p39_041 - 41. The difference between the electron-photon...

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Unformatted text preview: 41. The difference between the electron-photon scattering process in this problem and the one studied in the text (the Compton shift, see Eq. 39-11) is that the electron is in motion relative with speed v to the laboratory frame. To utilize the result in Eq. 39-11, shift to a new reference frame in which the electron is at rest before the scattering. Denote the quantities measured in this new frame with a prime ( ), and apply Eq. 39-11 to yield h 2h ∆ λ = λ − λ0 = (1 − cos π ) = , me c me c where we note that φ = π (since the photon is scattered back in the direction of incidence). Now, from the Doppler shift formula (Eq. 38-25) the frequency f0 of the photon prior to the scattering in the new reference frame satisfies c 1+β = f0 f0 = , λ0 1−β where β = v/c. Also, as we switch back from the new reference frame to the original one after the scattering c 1−β 1−β = . f =f 1+β λ 1+β We solve the two Doppler-shift equations above for λ and λ0 and substitute the results into the Compton shift formula for ∆λ : 1 1−β 1 1−β 2h . − = ∆λ = f 1+β f0 1 + β me c2 Some simple algebra then leads to −1 E = hf = hf0 2h 1+ me c2 1+β 1−β . ...
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