(3) Interaction of high energy photons with matter - LUTH

Compton scattering - energy transfer (1)
What is the change in wavelength of a photon scattering off an electron ?
We describe the collision with relativistic 4-vectors:
electron:
photon:
4-momentum conservation yields:
p = [ me c , me v] p ' = [ ' mec , ' m
e v ' ]
[
ℏ
k =
c
, ℏ k] i c
k' [
ℏ '
=
c
pk = p ' k '
Scalar products of 4-vectors are Lorentz invariant:
In the same way: p ' ⋅p ' = m e
Thus:
, ℏ ' '] ik c
p ⋅p = 2 2 2 2 2 2 2 c
me c − me v = me
2 −v 2
2
v
1−
c 2
= me 2 c 2 ; k⋅k = k ' ⋅k ' = 0
pk 2 = p ' k ' 2
p⋅p2p⋅kk⋅k = p ' ⋅p ' 2p ' ⋅k ' k ' ⋅k '
p⋅k = p ' ⋅k '
2 c 2
A. Zech, Instrumentation in High Energy Astrophysics
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