Fundamentals of Crystallography

by Giacovazzo, Carmelo; Monaco, H. L.; Viterbo, D.; Scordari, F.; Gilli, G.; Zanotti, G.; Catti, M.

• ISBN13: 9780198555780
• ISBN10: 0198555784
• Format: Paperback
• Copyright: 1992-10-22
• Publisher: Oxford University Press

Summary

In recent years, crystallographic techniques have found application in a wide range of subjects, and these applications in turn have lead to exciting developments in the field of crystallography itself. This new textbook offers an introduction to these developments, and for the first time provides, in a single volume, a comprehensive and accessible overview of modern crystallography. The text offers a rigorous treatment of theory and describes experimental applications in many fields. These are covered in chapters on crystal structure and methods of its determination, symmetry, crystallographic computing, X-ray diffraction, protein crystallography, and crystal physics.

Table of Contents

List of contributors	xiii
Symmetry in crystals	1	(60)
Carmelo Giacovazzo
The crystalline state and isometric operations	1	(2)
Symmetry elements	3	(3)
Axes of rotational symmetry	3	(2)
Axes of rototranslation or screw axes	5	(1)
Axes of inversion	5	(1)
Axes of rotoreflection	5	(1)
Reflection planes with translational component (glide planes)	6	(1)
Lattices	6	(1)
The rational properties of lattices	7	(2)
Crystallographic directions	7	(1)
Crystallographic planes	8	(1)
Symmetry restrictions due to the lattice periodicity and vice versa	9	(2)
Point groups and symmetry classes	11	(6)
Point groups in one and two dimensions	16	(1)
The Laue classes	17	(1)
The seven crystal systems	17	(1)
The Bravais lattices	18	(4)
Plane lattices	18	(1)
Space lattices	19	(3)
The space groups	22	(8)
The plane and line groups	30	(2)
On the matrix representation of symmetry operators	32	(3)
Appendices
The isometric transformations	35	(2)
Some combinations of movements	37	(4)
Wigner-Seitz cells	41	(2)
The space-group rotation matrices	43	(2)
Symmetry groups	45	(10)
Symmetry generalization	55	(5)
References	60	(1)
Crystallographic computing	61	(80)
Carmelo Giacovazzo
Introduction	61	(1)
The metric matrix	61	(2)
The reciprocal lattice	63	(2)
Basic transformations	65	(3)
Transformation from triclinic to orthonormal axes	68	(1)
Rotations in Cartesian systems	69	(4)
Some simple crystallographic calculations	73	(4)
Torsion angles	73	(1)
Best plane through a set of points	74	(1)
Best line through a set of points	75	(1)
Principal axes of a quadratic form	75	(2)
Metric considerations on the lattices	77	(10)
Niggli reduced cell	77	(3)
Sublattices and superlattices	80	(1)
Coincidence-site lattices	81	(2)
Twins	83	(4)
Calculation of the structure factor	87	(1)
Calculation of the electron density function	88	(2)
The method of least squares	90	(19)
Linear least squares	90	(2)
Reliability of the parameter estimates	92	(1)
Linear least squares with constraints	92	(1)
Non-linear (unconstrained) least squares	93	(1)
Least-squares refinement of crystal structures	94	(4)
Practical considerations on crystallographic least squares	98	(6)
Constraints and restraints in crystallographic least squares	104	(4)
Alternatives to the method of least squares	108	(1)
Rietveld refinement	109	(8)
The basis of the technique	109	(3)
Some practical aspects of Rietveld refinement	112	(5)
Analysis of thermal motion	117	(3)
The effect of thermal motion on bond lengths and angles	120	(2)
About the accuracy of the calculated parameters	122	(2)
Appendices
Some metric relations between direct and reciprocal lattices	124	(1)
Some geometrical calculations concerning directions and planes	125	(2)
Some transformation matrices	127	(1)
Reciprocity of F and I lattices	127	(1)
Transformations of crystallographic quantities in rectilinear spaces	128	(2)
Derivation of the normal equations	130	(1)
Derivation of the variance-covariance matrix Mx	131	(1)
Derivation of the unbiased estimate of Mx	131	(1)
The FFT algorithm and its crystallographic applications	131	(2)
Examples of twin laws	133	(4)
References	137	(4)
The diffraction of X-rays by crystals	141	(88)
Carmelo Giacovazzo
Introduction	141	(1)
Thomson scattering	142	(2)
Compton scattering	144	(1)
Interference of scattered waves	144	(2)
Scattering by atomic electrons	146	(1)
Scattering by atoms	147	(1)
The temperature factor	148	(2)
