Multidimensionally constrained relativistic Hartree-Bogoliubov study of spontaneous nuclear fission

Background: Recent microscopic studies, based on the theoretical framework of nuclear energy density functionals, have analyzed dynamic (least action) and static (minimum energy) fission paths, and it has been shown that in addition to the important role played by nonaxial and/or octupole collective degrees of freedom, fission paths crucially depend on the approximations adopted in calculating the collective inertia.

Purpose: To analyze effects of triaxial and octupole deformations, as well as approximations to the collective inertia, on the symmetric and asymmetric spontaneous fission dynamics, and compare with results of recent studies based on the self-consistent Hartree-Fock-Bogoliubov (HFB) method.

Methods: Deformation energy surfaces, collective potentials, and perturbative and nonperturbative cranking collective inertia tensors are calculated using the multidimensionally-constrained relativistic Hartree-Bogoliubov (MDC-RHB) model, with the energy density functionals PC-PK1 and DD-PC1. Pairing correlations are treated in the Bogoliubov approximation using a separable pairing force of finite range. The least-action principle is employed to determine dynamic spontaneous fission paths.

Results: The dynamics of spontaneous fission of ${}^{264}\mathrm{Fm}$ and ${}^{250}\mathrm{Fm}$ is explored. The fission paths, action integrals, and the corresponding half-lives predicted by the functionals PC-PK1 and DD-PC1 are compared and, in the case of ${}^{264}\mathrm{Fm}$, discussed in relation with recent results obtained using the HFB model based on the Skyrme functional ${\mathrm{SkM}}^{*}$ and a density dependent mixed pairing interaction.

Conclusions: The inclusion of nonaxial quadrupole and octupole shape degrees of freedom is essential for a quantitative analysis of fission dynamics. The action integrals and, consequently, the half-lives crucially depend on the approximation used to calculate the effective collective inertia along the fission path. The perturbative cranking approach underestimates the effects of structural changes at the level crossings and the resulting collective inertia varies relatively smoothly in the $({\beta }_{20},{\beta }_{22})$ and $({\beta }_{20},{\beta }_{30})$ planes. In contrast, the nonperturbative collective mass is characterized by the occurrence of sharp peaks on the surface of collective coordinates, that can be related to single-particle level crossings near the Fermi surface. This enhances the effective inertia, increases the values of the action integral, and results in longer fission half-lives.

Figure 1

RHB self-consistent triaxial quadrupole constrained energy surfaces of ${}^{264}\mathrm{Fm}$ in the $({\beta }_{20},{\beta }_{22})$ plane. In each panel energies are normalized with respect to the binding energy of the equilibrium minimum, and contours join points on the surface with the same energy (in MeV). The energy surfaces in (a) [(b)] are calculated with the density functionals PC-PK1 [60] (DD-PC1 [61]), and the pairing interaction Eq. (2).

Figure 2

The vibrational zero-point energies $E_{\mathrm{ZPE}}$ Eq. (14) of ${}^{264}\mathrm{Fm}$ in the $({\beta }_{20},{\beta }_{22})$ plane. Energy surfaces obtained with the PC-PK1 and DD-PC1 functionals are compared in (a) and (b), respectively. In each panel energies are normalized with respect to the the equilibrium minimum, and contours join points on the surface with the same energy (in MeV).

Figure 3

The ${\mathcal{M}}_{11}$ component of the inertia tensor (a), the binding energy (b), and the self-consistent deformation parameter ${\beta }_{40}$ (c) of ${}^{264}\mathrm{Fm}$ as functions of the deformation ${\beta }_{20}$. Axial symmetry ${\beta }_{22}=0$ is imposed, and the functional PC-PK1 is used in the RHB calculation.

Figure 4

Square-root determinants of the perturbative-cranking inertia tensor $|{\mathcal{M}}^{Cp}{|}^{1/2}$ (a), and nonperturbative-cranking inertia tensor $|{\mathcal{M}}^C{|}^{1/2}$ (b) (in $10\times {\hslash }^2{\mathrm{MeV}}^{-1}$), of ${}^{264}\mathrm{Fm}$ in the $({\beta }_{20},{\beta }_{22})$ plane. The functional PC-PK1 is used in the RHB calculation.

Figure 5

Same as described in the caption to Fig. 4 but for the functional DD-PC1.

Figure 6

Dynamic paths for spontaneous fission of ${}^{264}\mathrm{Fm}$ in the $({\beta }_{20},{\beta }_{22})$ plane, calculated with the functionals PC-PK1 (a) and DD-PC1 (b). The nonperturbative and perturbative cranking inertia are used, together with the DMP and RM techniques for the minimization of the collective action. The dotted and dash-dot-dot curves denote paths calculated with the perturbative-cranking inertia tensors using the DMP and RM, respectively, while the corresponding paths obtained with the nonperturbative-cranking inertia are plotted with the dash-dotted (solid) curves. The static path (dashed curve) is also shown for comparison.

Figure 7

RHB (DD-PC1 plus separable pairing) self-consistent constrained energy surfaces of ${}^{250}\mathrm{Fm}$ in the $({\beta }_{20},{\beta }_{30})$ (a) and $({\beta }_{20},{\beta }_{22})$ (b) planes. In each panel energies are normalized with respect to the binding energy of the equilibrium minimum, and contours join points on the surface with the same energy (in MeV).

Figure 8

The vibrational zero-point energies $E_{\mathrm{ZPE}}$ Eq. (14) of ${}^{250}\mathrm{Fm}$ in the $({\beta }_{20},{\beta }_{30})$ plane (a), and the $({\beta }_{20},{\beta }_{22})$ plane (b). Energies are normalized with respect to the the equilibrium minimum, and contours join points with the same energy (in MeV). The functional DD-PC1 is used in the RHB calculation.

Figure 9

Square-root determinants of the perturbative cranking inertia tensor $|{\mathcal{M}}^{Cp}{|}^{1/2}$ (a), and nonperturbative cranking inertia tensor $|{\mathcal{M}}^C{|}^{1/2}$ (b) (in $10\times {\hslash }^2{\mathrm{MeV}}^{-1}$), of ${}^{250}\mathrm{Fm}$ in the $({\beta }_{20},{\beta }_{30})$ plane. The calculation corresponds to axially symmetric but reflection asymmetric shapes.

Figure 10

Same as described in the caption to Fig. 9 but for the inertia tensor in the $({\beta }_{20},{\beta }_{22})$ plane. In this case the shape is allowed to be triaxial but reflection symmetry is assumed.

Figure 11

Dynamic paths for spontaneous fission of ${}^{250}\mathrm{Fm}$ in the $({\beta }_{20},{\beta }_{30})$ collective space with the perturbative and non-perturbative cranking inertia tensors. Both the dynamic-programming method and the Ritz method have been used to minimize the fission action integral. The static path (dashed curve) is also plotted for comparison. The results are obtained with the functionals PC-PK1 (a) and DD-PC1 (b), and axial symmetry is assumed.

Figure 12

Same as described in the caption to Fig. 11 but for the paths in the $({\beta }_{20},{\beta }_{22})$ plane. The shape is allowed to be triaxial but reflection symmetric.

Figure 13

Schematic diagram of the determination of the optimal trajectory using the dynamic-programming method.