Neutron drip line in the deformed relativistic Hartree-Bogoliubov theory in continuum: Oxygen to Calcium
Eun Jin In, Youngman Kim, Panagiota Papakonstantinou, Seung-Woo Hong
NNeutron drip line in the deformed relativistic Hartree-Bogoliubovtheory in continuum: Oxygen to Calcium
Eun Jin In , Youngman Kim , Panagiota Papakonstantinou , and Seung-Woo Hong Department of Energy Science, Sungkyunkwan University, Suwon 16419, Korea Rare Isotope Science Project, Institute for Basic Science, Daejeon 34000, Korea Department of Physics, Sungkyunkwan University, Suwon 16419, KoreaJuly 3, 2020
Abstract.
The location of the neutron drip line, currently known for only the lightest elements, remainsa fundamental question in nuclear physics. Its description is a challenge for microscopic nuclear energydensity functionals, as it must take into account in a realistic way not only the nuclear potential, but alsopairing correlations, deformation effects and coupling to the continuum. The recently developed deformedrelativistic Hartree-Bogoliubov theory in continuum (DRHBc) aims to provide a unified description ofeven-even nuclei throughout the nuclear chart. Here, the DRHBc with the successful density functionalPC-PK1 is used to investigate whether and how deformation influences the prediction for the neutrondrip-line location for even-even nuclei with 8 ≤ Z ≤
20, where many isotopes are predicted deformed. Theresults are compared with those based on the spherical relativistic continuum Hartree-Bogoliubov (RCHB)theory and discussed in terms of shape evolution and the variational principle. It is found that the Ne andAr drip-line nuclei are different after the deformation effect is included. The direction of the change is notnecessarily towards an extended drip line, but rather depends on the evolution of the degree of deformationtowards the drip line. Deformation effects as well as pairing and continuum effects treated in a consistentway can affect critically the theoretical description of the neutron drip-line location.
PACS.
XX.XX.XX No PACS code given
Nuclear masses are not only of great importance in nu-clear physics but are also of interdisciplinary interest no-tably for research on fundamental interactions and astro-physics [1]. While ongoing experiments reach ever fartheraway from the valley of stability, the neutron drip line hasonly been mapped for the lightest elements and theory isessential for providing qualitative and quantitative predic-tions and extrapolations. The theoretical research on theneutron drip line and associated exotic phenomena haslong been active [2,3,4,5,6,7,8,9,10]. The neutron-richnuclei show different structures and properties comparedto the stable nuclei: As the number of neutrons increases,the shape and size of nuclei may change in ways not ob-served in stable nuclei, leading to new phenomena likegiant haloes, soft and pygmy resonances, and new magicnumbers, in addition to neutron skin and new regions ofshape coexistence and isomerism [3,5,11,12,13,14,15,16,17,18,19,20], with implications for astrophysical and nu-cleosynthesis studies.In unstable, weakly-bound nuclei close to the neutrondrip line, continuum and pairing correlations play an im- a e-mail: [email protected] portant role [2,3,4,14,21,22,23,24]. The neutron Fermienergy is very close to the continuum, so in order to studytheoretically the neutron drip line, pairing correlationsand continuum effects must be taken into account to de-termine the neutron separation energy as precisely as pos-sible. Both pairing correlations and the treatment of con-tinuum effects can influence the predictions for the dripline.The Hartree-Fock-Bogoliubov (HFB) model has foundmany applications in the study of nuclear masses and thedrip lines [25,26,27]. The relativistic Hartree-Bogoliubovapproach (RHB) has been used in the study of haloes inspherical nuclei [23,28,29]. Recently, the Relativistic Con-tinuum Hartree-Bogoliubov (RCHB) theory [23,29] wasused to construct a complete mass table and explore nu-cleon drip-lines [30] by assuming spherical symmetry through-out. The PC-PK1 relativistic energy density functionalwas used, which has been shown to provide a good descrip-tion of infinite nuclear matter and finite nuclei, includingthe isospin dependence of the binding energy along iso-topic and isotonic chains [31]. The ground-state propertiesof nuclei with proton number 8 to 120 were investigated.With the effects of the continuum included, there are to-tally 9035 nuclei predicted to be bound. In that work,it was found that the coupling between the bound states a r X i v : . [ nu c l - t h ] J u l Eun Jin In et