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Preface XV
About the Editors XXI
List of Contributors XXIII
Part I Structure Determination 1
1 Structure Determination of Single Crystals 3
Sander van Smaalen
1.1 Introduction 3
1.2 The Electron Density 5
1.3 Diffraction and the Phase Problem 8
1.4 Fourier Cycling and Difference Fourier Maps 10
1.5 Statistical Properties of Diffracted Intensities 11
1.6 The Patterson Function 15
1.7 Patterson Search Methods 18
1.8 Direct Methods 19
1.9 Charge Flipping and Low-Density Elimination 21
1.10 Outlook and Summary 24
References 25
2 Modern Rietveld Refinement, a Practical Guide 27
Robert Dinnebier and Melanie M¨uller
2.1 The Peak Intensity 29
2.2 The Peak Position 30
2.3 The Peak Profile 31
2.4 The Background 38
2.5 The Mathematical Procedure 39
2.6 Agreement Factors 39
2.7 Global Optimization Method of Simulated Annealing 41
2.8 Rigid Bodies 44
2.9 Introduction of Penalty Functions 46
2.10 Parametric Rietveld Refinement 47
2.10.1 Parameterization of the Scale Factor Depending on Time for Kinetic Analysis 49
2.10.2 Parameterization of the Lattice Parameters Depending on Pressure for Determination of the Equations of State 50
2.10.3 Parameterization of Symmetry Modes Depending on Temperature for Determination of Order Parameters 53
References 58
3 Structure of Nanoparticles from Total Scattering 61
Katharine L. Page, Thomas Proffen, and Reinhard B. Neder
3.1 Introduction 61
3.2 Total Scattering Experiments 64
3.2.1 Using X-Rays 66
3.2.2 Using Neutrons 67
3.3 Structure Modeling and Refinement 69
3.3.1 Using a Particle Form Factor 69
3.3.2 Modeling Finite Nanoparticles 70
3.4 Examples 74
3.4.1 BaTiO3 74
3.4.2 CdSe/ZnS Core–Shell Particles 78
3.5 Outlook 82
References 83
Part II Analysis of the Microstructure 87
4 Diffraction Line-Profile Analysis 89
Eric J. Mittemeijer and Udo Welzel
4.1 Introduction 89
4.2 Instrumental Broadening 90
4.2.1 Determination of the Instrumental Profile Using a Reference (Standard) Specimen 92
4.2.2 Determination of the Instrumental Profile by Calculus 92
4.2.3 Subtraction/Incorporation of the Instrumental Broadening 93
4.3 Structural, Specimen Broadening 94
4.3.1 Measures of Line Broadening; Fourier Series Representation of Diffraction Lines 94
4.3.2 Column Length/Crystallite Size and Column-Length/Crystallite-Size Distribution 96
4.3.3 Microstrain Broadening 98
4.3.3.1 Assumptions in Integral-Breadth Methods 99
4.3.3.2 Assumptions in Fourier Methods 100
4.3.3.3 Microstrain-Broadening Descriptions Derived from a Microstructural Model 101
4.3.4 Anisotropic Size and Microstrain(-Like) Diffraction-Line Broadening 104
4.3.5 Macroscopic Anisotropy 106
4.3.6 Crystallite Size and Coherency of Diffraction 106
4.4 Practical Application of Line-Profile Analysis 111
4.4.1 Line-Profile Decomposition 111
4.4.1.1 Breadth Methods 111
4.4.1.2 Fourier Methods 115
4.4.1.3 Whole Powder-Pattern Fitting 116
4.4.2 Line-Profile Synthesis 116
4.4.2.1 General Strain-Field Method 117
4.4.2.2 Specific Microstructural Models: Whole Powder-Pattern Modeling (WPPM) and Multiple Whole-Profile Modeling/Fitting (MWP) 118
4.4.2.3 General Atomistic Structure: the Debye Scattering Function 120
4.5 Conclusions 122
References 123
5 Residual Stress Analysis by X-Ray Diffraction Methods 127
Christoph Genzel, Ingwer A. Denks, and Manuela Klaus
5.1 Introduction 127
5.2 Principles of Near-Surface X-Ray Residual Stress Analysis 129
5.2.1 Fundamental Relations 129
5.2.2 Concepts of Diffraction Data Acquisition: Angle-Dispersive and Energy-Dispersive Modes 130
5.2.3 Concepts of Strain Depth Profiling: LAPLACE and Real Space Approach 131
5.2.3.1 Definition of the Information Depth 131
5.2.3.2 Depth Profiling in the LAPLACE Space 133
5.2.3.3 Depth Profiling in Real Space 136
