Therefore, the 2h angles corresponding to these (hkl) planes are
sin h 100 ¼
k
2d 100
¼
k
2
Á
1
2
2 2 þ
0
2
3 2 þ
0
2
4 2
! 1=2
¼
1:54 Â 1
2 Â 2
or 2h 100 ¼ 2 sin
À1
k
2d 100
¼ 2 sin
À1 1:54 Â 1
2 Â 2
¼ 45:3
Similarly,
sin h 010 ¼
k
2d 010
¼
k
2
Á
0
2
2 2 þ
1
2
3 2 þ
0
2
4 2
! 1=2
¼
1:54 Â 1
2 Â 3
or 2h 010 ¼ 2 sin
À1
k
2d 010
¼ 2 sin
À1 1:54 Â 1
2 Â 3
¼ 29:7
sin h 001 ¼
k
2d 001
¼
k
2
Á
0
2
2 2 þ
0
2
3 2 þ
1
2
4 2
! 1=2
¼
1:54 Â 1
2 Â 4
or 2h 001 ¼ 2 sin
À1
k
2d 001
¼ 2 sin
À1 1:54 Â 1
2 Â 4
¼ 22:2
and
sin h 110 ¼
k
2d 110
¼
k
2
Á
1
2
2 2 þ
1
2
3 2 þ
0
2
4 2
! 1=2
¼
1:54 Â
ffiffiffiffiffi
13
p
2 Â 6
or 2h 110 ¼ 2 sin
À1
k
2d 110
¼ 2 sin
À1 1:54 Â
ffiffiffiffiffi
13
p
2 Â 6
¼ 55:1
Example 7 The first three S-values obtained from the powder pattern of a given
unknown cubic form of material are 56.8, 94.4 and 112.0 mm, respectively. If the
radius of the camera is 57.3 mm and the wavelength of the radiation used is 1.54 Å,
determine the crystal structure of the cubic form and its lattice parameter.
Solution Given: S-values: S 1 = 56.8 mm, S 2 = 94.4 mm, S 3 = 112.0 mm,
R = 57.3 mm, k = 1.54 Å, crystal structure = ?, a = ?
We know that the angular and linear relationship in a powder pattern is
4hðdegreesÞ ¼ SðmmÞ
or h ¼
S
4
9.1 Steps in Crystal Structure Determinations
347