Extended p _ 3 / 2 Neutron Orbital and the N = 32 Shell Closure in 52 Ca

The one-neutron knockout from 52 Ca in inverse kinematics onto a proton target was performed at ∼ 230 MeV/nucleon combined with prompt γ spectroscopy. Exclusive quasifree scattering cross sections to bound states in 51 Ca and the momentum distributions corresponding to the removal of 1 f 7 = 2 and 2 p 3 = 2 neutrons were measured. The cross sections, interpreted within the distorted-wave impulse approximation reaction framework, are consistent with a shell closure at the neutron number N ¼ 32 , found as strong as at N ¼ 28 and N ¼ 34 in Ca isotopes from the same observables. The analysis of the momentum distributions leads to a difference of the root-mean-square radii of the neutron 1 f 7 = 2 and 2 p 3 = 2 orbitals of 0.61(23) fm, in agreement with the modified-shell-model prediction of 0.7 fm suggesting that the large root-mean-square radius of the 2 p 3 = 2 orbital in neutron-rich Ca isotopes is responsible for the unexpected linear increase of the charge radius with the neutron number.

The one-neutron knockout from 52 Ca in inverse kinematics onto a proton target was performed at ∼230 MeV/nucleon combined with prompt γ spectroscopy.Exclusive quasifree scattering cross sections to bound states in 51 Ca and the momentum distributions corresponding to the removal of 1f 7=2 and 2p 3=2 neutrons were measured.The cross sections, interpreted within the distorted-wave impulse approximation reaction framework, are consistent with a shell closure at the neutron number N ¼ 32, found as strong as at N ¼ 28 and N ¼ 34 in Ca isotopes from the same observables.The analysis of the momentum distributions leads to a difference of the root-mean-square radii of the neutron 1f 7=2 and 2p 3=2 orbitals of 0.61 (23) fm, in agreement with the modified-shell-model prediction of 0.7 fm suggesting that the large root-mean-square radius of the 2p 3=2 orbital in neutron-rich Ca isotopes is responsible for the unexpected linear increase of the charge radius with the neutron number.DOI: 10.1103/PhysRevLett.129.262501Atomic nuclei can be described as neutrons and protons bound in a self-induced attractive mean field [1].In the shell-model picture, they occupy discrete energy levels, grouped into shells, while interacting with each other via nuclear residual interactions.Particularly stable configurations appear for full shells, associated to the so-called magic numbers of nucleons.The relative energies of singleparticle orbitals are dynamic over the nuclear chart, driven by the monopole part of the residual interaction [2,3].Far from stability the shell closures can disappear or weaken, e.g., at the neutron numbers N ¼ 8, 20, and 28 [4][5][6][7][8], and new magic numbers can emerge.In the neutron-rich pfshell nuclei, new shell closures at N ¼ 32 and 34 were revealed, corresponding to the filling of the 2p 1=2 and 2p 3=2 neutron orbitals, respectively.The N ¼ 32 shell closure in neutron-rich nuclei was claimed from a series of observations relying on first 2 þ excitation-energy [9][10][11][12][13][14], transition-probability [15,16], and mass measurements [17][18][19][20], correspondingly the N ¼ 34 shell closure from Eð2 þ Þ [21,22], mass [23], and neutron knockout cross-section [24] measurements.On the other hand, recent measurements along the Ca and K isotopic chains reveal an increase of the charge radii with a slope larger than expected from N ¼ 28 to N ¼ 32 and 33, respectively, and with no local minimum or inflection at N ¼ 32, which is usually considered as a sign of a neutron shell closure [25][26][27].This observation, not quantitatively reproduced by microscopic theories, was interpreted as challenging the doubly closedshell character of 52 Ca.An erosion of the proton shell closure is not supported by spectroscopic experiments [28,29], while the connection between charge radii evolution and the strength of a neutron shell closure has been recently questioned [30].As an alternative interpretation, an effective increase in size of shell-model valence p-wave neutron orbitals influencing the proton radial extension has been proposed to reproduce the observed increase of charge radii of Ca isotopes, while maintaining the doubly magic character of 52 Ca [31].A sizable difference of 0.7 fm was predicted between the root-mean-square (rms) radius of the 2p 3=2 and the 1f 7=2 neutron orbitals, in qualitative agreement with the matter radius trend of 49−51 Ca isotopes extracted from interaction cross sections [32].
In this Letter, we address the neutron shell closure of 52 Ca from direct neutron knockout (p; pn) and we provide a first determination of the spatial extension of the 1f 7=2 and 2p 3=2 neutron orbitals.
