Chapter 1. Principles of Powder Diffraction
Robert E. Dinnebier and Simon J.L. Billinge
1.1. Introduction ............................................... 1
1.2. Fundamentals ............................................... 1
1.3. Derivation of the Bragg Equation ........................... 3
1.4. The Bragg Equation in the Reciprocal Lattice ............... 6
1.5. The Ewald Construction .................................... 11
1.6. Taking Derivatives of the Bragg Equation .................. 15
1.7. Bragg's Law for Finite Size Crystallites .................. 17
Bibliography ................................................... 19
Chapter 2. Experimental Setups
Jeremy Karl Cockcroft and Andrew N. Fitch
2.1. Introduction .............................................. 20
2.2. Sources of X-ray Radiation ................................ 21
2.2.1. Laboratory X-ray Sources ........................... 21
2.2.2. Synchrotron X-ray Sources .......................... 25
2.3. X-ray Optics .............................................. 29
2.3.1. Filters ............................................ 29
2.3.2. Monochromators ..................................... 29
2.3.3. Mirrors ............................................ 30
2.4. X-ray Detectors ........................................... 31
2.4.1. Point Detectors .................................... 31
2.4.2. Linear Detectors ................................... 31
2.4.3. Area Detectors ..................................... 32
2.4.4. Detector Calibration ............................... 33
2.5. Laboratory Instrumental Configurations .................... 33
2.5.1. Reflection Geometry ................................ 33
2.5.2. Transmission Geometry .............................. 36
2.6. Synchrotron Instrumental Configurations ................... 37
2.6.1. Pre-sample Optics .................................. 37
2.6.2. Parallel-beam Instruments .......................... 38
2.6.3. Debye-Scherrer Geometry Instruments ................ 40
2.7. Measurements .............................................. 41
2.7.1. Sample Holders ..................................... 41
2.7.2. Standard Samples ................................... 43
2.7.3. Data Acquisition ................................... 44
2.8. Energy Dispersive Powder X-ray Diffraction ................ 45
2.9. Powder Neutron Diffraction ................................ 46
2.9.1. Properties of the Neutron .......................... 46
2.9.2. Sources of Neutrons ................................ 48
2.9.3. Detection of Neutrons .............................. 49
2.9.4. Monochromatic Techniques ........................... 50
2.9.5. Time-of-Flight Techniques .......................... 53
References ................................................ 56
Chapter 3. The Intensity of a Bragg Reflection
R.B. Von Dreele and J. Rodriguez-Carvajal
3.1. Introduction .............................................. 58
3.2. Single Atom Scattering Theory ............................. 58
3.2.1. X-ray Scattering ................................... 58
3.2.2. Neutron Scattering ................................. 62
3.3. Scattering from a Crystal Lattice ......................... 63
3.3.1. Thermal Motion Effects ............................. 65
3.3.2. The Lorentz Factor ................................. 66
3.3.3. Scattering from a Modulated Crystal Lattice ........ 67
3.3.4. Neutron Magnetic Moment Scattering ................. 71
3.4. Scattering from a Polycrystalline Powder .................. 83
3.4.1. Friedel Pair Overlap ............................... 84
3.4.2. Reflection Multiplicity ............................ 84
3.4.3. Texture Effects .................................... 84
3.4.4. Absorption Effects ................................. 86
Acknowledgements ............................................... 87
References ..................................................... 87
Chapter 4. General Data Reduction
Rudolf Allmann
4.1. Introduction .............................................. 89
4.2. Elimination of Fake Reflections (Outliers) ................ 90
4.3. Fitting and Subtraction of Background ..................... 91
4.4. Data Smoothing ............................................ 93
4.4.1. Smoothing by Sliding Polynomials (Savitzky-Golay
Method) ............................................ 93
4.4.2. Digital Low Pass Filters ........................... 96
4.5. Ka2-Stripping ............................................ 100
4.6. Peak Search Algorithms ................................... 105
4.6.1. Trend-oriented Peak Search ........................ 105
4.6.2. Peak Search by Second Derivatives ................. 107
