Prof dr brent fultz, prof dr james m howe (auth ) transmission electron microscopy and diffractometry of materials 2008

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

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

/>
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
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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


VI

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

VII

(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,


VIII

Preface to the First Edition

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


Preface to the First Edition

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

XIII

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

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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175
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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
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224
226
230
230


XIV

6.

Contents

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 . . . . . . . . . . . . . . . . .

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274
276
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282
284
284
290
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Contents

7.

XV

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
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369
378
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384
384
387
391
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XVI Contents

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
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459
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XVII

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

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Contents

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 Schră
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


Contents

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 . . . . . . . . . . . . . . . . . . . . .

XIX

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

e
D

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 )
.
(1.4)
2θhkl = 2 arcsin
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
=0,
2 sinθB + 2d cosθB
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 .