Smooth trends in fermium charge radii and the impact of shell effects

Experimental techniques

The long chain of fermium isotopes studied in this work was measured by combining different production schemes along with two laser spectroscopy techniques for the respective measurements. Spectroscopy of the isotopes ^{245,246,248,249,250,254}Fm was performed via the RADRIS technique^30,31, with the set-up located behind the velocity filter SHIP at the GSI Helmholtzzentrum für Schwerionenforschung in Darmstadt, Germany^35,36. Further details on RADRIS and the latest developments introduced to the set-up are given in refs. ^34,37,54. For the on-line measurements of fermium with RADRIS, a 1 × 0.025 mm² hafnium-strip filament was used for collection and neutralization of directly produced nuclei entering the buffer-gas cell filled with 95-mbar high-purity argon gas. A heat-pulse temperature of 1,450 °C was applied to desorb accumulated fermium ions from the filament as neutral atoms for subsequent laser spectroscopy. For the measurements on ²⁵⁵No, a 125–μm-diameter tantalum-wire filament and desorption temperatures of 1,100 °C were used.

The long-lived fermium isotopes ²⁵⁵Fm and ²⁵⁷Fm produced by neutron capture in the nuclear reactor became accessible by in-source hot-cavity laser spectroscopy at the RISIKO mass separator at Johannes Gutenberg-Universität Mainz^41,55,56. Here, the sample was placed in a heated reservoir, with a temperature of up to 1,600 K, and the atom vapour was probed by lasers for resonant ionization. The resulting ions were extracted by an electric potential of 30 kV and mass separated using a magnetic dipole to separate the species of interest from unwanted surface ions.

Laser set-up for in-gas-cell laser spectroscopy at RADRIS

Laser spectroscopy of fermium was performed by exciting from the 5f¹²7s^{2 3}H₆ atomic ground state to the known 5f¹²7s7p \(\genfrac{}{}{0ex}{}{5}{}{{\rm{G}}}_{5}^{{\rm{o}}}\) atomic level around 25,111.8 cm^−1 (refs. ^42,43). For nobelium, the excitation occurred from the 5f¹⁴7s^{2 1}S₀ ground state to the recently identified excited level 5f¹⁴7s7p \(\genfrac{}{}{0ex}{}{1}{}{{\rm{P}}}_{1}^{{\rm{o}}}\) at \({\mathrm{29,961.457}}_{-0.007}^{+0.041}\,{{\rm{cm}}}^{-1}\) for ²⁵⁴No (ref. ³²). A dye laser (Lambda Physics, FL series) pumped by a Xe:Cl excimer laser (Lambda Physik, LPX240) with 5-ns pulse length and 100-Hz repetition rate was used for laser spectroscopy with up to 500 μJ average energy per pulse and a spectral linewidth of 1.5 GHz using an intracavity etalon for narrow spectral linewidth operation. The laser wavelength was continuously monitored with a wavelength meter (HighFinesse-Ångstrom, WS/7-UVU) that was calibrated to an internal neon lamp. The laser light was transported to the buffer-gas cell using ultraviolet-grade optical fibres and was shaped to illuminate an area of about 3 cm² around the filament. The average energy of the laser pulse at the cell was kept in a range of 70–120 μJ for the scanning laser for fermium, matching the reported saturation intensity given in ref. ⁴², and about 10 μJ for nobelium, in accordance with the measurements presented in ref. ³³ on lighter nobelium isotopes. The pump laser for the first excitation step dye laser and the Xe:F excimer laser (Lambda Physik, LPX220), the latter providing the laser light for subsequent photoionization, were synchronized with excimer laser synchronization units (Lambda Physik, LPA 97). The ionizing laser featured about 30 mJ average energy per pulse at the cell after beam transport with mirrors. Both lasers had pulse lengths of about 18 ns.

