Continuum discretised BCS approach for weakly bound nuclei

arXiv:1510.03185v2 [nucl-th] 25 Feb 2016 Continuum discretized BCS approach for weakly bound nuclei J. A. Lay Dipartimento di Fisica e Astronomia “Galileo Galilei”, Universit`a di Padova, via Marzolo, 8, I-35131 Padova, Italy INFN, Sezione di Padova, via Marzolo, 8, I-35131 Padova, Italy E-mail: [email protected] C. E. Alonso Departamento de F´ısica At´omica, Molecular y Nuclear, Facultad de F´ısica, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain L. Fortunato and A. Vitturi Dipartimento di Fisica e Astronomia “Galileo Galilei”, Universit`a di Padova, via Marzolo, 8, I-35131 Padova, Italy INFN, Sezione di Padova, via Marzolo, 8, I-35131 Padova, Italy February 26, 2016 Abstract. The Bardeen-Cooper-Schrieffer (BCS) formalism is extended by including the single-particle continuum in order to analyse the evolution of pairing in an isotopic chain from stability up to the drip line. We propose a continuum discretized generalized BCS based on single-particle pseudostates (PS). These PS are generated from the diagonalization of the single-particle Hamiltonian within a Transformed Harmonic Oscillator (THO) basis. The consistency of the results versus the size of the basis is studied. The method is applied to neutron rich Oxygen and Carbon isotopes and compared with similar previous works and available experimental data. We make use of the flexibility of the proposed model in order to study the evolution of the occupation of the low-energy continuum when the system becomes weakly bound. We find a larger influence of the non-resonant continuum as long as the Fermi level approaches zero. Keywords: BCS, pairing, weakly-bound systems, continuum Submitted to: J. Phys. G: Nucl. Phys. Continuum discretized BCS approach for weakly bound nuclei 2 1. Introduction Nuclei far from the β-stability line are currently one of the most active fields in nuclear physics. New radioisotope ion beam facilities are nowadays producing short-lived nuclei at intermediate masses reaching the neutron drip-line, thus venturing into a hitherto unknown domain of nuclear phenomena. In this region of the nuclear chart, the ground state energy and the Fermi level approach the threshold for neutron emission, therefore, the proper inclusion of continuum states becomes progressively more and more important. The treatment of the continuum represents a challenge on both theoretical and computational grounds. Different representations of the continuum have been used and compared with few-body models [1–9]. These models are accurate in nuclei with a strong clusterization into few stable fragments as it occurs in the light region of the Segr`e Chart. When moving to the study of heavier isotopes, these models require some extension or additional ingredients [10–16]. On the other hand, the intermediate mass region of the Segr`e Chart is the main region of application of mean field approaches and the study of pairing. However, known isotopes in this region are strongly bound so that traditional formulations of many-body models do not include the continuum. In the field of many-body systems the first attempt to include the single particle continuum is the Continuum Shell Model (CSM) [17–19]. From this CSM, a BCS formulation was presented in [20] and different variations have been explored either using the complex energy plane [21, 22], diagonalizing in a box [23], or making use of continuum single particle level densities (CSPLD) [24, 25]. Other formulations, which have been generalized to include the continuum or at least its resonant part, are Hartree-Fock-BCS (HF-BCS) [26], Hartree- Fock-Bogoliubov (HFB) [27–30]. Another approach is the complex scaling method, in which a unitary transformation of the cut in the complex energy plane eases the treatment of resonances of the many-body system [31, 32]. These resonances can also be studied with the Gamow Shell Model (GSM) [33] together with its coupled-channel representation (GSM-CC) [34, 35]. Finally also configuration-space Monte Carlo method (CSMC) has been recently extended to the use of pairing within the continuum [36]. In some of these formulations, there is a separate treatment of resonant and non-resonant parts of the continuum. Also regarding the discretization method for the continuum, only box boundary condition or the Berggren representation in the complex energy plane are