Proof: The primitive translation vectors of a simple cubic lattice (2.18a) may be
written as:
a ¼ a ^ i; b ¼ b ^ j; c ¼ c ^ k
where a = b = c are lattice translations and ^ i, ^ j, ^ k are unit vectors. Volume of the
direct unit cell is given by
V ¼ a : b  c ¼ a ^ i : a ^ j  a ^ k ¼ a ^ i Á a
2 ^ i ¼ a
3
Therefore,
a
Ã
¼ 2p
b  c
a:b  c
¼
2p
a
^ i
b
Ã
¼ 2p
c  a
a:b  c
¼
2p
a
^ j
c
Ã
¼ 2p
a  b
a:b  c
¼
2p
a
^ k
The primitive reciprocal lattice translation vectors suggest that the reciprocal
lattice to simple cubic lattice is itself a simple cubic shown as the first BZ (2.18b)
with a lattice translation 2p/a. The volume of the reciprocal unit cell is given by
V
Ã
¼ a
Ã
:b
Ã
 c
Ã
¼
2p
a
^ i:
2p
a
^ j Â
2p
a
^ k ¼
2p
a
^ i:
2p
a
2
^ i ¼
2p
a
3
Example 9 Show that the reciprocal lattice to a body-centered cubic is a
face-centered cubic.
Fig. 2.18 a Direct and b reciprocal lattice of simple cubic
2.2 Construction of Reciprocal Lattice
61