Miller indices form a notation system in
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
for lattice planes in
crystal (Bravais) lattices.
In particular, a family of
lattice plane In crystallography, a lattice plane of a given Bravais lattice is any plane containing at least three noncollinear Bravais lattice points. Equivalently, a lattice plane is a plane whose intersections with the lattice (or any crystalline structure of ...
s of a given (direct) Bravais lattice is determined by three
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s ''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
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
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
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
,
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
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 by scattering, by a crystal lattice. T ...
, the measured scattered X-ray peak at each measured scattering vector
is marked by ''Miller indices''. By convention,
negative integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
s 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
X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...
. 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:
*the notation denotes the set of all planes that are equivalent to (hkℓ) by the symmetry of the lattice.
In the context of crystal ''directions'' (not planes), the corresponding notations are:
*
kℓ 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 kℓby symmetry.
Note, for Laue-Bragg interferences
* hkl lacks any bracketing when designating a reflection
Miller indices were introduced in 1839 by the British mineralogist William Hallowes Miller
Prof William Hallowes Miller FRS HFRSE LLD DCL (6 April 180120 May 1880) was a Welsh mineralogist and laid the foundations of modern crystallography.
Miller indices are named after him, the method having been described in his ''Treatise on Cry ...
, although an almost identical system (''Weiss parameters'') had already been used by German mineralogist Christian Samuel Weiss
Christian Samuel Weiss (26 February 1780 – 1 October 1856) was a German mineralogist born in Leipzig.
Following graduation, he worked as a physics instructor in Leipzig from 1803 until 1808. and in the meantime, conducted geological studies of ...
since 1817. The method was also historically known as the Millerian system, and the indices as Millerian, 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.
Definition
There are two equivalent ways to define the meaning of the Miller indices: via a point in the reciprocal lattice, or as the inverse intercepts along the lattice vectors. Both definitions are given below. In either case, one needs to choose the three lattice vectors a1, a2, and a3 that define the unit cell (note that the conventional unit cell may be larger than the primitive cell of the Bravais lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n_ ...
, as the examples below illustrate). Given these, the three primitive reciprocal lattice vectors are also determined (denoted b1, b2, and b3).
Then, given the three Miller indices h, k, ℓ, (hkℓ) denotes planes orthogonal to the reciprocal lattice vector:
:
That is, (hkℓ) simply indicates a normal to the planes in the basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
of the primitive reciprocal lattice vectors. Because the coordinates are integers, this normal is itself always a reciprocal lattice vector. The requirement of lowest terms means that it is the ''shortest'' reciprocal lattice vector in the given direction.
Equivalently, (hkℓ) denotes a plane that intercepts the three points a1/''h'', a2/''k'', and a3/''ℓ'', or some multiple thereof. That is, the Miller indices are proportional to the ''inverses'' of the intercepts of the plane, in the basis of the lattice vectors. If one of the indices is zero, it means that the planes do not intersect that axis (the intercept is "at infinity").
Considering only (hkℓ) planes intersecting one or more lattice points (the ''lattice planes''), the perpendicular distance ''d'' between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula: .
The related notation kℓdenotes the ''direction'':
:
That is, it uses the direct lattice basis instead of the reciprocal lattice. Note that kℓis ''not'' generally normal to the (hkℓ) planes, except in a cubic lattice as described below.
Case of cubic structures
For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted ''a''), as are those of the reciprocal lattice. Thus, in this common case, the Miller indices (hkℓ) and kℓboth simply denote normals/directions in Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
.
For cubic crystals with lattice constant ''a'', the spacing ''d'' between adjacent (hkℓ) lattice planes is (from above)
: .
Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:
*Indices in ''angle brackets'' such as ⟨100⟩ denote a ''family'' of directions which are equivalent due to symmetry operations, such as 00 10 01or the negative of any of those directions.
*Indices in ''curly brackets'' or ''braces'' such as denote a family of plane normals which are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.
For face-centered cubic and body-centered cubic lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.
Case of hexagonal and rhombohedral structures
With hexagonal
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
and rhombohedral lattice systems, it is possible to use the Bravais-Miller system, which uses four indices (''h'' ''k'' ''i'' ''ℓ'') that obey the constraint
: ''h'' + ''k'' + ''i'' = 0.
Here ''h'', ''k'' and ''ℓ'' are identical to the corresponding Miller indices, and ''i'' is a redundant index.
This four-index scheme for labeling planes in a hexagonal lattice makes permutation symmetries apparent. For example, the similarity between (110) ≡ (110) and (10) ≡ (110) is more obvious when the redundant index is shown.
In the figure at right, the (001) plane has a 3-fold symmetry: it remains unchanged by a rotation of 1/3 (2π/3 rad, 120°). The 00 10and the directions are really similar. If ''S'' is the intercept of the plane with the axis, then
: ''i'' = 1/''S''.
There are also ''ad hoc
Ad hoc is a Latin phrase meaning literally 'to this'. In English, it typically signifies a solution for a specific purpose, problem, or task rather than a generalized solution adaptable to collateral instances. (Compare with ''a priori''.)
Com ...
'' schemes (e.g. in the transmission electron microscopy
Transmission electron microscopy (TEM) is a microscopy technique in which a beam of electrons is transmitted through a specimen to form an image. The specimen is most often an ultrathin section less than 100 nm thick or a suspension on a g ...
literature) for indexing hexagonal ''lattice vectors'' (rather than reciprocal lattice vectors or planes) with four indices. However they don't operate by similarly adding a redundant index to the regular three-index set.
For example, the reciprocal lattice vector (hkℓ) as suggested above can be written in terms of reciprocal lattice vectors as . For hexagonal crystals this may be expressed in terms of direct-lattice basis-vectors a1, a2 and a3 as
:
Hence zone indices of the direction perpendicular to plane (hkℓ) are, in suitably normalized triplet form, simply