NNeutron-rich calcium isotopes within realistic Gamow shell modelcalculations with continuum coupling
J.G. Li, B.S. Hu, Q. Wu, Y. Gao, S.J. Dai, and F.R. Xu ∗ School of Physics, and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, China (Dated: August 28, 2020)
Abstract
Based on the realistic nuclear force of the high-precision CD-Bonn potential, we have performedcomprehensive calculations for neutron-rich calcium isotopes using the Gamow shell model (GSM)which includes resonance and continuum. The realistic GSM calculations produce well bindingenergies, one- and two-neutron separation energies, predicting that Ca is the heaviest boundodd isotope and Ca is the dripline nucleus. Resonant states are predicted, which provides usefulinformation for future experiments on particle emissions in neutron-rich calcium isotopes. Shellevolutions in the calcium chain around neutron numbers N = 32, 34 and 40 are understood bycalculating effective single-particle energies, the excitation energies of the first 2 + states and two-neutron separation energies. The calculations support shell closures at Ca ( N = 32) and Ca( N = 34) but show a weakening of shell closure at Ca ( N = 40). The possible shell closure at Ca ( N = 50) is predicted. ∗ [email protected] a r X i v : . [ nu c l - t h ] A ug . INTRODUCTION The long chain of calcium isotopes provides an ideal laboratory for both theoretical andexperimental investigations of unstable nuclei. With two typical doubly-magic isotopes Caand Ca, the calcium chain is speculated to be up to Ca, a possible third isotope of thedouble magicity. Current experiments have reached Ca [1], but theoretical calculations arevarious [2–9]. Further refined calculations are still in demand. Besides the N = 20 and 28magic numbers, experiments have also given evidences of additional shell closures at Ca( N = 32) [10] and Ca ( N = 34) [11]. It is still an open question whether the N = 40shell closure vanishes in the calcium chain. The spherical N = 40 shell closure remains inthe isotone Ni [12], while it disappears in the isotones Cr and Fe with the onsets ofdeformation and collectivity [13–17]. With advances in rare isotope beam facilities, moreand more structure data will be obtained for calcium isotopes, which attracts continuinginterests of theory [18].The N = 32 shell closure was observed in the early experiment [19], giving that Cahas a higher 2 +1 state by 1.5 MeV than in Ca. The precious mass measurements of Caand Ca at CERN [10] show that the trend of two-neutron separation energies supports theshell closure in Ca. The shell closure at N = 32 has also been found in nearby titaniumand chromium isotopes [20, 21]. The spectroscopic experiment has reached Ca, givingthat the 2 +1 state in Ca is at 2.0 MeV [11], slightly lower than in Ca, which providesan experimental signature of the shell closure in Ca. The precise mass measurementsof
Ca isotopes provide additional experimental evidences for the understanding of themagic nature in Ca [22]. To date, the mass measurements of calcium isotopes have beenup to Ca, while Ca is the heaviest calcium isotope for which spectroscopic data havebeen available. Ca is the neutron-richest calcium isotope obtained so far in experiment[1]. The experimental data provide valuable information to test theoretical calculations, andthen lead to more reliable predictions for dripline nuclei and beyond.The calcium region currently represents a frontier of theoretical calculations. With thephenomenological interactions, GXPF1A [23] and KB3G [24] for the pf shell, large-scaleshell-model calculations have been performed for calcium isotopes. The GXPF1A interactionresults in a strong shell gap at Ca ( N = 34), while KB3G does not give the shell gap.Based on a realistic interaction of the CD-Bonn potential [25], the realistic shell model2RSM) with empirical single-particle (s.p.) energies [3] has been applied to the spectra ofcalcium isotopes, predicting a weak shell closure at N = 34. Further RSM calculationsfor the whole isotopic chain of calcium were done in Refs. [5, 7]. The complex coupled-cluster (CC) model [4, 26] with