Wigner–Seitz cell

The Wigner–Seitz cell, named after Eugene Wigner and Frederick Seitz, is a primitive cell which has been constructed by applying Voronoi decomposition to a crystal lattice. It is used in the study of crystalline materials in crystallography. 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. 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 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. The concept of Voronoi decomposition was investigated by Peter Gustav Lejeune Dirichlet, leading to the name Dirichlet domain. Further contributions were made from Evgraf Fedorov, (Fedorov parallelohedron), Georgy Voronoy (Voronoi polyhedron), and Paul Niggli (Wirkungsbereich). The application to condensed matter physics was first proposed by Eugene Wigner and Frederick Seitz in a 1933 paper, where it was used to solve the Schrödinger equation for free electrons in elemental sodium. They approximated the shape of the Wigner–Seitz cell in sodium, which is a truncated octahedron, as a sphere of equal volume, and solved the Schrödinger equation exactly using periodic boundary conditions, which require at the surface of the sphere.