Crystal Structure Analysis Principles and Practice

by Main, Peter; William, Clegg; Blake, Alexander J.; Gould, Robert O.

• ISBN13: 9780198506188
• ISBN10: 019850618X
• Format: Hardcover
• Copyright: 2002-02-21
• Publisher: International Union of Crystallography

Summary

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 avoids undue emphasis on the mathematics involved, the book gives practical advice on topics suchas 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 final-year undergraduates, graduate students, postdoctoral andacademic staff in chemistry.

Table of Contents

A basic introduction to X-ray crystallography	1	(9)
X-ray scattering from electrons	1	(1)
X-ray scattering from atoms	1	(2)
X-ray scattering from the contents of a unit cell	3	(1)
The effects of the crystal lattice	3	(1)
X-ray scattering from the crystal	4	(1)
The structure factor equation	4	(1)
The electron density equation	5	(1)
A mathematical relationship	6	(1)
Bragg's law	6	(1)
Resolution	7	(1)
The phase problem	8	(1)
A flowchart for crystal structure determination	9	(1)
Crystal growth, evaluation and mounting	10	(14)
Crystal growth	10	(1)
Survey of methods	10	(7)
Solution methods	10	(5)
Sublimation	15	(1)
Fluid phase growth	15	(1)
Solid-state synthesis	16	(1)
General comments	16	(1)
Sample evaluation	17	(1)
Microscopy	17	(1)
X-ray photography	18	(1)
Diffractometry	18	(1)
Crystal mounting	18	(6)
Standard procedures	18	(2)
Air-sensitive crystals	20	(1)
References	21	(1)
Some references on crystal growth and handling	21	(1)
Exercises	22	(2)
Symmetry and space group determination	24	(18)
Introduction	24	(1)
Basic operations and point groups	25	(1)
External morphology	26	(3)
Diffraction symmetry and the amount of independent data	29	(1)
Internal symmetry and translational symmetry operations	29	(3)
Detection of symmetry elements from intensity statistics	32	(2)
Further notes on space group symbols	34	(3)
Symmetry restrictions on atoms in special positions	37	(5)
Exercises	38	(4)
Background theory for data collection	42	(9)
Introduction	42	(1)
The geometry of X-ray diffraction	42	(2)
The reciprocal lattice	44	(1)
Unit cell and orientation matrix on a diffractometer	45	(2)
Obtaining a matrix and cell from initially found reflections	47	(1)
Symmetry aspects of the diffraction pattern	48	(3)
References	49	(1)
Exercises	49	(2)
Data collection using four-circle diffractometers	51	(18)
Introduction	51	(1)
Experimental conditions	51	(2)
Radiation	51	(1)
Temperature	52	(1)
Other conditions	52	(1)
Getting started	53	(5)
Reflection searching	53	(2)
Indexing, orientation matrix and cell determination	55	(1)
Finding the correct cell	56	(1)
Obtaining a good orientation matrix	57	(1)
Obtaining the best unit cell dimensions	58	(1)
Preparing for data collection	58	(4)
Introduction	58	(1)
Parameters	59	(3)
Data collection	62	(1)
Gross systematic errors	63	(1)
Correction of intensity data	63	(6)
Absorption corrections	63	(2)
Decay corrections	65	(1)
Other possible corrections	65	(1)
Exercises	66	(3)
Area detectors	69	(8)
Introduction	69	(1)
Types of area detectors	69	(3)
Some characteristics of CCD area detector systems	72	(2)
A typical experiement	74	(3)
Crystal screening	74	(1)
Unit cell and orientation matrix determination	74	(2)
Data collection	76	(1)
Data reduction and corrections	76	(1)
Fourier syntheses	77	(13)
Fourier synthesis in 1 D	77	(1)
A 1 D example---iron pyrites	78	(2)
The 2D synthesis	80	(4)
The 3D synthesis	84	(1)
Uses of Fouriers	84	(1)
Weighted Fouriers	85	(5)
Exercises	87	(3)
Structure determination by Patterson methods	90	(12)
The heavy atom method	93	(3)
Patterson search techniques	96	(6)
Rotation search	96	(1)
Translation search	97	(3)
Exercises	100	(2)
Direct methods of crystal structure determination	102	(17)
Amplitudes and phases	102	(1)
The physical basis of direct methods	103	(1)
Constraints on the electron density	103	(6)
Discrete atoms	104	(1)
Non-negative electron density	104	(2)
Random atomic distribution	106	(1)
