Two-dimensional X-ray Diffraction

Written by one of the pioneers of 2D X-Ray Diffraction, this useful guide covers the fundamentals, experimental methods and applications of two-dimensional x-ray diffraction, including geometry convention, x-ray source and optics, two-dimensional detectors, diffraction data interpretation, and configurations for various applications, such as phase identification, texture, stress, microstructure analysis, crystallinity, thin film analysis and combinatorial screening. Experimental examples in materials research, pharmaceuticals, and forensics are also given. This presents a key resource to researchers in materials science, chemistry, physics, and pharmaceuticals, as well as graduate-level students in these areas. TWO-DIMENSIONAL X-RAY DIFFRACTION......Page 4 CONTENTS......Page 8 Preface......Page 16 1.1 X-Ray Technology and Its Brief History......Page 18 1.2 Geometry of Crystals......Page 19 1.2.1 Crystal Lattice and Symmetry......Page 20 1.2.2 Lattice Directions and Planes......Page 21 1.2.3 Atomic Arrangement in Crystal Structure......Page 26 1.2.4 Imperfections in Crystal Structure......Page 28 1.3.1 Bragg Law......Page 30 1.3.2 Diffraction Patterns......Page 31 1.4.1 Reciprocal Lattice......Page 33 1.4.2 The Ewald Sphere......Page 35 1.4.3 Diffraction Cone and Diffraction Vector Cone......Page 36 1.5.1 Diffraction Pattern Measured by Area Detector......Page 38 1.5.2 Two-Dimensional X-Ray Diffraction System and Major Components......Page 39 1.5.3 Summary......Page 40 References......Page 42 2.1 Introduction......Page 45 2.1.1 Comparison Between XRD(2) and Conventional XRD......Page 46 2.2.1 Diffraction Cones in Laboratory Coordinates......Page 47 2.2.2 Diffraction Vector Cones in Laboratory Coordinates......Page 50 2.3.1 Ideal Detector for Diffraction Pattern in 3D Space......Page 52 2.3.2 Diffraction Cones and Conic Sections with Flat 2D Detectors......Page 53 2.3.4 Pixel Position in Diffraction Space—Flat Detector......Page 54 2.3.5 Pixel Position in Diffraction Space—Curved Detector......Page 56 2.4.1 Sample Rotations and Translations in Eulerian Geometry......Page 59 2.4.2 Variation of Goniometer Geometry......Page 61 2.5 Transformation from Diffraction Space to Sample Space......Page 63 References......Page 66 3.1.1 X-Ray Spectrum and Characteristic Lines......Page 68 3.1.3 Focal Spot Brightness and Profile......Page 70 3.1.4 Absorption and Fluorescence......Page 72 3.2.1 Liouville's Theorem and Fundamentals......Page 73 3.2.2 X-Ray Optics in a Conventional Diffractometer......Page 76 3.2.3 X-Ray Optics in Two-Dimensional Diffractometer......Page 79 3.2.4 The β-Filter......Page 83 3.2.5 Crystal Monochromator......Page 85 3.2.6 Multilayer Mirrors......Page 87 3.2.7 Pinhole Collimator......Page 93 3.2.8 Capillary Optics......Page 96 References......Page 100 4.1 History of X-Ray Detection Technology......Page 102 4.2.1 Proportional Counters......Page 105 4.2.2 Scintillation Counters......Page 106 4.2.3 Solid-State Detectors......Page 107 4.3.1 Counting Statistics......Page 108 4.3.2 Detective Quantum Efficiency and Energy Range......Page 110 4.3.3 Detector Linearity and Maximum Count Rate......Page 111 4.3.4 Energy Resolution......Page 113 4.3.5 Detection Limit and Dynamic Range......Page 115 4.4.1 Geometry of Line Detectors......Page 117 4.4.2 Types of Line Detectors......Page 120 4.4.3 Characteristics of Line Detectors......Page 121 4.5 Characteristics of Area Detectors......Page 124 4.5.1 Geometry of Area Detectors......Page 125 4.5.2 Spatial Resolution of Area Detectors......Page 129 4.6 Types of Area Detectors......Page 131 4.6.1 Multiwire Proportional Counter......Page 132 4.6.2 Image Plate......Page 134 4.6.3 CCD Detector......Page 135 4.6.4 Microgap Detector......Page 139 4.6.5 Comparison of Area Detectors......Page 144 References......Page 147 5.1.1 Introduction......Page 150 5.1.2 Two-Circle Base Goniometer......Page 151 5.1.3 Sample Stages......Page 152 5.1.4 Sequence of the Goniometer Axes......Page 153 5.2.1 Sphere of Confusion......Page 155 5.2.2 Angular Accuracy and Precision......Page 158 5.3 Sample Alignment and Visualization Systems......Page 160 5.4.1 Domed High Temperature Stage......Page 162 5.4.2 Temperature Stage Calibration......Page 163 References......Page 166 6.2 Nonuniform Response Correction......Page 168 6.2.1 Calibration Source......Page 169 6.2.2 Nonuniform Response Correction Algorithms......Page 171 6.3.1 Fiducial Plate and Detector Plane......Page 173 6.3.2 Spatial Correction Algorithms......Page 175 6.4.1 Detector Position Tolerance......Page 180 6.4.2 Detector Position Calibration......Page 182 6.5.1 Definition of Frame Integration......Page 184 6.5.2 Algorithm of Frame Integration......Page 187 6.6.1 Lorentz......Page 192 6.6.2 Polarization......Page 193 6.6.3 Air Scatter and Be-Window Absorption......Page 197 6.6.4 Sample Absorption......Page 199 6.6.5 Combined Intensity Correction......Page 205 References......Page 206 7.1 Introduction......Page 