Introduction

The microscopic description of the fission process is one of the most challenging issues in nuclear structure. It represents an extreme example of the tunneling of the many-body system. There exist many descriptions of spontaneous fission based on the adiabatic assumption. While the microscopic-macroscopic models still offer the best fit to the data, during recent years, a number of self-consistent approaches rooted in the Adiabatic Time-Dependent Hartree-Fock method (such as the collective Schrödinger equation with microscopic mass tensor) have been introduced.

The spontaneous-fission half-life can be reliably estimated within the semi-classical WKB approximation. Here, the key quantity is the action integral

$$S=\int_{(s)} \left\{2\left[V(q)-E\right]\sum_{ij}B_{ij}(q) q_i^\prime q_j^\prime\right\}^{1/2}\,ds$$ (1)

along the trajectory $s$ in the multidimensional space of collective parameters $\{q\}$. In Eq. (1), $V({q})$ represents the collective potential given by the HF+BCS or HFB energy, $E$ is the ground-state energy, $B(q)$ is a second rank tensor of a collective mass, and the prime denotes a derivative with respect to $s$.

The total energy of the system as a function of collective coordinates represents a zero-order approximation to the potential energy surface for shape-vibrations, ${V}(q)$. However, before one can use ${V}(q)$ in calculations with the collective Hamiltonian, dynamic corrections have to be added. The reason is that the underlying states have a finite uncertainty in the collective deformation. As a consequence, the potential ${V}(q)$ contains contributions from quantum fluctuations, and these contributions need to be subtracted first before adding the energies associated with the true physical zero-point fluctuations, $E_0$, see Ref.[1]. The theoretical evaluation of these correction terms can be done in the framework of the cranking approximation (CRA), the generator coordinate method (GCM)[2], or the Gaussian overlap approximation to GCM (GOA)[3,4,5,6,7].

Having minimized the action $S$ in the collective space $\{q\}$, the spontaneous-fission half-life $T_{\rm sf}$ can be calculated from[8]

$$\log{T_{\rm sf}} = -20.54 +\log{[1+\exp{(2S_{\rm min})}]} - \log{(2E_0)}.$$ (2)

It immediately follows from Eq. (1) that both the potential $V(q)$ and the inertia tensor $B(q)$ determine the fission dynamics in the multidimensional collective space; hence the value of $T_{\rm sf}$.

In this work we discuss fission barriers, mass parameters, and zero-point energy (ZPE) corrections calculated in the Hartree-Fock+BCS model with the commonly used Skyrme SkM$^*$ and SLy4 energy density functionals. The Skyrme results are compared to those within the Gogny D1S Hartree-Fock-Bogolyubov model[9,7]. Another comparison is done between collective inertia obtained in the GOA and CRA approximations.