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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 size of the Ewald's sphere and hence the number of diffraction spots on the screen is controlled by the incident electron energy. From the knowledge of the reciprocal lattice models for the real space lattice can be constructed and the surface can be characterized at least qualitatively in terms of the surface periodicity and the point group.
Another helpful ingredient in the proof is the reciprocal lattice vectors. These are three vectors b 1, b 2, b 3 (with units of inverse length), with the property that a i · b i = 2π, but a i · b j = 0 when i ≠ j. (For the formula for b i, see reciprocal lattice vector.)
The wavevector takes on any value inside the Brillouin zone, which is a polyhedron in wavevector (reciprocal lattice) space that is related to the crystal's lattice. Wavevectors outside the Brillouin zone simply correspond to states that are physically identical to those states within the Brillouin zone.
The reciprocal lattice is easily constructed in one dimension: for particles on a line with a period , the reciprocal lattice is an infinite array of points with spacing /. In two dimensions, there are only five Bravais lattices. The corresponding reciprocal lattices have the same symmetry as the direct lattice.
The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.
The translational invariance of a crystal lattice is described by a set of unit cell, direct lattice basis vectors (contravariant [1] or polar) called a, b, and c, or equivalently by the lattice parameters, i.e. the magnitudes of the vectors, called a, b and c, and the angles between them, called α (between b and c), β (between c and a), and γ (between a and b).