Example 12 Show that the interplanar spacing d hkl in a real crystal is proportional
to 1=G(hkl)
j
j :
Proof: From the above proof, we know that G (hkl) is normal to the crystal plane
(hkl). Accordingly, we can write
G(hkl)
j
j^ n ¼ ha
Ã
þ kb
Ã
þ lc
Ã
ð
Þ
where ^ n is a unit vector, so that
^ n ¼
ha
Ã
þ kb
Ã
þ lc
Ã
ð
Þ
G(hkl)
j
j
In Fig. 2.21, the length of the interplanar spacing is
d hkl ¼
a
h
cos h
¼
a
h
^ n
¼
a
h
:
ha
Ã
þ kb
Ã
þ lc
Ã
ð
Þ
G(hkl)
j
j
¼
1
G(hkl)
j
j
Example 13 Show that the reciprocal of the reciprocal lattice is a direct lattice.
Proof: We know that the reciprocal lattice vector a* is given by
a
Ã
¼ 2p
b  c
a:b  c
Taking reciprocal of this, we can write
a
Ã
ð Þ
à ¼ 4p
2 b
Ã
 c
Ã
a à :b
Ã
 c Ã
Since a.a* = b.b* = c.c* = 2p, therefore, the above equation becomes
a
Ã
ð Þ
à ¼ 2pa : a
à b
Ã
 c
Ã
a à :b
Ã
 c Ã
¼ 2pa
a
Ã
:b
Ã
 c
Ã
a à :b
Ã
 c Ã
¼ 2pa
66
2 Unit Cell Construction