Short Topical Videos
Collisional Excitation Cross-sections
The Einstein analog:
So the Rate of Excitations is given by:
Suppose we have some distribution of relative velocities given by , where is the fraction of collisions occurring with relative velocities . Then:
where is the “collisional rate coefficient” . Then the Rate of de-excitation is given by:
We recognize now that is the rate of excitations of A using B moving at relative velocity . If we have detailed balance, then this has to be the same as the rate of de-excitation .
Where is the reduced mass . However many are created by collisional excitation, the same number are used for the reverse de-excitation. This is detailed balance.
Second, under thermal equilibrium, particles have a Maxwellian velocity distribution:
(Maxwellian velocity distribution) In thermal equilibrium,
Now, assuming detailed balance and thermal equilibrium,
This is the analog of the relationship between the Einstein coefficients and .
For a specific case, , ion with bound electron.
More on Einstein Analog
Forgot something for the Einstein Analog. Recall:
For the special case of being Maxwellian, then:
This has a Boltzmann factor, which makes you thing we’re assuming LTE, but we’re not.
Last time, we were talking about electron-ion collisional excitation, given by:
We can extend this for neutral-ion collisional excitation:
where scales as . Notice this means that for some neutral-ion collisional excitation, it is temperature independent. For neutral-neutral collisional excitation:
This is all we’ll talk about bound-bound transitions.