3-36 in modern physics 5th ed. This is a Compton scattering question. Suppose a

Question

3-36 in modern physics 5th ed.
This is a Compton scattering question.
Suppose a 0.511 MeV photon scatters at 110 degrees from a free electron. Relative to the initial direction of the 0.511 MeV photon, what is the direction of the recoiling electron?
Found: ?1 = 24.29x10^-13m, ?2 = 56.748x10^-13m            E1 = 0.511MeV, E2 = 0.218MeV, Ek(or energy of recoil electron) = 0.293MeV Cramster gives answer as tan(f)=[cot(?/2)]/[1+(E/mc^2)] Any clarification would greatly be appreciated. The only formula available in the book on compton scattering is ?2-?1=(h/mc)(1-cos?) 3-36 in modern physics 5th ed.
This is a Compton scattering question.
Suppose a 0.511 MeV photon scatters at 110 degrees from a free electron. Relative to the initial direction of the 0.511 MeV photon, what is the direction of the recoiling electron?
Found: ?1 = 24.29x10^-13m, ?2 = 56.748x10^-13m            E1 = 0.511MeV, E2 = 0.218MeV, Ek(or energy of recoil electron) = 0.293MeV Cramster gives answer as tan(f)=[cot(?/2)]/[1+(E/mc^2)] Any clarification would greatly be appreciated. The only formula available in the book on compton scattering is ?2-?1=(h/mc)(1-cos?)

Explanation / Answer

Given    energy of incident photon E = 0.511 MeV              wavelength of incident photon                          ? = h c / E                                  =   (6.625*10^ -34 ) ( 3*10^ 8 ) / ( 0.511*10^ 6 ) ( 1.6*10^ -19 J )                                = 24.30 *10^ -13 m              wave length of scattered photon                         ? ' = ? + ( h / mc ) ( 1 - cos ? )                               = (2.43*10^ -12 )+ ( 9.1*10^ 13 )(1- cos 110 )                              = (24.3*10^ -13 )+ ( 32.567*10^ -13 )                               = 56.847*10^ -13 m               energy of scattered photon                           E = h c /? '                                  = ( 6.635*10^ -34 ) ( 3*10^ 8 ) / ( 56.847*10^ -13 )                                  = 0.6587*10^ 10^ -13 J                                   = 0.218 MeV             energy of recoil electron                       E = 0.511 -0.218 = 0.293 MeV             direction of recoil electron               tan f =    cot (?/2) / [ 1+ E / mc^2 ) ]                         = cot (55) / ( 1 + (0.6587*10^ - 13/ 9.1*10^ -31 * 9*10^ 16)                   f   = 19.29 ^ o               

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