Reciprocal lattice

Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier transform of the lattice associated with the arrangement of the atoms. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system (usually a Bravais lattice). The reciprocal lattice exists in the mathematical space of spatial frequencies or wavenumbers k, known as reciprocal space or k space; it is the dual of physical space considered as a vector space. In other words, the reciprocal lattice is the sublattice which is dual to the direct lattice.

The reciprocal lattice is the set of all vectors ${\displaystyle \mathbf {G} _{m}}$, that are wavevectors k of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice ${\displaystyle \mathbf {R} _{n}}$. Each plane wave in this Fourier series has the same phase or phases that are differed by multiples of ${\displaystyle 2\pi }$ at each direct lattice point (so essentially same phase at all the direct lattice points).

The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively.

The Brillouin zone is a Wigner–Seitz cell of the reciprocal lattice.