Similarly,
sin h 001 ¼
k
2d 001
¼
k
2
Á
0
2
þ 0
2
4 2 þ
1
2
3 2
! 1=2
¼
1:54Â1
2 Â 3
or 2h 001 ¼ 2 sin
À1
k
2d 001
¼ 2 sin
À1 1:542
2 Â 3
¼ 29:7
sin h 110 ¼
k
2d 110
¼
k
2
Á
1
2
þ 1
2
4 2 þ
0
2
3 2
! 1=2
¼
1:54 Â
ffiffi ffi
2
p
2 Â 4
or 2h 110 ¼ 2 sin
À1
k
2d 110
¼ 2 sin
À1 1:54 Â
ffiffi ffi
2
p
2 Â 4
¼ 31:6
and
sin h 011 ¼
k
2d 011
¼
k
2
Á
0
2
þ 1
2
4 2 þ
1
2
3 2
! 1=2
¼
1:54 Â 5
2 Â 4 Â 3
or 2h 011 ¼ 2 sin
À1
k
2d 011
¼ 2 sin
À1 1:54 Â 5
2 Â 4 Â 3
¼ 37:5
Example 6 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
an orthorhombic structure with a = 2 Å, b = 3 Å and c = 4 Å.
Solution Given: k = 1.54 Å, orthorhombic structure with a = 2 Å, b = 3 Å, c = 4
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 an orthorhombic structure, the relationship between interplanar spacing ’d’
and lattice parameters “a,” “b” and “c” is
d hkl ¼
h
2
a 2 þ
k
2
b
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
a 2 þ
k
2
b
2
þ
l
2
c 2
! 1=2
We know that the structural conditions for a tetragonal structure are a 6 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 (010). 1 (001) and 2 (110)
346
9 Determination of Crystal Structure Parameters