Fundamentals of Crystallography

by Giacovazzo, C.; Monaco, H.L.; Artioli, G.; Viterbo, D.; Ferraris, G.; Gilli, G.; Zanotti, G.; Catti, M.

• ISBN13: 9780198509585
• ISBN10: 0198509588
• Edition: 2nd
• Format: Paperback
• Copyright: 2002-09-26
• Publisher: Oxford University Press

Summary

Crystallography and structure theory have recently received increasing interest due to their role in understanding biological structures, high-temperature superconductors, and effects on mineral properties related to changes in temperature and pressure. This book offers a comprehensive accountof the wide range of crystallography in many branches of science. The fundamentals, the most frequently used procedures and experimental techniques are all described in a detailed way. A number of appendices are devoted to more specialist aspects. The book is an updated and fully revised new editionwith emphasis on the wide range of topical applications and current areas of research. Ample illustrations help clarify the subject matter. To provide a better understanding of the basics of crystallography, a compact disk has been added to this new edition, offering the facilities of moderngraphics to simulate experiments, show complex images, and provide a number of exercises.

Table of Contents

List of contributors	xxi
Symmetry in crystals	1	(37)
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	(3)
Crystallographic directions	8	(1)
Crystallographic planes	8	(2)
Symmetry restrictions due to the lattice periodicity and vice versa	10	(2)
Point groups and symmetry classes	12	(6)
Point groups in one and two dimensions	18	(1)
The Laue classes	18	(1)
The seven crystal systems	18	(1)
The Bravais lattices	19	(5)
Plane lattices	19	(1)
Space lattices	20	(4)
The space groups	24	(8)
The plane and line groups	32	(1)
On the matrix representation of symmetry operators	32	(6)
Appendices	38	(99)
The isometric transformations	38	(2)
Direct movements	38	(1)
Opposite movements	39	(1)
Some combinations of movements	40	(4)
Wigner--Seitz cells	44	(1)
The space-group matrices	44	(3)
Symmetry groups	47	(12)
Sub groups	49	(1)
Cosets	50	(1)
Conjugate classes	51	(1)
Conjugate subgroups	52	(1)
Normal subgroups and factor groups	52	(2)
Maximal subgroups and minimal supergroups	54	(1)
Maximal subgroups and minimal supergroups for three-dimensional crystallographic point groups	55	(1)
Limiting groups in two and three dimensions	56	(1)
Representation of a group	57	(1)
Opposite movements	58	(1)
Symmetry generalization	59	(8)
The symmetry groups Gnm	59	(1)
The G1 groups	59	(1)
The G2 groups	59	(1)
The G3 groups	60	(3)
The G4 groups	63	(1)
The groups of colour symmetry	63	(1)
References	64	(3)
Crystallographic computing	67	(70)
Carmelo Giacovazzo
Introduction	67	(1)
The metric matrix	67	(2)
The reciprocal lattice	69	(3)
Basis transformations	72	(2)
Transformation from triclinic to orthonormal axes	74	(2)
Rotations in Cartesian systems	76	(4)
Some simple crystallographic calculations	80	(5)
Torsion angles	81	(1)
Best plane through a set of points	81	(1)
Best line through a set of points	82	(1)
Principal axes of a quadratic form	83	(2)
Metric considerations on the lattices	85	(6)
Niggli reduced cell	85	(3)
Sublattices and superlattices	88	(1)
Coincidence-site lattices	89	(2)
Calculation of the electron density function	91	(2)
Calculation of the structure factor	93	(2)
The method of least squares	95	(20)
Linear least squares	95
Linear least squares with constraints	9	(90)
Non-linear (unconstrained) least squares	99	(1)
Least-squares refinement of crystal structures	100	(4)
