Microscopic pairing in fission dynamics

TL;DR

This work advances spontaneous fission modeling by treating pairing correlations as dynamical degrees of freedom in a two-dimensional (quadrupole, pairing) collective space within self-consistent HFB theory using the Gogny D1S interaction. By computing the least-action path with a Dijkstra algorithm and contrasting it with static least-energy paths, the authors show that including pairing drastically lowers the action and markedly improves half-life predictions for many fermium isotopes and for $^{262}$Rf, despite higher barrier energies along the path. The study highlights the critical role of pairing dynamics in fission kinetics, demonstrates comparable results between the two pairing coordinates ($\Delta$ and $\langle \Delta N^2 \rangle$), and discusses limitations and avenues for future refinement, such as higher multipole degrees of freedom and improved inertia calculations. Overall, incorporating dynamical pairing yields a more accurate, physically motivated description of fission dynamics with tangible implications for predicting half-lives in heavy and super-heavy nuclei. $$S = \frac{1}{\hbar}\int_{in}^{out} \sqrt{2 B_{\rm eff}(s)(V(s)-E_{gs})}\, ds,$$ $$B_{\rm eff}(s)=\sum_{ij} B_{ij} \frac{dq_i}{ds} \frac{dq_j}{ds},$$ and $$t_{sf}=2.87\times 10^{-21}\left(1+\exp(2S)\right).$$

Abstract

Nuclear fission can be modelled as a quantum tunneling process driven by the interplay between the nuclear binding energy and the collective inertia. Within the Wentzel-Kramers-Brillouin formalism, spontaneous fission half-lives can be obtained by minimizing the action integral in the multidimensional space of collective degrees of freedom. Hence, including the relevant collective variables is crucial for properly describing spontaneous fission probabilities. Pairing correlations play an essential role in this evaluation since the collective inertia decreases as the inverse of the square of the pairing gap, and, therefore, they should be considered as a relevant degree of freedom on the same footing as deformation parameters. In this work, we show that the spontaneous fission half-lives in fermium isotopes can be reproduced in a microscopic theory by considering the least-action fission path in a two-dimensional space with constraints on the quadrupole moment and pairing correlations. We consider two microscopic quantities as degrees of freedom associated with pairing: the pairing gap parameter $Δ$, and the particle number fluctuations $\langle ΔN^2 \rangle$. Least-action paths, computed using the Dijkstra algorithm, are compared with minimum-energy paths, highlighting the importance of pairing correlations as a dynamical degree of freedom.

Figures (10)

• Figure 1: Pairing gap ${\Delta}$ as a function of $Q_{20}$ and $\langle \Delta N^2 \rangle$ for $^{262}$Rf obtained with the Gogny D1S parametrization. The white isolines correspond to the HFB energy.
• Figure 2: PES in MeV (left column) and $B_{22}$ inertia tensor component (right column) in the ($Q_{20}$, $\Delta$) plane (top row) and in the ($Q_{20}$, $\langle \Delta N^2 \rangle$) plane (bottom row) for $^{262}$Rf, obtained with the Gogny D1S parametrization. The red line corresponds to the least-energy path, while the blue one corresponds to the least-action path. The white dots show all the considered exit points from the fission barrier.
• Figure 3: The potential energy (top), the $B_{22}$ component of the collective inertia (middle), and the action integral (bottom) as a function of $Q_{20}$ along the least-action fission paths in a ($Q_{20}$, $\langle \Delta N^2 \rangle$) plane (red line) and in a ($Q_{20}$, $\Delta$) (blue line), and the least-energy (static) path (green line) in $^{262}$Rf. The $B_{22}$ component of the collective inertia for the least-energy (static) path is reduced by a factor of 10.
• Figure 4: PES (in MeV) of $^{242}$Fm (left column) and $^{244}$Fm (right column) in the ($Q_{20}$, $\Delta$) plane (top row) ($Q_{20}$, $\langle \Delta N^2 \rangle$) (bottom row). The red line corresponds to the least-energy path, while the blue line to the least-action path. The white dots show the considered exit points from the fission barrier.
• Figure 5: Same as Fig. \ref{['dynFm1']}, but for $^{246}$Fm and $^{248}$Fm.