# Unit cell

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Repeating unit formed by the vectors spanning the points of a lattice

In [geometry](/source/Geometry), [biology](/source/Biology), [mineralogy](/source/Mineralogy) and [solid state physics](/source/Solid_state_physics), a **unit cell** is a repeating unit formed by the vectors spanning the points of a lattice.[1] Despite its suggestive name, the unit cell (unlike a [unit vector](/source/Unit_vector), for example) does not necessarily have unit size, or even a particular size at all. Rather, the primitive cell is the closest analogy to a unit vector, since it has a determined size for a given [lattice](/source/Lattice_(group)) and is the basic building block from which larger cells are constructed. Geometrically, this is a special case of a [fundamental domain](/source/Fundamental_domain) of a group (the translation symmetries) acting on a topological space (Euclidean space).

The concept is used particularly in describing [crystal structure](/source/Crystal_structure) in two and three dimensions, though it makes sense in all dimensions. A lattice can be characterized by the geometry of its unit cell, which is a section of the tiling (a [parallelogram](/source/Parallelogram) or [parallelepiped](/source/Parallelepiped)) that generates the whole tiling using only translations.

There are two special cases of the unit cell: the **primitive cell** and the **conventional cell**. The primitive cell is a unit cell corresponding to a single [lattice point](/source/Lattice_point), it is the smallest possible unit cell.[2] In some cases, the full symmetry of a crystal structure is not obvious from the primitive cell, in which cases a conventional cell may be used. A conventional cell (which may or may not be primitive) is a unit cell with the full symmetry of the lattice and may include more than one lattice point. The conventional unit cells are [parallelotopes](/source/Parallelepiped#Parallelotope) in *n* dimensions.

## 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 n cells are counted as ⁠1/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 ⁠1/8⁠ of each of them.[3] An alternative conceptualization is to consistently pick only one of the n lattice points to belong to the given unit cell (so the other n-1 lattice points belong to adjacent unit cells).

The *primitive translation vectors* *a*→1, *a*→2, *a*→3 span a lattice cell of smallest volume for a particular three-dimensional lattice, and are used to define a crystal translation vector

- T → = u 1 a → 1 + u 2 a → 2 + u 3 a → 3 , {\displaystyle {\vec {T}}=u_{1}{\vec {a}}_{1}+u_{2}{\vec {a}}_{2}+u_{3}{\vec {a}}_{3},}

where *u*1, *u*2, *u*3 are integers, translation by which leaves the lattice invariant.[note 1] That is, for a point in the lattice **r**, the arrangement of points appears the same from **r′** = **r** + *T*→ as from **r**.[4]

Since the primitive cell is defined by the primitive axes (vectors) *a*→1, *a*→2, *a*→3, the volume *V*p of the primitive cell is given by the [parallelepiped](/source/Parallelepiped) from the above axes as

- V p = | a → 1 ⋅ ( a → 2 × a → 3 ) | . {\displaystyle V_{\mathrm {p} }=\left|{\vec {a}}_{1}\cdot ({\vec {a}}_{2}\times {\vec {a}}_{3})\right|.}

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.[5]

### Wigner–Seitz cell

Main article: [Wigner–Seitz cell](/source/Wigner%E2%80%93Seitz_cell)

In addition to the parallelepiped primitive cells, for every [Bravais lattice](/source/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](/source/Voronoi_cell). The Wigner–Seitz cell of the [reciprocal lattice](/source/Reciprocal_lattice) in [momentum space](/source/Momentum_space) is called the [Brillouin zone](/source/Brillouin_zone).

## Conventional cell

For each particular lattice, a conventional cell has been chosen on a case-by-case basis by crystallographers based on convenience of calculation.[6] These conventional cells may have additional lattice points located in the middle of the faces or body of the unit cell. The number of lattice points, as well as the volume of the conventional cell is an integer multiple (1, 2, 3, or 4) of that of the primitive cell.[7]

## Two dimensions

The [parallelogram](/source/Parallelogram) is the general primitive cell for the plane.

