Proof: The primitive translation vectors of the bcc lattice shown in Fig. 2.19 are
given by
a
0
¼
a
2
^ i þ ^ jÀ ^ k
À
Á
b
0
¼
a
2
À ^ i þ ^ j þ ^ k
À
Á
c
0
¼
a
2
^ iÀ ^ j þ ^ k
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 Â c
0 ¼
a
2
^ i þ ^ j À ^ k
À
Á :
a
2
À ^ i þ ^ j þ ^ k
À
Á Â
a
2
^ i À ^ j þ ^ k
À
Á
h
i
¼
a
2
i þ j À k
ð
Þ :
a
4
2
^ i
^ j
^ k
À1 1 1
1 À1 1
¼
a
8
3
i þ j À k
ð
Þ :
^ i
^ j ^ k
À1 1 1
0 0 2
¼
a
8
3 ^ i þ ^ j À ^ k
À
Á :2 ^ i þ ^ j
À
Á
¼
a
8
3
2 þ 2
ð
Þ
¼
a
2
3
Therefore, the primitive translation vectors of the reciprocal lattice are:
Fig. 2.19 Primitive translation vectors of bcc lattice
62
2 Unit Cell Construction