Compton scattering

Compton scattering is a particle scattering phenomenon between a Photon and an electron that involves the absorption and re-emission of a photon by the electron, deflecting it and changing its frequency. It essentially redirects electromagnetic radiation. It is a higher energy form of Thomson scattering.

Mechanism

In the language of particle physics, Compton scattering reads

γ+e−→γ+e−\gamma+ e^{-} \rightarrow \gamma +e^{-}

It is an elastic scattering process. Graphically it looks like this:

center

Before the scattering, the photon carries relativistic energy E0E_{0} and the electron only has its rest energy. The process can be explained starting from four-momentum conservation, which is valid in elastic scatterings. Elaboration through relativistic energy yields a final photon energy of

Eγ=11−cos⁡θmec2+1E0E_{\gamma}=\dfrac{1}{\dfrac{1-\cos\theta}{m_{e}c^{2}}+ \dfrac{1}{E_{0}}}

Applying Planck's formula Eγ=hν=hc/λE_{\gamma}=h\nu=hc/\lambda yields

hcλ=11−cos⁡θmec2+1E0\frac{hc}{\lambda}=\dfrac{1}{\dfrac{1-\cos\theta}{m_{e}c^{2}}+ \dfrac{1}{E_{0}}}

(λ\lambda is the final wavelength). Rearranging to extract λ\lambda and recognizing that E0=hc/λ0E_{0}=hc/\lambda_{0} where λ0\lambda_{0} is the starting wavelength yields

λ=hc(1−cos⁡θmec2+λ0hc)=λ0+hmec(1−cos⁡θ)\lambda=hc\left( \frac{1-\cos\theta}{m_{e}c^{2}}+ \frac{\lambda_{0}}{hc} \right)=\lambda_{0}+ \frac{h}{m_{e}c}(1-\cos \theta)

We define

λC=hmec\lambda_{C}=\frac{h}{m_{e}c}

which is known as the Compton wavelength (in this case of the electron). Then, the final wavelength depends on the starting one according to

λ=λ0+λC(1−cos⁡θ)\boxed{\lambda=\lambda_{0}+ \lambda_{C}(1-\cos \theta)}

The second term is an angular rescaling of the Compton wavelength based on the angle of scattering, meaning that there is a minimum and maximum variation that Compton scattering can do. Specifically:

  • If θ=0\theta=0, the electron continues straight and there is no change (cos⁡θ=1\cos \theta=1).
  • If θ=π\theta=\pi, the electron bounces back right where it came from and the change is maximized (cos⁡θ=−1\cos \theta=-1), being equal to 2λC2\lambda_{C}.

The Compton wavelength is an interesting quantity energetically speaking. The wavelength variation λ−λ0\lambda-\lambda_{0} is most important when λ0≃λC\lambda_{0}\simeq \lambda_{C}. The energy of a photon of wavelength λC\lambda_{C} is

EC=hν=hcλC=hchmec=mec2E_{C}=h\nu=\frac{hc}{\lambda_{C}}=\frac{hc}{h}m_{e}c=m_{e}c^{2}

which is precisely the rest energy of the electron. Evidently, the energy transfer is most efficient when the photon matches the energy of the electron (a resonance phenomenon).

By using energy conservation, it also follows that the energy transferred to the electron is:

Ee=EγEγmec2(1−cos⁡θ)1+Eγmec2(1−cos⁡θ)E_{e}=E_{\gamma} \dfrac{\dfrac{E_{\gamma}}{m_{e}c^{2}}(1-\cos\theta)}{1+\dfrac{E_{\gamma}}{m_{e}c^{2}}(1-\cos\theta)}

which, like the wavelength, varies between 0 and a maximum value when θ=π\theta=\pi, which is

Ee,max=Eγ2Eγmec21+2Eγmec2<EγE_{e,\text{max}}=E_{\gamma} \dfrac{\dfrac{2E_{\gamma}}{m_{e}c^{2}}}{1+\dfrac{2E_{\gamma}}{m_{e}c^{2}}}<E_{\gamma}

This is known as the Compton edge and, interestingly, it's guaranteed to be less than the photon's energy. But it must be so, because this is an elastic scattering event: photons have no mass, so if they were to transfer all their energy they'd just disappear and we'd be talking about the photoelectric effect instead. We know that the photoelectric effect can't happen in the vacuum, so there's no way Compton scattering can lead to complete energy transfer.

Finally, we know that the cross section of Compton scattering decreases with energy, meaning lower-energy photons are favored.

Compton scattering cross section is also known to go like ∼Z\sim Z, meaning that if the electron is bound to a nucleus, the larger the more likely Compton scattering is, linearly so to be specific.

In detectors

Unlike the photoelectric effect, Compton scattering is of limited utility for detectors. Since photons aren't detectable by themselves due to being electrically neutral, we need to want to use some other particle (an electron, say) as a proxy for its energy. In principle, since we know the relation between EeE_{e} and EγE_{\gamma}, we could measure the former and the calculate the latter. But there's two big problems:

  1. Unlike the photoelectric effect, which is entirely deterministic, Compton scattering is entirely random. The scattering angle is unpredictable and since the energy relation is dependent on it, we can't reliably use it. For a single electron, the energy can be anywhere between 0 and Ee,maxE_{e,\text{max}}, so we don't get any real information out of it besides a tenous lower bound, since Eγ>Ee,maxE_{\gamma}>E_{e,\text{max}}.
  2. Even if we try to measure a large number of photons to get a Probability distribution, the photons would have to all be the same energy in order to create electrons with the same energy distributions, which isn't terribly realistic. Still, this is a viable strategy if nothing better is available, as sampling the entire distribution gives us a better estimate of Ee,maxE_{e,\text{max}} and therefore a better lower bound for EγE_{\gamma}.

Inverse mechanism

It is sometimes possible that the energy exchange takes place in reverse, i.e. that a high-energy electron transfers energy to an impinging photon with less energy. This phenomenon is called inverse Compton scattering. It is rare in compact matter since electrons are confined to atoms, but under more extreme conditions, such as the accretion disk of a black hole, it is possible for a relativistic free electron to impinge on a photon with less energy, thus causing inverse Compton scattering.