Wigner-Seitz cell

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