Shell evolution of N=40 isotones towards 60 Ca: First spectroscopy of 62 Ti
M. L. Cortés, W. Rodriguez, P. Doornenbal, A. Obertelli, J. D. Holt, S. M. Lenzi, J. Menéndez, F. Nowacki, K. Ogata, A. Poves, T. R. Rodríguez, A. Schwenk, J. Simonis, S. R. Stroberg, K. Yoshida, L. Achouri, H. Baba, F. Browne, D. Calvet, F. Château, S. Chen, N. Chiga, A. Corsi, A. Delbart, J-M. Gheller, A. Giganon, A. Gillibert, C. Hilaire, T. Isobe, T. Kobayashi, Y. Kubota, V. Lapoux, H. N. Liu, T. Motobayashi, I. Murray, H. Otsu, V. Panin, N. Paul, H. Sakurai, M. Sasano, D. Steppenbeck, L. Stuhl, Y. L. Sun, Y. Togano, T. Uesaka, K. Wimmer, K. Yoneda, O. Aktas, T. Aumann, L. X. Chung, F. Flavigny, S. Franchoo, I. Gašparić, R.-B. Gerst, J. Gibelin, K. I. Hahn, D. Kim, T. Koiwai, Y. Kondo, P. Koseoglou, J. Lee, C. Lehr, B. D. Linh, T. Lokotko, M. MacCormick, K. Moschner, T. Nakamura, S. Y. Park, D. Rossi, E. Sahin, D. Sohler, P.-A. Söderström, S. Takeuchi, H. Toernqvist, V. Vaquero, V. Wagner, S. Wang, V. Werner, X. Xu, H. Yamada, D. Yan, Z. Yang, M. Yasuda, L. Zanetti
aa r X i v : . [ nu c l - e x ] D ec Shell evolution of N = 40 isotones towards Ca: First spectroscopy of Ti M. L. Cortés a,b, ∗ , W. Rodriguez c,a , P. Doornenbal a , A. Obertelli d,e , J. D. Holt f , S. M. Lenzi g , J. Menéndez h ,F. Nowacki i , K. Ogata j,k , A. Poves l , T. R. Rodríguez l , A. Schwenk e,m,n , J. Simonis o , S. R. Stroberg f,p , K. Yoshida q ,L. Achouri r , H. Baba a , F. Browne a , D. Calvet d , F. Château d , S. Chen s,a , N. Chiga a , A. Corsi d , A. Delbart d ,J-M. Gheller d , A. Giganon d , A. Gillibert d , C. Hilaire d , T. Isobe a , T. Kobayashi t , Y. Kubota a,h , V. Lapoux d ,H. N. Liu d,e,u , T. Motobayashi a , I. Murray v,a , H. Otsu a , V. Panin a , N. Paul d , H. Sakurai a,w , M. Sasano a ,D. Steppenbeck a , L. Stuhl h , Y. L. Sun d,e , Y. Togano x , T. Uesaka a , K. Wimmer w , K. Yoneda a , O. Aktas u ,T. Aumann e,y , L. X. Chung z , F. Flavigny v , S. Franchoo v , I. Gašparić aa,a , R.-B. Gerst ab , J. Gibelin r , K. I. Hahn ac ,D. Kim ac , T. Koiwai w , Y. Kondo ad , P. Koseoglou e,y , J. Lee ae , C. Lehr e , B. D. Linh z , T. Lokotko ae , M. MacCormick v ,K. Moschner ab , T. Nakamura ad , S. Y. Park ac , D. Rossi e , E. Sahin af , D. Sohler ag , P.-A. Söderström e , S. Takeuchi ad ,H. Toernqvist e,y , V. Vaquero ah , V. Wagner e , S. Wang ai , V. Werner e , X. Xu ae , H. Yamada ad , D. Yan ai , Z. Yang a ,M. Yasuda ad , L. Zanetti e a RIKEN Nishina Center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan b Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Legnaro, I-35020 Legnaro, Italy c Universidad Nacional de Colombia, Sede Bogota, Facultad de Ciencias,Departamento de Física, Bogotá, 111321, Colombia d IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France e Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany f TRIUMF, 4004 Wesbrook Mall, Vancouver BC V6T 2A3, Canada g Dipartimento di Fisica e Astronomia, Università di Padova and INFN, Sezione di Padova, Via F. Marzolo 8, I-35131 Padova, Italy h Center for Nuclear Study, The University