Masses of neutron-rich 52-54 Sc and 54,56 Ti nuclides: The N=32 subshell closure in scandium
X. Xu, M. Wang, K. Blaum, J. D. Holt, Yu. A. Litvinov, A. Schwenk, J. Simonis, S. R. Stroberg, Y. H. Zhang, H. S. Xu, P. Shuai, X. L. Tu, X. H. Zhou, F. R. Xu, G. Audi, R. J. Chen, X. C. Chen, C. Y. Fu, Z. Ge, W. J. Huang, S. Litvinov, D. W. Liu, Y. H. Lam, X. W. Ma, R. S. Mao, A. Ozawa, B. H. Sun, Y. Sun, T. Uesaka, G. Q. Xiao, Y. M. Xing, T. Yamaguchi, Y. Yamaguchi, X. L. Yan, Q. Zeng, H. W. Zhao, T. C. Zhao, W. Zhang, W. L. Zhan
aa r X i v : . [ nu c l - e x ] M a y Masses of neutron-rich
Sc and , Ti nuclides: The N = subshell closure in scandium X. Xu,
1, 2
M. Wang, ∗ K. Blaum, J. D. Holt, Yu. A. Litvinov,
A. Schwenk,
6, 7, 3
J. Simonis,
8, 6, 7
S. R. Stroberg,
Y. H. Zhang,
H. S. Xu, P. Shuai, X. L. Tu, X. H. Zhou, F. R. Xu, G. Audi, R. J. Chen, X. C. Chen, C. Y. Fu, Z. Ge, W. J. Huang, S. Litvinov, D. W. Liu, Y. H. Lam, X. W. Ma, R. S. Mao, A. Ozawa, B. H. Sun, Y. Sun, T. Uesaka, G. Q. Xiao, Y. M. Xing, T. Yamaguchi, Y. Yamaguchi, X. L. Yan, Q. Zeng, H. W. Zhao, T. C. Zhao, W. Zhang, and W. L. Zhan Key Laboratory of High Precision Nuclear Spectroscopy and Center for Nuclear Matter Science,Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, People’s Republic of China School of Science, Xi ′ an Jiaotong University, Xi ′ an, 710049, China Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany Institut für Kernphysik and PRISMA Cluster of Excellence,Johannes Gutenberg-Universität, 55099 Mainz, Germany Reed College, Portland, 97202 Oregon, USA State Key Laboratory of Nuclear Physics and Technology, School of Physics,Peking University, Beijing 100871, People’s Republic of China CSNSM-IN2P3-CNRS, Université de Paris Sud, 91405 Orsay, France Insititute of Physics, University of Tsukuba, Ibaraki 305-8571, Japan School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, People’s Republic of China Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China RIKEN Nishina Center, RIKEN, Saitama 351-0198, Japan Department of Physics, Saitama University, Saitama 338-8570, Japan (Dated: May 30, 2019)Isochronous mass spectrometry has been applied in the storage ring CSRe to measure the masses of theneutron-rich
Sc and , Ti nuclei. The new mass excess values ME ( Sc) = − ( ) keV, ME ( Sc) = − ( ) keV, and ME ( Sc) = − ( ) keV, deviate from the Atomic Mass Evaluation 2012 by2.3 σ , 2.8 σ , and 1.7 σ , respectively. These large deviations significantly change the systematics of the two-neutron separation energies of scandium isotopes. The empirical shell gap extracted from our new experimentalresults shows a significant subshell closure at N =
32 in scandium, with a similar magnitude as in calcium.Moreover, we present ab initio calculations using the valence-space in-medium similarity renormalization groupbased on two- and three-nucleon interactions from chiral effective field theory. The theoretical results confirmthe existence of a substantial N =
32 shell gap in Sc and Ca with a decreasing trend towards lighter isotones,thus providing a consistent picture of the evolution of the N =
32 magic number from the p f into the sd shell. I. INTRODUCTION
The particularly bound and enhanced stable nature of somespecial nuclei with certain configurations of protons and neu-trons led Mayer and Jensen to introduce the nuclear shellmodel [1, 2]. These are the well-known magic numbers as-sociated with proton or neutron numbers 2, 8, 20, 28, 50, 82,and neutron number 126. In the single-particle shell model,protons and neutrons occupy nuclear orbitals with differentquantum numbers. When the orbitals are fully filled, nuclidesare much more bound than the neighboring ones. The closed-orbit nuclei have typically spherical shapes. The magic num-bers were established for nuclei close to the valley of β sta-bility. However, the nuclear shell structure has been found tochange when moving towards the drip lines. For instance, anew shell gap at N =
