Reciprocal lattice in the context of "Hexagonal lattice"

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.

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.