Fundamentals of powder diffraction and structural characterization of materials

Requires no prior knowledge of the subject, but is comprehensive and detailed making it useful for both the novice and experienced user of the powder diffraction method.

Useful for any scientific or engineering background, where precise structural information is required.

Comprehensively describes the state-of-the-art in structure determination from powder diffraction data both theoretically and practically using multiple examples of varying complexity.

Pays particular attention to the utilization of Internet resources, especially the well-tested and freely available computer codes designed for processing of powder diffraction data.

Author Notes

Vitalij K. Pecharsky: Department of Materials Science and Engineering, Ames Laboratory of the U.S. Department of Energy Iowa State University, Ames
Peter Y. Zavalij: Department of Chemistry and Institute for Materials Research State University of New York at Binghamton, NY

Preface	p. xvii
1. Fundamentals of Crystalline State	p. 1
1.1 Introduction	p. 1
1.2 Crystalline state	p. 2
1.3 Crystal lattice and crystal structure	p. 4
1.3.1 Shape of the unit cell	p. 6
1.3.2 Content of the unit cell	p. 7
1.3.3 Asymmetric part of the unit cell	p. 8
1.4 Symmetry operations and symmetry elements	p. 10
1.5 Finite symmetry elements	p. 12
1.5.1 One-fold rotation axis and center of inversion	p. 16
1.5.2 Two-fold rotation axis and mirror plane	p. 16
1.5.3 Three-fold rotation axis and three-fold inversion axis	p. 17
1.5.4 Four-fold rotation axis and four-fold inversion axis	p. 18
1.5.5 Six-fold rotation axis and six-fold inversion axis	p. 18
1.6 Interaction of symmetry elements	p. 19
1.6.1 Symmetry groups	p. 21
1.6.2 Generalization of interactions between finite symmetry elements	p. 22
1.7 Fundamentals of group theory	p. 24
1.8 Crystal systems	p. 26
1.9 Stereographic projections	p. 27
1.10 Crystallographic point groups	p. 29
1.11 Laue classes	p. 31
1.12 Selection of a unit cell and Bravais lattices	p. 32
1.13 Infinite symmetry elements	p. 39
1.13.1 Glide planes	p. 40
1.13.2 Screw axes	p. 42
1.13.3 Interaction of infinite symmetry elements	p. 43
1.14 Crystallographic planes, directions and indices	p. 45
1.14.1 Indices of planes	p. 46
1.14.2 Lattice directions and indices	p. 49
1.15 Reciprocal lattice	p. 50
1.16 Crystallographic space groups	p. 53
1.16.1 Relationships between space and point groups	p. 53
1.16.2 Full international symbols of crystallographic space groups	p. 56
1.16.3 Visualization of space group symmetry in three dimensions	p. 58
1.16.4 Space groups in nature	p. 59
1.17 International Tables for Crystallography	p. 60
1.18 Equivalent positions	p. 65
1.18.1 General and special equivalent positions	p. 66
1.18.2 Special sites with points located on mirror planes	p. 66
1.18.3 Special sites with points located on rotation or inversion axes	p. 67
1.18.4 Special sites with points located on centers of inversion	p. 68
1.19 Symbolic description of symmetry operations	p. 69
1.19.1 Finite symmetry operations	p. 70
1.19.2 Infinite symmetry operations	p. 71
1.20 Algebraic treatment of symmetry operations	p. 72
1.20.1 Transformations of coordinates of a point	p. 72
1.20.2 Rotational transformations of vectors	p. 77
1.20.3 Translational transformations of vectors	p. 78
