Physics and Engineering of Radiation Detection

92 Chapter 2. Interaction of Radiation with Matter

θ (deg)

0 50 100 150 200 250 300 350

(fm)λ

∆

0

1

2

3

4

5

Figure 2.3.6: Angular distribution of

change in wavelength of a Compton-

scattered photon.

collision then some of the energy may also go into exciting the atom. A part of this
energy, which is equal to the binding energy of the electron, goes into helping the
electron break the potential barrier of the atom and get scattered. The remaining
excess energy may not be large enough for the atom to be emitted as some sort
of de-excitation process. To make life simple though, we can always assume that
this energy is very small as compared to the energy carried away by the photon.
In this case the energy carried away by the scattered electron can be estimated by
subtracting the scattered photon energyEγand the atomic binding energyEbof
the electron from the incident photon energy.

Ee≈Eγ 0 −Eγ−Eb=

E^2 γ 0

mc^2

[

1 −cosθ

1+mcEγ 20 (1−cosθ)

]

−Eb (2.3.18)

Another simplification to this equation can be made by noting that the binding
energy of electrons in low to moderateZelements is several orders of magnitude
smaller than the energy of theγ-ray photons emitted by most sources. In this case
one can simply ignore the termEbin the above equation and estimate the energy
of the electron from

Ee≈Eγ 0 −Eγ=

Eγ^20

mc^2

[

1 −cosθ

1+Emcγ 20 (1−cosθ)

]

. (2.3.19)

Up until now we have not said anything about the dependence of the cross section
on the scattering angleθ. Let us do that now. The differential cross section for
Compton scattering can be fairly accurately calculated from the so called Klein-
Nishina formula

dσc

dΩ

=

r^20

2

[

1+cos^2 θ

(1 +α(1−cosθ))^2

][

1+

4 α^2 sin^4 (θ/2)

(1 + cos^2 θ){1+α(1−cosθ)}

]

. (2.3.20)