Fundamentals of Powder Diffraction and Structural Characterization of Materials

by Pecharsky, Vitalij K.; Zavalij, Peter Y.

• ISBN13: 9781402073656
• ISBN10: 1402073658
• Format: Hardcover
• Copyright: 2003-04-01
• Publisher: Kluwer Academic Pub

Summary

Fundamentals of Powder Diffraction and Structural Characterization of Materials provides an in-depth introduction to the theories and applications of the powder diffraction method for structure determination. The emphasis is placed on powder diffraction data collected using conventional x-ray sources, which remain primary tools for thousands of researchers and students in their daily experimental work. The book is divided into two parts: chapters one though three give essential theoretical background, while

Author Biography

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

Table of Contents

Preface	p. xvii
Fundamentals of Crystalline State	p. 1
Introduction	p. 1
Crystalline state	p. 2
Crystal lattice and crystal structure	p. 4
Shape of the unit cell	p. 6
Content of the unit cell	p. 7
Asymmetric part of the unit cell	p. 8
Symmetry operations and symmetry elements	p. 10
Finite symmetry elements	p. 12
One-fold rotation axis and center of inversion	p. 16
Two-fold rotation axis and mirror plane	p. 16
Three-fold rotation axis and three-fold inversion axis	p. 17
Four-fold rotation axis and four-fold inversion axis	p. 18
Six-fold rotation axis and six-fold inversion axis	p. 18
Interaction of symmetry elements	p. 19
Symmetry groups	p. 21
Generalization of interactions between finite symmetry elements	p. 22
Fundamentals of group theory	p. 24
Crystal systems	p. 26
Stereographic projections	p. 27
Crystallographic point groups	p. 29
Laue classes	p. 31
Selection of a unit cell and Bravais lattices	p. 32
Infinite symmetry elements	p. 39
Glide planes	p. 40
Screw axes	p. 42
Interaction of infinite symmetry elements	p. 43
Crystallographic planes, directions and indices	p. 45
Indices of planes	p. 46
Lattice directions and indices	p. 49
Reciprocal lattice	p. 50
Crystallographic space groups	p. 53
Relationships between space and point groups	p. 53
Full international symbols of crystallographic space groups	p. 56
Visualization of space group symmetry in three dimensions	p. 58
Space groups in nature	p. 59
International Tables for Crystallography	p. 60
Equivalent positions	p. 65
General and special equivalent positions	p. 66
Special sites with points located on mirror planes	p. 66
Special sites with points located on rotation or inversion axes	p. 67
Special sites with points located on centers of inversion	p. 68
Symbolic description of symmetry operations	p. 69
Finite symmetry operations	p. 70
Infinite symmetry operations	p. 71
Algebraic treatment of symmetry operations	p. 72
Transformations of coordinates of a point	p. 72
Rotational transformations of vectors	p. 77
Translational transformations of vectors	p. 78
Combined symmetrical transformations of vectors	p. 79
Augmented matrices	p. 81
Algebraic representation of crystallographic symmetry	p. 82
Interactions between symmetry operations	p. 86
Non-conventional symmetry	p. 88
Commensurate modulation	p. 88
Incommensurate modulation	p. 90
Quasicrystals	p. 91
Additional reading	p. 94
Problems	p. 95
Fundamentals of Diffraction	p. 99
Introduction	p. 99
Properties and sources of radiation	p. 102
Nature and properties of x-rays	p. 102
Production of x-rays	p. 104
Conventional sealed x-ray sources	p. 105
Continuous and characteristic x-ray spectra	p. 107
Rotating anode x-ray sources	p. 110
Synchrotron radiation sources	p. 112
Other types of radiation	p. 113
Collimation and monochromatization	p. 115
Angular divergence and collimation	p. 116
Monochromatization	p. 119
Detection of x-rays	p. 128
Detector efficiency, linearity, proportionality and resolution	p. 128
Classification of detectors	p. 130
Point detectors	p. 132
