Ground-state configurations and theoretical soft-x-ray emission of highly charged actinide ions

It is well known that the lanthanide and actinide elements are formed by the filling of $4f$ and $5f$ subshells which occurs after the filling of $5d$ and $6d$ subshells, respectively, has begun. With increasing ionization one expects the energy levels to eventually regroup to their hydrogenic ordering, i.e., in terms of principal quantum number. In the lanthanides, the $4f$ electron binding energy overtakes that of $5p$ near the 6th or 7th ion stage and $5s$ near the 14th or 15th ion stage, leading to dramatic rearrangements of ground-state configurations. In this paper we report on the results of a study to explore the effects of increasing ionization on the ground-state configurations of actinide ions as a result of $5f$ and $6p$ or $6s$ level crossings. It is seen that the effects generally occur later and are more strongly influenced by spin-orbit splitting than in the lanthanides. The near degeneracies of $5f$ and $6l$ energies in these stages lead to configuration interaction (CI) amongst configurations with variable numbers of $5f$ and $6p$ electrons. The effects of CI on the level complexity are explored for ions along the Rn I sequence and are found to lead to the formation of “compound states” as predicted for the lanthanides. The extreme ultraviolet and soft x-ray spectra of medium and highly charged lanthanides are dominated by emission from unresolved transition arrays (UTAs) of the type $\mathrm{\Delta }n=0$, $4p^{6}4d^{N+1}-4p^{5}4d^{N+2}+4p^{6}4d^{N}4f$, which, in general, overlap in adjacent ion stages of a particular element. Here, the corresponding $\mathrm{\Delta }n=0$, $5p^{6}5d^{N+1}-5p^{5}5d^{N+2}+5p^{6}5d^{N}5f$ UTAs have been studied theoretically with the aid of Hartree-Fock with configuration interaction calculations. As well as predicting the wavelengths and spectral details of the anticipated features, the calculations show that the effects of configuration interaction are quite different for the two different families of $\mathrm{\Delta }n=0$ transitions and, once more, spin-orbit interactions play a major role.

Figure 1

Logarithmic dependence of the ratio $F^k\left(nlnl\right)/{\zeta }_{nl}$ on the spectrum number. Data points connected by dashed lines correspond to actinide ions, with diamonds and triangles representing $nl=5f$ and $6p$, respectively. Data points connected by unbroken lines correspond to lanthanide ions, with circles and squares representing $nl=4f$ and $5p$, respectively.

Figure 2

Lowest-lying energy levels of numerous configurations along the Rn-like isoelectronic sequence. Curves A–I correspond to the configurations $5f^8$, $6s5f^7$, $6s^{2}5f^6$, $6s^{2}6p5f^5$, $6s^{2}6p^{2}5f^4$, $6s^{2}6p^{3}5f^3$, $6s^{2}6p^{4}5f^2$, $6s^{2}6p^{5}5f$, and $6s^{2}6p^6$. The inset shows a continuation of curves A, B, and C up to $Z=109$, with a linear extrapolation being performed above $Z=109$.

Figure 3

Energies of levels belonging to Rn-like ${\mathrm{Cm}}^{10+}$. Configurations with variable $5f/6p$ occupancy are shown.

Figure 4

NNS distribution for the $J^{\pi }=5^{+}$ interacting set in Rn-like ${\mathrm{Cm}}^{10+}$. The dimensionless spacings $s$ are defined on the unfolded energy scale and the histogram is normalized to unity. A Brody distribution with the Brody parameter $q=0.932$ is shown in red (curved line).

Figure 5

Calculated positions of $\mathrm{\Delta }n=0$, $5p^{6}5d^{N+1}-5p^{5}5d^{N+2}+5p^{6}5d^{N}5f$ transitions for Ac, Cm, and Lr calculated using Cowan's suite of codes excluding CI (left) and including CI (right). Black denotes $5d-5f$ transitions and orange (gray) denotes $5p-5d$ transitions.

Figure 6

Calculated gA values versus wavelength data for ${\mathrm{Cm}}^{23+}$ for $\mathrm{\Delta }n=0$, $5p^{6}5d^5-5p^{5}5d^6+5p^{6}5d^{4}5f$ transitions shown in the form of binned scatter plots, where the color bar corresponds to the relative density of transition points within the bins. Each bin was of size $0.1\mathrm{nm}\times 0.1{\mathrm{s}}^{-1}$ [ln(gA)] for each scatter plot.

Figure 7

Calculated positions of the $5p-5d$ and $5d-5f$ transition arrays in the actinides along the Re I isoelectronic sequence [ground-state configuration $\left(\mathrm{Xe}\right)4f^{14}5p^{6}5d^5]$.