Maximum value of ∫ ρ3(x) dV	107	(1)
Equal atoms	108	(1)
Maximum entropy	108	(1)
Equal molecules and ρ(x) = constant	109	(1)
Structure invariants	109	(1)
Structure determination	109	(10)
Calculation of E values	110	(1)
Setting up phase relationships	111	(1)
Finding reflections for phase determination	111	(1)
Assignment of starting phases	112	(1)
Phase determination and refinement	112	(1)
Figures of merit	113	(1)
Interpretation of maps	114	(1)
Completion of the structure	114	(1)
References	114	(1)
Exercises	114	(3)
General bibliography	117	(2)
An introduction to maximum entropy	119	(6)
Entropy	119	(1)
Maximum entropy	119	(1)
Calculations with incomplete data	120	(2)
Forming images	122	(1)
Entropy and probability	122	(1)
Electron density maps	123	(2)
Least-squares fitting of parameters	125	(10)
Weighted mean	125	(1)
Linear regression	126	(1)
Variances and covariances	127	(1)
Restraints	128	(1)
Constraints	129	(2)
Non-linear least squares	131	(2)
Ill-conditioning	133	(1)
Computing time	134	(1)
Exercises	134	(1)
Practical aspects of structure refinement	135	(13)
Introduction	135	(1)
Data	136	(2)
Parameters	138	(2)
Constraints	140	(1)
Restraints	141	(1)
Refinement procedures	142	(2)
Disorder	144	(1)
Twinning	144	(1)
Absolute structure	145	(1)
Other problems	146	(2)
Exercises	147	(1)
The derivation of results	148	(21)
Introduction	148	(1)
Statistical background	149	(7)
Some basic mathematics and statistics	149	(4)
Errors, precision and accuracy	153	(1)
Estimated standard deviations/standard uncertainties in crystallographic results	154	(2)
Analysis of the agreement between observed and calculated data	156	(3)
Observed and calculated data	157	(2)
Significance testing	159	(1)
Geometry	159	(5)
Bond lengths, bond angles and torsion angles	160	(2)
Least-squares planes and dihedral angles	162	(1)
Conformations of rings and other molecular features	163	(1)
Hydrogen atoms and hydrogen bonding	163	(1)
Thermal motion	164	(5)
β B and U parameters	164	(1)
`The equivalent isotropic displacement parameter'	165	(1)
Models of thermal motion and geometrical corrections: rigid body motion	166	(1)
Temperature and atomic displacement parameters	167	(1)
References	167	(1)
Exercises	167	(2)
The interpretation of results	169	(15)
Introduction	169	(1)
Averages, comparisons and differences	169	(6)
Comparison of geometrical parameters	169	(2)
Averaging geometrical parameters	171	(1)
When is a set of atoms genuinely planar?	172	(1)
Comparing different structures	173	(2)
Interpretation of interatomic distances and bonds	175	(1)
The effects of errors on structural results	176	(5)
Systematic errors in the data	177	(1)
Data thresholds and weighting	178	(1)
Errors and limitations of the model	178	(3)
Assessment of a structure determination	181	(3)
References	182	(1)
Exercises	183	(1)
The presentation of results	184	(21)
Introduction	184	(1)
Graphics	184	(8)
Graphics programs	185	(1)
Underlying concepts	185	(2)
Drawing styles	187	(5)
Creating three-dimensional illusions	192	(1)
The use of colour	193	(1)
Textual information in drawings	194	(1)
Some hints for effective drawings	194	(2)
Tables of results	196	(1)
The content of tables	196	(2)
The format of tables	198	(1)
Hints on presentation	198	(3)
Archiving of results	201	(4)
References	202	(1)
Exercises	203	(2)
The Crystallographic Information File	205	(9)
Introduction	205	(1)
Basics	205	(2)
Uses of CIF	207	(1)
Some properties of the CIF format	207	(2)
Some practicalities	209	(5)
Strings	209	(1)
Text	209	(1)
Checking the CIF	210	(1)
References	210	(1)
Exercises	211	(3)
Crystallographic databases	214	(4)
Available structural databases	214	(1)
Contents of the Cambridge Structural Database	215	(1)
Searching the CSD	216	(2)
Other topics	218	(9)
Twinning	218	(2)
Anomalous dispersion	220	(3)
Sources of X-rays	223	(4)
References	226	(1)
Appendix 1 Useful mathematics and formulae	227	(10)
Appendix 2 A short crystallographic dictionary	237	(8)
Appendix 3 Answers to exercises	245	(18)
Index	263