Miller Indices

Rules for Miller Indices:

Determine the intercepts of the face along the crystallographic axes, in terms of unit cell dimensions.
Take the reciprocals
Clear fractions
Reduce to lowest terms

For example, if the x-, y-, and z- intercepts are 2, 1, and 3, the Miller indices are calculated as:

Take reciprocals: 1/2, 1/1, 1/3
Clear fractions (multiply by 6): 3, 6, 2
Reduce to lowest terms (already there)

Thus, the Miller indices are 3,6,2. If a plane is parallel to an axis, its intercept is at infinity and its Miller index is zero. A generic Miller index is denoted by (hkl).

If a plane has negative intercept, the negative number is denoted by a bar above the number. Never alter negative numbers. For example, do not divide -1, -1, -1 by -1 to get 1,1,1. This implies symmetry that the crystal may not have!

Some General Principles

If a Miller index is zero, the plane is parallel to that axis.
The smaller a Miller index, the more nearly parallel the plane is to the axis.
The larger a Miller index, the more nearly perpendicular a plane is to that axis.
Multiplying or dividing a Miller index by a constant has no effect on the orientation of the plane
Miller indices are almost always small.

Why Miller Indices?

Using reciprocals spares us the complication of infinite intercepts.
Formulas involving Miller indices are very similar to related formulas from analytical geometry.
Specifying dimensions in unit cell terms means that the same label can be applied to any face with a similar stacking pattern, regardless of the crystal class of the crystal. Face 111 always steps the same way regardless of crystal system.

Movie of aminated gif above. Click the pause button, then step through the planes