$\boldsymbol{N=32}$ shell closure below calcium: Low-lying structure of $^{50}$Ar

Low-lying excited states in the $N=32$ isotope $^{50}$Ar were investigated by in-beam $\gamma$-ray spectroscopy following proton- and neutron-knockout, multi-nucleon removal, and proton inelastic scattering at the RIKEN Radioactive Isotope Beam Factory. The energies of the two previously reported transitions have been confirmed, and five additional states are presented for the first time, including a candidate for a 3$^-$ state. The level scheme built using $\gamma\gamma$ coincidences was compared to shell-model calculations in the $sd-pf$ model space, and to ab initio predictions based on chiral two- and three-nucleon interactions. Theoretical proton- and neutron-knockout cross sections suggest that two of the new transitions correspond to $2^+$ states, while the previously proposed $4^+$ state could also correspond to a $2^+$ state.

Low-lying excited states in the N = 32 isotope 50 Ar were investigated by in-beam γ-ray spectroscopy following proton-and neutron-knockout, multi-nucleon removal, and proton inelastic scattering at the RIKEN Radioactive Isotope Beam Factory. The energies of the two previously reported transitions have been confirmed, and five additional states are presented for the first time, including a candidate for a 3 − state. The level scheme built using γγ coincidences was compared to shellmodel calculations in the sd − pf model space, and to ab initio predictions based on chiral two-and three-nucleon interactions. Theoretical proton-and neutron-knockout cross sections suggest that two of the new transitions correspond to 2 + states, while the previously proposed 4 + 1 state could also correspond to a 2 + state.

I. INTRODUCTION
Our understanding of the atomic nucleus has as one of its cornerstones the concept of shell structure, in which the location of single-particle orbitals defines shell closures and associated magic numbers. Experimental evi-dence collected in the past decades, particularly since the advent of radioactive ion beams, has shown that shell structure undergoes significant changes for isotopes far from stability [1]. Examples of this shell-evolution are the onset of N = 16 as a magic number for O isotopes [2][3][4] and the disappearance of the canonical magic number N = 20 around 32 Mg [5,6].
A particularly interesting case to study shell evolution is the region around the Ca isotopes between N = 28 and N = 40, where the development of shell closures for N = 32 and N = 34 has recently gained significant attention. In the Ca isotopes, the N = 32 sub-shell closure was first evidenced by its relatively high E(2 + 1 ) energy [7], and confirmed by two-proton knockout cross section [8] and mass measurements [9]. In turn, the first suggestion of the N = 34 shell closure on 54 Ca was also provided by the E(2 + 1 ) measurement [10] and confirmed by systematic mass measurements [11] and neutron-knockout reactions [12].
The persistence of these shell closures below and above Z = 20 has also been widely investigated. The preservation of the N = 32 shell closure above Ca has been determined in Ti and Cr via spectroscopy [13,14], reduced transition probabilities [15,16], and precision mass measurements [17]. On the other hand, the N = 34 shell closure has been suggested to disappear above Ca [18]. This is in contrast with the recently reported first spectroscopy measurement on 52 Ar, where the experimental value of E(2 + 1 ) suggests the conservation of the N = 34 shell closure for Z = 18 [19].
The first spectroscopy of 50 Ar showed a relatively high E(2 + 1 ) energy of 1178 (18) keV [20]. In that study, apart from the E(2 + 1 ), a E(4 + 1 ) was tentatively assigned, although the limited statistics prevented a firmer conclusion [20]. No further spectroscopic information is available for this very exotic nucleus. The increase of the E(2 + 1 ) with respect to neighboring isotopes has been interpreted as an indication of a sizable N = 32 gap along the Ar isotopic chain, therefore maintaining this sub-shell closure below 52 Ca [20].
