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The computer-generated reciprocal lattice of a fictional monoclinic 3D crystal. A two-dimensional crystal and its reciprocal lattice. Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid.
For instance, it is used in the statement of the Poisson summation formula, transference theorems provide connections between the geometry of a lattice and that of its dual, and many lattice algorithms exploit the dual lattice. For an article with emphasis on the physics / chemistry applications, see Reciprocal lattice. This article focuses on ...
Examples of determining indices for a plane using intercepts with axes; left (111), right (221) There are two equivalent ways to define the meaning of the Miller indices: [1] via a point in the reciprocal lattice, or as the inverse intercepts along the lattice vectors.
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
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 Ewald sphere is a geometric construction used in electron, neutron, and x-ray diffraction which shows the relationship between: the wavevector of the incident and diffracted beams, the diffraction angle for a given reflection, the reciprocal lattice of the crystal. It was conceived by Paul Peter Ewald, a German physicist and ...
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 ...
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 .
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