page_title: Miller index
section_title: Case of hexagonal and rhombohedral structures
hierarchical_section_title: Miller index / Case of hexagonal and rhombohedral structures
context_page_description: Miller indices form a notation system in crystallography for planes in crystal lattices.
  In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller indices. They are written, and denote the family of planes orthogonal to, where are the basis of the reciprocal lattice vectors. By convention, negative integers are written with a bar, as in 3 for −3. The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1. Miller indices are also used to designate reflections in X-ray crystallography. In this case the integers are not necessarily in lowest terms, and can be thought of as corresponding to planes spaced such that the reflections from adjacent planes would have a phase difference of exactly one wavelength, regardless of whether there are atoms on all these planes or not.
  There are also several related notations:
  the notation {hkℓ} denotes the set of all planes that are equivalent to by the symmetry of the lattice.
  In the context of crystal directions, the corresponding notations are:
context_section_description: With hexagonal and rhombohedral lattice systems, it is possible to use the Bravais-Miller system, which uses four indices (h k i ℓ) that obey the constraint
  h + k + i = 0.
  Here h, k and ℓ are identical to the corresponding Miller indices, and i is a redundant index.
  This four-index scheme for labeling planes in a hexagonal lattice makes permutation symmetries apparent.  For example, the similarity between (110) ≡ (1120) and (120) ≡ (1210) is more obvious when the redundant index is shown.
  In the figure at right, the (001) plane has a 3-fold symmetry: it remains unchanged by a rotation of 1/3 (2π/3 rad, 120°). The [100], [010] and the [110] directions are really similar. If S is the intercept of the plane with the [110] axis, then
  i = 1/S.
  There are also ad hoc schemes (e.g. in the transmission electron microscopy literature) for indexing hexagonal lattice vectors (rather than reciprocal lattice vectors or planes) with four indices.  However they don't operate by similarly adding a redundant index to the regular three-index set.
  For example, the reciprocal lattice vector (hkℓ) as suggested above can be written in terms of reciprocal lattice vectors as . For hexagonal crystals this may be expressed in terms of direct-lattice basis-vectors a₁, a₂ and a₃ as
  Hence zone indices of the direction perpendicular to plane (hkℓ) are, in suitably normalized triplet form, simply .  When four indices are used for the zone normal to plane (hkℓ), however, the literature often uses  instead.  Thus as you can see, four-index zone indices in square or angle brackets sometimes mix a single direct-lattice index on the right with reciprocal-lattice indices (normally in round or curly brackets) on the left.
  And, note that for hexagonal interplanar distances, they take the form
caption: Miller-Bravais indices