The basics of crystallography and diffraction (3rd Edition)

5.3 Indexing lattice planes—Miller indices 133
different point along OL—say Q—with coordinates 1 4 ,0, 1 2
, we would have obtained
the same result.
Now consider the direction SN. To find its direction symbol the origin must be shifted
from O to S. Proceeding as before (e.g. finding the coordinates of G with respect to the
origin at S) gives the direction symbol [1¯10] (pronounced one bar-one oh), the bar or
minus sign referring to a coordinate in the negative sense along the crystal axis.
Directions in crystals are, of course, vectors, which may be expressed in terms of
components on the three unit cell edge or ‘base’ vectors a, b and c. In the above example
the direction OL is written
r 102 = 1a + 0b + 2c.
The general symbol for a direction is [uvw] or, written as a vector,
r uvw = ua + vb + wc.
The direction symbols for the unit cell edge vectors a, b and c are [100], [010] and
[001], and very often these symbols are used in preference to the terms x-axis, y-axis
and z-axis.
5.3 Indexing lattice planes—Miller indices
First, for reasons which will be apparent shortly, the lattice plane whose index is to be
determined must not pass through the origin of the unit cell, or rather the origin must
be shifted to a corner of the cell which does not lie in the plane. Consider the unit cell
in Fig. 5.2 (identical to Fig. 5.1 but drawn separately to avoid confusion). We shall
determine the index of the lattice plane which is shaded and outlined by the letters RMS.
It is important first to realize that this plane extends indefinitely through the crystal; the
shaded area is simply that portion of the plane that lies within the unit cell of Fig. 5.2.
It is also important to realize that we are not just considering one plane but a whole
family of identical, parallel planes passing through the crystal. The next plane ‘up’ in
c
z
R
P
H
y
O
b
M
S
a
G
F
x
Fig. 5.2. Primitive unit cell (identical to Fig. 5.1), showing the first two planes, RMS and PGFH, in
the family. These planes are shaded within the confines of the unit cell.