Shell structure of 43 S and collapse of the N=28 shell closure
S. Momiyama, K. Wimmer, D. Bazin, J. Belarge, P. Bender, B. Elman, A. Gade, K. W. Kemper, N. Kitamura, B. Longfellow, E. Lunderberg, M. Niikura, S. Ota, P. Schrock, J. A. Tostevin, D. Weisshaar
SShell structure of S and collapse of the N = 28 shell closure S. Momiyama, K. Wimmer,
1, 2
D. Bazin, J. Belarge, P. Bender, B. Elman,
3, 4
A. Gade,
3, 4
K. W. Kemper, N. Kitamura, B. Longfellow,
3, 4
E. Lunderberg,
3, 4
M. Niikura, S. Ota, P. Schrock, J. A. Tostevin, and D. Weisshaar Department of Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Instituto de Estructura de la Materia, CSIC, E-28006 Madrid, Spain National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA Department of Physics, Florida State University, Tallahassee, Florida 32306, USA Center for Nuclear Study, University of Tokyo, Wako, Saitama 351-0198, Japan Department of Physics, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom (Dated: September 3, 2020)The single-particle structure of the N = 27 isotones provides insights into the shell evolution ofneutron-rich nuclei from the doubly-magic Ca toward the drip line. S was studied employing theone-neutron knockout reaction from a radioactive S beam. Using a combination of prompt anddelayed γ -ray spectroscopy the level structure of S was clarified. Momentum distributions wereanalyzed and allowed for spin and parity assignments. The deduced spectroscopic factors show thatthe S ground-state configuration has a strong intruder component. The results were confrontedwith shell model calculations using two effective interactions. General agreement was found betweenthe calculations, but strong population of states originating from the removal of neutrons from the2 p / orbital in the experiment indicates that the breakdown of the N = 28 magic number is morerapid than the theoretical calculations suggest. PACS numbers:
I. INTRODUCTION
The emergence of shell closures or their disappearancein exotic nuclei has been one of the main interests of thenuclear structure community since the advent of radioac-tive beam facilities. Islands of inversion and shape coexis-tence have been associated with the disappearance of theclassical shell closures on the neutron-rich side of the val-ley of stability [1]. In particular, the N = 28 shell closure,arising in a harmonic oscillator plus spin-orbit mean field,has recently attracted much interest [2]. Below the dou-bly magic nucleus Ca with 20 protons and 28 neutrons,the N = 28 nuclei show a variety of interesting features.Mass measurements [3], transfer [4], and nucleon knock-out reactions [5] support a strong N = 28 shell closure in Ar. Measurements of the reduced transition probabil-ity, B ( E S, the measurement of a large B ( E
2) value and its com-parison to theoretical calculations suggested a vibrationalcharacter of this nucleus [8]. The lowering of the excited0 +2 state from 3695 keV in Ar [9] to 1365 keV [10] in S indicates the onset of shape coexistence and a rapidweakening of the N = 28 shell closure. The measured E S 0 + states was interpreted asarising from the substantial mixing of spherical and pro-late configurations [11]. Theoretical calculations of thepotential energy surface using the symmetry-conservingconfiguration mixing method and the Gogny D1S interac-tion do not show distinct minima characteristic of shapecoexistence and rather suggest configuration mixing [12]. Later refinements of the theory and extended calculationsfind that the ground state of S has a collective wavefunction which is extended in the ( β, γ ) plane while theexcited 0 +2 is prolate, yet γ -soft [13]. Shell model calcu-lations using a newly derived SDPF-MU interaction [14]suggest that the evolution of collectivity along N = 28 isgoverned by the proton-neutron tensor force [15]. Here,the potential energy surface exhibits a minimum on theprolate side.The Si nucleus is well deformed, it exhibits a low ex-citation energy for the first 2 + state [16] and a large R / ratio [17]. Calculations with the SDPF-MU interactionpredict the ground state of Si to be strongly oblatedeformed [15]. Detailed spectroscopy of Si, however,questioned the 4 + assignment of Ref. [17] and proposedan excited 0 +2 state based on the observed populationcross section [18]. Approaching the drip-line [19], the last N = 28 nucleus with excited states known is Mg [20].The measured two-proton removal cross sections alongthe N = 28 isotones [17, 21] were interpreted as showinga change of the ground state deformation from prolate in S to oblate for Si, and back to prolate at Mg.Turning to the even-odd N = 27 nuclei, S has at-tracted special attention, both from the theoretical andexperimental side. In Ar the ground state is 7 / − ,as expected from the normal orbital filling. A low-lying J π = 3 / − state with a rather long lifetime [22] isstrongly populated in the ( d, p ) reaction adding a neu-tron to Ar [4] and