Title Details

Preface xiii

1. Introduction 1

1.1 X-Ray Technology and Its Brief History 1

1.2 Geometry of Crystals 2

1.2.1 Crystal Lattice and Symmetry 3

1.2.2 Lattice Directions and Planes 4

1.2.3 Atomic Arrangement in Crystal Structure 9

1.2.4 Imperfections in Crystal Structure 11

1.3 Principles of X-Ray Diffraction 13

1.3.1 Bragg Law 13

1.3.2 Diffraction Patterns 14

1.4 Reciprocal Space and Diffraction 16

1.4.1 Reciprocal Lattice 16

1.4.2 The Ewald Sphere 18

1.4.3 Diffraction Cone and Diffraction Vector Cone 19

1.5 Two-Dimensional X-Ray Diffraction 21

1.5.1 Diffraction Pattern Measured by Area Detector 21

1.5.2 Two-Dimensional X-Ray Diffraction System and Major Components 22

1.5.3 Summary 23

References 25

2. Geometry Conventions 28

2.1 Introduction 28

2.1.1 Comparison Between XRD2 and Conventional XRD 29

2.2 Diffraction Space and Laboratory Coordinates 30

2.2.1 Diffraction Cones in Laboratory Coordinates 30

2.2.2 Diffraction Vector Cones in Laboratory Coordinates 33

2.3 Detector Space and Detector Geometry 35

2.3.1 Ideal Detector for Diffraction Pattern in 3D Space 35

2.3.2 Diffraction Cones and Conic Sections with Flat 2D Detectors 36

2.3.3 Detector Position in the Laboratory System 37

2.3.4 Pixel Position in Diffraction Space—Flat Detector 37

2.3.5 Pixel Position in Diffraction Space—Curved Detector 39

2.4 Sample Space and Goniometer Geometry 42

2.4.1 Sample Rotations and Translations in Eulerian Geometry 42

2.4.2 Variation of Goniometer Geometry 44

2.5 Transformation from Diffraction Space to Sample Space 46

2.6 Summary of XRD2 Geometry 49

References 49

3. X-Ray Source and Optics 51

3.1 X-Ray Generation and Characteristics 51

3.1.1 X-Ray Spectrum and Characteristic Lines 51

3.1.2 Focal Spot and Takeoff Angle 53

3.1.3 Focal Spot Brightness and Profile 53

3.1.4 Absorption and Fluorescence 55

3.2 X-Ray Optics 56

3.2.1 Liouville’s Theorem and Fundamentals 56

3.2.2 X-Ray Optics in a Conventional Diffractometer 59

3.2.3 X-Ray Optics in Two-Dimensional Diffractometer 62

3.2.4 The b-Filter 66

3.2.5 Crystal Monochromator 68

3.2.6 Multilayer Mirrors 70

3.2.7 Pinhole Collimator 76

3.2.8 Capillary Optics 79

References 83

4. X-Ray Detectors 85

4.1 History of X-Ray Detection Technology 85

4.2 Point Detectors in Conventional Diffractometers 88

4.2.1 Proportional Counters 88

4.2.2 Scintillation Counters 89

4.2.3 Solid-State Detectors 90

4.3 Characteristics of Point Detectors 91

4.3.1 Counting Statistics 91

4.3.2 Detective Quantum Efficiency and Energy Range 93

4.3.3 Detector Linearity and Maximum Count Rate 94

4.3.4 Energy Resolution 96

4.3.5 Detection Limit and Dynamic Range 98

4.4 Line Detectors 100

4.4.1 Geometry of Line Detectors 100

4.4.2 Types of Line Detectors 103

4.4.3 Characteristics of Line Detectors 104

4.5 Characteristics of Area Detectors 107

4.5.1 Geometry of Area Detectors 108

4.5.2 Spatial Resolution of Area Detectors 112

4.6 Types of Area Detectors 114

4.6.1 Multiwire Proportional Counter 115

4.6.2 Image Plate 117

4.6.3 CCD Detector 118

4.6.4 Microgap Detector 122

4.6.5 Comparison of Area Detectors 127

References 130

5. Goniometer and Sample Stages 133

