The **reciprocal lattice** is a fundamental concept in the study of crystalline solids. A crystalline solid consists of a set of basis atoms dispersed in a Bravais lattice (see Bravais lattice). The *reciprocal lattice* of a particular Bravais lattice is defined as the lattice spanned by the following generating vectors:

**b **_{1} = 2π **a**_{2} x a_{3} / (a_{1} . (a_{2} x a_{3}))

**b **_{2} = 2π **a**_{3} x a_{1} / (a_{1} . (a_{2} x a_{3}))

**b **_{3} = 2π **a**_{1} x a_{2} / (a_{1} . (a_{2} x a_{3})),

where **a**_{n} is a basis vector of the particular Bravais lattice.

Given the above definition, it is easy to show that the reciprocal lattice has the following properties:

- The reciprocal lattice is itself a Bravais lattice.
**b**_{i} . a_{j} = 2πδ_{ij}
- For any Bravais lattice vector
**R** and any reciprocal lattice vector **K**, e^{iR.K} = 1.
- The reciprocal lattice of the reciprocal lattice is the original Bravais lattice.

Thus every Bravais lattice has a corresponding reciprocal Bravais lattice. This is a one-to-one correspondence. The reciprocal lattice of the simple cubic lattice is the simple cubic lattice. The reciprocal lattice of the face-centered cubic lattice is the body-centered cubic lattice (and vice-versa).

Notice that the third property of reciprocal lattices implies that any^{*} function V(**r**) with the same periodicity as a Bravais lattice can be expanded into the following Fourier series:

^{*} (to satisfy mathematicians, I should say *any physically meaningful*)

V(**r**) = Σ V_{K}e^{iK.r},

where the sum is taken over all reciprocal lattice vectors and V_{K} is a Fourier coefficient. It is often very useful to write the periodic potential in a solid as such a Fourier series.