including the continuum effect has calculated up to Ca.In the complex CC calculations [4], however, Ca is unbound, which is not consistent withthe recent experiment [1]. With a refined two- plus three-nucleon ∆NNLO interaction,the recent CC calculations extend the dripline beyond Ca [27]. Calcium isotopes havealso been investigated by the Green’s function (up to Ca) [6] and in-medium similarityrenormalization group (IM-SRG) (for even masses up to Ca) [8] giving unbound , , Ca.The nuclear density function theory (DFT) based on the Skyrme interaction predicts thatthe calcium two-neutron dripline should be at Ca [2, 28]. An early work by the relativisticmean field gave the dripline at Ca [29]. The theoretical calculations of neutron-rich calciumisotopes are still a challenge, which needs good understandings of the strong interaction,many-body correlation and coupling to scattering continuum.In the present paper, we give the comprehensive calculations of neutron-rich calciumisotopes using the Gamow shell model GSM with the CD-Bonn potential. The couplingto continuum is considered by using the complex-momentum (complex- k ) Berggren space.In Sec. II, we describe the Berggren basis which treats bound, resonant and continuumstates on equal footing. The effective Hamiltonian in the model space is derived fromthe realistic CD-Bonn interaction using the many-body perturbation theory (MBPT). Thedetailed calculations are given in Sec. III. Binding energies, one- and two-neutron separationenergies and excitation spectra are calculated and compared with existing data. The shellevolution in the calcium chain is discussed. II. THEORETICAL FRAMEWORK
The Gamow resonance is a time-dependent problem, associated with particle emissions.To solve a time-dependent Schr¨odinger equation is extremely difficult, especially for many-body problems. Berggren [30] generalized the Schr¨odinger equation to a complex- k planein which the eigen energy is written as a complex number, (cid:101) e i = e i − iγ i /
2, with γ i standingfor the resonance width (measuring the half-life of particle emission). The Berggren methodprovides an approach to solve a time-dependent problem in a time-independent way. In the3omplex- k plane, the Berggren ensemble contains three types of states: bound, resonant andscattering continuum.The shell model within the Berggren basis is called the Gamow shell model (GSM). Withphenomenological interactions, the GSM has been successfully applied to the systems of twovalence particles at first [31, 32] and more valence particles [33–38]. The GSM based onrealistic nuclear forces has also been developed [39–43]. To calculate heavy nuclei, an innercore is usually taken in shell-model calculations. The harmonic oscillator (HO) basis is oftenadopted in the conventional shell model, while the GSM usually uses the Woods-Saxon (WS)potential to create the Berggren basis [31–43].For calcium isotopes, the magic Z = 20 protons are well bound and hence can be treatedin the HO basis. For neutrons, we use the spherical WS potential V ( r ) = V / [1 + e ( r − R ) /a ]with a spin-orbit coupling V ls ( r ) = − χ r dVdr l · s , to create the neutron Berggren basis, where V = − V [1 − κ ( N − Z ) / ( N + Z )] and R = r A / . We fix r = 1 .
15 fm and a = 0 . Ca as the inner core, but forisotopes heavier than Ca the closed-shell Ca is taken as the core to reduce the modeldimension and computational cost. If we kept Ca as the core for the neutron-richestcalcium isotopes, the model dimension would be beyond the power of current computerswhen continuum channels are included. The parameters V , κ and χ are chosen such thatthe neutron orbits 1 p / , 1 p / , and 0 f / reproduce experimental s.p. energies obtained in Ca [45] and experimental one-neutron separation energy in Ca [22]. Table I lists theWS single particle (s.p.) energies for valence neutrons in the shell-model space, obtainedwith V = 62.8 MeV, κ = 0.738 and χ = 0 . p / , 1 p / and 0 f / orbits are bound, while1 d / and 0 g / are resonant. The obtained one-neutron separation energy in Ca is 1.40MeV which agrees with the experimental datum of 1 . .