of Tokyo, RIKEN campus, Wako, Saitama 351-0198, Japan i IPHC, CNRS/IN2P3, Université de Strasbourg, F-67037 Strasbourg, France j Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan k Department of Physics, Osaka City University, Osaka 558-8585, Japan l Departamento de Física Teórica and IFT-UAM/CSIC, Universidad Autónoma de Madrid, E-2804 Madrid, Spain m ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany n Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg Germany o Institut für Kernphysik and PRISMA Cluster of Excellence, Johannes Gutenberg-Universität, Mainz 55099, Germany p Department of Physics, University of Washington, Seattle WA, USA q Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan r LPC Caen, ENSICAEN, UniversitÃľ de Caen, CNRS/IN2P3, F-14050 Caen, France s State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, P.R. China t Department of Physics, Tohoku University, Sendai 980-8578, Japan u Department of Physics, Royal Institute of Technology, SE-10691 Stockholm, Sweden v IPN Orsay, CNRS and Université Paris Saclay, F-91406 Orsay Cedex, France w Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan x Department of Physics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 172-8501, Japan y GSI Helmoltzzentrum für Schwerionenforschung GmbH, Planckstr. 1, 64291 Darmstadt, Germany z Institute for Nuclear Science & Technology, VINATOM, P.O. Box 5T-160, Nghia Do, Hanoi, Vietnam aa Ruđer Bošković Institute, Bijenička cesta 54,10000 Zagreb, Croatia ab Institut für Kernphysik, Universität zu Köln, D-50937 Cologne, Germany ac Department of Science Education and Department of Physics, Ewha Womans University, Seoul 03760, Korea ad Department of Physics, Tokyo Institute of Technology, 2-12-1 O-Okayama, Meguro, Tokyo, 152-8551, Japan ae Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong af Department of Physics, University of Oslo, N-0316 Oslo, Norway ag Institute for Nuclear Research of the Hungarian Academy of Sciences (MTA Atomki), P.O. Box 51, Debrecen H-4001, Hungary ah Instituto de Estructura de la Materia, CSIC, E-28006 Madrid, Spain ai Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, China
Abstract
Excited states in the N = 40 isotone Ti were populated via the V ( p, p ) Ti reaction at ∼
200 MeV/nucleon at theRadioactive Isotope Beam Factory and studied using γ -ray spectroscopy. The energies of the +1 → +gs and +1 → +1 transitions, observed here for the first time, indicate a deformed Ti ground state. These energies are increasedcompared to the neighboring Cr and Fe isotones, suggesting a small decrease of quadrupole collectivity. The presentmeasurement is well reproduced by large-scale shell-model calculations based on effective interactions, while ab initioand beyond mean-field calculations do not yet reproduce our findings. The shell-model calculations for Ti show adominant configuration with four neutrons excited across the N = 40 gap. Likewise, they indicate that the N = 40 island of inversion extends down to Z = 20 , disfavoring a possible doubly magic character of the elusive Ca.
