Three-body Description of $2n$-Halo and Unbound $2n$-Systems: $^{22}$C and $^{26}$O

We study the two-neutron correlations in the ground state of the weakly-bound two-neutron halo nucleus $^{22}$C sitting at the edge of the neutron-drip line and also in the unbound nucleus $^{26}$O sitting beyond the neutron dripline. For the present study, we employ a three-body (core$+n+n$) structure model designed for describing the two-neutron halo system by explicit inclusion of unbound continuum states of the subsystem (core$+n$). We use either a density-independent or a density-dependent contact-delta interaction to describe the neutron-neutron interaction and its strength is varied to fix the binding energy. We report the configuration mixing in the ground state of these systems for different choices of pairing interactions.


Introduction
The new generation of radioactive ion beam facilities around the various parts of the globe has provided the access to the neutron-rich side of the nuclear chart. Due to this, there has been a rapidly increasing interest in the physics of the two-neutron (2n) halo nuclei sitting right on the top of neutron driplines and decays of 2n-unbound systems beyond the neutron dripline. These systems demand a three-body (3b) description with proper treatment of continuum, the conventional shellmodel assumptions being insufficient. The well established 2n-halo 6 He has long history of studies in 3b-framework and thus it can be safely remarked that the understanding of 3b-dynamics is fairly established for p-shell nuclei, whereas for the s-d shell nuclei still the situation is in progress stage [1]. In the present study we consider two different s-d shell nuclei, the 2n-halo 22 C and the 2nunbound system 26 O. Very recently a high precision measurement of the interaction cross-section for 22 C was made on a carbon target at 235 MeV/nucleon [2] and also the unbound nucleus 26 O has been investigated, using invariant-mass spectroscopy [3] at RIKEN. The structural spectroscopy of the two-body subsystem plays a vital role in the understanding the 3b-system. These high precision measurements and the sensitivity of the structural spectroscopy of subsystem with the structure of 3b-system (core+n + n), are the motivation for selecting these nuclei for the present study. We have studied the pairing collectivity in the ground state of the 2n-halo 22 C and in the the 2n-unbound system 26 O. For this study we have used our recently implemented 3b-structure model (core+n + n) for the ground and continuum states of the 2n-halo nuclei [4][5][6]. We have explored the role of different pairing interactions such as density independent (DI) contact-delta pairing interaction and density dependent (DD) contact-delta pairing interaction with the configuration mixing in the ground-state of these systems.

Model Formulation
In our approach we consider a 3b-system consisting of an inert core nucleus and two valence neutrons, which is specified by Hamiltonian where µ = A c m N /(A c + 1) is the reduced mass, and m N and A c are the nucleon mass and mass number of the core nucleus, respectively. The recoil term is neglected in the present study, as A c = 20 and 24 are large enough to ignore it. V core+n is the core-n potential and V 12 is n-n potential. The neutron single-particle unbound s-, p-, d-and f -wave continuum states of the subsystem 21 C and and 25 O are calculated in a simple shell model picture for the converged model parameter, bin width (∆E = 0.1), by using the Dirac delta normalization and are checked with a more refined phase-shift analysis. These core + n continuum wave functions are used to construct the two-particle states of the core + n + n system by proper angular momentum couplings. We use a density-independent (DI) contact-delta pairing interaction for simplicity, and its strength is the parameter which will be fixed to reproduce the ground-state energy. We have also used a density-dependent (DD) contact-delta pairing interaction. For a detailed formulation one can refer to [4][5][6].

Two-body unbound subsystems (core + n)
The investigation of the two-body (core + n) subsystem is crucial in understanding the three-body system (core + n + n). The interaction of the core with the valence neutron (n) plays a vital role in the binding mechanism of the core + n + n system. The elementary concern over the choice of a core + n potential is the scarce experimental information about the core-neutron systems. We employ the following core + n potential where R c = r 0 A 1 3 c with r 0 and a are the radius and diffuseness parameter of the Woods-Saxon potential.

