Ordering of the energy levels of the excited states of the alkali-metal atoms

The Madelung rules for the order of the filling of successive electron shells in the ground states of neutral atoms (and hence the properties of the resulting periodic table) suggest that the sum $n+l\equiv k$ is in the nature of an "energy-ordering" quantum number. Thus for neutral atoms, the levels of the outermost electron are filled in the order of increasing $k$ values as the atomic number $Z$ is increased. For a constant $k$, the shells are filled in the order of increasing $n$ or decreasing $l$ values. The general applicability of these ordering rules suggests that they may also be relevant to the structure of the excited-state energy spectrum of the valence electron of the alkali-metal atoms. With the exception of lithium, it was found that the excited-state levels are energy ordered according to the increasing value of $k$ ("$k$ ordering"). For states with the same value of $k$, the levels with increasing energy $E_{\mathrm{nl}}$ exhibit a sequence of $l$ values (the "$l$ pattern") which is characteristic of the alkali-metal atom considered (Na, K, Rb, and Cs), and is independent of $k$. Important additional evidence for the $k$ ordering and the constant $l$ patterns has been obtained from the energy spectra of the singly ionized alkaline-earth atoms, i.e., the ${\mathrm{Mg}}^{+}$, ${\mathrm{Ca}}^{+}$, ${\mathrm{Sr}}^{+}$, ${\mathrm{Ba}}^{+}$, and ${\mathrm{Ra}}^{+}$ ions. Several possible implications of the present results are discussed.