2d sin h 3 ¼ 3k
or sin h 3 ¼
3k
2d
¼
3
2
 0:518 ¼ 0:777
or h 3 ¼ sin
À1 3k
2d
¼ sin
À1 0:777
ð
Þ¼51
Example 2 When a Cu crystal is subjected to an X-ray beam the first-order
(111) plane appears at an angle of 21.7°. The lattice parameter of copper (fcc) is
found to be 3.61 Å. Find the wavelength of the X-ray used.
Solution: Given: Crystal is fcc, a ¼ 3:61 ˚
A, n = 1, plane 111
ð
Þ; h 1 ¼ 21:7
;
k = ?
For a cubic crystal the interplanar spacing is given by
d =
a
h
2 + k
2 + l
2
À
Á 1=2 ¼
3:61
ffiffi ffi
3
p
Further, the first-order Bragg’s equation is
2d sin h 1 ¼ k
or k ¼ 2d sin h 1 ¼
2 Â 3:61 Â sin 21:7
ffiffi ffi
3
p
¼ 1:54 ˚
A
Example 3 When a nickel crystal (fcc) is exposed to an X-ray of wavelength
1.54 Å, Bragg’s angle corresponding to (220) reflection is 38.2°. Determine its
lattice parameter.
Solution: Given: hkl
ð Þ 220
ð
Þ; structure is fcc, h ¼ 38:2
; k ¼ 1:54 ˚
A; a ¼ ?
For a cubic crystal the interplanar spacing is given by
d =
a
h
2 + k
2 + l
2
À
Á 1=2 =
a
2
2 + 2
2 + 0
2
À
Á 1=2 =
a
2
ffiffi ffi
2
p
Further, from Bragg’s equation, we have
2d hkl sin h ¼ k
or 2d hkl ¼
k
sin h
or
2a
2
ffiffi ffi
2
p ¼
1:54
sin 38:2
or a ¼
ffiffi ffi
2
p  1:54
sin 38:2
¼ 3:52 ˚
A
7.2 X-Ray Diffraction by Crystals
275