The experiment was performed at the Radioactive Isotope Beam Factory of RIKEN, operated jointly by the RIKEN Nishina Center and the Center for Nuclear Study, University of Tokyo.A 240-pnA 70 Zn primary beam at 345 MeV=nucleon impinged on a 10-mm-thick Be target for the production of the secondary cocktail beam.The beam-particle identification was done event by event using the BigRIPS separator [33] via magnetic-rigidity (Bρ), energy-loss (ΔE), and time-of-flight (TOF) measurements [34].The 52 Ca particles were produced with a mean intensity of 4.4 particles per second (2.3% purity).
The (p; pn) reaction was measured at the SAMURAI setup [35] using MINOS [36] and DALI2 þ [37].MINOS was composed of a 151(1) mm liquid-hydrogen target with a density of 73 mg=cm 3 surrounded by a 300-mm-long time projection chamber allowing a reaction vertex reconstruction with a resolution of 5 mm full-width-athalf-maximum (FWHM) [38].The prompt γ rays from the deexcitation of the 51 Ca fragments were measured with DALI2 þ , a high-efficiency array of 226 NaI(Tl) scintillation detectors surrounding MINOS and covering angles between 15°and 118°with respect to the target center.The response functions of DALI2 þ for γ-ray-source and inbeam measurements were simulated with the GEANT4 toolkit [39].The relative agreement between the simulated and measured γ-ray-source efficiencies is within 5%.The reaction fragments were analyzed by the large acceptance 2.7-T SAMURAI spectrometer.The fragment-particle identification was done by trajectory reconstruction using two multiwire drift chambers placed upstream and downstream of SAMURAI for the magnetic rigidity information, while the TOF and ΔE were provided by a 24 plastic scintillator-bar hodoscope for the fragment velocity and atomic number (Z) determination.A particle identification with 4.8 (7.2)σ separation in Z and 31 ð8.1Þσ separation in A=Q was achieved for the beam (fragmentlike) Ca isotopes.The 52 Ca beam particles had a kinetic energy at the reaction PHYSICAL REVIEW LETTERS 129, 262501 (2022) 262501-2 vertex between 190 and 270 MeV=nucleon with a midtarget energy of ∼230 MeV=nucleon.The (p; pn) reaction channel was tagged by gating on 52 Ca in BigRIPS, a highmomentum-transfer proton in MINOS, and 51 Ca in SAMURAI, in coincidence.A total of 37 000 such events were recorded.The detection efficiency of MINOS for 52 Caðp; pnÞ 51 Ca was 65(3)%, obtained from experimental data as in [38].Two plastic scintillator arrays, the NeuLAND [40] demonstrator and NEBULA [41], were placed after the SAMURAI magnet for neutron detection.The kinematics of the high-momentum transfer (p; pn) quasifree scattering reaction leads to neutrons recoiling at large scattering angles mainly outside the neutron detectors' acceptance.On the other hand, events of proton inelastic scattering followed by neutron evaporation, 005ð1Þ MeV [42] ] could be detected.These events were subtracted after efficiency correction to extract quasifree scattering cross sections.The efficiency correction was applied as a function of the relative energy of ( 51 Ca þ n) leading to a mean neutron detection efficiency of 32(4)%.
The parallel and perpendicular momentum distributions (PMDs) of the 51 Ca fragments relative to the beam were determined using the measured velocities and reconstructed angles at the reaction vertex.The differential PMDs dσ=dP jj and dσ=dP ⊥ to individual final states were constructed gating on each bin of the inclusive momentum and fitting the corresponding partial γ-ray spectra the same way as described for the total γ-ray spectrum.In this way the PMDs for the 7=2 − state were obtained, while the ground-state PMDs are the difference between the total and the excited-state PMDs.The other non-ground-state components were found negligible.The neutron-evaporation events were subtracted from all PMDs.The experimental PMDs are shown in Fig. 2.
The theoretical momentum distributions were calculated within the distorted-wave impulse approximation (DWIA) formalism [45][46][47][48][49].The folding potential (FP) [50] with the Melbourne G-matrix interaction [51] was used for incoming and outgoing nucleon scattering waves.The single-particle wave function of the knocked-out neutron was obtained as a bound state in the Bohr-Mottelson potential [52].The Woods-Saxon one-body potential was used with the radial extension r 0 ¼ 1.27 fm and diffuseness a 0 ¼ 0.67 fm parameters as a starting point, and with the depth of the potential adjusted to match the neutron effective separation energy throughout the study [53].The nonlocality correction was made to both the scattering and bound-state wave functions by using the Perey factor [54].The elementary p-n scattering process was described by the nucleon-nucleon effective interaction parametrized by Franey and Love [55]; the Möller factor [56] was introduced for treating the Lorentz transformation of the p-n cross section.The theoretical shapes were folded with the reaction energy profile and the experimental momentum resolution.The momentum profile of the direct 52 Ca beam contained the main information on the experimental momentum resolution of 49.5 MeV=c (76.5 MeV=c) for the parallel (perpendicular) component to which an additional degradation of the resolution of 1.5 MeV=c (7.5 MeV=c) originating from the vertex position uncertainty inside the target is considered.The momentum profile of the direct 52 Ca beam and the theoretical distributions for 51 Ca, for populating the ground state and the 3453-keV excited state by p 3=2 and f 7=2 neutron knockout, respectively, are also plotted in Fig. 2.