4.6.3. Peak Search with a Predefined Peak Shape .......... 110
4.7. Profile Fitting and Profile Shape Functions .............. 111
4.8. Detection and Correction of Systematic Errors ............ 119
4.8.1. External Standards ................................ 126
4.8.2. Internal Standards ................................ 127
4.8.3. Correction Together with the Refinement
of Lattice Constants .............................. 130
References .................................................... 131
Chapter 5. The Profile of a Bragg Reflection for Extracting
Intensities
Armel Le Bail
5.1. Introduction ............................................. 134
5.2. Overview of Contributions to the Peak
Profile Function ......................................... 135
5.3. Instrumental Aberrations ................................. 136
5.3.1. Largest Size Effect Ever Detected ................. 137
5.3.2. Monte Carlo Ray-tracing ........................... 138
5.4. Sample Broadening ........................................ 141
5.4.1. Crystallite Size .................................. 142
5.4.2. Lattice Strain .................................... 146
5.4.3. Anisotropic Sample Broadening: Faulting ........... 148
5.5. Individual Peak Fitting and Line Profile Analysis ........ 151
5.5.1. Peak Fitting for Intensity/Position Extraction-
With or without Cell Knowledge .................... 152
5.5.2. Using Individual Peaks for Size/Distortion
Extraction ........................................ 152
5.5.3. Further Approximations ............................ 152
5.6. Whole Powder Pattern Decomposition (WPPD) -
No Structure ............................................. 153
5.6.1. No Cell Restraint ................................. 153
5.6.2. Cell-restrained Whole Powder Pattern
Decomposition ..................................... 153
5.6.3. Main Applications of WPPD ......................... 156
5.7. Conclusions .............................................. 158
References .................................................... 159
Chapter 6. Instrumental Contributions to the Line Profile
in X-Ray Powder Diffraction. Example of the
Diffractometer with Bragg-Brentano Geometry
Alexander Zuev
6.1. Introduction ............................................. 166
6.2. Contributions to the Observed Profile .................... 169
6.3. General Description of the Method ........................ 171
6.4. Basic Equations .......................................... 173
6.4.1. Vector Equation of a Cone ......................... 173
6.4.2. Equation of a Conic ............................... 173
6.5. Diffractometer with Bragg-Brentano Geometry .............. 175
6.5.1. Coordinate Systems for Bragg-Brentano Geometry .... 175
6.5.2. Equation of a Conic in the Receiving Slit Plane
(Coordinate System CS) ............................ 176
6.5.3. Equation of a Conic in the Sample Surface Plane
(Coordinate System CS) ............................ 177
6.5.4. Case of the Degenerated Cone (2θ = 90°) ........... 177
6.5.5. Intersections of the Conic and Receiving Slit
Boundary .......................................... 178
6.5.6. Angle Between Two Planes .......................... 178
6.6. Application of the Method ................................ 179
6.6.1. Some Illustrative Examples of the Conic
in the Receiving Slit Plane ....................... 179
6.6.2. Specific Instrumental Function .................... 182
6.6.3. Total Instrumental Profile ........................ 192
6.7. About Misalignment, Soller Slits, Monochromator .......... 194
6.7.1. Misalignment ...................................... 194
6.7.2. Soller Slits ...................................... 194
6.7.3. Monochromator ..................................... 196
6.8. Plane Crystal Monochromator in the Diffracted
Beam ..................................................... 197
6.8.1. Setting of the Monochromator ...................... 197
6.8.2. Reflection Cones .................................. 198
6.8.3. Intersection of the Diffraction and Reflection
Conies in the Receiving Slit Plane ................ 199
6.9. Effect of the Plane Monochromator on Instrumental
Function ................................................. 200
6.9.1. Equatorial Aberration in the Presence
of the Monochromator .............................. 200
6.9.2. Axial Aberration in the Presence of the
Monochromator ..................................... 201
6.9.3. Total Instrumental Function in the Presence
of the Monochromator .............................. 201
6.10.Conclusions .............................................. 201
Acknowledgements .............................................. 203
References .................................................... 203