16) MeV [22].The completeness of the Berggren ensemble requires to include non-resonant continuumchannels described by contours ( L + ) in the complex- k plane [31, 32, 34, 42]. For a channelwith narrow resonant state(s), the continuum contour L + is chosen to contain the narrowresonant state(s) [42]. For a continuum channel without narrow resonant state, the contour L + is chosen to be a segment lying on the real-momentum x -axis (starting from the originof the coordinates) [42]. In numerical calculations, the continuum contour L + is discrete4 ABLE I. The WS neutron s.p. energies (in MeV) calculated with Ca or Ca, compared withexperimental s.p. energies extracted from Ca [45].Neutron WS Expt WSs.p. orbits ( Ca) ( Ca)1 p / − . − . − p / − . − . − f / − . − . − . d / . − i .
86 2 . − i . g / . − i .
01 2 . − i . using the Gauss-Legendre quadrature method [32, 34, 42]. For a continuum channel withoutnarrow resonant state, we set ten discretization points on the contour L + , which has beenwell tested to be sufficient to get convergence. In our previous publication for the sd -shellnuclei [42], we took eight discretization points that can give well converged results. The0 g / orbit has a small imaginary part of the eigen energy (only 10 keV, almost bound).This state is very close to the real-momentum x axis in the Berggren complex-momentumplane, thus a contour close to the x axis is chosen for the g / channel with 18 discretiza-tion points. The 1 d / orbit has a relatively large imaginary part of the energy. The d / continuum contour L + needs to include the 1 d / resonant state. For the d / contour, weset 44 discretization points. Note that a channel containing a resonant state which hasa significant imaginary energy needs more discretization points to reach the convergenceof the numerical calculation. We have tested that such discretizations above provide wellconverged calculations for the mass region investigated. The detail about the Berggrencontinuum contour and discretizing can be found in the previous paper [42] in which lessdiscretization points were taken for the sd -shell nuclei. In the present work, we focus onneutron-rich calcium isotopes heavier than Ca. Neutrons are treated in the Berggrencomplex- k basis. The active model space for the GSM calculations is the neutron { p / ,1 p / , 0 f / , 0 g / -resonant+continuum, 1 d / -resonant+continuum } with Ca as core, whileit is the neutron { f / , 0 g / -resonant+continuum, 1 d / -resonant+continuum } with Caas core. Effects from other partial waves (including the core polarization) are included viamany-body perturbation by the so-called nondegenerate ˆ Q -box folded diagrams [42].5he intrinsic Hamiltonian of an A -body system reads H = A (cid:88) i =1 p i m + A (cid:88) i 5. We have tested that such aHO truncation is sufficient to reach the convergences of the calculations.The interaction matrix elements obtained in the Berggren basis are complex and non-Hermitian. We employ the MBPT named the full ˆ Q -box folded-diagram method [50] to7onstruct the realistic GSM effective interaction in the defined model space. The complex- k Berggren basis states are non-degenerate, therefore a non-degenerate ˆ Q -box folded-diagramperturbation named the extended Kuo-Krenciglowa (EKK) method [51] has been used. Us-ing the MBPT, we first calculate the ˆ Q -box in the complex- k Berggren basis, as follows (cid:98) Q ( E )= P H P + P H Q E − QHQ QH P = P H P + P H Q E − QH Q QH P + ..., (9)where E is the starting energy. P and Q stand for the active model space and excludedspace, respectively, with P + Q = . The ˆ Q -box is composed of irreducible valence-linkeddiagrams [52, 53]. In the present calculation, ˆ Q -box diagrams are calculated up to the secondorder. The derivatives of the ˆ Q -box are defined as (cid:98) Q k ( E )= 1 k ! d k (cid:98) Q ( E ) dE k = ( − k P H Q E − QHQ ) k +1 QH P, (10)where k presents the k -th derivative.The effective Hamiltonian H eff can be constructed via [54] (cid:101) H eff = (cid:101) H BH ( E ) + ∞ (cid:88) k =1 (cid:98) Q k ( E ) (cid:101) H eff , (11)where (cid:101) H eff stands for (cid:101) H eff = H eff − E , while (cid:101) H BH ( E ) = H BH ( E ) − E is the Bloch-HorowitzHamiltonian shifted by an energy E , with H BH ( E )= P H P + (cid:98) Q ( E )= P H P + P H P + P H Q E − QHQ QH P. (12)The (cid:101) H eff is obtained by iterating Eq.