First glimpse of the N=82 shell closure below Z=50 from masses of neutron-rich cadmium isotopes and isomers
V. Manea, J. Karthein, 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, M. Mougeot, D. Neidherr, L. Schweikhard, A. Schwenk, J. Simonis, A. Welker, F. Wienholtz, K. Zuber
FFirst glimpse of the N = 82 shell closure below Z = 50 from masses of neutron-rich cadmium isotopes and isomers V. Manea,
1, 2, 3, ∗ J. Karthein ,
1, 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. Men´endez,
10, 11
M. Mougeot, ‡ D.Neidherr, L. Schweikhard, A. Schwenk,
13, 14, 2
J. Simonis,
15, 13, 14
A. Welker,
1, 4
F. Wienholtz,
1, 12, ¶ and K. Zuber CERN, 1211 Geneva 23, Switzerland Max-Planck-Institut f¨ur Kernphysik, 69117 Heidelberg, Germany Instituut voor Kern- en Stralingsfysica, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium Technische Universit¨at Dresden, 01069 Dresden, Germany IP2I Lyon, CNRS/IN2P3, Universit´e de Lyon, Universit´e 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, Universit´e Paris-Sud, 91406 Orsay, France GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, 64291 Darmstadt, Germany Center for Nuclear Study, The University of Tokyo, 113-0033 Tokyo, Japan Department de F´ısica Qu`antica i Astrof´ısica, Universitat de Barcelona, 08028 Barcelona, Spain Institut f¨ur Physik, Universit¨at Greifswald, 17487 Greifswald, Germany Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, 64291 Darmstadt, Germany Institut f¨ur Kernphysik and PRISMA Cluster of Excellence, Johannes Gutenberg-Universit¨at, 55099 Mainz, Germany (Dated: March 17, 2020)We probe the N = 82 nuclear shell closure by mass measurements of neutron-rich cadmiumisotopes with the ISOLTRAP spectrometer at ISOLDE-CERN. The new mass of Cd offers thefirst value of the N = 82, two-neutron shell gap below Z = 50 and confirms the phenomenon ofmutually enhanced magicity at Sn. Using the recently implemented phase-imaging ion-cyclotron-resonance method, the ordering of the low-lying isomers in
Cd and their energies are determined.The new experimental findings are used to test large-scale shell-model, mean-field and beyond-mean-field calculations, as well as the ab initio valence-space in-medium similarity renormalizationgroup.
The so-called magic numbers of protons and neutronsare associated with large energy gaps in the effectivesingle-particle spectrum of the nuclear mean field [1], re-vealing shell closures. As such, they are intimately con-nected to the nuclear interaction and represent essentialbenchmarks for nuclear models.Experiments with light radioactive beams have shownthat shell closures at N = 8 ,
20 and 28 are substantiallyweakened when the number of protons in the nuclear sys-tem is reduced (see [2, 3] for a review). New, but weakershell closures have also been found, e.g., N = 32 and34 [4–7]. In the shell model, this evolution results fromthe interplay between the monopole part of the valence-space nucleon-nucleon interaction that determines thesingle-particle spectrum and multipole forces that inducecorrelations [8]. Starting from realistic nuclear forces,the study of closed-shell nuclei provides benchmarks formicroscopic calculations of valence-space Hamiltonians,with their many-body contributions [9–13]. Despite ex-tensive work, significantly less is known for heavier nuclei,in particular for the magic N = 82.The doubly magic nature of Sn (with 50 protons and82 neutrons) was reconfirmed recently [14, 15]. But below Z = 50 the orbitals occupied by the Fermi-level protonschange, as does the proton-neutron interaction, whichdrives shell evolution. This means that without data for nuclides with Z <
50 and N ≈
