††thanks: Present address: Institut für Kernphysik, Technische Universität
Darmstadt, 64289 Darmstadt, Germany.††thanks: Present address: CERN, 1211,
Geneva 23, Switzerland.††thanks: Present address: Department of Oncology
Physics, Edinburgh Cancer Centre, Western General Hospital, Edinburgh, EH4
2XU, United Kingdom††thanks: Present address: Department of Physics,
University of Liverpool, Liverpool, L69 7ZE, United Kingdom††thanks: Present
address: Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay,
France.††thanks: Present address: Université Paris-Saclay, CNRS/IN2P3, IJCLab,
91405 Orsay, France.††thanks: Present address: Wako Nuclear Science Center
(WNSC), Institute of Particle and Nuclear Studies (IPNS), High Energy
Accelerator Research Organization (KEK), Wako, Saitama 351-0198,
Japan††thanks: Present address: ARC Centre of Excellence for Engineered
Quantum Systems, School of Physics, The University of Sydney, NSW 2006,
Australia.

# Charge radii, moments and masses of mercury isotopes across the $N=126$
shell closure

T. Day Goodacre<EMAIL_ADDRESS>Department of Physics and Astronomy,
School of Natural Science, The University of Manchester, Manchester, M13 9PL,
United Kingdom  CERN, CH-1211 Geneva 23, Switzerland  TRIUMF, Vancouver V6T
2A3, Canada  A.V. Afanasjev  Department of Physics and Astronomy, Mississippi
State University, MS 39762, USA  A.E. Barzakh  Petersburg Nuclear Physics
Institute, NRC Kurchatov Institute, Gatchina 188300, Russia  L. Nies  CERN,
CH-1211 Geneva 23, Switzerland  Universität Greifswald, Institut für Physik,
17487 Greifswald, Germany  B.A. Marsh  CERN, CH-1211 Geneva 23, Switzerland
S. Sels  CERN, CH-1211 Geneva 23, Switzerland  KU Leuven, Instituut voor Kern-
en Stralingsfysica, B-3001 Leuven, Belgium  U.C. Perera  Department of Physics
and Astronomy, Mississippi State University, MS 39762, USA  P. Ring  Fakultät
für Physik, Technische Universität München, D-85748 Garching, Germany  F.
Wienholtz  CERN, CH-1211 Geneva 23, Switzerland  Universität Greifswald,
Institut für Physik, 17487 Greifswald, Germany  A.N. Andreyev  Department of
Physics, University of York, York, YO10 5DD, United Kingdom  Advanced Science
Research Center (ASRC), Japan Atomic Energy Agency (JAEA), Tokai-mura, Japan
P. Van Duppen  KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001
Leuven, Belgium  N.A. Althubiti  Department of Physics and Astronomy, School
of Natural Science, The University of Manchester, Manchester, M13 9PL, United
Kingdom  Physics Department, Faculty of Science, Jouf University, Aljouf,
Saudi Arabia  B. Andel  KU Leuven, Instituut voor Kern- en Stralingsfysica,
B-3001 Leuven, Belgium  Department of Nuclear Physics and Biophysics, Comenius
University in Bratislava, 84248 Bratislava, Slovakia  D. Atanasov  Max-Planck-
Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  R.S.
Augusto  TRIUMF, Vancouver V6T 2A3, Canada  J. Billowes  Department of Physics
and Astronomy, School of Natural Science, The University of Manchester,
Manchester, M13 9PL, United Kingdom  K. Blaum  Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  T.E. Cocolios
Department of Physics and Astronomy, School of Natural Science, The University
of Manchester, Manchester, M13 9PL, United Kingdom  KU Leuven, Instituut voor
Kern- en Stralingsfysica, B-3001 Leuven, Belgium  J.G. Cubiss  Department of
Physics, University of York, York, YO10 5DD, United Kingdom  G.J. Farooq-Smith
Department of Physics and Astronomy, School of Natural Science, The University
of Manchester, Manchester, M13 9PL, United Kingdom  KU Leuven, Instituut voor
Kern- en Stralingsfysica, B-3001 Leuven, Belgium  D.V. Fedorov  Petersburg
Nuclear Physics Institute, NRC Kurchatov Institute, Gatchina 188300, Russia
V.N. Fedosseev  CERN, CH-1211 Geneva 23, Switzerland  K.T. Flanagan
Department of Physics and Astronomy, School of Natural Science, The University
of Manchester, Manchester, M13 9PL, United Kingdom  The Photon Science
Institute, The University of Manchester, Manchester, M13 9PL, United Kingdom
L.P. Gaffney  KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001
Leuven, Belgium  School of Computing, Engineering, and Physical Sciences,
University of the West of Scotland, Paisley PA1 2BE, United Kingdom  L. Ghys
KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium
Belgian Nuclear Research Center SCK$\bullet$CEN, Boeretang 200, B-2400 Mol,
Belgium  A. Gottberg  TRIUMF, Vancouver V6T 2A3, Canada  University of
Victoria, Department of Physics and Astronomy, Victoria, BC V8W 2Y2, Canada
M. Huyse  KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001 Leuven,
Belgium  S. Kreim  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1,
69117 Heidelberg, Germany  CERN, CH-1211 Geneva 23, Switzerland  P. Kunz
TRIUMF, Vancouver V6T 2A3, Canada  Department of Physics, Simon Fraser
University, Burnaby, BC, V5A 1S6, Canada  D. Lunney  CSNSM-IN2P3, Université
de Paris Sud, Orsay, France  K.M. Lynch  Department of Physics and Astronomy,
School of Natural Science, The University of Manchester, Manchester, M13 9PL,
United Kingdom  CERN, CH-1211 Geneva 23, Switzerland  V. Manea  Max-Planck-
Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Y.
Martinez Palenzuela  KU Leuven, Instituut voor Kern- en Stralingsfysica,
B-3001 Leuven, Belgium  CERN, CH-1211 Geneva 23, Switzerland  T.M. Medonca
CERN, CH-1211 Geneva 23, Switzerland  P.L. Molkanov  Petersburg Nuclear
Physics Institute, NRC Kurchatov Institute, Gatchina 188300, Russia  M.
Mougeot  CERN, CH-1211 Geneva 23, Switzerland  J.P. Ramos  CERN, CH-1211
Geneva 23, Switzerland  M. Rosenbusch  Universität Greifswald, Institut für
Physik, 17487 Greifswald, Germany  R.E. Rossel  CERN, CH-1211 Geneva 23,
Switzerland  Institut für Physik, Johannes Gutenberg-Universität, D-55099
Mainz, Germany  S. Rothe  CERN, CH-1211 Geneva 23, Switzerland  L. Schweikhard
Universität Greifswald, Institut für Physik, 17487 Greifswald, Germany  M.D.
Seliverstov  Petersburg Nuclear Physics Institute, NRC Kurchatov Institute,
Gatchina 188300, Russia  P. Spagnoletti  School of Computing, Engineering, and
Physical Sciences, University of the West of Scotland, Paisley PA1 2BE, United
Kingdom  C. Van Beveren  KU Leuven, Instituut voor Kern- en Stralingsfysica,
B-3001 Leuven, Belgium  M. Veinhard  CERN, CH-1211 Geneva 23, Switzerland  E.
Verstraelen  KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001
Leuven, Belgium  A. Welker  CERN, CH-1211 Geneva 23, Switzerland  Institut für
Kern- und Teilchenphysik, Technische Universität Dresden, Dresden 01069,
Germany  K. Wendt  Institut für Physik, Johannes Gutenberg-Universität,
D-55099 Mainz, Germany  R.N. Wolf  Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Universität Greifswald, Institut
für Physik, 17487 Greifswald, Germany  A. Zadvornaya  KU Leuven, Instituut
voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium  K. Zuber  Institut für
Kern- und Teilchenphysik, Technische Universität Dresden, Dresden 01069,
Germany

###### Abstract

Combining laser spectroscopy in a Versatile Arc Discharge and Laser Ion
Source, with Penning-trap mass spectrometry at the CERN-ISOLDE facility, this
work reports on mean-square charge radii of neutron-rich mercury isotopes
across the $N=126$ shell closure, the electromagnetic moments of 207Hg and
more precise mass values of 206-208Hg. The odd-even staggering (OES) of the
mean square charge radii and the kink at $N=126$ are analyzed within the
framework of covariant density functional theory (CDFT), with comparisons
between different functionals to investigate the dependence of the results on
the underlying single-particle structure. The observed features are defined
predominantly in the particle-hole channel in CDFT, since both are present in
the calculations without pairing. However, the magnitude of the kink is still
affected by the occupation of the $\nu 1i_{11/2}$ and $\nu 2g_{9/2}$ orbitals
with a dependence on the relative energies as well as pairing

##  I Introduction

The kink in the relative mean square charge radii ($\delta\langle
r^{2}\rangle$) at the $N=126$ shell closure has long been considered as a
benchmark for testing theoretical calculations. Traditionally the lead
isotopic chain was employed [1, 2, 3, 4, 5], but new experimental results in
this region revealing the systemics of other isotopic chains [6, 7, 8, 9],
mass measurements, and odd-even staggering (OES) in charge radii offer the
opportunity to broaden this benchmark. The droplet model is unable to
reproduce this kink because of the absence of single-particle degrees of
freedom [10]. Early non-relativistic mean field approaches were also incapable
of reproducing a kink at 208Pb [11], while conversely, relativistic mean field
approaches were demonstrated to be successful in doing so [2].

