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Summary
Summary
Powder diffraction is a widely used scientific technique in the characterization of materials with broad application in materials science, chemistry, physics, geology, pharmacology and archaeology. Powder Diffraction: Theory and Practice provides an advanced introductory text about modern methods and applications of powder diffraction in research and industry. The authors begin with a brief overview of the basic theory of diffraction from crystals and powders. Data collection strategies are described including x-ray, neutron and electron diffraction setups using modern day apparatus including synchrotron sources. Data corrections, essential for quantitative analysis are covered before the authors conclude with a discussion of the analysis methods themselves. The information is presented in a way that facilitates understanding the information content of the data, as well as best practices for collecting and analyzing data for quantitative analysis. This long awaited book condenses the knowledge of renowned experts in the field into a single, authoritative, overview of the application of powder diffraction in modern materials research. The book contains essential theory and introductory material for students and researchers wishing to learn how to apply the frontier methods of powder diffraction
Table of Contents
Chapter 1 Principles of Powder DiffractionRobert E. Dinnebier and Simon J. L. Billinge | |
1.1 Introduction | p. 1 |
1.2 Fundamentals | p. 1 |
1.3 Derivation of the Bragg Equation | p. 3 |
1.4 The Bragg Equation in the Reciprocal Lattice | p. 6 |
1.5 The Ewald Construction | p. 11 |
1.6 Taking Derivatives of the Bragg Equation | p. 15 |
1.7 Bragg's Law for Finite Size Crystallites | p. 17 |
Bibliography | p. 19 |
Chapter 2 Experimental SetupsJeremy Karl Cockcroft and Andrew N. Fitch | |
2.1 Introduction | p. 20 |
2.2 Sources of X-ray Radiation | p. 21 |
2.2.1 Laboratory X-ray Sources | p. 21 |
2.2.2 Synchrotron X-ray Sources | p. 25 |
2.3 X-ray Optics | p. 29 |
2.3.1 Filters | p. 29 |
2.3.2 Monochromators | p. 29 |
2.3.3 Mirrors | p. 30 |
2.4 X-ray Detectors | p. 31 |
2.4.1 Point Detectors | p. 31 |
2.4.2 Linear Detectors | p. 31 |
2.4.3 Area Detectors | p. 32 |
2.4.4 Detector Calibration | p. 33 |
2.5 Laboratory Instrumental Configurations | p. 33 |
2.5.1 Reflection Geometry | p. 33 |
2.5.2 Transmission Geometry | p. 36 |
2.6 Synchrotron Instrumental Configurations | p. 37 |
2.6.1 Pre-sample Optics | p. 37 |
2.6.2 Parallel-beam Instruments | p. 38 |
2.6.3 Debye-Scherrer Geometry Instruments | p. 40 |
2.7 Measurements | p. 41 |
2.7.1 Sample Holders | p. 41 |
2.7.2 Standard Samples | p. 43 |
2.7.3 Data Acquisition | p. 44 |
2.8 Energy Dispersive Powder X-ray Diffraction | p. 45 |
2.9 Powder Neutron Diffraction | p. 46 |
2.9.1 Properties of the Neutron | p. 46 |
2.9.2 Sources of Neutrons | p. 48 |
2.9.3 Detection of Neutrons | p. 49 |
2.9.4 Monochromatic Techniques | p. 50 |
2.9.5 Time-of-Flight Techniques | p. 53 |
References | p. 56 |
Chapter 3 The Intensity of a Bragg ReflectionR. B. Von Dreele and J. Rodriguez-Carvajal | |
3.1 Introduction | p. 58 |
3.2 Single Atom Scattering Theory | p. 58 |
3.2.1 X-ray Scattering | p. 58 |
3.2.2 Neutron Scattering | p. 62 |
3.3 Scattering from a Crystal Lattice | p. 63 |
3.3.1 Thermal Motion Effects | p. 65 |
3.3.2 The Lorentz Factor | p. 66 |