very weakly in the neutron removalreaction [5]. This confirms the vacancy of the 2 p / or-bital in both , Ar and the existence of a shell closureat N = 28. In S, an isomeric state with a lifetime of a r X i v : . [ nu c l - e x ] S e p / − and a level in-version compared to Ar was proposed. A measurementof the magnetic moment firmly assigned J π = 7 / − tothe isomeric state and, because its lifetime is only com-patible with an E J π = 3 / − [24]. The spherical nature of the7 / − isomeric state was questioned and the spectroscopicquadrupole moment, determined to be | Q s | = 23(3) efm ,was significantly larger than the expectation for a singlehole in the 1 f / orbital. While the state cannot be con-sidered spherical, shell model calculations do not predicta band structure built upon the isomeric state [25]. Theseresults triggered various theoretical discussions. Anti-symmetrized molecular dynamics (AMD) calculations in-dicate that the 7 / − isomer might be triaxial, and thatbands of prolate, oblate, and triaxial nature coexist atlow excitation energy [26]. The gap between neutronsingle-particle levels originating from the spherical 1 f / and 2 p / orbitals reduces as a function of the deforma-tion parameter β ; the two orbitals cross around a pro-late deformation with β ≈ . N = 28 shellgap disappears. A state at around 940 keV observedin a Coulomb excitation measurement [27] is suggestedas the 7 / − member of the prolate K π = 1 / − groundstate band with a negative decoupling parameter. Anoblate band built on the 3 / − state is also predicted. Ashell model study exploiting quadrupole rotational in-variants came to similar conclusions [28]. The calcu-lations based on the SDPF-U effective interaction [29]predict a third, prolate band with a dominant 2p-2h(1 f / ) − (2 p / ) configuration. Calculations using theSDPF-MU effective interaction [14] and the variation af-ter angular-momentum projection method show that theground state and the isomeric state are dominated by K = 1 / / K forbiddeness of the decay [15]. This interpre-tation also explains the occurrence of the long-lived 0 +2 and 4 +1 states in S [10, 11, 30, 31].Spectroscopic information on states in S beyond theground and isomeric state was obtained from nucleon re-moval reactions, however, placement in the level schemeproved difficult because of the presence of the isomericstate [32]. Most recently, excited state lifetimes in Swere measured. Using the proton knockout reaction from Cl several states were populated [33]. The level order-ing was reversed compared to the earlier study [32]. Itshould be noted that the level scheme and the interpreta-tion of Ref. [33] are at variance with the results presentedhere. In the present work, the neutron knockout reactionis measured with the additional capability to distinguishbetween decays to the isomer and to the ground state.In the present paper, we report on the measurement ofthe single-particle structure of S using the one-neutronknockout reaction from a fast radioactive S beam. Thecombination of prompt and delayed spectroscopy allowedfor an unambiguous construction of the level scheme and the extraction of spectroscopic factors using reac-tion model calculations. The results suggest an intruder-dominated configuration in the ground state of S. II. EXPERIMENT
The experiment was performed at the Coupled Cy-clotron Facility of the National Superconducting Cy-clotron Laboratory at Michigan State University [34].The secondary S beam was produced by projectile frag-mentation of a 140 A MeV Ca primary beam on a705 mg/cm
Be production target located at the en-trance of the A1900 separator [35]. The beam parti-cles were identified by their time-of-flight on an event-by-event basis. The secondary beam was separated andtransported to a 376(4) mg/cm
Be secondary targetlocated at the pivot point of the S800 spectrograph [36].The momentum acceptance of the A1900 separator wasset to 1%, resulting in a mid-target energy of 93.7 A MeVand an average S intensity and purity of about 1900 ppsand 98(1)%, respectively.The reaction residues were analyzed and identified inthe S800 spectrograph [36] as shown in Fig. 1. Parti- − − − − − − TOF (arb. units) ∆ E ( a r b . un i t s ) c o un t s FIG. 1. Particle-identification plot of reaction residues de-tected in the S800 spectrograph. A gate on incoming S ionsis applied. The dashed line is the outgoing S gate for thefurther analysis. cle identification was achieved by measuring the energyloss in an ionization chamber (∆ E ) in the focal plane ofthe S800 spectrograph and the time-of-flight (TOF) be-tween two plastic scintillators located before the targetand in the focal plane, respectively. Positions and anglesof reaction residues at the end of the S800 spectrographwere measured by two cathode-readout drift chambers(CRDC) and traced back to the secondary target by us-ing the ion optics code COSY Infinity [37]. This allowedthe determination of