Wigner–Seitz cell

The unique property of a crystal is that its atoms are arranged in a regular three-dimensional array called a lattice.

The primitive cell of the reciprocal lattice in momentum space is called the Brillouin zone.

[3] 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.

[4] 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

[6] These five parallelohedra can be used to classify the three dimensional lattices using the concept of a projective plane, as suggested by John Horton Conway and Neil Sloane.

[3] For example, the rectangular cuboid, right square prism, and cube belong to the same topological class, but are distinguished by different ratios of their sides.

This classification of the 24 types of voronoi polyhedra for Bravais lattices was first laid out by Boris Delaunay.

This implies that the cell spans the entire direct space without leaving any gaps or holes, a property known as tessellation.

[10] In two-dimensions only the lattice points that make up the 4 unit cells that share a vertex with the origin need to be used.

In three-dimensions only the lattice points that make up the 8 unit cells that share a vertex with the origin need to be used.

The Wigner–Seitz cell in the reciprocal space is called the Brillouin zone, which is used in constructing band diagrams to determine whether a material will be a conductor, semiconductor or an insulator.

Wigner–Seitz primitive cell for different angle parallelogram lattices.
Construction of a Wigner–Seitz primitive cell.
The construction process for the Wigner–Seitz cell of a hexagonal lattice.
The Wigner–Seitz cell of the primitive cubic lattice is a cube . In mathematics, it is known as the cubic honeycomb .
The Wigner–Seitz cell of the body-centered cubic lattice is a truncated octahedron . [ 9 ] In mathematics, it is known as the bitruncated cubic honeycomb .
The Wigner–Seitz cell of the face-centered cubic lattice is a rhombic dodecahedron . [ 9 ] In mathematics, it is known as the rhombic dodecahedral honeycomb .
The Wigner–Seitz cell of the body-centered tetragonal lattice that has lattice constants with is the elongated dodecahedron .
The Wigner–Seitz cell of the primitive hexagonal lattice is the hexagonal prism . In mathematics, it is known as the hexagonal prismatic honeycomb .