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

With a discussion of the effective extinction length and the effective tion parameter from dynamical diffraction, we extend the kinematical theory devia-as far devia-as it can go for electr

Transmission Electron Microscopy and Diffractometry of Materials

High resolution transmission electron microscope (HRTEM) image of a lead crystalbetween two crystals of aluminum (i.e., a Pb precipitate at a grain boundary inAl) The two crystals of Al have different orientations, evident from their differentpatterns of atom columns Note the commensurate atom matching of the Pb crystalwith 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

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

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Experimental methods for diffraction and microscopy are pushing the frontedge of nanoscience and materials science, and important new developmentsare covered in this third edition For transmission electron microscopy, a re-markable recent development has been a practical corrector for the sphericalaberration of the objective lens Image resolution below 1 ˚A can be achievedregularly now, and the energy resolution of electron spectrometry has alsoimproved dramatically Locating and identifying individual atoms inside ma-terials has been transformed from a dream of fifty years into experimentalmethods of today.

The entire field of x-ray spectrometry and diffractometry has benefitedfrom advances in semiconductor detector technology, and a large commu-nity of scientists are now regular users of synchrotron x-ray facilities Thedevelopment of powerful new sources of neutrons is elevating the field of neu-tron scattering research Increasingly, the most modern instrumentation formaterials research with beams of x-rays, neutrons, and electrons is becomingavailable through an international science infrastructure of user facilities thatgrant access on the basis of scientific merit

The fundamentals of scattering, diffractometry and microscopy remain asdurable as ever This third edition continues to emphasize the common theme

of how waves and wavefunctions interact with matter, while highlighting thespecial features of x-rays, electrons, and neutrons The third edition is notsubstantially longer than the second, but all chapters were updated and re-vised The text was edited throughout for clarity, often minimizing sources

of confusion that were found by classroom teaching There are significantchanges 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 tominimize 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 NationalScience 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 com-plete, but its favorable acceptance and quick sales prompted us to prepare thesecond 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 rected over the course of several critical re-readings The on-line solutionsmanual has been updated too

cor-The first edition arrived at a time of great international excitement innanostructured materials and devices, and this excitement continues to grow.The second edition shows better how nanostructures offer new opportunitiesfor transmission electron microscopy and diffractometry of materials Never-theless, the topics and structure of the first edition remain intact The aimsand 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 helpwith both editions Finally, we thank the National Science Foundation forsupport 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 cause of their properties such as electrical conductivity, strength, magnetiza-tion, toughness, chemical reactivity, and numerous others All these proper-ties originate with the internal structures of materials Structural features ofmaterials include their types of atoms, the local configurations of the atoms,and arrangements of these configurations into microstructures The charac-terization of structures on all these spatial scales is often best performed bytransmission electron microscopy and diffractometry, which are growing inimportance to materials engineering and technology Likewise, the internalstructures of materials are the foundation for the science of materials Much

be-of materials science has been built on results from transmission electron croscopy and diffractometry of materials

mi-This textbook was written for advanced undergraduate students and ginning graduate students with backgrounds in physical science Its goal is

be-to acquaint them, as quickly as possible, with the central concepts and somedetails of transmission electron microscopy (TEM) and x-ray diffractometry

(XRD) that are important for the characterization of materials The topics

in this book are developed to a level appropriate for most modern materialscharacterization research using TEM and XRD The content of this book hasalso been chosen to provide a fundamental background for transitions to morespecialized techniques of research, or to related techniques such as neutrondiffractometry The book includes many practical details and examples, but

it does not cover some topics important for laboratory work such as specimenpreparation methods for TEM

Beneath the details of principle and practice lies a larger goal of unifyingthe concepts common to both TEM and XRD Coherence and wave interfer-ence are conceptually similar for both x-ray waves and electron wavefunctions

In probing the structure of materials, periodic waves and wavefunctions shareconcepts of the reciprocal lattice, crystallography, and effects of disorder X-ray generation by inelastic electron scattering is another theme common toboth TEM and XRD Besides efficiency in teaching, a further benefit of an in-tegrated treatment is breadth – it builds strength to apply Fourier transformsand convolutions to examples from both TEM and XRD The book follows atrend at research universities away from courses focused on one experimentaltechnique, towards more general courses on materials characterization.The methods of TEM and XRD are based on how wave radiations interactwith individual atoms and with groups of atoms A textbook must elucidatethese 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 handyreference today On the other hand, high-resolution TEM (HRTEM), modernsynchrotron sources, and spallation neutron sources offer new ways for wave-matter interactions to probe the structures of materials A textbook mustintegrate 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 Chap-ters 3 and 4 by electron and x-ray interactions with atoms The atomic formfactor for elastic scattering, and especially the cross sections for inelasticelectron scattering, are covered with more depth than needed to understandChapters 5–7, which emphasize diffraction, crystallography, and diffractioncontrast In a course oriented towards diffraction and microscopy, it is pos-sible to take an easier path through only Sects 3.1, 3.2.1, 3.2.3, 3.3.2, andthe 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 Laueformulation to treat diffraction phenomena from crystalline materials withincreasing amounts of disorder The phase-amplitude diagram is used heav-ily in Chapter 7 for the analysis of diffraction contrast in TEM images ofdefects After a treatment of diffraction lineshapes in Chapter 8, the Pat-terson function is used in Chapter 9 to treat short-range order phenomena,

thermal diffuse scattering, and amorphous materials High-resolution TEMimaging and image simulation follow in Chapter 10, and the essentials of thedynamical 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 tion parameter from dynamical diffraction, we extend the kinematical theory

devia-as far devia-as it can go for electron diffraction We believe this approach is the rightone for a textbook because kinematical theory provides a clean consistencybetween diffraction and the structure of materials The phase-amplitude dia-gram, 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 valuableoutside the fields of diffraction and microscopy

