Experimental and theoretical Ritz–Rydberg analysis of the electronic structure of highly charged ions of lead and bismuth by optical spectroscopy

Intra-configuration fine-structure transitions in highly charged ions (HCIs) result in most cases from changes in the coupling of equivalent electrons. They are multipole forbidden to varying degrees and often occur within the optical range. In HCI with semi-filled nd and nf subshells, electrons can in principle couple to states which are energetically close but with very different total angular momenta, e.g. 0⩽J⩽10 . This gives rise to metastable states with very long lifetimes that exhibit particularly interesting clock transitions or, as in the case of orbital level crossings, acquire a considerable sensitivity to a possible variation of the fine-structure constant α. We investigate both experimentally and theoretically connecting the ground state and adjacent states of Pb XXII to Pb XXXIV and Bi X to Bi XV, respectively, covering complex couplings of electrons in the 4f and 5d shells. These and others such as nd and nf , which also contain equivalent electrons, are of interest for frequency metrology using quantum logic spectroscopy, for atomic structure and QED theory tests, and for the search for Dark Matter candidates. We infer their level structure, benchmark calculations from two different electronic structure codes, and lay the groundwork for inferring the wavelengths of forbidden transitions of higher multipolarity in the optical and VUV region.


Introduction
Frequency metrology of optical clock transitions is currently the most accurate experimental technique [1][2][3].Utilizing quantum logic spectroscopy (QLS) [4] or, e.g.electronshelving techniques, relative uncertainties below 10 −18 can be achieved in determining optical frequencies.In general, such experiments require fast-cycling transitions for laser cooling, which are unavailable in highly charged ions (HCIs).This problem was solved with the introduction of sympathetic cooling for such species [5].The recent implementation of QLS with HCI [6][7][8] has vastly increased the number of species available for frequency metrology since, for each element, many different charge states support forbidden transitions in their respective ground state configurations [9].For its application, QLS [4] requires such narrow ground-state transitions [10].As a consequence, frequency-stabilized lasers with sufficient power for both the narrow transition of the spectroscopy ion and the fast one of the logic ion used to cool the other sympathetically.Some HCI species provide transitions with extreme sensitivity for physics beyond the Standard Model, such as a possible time variation of the fine-structure constant α [9,11,12].Moreover, investigations of quantum electrodynamics (QEDs) contributions to the binding energy and isotopic shifts are also possible.As an additional advantage, the very low polarizability of HCI makes those transitions very insensitive to external electronic perturbations.Furthermore, an increased understanding of the electronic structure of HCI with high atomic numbers aids in the search for an electronic bridge excitation in highly charged thorium [13].
Most of the forbidden optical transitions possible in HCI [14] were never observed in the laboratory, and only a small fraction of them can be resolved against the bright backgrounds of other astrophysical sources-e.g. in the case of coronal lines during solar eclipses.Therefore, the foremost subject of this work is, after performing electronic structure calculations-having unavoidable uncertainties on the order of several percent of the transition energy-finding such transitions in the laboratory and determining their wavelengths with an accuracy improved by at least four orders of magnitude, to enable a future application of QLS.We thus systematically examine a series of Pb and Bi HCI, perform high-resolution emission spectroscopy on them, and search for ground-state transitions.Semi-filled subshells, like [Kr]4d 10 4f m , with m = 3 − 8, found in the ions Pb XXIX to Pb XXXIV, often display several optical transitions.The same holds for the semifilled d-subshells [Xe]5d m , with m = 1 − 6, in Bi X to Bi XV.We combine the results of calculations carried out using two different atomic-structure codes, FAC [15] and ambit [16] with in-EBIT spectroscopy to construct a conclusive electronic structure of the low-lying levels of these ions.A similar approach was applied in earlier work on Ir [17], Sn [18,19], and Pr [20] and Xe [21] ions.
Our study complements recent works from other groups that have investigated various interesting ionic species, for instance [22][23][24][25][26][27][28][29][30], using electron beam ion traps (EBITs).These works also emphasized heavy ions, orbital crossings, and hyperfine structure studies (e.g.[29] on I VIII).It is interesting to mention here that such hyperfine anomalies in neutrals (see [31] for a recent example, and [32] for a general review) can be tools for searching for New Physics, and that the same applies to HCI as well.Lifetime measurements of optical transitions in Mo XXII were reported in [33].Some more of those investigations are listed here: [34] reported W VI to W VIII spectra; [30] analyzed the spectra of Pm-like W XIV; [23] worked on W VIII; [26] re-assessed earlier work on W XII; [35] analyzed W XIII; W XIV was studied in [36]; in [27] observed spectra from ions W XXI up to W XXX were reported.Spectral data from Yb and W were recorded in [25], where the In-like Yb XXII and W XXVI and Sn-like W XXV were compared with detailed calculations.Fei et al [37] deals on Cd-like W XXVII.Most of the works on W HCI are also related to spectroscopic diagnostics of tokamak plasmas [38], e.g.[39].Magnetic-dipole optical transitions observed in tokamak plasmas were analyzed in [40], and optical lines from W HCI that are excited by charge-exchange processes populating high Rydberg states were also investigated there.
Numerous theoretical investigations have accompanied those works since intricacies of core-valence electronic correlations (e.g. in [41], which analyzes Yb XXIV and W XXVIII) still challenge theory.
Here, we cover two long series of isonuclear heavy HCI in a systematic way and with a higher resolution than earlier studies of such ions.This is required for both resolving the Zeeman components needed for unambiguous line assignments and obtaining wavelength uncertainties typically well below 1 part-per-million.Such accuracy is beneficial for the application of Ritz-Rydberg analysis to the level structure and for the future use of such transitions in optical clocks, since laser spectroscopic searches for forbidden transitions are very time-consuming.
With our choice of elements and ionization stages, we find a large number of ground state transitions, which in the case of Pb with its many isotopes can be of interest for King-plotbased searches for fifth forces [21,[42][43][44][45][46].Pb and Bi atoms were ionized in an EBIT [47,48], and prepared in charge-state distributions optimized for the observation of specific ions.An optical grating spectrometer attached to the Heidelberg EBIT [49,50] recorded the emission spectra of their ions in different charge states.These two elements with atomic number Z Pb = 82 and Z Bi = 83 are neighbors in the periodic table, and the scaling law for the fine-structure splitting (roughly ∝ Z 4 ) makes their isoelectronic ions display similar features.Therefore, the present study covers a broad type of HCI with semi-filled 4f and 5d subshells.Due to its nuclear spin I Bi = 9/2, Bi, unlike Pb, displays a hyperfine structure superimposed on the fine structure.From the hyperfine constants, information about the nucleus and its charge distribution can be acquired [51,52].When measuring these constants in HCI, theoreticians would be able to extract, e.g. the nuclear quadrupole moment Q [53,54] without relying on insufficiently accurate calculations of the electric field gradient of the neutral species.Unlike Th, Pb and Bi have convenient volatile compounds that can be used in the molecular beam injector as precursors for loading the EBIT.The present measurements also prepare planned investigations of 299 Th +35 outlined in [13], due to the similarities of the electronic structures.

