Chapters / Chapter 1 · Solid state / Topic 3 of 14
Symmetry and the seven crystal systems
Chapter 1 · Solid state and crystal structures
Symmetry elements
A symmetry operation is a movement of a crystal after which it looks exactly the same as before. The imaginary point, line or plane about which the operation is carried out is called a symmetry element.
- Proper rotation axes (n-fold). Rotation by 360°/n about an axis leaves the crystal unchanged. Only 1, 2, 3, 4 and 6-fold axes are possible in a periodic crystal. A 5-fold axis (and 7, 8-fold) can never exist in a crystal, because space cannot be filled by repeating a pentagon without leaving gaps.
- Plane of symmetry (m). An imaginary plane that divides the crystal into two halves, each a mirror image of the other.
- Centre of symmetry (i). A point such that any line drawn through it meets the crystal at equal distances on both sides. Also called a centre of inversion.
- Roto-inversion axes (Sn). Combined rotation followed by inversion through a point. These are improper symmetry operations.
Combining all possible symmetry elements gives 32 point groups, which fall into the seven crystal systems.
The seven crystal systems
| System | Bravais lattices | Cell | Symmetry |
|---|---|---|---|
| Cubic | 3 | a = b = c, α = β = γ = 90° | Four C3 axes |
| Tetragonal | 2 | a = b ≠ c, α = β = γ = 90° | One C4 axis |
| Orthorhombic | 4 | a ≠ b ≠ c, α = β = γ = 90° | Three perpendicular C2 axes |
| Monoclinic | 2 | a ≠ b ≠ c, α = γ = 90°, β ≠ 90° | One C2 axis |
| Hexagonal | 1 | a = b ≠ c, α = β = 90°, γ = 120° | One C6 axis |
| Rhombohedral | 1 | a = b = c, α = β = γ ≠ 90° | One C3 axis |
| Triclinic | 1 | a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90° | None (C1) |
The lattice counts add to 14 Bravais lattices in total: 3 + 2 + 4 + 2 + 1 + 1 + 1 = 14.