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,

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.

Reciprocal lattices of various crystals

Simple cubic lattice

We find that the reciprocal simple cubic Bravais lattice, with cubic primitive cell of side a, 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).

Face-centered cubic lattice

The reciprocal lattice to an FCC lattice is the BCC lattice.

Body-centered cubic lattice

The reciprocal lattice to an BCC lattice is the FCC lattice.

Mathematics of the dual lattice

There are actually two versions in mathematics of the abstract dual lattice concept, for a given lattice L in a real vector space V, of finite dimension.

The first, which generalises directly the reciprocal lattice construction, uses Fourier analysis. It may be stated simply in terms of Pontryagin duality. The dual group V^ to V is again a real vector space, and its closed subgroup L^ dual to L turns out to be a lattice in V^. Therefore L^ is the natural candidate for dual lattice, in a different vector space (of the same dimension).

The other aspect is seen in the presence of a quadratic form Q on V; if it is non-degenerate it allows an identification of the dual space V* of V with V. The relation of V* to V^ is not intrinsic; it depends on a choice of Haar measure (volume element) on V. But given an identification of the two, which is in any case well-defined up to a scalar, the presence of Q allows one to speak to the dual lattice to L while staying within V.