Miller Indices Calculator

Miller indices are a set of three integers (hkl), written in parentheses with no commas, that describe the orientation of a plane within a crystal lattice. They are found by taking the reciprocals of the plane's intercepts on the a, b, and c crystallographic axes, then clearing those reciprocals to the smallest possible set of integers.

Take the reciprocal of each axial intercept (measured in units of the lattice parameters a, b, c), then clear any fractions by multiplying all three reciprocals by their least common denominator, then divide by any common factor to reach the smallest integers. For example, intercepts of 1, 1, and infinity give reciprocals 1, 1, 0, which are already the smallest integers, so the plane is (110).

A bar over a digit, shown here as an overlined digit like 1 with a bar, means the plane's intercept on that axis is on the negative side of the origin rather than the positive side. It is read the same way as a normal digit but written with the bar instead of a leading minus sign, following standard crystallography notation.

A plane parallel to an axis never intersects it, so that intercept is treated as infinity. The reciprocal of infinity is 0, so that axis contributes a 0 to the Miller index. For example, a plane with intercepts 1, 1, and infinity (parallel to the c-axis) has Miller indices (110).

Miller indices describe an entire family of equally spaced parallel planes, not a single plane's exact position, so only the ratio between the reciprocals matters. Clearing fractions to the smallest integer ratio (for example turning 2, 1, and 1 into (211) from intercepts of 1/2, 1, and 1) keeps the notation compact and consistent across crystallography.

Take the reciprocal of each index. If an index is a nonzero integer h, the intercept on that axis is 1/h, in units of the lattice parameter. If an index is 0, the intercept on that axis is infinity, meaning the plane is parallel to that axis. This calculator's Find Intercepts mode performs this reverse calculation directly.

Parentheses (hkl) refer to one specific plane and its parallel family, in one fixed orientation. Curly braces {hkl} refer to a full family of symmetrically equivalent planes, all the planes related by the crystal's symmetry operations, such as (100), (010), and (001) together being written as {100} in a cubic system.

No, by definition Miller indices are always cleared to the smallest set of integers, including 0. If a calculation seems to produce a fraction, it means the clearing step (multiplying through by the least common denominator) was not completed. Note that this is different from the input intercepts, which are allowed to be fractional, like 1/2.

The three lowest-index planes in a cubic system are (100), the cube face, (110), the diagonal face, and (111), the plane cutting equally through all three axes. These three families are the most frequently discussed planes in introductory crystallography because of their high symmetry and their role in slip systems and surface energy calculations.

Miller indices identify which crystallographic plane is being discussed when describing slip systems in metal deformation, cleavage planes in minerals, the orientation of a diffraction peak in X-ray crystallography, and the surface exposed in thin-film growth or etching. Different (hkl) planes in the same crystal can have very different atomic packing, reactivity, and mechanical properties.

Standard Miller indices (hkl) use three integers and work for any crystal system, including cubic, tetragonal, and orthorhombic lattices. Hexagonal crystals commonly use Miller-Bravais indices (hkil) instead, a four-index system with a redundant third index (i = -(h+k)) that makes symmetrically equivalent hexagonal planes easier to recognize by inspection. This calculator computes standard three-index (hkl) Miller indices.