Crystal Structure Analysis

019921946X......Page 1
Contents......Page 10
1.3 X-ray scattering from atoms......Page 20
1.5 The effects of the crystal lattice......Page 21
1.6 X-ray scattering from the crystal......Page 22
1.7 The structure-factor equation......Page 23
1.8 The electron-density equation......Page 24
1.10 Bragg’s law......Page 25
1.11 Resolution......Page 26
1.12 The phase problem......Page 27
2.1 The relationship between a crystal structure and its diffraction pattern......Page 28
2.2 Translation symmetry in crystalline solids......Page 29
2.3 Symmetry of individual molecules, with relevance to crystalline solids......Page 31
2.4 Symmetry in the solid state......Page 35
2.5 Diffraction and symmetry......Page 37
2.6 Further points......Page 39
Exercises......Page 43
3.2 Protect your crystals......Page 46
3.4.1 Solution methods......Page 47
3.4.3 Fluid-phase growth......Page 52
3.4.5 General comments......Page 53
3.5.1 Microscopy......Page 54
3.6.1 Standard procedures......Page 55
3.6.2 Air-sensitive crystals......Page 57
3.6.3 Crystal alignment......Page 58
4.1 Introduction......Page 60
4.2 Prior knowledge and information other than from diffraction......Page 61
4.4 Unit cell contents......Page 62
4.5 Systematic absences......Page 63
4.6 The statistical distribution of intensities......Page 66
4.7 Other points......Page 67
4.8 A brief conducted tour of some entries in International Tables for Crystallography, Volume A......Page 69
Exercises......Page 71
5.2 A step-wise theoretical journey through an experiment......Page 72
5.3.1 Real-space considerations: Bragg’s law......Page 74
5.3.2 Reciprocal-space considerations: the Ewald sphere......Page 75
5.4.1 Indexing: a conceptual view......Page 77
5.4.2 Indexing procedure......Page 79
5.5 Relating diffractometer angles to unit cell parameters: determination of the orientation matrix......Page 81
5.6.1 Criteria for selecting which data to collect......Page 83
5.6.2 How best to measure data: the need for reflection scans......Page 84
5.7.1 Background subtraction......Page 86
5.7.3 Crystal and geometric corrections to data......Page 87
Exercises......Page 91
6.2 Collecting data with area-detector diffractometers......Page 92
6.3.1 Radiation......Page 94
6.3.2 Temperature......Page 95
6.4.1 Multiwire proportional chamber (MWPC)......Page 96
6.4.4 Charge-coupled device (CCD)......Page 97
6.5 Some characteristics of CCD area-detector systems......Page 99
6.5.4 Dark current......Page 100
6.6 Crystal screening......Page 101
6.6.1 Unit cell and orientation matrix determination......Page 103
6.6.3 Re-harvest the reflections......Page 105
6.6.6 Check for known cells......Page 106
6.7.1 Intensity level......Page 107
6.7.3 Crystal symmetry......Page 108
6.7.4 Other considerations......Page 109
Exercises......Page 110
7.2 Integration input and output......Page 112
7.3 Corrections......Page 113
7.5 A typical experiment?......Page 114
7.6 Examples of more problematic cases......Page 115
7.7 Twinning and area-detector data......Page 117
7.8 Some other special cases (in brief)......Page 118
Exercises......Page 120
8.1 Introduction......Page 122
8.2 Forward and reverse Fourier transforms......Page 123
8.3 Some mathematical and computing considerations......Page 126
8.4 Uses of different kinds of Fourier syntheses......Page 127
8.4.3 Full electron-density maps, using (8.2) or (8.3) as they stand......Page 128
8.4.4 Difference syntheses......Page 129
8.4.5 2F[sub(o)] – F[sub(c)] syntheses......Page 130
8.5 Weights in Fourier syntheses......Page 131
8.6 Illustration in one dimension......Page 132
8.6.4 Full F[sub(o)] synthesis......Page 133
Exercises......Page 134
9.1 Introduction......Page 136
9.2 What the Patterson synthesis means......Page 137
9.3.1 One heavy atom in the asymmetric unit of P1......Page 140
9.3.2 One heavy atom in the asymmetric unit of P2[sub(1)]/c......Page 141
9.3.4 One heavy atom in the asymmetric unit of Pbca......Page 143
9.3.6 Two heavy atoms in the asymmetric unit of P1 and other space groups......Page 144
9.4 Patterson syntheses giving more than one possible solution, and other problems......Page 145
9.5 Patterson search methods......Page 147
9.5.2 Translation search......Page 148
Exercises......Page 150
10.1 Amplitudes and phases......Page 152
10.2 The physical basis of direct methods......Page 153
10.3.1 Discrete atoms......Page 154
10.3.2 Non-negative electron density......Page 155
10.3.3 Random atomic distribution......Page 156
10.3.5 Equal atoms......Page 158
10.3.8 Structure invariants......Page 159
10.3.9 Structure determination......Page 160
10.3.12 Finding reflections for phase determination......Page 161
10.3.15 Figures of merit......Page 163
10.3.16 Interpretation of maps......Page 164
10.3.17 Completion of the structure......Page 165
Exercises......Page 166
11.1 Entropy......Page 168
11.2.1 Calculations with incomplete data......Page 169
11.2.3 Entropy and probability......Page 171
11.3 Electron-density maps......Page 172
12.1 Weighted mean......Page 174
12.2 Linear regression......Page 175
12.2.2 Restraints......Page 177
12.2.3 Constraints......Page 179
12.3 Non-linear least squares......Page 181
