Difference between revisions of "Collisional Excitations"

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dv^\prime \sigto$$
 
dv^\prime \sigto$$
 
$$\boxed{g_1v^2\sigot(v) = g_2v^{\prime2}\sigto(v^\prime)}$$
 
$$\boxed{g_1v^2\sigot(v) = g_2v^{\prime2}\sigto(v^\prime)}$$
This is the ``Einstein analog''.\par
+
This is the analog of the relationship between the Einstein coefficients $B_{12}$ and $B_{21}$.
 +
 
 
For a specific case, $B=e^-$, $A=$ion with bound electron.
 
For a specific case, $B=e^-$, $A=$ion with bound electron.
  

Revision as of 15:15, 1 October 2014

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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.