Crystallography

Terms Translations of nodes → Lattices A node is any sort of motif A property at the atomic level, not of crystal shapes Translations as a symmetry operation involves repeat distances The origin is arbitrary 1-D translations = a row

Symmetry Translations (Lattices) 1-D translations = a row A property at the atomic level, not of crystal shapes Symmetric translations involve repeat distances The origin is arbitrary 1-D translations = a row a  a is the repeat vector a

Lattices formed by 2-D translations

Lattices formed by 2-D translations Unit cell A Unit cell B Unit Cell: the basic repeat unit that, by translation only, generates the entire pattern

Choice of unit cell Two 2-D lattices Which unit cell should we choose?

Translations Which unit cell is correct ?? Conventions: 1. Cell edges should, whenever possible, coincide with symmetry axes or reflection planes 2. If possible, edges should relate to each other by lattice’s symmetry. 3. The smallest possible cell (the reduced cell) which fulfills 1 and 2 should be chosen

Choice of unit cell Arrows indicate the correct choice of unit cell based on the rules in the previous slide. Note that each of these is a primitive lattice i.e. each contains only one lattice node (if one adds up the partial nodes).

Translations The lattice and symmetry interrelate, because both are properties of the overall symmetry pattern. This is why 5-fold and >6 fold rotational symmetry won’t work

Glide Plane

Example of a screw operation screw axis Rotate 180° while translating ½ of the length of the axis Illustration of the operation of the screw 21

The operation of 31 and 32 screw axes are shown below, looking down the axis and showing the sense of rotation and the 1/3 2/3 axis axis 2/3 1/3 31 32 translation distance along the axis. The result is opposite senses of screw rotation; i.e. left-handed and right-handed threads as on a wood screw.

Screw Axes 21 31,3 2 41, 42, 4 3 61, 62, 63, 64, 65

The 14 Bravais lattices a=b=c, α = β = γ = 90° α = β = γ ≠ 90° a=b≠ c, α = β = 90°, γ = 120° a≠ b≠ c, α = γ = 90° β > 90° a≠ b≠ c, α ≠ β ≠ γ The 14 Bravais lattices

Of the 14 Bravais lattices, there are 5 distinct types Primitive (P) – Contains only one node within the lattice, but there is one node at each corner Body-centred (I from the German innern) – Contains two nodes, one in the centre of the lattice and one at the corners Face-centred (F) – Contains four nodes, nodes in the centre of each face and one at the corners End-centred (C, B or A) – Contains two nodes, one in two opposite faces and one at the corners

The previous four types may have more than one type of symmetry (all the crystal systems contain a primitive lattice, which defines the minimum symmetry of the crystal system), but the fifth type is the rhombohedral lattice: Rhombohedral (R) – Contains only one node at the corners, but is form by three axes of equal length separated by equal angles that are not 90˚

The 14 Bravais lattices a=b=c, α = β = γ = 90° α = β = γ ≠ 90° a=b≠ c, α = β = 90°, γ = 120° a≠ b≠ c, α = γ = 90° β > 90° a≠ b≠ c, α ≠ β ≠ γ The 14 Bravais lattices

3-D Symmetry Crystal Axes Axial convention: “right-hand rule” ß γ α +c +b Axial convention: “right-hand rule”

3-D Symmetry Unit-cell axes and angles are equal to crystal axes and angles

3-D Symmetry Unit-cell axes and angles are equal to crystal axes and angles

3-D Symmetry Unit-cell axes and angles are equal to crystal axes and angles

3-D Symmetry Crystal Axes Always 3-fold axes in the diagonals