Level Structures of $^{56,58}$Ca Cast Doubt on a doubly magic $^{60}$Ca

Gamma decays were observed in $^{56}$Ca and $^{58}$Ca following quasi-free one-proton knockout reactions from $^{57,59}$Sc beams at $\approx 200$ MeV/nucleon. For $^{56}$Ca, a $\gamma$ ray transition was measured to be 1456(12) keV, while for $^{58}$Ca an indication for a transition was observed at 1115(34) keV. Both transitions were tentatively assigned as the $2^+_1 \rightarrow 0^+_{gs}$ decays, and were compared to results from ab initio and conventional shell-model approaches. A shell-model calculation in a wide model space with a marginally modified effective nucleon-nucleon interaction depicts excellent agreement with experiment for $2^+_1$ level energies, two-neutron separation energies, and reaction cross sections, corroborating the formation of a new nuclear shell above the $N$ = 34 shell. Its constituents, the $0f_{5/2}$ and $0g_{9/2}$ orbitals, are almost degenerate. This degeneracy precludes the possibility for a doubly magic $^{60}$Ca and potentially drives the dripline of Ca isotopes to $^{70}$Ca or even beyond.

Understanding properties of atomic nuclei at the extremes, for example those with large proton-to-neutron imbalances, is of paramount importance in nuclear physics. In these systems, often called exotic nuclei, new features emerge [1] including those that can be traced back to facets of nuclear forces. For instance, the tensor force, which has been known for decades [2,3], can modify the spin-orbit energy splitting as a function of the proton number (Z) or the neutron number (N), resulting in changes of shell structures, i.e., shell evolution [4,5]. Examples have been found in several regions across the Segrè chart (see review papers, [6,7]). Among them, the Ca isotopes provides an exemplary case of shell evolution, with striking appearances of the new magic numbers N = 32 [8, 9,10] and N = 34 [11,12,13,14]. The discovery of new magic numbers is usually followed by the exploration of the new nuclear shell lying above them, which may yield precious hints of the whereabouts of the dripline [15,16,17]. This letter presents a finding along these lines based on stateof-the-art experimental and theoretical studies.
The Ca isotopes correspond to a complete filling of the Z = 20 shell, leading to a high sensitivity of the shell evolution according to the neutron number. Signatures of magicity or sub-shell closures have been observed in the Ca isotopes at N = 16, 20, 28, 32, and 34 based on the steep decrease of the two-neutron separation energies S 2n [10,13] and the enhancement of the excitation energy of the first excited state E(2 + 1 ) [8, 18,12,19]. The ground state of 54 Ca has been shown, by knockout reactions, to have a closed-shell configuration [14], supporting the N = 34 magicity. Having the N = 34 magic number thus confirmed, the nexus of interest is the shell above it. If the shell is composed only of the 0 f 5/2 orbital, the recently observed 60 Ca [20] may be doubly magic and become a dripline nucleus. However, if the orbitals above 0 f 5/2 contribute substantially, the dripline can be located deep into the terra incognita of the Segrè chart. The influence of the gds orbitals above the p f shell is often discussed in the literature when neutron-rich Ca, Ti, and Ni isotopes are addressed [21,22,23,24,25,26]. There are, however, no experimental data probing this shell in the Ca isotopes.
Theoretical predictions of the level structure of Ca isotopes beyond 54 Ca and the location of the dripline have been made by modern shell-model, ab initio, beyond mean field calculations, and energy density functionals [21,22,27,28,29,30,31,32,33,34,16,35]. There seems to be no sign of convergence or consistency of such predictions for the level structure of 56,58 Ca, as discussed later. In fact, the predicted values of E(2 + 1 ) for 56,58 Ca range from 0.5 to 2 MeV [27,31,32,30,28,33,34]. Such a large variance prevents useful insights or conclusions regarding the shell structure beyond the N = 34 (sub-)shell closure. Recent predictions of a newly developed fitted interaction within the f p-model space, tailored for the neutron-rich Ca isotopes, imply 60 Ca being doubly magic at a similar level to 68 Ni [34,36]. This prediction is, however, strongly depen-dent on the agreement to experimental data for [55][56][57][58][59]36]. The closest isotone along N = 40 with experimental information, 62 Ti, showed no indication for a new magic number [26], in agreement with the predictions presented in Ref. [37]. This letter reports on the first measurement of excitation energies of 56,58 Ca by means of in-beam γ-ray spectroscopy. Experimental data were confronted with modern shell-model and ab initio calculations combined with reaction theory.
