Examination of static fission properties of ${}^{236}$U and ${}^{233}$Th using Cassinian oval parametrization within the macroscopic-microscopic approach

We have conducted a thorough analysis of the potential energy surfaces (PES) in ${}^{236}$U and ${}^{233}$Th using the Cassini-ovals parameterization within the macro-micro approach. We employed the state-of-the-art immersion water flow (IWF) method to study the saddles on four-dimensional energy grids encompassing reflection-asymmetric shapes. For ${}^{233}$Th, we computed the adiabatic potential energy surfaces by minimizing configurations with one blocked neutron within ten levels below and above the Fermi level. Our results show satisfactory agreement with empirical and experimental estimates for both nuclei, specifically regarding the first and second fission barriers. This suggests that our method holds promise in efficiently describing non-compact shapes while reducing the dimensionality of the space without sacrificing accuracy. Interestingly, employing Cassinian oval parameterization fails to reveal a pronounced, hyper-deformed third minimum in the potential energy landscape. Instead, only a shallow third minimum is observed for ${}^{233}$Th, while in ${}^{236}$U, this minimum ultimately vanishes. This finding holds significant importance when considering the modeling of fission cross-sections.