Editing Reciprocal lattice (section)

{{Short description|Fourier transform of a real-space lattice, important in solid-state physics}}
[[File:Reciprocal_monoclinic_lattice.png|thumb|The computer-generated reciprocal lattice of a fictional [[monoclinic]] 3D crystal.]]
[[File:Rcprwrld2.png|thumb|A two-dimensional crystal and its 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 diffraction|X-ray]] and [[Electron diffraction|electron]] diffraction as well as the [[Electronic band structure|energies]] of electrons in a solid. It emerges from the [[Fourier transform]] of the [[lattice (group)|lattice]] associated with the arrangement of the atoms. The ''direct lattice'' or ''real lattice'' is a [[periodic function]] in [[Space (physics)|physical space]], such as a [[crystal system]] (usually a [[Bravais lattice]]). The reciprocal lattice exists in the [[mathematical space]] of [[spatial frequencies]] or [[wavenumber]]s ''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 [[Dual lattice|is dual]] to the direct lattice.

The reciprocal lattice is the set of all [[vector (geometric)|vectors]] <math>\mathbf{G}_m</math>, that are [[wave vector|wavevectors]] '''k''' of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice <math>\mathbf{R}_n</math>. Each plane wave in this Fourier series has the same phase or phases that are differed by multiples of {{nowrap begin}}<math>2\pi</math>,{{nowrap end}} 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 [[Covariance and contravariance of vectors|covariant and contravariant vectors]], respectively.

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