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Miller Indices

Direct space

Planes of a given family of lattice planes with Miller indices hkl make intercepts OP = C a/hOQ = C b/k, and OR = C c/l with the unit-cell axes OA = aOB = b, and OC = c (see Figure 1), where hkl are prime integers and C is a constant integer; the planes of the family are denoted (hkl). This property results from the law of rational indices.

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.

Bravais-Miller indices (hexagonal axes)

In the case of an hexagonal lattice, one uses four axes, a1a2a3c and four indices, (hkil), called Bravais-Miller indices, where hkil are again inversely proportional to the intercepts of a plane of the family with the four axes. The indices hki are cyclically permutable and are related by

\[h + k + i = 0\]

 

Behavior in a change of basis

In a change of basis the Miller indices hkl transform like the basis vectors abc and are for that reason covariant quantities.

Rhombohedral crystals

The Miller indices hRkRlR referred to rhombohedral axes are related to the corresponding indices, hHkHiH,lH referred 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' ABCABC" AA"BB" are (112) (111)(221)(110), respectively. These planes have AB, or  [1{\bar 1}0], 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.