Primitive cell

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

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

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

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

Primitive cell

See also