Laser set-up for in-source laser spectroscopy at RISIKO

The laser system for the hot-cavity in-source laser spectroscopy of fermium isotopes at RISIKO consisted of nanosecond-pulsed titanium:sapphire lasers, pumped by two frequency-doubled neodymium-doped yttrium aluminium garnet lasers with a 10-kHz repetition rate. The titanium:sapphire lasers can be equipped with either a grating or a birefringent-etalon combination as a frequency-selective element and featured an internal second harmonic generation. One titanium:sapphire laser with an average power of up to 1.2 W was used for photoionization. A high ionization efficiency was achieved by exploiting an auto-ionizing resonance at 52,166 cm^−1. For detailed laser spectroscopy of the first excitation step at 25,111.8 cm^−1 in ²⁵⁷Fm, one grating-tuned titanium:sapphire laser was equipped with an additional etalon, which reduced the spectral linewidth to about 1 GHz (ref. ⁵⁷), while the average laser power resulted in about 200 mW. Both laser beams were overlapped anti-collinearly with the ion beam in the hot cavity via a viewport at the bending magnet. For spectroscopy of ²⁵⁵Fm, the Perpendicularly-Illuminated Laser Ion Source Trap (PI-LIST) was employed using the atomic vapour effusing from the hot cavity and a perpendicular overlap of a narrow-linewidth laser to the atomic beam as discussed in more detail in ref. ⁵⁵. Here, an injection-locked titanium:sapphire laser, seeded by a continuous-wave titanium:sapphire laser⁵⁸ and equipped with an external single-pass second-harmonic-generation unit⁵⁹ provided laser light with a band spectral linewidth of 20 MHz and an average power of 100 mW. A laser pulse length of 40 ns was maintained for all lasers and pulse synchronization was achieved by external triggering of the pump lasers with a pulse delay generator. The laser wavenumber of the spectroscopic transition was monitored using two commercial wavelength meters (High Finesse, WS7 and WSU), which were regularly calibrated with a laser locked to a rubidium reference cell⁶⁰.

Isotope production

Different production schemes were applied to access the investigated isotopes in this work for laser spectroscopy studies.

Direct production on-line

The isotopes ²⁴⁵Fm (t_1/2 = 5.6 s) and ²⁴⁶Fm (t_1/2 = 1.54 s) were produced at the velocity filter SHIP, using the fusion-evaporation reactions ²⁰⁸Pb(⁴⁰Ar, 3n and 2n)^245,246Fm with reported cross-sections of 32 nb for ²⁴⁵Fm (ref. ⁶¹) and 10 nb for ²⁴⁶Fm (ref. ⁶²). An ⁴⁰Ar⁸⁺ primary beam featuring a macro-pulse structure of 5 ms beam-on and 15 ms beam-off periods, with a beam energy of 185 MeV for ²⁴⁶Fm and 193 MeV for ²⁴⁵Fm, and average intensities of 2 particle microampere (1.2 × 10¹³â€‰ions per second) was provided by the linear accelerator UNILAC. This primary beam impacted thin lead-sulfide (PbS) targets with an areal density of typically 470 μg cm^−2 for PbS on a 30 μg cm^−2 carbon backing and with a 10 μg cm^−2 carbon cover layer, the latter side facing SHIP. The targets were manufactured at the GSI target laboratory and mounted on a rotating target wheel to distribute the heat from the energy loss of the primary beam over a large area.

Indirect production on-line

The isotopes ²⁴⁸Fm (t_1/2 = 34.5 s), ²⁴⁹Fm (t_1/2 = 2.6 m), ²⁵⁰Fm (t_1/2 = 30 m) were obtained from the α-decay of the isotopes ^252,253,254No, directly produced in the fusion-evaporation reactions ^206,207,208Pb(⁴⁸Ca, 2n)^252,253,254No with respective cross-sections of 0.5 μb, 1.3 μb and 2 μb (ref. ⁶³). ²⁵⁴Fm was obtained from the radioactive decay of ²⁵⁴No using the 10% electron-capture branch to ²⁵⁴Md (t_1/2 = 10 m), which then decays exclusively by electron capture to ²⁵⁴Fm (t_1/2 = 3.24 h). The nobelium isotope ²⁵⁵No (t_1/2 = 3.52 m) was similarly obtained indirectly via the electron-capture branch (<30% (ref. ⁶⁴), and evaluation of previous data taken at SHIP published in ref. ⁶⁵) of ²⁵⁵Lr (t_1/2 = 31.1 s).

