# Maximum KE, Compton Scattering

I am asked to find the maximum KE value of the recoiling electron involved in a Compton scattering situation.

Obviously, the maximum final Kinetic Energy of the electron would result when as much momentum as possible is imparted on the electron as a result of the collision.
If the incident photon hit the electron head-on and was then reflected off in the opposite direction, in order to conserve momentum, the electron would need to recoil in the direction the photon was originally traveling. In this case, the maximum momentum (and thus KE) is imparted onto the electron.

Finding the change in wavelength of the reflected light is easy.
We know from the Compton equation that the change in the reflected photon’s wavelength is,
Δλ = λ_c (1 – cos (θ))
When θ = 180°, Δλ = 2 * λ_c = .0483 Å

Once we know the change in wavelength we can find the Kinetic Energy which can be imparted on the electron,
KE = (h * c * Δλ) / (λ(Δλ+λ))
Plugging in, Δλ = 2λ_c, we get,
KE = (h * c * 2λ_c) / (λ(2λ_c + λ))
Which can be simplified to,
KE = (h * f * 2λ_c) / (2λ_c + λ)
The (2λ_c + λ) term is just a fancy way of writing the final wavelength of the reflected light, so I can just as easily say,
KE = (h * f * 2λ_c) / λ_f

This last equation I would think to be my answer. However, the question specifically says to express the KE value in terms of h*f, and the rest mass energy of the electron.
I am almost to that point, if not for that pesky λ_f in the denominator.
I could, but I don’t yet see how it will help, inert the value for the relativistic Kinetic Energy of the recoiling photon in for KE,

KE = (γ – 1)mc^2 = (h * f * 2λ_c) / λ_f,
So I now have a rest mass energy term for the electron in the answer, but that didn’t really help anything.