Physical Chemistry , 1st ed.

Figure 21.18 shows the steps in determining the Miller indices of a plane in

a simple cubic unit cell. (The process is the same for other unit cells; it just

looks more complicated.) In the four unit cells shown, parallel planes of atoms

are indicated. How do we designate these parallel planes in terms of the unit

cell dimensions a,b, and c? We use the following steps:

1. Using any unit cell, pick a corner that will serve as a three-dimensional,
   Cartesian-axis origin. An example is indicated in Figure 21.18.


2. Determine the intercepts of the planes in terms of how many a,b, and c
   units the intercepts are from the origin. In Figure 21.18, the indicated
   plane intercepts the crystal axes at 1 aunit, 1 bunit, and 1 cunit of length
   in each of the crystal dimensions. The numbers we collect to represent
   the indicated plane are 1, 1, and 1—indicating the number of unit cell
   dimensions that represent the axes’ intercepts.


3. Take the reciprocal of each number: in this case, the reciprocals of 1, 1,
   and 1 are simply 1, 1, and 1 (but we will consider another example next).


4. Express the Miller indices as the three reciprocal numbers grouped to-
   gether inside parentheses, without punctuation: (111). The plane illus-
   trated in Figure 21.18 is referred to as the (111) plane for this simple
   cubic lattice.

The Miller indices for any plane of crystal positions can be determined by this

method. The general form of expressing Miller indices is (hk), where hrepre-

sents the Miller index along the aunit cell dimension,kis the Miller index along

the bunit cell dimension, and is for the cunit cell dimension. For cubic unit

cells, there are equivalences among planes that have the same Miller indices, such

that (100) is equivalent to (010), which is equivalent to (001), and so forth.

There are several pitfalls to watch out for. First, there are (maybe obvious)

planes that are in one of the planes of the unit cell itself: Figure 21.19 shows

several. None of these planes intersects the third crystal axis. In this case, the

intercepts are considered to occur at infinite dimensions of the unit cell, and

the reciprocal ofis zero: therefore, the Miller indices of the planes in Figure

21.19 are (100), (110), and (200), respectively. (You should satisfy yourself that

these designations are correct.)

Second, Miller indices can also be numbers greater than 1 (usually integers)

or less than 1 (usually a simple fraction). Figure 21.19c shows a plane of crys-

tal positions that includes an index greater than 1. Again, you should satisfy

yourself that the steps given above give corresponding Miller indices.

Finally, Miller indices are occasionally best described using negative num-

bers. But given the lack of punctuation in the expression (ABC) for a crystal

plane, using a minus sign might cause a problem with interpretation; consider

(A BC) as a label for a crystal plane! Rather, it is the convention to put a bar

over a Miller index to indicate that it is negative: rather than (A BC), we

write (ABC), where Bindicates that the Miller index (as determined from the

steps above) is actually B. Figure 21.20 shows an example of a crystal plane

defined this way.

Why do we go to such trouble? Because each and every definable plane can

act to diffract X rays to produce constructive interference of radiation. Since

different planes have different dspacings, planes having different Miller indices

will diffract the same X rays at different angles. As we use the phenomenon of

constructive radiation to understand the structures of crystals, we need a sys-

tem to keep track of the planes of atoms (molecules, ions, and so on) that are

reflecting the radiation. The concept of Miller indices shows that there are

21.6 Miller Indices 745

a

c
b
Figure 21.18 How to determine Miller indices
of parallel planes of atoms. See text for details.

(a) (100)

(b) (110)

(c) (200)

a

c

b

a

c

b

a

c

b

Figure 21.19 Miller indices of some planes in
cubic crystals.