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Assume we have a photon coming in with an energy $E$ in the positive $\hat{z}$ direction. It collides with an electron at rest that results in a scattered photon with energy $E_1$ in the $\hat{n}$ direction that is at an angle $\theta$ to $\hat{z}$ and a recoiled electron $(\beta,\gamma)$ that recoils at an angle $\phi$ to $\hat{z}$. I am trying to find an equation for the recoiled electrons kinetic energy in terms of $E$ and $\phi$.

I started with conservation of energy:

$E+mc^2=E_1+mc^2\gamma \rightarrow E-mc^2(\gamma-1)=E_1$

and conservation of momentum:

$\frac{E}{c}\hat{z}=\frac{E_1}{c}\hat{n}+mc\vec{\beta}\gamma$ $\rightarrow E\hat{z}-mc^2\vec{\beta}\gamma=E_1\hat{n}$

Squaring the CoE equation and the magnitude of CoM I got:

$E^2-2Emc^2(\gamma-1)+m^2c^4(\gamma -1)^2=E_1^2$

$E^2-2Emc^2\gamma\beta\cos\phi+m^2c^4\gamma^2\beta^2=E_1^2$

Setting these equal to each other, using some algebra, and exploiting this relationship:

$\gamma^2\beta^2=\gamma^2-1$

I was able to come up with this:

$K_e=mc^2(\gamma-1)=E[\gamma(\beta\cos\phi-1)+1]$

I can't seem to come up with a simpler form (i.e. removing $\gamma$ and $\beta$). Is there any way that would be possible? I'm struggling to relate those two variables to either $E$ or $\phi$

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This problem could be solved much easier when working with momentum 4-vectors of each particle. $p^2$ is known for both particles, and $p_1 \cdot p_2$ has simple form in the rest frames of both particles.

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