The experiment was carried out at the Radioactive Isotope Beam Factory, operated by the RIKEN Nishina Center and the Center for Nuclear Study, the University of Tokyo. Radioactive beams were produced by fragmentation of a 70 Zn beam at 345 MeV/nucleon on a 10-mm-thick 9 Be target. The 57,59 Sc isotopes were then separated and identified from focal plane F0 to F13 of the BigRIPS separator [38]. Afterwards, the secondary beams with intensities of 13.6 particles/s for 57 Sc and 0.3 particles/s for 59 Sc impinged on the MINOS liquidhydrogen (LH 2 ) target [39] to induce proton knockout reactions. Reaction residues, 56,58 Ca, were identified by the SAMU-RAI spectrometer [40]. Secondary beam energies at the target center were 209 MeV/nucleon for 57 Sc and 199 MeV/nucleon for 59 Sc, inducing considerable Doppler shifts for the emitted γ rays. The DALI2 + detector array [41] was used to measure the de-excitation γ rays. To overcome the large Doppler broadening partially caused by the long LH 2 target, Doppler corrections were performed using the reaction vertex information reconstructed by the MINOS time projection chamber. For further experimental details, the interested reader is referred to the supplemental material.
The Doppler-corrected γ-ray spectrum in coincidence with the 57 Sc(p,2p) 56 Ca reaction is shown in Fig. 1a. A single peak is observed at 1456(12) keV and tentatively assigned to the 2 + 1 → 0 + gs transition. Energy uncertainties are dominated by the fitting error and energy calibration. Lifetime effects on the measured energies were also evaluated. The heaviest Ca isotope with a known B(E2)↑ is 50 Ca [42]. Assuming the same transition strength, 37.5(10) e 2 fm 4 , for 56 Ca gives a lifetime of 17 ps. This lifetime value was adopted with an error of 100% and taken into account in the error determination.
Despite low statistics, the Doppler-corrected γ-ray energy spectrum of the 59 Sc(p,2p) 58 Ca, shown in Fig. 1b, revealed a peak-like structure in the energy range of 1000-1200 keV. To test the significance level of this peak, a maximum likelihood fit procedure was applied to the unbinned data (see bottom of Fig. 1b). The background in this spectrum was modeled from the 57 Sc(p,2p) 56 Ca reaction, with the amplitude normalized according to the event numbers. This procedure was validated with the 55 Sc(p,2p) 54 Ca data from the same experiment, which yielded a good description of the background. A significance of 2.8 σ, defined as the peak amplitude over the statistical uncertainty from the maximum likelihood fit, was obtained for the tentative 1115(34) keV γ-ray transition, including systematic errors from lifetime effects, and tentatively assigned to the 2 + 1 → 0 + gs decay of 58 Ca. The assumed lifetime is 66 ps based on the same assumption as 56 Ca. Noteworthy are the two counts observed at ∼1400 keV, comparable to the E(2 + 1 ) of 56  (2 + 1 ) 0 1456(12) 58 Ca g.s.
Poisson-statistics errors were adopted for the data points. Unbinned data are shown in the bottom of panel b for the 59 Sc(p,2p) 58 Ca channel. The response curve for an assumed 1400 keV γ-ray transition with a theoretical cross section of 0.25 mbarn is indicated for 58 Ca by the gray dashed line. ever, taking into account calculated theoretical cross sections, as discussed below, resulted in a poor overall agreement of the response function with the data, as evidenced by the gray dashed line in Fig. 1b. Further tests for the validity and the impact of the 58 Ca data is discussed in the supplemental material. The systematics of E(2 + 1 ) values as a function of neutron number presented in Fig. 2a evince the expected pattern for magic nuclei at N = 28: A sharp increase from N = 26 to 28 followed by a large reduction at N = 30. Similarly, a characteristic sharp increase from N = 30 to 32 exists for the N = 32 magic number, while the enhanced E(2 + 1 ) at N = 34 is indicative of magicity. The firmly established data point for 56 Ca and the tentative one for 58 Ca are as low as the N = 22, 24, 26, and 30 values, with a decrease from N = 36 to 38.