Although Fourier transforms are mentioned in Chapter 2 and used inChapter 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 Fouriertransforms at this level before reading Chapters 10 and 11 [now 10-12 in thethird edition] on HRTEM and dynamical theory The mathematical level isnecessarily higher for HRTEM and dynamical theory, which are grounded inthe 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

grad-uate students and advanced undergradgrad-uates at the California Institute of

Technology, and notes for the one-semester graduate courses MSE 703

Trans-mission Electron Microscopy and MSE 706 Advanced TEM, at the University

of Virginia Most of the students in these courses were specializing in terials science or applied physics, and had some background in elementarycrystallography and wave mechanics For a one-semester course (14 weeks)

ma-on introductory TEM, ma-one of the authors covers the sectima-ons: 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 stu-dents with a strong physics background, the other author has covered thefull book in 10 weeks by deleting about half of the “specialized” topics [Hehas never repeated this achievement, however, and typically manages to justtouch section 10.3.]

The choice of topics, depth, and speed of coverage are matters for the tasteand discretion of the instructor, of course To help with the selection of coursecontent, the authors have indicated with an asterisk, “*,” those sections of amore specialized nature The double dagger, “‡,” warns of sections contain-

ing a higher level of mathematics, physics, or crystallography Each chapterincludes several, sometimes many, problems to illustrate principles The textfor some of these problems includes explanations of phenomena that seemedtoo specialized for inclusion in the text itself Hints are given for some of the

VIII Preface to the First Edition

problems, and worked solutions are available to course instructors Exercisesfor an introductory laboratory course are presented in an Appendix.

When choosing the level of presentation for a concept, the authors facedthe conflict of balancing rigor and thoroughness against clarity and concise-ness Our general guideline was to avoid direct citations of rules, but instead

to provide explanations of the underlying physical concepts The ical derivations are usually presented in steps of equal height, and we try tohighlight the central tricks even if this means reviewing elementary concepts.The authors are indebted to our former students for identifying explanationsand calculations that needed clarification or correction

mathemat-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 Albertsonwith manuscript preparation N R Good and J Graetz performed much ofthe mathematical typesetting, and we are indebted to them for their carefulwork Prof P Rez suggested an approach to treat dynamical diffraction in aunified manner Both authors acknowledge the National Science Foundationfor financial support over the years

