# Primitive cell

In geometry, crystallography, mineralogy, and solid state physics, a **primitive cell** is a minimum volume cell (a unit cell) corresponding to a single lattice point of a structure with discrete translational symmetry. The concept is used particularly in describing crystal structure in two and three dimensions, though it makes sense in all dimensions. A lattice can be characterized by the geometry of its primitive cell.

The primitive cell is a *primitive unit*. A primitive unit is a section of the tiling (usually a parallelogram or a set of neighboring tiles) that generates the whole tiling using only translations, and is as small as possible.

The primitive cell is a fundamental domain with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller.

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

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

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 from the above axes as

A primitive cell is considered to contain 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]}

A Wigner–Seitz cell is a primitive cell centered on the single lattice point it contains. This is a type of Voronoi cell. The Wigner–Seitz cell of the reciprocal lattice in momentum space is called the Brillouin zone.

## 2-dimensional primitive cell

A 2-dimensional primitive cell is a parallelogram, which in special cases may have orthogonal angles, or equal lengths, or both.

Parallelogram (Monoclinic) |
Rhombus (Orthorhombic) |
Rectangle (Orthorhombic) |
Square (Tetragonal) |

## 3-dimensional primitive cell

A crystal can be categorized by its lattice and the atoms that lie in a primitive cell (the *basis*). A cell will fill all the lattice space without leaving gaps by repetition of crystal translation operations.

A 3-dimensional primitive cell is a parallelepiped, which in special cases may have orthogonal angles, or equal lengths, or both.

Parallelepiped (Triclinic) |
Clinorhombic prism (Monoclinic) |
Right parallelogrammic prism (Monoclinic) |
Right rhombic prism (Orthorhombic) |
Rectangular cuboid (Orthorhombic) |
Square cuboid (Tetragonal) |
Trigonal trapezohedron (Rhombohedral) |
Cube (Cubic) |

## See also

## Notes

- ↑ In n dimensions the crystal translation vector would be
- ↑ Kittel, Charles (2004).
*Introduction to Solid State Physics*(8th ed.). p. 4. - ↑ "DoITPoMS - TLP Library Crystallography - Unit Cell".
*Online Materials Science Learning Resources: DoITPoMS*. University of Cambridge. Retrieved 21 February 2015.