Two-Neutron Correlations in a Borromean $\varvec{^{20}\mathrm{C}+n+n}$ System: Sensitivity of Unbound Subsystems

Abstract

The structure of $^{22}$C plays a vital role in the new physics at subshell closure of $N=16$ in the neutron-rich region. We study the two-neutron correlations in the ground state of the weakly-bound Borromean nucleus $^{22}$C sitting at the edge of the neutron-drip line and its sensitivity to $\mathrm{core}$-n potential. For the present study, we employ a three-body ($\mathrm{core}+n+n$) structure model designed for describing the Borromean system by explicit coupling of unbound continuum states of the subsystem ($\mathrm{core}+n$). We use a density-independent contact-delta interaction to describe the neutron-neutron interaction and its strength is varied to fix the binding energy. Along with the ground-state properties of $^{22}$C, we investigate its electric-dipole and monopole responses, discussing the contribution of various configurations. Our results indicate more configuration mixing as compared to the previous studies in the ground state of $^{22}$C. However, they strongly depend upon the choice of the $^{20}\mathrm{C}$-n potential as well as the binding energy of $^{22}$C, which call for new precise measurements for the low-lying continuum structure of the binary system ($^{20}\mathrm{C}+n$) and the mass of $^{22}$C. These measurements will be essential to understand the Borromean three-body system $^{22}$C with more accuracy.

Appendix—Convergence with Model Parameters

Fig. 6

(Color online) a Convergence with continuum energy cut $E_{\mathrm{\,cut}}$ of the model space and b Convergence with energy spacing $\varDelta {E}$ of the model space

We calculate the continuum single-particle wave functions for different $E_{\mathrm{\,cut}}$’s. After fixing the convergence with $E_{\mathrm{\,cut}}$, the continuum single-particle wave functions, using the mid-point method for $E_{\mathrm{\,cut}}=5$ MeV, with $\varDelta {E}=1.0$, 0.5, 0.2, 0.1, 0.08, 0.06 and 0.05 MeV, the two-particle states are formed and the matrix elements of the pairing interaction are calculated. The convergence of the results is discussed on two different levels:

• 1. Convergence with the continuum energy cut ($E_{\mathrm{\,cut}}$) of the model space

  We check the convergence of our results (dominant contribution in the ground state of $^{22}$C) with different $E_{\mathrm{\,cut}}$’s. In the Fig. 6a, we show the variation of three dominant contributions ($s_{1/2}$, $d_{3/2}$ and $f_{7/2}$) in the ground state of $^{22}$C with different $E_{\mathrm{\,cut}}$’s for different values of the energy spacing ($\varDelta {E}$). One can clearly see that we achieve the convergence with different $E_{\mathrm{\,cut}}$’s but still there is significant variation in the percentage contribution with different $\varDelta {E}$s. In the fourth quadrant of the Fig. 6a, due to huge computational costs, we restrict ourselves to only 10 MeV. It is convincing that one can choose fairly well the $E_{\mathrm{\,cut}}=5$ MeV for this study and we will do all the calculations with $E_{\mathrm{\,cut}}=5$ MeV.

• 2. Convergence with the energy spacing ($\varDelta {E}$) of the model space

  As seen from the Fig. 6b, we study the convergence of our results (dominant contribution in the ground state of $^{22}$C) with inverse of $\varDelta {E}$ for $E_{\mathrm{\,cut}}=5$ MeV. It is clear from the Fig. 6b, the curves start getting flat from $\varDelta {E}=0.1$ MeV. Therefore, we adopt $\varDelta {E}=0.1$ MeV ($1/\varDelta {E}=10$ MeV) for the present calculations.