Brent Fultz and James Howe

Pasadena and Charlottesville

October, 2000

1 Diffraction and the X-Ray Powder Diffractometer . 1

1.1 Diffraction 1

1.1.1 Introduction to Diffraction 1

1.1.2 Bragg’s Law 3

1.1.3 Strain Effects 6

1.1.4 Size Effects 7

1.1.5 A Symmetry Consideration 9

1.1.6 Momentum and Energy 10

1.1.7 Experimental Methods 10

1.2 The Creation of X-Rays 13

1.2.1 Bremsstrahlung 14

1.2.2 Characteristic Radiation 16

1.2.3 Synchrotron Radiation 20

1.3 The X-Ray Powder Diffractometer 23

1.3.1 Practice of X-Ray Generation 23

1.3.2 Goniometer for Powder Diffraction 25

1.3.3 Monochromators, Filters, Mirrors 28

1.4 X-Ray Detectors for XRD and TEM 30

1.4.1 Detector Principles 30

1.4.2 Position-Sensitive Detectors 34

1.4.3 Charge Sensitive Preamplifier 36

1.4.4 Other Electronics 37

1.5 Experimental X-Ray Powder Diffraction Data 38

1.5.1 * Intensities of Powder Diffraction Peaks 38

1.5.2 Phase Fraction Measurement 45

1.5.3 Lattice Parameter Measurement 49

1.5.4 * Refinement Methods for Powder Diffraction Data 52

Further Reading 54

Problems 55

2 The TEM and its Optics 61

2.1 Introduction to the Transmission Electron Microscope 61

2.2 Working with Lenses and Ray Diagrams 66

2.2.1 Single Lenses 66

2.2.2 Multi-Lens Systems 69

2.3 Modes of Operation of a TEM 71

2.3.1 Dark-Field and Bright-Field Imaging 71

2.3.2 Selected Area Diffraction 76

2.3.3 Convergent-Beam Electron Diffraction 79

2.3.4 High-Resolution Imaging 81

2.4 Practical TEM Optics 85

2.4.1 Electron Guns 85

2.4.2 Illumination Lens Systems 87

2.4.3 Imaging Lens Systems 88

2.5 Glass Lenses 91

2.5.1 Interfaces 91

2.5.2 Lenses and Rays 92

2.5.3 Lenses and Phase Shifts 95

2.6 Magnetic Lenses 97

2.6.1 Focusing 97

2.6.2 Image Rotation 99

2.6.3 Pole Piece Gap 100

2.7 Lens Aberrations and Other Defects 102

2.7.1 Spherical Aberration 102

2.7.2 Chromatic Aberration 103

2.7.3 Diffraction 104

2.7.4 Astigmatism 104

2.7.5 Gun Brightness 108

2.8 Resolution 110

Further Reading 112

Problems 113

3 Scattering 119

3.1 Waves and Scattering 119

3.1.1 Wavefunctions 119

3.1.2 Coherent and Incoherent Scattering 122

3.1.3 Elastic and Inelastic Scattering 123

3.1.4 Wave Amplitudes and Cross-Sections 124

3.2 X-Ray Scattering 128

3.2.1 Electrodynamics of X-Ray Scattering 128

3.2.2 * Inelastic Compton Scattering 132

3.2.3 X-Ray Mass Attenuation Coefficients 134

3.3 Coherent Elastic Scattering 136

3.3.1 ‡ Born Approximation for Electrons 136

3.3.2 Atomic Form Factors – Physical Picture 141

3.3.3 ‡ Scattering of Electrons by Model Potentials 144

3.3.4 ‡ * Atomic Form Factors – General Formulation 148

3.4 * Nuclear Scattering 153

3.4.1 Properties of Neutrons 153

Contents XIII

3.4.2 Time-Varying Potentials and Inelastic Neutron

Scat-tering 155

3.4.3 * Coherent M¨ossbauer Scattering 158

Further Reading 160

Problems 160

4 Inelastic Electron Scattering and Spectroscopy 163

4.1 Inelastic Electron Scattering 163

4.2 Electron Energy-Loss Spectrometry (EELS) 165

4.2.1 Instrumentation 165

4.2.2 General Features of EELS Spectra 167

4.2.3 * Fine Structure 169

4.3 Plasmon Excitations 173

4.3.1 Plasmon Principles 173

4.3.2 * Plasmons and Specimen Thickness 175

4.4 Core Excitations 177

4.4.1 Scattering Angles and Energies – Qualitative 177

4.4.2 ‡ Inelastic Form Factor 180

4.4.3 ‡ * Double-Differential Cross-Section, d2σin/dφ dE 184

4.4.4 * Scattering Angles and Energies – Quantitative 186

4.4.5 ‡ * Differential Cross-Section, dσin/dE 187

4.4.6 ‡ Partial and Total Cross-Sections, σin 189

4.4.7 Quantification of EELS Core Edges 191

4.5 Energy-Filtered TEM Imaging (EFTEM) 193

4.5.1 Spectrum Imaging 193

4.5.2 Energy Filters 193

4.5.3 Chemical Mapping with Energy-Filtered Images 196

4.5.4 Chemical Analysis with High Spatial Resolution 197

4.6 Energy Dispersive X-Ray Spectrometry (EDS) 200

4.6.1 Electron Trajectories Through Materials 200

4.6.2 Fluorescence Yield 203

4.6.3 EDS Instrumentation Considerations 205

4.6.4 Thin-Film Approximation 208

4.6.5 * ZAF Correction 211

4.6.6 Artifacts in EDS Measurements 213

4.6.7 Limits of Microanalysis 215

Further Reading 217

Problems 217

5 Diffraction from Crystals 223

5.1 Sums of Wavelets from Atoms 223

5.1.1 Electron Diffraction from a Material 224

5.1.2 Wave Diffraction from a Material 226

5.2 The Reciprocal Lattice and the Laue Condition 230

5.2.1 Diffraction from a Simple Lattice 230

5.2.2 Reciprocal Lattice 231

5.2.3 Laue Condition 233

5.2.4 Equivalence of the Laue Condition and Bragg’s Law 233

5.2.5 Reciprocal Lattices of Cubic Crystals 234

5.3 Diffraction from a Lattice with a Basis 235

5.3.1 Structure Factor and Shape Factor 235

5.3.2 Structure Factor Rules 237

5.3.3 Symmetry Operations and Forbidden Diffractions 242

5.3.4 Superlattice Diffractions 243

5.4 Crystal Shape Factor 247

5.4.1 Shape Factor of Rectangular Prism 247

5.4.2 Other Shape Factors 252

5.4.3 Small Particles in a Large Matrix 252

5.5 Deviation Vector (Deviation Parameter) 256

5.6 Ewald Sphere 257

5.6.1 Ewald Sphere Construction 257

5.6.2 Ewald Sphere and Bragg’s Law 259

5.6.3 Tilting Specimens and Tilting Electron Beams 259

5.7 Laue Zones 262

5.8 * Effects of Curvature of the Ewald Sphere 262

Further Reading 266

Problems 266

6 Electron Diffraction and Crystallography 273

6.1 Indexing Diffraction Patterns 273

6.1.1 Issues in Indexing 274

6.1.2 Method 1 – Start with Zone Axis 276

6.1.3 Method 2 – Start with Diffraction Spots 279

6.2 Stereographic Projections and Their Manipulation 282

6.2.1 Construction of a Stereographic Projection 282

6.2.2 Relationship Between Stereographic Projections and Electron Diffraction Patterns 284

6.2.3 Manipulations of Stereographic Projections 284

6.3 Kikuchi Lines and Specimen Orientation 290

6.3.1 Origin of Kikuchi Lines 290

6.3.2 Indexing Kikuchi Lines 294

6.3.3 Specimen Orientation and Deviation Parameter 296

