Nature of Collision in Compton Scattering

[neilparker62]

TL;DR

Nature of photon/electron collision

I am just wondering how best to visualise a Compton scattering event. Since a photon has no mass, we can't exactly presume it's the same as two masses colliding even if at relativistic speeds.

Does the photon encounter some kind of force field as it approaches the stationary electron ? If so what is the nature of such ? Electric / magnetic / both / neither ?

Another question is whether there is an equation which governs the change in angular momentum experienced by the photon during scattering. Since it deflects, it must experience some kind of torque.

This question arises because of the equation $m \vec{a} \times \vec{s} = m \vec{v} \times \vec{u}$ applicable to projectile motion. Here a projectile moving in a gravitational field experiences torque (about launch point) and the right hand side of the above equation therefore corresponds to rate of change of angular momentum (as observed/proved by PF User Tsny). So I wondered if there is an analogous equation for a photon running into an electron's "force field".

 

[Nugatory]

Does the photon encounter some kind of force field as it approaches the stationary electron ? If so what is the nature of such ? Electric / magnetic / both / neither ?

When you're thinking the photon "approaches" the electron, you're thinking as if the photon is some sort of tiny object with a position and a speed and all of that. This mental model that is encouraged by the word "particle"; unfortunately in quantum physics that word doesn't mean anything like what it means in ordinary English, and the entire notion of the photon moving towards the electron and encountering forces that alter its trajectory and angular momentum is inapplicable. A photon doesn't even have a position in the traditional sense of the word.

 

[neilparker62]

Ok so let's stick to what can be measured/calculated then. In a Compton scattering experiment, we can measure an angle of deflection and we have a formula for calculating that. Hopefully measured agrees (more or less) with that calculated from theory. We also know/measure by some means the energy / (linear) momentum of the photon before and after its encounter with a stationary electron. Happy to be corrected if there are some incorrect perceptions here again. (I'm already puzzled by the quantity hf/c which is represented as a vector in Compton scattering diagrams - how do we multiply frequency by a constant and somehow produce a vector 'out of the hat' so to speak ? And then multiply by c to give energy which is suddenly a scalar again ??).

As is the case with ordinary projectile motion, not much is said (unfortunately) about angular momentum. So what can we say (if anything) about that in the case of Compton scattering ? Is there a 'before' and 'after' applicable in the same way as there's a 'before' and 'after' for the photon's linear momentum ? Or at quantum level is there a complete breakdown of the distinction between vector and scalar quantities ?

https://advances.sciencemag.org/content/2/9/e1600485

Quoting from above article abstract:

"During light-matter interaction, transfer of linear momentum leads to optical forces, whereas transfer of angular momentum induces optical torque."​

 

[vanhees71]

First forget about the very misleading idea photons were "particles" in any classical sense of the word. Photons cannot be localized, because they do not even have a position observable to begin with. While for massive particles, which have a non-relativistic limit, in some approximate sense you can think often in terms of particles about them, that's in no way possible for "photons".

Instead of talking about photons, we should talk about the electromagnetic field. If you want a classical picture, it's amazing how far you can get with the picture of a classical electromagnetic field. E.g., despite contrary claims even in university textbooks to lowest order perturbation theory the Compton effect can be understood in terms of the semiclassical approximation, i.e., treating the electromagnetic field as a classical field and only the electron with quantum theory (the same holds true for the photoelectric effect).

So the intuitive, classical picture, which is much more close to the full quantum picture, is that an electromagnetic wave hits the electron, which starts to get accelerated due to the electromagnetic force and thus itself produces also electromagnetic waves which are superimposed to the incoming electromagnetic wave. The net result is that both the electron and the electromagnetic waves are scattered on each other.

Quantum field theory, which is the only consistent theory of photons and other particles, in the relativistic realm provides a description of this scattering process in terms of the S-matrix, which describes the transition probability from an initial asymptotic free state (in this case a incoming quasifree photon and a quasifree electron) to a final asymptotic free state in terms of probabilities, and from these probabilities you can evaluate the cross section, which then can be measured in experiments an compared to the prediction.

The semiclassical result coincides with the leading order approximation of perturbation theory in QED. This shows that the classical-field picture for the em. wave, of which photons are a specific kind of quantum state in the quantized version of the theory, is not too far from what you get from the full quantum field theory and thus is a better (heuristic) picture of what's going on in such collisions than the naive particle picture of photons, which is never right in any approximate sense.

 

[neilparker62]

Thanks for the very comprehensive response above. Much appreciated - I understand it's difficult to try and explain concepts such as this (which really do require some fairly 'heavy duty' theory) to a lay person with minimal background in the requisite theory.

The reason I'm asking is because I'm writing an article on the "Tan Rule" (to add to the sine/cosine/area rules 'toolbox'!). Two applications thereof are "classical" Compton Scattering and determination of launch angle for a projectile (based on the angular momentum equation). I was struck by the similarity between these two situations which employ almost identical vector diagrams for solution. I hoped there might be a Compton scattering "analogue" of the angular momentum equation.

Will be posting shortly for review.