Therefore,
A
Ã
¼ 2p
B Â C
A:B Â C
¼ 2p
ffiffi
3
p ac
2
^ i þ ac
2
^ j
ffiffi
3
p a
2 c
2
¼
2p
a
^ i þ
2p
ffiffi ffi
3
p
a
^ j
Similarly,
B
Ã
¼ 2p
C Â A
A:B Â C
¼ 2p
ac
ffiffi
3
p a
2 c
2
^ j ¼
4p
ffiffi ffi
3
p
a
^ j
C
Ã
¼ 2p
A Â B
A:B Â C
¼ 2p
ffiffi
3
p a
2
2
^ k
ffiffi
3
p a
2 c
2
¼
2p
c
^ k
Comparing the direct and reciprocal lattice (Fig. 2.17), we observe that C and
C*-axes are parallel but A*-axis has rotated through an angle of 30° with respect to
A-axis in the hexagonal basal plane.
Further, the volume of the reciprocal unit cell is given by
V
Ã
¼ A
Ã
:B
Ã
 C
Ã
¼
2p
a
^ i þ
2p
ffiffi ffi
3
p
a
^ j
:
4p
ffiffi ffi
3
p
a
^ j Â
2p
c
k
¼
2p
a
^ i þ
2p
ffiffi ffi
3
p
a
^ j
:
8p
2
ffiffi ffi
3
p
ac
^ i
¼
2ð2pÞ
3
ffiffi ffi
3
p
a 2 c
Example 8 Show that the reciprocal lattice to a simple cubic lattice is also a simple
cubic.
Fig. 2.17 Primitive translation vectors a Direct lattice b Reciprocal lattice
60
2 Unit Cell Construction