3.3.3 Scattering from a Modulated Crystal Lattice | p. 67 |
3.3.4 Neutron Magnetic Moment Scattering | p. 71 |
3.4 Scattering from a Polycrystalline Powder | p. 83 |
3.4.1 Friedel Pair Overlap | p. 84 |
3.4.2 Reflection Multiplicity | p. 84 |
3.4.3 Texture Effects | p. 84 |
3.4.4 Absorption Effects | p. 86 |
Acknowledgements | p. 87 |
References | p. 87 |
Chapter 4 General Data ReductionRudolf Allmann | |
4.1 Introduction | p. 89 |
4.2 Elimination of Fake Reflections (Outliers) | p. 90 |
4.3 Fitting and Subtraction of Background | p. 91 |
4.4 Data Smoothing | p. 93 |
4.4.1 Smoothing by Sliding Polynomials (Savitzky-Golay Method) | p. 93 |
4.4.2 Digital Low Pass Filters | p. 96 |
4.5 K[alpha subscript 2]-Stripping | p. 100 |
4.6 Peak Search Algorithms | p. 105 |
4.6.1 Trend-oriented Peak Search | p. 105 |
4.6.2 Peak Search by Second Derivatives | p. 107 |
4.6.3 Peak Search with a Predefined Peak Shape | p. 110 |
4.7 Profile Fitting and Profile Shape Functions | p. 111 |
4.8 Detection and Correction of Systematic Errors | p. 119 |
4.8.1 External Standards | p. 126 |
4.8.2 Internal Standards | p. 127 |
4.8.3 Correction Together with the Refinement of Lattice Constants | p. 130 |
References | p. 131 |
Chapter 5 The Profile of a Bragg Reflection for Extracting IntensitiesArmel Le Bail | |
5.1 Introduction | p. 134 |
5.2 Overview of Contributions to the Peak Profile Function | p. 135 |
5.3 Instrumental Aberrations | p. 136 |
5.3.1 Largest Size Effect Ever Detected | p. 137 |
5.3.2 Monte Carlo Ray-tracing | p. 138 |
5.4 Sample Broadening | p. 141 |
5.4.1 Crystallite Size | p. 142 |
5.4.2 Lattice Strain | p. 146 |
5.4.3 Anisotropic Sample Broadening: Faulting | p. 148 |
5.5 Individual Peak Fitting and Line Profile Analysis | p. 151 |
5.5.1 Peak Fitting for Intensity/Position Extraction - With or without Cell Knowledge | p. 152 |
5.5.2 Using Individual Peaks for Size/Distortion Extraction | p. 152 |
5.5.3 Further Approximations | p. 152 |
5.6 Whole Powder Pattern Decomposition (WPPD) - No Structure | p. 153 |
5.6.1 No Cell Restraint | p. 153 |
5.6.2 Cell-restrained Whole Powder Pattern Decomposition | p. 153 |
5.6.3 Main Applications of WPPD | p. 156 |
5.7 Conclusions | p. 158 |
References | p. 159 |
Chapter 6 Instrumental Contributions to the Line Profile in X-Ray Powder Diffraction. Example of the Diffractometer with Bragg-Brentano GeometryAlexander Zuev | |
6.1 Introduction | p. 166 |
6.2 Contributions to the Observed Profile | p. 169 |
6.3 General Description of the Method | p. 171 |
6.4 Basic Equations | p. 173 |
6.4.1 Vector Equation of a Cone | p. 173 |
6.4.2 Equation of a Conic | p. 173 |
6.5 Diffractometer with Bragg-Brentano Geometry | p. 175 |
6.5.1 Coordinate Systems for Bragg-Brentano Geometry | p. 175 |
6.5.2 Equation of a Conic in the Receiving Slit Plane (Coordinate System CS) | p. 176 |
6.5.3 Equation of a Conic in the Sample Surface Plane (Coordinate System CS) | p. 177 |
6.5.4 Case of the Degenerated Cone (2[theta] = 90[degree]) | p. 177 |
6.5.5 Intersections of the Conic and Receiving Slit Boundary | p. 178 |
6.5.6 Angle Between Two Planes | p. 178 |
6.6 Application of the Method | p. 179 |
6.6.1 Some Illustrative Examples of the Conic in the Receiving Slit Plane | p. 179 |
6.6.2 Specific Instrumental Function | p. 182 |
6.6.3 Total Instrumental Profile | p. 192 |
6.7 About Misalignment, Soller Slits, Monochromator | p. 194 |
