Combined Analysis Book

Introduction

Acknowledgements

Chapter 1 Some Basic Notions About Powder Diffraction 1

1.1 Crystallite, grain, polycrystal and powder 1

1.2 Bragg's law and harmonic reflections 2

1.2.1 Bragg's law 2

1.2.2 Monochromator 3

1.2.3 Harmonic radiation components 3

1.3 Geometric conditions of diffraction, Ewald sphere 4

1.4 Imperfect powders 5

1.5 Main diffraction line profile components 6

1.5.1 Origin of g(x) 7

1.5.2 Origin of f(x) 9

1.5.3 Deconvolution-extraction of f(x) and g(x) 10

1.6 Peak profile parameters 11

1.7 Modeling of the diffraction peaks 11

1.7.1 Why do we need modeling? 11

1.7.2 Modeling of a powder diffraction pattern 12

1.8 Experimental geometry 22

1.8.1 Curved Position Sensitive detector, asymmetric reflection geometry 22

1.8.2 CCD or image plate detector, transmission geometry 23

1.8.3 Curved-area Position-Sensitive detector, transmission geometry 24

1.9 Intensity calibration (flat-field) 26

1.9.1 Counts and intensity 26

1.9.2 Flat-field 29

1.9.3 PSD detector 29

1.9.4 CAPS detector 30

1.10 Standard samples 32

1.10.1 Laboratory x-ray standards 32

1.10.2 Neutron texture standards 34

1.11 Probed thickness (penetration depth) 36

Chapter 2 Structure Refinement by Diffraction Profile Adjustment (Rietveld Method) 41

