Editing Wigner–Seitz cell

{{Short description|Primitive cell of crystal lattices with Voronoi decomposition applied}}
The '''Wigner–Seitz cell''', named after [[Eugene Wigner]] and [[Frederick Seitz]], is a [[primitive cell]] which has been constructed by applying [[Voronoi cell|Voronoi decomposition]] to a [[crystal lattice]]. It is used in the study of [[crystal]]line materials in [[crystallography]].

[[File:Wigner-Seitz Animation.gif|thumb|Wigner–Seitz primitive cell for different angle parallelogram lattices.]]

The unique property of a crystal is that its [[atom]]s are arranged in a regular three-dimensional array called a [[Lattice (group)|lattice]]. All the properties attributed to crystalline materials stem from this highly ordered structure. Such a structure exhibits [[Discrete mathematics|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 (mathematics)|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]].

==Overview==
===Background===
The concept of [[Voronoi diagram|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''),<ref>{{cite journal | last=Voronoi | first=Georges |author-link=Georgy Voronoy| title=Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs. | journal=Journal für die reine und angewandte Mathematik | publisher=Walter de Gruyter GmbH | volume=1908 | issue=134 | date=1908-07-01 | issn=0075-4102 | doi=10.1515/crll.1908.134.198 | pages=198–287| s2cid=118441072 |language=fr}}</ref><ref>{{cite journal | last=Voronoi | first=Georges |author-link=Georgy Voronoy| title=Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième Mémoire. Recherches sur les paralléloèdres primitifs. | journal=Journal für die reine und angewandte Mathematik | publisher=Walter de Gruyter GmbH | volume=1909 | issue=136 | date=1909-07-01 | issn=0075-4102 | doi=10.1515/crll.1909.136.67 | pages=67–182| s2cid=199547003 |language=fr}}</ref> and [[Paul Niggli]] (''Wirkungsbereich'').<ref name=Bohm>{{cite journal | last1=Bohm | first1=J. | last2=Heimann | first2=R. B. | last3=Bohm | first3=M. | title=Voronoi Polyhedra: A Useful Tool to Determine the Symmetry and Bravais Class of Crystal Lattices | journal=Crystal Research and Technology | publisher=Wiley | volume=31 | issue=8 | year=1996 | issn=0232-1300 | doi=10.1002/crat.2170310816 | pages=1069–1075}}</ref>

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]].<ref>{{cite journal|author1=E. Wigner|authorlink1=Eugene Wigner|author2=F. Seitz|authorlink2=Frederick Seitz|title=On the Constitution of Metallic Sodium|date=15 May 1933|journal=[[Physical Review]]|volume=43|issue=10|page=804|doi=10.1103/PhysRev.43.804|bibcode=1933PhRv...43..804W}}</ref> 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 <math>d \psi/d r=0</math> at the surface of the sphere. A similar calculation which also accounted for the non-spherical nature of the Wigner–Seitz cell was performed later by [[John C. Slater]].<ref>{{cite journal | last=Slater | first=J. C. |author-link=John C. Slater| title=Electronic Energy Bands in Metals | journal=Physical Review | publisher=American Physical Society (APS) | volume=45 | issue=11 | date=1934-06-01 | issn=0031-899X | doi=10.1103/physrev.45.794 | pages=794–801| bibcode=1934PhRv...45..794S }}</ref>

There are only five topologically distinct polyhedra which tile [[three-dimensional space]], {{math|ℝ<sup>3</sup>}}. These are referred to as the [[Parallelohedron|parallelohedra]]. They are the subject of mathematical interest, such as in higher dimensions.<ref>{{cite journal | last=Garber | first=A. I. | title=Belt distance between facets of space-filling zonotopes | journal=Mathematical Notes | publisher=Pleiades Publishing Ltd | volume=92 | issue=3–4 | year=2012 | issn=0001-4346 | doi=10.1134/s0001434612090064 | pages=345–355|arxiv=1010.1698| s2cid=13277804 }}</ref> 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]].<ref>{{cite web|archive-url=https://web.archive.org/web/20190103111048/http://www.ams.org/publicoutreach/feature-column/fc-2013-11|archive-date=2019-01-03|url-status=dead|url=http://www.ams.org/publicoutreach/feature-column/fc-2013-11|title=Fedorov's Five Parallelohedra|last=Austin|first=Dave|year=2011|publisher=American Mathematical Society}}</ref> However, while a topological classification considers any [[affine transformation]] to lead to an identical class, a more specific classification leads to 24 distinct classes of voronoi polyhedra with parallel edges which tile space.<ref name=Bohm /> 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]].<ref>{{cite journal | last1=Delone | first1=B. N. |author-link1=Boris Delaunay| last2=Galiulin | first2=R. V. | last3=Shtogrin | first3=M. I. | title=On the Bravais types of lattices | journal=Journal of Soviet Mathematics | publisher=Springer Science and Business Media LLC | volume=4 | issue=1 | year=1975 | issn=0090-4104 | doi=10.1007/bf01084661 | pages=79–156| s2cid=120358504 | doi-access=free }}</ref>