6.7.1 Misalignment | p. 194 |
6.7.2 Soller Slits | p. 194 |
6.7.3 Monochromator | p. 196 |
6.8 Plane Crystal Monochromator in the Diffracted Beam | p. 197 |
6.8.1 Setting of the Monochromator | p. 197 |
6.8.2 Reflection Cones | p. 198 |
6.8.3 Intersection of the Diffraction and Reflection Conics in the Receiving Slit Plane | p. 199 |
6.9 Effect of the Plane Monochromator on Instrumental Function | p. 200 |
6.9.1 Equatorial Aberration in the Presence of the Monochromator | p. 200 |
6.9.2 Axial Aberration in the Presence of the Monochromator | p. 201 |
6.9.3 Total Instrumental Function in the Presence of the Monochromator | p. 201 |
6.10 Conclusions | p. 201 |
Acknowledgements | p. 203 |
References | p. 203 |
Chapter 7 Indexing and Space Group DeterminationAngela Altomare and Carmelo Giacovazzo and Anna Moliterni | |
7.1 The Crystalline Lattice in Powder Diffraction | p. 206 |
7.2 Indexing of a Powder Pattern | p. 211 |
7.2.1 Introduction | p. 211 |
7.2.2 Figures of Merit | p. 213 |
7.2.3 Geometrical Ambiguities | p. 214 |
7.2.4 Historical Indexing Programs | p. 214 |
7.2.5 Evolved Indexing Programs | p. 217 |
7.3 Space Group Determination | p. 220 |
7.3.1 Introduction | p. 220 |
7.3.2 The DASH Procedure | p. 221 |
7.3.3 The EXPO2004 Procedure | p. 222 |
References | p. 225 |
Chapter 8 Crystal Structure DeterminationRocco Caliandro and Carmelo Giacovazzo and Rosanna Rizzi | |
8.1 Introduction | p. 227 |
8.2 The Patterson Function | p. 228 |
8.3 Direct Methods | p. 230 |
8.3.1 Scaling of the Observed Intensities and Normalization of the Structure Factors | p. 232 |
8.3.2 Estimate of Structure Invariants | p. 233 |
8.3.3 Tangent Formula | p. 238 |
8.3.4 A Typical Direct Methods Procedure | p. 239 |
8.3.5 Figure of Merit | p. 239 |
8.3.6 Completion of the Crystal Structure and Preliminary Refinement | p. 240 |
8.3.7 Solving Crystal Structures from Powder Neutron Data | p. 242 |
8.4 Direct-space Techniques | p. 243 |
8.4.1 Grid Search Methods | p. 245 |
8.4.2 Monte Carlo Methods | p. 245 |
8.4.3 Simulated Annealing Techniques | p. 249 |
8.4.4 Genetic Algorithm Techniques | p. 252 |
8.4.5 Hybrid Approaches | p. 254 |
8.4.6 Application to Real Structures | p. 257 |
8.4.7 Crystal Structure Prediction | p. 258 |
8.5 Conclusions and Outlook | p. 260 |
Symbols and Notation | p. 261 |
References | p. 261 |
Chapter 9 Rietveld RefinementR. B. Von Dreele | |
9.1 Introduction | p. 266 |
9.2 Rietveld Theory | p. 268 |
9.2.1 Least Squares | p. 268 |
9.3 Constraints and Restraints | p. 271 |
9.3.1 Introduction | p. 271 |
9.3.2 Rigid Body Refinement | p. 271 |
9.3.3 Rigid Body Refinement of Fe[OP(C subscript 6 H subscript 5) subscript 3 subscript 4]Cl[subscript 2]FeCl[subscript 4] | p. 274 |
9.3.4 Stereochemical Restraint Refinement | p. 277 |
9.3.5 Protein Powder Refinements | p. 279 |
Acknowledgement | p. 280 |
References | p. 280 |
Chapter 10 The Derivative Difference Minimization MethodLeonid A. Solovyov | |
10.1 Introduction | p. 282 |
10.2 Derivative Difference Minimization Principle | p. 283 |
10.3 DDM Decomposition Procedure | p. 285 |
10.4 Results and Discussion | p. 288 |
10.4.1 Tests on Simulated and Real Data | p. 288 |
10.4.2 Applications of DDM | p. 291 |
10.5 Conclusions | p. 295 |
References | p. 295 |
Chapter 11 Quantitative Phase AnalysisIan C. Madsen and Nicola V. Y. Scarlett | |
11.1 Introduction | p. 298 |
11.2 Phase Analysis | p. 299 |
