# Harvard Astronomy 201b

## CHAPTER: Ion-Neutral Reactions

In Book Chapter on March 7, 2013 at 3:20 pm

(updated for 2013)

In Ion-Neutral reactions, the neutral atom is polarized by the electric field of the ion, so that interaction potential is

$U(r) \approx \vec{E} \cdot \vec{p} = \frac{Z e} {r^2} ( \alpha \frac{Z e}{r^2} ) = \alpha \frac{Z^2 e^2}{r^4}$,

where $\vec{E}$ is the electric field due to the charged particle, $\vec{p}$ is the induced dipole moment in the neutral particle (determined by quantum mechanics), and $\alpha$ is the polarizability, which defines $\vec{p}=\alpha \vec{E}$ 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 $\mu' = m_1 m_2 / (m_1 + m_2)$ In the weak regime, the interaction energy is much smaller than the kinetic energy of the reduced mass:

$\frac{\alpha Z^2 e^2}{b^4} \ll\frac{\mu' v^2}{2}$.

In the strong regime, the opposite holds:

$\frac{\alpha Z^2 e^2}{b^4} \gg\frac{\mu' v^2}{2}$.

The spatial scale which separates these two regimes corresponds to $b_{\rm crit}$, the critical impact parameter. Setting the two sides equal, we see that $b_{\rm crit} = \big(\frac{2 \alpha Z^2 e^2}{\mu' v^2}\big)^{1/4}$

The effective cross section for ion-neutral interactions is

$\sigma_{ni} \approx \pi b_{\rm crit}^2 = \pi Z e (\frac{2 \alpha}{\mu'})^{1/2} (\frac{1}{v})$

Deriving an interaction rate is tricker than for neutral-neutral collisions because $n_i \ne n_n$ in general. So, let’s leave out an explicit n and calculate a rate coefficient instead, in ${\rm cm}^3 {\rm s}^{-1}$.

$k = <\sigma_{ni} v>$ (although really $\sigma_{ni} \propto 1/v$, so k is largely independent of v). Combining with the equation above, we get the ion-neutral scattering rate coefficient

$k = \pi Z e (\frac{2 \alpha}{\mu'})^{1/2}$

As an example, for $C^+ - H$ interactions we get $k \approx 2 \times 10^{-9} {\rm cm^{3} s^{-1}}$. This is about the rate for most ion-neutral exothermic reactions. This gives us

$\frac{{\rm rate}}{{\rm volume}} = n_i n_n k$.

So, if $n_i = n_n = 1$, the average time $\tau$ between collisions is 16 years. Recall that, for neutral-neutral collisions in the diffuse ISM, we had $\tau \sim 500$ years. Ion-neutral collisions are much more frequent in most parts of the ISM due to the larger interaction cross section.