Line and area detectors	p. 136
Scattering by electrons, atoms and lattices	p. 138
Scattering by electrons	p. 140
Scattering by atoms and scattering factor	p. 143
Scattering by lattices	p. 145
Geometry of diffraction by lattices	p. 146
Laue equations and Braggs' law	p. 147
Reciprocal lattice and Ewald's sphere	p. 149
Origin of the powder diffraction pattern	p. 153
Representation of powder diffraction patterns	p. 158
Understanding of powder diffraction patterns	p. 161
Positions of powder diffraction peaks	p. 164
Peak positions as a function of unit cell dimensions	p. 164
Other factors affecting peak positions	p. 167
Shapes of powder diffraction peaks	p. 171
Peak shape functions	p. 173
Peak asymmetry	p. 182
Intensity of powder diffraction peaks	p. 184
Integrated intensity	p. 185
Scale factor	p. 188
Multiplicity factor	p. 189
Lorentz-polarization factor	p. 190
Absorption factor	p. 193
Preferred orientation	p. 196
Extinction factor	p. 202
Structure factor	p. 203
Structure amplitude	p. 203
Population factor	p. 204
Temperature factor	p. 207
Atomic scattering factor	p. 212
Phase angle	p. 216
Effects of symmetry on the structure amplitude	p. 218
Friedel pairs and Friedel's law	p. 219
Friedel's law and multiplicity factor	p. 221
Systematic absences	p. 222
Space groups and systematic absences	p. 227
Fourier transformation	p. 237
Phase problem	p. 243
Patterson technique	p. 245
Direct methods	p. 249
Structure solution from powder diffraction data	p. 253
Additional reading	p. 256
Problems	p. 258
Experimental Techniques	p. 261
Introduction	p. 261
Brief history of the powder diffraction method	p. 262
Powder diffractometers	p. 267
Principles of goniometer design in powder diffractometry	p. 269
Goniostats with point detectors	p. 273
Goniostats with area detectors	p. 276
Safety	p. 279
Radiation quantities and terms	p. 280
Biological effects of ionizing radiation	p. 281
Exposure limits	p. 282
Radiation hazards of analytical x-ray systems	p. 283
Hazard control measures for analytical x-ray systems	p. 284
Sample preparation	p. 287
Powder requirements and powder preparation	p. 287
Powder mounting	p. 290
Sample size	p. 295
Sample thickness and uniformity	p. 297
Positioning the sample with respect to the goniometer axis	p. 298
Effects of sample preparation on powder diffraction data	p. 301
Data acquisition	p. 305
Wavelength selection	p. 305
Monochromatization	p. 306
Incident beam aperture	p. 309
Diffracted beam aperture	p. 313
Variable aperture	p. 316
Power settings	p. 317
Classification of powder diffraction experiments	p. 318
Step scan	p. 319
Continuous scan	p. 322
Scan range	p. 324
Quality of experimental data	p. 326
Quality of intensity measurements	p. 328
Factors affecting resolution	p. 331
Additional reading	p. 333
Problems	p. 335
Preliminary Data Processing and Phase Analysis	p. 339
Introduction	p. 339
Interpretation of powder diffraction data	p. 340
Preliminary data processing	p. 345
Background	p. 347
Smoothing	p. 352
K[alpha subscript 2] stripping	p. 354
Peak search	p. 356
Profile fitting	p. 360
Phase identification and analysis	p. 371
Crystallographic databases	p. 372
Phase identification and qualitative analysis	p. 377
Quantitative analysis	p. 384
Additional reading	p. 390
Problems	p. 392
Unit Cell Determination and Refinement	p. 399
Introduction	p. 399
The indexing problem	p. 399
Known versus unknown unit cell dimensions	p. 402
Indexing: known unit cell	p. 405
High symmetry indexing example	p. 407
Other crystal systems	p. 413
Reliability of indexing	p. 415
The F[subscript N] figure of merit	p. 418
The M[subscript 20] figure of merit	p. 419
Introduction to ab initio indexing	p. 420
Cubic crystal system	p. 422
Primitive cubic unit cell: LaB[subscript 6]	p. 425
Body-centered cubic unit cell: U[subscript 3]Ni[subscript 6]Si[subscript 2]	p. 427
Tetragonal and hexagonal crystal systems	p. 429
Indexing example: LaNi[subscript 4.85]Sn[subscript 0.15]	p. 433
