Physics and Engineering of Radiation Detection

88 Chapter 2. Interaction of Radiation with Matter

λ 0

Incident

Photon

Scattered Electron

λ

θ

φ

Scattered Photon

Figure 2.3.4: Compton scattering of a photon having energyEγ 0 =hc/λ 0

from a bound electron. Some of the energy of the incident photon goes into

knocking the orbital electron out of its orbit.

Example:

Derive equation 2.3.10.

Solution:

For this derivation we will assume that the electron is not only at rest before

the collision but is also not under the influence of any other potential. In other

words it is free to move around. As we saw before, these two assumption are

valid up to a good approximation even for electrons bound in an atom provided

the incident photon has high enough energy.

The scattering process is depicted in Fig.2.3.5 where the momenta before

and after collision have been broken down in their respective horizontal and

vertical components. This will aid us in taking thevector sumof the momenta

while applying the law of conservation of momentum.

Since electron is supposed to be at rest before the collision, thereforepe 0 =0

and the total momentum in horizontal direction before collision is simply pho-

ton’s momentumpγ 0. After the scattering there are two horizontal momenta

corresponding to both the electron and the photon. Application of conserva-

tion of momentum in horizontal direction then gives

pγ 0 =pγcosθ+pecosφ.

Rearranging and squaring of this equation gives

p^2 γcos^2 θ=(pγ 0 −pecosφ)^2 (2.3.11)

To apply the conservation of momentum in the vertical direction, we note that

before scattering there is no momentum in vertical direction. Hence we have

0=pγsinθ−pesinφ,

where the negative sign simply shows that the two momenta are in opposite

direction to each other. Rearranging and squaring of this equation gives

p^2 γsin^2 θ=p^2 esin^2 φ (2.3.12)