208 7.2.1 Multiplicity Factor......Page 210 7.2.2 Electron and Atomic Scattering......Page 211 7.2.3 Structure Factor......Page 213 7.3 Geometry and Resolution......Page 214 7.3.1 Detector Distance and Resolution......Page 215 7.3.2 Defocusing Effect......Page 216 7.3.3 Transmission Mode Diffraction......Page 218 7.4 Sampling Statistics......Page 219 7.4.1 Effective Sampling Volume......Page 220 7.4.2 Angular Window......Page 221 7.4.3 Virtual Oscillation......Page 222 7.4.4 Sample Oscillation......Page 223 7.5.1 Relative Intensity with Texture......Page 225 7.5.2 Intensity Correction on Fiber Texture......Page 228 References......Page 233 8.1 Introduction......Page 235 8.2 Pole Density and Pole Figure......Page 236 8.3.1 Pole Figure Angles......Page 239 8.3.2 Pole Density......Page 241 8.4.1 Single Scan......Page 242 8.4.2 Multiple Scan......Page 244 8.4.3 Comparison with Point Detector......Page 247 8.5.1 2θ Integration......Page 248 8.5.2 Absorption Correction......Page 251 8.5.4 Pole Figure Symmetry......Page 252 8.6.1 Eulerian Angles and Space......Page 254 8.6.2 ODF Calculation......Page 256 8.6.3 Calculated Pole Figures From ODF......Page 258 8.7.1 Pole Figures of Fiber Texture......Page 259 8.8 Other Advantages of XRD(2) for Texture......Page 261 8.8.2 Direct Observation of Texture......Page 262 References......Page 264 9.1 Introduction......Page 266 9.1.1 Stress......Page 267 9.1.2 Strain......Page 271 9.1.3 Elasticity and Hooke's Law......Page 273 9.1.4 X-Ray Elasticity Constants and Anisotropy Factor......Page 274 9.1.5 Residual Stresses......Page 275 9.2.1 Strain and Bragg Law......Page 277 9.2.2 Strain Measurement......Page 278 9.2.3 Stress Measurement......Page 280 9.2.4 Stress Measurement Without d(0)......Page 283 9.2.5 ψ-Tilt and Goniometer......Page 286 9.2.6 Sin(2)ψ Method with Area Detector......Page 287 9.3.1 2D Fundamental Equation for Stress Measurement......Page 289 9.3.2 Relationship Between Conventional Theory and 2D Theory......Page 293 9.3.3 2D Equations for Various Stress States......Page 295 9.3.4 True Stress-Free Lattice d-Spacing......Page 297 9.3.5 Diffraction Cone Distortion Simulation......Page 298 9.4.1 Instrument Requirements and Configurations......Page 305 9.4.2 Data Collection Strategy......Page 308 9.4.3 Data Integration and Peak Evaluation......Page 312 9.4.4 Stress Calculation......Page 316 9.4.5 Intensity Weighted Least Squares Regression......Page 317 9.5.1 Comparison Between 2D Method and Conventional Method......Page 320 9.5.2 Virtual Oscillation for Stress Measurement......Page 322 9.5.3 Stress Mapping on Weldment......Page 324 9.5.4 Residual Stresses in Thin Films......Page 327 9.5.5 Residual Stress Measurement with Multiple {hkl} Rings......Page 332 9.5.6 Gage Repeatability and Reproducibility Study......Page 333 Appendix 9.A Calculation of Principal Stresses from the General Stress Tensor......Page 337 Appendix 9.B Parameters for Stress Measurement......Page 340 References......Page 342 10.1 Introduction......Page 346 10.1.2 General Equation and Parameters in SAXS......Page 347 10.1.3 X-Ray Source and Optics for SAXS......Page 348 10.2 2D SAXS Systems......Page 350 10.2.1 SAXS Attachments......Page 351 10.2.2 Dedicated SAXS System......Page 353 10.2.3 Detector Correction and System Calibration......Page 354 10.2.4 Data Collection and Integration......Page 355 10.3.2 Scanning SAXS and Transmission Measurement......Page 358 10.4.1 Simultaneous Measurements of Transmission and SAXS......Page 360 10.4.2 Vertical SAXS System......Page 363 References......Page 364 11.1.1 Combinatorial Chemistry......Page 368 11.2 XRD(2) Systems for Combinatorial Screening......Page 369 11.2.1 Combinatorial Screening in Reflection Geometry......Page 370 11.2.2 Retractable Knife-Edge......Page 373 11.2.3 Combinatorial Screening in Transmission Geometry......Page 376 11.3 Combined Screening with XRD(2) and Raman......Page 381 References......Page 383 12.1.1 Introduction......Page 386 12.1.2 Comparison of Conventional XRD and XRD(2)......Page 387 12.1.3 Scatter Correction......Page 388 12.1.4 Internal and External Methods......Page 390 12.1.5 Full Method......Page 391 12.2.1 Introduction......Page 393 12.2.2 Line Broadening for Crystallite Size......Page 394 12.2.3 γ-Profile Analysis for Crystallite Size......Page 397 12.3 Retained Austenite......Page 404 References......Page 407 13.1 Introduction......Page 410 13.2.1 Working Principle......Page 411 13.2.2 Advantages of Scanning Line Detector......Page 413 13.3.1 The Third Dimension of a Detector......Page 415 13.3.2 Geometry of Three-Dimensional Detector......Page 416 13.3.3 Three-Dimensional Detector and Reciprocal Space......Page 418 13.4.1 Concept......Page 419 13.4.2 Pixel Diffraction Vector and Pixel Count......Page 420 13.4.3 PDD Analysis in Phase-ID, Texture, and Stress......Page 421 References......Page 423 Appendix A. Values of Commonly Used Parameters......Page 424 Appendix B. Symbols......Page 429 Index......Page 436