Theory
We perform electronic structure calculations to establish the level structure and ordering.They also help to assign the observed lines to the corresponding levels.We ascertained the outcome by means of the Ritz-Rydberg combination principle and observed the Zeeman splitting of the lines.For this purpose, we use two different ab initio codes: the Flexible Atomic Code (FAC) [15], and ambit [16].FAC has a strong focus on radiative and collisional processes related to plasma physics.Its fully relativistic approach combines Dirac-Hartree-Fock iterations with the configuration interaction method.It implements distorted-wave approximation to calculate crosssections for electron-impact ionization, collisional excitation, and radiative and autoionization processes.Its collisionalradiative model (CRM) allows for the generation of synthetic spectra to predict the brightest, thus detectable, lines of a given ion.Our CRM calculations do not include radiative excitation, photoionization, autoionization, radiative, or dielectronic recombination since they only play a minor role in optical transitions.For the FAC calculations, we specify an electronic core configuration that is assumed to remain unperturbed in combination with a ground state shell configuration to perform the radial optimization.Then, several shell configurations are added for the subsequent configuration interaction iterations.Our chosen configurations are given in section 4 for each ion [15].
The second code, ambit, combines configuration interaction with many-body perturbation theory (CI+MBPT) in different ways for core and valence electrons.Valence-valence interactions are treated with CI, and core-valence interactions are treated with MBPT to keep the calculations within reasonable limits.In addition to the energy levels and the electrical and magnetic multipole transition matrix elements, this package also provides isotope shifts and Landé g factors.These values are used to fit the line shapes resulting from Zeeman splitting.To use ambit, one specifies the ground state configuration with core and shell electrons separately, e.g. in the form core : shell m shell n , where the m + n shell electrons are partitioned into the relativistic electron levels.These specifications are given in section 4 for each ionic species individually, along with the leading configurations.To reduce the computational time, we restricted the possible configurations by using up to the 5g shell in Pb and the 8g shell in Bi, respectively, for the CI part of the calculations, while for MBPT contributions, we included such up to the 30h shell [16].We linked the corresponding levels of the FAC and ambit calculations by comparing their predicted energy E and total angular momentum J, although the level notation of the codes is different.The combination of Landé g factors and CRM-based synthetic spectra allowed for unambiguous identification of the lines, even in cases where the calculated transition energy is too imprecise.

Line shape model
The Zeeman splitting of the respective upper and lower levels of each of the lines at the B ext = 8 T field of the trap is of the order of µ B × B ext , i.e. 0.5 meV, as shown in figures 3-6.We convert our calibrated vacuum wavelengths λ vac = n air λ air in energies using E = hc/λ vac .We then assume multiple transition components with individual energy shifts depending on the Landé g factor for the total angular momentum J of initial g J,i and final state g J,f respectively, the z projection of the total angular momentum of initial m J,i and final state m J,f , and the Bohr magneton µ B .Line components with the same ∆m = m J,i − m J,f have known relative intensities, while ∆m = 0 components are polarized differently than ∆m = ±1, causing different detection efficiencies in the spectrometer.This leads to two intensity fitting parameters: I 0 for the central peaks with ∆m = 0, and I ± for the peripheral peaks with ∆m = ±1.There are no observable transitions for other values of ∆m.Therefore, we can calculate the relative intensities using the respective Clebsch-Gordan coefficients, where I 0/± refers to either I 0 or I ± , depending on ∆m.
In an EBIT, rapid ion heating by electronic collisions is only partially compensated by evaporative cooling [55,56] in the deep trapping potential acting on the HCI.This raises the temperature of the ions to the 10 6 K regime.For narrow entrance slits, the Doppler broadening of the lines is, therefore, greater than the instrumental resolution.Hence, we can fit the individual lines k as a Gaussian G(E µ , σ), with a joint width σ resulting from the convolution of Doppler width and instrumental function, and an individual central energy E µ = E k .These individual energies are calculated from the fitted central energy E 0 by E k = E 0 + ∆E k (g J,i , g J,i ).This function is weighted with the intensities calculated using the corresponding Clebsch-Gordan coefficients.The offset c is included to compensate for a possible flat underground.Together, this yields the function to model the experimental data.One of the bismuth lines additionally showed a clear hyperfine structure, which is modeled as where m I is the z component of the total angular momentum of the nucleus [57].The first term is the Zeeman effect; the second is the first-order hyperfine structure, which would be the sole term at even stronger magnetic fields.The third term is due to the incomplete Paschen-Back effect present at the 8 T magnetic field [58], which breaks the otherwise present symmetry in m J .Since all Landé g factors addressed hereafter relate to the total angular momentum, we omit the index J and write g = g J .

