Laser spectroscopy of neutron-rich $^{207,208}$Hg isotopes: Illuminating the kink and odd-even staggering in charge radii across the $N=126$ shell closure

The mean-square charge radii of $^{207,208}$Hg ($Z=80, N=127,128$) have been studied for the first time and those of $^{202,203,206}$Hg ($N=122,123,126$) remeasured by the application of in-source resonance-ionization laser spectroscopy at ISOLDE (CERN). The characteristic \textit{kink} in the charge radii at the $N=126$ neutron shell closure has been revealed, providing the first information on its behavior below the $Z=82$ proton shell closure. A theoretical analysis has been performed within relativistic Hartree-Bogoliubov and non-relativistic Hartree-Fock-Bogoliubov approaches, considering both the new mercury results and existing lead data. Contrary to previous interpretations, it is demonstrated that both the kink at $N=126$ and the odd-even staggering (OES) in its vicinity can be described predominately at the mean-field level, and that pairing does not need to play a crucial role in their origin. A new OES mechanism is suggested, related to the staggering in the occupation of the different neutron orbitals in odd- and even-$A$ nuclei, facilitated by particle-vibration coupling for odd-$A$ nuclei.

The mean-square charge radii of 207,208 Hg (Z = 80, N = 127, 128) have been studied for the first time and those of 202,203,206 Hg (N = 122, 123, 126) remeasured by the application of in-source resonanceionization laser spectroscopy at ISOLDE (CERN). The characteristic kink in the charge radii at the N = 126 neutron shell closure has been revealed, providing the first information on its behavior below the Z = 82 proton shell closure. A theoretical analysis has been performed within relativistic Hartree-Bogoliubov and non-relativistic Hartree-Fock-Bogoliubov approaches, considering both the new mercury results and existing lead data. Contrary to previous interpretations, it is demonstrated that both the kink at N = 126 and the odd-even staggering (OES) in its vicinity can be described predominately at the mean-field level, and that pairing does not need to play a crucial role in their origin. A new OES mechanism is suggested, related to the staggering in the occupation of the different neutron orbitals in odd-and even-A nuclei, facilitated by particle-vibration coupling for odd-A nuclei.
Experimental investigations of nuclear charge radii have revealed a rich abundance of regular patterns, abrupt changes and non-linear trends along isotopic chains across the nuclear chart [1,2]. Two near-universal features include kinks at neutron shell closures and oddeven staggering (OES), where an odd-N isotope has a smaller charge radius than the average of its two even-N neighbors [1,3]. The commonality of these features indicates that their origin is general and independent of local microscopic phenomena. As such, measurements of kinks and OES provide a particularly stringent benchmark for nuclear theory [4][5][6][7][8][9].
Historically, the majority of the discussion pertaining to the effect of shell closures on charge radii focused on the kink in the lead isotopic chain across N = 126 [4-6, 10, 11], which is shown in the inset of Fig. 1. A variety of theoretical approaches have been employed to investigate this kink. Relativistic mean field studies have described it with varying degrees of success [5,12]. By contrast, calculations in non-relativistic density func-arXiv:2012.13802v1 [nucl-ex] 26 Dec 2020 tional theories (NR-DFTs) based on conventional functionals were unable to reproduce the kink [4,8]. The differences between the descriptions in these theoretical frameworks are related to the relative occupations of the neutron ν1i 11/2 and ν2g 9/2 orbitals (located above the N = 126 shell closure), with the magnitude of the kink being driven by the occupation of the ν1i 11/2 orbital [11]. The extension of non-relativistic functionals, by the addition of gradient terms into the pairing interaction, has been demonstrated as a possible way to improve their description of OES and the kink [6,8]. An alternative approach is the inclusion of a density dependence in the spin-orbit interaction [7,13,14], derived from the chiral three-nucleon interaction by Kohno [15].
The optimal method of quantifying the kink at N = 126 considers isotopes with N = 124, 126, 128, to avoid contributions from either OES or possible deformation in isotopes further from the shell closure. However, in contrast to the regions near N = 50 and N = 82 [1], there are limited experimental data on charge radii behavior Ulm  across N = 126 (specifically for N = 128), with corresponding measurements available only for Z = 82 [18] and Z = 83 [19].
