Compton Scattering of Electron

[satchmo05]

Homework Statement

Suppose a 0.511[MeV] photon from a positron-electron annihilation scatters at 110 degrees from a free electron. What are the energies of the scattered photon and the recoiling electron? Relative to the initial direction of the 0.511[MeV] photon, what is the direction of the recoiling electron's velocity vector?

Homework Equations

λ₂ - λ₁ = (h/mc)(1-cos(theta))
change in energy = (hc/(delta(λ)) = h*delta(f)

The Attempt at a Solution

~ λ₂ - λ₁ = (h*c)/(m*c²)*(1 - cos(110)) = 0.00326[nm] = delta(λ)
~ variating change in energy formula --> delta(f) = c/(delta(λ)) = 9.2025e¹⁹ [Hz]
~ Multiplying frequency by Planck's constant, I get the change in energy = 6.1e^-19 [J] = 0.380732094 [MeV]

I am confused at what I do from here to determine the energies of either the scattered photon and the recoiling electron. Since the photon's initial energy is 0.511[MeV], and using change in energy answer from above, would this mean that the energy of recoiling electron be equal to (0.380732094 + 0.511) [MeV]?

Thank you for all help in advance. Hopefully this work I have shown in clear!

 

[vela]

Your equation for the change in energy is wrong. Use the fact that

$$E_\gamma = h\nu = \frac{hc}{\lambda}$$

 

[satchmo05]

Vela, thanks for the reply. There is nothing wrong with that formula that I used earlier. Planck's constant is in units of J*s, and frequency is in s^-1. Ergo, energy is in Joules, so my units are fluid. I am not understanding what you're saying. From what I see in your formula, your units for energy would be a Joule*meter, which does not work. Please explain. Thanks.

 

[vela]

The formulas I cited are correct. I'm not sure how you're getting joule-meter.

The units in your formula work okay, but algebraically, it's just wrong. The energy of the photons are given by E₁=hc/λ₁ and E₂=hc/λ₂, so the difference in energy is

$$\Delta E = E_2 - E_1 = \frac{hc}{\lambda_2} - \frac{hc}{\lambda_1} \ne \frac{hc}{\Delta \lambda}$$

 

[satchmo05]

Ah, I see what you're saying now. So what would I do with the change in energy value? That was my original question in the statement above. Thanks.

 

[vela]

Try looking in your text where it derives the formula for the change in wavelength. Compton scattering is an elastic collision between a photon and an electron.

 

[satchmo05]

Vela, I am not finding anything of use to me right now.

 

[satchmo05]

Yes, I just read over the text on Compton scattering, and no mention of a formula deriving the change in wavelength. Any help you would be able to give me? Thanks again.

 

[vela]

It's an elastic collision, so kinetic energy is conserved.