Reciprocal Lattice - Generalization of A Dual Lattice

Generalization of A 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.

In mathematics, the dual lattice of a given lattice L in an abelian locally compact topological group G is the subgroup L∗ of the dual group of G consisting of all continuous characters that are equal to one at each point of L. In discrete mathematics, a lattice is a locally discrete set of points described by all integral linear combinations of dim = n linearly independent vectors in Rn. The dual lattice is then defined by all points in the linear span of the original lattice (typically all of R^n) with the property that an integer results from the inner product with all elements of the original lattice. It follows that the dual of the dual lattice is the original lattice. Furthermore, if we allow the matrix B to have columns as the linearly independent vectors that describe the lattice, then the matrix

has columns of vectors that describe the dual lattice.