82, any predictions forthe N = 82 shell gap are rather uncertain. While decay-spectroscopy [16–18], laser-spectroscopy [19] and mass-spectrometry [20, 21] studies have been performed forthe neutron-rich cadmium isotopes, the energies of thelow-lying isomers in Cd and the N = 82 two-neutronshell gap remain unknown.The A ≈ r -process abundance peak has long beenconsidered an indication of a persistent N = 82 shell gapin various models. However, recent studies of r -processnucleosynthesis have underlined the importance of fis-sion recycling in certain scenarios, in which the A = 130abundance peak is primarily determined by the fission-fragment distribution of r -process actinides [22, 23].In this work, we present the first direct determinationof the N = 82 shell gap for Z <
50 with mass measure-ments of exotic cadmium isotopes and isomers between
Cd and
Cd. We exploit all mass-measurement tech-niques of the ISOLTRAP spectrometer, including thephase-imaging ion-cyclotron-resonance (PI-ICR) method[24–26]. The data are interpreted in comparison to thelarge-scale shell model and to new calculations made witha beyond-mean-field (BMF) approach [27, 28], as well asthe ab initio valence-space in-medium similarity renor-malization group (VS-IMSRG) [12, 29–33].The cadmium isotopes were produced at CERN’s a r X i v : . [ nu c l - e x ] M a r ISOLDE facility [34] by neutron-induced fission in auranium-carbide target. The neutrons were producedby 1.4-GeV protons accelerated by CERN’s Proton Syn-chrotron Booster and impinging on a tungsten rod,which reduced contaminants from proton-induced re-actions [35]. The neutral products diffused from the ≈ ◦ C target into a hot tantalum cavity where theresonance-ionization laser ion source [36] was used to pro-duce singly charged cadmium ions. A cold quartz line[37] greatly suppressed surface ionized cesium and bar-ium contaminants.The beam was accelerated to 50 keV, mass separatedby the ISOLDE High Resolution Separator and trans-ported to ISOLTRAP for accumulation in a segmented,linear radiofrequency quadrupole cooler and buncher[38]. The ion bunch was then injected into the multi-reflection time-of-flight mass spectrometer (MR-ToF MS)[39] where the cadmium ions were separated from con-taminants with a resolving power of ≈ . The sepa-rated ions were either detected using a secondary elec-tron multiplier for mass measurements, or purified [40]and transported to a tandem Penning-trap system, com-posed of a preparation trap for beam cooling and furtherpurification [41, 42] and a precision trap for measure-ments. C o u n t s / ( . n s ) Time of flight (μs) Ba + 132 Cs + 132 Cd + FIG. 1. MR-ToF spectrum (after 800 revolutions) of Cd + along with isobaric ions ( Ba + and Cs + ), with fits (inred) to Gaussian line shapes. In this work the masses of , Cd were determinedwith the MR-ToF MS (see Fig. 1) using a two-parametercalibration formula and hence requiring two referencemeasurements, as described in [5]. Its short measure-ment time of only about 27 ms and direct ion countingmade it the method of choice for the most exotic isotopes.Considering only singly charged ions, the mass m i,x ofthe ion of interest is related to the masses m i, and m i, of two reference ions by m / i,x = C T oF ∆ Ref + Σ Ref ,with ∆
Ref = m / i, − m / i, , Σ Ref = m / i, + m / i, and C T oF = (2 t x − t − t ) / [2( t − t )]. The quantities t , t and t x are the TOFs, measured in the same conditions,of the ions of mass m i,x , m i, and m i, , respectively, with m i, an isobar of the ion of interest.The masses of the other studied cadmium isotopeswere determined with the precision Penning trap, al-lowing typically a higher precision and resolving powerthan the MR-ToF MS, by measuring their cyclotronfrequency (as singly charged ions) in the trap, ν c,x = qB/ (2 πm i,x ) (where q is in our case the elementarycharge and B is the trap’s magnetic-field induction)[43]. The atomic mass m x can then be determined as m x = r ref,x ( m ref − m e ) + m e , where m e is the electronmass and r ref,x = ν c,ref /ν c,x is the measured cyclotron-frequency ratio between a singly charged reference ion ofatomic mass m ref and the ion of interest. The bindingenergy of the electron, neglected in the atomic-mass for-mula, is orders of magnitude smaller than the statisticaluncertainty.Penning-trap measurements of , , , Cdwere performed with the time-of-flight ion-cyclotron-resonance (ToF-ICR) method [44], including Ramsey-type excitations [45, 46]. -15 -10 -5 0 5 10 15
