d 100 ¼ a; d 110 ¼
a
ffiffi ffi
2
p and d 111 ¼
a
ffiffi ffi
3
p
Hence, their ratio is:
d 100 : d 110 : d 111 = 1 :
1
ffiffi ffi
2
p :
1
ffiffi ffi
3
p
Case II: Body-Centered Cubic System.
In a body-centered cubic system, the lattice points are situated at the eight corners
and at the body center of the unit cell. Because of the presence of an additional point
at the body center, the interplanar spacing for three low index planes (100), (110) and
(111) give slightly different results. There appears an additional plane halfway
between (100) and (111) planes as shown in Fig. 4.19, while no new planes appear
between (110) planes when compared to simple cubic system. Therefore, the
interplanar spacing for the low index planes in the body-centered cubic system is:
d 100 ¼
1
2
d 100
ð
Þsimple cubic lattice ¼
a
2
d 110 ¼ d 110
ð
Þsimple cubic lattice ¼
a
ffiffi ffi
2
p
d 111 ¼
1
2
d 111
ð
Þsimple cubic lattice ¼
a
2
ffiffi ffi
3
p
Fig. 4.19 Low index planes in bcc crystal: a (100) planes b (110) planes c (111) planes
4.4 Interplanar Spacing
159