5.2.3.4 ‘‘Fixed’’ versus ‘‘Variable Depth’’ Methods 139
5.3 Near-Surface X-Ray Residual Stress Analysis by Advanced and Complementary Methods 141
5.3.1 Residual Stress Depth Profiling in Multilayered Coating Systems 141
5.3.1.1 The ‘‘Equivalence Thickness’’ Concept 141
5.3.1.2 The ‘‘Stress Scanning’’ Method 144
5.3.2 Residual Stress Gradient Evaluation in Surface-Treated Bulk Samples 147
5.3.2.1 Fixed Depth Analysis in the Real Space: Direct Access to σ(z) 147
5.3.2.2 Residual Stress Evaluation in the LAPLACE Space: From σ(τ) to σ(z) 149
5.4 Final Remarks 151
References 153
6 Stress Analysis by Neutron Diffraction 155
Lothar Pintschovius and Michael Hofmann
6.1 Introductory Remarks 155
6.2 Fundamentals of the Technique 155
6.2.1 The d0-Problem 156
6.2.2 Macrostrains versus Microstrains 157
6.2.3 Strain Tensors 158
6.2.4 Reflection Line Broadenings 158
6.3 Instrumentation 159
6.3.1 Angle-Dispersive Instruments 159
6.3.1.1 Monochromators 159
6.3.1.2 Beam-Defining Optics 160
6.3.1.3 Detectors 161
6.3.1.4 Auxilliaries 162
6.3.2 Time-of-Flight Instruments 162
6.3.3 Special Instruments 164
6.4 Capabilities 164
6.4.1 Types of Materials 164
6.4.2 Spatial Resolution 164
6.4.3 Penetration Depth 165
6.4.4 Accuracy 166
6.4.5 Throughput 166
6.5 Examples 166
6.5.1 Railway Rail 166
6.5.2 Weldments 167
6.5.3 Ceramics 168
6.5.4 Composite Materials 170
References 170
7 Texture Analysis by Advanced Diffraction Methods 173
Hans-Rudolf Wenk
7.1 Introduction and Background 173
7.2 Synchrotron X-Rays 177
7.2.1 General Approach 177
7.2.2 Hard Synchrotron X-Rays 178
7.2.3 In situ High-Pressure Experiments 180
7.2.4 From Diffraction Images to Orientation Distribution 183
7.2.5 Opportunities with the Laue Technique 188
7.2.6 Synchrotron Applications 188
7.3 Neutron Diffraction 190
7.3.1 General Comments 190
7.3.2 Monochromatic Neutrons 193
7.3.3 Polychromatic Time-of-Flight (TOF) Neutrons 194
7.3.4 Special Techniques 197
7.3.5 Data Analysis for TOF Neutrons 198
7.3.6 Neutron Applications 202
7.3.6.1 Grain Statistics 202
7.3.6.2 Polymineralic Rocks 202
7.3.6.3 In situ Experiments and Phase Transformations 203
7.3.6.4 Magnetic Textures 204
7.4 Electron Diffraction 204
7.4.1 Transmission Electron Microscope 204
7.4.2 Scanning Electron Microscope (SEM) 205
7.4.3 EBSD Applications 209
7.4.3.1 Misorientations 209
7.4.3.2 In situ Heating 209
7.4.3.3 In situ Deformation 210
7.4.3.4 3D Mapping 211
7.4.3.5 Residual Strain Analysis 211
7.5 Comparison of Methods 212
7.6 Conclusions 213
Acknowledgments 214
References 214
8 Surface-Sensitive X-Ray Diffraction Methods 221
Andreas Stierle and Elias Vlieg
8.1 Introduction 221
8.1.1 Structure Determination by X-Ray Diffraction 223
8.2 X-Ray Reflectivity 224
8.3 Bragg Scattering in Reduced Dimensions (Crystal Truncation Rod Scattering) 227
8.3.1 Thin-Film Diffraction 227
8.3.2 Surface Diffraction from Half-Infinite Systems 230
8.3.2.1 Surface Relaxations 232
8.3.2.2 Surface Reconstructions and Fourier Methods 234
8.3.2.3 Surface Roughness 237
8.3.2.4 Vicinal Surfaces 239
8.3.2.5 Two-Layer Roughness Model for Growth Studies 240
8.3.2.6 Interface Diffraction 245
8.3.2.7 The Specular Rod 247
8.4 Grazing Incidence X-Ray Diffraction 249
8.5 Experimental Geometries 252
8.6 Trends 254
Acknowledgments 255
References 255
9 The Micro- and Nanostructure of Imperfect Oxide Epitaxial Films 259
Alexandre Boulle, Florine Conchon, and René Guinebretìere
9.1 The Diffracted Amplitude and Intensity 260
9.1.1 Diffracted Amplitude 260
9.1.2 Diffracted Intensity 261
9.2 The Correlation Volume 262
9.2.1 Crystallite Size and Shape 262
9.2.2 Crystallite Size Fluctuations 265
9.2.3 Crystallite Shape Fluctuations 267
9.3 Lattice Strain 269
9.3.1 Statistical Properties 269
9.3.2 Spatial Properties 272
