Example 11 Show that the reciprocal lattice vector G (hkl) is normal to the crystal
plane (hkl).
Proof: This can be proved if we can show that the scalar product of the reciprocal
lattice vector G (hkl) and any vector lying in the crystal plane (hkl) vanishes.
Let us consider the (hkl) plane as shown in Fig. 2.21. This plane intercepts the
a-axis at a/h, b-axis at b/k and c-axis at c/l, respectively. The vectors A, B and C lie
in the (hkl) plane. Now, let us take the scalar product of the vector C with G, we
obtain
C:G ¼
a
h
À
b
k
: ha
Ã
þ kb
Ã
þ lc
Ã
ð
Þ
¼
a
h
: ha
Ã
þ kb
Ã
þ lc
Ã
ð
Þ À
b
k
ha
Ã
þ kb
Ã
þ lc
Ã
ð
Þ
¼
h
h
þ 0 þ 0
À 0 þ
k
k
þ 0
¼ 0
Similarly, it can be shown that A.G = B.G = 0. This proves that the reciprocal
lattice vector G (hkl) is normal to the crystal plane (hkl).
Fig. 2.21 Intercepts of a plane on three axes and the interplanar spacing
2.2 Construction of Reciprocal Lattice
65