From a theoretical point of view, the tensor-forcedriven shell evolution has been used to explain the appearance of the N = 32 and N = 34 shell closures [21]. In this framework, the reduction of the attractive protonneutron interaction between the πf 7/2 and the νf 5/2 single-particle orbitals results in a separation between these levels and the formation of substantial neutron gaps. Calculations including this effect [22,23] successfully reproduce the E(2 + 1 ) of Ar isotopes [19,20] and suggest the magnitude of the N = 34 sub-shell closure in 52 Ar to be around 3 MeV.
The significance of three-nucleon forces (3NFs) in the description of neutron-rich isotopes has also been studied [24,25], and the relevance of this contribution to obtain an accurate description of the spectroscopic properties of Ca isotopes has been highlighted [26]. In particular, ab initio calculations with the valence-space inmedium similarity renormalization group (VS-IMSRG) method [27][28][29] including 3NFs have provided a satisfactory description of the E(2 + 1 ) along the Ar isotopic chain [19].
Our understanding of the nature of these sub-shell closures relies on the interpretation provided by the theoretical calculations. The validity of this picture can be further tested by studying its agreement with other nuclear properties, for example, the energies of low-lying states beyond the 2 + 1 . To get a better insight into the structure at the N = 32 shell closure below Ca, the present work reports low-lying states in 50 Ar populated following direct and indirect reactions.

II. EXPERIMENT
The experiment was performed at the Radioactive Isotope Beam Factory, operated by the RIKEN Nishina Center and the Center for Nuclear Study of the University of Tokyo. A 70 Zn beam with an energy of 345 MeV/nucleon and an average intensity of 240 pnA was fragmented on a 3-mm thick Be target to produce the secondary beam cocktail. Fragments of interest were selected by the BigRIPS separator [30] using the Bρ − ∆E − Bρ technique. Event-by-event identification was obtained by an energy-loss measurement in an ionization chamber, position and angle measurements with parallel plate avalanche counters, at different focal planes, and the time-of-flight measured between two plastic scintillators [30]. The selected isotopes were focused in front of the SAMURAI dipole magnet [31], where the 151.3(13)-mm long liquid hydrogen target of MINOS [32,33] was placed. Thanks to the use of a time projection chamber surrounding the target, it was possible to reconstruct the reaction vertex with a resolution of 2 mm (σ) [33]. Following the reactions in the target, ions were identified using the SAMURAI magnet and associated detectors. Positions and angles were measured at two multi-wire drift chambers placed in front and behind the magnet, the time-of-flight was obtained from a scintillator placed in front of the target and a hodoscope located downstream of SAMURAI, which also provided an energy loss measurement from which the atomic number was inferred [31].
The high-efficiency γ-ray detector array DALI2 + [34,35], composed of 226 NaI(Tl) detectors, was placed around MINOS to detect de-excitation γ rays. The array, which covered detection angles between ∼12 • and ∼118 • with respect to the center of the target, was calibrated in energy using standard 60 Co, 88 Y, 133 Ba, and 137 Cs sources. The full-energy-peak efficiency of the array, determined using a detailed GEANT4 [36] simulation, was 30% at 1 MeV with an energy resolution of 11% for a source moving at a velocity of 0.6c. Previous results and further details from the same experiment can be found in Refs. [12,19,37,38]. 50 Ar were populated by direct and indirect reactions. For each reaction channel inclusive cross sections were obtained using the effective transmission of 50 Ar (which includes the efficiency of the beam line detectors and the beam losses in the detectors and the target) measured to be 56.7(15)%, and the efficiency of MINOS for each reaction. Table I summarizes the number of events in each reaction channel, the mean incident beam energy, the experimental efficiency of MINOS, and the corresponding inclusive cross sections.