The primary ⁴⁸Ca¹⁰⁺ beam was delivered with average intensities of 0.8 particle microampere (5 × 10¹²â€‰ions per second), impinging on thin ^206,207,208PbS targets. The collection cycle of RADRIS was adapted to breed the fermium decay-daughter isotopes on the filament³⁴. Accumulation was done for 25 s (²⁴⁸Fm), 295 s (²⁴⁹Fm) and 355 s (²⁵⁰Fm), before evaporation of collected atoms from the filament followed by resonance ionization laser spectroscopy. For ²⁵⁴Fm, the long lifetime of the intermediate isotope ²⁵⁴Md necessitated a collection time of 3,600 s. This indirect isotope breeding reduced the total efficiency due to decay-branching ratios, recoil implantation into the filament material, and the half-life of mother and daughter nuclide, respectively. The effective yield was especially impacted in the case of ²⁴⁹Fm, which features a similar lifetime to its mother nuclide ²⁵³No (T_1/2 = 1.62 m) and an α-branching ratio of 55%. For the longer-lived ²⁵⁴Fm, a dedicated rotatable detection set-up consisting of three silicon detectors was used to enable longer counting times of accumulated laser ions parallel to a new collection of laser ions (see ref. ³⁷).

Production off-line

For the production of ²⁵⁷Fm, a ²⁴⁸Cm target was irradiated in the High Flux Isotope Reactor at Oak Ridge National Laboratory, USA^66,67. The fermium fraction of this sample, containing remaining einsteinium⁴⁰, was first used for studies at Florida State University and then made available for Mainz University for further investigations. For production of ²⁵⁵Fm, a sample of 290 pg (1.3 × 10¹⁴ atoms) ²⁵⁴Es provided by Oak Ridge National Laboratory and Florida State University, USA, was encapsulated in a quartz ampule inside a titanium cylinder and shipped to the high-flux research reactor at the Institut Laue-Langevin in Grenoble, France, for a neutron irradiation of 7 days duration. After a cool-down period of 4 days and shipping to Johannes Gutenberg-Universität Mainz, Germany, the sample contained 7.5 × 10¹⁰ atoms of ²⁵⁵Es (t_1/2 = 39.8 d) as determined by α-decay spectroscopy. This provided a generator system for the β^−-decay daughter ²⁵⁵Fm (t_1/2 = 20.1 h) present in secular equilibrium. A chemical separation of fermium was performed four times in appropriate intervals to allow ingrowth of ²⁵⁵Fm into the ²⁵⁵Es fraction between individual separations. This procedure was based on an α-hydroxyisobutyrate separation by cation exchange on a Mitsubishi CK10Y resin. The α-hydroxyisobutyrate complex was converted to a nitrate form; the final sample was obtained after cation exchange separation on an AG50WX8 column, placed on a zirconium metal foil of 10 × 10 mm², which promotes the release of neutral atoms⁶⁸, and evaporated to dryness. With this method, ²⁵⁵Fm samples of about 7 × 10⁸ atoms and one ²⁵⁷Fm sample with at most 5 × 10⁷ atoms were available for laser spectroscopy.

Data analysis

Events from resonant laser ionization were recorded as a function of the set wavenumber to analyse the respective transition resonance centre value and thus extract the isotope shift. In the on-line measurements, the α-decay events from the ions were registered and an α-energy range of interest was selected in the analysis. To account for unavoidable fluctuations in the primary beam intensity, extracted event rates were normalized to the accumulated primary beam charge integral on the beam dump of SHIP. For the off-line measurements, the ions were detected with an ion detector after acceleration and mass separation. Gating the signal on the time-of-flight structure of the resonantly produced ions improved the signal-to-noise ratio. To average over signal variations, the laser wavenumber was scanned slowly and repetitively over the resonance multiple times and the obtained counts were binned and normalized according to the time spent at the respective wavenumber. Observed laser resonances for fermium are presented in Extended Data Fig. 1 and the resonance of ²⁵⁵No is shown in Fig. 1 (bottom).

Centroid position

The centroid wavenumbers of the individual measured resonances in the obtained spectra were determined via a fit of a Voigt profile to the data for all even-A isotopes (A denoting the atomic mass number). The odd-mass-number isotopes feature a hyperfine structure splitting of more than 20 lines due to (tentatively assigned) nuclear spins of I = 7/2 for ^249,255Fm (refs. ^69,70) and I = 9/2 for ²⁵⁷Fm (ref. ⁷¹), which could be only partly resolved for ²⁵⁵Fm. This spectrum was analysed using the SATLAS Python package^72,73. Owing to the broadening mechanisms inherent in the spectral linewidth of ^245,249,257Fm from the environmental conditions of the gas cell and the hot cavity, the hyperfine structure could not be resolved. A Gaussian fit to the data for ²⁵⁷Fm and a Voigt fit for ^245,249Fm were used to extract the centre of the structures. The choice of fit profile was connected to the main broadening mechanisms dominating the lineshape in the respective measurement. For the analysis of ²⁵⁵No, with observed underlying hyperfine structure, a nuclear spin of I = 1/2 was assumed for the fit, as tentatively assigned from α-decay hindrance factor systematics⁷⁴. The results on transition resonance wavenumbers and extracted isotope shifts are summarized in Extended Data Table 1.

