Solution: Given: Simple cubic lattice with a = b = c and a ¼ b ¼ c ¼ 90
; the
primitive translation vectors of a simple cubic lattice 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 parallel to the cube
edges. Volume of the direct unit cell is given by
V ¼ a:b  c
j
j¼ a
3
Therefore,
a
Ã
¼ 2p
b  c
a:b  c
¼
2p
a
^ i ¼
2p
a
100
ð
Þ
b
Ã
¼ 2p
c  a
a:b  c
¼
2p
a
^ j ¼
2p
a
010
ð
Þ
c
Ã
¼ 2p
a  b
a:b  c
¼
2p
a
^ k ¼
2p
a
001
ð
Þ
This shows that the reciprocal lattice is a simple cubic lattice whose lattice
parameter is 2p/a. The first B-Z is constructed by drawing six perpendicular
bisector planes to the vectors a*, b* and c*. They are:
Æ
a
Ã
2
¼ Æ
p
a
^ i; Æ
b
Ã
2
¼ Æ
p
a
^ j; Æ
c
Ã
2
¼ Æ
p
a
^ k;
The space bounded by these six {100} planes (termed as zone faces) is also a
cube of side (2p/a) and is known as first B-Z of the simple cubic lattice (Fig. 2.24a).
Example 5 Construct B-Z corresponding to a bcc lattice with a = b = c and
a ¼ b ¼ c ¼ 90
: Show that the twelve equivalent zone faces are at a distance
ffiffi ffi
2
p
p=a
ð
Þ from the center of the zone.
Solution: Given: Body-centered cubic lattice with a = b = c and a ¼ b ¼ c ¼90
:
The primitive translation vectors of a bcc shown in Fig. 2.19 are given by:
a
0
¼
a
2
^ i þ ^ j À ^ 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 parallel
to the cube edges. Volume of the direct (primitive) unit cell is given by
2.3 Construction of Brillouin Zones
75