5.1 Goniometer and Sample Position 133

5.1.1 Introduction 133

5.1.2 Two-Circle Base Goniometer 134

5.1.3 Sample Stages 135

5.1.4 Sequence of the Goniometer Axes 136

5.2 Goniometer Accuracy 138

5.2.1 Sphere of Confusion 138

5.2.2 Angular Accuracy and Precision 141

5.3 Sample Alignment and Visualization Systems 143

5.4 Environment Stages 145

5.4.1 Domed High Temperature Stage 145

5.4.2 Temperature Stage Calibration 146

References 149

6. Data Treatment 151

6.1 Introduction 151

6.2 Nonuniform Response Correction 151

6.2.1 Calibration Source 152

6.2.2 Nonuniform Response Correction Algorithms 154

6.3 Spatial Correction 156

6.3.1 Fiducial Plate and Detector Plane 156

6.3.2 Spatial Correction Algorithms 158

6.4 Detector Position Accuracy and Calibration 163

6.4.1 Detector Position Tolerance 163

6.4.2 Detector Position Calibration 165

6.5 Frame Integration 167

6.5.1 Definition of Frame Integration 167

6.5.2 Algorithm of Frame Integration 170

6.6 Lorentz Polarization and Absorption Corrections 175

6.6.1 Lorentz 175

6.6.2 Polarization 176

6.6.3 Air Scatter and Be-Window Absorption 180

6.6.4 Sample Absorption 182

6.6.5 Combined Intensity Correction 188

References 189

7. Phase Identification 191

7.1 Introduction 191

7.2 Relative Intensity 193

7.2.1 Multiplicity Factor 193

7.2.2 Electron and Atomic Scattering 194

7.2.3 Structure Factor 196

7.2.4 Attenuation Factors 197

7.3 Geometry and Resolution 197

7.3.1 Detector Distance and Resolution 198

7.3.2 Defocusing Effect 199

7.3.3 Transmission Mode Diffraction 201

7.4 Sampling Statistics 202

7.4.1 Effective Sampling Volume 203

7.4.2 Angular Window 204

7.4.3 Virtual Oscillation 205

7.4.4 Sample Oscillation 206

7.5 Preferred Orientation Effect 208

7.5.1 Relative Intensity with Texture 208

7.5.2 Intensity Correction on Fiber Texture 211

References 216

8. Texture Analysis 218

8.1 Introduction 218

8.2 Pole Density and Pole Figure 219

8.3 Fundamental Equations 222

8.3.1 Pole Figure Angles 222

8.3.2 Pole Density 224

8.4 Data Collection Strategy 225

8.4.1 Single Scan 225

8.4.2 Multiple Scan 227

8.4.3 Comparison with Point Detector 230

8.5 Texture Data Process 231

8.5.1 2u Integration 231

8.5.2 Absorption Correction 234

8.5.3 Pole Figure Interpolation 235

8.5.4 Pole Figure Symmetry 235

8.5.5 Pole Figure Normalization 237

8.6 Orientation Distribution Function 237

8.6.1 Eulerian Angles and Space 237

8.6.2 ODF Calculation 239

8.6.3 Calculated Pole Figures From ODF 241

8.7 Fiber Texture 242

8.7.1 Pole Figures of Fiber Texture 242

8.7.2 ODF of Fiber Texture 244

8.8 Other Advantages of XRD2 for Texture 244

8.8.1 Orientation Relationship 245

8.8.2 Direct Observation of Texture 245

References 247

9. Stress Measurement 249

9.1 Introduction 249

9.1.1 Stress 250

9.1.2 Strain 254

9.1.3 Elasticity and Hooke’s Law 256

9.1.4 X-Ray Elasticity Constants and Anisotropy Factor 257

9.1.5 Residual Stresses 258

9.2 Principle of X-Ray Stress Analysis 260

9.2.1 Strain and Bragg Law 260

9.2.2 Strain Measurement 261

9.2.3 Stress Measurement 263

9.2.4 Stress Measurement Without d0 266

9.2.5 c-Tilt and Goniometer 269

9.2.6 Sin2c Method with Area Detector 270

9.3 Theory of Stress Analysis with XRD2 272

