# Wigner-Seitz cell

### From Online Dictionary of Crystallography

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

## Definition

The Wigner-Seitz cell is a 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 (Figure 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 (Figure 2) and the Wigner-Seitz cell of a face-centred cubic lattice *F* is a rhomb-dodecahedron (Figure 3). In reciprocal space this cell is the first Brillouin zone. Since the reciprocal lattice of 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 domain of influence by Delaunay (1933). It is also called 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

Section 9.1 of *International Tables of Crystallography, Volume A*

Section 1.5 of *International Tables of Crystallography, Volume B*

Sections 1.2 and 2.2 of *International Tables of Crystallography, Volume D*