Compton Scattering

In the Compton effect, a γ-ray photon of wavelength λ strikes a free, but initially stationary, electron of mass m. The photon is scattered an angle θ, and its scattered wavelength is  The electron recoils at an angle φ (see Figure 1.1).

 Figure 1.1

 a) Write the relativistic equations for momentum and energy conservation.

b) Find an expression for the change  in the photon wavelength for the special case θ = π/2.

SOLUTION

a) From momentum and energy conservation we can write

(1)

where  are the momenta and energies of the photon before and after the scattering, respectively,  are the final momentum and energies of the electron, and ε_e is its initial energy. We have for the electron

and for the photon

So we can rewrite (1) in the form

(2)

b) To solve these equations we can express the momentum of the recoil electron p_e in two ways

(3)

from (2).

and for a special case θ = π/2, cosθ = 0. We have

Dividing this equation by  we get

Taking into account that p = h/λ, we obtain the final result: