106). The electron distribution ρ(r) has a finite size, so the second term of (3.113) accounts for interferences of the scattered wavelets emitted from the different parts of the electron cloud. By itself, this second term of fel (Δk) has similarity to the form factor for x-ray scattering, fx (Δk). The total intensity, fel∗ fel , has a cross-term from the product of the nuclear and electronic scattering amplitudes in (3.113). This cross-term provides an interference between the coherent nuclear and electronic scatterings. It is therefore incorrect to add the intensities of the nuclear and electronic scattering, as was discussed in the context of (3.25). For x-ray scattering, the nucleus does not participate since it is too massive to accelerate. Only the atomic electrons participate in x-ray scattering, and each electron contributes to the scattered x-rays as in (3.32). The atomic form factor for x-rays therefore depends on the spatial extent of the electron charge density of the atom, ρ(r), as illustrated with Figs. 3.8 and 3.9. As in (3.87), the atomic form factor for x-ray scattering, fx (Δk), can be understood physically as a sum of phase factors, e−i∆k·rj , each associated with an outgoing wavelet from the position rj . Since the x-ray is scattered by motions of electrons, the amplitude of the wavelet emanating from the position rj is proportional to the electron density, ρ(r j ). With the Thompson scattering prefactor of (3.32), we have the following expression valid for high energy x-rays: e2 fx (Δk) = mc2

+∞

ρ(r)e−i∆k·r d3 r .

(3.114)

−∞

The atomic form factor for x-ray scattering, fx (Δk), is the Fourier transform of the electron density, ρ(r).14 When we compare (3.114) for x-ray scattering to (3.113) for electron scattering, we find the relationship: 14

This neglects the effects of anomalous x-ray scattering attributed to atomic resonances (section 3.2.1).

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3. Scattering

  2me2 mc2 f (Δk) , (3.115) Z − x 2 Δk 2 e2  3.779  (3.116) Z − 3.54 × 104 fx (Δk) , fel (Δk) = 2 Δk ˚, and Δk is in ˚ where the units for fel (Δk) and fx (Δk) are A A−1 . Since −1 a typical Δk in a diffraction problem is a few ˚ A , (3.116) shows that the electron form factor is typically 104 times stronger than the x-ray form factor. The shapes of V(r) for electron scattering and ρ(r) for x-ray scattering are different, causing different Δk dependencies of fel (Δk) and fx (Δk). Owing to the long range of the Coulomb potential, V(r) for electron scattering (e.g., (3.88) or (3.109)) is a smoother potential in real space than is the electron charge density, ρ(r), which is peaked more strongly at the center of the atom. The 1/r dependence of the Coulomb potential gives the factor 1/Δk 2 in (3.116), so the electron form factor falls off more strongly with Δk than does the x-ray form factor. A schematic comparison of these form factors is presented in Fig. 3.10. For comparison, note that neutron and M¨ ossbauer scattering (described in Sect. 3.4) have form factors that are nearly constant with Δk. Neutron and M¨ ossbauer scattering involve interactions with a nucleus, which is a scattering potential with a very small size. The Fourier transform of a point in real space is a constant in k-space.

fn 20 5

fx 10

fel 0 0

5

10

6k = 4/ sine h

15 –1

–1

[Å ]

0

Atomic Form Factor (electron) [Å]

Atomic Form Factor (x-ray) [A.U.]

fel (Δk) =

Fig. 3.10. The Δk-dependence of the atom form factors of Fe for neutrons (and γ-rays), x-rays, and electrons.

The detailed shapes of the atomic form factors for electron and x-ray scattering, fel (Δk) and fx (Δk), are determined by the details of the electron density about the atom ((3.113) and (3.114)). The atomic form factor shows clear features of the atomic shell structure. Consider the filling of the dshell across the periodic table as Z increases from 21 to 30 (Sc to Zn). The shell of d-electrons maintains about the same shape, but becomes a stronger scatterer as more d-electrons are added. This contributes to fel (Δk) mostly at small Δk because the radius is large for d-electrons. On the other hand, the increase of nuclear charge from Z=21 to 30 pulls the core electrons closer to

3.3 Coherent Elastic Scattering

151

the nucleus, contributing to increased scattering at larger Δk. The fel (Δk) from Sc to Zn has a tail that moves to larger Δk as Z increases. From Z= 31 to 36 (Ga to Kr), 4s and 4p electrons are added outside the radius of the d-shell. With the increasing nuclear charge from 31 to 36, the d-electrons reduce their radius as do the core electrons, and so contribute to fel (Δk) at somewhat larger values of Δk. It is challenging, but possible, to use measurements of x-ray atomic form factors, fx (Δk), to map the electron distributions of chemical bonds in crystals. Chemical bonding usually involves changes to only the outer electrons, so most of the x-ray scattering is unaffected and effects of bonding are small. The best possibilities for determining valence effects are at small Δk, where the outer parts of the atom are sampled most effectively. Electron atomic form factors, fel (Δk), are much more sensitive to effects of valence than are x-ray form factors, however. The sensitivity to chemical bonding of fel (Δk) occurs at small Δk, which can be understood easily by approximating the exponential in (3.113) as e−i∆k·r ≃ 1 − iΔk · r − 1/2(Δk · r)2 :  +∞ +∞

2me2 3 ρ(r) d r + iΔk · rρ(r) d3 r fel (Δk → 0) ≃ 2 2 Z −  Δk −∞

−∞

 +∞

1 2 3 + ρ(r)(Δk · r) d r . 2

(3.117)

−∞

We assume a spherically-symmetric atom or ion, so the third term in parentheses is zero. In the case of a neutral atom, the second term in parentheses equals Z, but we first allow for an ion having Z ′ electrons: ⎛ ⎞ +∞

1 2me2 ⎝ · fel (Δk → 0) ≃ 2 2 Z − Z ′ + ρ(r)(Δkr)2 (Δk r)2 d3 r ⎠ .  Δk 2 −∞

(3.118)

· r)2 : Using (9.162) for the angular average of (Δk ⎛ ⎞ +∞ 2 2 (Δk) 2me ρ(r) r2 d3 r ⎠ . fel (Δk → 0) ≃ 2 2 ⎝Z − Z ′ +  Δk 6

(3.119)

−∞

A neutral atom was assumed for the fel (Δk) shown in Fig. 3.10. In the A−1 ) case of an ion for which Z = Z ′ , at small Δk (typically Δk < 0.1 ˚ (3.119) can develop a large deviation from the curve of Fig. 3.10. Although a singularity in fel (Δk → 0) is predicted when Z ′ = Z owing to the Δk 2 in the denominator, this can occur only if ions are unscreened by compensating charge.15 In realistic calculations on neutral solids, we consider an average

15

This is a consequence of the long-range character of the Coulomb interaction around a non-neutral atom. A real crystal is electrically neutral, however. For

152

3. Scattering

Z ′ = Z, and variations in the scattering potential are treated as: me2 fel (Δk → 0) ≃ 32

+∞

ρ(r) r2 d3 r .

(3.120)

−∞

The value of fel at small Δk depends on the mean-squared radius of the atom, which is sensitive to the electron density at the outer parts of the atom. The electron form factor at small Δk is therefore a sensitive means of measuring changes in the bonding electrons in crystalline alloys and compounds. It turns out that the fel (Δk → 0) is proportional to the “mean inner potential” of the solid (see Problem 3.8 and Ref. [3.2]). Considerably more detail about the electron charge redistributions in chemical bonds (the ρ(r) in (3.113)), can be obtained from the crystallographic dependence of fel (Δk). This may be performed by analysis of HOLZ lines and other structures in CBED diffraction disks, for example. Another advantage of the CBED method is that it allows the experimenter to work with small regions of crystal that are essentially perfect. A successful use of precise form factor measurements and calculations is demonstrated in Fig. 3.11, which shows the difference between the electron density in a crystal of Cu2 O and that calculated for isolated ions. The elegant collar around the 3d3z2 −r2 orbitals at Cu ions is seen with clarity.

Fig. 3.11. Difference of electron density in crystalline Cu2 O and isolated Cu+ and O2− ions. After [3.3].

a pair of ions, one positive and one negative, it is straightforward to show that the prefactor in (3.118) changes from Δk−2 to Δk−1 . Furthermore, the Δk−1 divergence of fel is suppressed if there are alternating chains of +–+– and –+–+, or until Δk is so small that ΔkL ≃ 2π, where L is the size of the crystal along d Δk.

3.4 * Nuclear Scattering

153

3.4 * Nuclear Scattering Atoms scatter different types of radiation by different physical mechanisms, so experiments with different types of radiation can provide complementary information. As discussed in Sect. 3.2, the electric field of the incident xrays causes the electrons about an atom to oscillate, and their accelerations generate an outgoing wave. In electron diffraction, the charge of the incident electron interacts with the positively charged core of the atom, thus diverting the electron wavevector. Here we describe two nuclear scattering mechanisms that can be used for diffraction studies on materials. In neutron diffraction, the incident neutron interacts with the nuclei in the material, or with magnetic electrons. Finally, a nuclear γ-ray diffraction experiment can be designed around the M¨ ossbauer effect.16 3.4.1 Properties of Neutrons Free (i.e., unbound from the nucleus) neutrons are available from the core of a nuclear reactor. The fission of a 235 U nucleus provides 2–5 neutrons, with an average of 2.5 per fission. About 1.5 neutrons are required to sustain the reaction, so about 1 neutron per fission is available to experimenters. Nuclear spallation reactions are another source of neutrons. Nuclear spallation sources use a pulsed beam of protons with GeV energies to bombard a target of heavy elements. All sorts of particles are emitted, including neutrons, but the surroundings of the target are designed to pass the neutrons and attenuate the charged particles and photons. The free neutrons created by either fission or spallation reactions have high energies, and their kinetic energies must be reduced by a factor of about 108 so that their wavelengths are useful for diffraction and most scattering experiments. This “moderation” is accomplished by inelastic collisions with light atoms. After moderation, the “thermal” neutrons have a distribution of kinetic energies approaching that of a gas, a Maxwellian distribution, characteristic of the temperature of the moderator. Monochromatization can then be accomplished by Bragg diffraction from highly-oriented pyrolytic graphite, for example. Alternatively, since thermal neutrons have velocities of order 1,000 m/sec, monochromatization can also be accomplished by time-of-flight shuttering with “choppers” that open at precise times. Neutrons are heavy compared to electrons, and have a low energy for a given momentum. A neutron with a 1 ˚ A wavelength has a kinetic energy of about 0.082 eV, which is equivalent to a temperature of about 950 K. The scattering of a neutron by a non-magnetic atom occurs by interaction with the nucleus. There are two contributions to the scattering – potential scattering and resonance scattering. Potential scattering is best envisioned as 16

These two nuclear scattering mechanisms are also used for incoherent scattering experiments that provide energy spectra rather than diffraction patterns.

154

3. Scattering

the scattering of a neutron from a hard sphere. Potential scattering should increase with the cross-sectional area of the nucleus, so assuming the density of the nucleus is constant, the cross-section for potential scattering should increase as the atomic weight of the atom to the 2/3 power. This is only an approximate trend, however, owing to big effects from resonance scattering. Resonance scattering involves excitation and de-excitation of internal nuclear coordinates. The nucleus and the neutron sometimes combine into a compound nucleus for brief times before the neutron is re-emitted. Sometimes an energy level of the compound nucleus, Er , is low, and can be excited strongly by a neutron with energy of Er + Eb , where Eb is the binding energy of the compound nucleus. The scattering process is “resonant” because the change in phase of the scattered neutron wave is much as was discussed in Sect. 3.2.1 for the scattering of x-rays with energies above, at, and below the energy of an electronic resonance. For x-rays only a few of the electrons of the atom can be near resonance, but for neutrons the effects of resonance scattering can be larger. Resonance scattering of thermal neutrons is especially important for the isotopes: 1 H, 7 Li, 48 Ti, 51 V, 62 Ni, 64 Ni, Mn, 113 Cd, 149 Sm, 152 Sm, 162 Dy, 186 W – their scattering factors are negative. A negative scattering factor corresponds to an inversion in phase of the diffracted wave, as occurs when the resonance level is below the energy of the neutron. It is also interesting that the scattering factors for 113 Cd, 149 Sm, 152 Sm, and 157 Gd have large imaginary components, indicating that these isotopes absorb neutrons strongly. An interesting difference between x-ray and neutron resonance scattering and absorption is that for neutrons these processes depend on 1/v, where v is the velocity of the neutron – slower neutrons have more probability of interacting with the nucleus. The reader probably knows already that 235 U and 239 Pu are also strong absorbers of thermal neutrons. Transmutation of nuclei is a common phenomenon in neutron scattering experiments, and many materials become radioactive after exposure to the neutron beam. The magnetic moment of the neutron allows it to interact with unpaired electrons in magnetic materials. Control of the magnetic scattering can sometimes be performed by controlling the magnetic field at the sample, but sources of magnetically-polarized neutrons are also available. Neutron diffraction with polarized neutrons is a powerful probe of spin arrangements in magnetic crystals. In an antiferromagnetic material, for example, there is an alternation of magnetic alignment at sequential atom positions. Although the atoms are identical chemically, diffraction from the magnetic structure shows longer periodicities than the atomic repeat distance. The important nuclear parameter for neutron diffraction is the “coherent scattering length,” b [often in units of 10−12 cm]. The quantities b, fx (Δk), and fel (Δk) play equivalent roles in the coherent scattering of neutrons, x-rays, and electrons, respectively. It is possible to predict approximately neutron diffraction patterns from x-ray diffraction equations by simply sub-

3.4 * Nuclear Scattering

155

stituting |b|2 for |fx (Δk)|2 . As mentioned in Sect. 3.3.4, an important difference between neutron and x-ray scattering occurs because nuclei are essentially points. The neutron form factor therefore has no significant dependence on Δk, and very high-order diffractions can be measured at low temperatures. Thermal Debye–Waller factors suppress higher order neutron diffraction peaks in the same way as for x-ray diffraction, however (see Sect. 9.3). It is possible to perform x-ray diffraction experiments with only an occasional thought about coherence and incoherence, but not so for neutron diffraction. Neutron scattering can be elastic or inelastic, coherent or incoherent, and a single sample can have all four combinations of these processes (as mentioned in Sect. 3.1). With x-rays and electrons, inelastic scattering is usually incoherent, by which we mean that the phase of the scattered wave is not predictably related to the phase of the incident wave. Coherent inelastic neutron scattering is an important tool, however. When an individual phonon in the solid is created (or annihilated) in the scattering process, the neutron can lose (or gain) energy to the phonon. For coherent inelastic neutron scattering, the phase relationship between the incident and scattered neutron wave is predictable, and the wavelength of the excitation can be measured. Neutron scattering can also be both incoherent and elastic at the same time. Incoherent elastic scattering may occur for a crystal of Ti, for example, for two reasons. First, natural Ti contains a random mixture of isotopes. The phase of the scattered wave differs for each isotope, so unless there is an unnatural spatial regularity of the isotope distribution in the crystal, some incoherence is expected in a diffraction experiment. Second, Ti contains some isotopes having an atomic weight that is an odd number, and these have a nonzero nuclear spin. The scattering processes for different spin orientations of the same isotope do not have the same phase relationship between the incident and scattered neutron waves. 3.4.2 Time-Varying Potentials and Inelastic Neutron Scattering Time-Varying Potentials. Coherent inelastic neutron scattering is a powerful tool for studying the wavelengths and energies of elementary excitations in solids, such as phonons (vibrational waves) and magnons (spin waves). Neutron elastic scattering and neutron diffraction can be understood readily with analogies to x-ray and electron scattering and diffraction, but inelastic neutron scattering, especially coherent inelastic neutron scattering, requires additional concepts. A brief introduction is given here. Equations (3.82) and (3.83) were presented in the context of electron scattering, but nothing specific to electrons was used in obtaining them from the integral form of the Schr¨ odinger equation. They are repeated here (including the time-dependence of the outgoing wave):

 m ei(kf ·r−ωt) (3.121) V(r ′ ) e−i∆k·r d3 r ′ . Ψscatt (Δk, r, t) = − 2 2π |r|

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3. Scattering

To use (3.121) for neutron scattering, m denotes the mass of the neutron of course, and we need a potential, V(r), appropriate for neutron scattering. For nuclear scattering, we use the “Fermi pseudopotential,” which places all the potential at a point nucleus: 2 b δ(r) . (3.122) 2m Here b is a simple constant length (although sometimes it is a complex number). For thermal neutrons, the δ-function is an appropriate description of the shape of a nucleus.17 The next step is to place independent Fermi pseudopotentials at the positions {Rj }, of all atomic nuclei in the crystal. We also add one feature essential to inelastic scattering by atom vibrations – we allow the centers of the δ-functions to move with time. Our time-varying potential is: 2  V(r, t) = 4π bj δ(r − Rj (t)) . (3.123) 2m j Vnuc (r) = 4π

Substituting (3.123) into (3.121), we note the elegant cancellation of prefactors:

  ei(kf ·r−ω0 t) Ψsc (Q, r, t) = − bj δ(r ′ − Rj (t)) eiQ·r d3 r ′ . (3.124) |r| j In writing (3.124) we made the substitution Δk → −Q because this new symbol and sign are in widespread use for neutron scattering. The integration over the δ-functions of (3.124) fixes the exponentials at the nuclear positions {Rj (t)}: Ψsc (Q, r, t) = −

ei(kf ·r−ω0 t)  bj eiQ·Rj (t) . |r| j

(3.125)

It is convenient to separate the static and dynamic parts of the nuclear positions: Rj (t) = xj + uj (t) ,

(3.126)

so by substitution: Ψsc (Q, r, t) = −

ei(kf ·r−ω0 t)  bj eiQ·(xj +uj (t)) . |r| j

(3.127)

When Q · u is small, we can expand the exponential in (3.127) to obtain: 17

For magnetic scattering, however, an electron spin density is used, and this reflects the shape of the atom. Also, the potential for magnetic scattering has vector character.

3.4 * Nuclear Scattering

157

ei(kf ·r−ω0 t) |r|    1 × bj eiQ·xj 1 + iQ · uj (t) − (Q · uj (t))2 + . . . . (3.128) 2 j

Ψsc (Q, r, t) = −

Elastic Neutron Scattering. Neglecting the time-dependence of the scattering potential, i.e., setting uj (t) = 0 in (3.128), we recover the case of elastic scattering. The first term in parentheses in (3.128), the 1, involves el (Q, r), only the static part of the structure. We isolate this static term, Ψsc as the elastic part of the scattered wave: el (Q, r) = − Ψsc

ei(kf ·r−ω0 t)  bj eiQ·xj . |r| j

(3.129)

Because b is the neutron equivalent to f (Δk) for the coherent elastic scattering of electrons or x-rays, the further development of neutron diffraction takes the same path that follows (5.18) in Chapter 5. Phonon Scattering. The next term in (3.128), involving Q · uj (t), gives inelastic scattering. To calculate the inelastically-scattered neutron wavefunction, we first need the motions of all nuclei. For this we use the phonon expression for collective atom motions, uj (ω, q, t): U j (ω, q) i(q·xl −ωt) e . uj (ω, q, t) = 2Mj ω

(3.130)

Equation (3.130) has a typical form for an elementary excitation in a solid. In particular, this phonon excitation is specified by its combination of wavevector q and frequency ω. The phase factor, eiq·xj (t) , provides the long-range spatial modulation of uj at all atom positions. This spatial modulation has a “polarization,” U j that identifies the amplitude and direction of atom motions. The nuclear mass, Mj , is essentially the entire mass of the atom centered at Rj . After substitution of (3.130), the second term in (3.128) gives inel (Q, r): an inelastically-scattered wave, Ψsc i ei(kf ·r−ω0 t) |r|    bj × Q · U j (ω, q) eiQ·xj ei(q·xj −ωt) , 2Mj ω j

inel (Q, r, t) = − Ψsc

i eikf ·r −i(ω0 +ω)t e |r|    bj Q · U j (ω, q) ei(Q+q)·xj , × 2Mj ω j

(3.131)

inel Ψsc (Q, r, t) = −

(3.132)

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3. Scattering

Equation (3.132) identifies two important features about the phase of the neutron wavefunction after coherent inelastic scattering.18 First, the neutron wavefunction changes frequency from ω0 to ω0 + ω because the neutron gains an energy ω by annihilating the phonon of frequency ω. The wavevector in the phase factor for the neutron wavelet scattered from Rj is not the same Q as for elastic scattering, but is Q + q, equivalent to the momentum difference q. Energy and momentum are conserved, but are transferred between the crystal and the neutron. A spectrometer for inelastic neutron scattering measures the momentum and energy of scattered neutrons, and this may be enough information for the experimenter to deduce the frequencies and wavevectors of the elementary excitations in the sample. There are many additional considerations in such work, of course. 3.4.3 * Coherent M¨ ossbauer Scattering M¨ ossbauer γ-ray diffraction is a relatively new phenomenon. Of course γ-ray photons can be diffracted by the same electronic scattering mechanisms as xray photons, but “M¨ ossbauer diffraction” means that the γ-rays are absorbed and re-emitted by nuclei. During the absorption and re-emission processes, the nucleus makes transitions between a ground state and an excited state, which differ by 14.41 keV for the 57 Fe nucleus, for example. In practice, this absorption and reemission process requires the existence of the M¨ossbauer effect. In the M¨ ossbauer effect, a γ-ray can be emitted by one nucleus and absorbed by another if the two nuclei are embedded in solids, so no energy is lost to nuclear recoil. Rudolf M¨ ossbauer’s explanation of this effect was worthy of a Nobel Prize in 1961. M¨ ossbauer scattering can be useful for diffraction experiments when the collective nuclear excitation in a crystal (or “nuclear exciton”) decays to the ossbauer same ground state from which it originated.19 In this case the M¨ nuclear scattering has a precise phase relationship with the coherent x-ray scattering from the electrons of the atom, and the form factor of the atom is: m f (Δk, δεm i ) = −fx (Δk) + fM (Δk, δεi ) ,

(3.133)

where fx and fM are the x-ray and M¨ ossbauer form factors. We use the convention −fx (Δk) for the x-ray form factor because the γ-ray drives the atomic electrons at a frequency far above their resonance (ω ≫ ωr of Sect. 3.2.1). Here δεm i is the precise difference in energy between the incident photon and the energy of the nuclear excitation. The value of δεm i depends on the nuclear spin coordinates, m, and the chemical environment of the nucleus, 18

19

Other features that can be identified are the scaling of the phonon scattering inel∗ inel intensity, Ψsc Ψsc , with Q2 and with the factor b2 /Mj . Especially for single crystals, the orientation information Q · U j is also useful. If one nucleus were to decay to a ground state with a flipped spin, for example, it would be known that the excitation occurred at that particular nucleus, so interference cannot occur with wavelets emitted from different nuclei.

3.4 * Nuclear Scattering

159

parameterized by i. Typically this precise energy difference may range over 10−7 eV for 57 Fe, whose nuclear transition energy of 14.41 keV has a halfwidth, Γ , of only 4.55 × 10−9 eV. Notice the remarkable precisions of these nuclear energy levels, which permit measurements of tiny energy shifts such as from the hyperfine interactions discussed below. Simple diffraction patterns from pure bcc 57 Fe are presented in Fig. 3.12. The diffraction pattern at the top was obtained with the incident γ-rays tuned to the M¨ ossbauer resonance, so both nuclear and electronic scatterings were operating. The diffraction pattern in the middle was obtained with the incident γ-rays detuned from the nuclear resonance, so only x-ray electronic scattering was possible. The diffraction pattern at the bottom is the difference between the two. Notice how the M¨ossbauer nuclear scattering is much stronger than the x-ray electronic scattering for high-order diffraction peaks, largely because the x-ray form factor decreases strongly for high-order peaks, but the nuclear form factor does not (see Fig. 3.10).

Fig. 3.12. M¨ ossbauer diffraction patterns from bcc 57 Fe, acquired in several hours. After [3.4].

It is easy to drive the M¨ossbauer scattering below, at, or above resonance by tuning the energy of the incident photon. The functional form of the M¨ ossbauer scattering near resonance allows (3.133) to be rewritten as: f (Δk, δεm i ) = −fx(Δk) + 

−Gm i

δεm i Γ

 , +i

(3.134)

where the factor Gm i includes all information about the nuclear transition and the efficiency of the M¨ ossbauer effect for the chemical environment i. The denominator in (3.134) describes how the amplitude and phase of the M¨ ossbauer scattering are changed as the photon energy approaches the nuclear resom nance. Precisely at resonance (i.e., when δεm i = 0), f (Δk) = −fx (Δk)+iGi , ◦ and the M¨ ossbauer scattering is strongest but is 90 out-of-phase with the x-ray electronic scattering. In the case when the photon energy is far above

160

3. Scattering

resonance for both M¨ ossbauer and x-ray scattering, these two scatterings are nearly in phase.20 The wave fields from the two scatterings then add constructively. On the other hand, when the photon energy is well below the nuclear resonance, the two scatterings add destructively, suppressing scattering. Since Gm ossbauer energy spectrum can be i can be comparable to |fx (Δk)|, the M¨ altered significantly when there is coherent interference between M¨ ossbauer and x-ray scattering. Weak hyperfine interactions between the nucleus and its surrounding electrons allow the different chemical environments {i} to be identified in a M¨ ossbauer energy spectrum. These chemical shifts can be of order 10Γ . This provides for a unique chemical environment selectivity in M¨ ossbauer diffraction. The spectroscopic capabilities of the M¨ ossbauer effect can be used to tune the incident photon so that the diffraction pattern originates from 57 Fe atoms in the specific chemical environment, i. The phase of the M¨ ossbauerscattered wave is also highly sensitive to the local chemical environment of the 57 Fe atom.

Further Reading The contents of the following are described in the Bibliography. R. F. Egerton: Electron Energy-Loss Spectroscopy in the Electron Microscope, 2nd Ed. (Plenum Press, New York 1969). P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley, and M. J. Whelan: Electron Microscopy of Thin Crystals (R. E. Krieger, Malabar, Florida 1977). L. Reimer: Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, 4th Ed. (Springer–Verlag, New York 1997). G. L. Squires: Introduction to the Theory of Thermal Neutron Scattering (Dover, Mineola, New York 1996). J. C. H. Spence and J. M. Zuo: Electron Microdiffraction (Plenum, New York 1992). B. E. Warren: X-Ray Diffraction (Dover, Mineola, New York 1990).

Problems 3.1 Radiation can be scattered by the electrons or by the nuclei of atoms. Explain how these are important for the scattering of: (a) electrons, (b) x-rays, (c) neutrons. 20

Specifically, when δεm = +10Γ (i.e., δεm = 4.55 × 10−8 eV for 57 Fe), i i m m f (Δk, δεm ) = −f (Δk) − 0.099G + i0.0099G x i i i , and the phase difference between M¨ ossbauer and x-ray scattering is only 5.7◦ .

Problems

161

3.2 (a) Show that ψ= ψ0 e−ikx is a solution of the time-independent Schr¨ odinger equation in one dimension in a constant potential: −2 ∂ 2 ψ + U00 ψ = Eψ . 2me ∂x2

(3.135)

(b) For an electron with energy 100 keV, what is its (nonrelativistic) wavevector, k, when the potential, U00 , is zero (free space)? (c) The electron moves into a material, where the effective potential is attractive, changing the potential energy of the electron by 10 eV. Why does the electron wavelength change, and by how much? (Hint: Consider first the sign of the change in electron kinetic energy, then the corresponding changes in velocity and wavelength.) 3.3 (a) Using Appendix A.5, estimate the atomic scattering factor (amplitude) of 200 keV electrons at the angle of the (200) diffraction from Al and Ag. (Use aAl = 0.404 nm and aAg = 0.409 nm.) (b) Do the same for high-energy x-rays (Appendix A.3). 3.4 Calculate the fraction of x-rays transmitted through a quartz (SiO2 ) window of 0.6 cm thickness when the energies of the x-rays are: (a) 5.4 keV, and (b) 17.4 keV. (Quartz has a density of 2.65 g cm−3 . Use Appendix A.2.) 3.5 Starting with the tabulated x-ray form factor for aluminum, use the procedure explained in Appendix A.5 to calculate the electron form factor for Al for 300 keV electrons for s = 0.1, 0.5, 1, 2, 3, and 5 ˚ A−1 . 3.6 What energy of x-rays should be used so that the maximum change in their wavelength after Compton scattering is 1 percent? 3.7 Consider the anomalous scattering of x-rays using the classical model of Sect. 3.2.1. Near the K-edge of Cu (9 keV), two electrons are near resonant excitation and 27 are not. The 27 are driven at a frequency above resonance, but the two 1s electrons have a change in phase as the x-ray energy approaches 9 keV. Use (3.37) and (3.38) with the ratio β = 0.15ωr to answer this question. Here Eph = ω. (a) Graph the real part and imaginary part of fCu (Eph ). 2

(b) Graph the intensity, |fCu (Eph )| . (c) Take the square root of the minimum of the intensity, and estimate the maximum change in the effective scattering factor of Cu near resonance. (d) Describe the deficiencies of the classical model when Eph > 9 keV. 3.8 Show how the electron form factor as Δk → 0 can be used to measure the inner potential of a crystal, U00 of problem 3.2, and obtain U00 in terms of the electron charge density of the atom. (Hint: For a monoatomic crystal, assume the structure factor Fg = fel . Use fel (Δk → 0) of (3.120) and (11.18) and (11.41) with Ug−0 = U00 and ξg−0 = ξ00 .)

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3. Scattering

3.9 The Fourier transform of the screened Coulomb potential:

e−r/r0 3 d r, I (Δk, r0 ) ≡ e−i∆k·r r

(3.136)

all space

can be evaluated without the sine and cosine functions of (3.98) and the partial integrations needed for (3.100). Calculate (3.101) or (3.136) by direct attack on the z-integration of (3.97). (Hint: Manipulate the integrals of exponentials with complex arguments as you would exponentials with real arguments. Obtain real denominators by multiplying by complex conjugates.) 3.10 Use the first Born approximation to calculate the scattering form factor, f (Δk), from an exponential potential: V(r) = V0 e−r/r0 .

(3.137)

(Hint: This is a 3-dimensional problem, and is not simply the Fourier transform of a decaying exponential. Align Δk along the z-axis of the spherical coordinate system as in (3.92), and note the hints of problem 3.9.) 3.11 (difficult) Use the first Born approximation to show that the total scattering by a real potential that falls off as r−n is finite if and only if n > 2. (Hint: Note the use of the words “falls off.” Beware of singularities – the forward scattering may need to be treated separately.)

4. Inelastic Electron Scattering and Spectroscopy

4.1 Inelastic Electron Scattering Principles. This Chapter 4 first describes how high-energy electrons are scattered inelastically by materials, and then explains how electron energyloss spectrometry (EELS) is used in materials research. Inelastic scattering occurs by the processes listed below in order of increasing energy loss, E. Energy is conserved in these inelastic processes – the spectrum of energy gains by the sample is mirrored in the spectrum of energy losses of the highenergy electrons. Electrons undergoing energy losses to crystal vibrations, quantized as phonons with E ∼ 10−2 eV, are indistinguishable from elastically scattered electrons, given the present state-of-the-art for EELS in a TEM.

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4. Inelastic Electron Scattering and Spectroscopy

With modern instrumentation, it is possible to measure interband transitions of electrons from occupied valence bands to unoccupied conduction bands of semiconductors and insulators. With E ∼ 2 eV, these spectral features are quite close in energy to the intense zero-loss peak from elastic scattering, so resolving them has been a challenge. In many solids, especially metals, the bonding electrons can be understood as a gas of free electrons. When a high-energy electron suddenly passes through this electron gas, plasmons may be created. Plasmons are brief oscillations of the free electrons, giving broadened peaks in EELS spectra. Plasmon energies (E ∼ 10 eV) increase with electron density, so plasmon spectra can be used to estimate free electron density. Plasmon spectra are also useful for measuring the thickness of a TEM specimen because more plasmons are excited as the electron traverses a thicker specimen. Electrons that ionize atoms by causing core excitations are used for microchemical analysis. Chemical spectroscopy with EELS measures the intensities of “absorption edges,” which are jumps in spectral intensity at the threshold energies for ejecting core electrons from atoms in the material (102 < E < 104 eV). After a core electron has been excited from the atom, the remaining “core hole” decays quickly, often by the emission of a characteristic x-ray. Characteristic x-rays with energies from > 102 to > 104 eV are used in energy dispersive x-ray spectrometry (EDS) for chemical analysis. Methods. “Analytical transmission electron microscopy” uses EDS or EELS to identify the elements in a specimen, and to measure elemental concentrations or spatial distributions. To quantify chemical concentrations, a background is subtracted to isolate the heights of absorption edges (EELS) or the intensities of peaks in an x-ray energy spectrum (EDS). These isolated intensities are then compared for the different elements in the specimen, and often converted into absolute concentrations with appropriate constants of proportionality. The accuracy of quantification depends on the reliability of these constants, so significant effort has been devoted to understanding them. In this chapter, after brief descriptions of an EELS spectrometer and features of a typical EELS spectrum, plasmon energies are discussed with a simple model of a free electron gas. The section on “core excitations” provides a higher-level treatment of how a high-energy electron can cause a core electron to be ejected from the atom. It turns out that the probability of a core electron excitation is proportional to the square of the Fourier transform of the product of the initial and final wavefunctions of the excited electron. The cross-section for inelastic scattering also has an angular dependence that must be considered when making quantitative measurements with EELS. Some experimental aspects of EELS measurements are presented, including energy-filtered TEM imaging. This chapter then presents the principles of EDS in the TEM, which involves more physical processes than EELS. Interestingly, the cross-section for core ionization decreases with atomic number, but the cross-section for x-

4.2 Electron Energy-Loss Spectrometry (EELS)

165

ray emission increases with atomic number in an approximately compensating way. This gives EDS spectrometry a balanced sensitivity for most elements except the very lightest ones.

4.2 Electron Energy-Loss Spectrometry (EELS) 4.2.1 Instrumentation Spectrometer. After electrons have traversed a TEM specimen, a significant minority of them have lost energy to plasmons or core excitations, and exit the specimen with energies less than the energy of the incident electrons, E0 (E0 may be 200, 000 ± 0.5 eV, for example). To measure the energy spectrum of these losses, an EELS spectrometer can be mounted after the projector lenses of a TEM. The heart of a transmission EELS spectrometer is a magnetic sector, which serves as a prism to disperse electrons by energy. In the homogeneous magnetic field of the sector, Lorentz forces bend electrons of equal energies into arcs of equal curvature. Some electron trajectories are shown in Fig. 4.1. The spectrometer must allow an angular range for electrons entering the magnetic sector, both for reasons of intensity and for measuring how the choice of scattering angle, φ, affects the spectrum (cf., (4.44)). A well-designed magnetic sector provides good focusing action. Focusing in the plane of the paper (the equatorial plane) is provided by the magnetic sector of Fig. 4.1 because the path lengths of the outer trajectories are longer than the path lengths of the inner trajectories. It is less obvious, but also true, that the fringing fields at the entrance and exit boundaries of the sector provide an axial focusing action. With good electron optical design, the magnetic sector is “double-focusing” so that the equatorial and axial focus are at the same point on the right of Fig. 4.1. Since the energy losses are small in comparison to the incident energy of the electrons, the energy dispersion at the focal plane of typical magnetic sectors is only a few microns per eV. Electrons that lose energy to the sample move more slowly through the magnetic sector, and are bent further upwards in Fig. 4.1.1 In a “serial spectrometer,” a slit is placed at the focal plane of the magnetic sector, and a scintillation counter (see Sect. 1.4.1) is mounted after the slit. Intensity is recorded only from those electrons bent through the correct angle to pass through the slit. A range of energy losses is scanned by varying the magnetic field in the spectrometer. A “parallel spectrometer,” shown in the chapter title image, covers the focal plane of the magnetic sector with a scintillator and a position-sensitive photon detector such as a photodiode array. The postfield lenses Q1–Q4 magnify the energy dispersion before the electrons reach the scintillator. A parallel spectrometer has an enormous advantage over a 1

Their longer time in the magnetic field overcomes the weaker Lorentz forces.

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spectrometer entrance aperture `, entrance semi-angle

B

Fig. 4.1. Some electron trajectories through a magnetic sector with uniform magnetic field, B. The light curves are trajectories for lower-energy electrons (those with larger energy loss, E), and heavier curves are for higher-energy electrons. The collection semiangle of the spectrometer is β.

serial spectrometer in its rate of data acquisition, but it requires calibrations for variations in pixel sensitivity. The optical coupling of a magnetic sector spectrometer to the microscope usually puts the object plane of the spectrometer at the back focal plane of the final projector lens (Fig. 2.45). This back focal plane contains the diffraction pattern of the sample when the microscope is in image mode. When the microscope is operated in image mode, the spectrometer is therefore said to be “diffraction-coupled” to the microscope. With diffraction coupling, the collection angle, β, of the spectrometer is controlled by the objective aperture of the microscope. Alternatively, when the microscope is operated in diffraction mode, the back focal plane of the projector lens contains an image, and the spectrometer is said to be “image-coupled” to the microscope. With image-coupling, the collection angle, β, is controlled by an aperture at the entrance to the spectrometer (at top of Fig. 4.1). Monochromator. The typical energy resolution for EELS spectrometers was about 1 eV or so for many years, but recent developments have allowed energy resolutions better than 0.1 eV on commercial microscopes. This is accomplished by starting with a field emission gun, often a Schottky effect gun (Sect. 2.4.1), followed by an electron monochromator, often a Wien filter as described here. Electrons traveling through a Wien filter encounter a region with crossed electric and magnetic fields that induce competing forces on the electron. In a Wien filter these electric and magnetic forces are tuned to cancel for electrons of one velocity, v0 , which avoid deflection and pass through the exit aperture of the filter. Specifically, for an electron with velocity vz down the optic axis along z, a magnetic field oriented along y produces a force  in the , Fxmag = evz By . A Wien filter has an electric field along x along x same region, generating a force on the electron of Fxel = −eEx . The special condition of cancelling forces, Fxmag = −Fxel , can be true for electrons of only only one velocity, v0 :

4.2 Electron Energy-Loss Spectrometry (EELS)

ev0 By = eEx , Ex v0 = . By

167

(4.1)

Electrons with velocities differing from v0 are deflected, and do not pass through the exit aperture of the Wien filter. In practice, it is typical to operate the Wien filter at a voltage close to that of the electron gun itself, so the electron velocity through the filter will be slow enough that sub-eV resolution is possible with reasonable values of electric field, magnetic field, and aperture size. Biasing the Wien filter assembly near –100 or –200 keV can be challenging, however. The Wien filter first disperses electrons of different energies into different angles, and then allows electrons of only a selected energy to pass through its exit aperture. Monochromatization therefore discards a substantial fraction of electrons – perhaps 80% of the electrons are lost when monochromating to 0.1 eV. When operating in STEM mode, the electron current is also reduced substantially when forming the smallest electron probes. It is typical to make compromises between the brightness of the image, the electron monochromatization, and the size of the probe – an increase in one usually requires a decrease in another. Manufacturers are constantly trying to find ways to improve these aspects of microscope performance. 4.2.2 General Features of EELS Spectra A typical EELS spectrum is presented in Fig. 4.2. The enormous “zero-loss peak” is from electrons of 200,000 eV that passed through the specimen without any energy loss. The sharpness of this peak indicates that the energy resolution is about 1.5 eV. The next feature is at the energy loss E = 25 eV, from electrons having energies of 199,975 eV. It is the “first plasmon peak,” caused by the excitation of one plasmon in the sample. With thicker specimens there may also be peaks at multiples of 25 eV from electrons that excited two or more plasmons in the specimen. The small bump in the data at 68 eV is not a plasmon peak, but rather a core loss. Specifically it is a Ni M 2,3 absorption edge caused by the excitation of 3p electrons out of Ni atoms. An enormous feature is seen at an energy loss of about 375 eV, but it is an artifact of the serial data acquisition method, and not a feature of the material. (At 375 eV the detector operation was changed from measuring an analog current to the counting of individual electron events.) The background in the EELS spectrum falls rapidly with energy (the denominator of Δk 2 in (4.28) is partially responsible for this), and the next feature in the Ni spectrum of Fig. 4.2 is a core loss edge at 855 eV. This feature is caused by the excitation of 2p3/2 electrons out of the Ni atom, and is called the “L3 edge.” The L2 edge at 872 eV is caused by the excitation of 2p1/2 electrons out of the atom. Right at the L2 and L3 edges are sharp, intense peaks known as “white lines” that originate from the excitation of 2p

168

4. Inelastic Electron Scattering and Spectroscopy

Fig. 4.2. EELS spectrum of Ni metal, showing zero-loss peak, bulk plasmon, and L-edge with white lines at the edge. After [4.1].

electrons into unoccupied 3d states at a Ni atom. Such features are typical of transition metals and their alloys as described in Section 4.2.3. More generally, unoccupied states such as antibonding orbitals are often responsible for sharp peaks at core edges. Compared to plasmon excitations, the cross-sections for inner-shell ionizations are relatively small, and become smaller at larger energy losses. To obtain good intensities, for many elements it is preferable to use absorption edges at lower energy losses (e.g., L and M ). Some of the nomenclature of electronic transitions was given previously in Sect. 1.2.2. Figure 4.3 shows an orbital representation and associated nomenclature for EELS edges. N N 6,7 N2,3 4,5

Absorption Edges

N1 M4,5

M M1 2,3 L L 2,3 1

K 1 2

1 13 2 22

1 13 35 2 22 22

1 13 35 57 2 22 22 22

1s

2s 2p

3s 3p 3d

4s 4p 4d 4f

K

L

M State Notation

N

j n,l shell

Fig. 4.3. Possible edges of innershell ionizations and their nomenclature.

4.2 Electron Energy-Loss Spectrometry (EELS)

169

4.2.3 * Fine Structure Near-Edge Fine Structure. The region in an EELS spectrum around a core-loss edge often shows clear and reproducible structure that can be used to identify the local chemical environment. This “electron energy-loss near-edge structure” (ELNES) depends on the number and energy of unoccupied states at the excited atom. Chemists call these low-lying unoccupied states “lowest unoccupied molecular orbitals,” and they include antibonding orbitals. Physicists call them “states above the Fermi energy,” and they include the conduction band. A core electron can be excited into these unoccupied states, and the energy gained by the core electron during this transition is mirrored in the energy-loss spectrum of the high-energy electron. Simple metals with nearly-free electrons show core edges in EELS spectra that are smooth and without sharp features. On the other hand, materials with high densities of states just above the Fermi level, such as transition metals and rare-earth metals, have sharp features at their absorption edges associated with transitions into unoccupied d- and f -states, respectively. These features do not appear at all absorption edges, owing to the dipole selection rule where the angular momentum must change by ±1. This selection rule allows transition metals with unoccupied d-states to have intense white lines at their L2,3 edges, which involve excitations from core p-electrons, but not at their L1 edges, which involve excitations from s-electrons (see Fig. 4.2). The intensity of the white lines at the L2,3 edge of Ni in Fig. 4.2 can be understood with the inelastic cross section for core shell ionizations (4.37), where ψβ is an unoccupied 3d state and ψα is an occupied 2p core state, both centered at the Ni atom. The intensities of the white lines are larger when there are more unoccupied 3d states (the factor ρ(E) in (4.37)). If the integral in (4.37) is evaluated, integrated intensities of the white lines can be used to quantify ρ(E), the number of unoccupied 3d states at Ni atoms, and how this number changes with alloying or chemical bonding. Likewise, rare earth metals with unoccupied f -states have sharp features at their M4,5 edges, which involve core d-electrons (but not at their M2 or M3 edges, which involve p-electrons). Semiconductors and insulators usually show distinct structure at their absorption edges, owing to the excitation of core electrons into unoccupied states above the band gap. Because the number of unoccupied states is sensitive to the chemical and structural environment around the excited atom, ELNES can be used as a “fingerprint” of its local environment, even when the experimental systematics are not simple, or when electronic structure calculations are not possible. Figure 4.4 shows that the oxygen K-edge ELNES is sensitive to the local environment around the O atom in a variety of manganese oxides. The structure around 527-532 eV is dominated by the effects of chemical bonding on the density of electron states at the O atom, but the peak from 537-545 eV is more sensitive to the local positions of Mn atoms near the O atom – it is part of the “extended fine structure,” described below.

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4. Inelastic Electron Scattering and Spectroscopy

Fig. 4.4. Oxygen K-edges from various manganese oxides, showing a variety of ELNES spectral features. After [4.2].

Changes to the chemical environment around an atom alter the energy of the lowest unoccupied state, and therefore shift the onset energy of the core edge. Chemical shifts of absorption edges therefore reflect changes in the energies of the unoccupied states. More subtly, however, they also reflect changes in the energies of the core states. Any change to the outer atomic electrons, as caused by changes in chemical bonding for example, alters the intra-atomic electron-electron interactions. The energies of core electrons are therefore altered by changes in the outer electrons. For example, if an outer electron of a Li atom is transferred to a neighboring F atom, one may expect a lower-energy unoccupied state about the Li, and a shift of the Li K edge to lower energy. In fact, however, the loss of this electron in Li+ reduces the screening of the core 1s electrons, causing them to be more tightly bound to the nucleus. This causes the absorption edge to shift to higher energies. Lithium has only three electrons, so this effect is anomalous, but chemical shifts of absorption edges for all elements depend in part on the shifts in energies of the core electrons caused by intra-atomic screening. Finally, we note that the core hole itself alters the energies of the atomic electrons. It is sometimes assumed that the removal of a core electron serves to increase the effective nuclear charge from Z to Z +1, but the effect of a core hole on the energy levels of an unstable atom are not so easy to understand. Extended Fine Structure. Extended electron energy-loss fine structure (EXELFS) starts at energies where the outgoing electron state can be considered free of the atom, perhaps about 30 eV beyond the absorption edge. The state of the outgoing electron from the “central atom” is affected by the surrounding atoms, and self-interference occurs as the outgoing electron is backscattered from the nearest-neighbor shells of atoms. This process is illustrated schematically in Fig. 4.5. With changes in the wavelength of the outgoing electron, constructive and destructive interference occurs, causing

4.2 Electron Energy-Loss Spectrometry (EELS)

171

the EXELFS signal, χ, to be oscillatory:  Nj fj (k) 2 2 Nj fj (k) e−2rj /λ e−2σj k sin(2krj + δ0 + δj ) . (4.2) χ(k) = 2k r j j Equation (4.2) includes a number of different effects, and its factors are best justified one-by-one. The sine function is the oscillatory interference of the outgoing electron wavefunction with itself as it travels the distance 2rj from the central (excited) atom to a neighboring atom at rj and back again. The phase of this electron wave is shifted by the amount δj upon scattering by the neighboring atom at the distance rj , and by δ0 from the central atom. These phase shifts generally depend on the electron wavevector, and this kdependence must be known for quantitative work. The other factors in (4.2) are the number and backscattering strength of the neighboring atoms, Nj and fj (k), a qualitative decay factor to account for the finite lifetime of the outgoing electron state, e−2rj /λ (where λ is the electron mean-free-path), 2 2 and a Debye-Waller factor, e−2σj k , that attenuates χ(k). Here σj2 is a meansquared displacement of the central atom relative to its neighboring atoms, typically originating with temperature or disorder in the local structure. The sum in (4.2) is over the neighboring atoms, and typically includes the firstand second-nearest-neighbor (1nn and 2nn) shells around the central atom.

Fig. 4.5. Pictorial representation of the electron interference that gives rise to EXELFS. Crests of the electron wavefunction emanating from a central atom are drawn with an amplitude that diminishes with distance. For this particular wavevector and phase shifts, the electron wave crest backscattered from the four neighboring atoms is in phase with the wave crest emanating from the central atom, giving constructive interference and an enhanced probability for the emission of the electron.

Figure 4.6 shows some steps in a typical EXELFS analysis, in this case for the L2,3 edge from a slightly-oxidized sample of bcc Fe metal. Figure 4.6a shows the absorption edge after correction for the pre-edge background. The region of interest begins above the L3 and L2 edges. Unfortunately, the L1 edge (2s excitation) appears as a feature in the region of interest, so it is best to work with data at energies beyond the L1 edge. The useful data range did include the oscillations with broad peaks at about 920 and 1000 eV (barely visible in Fig. 4.6). Extracting these small oscillations from the monotonic decay characteristic of an isolated atom is usually done by fitting a cubic spline function through the EXELFS oscillations. Subtracting this spline fit reveals the oscillations in energy, which are converted to k-space as in Fig. 4.6b, using the wavevector dependence on energy above the absorption edge,

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4. Inelastic Electron Scattering and Spectroscopy

˚−1 ): Ea (where k is in A E − Ea =

2 k 2 = 3.81 k 2 [eV] , 2me

(4.3)

Real space periodicities are obtained from the data of Fig. 4.6b by taking their Fourier transform.2 The periodicities in real space are not affected significantly if χ(k) is multiplied by a power of k, and doing so helps to sharpen the peaks in the real space data. The real-space data of Fig. 4.6c, called a “pseudo-” or “raw-” radial distribution function, were obtained by taking the Fourier transform of kχ(k). The peak at 2.25 ˚ A corresponds approximately to the position of the 1nn shell of Fe atoms in bcc Fe, but this distance is not A because the phase shifts δj and δ0 of (4.2) quite the expected value of 2.02 ˚ were not included in the data analysis. For comparative work with similar specimens, however, this simple Fourier transform method may be adequate.

Fig. 4.6. (a) Fe L-edge from pure Fe metal at 97 K. Pre-edge background was subtracted, but no corrections were performed for plasmon excitations, which do not affect the gradual EXELFS structure. (b) Fe L2,3 edge EXELFS extracted from data in a. (c) Magnitude of Fourier transform of data in b. After [4.3]. 2

It is an approximation to ignore the slight phase difference between the L1 and L2 EXELFS oscillations, and to neglect the L1 EXELFS, but the approximation is not too bad.

4.3 Plasmon Excitations

173

Better-known than EXELFS is EXAFS (extended x-ray absorption fine structure) spectroscopy, performed with tuneable synchrotron radiation. EXAFS is identical to EXELFS, except that the excitation of the central atom is caused by a photon. The energy of the incident photon is tuned from below an absorption edge to well above it. The self-interference of the backscattered photoelectron is seen in the data as decreased or increased photon transmission through the sample (or electron yield in another variant of the EXAFS technique). The analysis of the χ(k) data is identical to that of EXELFS, and (4.2) was originally proposed for EXAFS. There is a stronger E-dependence for EXELFS spectra than for EXAFS spectra, causing EXELFS to be more practical than EXAFS for energies below about 2 keV. Nevertheless, EXAFS is more practical at higher energies, and higher energies have two advantages. Atomic levels at higher energies are better separated in energy, making it easier to obtain wide ranges of energy where the extended fine structure can be measured without interruption from other absorption edges. The second advantage of EXAFS is its ability to work with K-shell excitations of many elements, whose simpler structure allows their χ(k) to be interpreted more reliably. On the other hand, EXELFS can be performed readily on local regions of material identified in TEM images. Synchrotron beamline optics including x-ray mirrors and Fresnel zone plates now allow EXAFS measurements on areas as small as ∼ 1 μm, however.

4.3 Plasmon Excitations 4.3.1 Plasmon Principles A fast electron jolts the free electrons when it passes through a material. This displaced charge creates an electric field to restore the equilibrium distribution of electrons, but the charge distribution oscillates about equilibrium for a number of cycles. These charge oscillations are called “plasmons,” and are quantized in energy. Larger energy losses correspond to the excitation of more plasmons, not to an increase in the energy of a plasmon. In most EELS spectra, the majority of inelastic scattering events are plasmon excitations. To find the characteristic oscillation frequency of a plasmon, consider the rigid translation of a wide slab of electron density by a small amount, x, as in Fig. 4.7. At the bottom surface of the slab all the electrons are removed, but at the top the electron density is doubled. This charge disturbance therefore sets up the electric field, E, of a parallel-plate capacitor: E = 4πσs ,

(4.4)

where σs is the surface charge density equal to the electron charge, e, times ρ, the number of electrons per unit volume, times the displacement, x: σs = eρx .

(4.5)

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4. Inelastic Electron Scattering and Spectroscopy

Fig. 4.7. Displacement of a slab of electric charge, leading to doubling of the charge density at the top of the slab over thickness x, and depletion of charge at the bottom. A wide, flat slab idealizes the problem as one dimensional.

The field, E, provides the restoring force to move the electron slab back to its original position. The electrostatic restoring force per unit volume of slab is: F = −eρ E .

(4.6)

Substituting (4.5) into (4.4), and then into (4.6) provides: F = −e2 ρ2 4πx .

(4.7)

The Newtonian equation of motion per unit volume of the electron slab is: d2 x . dt2 Substituting (4.7) into (4.8) gives:   4πe2 ρ d2 x x. = − dt2 me F = ρ me

(4.8)

(4.9)

Equation (4.9) is the equation of motion for an undamped harmonic oscillator with the characteristic frequency:  4πe2 ρ √ = 5.64 × 104 ρ , (4.10) ωp = me where the units of ρ are [electrons cm−3 ], and ωp is [Hz]. With analogy to a mechanical oscillator, the electron density provides the stiffness. The higher the electron density, the higher the plasmon frequency. For metals, assuming an approximate free electron density of ρ = 1023 electrons cm−3 , ωp ≃ 2×1016 Hz.3 The characteristic energy of such an oscillation is the plasmon energy, given by: Ep =  ωp ,

(4.11)

and for our example Ep = (6.6×10−16 eV s) (2×1016 s−1 ) ≈ 13 eV. In EELS, intense plasmon peaks are prominent at energy losses of 10– 20 eV. Plasmons are not long-lived, however, often because they promote 3

The present one-dimensional approach is not necessarily reliable for relating the free electron density to the plasmon energy. A more general approach uses the imaginary component of the dielectric constant of the material, and the theory uses the mathematics discussed in Problem 4.6.

4.3 Plasmon Excitations

175

excitations of electrons near the Fermi energy. Plasmon peaks therefore tend to be broadened in energy.4 Free electron metals such as aluminum have sharper plasmon peaks than do alloys of transition metals, which have a high density of states at the Fermi energy. Compared to core electron excitations, however, plasmon excitations do not provide much detailed information about the individual atom species in the material. 4.3.2 * Plasmons and Specimen Thickness The characteristic length or “mean free path,” λ, over which a 100 keV electron excites one plasmon is about 100 nm in metals and semiconductors. This is an average length, so in a TEM specimen of even 50 nm, some electrons excite one, two, or more plasmons. The probability Pn for the excitation of n plasmons in a sample of thickness, t, is determined by the statistics of Poisson processes:  n In 1 t Pn = e−t/λ = , (4.12) n! λ It where In is the number of counts in the nth plasmon peak, and It is the number of counts in all plasmon peaks with n ≥ 0 (It includes the zero-loss peak for which n = 0). The EELS spectrum in Fig. 4.8 shows distinct plasmon peaks. After subtracting a background from other inelastic processes (originating with the Al L-edge and a contribution from oxides and the substrate), Pn is obtained as the fractional area of the nth plasmon peak.

1200

0

0

n=1

200

n=0

400

40

n=5

n=4

600

n=3

800

n=2

Counts

1000

80

Energy Loss [eV]

120

Fig. 4.8. Low-loss spectrum taken from a thick sample of ∼ 120 nm Al metal on C using 120 keV electrons and β = 100 mrad. Plasmon peaks are visible at energies of n × 15 eV, where n is the number of plasmons excited in the sample. After [4.4].

Good samples for TEM imaging are several times thinner than the sample used for Fig. 4.8, but plasmon peak areas still offer a practical way to 4

“Lifetime broadening” is understandable from the uncertainty principle: ΔEΔt ≃ . A short lifetime Δt comes with a large uncertainly in energy, ΔE.

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4. Inelastic Electron Scattering and Spectroscopy

determine thicknesses of thin samples. Setting n = 0 in (4.12), the thickness, t, is:   It t . (4.13) = ln I λ 0 Measurement of It and I0 (the zero-loss or the n = 0 plasmon peak) involves the choice of the energies ε, δ and Δ, which define the limits of integration, as illustrated in Fig. 4.9.5 The lower limit (−ε) of the zero-loss region can be taken anywhere to the left of the zero-loss peak where the intensity has decreased to zero, the separation point δ between the zero-loss and inelastic regions may be taken as the first minimum in intensity, and Δ ≈ 100 eV is usually sufficient to include most of the inelastic scattering in relatively thin, low Z materials (for high Z and/or thick specimens, several hundred eV should be used since the scattered intensity is shifted to higher energy loss). Equation (4.13) has been shown to give 10 % accuracy for relative thickness measurements on samples as thick as t = 5λ. Some deviations from the intensities of (4.12) are of course expected when the electron beam passes through regions of non-uniform thicknesses or composition, and when other contributions are present in the low-loss spectrum.

Fig. 4.9. The integrals and energies involved in measuring specimen thickness by the log-ratio method. After [4.5].

Absolute determinations of specimen thickness require values for the total inelastic mean free path. For materials of known composition, it is possible to calculate a value for the mean free path according to the semi-empirical equation [4.5]: λ≈

E0 106 F , ln(2βE0 /Em ) Em

(4.14)

where units for λ are [nm], β is the collection semiangle [mrad], E0 is the incident energy [keV], Em is a mean energy loss [eV] that depends on the composition of the sample, and F is a relativistic factor: F = 5

1 + E0 /1022 , (1 + E0 /511)2

(4.15)

If a collection aperture limits the angles recorded by the spectrometer to a maximum angle β, λ in (4.13) must be interpreted as an effective mean free path, λ(β).

4.4 Core Excitations

177

and F = 0.768 for E0 = 100 keV and 0.618 for E0 = 200 keV. For a specimen of average atomic number Z, Em can be obtained from the semi-empirical formula: Em ≈ 7.6Z 0.36 .

(4.16)

For large collection apertures, i.e., β > 20 mrad for E0 = 100 keV or β > 10 mrad at 200 keV, (4.14) becomes inapplicable and the mean free path saturates at a value independent of β. Specimen thickness measurement by this EELS plasmon technique has some advantages over other techniques for measuring specimen thickness (such as CBED) because it can be applied for a wide range of specimen thicknesses, including very thin specimens, and for specimens that are highly disordered or amorphous. Table 4.1 lists some values for calculated (with (4.10) and (4.11)) and measured plasmon energies Ep , widths of the plasmon peaks ΔEp , characteristic scattering angles for plasmons φEp , and calculated mean free paths λ, for 100 keV incident electrons. Table 4.1. Plasmon data for selected materials [4.5] Material

Ep (calc.)

Ep (expt.)

ΔEp

φ Ep

λ

(eV)

(eV)

(eV)

(mrad)

(nm)

Li

8.0

7.1

2.3

0.039

233

Be

18.4

18.7

4.8

0.102

102

Al

15.8

15.0

0.5

0.082

119

Si

16.6

16.5

3.7

0.090

115

4.4 Core Excitations 4.4.1 Scattering Angles and Energies – Qualitative When a high-energy electron undergoes inelastic scattering, its energy loss, E, is actually a transfer of energy to the sample. When this energy is transferred to an atomic electron, the atomic electron may find an unoccupied electron state about the same atom, or it may leave the atom entirely (i.e., the atom is ionized). The total energy and the total momentum are conserved, but the scattering redistributes the energy and momentum between the high-energy electron and the atomic electron. The two electrons have coupled behavior. In particular, the probabilities and energies for the allowed excitations of the atomic electron are mirrored in the spectrum of energy losses of the highenergy electron. Relevant energies and their notation are listed in Table 4.2.

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4. Inelastic Electron Scattering and Spectroscopy

Table 4.2. Energy notation variable E E0 T Eα Eαβ Ea Ep Em sign

definition energy transfer from incident electron to atomic electron energy of incident electron (T + mass energy) e.g., 100.00 keV incident kinetic energy (low E0 : T ≃ E0 ) (high E0 : T = mv 2 /2 < E0 ) energy of bound atomic electron difference in energy between atomic states α and β energy of atomic absorption edge (e.g., EK ), Ea ≃ −Eα plasmon energy mean energy loss all variables are positive except Eα

When a high-energy electron transfers energy to a core electron, the wavevector of the high-energy electron is changed in both magnitude and direction. The change in energy is obtained from the change in magnitude of the wavevector. The change in momentum is obtained from both the change in direction and the change in magnitude of the wavevector. Total momentum is conserved, and before scattering the total momentum is with the incident electron, p0 = me v 0 = k0 . After scattering, the momentum transfer to the atomic electron must be Δk = (k − k0 ). This same Δk ≡ k − k0 is used for elastic scattering (Fig. 1.5), but inelastic scattering has an extra degree of freedom because k = k0 . Figure 4.10a shows that increasing the scattering angle φ, larger values of Δk are possible for the same E, but momentum conservation requires that the head of the wavevector Δk lies along the circle of radius k. The scattering vector, Δk, can be zero only when both φ = 0 and E = 0.6 When E = 0 but φ = 0, Δk cannot be zero – this is the case for elastic scattering in diffraction experiments.

Fig. 4.10. Kinematics of inelastic electron scattering. (a) Definitions, with sphere of constant E. (b) Enlargement valid for small φ, or equivalently for small Δk. 6

This is the case for no scattering, or for elastic forward scattering, which involves a phase shift.

4.4 Core Excitations

179

We first consider general features of how inelastic scattering depends on E and Δk. For E only slightly larger than an absorption edge energy, Ea , the inelastic scattering is forward-peaked with a maximum intensity at the smallest Δk. Figure 4.10a shows that when φ = 0 and E = 0, there is a nonzero minimum value of Δk, corresponding to inelastic scattering in the  0 . In a particle model, these lowforward direction: Δkmin ≡ (|k| − |k0 |)k angle scatterings correspond to soft collisions with large impact parameters (meaning that the high-energy electron does not pass close to the center of the atom). The energy transfer is still large (E ≃ Ea ), unlike most soft classical collisions,7 but the outgoing core electron carries insignificant kinetic energy and Δk is small. For small Δk, the scattering is sensitive to the large r (long-range) features of the scattering potential. On the other hand, at larger energy losses (E ≫ Ea ), the scattering is at higher angles, corresponding to hard collisions with small impact parameters. The outgoing core electron carries significant kinetic energy (equal to E − Ea ), and the momentum transfer deflects the high-energy electron. For sufficiently large E, we expect the momentum transfer and energy transfer to be understandable by collisional kinematics, with little influence from the characteristics of the atom such as Ea . In fact, for larger energy transfers the inelastic intensity becomes concentrated around a specific value of Δk such that: √ 2mE . (4.17) ΔkB =  This peak in Δk corresponds to the momentum transfer in classical “elastic” scattering of a moving ball (electron) by another ball initially at rest. This peak is called the “Bethe peak,” and in a two-dimensional plot of inelastic scattering intensity versus Δk and E, these peaks become a “Bethe ridge” (cf., Fig. 4.11). Substituting into (4.17) a handy expression involving the Bohr radius, a0 , and the Rydberg energy, ER = 2 (2ma20 )−1 , we obtain for ΔkB : (ΔkB a0 )2 ≈

E , ER

for which the equivalent scattering angle for the Bethe ridge, φr , is:  E φr ≈ . E0

(4.18)

(4.19)

The results of (4.17)–(4.19) are valid for small φ and non-relativistic electrons. 7

A classical analogy can be contrived. Suppose a fast billiard ball passes near a second ball located in a pit, and some of the kinetic energy of the fast ball is used to lift the second ball out of the pit. If the second ball leaves the pit with minimal velocity, momentum conservation allows little change in direction of the fast ball as it slows down. Quantum mechanics uses the same energy and momentum arguments, but Sect. 4.4.2 provides the probabilistic mechanism for “lifting the ball out of the pit.” This mechanism provides an additional dependence on E and Δk.

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4. Inelastic Electron Scattering and Spectroscopy

Experimentally, we count electrons. The energy spectrum of these electrons, ρ(E) dE, varies with solid angle, Ω. With reference to Fig. 3.4, the three different dΩj will have different energy spectra. The most detailed experimental measurements would provide an energy spectrum at each differential solid angle, dΩ. The number of electrons detected in a range dΩ around Ω and a range dE around E is proportional to the “double-differential crosssection,” d2 σ/dΩ dE. In practice, there is often cylindrical symmetry around the forward beam, so we may need only the φ-dependence (where φ = 2θ in scattering angle). Experimental EELS spectra are measurements of intensity versus energy loss, I(E), over a finite range of scattering angles, φ. Theoretically, we calculate the probability that a transfer of energy, E, and momentum,  Δk, occurs between a high-energy electron and an atomic electron. To relate the theory to measured EELS spectra, we then need: • The variation of the inelastic scattering over the parameter space of (φ,E). This is given by a double-differential cross-section, d2 σin /dφ dE, described in Sect. 4.4.3 (dφ refers to rings of solid angle). This d2 σin /dφ dE includes the “generalized oscillator strength” of the specific atom. • EELS spectra, I(E), are measured over a range of φ, so we need to integrate d2 σin /dφ dE over angle to obtain the differential cross-section, dσin /dE, described in Sect. 4.4.5. • Compositional analysis by EELS uses total intensities, given by the total cross-section, σ (or more typically by partial cross-sections, corresponding to a finite range in energy), as described in Sect. 4.4.6. This total probability for ionizing an atom is also needed for understanding EDS spectra, which measure x-ray emissions after the atom is ionized. 4.4.2 ‡ Inelastic Form Factor Here we calculate the probability of an inelastic scattering process involving the excitation of a core electron. In this process, a high-energy electron excites a second electron from a bound atomic state into a state of higher energy. Since two electrons are involved, for conciseness we employ the Dirac bra and ket notation.8 The high-energy electron, “electron 1,” is initially in a plane wave state |k0 , and after scattering it is in the state |k . The atomic electron, “electron 2,” is initially in the bound state |α . After scattering, electron 2 is in the state |β , which may be either a bound state that is initially unoccupied, or a spherical (or plane) wave state if electron 2 is ejected from odinger the atom. For inelastic scattering, |k| =  |k0 | and α = β. The Schr¨ equation with the initial state is written as: 8

Recall that Dirac notation is free of spatial coordinates and explicit functional forms of wavefunctions, but these are obtained with the position operator for coordinate set 1, r 1 , as: r 1 |k =R ψ(r 1 ). Actual evaluations of integrals require expressions such as: a| H |a = ψα∗ Hψα d3 r. When |α is an eigenstate of H, α| H |α = Eα α|α = Eα , since the state functions are normalized. State functions are orthonormal, so α|β = 0 and α|α = 1.

4.4 Core Excitations

H0 |k0 , α = (E0 + Eα ) |k0 , α .

181

(4.20)

So long as the two electrons are far apart and therefore non-interacting, the two-electron system obeys the unperturbed Hamiltonian: 2 2 2 2 ∇1 − ∇ + V(r 2 ) . (4.21) 2me 2me 2 The coordinates of the high-energy electron 1 are r1 , and the coordinates of the atomic electron 2 are r 2 . With different coordinates, each Laplacian in (4.21) acts on only one of the two electrons, and the potential energy term involves only electron 2. In such problems we can express the initial state as a product of one-electron wavefunctions: |k0 , α = |k0 |α , and the final state as: |k, β = |k |β . When using a product wavefunction in (4.21), the factor for electron 2, |β , is a constant under the operations of ∇21 , and |k is a constant under the operations of ∇22 and V(r2 ). A “constant factor” does not affect the solution of the Schr¨ odinger equation for the other wavefunction of the product. The Hamiltonian of (4.21) is therefore equivalent to two independent Hamiltonians for two independent electrons. This is as expected when the two electrons have no interaction. As the high-energy electron approaches the atom, we must consider two perturbations of our two-electron system. One perturbation is the Coulombic interaction of electron 2 with the electron 1, which is +e2 / |r1 − r 2 |. The second perturbation is the interaction of the high-energy electron 1 with the potential from the rest of the atom,9 V(r 1 ). The perturbation Hamiltonian, H ′ , is: H0 = −

H′ =

e2 + V(r 1 ) . |r1 − r2 |

(4.22)

This perturbation H ′ couples the initial and final states of the system. The stronger the coupling, the more probable is the transition from the initial state |k0 |α to the final state |k |β . It is a result from time-dependent perturbation theory that the wavefunction of the scattered electron 1 is an outgoing spherical wave times the form factor, f (k, k0 ) (cf., (3.55)), where: −me β| k| H ′ |k0 |α . (4.23) f (k, k0 ) = 2π2 Substitution of (4.22) into (4.23) gives: 1 −me  2 e β| k| |k0 |α f (k, k0 ) = 2π2 |r1 − r 2 |  + β| k| V(r 1 ) |k0 |α . (4.24)

When evaluating the second term of (4.24), the coordinates of electron 2 appear only in the atomic wavefunctions |α and |β , so these wavefunctions are moved out of the integral involving the coordinates of electron 1: 9

For the potential from the rest of the atom, we could use the potential of an atom without a core electron, since we consider electron 2 separately.

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4. Inelastic Electron Scattering and Spectroscopy

f (k, k0 ) =

1 −me  2 |k0 |α e β| k| 2π2 |r1 − r 2 |  + β|α k| V(r 1 ) |k0 .

(4.25)

For inelastic scattering we have α = β, so the second term10 is zero by the orthogonality of the atomic wavefunctions. To be explicit in notation, we denote the inelastic contribution to f (k, k0 ) as fin(k, k0 ), and call it the “inelastic form factor.” To calculate fin(k, k0 ), we use spatial coordinate representations for our wavefunctions. The non-zero first term of (4.25) is: −me e2 fin(k, k0 ) = 2π2

+∞ +∞

−∞ −∞

e−ik·r1 eik0 ·r1

1 |r 1 − r2 |

× ψβ∗ (r 2 ) ψα (r 2 ) d3 r 2 d3 r 1 .

(4.26)

We change variables: r ≡ r 1 − r2 (so r 1 = r + r 2 ), and Δk ≡ k − k0 , and separate the integrations: −me e2 fin(k, k0 ) = 2π2

+∞

1 e−i∆k·r d3 r |r|

−∞

+∞

× e−i∆k·r2 ψβ∗ (r 2 ) ψα (r 2 ) d3 r 2 .

(4.27)

−∞

Equation (4.27) shows that the only dependence of fin on k and k0 is through their difference, Δk. The integral over r is 4π/Δk 2 (3.102): −2mee2 fin(Δk) = 2 2  Δk

+∞

e−i∆k·r2 ψβ∗ (r 2 )ψα (r 2 )d3 r2 .

