Effective pairing interactions with isospin density dependence

We perform Hartree-Fock-Bogoliubov (HFB) calculations for semi-magic calcium, nickel, tin, and lead isotopes and $N=20,28,50$, and 82 isotones using density-dependent pairing interactions recently derived from a microscopic nucleon-nucleon interaction. These interactions have an isovector component so that the pairing gaps in symmetric and neutron matter are reproduced. Our calculations well account for the experimental data for the neutron number dependence of binding energy, two-neutron separation energy, and odd-even mass staggering of these isotopes. This result suggests that by introducing the isovector term in the pairing interaction, one can construct a global effective pairing interaction that is applicable to nuclei in a wide range of the nuclear chart. It is also shown with the local density approximation that the pairing field deduced from the pairing gaps in infinite matter reproduces qualitatively well the pairing field for finite nuclei obtained with the HFB method.

Figure 1

The experimental odd-even mass staggering ${\Delta }_n^{(3)}$ given by Eq. (10) for the semi-magic Ca, Ni, Sn, and Pb isotopes. It is compared with the phenomenological fits ${\Delta }_n^{\mathrm{IS}}=13.3/A^{1/2}$ MeV and ${\Delta }_n^{\mathrm{IS}+\mathrm{IV}}=[7.2-44(1-2Z/A){}^2]/A^{1/3}$ MeV proposed in Ref. [27].

Figure 2

Comparison of the HFB calculations with the experimental binding energies, $B/A$. The solid line shows the results without pairing interaction (HF); the dotted, short dashed, and long dashed lines are obtained with the pairing interactions IS+IV bare, IS+IV induced, and IS bare, respectively. For each isotopic chain, we also plot the difference $\delta (B/A)=B(\mathrm{th})/A-B(\exp )/A$ between the theoretical and the experimental values for the binding energy. All units are given in MeV. See the text for more details.

Figure 3

Comparison between HFB and experiments for the two-neutron separation energies $S_{2n}$. The value $\delta (S_{2n})$ is defined as $\delta (S_{2n})=S_{2n}(\mathrm{th})-S_{2n}(\exp )$. See the caption of Fig. 2 and the text for details.

Figure 4

Comparison of the HFB pairing gaps ${\Delta }_n$ calculated with Eq. (11) with the OES given by ${\Delta }^{(3)}$. The value $\delta ({\Delta }_n)$ is defined as $\delta ({\Delta }_n)={\Delta }_n(\mathrm{th})-{\Delta }_n(\exp )$. See the caption of Fig. 2 and the text for details.

Figure 5

Comparison of the HFB pairing gaps ${\Delta }_p$ calculated with Eq. (11) with the OES given by ${\Delta }^{(3)}$. The value $\delta ({\Delta }_p)$ is defined as $\delta ({\Delta }_p)={\Delta }_p(\mathrm{th})-{\Delta }_p(\exp )$. See the caption of Fig. 2 and the text for details.

Figure 6

Pairing gaps in uniform matter obtained from the solution of the BCS equations with the pairing interactions IS+IV bare, IS+IV induced, and IS bare and the isoscalar interaction of Ref. [11] with ${\alpha }_s=1/2$ and ${\eta }_s=1$. Notice that the results of IS+IV bare are identical to those of IS bare in symmetric matter. See the text for more details.

Figure 7

Comparison of particle and pairing densities for the nucleus ${}^{150}\mathrm{Sn}$ obtained with several interaction sets from Ref. [11] and with the IS+IV bare interaction. See the text for more details.

Figure 8

The neutron Fermi momentum and the proton fraction, obtained with the HFB densities, for two mid-shell Sn nuclei, ${}^{110}\mathrm{Sn}$ (the solid line) and ${}^{150}\mathrm{Sn}$ (the dashed line).

Figure 9

Comparison of the pairing field ${\Delta }_n^{\mathrm{LDA}}(r)$ in the local density approximation (the dashed line) with that of the HFB calculations (the solid line) for the ${}^{110}\mathrm{Sn}$ and ${}^{150}\mathrm{Sn}$ nuclei. The pairing fields obtained with the different pairing interactions, IS+IV bare, IS+IV induced, and IS bare, are plotted separately.