# Klein-Nishina Formula for Polarized Photons

In Chapter 15 of *Intermediate Physics for Medicine and Biology*, Russ Hobbie and I discuss Compton Scattering. An incident photon scatters off a free electron, producing a scattered photon and a recoiling electron. We write

The inclusion of dynamics, which allows us to determine the relative number of photons scattered at each angle, is fairly complicated. Thequantum-mechanicalresult is known as theKlein-Nishinaformula(Attix 1986). The result depends on thepolarizationof the photons. For unpolarized photons, thecross sectionper unitsolid anglefor a photon to be scattered at angleθis

where

is theclassical radiusof the electron. [The variablexis the ratio of the incident photon’s energy to therest energyof 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

Unpolarized light means that you average over all angles *φ*, implying that factors of cos^2*φ* become ½. A bit of algebra should convince you that when the expression above is averaged over *φ* it’s equivalent to Eq. 15.16 in *IPMB*.

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 quantum-mechanical analysis of this behavior (Compton scattering) accounts for the momentum of the incident photon and the recoil of the electron. In the quantum case, some scattering occurs at *θ* = 90°, but it is suppressed unless the energy of the incident photon is much greater than the rest mass of the electron ( *x* >> 1).

## 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.

The figure below is similar to Fig. 15.6 in *IPMB*. The thick, solid lines indicate the amount of scattering (the differential cross section) for unpolarized light, as functions of *θ*. The thin dashed curves show the scattering for *φ* = 0 and the thin dash-dot curves show it for *φ* = 90°. The red curves are for a 10 keV photon, whose energy is much less than the 511 keV rest energy of an electron ( *x* << 1). The behavior is close to that of Thomson scattering. The light blue curves are for a 1 GeV photon (1,000,000 keV). For such a high energy ( *x* >> 1) almost all the energy goes to the recoiling electron, with little to the scattered photon. The dashed and dash-dot curves are present, but they overlap with the solid curve and are not distinguishable from it. Polarization makes little difference at high energies.

Why is there so little backscattering (*θ* = 180°) for high energy photons? It’s because the photon has too much momentum to have its direction reversed by a light electron. It would be like a truck colliding with a mosquito, and after the collision the truck recoils backwards. That’s extraordinarily unlikely. We all know what will happen: the truck will barrel on through with little change to its direction. Any scattering occurs at small angles.

Notice that Thomson scattering treats light as a wave and predicts what an oscillating electric field will do to an electron. Compton scattering treats light as a photon having energy and momentum, which interacts with an electron like two colliding billiard balls. That is wave-particle duality, which is at the heart of a quantum view of the world. Who says *IPMB* doesn’t do quantum mechanics?

*Originally published at **http://hobbieroth.blogspot.com**.*