Millerindices

Miller indices are a notation system in crystallography used to designate the orientation of crystal planes in a crystalline lattice. A plane in a crystal is assigned a set of integers (h, k, l) called Miller indices, derived from the reciprocals of the plane’s intercepts with the crystallographic axes. If a plane intersects the x, y, and z axes at a/h, b/k, and c/l respectively (a, b, c are the lattice constants along the three axes), the plane is indexed as (hkl). If the plane is parallel to an axis, the corresponding intercept is at infinity and the index for that axis is zero.

To determine hkl from intercepts, take the reciprocals of the normalized intercepts relative to the lattice constants and reduce to the smallest integer set. For example, a plane cutting the x-axis at a/2 and being parallel to the y and z axes has intercepts a/2, ∞, ∞, giving h = 2, k = 0, l = 0, i.e., the (200) plane.

All planes that share the same set of indices (hkl) are parallel and form a family of planes. Translation by a lattice vector does not change the indices, so the (hkl) designation identifies the entire family rather than a single plane.

The indices also relate to the spacing between parallel planes, d_hkl. In a general orthorhombic system, 1/d_hkl^2 = h^2/a^2 + k^2/b^2 + l^2/c^2; for cubic crystals this simplifies to d_hkl = a / sqrt(h^2 + k^2 + l^2). Hexagonal crystals use Miller-Bravais indices (hkil), with i = h − k, to reflect fourfold symmetry.

Miller indices are widely used to index crystal planes in X-ray diffraction, electron diffraction, and other crystallographic analyses. They are distinct from direction indices, which use square brackets [uvw] and refer to lattice directions rather than planes.