(4.28)

−∞

This inelastic form factor, fin (Δk), is the amplitude of the outgoing highenergy electron wavefunction along the direction k = k0 +Δk when the highenergy electron excites the atomic transition ψα → ψβ . The inelastic form factor has many similarities to the elastic form factor, fel (Δk), of (3.84). Specifically, the second term of (3.113) for fel (Δk), which describes elastic scattering from the atomic electron density, ρ(r), has the same form as (4.28). It is convenient to think of both the inelastic and elastic form factors in a common way. Along the direction k = k0 + Δk, wavelets are emitted from all sub-volumes, d3 r 2 , of the atom. Each wavelet has a relative phase 10

For elastic scattering there is no transfer of energy from the high-energy electron (electron 1) to the atomic electron (electron 2), so α = β. By the orthonormality of the atomic wavefunctions we know that α|α = 1, so this second term is nearly equal to the right hand side of (3.84). The difference is that the scattering potential from electron 2 is considered separately as the first term in (4.25), but together the two terms in (4.25) account for the scattering from the entire atom.

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183

e−i∆k·r2 , and its amplitude for elastic scattering is proportional to an electron density. The full wave is the coherent sum (integration) of wavelets from all volumes of the atom, weighted by an electron density. For elastic scattering the electron density is the usual electron density, ρ(r) = ψα∗ (r)ψα (r). For inelastic scattering, however, this “density” is the overlap of the initial and final wavefunctions, ρ′ (r) = ψβ∗ (r)ψα (r). Note the common prefactors of fel (Δk) of (3.113) and fin (Δk) of (4.28). Recall that the factor of Δk −2 originates with the Fourier transform of the Coulomb potential (3.112). Using the definition of the Bohr radius, a0 = 2 /(me e2 ), this prefactor is 2/(a0 Δk 2 ), which has dimensions of length. We now obtain the differential cross-section for inelastic scattering, dσin /dΩ, as ∗ fin (3.20)11 : fin 2  +∞    4  dσin (Δk) 3 −iΔk·r 2 ∗ (4.29) ψβ (r 2 )ψα (r 2 )d r2  . e = 2 4 dΩ a0 Δk   −∞

Although energy is transferred from the high-energy electron 1 to the atomic electron 2, the total energy is conserved. In the transition |k0 |α → |k |β , the total energy before scattering equals the total energy after scattering: E0 + Eα = (E0 − E) + Eβ , E = Eβ − Eα ≡ Eαβ .

(4.30) (4.31)

A spectrum of electron energy losses shows enhanced intensity when E = Eαβ . Owing to the Pauli principle, however, the state ψβ must be initially empty for it to be allowed as a final state for electron 2. The EELS spectrum usually shows a jump in intensity, or “edge jump,” when Eαβ = Ea , where Ea corresponds to the lowest energy of an unoccupied state ψβ . Enhanced intensity extends for E > Ea , because other unoccupied states of higher energy are available to the atomic electron 2. With actual wavefunctions for ψα and ψβ , we could use (4.29) to calculate the strength of this inelastic scattering,12 and the measured intensity of the electron energy-loss spectrum at the various energies Eαβ > Ea . To do this, however, we must first relate the experimental conditions to the cross-section of (4.29). Specifically, we need to know how experimental detector angles, φ, select Δk at various E. This is the topic of the next subsection. 11

12

A correction factor at high energy losses accounts for how the outgoing flux of scattered electrons is reduced when the electron is slowed (cf. (3.14)), but we safely ignore this effect for energy losses of a few keV. There is a subtle deficiency of (4.28) and (4.29). The excitation of a core electron changes the electronic structure of the atom. It is not necessarily true that atomic wavefunctions are appropriate for ψα or ψβ when a core hole is present. The atomic electrons change their positions somewhat is response to the core hole, so the second term in (4.25) may not be strictly zero by orthogonality.

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4.4.3 ‡ * Double-Differential Cross-Section, d2 σin /dφ dE In EELS, we measure the spectrum of energy losses from electrons in some range of Δk, set by the angle, β, of a collection aperture (see Fig. 4.1). To understand the intensity of core-loss spectra, we need to know how the inelastic scattering depends on both scattering angle, φ, and energy loss, E. This dependence of the intensity on φ and E is provided by a doubledifferential cross-section, d2 σin /dφ dE. We start with the φ-dependence for fixed E. For small Δk we can approximate, as shown in Fig. 4.10b: 2 Δk 2 = k 2 φ2 + Δkmin .

(4.32)

The increment in solid angle covered by an increment in φ (making a ring centered about k0 ) is: dΩ = 2π sinφ dφ .

(4.33)

By differentiating (4.32) (for fixed E, Δk min is a constant): Δk dΔk , k2 so for the small φ of interest:

(4.34)

φ dφ =

Δk dΔk . k2 Substituting (4.35) into (4.29), and re-defining r 2 → r, provides: dΩ = 2π

(4.35)

dσin (Δk) dσin dΩ = dΔk dΩ dΔk

 +∞ 2     8π −i∆k·r ∗ 3   e ψβ (r)ψα (r) d r  , = 2 3  a0 k 2 Δk  

(4.36)

−∞

where the right-hand side is averaged for all Δk of the detected electrons. When ψβ is a bound state of the atom, (4.36) can be used directly to obtain an EELS intensity at the energy corresponding to the transition α → β. In the more typical case, ψβ lies in a continuum of states, such as free electron states when the atomic electron leaves the atom with considerable energy, or a band of unoccupied antibonding states for energies E that are close to the absorption edge energy, Ea . We then need to scale the result of (4.36) by the number of states in the energy interval of the continuum, which is ρ(E)dE. Here ρ(E) is the “density of unoccupied states” available to the atomic electron when it is excited. Accounting for the density of states of ψβ gives the double-differential cross-section: 2  +∞     8π d2 σin (Δk,E) −i∆k·r ∗ 3   (4.37) ψβ (r)ψα (r) d r  . = 2 2 3 ρ(E)  e dΔk dE a0 k Δk   −∞

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185

The convention is to rewrite (4.37) to isolate the scattering properties of the atom. This is done by defining the “generalized oscillator strength,” GOS, or Gαβ (Δk,E):  +∞ 2    2me  −i∆k·r ∗ 3  Gαβ (Δk,E) ≡ Eαβ 2 2  e ψβ (r)ψα (r) d r  . (4.38)  Δk   −∞

Here Eαβ is the difference between the energies of the states ψα and ψβ . Using (4.38) in (4.37):13 1 2π4 d2 σin (Δk,E) = 2 2 ρ(E) Gαβ (Δk,E) . dΔk dE a0 me Eαβ T Δk

(4.41)

To make connection to experimental EELS spectra, we convert the Δkdependence of (4.41) into a dependence on the scattering angle φ of Fig. 4.10. We do so by arranging (4.34) as a relationship between dΔk and dφ, and substituting into (4.41): k02 φ 2π4 d2 σin (Δk,E) = 2 2 ρ(E) Gαβ (Δk,E) . dφ dE a0 me Eαβ T Δk 2

(4.42)

Figure 4.10b shows the definition of φE ≡ Δk min /k0 and the approximation:   Δk 2 ≃ k02 φ2 + φ2E , (4.43) from which we obtain a useful expression:

φ 2π4 d2 σin (φ,E) = 2 2 ρ(E) Gαβ (Δk,E) . 2 dφ dE a0 me Eαβ T φ + φ2E

(4.44)

From Newtonian mechanics we would expect φE , which is a ratio of kvectors, to depend on the energies in the collision problem as E/T . There is, however, a change in mass energy equivalent of the high-energy electron after scattering. This energy loss from the change in mass is significant, so the wavelength change is considerably smaller than the non-relativistic prediction. The result from relativistic kinematics is: E E ≃ . (4.45) φE = 2γT 2E0 As an example, for the C K-edge at 284 eV in a 200 kV microscope φE = 0.7 mrad. 13

For accuracy, we have written (4.41) with the incident kinetic energy, T , that differs from the incident energy, E0 , as: 1 E0 1 + γ T ≡ me v 2 = , (4.39) 2 2 γ2 owing to the relativistic correction: 1 E0 γ≡ p =1+ , (4.40) 2 2 mc 1 − (v/c) (γ ≈ 1.4 for 200 keV electrons).

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4.4.4 * Scattering Angles and Energies – Quantitative We revisit the angular dependence of the inelastic scattering. At lower energy losses (E slightly larger than Ea ), and at smaller scattering angles, the main angular dependence in (4.44) is from the Lorentzian factor, (φ2 + φ2E )−1 , peaked at φ = 0, with φE (4.45) as the half-width of the angular distribution. (The factor φ in the numerator of (4.44) merely accounts for the larger radius of a ring at larger φ.) We first compare this characteristic angle for inelastic scattering, φE , to the characteristic angle for elastic scattering, φ0 . The elastic angle φ0 is associated with the atomic form factor, which is a measure of the size of the atom. For convenience we select r0 , the Bohr radius of the Thomas–Fermi atom as this size (3.104), and obtain φ0 as: φ0 =

1 . k0 r0

(4.46)

Putting typical values into (4.45) and (4.46), we find that φE is generally a few tenths of a milliradian while φ0 is a few tens of milliradians, i.e., φ0 ≈ 100φE . The inelastic scattering is concentrated into a much smaller range of angles about the forward beam than the elastic scattering, especially when E ≃ Ea . Section 4.4.1 discussed the other extreme case where E ≫ Ea , and the collision kinematics are insensitive to the shape of the atom – recall that the intensity became bunched into angles characteristic of classical “billiardball” collisions. The generalized oscillator strength, Gαβ (Δk, E) of (4.38), helps complete the picture of how the inelastic intensity varies between these two extremes of E ≃ Ea and E ≫ Ea . The generalized oscillator strength, Gαβ (Δk, E), is the probability of the transition α → β, normalized by a factor related to the energy and momentum transfer. Figure 4.11 shows the GOS on the twodimensional space of {ln(φ), E} in a plot known as a “Bethe surface.” The individual curves in Fig. 4.11 show the angular dependence of the inelastic scattering for each energy loss above the carbon K-edge. Likewise, the energy dependence of the GOS may be obtained by taking sections through the Bethe surface at constant scattering angle. The Bethe ridge is marked on Fig. 4.11. Although distinct at large E, the Bethe peak is less well-defined at energy transfers closer to Ea (the C K-edge threshold at 0 eV in Fig. 4.11). In EELS measurements, an entrance aperture having an acceptance semiangle β is placed around the forward beam (Fig. 4.1). This aperture cuts off the scattering beyond a certain φ. The measured spectrum of intensity versus energy is therefore an integration of the scattering intensity over combinations of E and Δk that fall below this cutoff. At energies significantly above the edge, Fig. 4.11 shows that a substantial portion of the intensity is concentrated in the Bethe ridge at larger scattering angles. A relatively large objective aperture (> 10 mrad or so) is needed to include this intensity in the EELS spectrum. On the other hand, at energies just above the edge, a small

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Fig. 4.11. Bethe surface for K-shell ionization of C, calculated using a hydrogenic model. The GOS is zero for energy losses below the ionization threshold EK = Eαβ , or E < 0. The horizontal coordinate increases with scattering angle. The Bethe ridge is most distinct at large E towards the front of the figure. After [4.5].

aperture will collect most of the intensity. This small aperture may be useful for removing background intensity at large Δk that originates from tails of other elements with lower Ea . 4.4.5 ‡ * Differential Cross-Section, dσin /dE Ignoring any truncation of the scattered inelastic intensity caused by the spectrometer entrance aperture, β, we integrate (4.44) over all possible scattering angles, φ, from 0 to π. This provides the total inelastic differential cross-section, σin,αβ (E) for exciting an atomic electron from state |α to state |β : dσin,αβ (E) 2π4 = 2 2 ρ(E)Gαβ (Δk,E) dE a0 me Eαβ T

π 0

φ dφ . φ2 + φ2E

(4.47)

Here we have ignored the φ-dependence of the GOS, Gαβ (Δk,E). With the reasonable approximation that φE ≪ π, the integration of (4.47) gives:  2 π π4 dσin,αβ (E) ρ(E)Gαβ (Δk,E) ln 2 = 2 2 . (4.48) dE a0 me Eαβ T φE With (4.48) and (4.45) we obtain the inelastic differential cross-section:   2πγT 2π4 dσin,αβ (E) = 2 2 ρ(E)Gαβ (Δk,E) ln . (4.49) dE a0 me Eαβ T E Figure 4.12 shows a plot of the energy-differential cross-section for K-shell ionization of C (Ea = 284 eV) calculated for different collection semiangles

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β using hydrogenic wavefunctions.14 Logarithmic axes are used to illustrate the approximate behavior: dσin,αβ (E) ∝ E −r , (4.50) dE where r is the downward slope in Fig. 4.12 and is constant over various ranges in energy loss. The value of r depends on the size of the collection aperture. For large β, when most of the inner-shell scattering contributes to the energy loss spectrum, r is typically about 3 at the ionization edge, decreasing towards 2 with increasing energy loss. The asymptotic E −2 behavior occurs because for E ≫ Ea , practically all of the scattering lies within the Bethe ridge. It approximates Rutherford scattering from a free electron (3.106), for which dσin,αβ (E)/dE ∝ Δk −4 ∝ E −2 . For small β, r increases with increasing energy loss, the largest value (just over 6) corresponding to large E and small β. The breaks in slope in Fig. 4.12 correspond to the condition where E is large enough so that the Bethe ridge moves to angles outside the collection aperture. It is usually important to avoid this transition in experimental practice because it complicates the E-dependence of the measured intensity. It may be a good idea to calculate φr with (4.19), and use a collection angle, β, a few times larger than this, as mentioned in the context of (4.53).

Fig. 4.12. Energy-differential cross-section for K-shell ionization of C (Eαβ = EK = 284 eV) calculated for different collection semiangles β. After [4.5].

14

A hydrogenic atom uses the wavefunctions of a hydrogen atom, but with radial coordinates rescaled by a larger nuclear charge. There are no electron-electron interactions for a hydrogenic atom, but analytical expressions for the wavefunctions are available.

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189

4.4.6 ‡ Partial and Total Cross-Sections, σin In quantitative elemental analysis, the inelastic intensity measured with an aperture angle β is integrated over an energy range of width δ beyond an absorption edge. Assuming a thin specimen with negligible multiple scattering, the integrated intensity above Ea , is: Ia (Ea , δ, β) = N I0 σin,a(Ea , δ, β) ,

(4.51)

where N is the number of atoms per unit specimen area, and I0 is the integrated zero-loss intensity. In (4.51), the “partial cross-section” σin,a (Ea , δ, β) is the integral of (4.44) over the range of collection angle and energy: σin,a (Ea , δ, β) =

β E a +δ 0

d2 σin (φ, E) dE dφ . dφ dE

(4.52)

Ea

For numerical integration of d2 σin (φ, E)/dEdφ, it is sometimes convenient to use the power-law behavior of (4.50). Figure 4.13 shows the calculated angular dependence of K-shell partial cross-sections for the first-row (second-period) elements. The figure illustrates the dependence of the cross-sections on collection angle β, incident electron energy E0 , and ionization energy EK , for constant integration width δ. The cross-sections saturate at large values of β, i.e., above the Bethe ridge angle, φr , owing to the fall-off in Gαβ (Δk,E) at large Δk. The median scattering angle (for energy losses in the range Ea to Ea + δ) corresponds to a partial cross-section equal to one-half of the saturation value, and is typically 5 φE , where: EK + δ/2 , (4.53) φE = 2γT with γ ≡ (1−v 2 /c2 )−1/2 . Figure 4.13 shows that the saturation cross-sections decrease with increasing incident electron energy, although the low-angle values increase. This is because a small collection aperture accepts a greater fraction of the scattering when the incident energy is high and the scattering is more strongly forward-peaked. For a very large range of energy integration δ, the partial cross-section becomes the “integral cross-section” σin,K (EK , β) for inner shell scattering up to β and all permitted values of energy loss. By setting β = π, the integral cross-section becomes the “total cross-section” σin,K (EK ) for inelastic scattering from the K-shell. An approximate expression for σin,K (EK ) is the “Bethe asymptotic cross-section”:   cK T E2 , (4.54) σin,K (EK ) = 4πa20 NK bK R ln T EK EK where NK is the number of electrons in the K-shell (2, but for the L and M shells this would be 8 and 18, respectively), ER ≡ 2 (2me a20 )−1 ,

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Fig. 4.13. Partial cross-sections for Kshell ionization of second-period elements, calculated for an integration width δ equal to one-fifth of the edge energy, assuming hydrogenic wavefunctions and nonrelativistic kinematics. After [4.5].

bK ≈ fK /NK , cK ≈ 4EK / E , where fK ≈ 2.1 − Z/27 is the dipole oscillator strength for K-shell ionization and typically E ≈ 1.5EK . A similar expression, useful for calculating the integral cross-section as a function of collection angle β, is given in Problem 4.10. Computer programs are available to calculate differential cross-sections for K, L and M shell ionizations using various atomic models [4.5]. Figures 4.14a and 4.14b compare experimental N-K and Cr-L edges to those calculated with the widely-used SIGMAK and SIGMAL programs of Egerton [4.5]. These programs calculate inelastic cross-sections for isolated atoms, and cannot provide detailed information about local chemical effects in the near-edge region of the spectrum as discussed in Sect. 4.2.3. Their integrated intensities are generally reliable, however. As shown in Fig. 4.14a, the SIGMAK hydrogenic model does not provide the detailed fine structure above the edge, but accounts fairly well for the overall experimental shape. Likewise, Fig. 4.14b shows that the L-shell calculation with the SIGMAL program is also good on the average, but it cannot model the white line peaks at the edge onsets. It does estimate their average intensity, however, based on the number of unoccupied d-states of the element. * Dipole Approximation and X-Ray Absorption Edges. For energy losses near the absorption edge (small E − Ea ) where most of the intensity occurs with small φ and small Δk, it is sometimes convenient to use the

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191

Fig. 4.14. (a) Comparison of an experimental N K-edge and a hydrogenic fit to the edge using the SIGMAK program, and (b) comparison of an experimental Cr L2,3 -edge and a modified hydrogenic approximation to the edge using the SIGMAL program. After [4.6].

“dipole approximation” for the integral in (4.38). The dipole approximation is obtained by approximating e−i∆k·r ≃ 1 − iΔk · r, and recognizing that the integration of the first term of 1, i.e., β|1|α , is zero owing to the orthogonality of ψα and ψβ . The dipole approximation therefore amounts to replacing the factor e−i∆k·r in the integrand of (4.38) with the simpler factor −iΔk · r. Electric dipole radiation is the dominant transition process in EELS, but nondipole transitions are observed at large Δk when higher order terms must be considered in the expansion e−i∆k·r ≃ 1 − iΔk · r − (Δk · r)2 /2 + ... For atomic transitions induced by x-rays, the GOS for inelastic x-ray scattering differs from (4.38) in that the exponential, e−iΔk·r , is replaced by the dipole operator, er. For small values of Δk, the integral in (4.38) is identical for both electron and photon inelastic scattering, and x-ray and electron absorption edges look very similar. Although the dipole approximation provides the same selection rules for the allowed atomic transitions for both EELS and for inelastic x-ray scattering, the E-dependence of EELS spectra is significantly different from that for inelastic scattering spectra of photons. This difference originates from the nature of electron scattering by a Coulomb potential, whose Fourier transform causes (4.28) to decrease strongly with Δk. Since large energy losses, E, are associated with the larger Δk, it becomes difficult to acquire EELS spectra at E > 4 keV. In practice, inelastic x-ray scattering, using a synchrotron radiation source for example, is performed for energies from 5–50 keV or so, whereas EELS experiments measure energy losses less than 5 keV. 4.4.7 Quantification of EELS Core Edges The energies of absorption edges in EELS are quick and reliable indicators of the elements in a material, but quantifying the chemical composition requires more effort. The absorption edge must first be isolated from the background.

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The background originates primarily from other core edges, sometimes the high-energy tails of plasmon peaks, and occasionally from artifacts of the spectrometer. The background under an absorption edge is often modeled by a simple function such as AE −r , where A and r are constants obtained by fitting the pre-edge background. This background function is extrapolated under the absorption edge and subtracted from the data. The edge jump is often complicated by fine structure effects of chemical bonding, which while of interest in their own right, may interfere with quantification with the use of (4.55) below. After background removal, the intensity above an elemental absorption edge is integrated over an energy range, δ. This range, δ, is determined by two competing requirements. The energy range should be close to the edge, because the intensity is strongest and the background correction is most accurate. On the other hand, chemical effects in the near-edge structure may not be understood, especially in an unknown material. Furthermore, some edges display “delayed maxima,” where the maximum spectral intensity occurs at energies beyond the onset of the edge. Most quantification software uses an atomic wavefunction and a free electron wavefunction for the GOS of (4.38) (or fin(Δk) of (4.28)). Such software predicts smooth near-edge structures without solid-state or chemical effects (e.g., without the white lines of transition metals shown in Fig. 4.14b). Although it may seem desirable to start the range of energy integration above the near-edge region, it is common practice to start the energy integration at the edge, and ignore solid-state effects. Some solid-state effects average out when their peaks and valleys are averaged over a range in energy. An example of chemical analysis of BN is presented in Fig. 4.15. With the partial ionization cross-sections calculated from (4.52), the fractions of boron and nitrogen, cB and cN , were obtained from the integrated intensities, IBK (EK , δ, β) and INK (EK , δ, β), which are the shaded areas in Fig. 4.15. Using 1s wavefunctions for ψα , and outgoing plane waves for ψβ in (4.38), a software package (using 4.44 and (4.52)) was used to obtain a correction factor for the measured intensities: IBK (EK , δ, β) σin ,NK (EK , δ, β) cB . (4.55) = cN INK (EK , δ, β) σin ,BK (EK , δ, β) For specimens of even modest thickness, core edges in EELS spectra are distorted by effects of multiple scattering. The problem occurs when a highenergy electron undergoes an inelastic scattering from a core excitation plus a second inelastic scattering by a plasmon excitation. (Two core excitations are relatively unlikely.) The probability of a plasmon excitation can be determined by examining the low-loss part of the EELS spectrum in the region of the plasmon (energies from 0 to 30 eV in Fig. 4.2). By treating this low loss region as an “instrument function,” it is possible to use procedures analogous to those of Sect. 8.2 to deconvolute the effects of multiple scattering from the core loss spectrum. If only one plasmon peak is visible in the EELS spectrum

4.5 Energy-Filtered TEM Imaging (EFTEM)

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Fig. 4.15. Chemical composition determination of BN. Background fits are shown, as are the 50 eV ranges of energy integration, δ. Using a collection aperture angle of 20 mrad, the partial ionization cross-sections were calculated with (4.52). Agreement was within 4 % of the known composition of B–50 at.% N. After [4.7].

and its height is small relative to the zero-loss peak, the sample is probably thin enough so that multiple scattering can be neglected.

4.5 Energy-Filtered TEM Imaging (EFTEM) 4.5.1 Spectrum Imaging Chemical mapping of the element distributions in samples is possible when EELS is performed in STEM mode (Fig. 2.1). The electron beam is focused into a small probe and EELS spectra are acquired from a two-dimensional grid of points across the sample. Each “pixel” in the image can contain an entire EELS spectrum. The data set, called a “spectrum image,” contains a wealth of information on chemical variations across the specimen. Analysis of this chemical information follows procedures of EELS spectrometry described in Sects. 4.2–4.4. Unfortunately, it may take hours to acquire a complete spectrum image. Exposing a sample to a high-intensity probe beam for a long time often causes problems with contamination and radiation damage. Over long times the specimen may also drift in position, blurring the image. 4.5.2 Energy Filters Another method for chemical mapping has become popular, but it requires specialized instrumentation. Because an EELS spectrum consists of electrons that pass through the specimen and through the optical system of the TEM, the optical system can be used to make images of inelastically-scattered electrons. A conventional TEM uses all electrons that pass through the sample, but an instrument or an instrument modification known as an “energy filter” allows image formation with electrons that have undergone selected energy losses in the specimen. The technique of “energy-filtered TEM” (EFTEM), detects “chemical contrast” in specimens by adjusting an energy filter to pass

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electrons that have lost energy to core ionizations of selected elements.15 Under optimal conditions, these “energy-filtered images” (EFI) can reveal chemical contrast with sub-nanometer spatial resolution. Alternatively, an energy filter can pass only zero-loss electrons, thereby removing all inelastic scattering from a conventional image or diffraction pattern. By using pure elastically-scattered electrons, chromatic aberration and the inelastic background are eliminated, so improved contrast is possible for thicker specimens, and more reliable interpretations of both images and diffraction patterns are possible. Energy filters will probably find a wider use for chemical analysis at the near-atomic scale, however, so this is the focus of the present discussion. Figure 4.16a shows a modification of the magnetic prism EELS spectrometer shown previously in Fig. 4.1. As mentioned in Sect. 4.2.1, the magnetic sector operates as a focusing lens, and its optical analog is shown in Fig. 4.16b. By comparing the ray paths in Figs. 4.16a and b, it is evident that the magnetic sector effectively bends the ray diagrams of Fig. 4.16b. As in Fig. 4.1, the thin rays in Fig. 4.16a correspond to trajectories of electrons that have experienced energy losses, and the thick rays correspond to the zero-loss electrons. Notice how the thin and thick rays have the same pattern, but the thin rays are bent by an additional angle. The aperture at the image plane in Fig. 4.16a has been positioned to remove the zero-loss electrons. Additional lenses to the right of the energy-selecting slit in Fig. 4.16a (such as Q1–Q4 in the title image of Chapter 4) are used to project the energy-filtered image (EFI) onto an area detector such as a scintillator with a CCD camera. object plane energyselecting slit image plane lens

a

b

Fig. 4.16. (a) An energy filter based on a magnetic sector like that in Fig. 4.1. (b) Optical analogy for monochromatic electrons.

Chapter 2 described how lenses produce dispersions of positional information (images) and angular information (diffraction patterns). The energy filter of Fig. 4.16 also produces a dispersion of electron energies (an EELS spectrum). This particular energy filterfocuses an arrow on the object plane 15

Plasmon images may also be useful.

4.5 Energy-Filtered TEM Imaging (EFTEM)

195

(a diffraction pattern or an image at the back focal plane of the final projector lens) onto an arrow at the image plane. In this case the image plane happens to be the same plane as the energy dispersion plane (as in Fig. 4.1). Selecting a narrow range of energy therefore limits the field of view of the image, so at the energy-selecting slit it may be appropriate to form a small image with a low magnification (or a small diffraction pattern) and magnify it with subsequent lenses. Ensuring a good resolution of Δk, good spatial resolution, and high energy selectivity is a challenge for an energy filter, because it must also allow for measurements over wide ranges of these variables Δk, x, and E. Its performance is degraded by various types of aberrations. Recall (Sect. 2.7.1) that spherical aberration mixed angular and spatial information – spherical aberration caused errors in position (x) at the focal plane for rays leaving the specimen at different angles (Δk). An energy filter has aberrations that mix energy, angular, and spatial information. For example, not all locations on an EFI may correspond to the same energy loss. In analogy to the apertures used to suppress problems with spherical aberration, energy filters require apertures to limit their field of view, their angular acceptance, or their acceptance window of energy losses. To optimize imaging performance, it is best for electrons of only one energy to pass through the objective lens. Energy filters are integrated into the microscope electronics to achieve this by allowing the energy filter system to control the high voltage at the electron gun. For making images with electrons that have undergone an energy loss of eΔV , the high voltage is increased by the amount ΔV . With such control over the high voltage system, the focus can then be adjusted only once, and focus will be maintained for images of electrons that have undergone different losses in the sample. The change in incident energy of the electrons then requires that the condenser lens currents are also tuned by the electronics of the energy filter system. This ensures a consistent intensity of illumination on the specimen, important for quantitative work. Chemical analysis by EFTEM usually identifies elements by their core-loss spectra. For the thin specimens needed for EFTEM, however, only a small fraction of the incident electrons can ionize atoms in the sample. A central concern, therefore, is collecting as many inelastically-scattered electrons in as short a time as possible to minimize the effects of specimen drift, contamination, or radiation damage. This motivates the use of large apertures and wide energy windows, pushing the limits of filter performance. For example, more intensity is possible with a wider energy window, but a wider range of energies increases the effects of chromatic aberration, leading to a loss of spatial resolution. Fortunately, in spite of these challenges, many types of chemical analyses with sub-nanometer resolution are now possible with EFTEM.

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4.5.3 Chemical Mapping with Energy-Filtered Images The ability to quantify chemical information at a near-atomic scale (including light elements such as C, O and N) makes EFTEM an important tool for materials characterization. Chemical information is usually obtained from the increase (or “jump”) in EELS spectral intensity at an absorption edge. Unfortunately, absorption edges reside on a large, sloping backgrounds from plasmons or absorption edges of other elements. Chemical mapmaking therefore requires EFIs at energy losses both above and below an absorption edge. Data processing is then needed to isolate the chemical information from the background. For example, the chemical contrast in an image acquired above the absorption edge, the “post-edge” image, can be better seen by subtracting or dividing by an image of the background obtained from “pre-edge” EFIs.16 Two types of elemental maps are typically used: • a “jump-ratio image,” where a post-edge image is divided by a pre-edge background image, • a “three-window image,” where intensities in two pre-edge images are extrapolated to the energy of the post-edge image, and subtracted from it. Jump-ratio images have the advantage that variations in specimen thickness and diffraction contrast are largely cancelled by the division. The threewindow image provides better elemental quantification, however. Unfortunately, three-window images are generally noisier than jump-ratio images owing to the background subtraction procedure, and may require longer times for measurement. A first step in obtaining a chemical map with EFTEM is acquiring an EELS spectrum to locate the edges of interest, decide on placement of the energy windows, and determine the suitability of the specimen thickness. Another preparatory step is tilting the sample or incident beam to minimize the diffraction contrast in the bright-field TEM image. Since elastic scattering can be much stronger than inelastic scattering, diffraction contrast can dominate the appearance of energy-filtered images. To increase the relative amount of chemical contrast, the sample should be tilted away from strong diffraction conditions. Six EFIs of the same region are useful for making chemical maps. These images (with approximate energy windows) are: • • • • • •

Unfiltered (bright field) image (all energies), Zero-loss image (5–10 eV), Low-loss (plasmon) image (20 eV), Pre-edge image 1 (10–20 eV), Pre-edge image 2 (10–20 eV), Post-edge image (10–20 eV).

16

For example, the counts in each pixel of a background image could be subtracted from the counts in each corresponding pixel of the post-edge EFI.

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For thin samples, the intensity of the inelastic spectrum is proportional to the sample thickness. Variations in sample thickness do not distort ratios of element concentrations if they are obtained as ratios of images from different elemental edges. When examining the chemical map of one element, however, it is important to know how the thickness varies across the sample. Since the unfiltered image contains both the elastically- and inelastically-scattered electrons, while the zero-loss image contains only elastically-scattered electrons, it is possible to divide the unfiltered image by the zero-loss image and take the logarithm of the result to obtain a thickness map in units of t/λ (as described in Sect. 4.3.2). This thickness map can be used to identify irregular sample surfaces, and permits conversion from areal densities of atoms to absolute concentrations. Similarly, three-window images can be corrected for the effects of thickness by dividing by the low-loss image. For EFIs, the best specimen and microscope parameters are usually similar to those for EELS. For example, as in EELS, the specimen should be very thin, i.e., t/λ < 0.5, and ideally about half this thickness.17 A rule-of-thumb is that the plasmon peak should be no more than one-fifth the height of the zero-loss peak. Similarly, a small collection angle (5-10 mrad) is preferable because it usually increases the signal/background ratio of an edge (see Problem 4.9). Elements of atomic number greater than 12 allow a choice of edge energy for elemental analysis. It is best to use major edges, and those with threshold energies from 100 to 1000 eV. At lower energies diffraction contrast and a steep background complicate quantification. Spatial resolution is also impaired for energy losses below 50 eV because ionization becomes delocalized, not necessarily occurring at the atoms nearest the high-energy electron. At energies above 1000 eV the intensity becomes inconveniently low. This often requires a wider window for energy selection, leading to problems with chromatic aberration. Effects of chromatic aberration can be suppressed by using smaller objective apertures, but this restricts the range of Δk, and hence the spatial resolution (cf. (2.11)). 4.5.4 Chemical Analysis with High Spatial Resolution Figures 4.17 and 4.18 illustrate several aspects of making chemical maps by EFTEM imaging. The experiment confirmed that Ag enrichment was responsible for the conventional contrast of two planes at the interface between an Ω-phase precipitate and an Al-rich matrix. The flatness of this internal interface was helpful for detecting chemical contrast at high spatial resolution. The Ag M4,5 edge has a delayed maximum, peaking about 50 eV above the edge onset (Fig. 4.17). For best intensity, the post-edge window should incorporate the intensity maximum of the absorption edge, but this required the pre-edge 2 image and the post-edge image to be recorded with a fairly 17

Deconvolution of plural scattering is not possible in EFTEM because a full spectrum is not acquired.

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large energy separation. This is not optimal for EFTEM imaging. Better detectability and spatial resolution can be achieved when the windows labeled “pre-edge 2” and “post-edge” abut together at the onset of an abrupt absorption edge. Abrupt K-edges or intense, sharp white lines at the L-edges of transition metals offer this possibility. Additionally, spatial resolution can be improved by using energy windows narrower than the 30 eV windows used in this example, provided sufficient signal is available. Nevertheless, the research problem involved Ag, so the windows for the energy filter were chosen as in Fig. 4.17.

Fig. 4.17. A portion of an EELS spectrum showing the Ag M4,5 edge and the placement of the preedge 1, pre-edge 2, and post-edge energy windows used for the EFIs in Fig. 4.18. An extrapolated background and a background-stripped Ag M4,5 edge are also shown. After [4.8].

Figure 4.18 shows a set of EFTEM images of an Ω precipitate plate in an Al-Cu-Mg-Ag alloy. To suppress diffraction contrast, the sample was tilted off the exact [011] zone-axis of the α-phase matrix, but the α|Ω interfaces were still parallel to the electron beam. Two dark lattice-fringes can be seen at the α|Ω interfaces on both sides of the Ω plate in the zero-loss image (Figs. 4.18a,b). These fringes are the width of two {111} Al planes (0.46 nm). At each interface, both the Ag jump-ratio image and three-window image show high-intensity lines. The three-window image is noisier than the jump-ratio image, but provides a better estimate of the high enrichment of Ag. The background is featureless in both images, indicating that diffraction contrast and thickness effects are negligible. Two line profiles, acquired from the boxes shown in each image, are shown below the jump-ratio and three-window images. Both reveal segregation of Ag to the precipitate. These Ag layers are only 0.46 nm wide, showing the outstanding spatial resolution of the technique.

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Fig. 4.18. A set of EFIs from an edge-on Ω-phase plate-shaped precipitate in an Al-Cu-Ag-Mg alloy (α-phase), acquired with the sample tilted so the habit-plane interfaces were parallel to the electron beam, but the sample was not directly on the zone axis. Some energy windows are shown in Fig. 4.17. (a) zero-loss image, (b) enlargement of part a, (c) pre-edge 2 image, (d) post-edge image, (e) Ag M4,5 jump-ratio image, and (f) Ag M4,5 three-window chemical map. Line profiles corresponding to e and f should be aligned along the short edge of the boxes enclosing the α|Ω interfaces. (β = 18 mrad). After [4.8].

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4.6 Energy Dispersive X-Ray Spectrometry (EDS) 4.6.1 Electron Trajectories Through Materials To understand how a high-energy electron causes the emission of characteristic x-rays from a specimen, we first need to understand some features of electron trajectories through specimens, and then how to model the electron energy losses to inelastic processes. The simple model described here is useful for a physical understanding, but more complete models are available. In our simple model the electron trajectories through the solid are treated as straight lines between large-angle elastic scattering events. Along these straight paths we are interested in core electron excitations, since only these core ionization processes allow for subsequent x-ray emission. Large-angle scatterings of electrons are primarily elastic in origin.18 They occur when a high-energy electron passes close to an atomic nucleus. In these scatterings the shielding effects of the atomic electrons can be ignored, and the result is the Rutherford scattering cross-section, dσR /dΩ, of (3.107), written with 2θ ≡ φ as: 1 Z 2 e4 dσR = . dΩ 16T 2 sin4 (φ/2)

(4.56)

Equation (4.56) is also useful for understanding the occurrence of electron backscattering from the sample. “Backscattered electrons” are defined as electrons scattered by such a large angle that they reverse direction and go back out through the same surface they entered. Because of the T −2 dependence in (4.56), electron backscattering is relatively rare for electrons of several hundred keV passing through thin specimens.19 Electron trajectories are typically calculated individually with a Monte Carlo algorithm. The computer code allows for random occurrences of scattering events, consistent with a user-specified density of nuclei of charge Ze, electron energy, and Rutherford cross-section of (4.56). The electrons move along straight paths between these elastic collisions, which occur with randomness in the path length and scattering angle. Along the straight paths between the Rutherford scattering events, the electron is assumed to lose energy at random to inelastic processes, both core excitations and plasmons. The core electron excitations are the ionization events that enable the subsequent emission of x-rays. Sections 4.4.1–4.4.6 18

19

The discussion of ionization cross-sections in Sect. 4.4.4 showed that the electron energy-loss spectrum tends to be forward-peaked, especially at small energy losses, owing to the φ-dependence of (4.44). Backscattered electrons are much more common in scanning electron microscopy, which uses incident electrons of a few keV. Although these electrons tend to be multiply-scattered, backscattered electrons provide some chemical analysis capability to the SEM image; the factor of Z 2 in (4.56) causes the backscattered electron image (BEI) to be brighter in regions containing heavier elements.

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described how the probability for ionizing an atom depends on the scattering angle, φ, and energy loss, E, of the incident electron. To calculate x-ray emission, we need to integrate over all φ and E to obtain a total cross-section for inelastic scattering by core electron ionization, σin . This was obtained as (4.49) and (4.54), which depend on the energy of the incident electron as: 1/τ ln(τ ), where τ ≃ 2πγT /Eαβ ≫ 1. We therefore expect that as the high-energy electron loses energy in a thick specimen, the inelastic scattering processes become more frequent, at least until its kinetic energy, T , becomes too small. Monte Carlo codes have been developed to model electron trajectories in solids with all the physical phenomena mentioned in this section, and typical results from a Monte Carlo simulation are presented in Fig. 4.19.

Fig. 4.19. Monte Carlo simulations of electron trajectories (top), and assumed locations of x-ray emission (bottom). (In reality, most individual paths generate zero x-rays.) After [4.9].

A schematic map of the electron trajectories in a thick bulk specimen is shown in Fig. 4.20a. The deep penetration and lateral broadening of highenergy electrons in bulk material causes the region of x-ray emission to be approximately 1 μm in diameter. This is a typical spatial resolution of an electron microprobe, for example. Specimens used in TEM may be only tens of nm in thickness, however. A thin specimen, as depicted in Fig. 4.20b, lacks the bulk of the material where most broadening of the electron beam occurs. Spatial resolution in an analytical TEM is therefore much better than in an electron microprobe.20 As a rule of thumb, the spatial resolution is significantly smaller than the width of the probe beam plus the thickness of the sample. Monte Carlo simulations that implement the model of elastic– inelastic scattering described in this section provide an approximation for the beam broadening, b, in [cm]:  ρ t3 5 Z , (4.57) b = 6.25 × 10 E0 A 20

On the other hand, x-ray emission from the large volume on the left of Fig. 24 provides much greater intensity. This high intensity, and the higher current of the incident electron beam, allows electron microprobes to use wavelength dispersive x-ray spectrometers, which have low collection efficiency but excellent energy resolution.

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where A is the atomic weight of the element [g/mole], ρ is density [g cm−3 ], t is thickness [cm], and E0 is incident energy [eV].

Fig. 4.20. Differences in beam broadening in a bulk specimen (a), and a thin film (b). Part a shows regions of electron penetration, electron escape, and x-ray emission. For high-energy electrons, dimensions of regions of x-ray emission are typically a few microns, microns for backscattered electrons, tens of ˚ A for secondary electrons. The larger dimensions do not exist for the thin specimen in b.

“Secondary electron” emission is especially important in scanning electron microscopy (SEM). A secondary electron is an electron that is weakly bound to the sample and is ejected with a few (at most tens of) electron volts of energy. Since these electrons have little energy, they can traverse only short distances through a material (less than about 100 ˚ A), and therefore originate from the near-surface region. The detected secondary electrons are highly sensitive to surface topography, being more likely to emerge from the peaks than the valleys of the surface drawn in Fig. 4.21. Secondary electron imaging (SEI) is the main technique of SEM, and can be performed in much the same way in the TEM. The instrument is operated in scanning mode with a secondary electron detector attached to the microscope column as illustrated in Fig. 4.22. The number of secondary electrons emitted per incident electron is defined as the “secondary electron yield,” and can be either less than or greater than one. For incident electrons with energies less than 1 keV, the secondary electron yield increases with incident energy, but reaches a maximum (1–3 secondaries/incident electron) at an energy of order 1 keV. The yield is lower at higher energies because the incident electrons penetrate too deeply into the material, and the secondary electrons cannot escape.

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Fig. 4.21. The escape probability of a secondary electron depends on the surface topography.

Secondary Electron Trajectory Scintillator biased at +10 kV light pipe

Photomultiplier tube

Upper Objective Pole Piece

Specimen

Fig. 4.22. Everhart-Thornley detector and its configuration in a TEM. The secondary electron typically follows a spiral path along the magnetic field lines through the upper pole piece of the objective lens.

4.6.2 Fluorescence Yield After a core electron has been emitted from an atom, the ionized atom quickly decays from its excited state. This may occur by “radiative” or “nonradiative” processes, in which the atom emits either an x-ray or an Auger electron, respectively. Both processes (described in Sect. 1.2) compete for the atomic decay. For a K-shell ionization, for example, the “fluorescence yield,” ω K , is defined as the fraction of decays that occur by the emission of a K-shell x-ray. A calculation of ω K requires knowledge of the relative rates of decay of the atom by Auger and by x-ray processes. The rate of x-ray emission is calculated for an electric dipole transition between the two atomic states of the atom, |α and |β . The x-ray emission rate is proportional to factors like |α|er|β |2 . The rate of Auger electron emission involves two electrons, and is calculated for a Coulomb interaction between them. The Auger electron emission rate is proportional to factors like |k|β|e2 /(r1 − r2 )|α |γ |2 , where |α , |β , and |γ are atomic states. The state |k is that of a free electron with the Auger energy (the difference in energy between the states |α and |β , minus the binding energy of state |γ ). The fluorescence yield is the ratio of the x-ray rate to the total rate, where the total rate is the sum of x-ray plus Auger rates. Empirically, for a K-shell emission, ωK depends approximately on atomic number, Z, as:

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Z4 . (4.58) + Z4 Heavier elements tend to emit x-rays, and lighter elements tend to emit Auger electrons.21 The K-fluorescence yield of the elements is presented in Fig. 4.23. The fluorescence yield increases rapidly with Z. On the other hand, the Kshell ionization cross-section decreases strongly with Z. This decrease in total ionization cross-section, denoted QK but equal to σin of Sect. 4.4.6, can be obtained from (4.54), or can be calculated with actual wavefunctions as in (4.38) (substituted into (4.44) and (4.52)). It also can be approximated as Z −4 . This Z-dependence of QK is opposite to that of ω K in (4.58). The probability of generating an x-ray depends on the product of ω K and QK , and this product turns out to be relatively constant in the energy range from 1–20 keV. The EDS method therefore has a well-balanced sensitivity to the elements from Na to Rh. ωK =

106

Fig. 4.23. K-shell fluorescence yield of the elements. The difference, 1 − ω K , is the yield of Auger electrons. After [4.10].

The detection of x-ray fluorescence radiation is the most widely-used technique for microchemical analysis in a TEM. A solid state detector, whose characteristics were described in Sect. 1.4.1 (Fig. 1.18), is positioned near the specimen. The energy spectrum of the x-rays emitted from the specimen is acquired in a multichannel analyzer (Sect. 1.4.4, Fig. 1.20). A typical EDS 21

Approximately, the Auger emission probability is independent of Z, whereas the x-ray emission probability increases strongly with Z. Unfortunately, it is generally impractical to use a TEM for chemical analysis by measuring the energies of Auger electrons. Auger electrons lose a significant fraction of their energy through nanometer distances in a material. Auger energies characteristic of atomic transitions are obtained only for those few atoms at the very surface of a sample. Unfortunately, the vacuum in a TEM is not particularly good, and the sample is heated under the electron beam. The surfaces of a TEM specimen quickly become contaminated, even if they are not oxidized already.

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spectrum, in this case from SiC, is presented in Fig. 4.24. The widths of the peaks are set by the energy resolution of the detector, and not by the atoms in the specimen. The detector characteristics also affect the intensities of the peaks. Notice that in spite of the equiatomic stoichiometry of SiC, the intensity of the C peak in the spectrum is much less than that of the Si peak.

Fig. 4.24. EDS spectrum from SiC, acquired with a Si[Li] detector having an ultra-thin window. After [4.11].

Factors for converting x-ray intensities to elemental concentrations are a necessary part of quantitative EDS measurements. Fortunately, the thinness of a TEM sample simplifies the conversion process – for a particular sample geometry these conversion factors can often be regarded as a set of constants (Sect. 4.6.4). Simple constants of conversion are not appropriate when there is significant x-ray absorption and secondary x-ray fluorescence events in the sample, as illustrated in Fig. 4.20a, and this is typically the case for measurements on bulk samples in an electron beam microprobe or a scanning electron microscope. The thinness of the TEM sample minimizes problems with x-ray absorption and fluorescence (as illustrated in Fig. 4.20b), and quantitation is often straightforward. 4.6.3 EDS Instrumentation Considerations Beam-Specimen-Detector Geometry. In general, the largest detector, located as closely as possible to the specimen, has the best geometry for efficient x-ray detection. Another important geometrical parameter is the take-off angle, ψ, which is the angle between the specimen surface and the line taken by the x-rays to the center of the detector (see Fig. 4.20b). From (3.50), sometimes known as Beer’s law: I = e−(μ/ρ)ρx , I0

(4.59)

where I/I0 is the fraction of x-rays transmitted through a thickness, x, of a material with density ρ. Here μ/ρ is the mass absorption coefficient, which is tabulated as a function of Z and the energy of the x-ray (as in Appendix A.2). The likelihood of x-ray absorption in the sample depends on the length of

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the “escape path,” or “absorption path,” through the sample. The absorption path depends on two factors: 1) the depth of x-ray generation in the sample, t, and 2) the take-off angle, ψ (the larger ψ, the shorter the absorption path). Equation (4.59) becomes: I = e−(μ/ρ)ρt csc ψ . I0

(4.60)

This geometrical factor for x-ray escape probability is plotted in Fig. 4.25 for various realistic μt. A rule of thumb, perhaps consistent with Fig. 4.25, is that the x-ray emission increases from 0◦ to 30◦ and then levels off.22 In most microscopes, samples are typically tilted about 30◦ toward the xray detector, or the detector may be mounted at a high angle. For some horizontal detectors, the Si[Li] detector is tilted 20–30◦ toward the sample, so little specimen tilting is required.

Fig. 4.25. Fraction of x-rays that leave the sample, I/I0 , versus detector take-off angle, ψ, for various characteristic depths in the sample, μt, where t is the depth of the primary ionization. The full intensity is an integration of x-rays originating from all depths.

Probe Diameter, Current and Convergence Angle. The probe diameter, dmin , current, i, and convergence angle, α, all affect the x-ray emission process. Fortunately, elemental concentration data can be obtained with reasonable accuracy by comparisons of peak intensities in EDS spectra, but accurate quantitative work with EDS usually requires knowledge of these parameters. Techniques for determining them are described here. As suggested in Sect. 2.3.2 (and analyzed in Sect. 6.5.1), α can be measured directly from the diameter of the disks in the diffraction pattern. A straightforward method to measure the incident current, i, is with a Faraday cage (Fig. 4.26). For 22

Very thick samples may generate more x-rays with higher angles of tilt, since the angle of incidence between the electron beam and the sample surface affects the average depth of the interaction volume. The smaller this angle of incidence, the closer is the interaction volume to the surface, and the shorter the absorption path for emitted x-rays.

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some TEM’s, a Faraday cage is mounted on a removable specimen rod. Once calibrated against measurements with a Faraday cage, the beam current can be determined approximately from readings of the emission current meter on the instrument console. Another method of measuring the probe current is with an EELS spectrometer, by counting the total number of electrons in the spectrum acquired in a known time, and converting this to amperes.

Fig. 4.26. Faraday cage. Left: orientation for calibration of zero current and positioning the beam. Right: orientation for current measurement. After [4.9].

A common way to measure the probe size, dmin , is to record an image of the probe directly on a TEM negative (or better yet, a more linear detector such as a charge-coupled device (CCD) camera) at high magnification by forming a focused image at the eucentric height, and then focusing the probe on the viewing screen using the second condenser lens, C2. If the probe is assumed to be Gaussian, the size can be arbitrarily defined as the full-widthat-half-maximum (FWHM) containing 75 % of the current and indicated as Ip /2 in Fig. 4.27, or the full-width-at-tenth-maximum (FWTM) indicated as Ip /10 in the figure. The FWTM definition is probably a better criterion since this part of the beam contains most of the current, and the tails associated with very small probes may be non-Gaussian and quite wide. In the TEM, the probe size is determined by the current through the first condenser lens, C1 (often called the “spot size” control), and the convergence angle is determined primarily by the size of the C2 aperture (this is also accomplished by additional adjustment of the objective lens prefield or by so-called condenser “minilenses” in some microscopes). The probe current on the specimen in a typical TEM can be varied over two orders of magnitude, depending on the probe size. If spatial resolution is not the main consideration, then using a large probe size with a high current provides the best x-ray counting statistics. If high spatial resolution is needed, however, there is a trade-off between resolution and probe current. Theoretically, the probe size should be independent of the size of the C2 aperture. In practice, however, the C2 aperture affects the probe size because high convergence angles (> 10−2 rad) are typically used to form very small probes, and this can lead to wide tails in the probe that are truncated by the aperture.

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Fig. 4.27. A series of images showing the electron intensity distribution on the TEM screen at 100,000 X magnification in a Philips EM400T with a LaB6 filament for six different spot size settings. The FWHM probe sizes calculated from these images were 200, 100, 40, 20, 10 and 10 nm, respectively. After [4.9].

4.6.4 Thin-Film Approximation Cliff–Lorimer Factors. Quantitation of EDS spectra is helped considerably by the fact that TEM samples are so thin that the emitted x-rays are usually not absorbed within the sample (cf., Fig. 4.20). Microchemical analysis by EDS begins by removing the background from the measured x-ray spectrum. The background originates primarily from bremsstrahlung radiation, which we found in Sect. 1.2.1 to depend weakly on energy, especially for thin specimens where multiple scatterings of the high-energy electron are unlikely. In the analysis of an EDS spectrum such as that in Fig. 4.28, a power series in E is typically used to model the background. With two or more adjustable parameters, the background can be modeled well. Subtracting the background from the spectrum provides peaks that can be either integrated numerically (with the procedure of Fig. 1.27), or fit to analytical functions such as Gaussian functions. The peak areas can be treated individually, and this would be acceptable in the simple case of Fig. 4.28. When overlaps of peaks occur, it is preferable to work with sets of peaks (such as Kα, Kβ, L-series, etc.) with the energies and relative intensities expected for each element (including the sensitivity of the EDS spectrometer). Either method provides a set of peak intensities, {Ij }, where j denotes a particular chemical element. These {Ij } are converted to a set of elemental concentrations, {cj }, as described next. In thin foil specimens, it is unlikely that an x-ray emitted from one atom will be absorbed by a second atom. Such double-scattering processes are neglected in the “thin-film approximation.” This simplifies enormously the task of determining the {cj } from the {Ij }. In the thin-film approximation, the ratio of x-ray peak intensities from the elements A and B, IA /IB , is simply proportional to the corresponding weight-fraction ratio, cA /cB :

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Fig. 4.28. EDS spectrum of a Zr-based metallic glass, showing a fitted background and two peak areas above the background. After [4.12].

IA cA = kAB , cB IB

(4.61)

where kAB is a constant for a given accelerating voltage and a specific EDS spectrometer, and is independent of specimen thickness and composition. This constant kAB is often called a “Cliff–Lorimer factor.” It accounts for the efficiency of x-ray production for different accelerating voltages and the efficiency of the detector at the relevant x-ray energies. A convenient feature of EDS is that for a large number of elements, the kAB factor for their Kα x-ray peaks is approximately 1. The ratio of the peak intensities (or even peak heights) therefore gives a good approximation of the sample composition, making for a simple, semi-quantitative EDS analysis. This approximation holds for elements from about Mg to Zn. Below or above this range of atomic numbers the kAB factor gradually increases, but for elements of similar atomic number it is still reasonable to estimate their concentrations by comparing intensities of their Kα peaks. For samples containing elements of high and low atomic number, say ZrN or AgO, absorption of x-rays from the lighter element may distort any simple interpretation based on peak intensities, however. A normalization procedure:  cj = 1 , (4.62) j

is used to convert the ratios of the weight fractions to weight percentages (or, alternatively, atomic fractions to atomic percentages). That is, if kAB for elements A and B in a binary system is known, quantification is based on the measured ratio of IA and IB (4.61), and using (4.62): cA + cB = 1 .

(4.63)

For a ternary system with elements A, B and C, the following equations are used:

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IC cC = kCA , cA IA IC cC = kCB , cB IB cA + cB + cC = 1 .

(4.64) (4.65) (4.66)

For a ternary alloy we have one more unknown, but one more independent peak ratio and another equation (4.65). In general, as we add more elements we can still use a set of linear equations like (4.64) and (4.65), plus (4.62) to complete the alloy chemistry. The Cliff–Lorimer factors are mutually related. This is seen by dividing (4.65) by (4.64): kCB IC IA cA cC = . cC cB kCA IB IC

(4.67)

By the definition in (4.61), kCA = 1/kAC , so: IA cA = kAC kCB . cB IB

(4.68)

Comparing (4.61) and (4.68), we obtain a general relationship between the Cliff–Lorimer factors: kAB = kAC kCB .

(4.69)

Cliff–Lorimer factors, or “k-factors,” are often stored in a look-up table in EDS software. k-Factor Determination. Considerable effort is devoted to obtaining accurate Cliff–Lorimer factors, kAB , since the accuracy of the EDS analysis depends on them. The k-factors are a combination of specimen and detector properties. Consider a kAB coefficient for Kα x-ray emission from elements A and B. The thin film approximation assumes both types of x-rays originate in the same region, and take direct paths through the specimen. We therefore expect the kAB coefficient to be the ratio: kAB =

AA ωA aA QKA (μABe −μBBe )t e , AB ω B aB QKB

(4.70)

where Ai is the atomic weight of element i, (needed when the kAB are for determining mass fractions), ω i is its fluorescence yield, ai is its fraction of Kα emission (for which Kβ emission competes, but ai = 1 for Z < 19), and μiBe is the “effective” mass-absorption coefficient for the x-ray from element i and the detector window of effective thickness t (comprising, for example, the Be window, the Si dead layer, and the Au conductive film). The QKi are the K-shell ionization cross-sections (which could in principle be obtained from the total cross-section of (4.54), but better results are available). There are essentially three ways to determine kAB : 1) determine it experimentally using standards, 2) use values available in the literature, or

4.6 Energy Dispersive X-Ray Spectrometry (EDS)

211

3) calculate it from first principles. The first method is the most reliable. Experimental kAB values are determined for a specific microscope, detector and operating conditions. Use of calculated and/or experimental kAB values from the literature is possible, but errors are expected owing to differences in the characteristics of the specimen, microscope, detector, and experimental geometry (including the tilt of the sample). The agreement between experimental and calculated kAB values is typically good to 5 % for Z > 14, and for these elements it is often sufficient to calculate kAB values for a given detector and accelerating voltage. The disagreement between theory and experiment for low Z may be due to a combination of inadequate theory, absorption of low-energy x-rays within the specimen, contamination on the detector window, or the loss of light elements during electron irradiation. For routine analysis it is common to use the kAB values provided by the software of the EDS spectrometer system. Performing similar measurements on experimental standards of known composition can provide correction procedures to improve quantification for specimens of similar compositions. 4.6.5 * ZAF Correction In an EDS spectrum, the x-ray peaks from different elements have intensities that depend on: 1) the path and energy of the high-energy electron passing through the sample, 2) the ionization cross-sections of the elements, 3) the fluorescence yields, and 4) the probabilities that emitted x-rays are seen by the detector. The thin film approximation collects all these effects into a constant factor for each type of characteristic x-ray. In the thin-sample limit, all peaks in an EDS spectrum increase in intensity with increased sample thickness, but the ratios of peak intensities remain unchanged. This permits the use of (4.61) for samples of all thickness. For thicker samples, however, the peak intensity ratios are altered. In TEM, the attenuation of the incident electron beam provides only minor effects on the ionization cross-section, and has no effect on the fluorescence yield. The generation of characteristic x-rays from different elements is not altered by changes in the incident beam as it passes through a sample of moderate thickness. The thickness effects originate with the scattering of the characteristic x-rays by the different elements in the sample. As the samples become thicker and the x-ray exit paths through the sample become longer, these inelastic x-ray scattering processes involve a larger fraction of the x-rays, altering the ratios of peak intensities. Correction for these inter-element interactions is performed by considering the atomic number, Z, the absorption, A, and fluorescence, F , in procedures called “ZAF corrections.” * X-Ray Absorption Within the Specimen. X-ray absorption follows Beer’s Law (4.59). Since x-rays are generated throughout the foil thickness, evaluating the average absorption generally requires an integration of (4.60) over the sample thickness. Fortunately, for thin foils we can linearize the

212

4. Inelastic Electron Scattering and Spectroscopy

exponential in (4.60) as: e−x ≃ 1 − x, and take the average depth of x-ray emission as t/2, where t is the sample thickness. In this case absorption alters the x-ray intensity ratio IA /IB from the ratio recorded for an infinitely-thin specimen, IA0 /IB0 :   μA t IA0 1 − ρA 2 ρA csc ψ IA ≃ , (4.71) IB IB0 1 − μρBB 2t ρB csc ψ   t IA0 IA (4.72) 1 + (μB − μA ) csc ψ . ≃ IB IB0 2 Equation (4.72) shows the importance of the difference in absorption coefficients for the x-rays of elements A and B – if they have similar μ, the intensity ratios IA /IB are unaffected. Table 4.3 shows thicknesses at which the thin-film approximation is no longer valid due to absorption effects in specific materials.23 Table 4.3. Limits to the thin foil approximation caused by absorption. Thickness limit is for a 3 % error in the kAB factor [4.9] material

thickness [nm]

absorbed x-ray(s)

Al–7 %Zn NiAl Ag2 Al FeS FeP Fe–5 %Ni CuAu MgO Al2 O3 SiO2 SiC

94 9 10 50 34 89 11 25 14 14 3

Al Kα Al Kα Al Kα, Ag Lα S Kα P Kα Ni Kα Cu Kα, Au M α Mg Kα, O Kα Al Kα, O Kα Si Kα, O Kα Si Kα, C Kα

* Characteristic Fluorescence Correction. Characteristic x-rays from a heavier element can photoionize atoms of lighter elements, causing them to fluoresce. This enhances the number of x-rays detected from the light element, and suppresses the number from the heavier element. Fluorescence effects in thin foils are much weaker than in bulk samples (Fig. 4.20). Nevertheless when strong fluorescence does occur, e.g., Cr Kα fluorescence under Fe Kα radiation, quantitative microchemical analysis of TEM specimens may 23

To make an absorption correction, however, it is necessary to know the mean x-ray path length within the specimen, and this is difficult to determine from wedge-shaped or irregular specimens.

4.6 Energy Dispersive X-Ray Spectrometry (EDS)

213

require a fluorescence correction (Fig. 4.29). Several fluorescence corrections for thin films have been developed, and a successful model [4.13] uses an enhancement factor, XA , for the element, A, that undergoes fluorescence:  UB lnUB t  rA − 1 AA 0.923 − ln(μB t) , μBA (4.73) XA = cB ω KB rA AB UA lnUA 2

where Ui is the overvoltage ratio (ratio of incident electron energy to K-edge energy) for element i, Ai is its atomic weight, ci is its weight fraction, ri is its absorption-edge jump-ratio (the fractional countrate change across the EELS absorption edge), ω Ki is its fluorescence yield, and μBA is the mass absorption coefficient of element B in element A. For a sample in which the element B causes fluorescence of element A, the measured composition is corrected by: IB cB = kBA (1 + XA ) . cA IA

(4.74)

Fig. 4.29. Experimental data showing an increase in the apparent Cr concentration with thickness in thick specimens of a Fe-10% Cr alloy, owing to fluorescence of Cr Kα by Fe Kα x-rays. After [4.9].

4.6.6 Artifacts in EDS Measurements Ideally, when an electron probe is placed on a specimen for chemical analysis, the x-rays come only from the primary interaction of the beam and the specimen. Unfortunately, there are two other sources of spurious x-radiation in the TEM. One is x-ray generation in the illumination system of the microscope from electron interactions with the column components. If not eliminated by shielding and design, these x-rays (and/or stray electrons) “rain down” on the specimen through the upper objective lens pole piece. X-rays are also generated by electrons or x-rays that scatter from the specimen, and subsequently strike the objective lens components or specimen holder. These spurious x-ray signals must be subtracted from the measured spectrum to

214

4. Inelastic Electron Scattering and Spectroscopy

perform quantitative x-ray analyses. There are different ways of recognizing and dealing with these effects, depending on the type of sample. We describe three typical situations below. Manufacturers of TEM instruments are well aware of these issues, and go to great lengths in the design of their microscope columns and holders to minimize them, but this does not mean that spurious x-rays cannot be a problem for some measurements. In the case of an electropolished thin foil, where absorption and fluorescence can be neglected, the situation is straightforward. After measurements have been made on the specimen area of interest, the probe is placed just off the specimen in the hole and a spectrum is acquired under identical conditions. The x-ray signal from the hole, often called the “hole count,” is subtracted from the specimen spectra to effectively remove any spurious signal generated by the microscope or holder. For most modern instruments the hole count from thin specimens is almost negligible compared to the specimen spectra, so corrections are small and quantification is straightforward. It must be remembered, however, that sending the electron beam through the hole is not the same as sending it through the specimen, from which electrons and x-rays are emitted, enabling secondary x-ray generation. The hole count is not exactly the same as the actual background, but hopefully close to it. For thin foils, high-energy x-rays striking the specimen from above generally pass right through it and do not produce any significant background. At the other extreme, consider a typical sample prepared by focused ionbeam milling. Such samples comprise a thin membrane approximately 30 μm wide and extending 10 μm into the sample, surrounded by residual bulk material that is approximately 20 μm or more in thickness, supported on a Cshaped washer. In this case, there is ample opportunity for x-rays raining down on the sample, and for scattered electrons and x-rays generated by the sample, to strike bulk sample and support materials, generating many spurious x-rays. If this spurious x-radiation is coming from x-rays produced in the illumination system, the ratio of the Kα x-rays to the Lα x-rays for elements like Cu will be very high, because the high-energy x-rays will fluoresce the Kα x-rays much more than the Lα x-rays. As a rule, for electron excitation of x-rays from medium atomic number elements like Cu and Ni, the Lα peaks are usually as high or higher than the Kα peaks, so an abnormal ratio is an indication of x-ray fluorescence. It is always a good idea to acquire a hole count from such a specimen, but it is also important to realize that the signal generated from scattered electrons and x-rays in this type of specimen may be significantly greater when the probe is placed on the membrane than in the hole, so the hole count subtraction may not be so reliable as for thin-foil samples. It is often useful to place the probe on a C (carbon) layer at the edge of the specimen, or on a coating layer (e.g., if Pt was used to preserve the surface during milling), and compare these spectra to the hole count and specimen spectra to make the quantification process as accurate as possible.

4.6 Energy Dispersive X-Ray Spectrometry (EDS)

215

A common situation that falls in between these two extremes is small particles supported on a thin carbon film, which is supported on a metal grid (often Cu). Spurious x-rays from the grid can be eliminated by using Be or polymeric grids, which do not produce significant x-ray signals. If one stays away from the metal grid bars, x-rays produced from the stray radiation above the sample are generally not a problem, but spurious x-rays can still be produced from electron-specimen interactions. These are often seen as Cu peaks (assuming a Cu grid) in the x-ray spectra. Again, it is common practice to place the probe on the C support just next to the particle being analyzed, and subtract this from the specimen spectrum. Again this is an approximation, since electron scattering and x-ray generation from the C film are not the same as from the particle. Thought and judgement are needed, as always, for the best experimental work. 4.6.7 Limits of Microanalysis There are three quantifiable limits to microanalysis: 1) the absolute accuracy of quantification, 2) the minimum detectable mass (fraction), and 3) the spatial resolution. Of course there are other practical limits including contamination, insensitivity to low Z in EDS, and specimen preparation and geometry, but here we discuss the first two quantifiable limits. Limits on spatial resolution were discussed in Sects. 2.8 and 4.6.3. The accuracy of quantification is limited by the counting statistics of the x-ray spectra. For strong peaks on a weak background, the standard deviation, σ, is given by: √ (4.75) σ= N, where N is the number of counts in the peak after background subtraction.24 Once the standard deviation is known, different confidence limits can be set for the value of N , i.e. 68 % confidence that N will lie in N ±σ, 95 % in N ±2σ and 99 % in N ± 3σ. The value of 3σ, taken to be the 99 % confidence level in the value of IA , is often used to estimate the error in the peak intensity: √ N σ × 100 . (4.76) Error(%) = ±3 × 100 = ±3 N N The larger is N , the lower the error in the analysis. For a 1 % accuracy at the 99 % confidence level, one needs 105 counts in a peak, or 104 counts for 1 % accuracy at the 68 % confidence level. The error in IA /IB is: √  √  NA NB Error(%) = ± 3 × 100 . (4.77) +3 NA NB 24

When the background is a substantial fraction of the peak height, this argument is invalid for reasons stated in Problem 1.9. For weak peaks it is more accurate √ to use the background counts over the width of the peak, Nb , to obtain σ = Nb for use in (4.75).

216

4. Inelastic Electron Scattering and Spectroscopy

When using (4.61) for composition analysis, to the error of (4.77) we must add any error in kAB , which is again the sum of the errors in IA and IB for the standard. If Gaussian statistics are assumed, there is a simple statistical criterion that can be used to define the minimum mass fraction (MMF). A peak containing IB counts from element B in a matrix of A is considered statistically real and not a random fluctuation in the background intensity, IBb , when: (4.78) IB ≥ 3 2IBb . The MMF of B that can be detected in a binary material of elements A and B, cB (MMF) in at.%, is obtained using (4.61) and (4.78): cA kBA . (4.79) cB (MMF) = 3 2IBb b IA − IA

In practice, a MMF of approximately 0.1 wt.% can be obtained in EDS if enough counts are collected. Similarly, the minimum detectable mass (MDM) is predicted to be around 10−20 g for a range of Z from 10 to 40. These statistical analyses give the accuracy for quantification of a single measurement. In many cases, it is possible to obtain only a limited number of counts in a spectrum owing to factors such as beam damage or specimen drift. In such situations, it is possible to reduce the error in quantification (or at least assess it) by combining the results from n different measurements of the intensity ratio IA /IB . The total absolute error in IA /IB at a given confidence value is obtained using the Student-t distribution. In this approach, the error of the estimate E is given by: tα/2 S , E< √ N

(4.80)

where tα/2 is the Student-t value such that the normal curve area to its right equals α/2 with a probability of 1 − α, S is the standard deviation for n measurements of the intensity Ni , given by:   n  (Ni − Ni )2 , (4.81) S= n−1 i=1 which contain on average Ni counts. By increasing the number of measurements, one can reduce the error of measurement. In other words, if we estimate μ by means of a random sample size of n, we can assert with a probability of 1 − α (where 1 − α = 0.95 for a 95 % confidence level for example) √ that the error in the measurement E = |Ni − μ| is less than (tα/2 S)/ n, at least for sufficiently large values of n. Equation (4.80) can also be rearranged and solved for n to determine the number of measurements n that must be taken to achieve a mean Ni which is in error by less than E.

Problems

217

Further Reading The contents of the following are described in the Bibliography. C. C. Ahn, Ed.: Transmission Electron Energy Loss Spectrometry in Materials Science and the EELS Atlas 2nd Ed. (Wiley-VCH, Weinheim, 2004). C. C. Ahn and O. L. Krivanek: EELS Atlas (Gatan, Inc., Pleasanton, CA 1983). M. M. Disko, C. C. Ahn and B. Fultz, Eds.: Transmission Electron Energy Loss Spectroscopy in Materials Science (Minerals, Metals & Materials Society, Warrendale, PA 1992). R. F. Egerton: Electron Energy-Loss Spectroscopy in the Electron Microscope 2nd Ed. (Plenum Press, New York 1996). J. J. Hren, J. I. Goldstein and D. C. Joy, Eds.: Introduction to Analytical Electron Microscopy (Plenum Press, New York 1979). D. C. Joy, A. D. Romig, Jr. and J. I. Goldstein, Eds.: Principles of Analytical Electron Microscopy (Plenum Press, New York 1986). H. Raether: Excitations of Plasmons and Interband Transitions by Electrons (Springer-Verlag, Berlin and New York 1980). L. Reimer, Ed.: Energy-Filtering Transmission Electron Microscopy (SpringerVerlag, Berlin 1995). L. Reimer: Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, 4th Ed. (Springer-Verlag, New York 1997). P. Schattschneider: Fundamentals of Inelastic Electron Scattering (SpringerVerlag, Vienna, New York 1986). D. B. Williams: Practical Analytical Electron Microscopy in Materials Science (Philips Electron Instruments, Inc., Mahwah, NJ 1984). D. B. Williams and C. B. Carter: Transmission Electron Microscopy: A Textbook for Materials Science (Plenum Press, New York 1996).

Problems 4.1 Use Moseley’s laws (Sect. 1.2.2) to determine the Kα and Kβ x-ray energies for the elements Cu, Al, Mg, Zn, Be, Li and Ni. Which of these can be detected with a typical Be-window EDS detector? Explain. 4.2 (a) Using equations provided in the text, plot the mean free path, λ, for inelastic scattering of electrons in Cu with an accelerating potential of 200 kV as a function of collection angles β ranging from 0.1 to 20 mrad. (b) Using the same equations, plot the inelastic mean free path, λ, as a function of the average atomic number of a material Z. Explain the graphs.