===Definition===
The Wigner–Seitz cell around a lattice point is defined as the [[locus (mathematics)|locus]] of points in space that are closer to that lattice point than to any of the other lattice points.<ref name=Ashcroft>{{cite book|author1=Neil W. Ashcroft|authorlink1=Neil Ashcroft|author2=N. David Mermin|authorlink2=N. David Mermin|title=Solid State Physics|page=[https://archive.org/details/solidstatephysic00ashc/page/73 73–75]|isbn=978-0030839931|year=1976|publisher=Holt, Rinehart and Winston |url-access=registration|url=https://archive.org/details/solidstatephysic00ashc/page/73}}</ref>

It can be shown mathematically that a Wigner–Seitz cell is a [[primitive cell]]. This implies that the cell spans the entire [[Bravais lattice|direct space]] without leaving any gaps or holes, a property known as [[tessellation]].

==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]].]]
|}

{| 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
|
|
|
|
|style="background: lightgreen;" |Any
|-
! Face-centered cubic
|
|
|style="background: lightgreen;" |Any
|
|
|-
! Body-centered cubic
|style="background: lightgreen;" |Any
|
|
|
|
|-
! Primitive hexagonal
|
|
|
|style="background: lightgreen;" |Any
|
|-
! Primitive rhombohedral
|style="background: lightyellow;" | <math>\alpha > 90^\circ</math>
|
|style="background: lightyellow;" | <math>\alpha < 90^\circ</math>
|
|
|-
! Primitive tetragonal
|
|
|
|
|style="background: lightgreen;" |Any
|-
! Body-centered tetragonal
|style="background: lightyellow;" |<math>c/a < \sqrt{2}</math>
|style="background: lightyellow;" |<math>c/a > \sqrt{2}</math>
|
|
|
|-
! Primitive orthorhombic
|
|
|
|
|style="background: lightgreen;" |Any
|-
! Base-centered orthorhombic
|
|
|
|style="background: lightgreen;" |Any
|
|-
! Face-centered orthorhombic
|style="background: lightgreen;" |Any
|
|
|
|
|-
! 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>
|
|
|-
! Primitive monoclinic
|
|
|
|style="background: lightgreen;" |Any
|
|-
! 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>
|
|
|-
|style="background: lightyellow;" |<math> a > b</math>, <math> a^2 - b^2 < 2ac\cos\beta</math>
|style="background: lightyellow;" |<math>a=b</math>
|
|
|
|-
!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>
|
|
|}

==Composite lattices==
For [[composite lattice]]s, (crystals which have more than one vector in their [[Basis (crystal structure)|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.<ref>{{cite book|author1=Giuseppe Grosso|author2=Giuseppe Pastori Parravicini|title=Solid State Physics|page=54|isbn= 978-0123044600|date=2000-03-20}}</ref> For example, the [[Diamond cubic|diamond crystal structure]] contains a two atom basis. In diamond, carbon atoms have [[Tetrahedral molecular geometry|tetrahedral sp<sup>3</sup> bonding]], but since [[Tetrahedron packing|tetrahedra do not tile space]], the voronoi decomposition of the diamond crystal structure is actually the [[triakis truncated tetrahedral honeycomb]].<ref name=conway2008>{{cite book|last1=Conway|first1=John H.|last2=Burgiel|first2=Heidi|last3=Goodman-Strauss|first3=Chaim|title=The Symmetries of Things|page=332|year=2008|isbn=978-1568812205}}</ref> Another example is applying Voronoi decomposition to the atoms in the [[A15 phases]], which forms the [[Weaire–Phelan structure#Polyhedral approximation|polyhedral approximation of the Weaire–Phelan structure]].

==Symmetry==
The Wigner–Seitz cell always has the same [[Crystallographic point group|point symmetry]] as the underlying [[Bravais lattice]].<ref name=Ashcroft /> For example, the [[cube]], [[truncated octahedron]], and [[rhombic dodecahedron]] have point symmetry O<sub>h</sub>, since the respective Bravais lattices used to generate them all belong to the cubic [[lattice system]], which has O<sub>h</sub> point symmetry.

==Brillouin zone==
{{main|Brillouin zone}}
In practice, the Wigner–Seitz cell itself is actually rarely used as a description of [[Bravais lattice|direct space]], where the conventional [[unit cell]]s are usually used instead. However, the same decomposition is extremely important when applied to [[Reciprocal lattice|reciprocal space]]. 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 [[Electrical conductor|conductor]], [[semiconductor]] or an [[Insulator (electrical)|insulator]].

==See also==
* [[Delaunay triangulation]]
* [[Coordination geometry]]
* [[Crystal field theory]]
*[[Wigner crystal]]

==References==
{{reflist}}

{{DEFAULTSORT:Wigner-Seitz Cell}}
[[Category:Crystallography]]
[[Category:Mineralogy]]