Unit cell

Ingeometry,biology,mineralogyandsolid state physics,aunit cellis 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, 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 and is the basic building block from which larger cells are constructed.

The concept is used particularly in describingcrystal structurein 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 (aparallelogramorparallelepiped) that generates the whole tiling using only translations.

There are two special cases of the unit cell: theprimitive celland theconventional cell.The primitive cell is a unit cell corresponding to a singlelattice 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 areparallelotopesinndimensions.

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 as⁠1/ $n$ ⁠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 contain⁠1/8⁠of 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).

Theprimitive translation vectors $a \to 1$ , $a \to 2$ , $a \to 3$ span a lattice cell of smallest volume for a particular three-dimensional lattice, and are used to define a crystal translation vector

{\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 $r' = r + T \to$ as from $r$ .^[4]

Since the primitive cell is defined by the primitive axes (vectors) $a \to 1$ , $a \to 2$ , $a \to 3$ ,the volume $V p$ of the primitive cell is given by theparallelepipedfrom the above axes as

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

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 ofVoronoi cell.The Wigner–Seitz cell of thereciprocal latticeinmomentum spaceis called theBrillouin 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

For any 2-dimensional lattice, the unit cells areparallelograms,which in special cases may have orthogonal angles, equal lengths, or both. Four of the five two-dimensionalBravais latticesare 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

For any 3-dimensional lattice, the conventional unit cells areparallelepipeds,which in special cases may have orthogonal angles, or equal lengths, or both. Seven of the fourteen three-dimensionalBravais latticesare represented using conventional primitive cells, as shown below.

Conventional primitive cell
Shape name	Parallelepiped	Oblique rectangularprism	Rectangularcuboid	Squarecuboid	Trigonal trapezohedron	Cube	Right rhombicprism
Bravais lattice	PrimitiveTriclinic	PrimitiveMonoclinic	PrimitiveOrthorhombic	PrimitiveTetragonal	PrimitiveRhombohedral	PrimitiveCubic	PrimitiveHexagonal

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 rhombicprism	Right rhombicprism
Conventional cell
Bravais lattice	Base-centeredMonoclinic	Base-centeredOrthorhombic	Body-centeredOrthorhombic	Face-centeredOrthorhombic	Body-centeredTetragonal	Body-centeredCubic	Face-centeredCubic

Notes

^ In $n$ dimensions the crystal translation vector would be
${\vec {T}}=\sum _{i=1}^{n}u_{i}{\vec {a}}_{i},\quad {\mbox{where }}u_{i}\in \mathbb {Z} \quad \forall i.$
That is, for a point in the lattice $r$ ,the arrangement of points appears the same from $r' = r + T \to$ as from $r$ .

References

^Ashcroft, Neil W. (1976). "Chapter 4".Solid State Physics.W. B. Saunders Company. p. 72.ISBN 0-03-083993-9.
^Simon, Steven (2013).The Oxford Solid State Physics(1 ed.). Oxford University Press. p. 114.ISBN 978-0-19-968076-4.
^"DoITPoMS – TLP Library Crystallography – Unit Cell".Online Materials Science Learning Resources: DoITPoMS.University of Cambridge.Retrieved21 February2015.
^Kittel, Charles (11 November 2004).Introduction to Solid State Physics(8 ed.). Wiley. p.4.ISBN 978-0-471-41526-8.
^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:1806.07864.doi:10.1016/j matsci.2017.01.017.ISSN 0927-0256.S2CID 119490841.
^Aroyo, M. I., ed. (2016-12-31).International Tables for Crystallography.Chester, England: International Union of Crystallography. p. 25.doi:10.1107/97809553602060000114.ISBN 978-0-470-97423-0.
^Ashcroft, Neil W. (1976).Solid State Physics.W. B. Saunders Company. p. 73.ISBN 0-03-083993-9.

[first-4] In $n$ dimensions the crystal translation vector would be
${\vec {T}}=\sum _{i=1}^{n}u_{i}{\vec {a}}_{i},\quad {\mbox{where }}u_{i}\in \mathbb {Z} \quad \forall i.$
That is, for a point in the lattice $r$ ,the arrangement of points appears the same from $r' = r + T \to$ as from $r$ .

[1] Ashcroft, Neil W. (1976). "Chapter 4".Solid State Physics.W. B. Saunders Company. p. 72.ISBN 0-03-083993-9.

[2] Simon, Steven (2013).The Oxford Solid State Physics(1 ed.). Oxford University Press. p. 114.ISBN 978-0-19-968076-4.

[doitpoms-3] "DoITPoMS – TLP Library Crystallography – Unit Cell".Online Materials Science Learning Resources: DoITPoMS.University of Cambridge.Retrieved21 February2015.

[5] Kittel, Charles (11 November 2004).Introduction to Solid State Physics(8 ed.). Wiley. p.4.ISBN 978-0-471-41526-8.

[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:1806.07864.doi:10.1016/j matsci.2017.01.017.ISSN 0927-0256.S2CID 119490841.

[7] Aroyo, M. I., ed. (2016-12-31).International Tables for Crystallography.Chester, England: International Union of Crystallography. p. 25.doi:10.1107/97809553602060000114.ISBN 978-0-470-97423-0.

[8] Ashcroft, Neil W. (1976).Solid State Physics.W. B. Saunders Company. p. 73.ISBN 0-03-083993-9.

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