11.3 Mathematical Basis | p. 300 |
11.3.1 Reference Intensity Ratio (RIR) Methods | p. 303 |
11.3.2 Rietveld-based Methods | p. 304 |
11.4 Factors Limiting Accuracy | p. 308 |
11.4.1 Particle Statistics | p. 308 |
11.4.2 Preferred Orientation | p. 310 |
11.4.3 Microabsorption | p. 312 |
11.4.4 Precision, Accuracy and the Calculation of Error | p. 314 |
11.5 Examples of QPA via Powder Diffraction | p. 315 |
11.5.1 Application in Mineralogical Systems | p. 315 |
11.5.2 Applications in Industrial Systems | p. 322 |
11.6 Summary | p. 326 |
Acknowledgements | p. 326 |
Appendix A Derivation of Errors in Rietveld-based Quantitative Phase Analysis | p. 327 |
Relative Phase Abundances | p. 327 |
Absolute Phase Abundances | p. 327 |
Amorphous Content | p. 328 |
References | p. 329 |
Chapter 12 Microstructural Properties: Texture and Macrostress EffectsNicolae C. Popa | |
12.1 Texture | p. 332 |
12.1.1 The Orientation Distribution Function and the Pole Distributions | p. 332 |
12.1.2 Two Goals in Texture Analysis | p. 335 |
12.1.3 Dollase-March Model | p. 337 |
12.1.4 The Spherical Harmonics Approach | p. 339 |
12.2 Macroscopic Strain and Stress | p. 348 |
12.2.1 Elastic Strain and Stress in a Crystallite - Mathematical Background | p. 349 |
12.2.2 Strain and Stress in Polycrystalline Samples | p. 352 |
12.2.3 Status of the Strain/Stress Analysis by Diffraction | p. 355 |
12.2.4 Strain/Stress in Isotropic Samples - Classical Approximations | p. 357 |
12.2.5 Hydrostatic Pressure in Isotropic Polycrystals | p. 363 |
12.2.6 The Macroscopic Strain/Stress by Spherical Harmonics | p. 365 |
References | p. 373 |
Chapter 13 Microstructural Properties: Lattice Defects and Domain Size EffectsPaolo Scardi | |
13.1 Introduction | p. 376 |
13.2 Origin of Line Broadening | p. 377 |
13.2.1 Size Broadening | p. 377 |
13.2.2 Strain Broadening | p. 381 |
13.2.3 Other Sources of Line Broadening | p. 384 |
13.3 Traditional versus Innovative Methods | p. 387 |
13.3.1 Integral Breadth Methods | p. 387 |
13.3.2 Fourier Methods | p. 389 |
13.3.3 Profile Fitting and Traditional LPA Methods | p. 394 |
13.3.4 Whole Powder Pattern Modelling | p. 395 |
13.4 WPPM: Examples of Application | p. 396 |
13.4.1 Heavily Deformed Metal Powders | p. 396 |
13.4.2 Nanocrystalline Cerium Oxide Powder | p. 402 |
Acknowledgements | p. 405 |
List of Principal Symbols | p. 405 |
Appendix Fourier Transforms of Profile Components | p. 407 |
Instrumental Profile (IP) | p. 407 |
Domain Size (S) | p. 407 |
Faulting (F) | p. 408 |
Dislocations (D) | p. 408 |
Anti-phase Domain Boundaries (APB) | p. 410 |
Stoichiometry Fluctuation (C) | p. 410 |
References | p. 411 |
Chapter 14 Two-dimensional Diffraction Using Area DetectorsBernd Hinrichsen and Robert E. Dinnebier and Martin Jansen | |
14.1 Two-dimensional Detectors | p. 414 |
14.1.1 CCD Detectors | p. 415 |
14.1.2 Imaging Plate Detectors | p. 416 |
14.1.3 Flat Panel Detectors | p. 416 |
14.1.4 Hybrid Pixel Detectors | p. 417 |
14.2 Diffraction Geometry | p. 418 |
14.2.1 Resolution and FWHM in Two-dimensional Diffraction | p. 419 |
14.2.2 Diffraction Angle Transformation | p. 422 |
14.2.3 Incident Angle and Ray Distance Calculations | p. 426 |
14.2.4 General Transformations | p. 426 |
14.3 Intensity Corrections | p. 429 |
14.3.1 Lorentz Corrections | p. 430 |
14.3.2 Polarization Correction | p. 434 |
14.3.3 Incident Angle Correction | p. 435 |
References | p. 437 |
