Spherical Nuclei

Figure 7: Differences between pairing renormalization (RN) and regularization (RG) procedures for (a) total energies and neutron pairing energies, (b) two neutron separation energies, and (c) the average neutron gaps. The HFB+THO calculations are performed for the chain of the spherical Sn isotopes using the SkP Skyrme parameterization.

Figure 7 displays differences between the pairing renormalization and regularization procedures for the Sn isotopes. Calculations are performed with both volume and mixed pairing interactions. For the two-neutron separation energies, the maximum difference between the renormalization and regularization schemes is about 100 (300)keV for the volume (mixed) pairing. In the neutron gap, the corresponding difference is about 50 (100)keV, and in nuclear radii (not displayed) it is practically negligible (about 0.01fm). The largest differences show up in pairing energies - about 1 (3) MeV for the volume (mixed) pairing; however, total energy differences are much smaller - about 400 (800)keV.

Figure 8: Similar to Fig. 7 except for the deformed Dy isotopes. Quadrupole deformations are displayed in panel (c). Mixed pairing interaction was used.

Analyzing the total energies obtained in both methods, Fig. 7(a), one can see that the pairing renormalization procedure gives systematically more binding. The differences are negligible for stable nuclei and nuclei near the proton drip line. They increase in mid-shell nuclei near the two-neutron drip line where the pairing effects are the largest, and then decrease towards the closed-shell nucleus $^{176}$Sn located just at the two-neutron drip line. In general, both procedures give more similar results in the case of volume pairing than in the case of mixed pairing.

Recently, the pairing regularization procedure has been analyzed in the context of relativistic mean-field approximation [23]. In order to simulate the finite range contribution to the nuclear matter pairing gap coming from the Gogny pairing force, it was necessary to introduce strong density dependence in the pairing strength of the contact interaction.

Using the regularization procedure and calculating the Sn chain with both volume and newly constructed (surface) contact interaction, the authors of Ref. [23] have found differences in pairing energies of the order of 20MeV in the neutron-rich nuclei around $^{148}$Sn. In our work, for the same nuclei, the differences in pairing energies between volume and mixed pairing variants do not exceed 2.6MeV. This comparison shows that the density-dependent contact interaction proposed in Ref. [23] is questionable for finite nuclei, despite its agreement with the finite-range Gogny pairing force in the infinite nuclear matter.