Lattice parallelepiped

A lattice parallelepiped is a parallelepiped spanned by three linearly independent lattice vectors. If the vectors are a primitive basis of a Bravais lattice, translates of the corresponding half-open parallelepiped tile space without overlap, with one lattice point per cell under consistent boundary counting.

Its volume is $|{\mathbf{a}}_1\cdot ({\mathbf{a}}_2\times {\mathbf{a}}_3)|$. Choosing longer lattice vectors can instead produce a nonprimitive cell containing several lattice points; integer coordinates of its vertices alone do not prove primitiveness. The primitive version is useful for counting states and relating real-space volume to the volume of a reciprocal lattice cell.