Name: Powder Diffraction
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ISBN: 9780854042319

Principles of Powder Diffraction
Introduction	p. 1
Fundamentals	p. 1
Derivation of the Bragg Equation	p. 3
The Bragg Equation in the Reciprocal Lattice	p. 6
The Ewald Construction	p. 11
Taking Derivatives of the Bragg Equation	p. 15
Bragg's Law for Finite Size Crystallites	p. 17
Bibliography	p. 19
Experimental Setups
Introduction	p. 20
Sources of X-ray Radiation	p. 21
Laboratory X-ray Sources	p. 21
Synchrotron X-ray Sources	p. 25
X-ray Optics	p. 29
Filters	p. 29
Monochromators	p. 29
Mirrors	p. 30
X-ray Detectors	p. 31
Point Detectors	p. 31
Linear Detectors	p. 31
Area Detectors	p. 32
Detector Calibration	p. 33
Laboratory Instrumental Configurations	p. 33
Reflection Geometry	p. 33
Transmission Geometry	p. 36
Synchrotron Instrumental Configurations	p. 37
Pre-sample Optics	p. 37
Parallel-beam Instruments	p. 38
Debye-Scherrer Geometry Instruments	p. 40
Measurements	p. 41
Sample Holders	p. 41
Standard Samples	p. 43
Data Acquisition	p. 44
Energy Dispersive Powder X-ray Diffraction	p. 45
Powder Neutron Diffraction	p. 46
Properties of the Neutron	p. 46
Sources of Neutrons	p. 48
Detection of Neutrons	p. 49
Monochromatic Techniques	p. 50
Time-of-Flight Techniques	p. 53
References	p. 56
The Intensity of a Bragg Reflection
Introduction	p. 58
Single Atom Scattering Theory	p. 58
X-ray Scattering	p. 58
Neutron Scattering	p. 62
Scattering from a Crystal Lattice	p. 63
Thermal Motion Effects	p. 65
The Lorentz Factor	p. 66
Scattering from a Modulated Crystal Lattice	p. 67
Neutron Magnetic Moment Scattering	p. 71
Scattering from a Polycrystalline Powder	p. 83
Friedel Pair Overlap	p. 84
Reflection Multiplicity	p. 84
Texture Effects	p. 84
Absorption Effects	p. 86
Acknowledgements	p. 87
References	p. 87
General Data Reduction
Introduction	p. 89
Elimination of Fake Reflections (Outliers)	p. 90
Fitting and Subtraction of Background	p. 91
Data Smoothing	p. 93
Smoothing by Sliding Polynomials (Savitzky-Golay Method)	p. 93
Digital Low Pass Filters	p. 96
K[alpha subscript 2]-Stripping	p. 100
Peak Search Algorithms	p. 105
Trend-oriented Peak Search	p. 105
Peak Search by Second Derivatives	p. 107
Peak Search with a Predefined Peak Shape	p. 110
Profile Fitting and Profile Shape Functions	p. 111
Detection and Correction of Systematic Errors	p. 119
External Standards	p. 126
Internal Standards	p. 127
Correction Together with the Refinement of Lattice Constants	p. 130
References	p. 131
The Profile of a Bragg Reflection for Extracting Intensities
Introduction	p. 134
Overview of Contributions to the Peak Profile Function	p. 135
Instrumental Aberrations	p. 136
Largest Size Effect Ever Detected	p. 137
Monte Carlo Ray-tracing	p. 138
Sample Broadening	p. 141
Crystallite Size	p. 142
Lattice Strain	p. 146
Anisotropic Sample Broadening: Faulting	p. 148
Individual Peak Fitting and Line Profile Analysis	p. 151
Peak Fitting for Intensity/Position Extraction - With or without Cell Knowledge	p. 152
Using Individual Peaks for Size/Distortion Extraction	p. 152
Further Approximations	p. 152
Whole Powder Pattern Decomposition (WPPD) - No Structure	p. 153
No Cell Restraint	p. 153
Cell-restrained Whole Powder Pattern Decomposition	p. 153
Main Applications of WPPD	p. 156
Conclusions	p. 158
References	p. 159
Instrumental Contributions to the Line Profile in X-Ray Powder Diffraction. Example of the Diffractometer with Bragg-Brentano Geometry
Introduction	p. 166
Contributions to the Observed Profile	p. 169
General Description of the Method	p. 171
Basic Equations	p. 173
Vector Equation of a Cone	p. 173
Equation of a Conic	p. 173
Diffractometer with Bragg-Brentano Geometry	p. 175
Coordinate Systems for Bragg-Brentano Geometry	p. 175
Equation of a Conic in the Receiving Slit Plane (Coordinate System CS)	p. 176
Equation of a Conic in the Sample Surface Plane (Coordinate System CS)	p. 177
Case of the Degenerated Cone (2[theta] = 90[degree])	p. 177
Intersections of the Conic and Receiving Slit Boundary	p. 178
