Fission barriers of $^{256}$Fm, $^{258}$Fm, and $^{260}$Fm isotopes

The results of our calculations are presented in Figs. 1, 2, and 3, which display the total binding energy ( $E^{\mbox{\scriptsize {tot}}}$) and mass hexadecapole moment ($Q_{40}$) calculated along the static fission paths of $^{256}$Fm, $^{258}$Fm, and $^{260}$Fm, respectively. The fission paths were computed with a mass quadrupole moment ($Q_{20}$) used as a driving coordinate (constraint). Our study covers prolate shapes with $Q_{20}= 0\div 550$b.

A similar pattern of static fission trajectories was found for all investigated fermium isotopes. Beyond the region of the first fission barrier, at $Q_{20}\approx 150$b, a reflection-asymmetric path corresponding to elongated fragments (aEF) branches away from the symmetric valley. At $Q_{20}\approx 225$b, a reflection-symmetric path splits into two branches: one corresponding to a division into nearly spherical (compact) fragments (sCF) and the second branch corresponding to elongated fragments (sEF). It is worth mentioning that the identical pattern of static fission paths in $^{258}$Fm has been found recently in Ref. [15] within the same SHF+BCS framework, except that the axial-symmetry was enforced therein.

Figure 2: Similar to Fig. 1 except for $^{258}$Fm.

Furthermore, it appears that at b for $^{256}$Fm and at $Q_{20}\approx 470$b for $^{258,260}$Fm, the calculated asymmetric aEF solution becomes unstable and it falls back into the symmetric sCF one. For $^{256}$Fm, at this transition point the energy of the sCF configuration is equal to about $-$1950MeV, and is not shown in Fig. 1 in order not to extend the scale of the figure too much (see, however, Fig. 5 of Ref. [8]), and similarly holds for $^{258,260}$Fm. All these calculated points of instability correspond to scission configurations of fragments in the asymmetric channel. The analysis of nuclear shapes at these points [16] shows that one of the fragments is elongated while the other one is close to a sphere. This observation is in agreement with results obtained in Ref. [17] within the Hartree-Fock-Bogoliubov approach with the D1S Gogny interaction.

In the symmetric channels, the scission configuration for the sCF path (with two nearly spherical touching fragments) appears at $Q_{20}\approx 260$b (in all the investigated fermium isotopes), whereas for the sEF path, the scission configuration (with very elongated fragments) was reached at $Q_{20}\approx 810$b only in $^{258}$Fm.

Figure 3: Similar to Fig. 1 except for $^{260}$Fm.

Although the general pattern of static fission paths is fairly similar for the three fermium isotopes, the analysis of the binding energy along these paths reveals significant differences. In $^{256}$Fm, for instance, the outer barrier at $Q_{20}\approx 150$ b is very narrow along the asymmetric path aEF, whereas it is predicted to be broad for the two symmetric paths (sCF and sEF), and the saddle point is moved to larger deformations, $Q_{20}\approx 200$ b. While the quantitative analysis of the competition between different fission valleys obviously requires the proper treatment of fission dynamics, the calculated pattern for $^{256}$Fm shown in Fig. 1 is strongly indicative of the favored character of the asymmetric path aEF for this isotope.

In stark contrast to the situation in $^{256}$Fm, both symmetric paths are open for $^{258}$Fm, due to the disappearance of the outer fission barrier in sCF and sEF (Fig. 2). The sCF and sEF paths can be associated with the higher- and lower-TKE modes of the bimodal fission, respectively. Moreover, the less favorable aEF path may yield a small asymmetric contribution to the mass distribution of events with lower TKEs, as was postulated in Ref. [3].

In the case of $^{260}$Fm (Fig. 3), we find that there is no outer potential barrier along the sCF trajectory, and the sEF and aEF paths lie significantly higher in the outer region. Consequently, the spontaneous fission in this nucleus is expected to proceed along the symmetric sCF path corresponding to the higher-TKE mode. This transition from an asymmetric fission path in $^{256}$Fm to a compact-symmetric path in $^{260}$Fm is due to shell effects in the emerging fission fragments approaching the doubly magic nucleus Sn (see, e.g., discussion in Refs. [18,19,20]).