Nudged elastic band approach to nuclear fission pathways

Abstract

Background: The nuclear fission process is a dramatic example of the large-amplitude collective motion in which the nucleus undergoes a series of shape changes before splitting into distinct fragments. This motion can be represented by a pathway in the many-dimensional space of collective coordinates. Within a stationary framework rooted in a static collective Schr\"odinger equation, the collective action along the fission pathway determines the spontaneous fission half-lives as well as mass and charge distributions of fission fragments.Purpose: We study the performance and precision of various methods to determine the minimum-action and minimum-energy fission trajectories in the collective space.Methods: We apply the nudged elastic band method (NEB), grid-based methods, and the Euler-Lagrange approach to the collective action minimization in two- and three-dimensional collective spaces.Results: The performance of various approaches to the fission pathway problem is assessed by studying the collective motion along both analytic energy surfaces and realistic potential-energy surfaces obtained with the Skyrme-Hartree-Fock-Bogoliubov theory. The uniqueness and stability of the solutions is studied. The NEB method is capable of efficient determination of the exit points on the outer turning surface that characterize the most probable fission pathway and constitute the key input for fission studies. This method can also be used to accurately compute the critical points (i.e., local minima and saddle points) on the potential-energy surface of the fissioning nucleus that determine the static fission path. The dynamic programming method also performs quite well and it can be used in many-dimensional cases to provide initial conditions for the NEB calculations.Conclusions: The NEB method is the tool of choice for finding the least-action and minimum-energy fission trajectories. It will be particularly useful in large-scale static fission calculations of superheavy nuclei and neutron-rich fissioning nuclei contributing to the astrophysical $r$-process recycling.