Chapter 7. Indexing and Space Group Determination
Angela Altomare, Carmelo Giacovazzo and Anna
Moliterni
7.1. The Crystalline Lattice in Powder Diffraction ............ 206
7.2. Indexing of a Powder Pattern ............................. 211
7.2.1. Introduction ...................................... 211
7.2.2. Figures of Merit .................................. 213
7.2.3. Geometrical Ambiguities ........................... 214
7.2.4. Historical Indexing Programs ...................... 214
7.2.5. Evolved Indexing Programs ......................... 217
7.3. Space Group Determination ................................ 220
7.3.1. Introduction ...................................... 220
7.3.2. The DASH Procedure ................................ 221
7.3.3. The EXPO2004 Procedure ............................ 222
References .................................................... 225
Chapter 8. Crystal Structure Determination
Rocco Caliandro, Carmelo Giacovazzo and
Rosanna Rizzi
8.1. Introduction ............................................. 227
8.2. The Patterson Function ................................... 228
8.3. Direct Methods ........................................... 230
8.3.1. Scaling of the Observed Intensities
and Normalization of the Structure Factors ........ 232
8.3.2. Estimate of Structure Invariants .................. 233
8.3.3. Tangent Formula ................................... 238
8.3.4. A Typical Direct Methods Procedure ................ 239
8.3.5. Figure of Merit ................................... 239
8.3.6. Completion of the Crystal Structure and
Preliminary Refinement ............................ 240
8.3.7. Solving Crystal Structures from Powder
Neutron Data ...................................... 242
8.4. Direct-space Techniques .................................. 243
8.4.1. Grid Search Methods ............................... 245
8.4.2. Monte Carlo Methods ............................... 245
8.4.3. Simulated Annealing Techniques .................... 249
8.4.4. Genetic Algorithm Techniques ...................... 252
8.4.5. Hybrid Approaches ................................. 254
8.4.6. Application to Real Structures .................... 257
8.4.7. Crystal Structure Prediction ...................... 258
8.5. Conclusions and Outlook .................................. 260
Symbols and Notation .......................................... 261
References .................................................... 260
Chapter 9. Rietveld Refinement
R.B. Von Dreele
9.1. Introduction ............................................. 266
9.2. Rietveld Theory .......................................... 268
9.2.1. Least Squares ..................................... 268
9.3. Constraints and Restraints ............................... 271
9.3.1. Introduction ...................................... 271
9.3.2. Rigid Body Refinement ............................. 271
9.3.3. Rigid Body Refinement.
of Fe[OP(C6H5)3]4Cl2FeCl4 .......................... 274
9.3.4. Stereochemical Restraint Refinement ............... 277
9.3.5. Protein Powder Refinements ........................ 279
Acknowledgement ............................................... 280
References .................................................... 280
Chapter 10.The Derivative Difference Minimization Method
Leonid A. Solovyov
10.1.Introduction ............................................. 282
10.2.Derivative Difference Minimization Principle ............. 283
10.3.DDM Decomposition Procedure .............................. 285
10.4.Results and Discussion ................................... 288
10.4.1. Tests on Simulated and Real Data ................. 288
10.4.2. Applications of DDM .............................. 291
10.5.Conclusions .............................................. 295
References .................................................... 295
Chapter 11.Quantitative Phase Analysis
Ian C. Madsen and Nicola V.Y. Scarlett
11.1.Introduction ............................................. 298
11.2.Phase Analysis ........................................... 299
11.3.Mathematical Basis ....................................... 300
11.3.1.Reference Intensity Ratio (RIR) Methods ........... 303
11.3.2.Rietveld-based Methods ............................ 304
11.4.Factors Limiting Accuracy ................................ 308
11.4.1.Particle Statistics ............................... 308
11.4.2.Preferred Orientation ............................. 310
11.4.3.Microabsorption ................................... 312
11.4.4.Precision, Accuracy and the Calculation
of Error .......................................... 314
11.5.Examples of QPA via Powder Diffraction ................... 315
