Resonant laser ionization and spectroscopy are widely used techniques at radioactive ion beam facilities to produce pure beams of exotic nuclei and measure the shape, size, spin and electromagnetic multipole moments of these nuclei. However, in such measurements it is difficult to combine a high efficiency with a high spectral resolution. Here we demonstrate the on-line application of atomic laser ionization spectroscopy in a supersonic gas jet, a technique suited for high-precision studies of the ground- and isomeric-state properties of nuclei located at the extremes of stability. The technique is characterized in a measurement on actinium isotopes around the N=126 neutron shell closure. A significant improvement in the spectral resolution by more than one order of magnitude is achieved in these experiments without loss in efficiency.
The heaviest elements of the periodic table have intriguing atomic and nuclear properties1. Valence-electron configurations and ionization potentials are strongly influenced by relativistic effects directly affecting the element’s chemical behaviour and thus its position in the periodic table. In heavy nuclei, a competition between the short-range nuclear attraction and the long-range Coulomb repulsion determines their mere existence and properties. Answering questions such as where does Mendeleev’s table end and which positions in the periodic table do the heaviest elements occupy need intertwined chemical, atomic and nuclear physics studies.
Element Z=118 is thus far the heaviest element observed and some of its basic decay properties have been determined2. Owing to the extremely low production rates, crucial information on nuclear excitation energies, spins and parities, electromagnetic moments and charge radii is scarce for the transactinide isotopes (Z≥104). Even in the actinide region (Z≥89) such information only exists for selected, often long-lived isotopes. Copernicium (Z=112) is the heaviest element where chemical properties have firmly been established3, whereas lawrencium (Z=103) is the heaviest element where the ionization potential is measured4. Until recently, information on atomic transitions ended at fermium (Z=100)5 but has now been extended to nobelium (Z=102), where an optical ground-state transition has been observed6.
Resonant laser ionization is a well-established atomic physics technique to selectively produce radioactive ion beams and then deduce atomic properties such as ionization potentials, atomic levels, transition strengths, isomer and isotope shifts, and hypefine constants7,8,9. From these atomic observables, nuclear properties such as spins, magnetic dipole and electric quadrupole moments, and differences in mean-square nuclear charge radii can be deduced in a nuclear-model independent manner. However, current implementation of resonant laser ionization can be limited in the accessibility to all these nuclear observables due to a poor spectral resolution—in the GHz range. Furthermore, the applicability can be hampered for short-lived nuclei, even for certain elements, due to physico-chemical properties.
In this study we describe an approach that is efficient, chemically independent and applicable to short-lived isotopes (T1/2>0.1 s). This major expansion of the so-called in-gas laser ionization and spectroscopy (IGLIS) technique10,11 has been first examined offline on stable 63Cu isotopes12. We have validated this technique on-line in the actinide region by performing high-resolution laser-spectroscopy studies on the isotopes 214Ac (T1/2=8.2 s) and 215Ac (T1/2=0.17 s). Its projected spectral resolution for heavy elements of ∼100 MHz fulfils the requirement for accurate extraction of their ground-state nuclear properties. Moreover, this in-gas-jet laser ionization and spectroscopy method enables the production of extremely pure ion beams, including isomeric beams, which are suitable for further studies.
By choosing the neutron-deficient isotopes of actinium, the first and name-giving element of the actinide group, as a case study, we addressed all challenges for high-resolution resonance ionization spectroscopy of the heavy elements as follows: limited information on the atomic levels due to the absence of stable isotopes, limited production rates due to the necessity of using heavy-ion-induced fusion reactions on thin targets, large background from stronger reaction channels and the need to slow down the energetic radioactive ion beam and transfer it into a controlled ensemble of cooled atoms in a low-density environment, to minimize Doppler and collisional broadening.
Next to the characterization of the technique, we obtained nuclear spins, magnetic dipole and electric quadrupole moments, and differences in mean-square charge radii of neutron-deficient actinium isotopes around the N=126 shell closure by combining the spectroscopic data with atomic calculations. A comparison to large-scale nuclear shell-model calculations provides evidence for the extent of magicity in this region.
