Fission barriers at the end of the chart of the nuclides

We present calculated fission-barrier heights for 5239 nuclides for all nuclei between the proton and neutron drip lines with $171\le A\le 330$. The barriers are calculated in the macroscopic-microscopic finite-range liquid-drop model with a 2002 set of macroscopic-model parameters. The saddle-point energies are determined from potential-energy surfaces based on more than 5 000 000 different shapes, defined by five deformation parameters in the three-quadratic-surface shape parametrization: elongation, neck diameter, left-fragment spheroidal deformation, right-fragment spheroidal deformation, and nascent-fragment mass asymmetry. The energy of the ground state is determined by calculating the lowest-energy configuration in both the Nilsson perturbed-spheroid $\left(\epsilon \right)$ and the spherical-harmonic $\left(\beta \right)$ parametrizations, including axially asymmetric deformations. The lower of the two results (correcting for zero-point motion) is defined as the ground-state energy. The effect of axial asymmetry on the inner barrier peak is calculated in the $(\epsilon ,\gamma )$ parametrization. We have earlier benchmarked our calculated barrier heights to experimentally extracted barrier parameters and found average agreement to about 1 MeV for known data across the nuclear chart. Here we do additional benchmarks and investigate the qualitative and, when possible, quantitative agreement and/or consistency with data on $\beta$-delayed fission, isotope generation along prompt-neutron-capture chains in nuclear-weapons tests, and superheavy-element stability. These studies all indicate that the model is realistic at considerable distances in $Z$ and $N$ from the region of nuclei where its parameters were determined.

Figure 1

Calculated fission-barrier heights for 3282 nuclei. The highly variable structure is mostly due to ground-state shell effects. Ground-state shell effects are particularly strong in the deformed regions around ${}_{100}{}^{252}\mathrm{Fm}_{152}$ and ${}_{108}{}^{270}\mathrm{Hs}_{162}$ and in the nearly spherical region near the next doubly magic nuclide postulated to be at ${}_{114}{}^{298}\mathrm{Fl}_{184}$. Our strongest shell effects are slightly offset to the left with respect to this isotope.

Figure 2

Calculated fission-barrier heights for 2113 nuclei with generally lower proton and neutron numbers than those in Fig. 1. Because the macroscopic energy contributes the major part of the fission-barrier height for most nuclei in this region, and because of the different energy scale compared to Fig. 1, the only shell effects clearly visible come from the $N=126$ spherical neutron shell.

Figure 3

Calculated macroscopic and total potential energies for shape sequences leading to the touching configuration, at the long-dashed line, of spherical ${}^{78}\mathrm{Zn}$ and ${}^{208}\mathrm{Hg}$. To the left the calculations trace the energy for a single, joined shape configuration from oblate shapes through the spherical shape at $r=0.75$ to the touching configuration at $r=1.52$; to the right the calculations give the energy for separated spherical nuclei beyond the touching point. The continuous path through five-dimensional space from the ground state to the touching configuration is arbitrary; the key point is that the limiting shapes when approaching the line of touching from the left and right are identical, namely spherical ${}^{78}\mathrm{Zn}$ and ${}^{208}\mathrm{Hg}$ in contact. At a specific value of $r$ all curves are calculated for the same shape. To obtain continuity of the macroscopic energy at touching, a crucial feature in realistic models, it is essential that various model terms depend appropriately on nuclear shape, as is the case for the curves (a). The slight remaining discontinuity in the total fusion energy curve (solid line) arises because the Fermi surfaces of the nuclei readjust at contact and because pairing and spin-orbit strength parameters also undergo small discontinuous changes there. This figure and caption are adapted from Ref. [28].

Figure 4

A more detailed look at fission-barrier heights for 980 nuclei in the heaviest region. We have marked with black dots those nuclides for which spontaneous fission with a half-life under about 30 days has been observed.

Figure 5

Various energy concepts used in macroscopic-microscopic fission potential-energy calculations. The dotted line is the macroscopic “liquid-drop” energy along a specified path; the solid line is the total macroscopic-microscopic energy along a partially different shape sequence. So that the various energy concepts can be illustrated, the shapes for which the energies have been calculated are as follows. At $Q_2=0$ the energies are calculated for a spherical shape. For the shapes from the sphere to the ground-state shape, the shapes are the same for both curves and chosen so that they evolve continuously from the sphere to the calculated macroscopic-microscopic ground-state shape. From the ground state towards larger deformations the total-energy curve is along the optimal fission path that includes all minima and saddle points identified along this path in the five-dimensional deformation space; the liquid-drop-energy curve joins smoothly the macroscopic energy for the shape at the macroscopic-microscopic ground state (which is not the lowest macroscopic energy at this value of $Q_2$) to the liquid-drop-model saddle point. The energies are calculated for ${}^{232}\mathrm{Th}$. Some important shapes are also shown.

Figure 6

Difference between the saddle-point energy obtained from the five-dimensional macroscopic-microscopic potential-energy surface and the saddle-point energy obtained from a surface in the same deformation space using only the macroscopic model for 5139 nuclei.

Figure 7

Shape of ${}_{60}^{171}{\mathrm{Nd}}_{111}$ at the saddle point.

Figure 8

Calculated potential-energy surface for ${}_{108}{}^{298}\mathrm{Hs}_{190}$. Local minima are shown by black dots, while saddle points are shown by magenta-colored crosses. The ground state is located at $\gamma =60$ and ${\epsilon }_2=0.65$, because this is the minimum with the highest barrier with respect to fission; see text for further discussion. But with such a low barrier this isotope would not be observable.

Figure 9

Energy window [6] for neutron-induced fission. The black dots indicate even-$N$ nuclei for which $1.0\text{MeV}<S_{1\mathrm{n}}<2.0\text{MeV}$, the so-called r-process boulevard, the region of the chart where the r-process proceeds. If the plotted quantity is negative, the system is unstable with respect to thermal neutron-induced fission. The magenta arrows indicate prompt neutron-capture chains in nuclear-weapons tests; see text for further discussion of this and the single outlying dot at $Z=124$.

Figure 10

Illustration of quantities affecting $\beta$-delayed fission, namely the ratio $B_{\mathrm{f}}/Q_{\beta }$ (top panel) and $B_{\mathrm{f}}-Q_{\beta }$ (bottom panel). Only nuclides for which $B_{\mathrm{f}}-Q_{\beta }<2$ MeV are shown. The solid black dots are the same as in Fig. 9, illustrating the “r-process boulevard.”