Page 40 - Instant notes

Physical Chemistry 26

Fig. 3. Lattice rows in a two-
dimensional lattice.

indices. Taking the two-dimensional example, it is possible to ascribe a unique set of
intersect points for each set of rows. Thus in Fig. 3, the rows beginning at the top left-
hand edge passes through the next point at 1a and 1b. For other rows, the intersect is at,
for example, (3a, 2b). As the distances can always be written in terms of the length of the
unit cell, and the notation is simplified by referring to these intersects as (1,1) and (3,2)
respectively. The Miller indices are obtained by taking the reciprocal of the intersects.
Where the reciprocal yields fractional numbers, these numbers are multiplied up until
whole numbers are generated for the Miller indices. The Miller indices for a set of rows
whose reciprocal intercept is (⅓, ½), for example, is simplified to (2,3) by multiplying by
6.
The situation in three dimensions exactly parallels this argument, and the indices for
the lattice planes in Fig. 4 illustrate this point. Each Miller index, it should be noted,
refers not simply to one plane, but to the whole set of parallel planes with these indices.
For example, the (111) planes. By convention, these Miller indices are referred to as the
h, k, and l values respectively. Where an index has a negative value, this is represented by
a bar over the number, thus an index of (−112) is written as .