Why do we assume that a photon has encountered only a single collision with an electron in Compton scattering

For explaining the experimental results of Compton scattering theoretically we consider a collision between a photon and a free electron and then calculate the new wavelength of photon after collision which is dependent on the angle of deviation. Why do we assume here that the photon reaching the detector has encountered only one collision with an electron, it could have reached the detector after multiple collisions with different electrons which would give different $$\Delta\lambda$$ for same angle. Is it because a photon colliding with multiple electrons no its path is very unlikely? Also what is the reason for non zero intensity at $$\lambda$$ other than the two peak ones?

• It's not an assumption, it's an approximation (small-coupling expansion). Are you asking why it's a good approximation? Commented Jun 2, 2021 at 13:45
• @ChiralAnomaly can you recommend some resource regarding that small-coupling expansion and also why it is a good approximation? Commented Jul 10, 2023 at 5:09
• I don't have any references handy that treat collisions with multiple electrons, and now that I'm revisiting your question, I think my original comment about the small-coupling approximation may have missed the point of your question. The small-coupling expansion I referred to is an approximation we use in the case of a single electron. Going beyond that approximation for a single electron still doesn't (by itself) describe what happens with multiple electrons. Commented Jul 14, 2023 at 1:29

This is basically called the "thin target approximation". The experimental detection rate depends on the product of the photon flux, the target areal density, and the cross section. The achieve a suitable rate, it's better to have more photon flux than more target atoms. Indeed, if the target is too thick, there will be multiple scattering events, and the analysis will be more difficult.

• OP is asking about the Compton scattering, no thick target participates in such scattering. Are you sure you didn’t misinterpret the question? Commented Jun 2, 2021 at 16:05
• "Is it because a photon colliding with multiple electrons [on] its path is very unlikely?" = thin target approximation.
– JEB
Commented Jun 2, 2021 at 16:38
• Maybe I’m misunderstanding something, but isn’t the target a simple electron? Isn’t the right answer “this is just a leading order diagram, other diagrams contain more interaction vertices”? Commented Jun 2, 2021 at 17:43

Quoting one sentence from the wikipedia page,

The interaction between an electron and a photon results in the electron being given part of the energy (making it recoil), and a photon of the remaining energy being emitted in a different direction from the original, so that the overall momentum of the system is also conserved. If the scattered photon still has enough energy, the process may be repeated.

I suspect your confusion here is that the wavelengths received after multiple collisions are well outside the range of wavelengths received after a single scattering event. Keep in mind that, because Compton scattering is defined as acting on bound (or loosely bound) electrons, the event is inelastic. Some of the source photon's energy goes to breaking the electron bond; the remainder can be derived from momentum conservation rules.