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4. Inelastic Electron Scattering and Spectroscopy

4.3 Use the K, L, α, β, etc., notation to name the X-rays generated by the following electron transitions: (a) a hole in the K shell is filled by an electron from the LIII shell, (b) a hole in the K shell is filled by an electron from the MII shell, (c) a hole in the K shell is filled by an electron from the OIII shell, (d) a hole in the LIII shell is filled by an electron from the MI shell, (e) a hole in the LII shell is filled by an electron from the NIV shell, (f) a hole in the LI shell is filled by an electron from the OIII shell. 4.4 Find the thickness (in nm) of the Al sample in the EELS spectrum of Fig. 4.30. Assume 100 keV electrons.

Fig. 4.30. EELS low-loss spectrum of thin foil of Al for Problem 4.4. Assume the zero-loss peak is a triangle of 2.0 eV width at half-height.

4.5 A TEM specimen undergoes a type of radiation damage known as “knock-on damage” when a high-energy electron transfers enough energy to an atom to displace it from its crystallographic site. For a given electron energy, knock-on damage tends to be most severe for elements of low atomic number. (a) In a direct (“head-on”) collision between a high-energy electron and an atom, show that the energy transfer scales inversely with the atomic weight of the atom. (For simplicity, you may assume that the incident electron is scattered elastically by an angle of 180◦.) (b) If a Li atom requires 10 eV to leave its crystal site, calculate the threshold energy for an incident electron to induce knock-on damage. Do the same calculation for Al, Cu, and Au. 4.6 This problem presents two mathematical tricks for working with Dirac δ-functions. Calculations of electronic energies or scattering intensities often employ sums of Dirac δ-functions because δ-functions are handy for representing energy eigenvalues. For example, it is possible to write a distribution function for an energy spectrum, n(E), as:

Problems

n(E) =

N 1  δ(E − εα ) . N α

219

(4.82)

The idea behind this equation is that if N is large so there are numerous states (or transitions), each of energy εα , the discrete sum on the right becomes a continuum. To integrate the number of states up to some energy E ′ , each δ-function on the right side contributes 1 to the sum when E ′ > εα . A direct calculation of this type can be clumsy, however. Two expressions for δ-functions can be of assistance in mathematical work:   1 1 Im δ(E − εα ) = − lim , (4.83) δε→0 π E + iδε − εα

∞ 1 ei(E−εα )t dt . (4.84) δ(E − εα ) = 2π −∞

Prove, or convince yourself, that these two equations are appropriate ways to represent a δ-function. 4.7 Suppose that samples containing mixtures of elements A,B and A,C were used to obtain the Cliff–Lorimer constants, kAB and kAC . Suppose the x-ray intensity of element C was less reliably correlated to an independent determination of composition of the samples, and the error in kAB was estimated as 1 %, whereas the error in kAC was estimated as 10 %. (a) Estimate the error in a calculated kBC . (b) Estimate the error in composition of a material of A and B elements, with nominal compositions 10 % B, 50 % B, and 90 % B. 4.8 The EDS data in Figs. 4.31b–d below were obtained from the Al-Ag precipitate shown in a. The number of counts in the peaks and background beneath them in d are: Al Kα: 14,986 in peak, 1,969 in background Ag Kα: 10,633 in peak, 1,401 in background Given that kαAgAl = 2.3 for the microscope conditions used, and that the thin-film approximation is valid, what is the composition of the precipitate? 4.9 In an EELS experiment, suppose we seek to measure the near-edge region from an element with a low concentration in the specimen. To optimize the ratio of edge jump to background, is it better to use a large or small acceptance angle for the EELS spectrometer? Why? (Hint: Assume the angular dependence of the background is that of a single absorption edge that lies at lower energy than the edge of the dilute element.) (Further Hint: Look at the Bethe surface.)

220

4. Inelastic Electron Scattering and Spectroscopy

Fig. 4.31. a–d. EDS data (b–d) from extracted precipitate on holey C support film in (a) (for Problem 4.8). After [4.14].

4.10 The integral inelastic cross-section for a K edge, σin,K (β), as a function of collection angle, β, can be predicted with moderate accuracy using the formula [4.5]: !  2 " 2 β E , (4.85) σin,K (β) = 4πa20 R fK ln 1 + T E φE where φE = E (2γT )−1 , T = me v 2 /2, γ = (1 − v 2 /c2 )−1/2 , fK = 2.1 − Z/27, ER = h2 /(2me a0 )2 = 13.6 eV is the Rydberg energy, E = 1.5EK , EK is the K-edge energy, a0 = 52.92 × 10−12 m is the Bohr radius, me is the rest mass of the electron, v is the electron velocity, c is the speed of light, and Z is the atomic number. Using this equation, plot σin,K (β) versus β for carbon with a K-edge energy of 284 eV, for collection angles β ranging from 0.1 to 20 mrad, assuming an accelerating potential of 200 kV. 4.11 Consider the probability of inelastic scattering, pi , and elastic scattering, pe , through a thin layer of material. We set pe + pi = p, where p is the total probability of scattering from the incident beam. (a) For thin samples of n layers, show that the x-ray mass absorption factor, μ = (n/x)p, where x is the thickness of one layer. Subsequent layers have the same probabilities, so for n thin layers we expect:

Problems

(pe + pi )n = pn .

221

(4.86)

(b) For thin samples, show that the ratio of double inelastic scattering, p2i , to single inelastic scattering, pi , is: p2i /pi = (n/2)pi . (Hint: Perform a binomial expansion of (4.86) and consider the physical meaning of the individual terms.) 4.12 When a hole in the core shell of an atom decays by an Auger process, one electron falls into the core hole and a second electron carries energy from the atom. A proper treatment of the Auger effect accounts for the indistinguishability of the two electrons. For electrons of the same spin, this includes antisymmetrizing the two-electron wavefunction of the initial state:  1  (4.87) ψαγ = √ ψα (r1 )ψγ (r 2 ) − ψγ (r 1 )ψα (r 2 ) . 2

(a) Write integral expression(s) for the matrix element, βk|H ′ |αγ , with ∗ H ′ = e/(|r1 − r2 |), using (4.87) for ψαγ and a similar expression for ψβk . (b) Show that the rate of the Auger transition,

e 2π ∗ ψαγ d3 r 1 d3 r 2 , ψβk Γ = (4.88)  |r 1 − r 2 |

involves the difference of two matrix elements, one for the transition |αγ → |βk and the other for the “exchange transition,” |αγ → |kβ . (c) The inelastic scattering of a high-energy electron by a core electron transition involves two electrons, the perturbation H ′ = e/(|r1 − r 2 |), and the same calculation as in a and b (compare (4.24) and (4.88)). Is the exchange transition important for the scattering of a high-energy electron? Why?

5. Diffraction from Crystals

5.1 Sums of Wavelets from Atoms Chapters 5–7 are concerned with the angular dependence of the diffracted wave, ψ(Δk), emitted from different arrangements of atoms. The underlying mechanism is coherent elastic scattering from individual atoms, the topic of Chap. 3. Diffraction itself, however, is a cooperative phenomenon based on phase relationships between the wavelets1 scattered coherently by the individual atoms. This chapter explains how a translationally-periodic arrangement of atoms in a crystal permits strong constructive interferences between individual wavelets, creating the familiar Bragg diffractions. 1

We call the outgoing waves from individual atoms “wavelets,” to distinguish them from their coherent sum, the total diffracted wave, that is measured at the detector. The “wavelets” are in fact full wavefunctions, but each contributes a small amplitude to the total wave.

224

5. Diffraction from Crystals

The diffraction theory developed here is “kinematical theory.” As discussed in Chap. 3, the validity of kinematical theory for electron diffraction is contingent on the validity of the first Born approximation (presented as (3.74), leading to (3.82)). The assumption that the incident wave is scattered weakly by the material is also used when developing kinematical theories of x-ray and neutron diffraction. For the strong Coulomb interactions between incident electrons and atoms, however, kinematical theory must be used with caution. It is usually reliable for calculating the structure factor of the unit cell. For electron diffraction contrast from larger features such as crystal shapes and crystalline defects, however, kinematical theory is often only qualitative. Kinematical theory is more quantitative for x-ray diffraction because x-ray scattering is much weaker than electron scattering. Kinematical calculations can be highly reliable for neutron diffraction. For electron diffraction, kinematical theory can be improved considerably by redefining the extinction length as is done in Section 7.3, but quantitative results generally require the dynamical theory developed in Chap. 11 or the physical optics approach of Chap. 10. 5.1.1 Electron Diffraction from a Material Diffraction is a wave interference phenomenon. To form diffraction patterns, we must have more than one scattering center. Consider the geometrical array of scattering centers in Fig. 5.1. We use the same coordinates as in Fig. 3.5, but now we have a set of vectors {Rj }, which mark the centers of the atoms in the material. In Sect. 5.2 we impose the crystal symmetry on the vectors {Rj } (specifically, the translational periodicity), but this comes later.

{Rj }

Rn rʼ

rʼ– Rn

Fig. 5.1. Centers of atoms are at fixed coordinates, {Rj }. The independent vector r ′ ranges over all space, and the vector r ′ − R n is the distance of r ′ from the center of the nth atom.

Our scattered electron wave in the first Born approximation is (3.82):

 −m eik·r (5.1) ψscatt (Δk, r) = V (r′ ) e−i∆k·r d3 r′ . 2π2 |r|

An important step in calculating a diffracted wave from a group of atoms is choosing a suitable scattering potential, V (r′ ). For a group of atoms we use a sum of atomic potentials, Vat , each centered at an Rj that is an atom site:

5.1 Sums of Wavelets from Atoms

V (r ′ ) =

 Rj

Vat (r′ − Rj ) .

225

(5.2)

Note that whenever r ′ = Rj , one of the terms in (5.2) is Vat (0), and the potential V (r′ ) has a big contribution from the atom centered at Rj . Substituting (5.2) into (5.1):

  −m eik·r Vat (r ′ − Rj ) e−i∆k·r d3 r ′ . (5.3) ψscatt (Δk, r) = 2π2 |r| Rj

We ignore the r-dependent outgoing wave in front of the integral in (5.3), since we are not concerned with the dependence of intensity on 1/r2 (cf., (3.56), (3.57)). The trick now is to define the new coordinate: r ≡ r′ − Rj (so r′ = r + Rj ):2

 −m Vat,Rj (r) e−i∆k·(r+Rj ) d3 r . (5.4) ψ(Δk) = 2π2 Rj

Since the vectors r and {Rj } are independent, we remove each phase factor e−i∆k·Rj from the integral:    −m −i∆k·r 3 ψ(Δk) = (5.5) d r e−i∆k·Rj . Vat,Rj (r) e 2π2 Rj

The integral in parentheses in (5.5) involves the scattering potential of a single atom. It is the atomic form factor for electron scattering, fel (Rj , Δk) of (3.84), discussed in Sects. 3.3.2–3.3.4. We have written the atomic potential for electron scattering as Vat,Rj (r), using a subscript to remind us to keep track of the specific type of atom at position r ′ = Rj . As in (3.84) we define:

−m (5.6) Vat,Rj (r) e−i∆k·r d3 r . fel (Rj , Δk) ≡ 2π2 Because the atom is so much smaller than typical lengths of periodic crystals, diffraction effects from the crystal occur over a much smaller range in Δk than do effects from the shape of the atom. We can often understand the diffraction effects from the crystal without too much concern about the details of how fel (Rj , Δk) depends on Δk. For maximum simplicity, we sometimes treat fel as a number that depends only on the type of atom located at Rj . The scattered wave (5.5) from N atoms is written most simply as: ψ(Δk) =

N 

fel (Rj ) e−i∆k·Rj .

(5.7)

j=1

The diffracted wave is a sum of wavelets, each of amplitude fel (Rj ), emanating from atoms at all {Rj }. The exponential e−i∆k·Rj in (5.7) is not the 2

This substitution changes the exponentials from full phase factors of independent wavelets into relative phase factors of wavelets from the different atoms.

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5. Diffraction from Crystals

actual outgoing wavelet (which needs the prefactor in (5.3)), but this exponential gives the relative phase of the wavelet emitted from the atom at Rj . The phase relationships between the individual wavelets are our central concern because they determine the constructive or destructive interferences. To get the absolute intensity of the scattered wave at the detector, Iscatt (Δk, r), we must use the full prefactor of (5.3), and take the product of the wavefunction with its complex conjugate: ∗ Iscatt (Δk, r) = ψscatt (Δk, r) ψscatt (Δk, r) , 2   1 m ψ(Δk) 2 . Iscatt (Δk, r) = 2 4π 2 4 |r − r ′ |

(5.8) (5.9)

5.1.2 Wave Diffraction from a Material In the present derivation of the x-ray diffraction intensity from a material, we hide the mechanism of scattering and the r−2 intensity dependence of the outgoing wave.3 We assume each atom contributes to the scattered x-ray wave an amount proportional to its scattering factor, f . The physical picture is shown in Fig. 5.2. Figure 5.3 is essentially the same figure as Fig. 5.2, and uses the coordinates of Fig. 5.1, which are: • k0 • k • {Rj } • r

the incident wavevector, the scattered wavevector, the positions of the atoms in the material, the position of the x-ray detector.

First consider the coherent elastic scattering from one atom at Ri . For this first atom there are two waves to consider. There is an incident plane wave, Ψ0 , which reaches the atom at Ri at time t′ : Ψ0 (Ri , t′ ) = A ei(k0 ·Ri −ωt ) , 

(5.10)

and there is a coherently-scattered wave at the detector, Ψi , proportional to the amplitude, A, of the incident wave times the scattering factor, f (Ri ), which is unique to the type of atom at Ri . We also have to consider the phase of the wave as it travels from the atom at Ri to the detector during the time t:  (5.11) Ψi (r, Ri , t) = f (Ri ) Ψ0 (Ri , t′ ) ei[k·(r−Ri )−ω(t−t )] . Substituting (5.10) for Ψ0 into (5.11):   Ψi (r, Ri , t) = f (Ri ) A ei(k0 ·Ri −ωt ) ei[k·(r−Ri )−ω(t−t )] , Ψi (r, Ri , t) = f (Ri ) A ei[−(k−k0 )·Ri +k·r−ωt] .

3

(5.12) (5.13)

The derivation in this section pertains to any wave diffraction by a group of atoms. It begins differently than Sect. 5.1.1 because by starting Sect. 5.1.1 with (3.82), we had already considered the phase relation between the incident and outgoing electron waves.

5.1 Sums of Wavelets from Atoms

227

outgoing spherical waves

k

k0

Fig. 5.2. Instantaneous picture of a packet of 9 wave crests, incident from the left, when it has generated 3 wavelets from the leftmost atoms and 2 from the rightmost atoms.

incident plane wave

detector

r

k r – Ri

r – Rj

Ri k0 Rj source

Fig. 5.3. Coordinates for the x-ray (or wave) scattering problem.

We can ignore the frequency of the waves, ω, and time, t, as we work with only the spatial coordinates.4 With these simplifications and with the usual definition of the scattering vector, Δk: Δk ≡ k − k0 ,

(5.14)

(5.13) becomes: ψi (r, Ri ) = A f (Ri ) ei(−∆k·Ri +k·r) .

(5.15)

By an identical argument we can obtain the wavelet at the detector, ψj , scattered coherently from the atom at any Rj : ψj (r, Rj ) = A f (Rj ) ei(−∆k·Rj +k·r) . 4

(5.16)

Our final result can be multiplied by e−iωt if we so desire, or more generally by e−i(ωt−δ) to account for any phase lag, δ, between the scattered and incident wave.

228

5. Diffraction from Crystals

Now we sum the coherently-scattered wavelet amplitudes from all atoms in the material (cf., (3.9)). The total diffracted wave at the detector, ψ(r′ ), is just a sum over all N atoms: ′

ψ(r ) = A

N 

f (Rj ) ei(−∆k·Rj +k·r) .

(5.17)

j=1

In practice we never know the positions of the x-ray source and detector to within an x-ray wavelength. We therefore neglect the phase factors involving r.5 Absolute intensities are also difficult to measure, so we drop the A as well. The diffracted wave from a material is then: ψ(Δk) =

N 

f (Rj ) e−i∆k·Rj .

(5.18)

j=1

We write ψ as a function of Δk because the atom positions {Rj } are not adjustable, whereas Δk, the “scattering vector” (5.14), is controlled by the angle to the detector. Equation (5.18) states that: The diffracted wave is proportional to the Fourier transform of the scattering factor distribution in the material. Compare this statement to the italicized sentence at the end of Sect. 3.3.1 Also compare (5.18) to (5.7). Referring again to Fig. 5.3, note that we started with two wavevectors, k and k0 , but our result, (5.18), involves only one: Δk. Here is some justification for this change of variable. The wavevectors k and k0 have the same length (the scattering is elastic), so the difference in length of the wavevectors is not interesting. We are more interested in the angle between k and k0 , because this is the 2θ in the Bragg’s law construction of Fig. 1.1. This angle is contained in the difference Δk = k − k0 , as is information about the length of k0 . Referring to Fig. 5.4, we see that this relationship is: Δk = |k − k0 | = 2 |k| sinθ , (5.19) 4π sinθ . (5.20) Δk = λ The direction of Δk is the normal of the diffracting planes. The physical phenomenon is depicted in Fig. 5.5. Each of the four atoms emits an identical set of circular wave crests in prompt response to the crests of the incident wave, and with the same √ wavelength. The separation between the horizontal rows of atoms is 1/ 2 of this wavelength. This provides a Bragg angle, θB , of 45◦ . Notice the constructive interference of the outgoing waves at the angle 2θB from the incident wave direction. 5

We first note, however, that it was implicitly assumed that r is much larger than the {R j }. This way we can use the same diffracted wavevector, k, for all atoms without concern for how the rays from the different atoms make different angles into the detector. The same consideration applies to k 0 – the distance from the x-ray source is large compared to the size of the sample.

5.1 Sums of Wavelets from Atoms

229

Fig. 5.4. Relationship between Δk and θ for elastic scattering.

k0

k

λ crystal λ 2 Fig. 5.5. Wave interferences for many wave periods. Constructive interference occurs along ±k and ±k0 . The wavelength was matched to the atom spacing as shown √ in the lower left; an interplanary spacing of λ/ 2 provides a Bragg angle of 45◦ , so 2θB = 90◦ .

Each x-ray is assumed to be a plane wave, but it is not so over an extremely wide sample. The spherical shape of the incident x-ray from an x-ray tube causes a phase error between the edges and center of the specimen. Loss of coherency occurs when the difference between the crest of a plane wave and a spherical wave is less than 1 ˚ A. For equipment of 1 meter dimension, this “spatial incoherence” occurs over a width of 10 μm. Other sources of incoherence are more severe. In x-ray diffraction the source has a finite size, causing a convergence angle at a point on the sample of 10−4 – 10−3 radians. This limits the lateral coherence on the sample to 0.1 – 1.0 μm. There is also a loss of coherence due to the wavelength spread of the characteristic x-ray lines, typically 1 part in 104 . Although these deficiencies of a sealed tube

230

5. Diffraction from Crystals

source are usually acceptable, it should be noted that synchrotron sources have far superior coherence.

5.2 The Reciprocal Lattice and the Laue Condition 5.2.1 Diffraction from a Simple Lattice We know from Bragg’s law (1.1) that crystals give strong diffractions, so now we seek the analogous law for wavelet interference. The translational symmetry of unit cells on a lattice is the essential feature of the crystal that enables constructive interferences of the wavelets emitted by many atoms. Consider a simple crystal having only one species of atom, positioned with one atom per unit cell of the lattice. The form factor is unchanged under any lattice translation: f (r) = f (r + R). We seek to maximize the sum of complex exponentials in (5.7) and (5.18):   ψmax ∝ Max e−i∆k·R . (5.21) R

We discuss later the detailed shape of ψ(Δk), but now we seek only the condition for the maximum value of ψ. Our “primitive” (i.e., shortest possible) lattice translation vectors are: a1 , a2 , a3 .

Recall that all lattice sites are obtained by the translations {R} from a reference site at the origin: R =ma1 + na2 + oa3 ,

(5.22)

where {m, n, o} are independent integers. For our simple crystal having one atom per lattice site and the reference atom at the origin, the set of all R is the same as the positions of all atoms. The sum in (5.21) is:   e−i∆k·R = e−i∆k·(ma1 +na2 +oa3 ) , (5.23) R

 R

R

e

−i∆k·R

=

 m

n

e−i∆k·(ma1 +na2 +oa3 ) .

(5.24)

o

Each exponential phase factor (cf., (3.8)) is a complex number of modulus 1. The largest value we can expect for these sums of phase factors occurs when all phase factors have the same real and imaginary parts – this way all real parts and all imaginary parts add together without contributions of cancelling sign. The first term in our sum is for m = 0, n = 0, o = 0. This term is: e0 = 1. The maximum value of the sum occurs when all other terms are pure real numbers equal to 1. Because ei2πinteger = 1, the maximum wave amplitude occurs when:

5.2 The Reciprocal Lattice and the Laue Condition

Δk · (ma1 + na2 + oa3 ) = 2π · integer ,

231

(5.25)

for all possible combinations of the integers {m, n, o}. When this condition is satisfied, all terms in (5.24) are 1, so their sum equals N , the number of atoms in the crystal. This provides the largest possible intensity for the scattered wave of (5.7) and (5.18):  2 (5.26) Iscatt = ψ ∗ ψ = fat  N 2 . It may seem curious that the intensity at the optimal Δk grows as N 2 , rather than as N , the number of atoms in the crystal. This does not mean that an individual atom increases its scattering power when embedded in a larger crystal. What actually happens is that as N increases, the function ψ(Δk) becomes sharper (narrower width in Δk). The total diffracted intensity integrated over Δk then increases as N , not as N 2 . The total diffracted intensity per atom remains the same, as it must since the coherent cross-section is a property of the atom. This peak narrowing is related to the “shape factor,” discussed quantitatively in Sect. 5.4. 5.2.2 Reciprocal Lattice Suppose that (5.25) is true for the following 3 choices of m, n, o: (1, 0, 0), (0, 1, 0), (0, 0, 1). In this case, (5.25) is true for any R, since m, n, and o are integers. To ensure that (5.25) is true for all atoms in the crystal, therefore, we need only ensure that it is true for these individual translation vectors: a1 , a2 , and a3 , which are the primitive lattice translation vectors. That is, to ensure that (5.25) is true, we need not look at all possible m, n, o, but only ensure that, for an appropriate Δk′ : Δk′ · a1 = 2π · integer ,

(5.27)

′

Δk · a2 = 2π · integer , Δk′ · a3 = 2π · integer . Integer combinations of the three lattice translation vectors {a1 , a2 , a3 }, as in (5.22), account for all atoms in the crystal.6 Our quest for the conditions for strong diffraction becomes a search for three vectors to enumerate those {Δk} that satisfy (5.27), denoted Δk′ . That is, if we know the primitive translation vectors of the crystal lattice, {a1 , a2 , a3 }, it would be most handy to have a scheme for generating automatically all the values of {Δk′ } for which we expect strong diffraction. We want a “reciprocal lattice,” having a new set of translation vectors {a∗1 , a∗2 , a∗3 }, from which we obtain any Δk′ that satisfies (5.27) as: Δk′ = ha∗1 + ka∗2 + la∗3 ,

6

(5.28)

Note again we have assumed that the crystal has one atom per lattice site (i.e., no basis vectors other than (0, 0, 0)).

232

5. Diffraction from Crystals

where h, k, and l are integers. We call the set of shortest possible k-space vectors the “primitive translation vectors” of the reciprocal lattice: {a∗1 , a∗2 , a∗3 }. If the individual a∗1 , a∗2 , and a∗3 each satisfy (5.27), any integer combination of these three vectors (as in (5.28)) also satisfies (5.27), leading to strong diffraction. It is important for the primitive reciprocal lattice vectors to have the smallest possible lengths, so when forming linear combinations of these vectors we don’t miss any Δk′ that satisfies (5.27). To keep the {a∗1 , a∗2 , a∗3 } small, it is best for a vector such as a∗1 to have no component along the real space lattice vectors a2 or a3 . If there were such a component, it would either give a product in (5.27)b or (5.27)c that is not 2π · integer, making things confusing, or if it gave a product of 2π · integer or larger, we should have been able to make a∗1 smaller. One acceptable such choice of a∗1 , a∗2 , and a∗3 can be made with cross-products, since a cross-product is perpendicular to the two vectors in the cross-product: a2 × a3 a∗1 = 2π , (5.29) a1 · a2 × a3 a3 × a1 a∗2 = 2π , (5.30) a2 · a3 × a1 a1 × a2 . (5.31) a∗3 = 2π a3 · a1 × a2

The a∗1 , a∗2 , and a∗3 , defined in (5.29)–(5.31) are the primitive translation vectors of the reciprocal lattice. The dot products of these reciprocal lattice vectors and our primitive lattice translation vectors are: a∗i · aj = 2πδij ,

(5.32)

where the Kroneker delta function, δij , is zero unless i = j. The primitive translation vectors of the reciprocal lattice have dimensions of inverse length. Their three denominators, incidentally, are equal. They are scalar (or pseudoscalar) quantities, and are the volume of a parallelopipedon constructed with the edges a1 , a2 , a3 . This is shown in Fig. 5.6, where the area vector, A = a∗1 × a∗2 , is normal to a∗1 and a∗2 , and the projection of a∗3 along A gives a volume. a1*

A a*3

a*2 a2* a1*

a*2 sine

e a*1

Fig. 5.6. A parallelopipedon constructed from the three reciprocal lattice vectors, whose basal area is A = a∗1 × a∗2 = a∗1 a∗2 sin θˆ z , and volume is A · a∗3 .

Our prescription in (5.29)–(5.31) provides a set of primitive reciprocal lattice vectors, but there remains a question of their uniqueness. Did we have

5.2 The Reciprocal Lattice and the Laue Condition

233

to define them in this way? The answer is no, but alternative definitions are less convenient. For example, we can always permute cyclically the a∗1 , a∗2 , a∗3 among the various axes of cubic crystals. Less trivially, a hexagonal lattice can have different translation vectors in its basal plane. By swapping vectors, however, we lose the convenience of (5.32) (although equivalent relationships exist). Nevertheless, we cannot pick arbitrary lengths or directions for reciprocal lattice vectors and expect them to be related to diffractions from a physical crystal, and the relationship to diffraction is what motivated the concept of a reciprocal lattice in the first place. There are two conventions for what to do with the 2π in (5.29)–(5.31). Here the 2π was incorporated into the reciprocal lattice vector itself. Unfortunately, this convention leads to clumsy expressions such as “the (4π2π2π) diffraction intensity,” rather than “the (211) diffraction intensity.” For this reason, we now drop the 2π from the definition of the reciprocal lattice vector, and keep it only in the exponential, which is transformed from e−i∆k·R to e−i2π∆k·R . Simultaneously we must redefine Δk ≡ 1/λ, rather than Δk ≡ (2π/λ). Beware that both conventions are in common use.7 5.2.3 Laue Condition Upon comparing (5.27) and (5.32), we see that the reciprocal lattice vectors of (5.29)–(5.31) are appropriate Δk for satisfying (5.27). This must also be true for any integer combination of the {a∗1 , a∗2 , a∗3 }, as in (5.28). We arrive at the condition to satisfy (5.27), known as the Laue condition: Diffraction occurs when Δk is a vector of the reciprocal lattice. Denoting an arbitrary reciprocal lattice vector as g, where: g = ha∗1 + ka∗2 + la∗3 (so g is the desired Δk′ of (5.27) and (5.28)), the Laue condition for diffraction is: Δk = g .

(5.33)

5.2.4 Equivalence of the Laue Condition and Bragg’s Law The Laue condition is equivalent to the Bragg condition, as is readily demonstrated with the construction in Fig. 5.4. The wavevectors k and k0 in Fig. 5.4 lie along the rays that were used in Fig. 1.1, so the angle θ must be the same in both figures. From Fig. 5.4 we see that: Δk = 2k sinθ , 1 Δk = 2 sinθ . λ 7

(5.34) (5.35)

It is usually possible to determine an author’s convention by looking for the presence or absence of the 2π in the exponential. Physicists prefer e−i∆k·R , crystallographers prefer e−i2π∆k·R .

234

5. Diffraction from Crystals

Since Figs 1.1 and 5.4 show that Δk  d (where d is the distance vector between the diffracting planes), the Laue condition (5.33) with g = 1/d becomes: 1 Δk = . (5.36) d Equating the right sides of (5.35) and (5.36) gives Bragg’s law (1.1): 2d sinθ = λ .

(5.37)

Incidentally, (5.35) converts between 2θ and Δk, but the usual convention is to keep the 2π so that Δk = (4π/λ) sinθ, as in (5.20). 5.2.5 Reciprocal Lattices of Cubic Crystals The Laue condition (5.33) is powerful, and a few more facts about reciprocal lattices are helpful for using it effectively. In cubic, tetragonal, and orthorhombic lattices, the real and reciprocal lattice vectors are related as: ai  a∗i , ˆi ai = ∗ , |ai | ai ⊥ a∗j ,

(5.38) (5.39) (5.40)

when i = j.

a

a z

a2

y x

a3 a1

a2

a3 a1

Fig. 5.7. Cartesian axes at left and standard bcc unit cell in center. Translation vectors a1 , a2 , and a3 , access all atoms on the bcc lattice, and form the unit cell at right. (The cell on the right has half the volume of the cell in the center, and its volume is equal to the primitive cell constructed from the usual bcc primitive translation vectors: a/2(111), a/2(111), a/2(111).)

It is important to know that the reciprocal lattice of the bcc lattice is an fcc lattice, and vice-versa. We demonstrate this for the set of short translation vectors for the bcc lattice shown in Fig. 5.7: 

a1 = a [100] , a2 = a [010] , a3 = a 12 21 21 . (5.41) To get the reciprocal lattice vectors we apply the cross-product formulae. Equation (5.29) becomes:

5.3 Diffraction from a Lattice with a Basis

235



2πa2 [010] × 21 21 21

1 1 1 . (5.42) a3 [100] · [010] × 2 2 2     

ˆ (0 − 0) + z ˆ 0 − 21 , we get: ˆ 12 − 0 + y Evaluating: [010] × 12 21 21 = x   1 1 02 2 2π 4π » 1 1 –   = (5.43) 0 , a∗1 = a [100] · 1 0 1 a 2 2 a∗1 =

2

2

and similarly for (5.30) and (5.31):

4π » 1 1 – , (5.44) 0 a 22 4π [001] . (5.45) a∗3 = a These vectors {a∗1 , a∗2 , a∗3 } are drawn in Fig. 5.8. Notice that a∗1 and a∗2 are vectors to the centers of cube faces. Combinations of these vectors can be used to access all atom positions on an fcc reciprocal lattice. Hence, the fcc lattice is the reciprocal lattice of the bcc lattice, and vice-versa. (The standard primitive translation vectors of this particular fcc reciprocal lattice, incidentally, are {a∗1 , a∗2 , a∗1 + a∗2 + a∗3 }.) a∗2 =

a*3

a3

a3 a2

a2 a*1

a1

a1

a*2

4/

a 4/

a Fig. 5.8. Construction of the reciprocal lattice vectors of a bcc unit cell, using the relations (5.29)–(5.31) and the vectors of Fig. 5.7. The vectors a∗1 and a∗2 touch the centers of the cube faces.

5.3 Diffraction from a Lattice with a Basis 5.3.1 Structure Factor and Shape Factor In both real space and in reciprocal space, it is useful to divide a crystal composed of atoms at locations {r} into parts according to the prescription:

236

5. Diffraction from Crystals

crystal = lattice + basis + defect displacements , r = rg + r k + δr g,k ,

(5.46) (5.47)

but for a defect-free crystal the atom positions, R, are provided by vectors to each unit cell, {r g }, and vectors to the atom basis within the cell, {rk }: R = r g + rk .

(5.48)

The lattice is one of the 14 Bravais lattice types (the crystal typically has numerous unit cells on this lattice and numerous r g ). The basis is the atom group associated with each lattice site (the unit cell typically has a few r k ). Here we calculate the scattered wave, ψ(Δk), for the case of an infinitely large, defect-free lattice with a basis. From (5.7) or (5.18):  fat (R) e−i2π∆k·R . (5.49) ψ(Δk) = R

Substituting (5.48) into (5.49):  fat (r g + rk ) e−i2π∆k·(rg +rk ) . ψ(Δk) = rg

(5.50)

rk

Since the atom basis is identical for all unit cells, fat (r g + r k ) cannot depend on rg , so fat (rg + r k ) = fat (rk ):   e−i2π∆k·rg fat (r k ) e−i2π∆k·rk , (5.51) ψ(Δk) = rk

rg

ψ(Δk) = S(Δk) F (Δk) .

(5.52)

In writing (5.52) we have given formal definitions to the two summations in (5.51). The first sum, which is over all the lattice sites of the crystal (all unit cells), is known as the “Shape Factor,” S. The second sum, which is over the atoms in the basis (all atoms within the unit cell), is known as the “Structure Factor,” F : S(Δk) ≡ F (Δk) ≡

lattice 

e−i2π∆k·rg (shape factor),

(5.53)

rg

basis 

fat (r k ) e−i2π∆k·rk (structure factor).

(5.54)

rk

Since the structure factor of the unit cell is the same for all lattice points, it is usually convenient to write the diffracted wave as: ψ(Δk) =

lattice  rg

F (Δk) e−i2π∆k·rg .

(5.55)

The decomposition of the diffracted wave into the shape factor and the structure factor parallels the decomposition of the crystal into a lattice plus

5.3 Diffraction from a Lattice with a Basis

237

a basis. One can choose a large unit cell containing many atoms in its basis, but lattice sites that are far apart. Alternatively one can choose a small unit cell with fewer atoms in its basis, and lattice sites that are close together. For many problems it is more practical to choose the smallest possible unit cell that has orthogonal Cartesian translation vectors. This choice is convenient because working with orthonormal lattice translations simplifies the summations over the indices of R. For example, it is common practice to express a bcc crystal as a simple cubic lattice with a two-atom basis (an atom at a corner and an atom at the center of the cube). This is not the primitive unit cell of the bcc structure, however. The bcc structure is itself a Bravais lattice with a primitive unit cell containing one atom (a related 1-atom bcc unit cell is shown on the right in Fig. 5.7). Because the volume of the bcc standard cube is twice as large as the volume of the primitive bcc unit cell, it is not surprising that there are long-range periodicities of the standard cube that do not exist in the actual bcc structure. Many diffractions of the simple cubic crystal do not exist in the bcc diffraction pattern, and the systematic elimination of these non-bcc diffractions is performed with the “bcc structure factor rule” presented below. 5.3.2 Structure Factor Rules Structure Factor for sc Lattice. For a simple cubic (sc) lattice we show easily that strong diffraction occurs for any integer combination, (h, k, l). A general simple cubic reciprocal lattice vector, g, is (5.28): g =ha∗1 + ka∗2 + la∗3 .

(5.56)

For atoms located on the sites of a simple cubic lattice: {rg } = {ma1 + na2 + oa3 } where m, n, o are integers in all combinations , {r k } = {0a1 + 0a2 + 0a3 } one basis vector (of length zero) .

(5.57) (5.58)

We evaluate the structure and shape factors of (5.53) and (5.54) by imposing the Laue condition, Δk = g, and using the expression for g of (5.56). The arguments of the exponentials in (5.53) and (5.54) are: g · r g = (ha∗1 + ka∗2 + la∗3 ) · (ma1 + na2 + oa3 ) ,

g · rk =

(ha∗1

+

ka∗2

+

la∗3 )

· (0a1 + 0a2 + 0a3 ) ,

(5.59) (5.60)

using (5.32) (without the 2π): g · r g = hm + kn + lo = integer for any integers h, k, l , g · rk = 0 .

(5.61) (5.62)

Therefore, when the Laue condition, Δk = g, is satisfied for the simple cubic lattice:

238

5. Diffraction from Crystals

Ssc (Δk) = Fsc (Δk) =

N −1 

e−i2π(integer) =

r g =0 1 term

N −1 

1=N ,

(5.63)

r g =0

fat (r k ) e−i2π∆k·rk = fat (0) e−0

r k =(000)

= fat (Δk) ,

(5.64)

where we have explicitly written the Δk-dependence of the form factor in (5.64), and in this case when Δk = g: ψsc (Δk) = Ssc (Δk) Fsc (Δk) = N fat (Δk) .

(5.65)

The structure factor for the simple cubic lattice, Fsc (Δk), is the atomic form factor fat (Δk) for any and all integer combinations of h, k, and l, because there is only one atom per unit cell in the sc crystal. The same result holds for any primitive lattice – note that (5.61) and (5.62) do not require that a1 , a2 , a3 have the same lengths or lie along Cartesian axes. Structure Factor Rules for Other Lattices. The structure factor is more interesting when there is more than one atom in the basis of the unit cell. Interferences between wavelets scattered by the atoms in the basis lead to precise cancellations of some diffractions, disallowing certain combinations of h, k, l in the diffraction pattern. Prescriptions for enumerating the allowed diffractions are the “structure factor rules.” Before deriving the structure factor rule for a bcc crystal, we illustrate its origin with a specific physical example – the vanishing of the bcc (001) diffraction. Figure 5.9 compares simple cubic and bcc lattices. In the sc crystal, the atoms in the top plane contribute a phase factor e0 = 1 to the sum for the diffracted wave (5.49). Those from the next plane, located the distance a below, contribute the term ei2π = +1. Progressively lower planes contribute constructively to our phase factor sum as ei4π , ei6π . . . , all of which equal +1. Now compare the phase factor sum for the bcc crystal. Those atoms in the center of the unit cell are precisely halfway between the top and bottom atoms in the unit cell, a distance a/2, so the waves scattered from these central atoms are 180◦ out of phase with respect to the waves from the top atoms of each unit cell. Their contribution to the phase factor sum of (5.49) equals that of the plane above them times eiπ = −1. The waves scattered from the top and central atoms of each unit cell interfere destructively, and cancel in pairs for all unit cells. The (001) diffraction is therefore forbidden in bcc structures. This physical argument can be generalized: An identical plane of atoms halfway between two other planes causes destructive interference and absent diffractions. The general bcc structure factor rule is obtained by extending this line of reasoning to all combinations {h, k, l}, and to all atoms in the crystal. To obtain all atom sites of the bcc lattice from the sites of the sc lattice, we use (5.48) as:

5.3 Diffraction from a Lattice with a Basis

239

Fig. 5.9. With a (001) diffraction, constructive interference occurs between the top and bottom atom planes of a sc unit cell, but the center atoms of the bcc unit cell scatter out-of-phase by π with respect to the atoms immediately above them.

{rg } = {ma1 + na2 + oa3 } where m, n, o are integers in all combinations , # $ {r k } = 0a1 + 0a2 + 0a3 , 21 a1 + 12 a2 + 21 a3 two basis vectors, r k1 and rk2 .

(5.66) (5.67)

We have decomposed the bcc crystal into a simple cubic lattice ((5.66) is the same as (5.57)) with a basis of two atoms ((5.67) differs from (5.58)). The new basis vector for the bcc crystal is the atom site in the center of the simple cubic unit cell. The shape factor of our bcc crystal, which is a sum over all {r g }, is the same as Ssc (Δk) for the simple cubic crystal (5.63). The bcc structure factor, Fbcc (Δk), is different from the sc structure factor, Fsc (Δk), however. To calculate Fbcc(Δk), when Δk = g, we evaluate the effect of a two-atom basis on the different diffractions (h, k, l) for the sc lattice: 2 terms

Fbcc(Δk) =

fat (rk ) e−i2π∆k·rk .

(5.68)

r k1 ,r k2

For the k-space vectors, Δk, we use those of the simple cubic lattice, given in (5.56) or (5.66). When the Laue condition is satisfied for the simple cubic lattice, i.e., Δk = g, the dot products for our two atoms in the basis (5.67) are (after using (5.32)): Δk · r k1 = g · r k1 = h0 + k0 + l0 = 0 ,

Δk · r k2 = g · r k2 =

h 12

+

k 12

+

l 21

.

The two-term sum for the structure factor (5.68) is:   1 1 1 Fbcc(Δk) = fat (0) e0 + fat 21 , 12 , 21 e−i2π(h 2 +k 2 +l 2 ) .

(5.69) (5.70)

(5.71)

The structure factor takes on two values, depending on whether the sum h + k + l is odd or even:

240

5. Diffraction from Crystals

= fat (0) −

1, 2 1, 2



1, 2 1, 2

1, 2 fat 12 ,

Fbcc(Δk) = fat (0) + fat = fat (0) +



1, 2 fat 21 ,

Fbcc(Δk) = fat (0) + fat



integer 2

−i2π 1 2 e  1 h+ 2



k + l = odd number,

−i2πinteger 1 2 e  1 h+k+l 2

= even number.

(5.72)

(5.73)

When the same typeof atom is situated at both basis vectors (a true bcc crystal has fat (0) = fat 12 , 12 , 12 ), (5.72) shows that the structure factor equals zero when h + k + l = odd number. Unlike the case for the sc crystal, when we add a basis vector, a1 /2 + a2 /2 + a3 /2, to convert from sc to bcc, we obtain the bcc structure factor rule: The sum of the three integers h, k, l must be an even number. Consequently, for bcc W, for example, the lowest-order allowed diffractions (with reference to the simple cubic unit cell) are: (110), (200), (211), (220), (310), (222), (321), (400), (330), (411), (420), but other diffractions such as the (100), (111), (210) are forbidden. This same rule applies to the other centered lattices (denoted “I”): body-centered orthorhombic and body-centered tetragonal. (Note that (5.69)–(5.70) do not require that a1 , a2 , a3 form the edges of a cube.) A simple cubic lattice with a four-atom basis provides all atom positions in the fcc crystal: {r g } = {ma1 + na2 + oa3 }

where m, n, o are integers in all combinations ,

{r k } = { 0a1 + 0a2 + 0a3 , 0a1 + 21 a2 + 21 a3 , 1 2 a1

+ 0a2 + 21 a3 ,

1a 2 1

+ 21 a2 + 0a3 } .

(5.74)

(5.75)

This set of four r k provides the fcc structure factor rule: fcc structure factor rule: The three integers h, k, l must be all even, or all odd. The lowest-order diffractions from fcc Cu, for example, are: (111),(200),(220),(311),(222),(400),(331),(420), but other diffractions such as the (100), (110), (210), (211) are forbidden. This rule applies to the other face-centered lattice (denoted “F”): face-centered orthorhombic. For diamond cubic crystals, using a simple cubic lattice and an eight-atom basis set it can be shown: dc structure factor rule: if h, k, l are all even, then h+k+l = 4n, or h, k, l may be all odd integers. (This is the same as the fcc rule, except fewer diffractions are allowed when h, k, l are all even.) The lowest-order diffractions from dc Si, for example, are: (111), (220), (311), (400), (331).

5.3 Diffraction from a Lattice with a Basis

241

Scope and Usage of Structure Factor Rules. We have just seen how the use of a non-primitive unit cell gives rise to systematically absent diffractions. The enumeration of these absences by structure factor rules pertains to lattices of all sizes. For example, if identical groups of several atoms are themselves situated on the sites of a big bcc lattice, the bcc structure factor rule imposes systematic absences of mixed even-odd hkl indices for the big underlying sc lattice. (Nevertheless, the symmetrical positioning of atoms within these groups can produce additional systematic absences of diffractions.) Table 5.1 summarizes the systematic absences for Bravais lattices8 when the unit cell comprises two or more atoms having the Cartesian basis vectors in the second column. Table 5.1. Systematic absences of lattice types lattice type

convenient basis vectors

systematic absence

P (e.g., sc) A

0, 0, 0 0, 0, 0; 0, 21 ,

none k + l = 2n + 1

B

0, 0, 0; 21 , 0,

C

0, 0, 0; 12 , 21 , 0

h + k = 2n + 1

F (e.g., fcc)

0, 0, 0; 0, 21 , 12 ; 21 , 0, 12 ; 12 , 21 , 0

h, k, l neither all odd nor all even

I (e.g., bcc)

0, 0, 0; 12 , 21 ,

h + k + l = 2n + 1

R (hexagonal axes) R (rhombohedral axes)

0, 0, 0; 0, 0, 0

2 1 , , 3 3

1 2 1 2

1 2 1 1 2 2 ; , , 3 3 3 3

h + l = 2n + 1

−h + k + l = 3n ± 1 none

Finally, we emphasize that the structure factor rules are properties of the lattice, and do not depend on how the atoms are positioned within the volume of the unit cell. For example, we did not need to pick the two bcc basis vectors at the corner and center sites of the standard cube (as in (5.67)). Suppose these two basis vectors were offset from the origin of the underlying sc lattice by an arbitrary translation: ΔR = Aa1 + Ba2 + Ca3 . The new positions of the atoms (denoted by primes) are now {r′k1 , r′k2 } = {rk1 + ΔR, r k2 + ΔR}, which differ from (5.67): {r′k1 , r ′k2 } = { Aa1 + Ba2 + Ca3 , (A + 21 )a1 + (B + 21 )a2 + (C + 21 )a3 } , 8

(5.76)

Crystal system notation is: (a)triclinic, (m)monoclinic, (o)orthorhombic, (t)tetragonal, (h)trigonal, (h)hexagonal, (c)cubic. For hkl indexing based on the reciprocal lattices of these crystal systems, systematic absences of diffractions occur for the 14 Bravais lattices as listed in the rows of the table: row 1 (1 atom): aP, mP, oP, tP, hP, cP; rows 2, 3, 4 (2 atoms): mC, oC; row 5 (4 atoms): oF, cF; row 6 (2 atoms): oI, tI, cI; rows 7, 8 (3, 1 atoms): hR.

242

5. Diffraction from Crystals

′ and the structure factor is now Fbcc , instead of Fbcc of (5.71): ′ Fbcc (Δk) = fat e−i2π(Ah+Bk+Cl) 1

1

1

+ fat e−i2π[(A+ 2 )h+(B+ 2 )k+(C+ 2 )l] ,   1 1 1 ′ Fbcc (Δk) = e−i2π(Ah+Bk+Cl) fat e0 + fat e−i2π(h 2 +k 2 +l 2 ) .

(5.77)

′ Fbcc (Δk) = e−i2π(Ah+Bk+Cl) Fbcc(Δk) .

(5.79)

(5.78)

(When the crystal has the bcc structure, fat is the same for all atom sites.) ′ of (5.78) therefore differs from Fbcc of (5.71) by The structure factor Fbcc only a constant factor having modulus 1: −i2π(Ah+Bk+Cl)

The constant phase factor, e , does not alter the intensity of the ′ diffracted wave, which is proportional to Fbcc times its complex conjugate:

′∗ ′ ∗ Fbcc = e+i2π(Ah+Bk+Cl) Fbcc (Δk) e−i2π(Ah+Bk+Cl) Fbcc(Δk) , (5.80) Fbcc ′∗ ′ ∗ I(Δk) ∝ Fbcc Fbcc = Fbcc Fbcc .

(5.81)

Equation (5.81) shows that the diffracted intensity does not change if we start with the basis vectors {rk1 , rk2 } for the corner and center points of the sc unit cell as in (5.67), or if these basis vectors are displaced from the points of the sc lattice by an arbitrary displacement, as in (5.76). The result (5.81) requires, however, that the two basis vectors are translated by the same vector, ΔR. Unequal translations break the bcc symmetry. This causes incomplete phase cancellation for the h + k + l = odd diffractions, allowing some diffraction intensity. 5.3.3 Symmetry Operations and Forbidden Diffractions Specific diffractions can be eliminated when there are translational symmetry elements such as glide planes and screw axes in the space group of a crystal. Such absences of diffractions are restricted to one zone of planes9 in the case of a glide plane, or to one set of planes in the case of a screw axis. For example, the presence of an “a-glide plane” through the origin parallel  causes an atom at position ma1 + na2 + oa3 to be duplicated at to (001) m + 21 a1 + na2 − oa3 . This is a shift of the atoms below the glide plane by a1 /2. The term in the structure factor for atoms below the glide plane contains the factor: 1 (5.82) ei[2π( 2 h+hm+kn+lo)−2π(hm+kn+lo)] = eiπh . When h is odd, eiπh = −1, and this factor produces an absence in the hk0 diffractions. This occurs because this glide plane effectively halves the lattice spacing parallel to the x-axis for diffractions with Δk perpendicular to (001). All glide planes lead to absences of diffractions in the zone whose axis 9

A zone of planes comprises all planes having normals perpendicular to a given direction. This direction is the “zone axis.”

5.3 Diffraction from a Lattice with a Basis

243

is normal to the glide plane. Forbidden diffractions produced by all possible types of (001) glide planes are listed in Table 5.2. A complete list of conventional glide planes and their systematic absences is found in the International Tables for X-ray Crystallography [5.1]. Table 5.2. Systematic absences produced by glide planes parallel to (001) type of glide

translation

systematic absences in hk0 diffractions

a

a/2

h = 2n + 1

b

b/2

k = 2n + 1

n

(a + b)/2

h + k = 2n + 1

d

(a ± b)/4

h + k = 4n + 2 with h = 2n and k = 2n

The presence of a screw axis in a crystal also leads to forbidden diffractions. For example, a screw diad through the origin parallel to the zaxis causes an atom at position ma1 + na2 + oa3 to be duplicated at −ma1 − na2 + o + 21 a3 . The expression for the structure factor is similar to the a-glide plane considered above, and produces absences in the (00l) diffractions when l is odd. As with the glide plane, the screw diad effectively halves the lattice spacing parallel to the z-axis for the (00l) diffractions. Screw axes parallel to other crystallographic axes with different translations give rise to analogous absences. A list of absences produced by all possible types of screw axis parallel to [001] is given in Table 5.3. Table 5.3. Systematic absences produced by screw axes parallel to [001] screw axis

translation

systematic absences in 00l diffractions

21

c/2

l = 2n + 1

±c/4

l = 4n

41 and 43 42 31 and 32 61 and 65 62 and 64 63

c/2

±c/3

±c/6

±c/3

c/2

l = 2n + 1

l = 3n

l = 6n

l = 3n

l = 2n + 1

5.3.4 Superlattice Diffractions Consider a modification of the bcc unit cell in the lower left of Fig. 5.9, where the type of atom in the center of the unit cell is different from the

244

5. Diffraction from Crystals

type of atom on the cube corners. In the modified crystal, the waves in a (100) diffraction no longer cancel in pairs as in the lower right of Fig. 5.9. Such a crystal no longer has a bcc lattice, however. It has the sc lattice and the “B2” structure in the Strukturbericht designation (shown at top center of Fig. 5.11). The B2 structure is the ordered phase of CsCl. With different atomic scattering factors for atoms A and B, (5.72) becomes:   (5.83) fA (0) − fB 21 21 21 = 0 .

Instead of zero diffracted intensity, the (001) diffraction from B2-ordered FeCo has an intensity proportional to:  2 I(100) ∝ fCo − fFe  (weak) . (5.84) The (100) diffraction is called a “superlattice diffraction.” It reflects the periodicity of the sc lattice upon which the B2 structure is constructed using a basis of two different atoms. On the other hand, the allowed diffractions from bcc crystals, the “fundamental diffractions” (for example the (200)) have intensities from (5.73): 2  (strong) . (5.85) I(200) ∝ fCo + fFe 

As discussed in Sect. 3.2.1 for x-ray scattering, fat is nearly proportional to the atomic number Z, and as discussed in Sect. 3.3.3 for electron scattering, fat is sub-linear in Z. For x-ray diffraction we have the ratio: fFe /fCo ≃ 26/27. The strong “bcc fundamental” intensities from B2 FeCo (5.85) are very close to their corresponding intensities for pure bcc Fe (about 4 % stronger), and the weak “B2 superlattice” intensities (5.84) are very much weaker (by a factor of ≃ 2700). Table 5.4 lists these peaks and intensities. These B2 superlattice diffractions from FeCo are in fact so weak that they are barely detectable with most conventional x-ray diffractometers.10 An SAD pattern from a Ti-based alloy with the B2 structure is shown in Fig. 5.10. The zone axis, i.e., the normal to the plane of diffraction, is (001).  Three fundamental bcc diffractions are indexed, (110), 110 , and (020), and several superlattice diffractions are visible. For example, a weak diffraction is located halfway between the central (000) spot and the (020) spot. This is the (010) superlattice diffraction. Some weak {210} superlattice diffractions are also visible in Fig. 5.10. It is instructive to examine the real space and reciprocal space structures for the B2 (CsCl) structure. The reciprocal space structure of the B2 structure is the B1 (NaCl) structure, as we can show with the following argument. Figure 5.11 shows how the B2 structure is obtained as an ordered structure on the bcc lattice. By this we mean that all atoms are on bcc lattice sites, but there is a different type of atom on the center site than on the corner site. 10

One way to increase the scattering factor difference between Fe and Co is to use Co Kα radiation so that the anomalous scattering from Fe suppresses fFe and increases I(100) in (5.84).

5.3 Diffraction from a Lattice with a Basis

245

Table 5.4. Diffractions from the B2 structure (hkl)

h2 + k2 + l2

type

intensity

(100) (110) (111) (200) (210) (211) (220) (221) (300)

1 2 3 4 5 6 8 9 9

superlattice fundamental superlattice fundamental superlattice fundamental fundamental superlattice superlattice

weak strong weak strong weak strong strong weak weak

Fig. 5.10. SAD pattern from a Ti alloy having B2 chemical order. Superlattice diffraction spots such as (010) and (120) are the weak spots between the labeled bcc fundamental diffractions.

(Likewise, Fig. 5.11 shows that the B1 structure is derived from a sc lattice shown to its right.) Although the B2 structure is an ordered structure on the bcc lattice, it is accurate to regard the B2 structure as a simple cubic lattice with a two-atom basis. (Similarly, the B1 structure can be regarded as an fcc lattice with a two-atom basis.) To understand the diffraction intensities, consider the B2 structure as being intermediate between a bcc structure and a sc structure, as suggested by its position in the center of the top row of Fig. 5.11: • In one limiting case the scattering strength of the center atom vanishes – the real lattice is sc, so the reciprocal lattice is sc. All diffractions (h, k, l) are allowed. • In the other limiting case, the center atom has the same scattering strength as the corner atoms – the real lattice is bcc, so the reciprocal lattice is fcc. Some of the sc diffractions vanish (when h + k + l = odd integer). • In the intermediate case, the center atom has a different scattering strength than the corner atom. The real structure is B2, not bcc, so the strict phase cancellation of the h + k + l = odd diffractions is no longer true. Only partial cancellation occurs for these (h, k, l), shown as the small circles of

246

5. Diffraction from Crystals

the B1 structure. These superlattice diffractions appear because in the B2 structure the scattering strength of the center atom differs from that of the corner atom. bcc

A2

CsCl

B2

Po

sc

fcc

NaCl

Po

A1

B1

sc

Fig. 5.11. Top: Real space structures. Bottom: Their corresponding reciprocal space structures.

We can generalize this method. To obtain the superlattice diffractions of an ordered structure, first locate the fundamental diffractions of the underlying lattice (ignore the atom type). Next locate the diffractions from a modified lattice where one species of atoms is removed. The unit cell is now larger, so there are more diffractions. The superlattice diffractions occur at the locations of the new diffractions of this modified lattice. Figure 5.12a shows the L10 -ordered structure (CuAu prototype), which is derived from an fcc lattice. A pair of diffraction patterns from a TiAl alloy with L10 order are presented in Fig. 5.12b. The L10 structure is not symmetrical about all 001 directions. We expect only (001) superlattice diffractions, not (100) or (010) diffractions. These are present in the top diffraction pattern, showing the (001) diffraction, but not the (010). In the lower figure, neither the (100) nor the (010) are seen (the inner spots are {110} diffractions). In most samples containing precipitates with L10 order, the c-axes of different precipitates are parallel to all three [001], [010] and [100] directions of the matrix, forming three variants of the L10 structure. If all three variants are present simultaneously, then the (001)* section of the reciprocal lattice looks like the drawing in Fig. 5.12c. Unfortunately, this diffraction pattern is similar to the (001)* pattern for a crystal with

5.4 Crystal Shape Factor

247

the alternative L12 structure. We could distinguish between these two cases, however, by forming dark-field images using the spots labelled (010), (110) and (100) in the pattern. For each DF image, all precipitates would “light up” if they have L12 structure, while only one of the three variants would light up if they have the L10 structure.

Fig. 5.12. (a) The L10 structure. (b) Diffraction patterns from an L10 TiAl alloy from two zone axes, (100) top and (001) bottom, corresponding to indexing of unit cell in a. (c) Superposition of all three zone axes for an L10 structure. After [5.2].

5.4 Crystal Shape Factor 5.4.1 Shape Factor of Rectangular Prism Now we discuss the shape factor, S(Δk), of (5.53):  e−i2π∆k·rg . S(Δk) =

(5.86)

rg

For very large crystals, the shape factor gives little information about the crystal shape, and is therefore not very interesting. The argument near the end of Sect. 5.2.1 shows that for very large crystals the shape factor intensity becomes infinitely high and infinitesimally narrow – it is in fact a set of delta functions centered at the various values of Δk where Δk = g (g is a reciprocal lattice vector). The shape factor is most interesting for small crystals. For

248

5. Diffraction from Crystals

convenience, we sacrifice some generality by assuming that our small crystal ˆ, is a rectangular prism with Nx , Ny , and Nz unit cells along the directions x ˆ ) that can ˆ , az z ˆ , ay y ˆ , and z ˆ . Consider a set of short translation vectors (ax x y prescribe lattices with cubic, tetragonal, or orthorhombic unit cells: ˆ, ˆ + oaz z ˆ + nay y r g = max x ˆ + Δky y ˆ + Δkz z ˆ, Δk = Δkx x S(Δk) = S(Δk) =

y −1 Nz −1 N x −1 N  

m=0 n=0

N x −1

e

(5.87) (5.88)

e−i2π(Δkx ax m+Δky ay n+Δkz az o) ,

(5.89)

o=0

−i2πΔkx ax m

m=0

Ny −1



e−i2πΔky ay n

n=0

N z −1

e−i2πΔkz az o . (5.90)

o=0

Each sum in (5.90) is a truncated geometric series of the form: S = 1 + r + r2 + r3 + r4 + ... + rN −1 .

(5.91)

To evaluate such a series, note how it can be expressed as a difference of two infinite geometric series: S= S=

∞  j=0

∞  j=0

rj − j

∞ 

rj ,

(5.92)

j=N

r −r

N

∞ 

rj ,

(5.93)

j=0

∞   rj , S = 1 − rN

(5.94)

j=0

  1 − rN S= . (5.95) 1−r The evaluation of the infinite geometric series in the last step is a standard result.11 The form of (5.95) can be used to evaluate each of the three sums in (5.90). For example, using r = e−i2πΔkx ax we have: N x −1 m=0

 −i2πΔkx ax m 1 − e−i2πΔkx ax Nx . e = 1 − e−i2πΔkx ax

(5.96)

The diffracted intensity associated with the shape factor, S, is S ∗ S:

1 − e+i2πΔkx ax Nx 1 − e−i2πΔkx ax Nx . 1 − e+i2πΔkx ax 1 − e−i2πΔkx ax Multiplying the numerators and denominators: S ∗ S(Δkx ) =

11

You can confirm it with the mechanics of long division: 1/(1 − r).

(5.97)

5.4 Crystal Shape Factor

S ∗ S(Δkx ) =

249

2 − e−i2πΔkx ax Nx − e+i2πΔkx ax Nx . 2 − e−i2πΔkx ax − e+i2πΔkx ax

(5.98)

2 − 2 cos(2πΔkx ax Nx ) . 2 − 2 cos(2πΔkx ax )

(5.99)

Using the Euler relation: eiθ = cosθ + i sinθ : S ∗ S(Δkx ) =

Recall that: cos 2θ = 1 − 2 sin2 θ, so: 

2 − 2 1 − 2 sin2 (πΔkx ax Nx ) ∗

 , S S(Δkx ) = 2 − 2 1 − 2 sin2 (πΔkx ax ) S ∗ S(Δkx ) =

(5.100)

sin2 (πΔkx ax Nx ) . sin2 (πΔkx ax )

(5.101)

The function in (5.101) is the kinematical diffracted intensity from a column of Nx atoms in length.12 First we determine the conditions under which S ∗ S(Δkx ) is large. It is large when the denominator goes to zero, as it does when the argument of the sine function is equal to π or to some integral multiple of π. This corresponds to: Δkx ax = integer .

(5.102)

Since similar conditions are expected for the y- and z-sums, this condition requires that Δk is a reciprocal lattice vector. In other words, the kinematical intensity S ∗ S is large when the Laue condition is satisfied, just as expected. There is a subtlety here, however. When Δkx ax = integer, the numerator of (5.101) is also zero, so to evaluate it we need to apply l’Hˆ ospital’s rule twice. Instead, when Δkx ax = integer, it is easier to evaluate the intensity by returning to the diffracted wave. Consider the first sum in (5.90), and its associated intensity: S ∗ S(Δkx ax = integer ) = = =

N x −1

e+i2π(integer ) m

m=0

N x −1 m=0 Nx2 .

N x −1



e−i2π(integer) m , (5.103)

m =0

1

N x −1

1,

(5.104)

m =0

(5.105)

When Δk satisfies precisely the Laue condition, the diffraction intensity scales quadratically with the number of diffracting planes. When the number of diffracting planes is doubled, the net diffracted intensity increases by a factor of four. As shown below, however, when the number, N , of coherently diffracting planes is doubled, the width in Δk is halved. The net intensity 12

ˆ direction if the thickness The result applies to the diffraction intensity along the x of the small crystal is uniform in the x-direction ((5.101) does not apply to spheres, for example).

250

5. Diffraction from Crystals

in the diffracted Bragg peak therefore scales linearly with the number of diffracting planes. In Fig. 5.13 are plots of the function: S ∗ S(Δk) =

sin2 (πΔkaN ) , sin2 (πΔka)

(5.106)

for values of N =4, 8, and 12. The center positions of the main peaks of this function are controlled by the denominator, and are independent of N . The denominator varies slowly with respect to the numerator, so we make the approximation, valid near the center of the main peaks: S ∗ S(Δk) ≃

sin2 (πΔkaN ) (πΔka)

2

= sin2 (πΔkaN ) E(Δk) ,

(5.107)

where we have defined the envelope function, E(Δk), as: E(Δk) ≡

1

2

(πΔka)

.

(5.108)

This function, E(Δk), is the envelope of the satellite peaks situated near the main peaks. The envelope of the satellite peaks is independent of N . Since the heights of the main peaks grow as N 2 , however, the relative heights of the satellite peaks decrease as N becomes large. By examining the numerator, we see that the positions of the satellite peaks get closer to the main peak in proportion to (N a)−1 , and the position of the first minimum in the intensity is located on either side of the main peak at the position Δk = (N a)−1 . Similarly the widths of the main peaks and satellite peaks also decrease as (N a)−1 .

Fig. 5.13. Plots of (5.106) for N =4, 8, and 12.

The full three-dimensional expression for the kinematical shape factor of our prismatic crystal is:

5.4 Crystal Shape Factor

S ∗ S(Δk) =

251

sin2 (πΔkx ax Nx ) sin2 (πΔkx ax ) ×

sin2 (πΔky ay Ny ) sin2 (πΔkz az Nz ) , sin2 (πΔky ay ) sin2 (πΔkz az )

(5.109)

with which we can use (5.52) to write the diffracted intensity as: 2 2 sin2 (πΔkx ax Nx )   I (Δk) = ψ(Δk)  = F (Δk)  sin2 (πΔkx ax ) ×

sin2 (πΔky ay Ny ) sin2 (πΔkz az Nz ) . sin2 (πΔky ay ) sin2 (πΔkz az )

(5.110)

For experimental intensities in x-ray diffraction, we need to add the Lorentzpolarization and other factors as in (1.59) and (1.60). Specifically, (5.110) for 2 |ψ(Δk)| should replace [F ∗ (Δk) F (Δk)] in (1.59) and (1.60). An example of S ∗ S(Δk) for a two-dimensional rectangular crystal with Nx = 12 and Ny = 6 is shown in Fig. 5.14. The diffracted intensity is not uniform around the four main peaks, but extends in lines of local maxima along x and y. There are 10 such secondary maxima (and 11 minima) along x, and 4 maxima (5 minima) along y. Note that the maxima along y are stronger. Our rectangular crystal has minimal intensity along diagonal directions like x + y. The asymmetry of the primary maxima also originates from the rectangular shape of the crystal.

Fig. 5.14. Shape factor for a 2-D rectangular crystal with Nx = 12 and Ny = 6.

252

5. Diffraction from Crystals

5.4.2 Other Shape Factors In solid-state phase transformations, new crystalline phases frequently nucleate and grow as small particles with specific crystallographic orientations within the parent crystal. These small particles are usually shaped as spheres, plates or rods. When the particles have at least one dimension that is less than about 10 nm, their diffractions are distinctly broadened by the shape factor. Before we consider the typical situation where the particles are embedded in a matrix, we first consider the plot in k-space from the three-dimensional shape factor intensity |S|2 from isolated precipitates with a simple cubic crystal structure. Fig. 5.15. Left: Shapefactor-intensity-modified reciprocal lattice for large sc crystal. Right: Shape-factorintensity-modified reciprocal lattice for sc crystal shaped as a thin, wide disc with its normal pointing upwards.

On the left of Fig. 5.15 is the diffraction pattern from a very large, simple cubic crystal. To its right is the diffracted intensity from the same type of crystal, but now in the shape of a thin plate with its thin direction pointing up. By looking at the oblong shapes of the shape factor intensities, known as “rel-rods” (for reciprocal lattice rods), we can estimate that the plate is about 5 times wider than it is thick. We also estimate that the lengths of the rel-rods are about one-fifth of the distance between the points of the reciprocal lattice. Roughly, then, we deduce that the thickness of the plates is approximately 5 lattice spacings, and their widths are approximately 25 lattice spacings. Effects of shape factors occur in most diffraction patterns, and identifying these effects can be important. For example, the 2-dimensional diffraction intensity from a circular particle has rings of intensity around the main diffraction peaks, whose intensity falls away as the square of a Bessel function. Figure 5.16 shows some intensity distributions associated with crystals of different shapes. 5.4.3 Small Particles in a Large Matrix Now we calculate the effects of the shape factor when the small particles are embedded in a crystalline matrix. A classic example of this occurs in Al-Cu alloys, where very thin Cu-enriched precipitates form in an Al matrix. These precipitates, are shaped as thin disks whose normals lie along the 100 directions. The diffracted wave (5.51) is proportional to:

5.4 Crystal Shape Factor

253

Fig. 5.16. Approximate shape factor intensity distributions for various crystal shapes. Top: Crystal shapes – sphere, disk, rod. Bottom: Corresponding diffraction intensities, correctly aligned.

ψ(Δk) =



e−i2π∆k·rg



fat (rk ) e−i2π∆k·rk ,

(5.111)

rk

rg

but now we must account for the fact that fat is fAl in the matrix and is fAl-Cu in the disk: ψ(Δk) =

disk 

e−i2π∆k·rg

fAl-Cu (rk ) e−i2π∆k·rk

rk

rg

+



matrix 

e−i2π∆k·rg

rg

ψ(Δk) = FAl-Cu (Δk)



fAl (rk ) e−i2π∆k·rk ,

(5.112)

rk

disk  rg

matrix 

e−i2π∆k·rg + FAl (Δk)

e−i2π∆k·rg , (5.113)

rg

where we have used (5.54) for the structure factor in writing (5.113). The first term in (5.113) is not difficult to evaluate. The trick to evaluating the second term is to rewrite it as a sum over a full crystal of Al atoms, and then subtract the contribution from the disk-shaped precipitate: FAl (Δk)

matrix  rg

e−i2π∆k·rg = FAl (Δk) −FAl (Δk)

whole 

e−i2π∆k·rg

rg

disk  rg

e−i2π∆k·rg .

(5.114)

254

5. Diffraction from Crystals

We substitute (5.114) into (5.113), and combine the second term of (5.114) with the first term in (5.113): disk   e−i2π∆k·rg ψ(Δk) = FAl-Cu (Δk) − FAl (Δk) rg

whole 

+ FAl (Δk)

e−i2π∆k·rg .

(5.115)

rg

When small precipitates are present in low abundance,13 (5.115) predicts that the Al matrix provides strong, sharp diffraction spots whose intensity 2 is proportional to the number of atoms in the matrix times |fAl | . The diskshaped precipitates provide broadened diffraction spots whose intensity is proportional to the number of atoms in the precipitate times |fAl-Cu − fAl |2 . Here fAl-Cu is larger than fAl in proportion to the amount of Cu in the precipitate (to a maximum of fCu − fAl for a precipitate of pure Cu). The unit cell of the Al-Cu precipitate does not have the cubic symmetry of the Al matrix (see Problem 5.9), and the mismatch causes elastic energy. The mismatch is most severe perpendicular to the plane of the precipitate, so the energetic preference is for precipitates that are very thin. At low temperatures, the early “precipitates” are only monolayers in thickness, and are called “Guinier–Preston” (GP) zones. Two types of GP zones have been found. As illustrated in Fig. 5.17a, a GP(1) zone consists of a single layer of Cu atoms which have substituted for Al on a {100} plane, while Fig. 5.17b shows a GP(2) zone that contains two such Cu layers separated by three {100} planes of Al atoms. The experimental high-resolution TEM images of these two types of GP zones are interpretable in terms of the drawings in Fig. 5.17. For these HTREM images the atomic columns are white, and the Cu-rich layers appear as darker planes in the images. When the precipitate disks have thicknesses of only an atom or two, the precipitate diffractions appear as streaks rather than as discrete spots, as illustrated in Fig. 5.18. In Fig. 5.18a the streaks from the sample with GP(1) zones are practically continuous along the two 100 directions in the plane of the figure because the precipitate is essentially a monolayer of Cu atoms. In Fig. 5.18b, the streaks along 100 are no longer continuous, but have maxima at 1/4 100 positions in the diffraction pattern (arrows). This periodicity arises because the Cu planes in the GP(2) zone are spaced four {100} planes apart. This illustrates an important point: for every real-space periodicity in a specimen, there is a corresponding reciprocal-space intensity.

