Comparison of pairing models

Figures 1, 2, and 3 display total binding energies ( $E^{\mbox{\scriptsize {tot}}}$), mass hexadecapole moments ($Q_{40}$), and neutron/proton pairing gaps ($\Delta_{n/p}$) calculated along the static fission paths of $^{288}$Rf$_{184}$, $^{298}114_{184}$, and $^{310}126_{184}$. The fission paths were computed with a quadratic constraint[20] on the mass quadrupole moment ($Q_{20}$). Our study covers the oblate/prolate deformations of $Q_{20}= -30\div 300$b with a step of 10b.

Figure 2: Same as in Fig. 1 but now for $^{298}114_{184}$.

As in previous studies based on the SLy4 functional,[21,22] we found that all nuclei considered in this work have spherical shapes in their ground states. This is because $N$=184 appears to be the magic neutron number in most theoretical models based on the Skyrme approach.[23] Due to the magic character of $N$=184 isotones, ground-state neutron pairing gaps calculated in BCS vanish. The ground-state proton pairing gaps, on the other hand, show considerable variations with $Z$. They are large in the open-shell rutherfordium ($Z$=104), but in $Z$=114 the values of $\Delta_p$ are considerably reduced. As expected, proton pairing is much weakened for $Z$=126 which is predicted to be semi-magic.[21,22,23]

Figure 3: Same as in Fig. 1 but now for $^{310}126_{184}$.

The calculated static fission paths have reflection-symmetric shapes, i.e., $Q_{\lambda0}=0$ for all odd multipolarities $\lambda$. Furthermore, one can see that irrespective of the pairing interaction used, the hexadecapole moments are almost identical, and they gradually increase (from 0 up to about 80b$^2$) along the calculated static fission paths. In contrast to $Q_{40}$, the collective potentials (i.e., total energies $E^{\mbox{\scriptsize {tot}}}$ as functions of $Q_{20}$) differ depending on the pairing model employed. This is particularly evident for $^{288}$Rf$_{184}$ (Fig. 1A) where the fission barrier calculated within the SHF+BCS(G) model is significantly higher as compared to those obtained in the SHF+BCS($\delta$) variants. In the case of $^{298}114_{184}$ (Fig. 2A), the fission barriers calculated with the SHF+BCS(G) and SHF+BCS($\delta$) interactions are considerably closer to one another than in the case of $^{288}$Rf$_{184}$. Furthermore, for $^{310}126_{184}$ (Fig. 3A), all fission barriers calculated with both pairing models are almost identical. This can be attributed to the behavior of proton pairing along the fission paths. Indeed, in $^{288}$Rf$_{184}$ there is a large systematic difference between $\Delta_p$ values in SHF+BCS(G) and SHF+BCS($\delta$) variants, with the seniority-pairing model producing considerably larger pairing gaps. This difference decreases when going towards $^{310}126_{184}$ in which $\Delta_{n/p}$ obtained within the SHF+BCS(G) model are much closer to those obtained within the SHF+BCS($\delta$) model. This result indicates that the isospin dependence of seniority pairing strengths given by Eq. (1) is too strong and thus unrealistic. Another interesting observation is that neutron/proton pairing gaps (hence, corresponding potential energies) calculated within the SHF+BCS($\delta$) framework are very similar, regardless of the parameterization variant used (MIX, DDDI or DI), see Eq. (4).

Figure: The total binding energies $E^{\mbox{\scriptsize{tot}}}$ ( $\mbox{\large$\bullet$}$ - scales on the left-hand side) and mass hexadecapole moments $Q_{40}$ ($\circ$ - scales on the right-hand side) along the fission paths, calculated with the SLy4 interaction and the MIX parameterization of the $\delta$-interaction for the even-even $N$=184 isotones of 288$\le$$A$$\le$310. The differences between open and solid symbols illustrate the effects of triaxiality on the inner barriers.