Experiment
For ionization and trapping of the HCI, we use an EBIT [48,59].The Heidelberg-EBIT has an 8 T strong magnetic field that induces a measurable splitting of the Zeeman components [49].This field is necessary to compress the electron beam generated by a thermionic cathode and accelerated by static electric fields.Along the beam axis, the high electron density generates a negative space-charge potential responsible for trapping positive ions radially.At the same time, electronion interactions take place at a high rate.Starting from volatile compounds, we inject the elements of interest as a tenuous molecular beam which is collimated by a two-stage differential pumping system.Tetraethyl lead (CAS No. 78-00-2) is used for Pb and triphenyl bismuth (CAS No. 603-33-8) for Bi.At the trap center, the electron beam dissociates those molecules into atoms, ionizes these, and traps the resulting ions radially.An axial potential well formed by ring electrodes completes the approximately ∼200 µm wide, ∼5 cm long, cylindrical trapping region.Sequential electron-impact ionization proceeds to higher charge states as long as the electron-beam energy given by acceleration potential surpasses the corresponding ionization potentials.Competing electron-ion processes such as direct radiative recombination, dielectronic recombination, and other resonant processes, and charge exchange with residual gas result in a steady-state narrow charge-state distribution with about two dominating charge states.
Low-mass components of the injected compoundhydrogen, carbon, oxygen-give rise to ions that are less bound by the trapping potential since they can only reach lower charge states than those of the heavier Pb and Bi.Ionion Coulomb interactions thermalize the different ions in the trapped ensemble, causing the light ones to evaporate from the trap, efficiently cooling the remaining HCI [55,56].By carefully tuning the potential well, keeping only the heavy ions of interest is possible.Moreover, the ensuing cooling also reduces the Doppler broadening of the line, enhancing the spectral resolution.
Optical radiation naturally emerges from the beam-ion interaction as collisionally excited states relax by spontaneous emission.Mutual repulsion of the trapped HCI suppresses collisional quenching of metastable states.Electron impact acts in two directions, both quenching and populating metastable states.As a consequence, magnetic dipole transitions with lifetimes of tens of milliseconds can radiatively decay, generating the line emission recorded by the spectrograph.
Figure 1 outlines the optical setup.In-vacuum fused-silica lenses L1 and L2 relay the radiation emitted from the horizontal, cylindrical ion cloud to an intermediate image outside the experimental chamber.This horizontal image is then rotated by 90 degrees by a periscope (M1, M2, M3) and refocused by L3 and L4 onto the vertical entrance slit of the spectrometer.The position of L4 is adjustable to compensate for the wavelength-dependent chromatic aberration of the lenses.The polarization-dependent efficiency of all mirrors and the spectrometer grating requires an additional fit parameter as described in section 2.1.
Given the large number (order of millions) of trapped HCI projected into the entrance slit, single spectra of such transitions can be acquired with exposure times of several minutes even when the slit is set to values around 40 µm for good resolution.
We now estimate this figure.The fraction of the total solid angle that the spectrograph covers is approximately only ≈0.024%; reflection and transmission losses in lenses and mirrors amount to ≈50%, and with an average CCD quantum efficiency of ≈65% we arrive at a total photon-detection efficiency of ≈7.8 × 10 −5 .This number has to be multiplied by the number of ions seen by the entrance slit.A typical 500 eV, 40 mA electron beam implies a linear charge density of 1.9 × 10 8 e cm −1 .The trapped ions compensate for this negative charge density at the level of approximately 30%.If we assume an average ionic charge of q = −19 e, we arrive at around 3 million HCI per centimeter trap length.One centimeter is the approximate height of our entrance slit.Since for the charge states of interest, the width of the image of the trapped ion cloud on the slit is roughly five times broader than the slit itself, we collect the light emitted by approximately 600 000 HCIs with a detection efficiency of ≈7.8 × 10 −5 .Including cascades, electron-impact excitation rates per ion are of the order of several Hz, which translate into the same emission rates.In total, these figures yield up to hundreds of counts per second for strongly excited M1 transitions within the respective line profile.
For calibration, a hollow cathode lamp is used.An optical fiber transmits light from it to a retractable reflector that can be inserted at the plane of the intermediate image.This way, the grating illumination solid angle is essentially the same as when EBIT spectra are recorded.An entrance slit controls the radiation bandwidth admitted by the McPherson Model 2062 Czerny-Turner spectrometer.It is located at the source point of the curved mirror M4, which collimates the radiation and directs it to the grating.Three different gratings were used: one with 150 grooves mm −1 to take coarse overview spectra, one with 1800 grooves mm −1 for precision measurements between 400 nm and 800 nm, and one with 3600 grooves mm −1 for the range between 200 nm and 400 nm.The spectrally dispersed beam is refocused by a second curved mirror M5 to project the spectral image on the focal plane on an Oxford Instruments Newton 940 CCD camera cooled down to −80 • C to reduce thermal noise.The relative contribution of the readout noise was minimized by using long exposure times (T > 30 min).Background images were taken with the electron beam running but operating the trap with inverted potentials to empty it of the HCI of interest.This way, the background from stray light, mainly from the hot cathode, can be subtracted.Background images for the calibration are taken with the retractable reflector in place but the lamp turned off.Since the calibration lines are relatively bright, allowing shorter acquisition times, and are independent of EBIT operation, their background images depend less on ambient conditions and are therefore taken between measurement sequences.
Detrimental effects for thermal drifts of the spectrometer are mitigated by increasing the number of measurements and calibrations, whereby sequences of ion and calibration measurements with some additional empty-trap measurements are acquired.Background-corrected calibration lines are fitted individually for estimating residual thermal drifts.The resulting positions are fitted in a time-dependent manner with A • sin(ω(t − t 0 )) + p j with common amplitude A, phase shift t 0 and frequency ω for all and an individual mean position p j for each of the j calibration lines.This approach is justified by the daily cycle of temperature oscillations which affected the laboratory during the course of the present study.When the frequency ω could not be adequately determined-either because the overall duration of the sequence was too short or because there were not enough strong calibration lines for the wavelength range or a combination of both-ω = 2π /24 h = 0.262 h −1 , with an uncertainty of 0.05 h −1 , corresponding to a day cycle and consistent with the evaluation in [61], has been used.For very short sequences, the sine curve was approximated by a first-order polynomial.Taking the value of the fitted sine at the averaged time of the image acquisition gives the thermal shift for each spectrum, shown in figure 2 as an example.The shifted images are averaged and projected onto the dispersive axis in pixel units, delivering a single spectrum of the calibration lamp and a single spectrum of the HCI.After fitting each calibration line separately, a 2nd order polynomial fit is performed, yielding the dispersion curve, linking their nominal wavelengths [62] to their position in pixel units.
We fit the line profiles with Gaussian functions, which generally match them very well.Small line asymmetries can, however, arise from uneven grating illumination by the light entering through the slit.To ensure that this effect does not compromise the centroid position determination of the calibration lines due to the use of the removable reflector, an alternative fitting method was tested where applicable, e.g. if bright, non-overlapping lines are present.For this, we assumed a line profile resulting from that of the shape of the calibration light on the reflector, imaged by the lens system onto the entrance slit plane.This image is only partly transmitted through the entrance slit, and then convoluted with a Gaussian function.To analyze the spectral position of the lines, rather than their precise shape, we only want to treat their line wings where the properties of the convoluted Gaussian predominate.This is achieved by excluding data above 70% of the maximum line height.We then take the wings left and right and fit each of them a Gaussian of the same width σ G , obtaining a separation of the two of µ i .The mean of both central positions μ is then taken as the line position.Compared with this alternative method of fitting, the direct fitting of one single Gaussian to each transition proved more robust and was therefore retained.Any deviation between this procedure and a Gaussian fit is taken into account as a systematic error.The uncertainties of the line position in the Zeeman fit in pixels, of the pixelwavelength relation extracted from the dispersion fit, and of the thermal-shift compensation are the largest sources of statistical error in this experiment.Our experimental technique, as well as the image-processing method, were improved in earlier works of our group [20,21,46,60,61,[63][64][65].

