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 3 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 (2π), regardless of whether there are atoms on all these planes or not.
There are also several related notations:[1]
the notation denotes the set of all planes that are equivalent to by the symmetry of the lattice.
In the context of crystal directions (not planes), the corresponding notations are:
with square instead of round brackets, denotes a direction in the basis of the direct lattice vectors instead of the reciprocal lattice; and
similarly, the notation denotes the set of all directions that are equivalent to by symmetry.
Note, for Laue–Bragg interferences
lacks any bracketing when designating a reflection
Miller indices were introduced in 1839 by the British mineralogist William Hallowes Miller, although an almost identical system (Weiss parameters) had already been used by German mineralogist Christian Samuel Weiss since 1817.[2] The method was also historically known as the Millerian system, and the indices as Millerian,[3] although this is now rare.
The Miller indices are defined with respect to any choice of unit cell and not only with respect to primitive basis vectors, as is sometimes stated.
^Ashcroft, Neil W.; Mermin, N. David (1976). Solid state physics. New York: Holt, Rinehart and Winston. ISBN 0030839939. OCLC 934604.
^Weiss, Christian Samuel (1817). "Ueber eine verbesserte Methode für die Bezeichnung der verschiedenen Flächen eines Krystallisationssystems, nebst Bemerkungen über den Zustand der Polarisierung der Seiten in den Linien der krystallinischen Structur". Abhandlungen der physikalischen Klasse der Königlich-Preussischen Akademie der Wissenschaften: 286–336.
^Oxford English Dictionary Online (Consulted May 2007)
to Millerindex. IUCr Online Dictionary of Crystallography Millerindex description with diagrams Online tutorial about lattice planes and Miller indices...
111, a free-to-call non-emergency medical helpline in the UK (111) a Millerindex for the crystal face plane formed by cutting off the corner equally along...
Glycemic Index?". Angelika Christie. 2009-09-21. Archived from the original on 2016-03-03. Retrieved 2014-01-20. Foster-Powell K, Holt SH, Brand-Miller JC (July...
wafer's crystalline lattice, such as the [001] Millerindex of the film aligning with the [001] index of the substrate. In the simplest case, the epitaxial...
Vectors and planes in a crystal lattice are described by the three-value Millerindex notation. This syntax uses the indices h, k, and ℓ as directional parameters...
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atomic arrangement of the crystal: they are planes of relatively low Millerindex. This occurs because some surface orientations are more stable than others...
Hong Kong Kuala Lumpur Hospital (Malay: Hospital Besar Kuala Lumpur) Millerindex (hkℓ), describing crystal lattice planes This disambiguation page lists...
other one being the cube). In pyritohedral pyrite, the faces have a Millerindex of (210), which means that the dihedral angle is 2·arctan(2) ≈ 126.87°...
Wentworth Miller is British — BBC America". Retrieved May 11, 2017. "Index entry". FreeBMD. ONS. Retrieved January 28, 2017. "Being Wentworth Miller: The star...
to each face is plotted on the net. Each point is labelled with its Millerindex. The final plot allows the symmetry of the crystal to be established...
rhombohedral system, and the most common scalenohedron form has the Millerindex [2131]. Spar is a general term for transparent to translucent, generally...
The Index of Economic Freedom is an annual index and ranking created in 1995 by The Heritage Foundation and The Wall Street Journal to measure the degree...
com/news/flint/index.ssf/2009/12/convicted_killer_sharee_miller_2.html "Convicted killer Sharee Miller back on Facebook" mlive.com "BRUCE MILLER - SHAREE MILLER -...
atomic arrangement of the crystal: They are planes of relatively low Millerindex. This occurs because some surface orientations are more stable than others...
crystal in three dimensions. It is therefore indexed with direct lattice indices, instead of with Miller indices. High-symmetry zone axes through a crystal...
score is the higher. Disposition index Glycemic index Glycemic load Insulin oscillations Holt, Susanne H.A.; Brand-Miller, Janette Cecile; Petocz, Peter...
Kelley King Earnhardt Miller (née Earnhardt; formerly Earnhardt-Elledge; born August 28, 1972) is an American businesswoman. She is the chief executive...
The cephalic index or cranial index is a number obtained by taking the maximum width (biparietal diameter or BPD, side to side) of the head of an organism...
bipyramidal shape. The common crystal face Millerindex is {111}, But it can have smaller faces with Millerindex {010} and {110}. The density is 2.403 g/cm3...
V {\displaystyle V} with a basis B {\displaystyle B} of vectors indexed by an index set I {\displaystyle I} (the cardinality of I {\displaystyle I} is...
Reserve Bank of Minneapolis. "Consumer Price Index (estimate) 1800–". Retrieved February 29, 2024. "Miller Park". Projects. International Facilities Group...
characterized by a Millerindex. If the symmetry is high, e.g.: cubic or hexagonal it is usually not too hard to identify the index of each peak, even...