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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 ...
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 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.
Let denote a lattice in and the corresponding reciprocal lattice. The theorem of Petersen and Middleton [1] states that a function () that is wavenumber-limited to a set can be exactly reconstructed from its measurements on provided that the set does not overlap with any of its shifted versions + where the shift x is any nonzero element of the reciprocal lattice .
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.
While in optical goniometry net-plane normals (reciprocal lattice vectors) need to be successively aligned parallel to the reference direction of an optical goniometer in order to derive measurements of interfacial angles, the corresponding alignment needs to be done for zone axes (direct lattice vector) in transmission electron goniometry.
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 ...