Engineering Chemistry S Datta

SOLID STATE CHEMISTRY 483
and is multiplied throughout by the least common multiple to obtain integral values. Thus, the
Miller indices of the plane (2a : b : 2c) will be 1 :
2 1 : 1 i.e., 1 : 2 : 1 and this plane or face is
2
indicated as (1 2 1) face of the crystal.
Now, for a plane perpendicular to one axis and parallel to the other two, having intercepts
– a : ∞ b : ∞ c will have indices, as:
Weiss indices – 1 : ∞ : ∞
and Miller indices – 1 : 0 : 0 i.e., (100) plane.
If a plane produces an intercept on the negative side, say – a : – b : ∞ c, the Miller indices
for the plane would be (1 10), the bar above one indicates the intersection of the plane on the
negative side of the axis. The symbol 1 denotes minus unity.
Thus, if a face of a crystal makes intercepts OA, OB and OC on the three axes, then the
lengths of the intercepts may be expressed as OA/a, OB/b and OC/c where, a, b and c are unit
distances along three axes. The reciprocals of these lengths will be a/OA, b/OB and c/OC, and
these reciprocal intercepts are whole number or integers i.e.,
a/OA = h; b/OB = k and c/OC = l
where h, k, l are Miller indices of the face or plane of the crystal and the face is defined as
(h k l) face.
The distance between the parallel planes in a crystal is designated as d hkl
. For various
cubic lattices, these interplanar spacings are given by the formula:
a
d hkl
=
2 2 2
h + k + l
where ‘a’ is the length of the side of the cube and h, k, l are the Miller indices of the plane.
Some of the Miller indices in the case of cubic lattices are shown in Fig. 22.3 below.
X
O
Y
Z
(0,0,1)plane
(0,1,0)plane
(0,1,1)plane
(1,1,1)plane
Fig. 22.3 Miller indices in the case of cubic lattices.