the non-dispersive position and themomentum vector at the secondary target. In order toimprove the resolution for the momentum transfer, a par-allel plate avalanche counter (PPAC) was placed at theintermediate image plane upstream of the target. Here,the dispersive position is correlated with the momentumof the projectile, and the momentum of the incoming pro-jectile can thus be obtained. The momentum resolutionfor the incoming beam with the PPAC position correc-tion was deduced as 0.052 GeV/c.The secondary target was surrounded by theGamma Ray Energy Tracking In-beam Nuclear Array(GRETINA) [38, 39]. A GRETINA module consists offour high-purity germanium crystals, each 36-fold seg-mented. In the present experiment, four detector mod-ules were placed at 58 ◦ with respect to the beam axis andfour were placed at 90 ◦ . The signals were digitized andan online pulse-shape analysis algorithm allowed for thedetermination of γ -ray interaction points with energy andposition information. It was assumed that the hit withthe largest energy deposition was the first interaction,and its position was used for the Doppler correction. The γ -ray position information was also used in the trackinganalysis, where γ -ray interactions were added togetherwhen the difference between their emission angle withrespect to the target position was less than 25 ◦ . Thisadd-back analysis was adopted for the γ - γ coincidenceanalysis and for extracting the exclusive parallel momen-tum distributions. The energy and efficiency calibrationof GRETINA was done with standard radiation sourcesand the deviation from literature values were deduced tobe less than 1 keV. The efficiency of the whole array wasmeasured to be 5.9% at 1 MeV. The γ -ray yields weredetermined from a fit of simulated response functions tothe γ -ray energy spectrum. The experimental setup wasimplemented in a GEANT4 simulation [40] including theexperimentally determined thresholds and resolutions ofeach individual Ge crystal. In the χ fit, the γ -ray ener-gies and intensities were individually varied to reproducethe measured spectrum.Finally, the reaction residues were implanted into a6.35 mm thick Al plate at the back of the focal plane ofthe S800 spectrograph. Delayed γ rays emitted from thedecay of isomeric states were detected in IsoTagger [41]consisting of 32 CsI(Na) detectors. This allowed con-struction of the level scheme above the 320 keV isomericstate in S for the first time and deduction of the pop-ulation cross sections for all final states. The energy andefficiency calibration of IsoTagger was performed with astandard Y source. The efficiency at 898 keV was mea-sured to be 8.3%.The S nucleus has an isomeric 0 +2 state at 1365 keVwith a 2.619(26) µ s half-life [10, 11]. The beam can thusreach the secondary target in an excited state. In addi-tion to the direct E +1 state with a branch-ing ratio of 16.3(13)% [11]. The γ -ray transition from the2 +1 state to the ground state could have been observedin the IsoTagger, however, no transition at this energywas observed. The isomeric ratio of the 0 +2 state in S isthus assumed to be negligible for the extraction of crosssections.
III. RESULTS
Fig. 2 shows the prompt, Doppler-corrected γ -ray en-ergy spectrum measured with GRETINA gated on theone-neutron knockout reaction from S to S. Most of
500 1000 1500 2000 2500 3000 E γ (keV) c o un t s /2 k e V ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) databackgroundsimulationtotal FIG. 2. Prompt, Doppler-corrected γ -ray energy spectrumfor the one-neutron knockout reaction from S to S. Thepeaks are labeled with the transition energy and uncertaintyin keV. The background around 500 keV includes transitionsfrom neutron-induced reactions on Ge and Al. the previously observed γ rays [32, 33] were confirmedand their energies are shown in Fig. 2 together with theiruncertainty. The transition at 571(3) keV is newly ob-served in this work. For the error estimation of the γ -ray energy, the uncertainties of the energy calibrationof GRETINA, the velocity of S for the Doppler cor-rection, and a potential offset of the reaction target loca-tion along the beam axis were considered. The individualcontributions were, for example for the 2600 keV transi-tion, less than 0.5, 3, and 6 keV. To deduce the yieldof each prompt γ ray, a χ fit of the simulated responsefunctions to the experimental spectrum was performed.In this fitting procedure, background γ rays of neutron-induced reactions with the Ge detectors and surroundingmaterials were also considered. In the laboratory systemclear peaks around 600 keV are observed. The remainingcontinuous background was modeled as the sum of twoexponential functions connected to a linear function inthe lower energy region. The uncertainties for the γ -rayyields include, besides the statistical uncertainty, consid-eration of the deviation of the simulated efficiency fromthe measured one. This contribution was