Practical Electron Microscopy and Database

The orientation of crystal planes is normally defined by describing how the planes intersect the main crystallographic axes of the solid. The Miller Indices (three integers: h k l) are used to identify the planes in crystal (Bravais) lattices, and form a notation system and set the rules in crystallography. Each index denotes a plane orthogonal to a direction (h, k, ℓ) in the system of the reciprocal lattice vectors. The negative integers are conventionally written with a bar on top of the numbers. The integers are normally written in the lowest terms, and thus their greatest common divisor should be 1. Miller indices 100, 010, and 001 represent the planes orthogonal to directions h, k, and l, respectively. In general, low index planes have small values h, k and l.

There are several important, related notations:
        i) Plane(s):
           i.a) The notation (hkl) represents a plane.
           i.b) The notation {hkl} denotes the family of all the planes that are equivalent to (hkl) by the symmetry of the lattice.
        ii) Direction(s):
           ii.a) The notation [hkl] represents a direction that denotes the direction in the basis of the direct lattice vectors instead of the reciprocal lattice.
           ii.b) The notation <hkl> represents a family of the directions that are equivalent to [hkl] by symmetry.

Note that, the converting of hexagonal zone axes between Miller notation [U V W] and Miller-Bravais notation [u v t w] can be given by:
         U = u - t --------------------------------- [2506a]
         V = v - t --------------------------------- [2506b]
         W = w --------------------------------- [2506c]
And,
         u = (2U - V)/3 --------------------------------- [2506d]
         v = (2V - U)/3 --------------------------------- [2506e]
         t = - (u + v) = - (U + V)/3 --------------------------------- [2506f]
         w = W --------------------------------- [2506g]