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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.
The orientation of a plane with respect to the unit cell is most conveniently expressed in terms of Miller indices. For example, the set of Miller indices (110) describes the set of parallel planes (family of lattice planes) parallel to the z-axis and cutting the x- and the y-axis once, such that every unit cell is bisected by precisely one of ...
The Miller indices for a plane are integers with no common factors. Negative indices are indicated with horizontal bars, as in (1 2 3). In an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane. Considering only (hkℓ) planes intersecting one or more lattice ...
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.
Miller indices in parentheses such as (100) denote a plane of the crystal structure, and regular repetitions of that plane with a particular spacing. In the cubic system, the normal to the (hkl) plane is the direction [hkl], but in lower-symmetry cases, the normal to (hkl) is not parallel to [hkl].
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 ...
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.