Static Fission Properties of Even-Even Actinides within the Warsaw Macroscopic-Microscopic Model Using Fourier-over-Spheroid Parameterization

TL;DR

This work addresses the static fission properties of even-even actinides using the Warsaw macroscopic-microscopic framework with a five-dimensional Fourier-over-Spheroid (FoS) shape parameterization. A dense 5D deformation grid coupled with an interpolation-free Immersion Water Flow method yields robust fission barrier heights and reveals the PES topology, including a possible shallow hyperdeformed minimum in 230Th but not in heavier actinides. The calculated ground-state masses and barrier heights show good agreement with empirical data sets (Smirenkin1993, RIPL-3, RIPL-4), with inner barriers typically offset by about +0.46 MeV and outer barriers by about +0.39 MeV. The study highlights the critical role of a complete shape representation in predicting barrier systematics and sets the stage for future dynamical analyses, including triaxial effects and alternative macroscopic energy forms.

Abstract

A systematic study of fission barrier heights and static properties of even-even actinide nuclei from Th to Cf has been performed within the Warsaw macroscopic-microscopic model using the five-dimensional Fourier-over-Spheroid (FoS) shape parameterization. The use of a large deformation grid, containing about $1.3\times10^{8}$ points for each nucleus, allows for a refined and numerically complete exploration of the potential energy landscape without dividing the configuration space into subregions or applying interpolation. Barrier heights, extracted via the Immersion Water Flow method, show good agreement with empirical evaluations (including the new IAEA RIPL-4 dataset) with mean deviations below 1 MeV. Special attention is given to the long-debated third, hyperdeformed minimum. For Th isotopes, a shallow but distinct third well appears, whereas it's absent in heavier actinides (U, Pu).

Figures (6)

• Figure 1: Comparison of FoS shapes (black lines) with their equivalent representation in $\beta$ parameterization (red dashed lines) for selected equilibrium points (provided below each panel) for ^236U. The values of parameters $c$, $a_{3}$, $a_{4}$, $a_{5}$, and $a_{6}$ are given at each point, along with the values of $E_\text{total}$, $E_\text{macro}$, and $E_\text{micro}$, representing the total, macroscopic, and microscopic energy, respectively.
• Figure 2: Potential energy surfaces in the $\{c, a_4\}$ plane for ^230Th, ^236U, ^240Pu, and ^246Cm nuclei, obtained by minimization over the remaining parameters: $a_3$, $a_5$, and $a_6$. Total energy $E_{\text{total}}$ (in MeV) is given relative to the macroscopic energy of the spherical configuration.
• Figure 3: Differences between calculated and experimental ground-state mass excesses for even–even actinides, $\Delta M = M^\text{th}_\text{gs} - M^\text{exp}_\text{gs}$. Theoretical mass excesses are obtained within the Warsaw macroscopic–microscopic model employing the five-dimensional Fourier-over-Spheroid (FoS) parameterization. Empirical reference data are taken from the AME2020 mass evaluation Audi2020.
• Figure 4: The difference $B_{f}^{\rm{(I)}} - B_{f}^{\rm{(I)exp1}}$, allowing direct comparison with the INDC Smirenkin1993 evaluation ($B_{f}^{\rm{(I)exp1}}$) as a reference. Here, $B_{f}^{\rm{(I)}}$ is either the theoretical inner barrier height ($B_{f}^{\rm{(I)th}}$) from this work or the empirical value from RIPL-3 ($B_{f}^{\rm{(I)exp2}}$) Capote2009 or RIPL-4 ($B_{f}^{\rm{(I)exp3}}$) RIPL4. The INDC dataset is chosen as the reference point, as it contains all analyzed systems. See Tab. \ref{['tab:results']} for details. The theoretical values were obtained within the Warsaw macroscopic–microscopic model using the five-dimensional FoS parameterization (WMMM FoS 5D) without non-axial effects.
• Figure 5: Same as in Fig. \ref{['fig:inner_barrier']}, but for the outer fission barrier $B_{f}^{\rm{(II)}}$ heights for even–even actinide nuclei.