Keywords:
Shell evolution, Radioactive beams, Gamma-ray spectroscopyur understanding of atomic nuclei largely derives fromthe concept of nuclear shell structure. Within this picture,the arrangement of nucleons inside the nucleus can be ex-plained by the filling of discrete energy levels. Sizable gapsbetween these orbits disfavor the population of the higher-energy levels, and are interpreted as closed shells, whichgive rise to magic numbers. Such shell closures can be ev-idenced by a relatively high-lying first excited 2 + state, arelatively small electric quadrupole transition probabilityto the ground state, B ( E ↓ , and a steep decrease of theseparation energy. Experimental evidence collected in thelast decades, particularly since the advent of radioactiveion beams, has shown that shell structure undergoes signif-icant changes for isotopes far from stability [1]. Examplesof these changes are the appearance of new magic neutronnumbers at N = 32 , in the Ca isotopes and neighboringisotopic chains [2, 3, 4, 5, 6, 7, 8, 9], and at N = 16 forO isotopes [10, 11, 12], as well as the disappearance of theshell closure at N = 8 [13, 14, 15, 16], N = 20 [17, 18] and N = 28 [19, 20] in various neutron-rich isotopes.Given that N = 40 , which corresponds to the fillingof the neutron pf shells, is a harmonic oscillator magicnumber, the study of the structure of N = 40 isotones canprovide insight into the mechanisms governing shell evolu-tion. Indeed the characteristics of this isotonic chain varywith the number of protons. For Ni ( Z = 28 ), a high E (2 +1 ) energy and a low B ( E ↓ have been observed [21].However, due to the parity change between the pf shelland the g / orbit, the +1 state involves at least two neu-trons across N = 40 . Such a neutron-dominated excita-tion could result in a large E (2 +1 ) energy and low B ( E ↓ value without a large shell gap [22]. For the neutron-richFe ( Z = 26 ) and Cr ( Z = 24 ) isotopes, a monotonous de-crease of the E (2 +1 ) when approaching N = 40 and beyondhas been observed [23, 24, 25, 26]. This decrease indicatesa rapid development of collectivity when removing pro-tons from the f / shell. In contrast, the measurement ofthe E (2 +1 ) of Ti ( Z = 22 ) only showed a moderatedecrease towards N = 40 [27, 28]. The very exotic Ca( Z = 20 ), where the Ca isotopic chain meets the N = 40 isotones, is a key nucleus for shell evolution [29, 30], butdifficult to reach experimentally. Only recently its ex-istence has been established [31], supporting theoreticalpredictions for a bound Ca. However, the heaviest Caisotope with known spectroscopic information is Ca [4].Theoretical calculations in the shell-model framework [32]concluded that the development of collectivity in N = 40 nuclei is due to quadrupole correlations that give rise todeformed ground states, dominated by intruder neutronorbits beyond the pf shell. This leads to an island ofinversion below Ni, similar to the one formed around Mg [32]. These calculations predict an increase in the E (2 +1 ) energy of the more exotic N = 40 isotones Tiand Ca, while conserving the intruder character in the ∗ Corresponding author
Email address: [email protected] (M. L. Cortés) ground state. On the other hand, symmetry conservingconfiguration mixing calculations with the Gogny interac-tion predict a conservation of the N = 40 gap [33]. Theseresults agree with calculations performed using the five-dimension collective Hamiltonian, which suggest an energygap of about 4 MeV at N = 40 , predicting spherical Tiand Ca [34, 35]. It is noted that the beyond-mean-fieldand the shell model calculations provide similar results for Cr and Fe, while they substantially diverge for Caand Ti. Therefore, spectroscopy of Ti offers a crucialtest between the two different pictures. In addition, theproperties of Ca isotopes have been extensively studiedwith coupled-cluster theory [36] and valence-shell interac-tions [3, 37], in both cases using two-nucleon (NN) andthree-nucleon (3N) interactions from chiral effective fieldtheory. Such calculations agree well with experimental en-ergy levels and binding energies up to Ca, and predictthe drip line to be located around