Variance of the distributions of energy levels and of the transition arrays in atomic spectra. III. Case of spin-orbit-split arrays

In previous papers formulas for mean wave numbers and spectral widths of transition arrays have been presented. Here the method is extended to the cases ${\mathrm{nl}}^N$n’ $l_{j\mathcal{’}}^{\mathcal{’}}$-${\mathrm{nl}}^N$n’ ’ $l_{j\mathcal{’}\mathcal{’}}$ ${}_{}{}^{2\mathrm{sprime}}{}_{}{}^{}$, (${\mathrm{nl}}_j$ ${)}^N$ n’ ’ $l_{j\mathcal{’}\mathcal{’}}^{2\mathrm{sprime}}$, and (${\mathrm{nl}}_j$ ${)}^{N+1\mathrm{}}$-(${\mathrm{nl}}_j$ ${)}^N$n’ $l_{j\mathcal{’}}$ ${}_{}{}^{\mathcal{’}}{}_{}{}^{}$, i.e., when the spectrum exhibits several subarrays, due to the effects of large spin-orbit interactions. The first case is typical of the x-ray transitions between the internal subshells of the atom. The second and the third cases occur in the vuv and x-ray spectra of highly ionized heavy atoms. The evolution of the array 3$d^8$4s-3$d^8$4p along the isoelectronic sequence is presented as a first example of calculation, and criteria for the choice of the relevant formula are proposed. A second example is that of the 3$d^9$-3$d^8$4p array in the spectrum of tungsten. Formulas are given (in the Appendices) for the total intensities of subarrays and for the variances of the distribution of energy levels in subconfigurations of the types ${\mathrm{nl}}^N$n’ $l_{j\mathcal{’}}^{\mathcal{’}}$ and (${\mathrm{nl}}_j$ ${)}^N$(n’ $l_{j\mathcal{’}}^{\mathcal{’}}$ ${)}^{N\mathcal{’}}$.