Low-lying states in
Doppler corrected γ-ray spectra were obtained using the reaction vertex and the velocity of the fragment reconstructed with MINOS. Peak-to-total ratio and detection efficiency were improved by adding-up the energies of γ rays deposited in detectors up to 10 cm apart. To reduce the contribution of the low-energy atomic background, γ rays with energies below 100 keV in the laboratory frame of reference were not considered for the add-back. Figure 1a) shows the Doppler-corrected spectrum obtained following multinucleon removal reactions, when the γ-ray multiplicity (M γ ) was limited to a maximum of four. The spectrum was fitted with simulated response functions of the DALI2 + array and a double exponential function used to model the low-and high-energy background. The slopes of the two exponential functions were fixed by independent fits of the high-and low-energy regions. Six transitions at 826 (7)(8) (21)(27) keV provided the best fit to the spectrum. The first reported uncertainty corresponds to the statistical error from the fit, while the second is the systematic error arising from the calibration of the γ-ray detectors and the possible lifetime of the states. To place the observed transitions in a level scheme, γγ coincidences were investigated. Figure 1b) displays the γ-  (2) ray spectrum gated between 1090 keV and 1210 keV. A single background gate between 3000 keV and 4000 keV was used. Due to the many transitions observed in the spectrum it was not possible to place a more appropriate background gate. As a results, the transition where the gate was placed could not be completely removed by the background subtraction. Hence, the possibility of a doublet cannot be fully excluded. The best fit to the resulting spectrum, shown by the red line, was obtained by using the same response functions as in Fig 1a), suggesting that all the transitions are coincident with the one at ∼1150 keV. Calculations on the expected number of counts in the coincidence spectrum obtained based on the area of the gate and the efficiency of DALI2 + , are consistent with the observations, as shown by the blue line Fig. 1b). Figure 2a) shows the Doppler-corrected spectrum obtained for 50 Ar produced by the proton-knockout reaction. A total of four peaks provided the best fit to the spectrum. The transition energies deduced from   (6)(26) keV. The spectrum resembles the one observed for the multinucleon removal, and in fact all of the transitions observed seem to correspond within uncertainties to transitions also present in Fig. 1. In this case, however, the intensity of the transition around ∼ 1600 keV is smaller, while the transitions at ∼ 1900 keV and ∼ 2600 keV are more intense. The transition observed in Fig. 1 at ∼ 824 was observed with a significance below 1σ, and the one at ∼ 2230 keV was not visible in this spectrum. The projection of the γγ matrix gated around ∼ 1150 keV is shown in Fig. 2b). The best fit to the spectrum was obtained using the same four response functions used to fit the total spectrum, indicating that the transitions are coincident with the one at ∼ 1150 keV. As in the case of Fig. 1a), it was not possible to completely remove the transition where the gate was placed by background subtraction. It is noted that for the 51 K(p, 2p) 50 Ar and the multinucleon removal reactions, gates around ∼ 1600 keV, ∼ 1900 keV, and ∼ 2600 keV only showed the reciprocal coincidence of these transitions with the one at ∼ 1150 keV.
The Doppler-corrected spectrum corresponding to the 51 Ar(p, pn) 50 Ar reaction is displayed in Fig. 3a). Two peaks are visible in the spectrum with a significance above 2σ. The best fit yields transition energies of   Fig. 3b) clearly establishes the existence of the peak at ∼ 1600 keV, and shows that it is coincident with the one at ∼ 1150 keV. The energies observed in these spectrum are consistent with the ones obtained previously, suggesting the population of the same levels. Figure 4a) shows the Doppler-corrected spectrum obtained for the 50 Ar(p, p ′ ) 50 Ar reaction. Three transitions are visible and the transition energies obtained for this case are 1138(8)(11) keV, 1626(33)(16) keV, and 2890 (31)(29) keV. The background-subtracted coincidence spectrum, in Fig. 4b), shows that the transition at 2890 keV is coincident with the one at 1138 keV. No coincidence between the transitions at 1626 keV and 1138 keV was observed, which can be attributed to the reduced statistics.