X position (mm) -1050510 Y p o s i t i o n ( mm ) − + Cd + FIG. 2. PI-ICR ion projection image of Cd + with cen-ter ion spot measured separately (in black) and the 11 / − (blue) and 3 / + states (red) separated by the marked angleafter 106-ms phase accumulation at the modified cyclotronfrequency. For , Cd the beam was a mixture of ground andisomeric state ( J = 3 / + and J = 11 / − ) which in aprior attempt could not be separated by a long-excitationToF-ICR measurement [20] due to the short half-lives. Inthis work we used instead the recently developed PI-ICRmethod [24, 25], by which a radial frequency is deter-mined from the phase “accumulated” by the circular ionmotion in the trap in a given time t acc , using its projec-tion on a position-sensitive microchannel-plate detector(MCP). In PI-ICR MS one performs three ion-positionmeasurements: (1) the center of the radial ion trajec-tory 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 mag-netron orbit (at frequency ν − ), after evolving for thesame t acc . The cyclotron frequency is then given by ν c = [2 π ( n + + n − ) + φ ] / (2 πt acc ), where n + and n − arethe number of integer rotations performed by the ions insteps (2) and (3), respectively, while φ is the angle be-tween the ion positions measured in the two steps [24, 25].In the second step of the PI-ICR measurement, a re-solving power of about 2 × was achieved in only106 ms, allowing a clear separation of the two states asillustrated in Fig. 2 for Cd + . Their individual massescould thus be determined.The experimental results of this work are summarizedin Table I. During the Cd measurements the yield of(stable) Ba + remained constant, while a gradual in-crease in the yield of (radioactive) Cs + was observed.The data set for Cd was thus split, depending on whichisobaric reference dominated, resulting in two indepen-dent C T OF values. In this case, as well as for
Cd, theweighted averages of the new mass-excess values are usedfor the figures.The analysis of the ToF-ICR measurements followedthe procedure in [49]. For the MR-ToF MS spectra,Gaussian distributions were fit to the data (double-Gaussian for the Ba + / Cs + double peak) by thebinned maximum-likelihood method. When statisticallysignificant, shifts of the C T oF values from changing thefit range, data binning and number of ions simultane-ously stored in the MR-ToF MS were included in thetotal uncertainty.For the PI-ICR measurements, the unbinnedmaximum-likelihood fit of the ion-spot positions wasperformed using 2D Gaussian distributions. The effectof the number of ions simultaneously stored in the trapwas studied and, for the analysed data set, was withinstatistical uncertainties. The mass-dependent shift andsystematic uncertainty from [49] were quadraticallyadded to the total uncertainty.The spin assignments for the measured states in
Cdand
Cd are based on the fact that the high-spin iso-mers were systematically produced with higher yields,corroborated by a laser-spectroscopy study of cadmiumisotopes performed at ISOLDE [19] with the same pro-duction mechanism, where the yield ratios were deter-mined for , Cd. We conclude that the excited 11 / − state in Cd becomes the ground state in
Cd. The283(12)-keV excitation energy obtained for
Cd agreeswith the TITAN result using highly charged ions [21].The 343(8)-keV excitation energy of the 3 / + state in Cd is a new value.In a simple picture, the 3 / + and 11 / − states in Cdare formed by the odd neutron occupying the d / and h / orbitals, respectively, and allow probing the evolu-tion of the two states with proton number. This is shown in Fig. 3, where neutron binding energies, calculated asin [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 noticesthe larger slope of the 11 / − states, which changes moreabruptly for Z <
50, suggesting a stronger, attractivemonopole proton-neutron interaction for the high-spinstate.
