Coordinate space Hartree-Fock-Bogoliubov calculations for the zirconium isotope chain up to the two-neutron drip line

We solve the Hartree-Fock-Bogoliubov (HFB) equations for deformed, axially symmetric even-even nuclei in coordinate space on a two-dimensional (2-D) lattice utilizing the basis-spline expansion method. Results are presented for the neutron-rich zirconium isotopes up to the two-neutron drip line. In particular, we calculate binding energies, two-neutron separation energies, normal and pairing densities, mean square radii, quadrupole moments, and pairing gaps. Very large prolate quadrupole deformations $({\beta }_2=0.42,0.43,0.470)$ are found for the ${}^{102,104,112}\mathrm{Zr}$ isotopes, in agreement with recent experimental data. We compare 2-D basis-spline lattice results with the results from a 2-D HFB code that uses a transformed harmonic oscillator basis.

Figure 1

(Color online) Two-neutron separation energies for the neutron-rich zirconium isotopes. The drip line is located where the separation energy becomes zero. The ${}^{122}\mathrm{Zr}$ isotope is the last stable nucleus against two-neutron emission.

Figure 2

(Color online) Intrinsic quadrupole moments for protons and neutrons.

Figure 3

(Color online) Mass quadrupole parameter ${\beta }_2$ comparison for neutrons. Calculations by Lalazissis et al.. [19], HFB-2D-LATTICE, and Möller et al.. [18] (FRDM) (${\beta }_2$ total is shown).

Figure 4

(Color online) Root-mean-square radii for the chain of zirconium isotopes.

Figure 5

(Color online) Average neutron pairing gap for the chain of zirconium isotopes.

Figure 6

(Color online) Average proton pairing gap for the chain of zirconium isotopes.

Figure 7

(Color online) Contour plots of the ${}^{110}\mathrm{Zr}$ normal densities, for protons (left) and neutrons (right). Densities are shown as a function of the cylindrical coordinates $(r,z)$, where z is the symmetry axis. The scale ranges from $9.7\times {10}^{-2}$ fm${}^{-3}$ (dark red), through $5.0\hspace{1em}\times \hspace{1em}{10}^{-2}$ (light green), $3.0\times {10}^{-3}$ (light blue), to $3.4\times {10}^{-15}$ (dark blue).

Figure 8

(Color online) Contour plots of the ${}^{110}\mathrm{Zr}$ pairing densities, for protons (left) and neutrons (right). The densities are shown as a function of the cylindrical coordinates $(r,z)$, where $z$ is the symmetry axis. The scale ranges from $9.0\times {10}^{-3}$ fm${}^{-3}$ (dark red), through $4.9\hspace{1em}\times \hspace{1em}{10}^{-3}$ (light green), $8.0\times {10}^{-4}$ (light blue), to $9.3\times {10}^{-14}$ (dark blue).

Figure 9

(Color online) Logarithmic plot of the normal neutron density for the most deformed isotope ${}^{112}\mathrm{Zr}$ as a function of the distance $r=\sqrt{{\rho }^2+z^2}$.

Figure 10

(Color online) Linear plots of the normal neutron and proton densities for the drip line nucleus ${}^{122}\mathrm{Zr}$ as a function of distance $r=\sqrt{{\rho }^2+z^2}$. Comparison between the HFB-2D-THO code and the HFB-2D-LATTICE code.