smaller than4% over the whole energy range and thus smaller thanthe statistical uncertainties. The prompt γ -ray energiesand intensities are compiled in Table I. Fig. 3 shows thebackground subtracted γ - γ coincidence spectra gated onthe 1155, 625, 850 and 977 keV transitions. The threetransitions at 1155, 625, and 850 keV are emitted in cas-cade and the 977 and 185 keV transitions are in mutualcoincidence, but not with any of the other transitions.This is in agreement with the level scheme proposed inRef. [33] with a doublet of states at 1155 and 1162 keV. TABLE I. Observed γ ray energies, efficiency-corrected inten-sities, and coincidence information for S. The uncertaintiesof the γ ray energies include all systematic uncertainties whileyields include only the statistical errors.energy (keV) yield/ion (%) coincident γ rays level (keV)185(2) 5.8(3) 977 1162(4)228(2) 0.44(7) 228(2)320(2) 49(3) 1532 320571(3) 0.93(11)625(3) 3.6(2) 850, 1155 1780(5)720(3) 1.8(2)850(4) 3.6(2) 625, 1155 2628(6)977(4) 7.1(4) 185 977(4)1155(4) 13.2(6) 625, 850 1155(4)1209(4) 3.6(2) 1209(4)1469(7) 0.67(13) 2628(6)1532(5) 2.2(2) 320 1854(4)1856(7) 0.37(13) 1854(4)2600(8) 9.7(5) 2600(8)
050 gate 1155 keV (a)
625 850
025 gate 625 keV (b)
850 1155
500 1000 1500 2000 2500 E γ (keV)
625 1155
500 1000 1500 2000 2500 E γ (keV) . . . . . . . . . . . . c o un t s /10 k e V FIG. 3. Background subtracted γ - γ coincidence spectra mea-sured in GRETINA. Panels (a), (b), (c), and (d) show thespectra gated on the prompt 1155, 625, 850, and 977 keVtransitions. The high statistics obtained in the present work makes itpossible to determine the order of the γ -ray transitions inthe cascades by the comparison of the measured γ -ray in-tensities in Table I. These intensities confirm the order ofthe 850 → → →
977 keV cascades.The latter is opposite to the suggestion of Ref. [33] andthus challenges the result of the very similar lifetimes of the states at 185 and 1162 keV proposed in that work.The present ordering of the cascade is also consistent withearlier measurements of Coulomb excitation [27] assum-ing that the transition observed around 940 keV corre-sponds to the 977 peak observed in the present work. Infact, the transition energy is not determined accuratelyin Ref. [27], and the observed line could be composed ofseveral transitions within the limited energy resolution.The isobar Cl and the isotone Cl have transitionsat 943 and 928 keV which could have contaminated thespectrum. A recent Coulomb excitation experiment con-firmed the 977 keV state that is directly excited fromthe ground state [42]. The 850 keV transition is placedon top of the 625 keV one, since the former was not ob-served in the proton removal reaction [33]. The transitionat 1469 keV was placed to feed either the 1155 keV orthe 1162 keV state from the 2628 keV state based onthe matching energy sum. No coincidences were foundfor the 1209 and 2600 keV transitions. Based on theirintensities, coincidences should have been observed andthese transitions are therefore placed as direct groundstate decays. The transitions at 228, 571, and 720 keVcould not conclusively be placed in the level scheme dueto limited statistics. The 228 keV transition is placed asa direct ground state decay from the first excited stateat 228 keV, based on the comparison with theoreticalcalculations (see Section IV).Fig. 4 shows the γ -ray energy spectrum measured bythe IsoTagger in delayed coincidence with identified Sreaction residues. The decay of the known 320 keV iso- E γ (keV) c o un t s /4 k e V E γ (keV) c o un t s /8 k e V gate 320 keV in IsoTagger FIG. 4. Gamma-ray energy spectrum measured by IsoTag-ger. A gate on S has been applied. The isomeric de-cay of the 320 keV state is observed. The inset shows theprompt, Doppler-corrected γ -ray energy spectrum measuredwith GRETINA gated on the delayed 320 keV transition. meric state [23] is observed. The intensity of the 320 keVtransition was determined from a χ fit of a simulatedresponse function [41] to the spectrum in a similar man-ner as for the prompt spectrum. The background wasmodeled as the sum of two exponential functions. Theimplantation position distribution of the S ions was im-plemented in the simulation as described in Ref. [41].The position on the stopper plate was taken from the ex-perimental xy distribution measured by the CRDC de-tectors in the S800 focal plane and extrapolated to thestopper plate. The implantation depth, z coordinate,was estimated by the ATIMA code [43] using the ex-perimentally measured energy distribution of S ions.In order to extract the yield, the in-flight decay of theisomer between the secondary reaction target and thestopper plate needed to be taken into account. The half-life of the isomeric state has been previously measured( T / = 478(48) ns [23], 415(5) [24], and 200 +140 − ns [44]).In the present experiment, the half-life was determinedfrom the decay curve after implantation. The