1. Crystallography

2. 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

3. 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

4. Lattices formed by 2-D translations

5. 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

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

7. 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

8. 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).

9. 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

11. Glide Plane

13. 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

14. 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.

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

16. 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

17. 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

18. 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˚

19. 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

20. 3-D Symmetry Crystal Axes Axial convention: “right-hand rule” ß γ α +c+b
Axial convention:
“right-hand rule”

21. 3-D Symmetry
Unit-cell axes and angles are equal to crystal axes and angles

22. 3-D Symmetry
Unit-cell axes and angles are equal to crystal axes and angles

23. 3-D Symmetry
Unit-cell axes and angles are equal to crystal axes and angles

24. 3-D Symmetry
Crystal Axes
Always 3-fold axes in the diagonals