Miller planes

Planes in a crystal are described by notations called Miller indices
 Miller indices of a plane, indicated by h k l, are given by the reciprocal of the intercepts of the plane on the three axes.
 The plane, which intersects X axis at 1 (one lattice parameter) and is parallel to Y and Z axes, has Miller indices h = 1/1 = 1, k = 1/ = 0, l = 1/ = 0. It is written as (hkl) = (100).

 Crystal Planes

To find the Miller Indices of a plane, follow these steps:
 Determine the intercepts of the plane along the crystal axes
 Take the reciprocals
 Clear fractions
 Reduce to lowest terms and enclose in brackets () Ex: Intercepts on a, b, c : ¾, ½, ¼ (h k l) = (4/3, 2, 4) = (2 3 6)

Planes can also have negative intercept e.g. 1, -1/2, 1 h k l = 1 -2 1. This is denoted as ( ) 121

 Family of planes {hkl}
 Planes having similar indices are equivalent, e.g. faces of the cube (100), (010) and (001). This is termed as a family of planes and denoted as {100} which includes all the (100) combinations including negative indices.

Note the shift of origin from blue to red circle for the negative indices