48 50 52 54 56 58 60 62 64 66 68
Proton number −12−10−8−6−4−2 N e u t r o n b i n d i n g e n e r g y ( M e V ) N = 82 gap − − + + FIG. 3. Neutron binding energies of the low-lying nuclearstates of the N = 81 ( J π = 1 / + , / + , / − ) and N = 83( J π = 7 / − ) isotones. Experimental data are taken from[48, 50] and this work (open symbols). Figure 4 shows the difference in energy between the3 / + and 11 / − states for the odd cadmium isotopes.Shell-model calculations assuming a closed Sn (jj45pn[51, 52] and NA-14 [16–18, 53]) or allowing cross-shellexcitations (EPQQM [54]) predict the 11 / − state to be-come the ground state in Cd. For EPQQM, obtainingthe correct prediction required enhancing the monopoleinteraction between the πg / and νh / orbits [55].The mass of Cd allows addressing a broader rangeof models via the N = 82 two-neutron shell gap∆ n ( Z, N ) = S n ( Z, N ) − S n ( Z, N + 2) (where S n isthe two-neutron separation energy), a quantity involv-ing only even nuclei and the first such value below thedoubly magic Sn. This gap is shown as a function of Z in Fig. 5, with the new data (full circle) revealing apeak at the proton magic number Z = 50. This phe-nomenon called “mutually enhanced magicity” [56, 57]is known from other doubly-magic nuclei and was ex-plained by a BMF calculation using the SLy4 Skyrmeinteraction, within a symmetry-restored generator coor-dinate method (GCM) [27, 28]. In this work, we showthat this enhancement manifests also for Sn. TheBMF calculations were extended to Z = 46 and describethe peak at Z = 50. By contrast, results obtained withSLy4 just at the mean-field level (SLy4-MF) fail to re-produce the peak. It is by BMF correlations that the N = 80 ,
84 isotones gain binding with respect to N = 82, TABLE I. Frequency ratio ( r = ν c,ref /ν c ), time-of-flight ratio ( C ToF ) and mass excess of the cadmium isotopes measured inthis work. Mass excesses from the literature ([21] for
Cd, [20] for
Cd and AME2016 [47] for the rest) are given as well (
Cd). The yields, where available, are order-of-magnitude estimates of ion intensitieson the ISOLDE central beam line. Values between parentheses are total (statistical plus systematic) uncertainties.A J π Half-life Yield Method References Ratio r or C ToF
Mass excess (keV)(s) (Ions/s) this work literature124 0 + Cs + r = 0.9323743186(432) − . . − . . + Cs + r = 0.9474585581(503) − . . − . . / + ) 5 × PI-ICR Cs + r = 0.9550111122(922) − − . . / − × r = 0.9550133972(435) − . . − . . + × ToF-ICR Cs + r = 0.962547502(114) − − / − × PI-ICR Cs + r = 0.9701048175(432) − . . − / + × r = 0.9701075886(450) − . . / − × ToF-ICR Cs + r = 0.985217426(252) − − Cs + , Cs + C ToF = 0.4823166(126) − + Ba + , Cs + C ToF = 0.4592156(773) − − Cs + , Cs + C ToF = 0.460420(118) −
72 74 76 78 80 82
Neutron number −2000200 E ( / − ) − E ( / + ) ( k e V ) Theory
NA-14EPQQMjj45
Experiment
NUBASE16ISOLTRAP
FIG. 4. Energy difference between the J = 11 / − and J = 3 / + states in the odd cadmium isotopes. Experimen-tal data from [48] and this work are compared to theoreticalcalculations (EPQQM [54], NA-14 [18, 53], jj45pn [51] usingNUSHELLX [52]). lowering the empirical shell gap, while for Z = 50 theclosed proton shell maintains the high gap value. Thesame failure to produce the peak in more basic mean-field calculations is also found when using other inter-actions. Figure 5 illustrates this for the nonrelativisticHFB31 [58] and UNEDF0 [59] Skyrme interactions andthe relativistic DD-ME δ [60]. Calculations with HFB31include a collective-energy correction for BMF effects,which slightly enhances ∆ n around Z = 50. While thepeak is qualitatively described by BMF correlations, thesize of the drop of ∆ n below Z <
50 is not reproducedby any of these calculations.We also present VS-IMSRG calculations of ground-and 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
42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