Ca. This is in con-trast to density functional theories based on the mean fieldapproach which predict, depending on the selected interac-tion, Ca isotopes to be bound up to A = 68 − . Beyond N = 40 , coupled-cluster theory suggests the existence oftwo-neutron halos and Efimov states in Ca [38].Clearly, spectroscopic information on exotic isotopesaround Ca is necessary to deepen our understanding ofthe nuclear structure at N = 40 and to benchmark thetheoretical predictions towards the neutron drip line. Inthe present work, the first spectroscopy of Ti is pre-sented. This isotope represents the closest nucleus to Cafor which spectroscopic studies can be performed at exist-ing radioactive beam facilities.The experiment was carried out at the Radioactive Iso-tope Beam Factory, operated by the RIKEN Nishina Cen-ter and the Center for Nuclear Study of the Universityof Tokyo. A primary beam of Zn with an energy of345 MeV/nucleon and an average intensity of 240 pnA
Mass-to-Charge ratio A t o m i c nu m b e r Ti Figure 1: Particle identification plot for the outgoing fragments mea-sured with the SAMURAI dipole magnet and related detectors. In-coming V isotopes were selected with BigRIPS. Ti isotopes areindicated by the ellipse. V. The frag-ments of interest were selected with the Bρ − ∆ E − Bρ technique using two wedge-shaped aluminium degraderssituated at the dispersive focal planes of BigRIPS [39].Event-by-event identification was performed by an energyloss measurement in an ionization chamber, position andangle measurements in parallel plate avalanche countersat different focal planes, and the time-of-flight measuredbetween two plastic scintillators. The V isotopes weredelivered to the focus area in front of the SAMURAI dipolemagnet [40], with an average intensity of 3 pps and an av-erage energy of 239 MeV/nucleon. At this location the MI-NOS device [41], composed of a 151.3(13) mm long liquidhydrogen target surrounded by a Time Projection Cham-ber (TPC), was placed. The efficiency of MINOS to detectat least one proton was measured as 93(4)% and the res-olution for the vertex reconstruction was estimated to bebetter than 2 mm ( σ ) [42]. Following proton knockout re-actions in the liquid hydrogen target, the Ti fragmentshad an average energy of 154 MeV/nucleon and were iden-tified using the SAMURAI dipole magnet and associateddetectors [40]. Figure 1 shows the particle identificationobtained with SAMURAI when selecting V as incom-ing beam. A total of 1880 events corresponding to the V( p, p ) Ti reaction was reconstructed. The transmis-sion of the unreacted V beam along the beam line wasmeasured to be 50.9(11)% and the inclusive ( p, p ) crosssection was determined to be 4.0(1) mb.MINOS was surrounded by the high-efficiency γ -raydetector array DALI2 + , composed of 226 NaI(Tl) detec-tors covering angles between ∼ ◦ and ∼ ◦ with respectto the center of the target [43, 44]. The array was en-ergy calibrated using standard Co, Y, Ba, and
Cssources. The full-energy-peak efficiency of the array wasdetermined using a detailed GEANT4 [45] simulation andwas found to be 30% at 1 MeV with an energy resolutionof 11% for a source moving at 0.6c.Doppler corrected γ -ray spectra were obtained usingthe reaction vertex and the velocity of the fragment recon-structed with MINOS. Peak-to-total ratio and detectionefficiency improved by adding-up the energies of γ -raysdeposited in detectors up to 10 cm apart. To avoid thereconstruction of add-back events from the large atomicbackground, γ -rays with energies below 100 keV were nottaken into account in the analysis. The Doppler correctedspectrum obtained for the V( p, p ) Ti reaction is dis-played in Fig. 2a). Two peaks are clearly visible and the γ − γ coincidence analysis demonstrates their coincidence(Fig. 2b). Using a 2-dimensional χ minimization, the en-ergies of the transitions were deduced to be 683(10) keVand 823(20) keV. In this minimization procedure, the sim-ulated response of DALI2 + to transitions of different ener-gies were fitted in steps of 5 keV to the experimental dataand the χ value was obtained for each combination ofenergies. The simulation included the experimental reso-lution of each crystal and a double exponential background Energy (keV)500 1000 1500 2000 C oun t s ( k e V / b i n ) Energy (keV)