Two alternative modifications were suggested to correct this deficiency in
non-relativistic models. The first relies on the modification of the spin-
orbit interaction, either through a fitting procedure (see Refs. [12, 3]) or
via the introduction of a density dependence (see Refs. [13, 14]). This leads
to a reasonable reproduction of the experimental isotope shifts (see Refs. [3,
15]). The second approach (employing so-called Fayans functionals) introduces
gradient terms into the pairing and surface terms of the functional [4, 16,
17]. This significantly improves the general description of experimental data,
however, discrepancies are still apparent in the lead and tin isotopic chains
[18]. Moreover, pairing becomes a dominant contributor to the kink and OES
[18], in contradiction with the results obtained in relativistic Hartree-
Bogoliubov (RHB) calculations with the DD-ME2 functional and non-relativistic
Hartree-Fock-Bogoliubov (NR-HFB) calculations with the M3Py-P6a functional
presented in Ref. [9].

This work is an in-depth follow-up to Ref. [9], which reported on the
$\delta\langle r^{2}\rangle$ of mercury isotopes across $N$=126 and employed
these results, together with existing lead data, to compare RHB and NR-HFB
approaches. A new OES mechanism was additionally suggested, related to the
staggering in the occupation of the different neutron orbitals in odd- and
even-$A$ nuclei and facilitated by particle-vibration coupling (PVC) in
odd-$A$ nuclei. Here we report on the magnetic-dipole and electric-quadrupole
moments of 207Hg, new and improved mass measurements of 206-208Hg and a
detailed theoretical study within the RHB framework to better understand the
kinks and OES in lead and mercury isotopes. Multiple state-of-the-art
covariant energy density functionals (CEDFs) are employed (NL3* [19], DD-PC1
[20], DD-ME2 [21] and DD-ME$\delta$ [22]) to assess the dependence of the
theoretical results on the underlying single-particle structure. The global
performance of these functionals in describing ground state properties such as
masses and charge radii of even-even nuclei has been tested in Refs. [23] and
[24].

This article is arranged as follows. The experimental techniques are presented
in Sec. II. The radiogenic production of 207,208Hg in a molten target is
discussed in Sec. III. Experimental results on mean square charge radii, -
dipole and electric-quadrupole moments and masses are summarized in Sec. IV.
The discussion of experimental results is presented in Sec. V. Theoretical
formalism and theoretical analysis of the kinks and OES in charge radii are
presented in Sec. VI, together with their dependence on the underlying single-
particle structure and pairing and a comparison of experimental and calculated
binding energies. Finally, we give a brief summary in Sec. VII.

##  II Experimental technique

The neutron-rich mercury isotopes were studied at the ISOLDE facility [25] as
part of a wider experimental campaign which investigated both ends of the
isotopic chain [9, 26, 27]. The mercury nuclei were produced using the Isotope
Separator On-Line (ISOL) method [28, 29] and studied via in-source resonance
ionization spectroscopy [30, 31] as depicted in Fig. 1(a).

Figure 1: a) Overview of the experimental setup for the production and study
of mercury isotopes. b) Schematic of a VADLIS coupled to a molten lead target
via a temperature controlled chimney. c) The variation in the RILIS-mode and
VADIS-mode ion currents on mass A=197 for differing anode bias voltages. See
text for additional details and the definition of acronyms.

A molten lead target (thickness 170 g/cm2) was bombarded with 1.4-GeV protons,
resulting in a cocktail of reaction products which effused via a temperature
controlled chimney [32] into the anode volume of a Versatile Arc Discharge and
Laser Ion Source (VADLIS) [33]. The target and ion source were biased at 30 kV
and laser light from the ISOLDE Resonance Ionization Laser Ion Source (RILIS)
[34] was directed into the anode volume for multi-step resonance ionization
[35] of mercury isotopes. A
$\\{\lambda_{1}\left|\lambda_{2}\right|\lambda_{3}\\}=\\{254~{}$nm$|313~{}$nm$|$532
nm$\\}$ ionization scheme [36] was applied, with the first (254 nm) resonant
transition used to investigate the hyperfine structure (hfs) and isotope
shifts in the 5d106s2 1S0 $\rightarrow$ 5d106s6p 3P${}_{1}^{\circ}$ atomic
transition.

The ions were accelerated by the electric field resulting from the grounded
extraction electrode depicted in Fig. 1(a) and Fig. 1(b) to form a 30-keV
radioactive ion beam (RIB). The ISOLDE General Purpose Separator [25] was
employed for mass separation before the RIB was directed to either a Faraday
cup for direct ion current measurement or to the ISOLTRAP radio frequency
quadrupole cooler-buncher (RFQ C-B) [37]. Downstream of the RFQ C-B, the RIB
was injected into the Multi-Reflection Time-of-Flight Mass Spectrometer (MR-
ToF MS) [38] for either isobaric separation and subsequent detection [39] or
for mass measurements, either by measuring the time-of-flight of the ions [40]
or by utilizing the downstream Penning traps [41].

The lead target-VADLIS combination was required to avoid the overwhelming
isobaric francium contamination present on masses $A$=207, 208 when employing
a standard UC${}_{\text{x}}$ target with a hot cavity surface ion source for
the laser light-atom interaction region [42]. Alternative approaches have
struggled to suppress certain isotopes of francium, and would additionally be
expected to result in a factor of $\approx$20 reduction of the signal of
interest [43, 44].

A schematic of the VADLIS is presented in Fig. 1(b) together with the relative
bias of the components. In the standard Versatile Arc Discharge Ion Source
(VADIS)-mode of operation, atoms and molecules are ionized by electrons that
are emitted from the $\approx$2000 ∘C cathode and accelerated into the anode
volume by a relative anode voltage of 100-200 V [45]. The selective RILIS-mode
of operation was employed for this experiment, where the anode voltage is
optimized for laser-ion extraction while maintaining it below what is required
for significant electron impact ionization [33]. Figure 1(c) presents the on-
line optimization of the RILIS-mode with radiogenically produced 197Hg. The
RILIS-mode and VADIS-mode related signals were separated by blocking and
unblocking the laser light exciting the 254-nm transition. A clear maximum is
visible with a near-negligible background with the anode voltage set to
$\approx$8 V. The alternative, applying the RILIS lasers with a 100-200 V
anode bias (termed RILIS+VADIS-mode) would have resulted in significant
isobaric contamination and a reduced signal-to-noise ratio as a result of the
competing ionization processes.

The benefits of combining the RILIS-mode of operation with the ISOLTRAP MR-ToF
MS are highlighted in Fig. 2(a). Operating in RILIS-mode reduced the isobaric
208Pb background by seven orders of magnitude compared with RILIS+VADIS-mode.
This enabled the MR-ToF MS to be employed for selective detection and for
determining the mass of 208Hg. Time-of-flight spectra recorded on and off
resonance with the MR-ToF MS are shown in Fig. 2(b) and 2(c), respectively. By
applying time gates in the ToF spectra, it was possible to separate the 208Hg
signal from the remaining isobaric contamination.

Figure 2: a) Ion rate on mass $A$=208 measured by the MR-ToF MS detector for a
given frequency tripled laser frequency. The RILIS+VADIS-mode is dominated by
VADIS ionized 208Pb, the count rate was estimated based on Faraday cup
measurements. Further discussion in the text. b) and c) time-of-flight spectra
on mass $A$=208, measured downstream of the MR-ToF MS with the VADLIS in
RILIS-mode and the lasers on resonance and off-resonance, respectively. The
y-axis scales of b) and c) are identical.