16 has been observed establishing O ∗ Corresponding author: [email protected] † Corresponding author: [email protected] as a doubly magic nucleus [3]. The evolution of the nuclearshell structure at extreme proton-to-neutron ratios has becomeone of the key research quests [4].In the past decades, a lot of efforts have been made to studythe shell evolution of N =
32 and 34 subshells, where pro-tons ( π ) and neutrons ( ν ) p / - p / and f / - f / spin-orbitpartners determine the structure. A local maximum in the sys-tematics of the first 2 + excitation energies [ E ( + ) ] in even-even nuclei at N =
32 were reported in Ar [5], Ca [6], Ti [7], and Cr [8] isotopes, remarkably suggesting a newneutron shell closure at N =
32. Meanwhile, a local min-imum in the systematics of reduced transition probabilities B ( E
2; 0 + → + ) has also provided an evidence for the ex-istence of this sub-shell in Ti [9] and Cr [10] isotopes. Fur-thermore, a sizable subshell closure with a similar magnitudeas the N =
32 gap in Ca has been unambiguously demon-strated at N =
34 in Ca [11].The emergence and weakening of new subshell closures N =
32, 34 have been successfully elucidated within the shellmodel by the tensor force acting between protons in j = l ± and neutrons in j ′ = l ′ ± orbitals, where l and l ′ represent or-bital angular momenta of protons and neutrons, respectively.In the standard shell model picture in this mass region, thevalence protons in the π f / orbital have an attractive tensorforce with the valence neutrons in the ν f / orbital. As soonas the protons are removed from the π f / orbital, that is whengoing from Fe to Ca, the magnitude of the effect of theattractive π - ν tensor force decreases consequently resultingin an upshift of the ν f / orbital. If the π f / is completelyempty, a substantial energy gap may exist between the ν f / orbital and ν p / - ν p / spin-orbit partners leading to the for-mation of a new subshell at N =
34. Furthermore, the spin-orbit splitting of the partners results in a sizable energy gap,the N =
32 subshell. The determination of the upper bound-aries of these new sub-shells at N =
32, 34 provides informa-tion on the relative ordering of ν f / and ν p / - ν p / spin-orbit partners and leads to a better understanding of the role ofthe tensor force on the shell evolution in exotic neutron-richnuclei. The low-lying energy levels in Sc indicate a quiterapid reduction of the N =
34 sub-shell gap, even though onlyone proton is added to the π f / orbital [12]. The reductionof the N =
34 sub-shell gap in Ti [13] and the robustness ofthe N =
32 sub-shell in Ti [7, 9] reveals that the ν f / or-bital is still above the ν p / - ν p / partners but is quite closeto the ν p / orbital if two protons are added (for more de-tails see Fig. 1 and the related discussion in Ref. [11]). Wenote that the shell model can well reproduce the experimentalresults [5, 11].Due to the particularly strong binding nature of magicnuclei, the two-neutron separation energy, S n , defined as S n ( Z , N ) = BE ( Z , N ) − BE ( Z , N − ) , where BE is nuclearbinding energy, is a well-established signature of neutron shellgaps, when a sudden change in the slope of a smooth S n sys-tematics occurs. The advantage of this indicator is that it isapplicable not only in even- Z isotopic chains but also in theodd- Z ones. High-precision mass measurements [14, 15] haveconfirmed the existence of the N =
32 sub-shell closure in cal-cium. Furthermore, mass measurements for , K revealedthe persistence of the N =
32 shell gap in potassium belowthe proton magic number Z =
20 [16]. The overall picture isconsistent with nuclear spectroscopy data mentioned above.However, the S n behavior as a function of neutron number inthe N ≈
32 region becomes smooth for Ti [17], V [18],and Cr [19], indicating the reduction of the N =
32 subshellclosure.