For the (p; pn) reaction at ∼230 MeV/nucleon incident energy, the quasifree scattering approximation is proven to be suitable from the observed kinematics.The PMDs relate to the single-particle wave functions of the knocked-out neutrons, and therefore to their rms radii [57].We conducted a variation of r 0 and a 0 of the Woods-Saxon potential used for the calculation of the wave function of the knocked-out neutron.It was found that the PMDs are not sensitive to a 0 ; a change of a 0 by 10% (40%), causes a PMDs width variation by less than 1% (4%).Calculations with the Dirac phenomenology potential EDAD1 (Dirac) [58] were also performed to estimate the impact of the choice of potential on the PMDs; in this case, the nonlocality correction to the scattering waves was made by multiplying them by the Darwin factor [46,59] in the Dirac phenomenology.The folding and Dirac potentials lead to almost identical PMDs within 4.5% for all considered (r 0 ,a 0 ) combinations (Fig. 2) and therefore the choice of potential has no significant impact on the PMDs and rms radii study.The momentum distributions and singleparticle cross sections were calculated with FP for a range of r 0 values keeping a 0 ¼ 0.67 fm.A χ 2 criterion was used and a probability analysis assuming a Gaussian probability density function was performed in order to determine the rms radii of the individual orbitals within our framework.Figures 2(c  ) with associated spin-parity assignment (J π ) and the experimental cross sections (σ th −1n ) using r 0 ¼ 1.21ð5Þ fm, 1.35 (10) fm, and 1.27 fm (default) for the neutron knockout from f 7=2 , p 3=2 , and p 1=2 orbitals, respectively, together with the SM prediction for the excitation energies of 51 Ca (E SM ex ) and C 2 S SM .The theoretical cross sections σ th −1n are calculated using the shell model C 2 S SM and the DWIA single-particle cross section values, σ DWIA sp .The ratio of experimental and theoretical single-particle cross sections normalized to (2J þ 1) is given in the last column.the corresponding probability distribution as function of r 0 .The optimal r 0 and associated 1-σ uncertainty for neutron knockout from the p 3=2 and f 7=2 orbitals are 1.35(10) fm and 1.21(5) fm, respectively.The deduced r 0 values correspond to the rms radii of the single-particle wave functions of the knocked-out neutron of 4.74(18) fm for p 3=2 and 4.13 (14) fm for f 7=2 .The single-particle wave functions were also obtained from Hartree-Fock-Bogolyubov (HFB) calculations using the HFBRAD [60] code and the SKM Skyrme interaction [61].The SKM interaction was chosen for its best agreement to experimental data for the proton and matter radii.The rms radii of the single-particle wave function in this case were found at 4.49 fm for the p 3=2 orbital and 4.12 fm for the f 7=2 orbital, the rms radius of f 7=2 being in perfect agreement with the rms radius obtained with the optimal r 0 , while the p 3=2 radius is underestimated.The proton, neutron, and matter total density rms radii obtained with HFB calculations with the SKM interaction for 52 Ca are 3.46 fm, 3.74 fm, and 3.63 fm.Experimental data from isotopic shift measurements situate the charge distribution radius at 3.55 fm [25] and thus the proton rms radius at 3.46 fm [32].The "unexpectedly" large charge radius for 52 Ca found by [25] is explained by [31] proposing a "pronounced halo nature" of the p 3=2 and p 1=2 orbitals, 0.7 fm larger than the rms radii of the f 5=2 and f 7=2 orbitals.The rms radii difference between the p 3=2 orbital and the f 7=2 orbital obtained by the present analysis is 0.61 (23) fm, in agreement with this prediction.
