Klein-Nishina Formula for Polarized Photons
The inclusion of dynamics, which allows us to determine the relative number of photons scattered at each angle, is fairly complicated. The quantum-mechanical result is known as the Klein-Nishina formula (Attix 1986). The result depends on the polarization of the photons. For unpolarized photons, the cross section per unit solid angle for a photon to be scattered at angle θ is
where
is the classical radius of the electron. [The variable x is the ratio of the incident photon’s energy to the rest energy of the electron.]
What happens for polarized photons? In that case, the scattering may depend on the angle φ with respect to the direction of the electric field. The resulting scattering formula is
In order to analyze polarized photons, we must consider the two polarization states, φ = 0 and φ = 90°.
φ = 0
The incident and scattered photon directions define a plane. Assume the electric field associated with the incident photon lies in this plane, as shown in the drawing below. From a classical point of view, the electric field will cause the electron to oscillate, resulting in dipole radiation (a process called Thomson scattering). A dipole radiates perpendicular to its direction of oscillation, but not parallel to it. Therefore, you get scattering for θ = 0 and 180°, but not for θ = 90°.
A schematic diagram of Compton scattering for polarized light with φ = 0. |
φ = 90°
For Thomson scattering, if the electric field oscillates perpendicular to the scattering plane (shown below) then all angles θ are perpendicular to the dipole and therefore should radiate equally. This effect is also evident in a quantum analysis unless x >> 1.
A schematic diagram of Compton scattering for polarized light with φ = 90°. |
Source: http://hobbieroth.blogspot.com/2021/09/klein-nishina-formula-for-polarized.html
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