Impact of uncertainties of unbound 10 Li on the ground state of two-neutron halo 11

Recently, the energy spectrum of 10Li was measured upto 4.6 MeV, via d(9Li, p)11Li, one-neutron transfer reaction. Considering the ambiguities on the 10Li continuum spectrum with reference to new data, we report the configuration mixing in the ground state of the two-neutron halo nucleus 11Li for two different choices of the 9Li + n potential. For the present study, we employ a three-body (core + n + n) structure model developed for describing the two-neutron halo system by explicit coupling of unbound continuum states of the subsystem (core + n), and discuss the two-neutron correlations in the ground state of 11Li.


Introduction
The light dripline nuclei lying away from the strip of stability, have gained prodigious attention of the nuclear physics community over the past few decades and a significant progress has been made both on experimental and theoretical sides to understand their exotic nature [1]. The one of the eye-catching phenomenon in some light dripline nuclei is the formation of halo, which is linked to the small binding energy of one or two valence nucleons [2,3]. Particularly two-neutron (2n) halo systems, consisting of a core and two weakly bound valence neutrons, demand a three-body description with proper treatment of continuum. The stability of such three-body (core+n+n) system is linked to the continuum spectrum of the two-body (core+n) subsystem. In this context, to explore the sensitivity of choice of a core+n potential with the configuration mixing in the ground state of three-body systems (core + n + n), we will discuss the results of the 2n-halo 11 Li.
Although 11 Li is the first observed two-neutron halo four decades ago [3]. Since then a lot of experimental and theoretical studies have been reported on structure of the 11 Li. In order to understand the 11 Li structure, the information over low-lying spectrum of 10 Li is needed as a fundamental ingredient of three-body calculations. However, the 10 Li structure was studied by various techniques such as fragmentation [4], 11 Li(p, d) 10 Li transfer reaction at TRIUMF [5], multi-neutron transfer [6] and pion absorption reactions [7]. Maximum of these studies report the low-lying p 1/2 neutron resonance with peak lying in the range of 500-700 keV. Also few of these studies reported the presence of s-wave virtual state close to the threshold with a scattering length in the range from −20 to −30 fm [4] and not much information is available on neutron d-wave.
Recently, the 10 Li structure was investigated via d( 9 Li, p) 11 Li, one-neutron transfer reaction. This study reported 10 Li energy spectrum up to 4.6 MeV, with the existence of p 1/2 resonance at 0.45±0.03 MeV along with other two high lying structures at 1.5 and 2.9 MeV [8]. Also the role of 10 Li resonances is investigated in the halo structure of 11 Li via 11 Li(p, d) 10 Li transfer reaction at TRIUMF [5] and at the same facility the first conclusive evidence of a dipole resonance in 11 Li having an isoscalar character has been reported [9,10]. In view of these new measurements and ambiguities over the experimental data, we aim to explore the sensitivity of the 9 Li + n potential with the configuration mixing in the ground state of of three-body system ( 9 Li + n + n).
For this study, we use a three-body (core + n + n) structure model, developed for studying the weakly-bound ground and low-lying continuum states of Borromean systems sitting at the edge of neutron dripline [11]. In our approach, we start from the solution of the unbound subsystem (core + n) and the two-particle basis is constructed by explicit coupling of the two single-particle continuum wave functions. Initially, it was tested for the lightest 2n-halo 6 He [12,13], heaviest known 2n-halo 22 C [14] and 2n-unbound 26 O [15] and has been successful in explaining the ground-state properties and the electric-dipole and quadrupole responses.
In this contribution, Sec. 2 briefly describes the formulation of our three-body structure model. In Sec. 3 we analyze the subsystem 10 Li and fix the two different sets for 9 Li + n potential, consistent with available experimental information. Section 4 presents our results for the three-body system, 9 Li + n + n. Summary is made in Sec. 5.