##  III Radiogenic production of ${}^{207,208}\text{Hg}$ in a molten lead
target

Considering the natural lead (206-208Pb) target material used for this
experiment, the creation of mercury isotopes with $N\leq 126$ is comparatively
well understood as the result of spallation reactions induced by the incident
1.4 GeV proton beam. However, when going beyond $N=126$ (206Hg) the production
mechanism changes, and a range of other processes may become relevant [46, 47]
including secondary reactions induced by the light and energetic products of
the primary spallation reactions. The production of 207Hg via
208Pb($n,2p$)207Hg is a good example of such a process, and was first reported
at an ISOL facility in Ref. [48]. The mechanism for producing 208Hg is
significantly more exotic, as evidenced by a factor of 2400 decrease in the
measured ion rate between 207Hg and 208Hg. There are a number of potential
production channels including 208Pb(t,3$p$)208Hg, 208Pb($\alpha$,4$p$)208Hg,
or reactions with radiogenically produced 209Pb (t${}_{1/2}\approx$ 3 h) or
210Pb (t${}_{1/2}\approx$ 22 y) which build up within the target during the
experiment.

The in-target production of mercury isotopes was calculated via ABRABLA [49,
50], FLUKA [51, 52] and Geant4 [53, 54, 55] simulations. The results are
assessed by considering the isotope specific extraction and ionization
efficiencies ($\varepsilon$), determined by dividing the measured yield by the
calculated in-target production. Figure 3 presents the relationship between
$\varepsilon$ and half-life for the mercury isotopes measured with the MR-ToF
during this experimental campaign. For 202Hg (stable) and 203Hg
(t${}_{1/2}\approx$47 days) the half-lives are set at 1$\times 10^{5}$ s to
facilitate their inclusion. The data is fitted using Eq. (4) from Ref. [42],
with the hollow data points omitted from the fits to enable them to converge.
As expected, $\varepsilon$ generally increases with increasing half-life, and
stabilizes at a point where the half-life is sufficiently long compared to the
release time.

Figure 3: Ionization and release efficiency ($\varepsilon$) as a function of
the half-life of mercury isotopes from a molten lead target. The results from
ABRABLA, FLUKA and Geant4 simulations are compared. See text for details.

All of the results broadly agree with an extraction and ionization efficiency
of approximately 1% for sufficiently long-lived isotopes. ABRABLA [49] is not
capable of reproducing the secondary reactions required for the production of
207,208Hg, however, as it is commonly used for calculating in-target
production, it is useful for benchmarking the other codes for this
application. FLUKA [52] was found to reproduce some 207Hg production, though
based on Fig. 3 the rate appears to be underestimated. The Geant4
(geant4$-$10$-$07) simulations employing the Liege (QGSP$\\_$INCLXX$\\_$HP)
model [54] combined with the native de-excitation code were the most
successful in reproducing both 207Hg and 208Hg production, though with an
apparent overestimation of the latter. Discrepancies with the Geant4 results
may be a consequence of the necessity to scale from the simulation of a
reduced density target, which was required to enable a feasible simulation
time.

While 208Hg production was not present in the FLUKA results, the simulations
presented the possibility to investigate the 208Pb(t,3$p$)208Hg channel. The
results for tritium production (using FLUKA2021.0) were convoluted with cross-
section data from TENDL17 [56] over a 40-200 MeV interval. This resulted in an
in-target production rate of 110 atoms/µC for 208Hg, significantly below the
estimated rate of $\approx$56,000 atoms/µC calculated considering a 1$\%$
extraction and ionization efficiency. This suggests that 208Pb(t,3$p$)208Hg
reactions only contribute to a fraction of the observed 208Hg yields. Based on
this, we tentatively conclude that the observed 208Hg production is the result
of a combination of multiple reaction channels.

##  IV Experimental Results

###  IV.1 Laser spectroscopy of mercury isotopes across $N$=126

Mean square charge radii, magnetic-dipole and electric-quadrupole moments were
studied via the measurement of isotope shifts and hfs in the 254-nm
transition. Sample spectra are presented in Fig. 4(a), with the (substate
weighted) centroids indicated with solid black lines.

Reference scans of 198Hg were taken periodically to monitor the stability of
the experimental setup, with a 10 h interval. Multiple measurements of the hfs
of each isotope were taken (2$\times^{202,203}$Hg, 3$\times^{206,207}$Hg,
5$\times^{208}$Hg) and the fitting of the resulting spectra was cross-checked
using multiple software packages: Origin 2016 with a chi-squared minimization
[57] performed with a Levenberg-Marquardt algorithm [58, 59], the SATLAS open
source Python package [60] and a similar program written in ROOT [61]. The
results are presented in Table I, together with literature data
(202,203,206Hg) for comparison. Relative mean square charge radii, electric
dipole and magnetic quadrupole moments were extracted from the spectra by
applying standard methods, these are summarized in Appendix A.

Figure 4: (a) Hfs spectra of the measured isotopes, the (substate weighted) centroids are indicated with solid black lines. The y-axis represents the number of “counts per shot” from the RFQ C-B. (b) and c) Sample plots of the measured time-of-flight ion-cyclotron resonances. (b) Measurement of 206Hg using a Ramsey-type excitation scheme with an excitation time of $2\times$60\text{\,}\mathrm{ms}$$ separated by $480\text{\,}\mathrm{ms}$. (c) Measurement of 207Hg using a single-pulse excitation scheme with an excitation time of $1.2\text{\,}\mathrm{s}$. Table I: Extracted isotope shifts in the 254-nm line with respect to 198Hg, hyperfine $a$ and $b$ factors of the 5d106s6p 3P${}_{1}^{\circ}$ state and literature values recalculated from the compilations of [62] and [63]. The spin assignment of 207Hg is discussed in the text. Statistical uncertainties are listed in parentheses and the systematic uncertainties related to $F_{254}$ and $M$ are listed in curly brackets. Isotope | $I^{\pi}$ | $\delta v^{A,198}$ | $a$ | $b$ | $\delta\langle r^{2}\rangle^{A,198}$ | $\mu$ | $Q_{s}$ | Ref.
---|---|---|---|---|---|---|---|---
A |  | (MHz) | (MHz) | (MHz) | (fm2) | ($\mu_{N}$) | ($b$) |
202 | 0 | -10 100(180) |  |  | 0.197(3){14} |  |  | This work & [9]
|  | -10 102.4(4.2) |  |  | 0.1973(2){152} |  |  | [62]
203 | 5/2- | -11 870(200) | 5070(90) | -20(250) | 0.232(5){17} | 0.843(15) | 0.03(35) | This work & [9]
|  | -11 750(180) | 4991.33(4) | -249.2(3) | 0.2296(35) {180} | 0.8300(7) | 0.40(4) | [62, 63]
206 | 0 | -20 930(160) |  |  | 0.409(3){30} |  |  | This work & [9]
|  | -20 420(80) |  |  | 0.3987(16){308} |  |  | [62]
207 | (9/2+) | -25 790(190) | -4500(60) | 530(250) | 0.503(4){38} | -1.373(20) | -0.73(37) | This work & [9]
208 | 0 | -32 030(160) |  |  | 0.625(3){47} |  |  | This work & [9]

The nuclear spin of 207Hg could not be determined unambiguously because the
spectroscopic transition is between atomic states with electronic spins $J$=0
and $J$=1. $I^{\pi}=9/2^{+}$ was assumed for the analysis of the 207Hg
measurements based on Refs. [48, 64]. The extracted isotope shifts for
202,203Hg are in good agreement with literature. The same is true for the
neutron deficient isotopes that were re-measured during this experimental
campaign [26, 27]. The 500-MHz discrepancy between the $\delta\nu^{206,198}$
value of [65] and this work is discussed in [66]. The general agreement of the
extracted $\delta v^{A,198}$ and hyperfine $a$ and $b$ factors with the
previously published literature values further validates the method of in-
source resonance ionization spectroscopy with a VADLIS ion source.

###  IV.2 Mass spectrometry of 206-208Hg

The masses of 206-208Hg were measured, employing different techniques with
respect to earlier experiments that are referenced in the AME2020 [67]. A
short outline of the time-of-flight methods that were used in our experiment
can be found in Appendix B. A summary of the measured values is presented in
Table II and a comparison with the AME2020 is shown in Figure 5.

The atomic mass of 206Hg was previously deduced from $\alpha$-decay
measurements [68]. Our measurements of 206Hg, using the Time-of-Flight Ion-
Cyclotron-Resonance (ToF-ICR) technique, represent the first direct
determination of the mass of this nucleus. For the ToF-ICR measurements in a
Penning trap, two excitation schemes were employed: a single pulse excitation
of $400\text{\,}\mathrm{ms}$, as well as a Ramsey-type scheme [69], presented
in Figure 4b, where two excitation pulses of $60\text{\,}\mathrm{ms}$ were
applied, separated by a waiting time of $480\text{\,}\mathrm{ms}$.