Automatic ab initio indexing algorithms	p. 436
Trial-and-error method	p. 438
Zone search method	p. 439
Unit cell reduction algorithms	p. 440
Delaunay-Ito reduction	p. 441
Niggli reduction	p. 442
Automatic ab initio indexing: computer codes	p. 443
TREOR	p. 444
DICVOL	p. 447
ITO	p. 448
Selecting a solution	p. 449
Ab initio indexing examples	p. 451
Hexagonal indexing: LaNi[subscript 4.85]Sn[subscript 0.15]	p. 451
Monoclinic indexing: (CH[subscript 3]NH[subscript 3])[subscript 2]Mo[subscript 7]O[subscript 22]	p. 457
Triclinic indexing: Fe[subscript 7](PO[subscript 4])[subscript 6]	p. 460
Precise lattice parameters and linear least squares	p. 464
Linear least squares	p. 466
Precise lattice parameters from linear least squares	p. 469
Epilogue	p. 479
Additional reading	p. 481
Problems	p. 482
Crystal Structure Determination	p. 493
Introduction	p. 493
Ab initio methods of structure solution	p. 494
Conventional reciprocal space techniques	p. 495
Conventional direct space techniques	p. 495
Unconventional reciprocal and direct space strategies	p. 496
Validation and completion of the model	p. 499
The content of the unit cell	p. 500
Pearson's classification	p. 503
Structure factors from powder diffraction data	p. 504
Non-linear least squares	p. 507
Figures of merit in full pattern decomposition	p. 512
Structure solution from powder data	p. 515
Crystal structure of LaNi[subscript 4.85]Sn[subscript 0.15]	p. 516
Crystal structure of CeRhGe[subscript 3] from x-ray data	p. 530
Crystal structure of CeRhGe[subscript 3] from neutron data	p. 541
Crystal structure of Nd[subscript 5]Si[subscript 4]	p. 548
Crystal structure of NiMnO[subscript 2](OH)	p. 553
Crystal structure of tmaV[subscript 3]O[subscript 7]	p. 561
Crystal structure of ma[subscript 2]Mo[subscript 7]O[subscript 22]	p. 568
Crystal structure of Mn[subscript 7](OH)[subscript 3](VO[subscript 4])[subscript 4]	p. 571
Crystal structure of FePO[subscript 4]	p. 575
Empirical methods of solving crystal structures	p. 580
Crystal structure of Gd[subscript 5]Ge[subscript 4]	p. 583
Crystal structure of Gd[subscript 5]Si[subscript 4]	p. 585
Crystal structure of Gd[subscript 5]Si[subscript 2]Ge[subscript 2]	p. 587
Additional reading	p. 591
Problems	p. 594
Crystal Structure Refinement	p. 599
Introduction	p. 599
The Rietveld method	p. 601
Rietveld method basics	p. 603
Classes of Rietveld parameters	p. 606
Figures of merit and quality of refinement	p. 608
Termination of Rietveld refinement	p. 609
Rietveld refinement of LaNi[subscript 4.85]Sn[subscript 0.15]	p. 610
Scale factor and profile parameters	p. 611
Overall atomic displacement parameter	p. 614
Individual parameters, free and constrained variables	p. 614
Anisotropic atomic displacement parameters	p. 617
Multiple phase refinement	p. 617
Refinement results	p. 618
Different radiation	p. 619
Combined refinement using different diffraction data	p. 623
Rietveld refinement of CeRhGe[subscript 3]	p. 628
Refinement using x-ray diffraction data	p. 628
Refinement using neutron diffraction data	p. 632
Rietveld refinement of Nd[subscript 5]Si[subscript 4]	p. 635
Rietveld refinement using GSAS	p. 639
Completion of the model and Rietveld refinement of NiMnO[subscript 2](OH)	p. 643
Completion of the model and Rietveld refinement of tmaV[subscript 3]O[subscript 7]	p. 654
Rietveld refinement and completion of the ma[subscript 2]Mo[subscript 7]O[subscript 22] structure	p. 662
Rietveld refinement of Mn[subscript 7](OH)[subscript 3](VO[subscript 4])[subscript 4]	p. 669
Rietveld refinement of the monoclinic FePO[subscript 4]	p. 677
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
Gd[subscript 5]Ge[subscript 4]	p. 685
Gd[subscript 5]Si[subscript 4]	p. 687
Gd[subscript 5]Si[subscript 2]Ge[subscript 2]	p. 692
Epilogue	p. 697
Additional reading	p. 699
Problems	p. 700
Index	p. 703
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