Ab initio calculations using the valence-space in-medium similarity renormalization group (VS-IMSRG) havesuccessfully predicted binding energies of the nuclear groundstates in this mass region [17, 20, 21]. While these calcu-lations generally describe the overall trends pointing to newmagic numbers, signatures such as 2 + energies, and neutronshell gaps tend to be modestly overpredicted, as seen recentlyin the titanium isotopes [17], highlighting the need for futureimprovements in the many-body approach.Due to large uncertainties (about several hundred keV) ofnuclear masses for Sc isotopes around A ≈
53, it was difficultto give a definite conclusion on the N =
32 shell evolutionabove Z =
20, or in other words, on the ordering of ν f / and ν p / - ν p / partners. In this paper, we report direct massmeasurements of Sc and address the question of the upper
602 604 606 608 611.8 612 612.2 614 616 618 620 622 C oun t s / s Revolution time (ns) N - F - T i - N a - A l - P - C l - S c - V - A r- S - S i - M g - O - C - A s - G a - C u - N a - N - N e - V - C l - P - A l - C r- C o - F - S i - S c - T i - S c - FIG. 1: (Color online). Part of the measured revolution time spec-trum in the time window 603 ns ≤ t ≤
622 ns. The nuclides withwell-known masses were used for calibration (black color). The nu-clides whose masses were determined in this work are indicated withred color. The determination of the masses of the remaining isotopes(blue color) is outside of the scope of the present paper. boundary of the N =
32 subshell closure. In addition, massesof , Ti have been obtained. After a very preliminary anal-ysis reported in Ref. [22], this work presents the final experi-mental results and their theoretical interpretation.
II. EXPERIMENT
The experiment was conducted at the Heavy Ion ResearchFacility in Lanzhou (HIRFL) [23]. Primary Kr + beamswere accelerated to 460.65 MeV/u by the heavy-ion syn-chrotron CSRm. They were fast-extracted and focused upon a ≈
15 mm thick Be target placed in front of the in-flight frag-ment separator RIBLL2 [24]. At this relativistic energy, thereaction products from the projectile fragmentation of Kremerged the target predominantly as bare nuclei. They wereanalyzed [25] by their magnetic rigidities B ρ by the RIBLL2.A cocktail beam including the ions of interest was injectedinto the cooler storage ring (CSRe). The isochronous massspectrometry (IMS) technique [26–32] has been applied in theCSRe for precision mass measurements of the stored ions.The primary beam energy was selected according to theLISE++ simulations [33] such that after the target the Cr + ions had the most probable velocity with γ = γ t = . γ is the Lorentz factor and γ t is the set CSRe transi-tion point. For an optimal transmission of nuclides centeredat Cr, RIBLL2 and CSRe were set to a fixed magnetic rigid-ity of B ρ = . B ρ acceptance of ± .
2% of the RIBLL2-CSRe system have been transmitted and stored in the CSRe.The revolution times of the stored ions were measured witha time-of-flight (ToF) detector [34] installed inside the CSReaperture. At each revolution ions passed through a 19 µ g/cm carbon foil thereby releasing secondary electrons. The latterwere guided to a micro-channel plate (MCP) counter. The sig-nals from the MCP were directly recorded by an oscilloscope.The revolution frequencies of the ions were about 1.6 MHz.The resolution of the ToF detector was about 50 ps. For eachinjection, a measurement time of 200 µ s, triggered by theCSRe injection kicker, was acquired, which corresponds toabout 300 revolutions of the ions in the CSRe. The efficiencyvaried from 20% to 70% depending the charge of ion species.Because only about five ions were stored simultaneously ineach injection, the saturation effect of MCP did not occur. Thetypical efficiency for the nucleus of interest with ionic chargearound 20 was about 50%, see Refs. [34–36] . In total 10300injections were accomplished. The revolution time spectrumand the corresponding isotope identification were obtained asdescribed in Refs. [32, 35, 37–40]. A part of the measuredspectrum is shown in Fig. 1.Many of nuclides in Fig. 1 have well-known masses. Theirmass excess ( ME ) values from AME ′
12 [41] were used to fittheir mass-to-charge ratios m / q versus the corresponding rev-olution times T by employing a third-order polynomial func-tion. The mass calibration has been checked by redeterminingthe ME values of each of the N c reference nuclides ( N c = N c − χ n defined as: χ n = vuut N c N c ∑ i = [( mq ) i , exp − ( mq ) i , AME ] σ i , exp + σ i , AME , (1)was found to be χ n = .
97. This value is within the expectedrange of χ n = ± .
18 at 1 σ confidence level, indicating thatno additional systematic error has to be considered. Ti Ti Sc Sc M E M E C S R ( k e V ) AME’12 TOFI1 TOFI2 TOFI3 NSCL1 NSCL2 TITAN Sc FIG. 2: (Color online). Differences between ME values determinedin this work and other experiments (see legend, text, and Table I).The red shadings represent the uncertainties from this work whilethe error bars are uncertainties from other experiments. Since our preliminary values from Ref. [22] were included into the latestAtomic Mass Evaluation, AME ′
16 [42], we use the values from the pre-ceding AME ′
12 [41] for comparison.
