Estimate of the location of the neutron drip line for calcium isotopes from an exact Hamiltonian with continuum pair correlations

The eastern region of the calcium isotope chain of the nuclei chart is, nowadays, of great activity. The experimental assessment of the limit of stability is of interest to confirm or improve microscopic theoretical models. The goal of this work is to provide the drip line of the calcium isotopes from the exact solution of the pairing Hamiltonian which incorporates explicitly the correlations with the continuum spectrum of energy. The modified Richardson equations, which include correlations with the continuum spectrum of energy modeled by the continuum single particle level density, is used to solve the many-body system. Three models are used, two isospin independent models with core 40Ca and 48Ca, and one isospin dependent model. One and two-neutron separation energies and occupation probabilities for bound and continuum states are calculated from the solution of the Richardson equations. The one particle drip line is found at the nucleus 57Ca, while the two neutron drip line is found at the nucleus 60Ca from the isospin independent model and at 66Ca from the isospin dependent one.


I. INTRODUCTION
The lego-like construction of isotopes for a given atomic nucleus, sooner or later faces the particle continuum. For example, the last observed bound Fluorine is 31 F [1], while the last bound Oxygen is 24 O [2][3][4][5]. This simple comparison between two elements which defer only in a single proton, shows the complicated character of drip lines systems, posing a big challenge to nuclear structure models. Interaction [6], continuum [7] and many-body correlations [8], all together collude in this kingdom [9,10].
The isotopic chain of calcium is currently under scrutiny from both the theoretical and experimental aspects. A handful of nuclei 59 Ca and 60 Ca have been recently observed [11], they are the heaviest calcium isotopes discovery up to today and both were found to be bound. Their masses are not known yet, the more recent measured atomic mass is that of 57 Ca [12]. The calcium chain is also interesting because it allows the investigation for existence of doubly magic nuclei and the evolution of the charge radius [13][14][15][16][17][18][19][20].
This paper focuses on the stability limit of the calcium isotopes. We have to wait for updating [21] or finishing some facilities to get masses for isotopes of calcium beyond 57 Ca. For example, the Facility for Rare Isotope Beams (FRIB) [22] will measure the key nucleus 60 Ca, recently discovered at RIKEN [11]. Meanwhile, different theoretical approaches are implemented to predict the calcium drip line. Some formalism predicts it as soon as around 60 Ca [8,[23][24][25], while others predict the drip line at 68 Ca [26,27], or even 70 Ca [9,11,28].
Pairing encompasses an important part of the shortrange interaction between the neutrons [7,29]. Various approaches have been developed in the last fifty years [30] to incorporate pairing in finite nuclei. The Gorkov field theory approach [31,32] properly account for the pairing correlations in many-body systems. Its application to finite nuclei was recently developed [33,34] and applied to the calcium isotopes [25,35]. Exact results are important in many-body systems, the algebraic Gorkov solution for the separable interaction was given in Ref. [36], while in this paper we study the calcium chain from the exact solution of the pairing Hamiltonian [37][38][39]. The correlations with the continuum spectrum of energy is included through the continuum single particle level density [40].
In section II we give the theoretical tools used in this paper; with the outline of the method for solving exactly the many-body system with pairing in the continuum for even nuclei, in subsection II A. In subsection II B we relate the calculated magnitudes with the occupation probabilities and the binding energy for even and odd isotopes. In section III we develop the application to the calcium isotope chain. In subsection III A we deal with the isospin independent model, while in subsection III B the isospin dependent approach is used to determine the neutron drip line. The last section IV is reserved for discussions and conclusions.

A. Exact pairing solution
The Hamiltonian of a many-body system which includes pair scattering to the continuum may be written in terms of a set of negative and positive energy states, corresponding to bound and scattering states, respectively. For a constant pairing interaction the Hamiltonian is given by, where ε a are the discrete energies with degeneracŷ j 2 a = 2j a + 1, α = {a, m α } = {n a , l a , j a , m α } and c † α = (−1) ja−mα c † a,−mα . Following the derivation of Ref. [41], we may take the limit of the size of the spherical box to infinity, and keep only the physical relevant part of the single particle level density [42]. In this way, for a system with N particles, we end up with N pair = N/2 couple equations, which take into account continuum correlations [39,43], for i = 1, · · · , N pair , where ε b are the bound energy levels with quantum numbers {n b , l b , j b }, E i are the Richardson energies which are parameters of the formalism, related to the many-body energy E of the system [37,44] by, and g(ε) is the Continuum Single Particle Level Density (CSPLD) [42], where l max is an upper limit for the number of partial waves. Notice, in Eq. (2), that while the correlations between bound states are the same for all shells, the strength between continuum states is modulated by the CSPLD [39,40].
The solution of the N pair Richardson equations with the boundary conditions, with i = 1, · · · , N pair the lowest states, determine the ground-state energy of the pairing Hamiltonian of the N = 2N pair nucleus, where, the pair degeneracyĵ 2 i /2 of the level ε i must be taken into account [39]. For example, the isotope 44 Ca, considered as a core 40 Ca plus four neutrons, corresponds to solve two algebraic couple equations (2) with the boundary conditions, lim G→0 + E 1 = 2ε 1 and lim G→0 + E 2 = 2ε 1 , where ε 1 = ε f 7/2 . In this case, the single particle energy limits are the same because the pair degeneracy of the shell f 7/2 is four. Then, the ground-state energy is given by Eq. (3), i.e.
We will consider the independent and dependent isospin cases [45,46], where I = N −Z A .

