Reciprocal lattice

In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that

for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice.

For a three dimension lattice, defined by its primitive vectors , its reciprocal lattice can be determined by generating its three reciprocal primitive vectors, through the formula,

In particular, we find that the reciprocal simple cubic Bravais lattice, with cubic primitive cell of side , has for its reciprocal a simple cubic lattice with a cubic primitive cell of side . The cubic lattice is therefore said to be dual, having its reciprocal lattice being identical (up to a numerical factor).

The reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In X-ray diffraction, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The X-ray diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer the atomic arrangment of a crystal.

The Brillouin zone is a primitive unit cell of the reciprocal lattice.