According to the ratio of allowed reflections, this gives the diamond cubic
(DC) structure. Now, we can obtain the lattice parameter by using the expression
sin
2
h ¼
k
2
4a 2 h
2
þ k
2
þ l
2
À
Á ¼
k
2
4a 2 N
or a
2
1 ¼
k
2 N 1
4 sin
2
h 1
¼
0:71
ð
Þ
2 Â3
4 Â 0:0118
¼ 32:040 or a 1 ¼ 5:660 ˚
A
Similarly,
or a
2
2 ¼
k
2 N 2
4 sin
2
h 2
¼
0:71
ð
Þ
2 Â8
4 Â 0:0315
¼ 32:006 or a 2 ¼ 5:657 ˚
A
or a
2
3 ¼
k
2 N 3
4 sin
2
h 3
¼
0:71
ð
Þ
2 Â11
4 Â 0:0433
¼ 32:015 or a 3 ¼ 5:658 ˚
A
) The lattice parameter a = 5.66 Å
Example 9 In a diffraction experiment, the first reflection from an fcc crystal is
observed at 2h ¼ 84
when the X-ray of wavelength 1.54 Å is used. Determine the
indices of possible reflections and the corresponding interplanar spacing.
Solution Given: 2h ¼ 84
; that is, h ¼ 42
, n = 1, structure is fcc, k = 1.54 Å,
h 1 ; k 1 ; l 1 = ?, d 1 ¼ ?, h 2 ; k 2 ; l 2 = ?, d 2 ¼ ?
We know that the ratios of h
2
þ k
2
þ l
2
À
Á
for allowed reflections in fcc gives
are:
3 : 4 : 8 : 11 : 12 : 16 : 19 : 20
where
3 corresponds to the first reflection from (111) plane
4 corresponds to the second reflection from (200) plane
8 corresponds to the third reflection from (220) plane, and so on
Also, the Bragg’s equation for the first-order reflection is
2d sin h ¼ nk
ðiÞ
or d 111 ¼
k
2 sin h
¼
1:54
2 Â sin 42
¼ 1:15 ˚
A
Further, for cubic crystals, the relationship between d and a is
d hkl ¼
a
h
2 + k
2 + l
2
À
Á 1=2
ðiiÞ
9.1 Steps in Crystal Structure Determinations
349