Fundamentals of Crystallography

Fundamentals of
Crystallography
C. GIACOVAZZO, H. L MONACO, D. VITERBO
F. SCORDARI, G. GILLI, G. ZANOTTI, M. CATTI
Edited by
C. GIACOVAZZO
Dipartimento Geomineralogico, University of Bari, Italy
and Istituto di Ricerca per lo Sviluppo delle Metodologie
Cristallografiche,
CNR, Bari, Italy
INTERNATIONAL UNION OF CRYSTALLOGRAPHY
OXFORD UNIVERSITY PRESS
1992
Contents
List of contributors
1
Symmetry in crystals
Carmelo Giacovazzo
The crystalline State and isometric
Operations
Symmetry elements
Axes of rotational symmetry
Axes of rototranslation or screw axes
Axes of inversion
Axes of rotorefiection
Refiection planes with translational
component (glide planes)
Lattices
The rational properties of lattices
Crystallographic directions
Crystallographic planes
Symmetry restrictions due to the lattice
periodicity and vice versa
Point groups and symmetry classes
Point groups in one and two dimensions
The Laue classes
The seven crystal Systems
The Bravais lattices
Plane lattices
Space lattices
The space groups
The plane and line groups
On the matrix representation of symmetry
Operators
Appendices: L A The isometric
transformations
l.B Some combinations of
movements
l.C Wigner-Seitz cells
xiii
l.D The space-group
rotation matrices
l.E Symmetry groups
l.F Symmetry
generalization
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References
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Crystallographic Computing
Carmelo Giacovazzo
61
Introduction
The metric matrix
The reciprocal lattice
Basic transformations
Transformation from triclinic to
orthonormal axes
Rotations in Cartesian Systems
Some simple crystallographic calculations
Torsion angles
Best plane through a set of points
Best line through a set of points
Principal axes of a quadratic form
Metric considerations on the lattices
Niggli reduced cell
Sublattices and superlattices
Coincidence-site lattices
Twins
Calculation of the structure factor
Calculation of the electron density
function
The method of least Squares
Linear least Squares
Reliability of the parameter estimates
Linear least Squares with constraints
Non-linear (unconstrained) least Squares
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viii
Contents
Least-squares refinement of crystal
structures
Practical considerations on
crystallographic least Squares
Constraints and restraints in
crystallographic least Squares
Alternatives to the method of least
Squares
Rietveld refinement
The basis of the technique
Some practical aspects of Rietveld
refinement
Analysis of thermal motion
The effect of thermal motion on bond
lengths and angles
About the accuracy of the calculated
Parameters
Appendices: 2.A Some metric relations
between direct and
reciprocal lattices
2.B Some geometrical
calculations concerning
directions and planes
2.C Some transformation
matrices
2.D Reciprocity of F and I
lattices
2.E Transformations of
crystallographic
quantities in rectilinear
Spaces
2.F Derivation of the
normal equations
2.G Derivation of the
variance-covariance
matrix M^
2.H Derivation of the
unbiased estimate of
Mx
2.1 The FFT algorithm and
its crystallographic
applications
2.J Examples of twin laws
References
3
The diffraction of X-rays by crystals
Carmelo Giacovazzo
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131
Introduction
Thomson scattering
Compton scattering
Interference of scattered waves
Scattering by atomic electrons
Scattering by atoms
The temperature factor
Scattering by a molecule or by a unit cell
Diffraction by a crystal
Bragg's law
The reflection and the limiting spheres
Symmetry in reciprocal space
Friedel law
Effects of symmetry Operators in the
reciprocal space
Determination of the Laue class
Determination of reflections with
restricted phase values
Systematic absences
Unequivocal determination of the space
group
Diffraction intensities
Anomalous dispersion
The Fourier synthesis and the phase
problem
Modulated crystal structures
Appendices: 3.A Mathematical
background
3.B Scattering and related
topics
3.C Scattering of X-rays by
gases, liquids, and
amorphous solids
3.D About electron density
mapping
3.E Modulated structures
and quasicrystals
References
131
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137
Experimental methods in X-ray
crystallography
Hugo L. Monaco
141
X-ray sources
Conventional generators
Synchrotron radiation
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155
Contents
Monochromatization, collimation, and
focusing of X-rays
Data collection techniques for Single
crystals
The Weissenberg camera
The precession camera
The rotation (oscillation) method in
macromolecular crystallography
Densitometry
The single-crystal diffractometer
Area detectors
Data collection techniques for
polycrystalline materials
X-ray diffraction of polycrystalline
materials
Cameras used for polycrystalline
materials
Diffractometers used for polycrystalline
materials
Uses of powder diffraction
Data reduction
Lorentz correction
Polarization correction
Absorption corrections
Radiation damage corrections
Relative scaling
Appendices: 4.A Determination of the
number of molecules in
the unit cell of a crystal
References
Solution and refinement of crystal
structures
Davide Viterbo
Introduction
Statistical analysis of structure factor
amplitudes
The Patterson function and its use
The heavy atom method
Advanced Patterson methods
Direct methods
Introduction
Structure invariants and semi-invariants
Probability methods
Fixing the origin and the enantiomorph
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Phase determination procedures