Results
We first took overview spectra of all charge states using the coarse 150 grooves mm −1 grating.The electron beam energy was scanned in steps of 10 eV, from 600 eV to 2000 eV for Pb and from 80 eV to 500 eV for Bi, where the various charge states are optimally produced, allowing for preliminary line assignments.Lines of interest are remeasured with a fine 3600 grooves mm −1 grating (λ < 400 nm) or 1800 grooves mm −1 grating (λ > 400 nm), allowing up to subpm precision.Although the resolving power of this second grating is only half as much as that of the first one, we employ it where needed due to its wider wavelength coverage.We list Example of a sinusoidal fit for determining and correcting the thermal shift.Fitted centroids showing the daily drifts of calibration lines on the dispersion plane at different measurement times (red symbols).Dashed lines are sinusoidals fitted to the data, with a common amplitude, phase shift, and period for all calibration lines.Each plot features a line offset by its individual mean line position.The sinusodials feature a fixed cycle period of a day (blue) or treat the period as a fit parameter (green).We choose the second option as more accurate in this case and display as a green shaded area the confidence bounds of the fit.
the statistical uncertainty ∆λ stat and the systematic uncertainty ∆λ sys .
Figures 3-6 show a variety of Zeeman structures.As exemplary in these figures, the line model described in section 2.1 agrees very well with all the observed lines, except for those we have marked as only preliminarily identified.In these cases, either the spectral resolution or the calculation accuracy was insufficient to confirm the assignment unambiguously.
In order to estimate the effect of the ion temperature, we investigate the transition shown in figure 3, with only three clearly resolved Zeeman components.We fit to them a Gaussian profile, of which the width depends on our instrumental resolution and the Doppler broadening arising from the ion temperature.This combined line width was σ E = 8.092(25) × 10 −5 eV.The former can be determined by comparison with our calibration spectra, which are only broadened by the finite instrumental resolution and were thus much narrower, σ E = 2.986(43) × 10 −5 eV.By subtracting this contribution, we can extract from the measured line width σ E = 7.521(31) × 10 −5 eV an upper limit for the ion temperature (10) MK, where the mass of the ion is m, the speed of light c and the Boltzman constant k b .

Lead
The overview spectrum of lead, obtained by combining several measurements with the 150 grooves mm −1 grating, is shown in figure 7.For thirteen different lead ions, Pb XXII to Pb XXXIV, magnetic dipole transitions between 240 nm and 800 nm were identified.Thereby, we observe small systematic deviations of typically ∆λ s ⩽ 0.005 px, corresponding to uncertainties between 0.008 nm and 0.0017 nm depending on the spectral dispersion, which changes with grating and grating angle.Among those lines, we identified eleven ground state transitions.For Pb XXIX to Pb XXXIV, many Top: fit of the 1 → 0 line of PbXXXI at λ = (392.65739 ± 0.000 26 ± 0.000 06) nm, with the value given as λ ± ∆λstat ± ∆λsys.Cyan, purple, and red lines show the individual Zeeman components.As the total angular momentum J changes from 1 to 0, with a large initial Landé g factor of g i ≈ 1.5, the resulting three separate Zeeman components enable a precise determination of the line width.More details are presented in table 1. Bottom: closest calibration line (orange) for comparison with the ion spectrum (blue).This line was used to determine the apparatus' contribution to the resulting line shape.From that, we infer a value of T ≈ 1.27(1) MK as the upper limit of the ion temperature.more lines connecting other states were identified, as seen below.Table 1 shows them alongside their calculated values.In the next sections, we designate the ground shells with Fit of the 3 → 1 line of PbXXV.Cyan, purple, and red lines show its separate Zeeman components, the blue one their sum, and the green shaded area their confidence bounds.This is a J = 4 → 4 transition with Landé g factors of the initial and final state of g i ≈ 1.1 and g f ≈ 1.2, respectively.We found this line at λ = (252.17575 ± 0.000 25 ± 0.000 08) nm, see table 1, with the value given as λ ± ∆λstat ± ∆λsys.
Figure 5. Fit of the 4 → 2 line of PbXXXIII.Cyan, purple, and red lines show its separate Zeeman components, the blue one their sum, and the green shaded area their confidence bounds.This is a J = 1 → 2 transition with Landé g factors of the initial and final state of g i ≈ 0.2 and g f ≈ 0.9, respectively.We found this line at λ = (381.97750 ± 0.000 24 ± 0.000 06) nm, see table 1, with the value given as λ ± ∆λstat ± ∆λsys.
[Kr]4d 10 4f 14 5s 1 being the ground state, it is only comprised of a single level and does not partake in any optical transition.The transition listed in table 1 takes place between two [Kr]4d 10 4f 13 5s 2 levels.Since its decay to the ground state would require a ∆J = 3 transition, the lowest level L = 1 is metastable.

