Page 38 - Modern physical chemistry

2.4 Diffraction by Randomly Oriented Molecules 27

The beam has a wavelength determined by the momentum of a typical particle therein.
This in turn is determined by the kinetic energy of the particle. The jet scatters the beam
into a characteristic diffraction pattern.
Consider the scheme in figure 2.1. A potential between 30 and 70 kilovolts acceler-
ates the electrons emitted by a hot filament. The electrons travel in a vacuum tube to a
vapor stream containing the molecules to be studied. These diffract the beam. The result-
ing pattern is recorded on a photographic plate. Pertinent measurements are then made.
In analyzing these, we consider each atom in a target molecule to act as a scattering point.
Since molecular distances are very small compared to the dimensions of the apparatus, we
also consider the incident rays and the rays scattered at a certain angle by a molecule to be
parallel. For simplicity, we also limit ourselves initially to diatomic molecules, those of type
AB. A particular orientation of a scattering molecule then acts as shown in figure 2.2.
A reference Cartesian system is erected on atom A as shown. The corresponding
spherical coordinates of atom B are (r, a ,[3). Line AC is drawn perpendicular to the inci-
dent rays; line BD perpendicular to the scattered rays. Then points A and C would be at
the same phase. Also points B and D. But the ray scattered by A travels the extra length
(which may be either positive or negative)
£5 =AD-CB. [2.19]
From the figure and trigonometric definitions, we find that
AD = rsin a cos r = rsin a sin(,B + e), [2.20]

so
CB=rsinasin,B, [2.21]
£5 = r sin a[ Sin(,B + e) - Sin,B] = 2r sin ± sin a cos(,B + ± J [2.22]
e
e

In the coherent approximation, the electron wave has the amplitude
[2.23]

where N is a normalization constant, s the distance traveled, and t the time. Parameter
k is the wavevector
k = 27r [2.24]
).,'

To Vacuum
I
Photographic Plate

FIGORE:U Setup tor
studying' Ute diffraction of
electrol1$ .Pv randomly
orientedrnolecu1es·,