13

In this case, the cross-term in ψ ∗ ψ from the two terms in (5.115) serves only to contribute a little intensity to the matrix diffractions.

5.4 Crystal Shape Factor

255

Fig. 5.17. Drawing and HRTEM images of Guinier-Preston zones in Al-Cu taken along a 100 matrix direction in an Al-4wt.% Cu alloy: (a) GP(1) zone, (b) GP(2) zone. After [5.3].

Fig. 5.18. SAD patterns from Al-Cu as in Fig. 5.17 with: (a) GP (1) zones, (b) GP (2) zones. After [5.4].

256

5. Diffraction from Crystals

5.5 Deviation Vector (Deviation Parameter) To examine the shapes of diffraction peaks, it is convenient to employ a new notation for Δk. We express Δk as the difference of an exact reciprocal lattice vector, g, and a “deviation vector,” s: Δk = g − s ,

(5.116)

g = Δk + s ,

(5.117)

where the deviation vector, s, has the components: ˆ. ˆ + sz z ˆ + sy y s = sx x

(5.118)

Using the definition of the shape factor (5.53): lattice 

S(Δk) =

e−i2π∆k·rg ,

(5.119)

e−i2π(g−s)·rg ,

(5.120)

rg

lattice 

S(Δk) =

rg

and noting that g · r g = integer: lattice 

S(Δk) =

e−i2π integer e+i2πs·rg =

lattice 

e+i2πs·rg ,

(5.121)

rg

rg

S(Δk) = S(−s) .

(5.122)

We find the important result that the shape factor depends only on s, not g. Now examine the dependence of the structure factor F (Δk) on g and s. Consider the specific example of a bcc crystal with a two-atom basis in (5.68): F (Δk) =

2 terms

a(000), a(

fat (rk , Δk) e−i2πg·rk ei2πs·rk . )

(5.123)

1 1 1 2 2 2

Using (5.69) and (5.70): F (Δk) = fat (0, g) e0 e0   + fat 21 21 21 , g e−i2π(h+k+l)/2 ei2π(sx ax +sy ay +sz az )/2 .

(5.124)

h + k + l = odd number :   F (Δk) = fat (g) 1 − eiπ(sx ax +sy ay +sz az ) ≃ 0 ,

(5.125)

h + k + l = even number :   F (Δk) = fat (g) 1 + eiπ(sx ax +sy ay +sz az ) ≃ 2fat (g) ,

(5.126)

For a bcc crystal the structure factor takes on two values:

5.6 Ewald Sphere

F (Δk) ≃ F(g) .

257

(5.127)

Equations (5.125)–(5.127) are excellent approximations because s · rk is small.14 From (5.122) and (5.127), the shape factor, S(s), depends only on s, the structure factor, F (g), depends only on g. The diffracted intensity can be worked out with (5.121) in the same way that (5.86) was used to obtain (5.110). Comparing the forms of (5.86) and (5.121), we just replace Δk in (5.86) and (5.121) with −s and note that sin2 x is an even function in x. Doing so provides: 2 sin2 (πsx ax Nx ) 2   I (Δk) = ψ(Δk)  = F (Δk)  sin2 (πsx ax ) ×

sin2 (πΔsy ay Ny ) sin2 (πsz az Nz ) . sin2 (πsy ay ) sin2 (πsz az )

(5.128)

Equation (5.128) shows that the kinematical intensity distribution about any reciprocal lattice point is the same as that about the origin, a very important result. The intensity of S ∗ S is N 2 whenever s = 0 (see Fig. 5.13). If our crystal has a uniform thickness (no variations in Nx , for example), (5.128) also predicts that there are maxima and minima in diffracted intensity with increasing values of |s|.

5.6 Ewald Sphere 5.6.1 Ewald Sphere Construction The Laue condition for diffraction, Δk = g, can be implemented in a geometrical construction due to P. P. Ewald. In forming this construction (Fig. 5.19), we begin with a picture of our reciprocal lattice (the set {g}). In most of what follows, we assume that our reciprocal lattice is simple cubic, tetragonal, or orthorhombic. The “Ewald sphere” depicts the incident wavevector k0 , and all possible diffracted wavevectors k. The tip of the wavevector k0 is always placed at a point of the reciprocal lattice which serves as the origin.15 To obtain Δk = k − k0 , we would normally reverse the direction of k0 and place it tail-to-head with the vector k. In the Ewald sphere construction of Fig. 5.19, however, we draw k and k0 tail-to-tail. In elastic scattering, the length of k equals the length of k0 . The 14

15

Suppose, however, we defined the entire crystal as the unit cell. We could no longer calculate F with the approximation ei2πs·r k = 1, because we would have some large vectors r k . In what follows, however, we assume a small unit cell and a small s. The condition Δk = 0 always satisfies the Laue condition, so we always have a forward-scattered beam in the diffraction pattern.

258

5. Diffraction from Crystals

tips of all possible k vectors lie on an Ewald sphere, whose center is at the tails of k and k0 . The vector Δk is the vector from the head of k0 to the head of k. If the head of k touches any reciprocal lattice point, the Laue condition (Δk = g) is satisfied and diffraction occurs. We rephrase geometrically the Laue condition: Diffraction occurs whenever the Ewald sphere touches a point on the reciprocal lattice.

z y x

k0

k

_

100 000 100

k – k 0 = Δk

Ewald Sphere: acceptable surface for k

Fig. 5.19. Ewald sphere construction. The Laue condition is satisfied approximately for the ` ´ (100) and 100 diffractions.

The Ewald sphere is strongly curved for x-ray diffraction because |g| is comparable to |k0 |. Electron wavevectors, on the other hand, are much longer than the spacings in the reciprocal lattice (100 keV electrons have a wavelength of 0.037 ˚ A, whereas interatomic spacings in a crystal are ∼ 2 ˚ A). For high-energy electron diffraction, the three-dimensional Ewald sphere construction is analogous to putting a small molecular model (of cm dimensions) into an umbrella. For high-energy electrons, the Ewald sphere is rather “flat.” Consequently, Δk is nearly perpendicular to k0 . In practice, the diffraction intensity distribution, such as that of (5.110), is located in a finite volume around the reciprocal lattice points (Sect. 5.4), so Δk need not equal g ex2 actly in order for diffraction to occur. The shape factor intensity, |S| , effectively broadens the reciprocal lattice points. With a properly oriented crystal, the Ewald sphere goes through many of these small volumes around the reciprocal lattice points, and many diffractions occur. Figure 5.20 provides an example of Ewald sphere constructions for two orientations of a specimen with respect to k0 , together with their corresponding diffraction patterns. The top of Fig. 5.20 is drawn as a two-dimensional slice (the x-z plane) of Fig. 5.19. The crystal on the right is oriented precisely along a zone axis, but the crystal on the left is not. Two key facts to remember about the Ewald sphere and electron diffraction are:

5.6 Ewald Sphere

259

Fig. 5.20. Two orientations of the reciprocal lattice with respect to the Ewald sphere, and corresponding diffraction patterns.

• The diffraction, g, occurs when the Ewald sphere touches a reciprocal lattice point. (With shape factor broadening of the diffraction intensity, Δk need not equal exactly g, but it should be close.) • For high-energy electrons, approximately Δk⊥k0 because k0 is much larger than Δk (and |k| = |k0 |). (Equivalently, θBragg is small.) 5.6.2 Ewald Sphere and Bragg’s Law The Ewald sphere construction is a graphical implementation of the Laue condition, so it must be equivalent to Bragg’s law as shown previously in Sect. 5.2.4. This equivalence is easy to demonstrate. From the geometry of the Ewald sphere construction in Fig. 5.21: g/2 , (5.129) k and by definition of the reciprocal lattice vector and the wavevector: sinθ =

1 1 and k ≡ , d λ so Bragg’s law is recovered from (5.129): g=

(5.130)

2d sinθ = λ .

(5.131)

5.6.3 Tilting Specimens and Tilting Electron Beams Rules for Working with the Ewald Sphere. Many problems in the geometry of diffraction can be solved with an Ewald sphere and a reciprocal lattice. When working problems, remember: • The Ewald sphere and the reciprocal lattice are connected at the origin of the reciprocal lattice. Tilts of either the Ewald sphere or the reciprocal lattice are performed about this fixed pivot point.

260

5. Diffraction from Crystals

e 2e k0

k

g

0 k0 sine

Fig. 5.21. Relationship between Bragg angle and Ewald sphere construction. The vector +g could correspond to (100) in Fig. 5.19.

• The reciprocal lattice is affixed to the crystal. (For cubic crystals the reciprocal lattice directions are along the real space directions.) Tilting the specimen is performed by tilting the reciprocal lattice by the same angle and in the same direction. • The Ewald sphere surrounds the incident beam, and is affixed to it. Tilting the direction of the incident beam is performed by tilting the Ewald sphere by the same amount. These three facts are handy during practical work on a TEM. It is useful to think of the viewing screen as a section of the Ewald sphere, which shows a disk-shaped slice of the reciprocal space of the specimen. When you tilt the sample, the Ewald sphere and the viewing screen stay fixed, but the different points on the reciprocal lattice of the sample move into the viewing screen. For small tilts of the specimen, the diffraction pattern does not move on the viewing screen, but the diffraction spots do change in intensity. Alternatively, when you tilt the incident beam, you rotate the transmitted beam on the Ewald sphere. You could think of this operation as moving your disk-shaped viewing screen around the surface of the Ewald sphere, but it may be simpler to consider the movement of the forward beam on a fixed viewing screen. Tilted Illumination and Diffraction: How to Do Axial Dark-Field Imaging. As described in Sect. 2.3 and shown in Fig. 2.14, dark-field images with the best resolution are made when the diffracted rays travel straight down the optic axis.16 This requires that the direction of the incident beam (k0 ) be tilted away from the optic axis by an angle of 2θ as shown in Fig. 5.22. Tilting the illumination alters the positions and intensities of the diffraction spots. By tilting the illumination, we tilted the Ewald sphere about the origin of the reciprocal lattice – note the tilt of k0 at the bottom of Fig. 5.22. Tilting the Ewald sphere counterclockwise causes it to touch the −g beam. The −g diffraction becomes active, and its rays travel straight down the optic axis, 16

Otherwise, in the “dirty” dark-field technique, the strongly off-axis rays suffer from the spherical aberration of the objective lens, and their focus is imprecise.

5.6 Ewald Sphere

261

as needed for axial DF imaging. This tilt caused the diffraction g to move far from the optic axis and become weak. This procedure seems counterintuitive on the viewing screen. Before tilt, Fig. 5.22 shows a bright spot g to the right. Operationally, to tilt the transmitted beam counterclockwise, on the viewing screen we move the transmitted spot into the position of the initially bright spot g. We do not move the bright spot g into the center of the viewing screen to obtain an axial beam. This alternative does not work. If the active diffraction g were tilted clockwise onto the optic axis, the diffraction g would become weak, and the diffraction 3g would become strong.17 Since the diffraction g becomes weak, it is difficult to use it for making a dark-field image. We refer to this latter procedure as the “amateur mistake,” although it is used in the advanced technique of “weak-beam dark-field imaging.”

Fig. 5.22. Procedures for axial brightfield (BF) imaging, and axial dark-field (DF) imaging. The ray paths “reflect off the top of the crystal planes” in the left figure, and “off the bottom of the planes” in the right figure. As seen on the viewing screen – move the transmitted 0 beam into the position of the g diffraction, so the −g diffraction becomes strong. In the bottom drawings, note that the sphere and the two vectors are in identical orientations for the left and right drawings, but the k 0 vector switches from left to right.

17

See Fig. 7.36.

262

5. Diffraction from Crystals

5.7 Laue Zones Since the electron wavevector, k0 , is much larger than a typical reciprocal lattice vector, g, the surface of the Ewald sphere typically appears nearly flat over a few reciprocal lattice vectors. Nevertheless, its curvature over many reciprocal lattice vectors gives rise to diffractions from higher-order “Laue zones.” Laue zones are labeled in the top part of Fig. 5.23, showing an fcc reciprocal lattice with a vertical [001] crystal zone-axis orientation. Most diffractions in an electron diffraction pattern would come from the plane labeled “0,” which includes the origin of the reciprocal lattice (and the transmitted beam). Diffractions from this plane comprise the “zeroth-order Laue zone” (ZOLZ). Owing to curvature, however, the Ewald sphere could touch some reciprocal lattice points in higher-order Laue zones (HOLZ). This is illustrated in the bottom part of Fig. 5.23, which shows the zeroth-order Laue zone (ZOLZ) and the first-order Laue zone (FOLZ) for a sc crystal. Notice that in the lowest part of Fig. 5.23 (“top view” of the viewing screen), there is a gap between the FOLZ and the ZOLZ, and this gap is observable in the SAD spot pattern. The origin of this gap is seen in the middle part of Fig. 5.23 (“side view” of the specimen’s reciprocal lattice) where the Ewald sphere is intermediate between the zeroth and first layers of points in the reciprocal lattice. Examples of higher-order Laue zones are shown in Figs. 2.22 and 6.33. For a given crystal, the number of points in the ZOLZ increases as the electron wavelength decreases and the Ewald sphere “flattens.” The radii of the HOLZs are not equally separated – differences between these radii decrease with the order of the zone. The symmetry of the Laue zones about the transmitted beam can be used to monitor accurately the tilt of a crystalline specimen in the electron beam. Imagine starting with the symmetrical case in Fig. 5.23, and then tilting the specimen. Its reciprocal lattice rotates about the origin, which is best imagined by referring to the “side view” of Fig. 5.23. After tilting the specimen, one edge of the FOLZ becomes closer to the transmitted beam. This leads to a circular arc of bright spots such as in Fig. 5.24. The center of this circular arc does not coincide with the spot from the transmitted beam. The specimen orientation can be made symmetrical by tilting the specimen so as to “push” this arc of bright spots away from the transmitted beam. When the symmetrical orientation is attained (corresponding to a precise zone axis orientation), the center of the arc coincides with the bright spot from the transmitted beam. How would you determine the angle by which the Si sample of Fig. 5.24 deviates from the exact zone axis?

5.8 * Effects of Curvature of the Ewald Sphere The crystal shape factor, together with the curvature of the Ewald sphere, can distort the positions and symmetry of the diffraction spots in an SAD

5.8 * Effects of Curvature of the Ewald Sphere

263

Fig. 5.23. Top: Reciprocal lattice of a bcc crystal, showing Laue zones. Center and bottom: Intersections of the Ewald sphere with the reciprocal lattice of a sc crystal, and two zones of bright diffraction spots.

Fig. 5.24. Left: A specimen misoriented from a low-order zone axis produces an asymmetrical intersection of the Ewald sphere with the Laue zones. Right: Asymmetry in the pattern of bright spots from a Si single crystal, slightly off a 110 zone axis. After [5.5].

264

5. Diffraction from Crystals

pattern. Consider a plate of simple cubic crystal whose thin direction is along [101]. Suppose the crystal is tilted into a [100] zone axis. The reciprocal lattice is shown at the top of Fig. 5.25. The [101] rel-rods tilt along the face diagonals.

–g

–g

0

0

g

g

Fig. 5.25. Shape-factor-intensity modifications of sc reciprocal lattice, with rel-rods along [101]. Asymmetrical intersection of the rel-rods with the Ewald sphere causes the (001) spot to be closer to the (000) spot than the (001) spot.

The construction at the bottom of Fig. 5.25 shows two diffraction spots that intersect the Ewald sphere. Owing to the curvature of the Ewald sphere, the actual intersections with the sphere (solid lines) are shifted to the right of where they are expected. The SAD spot pattern is similarly shifted to the right. Note also that the shift of the +g (001) diffraction spot is greater than the shift of the −g (001) spot. Distortions of the diffraction spot positions caused by the curvature of the Ewald sphere may lead to errors in lattice parameter measurements by SAD. It is usually a good idea to obtain diffraction spot spacings from a SAD pattern by measuring the distance between the spots at −g and +g , and dividing by 2 (as opposed to measuring the distance of one spot from the origin, or the distance between two adjacent spots). Even these results may be distorted by unequal shifts of the two diffraction spots, however. The diffraction pattern in Fig. 5.26a is from a [001] zone axis of an AlAg alloy containing γ ′ precipitates. These thin hcp precipitates lie on all four {111} planes and their spots are streaked along the 111 directions. The shape factor intensity about each reciprocal lattice point is a set of relrods that are shaped a bit like children’s jacks, as in Fig. 5.26b. The spokes on the jacks, which originate from shape factor streaking, point along 111 directions. For diffractions that are away from the central beam, the curvature of the Ewald sphere causes the sphere to lie above the center of the jack, and it is intersected by four of the spokes. Tilting the sample can enhance this effect. In the diffraction pattern of Fig. 5.26a we can identify sets of four spots around diffraction spots such as (200) and (220). These sets of four

5.8 * Effects of Curvature of the Ewald Sphere

265

spots are rotated 45◦ with respect to the main pattern, as predicted by the drawing in Fig. 5.26b.

Fig. 5.26. Top: (a) SAD pattern from an fcc AlAg alloy with hcp γ ′ precipitates on all four {111} planes. The [001] diffraction pattern shows fine structure as sets of four spots around each higher-order diffraction spot. Bottom: (b) Origin of the fine structure. Note the four variants of 111 rel-rods about sites in the (001)* plane of the reciprocal lattice.

Rod-shaped precipitates lead to interesting features in diffraction patterns. By analogy with the real-space-plate/reciprocal-space-rod combination above, we expect the shape factor intensity from a thin rod precipitate to be plate-shaped (see Fig. 5.16). It is called a “rel-disk” (for reciprocal lattice disk). When the axis of the rod lies precisely along the z-axis, the diffraction spots appear as fat disks. When the axis of the rod lies off the z-axis, curious things happen, as illustrated in Fig. 5.27. It takes some three-dimensional visualization to understand how the observed SAD pattern from the tilted rod is obtained through the intersections of the tilted diffraction disks with the Ewald sphere, and the reader is encouraged to analyze this example.

Fig. 5.27. Streaking in diffraction pattern (lower right) caused by rod-shaped precipitates inclined to the zone axis (top left), producing rel-disks (lower left).

266

5. Diffraction from Crystals

Further Reading The contents of the following are described in the Bibliography. Leonid V. Az´aroff: Elements of X-Ray Crystallography (McGraw-Hill, New York 1968), reprinted by TechBooks, Fairfax, VA. J. W. Edington: Practical Electron Microscopy in Materials Science, 2. Electron Diffraction in the Electron Microscope (Philips Technical Library, Eindhoven 1975). C. Hammond: The Basics of Crystallography and Diffraction (International Union of Crystallography, Oxford University Press, Oxford 1977). P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley and M. J. Whelan: Electron Microscopy of Thin Crystals (Robert E. Krieger Publishing Company, Malabar FL 1977). M. H. Lorretto: Electron Beam Analysis of Materials (Chapman and Hall, London 1984). G. Thomas and M. J. Goringe: Transmission Electron Microscopy of Materials (John Wiley & Sons, New York 1979). D. B. Williams and C. B. Carter: Transmission Electron Microscopy: A Textbook for Materials Science (Plenum Press, New York 1996).

Problems 5.1 You are working with a novice electron microscopist who has obtained a selected-area diffraction pattern with a strong transmitted beam and one strong diffracted beam having the diffraction vector g. Your colleague prepares for axial dark-field imaging with this diffracted beam by tilting the illumination so that the diffraction spot g moves to the center of the viewing screen. He is dismayed to see the spot fade in intensity and another spot become bright. Use the Ewald sphere construction to show: (a) The diffraction condition before tilting the illumination. (b) The diffraction condition after he has tilted the illumination. Which diffracted beam has become active? (c) The correct tilting of illumination for the intended dark-field work. 5.2 Suppose that Fig. 5.28 is drawn to scale for 100 keV electrons. The crystal is simple cubic, and the reciprocal lattice planes are (100). The shape factor broadening is about 1/4g100 around each diffraction maximum. (The extent of the shape factor broadening is indicated with the dashed lines in the Fig. 5.28.) (a) What is the lattice constant of the crystal? (easy)

Problems

267

(b) Draw to scale the diffraction pattern for the zeroth-order Laue zone. Also, indicate the relative intensities of the spots. (c) By what angle must the incident beam be tilted to bring the transmitted beam back to the center of the pattern, producing symmetrical Laue zones? (This is trivial – just measure the angle in the Fig. 5.28 if you like.) (d) Draw the diffraction pattern for the zeroth-order Laue zone after the tilt of part c is performed.

Ewald Sphere

k0

reciprocal lattice planes

shape factor broadening

Fig. 5.28. Problem 5.2.

5.3 Calculate the structure factor for NaCl, and draw the reciprocal space structure for non-vanishing diffractions (Make a 3-D drawing in perspective. Use of a ruler is optional, but recommended.) Indicate the relative strengths of the diffracted intensities. (Hint: See Fig. 5.11.) 5.4 Take the origin of the face-centered cubic Bravais lattice for the diamond structure to be at its center of symmetry, and determine the expression for the structure factor. 5.5 An orthorhombic phase has only systematic absences in its diffraction pattern for 0kl diffractions when k + l is odd. What are the possible space groups for this structure? 5.6 Sketch to scale the (112)* reciprocal lattice planes for the following real space Bravais lattices: (a) simple cubic, (b) body-centered cubic and (c) face-centered cubic. 5.7 Sketch the (112)* reciprocal structure sections for the following real space structures: (a) L12 (Cu3 Au, Pm3m, cP4), (b) L10 (CuAu, P4/mmm, tP2) and (c) B2 (CsCl, Pm3m, cP2). 5.8 Indicate which beam is strongly diffracting under the following conditions: (a) in bright field, (200) is in the Bragg condition; in dark field, (200) is brought to the center of the optic axis,

268

5. Diffraction from Crystals

(b) in bright field, (200) is brought to the center of the (c) in bright field, (400) is brought to the center of the

in the Bragg condition; in dark field, (200) is optic axis, in the Bragg condition; in dark field, (200) is optic axis,

(d) in bright field, (600) is in the Bragg condition; in dark field, (200) is brought to the center of the optic axis. 5.9 The structure of the θ′ phase found in Al-Cu alloys is shown in Fig. 5.29, where the open circles represent Al and the filled circles Cu. (a) Calculate the structure factor of this phase for all values up to h2 + k 2 + l2 = 16. Leave your answer in terms of fAl and fCu . (b) Suppose that all of the atoms in the structure were the same. How would that affect the structure factor? Explain.

Fig. 5.29. Problem 5.9.

5.10 A certain phase structure: Al   Fe-C-Al  in the  system has the following   at (0, 0, 0), Fe at 21 , 12 , 0 , 21 , 0, 12 , 0, 21 , 21 and C at 12 , 12 , 12 . Its space group is Pm3m. The atomic scattering amplitudes for electrons of the three elements are shown in Fig. 5.30. (a) Derive an expression for the structure factor in terms of fAl , fFe and fC . (b) Calculate the relative intensity ratios for the following diffractions in an electron diffraction pattern: I001 /I002 and I011 /I002 . (c) Sketch the (100)* section of the reciprocal structure (intensity-weighted reciprocal lattice) for this Fe3 AlC phase, labelling the low-index diffractions and indicating relative intensities. 5.11 (a) Sketch the (010)* section of the reciprocal structure for the real space structure in Fig. 5.31, which is a derivative of a bcc structure. (b) Label the diffractions in terms of the lattice of the translation vectors a, b, c. Indicate which are fundamental diffractions of the bcc lattice (translation vectors a, b, c/2), and which are superlattice diffractions of the bcc lattice.

Problems

269

Fig. 5.30. Problem 5.10. After [5.6].

a = b = c/2

Fig. 5.31. Problem 5.11. After [5.6].

5.12 At elevated temperatures, a ternary alloy (A2 BC) has the fcc structure shown in Fig. 5.32a. As the temperature is lowered, all the C atoms go to the corners of the lattice while the A and B atoms randomly occupy the face centers, as shown in  Fig. 5.32b. At still lower temperatures, the B atoms  occupy the 21 , 12 , 0 sites, indicated in Fig. 5.32c. Sketch the (100)* and (001)* diffraction patterns for each of these structures, noting the positions and relative intensities of the different diffractions.

a

b

c

Fig. 5.32. Problem 5.12. After [5.6].

5.13 Intensity distributions for various crystal shapes are illustrated in Fig. 5.16. Using a piece of graph paper, construct a two-dimensional reciprocal space picture of a specimen containing needle shaped precipitates, according to the following directions.

270

5. Diffraction from Crystals

(a) The sample has a simple cubic crystal structure with a = 0.4 nm (similar to Al). Sketch a two-dimensional reciprocal lattice for the structure. (b) The TEM foil is 50 nm thick. Sketch the intensity distribution about the reciprocal lattice points for this thickness. If necessary, use an enlarged diagram of one point in order to keep the scales in perspective. (c) The foil contains coherent needle-shaped precipitates which are 2 nm in diameter and 20 nm long along the 100 directions. Assume the precipitates have the same crystal structure as the matrix. Sketch this intensity distribution in the same pattern. (d) Sketch the wavevector and Ewald sphere in the reciprocal space lattice for 120 keV electrons. Let the wavevector be vertical in the figure. (e) Indicate 15 mrad (0.85◦ ) of semi-angle of beam convergence by adding a second wavevector to the first, with this included angle. 5.14 A side-centered orthorhombic Bravais lattice is shown in Fig. 5.33. From the definition of the reciprocal lattice, obtain expressions for the three basis vectors of the reciprocal lattice of this real space lattice, in terms of the vectors aˆ x, bˆ y and cˆ z.

Fig. 5.33. Problem 5.14.

5.15 Particles of Ag in Al can have the shape of a tetra-kai-decahedron (14 sided, as shown in Fig. 5.34). What should be the shape of the diffuse scattering around the fundamental diffractions? Sketch it as precisely as possible.

Fig. 5.34. Problem 5.15. After [5.7].

Problems

271

5.16 A ternary in  has the following   atoms   a cubic unit cell: A at  phase (0, 0, 0), B at 12 , 12 , 0 and C at 12 , 0, 12 , 0, 21 , 12 . Calculate the following structure factors in terms of fA , fB and fC : (a) (001) and (002), and (b) (100) and (200). 5.17 Draw the reciprocal lattices of the three crystal structures in Fig. 5.35, following these instructions. The unit cells in Fig. 5.35 are drawn with the same scale. In drawing reciprocal lattices, there is no absolute scale factor relating distances in real space and reciprocal space. Nevertheless, use the same scale factor for all three reciprocal lattices. Try to be reasonably accurate in your drawings. (Hint: Draw the reciprocal structure for the L12 structure last – assume that the face-centered atoms scatter only weakly.) fcc

sc

L12

Fig. 5.35. Problem 5.17.

5.18 The kinematical sum for the diffracted wave from a 1-dimensional crystal with atoms at positions x = ma (where m is an integer): ψ(s) =

N 

f ei2πsma

(5.132)

m=1

is equal to N f when s = 0. (a) Calculate the first-order correction to ψ(s) for small s. (b) Give a physical justification for why the first-order correction is imaginary. (c) For small s, the diffracted intensity is not increased by the correction of part a. By how much must the real part of ψ be reduced for this to be true? 5.19 Neutron diffraction employs neutrons with relatively low velocities, so the diffraction condition can be altered by moving the crystal with respect to the neutron beam. For a neutron with wavevector k, the phase shift over the interatomic distance R′ in the crystal is no longer k · R′ , but is now:  ˆ v·k , (5.133) k · R′ 1 − vn where v is the velocity of the crystal and v n is the velocity of the neutron.

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ˆ = −vn , the phase of the neutron at the atom positions behaves (a) When v · k as if the neutron has a wavevector twice as large as for a stationary crystal. Why? ˆ > +vn , the phase shift is negative, but this is condition is (b) When v · k physically unimportant. Why? (c) By performing an analysis like that of Sect. 5.1.2 (starting with (5.11)), derive a new form of (5.18) that applies to neutron diffraction from a moving crystal. (d) A device called a “phase-space transform” chopper can be used to bunch neutrons in energy by using diffraction from a moving polycrystalline sample. Discuss the concept behind its operation.

6. Electron Diffraction and Crystallography

6.1 Indexing Diffraction Patterns Reciprocal lattices of crystals are spanned by three reciprocal-lattice vectors, so the diffraction patterns of materials are inherently three-dimensional. To obtain all available diffraction information, the diffraction intensity should be measured for all magnitudes and orientations of the three-dimensional diffraction vector, Δk. Appropriate spherical coordinates in k-space are Δk, θ, and

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φ. A practical approach to this fairly complicated problem is to separate the control over the magnitude of Δk from its orientation with respect to the sample. The goniometers described in Sect. 1.3.2 provide the required control  on the over the magnitude Δk, while maintaining a constant direction Δk sample. For isotropic polycrystalline samples, a single powder-diffraction pattern provides representative diffraction data because all crystal orientations are sampled. For specimens that are single crystals, however, it is also necessary to provide for the orientational degrees of freedom of the specimen (latitude and longitude angles, for example). A diffraction pattern (varying Δk) should then be obtained for each orientation within the selected solid angle (sinθ dθdφ) of reciprocal space. Diffraction experiments with single crystals require additional equipment for specimen orientation, and software to relate these data to the reciprocal space structure of the three-dimensional crystal. For publication and display of these data, however, it is typical to present the diffraction intensities as planar sections through the three-dimensional data. Diffraction data from the TEM are obtained as near-planar sections through k-space. The large electron wavevector provides an Ewald sphere that is nearly flat, allowing the handy approximation that a diffraction pattern from a single crystal is a picture of a plane in its reciprocal space.1 The magnitude of the diffraction vector, Δk, is obtained from the angle between the transmitted and diffracted beams. Two degrees of orientational freedom are required for the sample in a TEM. They are typically provided by a “double-tilt specimen holder,” which has two perpendicular tilt axes oriented perpendicular to the incident electron beam. A modern TEM provides two modes for obtaining diffraction patterns from individual crystallites. The oldest is selected area diffraction (SAD), which is useful for obtaining diffraction patterns from regions as small as 0.5 μm in diameter (see Problem 2.16). The second method is nanodiffraction, or convergent-beam electron diffraction (CBED), in which a focused electron probe beam is used to obtain diffraction patterns from regions as small as 10 ˚ A. Both techniques provide a two-dimensional pattern of diffraction spots, which can be highly symmetrical when a single crystal is oriented precisely along a crystallographic direction. The additional three-dimensional information available in CBED patterns is discussed in Sect. 6.5. 6.1.1 Issues in Indexing We now describe how to “index” the planar sections of single crystal diffraction patterns; i.e., label the individual diffraction spots with their appropriate values of h, k, and l. Indexing begins with the identification of the transmitted beam, or the (000) forward diffraction. This is usually the brightest spot 1

It is not quite the full picture, however, because the diffraction pattern measures the intensity of the diffracted wave and not the wave itself.

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275

in the center of the diffraction pattern. Next we need to index two independent (i.e., not co-linear) diffraction spots nearest the (000) spot. Once these two (short) vectors have been determined, linear combinations of them provide the positions and indices of all other diffraction spots. To complete the indexing of a diffraction pattern, we also specify the normal to the plane of the spot pattern – this normal is termed the “zone axis.” By convention the zone axis points towards the electron gun (i.e., upwards in most TEM’s). The indexing of a diffraction pattern is not unique. If a crystal has high symmetry, so does its reciprocal lattice. A high symmetry leads to a multiplicity of different, but equally correct, ways to index a diffraction pattern. For example, a vector cube axis can be called a [100], [010], or [001] vector. Once it is specified, however, the indices of all other directions must be consistent with it. There are two different approaches to indexing single-crystal diffraction patterns. You can either guess the zone axis first (Method 1), or you can determine the zone axis after you have labeled two or more diffraction spots (Method 2).2 We demonstrate both approaches. In either approach, most of the work involves measuring angles and distances between diffraction spots, then comparing these measurements to geometrical calculations of angles and distances. While indexing a diffraction pattern, you must remember that structure factor rules eliminate certain diffraction spots. For consistency you must also satisfy the “right-hand rule,” which is given by the cross-product  ×y z , or more physically by your right hand as shown in Fig. 6.1. relation: x The procedures are straightforward for low-index zone axes of simple crystal structures, but become increasingly difficult for crystal structures with low symmetry and for high-index zone axes, where many different combinations of interplanary spacings and angles provide diffraction patterns that look similar. In these cases, it is helpful to have a computer program to calculate the diffraction patterns.

Fig. 6.1. A right-handed coordinate system. (Don’t forget that your right hand differs from your left hand!)

The eye is able to judge distances to about 0.1 mm, particularly with the aid of a 10 X calibrated magnifier, so this is the typical measurement accu2

It is often important to identify quickly the zone axis. Experienced microscopists tend to recognize the zone axis from the symmetry of the spot pattern. Indexed diffraction patterns are available in Appendix A.6. It is a good idea to make copies of them and take them with you to the microscope.

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racy of spot spacings on a diffraction pattern. If a diffraction spot is 10 mm from the center of the pattern, expect a measurement error of a few percent. For highest accuracy in determining spot spacings, it is often preferable to measure the distance between sharper, higher order spots, and then divide by the number of spots between them (plus one). Unfortunately, this procedure can lead to errors if the Ewald sphere cuts the spot at an angle (as discussed in Sect. 5.8), or if there is a slight distortion of the diffraction pattern caused by the projector lens of the microscope. It is important to know the distortions and artifacts of your microscope, which can be assessed by measuring diffraction patterns of well-formed crystallites of a well-known material. Photographic printing can distort the spot spacings, so measurements should be performed directly on the negative if a digital optical system is not available. 6.1.2 Method 1 – Start with Zone Axis Indexing and its frustrations are best illustrated by example. Suppose we need to index the diffraction pattern in Fig. 6.2, and we know it is from an fcc crystal.

o

angles = 90

0.65 cm

1.10 cm

Fig. 6.2. An fcc diffraction pattern, ready for indexing.

The easy way to index this diffraction pattern is to look it up in Appendix A.6 of this book. Here, however, we index the pattern “by hand.” In the first method we “guess” the zone axis and its diffraction pattern, in the style of experienced microscopists (see footnote 2). This method is most useful when the diffraction pattern shows an obvious symmetry, such as a square or hexagonal array of spots for a cubic crystal.3 You should memorize the symmetries for fcc and bcc diffraction patterns listed in Table 6.1. We first note that our pattern in Fig. 6.2 is less symmetrical than those of Table 6.1. Nevertheless, the density of spots is reasonably high, so we expect that we have a fairly low-order zone axis. The lowest order zone axes are4 : 3 4

It is also useful when we do not know the camera length of the microscope, as in the present example. Remember: the structure factor rule for fcc (h, k, l all even or all odd) does not pertain to the choice of zone axis. For example, we can always tilt an fcc crystal so its [001] direction points up.

6.1 Indexing Diffraction Patterns

277

Table 6.1. Some symmetrical diffraction patterns of cubic crystals Zone Axis

[100]

[110]

[111]

Symmetry

square

hexagonal

Aspect Ratio

1:1

rectangular √ 1 : 2 for bcc, sc (almost hexagonal for fcc)

equilateral

[100] [110] [111] [210] [211] [310] [200] [220] [222] [300] First note that in defining a zone axis, the [100], [200], and [300] directions are the same. We therefore need only consider the lowest index [100] direction as a candidate zone axis. We eliminate the first 3 zone axes because the pattern does not have the required symmetry as listed in Table 1. At this point we make a guess and try the [210] zone axis. We could now compare our angles and distances in Fig. 6.2 to the diffraction pattern in Appendix A.6, but here we illustrate a systematic procedure to check the diffraction pattern. We seek the lowest order diffractions in the [210] diffraction pattern of an fcc crystal. Some allowed diffractions from fcc crystals (h, k, l are all even or all odd) are: (111) (200) (220) (311) (331) (420) (222) (400) (333) The allowed diffraction spots must be perpendicular to the5 [210] zone axis. To test for perpendicularity with [210], we seek dot products that are zero6 : [210] · [111] = 0 [210] · [002] = 0 [210] · [220] = 0 [210] · [113] = 0 [210] · [133] = 0 [210] · [240] = 0 We therefore expect the lowest order spots in the diffraction pattern to be (002) and (240). We must next confirm that the correct angle is made between the two lines running from the (000) spot to these two √ diffraction spots. We first need to normalize the vectors with the factor 1/ h2 + k 2 + l2 . After doing so, we check the normalization: 1 1 [002] · [002] = 1 , (6.1) 2 2 1 1 √ [240] · √ [240] = 1 . (6.2) 20 20 5 6

Remember: for high-energy electrons with small Bragg angles, Δk is nearly perpendicular to k0 . In so doing, it is necessary to try all orientations of the diffractions (e.g., [210] · [200], [210] · [020], [210] · [002]).

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The dot product of two normalized vectors equals the cosine of the angle between them. Here, with the [002] and [240] we could skip the normalization, since their dot product is exactly zero. This is consistent with the 90◦ angle on the diffraction pattern. So far, so good – the (002) and (240) diffractions seem promising because they are perpendicular to each other, satisfying our requirement that the angles between spots are 90◦ . The final step is to lay out the spots at the correct distances to make a diffraction pattern for the [210] fcc zone axis. We rearrange the camera equation, rd = λL (2.7), to obtain the measured distance, r, of a diffraction spot from the transmitted beam: λL 2 h + k2 + l2 . (6.3) r= a If we knew our camera constant, λL, it would be appropriate to work with absolute distances for the spot spacings. Here we work with relative spacings instead. Equation 6.3 √shows that the ratio of the spot distances must equal the ratio of the factors h2 + k 2 + l2 . We take the vertical spacing to the (002) spot as a reference distance (0.65 cm from Fig. 6.2). Doing so, we predict a spacing to the (240) spot shown in Fig. 6.3: (22 + 42 + 02 ) 0.65 cm = 1.45 cm . (6.4) (22 + 02 + 02 )

Since the answer should be closer to the 1.10 cm spacing of Fig. 6.2, the [210] zone axis must be wrong. Too bad. We have to try again. 002 if 0.65 cm 240

000

2

2

4 +2 then 0.65 = 1.45 cm 2

Fig. 6.3. A typical botched attempt at indexing the diffraction pattern in Fig. 6.2.

We make another guess – the [211] zone axis. Repeating the same procedure in abbreviated form: Expected Diffraction Normalized cosθ θ ◦ √1 [211] · √1 [111] √0 [211] · [111] = 0 90 6 3 18 1 1 1 √ [211] · √ [002] √ [211] · [002] = 0 65.9◦ 6 4 6 √1 [211] · √1 [022] √0 [211] · [022] = 0 90◦ 6 8 48 1 0 1 √ [211] · √ [113] √ [211] · [113] = 0 90◦ 6 11 66 1 5 1 √ [210] · √ [133] √ 59.1◦ [210] · [133] = 0 5 19 95 1 1 0 √ [210] · √ [240] √ [210] · [240] = 0 90◦ 5 20 100

6.1 Indexing Diffraction Patterns

279

In Fig. 6.4 we construct a diffraction pattern with the closest diffraction spots, [111] and [002], and calculate the distance ratio. 111 if 0.65 cm 000

then 0.65

022

8 3

= 1.06 cm

Fig. 6.4. Successful indexing of the diffraction pattern of Fig. 6.2.

Good, we got it. The 3.5 % accuracy seems okay, although it is a bit on the high side for this kind of work. Maybe we should remeasure our spot spacings, or perhaps if we look closely at the diffraction pattern we might see that the spots are asymmetrical, and there may be some distortion of the diffraction spots caused by the curvature of the Ewald sphere and an asymmetrical shape factor. Consistent indexing is a virtue. Once we have identified the diffraction pattern, we must ensure that all linear combinations of our reciprocal lattice vectors give the indices of all the other spots in the diffraction pattern. Our two shortest vectors in the pattern are [111] and [022]. Therefore the h, k, l, indices increase by [111] when we traverse a vertical column of spots, and increase by [022] when we traverse a horizontal row of spots as illustrated in Fig. 6.5. For example, when moving across the top row of spots in Fig. 6.5, the first index remains constant at −2, the second goes 0, 2, 4,. . . , and the third goes 4, 2, 0,. . . . In performing these row checks, we should confirm that we do not miss any spots or create any new spots. The zone axis should be consistent with a right-handed coordinate system. We confirm that the zone axis points up towards the electron gun, with the vector cross-product: x + (2 − 0) y + (0 + 2) z = [422]  [211] . [022] × [111] = (2 + 2) We are lucky – the vector [422] is parallel to our [211] zone axis. Since we originally guessed the [211] zone axis, we knew we would get either [211] or [211] from this cross-product. If we got the [211] result, we could reverse the direction of the [111] vector (make it a [111]), and relabel our diffraction pattern. 6.1.3 Method 2 – Start with Diffraction Spots In the second method for indexing a diffraction pattern, we index the spots first, and obtain the zone axis at the end of the procedure. This method is

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( h 204

222

240

113

111

131

022

000

022

131

111

113

240

222

204

( h

k

constant

k

l )

6 = +2 6 = x0 (extending below the plane of the paper). This halfplane of wavelet sources is at a set of distances of {r} away from the source, O, and at the distances {R} away from the point of observation, P. For different locations of the scattered wavelets in the half-plane (x > x0 , y, z = 0), the lengths of r and R are:

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10. High-Resolution TEM Imaging

Fig. 10.8. Geometry of an opaque halfplane (coming out of the plane of the paper) between a source of spherical waves and the observation point, P.

  x2 + y 2 r02 + x2 + y 2 ≃ r0 1 + , 2r02   x2 + y 2 2 2 2 . R = R0 + x + y ≃ R0 1 + 2R02 r=

(10.25) (10.26)

We need to integrate (10.11), the wavelet amplitude at point P, eikR dS , (10.27) R over all differential areas of the transparent half-plane. Using (10.18) for the incident wave and (10.25) and (10.26) for r and R, the integration of (10.27) over the transparent half-plane is:  

∞ ∞ exp ik r0 + (x2 + y 2 )/(2r0 ) i 0   Ψsc (P) = − Ψin 2 r0 1 + (x2 + y 2 )/(2r02 ) −∞ x0 

 exp ik R0 + (x2 + y 2 )/(2R0 )   × dxdy , (10.28) R0 1 + (x2 + y 2 )/(2R02 ) dΨsc (P) = −iA(2θ) Ψin

where we set A(2θ) equal to an average value of 1/2 because the integral converges with no special precautions. We continue to assume that x and y are small compared to r0 and R0 , so they can be neglected in the denominator. The phase in the numerator is sensitive to x and y, however. Rearranging: ∞ ∞   r + R  ik(r0 +R0 ) 0 0 0 e exp ikx2 Ψsc (P) = −i Ψin 2r0 R0 2r0 R0 −∞ x0

  r + R  0 0 × exp iky 2 dxdy . 2r0 R0

Normalized distances in the x-y plane are:

(10.29)

10.1 Huygens Principle

 r0 + R0 , X ≡x r0 R0  r0 + R0 Y ≡y , r0 R0  r0 R0 dx = dX , r0 + R0  r0 R0 . dy = dY r0 + R0 With (10.30)–(10.33) we re-write (10.29) as: ∞ ∞ 0 ik(r0 +R0 ) 2 2 −i Ψin e Ψsc (P) = eikX /2 eikY /2 dXdY . 2 (R0 + r0 )

529

(10.30) (10.31) (10.32) (10.33)

(10.34)

X0 −∞

Equation (10.34) does not have an analytic solution for arbitrary X0 . The real and imaginary parts of the two integrals are defined as Fresnel cosine and sine integrals, C(X) and S(X), so we write: ∞ ∞  0 ik(r0 +R0 )  −i Ψin e C(X) + i S(X) C(Y ) + i S(Y ) . Ψsc (P) = 2 (R0 + r0 ) X0 −∞ (10.35) The cosine and sine Fresnel integrals are tabulated. More commonly, however, these two Fresnel integrals are presented together in one plot in the complex plane. This plot of C(X) + i S(X) is called a “Cornu spiral” (Fig. 10.9).

Fig. 10.9. Cornu spiral. The points on the spiral are separated by increments of 0.1 units in X [J. C. Slater and N. C. Frank: Introduction to Theoretical Physics, (McGraw–Hill, New York 1933)]. Reproduced with the permission of The McGraw-Hill Companies. X 

It is easy to use the Cornu spiral to evaluate [C(X) + i S(X)]X  . First locate the limits of integration, X ′ and X ′′ , which are tick marks on the

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10. High-Resolution TEM Imaging

spiral. The limits −∞ and +∞ are at the ends of the spiral on the lower left and upper right, respectively, at ±1/2 (1 + i). For example, to evaluate the integral between the limits of −∞ and +∞ (the last factor of (10.35)), we take the difference: +1/2 (1 + i) − (−1/2) (1 + i) = 1 + i. To evaluate the first integral of C(X) + i S(X) from X0 to ∞, we measure the distance along the straight line from the point labeled “∞” in Fig. 10.10 to the point on the spiral marked with the value of X0 . Seven examples are presented in Fig. 10.10a. ∞ From the length of these straight lines it is evident that [C(X) + i S(X)]X0 is zero when X0 = ∞, has a maximum for X0 ≃ −1.2, and has a local minimum for X0 ≃ −1.9. For an increasingly negative √value of X0 , the integral oscillates about a value of 1+i, having amplitude of 2 (Fig. 10.10b and Problem 10.2).

Fig. 10.10. Use of Cornu spiral. (a) Seven integrals (“A”–“G”) over X from various X0 to +∞. (b) Graph of the seven amplitudes of part a, corresponding to the wave amplitude at point P for various edge positions proportional to X0 .

The physical position of the opaque edge in Fig. 10.8 is x0 , which is proportional to X0 (10.30). As the position of this opaque edge moves across the optic axis in Fig. 10.8, the wave amplitude at point P changes considerably, as shown in Fig. 10.10b. When x0 is positive and large, the intensity is zero, since the opaque half-plane blocks all paths for the wave. At the other extreme, when x0 is near −∞, the opaque half-plane is removed entirely. For this case we evaluate (10.35) with X0 = −∞, obtaining: Ψsc (P) =

0 ik(r0 +R0 ) −i Ψin e 2 (1 + i) , 2 (r0 + R0 )

(10.36)

Ψsc (P) =

0 ik(r0 +R0 ) e Ψin , for (x0 = −∞) . r0 + R0

(10.37)

Equation (10.37) shows that when no half-plane is present, the wave at point P is simply an unimpeded spherical wave. The interesting effects occur when X0 just moves across the optic axis (to negative x), and alternating bright and

10.1 Huygens Principle

531

dark fringes are seen. These “Fresnel fringes” are graphed in Fig. 10.10b, and experimental examples are shown in Fig. 10.11 and Fig. 2.44. Alternatively, we can achieve the same result by fixing the sharp edge and moving the point P, since this also causes the opaque edge to move with respect to the optic axis. This is the situation for a TEM image of a sharp edge on a specimen. The image is a map of the wave amplitude for all P in the x-y plane.

Fig. 10.11. Fresnel fringes near the edge of a hole. (a) Underfocus showing a prominent light fringe, (b) in focus, and (c) overfocus, showing a prominent dark fringe. Note the uniform circular nature of the fringes, indicating a lack of astigmatism (cf., Fig. 2.44d).

The spacing and visibility of Fresnel fringes depends on the focus of the microscope.6 From (10.30) we see that when R0 = 0, so the specimen is exactly in focus, X = ∞. In principle, Fresnel fringes are absent when the specimen is exactly in focus. With a zero denominator in (10.30), however, the image is highly sensitive to instrument imperfections that affect the focus. Obtaining minimum, uniform Fresnel fringes around a hole in a sample is a way to make an approximate correction for astigmatism, for example. With underfocus, a set of closely-spaced Fresnel fringes appears in the image near the edge of a hole, or around an opaque particle. In practical cases where r0 > R0 , the spacing between these fringes increases approximately as the square root of the underfocus (see Problem 10.2b). The visibility of Fresnel fringes also depends on the quality of the wave source (at O in Fig. 10.8). If this source is not a point, there is an effective distribution in the locations of optic axes (or equivalently, a distribution in the positions x0 of the opaque edge). This suppression of spatial coherency of the source washes out the fringe contrast. Modern illumination systems in the TEM using bright point sources such as field emission guns provide much better visibility of Fresnel fringes than do illumination systems with tungsten or LaB6 electron sources.

6

The fringe contrast also depends on the curvature of the incident wavefront on the specimen, but the effects of focus are easier to see.

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10.2 Physical Optics of High-Resolution Imaging This section develops a set of mathematical tools that are useful for calculating contrast in high-resolution images. Different mathematical functions correspond to wave propagation, lenses, and even materials. The mathematical operations are primarily Fourier transforms and convolutions of Gaussian functions and delta functions. Similar manipulations were performed in Sect. 9.4.2, and were first discussed in Sect. 8.2. In essence, an optical model with components of propagating wavefronts (pR ), specimens (qi ), and lenses (qlens ) is converted to a mathematical model of products or convolutions of real-space functions (q and p) or their Fourier transforms (Q and P ). Each function corresponds to a component of the model. The choice of a real-space function or a k-space function is usually made for the purpose of replacing an awkward convolution of two functions with a more convenient multiplication of their Fourier transforms. The presentation of the Huygens principle in the previous Sect. 10.1 motivates the definition of a wavefront propagator, which is a kernel of the Green’s function of the wave equation. This propagator, pR , expands a spherical wave outwards over the distance R. A lens function, qlens , provides the opposite action, and has the mathematical form to converge a plane wave into a point over the distance of one focal length, f . The specimen function, qi , discussed in Sect. 10.2.3, provides phase shifts (and also absorption) to the wave front. The set of mathematical tools presented in this Sect. 10.2 is well-suited for understanding the effects of lens defects on high-resolution TEM images. 10.2.1 ‡ Wavefronts and Fresnel Propagator In Sect. 10.1.2, all points on the surface of a spherical wavefront at r0 were assumed to be point emitters of spherical waves. This implementation of the Huygens principle predicted the correct forward propagation of the spherical wave. The actual work involved performing a convolution of the spherical wave propagator with the incident wavefront. It was essentially the procedure for solving the Schr¨ odinger wave equation with the method of Green’s functions. In both cases the Green’s function “kernel,” (10.7) or (10.38) below, is the spherical wave emitted by a single point on the wavefront. To calculate the total scattered wave, this point response was convoluted with the amplitude over the entire wavefront, (10.6) or (10.12). Here we define the Green’s function kernel, or “propagator” (of spherical waves), as: −i ikR e . (10.38) Rλ This p(R), convoluted with the surface of the wavefront in (10.12), provides the scattered wave amplitude at point P. Since R2 = x2 + y 2 + z 2 : p(R) ≡

10.2 Physical Optics of High-Resolution Imaging

533

−i ik(x2 +y2 +z2 )/R e . (10.39) Rλ The factor 1/λ is necessary to obtain the correct intensity when integrating over Fresnel zones as in Fig. 10.7. As explained following (10.24), waves with larger λ and smaller k have wider phase-amplitude spirals, and would have larger amplitudes unless we normalized by λ. The factor of −i compensates for the phase shift of the Fresnel integral, as explained after (10.24). , assume small angles of We now put the propagation direction along z scattering so that z ≃ R, and therefore ignore the z-dependence of p(x, y, z) in (10.39).7 We work instead with the “Fresnel propagator,” pR (x, y): p(x, y, z) =

−i ik(x2 +y2 )/R e . (10.40) Rλ This propagator is convoluted with a wavefront to move the wavefront forward by the distance R. As a first example, we apply the propagator to an incident spherical wavefront. Section 10.1.2 worked the details of this convolution of the propagator with a spherical wavefront, qsphr (x, y): pR (x, y) =

1 ik(x2 +y2 )/r e , r so from (10.24) we know the result: qsphr (x, y) =

(10.41)

2 2 1 eik(x +y )/(R+r) . (10.42) R+r Anticipating the multislice method of Sect. 10.2.3, we use the notation Ψi (x, y) for the incident wave, and Ψi+1 (x, y) for the wave after the operation of the propagator. In another example of the use of the Fresnel propagator, consider the wave emitted by a point source, qδ (x, y), which is a product of two Dirac delta functions:

Ψi+1 (x, y) = qsphr (x, y) ∗ pR (x, y) =

qδ (x, y) = δ(x) δ(y) .

(10.43)

The variables x and y are independent, so the convolution with   of (10.40)   each delta function of (10.43) simply returns exp ikx2 /R and exp iky 2 /R : Ψi+1 (x, y) = qδ (x, y) ∗ pR (x, y) , i ik(x2 +y2 )/R i ik(x2 +y2 )/R e e = . Ψi+1 (x, y) = δ(x) δ(y) ∗ Rλ Rλ The intensity is: ∗ Ψi+1 Ψi+1 = 7

1 λ2 R2

.

(10.44)

(10.45)

(10.46)

Note that exp(ikz 2 /R) ≃ exp(ikR), which has no effect on the intensity because exp(ikR) exp(−ikR) = 1.

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10. High-Resolution TEM Imaging

The point source wavefront of (10.43), convoluted with the propagator, gives a wave intensity that decreases as R−2 , as expected for a spherical wave. The factor of λ−2 was not obtained in the correct (10.42), however, even as we let the r in (10.41) go to zero. More deftness is required in performing the delta function convolutions than we used in (10.45). In most of what follows, however, we simply ignore the prefactor for the Fresnel propagator, and avoid the trouble of taking the delta function as a limit of a small spherical wavefront. 10.2.2 ‡ Lenses Figure 2.34 showed the essence of how to design a lens by considering phase shifts, and this concept is also shown in the center of Fig. 10.4 in the context of the Huygens principle. This section presents the lens as a mathematical phase shifter. The lens is considered to be a planar object, providing phase shifts across an x-y plane. An ideal lens of focal length f has the phase function: qlens (x, y) = e−ik(x

2

+y 2 )/f

.

(10.47)

The lens distorts the phases of a wavefront at its location, so the wavefront is multiplied by qlens (x, y) at the position of the lens. Note that the phase itself increases parabolically from the optic axis (as x2 + y 2 in (10.47)), consistent with (2.23) and our assumption of paraxial rays. Rules. The rules for working with lenses and propagators are: • Lenses (and materials), denoted “q(x, y),” are assumed infinitesimally thin, and their action is to make phase shifts in a wavefront. These objects multiply the wavefront at their locations in real space. (Lens distortions, however, are best parameterized k-space, where lens and material functions, Q(Δkx , Δky ), must be convoluted rather than multiplied.) . A • Propagators, denoted “p(x, y),” move the wavefront forward along z single point is propagated as a spherical wave, but the full wavefront must be convoluted with p(x, y) to move it forward. (When the wavefront can be expressed as a set of diffracted beams in k-space, the propagator, P (Δkx , Δky ), operates on the wavefront by multiplication rather than convolution.) Example One. Consider a plane wave that passes through a lens, and propagates a distance f , where f is the focal length of the lens. We know that the wave, Ψi+1 (x, y), must be focused to a point after these operations. The final wave is:8 8

Note the alternative k-space formulation of (10.48): Ψi+1 (Δk) = Ψi (Δk) ∗ Qlens (Δk) Pf (Δk).

10.2 Physical Optics of High-Resolution Imaging

Ψi+1 (x, y) = Ψi (x, y) qlens (x, y) ∗ pf (x, y) .

535

(10.48)

For simplicity we ignore the prefactors in (10.40), and work with the xdimension only. The wavefront of a plane wave has no variation with x, so we represent it as the factor 1. With (10.40) and (10.47), (10.48) becomes:   2 2 ψi+1 (x) = 1 e−ikx /f ∗ eikx /f . (10.49)

Section 8.1.3 (8.23) noted that the convolution of two Gaussians is another Gaussian. The breadths add in quadrature, even if they are complex numbers. For (10.49) we find a breadth, σ:  f f + =0. (10.50) σ= −ik ik A Gaussian of zero breadth is a delta function, so (10.49) becomes: ψi+1 (x) = δ(x) .

(10.51)

The function for the ideal lens (10.47) causes, as expected, a plane wave passing through the lens to be focused to a point at the distance f . Example Two. Consider a point source of illumination, propagated a distance d2 to the lens, passed through the lens, and propagated to a focal point at the distance d1 on the other side of the lens. This is the situation shown in Fig. 2.32. Our formalism for propagators and lens becomes9 : ψi+1 (x, y) = qδ (x, y) ∗ pd2(x, y) qlens (x, y) ∗ pd1(x, y) .

(10.52)

For simplicity, we work with one dimension only (x), and ignore the prefactor for the propagator in (10.40). Equation (10.52) becomes:

 2 2 2 (10.53) ψi+1 (x) = δ(x) ∗ eikx /d2 e−ikx /f ∗ eikx /d1 . We know from the lens formula (2.1) that for a point source to be focused to a point, the distance of propagation from the left and right are related as: 1 1 1 = − , (10.54) d2 f d1

so when we substitute (10.54) into (10.53),

 2 2 2 ψi+1 (x) = δ(x) ∗ eikx (1/f −1/d1 ) e−ikx /f ∗ eikx /d1 , ψi+1 (x) = δ(x) ∗ e−ikx

2

/d1

∗ eikx

2

/d1

.

(10.55) (10.56)

As discussed for (10.49) and (10.50), the second convolution is δ(x), so: ψi+1 (x) = δ(x) .

(10.57)

This second example showed how we can use phase shifts by lenses with propagators to take a point source of illumination through a lens and focus it to a point, given that the lens formula is satisfied. 9

Note the alternative k-space formulation of (10.52): Ψi+1 (Δk) Ψi (Δk) Pd2(Δk) ∗ Qlens (Δk) Pd1(Δk). For our point source, Ψi (Δk) = 1.

=

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10. High-Resolution TEM Imaging

Lens Distortions. The present formalism will be used in Sect. 10.3.2 for the analysis of non-ideal lenses. Lens defects modify the phase shift of the lens, and are included as a factor that multiplies the lens transfer function in   k-space. The essential features of this phase transfer function, exp iW(Δk) , are presented in k-space in Sect. 10.3.3. To work with the lens function of (10.47) transform of  in real  space, however, we convolute it with the Fourier ′ (x, y): exp iW(Δk) to obtain the performance of a real lens, qlens   2 2 ′ qlens (10.58) (x, y) = e−ik(x +y )/f ∗ F e−iW(Δk) .

In (10.58) we have written the phase transfer function as a function of Δk, which involves the angle made by an electron with respect to the optic axis as it enters the lens. Ideal lens performance is possible only if W(Δk) is a constant.10 We expect, however, that spherical aberration will cause W(Δk) to increase with Δk, and we evaluate this problem in detail in Sects. 10.3.1 ′ (x, y) by adjusting f . to 10.3.3, with emphasis on how to optimize qlens 10.2.3 ‡ Materials The present “physical optics approach” of wave propagators, wavefronts, and phase transfer functions of lenses is well-suited for computer simulations of high-resolution TEM images, as developed in Sect. 10.4. Consider the general  through N layers of mateexpression for the electron wave traveling along z rial. Each layer advances the phase of the wavefront by small amounts, and these amounts differ at various x, y over the layer (corresponding to atomic columns and channels). This phase advance through the layer is given by the multiplicative factor, qi (x, y), or symbolically, qi (x) or qi . (A layer of empty space has qi (x) = 1.) We have to convolute this new wavefront after the layer with a propagator pi (x) to move the wavefront to the next layer. The following expression for the wave just modified by the N th layer of material is simple if you first look at the zeroth layer in the center of the equation, using numbers below the brackets to match them in pairs: 





   ψN +1 (x) = qN (x) . . . q2 q1 q0 ∗ p0 ∗ p1 ∗ p2 . . . ∗ pN −1 (x) . N

3

2

1

1

2

3

N −1

N

(10.59)

In its alternative formulation in Fourier space, where Q(Δk) ≡ F −1 [q(x)] and P (Δk) ≡ F −1 [p(x)], this equation involves multiplications of the propagators instead of convolutions:





     ψN +1 (Δk) = QN (Δk) ∗ . . . Q2 ∗ Q1 ∗ Q0 P0 P1 P2 . . . PN −1 (Δk) . N

10

3

2

1

1

2

3

N −1

N

(10.60)

In this case, exp(−iW(Δk)) is a constant of modulus 1, so its Fourier transform is a δ-function. The convolution in (10.58) of this δ-function with the ideal lens function, exp{−ik[(x2 + y 2 )/f ]}, returns the ideal lens function.

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537

The propagators, pi (x), are assumed to be the same as in (10.40). In other words, the electron wavefront propagates between layers as if in a vacuum. The layers themselves are assumed infinitesimally thin, and provide only a phase shift, qi (x), and no propagation. We know the form of the free space propagators, but what is the meaning of qi (x) for the material? In general, qi (x, y) has the form: qi (x, y) = e−iσ φi(x,y)−μ(x,y) .

(10.61)

The first term in the exponent provides for a phase shift that varies with position, (x, y), and the second term provides for absorption. It is the role of the dynamical theory of diffraction to calculate q, starting with the Schr¨ odinger equation, and some aspects of a crystal as a “phase grating” are presented in Sect. 11.2.3. We can relate the phase distortion to the effective potential of the electron in the material. To do so, we make use of the fact that the electron wavevector in the crystal, k, differs from the wavevector in the vacuum, χ, because the potential energy for the electron in the crystal is −eV (the potential is attractive because the electron passes through positive ion cores). To conserve total energy, the kinetic energy of the electron in the crystal must increase by +eV to compensate for the potential energy, so while the wavevector in vacuum, χ, is:  2mE0 , (10.62) χ= 2 the wavevector in the crystal, k, is slightly larger:  2m (E0 + eV ) k= , (10.63) 2    eV 2mE0 1+ , (10.64) k≃ 2  2E0   eV k ≃ χ 1+ . (10.65) 2E0

At a snapshot in time at t′ , the wave ψ(kz − ωt′ ) has a phase, kz − ωt′ , that increases by the amount k dz over the distance interval from z to z + dz. Over this small  distance interval, the plane wave ψz = exp(ikz) changes into ψz+dz = exp ik(z + dz) = ψz exp(ik dz). After propagating in a material of average potential −eV from z to z + dz, the k of (10.65) gives the plane wave:  eV  dz . (10.66) ψz+dz ≃ ψz eiχ dz exp ik 2E0

The first exponential is as expected when the electron propagates through vacuum (cf., (10.38)). The second exponential in (10.66) is more interesting because V depends on position (since atoms are located at various x, y, z). The potential V is not homogeneous in x, y when atoms lie along columns, and we are interested in how electrons traveling down columns at different

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10. High-Resolution TEM Imaging

x, y experience different V . After a plane wave has propagated the thickness t, the new wavefront is found by summing (integrating) all phase shifts in the exponents of (10.66): ψz+t = ψz eiχt exp

  ike t V (x, y, z) dz . 2E0

(10.67)

0

The multislice calculational scheme of (10.59) and (10.60) assumes these phase shifts occur in layers infinitesimally thin, but spaced apart by the distance t. The phase shift and absorption of the infinitesimal layer is equal to that caused by a thickness, t, of material. The nth layer multiplies the wavefront by qn , where:   ike t qn (x, y) = exp V (x, y, z) dz . 2E0

(10.68)

0

Using this qn in (10.59) to represent the effect of a thin layer of material, the propagator of (10.40) then moves the wavefront by the distance, t, to the next layer. The choice of thickness, t, is discussed further in Sect. 10.4. Certainly this type of wave scattering calculation is accurate when t is subatomic, but much larger values of t (some fraction of the extinction distance) are acceptable in practice. To make further progress we need a “multislice” computer calculation code as described in Sect. 10.4. In principle, these calculations use expressions such as (10.68) for q and (10.40) for p. The multislice computer code performs a series of operations as in (10.59) and (10.60), where the phase distortion of a wave incident on the ith layer is calculated as a function of x and y, the th wave is propagated to the (i + 1) layer, and the process is repeated. Before we return to these issues in more detail, however, we next describe how the objective lens alters the phase of the electron wavefront.

10.3 Experimental High-Resolution Imaging 10.3.1 Defocus and Spherical Aberration The performance of the objective lens is the central issue in the method of HRTEM. We show in Sect. 10.3.2 that contrast in high-resolution images originates primarily with the phase shifts of the electron wavefront as it passes through the specimen. The objective lens is therefore best understood as a device that alters the phase of the electron wavefront. To focus the wavefront, Figs. 2.34 and 10.4 show that the phases of the off-axis rays must be advanced with respect to the on-axis ray. The phase advance must be done with great precision if the phase-contrast image is to provide meaningful information. Conspiring against this precision is the positive third-order

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539

spherical aberration of magnetic lenses (Sect. 2.7.1). A positive coefficient of spherical aberration, Cs , means that rays at larger angles to the optic axis will focus closer to the lens (see Fig. 2.38). The closer focus means that these off-axis rays have undergone an excessive amount of phase advance by the lens. It is unfortunate that all short solenoid magnetic lenses have a positive Cs , especially when they have a large bore and pole-piece gap. It is possible, however, to compensate in part for the errors caused by spherical aberration by adjusting the focus of the lens. Doing so optimizes the range of angles for which entering rays suffer acceptable phase distortions. The larger this range of angles, the larger the usable range of Δk for electrons diffracted from the sample. High values of Δk correspond to small distances in real space, so the image has better spatial resolution. The compensation of spherical aberration by defocus is not perfect, however, because defocus and spherical aberration depend differently on Δk. Optimizing the compensation provides the resolution limit of the microscope, a limit that is achieved regularly when skilled microscopists examine good specimens on well-maintained instruments. Effect of Defocus. The electron is assumed to originate from a point on the optic axis, and is assumed to make small angles with respect to the optic axis. These assumptions are good because the region examined is very small, and the diffraction angles are small too. We first calculate errors in bending angle, ε, as a function of R, the radius at which the ray enters the lens. Figure 10.12 shows the geometry for the error, εa , in bending angle caused by defocus. From the figure, the angle θ′ is: R . (10.69) θ′ = b

Fig. 10.12. The error in bend angle, εa , caused by defocus, Δf , is proportional to R, where R is the distance along the radius of the thin lens.

The ratio of defocus error εa to the angle θ′ is the same as the ratio of the distance Δb to the distance b, so: Δb θ′ . b Substituting (10.69) in (10.70): εa =

(10.70)

Δb R . (10.71) b2 We need to express εa in terms of the actual defocus, Δf , at the specimen on the left side of the lens in Fig. 10.12. Recall the lens formula, (2.1): εa =

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10. High-Resolution TEM Imaging

1 1 1 = + . f a b

(10.72)

For small differences in the lengths a and b (here Δa < 0 and Δb > 0), the lens formula is: 1 1 1 = + , (10.73) f a + Δa b + Δb     1 1 Δa 1 Δb ≃ 1− + 1− , (10.74) f a a b b 1 Δa 1 Δb 1 ≃ − 2 + − 2 . (10.75) f a a b b Substituting (10.72) into (10.75), we obtain: Δb Δa ≃− 2 . b2 a We substitute (10.76) into (10.71) for our error in angle:

(10.76)

Δa R. (10.77) a2 The objective lens is operated for high magnification, so b ≫ a, and a ≃ f from (10.72). The distance, Δa, is the defocus, Δf , so (10.77) becomes: εa = −

εa = −

Δf R . f2

(10.78)

Fig. 10.13. The error in bend angle caused by spherical aberration, εs , is proportional to R3 (see text).

Effect of Third-Order Spherical Aberration. Figure 10.13 shows the geometry for the error in bending angle caused by spherical aberration, εs . A perfect lens would focus the off-axis rays along the solid line, but positive spherical aberration causes the ray to follow the path of the dashed line.11 From Fig. 10.13, the angles θ and εs are: R , (10.79) θ= a Δr εs = . (10.80) b 11

By comparing Figs. 10.12 and 10.13 we can see immediately how defocus can be used to compensate for spherical aberration, at least for the one ray path at R.

10.3 Experimental High-Resolution Imaging

541

The distance, Δr, is proportional to both the spherical aberration, through a factor Cs θ3 , and the magnification, which is b/a: b . a Substituting (10.81) into (10.80): Δr = Cs θ3

(10.81)

Cs θ3 b/a . (10.82) b Using (10.79) for θ, and the approximation at high magnification that a ≃ f , (10.82) becomes: εs =

εs = Cs

R3 . f4

(10.83)

Compensate Errors of Spherical Aberration by Defocus. The errors in bending angle caused by defocus, εa , and spherical aberration, εs , add to give a total error in bending angle, ε: ε = εs + εa ,

(10.84)

and we substitute for εs and εa from (10.78) and (10.83): ε = Cs

R R3 − Δf 2 . 4 f f

(10.85)

We will show that this error in angle of bend is proportional to an error in phase. First, however, note from Fig. 10.14 that for proper focusing, the lens must bend the ray by the angle θ +θ′ . Another ray arriving at the lens further from the optic axis at the distance R + dR must bend more if it is to come to focus. With spherical aberration, however, the ray at the position R + dR is bent a bit too much. This excess is shown as the angle ε in Fig. 10.14. The excessive amount of path length traveled by the ray, dS, over the distance dR is: dS = ε dR .

(10.86)

The error in phase, dW , contributed over the radius dR at R, is therefore:

2π ε dR . (10.87) λ The total error in phase is obtained by integrating the contributions, dW , over all R. To do the integral, we need a reference phase that serves as the lower limit of integration. We assign zero phase to the ray along the optic axis. Integration of (10.87) is then performed from the center of the lens to R:

R 2π W (R) = ε dR , (10.88) λ dW =

0

542

10. High-Resolution TEM Imaging

Fig. 10.14. Geometry of the excessive angle of bend, ε, for a lens with positive spherical aberration.

and using (10.85) as the integrand:   R2 1 2π 1 R4 W (R) = Cs 4 − Δf 2 . λ 4 f 2 f

(10.89)

For high magnification: θ≃

R , f

(10.90)

so:

 π  Cs θ4 − 2 Δf θ2 . (10.91) 2λ The phase shift error is a function of the diffraction vector, Δk, since Δk = 4πθB λ−1 = 2kθB (see Fig. 5.4). The θ in W (θ) corresponds to twice the Bragg angle, θB , so for small θ = Δk/k:    4 2  Δk Δk k Cs . (10.92) − 2Δf W (Δk) = 4 k k W (θ) =

An electron wavelet traveling parallel to k0 + Δk undergoes a phase shift of W (Δk) when it comes to focus in a TEM image. Consider first the hypothetical case when W = 0 for all Δk. This is ideal for atomic resolution imaging. For spatial scales larger than atomic separations, however, the amplitudes of all scattered waves add in phase with the forward beam, so the image is indistinguishable from the case where no scattering occurs. When W = 0, which is reasonably achievable at small Δk, diffraction contrast in the image is weak.12 It is therefore useful to enhance the diffraction contrast by using an objective aperture as in bright-field or dark-field imaging. 12

When the scattering is incoherent or inelastic (both can be parameterized as “absorption”), some image contrast is expected when W = 0, however.