predominantly used. Since one of the main problems in these calculations is the computational cost, looking at a number of different basis sets in order to optimize the convergence is a very important factor. It can be also interesting to check the stability of the many-body approach versus different ways of discretizing the continuum. Therefore, we propose in this work the use of the Transformed Harmonic Oscillator (THO) basis for the discretization of the continuum into a Generalized BCS formalism. The prescription used here will be that of Karataglidis et al. [37] rather than that of the original work by Stoitsov et al. [38]. This THO basis has been shown to reduce Continuum discretized BCS approach for weakly bound nuclei 3 the computational cost in nuclear reactions within three- and four-body formalisms [8– 10, 39]. This, together with the flexibility of the THO basis shown in [5, 37] is crucial here for exploring the role of the continuum when a nucleus becomes more and more weakly bound. Another important fact is the possibility of treating both resonant and non-resonant part of the continuum in a natural way and on the same footing. This makes calculations simpler and more straightforward. A HFB formalism can give a more accurate description, solving some of the problems of the BCS plus continuum [40–42]. However, this choice implies an increase of the complexity of the calculation and, therefore, a more difficult analysis of the results. An alternative option is to keep the essence of the HFB equations in a simplified model as done by I. Hamamoto and B. R. Mottelson [29, 30]. The present work is structured as follows. The method is described in Sec. 2. First, we recall how to discretize the single-particle continuum with the THO basis in subsection 2.1. Later, we describe the Generalized BCS formalism using the THO pseudostates with a Density Dependent Delta Interaction in subsection 2.2. Finally, we apply the formalism to Oxygen and Carbon isotopes in Sec. 3. 2. Methodology 2.1. Discretization of the single-particle continuum In this subsection, we briefly review the features of the pseudo-states (PS) basis used in this work. The THO basis is meant to describe the states of a composite system consisting on two interacting inert fragments, such as a valence particle (proton/neutron) and a spherical and stable core. In this case, the core+valence Hamiltonian is simply given by: hs.p. =Tr+Vvc(r) (1) where ~r is the relative coordinate between the valence particle and the core, Trthe core-valence kinetic energy operator and Vvc(r) is the interaction between the valence particle and the core. The eigenstates of this Hamiltonian can be characterized by the excitation energy (ε) and the set of quantum numbers {ℓ, s, j, m}, which correspond to the orbital angular momentum (~ ℓ), the valence spin (~s) and their vector sum (~ j=~ ℓ+~s). For a central potential with, possibly, a spin-orbit term, these states can be written as: φε,ℓ,s,j,m(~r) = Rε,ℓ,j(r)Yℓsjm(ˆr) (2) where Yℓsjm(ˆr) = [Yℓ(ˆr)⊗χs]jm, with χsa spin function. The radial functions Rε,ℓ,j(r) can be obtained by solving the Schr¨odinger equation with appropriate boundary condition for bound (ε < 0) or unbound (ε > 0) states. Alternatively, these functions can be obtained by diagonalizing the Hamiltonian (1) in a discrete basis. Since any complete basis will be infinite, this procedure is not feasible in practice unless the basis is truncated. By doing so, one obtains a finite (and approximate) expansion Continuum discretized BCS approach for weakly bound nuclei 4 of the functions R(r) in the selected basis. If the basis functions are denoted by ϕn,ℓ,s,j,m(~r) = Rbasis n,ℓ (r)Yℓsjm(ˆr), we will have: Rα(r) = N X n=1 cα,nRbasis n,ℓ (r) (3) where α≡ {ε, ℓ, j}and Nis the number of states retained in the basis. As already mentioned, there are many possible choices for the basis functions {ϕn} (Gaussian, harmonic oscillator, Laguerre, etc). In this work we use the transformed harmonic oscillator (THO) basis, obtained from the harmonic oscillator basis with an appropriate local scale transformation (LST) as originally proposed by Stoitsov et al. [43, 44]. If the LST function is denoted by s(r), the THO states are obtained as RTHO n,ℓ (r) = rds drRHO n,ℓ [s(r)],(4) where RHO n,ℓ (s) is the radial part of the usual HO functions. With the criterion given above, the LST is indeed not unique. Here, we