# Wigner-Seitz cell

### From Online Dictionary of Crystallography

Synonym: Voronoi domain

Maille de Wigner-Seitz (*Fr*). Wigner-Seitz-Zelle, Wirkungsbereich (*Ge*). Cella di Wigner-Seitz (*It*). Celda de Wigner-Seitz (*Sp*).

## Definition

The Wigner-Seitz cell is a polyhedron obtained by connecting a lattice point *P* to all other lattice points and drawing the planes perpendicular to these connecting lines and passing through their midpoints (Fig. 1). The polyhedron enclosed by these planes is the Wigner-Seitz cell. This construction is called the Dirichlet construction. The cell thus obtained is a primitive cell and it is possible to fill up the whole space by translation of that cell.

The Wigner-Seitz cell of a body-centred cubic lattice *I* is a cuboctahedron (Fig. 2) and the Wigner-Seitz cell of a face-centred cubic lattice *F* is a rhomb-dodecahedron (Fig. 3). In reciprocal space this cell is the first Brillouin zone. Since the reciprocal lattice of a body-centred lattice is a face-centred lattice and reciprocally, the first Brillouin zone of a body-centred cubic lattice is a rhomb-dodecahedron and that of a face-centred cubic lattice is a cuboctahedron.

The inside of the Wigner-Seitz cell has been called the domain of influence by Delaunay (1933). It is also called the Dirichlet domain or Voronoi domain. The domain of influence of lattice point *P* thus consists of all points *Q* in space that are closer to this lattice point than to any other lattice point or at most equidistant to it (such that **OP** ≤ |**t** - **OP**| for any vector **t** ∈ *L*).

## See also

- Chapter 3.1.1.4 of
*International Tables for Crystallography, Volume A*, 6th edition - Chapter 1.5 of
*International Tables for Crystallography, Volume B* - Chapters 1.2 and 2.2 of
*International Tables for Crystallography, Volume D*