Editing Miller index (section)

==Case of hexagonal and rhombohedral structures==

[[Image:Miller-bravais.svg|thumb|Miller–Bravais indices]]

With [[hexagonal lattice system|hexagonal]] and [[rhombohedral lattice system|rhombohedral]] [[lattice system]]s, 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) ≡ (11{{overline|2}}0) and (1{{overline|2 }}0) ≡ (1{{overline|2 }}10) 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 (2{{pi}}/3 rad, 120°). The [100], [010] and the [{{overline|1}}{{overline|1}}0] directions are really similar. If ''S'' is the intercept of the plane with the [{{overline|1}}{{overline|1}}0] 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 do not 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 <math>h\mathbf{b}_1 + k\mathbf{b}_2 + \ell\mathbf{b}_3 </math>. For hexagonal crystals this may be expressed in terms of direct-lattice basis-vectors '''a'''<sub>1</sub>, '''a'''<sub>2</sub> and '''a'''<sub>3</sub> as

:<math>h\mathbf{b}_1 + k\mathbf{b}_2 + \ell \mathbf{b}_3 = \frac{2}{3 a^2}(2 h + k)\mathbf{a}_1 + \frac{2}{3 a^2}(h+2k)\mathbf{a}_2 + \frac{1}{c^2} (\ell) \mathbf{a}_3.</math>

Hence zone indices of the direction perpendicular to plane (''hkℓ'') are, in suitably normalized triplet form, simply <math>[2h+k,h+2k,\ell(3/2)(a/c)^2]</math>. When ''four indices'' are used for the zone normal to plane (''hkℓ''), however, the literature often uses <math>[h,k,-h-k,\ell(3/2)(a/c)^2]</math> instead.<ref>J. W. Edington (1976) ''Practical electron microscopy in materials science'' (N. V. Philips' Gloeilampenfabrieken, Eindhoven) {{ISBN|1-878907-35-2}}, Appendix 2</ref> 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
:<math>
d_{hk\ell} = \frac{a}{\sqrt{\tfrac{4}{3}\left(h^2+k^2+hk \right)+\tfrac{a^2}{c^2}\ell^2}}
</math>
:
:However, in general:
:<math>d_{hkl} = \frac{2\pi}{\sqrt{h^2 \textbf b_1^2 + k^2 \textbf b_2^2 + l^2 \textbf b_3^2 + 2hk \textbf b_1 \textbf b_2 \cos \gamma^* + 2kl\textbf b_2 \textbf b_3 \cos \alpha^* + 2lh\textbf b_1 \textbf b_3 \cos \beta^*}}
</math>