Similarly,
b
Ã
ð Þ
à ¼ 2pb
c
Ã
ð Þ
à ¼ 2pc
Example 14 Show that the volume of the reciprocal unit cell is inversely proportional to that of corresponding direct unit cell.
Proof: We know that the volume of the primitive unit cell of a direct lattice is
V ¼ a:b  c
Similarly, the volume of the primitive cell of reciprocal lattice is
V
Ã
¼ a
Ã
:b
Ã
 c
Ã
¼
b  c
a:b  c
!
:
c  ab
a:b  c
!
Â
a  b
a:b  c
!
¼
1
a:b  c
ð
Þ
3
b  c: c  a
ð
Þ a  b
ð
Þ
f
g
½
Š
¼
1
a:b  c
ð
Þ
3
b  c
ð
Þ c: a  b
ð
Þa À a: a  b
ð
Þ
f
g c
½
Š
¼
1
a:b  c
ð
Þ
3
c: a  b
ð
Þ a:b  c
ð
ÞÀ0
f
g
½
Š
¼
1
a:b  c
ð
Þ
3
a:b  c
ð
Þ
2
¼
1
a:b  c
ð
Þ
2.3 Construction of Brillouin Zones
A Brilloiun zone is defined as the region (in 1-D, 2-D or 3-D) in k-space (reciprocal
space) that the electrons/phonons occupy to move freely without being diffracted.
Because of its connection with k-space, the first Brillouin zone is equivalent to a
Wigner–Seitz unit cell in reciprocal space, that is,
B À Z ¼ W À S Cell of R À L
The first B-Z and all other (higher) zones are symmetrical about k = 0. The
construction of the first B-Z for any given (direct) lattice can be made according to
the following procedure.
2.2 Construction of Reciprocal Lattice
67