1 Fundamentals of Crystalline State and Crystal Lattice ........ 1
1.1 Crystalline State ....................................... 2
1.2 Crystal Lattice and Unit Cell ........................... 4
1.3 Shape of the Unit Cell .................................. 7
1.4 Crystallographic Planes, Directions, and Indices ........ 8
1.4.1 Crystallographic Planes .......................... 8
1.4.2 Crystallographic Directions ..................... 11
1.5 Reciprocal Lattice ..................................... 11
1.6 Additional Reading ..................................... 14
1.7 Problems ............................................... 14
2 Finite Symmetry Elements and Crystallographic Point
Groups ...................................................... 17
2.1 Content of the Unit Cell ............................... 17
2.2 Asymmetric Part of the Unit Cell ....................... 18
2.3 Symmetry Operations and Symmetry Elements .............. 19
2.4 Finite Symmetry Elements ............................... 22
2.4.1 Onefold Rotation Axis and Center of Inversion ... 25
2.4.2 Twofold Rotation Axis and Mirror Plane .......... 26
2.4.3 Threefold Rotation Axis and Threefold
Inversion Axis .................................. 26
2.4.4 Fourfold Rotation Axis and Fourfold Inversion
Axis ............................................ 27
2.4.5 Sixfold Rotation Axis and Sixfold Inversion
Axis ............................................ 28
2.5 Interaction of Symmetry Elements ....................... 29
2.5.1 Generalization of Interactions Between Finite
Symmetry Elements ............................... 31
2.5.2 Symmetry Groups ................................. 32
2.6 Fundamentals of Group Theory ........................... 33
2.7 Crystal Systems ........................................ 35
2.8 Stereographic Projection ............................... 36
2.9 Crystallographic Point Groups .......................... 38
2.10 Laue Classes ........................................... 40
2.11 Selection of a Unit Cell and Bravais Lattices .......... 41
2.12 Additional Reading ..................................... 47
2.13 Problems ............................................... 47
3 Infinite Symmetry Elements and Crystallographic Space
Groups ...................................................... 51
3.1 Glide Planes ........................................... 51
3.2 Screw Axes ............................................. 53
3.3 Interaction of Infinite Symmetry Elements .............. 54
3.4 Crystallographic Space Groups .......................... 56
3.4.1 Relationships Between Point Groups and Space
Groups .......................................... 57
3.4.2 Full International Symbols of Crystallographic
Space Groups .................................... 60
3.4.3 Visualization of Space-Group Symmetry in Three
Dimensions ...................................... 62
3.4.4 Space Groups in Nature .......................... 63
3.5 International Tables for Crystallography ............... 63
3.6 Equivalent Positions (Sites) ........................... 70
3.6.1 General and Special Equivalent Positions ........ 70
3.6.2 Special Sites with Points Located on Mirror
Planes .......................................... 71
3.6.3 Special Sites with Points Located on Rotation
and Inversions Axes ............................. 72
3.6.4 Special Sites with Points Located on Centers
of Inversion .................................... 73
3.7 Additional Reading ..................................... 73
3.8 Problems ............................................... 73
4 Formalization of Symmetry ................................... 77
4.1 Symbolic Representation of Symmetry .................... 77
4.1.1 Finite Symmetry Operations ...................... 77
4.1.2 Infinite Symmetry Operations .................... 78
4.2 Algebraic Treatment of Symmetry Operations ............. 79
4.2.1 Transformation of Coordinates of a Point ........ 79
4.2.2 Rotational Transformations of Vectors ........... 83
4.2.3 Translational Transformations of Vectors ........ 84
4.2.4 Combined Symmetrical Transformations of
Vectors ......................................... 85
4.2.5 Augmentation of Matrices ........................ 87
4.2.6 Algebraic Representation of Crystallographic
Symmetry ........................................ 88
4.2.7 Interaction of Symmetry Operations .............. 88
