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## Coulomb Focusing

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

${\frac {1}{2}}m_{r}v^{2}\approx {\frac {1}{2}}m_{e}v^{2}>h\nu _{21}\,\!$
Coulomb focusing gives ${1 \over v^{2}}$ cross-section.

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

$m_{e}v\,b=m_{e}v_{f}a_{0}\,\!$
We know that $v_{f}^{2}=v^{2}+v_{\perp }^{2}$, where $v_{\perp }$ is the velocity $\perp$ to the original electron velocity. This is a result of it falling toward the ion. Then:

${\frac {1}{2}}m_{e}v_{\perp }^{2}\sim {Ze^{2} \over a_{0}}\,\!$
$v_{f}^{2}=v^{2}+{Ze^{2} \over m_{e}a_{0}}\,\!$
$b={a_{0}v_{f} \over v}\,\!$
$\pi b^{2}={\pi a_{0}^{2} \over v^{2}}\left[v^{2}+{Ze^{2} \over m_{e}a_{0}}\right]\,\!$
$=\pi a_{0}^{2}\left[1+\underbrace {Ze^{2} \over m_{e}v^{2}a_{0}} _{Coulomb\ focusing \atop factor}\right]\,\!$
Generally, the Coulomb focusing factor $>1$ because we want to excite, not ionize. $a_{0}={\hbar ^{2} \over Ze^{2}m_{e}}$, so:

$\pi b^{2}={\pi \hbar ^{2} \over m_{e}v^{2}}\,\!$
$\sigma _{12}={\pi \hbar ^{2} \over m_{e}^{2}v^{2}}\overbrace {\left({\Omega (1,2) \over g_{1}}\right)} ^{quantum\ mechanical \atop correction\ factor}\,\!$
$\Omega$ is the “collisional strength”, and generally is 0 below the $v$ threshold, goes to 1 at the threshold, and decreases for increasing $v$, with some occasional spikes. Generally, it is of order 1, with some slight temperature dependency.

$q_{12}=\left\langle \sigma _{12}v\right\rangle \propto \left\langle {1 \over v}\right\rangle \propto {1 \over {\sqrt {T}}}\,\!$
2000 K gas. $v_{term}\sim {\sqrt {\gamma kT \over m}}$, so $v\sim {\sqrt {2000 \over 100}}42\cdot 1{km \over s}\approx 160{km \over s}$. Then

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