Compton scattering in a general medium

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The discussion focuses on the application of energy and momentum conservation equations in Compton scattering within a general medium. The user presents their equations for energy conservation and momentum conservation in both the x and y axes. They express confusion about deriving a quadratic equation in terms of f' while attempting to combine these equations, ultimately resulting in the Compton shift equation instead. The user seeks clarification on where their approach may have gone awry in the derivation process. Understanding the correct manipulation of these equations is crucial for accurately analyzing Compton scattering.
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Homework Statement
In a medium of refractive index 𝑛, the wavelength of light πœ†0 is related to its frequency 𝑓0 . Variables given: input photon frequency 𝑓, outgoing photon angle πœƒ, outgoing photon frequency 𝑓′, outgoing electron angle πœ™, 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′ = [βˆ’π΅ Β± √(𝐡^2 βˆ’ 4𝐴𝐢) ]/ (2𝐴) where you are to determine the unknowns 𝐴, 𝐡 and 𝐢 in terms of 𝑓, 𝑐, 𝑛, πœƒ, and electron rest mass π‘š. You may use the fact that for general (relativistic) electrons, the dispersion relation is 𝐸 = √[(𝑝𝑐)^2 + (π‘šπ‘^2)^2]
Relevant Equations
πœ†0 = 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?
 
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