Microscopic understanding of nuclear collective dynamics is a long-term goal of low-energy nuclear theory. Large amplitude collective motion (LACM), as seen in fission and fusion, provides a particularly important challenge. Those phenomena can be understood in terms of many-body tunneling involving the mixing of mean fields with different symmetries. We have yet to obtain a microscopic understanding of LACM that is comparable to what we have for ground states, excited states, and response functions.
For heavy, complex nuclei, the theoretical tool of choice is the self-consistent nuclear density functional theory (DFT) [1,2]. The advantage of DFT is that, while treating the nucleus as a many-body system of fermions, it provides an avenue for identifying the essential collective degrees of freedom and provides an excellent starting point for time-dependent extensions. The time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory appears, in principle, to provide a proper theoretical framework to describe the LACM. However, the main drawback of TDHFB, when applied to fission, is its inability to describe the quantum-mechanical motion under the collective barrier.
On the other hand, the adiabatic approximation to TDHFB (ATDHFB) has been successfully applied to the LACM [3,4,5,6,7,8,11,12,9,10]. The main assumption behind ATDHFB, well fulfilled in the context of spontaneous fission, is that the collective motion of the system is slow compared to the single-particle motion of individual nucleons [1,13]. According to the path formulation of the fission problem [14], ATDHFB provides the best framework to tackle the problem of nuclear dynamics under the barrier. Another advantage of ATDHFB is that it provides a connection between the microscopic many-body theory and phenomenological models based on collective shape variables.
The main theoretical input for an estimate of fission half-lives is collective inertia (mass tensor) and collective potential. ATDHFB provides the best framework to calculate mass tensor [14]. However, in most applications, various approximations are adopted. In the commonly used cranking expression, for instance, the derivatives with respect to collective coordinates (i.e., collective momenta) are evaluated using the perturbation theory, and the Thouless-Valatin self-consistent terms yielding time-odd fields are neglected. The resulting collective masses are known to be too small [6,9]; hence it is imperative to go beyond the perturbative cranking treatment.
In the self-consistent investigations of Ref. [9], based on the Gogny energy density functional, collective masses were calculated by explicitly evaluating the collective-coordinate derivatives appearing in the ATDHFB mass expression. The resulting collective mass obtained in such an approach turned out to exhibit appreciable variations along the collective path, suppressed in the perturbative cranking treatment. Furthermore, they noted that ATDHFB cranking mass could be an order of magnitude greater than the perturbative cranking mass. As noted in Ref. [9], the enhanced masses obtained in the improved analysis can significantly impact the calculated fission lifetimes.
The main goal of this work is to investigate the ATDHFB cranking mass
using the nuclear DFT approach with Skyrme energy functionals. The
paper is organized as follows. Section 2 summarizes the
basic ATDHFB expressions for collective inertia obtained in
Ref. [6]. The approximate cranking, perturbative
cranking, and Gaussian Overlap Approximation (GOA)
formulations are given in Sec. 3. The
illustrative examples of calculations are contained in
Sec. 4, where the results are presented for
Fm. Finally, the main results are given in
Sec. 5.