adopted a parametric form for the LST from Karataglidis et al. [37] s(r) =   1 1 rm+1 γ√rm  1 m ,(5) that depends on the parameters mand γ. The extension of RHO n,ℓ (s) will also depend on the oscillator length b. Note that, asymptotically, the function s(r) behaves as s(r)∼γ√rand hence the functions obtained by applying this LST to the HO basis behave at large distances as exp(−γ2r/(2b2)) = exp(−keffr). Therefore, the ratio γ/b can be related to an effective linear momentum, keff =γ2/(2b2), which governs the asymptotic behavior of the THO functions. As the ratio γ/b increases, the radial extension of the basis decreases and, consequently, the eigenvalues obtained upon diagonalization of the Hamiltonian in the THO basis tend to concentrate at higher excitation energies. Therefore, γ/b determines the density of eigenstates as a function of the excitation energy. In all the calculations presented in this work, the power mhas been taken as m= 4. This choice is discussed in Ref. [37] where the authors found that the results are weakly dependent on m. Note that, by construction, the family of functions RTHO n,ℓ (r) constitute a complete orthonormal set. Moreover, they decay exponentially at large distances, thus ensuring the correct asymptotic behaviour for the bound wave functions. In practical calculations a finite set of functions (4) is retained, and the single-particle Hamiltonian is diagonalized in this truncated basis with Nstates, giving rise to a set of eigenvalues and their associated eigenfunctions, denoted respectively by {εn}and {ϕ(N) n,ℓ (r)}(n= 1,...,N). As the basis size is increased, the eigenstates with negative energy will tend to the exact bound states of the system, while those with positive eigenvalues can be regarded as a finite representation of the unbound states. Continuum discretized BCS approach for weakly bound nuclei 5 Generalizations of this basis can be found for three-body systems [8] and for twobody systems with core excitations [39] with great success. Nevertheless, this is the first time this basis is applied to the study of pairing in the continuum. 2.2. Generalized BCS with pseudostates Now we solve the many-body Hamiltonian of Nneutrons moving in a mean field. In second quantization this Hamiltonian reads [45]: H=Xhs.p. +X klk′l′ vklk′l′a† ka† lak′al′.(6) In BCS the two-body interaction is reduced to those situations where two nucleons form a strongly correlated pair of time reversal states. Therefore, if we define the pair creation (and corresponding annihilation) operator: P† j=X m>0 (−1)j−ma† jma† j−m,(7) the previous Hamiltonian reads: H=Xhs.p. −X jj′ Gjj′P† jPj′,(8) where Gjj′is a state-dependent coefficient that replaces vklk′l′. One should add to the sum over the different bound states, an integral over the different momentum states defined in the single-particle continuum . However, with the use of a discretized basis, we can turn this integral into a sum over different pseudostates. Notice that the pair operator is not uniquely defined with j. For each jwe would have Npseudostates and, therefore, Ndifferent pair operators defined by ν→ {n, j}, where nis the label ordering in energy the pseudostates. From these Npair operators, we will have one operator with associated negative single particle energy per each bound state present in hs.p. and the rest up to Nwill have an associated positive single particle energy so that they will create a Cooper pair in the continuum. The final number will be smaller than Ntimes the number of angular momenta considered since we have also imposed a maximum energy cut off. Therefore, we can continue with the traditional definition of the BCS function: |BCSi= Πν(uν+vνP† ν)|0i.(9) Finally, we can obtain the corresponding BCS equations by minimizing the Hamiltonian Hwith constraint on the number of particles, i.e.: hBCS|H−λN|BCSi,(10) where λis the Fermi level. The problem can be recast into solving the following set of non-linear coupled equations: Continuum discretized BCS approach for weakly bound nuclei 6      ∆ν=1 4Pν′ (2j′+1)Gνν′∆ν′ √(εν′−λ)2+∆2 ν′ N=Pν(2j+ 1)v2 ν, (11) where ενis the single-particle energy and the coefficients uand vare given by: v2 ν=1 2 1−εν−λ p(εν−λ)2+ ∆2 ν!,(12) u2 ν=1 2 1 + εν−λ p(εν−λ)2+ ∆2 ν!.