Nd-like Pb XXIII.
According to NIST, Nd-like lead Pb XXIII has an electronic ground state configuration of [Kr]4d 10 4f 13 5s 1 [62].However, both FAC and ambit converged on [Kr]4d 10 4f 14 , and extended calculations were performed using that configuration instead.We ran FAC with a core configuration of [Kr] and used 4d 10 4f 14 for the radial optimization.As possible shell configurations, we included 4d 10 4f 14 , 4d 10 4f 13 5s 1 and 4d 10 4f 13 5p 1 .For ambit, we gave the ground state configuration as [Kr] 4d 10 : 4 f6 4f 8 and the leading configurations 4f 14 and 4f 13 5s 1 .Here, we also allow up to two electron excitations and one hole excitation.The ground state [Kr]4d 10 4f 14 is not involved in the observed optical transition listed in table 1, which takes place between two [Kr]4d 10 4f 13 5s 1 levels.As in the case of Pb XXII, the lower level with L = 1 is a long-lived metastable state that only relaxes to the ground state by a ∆J = 4 electric hexadecapole transition.

Pr-like Pb XXIV.
For Pr-like lead Pb XXIV, NIST lists a ground state configuration [Kr]4d 10 4f 13 [62].Both FAC and ambit calculation results confirm this.In FAC included a core configuration of [Kr] and used 4d 10 4f 13 for the radial optimization.For the interacting configurations, we included 4d 10 4f 13 , 4d 10 4f 12 5s 1 , 4d 10 4f 12 5p 1 , 4d 9 4f 14 and 4d 9 4f 13 5s 1 .In ambit, we selected the ground state configuration [Kr] 4d 10 : 4 f5.574f 7.43 and for the leading configuration 4f 13 .As in the previous cases, we allow up to two electron excitations.However, we do not foresee any hole excitation based on the FAC results.All transitions measured and listed Figure 7. Spectra of Pb ions in the charge states dominating the distribution as a function of the electron beam energy obtained at low resolution with a 150 grooves mm −1 grating.Main panel: two-dimensional map of the spectra at different electron-beam energies.Projected spectra corresponding to the abundances of the ions are superimposed at the energy of their highest abundance.Right panel: charge-state distribution extracted from the intensity of the brightest lines of each charge state as a function of the electron beam energy.Optical spectra and charge-state distribution were extracted from the two-dimensional scan using the non-negative matrix factorization method, as described in [18].
in table 1 occur between different levels of the [Kr]4d 10 4f 13 configuration.
All measured transition listed in table 1 occur between different levels of the [Kr]4d 10 4f 12 configuration.Figure 4 shows one of the found transitions in greater detail regarding its Zeeman structure.

La-like Pb XXVI.
According to NIST, La-like lead Pb XXVI has an electronic ground state configuration of [Kr]4d 10 4f 11 [62].As FAC and ambit agree with this assessment, we used it as the ground state in our calculations.Thus, we ran FAC with a core configuration of [Kr] and used 4d 10 4f 11  for the radial optimization.As possible shell configurations, we included 4d 10 4f 11 , 4d 10 4f 10 5s 1 , 4d 10 4f 10 5p 1 , 4d 9 4f 12 and 4d 9 4f 11 5s 1 .For ambit, we gave the ground state configuration as [Kr] 4d 10 : 4 f4.71 4f 6.29 and the leading configuration 4f 11 .We have included up to two electron excitations and no hole excitation.All transition measured and listed in table 1 occur between different levels of the [Kr]4d 10 4f 11 configuration.According to NIST, Ba-like lead Pb XXVII has an electronic ground state configuration of [Kr]4d 10 4f 10 [62].As FAC and ambit agree with this assessment, it was used for our calculations.Thus, we ran FAC with a core configuration of [Kr] and used 4d 10 4f 10 for the radial optimization.As possible shell configurations, we included 4d 10 4f 10 , 4d 10 4f 9 5s 1 , 4d 10 4f 9 5p 1 , 4d 9 4f 11 and 4d 9 4f 10 5s 1 .For ambit, we gave the ground state configuration as [Kr] 4d 10 : 4 f4.29 4f 5.71 and the leading configuration 4f 10 .We have included up to two electron excitations and no hole excitation.All transition measured and listed in table 1 occur between different levels of the [Kr]4d 10 4f 10 configuration.

Xe-like Pb XXIX.
According to NIST, Xe-like lead Pb XXIX has an electronic ground state configuration of [Kr]4d 10 4f 8 [62].As FAC and ambit both agree with this assessment, it was used for our calculations.Thus, we ran FAC with a core configuration of [Kr] and used 4d 10 4f 8 for the radial optimization.As possible shell configurations, we included 4d 10 4f 8 , 4d 10 4f 7 5s 1 , 4d 10 4f 7 5p 1 , 4d 9 4f 9 and 4d 9 4f 8 5s 1 .For ambit, we gave the ground state configuration as [Kr] 4d 10 : 4 f3.43 4f 4.57 and the leading configuration 4f 8 .We have included up to two electron excitations and no hole excitation.All transition measured and listed in table 1 occur between different levels of the [Kr]4d 10 4f 8 configuration.Those transitions are solid black lines in figure 8.A potentially optical E2 clock transition, 1 → 0, with λ ≈ 595 nm, as calculated by ambit, is denoted green.A level predicted at E ≈ 9.3 eV, J = 4, shown in gray, could be linked to L = 1 with a λ ≈ 170 nm and to L = 2 with a λ ≈ 190 nm transition in future VUV experiments.This would determine the E2 clock transition energy.
4.1.9.I-like Pb XXX.According to NIST, I-like lead Pb XXX has an electronic ground state configuration of [Kr]4d 10 4f 7 [62].As FAC and ambit both agree with this assessment, it was used for our calculations.Thus, we ran FAC with a core  (bottom).Solid black lines between the levels mark identified transitions.Energy levels linked by observed transitions are shown over their total angular momentum J.For transitions not linked to the ground state, the lowest energy level has been taken from ambit calculations.A predicted energy level, shown in gray, is connected by two VUV transitions (in purple) that were not yet measured.The green line marks a highly forbidden optical ground state transition, a promising candidate for a clock transition.The inlay in the bottom panel shows the Pb XXX 3 → 1 line and its Zeeman components.configuration of [Kr] and used 4d 10 4f 7 for the radial optimization.As possible shell configurations, we included 4d 10 4f 7 , 4d 10 4f 6 5s 1 , and 4d 9 4f 8 .
For ambit, we gave the ground state configuration as [Kr] 4d 10 : 4 f3 4f 4 and the leading configuration 4f 7 .We have included up to two electron excitations and no hole excitation.
All transition measured and listed in table 1 occur between different levels of the [Kr]4d 10 4f 7 configuration.The calculations show that the first excited level is too high for optical ground state transitions.However, we have found some transitions between excited levels, depicted as solid black lines in figure 8.

