Fission barriers for the even-even $N$=184 isotones with $\delta$-pairing interaction

Figure 4 compares the total binding energies ( $E^{\mbox{\scriptsize {tot}}}$) and the mass hexadecapole moments ($Q_{40}$) calculated along the fission paths for twelve even-even $N$=184 isotones. Here we use the MIX parameterization of the $\delta$-interaction. We found that all of the super-heavy elements (SHE) studied here have reflection-symmetric static fission paths and are spherical in the ground states, and the $Q_{40}$ moments calculated along the static fission paths follow the same pattern; i.e., their values continuously increase from 0 up to about 80b$^2$.

Figure: The total binding energies $E^{\mbox{\scriptsize{tot}}}$ ( $\mbox{\large$\bullet$}$, scales on the left-hand side) and the spectral neutron $\langle\Delta_{n}\rangle$ ($\circ$) and proton $\langle\Delta_{p}\rangle$ ($\circ$) pairing gaps (scales on the right-hand side) for the $N$=184 isotones shown in Fig. 4.

All collective potentials shown in Fig. 4 have two-humped shapes and similar widths. Only in the case of $^{288}$Rf$_{184}$, can one see an additional small third external barrier. The outer barrier is systematically reduced with $Z$. The differences between open and solid symbols illustrate the effect of triaxiality on the inner barriers. The strongest effect is predicted for the nucleus $^{310}126_{184}$, where the barrier is lowered by more than 3MeV due to triaxiality. However, for lighter isotones with $Z\leq 114$, the role of triaxiality is modest.

For completeness, it is interesting to look at the neutron $\langle\Delta_{n}\rangle$ and proton $\langle\Delta_{p}\rangle$ spectral pairing gaps calculated along the static fission paths shown in Fig. 4. They are shown in Fig. 5. As already mentioned, the ground-state neutron pairing gaps $\langle\Delta_{n}\rangle$ $(\circ)$ are equal to 0 in all the $N$=184 isotones, whereas the proton gaps $\langle\Delta_{p}\rangle$ $(\circ)$ change from 1.3MeV in $^{288}$Rf$_{184}$) to less than 0.5MeV in $^{288}126_{184}$. In the sub-barrier regions of the fission paths (shadowed regions in the plot), neutron and proton spectral pairing gaps fluctuate around 1MeV.

The size of a fission barrier is a measure of stability of the nucleus against spontaneous fission. The static fission barriers shown in Fig. 4 suggest a possible increase of the stability against spontaneous fission for $118\leq Z \leq 124$. This predicted increase of the stability supports the results obtained in our previous paper,[10] where the same nuclei were considered within the SHF+BCS(G) framework, but with the pairing strengths of Eq. (1) scaled to reproduce the pairing gaps of the finite-range droplet model (FRDM).[24]