9.3.1 2D Fundamental Equation for Stress Measurement 272

9.3.2 Relationship Between Conventional Theory and 2D Theory 276

9.3.3 2D Equations for Various Stress States 278

9.3.4 True Stress-Free Lattice d-Spacing 280

9.3.5 Diffraction Cone Distortion Simulation 281

9.4 Process of Stress Measurement with XRD2 288

9.4.1 Instrument Requirements and Configurations 288

9.4.2 Data Collection Strategy 291

9.4.3 Data Integration and Peak Evaluation 295

9.4.4 Stress Calculation 299

9.4.5 Intensity Weighted Least Squares Regression 300

9.5 Experimental Examples 303

9.5.1 Comparison Between 2D Method and Conventional Method 303

9.5.2 Virtual Oscillation for Stress Measurement 305

9.5.3 Stress Mapping on Weldment 307

9.5.4 Residual Stresses in Thin Films 310

9.5.5 Residual Stress Measurement with Multiple {hkl} Rings 315

9.5.6 Gage Repeatability and Reproducibility Study 316

Appendix 9.A Calculation of Principal Stresses from the General Stress Tensor 320

Appendix 9.B Parameters for Stress Measurement 323

References 325

10. Small-Angle X-Ray Scattering 329

10.1 Introduction 329

10.1.1 Principle of Small-Angle Scattering 330

10.1.2 General Equation and Parameters in SAXS 330

10.1.3 X-Ray Source and Optics for SAXS 331

10.2 2D SAXS Systems 333

10.2.1 SAXS Attachments 334

10.2.2 Dedicated SAXS System 336

10.2.3 Detector Correction and System Calibration 337

10.2.4 Data Collection and Integration 338

10.3 Application Examples 341

10.3.1 Particles in Solutions 341

10.3.2 Scanning SAXS and Transmission Measurement 341

10.4 Some Innovations in 2D SAXS 343

10.4.1 Simultaneous Measurements of Transmission and SAXS 343

10.4.2 Vertical SAXS System 346

References 347

11. Combinatorial Screening 351

11.1 Introduction 351

11.1.1 Combinatorial Chemistry 351

11.1.2 Combinatorial Screening 352

11.2 XRD2 Systems for Combinatorial Screening 352

11.2.1 Combinatorial Screening in Reflection Geometry 353

11.2.2 Retractable Knife-Edge 356

11.2.3 Combinatorial Screening in Transmission Geometry 359

11.3 Combined Screening with XRD2 and Raman 364

References 366

12. Quantitative Analysis 369

12.1 Percent Crystallinity 369

12.1.1 Introduction 369

12.1.2 Comparison of Conventional XRD and XRD2 370

12.1.3 Scatter Correction 371

12.1.4 Internal and External Methods 373

12.1.5 Full Method 374

12.2 Crystal Size 376

12.2.1 Introduction 376

12.2.2 Line Broadening for Crystallite Size 377

12.2.3 g-Profile Analysis for Crystallite Size 380

12.3 Retained Austenite 387

References 390

13. Innovation and Future Development 393

13.1 Introduction 393

13.2 Scanning Line Detector for XRD2 394

13.2.1 Working Principle 394

13.2.2 Advantages of Scanning Line Detector 396

13.3 Three-Dimensional Detector 398

13.3.1 The Third Dimension of a Detector 398

13.3.2 Geometry of Three-Dimensional Detector 399

13.3.3 Three-Dimensional Detector and Reciprocal Space 401

13.4 Pixel Direct Diffraction Analysis 402

13.4.1 Concept 402

13.4.2 Pixel Diffraction Vector and Pixel Count 403

13.4.3 PDD Analysis in Phase-ID Texture and Stress 404

References 406

Appendix A. Values of Commonly Used Parameters 407

Appendix B. Symbols 412

Index 419