Symmetry of Crystals and Molecules

Content: 1. Symmetry everywhere --
1.1. Introduction --
1.2. Looking at symmetry --
1.3. Some symmetrical objects --
1.4. Denning symmetry --
1.5. Symmetry in science --
1.6. Symmetry in music --
1.7. Symmetry in architecture --
1.8. Summary and notation --
1.8.1. Introducing symmetry notation --
References 1 --
Problems 1 --
2. Geometry of crystals and molecules --
2.1. Introduction --
2.2. Reference axes --
2.2.1. Crystallographic axes --
2.3. Equation of a plane --
2.4. Miller indices --
2.4.1. Miller --
Bravais indices --
2.5. Zones --
2.5.1. Weiss zone equation --
2.5.2. Addition rule for crystal planes --
2.6. Projection of three-dimensional features --
2.6.1. Stereographic projection --
2.6.2. Calculations in stereographic projections --
2.6.3. Axial ratios and interaxial angles --
2.7. Molecular geometry: VSEPR theory --
2.8. Molecular geometry: experimental determination --
2.8.1. Interatomic distances and angles --
2.8.2. Conformational parameters 2.8.3. Internal coordinates --
2.8.4. Errors and precision --
2.9. Molecular geometry: theoretical determination --
2.9.1. The Schrodinger equation --
2.9.2. Atomic orbitals --
2.9.3. Normalization --
2.9.4. Probability distributions --
2.9.5. Atomic\'s and p orbitals --
2.9.6. Chemical species and molecular orbitals --
2.10. Crystal packing --
References 2 --
Problems 2 --
3. Point group symmetry --
3.1. Introduction --
3.2. Symmetry elements, symmetry operations and symmetry operators --
3.3. Point groups --
3.4. Symmetry in two dimensions --
3.4.1. Rotation symmetry --
3.4.2. Reflection symmetry --
3.4.3. Combinations of symmetry operations in two dimensions --
3.4.4. Two-dimensional systems and point group notation --
3.4.5. Subgroups --
3.5. Three-dimensional point groups --
3.5.1. Rotation symmetry in three dimensions --
3.5.2. Reflection symmetry in three dimensions --
3.5.3. Roto-inversion symmetry --
3.5.4. Stereogram representations of three-dimensional point groups --
3.5.5. Crystallographic point groups --
3.5.6. Crystal classes 3.5.7. Crystal systems --
3.6. Derivation of point groups --
3.6.1. Ten simple point groups --
3.6.2. Combinations of symmetry operations in three dimensions --
3.6.3. Euler\'s construction --
3.6.4. Centrosymmetric point groups (Laue groups) and Laue classes --
3.6.5. Projected symmetry --
3.7. Point groups and physical properties of crystals and molecules --
3.7.1. Enantiomorphism and chirality --
3.7.2. Optical properties --
3.7.3. Pyroelectricity and piezoelectricity --
3.7.4. Dipole moments --
3.7.5. Infrared and Raman activity --
3.8. Point groups and chemical species --
3.8.1. Point groups R --
3.8.2. Point groups R --
3.8.3. Point groups R1 --
3.8.4. Point groups R2 --
3.8.5. Point groups Rm --
3.8.6. Point groups Rm --
3.8.7. Point groups R2 and 1 --
3.9. Non-crystallographic point groups --
3.10. Hermann --
Mauguin and Schonflies point group symmetry notations --
3.10.1. Roto-reflection (alternating) axis of symmetry --
3.10.2. The two symmetry notations compared --
3.11. Point group recognition --
3.12. Matrix representation of point group symmetry operations 3.12.1. Rotation matrices --
3.13. Non-periodic crystals --
3.13.1. Quasicrystals --
3.13.2. Buckyballs --
3.13.3. Icosahedral symmetry --
References 3 --
Problems 3 --
4. Lattices --
4.1. Introduction --
4.2. One-dimensional lattice --
4.3. Two-dimensional lattices --
4.3.1. Choice of unit cell --
4.3.2. Nets in the oblique system --
4.3.3. Nets in the rectangular system --
4.3.4. Square and hexagonal nets --
4.3.5. Unit cell centring --
4.4. Three-dimensional lattices --
4.4.1. Triclinic lattice --
4.4.2. Monoclinic lattices --
4.4.3. Orthorhombic lattices --
4.4.4. Tetragonal lattices --
4.4.5. Cubic lattices --
4.4.6. Hexagonal lattice --
4.4.7. Trigonal lattices --
4.5. Lattice directions --
4.6. Law of rational intercepts: reticular density --
4.7. Reciprocal lattice --
4.8. Rotational symmetry of lattices --
4.9. Lattice transformations --
4.9.1. Bravais lattice unit cell vectors --
4.9.2. Zone symbols and lattice directions --
4.9.3. Coordinates of points in the direct unit cell --
4.9.4. Miller indices --
4.9.5. Reciprocal unit cell vectors 4.9.6. Volume relationships --
4.9.7. Reciprocity of F and I unit cells --
4.9.8. Wigner-Seitz cells --
References 4 --
Problems 4 --
5. Space groups --
5.1. Introduction --
5.2. One-dimensional space groups --
5.3. Two-dimensional space groups --
5.3.1. Plane groups in the oblique system --
5.3.2. Plane groups in the rectangular system --
5.3.3. Limiting conditions on X-ray reflections --
5.3.4. Plane groups in the square and hexagonal systems --
5.3.5. The seventeen plane groups summarized --
5.3.6. Comments on notation --
5.4. Three-dimensional space groups --
5.4.1. Triclinic space groups --
5.4.2. Monoclinic space groups --
5.4.3. Space groups related to point group 2 --
5.4.4. Screw axes --
5.4.5. Space groups related to point group m: glide planes --
5.4.6. Space groups related to point group 2/m --
5.4.7. Summary of the monoclinic space groups --