Model Formulation
The three-body wave function for the 9 Li + n + n system is specified by the Hamiltonian where µ = A c m N /(A c + 1) is the reduced mass, and m N and A c = 9 are the nucleon mass and mass number of the core nucleus, respectively. V core+n is the core-neutron potential and V 12 is n-n potential. The neutron single-particle unbound s-, p-, and d-wave continuum states of the subsystem ( 10 Li) are calculated in a simple shell model picture for different continuum energy E C by using the Dirac-delta normalization and are checked with a more refined phase-shift analysis. Each single-particle continuum wave function of 10 Li is given by We use the mid-point method to discretize the continuum. The convergence of the results will be checked with the continuum energy cut E cut and ∆E. These core + n continuum wave functions are used to construct the two-particle 11 Li states by proper angular momentum couplings and taking contribution from different configurations. The combined tensor product of these two continuum states is given by We use a density-dependent (DD) contact-delta pairing interaction [16], given by .
The first term in Eq. (4) with v 0 simulates the free n-n interaction, which is characterized by its strength and the second term in Eq. (4) represents density-dependent part of the interaction. The strengths v 0 and v ρ are scaled with the ∆E by following relation from Ref. [14]. The v ρ is the parameter which will be fixed to reproduce the ground-state energy. For a detailed formulation and calculation procedure one can refer to Refs. [11][12][13]17].
3 Two-body unbound subsystem (core + n) The investigation of the two-body (core + n) subsystem is crucial in understanding the threebody system (core + n + n). The interaction of the core with the valence neutron (n) plays a fundamental role in the binding mechanism of the three-body system. The elementary concern over the choice of a core+n potential is the ambiguities in the experimental information about the core + n system. 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. The values of r 0 = 1.27 fm and a = 0.67 fm are adopted from Refs. [16,18]. Table 1: Parameter sets of the core-n potential for = 0, 1, 2 states of a 9 Li + n system. The possible resonances with resonance energy E R and decay width Γ in MeV are also tabulated.   Table. 1 For the present calculations we ignore the spin of the core 9 Li. The neutron number 6 is assumed for the neutron core configuration given by (0s 1/2 ) 2 (0p 3/2 ) 4 . The four valence neutron continuum orbits, i.e., p 1/2 , d 5/2 , s 1/2 and d 3/2 are considered in the present calculations for 10 Li. 10 Li is interesting in the sense that it shows inversion of s 1/2 and p 1/2 levels.
The scattering length of the virtual s-state, position and width of low-lying p-resonance along with higher lying = 2 resonance vary from experiment to experiment. In the view of the new experimental measurements [5,8], we use two different potential sets for core + n potential, which are tabulated in Table 1. The only difference between our two sets A and B is we use different s-wave depth (V 0 0 ), leading to different scattering length of the s 1/2 virtual state, which further effect the s-wave component in ground state of 11 Li. In our set A the s-wave potential is deep enough to increase the s-component dominance in the ground state of 11 Li in comparison to set B. Our both sets reproduces the observed p 1/2 resonance at 0.45 MeV consistent with Ref. [8] and the d 5/2 resonance, that lies at higher energy around 2.98 MeV, this position is consistent with the high-lying structure of 10 Li reported in Ref. [8]. The phase-shifts corresponding to these resonances are shown in Fig. 1. Similar potentials are used also in Refs. [16,18].