500 1000 1500 C oun t s ( k e V / b i n ) a) b) Figure 2: a) Doppler corrected γ -ray spectrum of Ti obtainedfrom proton knockout from V. The spectrum was fitted by theconvolution of the simulated response of DALI2 + to the observedtransitions and a double exponential background. Two additionaltransitions are included to improve the fit (see text for details). b)Coincidence spectrum obtained when applying the gate indicated bythe blue area. was assumed for the fit. The parameters of these exponen-tial functions were chosen based on a consistent analysis ofthe spectra of proton knockout reactions producing Arand Ti. The errors on the transition energies include thestatistical error from the fit, as well as the systematic errorarising from the calibration of the γ -ray detectors and thepossible lifetime of the states. Given that global system-atic fits [46] suggest a lifetime of the +1 state below 30 ps,an uncertainty of ± ps was considered for the decayof the +1 , while the +1 was considered short lived. Thebest total fit as well as the individual response functions ofDALI2 + are shown in Fig. 2. The relative intensities of thepeaks suggest the tentative assignment of the 683(10) keVand the 823(20) keV peaks to the +1 → +gs and +1 → +1 transitions, respectively.A structure in the γ -ray spectrum above the estimatedbackground was observed between 1000 and 1500 keV.Two additional transitions at energies of 1222(37) keV and1328(45) keV, were used to reproduce this structure. Thesignificance levels of these peaks are σ and σ , respec-tively. The inclusion of more transitions did not provideany further improvement on the χ of the fit. A structureat 320 keV was observed with a significance level of 1 σ .The existence of this peak could not be firmly established,therefore it was not considered, and its possible contribu-tion to the partial cross section was assumed to be withinthe error bars of the analysis. These possible transitionsindicate the presence of different states being populated inthe reaction, but the limited resolution of DALI2 + and thelow statistics did not allow to identify them nor to performa coincidence analysis. The existence of such transitions,which potentially feed the +1 or +1 states, implies a frag-3 .51.01.52.02.53.03.54.0 20 22 24 26 28 30 32 N= E (2 ) +1 E (4 ) +1 E (2 ): +1 E (4 ): +1 E n e r gy ( M e V ) Atomic number
LiteratureThis workLSSMSCCMVS-IMSRG LiteratureThis workLSSMSCCMVS-IMSRG
Figure 3: Systematics of E (2 +1 ) (filled symbols) and E (4 +1 ) (opensymbols) for even-even N = 40 isotones. The circles representthe present measurement. The black, blue, and red lines representLSSM, SCCM, and VS-IMSRG calculations, respectively (see textfor details). mented spectroscopic strength.Exclusive cross sections to populate the ( +1 ) and ( +1 )states, from which additional feeding should be subtracted,were calculated based on the fitted γ -ray intensities, thetotal transmission of the isotopes and the efficiency of MI-NOS. Cross sections of 1.5(3) mb and 0.8(1) mb were ob-tained for the ( +1 ) state and the ( +1 ) state, respectively.The cross sections measured for the possible transitionsat 1222(37) keV and 1328(45) keV were determined to be0.2(1) mb and 0.3(1) mb, respectively. As no firm state-ment can be made regarding these transitions, we limitthe interpretation to their possible direct feeding to the +1 state. For this, the average value between 100% feed-ing and no feeding was considered and the error increasedto cover both possibilities, giving a exclusive cross sectionof 1.3(4) mb for the ( +1 ) state.The evolution of measured E (2 +1 ) and E (4 +1 ) energiesfor the N = 40 isotones between Ti and Ge [47] is pre-sented in Fig. 3. The E (2 +1 ) and E (4 +1 ) reported in thisLetter for Ti have a similar value than the ones measuredfor Fe, higher than those of Cr. It is pointed out that Cr, with a E (2 +1 ) of 420 keV, has the largest quadrupoledeformation observed in the region [26, 48]. Our resultsshow the first increase of E (2 +1 ) along the N = 40 