1 Introduction to diffraction .................................. 1
1.1 Introduction ............................................ 1
1.2 X-ray scattering from electrons ......................... 1
1.3 X-ray scattering from atoms ............................. 1
1.4 X-ray scattering from a unit cell ....................... 2
1.5 The effects of the crystal lattice ...................... 2
1.6 X-ray scattering from the crystal ....................... 3
1.7 The structure-factor equation ........................... 4
1.8 The electron-density equation ........................... 5
1.9 A mathematical relationship ............................. 6
1.10 Bragg's law ............................................. 6
1.11 Resolution .............................................. 7
1.12 The phase problem ....................................... 8
2 Introduction to symmetry and diffraction ..................... 9
2.1 The relationship between a crystal structure and its
diffraction pattern ..................................... 9
2.2 Translation symmetry in crystalline solids ............. 10
2.3 Symmetry of individual molecules, with relevance to
crystalline solids ..................................... 12
2.4 Symmetry in the solid state ............................ 16
2.5 Diffraction and symmetry ............................... 18
2.6 Further points ......................................... 20
Exercises ................................................... 24
3 Crystal growth and evaluation ............................... 27
3.1 Introduction ........................................... 27
3.2 Protect your crystals .................................. 27
3.3 Crystal growth ......................................... 28
3.4 Survey of methods ...................................... 28
3.5 Evaluation ............................................. 35
3.6 Crystal mounting ........................................ 36
4 Space-group determination ................................... 41
4.1 Introduction ........................................... 41
4.2 Prior knowledge and information other than from
diffraction ............................................ 42
4.3 Metric symmetry and Laue symmetry ...................... 43
4.4 Unit cell contents ..................................... 43
4.5 Systematic absences .................................... 44
4.6 The statistical distribution of intensities ............ 47
4.7 Other points ........................................... 48
4.8 A brief conducted tour of some entries in
International Tables for Crystallography, Volume A ..... 50
Exercises ................................................... 52
5 Background theory for data collection ....................... 53
5.1 Introduction ........................................... 53
5.2 A step-wise theoretical journey through an
experiment ............................................. 53
5.3 The geometry of X-ray diffraction ...................... 55
5.4 Determining the unit cell: the indexing process ........ 58
5.5 Relating diffractometer angles to unit cell
parameters: determination of the orientation matrix .... 62
5.6 Data-collection procedures and strategies .............. 64
5.7 Extracting data intensities: data integration and
reduction .............................................. 67
Exercises ................................................... 72
6 Practical aspects of data collection ........................ 73
6.1 Introduction ............................................ 73
6.2 Collecting data with area-detector diffractometers ..... 73
6.3 Experimental conditions ................................ 75
6.4 Types of area detector ................................. 77
6.5 Some characteristics of CCD area-detector systems ...... 80
6.6 Crystal screening ...................................... 82
6.7 Data collection ........................................ 88
Exercises ................................................... 91
7 Practical aspects of data processing ........................ 93
7.1 Data reduction and correction .......................... 93
7.2 Integration input and output ........................... 93
7.3 Corrections ............................................ 94
7.4 Output ................................................. 95
7.5 A typical experiment? .................................. 95
7.6 Examples of more problematic cases ..................... 96
7.7 Twinning and area-detector data ........................ 98
7.8 Some other special cases (in brief) .................... 99
Exercises .................................................. 101
8 Fourier syntheses .......................................... 103
8.1 Introduction .......................................... 103
8.2 Forward and reverse Fourier transforms ................ 104
8.3 Some mathematical and computing considerations ........ 107
8.4 Uses of different kinds of Fourier syntheses .......... 108
8.5 Weights in Fourier syntheses .......................... 112
8.6 Illustration in one dimension ......................... 113
Exercises .................................................. 115
9 Patterson syntheses for structure determination ............ 117
9.1 Introduction .......................................... 117
9.2 What the Patterson synthesis means .................... 118
9.3 Finding heavy atoms from a Patterson map .............. 121
9.4 Patterson syntheses giving more than one possible
solution, and other problems .......................... 126
9.5 Patterson search methods .............................. 128
Exercises .................................................. 131
10 Direct methods of crystal-structure determination .......... 133
10.1 Amplitudes and phases ................................. 133
10.2 The physical basis of direct methods .................. 134
10.3 Constraints on the electron density ................... 135
Exercises .................................................. 147
11 An introduction to maximum entropy ......................... 149
11.1 Entropy ............................................... 149
11.2 Maximum entropy ....................................... 150
11.3 Electron-density maps ................................. 153
12 Least-squares fitting of parameters ........................ 155
12.1 Weighted mean ......................................... 155
12.2 Linear regression ..................................... 156
12.3 Non-linear least squares .............................. 162
12.4 Ill-conditioning ...................................... 164
12.5 Computing time ........................................ 165
Exercises .................................................. 167
13 Refinement of crystal structures ........................... 169