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543

High-resolution TEM requires that Δk be as large as possible, so it is important to understand image contrast in realistic cases where W (Δk) is not small. The waves diffracted by the various Δk must have their phase multiplied by a phase transfer function of the objective lens, QPTF (Δk): QPTF (Δk) = e−iW(Δk) .

(10.93)

Since this function QPTF (Δk) is in k-space, and our specimen function qi (x, y) of (10.61) is in real space, we should either transform (10.93) into real space, or transform the specimen function qi (x, y) into k-space. Our interest is in how the lens alters the contrast from various periodicities of the sample, so we take the k-space approach. 10.3.2 ‡ Lenses and Specimens Lattice Fringe Imaging. A simple example shows how the phase transfer function of the objective lens, QPTF (Δk) of (10.93), affects a high resolution image. Here the electron wavefunction through the specimen is represented with only the forward beam and one diffracted beam. High-resolution imaging is phase coherent imaging, so we add the amplitudes of the two beams:   Δz i(k0 +g)·r iW(g) ik0 ·r iW(0) . (10.94) e e e +i ψtot = φ0 e ξg The phases exp(ik0 · r) and exp(i (k0 + g)·r) of the forward and diffracted beams, respectively, are altered by the QPTF (Δk) of the objective lens. These forward and diffracted beams have specific Δk, so the phase alterations by the lens are W (0) and W (g). Note that W (0) ≡ 0, so exp(iW (0)) = 1 for the forward beam. The constant prefactor of the diffracted beam, iφ0 Δz/ξg , is derived in Chapter 11. It includes the incident wave amplitude, φ0 , times the scattering strength of an increment of material. The scattering strength naturally depends on the ratio of thickness, Δz, to extinction length, ξg , and we assume Δz ≪ ξg . The intensity of the electron wavefunction at the image ∗ ψtot : is, as usual, ψtot   Δz −i(k0 +g)·r −iW(g) ∗ −ik0 ·r e e Itot = φ0 e −i ξg   Δz i(k0 +g)·r iW(g) ik0 ·r × φ0 e , (10.95) +i e e ξg  Δz ig·r iW(g) 2 Itot = |φ0 | 1 + i e e ξg  Δz −ig·r −iW(g)  Δz 2 . (10.96) e e + −i ξg ξg We have already assumed that the sample is very thin and the scattering is weak. The last term in (10.96), which is of second order in the scattering, can therefore be neglected:

544

10. High-Resolution TEM Imaging 2

Itot = |φ0 |

2

    2Δz , sin g·r + W (g) 1− ξg 2

Itot = |φ0 | − |φ0 |

   2Δz sin(g·r) cos W (g) ξg    + cos(g·r) sin W (g) .

(10.97)

(10.98)

The larger first term in (10.98) is from the forward beam. The second term is proportional to the scattering, 1/ξg , and predicts contrast known as “lattice fringes.” These fringes lie perpendicular to g, and have a periodicity 2π/g. Both the sin(g·r) and cos(g·r) terms provide fringes of the same periodicity, but with displaced positions on the image. The precise position of the observed fringes depends on the phase error, W (g), for the diffracted beam. For an image obtained with no defocus (Δf = 0), and a small g for the diffracted beam (small Δk), from (10.92) this phase error is expected to be near zero, so the sin(g·r) term in braces in (10.98) would dominate. On the other hand, as discussed below, the best resolution of the microscope is often obtained when W (g) is approximately −π/2, so the cos(g·r) term often dominates in a high-resolution image. When only one set of fringes is visible in an image, it is rarely important to know exactly where the fringes are positioned. On the other hand, an image showing only one set of fringes is not very informative about the atomic structure of the sample, since this information can be obtained from a diffraction pattern (at least when the crystal is large). A more substantial HRTEM research project may seek the interface structure when two crystals are in physical contact with near-atomic registry. Suppose it is possible to obtain lattice fringe images from both crystals, and suppose further that the fringes from both crystals touch each other. It might be tempting to claim from inspection of the image that the atomic planes are in alignment across the interface. Such an interpretation could be na¨ıve, however. The phase errors caused by the objective lens, W (g), may not be the same for both sets of lattice fringes. Any difference would affect the weights of the cos(W (g)) and sin(W (g)) terms in (10.98), so the fringes from the two crystals could be shifted differently. To obtain reliable information about the structure of the interface, further analysis of the image is generally required. A structural image is a high-resolution image made with several diffracted beams. Intersecting sets of fringes are obtained in structural images, producing sets of black or white dots, as in Figs. 2.3, 2.23, 2.26 and 2.27. The phase error is generally different for each diffraction used in the image, however, owing to differences in how W (Δk) depends on the order of the diffraction and on the defocus. It is not obvious, for example, if columns of atoms should appear as white or black dots, and this appearance can change with the defocus of the objective lens and the thickness of the sample.

10.3 Experimental High-Resolution Imaging

545

Weak Phase Object Approximation. The physical issues of high-resolution imaging can be understood better by considering a real-space phase function of the sample, qi (x, y) of (10.61), which also includes absorption. To understand how the specimen interacts with the phase transfer function of the objective lens, QPTF (Δk) of (10.93), we first take the Fourier transform of (10.61):   (10.99) Qi (Δkx , Δky ) = F e−iσ φ(x,y) e−μ(x,y) .

The weak phase object (WPO) approximation assumes that the specimen is very thin, so σ φ(x, y) and μ(x, y) are very small. The exponentials in (10.99) can therefore be linearized:    Qi (Δkx , Δky ) = F 1 − i σ φ(x, y) 1 − μ(x, y) . (10.100) Likewise we can neglect the small second-order product i σ φ(x, y) μ(x, y), so Qi (Δkx , Δky ) = F [1 − i σ φ(x, y) − μ(x, y)] .

(10.101)

Qi (Δkx , Δky ) = δ(Δkx , Δky ) − F [μ(x, y)] − iσF [φ(x, y)] .

(10.102)

Fourier transforms are distributive, and F [1] = δ(Δkx , Δky ), so This k-space representation of the phase of the electron wavefunction through the sample now can be multiplied conveniently by (10.92), the phase transfer function of the objective lens. This gives the “phase transfer modified” electron wavefunction, Q′i (Δkx , Δky ):   Q′i (Δkx , Δky ) = δ(Δkx , Δky ) − F [μ(x, y)] − iσF [φ(x, y)] × eiW(Δkx ,Δky ) .

(10.103)

The important quantity for image formation is of course the intensity. The intensity in a real-space image is qi′∗ (x, y) qi′ (x, y). Calculating the complementary k-space intensity function, Itot (Δkx , Δky ), requires a convolution in the Fourier transform representation of the wavefunction and its complex ′ conjugate. We calculate Q∗′ i (Δkx , Δky ) ∗ Qi (Δkx , Δky ):  Itot (Δkx , Δky ) = δ ∗ (Δkx , Δky ) − F ∗ [μ(x, y)]  +iσF ∗ [φ(x, y)] e−iW(Δkx ,Δky )  ∗ δ(Δkx , Δky ) − F [μ(x, y)]  −iσF [φ(x, y)] eiW(Δkx ,Δky ) . (10.104)

Equation (10.104) includes nine convolutions. Again, however, for thin samples the convolutions, F ∗ [μ(x, y)] ∗ F [μ(x, y)], F ∗ [μ(x, y)] ∗ σF [φ(x, y)], σF ∗ [φ(x, y)] ∗ F [μ(x, y)], and σ 2 F ∗ [φ(x, y)] ∗ F [φ(x, y)], are of second order and are neglected. The remaining five convolutions involve delta functions, and can be performed by inspection13 of (10.104): 13

Note that δ(Δkx , Δky ) e−iW(Δkx ,Δky ) = δ(Δkx , Δky ) e−iW(0,0) = δ(Δkx , Δky ).

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10. High-Resolution TEM Imaging

Itot (Δkx , Δky ) = δ(Δkx , Δky ) −F ∗ [μ(x, y)] e−iW(Δkx ,Δky ) − F [μ(x, y)] eiW(Δkx ,Δky )

+iσF ∗ [φ(x, y)] e−iW(Δkx ,Δky ) − iσF [φ(x, y)] eiW(Δkx ,Δky ) . (10.105)

When the crystal is centro-symmetric, we can set F ∗ [φ(x, y)] = F [φ(x, y)] and F ∗ [μ(x, y)] = F [μ(x, y)], so: Itot (Δkx , Δky ) = δ(Δkx , Δky ) − 2F [μ(x, y)] cos(W (Δkx , Δky ))

+ 2σF [φ(x, y)] sin(W (Δkx , Δky )) , (10.106)

and Itot is real. The large first term in (10.106) is the forward beam, peaked at Δk = 0. The second term is the amplitude contrast term, which depends on the absorption of the sample. The third term involves the phase shift of the electron wavefront, as provided by the projected potential (cf., (10.68)). Again, as in (10.98) for the two-beam case, the intensity depends on details of the phase error, W (Δkx , Δky ) of (10.92). Real lens characteristics are presented in the next section, but it is often assumed that these characteristics can be loosely approximated as W (Δkx , Δky ) = −π/2, so cos(W (Δkx , Δky )) = 0 and sin(W (Δkx , Δky )) = −1. In many cases absorption is small, also suppressing the second term in (10.106). Equation (10.106) can then be approximated as: Itot (Δkx , Δky ) ≃ δ(Δkx , Δky ) − 2σF [φ(x, y)] .

(10.107)

The contrast in the image is therefore approximated as originating from the phase shift of the electron wavefunction through the sample (as caused by (10.68), for example). We say that the specimen is a “weak phase object,” or WPO. This approximation is handy for explaining the origin of contrast in a high-resolution image. Unfortunately, however, samples are rarely thin enough for this approximation to be valid, and the characteristics of the objective lens cannot be approximated reliably as W (Δkx , Δky ) = −π/2. 10.3.3 Lens Characteristics Lens Phase Errors. Figure 10.15 shows the phase shift error of a particular microscope, W (Δk) of (10.92), for various values of defocus, Δf . The forwardscattered beam at Δk = 0 has a reference phase of 0 for all curves. For the smallest values of defocus, there is a rather small error in phase for Δk below about 10 nm−1 , corresponding to a real-space distance of 2π/Δk = 0.6 nm. Most samples have many features larger than 0.6 nm. For small values of defocus, these larger features show little contrast in the image, since electrons scattered at small Δk are recombined accurately in phase with the forward beam.14 This fact is useful for adjusting the microscope for peak performance. 14

So although bright-field and dark-field images have resolution limitations owing to the finite size of the objective aperture, these conventional methods are a good choice for making images of features larger than those on the atomic scale.

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547

As the focus is adjusted precisely, the image should “go out of contrast,” where many features disappear.15

2

80

60

40

0

defocus (nm)

20

+/ +//2

100

phase 0

φcrit . (Unfortunately, the spatial resolution of spectroscopic methods such as EELS are more seriously impaired when α > φcrit .) Nevertheless, a large α offers other advantages. Section 11.7.1 shows that three-dimensional imaging is possible with large α. In essence, with large α the depth of field becomes small, offering the possibility of vertical resolution on the order of nm. 11.2.3 * Tunneling Between Columns

From (11.10) we can understand another phenomenon that limits the quality of electron channeling down atomic columns – tunneling of the electron between adjacent atom columns. Tunneling will be severe for the critical glancing angle of Fig. 11.3, because the electron wavefunction is a constant through the interstitial region and therefore appears with full amplitude in the next column of atoms. Tunneling is suppressed if the glancing angle φ is smaller, but it will occur even when φ = 0. When φ = 0, it is convenient to rebalance the potential between U0 and U (x) so that U (x) = 0 in the atom columns, consistent with no x-component of the wavevector in this region. In the interstitial region, U (x) then becomes a positive potential, presenting a barrier that helps confine the electron wavefunction to the atom columns. The barrier penetration problem has the solution of (11.10), but with positive U (x) the square root is an imaginary number, cancelling the i in the exponent. A damped wavefunction, the standard solution for barrier penetration problems in quantum mechanics, is found: 4

See Fig. 2.8 for the definition of α. For the STEM mode of operation, the ray paths are from right to left in Fig. 2.8, forming a smaller probe beam when α is larger.

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√ 1 2 ψx (x) = √ e− 2mU/ L

x

1 = √ e−x/x . L

(11.13)

There is negligible tunneling when the columns are far apart (large x), or separated by a large potential barrier (large U ) in the interstitial region. Nevertheless when U = +10 eV, the characteristic tunneling length, x = 0.6 ˚ A. For a typical separation of 1 ˚ A to the next atomic column, the fraction of tunneled amplitude at the next column is e−1.0/0.6 = 0.19. We next calculate the frequency of quantum mechanical tunneling of the high-energy electron between atom columns, and then find the distance the electron travels down one column by dividing its speed by the tunneling frequency. The key parameter is the transition matrix element 2|U |1 , where the states 1 and 2 have the forms of φx (x) φz (z), where φx (x) and φz (z) are solutions to the Schrdinger equation in the absence of tunneling. The φx (x) has the form of (11.13) in the interstitial region. There is another step in the analysis, however, which is required because the wavefunctions in the different atomic columns have exactly the same energy. This step produces two solutions that mix the wavefunctions in the two columns, but the two new solutions have constant amplitudes throughout the crystal.5 The solutions differ in energy by ±2|U |1 . This integral is the overlap of the tails of the wavefunctions in the region of the interstitial potential, U :

a −x/x e−(a−x)/x e √ √ U dx , (11.14) 2|U |1 = a a 0

a 1 (11.15) e−a/x U dx , = a 0 = U a e−a/x .

(11.16)

In (11.14) we overlapped two exponential tails that penetrate into each other. Note that one is offset by the width of the interstitial region a. The integration is over the width of the interstitial region, 0 to a, where the potential is U . Using the typical numbers following (11.13), we find 2|U |1 = 1.9 eV. The characteristic frequency of tunneling ω = 2 × 1.9 eV/, and the characteristic distance of tunneling for a fast 200 keV electron is 300 ˚ A. The precise results are exponentially sensitive to intercolumn distances, and we worked only a one-dimensional example. Nevertheless, we can now see why channeling is reasonably effective in confining electrons to a column of atoms, facilitating the resolution of atomic columns in HAADF imaging. 5

The analysis is known as “first-order degenerate perturbation theory.” It is used for the Bloch waves of Chapter 12, which are combinations of diffracted beams. It is also used in physical chemistry for bonding and antibonding orbitals that differ in energy by ±E, where E is the matrix element coupling the two atomic states.

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11.3 Scattering of Channeled Electrons 11.3.1 Elastic Scattering of Channeled Electrons High-energy electrons that travel down atom columns have wavefunctions with many periods along the z-direction. The electron wavelengths are much smaller than the distance between atoms in the column, but the wavefunctions nevertheless lock in to the crystal periodicity. Bloch waves are the electron eigenfunctions in a periodic crystal, and these tend to maximize or minimize their overlap with the atom cores as discussed later (cf., Fig. 12.11). Disruptions in the atom periodicity cause scattering from these Bloch wave states. These disruptions in periodicity were discussed in Chapter 9 as displacement disorder and chemical disorder. By definition, high-angle annular dark-field imaging involves large scattering angles and large Δk. Chemical disorder is negligible because its contribution decreases with Δk. On the other hand, Sect. 9.2.1 showed that diffuse scattering from displacement disorder 2 2 increases with Δk as [1 − e−(Δk) x  ]. Differences in atomic size disorder can make a contribution to the HAADF image, and HAADF contrast could perhaps be used to measure this type of disorder. In the present section, however, we discuss the thermal contribution to the high-angle electron scattering. A systematic way to account for thermal vibrations of all atoms in a crystal is with a phonon analysis, because phonon modes act as independent degrees of freedom in crystalline solids. The energy and atom displacements can be analyzed independently for each phonon of angular frequency ω. A full phonon analysis is a complex undertaking, but the thermal energy of a moving atom, Etherm , can be as simple as: 1 (11.17) Etherm = M ω 2 u2 , 2 where the atom mass is M , and its mean-squared displacement is u2 .6 Phonons are quantized with energy intervals of ω, so the number of phonons in a phonon mode is simply n: Ethermal . (11.18) n= ω Suppose that one atom of mass M takes up the recoil of high-angle scattering,7 which has momentum p = Δk. The recoil energy is: 2 (Δk)2 , (11.19) 2M From (11.17) and (11.19) we can obtain the quantity (Δk)2 u2 , which is central to the Debye–Waller factor, D(Δk) = exp(−(Δk)2 u2 ), of (9.59).8 Erecoil =

6

7 8

p Substituting the elementary result for a harmonic oscillator, ω = k/M , gives the potential energy of a fully compressed spring, Etherm = 1/2 k u2 . This supposition implies an Einstein model of atom vibrations in a crystal. Recall that the coherent (Bragg) scattering is diminished by the factor D(Δk), whereas the thermal diffuse scattering grows as 1 − D(Δk).

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By obtaining u2 from (11.17) with (11.18), and (Δk)2 from (11.19), we obtain the relationship: (Δk)2 u2 = 4

Erecoil n. Ephonon

(11.20)

Consider some typical numbers for (11.20). For 200 keV electrons scattered at 2 100 mrad, using a typical u2 = 0.03 ˚ A , (Δk)2 u2 = 30. A typical Erecoil = 0.05 eV, and a typical Erecoil /Ephonon = 2. With (11.20) we find that n = 4. Although exceptions are indeed possible (especially with differences in atom mass and u2 ), the scattering of 200 keV electrons into high angles typically involves the creation or annihilation of multiple phonons. Another important result from this analysis is that when (Δk)2 u2 ≫ 1, the Debye–Waller factor, exp[−(Δk)2 u2 ] ≪ 1, suppresses strongly the coherent scattering, justifying the assumption of incoherent imaging in HAADF measurements. Incoherent imaging can be interpreted as scattering from individual atoms without phase relationships between them. The individual atoms do have their own form factors, however, and these must be considered when accounting for the intensity of the high-angle scattering. Combining this factor with the thermal diffuse intensity discussed 2 2 above, [1 − e−(Δk) x  ], the intensity of the high-angle incoherent scattering depends on Δk as:  2 2  (11.21) IHAADF = |fat (Δk)|2 1 − e−(Δk) x  .

The atomic form factor for electrons, fat , approaches the limit of Rutherford scattering at large Δk, given in (3.106): 2 2  4Z 2  1 − e−(Δk) x  . (11.22) IHAADF = 2 4 a0 Δk The thermal diffuse intensity, the factor in the square braces in (11.21), approaches 1 for large Δk: IHAADF ≃

4Z 2 . Δk 4

a20

(11.23)

The characteristic feature of the material probed by high-angle annular darkfield imaging is the atomic number, Z, and hence the name “Z-contrast imaging.” In summary, for accurate crystal orientations with a small incident probe beam, the attractive potential of atom cores channels the electrons along columns of atom cores. There is some tunneling probability to adjacent columns, depending on the variation of the crystal potential and the beam tilt. The scattering to high angles is caused largely by the displacement disorder of thermal diffuse scattering. This is an incoherent process that becomes independent of Δk in the limit of large Δk. The scattering is essentially elastic, however, with only small losses of energy to phonons. High-angle elastic

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scattering is in the limit of Rutherford scattering, which depends only on Z 2 , but decreases strongly with Δk. 11.3.2 * Inelastic Scattering of Channeled Electrons The inelastic form factor for core electron excitations, fin (Δk), was worked out in Sect. 4.4.2 for the case when the incident high-energy electron was a plane wave. In brief, conservation of energy and momentum required that the excitation of the atomic core electron and the change in direction and energy of the incident electron were treated as a coupled system. The interaction between the two electrons was through the Coulomb interaction, +e2 /|r1 − r 2 |. A key step between (4.26) and (4.27) was the substitution of r ≡ r1 − r 2 . This simplified the interaction to +e2 /r, which was Fourier-transformed to give the standard result 4π(Δk)−2 . The coordinates of the atomic electron were arranged to be unchanged, but their Fourier transformation occurred when the exponentials were transformed as r 1 = r+r 2 , so r 2 appeared in the exponential after substitution. This latter step can be arranged for the case of a channeled electron with coordinates r1 . The second integral in (4.27) is therefore recovered if the incident electron is a channeled one, so the shapes of core edges are similar if the incident electrons are channeled, or are plane waves. The Coulomb interaction has a long range, so the shapes of core edges should be preserved even if the channeled electron passes near the atom, and not through it. The question immediately arises as to the localization of excitations caused by channeled electrons. This is not a simple problem, in part because when the channeled electron passes some distance from the atom, other electrons can intervene, and their screening will reduce the strength of the interaction. Nevertheless, even without these additional interactions, the first integral in (4.27) is substantially different for channeled electrons than for plane waves. Consider the possible magnitudes of r ≡ r 1 − r 2 when the channeled electron number 1 misses the atomic electron 2. For this channeled electron away from the ionized atom, the distance r 1 −r2 does not go to zero, and there is a minimum value of r. The range of integration for the first integral in (4.27) therefore has a hole in it around r = 0, where the Coulomb potential is singular. The singularity of the Coulomb potential makes an important contribution to the first integral in (4.27). The result is that this integral does not evaluate to 4π(Δk)−2 , but to a smaller value (see Problem 11.5). The inelastic form factor therefore has a diminished amplitude when the channeled electron does not pass close to the atom being ionized. The experimental evidence is that the channeled electrons tend to ionize primarily the atoms in their own column. Delocalization effects appear to be small [11.4]. For core electron excitations, both EELS and EDS spectroscopies have spatial resolutions characteristic of the channeled electrons.

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11.4 * Comparison of HAADF and HRTEM Imaging In HAADF imaging the scattering from the sample is incoherent, but the incident beam is highly coherent. A coherent, bright source is necessary to form a narrow probe beam on the sample. This is an optical challenge analogous to the formation of a phase contrast image in HRTEM. A comparison between the two methods can be developed using (10.108). For HAADF imaging, we start with the electron probe amplitude P (x) on (or in) the specimen. This probe is small if the objective lens can work with off-axis rays of high angle, corresponding to short spatial periodicities. We modify (10.108) by adding an objective aperture to truncate some of the off-axis rays, multiplying the integrand by the function A(Δk) as in (10.120): P (x) =

∞

eiΔkx eiW(Δk) A(Δk) dΔk .

(11.24)

−∞

As for an ideal TEM, the ideal STEM instrument would have W(Δk) = 0 and A(Δk) = 1. The loss of resolution caused by the W(Δk) of (10.109) was discussed previously for coherent HRTEM imaging, as was a strategy to optimize resolution by defocusing the objective lens. In HAADF imaging, however, we need coherence to form the probe beam, but for incoherent scattering the important quantity is the intensity of the electron beam. Once the beam is formed, we can ignore the phase issues, such as the phase of P (x) in (11.24). We work instead with the intensity |P (x)|2 , whose Fourier transform is the actual lens transfer function T (Δk) for incoherent imaging. The product in real space, |P (x)|2 , corresponds to a convolution in k-space: T (Δk) =

∞

−∞





e−iW(Δk ) A(Δk ′ ) eiW(Δk −Δk) A(Δk ′ −Δk) dΔk ′ . (11.25)

An important difference between the coherent transfer function eiW(Δk) A(Δk), and the “incoherent” transfer function T (Δk) of (11.25), is that T (Δk) is a convolution of this coherent transfer function, so it extends over a wider range of k-space. The spatial resolution of incoherent imaging is therefore better than for coherent imaging, assuming equivalent lens characteristics. Another favorable characteristic of HAADF imaging compared to HRTEM imaging is that resolution in HAADF imaging is less disturbed by those microscope defects responsible for the damping of high frequencies in the coherent contrast transfer function (see Figs. 10.19 and 10.20). Figure 10.19 shows how the coherent addition of different CTFs causes a loss of amplitude at higher Δk. Notice that this suppression of amplitude occurs where the CTF oscillates in sign. Much of the damping of the CTF originates with fluctuations in lens currents or fluctuations in the high voltage of the microscope. These instabilities cause differences in phase at large Δk for individual electrons. Each electron, however, has an intensity that is positive definite, so

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595

these wave cancellation effects do not occur for the incoherent scattering of HAADF imaging. For incoherent scattering, the important quantity is the intensity of the electron distribution in the probe beam. The information limit for HAADF imaging is therefore larger than for coherent HRTEM imaging.9 The comparable resolution of HAADF and HRTEM images is expected from a general principle. There exists a theorem of reciprocity in optics that provides a useful comparison between a TEM and a STEM. Consider the STEM as a TEM operated backwards, with the source and the detector interchanged. For example, in a STEM the lens is before the specimen, and the ray paths from the lens to viewing screen in a TEM become the ray paths between the lens and the source of illumination in the STEM. At the STEM detector the outgoing electrons are essentially plane waves, in analogy to the incident plane waves from the source to the sample in a TEM. In its simplest form, the reciprocity theorem states that if we swap a point source of illumination between the TEM and the STEM configurations, the intensity at the detector will be the same. This is expected from a ray diagram by just tracing backwards along the ray paths from a point object to a point image.10 In STEM, the focusing of incident electrons into a small spot requires converging ray paths, in analogy to the requirement in TEM that rays are included that have undergone scattering to large Δk and contain information on short spatial periodicities. There is a difference between HAADF and HRTEM imaging modes as discussed above, because the resolution in incoherent HAADF imaging is less affected by phase errors induced by the objective lens.

11.5 HAADF Imaging with Atomic Resolution 11.5.1 * Effect of Defocus Changing the focus of the objective lens alters the intensity distribution of the probe on the specimen surface. The effect of defocus is illustrated by the series of simulated HAADF images for 110 Si in Fig. 11.4. Near Scherzer defocus, 9

10

Here is a more physical argument. Recall that the electron wavefunction interferes coherently with itself in making a HRTEM image, especially between the forward beam and the diffracted beams, requiring accurate phase relationships between all these beams. The image is made up of many such interferences between different electrons, so all electrons should have the same phase contrast conditions if the image is to have sharp detail. For HAADF imaging, the incoherent scattering depends on the presence of one real electron at a point in space, and the electron density is less sensitive to the fluctuations of electron phase caused by microscope instabilities. A more powerful reciprocity theorem exists, however. It equates the amplitudes (and hence phases) of the waves between interchanged source and detector, not just their intensities. This originates with the symmetry of incident and scattered wavefunctions in (3.73), for example.

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the central image at −700 ˚ A defocus, the probe most closely resembles a diffraction-limited Airy disk. At smaller defocus the central peak broadens. At larger defocus the central peak sharpens, but more intensity appears in subsidiary maxima. While these conditions result in significantly different images, the contrast away from Scherzer defocus is reduced substantially in both cases, so in practice the Scherzer defocus can be found readily. Also notice that although the expected Si dumbbells are not resolved in these 110 images, by knowing that atoms act as sharp objects, one can infer from the oval shape of the bright spots that at least two atomic columns must be present close together.

Fig. 11.4. Simulated defocus series for Si 110 with corresponding probe intensity profiles (100 kV, Cs = 1.3 mm, optimum objective aperture semiangle 10.3 mrad), giving a probe size of 0.22 nm at the optimum Scherzer defocus of −69.3 nm. After [11.2].)

A narrow probe beam depends on the coherent nature and large angular spread of the incident beam. This narrow lateral coherence can be understood for a wave along a forward direction, and a second wave with wavevector deviating by the angle α with respect to the first. In this case, coherence occurs over lateral dimensions of approximately λ/α.11 Although the formation of a narrow probe requires the convergence of waves of relatively large α (analogous to HRTEM imaging requiring diffractions of large Δk), in practice α is of order 0.01 radian. Lateral coherence with high-energy electrons is therefore only about 0.1 nm, so the incident electron wavefunction has an appropriate width to channel along atomic columns. The results of Fig. 11.4 also hold for thicker specimens. More complete dynamical calculations of electron propagation through specimens have shown that in a zone-axis orientation, the STEM probe forms strong peaks on the atomic columns with a width of about 0.1 nm, even as it propagates deeper into the specimen. Channeling 11

This can be derived by recognizing that α = Δk/k, and substituting Δk = 2πα/λ into l Δk = π of (9.159).

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597

was described with a simple model in Sect. 11.2, and HAADF contrast was described as originating from Rutherford scattering. This contrast continues until significant absorption occurs and ultimately, in very thick crystals, there is no longer a high-resolution image. 11.5.2 Experimental Examples An example of the compositional sensitivity of HAADF imaging is shown in Fig. 11.5, which compares 110 experimental and simulated images of interfacial ordering in a (Si4 Ge8 )24 superlattice grown on Ge 001 . In this image, a different ordered arrangement is seen at each interface: 2 × n interfacial ordering at the top Si-on-Ge interface, a {111} planar structure in the central Si layer with Ge threading through to the next Ge layer, and cross-like structures in the lowest Si layer. It is also apparent that much Ge is present in the Si layers, while there is little Si in the Ge layers, and the Si-on-Ge interfaces are generally much broader than the Ge-on-Si interfaces. These features are inconsistent with strain-induced interdiffusion and suggest that the chemical mixing was a result of the growth process itself.

Fig. 11.5. 110 experimental and simulated images of interfacial ordering in a (Si4 Ge8 )24 superlattice grown on Ge 001 at 350◦ C, with interpretation of the structure based on an atom pump model. Shaded circles represent alloy columns, solid circles Si, and open circles Ge. After [11.1].

Figure 11.6 is a HAADF image that resolves individual dopant atoms inside a silicon crystal. As silicon-based electronic devices become smaller, doping must provide higher densities of charge carriers. At higher dopant concentrations the dopant atoms form nanoclusters, suppressing the number of available carriers. The sample of antimony-doped silicon used for Fig. 11.6

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Fig. 11.6. HAADF image of a cross-section through an interface between zones of Sb-doped Si (left), and undoped Si (right). The bright dots on the left are from atomic columns containing one or more Sb atoms. The undoped region on the right contains no bright columns. The image was formed with 200 keV electrons, a 10 mrad half-angle for probe convergence, and a 50 mrad inner angle for the annular detector. Probe size was ∼0.15 nm. The image has been smoothed, and a slowlyvarying background was subtracted. Image width is 12 nm. Image provided by P. M. Voyles and D. A. Muller, after [11.5].

was very thin, < 5 nm, very smooth, and without surface oxide. With so few atoms in a column through the sample, it is possible for an individual Sb atom with Z = 51 to cause a change of 25 % or more in the intensity of a column of Si atoms (for Si, Z = 14). Figure 11.6 therefore shows bright spots from columns containing Sb atoms. These are distinctly more numerous in the region at left that was doped with Sb. The researchers showed that the nanocluster responsible for the reduced number of carriers consists of SbSb pairs of atoms, perhaps involving a vacant Si site. The fraction of these pairs was shown to account for the fractional decrease in the effective carrier concentration.

11.6 * Lens Aberrations and Their Corrections

599

11.6 * Lens Aberrations and Their Corrections 11.6.1 Cs Correction with Magnetic Hexapoles The positive spherical aberration of an objective lens was discussed in Sect. 2.7.1. Its deleterious effects on high-resolution images were explained in the context of HRTEM imaging in Sects. 2.8 and 10.3.3, but (11.1) shows that spatial resolution in STEM mode is similarly impaired by Cs . Recall that the positive sign of Cs means that off-axis rays converge excessively compared to paraxial rays, so the off-axis rays come to focus closer to the magnetic lens. It turns out that it is impossible to eliminate the spherical aberration of a magnetic lens built as a short solenoid. One reason is that the off-axis electrons spend a bit more time in the lens, and have larger deflections during this time. Other problems come from non-ideal magnetic field distributions in short solenoids, such as a higher Bz away from the optic axis. Small, compact lenses with high magnetic fields have proved effective in minimizing Cs , although not eliminating it. In a significant and recent development in TEM instrumentation, it is now possible to eliminate Cs . This is done with a “Cs corrector” system in series with the objective lens. This device adds extra divergence to the offaxis rays, compensating for the excessive convergence of off-axis rays caused by the spherical aberration of the objective lens. One type of Cs corrector system, described here, uses two hexapole magnetic lenses. The fields and forces on electrons in one hexapole lens are depicted in Fig. 11.7. The electron trajectories travel down through the plane of the paper from above, and experience the Lorentz forces, F mag = −e v × B, where we consider B to be in the plane of the paper. By symmetry, the fields and forces are zero for electrons that pass through the exact center of the lens, and the forces increases with radius, r. At the smallest r, the forces are dominated by terms having the lowest power of r allowed by symmetry. The linear term is disallowed since the fields and forces along the important horizontal direction in Fig. 11.7 do not change sign across the center of the lens. The quadratic term is allowed, so F increases in proportion to r2 , at least for small r. The distance over which the electrons accelerate under this force, the deflection through the lens, therefore scales with r2 .12 The direction of this deflection is shown in Fig. 11.7b. Figures 11.7b and 11.7c show a lateral cut of electron entry points in the hexapole field. At the center of the lens, the field, force, and deflection are all zero. Importantly, for electron trajectories to either the left or right sides of the lens center, all deflections are to the left. This reflects the symmetry of the hexapole, where North and South poles are diametrically opposed. The 12

Recall that distance, d = 1/2at2 and a = F/m, where a is acceleration and t is time allowed for acceleration. The time, t, is assumed the same for paths through the lens, or at least for all positions along a line like the one selected in Fig. 11.7b.

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Fig. 11.7. (a) Magnetic poles and field directions for a magnetic hexapole lens. Thicker lines denote higher flux density. (b) Forces on an electron traveling down through the plane of the paper from above. The larger arrowheads denote larger forces, and the dots at the ends of the arrowheads indicate the deflections of the electrons that entered the lens on the dashed circles. (c) A set of deflections for electron trajectories across the center of the hexapole lens, identified with a rectangular box in part (b). Note that all deflections are to the left, but are larger further from the center of the lens.

leftwards deflections are larger away from the center of the lens, as depicted by the solid curve in Fig. 11.7c. Deflecting off-axis rays in the same direction is interesting, but this does not give the divergence needed for a Cs corrector. In addition to the hexapole field pattern, the Cs corrector requires a “long” hexapole. With reference to Fig. 11.7b, we see that the leftmost electron moves into a region of stronger field, whereas the rightmost electron moves into a region of weaker field. Assuming the hexapole field has the same pattern some distance below the plane of the paper, the leftmost electron path will experience an increasingly larger leftwards deflection as it moves through the lens. The rightmost electron will be deflected less because it moves into a weaker field. These deflections are depicted in the dashed line of Fig. 11.7c. This gives divergence. Electron paths to the left and right of center of the lens will both be bent to the left, but the leftmost electron will be bent further, giving a divergence between left and right rays. Those electrons further from the optic axis will diverge more. This is just what is needed to compensate for the excessive convergence of the objective lens from its positive Cs . Although this single hexapole has a negative Cs , it causes some distortions itself. Notice in Fig. 11.7b that there is a 3-fold distortion to the electron trajectories. Electron paths bunch up at 1, 5, and 9 o’clock around the center. Correction for this requires a second hexapole lens, essentially identical to the first, but operated out of phase. If the two lenses were adjacent, interchange of the North and South poles would successfully cancel this 3-fold distortion. Practical Cs corrector systems use a transfer lens system comprising two conventional lenses as shown in Fig. 11.8. The transfer lenses in the center serve to project the output from the first hexapole onto the input of the

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601

second, with one inversion. With the hexapoles inactive, the dashed lines show how the two transfer lenses transform two parallel rays on the left into the two parallel rays on the right. With the hexapole lenses active, the overall divergence from the Cs corrector system is seen in the two solid rays emerging to the right of the system. With the inversion provided by the transfer lenses, the 3-fold distortion is cancelled when the two hexapoles are identical. In practice, additional lenses are used in the Cs corrector system. If the system is for microbeam formation in STEM mode, and is installed between the illumination system and the objective lens, a lens is needed to couple the illumination system to the first hexapole, and another lens projects the output of the second hexapole into the objective lens.

Fig. 11.8. Ray paths through a dual-hexapole Cs corrector system. Dashed lines are the reference for parallel rays with the hexapoles off. The divergence from the hexapoles is shown in the two solid rays from left to right.

An alternative optical design for a Cs corrector system, which may have advantages for STEM work, is based on a set of quadrupole, octupole, and quadrupole lenses [11.6]. Like the twin hexapole design, this concept for a Cs corrector system is not new. Early efforts to build such systems were unsuccessful because of the complexity of optimizing all the free parameters of the system.13 With computer control and computer image recognition, however, it is now possible to optimize such complex systems. Any efficient and robust tuning of a Cs corrector system needs a method for characterizing the lens aberrations, and a method for using this information to do the corrections. There are presently two approaches to identifying the aberrations. The first approach, well-suited for HRTEM imaging, makes use of the diffraction information shown in the ring-like patterns below each phase contrast image in Fig. 10.29. The images of Fig. 10.29 were acquired with the incident beam aligned along the optic axis, minimizing the effects of lens aberrations. By tilting the beam off the optic axis, lens aberrations become apparent in the Fourier transforms of phase contrast images. A set of such patterns, acquired from phase contrast images for various beam tilts 13

With a bare minimum of 6 lenses, the multiple degrees of freedom of the hexapole lens currents, and a high degree of accuracy needed for compensating mechanical misalignments, selecting the optimal lens currents for the Cs corrector of Fig. 11.8 is not simple.

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around the optic axis, is called a “Zemlin tableau.” The distortions of the circular patterns in the Zemlin tableau are useful for characterizing the aberrations [11.7]. The second method for measuring aberrations involves measuring “Ronchigrams” that are acquired in the mode for convergent beam electron diffraction. Unlike the CBED pattern of Fig. 2.21, a Ronchigram is made using a large aperture angle, so the diffraction disks overlap considerably (imagine enlarging α in Fig. 6.34 by a factor of ten, for example). Phase interferences of the different beams occur in the Ronchigram, and these vary with focus. A set of Ronchigrams acquired at different focus settings can characterize the lens aberrations [11.8]. Methods for converting the information in a Zemlin tableau or a set of Ronchigrams into lens currents for correcting lens aberrations require software algorithms far beyond the scope of this book. It may prove true that the physical lens configurations of Cs correctors are secondary in importance to the robustness of the software algorithms for their operation. The commercial products of today work impressively well, essentially reducing Cs to zero, and even allowing it to go negative. Point-to-point spatial resolution has fallen to below 1 ˚ A, and continues to decrease. Now that the third-order spherical aberration can be eliminated from the objective lens, higher-order lens aberrations are the subject of attention. A new era of atomic-resolution imaging has arrived. 11.6.2 ‡ Higher-Order Aberrations and Instabilities The availability of Cs corrector systems has increased the attention on all lens aberrations in TEM instrumentation. We continue the discussion of the phase transfer function of the objective lens, QPTF (Δk), presented previously as (10.93): QPTF (Δk) = e−iW(Δk) .

(11.26)

In general, the distortion of paraxial rays depends on Δk, which sets their angle from the optic axis, and their azimuthal angle, or the angle around the optic axis. A circularly-symmmetric lens would not have any azimuthal dependence of W , but a general treatment of lens distortions allows for phase distortions that depend on 2πm(φ − φij ), where m is an integer and 2πm determines the rotational (azimuthal) symmetry of the aberration. The angle φ is the azimuthal angle around the optic axis, and each aberration, designated with i and j, has an angular offset φij . A general expression for W (Δk) is a sum of many types of aberrations: W (Δk) =

j+1 ∞

 2π   Cjm (λ Δk)j+1 cos 2π m(φ − φmj ) . λ m=0 j=1 j + 1

Some important points about (11.27) are:

(11.27)

11.6 * Lens Aberrations and Their Corrections

603

• For larger j, and hence a larger power-law dependence on Δk, there are more types of terms with different rotational symmetries set by m. • All terms in (11.27) are real, so there is no absorption caused by the lens (QPTF does not have an exponential decay with Δk). • All constants Cjm have dimensions of length so that (11.27) is dimensionless. Here λ is the wavelength of the incident electron. • The rotational offsets, φmj , could be different for each term in (11.27). • Some Cjm are identically zero because their rotational symmetry, denoted by k, is inconsistent with the (j + 1)-power of Δk in (11.27).14 All higher-order terms in (11.27) are aberrations with their own recognizable characteristics. We have already encountered C10 and C30 , which are the defocus, and the third-order spherical aberration coefficient. Note that these two aberrations have circular symmetry (their second subscript is zero), and they depend on Δk as (Δk)2 and (Δk)4 , respectively, as determined by their first subscript. A more complete listing of aberrations is presented in Table 11.1. The second column provides a notation that is arguably more compatible with the names in the last column. The full W (Δk) is a sum of all these contributions as in (11.27). Nevertheless, if there were only one nonzero Cjm that limited W (Δk) to π/4 to Δk = 7 nm−1 , the length of what this Cjm would be is listed in the fifth column for 200 keV electrons [11.7]. Table 11.1. Axial Aberration Coefficients Azimuthal Symmetry 1 2 3 4 5 6 ∞ ∞ ∞

Δk = 7 nm−1 limit

A0 A1 A2 A3 A4 A5 C1 C3 C5

Power of Δk 1 2 3 4 5 6 2 4 6

B2 B4 S3 S5 D4 D5

3 5 4 6 5 6

1 1 2 2 3 4

58 nm 1.9 nm 3.3 μm

Coeff.

Notation

C01 C12 C23 C34 C45 C56 C10 C30 C50 C21 C41 C32 C52 C43 C54 14

2.0 nm 0.17 μm 13 μm 0.94 nm 64 nm 2.0 nm 13 μm 64 nm

1.9 nm

Name image shift twofold astigmatism threefold astigmatism fourfold astigmatism fivefold astigmatism sixfold astigmatism defocus spherical aberration 5th order spherical aberration axial coma axial coma axial star axial star 3-lobe aberration 4-lobe aberration

For example, a second-order spherical aberration coefficient C20 would be circularly-symmetric by m = 0, but this is inconsistent with a cubic dependence on Δk, where inversion across the optic axis requires a change in sign.

604

11. High-Resolution STEM Imaging

Now that the third-order spherical aberration C3 can be eliminated, the discussion and design of Cs corrector systems is focused on higher-order aberrations that degrade image quality. With the present directions of this work, it is possible that the aberrations C5 , A5 and D4 may soon be eliminated, or substantially minimized by good design of the corrector system [11.9]. Even if these are not eliminated, techniques may be developed to minimize their impact on phase distortions. For example, with a corrector to eliminate C3 , it may be possible to balance the residual C5 against C3 to minimize phase distortions. Minimizing coma (which causes streaking sometimes in the shape of a comet) is generally accomplished by careful microscope alignment. There are criteria for quantifying the effects of coma, but alignment procedures are instrument-specific. Considerable effort is being devoted to extending the information limit of the instrument. Improvements in the stability of the high-voltage systems and lens currents are underway, for example. Reducing the effects of chromatic aberration by monochromatization of the electron beam is a new development, and the energy spread of the incident electrons has decreased from approximately 1 eV to less than 0.1 eV.15 The mechanical stability of the sample stage is becoming more critical, and the sample region requires a better vacuum than in the past. These new capabilities bring new costs. Besides higher prices for instrumentation and service agreements, installing a modern TEM requires more attention than ever for minimizing vibrations, stray electrical and magnetic fields, and air turbulence around the instrument.

11.7 Examples of Cs -Corrected Images Materials science investigations with microscopes having Cs corrector systems are highly varied, and practices are evolving to take profit of the new instrumentation. In the instrument configuration for HRTEM, the Cs corrector system is located below the sample in the imaging lens system. Using the rules of Sect. 10.3.3 such as (10.110), if Cs were zero, we should select zero defocus for optimal resolution in HRTEM. High resolution is indeed possible this way, but contrast is minimized owing to the coherent recombination of all wavelets scattered from the specimen. This minimum contrast condition is quite sensitive to defocus, and techniques are evolving to use this to advantage by emphasizing contrast from planes at different heights in the sample. Another possible technique for HRTEM imaging is enabled by negative spherical aberration, which is possible with some Cs corrector systems. In this case all curves in Fig. 10.15 are inverted vertically, and positive defocus 15

This causes some loss of intensity, but energy monochromatization is of special interest for improving the energy resolution of EELS spectrometry.

11.7 Examples of Cs -Corrected Images

605

will optimize the spatial resolution. The phase shift error W (Δk) in (10.92) will change sign because both of its terms change sign. The W (Δk) for positive defocus and negative Cs will first increase with Δk in Fig. 10.15, then decrease. In principle, the Scherzer resolution is obtained the same way as before. Nevertheless, there is a difference in contrast that may be advantageous in imaging light atoms with small atomic form factors. First consider the change in contrast in the weak phase object approximation of Sect. 10.3.2. Changing the sign of W (Δk) will change the sign of the second term in (10.107). For a weak phase object, the intensity change along an atom column is reversed when Cs and defocus are both changed in sign. Although it is not obvious why this would enhance the image contrast, there is evidence that contrast is indeed enhanced [11.10]. This must involve nonlinear intensity responses that are not present in (12.114). A Cs corrector system installed between the electron gun and the sample is used for the STEM mode of operation, and this seems to be a particularly popular instrument configuration. Examples of materials science investigations using such instrumentation follow below. 11.7.1 Three-Dimensional Imaging A Cs corrector system allows for larger angles of illumination of the electron probe beam. The most compelling advantage of this large angle α is a probe beam of smaller diameter. (This is especially true for a high brightness electron source, but brightness is not a limiting quantity for the field emission guns that are used with Cs corrector systems.) Section 2.7.3 showed that the beam diameter decreases as α−1 . Lens aberrations set the maximum value of α, and minimizing them with a Cs corrector gives an immediate benefit to probe diameter and spatial resolution in the STEM mode of operation. Three-dimensional imaging is a new capability that is emerging as a byproduct of a large illumination angle α. Recall (Sect. 2.4.3) that the depth of field is the range of distances in the object that are imaged in focus. The depth of field decreases with α. The ray paths of Fig. 2.27 are useful for the STEM mode of operation if you imagine the electrons going backwards from the electron source at right to the sample at the left. The d at the right in this figure is set by characteristics of the electron gun. With larger aperture angle, the range of D1 at left becomes smaller as α−1 (cf., (2.12)). The depth of field contracts the electron beam vertically, and a further contraction is obtained as the beam is rescaled to smaller diameter. The net effect of the illumination angle on the vertical resolution, Δz, is: λ , (11.28) α2 where λ is the electron wavelength. Although (11.28) is a bit simplified, it is supported by more detailed analysis of the physical optics of electrons [11.11]. As a realistic example with a Cs corrector, for a 300 keV electron Δz ≃

606

11. High-Resolution STEM Imaging

with illumination angle α = 0.02 rad, (11.28) predicts a vertical resolution dz = 50 ˚ A. If future Cs corrector systems serve to minimize both C5 and C3 , it is expected that this vertical resolution may become of order 1 nm. Results from an early example of a 3D imaging experiment are shown in Fig. 11.9, which shows five Hf atoms in an SiO2 interlayer in a semiconductor device. The specimen is a good one for microbeam 3D imaging because Hf atoms scatter electrons much more strongly than the amorphous SiO2 in which they were embedded. A series of images was taken at 41 values of defocus, with focal increments of 5 ˚ A. These two-dimensional images were processed with volume rendering software to produce a three-dimensional model of the sample. Figure 11.9 shows only two views of this structure. Nevertheless, with this information it is clear that the Hf atoms are distributed randomly in volume of the amorphous SiO2 layer, and are not confined to interface positions, for example.

Fig. 11.9. Angular dark-field microscopy images with different defocus were used for making a three-dimensional reconstruction of an interface HfO2 /SiO2 /Si. Two views of the reconstruction are shown: plan view (left) and side view (right). Details of the HfO2 (left of images) and SiO2 (in the interface) have been replaced with uniform gray, but Si columns are shown. Five Hf atoms at the interface are seen as black rods. Vertical resolution of the Hf atoms is 0, • forward beam is mostly Bloch wave 2 – Bloch wave 2 is mostly forward beam. Bloch wave 2 is excited most strongly since φ0 = 1, φg = 0 at the top of the sample, • diffracted beam is mostly Bloch wave 1 – Bloch wave 1 is mostly diffracted beam. These rules are reversed for the −g diffraction.

12.7 Dynamical Diffraction Contrast from Crystal Defects

655

It is instructive to analyze the symmetrical case when s = 0, for which (12.129) shows that β = π/2. We can then use (12.143) with coefficients evaluated for β = π/2 to obtain the amount of forward and diffracted beams in the two Bloch waves: ⎤⎡ ⎤ ⎤⎡ ⎡ ⎤ ⎡ (1) φ0 (z) ψ (1) eiγ z 0 sin π4 cos π4 ⎦⎣ ⎦ , ⎦⎣ ⎣ ⎦=⎣ (12.176) (2) ψ (2) 0 eiγ z cos π4 −sin π4 φg (z) ⎡ ⎤ ⎡ ⎤ √ iγ (1) z (1) iγ (2) z (2) φ0 (z) e ψ + e ψ ⎣ ⎦= 2⎣ ⎦ , (12.177) 2 eiγ (1) z ψ (1) − eiγ (2) z ψ (2) φg (z) which confirms (12.78) and (12.79). A point to notice from (12.177) is that with either ψ (1) or ψ (2) alone, there exists both a diffracted beam and a forward beam, φ0 and φg . For the case 2 2 when ψ (2) alone is present, the intensities of the beams are |φ0 | = |φg | =  (2) 2 1/2 ψ  . Interestingly, without ψ (1) the beam intensities are independent of depth. We need both Bloch waves to have the beats that give depth dependence to the beams φ0 and φg . Absorption of Bloch wave 1 can therefore eliminate the beat pattern. Finally, we note that in Fig. 12.17 the two dots on the dispersion curves and the two dots on the top ends of the beam wavevectors are all four in vertical alignment. Equal values of kx ensure that all wave crests match in the x-direction, or in other words that the x-component of momentum is conserved across the crystal-vacuum interface. A similar condition holds for the y-component.

12.7 Dynamical Diffraction Contrast from Crystal Defects 12.7.1 Dynamical Diffraction Contrast Without Absorption The two-beam dynamical theory can explain many types of diffraction contrast in BF and DF TEM images. It is usually most reliable when s = 0 for only one diffracted beam, because this g-beam is then strong compared to the other diffracted beams. The two-beam dynamical theory is most reliable when the crystal has a thickness of several extinction distances. For very thin crystals (less than 0.1ξg ), or s ≫ 1/ξg , multibeam kinematical theory is sometimes more reliable than a dynamical theory that uses only two beams. For thicker crystals (of order 10ξg ), the coherence of the electron wavefunction is lost through the cumulative effects of incoherent scatterings, and “absorption” is required for understanding some features of the observed contrast. The present section treats diffraction contrast from defects using dynamical theory without absorption. The effects of absorption are discussed in Sect. 12.7.3.

656

12. Dynamical Theory

Consider a defect that runs from top to bottom of a thin crystal as shown in Fig. 12.18. The defect may be a linear defect (i.e., dislocation) or a planar defect (e.g., a stacking fault or interface), but we assume for the moment that the crystal below the defect is in the same orientation as the crystal above, but displaced slightly along the direction of the active diffraction. At the top of the crystal, the incident beam is resolved into two Bloch wave states “ψ (1) + ψ (2) .” Bloch wave states are in fact acceptable solutions below the defect too. The Bloch wavefunctions do not undergo any abrupt kink or major rearrangement immediately at the defect because this would be costly in energy. Below the defect, however, the atom positions are shifted with respect to the upper crystal, so the atom cores are shifted with respect to the electron densities in the two Bloch waves. The Bloch wave 1, which had its electron density on the atom cores (Fig. 12.11) is shifted off the atom cores below the defect. Figure 12.19 shows the special case of a defect with the displacement of +a/2 x, where there is a complete switch in roles of the two Bloch waves below the defect. This switch in position requires a switch in wavevectors, with Bloch wave 2 having the larger wavevector below the plane of the defect. The beat pattern of the Bloch waves is usually altered by the defect.

Fig. 12.18. Geometry of a defect running from top to bottom of a sample. Notation is described in the text. Transformation of the forward-scattered and diffracted waves into the coordinate system below the defect is denoted with a circular arrow. (The horizontal image width is the width of the defect projected onto the x-y plane.)

Consider the special case where the defect is precisely halfway down the sample, the defect displacement is exactly a/2 along the direction of the active diffraction, and s = 0. In this case, the two Bloch waves switch roles and wavevectors exactly, as in Fig. 12.19. Furthermore, an electron in one of these states travels exactly the same distance with both wavevectors k(1)

12.7 Dynamical Diffraction Contrast from Crystal Defects

657

(1)

^ (r) (2) ^ (r)

br

defect plane r'=r+br (2)

^ (r') (1) ^ (r')

Fig. 12.19. Defect displacement of atom columns by +a/2b x, causing the Bloch waves to switch roles below the plane of the defect.

and k(2) . The net effect is for the bottom half of the crystal to null the beat pattern from the top half. The beam condition at the top of the crystal is therefore recovered at the bottom of the sample – only the forward beam leaves the bottom of the crystal (no diffracted intensity). On the other hand, the region of crystal away from this defect has a diffraction intensity that depends on the sample thickness and effective extinction distance, ξeff . When the defect plane is above or below the center of the sample, the cancellation of beats is usually incomplete. Exceptions occur when the defect plane is offcenter by integral numbers of ξeff . Columns of crystal intersecting the defect at these depths differ by a full beat period, which means they have the same intensities in their diffracted beam (in this case, zero). For defects inclined to the plane of the specimen, the diffracted intensity varies with the depth of the defect, giving rise to wiggles in the image. The number of fringes equals the sample thickness in units of ξeff . A more general analysis of defect contrast requires a thorough analysis of the change in Bloch waves across the plane of the defect. We saw in Sect. 7.11.1 and Fig. 7.44 that the defect displacement is handled conveniently with a kink in the phase-amplitude diagram. This amounts to multiplying the beams by phase factors such as exp(iα), where α ≡ 2πg · δr (cf., Eq. 7.46). This operation is performed by resolving the Bloch waves into diffracted beams just above the defect, performing the phase shift on the diffracted beam across the defect by multiplying it by a factor like exp(iα), and then resolving the beams into a new set of Bloch waves below the defect. These new Bloch waves then propagate to the bottom of the specimen. Figure 12.18 illustrates all these steps. Along the middle and the right columns in Fig. 12.18, the electron probability at the bottom of the sample is resolved differently into forward and diffracted beams. The difference depends on whether the beat pattern of the two Bloch waves has produced a forward beam or a diffracted beam at the depth of the defect. When sg = 0, this depth oscillation occurs with a pe-

658

12. Dynamical Theory

riodicity of the extinction distance, ξg . The diffraction contrast, either in bright- or dark-field images, follows this periodicity, showing either maxima or minima along the length of the defect. The result is a series of wiggles in the diffraction contrast from the defect. The spacing of the wiggles is the horizontal projection of the vertical beat period of the Bloch waves along the line of the defect. The number of wiggles depends on the thickness of the sample, and for s = 0 their number increases with s. The wiggles could be fringes for a planar defect, or variations in width for an image of a dislocation. This effect is illustrated for several perfect dislocations in an Al-Ag-Mg alloy labeled A through D in Fig. 12.20 (dislocations E and F are out-of-contrast for this g-vector). Compare the contrast from these inclined defects with that illustrated schematically in Fig. 12.18.

Fig. 12.20. Several inclined dislocations, labeled A-D, with Burgers vector a/2 110 in an Al-AgMg alloy taken under two-beam BF conditions with g = 111. After [12.1].

We now recast the kink in the phase-amplitude diagram of Sect 7.11.1 into the mathematical form used in the next section on stacking fault contrast. The forward and diffracted beams above the defect (superscript “a”) are: 1 Φa0 (r) = √ φ0 (z) eik0 ·r , V 1 a Φg (r) = √ φg (z) ei(k0 +g)·r . V

(12.178) (12.179)

Below the defect, the diffracted beams are in a new coordinate system. If the atom positions below the defect are shifted by δr, the coordinate system also moves by δr. The forward and diffracted beams below the defect (superscript “b”) are:

12.7 Dynamical Diffraction Contrast from Crystal Defects

659

1 (12.180) Φb0 (r) = √ φ0 (z) eik0 ·(r+δr) , V 1 (12.181) Φbg (r) = √ φg (z) ei(k0 +g)·(r+δr) . V We rearrange the phase factors for the beams below the defect to obtain:  1  ik0 ·δr Φb0 (r) = √ e φ0 (z) eik0 ·r , (12.182) V  1  ik0 ·δr ig·δr e Φbg (r) = √ e φg (z) ei(k0 +g)·r . (12.183) V We incorporate these phase factors into the beam amplitudes as follows: φb0 (z) = eik0 ·δr φa0 (z) , φbg

(z) =

eik0 ·δr eig·δr φag

(12.184) (z) .

In vector form, (12.184) and (12.185) are: ⎤a ⎤⎡ ⎡ ⎤b ⎡ φ (z) 1 0 φ0 (z) 0 ⎦ . ⎦⎣ ⎦ = eik0 ·δr ⎣ ⎣ φg (z) 0 eig·δr φg (z)

(12.185)

(12.186)

The inverse transformation, which can be verified by multiplication, is: ⎤⎡ ⎡ ⎡ ⎤b ⎤a φ0 (z) 1 0 φ (z) 0 ⎣ ⎦⎣ ⎦ = e−ik0 ·δr ⎣ ⎦ . (12.187) 0 e−ig·δr φg (z) φg (z)

With (12.186), we can take the forward and diffracted beams above the defect, Φa0 (r) and Φag (r), and transform them into beams in the coordinate system below the defect, Φb0 (r) and Φbg (r). After this transformation, we can resolve the beams into Bloch waves below the defect and propagate the electron amplitudes to the bottom of the sample. The mathematical steps in the dynamical theory of stacking fault contrast are illustrated in Fig. 12.21. At three depths the electron is resolved into the beam representation – at the top, at the fault, and at the bottom of the sample. The phase shift across the fault is conveniently performed with beams as shown in (12.187). To reach the fault, the beams at the top of the sample are propagated by the matrix P (t1 ), and a similar matrix, P (t2 ), is used to propagate the beams from the fault to the bottom of the sample. The matrix P (t1 ) is obtained by resolving the beams into Bloch waves, and then into beams again after the distance t1 . The form of P (t) is obtained in the next section. Absorption makes important modifications to the present argument. We have assumed that the two Bloch waves propagate through the crystal without absorption, or that the absorption is equal for the two Bloch states. We know, however, that Bloch wave 1 is absorbed more strongly than Bloch wave 2. This alters the balance of the two Bloch waves with depth, and

660

12. Dynamical Theory

q0

Incident Beam at Top

0

Bloch Wave Propagation of Beams Through t1 Phase Shifts of above “a” Beams at Fault below “b”

_P ik0.br 1 0

e

Bloch Wave Propagation of Beams Through t 2 Exit Beams at Bottom

a

b

(t1 )

0 ig.br

e

_P q0 qg

a

br

b (t2)

q0 qg

a

Fig. 12.21. Flow chart for two-beam dynamical theory of stacking fault contrast. Flow is from top to bottom of the sample.

thereby alters the balance of the electron wavefunction between the forward and diffracted waves. Before we present a qualitative description of the effect of absorption on images of stacking faults, the next section develops the formal two-beam dynamical theory for stacking fault contrast. 12.7.2 ‡ * Two-Beam Dynamical Theory of Stacking Fault Contrast This section uses two-beam dynamical theory to calculate the diffraction contrast from a stacking fault. The goal is to find the amplitudes of the forward and diffracted beams at the bottom of a sample containing a stacking fault. This section follows the now classic approach of Hirsch, et al. [12.2], but with notation consistent with the previous Sect. 12.4. and Figs. 12.11 and 12.21. The first task is to build a tool, a “Bloch-wave propagator,” P (z), that takes the forward and diffracted beams through a region of perfect crystal of thickness z. The vector expression for the wavefunction in the beam representation is: ⎤ ⎡ φ0 (z) ⎦ . ⎣ (12.188) φg (z)

The matrix expression for resolving beams into Bloch waves was given in (12.145), which we write in concise form:

12.7 Dynamical Diffraction Contrast from Crystal Defects

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ (1) e−iγ z φ0 (z) ψ (1) 0 −1 ⎦C ⎣ ⎦ . ⎣ ⎦=⎣ (2) 0 e−iγ z φg (z) ψ (2)

661

(12.189)

Similarly, the expression for resolving Bloch waves into beams was given in (12.75) and (12.142), which we write concisely as: ⎤⎡ ⎤ ⎡ ⎤ ⎡ (1) ψ (1) eiγ z 0 φ0 (z) ⎦⎣ ⎦ . ⎦=C⎣ ⎣ (12.190) (2) ψ (2) 0 eiγ z φg (z)

 A simple substitution of ψ (1) , ψ (2) from (12.189) into (12.190) converts the beams into Bloch waves, and back into beams again at the same depth, z. It is also possible, and far more useful, to use the intermediate Bloch wave representation to propagate the beams over a distance, z. This can be performed by starting with the beams at any location because the Bloch wave  amplitudes, ψ (1) , ψ (2) , are independent of depth. The simplest approach is 

to evaluate ψ (1) , ψ (2) where z = 0, i.e., at the top of the sample.11 When     z = 0, the factors in (12.189) are: exp −iγ (1) 0 = exp −iγ (2) 0 = 1, and (12.189) is simply: ⎡ ⎤ ⎤ ⎡ ψ (1) φ (0) 0 ⎣ ⎦ . ⎦ = C −1 ⎣ (12.191) (2) ψ φg (0)

Now that we have the Bloch wave amplitudes from (12.191), we can evaluate the beam amplitudes at depth, z, using (12.190). Substituting (12.191) into (12.190): ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ (1) φ0 (z) φ (0) eiγ z 0 ⎣ ⎦=C⎣ ⎦ C −1 ⎣ 0 ⎦ . (12.192) iγ (2) z φg (z) φg (0) 0 e Equation (12.192) performs the operation of moving the beams from the depth zero (on right-hand-side) to the depth, z (on left-hand-side). We write (12.192) as: ⎤ ⎤ ⎡ ⎡ φ0 (0) φ0 (z) ⎦ = P (z) ⎣ ⎦ , ⎣ (12.193) φg (z) φg (0) where we define the Bloch wave propagator, P (z), as: ⎡ ⎤ (1) eiγ z 0 ⎦ C −1 . P (z) ≡ C ⎣ (2) 0 eiγ z

(12.194)

Using P (z) of (12.194), the forward and diffracted beams can be brought to the depth of the stacking fault. The next steps are to transfer them across the fault, and propagate them to the bottom of the crystal. 11

This same trick was used when solving for the Cij ’s in (12.146).

662

12. Dynamical Theory

Across the fault, the phase shifts of the beams were given by (12.186). With the superscript “a” to denote the beams above the fault, and “b” to denote the beams in the coordinate system below the fault: ⎤a ⎡ ⎤⎡ ⎤b ⎡ (t ) 1 0 φ φ0 (t1 ) 0 1 ⎦ . ⎦⎣ ⎦ = eik0 ·δr ⎣ ⎣ (12.195) φg (t1 ) φg (t1 ) 0 eig·δr We next take the beams below the fault (the left-hand-side of (12.195)), and propagate them to the bottom of the sample. Again we use P (t2 ) of (12.194), but in the coordinate system “b” below the fault: ⎡ ⎤ ⎡ ⎡ ⎤b ⎤b (1) eiγ t2 φ φ0 (t) 0 (t ) 0 1 ⎣ ⎦ C −1 ⎣ ⎦ =C⎣ ⎦ . (12.196) (2) φg (t) 0 eiγ t2 φg (t1 )

The final step is to take φb (t) from (12.196), and convert it to φa (t) in the original coordinate system. We need to return to the coordinate system of the crystal above the fault (superscript “a”) because the condition φ0 (0) = 1 at the top of the sample was defined in these coordinates. (In the coordinate system of the crystal below the fault, the incident beam at the top of the sample is a mix of forward and diffracted beams.) This transformation is accomplished with (12.187): ⎤a ⎡ ⎤⎡ ⎤b ⎡ (t) 1 0 φ φ0 (t) ⎦⎣ 0 ⎦ . ⎦ = e−ik0 ·δr ⎣ ⎣ (12.197) −ig·δr φg (t) φg (t) 0 e

The left-hand-side of (12.197) is the answer we seek. It is evaluated by multiplying the chain of matrices and vectors of (12.192) (where we set z = t1 ), (12.195), (12.196), and (12.197). (In doing so, we see immediately the cancellation of the constant phase factors exp(ik0 · δr) and exp(−ik0 · δr).) Specific forms for the Bloch wave propagators are obtained with (12.143) and (12.144) for the matrices C and C −1 . These forms, and the forms for the forward and diffracted beams, are complicated. They usually require numerical evaluation for each combination of g · δr, β, Δk, t1 and t2 . An important special case for sg = 0 (β = π/2) provides (after some algebra):     φ0 (t) = cos t1 /(2ξg ) cos t2 /(2ξg )     −eig·δr sin t1 /(2ξg ) sin t2 /(2ξg ) , (12.198)     φg (t) = i sin t1 /(2ξg ) cos t2 /(2ξg )     + e−ig·δr cos t1 /(2ξg ) sin t2 /(2ξg ) ,

(12.199)

In cases where g · δr = 2π integer, we have exp(ig · δr) = exp(−ig · δr) = 1 in (12.198) and (12.199), so:

12.7 Dynamical Diffraction Contrast from Crystal Defects

  φ0 (t) = cos t/(2ξg )   φg (t) = i sin t/(2ξg )

663

(g · δr = 2π integer ) ,

(12.200)

(g · δr = 2π integer ) .

(12.201)

These equations (12.200) and (12.201) for φ0 (t) and φg (t) are identical to (12.56) and (12.57), respectively, for the forward and diffracted beams from a perfect crystal of thickness, t. The fault is therefore invisible when g · δr = 2π integer. Diffraction contrast from the fault requires that g·δr = 2π integer. In this case, variations in the depth of the fault (variations in t1 ) cause changes in the beat pattern of the Bloch waves, leading to periodic fringes in the bright and dark-field images. For completeness, the expressions for the stacking fault contrast for all β are: φ0 (t) e−isg t/2 = cos(Δkt/2) − i cosβ sin(Δkt/2)   1 + sin2 β eig·δr − 1 cos(Δkt/2) 2   1 − sin2 β eig·δr − 1 cos(Δkt′ ) , 2

(12.202)

φg (t) e−isg t/2 = i sinβ sin(Δkt/2)  

 1 + sinβ 1 − e−ig·δr cosβ cos(Δkt/2) − i sin(Δkt/2) 2   

1 − sinβ 1 − e−ig·δr cosβ cos(Δkt′ ) − i sin(Δkt′ ) , (12.203) 2 where, following convention [12.2], we have defined the quantity, Δk ≡ seff , of (12.158), and t′ : 2 1 + (ξg sg ) (1) (2) Δk ≡ γ − γ = = seff , (12.204) ξg t1 − t 2 t′ ≡ . (12.205) 2 To understand the diffraction contrast from a stacking fault, we need to know the following parameters: 1) the fault vector, δr, 2) the thickness of the sample, t, 3) the deviation from the Bragg condition, sg (often sg = 0 is used to ensure a good two-beam condition), 4) the extinction distance, ξg , and 5) the g-vector of the diffraction. Since we can set up specific conditions in the microscope to establish 3) and 5), and use a technique such as CBED to determine 2) and 4), we can determine the fault vector in a material, i.e., 1). A specific procedure to do this was explained in Sects. 7.12.2 and 7.12.3. It can be shown that the intensities of (12.202) and (12.203) are normal2 2 ized so that: |φ0 (t)| + |φg (t)| = 1. The fringes in bright-field and dark-field images are therefore complementary. This is a consequence of neglecting absorption. To treat more realistically a crystal with absorption, a complex Δk is used. At the exact Bragg angle, β = π/2, so:

664

12. Dynamical Theory

Δk →

i 1 + ′ , ξg ξg

(12.206)

where ξg′ is the absorption length. In this case we can multiply the amplitudes   of the forward and diffracted beams, φ0 and φg , by the factor exp −πt/ξg′ to attenuate them. Typically ξg′ is about 10 times ξg . The effect of absorption is discussed qualitatively in the next section. 12.7.3 Dynamical Diffraction Contrast with Absorption Absorption of Bloch Wave 1 and Attenuation of Beats. We pause to summarize the two Bloch waves in two-beam dynamical theory for the diffraction condition sg = 0: Bloch Wave 1, Ψ (1) (r) • Its electron density is highest at the ion cores (see Fig. 12.11), so it has lower potential energy, higher kinetic energy, and a larger wavevector than Bloch wave 2 (or in vacuum). • It has stronger interactions with the electrons in the crystal, so there is more absorption of Bloch wave 1 than Bloch wave 2. With more absorption, Bloch wave 1 does not travel so far into the crystal as we would expect from the average inelastic scattering cross-section. Bloch Wave 2, Ψ (2) (r) • Its electron density is highest in the interstitial regions between the ion cores, so it has higher potential energy, lower kinetic energy, and a smaller wavevector than Bloch wave 1. • It has weaker interactions with the electrons in the crystal, so there is less absorption of Bloch wave 2 than Bloch wave 1. With less absorption, Bloch wave 2 travels farther into the crystal than we would predict from the average inelastic scattering cross-section. Section 12.4.1 showed the essence of how the depth-dependence of the forward and diffracted beams is understood as beats of Bloch waves with slightly  and k + γ (2) z . This beat pattern provides different wavevectors, k + γ (1) z different amounts of diffracted beam, {φg }, in samples of different thickness, as worked out in detail in (12.161) (using (12.159)). The depth periodicity of these beats of Bloch waves produces thickness contours in a wedge-shaped sample, as described in Sect. 7.5.2 with the kinematical theory. Although (7.12) and (12.161) are essentially the same, and predict the same behavior, the reader should be aware that the physical explanation from kinematical theory breaks down even for rather small thicknesses. At greater thicknesses approaching 10 extinction distances, however, (12.161) also becomes unreliable because of incoherent scattering. Two Bloch waves are needed to form beats, but Bloch wave 1 is attenuated rapidly with depth in the specimen.