The RADRIS measurements were performed inside a cell filled with 95-mbar argon buffer gas and are thus affected by a pressure shift and broadening. The broadening is effectively taken into account in the fitting routine, while the pressure shift, equivalent across all RADRIS measurements, effectively does not contribute to the isotope-shift measurement. For comparison with the off-line measurements, the pressure shift had to be evaluated. In the element erbium, a pressure shift of 4(1) MHz mbar^−1 was recently reported⁷⁵, which is in line with observations in actinium⁷⁶. Therefore, a shift of about −400(300) MHz can be inferred for the in-gas-cell laser spectroscopy measurements, with a 3 times larger uncertainty assumed for the application to fermium.

The off-line measurements were performed inside the hot cavity with an anti-collinearly propagating laser beam for ²⁵⁷Fm and ²⁵⁵Fm, and with a perpendicular arrangement of the laser beam and the atomic beam for ²⁵⁵Fm. The latter measurement corresponds to the rest frame of the atom in vacuum conditions. The Doppler shift from the moving ensemble of atoms in the hot cavity can be determined from the ²⁵⁵Fm measurements (anti-collinear and perpendicular) to −100(100) MHz, in agreement with observations reported in californium⁵⁵. For comparison with the gas-cell measurements, with ²⁵⁰Fm being the reference for isotope-shift measurements, the obtained resonance frequencies for ²⁵⁵Fm and ²⁵⁷Fm, which were measured in vacuum conditions, have therefore to be shifted by −400(300) MHz and −300(400) MHz, respectively.

The individual experimental cycle of the RADRIS technique (for details, see refs. ^30,31) adapted to each on-line studied isotope leads to a suppression of known isomers in isotopes ^248,250Fm, which are shorter lived than the ground state. This ensured that purely the nuclear ground state was probed.

The accuracies of the extracted centroid wavenumbers are mainly limited by broadening processes. Pressure broadening is the dominant mechanism for all measurements performed in the gas cell. For ²⁵⁷Fm, Doppler broadening owing to the high temperature in the hot-cavity environment needs to be considered. Power broadening mechanisms only had a role in the case of ²⁵⁰Fm, for which an increased laser power of 150 μJ per pulse was utilized. The origins of effects contributing to the isotope-shift uncertainty are summarized in Extended Data Table 2. This includes the accuracy in the wavenumber measurement. The fit uncertainty for extraction of the centroid position in the analysis is included. As the granularity of data points and counting statistics is small for the on-line investigated isotopes, an uncertainty of half the mean-step size in these measurements can be considered instead, to avoid underestimating the centroid uncertainty. Both factors are included in the table; the respective larger contribution was considered for the total accuracy of the isotope shift.

To account for the model uncertainty for the odd–even isotopes ^245,249,257Fm by choosing a single line profile for the extraction of the resonance centroid, one-third of the full-width at half-maximum of the fit profile was considered in the uncertainty analysis. For ²⁵⁵Fm, the hyperfine structure was resolved, and thus the uncertainty in the determination of the hyperfine parameters was used to determine the model uncertainty in the centre of gravity.

Determination of δ⟨r
²âŸ©

The changes in the mean-square charge radius δ⟨r²âŸ©^A,A′ relative to a reference isotope A can be extracted from the measured isotope shifts via the relation

with the measured isotope shift δν^A,A′ = ν^A′ − ν^A in the atomic transition of isotopes with mass number A and A′, the mass-shift constant M = M_NMS + M_SMS, with normal mass shift (NMS) and specific mass shift (SMS), and the field-shift constant F.

Recently published results from atomic model calculations provided the field-shift constant F for fermium⁴⁴ to evaluate the changes in the nuclear mean-square charge radii. This was performed analogously for nobelium in ref. ³³, which was also used for the evaluation of ²⁵⁵No. Here, an uncertainty of 0.007 cm^−1 of the reference isotope ²⁵⁴No in the argon buffer-gas atmosphere as stated in ref. ³² was assumed to contribute to the isotope-shift measurement’s uncertainty.