(11), which is equivalently to calculate the foldeddiagrams with including high-order contributions by summing up the subsets of diagramsto finite order. The effective Hamiltonian is given by H eff = (cid:101) H eff + E , and the effectiveinteraction is obtained by V eff = H eff − P H P .We choose Ca as the inner core and { p / , p / , f / , 0 g / -resonance + continuum,1 d / -resonance + continuum } as the model space for valence neutrons outside the Ca core.For isotopes heavier than Ca, we take the closed-shell Ca as the inner core and the neutron { f / , g / -resonance + continuum, 1 d / -resonance + continuum } as the model space to8educe the model dimension and computational task. The non-Hermitian GSM Hamiltonianis diagonalized in the model space by using the Lanczos method in the m scheme. Due to thefact that the scattering continuum states are included in the shell-model space, the modeldimension increases dramatically with increasing the number of particles in the continuumstates [34]. Similar to our previous calculations [42], we allow at most two particles in thecontinuum. It has been tested that such truncation can give a good convergence of thecalculation. III. CALCULATIONS AND DISCUSSIONS E gs (MeV) M a s s N u m b e r A FIG. 1. Calculated ground-state energies with respect to Ca, compared with experimentaldata and other theoretical calculations: the complex CC with N LO(NN)+3NF eff [4], RSM withN LO(NN)+NNLO(3NF) [7], IM-SRG with N LO(NN) + NNLO(3NF) [9] and SV-min DFT [55].The data for Ca have been collected in AME2016 [56], while the data for Ca are takenfrom the recent experiment [22]. The Ca datum takes the evaluation given in AME2016 [56].The CD-Bonn interaction is renormalized by V low- k with Λ = 2 . − . Ca [1], andmass measurements up to Ca [22]. We have made detailed calculations for isotopes up to Ca, using the GSM with the core and the corresponding model spaces described above.Figure 1 shows the calculated ground-state energies, compared with experimental data [22,56] and other theoretical calculations [4, 7, 9, 55]. There have existed several theoreticalinvestigations within mean field (e.g., in [28, 55]) and ab initio (e.g., [4–9]) models. Therecent calculation based on the Skyrme-type DFT with the Bayesian statistical correctionpredicts that the neutron dripline would be at Ca [28]. The complex CC with chiral two-nucleon (NN) and density-dependent 3N forces has calculated isotopes up to Ca [4]. Theshell model with the Hamiltonian derived by MBPT based chiral NN and normal-ordered 3Nforces has investigated the whole chain, giving slight decreases in binding energies beyond Ca [7]. The IM-SRG with a chiral interaction has computed even-mass isotopes up to Ca [9]. We see in Fig. 1 that the calculations lead to overall agreements in energies withexisting data. The maximum discrepancy between the present calculation and experimentalenergy is about 2.5 MeV happening in Ca.Figure 2 displays one- and two-neutron separation energies, compared with data [22, 56],DFT [55] and IM-SRG [8] calculations. The calculated one-neutron separation energies showthat Ca is the heaviest odd-mass calcium isotope which is bound against neutron emission.This is consistent with the MBPT calculations [7]. Ca is weakly