Coulomb collision

A Coulomb collision is a binary elastic collision between two charged particles interacting through their own electric field. As with any inverse-square law, the resulting trajectories of the colliding particles is a hyperbolic Keplerian orbit. This type of collision is common in plasmas where the typical kinetic energy of the particles is too large to produce a significant deviation from the initial trajectories of the colliding particles, and the cumulative effect of many collisions is considered instead.
Mathematical treatment for plasmas.
In a plasma, a Coulomb collision rarely results in a large deflection. The cumulative effect of the many small angle collisions, however, is often larger than the effect of the few large angle collisions that occur, so it is instructive to consider the collision dynamics in the limit of small deflections.
We can consider an electron of charge formula_1 and mass formula_2 passing a stationary ion of charge formula_3 and much larger mass at a distance formula_4 with a speed formula_5. The perpendicular force is formula_6 at the closest approach and the duration of the encounter is about formula_7. The product of these expressions divided by the mass is the change in perpendicular velocity:
Note that the deflection angle is proportional to formula_9. Fast particles are "slippery" and thus dominate many transport processes. The efficiency of velocity-matched interactions is also the reason that fusion products tend to heat the electrons rather than (as would be desirable) the ions. If an electric field is present, the faster electrons feel less drag and become even faster in a "run-away" process.
In passing through a field of ions with density formula_10, an electron will have many such encounters simultaneously, with various impact parameters (distance to the ion) and directions. The cumulative effect can be described as a diffusion of the perpendicular momentum. The corresponding diffusion constant is found by integrating the squares of the individual changes in momentum. The rate of collisions with impact parameter between formula_4 and formula_12 is formula_13, so the diffusion constant is given by
Coulomb logarithm.
The integral of formula_15 thus yields the logarithm of the ratio of the upper and lower cut-offs. This number is known as the Coulomb logarithm and is designated by either formula_16 or formula_17. It is the factor by which small-angle collisions are more effective than large-angle collisions. For many plasmas of interest it takes on values between formula_18 and formula_19. (For convenient formulas, see pages 34 and 35 of the "NRL Plasma formulary".) The limits of the impact parameter integral are not sharp, but are uncertain by factors on the order of unity, leading to theoretical uncertainties on the order of formula_20. For this reason it is often justified to simply take the convenient choice formula_21. The analysis here yields the scalings and orders of magnitude.