Chapter 15 Powder Diffraction under Non-ambient ConditionsPoul Norby and Ulrich Schwarz | |
15.1 Introduction | p. 439 |
15.2 In Situ Powder Diffraction | p. 440 |
15.2.1 Techniques and Instrumentation | p. 442 |
15.3 Powder Diffraction at High Pressure | p. 450 |
15.3.1 Introduction | p. 450 |
15.3.2 The Diamond Anvil Cell | p. 451 |
15.3.3 Pressure Media | p. 453 |
15.3.4 Diffraction Measurements | p. 454 |
15.3.5 Pressure Measurement | p. 457 |
15.3.6 Thermodynamic Considerations | p. 459 |
Selected Reviews | p. 461 |
In-situ diffraction | p. 461 |
High-pressure Diffraction | p. 461 |
References | p. 462 |
Chapter 16 Local Structure from Total Scattering and Atomic Pair Distribution Function (PDF) AnalysisSimon Billinge | |
16.1 Introduction | p. 464 |
16.2 Theory | p. 470 |
16.2.1 Single Component Systems | p. 470 |
16.2.2 Multicomponent Systems | p. 473 |
16.3 Experimental Methods | p. 479 |
16.4 Structural Modeling | p. 481 |
16.4.1 Model Independent Structural Information from the PDF | p. 481 |
16.4.2 Modeling the PDF | p. 482 |
16.4.3 Modeling Total Scattering in Reciprocal Space | p. 485 |
16.4.4 Emerging Modeling Approaches | p. 486 |
References | p. 491 |
Chapter 17 Computer Software for Powder DiffractionLachlan M. D. Cranswick | |
17.1 Introduction | p. 494 |
17.2 Finding and Testing Software | p. 494 |
17.2.1 Locating New Software | p. 494 |
17.2.2 Selecting Software | p. 495 |
17.2.3 Re-locating Software on the Internet | p. 495 |
17.3 Available Software | p. 495 |
17.3.1 Third-party Diffractometer Control Software | p. 495 |
17.3.2 Phase Identification and Search-match Software | p. 496 |
17.3.3 Crystal Structure Databases | p. 498 |
17.3.4 Powder Data Conversion | p. 500 |
17.3.5 Structure Data Conversion and Transformation | p. 503 |
17.3.6 Powder Diffraction Pattern Viewing and Processing | p. 504 |
17.3.7 Peak Finding and Peak Profiling | p. 510 |
17.3.8 Powder Indexing | p. 510 |
17.3.9 Space Group Assignment | p. 521 |
17.3.10 Space Group Information Software and Databases | p. 521 |
17.3.11 Unit Cell Refinement | p. 522 |
17.3.12 Full Profile Fitting (Pawley, Le Bail) | p. 523 |
17.3.13 Texture Analysis Software | p. 528 |
17.3.14 Size Strain Analysis | p. 528 |
17.3.15 Single Crystal Suites useful to Powder Diffraction | p. 530 |
17.3.16 Powder Diffraction Suites | p. 531 |
17.3.17 Structure Solution Software Specifically for Powder Diffraction | p. 531 |
17.3.18 Structure Solution Using Single Crystal Software | p. 534 |
17.3.19 2D to 3D Molecular Model Generation | p. 534 |
17.3.20 Single Crystal Refinement Programs and Helper Programs to Assist in Building up the Structure | p. 538 |
17.3.21 Rietveld Structure Refinement | p. 541 |
17.3.22 Pair Distribution Function Software | p. 541 |
17.3.23 Hydrogen Placement Using Single Crystal and Ancillary Software | p. 541 |
17.3.24 Free Standing Powder and Single Crystal Fourier Map Generation and Display Software | p. 541 |
17.3.25 Quantitative Phase Analysis | p. 548 |
17.3.26 Powder Pattern Calculation | p. 548 |
17.3.27 Structure Validation | p. 548 |
17.3.28 Crystallographic Structure Visualization: During Structure Solution and Refinement | p. 554 |
17.3.29 Visualization and Photo Realistic Rendering of Crystal Structures | p. 555 |
17.3.30 Miscellaneous Resources | p. 562 |
Appendix 1 Internet links for Cited Software and Resources | p. 562 |
Subject Index | p. 571 |