Angular Momentum Distributions in the Thomas-Fermi-Dirac Model of the Atom

According to the Thomas-Fermi and the Thomas-Fermi-Dirac models of the atom, the electrons in an atom of a given $Z$ have a given continuous distribution in angular momentum. Therefore, the question of how many $s$, $p$, $d$, or $f$ electrons are to be found in an atom of a given $Z$ cannot be answered by the statistical model as it stands. However, this question can be answered by assigning all the electrons which have values of angular momentum within a certain interval to a definite angular momentum quantum number $l$. There is, of course, some arbitrariness involved in selecting the method of angular momentum assignment. Fermi first calculated a set of curves for the numbers $\nu (l, Z)$ of $s$, $p$, $d$, and $f$ electrons in an atom as a function of $Z$ on the basis of a particular angular momentum assignment, and using the Thomas-Fermi statistical theory. Since the completion of Fermi's work, Dirac has modified the Thomas-Fermi theory to include the exchange effects of the electrons. In the present work, curves for $\nu (l, Z)$ have been calculated on the basis of the Thomas-Fermi-Dirac theory in order to investigate the effect on such curves of the exchange interaction between the electrons. Complete sets of curves have been found using the angular momentum assignment proposed by Fermi, and also using an angular momentum assignment proposed later by Jensen and Luttinger. Some isolated points on the $\nu (l, Z)$ curve have been plotted using a third angular momentum assignment. On the basis of the analysis of the graphs so obtained and their comparison with the empirical data, it is concluded that the exchange effects are not negligible for this calculation, that no simple angular momentum assignment such as the ones proposed by Fermi and by Jensen and Luttinger agrees well with the theory, and that it is probably possible by sufficient juggling to find an angular momentum assignment which would fit the empirical data reasonably well.