Finite Electronic Partition Function from Screened Coulomb Interactions

A definition of a finite partition function for bound electronic states is presented for a hydrogenic ion, with the associated problems of the fall in intensity of spectral lines and the lowering of the effective ionization potential. The partition function is partly based on numerical solutions of the Schrödinger equation (SE) with the complete screened Coulomb potential (CSCP), where the $1s$, $2s$, $2p$, $3s$, $3p$, and $3d$ states are considered. The CSCP is given by $\left\{V\right(r)\mathrm{},=V_i\mathrm{}(r)=-Z{e}^2\left(\frac{1}{r}-\frac{1}{D+A}\right), 0<~r<~A\}\{=V_0\mathrm{}(r)=-Z{e}^2\frac{D}{D+A}\frac{\exp \left[\frac{\mathrm{}(A-r)\mathrm{}}{D}\right]}{r}, r>~A\}$ where $D$ is the screening radius and $A$ is the mean minimum radius of the ion atmosphere. The standard transformations $x=\frac{2Zr}{\lambda a_0}$ and $E_{\lambda }=-\frac{Z^2\mu e^4}{2{\hslash }^2{\lambda }^2}$, where $\lambda$ is the CSCP quantum number, yield the standard form of the SE equation with $\lambda$ in place of $n$. The numerical solutions are obtained with a nonlinear method that is both accurate and stable. Since the accurate numerical solutions do not yield an explicit maximum-bound principal quantum number, another property of the screened solutions is used to define a decreasing probability of an electron occupying a particular Coulomb eigenstate. The quantity derived is the relative probability of the screened to the unscreened Coulomb state, given by ${\Phi }_{n,l}=\frac{{\lambda }^{2l+1}N(\lambda , l, d, a)}{n^{2l+1}N(n, l)},$ where the $N\text{'}\mathrm{s}$ are the normalizations in $x$ space, calculated with variations of $D$ and $A$ only. This relative occupation probability of the bound state is used to modify the Boltzmann factor in the standard expression for the electronic partition function. Correlations with observations are discussed: there is excellent agreement with the fall in intensity of hydrogen lines in the solar atmosphere and good agreement with lines from laboratory hydrogen at 21°K. The effective ionization potential of hydrogen is calculated using the maximum detected level of hydrogen observed in the solar chromosphere. A simple analytical fit to the $\Phi$ function and a useful approximate analytical expression for the screened Coulomb partition function are given.