For any 2-dimensional lattice, the unit cells are [parallelograms](/source/Parallelogram), which in special cases may have orthogonal angles, equal lengths, or both. Four of the five two-dimensional [Bravais lattices](/source/Bravais_lattice) are represented using conventional primitive cells, as shown below.

Conventional primitive cell Shape name Parallelogram Rectangle Square Rhombus Bravais lattice Primitive Oblique Primitive Rectangular Primitive Square Primitive Hexagonal

The centered rectangular lattice also has a primitive cell in the shape of a rhombus, but in order to allow easy discrimination on the basis of symmetry, it is represented by a conventional cell which contains two lattice points.

Primitive cell Shape name Rhombus Conventional cell Bravais lattice Centered Rectangular

## Three dimensions

A [parallelepiped](/source/Parallelepiped) is a general primitive cell for 3-dimensional space.

For any 3-dimensional lattice, the conventional unit cells are [parallelepipeds](/source/Parallelepiped), which in special cases may have orthogonal angles, or equal lengths, or both. Seven of the fourteen three-dimensional [Bravais lattices](/source/Bravais_lattice) are represented using conventional primitive cells, as shown below.

Conventional primitive cell Shape name Parallelepiped Oblique rectangular prism Rectangular cuboid Square cuboid Trigonal trapezohedron Cube Right rhombic prism Bravais lattice Primitive Triclinic Primitive Monoclinic Primitive Orthorhombic Primitive Tetragonal Primitive Rhombohedral Primitive Cubic Primitive Hexagonal

The other seven Bravais lattices (known as the centered lattices) also have primitive cells in the shape of a parallelepiped, but in order to allow easy discrimination on the basis of symmetry, they are represented by conventional cells which contain more than one lattice point.

Primitive cell Shape name Oblique rhombic prism Right rhombic prism Conventional cell Bravais lattice Base-centered Monoclinic Base-centered Orthorhombic Body-centered Orthorhombic Face-centered Orthorhombic Body-centered Tetragonal Body-centered Cubic Face-centered Cubic

## See also

- [Wigner–Seitz cell](/source/Wigner%E2%80%93Seitz_cell)

- [Bravais lattice](/source/Bravais_lattice)

- [Wallpaper group](/source/Wallpaper_group)

- [Space group](/source/Space_group)

## Notes

1. **[^](#cite_ref-first_4-0)** In n dimensions the crystal translation vector would be 1. T → = ∑ i = 1 n u i a → i , where u i ∈ Z ∀ i . {\displaystyle {\vec {T}}=\sum _{i=1}^{n}u_{i}{\vec {a}}_{i},\quad {\mbox{where }}u_{i}\in \mathbb {Z} \quad \forall i.} That is, for a point in the lattice **r**, the arrangement of points appears the same from **r′** = **r** + *T*→ as from **r**.

## References

1. **[^](#cite_ref-1)** Ashcroft, Neil W. (1976). "Chapter 4". *Solid State Physics*. W. B. Saunders Company. p. 72. [ISBN](/source/ISBN_(identifier)) [0-03-083993-9](https://en.wikipedia.org/wiki/Special:BookSources/0-03-083993-9).

1. **[^](#cite_ref-2)** Simon, Steven (2013). *The Oxford Solid State Physics* (1 ed.). Oxford University Press. p. 114. [ISBN](/source/ISBN_(identifier)) [978-0-19-968076-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-968076-4).

1. **[^](#cite_ref-doitpoms_3-0)** ["DoITPoMS – TLP Library Crystallography – Unit Cell"](http://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.php). *Online Materials Science Learning Resources: DoITPoMS*. University of Cambridge. Retrieved 21 February 2015.

1. **[^](#cite_ref-5)** Kittel, Charles (11 November 2004). *[Introduction to Solid State Physics](/source/Introduction_to_Solid_State_Physics)* (8 ed.). Wiley. p. [4](https://archive.org/details/isbn_9780471415268/page/4). [ISBN](/source/ISBN_(identifier)) [978-0-471-41526-8](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-41526-8).

