In geometry, solid state physics and mineralogy, particularly in describing crystal structure, a primitive cell, is a minimum cell corresponding to a single lattice point of a structure with translational symmetry in 2D, 3D, 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, <math> \vec T </math>, and also gives a lattice cell of smallest volume for a particular lattice. The lattice and translation vectors <math> \vec a_1 </math>, <math> \vec a_2 </math>, and <math> \vec a_3 </math> are primitive if the atoms look the same from any lattice points using integers <math> u_1 </math>, <math> u_2 </math>, and <math> u_3 </math>.
- <math> \vec T = u_1\vec a_1 + u_2\vec a_2 + u_3\vec a_3 </math>
The primitive cell is defined by the primitive axes (vectors) <math> \vec a_1 </math>, <math> \vec a_2 </math>, and <math> \vec a_3 </math>. The volume, <math> V_c </math>, of the primitive cell is given by the parallelepiped from the above axes as,
- <math> V_c = | \vec a_1 \bullet \vec a_2 \times \vec a_3 | </math>.
A Wigner-Seitz cell is an example of another kind of primitive cell. In certain circumstances, primitive cell is synonymous with unit cell. However, the conventional description of cubic lattices, such as body centered cubic (BCC) and face centered cubic (FCC) lattices, relies on a cubic unit cell. In the BCC and FCC cases, the primitive cell is distinct from this conventional unit cell. The general mathematical concept behind the primitive cell is termed the fundamental domain or the Voronoi cell. The primitive cell of the reciprocal lattice in momentum space is called the Brillouin zone.