The E(2 + 1 ) systematics of Ca isotopes were compared to conventional shell-model calculations with the GXPF1Bs Hamiltonian in the model space of the full p f shell [14,22], and two state-of-the-art ab initio approaches: The valence-space in-medium similarity renormalization group (VS-IMSRG) [43,44,45,46] and the coupled-cluster theory (CC) [47], both employing the two-(NN) and three-nucleon (3N) interaction 1.8/2.0 (EM) [48], derived from chiral effective field theory [49]. Details of these theoretical approaches are provided in the supplemental material. Figure 2a shows the theoretical calculations well describe the E(2 + 1 ) excitation energies up to N = 34, and the GXPF1Bs Hamiltonian also provides a good agreement with the present experimental value for 56 Ca. However, all these calculations predict a flat behavior from N = 36 to 38.
A more general discussion provides an instructive viewpoint of the E(2 + 1 ) values of 56,58 Ca. If the 0 f 5/2 orbital is isolated from the other orbitals, N = 36 corresponds to a system of two neutrons solely occupying the 0 f 5/2 orbital. Likewise, N = 38 would be four neutrons in the 0 f 5/2 orbital, or, equivalently, two neutron holes of the fully occupied 0 f 5/2 orbital. The two-body interaction is invariant between particle and hole systems, but the single-particle energies can vary with neutron number. Such changes of single-particle energies do not affect excitation level energies, because only one orbital is relevant. Thus, assuming the 0 f 5/2 neutron orbital is marginally modified between 56 Ca and 58 Ca, the E(2 + 1 ) value should be identical between N = 36 and N = 38 as a consequence of this particle-hole symmetry. It is emphasized that this consequence is independent of the choice of the two-body interaction.
The present results indicate a decrease of E(2 + 1 ) from N = 36 to 38 by several hundred keV. This observation conflicts with the arguments above, implying a non-isolated 0 f 5/2 orbital. Since all experimental evidence supports an N = 34 magic number in the Ca isotopes, the 0 f 5/2 orbital is considered as isolated from lower-energy orbitals. This points to the other possibility that the 0 f 5/2 orbital is coupled to higher orbitals, suggesting a shell comprising the 0 f 5/2 orbital and at least one higher orbital. This new shell has never been discussed and its appearance excludes the N = 40 magic number in Ca isotopes.
A previous theoretical study has discussed an "sdg" shell built on an inert 60 Ca core [24]. This approach proved valid for the 78 Ni region, but remains untested for the Ca isotopes. In the present work, only the characteristics of the 0g 9/2 and 1d 5/2 orbitals can be constrained by experiment. As the protons can be assumed to form a Z = 20 closed shell, only neutrons above N = 20 are treated as valence nucleons in the calculations.
The existing effective A3DA-m [23] NN interaction, defined for a model space comprising the full p f shell, the 0g 9/2 , and 1d 5/2 orbitals, is used as a starting point. It has been successfully used for the systematic descriptions of Ni (Z = 28) [23] and Cu (Z = 29) [50] isotopes. Figure 2b shows that the observed E(2 + 1 ) values are well reproduced by the A3DA-m interaction up to N = 34, and substantial deviations for N = 36 and 38. The excitation-energy lowering from N = 36 to 38 is well reproduced by the A3DA-m interaction, in contrast to trends shown in Fig. 2a. This suggests a minor revision of the interaction may be sufficient to reproduce the experimental data. Figure 2c shows the S 2n values for the Ca isotopes. There is no notable deviation for the nuclei where experimental data are available, implying the validity of the A3DA-m interaction.
