Isoscalar and isovector density dependence of the pairing functional determined from global fitting

We optimize the density dependence of a local energy density functional for pairing correlations (pair-DF), taking into account both the isoscalar and the isovector densities. All the experimental pairing gaps in even-even nuclei with $N,Z>8$ are analyzed by performing the Hartree-Fock-Bogoliubov calculation. The new pairing energy density functionals give the best fit for all the pairing gaps with the rms deviation of ${\sigma }_{\text{tot}}=(260$–360) keV, depending on adopted Skyrme interactions in the particle-hole channel. It is shown that the isoscalar-density dependence in the pair-DF can be determined almost uniquely for the several Skyrme forces by adjusting the neutron and proton pairing gaps. An empirical relation among the parameters in the isoscalar part of pair-DF is extracted. The uncertainty of the isovector part is also examined by calculating the pairing gaps of neutron-rich and proton-rich nuclei in a wider region of the nuclear chart. We point out also that the linear isovector-density dependence in the pair-DF can mimic well the Coulomb effect for pairing correlations in neutron-rich nuclei. We discuss finally the predictive power of the pair-DF for pairing properties around drip-line regions.

Figure 1

Experimental pairing gaps of even-even nuclei are grouped into zones 1, 2, and 3. The subset of zone 1, which we call zone 0, is also indicated. See text and Table for the definitions.

Figure 2

(Left) Distribution of experimental pairing gaps in each zone is shown as a function of $A$. (Right) The same data sets are plotted as a function of $\alpha$.

Figure 3

The ${\sigma }_{\text{tot}}^{\left(\text{all}\right)}$ is shown as a function of ${\eta }_2$. Here ${\eta }_0=0.75$ is fixed. The parameters ${\eta }_1$ and $V_0$ for SLy4 plus PDF-IV2 are optimized for each ${\eta }_2$. See text for details.

Figure 4

(Left) ${\sigma }_{\text{tot}}^{\left(\text{zone}\text{1}\right)}$ is shown as a function of $V_0$. The results with ${\eta }_1=0.270$ and $0.390$ are compared. Here ${\eta }_0=0.75$ and ${\eta }_2=2.5$ are fixed, and $V_0^{\left(\text{two-step}\right)}=-396.47$ MeV fm${}^{-3}$. The filled box and circle represent the minimum values of ${\sigma }_{\text{tot}}^{\left(\text{zone}\text{1}\right)}$ in the two-step and direct procedures respectively. The Skyrme SLy4 and PDF-IV2 are used. (Right) The ${\sigma }_{\text{tot}}^{\left(\text{zone}\text{1}\right)}$ is shown as a function of ${\eta }_1$. The $V_0$ is optimized in zone 1 for each ${\eta }_1$.

Figure 5

${\sigma }_{\text{tot}}^{\left(\text{all}\right)}$, ${\sigma }_n^{\left(\text{all}\right)}$, and ${\sigma }_p^{\left(\text{all}\right)}$ are shown as functions of ${\eta }_0$. The SLy4 and the PDF-IV2 determined by the two-step procedure are used to draw the curves. The direct procedure is also performed at ${\eta }_0=0.625$, $0.750$, and $0.875$ and the results are marked by the open circles. The results with PDF-IS are also shown.

Figure 6

Comparison of ${\sigma }_{\text{tot}}^{\left(\text{all}\right)}$ with SLy4, SkM*, LNS, and SkP. The PDF-IV2 determined by the two-step procedure is used.

Figure 7

(a) ${\sigma }_{\text{tot}}^{\left(\text{zone}\text{1}\right)}$, ${\sigma }_{\text{tot}}^{\left(\text{zone}\text{2}\right)}$, and ${\sigma }_{\text{tot}}^{\left(\text{zone}\text{3}\right)}$ are shown as a function of ${\eta }_0$. Here ${\eta }_2=2.5$ is fixed. The parameters ${\eta }_1$ and $V_0$ for SLy4 plus PDF-IV2 are determined by the two-step procedure. (b) The same figure with Fig. 7a but plotted as a function of ${\eta }_2$. Here ${\eta }_0=0.75$ is fixed.

Figure 8

The difference of pairing gaps obtained with ${\eta }_0=0$ and ${\eta }_0=0.75$ for each nucleus. The result in zone 3 is plotted as a function of $A$. The SLy4 and PDF-IV2 determined by the two-step procedure are used.

Figure 9

The neutron pairing gaps in Ca, Ni, Sn, and Pb isotopes (left) and the proton pairing gaps in $N=20$, 28, 50, and 82 isotones (right). The results obtained with PDF-IS, PDF-IV1, and PDF-IV2 at ${\eta }_0=0.75$ are compared. The two-step procedure is performed. The experimental pairing gaps are also shown.

Figure 10

The same as in Fig. 9, but the results obtained by the PDF-IV2 with the optimized strengths under the assumption $V_p=V_n$ and $V_p=0.9V_n$, and the fixed $V_p=V_n=V_{\text{vac}}$ are compared. See the text for details.

Figure 11

The parameters $V_0$ of PDF-IV2 for SLy4, SkM*, LNS, and SkP are shown as functions of ${\left({\eta }_0\right)}^2$. The two-step procedure is performed.

Figure 12

The optimal values of $V_0$ at ${\eta }_0=0.25$, $0.5$, $0.75$, and $1.0$ are shown as functions of $m_s^{*}/m$. The two-step procedure is performed. The dashed lines represent the extracted relation $|V_0|=260.58{\left({\eta }_0\right)}^2-255.18m_s^{*}/m+418.59$ MeV fm${}^{-3}$. The shaded region indicates the uncertainty of parameters in the isoscalar part of PDF-IV2. ${\sigma }_{\text{tot}}^{\left(\text{all}\right)}$ differs less than 0.05 MeV from the minimum value for each Skyrme force if $(V_0,{\eta }_0)$ is in the shaded region. See the text for details.