Physics and Engineering of Radiation Detection

2.3. Interaction of Photons with Matter 91

Case-3 (θ= 180^0 ):It is obvious from equation 2.3.15 that a photon scattered

atθ= 180^0 will carry the minimum possible energy (since 1−cosθis maximum

atθ= 180^0 ). In this case, substituting 1−cosθ= 2 in equation 2.3.15 yields

Eγmin=Eγ 0

[

1+

2 Eγ 0

m 0 c^2

]− 1

. (2.3.16)

This equation can also be written as

Eγmin=

m 0 c^2

2

[

1+

m 0 c^2

2 Eγ 0

]− 1

. (2.3.17)

To obtain a numerical result independent of the incident photon energy, let us

assume that the incident photon energy is much higher than half the electron

rest energy, such asEγ m 0 c^2 /2. There is nothing magic abouthalfthe

electron rest energy. It’s been chosen because we want to eliminate the term

containingm 0 c^2 / 2 Eγ0. In this situation the above equation reduces to

Eγmin ≈

m 0 c^2

2

= 255keV.

This is a very interesting result because it tells us that the electron will carry

the maximum energy it could at any angle. This process resembles the simple

head-on collision of two point masses in which the incident body completely

reverses its motion and the target body starts moving forward. To determine

the energy of the electron, we assume thatmostof the energy of the incident

photon is distributed between the scattered photon and the electron. Hence

the maximum energy of the scattered electron can be calculated from

Emaxe ≈Eγ− 255 keV.

This implies that in aγ-ray spectroscopy experiment one should see a peak at

the energyEγ− 255 keV. Such a peak is actually observed and is so promi-

nent that it has gotten a name of its own: the Compton edge(see Fig.2.3.7).

A consequence of this observation is that our assumption that even though

the scattering is inelastic, however the energy imparted to the atom is not

significantly high.

It is also instructive to plot the dependence of change in wavelength of the photon
on the scattering angle. Fig.2.3.6 shows such a plot spanning the full 360^0 around the
target electron. As expected, the largest change in wavelength occurs atθ= 180^0 ,
which corresponds to the case-3 we discussed above. The photon scattered at this
angle carries the minimum possible energyEγminas allowed by the Compton process.
A fair question to ask at this point ishow can the energy that the electron car-
ries with it be calculated? Though at first sight it may seem trivial to answer this
question, the reality is that theinelasticbehavior of this scattering process makes
it a bit harder than merely subtracting the scattered photon energy from the inci-
dent photon energy. The reason is that if the electron is in an atomic orbit before