Contents lists available atScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Restoration of the natural E(1/2 + 1 ) - E(3/2 + 1 ) energy splitting in odd-K isotopes towards N = 40
Y.L. Sun
a,b,∗, A. Obertelli
a,b,c, P. Doornenbal
c, C. Barbieri
d, Y. Chazono
e, T. Duguet
b,f, H.N. Liu
a,b,g, P. Navrátil
h, F. Nowacki
i,j, K. Ogata
e,k, T. Otsuka
l,m, F. Raimondi
d,n, V. Somà
b, Y. Utsuno
o, K. Yoshida
o, N. Achouri
p, H. Baba
c, F. Browne
c, D. Calvet
b, F. Château
b, S. Chen
q,c,r, N. Chiga
c, A. Corsi
b, M.L. Cortés
c, A. Delbart
b, J.-M. Gheller
b, A. Giganon
b, A. Gillibert
b, C. Hilaire
b, T. Isobe
c, T. Kobayashi
s, Y. Kubota
c,m, V. Lapoux
b, T. Motobayashi
c, I. Murray
c, H. Otsu
c, V. Panin
c, N. Paul
b, W. Rodriguez
t,c, H. Sakurai
c,l, M. Sasano
c, D. Steppenbeck
c, L. Stuhl
m, Y. Togano
u, T. Uesaka
c, K. Wimmer
l, K. Yoneda
c, O. Aktas
g, T. Aumann
a,v, L.X. Chung
w, F. Flavigny
p, S. Franchoo
p, I. Gašpari ´c
x,c,
R.-B. Gerst
y, J. Gibelin
z, K.I. Hahn
aa, D. Kim
aa,c, T. Koiwai
l, Y. Kondo
ab, P. Koseoglou
a,v, J. Lee
r, C. Lehr
a, B.D. Linh
w, T. Lokotko
r, M. MacCormick
p, K. Moschner
y, T. Nakamura
ab, S.Y. Park
aa,c, D. Rossi
a, E. Sahin
ac, D. Sohler
ad, P.-A. Söderström
a, S. Takeuchi
ab,
H. Törnqvist
a,v, V. Vaquero
ae, V. Wagner
a, S. Wang
af, V. Werner
a, X. Xu
r, H. Yamada
ab, D. Yan
af, Z. Yang
c, M. Yasuda
ab, L. Zanetti
aaInstitutfürKernphysik,TechnischeUniversitätDarmstadt,D-64289Darmstadt,Germany bIRFU,CEA,UniversitéParis-Saclay,F-91191Gif-sur-Yvette,France
cRIKENNishinaCenter,2-1Hirosawa,Wako,Saitama351-0198,Japan dDepartmentofPhysics,UniversityofSurrey,GuildfordGU27XH,UK
eResearchCenterforNuclearPhysics(RCNP),OsakaUniversity,Ibaraki567-0047,Japan fKULeuven,InstituutvoorKern- enStralingsfysica,3001Leuven,Belgium
gDepartmentofPhysics,RoyalInstituteofTechnology,SE-10691Stockholm,Sweden hTRIUMF,4004WesbrookMall,Vancouver,BritishColumbiaV6T2A3,Canada iUniversitédeStrasbourg,IPHC,67037StrasbourgCedex,France
jCNRS,UMR7178,67037Strasbourg,France
kDepartmentofPhysics,OsakaCityUniversity,Osaka558-8585,Japan
lDepartmentofPhysics,UniversityofTokyo,7-3-1Hongo,Bunkyo,Tokyo113-0033,Japan mCenterforNuclearStudy,UniversityofTokyo,RIKENcampus,Wako,Saitama351-0198,Japan nESNT,CEA,UniversitéParis-Saclay,F-91191Gif-sur-Yvette,France
oJapanAtomicEnergyAgency,Tokai,Ibaraki319-1195,Japan
pInstitutdePhysiqueNucléaire,CNRS-IN2P3,UniversitéParis-Sud,UniversitéParis-Saclay,91406OrsayCedex,France qStateKeyLaboratoryofNuclearPhysicsandTechnology,PekingUniversity,Beijing100871,PRChina
rDepartmentofPhysics,TheUniversityofHongKong,Pokfulam,HongKong sDepartmentofPhysics,TohokuUniversity,Sendai980-8578,Japan
tUniversidadNacionaldeColombia,SedeBogota,FacultaddeCiencias,DepartamentodeFísica,Bogotá111321,Colombia uDepartmentofPhysics,RikkyoUniversity,3-34-1Nishi-Ikebukuro,Toshima,Tokyo172-8501,Japan
vGSIHelmoltzzentrumfürSchwerionenforschungGmbH,Planckstr.1,64291Darmstadt,Germany wInstituteforNuclearScience&Technology,VINATOM,P.O.Box5T-160,NghiaDo,Hanoi,VietNam xRu ¯derBoškovi´cInstitute,Bijeniˇckacesta54,10000Zagreb,Croatia