unbound with a smallone-neutron separation energy of − 326 keV in the present calculation. The experiment [1]observed a bound Ca. Theoretical predictions are various. The DFT calculation with theBayesian statistical correction predicts that Ca is bound and Ca has a ∼ 50% probabilitybeing bound [28], while the relativistic mean-field calculation gives that the heaviest boundodd isotope is Ca [57]. The IM-SRG calculations [9] show that the heaviest bound oddisotope would be in Ca.Figure 2(b) gives two-neutron separation energies in even calcium isotopes. To seewhether the different choices of the shell-model core give consistent results, we have per-formed two kinds of calculations with the Ca or Ca core for , , Ca. We see in Fig.2(b) that the resulted two-neutron separation energies are well similar. The calculated two-neutron separation energies show an overall agreement with experimental data and othertheoretical calculations, e.g., by DFT [55] and IM-SRG [8]. The large two-neutron separa-tion energies at N = 32 and 34 indicate the subshell closures which have been suggested in10 Sn (MeV) E x p t G S M S V - m i n D F T ( a ) S2n (MeV) M a s s N u m b e r A E x p t G S M ( c o r e C a ) G S M ( c o r e C a ) S V - m i n D F T I M - S R G ( b ) FIG. 2. Calculated one- (a) and two-neutron (b) separation energies, compared with data [22, 56]and calculations by SV-min DFT [55] and multi-reference IM-SRG (only S n calculated) [8]. The S n calculations stop at Ca because odd isotopes heavier than Ca become unbound in calculations. experiments [10, 11, 19, 22, 58] and theories [3–5, 7, 8]. From the calculated two-neutronseparation energies, we predict that the two-neutron dripline of the calcium chain shouldlocate at Ca. This agrees with the recent mean-field calculation [28].The shell evolution in the calcium chain around the neutron numbers N = 32, 34 and11 ESPE (MeV) N e u t r o n N u m b e r N p p f d g FIG. 3. Neutron effective single-particle energies (ESPE) with respect to the Ca core, as afunction of neutron number. The V low- k Λ = 2 . − CD-Bonn interaction is used. 40 is an interesting topic [10, 11, 22]. With the V low- k Λ = 2 . − CD-Bonn interaction,we have estimated the effective single-particle energy (ESPE) defined in Ref. [59]. Figure3 shows the evolutions of the valence neutron ESPEs with increasing the neutron number.We see that significant shell gaps exist between 1 p / and 1 p / and between 1 p / and 0 f / ,indicating shell closures at N = 32 and 34, respectively. This is consistent with experimentalobservations [10, 11, 19, 22, 58] and theoretical calculations [3–5, 7, 8]. The shell gap abovethe 0 f / orbit is reduced around N = 40, implying a weakening of the N = 40 shell closurein the calcium chain. In the isotone Ni the spherical N = 40 shell closure exists [12], whilethe shell closure vanishes in the isotones Cr and Fe with the onset of deformation andcollectivity [13–17]. The N = 40 shell closure is eroded due to the intrusion of the 0 g / orbit,as illustrated in Fig. 3. The 0 g / orbit can drive the nucleus to be deformed. However, theonset of deformation depends upon how much 0 g / component appears in the state. Thederived effective interaction gives a strong monopole attraction between the 0 f / and 0 g / orbits, which results in the drop of the 0 g / orbit with increasing the neutron number. The12ccupation of the 0 g / orbit leads to the phenomenon of the so-called island of inversion predicted around N = 40 in Cr and Fe isotopes [60]. The 0 g / orbit becomes bound at N ≥ 40, which can enhance the stability of heavy calcium isotopes. The experimentaldiscovery of the Ca ( N = 40) [1] may be an indication of the enhanced stability. TheESPEs in Fig. 3 show a clear shell gap at N = 50, implying a shell closure there. Ex (2+1) (MeV) N e u t r o n N u m b e r N E x p t G S M FIG. 4. Calculated excitation energies of the 2 +1 excited states for calcium isotopes, compared withdata [11, 61, 62]. The excitation energy of the first 2 + excited state in even-even nuclei can be used toanalyze the shell gap. With the present GSM based on the V low- k CD-Bonn interaction, wehave calculated the excitation energies E x (2 +1 ) for neutron-rich calcium isotopes, shown inFig. 