The quantification of pairing interaction in a two-neutron transfer through the intermediary continuum

An unbound intermediate system (A+1) with the presence of a resonance, can aid in the pairing enhancement in a two-neutron transfer reaction from a bound system A to another bound system (A+2). This enhancement is a consequence of the constructive interference via the several reaction channels available in the (A+1) system. We test this feature through our study in 6He, modeled as two neutrons in the orbitals of an intermediate 5He nucleus. Weighing up the natural case of an unbound 5He with a hypothetically bound 5He, we find that the inclusion of a properly modeled continuum favours the pairing correlations leading to enhanced two-neutron transfer cross-sections.


Introduction
Impressive advancements in experimental facilities the world over have made it possible to study the nuclear systems away from the valley of stability.They are interesting because their characteristics are peculiar when compared to their well known, stable siblings and are thus, called exotic.Despite their small life times, they are formed in nature and are able to exist due a plethora of phenomena which deviate from naturally found isotopes, like modulation of the usual shell gaps, the variation in shapes leading to possible deformation and/or the halo or bubble characteristics [1,2,3,4].However, they are still significant because they are the gateways to the pathways for stellar nucleosynthesis [5].In fact, low mass neutron rich nuclei, when included in reaction network calculations for the r-process, are known to significantly affect the abundances of stellar nuclei [6,7].
As these nuclei are extremely short-lived and radioactive, transfer reactions, which are a combination of theory and experiment, are widely used to produce these nuclei and examine their genesis, reaction and structural properties.For example, two-neutron transfer reactions are an elegant way to analyse the pairing between valence neutrons in a nucleus and hence, study neutron-neutron correlations [8,9].This is interesting when seen in light of the Borromean halos, where the (A+2) system is bound but the (A+1) is not.Two-neutron transfers are even known to have some affect on the low temperature r-process seed nuclei production in a neutron rich environment [10].
In view of this, we present here our calculations and results for the two-neutron transfer reaction 18 O( 4 He, 6 He) 16 O.The aim is to successively add two neutrons to 4 He and form the Borromean halo 6 He and in the process, make a pedagogical study of the role of the intermediate 5 He continuum to pairing enhancement.This is an interesting aspect because it is known that the couplings amongst continuous positive energy states and pairing correlations strongly enhance two particle transfer [8,9].The intermediate continuum plays a major role here by providing the various paths which interfere coherently and contribute to this pairing enhancement.The information of these pairing correlations can be obtained, in turn, from the initial and final state wave functions (or the single particle form factors thus computed).
Such a study, trying to quantify the role of the continuum, was done in Ref. [11], where we studied the role of the intermediate continuum on pairing enhancement, fixing the ground state of the final nucleus 6 He at the usual 0.975 MeV below zero.The pairing was introduced as a perturbation to the system and the entire premise was compared with a hypothetical case where the intermediate 5 He nucleus was considered to be bound.It was found that in the naturally occuring unbound case, the continuum provided significant enhancement to the two-neutron transfer cross-section.Since continuum wave functions are difficult to work with, the continuum was discretised using a transformed harmonic oscillator (THO) basis [2,11] and the resonance state for the continuum case (the third state in our basis) was fixed at 0.69 MeV.
Following the same procedure while working in the prior-prior picture [12] and using a modified version of the Transfer Form Factors (TFF) code [13], we provide here the results for the probability contribution of each of the configurations in the basis, for three different scenarios.The naturally unbound case of 5 He is compared with two hypothetical cases, where the intermediate 5 He nucleus is considered bound by S n = 1 MeV and S n = 0.1 MeV.However, unlike the study in Ref. [11], where we varied the pairing strength ∆ (described using a contact delta interaction of the form −gδ( r 1 − r 2 ), g being the coupling constant.The value of g is what modifies the perturbation to the uncompromised Hamiltonian.),we fix it here and allow instead, the final ground state of 6 He to vary.Therefore, the same pairing strength, when applied to all the three cases of interest, results in different ground state energies of 6 He.For the purpose of simplicity, we took ∆ = 1 MeV.This would mean that in the unbound 5 He case, the energy of the final state of 6 He is positive because the pairing energy of 1 MeV considered lowers the total energy of the two single particle resonance basis states from 1.38 MeV (0.69 + 0.69) to 0.38 MeV.It is also noteworthy that different number of states were chosen to represent each of the three cases under study.The S n = 1 MeV case was populated using 8 states of the THO basis, while the S n = 0.1 MeV and the unbound cases were analysed using 10 and 9 basis states, respectively.Further, we studied this phenomenon at a lab beam energy of 100 MeV to enable us to populate the higher lying continuum states as well as minimize the effect of Q-values on the reaction.We must also mention that no excitation or de-excitation of Oxygen or Helium isotopes was considered and since the projectile and target (in both the initial and final channels) had spin parities of 0 + , the simultaneous transfer contribution was cancelled by the non-orthogonality term [12].

Results and discussion
The probability contributions of each of the configurations considerd in the basis for the three cases under consideration are shown in Fig. 1.One can see that for the bound cases, i.e., for the S n = 1 MeV and S n = 0.1 MeV, the contributions of the bound state are expectedly most dominating, being 99.6% and 97%, respectively.In fact, even the configurations where one of the particles is in the ground state, while the other is in any of the excited states of the continuum (|1, j configurations, see Ref. [11] for details), the contributions are small, but perhaps still measureable.The continuum states offer little to no contribuion in these cases.
On the other hand, in the unbound case shown in the lowest panel in Fig. 1, the continuum states all couple amongst themselves and contribute significantly.In fact, the resonant state has the highest contribution among all, nevertheless, all the basis configurations with one of the  states as the resonant state (|1, 3 , |2, 3 and |3, j configurations) offer more contributions than others.The entire contribution of the continuum in this case was ∼ 55%, much higher than in the bound cases.It is, thus, seen that the pairing results in significant probability contribution in the continuum, even for a fixed pairing correlation between the valence neutrons.This would mean significant enhancement of the two-neutron transfer cross-sections, via the continuum when compared with unperturbed scenario where the pairing strength is set to zero.The result is similar to the one presented in Ref. [11].This rise in the cross-sections due to the intermediary continuum can in a way supplement and aid the formation of Borromean halos in stellar plasma.

Conclusions
Performing two-neutron transfer calculations for an even (A+2) system, we have compared the role of pairing in cases when the intermediate (A+1) system is bound and unbound.We chose 6 He as our candidate nucleus as it is the lightest Borromean halo near the dripline.We conclude that approaching the driplines, it is vital that the continnum contribution be considered for two-neutron transfers, especially when the (A+1) system is unbound.This highly correlated case offers many paths through which transfer can occur.Such paths interfere coherently, giving significant enhancement to the transfer cross-sections.This seems to be, in a way, suggestive of the probability of formation of Borromean systems, where the (A+2) system exists despite the non-existence of a bound (A+1) system.

Figure 1 .
Figure1.The cases of study for the sequential two-neutron transfer reaction 18 O( 4 He,6 He)16 O at 100 MeV beam energy showing the probability contribution of each of the configurations through their respective bases.The pairing configurations |i, i give the highest probability for particle occupation.For the bound systems with S n = 1 Mev and 0.1 MeV, the bound state contributes about entirely via the ground state configuration, while for the continuum case, each basis state couples to others giving considerable contributions.Further, in the lowest panel, the dominant contribution of the basis state closest to the resonance energy is clearly visible.