Local HFB densities and mean-field potentials in neutron-rich nuclei

Local p-h and p-p densities are basic elements of the HFB-Skyrme theory: they determine self-consistent fields; hence the static properties of the nucleus such as the binding energy, radius, and shape. The particle and pairing local HFB+SLy4 neutron densities $\rho_n(r)$ and $\tilde{\rho}_n(r)$calculated for several values of $\alpha$are displayed in Fig. 2 for ¹⁵⁰Sn.

With decreasing $\alpha$, the p-p density $\tilde{\rho}_n(r)$ develops a long tail extending towards large distances. This is a direct consequence of the attractiveness of DDDI at low densities when $\alpha$ is small. While in the nuclear interior, the p-h density $\rho_n(r)$ depends extremely weakly on the actual form of the pairing interaction; its asymptotic values are significantly increased when $\alpha$ gets small (see inset). Moreover, one observes a clear development of a halo structure, i.e., a smooth exponential decrease, that for $\alpha$=1 starts at r$\simeq$6fm, for small $\alpha$ is interrupted at r$\simeq$9fm, and replaced by a significantly slower decrease of the density.

This is a direct consequence of the self-consistent coupling between p-h and p-p parts of the HFB Hamiltonian. That is, the increased probability of finding a correlated pair of neutrons at large distances impacts the probability of finding a single neutron in the halo region. It is to be noted that the `halo' effect seen for $\alpha$=1/6 solely results from pairing, and it is not related to reduced binding energy. In contrast, as it is shown below, increased pairing correlations lead to greater separation energies and lower chemical potentials, i.e., to increased particle stability.

As emphasized in Ref. [2], pairing correlations can be enhanced in weakly bound nuclei due to increased surface effects and the closeness of the particle continuum. In turn, pairing can influence quite dramatically the asymptotic properties of density distributions in drip-line systems. This is nicely illustrated in Fig. 3 which compares the HFB+SLy4 densities calculated with $\alpha$=1/2 for ¹²⁰Sn (well bound), ¹⁵⁰Sn (weakly bound), and ¹⁷⁰Sn (very weakly bound) drip-line nuclei. One can see that adding neutrons results in a simultaneous increase of density both in the nuclear interior and in the surface region. At very large distances, the asymptotic behavior of $\rho_n(r)$reflects the gradual rise of the Fermi level $\lambda$ with a neutron number. However, this effect is much stronger for the pairing density [2]. Indeed, as seen in Fig. 3, as compared to ¹²⁰Sn, there is a dramatic increase in $\tilde{\rho}_n(r)$ in the outer regions of weakly bound nuclei ¹⁵⁰Sn and, in particular, ¹⁷⁰Sn. These calculations indicate that for small values of $\alpha$the box size should be chosen as very large if one aims for a very accurate description of HFB densities at large distances.

The structure of HFB densities determines the behavior of the self-consistent p-h and p-p potentials. Figure 4 shows the behavior of $U(r)\equiv\Gamma(r,r)$(6) (local part only - see discussion in Ref. [2]) and $\tilde{U}(r)\equiv \tilde{h}(r,r)$(7) obtained for neutrons in the weakly bound nucleus ¹⁵⁰Sn in the HFB+SLy4 model for several values of $\alpha$. The behavior of the pairing potential $\tilde{U}(r)$ is consistent with the pattern shown in Fig. 1. Indeed, for DDDI, the pairing potential is proportional to the product of the pairing density $\tilde{\rho}$ and the pairing strength factor $f_{\rm pair}$ (11). Consequently, $\tilde{U}(r)$ is essentially peaked around the nuclear surface, and both its minimum and range shift towards larger values of r with decreasing $\alpha$. For $\alpha$=1/6, the pairing potential is still sizeable at large distances reaching 14fm (i.e., twice the nuclear radius). The central p-h potential U(r) very weakly depends on the form of the pairing interaction. It is only for small values of $\alpha$ that, due to a direct contribution from the pairing density to the p-h potential [see Eq. (A.5a) in Ref. [19]], a small barrier develops just beyond the nuclear surface. That is, the central neutron potential becomes slightly repulsive at r$\simeq$9fm. However, this effect is more than compensated by the increased pairing potential and the total binding energy decreases.

Figure 5 compares the HFB+SLy4 HFB potentials calculated with $\alpha$=1/2 for ^120,150,170Sn. With increasing neutron number, the radius of the p-h potential increases, and the potential becomes more wide in the outer region (i.e., it becomes more diffused). The p-p potential becomes more surface-peaked and its range increases. By analyzing Figs. 4 and 5, one can conclude that it is in N-rich weakly bound nuclei that the density dependence of the pairing interaction is most important.