Editing Wigner–Seitz cell (section)

==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]].