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.
problems.proof.stepOf
problems.proof.prevStepproblems.proof.nextStep
λ′=λ+hmec(1−cos⁡θ),Ke=hc(1/λ−1/λ′)\lambda\prime=\lambda+\frac{h}{m_ec}(1-\cos\theta),\quad K_e=hc(1/\lambda-1/\lambda\prime)
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Substitute E=hc/λE=hc/\lambda and E′=hc/λ′E\prime=hc/\lambda\prime and cancel common factors to obtain the Compton shift. Electron kinetic energy is the photon energy difference.

problems.proof.pitfall. Pitfall: using nonrelativistic kinetic-energy conservation for the recoil electron. Its kinetic energy is total energy minus mec2m_ec^2, equal to the photon’s energy loss.