The occupancies can be tested using neutron knockout cross sections and we use the ratio between the experimental and the single-particle cross sections normalized to (2J þ 1), R S ¼ σ −1n =ð2J þ 1Þσ sp in comparing 52 Ca relative to 48 Ca and 54 Ca, considered as doubly closed-shell neutron-rich Ca isotopes [21][22][23][24]63].For the systematic comparison, the theoretical single-particle cross sections obtained with the best-fit r 0 values are used for 52 Ca.The 48 Caðp; pnÞ triple-differential cross section (TDX) was studied at 149.5 MeV=nucleon by [64].The theoretical single-particle TDX for the 48 Caðp; pnÞ reaction was calculated both with the FP, as for 52 Ca, and Dirac.At higher reaction energies, as in the present experiment, the difference in cross sections between FP and Dirac are below 5%, but at ∼150 MeV=nucleon as for 48 Ca from [64], the differences become significant.Dirac is more suitable for the stable 48 Ca. Th following values were obtained for the proportionality factors between the experimental and calculated TDX after the normalization to (2J þ 1), equivalent to R S : 1.49(17) (FP) and 1.19 (13) (Dirac) for the f 7=2 orbital and 1.02(23) (FP) and 0.78 (19) (Dirac) for the d 3=2 orbital.The ratio between experimental and theoretical cross sections for 54 Ca from [24], reporting on data obtained from the same measurement as the present work, is used for the comparison.The values for R S for 48 Ca (with FP and Dirac), 52 Ca, and 54 Ca for the orbitals below and above the neutron Fermi level are plotted in Fig. 3 together with the C 2 S SM .The three Ca isotopes exhibit a consistent pattern: a ratio close to unity below the Fermi level and a very small ratio above.The N ¼ 32 shell closure in 52 Ca thus proves to be as strong as N ¼ 28 and N ¼ 34 in Ca isotopes.This finding justifies the use of single-particle wave functions in this work.
To summarize, the (p; pn) one-neutron knockout from 52 Ca at ∼230 MeV=nucleon was measured.Exclusive cross sections to bound final states in 51 Ca and the momentum distributions corresponding to the removal of 1f 7=2 and 2p 3=2 neutrons were measured and analyzed within the DWIA framework.A consistent shell structure for 48;52;54 Ca was obtained from the ratio of experimental and singleparticle cross sections.The agreement with shell-model predictions places 52 Ca among the doubly magic Ca isotopes.In addition, the measured momentum distributions with high statistics allowed to access the rms radii of FIG. 3. The ratio of experimental neutron knockout cross sections and theoretical single-particle cross sections normalized to (2J þ 1) for 48;52;54 Ca (gray, blue, and turquoise, respectively) below and above the corresponding shell closures.Experimental data are from [24,64] and this work.The error bars contain experimental cross-section uncertainties.For 52 Ca, theoretical uncertainties from the r 0 sensitivity study are added quadratically.
the 1f 7=2 and 2p 3=2 neutron orbitals at 4.13 (14) fm and 4.74 (18) fm, respectively.With this result, the p 3=2 neutron single-particle orbital rms radius, 0.61 (23) fm larger than 1f 7=2 , supports the prediction of [31] where the large spatial extension of p neutron orbitals in neutron-rich Ca isotopes is proposed to be responsible for the linear increase of their charge radii beyond 48 Ca.The present result calls for a systematic extension of the method to several isotopic chains, complementary to ongoing efforts to explore the neutron radial extension in radioactive nuclei [65][66][67][68][69], relevant to the nuclear equation of state and the physics of neutron stars [70][71][72][73].

FIG. 1 .
FIG.1.Top: single-particle neutron configurations of51 Ca corresponding to the ground state, 3.4 MeV, and 1.7 MeV excited states, from left to right.Bottom: the γ-ray decay spectrum of51 Ca via the (p; pn) reaction after Doppler-shift and add-back correction (blue circles), including the neutron-evaporation contribution (gray line).The experimental γ spectrum is fitted (dark-blue line) with the simulated response functions (dotted red line) and a double-exponential background (dashed black line).The lefthand-side inset shows the low-energy region of the spectrum containing the transition at 691(4) keV, where the response function is shifted vertically (þ130) for visualization.The level and decay scheme of51 Ca is summarized on the right.
FIG.2.Experimental parallel (a) and perpendicular (b) momentum distributions of the52 Ca direct beam (dotted black line),51 Ca ground-state (red squares), and 3453-keV state (blue circles) population together with the theoretical curves for p-wave (red) and f-wave (blue), with a binning of 40 MeV=c.The calculations were performed using a folding potential (solid lines) and the Dirac phenomenology potential (dashed lines) with a 0 ¼ 0.67 fm and the optimal r 0 values: 1.35 fm (p wave) and 1.21 fm (f wave).The statistical errors are marked with crosses and the systematic errors on the absolute normalization with boxes.The (c) and (d) panels show the reduced χ 2 (upper panels), i.e., χ 2 =NDF (NDF being the number of degrees of freedom), and the probability distribution (lower panels) for the f 7=2 and p 1=2 orbitals as a function of the parameter r 0 .Study performed for vertex kinetic energies between 190 and 270 MeV=nucleon.See text for details.

TABLE I .
Experimental excitation energies (E exp ex