In this Letter we report the first study of charge radii across N = 126 in the mercury (Z = 80) isotopic chain, thus enabling the Z-dependence of the kink at N = 126 to be probed and providing the first benchmark for theory in the region below the Z = 82 proton shell closure. These new data motivated us to undertake a comparative theoretical investigation of the kinks and OES in lead and mercury charge radii across N = 126. By applying both spherical relativistic Hartree-Bogoliubov (RHB) [20] and spherical non-relativistic Hartree-Fock-Bogoliubov (NR-HFB) [21] approaches, we explored whether an alternative explanation of the kink and OES is possible. This was also the first study of OES within a relativistic framework.
The experimental data originates from the same measurement campaign as described in Refs. [17,22], where neutron-deficient mercury isotopes were also studied ( Fig. 1). Therefore, only a brief experimental overview is included here. Mercury isotopes were produced at the CERN-ISOLDE facility [23] by impinging a 1.4-GeV proton beam from the Proton Synchrotron Booster on to a molten-lead target. The neutral reaction products effused into the anode cavity of a Versatile Arc Discharge and Laser Ion Source (VADLIS) [24]. Laser light from the ISOLDE Resonance Ionization Laser Ion Source (RILIS) complex [25] was used to excite three sequential atomic transitions, for the resonance ionization of the mercury isotopes [26]. The photo-ions were extracted and mass separated by the ISOLDE general purpose separator and then directed to either a Faraday cup for direct photoion detection, or to the ISOLTRAP Paul trap [27] and multi-reflection time-of-flight mass spectrometer (MR-ToF MS) [28] for single-ion counting and discrimination from isobaric contamination.
The first of the three atomic transitions (5d 10 6s 2 1 S 0 → 5d 10 6s6p 3 P • 1 , 253.65 nm) was probed by scanning a frequency-tripled titanium-sapphire laser (full width at half maximum bandwidth <1 GHz before tripling) [29]. Isotope shifts (IS), δν A,A = ν A − ν A , in the frequency of this transition were measured for mercury nuclei with A = 202, 203, 206, 207, 208 relative to the stable reference isotope with A = 198. Sample spectra are presented in Fig. 2. Details of the scanning and fitting procedures can be found in Refs. [22,30] and Refs. [22,31], respectively, with further information on the data analysis in Refs. [32,33]. The relative changes in the mean-square charge radii, δ r 2 A,A , were extracted from the measured δν A,A values via standard methods described in the supplemental material [34].
The extracted δν A,A and δ r 2 A,A values are presented in Table I  for 202,203 Hg, and the remeasured neutron-deficient isotopes [17,22] from the same experimental campaign, are in good agreement with literature. For 206 Hg there is a ≈ 500-MHz discrepancy between this work and the thesis value of Ref. [35] cited in [16], which is discussed in Ref. [32]. Given the agreement of the results from the present experimental campaign with the other literature data, the δν 206,198 value from this work is used for the following discussion.
To interpret the data, a new RHB code was developed which enables the blocking of selected single-particle orbitals and allows for fully self-consistent calculations of the ground and excited states in odd-A nuclei. A separable version of the Gogny pairing is used [36], with the pairing strength of Ref. [37]. The NL3*, DD-PC1, DD-ME2 and DD-MEδ covariant energy density functionals (CEDF) were employed, the global performance of which was tested in Ref. [37]. The functionals achieved a comparable description of the kink and the OES, thus only the DD-ME2 results are discussed below. The results for the other functionals will be included in a follow-up paper [38]. The NR-HFB calculations were performed assuming spherical symmetry with the semi-realistic M3Y-P6a interaction, the spin-orbit properties of which were modified [13] to improve the description of the charge radii of proton-magic nuclei [7,13,14]. We applied it here for the first time to the mercury isotopic chain. For N ≤ 126, isotopes with β 2 2 1/2 < 0.1 were considered, where β 2 2 is the mean-square deformation deduced from δ r 2 using the droplet model [3,39]. This restriction corresponds to N ≥ 116 and N ≥ 121 for lead and mercury isotopes, respectively.
The importance of a simultaneous agreement of energetic and geometric nuclear observables in such investigations has been highlighted previously [6]. Thus, in addition to calculating the charge radii, we also checked the quality of the binding energy description. There is a good agreement between the DD-ME2 and experimental binding energies for both lead and mercury isotopes [40]. The rms deviation is 1.3 MeV and 1.1 MeV for 198−214 Pb and 201−208 Hg, respectively, a comparable performance to the widely used NR-DFT functional UNEDF1 [41] (1.4 MeV and 1.0 MeV for lead and mercury, respectively [42]). The binding and separation energy descriptions of all of the employed CEDFs will be discussed in detail in Ref. [38].
