hkl

In crystallography, hkl denotes Miller indices used to label lattice planes and directions in a crystal. The integers h, k, and l describe the orientation of a plane or a direction relative to the crystal axes, and they can be zero or negative.

For a crystal with lattice constants a, b, c along the x, y, z axes, a plane with Miller indices (hkl) intercepts the axes at a/h, b/k, c/l, provided the indices are nonzero. Negative indices indicate intercepts on the opposite axes. Planes with the same (hkl) define a family of parallel planes.

The spacing d_hkl between parallel planes with indices (hkl) depends on the lattice parameters and symmetry. In cubic crystals, d_hkl = a / sqrt(h^2 + k^2 + l^2). In general, d_hkl is determined from the metric of the crystal lattice or from reciprocal-space considerations.

In addition to planes, Miller indices can label directions. The set of indices (hkl) is perpendicular to the direction [hkl] in many common systems, with the directional vector [uvw] indicating a line parallel to the plane’s normal. In hexagonal crystals, a four-index notation hkil is used, where i = -(h + k); thus h + k + i = 0.

Applications of hkl indices include X-ray and electron diffraction, indexing diffraction patterns, and crystal-structure determination. They provide a concise, symmetry-aware framework for describing the orientations of planes and directions within crystalline materials.