Miller Indices Calculator

Miller indices label the orientation of a lattice plane by the reciprocals of the intercepts it makes on the crystal axes, cleared to the smallest whole numbers. Reciprocals are used because a plane parallel to an axis has an infinite intercept, which becomes a well-behaved zero, and because the resulting integers are proportional to the reciprocal lattice vector normal to the plane. Round brackets (hkl) denote one plane, curly braces {hkl} the full symmetry-equivalent family, and square brackets [uvw] a direction. In cubic crystals the family multiplicities are 6 for {100}, 12 for {110}, 8 for {111}, 24 for {hk0} and 48 for a general {hkl}. Interplanar spacing follows 1/d2 = h2/a2 + k2/b2 + l2/c2, which for silicon with a = 5.431 angstrom gives d(111) = 3.135 angstrom. The practical use is diffraction. Bragg reflection needs sin theta at most 1, so with copper K-alpha radiation at 1.5406 angstrom no plane spaced closer than about 0.77 angstrom can be observed, setting the resolution limit of a laboratory powder pattern. Lattice centring removes reflections in systematic patterns: body-centred lattices show peaks only when h + k + l is even, and face-centred lattices only when h, k and l are all odd or all even. Two limitations apply to the simple spacing formula. It is valid only for mutually orthogonal axes, so monoclinic and triclinic cells need the full reciprocal-metric expression, and hexagonal crystals conventionally use four Miller-Bravais indices (hkil) with i equal to minus h plus k.

Notation for crystal faces grew out of early nineteenth century mineralogy. Rene Just Hauy had established the law of rational intercepts, showing that crystal faces cut the axes in simple whole-number ratios, and Christian Samuel Weiss and others then devised parameter-based symbols to record those ratios. William Whewell proposed a reciprocal-index scheme at Cambridge in the 1820s, but it was his student and successor William Hallowes Miller who developed and popularized it in A Treatise on Crystallography, published in 1839. Miller held the chair of mineralogy at Cambridge, and his notation won out because reciprocal integers made zone relationships and face symmetry easy to compute, replacing a confusion of competing regional symbol systems. For most of a century the indices described only external face geometry. That changed with the discovery of X-ray diffraction by Max von Laue and colleagues in 1912 and the structure determinations by William Henry Bragg and William Lawrence Bragg in 1913, which showed that each diffraction spot corresponds to reflection from a set of internal planes with a measurable spacing. Miller indices thus became labels for physically observed reflections rather than an abstract bookkeeping device. Paul Ewald formalized the reciprocal-space picture in 1913, and the systematic tabulation of allowed reflections, multiplicities and extinction conditions for all 230 space groups appeared in the International Tables for Crystallography, first issued in the 1930s and still the reference used for indexing patterns today.