11.5.1.Application in Mineralogical Systems .............. 315
11.5.2.Applications in Industrial Systems ................ 322
11.6.Summary .................................................. 326
Acknowledgements .............................................. 326
Appendix A: Derivation of Errors in Rietveld-based
Quantitative Phase Analysis .............................. 327
Relative Phase Abundances ................................ 327
Absolute Phase Abundances ................................ 327
Amorphous Content ........................................ 328
References .................................................... 329
Chapter 12.Microstructural Properties: Texture and
Macrostress Effects
Nicolae С. Рора
12.1.Texture .................................................. 332
12.1.1.The Orientation Distribution Function and the
Pole Distributions ................................ 332
12.1.2.Two Goals in Texture Analysis ..................... 335
12.1.3.Dollase-March Model ............................... 337
12.1.4.The Spherical Harmonics Approach .................. 339
12.2.Macroscopic Strain and Stress ............................ 348
12.2.1.Elastic Strain and Stress in a Crystallite-
Mathematical Background ........................... 349
12.2.2.Strain and Stress in Polycrystalline Samples ...... 352
12.2.3.Status of the Strain/Stress Analysis by
Diffraction ....................................... 355
12.2.4.Strain/Stress in Isotropic Samples - Classical
Approximations .................................... 357
12.2.5.Hydrostatic Pressure in Isotropic Polycrystals .... 363
12.2.6.The Macroscopic Strain/Stress by Spherical
Harmonics ......................................... 365
References .................................................... 373
Chapter 13.Microstructural Properties: Lattice Defects and
Domain Size Effects
Paolo Scardi
13.1.Introduction ............................................. 376
13.2.Origin of Line Broadening ................................ 377
13.2.1.Size Broadening ................................... 377
13.2.2.Strain Broadening ................................. 381
13.2.3.Other Sources of Line Broadening .................. 384
13.3.Traditional versus Innovative Methods .................... 387
13.3.1.Integral Breadth Methods .......................... 387
13.3.2.Fourier Methods ................................... 389
13.3.3.Profile Fitting and Traditional LPA Methods ....... 394
13.3.4.Whole Powder Pattern Modelling .................... 395
13.4.WPPM: Examples of Application ............................ 396
13.4.1.Heavily Deformed Metal Powders .................... 396
13.4.2.Nanocrystalline Cerium Oxide Powder ............... 402
Acknowledgements .............................................. 405
List of Principal Symbols ..................................... 405
Appendix: Fourier Transforms of Profile Components ............ 407
Instrumental Profile (IP) ................................ 407
Domain Size (S) .......................................... 407
Faulting (F) ............................................. 408
Dislocations (D) ......................................... 408
Anti-phase Domain Boundaries (APB) ....................... 410
Stoichiometry Fluctuation (C) ............................ 410
References .................................................... 411
Chapter 14.Two-dimensional Diffraction Using Area Detectors
Bernd Hinrichsen, Robert E.Dinnebier and
Martin Jansen
14.1.Two-dimensional Detectors ................................ 414
14.1.1.CCD Detectors ..................................... 415
14.1.2.Imaging Plate Detectors ........................... 416
14.1.3.Flat Panel Detectors .............................. 416
14.1.4.Hybrid Pixel Detectors ............................ 417
14.2.Diffraction Geometry ..................................... 418
14.2.1.Resolution and FWHM in Two-dimensional
Diffraction ....................................... 419
14.2.2.Diffraction Angle Transformation .................. 422
14.2.3.Incident Angle and Ray Distance Calculations ...... 426
14.2.4.General Transformations ........................... 426
14.3.Intensity Corrections .................................... 429
14.3.1.Lorentz Corrections ............................... 430
14.3.2.Polarization Correction ........................... 434
14.3.3.Incident Angle Correction ......................... 435
References .................................................... 437
Chapter 15.Powder Diffraction under Non-ambient
Conditions
Poul Nor by and Ulrich Schwarz
15.1.Introduction ............................................. 439
