# Reciprocal space¶

Reciprocal space (also called "k-space") 1 is the space in which the Fourier transform of a spatial function is represented. A Fourier transform takes us from "real space" to reciprocal space or vice versa.

## Reciprocal Lattice¶

In crystallography, the reciprocal lattice 2 represents the Fourier transform of an original periodic Bravais lattice in the real physical space (the direct lattice). This reciprocal lattice exists in the so-called reciprocal space (also called "k-space"), which is in general the space in which the Fourier transform of a spatial function is represented.

It follows from its definition that the reciprocal lattice of a reciprocal lattice is nothing but the original direct lattice again, since the two lattices are reciprocally related to each other by Fourier transforms. The reciprocal lattice plays a fundamental role in the analytic study of periodic structures, particularly in the theory of X-ray and neutron diffraction.

Another important concept within the realm of reciprocal space is that of the Brillouin zone of a crystal, which consists in the Wigner-Seitz cell of the crystal's reciprocal lattice.

## Some important properties¶

The properties below are frequently considered in the computational studies of crystalline materials: