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ck99
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Homework Statement
Show that, for low energy photons scattered by ultrarelativistic electrons, the cange in frequency of the photon is given by
(v'-v) / v = [(Ω'-Ω).β] / [1-Ω'.β]
Homework Equations
The full/general form of Compton scattering is given by
v'/v = (1-Ω.β) / [(1-Ω'β) + hv/(γmc2) (1 - Ω.Ω') ]
where v is photon frequency
m is electron mass
β is electron velocity divided by c
c is speed of light
γ is Lorentz factor
Ω is unit vector of propagation of the photon
and primed quantities are those quantities after scattering
The Attempt at a Solution
I have attempted the following. For low energy photons, hv << mc2 so that reduces the equation to
v'/v = (1-Ω.β) / (1-Ω'β)
or (v'-v)/v = [ (1-Ω.β) / [(1-Ω'β) ] - 1
For ultra-relativistic electrons, velocity is almost c, so β = 1 but looking at the target answer it is not helpful to remove β from the equation.
I think maybe I am missing something to do with vectors. How do I properly evaluate (Ω'-Ω).β ?
Is it just (Ω'-Ω).β = Ω'.β - Ω.β ?