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Concept
Miller index
Summary
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 , where 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 because the direct lattice vectors need not be mutually orthogonal.) 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 are coincident with parallel lattice planes of the original lattice. Since a measured scattering vector in X-ray crystallography, with as the outgoing (scattered from a crystal lattice) X-ray wavevector and as the incoming (toward the crystal lattice) X-ray wavevector, is equal to a reciprocal lattice vector as stated by the Laue equations, the measured scattered X-ray peak at each measured scattering vector is marked by Miller indices. By convention, negative integers are written with a bar, as in for −3. The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1. Miller indices are also used to designate reflections in X-ray crystallography. In this case the integers are not necessarily in lowest terms, and can be thought of as corresponding to planes spaced such that the reflections from adjacent planes would have a phase difference of exactly one wavelength (2pi), regardless of whether there are atoms on all these planes or not.
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Related concepts (26)
Miller index
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 , where are the basis or primitive translation vectors of the reciprocal lattice for the given Bravais lattice.
Reciprocal lattice
In physics, the reciprocal lattice represents the Fourier transform of another lattice. The direct lattice or real lattice is a periodic function in physical space, such as a crystal system (usually a Bravais lattice). The reciprocal lattice exists in the mathematical space of spatial frequencies, known as reciprocal space or k space, where refers to the wavevector. In quantum physics, reciprocal space is closely related to momentum space according to the proportionality , where is the momentum vector and is the reduced Planck constant.
Cubic crystal system
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of these crystals: Primitive cubic (abbreviated cP and alternatively called simple cubic) Body-centered cubic (abbreviated cI or bcc) Face-centered cubic (abbreviated cF or fcc) Note: the term fcc is often used in synonym for the cubic close-packed or ccp structure occurring in metals.
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