Editing Unit cell (section)

==Primitive cell==
A primitive cell is a unit cell that contains exactly one lattice point.  For unit cells generally, lattice points that are shared by {{mvar|n}} cells are counted as {{sfrac|1|{{mvar|n}}}} of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain {{sfrac|1|8}} of each of them.<ref name=doitpoms>{{cite web|title=DoITPoMS – TLP Library Crystallography – Unit Cell|url=http://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.php|website=Online Materials Science Learning Resources: DoITPoMS|publisher=University of Cambridge|access-date=21 February 2015|ref=doitpoms}}</ref> An alternative conceptualization is to consistently pick only one of the {{mvar|n}} lattice points to belong to the given unit cell (so the other {{mvar|n-1}} lattice points belong to adjacent unit cells).

The ''primitive translation vectors'' {{math|{{vec|''a''}}<sub>1</sub>}}, {{math|{{vec|''a''}}<sub>2</sub>}}, {{math|{{vec|''a''}}<sub>3</sub>}} span a lattice cell of smallest volume for a particular three-dimensional lattice, and are used to define a crystal translation vector

:<math> \vec T = u_1\vec a_1 + u_2\vec a_2 + u_3\vec a_3, </math>

where {{math|''u''<sub>1</sub>}}, {{math|''u''<sub>2</sub>}}, {{math|''u''<sub>3</sub>}} are integers, translation by which leaves the lattice invariant.{{refn|
group=note|name=first|
In {{mvar|n}} dimensions the crystal translation vector would be
:<math> \vec T = \sum_{i=1}^{n} u_i\vec a_i, \quad \mbox{where }u_i \in \mathbb{Z} \quad \forall i.</math>
That is, for a point in the lattice {{math|'''r'''}}, the arrangement of points appears the same from {{math|'''r′''' {{=}} '''r''' + {{vec|''T''}}}} as from {{math|'''r'''}}.}} That is, for a point in the lattice {{math|'''r'''}}, the arrangement of points appears the same from {{math|'''r′''' {{=}} '''r''' + {{vec|''T''}}}} as from {{math|'''r'''}}.<ref>{{cite book |last=Kittel |first=Charles |title=[[Introduction to Solid State Physics]] |date=11 November 2004 |publisher=Wiley |page=[https://archive.org/details/isbn_9780471415268/page/4 4] |isbn=978-0-471-41526-8 |edition=8 }}</ref>

Since the primitive cell is defined by the primitive axes (vectors) {{math|{{vec|''a''}}<sub>1</sub>}}, {{math|{{vec|''a''}}<sub>2</sub>}}, {{math|{{vec|''a''}}<sub>3</sub>}}, the volume {{math|''V''<sub>p</sub>}} of the primitive cell is given by the [[parallelepiped]] from the above axes as

:         <math> V_\mathrm{p} = \left| \vec a_1 \cdot ( \vec a_2 \times \vec a_3 ) \right|.</math>

Usually, primitive cells in two and three dimensions are chosen to take the shape parallelograms and parallelepipeds, with an atom at each corner of the cell. This choice of primitive cell is not unique, but volume of primitive cells will always be given by the expression above.<ref>{{cite journal | last1=Mehl | first1=Michael J. | last2=Hicks | first2=David | last3=Toher | first3=Cormac | last4=Levy | first4=Ohad | last5=Hanson | first5=Robert M. | last6=Hart | first6=Gus | last7=Curtarolo | first7=Stefano | title=The AFLOW Library of Crystallographic Prototypes: Part 1 | journal=Computational Materials Science | publisher=Elsevier BV | volume=136 | year=2017 | issn=0927-0256 | doi=10.1016/j.commatsci.2017.01.017 | pages=S1–S828| arxiv=1806.07864 | s2cid=119490841 }}</ref> 

===Wigner–Seitz cell===
{{main|Wigner–Seitz cell}}
In addition to the parallelepiped primitive cells, for every Bravais lattice there is another kind of primitive cell called the Wigner–Seitz cell. In the Wigner–Seitz cell, the lattice point is at the center of the cell, and for most Bravais lattices, the shape is not a parallelogram or parallelepiped. This is a type of [[Voronoi cell]]. The Wigner–Seitz cell of the [[reciprocal lattice]] in [[momentum space]] is called the [[Brillouin zone]].