At the borderline of shape coexistence: Mo and Ru

TL;DR

This study uses the interacting boson model with configuration mixing to reproduce and analyze the spectroscopic properties of Mo and Ru isotopes, revealing shape coexistence effects and quantum phase transitions around neutron number 60.

Contribution

It provides a detailed theoretical analysis of shape coexistence and quantum phase transitions in Mo and Ru isotopes using configuration mixing in the interacting boson model.

Findings

Shape coexistence significantly influences Mo isotopes at neutron number 60.

In Ru isotopes, intruder states are higher in energy but also lead to a quantum phase transition at neutron number 60.

The model achieves good agreement with experimental data for energies and transition rates.

Abstract

Background Even-even isotopes of Mo ($Z=42$) and Ru ($Z=44$) are nuclei close to the subshell closure at $Z=40$, where shape coexistence plays a significant role. As a result, their spectroscopic properties are expected to resemble those of Sr ($Z=38$) and Zr ($Z=40$). Exploring the evolution of these properties as they move away from the subshell closure is of great interest. Purpose The purpose of this study is to reproduce the spectroscopic properties of even-even $^{96-110}_{\phantom{961-}42}$Mo and $^{98-114}_{\phantom{961-}44}$Ru isotopes and to determine the influence of shape coexistence. Method We have employed the interacting boson model with configuration mixing as the framework to calculate all the observables for Mo and Ru isotopes. We have considered two types of configurations: 0-particle-0-hole and 2-particle-2-hole excitations. The model parameters have been determined using a least-squares fitting to match the excitation energies and the $B(E2)$ transition rates. Results We have obtained the excitation energies, $B(E2)$ values, two-neutron separation energies, nuclear radii, and isotope shifts for the entire chain of isotopes. Our theoretical results have shown good agreement with experimental data. Furthermore, we have conducted a detailed analysis of the wave functions and obtained the mean-field energy surfaces and the nuclear deformation parameter, $\beta$, for all considered isotopes. Conclusions Our findings reveal that shape coexistence plays a significant role in Mo isotopes, with the crossing of intruder and regular configurations occurring at neutron number $60$ ($A=102$), which induces a quantum phase transition. In contrast, in Ru isotopes, the intruder states have minimal influence, remaining at higher energies. However, at neutron number $60$, also a quantum phase transition occurs in Ru isotopes.