11.9: Miller Indices

Miller indices were first developed in 1825 by W. Whewell, a professor of mineralogy at Cambridge University. We use them to describe the orientation of crystal faces, and also the orientations of cleavages and other planar properties. They are named after W. H. Miller, a student of Whewell’s, who promoted and popularized their use in 1839. The general symbol for a Miller index is (hkl), in which the letters h, k, and l each stand for an integer. Parentheses enclose the resulting Miller index. As with directions, bars above numbers show negative values and we do not include commas unless numbers have more than one digit.

We calculate Miller indices for a plane from its axial intercepts (Figure 11.64). The procedure is as follows:

• First, we invert axial intercept values. (∞ becomes zero after inversion.)
• Then we clear all fraction by multiplying by a constant.
• And we divide by a constant to eliminate common denominators.

Consider the plane in Figure 11.64b. It has intercepts 1, 1, ∞. Inverting these values gives us (110), the Miller index of the plane. And the plane in Figure 11.64c has axial intercepts ∞, ∞, 3. Inverting these values gives us 0, 0, ¹/₃. Clearing the fraction gives us (001), the Miller index. We articulate it as “oh-oh-one.”

Figure 11.64d contains two planes. One has intercepts 1, 1, 2. The other has intercepts 3, 3, 6. Inverting intercepts for the first plane gives us 1, 1, ¹/₂. Clearing fractions yields the Miller index (221). Inverting intercepts for the second plane gives ¹/₃, ¹/₃, ¹/₆. Clearing fractions yields (221). So we see that parallel planes have the same Miller index. If the planes shown in this figure were crystal faces, we would call them the “two two one face” no matter their size or shape. Thus the Miller index describes the orientation of a crystal face with respect to crystallographic axes, but not the absolute size or location of the face. The relationship between planes and directions in a crystal depends on the crystal system. Except in the cubic system, the direction [uvw] is neither perpendicular nor parallel to planes with the Miller index (uvw).

Because crystal faces are parallel to rows of lattice points, they are parallel to planes that intercept crystal axes at an integral number of unit cells from the origin. Consequently, inversion of intercepts and clearing fractions always yields integers. This observation is known as the Law of Rational Indices. Another observational law, called Haüy’s Law:
blank■ Miller indices of faces generally contain low numbers.

For example, (111) is a common face in crystals, while (972) is not. Haüy’s law is really a corollary to the Law of Bravais, already discussed, which states that faces form parallel to planes of high lattice point density; planes with low values in their Miller index have the greatest lattice point density.

As mentioned previously, crystallographers have in the past used four axes for crystals in the hexagonal system. This yields a Miller index with four numbers (hkil). One of the first three values h, k, or i is redundant because we can always describe the location of a plane in three-dimensional space with three variables. In all cases: h + k + i = 0. Because of the redundancy, and to be consistent with other crystal systems, many crystallographers today use only three indices for hexagonal minerals.

For further discussion of Miller indices with examples, watch the videos linked here:

Video 11-5: Miller indices example (2 minutes)

Video 11-6: Miller indices of 0 (5 minutes)

Video 11-7: Miller indices for an octahedron (9 minutes)

Video 11-8: Miller indices for hexagonal minerals (4 minutes)