The error on δ⟨r²âŸ©^A,A′ results from a propagation of the isotope-shift uncertainty, whereas the uncertainty from the atomic coupling factors is included as a systematic uncertainty. For the field-shift constant, which was predicted with F = −3.14 cm^−1 fm^−2, the uncertainty evaluated from the atomic calculations amounts to 10%.

Although the mass shift can be neglected for the calculation of δ⟨r²âŸ©^A,A′, a contribution of the mass shift to the final uncertainty is nevertheless considered.

The NMS can be calculated to M_NMS = m_eν ≈ 0.4 THz × u with the transition frequency ν and the electron mass m_e. So far, the SMS contribution can be only estimated. For s–p transitions as in our case, the SMS is usually on the order of the NMS. For transitions including orbitals with a higher main quantum number, it can be more than ten times larger. Therefore, a value of 2 THz × u, 5 times larger than the NMS, is considered as a conservative estimate for the contribution to the total systematic uncertainty in the change in mean-square charge radius⁷⁷.

The additional effect of the isotope shift depending on the nuclear deformation proposed in ref. ⁴⁴ was investigated. With the available information on the expected deformation change, this proposed additional effect amounts to −0.003 fm², which is small compared with the uncertainties and is thus neglected.

Nuclear-structure models

Below, we provide a brief description of the models used to interpret experimental findings. All our models are based on the nuclear EDF approach. For a detailed discussion of EDF, see refs. ^78,79.

In this study, calculations using six different EDF models were performed: Fy(IVP) (P.-G.R. & W.N., manuscript in preparation), D1M⁸⁰, BSkG2⁸¹, SV-min⁸² and SLyMR1⁸³ (note that this model is called SLyMR1_3b in ref. ⁸³). This selection aims to represent current EDF models. Concerning the functional form, SV-min and BSkG2 are based on standard Skyrme functionals and density-dependent contact pairing interactions. SLyMR1 uses an extended Skyrme functional, where the density dependences are replaced by an explicit three-body interaction⁸⁴. D1M is based on the Gogny functional. Finally, Fy(IVP) uses the Fayans functional^85,86. SV-min and Fy(IVP) are based on a single-reference approach, whereas D1M and BSkG2 also include approximated beyond-mean-field corrections. SLyMR1 involves explicit configuration mixing and restoration of particle-number and angular-momentum symmetries^87,88,89, hence it is a multi-reference approach. The models were calibrated on experimental ground-state properties of finite nuclei but differ in the choice of the calibration data. SV-min and Fy(IVP) include a statistical analysis of the underlying parameterizations. This allows predictions to be given together with a statistical calibration^47,48.

To assess the predictive power of the theory frameworks, we computed three-point differences

in the binding energy (\({\mathcal{O}}=E\)) or squared charge radius (\({\mathcal{O}}={r}^{2}\)). For N = 82, N = 126 and N = 152, Extended Data Fig. 2 shows predictions of the SV-min and Fy(IVP) models compared with experimental values.

Both EDF frameworks reproduce the trends in the shell gaps and agree with the data that the N = 152 gap is weak. As discussed in ref. ³, the size of this shell gap strongly depends on model details, see, for example, ref. ¹⁸ for the predictions of this subshell with different models. In particular, the Fy(IVP) model reproduces the experimental values for ¹³²Sn and ²⁰⁸Pb, giving confidence also in the value for ²⁵²Fm. This is in agreement with the reported smooth trends along the isotopic chains.

Treatment of odd-A nuclei

EDF calculations for odd-mass nuclei are not straightforward: solving the self-consistent equations requires the creation of a one-quasiparticle excitation on top of a reference state that is typically associated with an even–even nucleus. For each iteration, identifying the physically relevant quasiparticle while guaranteeing convergence of the self-consistent procedure is a non-trivial task. Most calculations for odd-mass nuclei in this study concern the predicted ground state. In the case of SV-min and Fy(IVP), all one-quasiparticle configurations below 1-MeV excitation energy have been investigated, and the states with lowest energy for each angular-momentum projection K have been examined. It became apparent that the radii vary little with K (variance about 0.001 fm), such that taking the minimal-energy state is an acceptable choice. The other exception is SLyMR1: among the several many-body states with different angular momenta that result from symmetry restoration, we select those states matching the (often tentative) experimental quantum numbers. Although the blocking strategy is common to all models, they differ in their treatment of the blocked quasiparticle. D1M relies on the equal filling approximation⁹⁰, a computationally efficient approximation that includes the blocking effect of the odd nucleon(s), but ignores the effect of any time-odd currents or densities that might develop due to the polarization effects⁹¹. The calculations with BSkG2, Fy(IVP), SLyMR1 and SV-min invoke no approximations in this respect.

