Compton Effect (scattered photon angle)

[BAlexanderSUS]

Homework Statement

Show that for Compton Effect: tan(ϴ) = (sinφ)/[(Δλ/λ) + (1-cosφ)]

Where,
ϴ = angle that the scattered photon makes with the incident photon
φ = angle that hte scattered electron makes with the incident photon

Homework Equations

For Photon:

initial;
E=hv
p=hv/c
(where v is frequency)

final;
E=hvʹ
p=hvʹ/c

For Electron;

initial;
E=mc^2
p=0

final;
p=p
E=sqrt(m^2c^4+p^2c^2)

note* (changed the v/c to 1/λ)

The Attempt at a Solution

Breaking the problem into components and solving for ϴ such that tanϴ=sinϴ/cosϴ:

X; (h/λ) = (h/λʹ)cosϴ +Pcosφ
Y; 0=(h/λʹ)sinϴ-Psinφ

so,

cosϴ=[((h/λ)-Pcosφ)λʹ]/h
sinϴ=Psinφ(λʹ)/h

tanϴ=sinϴ/cosϴ

so,

tanϴ=Psinφ/((h/λ)-Pcosφ)

from here I am lost, I'm assuming that there must be an property of momentum that I am not taking into account. Any help would be appreciated! thank you!

 

[mark726]

Thank you for your post. I am a scientist and I would like to help you solve this problem. First of all, you have the right idea by breaking the problem into components and using the equations for photon and electron energy and momentum. However, there are a few corrections and additions that need to be made to your attempt at a solution.

1. The equation for the scattered photon's energy should be E = hc/λ', not E = hv'/c. This is because in Compton scattering, the energy of the photon changes due to the interaction with the electron, but the frequency remains the same.

2. The equation for the scattered electron's energy should be E = sqrt(m^2c^4 + p^2c^2), not E = mc^2. This is because the electron's energy also changes due to the interaction with the photon, and its momentum is not zero in the final state.

3. In your attempt at a solution, you have used the same momentum value for the initial and final state of the electron. However, in Compton scattering, the momentum of the electron changes due to the interaction with the photon. The correct equation for the electron's momentum in the final state is p = sqrt((h/λ')^2 + (Pcosφ)^2 - 2(h/λ')(Pcosφ)cosϴ).

4. In the equation for cosϴ, you have used (h/λ) as the momentum value for the initial state of the photon. However, this should be (h/λ') since the photon's momentum changes after the interaction with the electron.

With these corrections and additions, you should be able to obtain the correct expression for tanϴ. Remember to use the fact that the total energy and momentum must be conserved in the scattering process. I hope this helps. Good luck!