How much energy, momentum, and/or angular momentum are lost to radiation in Rutherford scattering?

Question

Inspired by this question:

Consider two charged particles, of masses $m_1$ & $m_2$ and charges $q_1$ and $q_2$. They approach each other from a great distance, interact via their electromagnetic fields, and end up going in different directions. (For simplicity, let's work in the CM frame.) Usually when we analyse this problem, we assume that the two particles interact via a potential energy $$ V(\vec{r}_1, \vec{r}_2) = -\frac{q_1 q_2}{4 \pi \epsilon_0 r_{12}}. $$ Under these assumptions, via standard arguments, the total mechanical energy $E_\text{mech} = \sum_i \frac{1}{2} m_i \vec{v}_i^2$, mechanical momentum $\vec{p}_\text{mech} = \sum_i m_i \vec{v}_i$, and mechanical angular momentum $\vec{L}_\text{mech} = \sum_i m_i \vec{r}_i \times \vec{v}_i$ of the two particles are conserved in the scattering event.

However, we also know that these charges are accelerating during their collision, and accelerating charges (can) radiate. This radiation can in principle carry energy, linear momentum, and angular momentum away from the particles.

1. Do scattering charges radiate net energy? If so, how much?
2. Does the total mechanical momentum of the charges change in the scattering process? If so, what is $\Delta \vec{p}_\text{mech}$?
3. Does the total mechanical angular momentum of the charges change in the scattering process? If so, what is $\Delta \vec{L}_\text{mech}$?

It seems like this is a natural enough question that someone should have already addressed it, so pointers to the literature (in lieu of a full answer here) would still be helpful. I have a nagging suspicion that there is a nice symmetry argument to be made for question 2 (and possible question 3 as well), but I can't quite put my finger on it.

I'm most interested in an answer couched in the language of classical electrodynamics, though if insight can be gleaned from quantum probabilities I'd be happy to hear about it.

Answer 1

These questions are covered in chapter 15 of the classic text Classical Electrodynamics by Dr. Jackson. I'm not qualified to explain exactly how everything works (I have not gotten to that chapter yet), but I can provide relevant equations and pointers to the text. Also, I'm going to assume you are working non-relativistically (one never knows what another thinks 'classical' means, ha).

1. Yes, scattering charges radiate energy. According to pages 717 and 718 of the aforementioned book, the total energy lost in radiation by a particle with charge $ze$ and mass $M$ traversing unit thickness of matter containing one fixed charge with a charge $Ze$ per unit volume is $$\frac{dE_{rad}}{dx}=\frac{16}{3}Z\left(\frac{Ze^2}{\hbar c}\right)\frac{z^4e^4}{Mc^2}\int_0^1\mathrm{d}x\ln\left(\frac{1+\sqrt{1-x}}{\sqrt{x}}\right)$$
2. I believe the change in momentum is $$(\Delta p)^2=2p^2(1-\cos\theta')$$ as given on page 714. $\theta'$ is the scattering angle.
3. I assume that, since these are both point particles, that one could calculate the change in angular momentum from the change in linear momentum and choice of a suitable origin.

Answer 2

As mentioned in one of the comments, the right keyword here is bremsstrahlung. Try electron-electron bremsstrahlung for example in your favorite search engine. Add "classical" if you want to avoid quantum treatments. This effect has been widely explored. An example of a paper to get you started: https://iopscience.iop.org/article/10.1086/303901/fulltext/34481.text.html This paper states some results and refers to Jackson and papers that you may have access to. I don't have access to my copy of Jackson now, but this seems like the type of things that would be explored in this classic, maybe in a problem if not in an example. I doubt that any recent (less than 50 years old) paper gives a derivation in classical (non-quantum) theory.