4. We see that at N = 32 and 34 the obtained 2 +1 energies are significantly larger than inneighboring isotopes, which implies shell closures at N = 32 and 34. This is consistent withthe experimental [10, 11, 19, 22, 58] and theoretical [3–5, 7, 8] conclusions. The calculated2 +1 excitation energies around N = 40 are lower than at N = 32 and 34, which implicatesa reduction of shell gap at N = 40 in the calcium chain. The spherical N = 40 shellclosure was suggested experimentally in the isotone Ni with E x (2 +1 ) = 2033 keV [12], while13he shell closure vanishes in the isotones, Fe with E x (2 +1 ) = 573 keV [63] and Cr with E x (2 +1 ) = 420 keV [64]. Experiments [13–17] show more collectivity in Fe and Cr. The2 +1 state in the lighter isotone Ti has not been detected in experiment. However, theexperiment observed a 2 +1 state in Ti ( N = 38) at an energy of 850 keV [65] which isabout twice the excitation energy of the 2 +1 state in the isotone Cr. This would implicatepossible higher 2 +1 excitation energies in Ti and Ca than in the isotones Fe and Cr.Indeed, the shell-model calculations [60] give higher 2 +1 energies in Ca and Ti than in theisotones Cr and Fe. Our calculation shown in Fig. 4 gives a 2 +1 state around 1.6 MeVfor Ca. This result is consistent with the shell-model calculations in Refs. [5, 60]. Sucha 2 +1 excitation energy is remarkably higher than in the isotones Cr and Fe which havethe 2 +1 energies around 500 keV observed experimentally [63, 64]. Though the N = 40 shellgap is reduced compared with the N = 32 and 34 gaps in the calcium chain, the sizable2 +1 excitation energy in Ca would indicate an enhancement in the stabilities of Ca andheavier isotopes.The GSM calculation gives that the dominant configurations of the Ca ground stateare ν { (1 p / ) (1 p / ) (0 f / ) (0 g / ) } (50%) and ν { (1 p / ) (1 p / ) (0 f / ) } (30%), wherethe percentage indicates the proportion of the component. We see that there is a 50%probability of one pair of neutrons occupying the 0 g / intruder orbit. However, such anoccupation in 0 g / should not be able to lead to a stable deformation in Ca which has aspherical proton magicity of Z = 20. In Fig. 4, we see a high 2 +1 excitation energy at N =50, which should indicate a shell closure there. The result is consistent with the shell-modelcalculation in Ref. [5]. Combining the two-neutron separation energy given in Fig. 2(b), wepredict a doubly magic dripline nucleus of Ca for the calcium chain.Spectroscopic calculations can provide further information on nuclear structures. Theexperimental spectroscopy has reached Ca [11]. In Fig. 5, we show the GSM calculationsof excitation spectra for Ca, compared with experimental spectra available. To see theeffect from the continuum, we have also made conventional shell-model calculations withinthe HO basis, denoted by RSM as in Refs. [5, 60]. The same V low- k Λ = 2 . − CD-Bonnpotential is used. The RSM space for valence neutrons is { p / , p / , f / , g / , d / } which is the same as in the GSM calculation, except that the g / and d / continuumpartial waves are not able to be included in the discrete HO basis. We see that low-lyingexcited states given by GSM and RSM are similar and agree with experimental data. This14 Excitation Energy (MeV) - - - - - - - - - - - - - - - - - - - - - - - - - - + + - - + + C a + + + + + + + + + - + + + + + + + + + + C a R S MG S