The masses of 207Hg and 208Hg have been determined previously by storage-ring
measurements at GSI using Schottky mass spectrometry [70, 71, 72]. In the
present experiment, ToF-ICR measurements were performed for 207Hg with a
single-pulse excitation of $1.2\text{\,}\mathrm{s}$ (see Figure 4c). For
208Hg, the hyperfine structure was measured in five different laser scans,
using ISOLTRAP’s MR-ToF MS as mass separator and ion counter. Mass data were
extracted from the scans by summing individual binned data per scan step into
a single histogram, resulting in one histogram per scan. The ToF-distributions
corresponding to singly charged ions of the isotope of interest, 208Hg+, and
to an on-line reference ion, 208Pb+, were aligned as outlined in [73] and
fitted by employing an unbinned maximum-likelihood estimation where the fit
function was constructed as an Exponential-Gaussian-Hybrid (EGH) to account
for the tails of the ToF-distributions towards longer flight times. The mass
was extracted by calculating the average $C_{\text{ToF}}$ of the five scans,
using the 208Pb+ present in the RIB and 133Cs+ from an off-line ion source as
reference ions. The results we report are in general agreement with the
literature and improve upon the precision.

Table II: Mass-measurement results for the mercury isotopes, given either as the ratio $R$ of cyclotron frequencies from ToF-ICR or as the C${}_{\text{ToF}}$ from the MR-ToF MS. The computed mass excess values M${}_{\text{exc}}$ from this work are compared to the literature values found in [67]. Additionally, half-lives T1/2 from [67] are given as well as the reference ions that were used to extract the mass values. The mass excesses are given in terms of energy divided by the square of the speed of light $\mathrm{\text{ $c_{0}$ }}$. Isotope |  T1/2 | Ref. ions |  C${}_{\text{ToF}}$ | R |  M${}_{\text{exc, ISOLTRAP}}$ |  M${}_{\text{exc, AME20}}$ | $|\Delta_{\text{TRAP-AME20}}|$ |
---|---|---|---|---|---|---|---|---
|  ($\mathrm{min}$) |  |  |  |  ($\mathrm{k}\mathrm{eV}\mathrm{/}\mathrm{\text{ $c_{0}$ }}\mathrm{{}^{2}}$) |  ($\mathrm{k}\mathrm{eV}\mathrm{/}\mathrm{\text{ $c_{0}$ }}\mathrm{{}^{2}}$) |  ($\mathrm{k}\mathrm{eV}\mathrm{/}\mathrm{\text{ $c_{0}$ }}\mathrm{{}^{2}}$) |
206Hg | $8.32(7)\text{\,}$ |  133Cs+ | / | $1.549\,807\,228\,2(661)\text{\,}$ | $-20\,932.1(82)\text{\,}$ | $-20\,946(20)\text{\,}$ | $13(22)\text{\,}$ |
207Hg | $2.9(2)\text{\,}$ |  133Cs+ | / | $1.557\,367\,639\,6(489)\text{\,}$ | $-16\,446.2(61)\text{\,}$ | $-16\,487(30)\text{\,}$ | $41(31)\text{\,}$ |
208Hg | $42(5)\text{\,}$ |  208Pb+, 133Cs+ | $0.500\,108\,969(255)\text{\,}$ | / | $-13\,279(20)\text{\,}$ | $-13\,270(30)\text{\,}$ | $9(36)\text{\,}$ |
Figure 5: Difference between the ISOLTRAP mass measurements of this work and
the corresponding AME20 values (black), including the AME20 error band (grey).

##  V Discussion

###  V.1 Magnetic moment of 207Hg

A value of $g$(207Hg)=$-$0.305(6) was deduced based on the results presented
in Table I. The $g$-factors of the $\nu g_{9/2}$ isotones 209Pb, 211Po are
plotted in Fig. 6 together with the Schmidt value, the energies of the first
excited $2^{+}$ states ($E(2^{+})^{-2}$) of the $N$=126 cores and
$g(^{210}$Bi) calculated from the measured magnetic moments of the 5–, 7– and
9– isomeric states [74, 75] using the additivity relation and assuming a pure
$[\pi h_{9/2}\otimes\nu g_{9/2}]$ configuration for these states. The inverse
square of $E$(2+) is of particular relevance due to its approximate
proportionality to the second-order perturbation theory correction [76, 77].

Figure 6: Squares: experimental $g$-factors for $N=127$ isotones ($\nu
g_{9/2}$ [74, 75, 78, 79] and this work). The $g$-factor of 210Bi is deduced
from measurements of isomer states, see text for details. Dashed line shows
the Schmidt value for $\nu g_{9/2}$ shell. Triangles: $E$(2+)-2 values of the
corresponding $N=126$ even-even nuclei (see the corresponding scale on the
right-hand side of the figure) experimental data from [80].

The experimentally determined $g$-factors differ significantly from the
Schmidt value $g_{Schmidt}$=$-$0.425 and there is a noticeable $Z$-dependence.
The deviation of the $g$-factor of the magic 209Pb isotope was explained in
non-relativistic [81, 82] and relativistic [83] approaches by taking into
account corrections for the meson exchange current and first and second order
core polarization (CP). The configuration admixture contributing to the
magnetic dipole moment in the first order of perturbation theory corresponds
to a particle-hole excitation from an orbit $j=l+1/2$ to its spin-orbit
partner $j=l–1/2$ (CP1 correction [84]). In the second order of perturbation
theory, the most important magnetic moment correction stems from the odd-
particle coupling with the lowest 2+ excitation of the core (CP2 correction
[76, 77]). In the vicinity of the doubly magic 208Pb, the most important
$Z$-dependent CP1 correction corresponds to the proton
($h^{-1}_{11/2}h_{9/2}$) particle-hole excitation. A corresponding increase in
the occupancy of the $\pi h_{9/2}$ orbital with the increase of $Z$, reduces
the probability of proton ($h^{-1}_{11/2}h_{9/2}$) core excitations, thereby
decreasing the magnitude of the CP1 correction. The opposite is apparent in
Fig. 6, where the deviation from the Schmidt value increases between
${}^{209}_{82}$Pb and ${}^{211}_{84}$Po, thus CP1 corrections do not appear to
be the dominant driver for the $g$-factor $Z$-dependence. Additionally, meson
exchange corrections have been shown to have a weak $A$ dependence in the
vicinity of 208Pb [77, 85]. It could then be suggested that CP2 corrections
are primarily responsible for the $Z$-dependence of the discrepancy with the
Schmidt value.

The same mechanism (particle-quadrupole-vibration coupling) was used to
explain the magnetic moment evolution in the vicinity of the magic numbers
[77, 76]. This explanation is supported by the apparent correspondence of the
energies of the first excited $2^{+}$ states of the $N$=126 cores and the
$g$-factors of the $N$=127 isotones in Fig. 6. Considering that
$E$($2^{+},^{208}$Pb)$\approx$4.1 MeV, $E$($2^{+},^{206}$Hg)$\approx$1.1 MeV
and $E$($2^{+},^{210}$Po)$\approx$1.2 MeV [80], the admixture of the (2+,$\nu
i_{13/2}$)9/2+ should be larger for 207Hg and 211Po than for 209Pb, resulting
in an increased CP2 correction. It is worth noting that this interpretation
indicates the importance of particle-vibration coupling in the description of
the ground-state properties of the odd-$A$ nuclei in the vicinity of shell
closure. This same mechanism proves to be decisive for explanation of the
charge radii behavior in this region of the chart of the nuclides (see Sec.
VI).

###  V.2 Quadrupole moment of 207Hg

Quadrupole moments near the closed proton and neutron shells have a
predominantly single-particle nature and are usually well described by the
shell-model formula:

$Q=-e_{eff}\frac{2j-1}{2j+1}\left\langle r^{2}\right\rangle_{j}$ (1)

where $j$ is the spin of the odd particle, $\langle r^{2}\rangle_{j}$ is its
mean square radius and $e_{eff}$ is an effective charge. Taking $\langle
r^{2}\rangle_{j}$ from Ref. [86], one obtains: $e_{eff}(^{207}$Hg$;\nu
2g_{9/2})$=2.4(13) $e$. This neutron effective charge is noticeably larger
than the value of the “universal” neutron $e_{eff}=0.95~{}e$ which describes
fairly well the measured quadrupole moments for all closed-shell $\pm 1$
nuclear states in the vicinity of 208Pb [87]. This points to the rapid
increase of a quadrupole core polarization when moving away from the magic
$Z$=82, similar to that observed for the proton $e_{eff}$ when moving away
from the magic $N$=126 [87].

##  VI Theoretical interpretation

###  VI.1 Theoretical framework and the details of the calculations

A theoretical interpretation of experimental data is performed within the
framework of covariant density functional theory [88] using the RHB computer
code for spherical nuclei first employed in Ref. [9]. This code enables the
blocking of selected single-particle orbitals and allows for fully self-
consistent calculations of the ground and excited states in even-even and
odd-$A$ spherical nuclei. In the pairing channel, the RHB code employs a
separable version of the Gogny pairing [89] with the pairing strength defined
in Ref. [23].

