Article — Miller Indices Calculator
Miller Indices Calculator: From Plane Intercepts to (hkl) and d-Spacing
Miller indices (hkl) are three integers that describe a plane in a crystal lattice. They are the reciprocals of the plane's intercepts with the crystallographic axes, cleared of fractions and reduced to lowest terms. For a cubic crystal, the spacing between (hkl) planes is d = a / sqrt(h² + k² + l²).
William Hallowes Miller introduced the notation in 1839, and it has been the standard language of crystallography ever since. Every X-ray diffraction peak, every semiconductor wafer specification, every slip system in metallurgy is written using Miller indices.
What Miller indices describe
A plane in a crystal can cut the three crystallographic axes at any point. The Miller index notation captures the orientation of that plane with three small integers (h k l), written without commas inside parentheses. The trick is that the indices are derived from reciprocals of the intercepts, not the intercepts themselves. This handles the awkward case of planes parallel to an axis: instead of an infinite intercept, you get a clean zero index.
A plane labelled (100) cuts the x-axis at 1 unit and runs parallel to the y and z axes. A plane labelled (111) cuts all three axes at equal distances. The (000) notation is forbidden because it describes no plane at all.
How to find Miller indices from intercepts
The procedure is three short steps. First, write down the intercepts in units of the lattice constants a, b, c. Second, take the reciprocal of each intercept; an intercept of infinity (the plane is parallel to that axis) gives a reciprocal of zero. Third, multiply through by the smallest integer that clears all fractions, and reduce by the greatest common divisor.
1, 1, 1 → (111)2, 3, infinity → 1/2: 1/3: 0 → (320)1, 2, infinity → 1: 1/2: 0 → (210)infinity, 1, infinity → (010)The intercepts (2, 3, infinity) become reciprocals (1/2, 1/3, 0). Multiplying by 6 clears fractions to (3, 2, 0). There is no common factor greater than 1, so the Miller indices are (320).
Miller indices and d-spacing
The most useful quantity derived from Miller indices is the interplanar spacing d, the perpendicular distance between adjacent parallel planes in the same family. For a cubic crystal with lattice constant a:
d_hkl = a / sqrt(h² + k² + l²)
Aluminum has a = 4.046 A. The (111) spacing is 4.046 / sqrt(3) = 2.336 A. The (200) spacing is 4.046 / 2 = 2.023 A. These distances are what X-rays sense in diffraction experiments.
The atomic spacing in metals is on the order of 2-3 Angstroms. X-rays with wavelengths of about 1.5 A (copper K-alpha) match this scale perfectly, which is why X-ray diffraction is the workhorse tool for measuring crystal structures. Visible light, at 5000 A, simply walks past the lattice without seeing it.
Miller indices versus crystal directions
A common source of confusion: (hkl) with parentheses is a plane, but [hkl] with square brackets is a direction. A direction is just a vector with components h, k, l in the lattice basis. In cubic crystals only, the direction [hkl] is perpendicular to the plane (hkl). In non-cubic systems they point in different directions.
Curly braces {hkl} mean the whole family of planes related by the crystal's point-group symmetry. The {100} family of a cubic crystal contains six equivalent planes: (100), (010), (001), and their three negative twins. Angle brackets <hkl> play the same role for directions.
Negative Miller indices and bar notation
If an intercept is on the negative side of an axis, the corresponding index is negative. Convention writes the minus sign as a bar over the digit, read aloud as "bar". So the plane through x = -1 with the other intercepts at infinity is (1̄00), pronounced "bar one zero zero". In typed text, a leading minus or a bar-using Unicode combining mark are both seen.
The indices (2 4 6) reduce to (1 2 3); same plane family. But (2 4 -6) reduces to (1 2 -3), not (1 2 3). Negative indices change the geometry. Always factor out the absolute GCD and keep the original signs.
Miller indices in X-ray diffraction
The Bragg equation nλ = 2 d_hkl sinθ ties (hkl) to a measurable diffraction angle. Each peak in an XRD pattern is labeled with the (hkl) of the planes that produced it. The intensities depend on what atoms sit in the cell, but the angles are fixed by Miller indices and the lattice constant.
For face-centered cubic metals like copper and aluminum, only (hkl) with all three even or all three odd diffract; mixed-parity reflections like (100) and (110) are forbidden by structure-factor cancellation. Body-centered cubic metals show the opposite rule: h + k + l must be even.
Common Miller index families
- (100) = standard silicon wafer for CMOS microelectronics
- (111) = close-packed plane in FCC; slip occurs along it
- (110) = primary slip plane in BCC iron and steel
- {100} = 6 planes in cubic symmetry
- {110} = 12 planes in cubic symmetry
- {111} = 8 planes in cubic symmetry
- (0001) = basal plane of hexagonal crystals (graphite, sapphire)
Miller indices calculation mistakes
The single most common error is treating intercepts directly as indices: writing (2 3 1) for intercepts (2, 3, 1) instead of taking reciprocals first. The reciprocals of (2, 3, 1) clear to (3 2 6), a completely different plane. Always reciprocate before normalizing.
Other slips: forgetting that an intercept of infinity gives an index of zero, not omitting the index entirely; reducing across signs and losing a minus; mixing parentheses (plane) with brackets (direction); and assuming directions and planes coincide in non-cubic systems where they do not.
For hexagonal crystals, four-index Miller-Bravais notation (hkil) is preferred, where i = -(h + k). This makes equivalent planes look symmetrically related on paper, the way they actually are by hexagonal symmetry. Graphite (0001), sapphire (11-20), and other hexagonal materials are almost always written in the four-index form.