yInstitutfürKernphysik,UniversitätzuKöln,D-50937Cologne,Germany zLPCCaen,ENSICAEN,UniversitédeCaen,CNRS/IN2P3,F-14050Caen,France
aaDepartmentofScienceEducationandDepartmentofPhysics,EwhaWomansUniversity,Seoul03760,RepublicofKorea abDepartmentofPhysics,TokyoInstituteofTechnology,2-12-1O-Okayama,Meguro,Tokyo,152-8551,Japan
acDepartmentofPhysics,UniversityofOslo,N-0316Oslo,Norway adMTAAtomki,P.O.Box51,DebrecenH-4001,Hungary
aeInstitutodeEstructuradelaMateria,CSIC,E-28006Madrid,Spain afInstituteofModernPhysics,ChineseAcademyofSciences,Lanzhou,PRChina
*
Correspondingauthorat:InstitutfürKernphysik,TechnischeUniversitätDarmstadt,D-64289Darmstadt,Germany.E-mailaddress:ysun@ikp.tu-darmstadt.de(Y.L. Sun).
https://doi.org/10.1016/j.physletb.2020.135215
0370-2693/©2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
a r t i c l e i n f o a b s t ra c t
Articlehistory:
Received9September2019
Receivedinrevisedform26December2019 Accepted10January2020
Availableonline16January2020 Editor:D.F.Geesaman
Keywords:
Spectroscopy Shellevolution 51K
53K
We report on the first γ-ray spectroscopy of 51,53K produced via the 52,54Ca(p,2p) reactions at
∼250 MeV/nucleon.Unambiguousfinal-state angular-momentumassignmentswereachievedforbeam intensities down to few particles per second by using a new technique based on reaction vertex tracking combined with a thick liquid-hydrogen target. Through γ-ray spectroscopy and exclusive parallelmomentumdistributionanalysis,3/2+groundstatesand1/2+firstexcitedstatesin51,53Kwere establishedquantifyingthenaturalorderingofthe1d3/2 and2s1/2proton-holestatesthatarerestored at N = 32 and 34. State-of-the-art abinitio calculations and shell-model calculations with improved phenomenologicaleffectiveinteractionsreproducethepresentdataandpredictconsistentlytheincrease oftheE(1/2+1)- E(3/2+1)energydifferencestowardsN =40.
©2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
The atomic nucleus is a strongly interacting many-body sys- temthat,however,showsfeaturesofindependentnucleonmotion nearthe Fermi surface [1]. The first predictive realization ofthe independent-particlemodelwasachievedbyMayerandJensen[2, 3], who established the conventional nuclear shell structure by introducing a strong spin-orbit coupling to explain the “magic numbers” andtheground-statespins ofstableodd-mass-number nuclei. In the modern shell-modelapproach, the variation ofthe nuclear structure is mainly driven by the monopole component oftheinteraction,which isresponsibleforthesingle-particle be- havior, and further modified by multiple correlations [4,5]. The correlation effects vanish in nuclei with a one-particle or one- hole configuration with respect to closed shells, resulting in the so-calledmonopoleenergyshiftalong thecorresponding isotopic orisotonic chains [6,7]. In thissituation thesingle-particle ener- gies ofthe odd nucleon are solely determined by the monopole interactionandchangeaslinearfunctionsofthenucleon number oftheothertype.