Practical considerations on crystallographic least squares	104	(7)
Constraints and restraints in crystallographic least squares	111	(4)
Alternatives to the method of least squares	115	(5)
Maximum likelihood refinement	116	(2)
Gradient methods	118	(2)
Rietveld refinement	120	(10)
Preliminary steps	120	(2)
The basis of the Rietveld refinement	122	(3)
Some practical aspects of Rietveld refinement	125	(5)
Analysis of thermal motion	130	(3)
The effect of thermal motion on bond lengths and angles	133	(2)
About the accuracy of the calculated parameters	135	(2)
Appendices	137	(50)
Some metric relations between direct and reciprocal lattices	137	(1)
Some geometrical calculations concerning directions and planes	138	(3)
Some transformation matrices	141	(1)
Reciprocity of F and I lattices	141	(1)
Transformations of crystallographic quantities in rectilinear spaces	142	(2)
Derivation of the normal equations	144	(1)
Derivation of the variance--covariance matrix Mx	145	(1)
Derivation of the unbiased estimate of Mx	145	(1)
The FFT algorithm and its crystallographic applications	146	(7)
References	147	(6)
The diffraction of X-rays by crystals	153	(34)
Carmelo Giacovazzo
Introduction	153	(1)
Basic properties of X-rays	153	(1)
Thomson scattering	154	(2)
Compton scattering	156	(1)
Interference of scattered waves	157	(1)
Scattering by atomic electrons	158	(2)
Scattering by atoms	160	(1)
The temperature factor	161	(2)
Scattering by a molecule or by a unit cell	163	(1)
Diffraction by a crystal	164	(4)
Bragg's law	168	(1)
The reflection and the limiting spheres	168	(1)
Symmetry in reciprocal space	169	(7)
Friedel law	170	(1)
Effects of symmetry operators in the reciprocal space	170	(1)
Determination of the Laue class	170	(1)
Determination of reflections with restricted phase values	171	(2)
Systematic absences	173	(3)
Unequivocal determination of the space group	176	(1)
Diffraction intensities	176	(4)
Anomalous dispersion	180	(5)
The Fourier synthesis and the phase problem	185	(2)
Appendices	187	(91)
Mathematical background	187	(15)
Dirac delta function	187	(2)
A mathematical model for the lattice	189	(1)
Convolutions	190	(3)
Some properties of convolutions	193	(1)
The Fourier transform	194	(2)
Some examples of Fourier transform	196	(4)
Fourier transform of spherically symmetric functions	200	(1)
Deconvolution of spectra	201	(1)
Scattering and related topics	202	(18)
Compton scattering	202	(1)
The anisotropic temperature factor	202	(3)
Symmetry restrictions on the anisotropic temperature factors	205	(3)
The Renninger effect and experimental phase determination by means of multiple diffraction experiments	208	(5)
Electron diffraction	213	(3)
Neutron scattering	216	(4)
About electron density mapping	220	(7)
References	224	(3)
Beyond ideal crystals	227	(51)
Carmelo Giacovazzo
Introduction	227	(1)
Ordering types	228	(1)
Crystal twins	229	(7)
Diffuse scattering	236	(8)
Thermal diffuse scattering	237	(2)
Disorder diffuse scattering	239	(5)
Modulated crystal structures	244	(3)
Quasicrystals	247	(9)
Introductory remarks	247	(1)
A mathematical basis	248	(2)
Aperiodic tiling and quasicrystals	250	(3)
Embedding quasicrystals in higher-dimensional space	253	(3)
Liquid crystals (or mesomorphic phases)	256	(4)
The paracrystal	260	(2)
Amorphous and liquid states	262	(8)
Diffraction from a finite statistically homogeneous object	263	(2)
Diffraction from a finite statistically homogeneous object with equal atoms	265	(3)
Diffraction from an isotropic statistically homogeneous object	268	(1)
The Debye formula	268	(2)
Diffraction by gases	270	(2)
Diffraction by liquids and amorphous bodies	272	(2)