B. One and two-neutron separation energies
The drip line becomes defined by the conditions S n ≤ 0 and S 2n ≤ 0. Let us consider A = A core + N , where A core is the inert core from where the mean-field Hamiltonian is set up, and N = 2N pair . Then, the two-neutron separation energy from the Richardson formalism is given by, with E(N pair ) from Eq. (3). While the one-neutron separation energy is calculated from the approximate equation [47,48], with λ F and ∆ the Fermi level and pairing gap, respectively, calculated in the blocking approximation, i.e. λ F (2N pair ) and ∆(2N pair ); while ∂λF ∂N is calculated in Sec. III B 3.
From the Richardson formalism, the Fermi level and the pairing gap can be calculated by combining the BCS equations with continuum spectrum [40], and the occupation probabilities, were we have extended the definition [38] to the continuum spectrum of energy, and we have introduced a cutoff ε max . For a given nucleus N = 2N pair , we solve the Richardson equations (2) for many strengths G. Then, from Eqs. (11) and (12) we calculated the occupation probabilities by finite differences. By substituting these results in Eq. (9), we obtain the pairing gap. Finally, with this value of ∆, we fit λ F from Eq. (10). In this way, the Fermi level and the pairing gap have been obtained for each even nucleus. Using these parameters in Eq. (8) we get the one-neutron separation energy S n for the A + 1 nucleus.
In the applications we also will show binding energy for the even A = A core + 2N pair and odd A + 1 isotopes, given by, were E Bin (A core ) will be taken from experimental data.

A. Isospin independent model
We begin with the calculation of the drip line for the calcium isotopes in the isospin independent approximation.

Single particle representation
Even when the solution of the reduced pairing Hamiltonian does not require the single-particle wave function of the mean-field but only the energies, in our formulation we make use of the single particle density Eq. (4), which requires the continuum eigenfunctions, and so, we need to define a mean-field. The Woods-Saxon and spinorbit parameters were constrained by experimental data and χ 2 optimization.
In this section we consider fixed strengths for the meanfield of the cores 40 Ca and 48 Ca. We will take the same reduced radius and diffuseness for both cores, in preparation for section III B, where the strengths of both cores will be joined smoothly. The reduced radius r 0 = 1.28 fm is extracted from the experimental neutron root-meansquare r n = 3.555 fm for 48 Ca [49] and the relation r n = 3/5R. For the diffuseness we take a = 0.75 fm in order to get into consideration the enhancement of the nuclear size reported in Ref. [17] which is justified by and increase in the surface diffuseness of the neutron density distribution. Finally, the strengths are optimized by χ 2 using the Levenberg-Marquardt algorithm [50]. Due to the fragmentation of the single particle states in the nuclei 41 Ca and 49 Ca, we take as experimental energies, the average of the fragmented levels weighted with its respective spectroscopic factor [51]. The optimized strength with their errors are shown in Table I. The left and center panels of Fig. 1 compare the average experimental neutron levels of 41 Ca and 49 Ca [52], with that calculated using the code GAMOW [53] with the parameters of Table I. The right panel shows the continuum single-particle level density g(ε) Eq. (4), with l max = 15. The scattering states were calculated using the code ANTI [54,55]. The peaks are manifestation of the single particle resonances, which are labeled following the usual convention for bound-state shells. We observe that resonances move to the continuum threshold while they became narrower when changing from 41 Ca to 49 Ca. The figure shows a near degeneracy of the levels 1g 9/2 and 2d 5/2 for both nuclei [56], which manifest as a single peak in the 49 Ca. In section III B we will show the evolution of the single particle levels with A.

