The method devised by the British mineralogist, William H. Miller, in 1839 is still used to denote crystallographic planes and directions. That is why the method named after him uses the Miller indices labeling technique. Besides closed-packed planes, there could be a number of crystallographic planes of interest. Their orien­tation, arrangement, and atom density could be different. It may be cumbersome to name a crystallographic plane as “cube face plane,” “octahedral plane,” and so on. The same may be applicable to the issue of defining crystallographic directions. One would soon run out of names just to refer to the directions as “cube edge,” “face diagonal,” “body diagonal,” and so on. This type of nomenclature also lacks adequate specificity to be seriously considered. That is why a more convenient and systematic technique, such as Miller indices, is used to denote crystallographic planes and directions. As a matter of fact, each point on a crystal can be reached by the translation vector composed of the sum of the multiples of the crystal lattice vectors.

For labeling a plane in a crystal with Miller indices, the following general proce­dure needs to be followed:

1) Select an origin at a lattice point that is not on the crystallographic plane to be indexed.

2) Fix the three orthogonal axes (a, b, and c or x, y, and z) from the selected origin.

3) Find the intercepts (in multiples of the unit lattice vector) that the plane makes on the three coordinate axes.

4) Take reciprocals of these multiples.

5) Convert the fractions (if any) to a set of integers and reduce the integers by dividing by a common integer factor. However, care should be exercised so that the atom configuration of the original plane remains the same.

6) Enclose the final numbers in parentheses such as (hkl), which is the Miller index of the particular plane. A family of equivalent planes is given by numbers enclosed in curly brackets {hkl}. For example, the faces of a cube are given by {100}, whereas an individual plane is denoted by (100), (010), and so on. A nega­tive intercept is denoted by a “bar” on the index such as (100).

Figure 2.10a-d shows four planes in a cubic unit cell. In Figure 2.10a, the hatched plane makes an intercept on the x-axis for a one lattice vector (take it as a unity 1 in place of lattice constant a), however it does not intersect the y — and z-axes making the intercept of 1. Therefore, the reciprocals of the intercepts become 1/1, 1/i, and 1/i. Hence, the Miller index of the plane is (100). Similarly, for the plane shown in Figure 2.10b, the intercepts are 1, 1, and 1 on the x-, y-, and z-axes respectively, and therefore, the reciprocals to the intercepts become 1, 1, and 0. Thus, the Miller index of the plane becomes (110). In Figure 2.10c, the plane with hatch marks portends positive intercepts of 1 each on all the three axes. When the reciprocals of the intercepts are taken, they remain the same, and thus the Miller index of the plane becomes (111). In Figure 2.10d, the plane denoted by the hatch

marks cuts the x-, y-, and z-axes creating positive intercepts of 1, 1, and 1/2, respec­tively. We take the reciprocals of these intercepts to obtain 1, 1, and 2. Thus, the Miller index of the plane would be (112). Even though we have not given a direct example where one needs to reduce the Miller index to lower integers, the issue merits some discussion. This is related to the step 5 in the above-mentioned proce­dure. The operation needs to be carried out depending on the specific case. For example, (220) and (110) are equivalent planes in an FCC crystal, so be treated so. On the other hand, even though (200) and (100) planes in FCC and BCC crystals are equivalent, that is not the case in a simple cubic crystal. Hence, in these types of cases, individual attention needs to be paid to the atom configurations of the planes in question to ascertain whether further reduction is going to be permitted or not.

Another important issue for denoting crystallographic planes is the situation where negative indices become essential. This can be easily illustrated referring to Figure 2.10a. Let us say one wishes to index the leftmost cube face in Figure 2.10a. As the plane passes through the previous origin, it needs to be shifted to some other position, say shifted to the right side by one lattice spacing. In this case, the intercepts the plane is making to x-, y-, and z-axes can be interpreted as 1, —1, and 1, respectively. So by taking reciprocals, we get 0, —1, and 0. This means the Miller

index of the plane can be represented as (010) where 1 is written in place of —1, and called “bar 1.”

So we understand pretty much how to label a plane using Miller indices. We now turn our attention to labeling a crystallographic direction with a Miller index. Let us label the “direction A” in the cubic unit cell on the right-hand side in Figure 2.11. For denoting directions, the direction must pass through the origin (O). In a situa­tion, where the original direction does not go through the origin, one needs to a draw a parallel line that passes through the origin. This parallel line will have the same Miller index as the original one. For direction A, the components along the x-, y-, and z-axes need to be resolved. In this case, direction A can be resolved half the lattice constant along X-axis, a unit lattice vector distance along Y-axis, and no com­ponent (i. e., 0) along the Z-axis. Now these components need to be converted to the smallest whole integers and then put into the square brackets to obtain the Miller index of the direction. Hence, the Miller index of direction A would be [1/2 1 0], that is, [120]. General notation used for a crystallographic direction is [uvw]. The Miller index of “direction B” on the left-hand side unit cell can be found out in the same way. Note that the component of “direction B” can be resolved into a positive unit distance along the X-axis (i. e., 1), a lattice vector in the negative Y-axis (i. e., —1), and no component (i. e., 0) along the Z-axis. Thus, the Miller index of “direction B” would be [110]. Note that the face diagonal of the base face of the cube on the right — hand side unit cell is parallel to “direction B” and will have the same Miller index [110]. A class of equivalent directions is called a “family of directions” and denoted by {uvw). In the case of face diagonal of the cubic unit cell, all face diagonals belong to a single family of directions, {110), meaning directions [110], [101], [011], [110], and so on.

An alternative procedure is quite commonly used where we first determine the coordinates of two points on the direction; the points are specified with respect to a defined set of orthogonal coordinate system (x, y z or a, b, c), as shown in Figure 2.12a. Next, subtract the coordinates of the “tail” from the “head” and clear the fractions and/or reduce to lowest integers. Enclose the numbers in brack­ets [ ] with the negative sign by “bar” above the number. Three examples are

illustrated in Figure 2.12b. The Miller indices of the directions A, B, and C in Figure 2.12b are thus [100], [111], and [122], respectively.

We note that the definitions of the Miller indices for planes and directions in cubic structures are such that the direction with Miller index [hkl] is normal to the plane (hkl).