In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones.

A Brillouin zone is defined as a Wigner-Seitz primitive cell in the reciprocal lattice. The first Brillouin zone is the smallest volume entirely enclosed by planes that are the perpendicular bisectors of the reciprocal lattice vectors drawn from the origin.

The different Brillouin zones correspond to primitive cells of a different type that come up in the theory of electronic levels in a periodic potential. The first Brillouin zone is considered as the Wigner-Seitz (WS) primitive cell in the reciprocal lattice.

Brillouin Zones are fundamental for understanding the electronic properties of crystalline solids because they delineate regions of allowed and forbidden energy states for electrons. Each zone corresponds to specific energy levels determined by the crystal's periodic structure.

Reciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch’s theorem has the following form in two and three dimensions: k(r +R) =e 2 ik R k(r). In this expression, R is a lattice vector between a pair of unit cells: R =ua +vb+wc;

This region, which is chosen to be the Wigner-Seitz cell of the reciprocal lattice point \(\mathbf{K}_{000} = \mathbf{0}\), is called the first Brillouin zone. These regions are illustrated below for various 1-d, 2-d, and 3-d reciprocal lattices.

The Brillouin zone is a very important concept in solid state physics; it plays a major role in the theoretical understanding of the elementary ideas of electronic energy bands. The first Brillouin zone is defined as the Wigner–Seitz primitive cell of the reciprocal lattice.

The Brillouin Zone Slide 14 The Brillouin zone is the Wigner‐Seitz unit cell constructed from the reciprocal lattice. It is essentially a map of the periodicity of the lattice as a function of direction. Shown here is the Brillouin zone for a face‐ centered cubic lattice. It is a truncated

Brillouin Zone construction. The reciprocal lattice basis vectors span a vector space that is commonly referred to as reciprocal space, or often in the context of quantum mechanics, k space. This section covers the construction of Brillouin zones in two dimensions.

A Brillouin zone is a particular choice of the unit cell of the reciprocal lattice. It is defined as the Wigner-Seitz cell (also called Dirichlet or Voronoi domain of influence) of the reciprocal lattice.