Miller indices

Indices de Miller (Fr); Millersche Indizes (Ge); Indices de Miller (Sp); Indici di Miller (It); Индексы Миллера (Ru); ミラー指数 (Ja)

Definition

Direct space

The law of rational indices states that the intercepts, OP, OQ, OR, of the natural faces of a crystal form with the basis vectors OA' = a, OB = b, and OC = c are inversely proportional to prime integers, h, k, l, called Miller indices of the face. This definition can be extended to any reticular plane whose intercepts with the basis vectors are OP = C a/h, OQ = C b/k, and OR = C c/l (see Figure 1). When the lattice is indexed with respect to a primitive basis, h, k, l are co-prime integers; this restriction does not hold instead when a centred unit cell is chosen. C is an integer which, when made variable, identifies a specific lattice plane of the infinite set having the same orientation with respect to the basis vector. This infinite set of planes defines a family of lattice planes, denoted by the Miller indices in parentheses: (hkl). The Miller indices of the equivalent faces of a crystal form are denoted by {hkl}. The variation of the orientation of the planes with the ratios of the Miller indices is illustrated in the attached examples. The equation of the planes of the family is:

hx + ky + lz = C

Reciprocal space

The reciprocal lattice vector associated to the family of lattice planes is OH = h a* + k b* + l c*, where a*, b*, c* are the reciprocal lattice basis vectors. OH is perpendicular to the family of lattice planes and OH = 1/d where d is the lattice spacing of the family. When a centred unit cell is used in direct space, integral reflection conditions are observed in the reciprocal space which correspond to the non-coprime Miller indices of the family of lattice planes.

Bravais-Miller indices (hexagonal axes)

In the case of an hexagonal lattice, one uses four axes, a₁, a₂, a₃, c and four indices, (hkil), called Bravais-Miller indices, where h, k, i, l are again inversely proportional to the intercepts of a plane of the family with the four axes. The indices h, k, i are cyclically permutable and are related by

h + k + i = 0

Behaviour in a change of basis

In a change of basis the Miller indices h, k, l transform like the basis vectors a, b, c and are for that reason covariant quantities.

Rhombohedral crystals

The Miller indices h_R, k_R, l_R referred to rhombohedral axes are related to the corresponding indices, h_H, k_H, i_H,l_H reffered to hexagonal axes by:

hH = kR - lR	;	hR = ⅓(- kH + iH + lH)
kH = lR - hR	;	kR = ⅓(hH - iH + lH)
iH = hR - kR	;	lR = ⅓(- hH + kH + lH)
lH = hR + kR + lR

Example

Orientation of lattice planes depending on their Miller indices

The Miller indices of the planes ABC' , ABC, ABC" , AA"BB" are (112) , (111), (221), (110), respectively. These planes have AB , or [math] [1{\bar 1}0][/math], as common zone axis.

History

The Miller indices were first introduced, among others, by W. Whewell in 1829 and developed by W.H. Miller, his successor at the Chair of Mineralogy at Cambridge University, in his book A treatise on Crystallography (1839) - see Historical Atlas of Crystallography (1990), edited by J. Lima de Faria, published for the International Union of Crystallography by Kluwer Academic Publishers, Dordrecht.

See also

• law of rational indices
• Laue indices
• reciprocal lattice
• Weiss parameters
• Discussion about non-coprime Miller indices
• Miller's book at archive.org