21 C
For 21 C, not much is known beyond that it is unbound. The only available experimental study using the single-proton removal reaction reported the limit to the scattering length |a 0 |< 2.8 fm and due to the low statistics of this experimental data at low energies, the possibility of low-lying resonance states can not be ruled out [7]. In the view of exploring the sensitivity of the core-n potential to the possible resonances and configuration mixing in the ground state of 22 C, very recently we examined in detail the four different potential sets (for details see text and Table 1 of Ref. [6]). Here we will discuss the results corresponding to potential Set-1, which is adopted from the literature [9], beacuse of its acceptance by different 3-body models leading to good explanation of the observed properties for 22 C. The subshell closure of the neutron number 14 is assumed for the core configuration given by (0s 1/2 ) 2 (0p 3/2 ) 4 (0p 1/2 ) 2 (0d 5/2 ) 6 . The seven valence neutron continuum orbits, i.e., s 1/2 , d 3/2 , f 7/2 , p 3/2 , f 5/2 , p 1/2 and d 5/2 are considered in the present calculations for 21 C.

25 O
In the recent measurement conducted at RIKEN [3], along with high accuracy measurement of ground state of 26 O, they have also reported the d 3/2 resonance state at 749(10) keV with width of 88(6) keV for 25 O. This information will serve as input for fixing the core+n potential parameters. For 25 O, we adopt the same value for the diffuseness parameter (a) and the radius parameter (r 0 ), as in Ref. [10]. For the Wood-Saxon depth parameter (V 0 ) and the strength of spin-orbit potential (V ls ) parameter tabulated in Table I, we use the information for the energy of unbound d 3/2 state. Our parameters for V 0 and V ls are consistent with the one reported in Ref. [10]. The neutron number 16 is assumed for the core configuration given by (0s 1/2 ) 2 (0p 3/2 ) 4 (0p 1/2 ) 2 (0d 5/2 ) 6 (1s 1/2 ) 2 . The three Table I. Parameter sets of the core-n potential for l = 1, 2, 3 states of a 24 O+n system. The possible resonances with resonance energy E R and decay width Γ in MeV are also tabulated.

Results and Discussions
The 3b-model with two non-interacting particles in the above single-particle levels of 21 C and 25 O produces different parity states, when two neutrons are placed in different unbound orbits. The seven configurations, (s 1/2 ) 2 , (p 1/2 ) 2 , (p 3/2 ) 2 , (d 3/2 ) 2 , (d 5/2 ) 2 , ( f 5/2 ) 2 and ( f 7/2 ) 2 couple to J π = 0 + for 22 C and three configurations (d 3/2 ) 2 , (p 3/2 ) 2 and ( f 7/2 ) 2 couple to J π = 0 + for 26 O. In the 3bcalculations, along with the core-n potential the other important ingredient is the n-n interaction. An attractive contact-delta pairing interaction is used, gδ( r 1 − r 2 ) for simplicity, with the only adjustable parameter being g. We also use the DD contact-delta pairing interaction to explore the role of different pairing interactions. In the DD contact-delta pairing interaction (defined by Eq. (8)  We found that configuration mixing in the ground state of 22 C does not change much with the choice of n − n interaction. We present the numbers for our potential Set 1 in Table II which are consistent with results of Ref. [9], and the same behavior is observed for the other sets [6]. Whereas for the 26 O case our results report different magnitude of configuration mixing for the different interactions and the results corresponding to the DD part that are consistent with the results of Ref. [10], where they have also used the DD interaction. One possible reason for this difference is that 26 O is unbound by 18 keV whereas 22 C is bound by 0.140 MeV. In order to reach a final conclusion we need more analysis, which will be reported elsewhere. The two particle density of 22 C and 26 O as a function of two radial coordinates, r 1 and r 2 , for valence neutrons, and the angle between them, θ 12 in the LScoupling scheme is calculated by following Refs. [5,10]. The distribution at smaller and larger θ 12 are referred to as "di-neutron"and "cigar-like"configurations, respectively. One can see in Fig. 1 that the two-particle density is well concentrated around θ 12 ≤ 90 • , which is the clear indication of the di-neutron correlation and the di-neutron component has a relatively higher density in comparison to the small cigar-like component for both 22 C and 26 O. The reflection of dominance of s-component in ground state of 22 C can be seen in left panel of Fig. 1 showing extended di-neutron component in comparison to 26 O (in right panel of Fig. 1), which has sharper dineutron component due to the mixing of l > 0 components in its ground-state.

Conclusions
In the present study we present the emergence of bound 2n-halo ground state of 22 C from the coupling of seven unbound spd f -waves in the continuum of 21 C and 2n-unbound ground state of 26 O from the coupling of three unbound pd f -waves in the continuum of 25 O due to presence of   Table II . pairing interaction. Contribution of different configurations has been presented along with the 2ncorrelations. More investigations are needed to comment on the role of different pairing interactions and to explain in detail the role of each wavefunction component.