Density dependence of the pairing interaction and pairing correlation in unstable nuclei

This work aims at a global assessment of the effect of the density dependence of the zero-range pairing interaction. Systematic Skyrme-Hartree-Fock-Bogoliubov calculations with the volume, surface, and mixed pairing forces are carried out to study the pairing gaps in even-even nuclei over the whole nuclear chart. Calculations are also done in coordinate representation for unstable semimagic even-even nuclei. The calculated pairing gaps are compared with empirical values from four different odd-even staggering formulas. Calculations with the three pairing interactions are comparable for most nuclei close to the $\beta$-stability line. However, the surface interaction calculations predict neutron pairing gaps in neutron-rich nuclei that are significantly stronger than those given by the mixed and volume pairing. On the other hand, calculations with volume and mixed pairing forces show noticeable reduction of neutron pairing gaps in nuclei far from stability.

Figure 1

Neutron pairing gaps calculated for ${\Delta }^{\left(3\right)}$ (a), ${\Delta }_C^{\left(3\right)}$ (b), ${\Delta }^{\left(4\right)}$ (c), and ${\Delta }^{\left(5\right)}$ (d) for all known even-even nuclei.

Figure 2

Neutron pairing gap ${\Delta }_C^{\left(3\right)}$ for even (odd) number of protons and even (even) number of neutrons, red circles (black squares). Green diamonds are proton-neutron interactions. Solid (dashed) lines are the mean values $\left(\pm \sigma \right)$. Gap data with errors more than 100 keV are excluded.

Figure 3

hfbtho calculations with the mixed pairing interaction for the Fermi level ${\lambda }_n$ (b), two-neutron separation energy $S_{2n}=B(Z,N-2)-B(Z-N)$ (a), mean neutron pairing gap ${\Delta }_n$ (c), and the deformation $\beta$ (d).

Figure 4

hfbtho calculations with the volume pairing interaction for the Fermi level ${\lambda }_n$ (b), two-neutron separation energy $S_{2n}=B(Z,N-2)-B(Z-N)$ (a), mean neutron pairing gap ${\Delta }_n$ (c), and the deformation $\beta$ (d).

Figure 5

hfbtho calculations with the surface pairing interaction for the Fermi level ${\lambda }_n$ (b), two-neutron separation energy $S_{2n}$ (a), mean neutron pairing gap ${\Delta }_n$ (c), and the deformation $\beta$ (d).

Figure 6

The neutron drip lines as defined by $\lambda =0$ (a) and $S_{2n}=0$ (b) given by the hfbtho calculations with different pairing interactions.

Figure 7

hfbtho calculations with mixed pairing (green circles), volume pairing (blue diamonds), and surface pairing (black squares) for the neutron pairing gaps in semimagic nuclei. The values of ${\Delta }_{\mathrm{LCS}}$ (open markers) are connected by dashed lines while the average gaps ${\Delta }_{\mathrm{mean}}$ (filled markers) are linked by solid lines. The red triangles correspond to the empirical pairing gaps ${\Delta }_C^{\left(N\right)}$.

Figure 8

hfbrad calculations with mixed pairing (green circles), volume pairing (blue diamonds), and surface pairing (black squares) for the neutron pairing gaps in semimagic nuclei. The values of ${\Delta }_{\mathrm{LCS}}$ (open markers) are connected by dashed lines while the average gaps ${\Delta }_{\mathrm{mean}}$ (filled markers) are linked by solid lines. The red triangles correspond to the empirical pairing gaps ${\Delta }_C^{\left(N\right)}$.

Figure 9

SHF calculations with the Sly4 force on the evolution of the single-particle energies in the neutron-rich $N=52$ isotones. With no pairing considered, the spurious $s_{1/2}$ states with positive energies have no physical meaning and are shown only to illustrate the tendency.

Figure 10

Two-particle wave function ${\Psi }^{\left(2\right)}$ for ${}^{82}\mathrm{Ni}.$ Calculations with the volume, mixed, and surface interactions are shown at the top, middle and bottom panels, respectively.