Production and detection of actinium isotopes
We produced neutron-deficient actinium isotopes in a complete fusion reaction of 20Ne or 22Ne projectiles on a 197Au target at the Leuven Isotope Separator On Line facility13 coupled to the CYCLONE accelerator of the Centre de Ressources du Cyclotron (Louvain-la-Neuve, Belgium). The reaction products were thermalized in a gas cell (see Fig. 1), filled with 350 mbar purified argon and were evacuated together with the buffer gas through a de Laval (convergent–divergent) nozzle resulting in a collimated supersonic gas jet. In contrast to former in-gas-cell laser-spectroscopy studies10,11, the laser beams were now overlapped with the gas jet outside the gas cell, leading to the resonant ionization of actinium in the low-density, low-temperature supersonic gas-jet expansion (see Methods section). The photoions were then captured and transported in a radiofrequency ion guide, accelerated, mass separated and subsequently implanted into an α-decay detection setup. By recording the α-decay spectra as a function of the laser wavelength of the excitation step laser, spectra of the hyperfine structure (hfs) were obtained.
Determination of atomic transitions in actinium
Given the lack of information on atomic transitions in actinium, we first performed offline laser-spectroscopic studies using the long-lived isotope 227Ac (T1/2=21.8 y) to identify an efficient two-step ionization scheme featuring suitable properties for high-resolution spectroscopy studies, that is, a measurable hyperfine splitting and sensitivity to changes in the charge radius (see Methods). As a result of these experiments, we selected the ionization scheme shown in Fig. 2a comprising six multiplets (numbered from I to VI) as the best choice to perform the on-line studies. It is noteworthy that the angular momentum for the level at 22,801 cm−1 was identified in these measurements to be J=5/2, which is in contrast to the reported literature value14. The assignment of the atomic term as is supported by Multi Configuration Dirac Hartree Fock (MCDHF) calculations (see Methods).
Study of 212–215Ac in the gas cell
To identify the corresponding hfs of short-lived 212–215Ac, we then used the in-gas-cell method10,11 to perform a broad-band spectroscopy search, which enabled us to narrow down the scanning ranges for the high-resolution spectroscopy study. An example of such a scan for 215Ac is shown in Fig. 2b. The spectral linewidth of 5.8(2) GHz (full width at half maximum (FWHM)) mainly resulted from collisional and temperature-associated (Doppler) broadening and made it possible to scan efficiently the full ∼80 GHz hfs of the excited state. However, such a linewidth completely masked the ground-state hyperfine splitting, essential for the spin and quadrupole moment determination.
Study of 214,215Ac in the gas jet
Finally, low-temperature and low-pressure conditions were obtained in the supersonic jet, where laser ionization and spectroscopy studies on the 214,215Ac isotopes were performed. The hfs of 215Ac obtained in the jet experiments is compared with the in-gas-cell results in Fig. 2b, whereas the parameters and performances of both approaches are summarized in the second and third column in Table 1. The reduction of both collisional and temperature-associated broadening in the jet experiments led to a 15-fold improvement of the spectral resolution, which enabled a precise fitting of the ground-state hfs, as shown in Fig. 2c for the triplet (IV) and Fig. 2d for the doublet (V) transitions in 214Ac and 215Ac, respectively. Owing to technical constraints, the temporal overlap of the laser beams with the atoms in the jet at the given laser repetition rate of 10 kHz (ref. 15) could not be optimized and only a fraction of 1/14 of the available atoms was irradiated (see Methods). Even with this small duty cycle, the efficiency was similar to that in the gas-cell studies.
Ground-state properties of actinium
The magnetic-dipole-interaction hfs constant a, the electric-quadrupole-interaction hfs constant b and the isotope shift δνA,215 were extracted from the obtained spectra and are given in Table 2. The experimental systematic uncertainties originate from the instability and nonlinearity of the different wavelength meters used in the three experiments and from the pressure dependence in case of the in-gas-cell data. Out of these isotope-dependent atomic observables, we could deduce a number of nuclear ground-state properties for the different isotopes, also given in Table 2. The spectral resolution obtained in the offline and in-gas-jet measurements, made it possible to unambiguously determine the nuclear spins I of the isotopes investigated. This results in I=3/2 for 227Ac and confirms the 1951 measurement16, I=9/2 for 215Ac substantiating the previous assignment based on the observation of favoured α-decay to 211Fr ( http://www.nndc.bnl.gov/ensdf/) and a spin value of I=5 for 214Ac, for which no previous assignment is given in the National Nuclear Data Center data base. The in-gas-cell data for 212,213Ac does not enable a firm spin assignment. A spin I=(7) for 212Ac is tentatively proposed based on the result from the additivity rule and shell-model calculation, which places the I=7+ as the ground state ∼200 keV below the I=5+ and 6+ states (see Methods). Based on the observed experimental trend in the magnetic moments of 217,215,213Ac (see Fig. 3a) a spin I=(9/2) can tentatively be assigned to 213Ac.
