sin h 311 ¼
k
2d 311
¼
3
2
þ 1
2
þ 1
2
ð
Þ
1=2
2a
¼
ffiffiffiffiffi
11
p  1:54
2 Â 4
or 2h 311 ¼ 2 sin
À1
k
2d 311
¼ 2 sin
À1
ffiffiffiffiffi
11
p  1:54
2 Â 4
¼ 79:4
Example 5 Determine the (hkl) and 2h values of the first four lines on the powder
patterns obtained by using CuKa radiation of wavelength 1.54 Å corresponding to a
simple tetragonal structure with a = b = 4 Å and c = 3 Å.
Solution Given: k = 1.54 Å, simple tetragonal structure with a = b = 4 Å, c = 3 Å,
(hkl) and 2h values of the first four lines = ?
We know that the Bragg’s equation is given by
2d sin h ¼ nk
ðiÞ
For a tetragonal structure, the relationship between interplanar spacing ’d’ and
lattice parameters “a” and “b” is
d hkl ¼
h
2
þ k
2
a 2 þ
l
2
c 2
! À1=2
ðiiÞ
Combining these two equations, we have
sin h ¼
k
2d hkl
¼
k
2
:
h
2
þ k
2
a 2 þ
l
2
c 2
! 1=2
We know that the structural conditions for tetragonal structure are a ¼ b 6 ¼ c
and a ¼ b ¼ c ¼ 90
, therefore the first four lines and their corresponding planes
are: 1 (100), 1 (001). 2 (110) and 2 (011)
Therefore, the 2h angles corresponding to these (hkl) planes are
sin h 100 ¼
k
2d 100
¼
k
2
Á
1
2
þ 0
2
4 2 þ
0
2
3 2
! 1=2
¼
1:54 Â 1
2 Â 4
or 2h 100 ¼ 2 sin
À1
k
2d 100
¼ 2 sin
À1 1:54 Â 1
2 Â 4
¼ 22:2
9.1 Steps in Crystal Structure Determinations
345