Third minima in thorium and uranium isotopes in a self-consistent theory

Background: Well-developed third minima, corresponding to strongly elongated and reflection-asymmetric shapes associated with dimolecular configurations, have been predicted in some non-self-consistent models to impact fission pathways of thorium and uranium isotopes. These predictions have guided the interpretation of resonances seen experimentally. On the other hand, self-consistent calculations consistently predict very shallow potential-energy surfaces in the third minimum region.

Purpose: We investigate the interpretation of third-minimum configurations in terms of dimolecular (cluster) states. We study the isentropic potential-energy surfaces of selected even-even thorium and uranium isotopes at several excitation energies. In order to understand the driving effects behind the presence of third minima, we study the interplay between pairing and shell effects.

Methods: We use the finite-temperature superfluid nuclear density functional theory. We consider two Skyrme energy density functionals: a traditional functional SkM${}^{*}$ and a recent functional UNEDF1 optimized for fission studies.

Results: We predict very shallow or no third minima in the potential-energy surfaces of ${}^{232}$Th and ${}^{232}$U. In the lighter Th and U isotopes with $N=136$ and 138, the third minima are better developed. We show that the reflection-asymmetric configurations around the third minimum can be associated with dimolecular states involving the spherical doubly magic ${}^{132}$Sn and a lighter deformed Zr or Mo fragment. The potential-energy surfaces for ${}^{228,232}$Th and ${}^{232}$U at several excitation energies are presented. We also study isotopic chains to demonstrate the evolution of the depth of the third minimum with neutron number.

Conclusions: We show that the neutron shell effect that governs the existence of the dimolecular states around the third minimum is consistent with the spherical-to-deformed shape transition in the Zr and Mo isotopes around $N=58$. We demonstrate that the depth of the third minimum is sensitive to the excitation energy of the nucleus. In particular, the thermal reduction of pairing, and related enhancement of shell effects, at small excitation energies help to develop deeper third minima. At large excitation energies, shell effects are washed out and third minima disappear altogether.