1.20.4 Combined symmetrical transformations of vectors	p. 79
1.20.5 Augmented matrices	p. 81
1.20.6 Algebraic representation of crystallographic symmetry	p. 82
1.20.7 Interactions between symmetry operations	p. 86
1.21 Non-conventional symmetry	p. 88
1.21.1 Commensurate modulation	p. 88
1.21.2 Incommensurate modulation	p. 90
1.21.3 Quasicrystals	p. 91
1.22 Additional reading	p. 94
1.23 Problems	p. 95
2. Fundamentals of Diffraction	p. 99
2.1 Introduction	p. 99
2.2 Properties and sources of radiation	p. 102
2.2.1 Nature and properties of x-rays	p. 102
2.2.2 Production of x-rays	p. 104
2.2.3 Conventional sealed x-ray sources	p. 105
2.2.4 Continuous and characteristic x-ray spectra	p. 107
2.2.5 Rotating anode x-ray sources	p. 110
2.2.6 Synchrotron radiation sources	p. 112
2.2.7 Other types of radiation	p. 113
2.3 Collimation and monochromatization	p. 115
2.3.1 Angular divergence and collimation	p. 116
2.3.2 Monochromatization	p. 119
2.4 Detection of x-rays	p. 128
2.4.1 Detector efficiency, linearity, proportionality and resolution	p. 128
2.4.2 Classification of detectors	p. 130
2.4.3 Point detectors	p. 132
2.4.4 Line and area detectors	p. 136
2.5 Scattering by electrons, atoms and lattices	p. 138
2.5.1 Scattering by electrons	p. 140
2.5.2 Scattering by atoms and scattering factor	p. 143
2.5.3 Scattering by lattices	p. 145
2.6 Geometry of diffraction by lattices	p. 146
2.6.1 Laue equations and Braggs' law	p. 147
2.6.2 Reciprocal lattice and Ewald's sphere	p. 149
2.7 Origin of the powder diffraction pattern	p. 153
2.7.1 Representation of powder diffraction patterns	p. 158
2.7.2 Understanding of powder diffraction patterns	p. 161
2.8 Positions of powder diffraction peaks	p. 164
2.8.1 Peak positions as a function of unit cell dimensions	p. 164
2.8.2 Other factors affecting peak positions	p. 167
2.9 Shapes of powder diffraction peaks	p. 171
2.9.1 Peak shape functions	p. 173
2.9.2 Peak asymmetry	p. 182
2.10 Intensity of powder diffraction peaks	p. 184
2.10.1 Integrated intensity	p. 185
2.10.2 Scale factor	p. 188
2.10.3 Multiplicity factor	p. 189
2.10.4 Lorentz-polarization factor	p. 190
2.10.5 Absorption factor	p. 193
2.10.6 Preferred orientation	p. 196
2.10.7 Extinction factor	p. 202
2.11 Structure factor	p. 203
2.11.1 Structure amplitude	p. 203
2.11.2 Population factor	p. 204
2.11.3 Temperature factor	p. 207
2.11.4 Atomic scattering factor	p. 212
2.11.5 Phase angle	p. 216
2.12 Effects of symmetry on the structure amplitude	p. 218
2.12.1 Friedel pairs and Friedel's law	p. 219
2.12.2 Friedel's law and multiplicity factor	p. 221
2.12.3 Systematic absences	p. 222
2.12.4 Space groups and systematic absences	p. 227
2.13 Fourier transformation	p. 237
2.14 Phase problem	p. 243
2.14.1 Patterson technique	p. 245
2.14.2 Direct methods	p. 249
2.14.3 Structure solution from powder diffraction data	p. 253
2.15 Additional reading	p. 256
2.16 Problems	p. 258
3. Experimental Techniques	p. 261
3.1 Introduction	p. 261
3.2 Brief history of the powder diffraction method	p. 262
3.3 Powder diffractometers	p. 267
3.3.1 Principles of goniometer design in powder diffractometry	p. 269
3.3.2 Goniostats with point detectors	p. 273
3.3.3 Goniostats with area detectors	p. 276
3.4 Safety	p. 279
3.4.1 Radiation quantities and terms	p. 280
3.4.2 Biological effects of ionizing radiation	p. 281
3.4.3 Exposure limits	p. 282