To obtain the magnetic dipole moment μ or the spectroscopic-quadrupole moment Q from the measured hfs constants, a and b, through the equations a=μ·B0/(I·J) or b=eQ·Vzz, the magnetic field B0 and the electric field gradient Vzz created by the atomic electrons at the site of the nucleus must be known. As B0 and Vzz depend only on the electronic configuration and not on the specific isotope, one can usually circumvent this by means of a scaling relation with experimentally measured hyperfine-splitting constants and the independently measured magnetic dipole or electric quadrupole moment of a known isotope. However, the only ground-state dipole- and quadrupole-moments available in the literature are, as quoted by the authors, the preliminary values μ=1.1(1) μN and Q=1.7(2) eb for 227Ac (ref. 17) (it is noteworthy that these values from 1955 to 58 stem from hfs measurements and atomic calculations rather than from independent μ and Q measurements) and the magnetic dipole moment for the very short-lived (T1/2=69 ns) isotope 217Ac, inaccessible to optical spectroscopy measurements. Therefore, we performed advanced calculations of the a/μ and b/Q values (see Methods) for several atomic levels applying the MCDHF method as implemented in the GRASP2k programme18 and previously applied to other nuclei19,20.
These calculations, combined with the a and b constants of the corresponding levels measured in 227Ac, resulted in a magnetic dipole moment μ=1.07(18)μN and a spectroscopic electric-quadrupole moment Q=1.74(10) eb for the 227Ac ground state. The uncertainty on these values reflects the s.d. of the results from the eight transitions used in the evaluation. Although the obtained values are in agreement with the literature values and have similar error bars, they are more reliable as they are based on superior experimental input and advanced atomic model calculations. Using these values as reference, the magnetic dipole moments were deduced for 212–215Ac from the in-gas-cell measurements and for 214,215Ac, with a higher precision, from the in-gas-jet measurements. Only with the in-gas-jet method could the electric-quadrupole moments for 214,215Ac be obtained. The statistical and systematic errors for μ and Q quoted in Table 2 arise from the uncertainties in the experimental a and b constants. The errors from the 227Ac reference values are not propagated, leaving the possibility for later improvements of the obtained results once the moments of 227Ac can be measured in an independent way or the precision of atomic model calculations is improved.
In Fig. 3a our experimentally obtained magnetic dipole moments are shown together with the known ground-state magnetic moment of 217Ac (ref. 21). We compare these results with shell-model calculations involving 208Pb (N=126, Z=82) as core nucleus. For the odd–odd 212,214Ac isotopes, the results are also compared with values from the additivity relation using the magnetic moment of 211,213Ra (refs 22, 23), respectively, and the magnetic moment of 215Ac. Both calculations are further explained in the Methods section. The contribution in our data points of the 17% systematic error on the calculated magnetic moment for the 227Ac ground state is shown with a shaded band. Taking this error into account, our experimental values for 212–215Ac, which are in good agreement with the results from the additivity relation, do overlap with the shell-model calculations but are systematically lower by ∼13%. A similar discrepancy is present when comparing our g-factor of the 215Ac ground state with the experimentally known g-factors of all -based states in the N=126 isotones24, with n representing the number of protons occupying the h9/2 proton shell (nmax=10). For pure states, it is expected from angular momentum algebra that g (h9/2)=g(h9/2)n (ref. 25). For example, in 213Fr the g-factors of the 9/2−, 17/2− and 21/2− states are within experimental uncertainty identical24, whereas our value of g(9/2−,215Ac) =0.8056(16) differs considerably from the g-factors of the I=17/2− and 21/2− high-spin isomers (g(21/2−,215Ac)=g(17/2−,215Ac)=0.920(19))24. Further developments on atomic theory or an independent measurement of the magnetic dipole moment in 227Ac are needed to clarify this discrepancy.
