Saha equation partition function for Argon?

[randombill]

This question is in regards to the degeneracy of states for an Argon atom with just one missing electron. For hydrogen the problem of finding the partition function depends on finding the the ionized state of hydrogen divided by the non-ionized state...

(please see Saha equation -> en.wikipedia.org/wiki/Saha_ionization_equation where
they use g_i+1/ g_i but most books use the following)

g_i/ g_a where g_a for hydrogen is 2 because of the number of spins for a proton (I guess) but what about Argon's g_a partition, would this require trying to find all the possible configurations (or degenerate states) down to the core (ground state)?

 

[Drakkith]

Hi randombill,

Is this related to astronomy somehow? If not, I can move it to the appropriate forum.

Thanks.

 

[Ken G]

This question is in regards to the degeneracy of states for an Argon atom with just one missing electron. For hydrogen the problem of finding the partition function depends on finding the the ionized state of hydrogen divided by the non-ionized state...

(please see Saha equation -> en.wikipedia.org/wiki/Saha_ionization_equation where
they use g_i+1/ g_i but most books use the following)

g_i/ g_a where g_a for hydrogen is 2 because of the number of spins for a proton (I guess) but what about Argon's g_a partition, would this require trying to find all the possible configurations (or degenerate states) down to the core (ground state)?

It sounds like you are asking if the partition function for the one-electron-removed Argon should include only the ground state, or all possible excited states. In principle, the latter, but in practice, the former is often all you need. The same can be said about the Argon without the missing electron-- all its excited states should be included too, but might not need to be. This is even true of hydrogen-- if you want to find the probability that a given hydrogen atom will be ionized, then you need to include all the excited states in the non-ionized hydrogen. But we generally would not bother to do that, and indeed the classical Saha equation doesn't do that. The reason is, the energy levels are usually either closely enough spaced that we can consider them to be degenerate with the ground state, or far enough spaced that the Boltzmann factor we'd associate with those states would make their contributions too small to matter. The only time we run into problems is when there are spacings that are of order kT, and in that case we must incude all such states explicitly in the partition function, and not just in some degenerate g factor. I don't know the energy levels of Argon well enough to say at what kT this could be a concern, so you'd have to check on that, but it is a normal assumption to make the "degenerate versus unimportant" distinction and just call it a g instead of a full partition function.

 

[randombill]

Hi randombill,

Is this related to astronomy somehow? If not, I can move it to the appropriate forum.

Thanks.

It's loosely based on astronomy, actually mostly astrophysics, I assume you know that the Saha equation is used for solar spectroscopy, albeit not for argon gas. Its fine if you move it but I cannot see an other category that would fit this question better which is why I choose this forum.

 

[randombill]

It sounds like you are asking if the partition function for the one-electron-removed Argon should include only the ground state, or all possible excited states. In principle, the latter, but in practice, the former is often all you need. The same can be said about the Argon without the missing electron-- all its excited states should be included too, but might not need to be. This is even true of hydrogen-- if you want to find the probability that a given hydrogen atom will be ionized, then you need to include all the excited states in the non-ionized hydrogen. But we generally would not bother to do that, and indeed the classical Saha equation doesn't do that. The reason is, the energy levels are usually either closely enough spaced that we can consider them to be degenerate with the ground state, or far enough spaced that the Boltzmann factor we'd associate with those states would make their contributions too small to matter. The only time we run into problems is when there are spacings that are of order kT, and in that case we must incude all such states explicitly in the partition function, and not just in some degenerate g factor. I don't know the energy levels of Argon well enough to say at what kT this could be a concern, so you'd have to check on that, but it is a normal assumption to make the "degenerate versus unimportant" distinction and just call it a g instead of a full partition function.

Alright, I'll just use the model for hydrogen then with the ionization energy for Argon with a single electron missing from the outer shell, thanks!

 

[Drakkith]

It's loosely based on astronomy, actually mostly astrophysics, I assume you know that the Saha equation is used for solar spectroscopy, albeit not for argon gas.

Well, I know now!