III. RESULTS AND DISCUSSION
Figure 2 presents the differences between ME values de-termined in this work and their previously known literaturevalues. The obtained results are listed in Table I. Owing to thelarge uncertainties of the ToF- B ρ measurements establishedat a radioactive beam line, results of all experiments seem tobe in general consistent at 3 σ confidence level. The excellentagreement between our results and the precision mass mea-surements from MR-ToF at TITAN for Ti [17], which is atthe edge of the isochronous window [35], proves the reliabilityof our measurements. All previous measurements were evalu-ated in the AME ′
12 yielding recommended values, which arealso illustrated in Fig. 2. It is striking that, except for the pre-cision value of Ti, our new results significantly deviate fromAME ′
12 values, namely by 2.3, 2.8, 1.7, and 2.5 standard de-viations, respectively, for , , Sc and Ti nuclei. We notethat unpublished measurements from GSI [43] are in overallgood agreement with our results. However, they were dis-carded in the AME ′ S n of the scandium isotopic chain as a function of neutronnumber N . As can be seen from Fig. 3, the S n ( Sc) as wellas S n ( Sc) are now significantly larger than assumed pre-viously, and consequently, a kink at N =
32 emerges clearly.This behavior is in line with the recently established trends forcalcium [15] and potassium [16] isotopic chains. Our resultsundoubtedly indicate the persistence of the sub-shell N =
28 30 32 3448121620 T w o - neu t r on s epa r a t i on ene r g y ( M e V ) Neutron number AME’12 this work VS-IMSRG
FIG. 3: (Color online). S n values for K, Ca, Sc, and Ti isotopicchains (see legend). The remarkable agreement between the experi-mental data and VS-IMSRG calculations is clearly seen. The strength of neutron subshell/shell closures can be eval-uated via the empirical neutron shell gap energy, defined asthe difference of two-neutron separation energies ∆ n ( N , Z ) = S n ( N , Z ) − S n ( N + , Z ) . As seen in Figure 4, the N = N =
28 shell gap, suggestingthe robustness of a prominent N =
32 subshell closure. Weemphasize that a rapid reduction of the N =
32 shell gap is
TABLE I: Mass excess ( ME ) values in keV of Sc and , Ti from the present work, three ToF-B ρ measurements at TOFI-LosAlamos [44–46], two ToF-B ρ measurements at NSCL-MSU [47, 48], and a MR-ToF measurement at TITAN-TRIUMF [17]. The ME values from the AME ′
12 [41] and their deviations from our new results taking into account both error bars, ∆ / δ = | ME CSRe − ME AME ′ | / p δ ( ME CSRe ) + δ ( ME AME ′ ) , are given in the last two columns.Atom ME CSRe ME TOFI1 [44] ME TOFI2 [45] ME TOFI3 [46] ME NSCL1 [47] ME NSCL2 [48] ME TITAN [17] ME AME ′ [41] ∆ / δ Sc − ( ) − ( ) − ( ) − ( ) - − ( ) - − ( ) Sc − ( ) − ( ) − ( ) − ( ) − ( ) − ( ) - − ( ) Sc − ( ) − ( ) − ( ) − ( ) − ( ) − ( ) - − ( ) Ti − ( ) − ( ) − ( ) − ( ) - - − ( ) − ( ) Ti − ( ) − ( ) − ( ) − ( ) - - - − ( )
24 26 28 30 320123456 E m p i r i c a l s he ll gap ( M e V ) Neutron number K Ca Sc Ti
FIG. 4: (Color online). Empirical shell gap for K, Ca, Sc, and Tiisotopic chains. The shell gap at N =
28 is nicely seen in all fourelements. The shell gap at N =
32 is well pronounced in scandiumand calcium and is strongly reduced in titanium. confirmed experimentally in titanium [17] and beyond, wheretwo or more protons occupy the π f / orbital.In VS-IMSRG calculations, we have found a strong reduc-tion in the N =
32 shell gap only at the vanadium isotopes,overpredicting the shell gap in titanium compared to experi-ment. In this work, we have performed calculations for thescandium, calcium, and potassium isotopes to determine theevolution of the shell gap across Z =
20. In particular, we usea VS-IMSRG approach [49–51], where an approximate uni-tary transformation [52, 53] is constructed to first decouplethe Ca core, as well as a standard p f valence-space Hamil-tonian. This interaction is subsequently diagonalized usingthe
NUSHELLX @ MSU shell-model code [54]. We further cap-ture the effects of three-nucleon (3 N ) forces between valencenucleons through the ensemble normal ordering [55], whichgives a unique valence-space Hamiltonian for each nucleus.We are then able to test nuclear forces in essentially all open-shell systems accessible to the nuclear shell model with a levelof accuracy comparable to large-space ab initio methods [55].We use the EM(1.8/2.0) NN + N interactions of Refs. [56,57], which begins from the chiral NN N LO potential ofRef. [58] combined with a non-local 3 N force fit in A =
