Solution: Given: Face-centered cubic lattice with a = b = c and a ¼ b ¼ c ¼ 90
:
The primitive translation vectors of an fcc 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
j
j¼
a
4
3
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
^ i þ ^ j À ^ k
À
Á ¼
2p
a
ð11 1Þ
Similarly,
b
0
ð Þ
à ¼ 2p
c
0
 a
0
a 0 :b
0
 c 0 ¼
2p
a
À ^ i þ ^ j þ ^ k
À
Á ¼
2p
a
ð 111Þ
c
0
ð Þ
à ¼ 2p
a
0
 b
0
a 0 :b
0
 c 0 ¼
2p
a
^ i À ^ j þ ^ k
À
Á ¼
2p
a
ð1 11Þ
Fig. 2.26 Brillouin zone of
bcc lattice
2.3 Construction of Brillouin Zones
77