Two different procedures labeled as "LES" and "EGS" are used for the blocking in odd-A nuclei and the results of respective calculations are labeled by the "Functional-Procedure" labels (for example, DD-ME2-EGS). In the LES procedure, the lowest in energy configuration is used: this is similar to all earlier calculations of OES [6,43]. In the EGS procedure, the configuration with the spin and parity of the blocked state corresponding to those of the experimental ground state is employed, although it is not necessarily the lowest in energy. For example, in the RHB(DD-ME2) calculations the EGS configurations with a blocked ν2g 9/2 state are located at excitation energies of 137 keV, 122 keV and 96 keV above the ground state configurations with a blocked ν1i 11/2 state in 209,211,213 Pb. At first glance, this contradicts experimental findings that the ground state is based on the ν2g 9/2 orbital in odd-A lead isotopes with N > 126. However, particle-vibration coupling (PVC) lowers the energy of this state below that of the ν1i 11/2 one (see Fig.  5 in Ref. [44]) so that it becomes the lowest in energy in the PVC calculations. Note that PVC significantly improves the accuracy of the description of the energies of experimental states in model calculations [44,45]. However, it is neglected in the present study since its impact on charge radii is still an open theoretical question.
The results of the RHB and NR-HFB calculations are presented in Fig. 3, together with the experimental results for the lead and mercury chains. In both cases, the kink at N = 126 is visibly better reproduced in the RHB (1) To quantify the shell effect at N = 126, the kink indicator of Ref. [8] is used which considers the isotope's next-tonearest neighbors and is defined as where R(A) = r 2 1/2 (A) is the charge radius of the isotope with mass A of the element under consideration. Note that the kink indicator is independent of the blocking procedure in odd-A nuclei and therefore we omit the corresponding specifications (LES or EGS) in the discussion of the kink. ∆ r 2 (3) (A) and ∆R (3) (A) values calculated from the experimental results and theoretical calculations for both lead and mercury are presented in Fig. 4, and the ∆R (3) (A) values are listed in the supplemental material [34]. It is evident in Figs. 4 (a) and 4 (b) that the magnitude of the kinks in the isotopic chains are comparable, suggesting that the kink at N = 126 is broadly insensitive to the change of the occupied proton states when crossing Z = 82 (π2d 3/2 in mercury and π3s 1/2 in lead). In addition, the RHB(DD-ME2) calculations best reproduce the kink, while the NR-HFB(M3Y-P6a) and NR-HFB(Fy(∆r) [8]) results underestimate and overestimate its magnitude, respectively. It is worth noting that both the RHB(DD-ME2) and NR-HFB(M3Y-P6a)  (N = 126), respectively. Experimental mercury data: this work and [16], experimental lead data: [18], the statistical uncertainties are smaller than the data points. RHB(DD-ME2) results: this work, NR-HFB(M3Y-P6a) results: this work (mercury) and [14] (lead), NR-HFB(UNEDF1) results: [42].
approaches are reasonably successful in the reproduction of the absolute charge radius values for 206 Hg and 208 Pb, the details are included in the supplemental material [34].
In both approaches, the OES is best reproduced if the EGS procedure is applied (see Figs. 4 (c), (d) and Fig. 5). If the LES procedure is applied, the experimental OES is significantly underestimated for all nuclei under study in the RHB calculations and for N < 126 nuclei in the NR-HFB calculations. Note that for simplicity we show only NR-HFB results with both procedures in Fig. 4 (c) and (d).
For a better understanding of the underlying mechanisms of both the kink and OES, we also performed RHB calculations without pairing for lead isotopes. The labels identifying such results contain "np". Significantly, a kink is still present in the results as depicted in Fig. 4 (a), due to the occupation of the ν1i 11/2 orbital. This indicates a mechanism alternative to the one based on gradient terms in pairing interactions [6,8].
The RHB results with and without pairing are compared via ∆ r  (c), respectively, and mercury (b) and (d), respectively. Experimental values are taken from this work and from Ref. [16] (mercury) and Ref. [18] (lead). The RHB(DD-ME2) and NR-HFB(M3Y-P6a) results are obtained in this work and the NR-HFB results with Fy(∆r) and UNEDF1 are taken from Refs. [8] and [42], respectively. Experimental uncertainty is depicted as translucent gray bars in (a) and (b), and as error bars in (c) and (d). calculations (the curves labeled as "DD-ME2-EGS" and "DD-ME2-np-EGS") under the condition that, in odd-A nuclei, the EGS procedure is used. One can see that the inclusion of pairing somewhat reduces this effect. However, OES is mostly absent if the LES procedure is used in odd-A nuclei.
