Partition Function of a Hydrogen Atom - How it is derived?

Question

It is well-known that the Partition Function of Hydrogen Atom diverges if we calculate in naive manner. And I could find the partition function named Brillouin-Planck-Larkin partition function which gives proper physical effect (from SE):

• Partition Function of a Hydrogen Atom
• Partition function of a hydrogen gas $$Z_{BPL}=\sum_{n=1}^\infty n^2 [\exp (-\beta E_n)-1+\beta E_n]\; , $$

where $E_n=R(1-1/n^2)$ (the ground state is at zero energy) and $R$ is the ionization energy.

But when I tried to follow how it was derived, too much of advanced concepts popped up. Currently I am stuck in the papers discussing them:

• https://articles.adsabs.harvard.edu/pdf/1986apj...310..723r
• https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.201200728

As I currently have no access to the scientific papers which is cited, and as my knowledge about statistical mechanics is limited to basic freshman graduate level, I could not follow the argument successfully. So, my questions are:

1. What assumption is made for deriving this partition function?
2. What physical approximation is made while deriving this equation?
3. How 1,2 is combined to derive that partition function?
4. Is $[\exp (-\beta E_n)-1+\beta E_n]/Z_{BPL}$ can be used as a probability of a bound hydrogen electron in (n,l,m) state?

Answer 1

Here is a late answer if you still need one.

The BPL partition function is not a true partition function. It is just part of the second virial coefficient (the virial expansion is valid at low density, so this answers your second question) of hydrogen plasma (which is a system consisting of equal number of electrons and identical nucleus; this answers your first question). Therefore, you cannot (directly) use it to find occupation probability of the $(n,l,m)$ state (this answers your fourth question); instead, the first paper you cited gives you the correct occupation numbers (see its equation 24). For your third question, the second paper you cited gives a fair summary (you can also read Ebeling 2011, which it also cited, for a more detailed but not technical derivation; for technical details, you need to look at papers cited by Ebeling). You can also read Larkin's original paper in 1960, which should be the first convincing derivation of the BPL partition function (it should be, as he is the L in BPL).

I think your main misunderstanding is confusing between a single hydrogen atom and a hydrogen plasma. The main difference is that a hydrogen plasma has many particles to screen the Coulomb potential so that we can get a finite second virial coefficient. Therefore, the probability that a single hydrogen atom is in a certain state is not the same as the occupation fraction of atoms in that state in a plasma. Here is another way to understand this. Remember I said the derivation of BPL partition function assumed low density so that the virial expansion is valid. I didn't talk about the temperature because for any given temperature we can always go to low enough density. However, when the temperature is too low (e.g., room temperature, which is quite an untypical temperature for plasma), the density can never be low enough because even a single hydrogen atom inside the whole universe is too high a density. You can read Miranda 2001 (it is short and easy to understand; it did not consider scattering states, but their contribution is negligible, which you can work out as a simple exercise), which I found in the comments of the first SE question you linked; it derives that a single hydrogen atom in the outerspace has only $10^{-176}$ probability NOT to be in the ground state. This is the opposite of a low-density plasma, where a large portion of atoms is ionized.

All the links are directly PDF, so don't worry about accessing them.