Small-angle scattering	274	(4)
Appendices	278	(110)
Examples of twin laws	278	(3)
Cubic system	278	(1)
Tetragonal system	278	(1)
Hexagonal and tragonal systems	278	(2)
Orthorhombic system	280	(1)
Monoclinic system	280	(1)
Triclinic system	280	(1)
How to recognize and treat twins	281	(4)
Embedding of modulated structures in higher-dimensional space	285	(4)
About Fibonacci numbers and sequences	289	(6)
References	291	(4)
Experimental methods in X-ray and neutron crystallography	295	(93)
Hugo L. Monaco
Gilberto Artioli
Introduction	295	(1)
X-ray sources	295	(18)
Conventional generators	295	(5)
Synchrotron radiation	300	(6)
X-ray optics: monochromatization, collimation, and focusing of X-rays	306	(7)
Neutron sources	313	(5)
Nuclear reactors	314	(1)
Pulsed neutron sources	315	(2)
Neutron optics	317	(1)
X-ray and neutron detectors	318	(11)
X-ray detectors	320	(8)
Neutron detectors	328	(1)
Data collection techniques for single crystals	329	(19)
The Laue method	330	(1)
The single crystal cameras	331	(4)
The single crystal diffractometer	335	(13)
Data collection techniques for polycrystalline materials	348	(21)
Diffraction of polycrystalline materials	348	(4)
Cameras used for polycrystalline materials	352	(5)
Diffractometers used for polycrystalline materials	357	(5)
Applications of powder diffraction	362	(7)
In situ measurements in non-ambient conditions	369	(7)
High-temperature polycrystalline diffraction	372	(1)
Low-temperature single crystal diffractometry	373	(1)
High-pressure experiments	374	(2)
Data Reduction	376	(12)
Lorentz correction	377	(1)
Polarization correction	378	(2)
Absorption correction	380	(3)
Radiation damage correction	383	(3)
Relative scaling	386	(2)
Appendices	388	(88)
Determination of the number of molecules in the unit cell of a crystal	388	(2)
The cylindrical film cameras geometry	390	(6)
The precession camera geometry	396	(3)
The rotation method geometry	399	(14)
References	403	(10)
Solution and refinement of crystal structures	413	(63)
Davide Viterbo
Introduction	413	(2)
Statistical analysis of structure--factor amplitudes	415	(4)
The Patterson function and its use	419	(13)
The heavy atom method	424	(8)
Direct methods	432	(33)
Introduction	432	(2)
Structure invariants and semi-invariants	434	(4)
Probability methods	438	(6)
Fixing the origin and the enantiomorph	444	(4)
Phase determination procedures	448	(17)
Completing and refining the structure	465	(11)
Difference Fourier method	465	(2)
Least-squares method	467	(8)
Absolute configuration	475	(1)
Appendices	476	(270)
Structure--factor probability distributions	476	(3)
Patterson vector methods	479	(7)
Probability formulae for triplet invariants	486	(3)
Pseudotranslational symmetry	489	(1)
Magic integers	490	(1)
New multisolution techniques	491	(2)
Procedures for completing a partial model	493	(10)
Weights for Fourier syntheses	493	(2)
Syntheses for completing a partial model	495	(1)
References	496	(7)
Mineral and inorganic crystals	503	(82)
Giovanni Ferraris
Introduction	503	(2)
Bonding aspects	505	(33)
Chemical bond and solid-state properties	505	(1)
Melting	506	(1)
Cleavage	507	(1)
Structure and morphology	508	(1)
Morphology and optical properties	509	(1)
Representing crystal structures	509	(1)
The ionic radii	510	(2)
Packing of spheres	512	(4)
Coordination polyhedra	516	(1)
Interstitial sites	516	(2)
Ionic radii and coordination polyhedra	518	(3)
Electrostatic bond strength and Pauling's rules	521	(3)
Bond strength versus bond length	524	(1)
The charge distribution (CD) method	525	(3)
Applications of CD and ECoN	528	(2)
Bond valence and hydrogen bond	530	(1)
Bond valence and hydrates	531	(1)
