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How many photons does a free electron (in sunlight say) interact with per second?

I did a rough calculation assuming the electron interacts with any photon that enters through an area the size of the Thomson cross-section, and that the light is monochromatic with angular frequency 2x10^14 Hz and electric field of 1000 Vm^-1. And the power through this area is the Poynting vector times the area. Then the rate of interactions is the power divided by the energy of one photon:

$$ \text{rate} = \frac{\epsilon_0 c E_0^2}{2 \hbar \omega} \frac{8 \pi}{3} r_e^2 \approx 4\times10^{-6}s^{-1}$$

where $r_e$ is the classical electron radius. But this means it takes 3 days to interact once which doesn't sound right. I may have made some bad assumption or something here.

I'd like to know if this approach is valid. If not, where, conceptually, does this approach fail and how should it be improved?

A possible avenue to answer the question could be to describe a known calculation of the interaction rate between free electrons and light and point out how to connect it to this problem.

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    $\begingroup$ Hi Matt Majic and welcome to Physics.SE! Please note that this is not a homework help site. See How do I ask homework questions on Physics Stack Exchange? and Should any check my work questions be made on topic? posts on meta for more information. $\endgroup$ – Gonenc Aug 8 '15 at 13:07
  • $\begingroup$ Yes its not a homework question $\endgroup$ – Matt Majic Aug 8 '15 at 23:01
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    $\begingroup$ It is a check-my-work-type of question. $\endgroup$ – Gonenc Aug 8 '15 at 23:02
  • $\begingroup$ Well actually the only question is in the first line but ok should I remove some of the later part? $\endgroup$ – Matt Majic Aug 8 '15 at 23:26
  • $\begingroup$ If you have a conceptual question go ahead but I don't know how you can manage to do that. If you remove the later part then it might be a derive-the-equation-for-me-type of question. Try to mould it in such a way that it will be a conceptual question ;) $\endgroup$ – Gonenc Aug 8 '15 at 23:30
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The electron doesn't absorb the photons, it scatters them. The energy absorbed by an interaction depends on the scattering angle - this can be determined using the Compton formula for the wavelength of the scattered photon. And the electron tends to scatter more at different angles (proportional to the Thomson differential cross section). From this I found the rate of interactions to be more like once every 30 seconds.

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