The Fourteen Bravais Lattices

Although for simplicity we have so far chosen to discuss only a two dimensional space lattice, the extension of these concepts to three dimensions apply equally well. If the seven crystal systems discussed in  the table, are represented by their primitive unit cells, then we shall have seven possible lattice types. But sometimes the smallest primitive unit cell does not display the full symmetry of the lattice. In such a case, unit cell with the highes symmetry is chosen. For illustration, we may take the face centered cubic structure (FCC) of silver.

In the face-centred representation there are four silver atoms associated with this unit cell. (Eight corner atoms = 8 x 1/8 = 1 atom; Six face atoms = 6 x 1/2 = 3 atoms; Total = 4 atoms.)

We may now choose a primitive unit cell as shown by heavy lines. It contains only one silver atom per unit cell. Although this representation has the smallest number of atoms per unit cell, but them it does not display the full summetry of structure. The face-structure ( thin lines ) represents cubic symmetry, whereas, the primitive cell unit cell represents a rhombohedral. The unit cell for these reasons is chosen as face-centered cubic.

It was showm by A. Bravais in 1848 that all possible three dimensional space lattice are of fourteen distinct types. These fourtenn lattice types ( also known as Bravais lattices ) are derived from seven crystal systems. The unit cells for these fourteen Bravais lattices are show in the figure.

Triclinic

Monoclinic

Orthorhombic

Tetragonal

Rhombohedral

Hexagonal

Cubic

All crystalline solids can be represented by one of these lattice structures. Simple unit cells are labelled as P (primitive). Non primitive unit cells are of three type and described as:

• body centered (I);
• face centered (F);
• base centered (C).

Cubic system is represented by three type of untis cells. The primitive, also known as simple cubic has an atom or molecule at each point at corner. The number of atmos or molecules per unit cell is one. The other type of unit cell representing cubic symmetry is body centered cubic. It is made up of point at the corners of a cube with an additional point in the centre of the cube. In face centered cubic symmetry there are points at the corners of the cube with additional points at the centre of each face. The number of atoms or molecules per unit cell are four. The unit cell dimensions and angles of fourteen Bravais lattices are given in the table.

N°	Crystal System	Unit cell dimensions and angles	Bravais lattices
1	cubic	a=b=c ; α=β=γ=90°	Primitive Face centered Body centered
2	Orthorhombic	a≠b≠c; α=β=γ=90°	Primitive Face centered Body centered Base centered
3	Tetragonal	a=b≠c; α=β=γ=90°	Primitive Body centered
4	Monoclinic	a≠b≠c; α=γ=90° β≠90°	Primitive Base Centered
5	Rhombohedral	a=b=c; α=β=γ≠90°	Primitive
6	Triclinic	a≠b≠c; α≠β≠γ≠90°	Primitive
7	Hexagonal	a=b≠c; α=β=90° γ=120°	Primitive