19 20 21 22 23 24 252345 E m p i r i c a l s he ll gap ( M e V ) Proton number other experiments this work TITAN + this work VS-IMSRG
FIG. 5: (Color online). Empirical shell gap values for N = ′
12 [41] while colorpoints are from recent mass measurements by different labs. Asignificant shell gap in scandium is well reproduced by the theory aswell as the decrease of the shell gap towards heavier elements. S n values are also very well reproduced along thechain, including the sharp drops in S n at N =
28 and N = ′
12 values at N =
31, 33. Figure 5illustrates the experimental and calculated empirical shell gapfor N =
32 isotones. The theoretical values describe reason-ably the trend of experimentally determined shell gaps frompotassium to manganese, including the sharp peaks at calciumand scandium. However, we note that in general the shell gapsare overpredicted by several hundred keV, but are particularlyhigh in titanium and calcium, as first noted in Ref. [17]. Theorigin of this deviation is not yet fully understood, but signa-tures of shell closures are often modestly overestimated bythe current level of many-body truncations implemented inthe VS-IMSRG [20]. This overprediction is also consistentlyseen in first excited 2 + energies, where predictions are 300-400 keV too high in both Ca and Ti. From benchmarkswith coupled-cluster theory [59], it is expected that improvedtreatments for currently neglected three-body operators in theVS-IMSRG will improve these predictions.
IV. SUMMARY AND CONCLUSIONS
In summary, the masses of
Sc and , Ti nuclides havebeen directly measured in the heavy ion storage ring CSRe inLanzhou by employing isochronous mass spectrometry. Withthe new mass values the previously known mass surface hasbeen significantly modified. The existence of a strong N = N =
32 shell gap is the largest in scandium.Furthermore, our new data confirm the absence of a signif-icant shell gap in titanium. The ab initio calculations usingthe VS-IMSRG approach with NN and 3 N interactions fromchiral effective field theory confirm the experimental obser-vations for calcium and scandium, but predict a persistenceof a large N =
32 gap in titanium, at odds with these andother experimental measurements. Work is currently under-way to improve the IMSRG approach to include the physicsof neglected three-body operators, likely the origin of this dis-crepancy. The understanding of shell closures in neutron-richnuclei is not only important for nuclear structure but is alsocritical for the reliable modeling of the structure of compactstellar objects as well as for nucleosynthesis.
Acknowledgments
We thank the staffs of the accelerator division of the IMPfor providing stable beam. This work is supported in partby the National Key R&D Program of China (Grant No.2018YFA0404401 and No. 2016YFA0400504), the NSFC(Grants No. 11605249, No. 11605248, No. 11605252, No.11505267, No. 11575112, and No. 11575007), the CAS Ex-ternal Cooperation Program (Grant No. GJHZ1305), the CAS"Light of West China" Program, the CAS visiting professor-ship for senior international scientists (Grant No. 2009J2-23), the CAS through the Key Research Program of FrontierSciences (Grant No. QYZDJ-SSW-SLH005), the Helmholtz-CAS Joint Research Group (HCJRG-108), the HGF Nu-clear Astrophysics Virtual Institute (NAVI), the DeutscheForschungsgemeinschaft (DFG, German Research Founda-tion) – Projektnummer 279384907 – SFB 1245, the BMBF(Contract No. 05P18RDFN1), the DAAD PPP program withChina (Project-ID 57389367), the National Research Coun-cil of Canada and NSERC, the DFG through the Clusterof Excellence PRISMA, and the European Research Coun-cil (ERC) (ERC-StG 307986 "STRONGINT" and ERC-CG682841 "ASTRUm"), A.S. is supported by the Max PlanckSociety. X.L.T. acknowledges the support from the MaxPlanck Society through the Max-Planck Partner Group. Fur-thermore computations were performed with an allocationof computing resources at the Jülich Supercomputing Center(JURECA). [1] M. G. Mayer, Phys. Rev. , 1969 (1949).[2] M. G. Mayer and J. H. D. Jensen, Elementary Theory of NuclearShell Structure (Wiley, New York, 1955).[3] R. Kanungo et al. , Phys. Rev. Lett. , 152501 (2009).[4] R. Kanungo, Phys. Scr.,
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