al.: Neutron drip line in DRHBc: O to Ca and the continuum due to the pairing correlations playsan essential role in extending the nuclear landscape.Most known nuclei are deformed. In order to properlydescribe deformed exotic nuclei, the deformed relativisticHartree-Bogoliubov theory in continuum (DRHBc) wasdeveloped, initially based on the meson-exchange densityfunctional [32,33] and extended to the version with density-dependent meson-nucleon couplings [34] and to includethe blocking effect [35]. Continuum effects are includedby solving the deformed RHB equations in a Dirac Woods-Saxon basis [36]. Recently, the DRHBc with point-couplingdensity functionals was developed [37]. The aim is to pro-vide a unified description of even-even nuclei through-out the nuclear chart by including all important effects,namely continuum, pairing, and deformation, presently re-stricted to axial symmetry. The DRHBc theory in its var-ious iterations has been applied in the study of exotic Mgisotopes, predicting a decoupling between the core andthe halo in , Mg [32,33], in resolving the puzzle con-cerning the radius and configuration of valence neutronsin C [38], and a study of particles in the classically for-bidden regions of Mg isotopes [39].In the present investigation, we are interested in howthe inclusion of deformation degrees of freedom can affectthe predictions for the drip line. The theoretical questionposed is whether the additional degrees of freedom rep-resented by deformation and generally leading to morebinding can lead to a sizable extension of the nuclear land-scape. To this aim, we compare new results obtained withthe DRHBc with the RCHB predictions of Ref. [30] and ofRef. [40], which focuses on the region from Oxygen to Ti-tanium. In both the DRHBc and the RCHB calculations,the PC-PK1 functional is used. We focus on even-even nu-clei in the nuclear-chart region from Oxygen to Calcium.This paper is structured as follows. In Sec. 2 the DRHBcformalism and numerical implementation are presentedbriefly. In Sec. 3 we present and discuss our results. First,we examine how the deformation parameter evolves alongthe isotopic chains. Second, we report how the drip-linepredictions change with respect to RCHB predictions, i.e.,when we include deformation. Finally, we inspect the evo-lution of separation energies towards the drip line in Ne,Mg, and Ar isotopic chains in order to interpret and dis-cuss the results. We conclude in Sec. 4.
To describe the finite nuclear system, the starting pointis presently a Lagrangian density of the point-coupling model [41], L = ¯ ψ ( iγ µ ∂ µ − m ) ψ − α S (cid:0) ¯ ψψ (cid:1) (cid:0) ¯ ψψ (cid:1) − α V (cid:0) ¯ ψγ µ ψ (cid:1) (cid:0) ¯ ψγ µ ψ (cid:1) − α T V (cid:0) ¯ ψ τ γ µ ψ (cid:1) (cid:0) ¯ ψ τ γ µ ψ (cid:1) − β S (cid:0) ¯ ψψ (cid:1) − γ S (cid:0) ¯ ψψ (cid:1) − γ V (cid:2)(cid:0) ¯ ψγ µ ψ (cid:1) (cid:0) ¯ ψγ µ ψ (cid:1)(cid:3) − δ S ∂ ν (cid:0) ¯ ψψ (cid:1) ∂ ν (cid:0) ¯ ψψ (cid:1) − δ V ∂ ν (cid:0) ¯ ψγ µ ψ (cid:1) ∂ ν (cid:0) ¯ ψγ µ ψ (cid:1) − δ T V ∂ ν (cid:0) ¯ ψ τ γ µ ψ (cid:1) ∂ ν (cid:0) ¯ ψ τ γ µ ψ (cid:1) − F µν F µν − e − τ ψγ µ ψA µ , (1)where M is the nucleon mass, α S , α V and α T V representthe coupling constants for four-fermion contact terms, β S , γ S and α V are those for the higher-order terms which areresponsible for the effects of medium dependence, and δ S , δ V and δ T V represent the gradient terms which are in-cluded to simulate the finite range effects. A µ and F µν arerespectively, the four-vector potential and field strengthtensor of the electromagnetic field. The subscripts S, V and
T V stand for scalar, vector and isovector, respectively.By applying the mean-field theory to the Lagrangiandensity (1), the Hamiltonian density can be obtained bythe Legendre transformation. The RHB equation for thenucleons can be derived from the variational principle asfollows [22], (cid:18) h D − λ ∆ − ∆ ∗ − h ∗ D + λ (cid:19) (cid:18) U k V k (cid:19) = E k (cid:18) U k V k (cid:19) (2)where E k is the quasiparticle energy, U k and V k are thequasiparticle wave functions, and λ is the Fermi energy.The Dirac Hamiltonian h D is h D = α · p + β ( M + S ( r )) + V ( r ) , (3)where the scalar and vector potentials are, respectively, S ( r ) = α S ρ S + β S ρ S + γ S ρ S + δ S ∆ρ S , (4a) V ( r ) = α V ρ V + γ V ρ V + δ V ∆ρ V + eA + α T V τ ρ T V + δ T V τ ∆ρ T V . (4b)The local densities in Eqs.