Regularization of the pairing field

In this section, we present results obtained by using the regularization method proposed by A. Bulgac and Y. Yu [9]. Calculations have been made with the same box size and integration step as in the previous section, and the partial waves have been included up to $J=43/2$. The effective force used is SLy4, see table 1, combined with the mixed pairing force with parameters given in table 2. When evaluating the densities, contributions of quasiparticle states are included up to the maximum equivalent energy $E_{\mathrm{max}}$, see Eq. (47), but once this maximum energy is high enough the global properties of the nucleus do not depend on it.

Figure 6: Modulus of the difference between the total energy $E$ and its mean value $E^{(a)}$ (solid lines) and between the average neutron gap and its mean value (dashed lines) as functions of the energy $E_{\mathrm{max}}$ (see text) for $^{120}$Sn (left) and $^{150}$Sn (right). The mean values are calculated over the interval $60< E_{\mathrm{max}}<80$ MeV.

$\begin{matrix} \mbox{\includegraphics*[scale=0.47,bb=52 60 464 348,clip]{figcv... ...{\includegraphics*[scale=0.47,bb=52 60 464 348,clip]{figcvec2.ps}} \end{matrix}$

It appears that the stability of results is very satisfying, even for a very exotic nucleus. This is shown in Figure 6, where the total energy and mean neutron gap are displayed as functions of $E_{\mathrm{max}}$ for $^{120}$Sn and $^{150}$Sn. For $E_{\mathrm{max}}$ greater than 60 MeV, where the total energy does not show any significant evolution, we have evaluated its asymptotic limit $E^{(a)}$, and the analogous limit of the neutron mean pairing gap $\langle\Delta\rangle^{(a)}$, by averaging their respective values over the interval ranging from 60 to 80 MeV. The results are $E^{(a)}=1\,018.529$ MeV and $\langle\Delta\rangle^{(a)}=1.245$ MeV for $^{120}$Sn, and $E^{(a)}=1\,131.492$ MeV and $\langle\Delta\rangle^{(a)}=1.499$ MeV for $^{150}$Sn. In this interval, the energies are scattered within $\pm 16$ keV and the gaps within $\pm 7$ keV. Since the increase of $E_{\mathrm{max}}$ from 60 MeV to 80 MeV does not change the results in a significant way, the choice of $E_{\mathrm{max}}=60$ MeV has been made for the rest of this study. In principle, this value should be readjusted in other mass region or when using a different effective force.

Figure: Left figure: Binding energy per nucleon in tin isotopes. The solid line correspond to calculations with a cut-off energy of $E_{\mathrm{cut}}=60$ MeV in the equivalent spectrum and the diamond symbols to the regularization of the energy with the summation of densities up to $E_{\mathrm{max}}=60$ MeV (see text). The difference between the two sets of results is plotted in the inset. Right figure: Differences between the calculated and measured binding energies shown for those tin isotopes for which the data are known; solid line corresponds to the introduction of a cut-off and the dashed line to the regularization.

Figure 8: Left figure: Mean neutron gap in tin isotopes, the solid line corresponds to the introduction of a cut-off and the dashed line to the regularization. Right figure: Differences between the rms radii of neutrons and protons in tin isotopes compared with the same quantity for $^{132}$Sn, the solid line corresponds to the introduction of a cut-off and the diamond symbol to the regularization, while the difference between the two sets is plotted in the inset.

Within the cut-off prescription and regularization scheme we have calculated the series of even-even tin isotopes by using the SLy4 $^{(\rho+\delta\rho)}$ force. The left part of Fig. 7 displays the binding energies per particle and the right part the deviation between the calculated binding energies and the experimental ones [25]. One can see that both methods give very similar results. The neutron mean gap are plotted on the left part of Fig. 8. In the two sets of calculation, the strengths of the pairing force have been adjusted in order to give the same gap in $^{120}$Sn. Again, we do not observe here any significant change when using or not the regularization scheme, although the gap is slightly reduced in heavy tin isotopes. Finally, the right part of Fig. 8 compares the differences between the neutron and proton rms radii plotted with respect to that in $^{132}$Sn; once again the two methods give extremely similar results.