4.3 Additional Reading ..................................... 93
4.4 Problems ............................................... 94
5 Nonconventional Symmetry .................................... 97
5.1 Commensurate Modulation ................................ 98
5.2 Incommensurate Modulation .............................. 99
5.3 Composite Crystals .................................... 100
5.4 Symmetry of Modulated Structures ...................... 101
5.5 Quasicrystals ......................................... 103
5.6 Additional Reading .................................... 105
5.7 Problems .............................................. 105
6 Properties, Sources, and Detection of Radiation ............ 107
6.1 Nature of X-Rays ...................................... 109
6.2 Production of X-Rays .................................. 110
6.2.1 Conventional Sealed X-Ray Sources .............. 111
6.2.2 Continuous and Characteristic X-Ray Spectra .... 113
6.2.3 Rotating Anode X-Ray Sources ................... 116
6.2.4 Synchrotron Radiation Sources .................. 117
6.3 Other Types of Radiation .............................. 119
6.4 Detection of X-Rays ................................... 121
6.4.1 Detector Efficiency, Linearity,
Proportionality and Resolution ................. 121
6.4.2 Classification of Detectors .................... 123
6.4.3 Point Detectors ................................ 125
6.4.4 Line and Area Detectors ........................ 128
6.5 Additional Reading .................................... 131
6.6 Problems .............................................. 131
7 Fundamentals of Diffraction ................................ 133
7.1 Scattering by Electrons, Atoms and Lattices ........... 134
7.1.1 Scattering by Electrons ........................ 136
7.1.2 Scattering by Atoms and Atomic Scattering
Factor ......................................... 138
7.1.3 Scattering by Lattices ......................... 140
7.2 Geometry of Diffraction by Lattices ................... 142
7.2.1 Laue Equations ................................. 142
7.2.2 Braggs' Law .................................... 142
7.2.3 Reciprocal Lattice and Ewald's Sphere .......... 144
7.3 Additional Reading .................................... 148
7.4 Problems .............................................. 148
8 The Powder Diffraction Pattern ............................. 151
8.1 Origin of the Powder Diffraction Pattern .............. 152
8.2 Representation of Powder Diffraction Patterns ......... 157
8.3 Understanding of Powder Diffraction Patterns .......... 159
8.4 Positions of Powder Diffraction Peaks ................. 162
8.4.1 Peak Positions as a Function of Unit Cell
Dimensions ..................................... 163
8.4.2 Other Factors Affecting Peak Positions ......... 165
8.5 Shapes of Powder Diffraction Peaks .................... 168
8.5.1 Peak-Shape Functions ........................... 170
8.5.2 Peak Asymmetry ................................. 179
8.6 Intensity of Powder Diffraction Peaks ................. 182
8.6.1 Integrated Intensity ........................... 182
8.6.2 Scale Factor ................................... 185
8.6.3 Multiplicity Factor ............................ 186
8.6.4 Lorentz-Polarization Factor .................... 187
8.6.5 Absorption Factor .............................. 188
8.6.6 Preferred Orientation .......................... 194
8.6.7 Extinction Factor .............................. 199
8.7 Additional Reading .................................... 201
8.8 Problems .............................................. 201
9 Structure Factor ........................................... 203
9.1 Structure Amplitude ................................... 203
9.1.1 Population Factor .............................. 204
9.1.2 Temperature Factor (Atomic Displacement
Factor) ........................................ 206
9.1.3 Atomic Scattering Factor ....................... 211
9.1.4 Phase Angle .................................... 215
9.2 Effects of Symmetry on the Structure Amplitude ........ 217
9.2.1 Friedel Pairs and Friedel's Law ................ 218
9.2.2 Friedel's Law and Multiplicity Factor .......... 220
9.3 Systematic Absences ................................... 220
9.3.1 Lattice Centering .............................. 221
9.3.2 Glide Planes ................................... 222
