A linear accelerator is a machine used to deliver radiation therapy to cancer patients. It shoots very precise beams of high energy photons into the patient's body, and those photons interact with electrons and atoms in the cancer cells to kill it. There are several different ways a photon can interact with electrons and atoms. In this question, we're going to explore a Compton interaction between a photon and an electron. Historically accepted relativistic equations are provided¹.
The electron is originally at rest, has a mass mo and energy moc2. An incoming photon with energy λhc and momentum λh, where λ is the wavelength of the photon, collides with this electron and transfers some of its energy, which causes the electron and the photon to scatter in different directions. The ejected electron is now travelling at speed v and has a mass m=1−c2v2mo and energy mc2.
Derive an equation that relates the wavelength (λ) of the incoming photon with the wavelength (λ') of the outgoing scattered photon and its scattering angle (θ).
Hint: sin2θ+cos2θ=1
[Figure: Shows before and after collision diagrams with incoming photon, electron at rest, scattered photon at angle θ, and scattered electron at angle φ]
¹It is important to note that the historically accepted equations for the relativistic mass of an electron that are provided here are not properly correct. They imply that mass changes with velocity, when in fact mass is invariant. The proper equations of relativistic mass are much more complex and are not needed to solve this problem; the correct solution can be found with the equations provided.