Recently, significant experimental and theoretical efforts have focused on the monopole shift of proton-hole states in the neutron-richodd-Kisotopes[8–14],whereZ =20remainsashell closureawayfromstability [15,16].From39K(N =20)to47K(N = 28),the energylevel splittingbetween the1/2+1 and3/2+1 states drops rapidly and the 1/2+1 state becomes lower in energy than the3/2+1 state in47K[17–19], whichis interpreted asan energy inversionof the
π
1d3/2 andπ
2s1/2 orbitalsatN =28 [9–11,18].The energysplitting decreases almost in proportionto the num- ber of valence neutrons in the f7/2 orbital, reflecting the robust feature of the monopole interaction [20]. For K isotopes beyond N = 28wherethevalenceneutrons startfillingthe p3/2 andp1/2 orbitals, data are scarce. The only known energy splitting is for 49K (N = 30), in whichthe 1/2+ ground state and the 3/2+1 ex- cited state become nearly degenerate [13,21]. Predictions from phenomenological shell-model calculations using different effec- tive interactionsreproduce welltheenergysplittings from39Kto 49Kbutdiffersignificantly atN =32and34[14].Aground-state spinI = 3/2 hasbeenrecentlyestablishedin 51K[13] using laser spectroscopy, indicating that the
π
1d3/2 andπ
2s1/2 orbitalsre- storenaturalorderatN = 32.However,theenergysplittingin51K was still unknown, since the 1/2+1 excitation energy in 51K has not been measured. For 53K and beyond, no spectroscopic infor- mation was known.Themeasurements towards 59K(N = 40)are essential to understand the development of collectivity towards thepotentially doublymagic nucleus60Ca discoverednewly [22].More recently,the protonstructure of Kand Ca isotopes beyond N =28hasattractedparticularinterestwithasteepraiseofcharge radiiupto51K [23] and52Ca [24],whichcanbeunderstoodasdue tothehalo-likenatureofthe
ν
2p3/2 orbital [25].InthisLetter,we reportonthefirstmeasurementoflow-lyingstatesin51,53Kpopu-latedbyone-protonremovalfromdoublymagicnuclei52,54Caand the unambiguous angular-momentum assignments of the popu- lated1/2+1 and3/2+1 states.
TheexperimentwasperformedattheRadioactiveIsotopeBeam Factory operatedby theRIKEN NishinaCenter andthe Centerfor Nuclear StudyoftheUniversityofTokyo.The radioactiveisotopes were producedby fragmentationofa345 MeV/nucleon70Znpri- mary beam on a Be target with an average beam intensity of 240 pnA. The secondary cocktail beam was identified using the magnetic rigidity (B
ρ
), energyloss (E)and time-of-flight (TOF) information in the BigRIPS two-stage fragment separator [26,27].Themeanintensitiesofthe52Caand54Ca secondarybeamswere 4.4and7.3particlespersecond,respectively.
The 52Ca and 54Ca secondary beams withaverage kinetic en- ergies of 266 and251 MeV/nucleon, respectively, impinged on a 151(1)-mm-thickliquidhydrogen(LH2)target [30] toinduceone- protonknockoutreactions. Theincidentangleandpositionofthe projectiles were determined by two multi-wire drift chambers (MWDCs). The heavy fragments were measured by the SAMU- RAI spectrometer [31] witha central magnetic field of2.7 Tesla.
Trajectories of the charged fragments were determined by two MWDCs located at the entrance andexit of the SAMURAI mag- net. The E and TOF of the fragments were measured by a 24-element 10-mm-thick plastic scintillator hodoscope. 51,53K from the 52,54Ca(p,2p) reactions could be clearly identified using theB
ρ
-E-TOFmethod [32] andwerefullywithintheacceptance ofthespectrometer.The LH2 target was surrounded by the MINOS time projec- tion chamber[33] totracktheoutgoingprotons forreactionver- tex reconstruction. The estimated vertex resolution was ∼4 mm (FWHM)[34]. Velocitiesoftheprojectilesandresiduesatthereac- tionvertexwerededucedbytakingintoaccounttheenergylossin the materialsalong their trajectories.The efficienciesofdetecting at leastone protonfor52Ca(p,2p)51Kand54Ca(p,2p)53K reactions weredeterminedtobe90(6)%and88(5)%,respectively.