9.3.3 Screw Axes ..................................... 223
9.4 Space Groups and Systematic Absences .................. 225
9.5 Additional Reading .................................... 235
9.6 Problems .............................................. 236
10 Solving the Crystal Structure .............................. 239
10.1 Fourier Transformation ................................ 239
10.2 Phase Problem ......................................... 245
10.2.1 Patterson Technique ............................ 246
10.2.2 Direct Methods ................................. 250
10.2.3 Structure Solution from Powder Diffraction
Data ........................................... 253
10.3 Total Scattering Analysis Using Pair Distribution
Function .............................................. 255
10.4 Additional Reading .................................... 261
10.5 Problems .............................................. 262
11 Powder Diffractometry ...................................... 263
11.1 Brief History of the Powder Diffraction Method ........ 264
11.2 Beam Conditioning in Powder Diffractometry ............ 269
11.2.1 Collimation .................................... 271
11.2.2 Monochromatization ............................. 274
11.3 Principles of Goniometer Design in Powder
Diffractometry ........................................ 280
11.3.1 Goniostats with Strip and Point Detectors ...... 283
11.3.2 Goniostats with Area Detectors ................. 287
11.4 Nonambient Powder Diffractometry ...................... 292
11.4.1 Variable Temperature Powder Diffractometry ..... 292
11.4.2 Principles of Variable Pressure Powder
Diffractometry ................................. 294
11.4.3 Powder Diffractometry in High Magnetic
Fields ......................................... 296
11.5 Additional Reading .................................... 299
11.6 Problems .............................................. 299
12 Collecting Quality Powder Diffraction Data ................. 301
12.1 Sample Preparation .................................... 301
12.1.1 Powder Requirements and Powder Preparation ..... 301
12.1.2 Powder Mounting ................................ 304
12.1.3 Sample Size .................................... 310
12.1.4 Sample Thickness and Uniformity ................ 311
12.1.5 Sample Positioning ............................. 313
12.1.6 Effects of Sample Preparation on Powder
Diffraction Data ............................... 314
12.2 Data Acquisition ...................................... 318
12.2.1 Wavelength ..................................... 318
12.2.2 Monochromatization ............................. 320
12.2.3 Incident Beam Aperture ......................... 322
12.2.4 Diffracted Beam Aperture ....................... 325
12.2.5 Variable Aperture .............................. 329
12.2.6 Power Settings ................................. 330
12.2.7 Classification of Powder Diffraction
Experiments .................................... 331
12.2.8 Step Scan ...................................... 331
12.2.9 Continuous Scan ................................ 334
12.2.10 Scan Range .................................... 336
12.3 Quality of Experimental Data .......................... 338
12.3.1 Quality of Intensity Measurements .............. 339
12.3.2 Factors Affecting Resolution ................... 342
12.4 Additional Reading .................................... 343
12.5 Problems .............................................. 344
13 Preliminary Data Processing and Phase Analysis ............. 347
13.1 Interpretation of Powder Diffraction Data ............. 348
13.2 Preliminary Data Processing ........................... 353
13.2.1 Background ..................................... 355
13.2.2 Smoothing ...................................... 359
13.2.3 Kα2 Stripping .................................. 361
13.2.4 Peak Search .................................... 363
13.2.5 Profile Fitting ................................ 366
13.3 Phase Identification and Quantitative Analysis ........ 377
13.3.1 Crystallographic Databases ..................... 377
13.3.2 Phase Identification ........................... 382
13.3.3 Quantitative Analysis .......................... 390
13.3.1 Phase Contents from Rietveld Refinement ........ 394
13.3.5 Determination of Amorphous Content or Degree
of Crystallinity ............................... 395
