Variation after particle-number projection for the Hartree-Fock-Bogoliubov method with the Skyrme energy density functional

Variation after particle-number restoration is incorporated for the first time into the Hartree-Fock-Bogoliubov (HFB) framework employing the Skyrme energy density functional with zero-range pairing. The resulting projected HFB equations can be expressed in terms of the local gauge-angle-dependent densities. Results of projected calculations are compared with those obtained within the Lipkin-Nogami method in the standard version and with the Lipkin-Nogami method followed by exact particle-number projection.

Figure 1

The neutron equivalent single-particle energies (84) for $N=70$ and $Z=50$ obtained in the HFB + LN method (first spectrum), HFB + VAPNP method using the average quasiparticle energies $E_n$ (second spectrum) (85), and by using the “quasiparticle” energies $E_k^N$ calculated in the HFB + VAPNP method for different values of the intrinsic neutron $\overline{N}$ and proton $\overline{Z}$ numbers (the remaining five spectra). The dashed line indicates the position of the LN Fermi energy λ.

Figure 2

Comparison between the LN, PLN, and VAPNP results for the chain of Ca isotopes. The upper panel shows the neutron pairing energies, whereas 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.

Figure 4

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

Figure 5

Similar to Fig. 4, except for nuclei near ${}^{132}\mathrm{Sn}$.

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 α.