# Coulomb Focusing

## Coulomb Focusing

Imagine an incident electron has kinetic energy ${\displaystyle >h\nu _{21}}$.

${\displaystyle {\frac {1}{2}}m_{r}v^{2}\approx {\frac {1}{2}}m_{e}v^{2}>h\nu _{21}\,\!}$

Coulomb focusing gives ${\displaystyle {1 \over v^{2}}}$ cross-section.

We want to know how far away an electron with ${\displaystyle v}$ can be aimed and still hit the ${\displaystyle a_{0}}$ radius cloud around the ion. This is ${\displaystyle b}$, the impact parameter. Our collision cross-section ${\displaystyle =\pi b^{2}}$. Our angular momentum is conserved, so

${\displaystyle m_{e}v\,b=m_{e}v_{f}a_{0}\,\!}$

We know that ${\displaystyle v_{f}^{2}=v^{2}+v_{\perp }^{2}}$, where ${\displaystyle v_{\perp }}$ is the velocity ${\displaystyle \perp }$ to the original electron velocity. This is a result of it falling toward the ion. Then:

${\displaystyle {\frac {1}{2}}m_{e}v_{\perp }^{2}\sim {Ze^{2} \over a_{0}}\,\!}$
${\displaystyle v_{f}^{2}=v^{2}+{Ze^{2} \over m_{e}a_{0}}\,\!}$
${\displaystyle b={a_{0}v_{f} \over v}\,\!}$
${\displaystyle \pi b^{2}={\pi a_{0}^{2} \over v^{2}}\left[v^{2}+{Ze^{2} \over m_{e}a_{0}}\right]\,\!}$
${\displaystyle =\pi a_{0}^{2}\left[1+\underbrace {Ze^{2} \over m_{e}v^{2}a_{0}} _{Coulomb\ focusing \atop factor}\right]\,\!}$

Generally, the Coulomb focusing factor ${\displaystyle >1}$ because we want to excite, not ionize. ${\displaystyle a_{0}={\hbar ^{2} \over Ze^{2}m_{e}}}$, so:

${\displaystyle \pi b^{2}={\pi \hbar ^{2} \over m_{e}v^{2}}\,\!}$
${\displaystyle \sigma _{12}={\pi \hbar ^{2} \over m_{e}^{2}v^{2}}\overbrace {\left({\Omega (1,2) \over g_{1}}\right)} ^{quantum\ mechanical \atop correction\ factor}\,\!}$

${\displaystyle \Omega }$ is the “collisional strength”, and generally is 0 below the ${\displaystyle v}$ threshold, goes to 1 at the threshold, and decreases for increasing ${\displaystyle v}$, with some occasional spikes. Generally, it is of order 1, with some slight temperature dependency.

${\displaystyle q_{12}=\left\langle \sigma _{12}v\right\rangle \propto \left\langle {1 \over v}\right\rangle \propto {1 \over {\sqrt {T}}}\,\!}$

2000 K gas. ${\displaystyle v_{term}\sim {\sqrt {\gamma kT \over m}}}$, so ${\displaystyle v\sim {\sqrt {2000 \over 100}}42\cdot 1{km \over s}\approx 160{km \over s}}$. Then

${\displaystyle \sigma _{12}\sim 10^{-14}cm^{2}\left({\Omega (1,2) \over g_{1}}\right)\,\!}$
${\displaystyle \sigma _{12}{\big |}_{osterbrock}\sim 10^{-15}cm^{2}\,\!}$