Editing Wigner–Seitz cell (section)

{{Short description|Primitive cell of crystal lattices with Voronoi decomposition applied}}
The '''Wigner–Seitz cell''', named after [[Eugene Wigner]] and [[Frederick Seitz]], is a [[primitive cell]] which has been constructed by applying [[Voronoi cell|Voronoi decomposition]] to a [[crystal lattice]]. It is used in the study of [[crystal]]line materials in [[crystallography]].

[[File:Wigner-Seitz Animation.gif|thumb|Wigner–Seitz primitive cell for different angle parallelogram lattices.]]

The unique property of a crystal is that its [[atom]]s are arranged in a regular three-dimensional array called a [[Lattice (group)|lattice]]. All the properties attributed to crystalline materials stem from this highly ordered structure. Such a structure exhibits [[Discrete mathematics|discrete]] [[translational symmetry]]. In order to model and study such a periodic system, one needs a mathematical "handle" to describe the symmetry and hence draw conclusions about the material properties consequent to this symmetry. The Wigner–Seitz cell is a means to achieve this.

A Wigner–Seitz cell is an example of a [[primitive cell]], which is a [[unit cell]] containing exactly one lattice point. For any given lattice, there are an infinite number of possible primitive cells. However there is only one Wigner–Seitz cell for any given lattice. It is the [[locus (mathematics)|locus]] of points in space that are closer to that lattice point than to any of the other lattice points.

A Wigner–Seitz cell, like any primitive cell, is a [[fundamental domain]] for the discrete translation symmetry of the lattice. The primitive cell of the [[reciprocal lattice]] in [[momentum space]] is called the [[Brillouin zone]].