Fission properties of superheavy nuclei for $r$-process calculations

We computed a new set of static fission properties suited for $r$-process calculations. The potential energy surfaces and collective inertias of 3640 nuclei in the superheavy region are obtained from self-consistent mean-field calculations using the Barcelona-Catania-Paris-Madrid energy density functional. The fission path is computed as a function of the quadrupole moment by minimizing the potential energy and exploring octupole and hexadecapole deformations. The spontaneous fission lifetimes are evaluated employing different schemes for the collective inertias and vibrational energy corrections. This allows us to explore the sensitivity of the lifetimes to those quantities together with the collective ground-state energy along the superheavy landscape. We computed neutron-induced stellar reaction rates relevant for $r$-process nucleosynthesis using the Hauser-Feshbach statistical approach and study the impact of collective inertias. The competition between different reaction channels including neutron-induced rates, spontaneous fission, and $\alpha$ decay is discussed for typical $r$-process conditions.

Figure 1

Collective fission properties of ${}^{232}\mathrm{Th}$, ${}^{262}\mathrm{No}$, ${}^{290}\mathrm{No}$, and ${}^{316}\mathrm{Ds}$ as a function of the quadrupole moment. Bottom panels show different contributions to the potential energy surface $\mathcal{V}\left(Q_{20}\right)$ of Eq. (5). Top panels show collective inertias $\mathcal{M}\left(Q_{20}\right)$ computed with the ATDHFB (dashed line), GCM (solid line), and semiempirical inertia formula (dotted line), renormalized to the reduced inertia $\tilde{\mathcal{M}}$.

Figure 2

Region of the nuclear landscape explored in this work. Nuclei present in the AME2012 mass table evaluation [46] are depicted with orange squares. Nuclei with experimentally measured fission barriers and spontaneous fission lifetimes are marked with open circles and crosses, respectively. Nuclei for which the BCPM interaction predicts an oblate-deformed ground state are depicted with solid circles. Dashed and dot-dashed lines represent the heaviest isotope of each element with $S_n\gtrsim 2$ and $S_n\gtrsim 0$ MeV, respectively. Contour lines show the highest predicted fission barrier, in MeV.

Figure 3

Predicted two-neutron separation energies in MeV for nuclei with $84\le Z\le 120$ as a function of neutron number for three different models: BCPM (top panel), HFB21 (middle panel), and FRDM (bottom panel). Isotopic chains are connected by solid lines. Nuclei with a proton number equal to 90, 100, 110, and 120 are depicted with blue thick lines.

Figure 4

Energy gap ${\mathrm{\Delta }}_{2n}=S_{2n}(Z,N)-S_{2n}(Z,N+2)$ in MeV for nuclei with $N=184$ as a function of proton number predicted by BCPM (open circles), FRDM (triangles), and HFB21 (squares).

Figure 5

Inner fission barrier height $B_I$ (top panel), outer fission barrier height $B_{II}$ (middle panel), and isomer excitation energy $E_{II}$ (bottom panel) of eight isotopes computed with BCPM (lines) and compared with experimental data of Bjørnholm [52] (open squares) and Capote [53] (circles).

Figure 6

Spontaneous fission lifetimes $t_{\mathrm{sf}}$ as a function of the fissibility parameter. Experimental data (black solid symbols) is compared with different inertia schemes: ATDHFB (top panels), GCM (middle panels), and semiempirical formula (bottom panels). The results obtained with the renormalized collective inertia (see Table 1) are depicted with colored full symbols.

Figure 7

Calculated fission barrier heights in MeV in the region $84\le Z\le 120$ and $118\le N\le 250$ for three different mass models: BCPM (this study, top panel), HFB14 [16] (middle panel), and FRLDM [13] (bottom panel). Drip lines are represented by dashed black lines. The solid black lines show the $r$-process path, given by the heaviest isotope of each nuclei with $S_n\ge 2\mathrm{MeV}$.

Figure 8

Energy window for the neutron-induced fission $B_f-S_n$ computed with the BCPM EDF. The solid black line represents the $r$-process path, given by the heaviest isotope of each nuclei with $S_n\ge 2\mathrm{MeV}$. The drip line predicted by the BCPM EDF is represent by dashed line.

Figure 9

Decimal logarithm of the spontaneous fission lifetimes computed with the ATDHFB-r (top panel), GCM-r (middle panel), and SEMP-r (bottom panel) schemes (see Table 1 for the renormalization coefficients). Dotted regions represent nuclei with $t_{\mathrm{sf}}\le 3\mathrm{s}$ which may be relevant for $r$-process calculations. All the lifetimes were obtained with $E_0=0.5$ MeV.

Figure 10

Logarithm of the ratio of the spontaneous fission lifetimes for different combinations of collective inertias: $R_{\mathrm{GCM}}^{\mathrm{ATDHFB}}$ (top panel); $R_{\mathrm{SEMP}}^{\mathrm{GCM}}$ (middle panel), and $R_{\mathrm{SEMP}}^{\mathrm{ATDHFB}}$ (bottom panel).

Figure 11

Sensitivity of the spontaneous fission lifetimes to different values of the collective ground-state energy ${log}_{10}\left[t_{\mathrm{sf}}(E_0=0.5\mathrm{MeV})/t_{\mathrm{sf}}(E_0=1.5\mathrm{MeV})\right]$ computed with different collective inertias: ATDHFB (top panels), GCM (middle panels), and semiempirical inertia formula (bottom panels).

Figure 12

Sensitivity of the spontaneous fission lifetimes to different values of the collective ground-state energy ${log}_{10}[t_{\mathrm{sf}}\left({\epsilon }_{\mathrm{vib}}^{\mathrm{ATDHFB}}\right)/t_{\mathrm{sf}}\left({\epsilon }_{\mathrm{vib}}^{\mathrm{GCM}}\right)]$ computed using the semiempirical collective inertias.

Figure 13

Dominating channel between $\alpha$ decay and spontaneous fission for different collective inertias. For comparison, the top panel shows the dominating channel for nuclei experimentally observed [46].

Figure 14

Dominating decay channels: spontaneous fission, $\alpha$-decay, neutron-capture, neutron-induced $\alpha$ emission, neutron-induced fission, and two-neutron emission computed using different collective inertias schemes.