Editing Wigner–Seitz cell (section)

==Constructing the cell==
[[Image:Wigner–Seitz cell.svg|thumb|right|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]].
[[File:Hex bz blue2.gif|thumb|The construction process for the Wigner–Seitz cell of a hexagonal lattice.]]
The cell may be chosen by first picking a [[Lattice (group)|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 [[Surface normal|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.<ref>{{Cite journal|last1=Hart|first1=Gus L W|last2=Jorgensen|first2=Jeremy J|last3=Morgan|first3=Wiley S|last4=Forcade|first4=Rodney W|date=2019-06-26|title=A robust algorithm for k-point grid generation and symmetry reduction|journal=Journal of Physics Communications|volume=3|issue=6|pages=065009|doi=10.1088/2399-6528/ab2937|arxiv=1809.10261|bibcode=2019JPhCo...3f5009H|issn=2399-6528|doi-access=free}}</ref> 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.

{|
|[[File:HC-P2.png|thumb|left|The Wigner–Seitz cell of the [[Cubic crystal system|primitive cubic]] lattice is a [[cube]]. In mathematics, it is known as the [[cubic honeycomb]].]]
|[[File:Truncated octahedra.png|thumb|left|The Wigner–Seitz cell of the [[Cubic crystal system|body-centered cubic]] lattice is a [[truncated octahedron]].<ref name=Ashcroft /> In mathematics, it is known as the [[bitruncated cubic honeycomb]].]]
|[[File:Rhombic dodecahedra.png|thumb|left|The Wigner–Seitz cell of the [[Cubic crystal system|face-centered cubic]] lattice is a [[rhombic dodecahedron]].<ref name=Ashcroft /> In mathematics, it is known as the [[rhombic dodecahedral honeycomb]].]]
|[[File:Rhombo-hexagonal dodecahedron tessellation.png|thumb|left|The Wigner–Seitz cell of the [[Tetragonal crystal system|body-centered tetragonal]] lattice that has [[lattice constant]]s with <math>c/a > \sqrt{2}</math> is the [[elongated dodecahedron]].]]
|[[File:Hexagonal prismatic honeycomb.png|thumb|left|The Wigner–Seitz cell of the [[Hexagonal crystal family|primitive hexagonal]] lattice is the [[hexagonal prism]]. In mathematics, it is known as the [[hexagonal prismatic honeycomb]].]]
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{| class="wikitable" style="text-align: center;"
|+ The shape of the Wigner–Seitz cell for any Bravais lattice takes the form of one of the 24 Voronoi polyhedra.<ref name=Bohm /><ref>{{cite conference | last1=Lulek | first1=T | last2=Florek | first2=W | last3=Wałcerz | first3=S | title=Symmetry and Structural Properties of Condensed Matter|chapter=Bravais classes, Vonoroï cells, Delone symbols | publisher=World Scientific | year=1995 | isbn=978-981-02-2059-4 | doi=10.1142/9789814533508 | pages=279–316|chapter-url=https://www.ihes.fr/~vergne/LouisMichel/publications/SymmStr.1995_279.pdf}}</ref> For specifying additional constraints, <math>a, b, c, \alpha, \beta</math> are the unit cell parameters, and <math>\vec a_1, \vec a_2, \vec a_3, \vec a_4</math> are the basis vectors.
!rowspan=2 colspan=2|
!colspan=5|Topological class (the affine equivalent [[parallelohedron]])
|-
! Truncated octahedron <!-- I-->
! Elongated dodecahedron <!-- II-->
! Rhombic dodecahedron <!-- III-->
! Hexagonal prism <!-- IV-->
! Cube <!-- V-->
|-
! rowspan=15|[[Bravais lattice]]
! Primitive cubic
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|style="background: lightgreen;" |Any
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! Face-centered cubic
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|style="background: lightgreen;" |Any
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! Body-centered cubic
|style="background: lightgreen;" |Any
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! Primitive hexagonal
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|style="background: lightgreen;" |Any
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! Primitive rhombohedral
|style="background: lightyellow;" | <math>\alpha > 90^\circ</math>
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|style="background: lightyellow;" | <math>\alpha < 90^\circ</math>
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|-
! Primitive tetragonal
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|style="background: lightgreen;" |Any
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! Body-centered tetragonal
|style="background: lightyellow;" |<math>c/a < \sqrt{2}</math>
|style="background: lightyellow;" |<math>c/a > \sqrt{2}</math>
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! Primitive orthorhombic
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|style="background: lightgreen;" |Any
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! Base-centered orthorhombic
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|style="background: lightgreen;" |Any
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! Face-centered orthorhombic
|style="background: lightgreen;" |Any
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! Body-centered orthorhombic
|style="background: lightyellow;" |<math>c^2 < a^2 + b^2</math>
|style="background: lightyellow;" |<math>c^2 > a^2 + b^2</math>
|style="background: lightyellow;" |<math>c^2 = a^2 + b^2</math>
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! Primitive monoclinic
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|style="background: lightgreen;" |Any
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! rowspan=2|Base-centered monoclinic
|style="background: lightyellow;" | <math> a < b</math><br> 
|style="background: lightyellow;" |<math> a > b</math>, <math> a^2 - b^2 > 2ac \cos\beta</math>
|style="background: lightyellow;" |<math> a > b</math>, <math> a^2 - b^2 = 2ac\cos\beta</math>
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|style="background: lightyellow;" |<math> a > b</math>, <math> a^2 - b^2 < 2ac\cos\beta</math>
|style="background: lightyellow;" |<math>a=b</math>
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!Primitive triclinic
|style="background: lightyellow;" |<math> \vec a_i \cdot \vec a_j \neq 0</math><br><math> i, j \in \{1,2,3,4\}</math><br>where <math>i \neq j</math>
|style="background: lightyellow;" |<math> \vec a_i \cdot \vec a_j = 0</math><br>one time
|style="background: lightyellow;" |<math> \vec a_i \cdot \vec a_j = 0 =\vec a_k \cdot \vec a_l</math><br><math> i, j, k, l \in \{1,2,3,4\}</math><br>where <math>i \neq j \neq k \neq l</math>
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