Wigner–Seitz cell

Construction of a Wigner–Seitz primitive cell.

The general mathematical concept embodied in a Wigner–Seitz cell is more commonly called a Voronoi cell, and the partition of the plane into these cells for a given set of point sites is known as a Voronoi diagram.

The construction process for the Wigner–Seitz cell of a hexagonal lattice.

The cell may be chosen by first picking a lattice point. After a point is chosen, lines are drawn to all nearby lattice points. At the midpoint of each line, another line is drawn normal to each of the first set of lines. The smallest area enclosed in this way is called the Wigner–Seitz primitive cell.

For a 3-dimensional lattice, the steps are analogous, but in step 2 instead of drawing perpendicular lines, perpendicular planes are drawn at the midpoint of the lines between the lattice points.

As in the case of all primitive cells, all area or space within the lattice can be filled by Wigner–Seitz cells and there will be no gaps.

Nearby lattice points are continually examined until the area or volume enclosed is the correct area or volume for a primitive cell. Alternatively, if the basis vectors of the lattice are reduced using lattice reduction only a set number of lattice points need to be used.[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 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 ${\displaystyle c/a>{\sqrt {2}}}$ 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.

The shape of the Wigner–Seitz cell for any Bravais lattice takes the form of one of the 24 Voronoi polyhedra.[3][11] For specifying additional constraints, ${\displaystyle a,b,c,\alpha ,\beta }$ are the unit cell parameters, and ${\displaystyle {\vec {a}}_{1},{\vec {a}}_{2},{\vec {a}}_{3},{\vec {a}}_{4}}$ are the basis vectors.

		Topological class (the affine equivalent parallelohedron)
		Truncated octahedron	Elongated dodecahedron	Rhombic dodecahedron	Hexagonal prism	Cube
Bravais lattice	Primitive cubic					Any
	Face-centered cubic			Any
	Body-centered cubic	Any
	Primitive hexagonal				Any
	Primitive rhombohedral	${\displaystyle \alpha >90^{\circ }}$		${\displaystyle \alpha <90^{\circ }}$
	Primitive tetragonal					Any
	Body-centered tetragonal	${\displaystyle c/a<{\sqrt {2}}}$	${\displaystyle c/a>{\sqrt {2}}}$
	Primitive orthorhombic					Any
	Base-centered orthorhombic				Any
	Face-centered orthorhombic	Any
	Body-centered orthorhombic	${\displaystyle c^{2}<a^{2}+b^{2}}$	${\displaystyle c^{2}>a^{2}+b^{2}}$	${\displaystyle c^{2}=a^{2}+b^{2}}$
	Primitive monoclinic				Any
	Base-centered monoclinic	${\displaystyle a<b}$	${\displaystyle a>b}$, ${\displaystyle a^{2}-b^{2}>2ac\cos \beta }$	${\displaystyle a>b}$, ${\displaystyle a^{2}-b^{2}=2ac\cos \beta }$
		${\displaystyle a>b}$, ${\displaystyle a^{2}-b^{2}<2ac\cos \beta }$	${\displaystyle a=b}$
	Primitive triclinic	${\displaystyle {\vec {a}}_{i}\cdot {\vec {a}}_{j}\neq 0}$ ${\displaystyle i,j\in \{1,2,3,4\}}$ where ${\displaystyle i\neq j}$	${\displaystyle {\vec {a}}_{i}\cdot {\vec {a}}_{j}=0}$ one time	${\displaystyle {\vec {a}}_{i}\cdot {\vec {a}}_{j}=0={\vec {a}}_{k}\cdot {\vec {a}}_{l}}$ ${\displaystyle i,j,k,l\in \{1,2,3,4\}}$ where ${\displaystyle i\neq j\neq k\neq l}$

For composite lattices, (crystals which have more than one vector in their basis) each single lattice point represents multiple atoms. We can break apart each Wigner–Seitz cell into subcells by further Voronoi decomposition according to the closest atom, instead of the closest lattice point.[12] For example, the diamond crystal structure contains a two atom basis. In diamond, carbon atoms have tetrahedral sp³ bonding, but since tetrahedra do not tile space, the voronoi decomposition of the diamond crystal structure is actually the triakis truncated tetrahedral honeycomb.[13] Another example is applying Voronoi decomposition to the atoms in the A15 phases, which forms the polyhedral approximation of the Weaire–Phelan structure.

The Wigner–Seitz cell always has the same point symmetry as the underlying Bravais lattice.[9] For example, the cube, truncated octahedron, and rhombic dodecahedron have point symmetry O_h, since the respective Bravais lattices used to generate them all belong to the cubic lattice system, which has O_h point symmetry.

In practice, the Wigner–Seitz cell itself is actually rarely used as a description of direct space, where the conventional unit cells are usually used instead. However, the same decomposition is extremely important when applied to reciprocal space. The Wigner–Seitz cell in the reciprocal space is called the Brillouin zone, which contains the information about whether a material will be a conductor, semiconductor or an insulator.

• Delaunay triangulation
• Coordination geometry
• Crystal field theory
• Wigner crystal