I am trying to understand the difference between the 3 above mentioned terms but i have it hard to picture each of them. I know that the geometrical cross section would be the physical area for which we would have a collision (billiard spheres) and i know that the scattering cross section if we have a charged nucleus and an electron colliding with it, will be bigger then the physical cross section of the nucleus. But what i cannot understand are the following 3 things:

How does the total cross section in case of a scattering event looks like? Like how to visualize that.

How is it related to the differential cross section (geometrically). I assume if i integrated in spherical coordinates the diff. cross section I should get the total one.

And most importantly ,we were given the following formula without actually showing the logic of how it was derivated:

Total rate $$W_r$$ of scattering events $$dN_s$$ per unit of time:

$$W_r = dN_s/dt = J \cdot N_t \cdot \sigma_{tot}$$, where

J is the flux density, $$N_t$$ is the number of the nuclei of the target object and $$\sigma_{tot}$$ is the total cross section.

If there are links/ docs with detailed explanation to the above 3 questions (specially the last one) and about the Rutherford scattering, that would also help!

Thanks

Difference between geometrical cross section / total cross section and differential cross section (scattering)

I am trying to understand the difference between the 3 above mentioned terms but i have it hard to picture each of them. I know that the geometrical cross section would be the physical area for which we would have a collision (billiard spheres) and i know that the scattering cross section if we have a charged nucleus and an electron colliding with it, will be bigger then the physical cross section of the nucleus. But what i cannot understand are the following 3 things:

How does the total cross section in case of a scattering event looks like? Like how to visualize that.

How is it related to the differential cross section (geometrically). I assume if i integrated in spherical coordinates the diff. cross section I should get the total one.

And most importantly ,we were given the following formula without actually showing the logic of how it was derivated:

Total rate $$W_r$$ of scattering events $$dN_s$$ per unit of time:

$$W_r = dN_s/dt = J \cdot N_t \cdot \sigma_{tot}$$, where

J is the flux density, $$N_t$$ is the number of the nuclei of the target object and $$\sigma_{tot}$$ is the total cross section.

If there are links/ docs with detailed explanation to the above 3 questions (specially the last one) and about the Rutherford scattering, that would also help!

Thanks