Wigner–Seitz unit cell

The Wigner–Seitz cell, named after Eugene Wigner and Frederick Seitz, is a type of Voronoi cell used in the study of crystalline material in solid-state physics.

The unique property of a crystal is that its atoms are arranged in a regular three-dimensional array called a lattice. All the properties attributed to crystalline materials stem from this highly ordered structure. Such a structure exhibits 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. It is the 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 discete translation symmetry of the lattice. The primitive cell of the reciprocal lattice in momentum space is called the Brillouin zone.

The Wigner–Seitz cell around a lattice point is defined as the locus of points in space that are closer to that lattice point than to any of the other lattice points.

It can be shown mathematically that a Wigner–Seitz cell is a primitive cell spanning the entire direct space without leaving any gaps or holes.

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