Miller Index - Case of Hexagonal and Rhombohedral Structures

Case of Hexagonal and Rhombohedral Structures

With hexagonal and rhombohedral lattice systems, it is possible to use the Bravais-Miller index which has 4 numbers (h k i ℓ)

i = −(h + k).

Here h, k and ℓ are identical to the Miller index, 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, and the directions are really similar. If S is the intercept of the plane with the 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 as ha*+kb*+ℓc*if the reciprocal-lattice basis-vectors are a*, b*, and c*. For hexagonal crystals this may be expressed in terms of direct-lattice basis-vectors a, b and c 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.