Bond strength of the O . . . O hydrogen bond	532	(1)
Polymorphism	533	(1)
Solid solutions	534	(2)
Solid solutions, order/disorder and crystal-chemical formula	536	(2)
Structure types	538	(14)
Closest- and close-packing type structures	538	(1)
Packing spheres only	538	(1)
Filling tetrahedral sites	539	(4)
Filling octahedral sites	543	(6)
Filling octahedral and tetrahedral sites	549	(1)
More cp structures	550	(2)
Structures with complex anions	552	(9)
Orthosilicates	552	(2)
Disilicates and ring silicates	554	(1)
Chain silicates (inosilicates)	555	(1)
Layer (sheet) silicates (phyllosilicates)	556	(3)
Tectosilicates	559	(2)
More materials of technological interest	561	(1)
Modular structures	561	(14)
Polytypism	561	(1)
Modelling the structure of OD polytypes	562	(2)
Identification of long-period polytypes	564	(3)
Polysomatic series	567	(1)
Modelling structures of polysomes	568	(3)
Modular series and homology	571	(2)
Intersecting modular series	573	(1)
Miscellany of modular structures	574	(1)
Modulated structures	575	(1)
Real structures	575	(10)
Symmetry domains	576	(2)
Unmixing phenomena	578	(2)
References	580	(5)
Molecules and molecular crystals	585	(82)
Gastone Gilli
Chemistry and X-ray crystallography	585	(3)
Crystal and molecular structure	585	(2)
The growth of structural information	587	(1)
The nature of molecular crystals	588	(22)
Intermolecular forces	588	(8)
A more detailed analysis of intermolecular forces	596	(4)
Thermodynamics of molecular crystals	600	(3)
Free and lattice energy of a crystal from atom--atom potentials	603	(3)
Polymorphism	606	(1)
The prediction of the crystal structures	606	(4)
Elements of classical stereochemistry	610	(15)
Structure: constitution, configuration, and conformation	610	(2)
Isomerism	612	(5)
Ring conformations	617	(1)
Ring conformation and group theory	618	(6)
Computation of puckering coordinates	624	(1)
Molecular structure and chemical bond	625	(14)
Introduction	625	(1)
Quantum-mechanical methods	626	(2)
Qualitative bonding theories	628	(2)
The VSEPR theory	630	(2)
VB theory	632	(2)
Molecular mechanics (MM)	634	(3)
Molecular mechanics, force fields, and molecular modelling	637	(2)
Molecular hermeneutics: the interpretation of molecular structures	639	(28)
Correlation methods in structural analysis	639	(2)
Some three-centre-four-electron linear systems	641	(1)
Nucleophilic addition to organometallic compounds	642	(2)
Nucleophilic addition to the carbonyl group	644	(1)
Conformational rearrangements	645	(5)
Advances in hydrogen bond theory by structure-correlation methods	650	(7)
References	657	(10)
Protein crystallography	667	(79)
Giuseppe Zanotti
Introduction	667	(2)
Biological macromolecules	669	(14)
Globular proteins	669	(1)
Protein folding: general rules	670	(1)
Levels of organization of proteins: secondary structure	671	(5)
Representation of the polypetide chain conformation	676	(1)
Higher levels of organization: tertiary and quaternary structure, domains, and subunits	676	(2)
The influence of the medium	678	(1)
Groups other than amino-acids	679	(2)
Protein classification	681	(2)
Nucleic acids	683	(1)
Protein crystals	683	(11)
Principles of protein crystallization	685	(1)
Crystallization methods	686	(1)
Testing the conditions: factorial approaches	687	(2)
The solvent content of protein crystals	689	(2)
Cryotechniques	691	(1)
Preparation of isomorphous heavy-atom derivatives	692	(1)
How isomorphous are isomorphous derivatives?	693	(1)
The solution of the phase problem	694	(34)
The isomorphous replacement method	694	(1)
The determination of heavy-atom positions	695	(1)