Based on the γγ analysis discussed above, the tentative level scheme shown in Fig. 5a) was constructed. The energies of low-lying states in 50 Ar were calculated as the weighted average of the values obtained from the different reactions, when applicable. The weights were determined based only on the statistical uncertainty, and the systematic error was added in quadrature. Being the one with the highest intensity, the 1150(12) keV transition was placed decaying directly into the ground state. This transition agrees, within error bars, with the , and 54 Ti [42], that proton inelastic scattering populates preferentially 2 + , 4 + , and 3 − states, therefore a 3 − spin and parity can be reasonably assigned to this level. The spin assignment for the 2744(20) keV level, also observed in this reaction, could then be either 2 + or 4 + . Further discussion on the possible spin and parity assignments for the levels obtained in this work will be presented below.
For the direct reactions, exclusive cross sections to populate each observed state were obtained from the fitted γ-ray intensities. Table II summarizes the adopted level energies and exclusive cross sections obtained in this work. Based on simulated angular distributions of the γ-rays, an additional uncertainty of 4% has been in-  (44) -cluded to account for possible alignment of the states. The ground state cross section was calculated by subtracting the exclusive cross sections from the inclusive one reported in Table I. The high background level, low statistics and limited resolution of DALI2 + could prevent the observation of low-intensity, high-energy transitions feeding directly the 0 + gs state, therefore the ground-state cross section is prone to be overestimated. In addition, it was not possible to disentangle between the direct 51 Ar(p, pn) 50 Ar reaction and the scattering followed by neutron emission, 51 Ar(p, p ′ ) 51 Ar→ 50 Ar + n, therefore, all the cross sections for this channel are to be considered as an upper limit.

IV. DISCUSSION
Predictions for the energies of low-lying states in 50 Ar were obtained within the shell-model framework using the SDPF-MU effective interaction [43] and considering the full sd and pf model space for protons and neutrons. The original Hamiltonian was modified [44] using experimental data on exotic Ca [10] and K [45] isotopes. These calculations have previously provided good agreement with the experimental E(2 + 1 ) and E(4 + 1 ) energies in neutron-rich Ar isotopes [20] and suggest a N = 32 gap of ∼ 3 MeV for 50 Ar. Although this gap is predicted to be of similar magnitude as for 52 Ca, the wave function of the 2 + 1 state for 50 Ar turns out to be more mixed than the one for 52 Ca, making the effect of this shell closure less evident [20].
Calculations were also performed using the ab initio VS-IMSRG approach using the chiral NN+3N interaction labeled 1.8/2.0 (EM) in Refs. [46,47]. This NN+3N interaction is based on chiral effective field theory [48,49], a low-energy effective theory of quantum chromodynamics, with low-energy constants fitted to the properties of the lightest nuclei up to 4 He. The same chiral interaction has been successfully used to study E(2 + 1 ) in the Ar isotopic chain [19], as well as excitation spectra from oxy- gen [50] to nickel [51] and tin [52] isotopes. For the model spaces, the sd space was considered for the protons, and the pf for the neutrons, preventing the calculation of negative parity states. As in previous works [19,[50][51][52] the IMSRG(2) approximation, where all induced operators are truncated at the two-body level, was employed. The VS-IMSRG was used to decouple a valence-space Hamiltonian, which captures 3N forces between valence nucleons via an ensemble normal ordering, for each nucleus of interest [53]. Spectroscopic factors, C 2 S, were calculated within each model. For the case of the 51 K(p, 2p) 50 Ar reaction, the J π = 3/2 + ground state for 51 K was employed [45,54] and knockout from the sd shell was considered, leading to the population of positive parity states exclusively. For the case of the 51 Ar(p, pn) 50 Ar reaction the predicted ground-state spin of 1/2 − for 51 Ar was assumed. Figs. 5b) and 5c) show the level scheme obtained from the calculations where only positive-parity states with calculated C 2 S ≥ 0.1 for the 51 K(p, 2p) 50 Ar or 51 Ar(p, pn) 50 Ar reactions are displayed. The predictions for the 3 − state based on the SDPF-MU Hamiltonian will be discussed afterwards.
