New efficient method for performing Hartree-Fock-Bogoliubov calculations for weakly bound nuclei

We propose a new method to solve the Hartree-Fock-Bogoliubov equations for weakly bound nuclei, which works for both spherical and axially deformed cases. In this approach, the quasiparticle wave functions are expanded in a complete set of analytical Pöschl-Teller-Ginocchio and Bessel/Coulomb wave functions. Correct asymptotic properties of the quasiparticle wave functions are endowed in the proposed algorithm. Good agreement is obtained with the results of the Hartree-Fock-Bogoliubov calculation using box boundary condition for a set of benchmark spherical and deformed nuclei.

Figure 1

The shifted PTG potential, the HF potential calculated with the SLy4-force, and the unshifted PTG potential for neutrons in ${}^{84}\mathrm{Ni}$. HF and shifted PTG potentials to which the centrifugal part is added are provided as well, and the energies of $0g_{7/2}$ levels for each potential are indicated. All data respectively associated to HF, shifted and unshifted PTG potentials are respectively shown in solid, dashed, and dotted lines.

Figure 2

The PTG (dashed lines) and GHF (solid lines) wave functions for various resonance states.

Figure 3

The PTG (dashed lines), GHF (solid lines), and HO (dotted lines) wave functions including the asymptotic region for the bound $0s_{1/2},1s_{1/2}$, and $2s_{1/2}$ neutron states both in normal scale (lower panel) and logarithmic scale (upper panel).

Figure 4

The PTG (dashed lines) and GHF (solid lines) wave functions of the neutron continuum $s$-states calculated with energies of 0.118 MeV, 9.996 MeV, and 66.119 MeV.

Figure 5

The PTG (dashed lines) and the GHF (solid lines) wave functions for the neutron continuum $d_{3/2}$-states calculated at the same energies as in Fig. 4

Figure 6

Dependence on $k_{\max }$ of the neutron density ${\rho }_n$ and the neutron pairing density ${\tilde{\rho }}_n$ calculated for ${}^{84}\mathrm{Ni}$ by the HFB/PTG method.

Figure 7

The neutron densities ${\rho }_n$ and proton densities ${\rho }_p$ both in normal (left-hand side) and logarithmic (right-hand side) scales. Results of the HFB/Box calculation are displayed by solid lines, while those of the HFB/PTG calculations by open circles and dashed lines. The HFB/HO densities are also indicated by dotted lines in the right panels for comparison.

Figure 8

The neutron pairing densities ${\tilde{\rho }}_n$ in normal (left-hand side) and logarithmic (right-hand side) scales. There are no pairing correlations in the proton channel. Results of the HFB/Box and HFB/PTG calculations are displayed both by solid lines, as they are almost indistinguishable, while HFB/THO pairing densities are represented by dashed lines.

Figure 9

The neutron and proton densities of the prolately deformed nucleus ${}^{110}\mathrm{Zr}$ ($\beta =0.40$), respectively calculated by the HFB/PTG (solid and dashed lines, respectively) and HFB/THO (circles) methods in normal (top left) and logarithmic (top right) scale. They are given along the long and short axes of deformation, easily identified from the figure. The neutron and proton densities of ${}^{40}\mathrm{Mg}$ calculated by the HFB/PTG method for two states with different deformations (oblate $\beta =-0.09$ and prolate $\beta =0.26$) in normal (middle and bottom left) and logarithmic (middle and bottom right) scale are also provided with the same line convention.

Figure 10

Same as in Fig. 9 but for pairing densities and without HFB/THO results. Proton pairing density is not represented for ${}^{110}\mathrm{Zr}$ as it is negligible therein.