Unit cell

The concept is used particularly in describing crystal structure in two and three dimensions, though it makes sense in all dimensions. A lattice can be characterized by the geometry of its unit cell, which is a section of the tiling (a parallelogram or parallelepiped) that generates the whole tiling using only translations.

There are two special cases of the unit cell: the primitive cell and the conventional cell. The primitive cell is a unit cell corresponding to a single lattice point; it is the smallest possible unit cell. In some cases, the full symmetry of a crystal structure is not obvious from the primitive cell, in which case a conventional cell may be used. A conventional cell (which may or may not be primitive) is the smallest unit cell with the full symmetry of the lattice and may include more than one lattice point. The conventional unit cells are parallelotopes in n dimensions.

Primitive cell

A primitive cell is a unit cell that contains exactly one lattice point. For unit cells generally, lattice points that are shared by n cells are counted as ⁠1/n⁠ of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain ⁠1/8⁠ of each of them. An alternative conceptualization is to consistently pick only one of the n lattice points to belong to the given unit cell (so the other n-1 lattice points belong to adjacent unit cells).

The primitive translation vectors a→₁, a→₂, a→₃ span a lattice cell of smallest volume for a particular three-dimensional lattice, and are used to define a crystal translation vector

${\displaystyle \overrightarrow{T}=u_1{\overrightarrow{a}}_1+u_2{\overrightarrow{a}}_2+u_3{\overrightarrow{a}}_3,}$

where u₁, u₂, u₃ are integers, translation by which leaves the lattice invariant. That is, for a point in the lattice r, the arrangement of points appears the same from r′ = r + T→ as from r.

Since the primitive cell is defined by the primitive axes (vectors) a→₁, a→₂, a→₃, the volume V_p of the primitive cell is given by the parallelepiped from the above axes as

${\displaystyle V_{\mathrm{p}}=\left|{\overrightarrow{a}}_1\cdot ({\overrightarrow{a}}_2\times {\overrightarrow{a}}_3)\right|.}$

Usually, primitive cells in two and three dimensions are chosen to take the shape parallelograms and parallelepipeds, with an atom at each corner of the cell. This choice of primitive cell is not unique, but volume of primitive cells will always be given by the expression above.

Wigner–Seitz cell

In addition to the parallelepiped primitive cells, for every Bravais lattice there is another kind of primitive cell called the Wigner–Seitz cell. In the Wigner–Seitz cell, the lattice point is at the center of the cell, and for most Bravais lattices, the shape is not a parallelogram or parallelepiped. This is a type of Voronoi cell. The Wigner–Seitz cell of the reciprocal lattice in momentum space is called the Brillouin zone.