Compton Scattering
The derivation of Compton Scattering informs many uses of both the
result and the technique used to derive it.
We start with the knowledge that a photon has the momentum
\[p = \frac{E}{c}\]
and we must apply special relativity to that, as an electron. Thus the
kinetic energy of that photon is
\[ \begin{align}\begin{aligned}T_{\gamma}=\sqrt{p_{e}^{2}c^{2}+m_{e}^{2}c^{4}}-m_{e}c^{2}\\It should be noted that the last two terms will cancel each other out.\end{aligned}\end{align} \]
We then apply elastic scattering, knowing that the total energy is the
initial energy of the photon and the sum of its kinetic energy and the
secondary energy of the electron
\[E=E^{'}+T\]
Now we get into the real math (algebra). We find the momentum of the
scattered photon and the momentum of the electron.
\[\left(p^{'}\right)^{2}=\left|\vec{p}-\vec{p}_{e}\right|^{2}-2pp_{e}\cos\theta_{r}\]
\[ \begin{align}\begin{aligned}\left(p_{e}\right)^{2}=\left|\vec{p}-\vec{p}^{'}\right|^{2}=p^{2}+\left(p^{'}\right)^{2}-2pp^{'}\cos\theta_{s}\\Note that the last terms are from the law of cosines from the\end{aligned}\end{align} \]
scattering angles.
We then continue with our balances, this time doing an energy balance on
the two particles.
\[\left(E^{'}\right)^{2}=E^{2}+T\left(T+2m_{e}c^{2}\right)-2E\sqrt{T\left(T+2m_{e}c^{2}\right)}\cos\theta_{r}\]
\[ \begin{align}\begin{aligned}T\left(T+2m_{e}c^{2}\right)=E^{2}+\left(E^{'}\right)^{2}-2EE^{'}\cos\theta_{s}\\and our energy conservation is simply stated earlier, and restated as\end{aligned}\end{align} \]
\[T=E-E^{'}\]
Performing this energy balance, we get
\[E^{2}+\left(E^{'}\right)^{2}-2EE^{'}\cos\theta_{s} =\left(E-E^{'}\right)\left(E-E^{'}+2m_{e}c^{2}\right)\]
\[ \begin{align}\begin{aligned}\left[\left(1-\cos\theta_{s}\right)E+m_{e}c^{2}\right]E^{'} =m_{e}c^{2}E\\which gives us\end{aligned}\end{align} \]
\[E^{'}=\frac{E}{1+\left(\frac{E}{m_{e}c^{2}}\right)\left(1-\cos\theta_{s}\right)},\quad0\leq\theta_{s}\leq\pi\]
Finally, we complete this using the momentum equations we wrote earlier
\[E^{2}+T\left(T+2m_{e}c^{2}\right)-2E\sqrt{T\left(T+2m_{e}c^{2}\right)}\cos\theta_{r}=\left(E-T\right)^{2}\]
\[ \begin{align}\begin{aligned}\left[\left(E+m_{e}c^{2}\right)-E^{2}\cos^{2}\theta_{r}\right]T=2m_{e}c^{2}E^{2}\cos^{2}\theta_{r}\\which yields the result\end{aligned}\end{align} \]
\[T=\frac{2m_{e}c^{2}E^{2}\cos^{2}\theta_{r}}{\left(E+m_{e}c^{2}\right)^{2}-E^{2}\cos^{2}\theta_{r}},\quad0\leq\theta_{r}\leq\frac{\pi}{2}\]