Scattering by a molecule or by a unit cell	150	(1)
Diffraction by a crystal	151	(3)
Bragg's law	154	(1)
The reflection and the limiting spheres	154	(1)
Symmetry in reciprocal space	155	(6)
Friedel law	155	(1)
Effects of symmetry operators in the reciprocal space	156	(1)
Determination of the Laue class	156	(1)
Determination of reflections with restricted phase values	157	(2)
Systematic absences	159	(2)
Unequivocal determination of the space group	161	(1)
Diffraction intensities	161	(4)
Anomalous dispersion	165	(4)
The Fourier synthesis and the phase problem	169	(2)
Modulated crystal structures	171	(2)
Appendices
Mathematical background	173	(12)
Scattering and related topics	185	(16)
Scattering of X-rays by gases, liquids, and amorphous solids	201	(15)
About electron density mapping	216	(5)
Modulated structures and quasicrystals	221	(5)
References	226	(3)
Experimental methods in X-ray crystallography	229	(90)
Hugo L. Monaco
X-ray sources	229	(16)
Conventional generators	229	(5)
Synchrotron radiation	234	(7)
Monochromatization, collimation, and focusing of X-rays	241	(4)
Data collection techniques for single crystals	245	(42)
The Weissenberg camera	247	(7)
The precession camera	254	(5)
The rotation (oscillation) method in macromolecular crystallography	259	(9)
Densitometry	268	(5)
The single-crystai diffractometer	273	(8)
Area detectors	281	(6)
Data collection techniques for polycrystalline materials	287	(14)
X-ray diffraction of polycrystalline materials	287	(2)
Cameras used for polycrystalline materials	289	(4)
Diffractometers used for polycrystalline materials	293	(4)
Uses of powder diffraction	297	(4)
Data reduction	301	(11)
Lorentz correction	301	(2)
Polarization correction	303	(1)
Absorption corrections	304	(4)
Radiation damage corrections	308	(2)
Relative scaling	310	(2)
Appendices
Determination of the number of molecules in the unit cell of a crystal	312	(2)
References	314	(5)
Solution and refinement of crystal structures	319	(84)
Davide Viterbo
Introduction	319	(2)
Statistical analysis of structure factor amplitudes	321	(3)
The Patterson function and its use	324	(11)
The heavy atom method	328	(7)
Advanced Patterson methods	335	(1)
Direct methods	335	(30)
Introduction	335	(2)
Structure invariants and semi-invariants	337	(3)
Probability methods	340	(6)
Fixing the origin and the enantiomorph	346	(5)
Phase determination procedures	351	(14)
Completing and refining the structure	365	(10)
Difference Fourier method	366	(1)
Least-squares method	367	(7)
Absolute configuration	374	(1)
Appendices
Structure factor probability distributions	375	(2)
Patterson vector methods	377	(7)
Two examples of deriving phase information from positivity	384	(1)
Probability formulae for triplet invariants	385	(2)
Pseudotranslational symmetry	387	(1)
Magic integers	388	(2)
New multisolution techniques	390	(3)
Procedures for completing a partial model	393	(4)
References	397	(6)
Ionic crystals	403	(62)
Fernando Scordari
The structure of the atom	403	(3)
Atoms with a single electron	403	(1)
Atoms with more than one electron	404	(2)
Interactions between ions	406	(12)
Notes on chemical bonds	406	(3)
Ionic crystals	409	(1)
Lattice energy: the contributions of attractive and repulsive terms	410	(4)
Lattice energy: CFSE contribution	414	(3)
Applications of lattice energy calculations	417	(1)
Ionic radius	418	(6)
Maximum filling principle	424	(9)
Coordination polyhedra	425	(1)
Radius ratio rule	425	(2)
Applications of the concept of ionic radius	427	(2)
Closest packings	429	(4)
Pauling's rules	433	(3)
Pauling's first rule	433	(1)
Pauling's second rule	433	(2)
Pauling's third rule	435	(1)
Pauling's fourth rule	436	(1)
Pauling's fifth rule	436	(1)
Ideal and defect structures	436	(5)
MX structures	437	(1)
MX2 and M2X structures	438	(2)
MX3 and M2X structures	440	(1)
Am, BnXp structures	441	(4)
On the classification of silicates	445	(11)
Liebau's crystallochemical classification	447	(6)
Structural formulae	453	(1)
Relationship between classification parameters and properties of the cations	453	(3)
Appendices
Application of the concept of the packing coefficient (ci)	456	(3)
Structural inferences from crystallochemical parameters	459	(4)
References	463	(2)
Molecules and molecular crystals	465	(70)
Gastone Gilli
Chemistry and X-ray crystallography	465	(3)
Crystal and molecular structure	465	(2)
The growth of structural information	467	(1)
