Pecharsky V.K. Fundamentals of powder diffraction and structural characterization of materials (New York, 2009). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаPecharsky V.K. Fundamentals of powder diffraction and structural characterization of materials / V.K.Pecharsky, P.Y.Zavalij. - 2nd ed. - New York: Springer, 2009. - xxiii, 741 p.: ill. - ISBN 978-0-387-09578-3
 

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Оглавление / Contents
 
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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