(13) 2.3. The pairing interaction In order to calculate the values of Gνν′we make use of a Density Dependent Delta Interaction (DDDI). For a general radial dependence of the delta interaction: Vres(~r1−~r2) = V(r)δ(~r1−~r2),(14) the final values of Gνν′will be: Gνν′=−1 2(4π)Z∞ 0|Rν(r)|2|Rν′(r)|2V(r)r2dr, (15) where Rν(r) are radial single-particle wavefunctions. The minus sign ensures that the values of Gare defined positive for any attractive interaction. In our case, the radial form of the interaction is: V(r) = "V0−VI ρ0 1 + exp r−R0 a0!η#,(16) which can be accomodated to the more traditional form: V(r) = V01−ρ(r) ρ0η,(17) for a certain selection of the parameters. We can compare these results with the use of a constant pairing G. In [23–25], it has already been shown that, in order to keep a constant G, a renormalization of the pairing strength in the continuum is needed. In that sense, the results from [23–25] cannot be directly compared with the use of a constant Ghere. With the use of a DDDI we hope to recover this renormalization in a natural way without distinguishing resonant and non-resonant continuum. This represents a major advantage of the present approach. Finally, it should be said that the density ρ(r) is taken from a phenomenological Woods-Saxon form adjusted to each case. No self-consistent recalculation of the density is considered. Continuum discretized BCS approach for weakly bound nuclei 7 16 18 20 22 24 26 28 30 A -5 0 5 10 15 S2n (MeV) Experimental S2n Present Calculation AO Figure 1. (Color online) Two-neutron separation energies for the even neutron rich oxygen isotopes. Experimental points contain error bars (sometimes invisible at this scale). 3. Results 3.1. Oxygen isotopes The isotopic chain of oxygen is the heaviest in atomic number with a well-known position of the neutron drip-line. There are two isotopes, 26O and 28O, with experimentally measured negative two-neutron separation energies (S2n), meaning that both nuclei are unbound with respect to two-neutron emission [46]. In our calculations, we took a single-particle Hamiltonian able to reproduce the spectrum of 17O taking the n-16O potential from Ref. [47]. We will include all partial waves with ℓ≤2. For constraining the pairing interaction, we use a volume term with the geometry of this potential and a strength that reproduces the S2nin 18O. The strength of VIis later renormalized for the following even oxygen isotopes so that it follows the known 1/A dependence. The energy cut-off is set at 20 MeV. Therefore, all free parameters are tied to the knowledge of stable isotopes. We then add one, two, three, four, five and six pairs of neutrons to the 16O core in order to reach 28O. The theoretical values obtained within the present framework for the two-neutron separation energies compared with the experimental ones taken from [46] are shown in Fig. 1. Recently, within the coupled-cluster theory [48] it has been possible to look at the binding energies and the spectra of Oxygen (and also Carbon) isotopes. In our formalism we concentrate on binding energies focusing on the evolution of superfluidity and the role of continuum. In this calculation, the structure of 24O is not superfluid. Last bound orbitals 1d5/2and 2s1/2are fully occupied, so that the continuum and also the d3/2resonance are not significantly populated. Therefore we choose the closest isotope 22O where the unoccupied 2n-hole in the 1d5/2or 2s1/2orbits enhances the effect of pairing, thus increasing the population of the continuum. This can be understood by looking at the ∆νfound for the present value of the pairing strength as shown in Fig. 2 for 22O and in Continuum discretized BCS approach for weakly bound nuclei 8 0 1 2 G1d5/2-1d5/2 (MeV) 0 1 2 3 4 ∆1d5/2 (MeV) No Continuum Continuum up to 20 MeV 22O Figure 2. (Color online) Value of ∆ for the 1d5/2orbit versus pairing strength for Cooper pairs in the same 1d5/2orbit for 22O with (dashed line) and without (solid line) inclusion of the continuum. The red arrow show the pairing strength used in the present calculations for 22O. 0 1 2 G1d5/2-1d5/2 (MeV) 0 1 2 3 4 ∆1d5/2 (MeV) No cont. (full) Continuum up to 20 MeV 24O Figure 3. (Color online) Value of ∆ for the 1d5/2orbit versus pairing strength for Cooper pairs in the same 1d5/2orbit for 24O with (dashed line) and without (solid line) inclusion of the continuum. The red arrow show the pairing strength used in the present calculations for 24O. Fig. 3 for 24O. In these figures we plot ∆νversus Gνν of the single-particle ground state 1d5/2. The red arrow in these figures indicates the pairing strength used in the present calculations for these nuclei. We see that in 22O the arrow indicates a point located after the transition to superfluidity, whereas in 24O the arrow is still far from the critical point. 