Physics and Engineering of Radiation Detection

2.3. Interaction of Photons with Matter 89

By adding equations 2.3.11 and 2.3.11 together and using the trigonometric

identity sin^2 θ+cos^2 θ=1weget

p^2 e=p^2 γ 0 +p^2 γ− 2 pγ 0 pγcosθ (2.3.13)

To eliminatepefrom this equation we apply the conservation of total energy,

that is we require that the total energy of the system before and after mo-

mentum must be equal. The electron, being initial at rest, does not have

any kinetic energy and its total energy is only its rest mass energym 0 c^2 .Af-

ter scattering it attains a momentum, which according to special relativity is

related to its energy by the relation

E^2 e=p^2 ec^2 +m^20 c^4.

the energy of a photon in terms of its momentum is given by

Eγ=pγc.

Application of the conservation of energy to the scattering process and using

the above two energy relations gives

Eγ 0 +Ee 0 = Eγ+Ee

⇒pγ 0 c+m 0 c^2 = pγc+

√

p^2 ec^2 +m^20 c^4

⇒p^2 e = p^2 γ 0 +p^2 γ− 2 pγ 0 pγ+2m 0 c(pγ 0 −pγ) (2.3.14)

Equating 2.3.13 with 2.3.14 and some rearrangements yields

pγ 0 −pγ=pγ 0 pγ(1−cosθ)

Usingpγ=h/λin the above equation we get the required relation

λ=λ 0 +

h

m 0 c

(1−cosθ)

In terms incident and scattered photon energies, equation 2.3.10 can be written
as

Eγ=Eγ 0

[

1+

Eγ 0

m 0 c^2

(1−cosθ)

]− 1

, (2.3.15)

where we have used the energy-wavelength relationEγ=hc/λ.
This relation shows that energy of the scattered photon depends not only on
the incident photon energy but also on the scattering angle. In other words the
scattering process is in no way isotropic. We will see later on that this directional
dependence is actually a good thing for spectroscopic purposes. Let us now have a
look at the dependence of scattered photon energy on three extreme angles: 0^0 ,90^0 ,
and 180^0.

Case-1 (θ=0):In this case cosθ= 1 and therefore equation 2.3.15 gives

Eγ=Eγmax=Eγ 0.