(4a) and (4b) are defined by ρ S ( r ) = (cid:88) k> ¯ V k ( r ) V k ( r ) , (5a) ρ V ( r ) = (cid:88) k> V † k ( r ) V k ( r ) , (5b) ρ T V ( r ) = (cid:88) k> V † k ( r ) τ V k ( r ) , (5c)which are constructed by the quasiparticle wave functions.The pairing potential for particle-particle channel reads ∆ kk (cid:48) ( r , r (cid:48) ) = − (cid:88) ˜ k ˜ k (cid:48) V kk (cid:48) , ˜ k ˜ k (cid:48) ( r , r (cid:48) ) κ ˜ k ˜ k (cid:48) ( r , r (cid:48) ) , (6) un Jin In et al.: Neutron drip line in DRHBc: O to Ca 3 which depends on the pairing tensor κ = U ∗ V T [42]. Weuse the density-dependent delta pairing force V pp ( r , r (cid:48) ) = V − P σ ) δ ( r − r (cid:48) ) (cid:18) − ρ ( r ) ρ sat (cid:19) , (7)where ρ sat is the nuclear matter saturation density. Thetotal energy of a nucleus is calculated by [41,29] E = (cid:88) k> ( λ τ − E k ) v k − E pair + E c . m . − (cid:90) d r (cid:20) α S ρ S + 12 α V ρ V + 12 α T V ρ + 23 β S ρ S + 34 γ S ρ S + 34 γ V ρ V + 12 ( δ S ρ S ∆ρ S + δ V ρ V ∆ρ V + δ T V ρ ∆ρ + ρ p eA ) (cid:21) . (8)The zero-range pairing force results in a local pairing field ∆ ( r ). The pairing energy is given by E pair = − (cid:90) d r κ ( r ) ∆ ( r ) . (9)The center-of-mass (c.m.) correction is calculated micro-scopically from the expectation value of the total momen-tum in the c.m. frame, E c . m . = −(cid:104) ˆ P (cid:105) / mA. (10)Axial deformation with spatial reflection symmetry istaken into account by expanding the potential ( S ( r ), V ( r ))and densities ( ρ S ( r ), ρ V ( r ), ρ T V ( r )) in terms of the Leg-endre polynomials [43], f ( r ) = (cid:88) λ f λ ( r ) P λ (cos θ ) , λ = 0 , , , · · · . (11)For the description of nuclei close to the drip line it is im-portant to include consistently both the continuum anddeformation effects and the coupling among all these fea-tures. A full treatment of the continuum by solving theequations in coordinate space, as done in the sphericalcase within the RCHB model [23,29], is not feasible atpresent. Instead, in the deformed case the continuum istaken into account by expanding the wavefunctions in theDirac Wood-Saxon (WS) basis [36]. The present numer-ical implementation is therefore called DRHBc, standingfor Deformed Relativistic Hartree-Bogoliubov theory incontinuum [33]. The numerical code for applying DRHBc and various tests,including convergence tests with respect to the basis andthe Legendre expansion, are presented in Ref. [37]. Fol-lowing those tests, for convergence of the DRHBc calcula-tions, we use the angular momentum cutoff J max = 23 / (cid:126) and the expansion order λ max = 6. The energy cutofffor the Dirac WS basis is taken as E cut = 300 MeV. At present we use the density functional PC-PK1 [31] for theparticle-hole channel. For the particle-particle channel, thezero-range pairing force with saturation density ρ sat =0 .
152 fm − [30] and the strength V = −
325 MeV fm isused. Since we use a zero-range pairing force, we have tointroduce a pairing cutoff over quasiparticle space. Here, asharp cutoff is adopted with a pairing window of 100 MeV.The pairing strength was determined so as to reproduceexperimental even-odd mass differences within the above-defined model space [37].For each isotope, unconstrained calculations are per-formed by using different initialization conditions, in par-ticular, different initial deformation parameter values β = − . , − . , . . . . − . , while the angular momentum cut-off was set to 19 / (cid:126) . Thus comparisons between DRHBcand RCHB results will reflect the deformation and pairingeffects combined. In order to test overall consistency, weverified that total binding energies of closed-shell nucleifrom DRHBc and those from RCHB are the same (withintens of keV). The main purpose of the present work is to investigate theeffect of the deformation degree of freedom on the neu-tron drip line location for even-even isotopes from O toCa. We begin by inspecting the evolution of deformationalong the isotopic chains for N ≥ Z . The O and Ca iso-topes are predicted spherical by DRHBc and therefore nochanges are expected with respect to RCHB predictions.The quadrupole deformation parameters for the Ne, Mg,Si, S, and Ar isotopic chains are shown in Fig. 1 up to oneisotope beyond the predicted drip-line nucleus. Many iso-topes, including very neutron-rich ones, are predicted de-formed. The proton and neutron deformation parameters( β ,p and β ,n ) are similar to each other (though not equal)in all the cases shown here, and we show the weighted av-erage deformation parameter given by β = ( Z/A ) β ,p +( N/A ) β ,n . All the experimental data are taken from theNNDC [44]. Eun Jin In et al.: Neutron drip line in DRHBc: O to Ca -0.40.00.40.8 -0.40.00.40.8-0.4-0.20.00.20.4 -0.4-0.20.00.20.4 -0.4-0.20.00.20.4 Ne (a) Mg (b) D e f o r m a t i o n Si (c) S (d) N-Z
DRHBc
Exp (e) Ar Fig. 1.