Te-like Pb XXXI.
According to NIST, Te-like lead Pb XXXI has an electronic ground state configuration of [Kr]4d 10 4f 6 [62].As FAC and ambit both agree with this assessment, it was used for our calculations.Thus, we ran FAC with a core configuration of [Kr] and used 4d 10 4f 6 for the radial optimization.As possible shell configurations, we included 4d 10 4f 6 , 4d 10 4f 5 5s 1 , and 4d 9 4f 7 .For ambit, we gave the ground state configuration as [Kr] 4d 10 : 4 f2.574f 3.43 and the leading configuration 4f 6 .We have included up to two electron excitations and no hole excitation.All transition measured and listed in table 1 occur between different levels of the [Kr]4d 10 4f 6 configuration.They are all connected, as shown figure 9. Therefore, we obtained the precise energies of all involved levels.The ground state transition is shown with a detailed Zeeman structure in figure 3.

Sb-like Pb XXXII.
According to NIST, Sb-like lead Pb XXXII has an electronic ground state configuration of [Kr]4d 10 4f 5 [62].As FAC and ambit both agree with this assessment, it was used for our calculations.Thus, we ran FAC with a core configuration of [Kr] and used 4d 10 4f 5 for the radial optimization.As possible shell configurations, we included 4d 10 4f 5 , 4d 10 4f 4 5s 1 , and 4d 9 4f 6 .For ambit, we gave the ground state configuration as [Kr] 4d 10 : 4 f2.14 4f 2.86 and the leading configuration 4f 5 .We have included up to two electron excitations and no hole excitation.All transition measured and listed in table 1 occur between different levels of the [Kr]4d 10 4f 5 configuration.They are all connected, as shown in figure 9. Therefore, we obtained the precise energies of all involved levels.The measured Zeeman structure of the ground state transition is shown in figure 6.
4.1.12.Sn-like Pb XXXIII.According to NIST, Sn-like lead Pb XXXIII has an electronic ground state configuration of [Kr]4d 10 4f 4 [62].As FAC and ambit both agree with this assessment, it was used for our calculations.Thus, we ran FAC with a core configuration of [Kr] and used 4d 10 4f 4 for the radial optimization.As possible shell configurations, we included 4d 10 4f 4 , 4d 10 4f 3 5s 1 , and 4d 9 4f 5 .For ambit, we gave the ground state configuration as [Kr] 4d 10 : 4 f1.71 4f 2.29 and the leading configuration 4f 4 .We have included up to two electron excitations and no hole excitation.All transition measured and listed in table 1 occur between different levels of the [Kr]4d 10 4f 4 configuration.The measured transitions are seen as solid black lines in figure 10.As an example, the Zeeman structure of the 4 → 2 transition is shown in figure 5.
The level structure features many promising pathways for VUV spectroscopy, most notably 6 → 0, denoted purple.This ground state transition features a strong relative CRM intensity I F = 1.26 at a wavelength λ ≈ 130 nm.Observing this transition would allow us to determine the energy of the 2 → 0 E2 clock transition predicted at λ ≈ 225 nm with ambit, which will be accessible with VUV frequency combs [66][67][68].According to NIST, In-like lead Pb XXXIV has an electronic ground state configuration of [Kr]4d 10 4f 3 [62].As FAC and ambit agree with this assessment, it was used for our calculations.Thus, we ran FAC with a core configuration of [Kr] and used 4d 10 4f 3 for the radial optimization.As possible shell configurations, we included 4d 10 4f 3 , 4d 10 4f 2 5s 1 , and 4d 9 4f 4 .For ambit, we gave the ground state configuration as [Kr] 4d 10 : 4 f1.29 4f 1.71 and the leading configuration 4f 3 .We have included up to two electron excitations and no hole excitation.All transition measured and listed in table 1 occur between different levels of the [Kr]4d 10 4f 3 configuration.The calculations show that the first excited level is too high for optical ground state transitions.We have found some transitions between excited levels (compare figure 10).

Bismuth
We found 19 magnetic dipole (M1) transitions in Bi X to Bi XV.Here, the systematic error is within the uncertainties for narrow entrance slits, especially for the measurement with finer gratings.This effect gets more significant in lower precision measurements with a coarser grating and wider entrance slit.Figures 12-14 show these results, which are again summarized in table 1.To designate the ground shells in the following subsections, the abbreviation [Xe] = 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 5s 2 5p 6 , analogous to the NIST database [62], is used.
An overview spectrum by combined measurements with the 150 grooves mm −1 grating is shown in figure 11.The measured transitions are additionally marked with an arrow, where their filling indicates their diffraction order.This does not mean that all of these higher orders were seen, as some are too weak.A strong overlap of the different charge states is observed due to the small differences in ionization energies.

