Crystallography H. K. D. H. Bhadeshia Introduction and point groups

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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 https://edge.edx.org/courses/MSM/C6/2013_Winter/about

https://edge.edx.org/courses/MSM/C6/2013_Winter/about

Introduction

Liquid Crystals (Z. Barber)

Form

Anisotropy (elastic modulus, MPa) Ag Mo

Polycrystals

2D lattices

The Lattice

Graphene, nanotubes

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 6d 2s 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)

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 www.msm.cam.ac.uk/phase-trans/teaching.html 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 0,0,0 C 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 0,0,0 S 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

without high order axes

without order axes

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