It is often convenient to identify a plane or direction in a crystal by its indices. Note that all parallel planes have the same indices.

Planar Indices

The rules for determining the Miller indices of planes are as follows:

1. Write the intercepts of the plane on the three axes in order (a1, a2, and a3).

2. Take the reciprocals of these.

3. Reduce to the lowest set of integers with the same ratios.

4. Enclose in parentheses (hkl).

Commas are not used except in the rare case that one of the integers is larger than one digit. (This is rare because we are normally interested only in planes with low indices.) If a plane is parallel to an axis, its intercept is taken as ∞ and its reciprocal as 0. If the plane contains one of the axes or the origin, either analyze a parallel plane or translate the axes before finding indices. This is permissible since all parallel planes have the same indices. A negative index is indicated by a bar over the index rather than a negative sign, e.g., (11). Figure A2.1 shows several examples.

Direction Indices

The indices of a direction are the translations parallel to the three axes that produce the direction under consideration. The rules for finding direction indices are as follows:

1. Write the components of the direction parallel to the three axes in order.

2. Reduce to the lowest set of integers with the same ratios.

3. Enclose in brackets [uvw].