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
ghk\ell=hb1+kb2+\ellb3
bi
ha1+ka2+\ella3
g
\Deltak=kout-kin
kout
kin
g
\Deltak
There are also several related notations:[1]
(hk\ell)
[hk\ell],
\langlehk\ell\rangle
[hk\ell]
hk\ell
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.
There are two equivalent ways to define the meaning of the Miller indices:[1] 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, 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,\ell,(hk\ell)
ghk\ell=hb1+kb2+\ellb3.
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:
d=2\pi/|gh|
The related notation [hkℓ] denotes the direction:
ha1+ka2+\ella3.
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 [''hkℓ''] both simply denote normals/directions in Cartesian coordinates.
For cubic crystals with lattice constant a, the spacing d between adjacent (hkℓ) lattice planes is (from above)
dhk=
a | |
\sqrt{h2+k2+\ell2 |
}
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:
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.
With hexagonal 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 (2/3 rad, 120°). The [100], [010] and the [{{overline|1}}{{overline|1}}0] directions are really similar. If S is the intercept of the plane with the [{{overline|1}}{{overline|1}}0] axis, then
i = 1/S.
There are also ad hoc schemes (e.g. in the transmission electron microscopy 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
hb1+kb2+\ellb3
hb1+kb2+\ellb3=
2 | |
3a2 |
(2h+k)a1+
2 | |
3a2 |
(h+2k)a2+
1 | |
c2 |
(\ell)a3.
Hence zone indices of the direction perpendicular to plane (hkℓ) are, in suitably normalized triplet form, simply
[2h+k,h+2k,\ell(3/2)(a/c)2]
[h,k,-h-k,\ell(3/2)(a/c)2]
And, note that for hexagonal interplanar distances, they take the form
dhk\ell=
a | |
\sqrt{\tfrac{4 |
{3}\left(h2+k2+hk\right)+\tfrac{a2}{c2}\ell2}}
Crystallographic directions are lines linking nodes (atoms, ions or molecules) of a crystal. Similarly, crystallographic planes are planes linking nodes. Some directions and planes have a higher density of nodes; these dense planes have an influence on the behavior of the crystal:
in condensed matter, light "jumps" from one atom to the other with the Rayleigh scattering; the velocity of light thus varies according to the directions, whether the atoms are close or far; this gives the birefringence
the condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species; the surface tension of an interface thus varies according to the density on the surface
For all these reasons, it is important to determine the planes and thus to have a notation system.
Ordinarily, Miller indices are always integers by definition, and this constraint is physically significant. To understand this, suppose that we allow a plane (abc) where the Miller "indices" a, b and c (defined as above) are not necessarily integers.
If a, b and c have rational ratios, then the same family of planes can be written in terms of integer indices (hkℓ) by scaling a, b and c appropriately: divide by the largest of the three numbers, and then multiply by the least common denominator. Thus, integer Miller indices implicitly include indices with all rational ratios. The reason why planes where the components (in the reciprocal-lattice basis) have rational ratios are of special interest is that these are the lattice planes: they are the only planes whose intersections with the crystal are 2d-periodic.
For a plane (abc) where a, b and c have irrational ratios, on the other hand, the intersection of the plane with the crystal is not periodic. It forms an aperiodic pattern known as a quasicrystal. This construction corresponds precisely to the standard "cut-and-project" method of defining a quasicrystal, using a plane with irrational-ratio Miller indices. (Although many quasicrystals, such as the Penrose tiling, are formed by "cuts" of periodic lattices in more than three dimensions, involving the intersection of more than one such hyperplane.)