13.4 Additional Reading .................................... 399
13.5 Problems .............................................. 400
14 Determination and Refinement of the Unit Cell .............. 407
14.1 The Indexing Problem .................................. 407
14.2 Known Versus Unknown Unit Cell Dimensions ............. 410
14.3 Indexing: Known Unit Cell ............................. 412
14.3.1 High Symmetry Indexing Example ................. 414
14.3.2 Other Crystal Systems .......................... 420
14.4 Reliability of Indexing ............................... 421
14.4.1 The FN Figure of Merit ......................... 424
14.4.2 The M20(MN) Figure of Merit .................... 425
14.5 Introduction to Ab Initio Indexing .................... 426
14.6 Cubic Crystal System .................................. 428
14.6.1 Primitive Cubic Unit Cell: LaB6 ................ 430
14.6.2 Body-Centered Cubic Unit Cell: U3Ni6Si2 ........ 432
14.7 Tetragonal and Hexagonal Crystal Systems .............. 434
14.7.1 Indexing Example: LaNi4.85Sn0.15 ................ 437
14.8 Automatic Ab Initio Indexing Algorithms ............... 440
14.8.1 Indexing in Direct Space ....................... 441
14.8.2 Indexing in Reciprocal Space ................... 444
14.9 Unit Cell Reduction Algorithms ........................ 447
14.9.1 Delaunay-Ito Transformation .................... 448
14.9.2 Niggli Reduction ............................... 449
14.10 Automatic Ab Initio Indexing: Computer Codes ......... 450
14.10.1 TREOR ......................................... 451
14.10.2 DICVOL ........................................ 453
14.10.3 ITO ........................................... 454
14.10.4 Selecting a Solution .......................... 455
14.11 Ab Initio Indexing Examples .......................... 457
14.11.1 Hexagonal Indexing: LaNi4.85Sn0.15 ............. 457
14.11.2 Monoclinic Indexing: (CH3NH3)2Mo7O2 ........... 462
14.11.3 Triclinic Indexing: Fe7(PO4)6 ................. 466
14.11.4 Pseudo-Hexagonal Indexing: LiB(C2O4)2 ......... 470
14.12 Precise Lattice Parameters and Linear Least
Squares ............................................... 473
14.12.1 Linear Least Squares .......................... 475
14.12.2 Precise Lattice Parameters from Linear Least
Squares ....................................... 477
14.13 Concluding Remarks ................................... 485
14.14 Additional Reading ................................... 485
14.15 Problems ............................................. 486
15 Solving Crystal Structure from Powder Diffraction Data ..... 497
15.1 Ab Initio Methods of Structure Solution ............... 497
15.1.1 Conventional Reciprocal Space Methods .......... 498
15.1.2 Conventional Direct Space Modeling ............. 499
15.1.3 Unconventional Direct, Reciprocal, and Dual
Space Methods .................................. 500
15.1.4 Validation and Completion of the Model ......... 505
15.2 The Content of the Unit Cell .......................... 506
15.3 Pearson's Classification .............................. 509
15.4 Finding Structure Factors from Powder Diffraction
Data .................................................. 510
15.5 Nonlinear Least Squares ............................... 513
15.6 Quality of Profile Fitting ............................ 517
15.6.1 Visual Assessment of the Quality of Profile
Fitting ........................................ 518
15.6.2 Figures of Merit ............................... 521
15.7 The Rietveld Method ................................... 524
15.7.1 Fundamentals of the Rietveld Method ............ 527
15.7.2 Classes of Rietveld Refinement Parameters ...... 529
15.7.3 Restraints, Constraints, and Rigid-Bodies ...... 531
15.7.4 Figures of Merit and Quality of Rietveld
Refinement ..................................... 538
15.7.5 Common Problems and How to Deal with Them ...... 539
15.7.6 Termination of Rietveld Refinement ............. 542
15.8 Concluding Remarks .................................... 543
15.9 Additional Reading .................................... 544
16 Crystal Structure of LaNi4.85Sn0.15 ......................... 547
16.1 Full Pattern Decomposition ............................ 549
16.2 Solving the Crystal Structure ......................... 556
16.3 Rietveld Refinement Using Сu Кα1,2 Radiation .......... 560
16.3.1 Scale Factor and Profile Parameters ............ 561
