# How to calculate the $g$ degeneracy factor for alkali metals and their singly ionized species?

$$\frac{n(X_{i+1})}{n(X_{i})} = \frac{(2\pi m k T)^{1.5}}{n_e h^3}\frac{2g_{i+1}}{g_{i}}e^{-\chi/kT}$$

where $\chi$ is the energy difference between the two ionization states, $n_e$ is the number density of electrons, $k$ is the Boltzmann constant, $h$ is the Planck constant, $T$ is the temperature, $X_i$ is an atom $X$ in the $i$th ionization state, and $g_i$ is the degeneracy factor corresponding to the $i$th ionization state.

My question is how to calculate the degeneracy factor $g_i$ (and $g_{i+1}$) for an atom/ion in a given ionization state. For ground state hydrogen $g_i=2$ (since the ground state electron has two quantum states accessible to it, spin up or spin down) while ionized hydrogen $H^+$ has $g_{i+1}=1$. For helium the ground state has a $g$-factor of 1 since its $1s$ shell is filled and there is only one way to fill a shell, while singly ionized helium has a $g$-factor of 2 for the same reason neutral hydrogen has a $g$-factor of 2.

I specifically would like to calculate the $g$-factor associated with the alkali metals lithium, sodium, potassium, and rubidium. My guess is that for $g_i$ (corresponding to neutral atoms) the $g$-factor will equal 2 because the outermost electron is in an $s$ orbital with two possible configurations, while for $g_{i+1}$ (corresponding to the singly ionized species) it will equal 1 because singly ionized alkali metals have filled valence shells as their outermost electrons, which, since they are completely filled, can only be filled one way: filled. However, my professor has indicated it is more complex that that.

Any links to papers treating this topic will be appreciated as well.

If it is simple to generalize your answer to work for any atoms and their ions then please do so in order to make my question and its answers as generally applicable as possible.

• well as is said the equation is derived from quantum and statistical mechanics, the degeneracy of an energy state, measures the number of essentialy different configurations coresponding exactly to the same eigen-energy. One can use group theory (a-la QM) or statistical methods to calculate the degeneracy factor of a state. – Nikos M. Oct 22 '14 at 17:18
• – Nikos M. Oct 22 '14 at 17:20
• @NikosM. I have already looked at both the articles you suggested and they didn't answer my question. The articles provide general definitions of the degeneracy factor, I want to know how to apply it specifically to atoms and ions. – NeutronStar Oct 22 '14 at 17:28
• hmm, are you looking for the specific calculations? – Nikos M. Oct 22 '14 at 17:30
• For atoms and ions above the hydrogen, approximations are used (e.g quantum chemistry, there is no exact solution for complex atomic systems). If that is the case, this would be a very difficult question (involving many parts) – Nikos M. Oct 22 '14 at 17:31