Binding energy
Using the two single particle model spaces for the cores 40 Ca and 48 Ca, formed by the bound and continuum states of Fig. 1, we solve the Richardson equations (2) for the calcium isotopes. Then, using Eqs. (3) and (13) we calculate the binding energy of the even isotopes.
The pairing strength G is parametrized by Eq. (6) with χ 2 = 0. The reduced pairing strength χ 1 , for each core, were fixed in order to reproduce the experimental binding energy of two nuclei, one for the core 40 Ca, and another for the core 48 Ca. Table II shows the value of the parameter χ 1 and compare the calculated and the experimental binding energy of the nuclei 50 Ca and 54 Ca used as reference.
Using the reduced pairing strength χ 1 of Table II Figure 2. The two-neutron separation energy, calculated using Eq. (7), is shown in the inset. The results of both model spaces follow the experimental energy till the nucleus 54 Ca, and then, the solutions using the core 40 Ca does a better job. Both model spaces found the two-neutron drip line at the nucleus 60 Ca, in concordance with Refs. [8,[23][24][25]58].
Since the selection of the nuclei 50 Ca and 54 Ca (Table II) to fix the reduced pairing strength was arbitrary, we considered a second pair of reference nuclei, 44 Ca and 52 Ca, for the model spaces with core 40 Ca and 48 Ca, respectively. With the new pair of reduced pairing strengths χ 1 we calculate the binding energy, and compare them with the previous one in Fig. 3. The new calculations found the two-neutron drip line at 60 Ca, for both model spaces, in agreement with the previous parametrization.
Motivated by the analysis of Ref. [11] and other theoretical predictions [27,28], we consider, in the next section, the dependence of isospin on the mean field and on the pairing force for the determination of the drip line.

B. Isospin dependent model
In this section we will consider the solution of the Richardson equations from the core 48 Ca, with an isospin dependent single-particle model space and isospin dependent pairing strength.  Table II for each one of the two model spaces. The experimental data were taken from [57]. The inset shows the two-neutron separation energies.

Single particle representation
The single particle bound states and the CSPLD will change smoothly from isotope to isotope according to the following isospin dependent Woods-Saxon and spin-orbit strengths [59], with I = N −Z A . The four parameters η, shown in Table  III, were fixed using the four strengths of Table I optimized by χ 2 minimization in the previous section.
The evolution of the bound levels of Fig. 1 Fig. 4. They were calculated using the code GAMOW [53], with r 0 = 1.28 fm, a = 0.75 fm and the isospin dependent strength Eqs. (15) and (16), with the parameters of Table III. From the figure can be appreciated the inversion [23] and the near degeneracy [56] of the levels 2d 5/2 and 1g 7/2 . The figure shows the transition of the state g 9/2 from a resonance to a bound state. This behavior seems to be a consequence of the increasing of the effective spin-orbit strength with l, and the enhancement of the centrifugal barrier, which is proportional to l(l + 1). These two factors are more pronounced for the g 9/2 shell. Figure 4 also shows two gaps between the shells 2p 3/2 -2p 1/2 and 2p 1/2 -1f 5/2 , which are consistent with the shell closure of the nuclei 52 Ca and 54 Ca [12,18,19,28,60]. The weakening of shell closure at 60 Ca, due to the tendency of the shell g 9/2 , is in agreement with Ref. [28], but we do not find a shell closure at 70 Ca [28]. The continuum spectrum of energy enters the manybody calculation through the continuum single particle level density, which also smoothly changes from isotope to isotope. In Fig. 5 we show, as an example, how the  The mean-field is the same as that used to construct Fig. 4.

Two-neutron separation energy
In this section we solve the Richardson equations (2) for the even isotopes from 50 Ca to 74 Ca. The core is taken to be the nucleus 48 Ca, with the model space as described in the previous section III B 1. The isospin pairing strength G is modeled by Eq. (6), with the parameters χ 1 and χ 2 optimized to reproduce the experimental binding energy of the nuclei 54 Ca and 58 Ca, Table IV. The calculated binding energy of the even isotopes and the two-neutron separation energy is shown in Fig. 6. The figure shows that the last even isotope is 66 Ca. This result is consistent with that of Ref. [27] which found the nuclei 66 Ca and 68 Ca to be bound, with a probability 67%− 84%. Reference [28] finds a pronounced smoothing of the binding energy for the isotopes 66 Ca-70 Ca, with the drip line at the nucleus 70 Ca. Since the pair of nuclei used to fix the reduced pairing strengths χ 1 and χ 2 have nothing of particular, we repeated the calculation fixing the reduced pairing strengths using the experimental binding energy of the nuclei 52 Ca and 56 Ca. Figure 6 shows the calculation with the new pair of χ 1 and χ 2 . We observe a difference in the binding energy using the two different set of parameters, while there is a good agreement for twoneutron separation energy. The second parametrization also finds the drip line at the nucleus 66 Ca. By solving the Richardson equations for the pairing  strength with the parameters of Table IV, we calculate the occupation probability v 2 b and v 2 (ε) for the bound and continuum states from Eqs. (11) and (12), respectively. Figure 7 shows some examples; it can be observed how the occupation probabilities of the continuum levels, ε > 0, monotonically increase as the number of particles increases. With the calculated occupation probabilities we get the discrete ∆ b and continuum ∆ c gap parameters from Eq. (9) with ε max = 100 MeV. Figure 8 shows the total gap ∆ discriminate by the discrete and continuum parts. The profile of the total gap is the usual for a strong pairing. Using the three-point formula [62], with the experimental binding energies from Ref. [57], we calculate the experimental gap ∆ exp ; except for the nucleus 52 Ca, our gap are greater than the experimental one. The figure shows that ∆ c increases while ∆ b remains more or less constant up to the isotope 64 Ca, where both suddenly change, but, the total pairing gap ∆ remains smooth. The abrupt change of ∆ b and ∆ c is due that the state 1g 9/2 becomes a bound state, as it can be seen in Fig. 4.

