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 ) |.

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