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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