# Compton Scattering

Compton scattering is the inelastic scattering of a photon off of an electron. This is the quantum mechanical or high-energy extension to Thomson Scattering.

If the photon loses energy, its wavelength will increase and this is called Compton Scattering. If the electron has sufficient initial kinetic energy, the photon the photon can gain energy, this is called Inverse Compton Scattering, or “Compton Up-Scattering”. An alteration of the photon spectrum (either up-scattering or down-scattering) due to interactions with electrons is called “comptonization”

Some examples of applications of Compton scattering are:

• Compton exchange keeps electrons in thermal equilibrium with photons at redshifts ${\displaystyle z\geq 10^{3}}$.
• The spectra of AGN and xray binaries are altered by Compton Scattering (e.g. radio emission to optical wavelengths).
• CMB photons get upscattered by galaxy cluster plasma. This is called the Sunyaev-Zeldovich effect.
• Inverse Compton scattering has been proposed as a likely emission mechanism for gamma ray bursts.

The Compton effect was first observed in the 1920’s and Arthur Holly Comton was awarded the 1927 Nobel prize for the discovery. The effect is often regarded as some of the first concrete experimental evidence for quantum mechanics.

## 1 Mathematical Derivation of Compton Scattering

To find the final energy of a scattered photon and the Compton shift (the change in the photon wavelength), we use the conservation of energy and momentum.

In the standard derivation of Compton scattering, the electron is assumed to be free and at rest. This is a good approximation considering the photon energies for which this process is significant are much larger than relevant electron binding energies.

Compton scattering geometry.

A photon with initial energy ${\displaystyle E_{\gamma _{i}}=h\nu _{i}}$ travelling in the ${\displaystyle {\hat {x}}}$ direction scatters of an electron at rest (${\displaystyle E_{e_{i}}=m_{e}c^{2}}$). After the scatter, the photon has energy ${\displaystyle E_{\gamma _{f}}=h\nu _{f}}$ and is travelling at an angle ${\displaystyle \theta }$ relative to the original ${\displaystyle {\hat {x}}}$ direction. The electron has energy given by ${\displaystyle E_{e_{f}}={\sqrt {p_{e}^{2}c^{2}+m_{e}^{2}c^{4}}}}$ and has scattered at a different angle. See figure below for a depiction of the relevant variables.

The conservation of energy tells us

{\displaystyle {\begin{aligned}E_{\gamma _{i}}+E_{e_{i}}&=E_{\gamma _{f}}+E_{e_{f}}\\E_{\gamma _{i}}+m_{e}c^{2}&={\sqrt {p_{e}^{2}c^{2}+m_{e}^{2}c^{4}}}+E_{\gamma _{f}}\end{aligned}}\,\!}

Rearranging and squaring both sides gives

${\displaystyle (E_{\gamma _{i}}-E_{\gamma _{f}}+m_{e}c^{2})^{2}=p_{e}^{2}c^{2}+m_{e}^{2}c^{4}.\,\!}$

We will use the conservation of momentum to write the ${\displaystyle p_{e}^{2}c^{2}}$ factor in terms of the photon energies, so we will come back to this equation.

The conservation of momentum tells us

${\displaystyle {\vec {p_{\gamma _{i}}}}={\vec {p_{\gamma _{f}}}}+{\vec {p_{e_{f}}}}\,\!}$
${\displaystyle ({\vec {p_{\gamma _{i}}}}-{\vec {p_{\gamma _{f}}}})^{2}={\vec {p_{e_{f}}}}^{2}\,\!}$
${\displaystyle p_{\gamma _{i}}^{2}+p_{\gamma _{f}}^{2}-2p_{\gamma _{i}}p_{\gamma _{f}}\cos \theta =p_{e_{f}}^{2}\,\!}$

where ${\displaystyle \theta }$ is the angle between the initial photon direction ${\displaystyle {\hat {x}}}$ and the final photon direction.

Multiplying the above equation by ${\displaystyle c^{2}}$ and using the relation ${\displaystyle E_{\gamma }=p_{\gamma }c}$, we can rewrite it as

${\displaystyle E_{\gamma _{i}}^{2}+E_{\gamma _{f}}^{2}-2E_{\gamma _{i}}E_{\gamma _{f}}\cos \theta =p_{e}^{2}c^{2}.\,\!}$

Now, we can plug this into our final expression for the conservation of energy above and expand the squared brackets on the left side at the same time:

${\displaystyle {E_{\gamma _{i}}^{2}}+{E_{\gamma _{f}}^{2}}+{m_{e}^{2}c^{4}}+{2}E_{\gamma _{i}}m_{e}c^{2}-{2}E_{\gamma _{f}}m_{e}c^{2}-{2}E_{\gamma _{i}}E_{\gamma _{f}}={E_{\gamma _{i}}^{2}}+{E_{\gamma _{f}}^{2}}-{2}E_{\gamma _{i}}E_{\gamma _{f}}\cos \theta +{m_{e}^{2}c^{4}}.\,\!}$

