Compton effect question please help?
Gamma rays emitted by radioactive nuclei exhibit measureable compton scattering. Suppose a 0.511-MeV photon from a positron electron annihilation scatters at 110 degrees from a freee electron. What are the energies of the scattered photon and the recoiling electron? Relative to the initial direction of the 0.511-MeV photon, what is the direction of the recoiling electron's velocity vector?
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Note: KE = energy, P = momentum, X axis is initial photon direction.
Θ is photon scatter angle (given), Φ is electron scatter angle.
Suffixes 0, 1, e = incoming photon, scattered photon, scattered electron.
lambdaC is Compton electron wavelength = h/(m(e)c) = 2.42631021332511D-12 m
The Compton equation is lambda1-lambda0 = lambdaC(1-cosΘ).
arctan2 is a 4-quadrant arctangent function that takes separate y and x arguments.
dlambda = lambdaC(1-cos(Θ)) = 3.25615718023909D-12 m
KE0 (given) = 511000 eV
lambda0 = hc/KE0 = 2.42630503782974D-12 m
lambda1 = lambda0+dlambda = 5.68246221806883D-12 m
KE1 = hc/lambda1 = 218187.438253193 eV
KEe = KE0-KE1 = 292812.561746807 eV
PX0 = KE0/c = 511000 eV/c
P1 = KE1/c = 218187.438253193 eV/c
PX1 = P1*cosΘ = -74624.4989032173 eV/c
PY1 = P1*sinΘ = 205029.125674707 eV/c
PXe = PX0-PX1 = 585624.498903217 eV/c
PYe = -PY1 = -205029.125674707 eV/c
Φ = arctan2(PYe,PXe) = 5.94641837877501 rad, 340.704676322834 deg
Θ = arctan2(PY1,PX1) = 1.91986217719376 rad, 110 deg
The 1st ref. below explains the theory of Compton scattering analysis, and the 2nd has a link to a graphic calculator where you can enter the photon angle and wavelength lambda0, and measure the resulting Φ.