2.1 Principle of the Rietveld method 41

2.2 Rietveld-based codes 43

2.3 Parameter modeling 44

2.3.1 Background modeling 44

2.3.2 Structure factor 47

2.3.3 Crystallites' preferred orientation (texture) corrections 53

2.3.4 Peak asymmetry 57

2.3.5 Peak displacements 59

2.3.6 Lorentz-polarization correction 60

2.3.7 Volume, absorption, thickness corrections 62

2.3.8 Localization corrections 66

2.3.9 Microabsorption/roughness corrections 68

2.3.10 Wavelength 70

2.4 Crystal structure databases 71

2.5 Reliability factors in profile refinements 71

2.6 Parameter exactness 75

2.7 The Le Bail method 75

2.8 Refinement procedures 76

2.8.1 Least squares 76

2.8.2 Genetic or evolutionary algorithms 78

2.8.3 Derivative difference minimization (DDM) 80

2.8.4 Simulated annealing 80

2.9 Refinement strategy 81

2.10 Structural determination by diffraction 82

2.10.1 The phase problem in diffraction 82

2.10.2 Patterson function 83

2.10.3 Direct methods 84

2.10.4 Direct space methods 87

2.10.5 Fourier difference map 87

2.10.6 Extension to aperiodic structures 88

Chapter 3 Automatic Indexing of Powder Diagrams 91

3.1 Principle 91

3.2 Dichotomy approach 92

3.3 Criterions for quality 93

Chapter 4 Quantitative Texture Analysis 95

4.1 Classic texture analysis 95

4.1.1 Qualitative aspects of texture analysis 95

4.1.2 Effects on diffraction diagrams 98

4.1.3 Limitations of classic diagrams 102

4.1.4 The Lotgering factor 106

4.1.5 Representations of textures: pole figures 107

4.1.6 Localization of crystallographic directions from pole figures 121

4.1.7 Texture types 126

4.2 Orientation distribution (OD) or orientation distribution function (ODF) 131

4.2.1 Pole figures and orientation spaces 131

4.2.2 The orientation space H 132

4.2.3 Euler angle conventions 133

4.2.4 Orientations and pole figures 135

4.2.5 Choice for the sample co-ordinate system KA 136

4.2.6 Pole figure, crystal, texture and sample symmetries 139

4.2.7 Orientation distance 139

4.3 Distribution density and normalization 140

4.4 Direct and normalized pole figures 140

4.4.1 Direct experimental pole figures 140

4.4.2 Normalized pole figures 141

4.5 Reduced pole figures 143

4.6 Fundamental equation of quantitative texture analysis 143

4.6.1 Fundamental equation 143

4.6.2 Typical OD components 144

4.6.3 OD plotting 147

4.6.4 Finding an orientation component in the OD 149

4.7 Resolution of the fundamental equation 150

4.7.1 ODF and OD 150

4.7.2 Generalized spherical harmonics 150

4.7.3 Vector method 155

4.7.4 Williams-Imhof-Matthies-Vinel (WIMV) method 155

4.7.5 Arbitrarily-defined cells (ADC) method 157

4.7.6 Entropy maximization method 157

4.7.7 Component method 158

4.7.8 Exponential harmonics 159

4.7.9 Radon transform and Fourier analysis 160

4.7.10 Orientation space coverage 160

4.8 OD refinement reliability estimators 161

4.8.1 RP factors 161

4.8.2 RPw surface weighted factors 163

4.8.3 RB Bragg-like factors 164

4.8.4 RBw Bragg-like weighted factors 165

4.8.5 Rw weighted factors 165

4.8.6 Visual inspection 166

4.9 Inverse pole figures 168

4.9.1 Definition 168

4.9.2 Inverse pole figure sectors 168

4.10 Texture strength factors 170

4.10.1 Texture index 170

4.10.2 Texture entropy 171

4.10.3 Pole figure and ODF strengths 171

4.10.4 Correlation between F2 and S 171

4.11 Texture programs 173

4.11.1 Berkeley texture package (BEARTEX) 173

4.11.2 Material analysis using diffraction (MAUD) 173

4.11.3 General structure analysis system (GSAS) 173

4.11.4 Preferred orientation package, Los Alamos (popLA) 174

4.11.5 Texture analysis software (Labo Tex) 174

4.11.6 Pole figure interpretation (POFINT) 174

4.11.7 Strong textures (STROTEX and Phiscans) 175

4.11.8 STEREOPOLE 175

4.11.9 MTEX 175

4.12 Limits of the classic texture analysis 176

4.13 Magnetic quantitative texture analysis (MQTA) 178

4.13.1 Magnetization curves and magnetic moment distributions 178

4.13.2 A simple sample holder for MQTA 179

4.13.3 Methodology 179

4.13.4 From magnetic-scattering to the MODF and magnetic moment distributions 186

4.13.5 One example 186

4.14 Reciprocal space mapping (RSM) 189

Chapter 5 Quantitative Microstructure Analysis 191

5.1 Introduction 191

5.2 Microstructure modeling (classic) 192

5.2.1 Integral Breadth, FWHM, volume- and area-weighted sizes 192

5.2.2 Scherrer approach 195

5.2.3 Stokes and Wilson microstrains 196

5.2.4 Williamson-Hall approach 196

5.3 Bertaut-Warren-Averbach approach (Fourier analysis) 197

5.3.1 Instrumental contribution removal 197

5.3.2 Broadening due to crystallite size 198

5.3.3 Crystallite size and microdistortion broadening 199

5.3.4 Fourier analysis to integral breadths 201

5.3.5 Integral breadths to distributions, sizes and microstrains 202

5.3.6 Relationships between <RA> and <RV> 203

5.4 Anisotropic broadening: the Popa approach 204

5.4.1 Anisotropic broadening 204

5.4.2 Anisotropic crystallite sizes 205

5.4.3 Anisotropic microstrains 211

5.5 Stacking and twin faults 212

5.5.1 From Line shifts and Fourier analysis 212

5.5.2 Popa approach 214

5.6 Dislocations 214

5.6.1 Dislocation density 215

5.6.2 Wilkens' model and Fourier analysis 215

5.7 Crystallite size distributions 217

5.7.1 Normal size distribution function 217

5.7.2 Lognormal distribution function 217

5.7.3 Gamma distribution function 218

5.7.4 Anisotropic distribution functions 218

5.8 Rietveld approach 219

5.8.1 Constant wavelength data 219

5.8.2 Time of flight (TOF) neutrons 220

Chapter 6 Quantitative Phase Analysis 221

6.1 Standardized experiments 221

6.2 Polycrystalline samples 221

6.3 Amorphous-crystalline aggregates 223

6.3.1 Crystallinity fraction 223

6.3.2 Amorphous modeling 224

6.4 Detection Limit 225

Chapter 7 Residual Strain-Stress Analysis 227

7.1 Strain definitions 227

7.2 ε33 strain determination 229

7.2.1 Isotropic polycrystalline sample 229

7.2.2 Single crystal 229

7.3 Complete strain tensor determination 230

7.3.1 Isotropic polycrystalline samples 230

7.3.2 Single crystal samples 231

7.4 Textured samples 232

7.4.1 Introduction 232

7.4.2 Non-linear least-squares fit 233

7.4.3 Strain and stress distribution functions 234

Chapter 8 X-Ray Reflectivity 235

8.1 Introduction 235

8.1.1 Definition of the reflectivity 235

8.1.2 Specular and off-specular reflectivity 236

8.1.3 Combined specular and off-specular scans 237

8.2 X-rays and neutrons refractive index 238

8.2.1 X-rays 238

8.2.2 Neutrons 239

8.3 The critical angle of reflection 240

8.3.1 X-rays 240

8.3.2 Neutrons 241

8.4 Fresnel formalism (specular reflectivity) 241

8.4.1 Reflection coefficient and reflectivity 241

8.4.2 Transmission coefficient 244

8.4.3 Yoneda wings 245

8.5 Surface roughness 246

8.5.1 Roughness representation 246

8.5.2 Single layer on a substrate 249

8.6 Matrix formalism (specular reflectivity) 250

8.7 Born approximation 251

8.8 Electron density profile 251

8.9 Multilayer reflectivity curves 252

8.10 Instrumental corrections 253

8.10.1 Correction for an irradiated area 253

8.10.2 Imperfectly parallel beam 254

Chapter 9 Combined Structure-Texture-Microstructure-Stress-Phase Reflectivity Analysis 257

9.1 Initial queries 257

9.2 Implementation 261

9.3 Experimental set-up 264

9.4 Instrument calibration 264

9.4.1 Peak broadenings 266

9.4.2 Peak shifts 269

9.4.3 Background variations 269

9.5 Refinement strategy 269

9.5.1 Global scheme 269

9.5.2 Solution examination 271

9.6 Examples 272

9.6.1 QTA of single-phased materials 272

9.6.2 QTA and isotropic QMA 297

9.6.3 Anisotropic crystallite shape, texture, cell parameters, and thickness 300

9.6.4 Layering, isotropic shape, microstrains, texture, and structure 307

9.6.5 Phase and texture 311

9.6.6 Texture of modulated structures 334

9.6.7 Texture, residual stresses and layering 341

9.6.8 Texture and structure 346

Chapter 10 Macroscopic Anisotropic Properties 363

10.1 Aniso- and isotropic samples and properties 363

10.2 Macroscopic/microscopic properties 364

10.2.1 ™ and T tensors 364

10.2.2 Microscopic properties 364

10.2.3 Macroscopic properties anisotropy and modeling 413

Bibliography 441

Glossary 483

Abbreviations 487

Mathematical Operators 489

Index 491