Example 10 Show that the reciprocal lattice to a face-centered cubic is a
body-centered cubic.
Proof: The primitive translation vectors of the fcc lattice shown in Fig. 2.20 are
given by
a
0
¼
a
2
^ i þ ^ j
À
Á
b
0
¼
a
2
^ j þ ^ k
À
Á
c
0
¼
a
2
^ k þ ^ i
À
Á
where a is the side of the conventional unit cube and ^ i, ^ j, ^ k are unit vectors. Volume
of the direct (primitive) unit cell is given by
V ¼ a
0
:b
0
 c
0
¼
a
2
^ i þ ^ j
À
Á :
a
2
^ j þ ^ k
À
Á Â
a
2
^ k þ ^ i
À
Á
h
i
¼
a
2
i þ j
ð
Þ:
a
4
2
^ i ^ j ^ k
0 1 1
1 0 1
¼
a
8
3 ^ i þ ^ j
À
Á : ^ i þ ^ j À ^ k
À
Á
¼
a
8
3
1 þ 1
ð
Þ
¼
a
4
3
Therefore, the primitive translation vectors of the reciprocal lattice are:
a
0
ð Þ
à ¼ 2p
b
0
 c
0
a 0 :b
0
 c 0 ¼ 2p
a
4
2 ^ i þ ^ j À ^ k
À
Á
a
4
3
¼
2p
a
^ i þ ^ j À ^ k
À
Á
Similarly,
b
0
ð Þ
à ¼
2p
a
À ^ i þ ^ j þ ^ k
À
Á
c
0
ð Þ
à ¼
2p
a
^ i À ^ j þ ^ k
À
Á
These reciprocal lattice vectors are found to be the primitive vectors of bcc
lattice (Fig. 2.19).
64
2 Unit Cell Construction