Crystallography Lecture notes Many other things Crystallography H. K. D. H. Bhadeshia Introduction and point groups
Stereographic projections Low symmetry systems Space groups Deformation and texture Interfaces, orientation relationships Martensitic transformations Introduction Liquid Crystals (Z. Barber) Form Anisotropy (elastic modulus, MPa)
Ag Mo Polycrystals The Lattice Centre of symmetry and inversion Bravais Lattices Triclinic P Monoclinic P & C Orthorhombic P, C, I & F
Tetragonal P & I Hexagonal Trigonal P Cubic P, F & I Bravais Lattices body-centred cubic (ferrite)
face-centred cubic (austenite) Bundy (1965) Fe Ru 6d2s Os Hs Cohesive energy (eV/atom) Pure iron
-65 -55 -45 -35 Cubic-P Cohesive energy (eV/atom) Diamond cubic Pure iron Hexagonal-P b.c.c c.c.p h.c.p 0.8 1.0 1.2 1.4 1.6 Normalised volume Paxton et al. (1990) 2D lattices Graphene, nanotubes Amorphous - homogeneous, isotropic
Crystals - long range order, anisotropic Crystals - solid or liquid Crystals - arbitrary shapes Polycrystals Lattice, lattice points Unit cell, space filling Primitive cell, lattice vectors Bravais lattices Directions, planes Weiss zone rule Symmetry Crystal structure Point group symmetry Point group symbols Examples Crystal Structure 1/2 1/2 1/2 1/2 lattice + motif = structure
primitive cubic lattice motif = Cu at 0,0,0 Zn at 1/2, 1/2, 1/2 Lattice: face-centred cubic Motif: C at 0,0,0 C at 1/4,1/4,1/4
3/4 1/4 3/4 1/4 3/4 1/4 3/4 1/4 Lattice: face-centred cubic Motif: C at 0,0,0C at 1/4,1/4,1/4 3/4 1/4 1/4 3/4 Lattice: face-centred cubic Motif: Zn at 0,0,0 S at 1/4,1/4,1/4
3/4 1/4 1/4 3/4 Lattice: face-centred cubic Motif: Zn at 0,0,0S at 1/4,1/4,1/4 fluorite 2 diad 3 triad 4 tetrad 6 hexad
Rotation axes 2 diad 3 triad 4 tetrad 6 hexad Point groups 2m Water and sulphur tetrafluoride have same point symmetry and hence same number of vibration modes - similar spectra Sulphur tetraflouride Gypsum 2/m Epsomite 222 4/m mm or 4/mmm first number c-axis second number normal to c-axis some exceptions If a direction [uvw] lies in a plane (hkl) then uh+vk+wl = 0
Weiss Law If a direction [uvw] lies in a plane (hkl) then uh+vk+wl = 0 [uvw] (hkl) [110] (110) x y z y x z