ME x p t - - - - - - - - - - - - - - + + - - - + - + - C a + + + + + + + + + + + + - C a R S MG S ME x p t FIG. 5. Calculated excited states for Ca. The same V low- k Λ = 2 . − CD-Bonn interactionis used in GSM and RSM calculations. Data are from [11, 61, 62]. Particle continua above emissionthresholds are marked by purple shadowing, while resonant states are indicated by red shadowing. can be understood by the fact that the continuum effect is not significant in well-boundstates. Resonances are seen in the GSM calculations around E x ∼ . ∼ . ∼ . ∼ . Ca and Ca,respectively. The resonant 5 / +1 and 9 / +1 excited states in the odd Ca isotopes reflect the15esonant single-particle orbits of 1 d / and 0 g / . The ordering of the 5 / +1 and 9 / +1 excitedstates is consistent with the order of the 1 d / and 0 g / orbits (as shown in Fig. 3). TableII gives the 5 / +1 and 9 / +1 resonant excited states predicted for the odd Ca isotopes. The9 / + state has a large l = 4 centrifugal barrier and hence a weak coupling to the continuum,giving a narrow resonance (or called quasi-bound state) [4]. By contrast, the 5 / + state hasa stronger coupling to continuum with a lower l = 2 centrifugal barrier, resulting in a wideresonance. TABLE II. Predicted 5 / +1 and 9 / +1 resonant states in odd calcium isotopes , , , Ca by theGSM with the V low- k Λ = 2 . − CD-Bonn interaction. The excitation energy is defined by (cid:101) E = E − i Γ / 2, where the real part ( E ) of the energy gives the level position while the imaginarypart defines the resonance width Γ. Both energy and width are in MeV.Nuclei Ca Ca Ca Ca E Γ E Γ E Γ E Γ5 / +1 / +1 Figure 6 predicts low-lying excitation spectra in Ca, which should be useful for nearfuture spectroscopic experiments. In heavy isotopes, the continuum effect becomes moresignificant. As described above, the Ca core is used in the calculations of isotopes lighterthan Ca. In Ca, the 5 / − ground state is governed by the odd neutron occupying the0 f / orbit above the Ca Fermi level. For the negative-parity 1 / − and 3 / − excited states,the dominant configuration is the odd 0 f / neutron coupling to the first 2 + excited state of Ca. The positive-parity 5 / + and 9 / + states have the odd neutron being excited to the1 d / and 0 g / orbits, respectively. In Ca, the first 2 + and 4 + excited states are below theneutron emission threshold, with the dominant configuration of ( ν f / ) ⊗ Ca. In Ca,the 5 / − ground state, 3 / − and 9 / − excited states are dominated by the ( ν f / ) ⊗ Caconfiguration. The positive-parity 9 / + and 5 / + excited states have a character of single-particle excitation, with the odd neutron being excited to the 0 g / and 1 d / , respectively.In Ca, the 0 + ground state and the first 2 + , 4 + excited states contain two dominateconfigurations of ( ν f / ) ⊗ Ca and ( ν f / ) ( ν g / ) ⊗ Ca with the intruder 0 g / orbitinvolved. We see that there exist low-lying resonant 5 / + excited states in the odd isotopes16 Excitation Energy (MeV) C a ( 5 / 2 - ) 5 / 2 - - - - + + - - + - + C a + + + + + ( 0 + ) C a ( 5 / 2 - ) 5 / 2 - - - + + + - - + + E x p t G S M R S M E x p t G S M R S M C a + + + + + + ( 0 + ) FIG. 6. Similar to Fig. 5, but for Ca as the predictions of low-lying spectra. Note that theexisting experimental data give large uncertainties in particle emission thresholds, indicated byerror bars [22, 56]. , Ca.At the end, we test how sensitive the predictions are to the choice of interaction, byperforming similar calculations but using different effective interactions: a softer CD-Bonnand the chiral N LO [66, 67] with Λ = 2 . − in V low- k . With the Ca core, the CD-17onn GSM calculations with Λ = 2 . − give that the ground states become slightlymore bound by 0.4 − Ca to Ca, compared with the calculations at Λ = 2 . − . It has been