Example 5 A transmission Laue pattern of a cubic crystal with a = 3 Å is obtained
with a horizontal X-ray beam in the following orientation. The [010] axis of the
crystal points along the beam away from the X-ray tube, [001] points vertically
upwards and [100] is horizontal and parallel to the X-ray film. The crystal to film
distance is 5 cm. Determine the wavelengths of the radiation diffracted from the
(01 2) planes and the location of the diffraction spot on the X-ray film.
Solution: Given: Three crystallographic axes for beam and film orientation
(Fig. 9.3), a = 3 Å, D = 5 cm, diffraction plane = (01 2), the wavelengths and
location of the diffraction spot on the X-ray film = ?
For a cubic system, the relationship between the interlayer spacing and the lattice
parameter is
d 01 2 ¼
a
h
2
þ k
2
þ l
2
À
Á 1=2 ¼
a
0 2 þ 1 2 þ 2 2
ð
Þ
1=2
¼
3
ffiffi ffi
5
p ¼ 1:34 ˚
A
Now from the figure, the angle between the directions [010] and [021] is
cos h ¼
0 þ 2 þ 0
1: 0 þ 4 þ 1
ð
Þ
1=2
¼
2
ffiffi ffi
5
p
or h ¼ cos
À1
2
ffiffi ffi
5
p
¼ 26:56
From Bragg’s law, we have
for n ¼ 1; k ¼ 2d 01 2 sin h ¼ 2 Â 1:34 Â sin 26:56 ¼ 1:20 ˚
A
Similarly,
for n ¼ 2; 2k ¼ 2d 01 2 sin h ¼ 2 Â 1:34 Â sin 26:56 ¼ 1:20 ˚
A; and k ¼ 0:60 ˚
A
for n ¼ 3; 3k ¼ 2d 01 2 sin h ¼ 2 Â 1:34 Â sin 26:56 ¼ 1:20 ˚
A; and k ¼ 0:40 ˚
A
for n ¼ 4; 4k ¼ 2d 01 2 sin h ¼ 2 Â 1:34 Â sin 26:56 ¼ 1:20 ˚
A; and k ¼ 0:30 ˚
A
Fig. 9.3 Crystal axes and
X-ray beam
9.1 Steps in Crystal Structure Determinations
359