6.3.4 The Sign of s 299

6.3.5 Kikuchi Maps 299

6.4 Double Diffraction 302

6.4.1 Occurrence of Forbidden Diffractions 302

6.4.2 Interactions Between Crystallites 303

6.5 * Convergent-Beam Electron Diffraction 304

6.5.1 Convergence Angle of Incident Electron Beam 306

6.5.2 Determination of Sample Thickness 307

Contents XV

6.5.3 Measurements of Unit Cell Parameters 309

6.5.4 ‡ Determination of Point Groups 314

6.5.5 ‡ Determination of Space Groups 325

6.6 Further Reading 330

Problems 330

7 Diffraction Contrast in TEM Images 337

7.1 Contrast in TEM Images 337

7.2 Diffraction from Crystals with Defects 339

7.2.1 Review of the Deviation Parameter, s 339

7.2.2 Atom Displacements, δr 340

7.2.3 Shape Factor and t 341

7.2.4 Diffraction Contrast and{s, δr, t} 342

7.3 Extinction Distance 342

7.4 The Phase-Amplitude Diagram 345

7.5 Fringes from Sample Thickness Variations 347

7.5.1 Thickness and Phase-Amplitude Diagrams 347

7.5.2 Thickness Fringes in TEM Images 348

7.6 Bend Contours in TEM Images 353

7.7 Diffraction Contrast from Strain Fields 357

7.8 Dislocations and Burgers Vector Determination 359

7.8.1 Diffraction Contrast from Dislocation Strain Fields 359

7.8.2 The g ·b Rule for Null Contrast 362

7.8.3 Image Position and Dislocation Pairs or Loops 368

7.9 Semi-Quantitative Diffraction Contrast from Dislocations 369

7.10 Weak-Beam Dark-Field (WBDF) Imaging of Dislocations 378

7.10.1 Procedure to Make a WBDF Image 378

7.10.2 Diffraction Condition for a WBDF Image 379

7.10.3 Analysis of WBDF Images 380

7.11 Fringes at Interfaces 384

7.11.1 Phase Shifts of Electron Wavelets Across Interfaces 384

7.11.2 Moir´e Fringes 387

7.12 Diffraction Contrast from Stacking Faults 391

7.12.1 Kinematical Treatment 391

7.12.2 Results from Dynamical Theory 397

7.12.3 Determination of the Intrinsic or Extrinsic Nature of Stacking Faults 399

7.12.4 Partial Dislocations Bounding the Fault 399

7.12.5 An Example of a Stacking Fault Analysis 400

7.12.6 Sets of Stacking Faults in TEM Images 402

7.12.7 Related Fringe Contrast 403

7.13 Antiphase (π) Boundaries and δ Boundaries 404

7.13.1 Antiphase Boundaries 404

7.13.2 δ Boundaries 405

7.14 Contrast from Precipitates and Other Defects 407

7.14.1 Vacancies 407

7.14.2 Coherent Precipitates 408

7.14.3 Semicoherent and Incoherent Particles 413

Further Reading 413

Problems 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 457

9.1 The Patterson Function 457

9.1.1 Overview 457

9.1.2 Atom Centers at Points in Space 458

9.1.3 Definition of the Patterson Function 459

9.1.4 Properties of Patterson Functions 461

9.1.5 ‡ Perfect Crystals 463

9.1.6 Deviations from Periodicity and Diffuse Scattering 467

9.2 Diffuse Scattering from Atomic Displacements 469

9.2.1 Uncorrelated Displacements – Homogeneous Disorder 469 9.2.2 ‡ Temperature 472

9.2.3 * Correlated Displacements – Atomic Size Effects 477

9.3 Diffuse Scattering from Chemical Disorder 481

9.3.1 Uncorrelated Chemical Disorder – Random Alloys 481

9.3.2 ‡ * SRO Parameters 485

9.3.3 ‡ * Patterson Function for Chemical SRO 487

9.3.4 Short-Range Order Diffuse Intensity 488

9.3.5 ‡ * Isotropic Materials 488

9.3.6 * Polycrystalline Average and Single Crystal SRO 490

9.4 * Amorphous Materials 491

9.4.1 ‡ One-Dimensional Model 491

9.4.2 ‡ Radial Distribution Function 495

Contents XVII

9.4.3 ‡ Partial Pair Correlation Functions 500

9.5 Small Angle Scattering 502

9.5.1 Concept of Small Angle Scattering 502

9.5.2 * Guinier Approximation (small Δk) 504

9.5.3 * Porod Law (large Δk) 508

9.5.4 ‡ * Density-Density Correlations (all Δk) 510

Further Reading 512

Problems 513

10 High-Resolution TEM Imaging 517

10.1 Huygens Principle 518

10.1.1 Wavelets from Points in a Continuum 518

10.1.2 Huygens Principle for a Spherical Wavefront – Fresnel Zones 523

10.1.3 ‡ Fresnel Diffraction Near an Edge 527

10.2 Physical Optics of High-Resolution Imaging 532

10.2.1 ‡ Wavefronts and Fresnel Propagator 532

10.2.2 ‡ Lenses 534

10.2.3 ‡ Materials 536

10.3 Experimental High-Resolution Imaging 538

10.3.1 Defocus and Spherical Aberration 538

10.3.2 ‡ Lenses and Specimens 543

10.3.3 Lens Characteristics 546

10.4 * Simulations of High-Resolution TEM Images 555

10.4.1 Principles of Simulations 555

10.4.2 Practice of Simulations 561

10.5 Issues and Examples in High-Resolution TEM Imaging 562

10.5.1 Images of Nanostructures 562

10.5.2 Examples of Interfaces 565

10.5.3 * Specimen and Microscope Parameters 568

10.5.4 * Some Practical Issues for HRTEM 576

Further Reading 580

Problems 581

11 High-Resolution STEM Imaging 583

11.1 Characteristics of High-Angle Annular Dark-Field Imaging 583

11.2 Electron Channeling Along Atomic Columns 586

11.2.1 Optical Fiber Analogy 586

11.2.2 ‡ Critical Angle 588

11.2.3 * Tunneling Between Columns 589

11.3 Scattering of Channeled Electrons 591

11.3.1 Elastic Scattering of Channeled Electrons 591

11.3.2 * Inelastic Scattering of Channeled Electrons 593

11.4 * Comparison of HAADF and HRTEM Imaging 594

11.5 HAADF Imaging with Atomic Resolution 595

11.5.1 * Effect of Defocus 595

11.5.2 Experimental Examples 597

11.6 * Lens Aberrations and Their Corrections 599

11.6.1 CsCorrection with Magnetic Hexapoles 599

11.6.2 ‡ Higher-Order Aberrations and Instabilities 602

11.7 Examples of Cs-Corrected Images 604

11.7.1 Three-Dimensional Imaging 605

11.7.2 High Resolution EELS 606

Further Reading 607

Problems 608

12 Dynamical Theory 611

12.1 Chapter Overview 611

12.2 ‡ * Mathematical Features of High-Energy Electrons in a Periodic Potential 613

12.2.1 ‡ * The Schr¨odinger Equation 613

12.2.2 ‡ Kinematical and Dynamical Theory 619

12.2.3 * The Crystal as a Phase Grating 621

12.3 First Approach to Dynamical Theory – Beam Propagation 623

12.4 ‡ Second Approach to Dynamical Theory – Bloch Waves and Dispersion Surfaces 627

