# Scattering cross section in terms of current density

I was struggling to understand what a scattering cross section is (in a classical setting), and I found this document:

https://www.ippp.dur.ac.uk/~krauss/Lectures/IntoToParticlePhysics/2010/Lecture4.pdf

If I understand correctly from page 3 if we use a "particle current density" $\vec J (\vec x,t)$ (analogously to electromagnetism) then the differential cross section is given by:

$$\frac{d\sigma}{d \Omega}=\frac{R^2 \vec J_{\infty} \cdot \hat r}{\vec J_0 \cdot \hat n}$$

Where $\vec J_{\infty}$ is the current when the interaction is over and $\vec J_0$ is the current before the interaction happens. This resembles some expressions I've seen regarding scattering of electromagnetic waves, so I thought it might at least make some sense. When we plug in $\vec J$ though problems arise. If we write $\vec J$ for $N$ point particles:

$$\vec J (\vec x, t)=\sum_{i=1}^N \vec v_i \delta (\vec x-\vec x_i (t))$$

Then we have delta functions all over the place and I do not know how to give an interpretation to that thing anymore.

My question is: am I completely off target about this? Is there a way to give meaning to what I wrote and use it to actually calculate a "toy" cross section (for example, Rutherford scattering)?

• I'm a little irritated that Rutherford scattering would be considered a 'toy' problem, particularly since it works for all $1/r^{2}$ potentials. Jan 8, 2016 at 20:12