15.2.In Situ Powder Diffraction ............................... 440
15.2.1.Techniques and Instrumentation .................... 442
15.3.Powder Diffraction at High Pressure ...................... 450
15.3.1.Introduction ...................................... 450
15.3.2.The Diamond Anvil Cell ............................ 451
15.3.3.Pressure Media .................................... 453
15.3.4.Diffraction Measurements .......................... 454
15.3.5.Pressure Measurement .............................. 457
15.3.6.Thermodynamic Considerations ...................... 459
Selected Reviews .............................................. 461
In-situ diffraction ...................................... 461
High-pressure Diffraction ................................ 461
References .................................................... 462
Chapter 16.Local Structure from Total Scattering and
Atomic Pair Distribution Function (PDF) Analysis
Simon Billinge
16.1.Introduction ............................................. 464
16.2.Theory ................................................... 470
16.2.1.Single Component Systems .......................... 470
16.2.2.Multicomponent Systems ............................ 473
16.3.Experimental Methods ..................................... 479
16.4.Structural Modeling ...................................... 481
16.4.1.Model Independent Structural Information from
the PDF ........................................... 481
16.4.2.Modeling the PDF .................................. 482
16.4.3.Modeling Total Scattering in Reciprocal Space ..... 485
16.4.4.Emerging Modeling Approaches ...................... 486
References .................................................... 491
Chapter 17.Computer Software for Powder Diffraction
Lachlan M.D. Cranswick
17.1.Introduction ............................................. 494
17.2.Finding and Testing Software ............................. 494
17.2.1.Locating New Software ............................. 494
17.2.2.Selecting Software ................................ 495
17.2.3.Re-locating Software on the Internet .............. 495
17.3.Available Software ....................................... 495
17.3.1.Third-party Diffractometer Control Software ....... 495
17.3.2.Phase Identification and Search-match Software .... 496
17.3.3.Crystal Structure Databases ....................... 498
17.3.4.Powder Data Conversion ............................ 500
17.3.5.Structure Data Conversion and Transformation ...... 503
17.3.6.Powder Diffraction Pattern Viewing and
Processing ........................................ 504
17.3.7.Peak Finding and Peak Profiling ................... 510
17.3.8.Powder Indexing ................................... 510
17.3.9.Space Group Assignment ............................ 521
17.3.10.Space Group Information Software and Databases ... 521
17.3.11.Unit Cell Refinement ............................. 522
17.3.12.Full Profile Fitting (Pawley, Le Bail) ........... 523
17.3.13.Texture Analysis Software ........................ 528
17.3.14.Size Strain Analysis ............................. 528
17.3.15.Single Crystal Suites useful to Powder
Diffraction ....................................... 530
17.3.16.Powder Diffraction Suites ........................ 531
17.3.17.Structure Solution Software Specifically
for Powder Diffraction ............................ 531
17.3.18.Structure Solution Using Single Crystal
Software .......................................... 534
17.3.19.2D to 3D Molecular Model Generation .............. 534
17.3.20.Single Crystal Refinement Programs and Helper
Programs to Assist in Building up
the Structure .................................... 538
17.3.21.Rietveld Structure Refinement .................... 541
17.3.22.Pair Distribution Function Software .............. 541
17.3.23.Hydrogen Placement Using Single Crystal and
Ancillary Software ............................... 541
17.3.24.Free Standing Powder and Single Crystal Fourier
Map Generation and Display Software .............. 541
17.3.25.Quantitative Phase Analysis ...................... 548
17.3.26.Powder Pattern Calculation ....................... 548
17.3.27.Structure Validation ............................. 548
17.3.28.Crystallographic Structure Visualization:
During Structure Solution and Refinement .......... 554
17.3.29.Visualization and Photo Realistic Rendering
of Crystal Structures ............................. 555
17.3.30.Miscellaneous Resources .......................... 562
Appendix 1: Internet links for Cited Software and Resources ... 562
Subject Index ................................................. 571
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