5.4.8. Half-shift rule --
5.4.9. Orthorhombic space groups --
5.4.10. Change of origin --
5.4.11. Standard and alternative settings of space groups --
5.4.12. Tetragonal space groups 5.4.13. Space groups in the trigonal and hexagonal systems --
5.4.14. Cubic space groups --
5.4.15. Space groups and crystal structures --
5.5. Matrix representation of space group symmetry operations --
5.6. Black-white and colour symmetry --
5.6.1. Black-white symmetry: potassium chloride --
5.6.2. Colour symmetry --
5.7. The international tables and other crystallographic compilations --
5.7.1. The international tables for crystallography, Vol. A --
References 5 --
Problems 5 --
6. Symmetry and X-ray diffraction --
6.1. Introduction --
6.2. X-ray diffraction --
6.3. Recording X-ray diffraction spectra --
6.4. Reciprocal lattice and Ewald\'s construction --
6.5. X-ray intensity data collection --
6.5.1. Laue X-ray photography --
6.5.2. Laue projection symmetry --
6.5.3. X-ray precession photography --
6.5.4. Diffractometric and image plate recording of X-ray intensities --
6.6. X-ray scattering by a crystal: the structure factor --
6.6.1. Limiting conditions and the structure factor --
6.6.2. Geometrical structure factor for a centrosymmetric crystal 6.6.3. Geometrical structure factor for an I centred unit cell --
6.6.4. Geometrical structure factor for space group P21/c --
6.6.5. Geometrical structure factor for space group Pmd2 --
6.6.6. Geometrical structure factor for space group P63/m --
6.7. Using X-ray diffraction information --
References 6 --
Problems 6 --
7. Elements of group theory --
7.1. Introduction --
7.2. Group requirements --
7.3. Group definitions --
7.4. Examples of groups --
7.4.1. Group multiplication tables --
7.4.2. Reference axes in group theory --
7.4.3. Subgroups and cosets --
7.4.4. Similarity transformations, conjugates and symmetry classes --
7.5. Representations and character tables --
7.5.1. Representations on position vectors --
7.5.2. Representations on basis vectors --
7.5.3. Representations on atom vectors --
7.5.4. Representations on functions --
7.6. A first look at character tables --
7.6.1. Transformation of atomic orbitals --
7.6.2. Orthonormality and orthogonality --
7.6.3. Notation for irreducible representations --
7.6.4. Complex characters 7.6.5. Linear groups --
7.6.6. Some properties of character tables --
7.7. The great orthogonality theorem --
7.8. Reduction of reducible representations --
7.9. Constructing a character table --
7.9.1. Summary of the properties of character tables --
7.9.2. Constructing the character table for point group D3h --
7.9.3. Handling complex characters --
7.10. Direct products --
7.10.1. Representations on direct product functions --
7.10.2. Formation of a character table by direct products --
7.10.3. How the direct product has been used --
References 7 --
Problems 7 --
8. Applications of group theory --
8.1. Introduction --
8.2. Structure and symmetry in molecules and ions --
8.2.1. Application of models --
8.2.2. Application of diffraction studies --
8.2.3. Application of theoretical studies --
8.2.4. Monte Carlo and molecular dynamics techniques --
8.2.5. Symmetry adapted molecular orbitals --
8.2.6. Transition metal compounds: crystal-field and ligand-field theories --
8.2.7. The hexacyanoferrate(II) ion --
8.3. Vibrational studies 8.3.1. Symmetry of normal modes --
8.3.2. Boron trifluoride --
8.3.3. Selection rules for infrared and Raman activity: dipole moment and polarizability --
8.3.4. Harmonics and combination vibrations --
8.4. Group theory and point groups --
8.4.1. Cyclic point groups --
8.4.2. Dihedral point groups --
8.4.3. Cubic rotation point groups --
8.4.4. Point groups from combinations of operators --
8.5. Group theory and space groups --
8.5.1. Triclinic and monoclinic space groups --
8.5.2. Orthorhombic space groups --
8.5.3. Tetragonal space groups --
8.5.4. Cubic space groups --
8.6. Factor groups --
8.6.1. Factor group analysis of iron(II) sulphide --
8.6.2. Symmetry ascent and correlation --
8.6.3. Site group and factor group analyses --
References 8 --
Problems 8 --
9. Computer-assisted studies --
9.1. Introduction 9.2. Derivation of point groups --
9.3. Recognition of point groups --
9.4. Internal and Cartesian coordinates --
9.5. Molecular geometry --
9.6. Best-fit plane --
9.7. Reduction of a representation in point group D6h --
9.8. Unit cell reduction --
9.9. Matrix operations --
9.10. Zone symbol or Miller indices --
9.11. Linear least squares --
Reference 9 --
A1. Stereoviews and crystal models --
A1.1. Stereoviews and stereoviewing --
A1.2. Crystal models --
References --
A2. Analytical geometry of direction cosines --
A2.1. Direction cosines of a line --
A2.2. Angle between two lines --
A3. Vectors and matrices --
A3.1. Introduction --
A3.2. Vectors --
A3.3. Volume of a parallelepiped --
A3.4. Matrices --
A3.5. Normal to a plane (hkl) --
A3.6. Solution of linear simultaneous equations --
A3.7. Useful matrices --
A4. Stereographic projection of a circle is a circle