Editing Miller index (section)

{{Short description|Notation system for crystal lattice planes}}
[[Image:Miller Indices Felix Kling.svg|thumb|300px|Planes with different Miller indices in cubic crystals]]
[[Image:Indices miller direction exemples.png|thumb|300px|Examples of directions]]
'''Miller indices''' form a notation system in [[crystallography]] for lattice planes in [[Bravais lattice|crystal (Bravais) lattices]].

In particular, a family of [[lattice plane]]s of a given (direct) Bravais lattice is determined by three [[integer]]s ''h'', ''k'', and&nbsp;''ℓ'', the ''Miller indices''. They are written (''hkℓ''), and denote the family of (parallel) lattice planes (of the given Bravais lattice) orthogonal to <math>\mathbf{g}_{hk\ell} = h\mathbf{b}_1 + k\mathbf{b}_2 + \ell\mathbf{b}_3 </math>, where <math>\mathbf{b}_i</math> are the [[Basis (linear algebra)|basis]] or [[Bravais lattice|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 <math>h\mathbf{a}_1 + k\mathbf{a}_2 + \ell\mathbf{a}_3 </math> because the direct lattice vectors need not be mutually orthogonal.) This is based on the fact that a reciprocal lattice vector <math>\mathbf{g}</math> (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]], <math>\Delta\mathbf{k}= \mathbf{k}_{\mathrm{out}} - \mathbf{k}_{\mathrm{in}}</math> with <math>\mathbf{k}_{\mathrm{out}}</math> as the outgoing (scattered from a crystal lattice) X-ray wavevector and <math>\mathbf{k}_{\mathrm{in}}</math> as the incoming (toward the crystal lattice) X-ray wavevector, is equal to a reciprocal lattice vector <math>\mathbf{g}</math> as stated by the [[Laue equations]], the measured scattered X-ray peak at each measured scattering vector <math>\Delta\mathbf{k}</math> is marked by ''Miller indices''. By convention, [[negative integer]]s are written with a bar, as in {{overline|3}} for&nbsp;−3. The integers are usually written in lowest terms, i.e. their [[greatest common divisor]] should be&nbsp;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 (2{{pi}}), regardless of whether there are atoms on all these planes or not.

There are also several related notations:<ref name="Ash">{{Cite book|url=https://archive.org/details/solidstatephysic00ashc|title=Solid state physics|last1=Ashcroft|first1=Neil W.|date=1976|publisher=Holt, Rinehart and Winston|last2=Mermin|first2=N. David|isbn=0030839939|location=New York|oclc=934604|url-access=registration}}</ref>
*the notation <math display="inline"> \{hk\ell\} </math> denotes the set of all planes that are equivalent to <math> (hk\ell) </math> by the symmetry of the lattice.
In the context of crystal ''directions'' (not planes), the corresponding notations are:
* <math> [hk\ell],</math> 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 <math> \langle hk\ell\rangle </math> denotes the set of all directions that are equivalent to <math> [hk\ell] </math> by symmetry.
Note, for Laue–Bragg interferences
* <math> hk\ell </math> 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.<ref>{{cite journal |last1=Weiss |first1=Christian Samuel |title=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 |journal=Abhandlungen der physikalischen Klasse der Königlich-Preussischen Akademie der Wissenschaften |date=1817 |pages=286–336 |url=https://archive.org/stream/abhandlungenderp16akad#page/286/mode/2up}}</ref> The method was also historically known as the Millerian system, and the indices as Millerian,<ref>[http://dictionary.oed.com Oxford English Dictionary Online] (Consulted May 2007)</ref> 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.