# Primitive cell

The parallelogram is the general primitive cell for the plane.
A parallelepiped is a general primitive cell for 3-dimensional space.

A primitive cell is a unit cell built on the primitive basis of the direct lattice, namely a crystallographic basis of the vector lattice L such that every lattice vector t of L may be obtained as an integral linear combination of the basis vectors, a, b, c.

Used predominantly in geometry, solid state physics, and mineralogy, particularly in describing crystal structure, a primitive cell is a minimum volume cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its primitive cell.

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

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.

Primitive translation vectors are used to define a crystal translation vector, ${\displaystyle {\vec {T}}}$, and also gives a lattice cell of smallest volume for a particular lattice. The lattice and translation vectors ${\displaystyle {\vec {a}}_{1}}$, ${\displaystyle {\vec {a}}_{2}}$, and ${\displaystyle {\vec {a}}_{3}}$ are primitive if the atoms look the same from any lattice points using integers ${\displaystyle u_{1}}$, ${\displaystyle u_{2}}$, and ${\displaystyle u_{3}}$.

${\displaystyle {\vec {T}}=u_{1}{\vec {a}}_{1}+u_{2}{\vec {a}}_{2}+u_{3}{\vec {a}}_{3}}$

The primitive cell is defined by the primitive axes (vectors) ${\displaystyle {\vec {a}}_{1}}$, ${\displaystyle {\vec {a}}_{2}}$, and ${\displaystyle {\vec {a}}_{3}}$. The volume, ${\displaystyle V_{p}}$, of the primitive cell is given by the parallelepiped from the above axes as

${\displaystyle V_{p}=|{\vec {a}}_{1}\cdot ({\vec {a}}_{2}\times {\vec {a}}_{3})|.}$