Reciprocal lattice in the context of Wave vector

Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices.

In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers h, k, and ℓ, the Miller indices. They are written (hkℓ), and denote the family of (parallel) lattice planes (of the given Bravais lattice) orthogonal to ${\displaystyle \mathbf {g} _{hk\ell }=h\mathbf {b} _{1}+k\mathbf {b} _{2}+\ell \mathbf {b} _{3}}$, where ${\displaystyle \mathbf {b} _{i}}$ are the basis or primitive translation vectors of the reciprocal lattice for the given Bravais lattice. (Note that the plane is not always orthogonal to the linear combination of direct or original lattice vectors ${\displaystyle h\mathbf {a} _{1}+k\mathbf {a} _{2}+\ell \mathbf {a} _{3}}$ because the direct lattice vectors need not be mutually orthogonal.) This is based on the fact that a reciprocal lattice vector ${\displaystyle \mathbf {g} }$ (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 are coincident with parallel lattice planes of the original lattice. Since a measured scattering vector in X-ray crystallography, ${\displaystyle \Delta \mathbf {k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }}$ with ${\displaystyle \mathbf {k} _{\mathrm {out} }}$ as the outgoing (scattered from a crystal lattice) X-ray wavevector and ${\displaystyle \mathbf {k} _{\mathrm {in} }}$ as the incoming (toward the crystal lattice) X-ray wavevector, is equal to a reciprocal lattice vector ${\displaystyle \mathbf {g} }$ as stated by the Laue equations, the measured scattered X-ray peak at each measured scattering vector ${\displaystyle \Delta \mathbf {k} }$ is marked by Miller indices.

The hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types. 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 hexagonal lattice in reciprocal space with orientation changed by 90° and primitive lattice vectors of length

Kikuchi lines are patterns of electrons formed by scattering. They pair up to form bands in electron diffraction from single crystal specimens, there to serve as "roads in orientation-space" for microscopists uncertain of what they are looking at. In transmission electron microscopes, they are easily seen in diffraction from regions of the specimen thick enough for multiple scattering. Unlike diffraction spots, which blink on and off as one tilts the crystal, Kikuchi bands mark orientation space with well-defined intersections (called zones or poles) as well as paths connecting one intersection to the next.

Experimental and theoretical maps of Kikuchi band geometry, as well as their direct-space analogs e.g. bend contours, electron channeling patterns, and fringe visibility maps are increasingly useful tools in electron microscopy of crystalline and nanocrystalline materials. Because each Kikuchi line is associated with Bragg diffraction from one side of a single set of lattice planes, these lines can be labeled with the same Miller or reciprocal-lattice indices that are used to identify individual diffraction spots. Kikuchi band intersections, or zones, on the other hand are indexed with direct-lattice indices i.e. indices which represent integer multiples of the lattice basis vectors a, b and c.

In crystallography and solid state physics, the Laue equations relate incoming waves to outgoing waves in the process of elastic scattering, where the photon energy or light temporal frequency does not change upon scattering by a crystal lattice. They are named after physicist Max von Laue (1879–1960).

The Laue equations can be written as ${\displaystyle \mathbf {\Delta k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {G} }$ as the condition of elastic wave scattering by a crystal lattice, where ${\displaystyle \mathbf {\Delta k} }$ is the scattering vector, ${\displaystyle \mathbf {k} _{\mathrm {in} }}$, ${\displaystyle \mathbf {k} _{\mathrm {out} }}$ are incoming and outgoing wave vectors (to the crystal and from the crystal, by scattering), and ${\displaystyle \mathbf {G} }$ is a crystal reciprocal lattice vector. Due to elastic scattering ${\displaystyle |\mathbf {k} _{\mathrm {out} }|^{2}=|\mathbf {k} _{\mathrm {in} }|^{2}}$, three vectors. ${\displaystyle \mathbf {G} }$, ${\displaystyle \mathbf {k} _{\mathrm {out} }}$, and ${\displaystyle -\mathbf {k} _{\mathrm {in} }}$ , form a rhombus if the equation is satisfied. If the scattering satisfies this equation, all the crystal lattice points scatter the incoming wave toward the scattering direction (the direction along ${\displaystyle \mathbf {k} _{\mathrm {out} }}$). If the equation is not satisfied, then for any scattering direction, only some lattice points scatter the incoming wave. (This physical interpretation of the equation is based on the assumption that scattering at a lattice point is made in a way that the scattering wave and the incoming wave have the same phase at the point.) It also can be seen as the conservation of momentum as ${\displaystyle \hbar \mathbf {k} _{\mathrm {out} }=\hbar \mathbf {k} _{\mathrm {in} }+\hbar \mathbf {G} }$ since ${\displaystyle \mathbf {G} }$ is the wave vector for a plane wave associated with parallel crystal lattice planes. (Wavefronts of the plane wave are coincident with these lattice planes.)