result of T / = 391(14) ns is slightly lower than the most precisevalue but consistent. Considering the trajectory and thevelocity of S behind the secondary target, 79.4(23)% ofthe isomeric state initially produced at the target reachedthe stopper. For the uncertainty estimation on the yield,the deviation of the present half-life from the previousmeasurement of 415(5) ns [24], the velocity distributionof the S reaction products, and the effect of the uncer-tainty of the simulated implantation depth on the effi-ciency of IsoTagger (2% at 320 keV) were considered.Fig 4 also shows the prompt γ rays detected inGRETINA in delayed coincidence with the decay of the320 keV isomer. The 1532 keV transition is clearly incoincidence with the isomeric transition and the energysum matches the 1856 keV transition. This establishes anew state at 1854(4) keV using the weighted average ofthe energies. Looking for coincidences with the 1532 keVtransition in GRETINA does not reveal another γ -raytransition as a candidate for a transition on top of theisomer.The level scheme of S, determined in the presentwork, is shown in Fig. 5. The order of the transitionsof a γ -ray cascade was determined by comparing the ob-served yields. The 1469 and the 1856 keV transitionswere placed in the level scheme solely based on energy dif-ferences. Two states are located close to the neutron sep-aration energy S n = 2629 keV [45]. The 2600 keV statedecays directly to the ground state, while the 2628 keVstate decays via a cascade. The fact that the 2600 keVtransition was not observed in the fragmentation reac-tion of Cl [32] nor in the proton knockout reaction [33]from Cl supports the presence of two different states.The very different momentum distributions (see below)for the 2600 and 2628 keV states further confirm theexistence of two close-lying states near the neutron sep-aration energy.Using the level scheme presented in Fig. 5 the final-state exclusive cross sections were determined. They arepresented in Table II and Fig. 5. The inclusive crosssection to bound states in S was determined from thenumber of particles identified in the S800 spectrographand amounts to 91(4) mb, slightly larger than but consis-tent with the previous measurement of the same reactionof 79(7) mb [32]. The uncertainties include, in additionto statistical sources, the selection of the particle identi- E x c it a ti on e n e r gy ( k e V ) n S 228 320 - - (1/2 - - - , 1/2 - - , 1/2 - - , 1/2 - + , 5/2 + - , 5/2 - Experiment p J (mb) s - - - - - - - - - - - SDPF-U - - - - - - - - - - - - - - SDPF-MU
FIG. 5. Level scheme of S determined from the present ex-perimental results and predicted by shell model calculations.The width of the arrows reflects the measured γ -ray yields.Gray, dashed transitions are place based on the energy differ-ences of established levels, the 228 keV state is placed basedon the comparison to the shell model calculations. The levelsare labeled with the spin and parity assignments derived fromthe measured momentum distributions and the partial crosssection for each state (in mb). Spins and parities of predicted1 / − , 3 / − , 5 / − , and 7 / − states are indicated in black,red, green, and blue, respectively. Theoretical cross sections(in mb) include the calculated spectroscopic factors and thereaction model calculations for the single-particle cross sec-tions (see text for details). fication gate, the purity and intensity fluctuation of theincoming S beam, uncertainties related to the transmis-sion of the analysis line of the S800, and the thickness ofthe secondary target. The one-neutron reaction from Swas fully within the acceptance of the S800 spectrographso that corrections were not necessary.The parallel momentum distributions for several finalstates populated in S are shown in Fig. 6. In each casegates on the depopulating γ -ray transitions were applied,and feeding from the higher-lying states was subtractedusing the level scheme of S and the efficiency of the γ -ray detectors at the respective energies. The data arecompared to theoretical calculations of neutron knockoutfrom the l = 1, 2, and 3 single-particle orbits using theeikonal reaction model [46, 47]. In this approach the pro-jectile and target densities, taken from a Skyrme Hartree- TABLE II. Inclusive and exclusive cross sections to boundfinal states. ( nlj ) refers to the quantum numbers used inthe calculation of the single-particle cross section, σ sp , in theeikonal reaction theory. E (keV) J π σ exp (mb) ( nlj ) σ sp (mb) C S exp / − p / / − ) 0.4(1) 2 p / / − f / / − / − (, 1 / − ) 8.2(9) 2 p / / − (, 1 / − ) 5.3(3) 2 p / / − (, 1 / − ) 3.3(3) 2 p / / +1 (, 5 / + ) 8.8(4) 1 d / / − (, 5 / − ) 3.9(3) 1 f / c o un t s / ( M e V / c ) c o un t s / ( M e V / c ) l = 1l = 2l = 3020004000 c o un t s / ( M e V / c ) c o un t s / ( M e V / c ) .
00 17 .
25 17 .
50 17 .
75 18 . p || (GeV/c) c o un t s / ( M e V / c ) .
00 17 .
25 17 .
50 17 .
