Physics and Engineering of Radiation Detection

106 Chapter 2. Interaction of Radiation with Matter

few deflected at very large angles. This experiment proved that most of the space
in the atom is empty and the positive charge is concentrated into a small space,
which we now call nucleus. This pioneering work by Rutherford changed the way
the atoms are visualized forever.

θ

Detector

Gold Foil

Collimator

Source

To Vaccuum

Pump

Detector

Output

Figure 2.4.1: A simple setup to determine the angular distri-

bution of Rutherford scattering.

Rutherford scattering is sometimes referred to as Coulomb scattering since the
Coulomb force between the incident particle and the target nucleus is responsible for
the interaction. A simplified diagram of a Rutherford scattering experiment is shown
in Fig.2.4.1. A very thin gold foil is bombarded byα-particles from a radioactive
source, such as Americium-241. The scattered particles are detected by a particle
detector, which can be rotated at different angles. Such an experiment yields the
following dependence of the number of scattered particlesNsto the scattering angle
θ.

Ns∝

1

sin^4 (θ/2)

(2.4.1)

This relationship is plotted in Fig.2.4.2 forθ=0toθ= 180^0. It is apparent that
most of the incident particles pass through the target undeflected, hinting to the
vast emptiness of atomic space.
The differential cross section of this process is given by the so calledRutherford
formula
dσ
dΩ

=

Zi^2 Zt^2 re^2

4

[

mec

βp

] 2

1

sin^4 (θ/2)

. (2.4.2)

HereZiandZtare the atomic numbers of the incident and target particles respec-
tively andθis the scattering angle. The impact parameterband the scattering
angleθare as shown in Fig.2.4.3. These and other factors in the above definition of
the differential cross section can be calculated from the following relations.

p =

2 Z 1 Z 2 e^2

βcbθ

b = rsinφ

β =

(

1 −

v^2

c^2

) 1 / 2