16.3.2 Overall Atomic Displacement Parameter .......... 563
16.3.3 Individual Parameters, Free and Constrained
Variables ...................................... 564
16.3.4 Anisotropic Atomic Displacement Parameters ..... 567
16.3.5 Multiple Phase Refinement ...................... 567
16.3.6 Refinement Results ............................. 568
16.4 Rietveld Refinement Using Mo Кα1,2 Radiation .......... 569
16.5 Combined Refinement Using Different Sets of
Diffraction Data ...................................... 573
17 Crystal Structure of CeRhGe3 ............................... 579
17.1 Full Pattern Decomposition ............................ 579
17.2 Solving the Crystal Structure from X-Ray Data ......... 583
17.2.1 Highest Symmetry Attempt ....................... 584
17.2.2 Low-Symmetry Model ............................. 586
17.3 Solving the Crystal Structure from Neutron Data ....... 589
17.4 Rietveld Refinement ................................... 595
17.4.1 X-Ray Data, Correct Low Symmetry Model ........ 595
17.4.2 X-Ray Data, Wrong High-Symmetry Model .......... 598
17.4.3 Neutron Data ................................... 599
18 Crystal Structure of Nd5Si4 ................................ 603
18.1 Full Pattern Decomposition ............................ 603
18.2 Solving the Crystal Structure ......................... 604
18.3 Rietveld Refinement ................................... 607
19 Empirical Methods of Solving Crystal Structures ............ 611
19.1 Crystal Structure of Gd5Ge4 ........................... 612
19.2 Crystal Structure of Gd5Si4 ........................... 615
19.3 Crystal Structure of Gd5Si2Ge2 ........................ 616
19.4 Rietveld Refinement of Gd5Ge4,Gd5Si4, and Gd5Si2Ge2 ... 620
19.4.1 Gd5Ge4 ......................................... 620
19.4.2 Gd5Si4 ......................................... 623
19.4.3 Gd5Si2Ge2 ...................................... 627
19.5 Structure-Property Relationships ...................... 630
20 Crystal Structure of NiMnO2(OH) ............................ 633
20.1 Observed Structure Factors from Experimental Data ..... 636
20.2 Solving the Crystal Structure ......................... 640
20.3 A Few Notes About Using GSAS .......................... 643
20.4 Completion of the Model and Rietveld Refinement ....... 643
20.4.1 Initial Refinement Steps ....................... 647
20.4.2 Where Is Mn and Where Is Ni? ................... 648
20.4.3 Finalizing the Refinement of the Model
Without Hydrogen ............................... 648
20.4.4 Locating Hydrogen .............................. 648
20.4.5 Combined Rietveld Refinement ................... 650
21 Crystal Structure of tma V3O7 .............................. 655
21.1 Observed Structure Factors ............................ 656
21.2 Solving the Crystal Structure ......................... 658
21.3 Completion of the Model and Rietveld Refinement ....... 561
21.3.1 Unrestrained Rietveld Refinement ............... 662
21.3.2 Rietveld Refinement with Restraints ............ 665
22 Crystal Structure of ma2Мо7О22 ............................. 669
22.1 Possible Model of the Crystal Structure ............... 669
22.2 Rietveld Refinement and Completion of the Model ....... 672
23 Crystal Structure of Mn7(OH)3(VO4)4 ........................ 679
23.1 Solving the Crystal Structure ......................... 680
23.2 Rietveld Refinement ................................... 682
23.3 Determining Chemical Composition ...................... 685
24 Crystal Structure of FePO4 ................................. 691
24.1 Building and Optimizing the Model of the Crystal
Structure ............................................. 692
24.2 Rietveld Refinement ................................... 696
25 Crystal Structure of Acetaminophen, C8H9NO2 ................ 703
25.1 Ab Initio Indexing and Le Bail Fitting ................ 705
25.2 Solving the Crystal Structure ......................... 709
25.2.1 Creating a Model ............................... 709
25.2.2 Optimizing the Model (Solving the Structure) ... 713
25.3 Restrained Rietveld Refinement ........................ 717
25.4 Chapters 15-25: Additional Reading .................... 721
25.5 Chapters 15-25: Problems .............................. 723
Index ......................................................... 729
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