De-excitation
γ
-raysfromthereactionresiduesweremeasured by the upgraded DALI2+ array [32,35], which consisted of 226 NaI(Tl)detectorswithanaveragethresholdsettoaround50 keV.The gainoftheNaI(Tl)detectorswas settomeasure
γ
-rayswith energies upto∼6MeVafterDopplercorrection.Thewholearray was calibratedwith60Co,137Cs, 88Y,and133Ba sources,yielding a calibration uncertaintyof4 keV. The measured energyresolution forthe662 keVγ
-raypeakof137Cswas9.1%(FWHM).Toincrease thefull energypeak efficiency,add-backanalysiswasadoptedforγ
-ray hits in detectors located within 12 cm of each other. For 1-MeVγ
-raysemittedfromnucleiwithavelocityof0.6c,thesim- ulated photopeak efficiency andenergy resolution were 30% and 11%(FWHM),respectively.TheDoppler-correctedγ
-rayspectraof 51K(53K) from one-protonremoval of52Ca(54Ca) are displayed inFig. 1. Doppler-correctedγ-rayspectraof51K(upperpanel)and53K(lowerpanel).
Theblacksolidlinesshowthefitresultsofthesimulatedresponsefunctionsforthe observedtransitions(reddottedlines)anddoubleexponentialbackgrounds(blue dashedlines).Theinsetsshowthespectraupto5MeV.Thededucedexperimen- tallevelschemesareshowntotherightofthespectra.Theexperimentalneutron separationenergies(Sn)aretakenfromRefs. [28,29].
Fig. 1. The spectra were fitted with GEANT4 [36] simulated re- sponse functions on top of a double-exponential background to determine the transition energies and intensities. For 51K, three transitions at 737(5), 1950(16), 2249(25) keV were clearly ob- served and one structure at 4281(47) keV was observed with a significancelevelof3.0standarddeviations(
σ
).Inthecaseof53K, onlyoneclearγ
-raytransitionat837(5) keVwasobserved.Allthe transitionswereattributedtodirectdecaystothegroundstatesas noγ
-γ
coincidenceswereobserved.Inthiswork,wefocusonthe firsttwolow-lyingstatesin51,53K,whoseextractedcrosssections arelistedinTable1.Thequoteduncertaintiesfortheexcitedstates include contributions from statistics, MINOS efficiency (7%) andγ
-ray detectionefficiency(5%).The reactionlossesof52,54Ca and 51,53K in the materials along their trajectories were determined bymeasuring thecorresponding unreacted beamandweretaken intoaccount in thecross-section deduction. The measured inclu- sive cross sections of 52Ca(p,2p)51K and 54Ca(p,2p)53K reactions were9.0(6) mband5.3(3) mb,respectively.Onlyone gamma-ray transitionwasobservedin53K.Whilethreeothertransitionswere observedin51Kinadditiontothetransitionfromits firstexcited state,amountingtoacrosssectionof2.1(3)mbtothecorrespond- ingexcitedstates.Theground-statepopulationcrosssectionswere deducedaftersubtractingtheexcited-statecontributionsfromtheinclusive yields assuming that there are no higherexcited states decayingbyemitting
γ
-raysbeyond5MeV,sincetheneutronsep- arationenergiesof51K(53K)is4.86MeV[28] (3.23MeV[29]).Themeasuredpartialcrosssectionsarecomparedtotheprod- uctsofthe single-particle crosssections(
σ
sp) obtainedusing the distorted-waveimpulse-approximation(DWIA) modelofRefs. [37, 38] andthe spectroscopic factors (SFs)obtainedusing eitherthe self-consistentGreen’sfunction(SCGF)approachinthethird-order algebraic diagrammatic construction [ADC(3)] approximation [39]with the chiral interaction N N+3N(lnl) [40] or large-scale shell- model(LSSM) calculationswithphenomenologicaleffectiveinter- actionsdiscussedlater inthetext. TheDWIAmodelhasbeenap- pliedtocalculate
σ