Angle Between Two Planes	p. 178
Application of the Method	p. 179
Some Illustrative Examples of the Conic in the Receiving Slit Plane	p. 179
Specific Instrumental Function	p. 182
Total Instrumental Profile	p. 192
About Misalignment, Soller Slits, Monochromator	p. 194
Misalignment	p. 194
Soller Slits	p. 194
Monochromator	p. 196
Plane Crystal Monochromator in the Diffracted Beam	p. 197
Setting of the Monochromator	p. 197
Reflection Cones	p. 198
Intersection of the Diffraction and Reflection Conics in the Receiving Slit Plane	p. 199
Effect of the Plane Monochromator on Instrumental Function	p. 200
Equatorial Aberration in the Presence of the Monochromator	p. 200
Axial Aberration in the Presence of the Monochromator	p. 201
Total Instrumental Function in the Presence of the Monochromator	p. 201
Conclusions	p. 201
Acknowledgements	p. 203
References	p. 203
Indexing and Space Group Determination
The Crystalline Lattice in Powder Diffraction	p. 206
Indexing of a Powder Pattern	p. 211
Introduction	p. 211
Figures of Merit	p. 213
Geometrical Ambiguities	p. 214
Historical Indexing Programs	p. 214
Evolved Indexing Programs	p. 217
Space Group Determination	p. 220
Introduction	p. 220
The DASH Procedure	p. 221
The EXPO2004 Procedure	p. 222
References	p. 225
Crystal Structure Determination
Introduction	p. 227
The Patterson Function	p. 228
Direct Methods	p. 230
Scaling of the Observed Intensities and Normalization of the Structure Factors	p. 232
Estimate of Structure Invariants	p. 233
Tangent Formula	p. 238
A Typical Direct Methods Procedure	p. 239
Figure of Merit	p. 239
Completion of the Crystal Structure and Preliminary Refinement	p. 240
Solving Crystal Structures from Powder Neutron Data	p. 242
Direct-space Techniques	p. 243
Grid Search Methods	p. 245
Monte Carlo Methods	p. 245
Simulated Annealing Techniques	p. 249
Genetic Algorithm Techniques	p. 252
Hybrid Approaches	p. 254
Application to Real Structures	p. 257
Crystal Structure Prediction	p. 258
Conclusions and Outlook	p. 260
Symbols and Notation	p. 261
References	p. 261
Rietveld Refinement
Introduction	p. 266
Rietveld Theory	p. 268
Least Squares	p. 268
Constraints and Restraints	p. 271
Introduction	p. 271
Rigid Body Refinement	p. 271
Rigid Body Refinement of Fe[OP(C subscript 6 H subscript 5) subscript 3 subscript 4]Cl[subscript 2]FeCl[subscript 4]	p. 274
Stereochemical Restraint Refinement	p. 277
Protein Powder Refinements	p. 279
Acknowledgement	p. 280
References	p. 280
The Derivative Difference Minimization Method
Introduction	p. 282
Derivative Difference Minimization Principle	p. 283
DDM Decomposition Procedure	p. 285
Results and Discussion	p. 288
Tests on Simulated and Real Data	p. 288
Applications of DDM	p. 291
Conclusions	p. 295
References	p. 295
Quantitative Phase Analysis
Introduction	p. 298
Phase Analysis	p. 299
Mathematical Basis	p. 300
Reference Intensity Ratio (RIR) Methods	p. 303
Rietveld-based Methods	p. 304
Factors Limiting Accuracy	p. 308
Particle Statistics	p. 308
Preferred Orientation	p. 310
Microabsorption	p. 312
Precision, Accuracy and the Calculation of Error	p. 314
Examples of QPA via Powder Diffraction	p. 315
Application in Mineralogical Systems	p. 315
Applications in Industrial Systems	p. 322
Summary	p. 326
Acknowledgements	p. 326
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
Microstructural Properties: Texture and Macrostress Effects
Texture	p. 332
The Orientation Distribution Function and the Pole Distributions	p. 332
Two Goals in Texture Analysis	p. 335
Dollase-March Model	p. 337
The Spherical Harmonics Approach	p. 339
Macroscopic Strain and Stress	p. 348
Elastic Strain and Stress in a Crystallite - Mathematical Background	p. 349
Strain and Stress in Polycrystalline Samples	p. 352
Status of the Strain/Stress Analysis by Diffraction	p. 355
Strain/Stress in Isotropic Samples - Classical Approximations	p. 357
Hydrostatic Pressure in Isotropic Polycrystals	p. 363
The Macroscopic Strain/Stress by Spherical Harmonics	p. 365
