First glimpse of the N = 82 shell closure below Z = 50 from masses of neutron-rich cadmium isotopes and isomers

V. Manea, 2, 3, ∗ J. Karthein , 2, † D. Atanasov, ‡ M. Bender, K. Blaum, T. E. Cocolios, S. Eliseev, A. Herlert, J. D. Holt, W. J. Huang, § Yu. A. Litvinov, D. Lunney, J. Menéndez, 11 M. Mougeot, ‡ D. Neidherr, L. Schweikhard, A. Schwenk, 14, 2 J. Simonis, 13, 14 A. Welker, 4 F. Wienholtz, 12, ¶ and K. Zuber CERN, 1211 Geneva 23, Switzerland Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany Instituut voor Kernen Stralingsfysica, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium Technische Universität Dresden, 01069 Dresden, Germany IP2I Lyon, CNRS/IN2P3, Université de Lyon, Université Claude Bernard Lyon 1, F-69622, Villeurbanne, France FAIR GmbH, 64291 Darmstadt, Germany TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada CSNSM-IN2P3-CNRS, Université Paris-Sud, 91406 Orsay, France GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany Center for Nuclear Study, The University of Tokyo, 113-0033 Tokyo, Japan Department de F́ısica Quàntica i Astrof́ısica, Universitat de Barcelona, 08028 Barcelona, Spain Institut für Physik, Universität Greifswald, 17487 Greifswald, Germany Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany Institut für Kernphysik and PRISMA Cluster of Excellence, Johannes Gutenberg-Universität, 55099 Mainz, Germany (Dated: January 16, 2020)

We probe the N = 82 nuclear shell closure by mass measurements of neutron-rich cadmium isotopes with the ISOLTRAP spectrometer at ISOLDE-CERN. The new mass of 132 Cd offers the first value of the N = 82, two-neutron shell gap below Z = 50 and confirms the phenomenon of mutually enhanced magicity at 132 Sn. Using the recently implemented phase-imaging ion-cyclotronresonance method, the ordering of the low-lying isomers in 129 Cd and their energies are determined. The new experimental findings are used to test large-scale shell-model, mean-field and beyondmean-field calculations, as well as the ab initio valence-space in-medium similarity renormalization group.
The so-called "magic" numbers of protons and neutrons are associated with large energy gaps in the effective single-particle spectrum of the nuclear mean field [1], revealing shell closures. As such, they are intimately connected to the nuclear interaction and represent essential benchmarks for nuclear models.
Experiments with light radioactive beams have shown that shell closures at N = 8, 20 and 28 are substantially weakened when the number of protons in the nuclear system is reduced (see [2,3] for a review). New, but weaker shell closures have also been found, e.g., N = 32 and 34 [4][5][6][7]. In the shell model, this evolution results from the interplay between the monopole part of the valencespace nucleon-nucleon interaction that determines the single-particle spectrum and multipole forces that induce correlations [8]. Starting from realistic nuclear forces, the study of closed-shell nuclei provides benchmarks for microscopic calculations of valence-space Hamiltonians, with their many-body contributions [9][10][11][12][13]. Despite extensive work, significantly less is known for heavier nuclei, in particular for the magic N = 82.
The doubly-magic nature of 132 Sn (with 50 protons and 82 neutrons) was reconfirmed recently [14,15]. But below Z = 50 the orbitals occupied by the Fermi-level protons change, as does the proton-neutron interaction, which drives shell evolution. This means that without data for nuclides with Z < 50 and N ≈ 82, any predictions for the N = 82 shell gap are rather uncertain. While decayspectroscopy [16][17][18], laser-spectroscopy [19] and massspectrometry [20,21] studies have been performed for the neutron-rich cadmium isotopes, the energies of the low-lying isomers in 129 Cd and the N = 82 two-neutron shell gap remain unknown.