sp for(p,2p)reactionsinrecentworks[41–43].Here, the proton single-particle wave functionswere determined by a mean-field Hartree-Fock-Bogoliubov approachwith theSLy4 interaction[44].Theenergydependenceofthecrosssectionswas consideredbytakingthe
σ
sp averagedoverthebeamenergyalong thethick target.TheSCGFSFsaresystematicallysmallerthan the shell-modelSFs becauseof the particle-vibrationcouplingeffects that cannot be accounted even in LSSM valence spaces [45]. As shownin Table 1,the experimental cross sectionsto the ground states are 3.1(4) and 2.5(4) times of the cross sections to the first excited states for 51K and53K,respectively, while the theo- retical ratios of these two states are around 2.0 due to the fact thattheoreticalcalculationsoverestimatethecrosssectionsforthe 1/2+1 states. Nevertheless,the experimental ground-state popula- tioncrosssectionsarehigherthanthoseofthefirstexcitedstates, consistent withthe removalofprotonin1d3/2 and2s1/2 orbitals withoccupancyof2j+1=4and2,respectively.Inadditiontothis firstindication,thespin-parityassignmentsaredeterminedunam- biguouslyusingtheparallelmomentumdistribution(PMD) ofthe residualnuclei,whichlinksitsshapedirectlytotheorbitalangular momentumoftheknocked-outproton [46].Fig.2displays thePMDsof51,53Kfromthe52,54Ca(p,2p) reac- tionswhichwereobtainedbytransformingthemeasuredmomen- tum of the residues to the beam-at-rest frame to eliminate the momentumspreadoftheincidentbeam.Parallelmomentumres- olutions of
σ
= 37 MeV/c and 43 MeV/c were achieved for the 52Ca(p,2p)51K and54Ca(p,2p)53Kreactions, respectively, by taking into account the measured momentum resolution using the un- reacted beamand the momentum spreaddue to the vertexres- olution.Theangular-momentumassignments weredeterminedby comparingthemeasuredPMDstoDWIApredictions[37] assuming removal of the proton fromdifferent single-particle orbitals and folded with the experimental momentum resolutions. The PMDs ofthe 737-keV statein 51K andthe 837-keV state in53Kcanbe wellreproducedbythetheoreticalcurvesassumingremovalofan s-wave protonfrom 52Ca and54Ca, respectively.The assignments are strongly supported by the log10 scaled Bayes factors [47] of s-wave over d-wave which are 3 and7 for 51K and53K,respec-Table 1
Themeasuredexcitationenergies(Eexp),spin-parity(Jπ)andcrosssections(σexp)fromthe52,54Ca(p,2p)51,53Kreactions.Onlythe1/2+1 and3/2+1 statesarelisted.Theoretical excitationenergies(Eth),Jπ,andspectroscopicfactors(SFth)areobtainedfromabinitio SCGFcalculationswiththeN N+3N(lnl) interaction(Th.1)orshell-modelcalculations withtheSDPF-Umodinteraction(Th.2)ortheSDPF-MUsinteraction(Th.3).Theoreticalpartialcrosssections(σth)aretheproductsoftheSFthandthesingle-particlecross sections(σsp)calculatedusingtheDWIAmodel.Seetextfordetails.
Eexp(keV) Jπ σexp(mb) Eth(keV) Jπ SFth σsp(mb) σth(mb)
Th.1 Th.2 Th.3 Th.1 Th.2 Th.3 Th.1 Th.2 Th.3
51K 0 3/2+ 5.2(4) 0 0 0 3/2+ 2.76 3.65 3.53 1.73 4.77 6.31 6.11
737(5) 1/2+ 1.7(2) 883 747 846 1/2+ 1.27 1.68 1.59 1.88 2.39 3.16 2.99
53K 0 3/2+ 3.8(3) 0 0 0 3/2+ 2.90 3.80 3.71 1.49 4.32 5.66 5.53
837(5) 1/2+ 1.5(2) 717 560 794 1/2+ 1.36 1.74 1.66 1.64 2.23 2.85 2.72
Fig. 2. Individual parallelmomentumdistributionsofthe51,53Kresiduesfromthe52,54Ca(p,2p)reactionscomparedtotheDWIAcalculatedmomentumdistributionsassuming removalofa2s1/2or1d3/2proton.Theoreticalcurveshavebeenconvolutedwiththeexperimentalresolutionsandnormalizedtotheexperimentalcrosssections.Theright mostpanelshowsthemomentumdistributionof53Kwithoutvertexandthemomentumdistributionfromunreacted54Cabeam.