75 18 . p || (GeV/c) c o un t s / ( M e V / c )
320 keV (f)
FIG. 6. Parallel momentum distributions of the one-neutronknockout reaction for several states in S. Each panel showsthe experimental parallel momentum distribution obtained bygating on γ -ray transitions in black, compared to theoreticaleikonal reaction model calculations for removal of a neutronfrom the p (red), d (green), and f (blue) orbital. Panel (f)is the momentum distribution extracted in coincidence withthe isomeric transition measured in IsoTagger and all oth-ers are obtained by gating prompt transitions measured inGRETINA. Fock calculation for the projectile and assuming a Gaus-sian distribution for the light target, are used to constructthe eikonal S matrices for the ejectile- and nucleon-targetinteraction. The radial wave functions of the removednucleon from each of the active orbitals are calculated inWoods-Saxon potentials with geometries constrained by the rms radius of the orbital from the Hartree-Fock calcu-lation. The calculated parallel momentum distributionswere transformed into the laboratory system and foldedwith the experimental momentum resolution that was ob-tained from dedicated calibration runs. The theoreticalcalculations are normalized to the experimental counts inthe 17.3 to 17.8 GeV/c momentum region. This momen-tum region was selected to eliminate the lower momen-tum tail region which is not reproduced by the eikonalreaction theory. The states at 1155, 1162, and 1209 keVare well explained by neutron knockout from a l = 1 p or-bital, probing the occupation of neutron orbits above the N = 28 shell gap in the ground state of S. On the otherhand, the momentum distributions for the state at 2628and the isomeric state at 320 keV are consistent withneutron knockout from the l = 3 orbit. Thus, the spin-parity of the isomeric state of S, already established as7 / − [24], is confirmed in the present work. It is interest-ing to note that the momentum distribution of the stateat 2600 keV, shown in Fig. 6 (b), can only be reproducedby assuming removal of a neutron from the 1 d / orbitalwith l = 2. This state is located close to the neutronseparation energy [45] and a candidate for a hole state inthe 1 d / orbital below N = 20. Such a state would notbe populated in proton removal reactions in agreementwith its non-observation [32, 33]. By subtracting the dis-tributions of all excited states from the inclusive one, themomentum distribution and cross section directly pop-ulating the ground state of S via one-neutron knock-out reaction was extracted. Due to ambiguities in thelevel scheme and the unplaced prompt γ rays, the dis-tinction between the neutron knockout from the f and p orbits is less clear, but the momentum distribution iswell described by knockout from the 2 p / orbital. Inthe following discussion, the spin-parity of the groundstate of S is assumed to be 3 / − , which was suggestedfrom the transition rate from the 7 / − isomeric state tothe ground state [24]. The momentum distribution forthe 977 and 1856 keV states are asymmetric and verybroad, suggesting the population via a non-direct pro-cess. This would be expected from a collective rotationalband member [27, 42].Using the eikonal reaction model calculations, thesingle-particle cross sections σ sp were calculated (see Ta-ble II). These depend on the effective separation energyfor the final state and the quantum numbers of the or-bital the nucleon was removed from. Using the spin andparity assignments shown in Fig. 5 the spectroscopic fac-tors, C S exp = σ exp /σ sp , for each state were obtained.They are listed in Table II. IV. DISCUSSION
The experimental level scheme is compared to the re-sults of shell model calculations in Fig. 5. Two effec-tive interactions in the full proton sd and neutron f p model space were used to calculate the excitation ener- TABLE III. Results of the shell model calculations with theSDPF-U [29] and SDPF-MU [14] effective interactions. Inaddition to the bound states populated in the one-neutronknockout reaction, the members of rotational bands discussedin the text are listed.SDPF-U SDPF-MU E (keV) J π C S band E (keV) J π C S band0 3 / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − (b)2366 9 / − (b) 2196 3 / − / − / − / − / − / − (b) 2496 5 / − / − / − / − (b) gies, transition probabilities, and spectroscopic factors.Effective charges ( e p = 1 . , e n = 0 .
35) and g factorssuggested in Ref. [28] have been used. The SDPF-U [29]and SDPF-MU [14] interactions have been previously ap-plied to , S [14, 15, 25, 28, 33] and predict, at firstglance, very similar level schemes shown in Fig. 5. Thecalculated energies, spectroscopic factors, and band as-signments are listed in Table III.In both cases, three rotational bands are predicted andthese are labeled (a), (b), (c) in Table III. For the caseof the SDPF-U interaction the band structure in , Sis extensively discussed in Ref. [28]. The collective 7 / − state at 977 keV is a member of the ground state band(a). Based on the comparison with the shell model calcu-lation the 228 keV transition is a candidate for the decayfrom the 1 / − state. In the shell model calculations, the5 / − state is predicted to decay to the 1 / − state with alarge B ( E
