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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 = a2 x a3 / (a1 . (a2 x a3))
b 2 = a3 x a1 / (a1 . (a2 x a3))
b 3 = a1 x a2 / (a1 . (a2 x a3)),

where an 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.
  • bi . aj = 2πδij
  • For any Bravais lattice vector R and any reciprocal lattice vector K, eiR.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 VKeiK.r,

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