12.4.1 Diffracted Beams,{Φg}, are Beats of Bloch Waves,{Ψ(j) } 627

12.4.2 Crystal Periodicity and Dispersion Surfaces 633

12.4.3 Energies of Bloch Waves in a Periodic Potential 637

12.4.4 General Two-Beam Dynamical Theory 640

12.5 Essential Difference Between Kinematical and Dynamical Theories 646

12.6 ‡ Diffraction Error, sg, in Two-Beam Dynamical Theory 651

12.6.1 Bloch Wave Amplitudes and Diffraction Error 651

12.6.2 Dispersion Surface Construction 653

12.7 Dynamical Diffraction Contrast from Crystal Defects 655

12.7.1 Dynamical Diffraction Contrast Without Absorption 655

12.7.2 ‡ * Two-Beam Dynamical Theory of Stacking Fault Contrast 660

12.7.3 Dynamical Diffraction Contrast with Absorption 664

12.8 ‡ * Multi-Beam Dynamical Theories of Electron Diffraction 669

Further Reading 672

Problems 672

Bibliography 677

Further Reading 677

References and Figures 682

Contents X X

A Appendix 691

A.1 Indexed Powder Diffraction Patterns 691

A.2 Mass Attenuation Coefficients for Characteristic Kα X-Rays 692 A.3 Atomic Form Factors for X-Rays 693

A.4 X-Ray Dispersion Corrections for Anomalous Scattering 697

A.5 Atomic Form Factors for 200 keV Electrons and Procedure for Conversion to Other Voltages 698

A.6 Indexed Single Crystal Diffraction Patterns: fcc, bcc, dc, hcp 703 A.7 Stereographic Projections 713

A.8 Examples of Fourier Transforms 717

A.9 Debye–Waller Factor from Wave Amplitude 720

A.10 Review of Dislocations 721

A.11 TEM Laboratory Exercises 728

A.11.1 Preliminary – JEOL 2000FX Daily Operation 728

A.11.2 Laboratory 1 – Microscope Procedures and Calibration with Au and MoO3 732

A.11.3 Laboratory 2 – Diffraction Analysis of θ  Precipitates 735 A.11.4 Laboratory 3 – Chemical Analysis of θ  Precipitates 739

A.11.5 Laboratory 4 – Contrast Analysis of Defects 740

A.12 Fundamental and Derived Constants 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

crys-tallography

I

of atoms in the range from 10−8 to 10−4cm, bridging from the unit cell of

the crystal to the microstructure of the material There are many differentmethods for for measuring structure across this wide range of distances, butthe more powerful experimental techniques involve diffraction To date, most

of our knowledge about the spatial arrangements of atoms in materials hasbeen gained from diffraction experiments In a diffraction experiment, anincident wave is directed into a material and a detector is typically movedabout 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 structive or destructive interference then occurs along different directions asscattered waves are emitted by atoms of different types and positions There

Con-is a profound geometrical relationship between the directions of waves thatinterfere constructively, which comprise the “diffraction pattern,” and thecrystal structure of the material The diffraction pattern is a spectrum of realspace periodicities in a material.1 Atomic periodicities with long repeat dis-tances cause diffraction at small angles, while short repeat distances (as fromsmall interplanar spacings) cause diffraction at high angles It is not hard toappreciate that diffraction experiments are useful for determining the crystalstructures of materials Much more information about a material is contained

in its diffraction pattern, however Crystals with precise periodicities overlong distances have sharp and clear diffraction peaks Crystals with defects(such as impurities, dislocations, planar faults, internal strains, or small pre-cipitates) are less precisely periodic in their atomic arrangements, but theystill have distinct diffraction peaks Their diffraction peaks are broadened,distorted, and weakened, however, and “diffraction lineshape analysis” is animportant method for studying crystal defects Diffraction experiments arealso used to study the structure of amorphous materials, even though theirdiffraction patterns lack sharp diffraction peaks

In a diffraction experiment, the incident waves must have wavelengthscomparable to the spacings between atoms Three types of waves have proveduseful for these experiments X-ray diffraction (XRD), conceived by von Laueand the Braggs, was the first The oscillating electric field of an incident x-raymoves the atomic electrons and their accelerations generate an outgoing wave

In electron diffraction, originating with Davisson and Germer, the charge ofthe incident electron interacts with the positively-charged core of the atom,generating an outgoing electron wavefunction In neutron diffraction, pio-neered by Shull, the incident neutron wavefunction interacts with nuclei orunpaired electron spins These three diffraction processes involve very differ-ent physical mechanisms, so they often provide complementary informationabout atomic arrangements in materials Nobel prizes in physics (1914, 1915,

1937, 1994) attest to their importance As much as possible, we will size the similarities of these three diffraction methods, with the first similaritybeing Bragg’s law

empha-1 Precisely and concisely, the diffraction pattern measures the Fourier transform

of an autocorrelation function of the scattering factor distribution The previoussentence is explained with care in Chap 9 More qualitatively, the crystal can

be likened to music, and the diffraction pattern to its frequency spectrum Thisanalogy illustrates another point Given only the amplitudes of the differentmusical frequencies, it is impossible to reconstruct the music because the timing

or “phase” information is lost Likewise, the diffraction pattern alone may beinsufficient to reconstruct all details of atom arrangements in a material

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.)

ee

e ee

AB

CD

d sine d sined

fronts of equalphase

q

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:

The right hand side is sometimes multiplied by an integer, n, since this

con-dition 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

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, d hkl, of planes

labeled by Miller indices (hkl) are:

4 1 Diffraction and the X-Ray Powder Diffractometer

d hkl= √ a0

(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 2θ hkl:

of an indexed diffraction pattern is shown in Fig 1.2 Notice that the tensities of the different diffraction peaks vary widely, and are zero for some

in-combinations of h, k, and l For this example of polycrystalline silicon, tice the absence of all combinations of h, k, and l that are mixtures of even

no-and odd integers, no-and the absence of all even integer combinations whosesum is not divisible by 4 This is the “diamond cubic structure factor rule,”discussed in Sect 5.3.2

Fig 1.2 Indexed

pow-der diffraction patternfrom polycrystalline sil-icon, obtained with Co

Kα radiation.