Deformation parameters β as a function of the differ-ence between the neutron and proton numbers N − Z for (a)Ne, (b) Mg, (c) Si, (d) S and (e) Ar isotopes. The experimentaldata are taken from the NNDC [44]. Filled circles denote thefirst isotope beyond the drip line. We note that only the absolute value, not the sign,of the deformation parameter is determined experimen-tally. In addition, the definitions of the experimental andthe calculated parameters are different: The experimental β parameter is determined from the electric quadrupoleexcitation spectrum while the theoretical one quantifiesthe degree of the intrinsic deformation of the mean field.Therefore, the values do not necessarily agree [45]. Note that the experimental β values of intrinsically sphericalnuclei are not necessarily zero.In some cases the deformation parameter seems to os-cillate at random between neighboring isotopes, for exam-ple, around Ne or S . It is because we show only thedeformation parameter of the very lowest-energy solutionin each case. However, the 8 < Z <
20 region examinedhere includes several candidates for shape coexistence orsoftness against deformation, which is borne out of our cal-culations. We find several isotopes with near-degeneratelocal energy minima (with energy difference of up to a fewhundred keV) for different deformation parameters. (Westress that there is no ambiguity for the predicted locationof the drip line as a result.) We show in Fig. 2 the energiesof some representative isotopes as a function of deforma-tion parameter, obtained by constrained DRHBc calcula-tions. For example, although Si is indicated as oblatelydeformed in Fig. 1(c), it is revealed that the energy of thespherically symmetric solution is within about 300 keVfrom the oblate solution. The energies of the local minimanear β = ± . Ar differ by less than 100 keV.The examples shown in Fig. 2 include very neutron-richAr isotopes for the discussion in Sec. 3.3. The theoretical prediction for the location of the neutrondrip line is determined by inspecting both the two neutronseparation energy S n and the neutron Fermi energy λ n .The former is defined as follows, S n ( Z, N ) = B ( Z, N ) − B ( Z, N − , (12)where B ( Z, N ) = − E ( Z, N ) is the binding energy of thenucleus with atomic number Z and neutron number N .When the two neutron separation energy is found positiveand the neutron Fermi energy is found negative, the nu-cleus is classified as bound. The last bound isotope foundalong an isotopic chain defines the location of the neutrondrip line. The heaviest even isotopes of the O–Ca elementsdiscovered so far are O (somewhat unbound), Ne , Mg , Si , S , Ar , and Ca [46,47,48,49,50,51,52,53,54].In Ref. [30] it was shown that the coupling of boundstates with continuum states due to pairing correlationscan extend considerably the theoretically predicted nu-clear landscape. In Ref. [40], which focused on the regionfrom Oxygen to Titanium, RCHB results were comparedwith those of various mass models with no explicit treat-ment of the continuum. It was concluded that including aproper description of the continuum can extend the neu-tron drip line by several isotopes for each element.Here we explore the deformation effects on the neu-tron drip line location by comparing the results from theDRHBc theory with those from the RCHB theory [30].The predicted location of the drip line will of course de-pend on the density functional parameterization used (here,PC-PK1). In this work, we are mainly interested in pos-sible deformation effects on the neutron drip lines in arelative sense. un Jin In et al.: Neutron drip line in DRHBc: O to Ca 5 -268-264-260-256 -364-360-356-424-422-420 -426-423-420 -0.4 -0.2 0.0 0.2 0.4 0.6 -426-423-420 Si (a)(b) 48 S (c) 62 Ar Ar (d) 72 Ar (e) E ( M e V ) Deformation
Contrained Unconstrained
Fig. 2.
Energy as a function of the deformation parameter β obtained via constrained DRHBc calculations for selectedisotopes: (a) Si , (b) S , (c) Ar , (d) Ar , and (e) Ar (unbound). The solution obtained by unconstrained cal-culations is also indicated. The results are summarized and visualized in Fig. 3.In all the cases the predicted drip line lies beyond the iso-topes experimentally discovered so far. The drip-line lo-cations are not affected with the inclusion of deformation,except for the Ne and Ar isotopic chains. As expected, thedrip O and Ca isotopes are predicted the same in bothcases, because all those isotopes are predicted spherical.We also find that the last two or three bound S and Siisotopes are predicted spherical (see Fig. 1). On the con-
NRCHBDRHBcONeMgSiSArCa 20 28 36 44 52 60
Element (Z) neutron number (N)RCHB DRHBcO (8) 20 20Ne (10) 32 28Mg (12) 34 34Si (14) 38 38S (16) 40 40Ar (18) 44 52Ca (20) 60 60
Fig. 3.
