Physic Labs

Problem 3

A photon of wavelength λ\lambda scatters from an initially stationary electron; the scattered photon makes angle θ\theta with the incident direction. Find the wavelength shift Δλ\Delta\lambda and recoil-electron kinetic energy. Use electron mass mem_e, speed of light cc, and Planck constant hh. (a) Combine energy and momentum conservation with the electron’s relativistic relation to derive Compton shift Δλ\Delta\lambda. (b) Find the photon energy loss and identify it with recoil-electron kinetic energy. (c) Evaluate forward scattering θ=0\theta=0 and backscattering θ=π\theta=\pi, and state which gives the greatest recoil.
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mec2(Eγ−Eγ′)=EγEγ′(1−cos⁡θ)m_ec^2(E_\gamma-E_\gamma\prime)=E_\gamma E_\gamma\prime(1-\cos\theta)
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Combine energy conservation with Ee2=me2c4+pe2c2E_e^2=m_e^2c^4+p_e^2c^2 to eliminate electron momentum. Rearrangement gives mec2(E−E′)=EE′(1−cos⁡θ)m_ec^2(E-E\prime)=EE\prime(1-\cos\theta).