iso-tones towards Ca. This increase establishes a parabolictrend and suggests a decrease in quadrupole collectivity.This, in turn, could be interpreted as a sign of a signifi-cant N = 40 shell gap, and gives the possibility of a doublymagic character for Ca.Large Scale Shell Model (LSSM) calculations, shownby the black lines in Fig. 3, were carried out with theLNPS interaction [32] using a Ca core and a valencespace which included the full pf shell for protons and the f / , p / , p / , g / , and d / orbits for neutrons.This interaction has already successfully reproduced the E (2 +1 ) of the heavier N = 40 isotones [32]. The LSSMcalculations reproduce very accurately the data for boththe E (2 +1 ) and E (4 +1 ) of the N = 40 isotones includingour values for Ti. This agreement strengthens the tenta-tive spin and parity assignment for these states. As shownin Ref. [32], the calculations predict a reduction of the f / − g / gap when going from Ni to Ca, as well asthe closeness of the quadrupole partner orbits g / and d / . Due to this proximity, quadrupole correlations pro-duce a gain in energy that largely overcomes the cost ofexciting neutrons across the N = 40 gap, thereby favor-ing many-particle-many-hole configurations. This situa-tion resembles the behavior at N = 20 and suggests anisland of inversion for N = 40 isotones below Ni. For Ti, a gap of about 1 MeV is predicted, with a resultingwave function dominated by 4p-4h excitations (63%) anda significant 6p-6h component (22%) [32]. Furthermore,a ground-state deformation parameter β = 0 . for Tiis obtained. The agreement with the measured energiesof the N = 40 isotones, including Ti, indicates that theisland of inversion in this region extends down to Ca.It is particularly remarkable that although the E (2 +1 ) for Ca is predicted to be 1.35 MeV, which represents an in-crease with respect to the neighboring isotones, a 4p-4hconfiguration dominance (59%) prevails [32].Symmetry conserving configuration mixing (SCCM) cal-culations using the Gogny D1S effective interaction [49, 50]were performed for Ti, Cr, and Fe, and are indicatedby the blue lines in Fig. 3. For the calculations, each in-dividual nuclear state was defined as the linear combina-tion of multiple intrinsic many-body states with differentquadrupole (axial and triaxial) shapes [51, 33]. Crankedor octupole deformed states were not included, therefore,a systematic stretching of the levels with respect to theexperimental values is expected [52, 53]. The E (2 +1 ) pre-dicted for Cr and Fe lie very close to the LSSM pre-dictions, and are in fair agreement with the experimentaldata. However, when going to Ti, a more abrupt in-crease of the E (2 +1 ) is obtained. For the E (4 +1 ) energies,the calculations overestimate the experimental values byabout 500 keV, although the minimum value for Cr ismaintained. It is noted that for Cr and Fe, where thedeformation is well described by the model, the inclusionof cranking would further improve the agreement with theexperimental data. Within this model, the energy gap at N = 40 is conserved, leading to a ground state of Tihighly mixed with the spherical configuration. This is alsothe case for Ca, which is predicted as a doubly magicnucleus with an E (2 +1 ) of 4.73 MeV [53]. It is noted thatalthough this calculation yields a spherical ground statefor Ti, the +1 and +1 states belong to a deformed bandstarting at the +2 state. This band can correspond to thepredictions of the LSSM calculations and indicate that theSCCM calculations overestimate the N = 40 gap in thisregion.Ab initio valence-space in-medium similarity renormal-ization group (VS-IMSRG) [54, 55, 56, 57, 58] calculations4ere also performed for Ti, Cr, and Fe, as shownby the red lines in Fig. 3. The chiral NN+3N interactionlabeled 1.8/2.0 (EM) in Refs. [59, 60] was used, which isbased on the NN potential from Ref. [61] and 3N forcesfitted to light systems up to He only. With this NN+3Ninteraction, ground-state energies up to Sn [58, 59, 62, 63]are generally well reproduced. As the VS-IMSRG cap-tures 3N forces between valence nucleons via an ensemblenormal ordering [57], a separate valence-space interac-tion