www msm cam ac ukphasetransteaching html Crystallography Lecture

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www. msm. cam. ac. uk/phase-trans/teaching. html Crystallography Lecture notes Many other things

www. msm. cam. ac. uk/phase-trans/teaching. html Crystallography Lecture notes Many other things

Crystallography Introduction and point groups Stereographic projections Low symmetry systems Space groups Deformation and

Crystallography Introduction and point groups Stereographic projections Low symmetry systems Space groups Deformation and texture Interfaces, orientation relationships Martensitic transformations H. K. D. H. Bhadeshia

Introduction

Introduction

Liquid Crystals (Z. Barber)

Liquid Crystals (Z. Barber)

Form

Form

Anisotropy (elastic modulus, MPa) Ag Mo

Anisotropy (elastic modulus, MPa) Ag Mo

Polycrystals

Polycrystals

The Lattice

The Lattice

Centre of symmetry and inversion

Centre of symmetry and inversion

Bravais Lattices • Triclinic P • Monoclinic P & C • Orthorhombic P, C,

Bravais Lattices • Triclinic P • Monoclinic P & C • Orthorhombic P, C, I & F • Tetragonal P & I • Hexagonal • Trigonal P • Cubic P, F & I

Bravais Lattices

Bravais Lattices

body-centred cubic (ferrite) face-centred cubic

body-centred cubic (ferrite) face-centred cubic

Bundy (1965)

Bundy (1965)

Fe Ru Os Hs 6 d 2 s

Fe Ru Os Hs 6 d 2 s

Cohesive energy (e. V/atom) -35 Cubic-P Diamond cubic Pure iron -45 Hexagonal-P -55 -65

Cohesive energy (e. V/atom) -35 Cubic-P Diamond cubic Pure iron -45 Hexagonal-P -55 -65 b. c. c. p h. c. p 0. 8 1. 0 1. 2 1. 4 1. 6 Normalised volume Paxton et al. (1990)

2 D lattices

2 D lattices

Graphene, nanotubes

Graphene, nanotubes

Amorphous - homogeneous, isotropic Crystals - long range order, anisotropic Crystals - solid or

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

1/2 1/2 Crystal Structure

1/2 1/2 Crystal Structure

lattice + motif = structure primitive cubic lattice motif = Cu at 0, 0,

lattice + motif = structure primitive cubic lattice motif = Cu at 0, 0, 0 Zn at 1/2, 1/2

1/4 3/4 1/4 Lattice: face-centred cubic Motif: C at 0, 0, 0 C at

1/4 3/4 1/4 Lattice: face-centred cubic Motif: C at 0, 0, 0 C at 1/4, 1/4

3/4 1/4 3/4

3/4 1/4 3/4

3/4 1/4 3/4 Lattice: face-centred cubic Motif: Zn at 0, 0, 0 S at

3/4 1/4 3/4 Lattice: face-centred cubic Motif: Zn at 0, 0, 0 S at 1/4, 1/4

fluorite

fluorite

Rotation axes 2 diad 3 triad 4 tetrad 6 hexad

Rotation axes 2 diad 3 triad 4 tetrad 6 hexad

Point groups 2 m

Point groups 2 m

Water and sulphur tetrafluoride have same point symmetry and hence same number of vibration

Water and sulphur tetrafluoride have same point symmetry and hence same number of vibration modes - similar spectra

Sulphur tetraflouride

Sulphur tetraflouride

Gypsum 2/m

Gypsum 2/m

222 Epsomite

222 Epsomite

4/m mm or 4/mmm

4/m mm or 4/mmm

first number c-axis second number normal to c-axis some exceptions

first number c-axis second number normal to c-axis some exceptions

Weiss Law If a direction [uvw] lies in a plane (hkl) then [uv w]

Weiss Law If a direction [uvw] lies in a plane (hkl) then [uv w] uh+vk+wl = 0 (hkl)

z ) (110 y x [11 0] z y x

z ) (110 y x [11 0] z y x