As shown in Fig. 3b, the experimental electric-quadrupole moments of 214,215Ac and literature values for the moments of the neighbouring odd-Z isotones agree with shell-model calculations. The linear increase of the quadrupole moments for N=126 and 125 with increasing proton occupation demonstrates a seniority-type dependence26 and for N=125 its coupling to a non-contributing νp1/2 hole (see Methods). The validation of seniority as good quantum label is indicating a robust N=126 gap up to Z=89. A significant weakening of the N=126 shell closure in uranium (Z=92) is inferred from a recent analysis of the reduced α widths in the 221,222U isotopes27. No effect is observed in the present data even though increased cross-shell interactions would have a dramatic impact on the small quadrupole moments at πh9/2 mid-shell (n=5).
The centre of gravity of the hfs extracted from the fits for 212–214,227Ac, relative to that for 215Ac (N=126) defines the isotope shift δνA,215. The isotope shift originates from the change in nuclear mass (A) and nuclear charge distribution. From the isotope shift, the difference in mean-square charge radius δ<r2>A,215 of the different isotopes can be deduced using the calculated parameters M and F (see Table 2 and Methods section). Our measured values exhibit a similar behaviour as those in lighter neighbouring isotopic chains with Z=82–88 (refs 9, 28). This will be discussed in detail in a forthcoming publication.
Spatial constraints and limitations of the pumping system in the present setup prevented a high-quality jet formation and, as a consequence, an optimal laser-atom overlap in space and time. These will be overcome in future experiments when dedicated IGLIS setups are in operation at new-generation radioactive beam facilities (see for example, ref. 29). Table 1 compares the presently obtained in-gas-jet performance (third column) with the projected performance of the technique under optimal conditions (fourth column), which also include a new gas-cell design with better transport and extraction characteristics30. Thus, assuming a duty cycle adapted to the supersonic expansion through a de Laval nozzle with a modest Mach number 10, tuning the background pressure for a pencil-like jet and optimizing the laser interaction zone through a multi-pass mirror system, a final spectral resolution of δν/ν∼1 × 10−7 (∼100 MHz FWHM) and an overall efficiency beyond 10% can be obtained.
Combined with the fast and chemically insensitive stopping and transport using a purified noble gas, it becomes possible to measure nuclear properties such as charge radii, electric and magnetic moments, and nuclear spins of all isotopes with half-lives above 100 ms and produced at a rate of only 1 atom every 10 s. As the technique is based on laser spectroscopy, fundamental atomic properties such as ionization potentials, transition energies and rates can also be determined. The advantages of this, now proven, approach are obvious when compared with the low-resolution in-source7,31 or in-gas-cell8,10 approach but are also evident when compared with the collinear laser spectroscopy technique9,32, which has a superior spectral resolution but is ultimately limited by the production mechanism of the desired radioactive ion beam and needs at least 100 ions per second.
All radioactive isotopes with published optical data are shown with orange squares on the nuclear chart in Fig. 4. One notices a rather sudden dearth of radionuclides which have been probed as one enters the actinide region and beyond. This echoes a number of significant obstacles in accessing the heaviest of elements including low production cross-sections, strongly competing reaction channels and limited atomic information mainly due to a lack of stable or long-lived isotopes. The reach of the IGLIS technique, based on the projected performance of the in-gas-jet method (see Table 1) and on experimental and estimated cross sections (blue and green squares, respectively), can be seen in the inset of Fig. 4 by the number of new isotopes that can be studied. Furthermore, in the medium-mass region the in-gas-jet laser ionization and spectroscopy method can also be used to study those nuclei, which are inaccessible to conventional laser-spectroscopy techniques, owing to the physico-chemical properties of the involved elements, such as the refractory elements around molybdenum and tantalum.