The neutron drip-line location predicted in the presentwork by the DRHBc theory (with deformation) in comparisonwith the RCHB results (without deformation) [30] for even-even isotopes with 8 ≤ Z ≤
20. The arrows show the directionof the changes with the inclusion of deformation. The neutronnumbers at the drip line are tabulated below the figure. trary, nuclei in the vicinity of the drip line are predicteddeformed for Ne, Mg, and Ar.It is especially interesting that, while the Ar isotopicchain is extended to higher N when deformation is in-cluded, augmenting the continuum effects, the neutrondrip line shrinks in the case of Ne isotopes. There is nochange in Mg due to the inclusion of deformation. Wetherefore proceed to inspect these isotopes in more detail. For Ne, Mg, and Ar isotopes we inspect the quantities S n and λ n which define the location of the drip line. Figure 4(a) shows two neutron separation energy ( S n ) of Ne iso-topes with respect to the neutron number. In the RCHBcalculations (green lines), two neutron separation energies( S n ) remain positive until the neutron number 32. How-ever, the value of S n from the DRHBc calculations (redlines) remain positive up to the neutron number 28. Fig-ure 4 (b) shows neutron Fermi energies ( λ n ) of Ne isotopesand they stay negative up to the neutron number 32 in theRCHB calculation. However, they stay negative up to 30in the DRHBc calculations. Therefore, the neutron drip-line location of Ne isotopes is predicted to be Ne insteadof Ne.Figure 5 shows the same quantities for Mg isotopes.There are sizable differences in the predictions towards
Eun Jin In et al.: Neutron drip line in DRHBc: O to Ca -24-16-80 (a) Ne S n ( M e V ) DRHBc
RCHB
Exp
PC-PK1 (b) n ( M e V ) Neutron number N
Fig. 4. (a) Two neutron separation energy ( S n ) and (b)neutron Fermi energy ( λ n ) of Ne isotopes. The red lines withempty circles are for DRHBc calculations and the green lineswith empty diamonds are for RCHB calculations. The experi-mental data taken from the NNDC [44] are shown by the blackfilled squares. the drip line but not for the drip nucleus, N = 34. Inaddition, the next isotope, unbound with N = 36, is pre-dicted spherical. Therefore, the situation turns out to besimilar to the Si, S cases.Figure 6 shows the same quantities for Ar isotopes. Inthe RCHB calculations S n remains positive until the neu-tron number becomes 44, while in the DRHBc calculations S n is positive until the neutron number 52. Similarly, λ n values of Ar isotopes stay negative up to the neutronnumber 46 in the RCHB calculations and up to 52 in theDRHBc calculations. Therefore, the neutron drip-line lo-cation of Ar isotopes is predicted to be Ar instead of Ar.For both Ne and Ar the differences between the twomodels appear marginal at large N , but they are suffi-cient to shift the drip-line location. Furthermore, the dif-ferences in the proximity of the drip line have oppositesigns for Ne and Ar leading to the shrinking of the dripline in the former case when deformation is taken into ac-count and to extension in the latter. The question arises asto whether such results are consistent with the generallystronger binding expected from the DRHBc calculations.Indeed, when deformation is included in the calcula-tions, but everything else (interaction, cut-off parameters,etc.) kept the same as in the calculations in spherical sym-metry and if a deformed solution is found to be the groundstate, the deformed state must be more bound than the -24-16-80 (a) Mg S n ( M e V ) DRHBc
RCHB
Exp
PC-PK1 (b) n ( M e V ) Neutron number N
Fig. 5.
Same as Fig. 4 but for Mg. -24-120 (a) Ar S n ( M e V ) DRHBc
RCHB
Exp
PC-PK1 (b) n ( M e V ) Neutron number N
Fig. 6.
