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.

 

[paranormal]

I would suggest approaching this problem by considering the relativistic effects that come into play when dealing with moving electrons in Compton scattering. While the derivation of the Compton scattering formula assumes that the target electrons are at rest, in reality, they are in motion due to their orbit around the nucleus. This introduces additional factors such as the electron's momentum and kinetic energy, which must be taken into account in the calculation.

To properly calculate the wavelength shift in this scenario, one could use the conservation of momentum and energy equations as stated in the problem, but also include the relativistic corrections for the moving electron. This would involve using the relativistic momentum equation p = \gamma m v, where \gamma is the Lorentz factor and v is the velocity of the electron. Additionally, the kinetic energy of the electron would need to be included in the energy equation, as stated in the problem.

By considering these additional factors, one can properly calculate the wavelength shift of the scattered photon and compare it to the expected Compton shift. Neglecting terms of order K, K^2, or the electron momentum squared may not give an accurate result in this case, as these factors play a significant role in the relativistic effects of the moving electron. Therefore, it would be more accurate to include them in the calculations.