The A3DA-m interaction is revised by varying only the two-body-matrix-elements (TBME) in the linear combination Table 1: Observed excitation energies (E exp ) in keV and cross sections (σ exp ) in mbarn from the 57 Sc(p,2p) 56 Ca and 59 Sc(p,2p) 58 Ca reaction channels compared to theoretical values (σ th ) using the DWIA calculated single-particle cross sections (σ sp ) and spectroscopic factors (C 2 S ) from VS-IMSRG, GXPF1Bs, and A3DA-t. Predicted spin-parities (J π ), associated proton-removal orbitals (nl j ), and excitation energies (E x ) are also provided. The revised interaction is labeled "A3DA-t" hereafter. Figure 2b depicts the E(2 + 1 ) values obtained with the A3DA-m and the A3DA-t interactions from 42 Ca to 74 Ca, with A3DA-t reproducing the E(2 + 1 ) values of 56,58 Ca. The E(2 + 1 ) value remains almost constant until 68 Ca, where the value for 70 Ca rises due to the filled 0 f 5/2 -plus-0g 9/2 shell and the necessity of neutron excitations to the high-lying 1d 5/2 orbital. As orbitals above this are not included, the present work cannot describe excitation energies much beyond 70 Ca. Figure 2c shows S 2n values up to 76 Ca, the last possible nucleus in the present model space. A plateau is formed from 56 Ca to 70 Ca. Beyond this, S 2n becomes negative, implying the dripline is located at 70 Ca, close to some predictions [53, 15], beyond others [54] or within argued ranges [20,16,35]. Inclusion of higher sdg orbitals may slant the dripline even further due to quadrupole correlations [24,25]. As noted in Ref. [24], neutrons in higher sdg orbitals can enhance quadrupole collectivity, which may lead to a well-deformed ground state of 70 Ca.
The sensitivity of the neutron 0g 9/2 single-particle-energy (SPE) was characterised by varying it up to ±2 MeV with respect to the original value of the A3DA-t interaction. Results of this can be seen in Fig. 2b for the E(2 + 1 ) and in Fig. 2c for the S 2n . Of particular interest is the difference of E(2 + 1 ), defined as ∆E = E(2 + A ) -E(2 + A−2 ) and shown in the inset of Fig. 2b. The larger the neutron 0g 9/2 SPE, the larger the drop from 54 Ca to 56 Ca, producing a local E(2 + 1 ) maximum for 60 Ca and shifting the dripline to 62 Ca. A positive shift of +1 or +2 MeV can be excluded from the experimental E(2 + 1 ) and S 2n of 56 Ca. Conversely, a too low 0g 9/2 SPE value quenches the experimentally established magicity at N = 34 [12, 13,14], resulting in a high neutron 0g 9/2 occupation number not observed in 54 Ca [14]. Our results obtained from 56 Ca challenge the notion of an N = 40 magicity at 60 Ca and are reinforced by the tentative experimental value for 58 Ca, as all ∆E remain negative except for the 0g 9/2 SPE shifted by +2 MeV.
The occupation number of each single-particle orbital is shown in Fig. 3a. Likewise, the ESPE [7] are displayed in Fig. 3b. The shell structure above N = 34 is clearly characterized by two orbitals, 0 f 5/2 and 0g 9/2 , which remain almost degenerate across the range shown in Fig. 3b. Thus, the emergence of a new shell above N = 34 comprising 0 f 5/2 , 0g 9/2 orbitals and some others, like 1d 5/2 , is evident. The N = 34 magic gap decreases beyond A = 60, but has little affect because the 1p 1/2 orbital remains almost completely occupied. This degeneracy is lifted for isotopes with Z > 20 due to the strong monopole attraction between a proton in 0 f 7/2 and a neutron in 0 f 5/2 [7].
Further discussions are concentrated on the cross sections. Inclusive cross sections for the 57 Sc(p,2p) 56 Ca and 59 Sc(p,2p) 58 Ca reactions were measured to be 1.23(5) and 1.14(15) mb, respectively. Partial cross sections of the excited states were extracted using the efficiency-corrected γ-ray intensities, and those to the ground-states deduced by subtraction. All measured cross sections are summarized in Tab. 1. Inclusive and partial cross sections were comparable between both nuclei, lending support to the assignment of a peak in 58 Ca.