Same as Fig. 4 but for Ar.un Jin In et al.: Neutron drip line in DRHBc: O to Ca 7 spherical solution by definition. To put it differently, by in-cluding more degrees of freedom (deformation) we expectequal or lower energies by virtue of the variational prin-ciple. Then generally the DRHBc calculations should givemore (or equally) bound solutions than the RCHB calcu-lations. (Exceptions may still occur, especially for near-spherical nuclei, owing to the different numerical imple-mentations and different pairing parameters used in thetwo types of calculations.) As s result, an extension ofthe drip line might be expected rather than a shrinking.On the other hand, it is not the value of the binding en-ergy that matters, but the difference between the bindingenergies of neighboring isotopes. The difference betweenthe two-neutron separation energy predicted by DRHBc, S n (DRHBc), and that predicted by the spherical modelRCHB, S n (RCHB), is given by ∆S n ≡ S n (DRHBc) − S n (RCHB)= ∆B ( Z, N ) − ∆B ( Z, N − , (13)where we have denoted by ∆B = B (DRHBc) − B (RCHB)the difference in the binding energies of each model. Eventhough both ∆B ( Z, N ) and ∆B ( Z, N −
2) are expectedto be positive, the difference ∆S n can be either positiveor negative. If ∆B ( Z, N ) is smaller than ∆B ( Z, N − N . Let us thereforeinvestigate ∆B and ∆S n .The difference in binding and separation energies forthe three isotopic chains Ne, Mg, and Ar is shown inFig. 7. As expected from Eq. (13) ∆S resembles a differ-ential of ∆B : an increase (decrease) of ∆B with N leadsto a positive (negative) value for ∆S . For Ar, the sep-aration energy difference increases towards the drip line.Thus, for example, ∆B ( Z = 18 , N = 46) is larger than ∆B ( Z = 18 , N = 44). The increase in the gain of bind-ing owing to deformation is consistent with the increasein the (absolute value of the) deformation parameter. (SeeFig. 1(e).) The evolution of the Ar energy landscape to-wards and beyond the neutron drip line is also exemplifiedby Fig. 2(c),(d), and (e). Ar represents a transitionalsituation between the spherical , Ar isotopes and theheavier oblate isotopes, whose absolute deformation para-mater increases with N until N becomes equal to 52 andthe drip line is reached. Such a trend can lead to an ex-tension of the drip line. On the contrary, for Ne the trendof the separation energy difference is to drop towards thedrip line. Thus ∆B ( Z = 10 , N = 28) is smaller than ∆B ( Z = 10 , N = 26). The drop in the gain in bindingowing to deformation is consistent with the decrease inthe deformation parameter. (See Fig. 1(a).) Such a trendcan lead to the shrinking of the drip line. As obvious fromthe Mg case, the above are not sufficient conditions forthe respective results to occur. However, the condition ∆S n < ∆B becomes negative for someneutron numbers, in which case the DRHBc solutions areless bound than the RCHB ones. First, as already men- -20246810 6 12 18 24 30 36 42 48-8-4048 B ( M e V ) Ne Mg Ar (a)(b) S n ( M e V ) Neutron number N
Fig. 7.
The differences ∆B and ∆S n between results ofthe DRHBc calculations (with V = −
325 MeV fm ) and theRCHB calculations (with V = − . ). tioned, some minor numerical discrepancies are expectedfrom the model space cut-off parameters whose optimalvalues differ for each of the numerical implementationsand are related to the WS basis treatment of continuumstates. More importantly, the optimal pairing parameterfor DRHBc is found weaker than in RCHB leading to lessbinding coming from pairing correlations. In order to in-vestigate this effect, we have performed the same calcula-tions and comparisons by using the same pairing strengthas in RCHB, V = − . . We found that thegeneral trends for ∆B , ∆S n do not change. The maindifference is a small positive shift for ∆B , leading to bet-ter compliance with the variational principle. Even-even neutron-rich isotopes from O to Ca were in-vestigated by using the DRHBc theory with the PC-PK1functional. The neutron drip-line location was determinedby calculating the two-neutron separation energies andthe neutron Fermi energies. In order to investigate thedeformation effect, on the neutron drip-line location, wecompared the present results with those predicted by theRCHB theory with spherical symmetry. We found that theNe and Ar drip line nuclei are different when the deforma-tion effect is included. The direction of the change in theneutron drip line is not necessarily towards an extendeddrip line. It rather appears dependent upon the evolutionof the degree of deformation (magnitude of the deforma-
Eun Jin In et al.: Neutron drip line in DRHBc: O to Ca tion parameter) towards the drip line: When the drip linenuclei are predicted spherical, the drip line doesn’t change;for Ne the deformation decreases towards the drip line andthe drip line “shrinks”; the opposite is seen for Ar. We con-clude that taking into account deformation effects as wellas pairing and continuum effects in a consistent way canaffect critically the theoretical description of the neutrondrip-line location.It would be interesting to see if similar trends are ob-served in different regions of the nuclear chart. Also shapecoexistence and isomerism remain important topics for fu-ture work. Several investigations are currently in progresswithin the broader DRHBc mass table collaboration [37,38,39,55].
Acknowledgments
We thank the members of the DRHBc Mass Table Col-laboration for useful discussions. The work was supportedpartly by the Rare Isotope Science Project of Institute forBasic Science funded by Ministry of Science, ICT and Fu-ture Planning, and NRF of Korea (2013M7A1A1075764).EJI and SWH were supported in part by the Korea gov-ernment MSIT through the National Research Foundation(2018M2A8A2083829).
References
1. D. Lunney, J. M. Pearson, and C. Thibault. Recent trendsin the determination of nuclear masses.
Rev. Mod. Phys. ,75:1021–1082, 2003.2. J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Long,and L.S. Geng. Relativistic continuum Hartree Bogoliubovtheory for ground-state properties of exotic nuclei.
Prog.Part. Nucl. Phys. , 57(2):470 – 563, 2006.3. J. Meng and S.G. Zhou. Halos in medium-heavy and heavynuclei with covariant density functional theory in contin-uum.
J. Phys. , G42:093101, 2015.4. Shan-Gui Zhou. Multidimensionally constrained covariantdensity functional theories—nuclear shapes and potentialenergy surfaces.
