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The reciprocal lattice is a term associated with solids with translational symmetry, and plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier transform of the lattice associated with the arrangement of the atoms. The direct lattice or real lattice is ...
That is, (hkℓ) simply indicates a normal to the planes in the basis of the primitive reciprocal lattice vectors. Because the coordinates are integers, this normal is itself always a reciprocal lattice vector. The requirement of lowest terms means that it is the shortest reciprocal lattice vector in the given direction.
In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries of this cell are given ...
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.)
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.
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.
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 hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types. [1] The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° and are of equal lengths, The reciprocal lattice of the hexagonal lattice is a ...