Assessing theoretical uncertainties in fission barriers of superheavy nuclei

Theoretical uncertainties in the predictions of inner fission barrier heights in superheavy elements have been investigated in a systematic way for a set of state-of-the-art covariant energy density functionals which represent major classes of the functionals used in covariant density functional theory. They differ in basic model assumptions and fitting protocols. Both systematic and statistical uncertainties have been quantified where the former turn out to be larger. Systematic uncertainties are substantial in superheavy elements and their behavior as a function of proton and neutron numbers contains a large random component. The benchmarking of the functionals to the experimental data on fission barriers in the actinides allows reduction of the systematic theoretical uncertainties for the inner fission barriers of unknown superheavy elements. However, even then, on average they increase on moving away from the region where benchmarking has been performed. In addition, a comparison with the results of nonrelativistic approaches is performed in order to define full systematic theoretical uncertainties over the state-of-the-art models. Even for the models benchmarked in the actinides, the difference in the inner fission barrier height of some superheavy elements reaches $5–6$ MeV. This uncertainty in the fission barrier heights will translate into huge (many tens of the orders of magnitude) uncertainties in the spontaneous fission half-lives.

Figure 1

The heights of inner fission barriers in selected nuclei as obtained in axially symmetric RHB calculations with indicated CEDFs.

Figure 2

The heights of inner fission barriers (in MeV) obtained in axially symmetric RHB calculations as a function of proton and neutron numbers. The results of the calculations with the indicated CEDFs are shown from the two-proton drip line up to $N=196$.

Figure 3

The spreads $\mathrm{\Delta }{E}^B$ of the heights of inner fission barriers as a function of proton and neutron numbers. $\mathrm{\Delta }{E}^B(Z,N)=|E_{max}^B(Z,N)-E_{min}^B(Z,N)|$, where, for given $Z$ and $N$ values, $E_{max}^B(Z,N)$ and $E_{min}^B(Z,N)$ are the largest and smallest heights of inner fission barriers obtained with the set of functionals NL3*, DD-ME2, $\mathrm{DD}-\mathrm{ME}\delta$, DD-PC1, and PC-PK1. Panel (a) shows the results for all five functionals, while DD-ME2 and $\mathrm{DD}-\mathrm{ME}\delta$ are excluded in the results shown in panel (b).

Figure 4

The spreads in the binding energies of the ground states (a) and the energies of saddle points (b) as obtained with five employed CEDFs. Panels (c) and (d) show the same results but with the energy color map used in Fig. 3.

Figure 5

The same as Fig. 4 but for the case when DD-ME2 and $\mathrm{DD}-\mathrm{ME}\delta$ CEDFs are excluded from consideration.

Figure 6

Statistical uncertainties in the deformation energy curves of the ${}^{296}112$ nucleus. The mean potential energy curve is shown by a solid line. The red-colored region shows the standard deviations in energy.

Figure 7

Potential energy surfaces of the ${}^{300}120$ nucleus as obtained in the calculations with indicated CEDFs. The energy difference between two neighboring equipotential lines is equal to 0.5 MeV. The Ax, Ax-Tr, Tr-A, and Tr-B saddles are shown by blue or red circles, diamonds, triangles, and squares, respectively. The PES are shown in the order of decreasing height of inner fission barrier.

Figure 8

The same as in Fig. 7 but for the ${}^{284}112$ nucleus.

Figure 9

The spreads $\mathrm{\Delta }{E}^S$ of the energies of axial [panel (a)], triaxial [panels (b), (c), (d)], and the lowest in energy [panel (d)] saddles for a selected set of the $Z=112–120$ nuclei as a function of proton and neutron number. $\mathrm{\Delta }{E}^S(Z,N)=|E_{max}^S(Z,N)-E_{min}^S(Z,N)|$, where, for given $Z$ and $N$ values, $E_{max}^S(Z,N)$ and $E_{min}^S(Z,N)$ are the largest and smallest energies of the saddles obtained with the set of functionals NL3*, DD-ME2, $\mathrm{DD}-\mathrm{ME}\delta$, DD-PC1, and PC-PK1. Note that the same color map as in Fig. 3 is used here.

Figure 10

The heights of inner fission barriers in selected nuclei as obtained in the TRHB calculations with indicated CEDFs. The style of the symbol filling indicates the type of the lowest in energy saddle. Note that the TRHB results in a few $N\sim 166$ and $N=172$ nuclei (see Fig. 9 below) and the trends of the evolution of PES with particle number allow us to firmly establish the axially symmetric nature of the lowest saddle in the $Z=112$ and 114 nuclei (as well as in $Z=116$ nuclei for the NL3* and DD-ME2 functionals) for neutron numbers between 164 and approximately $N=172$. For some of these nuclei, we use axial RHB results when the TRHB results are not available.

Figure 11

The same as in Fig. 9 but for the case when the DD-ME2 and $\mathrm{DD}-\mathrm{ME}\delta$ CEDFs are excluded from consideration.

Figure 12

Inner fission barrier heights $B_f$ as a function of the neutron number $N$. The position of the inner fission barrier saddle in deformation space varies as a function of particle number. Thus, the lowest saddles are labeled by Ax, Ax-Tr, Tr-A, and Tr-B (see text for details). The results of triaxial RMF + BCS calculations are taken from Ref. [11]. The results of Skyrme DFT calculations with SkM* have been taken from Ref. [40]. The results of the MM calculations are taken from Ref. [59] [labeled as MM (Möller)] and Ref. [60] [labeled as MM (Kowal)]. Note that the style of Fig. 1 is used here for easy comparison between two figures.