Reciprocal lattice

The reciprocal lattice is a term associated with solids with translational symmetry, and 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, known as reciprocal space or k space, which is the dual of physical space considered as a vector space, and the reciprocal lattice is the sublattice of that space that is dual to the direct lattice.

In quantum physics, reciprocal space is closely related to momentum space according to the proportionality ${\displaystyle \mathbf {p} =\hbar \mathbf {k} }$, where ${\displaystyle \mathbf {p} }$ is the momentum vector and ${\displaystyle \hbar }$ is the reduced Planck constant. 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 reciprocal lattice is the set of all vectors ${\displaystyle \mathbf {G} _{m}}$, that are wavevectors 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 Brillouin zone is a Wigner–Seitz cell of the reciprocal lattice.