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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 ...
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.
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 .
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 ...
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 ...
with the Fourier series coefficients being the K numbers subscripted by m, n respectively, and the reciprocal lattice vector given by . In real modeling, the range of components considered will be reduced to just ± N max {\displaystyle \pm N_{\max }} instead of the ideal, infinite wave.
Consider an FCC compound unit cell. Locate a primitive unit cell of the FCC, i.e., a unit cell with one lattice point. Now take one of the vertices of the primitive unit cell as the origin. Give the basis vectors of the real lattice. Then from the known formulae you can calculate the basis vectors of the reciprocal lattice.