The nature of molecular crystals	468	(16)
Generalities	468	(5)
A more detailed analysis of intermolecular forces	473	(5)
Thermodynamics of molecular crystals	478	(2)
Free and lattice energy of a crystal from atom-atom potentials	480	(2)
Polymorphism and the prediction of crystalstructures	482	(1)
Effect of crystal forces on molecular geometry	483	(1)
Elements of classical stereochemistry	484	(15)
Structure: constitution, configuration, and conformation	484	(2)
Isomerism	486	(4)
Ring conformations	490	(2)
Ring conformation and group theory	492	(6)
Computation of puckering coordinates	498	(1)
Molecular geometry and the chemical bond	499	(12)
An overview of bond theories	499	(2)
The VSEPR theory	501	(1)
Valence bond (VB) theory	502	(2)
Hybridization. The machinery	504	(2)
Molecular mechanics	506	(5)
Molecular hermeneutics: the interpretation of molecular structures	511	(18)
Correlative methods in structural analysis	511	(1)
Some three-centre-four-electron linear systems	512	(2)
Nucleophilic addition to organometallic compounds	514	(1)
Nucleophilic addition to the carbonyl group	515	(1)
A case of conformational rearrangement	516	(5)
Resonance assisted hydrogen bonding (RAHB)	521	(8)
References	529	(6)
Protein crystallography	535	(64)
Giuseppe Zanotti
Introduction	535	(1)
Protein crystals	536	(4)
Principles of protein crystallization	536	(2)
The solvent content of protein crystals	538	(1)
Preparation of isomorphous heavy-atom derivatives	538	(2)
How isomorphous are isomorphous derivatives?	540	(1)
The solution of the phase problem	540	(22)
The isomorphous replacement method	540	(4)
Anomalous scattering: a complementary (or alternative) approach to the solution of the phase problem	544	(1)
The use of anomalous scattering in the determination of the absolute configuration of the macromolecule	545	(1)
The treatment of errors	546	(3)
The refinement of heavy-atom parameters	549	(2)
Picking up minor heavy-atom sites: the difference-Fourier synthesis	551	(1)
A third approach to the solution of the phase ambiguity: real-space filtering	551	(1)
Rotation and translation functions and the molecular replacement method	552	(8)
Direct methods and the maximum- entropy principle in macromolecular Crystallography	560	(2)
The interpretation of electron density maps and the refinement of the model	562	(10)
The interpretation of electron density maps	562	(1)
Interactive computer graphics and model building	562	(1)
The refinement of the structure	563	(9)
Protein structure	572	(15)
General aspects	573	(1)
Levels of organization of proteins: secondary structure	574	(3)
Polypeptide chain description	577	(1)
Higher levels of organization: tertiary and quaternary structure, domains, and subunits	578	(4)
Groups other than amino acids	582	(1)
Thermal parameters and disordered structures	583	(1)
Solvent structure	583	(1)
The influence of crystal packing	584	(1)
Protein classification	585	(2)
Appendices
Some formulae for isomorphous replacement and anomalous dispersion	587	(1)
Translation functions	588	(2)
Macromolecular least-squares refinement and the conjugate-gradient algorithm	590	(1)
Conventions and symbols for amino acids and peptides	591	(3)
References and further reading	594	(5)
Physical properties of crystals	599	(43)
Michele Catti
Introduction	599	(1)
Crystal anisotropy and tensors	600	(5)
Tensorial quantities	600	(3)
Symmetry of tensorial properties	603	(2)
Overview of physical properties	605	(1)
Electrical properties of crystals	605	(3)
Pyroelectricity	606	(1)
Dielectric impermeability and optical properties	607	(1)
Elastic properties of crystals	608	(11)
Crystal strain	609	(2)
Inner deformation	611	(2)
Stress tensor	613	(1)
Elasticity tensor	614	(3)
Examples and applications	617	(2)
Piezoelectricity	619	(3)
Symmetry properties of the piezoelectric tensor	620	(2)
Crystal defects	622	(1)
Experimental methods	623	(2)
Planar defects	625	(3)
Line defects: dislocations	628	(7)
The Burgers circuit	629	(1)
X-ray topography of dislocations	630	(2)
Energy of a dislocation	632	(1)
Motion and interaction of dislocations	633	(1)
Partial dislocations	634	(1)
Small-angle grain boundaries	635	(1)
Point defects	635	(1)
Thermal distribution of defects	636	(1)
Diffusion	637	(2)
Ionic conductivity	639	(1)
Appendix
Properties of second-rank tensors	640	(2)
Further reading	642	(3)
Index	645