12.4 Ill-conditioning......Page 183
12.5 Computing time......Page 184
Exercises......Page 186
13.1 Equations......Page 188
13.1.3 Electron density from the structure amplitude and phase......Page 189
13.2.1 To improve phasing so that computed electron density maps more closely represent the actual electron density......Page 191
13.2.2 To try to verify that the structure is ‘correct’......Page 192
13.3.1 Resolution......Page 194
13.3.4 Weak reflections and systematic absences......Page 195
13.4 Refinement fundamentals......Page 196
13.4.2 γ[sub(1)], the observations......Page 197
13.4.3 γ[sub(2)], the calculations......Page 198
13.5 Refinement strategies......Page 199
13.6.1 Under-parameterization......Page 201
13.7 Pseudo-symmetry, wrong space groups and Z' > 1 structures......Page 202
13.8 Conclusion......Page 203
Exercises......Page 205
14.1 Introduction......Page 208
14.2 Disorder......Page 209
14.2.1 Site-occupancy disorder......Page 210
14.2.2 Positional disorder......Page 211
14.2.3 Limits of Bragg diffraction......Page 212
14.3 Phase transitions......Page 213
14.4 Structure validation......Page 214
14.5 Case history 1 – BiMg[sub(2)]VO[sub(6)]......Page 215
14.6 Case history 2 – Mo[sub(2)]P[sub(4)]O[sub(15)]......Page 218
Exercises......Page 222
15.2.1 Fractional and Cartesian co-ordinates......Page 224
15.2.2 Bond distance and angle calculations......Page 226
15.2.4 Transforming co-ordinates......Page 227
15.2.5 Standard uncertainties......Page 228
15.3 Least-squares planes and dihedral angles......Page 230
15.4 Hydrogen atoms and hydrogen bonding......Page 232
15.5 Displacement parameters......Page 233
15.5.2 ‘The equivalent isotropic displacement parameter’......Page 234
15.5.3 Symmetry and anisotropic displacement parameters......Page 235
15.5.4 Models of thermal motion and geometrical corrections: rigid-body motion......Page 236
15.5.5 Atomic displacement parameters and temperature......Page 237
Exercises......Page 238
16.1 Random and systematic errors......Page 240
16.2.2 Describing data......Page 241
16.2.3 Theoretical distributions......Page 244
16.2.4 Expectation values......Page 246
16.3 Taking averages......Page 248
16.3.1 Testing for normality using a histogram......Page 249
16.3.2 The χ[sup(2)] test for normality......Page 250
16.4 Weighting schemes......Page 251
16.4.1 Weights used in least-squares refinement with single-crystal diffraction data......Page 252
16.4.2 Robust-resistant weighting schemes and outliers......Page 253
16.4.3 Assessing weighting schemes......Page 254
16.5.1 R factors......Page 257
16.5.2 Significance testing......Page 258
16.6.1 Correlation and covariance......Page 259
16.7 Systematic errors......Page 261
16.7.1 Systematic errors in the data......Page 262
16.7.3 Errors and limitations of the model......Page 263
16.7.4 Assessment of a structure determination......Page 266
Exercises......Page 269
17.1 Introduction to powder diffraction......Page 270
17.2 Powder versus single-crystal diffraction......Page 271
17.3 Experimental methods......Page 273
17.4.1 Phase identification......Page 277
17.4.2 Quantitative analysis......Page 278
17.4.3 Peak-shape information......Page 279
17.5 Rietveld refinement......Page 280
17.6 Structure solution from powder diffraction data......Page 283
17.7 Non-ambient studies......Page 284
Exercises......Page 287
18.2 A simple model for twinning......Page 290
18.3 Twinning in crystals......Page 291
18.4 Diffraction patterns from twinned crystals......Page 293
18.5 Inversion, merohedral and pseudo-merohedral twins......Page 295
18.6 Derivation of twin laws......Page 298
18.7 Non-merohedral twinning......Page 299
18.8 The derivation of non-merohedral twin laws......Page 301
18.9 Common signs of twinning......Page 302
18.10 Examples......Page 304
Exercises......Page 315
19.1 Introduction......Page 318
19.3 Graphics programs......Page 319
19.4 Underlying concepts......Page 320
19.5 Drawing styles......Page 321
19.6 Creating three-dimensional illusions......Page 325
19.8 Textual information in drawings......Page 326
19.9 Some hints for effective drawings......Page 327
19.10 Tables of results......Page 328
19.11.1 Selected results......Page 329
19.11.3 Additional entries......Page 330
19.13.1 In research journals......Page 331
19.13.4 As oral presentations......Page 332
19.13.5 On the web......Page 333
19.14 Archiving of results......Page 334
20.2 Basics......Page 338
20.4 Some properties of the CIF format......Page 340
20.5.1 Strings......Page 342
20.5.2 Text......Page 343
20.5.3 Checking the CIF......Page 344
21.2 What types of search are possible?......Page 346
21.6 Short descriptions of crystallographic databases......Page 347
22.2 Laboratory X-ray sources......Page 352
22.3 Synchrotron X-ray sources......Page 354
22.4 Neutron sources......Page 358
A.2 Trigonometry......Page 362
A.3 Complex numbers......Page 363
A.4 Waves and structure factors......Page 364
A.5 Vectors......Page 365
A.7 Matrices......Page 367
A.8 Matrices in symmetry......Page 368
A.9 Matrix inversion......Page 369
A.10 Convolution......Page 370
B: Appendix B: Questions and answers......Page 372
E......Page 404
S......Page 405
Z......Page 406