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Miller–Bravais indices. 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.
Miller indices of a plane (hkl) and a direction [hkl].The intercepts on the axes are at a/ h, b/ k and c/ l. The International Union of Crystallography (IUCr) gives the following definition: "The law of rational indices states that the intercepts, OP, OQ, OR, of the natural faces of a crystal form with the unit-cell axes a, b, c are inversely proportional to prime integers, h, k, l.
() defines a reciprocal lattice point at (,,) which corresponds to the real-space plane defined by the Miller indices (see Bragg's law). F h k ℓ {\displaystyle F_{hk\ell }} is the vector sum of waves from all atoms within the unit cell.
Miller projection with 1,000 km indicatrices of distortion. The Miller cylindrical projection is a modified Mercator projection , proposed by Osborn Maitland Miller in 1942. The latitude is scaled by a factor of 4 ⁄ 5 , projected according to Mercator, and then the result is multiplied by 5 ⁄ 4 to retain scale along the equator. [ 1 ]
The attitude of a lattice plane is the orientation of the line normal to the plane, [12] and is described by the plane's Miller indices. In three-space a family of planes (a series of parallel planes) can be denoted by its Miller indices (hkl), [13] [14] so the family of planes has an attitude common to all its constituent planes.
If, on the other hand, the Miller indices are not relative prime, the family of planes defined by them is not a family of lattice planes, because not every plane of the family then intersects lattice points. [2] Conversely, planes that are not lattice planes have aperiodic intersections with the lattice called quasicrystals; this is known as a ...
where is the lattice spacing of the cubic crystal, and h, k, and ℓ are the Miller indices of the Bragg plane. Combining this relation with Bragg's law gives: Combining this relation with Bragg's law gives:
In a haxagonal system hkl miller indices are enough to define a plane or a direction, however, You will prefer to use four hkil indices, because this way equivalent planes/directions will have similar indices. E.g. the unit cell defines 3 lattice unit vectors a, b, and c. Between a and b there is a 120° angle.