Nuclear-matter properties of EDF-based models

The leading properties of our models can be characterized in terms of the infinite nuclear-matter properties shown in Extended Data Table 3: saturation density ρ_sat, binding energy per particle E/A, incompressibility K, (isoscalar) effective mass m*/m, symmetry energy at saturation J, and slope of symmetry energy L. The isoscalar effective mass m*/m shows the largest variation. This parameter impacts the single-particle level density around the Fermi level and hence the magnitude of shell effects. The other matter parameters show fewer variations.

Predicted charge radii

In SV-min and Fy(IVP), the charge radii were calculated directly from the charge form factor that contains the proton form factor folded with the intrinsic form factors of the free nucleons, relativistic corrections and the centre-of-mass correction⁹². A similar procedure is followed for BSkG2 and D1M but without relativistic corrections. It is noted that D1M also adds a quadrupole correction estimated by solving the collective Schrödinger equation with the five-dimensional collective Hamiltonian to the absolute charge radius, as described in ref. ⁸⁰. In SLyMR1 calculations, the charge radii were computed from the expectation value of the squared point-proton radius operator at the beyond-mean-field level corrected for the finite size of the protons and neutrons⁸⁸.

To check that our models produce sensible results for the total charge radii in the heavy actinides, Extended Data Table 4 compares our predictions with the measured radii for ²³²Th, ²³⁸U and ²⁴⁴Pu. Given the high computational cost of multi-reference calculations, for SLyMR1, we report only the value for ²³⁸U. The errors given for SV-min and Fy(IVP) are the estimated extrapolation errors from statistical analysis of the underlying χ² fits. The predictions are in good agreement with available data within uncertainties. This instills confidence in the validity of predictions for fermium and nobelium isotopes.

The predicted root-mean-square charge radii for fermium and nobelium isotopes are shown in Extended Data Fig. 3. Prediction uncertainties are indicated for SV-min. Unlike for differential radii, the results for the total radii show a larger spread between the models. This complies with the observation that the isoscalar matter for parameters in Extended Data Table 3 differ outside error bands. Following the discussion in ref. ⁹³, one would expect that the radii would be sorted according to saturation densities in Extended Data Table 3. This is not necessarily the case as the data on charge radii were used in the calibration of individual models and this spoils the correlation⁹³. Still, the inter-model similarity of charge radii is related to similar saturation densities of our models.

Multipole decomposition of densities

The fermium isotopes under consideration are all deformed in shape, and hence their intrinsic densities are non-spherical. To make an inter-model comparison of proton densities ρ_p, we carry out a multipole decomposition. To this end, we define a radial proton density ρ_p,ℓm(r) as an angular average:

Here Y_lm is the spherical harmonics of degree l and order m and Î© represents angular coordinates. For an axially deformed nucleus, m = 0 and we denote ρ_p,ℓ ≡ ρ_p,ℓm=0. These radial densities are related by a Fourier transformation to the radial scattering form factors typically discussed in the context of electron scattering⁹⁴. The root-mean-square point-proton radius can be obtained from the monopole component of the proton density:

where A is the mass number of the nucleus.

Higher-multipolarity radial densities define axial shape deformation parameters:

where R = 1.2A^1/3 fm. The calculated quadrupole (ℓ = 2) and hexadecapole (ℓ = 4) radial densities are shown in Fig. 3.

For the nuclei we study here, single-reference models predict nuclear densities that are deformed but retain both reflection symmetry and axial symmetry. The multi-reference techniques used in SLyMR1 render the comparison with the other models slightly more intricate. For comparison purposes, we use the multipole decomposition of the density of the deformed reference state with lowest particle-number restored energy. This deformed state also breaks axial symmetry. However, the triaxial components are small compared with the axial ones, as also found within the D1M and BSkG2 calculations.