24O appears, therefore, as a closed sub-shell nucleus. In [36], they applied the CSMC method also to 24O obtaining a really small impact of the continuum on the total binding energy in agreement with what is found here. We also compare with calculations without the continuum in both figures. Strong deviations from this no-continuum reference are Continuum discretized BCS approach for weakly bound nuclei 9 10 15 20 25 30 35 Av. density of states (MeV-1) 1.83 1.84 1.85 1.86 1.87 1.88 1.89 ∆av. (MeV) 22O Figure 4. Convergence of ∆av.with the size of the basis. The X axis represents the number of levels in the calculation divided by the cut off energy. The different points correspond to N={60,70,80,90,100,150,200}. not seen unless one forces the strength of the pairing well beyond its commonly accepted values. This is something to be expected, if we take into account that the continuum threshold is at 4.14 MeV in 17O, and this is a gap to be overcome with the help of pairing. One should keep in mind that the single-particle spectrum varies along the isotopic chain, so that by doing HF or similar calculations we expect this continuum threshold to be reduced. Therefore, there is still a chance for 24O to be superfluid or at least closer to superfluidity. In order to study the convergence properties, we show, for instance, the average ∆av.in 22O with respect to the size of the basis. This quantity is defined as the weighted sum over the corresponding occupations [40]: ∆av.=Pν(2j+ 1)∆νv2 ν Pν(2j+ 1)v2 ν =Pν(2j+ 1)∆νv2 ν N.(18) Since Nis not a direct measurement of the final number of states considered in the calculation, we define an average density of states. For each partial wave we keep Nj≤N pseudostates, those with energies smaller than the energy cut off Ecut. Therefore, a way to measure an average density of states is: Av.density of states = PjNj Ecut .(19) The result is shown in Fig. 4. One can see that the convergence is very good. These results have been obtained for the following parameters of the basis: γ= 2.1 fm1/2and b= 2.1 fm. For the sake of convenience, we use N=100 for all the remaining calculations in this work. We show in Fig. 5 the occupancies v2 νin the continuum for the different partial waves. We clearly see the dominant role of the low-lying d3/2resonance creating a peak around 1 MeV with an occupation several orders of magnitude larger than those of the non-resonant continuum. In a discrete set the direct sum of these values should give Continuum discretized BCS approach for weakly bound nuclei 16 Figure 16. (Color online) Segr`e chart with the total sum of the product uv for the different Oxygen and Carbon isotopes. Stable nuclei are shown in black and those nuclei which are stable with respect to particle emission, in light grey. pseudostates for BCS including only the resonant part. The selection of the pseudoestate for representing the resonance should fit the stabilization method, see Ref. [11, 50, 51]. As already mentioned, the main advantage of the discretization with pseudostates is the natural treatment of the continuum where resonant and non-resonant continuum appears on the same footing. To conclude this results section, we compare both isotopic chains in Fig. 16 by means of total uv product. As we have already mentioned, this quantity is related to the pair transition density reflecting the collectivity of the nucleus. We note in both chains a maximum around N= 12. For N= 16 there is significant difference between both chains, 24O behaves like a closed sub-shell nucleus whereas in 22C the product of uv is still far from zero. This quantity is zero by construction for 16O and 12C. 4. Conclusions We have demonstrated that the THO basis can be profitably used in the solution of the BCS equations with a discretized single-particle continuum. The single-particle Hamiltonian is constructed with a Wood-Saxon potential with parameters adjusted to just one isotope for each chain in order to reproduce main bound states and first resonances. This Hamiltonian is discretized in the THO basis. Once discretized, Generalized BCS equations can be solved following the traditional