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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.
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.
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.
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 ...
The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis. [1] The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb point set can be seen as the union of two offset hexagonal lattices. In nature, carbon atoms of the two-dimensional material graphene are arranged in a honeycomb ...
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.
For each Bravais lattice vector we define a translation operator ^ which, when operating on any function () shifts the argument by : ^ = (+) Since all translations form an Abelian group, the result of applying two successive translations does not depend on the order in which they are applied, i.e. ^ ^ = ^ ^ = ^ + In addition, as the Hamiltonian ...