2) value, but no such state was observed in thepresent work. The small cross sections for the 1 / − and7 / − states suggest that they are not of single-particlecharacter, in agreement with the calculations.In the present work, we have for the first time identifieda state built on top of the isomer. The state at 1854 keVdecays to the 7 / − isomer via the 1532 keV transition(see Fig. 4). The 1856 keV transition has been tentativelyassigned to a ground state decay. This would limit thespin and parity values to J π = (3 / , / , / − . The mo-mentum distribution for this state is rather broad, but noconclusion can be drawn. The state could be a candidatefor the oblate 3 / − band head predicted by the AMDcalculations whose main decay branch is to the triaxial7 / − isomeric states [26]. The shell model calculationsdo not predict a candidate for a corresponding state, but rather states with a J π = 7 / − , / − , / − sequence(band (b)) are predicted, where the 9 / − state is con-nected by strong M µ N ) and E fm for theSDPF-MU interaction to the 7 / − isomer. For SDPF-Uthe values are similar (see [28]). If the 1856 keV transi-tion is placed elsewhere in the level scheme, the 1854 keVstate could be a natural candidate for the 9 / − state. Afirm spin and parity assignment for the 1854 keV stateis required in order to draw further conclusions. Finally,a third band-like structure is built on the 3 / − state at1155 keV. The 7 / − state at 2628 keV decays to the stateat 1780 keV via the 850 keV transition, as well as to the1155 or 1162 keV state by emission of a 1469 keV γ ray.The 1780 keV state is not populated directly, it decaysvia the 625 keV transition, a likely spin assignment isthus 5 / − . The 3 / − and 7 / − states can be associ-ated with the shell model states at 1405 (875) and 2479(2466) or 3093 (2722) keV in the SDPF-U (SDPF-MU)results, based on the comparison of the spectroscopic fac-tors. However, none of the shell model states shows a de-cay pattern similar to the experimentally observed one.The decay of these is fragmented to several states belowwith individual B ( E
2) values around 1-100 e fm . The7 / − member of the second prolate band (c) at 2479 keVpredicted by the SDPF-U calculations [28], for example,has a strong B ( E
2) value for the decay to the 5 / − bandhead (1990 keV), but the predicted branching ratio isonly 15.2 % owing to the higher energy difference for theother possible decays to lower lying states. Furthermore,the SDPF-MU calculations, in contrast with those us-ing SDPF-U, predict a strong transition from the 5 / − state to the 3 / − state suggesting a 3 / − band head in-stead, more in line with the results from the AMD calcu-lations [26]. Clearly, more experimental investigation isrequired to establish the band structure and determineits deformation characteristics.The neutron knockout cross sections to the bound,shell-model final states in S have been calculated usingthe theoretical spectroscopic factors C S and the single-particle cross sections σ sp σ ( J π ) = (cid:18) AA − (cid:19) N C S ( J π ) σ sp ( nlj, S n + E ( J π )) . They are compared to the experimental results inFig. 5. The inclusive theoretical cross section was cal-culated by summing the contributions of all states up tothe experimental neutron separation energy S n ( S) =2629 keV [45]. The inclusive cross section amounts to94.3 (91.7) mb for the SDPF-U (SDPF-MU) interactions.Experimentally the cross section populating positive par-ity states by sd -shell neutron removal, which are outsideof the model space of the calculations, amounts to atleast 8.8(4) mb. An estimate for the reduction factor R S [47, 48] is thus given by the ratio of the cross sec-tion to f p states to the theoretical value and amounts to0.87(0.90) for the two effective interactions, in line withthe systematics [47, 48].The isomeric 7 / state carries the major fraction ofthe single-particle strength, but still significantly lessthan expected from a pure ν ( f / ) − configuration. Thisis in agreement with the interpretation of the electricquadrupole moment of this state [25], which is signifi-cantly larger than expected for a single hole in the 1 f / orbital. The shell model calculations predict that a largefraction of the 1 f / strength is located close to the neu-tron separation energy. Experimentally, the strength tounbound states is inaccessible in the present setup, there-fore, part of the 1 f / strength could be missed in the ex-periment. Three states with significant l = 1 strength areobserved around 1200 keV. This is not reproduced by theshell model calculations which predict only one excited3 / − state in this energy region. The 1162 keV state de-cays to the 977 keV J π = 7 / − state. The lifetime of thisstate, if the present level ordering is adopted, amountsto 15(2) ps [33]. Such a state is not found in the shellmodel calculations. The spectroscopic factors for 1 / − and 5 / − states are small as it is expected that the occu-pation of the 2 p / and 1 f / orbitals in the ground stateof S is small.The spectroscopic factors can also be compared tothe N = 28 isotones. Ca has been studied in detailby pickup transfer reactions using Ca targets. Thespectroscopic factor for the ground state amounts to C S = 6 .
