Deformed coordinate-space Hartree-Fock-Bogoliubov approach to weakly bound nuclei and large deformations

The coordinate-space formulation of the Hartree-Fock-Bogoliubov (HFB) method enables the self-consistent treatment of mean field and pairing in weakly bound systems whose properties are affected by the particle continuum space. Of particular interest are neutron-rich, deformed drip-line nuclei, which can exhibit novel properties associated with neutron skin. To describe such systems theoretically, we developed an accurate two-dimensional lattice Skyrme-HFB solver HFB-AX based on basis (or B)-splines. Compared to previous implementations, ours incorporated a number of improvements aimed at boosting the solver's performance. These include the explicit imposition of axiality and space inversion, use of the modified Broyden method to solve self-consistent equations, and a partial parallelization of the code. HFB-AX has been compared with other HFB codes, both spherical and deformed, and the accuracy of the B-spline expansion was tested by employing the multiresolution wavelet method. Illustrative calculations are carried out for stable and weakly bound nuclei at spherical and very deformed shapes, including constrained fission pathways. In addition to providing new physics insights, HFB-AX can serve as a useful tool to assess the reliability and applicability of coordinate-space and configuration-space HFB frameworks, both existing and in development.

Figure 1

Comparison between linear mixing (squares) and modified Broyden method (circles) in HFB-AX for ${}^{22}\mathrm{O}$. The largest element of $\left|{\mathbf{F}}^{(m)}\right|$ is shown as a function of the number of iterations $m$. See text for details.

Figure 2

Two-center inverted-cosh potential of Eq. (20) as a function of $z$ at $\rho =0$. The two centers are 15 fm apart ($\zeta =7.5$ fm) and $V_0=-50$ MeV, $R_0=2$ fm, and $a=1$ fm.

Figure 3

Eigenvalues of a two-center inverted-cosh potential with the spin-orbit term calculated in HFB-AX as functions of the distance between two centers $2\zeta$.

Figure 4

Proton (top) and neutron pairing (bottom) densities calculated in HFB-AX for ${}^{84,86,88,90}\mathrm{Ni}$. Proton pairing is zero.

Figure 5

Contour plots of proton and neutron density distributions in the ($\rho ,z$) plane for the deformed ground state of ${}^{110}\mathrm{Zr}$ calculated in HFB-AX. The densities are in nucleons/${\mathrm{fm}}^{-3}$.

Figure 6

Particle (top) and pairing (bottom) ground-state densities in ${}^{110}\mathrm{Zr}$ along $\rho =0$ and $z=0$. The size of the box is $R=19.2$ fm.

Figure 7

Top: axial, reflection-symmetric fission path of ${}^{240}\mathrm{Pu}$ calculated with HFB-AX and HFBTHO (in a spherical and stretched HO basis) as a function of the mass quadrupole moment $Q_{20}$ at the self-consistent solution. Bottom: the difference between HFBTHO and HFB-AX results (normalized to zero at the ground-state configuration). The minima and maxima of energy are marked: ground state, g.s.; first barrier, b(1); fission isomer, f.i.; second barrier, b(2).