The A ≈ 130 r-process abundance peak has long been considered an indication of a persistent N = 82 shell gap in various models. However, recent studies of r-process nucleosynthesis have underlined the importance of fission recycling in certain scenarios, in which the A = 130 abundance peak is primarily determined by the fissionfragment distribution of r-process actinides [22,23].
In this work, we present the first direct determination of the N = 82 shell gap for Z < 50 with mass measurements of exotic cadmium isotopes and isomers between 124 Cd and 132 Cd. We exploit all mass-measurement techniques of the ISOLTRAP spectrometer, including the phase-imaging ion-cyclotron-resonance (PI-ICR) method [24][25][26]. The data are interpreted in comparison to the large-scale shell model and to new calculations made with a beyond-mean-field (BMF) approach [27,28], as well as the ab initio valence-space in-medium similarity renormalization group (VS-IMSRG) [12,[29][30][31][32][33].
The cadmium isotopes were produced at CERN's arXiv:2001.05075v1 [nucl-ex] 14 Jan 2020 ISOLDE facility [34] by neutron-induced fission in a uranium-carbide target. The neutrons were produced by 1.4 GeV protons accelerated by CERN's Proton Synchrotron Booster and impinging on a tungsten rod, which reduced contaminants from proton-induced reactions [35]. The neutral products diffused from the ≈ 2000 • C target into a hot tantalum cavity where the Resonance-Ionization Laser Ion Source [36] was used to produce singly-charged cadmium ions. A cold quartz line [37] greatly suppressed surface ionized cesium and barium contaminants. The beam was accelerated to 50 keV, mass separated by the ISOLDE High Resolution Separator and transported to ISOLTRAP for accumulation in a segmented, linear radiofrequency quadrupole cooler/buncher [38]. The ion bunch was then injected into the multi-reflection time-of-flight mass separator (MR-ToF MS) [39] where the cadmium ions were separated from contaminants with a resolving power of ≈ 10 5 .The separated ions were either detected using a secondary electron multiplier for mass measurements, or purified [40] and transported to a tandem Penning-trap system, composed of a preparation trap for beam cooling and further purification [41,42] and a precision trap for measurements. In this work the masses of 131,132 Cd were determined with the MR-ToF MS using a two-parameter calibration formula and hence requiring two reference measurements, as described in [5]. Its short measurement time of only about 27 ms and direct ion counting made it the method of choice for the most exotic isotopes. Considering only singly charged ions, the mass m i,x of the ion of interest is related to the masses m i,1 and m i,2 of two reference ions by m and t x are the TOFs, measured in the same conditions, of the ions of mass m i,x , m i,1 and m i,2 , respectively, with m i,1 a contaminating isobar of the ion of interest.
The masses of the other studied cadmium isotopes were determined with the precision Penning trap, allowing typically a higher precision and resolving power than the MR-ToF MS, by measuring their cyclotron frequency (as singly charged ions) in the trap, ν c,x = qB/(2πm i,x ) (where q is in our case the elementary charge and B is the trap's magnetic-field induction) [43]. The atomic mass m x can then be determined as where m e is the electron mass and r ref,x = ν c,ref /ν c,x is the measured cyclotronfrequency ratio between a singly-charged reference ion of atomic mass m ref and the ion of interest. The binding energy of the electron, neglected in the atomic-mass formula, is orders of magnitude smaller than the statistical uncertainty.
For 127,129 Cd the beam was a mixture of ground and isomeric state (J = 3/2 + and J = 11/2 − ) which in a prior attempt could not be separated by a long-excitation ToF-ICR measurement [20] due to the short half-lives. In this work we used instead the recently developed PI-ICR method [24,25], by which a radial frequency is determined from the phase "accumulated" by the circular ion motion in the trap in a given time t acc , using its projection on a position-sensitive microchannel-plate detector (MCP). In PI-ICR one performs three ion-position measurements: (1) the center of the radial ion trajectory by ejection without preparing a radial motion; (2) for ions prepared on a cyclotron orbit (at frequency ν + ) after evolving for t acc ; (3) for ions prepared on a magnetron orbit (at frequency ν − ), after evolving for the same t acc . The cyclotron frequency is then given by ν c = [2π(n + + n − ) + φ]/(2πt acc ), where n + and n − are the number of integer rotations performed by the ions in steps (2) and (3), respectively, while φ is the angle between the ion positions measured in the two steps [24,25].