formulation for bound states. The different pairing couplings are calculated with a density-dependent delta interaction (DDDI). The Locale Scale Transformation (LST) applied to the HO wavefunctions allows us to distribute the discretized states along the single-particle continuum almost on demand. This characteristic makes the procedure explained here an easy and practical Continuum discretized BCS approach for weakly bound nuclei 17 tool in the search for threshold effects related to pairing within weakly bound systems. Such malleability is not present in the bases commonly used for the same scope. This method has been applied to Oxygen and Carbon isotopic chains. For the oxygen case we use the potential from Ref. [47] that reproduces the low-lying spectrum of 17O. The DDDI is fixed to reproduced the two-neutron separation energy in 18O and later renormalized for each even isotope according the well known factor 1/A. With this set of parameters, the overall trend of the binding energies is reproduced and the neutron drip-line appears in the right place. However, a small effect of the continuum is found, as expected regarding the large separation between the last bound state and the continuum threshold. In order to investigate how the system and the importance of the continuum evolves when this continuum threshold is reduced, we focused on 22O and reduced the binding potential. An increase of the occupation near the threshold is found for s−and d5/2−waves whereas a decrease is found for p−waves. We understand this is due to the presence of a low-lying bound state with same spin and parity. The pairing of the low-lying continuum to this last bound state is strong and increase as long as the Fermi level approaches the continuum threshold. This effect was previously found in more complex calculation for different cases, but only for s−states [28, 30]. Moreover, when the separation between the Fermi level and the continuum threshold goes to zero, the contribution of the d3/2resonance to the total occupation is reduced in favor of these threshold contributions. This opens the possibility of situations where the presence of the low-lying continuum is more important than the resonances in the ground state of a many-body system. However, it seems to happen only for extreamely weakly bound situations. This is the most interesting contributions of the present manuscript. This conclusion is also based in the fact that the present model is able to treat resonant and non-resonant continuum on the same footage. Same procedure has been applied to Carbon isotopes, fitting the parameters following a similar work by Id Betan [24, 25] in order to compare both methods. Our results for the binding energies of the different isotopes coincide with this former work. The drip-line is also found at the same place. However, experimental binding energies are larger. As already stated in [24, 25], this should be attributed to the deformation. The distributions of the occupation in 20C and 22C show similar features to those found for 22O when reducing the binding energy of the single-particle states. In 22C, the importance of the scontinuum is almost of the same order than the d3/2resonance. However, deformation can change this situation, so that a BCS description based on Nilsson rather than spherical states might be more accurate. We are aware of the fact that, in coordinate space, densities obtained with a continuum formulation of the BCS show certain drawbacks which could be connected with the presence of an unphysical gas of neutrons. However, the present formalism satisfies a compromise between accuracy and simplicity needed to pinpoint, control and separate the different ingredients governing pairing in weakly-bound systems. In conclusion, the simplicity of the present model has given us the flexibility required REFERENCES 18 for exploring the evolution of many-body systems when becoming weakly bound. We found situations where the non-resonant continuum cannot be neglected in favor of resonances as previously commonly stated. However, these facts should be explored increasing the complexity step by step in order to have a clear view of the behavior of weakly bound system. The ability of the THO basis to concentrate states at the really low-energy continuum, reducing the overall computational demand, might be of great help in future attempts. Acknowledgments This