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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 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.
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.
Every crystal is a periodic structure which can be characterized by a Bravais lattice, and for each Bravais lattice we can determine the reciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors (b 1, b 2, b 3).
Examples of determining indices for a plane using intercepts with axes; left (111), right (221) There are two equivalent ways to define the meaning of the Miller indices: [1] via a point in the reciprocal lattice, or as the inverse intercepts along the lattice vectors.
Fig. 1: A hexagonal sampling lattice and its basis vectors v 1 and v 2 Fig. 2: The reciprocal lattice corresponding to the lattice of Fig. 1 and its basis vectors u 1 and u 2 (figure not to scale). The concept of a bandlimited function in one dimension can be generalized to the notion of a wavenumber-limited function in higher dimensions.
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 ...