is decoupled for each nucleus of interest. Here, thesame model space as the LNPS Hamiltonian is considered(adding the s / neutron orbital for Ti). Using theMagnus formulation of the IMSRG [64], operators at thetwo-body level are truncated in the so-called IMSRG(2)approximation. The VS-IMSRG interaction is diagonal-ized with the code ANTOINE [65], including, for the firsttime in the VS-IMSRG, both intruder quadrupole part-ners, such as g / – d / [66]. The VS-IMSRG overesti-mates the E (2 +1 ) and E (4 +1 ) excitation energies in Ti, Cr, and Fe, predicting all states as spherical. Cross-shell excitations to the g / – d / orbits stay at the 1p-1hlevel because of the substantial N = 40 shell gap, 3.7 MeVin Ti. Within this model, a E (2 +1 ) of around 7 MeVis predicted for Ca, an overestimation which is also ob-served at other shell closures with the VS-IMSRG [59, 63,67]. This limitation has been related to the IMSRG(2)truncation [66], which may not fully capture correlationsassociated with cross-shell excitations. Preliminary com-parisons with coupled-cluster theory indicate that keepingoperators at the three-body level will improve the results.Also, choosing a deformed reference state, instead of spher-ical as in the present work, may capture quadrupole cor-relations more efficiently [68, 69].Single-particle theoretical cross sections were computedin the DWIA framework [70]. The single-particle wavefunctions and the nuclear density were obtained by theBohr-Mottelson single-particle potential [71]. The opti-cal potentials for the distorted waves in the initial andfinal channels were constructed by the microscopic foldingmodel [72] with the Melbourne G-matrix interaction [73]and with the calculated nuclear density. The spin-orbitpart of each distorting potential was disregarded. As forthe transition interaction, the Franey-Love effective proton-proton interaction was adopted [74]. Cross sections at dif-ferent beam energies, from 240 MeV/nucleon at the en-trance of the target to 154 MeV/nucleon at the exit, werecalculated and weighted according to the energy loss in thetarget. Theoretical cross sections ( σ theo ) were obtained byweighting the single particle cross sections by the calcu-lated spectroscopic factors.The spin and parity of the ground state of V arenot known experimentally. The LSSM calculation sug-gests it to be / − , although states with spin and parityof / − and / − appear very close in energy, suggestingthe presence of isomeric states. No experimental evidenceof such states has been reported so far and available data + +1 +1 +2 +3 +1 +2 +3 C r o ss s e c t i on ( m b ) Energy (keV)Theory. J π = 7/2 - + +1 +1 +2 +3 +1 +3 C r o ss s e c t i on ( m b ) Theory. J π = 5/2 - + +1 +1 +2 C r o ss s e c t i on ( m b ) Theory. J π = 3/2 - + (2 +1 ) (4 +1 ) C r o ss s e c t i on ( m b ) Experiment
Figure 4: Partial proton removal cross sections for the V ( p, p ) Tireaction. Panel a) shows the experimental results. Panels b) to d)show LSSM calculations using the LNPS interaction assuming theground state of V as / − , / − and / − , respectively. are consistent with a / − assignment [75]. Results ofthe calculations for the three cases are shown in Table 1,and displayed in Fig. 4, together with the experimentalresults. It can be seen that neither the absolute value orthe general trend shown by the data are reproduced by thecalculation in any scenario. The calculation for the groundstate of J π = 3 / − resembles better the experimental datain terms of the number of states that are populated, whilefor the cases of J π = 5 / − and J π = 7 / − a considerablepopulation of the +1 state would be expected. In particu-lar for the case of J π = 7 / − a population of the +1 statehigher than the one of the +1 state would be expected,at odds with the experimental result. It is noted that thecalculated spectroscopic factors add up to less than half ofthe total strength in the three cases. Therefore, popula-tion of higher lying states is expected by the calculations.Such a scenario would lead to unobserved transitions feed-ing the +1 or the +1 states directly, which can accountfor the excess of the measured cross section in