Lu-like Bi XIII.
Lu-like bismuth Bi XIII has an electronic ground state configuration of [Xe]4f 14 5d 3 [62].FAC and ambit both agree with this assessment.Thus, we ran FAC with a core configuration of [Xe]4f 14 and used 5d 3  for the radial optimization.As possible shell configurations, we included 5d 3 , 5d 2 5f 1 , 5d 2 5g 1 , 5d 2 6s 1 , 5d 2 6p 1 , 5d 2 7s 1 , and 5d 2 7p 1 .For ambit, we gave the ground state configuration as [Xe] 4f 14 : 5 d1.2 5d 1.8 and the leading configuration 5d 3 .We have included up to two electron excitations and no hole excitation.Five transitions between different levels of the [Xe] 4f 14 5d 3 configuration have been found, compare  (bottom).Solid black lines between the levels indicate identified transitions.Dotted lines denote preliminary identifications which require further measurements to be unambiguously identified.Energy levels linked by measured transitions are shown over the total angular momentum J.For transitions not linked to the ground state, the lowest energy level has been taken from ambit calculations.figure 13.Except for the two ground state transitions, identifications are preliminary.Four transitions are connected, allowing us to infer the energy of four levels relative to the ground state.Finding a link to the other transition-using VUV spectroscopy-may provide the higher optical ground state transition 3 → 0 with ∆J = 2. 4.2.5.Yb-like Bi XIV.Yb-like bismuth Bi XIV has an electronic ground state configuration of [Xe]4f 14 5d 2 [62].FAC and ambit both agree with this assessment.Thus, we ran FAC with a core configuration of [Xe]4f 14 and used 5d 2 for the radial optimization.As possible shell configurations, we included 5d 2 , 5d 1 5f 1 , 5d 1 5g 1 , 5d 1 6s 1 , 5d 1 6p 1 , 5d 1 7s 1 , Figure 14.Grotrian diagram showing observed transitions in Bi XIV.Solid black lines between the levels indicate identified transitions.Dotted lines denote preliminary identifications that require further measurements for confirmation.Energy levels linked by measured transitions are shown over the total angular momentum J.For transitions not linked to the ground state, the lowest energy level has been taken from ambit calculations.The line marked in red is too low in intensity to be detectable with our setup, and the one shown in purple has an energy too high for it.The green line shows a promising candidate for a higher-multipole order optical ground state transition.and 5d 1 7p 1 .For ambit, we gave the ground state configuration as [Xe] 4f 14 : 5 d0.8 5d 1.2 and the leading configuration 5d 2 .We have included up to two electron excitations and no hole excitation.Six transitions between different levels of the [Xe] 4f 14 5d 2 configuration have been identified, involving seven of the eight levels predicted by FAC.However, it is divided in the ground state branch L ∈ 0, 2, 4, 6 and the branch L ∈ 1, 3, 5, 7, which are not interconnected yet, compare figure 14.The transition 3 → 0, marked violet, is out of range of our measurement setup, and the 3 → 2 transition, marked red, is too weak.It might be possible to find a link between these branches using VUV or XUV spectroscopy in future experiments.Connecting the branches would reveal a potentially interesting optical E2 clock transition marked green.
4.2.6.Tm-like Bi XV.Tm-like bismuth Bi XV has an electronic ground state configuration of [Xe]4f 14 5d 1 [62] FAC and ambit both agree with this assessment.Thus, we ran FAC with a core configuration of [Xe]4f 14 and used 5d 1 for the radial optimization.As possible shell configurations, we included 5d 1 , 5f 1 , 5g 1 , 6s 1 , 6p 1 , 7s 1 , and 7p 1 .For ambit, we gave the ground state configuration as [Xe] 4f 14 : 5 d0.4 5d 1.6 and the leading configuration 5d 1 .We have included up to two electron excitations and no hole excitation.A single ground state transition has been found inside the [Xe] 4f 14 5d 1 configuration, which was measured in third order at 802.7 nm.While the transition itself is unremarkable, its high transition rate makes it an easy target to observe with the highest possible resolution.In particular, we could observe the effects of the Observed wavelength λ of an optical, magnetic dipole (M1) transitions in bismuth and lead.Wavelength uncertainties are given as statistical uncertainty (∆λstat) and a systematic uncertainty (∆λsys).The initial (L i ) and final (L f ) levels are given, as well as the initial (J i ) and final (J f ) total angular momenta.Calculations with ambit include the calculated wavelength λ A , the g factor of the initial (g Ai ) and final (g Af ) Level, as well as transition rates A ki /s −1 for bismuth.Relative intensities I F from the CRM calculated with FAC are given for lead.hyperfine components and fit them accordingly due to bismuth's large nuclear spin of I = 9/2.Figure 15 shows the fitted Zeeman-and hyperfine effects to the Bi XV transition in the third order.The fitting was based on the energy splitting described in section 2.1 and equation ( 1), where initial parameters were calculated with ambit.The initial ambit parameters and fitting results are listed in table 2. The calculated parameters typically deviate less than 3% from the measurement results.Only the hyperfine constants A deviate up to 40%, which might be because of the poorly known nuclear properties of the atom.

Conclusion
In this paper, we present a systematic series of accurate wavelength measurements of optical M1 transitions in HCIs of lead and bismuth.Our line-identification scheme proved suitable for the complex case of lead ions with semi-filled f-shell, and several ground-state transitions were found.In Pb XXXI and Pb XXXII, we have fully reconstructed the level structure of the lowest energy levels accessible to optical spectroscopy.Performing, in addition, VUV spectroscopic measurements promise a similar reconstruction of the structures of Pb XXIX and Pb XXXIII, where interesting optical ground state transitions of higher-multipole-order are expected.In Pb XXX and Pb XXXIV, well-resolved EUV spectra would also allow to complete the electronic structure of the levels of interest close to the ground state.Charge states between Bi X to Bi XV were investigated in bismuth, and their ground-state transitions were found.In particular, Bi XV was investigated in close detail to obtain the hyperfine constants A. Due to the larger hyperfine interaction in the highly charged species and the comparatively simpler system, theory is more accurate here than in neutral or singly charged Bi.Linking the level branches in Bi XIV also promises to determine the wavelength of a candidate E2 clock transition.Its very long lifetime, in combination with the heavy mass of the ion that reduces the Doppler shift, could enable a very stable optical clock operating in the VUV or XUV range with a faster averaging time than the currently used M1 transitions [8].Recent E2 lifetime measurements of VUV-excitable levels [69] confirm these possibilities.The present systematic compilation contains several attractive candidates for clock transitions, which can also potentially be used in the search for physics beyond the Standard Model.The lead and bismuth ions studied here, with their high nuclear charge, have large relativistic, nuclearsize, and QED contributions to the electronic binding energy, conditions that are particularly interesting not only for such searches [9] but also for testing, e. g., calculations of the nuclear recoil effect in many-electron ions (for a recent example, see [70]).Moreover, the very large hyperfine splitting in the case of bismuth opens a portal for the study of hadronic contributions beyond the Standard Model through its strong coupling with the electronic energy scale.