known that a soft interaction without 3NF invoked can lead to overbindingenergies. The 3NF effect can be reduced by choosing a large Λ cutoff in the V low- k procedure.The induced 3NF usually provides a repulsive effect on the binding energy, and the effectbecomes larger as the number of valence particles increases. However, we find that theneutron separation energies which are the differences of binding energies do not changemuch from Λ = 2 . − . The conclusions remain unchanged with Λ = 2 . − , e.g., the heaviest bound odd isotope is Ca, and Ca remains the dripline nucleus.The calculations with Λ = 2 . − give almost the same 2 +1 excitation energy in Ca (with a difference of only 0.1 MeV). The calculations using the chiral N LO softenedwith Λ = 2 . − give the binding energies within 0.6 MeV of the Λ = 2 . − CD-Bonn results. However, the heaviest bound odd isotope is Ca with a small one-neutronseparation energy of only 0.08 MeV. This result seems to be consistent with the experiment[1] and the mean-field calculations [28, 55]. Ca is still the neutron dripline nucleus, witha 2 +1 excitation energy of 2.3 MeV which is 0.4 MeV lower than that in the Λ = 2 . − CD-Bonn calculation. IV. SUMMARY Using the Gamow shell model with the high-precision charge-dependent Bonn nucleon-nucleon interaction renormalized by the V low- k technique, we have performed comprehensivecalculations for neutron-rich calcium isotopes up to beyond the neutron dripline. The cou-pling to continuum is included in the Gamow shell model by using the complex-momentumBerggren basis in which bound, resonant and continuum states are treated on equal footing.The Gamow shell model calculations can well describe the resonant properties of particleemission states in weakly-bound or unbound nuclei. Nuclear binding energies and neutronseparation energies are calculated up to Ca, predicting that the heaviest odd bound iso-tope is Ca and the dripline locates at Ca. The calculations of the 2 +1 excitation andeffective single-particle energies, combined with two-neutron separation energies, show theshell closures at N = 32, 34 and 50 and a shell weakening at N = 40. Calculated low-lyingexcitation spectra in Ca agree well with existing data. As predictions for near future18pectroscopic experiments, we have calculated low excited states for Ca, providing use-ful information about the configurations of the low-lying states. Resonant excited statesemerge in odd isotopes , , , Ca, which involve heavily the widely-resonant neutron 1 d / orbit. The continuum effect is seen by the comparison between the Gamow and conventionalshell-model calculations. ACKNOWLEDGMENTS Valuable discussions with Z.H. Sun, N. Michel, M. Hjorth-Jensen, L. Coraggio, S.M.Wang, Y.Z. Ma and J.C. Pei are gratefully acknowledged. This work has been supported bythe National Key R&D Program of China under Grant No. 2018YFA0404401; the NationalNatural Science Foundation of China under Grants No. 11835001 and No. 11921006; theState Key Laboratory of Nuclear Physics and Technology, Peking University under GrantNo. NPT2020ZZ01; and the CUSTIPEN (China-U.S. Theory Institute for Physics withExotic Nuclei) funded by the U.S. Department of Energy, Office of Science under GrantNo. de-sc0009971. We acknowledge the High-performance Computing Platform of PekingUniversity for providing computational resources. [1] O. B. Tarasov, D. S. Ahn, D. Bazin, N. Fukuda, A. Gade, M. Hausmann, N. Inabe, S. Ishikawa,N. Iwasa, K. Kawata, T. Komatsubara, T. Kubo, K. Kusaka, D. J. Morrissey, M. Ohtake,H. Otsu, M. Portillo, T. Sakakibara, H. Sakurai, H. Sato, B. M. Sherrill, Y. Shimizu,A. Stolz, T. Sumikama, H. Suzuki, H. Takeda, M. Thoennessen, H. Ueno, Y. Yanagisawa,and K. Yoshida, Phys. Rev. Lett. , 022501 (2018).[2] J. Erler, N. 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