In the second step of the PI-ICR measurement, a resolving power of about 2 × 10 6 was achieved in only 106 ms, allowing a clear separation of the two states as illustrated in Fig. 2 for 129 Cd + . Their individual masses could thus be determined.
The experimental results of this work are summarized in Table I. During the 132 Cd measurements the yield of (stable) 132 Ba + remained constant, while a gradual increase in the yield of (radioactive) 132 Cs + was observed. The data set for 132 Cd was thus split, depending on which isobaric reference dominated, resulting in two independent C T OF values.
The analysis of the ToF-ICR measurements followed the procedure in [49]. For the MR-ToF MS spectra, Gaussian distributions were fit to the data (double-Gaussian for the 132 Ba + / 132 Cs + double peak) by the binned maximum-likelihood method. When statistically significant, shifts of the C T oF values from changing the fit range, data binning and number of ions simultaneously stored in the MR-ToF MS were included in the total uncertainty.
For the PI-ICR measurements, the unbinned maximum-likelihood fit of the ion-spot positions was performed using 2D Gaussian distributions. The effect of the number of ions simultaneously stored in the trap was studied and, for the analysed data set, was within statistical uncertainties. The mass-dependent shift and systematic uncertainty from [49] were quadratically added to the total uncertainty.
The spin assignments for the measured states in 127 Cd and 129 Cd are based on the fact that the high-spin isomers were systematically produced with higher yields, corroborated by a laser-spectroscopy study of cadmium isotopes performed at ISOLDE [19] with the same production mechanism, where the yield ratios were determined for 127,129 Cd. We conclude that the excited 11/2 − state in 127 Cd becomes the ground state in 129 Cd. The 283(12)-keV excitation energy obtained for 127 Cd agrees with the TITAN result using highly charged ions [21]. The 343(8)-keV excitation energy of the 3/2 + state in 129 Cd is a new value.
In a simple picture, the 3/2 + and 11/2 − states in 129 Cd are formed by the odd neutron occupying the d 3/2 and h 11/2 orbitals, respectively, and allow probing the evolution of the two states with proton number. This is shown in Fig. 3, where neutron binding energies, calculated as in [2] for the low-lying states in the even-Z, N = 81 and N = 83 isotones, are plotted as a function of Z. For Z = 48 they are obtained from this work. One notices the larger slope of the 11/2 − states, which changes more abruptly for Z < 50, suggesting a stronger, attractive monopole proton-neutron interaction for the high-spin state.  Figure 4 shows the difference in energy between the 3/2 + and 11/2 − states for the odd cadmium isotopes. Shell-model calculations assuming a closed 132 Sn (jj45pn [51,52] and NA-14 [16][17][18]53]) or allowing cross-shell excitations (EPQQM [54]) predict the 11/2 − state to become the ground state in 129 Cd. For EPQQM, obtaining the correct prediction required enhancing the monopole interaction between the πg 9/2 and νh 11/2 orbits [55]. Energy difference between the J = 11/2 − and J = 3/2 + states in the odd cadmium isotopes. Experimental data from [48] and this work are compared to theoretical calculations (EPQQM [54], NA-14 [18,53], jj45pn [51] using NUSHELLX [52]).