9.4 Example 274
9.5 Strain Gradients 277
9.5.1 Background 277
9.5.2 Strain Profile Retrieval 277
9.5.3 Example 278
9.6 Conclusions 279
References 281
Part III Phase Analysis and Phase Transformations 283
10 Quantitative Phase Analysis Using the Rietveld Method 285
Ian C. Madsen, Nicola V.Y. Scarlett, Daniel P. Riley, and Mark D. Raven
10.1 Introduction 285
10.2 Mathematical Basis 286
10.2.1 Rietveld-Based Methods 286
10.2.2 Improving Accuracy 290
10.2.3 Correlation with Thermal Parameters 292
10.3 Applications in Minerals and Materials Research 295
10.3.1 Crystallization from Hydrothermal Solutions 295
10.3.2 Energy-Dispersive Diffraction 298
10.3.2.1 Application of EDD to the Study of Inert Anodes for Light Metal Production 301
10.3.3 Quantitative Phase Analysis in Mineral Exploration 304
10.3.3.1 Particle Statistics 306
10.3.3.2 Preferred Orientation 306
10.3.3.3 Microabsorption 306
10.3.3.4 Identification of Mineral Types and Polytypes 307
10.3.3.5 Element Substitution and Solid Solution 307
10.3.3.6 Severe Peak Overlap 308
10.3.3.7 Poorly Crystalline Components 309
10.3.3.8 Clay and Disordered Structures 309
10.3.4 The Reynolds Cup 310
10.3.5 Use of QPA-Derived Kinetics in the Design of Novel Materials 312
10.3.5.1 Methodologies for Synthesis Optimization Using QPA 312
10.3.5.2 Design and Synthesis Optimization of Novel Materials: Mn+1AXn Phases 312
10.3.5.3 In situ Differential Thermal Analysis (DTA) Using QPA 316
10.4 Summary 318
Acknowledgments 318
References 318
11 Kinetics of Phase Transformations and of Other Time-Dependent Processes in Solids Analyzed by Powder Diffraction 321
Andreas Leineweber and Eric J. Mittemeijer
11.1 Introduction 321
11.2 Kinetic Concepts 323
11.2.1 Process Rates 323
11.2.2 The Temperature Dependence of the Process Rate 327
11.2.2.1 Arrhenius-Type Temperature Dependence of the Rate Constant k(T) 327
11.2.2.2 Non-Arrhenius-Type Process Kinetics 328
11.2.3 Rate Laws for Isothermally Conducted Processes 330
11.2.3.1 mth-Order Kinetics of Homogeneous Processes 330
11.2.3.2 Johnson-Mehl-Avrami-Kolmogorov Kinetics of Heterogeneous Phase Transformations 331
11.2.3.3 Grain Growth and Ostwald Ripening 332
11.2.3.4 Volume-Diffusion-Controlled Processes 333
11.2.3.5 Order-Disorder-Related Processes 333
11.2.4 Rate Laws for Nonisothermally Conducted Processes 336
11.3 Tracing the Process Kinetics by Powder Diffraction 337
11.4 Mode of Measurement: In Situ versus Ex Situ Methods 339
11.5 Types of Kinetic Processes and Examples 342
11.5.1 Local Composition in Solid is Retained 342
11.5.1.1 Reconstructive, Polymorphic Transformations α → β 342
11.5.1.2 Polymorphic Transformations of Order–Disorder Character and Related Processes 346
11.5.1.3 Polymorphic Transformations of Polytypic Character 347
11.5.1.4 Grain Growth 349
11.5.2 Local Concentration Variations within Isolated Solid Systems 350
11.5.2.1 Precipitation Processes 350
11.5.2.2 Solid-State Reaction between Different Phases 351
11.5.3 Composition Changes in Solids by Reaction with Fluid Matter 352
11.6 Concluding Remarks 354
References 354
Part IV Diffraction Methods and Instrumentation 359
12 Laboratory Instrumentation for X-Ray Powder Diffraction: Developments and Examples 361
Udo Welzel and Eric J. Mittemeijer
12.1 Introduction: Historical Sketch 361
12.2 Laboratory X-Ray Powder Diffraction: Instrumentation 365
12.2.1 Overview 365
12.2.2 Laboratory X-Ray Sources; Monochromatization 365
12.2.2.1 X-Ray Sources 365
12.2.2.2 Monochromatization/Filtering 368
12.2.3 Debye–Scherrer (−Hull) Geometry 370
12.2.4 Monochromatic Pinhole Techniques 371
12.2.5 (Para-)Focusing Geometries 371
12.2.5.1 Seemann–Bohlin Geometry 372
12.2.5.2 Bragg–Brentano Geometry 373
12.2.6 Instrumental Aberrations of (Para-)Focusing Geometries 376