The E(2 + 1 ) of 50 Ar is accurately reproduced by both calculations, and a 0 + 2 is predicted to be the next excited state. The experimental level at 1976(15) keV has a good agreement with this state. It is noted that the SPDF-MU calculations predict the 0 + 2 state of 56 Cr to be 1982.1 keV, in fair agreement with the tentative experimental level at 1674.5(4) keV [55]. The structure at higher energies also presents many similarities: The next levels predicted to be populated are the 4 + 1 , 1 + 1 , 2 + 3 and 2 + 4 . However, the energies predicted in the VS-IMSRG approach are modestly higher than in the SDPF-MU cal-culations. By enlarging the configuration space of this theoretical framework to include the sd − pf orbitals for protons and neutrons, additional excited states may appear at lower energies. The SDPF-MU calculations also predict significant population of more levels, in particular of the 2 + 2 and states with spin and parity 1 + and 4 + . To get an insight on the spin and parity of the observed levels, single-particle theoretical cross sections were computed in the DWIA framework [56] using the Bohr-Mottelson single-particle potential [57]. For the optical potentials of the distorted waves, the microscopic folding model [58] with the Melbourne G-matrix interaction [59] and calculated nuclear density was employed. The Franey-Love effective proton-proton interaction was adopted [60] and the spin-orbit part of each distorting potential was disregarded. Cross sections at different beam energies were calculated to take into account the energy loss of the beam in the thick target. The calculated single-particle cross sections were multiplied by the spectroscopic factors calculated for the reactions in each theoretical framework. Tables III and IV show the obtained results for the 51 K(p, 2p) 50 Ar and 51 Ar(p, pn) 50 Ar reactions, respectively.
The calculated ground state cross section for both reactions is much lower than the experimental values. As already mentioned this is due to the non-observation of states decaying directly to the 0 + 1 state, which results in an over-estimation of the experimental cross section. For the case of the 51 K(p, 2p) 50 Ar reaction, the SDPF-MU and VS-IMSRG calculations predict a cross section to the 2 + 1 state of 1.0 mb and 0.62 mb, respectively, in reasonable agreement with the experimental value of 0.8(2) mb. At higher energies the SPDF-MU calculation suggest the population of the 2 + 3 , 2 + 4 , and 2 + 5 states.  Although high cross sections are also predicted for the 1 + 2 and 1 + 4 states, they would decay preferentially to the ground-state, therefore its correspondence to any experimental level is unlikely. They may, however, account for the seeming too high experimental population of the ground state when compared to calculated cross sections. The VS-IMSRG calculation, on the other hand, only indicates the population of the 2 + 4 and 4 + 1 states. The fact that the VS-IMSRG calculations only predicts two states with sizable sd-proton cross-sections is related with the reduced model space, which prevents proton pf −sd excitations. This in turn, highlights the importance of such excitations in the population of low-lying states. They will be investigated in the future with a newly developed cross-shell VS-IMSRG approach [61]. In spite of the differences between the models, they both point out that the 51 K(p, 2p) 50 Ar reaction mostly populates 2 + states. The experimental levels at 2744 (20) keV, 3053(23) keV, and 3771(34) keV, observed in this reaction, are in fair agreement with the predictions for the 2 + 3 , 2 + 4 , and 2 + 5 states in the SDPF-MU model. We therefore tentatively assign this spin and parity to these states. The level at 2744 (20) keV has been previously suggested to be the 4 + 1 [20]. Although the SDPF-MU calculations favors a 2 + assignment, the comparison with the VS-IMSRG results make it also compatible with the 4 + 1 . Furthermore, the population of this state in the 50 Ar(p, p ′ ) 50 Ar reaction favors a 4 + 1 assignment. Therefore a (2 + , 4 + ) assignment is left open for this state. It is worth mentioning that the state at 1976(15) keV has a negligible cross section for the 51 K(p, 2p) 50 Ar reaction, which is consistent with the theoretical predictions for the 0 + 2 state. The agreement between the SDPF-MU and VS-IMSRG calculations on the energy and spectroscopic factor of the 0 + 2 state suggest that it is spherical in nature, as the VS-IMSRG does not properly account for deformed low-lying states [37,51].
