Page 116 - Materials Science and Engineering An Introduction

88 • Chapter 3 / The Structure of Crystalline Solids

in such a way that they traverse different paths. The phase relationship between the
scattered waves, which depends upon the difference in path length, is important.
One possibility results when this path length difference is an integral number of
wavelengths. As noted in Figure 3.21a, these scattered waves (now labeled 1 and 2 )
are still in phase. They are said to mutually reinforce (or constructively interfere
with) one another; when amplitudes are added, the wave shown on the right side of
diffraction the figure results. This is a manifestation of diffraction, and we refer to a diffracted
beam as one composed of a large number of scattered waves that mutually reinforce
one another.
Other phase relationships are possible between scattered waves that will not
lead to this mutual reinforcement. The other extreme is that demonstrated in
Figure 3.21b, in which the path length difference after scattering is some integral
number of half-wavelengths. The scattered waves are out of phase—that is, corre-
sponding amplitudes cancel or annul one another, or destructively interfere (i.e., the
resultant wave has zero amplitude), as indicated on the right side of the figure. Of
course, phase relationships intermediate between these two extremes exist, resulting
in only partial reinforcement.

X-Ray Diffraction and Bragg’s Law
X-rays are a form of electromagnetic radiation that have high energies and short
wavelengths—wavelengths on the order of the atomic spacings for solids. When a beam
of x-rays impinges on a solid material, a portion of this beam is scattered in all direc-
tions by the electrons associated with each atom or ion that lies within the beam’s path.
Let us now examine the necessary conditions for diffraction of x-rays by a periodic
arrangement of atoms.
Consider the two parallel planes of atoms A–A and B–B in Figure 3.22, which
have the same h, k, and l Miller indices and are separated by the interplanar spacing d hkl .
Now assume that a parallel, monochromatic, and coherent (in-phase) beam of x-rays
of wavelength l is incident on these two planes at an angle u. Two rays in this beam,
labeled 1 and 2, are scattered by atoms P and Q. Constructive interference of the scat-
tered rays 1 and 2 occurs also at an angle u to the planes if the path length difference
between 1–P–1 and 2–Q–2 (i.e., SQ + QT) is equal to a whole number, n, of wave-
lengths—that is, the condition for diffraction is

nl = SQ + QT (3.20)

Figure 3.22 Diffraction of 1 1'
x-rays by planes of atoms Incident Diffracted
(A–A and B–B ). beam beam
2 2'

P

A A'
d hkl
S T
B B'
Q