Coulomb Focusing

Short Topical Videos

Coulomb Focusing (Aaron Parsons)

Reference Material

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.