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Crystallography

Lecture notes

Many other things

Introduction and point groups

Stereographic projections

Low symmetry systems

Space groups

Deformation and texture

Interfaces, orientation relationships

Martensitic transformations

Crystallography

H. K. D. H. Bhadeshia

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

Os

Hs

6d 2s

1.61.41.21.00.8-65

-55

-45

-35

Normalised volume

Diamond cubicCubic-P

Hexagonal-P

b.c.cc.c.ph.c.p

Coh

esiv

e en

ergy

(eV

/ato

m)

Paxton et al. (1990)

Pure iron

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

1/2 1/2

1/21/2

Crystal Structure

lattice + motif = structure

primitive cubic lattice

motif = Cu at 0,0,0

Zn at 1/2, 1/2, 1/2

1/4

1/4

1/4

1/4

1/43/4

3/4

3/4

3/4

Lattice: face-centred cubic

Motif: C at 0,0,0 C at 1/4,1/4,1/4

1/4

1/43/4

3/4

1/4

1/43/4

3/4

Lattice: face-centred cubic

Motif: Zn at 0,0,0 S at 1/4,1/4,1/4

fluorite

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

Epsomite222

4/m mm or 4/mmm

first number c-axis

second number normal to c-axis

some exceptions

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