comparisonwith the calculations. Although not in good agreement,the low measured and calculated partial cross sections, aswell as the apparent fragmentation of the spectroscopic5 able 1: Experimentally deduced excitation energies and cross sections for Ti following the V ( p, p ) Ti reaction, and comparison withtheoretical cross sections obtained with the LSSM calculation. The spectroscopic factors and corresponding cross sections are shown forthe three possible values of the spin and parity of the ground state of V. The experimental ground-state cross section was calculated bysubtracting the cross sections of the measured transitions from the inclusive cross section. E (keV) σ exp (mb) E (keV) J π l j σ s.p (mb) J π = 3 / − J π = 5 / − J π = 7 / − C S σ theo (mb) C S σ theo (mb) C S σ theo (mb)0 1.4(4) 0 0 +1 p / f / +1 p / f / +1 p / f / Ti ground state discussed in this work. However, thelarge error bars prevent a firmer conclusion.In summary, first spectroscopy of Ti was obtained bymeans of the V ( p, p ) Ti reaction at ∼
200 MeV/nucleon.Transitions at 683(10) keV and 823(20) keV were assignedto the decay of the +1 and +1 states at 683(10) keV and1506(22) keV, respectively. Our result shows for the firsttime an increase of the E (2 +1 ) for N = 40 isotones to-wards Ca. LSSM calculations were in good agreementwith the experimental findings. The calculations suggestthat although the collectivity decreases approaching Ca,with an ensuing increase of E (2 +1 ) , quadrupole correla-tion contributions remain and lead to the extension of the N = 40 island of inversion down to Ca. SCCM cal-culations overestimate the measured E (2 +1 ) and E (4 +1 ) of Ti, predicting a doubly magic character of Ca and aweakly deformed ground state in Ti, at variance withthe LSSM calculations. For these calculations the N = 40 spherical gap is too large to produce the inversion betweenthe quasi-spherical and deformed + states. VS-IMSRGcalculations, which provide a good description of excitedstates in Ca isotopes, largely overestimate the E (2 +1 ) and E (4 +1 ) energies of Ti, even after the inclusion of the neu-tron g / , d / and s / orbitals. The spectroscopicinformation presented in this Letter offers an importantbenchmark for our understanding of nuclear structure ap-proaching Ca and the location of the neutron drip line.We thank the RIKEN Nishina Center accelerator staffand the BigRIPS team for the stable operation of the high-intensity Zn beam and for the preparation of the secondarybeam setting. K.O. acknowledges the support by Grant-in-Aid for Scientific Research JP16K05352. A.P. is sup-ported in part by the Ministerio de Ciencia, Innovaciony Universidades (Spain), Severo Ochoa Programme SEV-2016-0597 and grant PGC-2018-94583. F.B. is supportedby the RIKEN Special Postdoctoral Researcher Program.L.X.C. and B.D.L would like to thank MOST for its sup-port through the Physics Development Program GrantNo.ÐTÐLCN.25/18. I.G. has been supported by HIC forFAIR and Croatian Science Foundation under projects no. 1257 and 7194. D.So. was supported by projects No.GINOP-2.3.3-15-2016-00034 and No. K128947. V.V. ac-knowledges support from the Spanish Ministerio de EconomÃŋay Competitividad under Contract No. FPA2017-84756-C4-2-P. K.I.H., D.K. and S.Y.P. acknowledge the supportfrom the NRF grant funded by the Korea government(No. 2018R1A5A1025563 and 2019M7A1A1033186). Thedevelopment of MINOS was supported by the EuropeanResearch Council through the ERC Grant No. MINOS-258567. This work was also supported by the NKFIH(128072), the JSPS KAKENHI Grant No. 18K03639, MEXTas âĂIJPriority issue on post-K computerâĂİ (Elucida-tion of the fundamental laws and evolution of the uni-verse), JICFuS, the CNS-RIKEN joint project for large-scale nuclear structure calculations, NSERC, the DeutscheForschungsgemeinschaft – Projektnummer 279384907 – SFB1245, the PRISMA Cluster of Excellence, and the BMBFunder Contracts No. 05P18RDFN1 and 05P19RDFN1. TRI-UMF receives funding via a contribution through the Na-tional Research Council Canada. Computations were per-formed at the Jülich Supercomputing Center (JURECA).
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