22 [49]. Using the typical reduction R ≈ . C S = 8 for the 1 f / orbital. For the 3 / − state at 2014 keV only a smallspectroscopic factor of C S = 0 . Ar nucleus a measurement of the neutronknockout reaction from Ar also found a small spectro-scopic factor for the first excited 3 / − state of 0 . C S = 4 . N = 28 is a goodshell closure in Ca and Ar nuclei.In the present work, the spectroscopic strength for thepopulation of the first 7 / − state amounts to 3.00(21),significantly lower than for the heavier isotones. Thespectroscopic factor for the 3 / − ground state is 0.55(17).However, several other states are populated by the re-moval of a neutron from the p orbitals, as evidenced fromthe momentum distributions shown in Fig. 6. While thepresent experiment cannot distinguish between removalof a 2 p / and a 2 p / neutron, the latter is unlikely asthe 2 p / is expected to lie higher in energy. The shellmodel calculations also do not predict large spectroscopicfactors for the J π = 1 / − states (see Table III). Theobserved fragmentation of the 2 p / strength is not pre-dicted by the shell model calculations. If the experimen-tal spectroscopic factors for the 2 p states are added, and normalized using the reduction factor [47, 48], R S , asdetermined from other nuclei as a function of the sep-aration energies, the summed normalized spectroscopicstrength can be used as a indicator for the occupationnumber. In the present case the sum amounts to 1.8(4)where the uncertainty is dominated by a systematic un-certainty of R S which has been assumed to be 20%. Thissuggests that the ground-state configuration of S iscomposed of up to two neutrons in the 2 p / orbital.The shell model calculation for the summed spectroscopicstrengths amount to (cid:80) C S (1 f / ) = 4 .
89 (4.35) and (cid:80) C S (2 p / ) = 0 .
89 (1.01) for the SDPF-U (SDPF-MU) interactions. The occupation numbers for the 2 p / orbital in the ground state of S are 1.18 and 1.38, re-spectively. If the cross-shell πsd − νf p tensor compo-nent of the SDPF-MU matrix elements is removed, thesummed spectroscopic strength, up to S n , increases to5.26 for the 1 f / orbital. This is in line with the in-terpretation that the proton-neutron tensor interactionis driving the shell evolution in this exotic region of thenuclear chart [14].The location of the 7 / − and 3 / − states in S alreadysuggested the inversion of the normal (1 f / ) − and in-truder (2 p / ) neutron configurations. The present ex-periment proves for the first time an intruder dominanceof the ground state of S. This is significantly differentfrom the less exotic isotones, and the increase in 2 p / configurations in the ground state of S compared to Ar is abrupt. In the even more exotic isotone Si onlyone transition was observed [50], however, many morelow-lying states are expected based on the shell modelcalculations. Si would be an ideal testing ground forthe shell model calculations, since there the SDPF-U(-SI)and SDPF-MU interactions predict very different spec-troscopic factors for the one-neutron removal reactionfrom Si.Finally, in the present experiment the population of apositive parity state at 2600 keV was observed. The spec-troscopic factor amounts to 0.83(4) assuming removal ofa neutron from the 1 d / orbital. This value can be com-pared to the isotone Ca, where the 3 / +1 state is lo-cated at 2580 keV and has a deduced spectroscopic factorof 1.18 [49], determined from the ( d, t ) transfer reactionmeasurement. This state lies outside of the model spaceand is not described with the present shell model calcu-lations. V. SUMMARY AND OUTLOOK
In summary, we have performed spectroscopy of Susing the one-neutron knockout reaction from S. Usingprompt and delayed γ -ray spectroscopy in coincidence,the level scheme of S was constructed. Previously, thiswas beset with ambiguities due to the presence of a long-lived isomeric state in S. Final-state exclusive momen-tum distributions of the residue allowed for firm spin andparity assignments. The level ordering and assignmentsof a recent lifetime measurement [33] were revised. Astate above the isomer was identified for the first time,but its properties could not be reproduced using shellmodel calculations. Coulomb excitation measurementsusing an isomeric S beam could help in resolving thisissue. A band-like structure built on a 3 / − state was ob-served, but further experimental investigation is requiredto confirm a band and determine its properties. The crosssections for the population of states originating from theremoval of a 2 p / neutron from the S ground statewere found to be large. This is a direct measure of theamount of intruder configuration in the ground state of S and quantifies the N = 28 shell quenching in thisexotic nucleus. ACKNOWLEDGMENTS
We would like to thank the NSCL staff for the prepa-ration of the radioactive beam at the Coupled CyclotronFacility. This work was supported by the U.S NationalScience Foundation under Grant No. PHY-1306297,PHY-1102511, and PHY-1565546, by the U.S. Depart-ment of Energy (DOE) National Nuclear Security Ad-ministration through the Nuclear Science and SecurityConsortium under Award No. DE-NA0003180, and bythe DOE-SC Office of Nuclear Physics under Grants No.DE-SC002045. GRETINA was funded by the DOE, Of-fice of Science. Operation of the array at NSCL was sup-ported by the DOE under Grants No. DE-SC0014537(NSCL) and No. DE-AC02-05CH11231 (LBNL). SM ac-knowledges support from JSPS Grant-in-Aid for JSPSResearch Fellow Grant Number JP15J08882. KW ac-knowledges support from the Spanish Ministerio deEconom´ıa y Competitividad RYC-2017-22007. JAT ac-knowledges support from the Science and Technology Fa-cilities Council (U.K.) Grant No. ST/L005743/1. [1] A. Gade and S. N. Liddick, Jour. Phys. 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