12.2.7 Parallel-Beam Geometry 377
12.2.7.1 Polycapillary Collimators 378
12.2.7.2 X-Ray Mirrors 379
12.2.7.3 X-Ray Mirrors versus X-Ray Lenses; Comparative Discussion 381
12.2.7.4 Instrumental Aberrations of Parallel-Beam Geometry 383
12.2.8 Further, Recent Developments 384
12.2.8.1 Two-Dimensional Detectors 384
12.2.8.2 Microdiffraction 387
12.2.8.3 Energy-Dispersive Diffraction 388
12.3 Examples 388
12.3.1 Parallel-Beam Diffraction Methods 388
12.3.1.1 High Brilliance, Parallel-Beam Laboratory X-Ray Source 388
12.3.1.2 Applications 389
12.3.2 Two-Dimensional Diffraction Methods 391
Acknowledgments 394
References 394
13 The Calibration of Laboratory X-Ray Diffraction Equipment Using NIST Standard Reference Materials 399
James P. Cline, David Black, Donald Windover, and Albert Henins
13.1 Introduction 399
13.2 The Instrument Profile Function 400
13.3 SRMs, Instrumentation, and Data Collection Procedures 411
13.4 Data Analysis Methods 418
13.5 Instrument Qualification and Validation 423
13.6 Conclusions 436
References 437
14 Synchrotron Diffraction: Capabilities, Instrumentation, and Examples 439
Gene E. Ice
14.1 Introduction 439
14.2 The Underlying Physics of Synchrotron Sources 441
14.2.1 Storage Ring Sources 441
14.2.2 Free-Electron Lasers and Other Emerging X-Ray Sources 445
14.3 Diffraction Applications Exploiting High Source Brilliance 445
14.3.1 Microdiffraction 446
14.3.1.1 Microdiffraction Example 1: Stress-Driven Sn Whisker Growth 449
14.3.1.2 Microdiffraction Example 2: Damage in Ion-Implanted Si 451
14.3.1.3 Other Microdiffraction Applications 452
14.3.2 Surface and Interface Diffraction 452
14.3.2.1 Surface Diffraction Example 1: Truncation Rod Scattering (TRS) 453
14.3.2.2 Surface Diffraction Example 2: Surface Studies of Phase Transformations in Langmuir–Blodgett Films 455
14.4 High Q-Resolution Measurements 456
14.5 Applications of Tunability: Resonant Scattering 456
14.5.1 Resonant Scattering Example 1: Multiple Anomalous Diffraction, MAD 458
14.5.2 Resonant Scattering Example 2: 3λ Determination of Local Short-Range Correlation in Binary Alloys 461
14.5.3 Resonant Scattering Example 3: Determination of Magnetic Structure and Correlation Lengths 464
14.6 Future: Ultrafast Science and Coherence 465
14.6.1 Coherent Diffraction 466
14.6.2 Ultrafast Diffraction 466
References 467
15 High-Energy Electron Diffraction: Capabilities, Instrumentation, and Examples 469
Christoph T. Koch
15.1 Introduction 469
15.2 Instrumentation 470
15.2.1 Fundamentals 470
15.2.2 Diffraction Modes in a TEM 472
15.2.3 Femtosecond Electron Diffraction 474
15.3 Electron Diffraction Methods in the TEM 474
15.3.1 Precession Electron Diffraction (PED) 474
15.3.2 Quantitative Convergent-Beam Electron Diffraction (QCBED) 476
15.3.3 Large-Angle Convergent-Beam Electron Diffraction (LACBED) 477
15.3.4 Large-Angle Rocking-Beam Electron Diffraction (LARBED) 478
15.3.5 Diffraction Tomography 482
15.3.6 Real-Space Crystallography 482
15.3.7 Coherent Diffractive Imaging (CDI) with Electrons 483
15.3.8 Mapping Strain by Electron Diffraction 485
15.4 Summary and Outlook 486
Acknowledgment 486
References 486
16 In Situ Diffraction Measurements: Challenges, Instrumentation, and Examples 491
Helmut Ehrenberg, Anatoliy Senyshyn, Manuel Hinterstein, and Hartmut Fuess
16.1 Introduction 491
16.2 Instrumentation and Experimental Challenges 492
16.2.1 General Considerations 492
16.2.2 Absorption 493
16.2.3 Detection Challenges 494
16.3 Examples 497
16.3.1 Electrochemical In Situ Studies of Electrode Materials and In Operando Investigations of Li-Ion Batteries 497
16.3.2 In situ Studies of Piezoceramics in Electric Fields 502
Acknowledgment 515
References 515
Index 519
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