Phys. Scr. , 91:063008, 2016.5. Raj K. Gupta, M. Balasubramaniam, Rajeev K. Puri, andWerner Scheid. The Halo structure of neutron-drip linenuclei: (Neutron) cluster-core model.
J. Phys. , G26:L23–L32, 2000.6. Masayuki Matsuo and Takashi Nakatsukasa. Open prob-lems in nuclear structure near drip lines.
J. Phys. ,G37:064017, 2010.7. Ikuko Hamamoto. Neutron shell structure and deforma-tion in neutron-drip-line nuclei.
Phys. Rev. , C85:064329,2012.8. A. V. Afanasjev, S. E. Agbemava, D. Ray, and P. Ray.Neutron drip line: Single-particle degrees of freedom andpairing properties as sources of theoretical uncertainties.
Phys. Rev. , C91:014324, 2015.9. Rui Wang and Lie-Wen Chen. Positioning the neutrondrip line and the r-process paths in the nuclear landscape.
Phys. Rev. , C92:031303, 2015. 10. L´eo Neufcourt, Yuchen Cao, Witold Nazarewicz, ErikOlsen, and Frederi Viens. Neutron drip line in the Caregion from Bayesian model averaging.
Phys. Rev. Lett. ,122:062502, 2019.11. D. Savran, T. Aumann, and A. Zilges. Experimental stud-ies of the Pygmy Dipole Resonance.
Prog.Part.Nucl. Phys. ,70:210 – 245, 2013.12. Takaharu Otsuka, Alexandra Gade, Olivier Sorlin, ToshioSuzuki, and Yutaka Utsuno. Evolution of shell structurein exotic nuclei.
Rev. Mod. Phys. , 92:015002, 2020.13. I. Tanihata. Nuclear structure studies from reaction in-duced by radioactive nuclear beams.
Prog. Part. Nucl.Phys. , 35:505 – 573, 1995.14. J. Meng and P. Ring. A Giant halo at the neutron dripline.
Phys. Rev. Lett. , 80:460, 1998.15. P. Egelhof et al. Nuclear-matter distributions of halo nucleifrom elastic proton scattering in inverse kinematics.
Eur.Phys. J. A , 15:27–33, 2002.16. Isao Tanihata and Rituparna Kanungo. Halo and skinnuclei.
Comptes Rendus Physique , 4(4-5):437–449, 2003.17. M. V. Stoitsov, J. Dobaczewski, W. Nazarewicz, S. Pittel,and D. J. Dean. Systematic study of deformed nuclei atthe drip lines and beyond.
Phys. Rev. , C68:054312, 2003.18. Wen Hui Long, Peter Ring, Jie Meng, Nguyen Van Giai,and Carlos A. Bertulani. Nuclear halo structureand restoration of relativistic symmetry.
Phys. Rev. ,C81:031302, 2010.19. Virender Thakur and Shashi K. Dhiman. A Study ofCharge Radii and Neutron Skin Thickness near NuclearDrip Lines.
Nucl. Phys. , A992:121623, 2019.20. G. Saxena, M. Kumawat, and Mamta Aggarwal. Searchfor exotic features in the ground state light nuclei with10 ≤ Z ≤
18 from stable valley to drip lines.
Int. J. ofMod. Phys. E , 28(11):1950101, 2019.21. P. G. Hansen and B. Jonson. The Neutron halo of ex-tremely neutron-rich nuclei.
Europhys. Lett. , 4:409–414,1987.22. H. Kucharek and P. Ring. Relativistic field theory of su-perfluidity in nuclei.
Z. Phys. , A339(1):23–35, 1991.23. J. Meng and P. Ring. Relativistic Hartree-Bogoliubov De-scription of the Neutron Halo in 11Li.
Phys. Rev. Lett. ,77:3963–3966, 1996.24. J. Meng, H. Toki, J. Y. Zeng, S. Q. Zhang, and S.-G. Zhou.Giant halo at the neutron drip line in Ca isotopes in rela-tivistic continuum Hartree-Bogoliubov theory.
Phys. Rev. ,C65:041302, 2002.25. S. Goriely, N. Chamel, and J. M. Pearson. Skyrme-Hartree-Fock-Bogoliubov nuclear mass formulas: Crossingthe 0.6 MeV accuracy threshold with microscopically de-duced pairing.
Phys. Rev. Lett. , 102:152503, 2009.26. S. Goriely, S. Hilaire, M. Girod, and S. P´eru. First Gogny-Hartree-Fock-Bogoliubov nuclear mass model.
Phys. Rev.Lett. , 102:242501, 2009.27. Jochen Erler et al. The limits of the nuclear landscape.
Nature , 486:509–512, 2012.28. W. Poschl, D. Vretenar, G. A. Lalazissis, and P. Ring.Relativistic Hartree-Bogolyubov theory with finite rangepairing forces in coordinate space: Neutron halo in lightnuclei.
