and so on. Thus, the incident beam contains these wavelengths. Now, from the Laue
arrangement, we obtain
r ¼ D tan 2h ¼ 5 tan 52:12 ¼ 6:66 cm
Example 6 A back-reflection Laue pattern of an aluminum crystal (fcc) with
a = 4.05 Å is obtained from (111) planes making an angle of 85° with the incident
X-ray beam. Assuming that the wavelengths above 2.5 Å are extremely weak and
they get absorbed in air. What orders of (111) reflection are present if the tube
voltage is 60 kV ?
Solution: Given: Lattice parameter a = 4.05 Å, diffraction plane (111), Bragg’s
angle h 111 ¼ 85
; V = 60 kV, orders of (111) reflection possible = ?
Minimum wavelength of the X-ray produced at the given electrode potential is
given by
k min ¼
hc
eV
¼
6:626 Â 10
À34
 3  10
8
1:6 Â 10 À19 Â 6 Â 10 4
¼ 2:1 Â 10
À11 m ¼ 0:21 ˚
A
For a cubic system, the relationship between the interlayer spacing and the lattice
parameter is
d 111 ¼
a
h
2
þ k
2
þ l
2
À
Á 1=2 ¼
a
1 2 þ 1 2 þ 2 2
ð
Þ
1=2
¼
4:05
ffiffi ffi
3
p ¼ 2:34 ˚
A
Further, from Bragg’s equation, we have
k ¼ 2d 200 sin h
for n ¼ 1; sin h 1 ¼
k min
2d 111
¼
0:21
2 Â 2:34
; and h 1 ¼ sin
À1 0:21
4:68
¼ 2:57
Similarly,
for n ¼ 10; sin h 10 ¼
k min
2d 111
¼
10 Â 0:21
2 Â 2:34
; and h 1 ¼ sin
À1 2:1
4:68
¼ 26:66
for n ¼ 20; sin h 20 ¼
k min
2d 111
¼
20 Â 0:21
2 Â 2:34
; and h 1 ¼ sin
À1 4:2
4:68
¼ 63:82
for n ¼ 22; sin h 22 ¼
k min
2d 111
¼
22 Â 0:21
2 Â 2:34
; and h 1 ¼ sin
À1 4:62
4:68
¼ 80:82
360
9 Determination of Crystal Structure Parameters