Example 7 Construct B-Z corresponding to a simple hexagonal lattice with a = b
6 ¼ c and a ¼ b ¼ 90
and c ¼ 120
: Show that six vertical planes of B-Z are at a
distance of
2 ffiffi
3
p
p
a
À Á
and two horizontal planes are at a distance p=c
ð Þ from the center
of the zone.
Solution: Given: Simple hexagonal lattice with a = b 6 ¼ c and a ¼ b ¼ 90
and c ¼
120
: The primitive translation vectors of simple hexagonal lattice shown in
Fig. 2.16 are given by:
a ¼ a ^ i; b ¼
a
2
^ i þ
ffiffi ffi
3
p
a
2
^ j; c ¼ c ^ k
where ^ i, ^ j, ^ k are the orthogonal unit vectors. Volume of the direct (primitive) unit
cell is given by
V ¼ a ^ i:
a
2
^ i þ
ffiffi ffi
3
p
a
2
^ j
 c ^ k
¼ a ^ i: À
ac
2
^ j þ
ffiffi ffi
3
p
ac
2
^ i
¼
ffiffi ffi
3
p
a
2 c
2
The primitive translation vectors of the reciprocal lattice are:
a
Ã
¼ 2p
b  c
a:b  c
¼
2p
a
^ i À
2p
ffiffi ffi
3
p
a
^ j
Similarly,
Fig. 2.27 Brillouin zone of a
fcc lattice
2.3 Construction of Brillouin Zones
79