# Wigner-Seitz cell

(Difference between revisions)
 Revision as of 12:09, 8 February 2012 (view source)m← Older edit Revision as of 17:02, 11 April 2017 (view source)m (→See also: ITA 6th edition)Newer edit → Line 17: Line 17: == See also == == See also == + *Section 3.1.1.4 of ''International Tables of Crystallography, Volume A'', 6th edition + *Section 1.5 of ''International Tables of Crystallography, Volume B'' + *Sections 1.2 and 2.2 of ''International Tables of Crystallography, Volume D'' - Section 9.1 of ''International Tables of Crystallography, Volume A''
+ [[Category:Fundamental crystallography]] - Section 1.5 of ''International Tables of Crystallography, Volume B''
+ - Sections 1.2 and 2.2 of ''International Tables of Crystallography, Volume D''
+ - + - [[Category:Fundamental crystallography]]
+

## Revision as of 17:02, 11 April 2017

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 tL).

• Section 3.1.1.4 of International Tables of Crystallography, Volume A, 6th edition
• Section 1.5 of International Tables of Crystallography, Volume B
• Sections 1.2 and 2.2 of International Tables of Crystallography, Volume D