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