The single isomorphous replacement (SIR) method	696	(2)
The classical solution of the problem of phase ambiguity: the MIR technique	698	(1)
Anomalous scattering: a complementary (or alternative) approach to the solution of the phase problem	699	(2)
The optimal choice of wavelength and the multiple anomalous dispersion (MAD) technique	701	(2)
The use of anomalous scattering in the determination of the absolute configuration of the macromolecule	703	(2)
The treatment of errors	705	(4)
The refinement of heavy atoms parameters	709	(2)
Maximum-likelihood and Bayesian estimates: an alternative approach in phase refinement	711	(1)
Picking up minor heavy-atom sites: the difference-Fourier synthesis	712	(1)
Density-modification: how to solve the phase ambiguity and improve the electron density map	713	(3)
Rotation and translation functions and the molecular replacement method	716	(2)
The first step in molecular replacement: the rotation function	718	(2)
The rotation matrix C and the choice of variables	720	(1)
Translation functions	720	(3)
Self-rotation and self-translation functions: improving the electron density maps	723	(2)
Practical hints in molecular replacement	725	(1)
The direct methods in macromolecular crystallography	725	(3)
The interpretation of the electron density maps and the refinement of the model	728	(18)
The interpretation of the electron density maps	728	(1)
Interactive computer graphics and model building	729	(2)
The refinement of the structure	731	(1)
Constrained versus restrained least-squares	732	(3)
Restrained and constrained least-squares	735	(1)
Crystallographic refinement by molecular dynamics	736	(1)
The strategy of the refinement of protein structures	737	(2)
R factor and Rfree: structure validation	739	(2)
Thermal parameters and disordered structures	741	(2)
The organization of solvent	743	(1)
The influence of crystal packing	744	(1)
Dynamical studies: time-resolved crystallography	744	(2)
Appendices	746	(64)
Some formulae for isomorphous replacement and anomalous dispersion	746	(2)
Translation functions	748	(1)
Macromolecular least-squares refinement and the conjugate-gradient algorithm	749	(2)
Conventions and symbols for amino acids and peptides	751	(8)
References	752	(7)
Physical properties of crystals: Phenomenology and modelling	759	(51)
Michele Catti
Introduction	759	(1)
Crystal anisotropy and tensors	760	(5)
Tensorial quantities	761	(2)
Symmetry of tensorial properties	763	(2)
Overview of physical properties	765	(1)
Electrical properties of crystals	766	(3)
Pyroelectricity	767	(1)
Dielectric impermeability and optical properties	768	(1)
Elastic properties of crystals	769	(12)
Crystal strain	770	(3)
Inner deformation	773	(2)
Stress tensor	775	(1)
Elasticity tensor	775	(4)
Examples and applications	779	(2)
Piezoelectricity	781	(4)
Symmetry properties of the piezoelectric tensor	783	(2)
Modelling of structural and elastic behaviour	785	(6)
Atomistic potential functions	785	(2)
Athermal equation of state	787	(2)
Elastic constants	789	(2)
Crystal defects	791	(1)
Experimental methods	792	(2)
Planar defects	794	(3)
Line defects: dislocations	797	(8)
The Burgers circuit	799	(1)
X-ray topography of dislocations	799	(2)
Energy of a dislocation	801	(1)
Motion and interaction of dislocations	802	(1)
Partial dislocations	803	(1)
Small-angle grain boundaries	804	(1)
Point defects	805	(1)
Thermal distribution of defects	806	(1)
Diffusion	807	(1)
Ionic conductivity	808	(2)
Appendices	810	(5)
Properties of second-rank tensors	810	(1)
Eigenvalues and eigenvectors	810	(1)
Representation surfaces and their properties	811	(4)
References	813	(2)
Index	815