Compton

Peter Fisher

11/26/08

8.033

The scattering of a photon by an electron is called Compton scattering after A.H. Compton who first observed the scattering photon has a longer wavelength than the incident photon. In recitation, we used momentum energy conservation to relate the shift in wavelength to the scattering angle. The calculation took four blackboards as was somewhat tedious. This post shows how to carry out the calculation much more efficiently using four vectors. The initial photon is denoted by the four vector K, the initial electron by P. After the collision, K’ and P’ denote the photon and electron, respectively. Then, energymomentum conservation says P+K=P’+K’ which may be written P-P’=K’-K. Then, square both sides:

( P − P ')

2

2

= ( K '− K ) 2 2 P − 2PP '+ P ' = K 2 − 2KK '+ K '2 2m 2 c 4 − 2PP ' = −2KK '

The last step makes use of P2=P’2=m2c4 and K2=K’2=0. To work out PP’ and KK’, we need to write out the four vectors:  0   kc   k 'c cosθ   p 'c cos φ          0  0 k 'csin θ  p 'csin φ      P= K= K'= P' =  0  0  0    0  2        mc   kc   k 'c   E' 

Notice I have used k to denote the momentum of the photon (not the wave number). Then, PP ' = E ' mc2 and KK ' = k ' kc2 − k ' kc2 cosθ and we have m 2 c 4 − E ' mc2 = −kk 'c 2 (1 − cosθ ) . Energy conservation gives E ' = mc2 + kc − k 'c and we have m 2 c 4 − m 2 c 4 − mc 3 ( k − k ') = −kk 'c 2 (1 − cosθ ) → ( k − k ') = Finally, use k =

h h to obtain λ '− λ = (1 − cosθ ) . λ mc

kk ' (1 − cosθ ) . mc