In conclusion, the feasibility and impact of the in-gas-jet laser ionization and spectroscopy method have been demonstrated on-line by measuring nuclear and atomic properties of the short-lived isotopes 214Ac and 215Ac. The resulting magnetic and quadrupole moments are compared with shell-model calculations and witness a stabilising effect of the N=126 shell up to the actinium isotopes. The obtained efficiency and spectral resolution demonstrate that basic ground- and isomeric-state nuclear properties of heavier actinides and eventually super-heavy elements, as well as their atomic properties, can be determined to high precision. The technique presented has also far reaching consequences for the exploration of the refractory elements, so far hardly accessible to high-resolution laser spectroscopy techniques. In addition, the highly selective ionization enables the production of high-quality, high-purity isotopic and isomeric radioactive ion beams that can be used for other applications in nuclear physics, chemistry and astrophysics, as well as in atomic physics.
To obtain an optimal ionization scheme, we performed offline experiments at the University of Mainz using a number of samples, each containing about 1011 atoms, of the T1/2=21.8 y isotope 227Ac. An atomic vapour of 227Ac was produced by resistively heating the samples in a hot cavity and then step-wise ionized by pulsed Ti:sapphire lasers33 resulting in an overall efficiency of 10−3%. We chose the transition from the ground state 6d 7s2 at 438.58 nm for the first-step excitation in combination with two transitions to autoionising states at 434.51 nm34 and 424.69 nm for subsequent ionization. We characterized in detail the hfs of eight transitions of actinium including that at 438.58 nm35.
The short-lived 212,213Ac and 214,215Ac isotopes were produced at the Leuven Isotope Separator On Line facility in the fusion reaction of 0.16 pμA 20Ne and 22Ne projectiles, respectively, impinging with a total energy of ∼105 MeV on a 197Au target of 1.931 mg cm−2 thickness. For the determination of the efficiency of the in-gas-cell and in-gas-jet technique, the experimental cross-sections from refs 29, 36 were used. The change in cross-section as a function of the energy loss of the neon beam in the target was taken into account to calculate the production rate. More information on the preparation of the experiments can be found in ref. 15.
We performed low-resolution laser spectroscopy in the gas cell employing a dual-chamber gas cell37 and a tunable excimer-pumped dye-laser system at a maximum repetition rate of 200 Hz (ref. 13). To characterize the frequency shift due to atom–atom interactions in the high-pressure gas-cell environment, we scanned the singlet hyperfine transition in 215Ac (number VI in Fig. 2a) for different stagnation pressures. From a linear fit of the centroids and the widths we obtained a collisional shift coefficient γsh=−3.7(9) MHz mbar−1, that was used to deduce the position of the hyperfine peaks in the absence of collisions, and a collisional broadening coefficient γcoll=11.5(10) MHz mbar−1, leading to a Lorentz contribution in the total linewidth of 4,000(400) MHz (see Table 2). The Gaussian linewidth contribution reported in Table 2 was calculated from the gas temperature and the laser linewidth (1.2(1) GHz).
To realize high-resolution laser spectroscopy in the quasi-collisional free and cold atomic jet we installed at the gas-cell exit a convergent–divergent nozzle designed for Mach number ∼6 with a throat diameter of 1 mm. Under optimal conditions, that is, with a matching of the jet and the background pressures, reached at the dedicated offline laboratory in KU Leuven30, this leads to a gas jet of about 3 mm diameter, quasi-parallel over >200 mm. In the on-line experiment, where the background pressure could not be controlled sufficiently and where a radiofrequency ion guide was placed at 11 mm distance from the exit of the nozzle, the optimal conditions could not be realized. The supersonic jet, moving with a velocity ν∼550 m s−1, was irradiated at normal incidence by a laser system, similar to that employed during the offline studies of 227Ac, comprising a narrow bandwidth (10–20 MHz) injection-locked Ti:saphire laser with up to 10 μJ available average energy per pulse used for the first-step excitation at 439 nm. Dedicated spectra of the triplet, doublet and singlet multiplets (see Fig. 2a) were acquired varying the average energy per pulse of the excitation step laser (from 0.08 to 1 μJ), to investigate the contribution of the different broadening mechanisms to the line shape. This enabled to separate the contributions of collisional broadening and the natural linewidth (Lorentz) from temperature-associated, laser linewidth and gas-jet divergence broadening (Gauss), see Table 2. For the spectroscopic measurements, the energy per pulse of the