Identification of the Predicted $5s\text{−}4f$ Level Crossing Optical Lines with Applications to Metrology and Searches for the Variation of Fundamental Constants

We measure optical spectra of Nd-like W, Re, Os, Ir, and Pt ions of particular interest for studies of a possibly varying fine-structure constant. Exploiting characteristic energy scalings we identify the strongest lines, confirm the predicted $5s\text{−}4f$ level crossing, and benchmark advanced calculations. We infer two possible values for optical $M2/E3$ and $E1$ transitions in ${\mathrm{Ir}}^{17+}$ that have the highest predicted sensitivity to a variation of the fine-structure constant among stable atomic systems. Furthermore, we determine the energies of proposed frequency standards in ${\mathrm{Hf}}^{12+}$ and ${\mathrm{W}}^{14+}$.

Figure 1

Grotrian diagrams based on our Fock space coupled cluster calculations of the (a) $4f^{12}5s^2$, and (b) $4f^{13}5s^1$ configurations of Nd-like W, Re, Os, Ir, and Pt ($Z=74–78$). A smooth energy scaling depending on the atomic number, shown here relative to the levels (a) ${{}^{3}F}_3$ and (b) ${{}^{3}F}_4^o$, is observable. In (c), the energy separation between the lowest energy states in each configuration, $4{f^{14}}^1{S}_0$, $4f^{13}5{s^1}^3{F}_4^o$, and $4f^{12}5{s^2}^3{H}_6$, is plotted. The three configurations cross near ${\mathrm{Ir}}^{17+}$ resulting in several $\alpha$-sensitive optical transitions in this species.

Figure 2

Identification of transitions by their characteristic energy scaling for Nd-like W, Re, Os, Ir, and Pt ions. In (a), their measured optical spectra (black lines) are shown; transitions belonging to other charge states are depicted in gray. Several transition energies $E_i$ (symbols) are found to follow a function given by Eq. (1) (colored lines). A comparison between the fitted offsets $A_i$ and the linear coefficients $B_i$ (filled symbols) and the corresponding parameters predicted by Fock space coupled cluster calculations (open symbols) is shown in (b). Uncertainties on the fitted parameters $A_i$ and $B_i$ are smaller than the symbol. Not all possible transitions could be observed and two theoretically expected scalings (black crosses) have not been identified at the current sensitivity. Through a ${\chi }^2$ minimization of theoretical and experimental ($A_i,B_i,C_i$)-parameter assignments, spectral lines along the measured isoelectronic sequence can be clearly identified with their transitions.

Figure 3

Identification of highly resolved ${\mathrm{Ir}}^{17+}$ spectral lines between odd(even)-parity states belonging to the $4f^{13}5s^1$ ($4f^{12}5s^2$) configurations using their Zeeman components. Their characteristic splitting (measured, black lines) is fitted with a model function [red line, see Eq. (2)] using calculated Zeeman components (black squares: $\mathrm{\Delta }m=+1$, blue triangles: $\mathrm{\Delta }m=0$, red circles: $\mathrm{\Delta }m=-1$).

Figure 4

Left: Comparison between the experimental spectrum of the Nd-like ${\mathrm{Ir}}^{17+}$ ion and different atomic calculations performed in this work [except (e)] using the same symbols and colors for the transitions as in Fig. 2. For theories, only $M1$ transitions in the $4f^{12}5s^2$ and $4f^{13}5s^1$ configuration are displayed, using Einstein coefficients as amplitudes. (a) Fock space coupled cluster, (b) configuration interaction Dirac-Fock Sturmian (CIDFS) [41, 42], (c) cowan code [43], (d) Flexible Atomic Code (fac) [44], (e) configuration interaction (CI) [2, 45]. Einstein coefficients were not available for the FSCC and CI methods; the fac ones were used instead. The identified $M1$ transitions (symbols) in the data are connected across the spectra. Right: Mean difference (symbols) and standard deviation (error bars) between theories and experiment. The average separation between the measured transitions of 0.1 eV is indicated with the green band.