We know that the Bragg’s equation is given by
2d sin h ¼ nk
ðiÞ
For a cubic structure, the relationship between interplanar spacing “d” and lattice
parameter “a” is
d hkl ¼
a
h
2
þ k
2
þ l
2
À
Á 1=2
ðiiÞ
Combining these two equations, we have
sin h ¼
k
2d hkl
¼
h
2
þ k
2
þ l
2
À
Á 1=2
2a
From Table 9.1, we know that the first four lines and their corresponding planes
for body-centered cubic structure are: 2 (110), 4 (200), 6 (211) and 8 (220).
Therefore, the 2h angles corresponding to these (hkl) planes are
sin h 110 ¼
k
2d 110
¼
1
2
þ 1
2
þ 0
2
ð
Þ
1=2
2a
¼
ffiffi ffi
2
p  1:54
2 Â 4
or 2h 110 ¼ 2 sin
À1
k
2d 110
¼ 2 sin
À1
ffiffi ffi
2
p  1:54
2 Â 4
¼ 31:6
Similarly,
sin h 200 ¼
k
2d 200
¼
2
2
þ 0
2
þ 0
2
ð
Þ
1=2
2a
¼
2 Â 1:54
2 Â 4
or 2h 200 ¼ 2 sin
À1
k
2d 200
¼ 2sin
À1 2 Â 1:54
2 Â 4
¼ 45:3
sin h 211 ¼
k
2d 211
¼
2
2
þ 1
2
þ 1
2
ð
Þ
1=2
2a
¼
ffiffi ffi
6
p  1:54
2 Â 4
or 2h 211 ¼ 2 sin
À1
k
2d 211
¼ 2 sin
À1
ffiffi ffi
6
p  1:54
2 Â 4
¼ 56:3
and
sin h 220 ¼
k
2d 220
¼
2
2
þ 2
2
þ 0
2
ð
Þ
1=2
2a
¼
ffiffi ffi
8
p  1:54
2 Â 4
or 2h 220 ¼ 2 sin
À1
k
2d 220
¼ 2 sin
À1
ffiffi ffi
8
p  1:54
2 Â 4
¼ 66:0
9.1 Steps in Crystal Structure Determinations
343