Reciprocal lattice

Reciprocal lattice is a concept associated with solids with <a href="/facts/Translational_symmetry/PYVMaqWb">translational symmetry</a> which plays a major role in many areas such as <a href="/facts/X-ray_diffraction/WtI0F5DN">X-ray</a> and <a href="/facts/Electron_diffraction/IlFzQyVu">electron</a> diffraction as well as the <a href="/facts/Electronic_band_structure/W5o7oE2R">energies</a> of electrons in a solid. It emerges from the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of the <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice</a> associated with the arrangement of the atoms. The direct lattice or real lattice is a <a href="/facts/Periodic_function/Sm8lJQsF">periodic function</a> in <a href="/facts/Space_(physics)/tq6LUM3C">physical space</a>, such as a <a href="/facts/Crystal_system/dMQLk1q4">crystal system</a> (usually a <a href="/facts/Bravais_lattice/7gQKTsXk">Bravais lattice</a>). The reciprocal lattice exists in the <a href="/facts/Mathematical_space/G6rt4bQ8">mathematical space</a> of <a href="/facts/Spatial_frequencies/b0un3CNv">spatial frequencies</a> or <a href="/facts/Wavenumber/cLaLAeW5">wavenumbers</a> k, known as reciprocal space or k space; it is the dual of physical space considered as a <a href="/facts/Vector_space/M1pxMLx2">vector space</a>. In other words, the reciprocal lattice is the <a href="/facts/Sublattice/5ak6ufky">sublattice</a> which <a href="/facts/Dual_lattice/56zIMYcA">is dual</a> to the direct lattice.
The reciprocal lattice is the set of all <a href="/facts/Vector_(geometric)/bCLrdksy">vectors</a> 
 
 
 
 
 
 G
 
 
 m
 
 
 
 
 {\displaystyle \mathbf {G} _{m}}
 
, that are <a href="/facts/Wave_vector/Jk8wUEft">wavevectors</a> k of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbf {R} _{n}}
 
. Each plane wave in this Fourier series has the same phase or phases that are differed by multiples of 
 
 
 
 2
 π
 
 
 {\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 <a href="/facts/Covariance_and_contravariance_of_vectors/s3HGMkhz">covariant and contravariant vectors</a>, respectively.
The <a href="/facts/Brillouin_zone/V8unhAAl">Brillouin zone</a> is a <a href="/facts/Wigner%E2%80%93Seitz_cell/qU6azUs6">Wigner–Seitz cell</a> of the reciprocal lattice.

Reciprocal lattice open-in-new

Reciprocal lattice