Cancelling off similar terms on both sides gives us

${\displaystyle E_{\gamma _{i}}m_{e}c^{2}-E_{\gamma _{f}}m_{e}c^{2}-E_{\gamma _{i}}E_{\gamma _{f}}=-E_{\gamma _{i}}E_{\gamma _{f}}\cos \theta .\,\!}$

Solving for the final photon energy gives

${\displaystyle E_{\gamma _{f}}={\frac {E_{\gamma _{i}}}{1+{\frac {E_{\gamma _{i}}}{m_{e}c^{2}}}(1-\cos \theta )}},\,\!}$

and the change in the photon wavelength, referred to as the Compton shift, is found to be

${\displaystyle {\frac {1}{E_{\gamma _{f}}}}-{\frac {1}{E_{\gamma _{i}}}}={\frac {1}{m_{e}c^{2}}}(1-\cos \theta )\,\!}$
${\displaystyle \lambda _{f}-\lambda _{i}={\frac {h}{m_{e}c}}(1-\cos \theta )\,\!}$

The prefactor ${\displaystyle \lambda ={\frac {h}{m_{e}c}}}$ is called the Compton wavelength. ${\displaystyle \lambda _{c}=0.02}$ A gives the order of change in wavelength during a Compton scatter interaction; therefore, Compton scattering is only relevant for high energy (low wavelength) photons.

• ${\displaystyle \lambda _{1}-\lambda >0}$: the shift here is tiny. The maximum possible value is ${\displaystyle \lambda _{1}-\lambda =2hm_{e}c=0.04}$A. The momentum tends to be shared between the electron and photon, but no so much the energy.
• Remember this is scattering, not absorption. Photon # is conserved.
• If ${\displaystyle \lambda _{i}\gg \lambda _{c}}$ (or ${\displaystyle h\nu \ll m_{e}c^{2}}$), the scattering can be approximated as elastic (Thomson scattering) and ${\displaystyle \Delta E_{\gamma }=0}$.
• The derivation for the cross section of Compton scattering, which is given by the Klein-Nishina formula, is “outside of the scope of [R & L]”. An important thing to know is that the scattering angle is anisotropic and depends on energy. In the limit that ${\displaystyle h\nu \gg m_{e}c^{2}}$, then ${\displaystyle \sigma \sim \sigma _{T}\left({\frac {m_{e}c^{2}}{h\nu }}\right)}$, and this additional term is called the Klein-Nishina correction.

## 2 Compton Scattering for a Moving Electron

What we’ve done so far was for a stationary electron. For a moving electron, we need to consider the dependence on the angle at which the photon is coming in with respect to the direction of velocity (${\displaystyle \theta }$). To make this situation similar to the one we just considered, we need to be in the frame of the electron. In this frame, the photons has a new energy as a result of time dilation:

${\displaystyle E^{\prime }=\gamma E(1-{v \over c}\cos \theta )\,\!}$

The ${\displaystyle v \over c}$ term is just the classic Doppler shift of the photon. Now suppose that the photon (which entered at angle ${\displaystyle \theta ^{\prime }}$ in the electron frame) rebounds at an angle ${\displaystyle \theta _{1}^{\prime }}$. To relate this to our previous derivation, we want to find ${\displaystyle \phi }$. So note that ${\displaystyle \theta _{1}^{\prime }-\phi ^{\prime }=\theta ^{\prime }}$. Thus, in the electron’s frame:

${\displaystyle E_{1}^{\prime }={E^{\prime } \over 1+{E^{\prime } \over m_{e}c^{2}}(1-\cos \phi ^{\prime })}\,\!}$

Transforming this back into the lab frame:

${\displaystyle E_{1}=E_{1}^{\prime }\gamma (1+{v \over c}\cos \theta _{1}^{\prime })\,\!}$

This generally follows Rybicki & Lightman. However, it might help to know that in R&L, 7.7b follows from 7.8a, which follows from 7.7a.

• If ${\displaystyle E^{\prime }\ll m_{e}c^{2}}$, then:
{\displaystyle {\begin{aligned}E_{1}&\approx E^{\prime }\gamma (1+{v \over c}\cos \theta _{1}^{\prime })\\&\approx \gamma ^{2}E(1+{v \over c}\cos \theta _{1}^{\prime })(1-{v \over c}\cos \theta )\\\end{aligned}}\,\!}
In a “typical” collision, ${\displaystyle \theta \sim \theta _{1}^{\prime }\sim {\pi \over 2}}$, so ${\displaystyle E_{1}\sim \gamma ^{2}E}$.
• If ${\displaystyle E^{\prime }\gg m_{e}c^{2}}$, then:
{\displaystyle {\begin{aligned}E_{1}^{\prime }&={E^{\prime } \over 1+{E^{\prime } \over m_{e}c^{2}}(1-\cos \phi ^{\prime })}\approx m_{e}c^{2}\\E_{1}&=E_{1}^{\prime }\gamma (1+{v \over c}\cos \theta _{1}^{\prime })\\&\approx m_{e}c^{2}\gamma (1+{v \over c}\cos \theta _{1}^{\prime })\approx \gamma m_{e}c^{2}\\\end{aligned}}\,\!}