For (110) planes in a cubic crystal, the interplanar spacing is given by
d 110 =
a
h
2 + k
2 + l
2
À
Á 1=2 =
a
1
2 + 1
2 + 0
2
À
Á 1=2 =
2:87
ffiffi ffi
2
p
Further, from Bragg’s equation, we have
k ¼ 2d sin h ¼
2 Â 2:87 Â sin 22:35
ð
Þ
ffiffi ffi
2
p
¼ 1:543 ˚
A
Example 7 Show that the greater is the angle of diffraction, greater is the accuracy
in determining the lattice parameters.
Solution: Let us start with the Bragg’s equation with interplanar spacing d = a (also
lattice parameter as in a simple cubic crystal) as
2a sin h ¼ nk
or a sin h ¼
nk
2
Differentiating this equation, we obtain
a cos h dh þ sin h da ¼ 0
or
da
a
¼ À cot h dh
It is evident that as h !
p
2 ; coth ! 0; that is, the error in the lattice parameter
approaches zero. This proves the above statement.
Example 8 Determine the longest wavelength that can be analyzed by a rock-salt
crystal with interplanar spacing 2.82 Å in the first and second orders of X-ray
diffraction.
Solution: Given: Interplanar spacing, d = 2.82 Å, n = 1, 2, k max ¼ ?
We know that the Bragg’s equation is given by
2d sin h ¼ nk
Here, for the k to be maximum, sin h must be maximum, that is, sin h
ð
Þ max ¼ 1
This gives us
7.2 X-Ray Diffraction by Crystals
277