Figure 1 .
Figure 1.Schematic of the setup.In the trap region shown on the left, electron-impact excited HCIs emit fluorescence light.In-vacuum lenses L1 and L2 form an intermediate image outside the EBIT vacuum chamber.At that position, a movable reflector can couple into the optical path calibration light transmitted by an optical fiber.Mirrors M1, M2, and M3 inside a periscope box rotate the image of the ion cloud from horizontal to vertical such that lenses L3 and L4 can efficiently project it onto the vertical entrance slit.Inside the Czerny-Turner spectrometer, two spherical mirrors, M4 and M5, image the entrance slit onto the CCD camera where it is recorded.Between them, a grating acts as a dispersive element.Adapted with permission from [60].

Figure 2 .
Figure 2.Example of a sinusoidal fit for determining and correcting the thermal shift.Fitted centroids showing the daily drifts of calibration lines on the dispersion plane at different measurement times (red symbols).Dashed lines are sinusoidals fitted to the data, with a common amplitude, phase shift, and period for all calibration lines.Each plot features a line offset by its individual mean line position.The sinusodials feature a fixed cycle period of a day (blue) or treat the period as a fit parameter (green).We choose the second option as more accurate in this case and display as a green shaded area the confidence bounds of the fit.

Figure 3 .
Figure3.Top: fit of the 1 → 0 line of PbXXXI at λ = (392.65739 ± 0.000 26 ± 0.000 06) nm, with the value given as λ ± ∆λstat ± ∆λsys.Cyan, purple, and red lines show the individual Zeeman components.As the total angular momentum J changes from 1 to 0, with a large initial Landé g factor of g i ≈ 1.5, the resulting three separate Zeeman components enable a precise determination of the line width.More details are presented in table 1. Bottom: closest calibration line (orange) for comparison with the ion spectrum (blue).This line was used to determine the apparatus' contribution to the resulting line shape.From that, we infer a value of T ≈ 1.27(1) MK as the upper limit of the ion temperature.

Figure 4 .
Figure 4. Fit of the 3 → 1 line of PbXXV.Cyan, purple, and red lines show its separate Zeeman components, the blue one their sum, and the green shaded area their confidence bounds.This is a J = 4 → 4 transition with Landé g factors of the initial and final state of g i ≈ 1.1 and g f ≈ 1.2, respectively.We found this line at λ = (252.17575 ± 0.000 25 ± 0.000 08) nm, see table 1, with the value given as λ ± ∆λstat ± ∆λsys.

Figure 8 .
Figure 8. Grotrian diagrams showing observed transitions in Pb XXIX (top) and Pb XXX(bottom).Solid black lines between the levels mark identified transitions.Energy levels linked by observed transitions are shown over their total angular momentum J.For transitions not linked to the ground state, the lowest energy level has been taken from ambit calculations.A predicted energy level, shown in gray, is connected by two VUV transitions (in purple) thatwere not yet measured.The green line marks a highly forbidden optical ground state transition, a promising candidate for a clock transition.The inlay in the bottom panel shows the Pb XXX 3 → 1 line and its Zeeman components.

Figure 9 .
Figure 9. Grotrian diagrams showing observed transitions in Pb XXXI (top) and Pb XXXII (bottom).Solid black lines between the levels indicate identified transitions.Energy levels linked by measured transitions are shown over the total angular momentum J.The inlay in the top panel shows the Pb XXXI 2 → 1 line, and the one in the bottom panel shows the Pb XXXII 1 → 0 line, with their respective Zeeman components.

Figure 10 .
Figure 10.Grotrian diagrams showing observed transitions in Pb XXXIII (top) and Pb XXXIV (bottom).Solid black lines between the levels indicate identified transitions.Dotted lines denote preliminary identifications which require further measurements to be unambiguously identified.Energy levels linked by measured transitions are shown over the total angular momentum J.For transitions not linked to the ground state, the lowest energy level has been taken from ambit calculations.A predicted, bright VUV transition is shown in purple.The green line is a promising candidate for a higher-order optical ground state transition.The inlay in the top panel shows the Pb XXXIII 1 → 0 line, and the one in the bottom panel shows the Pb XXXIV 3 → 2 line, with their respective Zeeman components.

Figure 11 .
Figure 11.Composite map of optical spectra of Bi ions collected as a function of the electron-beam energy, focusing on the range of the transitions of interest.White lines show the charge states with arrows pointing out the measured transitions, where filled arrows indicate the first diffraction order, half-filled the second order, and empty symbols indicate the third-order position of the line within that charge state.The charge-state distribution is shown on the right side, normalized to their respective intensities.Note that a moving average was applied to the spectral curves for clarity.

Figure 12 .
Figure 12.Grotrian diagram showing observed transitions in Bi X (top) and Bi XI (bottom).Solid black lines between the levels indicate identified transitions.Dotted lines denote preliminary identifications that require further measurements to be unambiguously identified.Energy levels linked by measured transitions are shown over the total angular momentum J.

Figure 13 .
Figure 13.Grotrian diagram showing observed transitions in Bi XII (top) and Bi XIII(bottom).Solid black lines between the levels indicate identified transitions.Dotted lines denote preliminary identifications which require further measurements to be unambiguously identified.Energy levels linked by measured transitions are shown over the total angular momentum J.For transitions not linked to the ground state, the lowest energy level has been taken from ambit calculations.

Figure 15 .
Figure 15.Fit of Bi XV transition in third order with visible Zeeman-and hyperfine-effects.Vertical lines with triangles mark Zeeman components with ∆m = 0 in magenta, ∆m = +1, −1 in red, and respectively cyan.Observed asymmetries are due to the incomplete Paschen-Back effect.

Table 2 .
Comparison between parameters calculated by ambit and fit results for Bi XV.