Difference between revisions of "Coulomb Focusing"

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(Created page with '===Short Topical Videos=== * [http://youtu.be/LXGBGNR5JxI Coulomb Focusing (Aaron Parsons)] ===Reference Material=== * <latex> \documentclass[11pt]{article} \def\inv#1{{1 \ove…')
 
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\begin{document}
 
\begin{document}
 
\section*{ Coulomb Focusing}
 
\section*{ Coulomb Focusing}
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Imagine an incident electron has kinetic energy $>h\nu_{21}$.
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$$\frac12 m_rv^2 \approx \frac12 m_ev^2 > h\nu_{21}$$
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\item  Coulomb focusing gives $\inv{v^2}$ cross-section.
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\end{itemize}
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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
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{\bf impact parameter}.  Our collision cross-section $=\pi b^2$.  Our
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angular momentum is conserved, so
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$$m_ev\,b=m_ev_fa_0$$
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We know that $v_f^2 = v^2 + v_\perp^2$, where $v_\perp$ is the velocity
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$\perp$ to the original electron velocity.  This is a result of it falling
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toward the ion.  Then:
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$$\frac12 m_ev_\perp^2 \sim {Ze^2\over a_0}$$
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$$v_f^2 = v^2 + {Ze^2\over m_ea_0}$$
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$$b={a_0v_f\over v}$$
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$$\pi b^2 = {\pi a_0^2\over v^2}\left[v^2+{Ze^2\over m_ea_0}\right]$$
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$$= \pi a_0^2\left[1+\underbrace{Ze^2\over m_ev^2a_0}_{Coulomb\ focusing\atop
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factor}\right]$$
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Generally, the Coulomb focusing factor $>1$ because we want to excite, not
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ionize.  $a_0={\hbar^2\over Ze^2m_e}$, so:
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$$\pi b^2 = {\pi\hbar^2\over m_ev^2}$$
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$$\sigot = {\pi\hbar^2\over m_e^2v^2}
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\overbrace{\left({\Omega(1,2)\over
 +
g_1}\right)}^{quantum\ mechanical\atop correction\ factor}$$
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$\Omega$ is the ``collisional strength'', and generally is 0 below the
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$v$ threshold, goes to 1 at the threshold, and decreases for increasing
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$v$, with some occasional spikes.  Generally, it is of order 1, with some
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slight temperature dependency.
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$$\qot=\mean{\sigot v} \propto \mean{\inv{v}} \propto \inv{\sqrt{T}}$$
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2000 K gas.  $v_{term} \sim \sqrt{\gamma kT\over m}$, so
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$v\sim \sqrt{2000\over 100}42\cdot 1{km\over s} \approx 160{km\over s}$. Then
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$$\sigot \sim 10^{-14}cm^2\left({\Omega(1,2)\over g_1}\right)$$
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$$\sigot\eval{osterbrock} \sim 10^{-15} cm^2$$
  
  
 
\end{document}
 
\end{document}
 
</latex>
 
</latex>

Revision as of 15:21, 2 October 2013

Short Topical Videos

Reference Material

Coulomb Focusing

Imagine an incident electron has kinetic energy .

Coulomb focusing gives cross-section.

We want to know how far away an electron with can be aimed and still hit the radius cloud around the ion. This is , the impact parameter. Our collision cross-section . Our angular momentum is conserved, so

We know that , where is the velocity to the original electron velocity. This is a result of it falling toward the ion. Then:

Generally, the Coulomb focusing factor because we want to excite, not ionize. , so:

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

2000 K gas. , so . Then