Proton number T w o - n e u t r o n s h e ll g a p ( M e V ) Experiment
AME2016this work
Theory
HFB31UNEDF0DD-ME δ SLy4-MFSLy4-GCMVS-IMSRG
FIG. 5. Experimental two-neutron shell gap of the N = 82isotones from the AME2016 [47] and this work, compared topredictions of different calculations (for details, see text). Thedashed line corresponds to the VS-IMSRG results shifted tomatch the Z = 50 value. valence-space Hamiltonian, we refer to Refs. [12, 29–33]. When this ab initio valence-space Hamiltonian isdiagonalized (here with the shell-model code ANTOINE[8]) some subset of eigenvalues of the full Hamiltonianshould be reproduced when no IMSRG approximationsare made. In this work we use the IMSRG(2) approxi-mation, where all induced operators are truncated at thetwo-body level, typically giving binding energies closerthan 1% to full-space ab initio results [12]. We beginfrom the 1.8/2.0(EM) chiral interaction of Refs. [61, 62],used successfully throughout the medium- to heavy-massregion [13, 63, 64]. For heavier systems, achieving con-vergence with respect to the E cut on 3 N matrixelements is however a key limitation. The resulting ∆ n values are presented in Fig. 5. The calculations overesti-mate data by almost 3 MeV, but are not fully convergedwith respect to the 3N matrix elements included, hereup to E = 18 excitations in a harmonic oscillatorbasis. In contrast, the relative trend of ∆ n , which issafely converged up to ∼
50 keV, is well described. Thisis illustrated by the dashed lines in Fig. 5, which showthe IMSRG results translated to match the ∆ n value at Z = 50.In summary, we have measured the masses of neutron-rich cadmium isotopes and isomers across the N = 82shell closure. The PI-ICR technique allowed establish-ing the inversion of the 11 / − and 3 / + states in Cd,showing that the h / neutron orbital is key for the evo-lution of the N = 82 shell gap towards Z = 40. The trendof the N = 82 shell gap was determined below Z = 50with the mass of Cd, showing a large drop, which con-firms the mutually enhanced magicity of
Sn. A BMFmodel reproduces the effect, but underestimates its size,whereas the VS-IMSRG approach shows an offset to ex-periment, but describes it qualitatively.V.M. and J.K. contributed equally to this work.We thank D.T. Yordanov for the helpful communica-tion regarding the spin assignment in
Cd and R.Stroberg for fruitful discussions on the VS-IMSRG frame-work. We thank the ISOLDE technical group and theISOLDE Collaboration for their support and the excel-lent quality of the neutron-rich beams. We acknowl-edge support by the Max-Planck Society, the GermanFederal Ministry of Education and Research (BMBF,contracts 05P12HGCI1, 05P12HGFNE, 05P15ODCIA,05P15HGCIA, 05P18HGCIA and 05P18RDFN1), theEuropean Union 7th framework through ENSAR2 (con-tract no. 262010), the French IN2P3 and FWO Vlaan-deren (Belgium). J. Karthein and A. Welker acknowl-edge support by a Wolfgang Gentner Ph.D. Scholar-ship of the BMBF (05E15CHA). W. J. Huang acknowl-edges the support by the China Scholarship Council (No.201404910496). J. Men´endez acknowledges support fromthe JSPS KAKENHI Grant No.18K03639, MEXT as Pri-ority issue on post-K computer (Elucidation of the funda-mental laws and evolution of the universe), JICFuS, theCNS-RIKEN joint project for large-scale nuclear struc-ture calculations, and the Ramn y Cajal program RYC-2017-22781 of the Spanish Ministry of Science, Innova-tion and Universities. ∗ Corresponding author: [email protected] † This article contains data from the Ph.D thesis workof Jonas Karthein, enrolled at the Ruprecht-Karls-Universit¨at Heidelberg ‡ Present address: CERN, 1211 Geneva 23, Switzerland § Present address: Max-Planck-Institut f¨ur Kernphysik,69117 Heidelberg, Germany ¶ Present address: Institut f¨ur Kernphysik, TechnischeUniversit¨at Darmstadt, 64289 Darmstadt, Germany[1] M. G. Mayer and J. H. D. Jensen,
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