Compton Scattering of of monochromatic light

[eep]

I'm working out a problem from a text concerning the scattering of monochromatic light by free electrons (Compton effect) which asks me to derive expressions for the wavelength shift, electron momentum, and electron scattering angle in terms of the photon scattering angle assuming that the electron is initially at rest.
I've managed to derive the standard wavelength shift formula no problem, it depends only on the scattering angle of the photon. However, I can't manage to derive anything for the momentum of the electron without it also depending on the initial wavelength of the photon. Is there an expression for the momentum of the electron which ONLY depends on the scattering angle of the photon?
My hunch is no, since if one looks at the following formula:
$$\Delta E = h(\nu - \nu^{\prime}) = h(\frac{c}{\lambda} - \frac{c}{\lambda^{\prime}}) = hc(\frac{\lambda^{\prime} - \lambda}{\lambda\lambda^{\prime}})$$
The change in energy (which will lead you to the electron momentum) depends on the product of $\lambda$ and $\lambda^{\prime}$. Am I missing something obvious?

 

[LURCH]

You're in way over my head here, but just a suggestion;
Wouldn't the electron's momentum depend entirely on the change in energy of the photon? So the solution for momentum (of the electron) should only depend on the difference between the initial and the final wavelength (of the photon), regardless of what those values are, shouldn't it?

 

[inha]

Yes and you can express the change in energy in terms of the scattering angle of the photon so the momentum can be solved with some algebra. It's manipulated to the angle dependent form at scienceworld for example.

 

[eep]

Yes and you can express the change in energy in terms of the scattering angle of the photon so the momentum can be solved with some algebra. It's manipulated to the angle dependent form at scienceworld for example.

I can't seem to find it at scienceworld. Doesn't the change in energy depend not only on the wavelength shift, but also on the product of the initial and final wavelength? I don't see how to get rid of the $\lambda\lambda^\prime$...

 

[inha]

Basically you'll have to solve the electron momentum squared twice. Once from conservation of momentum and once from conservation of energy and then equate them. The details are long to type out so I'll just refer you to the scienceworld page here: http://scienceworld.wolfram.com/physics/ComptonEffect.html

 

[eep]

I think you misunderstood what I was looking for. I was looking for an expression for the electron momentum which does not depend on wavelength. What you gave me was a method of solving for the wavelength shift $\lambda^\prime - \lambda$. Thank you for your help, anyways though. I came to my own conlclusion that the electron momentum *must* depend on the initial wavelength.

 

[inha]

Whoops, sorry about that. You're correct about the energy transfer having to depend on the incoming photon's energy. You can solve for the ratio of incoming and outgoing photon energy though but that won't help you get forward with the assignment.

 

[DaTario]

I think the dependence on the initial wavelenght is something that could be realized from the very begining. The angle of scattering, due to its geometrical nature, is to be understood as an indicator of the proportion (percentage) of the total momentum which will be taken from the electron. The rest of the information lies in the initial momentum itself.

Best Regards

DaTario