A critical assessment of the current implementations of the Generator Coordinate Method

TL;DR

This work critiques current Generator Coordinate Method (GCM) implementations for non-equilibrium, dissipative nuclear dynamics and introduces an enhanced GCM (eGCM) that integrates Time-Dependent Density Functional Theory (TDDFT) trajectories as generator coordinates. It argues that Reinhard et al.'s TDGCM prescription imprudently enforces simultaneous arrival of mean-field trajectories, suppressing natural mixing driven by Hamiltonian overlaps and internal excitations. The eGCM formalism orthogonalizes a continuum of TDDFT-generated configurations, yielding a configuration-interaction–like framework capable of describing interference and entanglement in fission and heavy-ion reactions. Feasibility analyses and illustrative TDDFT+eGCM results (e.g., $^{48}$Ca+$^{208}$Pb) show significantly richer final-state structure and entropy growth, indicating eGCM’s potential to deliver microscopic, dissipation-inclusive cross sections for LACM phenomena. Overall, eGCM emerges as a physically well-motivated, computationally viable path beyond conventional GCM for static and time-dependent microscopic nuclear dynamics.

Abstract

The generator coordinate method (GCM) was introduced in nuclear physics by Wheeler and independently by Peierls and their collaborators in 1950's and it is still one of the mostly used approximations for treating nuclear large amplitude collective motion (LACM). GCM was inspired by similar methods introduced in molecular and condensed matter physics in the late 1920's, after the Schrödinger equation became the tool of choice to describe quantum phenomena. The interest in the 1983 extension of GCM suggested by Reinhard, Cusson and Goeke, which includes the internal excitations (absent in the initial GCM formulation), was revived in recent years. Unfortunately this newer version of time-dependent GCM (TDGCM) framework has flaws, which prevents it from describing correctly many anticipated features, in particular interference and entanglement, which can play an important role in fission and many-nucleon transfer reactions. I present here an alternative formulation, the enhanced GCM (eGCM), which is free of difficulties encountered in previous GCM implementations and which is relevant for fission and many-nucleon transfer in heavy-ion reactions, and which can be used in either static or time-dependent eGCM formulations. In the eGCM framework the corresponding many-body waves functions have a much more complex structure and this framework is equivalent to a configuration interaction (CI) approach in the continuum for nuclear reactions. eGCM is aimed to be used in the microscopic description of heavy-ion reactions and fission in particular.

Figures (5)

• Figure 1: This open access figure from Refs. Bender:2020Schunck:2020, courtesy of N. Schunck, vividly illustrates the complexity of the three main stages of induced fission. In the initial state and in the fission isomer stage a fissioning nucleus spends approximately $10^{-15}$ seconds Gonnenwein:2014, while in the third stage, the non-adiabatic stage, before scission the nucleus evolves during approximately $10^{-20}$ seconds Gonnenwein:2014Bulgac:2016Bulgac:2019cBulgac:2020, and the neck rupture and the separation of the fissioning nucleus into FFs happens in about $10^{-21}$ seconds Abdurrahman:2024. Subsequent processes, such as prompt neutron and light charge particle emission Madland:2006, $\gamma$, and $\beta$ decay happen at much longer time scales Gonnenwein:2014.
• Figure 2: These are five of the infinitely many paths available in case of neutron induced fission from point A (the time of impinging neutron impact) to point B (detector) at a later time. This figure with added interaction times at random positions (red dots) is reproduced from Wikipedia entry Path integral formulation, is a pictorial representation of a many-body quantum propagator $K(\xi_1\ldots\xi_A,\tau| \xi'_1\ldots\xi'_A,\tau')$ from point ${\rm A}$ at time $\tau'$ to point ${\rm B}$ at time $\tau$.
• Figure 3: The proton (solid lines) and neutron (dashed lines) absolute error in particle numbers if only a reduced number of canonical quasi-particle wave functions $N_{cwfs}$ are used at different times, along an induced fission trajectory of $^{238}$U, to evaluate the total nucleqers.
• Figure 4: In the impact parameter plane $({\rm b}_x,{\rm b}_y,0)$ only the impact parameter points with centers inside the orange ring, with $|{\bf b}|\approx 5-6$ fm, were considered. For larger impact parameters the two nuclei do not form a neck and for a smaller parameters the formed "compound nucleus" fails to separate for very large simulations times.
• Figure 5: The Inverse Participation Ratio (IPR) for all corresponding solutions to the GMR$_R$ and the present eGCM implementations for the corresponding eigenfunctions of the norm overlaps Eqs. (\ref{['eq:GCM00']}, \ref{['eq:GCM1']}). The eGCM IPR values are consistently well below the corresponding GCM$_R$ IPR values, thus with a significantly larger delocalization in eGCM, corresponding to eGCM many-body wave functions with a significantly more complex structure than the GCM$_R$ many-body wave functions.