(c) Three-Dimensional (Simple cubic) Lattice
Again for simplicity of the problem, let us take the case of a simple cubic lattice
with a = b = c and a ¼ b ¼ c ¼ 90
: The components of the reciprocal lattice
vector G and the wave vector k are given by
G ¼
2p
a
^ i n x þ ^ j n y þ ^ kn z
À
Á
and k ¼ ^ i k x þ ^ j k y þ ^ k k z
À
Á
Substituting these values in Eq. 2.1, we obtain
2p
a
^ i n x þ ^ j n y þ ^ k n z
À
Á
! 2
þ 2 ^ i k x þ ^ j k y þ ^ k n z
À
Á :
2p
a
^ i n x þ ^ j n y þ ^ k n z
À
Á
!
¼ 0
or
4p
2
a 2 n
2
x þ n
2
y þ n
2
z
þ
4p
a
n x k x þ n y k y þ ^ kk z
À
Á ¼ 0
where ^ i : ^ i ¼ ^ j : ^ j ¼ ^ k : ^ k ¼ 1 and ^ i : ^ j ¼ ^ j : ^ k ¼ ^ k : ^ i ¼ 0:
Further simplifying this, we obtain
n x k x þ n y k y þ n z k z ¼ À
p
a
n
2
x þ n
2
y þ n
2
z
ð2:4Þ
This equation could have been obtained by simply generalizing the Eq. 2.3
obtained in 2-D case. However, from Eq. 2.4, it follows that the first zone for a
simple cubic lattice is a simple cube whose walls intersect the k x , k y and k z axes
at the Æp=a as shown in Fig. 2.24a. The second zone is obtained by adding a
pyramid (like a triangle in 2-D) to each face of the cube of the first zone. The
resulting diagram is shown in Fig. 2.24b.
Fig. 2.24 First and second Brillouin zone of simple cubic lattice
2.3 Construction of Brillouin Zones
71