One important use of x-ray powder diffractometry is for identifying known crystals in a sample of material The idea is to match the positionsand the intensities of the peaks in the observed diffraction pattern to a knownpattern of peaks from a standard sample or from a calculation There should

un-be a one-to-one correspondence un-between the observed peaks and the indexedpeaks 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 toChap 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

iter-ative 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 unitcell, it becomes increasingly impractical to try to index the diffraction pat-tern by hand In practice, two approaches are used The oldest and mostreliable is a “fingerprinting” method The International Centre for Diffrac-tion Data (ICDD, formerly the Joint Committee on Powder Diffraction Stan-dards, JCPDS) maintains a database of diffraction patterns from more thanone hundred thousand inorganic and organic materials [1.1] For each ma-terial the data fields include the observed interplanar spacings for all ob-

served diffraction peaks, their relative intensities, and their hkl indexing.

Software packages are available to identify peaks in the experimental tion pattern and then search the ICDD database to find candidate materials.Computerized fingerprint searches are particularly valuable when the samplecontains a mixture of phases, and their chemical compositions are uncertain.When the chemical compositions of the crystallographic phases are knownwith some accuracy, however, the indexing of diffraction patterns is consider-ably easier Phase determination is facilitated by finding candidate phases inhandbooks of phase diagrams, and their diffraction patterns from the ICDDdatabase The problem is usually more difficult when multiple phases arepresent in the sample, but sometimes it is easy to distinguish individualdiffraction patterns The diffraction pattern in Fig 1.3 was measured to de-termine if the surface of a glass-forming alloy had crystallized The amor-

diffrac-phous phase has two very broad peaks centered at 2θ = 38 ◦and 74◦ Sharp

diffraction peaks from crystalline phases are distinguished easily from theamorphous peaks Although this crystalline diffraction pattern has not beenindexed, the measurement was useful for showing that the solidification con-ditions were inadequate for obtaining a fully amorphous solid

Beyond fingerprinting, another approach to structural determination bypowder diffractometry is to calculate diffraction patterns from candidate crys-tal 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 diffractionpatterns (e.g., Fig 1.2) can often be calculated readily, but structure factorsfor 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 powderdiffraction peaks Such software is straightforward to use In a more sophis-ticated 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-castZr-Cu-Ni-Al alloy Thesmooth intensity withbroad peaks around

2θ = 38 ◦ and 74◦, isthe contribution from theamorphous phase Thesharp peaks show somecrystallization at the sur-face of the sample thatwas in contact with thecopper mold

lattice parameters, are treated as adjustable parameters These parametersare adjusted or “refined” as the software seeks the best fit between a calcu-lated 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-raydiffraction peaks The simplest type of strain is a uniform dilatation If allparts 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):

crys-Bragg angles are broadened more This same argument applies when the teratomic separation depends on chemical composition – diffraction peaks arebroadened when the chemical composition of a material is inhomogeneous.

in-1.1.4 Size Effects

The width of a diffraction peak is affected by the number of crystallographicplanes 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:

If we had only two diffracting planes, as shown in Fig 1.1,

partially-constructive wave interference occurs even for large deviations of θ from the correct Bragg angle, θB In fact, for two scattered waves, errors in phasewithin 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:

6 4 2

0

Phase Angle (radians)

unshifted

sum

(bottom) of two waves

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 deviationfor which constructive interference occurs:

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 θ

8 1 Diffraction and the X-Ray Powder Diffractometer

number of diffracting planes, however Consider the situation with 4 ing planes as shown in Fig 1.5b The total distance between the top planeand bottom plane is now 3 times larger For the same path length error as inFig 1.5a, the error in diffraction angle must be about 3 times smaller For

diffract-N diffracting planes (separated by a distance d = a(diffract-N − 1)) we have instead

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 planesfor typical x-rays, so precise diffraction angles are possible for high-qualitycrystals

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 thevery 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:

Δθ

θB  0.9

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

number of diffracting planes is nearly equal to the ratio of the angle of thediffraction peak to the width of a diffraction peak The widths of x-ray diffrac-tion peaks are handy for determining crystallite sizes in the range of severalnanometers (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 thescattering plane Along a continuous plane, there is a continuous range ofseparations between O and P, so there is as much destructive interference asconstructive 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 generalprinciple 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 spacingsalong Δk.4

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

The crystal that does the diffraction must gain an equal but opposite mentum – momentum is always conserved This momentum is eventuallytransferred to Earth, which undergoes a negligible change in its orbit.Any transfer of energy to the crystal means that the scattered x-ray willhave somewhat less energy than the incident energy, which might impairdiffraction experiments Consider two types of energy transfers

mo-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 kineticenergy is transferred to all atoms in the crystal, or at least those atoms withinthe spatial range described in Sect 1.1.4

Second, energy may be transferred to a single atom, such as by movingthe nucleus (causing atom vibrations), or by causing an electron of the atom

to escape, ionizing the atom A feature of quantum mechanics is that theseevents happen to some x-rays, but not to others In general, the x-rays thatundergo these “inelastic scattering” processes6 are “tagged” by one atom,and cannot participate in diffraction from a full crystal

