Least action description of spontaneous fission in fermium and nobelium nuclei based on the Gogny energy density functional

The systematic of the spontaneous fission half-lives for the nuclei ${}^{242–262}\mathrm{Fm}$ and ${}^{250–260}\mathrm{No}$ is analyzed, within a least action scheme, with the parametrization D1M of the Gogny energy density functional. The properties of the dynamic (least action) fission paths are analyzed and compared to those of the static (minimal energy) ones. The constrained Hartree-Fock-Bogoliubov approximation is used to compute deformed mean-field configurations, zero-point quantum corrections, and collective inertias. It is shown that a cumbersome full variational search of the least action fission path, within the space of Hartree-Fock-Bogoliubov (HFB) states, might not be required if the relevant degrees of freedom are taken into account in the minimization of the Wentzel-Kramers-Brillouin action. The action is minimized in terms of pairing fluctuations that explore the pairing content of the HFB states along the fission paths of the considered nuclei. It is found that, for a given shape, the minimum of the action in fermium and nobelium nuclei corresponds to a value of the pairing fluctuations larger than the one associated with the minimal energy solution for the same shape. The reduction of the action, via larger pairing correlations, has a significant impact on the predicted spontaneous fission half-lives, improving their comparison with experiments by several orders of magnitude.

Figure 1

The HFB plus rotational correction energy corresponding to the static fission path is plotted, as a function of the quadrupole moment $Q_{20}$, for the nucleus ${}^{252}\mathrm{No}$.

Figure 2

The HFB plus rotational correction energies corresponding to the dynamic ATDHFB (dashed line) and GCM (dotted line) fission paths are plotted, as functions of the quadrupole moment $Q_{20}$, for the nucleus ${}^{252}\mathrm{No}$. The energies corresponding to the static path (full line) are also included in the plot. For details, see the main text.

Figure 3

The intrinsic HFB energies (full lines) and the HFB plus rotational correction energies (dashed lines) [panel (a)], GCM and ATDHFB collective masses [panels (b) and (c)], as well as the GCM and ATDHFB integrand of the action (2) [panels (d) and (e)] are plotted as functions of the particle number fluctuations $\langle \mathrm{\Delta }{\widehat{N}}^2\rangle$. Results are shown for the quadrupole moments $Q_{20}=14$, 26, 44, and 70 b. For more details, see the main text.

Figure 4

The HFB plus rotational correction energies corresponding to the dynamic ATDHFB (dashed line) and GCM (dotted line) fission paths are plotted as functions of the quadrupole moment $Q_{20}$ for the nuclei ${}^{242,246,250,254,258,262}\mathrm{Fm}$. The energies corresponding to the static paths (full line) are also included in the plot. Starting from the nucleus ${}^{246}\mathrm{Fm}$, the curves have been shifted by 50 MeV in order to accommodate them in a single plot. For details, see the main text.

Figure 5

The HFB plus rotational correction energies corresponding to the dynamic ATDHFB (dashed line) and GCM (dotted line) fission paths are plotted as functions of the quadrupole moment $Q_{20}$ for the nuclei ${}^{250,252,256,260}\mathrm{No}$. The energies corresponding to the static paths (full line) are also included in the plot. Starting from the nucleus ${}^{252}\mathrm{No}$, the curves have been shifted by 50 MeV in order to accommodate them in a single plot. For details, see the main text.

Figure 6

The inner barrier heights corresponding to the dynamic GCM and ATDHFB fission paths in ${}^{242–262}\mathrm{Fm}$ and ${}^{250–260}\mathrm{No}$ are plotted in panels (a) and (b), respectively, as functions of the neutron number. The inner barrier heights corresponding to the static paths in those nuclei are also included in the plot.

Figure 7

The spontaneous fission half-lives predicted within the dynamic GCM and ATDHFB schemes for the isotopes ${}^{242–262}\mathrm{Fm}$ are depicted as functions of the neutron number $N$. Calculations have been carried out with $E_0=0.5$ MeV. Results corresponding to the static GCM and ATDHFB schemes [34] are also included in the plot. The experimental $t_{SF}$ values are taken from Ref. [7].

Figure 8

The spontaneous fission half-lives predicted within the dynamic GCM and ATDHFB schemes for the isotopes ${}^{250–260}\mathrm{No}$ are depicted as functions of the neutron number $N$. Calculations have been carried out with $E_0=0.5$ MeV. Results corresponding to the static GCM and ATDHFB schemes [38] are also included in the plot. The experimental $t_{SF}$ values are taken from Ref. [7].