What is the maximum recoil energy of photons in Inverse Compton Scattering?

[Matt atkinson]

Hello all, I'm just doing some practice for an upcoming exam and came upon this question in my notes:

One experimental way to generate very high energy photons is to ”collide” a laser beam against an electron beam, the photons that recoil in the direction parallel to the electron beam will have large energy. This is called ”Inverse Compton scattering”. Calculate the maximum recoil energy of the photons, assuming the initial energy of the photons is 1 eV and the electrons in the beam have energy E = 50 MeV.

Now I'm really stuck at how I should approach the question just drawing diagram wise...
i've had a few attempts where I set the electron and photon to move against each other on the x direction and then afterwards the photon recoils back 180 degrees from its intial momentum. But i just can't get a reasonable answer...

 

[Ibix]

So what answers have you got? Please show your working. The Latex primer linked below the reply box may be of interest.

 

[Matt atkinson]

Sorry i'll do that now!
its more the issue of angles I'm struggling with for the maximum value.

I found a helpful page on the inverse compton scattering which led me to derive the following equations (electron rest frame is the primed variables):
The energy of the final photon in the rest frame of the electron.
$E'_{\gamma_f}=\frac{E'_{\gamma_i}}{1+\frac{E^,_{\gamma_i}}{m_ec^2}(1-cos(\theta'))}\\ E^,_{\gamma_i}=E_{\gamma_i}\gamma(1-\beta cos(\theta')$
I then used the relativistic dopper shift formula back to the lab frame:
$E_{\gamma_f}=E'_{\gamma_f}\gamma(1+\beta cos(\phi))\\$
where i understand that I've been given $\gamma$ from the electron energy, and from that i can work out $\beta$ but I am just struggling with what angles to use for maximum value of $E_{\gamma_f}$.

 

[Matt atkinson]

I've actually just figured it out I am very sorry!