Modern X-Ray Analysis on Single Crystals (eBook)

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Peter Luger, Free University Berlin, Germany

1 Introduction 13
1.1 Historical remarks 13
1.2 The crystal lattice: Basic definitions 14
1.2.1 Periodicity, lattice constants 14
1.2.2 Lattice points, lattice planes, reciprocal lattice 17
1.3 Sample structures 25
2 Fundamental results of diffraction theory, X-radiation 26
2.1 Electron density and related functions 26
2.2 Diffraction conditions for single crystals 29
2.3 X-rays 33
2.3.1 Generation of X-rays 33
2.3.2 Absorption 38
2.3.3 Filters, monochromators 43
2.3.4 X-ray tubes 46
2.3.5 Synchrotron radiation 51
3 Preliminary experiments 54
3.1 Film methods 54
3.1.1 The rotation method 54
3.1.2 Zero level Weissenberg method 58
3.1.3 Upper level Weissenberg – normal beam and equi-inclination method 62
3.1.4 Precession technique 66
3.2 Practicing film techniques 73
3.2.1 Choice of experimental conditions 73
3.2.2 Rotation and Weissenberg photographs of KAMTRA and SUCROS 75
4 Crystal symmetry 80
4.1 Symmetry operations in a crystal lattice 80
4.1.1 Introduction 80
4.1.2 Basic symmetry operations 81
4.1.3 Crystal classes and related coordinate systems 85
4.1.4 Translational symmetry, lattice types and space groups 94
4.2 Crystal symmetry and related intensity symmetry 106
4.2.1 Representation of ? and F as Fourier series 106
4.2.2 Thermal motion, displacement parameters 110
4.2.3 Intensity symmetry, asymmetric unit 115
4.2.4 Systematic extinctions 122
4.2.5 Quasicrystals 124
4.3 Space group determination 127
4.3.1 General considerations, practical aspects 127
4.3.2 Space groups of KAMTRA and SUCROS from film exposures 132
5 Diffractometer measurements 138
5.1 Point detector data collection on a four-circle diffractometer 138
5.1.1 Eulerian cradle geometry 139
5.1.2 Choice of experimental conditions 143
5.1.3 Precise determination of lattice constants 146
5.1.4 Intensity measurements 147
5.2 Area detector diffractometers 153
5.2.1 Imaging plates 154
5.2.2 CCD diffractometers 156
5.2.3 Data processing for area detector measurements 158
5.2.4 Data collections for B12 and C60F18 159
6 Computer programs 162
7 Solution of the phase problem 166
7.1 Data reduction 166
7.2 Fourier methods 171
7.2.1 Interpretation of the Patterson function 171
7.2.2 Heavy atom methods, principle of difference electron density 173
7.2.3 Harker sections, applications to KAMTRA 177
7.2.4 Numerical calculation of Fourier syntheses 182
7.3 Direct methods 184
7.3.1 Normalization 184
7.3.2 Fundamental formulae 192
7.3.3 Origin definition, choice of starting set 199
7.3.4 Application of direct methods, the examples of SUCROS and C60F18 207
7.4 Phase determination for macromolecules 209
8 Refinements 215
8.1 Theoretical aspects 215
8.1.1 Model versus experiment, R-value 215
8.1.2 Theory of least-squares refinement 216
8.2 Practicing least-squares methods 224
8.2.1 Aspects of numerical calculations 224
8.2.2 Execution of a complete refinement process 224
8.2.3 Corrections to be applied during refinement 226
8.3 Analysis and representation of results 234
8.3.1 Geometrical data 234
8.3.2 Graphical representations 236
8.3.3 Archiving data, crystallographic information file (CIF) 238
8.4 Applications to the test structures 241
8.4.1 KAMTRA and SUCROS: Completion and refinement 241
8.4.2 Refinement of C60F18 247
8.4.3 Structure solution and refinement of B12 248
9 Structure analysis at non-routine conditions 252
9.1 Low temperature X-ray diffraction experiments 252
9.2 Neutron diffraction 256
9.3 Structure analysis at high pressures 259
9.4 Electron density 261
9.4.1 Multipole model, experimental conditions 261
9.4.2 Topological analysis using QTAIM 265
9.4.3 Examples in the Life Sciences 274
10 Concluding remarks and outlook 282
Appendix: Mathematics 284
A.1 Matrices, vectors 284
A.1.1 Introduction 284
A.1.2 Matrices, determinants, linear equations 284
A.1.3 Vector algebra 293
A.1.4 Linear independency, bases, reciprocal bases 300
A.1.5 Basis transformations 308
A.1.6 Lines and planes 312
A.2 Fourier transforms and convolution operations 317
A.3 Counter statistics, mean value, standard deviation 322
Bibliography 327
Index 335

lt;P>"This completely revised edition is a guide for practical work in X-ray analysis. An introduction to basic crystallography moves quickly to a practical and experimental treatment of structure analysis. Emphasis is placed on understanding results and avoiding pitfalls. Essential reading for researchers from the student to the professional level interested in understanding the structure of molecules."
INIS Database, DE18F3079

"Throughout the book, the mathematical bases of the concepts being described are given in an easily readable presentation. One particularly useful feature is the Appendix on mathematics, which deals with the main mathematical tools used in crystallography, namely matrix and vector algebra, basis transformation and Fourier transforms. The very clear presentation means that this section should be very helpful to the student with a weaker mathematical background.[…] This text is very attractively presented, with many fine & clear illustrations. It can strongly be recommended as an introduction to small molecule crystallography."
Louis J. Farrugia, University of Glasgow

"In conclusion, this comprehensive book is very well written and illustrated; it is a book written by a colleague widely recognized in the crystallography community. I recommend it to all structural scientists who are starting out in crystallography and want to understand crystal structure determinations of organic and organometallic structures; it is also an excellent book for professional crystallographers."
Claude Lecomte in: Acta Cryst. 72/2016