Pairing in the continuum
Finally, with the calculated gap ∆, we determine the corresponding Fermi level by optimizing the parameter λ F in Eq. (10) using Levenberg-Marquardt algorithm [63]. Figure 9 shows the optimized values with their errors.
For the determination of the one-neutron separation energy Eq.

Gap [MeV]
FIG. 8: Pairing gap (9) discriminated by bound ∆ b and continuum ∆ c contributions. The experimental gap was calculated using the three-point formula [62], with binding energies from Ref. [57]. N , it shows that this magnitude is smaller approaching to the drip line.

One-neutron separation energy
To complete the determination of the drip line we will calculate the one-neutron separation energy from Eqs. (8) and (17) and the magnitudes of the previous sub-section. Then, using Eq. (14) we evaluate the binding energy for the odd calcium isotopes, which is shown in Fig. 10. The usual staggering, mounting onto the parabola-like curve, can be observed. The inset shows the one-neutron separation energy, the comparison with the experimental data shows a good agreement. We found that the one-neutron drip line happens to be at 57 Ca, in agreement with ab initio models [8,23], and the Gamow Shell Model [28], but in disagreement with the experimental result of Ref. [11] which found that the nucleus 59 Ca is also bound. FIG. 10: Binding energy of the even and odd calcium isotopes. The experimental data was taken from [57].

IV. DISCUSSION AND CONCLUSIONS
We have calculated the one-and two-neutron separation energy of the calcium isotopes from the exact solution of the pairing Hamiltonian. While the two-neutron separation energy is obtained straightforward from the Richardson solution, the one-neutron separation energy was calculated using the pairing gap ∆ and Fermi level λ F , borrowed from the BCS formalism. The occupation probabilities needed to calculate ∆ and λ F were obtained from the exact solution of the pairing Hamiltonian by finite difference. The correlations with the continuum spectrum of energy was taken into account through the single-particle density. Outcomes from isospin independent and dependent mean-field and pairing were investigated.
The evolution of the single particle levels shows an inversion of the shells 2d 5/2 and 1g 9/2 at the beginning of the chain, as reported in Ref. [23]; and then, a near degeneracy, as the one reported in [56] for deformed nuclei. Finally, the original order is reversed to the usual shell ordering, with the shell 1g 9/2 becoming a bound state, at the time that the shell 2d 5/2 remains in the continuum. The displacement of the single particle levels shows a shell closure for the calcium isotopes with N = 32 and N = 34. The intrusion of the shell 1g 9/2 from the continuum slightly hinders a closure for N = 40, and prevents a closure at N = 50. The influence of deformations upon the level structures in very rich neutron nuclei is expected to be important, in particular, more experimental structure information on the calcium isotopes is expected in the near future.
Our calculation found the nucleus 57 Ca as the last bound odd isotope, in agreement with [28], but in dis-agreement with the experimental finding reported in Ref. [11], probably due and overestimation of the pairing gap.
The results from the isospin independent formulation shows that 60 Ca is the last bound calcium isotope. Similar result was found using Bogoliubov perturbation formalism [24] and self-consistent Green's function [25]. By including the isospin dependence in the mean-field and pairing strength, drip line is extended to 66 Ca. This result is smaller than the predictions from the Bayesian Model Averaging [27] and the Gamow Shell Model [28], which allocate the drip line around 68 Ca-70 Ca.
The outcome of this paper shows the ability of the exact pairing formalism to describe one and two neu-tron drip lines in the calcium isotope chain. In the current state of knowledge, all three models reproduce the known data equally well. This places an uncertainty in our prediction for the two neutron drip line, with 60 Ca or 66 Ca depending on whether the independent or dependent isospin model is considered.