Several restrictions/constraints are employed in the present paper. First, we
consider only spherical nuclei (i.e. for which
$\left<\beta_{2}^{2}\right>^{1/2}<0.1$ where
$\left<\beta_{2}^{2}\right>^{1/2}$ is the mean-square deformation deduced from
experimental $\delta\left<r^{2}\right>$ using the droplet model (see Refs.
[90, 91]).) This restriction corresponds to $N\geq 116$ and $N\geq 121$ for
lead and mercury isotopes, respectively, and it was already used in our
earlier paper [9].

Second, two different procedures labeled as “LES” and “EGS” are used for the
blocking in odd-$A$ nuclei, and the results of the respective calculations are
labeled by these abbreviations. In the LES (lowest in energy solution)
procedure, the lowest in energy configuration is used, which is similar to all
earlier calculations of OES in non-relativistic DFTs [4, 17]. In the EGS
(experimental ground state) 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. The
need for this procedure is due to the following considerations. First, the
nodal structure of the wavefunctions and neutron radius of the single-neutron
orbital depend on its quantum numbers such as total and orbital angular
momenta (see Refs. [3, 5, 92]). Thus, the occupation of different neutron
single-particle states impacts the resulting charge radii (see Refs. [3, 5]
and the discussion of Fig. 7(a) below). Second, the structure of the
experimental ground states in odd-$A$ nuclei is reproduced globally only in
approximately 40% of the nuclei in the non-relativistic and relativistic DFTs
[93, 94]111The inclusion of particle-vibrational coupling increases the
accuracy of the description of the single-particle configurations in odd-$A$
nuclei (see Refs. [95, 96])..

If there is a mismatch in the structure of experimental and calculated ground
states the impact of the blocked orbital on the physical observable of
interest (charge radius, deformations, binding etc.) in odd-$A$ is expected to
deviate from the value observed in experiment. The consequences of this
mismatch exist for the deformations of one-quasiparticle states (see Ref.
[94]), odd-even staggerings in charge radii (see Ref. [9]) and binding
energies (see Ref. [97]). Note that the latter is used to define pairing
indicators. In such a situation it is safer to use the blocked solution with
the structure corresponding to experimental ground state (even if it leads to
an excited solution) for the description of charge radii since moderate shift
in the energy of single-particle state has negligible effect on its neutron
single-particle rms radius $\left<r^{2}\right>_{sp}$.

###  VI.2 Charge radii and related indicators

The charge radii were calculated from the corresponding point proton radii as

$r_{ch}=\sqrt{\left<r^{2}\right>+0.64}\,\,\,\,{\rm fm}$ (2)

where $\left<r^{2}\right>$ stands for mean square radius of proton density
distribution and the factor 0.64 accounts for the finite-size of the proton.

Figure 7: The $\delta\left<r^{2}\right>^{N,126}$ values of the Pb and Hg
isotopes obtained in the RHB calculations with indicated CEDFs. The
experimental data are taken from Refs. [9, 62, 78]. The
$\delta\left<r^{2}\right>^{N,126}$ values, obtained in the calculations with
CEDF NL3* without pairing under the condition that either only $\nu 1i_{11/2}$
or only $\nu 2g_{9/2}$ orbitals are occupied in the $N>126$ nuclei, are shown
by blue dashed and dash-dotted lines, respectively.

Three indicators are commonly used to facilitate the quantitative comparison
of the experimental results with those from theoretical calculations. The
first is differential mean-square charge radius222This quantity is frequently
written as a function of mass number $A$. However, we prefer to define it as a
function of neutron number $N$ since this allows to see the behavior of the
$\delta\left<r^{2}\right>^{N,N^{\prime}}$ curves at neutron shell closures for
different isotopic chains.

$\displaystyle\delta\left<r^{2}\right>^{N,N^{\prime}}=\left<r^{2}\right>(N)-\left<r^{2}\right>(N^{\prime})=r^{2}_{ch}(N)-r^{2}_{ch}(N^{\prime})$
(3)

where $N^{\prime}$ is the neutron number of the reference nucleus.

The second one is the $\xi_{even}$ indicator

$\displaystyle\xi_{even}=\frac{\delta\langle
r^{2}\rangle^{128,126}}{\delta\langle r^{2}\rangle^{126,124}},$ (4)

introduced in Ref. [8, 66]. It provides a quantitative measure of the change
of the slope (i.e. $\delta\left<r^{2}\right>^{N,126}/\delta N$) of
differential charge radii as a function of neutron number $N$ at $N=126$. The
applicability of $\xi_{even}$ is restricted by the limited availability of the
data for the $N$=128 isotones because for $84\leq Z\leq 88$ they have half-
lives ($t_{1/2})<$300 µs [98] which limits the potential for laser
spectroscopy measurements. Thus, in these cases the $\xi_{even}$ indicator is
replaced by

$\displaystyle\xi^{*}_{even}=\frac{2}{N_{0}-126}\,\,\frac{\delta\langle
r^{2}\rangle^{N_{0},126}}{\delta\langle r^{2}\rangle^{126,124}},$ (5)

where $N_{0}$ is the lowest even neutron number at $N>126$ with measured
isotopic shift: $N_{0}=132$ for 84Po [99, 6], 85At [100], 86Rn [90, 101], 87Fr
[102, 103, 104], 88Ra [105], and $N_{0}=138$ for 89Ac [106]. For the 82Pb and
83Bi isotopes, the data for the nuclei with $N=124,126,128$ were taken from
[78] and [8], respectively. Note that for the Bi and Pb isotopes, for which
the $N=128$ data are available, $\frac{2}{N_{0}-126}\,\,\delta\langle
r^{2}\rangle^{N_{0},126}\approx\delta\langle r^{2}\rangle^{128,126}$
($N_{0}=130$ for Bi and $N_{0}=132$ for Pb) since differential radii increase
nearly linearly for the $N>126$ nuclei under study.

The third is the three-point indicator

$\displaystyle\Delta\langle r^{2}\rangle^{(3)}(N)$ $\displaystyle=$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left[\langle
r^{2}\rangle(N-1)+\langle r^{2}\rangle(N+1)-2\langle r^{2}\rangle(N)\right]=$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[r_{ch}^{2}(N-1)+r_{ch}^{2}(N+1)-2r_{ch}^{2}(N)\right]$
(6)

which quantifies OES in charge radii.

###  VI.3 The kink in charge radii and its relation to underlying single-
particle structure and pairing

Figure 8: The energies of neutron single-particle states at spherical shape in
208Pb obtained in the calculations without pairing with the indicated CEDFs.
Solid and dashed connecting lines are used for positive- and negative-parity
states. Spherical gaps are indicated.

The differential charge radii of the Pb and Hg isotopes are shown in Fig. 7.
One can see that all of the employed CEDFs generate a kink at $N=126$ and that
the best description is provided by the CEDF DD-ME2. Thus, it is important to
understand which physical features determine the differences between the
functionals. For that we look at the energies of the neutron single-particle
states and their occupation probabilities.

The energies of neutron single-particle states obtained in the 208Pb nucleus
with the employed CEDFs are shown in Fig. 8. One can see close similarities in
the predictions of the energies and relative positions of the $2f_{5/2}$,
$3p_{3/2}$ and $3p_{1/2}$ states occupied in the $N\leq 126$ nuclei. In
contrast, the differences are more pronounced for the $1i_{11/2}$ and
$2g_{9/2}$ orbitals located above the $N=126$ shell closure. In all
functionals, the $1i_{11/2}$ orbital is the lowest in energy333The order of
these two orbitals is inverted in the majority of non-relativistic functionals
(see Refs. [3, 5]) and in many of them this creates a problem in the
description of the kink in charge radii at $N=126$. but the energy gap between
these two orbitals strongly depends on the functional. It is smallest in DD-
ME2, gets larger in NL3* and DD-PC1 and become extremely large in DD-
ME$\delta$. Because of this feature, and the fact that the kink in charge
radii at $N=126$ is defined by the interplay of the occupation of these two
orbitals, we focus our discussion on the $N>126$ nuclei.