tively.1Thereby,thespinsandparitiesofthefirstexcitedstatesof 51,53Kareassignedas1/2+.Topindowntheangularmomentaof thegroundstates,wesubtractedthefirst-excited-statePMDsfrom theinclusivespectra.Theresultingdistributionsaredominatedby theground-state contribution(71% in51Kand100% in 53K), and canbewellreproducedbytheDWIAcalculationassumingremoval of d-wave protons. For the ground states the log10 scaled Bayes factorsofd-waveovers-waveare100and77for51Kand53K,re- spectively.Spinsandparitiesof3/2+ arethereforeestablishedfor thegroundstatesof51,53K.Noboundfp-shellstatesin51,53Kcould bepopulatedvia (p,2p)reactions basedontheabinitio SCGFcal- culations,whichisinlinewiththedoublymagicnatureof52,54Ca.
Notethat themomentumresolutionwithoutusing thevertexin- formationwouldnotbesufficienttodisentanglethes- andd-wave contributionsasillustratedintherightmostpanelofFig.2.Thisis thefirsttimethePMDsareextractedwithenoughresolvingpower todisentangleprotonremovalfromdifferentorbitalswhenusing thicktargets,pushingthesensitivityfrontiersforthespectroscopy ofveryexoticnucleiproducedatonlyfewparticlespersecond.
We now discuss the E(1/2+1) - E(3/2+1) systematics in odd-K isotopes. As displayed in Fig. 3, the reinversion of the 3/2+1 and 1/2+1 proton-holestates in51Kis confirmed andits continuation to 53K is established for the first time. We compare the experi- mental energy splittings to 0h¯
ω
LSSM calculations performed in thesd-pf valencespaceemployingtheSDPF-Uinteraction[9] and effectiveinteractionsbasedontheSDPF-MUHamiltonian[11].The shell-modelcalculationsshowexcellentagreementwiththeexper- imentalresultsfrom39Kto 49Kandstarttodiffersignificantly at 51K. The SDPF-U calculation underestimates the energy splitting by ∼0.3 MeV in 51K and by ∼0.7 MeV in 53K. To improve the SDPF-U interaction in the N > 30 region, theπ
1d3/2-ν
2p3/2p1/2 andπ
2s1/2-ν
2p3/2p1/2 monopolesneedtobetuned by+37.5 keV and-75 keV, respectively, leading to the SDPF-Umodinteraction.The predicted energy splittings in 51,53K using the SDPF-Umod interaction are much closer to the data, increasing by 277 keV and 410 keV, respectively, demonstrating a strong sensitivity to thesespecificproton-neutroncross-shellinteractions.Notethatthe modifiedSDPF-Uinteractioncanwell reproducethemeasured 2+1 systematics inCa isotopes andthe single particle spectra inN = 21 isotones [9]. Regarding the SDPF-MU Hamiltonian [11], it has been successfullyapplied to the presentmass region. It was de- rivedbasedon theVMU interactionwhichis suitablefora global descriptionasitssixparametersofGaussiancentralforcewerede- terminedfromG-matrix andeffectiveinteractions in thesd- and p f -shells[48]. TheSDPF-MUr[49] isarevisionmadefor49K[13, 21] and54Ca[16].51,53Kprovideapreciousopportunitytofurther tune certain two-bodymatrix elements beyondthe globaldeter-
1 Onehypothesishasdecisiveevidenceagainsttheotheroneiftheirlog10scaled Bayesfactorislargerthan2.
Fig. 3. Energy splitting betweenthe1/2+1 and 3/2+1 statesintheodd-Kisotopes comparedtoshell-model calculationsusingtheSDPF-U[9],the SDPF-Umod,the SDPF-MU[11],theSDPF-MUr[49],theSDPF-MUseffectiveinteractionsandtheab initio SCGF calculationemployingtheNNLOsat [54] andthe N N+3N(lnl) [40] in- teractions.Therectangularregionsfor51,53Krepresentestimatedtheoreticaluncer- taintiesduetomany-bodytruncations.ExperimentaldataaretakenfromRef. [21]
andthiswork.