12.7 Dynamical Diffraction Contrast from Crystal Defects

665

The result is that the beats diminish with depth. Consider the special case where s = 0 so both Bloch waves are excited equally at the top of the sample, but Bloch wave 1 is absorbed quickly with depth. In this case, at the bottom of the sample the amplitude of the diffracted beam remains in Bloch wave 2 only. The wave amplitude is thus 1/2 the amplitude at the top of the sample, and the intensity is 1/4. Such behavior is shown in Fig. 7.12, where the spacing between fringes is approximately ξg−1 when s = 0. Important effects in diffraction contrast are caused by differences in the absorption of the two Bloch waves. The different attenuations of the two Bloch waves are indicated qualitatively on the far left of Fig. 12.22, along columns of perfect crystal. Both Bloch waves must be present to have beats, but deep in the sample only Bloch wave 2 has significant amplitude. Near a defect, however, a phase shift of the lattice periodic potential serves to transfer electron amplitude from Bloch wave 2 into Bloch wave 1, making beats possible again. Absorption of Bloch Wave 1 and Bend Contour Asymmetry. Bend contours were discussed in Sect. 7.6, and Figs. 7.13 and 7.14 are needed for the present discussion of dynamical theory with absorption. The kinematical (7.12) and the dynamical (12.161) provide qualitatively similar results for the appearance of bend contours with intensity oscillations12 in BF and DF images. Absorption, however, causes a loss of symmetry of the BF image. This can be understood with reference to Figs. 7.14 and Fig. 12.17, and the strong absorption of Bloch wave 1. Figure 7.14 shows that in the middle of the bend contour, s < 0 for both diffractions ±g. Figure 12.17 shows that when s < 0, and the tops of the arrows are to the left of center, the forward beam is composed mostly of Bloch wave 1. This Bloch wave 1 is absorbed strongly, so the center of the bend contour is generally dark in bright-field. As shown in Fig. 7.14, however, the +g and −g diffractions are in the condition s = 0 on the left and right sides of the bend contour. For s = 0, the forward beam is composed more of Bloch wave 2, and is less absorbed. For most samples, however, diffraction is very strong and the bend contour is darkest in a BF image near these locations where s = 0 on the left and right sides of the bend contour. Further out from the center, for s > 0, the forward beam is composed even more of Bloch wave 2, and the BF image is generally brighter than inside the bend contour. The diffracted beam, on the other hand, originates with the beating of both Bloch waves 1 and 2. The diffracted beam is strongest when s = 0, when Bloch waves 1 and 2 are excited equally. The diffracted intensity is expected to be symmetrical about s = 0, since the intensity of beating is reduced for either +s or −s. These features can be seen in Fig. 12.15b, for example. When viewing such images, it is important to check the precise position of the bright band in the DF image by noting the positions of the 12

Spacings of the prominent intensity oscillations are difficult to quantify because they depend on the details of how the specimen is bent.

666

12. Dynamical Theory

θ′ precipitates in both BF and DF images. The bright band in the DF image defines the locations where s = 0. The BF image is symmetrical about the overall center of the bend contour, but not about the locations where s = 0 for either diffraction. The BF image tends to be asymmetrically darker inside the bend contour than outside, owing to the larger fraction of Bloch wave 1 inside the bend contour. Absorption of Bloch Wave 1 and Stacking Fault Asymmetry. Consider the case of a stacking fault inclined from top to bottom of the sample as in Fig. 12.22. In general, and especially when s = 0, both Bloch waves Ψ (1) and Ψ (2) are excited at the top of the specimen. As shown at the left of Fig. 12.22, the Bloch wave amplitude ψ (1) is attenuated rapidly with depth. Along column 1 (indicated at the top of the figure) only Bloch wave Ψ (2) reaches the stacking fault, which is near the bottom of the sample along column 1. The shift of atom positions below the fault puts the crests of Bloch wave Ψ (2) partly on the atom centers and partly off, in effect making the transformation Ψ (2) → αΨ (1) + βΨ (2) in the coordinate system below the fault.13 The presence of both Bloch waves Ψ (1) and Ψ (2) causes beats to form, so the amplitude of the diffracted beam, φg , below the fault varies with distance to the bottom of the sample.

Fig. 12.22. Dynamical diffraction contrast from a stacking fault in a material with absorption. Notice that the diffraction contrast is not complementary in the brightand dark-field images. 13

Except for the special case where the fault displacement is such that δr = m (2g)−1 b g , where g is the active diffraction and m is an integer.

12.7 Dynamical Diffraction Contrast from Crystal Defects

667

For column 2 in Fig. 12.22, the distances from the fault to either the top or the bottom of the sample are so long that the Bloch wave Ψ (1) is absorbed substantially, and Ψ (2) dominates at the bottom of the sample. Beats are suppressed owing to the absence of Ψ (1) , and the diffraction contrast is weak from this part of the fault near the center of the sample. Column 3, however, has strong beats of Ψ (1) and Ψ (2) in the crystal above the fault, causing the diffracted intensity to vary with the depth of the fault. Below the fault, the beam amplitudes φ0 and φg are resolved into Bloch wave amplitudes ψ (1) and ψ (2) . In the new coordinate system below the fault, the amplitudes ψ (1) and ψ (2) vary with the phase shift of the beams at the fault, and with the amplitudes of the beams, φ0 and φg , at the fault. The phase shifts across the fault are the same for any depth of the fault, but the amplitudes φ0 and φg are not. The amplitudes ψ (1) and ψ (2) below the fault therefore vary with the depth of the fault. Only Ψ (2) travels the long distance to the bottom of the sample below the fault, however. Further beats of Ψ (1) and Ψ (2) do not occur over this long distance. At the bottom of the sample, the resolution of Ψ (2) into φ0 and φg depends on the amplitude of φ0 and φg at the depth of the fault. The important point is that the amplitude of Ψ (2) in the crystal below the fault depends on the beat pattern above the fault. We know from (12.177) that Ψ (2) then provides components of both φ0 and φg at the bottom of the sample. Without a Ψ (1) at the bottom of the sample, both φ0 and φg are proportional to the amplitude Ψ (2) (a point discussed after (12.177)). The intensities of the forward beam and the diffracted beam therefore have the same pattern – they are not complementary to each other for column 3. Central to this argument was the role of absorption. Absorption of Bloch wave 1 breaks the complementarity of BF and DF images at the top and bottom of the sample, and dampens the contrast near the center. It is instructive to look again at Fig. 7.58 showing contrast from inclined, planar hcp γ ′ precipitates that exhibit stacking-fault contrast. In the BF image in Fig. 7.58a, the displacement fringes are symmetric with respect to the center of the thin foil whereas in the DF image in Fig. 7.58b, the fringes are asymmetric in their overall intensity, as in Fig. 12.22. According to the rules previously outlined for stacking faults in Sect. 7.12.2, the phase angle equals −2π/3 for this γ ′ plate since the outer fringes are both dark in the BF image. Equations like (12.202)–(12.205) were used to calculate the forward and diffracted beam intensities for these γ ′ precipitates, taking into account the actual sample thickness, deviation parameter, absorption, and other experimental conditions. Figure 12.23 compares these calculated results to the intensities in BF and DF images perpendicular to the displacement fringes. The simulated and experimental intensity profiles match relative intensities to within 90 %.14 Comparison between the experimental and calculated im14

Most of the irregularities in the experimental traces are caused by noise or dirt on the TEM negatives, although the doublet in the intensity of the first DF fringe is a real effect.

668

12. Dynamical Theory

ages shows that the BF and DF fringes are nearly complementary at the bottom but not at the top of the thin foil sample, just like the stacking fault shown schematically in Fig. 12.22. Note that similar features are present in the stacking fault calculations shown in Fig. 7.55 and the experimental images in Figs. 7.56 and 7.57, although they are quantitatively different owing to differences in experimental conditions.

Fig. 12.23. Intensity profiles ( BF, —— DF) across the face of the upper γ ′ precipitate in Fig. 7.58, plotted as a function of depth of the bottom surface of the precipitate: (a) calculated profile and (b) experimental profile. After [12.3].

Absorption and Spectroscopy (ALCHEMI). The treatments of inelastic scattering in Chapter 4 neglected some interesting effects of lattice periodicity. Since Bloch wave 1 undergoes a greater amount of inelastic scattering than Bloch wave 2, changing the tilt of the crystal (altering sg ) changes the amount of inelastic scattering because the tilt changes the relative fractions of Bloch waves 1 and 2. The number of inelastic scattering events therefore depends on the diffraction condition, sg , and not just the number of atoms in the material. The experimental method of Atom Location by CHanneling Electron MIcroanalysis, ALCHEMI, uses the sensitivity of inelastic scattering to the tilt of the sample. In its simplest form, ALCHEMI can be used to determine site occupancies of atom species by measuring the intensities of their x-ray emissions. Consider, for example, a crystal in which atoms A and B occupy the same crystallographic site, different from that of atom C. Pick a diffraction, g, which involves the partially-destructive interference between these two sites. For example, the diffraction could be the (100) superlattice diffraction of the B2 structure (Fig. 5.11), where the center sites of the underlying bcc lattice are occupied by A and B atoms, and the corner sites are occupied by C. By tilting the sample to change the sign of the sg for this (100) diffraction, the dominant Bloch wave changes from having electron density maxima on the A, B sites, to having maxima on the C sites. The characteristic x-ray spectrum therefore shows an enhancement of either the A and B fluorescence, or the

12.8 ‡ * Multi-Beam Dynamical Theories of Electron Diffraction

669

C fluorescence. With such information, some interpretations of ALCHEMI data can be very easy. For example, with only the knowledge that the x-ray emissions from the A and B atoms track each other for different tilts, it is possible to tell that the B-atoms occupy the same crystallographic site as the A-atoms. An illustration of this effect in a NiAl-Ti alloy is shown in Fig. 12.24. Additions of Ti increase the strength and creep resistance of NiAl alloys. To understand this effect, it is necessary to first determine whether Ti atoms occupy the Ni or Al sites in the B2 structure. The x-ray spectra shown in Fig. 12.24 were obtained in a [001] orientation under two-beam conditions with s > 0 and s ≪ 0 for the (100) superlattice diffraction, which localizes the electron intensity on either the Al (light) or Ni (heavy) columns of atoms, respectively. The intensity is scaled so the Ni Kα peaks of both spectra are of equal intensity. There is a dramatic, simultaneous change in the relative intensities of both the Al Kα and Ti Kα peaks when s ≪ 0. Since the Al and Ti signals show the same trend, the Ti atoms are located on the Al atom sites.

Fig. 12.24. EDS spectra obtained from a NiAl4.3Ti alloy with g = 100 and s > 0 or s  0. After [12.4].

12.8 ‡ * Multi-Beam Dynamical Theories of Electron Diffraction A high-resolution TEM image is made by the wave interference of the forward beam and a few low-order diffracted beams (Chapter 10). HRTEM cannot make use of diffractions at large Δk because their phases are disturbed by the spherical aberration of the objective lens and the damping of the contrast transfer function owing to instrument instabilities. Nevertheless, high-order diffractions are required in any realistic calculation of the electron transmission through the sample. In a sample of modest thickness, the electron wave

670

12. Dynamical Theory

amplitude in the low-order beams has been transferred back-and-forth between beams of many different orders before emerging from the bottom of the sample. Accurate image simulations therefore require general n-beam solutions of the dynamical equations (12.20) or (12.34), even when only a few low-order beams are used to form the image. For further formal manipulations, it is elegant and expedient to rewrite (12.34) in matrix form [12.5]. In rewriting (12.34), the components are the amplitudes of the actual diffracted waves: ∂φ(z) = iA φ(z) . ∂z The vector φ(z) is: ⎤ ⎡ φ0 (z)  ⎢  ⎢ ⎢ φg (z)   , ⎢ φ(z) = ⎢  ⎢φ2g (z) ⎦ ⎣ .

(12.207)

(12.208)

whose components are the amplitudes of the diffracted beams. The matrix, A, is: ⎡ ⎤ 1 ... 0 2ξ1−g 2ξ−2g  ⎢ ⎢ 1  ⎢ 2ξg sg 2ξ1−g . . .  . A=⎢ (12.209) ⎢ 1  1  ⎢ 2ξ s . . . 2g ⎦ ⎣ 2g 2ξg . The formal solution to the matrix equation of (12.207) is simply: φ(z) = eiAz φ(z = 0) ,

(12.210)

where the boundary condition with the incident beam at the top of the crystal is: ⎡ ⎤ 1 ⎢  ⎢  ⎢0  (12.211) φ(z = 0) = ⎢ ⎢  . ⎢0 ⎣ ⎦ . Equation (12.210) shows how the incident beam in (12.211) evolves into a set of diffracted beams. There are now two reasonable approaches to develop a dynamical theory. They are:

12.8 ‡ * Multi-Beam Dynamical Theories of Electron Diffraction

671

• Set up an eigenvalue equation: A C (j) = γ (j) C (j) .

(12.212)

The components of the eigenvectors C (j) are weighted sums of the diffracted beams. They are Bloch wave states for the high-energy electron. Section 12.4 developed this eigenvalue equation for the simplest case of two-beam dynamical theory. The 2 × 2 matrix, A, gave a quadratic secular equation that was solved easily for the two γ (j) . This eigenvalue approach becomes more difficult as the number of beams, n, becomes larger, since A becomes an n × n matrix (although its largest elements are near the diagonal). • Work with (12.210) directly: n

(12.213) eiAz = eiAΔz , (where z = nΔz). The exponent, n, indicates that the operator, eiAΔz , must be applied n times to the vector φ(z = 0) to generate the diffracted beam amplitudes at depth z. Each operation accounts for an increase in depth, Δz.

In the development of the second method, it is more convenient to write the operator in (12.213) as: eiAΔz = eisΔz eiσU Δz , where the new matrices are defined as: ⎤ ⎡ 0 0 0 ...  ⎢  ⎢ ⎢ 0 sg 0 . . .   , ⎢ s=⎢  ⎢ 0 0 s2g . . .  ⎦ ⎣ . ⎤ ⎡ 0 U−g U−2g . . .  ⎢  ⎢ ⎢ Ug 0 U−g . . .  , ⎢ U =⎢  ⎢U2g Ug 0 . . . ⎦ ⎣ .

(12.214)

(12.215)

(12.216)

and the extinction distances, ξg , are related to the Fourier components of the crystal potential, Ug , by the constant (cf., (12.18)): me σ=− 2 . (12.217) h k The two factors in (12.214) are the propagator, eisΔz , and the phase grating, eiσU Δz . We discussed the phase grating in 12.6. It is a convolution of the diffracted beams with the Fourier components of the potential, as described after (12.25). This convolution in k-space is performed more conveniently in

672

12. Dynamical Theory

real space, where it becomes a multiplication. The propagator, however, is easier to use in k-space. For each increment in depth, Δz, the direct approach of (12.213) requires one forward Fourier transformation, and one inverse Fourier transformation. Although cumbersome, high performance algorithms for these operations are available, and the direct approach of (12.214) is the preferred method for calculating the dynamical behavior of high-energy electrons in solids. This method is called the “physical optics approach,” or the “Cowley-Moodie method,” described in concept in Sect. 10.2, and in practice in Sect. 10.4.

Further Reading The contents of the following are described in the Bibliography. S. Amelinckx, R. Gevers and J. Van Landuyt: Diffraction and Imaging Techniques in Materials Science (North–Holland, Amsterdam 1978). J. M. Cowley: Diffraction Physics, 2nd edn. (North–Holland, Amsterdam 1975). P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley, and M. J. Whelan: Electron Microscopy of Thin Crystals (R. E. Krieger, Malabar, Florida 1977). A. J. F. Metherell: ‘Diffraction of Electrons by Perfect Crystals,’ in Electron Microscopy in Materials Science II, ed. by U. Valdre and E. Ruedl (CEC Brussels 1975) pp. 387. L. Reimer: Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, 4th edn. (Springer–Verlag, New York 1997). J. C. H. Spence and J. M. Zuo: Electron Microdiffraction (Plenum Press 1992). G. Thomas and M. J. Goringe: Transmission Electron Microscopy of Materials (Wiley–Interscience, New York 1979). D. B. Williams and C. B. Carter: Transmission Electron Microscopy: A Textbook for Materials Science (Plenum Press, New York 1996).

Problems 11.1 (a) Show that the product of γ (1) and γ (2) gives the equation of a hyperbola. (b) Show that the sum of γ (1) and γ (2) is a constant. (c) Which of the results of part a or b correspond to a: material parameter? diffraction condition? 11.2 Confirm the normalization factor, 1/V , for (12.93) and (12.95).

Problems

673

11.3 Using Dirac notation, the beams, {|0 , |g }, and the Bloch waves, {|1 , |2 }, are related in two-beam dynamical theory as: (2)

(1)

|0 = C0 |1 + C0 |2 ,

|g =

Cg(1) |1

+

Cg(2) |2

(12.218)

,

(12.219)

We have the normalizations:      (1) 2  (2) 2 C0  + C0  = 1 ,      (1) 2  (2) 2 Cg  + Cg  = 1 ,

(12.220) (12.221)

And the Bloch waves are orthonormal: (1) | (1) = 1 ,

(12.222)

(2) | (2) = 1 , (1) | (2) = 0 ,

(12.223) (12.224)

(2) | (1) = 0 ,

(12.225)

From this information only, use Dirac notation to prove the following: (a) (0) | (0) = 1, (g) | (g) = 1. (1)∗

(b) If (0) | (g) = 0, then C0

(1)

(2)∗

Cg + C0

(2)

Cg = 0.

11.4 For two-beam dynamical theory: (a) using the expressions for the amplitude of the diffracted beam:  s z g 1 + (sg ξg )−2 , (12.226) φg = ieisg z/2 sinβ sin 2 and the forward beam (12.155):    s z  sg z g isg z/2 −2 −2 − i cosβ sin 1 + (sg ξg ) 1 + (sg ξg ) cos φ0 = e , 2 2 (12.227) show that: 2

2

|φ0 | + |φg | = 1 ,

(12.228)

(b) Obtain (12.155) [(12.227)] for φ0 , starting with (12.142). 11.5 Starting with (12.154) for φg :  s z g φg = ieisg z/2 sinβ sin 1 + (sg ξg )−2 , 2

(12.229)

(a) Calculate dφg /dz for the case sg = seff , which is the kinematical limit. (b) Show how the result can be used to generate the phase-amplitude diagram of kinematical theory.

674

12. Dynamical Theory

K

–

0

K

k

+

–

x1

0

x2

+

Fig. 12.25. Problem 11.6.

11.6 Two simple harmonic oscillators, each having mass, m, and spring constant, K, are coupled together with a weak spring having spring constant, k, as shown in Fig. 12.25. Assume 1-dimensional horizontal motion only. When both masses are at rest at x1 = 0 and x2 = 0, the forces in all 3 springs are zero. The equations of motion for x1 (t) and x2 (t) are: d2 x1 = −Kx1 − k (x1 − x2 ) , dt2 d2 x2 m 2 = −Kx2 − k (x2 − x1 ) . dt (a) Assuming solutions of the form: ⎡ ⎤ ⎡ ⎤ (j) x1 (t) X1 (j) ⎣ ⎦=⎣ ⎦ eiω t , (j) x2 (t) X2 m

(j)

(12.230) (12.231)

(12.232)

(j)

where X1 and X2 are constants, solve for the two values of ω (j) . (Hint: Solve for mω 2 in the secular equation, and also read part d now.) (b) From your work in part a, which terms play the roles analogous to the following three terms in two-beam dynamical theory: γ (j) , (2ξg )−1 , and (in part) sg ? (j)

(j)

(c) For each value of ω (j) from part a, solve for X1 and X2 . (d) Make the approximation, valid for k ≪ K:    k K + 2k K (2) ≃ 1+ , (12.233) ω = m m K and solve for the beat period of the coupled variables x1 (t) and x2 (t). (Hint: : Make a linear combination of the solutions: ⎤ ⎤ ⎡ ⎡ ⎤ ⎡ (2) (1) x X (t) X1 (2) (1) 1 1 ⎦ , (12.234) A ⎣ (1) ⎦ eiω t + B ⎣ (2) ⎦ eiω t = ⎣ x2 (t) X2 X2

and choose A and B so that at for the boundary condition at t = 0: ⎡ ⎤ ⎡ ⎤ x1 (t = 0) 1 ⎣ ⎦ = ⎣ ⎦a , (12.235) 0 x2 (t = 0) where a is the initial displacement of mass 1.)

Problems

675

11.7 (a) With (12.198) and (12.199), obtain general expressions for BF and DF stacking fault contrast. (b) Using a graphics-based mathematics software package, graph the BF and DF intensities of part a versus t1 for the case where g · δr = −2π/3. In doing so, include absorption through (12.206). Choose ξg = 7ξg and set t2 + t2 = 10ξg . (Hint: You may need to rewrite cosine and sine functions as: sin x = 1/(2i)[exp(−ix) − exp(ix)] and cos x = 1/2[exp(−ix) + exp(ix)].)

Please examine Fig. 7.55.

11.8 (harder) Use the phase grating approximation: Ψ (x, y, z) = e

ik·r

  t m V (x, y, z) dz , exp i h2 k

(12.236)

0

to obtain moir´e fringes for the g diffraction along x from two superposed crystals, each of thickness t, which have potentials along x given by: V0 + V1 cos(g 1 · x) , V0 + V1 cos(g 2 · x) .

(12.237) (12.238)

(Hint: Write the potential as a product of cosine functions having an average g and a difference Δg.) 

11.9 (hard) A high-energy electron is incident on an fcc crystal near 111 , such that the exact Laue condition occurs for both the (220) and 202 diffractions. (a) Derive expressions for the Bloch waves set up in the crystal, including the electron densities around the atoms.

(Hint: Owing to geometrical symmetry, show that the three diagonal elements are equal in the 3 × 3 matrix equivalent of (12.116).) (b) Compare the results to those of the two-beam case with only the (220) diffraction.

Bibliography

Further Reading C. C. Ahn, Ed.: Transmission Electron Energy Loss Spectroscopy in Materials Science and the EELS Atlas (Wiley–VCH, Weinheim 2004). An updated 2nd edition of the Disko, Ahn and Fultz book by the same name. A practical reference covering EELS instrumentation, quantification, fine structure, and applications to the different classes of materials. Includes a CD ROM with the EELS Atlas. C. C. Ahn and O. L. Krivanek: EELS Atlas (Gatan, Inc., Pleasanton, CA 1983). The standard reference presenting EELS spectra of nearly all the elements in the periodic table and some compounds. S. Amelinckx, R. Gevers and J. Van Landuyt: Diffraction and Imaging Techniques in Materials Science (North–Holland, Amsterdam 1978). Excellent chapters on kinematical and dynamical electron diffraction, the WBDF technique, computed electron micrographs, Kikuchi diffraction and defects in materials. Leonid V. Az´aroff: Elements of X-Ray Crystallography (McGraw–Hill, New York 1968), reprinted by TechBooks, Fairfax, VA. Emphasizes crystal structure and symmetry determination by x-ray diffractometry. B. W. Batterman and H. Cole: Rev. Mod. Phys. 36, 681-717 (1964). A systematic presentation of the dynamical theory of x-ray diffraction based on Maxwell’s equations. J. M. Cowley: Diffraction Physics, 2nd edn. (North–Holland Publishing, Amsterdam 1975). Thorough but concise presentation of the physical optics approach to diffraction and imaging, scattering of radiation by atoms and crystals, kinematical and dynamical diffraction, and applications to selected topics. B. D. Cullity and S. R. Stock: Elements of X-Ray Diffraction (Prentice– Hall, Upper Saddle River, NJ 2001). A popular introductory text on x-ray diffraction – provides physical explanations of many topics. M. De Graef: Introduction to Conventional Transmission Electron Microscopy (Cambridge University Press, Cambridge 2003). Much more than an introduction, this textbook provides excellent and thorough coverage of electron

678

Bibliography

optics, crystallography, and defect contrast in dynamical theory. Computational methods are presented with an accompanying website. M. M. Disko, C. C. Ahn and B. Fultz, Eds.: Transmission Electron Energy Loss Spectroscopy in Materials Science (Minerals, Metals & Materials Society, Warrendale, PA 1992). A practical reference covering EELS instrumentation, quantification, fine structure, and applications to the different classes of materials. J. A. Eades: ‘Convergent-Beam Diffraction’. In: Electron Diffraction Techniques, Volume 1, ed. by J. M. Cowley (International Union of Crystallography, Oxford University Press, Oxford 1992). Good overall review of the subject. J. W. Edington: Practical Electron Microscopy in Materials Science, 1. The Operation and Calibration of the Electron Microscope (Philips Technical Library, Eindhoven 1974). Easy to understand discussion of the optics, alignment and calibration of the TEM. J. W. Edington: Practical Electron Microscopy in Materials Science, 2. Electron Diffraction in the Electron Microscope (Philips Technical Library, Eindhoven 1975). Thorough discussion of electron diffraction patterns, Kikuchi lines, and their use in the TEM. Has a good appendix on stereographic projections. J. W. Edington: Practical Electron Microscopy in Materials Science, 3. Interpretation of Transmission Electron Micrographs (Philips Technical Library, Eindhoven 1975). Excellent discussion of diffraction contrast and quantitative defect analysis in the TEM with many useful examples. J. W. Edington: Practical Electron Microscopy in Materials Science, 4. Typical Electron Microscope Investigations (Philips Technical Library, Eindhoven 1976). A number of illustrative examples of diffraction and imaging analyses in the TEM. T. Egami and S. J. L. Billinge: Underneath the Bragg Peaks: Structural Analysis of Complex Materials (Pergamon Materials Series, Elsevier, Oxford 2003). A book on modern powder diffraction experiments, with emphasis on total scattering measurements and pair distribution function analysis. Clear and thorough coverage of theory and practice of experiments with synchrotron radiation and neutron scattering for identifying nanoscale structures and disorder in hard condensed matter. R. F. Egerton: Electron Energy-Loss Spectroscopy in the Electron Microscope, 2nd edn. (Plenum Press, New York 1996). Thorough, scholarly and rigorous coverage of EELS instrumentation, electron scattering theory, quantitative EELS analysis, and examples in materials research. C. T. Forwood and L. M. Clarebrough: Electron Microscopy of Interfaces in Metals and Alloys (Adam Hilger IOP Publishing Ltd., Bristol 1991). Excellent reference on computed electron micrographs of interfaces in materials.

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P. J. Goodhews and F. J. Humphreys: Electron Microscopy and Microanalysis (Taylor & Francis Ltd., London 1988). Easy-to-follow discussions of electron optics in the TEM, electron beam-specimen interactions, electron diffraction and imaging, and microanalysis. P. Grivet: Electron Optics, revised by A. Septier, translated by P. W. Hawkes (Pergamon, Oxford, 1965). The electromagnetics of electron optics, with emphasis on electron lenses and the TEM. C. Hammond: The Basics of Crystallography and Diffraction (International Union of Crystallography, Oxford University Press, Oxford 1977). Simple and understandable introduction to crystallography and diffraction techniques, with worked examples of structure factor calculations and diffraction analyses. A. K. Head, P. Humble, L. M. Clarebrough, A. J. Morton and C. T. Forwood: Computed Electron Micrographs and Defect Identification (North–Holland Publishing Company, Amsterdam 1973). Excellent reference on computed electron micrographs based on the Howie-Whelan two-beam theory of diffraction, including applications and limitations of the technique. P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley, and M. J. Whelan: Electron Microscopy of Thin Crystals (R. E. Krieger, Malabar, Florida 1977). A reprinted early book on conventional TEM. Excellent discussions of kinematical and dynamical electron diffraction theory and application to defect analysis in materials. It offers a broad coverage of experimental technique, and for many years was the essential text on the subject. Includes worked problems. J. J. Hren, J. I. Goldstein and D. C. Joy, Eds.: Introduction to Analytical Electron Microscopy (Plenum Press, New York 1979). Good overall book on TEM, providing treatment of electron optics, EDS, EELS, CBED, STEM. The International Union of Crystallography publishes the International Tables for X-ray Crystallography (Kynock Press, Birmingham, England, 1952-), which contain the standard tables of crystal symmetry plus a wealth of tabulated data on scattering factors, dispersion corrections, and other details and principles of x-ray data analysis. O. Johari and G. Thomas: The Stereographic Projection and Its Applications (Interscience Publishers, John Wiley & Sons, New York 1969). Provides stereographic projections and presents their applications to problems in materials science. D. C. Joy, A. D. Romig, Jr. and J. I. Goldstein, Eds.: Principles of Analytical Electron Microscopy (Plenum Press, New York 1986). Provides a good introduction to electron scattering and electron optics, with emphasis on EDS and EELS spectroscopy. Contains worked examples. R. J. Keyse, A. J. Garratt-Reed, P. J. Goodhew and G. W. Lorimer: Introduction to Scanning Transmission Electron Microscopy (Springer BIOS Scientific

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Publishers Ltd., New York 1998). Practical explanation of optics, diffraction, imaging and microanalysis – specifically for the STEM. Harold P. Klug and Leroy E. Alexander: X-Ray Diffraction Procedures (Wiley–Interscience, New York 1974). Provides an encyclopedic coverage of experimental methods and many principles of x-ray diffraction. M. A. Krivoglaz: Theory of X-Ray and Thermal Neutron Scattering by Real Crystals (Plenum, New York 1969). An elegant and formal treatment of scattering from fluctuations with analysis of their correlation functions. M. H. Lorretto: Electron Beam Analysis of Materials (Chapman and Hall, London 1984). Concise discussion of most TEM topics, including electron diffraction and imaging, CBED, and microanalysis. A. J. F. Metherell: ‘Diffraction of Electrons by Perfect Crystals’, in Electron Microscopy in Materials Science II, ed. by U. Valdre and E. Ruedl (CEC Brussels 1975) pp. 387. This is probably the most detailed and comprehensive article written on materials analysis using the Bloch wave approach to dynamical electron diffraction. I. C. Noyan and J. B. Cohen: Residual Stress (Springer–Verlag, New York 1987). A thorough development of the experiment and theory connecting continuum mechanics to x-ray diffractometry. Includes x-ray lineshape analysis. S. J. Pennycook, D. E. Jesson, M. F. Chisholm, N. D. Browning, A. J. McGibbon, and M. M. McGibbon: ‘Z-Contrast Imaging in the Scanning Transmission Electron Microscope’, J. Micros. Soc. Amer. 1, 234 (1995). An overview of the principles and practice of Z-contrast imaging in the STEM, with emphasis on chemical and structural information on the atomic scale. H. Raether: Excitations of Plasmons and Interband Transitions by Electrons (Springer–Verlag, Berlin and New York 1980). An in-depth treatment of the low-loss part of EELS spectra. L. Reimer, Ed.: Energy-Filtering Transmission Electron Microscopy (Springer– Verlag, Berlin 1995). Contains detailed theoretical discussions of electronspecimen interactions, EELS instrumentation, spectroscopic diffraction and imaging techniques. L. Reimer: Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, 4th edn. (Springer–Verlag, New York 1997). Comprehensive, scholarly, and rigorous coverage of TEM instrumentation, imaging and diffraction techniques. Strong emphasis on the underlying physics. Extensive references to recent research. P. Schattschneider: Fundamentals of Inelastic Electron Scattering (Springer– Verlag, Vienna, New York 1986). The theoretical physics of low-loss EELS spectra, with emphasis on the quantum mechanics of scattering using manybody theory. L. H. Schwartz and J. B. Cohen: Diffraction from Materials (Springer–Verlag, Berlin 1987). Provides a thorough treatment of x-ray theory and experiment, including crystallography.

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F. G. Smith and J. H. Thomson: Optics, 2nd edn. (John Wiley & Sons, New York 1988). Although this book is concerned with light optics, it provides excellent coverage on the subjects of wave propagation, geometrical optics, interference and diffraction, resolution, and phase-amplitude diagrams. J. C. H. Spence: Experimental High-Resolution Electron Microscopy, 2nd edn. (Oxford Univ. Press, New York 1988). Wide-ranging coverage of the theory and practice of TEM, emphasizing HRTEM. J. C. H. Spence and J. M. Zuo: Electron Microdiffraction (Plenum Press, New York 1992). Excellent discussion of dynamical electron diffraction and convergent-beam electron diffraction. G. L. Squires: Introduction to the Theory of Thermal Neutron Scattering (Dover, Mineola, New York 1996). A broad coverage of the theoretical physics of neutron scattering, developed elegantly and concisely. J. W. Steeds: ‘Convergent Beam Electron Diffraction’. In: Introduction to Analytical Electron Microscopy, ed. by J. J. Hren, J. I. Goldstein, D. C. Joy (Plenum Press, New York 1979) p. 401. Good overall discussion of CBED technique and application to materials. J. W. Steeds and R. Vincent: ‘Use of High-Symmetry Zone Axes in Electron Diffraction in Determining Crystal Point and Space Groups’, J. Appl. Cryst. 16, 317 (1983). Provides a useful sequence of steps for determining crystal point and space groups from high-symmetry zone axes. M. Tanaka and M. Terauchi: Convergent-Beam Electron Diffraction (JEOL Ltd., Nakagami, Tokyo 1985). M. Tanaka, M. Terauchi and T. Kaneyama, Convergent-Beam Electron Diffraction II (JEOL Ltd., Musashino 3-chome, Tokyo 1988). These compilations provide a thorough summary of CBED procedures such as point and space group determination, lattice parameter measurement, etc. G. Thomas and M. J. Goringe: Transmission Electron Microscopy of Materials (Wiley–Interscience, New York 1979). Good general discussion of TEM techniques, including kinematical and dynamical electron diffraction and imaging. Many examples of TEM images of defects in materials with discussion of practice. Includes worked problems. B. E. Warren: X-Ray Diffraction (Addison–Wesley, Reading, MA 1969), is now a best buy as a Dover reprint (Dover, New York, 1990). It provides a rigorous coverage of concepts in x-ray powder diffractometry of imperfect crystals. D. B. Williams: Practical Analytical Electron Microscopy in Materials Science (Philips Electron Instruments, Inc., Mahwah, NJ 1984). In-depth discussion of alignment and calibration of the TEM, quantitative x-ray microanalysis and EELS spectrometry with many useful examples. D. B. Williams and C. B. Carter: Transmission Electron Microscopy: A Textbook for Materials Science (Plenum Press, New York 1996). Probably the

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most comprehensive current book on TEM available, covering almost all aspects of the technique. Includes both theory and practical examples.

References and Figures Chapter 1 title photograph of Inel Corp. CPS-120 x-ray diffractometer with large-angle position-sensitive detector. Radiation shielding not shown. 1.1 1.2 1.3 1.4

1.5

1.6

1.7 1.8

International Centre for Diffraction Data, 12 Campus Boulevard Newtown Square, PA 19073-3273 USA. http://www.icdd.com H. G. J. Moseley: Philos. Mag. 27, 713 (1914). F. Richtmyer and E. Kennard: Introduction to Modern Physics (McGraw– Hill, New York 1947). A partial list of web sites for synchrotron sources includes (prefixed with http:// ): aps.anl.gov/, www.esrf.eu/, www.spring8.or.jp/, www-hasylab.desy.de/, slac.stanford.edu/, www.srs.ac.uk/srs/ www.bessy.de/, www.nsls.bnl.gov/, www.als.lbl.gov/, ssrc.inp.nsk.su/ Leonid V. Az´ aroff: Elements of X-Ray Crystallography (McGraw–Hill, New York 1968). Figure reprinted with the courtesy of TechBooks, Fairfax, VA. National Institute of Standards and Technology, Standard Reference Materials Program, Bldg. 202, Rm 204, Gaithersburg, MD 20899. http://ts.nist.gov/srm J. Nelson and D. Riley: Proc. Phys. Soc. (London) 57, 160 (1945). Harold P. Klug and Leroy E. Alexander: X-Ray Diffraction Procedures (Wiley–Interscience, New York 1974). Figure reprinted with the courtesy of John Wiley–Interscience.

Chapter 2 title drawing of JEOL JEM-2010F. Figure reprinted with the courtesy of JEOL Ltd., Tokyo. 2.1 2.2

2.3

2.4 2.5

B. Demczyk: Ultramicros., 47, 433 (1993). Figure reprinted with the courtesy of Elsevier Science Publishing B.V. J. M. Howe, W. E. Benson, A. Garg and Y. C. Chang: Mater. Sci. Forum, 189-90, 255 (1995). Figure reprinted with the courtesy of Trans Tech Publications Ltd. Near the year 2007, manufacturers of TEM instruments include JEOL, FEI, Hitachi and Zeiss. A partial list of web sites for manufacturers of TEM instruments includes: www.jeol.com/, www.fei.com/, www.hitachi-hta.com/, www.smt.zeiss.com/ Figure reprinted with the courtesy of FEI Company. P. J. Goodhew and F. J. Humphreys: Electron Microscopy and Analysis, 2nd edn. (Taylor & Francis, Ltd., London 1975). Figure reprinted with the courtesy of Taylor & Francis, Ltd.

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2.6 2.7

Figure reprinted with the courtesy of Prof. M. K. Hatalis. M. Bilaniuk and J. M. Howe: Interface Sci., 6, 328 (1998). Figure reprinted with the courtesy of Kluwer Academic Publishers. 2.8 D. B. Williams: Practical Analytical Electron Microscopy in Materials Science (Philips Electron Optics Publishing Group, Mahwah, NJ 1984). Figure reprinted with the courtesy of FEI Company. 2.9 Figure courtesy of Dr. Simon Nieh. 2.10 L. Reimer: Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, 4th edn. (Springer–Verlag, New York 1997). Figure reprinted with the courtesy of Springer–Verlag. 2.11 F. W. Sears and M. W. Zemansky: University Physics, 4th edn. (Addison– Wesley–Longman Publishing, Reading, MA 1973). Figure reprinted with the courtesy of Addison–Wesley–Longman Publishing. 2.12 J. W. Edington: Practical Electron Microscopy in Materials Science, 1. The Operation and Calibration of the Electron Microscope (Philips Technical Library, Eindhoven 1975). Figure reprinted with the courtesy of FEI Company. Chapter 3 title image conveys the important concept of Fig. 3.7. 3.1 3.2 3.3

3.4

Figure reproduced with the courtesy of the Huntington Library, Art Collections, and Botanical Gardens, San Marino, CA. J. C. H. Spence: Acta Cryst. A49, 231 (1993). J. M. Zuo, M. Kim, M. O’Keefe, and J. C. H. Spence: Nature 401, 49 (1999). Figure reproduced with the courtesy of N ature and J. C. H. Spence. U. Kriplani: Kinematical M¨ ossbauer Diffraction from Polycrystalline 57 Fe. Ph.D. Thesis, California Institute of Technology, California (2000).

Chapter 4 title drawing of Gatan 666 EELS spectrometer. Figure reprinted with the courtesy of Dr. C. C. Ahn. 4.1

4.2

4.3

D. H. Pearson: Measurements of White Lines in Transition Metals and Alloys using Electron Energy Loss Spectrometry. Ph.D. Thesis, California Institute of Technology, California (1991). Figure reprinted with the courtesy of Dr. D. H. Pearson. M. M. Disko: ‘Transmission Electron Energy-Loss Spectrometry in Materials Science’. In: Transmission Electron Energy Loss Spectroscopy in Materials Science, ed. by M. M. Disko, C. C. Ahn and B. Fultz (Minerals, Metals & Materials Society, Warrendale, PA 1992). Reprinted with courtesy of The Minerals, Metals & Materials Society. J. K. Okamoto: Temperature-Dependent Extended Electron Energy Loss Fine Structure Measurements from K, L23 , and M45 Edges in Metals, Intermetallic Alloys, and Nanocrystalline Materials. Ph.D.

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4.4

4.5

4.6

4.7

4.8 4.8

4.10

4.11 4.12 4.13

4.14

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Thesis, California Institute of Technology, California (1993). Figure reprinted with the courtesy of Dr. J. K. Okamoto. A. Hightower: Lithium Electronic Environments in Rechargeable Battery Electrodes. Ph.D. Thesis, California Institute of Technology, California (2000). R. F. Egerton: Electron Energy-Loss Spectroscopy in the Electron Microscope, 2nd edn. (Plenum Press, New York 1996). Figures reprinted with the courtesy of Plenum Press. D. B. Williams and C. B. Carter: Transmission Electron Microscopy: A Textbook for Materials Science (Plenum Press, New York 1996). Figure reprinted with the courtesy of Plenum Press. R. D. Leapman: ‘EELS Quantitative Analysis’. In: Transmission Electron Energy Loss Spectroscopy in Materials Science, ed. by M. M. Disko, C. C. Ahn and B. Fultz (Minerals, Metals & Materials Society, Warrendale, PA 1992). Reprinted with courtesy of The Minerals, Metals & Materials Society. Figure reprinted with the courtesy of K. T. Moore. D. B. Williams: Practical Analytical Electron Microscopy in Materials Science (Philips Electron Optics Publishing Group, Mahwah, NJ 1984). Figure reprinted with the courtesy of FEI Company. E. H. S. Burhop: The Auger Effect and Other Radiationless Transitions (Cambridge University Press 1952). Figure reprinted with the permission of Cambridge University Press. Figure reprinted with the courtesy of Dr. K. M. Krishnan. Figure reprinted with the courtesy of C. M. Garland. C. Nockolds, M. J. Nasir, G. Cliff and G. W. Lorimer, In: Electron Microscopy and Analysis - 1979, ed. by T. Mulvey (The Institute of Physics, Bristol and London, 1980) p. 417. J. M. Howe and R. Gronsky: Scripta Metall., 20, 1168 (1986). Figure reprinted with the courtesy of Elsevier Science Ltd.

Chapter 5 title image of electron diffraction pattern from precipitates in an Al-Cu-Li alloy. 5.1 5.2 5.3

5.4

The International Union of Crystallography: International Tables for X-ray Crystallography (Kynock Press, Birmingham, England, 1952-). Figure reprinted with the courtesy of Dr. S. R. Singh. Y. C. Chang: Crystal Structure and Nucleation Behavior of {111} Precipitates in an Al-3.9Cu-0.5Mg-0.5Ag Alloy. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA (1993). Figure reprinted with the courtesy of Dr. Y. C. Chang. R. J. Rioja and D. E. Laughlin: Metall. Trans., 8A, 1259 (1977). Figure reprinted with the courtesy of The Minerals, Metals and Materials Society.

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5.6 5.7

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G. Thomas and M. J. Goringe: Transmission Electron Microscopy of Materials (Wiley–Interscience, New York 1979). Figure reprinted with the courtesy of Wiley–Interscience. Figure and problem reprinted with the courtesy of Prof. D. E. Laughlin. F. K. LeGoues, H. I. Aaronson, Y. W. Lee and G. J. Fix: In: Proceedings of the International Conference on Solid-Solid Phase Transformations ed. by H. I. Aaronson, D. E. Laughlin, R. F. Sekerka and C. M. Wayman (TMS-AIME, Warrendale, PA 1982) p. 427. Figure reprinted with the courtesy of The Minerals, Metals and Materials Society.

Chapter 6 title image of Kikuchi map of bcc crystal. G. Thomas and M. J. Goringe: Transmission Electron Microscopy of Materials (Wiley–Interscience, New York 1979). Figure reprinted with the courtesy of Wiley–Interscience. 6.1

G. Thomas and M. J. Goringe: Transmission Electron Microscopy of Materials (Wiley–Interscience, New York 1979). Figure reprinted with the courtesy of Wiley–Interscience. 6.2 J. W. Edington: Practical Electron Microscopy in Materials Science, 2. Electron Diffraction in the Electron Microscope (Philips Technical Library, Eindhoven 1975). Figure reprinted with the courtesy of FEI Company. 6.3 Dr. J.-S. Chen, unpublished results. 6.4 M. Tanaka and M. Terauchi: Convergent-Beam Electron Diffraction (JEOL Ltd., Nakagami, Tokyo 1985). Figures reprinted with the courtesy of JEOL, Ltd. Worked thickness example on pp. 38-39. 6.5 R. Ayer: J. Electron Micros. Tech. 13, 16 (1989). Figure reprinted with the courtesy of Alan R. Liss, Inc. 6.6 S. J. Rozeveld: Measurement of Residual Stress in an Al-SiCw Composite by Convergent-Beam Electron Diffraction, Ph.D. Thesis, CarnegieMellon University, Pittsburgh, PA (1991). Figure reprinted with the courtesy of Dr. S. J. Rozeveld. 6.7 B. F. Buxton, et al.: Proc. Roy. Soc. London A281, 188 (1976). B. F. Buxton, et al.: Phil. Trans. Roy. Soc. London, A281, 171 (1976). Tables reprinted with the courtesy of The Royal Society, London. 6.8 M. Tanaka, H. Sekii and T. Nagasawa: Acta Cryst. A39, 825 (1983). Figure reprinted with the courtesy of the International Union of Crystallography. 6.9 M. Tanaka, R. Saito and H. Sekii: Acta Cryst. A39, 359 (1983). Figure reprinted with the courtesy of International Union of Crystallography. 6.10 J. M. Howe, M. Sarikaya and R. Gronsky: Acta Cryst. A42, 371 (1986). Figure reprinted with the courtesy of International Union of Crystallography. 6.11 The International Union of Crystallography: International Tables for X-ray Crystallography (Kynock Press, Birmingham, England, 1952-).

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6.12 J. W. Steeds and R. Vincent: ‘Use of High-Symmetry Zone Axes in Electron Diffraction in Determining Crystal Point and Space Groups’, J. Appl. Cryst. 16 317 (1983). 6.13 J. Gjønnes and A. F. Moodie: Acta Cryst. 19, 65 (1965). 6.14 M. J. Kaufman and H. L. Fraser: Acta Metall. 33, 194 (1985). Figure reprinted with the courtesy of Elsevier Science Ltd. Chapter 7 title image of dislocations in Al. 7.1

7.2

7.3 7.4

7.5 7.6 7.7 7.8

7.9

7.10

7.11

7.12 7.13 7.14 7.15

P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley, and M. J. Whelan: Electron Microscopy of Thin Crystals (R. E. Krieger, Malabar, Florida 1977). Figure reprinted with the courtesy of R. E. Krieger. J. W. Edington: Practical Electron Microscopy in Materials Science, 3. Interpretation of Transmission Electron Micrographs (Philips Technical Library, Eindhoven 1975). Figure reprinted with the courtesy of FEI Company. Figure reprinted with the courtesy of Dr. Y. C. Chang. G. Thomas and M. J. Goringe: Transmission Electron Microscopy of Materials (Wiley–Interscience, New York 1979). Figure reprinted with the courtesy of Wiley–Interscience. Figure reprinted with the courtesy of Dr. S. R. Singh. J. M. Howe, H. I. Aaronson and R. Gronsky: Acta Metall. 33, 641 (1985). Figure reprinted with the courtesy of Elsevier Science Ltd. P. B. Hirsch, A. Howie and M. J. Whelan: Phil. Trans. Royal Soc. (London) 252A, 499 (1960). D. J. H. Cockayne, I. L. F. Ray, and M. J. Whelan: Philos. Mag. 20, 1265 (1969). D. J. H. Cockayne, M. L. Jenkins, and I. L. F. Ray: Philos. Mag. 24, 1383 (1971). L. Reimer: Transmission Electron Microscopy: Physics of Image Formation and Microanalysis, 4th edn. (Springer–Verlag, New York 1997). Figure reprinted with the courtesy of Springer–Verlag. A. Garg and J. M. Howe: Acta Metall. Mater. 39, 1934 (1991). A. Garg, Y. C. Chang and J. M. Howe: Acta Metall. Mater. 41, 240 (1993). Figures reprinted with the courtesy of Elsevier Science Ltd. J. W. Edington: Practical Electron Microscopy in Materials Science, 3. Interpretation of Transmission Electron Micrographs (Philips Technical Library, Eindhoven 1975) p. 40. R. Gevers, A. Art and S. Amelinckx: Phys. Stat. Sol. 3, 1563 (1963). N. Prabhu and J. M. Howe: Philos. Mag. A 63, 650 (1991). Figure reprinted with the courtesy of Taylor & Francis, Ltd. M. F. Ashby and Brown: Philos. Mag. 8, 1083 (1963). H. P. Degischer: Philos. Mag. 26, 1147 (1972). Figure reprinted with the courtesy of Taylor & Francis, Ltd. M. Hwang, D. E. Laughlin and I. M. Bernstein: Acta Metall. 28, 629 (1980). Figure reprinted with the courtesy of Elsevier Science Ltd.

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7.16 Figure reprinted with the courtesy of Dr. A. Garg. Chapter 8 title figure of (400)fcc diffraction from a nanocrystalline iron alloy (Mo Kα radiation). 8.1 8.2 8.3 8.4

H. P. Klug and L. E. Alexander: X-Ray Diffraction Procedures (Wiley– Interscience, New York 1974) pp. 687-692. H. P. Klug and L. E. Alexander: X-Ray Diffraction Procedures (Wiley– Interscience, New York 1974) pp. 655-665. B. E. Warren: X-Ray Diffraction (Dover, New York, 1990) pp. 251-275. H. Frase: Vibrational and Magnetic Properties of Mechanically Attrited Ni3 Fe Nanocrystals. Ph.D. Thesis, California Institute of Technology, California (1998).

Chapter 9 title image conveys the important concept of Fig. 9.3. 9.1 9.2

B. E. Warren: X-Ray Diffraction (Dover, New York, 1990) pp. 178-193. F. Ducastelle: Order and Phase Stability in Alloys (North–Holland, Amsterdam 1991) pp. 439-442. This “relaxation energy” is important for the thermodynamics of many alloys. 9.3 J. A. Rodriguez, S. C. Moss, J. L. Robertson, J. R. D. Copley, D. A. Neumann and J. Major: Phys. Rev. B 74, 104115 (2006). 9.4 B. E. Warren: X-Ray Diffraction (Dover, New York, 1990) pp. 206-250. 9.5 L. H. Schwartz and J. B. Cohen: Diffraction from Materials (Springer– Verlag, Berlin 1987) pp. 407-409. 9.6 J. M. Cowley: Diffraction Physics, 2nd edn. (North–Holland Publishing, Amsterdam 1975) pp. 152-154. 9.7 A. Williams: Atomic Structure of Transition Metal Based Metallic Glasses. Ph.D. Thesis, California Institute of Technology, California (1981). 9.8 H. P. Klug and L. E. Alexander: X-Ray Diffraction Procedures (Wiley– Interscience, New York 1974) pp. 791-859. 9.9 T. Egami: ‘PDF Analysis Aplied to Crystalline Materials’, in: Local Structure from Diffraction, ed. by S. J. L. Billinge and M. F. Thorpe (Plenum, New York 1998) pp. 1-21. 9.10 A. Guinier: X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies (Dover, New York 1994) pp. 344-349. Chapter 10 title image of Pb precipitate in Al. Figure reprinted with the courtesy of U. Dahmen. 10.1 J. M. Cowley and A. F. Moodie: Acta Cryst. 10, 609 (1957). Ibid. 12, 353, 360, 367 (1959).

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10.2 M. A. O’Keefe: ‘Electron image simulation; a complementary processing technique’. In: Proceedings of the 3rd Pfeffercorn Conference on Electron Optical Systems, Ocean City, MD ed. by J. J. Hren, F. A. Lenz, E. Munro, P. B. Sewell, and S. A. Bhatt (Scanning Electron Microscopy, Inc., Illinois 1984) pp. 209-220. 10.3 R. R. Meyer, J. Sloan, R. E. Dunin-Borkowski, A. I. Kirkland, M. C. Novotny, S. R. Bailey, J. L. Hutchison and M. L. H. Green: Science 289, 1324 (2000). Figure reproduced with the courtesy of J. L. Hutchison and the American Association for the Advancement of Science. 10.4 S. D. Hudson, H. T. Jung, V. Percec, W. D. Cho, G. Johansson, G. Ungar, V. S. K. Balagurusamy: Science 278, 449 (1997). Figure reproduced with the courtesy of S. D. Hudson and the American Association for the Advancement of Science. 10.5 S. R. Singh and J. M. Howe: Philos. Mag. A 66, 746 (1992). Figure reprinted with the courtesy of Taylor & Francis, Ltd. 10.6 S. Das, J. M. Howe and J. H. Perepezko: Metall. Mater. Trans. 27A, 1627 (1996). Figure reprinted with the courtesy of The Minerals, Metals & Materials Society. 10.7 G. Rao, J. M. Howe and P. Wynblatt, unpublished research. 10.8 U. Dahmen: Micros. Soc. Amer. Bull. 24, 341 (1994). Figure reprinted with the courtesy of Microscopy Society of America. 10.9 Figure reprinted with the courtesy of R. Gronsky and D. Acklund. 10.10 J. M. Howe and S. J. Rozeveld: J. Micros. Res. Tech. 23, 233 (1992). Reprinted with the courtesy of Wiley–Liss, Inc. 10.11 M. M. Tsai: Determination of the Growth Mechanisms of TiH in Ti Using High-Resolution and Energy-Filtering Transmission Electron Microscopy. Ph.D. Thesis, University of Virginia, Charlottesville, VA (1997). Figure reprinted with the courtesy of Dr. M. M. Tsai. 10.14 such as Gatan Digital MicrographTM or NIH Image. 10.15 B. Laird and J. M. Howe, unpublished research. 10.16 R. Kilaas and R. Gronsky: Ultramicros. 16, 193 (1985). Figure reprinted with the courtesy of Elsevier Science Publishing B.V. 10.17 J.O. Malm and M.A. O’Keefe: Ultramicros. 68, 13 (1997). 10.18 S.-C.Y. Tsen, P.A. Crozier and J. Liu: Ultramicros. 98, 63 (2003). Chapter 11 title figure shows HAADF images acquired with a Cs -corrected instrument. The images were acquired at different values of defocus as labeled. Together with other measurements and computational support, the images show how that La atoms segregate to sites on the surfaces of an Al2 O3 crystal, which correspond to defocus values of 0 and −8 nm. Bar length is 1 nm. After S. Wang, A.Y. Borisevich, S.N. Rashkeev, M.V. Glazoff, K. Sohlberg, S.J. Pennycook, and S.T. Pantelides: Nature Materials 3, 143 (2004).

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11.1 After N. D. Browning, D. J. Wallis, P. D. Nellist and S. J. Pennycook: Micron 28, 334 (1997). Reprinted with the courtesy of Elsevier Science Ltd. 11.2 S. J. Pennycook, D. E. Jesson, M. F. Chisholm, N. D. Browning, A. J. McGibbon, and M. M. McGibbon: J. Micros. Soc. Amer. 1, 234 (1995). Reprinted with the courtesy of Microscopy Society of America. 11.3 A. Amali and P. Rez: Microsc. and Microanal. 3, 28 (1997). 11.4 A.R. Lupini and S. J. Pennycook: Ultramicroscopy 96, 313 (2003). 11.5 P. M. Voyles and D. A. Muller, private communication. See also P. M. Voyles, D. A. Muller, J. L. Grazul, P. H. Citrin, and H-.J. L. Gossmann: Nature 416, 826 (2002). 11.6 O.L. Krivanek, N. Dellby, and A.R. Lupini: Ultramicroscopy 78, 1 (1999). 11.7 S. Uhlemann and M. Haider: Ultramicroscopy 72, 109 (1998). 11.8 Q.M. Ramasse and A.L. Bleloch, Ultramicroscopy 106, 37 (2005). 11.9 H. M¨ uller, S. Uhlemann, P. Hartel and M. Haider: Microsc. and Microanal. 12, 442 (2006). 11.10 M. Lentzen: Microsc. Microanal. 12, 191 (2006). 11.11 A.Y. Borisevich, A.R. Lupini and S.J. Pennycook: Proc. Nat. Acad. Sci. 103, 3044 (2006). 11.12 K. van Benthem, A.R. Lupini, M. Kim, K.-S. Baik, S. Doh, J.-H. Lee, M.P. Oxley, S.D. Findlay, L.J. Allen, J.T. Luck and S. J. Pennycook: Appl. Phys. Lett. 87, 034104 (2005). Reprinted with the courtesy of the American Institute of Physics. 11.13 N.D. Browning, R.P. Erni, J.C. Idrobo, A. Ziegler, C.F. Kisielowski, R.O. Ritchie: Microsc. Microanal. 11 (Suppl 2), 1434 (2005). Chapter 12 title figure is an enlargement of Fig. 12.15. 12.1 Figure reprinted with the courtesy of Dr. Y. C. Chang. 12.2 P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley, and M. J. Whelan: Electron Microscopy of Thin Crystals (R. E. Krieger, Malabar, Florida 1977) pp. 222-242. 12.3 N. Prabhu and J. M. Howe: Philos. Mag. A 63, 650 (1991). Figure reprinted with the courtesy of Taylor & Francis, Ltd. 12.4 A. W. Wilson: Microstructural Examination of NiAl Alloys. Ph.D. Thesis, University of Virginia, Charlottesville, VA (1999). Figure reprinted with the courtesy of Dr. A. W. Wilson. 12.5 Peter Rez, private communication of academic course notes.

A. Appendix

A.1 Indexed Powder Diffraction Patterns

Fig. A.1. Indices of peaks in powder diffraction patterns from simple cubic, facecentered cubic, body-centered cubic, diamond cubic, and hexagonal close-packed crystals.

692

A. Appendix

Table A.1. Mass attenuation coefficients for characteristic Kα x-rays [cm2 /g] Z 1H 2 He 3 Li 4 Be 5B 6C 7N 8O 9F 10 Ne 11 Na 12 Mg 13 Al 14 Si 15 P 16 S 17 Cl 18 Ar 19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn 31 Ga 32 Ge 33 As 34 Se 35 Br 36 Kr 37 Rb 38 Sr 39 Y 40 Zr 41 Nb 42 Mo 43 Tc 44 Ru 45 Rh 46 Pd 47 Ag

Cr 0.412 0.498 1.30 3.44 7.59 15.0 24.7 37.8 51.5 74.1 94.9 126 155 196 230 281 316 342 421 490 516 590 74.7 86.8 97.5 113 124 144 153 171 183 199 219 234 260 277 303 328 358 386 416 442 474 501 536 563 602

Co 0.397 0.343 0.693 1.67 3.59 7.07 11.7 18.0 24.7 35.8 46.2 61.9 76.4 97.8 115 142 161 176 218 255 269 291 325 408 393 57.2 63.2 73.5 78.0 87.1 93.4 102 112 120 133 142 156 170 185 200 216 230 247 262 280 295 316

Cu 0.391 0.292 0.500 1.11 2.31 4.51 7.44 11.5 15.8 22.9 29.7 40.0 49.6 63.7 75.5 93.3 106 116 145 170 180 200 219 247 270 302 321 48.8 51.8 57.9 62.1 67.9 74.7 80.0 89.0 95.2 104 113 124 139 145 154 166 176 189 199 213

Mo 0.373 0.202 0.198 0.256 0.368 0.576 0.845 1.22 1.63 2.35 3.03 4.09 5.11 6.64 7.97 9.99 11.5 12.8 16.2 19.3 20.8 23.4 26.0 29.9 33.1 37.6 41.0 46.9 49.1 54.0 57.0 61.2 66.1 69.5 75.6 79.3 85.1 90.6 97.0 16.3 17.7 18.8 20.4 21.7 23.3 24.7 26.5

Z 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94

Cd In Sn Sb Te I Xe Cs Ba La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu

Cr 626 663 691 723 740 796 721 760 570 225 238 238 251 294 279 309 298 332 325 347 352 386 387 431 425 432 457 501 499 520 541 551 541 597 643 666 691 680 734 758 743 739 768 738 766 800 760

Co 329 349 364 383 394 425 440 465 480 507 535 565 505 400 440 153 161 180 176 187 191 206 206 229 227 231 246 268 268 278 276 295 273 331 343 355 370 363 392 403 398 461 406 394 420 430 408

Cu 222 236 247 259 267 288 299 317 325 348 368 390 404 426 434 434 403 321 362 129 132 140 142 156 155 158 168 187 184 191 188 201 188 226 235 244 254 248 267 277 273 317 306 271 288 314 280

Mo 27.8 29.5 31.0 32.7 33.8 36.7 38.2 40.7 42.3 44.9 47.7 50.7 53.0 56.3 57.8 60.9 62.6 65.8 68.3 71.3 74.4 77.9 80.4 84.0 86.9 90.4 93.8 97.4 100 104 107 112 115 118 122 126 132 117 108 87.0 88.0 90.8 96.5 101 102 42.2 39.9

Example: calculate the fraction, I/I0 , of Mo Kα x-rays transmitted through 0.01 cm of metallic Ag (having density 10.5 g cm−3 ): I/I0 = exp(−26.5 cm2 g−1 10.5 g cm−3 0.01 cm) = e−2.78 = 0.062 .