first step was set around 0.8μ J. Two broad-bandwidth (∼4 GHz) pulsed Ti:sapphire lasers were used simultaneously for the second ionization step at 425 and 435 nm with up to 180 μJ pulse energy (for a detailed description of the laser system see ref. 15). Operated at 10 kHz pulse repetition rate, the lasers with an effective beam spot area of 4 mm illuminated the atomic jet immediately at the nozzle exit. With these ionization conditions, fixed by limitations in the present experimental setup, only a fraction (∼1/14) of the atoms in the jet could be irradiated, resulting in an overall efficiency of 0.40(13)% and a selectivity of 121(27). The efficiency in the in-gas-cell experiments, where in contrast all atoms are irradiated, is comparable with the in-gas-jet result. This points to collisional de-excitation losses in the ionization process due to the high-pressure environment. The overall in-gas-jet efficiency depends on different factors, namely stopping of the reaction products (100% in this case), transport of the atoms towards the de Laval nozzle (35% diffusion losses, 17% decay losses for the 170 ms 215Ac), the availability of actinium in its atomic ground state (losses due to non-neutralized ions (>10%) and to molecule formation) and the efficiency of the laser ionization. Increasing the average pulse energy of the first step laser to 9 μJ resulted in a 1.9(2) gain in overall efficiency but the spectral resolution deteriorated. Optimizing the gas-jet formation and shaping the laser beam spot area to a sheet 55 mm long and 3 mm wide would result in a 100% duty cycle12,30. In this way, an overall efficiency of >10% and a spectral resolution of ∼100 MHz (FWHM) can be obtained.
For the high-resolution data obtained in the offline and in the gas-jet experiments, the fitting procedure converged only for a particular spin value. This made an unambiguous spin assignment for the isotopes 214,215,227Ac possible. We obtained the hfs constants, a and b, and the centre of gravity of the hfs for each isotope from least-square minimization fits of a 12-peak Voigt profile to the data points. The ratios au/a1 and bu/b1 between the hyperfine constants of the excited (upper) and ground-state (lower) constants were fixed in all the fits to those deduced for 227Ac from the fully resolved spectra obtained in the offline experiments, neglecting the effect of a possible differential hyperfine anomaly. This effect is generally <1% (see ref. 38) and its influence on the final results can be neglected in comparison with the experimental uncertainties.
We deduced the magnetic dipole moment of the short-lived actinium isotopes from the scaling relation
using our experimental a factors and the calculated value for 227Ac, and assuming that any possible contribution arising from the presence of hyperfine anomaly between 227Ac and the lighter actinium isotopes is masked by the large theoretical systematic uncertainty associated with the μ227 value.
In a similar way, we deduced the spectroscopic quadrupole moments using the scaling relation
where the calculated quadrupole moment of 227Ac is used as reference.
The measured isotope shift δνA,215 of two nuclei with mass A and 215 originates from the change in nuclear mass and in nuclear charge distribution and is related to the change in the mean-square charge radius by39
with both the mass shift constant M and the field shift constant F obtained by calculations.
Atomic structure calculations
The MCDHF method was applied to study the atomic structure of neutral actinium. The experimentally observed levels were identified and subsequently their hyperfine coupling constants a and b were calculated. Furthermore, we determined the isotope-shift parameters M and F for the transition at 438.58 nm (22,801.1 cm−1).
The wave functions were generated using the relativistic atomic structure package GRASP2k18. Virtual excitations from a set of reference configurations to a systematically enlarged set of up to five layers of correlation orbitals with angular momenta up to g are used to generate the multiconfiguration basis. Here we took single, double and triple excitations from the valence shells into account. Core-valence correlation was accounted for by performing single excitations from the 6p-shell together with a second excitation from one of the valence shells. Furthermore, core polarization was also included in our calculations by including single excitations from all core orbitals into the active space.
In a subsequent step, the obtained wave functions were used to extract the isotope-shift parameters and the hyperfine coupling constants a and b. By scaling the calculated hfs interaction constants for different energy levels in 227Ac with those measured in our offline studies, we obtained the values μ=1.07(18)μN and Q=1.74(10) eb for the magnetic dipole and electric quadrupole moments of 227Ac, respectively.
