Plasmapy: Create Coulomb collision cross-section function

[lemmatum]

Create a cross-section function for Coulomb collision under collisions.py. This should effectively be sigma under collision_frequency. Replace instance of sigma with this new function.

[lemmatum]

Note that the sigma ~ pi * b_perp ** 2 definition of the Coulomb collision cross-section is only valid for large angle collisions (e.g. 90 degree collisions based on b_perp).

• [x] Document this point clearly
• [ ] Check if there is a useful definition for small angle collisions! (perhaps using the Coulomb logarithm?)
• [x] Change docs to define sigma = pi * (2 * b_perp) ** 2 instead of sigma = pi * b_perp ** 2

[ritiek]

Note that the sigma ~ pi * b_perp ** 2 definition of the Coulomb collision cross-section is only valid for large angle collisions

Just wanted to clear some confusion. This definition matches the one in the docs but is there a reason for (2 * bPerp) in the source code?

https://github.com/PlasmaPy/PlasmaPy/blob/166747196434232946086a36ae916fbfc65d6bc6/plasmapy/physics/transport/collisions.py#L717

[lemmatum]

@ritiek Note that sigma ~ pi * b_perp ** 2 is using ~ (similar) rather than = (equals). When we use ~ in physics the tendency is to keep variables but drop coefficients, and equations written in this form are there to give an idea of what are the important effects/parameters rather than to be precise.

sigma = pi * (2 * b_perp) ** 2 is correct, and it follows the general definition for a collisional cross-section between two particles. b_perp is essentially the radius of a particle, but one must account for the radii of both particles in the interaction, as illustrated here.

[lemmatum]

@ritiek we can of course replace the equation in the docs with the more precise version, to eliminate any ambiguity.

EDIT: the docs are wrong, because they use an equality. This needs to be changed. I've added this to the TO DO list.

[lemmatum]

If we have time, we should do this for 0.1 (as it is quite easy), if not it can easily be put off until later.

[StanczakDominik]

This is what I actually closed in #388.

Holy wow, we're at 389 already?!