Results and Discussions
The three-body model with two non-interacting particles in the above single-particle levels of 10 Li, produces different parity states, when two neutrons are placed in different unbound orbits mentioned in Sec. 3 (for details see Table. 2). The corresponding oscillatory singleparticle continuum wave functions for s 1/2 , p 1/2 , d 5/2 , and d 3/2 states are plotted in Fig. 2. The four configurations (s 1/2 ) 2 , (p 1/2 ) 2 ,(d 5/2 ) 2 , (d 3/2 ) 2 couple to J π = 0 + for 11 Li.  The continuum single-particle wavefunctions are calculated with energies from 0.0 to 5.0 MeV and normalized to a delta for the spd-states of 10 Li on a radial grid which varies from 0.1 to 100.0 fm with the 9 Li+n potential discussed in Sec. 3. In the three-body calculations, along with the core + n potential the other important ingredient is the n-n interaction. We use the DD contact-delta pairing interaction, with the only adjustable parameter being v ρ . The two particle states are formed using mid-point method with an energy spacing of 2.0, 0.5, 0.25 and 0.1 MeV corresponding to block basis dimensions of N = 5, 10, 20 and 50, respectively, and the matrix elements of the pairing interaction are calculated. In Fig. 3, the eigenspectrum for J = 0 + case is presented and from figure it is clear that with increase in basis dimensions the superflous bound states moves into the continuum. The biggest adopted basis size gives a fairly dense continuum in the region of interest.  Figure 3: Eigenspectrum of the interacting two-particle case for J = 0 + for increasing basis dimensions, N . The parameter of pairing interaction v ρ , has been adjusted each time to reproduce the two-neutron separation energy (S 2n ).
In the DD contact-delta pairing interaction (defined by Eq. (4)), the strength of the DI part is given as v 0 = 2π 2 2 m N 2ann π−2kcann , where a nn is the scattering length for the free neutron-neutron scattering and k c is related to the cutoff energy, e c , as k c = m N ec 2 . We use a nn = −15 fm and e c = 30 MeV [16], which leads to v 0 = 857.2 MeV fm 3 . For the parameters of the DD part, we determine them so as to reproduce the two-neutron separation energy of 11 Li, S 2n = −0.369 MeV [19]. The values of the parameters that we employ are R ρ = 1.25×A 1 3 c (A c = 9) and v ρ = 862.5 and 861.75 MeV fm 3 for set A and B, respectively.
We report the percentage configuration mixing in the ground state of 11 Li in Table 3. We found that for Set A for which V 0 0 is deeper shows dominance of (s 1/2 ) 2 configuration in the ground state leading to formation of s-neutron halo. Whereas for Set B for which V 0 0 is shallower shows dominance of (p 1/2 ) 2 configuration in the ground state leading to formation of p-neutron halo. The preliminary numbers for calculated matter radii with these potential sets are 3.53 and 3.24 fm for Set A and B, respectively. These results of configuration mixing and matter radii are consistent with the results of Refs. [16,20] for 11 Li. The detailed investigation of the configuration mixing with inclusion of core spin is in progress.   and Set B (lower-panel) as a function r 1 = r 2 = r and the opening angle between the valence neutrons θ 12 for settings mentioned in caption of Table 3 .
The two particle density of 11 Li as a function of two radial coordinates, r 1 and r 2 , for valence neutrons, and the angle between them, θ 12 in the LS-coupling scheme is given by ρ(r 1 , r 2 , θ 12 ) = ρ S=0 (r 1 , r 2 , θ 12 ) + ρ S=1 (r 1 , r 2 , θ 12 ) The explicit expression for S = 0 component is given by [16,21] ρ S=0 (r 1 , r 2 , θ 12 ) = 1 8π whereˆ = √ 2 + 1 and ψ j (r 1 , r 2 ) is the radial part of the two-particle wave function which is determined from Eq. (3) by making use of Eqs. (5) and (6) of [13]. Figure 4 shows the two-particle density plotted as a function of the radius r 1 = r 2 = r and their opening angle θ 12 , with a weight factor of 4πr 2 · 2πr 2 sinθ 12 for both Sets A (upper panel) and B (lower panel). The distribution at smaller and larger θ 12 are referred to as "di-neutron"and "cigar-like"configurations, respectively. One can see in Fig. 4 that the twoparticle density is well concentrated around θ 12 ≤ 90 • for both Sets A (upper panel) and B (lower panel), which is the clear indication of the di-neutron correlation. The di-neutron component has a relatively higher density in comparison to the small cigar-like component for both sets in the ground state of 11 Li. The two peak structure in the two-particle density is attributed to the mixing of the s-and p-wave components ( ≤ 1) in the ground state of 11 Li.

Summary
In the present study we report the emergence of bound 2n-halo ground state of 11 Li from the coupling of four unbound spd-waves in the continuum of 10 Li due to the presence of pairing interaction. The configuration mixing in the ground state of 11 Li has been reported for the two particular choices of core+n potential, fixed in the view of the available recent experimental data. Also, the 2n-neutron correlation for this system showing prominence of the di-neutron component is discussed. Investigations with different choices of pairing interactions and inclusion of spin of core ( 9 Li) are in progress and will be reported elsewhere.