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Planes with different Miller indices in cubic crystals Examples of directions. Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers h, k, and ℓ, the Miller indices.
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 ...
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.
A family of lattice planes is a collection of equally spaced parallel lattice planes that, taken together, intersect all lattice points. Every family of lattice planes can be described by a set of integer Miller indices that have no common divisors (i.e. are relative prime). Conversely, every set of Miller indices without common divisors ...
The structure of a crystal is often represented by the pole figure of its crystallographic planes. 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 ...
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 ...
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 ...