where k is the incident wave vector and G is the reciprocal lattice vector
(Fig. 7.6).
(c) The Laue equations are more general as compared to Bragg’s equation. They
are derived on the basis of a simple static atomic model of a crystal. Each row
of atoms of a one-dimensional crystal gives rise to diffraction cones of various
orders governed by the Laue equation
aðcos aÀ cos a 0 Þ ¼ ek
where a 0 and a are incident and diffracted angles with respect to a row of atoms
and e is the order of diffraction. In a similar manner, for a space lattice three
such equations (for three crystallographic axes) can be obtained. They are:
aðcos a À cos a 0 Þ ¼ ek
bðcos b À cos b 0 Þ ¼ fk
cðcos c À cos c 0 Þ ¼ gk
ð7:3Þ
where b 0 ; b; c 0 ; c; f and g have the same meaning for other rows of atoms:
Solved Examples
Example 1 When a crystal is subjected to a monochromatic X-ray beam, the
first-order diffraction is observed at an angle of 15°. Determine the angles for
second and third orders when the same X-ray beam is used.
Solution: Given: n = 1, h 1 ¼ 15
; k = fixed, h 2 ¼ ?, and h 3 ¼ ?
For first-order diffraction, the Bragg’s equation is
2d sin h 1 ¼k
or
k
d
¼ 2 sin h 1 ¼ 2 sin 15
¼ 0:518
For second-order diffraction, n = 2, the Bragg’s equation becomes
d sinh 2 ¼k
or sinh 2 ¼
k
d
¼ 0.518
or h 2 ¼sin
À1 kk
d
¼sin
À1 0.518
ð
Þ¼31.2
Similarly, for third-order diffraction, n = 3, the Bragg’s equation becomes
274
7 Diffraction of Waves and Particles by Crystal