“Debye–Scherrer” method, uses monochromatic radiation, but uses a bution of crystallographic planes as provided by a polycrystalline sample.Another approach, the “Laue method,” uses the distribution of wavelengths

distri-in polychromatic or “white” radiation, and a sdistri-ingle crystal sample The bination of white radiation and polycrystalline samples produces too manydiffractions, so this is not a useful technique On the other hand, the study ofsingle crystals with monochromatic radiation is an important technique, espe-cially for determining the structures of minerals and large organic molecules

com-in crystallcom-ine form

5

This is consistent with a photon momentum of p = b k E/c = b k ω/c =  k, where

6

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

Table 1.1 Experimental methods for diffraction

Radiation

single crystal single crystal methods Laue

The “Laue Method” uses a broad range of x-ray wavelengths with mens that are single crystals It is commonly used for determining the orienta-tions of single crystals With the Laue method, the orientations and positions

speci-of both the crystal and the x-ray beam are stationary Some speci-of the incidentx-rays have the correct wavelengths to satisfy Bragg’s law for some crystalplanes In the Laue diffraction pattern of Fig 1.7, the different diffractionspots along a radial row originate from various combinations of x-ray wave-lengths and crystal planes having a projected normal component along therow It is not easy to evaluate these combinations (especially when there aremany orientations of crystallites in the sample), and the Laue method willnot be discussed further

Fig 1.7 Backscatter Laue

diffraction pattern from Si in[110] zone orientation Noticethe high symmetry of thediffraction pattern

The “Debye–Scherrer” method uses monochromatic x-rays, and

equip-ment 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 lites are irradiated by the beam, the crystallites diffract the x-rays into a set

crystal-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

sam-ple

Debye–Scherrer diffraction patterns are also obtained by diffraction ofmonochromatic electrons from polycrystalline specimens Two superimposedelectron diffraction patterns are presented in Fig 1.9 The sample was acrystalline 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 Thesecones were oriented to intersect a sheet of film in the transmission electronmicroscope, thus forming an image of “diffraction rings.” In addition to thediffraction 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

diffrac-tion patterns from polycrystalline Ni-Zrand single crystal NaCl

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

vide only one degree of freedom in changing the diffraction conditions,

corre-sponding 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 gle crystal diffraction experiments with monochromatic radiation Althoughdiffractions from single crystals are more intense, these added parametricdimensions require a considerable increase in data measurement time Suchmeasurements are possible with equipment in a small laboratory, but brightsynchrotron radiation sources have enabled many new types of single crystaldiffraction experiments

sin-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 transmissionelectron microscope Some relevant electron-atom interactions are summa-rized in Fig 1.10 Figure 1.10a shows the process of elastic scattering wherethe electron is deflected, but no energy loss occurs Elastic scattering is thebasis for electron diffraction Figure 1.10b is an inelastic scattering where thedeflection of the electron results in radiation The acceleration during the de-flection of a classical electron would always produce radiation, and hence noelastic scattering In quantum electrodynamics the radiation may or may notoccur (compare Figs 1.10a and 1.10b), but the average over many electronscatterings corresponds to the classical radiation field

Figure 1.10c illustrates two processes involving energy transfer betweenthe 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 theatom An outer electron of more positive energy falls into this core hole, butthere are two ways to dispose of its excess energy: 1) an x-ray can be emitteddirectly from the atom, or 2) this energy can be used to eject another outerelectron from the atom, called an “Auger electron.” The “characteristic x-ray” 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 kineticenergy of the emitted Auger electron is this energy difference minus its initialbinding energy After either decay process of Fig 1.10c, there remains anempty electron state in an outer shell of the atom, and the process repeatsitself 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:

14 1 Diffraction and the X-Ray Powder Diffractometer

c

elastic scattering

inelastic scattering EELS background

bremsstrahlung EDS background

characteristic x-ray

Auger electron

high-energy

secondary

decaychannels

Imaging

EDS,WDSAESEELS

Fig 1.10 a–c Some

processes of tion between a high-energy electron and an

interac-atom: (a) is useful for

diffraction, whereasthe ejection of a core

electron in (c) is the

basis for chemicalspectroscopies Twodecay channels for thecore hole in c are indi-cated 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 strongdeflection as depicted in Fig 1.10b, because the deflection causes an accel-eration 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:

transform of the time dependence of the electron acceleration, a(t) The

pas-sage of each electron through an atom provides a brief, pulse-type tion The average over many electron-atom interactions provides a broadbandx-ray energy spectrum Electrons that pass closer to the nucleus undergostronger accelerations, and hence radiate with a higher probability Theirspectrum, however, is the same as the spectrum from electrons that traverse

accelera-the outer part of an atom In a thin specimen where only one sharp eration of the electron can take place, the bremsstrahlung spectrum has anenergy distribution shown in Fig 1.11a; a flat distribution with a cutoff of

accel-40 keV for electrons of accel-40 keV

The general shape of the wavelength distribution can be understood asfollows The energy-wavelength relation for the x-ray is:

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) = ch I(E)

Figure 1.11b is the wavelength distribution (1.23) that corresponds to theenergy 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 verythin specimen The anode of an x-ray tube is rather thick, however Mostelectrons do not lose all their energy at once, and propagate further intothe anode When an electron has lost some of its initial energy, it can re-

radiate again, but with a smaller Emax (or larger λmin) Deeper within theanode, these multiply-scattered electrons emit more bremsstrahlung of longerwavelengths The spectrum of bremsstrahlung from a thick sample can be un-derstood by summing the individual spectra from electrons of various kineticenergies 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

accel-erations of the electrons Atoms of larger atomic number, Z, have stronger

16 1 Diffraction and the X-Ray Powder Diffractometer

E0

Fig 1.11 (a) Energy distribution for single bremsstrahlung process (b) wave-

length distribution for the energy

distri-bution of part a (c) coarse-grained sum

of wavelength distributions expectedfrom 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 V2Z2

