Brillouin zone

The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner–Seitz cell). Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane. Equivalently, this is the Voronoi cell around the origin of the reciprocal lattice.

There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used less frequently. As a result, the first Brillouin zone is often called simply the Brillouin zone. In general, the n-th Brillouin zone consists of the set of points that can be reached from the origin by crossing exactly n − 1 distinct Bragg planes. A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice (point group of the crystal).

The concept of a Brillouin zone was developed by Léon Brillouin (1889–1969), a French physicist.

Within the Brillouin zone, a constant-energy surface represents the loci of all the ${\displaystyle \overrightarrow{k}}$-points (that is, all the electron momentum values) that have the same energy. The Fermi surface is a special constant-energy surface that separates the unfilled orbitals from the filled ones at zero Kelvin.

Critical points

Several points of high symmetry are of special interest – these are called critical points.

Other lattices have different types of high-symmetry points. They can be found in the illustrations below for both three- and two-dimensional lattices.

Brillouin zone types in 3D

Brillouin zone types in 2D

Sources:

In two dimensions, the number of Bravais lattices reduces to five. In the following illustration, the corresponding Brillouin zones are delimited by red lines.

For each lattice, the irreducible Brillouin zone is highlighted (blue) in accordance with the definitions of the symmetry points given in Refs. and .