mination. In order to describe the measured energysplittings, a modification is required to shift the original T = 0 central force monopole strength between
π
2s1/2 andν
2p1/2 (-1.27 MeV)and betweenπ
2s1/2 andν
2p3/2 (-1.29 MeV) by -0.2 MeV, meaning strengthening the attraction.In addition, theν
1f25/2 pairing ma- trix element isshifted by -0.4 MeV to better describe the2+ in Ca isotopes, which has no direct relevance to the present case.The derived SDPF-MUs interaction with the above modifications can reproduce theobserved splittings from39Kto 53K.The orig- inal SDPF-MU interaction produces the T = 0 monopole strength between
π
2s1/2 andν
2p1/2andthatbetweenπ
2s1/2 andν
2p3/2, bothlargerinmagnitudecomparedtoothercasesbecausethatthe relativewavefunctionscontainsmorecomponentswithlowerrel- ativeorbital angularmomentum,such astherelatives-wave,and theemployedGaussiancentralforceproducesaflatcontributionat shortdistances[48].Thisshort-rangebehavior oftheeffectiveNN interaction doesnot show up in mostother monopolestrengths.Such informationis aunique outcome ofthepresentexperiment andwillbe very usefulin obtainingmore insightoftheeffective NN interaction, including global ones like SDPF-MU’s. Note that the above-discussed modifications of interactions have negligible impact onthetotal bindingenergy,whichinthe shell-modelap- proachinvolvesallmonopoleinteractionsanddependsonthesum
oftheir absolutevalues,as illustrated in the parameterizationof theDuflo-Zukershell-modelmassformula[50,51].
Inthe shell-model calculations, the increasedenergy splitting from49Kto51,53Kisaconsequenceoftherestorationofasizeable Z =16 sub-shell gap in51Kand 53K, whichare calculated to be 0.95 MeVand1.38 MeV,respectively,whentheSDPF-MUsinterac- tionisemployed. Theneardegeneracyofthe
π
1d3/2 andπ
2s1/2 orbitalsaround N = 28has beensuggested toplay a key role in thedevelopment ofcollectivityinneutron-richsilicon,sulfur and argonisotopes [11,52,53].The restoration oftheZ = 16sub-shell gapwouldthereforehavethe profoundconsequenceofsuppress- ing thecollectivityinducedbyprotonexcitationsforZ =14,16,18 nucleilyingbeyondN =30.Indeed,anarrowedshellgapwillfree nucleons to participate in collective motions, while a large shell gapwillquenchcorrelationsbyenlargingtheenergycostforexci- tationsacrossthegaps.Theenergysplittings along theodd-Kisotopic chainalsopro- vide a testingground to validate the chiral effectivefield theory (EFT) interactions in the abinitio many-body calculations, which haveextendedtheir reachtoentiremedium-mass isotopicchains veryrecently.In particular,we comparetheexperimental datato theinitialADC(2)Gorkov-SCGFcalculations[55,56] performedina modelspaceofupto14harmonicoscillatorshellsincludingthree- nucleon(3N) interactions limitedto basis states withN1 + N2 + N3 ≤16whereN =2n +l.Weemployedtwosetsofstate-of-the- art EFT interactions: the NNLOsat [54] and the newly developed N N+3N(lnl) [40]. Uncertaintiesassociatedwiththe approximated many-bodyschemewereestimatedbydifferencesbetweenADC(2) and available ADC(3) results. With the most recent N N+3N(lnl) interaction, SCGF total binding energies in this mass region are within 3% (1%) from experiment when computed in the ADC(2) [ADC(3)] scheme [57]. As shown in Fig. 3, the NNLOsat calcula- tionssystematicallyunderestimatethesplittingsinodd-Kisotopes by ∼1 MeV, although the same interaction successfully repro- ducescharge radii, binding energies andspectroscopicproperties oflightermedium-massnuclei [54,58].Incontrast,theN N+3N(lnl) calculationsshow better overall agreement with the experimen- taldata,consistentwithitsapplicationonthemasspredictionof 48−56Ti[40].
