WignerSeitz

The Wigner-Seitz cell is a fundamental construct in crystallography and solid-state physics. Named after Eugene Wigner and Frederick Seitz, it is the primitive cell of a lattice, defined as the Voronoi region around a lattice point. In practical terms, it is the region of space consisting of all points closer to a given lattice point than to any other lattice point.

Construction is geometric: for a Bravais lattice with lattice vectors R, the Wigner-Seitz cell around the origin contains all points x such that the distance |x| is less than or equal to |x − R| for every nonzero lattice vector R. The faces of the cell lie on the perpendicular bisectors of lines joining the origin to its nearest neighboring lattice points. The resulting polyhedron is a primitive cell that, when translated by all lattice vectors, tiles space without gaps or overlaps.

In three dimensions, the shape of the Wigner-Seitz cell depends on the lattice type. For a simple cubic lattice it is a cube; for a body-centered cubic lattice it is a truncated octahedron; and for a face-centered cubic lattice it is a rhombic dodecahedron. The concept generalizes to any dimension and lattice.

In reciprocal space, the Wigner-Seitz construction yields the first Brillouin zone of the reciprocal lattice. This zone plays a central role in electronic structure calculations, as it defines the natural domain for plotting band structures and for sampling wavevectors in solids.

Overall, the Wigner-Seitz cell provides a concise description of the local geometric environment around a lattice point and underpins many practical methods in crystallography and solid-state physics.