References	p. 373
Microstructural Properties: Lattice Defects and Domain Size Effects
Introduction	p. 376
Origin of Line Broadening	p. 377
Size Broadening	p. 377
Strain Broadening	p. 381
Other Sources of Line Broadening	p. 384
Traditional versus Innovative Methods	p. 387
Integral Breadth Methods	p. 387
Fourier Methods	p. 389
Profile Fitting and Traditional LPA Methods	p. 394
Whole Powder Pattern Modelling	p. 395
WPPM: Examples of Application	p. 396
Heavily Deformed Metal Powders	p. 396
Nanocrystalline Cerium Oxide Powder	p. 402
Acknowledgements	p. 405
List of Principal Symbols	p. 405
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
Two-dimensional Diffraction Using Area Detectors
Two-dimensional Detectors	p. 414
CCD Detectors	p. 415
Imaging Plate Detectors	p. 416
Flat Panel Detectors	p. 416
Hybrid Pixel Detectors	p. 417
Diffraction Geometry	p. 418
Resolution and FWHM in Two-dimensional Diffraction	p. 419
Diffraction Angle Transformation	p. 422
Incident Angle and Ray Distance Calculations	p. 426
General Transformations	p. 426
Intensity Corrections	p. 429
Lorentz Corrections	p. 430
Polarization Correction	p. 434
Incident Angle Correction	p. 435
References	p. 437
Powder Diffraction under Non-ambient Conditions
Introduction	p. 439
In Situ Powder Diffraction	p. 440
Techniques and Instrumentation	p. 442
Powder Diffraction at High Pressure	p. 450
Introduction	p. 450
The Diamond Anvil Cell	p. 451
Pressure Media	p. 453
Diffraction Measurements	p. 454
Pressure Measurement	p. 457
Thermodynamic Considerations	p. 459
Selected Reviews	p. 461
In-situ diffraction	p. 461
High-pressure Diffraction	p. 461
References	p. 462
Local Structure from Total Scattering and Atomic Pair Distribution Function (PDF) Analysis
Introduction	p. 464
Theory	p. 470
Single Component Systems	p. 470
Multicomponent Systems	p. 473
Experimental Methods	p. 479
Structural Modeling	p. 481
Model Independent Structural Information from the PDF	p. 481
Modeling the PDF	p. 482
Modeling Total Scattering in Reciprocal Space	p. 485
Emerging Modeling Approaches	p. 486
References	p. 491
Computer Software for Powder Diffraction
Introduction	p. 494
Finding and Testing Software	p. 494
Locating New Software	p. 494
Selecting Software	p. 495
Re-locating Software on the Internet	p. 495
Available Software	p. 495
Third-party Diffractometer Control Software	p. 495
Phase Identification and Search-match Software	p. 496
Crystal Structure Databases	p. 498
Powder Data Conversion	p. 500
Structure Data Conversion and Transformation	p. 503
Powder Diffraction Pattern Viewing and Processing	p. 504
Peak Finding and Peak Profiling	p. 510
Powder Indexing	p. 510
Space Group Assignment	p. 521
Space Group Information Software and Databases	p. 521
Unit Cell Refinement	p. 522
Full Profile Fitting (Pawley, Le Bail)	p. 523
Texture Analysis Software	p. 528
Size Strain Analysis	p. 528
Single Crystal Suites useful to Powder Diffraction	p. 530
Powder Diffraction Suites	p. 531
Structure Solution Software Specifically for Powder Diffraction	p. 531
Structure Solution Using Single Crystal Software	p. 534
2D to 3D Molecular Model Generation	p. 534
Single Crystal Refinement Programs and Helper Programs to Assist in Building up the Structure	p. 538
Rietveld Structure Refinement	p. 541
Pair Distribution Function Software	p. 541
Hydrogen Placement Using Single Crystal and Ancillary Software	p. 541
Free Standing Powder and Single Crystal Fourier Map Generation and Display Software	p. 541
Quantitative Phase Analysis	p. 548
Powder Pattern Calculation	p. 548
Structure Validation	p. 548
Crystallographic Structure Visualization: During Structure Solution and Refinement	p. 554
Visualization and Photo Realistic Rendering of Crystal Structures	p. 555
Miscellaneous Resources	p. 562
Internet links for Cited Software and Resources	p. 562
Subject Index	p. 571
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Powder Diffraction

9780854042319

0854042318

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