Sample results

To illustrate the Skyrme HFB+VAPNP procedure, we carried out calculations for the complete chain of calcium isotopes, from the proton drip line to the neutron drip line, and for the chain of tin isotopes with $70\leq{N}\leq90$. We used the Sly4 Skyrme force parameterization [37] and the mixed delta pairing [38,39]. The calculations were performed in the basis of 20 major HO shells. We took $L$=13 gauge-angle points, and this practically ensures exact projection for all considered nuclei. We have found that the HFB+VAPNP procedure is just $L$-times slower compared to the PLN method.

In our standard HFB calculations [35,36], the strength of the pairing force (assumed identical for protons and neutrons) is usually adjusted at a given cut-off energy $\epsilon_{\text{cut}}=60$MeV to the experimental value of the average neutron gap $\tilde{\Delta}_n$=1.245MeV in $^{120}$Sn. In the present study, we used this procedure to fix the pairing force for all LN and PLN calculations. However, it is well known that the PNP method requires another strength of the pairing force. Unfortunately, the average pairing gap $\tilde{\Delta}_n$ is not defined within the VAPNP approach, and the standard procedure for adjusting the pairing strength is no longer applicable. In this study, we adjusted the VAPNP pairing strength to the total energy of the $^{44}$Ca nucleus calculated in HFB+PLN. A much more consistent way of fitting the pairing strength should be based on calculating the mass differences of the odd-mass and even-even nuclei, all obtained within the VAPNP method. We intend to adopt such a procedure in future applications.

Figure 2: Comparison between the LN, PLN, and VAPNP results for the chain of Ca isotopes. The upper panel shows the neutron pairing energies while the lower panel shows the total LN and PLN energies relative to the VAPNP values.

Figure 3: Similar to Fig. 2, except for the chain of Sn isotopes.

A measure of pairing correlations in a nucleus is the particle-particle energy (pairing energy) given by the second term in Eq. (44). The energy of proton pairing correlations is about 2-3MeV and it changes smoothly with $N$ along the isotopic chains. On the other hand, the neutron pairing is significantly affected by the shell structure. As seen in Figs. 2 and 3, upper panels, the neutron pairing energies obtained within the LN, PLN, and VAPNP methods (and with pairing strengths adjusted as described above) are quite similar to one another.

The lower panels of Figs. 2 and 3 show differences between the total energies obtained in the LN and PLN methods and those obtained in VAPNP. The LN or PLN results are fairly close to VAPNP for mid-shell nuclei, where the neutron pairing correlations are large and static in character. Near closed shells, pairing is dynamic in nature, and the LN/PLN results deviate from those obtained in VAPNP. For open-shell nuclei, the PLN approximation is particularly good; in the calcium isotopes, the deviations from the HFB+VAPNP method usually do not exceed 250keV. For the closed-shell nuclei, on the other hand, the LN method is not appropriate [20,40,19], and the energy differences increase to more than 1MeV. Figures 2 and 3 also show that the PLN method always leads to a considerable improvement over LN, often reducing the deviation of the total energy with respect to VAPNP by about 1MeV.

As suggested in Refs. [20,21], one can further improve the PLN approximation around magic nuclei by applying the PNP to the LN solutions obtained in the neighboring nuclei. This procedure is illustrated in Figs. 4 and 5 for the magic nuclei $^{48}$Ca and $^{132}$Sn, respectively. It is seen that while the projection from $^{46}$Ca nicely reproduces the VAPNP result in $^{48}$Ca, the approximation fails when projecting from $^{50}$Ca. Similarly, projection from the LN solution in $^{130}$Sn ($^{134}$Sn) gives a better (worse) result than the projection of the LN solution obtained in $^{132}$Sn. We observe a similar pattern of results in other cases near closed shells; however, the improvement gained by projecting from isotopes below closed shells is not sufficient to replace the full VAPNP calculations at closed shells.

Figure 4: The total binding energy (with respect to a linear reference) as a function of $N$ for even-even nuclei around $^{48}$Ca, calculated in the LN, PLN and VAPNP methods. Crosses indicate the PLN results for $^{48}$Ca obtained by projecting from the LN solutions in neighboring nuclei $^{46}$Ca and $^{50}$Ca as indicated by arrows.

Figure 5: Similar to Fig. 4, except for nuclei near $^{132}$Sn.

In order to discuss the quality of prescription to calculate the LN parameter $\lambda_2$ presented in Sec. 3.2, we have repeated all our LN and PLN calculations with the effective pairing strengths $G'_{\text{eff}}=\alpha G_{\text{eff}}$ scaled by factors of $\alpha$=0.9 or 1.1 with respect to those given by Eq. (47). In this way, we tested whether our results are sensitive to this phenomenological prescription. The results obtained for the chains of Ca and Sn isotopes are shown in Fig. 6. While the LN energies (45) uniformly depend on the scaling factor $\alpha$, the PLN energies are almost independent of the scaling factor. This shows that the PNP components of the LN states weakly depend on $\lambda_2$ and can be obtained without paying too much attention to the way in which $\lambda_2$ is calculated. A rough estimate given by our phenomenological prescription is good enough to obtain reliable PLN results. On the other hand, deviations between the LN/PLN and VAPNP energies depend mostly on the local shell structure and visibly cannot be corrected by modifications of the prescription used to calculate $\lambda_2$. In large part, these deviations stem from the inapplicability of the LN/PLN method to closed-shell nuclei, where the total energy in function of particle number cannot be well approximated by the quadratic Kamlah expansion. Altogether, we conclude that the PLN method gives a fair approximation of the full VAPNP results, but fails in reproducing detailed values, especially near closed shells.

Figure 6: Total LN and PLN energies relative to the VAPNP values, calculated in the Ca (upper panel) and Sn (lower panel) isotopes with the effective pairing strengths scaled by a factor $\alpha$.