13.1 Equations ............................................. 169
13.2 Reasons for performing refinement ..................... 172
13.3 Data quality and limitations .......................... 175
13.4 Refinement fundamentals ............................... 177
13.5 Refinement strategies ................................. 180
13.6 Under- and over-parameterization ...................... 182
13.7 Pseudo-symmetry, wrong space groups and Z' > 1
structures ............................................ 183
13.8 Conclusion ............................................ 184
Exercises .................................................. 186
14 Analysis of extended inorganic structures .................. 189
14.1 Introduction .......................................... 189
14.2 Disorder .............................................. 190
14.3 Phase transitions ..................................... 194
14.4 Structure validation .................................. 195
14.5 Case history 1 - BiMg2VO6 ............................. 196
14.6 Case history 2 - Mo2P4O15 .............................. 199
Exercises .................................................. 203
15 The derivation of results .................................. 205
15.1 Introduction .......................................... 205
15.2 Geometry calculations ................................. 205
15.3 Least-squares planes and dihedral angles .............. 211
15.4 Hydrogen atoms and hydrogen bonding ................... 213
15.5 Displacement parameters .............................. 214
Exercises .................................................. 219
16 Random and systematic errors ............................... 221
16.1 Random and systematic errors .......................... 221
16.2 Random errors and distributions ....................... 222
16.3 Taking averages ....................................... 229
16.4 Weighting schemes ..................................... 232
16.5 Analysis of the agreement between observed and
calculated data ....................................... 238
16.6 Estimated standard deviations and standard
uncertainties of structural parameters ................ 240
16.7 Systematic errors ..................................... 242
Exercises .................................................. 250
17 Powder diffraction ......................................... 251
17.1 Introduction to powder diffraction .................... 251
17.2 Powder versus single-crystal diffraction .............. 252
17.3 Experimental methods .................................. 254
17.4 Information contained in a powder pattern ............. 258
17.5 Rietveld refinement ................................... 261
17.6 Structure solution from powder diffraction data ....... 264
17.7 Non-ambient studies ................................... 265
Exercises .................................................. 268
18 Introduction to twinning ................................... 271
18.1 Introduction .......................................... 271
18.2 A simple model for twinning ........................... 271
18.3 Twinning in crystals .................................. 272
18.4 Diffraction patterns from twinned crystals ............ 274
18.5 Inversion, merohedral and pseudo-merohedral twins ..... 276
18.6 Derivation of twin laws ............................... 279
18.7 Non-merohedral twinning ............................... 280
18.8 The derivation of non-merohedral twin laws ............ 282
18.9 Common signs of twinning .............................. 283
18.10 Examples ............................................. 285
Exercises .................................................. 296
19 The presentation of results ................................ 299
19.1 Introduction .......................................... 299
19.2 Graphics .............................................. 300
19.3 Graphics programs ..................................... 300
19.4 Underlying concepts ................................... 301
19.5 Drawing styles ........................................ 302
19.6 Creating three-dimensional illusions .................. 306
19.7 The use of colour ..................................... 307
19.8 Textual information in drawings ....................... 307
19.9 Some hints for effective drawings ..................... 308
19.10 Tables of results .................................... 309
19.11 The content of tables ................................ 310
19.12 The format of tables ................................. 312
19.13 Hints on presentation ................................ 312
19.14 Archiving of results ................................. 315
20 The crystallographic information file (CIF) ................ 319
20.1 Introduction .......................................... 319
20.2 Basics ................................................ 319
20.3 Uses of CIF ........................................... 321
20.4 Some properties of the CIF format ..................... 321
20.5 Some practicalities ................................... 323
21 Crystallographic databases ................................. 327
21.1 What is a database? ................................... 327
21.2 What types of search are possible? .................... 327
21.3 What information can you get out? ..................... 328
21.4 What can you use databases for? ....................... 328
21.5 What are the limitations? ............................. 328
21.6 Short descriptions of crystallographic databases ...... 328
22 X-ray and neutron sources .................................. 333
22.1 Introduction .......................................... 333
22.2 Laboratory X-ray sources .............................. 333
22.3 Synchrotron X-ray sources ............................. 335
22.4 Neutron sources ....................................... 339
A Appendix A: Useful mathematics and formulae ................ 343
A.l Introduction .......................................... 343
A.2 Trigonometry .......................................... 343
A.3 Complex numbers ....................................... 344
A.4 Waves and structure factors ........................... 345
A.5 Vectors ............................................... 346
A.6 Determinants .......................................... 348
A.7 Matrices .............................................. 348
A.8 Matrices in symmetry .................................. 349
A.9 Matrix inversion ...................................... 350
A.10 Convolution ........................................... 351
В Appendix B: Questions and answers .......................... 353
Index ......................................................... 385
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