Fission barriers in the neutron-proton isospin plane for heavy neutron-rich nuclei

We discuss the sensitivity of fission barrier for heavy neutron-rich nuclei to fission paths in the two-dimensional neutron-proton quadrupole plane. To this end, we use the constrained Skyrme-Hartree-Fock + BCS method, and examine the difference of fission barriers obtained with three constraining operators, that is, the neutron, proton, and mass quadrupole operators. We investigate ${}^{220}\mathrm{U}$, ${}^{236}\mathrm{U}$, and ${}^{266}\mathrm{U}$, that is relevant to r-process nucleosynthesis. We find that the fission barrier heights are almost the same among the three constraining operators even for neutron-rich nuclei, indicating that the usual way to calculate fission barriers with the mass quadrupole operator is well justified. We also discuss the difference between proton and neutron deformation parameters along the fission paths.

Figure 1

Schematic pictures of the constrained Hartree-Fock method. Fig. 1a is for the total constraint, in which the energy is minimized along the lines of $Q_p+Q_n=\mathrm{const}$. shown by the dashed line. Fig. 1b is for the proton constraint, in which the energy is minimized along the lines of $Q_p=\mathrm{const}$. The corresponding paths are shown by the thick solid lines in both the figures.

Figure 2

The energy along the paths shown in Fig. 1. The solid and the dotted lines are for the total and the proton constraints, respectively.

Figure 3

The fission barrier for ${}^{236}\mathrm{U}$ as a function of deformation parameter ${\beta }_t$ obtained with the total constraint scheme (Fig. 3a). The dashed line in the Fig. 3b shows the difference of the fission barrier obtained with the proton constraint and that with the total constraint, while the dotted line denotes the difference obtained with the neutron constraint.

Figure 4

Same as Fig. 3, but for ${}^{266}\mathrm{U}$.

Figure 5

Proton single particle levels near the Fermi energy for ${}^{236}\mathrm{U}$ (the top panel) and ${}^{266}\mathrm{U}$ (the bottom panel) as a function of the total deformation parameter. The solid and the dashed lines are the results for the total and the proton constraints, respectively.

Figure 6

Difference between the proton and the neutron deformation parameters, ${\beta }_p-{\beta }_n$, for ${}^{220,236,266}\mathrm{U}$ nuclei as a function of the total deformation parameter. The solid, dashed, and dotted lines are the results for the total, proton, and neutron constraints, respectively.

Figure 7

Difference of the proton and the neutron density distributions, ${\rho }_p-{\rho }_n$, for ${}^{236}\mathrm{U}$ (Fig. 7a) and ${}^{266}\mathrm{U}$ (Fig. 7b) at ${\beta }_t=2.0$ obtained with the total constraint. The densities are axial symmetric around the $z$-axis.