Theoretical cross sections were obtained by combining single-particle cross sections σ sp calculated from the distortedwave impulse approximation (DWIA) and the spectroscopic factors C 2 S from the GXPF1Bs and A3DA-t Hamiltonians, and VS-IMSRG approach described above [55]. They are listed in Tab. 1. The beams of ground-state 57,59 Sc have assumed J π = 7/2 − . Only removal from the proton 0 f 7/2 orbital was considered, as higher-lying proton orbitals contributed only a few percent to the final states. Negligible cross sections were calculated to states other than the listed 0 + g.s. , 2 + 1 , and 4 + 1 states. Similar inclusive cross sections for both reaction channels were predicted, as observed experimentally, and with σ exp -toσ th ratios ∼0.75, agreeing with previous values obtained in the region [26,56,57] and for stable nuclei [58]. This signifies a low occupation number of protons across the Z = 20 shell in the ground states of 57,59 Sc, hence a good proton shell closure. A different picture is observed for the partial cross sections to  Figure 2: Comparison of calculated E(2 + 1 ) and S 2n values with experimental data. a, E(2 + 1 ) systematics in even-even Ca isotopes confronted with theoretical approaches: The shell model using the GXPF1Bs Hamiltonian, the VS-IMSRG method, and CC calculations. b, E(2 + 1 ) systematics in even-even Ca isotopes, and their differences (inset). Experimental points are the same as a. The calculated values are obtained by the original A3DA-m Hamiltonian as well as its revised one (A3DA-t) c, S 2n systematics in even-even Ca isotopes. Also shown in b and c are the effect of shifting the neutron 0g 9/2 orbital for predictions of A3DA-t. the 2 + 1 states. While the σ exp -to-σ th ratio holds for 56 Ca, despite considerable uncertainties, the experimental partial cross section for 58 Ca is two times larger than the value predicted by the GXPF1Bs and VS-IMSRG calculations. In contrast, the A3DA-t Hamiltonian gives results in good agreement with experiment: Partial cross sections change from N = 36 to 38 in a consistent manner with experiment.
In conclusion, the first spectroscopy measurements for 56,58 Ca following the 1p-knockout reactions from scandium isotopes were carried out. A γ ray transition associated with the 2 + 1 → 0 + gs decay was assigned for 56 Ca and an indication of this transition was observed for 58 Ca. A comparison with standard shell-model and ab initio theoretical calculations exhibits a notable deficiency in their descriptions of nuclear structure around N = 40. The particle-hole symmetry argument robustly leads to a new shell comprising at least the 0 f 5/2 and 0g 9/2 orbitals above N = 34. The fitted A3DA-t interaction, introduced in this work, shows an excellent description of so far known experimental data, and predicts the Ca dripline at N = 50, because of substantial correlation energies from the pairing between the 0 f 5/2 and 0g 9/2 orbitals. Thus, the picture of the new magic number N = 34 [11,12,13,14] becomes more complete with a shell built atop of it. More detailed structure information of 56,58 Ca can be obtained by future measurements with Ge detector arrays at next-generation facilities like FRIB [59], notably validation of the tentative transition at 1115(34) keV. Additional experimental investigations of neutron-rich Ca isotopes, such as particle states in 55 Ca from neutron pickup reactions and the spectroscopy of 60 Ca, not believed to be doubly magic from the assessment of results presented here and Ref. [26], would provide key information to further characterize the new shell.
We would like to express our gratitude to the RIKEN Nishina Center accelerator staff for providing the stable and highintensity primary beam and to the BigRIPS team for operating the secondary beams.  hp190160, hp200130,  hp210165). This work was supported in part by MEXT as "Priority Issue on Post-K computer" (Elucidation of the Fundamental Laws and Evolution of the Universe) and "Program for Promoting Researches on the Supercomputer Fugaku" (Simulation for basic science: from fundamental laws of particles to creation of nuclei) and by JICFuS. This work was supported in part by MEXT KAKENHI Grant No. JP19H05145.