Let us analyze the slope of differential radii defined as
$\delta\left<r^{2}\right>^{N,N^{\prime}}/\delta N$. Note that in such an
analysis we consider only even-even nuclei. The results of the calculations
without pairing indicate that this slope is almost the same for $N<126$444The
averaged slopes for the $N<126$ nuclei are almost the same in the calculations
with and without pairing (see Ref. [92] for detail). and $N>126$ nuclei when
only the $2g_{9/2}$ states are occupied above $N=126$ (see Fig. 7(a) for Pb
isotopes and Fig. 7(b) for Hg isotopes). As a consequence, there would be
either no kink or a very small kink in the charge radii at $N=126$. However,
the situation drastically changes when only $1i_{11/2}$ states are occupied
above $N=126$. This leads to a substantial increase of the
$\delta\left<r^{2}\right>^{N,N^{\prime}}/\delta N$ slope and as a result to a
creation of large kink in the charge radii at $N=126$. This is related to the
fact that the neutron $1i_{11/2}$ orbital is the $n=1$ orbital which overlaps
more strongly with the majority of the proton orbitals than the $n=2$ neutron
$2g_{9/2}$ orbital [5]. As a consequence, it provides a larger pull of the
$1i_{11/2}$ neutron states on proton orbitals via the symmetry energy. This is
despite the fact that in 208Pb the rms radius of the neutron $1i_{11/2}$
orbital (for example, $r_{ch}=6.4131$ fm in DD-ME2) is smaller than that of
the neutron $2g_{9/2}$ one (for example, $r_{ch}=7.0227$ fm in DD-ME2) by
$\approx 0.6$ fm in all employed CEDFs.

Figure 9: (a) The evolution of the cumulative occupation probabilities
$\text{v}^{2}_{state}$ of the neutron $2g_{9/2}$ and $1i_{11/2}$ orbitals as a
function of neutron number in the $N\geq 126$ Pb nuclei for the indicated
CEDFs. (b) The evolution of the energies of these single-particle states as a
function of neutron number. Note that both types of physical quantities are
calculated in canonical basis. The neutron shell closure at $N=126$ and the
energy gap between the $2g_{9/2}$ and $1i_{11/2}$ orbitals at $N=138$ are
indicated. Note that this figure is based on the results of the calculations
of even-even nuclei. Dashed ellipses envelope the states with the same
structure.

The inclusion of pairing modifies the situation in the $N>126$ nuclei in such
a way that both of these orbitals become partially occupied (see Fig. 9(a)).
Note that we consider a cumulative occupation probability ${\rm
v}^{2}_{state}$ which provides information on the filling of a given
$j$-subshell: ${\rm v}^{2}_{state}$ could take any value between 0 (unoccupied
subshell) and $2j+1$ (fully occupied $j$-subshell). Pairing also leads to a
partial occupation of the single-particle states located above the $N=148$ gap
(see Fig. 8), but their occupation probabilities are relatively small because
of the presence of this gap. Thus, for the sake of simplicity we focus our
discussion on the interplay of the occupation of the $2g_{9/2}$ and
$1i_{11/2}$ states and the consequences of this interplay on the
$\delta\left<r^{2}\right>^{N,N^{\prime}}/\delta N$ slope.

The large energy gap of 1.27 MeV between the $2g_{9/2}$ and $1i_{11/2}$ states
in the DD-ME$\delta$ functional (see Fig. 8) is responsible for a predominant
occupation of the lower lying $1i_{11/2}$ states (see Fig. 9(a)). Considering
the $N=134$ nucleus as an example, the eight neutrons outside of the $N=126$
shell closure are located almost entirely in the $1i_{11/2}$ subshell (${\rm
v}^{2}_{1i_{11/2}}\approx 7.1$) with only a small portion occupying $2g_{9/2}$
(${\rm v}^{2}_{2g_{9/2}}\approx 0.5$). This leads to a large
$\delta\left<r^{2}\right>^{N,N^{\prime}}/\delta N$ slope (see Fig. 7). Note
that this slope is the largest among the considered functionals and it is not
far away from the one obtained in the calculations without pairing, when only
the $1i_{11/2}$ states are occupied in the nuclei with $N>126$ (see Fig. 7).
The reduction of the gap between the $2g_{9/2}$ and $1i_{11/2}$ orbitals in
the DD-PC1, NL3* and, especially, DD-ME2 functionals (see Figs. 8 and 9(b)) is
responsible for a decrease of the difference in the occupation of these
orbitals (see Fig. 9). For these three functionals, the eight neutrons outside
of the $N=126$ shell closure in the $N=134$ nucleus are still located
predominantly in the $1i_{11/2}$ subshell (${\rm v}^{2}_{1i_{11/2}}\approx
4.7$) but with a significant portion also found in $2g_{9/2}$ (${\rm
v}^{2}_{2g_{9/2}}\approx 2.5$). This leads a reduction of the
$\delta\left<r^{2}\right>^{N,N^{\prime}}/\delta N$ slope as compared with the
case of the DD-ME$\delta$ functional. One can see in Fig. 7 that the
experimental $\delta\left<r^{2}\right>^{N,N^{\prime}}/\delta N$ slope in the
$N>126$ nuclei is reproduced with comparable accuracy by the DD-ME2, NL3* and
DD-PC1 functionals. Note that with increasing neutron number $N$ the single-
particle states become more bound, but for a given functional, the relative
energies of the $2g_{9/2}$ and $1i_{11/2}$ states change only slightly (see
Fig. 9(b)). As a consequence, the occupation probabilities of the single-
particle states behave as almost linear functions of neutron number (see Fig.
9(a)).

The magnitude of the kink in charge radii at $N=126$ is better quantified by
the $\xi_{even}$ and $\xi^{*}_{even}$ indicators defined in Eqs. (4) and (5),
respectively. Experimentally available values of these indicators (both for
even-even and odd nuclei) and calculated (only for even-even nuclei) are
compared in Fig. 10. Mercury is the first element below $Z$=82 and only the
second even-$Z$ element for which $\xi_{even}$ is experimentally determined;
the kink in the mercury charge radii is of a similar magnitude to that of
lead. Only experimental $\xi^{*}_{even}$ indicators are available for $Z>83$
nuclei. Note that experimental $\xi_{even}$ and $\xi^{*}_{even}$ indicators
form a smooth curve which indicates that neither addition nor subtraction of
proton(s) from the $Z=82$ nuclei affects drastically a kink in charge radii at
$N=126$. Figure 10 clearly shows that the best reproduction of this trend is
achieved by the CEDF DD-ME2. Other functionals (including the Fayans
Fy($\Delta r$) functional) deviate somewhat from experimental data.

This analysis clearly indicates that the evolution of charge radii with
neutron and proton numbers is sensitive to the details of underlying singe-
particle structure and the occupation probabilities of these states. Note that
the latter depends on the type of pairing interaction employed in the
calculations and its strength. It also indicates that the magnitude of the
kink in charge radii at $N=126$ depends on the relative balance of the
occupation of the $2g_{9/2}$ and $1i_{11/2}$ orbitals. In general, the
substantial occupation of the $1i_{11/2}$ orbital and the kink can be obtained
even when the $2g_{9/2}$ orbital is located lower in energy than the
$1i_{11/2}$ orbital (but still within its vicinity).This, for example, takes
place in the non-relativistic HFB (NR-HFB) calculations with the semirealistic
M3Y-P6a interaction, the spin-orbit properties of which were modified [13] to
improve the description of the charge radii of proton-magic nuclei [15, 13,
14]. However, these calculations underestimate the kink. A similar situation
exists in the Skyrme DFT calculations with the SLy4mod functional presented in
Ref. [5].

Figure 10: Experimental and calculated $\xi_{even}$ (for $Z\leq 83$) and
$\xi^{*}_{even}$ (for $Z\geq 84$) indicators. Note that the calculated results
are presented only for even-even nuclei. The experimental values are
determined using Eq. (4) and Eq. (5), using data from [8, 9, 62, 78] and the
references listed in the explanation of Eq. (5), while the calculated ones are
from the charge radii defined in the present paper and in Ref. [23] which is
publicly available at Ref. [107]. The $\xi_{even}$(Pb) value for Fayans
Fy$(\Delta r)$ functional was extracted from the data presented in Ref. [18]).

###  VI.4 Odd-even staggering in charge radii

It was shown in Ref. [9] both in the RHB calculations with DD-ME2 and in the
NR-HFB studies with semi-realistic M3Y-P6a interaction that OES in charge
radii is best reproduced when the EGS procedure is applied in odd-$A$ nuclei.
In contrast, the experimental OES is significantly underestimated when the LES
procedure is used in odd-$A$ nuclei in the RHB framework for all nuclei under
study and for $N<126$ nuclei in the NR-HFB approach. The same behavior is
observed also in the results of RHB calculations with CEDFs DD-ME$\delta$, DD-
PC1 and NL3*, the results of which are shown in Fig. 11. This figure shows
also that there is some dependence of the magnitude of the
$\Delta\left<r^{2}\right>^{(3)}(N)$ values on the employed functional which
comes from the differences in the energies of the single-particle states and
their occupations (see Figs. 8 and 9).

