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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–Bravais index for HCP lattice. Crystallographic features of HCP systems, such as vectors and atomic plane families, can be described using a four-value Miller index notation ( hkil) in which the third index i denotes a degenerate but convenient component which is equal to −h − k.
One result of this, as illustrated in the figure above, is that "low-index" zones are generally perpendicular to "low-Miller index" lattice planes, which in turn have small spatial frequencies (g-values) and hence large lattice periodicities (d-spacings). A possible intuition behind this is that in electron microscopy, for electron beams to be ...
A plane is chosen as the equator, usually the (001) or (011) plane; its pole is the center of the figure. Then, the poles of the other planes are placed on the figure, with the Miller indices for each pole. The poles that belong to a zone are sometimes linked with the related trace.
In crystallography, a lattice plane of a given Bravais lattice is any plane containing at least three noncollinear Bravais lattice points. Equivalently, a lattice plane is a plane whose intersections with the lattice (or any crystalline structure of that lattice) are periodic (i.e. are described by 2d Bravais lattices). [1]
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.
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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 ]