Compton Scattering: Finding Angle from only Initial Energy

[1saac]

Homework Statement

Gamma rays of energy 1.02 MeV are scattered from electrons that are initially at rest. If the
scattering is symmetric, that is, if θ = ϕ in Fig. 1, find
i. the scattering angle θ
ii. the energy of the scattered photons.

Homework Equations

λ' - λ = (h/mc)[1 - cos(θ)]

The Attempt at a Solution

I'm not entirely sure where to go on this one. The first thing I did was take the energy of the gamma ray and find its wavelength via the relationship:

E = hf

E = hc/λ

λ = hc/E

λ = 1.217E-22 m

I've been scouring every resource I can but I cannot find a relationship for scattering angle dealing only with energy. I am certain that the fact that the scattering is symmetric has something to do with it, but I do not know how to apply that fact.

Does anyone know how I can approach this problem?

 

[diazona]

I would try starting from conservation of energy and momentum.

 

[1saac]

So I tried applying the conservation of momentum with a bit of help from a book I found in the study room. This was my attempt to solve for the angle, however I ended up with sec(theta) = 0, which is impossible. I know I must be missing a term or something somewhere.

 

[vela]

Why are you using the same angle for both the electron and the photon? Also, one of the terms in the y-equation should be negative, right?

 

[1saac]

I'm using the same angle because the problem states that "the scattering is symmetric, that is, θ = ϕ in Fig. 1". Am I interpreting this part incorrectly?

I realized that the term describing the momentum of the scattered photon should be negative. It helped a bit, but I now arrive at a point where I cannot eliminate either the recoiling electron's velocity and the scattered photon's wavelength at the same time.

 

[diazona]

It looks like you haven't used conservation of energy...

 

[1saac]

I've solved the problem. I friend of mine told me that you only need to use conservation of momentum in the x-direction, and the Compton shift equation. Then you solve both for scattered photon wavelength and set them equal.

I hate it when I look for help with a problem on Google and find a thread where someone says they solved their problem without showing how, so here's my workings for all you people who found this thread via Google: (although I forgot to rotate it so you'll have to either tilt your head or rotate it yourself)

 

[diazona]

Yeah, that works because the Compton scattering equation incorporates conservation of both momentum and energy already.