3.4.4 Radiation hazards of analytical x-ray systems	p. 283
3.4.5 Hazard control measures for analytical x-ray systems	p. 284
3.5 Sample preparation	p. 287
3.5.1 Powder requirements and powder preparation	p. 287
3.5.2 Powder mounting	p. 290
3.5.3 Sample size	p. 295
3.5.4 Sample thickness and uniformity	p. 297
3.5.5 Positioning the sample with respect to the goniometer axis	p. 298
3.5.6 Effects of sample preparation on powder diffraction data	p. 301
3.6 Data acquisition	p. 305
3.6.1 Wavelength selection	p. 305
3.6.2 Monochromatization	p. 306
3.6.3 Incident beam aperture	p. 309
3.6.4 Diffracted beam aperture	p. 313
3.6.5 Variable aperture	p. 316
3.6.6 Power settings	p. 317
3.6.7 Classification of powder diffraction experiments	p. 318
3.6.8 Step scan	p. 319
3.6.9 Continuous scan	p. 322
3.6.10 Scan range	p. 324
3.7 Quality of experimental data	p. 326
3.7.1 Quality of intensity measurements	p. 328
3.7.2 Factors affecting resolution	p. 331
3.8 Additional reading	p. 333
3.9 Problems	p. 335
4. Preliminary Data Processing and Phase Analysis	p. 339
4.1 Introduction	p. 339
4.2 Interpretation of powder diffraction data	p. 340
4.3 Preliminary data processing	p. 345
4.3.1 Background	p. 347
4.3.2 Smoothing	p. 352
4.3.3 K[alpha subscript 2] stripping	p. 354
4.3.4 Peak search	p. 356
4.3.5 Profile fitting	p. 360
4.4 Phase identification and analysis	p. 371
4.4.1 Crystallographic databases	p. 372
4.4.2 Phase identification and qualitative analysis	p. 377
4.4.3 Quantitative analysis	p. 384
4.5 Additional reading	p. 390
4.6 Problems	p. 392
5. Unit Cell Determination and Refinement	p. 399
5.1 Introduction	p. 399
5.2 The indexing problem	p. 399
5.3 Known versus unknown unit cell dimensions	p. 402
5.4 Indexing: known unit cell	p. 405
5.4.1 High symmetry indexing example	p. 407
5.4.2 Other crystal systems	p. 413
5.5 Reliability of indexing	p. 415
5.5.1 The F[subscript N] figure of merit	p. 418
5.5.2 The M[subscript 20] figure of merit	p. 419
5.6 Introduction to ab initio indexing	p. 420
5.7 Cubic crystal system	p. 422
5.7.1 Primitive cubic unit cell: LaB[subscript 6]	p. 425
5.7.2 Body-centered cubic unit cell: U[subscript 3]Ni[subscript 6]Si[subscript 2]	p. 427
5.8 Tetragonal and hexagonal crystal systems	p. 429
5.8.1 Indexing example: LaNi[subscript 4.85]Sn[subscript 0.15]	p. 433
5.9 Automatic ab initio indexing algorithms	p. 436
5.9.1 Trial-and-error method	p. 438
5.9.2 Zone search method	p. 439
5.10 Unit cell reduction algorithms	p. 440
5.10.1 Delaunay-Ito reduction	p. 441
5.10.2 Niggli reduction	p. 442
5.11 Automatic ab initio indexing: computer codes	p. 443
5.11.1 TREOR	p. 444
5.11.2 DICVOL	p. 447
5.11.3 ITO	p. 448
5.11.4 Selecting a solution	p. 449
5.12 Ab initio indexing examples	p. 451
5.12.1 Hexagonal indexing: LaNi[subscript 4.85]Sn[subscript 0.15]	p. 451
5.12.2 Monoclinic indexing: (CH[subscript 3]NH[subscript 3])[subscript 2]Mo[subscript 7]O[subscript 22]	p. 457
5.12.3 Triclinic indexing: Fe[subscript 7](PO[subscript 4])[subscript 6]	p. 460
5.13 Precise lattice parameters and linear least squares	p. 464
5.13.1 Linear least squares	p. 466
5.13.2 Precise lattice parameters from linear least squares	p. 469
5.14 Epilogue	p. 479
5.15 Additional reading	p. 481
5.16 Problems	p. 482
6. Crystal Structure Determination	p. 493
6.1 Introduction	p. 493
6.2 Ab initio methods of structure solution	p. 494
