In two previous papers, the quantum number $k\ensuremath{\equiv}n+l$ was considered as an energy-ordering quantum number for the excited states of 42 atoms and ions, which can be regarded as being composed of a single excited valence electron outside closed shells. It was shown that the energy levels of the valence electron are energy ordered according to the (increasing) values of $k$, and moreover within each $k$ group or $k$ band, there is a definite and generally constant sequence of $l$ values for each atom or ion, as the electronic energy is increased within the $k$ band; this sequence has been called the "$l$ pattern." Altogether 42 spectra involving 1155 levels were included in the previous investigation. Of these 1155 levels, 120 show a small deviation from $k$ ordering, and these have been called "$k+\ensuremath{\lambda}$ exceptions." In the present work, a thorough analysis of these $k+\ensuremath{\lambda}$ exceptions has been made, and it has been found that in the vast majority of the cases, the $k+\ensuremath{\lambda}$ exceptions arise from the fact that the angular momentum $l$ is too large, and is close to or larger than the limiting angular momentum ${l}_{1}$ for which a "phase transition" occurs from $k$ ordering to hydrogenic (H) ordering, which is similar to the phase transition which arises when the ionicity $\ensuremath{\delta}\ensuremath{\equiv}Z\ensuremath{-}N$ ($N$ being the number of electrons) is too large, and exceeds the "limiting ionicity" ${\ensuremath{\delta}}_{1}$. A diagram showing the regions of validity of $k$ and H ordering as a function of the angular momentum $l$ and the atomic number $Z$, i.e., the curve of ${l}_{1}$ vs $Z$, has been obtained. A tentative explanation of the phase diagrams of $\ensuremath{\delta}$ vs $Z$ and $l$ vs $Z$ is discussed, i.e., the reason why too large a value of $\ensuremath{\delta}$ or $l$ will destroy the $k$ ordering phase of the spectrum.