For the 51 Ar(p, pn) 50 Ar reaction the theoretical models predict a 2 + 1 state cross section of ∼ 7 − 8 mb, while the experimental value is 15(4) mb. In this case, the experimental over-estimation comes from the impossibility to distinguish between the direct and indirect reactions in this channel. The next states with the higher predicted cross section are the 1 + 1 , 4 + 1 , and 2 + 3 states in both calculations. In the VS-IMSRG, population to the 3 + 3 state at 4428 keV is also predicted. As previously noted, the 1 + 1 state would most probably decay directly to the groundstate. Furthermore, the 3 + 3 is not predicted by the VS-IMSRG to be populated in the 51 K(p, 2p) 50 Ar reaction, so it is improbable that it corresponds to an experimental level. The ambiguity between the 2 + and 4 + characters for the state at 2744 (20) keV observed in this reaction is therefore maintained.
Finally, theoretical predictions of the systematic of 3 − 1 states for the N = 32 isotones have been obtained using the SDPF-MU calculations and confronted to available data [42,62] including the state at 4040(44) keV obtained in this work, as shown in Fig. 6. The E(3 − 1 ) for 50 Ar is comparable in magnitude to the one of 52 Ca, and the theoretical predictions show a good agreement with both isotopes, reinforcing the spin and parity assignments. However, the calculations overestimate the E(3 − 1 ) as Z increases. For nuclei around Ca, the Fermi surface is located near the Z = 20 shell gap, therefore proton excitations require less energy. This is reflected in the calculations by the low 3 − levels predicted for 50 Ar, 52 Ca, and 54 Ti, where the calculations show a good agreement with the data. Going towards the Si isotopes, the excitation from the p shell to the sd shell becomes likely. This possible excitation is not taken into account in the calculations, which in turn increases the predicted E(3 − 1 ) energies. On the other side, towards higher Z, the experimental levels are rather stable around 4 MeV, but the calculations are not able to reproduce them. In Ni, proton excitations from the sd to the pf shells, as well as neutron excitations from the pf to the sdg shells may contribute. In particular, it has been reported that the neutron excitations from the pf to the sdg shell are not well reproduced due to a too large shell gap between pf and sdg shells, and it has been suggested that it is necessary to lower the sdg shell by 1 MeV to reproduce the negative parity states of Ni isotopes [63,64].

V. SUMMARY
Low-lying levels of 50 Ar have been investigated by proton-and neutron-knockout reactions, inelastic proton scattering, and multinucleon removal reactions. Based on the γγ analysis, a level scheme was constructed, including five newly observed transitions among which a candidate for the 3 − state has been reported for the first time. The experimental level scheme was compared to theoretical calculations performed in the SDPF-MU shell model, as well as the ab initio VS-IMSRG approach. Both calculations predict similar level schemes for 50 Ar. Theoretical cross sections for the 51 K(p, 2p) 50 Ar and 51 Ar(p, pn) 50 Ar were compared to the observed ones, to infer the spin and parity of the states. Two of the newly observed states were tentatively assigned a (2 + ) spin and parity, and it was shown that the state previously suggested to be the 4 + 1 could also correspond to a 2 + state.
Overall, both theoretical calculations provide consistent results and a relatively good agreement with the experimental data for both the 51 K(p, 2p) 50 Ar and 51 Ar(p, pn) 50 Ar reactions. This emphasizes the sub-shell closure at N = 32 and our understanding of shell evolution in this region. The remaining differences among calculations most likely arise from the reduced proton and neutron spaces employed in the VS-IMSRG and highight their importance in the understanding of the low-lying structure of 50 Ar.