Multidimensionally-constrained relativistic mean-field study of spontaneous fission: Coupling between shape and pairing degrees of freedom

Background: Studies of fission dynamics, based on nuclear energy density functionals, have shown that the coupling between shape and pairing degrees of freedom has a pronounced effect on the nonperturbative collective inertia and, therefore, on dynamic (least-action) spontaneous fission paths and half-lives.

Purpose: The aim is to analyze the effects of particle-number fluctuation degrees of freedom on symmetric and asymmetric spontaneous fission (SF) dynamics, and to compare the findings with the results of recent studies based on the self-consistent Hartree-Fock-Bogoliubov (HFB) method.

Methods: Collective potentials and nonperturbative cranking collective inertia tensors are calculated using the multidimensionally-constrained relativistic-mean-field (MDC-RMF) model. Pairing correlations are treated in the BCS approximation using a separable pairing force of finite range. Pairing fluctuations are included as a collective variable using a constraint on particle-number dispersion. Fission paths are determined with the dynamic programming method by minimizing the action in multidimensional collective spaces.

Results: The dynamics of spontaneous fission of ${}^{264}\mathrm{Fm}$ and ${}^{250}\mathrm{Fm}$ are explored. Fission paths, action integrals, and corresponding half-lives computed in the three-dimensional collective space of shape and pairing coordinates, using the relativistic functional DD-PC1 and a separable pairing force of finite range, are compared with results obtained without pairing fluctuations. Results for ${}^{264}\mathrm{Fm}$ are also discussed in relation with those recently obtained using the HFB model.

Conclusions: The inclusion of pairing correlations in the space of collective coordinates favors axially symmetric shapes along the dynamic path of the fissioning system, amplifies pairing as the path traverses the fission barriers, significantly reduces the action integral, and shortens the corresponding SF half-life.

Figure 1

Effective collective potential $V_{\mathrm{eff}}$ of ${}^{264}\mathrm{Fm}$ in the $({\beta }_{20},{\beta }_{22})$ plane for ${\lambda }_2=0$ (a), and in the $({\beta }_{20},{\lambda }_2)$ plane for ${\beta }_{22}=0$ (b). In each panel energies are normalized with respect to the corresponding value at the equilibrium minimum, and contours join points on the surface with the same energy (in MeV). The energy surfaces are calculated with the relativistic density functionals DD-PC1 [30] and the pairing interaction Eq. (1).

Figure 2

Cubic root determinants of the nonperturbative-cranking inertia tensor $|{\mathcal{M}}^C{|}^{1/3}$ (in $10\times {\hslash }^2{\mathrm{MeV}}^{-1}$) of ${}^{264}\mathrm{Fm}$ in the $({\beta }_{20},{\beta }_{22})$ plane for ${\lambda }_2=0$ (a), and in the $({\beta }_{20},{\lambda }_2)$ plane for ${\beta }_{22}=0$ (b).

Figure 3

Projections of the 3D dynamic path (solid curves) for the spontaneous fission of ${}^{264}\mathrm{Fm}$ on the $({\beta }_{20},{\beta }_{22})$ plane for ${\lambda }_2=0$ (a), and the $({\beta }_{20},{\lambda }_2)$ plane for ${\beta }_{22}=0$ (b), calculated using the dynamic programming method. The dash-dot-dot curve denotes the 2D path computed without the inclusion of dynamic pairing correlations.

Figure 4

Same as Fig. 1, but for the nucleus ${}^{250}\mathrm{Fm}$.

Figure 5

Same as Fig. 2, but for the nucleus ${}^{250}\mathrm{Fm}$.

Figure 6

Projections of the 3D dynamic fission path (solid curves) of ${}^{250}\mathrm{Fm}$ on the $({\beta }_{20},{\beta }_{22})$ plane for ${\lambda }_2=0$ (a), and the $({\beta }_{20},{\lambda }_2)$ plane for ${\beta }_{22}=0$ (b), The dash-dot (red) curve denotes the 2D path computed without the inclusion of dynamic pairing correlations. The minimum-action paths connect the inner turning point and the fission isomer.

Figure 7

Effective collective potential $V_{\mathrm{eff}}$ of ${}^{250}\mathrm{Fm}$ in the $({\beta }_{20},{\beta }_{30})$ plane for ${\lambda }_2=0$ (a) and the $({\beta }_{20},{\lambda }_2)$ plane for ${\beta }_{30}=0$ (b). In each panel energies are normalized with respect to the corresponding value of the equilibrium minimum, and contours join points on the surface with the same energy (in MeV). The energy surfaces are calculated with the density functionals DD-PC1 [30], and the pairing interaction Eq. (1).

Figure 8

Projections of the 3D dynamic fission path (solid curves) of ${}^{250}\mathrm{Fm}$ on the $({\beta }_{20},{\beta }_{30})$ plane for ${\lambda }_2=0$ (a), and the $({\beta }_{20},{\lambda }_2)$ plane for ${\beta }_{30}=0$ (b). The dash-dot (red) curve denotes the 2D path computed without the inclusion of dynamic pairing correlations.