The mass of 132 Cd allows addressing a broader range of models via the N = 82 two-neutron shell gap ∆ 2n (Z, N ) = S 2n (Z, N ) − S 2n (Z, N + 2), a quantity involving only even nuclei and the first such value below the doubly magic 132 Sn. This gap is shown as a function of Z in Fig. 5, with the new data (full circle) revealing a peak at the proton magic number Z = 50. This phenomenon called "mutually enhanced magicity" [56] is known from other doubly-magic nuclei and was explained by a BMF calculation using the SLy4 Skyrme interaction, within a symmetry-restored generator coordinate method (GCM) [27,28]. In this work, we show that this enhancement manifests also for 132 Sn. The BMF calculations were extended to Z = 46 and describe the peak at Z = 50. By contrast, results obtained with SLy4 just at the meanfield level (SLy4-MF) fail to reproduce the peak. It is by BMF correlations that the N = 80, 84 isotones gain binding with respect to N = 82, lowering the empirical shell gap, while for Z = 50 the closed proton shell maintains the high gap value. The same failure to produce the peak in more basic mean-field calculations is also found when using other interactions. Figure 5 illustrates this for the non-relativistic HFB31 [57] and UNEDF0 [58] Skyrme interactions and the relativistic DD-MEδ [59]. Calculations with HFB31 include a collective-energy correction for BMF effects, which slightly enhances ∆ 2n around Z = 50. While the peak is qualitatively described by BMF correlations, the size of the drop of ∆ 2n below Z < 50 is not reproduced by any of these calculations.
We also present VS-IMSRG calculations of groundand two-neutron separation energies of cadmium, tin, and tellurium isotopes across the N = 82 shell gap. For details on the VS-IMSRG decoupling to derive the valence-space Hamiltonian, we refer to Refs. [12,[29][30][31][32][33]. When this ab initio valence-space Hamiltonian is diagonalized (here with the shell-model code Antoine [8]) some subset of eigenvalues of the full Hamiltonian should be reproduced when no IMSRG approximations are made. In this work we use the IMSRG(2) approximation, where all induced operators are truncated at the two-body level, typically giving binding energies closer than 1% to full-space ab initio results [12]. We begin from the 1.8/2.0(EM) chiral interaction of Refs. [60,61], used successfully throughout the medium-to heavy-mass region [13,62,63]. For heavier systems, achieving convergence with respect to the E 3max cut on 3N matrix elements is however a key limitation. The resulting ∆ 2n values are presented in Fig. 5. The calculations overestimate data by almost 3 MeV, but are not fully converged with respect to the 3N matrix elements included, here up to E 3max = 18 excitations in a harmonic oscillator basis. In contrast, the relative trend of ∆ 2n , which is safely converged up to ∼ 50 keV, is well described. This is illustrated by the dashed lines in Fig. 5, which show the IMSRG results translated to match the ∆ 2n value at Z = 50.
In summary, we have measured the masses of neutronrich cadmium isotopes and isomers across the N = 82 shell closure. The PI-ICR technique allowed establishing the inversion of the 11/2 − and 3/2 + states in 129 Cd, showing that the h 11/2 neutron orbital is key for the evolution of the N = 82 shell gap towards Z = 40. The trend of the N = 82 shell gap was determined below Z = 50 with the mass of 132 Cd, showing a large drop, which confirms the mutually enhanced magicity of 132 Sn. A BMF model reproduces the effect, but underestimates its size, whereas the VS-IMSRG approach shows an offset to experiment, but describes it qualitatively.
V.M. and J.K. contributed equally to this work. We thank D.T. Yordanov for the helpful communication regarding the spin assignment in 129 Cd and R. Stroberg for fruitful discussions on the VS-IMSRG framework. We thank the ISOLDE technical group and the ISOLDE Collaboration for their support and the excellent quality of the neutron-rich beams. We acknowledge support by the Max-Planck Society, the German Federal Ministry of Education and Research (BMBF, contracts 05P12HGCI1, 05P12HGFNE, 05P15ODCIA, 05P15HGCIA, 05P18HGCIA and 05P18RDFN1), the European Union 7th framework through ENSAR2 (contract no. 262010), the French IN2P3 and FWO Vlaanderen (Belgium). J. Karthein