= 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.
Back to article page
