Fission barriers for deformed SHE with $100\leq Z \leq 110$

The nuclei from this region are well deformed in their ground states. It is worth noting that their stability is determined by the deformed subshell closures at neutron numbers $N$=152 for the isotopes of fermium ($Z$=100) and nobelium ($Z$=102), and $N$=162 for the isotopes of hassium ($Z$=108).

Figure 1: The total binding energies $E^{\mbox{\scriptsize{tot}}}$ ( $\mbox{\large$\bullet$}$, scales on the left-hand sides) and the mass octupole moments $Q_{30}$, scales on the right-hand sides) calculated along the lowest static fission paths for the even-even fermium isotopes. The differences between the open and solid symbols in the region of the first barrier represent the energy reduction due to the presence of triaxial distortions.

Figure 2: Similar as in Fig. 1 except for the even-even nobelium isotopes.

Figures 1 and 2 show the total binding energies ( $E^{\mbox{\scriptsize {tot}}}$) and mass octupole moments ($Q_{30}$) calculated for the even-even fermium and nobelium isotopes, respectively, as functions of the mass quadrupole moment $Q_{20}$. The predicted static barriers have a two-humped shape with an inner barrier greater than an outer one. The sizes of fission barriers are well correlated with the `magic' deformed neutron number $N$=152. Indeed, one can see that the barriers reach their local maxima for $^{252}$Fm and $^{254}$No. For the heavier isotopes with $N$$>$152, the second barriers collapse and practically disappear for $N$=162 (in Fm isotopes) and for $N$=160 (in No isotopes).

The disappearance of the second barrier is related to the transition from the reflection-symmetric ($Q_{30}= 0$) fission path to the reflection-asymmetric fission path ($Q_{30}\neq 0$). For the heavier Fm and No isotopes, a switch from the reflection-symmetric to the reflection-asymmetric nuclear shapes occurs at greater values of $Q_{20}$. In the extreme cases of $^{264}$Fm (when a fission into the two doubly magic $^{132}$Sn nuclei is expected) and $^{264}$No, the static fission paths appear to be reflection-symmetric. The reduction of the second barrier plays a crucial role in the standard interpretation of the experimentally known rapid decrease of the spontaneous fission half-lives in the heavy Fm and No isotopes.

The influence of triaxiality ($Q_{22} \neq 0$) on the first fission barrier is given by the difference between the open and solid symbols in Figs. 1 and 2. The effect of triaxiality increases with the neutron number, reaching a peak value of about 3MeV.

Figure 3: The total binding energies $E^{\mbox{\scriptsize{tot}}}$ ( $\mbox{\large$\bullet$}$, scales on the left-hand sides) and the mass hexadecapole moment $Q_{40}$, scales on the right-hand sides) as functions of $Q_{20}$ for the even-even isotopes of rutherfordium (A) and seaborgium (B). The differences between the open and solid symbols represent the effects of triaxiality on the first barrier.

The total binding energies ( $E^{\mbox{\scriptsize {tot}}}$) and mass hexadecapole moments ($Q_{40}$) calculated along the static fission paths in the even-even rutherfordium ($Z$=104), seaborgium ($Z$=106), hassium ($Z$=108), and darmstadtium ($Z$=110) isotopes are shown in Figs. 3 and 4. We have found that almost all of these nuclei have purely reflection-symmetric static fission paths. Only in the case of $^{254}$Rf, $^{256}$Rf, and $^{258}$Sg have the reflection-asymmetric paths been observed (results not shown). One can see that the hexadecapole moments gradually increase along the paths to fission. It is worth noting that a similar behavior of $Q_{40}$ has been also observed for Fm and No isotopes. The reduction of fission barriers in Figs. 3 and 4 due to triaxiality is similar to that calculated for $Z$=100 and 102.

Figure 4: Similar as in Fig. 3 except for the even-even hassium (A) and darmstadtium (B) isotopes.

With the increase of proton number from $Z$=100 to 110, one can notice two effects, namely, (i) the disappearance of the outer barriers, and (ii) the decrease of the inner barriers. The first of these effects can be seen in Figs. 3B and 4 which display one-humped, narrow barriers predicted for Sg, Hs, and Ds. The second effect is particularly evident for the heaviest $^{276,278}$Ds isotopes having barriers reduced by a factor of two as compared to $^{252}$Fm.