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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 ...
For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a); similarly for the reciprocal lattice. So, in this common case, the Miller indices ( ℓmn ) and [ ℓmn ] both simply denote normals/directions in Cartesian coordinates .
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 ...
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).
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.
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.
Ray diagram of Von Laue formulation. In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, , at right angles. [1] The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography.