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Reciprocal space (also called k-space) provides a way to visualize the results of the Fourier transform of a spatial function. It is similar in role to the frequency domain arising from the Fourier transform of a time dependent function; reciprocal space is a space over which the Fourier transform of a spatial function is represented at spatial frequencies or wavevectors of plane waves of the ...
The reciprocal lattices (dots) and corresponding first Brillouin zones of (a) square lattice and (b) hexagonal lattice. In mathematics and solid state physics , the first Brillouin zone (named after Léon Brillouin ) is a uniquely defined primitive cell in reciprocal space .
For example, in a crystal's k-space, there is an infinite set of points called the reciprocal lattice which are "equivalent" to k = 0 (this is analogous to aliasing). Likewise, the " first Brillouin zone " is a finite volume of k -space, such that every possible k is "equivalent" to exactly one point in this region.
This is based on the fact that a reciprocal lattice vector (the vector indicating a reciprocal lattice point from the reciprocal lattice origin) is the wavevector of a plane wave in the Fourier series of a spatial function (e.g., electronic density function) which periodicity follows the original Bravais lattice, so wavefronts of the plane wave ...
In the Figure the red dot is the origin for the wavevectors, the black spots are reciprocal lattice points (vectors) and shown in blue are three wavevectors. For the wavevector k 1 {\displaystyle \mathbf {k_{1}} } the corresponding reciprocal lattice point g 1 {\displaystyle \mathbf {g_{1}} } lies on the Ewald sphere, which is the condition for ...
The Laue equations can be written as = = as the condition of elastic wave scattering by a crystal lattice, where is the scattering vector, , are incoming and outgoing wave vectors (to the crystal and from the crystal, by scattering), and is a crystal reciprocal lattice vector.
In a one-dimensional lattice the number of reciprocal lattice vectors that determine the bands in an energy interval is limited to two when the energy rises. In two and three dimensional lattices the number of reciprocal lattice vectors that determine the free electron bands E n ( k ) {\displaystyle E_{n}(\mathbf {k} )} increases more rapidly ...
An example of the tetragonal crystals, wulfenite Two different views (top down and from the side) of the unit cell of tP30-CrFe (σ-phase Frank–Kasper structure) that show its different side lengths, making this structure a member of the tetragonal crystal system.