Crystal Structure Analysis: Principles and Practice (International Union of Crystallography Texts on Crystallography, 13)

This text focuses on the practical aspects of crystal structure analysis, and provides the necessary conceptual framework for understanding and applying the technique. By choosing an approach that does not put too much emphasis on the mathematics involved, the book gives practical advice on topics such as growing crystals, solving and refining structures, and understanding and using the results. The technique described is a core experimental method in modern structural chemistry, and plays an ever more important role in the careers of graduate students, postdoctoral and academic staff in chemistry, and final-year undergraduates. Much of the material of the first edition has been significantly updated and expanded, and some new topics have been added. The approach to several of the topics has changed, reflecting the book's new authorship, and recent developments in the subject. Booknews Explains the theory and methods behind the techniques of x-ray scattering structure analysis of crystals. Downplaying, but not eliminating, the mathematics involved, chapters discuss: crystal growth, evaluation, and mounting; symmetry and space group determination, data collection using four-circle diffractometers, area detectors, Fourier syntheses, structure determination by Patterson methods, direct methods of crystal structure determination, maximum entropy, least-squares fitting of parameters, structure refinement, the derivation and interpretation of results, the crystallographic information file, and crystallographic databases. Originating in an intensive course in x-ray structure analysis organized on behalf of the Chemical Crystallography Group of the British Crystallographic Association, this text assumes basic knowledge of crystal structure determination. Annotation c. Book News, Inc., Portland, OR (booknews.com) 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 This text focuses on the practical aspects of crystal structure analysis and provides the necessary conceptual framework for understanding and applying the technique. Many worked examples, problems with answers, and illustrations are included throughout to reinforce the material presented. Significantly updated from the first edition.