B
Ã
¼ 2p
C Â A
A:B Â C
¼ 2p
ac
2
^ j À
ffiffi
3
p ac
2
^ i
ffiffi
3
p a
2 c
2
¼ À
2p
a
^ i þ
2p
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
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
: À
2p
a
^ i þ
2p
ffiffi ffi
3
p
a
^ j
Â
2p
c
^ k
¼
2p
a
^ i þ
2p
ffiffi ffi
3
p
a
^ j
:
4p
2
ac
^ j þ
4p
2
ffiffi ffi
3
p
ac
^ i
¼
8p
3
ffiffi ffi
3
p
a 2 c
þ
8p
3
ffiffi ffi
3
p
a 2 c
¼
2ð2pÞ
3
ffiffi ffi
3
p
a 2 c
Example 7 The primitive translation vectors of a simple hexagonal space lattice
may be taken as
A ¼ a ^ i; B ¼ À
a
2
^ i þ
ffiffi ffi
3
p
a
2
^ j
and C ¼ c ^ k
where ^ i, ^ j and ^ k are unit vectors.
Determine the primitive translation vectors of the reciprocal lattice and show that
they are rotated with respect to C or C* axis through an angle of 30°. Also,
determine reciprocal unit cell volume.
Solution: Given: Direct lattice parameters A, B and C; determine the reciprocal
lattice parameters A*, B*, C* and volume V*. Volume of the direct unit cell is
given by
V ¼ A:B Â C ¼ a ^ i: À
a
2
^ i þ
ffiffi ffi
3
p
a
2
^ j
 c ^ k
¼ a ^ i:
ac
2
^ j þ
ffiffi ffi
3
p
ac
2
^ i
¼
ffiffi ffi
3
p
a
2 c
2
2.2 Construction of Reciprocal Lattice
59