Figure 11:  Comparison of experimental and theoretical
$\Delta\left<r^{2}\right>^{(3)}(N)$ values for isotopes of lead (a) and
mercury (b) , respectively. The EGS procedure is employed in odd-$A$ nuclei.
Experimental data are taken from Refs. [62, 108, 9]. Vertical orange dot-dot-
dashed lines indicate $N=126$.

Particle-vibration coupling (PVC) plays a critical role in the emergence of
such significant OES in charge radii because it leads to the nucleonic
configuration with the blocked state corresponding to the experimental ground
state (see Ref. [9] for details). Let us illustrate that with the case of the
$N>126$ Pb nuclei. In the odd-$A$ isotopes, the PVC coupling lowers the
$2g_{9/2}$ state below the $1i_{11/2}$ state making it the ground state
despite the fact that at the mean field level (as well as in even-even nuclei)
the energy of the $2g_{9/2}$ state is higher than that of the $1i_{11/2}$
state (see Fig. 5 of Ref. [95]). Although these results were obtained with
NL3*, a comparable effect of PVC on relative energies of these states is
expected in other functionals. However, the feasibility of such a scenario
also crucially depends on the relative energies of the single-particle
orbitals of interest at the mean field level. For example, if the energy gap
between the $2g_{9/2}$ and $1i_{11/2}$ states is too large (like in the case
of the DD-ME$\delta$ functional, see Figs. 8 and 9(b)), the impact of the PVC
would most likely not be enough to make the $2g_{9/2}$ state as a ground
state. This functional however suffers from some significant deficiencies in
the $Z>82$ region which probably are the consequences of the overly large
energy gap between these two states. For example, it does not predict octupole
deformed actinides [109] and predicts fission barriers in superheavy nuclei
which are too small to make them relatively stable [110]. One has to keep in
mind, however, that the interaction DD-ME$\delta$ is different from the other
interactions discussed here. It is the most microscopic one among considered
functionals: only four parameters at the saturation density are fitted to
finite nuclei and the full density dependence of the parameters is derived
from ab-initio calculations. On the contrary, the other interactions contain
an additional 2 (NL3*), 4 (DD-ME2) or 6 (DD-PC1) phenomenological parameters
for the fine-tuning of the density dependence.

###  VI.5 Binding energies and $\Delta^{(3)}_{E}$ indicators

Empirical approaches considering binding energies can help with understanding
the interplay between nucleons. With exception of the DD-ME$\delta$
functional, there is a good agreement between the calculated and the
experimental binding energies for both lead and mercury isotopes. For
198-214Pb and 201-208Hg nuclei, the rms deviations from experiment are 1.3 MeV
and 0.6 MeV for NL3* CEDF, 1.3 MeV and 1.1 MeV for DD-ME2, 1.8 MeV and 2.1 MeV
for DD-PC1, and 4.0 MeV and 3.9 MeV for DD-ME$\delta$, respectively. To
highlight the odd-even staggering of binding energies along the isotopic
chains, an indicator in the form of

$\displaystyle\Delta^{(3)}_{E}(Z,N)=\frac{1}{2}\left[B(Z,N-1)-2B(Z,N)+B(Z,N+1)\right]$
(7)

was used. Where $B(N,Z)$ is the binding energy for a nucleus with proton
number $Z$ and neutron number $N$. The odd-even staggerings in the binding
energies555This quantity is frequently used as a pairing indicator. However,
in no way it should be considered as a clean measure of pairing correlations
since it is polluted by time-odd mean fields and particle-vibration coupling
in odd-$A$ nuclei (see detailed discussion in Ref. [97]). $\Delta^{(3)}_{E}$
are reproduced reasonably well (see Fig. 12), especially when the LES are used
in odd-$A$ nuclei. The level of agreement is comparable with that provided by
the Fayans Fy($\Delta r$) functionals in Ref. [17].

Figure 12 also illustrates the challenges faced by all existing theories.
Polarization effects in deformation/radial density distributions and pairing
depend on blocked state in odd-$A$ nuclei. Thus, on the one hand, the
$\Delta^{(3)}_{E}$ indicators are distorted by incorrect polarizations effects
when a wrong (as compared with experiment) state is used for the ground state
of odd-$A$ nucleus. The related uncertainties in binding energies due to
polarization effects in the pairing channel are on the level of 150 keV and
those due to deformation/radial density distribution polarization effects are
more difficult to estimate but are expected typically to be less than 100 keV.
On the other hand, large theoretical uncertainties in the predictions of
single-particle energies (see Refs. [95, 96, 94] and Fig. 8) reveal themselves
when the $\Delta^{(3)}_{E}$ indicators are defined using the EGS procedure in
odd-$A$ nuclei. This is especially pronounced in the calculations for the
$N>126$ nuclei with the DD-ME$\delta$ functional (see Fig. 12) which is
characterized by a large energy gap between the $2g_{9/2}$ and $1i_{11/2}$
subshells (see Fig. 8). Comparing these uncertainties, it is safer to use the
LES procedure in the definition of the $\Delta^{(3)}_{E}$ indicators and their
association with pairing indicators (see detailed discussion in Ref. [97]).
Note that the calculations somewhat overestimate experimental
$\Delta^{(3)}_{E}$ indicators. The PVC provides additional binding (on the
level of few 100 keV) to the ground states of odd-$A$ nuclei. Thus, its
inclusion into the calculations is expected to decrease the calculated
$\Delta^{(3)}_{E}$ and, as a consequence, to improve the description of
experimental data.

Figure 12:  Experimental and calculated odd-even staggerings in binding
energies $\Delta^{(3)}_{E}$ of the Pb and Hg isotopes under study. The
calculated results are defined for the EGS and LES procedures in odd-$A$
nuclei.

##  VII Conclusion

The kink in the $\delta\langle r^{2}\rangle$ systematics of the mercury
isotopic chain has been analyzed considering a variety of dimensionless
parameters and employed in the comparison of the results of a range of
covariant energy density functionals. The results of mass measurements of
206-208Hg are presented which improve upon the precision of previously
measured values. The observed $Z$ dependence of the $g$-factors of $I$=9/2,
$N$=127 isotones is interpreted as the result of CP2 corrections. The
magnitude of the extracted neutron effective charge for 207Hg suggests a
rapidly increasing quadrupole core polarization when moving away from the
$Z$=82 proton shell.

The theoretical analysis of charge radii and related indicators in the Pb and
Hg isotopic chain has been performed within the RHB framework using several
CEDFs characterized by different single-particle properties. This analysis
supports the conclusion that the kink at $N=126$ in
$\delta\left<r^{2}\right>^{N,N^{\prime}}$ originates from the occupation of
the $\nu 1i_{11/2}$ orbital located above the $N=126$ shell gap. The pairing
effect does not play a critical role here since the kink is present also in
the calculations without pairing. This is in contrast to non-relativistic
Skyrme and Fayans functionals, in which the pairing becomes a dominant
contributor to the kink and OES [18]. However, the pairing plays an important
role in defining the magnitude of the kink which depends on the balance of the
occupation of the $1i_{11/2}$ and $2g_{9/2}$ orbitals. This balance
sensitively depends on the relative energies of these two orbitals. The DD-ME2
functional provides the best description of kinks at $N=126$ not only in the
Pb and Hg isotopes but also in all isotopic chains for which experimental data
is available. A reasonable description of OES in charge radii has been
achieved with all employed functionals. This confirms a new mechanism of OES
suggested in Ref. [9] which is related to the staggering in the occupation of
neutron orbitals between odd and even isotopes facilitated by PVC in odd-mass
nuclei.

##  VIII Acknowledgements

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 (LA3NET) and 267216
(PEGASUS). This project has received funding from the European Union’s Horizon
2020 research and innovation programme grant agreement No 654002 (ENSAR2).
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 Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
under Germany’s Excellence Strategy EXC-2094-390783311, ORIGINS. 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 according to the research project N 19-02-00005; the
ERC Consolidator Grant No. 648381; the IUAP-Belgian State Science Policy (BRIX
network P7/12), FWO-Vlaanderen (Belgium) and GOA’s 10/010 and 10/05 and
starting grant STG 15/031 from KU Leuven; the Science and Technology
Facilities Council Consolidated Grant Nos. ST/F012071/1 and ST/P003885/1,
Continuation Grant No. ST/J000159/1 and Ernest Rutherford Grant No.
ST/L002868/1; the Slovak Research and Development Agency, contract No.
APVV-14-0524; the French IN2P3, the BMBF (German Federal Ministry for
Education and Research) grants Nos. 05P12HGCI1, 05P15HGCIA, 05E18CHA, and
05P18HGCIA.