The fundamentals, theory, and wide-ranging applications of two-dimensional X-ray diffraction

Two-Dimensional X-Ray Diffraction is proving itself as an ideal non-destructive, analytical method for measuring the atomic arrangement of materials and extracting an array of information beyond the limitations of conventional X-ray diffraction. Researchers in materials science, chemistry, physics, pharmaceuticals, and related fields will find this introductory reference invaluable in understanding and applying two-dimensional X-ray diffraction for examining a broad range of samples.

Two-Dimensional X-Ray Diffraction shows how two-dimensional X-ray diffraction can be a useful tool for the examination of metals, polymers, ceramics, semiconductors, thin films, coatings, paints, biomaterials and composites for material science researches, molecular structure determination and polymorphism study for drug discovery and processing, and samples with micro volume or micro-area for forensic analysis, and archaeology analysis, to name just a few of the method's applications.

The text covers:

• The fundamentals of X-ray diffraction and its extension to two-dimensional X-ray diffraction
• The geometry conventions and diffraction vector approach for diffraction data interpretation, data correction, and process algorithms for various applications
• Instrumentation technologies, including the critical components, such as X-ray source and optics, two-dimensional detectors, goniometer, and sample stages
• The configurations of the two-dimensional X-ray diffraction systems for various applications, such as phase identification, texture, stress, microstructure analysis, crystallinity, thin film analysis, and combinatorial screening
• Experimental examples in materials research, pharmaceuticals, materials processing, and quality control

Written by one of the pioneers in the field, Two-Dimensional X-Ray Diffraction brings readers up to speed on a fast-rising, state-of-the-art method for materials characterization.

"Two-Dimensional X-Ray Diffraction shows how two-dimensional X-ray diffraction can be a useful tool for the examination of metals, polymers, semiconductors, thin films, coatings, paints, biomaterials and composites for material science researches, molecular structure determination and polymorphism study for drug discovery and processing, and samples with micro volume or micro-area for forensic analysis, and archaeology analysis, to name just a few of the method's applications. Researchers in materials science, chemistry, physics, pharmaceuticals, and related fields will find this introductory reference invaluable in understanding and applying two-dimensional X-ray diffraction for examining a broad range of samples."--Jacket