Compton scattering, angle of recoiled electron

[Dawei]

Trying to find the angle of the velocity vector from a recoiled electron after an impact from a photon.

I have already found the initial energy of the photon, and the energy of the scattered photon, and so can calculate the momentums. I know the photon is scattered at an angle of 110 degrees, and I know from the Compton effect and the difference in the wavelengths what the electron's energy is.

Here is a relevant sketch:
http://upload.wikimedia.org/wikipedia/commons/e/e3/Compton-scattering.svg

Note that theta here is the 110 degrees. I want to know the angle of the velocity vector of that recoiled electron.

Please, if anyone can point a finger at where to go I'd be happy. I've been working on this for quite a long time, trying to turn the initial and final momentum into i and j components, but it is getting me nowhere. I have the feeling the answer is a lot more simple than what I've made it out to be!

 

[vela]

In the derivation of the formula relating the scattering angle and the shift in wavelength, the electron is assumed to be at rest initially, so its final momentum will just be the difference of the photon's initial and final momenta. In the figure, the initial momentum is in the +x-direction, and you can calculate the components of the photon's final momentum since you know its energy and angle. I'm not sure what you're doing to make the problem so hard. Is there some specific form of the answer you're looking for?

 

[Dawei]

I just keep getting impossible answers for the cosine of theta (where theta is not the 'theta' in the above drawing, but rather the angle the electron makes with the horizontal).

I start out with initial photon momentum, equal to the final momentum in the x direction. That is,

hf₁/c = (hf₂/c)*(cos110) + (m_ev_e)*(cosθ)

And for the y direction:

0 = (hf₂/c)*(sin110) + (m_ev_e)*(sinθ)

I don't know if this is right, and even if it is I can't figure out how to solve it.

Also, should hf₂/c be negative?

 

[vela]

I just keep getting impossible answers for the cosine of theta (where theta is not the 'theta' in the above drawing, but rather the angle the electron makes with the horizontal).

I start out with initial photon momentum, equal to the final momentum in the x direction. That is,

hf₁/c = (hf₂/c)*(cos110) + (m_ev_e)*(cosθ)

And for the y direction:

0 = (hf₂/c)*(sin110) + (m_ev_e)*(sinθ)

I don't know if this is right, and even if it is I can't figure out how to solve it.

Also, should hf₂/c be negative?

The equations are right. You're using hf₂/c as the magnitude of the scattered photon's momentum, so it should be positive. The sign is taken care of by the sines and cosines. (In particular, cos 110<0.)

To solve for theta, move the hf₂/c term to the other side in each equation, then divide one equation by the other.