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 constructed so that it contains only one lattice point (each vertex of the cell sits on a lattice point which is shared with the surrounding cells, each lattice point is said to contribute 1/n to the total number of lattice points in the cell where n is the number of cells sharing the lattice point). A primitive cell is 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.

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.

2-dimensional primitive cells

Parallelogram

Rectangle

Rhombus

Square

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.

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.

3-dimensional primitive cells

Triclinic

Monoclinic

Rhombohedral

Orthorhombic

Tetragonal

Cubic

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

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

See also

• Bravais lattice
• Wallpaper group
• Space group