Completing and refining the structure
Difference Fourier method
Least-squares method
Absolute config uration
Appendices: 5.A Structure factor
probability
distributions
5.B Patterson vector
methods
5.C Two examples of
deriving phase
Information from
positivity
5.D Probability formulae
for triplet invariants
5.E Pseudotranslational
symmetry
5.F Magic integers
5.G New multisolution
techniques
5.H Procedures for
completing a partial
model
References
ix
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397
lonic crystals
Fernando Scordari
403
The structure of the atom
Atoms with a Single electron
Atoms with more than one electron
Interactions between ions
Notes on chemical bonds
lonic crystals
Lattice energy: the contributions of
attractive and repulsive terms
Lattice energy: CFSE contribution
Applications of lattice energy
calculations
lonic radius
Maximum Alling principle
Coordination polyhedra
Radius ratio rule
Applications of the concept of ionic
radius
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x I Contents
Closest packings
Pauling's rules
Pauling's first rule
Pauling's second rule
Pauling's third rule
Pauling's fourth rule
Pauling's fifth rule
Ideal and defect structures
MX structures
MX 2 and M2X structures
MX 3 and M 2 X 3 structures
A m B„X p structures
On the Classification of Silicates
Liebau's crystallochemical Classification
Structural formulae
Relationship between Classification
Parameters and properties of the cations
Appendices: 6.A Application of the
concept of the packing
coefficient (c,)
6.B Structural inferences
from crystallochemical
Parameters
References
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440
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453
Molecules and molecular crystals
Gastone Gilli
465
Chemistry and X-ray crystallography
Crystal and molecular structure
The growth of structural Information
The nature of molecular crystals
Generalities
A more detailed analysis of
intermolecular forces
Thermodynamics of molecular crystals
Free and lattice energy of a crystal from
atom-atom potentials
Polymorphism and the prediction of
crystal structures
Effect of crystal forces on molecular
geometry
Elements of classical stereochemistry
Structure: Constitution, configuration,
and conformation
Isomerism
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486
Ring conformations
Ring conformation and group theory
Computation of puckering coordinates
Molecular geometry and the chemical
bond
An overview of bond theories
The VSEPR theory
Valence bond (VB) theory
Hybridization. The machinery
Molecular mechanics
Molecular hermeneutics: the
Interpretation of molecular structures
Correlative methods in structural
analysis
Some three-centre-four-electron linear
Systems
Nucleophilic addition to organometallic
Compounds
Nucleophilic addition to the carbonyl
group
A case of conformational rearrangement
Resonance assisted hydrogen bonding
(RAHB)
References
Protein crystallography
Giuseppe Zanotti
Introduction
Protein crystals
Principles of protein crystallization
The solvent content of protein crystals
Preparation of isomorphous heavy-atom
derivatives
How isomorphous are isomorphous
derivatives?
The Solution of the phase problem
The isomorphous replacement method
Anomalous scattering: a complementary
(or alternative) approach to the Solution
of the phase problem
The use of anomalous scattering in the
determination of the absolute
configuration of the macromolecule
The treatment of errors
Contents I xi
The refinement of heavy-atom
Parameters
Picking up minor heavy-atom sites:
the difference-Fourier synthesis
A third approach to the resolution of
the phase ambiguity: real-space filtering
Rotation and translation functions and
the molecular replacement method
Direct methods and the maximumentropy principle in macromolecular
crystallography
The Interpretation of electron density
maps and the refinement of the model
The interpretation of electron density
maps
Interactive Computer graphics and
model building
The refinement of the structure
Protein structure
General aspects
Levels of Organization of proteins:
secondary structure
Polypeptide chain description
Higher levels of Organization: tertiary
and quaternary structure, domains, and
subunits
Groups other than amino acids
Thermal parameters and disordered
structures
Solvent structure
The influence of crystal packing
Protein Classification
Appendices: 8.A Some formulae for
isomorphous
replacement and
anomalous dispersion
8.B Translation functions
8.C Macromolecular leastsquares refinement and
the conjugate-gradient
algorithm
8.D Conventions and
symbols for amino
acids and peptides
References and further reading
594
Physical properties of crystals
Michele Catti
599
Introduction
Crystal anisotropy and tensors
Tensorial quantities
Symmetry of tensorial properties
Overview of physical properties
Electrical properties of crystals
Pyroelectricity
Dielectric impermeability and optical
properties
Elastic properties of crystals
Crystal strain
Inner deformation
Stress tensor
Elasticity tensor
Examples and applications
Piezoelectricity
Symmetry properties of the piezoelectric
tensor
Crystal defects
Experimental methods
Planar defects
Line defects: dislocations
The Burgers circuit
X-ray topography of dislocations
Energy of a dislocation
Motion and interaction of dislocations
Partial dislocations
Small-angle grain boundaries
Point defects
Thermal distribution of defects
Diffusion
Ionic conductivity
Appendix: 9.A Properties of second-rank
tensors
Further reading
Index
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