0.0

2.00 2.00 3.00 2.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 9.00 10.00 10.00 10.00 11.00 10.00 12.00 10.00 13.00 14.00 15.00 16.00 17.00 18.00 18.00 18.00 19.00 18.00 20.00 21.00 18.00 22.00 18.00 23.00 20.00

s

He Li+1 Li Be+2 Be B C N O O−1 O−2 F F−1 Ne Na+1 Na Mg+2 Mg Al+3 Al Si P S Cl Cl−1 Ar K+1 K Ca+2 Ca Sc Ti+4 Ti V+5 V Cr+4

1.96 1.98 2.71 1.99 3.71 4.73 5.75 6.78 7.80 8.71 9.59 8.82 9.73 9.83 9.88 10.57 9.91 11.51 9.93 12.44 13.44 14.46 15.48 16.51 17.36 17.54 17.65 18.21 17.72 19.09 20.13 17.81 21.17 17.84 22.21 19.80

0.05

1.84 1.94 2.22 1.97 3.07 4.06 5.12 6.18 7.25 7.92 8.54 8.30 9.02 9.35 9.55 9.76 9.66 10.48 9.74 11.23 12.15 13.14 14.18 15.24 15.76 16.30 16.68 16.74 16.94 17.34 18.36 17.26 19.41 17.37 20.48 19.23

0.1

1.66 1.86 1.90 1.93 2.47 3.32 4.33 5.39 6.47 6.89 7.22 7.56 8.04 8.65 9.03 9.03 9.27 9.51 9.43 10.06 10.78 11.63 12.58 13.60 13.81 14.66 15.30 15.25 15.78 15.73 16.65 16.42 17.64 16.63 18.66 18.34

0.15

1.46 1.76 1.74 1.87 2.06 2.70 3.57 4.57 5.63 5.84 5.96 6.71 6.98 7.81 8.38 8.34 8.76 8.74 9.02 9.16 9.68 10.33 11.11 12.00 12.02 12.96 13.77 13.74 14.42 14.31 15.14 15.36 16.05 15.70 17.00 17.23

0.2

1.26 1.65 1.63 1.80 1.83 2.27 2.96 3.83 4.81 4.89 4.90 5.86 5.98 6.93 7.65 7.62 8.16 8.08 8.53 8.47 8.86 9.34 9.93 10.64 10.59 11.45 12.29 12.28 13.03 12.97 13.74 14.20 14.58 14.65 15.47 15.98

0.25 1.06 1.52 1.51 1.73 1.69 1.98 2.50 3.22 4.09 4.10 4.06 5.06 5.09 6.09 6.90 6.89 7.52 7.45 7.98 7.88 8.24 8.60 9.04 9.58 9.53 10.23 10.98 10.99 11.72 11.72 12.43 13.01 13.21 13.54 14.03 14.68

0.3 0.89 1.40 1.39 1.64 1.60 1.80 2.18 2.75 3.49 3.47 3.42 4.36 4.35 5.31 6.17 6.16 6.86 6.82 7.41 7.32 7.70 8.03 8.38 8.79 8.75 9.28 9.91 9.92 10.59 10.60 11.25 11.88 11.96 12.46 12.71 13.42

0.35 0.74 1.27 1.27 1.55 1.52 1.68 1.95 2.40 3.01 2.98 2.94 3.76 3.74 4.63 5.48 5.48 6.22 6.20 6.83 6.77 7.21 7.55 7.86 8.19 8.16 8.57 9.06 9.07 9.64 9.66 10.24 10.85 10.86 11.43 11.54 12.25

0.4

Table A.2. Atomic form factors for high-energy x-rays 0.5 0.51 1.03 1.03 1.37 1.36 1.53 1.69 1.94 2.34 2.32 2.30 2.88 2.85 3.54 4.30 4.30 5.03 5.04 5.70 5.70 6.25 6.68 7.02 7.31 7.31 7.58 7.89 7.90 8.27 8.28 8.70 9.20 9.16 9.70 9.67 10.27

0.6 0.35 0.82 0.83 1.18 1.20 1.40 1.54 1.70 1.95 1.94 1.93 2.31 2.29 2.80 3.40 3.40 4.05 4.07 4.70 4.72 5.32 5.84 6.26 6.60 6.61 6.88 7.13 7.13 7.40 7.40 7.69 8.05 8.02 8.44 8.38 8.83

0.7 0.25 0.64 0.65 1.01 1.03 1.28 1.43 1.55 1.72 1.71 1.71 1.96 1.95 2.30 2.76 2.76 3.29 3.30 3.87 3.89 4.48 5.03 5.51 5.92 5.93 6.26 6.53 6.53 6.77 6.77 7.00 7.26 7.25 7.55 7.51 7.84

0.8 0.18 0.51 0.51 0.85 0.88 1.15 1.32 1.45 1.57 1.57 1.57 1.74 1.73 1.97 2.31 2.31 2.73 2.73 3.21 3.23 3.76 4.29 4.80 5.25 5.26 5.65 5.97 5.97 6.24 6.23 6.47 6.69 6.68 6.92 6.90 7.14

0.13 0.40 0.41 0.72 0.74 1.02 1.22 1.35 1.46 1.46 1.47 1.59 1.59 1.76 2.00 2.00 2.32 2.32 2.71 2.72 3.17 3.66 4.15 4.62 4.62 5.04 5.41 5.41 5.73 5.72 5.98 6.22 6.21 6.43 6.41 6.62

0.9 0.10 0.32 0.32 0.60 0.62 0.90 1.12 1.27 1.38 1.38 1.38 1.48 1.48 1.61 1.79 1.79 2.03 2.03 2.33 2.34 2.71 3.13 3.58 4.03 4.03 4.47 4.87 4.87 5.22 5.22 5.51 5.78 5.76 6.00 5.98 6.19

1.0 0.05 0.20 0.21 0.43 0.44 0.69 0.92 1.09 1.22 1.22 1.22 1.33 1.32 1.42 1.53 1.53 1.66 1.66 1.85 1.84 2.08 2.37 2.71 3.08 3.08 3.47 3.86 3.86 4.24 4.24 4.58 4.91 4.88 5.19 5.15 5.40

1.2

1.4 0.03 0.13 0.14 0.30 0.31 0.53 0.74 0.92 1.07 1.07 1.07 1.19 1.19 1.28 1.37 1.37 1.46 1.46 1.58 1.57 1.72 1.91 2.14 2.41 2.41 2.72 3.05 3.05 3.40 3.40 3.73 4.08 4.05 4.39 4.34 4.63

1.6 0.02 0.09 0.09 0.22 0.22 0.40 0.59 0.77 0.93 0.92 0.92 1.06 1.05 1.16 1.25 1.25 1.33 1.33 1.41 1.41 1.51 1.63 1.78 1.97 1.97 2.20 2.46 2.46 2.74 2.74 3.03 3.35 3.32 3.65 3.61 3.91

1.8 0.01 0.06 0.06 0.16 0.16 0.30 0.47 0.64 0.79 0.79 0.79 0.93 0.93 1.04 1.14 1.14 1.22 1.22 1.29 1.29 1.37 1.45 1.56 1.69 1.69 1.85 2.04 2.04 2.26 2.25 2.49 2.76 2.74 3.03 3.00 3.28

2.0 0.01 0.04 0.05 0.12 0.12 0.23 0.37 0.53 0.68 0.67 0.67 0.81 0.81 0.93 1.03 1.03 1.12 1.12 1.20 1.20 1.27 1.34 1.41 1.50 1.50 1.62 1.75 1.75 1.91 1.91 2.10 2.31 2.30 2.54 2.51 2.75

2.5 0.00 0.02 0.02 0.06 0.06 0.13 0.22 0.33 0.44 0.44 0.44 0.57 0.56 0.68 0.79 0.79 0.89 0.89 0.98 0.98 1.06 1.12 1.18 1.24 1.24 1.30 1.37 1.37 1.45 1.45 1.54 1.64 1.64 1.77 1.76 1.90

3.0 0.00 0.01 0.01 0.03 0.03 0.07 0.13 0.21 0.29 0.29 0.29 0.39 0.39 0.49 0.59 0.59 0.69 0.69 0.78 0.78 0.87 0.94 1.01 1.07 1.07 1.12 1.18 1.18 1.23 1.23 1.28 1.34 1.34 1.41 1.41 1.48

4.0 0.00 0.00 0.00 0.01 0.01 0.03 0.05 0.09 0.14 0.13 0.13 0.19 0.19 0.25 0.33 0.32 0.40 0.40 0.48 0.48 0.56 0.63 0.71 0.77 0.77 0.84 0.90 0.89 0.95 0.95 1.00 1.04 1.04 1.09 1.09 1.13

5.0 0.00 0.00 0.00 0.01 0.01 0.01 0.02 0.04 0.07 0.07 0.07 0.10 0.10 0.14 0.18 0.19 0.23 0.24 0.29 0.29 0.35 0.42 0.48 0.54 0.54 0.60 0.66 0.66 0.72 0.72 0.78 0.82 0.83 0.87 0.87 0.91

6.0 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.02 0.04 0.04 0.03 0.06 0.06 0.08 0.11 0.11 0.14 0.14 0.18 0.18 0.22 0.27 0.32 0.37 0.37 0.43 0.48 0.48 0.53 0.53 0.58 0.64 0.63 0.69 0.68 0.73

A.3 Atomic Form Factors for X-Rays 693

0.0

24.00 23.00 25.00 24.00 26.00 25.00 27.00 26.00 28.00 27.00 29.00 28.00 30.00 31.00 32.00 33.00 34.00 35.00 36.00 36.00 36.00 37.00 36.00 38.00 36.00 39.00 36.00 40.00 36.00 41.00 36.00 42.00 44.00 45.00 44.00 46.00 45.00 47.00

s

Cr Mn+2 Mn Fe+2 Fe Co+2 Co Ni+2 Ni Cu+2 Cu Zn+2 Zn Ga Ge As Se Br Br−1 Kr Rb+1 Rb Sr+2 Sr Y+3 Y Zr+4 Zr Nb+5 Nb Mo+6 Mo Ru Rh Pd+2 Pd Ag+2 Ag

23.33 22.71 24.28 23.71 25.30 24.72 26.33 25.72 27.36 26.73 28.38 27.73 29.40 30.30 31.28 32.27 33.27 34.29 35.12 35.31 35.44 35.95 35.53 36.80 35.59 37.82 35.64 38.85 35.68 39.97 35.72 41.00 43.06 44.09 43.46 45.23 44.46 46.14

0.05

21.79 21.87 22.61 22.89 23.68 23.90 24.75 24.92 25.81 25.94 26.87 26.96 27.93 28.67 29.53 30.46 31.44 32.45 32.91 33.47 33.90 33.91 34.20 34.47 34.44 35.37 34.62 36.36 34.77 37.59 34.90 38.64 40.77 41.84 41.93 43.18 42.95 43.97

0.1

20.02 20.63 20.76 21.65 21.83 22.67 22.90 23.70 23.98 24.74 25.05 25.77 26.13 26.79 27.50 28.30 29.16 30.09 30.24 31.07 31.75 31.69 32.29 32.18 32.72 32.91 33.08 33.76 33.37 34.90 33.62 35.89 37.96 39.02 39.67 40.37 40.70 41.17

0.15

Table A.2. (continued)

0.2

18.25 19.13 19.01 20.14 20.05 21.17 21.10 22.20 22.16 23.24 23.22 24.29 24.30 24.94 25.57 26.23 26.95 27.75 27.73 28.61 29.41 29.38 30.10 30.00 30.70 30.65 31.21 31.37 31.65 32.30 32.03 33.18 35.09 36.09 37.02 37.31 38.03 38.17

0.25

16.56 17.52 17.36 18.51 18.35 19.52 19.37 20.54 20.40 21.58 21.44 22.62 22.49 23.19 23.80 24.39 25.00 25.66 25.61 26.38 27.16 27.16 27.91 27.88 28.59 28.50 29.21 29.16 29.76 29.89 30.25 30.66 32.36 33.28 34.27 34.31 35.23 35.22

0.3 14.97 15.92 15.81 16.87 16.75 17.85 17.71 18.85 18.70 19.86 19.71 20.89 20.74 21.50 22.15 22.74 23.30 23.87 23.82 24.47 25.17 25.18 25.88 25.89 26.57 26.51 27.23 27.11 27.84 27.71 28.41 28.39 29.88 30.69 31.62 31.55 32.51 32.44

0.35 13.52 14.42 14.36 15.32 15.24 16.24 16.15 17.20 17.09 18.17 18.06 19.17 19.05 19.87 20.58 21.20 21.77 22.30 22.27 22.84 23.44 23.45 24.08 24.11 24.74 24.70 25.39 25.27 26.01 25.78 26.61 26.38 27.68 28.39 29.22 29.11 30.01 29.94

0.4 12.24 13.06 13.04 13.89 13.85 14.75 14.70 15.65 15.59 16.57 16.50 17.52 17.44 18.30 19.07 19.75 20.35 20.89 20.88 21.41 21.95 21.95 22.52 22.54 23.12 23.10 23.72 23.63 24.33 24.11 24.93 24.64 25.77 26.39 27.10 26.99 27.79 27.73

0.5 10.19 10.84 10.85 11.50 11.51 12.22 12.22 12.97 12.97 13.77 13.76 14.60 14.58 15.43 16.25 17.01 17.71 18.33 18.33 18.89 19.41 19.41 19.92 19.92 20.43 20.43 20.95 20.92 21.48 21.37 22.02 21.82 22.73 23.20 23.71 23.65 24.23 24.20

0.6 8.77 9.24 9.25 9.75 9.76 10.30 10.32 10.91 10.92 11.56 11.57 12.25 12.25 13.02 13.79 14.56 15.29 15.98 15.99 16.61 17.19 17.19 17.72 17.72 18.22 18.23 18.71 18.72 19.19 19.18 19.67 19.62 20.43 20.83 21.22 21.21 21.63 21.63

0.7 7.80 8.14 8.15 8.51 8.52 8.93 8.94 9.39 9.40 9.90 9.91 10.45 10.46 11.09 11.76 12.46 13.17 13.86 13.86 14.53 15.15 15.15 15.73 15.72 16.27 16.27 16.77 16.80 17.25 17.30 17.71 17.75 18.58 18.96 19.32 19.32 19.68 19.68

0.8 7.12 7.37 7.38 7.65 7.65 7.96 7.96 8.30 8.31 8.69 8.70 9.11 9.12 9.62 10.17 10.76 11.38 12.02 12.02 12.67 13.29 13.29 13.90 13.89 14.47 14.47 15.01 15.03 15.51 15.56 15.99 16.06 16.95 17.36 17.73 17.74 18.08 18.09

0.9 6.61 6.81 6.81 7.03 7.03 7.26 7.27 7.52 7.53 7.81 7.82 8.14 8.14 8.52 8.95 9.43 9.94 10.50 10.50 11.08 11.67 11.67 12.25 12.25 12.83 12.83 13.38 13.39 13.90 13.95 14.41 14.47 15.44 15.88 16.28 16.30 16.67 16.67

1.0 6.18 6.37 6.37 6.55 6.55 6.75 6.75 6.95 6.95 7.18 7.18 7.42 7.43 7.71 8.04 8.41 8.82 9.28 9.28 9.77 10.29 10.29 10.82 10.83 11.37 11.37 11.91 11.91 12.43 12.46 12.94 12.99 14.00 14.47 14.92 14.93 15.33 15.34

1.2 5.38 5.59 5.59 5.78 5.78 5.96 5.96 6.13 6.12 6.29 6.29 6.46 6.46 6.64 6.84 7.06 7.31 7.59 7.59 7.91 8.27 8.27 8.65 8.66 9.07 9.07 9.51 9.51 9.97 9.97 10.43 10.45 11.42 11.90 12.37 12.38 12.83 12.84

1.4 4.61 4.86 4.86 5.08 5.08 5.28 5.28 5.46 5.46 5.62 5.63 5.78 5.78 5.93 6.08 6.24 6.40 6.58 6.58 6.78 7.01 7.01 7.26 7.26 7.54 7.54 7.85 7.85 8.19 8.19 8.55 8.56 9.35 9.77 10.21 10.21 10.64 10.64

1.6 3.88 4.15 4.15 4.39 4.40 4.62 4.62 4.82 4.83 5.01 5.02 5.19 5.19 5.35 5.50 5.64 5.78 5.92 5.92 6.06 6.22 6.22 6.38 6.38 6.57 6.57 6.78 6.78 7.01 7.01 7.26 7.26 7.85 8.17 8.52 8.52 8.89 8.89

1.8 3.25 3.51 3.51 3.76 3.76 3.99 4.00 4.22 4.22 4.42 4.43 4.61 4.62 4.80 4.97 5.12 5.27 5.41 5.41 5.54 5.66 5.66 5.79 5.79 5.93 5.93 6.07 6.07 6.23 6.23 6.40 6.41 6.81 7.05 7.31 7.31 7.58 7.58

2.0 2.73 2.97 2.97 3.20 3.20 3.43 3.43 3.65 3.66 3.86 3.87 4.07 4.07 4.27 4.45 4.63 4.79 4.94 4.94 5.08 5.21 5.21 5.33 5.33 5.45 5.45 5.57 5.57 5.69 5.69 5.82 5.82 6.11 6.28 6.46 6.46 6.66 6.66

2.5 1.89 2.04 2.04 2.20 2.20 2.37 2.37 2.55 2.55 2.73 2.73 2.91 2.92 3.11 3.29 3.48 3.67 3.84 3.84 4.01 4.18 4.18 4.33 4.33 4.47 4.47 4.60 4.60 4.73 4.72 4.84 4.83 5.05 5.15 5.25 5.25 5.36 5.36

3.0 1.48 1.57 1.57 1.66 1.66 1.77 1.77 1.88 1.88 2.01 2.01 2.14 2.14 2.28 2.43 2.59 2.75 2.92 2.92 3.08 3.25 3.25 3.41 3.41 3.57 3.57 3.72 3.72 3.87 3.86 4.01 4.00 4.25 4.36 4.47 4.47 4.57 4.57

4.0 1.13 1.17 1.17 1.22 1.21 1.26 1.26 1.31 1.31 1.36 1.36 1.42 1.42 1.48 1.55 1.62 1.71 1.80 1.80 1.89 2.00 2.00 2.11 2.10 2.22 2.22 2.34 2.34 2.47 2.46 2.59 2.59 2.84 2.96 3.09 3.09 3.22 3.21

5.0 0.92 0.96 0.96 1.00 1.00 1.03 1.04 1.07 1.07 1.11 1.11 1.14 1.14 1.18 1.22 1.26 1.30 1.34 1.34 1.39 1.44 1.44 1.50 1.50 1.56 1.56 1.62 1.63 1.70 1.70 1.77 1.77 1.94 2.03 2.12 2.12 2.21 2.21

6.0 0.73 0.78 0.78 0.82 0.82 0.86 0.86 0.90 0.90 0.94 0.94 0.97 0.97 1.00 1.03 1.07 1.10 1.13 1.13 1.16 1.19 1.19 1.23 1.23 1.27 1.26 1.31 1.30 1.35 1.34 1.39 1.38 1.48 1.54 1.60 1.59 1.66 1.66

694 A. Appendix

0.0

46.00 48.00 49.00 50.00 51.00 52.00 53.00 54.00 54.00 54.00 55.00 54.00 56.00 54.00 57.00 54.00 58.00 56.00 59.00 57.00 60.00 59.00 62.00 60.00 63.00 61.00 64.00 65.00 66.00 67.00 68.00 69.00 70.00 71.00 72.00 73.00 74.00 75.00

s

Cd+2 Cd In Sn Sb Te I I−1 Xe Cs+1 Cs Ba+2 Ba La+3 La Ce+4 Ce Pr+3 Pr Nd+3 Nd Sm+3 Sm Eu+3 Eu Gd+3 Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re

45.47 47.09 47.97 48.92 49.90 50.89 51.90 52.69 52.92 53.09 53.53 53.21 54.35 53.30 55.35 53.37 56.39 55.31 57.44 56.32 58.47 58.34 60.53 59.35 61.55 60.36 62.55 63.60 64.63 65.65 66.68 67.70 68.72 69.70 70.72 71.74 72.76 73.78

43.98 44.81 45.52 46.33 47.21 48.14 49.13 49.50 50.14 50.64 50.61 51.03 51.13 51.34 51.98 51.60 53.05 53.40 54.29 54.43 55.34 56.50 57.46 57.53 58.52 58.56 59.41 60.64 61.69 62.75 63.81 64.86 65.91 66.78 67.74 68.73 69.74 70.76

0.05 0.1

41.74 41.94 42.60 43.29 44.02 44.81 45.69 45.75 46.61 47.36 47.31 48.00 47.85 48.54 48.53 49.00 49.58 50.62 50.96 51.67 52.02 53.77 54.15 54.83 55.22 55.88 55.98 57.37 58.44 59.51 60.59 61.66 62.73 63.47 64.30 65.19 66.11 67.06

39.06 38.95 39.66 40.31 40.95 41.61 42.34 42.28 43.11 43.92 43.91 44.67 44.61 45.36 45.23 45.97 46.24 47.41 47.62 48.45 48.66 50.56 50.77 51.63 51.83 52.69 52.57 53.98 55.06 56.14 57.23 58.31 59.39 60.11 60.86 61.64 62.46 63.31

0.15 0.2

Table A.2. (continued) 33.43 33.28 34.09 34.83 35.50 36.11 36.70 36.66 37.28 37.92 37.93 38.59 38.62 39.28 39.24 39.96 40.13 41.11 41.26 42.07 42.21 44.03 44.16 45.03 45.16 46.05 45.96 47.20 48.24 49.29 50.35 51.41 52.48 53.31 54.08 54.81 55.53 56.25

0.25 0.3

36.21 36.03 36.80 37.49 38.12 38.72 39.34 39.29 39.99 40.73 40.74 41.47 41.48 42.19 42.10 42.88 43.06 44.16 44.33 45.17 45.34 47.23 47.39 48.28 48.43 49.33 49.20 50.54 51.61 52.69 53.77 54.85 55.93 56.70 57.44 58.17 58.92 59.70 28.53 28.49 29.28 30.05 30.78 31.47 32.11 32.11 32.70 33.27 33.27 33.83 33.84 34.41 34.43 35.00 35.17 35.94 36.05 36.75 36.86 38.45 38.55 39.33 39.42 40.22 40.21 41.24 42.17 43.13 44.09 45.07 46.07 46.95 47.76 48.52 49.24 49.94

0.35 0.4 30.84 30.75 31.57 32.34 33.05 33.71 34.31 34.30 34.88 35.46 35.47 36.07 36.09 36.69 36.69 37.33 37.50 38.37 38.51 39.25 39.38 41.09 41.20 42.04 42.14 42.99 42.95 44.08 45.08 46.08 47.10 48.13 49.16 50.04 50.83 51.58 52.29 52.99

0.5 24.78 24.81 25.45 26.12 26.81 27.50 28.17 28.18 28.81 29.41 29.41 29.98 29.98 30.53 30.55 31.07 31.18 31.82 31.88 32.51 32.56 33.95 34.00 34.70 34.76 35.48 35.48 36.33 37.15 37.99 38.84 39.72 40.60 41.44 42.24 43.01 43.75 44.45

0.6 22.05 22.09 22.57 23.10 23.67 24.26 24.88 24.88 25.50 26.10 26.10 26.69 26.68 27.25 27.26 27.79 27.82 28.38 28.38 28.98 28.98 30.22 30.23 30.87 30.88 31.54 31.55 32.25 32.97 33.70 34.45 35.22 36.00 36.77 37.54 38.29 39.02 39.72

0.7 20.03 20.05 20.43 20.83 21.27 21.74 22.25 22.25 22.78 23.33 23.33 23.88 23.88 24.43 24.43 24.97 24.93 25.44 25.40 25.96 25.92 27.05 27.02 27.62 27.59 28.20 28.21 28.79 29.41 30.06 30.72 31.39 32.08 32.79 33.51 34.22 34.93 35.62

0.8 18.42 18.42 18.76 19.09 19.44 19.81 20.21 20.21 20.64 21.10 21.10 21.57 21.57 22.06 22.06 22.56 22.49 22.94 22.89 23.40 23.35 24.35 24.31 24.84 24.81 25.35 25.36 25.84 26.39 26.95 27.53 28.12 28.73 29.37 30.03 30.70 31.37 32.04

0.9 17.02 17.02 17.35 17.66 17.98 18.29 18.62 18.62 18.96 19.33 19.33 19.72 19.72 20.13 20.13 20.56 20.51 20.89 20.85 21.28 21.24 22.10 22.06 22.52 22.49 22.96 22.96 23.38 23.85 24.33 24.83 25.34 25.87 26.45 27.05 27.65 28.28 28.90

1.0 15.73 15.72 16.07 16.40 16.72 17.02 17.31 17.31 17.61 17.92 17.92 18.24 18.25 18.58 18.58 18.94 18.90 19.23 19.21 19.56 19.54 20.26 20.23 20.62 20.59 20.99 21.00 21.35 21.75 22.16 22.58 23.03 23.48 23.98 24.51 25.05 25.61 26.18

1.2 13.28 13.27 13.69 14.08 14.45 14.79 15.11 15.11 15.41 15.70 15.70 15.97 15.97 16.25 16.25 16.52 16.51 16.77 16.77 17.03 17.02 17.55 17.55 17.82 17.81 18.09 18.09 18.36 18.65 18.94 19.25 19.56 19.89 20.25 20.63 21.03 21.45 21.89

1.4 11.08 11.08 11.51 11.94 12.34 12.73 13.10 13.10 13.45 13.78 13.78 14.09 14.09 14.38 14.38 14.66 14.65 14.91 14.91 15.16 15.16 15.63 15.63 15.86 15.86 16.09 16.09 16.32 16.54 16.77 17.00 17.24 17.48 17.73 18.00 18.28 18.58 18.89

1.6 9.27 9.27 9.66 10.05 10.45 10.85 11.24 11.24 11.61 11.98 11.98 12.33 12.33 12.66 12.66 12.97 12.96 13.25 13.24 13.52 13.52 14.03 14.03 14.27 14.27 14.50 14.50 14.72 14.93 15.14 15.35 15.55 15.75 15.96 16.17 16.38 16.60 16.82

1.8 7.88 7.88 8.20 8.53 8.88 9.23 9.60 9.60 9.96 10.32 10.32 10.68 10.68 11.03 11.03 11.37 11.36 11.67 11.65 11.97 11.96 12.53 12.52 12.79 12.79 13.04 13.04 13.28 13.51 13.74 13.95 14.16 14.36 14.55 14.75 14.94 15.12 15.31

2.0 6.88 6.88 7.12 7.38 7.66 7.95 8.26 8.26 8.57 8.90 8.90 9.23 9.23 9.57 9.57 9.90 9.89 10.20 10.19 10.51 10.49 11.10 11.09 11.38 11.37 11.65 11.65 11.91 12.16 12.41 12.64 12.87 13.08 13.29 13.50 13.70 13.89 14.08

2.5 5.47 5.47 5.59 5.71 5.85 5.99 6.15 6.15 6.33 6.51 6.52 6.72 6.72 6.93 6.93 7.16 7.16 7.39 7.38 7.63 7.62 8.12 8.11 8.38 8.37 8.63 8.63 8.88 9.14 9.39 9.65 9.90 10.14 10.39 10.64 10.88 11.11 11.34

3.34 3.34 3.45 3.57 3.68 3.79 3.90 3.90 4.00 4.09 4.09 4.19 4.18 4.27 4.27 4.36 4.35 4.44 4.44 4.52 4.52 4.68 4.67 4.76 4.75 4.84 4.83 4.92 5.00 5.09 5.18 5.27 5.37 5.47 5.58 5.69 5.81 5.94

2.31 2.31 2.42 2.52 2.63 2.73 2.84 2.84 2.95 3.05 3.05 3.15 3.16 3.26 3.26 3.36 3.36 3.46 3.46 3.55 3.55 3.73 3.74 3.82 3.82 3.90 3.91 3.99 4.07 4.15 4.22 4.30 4.37 4.44 4.51 4.58 4.65 4.73

1.72 1.72 1.79 1.86 1.93 2.01 2.09 2.09 2.18 2.27 2.27 2.36 2.36 2.45 2.45 2.55 2.54 2.64 2.63 2.73 2.73 2.92 2.91 3.01 3.01 3.10 3.10 3.19 3.28 3.37 3.45 3.54 3.62 3.70 3.78 3.85 3.93 4.00

3.0 4.0 5.0 6.0 4.67 4.67 4.77 4.86 4.95 5.05 5.14 5.14 5.24 5.34 5.34 5.45 5.45 5.57 5.57 5.69 5.69 5.83 5.83 5.97 5.97 6.28 6.28 6.45 6.45 6.62 6.63 6.81 7.00 7.19 7.39 7.60 7.81 8.03 8.25 8.47 8.70 8.93

A.3 Atomic Form Factors for X-Rays 695

Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U

71.79 72.85 74.08 75.15 76.03 76.69 77.46 78.28 79.16 80.16 81.18 81.68 82.22 82.98 83.84 85.04 86.08

0.05 0.1

74.80 75.83 76.91 77.94 78.90 79.75 80.67 81.63 82.61 83.63 84.65 85.29 86.11 87.07 88.07 89.13 90.16

64.20 65.14 66.32 67.32 68.11 68.86 69.52 70.16 70.79 71.51 72.29 73.06 73.76 74.35 74.93 76.11 77.04

0.15 0.2

68.04 69.07 70.34 71.40 72.22 72.88 73.54 74.23 74.96 75.84 76.76 77.45 78.01 78.61 79.27 80.54 81.54

56.98 57.75 58.60 59.43 60.21 61.02 61.79 62.49 63.15 63.76 64.37 65.03 65.73 66.39 67.00 67.84 68.61

0.25 0.3

60.49 61.34 62.35 63.27 64.06 64.86 65.57 66.23 66.85 67.49 68.14 68.87 69.61 70.24 70.83 71.84 72.70 50.62 51.29 51.93 52.62 53.36 54.12 54.88 55.63 56.36 57.03 57.67 58.29 58.92 59.56 60.19 60.79 61.43

0.35 0.4 53.69 54.40 55.12 55.87 56.64 57.43 58.21 58.95 59.65 60.29 60.90 61.53 62.18 62.83 63.46 64.15 64.85

0.5 45.12 45.77 46.35 46.97 47.64 48.31 49.01 49.71 50.42 51.12 51.81 52.46 53.09 53.73 54.36 54.87 55.44

40.40 41.05 41.65 42.25 42.87 43.48 44.10 44.73 45.37 46.04 46.70 47.36 47.99 48.62 49.25 49.76 50.31

0.6 0.7 36.30 36.95 37.58 38.19 38.79 39.38 39.95 40.53 41.11 41.71 42.32 42.94 43.55 44.16 44.76 45.29 45.83

0.8 32.70 33.35 34.00 34.62 35.22 35.80 36.36 36.92 37.46 38.02 38.58 39.14 39.71 40.27 40.84 41.38 41.92

0.9 29.54 30.17 30.81 31.43 32.02 32.60 33.17 33.72 34.26 34.79 35.32 35.85 36.38 36.91 37.44 37.98 38.50

1.0 26.77 27.36 27.97 28.57 29.15 29.73 30.30 30.85 31.39 31.92 32.43 32.94 33.45 33.96 34.46 34.99 35.50

1.2 22.34 22.82 23.32 23.83 24.34 24.86 25.39 25.92 26.44 26.96 27.47 27.97 28.46 28.95 29.43 29.94 30.43

1.4 19.22 19.57 19.93 20.32 20.72 21.14 21.58 22.03 22.48 22.95 23.42 23.89 24.35 24.82 25.29 25.76 26.23

1.6 17.06 17.31 17.57 17.85 18.14 18.45 18.78 19.12 19.48 19.86 20.25 20.65 21.05 21.47 21.88 22.31 22.73

1.8 15.50 15.70 15.90 16.10 16.32 16.55 16.79 17.04 17.31 17.59 17.89 18.20 18.52 18.86 19.20 19.55 19.92

2.0 14.26 14.44 14.61 14.79 14.97 15.15 15.34 15.53 15.73 15.94 16.17 16.40 16.65 16.91 17.18 17.45 17.74

2.5 11.56 11.78 11.99 12.19 12.38 12.57 12.74 12.92 13.08 13.25 13.40 13.56 13.72 13.88 14.04 14.20 14.36

3.0 9.16 9.39 9.62 9.85 10.07 10.29 10.50 10.71 10.91 11.11 11.30 11.49 11.66 11.84 12.00 12.16 12.31

4.80 4.87 4.94 5.02 5.10 5.19 5.27 5.37 5.46 5.56 5.66 5.77 5.89 6.01 6.14 6.26 6.40

4.07 4.14 4.21 4.28 4.34 4.41 4.47 4.54 4.60 4.67 4.73 4.80 4.87 4.95 5.02 5.10 5.18

4.0 5.0 6.0 6.07 6.21 6.35 6.50 6.66 6.82 6.98 7.16 7.33 7.52 7.70 7.89 8.08 8.27 8.46 8.65 8.84

This table of x-ray atomic form factors, fx (s), for elements and some ions was obtained from calculations with a Dirac-Fock method by D. Rez, P. Rez and I. Grant: Acta Cryst. A50, 481 (1994). The column headings are s ≡ sinθ/λ, in units of ˚ A−1 . This diffraction vector, s, is converted to the Δk used in the text by multiplication by 4π. The tabulated values of fx (s) are in electron units. Conversion to units of cm is performed by multiplying them by the “classical electron radius,” e2 m−1 c−2 = 2.81794 × 10−13 cm.

0.0

76.00 77.00 78.00 79.00 80.00 81.00 82.00 83.00 84.00 85.00 86.00 87.00 88.00 89.00 90.00 91.00 92.00

s

Table A.2. (continued)

696 A. Appendix

A.4 X-Ray Dispersion Corrections for Anomalous Scattering

A.4 X-Ray Dispersion Corrections for Anomalous Scattering

697

698

A. Appendix

A.5 Atomic Form Factors for 200 keV Electrons and Procedure for Conversion to Other Voltages Electron form factors can be obtained from the x-ray atomic form factors, fx (s), with the Mott formula (3.113) as: fel0 (s) =

1 (Z − fx (s)) , s2

where the fx (s) are the values listed in the preceding table. Conversion of fel0 (s) to units of ˚ A requires multiplication by the factor given in (3.113): 2me2 = 2.3933 × 10−2 , (4π)2 where the extra factor of (4π)−2 originates with the definition s ≡ sinθ/λ (s is converted to the Δk used in the text by multiplication by 4π). For an incident electron with velocity, v, it is necessary to multiply fel0 (s) by the relativistic mass correction factor, γ: 1 , γ≡ (1 − (v/c)2

so that:

  fel (s) = 2.3933 × 10−2 γfel0 (s) .

For high-energy electrons of known energy E, the following expression is usually more convenient: γ =1+

E E[keV] ≃1+ m e c2 511

Form factors for 200 keV electrons are given in the following table. They were derived from the previous table of x-ray atomic form factors, fx (s), calculated with a Dirac-Fock method by D. Rez, P. Rez and I. Grant: Acta Cryst. A50, 481 (1994). Form factors at other electron energies can be obtained from x-ray form factors by the procedure above. More conveniently, electron form factors for other accelerating voltages can be obtained from the values in the following table for 200 keV electrons by multiplying by the ratio of relativistic factors. For example, for 100 keV electrons the values in the table should be multiplied by the constant factor: 1 + 100/511 γ100 = 0.859 , = γ200 1 + 200/511 so the values for 100 keV electrons are smaller than those in the table. The column headings in the table are s ≡ sinθ/λ, in units of ˚ A−1 , Δk ≡ 4πs.

0.0

0.581 – 4.530 – 4.227 3.875 3.438 3.066 2.760 – – 2.507 – 2.295 – 6.593 – 7.204 – 8.162 8.005 7.616 7.185 6.757 – 6.360 – 12.38 – 13.69 12.87 – 12.14 – 11.50 –

s

He Li+1 Li Be+2 Be B C N O O−1 O−2 F F−1 Ne Na+1 Na Mg+2 Mg Al+3 Al Si P S Cl Cl−1 Ar K+1 K Ca+2 Ca Sc Ti+4 Ti V+5 V Cr+4

0.569 13.54 3.885 26.75 3.895 3.660 3.298 2.970 2.692 -9.391 -21.17 2.455 -9.784 2.255 14.87 5.742 27.78 6.544 40.84 7.461 7.467 7.209 6.872 6.512 -4.833 6.165 17.99 10.57 30.34 12.08 11.55 55.81 11.02 68.76 10.53 55.92

0.05

0.540 3.542 2.609 6.772 3.106 3.123 2.940 2.721 2.512 0.250 -1.790 2.322 -0.060 2.153 4.837 4.120 7.781 5.076 10.86 5.887 6.177 6.191 6.070 5.875 4.142 5.652 7.722 7.533 10.20 8.870 8.795 15.78 8.617 18.75 8.404 15.88

0.1

0.498 1.686 1.621 3.070 2.272 2.488 2.479 2.383 2.259 1.636 1.149 2.128 1.426 2.002 2.918 2.916 4.044 3.691 5.288 4.347 4.767 4.983 5.056 5.032 4.721 4.950 5.470 5.550 6.245 6.319 6.442 8.264 6.459 9.421 6.423 8.373

0.448 1.030 1.047 1.773 1.614 1.913 2.020 2.024 1.977 1.800 1.697 1.905 1.682 1.823 2.183 2.215 2.701 2.714 3.317 3.195 3.597 3.887 4.070 4.166 4.145 4.196 4.354 4.381 4.644 4.734 4.880 5.526 4.957 6.075 4.993 5.639

0.15 0.2

0.397 0.720 0.732 1.170 1.157 1.457 1.620 1.688 1.699 1.659 1.652 1.676 1.611 1.633 1.784 1.799 2.045 2.087 2.383 2.414 2.737 3.016 3.234 3.389 3.414 3.489 3.577 3.581 3.715 3.745 3.869 4.158 3.956 4.451 4.014 4.275 0.347 0.546 0.550 0.841 0.853 1.117 1.295 1.397 1.446 1.444 1.457 1.458 1.445 1.448 1.516 1.521 1.658 1.683 1.856 1.895 2.133 2.366 2.574 2.744 2.765 2.876 2.966 2.965 3.062 3.065 3.170 3.326 3.254 3.499 3.319 3.447

0.25 0.3 0.302 0.436 0.436 0.641 0.652 0.870 1.040 1.155 1.225 1.233 1.244 1.262 1.264 1.275 1.314 1.315 1.396 1.407 1.519 1.543 1.712 1.893 2.071 2.232 2.243 2.370 2.471 2.469 2.559 2.555 2.649 2.751 2.730 2.867 2.797 2.875 0.262 0.361 0.360 0.509 0.516 0.691 0.843 0.958 1.039 1.046 1.053 1.090 1.095 1.119 1.149 1.149 1.203 1.207 1.285 1.296 1.413 1.550 1.694 1.834 1.839 1.964 2.068 2.067 2.156 2.152 2.240 2.320 2.318 2.407 2.385 2.446

0.35 0.4

Table A.3. Atomic form factors for 200 keV electrons 0.5 0.198 0.263 0.262 0.351 0.351 0.463 0.575 0.673 0.754 0.757 0.759 0.815 0.819 0.860 0.893 0.892 0.928 0.927 0.973 0.972 1.033 1.108 1.196 1.291 1.291 1.388 1.479 1.479 1.562 1.561 1.639 1.706 1.711 1.771 1.776 1.828

0.6 0.152 0.202 0.201 0.261 0.259 0.333 0.413 0.490 0.560 0.561 0.562 0.619 0.620 0.666 0.703 0.703 0.735 0.734 0.768 0.766 0.803 0.847 0.901 0.962 0.961 1.029 1.098 1.098 1.166 1.166 1.231 1.291 1.294 1.347 1.352 1.403

0.7 0.119 0.160 0.160 0.203 0.202 0.253 0.311 0.370 0.427 0.427 0.427 0.479 0.479 0.523 0.560 0.560 0.592 0.591 0.621 0.619 0.647 0.678 0.713 0.753 0.752 0.798 0.847 0.848 0.899 0.899 0.951 1.001 1.003 1.050 1.052 1.099

0.8 0.095 0.130 0.129 0.164 0.162 0.200 0.243 0.289 0.335 0.335 0.335 0.378 0.378 0.418 0.452 0.452 0.482 0.482 0.509 0.508 0.533 0.557 0.583 0.611 0.611 0.643 0.678 0.678 0.716 0.716 0.756 0.796 0.797 0.837 0.838 0.877

0.9 0.077 0.107 0.107 0.135 0.134 0.164 0.197 0.232 0.269 0.269 0.269 0.305 0.305 0.339 0.370 0.370 0.398 0.398 0.423 0.423 0.445 0.466 0.487 0.509 0.509 0.533 0.559 0.559 0.587 0.587 0.617 0.649 0.649 0.681 0.682 0.715

1.0 0.063 0.089 0.089 0.113 0.112 0.136 0.163 0.191 0.221 0.221 0.221 0.250 0.250 0.279 0.307 0.307 0.332 0.332 0.355 0.355 0.376 0.395 0.414 0.432 0.432 0.451 0.471 0.471 0.492 0.492 0.516 0.540 0.541 0.566 0.567 0.593

1.2 0.045 0.065 0.065 0.083 0.082 0.100 0.118 0.137 0.157 0.157 0.157 0.177 0.177 0.198 0.219 0.219 0.239 0.239 0.258 0.258 0.276 0.292 0.307 0.322 0.322 0.336 0.350 0.350 0.364 0.364 0.380 0.395 0.396 0.412 0.413 0.430

1.4 0.033 0.049 0.049 0.063 0.063 0.076 0.089 0.103 0.118 0.118 0.118 0.133 0.133 0.148 0.164 0.164 0.179 0.179 0.194 0.194 0.209 0.222 0.236 0.248 0.248 0.260 0.271 0.271 0.282 0.282 0.293 0.305 0.305 0.316 0.317 0.329

1.6 0.026 0.038 0.038 0.049 0.049 0.060 0.070 0.081 0.092 0.092 0.092 0.103 0.103 0.115 0.127 0.127 0.139 0.139 0.151 0.151 0.163 0.174 0.185 0.195 0.195 0.206 0.215 0.215 0.225 0.225 0.234 0.243 0.243 0.252 0.252 0.261

1.8 0.020 0.030 0.030 0.039 0.039 0.048 0.057 0.065 0.074 0.074 0.074 0.083 0.083 0.092 0.101 0.101 0.111 0.111 0.120 0.120 0.130 0.139 0.148 0.157 0.157 0.166 0.174 0.174 0.182 0.182 0.190 0.198 0.198 0.205 0.206 0.213

2.0 0.017 0.025 0.025 0.032 0.032 0.040 0.047 0.054 0.061 0.061 0.061 0.068 0.068 0.076 0.083 0.083 0.091 0.091 0.098 0.098 0.106 0.114 0.121 0.129 0.129 0.136 0.144 0.144 0.151 0.151 0.157 0.164 0.164 0.170 0.171 0.177

2.5 0.011 0.016 0.016 0.021 0.021 0.026 0.031 0.036 0.040 0.040 0.040 0.045 0.045 0.050 0.054 0.054 0.059 0.059 0.064 0.064 0.069 0.074 0.079 0.084 0.084 0.089 0.094 0.094 0.099 0.099 0.104 0.108 0.108 0.113 0.113 0.118

3.0 0.007 0.011 0.011 0.015 0.015 0.018 0.022 0.025 0.029 0.029 0.029 0.032 0.032 0.035 0.039 0.039 0.042 0.042 0.045 0.045 0.049 0.052 0.055 0.059 0.059 0.062 0.066 0.066 0.069 0.069 0.073 0.076 0.076 0.080 0.080 0.083

4.0 0.004 0.006 0.006 0.008 0.008 0.010 0.012 0.014 0.016 0.016 0.016 0.018 0.018 0.020 0.022 0.022 0.024 0.024 0.026 0.026 0.028 0.030 0.032 0.034 0.034 0.036 0.038 0.038 0.040 0.040 0.042 0.044 0.044 0.046 0.046 0.048

6.0 0.002 0.003 0.003 0.004 0.004 0.005 0.006 0.006 0.007 0.007 0.007 0.008 0.008 0.009 0.010 0.010 0.011 0.011 0.012 0.012 0.013 0.014 0.015 0.015 0.015 0.016 0.017 0.017 0.018 0.018 0.019 0.020 0.020 0.021 0.021 0.022

A.5 Atomic Form Factors for 200 keV Electrons 699

0.0

9.676 – 10.40 – 9.934 – 9.503 – 9.108 – 8.744 – 8.408 9.936 10.26 10.25 10.11 9.851 – 9.574 – 16.24 – 18.09 – 17.52 – 16.85 – 14.89 – 14.31 13.29 12.83 – 10.52 – 12.02

s

Cr Mn+2 Mn Fe+2 Fe Co+2 Co Ni+2 Ni Cu+2 Cu Zn+2 Zn Ga Ge As Se Br Br−1 Kr Rb+1 Rb Sr+2 Sr Y+3 Y Zr+4 Zr Nb+5 Nb Mo+6 Mo Ru Rh Pd+2 Pd Ag+2 Ag

8.946 30.55 9.649 30.49 9.261 30.42 8.899 30.35 8.562 30.28 8.248 30.21 7.955 9.263 9.654 9.732 9.664 9.473 -1.554 9.251 20.83 13.98 32.96 15.92 45.40 15.74 58.05 15.34 70.83 13.77 83.70 13.33 12.52 12.13 33.89 10.20 33.78 11.43

0.05

7.373 10.41 7.950 10.36 7.726 10.31 7.505 10.25 7.290 10.18 7.084 10.12 6.886 7.754 8.217 8.450 8.541 8.505 6.972 8.413 10.33 10.28 12.64 11.77 15.20 12.09 17.91 12.13 20.74 11.34 23.66 11.18 10.76 10.54 13.57 9.388 13.49 10.08

0.1

5.896 6.471 6.270 6.442 6.172 6.403 6.064 6.357 5.953 6.308 5.839 6.256 5.724 6.238 6.658 6.961 7.163 7.264 7.049 7.301 7.768 7.856 8.450 8.611 9.289 9.009 10.24 9.233 11.28 9.026 12.40 9.044 8.947 8.854 9.366 8.327 9.320 8.626

4.783 4.891 4.986 4.879 4.958 4.857 4.916 4.829 4.866 4.795 4.810 4.758 4.749 5.042 5.354 5.634 5.867 6.036 6.049 6.156 6.321 6.341 6.574 6.659 6.911 6.955 7.319 7.184 7.785 7.244 8.301 7.342 7.421 7.416 7.475 7.235 7.465 7.348

0.15 0.2

Table A.3. (continued) 3.342 3.358 3.401 3.377 3.424 3.385 3.436 3.387 3.440 3.382 3.436 3.372 3.427 3.516 3.644 3.797 3.960 4.119 4.137 4.266 4.378 4.375 4.485 4.480 4.598 4.623 4.725 4.768 4.868 4.917 5.029 5.037 5.225 5.294 5.319 5.345 5.361 5.387

0.25 0.3

3.965 3.986 4.069 3.991 4.074 3.986 4.067 3.973 4.051 3.955 4.029 3.932 4.000 4.162 4.369 4.587 4.794 4.975 5.005 5.126 5.241 5.240 5.377 5.392 5.546 5.592 5.751 5.778 5.991 5.921 6.262 6.041 6.201 6.245 6.251 6.227 6.272 6.279 2.449 2.485 2.490 2.521 2.529 2.549 2.559 2.571 2.583 2.586 2.601 2.597 2.614 2.643 2.691 2.758 2.841 2.936 2.939 3.038 3.133 3.132 3.222 3.217 3.306 3.310 3.387 3.406 3.469 3.516 3.553 3.613 3.794 3.873 3.934 3.956 3.998 4.010

0.35 0.4 2.849 2.875 2.893 2.904 2.926 2.924 2.949 2.937 2.965 2.943 2.974 2.945 2.978 3.027 3.104 3.207 3.325 3.451 3.459 3.578 3.686 3.683 3.783 3.776 3.877 3.888 3.973 4.005 4.074 4.136 4.183 4.246 4.436 4.515 4.562 4.591 4.619 4.639

0.5 1.839 1.887 1.885 1.931 1.930 1.969 1.969 2.001 2.002 2.029 2.030 2.051 2.054 2.074 2.098 2.130 2.170 2.221 2.220 2.279 2.343 2.343 2.409 2.408 2.474 2.474 2.538 2.542 2.601 2.615 2.662 2.688 2.834 2.904 2.970 2.977 3.033 3.036

0.6 1.409 1.458 1.457 1.504 1.502 1.544 1.543 1.581 1.580 1.614 1.613 1.642 1.642 1.663 1.684 1.706 1.730 1.759 1.759 1.793 1.832 1.833 1.876 1.876 1.922 1.921 1.970 1.968 2.018 2.018 2.066 2.070 2.180 2.236 2.292 2.293 2.347 2.347

0.7 1.101 1.146 1.145 1.189 1.188 1.228 1.227 1.265 1.264 1.298 1.297 1.329 1.328 1.353 1.375 1.396 1.416 1.437 1.436 1.459 1.485 1.485 1.514 1.514 1.545 1.544 1.579 1.577 1.614 1.611 1.651 1.648 1.727 1.770 1.813 1.813 1.857 1.857

0.8 0.878 0.917 0.917 0.955 0.955 0.991 0.991 1.025 1.024 1.057 1.056 1.087 1.086 1.112 1.136 1.157 1.177 1.196 1.195 1.214 1.233 1.233 1.254 1.254 1.276 1.276 1.301 1.299 1.326 1.324 1.353 1.350 1.407 1.438 1.471 1.470 1.505 1.504

0.9 0.715 0.748 0.748 0.780 0.780 0.811 0.811 0.842 0.842 0.871 0.871 0.899 0.899 0.924 0.948 0.969 0.989 1.007 1.007 1.025 1.042 1.042 1.059 1.059 1.076 1.076 1.094 1.094 1.114 1.112 1.134 1.132 1.174 1.197 1.222 1.221 1.247 1.247

1.0 0.593 0.620 0.621 0.648 0.648 0.674 0.674 0.701 0.701 0.727 0.727 0.752 0.752 0.775 0.798 0.819 0.838 0.857 0.857 0.873 0.889 0.889 0.905 0.905 0.920 0.920 0.936 0.935 0.951 0.950 0.968 0.966 0.999 1.017 1.035 1.035 1.054 1.054

1.2 0.431 0.449 0.449 0.468 0.468 0.487 0.487 0.506 0.506 0.525 0.525 0.544 0.544 0.563 0.582 0.600 0.617 0.634 0.634 0.650 0.664 0.664 0.679 0.679 0.692 0.692 0.705 0.705 0.718 0.717 0.730 0.730 0.753 0.765 0.778 0.778 0.790 0.790

1.4 0.329 0.342 0.342 0.355 0.355 0.369 0.369 0.383 0.383 0.397 0.397 0.411 0.411 0.426 0.440 0.455 0.469 0.483 0.483 0.496 0.510 0.510 0.522 0.522 0.534 0.534 0.546 0.546 0.557 0.557 0.568 0.568 0.589 0.598 0.608 0.608 0.618 0.618

1.6 0.262 0.271 0.271 0.281 0.281 0.291 0.291 0.301 0.301 0.312 0.312 0.323 0.323 0.334 0.345 0.356 0.367 0.378 0.378 0.389 0.400 0.400 0.411 0.411 0.422 0.422 0.432 0.432 0.442 0.442 0.452 0.452 0.470 0.479 0.488 0.488 0.496 0.496

1.8 0.213 0.221 0.221 0.229 0.229 0.236 0.236 0.244 0.244 0.253 0.253 0.261 0.261 0.269 0.278 0.286 0.295 0.304 0.304 0.313 0.322 0.322 0.331 0.331 0.340 0.340 0.349 0.349 0.357 0.357 0.366 0.366 0.382 0.390 0.398 0.398 0.405 0.405

2.0 0.177 0.183 0.183 0.190 0.190 0.196 0.196 0.203 0.203 0.209 0.209 0.216 0.216 0.223 0.229 0.236 0.243 0.250 0.250 0.257 0.265 0.265 0.272 0.272 0.279 0.279 0.287 0.287 0.294 0.294 0.301 0.301 0.315 0.322 0.329 0.329 0.336 0.336

2.5 0.118 0.122 0.122 0.127 0.127 0.131 0.131 0.136 0.136 0.140 0.140 0.144 0.144 0.149 0.153 0.157 0.162 0.166 0.166 0.170 0.175 0.175 0.179 0.179 0.184 0.184 0.189 0.189 0.193 0.193 0.198 0.198 0.208 0.212 0.217 0.217 0.222 0.222

3.0 0.083 0.087 0.087 0.090 0.090 0.093 0.093 0.097 0.097 0.100 0.100 0.103 0.103 0.106 0.109 0.113 0.116 0.119 0.119 0.122 0.125 0.125 0.128 0.128 0.131 0.131 0.134 0.134 0.137 0.137 0.141 0.141 0.147 0.150 0.154 0.154 0.157 0.157

4.0 0.048 0.050 0.050 0.052 0.052 0.054 0.054 0.056 0.056 0.058 0.058 0.059 0.059 0.061 0.063 0.065 0.067 0.069 0.069 0.071 0.073 0.073 0.075 0.075 0.077 0.077 0.078 0.078 0.080 0.080 0.082 0.082 0.086 0.087 0.089 0.089 0.091 0.091

6.0 0.022 0.022 0.022 0.023 0.023 0.024 0.024 0.025 0.025 0.026 0.026 0.027 0.027 0.028 0.029 0.030 0.030 0.031 0.031 0.032 0.033 0.033 0.034 0.034 0.035 0.035 0.036 0.036 0.037 0.037 0.038 0.038 0.039 0.040 0.041 0.041 0.042 0.042

700 A. Appendix

0.0

– 12.80 14.74 15.36 15.55 15.55 15.28 – 14.98 – 22.75 – 25.20 – 24.63 – 24.06 – 23.46 – 22.94 – 21.98 – 21.52 – 21.23 20.66 20.25 19.86 19.48 19.10 18.75 18.76 18.39 17.99 17.59 17.20

s

Cd+2 Cd In Sn Sb Te I I−1 Xe Cs+1 Cs Ba+2 Ba La+3 La Ce+4 Ce Pr+3 Pr Nd+3 Nd Sm+3 Sm Eu+3 Eu Gd+3 Gd Tb Dy Ho Er Tm Yb Lu Hf Ta W Re

33.67 12.16 13.76 14.41 14.69 14.77 14.60 4.06 14.38 25.48 19.57 37.22 22.00 49.34 21.94 61.69 21.51 49.09 20.76 48.96 20.36 48.69 19.62 48.56 19.27 48.44 19.28 18.59 18.26 17.94 17.63 17.33 17.04 17.25 17.09 16.84 16.57 16.28

13.40 10.64 11.60 12.22 12.62 12.86 12.90 11.66 12.87 14.50 14.61 16.54 16.21 18.84 16.70 21.33 16.50 18.65 15.70 18.54 15.50 18.32 15.11 18.21 14.92 18.11 15.29 14.53 14.34 14.15 13.97 13.79 13.61 14.04 14.19 14.23 14.19 14.11

0.05 0.1

9.263 8.97 9.47 9.93 10.33 10.64 10.82 10.73 10.94 11.30 11.39 11.84 12.06 12.53 12.54 13.32 12.46 12.41 11.90 12.33 11.81 12.18 11.61 12.09 11.52 12.01 11.87 11.30 11.19 11.08 10.97 10.86 10.75 11.14 11.39 11.56 11.68 11.74

7.443 7.535 7.779 8.071 8.369 8.649 8.878 8.921 9.067 9.226 9.233 9.431 9.485 9.694 9.802 10.01 9.787 9.649 9.478 9.611 9.441 9.520 9.347 9.469 9.297 9.417 9.517 9.172 9.106 9.038 8.970 8.901 8.831 9.066 9.277 9.458 9.608 9.729

0.15 0.2

Table A.3. (continued) 5.392 5.448 5.517 5.614 5.736 5.878 6.032 6.044 6.188 6.320 6.317 6.441 6.430 6.557 6.572 6.674 6.612 6.618 6.562 6.635 6.582 6.649 6.600 6.648 6.602 6.643 6.674 6.585 6.570 6.552 6.532 6.509 6.484 6.545 6.632 6.731 6.834 6.937

0.25 0.3

6.279 6.378 6.500 6.668 6.864 7.073 7.276 7.307 7.465 7.605 7.600 7.742 7.737 7.889 7.939 8.055 7.958 7.907 7.816 7.900 7.813 7.868 7.785 7.845 7.765 7.819 7.886 7.702 7.665 7.626 7.584 7.541 7.496 7.618 7.758 7.899 8.032 8.154 4.053 4.060 4.105 4.153 4.208 4.272 4.348 4.348 4.434 4.523 4.523 4.613 4.611 4.701 4.698 4.787 4.752 4.799 4.775 4.839 4.816 4.902 4.882 4.927 4.908 4.948 4.952 4.946 4.959 4.968 4.975 4.979 4.981 5.006 5.045 5.095 5.152 5.216

0.35 0.4 4.665 4.689 4.739 4.801 4.879 4.972 5.081 5.084 5.199 5.312 5.310 5.419 5.412 5.521 5.522 5.619 5.572 5.608 5.571 5.639 5.605 5.684 5.653 5.699 5.670 5.710 5.722 5.686 5.688 5.686 5.682 5.674 5.665 5.699 5.754 5.823 5.900 5.982

0.5 3.093 3.089 3.137 3.180 3.222 3.263 3.307 3.306 3.355 3.408 3.409 3.465 3.466 3.525 3.523 3.587 3.572 3.620 3.612 3.662 3.654 3.736 3.729 3.769 3.762 3.799 3.799 3.819 3.843 3.864 3.884 3.901 3.916 3.937 3.963 3.994 4.030 4.069

0.6 2.400 2.397 2.444 2.488 2.528 2.566 2.601 2.601 2.636 2.673 2.673 2.711 2.712 2.752 2.751 2.794 2.792 2.832 2.832 2.870 2.869 2.939 2.939 2.972 2.971 3.002 3.002 3.029 3.056 3.080 3.103 3.125 3.145 3.166 3.188 3.211 3.236 3.264

1.901 1.900 1.942 1.982 2.020 2.056 2.090 2.090 2.121 2.152 2.152 2.183 2.183 2.214 2.214 2.245 2.248 2.281 2.284 2.313 2.316 2.375 2.377 2.405 2.406 2.433 2.432 2.461 2.486 2.511 2.534 2.556 2.577 2.597 2.616 2.635 2.655 2.676

0.7 0.8 1.539 1.539 1.574 1.608 1.642 1.675 1.706 1.706 1.736 1.764 1.764 1.791 1.791 1.818 1.818 1.844 1.847 1.876 1.879 1.904 1.907 1.959 1.961 1.985 1.987 2.011 2.011 2.037 2.061 2.084 2.106 2.127 2.147 2.166 2.184 2.201 2.218 2.235

0.9 1.273 1.274 1.301 1.329 1.358 1.386 1.413 1.413 1.440 1.466 1.466 1.491 1.491 1.516 1.516 1.539 1.541 1.567 1.568 1.592 1.593 1.641 1.642 1.664 1.666 1.687 1.687 1.711 1.733 1.754 1.775 1.795 1.814 1.831 1.848 1.864 1.880 1.895

1.0 1.075 1.075 1.096 1.119 1.142 1.165 1.188 1.188 1.212 1.235 1.235 1.257 1.257 1.279 1.279 1.301 1.302 1.324 1.325 1.347 1.347 1.390 1.391 1.411 1.412 1.432 1.432 1.454 1.474 1.493 1.512 1.531 1.549 1.566 1.581 1.597 1.611 1.626

1.2 0.803 0.803 0.817 0.831 0.845 0.860 0.876 0.876 0.892 0.909 0.909 0.926 0.926 0.942 0.942 0.959 0.959 0.977 0.977 0.994 0.994 1.028 1.028 1.045 1.045 1.062 1.062 1.079 1.095 1.111 1.127 1.143 1.159 1.174 1.188 1.202 1.215 1.228

1.4 0.627 0.627 0.637 0.647 0.657 0.667 0.678 0.678 0.689 0.700 0.700 0.712 0.712 0.724 0.724 0.736 0.736 0.749 0.749 0.762 0.762 0.788 0.788 0.801 0.801 0.814 0.814 0.827 0.840 0.853 0.866 0.879 0.892 0.905 0.917 0.930 0.942 0.953

1.6 0.504 0.504 0.512 0.520 0.527 0.535 0.543 0.543 0.551 0.560 0.560 0.568 0.568 0.577 0.577 0.586 0.586 0.595 0.595 0.605 0.605 0.624 0.624 0.634 0.634 0.644 0.644 0.654 0.664 0.675 0.685 0.695 0.706 0.716 0.726 0.737 0.747 0.757

1.8 0.412 0.412 0.419 0.426 0.433 0.440 0.446 0.446 0.453 0.459 0.459 0.466 0.466 0.472 0.472 0.479 0.479 0.486 0.487 0.494 0.494 0.508 0.509 0.516 0.516 0.524 0.524 0.532 0.539 0.547 0.556 0.564 0.572 0.580 0.588 0.597 0.605 0.613

2.0 0.342 0.342 0.349 0.355 0.361 0.367 0.372 0.372 0.378 0.384 0.384 0.389 0.389 0.395 0.395 0.400 0.401 0.406 0.406 0.412 0.412 0.424 0.424 0.430 0.430 0.436 0.436 0.442 0.448 0.455 0.461 0.467 0.474 0.480 0.487 0.494 0.500 0.507

2.5 0.227 0.227 0.231 0.236 0.241 0.245 0.250 0.250 0.254 0.258 0.258 0.263 0.263 0.267 0.267 0.271 0.271 0.275 0.275 0.279 0.279 0.287 0.287 0.291 0.291 0.295 0.295 0.299 0.303 0.307 0.311 0.315 0.319 0.323 0.327 0.331 0.335 0.339

3.0 0.160 0.160 0.164 0.167 0.170 0.174 0.177 0.177 0.180 0.184 0.184 0.187 0.187 0.190 0.190 0.194 0.194 0.197 0.197 0.200 0.200 0.206 0.206 0.209 0.209 0.212 0.212 0.215 0.218 0.221 0.224 0.227 0.230 0.233 0.236 0.239 0.242 0.244

4.0 0.093 0.093 0.095 0.097 0.098 0.100 0.102 0.102 0.104 0.106 0.106 0.108 0.108 0.110 0.110 0.112 0.112 0.114 0.114 0.115 0.115 0.119 0.119 0.121 0.121 0.123 0.123 0.125 0.127 0.129 0.131 0.133 0.135 0.136 0.138 0.140 0.142 0.144

6.0 0.043 0.043 0.044 0.045 0.045 0.046 0.047 0.047 0.048 0.049 0.049 0.050 0.050 0.050 0.050 0.051 0.051 0.052 0.052 0.053 0.053 0.055 0.055 0.055 0.055 0.056 0.056 0.057 0.058 0.059 0.060 0.061 0.061 0.062 0.063 0.064 0.065 0.066

A.5 Atomic Form Factors for 200 keV Electrons 701

Os Ir Pt Au Hg Tl Pb Bi Po At Rn Fr Ra Ac Th Pa U

14.00 13.82 13.04 12.83 13.23 14.34 15.13 15.70 16.12 16.11 16.05 17.72 19.26 20.06 20.53 19.86 19.72

0.05 0.1

15.99 15.64 14.47 14.14 14.64 16.71 17.69 18.24 18.55 18.26 17.96 22.76 25.18 25.67 25.68 24.88 24.52

9.824 9.871 9.727 9.722 9.895 10.11 10.39 10.69 11.00 11.23 11.42 11.60 11.86 12.20 12.55 12.40 12.45

0.15 0.2

11.77 11.74 11.34 11.25 11.52 12.01 12.52 12.97 13.37 13.56 13.68 14.13 14.79 15.38 15.88 15.49 15.48

7.036 7.121 7.179 7.241 7.322 7.391 7.479 7.588 7.715 7.857 8.005 8.128 8.239 8.367 8.510 8.571 8.654

0.25 0.3

8.263 8.342 8.339 8.380 8.493 8.601 8.752 8.934 9.136 9.331 9.516 9.658 9.799 9.994 10.21 10.21 10.28 5.283 5.350 5.426 5.491 5.545 5.595 5.644 5.696 5.753 5.821 5.896 5.974 6.053 6.127 6.204 6.287 6.362

0.35 0.4 6.065 6.143 6.220 6.287 6.351 6.407 6.467 6.537 6.619 6.717 6.824 6.925 7.020 7.114 7.216 7.298 7.379

0.5 4.113 4.160 4.216 4.267 4.310 4.354 4.395 4.434 4.473 4.512 4.554 4.600 4.650 4.699 4.747 4.812 4.869

0.6 3.293 3.326 3.363 3.400 3.435 3.471 3.506 3.540 3.573 3.604 3.635 3.667 3.701 3.735 3.769 3.814 3.856

0.7 2.698 2.722 2.747 2.773 2.800 2.829 2.858 2.886 2.915 2.942 2.968 2.995 3.021 3.048 3.075 3.106 3.137

0.8 2.253 2.271 2.289 2.309 2.330 2.352 2.375 2.398 2.421 2.445 2.468 2.490 2.513 2.535 2.558 2.582 2.606 1.910 1.925 1.940 1.956 1.972 1.990 2.007 2.026 2.045 2.064 2.084 2.103 2.122 2.142 2.161 2.180 2.199

0.9 1.639 1.653 1.666 1.679 1.693 1.707 1.722 1.737 1.752 1.768 1.784 1.800 1.816 1.833 1.850 1.865 1.881

1.0 1.241 1.253 1.265 1.276 1.287 1.298 1.309 1.320 1.331 1.342 1.354 1.365 1.377 1.389 1.401 1.412 1.424

1.2

1.4 0.965 0.976 0.987 0.997 1.007 1.017 1.027 1.036 1.045 1.054 1.063 1.072 1.081 1.090 1.099 1.108 1.117

1.6 0.767 0.776 0.786 0.795 0.805 0.814 0.822 0.831 0.839 0.847 0.855 0.863 0.871 0.878 0.886 0.894 0.901

1.8 0.622 0.630 0.638 0.646 0.654 0.662 0.670 0.678 0.685 0.693 0.700 0.707 0.714 0.721 0.728 0.734 0.741

2.0 0.514 0.521 0.528 0.535 0.541 0.548 0.555 0.562 0.568 0.575 0.581 0.588 0.594 0.600 0.606 0.612 0.618

2.5 0.343 0.348 0.352 0.356 0.360 0.365 0.369 0.373 0.378 0.382 0.387 0.391 0.396 0.400 0.405 0.409 0.414

3.0 0.247 0.250 0.253 0.256 0.259 0.262 0.265 0.267 0.270 0.273 0.276 0.279 0.282 0.286 0.289 0.292 0.295

4.0 0.146 0.147 0.149 0.151 0.153 0.154 0.156 0.158 0.160 0.161 0.163 0.165 0.166 0.168 0.170 0.171 0.173

6.0 0.067 0.067 0.068 0.069 0.070 0.071 0.072 0.073 0.073 0.074 0.075 0.076 0.077 0.078 0.079 0.079 0.080

A. The column headings are s ≡ sinθ/λ, in units of ˚ Table entries are for 200 keV electrons. The units for all entries are ˚ A−1 . This diffraction vector, s, is converted to the Δk used in the text by multiplication by 4π.

0.0

16.82 16.39 15.06 14.67 15.21 17.81 18.83 19.33 19.57 19.13 18.72 25.81 28.37 28.48 28.11 27.33 26.84

s

Table A.3. (continued)

702 A. Appendix

A.6 Indexed Single Crystal Diffraction Patterns

703

704

A. Appendix

A.6 Indexed Single Crystal Diffraction Patterns

705

706

A. Appendix

A.6 Indexed Single Crystal Diffraction Patterns

707

708

A. Appendix

A.6 Indexed Single Crystal Diffraction Patterns

709

710

A. Appendix

A.6 Indexed Single Crystal Diffraction Patterns

711

712

A. Appendix

A.7 Stereographic Projections

713

714

A. Appendix

A.7 Stereographic Projections

715

716

A. Appendix

A.8 Examples of Fourier Transforms

717

718

A. Appendix

A.8 Examples of Fourier Transforms

719

720

A. Appendix

A.9 Debye–Waller Factor from Wave Amplitude Another approach to calculating the Debye–Waller factor, perhaps simpler than that of Chapter 9, makes use of the phase relationships in the diffracted wave. The instantaneous positions of the atom centers are {ri + δ i }, and the intensity, I(Δk), is written as ψ ∗ ψ :   I(Δk) = fj e−i∆k·(rj +δj ) , (A.1) fi∗ e+i∆k·(ri +δ i ) j

i

I(Δk) =

 i

fi∗ fj e+i∆k·(ri −r j ) e+i∆k·(δi −δ j ) .

(A.2)

j

We confine our attention to Bragg peaks where Δk · (r i − r j ) = 2πinteger, so the first exponential in (A.2) is 1:  fi∗ fj ei∆k·(δi −δj ) . (A.3) I(Δk) = i

j

We assume the displacements are small, and expand the exponential in (A.3):   2  1

. (A.4) fi∗ fj 1 + iΔk · (δ i − δ j ) − Δk · (δ i − δ j ) I(Δk) = 2 i j We simplify further by assuming that the differences, δ i − δ j , average to zero when summed over all pairs separated by r i − r j :  2  1

2 . (A.5) I(Δk) = |f | 1 − Δk · (δ i − δ j ) 2 i j

From (9.162) the isotropic average of [Δk · (δ i − δ j )]2 is 1/3Δk2 (δ i − δ j )2 so:   1 I(Δk) = N 2 |f |2 1 − Δk2 (δ i − δ j )2 . (A.6) 6 Following Sect. 9.2.2, we assume that the displacements of the atom centers, δ i and δ j , are isotropic random variables with a Gaussian distribution and a characteristic √ range, δ. The difference, δ i − δ j , will therefore have an average range of 2 δ, allowing us to simplify (A.6) as:   1 (A.7) I(Δk) = N 2 |f |2 1 − Δk2 δ 2 . 3 Approximately, the third factor in (A.7) is the exponential function: I(Δk) = N 2 |f |2 e−1/3∆k

2 2

δ

.

(A.8)

The exponential factor in (A.8) is the Debye–Waller factor. It is essentially the same as (9.59), but with an additional factor of 1/3 in the exponent. The % & derivation of (9.59) was performed in one dimension, so the x2 in (9.59) corresponds to the average value of x2 along the direction Δk. Equation (A.8) refers to the average of the mean-squared displacement over all directions in space, δ 2 . It can be important to specify which average is being reported.

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721

A.10 Review of Dislocations Structure of a Dislocation A dislocation is the only line defect in a solid. A large body of knowledge has formed around dislocations because their movement is the elementary mechanism of plastic deformation of many crystalline materials. In addition, dislocations in semiconducting crystals are sinks for charge carriers. More than any other experimental technique, TEM has revealed the structures and interactions of dislocations. There are two types of “pure” dislocations. An edge dislocation is the easiest to illustrate. In Fig. A.2, notice how an extra half-plane of atoms has been inserted in the upper half of the simple cubic crystal. This extra halfplane terminates at the “core” of the edge dislocation line. On the figure is drawn a circuit of 5 × 5 × 5 × 5 atoms. This circuit, known as a “Burgers circuit,” does not close perfectly when it encloses the dislocation line. (It does close in a perfect simple cubic crystal, of course, and it also closes perfectly when it is drawn in a dislocated crystal around a region that does not contain the dislocation core.) The vector from the end to the start of the circuit is defined as the “Burgers vector” of the dislocation, b. Dislocations are characterized by their Burgers vector and the direction of their dislocation line. The magnitude of the Burgers vector parameterizes the strength of the dislocation – dislocations with larger Burgers vectors cause larger crystalline distortions. The “character” of the dislocation is determined by the direction of the Burgers vector with respect to the direction of the dislocation line. In Fig. A.2 the Burgers vector is perpendicular to the dislocation line. This is an “edge dislocation.” The other type of “pure” dislocation has its Burgers vector parallel to the dislocation line. It is a “screw dislocation,” and is illustrated in Fig. A.3. Around the core of a screw dislocation, the crystal planes form a helix. When we complete a Burgers circuit in the x-y plane in Fig. A.3, the vector from . For a screw dislocation, b is parallel to the line of finish to start lies along z the dislocation. In general, dislocations are neither pure edge dislocations nor pure screw dislocations, but rather have their Burgers vectors at some intermediate angle to the line of their cores. These are “mixed dislocations.” Whenever a dislocation line is curved, part of the dislocation must have mixed character. An example of a curved dislocation line is shown in Fig. A.4, with labels indicating the pure edge and screw parts. All other parts of the dislocation are of mixed character. Notice how the dislocation was made. The crystal was cut in the lower right corner, and the top (gray) atoms were pushed to the left with respect to the lower (black) atoms. The edge of the cut is the dislocation line. A Burgers circuit around any part of this dislocation line always gives the same Burgers vector. Since the dislocation line changes direction, however, the character of the dislocation changes along its line.

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Fig. A.2. Edge dislocation in a cubic crystal. Dislocation line is parb , b = a 100, and b is perallel to y pendicular to the dislocation line.

Fig. A.3. Screw dislocation in a cylinder of cubic crystal. Dislocab, b = tion line is parallel to z a 001, and b is parallel to the dislocation line.

A dislocation loop, which is mostly of mixed character, is illustrated in Fig. A.5. A planar circular cut is made inside a block of material. The atoms across this cut are sheared as shown in the figure. The edge of the cut is the dislocation line. On the left and right edges of this dislocation loop we have edge dislocations (with b of opposite signs). On the front and back, the dislocation loop has pure screw character (again with b of opposite signs). Everywhere else the dislocation has mixed character. Strain Energy of a Dislocation (Self Energy) A dislocation generates large elastic strains in the surrounding crystal, as is evident from Figs. A.2-A.4. The strain in the material in the dislocation core

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723

pure screw

pure edge

displacement (and b)

Fig. A.4. Mixed dislocation in a cubic crystal. Quartercircle of cut plane is in the lower right. All atoms across the cut are displaced to the left by b. Fig. A.5. Left: dislocation loop in a cube of crystal. All atoms across the cut are displaced to the left by b. Right: top view of loop.

(usually considered to be cylinder of radius 5b) is so large that its excess energy cannot be accurately regarded as elastic energy. Sometimes this “core energy” is estimated from the heat of fusion of the crystal. Outside the core region, however, it is reasonable to calculate the energy by linear elasticity theory. It turns out that this total elastic energy in the surrounding crystal is typically an order-of-magnitude larger than the energy of the core region. Approximately, therefore, the energy cost of making a unit length of dislocation line is equal to the elastic energy per unit length of the dislocation. We have seen how dislocations can be created by a cut-and-shear process. The dislocation line is located at the edge of the cut, and the Burgers vector is the vector of the shear displacement. We seek the energy needed to make the dislocation this way. First note that the cut itself requires no energy, since the atoms across the cut are properly reconnected after the dislocation is made. The energy needed to make the dislocation is the energy required to make the shear across the cut surface. Think of the cut crystal as a spring. An elastic restoring force opposes the shear, and this restoring force is proportional to the shear times the shear modulus, G. The distance of displacement across the cut is b. The elastic energy stored in the crystal is obtained by integrating the force over the distance, x, of shear for 1 cm of dislocation line:

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Eelastic ∝

1cm b

0

G x dxdz .

(A.9)

0

Eelastic = Gb2 K [J/cm] .

(A.10)

Here K is a geometrical constant that depends on the size and shape of the crystal (and somewhat on the dislocation character). Neglecting the core energy (which is often small), the energy cost of creating a unit length of edge dislocation is the Eelastic of (A.10). Dislocation Reactions Because the self-energy of a dislocation increases as b2 , dislocations Burgers vectors that are as small as possible. Figure A.6 shows how to accommodate two extra half-planes with either one dislocation of b = 2a, or two dislocations, each of b = a. The total elastic energy of a crystal with the two separate, smaller dislocations is half as large, however. Big dislocations therefore break into smaller ones, so single dislocations have the smallest possible Burgers vector. The lower limit to the Burgers vector is set by the requirement that the atoms must match positions across the cut in the crystal. This lower limit is typically the distance between nearest-neighbor atoms. Smaller Burgers vectors are usually not possible, but an exception occurs for fcc and hcp crystals.

a

Fig. A.6. Accommodation of the same slip by two dislocations or by one dislocation.

Stacking Faults in fcc Crystals A special dislocation reaction occurs for dislocations on {111} planes in fcc crystals. Figure A.7 shows how the stacking of close-packed planes determines whether the crystal is fcc or hcp. The “perfect dislocation” in the fcc crystal has a Burgers vector of the nearest-neighbor separation, b = 1/2[110]. The shifts between the adjacent layers of the fcc structure are smaller than this, however, and we can obtain these shifts by creating a “stacking fault” in the fcc crystal. Specifically, assume that we interrupt the ABCABCABC stacking of the fcc crystal and make a small shift of a {111} plane as: ABCAB|ABCABC. Here we have

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725

C B A

a b c d A B A

Fig. A.7. (a) fcc stacking of close-packed (111) planes; perspective view of three layers, with the cubic face marked with the square. (b) Stacking of the three types of (111) planes seen from above. The next layer will be an A-layer, and will locate directly above the dark A-layer at the bottom. (c) hcp stacking of close-packed (0001) basal planes; perspective view of three layers. (d) Stacking of the two types of close-packed planes seen from above. The next layer will be an A-layer, and will locate directly above the dark A-layer at the bottom.

errored in the stacking by placing an A-layer to the immediate right of a B-layer. The structure is still close packed, but there is a narrow region of hcp crystal (. . . AB|AB. . . ). This region of hcp crystal need not extend to the edge of the crystal, however. At the boundary of the hcp region we can insert a “Shockley partial” dislocation, which has a Burgers vector equal to the shift between an A and a B-layer. This shift is a vector of the type: a/6112 . Consider a specific dislocation reaction for which the total Burgers vectors across the arrow are equal1 : a/2[110] → a/6[121] + a/6[211] .

(A.11)

The energy, proportional to the square of the Burgers vector, is smaller for the two Shockley partials on the right than the single perfect dislocation on the left, as we verify by calculating the energies (A.10):   KGa2 /4 12 + 12 + 0 3 Eperfect = .  (A.12) = E2partials 2 2KGa2 /36 12 + 22 + 12

Equation (A.12) shows that it is energetically favorable for a perfect dislocation in an fcc crystal to split into two Shockley partial dislocations, which then repel each other elastically (as discussed in the next subsection). There is, however, a thin region of hcp crystal between these two Shockley partials (the stacking fault), and the stacking fault energy tends to keep the partials 1

The conservation of Burgers vector is equivalent to the fact that a dislocation line cannot terminate in the middle of a crystal, but must extend to the surface or form a loop.

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from getting too far apart. Equilibrium separations of Shockley partial dislocations, measured by TEM, are a means of determining the stacking fault energy of fcc crystals. This stacking fault energy is qualitatively related to the free energy difference between the fcc and hcp crystal structures. Stable Arrays of Dislocations Look again at the atom positions around the dislocation core in Fig. A.2. Inserting an extra half-plane of atoms in the top half of the crystal causes compressive stresses above, and tensile stresses below the dislocation line. An edge dislocation line, seen on end in Fig. A.8, is marked with a “⊥” symbol. The circles are lines of constant strain.

Fig. A.8. Compression and tension fields around an edge dislocation.

