Compton Scattering with Moving Electron

[jowens1988]

Homework Statement

Compton's derivation of his scattering formula:
$$\delta \lambda \equiv \lambda' - \lambda = \frac{hc}{m_e c^2} (1 - cos\theta)$$
assumed that that the target electrons were at rest. In reality, they are orbiting around nuclei. In a material like lead, the electrons have a kinetic energy as large as K ~ 200 eV.

Suppose that the atomic electron was heading directly toward the incoming X-ray photon. Calculate the wavelength of the scattered photon $$\lambda'$$. By how much is this different from the expected Compton shift? Neglect terms of order K,K^2, or the electron momentum squared.

Homework Equations

Conservation of Momentum:
x-direction: $$p_1 c - p_{e1}c = p_2 c cos(\theta) + p_{e2}c cos\phi$$
and
y-direction: $$p_2 c sin\theta = p_{e2}c sin\phi$$

Conservation of Energy:
$$p_1 c + \sqrt((p_{e1}c)^2 + (m_e c^2)^2) = p_2 c + \sqrt((p_{e2}c)^2 + (m_e c^2)^2)$$

The Attempt at a Solution

I think I am setting it up correctly, just adding the kinetic energy of the electron to the energy equations and an initial momentum to the x-direction of the momentum equations.

But if I neglect the terms that it tells me to neglect, then I get $$p_1 = p_2$$, which would imply there is no wavelength shift at all, which doesn't seem right.

Would it make more sense to view the post collision frame in a center of momentum frame?

 

[vela]

Are you sure the problem said to ignore terms of order K?

I'd probably try to Lorentz boost to the frame where the electron is at rest, use the regular Compton scattering formula, and then transform back to the original lab frame.

 

[jowens1988]

Thanks for getting back to me.

That's what I ended up doing, and I think it worked out.

I suppose the ignore order K on the sheet may have been a mistake...otherwise, it seems the problem reduces too much. If I did want to go about it the same way Compton did, was I on the right track, though?

 

[vela]

Well, I think Compton would have done it the easy way by transforming to the electron rest frame.

In your lab-frame equations, I think you have a sign error in the conservation of momentum equation for the x-direction.