Sofar,state-of-the-artshell-modelandabinitio calculationsre- produce experimental energy splittings at all the shell closures includingN =20,28,32,34.Theabinitio N N+3N(lnl) calculations predictthattheenergysplittingincreaseslinearlyfrom0.59 MeV in 53K to 1.35 MeV in 59K. The two modified phenomenological effectiveinteractions SDPF-Umod and SDPF-MUs anticipate simi- lartrendsasshowninFig.3,reaching1.88 MeVand1.38 MeVat 59K,respectively,equivalent tothecorresponding effectivesingle- particle energy differences between the
π
1d3/2 andπ
2s1/2 or- bitals.Since the valenceneutrons inthe shell-model calculations are restricted to pf -shells below N = 40, the consistent energy- splitting increase towards 59K, still to be proven experimentally, supportstheN = 40shellclosureassumedinallthe aboveshell- modelcalculations.Insummary,wehavereportedonthefirstmeasurementofthe low-lying states in 51,53K populated fromthe 52,54Ca(p,2p) reac- tions at ∼250 MeV/nucleon. We implemented a new technique based on reaction vertex tracking to achieve momentum resolu- tions of ∼40 MeV/c when using a thick liquid-hydrogen target.
The1/2+1 →3/2+1 transitionsof51,53Kweremeasuredforthefirst time and the spins-parities were unambiguously assigned based onthemeasured crosssectionsandparallel momentumdistribu- tions.The measured E(1/2+1) - E(3/21+) energysplittings in51,53K providea stringentconstraintonthe
π
2s1/21d3/2-ν
2p1/2p3/2 ma- trixelements. Therestoration ofthenaturalorderingofthe1/2+1 and3/2+1 proton-holestates in51,53Kisinterpreted asa restora-tionofasizeableZ =16sub-shellgapbeyondN =30,havingasa consequencethesuppressionofproton-inducedcollectivityinthe region.State-of-the-artshellmodelcalculationsandabinitio calcu- lationsall predictconsistently thecontinuationandenhancement oftherestorationtowardsN =40.
Acknowledgements
We are very gratefulto the RIKEN Nishina Center accelerator staff for providing the stable and high-intensity zinc beam and to the BigRIPS team for the smooth operation of the secondary beams. The development of MINOS has been supported by the European Research Council through the ERC Grant No. MINOS- 258567. Green’s function calculationswere performed using HPC resources from GENCI-TGCC, France (Projects A0030507392 and A0050507392) and from the DiRAC Data Intensive service at Leicester, UK (funded by the UK BEIS via STFC capital grants ST/K000373/1andST/R002363/1andSTFCDiRACOperationsgrant ST/R001014/1).Thiswork(C. B.)wasalsosupportedbytheUnited Kingdom Science and Technology Facilities Council (STFC) under GrantsNo.ST/P005314/1andNo.ST/L005816/1.K. O.acknowledges the support by Grant-in-Aid for Scientific Research JP16K05352.
Y. L. S. acknowledges the support ofMarie Skłodowska-Curie In- dividualFellowship(H2020-MSCA-IF-2015-705023)fromtheEuro- peanUnionandthesupportfromtheHelmholtzInternationalCen- terforFAIR.The valuablediscussionswithC.Qiaregratefullyac- knowledged.H. N. L.acknowledgesthesupportfromtheEnhanced Eurotalents program (PCOFUND-GA-2013-600382) co-funded by CEAandtheEuropean Union.H. N. L.andA.O.acknowledge the supportfromtheDeutscheForschungsgemeinschaft(DFG,German Research Foundation) - Project No. 279384907-SFB 1245. Y. L. S.
andA.O.acknowledgethesupportfromtheAlexandervon Hum- boldt Foundation. L. X. C. andB. D. L wouldlike to thankMOST for its support through the Physics Development Program Grant No. ÐTÐLCN.25/18. I.G. has been supported by HIC forFAIR and HRZZunderprojectNo. 1257and7194.K. I. H., D. K.andS. Y. P.
acknowledgethesupportfromtheNRFgrantfundedbytheKorea government (No. 2017R1A2B2012382 and 2019M7A1A1033186).
F. B.acknowledgethesupportfromtheRIKENSpecialPostdoctoral Researcher Program. D.S. was supported by projects No. GINOP- 2.3.3-15-2016-00034andNo.K128947.V. V.acknowledgessupport fromtheSpanishMinisteriodeEconomíayCompetitividadunder Contract No. FPA2017-84756-C4-2-P. V. W. acknowledges support fromBMBF grants 05P15RDFN1,05P19RDFN1 andDFG grant SFB 1245.P. K.acknowledgessupport fromHGS-HIReandBMBFgrant 05P19RDFN1.ThisworkwasalsosupportedbyNKFIH(128072).
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