The mass-shift constant M was obtained by calculating the expectation value of the relativistic recoil operator in the generated eigenfunctions as described in ref. 40. A series of configuration interaction calculations was performed to obtain the isotope-shift between different isotopes for a model nucleus. Subsequently, the field-shift was extracted from these differences in the transition energy. This method, applied to other elements as heavy as polonium41,42, resembles the experimental approach and we refer to ref. 43 for more details.
The mass-shift constant was found to be M=500(180) GHz amu, which results in a very small contribution compared with the total isotope shift (see equation (3)) and a field-shift constant of F=−39(2) GHz fm−2 was obtained. Several independent calculations were carried out and the resulting differences were used to estimate the uncertainties.
Nuclear-shell model calculations and additivity rule
We performed shell-model calculations for ground and excited states in the region northwest of 208Pb that resulted in the magnetic dipole and electric quadrupole moments for the N=123–126 actinium isotopes. We have used the π(h9/2,f7/2,i13/2), ν(f5/2,p3/2, p1/2,i13/2) model space and the PBPKH interaction44 from the OXBASH package45. The interaction comprises the Kuo–Herling two-body matrix elements for ππ and νν, and for πν the two-body matrix element from the H7B potential46. The large numbers of valence particles and orbits in the model space made it necessary to apply truncation, to make calculations feasible for Z≥87 and N<126. This in turn implied tuning of the pairing strength, to describe the evolution of occupation and single-particle energies with increasing distance from the core nucleus 208Pb in a satisfactory way. The degree of truncation was benchmarked using the francium isotopes where experimental data are known. In particular, the high spin π(i13/2), ν(h9/2, f7/2, i13/2) and the low-spin π(p3/2, p1/2) were blocked, which leaves a π(h9/2, f7/2), ν(f5/2, p3/2, p1/2) model space. The effective nucleon charges and g-factors used in these calculation that describe the whole region very well are the E2 operators ep=1.5, en=0.85 and the M1 operators ; , .
The g-factors of the ground state (9/2−) and high-spin (17/2− and 21/2−) isomers that are based on a proton h9/2 configuration in 215Ac have been calculated to be 0.912, 0.925 and 0.969, respectively. Only the value for both isomers agree with the experimental value 0.920(19)24.
Semi-magic nuclei can be described in a seniority scheme47. Hereby, a jn configuration with spin I is additionally labelled by the seniority ν, which counts the number of unpaired nucleons. In this framework, the E2 matrix element is given by and is linear in n and changes sign in the half-filled orbit. The continuous linear trend in the ground-state quadrupole moments of the N=126 isotones (seniority ν=1) from Z=83 to 89 classifies these nuclei as semi-magic.
In the analysis of the odd–odd nuclei we compare our results for the magnetic moments with those provided by the additivity rule48 for different spin values. For both isotopes 212,214Ac the experimental values are predicted by simple coupling rules in combination with the suitable nucleon configuration involving only the experimental magnetic moments of the neighbouring odd-A nuclei, namely 211,213Ra (refs 22, 23) for the neutrons and the in-gas-jet result of 215Ac for the protons (see Table 2).
All the relevant data supporting the findings of these studies are available from the corresponding author upon request.
How to cite this article: Ferrer, R. et al. Towards high-resolution laser ionization spectroscopy of the heaviest elements in supersonic gas jet expansion. Nat. Commun. 8, 14520 doi: 10.1038/ncomms14520 (2017).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank Nancy Postiau and the operators of the CRC Louvain-la-Neuve for the technical support and the preparation of the stable neon beams. This work was supported by FWO-Vlaanderen (Belgium), by GOA/2010/010 (BOF KU Leuven), by the IAP Belgian Science Policy (BriX network P7/12) and by a Grant from the European Research Council (ERC-2011-AdG-291561-HELIOS). S.S. acknowledges a Ph.D. Grant of the Belgian Agency for Innovation by Science and Technology (IWT). L.P.G. acknowledges FWO-Vlaanderen (Belgium) as an FWO Pegasus Marie Curie Fellow. S.F. and R.B. acknowledge the support by the German Ministry for Education and Research (BMBF) under contract No. 05P15SJCIA.
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/