Compton scattering in a general medium

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Homework Statement

In a medium of refractive index 𝑛, the wavelength of light 𝜆0 is related to its frequency 𝑓0 . Variables given: input photon frequency 𝑓, outgoing photon angle 𝜃, outgoing photon frequency 𝑓′, outgoing electron angle 𝜙, outgoing electron energy 𝐸 and outgoing electron momentum 𝒑. In this problem, the momenta of all particles, before and after, lie entirely in the plane of the paper.

Part 1: Write the energy conservation equation and the momentum conservation equations for Compton scattering for incident and outgoing photons in a medium of refractive index n

Part 2: Solve the equations to obtain the outgoing photon frequency 𝑓′ in the following form:
𝑓′ = [−𝐵 ± √(𝐵^2 − 4𝐴𝐶) ]/ (2𝐴) where you are to determine the unknowns 𝐴, 𝐵 and 𝐶 in terms of 𝑓, 𝑐, 𝑛, 𝜃, and electron rest mass 𝑚. You may use the fact that for general (relativistic) electrons, the dispersion relation is 𝐸 = √[(𝑝𝑐)^2 + (𝑚𝑐^2)^2]

Relevant Equations

𝜆0 = 𝑐/(𝑛𝑓0) where 𝑐 is the speed of light in vacuum

For part one, my energy conservation equation is nhf₀ + mc² = nhf' + E

my momentum conservation in x-axis is nhf₀= nhf' cos(theta) + c𝒑 cos(fi)

My momentum conservation in y-axis is nhf' sin(theta) = c𝒑 sin(fi)

For part 2 I understand that I am supposed to get a qudratic equation in terms of f' but when I tried combining all three equations but all I did was derive Compton shift equation. Where did I go wrong?