(updated for 2013)
where is the electric field due to the charged particle, is the induced dipole moment in the neutral particle (determined by quantum mechanics), and is the polarizability, which defines for a neutral atom in a uniform static electric field. See Draine, section 2.4 for more details.
This interaction can take strong or weak forms. We distinguish between the two cases by considering b, the impact parameter. Recall that the reduced mass of a 2-body system is In the weak regime, the interaction energy is much smaller than the kinetic energy of the reduced mass:
In the strong regime, the opposite holds:
The spatial scale which separates these two regimes corresponds to , the critical impact parameter. Setting the two sides equal, we see that
The effective cross section for ion-neutral interactions is
Deriving an interaction rate is tricker than for neutral-neutral collisions because in general. So, let’s leave out an explicit n and calculate a rate coefficient k instead, in .
(although really , so k is largely independent of v). Combining with the equation above, we get the ion-neutral scattering rate coefficient
As an example, for interactions we get . This is about the rate for most ion-neutral exothermic reactions. This gives us
So, if , the average time between collisions is 16 years. Recall that, for neutral-neutral collisions in the diffuse ISM, we had years. Ion-neutral collisions are much more frequent in most parts of the ISM due to the larger interaction cross section.