##  Appendix A Nuclear observables in the hyperfine structure

Nuclear observables were extracted from the experimentally measured hfs
through the application of standard methods [90]. The substate weighted
centroid $\nu_{0}$ and the hyperfine $a$ and $b$ parameters were extracted
from the fitted spectra, where the shift $\Delta\nu^{F}$ of the state $F$ of
the hyperfine multiplet from $\nu_{0}$ is given as

$\Delta\nu^{F}=a\frac{K}{2}+b\frac{\frac{3}{4}K(K+1)-I(I+1)J(J+1)}{2(2I-1)(2J-1)IJ},$
(8)

where $K$=$(F(F+1)-I(I+1)-J(J-1))$, $I$ is the nuclear spin and $J$ is the
atomic angular momentum. The extraction of $\nu_{0}$ enabled the calculation
of $\delta\nu^{A,A^{\prime}}$, the isotope shift between $A$ the isotope under
investigation and $A^{\prime}$ a reference isotope. $\delta\langle
r^{2}\rangle^{A,A^{\prime}}$, the change in the mean square charge radius of
$A$ with respect $A^{\prime}$ was extracted as

$\delta\nu^{A,A^{\prime}}=F_{\lambda}K(Z)\delta\langle
r^{2}\rangle^{A,A^{\prime}}+M\times\frac{A-A^{\prime}}{AA^{\prime}},$ (9)

where for the spectroscopic transition $F_{254nm}=-55.36$ GHz fm-2 [62],
$K(Z=80)=0.931$ (taking into account [111, 112]) is a calculable correction
factor and $M$ is the mass shift factor representing the sum of the normal
mass shift ($M_{NMS}$) and the specific mass shift ($M(Z=80)=(1\pm 0.5)\cdot
M_{NMS}$ [62]).

The magnetic moments $\mu_{A}$ were calculated as

$\mu_{A}=\mu_{ref}\times\dfrac{I_{A}}{I_{ref}}\times\dfrac{a_{A}}{a_{ref}}\times(1+^{ref}\Delta^{A})$
(10)

where the isomer 199mHg ($I=13/2$) was used as a reference nucleus with
$\mu(^{199m}Hg)=-1.0147(8)~{}\mu_{N}$ [113] and $a(^{199m}Hg)=-2298.3(2)$ MHz
[113]. An additional correction for the hyperfine anomaly (HFA) is included
for the $\mu$ values presented in Table I. Moskowitz and Lombardi [114]
demonstrated that for mercury isotopes the “Bohr-Weisskopf” component
(${}^{A_{1}}\Delta^{A_{2}}_{BW}$) [115] is dominant, thus the “Breit-
Rosenthal” component (${}^{A_{1}}\Delta^{A_{2}}_{BR}$) [116] can be ignored.
The following relation between the magnetic moments and the HFA, the so called
Moskowitz-Lombardi (ML) rule was used, determining
${}^{A_{1}}\Delta^{A_{2}}_{BW}$ [115] as

${}^{A_{1}}\Delta^{A_{2}}_{BW}=\pm\alpha\times\Bigg{(}\dfrac{1}{\mu_{1}}-\dfrac{1}{\mu_{2}}\Bigg{)},I=l\pm\dfrac{1}{2},$
(11)

where $\alpha=1\times 10^{-2}$ and $l$ is the orbital momentum of the last
neutron. The ML rule was verified later by the microscopic theory [117]. The
application of the ML rule to estimate the Bohr-Weisskopf correction of the
magnetic moment of 207Hg is justified based on the successful reproduction of
the experimental HFA by this rule for neutron single particle states in
mercury nuclei across a rather large range of masses [114].

For previously measured isotopes and isomers the maximum deviation of the
experimental ${}^{A_{1}}\Delta^{199}_{BW}$ from the ML calculation is equal to
2$\times 10^{-3}$. Correspondingly, we conservatively estimated the error of
the ML prediction for ${}^{207}\Delta^{199}_{BW}$ as 5$\times 10^{-3}$. The
uncertainty of this correction was estimated based on the omitted
${}^{A_{1}}\Delta^{A_{2}}_{BR}$ correction. It was shown in [118] that
${}^{A_{1}}\Delta^{A_{2}}_{BR}$ is proportional to $\delta\langle
r^{2}\rangle^{A_{1},A_{2}}$. Thus ${}^{207}\Delta^{199}_{BR}$ for 207Hg can be
estimated by scaling the calculated ${}^{201}\Delta^{199}_{BR}$ [119]. It
should be noted that ${}^{205}\Delta^{203}_{BR}$ for thallium isotopes,
calculated in [119] by solving the one-electron Dirac equation, practically
coincides with that calculated in [118] by the Dirac-Fock approach. Taking
into account the independence of the ${}^{A_{1}}\Delta^{A_{2}}_{BR}$
correction on the details of the atomic calculations and the uncertainty of
$\delta\langle r^{2}\rangle$, we estimate the uncertainty of this correction
as 10%.

The spectroscopic quadrupole moments $Q^{A}_{s}$ were calculated using the
relation

$Q^{A}_{s}=\dfrac{b_{A}}{b_{{}^{201}\text{Hg}}}Q^{{}^{201}\text{Hg}}_{s},\\\ $
(12)

where $Q^{{}^{201}\text{Hg}}_{s}$=$0.387(6)$ b (from [120]) and
$b_{{}^{201}\text{Hg}}$=$-280.107(5)$ MHz [121]. The results are presented in
Table I.

##  Appendix B Mass spectrometry methods

The mass measurements were performed with the ISOLTRAP mass-spectrometer
[122]. As described in parts above, the setup consists of four ion traps for
beam preparation and mass measurements. First, a linear radio-frequency
quadrupole trap was used to accumulate, cool and bunch the quasi-continuous
radioactive ion beam delivered by ISOLDE [37]. Using a pulsed drift tube, the
energy of the bunched beam is then reduced from the initial
$30\text{\,}\mathrm{keV}$ to $3.2\text{\,}\mathrm{keV}$.

Ions are captured by the Multi-Reflection Time-of-Flight Mass
Spectrometer/Mass Separator (MR-ToF MS) [38] using the in-trap lift electrode
[123], following this, they undergo a certain number of revolutions in-between
the mirror electrodes of the device. Here, the time-of-flight is given as
$t=a\sqrt{m}+b$, where $a$ and $b$ are device-dependent parameters. Due to the
mass-dependence of the trapped ions moving at the same kinetic energy, the
different isobaric species separate in time-of-flight. In mass-spectrometry
mode, the MR-ToF MS can be used to determine the mass of the ion of interest
by using well known reference masses to account for the calibration parameters
$a$ and $b$. This is done by expressing the mass $m$ using the so-called
$C_{ToF}$-value [40]:

$\displaystyle
m^{1/2}=C_{ToF}\left(m_{1}^{1/2}-m_{2}^{1/2}\right)+\frac{1}{2}\left(m_{1}^{1/2}+m_{2}^{1/2}\right),$
(13)

where $m_{1},m_{2}$ and $t_{1},t_{2}$ are the masses and time-of-flights of
the two reference species, respectively, and
$C_{ToF}=(2t-t_{1}-t_{2})/\left[2(t_{1}-t_{2})\right]$. An example of a time-
of-flight spectrum is presented in Figs. 2(b) and 2(c).

In mass-separation mode, the MR-ToF MS selectively ejects the ions of interest
towards the Penning traps located downstream of the setup. In the so-called
preparation Penning trap, the ions are captured and cooled using a mass-
selective buffer-gas method to improve beam emittance [124] and to further
reduce contamination. Subsequently, the ions are ejected and recaptured in the
precision Penning trap in which the high precision mass measurement is
performed. In the present measurement, this is achieved by employing the Time-
of-Flight Ion Cyclotron Resonance technique (ToF-ICR), which determines the
cyclotron frequency of the trapped ions by scanning the frequency of an
applied quadrupolar electric field [125]. This frequency can be written as
$\omega_{c}=\frac{qB}{m}$, where $q$ is the electric charge of the ion, $B$ is
the magnetic field, and $m$ is the ion mass. By performing cross-reference
measurements with known mass calibrants, the mass $m$ of the ion of interest
can be extracted by comparing its cyclotron frequency with that of a well-
known mass. Expressed as a frequency ratio

$\displaystyle R=\frac{\omega_{c,ref}}{\omega_{c}}=\frac{m}{m_{ref}},$ (14)

the magnetic field and the charge cancel out. For the atomic mass, the
electron mass is added to the measured ion mass. The ionization energy is
negligible.

Expressing the ion masses in MR-ToF MS and ToF-ICR measurements via the
$C_{ToF}$-value and the frequency ratio $R$, respectively, facilitates an easy
recalculation of the mass of interest in case one of the employed reference
masses are measured more precisely in the future.

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