1.2.2 Characteristic Radiation

In addition to the bremsstrahlung emitted when a material is bombardedwith high-energy electrons, x-rays are also emitted with discrete energiescharacteristic of the elements in the material, as depicted in Fig 1.10c (toppart) The energies of these “characteristic x-rays” are determined by thebinding energies of the electrons of the atom, or more specifically the differ-ences 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 thetime-independent Schr¨odinger equation for the electron wavefunction:

−2

2m ∇2ψ(r, θ, φ) − Ze2

r ψ(r, θ, φ) = E ψ(r, θ, φ) (1.24)

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 simpleform:

Since E is a constant, acceptable expressions for ψ(r) must provide an E that

is independent of r Two such solutions are:

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):

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 ψ1 , 2 for ψ2 , etc It is well-known that there are

other solutions for ψ that are not spherically-symmetric, for example, ψ2 ,

ψ3 , and ψ3 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 nottrue 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:

ΔE = E2− E1=−

1

(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

Charac-teristic x-ray energies

of the elements Thex-axis of plot was orig-inally the square root

Moseley’s laws Moseley’s laws are modifications of (1.30) For Kα and Lα

x-rays, they are:

E Kα = (Z − 1)2ER

1

Equations (1.31) and (1.32) are good to about 1 % accuracy for x-rays withenergies 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

elec-trons 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 tran-

sition 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]

Notice that Table 1.2 and Fig 1.13 do not include the transition 2s → 1s.

This transition is forbidden The two wavefunctions, ψ1 (r) and ψ2 (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 ψ2 (r) has zero net overlap with ψ1 (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 ing World War I The British response to this loss was to assign scientists tononcombatant duties during World War II

dur-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 Kα1x-ray is slightly more energetic than the Kα2x-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 Lα1and Lβ1x-rays are differentiated inthis 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.

ca-or international labca-oratca-ories.10 These facilities are centered around an tron (or positron) storage ring with a circumference of about one kilometer

elec-10Three premier facilities are the European Synchrotron Radiation Facility inGrenoble, France, the Advanced Photon Source at Argonne, Illinois, USA, andthe Super Photon Ring 8-GeV, SPring-8 in Harima, Japan [1.4]

The electrons in the storage ring have energies of typically 7× 109eV, andtravel close to the speed of light The electron current is perhaps 100 mA,but the electrons are grouped into tight bunches of centimeter length, eachwith a fraction of this total current The bunches have vertical and horizontalspreads of tens or hundreds of microns.

The electrons lose energy by generating synchrotron radiation as they arebent around the ring The electrical power needed to replenish the energy

of the electrons is provided by a radiofrequency electric field This cyclicelectric field accelerates the electron bunches by alternately attracting andrepelling them as they move through a dedicated section of the storage ring.(Each bunch must be in phase with the radiofrequency field.) The ring iscapable of holding a number of bunches equal to the radiofrequency times thecycle time around the ring For example, with a 0.3 GHz radiofrequency, anelectron speed of 3×105km/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 “insertiondevices,” their accelerations cause photon emission X-ray emission there-fore occurs in pulsed bursts, or “flashes.” The flash duration depends on theduration of the electron acceleration, but this is shortened by relativistic con-traction The flash duration depends primarily on the width of the electronbunch, and may be 0.1 ns In a case where every fiftieth bucket is filled inour hypothetical ring, these flashes are separated in time by 167 ns Someexperiments based on fast timing are designed around this time structure ofsynchrotron radiation

Although the energy of the electrons in the ring is restored by the highpower radiofrequency system, electrons are lost by occasional collisions withgas atoms in the vacuum The characteristic decay of the beam current overseveral hours requires that new electrons are injected into the bunches

Undulators. Synchrotron radiation is generated by the dipole bending nets used for controlling the electron orbit in the ring, but all modern “thirdgeneration” synchrotron radiation facilities derive their x-ray photons from

mag-“insertion devices,” which are magnet structures such as “wigglers” or lators.” Undulators comprise rows of magnets along the path of the electronbeam The fields of these magnets alternate up and down, perpendicular tothe direction of the electron beam Synchrotron radiation is produced whenthe electrons accelerate under the Lorentz forces of the row of magnets Themechanism of x-ray emission by electron acceleration is essentially the same

“undu-as that of bremsstrahlung radiation, which w“undu-as described in Fig 1.10 andSect 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 sitioned precisely so that the photon field is built by the constructive inter-ference of radiation from a row of accelerations The x-rays emerge from the

po-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 thesquare of the number of coherent magnetic periods (typically tens) Again inanalogy with Bragg diffraction, there is a corresponding decrease in the an-gular 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 electronpath, the electron oscillation frequency is enhanced by the relativistic factor2(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 ings of the magnets are 3 cm, a distance traversed by light in 10−10sec The

spac-relativistic enhancement brings the frequency to 1018Hz, 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-raybeam emerging from an undulator may have an angular spread of microradi-ans, diverging by only a millimeter over distances of tens of meters A smallbeam divergence and a small effective source area for x-ray emission makes anundulator beam an excellent source of x-rays for operating a monochromator

Brightness. Various figures of merit describe how x-ray sources provideuseful photons The figure of merit for operating a monochromator is pro-portional to the intensity (photons/s) per area of emitter (cm−2), but an-

other factor also must be included For a highly collimated x-ray beam, themonochromator 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 monochromatoroperation is “brightness” (often called “brilliance”), which is normalized bythe solid angle of the beam Brightness has units of [photons (s cm2sr)−1].

The brightness of an undulator beam can be 109times that of a conventionalx-ray tube Brightness is also a figure of merit for specialized beamlines thatfocus an x-ray beam into a narrow probe of micron dimensions Finally, thex-ray intensity is not distributed uniformly over all energies The term “spec-tral 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 energyrange 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 undulatorbeam 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 goniometersneeded for synchrotron radiation experiments are located in a “beamline,”which is along the forward direction from the insertion device These com-ponents are typically mounted in lead-lined “hutches” that shield users fromthe lethal radiation levels produced by the undulator beam