6.2.1 Conventional reciprocal space techniques	p. 495
6.2.2 Conventional direct space techniques	p. 495
6.2.3 Unconventional reciprocal and direct space strategies	p. 496
6.2.4 Validation and completion of the model	p. 499
6.3 The content of the unit cell	p. 500
6.4 Pearson's classification	p. 503
6.5 Structure factors from powder diffraction data	p. 504
6.6 Non-linear least squares	p. 507
6.7 Figures of merit in full pattern decomposition	p. 512
6.8 Structure solution from powder data	p. 515
6.9 Crystal structure of LaNi[subscript 4.85]Sn[subscript 0.15]	p. 516
6.10 Crystal structure of CeRhGe[subscript 3] from x-ray data	p. 530
6.11 Crystal structure of CeRhGe[subscript 3] from neutron data	p. 541
6.12 Crystal structure of Nd[subscript 5]Si[subscript 4]	p. 548
6.13 Crystal structure of NiMnO[subscript 2](OH)	p. 553
6.14 Crystal structure of tmaV[subscript 3]O[subscript 7]	p. 561
6.15 Crystal structure of ma[subscript 2]Mo[subscript 7]O[subscript 22]	p. 568
6.16 Crystal structure of Mn[subscript 7](OH)[subscript 3](VO[subscript 4])[subscript 4]	p. 571
6.17 Crystal structure of FePO[subscript 4]	p. 575
6.18 Empirical methods of solving crystal structures	p. 580
6.18.1 Crystal structure of Gd[subscript 5]Ge[subscript 4]	p. 583
6.18.2 Crystal structure of Gd[subscript 5]Si[subscript 4]	p. 585
6.18.3 Crystal structure of Gd[subscript 5]Si[subscript 2]Ge[subscript 2]	p. 587
6.19 Additional reading	p. 591
6.20 Problems	p. 594
7. Crystal Structure Refinement	p. 599
7.1 Introduction	p. 599
7.2 The Rietveld method	p. 601
7.2.1 Rietveld method basics	p. 603
7.2.2 Classes of Rietveld parameters	p. 606
7.2.3 Figures of merit and quality of refinement	p. 608
7.2.4 Termination of Rietveld refinement	p. 609
7.3 Rietveld refinement of LaNi[subscript 4.85]Sn[subscript 0.15]	p. 610
7.3.1 Scale factor and profile parameters	p. 611
7.3.2 Overall atomic displacement parameter	p. 614
7.3.3 Individual parameters, free and constrained variables	p. 614
7.3.4 Anisotropic atomic displacement parameters	p. 617
7.3.5 Multiple phase refinement	p. 617
7.3.6 Refinement results	p. 618
7.3.7 Different radiation	p. 619
7.3.8 Combined refinement using different diffraction data	p. 623
7.4 Rietveld refinement of CeRhGe[subscript 3]	p. 628
7.4.1 Refinement using x-ray diffraction data	p. 628
7.4.2 Refinement using neutron diffraction data	p. 632
7.5 Rietveld refinement of Nd[subscript 5]Si[subscript 4]	p. 635
7.6 Rietveld refinement using GSAS	p. 639
7.7 Completion of the model and Rietveld refinement of NiMnO[subscript 2](OH)	p. 643
7.8 Completion of the model and Rietveld refinement of tmaV[subscript 3]O[subscript 7]	p. 654
7.9 Rietveld refinement and completion of the ma[subscript 2]Mo[subscript 7]O[subscript 22] structure	p. 662
7.10 Rietveld refinement of Mn[subscript 7](OH)[subscript 3](VO[subscript 4])[subscript 4]	p. 669
7.11 Rietveld refinement of the monoclinic FePO[subscript 4]	p. 677
7.12 Rietveld refinement of Gd[subscript 5]Ge[subscript 4], Gd[subscript 5]Si[subscript 4] and Gd[subscript 5]Si[subscript 2]Ge[subscript 2]	p. 684
7.12.1 Gd[subscript 5]Ge[subscript 4]	p. 685
7.12.2 Gd[subscript 5]Si[subscript 4]	p. 687
7.12.3 Gd[subscript 5]Si[subscript 2]Ge[subscript 2]	p. 692
7.13 Epilogue	p. 697
7.14 Additional reading	p. 699
7.15 Problems	p. 700
Index	p. 703

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