Modern Diffraction Methods Book

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