Let us consider the lead isotopes with N ≥ 126 for a more detailed discussion of the origin of OES in the calculations without pairing. By designating the ground state of 208 Pb as a "core" and noting that PVC lowers the energy of the ν2g 9/2 state below ν1i 11/2 in odd-A nuclei [44], the sequence of the ground states in the N ≥ 126 nuclei can be described as "core" ( 208 Pb), "core"⊗ν(2g 9/2 ) 1 ( 209 Pb), "core"⊗ν(2i 11/2 ) 2 ( 210 Pb), "core"⊗ν(2i 11/2 ) 2 (2g 9/2 ) 1 ( 211 Pb) and so on in the relativistic calculations without pairing. The ν1i 11/2 orbital has a smaller rms-radius than the ν2g 9/2 orbital. However, because of the isovector nature of nuclear forces its occupation leads to a larger charge radius as compared with the occupation of ν2g 9/2 orbital. Thus, the staggering in their occupation between odd and even isotopes results in the OES seen in Fig. 5.
On the contrary, in the majority of conventional nonrelativistic functionals, the ν2g 9/2 orbital is lower in energy than the ν1i 11/2 orbital. This is in agreement with experimental data on the structure of the ground states in odd-mass nuclei, but creates a problem in the description of the kinks. In addition, in calculations with and without pairing this leads to the sole or predominant occupation of the ν2g 9/2 state in even-even and odd-even nuclei with N > 126 and thus to a negligible or comparatively small OES. To address this, several prescriptions have been suggested over the years to increase the occupation of the ν1i 11/2 orbital in the N > 126 lead nuclei. One approach includes a modification of the spin-orbit interaction leading either to the inversion of the relative energies of these two states or to their proximity in energy [11,[46][47][48][49][50]. The NR-HFB results with M3Y-P6a shown in Figs. 3 and 4 are also based on a modification of the spin-orbit interaction, with the inclusion of a density dependent term in the spin-orbit channel. Alternatively, the so-called Fayans functionals employ a specific form of the pairing interaction containing a gradient term [6,8,43,51]. Although this improves the general description of experimental data, discrepancies between theory and experiment still exist in the lead and tin isotopic chains [8]. Moreover, pairing becomes a dominant contributor to the kink and OES [8].
The present RHB interpretation of the kinks and OES differs from that suggested in [8], which is based on non-relativistic Skyrme and Fayans functionals. In the RHB approach, the kink and OES are already present in the calculations without pairing. Thus, the evolution of charge radii with neutron number depends significantly on the mean-field properties. Pairing acts only as an additional tool which mixes different configurations and somewhat softens the evolution of charge radii as a function of neutron number.
In conclusion, the determination of the δ r 2 A,A of 207,208 Hg has revealed a kink at N = 126 in the mercury nuclear charge radii systematics, with a magnitude comparable to that in the lead isotopic chain. These new data have been analyzed via both RHB and NR-HFB approaches, together with the traditional magic-Z theoretical benchmark of the lead isotopic chain. We demonstrate that the kinks at the N = 126 shell closure and the OES in the vicinity, are currently best described in the RHB approach without any readjustment of the parameters defined in Ref. [52]. According to the RHB calculations, the kink at N = 126 in δ r 2 A,A originates from the occupation of the ν1i 11/2 orbital located above the N = 126 shell gap. A new mechanism for OES is suggested, related to the staggering in the occupation of neutron orbitals between odd and even isotopes and facilitated by PVC in odd-mass nuclei. Thus, contrary to previous interpretations, it is determined that both the kink and OES in charge radii can be defined predominantly in the particle-hole channel.
This project has received funding through the European Union's Seventh Framework Programme for Research and Technological Development under grant agreements 267194 (COFUND), 262010 (ENSAR), 289191 (LA 3 NET) and 267216 (PEGASUS). This project has received funding from the European Union's Horizon 2020 research and innovation programme grant agreement No 654002. This material is based upon work supported by the US Department of Energy, Office of Science, Office of Nuclear Physics under Award No. DE-SC0013037, and by the DFG cluster of excellence "Origin and Structure of the Universe" (www.universecluster.de). S. S. acknowledges a SB PhD grant from the former Belgian Agency for Innovation by Science and Technology (IWT), now incorporated in FWO-Vlaanderen. This work was supported by the RFBR