Dislocations interact with each other through their elastic fields, so groups of dislocations are frequently found in special arrangements. For example, two edge dislocations with the same Burgers vector repel each other when they are situated on the same glide plane. When they are close together, their compression and tensile strains add. The elastic energy increases quadratically with the strain field. It is therefore favorable for the dislocations to move apart as in Fig. A.9 (cf., Fig. A.6), so there is an elastic repulsion between these two edge dislocations. repulsion

Fig. A.9. Elastic repulsion of two edge dislocations on the same glide plane.

The six dislocations on the left of Fig. A.10 are in a stable configuration, however, since the compressive stress above each dislocation cancels partially the tensile stress below its neighboring dislocation. Perturbing the dislocations out of this linear array increases the elastic energy. The right side of Fig. A.10 shows in more detail the extra half-planes of the six edge dislocations in a simple cubic crystal. This dislocation array creates a low-angle tilt boundary between two perfect crystals. This particular example is a symmetric tilt boundary. Other types of tilt boundaries are possible, as are twist boundaries comprising arrays of screw dislocations. Arrays of (1-dimensional) dislocations are common at 2-dimensional interfaces between different phases in a material.

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Fig. A.10. Stable dislocation structure constituting a small angle tilt boundary.

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A. Appendix

A.11 TEM Laboratory Exercises This appendix presents the content of a university laboratory course designed to familiarize the new user with the practice of microscope calibration, conventional diffraction and imaging techniques, and energy-dispersive x-ray spectroscopy. In such a course, students have access to the instrument in three hour sessions. Each laboratory requires 3–4 sessions to complete. Additional time is required for developing photographic plates, data analysis, report writing, and perhaps specimen preparation. The alignment, Au, and MoO3 exercises in Laboratory 1 require 2–3 sessions. Sample tilting is needed in Laboratory 2 on DF imaging of θ′ precipitates, and tilting requires some practice. Laboratory 3 on EDS of θ′ precipitates is straightforward, and could perhaps be performed before Laboratory 2. Laboratory 4 on dislocations and stacking faults in stainless steel is typical of physical metallurgy research with conventional TEM. The instructor may consider substituting another laboratory on a material more relevant to the research interests of the student. The authors often modify the laboratories – some variations are given in the Specimen or Procedures sections. Similarly, the alignment procedures must be adapted for a particular microscope. The format of these alignment procedures, condensed versions of instructions usually found in the microscope manufacturers’ manuals, are handy references in the laboratory. Please read the manufacturer’s manuals, however – they are generally well written and rich in information. A.11.1 Preliminary – JEOL 2000FX Daily Operation Beginning the Session 1. Be sure the column vacuum is 1 × 10−4 Pa (blue scale) on the ion gauge behind the microscope. Alternatively, the “PEG” gauge setting for the vacuum meter in the lower left panel should be in the green range. If any red lights to the right of the vacuum meter are on, please get help. The “Accel Voltage” and “Filament Ready” lights (left panel) should both be green. 2. Adjust the brightness and contrast knobs to the right of the CRT to view the CRT. Remove objective and intermediate apertures from the column. 3. Before inserting a sample into the column, make sure the goniometer is locked in the zero position and the X and Y translates (page 2 on the CRT screen) are on zero. Now insert the specimen into the goniometer. The red light will come on and then turn off. Let this light cycle repeat at least six times before inserting the specimen into the column. 4. If necessary, fill the cold-trap with liquid nitrogen using the small plastic funnel and a styrofoam cup. It holds about two cupfuls of LN. Do not attempt to lift the LN dewar up to the cold-trap. It is usually necessary to use the LN dewar only during the summer when the humidity is high.

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Obtaining a Beam 1. Set accelerating voltage to 200 kV using the white toggle switch on left panel (page 1 on CRT screen) and depress “HT ON”. The HT will step up to 200 kV and the “Beam Current” should read approximately 100 μA (dark current). If you need to adjust the high tension, use the command to set the high tension step and use the white toggle switch to set the value. 2. Increase filament heating to “3”; wait 3 minutes. Reach filament saturation by increasing heating at 0.5 step/minute. The filament bias should be about 56 as of 8/26/99. Adjust the fine “Bias” for a beam current ≈115 μA with a just-saturated filament (about 15 μA above dark current). Alignment of the Illumination System 1. Use the “Mag 1” button and white toggle “Selector” switch on the right panel to select a convenient magnification, maybe 15 kX. Use the “Brightness” knob on the left panel to adjust the beam intensity and the “Shift” knobs on the left and right panels to center the beam on the screen. Focus the image using the “Obj Focus” control. The step switch adjusts the strength of the focus; 1 is the smallest increment. 2. Establish the eucentric position as follows. Set the objective lens current on page 4 of the CRT to 7.00 (for AHP20L pole piece). Use the small knob (Z control) knob under the rod on the goniometer to focus the specimen. This should be very close to the eucentric position. To further converge on the eucentric position: 1) release the lock on the goniometer and set it to zero, 2) tilt the specimen a few degrees and bring the image back to the center with the small knob, and 3) return to zero and adjust the focus. Repeat steps 1)-3) until the image does not move with tilt in either direction. 3. Align filament at about 60 kX with a slightly undersaturated filament image using “Gun” button and “Def” knobs (tilt) in right drawer under “Deflector” row of buttons and center the filament image on the screen with the “Shift” knobs (translate) in drawer. The filament image should be symmetrical. Saturate the filament after aligning. 4. Align illumination down the optic axis of the condenser lens system by switching between “Spot Size 5” and “Spot Size 1” at about 50 kX and centering the beam. Use the “Brightness Shift” knobs to center the beam with “Spot Size 5,” and the “Gun” button and “Shift” knobs in the right drawer for “Spot Size 1”. Return to spot size 2 when aligned and use “Spot” button and “Shift” knobs in right drawer for final adjustment at spot size 2. The spot size is shown on page 1 of the screen and can be changed by the toggle switch “Spot Size” on the left panel. 5. Using “Spot Size 2” and condenser aperture 2 or 3, center condenser aperture and correct condenser astigmatism. Use “Con Stigmator” button and “Def” knobs in right drawer. Use the detail in a slightly undersaturated filament image at >60 kX, and return to saturation after astigmatism correction.

730

A. Appendix

6. Wobble beam tilts by depressing “Image X, Y Wobbler” buttons (either singly or together) on right panel. Center beam by depressing “Tilt” button in right drawer (under Cond/Def/Adj) and using “Shift” and “Def” buttons to make the filament image stationary. Alignment of the Imaging System 1. Find an area of interest and focus the specimen. 2. Establish the voltage center by depressing the “Brit Tilt” button on the left panel, pushing “HT Wobbler” on the right panel, and adjusting the “Def” knobs on the left and right panels to obtain a stationary image with the beam centered on the optic axis. Use the highest magnification possible, at least 250 kX. Turn off the “Brit Tilt” button after adjustment to save the setting. 3. If desired, turn on the TV and the Gatan CCD camera control. To use the TV, flip the “Camera In” button on the CCD camera control unit and focus the image. Use “Auto Contrast” except for unusual cases. If the TV is white, spread the beam to reduce the electron intensity on the TV camera. The TV is useful for correcting the objective lens astigmatism. NOTE: Because focusing accuracy is better on the large or small phosphorus screens, you must use them for your negatives to be in focus. 4. Correct the objective lens astigmatism by depressing the “Obj Stig” button on the left panel and adjusting the “Def” knobs. The X and Y stigmator settings can be read on page 6 of the screen. Typical settings for the X and Y stigmators are about 0.1 and −0.25, respectively. It is convenient to use the TV at >500 kX. Adjust objective astigmatism by: 1) focussing the image, 2) using one “Obj Stig” knob to make the image as sharp as possible, and 3) using the other “Obj Stig” knob to make the image as sharp as possible. Repeat 1–3 until the image is as sharp as possible and goes in and out of focus symmetrically as the objective lens is over and underfocused. You may then correct astigmatism by seeking the minimum contrast condition using the amorphous region near the edge of a hole (using the same steps 1–3 when the objective aperture is out). 5. Make final adjustments to the specimen tilt, alignment, insert objective aperture, touch up astigmatism, etc. Alignment of Diffraction System 1. Adjust the beam to fill the screen and insert the selected area aperture (bottom aperture drive) at about 30 kX. Go to diffraction mode by pushing “Diff” button on right panel. Adjust camera length to a reasonable value (≈100 cm) using white toggle “Selector” switch on right panel. Center the illumination and adjust the intensity. Remove the objective aperture. 2. Focus the diffraction pattern by inserting the objective aperture and use the “Diff Focus” knob (right panel) to focus the aperture edge. 3. Depress the “Proj” button in the right drawer and use the “Shift” knobs to center the transmitted beam onto the optic axis.

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Taking Photographs 1. Select automatic or manual exposure time on Page 1 of screen. The full screen or the small screen can be used to estimate the exposure time. Insert the small screen by pulling the lever located on the right side of the viewing chamber. 3. For automatic plate advance, press “Auto” on the right panel. For manual plate advance, press “Photo” on the right panel after each picture to advance the next negative. 4. The “Photo” light will illuminate when a plate has advanced. When the plate is in position press “Photo” to start the exposure. The film will drop into the receiving chamber after the exposure. Finishing the Session 1. Return magnification to a relatively low setting (maybe 1 kX), slightly defocus beam and remove objective and intermediate apertures. 2. Turn off the TV and the Gatan CCD camera control. 3. Turn down the filament slowly (about one step every 10 sec) until “3,” then turn down continuously to “Off”. Release the HT button. 4. If you have adjusted the filament bias, reset it to its original value, slightly lower than at the end of the session. 5. Reset X and Y goniometers to zero and center the specimen. The specimen position (X and Y in 0.001 mm up to 1.0 mm) is shown on page 2 of the CRT. Remove the specimen from the column. 6. Exchange film cassettes. It takes about 20 minutes for the vacuum to recover after the plates are exchanged, provided the new cassettes have been thoroughly evacuated. Please allow time for this in your session. 7. Reset number of unexposed plates by typing . 8. Sign the log book and note any problems, etc. 9. Turn down the CRT intensity (right panel), panel light (left panel) and keyboard light (right drawer). Other Useful Information The binoculars are helpful for focussing, correcting the objective lens astigmatism, focussing the objective aperture edge, etc. 1. First insert the small screen using the lever to the right of the viewing chamber. 2. Adjust the distance between the eyepieces to form a single image. 3. Close your right eye and focus your left eye on the screen (focus on the pointer or dirt on the screen) by rotating the ring at the base of the eyepiece. 4. Look through both eyes and focus the right eye on the screen by rotating the ring at the base of the eyepiece. Aperture Drives

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A. Appendix

All the aperture drives work the same. An aperture drive is inserted by switching the lever to the left. There are three apertures in a drive (six for the condenser aperture). A particular aperture is selected by turning the largest outer sleeve CW for smaller apertures and CCW for larger ones. The small knob on the right and the inner sleeve are the X-Y translates for the apertures. If You Get Lost If you are in the image (or diffraction) mode and get lost (i.e., everything goes black), reduce the magnification (to 2–10 kX), remove the objective and SAD apertures, spread out the illumination and look for the beam. (Translate the specimen if it has grid bars.) If you still can’t find the beam, please get help. A.11.2 Laboratory 1 – Microscope Procedures and Calibration with Au and MoO3 The principles of operation and alignment of the transmission electron microscope should be learned as soon as possible. This laboratory exercise covers basic imaging and diffraction, and provides calibration information needed in the later laboratories. The Au and MoO3 exercises are often the first rewarding experiences with a TEM. A. Camera Constant Determination Specimen. Polycrystalline Au film evaporated onto a holey carbon film supported on a 200 mesh copper grid. (Such Au samples are available from vendors of microscope supplies.) Measurements. (a) With the microscope at 200 kV and the specimen in the eucentric position, obtain two focused bright-field (BF) images of the same specimen area at a medium magnification (∼ 60 kX) using the largest and smallest objective lens apertures. Photograph the corresponding electron diffraction patterns (with a camera length of ∼ 100 cm) using the doubleexposure technique with the objective aperture in to record also the sizes and positions of the two different objective apertures. Explain why the size and position of the objective aperture affects the contrast in the image. (b) Photograph two selected-area diffraction (SAD) patterns (with a camera length of ∼ 100 cm) from the same specimen area using the largest and smallest intermediate apertures. Photograph the corresponding BF images, again using a double exposure with the intermediate aperture in and the objective aperture out (and an appropriate magnification) to record the sizes and positions of the intermediate apertures. Also record the objective, intermediate and projector lens currents. Calculate the microscope camera constant (λL) from these results.

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733

Explain why the size of the intermediate aperture affects the appearance of the diffraction pattern. Procedures for taking images and SAD patterns. (written for the JEOL 2000FX) Starting with a properly aligned TEM in the magnification (Mag) mode, and the specimen in the eucentric position: • • • • • • • • • • • •

Focus the image using the objective lens (focus) controls. Insert the desired SAD aperture and center it. Go to the SAD diffraction mode (Diff). Remove the objective aperture (if it was in). Center the illumination and spread the beam to obtain sharp diffraction spots. Focus the spot pattern using the diffraction focus knob. (You can insert the objective aperture and focus the aperture edge to confirm that the spot pattern is in focus.) Center the diffraction pattern on the screen using the projector alignment knobs in the right drawer. Set the exposure to approximately 1/3–1/4 of the full-screen meter reading, and photograph the diffraction pattern. (Alternatively, you may use about 3/4 of the small screen reading as an exposure estimate). Insert and center the desired objective lens aperture. Return to the magnification (Mag) mode. Focus and stigmate the image using the objective lens stigmator controls (stigmation is required only on the first image). Using the meter reading, photograph the image. (Repeat for all magnifications and diffraction patterns.)

Taking Double Exposures. For double exposures, press the photo button to start the exposure process, and then press it a second time while the screen is raising. This prevents the film from advancing after the first exposure. When the first exposure is complete, the photo button light comes back on. Press the button again for the second exposure (after setting the desired exposure time). B. Astigmatism Correction Specimen. Same evaporated Au as above. Procedures. Find a small hole in the holey carbon film that is not covered with gold, i.e., the carbon is exposed around the edge of the hole. Go to a high magnification (∼ 500 kX) so the granular features in the carbon film are visible. (You may want to insert a medium-size objective aperture to increase the contrast from the amorphous carbon. Make sure the aperture is centered!) View the image on the TV rate camera and correct the astigmatism using the stigmator knobs on the microscope. Remove the TV-rate camera, and use

734

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the CCD camera with a simultaneous live FFT display to perform a final correction of the astigmatism. When the astigmatism is corrected, record three images on the CCD in overfocused, minimum contrast, and underfocused conditions. Print these images and their corresponding FFTs, and discuss their features. C. Rotation Calibration (written for the Philips EM400T) Specimen. Molybdenum trioxide on carbon substrates. (MoO3 is formed by heating a Mo wire with an oxygen-acetylene torch in air. Carbon substrates supported on 200 mesh copper grids are passed through the smoke to collect the MoO3 crystals.) Experimental Measurements. (a) Find a small crystallite of MoO3 with well-defined facets. With the magnification (M) and diffraction (D) modes, use the double exposure method to record superimposed BF images of the specimen and its corresponding SAD diffraction pattern. Repeat this procedure on the same crystallite for each magnification (intermediate lens current) in the M mode – magnifications of 10, 13, 17, 22, 28, 36, 46, 60, 80 and 100 kX (10 total). (Note: The most common camera lengths are typically 575 and 800 mm.) (b) Record the currents of the objective, diffraction, intermediate and projector lenses (P1 and P2 ) for each magnification in the M mode, and for the diffraction patterns in the D mode, using the display selector knob in the back panel. Data Analysis. (a) Using the superimposed BF/SAD images, graph the magnitude and direction of the image rotation as a function of magnification. Comment on the important features of this plot. The crystallography of the MoO3 crystal and its relationship to the diffraction pattern are illustrated in Fig. A.11 for a JEOL 100CX microscope. There are errors in these features in all four references below, so be careful! (b) Measure the width of the MoO3 crystal and plot the crystal width as a function of the dial magnification. (A small crystal is required if its edges are to remain in the field of view at high magnification.) (c) On two separate graphs, plot the objective, diffraction, intermediate and projector lens currents for the magnification (M) and diffraction (D) modes as a function of the dial magnification. Discuss the significance of these graphs for image magnification and accuracy in SAD. References for Laboratory 1 1. J. W. Edington: Practical Transmission Electron Microscopy in Materials Science - 1. Operation and Calibration of the TEM (Philips Technical Library, Eindhoven, Netherlands 1974).

A.11 Laboratory 2 – Diffraction of θ′ Precipitates

735

Fig. A.11. Image rotation calibration of JEOL 120CX microscope operated at 120 kV. Note abrupt change in image rotation at 40 kX.

2. J. W. Edington: Practical Transmission Electron Microscopy in Materials Science - 2. Electron Diffraction in the Electron Microscope, (Philips Technical Library, Eindhoven, Netherlands 1974) pp. 11-16. 3. G. Thomas and M. J. Goringe: Transmission Electron Microscopy of Materials (John Wiley and Sons, NY 1979) pp. 28-33. 4. D. B. Williams: Practical Analytical Electron Microscopy in Materials Science (Philips Electron Instruments, Inc. Mahwah, NJ 1984) pp. 26-30. A.11.3 Laboratory 2 – Diffraction Analysis of θ  Precipitates This experiment introduces the important methods of electron diffraction and dark-field imaging to determine the identity and orientation relationship of precipitates in a matrix. For an introductory laboratory, θ′ precipitates have proved convenient in size and contrast against the Al matrix. This exercise also provides experience with sample tilt, which may require a prior session of practice. Laboratory 2 couples well with the energy-dispersive x-ray analysis in Laboratory 3, but the two can be performed independently. Background. The θ′ phase is a metastable precipitate that often forms during aging of Al-Cu base alloys. It has a tetragonal crystal structure with space group symmetry I4/mmm and a = 0.404 nm and c = 0.58 nm. A perspective drawing of the unit cell of the θ′ phase is shown in Fig. A.12. The

736

A. Appendix

unit cell contains four atoms of Al and two atoms of Cu. The θ′ precipitates form as thin plates on the 100 planes in the Al matrix with the orientation relationship (001)θ  (001)Al and [100]θ  [100]Al .

Fig. A.12. Left: Labeled crystal structure of θ′ precipitate. Right: Orientations of three variants of θ′ plates in the fcc Al matrix.

The θ′ phase forms as thin plates on all three {001}Al matrix planes. When a thin foil is viewed along a 001 Al orientation, one variant of θ′ phase is face-on, while the other two variants are edge-on and perpendicular to each another (see Fig. A.12). The Al matrix and each variant of θ′ phase each produce a different diffraction pattern. When all three variants are present within the selected area aperture, all of these diffraction patterns are superimposed. If a small selected area aperture is used, however, it may be possible to obtain diffraction patterns from only one or two variants of precipitate. Figure A.13 shows diffraction patterns for the Al matrix in a 001 orientation, and two variants of the θ′ phase, one face-on along [001]θ and the other edge-on along [100]θ . (The diffraction pattern for the third variant of θ′ can be obtained by rotating the [100]θ pattern on the lower right by 90◦ .) All three of these patterns can then be superimposed to obtain the composite diffraction pattern in Fig. A.14. An experimental 001 Al SAD pattern containing all three precipitate variants (and also double-diffraction spots) is also shown in Fig. A.14. The different variants of precipitate can be identified by bringing each of the precipitate diffractions labeled 1, 2 and 3 in the composite pattern onto the optic axis within a small objective aperture, and making a dark-field (DF) image. Specimen. Electropolished thin foils of Al–4.0 wt% Cu alloy. A sheet of polycrystalline alloy about 150 μm thick was solution treated for 1 h at 550◦ C, quenched into water and aged for 12 h at 300◦C to produce well-developed θ′ precipitate plates. Disks 3 mm in diameter were punched from the sheet and electropolished in a twin-jet Fischione apparatus using a 25 %HNO3 – methanol solution at about –40◦ C and 15 V.

A.11 Laboratory 2 – Diffraction of θ′ Precipitates

737

Fig. A.13. Indexed 001 diffraction patterns from fcc Al matrix (left), and two variants of θ′ precipitates within the Al matrix (right).

Fig. A.14. Composite diffraction pattern from all three variants of θ′ precipitate in Al matrix in [100] zone axis. Left: schematic, Right: experimental SAD.

738

A. Appendix

(Alternative samples: carbon extraction replicas from a medium carbon steel, or pieces of aluminum beverage cans.) Procedures (a) Before going to the microscope, photocopy and enlarge the low index fcc diffraction patterns in the Appendix of this book. On a second set of diffraction patterns you should prepare a set of Kikuchi line patterns. To do so, draw straight lines through the low index spots. The line through the  (the direction of spot g should be oriented perpendicularly to the direction g the spot from the origin). You may want to plot other low-index diffraction patterns for the θ′ phase using a computer program, if available. Please read some of the four references below. They contain information about the crystal structure, morphology, interfacial structure, and growth kinetics of the θ′ phase. (b) Obtain SAD patterns of the matrix and precipitates by tilting the specimen to low-index orientations such as 001 Al , 011 Al or 112 Al . Use Kikuchi line patterns and indexed diffraction patterns to help you. The 001 Al zone axis is the easiest to interpret, so you should try to obtain this orientation. Orient the specimen so that the pattern is exactly on the zone axis. Spread the illumination and take long exposures when photographing diffraction patterns so the faint precipitate spots will be sharp and visible. You might try several different exposures until you get a feel for the best exposure (typically about 1/4 of the automatic exposure reading). Don’t forget to focus the diffraction pattern! (c) To identify the precipitates in the intermediate aperture that contributed to the SAD pattern, photograph the corresponding BF images using the double-exposure technique. You may want to experiment with different size apertures, using a large aperture to obtain a pattern from all three θ′ variants, using a smaller aperture to obtain diffraction patterns from only one or two variants. (d) Photograph DF images of each of the θ′ variants on the three {100}Al planes. Do this by tilting the incident beam into the position of the precipitate diffraction spot, so the −g diffraction appears on the optic axis. (Avoid the “amateur mistake.”) Also photograph the corresponding diffraction patterns. Record the precipitate diffraction that was used to form the DF image. This can be done by either photographing the beam-stop, or using the double-exposure technique with the objective aperture superimposed on the diffraction pattern for one of the exposures. This record is needed to positively identify each precipitate variant. (e) Identify the θ′ precipitates by fully indexing the diffraction patterns and correlating them to the particle morphologies and orientations in the BF and DF images. Your rotation calibration from the previous lab will be useful here. Also determine the lattice spacings for the θ′ phase by using the Al

A.11 Laboratory 3 – EDS of θ′ Precipitates

739

diffraction pattern as a standard, with crystallographic data for this phase provided in the references. (f) On a 001 stereographic projection, show the orientation relationship between the θ′ precipitate and matrix. Mark most of the low-index poles for the precipitate and matrix phases. Diffraction programs that also plot stereographic projections are very useful for this. References for Laboratory 2 1. J. M. Silcock and T.J. Heal: Acta Cryst. 9, 680 (1956). 2. G. C. Weatherly and R. B. Nicholson: Philos. Mag. A 17, 801 (1968). 3. U. Dahmen and K. H. Westmacott: Phys. Stat. Sol. (a) 80, 248 (1983). 4. G. W. Lorimer: in Precipitation Processes in Solids (TMS-AIME, Warrendale, PA 1978) p. 87. A.11.4 Laboratory 3 – Chemical Analysis of θ  Precipitates This laboratory could be performed simultaneously with laboratory 2, since it uses the same specimens of θ′ precipitates in Al–Cu. The present laboratory demonstrates microbeam chemical analysis with EDS spectroscopy. Specimen. Same electropolished thin foils of Al–4.0 wt% Cu alloy used in Laboratory 2. Procedures. (a) Using the same basic probe conditions as in b below, but with the beam spread over a large area near the edge of the foil, acquire an EDS spectrum with at least 100,000 counts in the Al Kα peak. Assuming this spectrum represents the average alloy composition, use this spectrum to determine the k-factor for Al and Cu. (b) Obtain EDS spectra from about 6 different edge-on θ′ plates using the same probe and counting conditions. Try a small spot size (say 8) for 60 sec and work near the edge of the foil, i.e., thin-film conditions. If you need more counts, switch to a larger spot size (maybe 6) or a longer counting time. Use the second or third condenser aperture to obtain a well-defined probe. (c) Take bright-field images of each θ′ plate. Use the double exposure technique to show the size and position of the probe on the plate. Use a magnification of around 100 kX. (d) Find three edge-on θ′ plates in about the same area (same specimen thickness) but with different plate thicknesses. How do their EDS spectra compare? (e) Choose three plates, one very near the edge of the foil, one slightly further in, and the third even further in. How do their spectra compare and why? (f) If you have time, obtain three more spectra on the same precipitate in a relatively thin area with spot sizes of 2, 4, 6 and 8. How does the spot size affect the spectra and why?

740

A. Appendix

(g) If you have time, obtain three spectra along the length of the same precipitate using the same spot size as in b above. What causes the variation among the spectra? (h) If you still have time, use a spot size of 8 and take a composition profile across the precipitate/matrix interface. You will need a high magnification to do this. References for Laboratory 3 same as for Laboratory 2 A.11.5 Laboratory 4 – Contrast Analysis of Defects This experiment gives experience in defect identification using contrast analysis. The defect type, plane and displacement vector as well as the Burgers vectors of isolated perfect dislocations partial dislocations bounding stacking faults will be determined. It is more challenging to attempt a full stacking fault analysis as in Sect. 7.12.5. Specimen. Electropolished thin foils of AISI Type 302 (or 309) fcc stainless steel, annealed and lightly cold-rolled. Disks 3 mm in diameter were punched from the rolled sheet and electropolished in a twin-jet Fischione apparatus using a 10 % perchloric acid–ethanol solution at about –15◦ C and 30 V. (Alternative samples: Cu–7 % Al sample deformed approximately 5 % in tension, interfacial dislocations on the θ′ plates used in Laboratory 3, misfit dislocations in Si-Ge heterostructures, dislocations in NiAl deformed a few percent in tension.) Procedures (a) Before going to the microscope, prepare contrast analysis (g · b) tables for defect visibility, paying particular attention to low-index orientations such as 110 , 100 , 112 , and 111 . Examples of contrast tables are presented in Sect. 7.8. The 110 orientation is particularly good for analysis since many different g vectors are available in this orientation. Other microscopists like to start with a 100 orientation, since it is also a convenient starting place for tilting into other zone axes. To identify uniquely the dislocation line direction or Burgers vector, you will need at least two zone axes. (b) Locate isolated planar defects in the foil (either singly or in groups) and image the same area in a strong two-beam, bright-field (BF) and centered dark-field (DF) condition with s = 0. Try to ensure that the deviation parameter s is identical for the BF and DF images by tilting the foil so that the relevant extinction contour passes through the defect(s) to be analyzed. Record the corresponding SAD patterns. Check the crystallographic orientation on either side of the planar defect. If it is different, record both patterns.

A.11 Laboratory 4 – Defect Analysis

741

(c) Continue to image the same defect region under other two-beam BF conditions indicated by the contrast tables prepared in a above. Again, pay particular attention to the deviation parameter to ensure that s ≥ 0. Look for evidence of bounding partial dislocations. Record the corresponding SAD patterns. (d) Using additional diffraction conditions (as identified in your contrast table), image isolated slip dislocations or dislocation pile-ups present in the foil. Record the corresponding SAD pattern. (e) By trace analysis on an appropriate stereographic projection, identify the defect planes and slip planes. Arrange the data to show the nature of the defects and determine the Burgers vectors of all dislocations. References for Laboratory 4 1. J. W. Edington: Practical Electron Microscopy in Materials Science Volume 3 - Interpretation of Transmission Electron Micrographs (Philips Technical Library, Eindhoven 1975) pp. 10-55. 2. G. Thomas and M. J. Goringe: Transmission Electron Microscopy of Materials (John Wiley and Sons, New York 1979) pp. 142-169. 3. P. B. Hirsch, et al.: Electron Microscopy of Thin Crystals (R. E. Krieger Pub. Co., Malabar, FL 1977) pp. 141-147, 162-193, 222-275, 295-316. 4. P. H. Humphrey and K. M. Bowkett: Philos. Mag. 24, 225 (1971). 5. J. M. Silcock and W. J. Tunstall: Philos. Mag. A 10, 361 (1965).

742

A. Appendix

A.12 Fundamental and Derived Constants Fundamental Constants  = 1.0546 × 10−27 erg·sec = 6.5821 × 10−16 eV·sec kB = 1.3807 × 10−23 J/(atom·K) = 8.6174 × 10−5 eV/(atom·K) R = 0.00198 kcal/(mole·K) = 8.3145 J/(mole·K) (gas constant) c = 2.998 × 1010 cm/sec (speed of light in vacuum) me = 0.91094 × 10−27 g = 0.5110 MeV·c−2 (electron mass) mn = 1.6749 × 10−24 g = 939.55 MeV·c−2 (neutron mass) NA = 6.02214 × 1023 atoms/mole (Avogadro constant) e = 4.80 × 10−10 esu = 1.6022 × 10−19 coulomb μ0 = 1.26 × 10−6 henry/m ε0 = 8.85 × 10−12 farad/m

a0 = 2 /(me e2 ) = 5.292 × 10−9 cm (Bohr radius) e2 /(me c2 ) = 2.81794 × 10−13 cm (classical electron radius) e2 /(2a0 ) = R (Rydberg) = 13.606 eV (K-shell energy of hydrogen) e/(2mec) = 0.9274 × 10−20 erg/oersted (Bohr magneton) 2 /(2me ) = 3.813 × 10−16 eV cm2

Definitions 1 becquerel (B) = 1 disintegration/second 1 Curie = 3.7 × 1010 disintegrations/second

radiation dose: 1 roentgen (R) = 0.000258 coulomb/kilogram Gray (Gy) = 1 J/kG Sievert (Sv) is a unit of “radiation dose equivalent” (meaning that doses of radiation with equal numbers of Sieverts have similar biological effects, even when the types of radiation are different). It includes a dimensionless quality factor, Q (Q∼1 for x-rays, 10 for neutrons, and 20 for α-particles), and energy distribution factor, N. The dose in Sv for an energy deposition of D in Grays [J/kG] is: Sv = Q×N×D [J/kG] Rad equivalent man (rem) is a unit of radiation dose equivalent approximately equal to 0.01 Sv for hard x-rays. 1 1 1 1 1 1 1 1

joule = 1 J = 1 W·s = 1 N·m = 1 kg·m2 ·s−2 joule = 107 erg newton = 1 N = 1 kg·m·s−2 dyne = 1 g·cm·s−2 = 10−5 N erg = 1 dyne·cm = 1 g·cm2 ·s−2 Pascal = 1 Pa = 1 N·m−2 coulomb = 1 C = 1 A·s ampere = 1 A = 1 C/s

A.12 Fundamental and Derived Constants

1 1 1 1 1

743

volt = 1 V = 1 W·A−1 = 1 m2 ·kg·A−1 ·s−3 ohm = 1 Ω = 1 V·A−1 = 1 m2 ·kg·A−2 ·s−3 farad = 1 F = 1 C·V−1 = 1 m−2 ·kg−1 ·A2 ·s4 henry = 1 H = 1 Wb·A−1 = 1 m2 ·kg·A−2 ·s−2 tesla = 1 T = 10, 000 gauss = 1 Wb·m−2 = 1 V·s·m−2 = 1 kg·s−2 ·A−1

Conversion Factors 1 1 1 1 1 1 1 1

˚ A = 0.1 nm = 10−4 μm = 10−10 m b (barn) = 10−24 cm2 eV = 1.6045 × 10−12 erg eV/atom = 23.0605 kcal/mole = 96.4853 kJ/mole cal = 4.1840 J bar= 105 Pa torr = 1 T = 133 Pa kG = 5.6096 × 1029 MeV·c−2

Useful Facts ˚ photon = 12.3984 keV energy of 1 A hν for 1012 Hz = 4.13567 meV 1 meV = 8.0655 cm−1 temperature associated with 1 eV = 11, 600 K A lattice parameter of Si (in vacuum at 22.5◦ C) = 5.431021 ˚ Neutron Wavelengths, Energies, Velocities En = 81.81 λ−2 (energy-wavelength relation for neutrons [meV, ˚ A]) A, m/s]) λn = 3955.4/vn (wavelength-velocity relation for neutrons [˚ En = 5.2276 × 10−6 vn2 (energy-velocity relation for neutrons [meV, m/s]) Some X-Ray Wavelengths [˚ A] Element

Kα

Kα1

Kα2

Kβ1

Cr

2.29092

2.28962

2.29351

2.08480

Co

1.79021

1.78896

1.79278

1.62075

Cu

1.54178

1.54052

1.54433

1.39217

Mo

0.71069

0.70926

0.71354

0.632253

Ag

0.56083

0.55936

0.56377

0.49701

744

A. Appendix

Relativistic Electron Wavelengths For an electron of energy E [keV] and wavelength λ [˚ A]:   λ = h 2me E 1 +

0.3877 E −1/2 = 1/2 1/2 2me c2 E (1 + 0.9788 × 10−3 E)

kinetic energy ≡ T = 21 me v 2 = 12 E 1+γ γ2 Table A.4. Parameters of high-energy electrons E [keV]

λ [˚ A]

γ

v [c]

T [keV]

100

0.03700

1.1957

0.5482

76.79

120

0.03348

1.2348

0.5867

87.94

150

0.02956

1.2935

0.6343

102.8

200

0.02507

1.3914

0.6953

123.6

300

0.01968

1.587

0.7765

154.1

400

0.01643

1.7827

0.8279

175.1

500

0.01421

1.9785

0.8628

190.2

1000

0.008715

2.957

0.9411

226.3

Index

aberrations, 102, 195, 197, 602, 603 – allowed by symmetry, 599 – higher order, 602 absorption (electron incoherence), 613, 627, 659 absorption and thin-film approximation (table), 212 absorption correction, 43, 211 – flat specimen, 43 – granularity, 47 – validity, 47 absorption edge, 167 accidental degeneracy, 17 Ag-Cu interface, 566 Aharonov–Bohm effect, 65 Al12 Mn, 78 Al-4wt.% Cu alloy, 255 Al-Cu, 735 Al-Cu θ′ phase, 268 Al-Ge interface, 566 Al-Li alloy, 74 α boundary, 405 amateur mistake, 261, 266, 378, 738 amorphous material, 491 – one-dimensional, 491 analytical TEM, 62, 164 anisotropy, elastic, 377, 411, 429 anomalous scattering, 132, 462 – partial pair correlations, 501 antibonding orbitals, 168 antiphase boundary, 404 – superlattice diffraction, 405 aperture angle, 69, 89, 115, 605 – optimum, 111 apertureless image, 72 artificial rays, 69 Ashby-Brown contrast, 409 astigmatism, 104, 531, 570, 603 – correction procedure, 106, 531 – salt and pepper contrast, 107 atom, 1

– as point, 458 atomic displacement disorder, 469 atomic form factor, 225 – dependence on Δk, 150 – destructive interference at angles, 142 – effective Bohr radius, 146 – electron, table of, 698 – electrons and x-rays, 149 – model potentials, 144 – Mott formula, 149 – physical picture, 141 – Rutherford, 146 – screened Coulomb potential, 144 – sensitivity to bonding electrons, 151 – shape of atom, 141, 150 – shapes of V (r), 150 – Thomas–Fermi, 146 – x-ray, table of, 693 atomic periodicities, resolution of, 82 atomic size effect, 477 Auger effect, 13, 221 autocorrelation function, 2, 459 average potential of solid, 613 Avogadro constant, 742 axial dark-field imaging, 74, 261, 378 axial divergence, 26, 430 B2 structure, 244, 567 back focal plane, 68, 71 background, 57 – subtraction and integration, 48 backscattered electron image (BEI), 200 backscattered electrons, 200 bar, 743 barn, 743 barrier penetration, 589 basis vectors, 235 beam propagation, 623 beam representation, 612 beam tilt

746

Index

– coils, 572 – dislocation position, 582 – HRTEM, 553, 570 beams and Bloch waves, 628 – normalization, 630 beats – acoustic, 627 – mathematical analysis, 632 – pattern, 632 – physical picture, 627 becquerel, 742 Beer’s law, 205 bend contour, 353, 414 – Cu-Co, 357 – diffraction patterns, 356 bending magnets, 21 Bethe asymptotic cross-section, 189 Bethe ridge, 179 Bethe surface, 186, 187 biology, 73 biprism, 65 black cross, 326 Bloch waves, 628 – amplitudes and dispersion surface, 654 – change across defect, 657 – channeled, 591 – characteristics, 638 – energies, 637 – orthogonality, 635 – propagator, 660 – representation, 612 – weighting coefficients, 630 block diagram of a TEM, 62 Blue Boy, 135 blue sky, 129 Boersch effect, 103 Bohr magneton, 742 Bohr radius – dependence on Z, 146 – effective, 146 Born approximation, 139, 224, 619 – first, 139 – higher order, 139 Bose–Einstein statistics, 476 boundary conditions, 644 Bragg’s law, 3 Bragg–Brentano geometry, 26 bremsstrahlung – coherent, 59 – intensity, 16, 24 bright-field (BF) imaging, 68, 71, 337, 353, 542, 547, 649

brightness, 22, 108 – compromises, 167 – conservation of, 109 – electron gun, 113, 605 brilliance, 22 broadening of x-ray peaks, 2, 6 – complement of TEM, 452 – dislocation, 452 – meaning of size and strain, 452 – stacking faults, 444 buckled specimen, 353, 420 Burgers circuit, 721 – in HRTEM image, 84 Burgers vector, 362, 565, 721 – conservation of, 725 – fcc, 366 calorie, 743 camera constant, 78 – calibration, 78 – determination of, 732 camera equation, 78 camera-length, 78, 88 carrier, 83 catalyst, 562, 563 Cauchy function, 432 CCD cameras, 378, 578 CdSe, 606 center of gravity, 505 center of the goniometer, 50 channeling, 586 characteristic x-ray, 13, 16, 24 chemical bonding, 170 chemical disorder, 469, 481 chemical map, 63 chemical short-range order, 481, 485 children’s jacks, 264 chromatic aberration, 103, 195, 197, 604 – importance of thin specimens, 103 classical electron radius, 129, 742 Cliff–Lorimer factor, 208 – calculation, 211 – experimental determination, 210 coherence, 122 coherent bremsstrahlung, 59 coherent elastic scattering, 123 coherent imaging, 583 coherent scattering, 119 – forward direction, 142, 503 – inelastic, 123, 155 – phases, 128 column approximation, 446 column lengths

Index – distribution, 448 – neighbor pairs in column, 450 – random termination, 449 coma, 603 complementarity of BF and DF, 74 Compton scattering, 132 – incoherence, 133 computer control, 601 condenser lens – aperture, 88 – convergence (C2), 88 – spot size (C1), 88 constants, 742 constructive interference, 3, 119 contrast transfer function, 550 – damping of, 554 – incoherent vs. coherent, 594 conventional modes, 71 conventional TEM, 79 convergence angle control, 207 convergent-beam electron diffraction (CBED), 79, 304 – α-Ti, 321, 328, 329 – BF disk symmetry, 316 – black cross, 326 – DF disk symmetry, 316 – diffraction group, 316, 317 – disk and crystal symmetry, 317 – disk intensity nonuniformity, 81 – Ewald sphere, 305 – FeS2 , 327 – Friedel’s law, 315 – G disk symmetry, 317 – Gjønnes–Moodie lines, 325 – glide plane, 325 – HOLZ lines and lattice parameter, 312 – HOLZ radius Gn , 310 – illumination, 80 – intensity oscillations in disk, 307 – point group, 314 – positions of disks, 311 – – orthorhombic examples, 312 – projection diffraction group, 316 – sample thickness determination, 307 – screw axis, 325 – semi-angle of convergence, 306 – space group, 322, 325, 327 – special positions, 325 – symmetric many-beam, 325 – unit cell, 309 – whole pattern symmetry, 316 conversion factors, 743

747

convolution – commutative property, 464 – defined, 431 – delta function, 459, 464 – example, 432 – Gaussians, 432 – Lorentzians, 432 – of potential and beams, 618 – theorem, 435 – Voigt function, 433 core excitations, table of energies, 178 core hole, 170 – decay, 164 Cornu spiral, 529, 581 correlations, 485 – short-range, 486 costs, 604 Coulombic interaction, 181, 593 coupled harmonic oscillators, 624, 626, 674 Cowley-Moodie method, 672 critical angle, 588 crystal potential – inversion symmetry, 619 – real, 619 crystal symmetry elements, 317 crystal system notation, 241 crystallite sizes – distribution and TEM, 453 – Patterson function, 467 – TEM and x-ray, 453 Cu2 O, 152 Cu-Co, 412 Curie, 742 d-orbitals (shapes), 152 damping function, 560 dark-field (DF) imaging, 68, 71, 72, 337, 542, 547, 649 dead time, 30 Debye model, 475 Debye–Scherrer, 10 Debye–Waller factor, 171, 472, 585, 720 – calculation of, 475 – concept, 474 – conventions, 475 deconvolution, 433 – Fourier transform procedure, 433 – frequency filter, 439 – procedure with noise, 438 defects, 338 δ boundary, 405 delta function, 459 density, 44

748

Index

density heterogeneity, 497 density of unoccupied states, 169 density-density correlations, 510 depth of field, 89, 115, 604, 605 depth of focus, 89, 115 detector – analytic TEM, 34 – annular dark-field, 584 – beryllium window, 33 – calorimetric, 34 – charge sensitive preamplifier, 36 – count rates, 31 – dead layer, 32 – energy resolution, 30 – escape peak, 32 – gas-filled proportional counter, 31 – intrinsic semiconductor, 32 – position-sensitive, 34 – quantum efficiency, 30 – scintillation counter, 31 – Si[Li], 34 – silicon drift, 33 – solid state, 32 – table of characteristics, 31 – x-ray, 30 deviation parameter, s, 256, 339, 343 – effective, 646 – Kikuchi lines, 299 deviation vector, s, 256, 339 – in dynamical theory, 617 differential scattering cross-section, 126 – inelastic, 183 diffraction – beams across defect, 659 – coherence, 229 – Δk and θ, 229 – effect of apertures, 104 – electron, 224 – fine structure, 264 – forbidden diffractions, 238 – Fourier transform of potential, 228 – frequency and time, 227 – incident wave, 226 – line broadening, 423 – rel-rods, 252 – shape factor, 235 – structure factor, 235 – structure factor rules, 237 – translational invariance in plane, 9 – vectors and coordinates, 226 – wave, 226 – wavevectors, 228 diffraction contrast, 62, 72, 337

– dynamical – – dislocation, 656 – – interface, 656 – – stacking fault, 656 – dynamical without absorption, 655 – null contrast, 359 – strain fields, 358 diffraction coupling, 166 diffraction lens, 88 diffraction mode, 70 diffraction pattern – background, 57 – bcc, 705, 706 – chemical composition, 7 – crystallite sizes, 9 – dc, 707, 708 – fcc, 703, 704 – hcp, 709–712 – indexed powder, 691 – internal strains, 6, 427 – internal stresses, 428 – peak broadening, 6 – silicon, 4 – size effect broadening, 7, 425 diffraction vector, 81 diffuse scattering, 457 – chemical disorder, 484, 591 – displacement disorder, 477 – Huang, 478 – short-range order (SRO), 488 – thermal, 474, 592 dilatation, 427 dipole approximation, 191 dipole oscillator, 128 Dirac δ-function, 218, 459 Dirac equation, 17 Dirac notation, 180 dirty dark-field technique, 74 disk of least confusion, 102 – resolution, 110 dislocation, 337, 360, 721 – Burgers vector, 721, 741 – charge sinks, 721 – contrast tables, 740 – core, 723 – dipole, 368 – double image, 374 – dynamical contrast, 376, 658 – edge, 362, 721 – fcc and hcp, 724 – g · b analysis, 740 – groups of, 726 – image width, 374

Index – interactions, 726 – loop, 722 – mixed, 721 – partial, 725 – phase-amplitude diagram, 361 – plastic deformation, 721 – position of image, 361, 373 – reactions, 724 – screw, 364, 721 – self energy, 722 – strain field, 726 – superdislocation, 368 – tilt boundary, 726 – weak-beam dark-field method, 378 dispersion corrections, 697 dispersion surface, 634, 653 displacement disorder – dynamic, 469 – static, 469 divergence – thick hexapole, 599 DO19 structure, 566 dopant, 597 double diffraction – forbidden diffractions, 302 – – tilting experiment, 302 double exposures, 733 double-differential cross-section, 184 double-tilt holder, 274 drift of sample, 378 Duane–Hunt rule, 14 dynamical absences – space group, 327 dynamical theory – boundary conditions, 644 – eigenvalue problem, 671 – extinction distances, 671 – intuitive approach, 611 – multibeam, 618, 669 – multibeam and HRTEM, 669 – multislice method, 672 – phase grating, 671 – propagator, 671 – vs. kinematical theory, 619, 623 effective deviation parameter, 343 effective extinction distance, 343 eigenfunctions for electrons, 612 elastic anisotropy, 444, 480 elastic cross-section – Rutherford, 147 elastic scattering, 123 electric dipole radiation, 128

749

electric dipole selection rule, 19, 59 electron – holography, 65 electron coherence length, 88 electron energy-loss near-edge structure (ELNES), 169 electron energy-loss spectrometry (EELS), 62, 593, 606 – background in spectrum, 167 – chemical analysis, 191 – energy filter, 577 – experimental intensities, 185 – fine structure, 169, 170 – M4,5 edge, 198 – magnetic prism, 194 – Ni spectrum, 167 – nomenclature for edges, 168 – partial cross-section, 189 – plasmon peak, 167 – spectrometer, 165, 586 – – monochromator, 606 – – aperture, 188 – – diffraction-coupled, 166 – – entrance aperture, 166, 186 – – image-coupled, 166 – – parallel or serial, 165 – spectrum – – background, 191 – – edge jump, 219 – – multiple scattering, 192 – thickness gradients, 578 – typical spectrum, 167 – white lines, 167, 171 – zero-loss peak, 167, 607 electron form factors, table of, 698 electron gun – brightness, 108 – filament saturation, 86 – self-bias design, 85 – thermionic triode, 85 electron interaction parameter, 558 electron mass, 742 electron microprobe, 201, 205 electron probe size, 207, 208 electron scattering – Born approximation, 136 – coherent elastic, 136 – Green’s functions, 138 electron wave probability, 136 electron wavelengths, table of, 743 electron-atom interactions, 13 electronic transition nomenclature, 168 electropolishing, 736

750

Index

elegant collar, 135, 152 elemental mapping, 38 energy, 123 energy transfer, 10 energy-dispersive x-ray spectrometry (EDS), 30, 62, 200, 593 – artifacts, 213 – background, 208 – compositional accuracy, 216 – confidence level, 216 – detector take-off angle, 206 – electron trajectories in materials, 200 – escape path, 206 – hole count, 214 – k-factor determination, 739 – microchemical analysis, 204 – minimum detectable mass (MDM), 216 – minimum mass fraction (MMF), 216 – practice, 739 – quantification, 206 – sensitivity versus Z, 165 – spectrometer, 205 – spurious x-rays, 214 – statistical analyses, 216 – Student-t distribution, 216 – typical spectrum, 205, 219 energy-filtered TEM (EFTEM), 577 – chemical mapping, 196 – diffraction contrast, 196 – energy-filtered TEM imaging, 193 – instrumentation, 193 – spatial resolution, 198 equatorial divergence, 26 eucentric tilt, 101 Everhart-Thornley detector, 203 Ewald sphere – and Bragg’s Law, 259 – axial dark-field imaging, 261 – construction, 257, 258 – curvature, 258 – dynamical theory, 651 – Laue condition, 258 – manipulations, 259 excitation error, sg , 616 – in dynamical theory, 617 extended electron energy-loss fine structure (EXELFS), 170 extended x-ray absorption fine structure (EXAFS), 173 extinction distance, 343, 612, 616 – and structure factor, 620 – effective, 343

– table of, 344 extracted particle, 76 factors of 2π, 233, 458, 646 Faraday cage, 207 fast Fourier transform, 561 – deconvolution, 455 Fe3 Al, 406 Fe-Cu (grain boundaries), 507 FeCo, 244 field effect transistor, 36 field emission gun, 86 – cold, 86 – Schottky, 86 filament lifetime, 85 fingerprinting, 5 first-order Laue zone (FOLZ), 262, 311 fission, 153 fluorescence correction, 212 fluorescence yield, 203 flux (in scattering), 125 focused ion-beam milling, 214 focusing circle, 27 focusing strength, 67 forbidden diffractions, 238, 242 – double diffraction, 302 forbidden transitions, 19 form factor – electron, 140 – – table of, 698 – physical picture, 141 – x-ray, 131 – – table of, 693 forward scattering (coherence), 503 Fourier transform – bare Coulomb, 146 – complex, 436 – cutoff oscillations, 438 – decaying exponential, 145 – deconvolution, 433 – Gaussian, 438 – Lorentzian, 145, 438 – low-pass filter, 438 – scattered wave, 140 – table of pairs, 717 Frank interstitial loop – HRTEM image of, 84 Fraunhofer region, 520 Fresnel fringes, 581 – astigmatism, 107 – at edge, 530 – focus, 107, 531, 581 – spacing, 531 Fresnel integrals, 529

Index Fresnel propagator, 533 Fresnel region, 520 Fresnel zones, 524 Friedel’s law, 448, 461 – CBED, 315 g · b rule, 362 GaAs, 83 gas gain, 31 Gaussian damping function, 560 Gaussian focus, 547, 569 Gaussian function, 451 Gaussian image plane, 102 Gaussian thermal displacements, 720 Geiger, 147 generalized oscillator strength (GOS), 185, 186 geometrical optics, 66 Gjønnes–Moodie (GM) lines, 325 glass lens, 91 – concave, 95 – Fermat’s principle, 96 – phase shifts, 95 – shape of surface, 94 – spherical surface, 95 goniometer, 25, 274 – Bragg–Brentano, 26 – circle, 26 – TEM sample, 66 grain boundary, 407, 565 Gray, 742 Green’s function, 138 – spherical wavelet, 519 – wave equation, 532 growth ledges, 415 Guinier approximation, 504, 506 Guinier radius, 506 Guinier–Preston zones, 254 HAADF imaging, 62, 583 – defocus, 595 – electron channeling, 586, 596 – electron scattering, 591 – electron tunneling, 589 – resolution, 594 – sample drift, 586 – source of incoherence, 584, 585 – vs. HRTEM images, 594 half-width-at-half-maximum (HWHM), 424 hexagonal close packed – interplanar spacings, 55 – structure factor rule, 55

751

hexapole, 598 Hf, 605 high-resolution TEM (HRTEM), 81 – as interference patterns, 83 – compensate aberration with defocus, 541 – effect of defocus, 539 – effect of spherical aberration, 540 – experimental, 538 – image matching, 555 – lens characteristics, 546 – microscope parameters, 559 – simple interpretations, 567 – specimen parameters, 556 – total error in phase, 541 high-resolution TEM practice – anomalous spot intensities, 578 – beam tilt effects, 572 – defocus, 569 – doubling of spot periodicities, 573 – FFTs from local regions, 577 – minimum contrast condition, 569 – sample thickness, 575 – surface layers, 578 – use of EELS, 576 high-resolution TEM simulations – beam convergence, 560 – diffuse scattering, 561 – measurement of parameters, 561 – microscope instabilities, 560 – other helpful programs, 576 – procedure, 556 – quantifying parameters, 560 – size of array and unit cell, 561 – specimen and microscope, 568 higher-order Laue zone (HOLZ), 262, 311 – dynamical absences, 328 – excess and deficit lines, 313 – lines and lattice parameter, 312 hole count, 214 holography, 65 homogeneous medium, plane wave in, 518 H¨ onl dispersion corrections, 131 Howie–Whelan–Darwin equations, 617 Huang scattering, 478 Huygens principle, 522 – spherical wave analysis, 523 hydrogenic atom, 188 ideal gas, 503 illumination angle, 88 illumination system

752

Index

– convergence (C2), 88 – lenses, 85, 87 – point source, 87 – spot size (C1), 88 image coupling, 166 image shift, 603 imaging lens system, 88 – cross-overs, 100 – image inversions, 88 imaging mode, 70 imaging plates, 35, 378 in-situ studies, 64 incident plane wave, 136 incoherence, 119, 122 incoherent elastic scattering, 123 incoherent imaging, 583 incoherent inelastic scattering, 123 incoherent scattering, 122, 583 index of refraction, 91 indexing diffraction patterns – concept, 4, 274 – easy way, 276 – indexed patterns, 691 – row and column checks, 279 – start with diffraction spots, 279 – start with zone axis, 276 inelastic, 123 inelastic electron scattering, 593 inelastic form factor, 182, 593 inelastic scattering, 10, 123 information limit, 551, 604 – HAADF imaging, 594 insertion device, 21 instrument function, 435 instrumental broadening, 430 integral cross-section, 190 integral inelastic cross-section, 220 interband transition, 606 interface – coherent, 565 – crystal-liquid, 576 – incoherent, 566 – semicoherent, 566 intermediate aperture, 76 intermediate lens, 70, 88 internal interfaces – displacement vector, 384 – phase shifts, 384 – phase-amplitude diagram, 388 internal stress, 428 International Centre for Diffraction Data, 5 interphase boundaries, 565

interstitial loop, 407 ionization, 13 – cross-section, 204 isomorphous substitutions, 462 isotopic substitutions, 462 isotropic averages, 488 JEOL 2000FX, 728 JEOL 200CX, 570 JEOL 2010F, 61 JEOL 4000EX, 553, 570 Johansson crystals, 27 jump-ratio image, 196 K-B mirror, 29 Kikuchi lines – deviation parameter, 299 – indexing, 294 – Kikuchi maps, 299 – Kossel cones, 293 – measure of s, 340, 379 – origin, 291 – sign of s, 299 – specimen orientation, 296 – visibility, 293 kinematical theory – disorder, 457 – validity, 224, 339, 347 – vs. dynamical theory, 619, 623, 673 kinematics of inelastic scattering, 178 knock-on damage, 218 Kossel cones, 293 l’Hˆ ospital’s rule, 249 L10 structure, 566 LaB6 thermionic electron source, 85 laboratory exercises, 728 lattice fringe imaging, 543 lattice parameter measurement, 49 lattice translation vectors – primitive, 230 Laue condition, 233 – and Bragg’s law, 233 – Ewald sphere, 258 Laue method, 10 – backscatter Laue of Si, 11 Laue monotonic scattering, 484, 488 Laue zones, 262, 311 – symmetry and specimen tilt, 262 ledges, 565 lens, 194, 534 – aberrations, 102, 602 – as phase shifter, 534 – curvature of glass, 93

Index – double convex, 93 – glass, 92 – ideal phase function, 534 – magnetic, 97 – performance criteria, 102, 602 – phase transfer function, 543 – transfer, 600 lens and propagator rules, 534 lens design – phase shifts, 95 – ray tracing, 93 lens formula, 68, 116, 535 light in transparent medium, 521 line of no contrast, 409, 410 liquid crystal, 563, 564 lobe aberration, 603 Lorentz factor, 40, 42 Lorentz force, 599 Lorentz microscopy, 63 Lorentzian function, 448, 451 – second moment divergence, 456 magnetic lens – electron trajectory, 98 – focusing action, 99 – image rotation, 99, 734 – Lorentz forces of solenoid, 98 – pole pieces, 97 – post-field, 99 – rotation calibration with MoO3 , 100, 734 main amplifier, 37 manufacturers (TEM), 65, 167, 214 Marsden, 147 mass attenuation coefficients, 134 – x-ray, table of, 692 mass-thickness contrast, 73, 338 materials, 1 – chemical compositions, 1 – crystal structure, 1 – diffraction pattern, 2 – microstructures, 1 matrix C or C −1 , 631 mean inner potential, 152 measured intensities, 45 metallic glass, 5, 495 metals, cold-worked, 452 microchemical analysis, 164 microstructure, 1, 61, 337 Miller index, 3 minimum contrast condition, 569 modulation, 83 moir´e fringes, 389, 416

753

– parallel, 389 – rotational, 390 momentum transfer, 10 monochromatic radiation, 10 monochromator, 27 – asymmetrically-cut crystal, 28 – diffracted beam, 29 – electron, 166 – incident beam, 29 Monte Carlo, 200 Moseley’s laws, 18, 217 M¨ ossbauer diffraction, 158 – chemical sensitivity, 160 – form factors, 158 – interference with x-ray scattering, 160 – resonance and phase, 159 M¨ ossbauer spectroscopy, 160 Mott formula, 149 multi-body spatial correlations, 500 multi-lens systems, 69 multichannel analyzer, 38 multiphonon scattering, 585, 592 multiplicity, 44 multislice method – accuracy, 581 – defocus, 559 – deviation parameter, 581 – in k-space, 558 – microscope parameters, 559 – phase shifts in, 538 – projected potential, 558 – slice thickness, 556 nanocrystal – CeO2 and Pd, 562, 563 – Fe-Cu, 445 – KI, 563, 564 – Ni3 Fe, 444 nanodiffraction, 76 nanostructure, 562, 597 nanotube – single-wall carbon, 562, 564 nearest-neighbor shells, 490 Nelson–Riley lattice parameter determination, 51 neutron – chopper, 153 – coherent scattering length, 154 – magnetic scattering, 154 – mass, 742 – moderation, 153 – polarized, 154 – potential or resonance scattering, 154

754

Index

– reactor source, 153 – spallation source, 153 – time-of-flight monochromator, 153 – transmutation of samples, 154 – wavelength, 742 NIST SRM, 45 Nobel prizes, 2 nomenclature – EELS edges, 168 – electronic transitions, 168 – x-ray, 17, 19 non-dipole transitions, 191 normal stress, 428 normalization of vectors, 277 nuclear exciton, 158 null contrast condition, 359 objective aperture, 68 objective lens, 66 – construction, 88 – pole pieces, 88 optical fiber principle, 586 ordering, 486 orientation for diffraction, 38 orientation relationship – crystallographic, 290 – image and diffraction pattern, 89 orthogonality condition, 434 orthogonality relationships, 615 osmium staining, 73 pair distribution function, 496 – synchrotron source, 500 pair probability (conditional), 486 partial cross-section, 189 partial dislocation, 391, 725 – Frank, 391 – Shockley, 391 partial pair correlations, 500 Patterson function, 446, 457 – atomic displacement disorder, 470 – average crystal, 469 – chemical disorder, 483 – definition of, 459 – deviation crystal, 469 – example, 466 – graphical construction, 462 – homogeneous disorder, 469 – infinite δ series, 464 – perfect crystal, 463 – random displacements, 471 – SRO, 488 – thermal spread, 474

Pauli principle, 183 peak width vs. Δk method, 441 Pearson VII function, 53 Peltier cooler, 34 pendell¨ osung, 624 periodic boundary conditions, 561 perturbation theory, 590, 637 phase – and materials, 536 – of electron wavefront, 517 – velocity, 120 phase contrast, 62, 338 phase errors, 83 – constructive interference, 548 – lens accuracy, 95 phase fraction determination, 45 – integrated areas, 48 – internal standard method, 48 – retained austenite, 48 phase grating, 558, 621 – approximation, 675 phase problem, 462 – anomalous scattering, 462 phase relationships, 83, 119, 447 phase transfer function, 536 phase-amplitude diagram, 339, 345, 346, 673 – bend contour, 353 – dislocation, 361 – Fresnel zones, 526 – in dynamical theory, 623 – interfaces, 384 – moir´e fringes, 388 – of white noise, 437 – screw dislocation, 369, 373 – stacking fault, 395 – thickness fringes, 348 phase-space transform chopper, 272 Philips EM400T, 208 Philips EM430, 547 phonon, 157, 476, 591 – multiphonon scattering, 585, 592 – scattering, 123, 157 photoelectric scattering, 131 π boundary, 405 Planck’s constant, 10, 116, 742 plasmon, 167, 173 – data, table of, 177 – lifetime, 175 – mean free path, 175, 217 – specimen thickness, 175, 218 point resolution, 548 Poisson ratio, 429

Index polar net, 285 polarization correction, 43 polarized incident radiation, 47 pole-zero cancellation, 37 poly-DCH polymer, 78 polychromatic radiation, 10 polycrystalline Au, 732 polymer (liquid crystal), 563, 564 Porod law, 508, 515 Porod plot, 510 – fractal particles, 510 – surface area, 510 position-sensitive detector, 27 – area detector, 35 – charge-coupled-device, 35, 578 – delay line, 35 – imaging plates, 35, 578 – measured intensities, 45 – pixellated diodes, 35 – resistive wire, 34 powder average for x-ray diffractometry, 46 powder method, 12 precipitate – coherency, 408 – fringe contrast, 403 – image of coherent, 412 – incoherent, 413 – orientation relationship, 739 – semi-coherent, 413 – variants, 736 principal quantum number, 17 principal strains, 428 projected potential, 558 projector lens, 70, 88 – distortion, 276 propagator, 532, 558 pseudo-Voigt function, 53, 433 quadrupole lens, 106 quantum dot, 606 quantum efficiency, 30 quantum electrodynamics, 13 quantum mechanics, 10, 17 quantum numbers, 17 quasi-elastic, 429 radial distribution function, 172, 495, 511 – small-angle scattering, 512 radio analogy for HRTEM, 83 radius of gyration, 506 ray diagram, 66

755

– for TEM, 114 ray tracing, 71, 93 real image, 66 receiving slit, 26 reciprocal lattice, 231 – dimensionality, 273 – primitive translation vectors, 232 reciprocal lattice vectors – fcc, bcc, sc, 234 – uniqueness, 232 reciprocity – in optics, 595 reduced diffraction intensity, 498 reduced x-ray interference function, 501 refinement methods, 52 – constraints, 54 – parameters, 52 – peak shape, 53 reflected waves, 520 refractive index, 91 rel-disk, 265 rel-rods, 252 relativistic correction, 116, 743 relaxation energy, 477 representations in quantum mechanics, 612, 633 residual contrast, 364, 369 resolution, 110 – energy, 166 – limit in HRTEM, 112 – limit in STEM, 584 – optimal, 548 – point, 548 – point-to-point, 550 – state-of-the-art in 2007, 84 – vertical, 605 resonance scattering, 154 Richardson’s constant, 113 Rietveld refinement, 52 right-hand rule, 275 – zone-axis convention, 279 roentgen, 742 Ronchigram, 602 rotating anode source, 24 Rutherford cross-section, 147 Rutherford scattering, 200 – in HAADF imaging, 586, 592 Rydberg, 17, 742 sample shape for x-ray diffractometry, 46 Sb in Si, 597 scanning electron microscopy (SEM), 200, 202, 205

756

Index

scanning transmission electron microscopy (STEM), 62, 583, 584 scattered wave, 519 scattering – complementarity of different methods, 153 – differential cross-section, 126 – phase lag, 522 – total cross-section, 127 scattering factor – electron, 557 scattering potential, 224 – time-varying, 155 Scherrer equation, 426 Scherzer defocus, 550, 570, 582 Scherzer resolution, 548, 550 – in HAADF imaging, 584 Schr¨ odinger equation, 16, 518, 614 – Green’s function, 138, 519 secondary electron imaging (SEI), 202 secondary electrons, 202 Seemann–Bohlin diffractometer, 27 selected-area diffraction (SAD), 76 – spherical aberration, 117 selection rule, 59 semiconductor device, 605 shape factor, 339, 446, 466, 502 – and s, 257 – column of atoms, 446 – definition, 236 – envelope function, 250 – intensity, 341, 466 – rectangular prism, 247 – rel-rods, 252 – sphere, disk, rod, 253 shear strain, 428 shielding by core electrons, 19 Shockley partial dislocation, 725 short-range order (SRO), 481, 485 – single crystal, 490 – Warren–Cowley parameters, 486 Si, 4, 83, 597 Si dumbbells, 596 Si-Ge superlattice, 597 side-centered orthorhombic lattice, 270 side-entry stage, 101 sideband, 83 Sievert, 742 SIGMAK, SIGMAL, 190 sign of s, 299 signal-to-noise ratio, 30 simultaneous strain and size broadening, 440

single channel analyzer, 37 single crystal methods, 10 single-wall carbon nanotube, 562 SiO2 , 605 size broadening, 424, 446, 456, 467 skilled microscopist, 62, 106, 215, 539, 572, 586 slit width, 39, 430 small-angle scattering, 502 – concept, 502 – from continuum, 503 – Guinier radius, 506 – neutron (SANS), 512 – Porod plot, 510 – x-ray (SAXS), 512 solid mechanics, 430 solid-solid interfaces by HRTEM, 565 Soller slits, 26, 430 space group (CBED), 327 spectral brilliance, 22 spectrum image, 193 spherical aberration, 102, 598, 599, 602 – and defocus in HRTEM, 102, 539 – and underfocus for SAD, 76 – correction, 578, 598 – effect on SAD, 117 – negative, 604 – phase distortion, 84 spin, 17 spin wave scattering, 123 spin-orbit splitting, 19 spot size control (C1), 207 stacking fault, 391, 444 – analysis example, 400 – asymmetry of images, 666 – bounding partials, 397, 399 – diffraction peak broadening, 444 – diffraction peak shifts, 446 – dynamical theory, 660, 663 – dynamical treatment, 397 – energy, 726 – extrinsic/intrinsic rule, 399 – graphite, 414 – HRTEM image of, 84 – kinematical treatment, 391, 395 – tetrahedra, 407 – top of specimen, 399, 400 – visibility, 397 – widths in images, 402 staining, 73 star aberration, 603 statistical scatter, 30, 52, 58, 436 stereographic projection, 713–716

Index – construction, 282 – electron diffraction patterns, 284 – examples, 286 – Kurdjumov-Sachs relationship, 290 – polar net, 285 – poles, 282 – rules for manipulation, 284, 285 – twinning, 288 – Wulff net, 285, 713, 716 stigmation, 105 – procedure, 531, 547, 733 – stigmator, 106 Stokes correction, 433 storage ring, 20 strain broadening – distribution of strains, 427, 441, 477 – heterogeneity of strains, 477 – origin, 426 strain fields, 340, 358 stray fields, 604 strip chart recorder, 455 structural image, 544 structure factor, 339 – and s, 257 – and extinction distance, 620 – bcc, 240 – dc, 4, 240 – definition, 236 – fcc, 240 – hcp, 55 – lattice, 241 – phase factor, 230 – sc, 237 – simple lattice, 230 sum peak, 37 supercell, 556 superlattice diffractions, 243, 246 – B1 structure, 244 – B2 table of, 245 – L10 -ordered structure, 246 – L12 -ordered structure, 247 symmetry elements and diffraction groups, 318 synchrotron radiation, 20, 191 – beamlines, 22 – pair distribution function, 500 – user and safety programs, 23 systematic absences – glide planes, 243 – screw axes, 243 take-off angle, 25 TEM laboratory practice

757

– alignments, 729 – apertures, 731, 738 – eucentric height, 729 – film plates and vacuum, 731 – JEOL 2000FX, 728 – laboratory exercises, 728 – preparation, 738 – procedures, 732 – shutdown, 731 – startup, 728 – stigmation correction, 730 – voltage center, 730 – wobbler, 730 thermal diffuse scattering, 469, 472 thermal field emission gun, 86 thermal vibrations, 513 thermionic electron gun, 85 θ′ precipitate, 735 thickness contours, 349 – effect of absorption, 352 – wedge-shaped specimen, 351 thin-film approximation, 208 Thomas Gainsborough, 135 Thompson scattering, 129 three dimensional imaging, 605 three-window image, 196 through-focus series, 562, 570, 573 Ti-Al, 366, 565 Ti-Al-Mo alloy, 567 tilt of beam or crystal, 559 torr, 743 total internal reflection, 587 total scattering cross-section, 127 transfer lens, 600 transparency broadening, 430 tungsten filament, 85 tunneling, 589 turbulence of air, 604 twin, 416 – boundary, 407 two-beam BF images, 342 – antiphase boundary, 406 – contrast of dislocation, 376 – dislocation, 363, 370, 371 – moir´e fringes, 391 – stacking fault, 397, 402 – twin boundary, 407 two-beam dynamical theory, 625, 630, 640 two-lens system, 70 undulator, 21 uniform strain, 477 unmixing, 486

758

Index

vacancy, 407 – loop, 407 valence electrons, 151, 170 vector ψ or φ, 632 Vegard’s law, 478 vertical resolution, 605 vibrations, 604 videorecording for kinetics, 65 void, 408 – Fresnel effect, 408 Voigt function, 433 – second moment divergence, 456 voltage center alignment, 571 Warren–Cowley SRO parameters, 486 wave amplitudes, 122 wave crests, 120 – match at interface, 92 wave equation – Green’s function, 532 wavefront modulations, 621 wavelengths – electron, table of, 743 – x-ray, table of, 743 wavelet (defined), 119, 223 wavevector of electron in solid, 614 weak phase object, 545 weak-beam dark-field method, 378 – analysis of, 380 – deviation parameter, s, 380 – dislocations in Si, 384 – g-3g, 378 – Kikuchi lines, 379 – stationary phase, 381 Wehnelt electrode, 85 white lines, 167, 171 white noise, 437 Wien filter, 166 wiggler, 21 window discriminator, 37 wobbling, 572 Wulff net, 285, 713, 716 x-ray – absorption, 43 – absorption coefficients, table of, 692

– anomalous scattering, 132, 161 – – chart, 697 – bremsstrahlung, 14 – characteristic, 13 – characteristic depth, 134 – classical electrodynamics of scattering, 128 – coherent bremsstrahlung, 59 – Compton scattering, 132 – detector, 30 – dispersion corrections, 131 – electric dipole radiation, 128 – energy spectrum, 38 – energy-wavelength relation, 15 – form factors, table of, 693 – generation, 13 – line broadening, 423 – mapping, 38 – mass attenuation (absorption), 134 – mirror, 29 – near-resonance scattering, 130 – notation, 19, 20 – photoelectric scattering, 131 – scattering, 128 – scattering dependence on atomic number, 131 – spectrometer, 34 – spectroscopy system, 37 – spurious, 213 – synchrotron radiation, 20 – tube, 23 – wavelength distribution, 15 – wavelengths, table of, 743 Young’s modulus, 429, 444 Z-contrast imaging, see HAADF imaging, 338, 583 ZAF correction, 211 Zemlin tableau, 601 zero-loss peak, 167, 607 zero-order Laue zone, ZOLZ, 262 zero-point vibrations – diffuse scattering from, 476 zone axis, 275