work has been supported by MIUR research fund PRIN 2009TWL3MX, by the Spanish Ministerio de Econom´ıa y Competitividad under Project FIS2014-53448-C2-1- P by Junta de Andaluc´ıa under group number FQM-160 and Project P11-FQM-7632, and by the Consolider-Ingenio 2010 Program CPAN (CSD2007-00042). The research leading to these results has received funding from the European Commission, Seventh Framework Programme (FP7/2007-2013) under Grant Agreement n 600376. J.A.L. is a Marie Curie Piscopia fellow at the University of Padova. References [1] Dasso C H and Vitturi A 2009 Phys. Rev. C 79 064620 [2] Deltuva A, Moro A M, Cravo E, Nunes F M and Fonseca A C 2007 Phys. Rev. C 76 064602 [3] Moro A M, P´erez-Bernal F, Arias J M and G´omez-Camacho J 2006 Phys. Rev. C 73 044612 [4] Moro A M, Arias J M, G´omez-Camacho J and P´erez-Bernal F 2009 Phys. Rev. C 80 054605 [5] Lay J A, Moro A M, Arias J M and Gomez-Camacho J 2010 Phys. Rev. C 82 024605 [6] Druet T, Baye D, Descouvemont P and Sparenberg J 2010 Nucl. Phys. A 845 88 [7] Fortunato L, Chatterjee R, Singh J and Vitturi A 2014 Phys. Rev. C 90 064301 [8] Rodr´ıguez-Gallardo M, Arias J M, G´omez-Camacho J, Johnson R C, Moro A M, Thompson I J and Tostevin J A 2008 Phys. Rev. C 77 064609 [9] Casal J, Rodr´ıguez-Gallardo M and Arias J M 2013 Phys. Rev. C 88 014327 [10] de Diego R, Arias J M, Lay J A and Moro A M 2014 Phys. Rev. C 89 064609 [11] Moro A M and Lay J A 2012 Phys. Rev. Lett. 109 232502 [12] Lay J A, Moro A M, Arias J M and Kanada-En’yo Y 2014 Phys. Rev. C 89 014333 [13] Deltuva A 2013 Phys. Rev. C 88 011601 [14] Capel P 2014 Jour. of Phys. G 41 094002 REFERENCES 19 [15] Potel G, Idini A, Barranco F, Vigezzi E and Broglia R 2014 Physics of Atomic Nuclei 77 941–968 [16] Descouvemont P and Hussein M S 2013 Phys. Rev. Lett. 111 082701 [17] Mahaux C and Weidenm¨uller H A 1969 Shell Model Approach to Nuclear Reactions (North-Holland, Amsterdan) [18] Esbensen H, Bertsch G F and Hencken K 1997 Phys. Rev. C 56 3054 [19] Volya A and Zelevinsky V 2006 Phys. Rev. C 74 064314 [20] Sandulescu N S, Liotta R J and RWyss 1997 Phys. Lett. B 394 6 [21] Id Betan R, Sandulescu N and Vertse T 2006 Nucl. Phys. A 771 93 [22] Dussel G G, Id Betan R, Liotta R J and Vertse T 2007 Nucl. Phys. A 789 182 [23] Id Betan R, Dussel G G and Liotta R J 2008 Phys. Rev. C 78 044325 [24] Id Betan R 2012 Nucl. Phys. A 879 14 – 24 [25] Id Betan R 2012 Phys. Rev. C 85 064309 [26] Sandulescu N, VanGiai N and Liotta R J 2000 Phys. Rev. C 61 061301(R) [27] Zhang Y, Matsuo M and Meng J 2011 Phys. Rev. C 83 054301 [28] Grasso M, Sandulescu N, VanGiai N and Liotta R J 2001 Phys. Rev. C 64 064321 [29] Hamamoto I and Mottelson B R 2003 Phys. Rev. C 68 034312 [30] Hamamoto I and Mottelson B R 2004 Phys. Rev. C 69 064302 [31] Myo T, Kato K and Ikeda K 2007 Phys. Rev. C 76 054309 [32] Myo T, Kikuchi Y and Kato K 2012 Phys. Rev. C 85 034338 [33] Michel N, Nazarewicz W, P loszajczak M and Bennaceur K 2002 Phys. Rev. Lett. 89 042502 [34] Jaganathen Y, Michel N and P loszajczak M 2014 Phys. Rev. C 89 034624 [35] Fossez K, Michel N, P loszajczak M, Jaganathen Y and Id Betan R M 2015 Phys. Rev. C 91 034609 [36] Lingle M and Volya A 2015 Phys. Rev. C 91 064304 [37] Karataglidis S, Amos K and Giraud B G 2005 Phys. Rev. C 71 064601 [38] Stoitsov M V, Nazarewicz W and Pittel S 1998 Phys. Rev. C 58 2092 [39] Lay J A, Moro A M, Arias J M and G´omez-Camacho J 2012 Phys. Rev. C 85 054618 [40] Dobaczewski J, Nazarewicz W, Werner T R, Berger J F, Chinn C R and Decharg´e J 1996 Phys. Rev. C 53 2809–2840 [41] Kruppa A T, Heenen P H and Liotta R J 2001 Phys. Rev. C 63 044324 [42] Dean D J and Hjorth-Jensen M 2003 Rev. Mod. Phys. 75 607–656 [43] Stoitsov M V and Petkov I Z 1988 Ann. Phys. (N. Y.) 184 121 [44] Petkov I Z and Stoitsov M V 1991 Nuclear Density Functional Theory, Oxford Studies in Physics (Clarendon, Oxford) REFERENCES 20 [45] Ring P and Schuck P 1980 The Nuclear Many-Body Problem (Springer-Verlag) [46] National nuclear data center national Nuclear Data Center, www.nndc.bnl.gov [47] Sparenberg J M, Baye D and Imanishi B 2000 Phys. Rev. C 61 054610 [48] Jansen G R, Engel J, Hagen G, Navratil P and Signoracci A 2014 Phys. Rev. Lett. 113 142502 [49] Pastore A, Margueron J, Schuck P and Vi˜nas X 2013 Phys. Rev. C 88 034314 [50] Hazi A U and Taylor H S 1970 Phys. Rev. A 11109 [51] Lay J A, Arias J M, G´omez-Camacho J and Moro A M 2012 AIP Conf. Proc. 1488 436–440