Phys. Rev. Lett. , 79:3841–3844, 1997.29. Jie Meng. Relativistic continuum Hartree-Bogoliubov the-ory with both zero range and finite range Gogny force andtheir application.
Nucl. Phys. , A635:3–42, 1998.un Jin In et al.: Neutron drip line in DRHBc: O to Ca 930. X.W. Xia et al. The limits of the nuclear landscape ex-plored by the relativistic continuum Hartree–Bogoliubovtheory.
Atomic Data and Nuclear Data Tables , 121:1–215,2018.31. P. W. Zhao, Z. P. Li, J. M. Yao, and J. Meng. Newparametrization for the nuclear covariant energy densityfunctional with point-coupling interaction.
Phys. Rev. ,C82:054319, 2010.32. Shan-Gui Zhou, Jie Meng, P. Ring, and En-Guang Zhao.Neutron halo in deformed nuclei.
Phys. Rev. , C82:011301,2010.33. Lulu Li, Jie Meng, P. Ring, En-Guang Zhao, and Shan-GuiZhou. Deformed relativistic Hartree-Bogoliubov theory incontinuum.
Phys. Rev. , C85:024312, 2012.34. Ying Chen, Lulu Li, Haozhao Liang, and Jie Meng.Density-dependent deformed relativistic hartree-bogoliubov theory in continuum.
Phys. Rev. , C85:067301,2012.35. Lu-Lu Li, Jie Meng, P. Ring, En-Guang Zhao, and Shan-Gui Zhou. Odd systems in deformed relativistic hartreebogoliubov theory in continuum.
Chin. Phys. Lett. ,29(4):042101, 2012.36. Shan-Gui Zhou, Jie Meng, and Peter Ring. Spherical rel-ativistic Hartree theory in a Woods-Saxon basis.
Phys.Rev. , C68:034323, 2003.37. Kaiyuan Zhang et al. Toward a nuclear mass table withthe continuum and deformation effects: even-even nucleiin the nuclear chart. 2020. arXiv:2001.06599.38. Xiang-Xiang Sun, Jie Zhao, and Shan-Gui Zhou. Shrunkhalo and quenched shell gap at N=16 in 22C: Inversion ofsd states and deformation effects.
Phys. Lett. B , 785:530– 535, 2018.39. K. Y. Zhang, D. Y. Wang, and S. Q. Zhang. Effects ofpairing, continuum, and deformation on particles in theclassically forbidden regions for Mg isotopes.
Phys. Rev. ,C100:034312, 2019.40. XiaoYing Qu et al. Extending the nuclear chart by con-tinuum: From oxygen to titanium.
Sci. China Phys. Mech.Astron. , 56:2031 – 2036, 2013.41. Jie Meng.
Relativistic density functional for nuclear struc-ture . World Scientific, 2016.42. Peter Ring and Peter Schuck.
The nuclear many-body prob-lem . Springer Science & Business Media, 2004.43. C. E. Price and G. E. Walker. Self-consistent Hartree De-scription of Deformed Nuclei in a Relativistic QuantumField Theory.
Phys. Rev.
Atomic Data and Nuclear Data Tables ,78:1–128, 2001.46. E. Lunderberg et al. Evidence for the ground-state reso-nance of O. Phys. Rev. Lett. , 108:142503, 2012.47. M. Notani et al. New neutron-rich isotopes, Ne, Naand Si, produced by fragmentation of a 64 A MeV Cabeam.
Phys. Lett. B , 542(1):49 – 54, 2002.48. D. S. Ahn et al. Location of the neutron dripline at Fluo-rine and Neon.
Phys. Rev. Lett. , 123:212501, 2019.49. T. Baumann et al. Discovery of Mg and Al suggestsneutron drip-line slant towards heavier isotopes.
Nature ,449(7165):1022–1024, 2007. 50. O. B. Tarasov et al. New isotope Si and systematicsof the production cross sections of the most neutron-richnuclei.
Phys. Rev. C , 75:064613, 2007.51. M. Lewitowicz et al. First observation of the neutron-richnuclei Si, , P, S, and Cl from the interaction of 44MeV/u Ca + Ni.
Z. Phys. A , 335:117–118, 1990.52. O. B. Tarasov et al. Discovery of Ca and implications forthe stability of Ca.
Phys. Rev. Lett. , 121:022501, 2018.53. M. Thoennessen. Discovery of isotopes with Z ≤ Atomic Data and Nuclear Data Tables , 98:43 – 62, 2012.54. M. Thoennessen. Discovery of the isotopes with 11 ≤ Z ≤ Atomic Data and Nuclear Data Tables , 98:933 – 959,2012.55. Cong Pan, Kaiyuan Zhang, and Shuangquan Zhang. Mul-tipole expansion of densities in the deformed relativisticHartree-Bogoliubov theory in continuum.