1. **[^](#cite_ref-6)** Mehl, Michael J.; Hicks, David; Toher, Cormac; Levy, Ohad; Hanson, Robert M.; Hart, Gus; Curtarolo, Stefano (2017). "The AFLOW Library of Crystallographic Prototypes: Part 1". *Computational Materials Science*. **136**. Elsevier BV: S1–S828. [arXiv](/source/ArXiv_(identifier)):[1806.07864](https://arxiv.org/abs/1806.07864). [doi](/source/Doi_(identifier)):[10.1016/j.commatsci.2017.01.017](https://doi.org/10.1016%2Fj.commatsci.2017.01.017). [ISSN](/source/ISSN_(identifier)) [0927-0256](https://search.worldcat.org/issn/0927-0256). [S2CID](/source/S2CID_(identifier)) [119490841](https://api.semanticscholar.org/CorpusID:119490841).

1. **[^](#cite_ref-7)** Aroyo, M. I., ed. (2016-12-31). *International Tables for Crystallography*. Chester, England: International Union of Crystallography. p. 25. [doi](/source/Doi_(identifier)):[10.1107/97809553602060000114](https://doi.org/10.1107%2F97809553602060000114). [ISBN](/source/ISBN_(identifier)) [978-0-470-97423-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-470-97423-0).

1. **[^](#cite_ref-8)** Ashcroft, Neil W. (1976). [*Solid State Physics*](https://archive.org/details/solidstatephysic00ashc?q=just+fills+all+of+space+without+either+overlapping+itself+or+leaving+voids+is+called+a+primitive+cell). W. B. Saunders Company. p. 73. [ISBN](/source/ISBN_(identifier)) [0-03-083993-9](https://en.wikipedia.org/wiki/Special:BookSources/0-03-083993-9).

v t e Crystallography Key concepts Timeline of crystallography Crystallographers Metallurgy Biocrystallography Structure Unit cell Bravais lattice Miller index Point group Reciprocal lattice Restriction theorem Periodic table Structure prediction Systems Cubic Hexagonal Monoclinic Orthorhombic Tetragonal Triclinic Growth Crystallite Equiaxed Twinning Fiveling Aperiodic crystal Quasicrystal Phase transition Phase diagram Eutectic Miscibility gap Polymorphism Liquid crystal Phase transformation crystallography Precipitation Segregation Spinodal decomposition Supersaturation GP-zone Ostwald ripening Defects Grain boundary Disclination CSL Growth Abnormal growth Perfect crystal Stacking fault Dislocation Burgers vector Partial dislocation Kink Cross slip Frank–Read source Cottrell atmosphere Peierls stress GND Lomer–Cottrell junction Slip Slip bands Interstitials Bjerrum defect Frenkel defect Wigner effect Vacancy Schottky defect F-center Stone–Wales defect Defects in diamond Laws Bragg's law Friedel's law Steno's law (constancy of interfacial angles) Law of rational indices Law of symmetry Bragg plane Ewald's sphere Hermann–Mauguin notation Structure factor Thermal ellipsoid Characterisation Electron Diffraction Scattering Neutron Diffraction Scattering Nuclear magnetic resonance X-ray Diffraction Scattering Algorithms Direct methods Isomorphous replacement Molecular replacement Molecular dynamics Patterson map Phase retrieval Gerchberg–Saxton Single particle analysis Software CCP4 Coot CrystalExplorer DSR JANA2020 MTEX OctaDist Olex2 SHELX Databases Bilbao Crystallographic Server CCDC CIF COD ICSD ICDD PDB Journals Crystal Growth & Design Crystallography Reviews Journal of Chemical Crystallography Journal of Crystal Growth Kristallografija Zeitschrift für Kristallographie – Crystalline Materials Zeitschrift für Kristallographie – New Crystal Structures Awards Carl Hermann Medal Ewald Prize Gregori Aminoff Prize History Timeline of crystallography History of crystallography before X-rays Chemical Geometrical Physical Organisation IUCr IOBCr RAS DMG Associations Europe France Germany UK US Japan Category Commons

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Adapted from the Wikipedia article [Unit cell](https://en.wikipedia.org/wiki/Unit_cell) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Unit_cell?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.