Reciprocal lattice

In physics, the reciprocal lattice emerges from the Fourier transform of another lattice. 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 reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In neutron, helium and X-ray diffraction, due to the Laue conditions, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer the atomic arrangement of a crystal.

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