Nuclear shapes of Nb isotopes

Abstract

The study of the structure of odd-mass nuclei in regions characterized by the interplay of multiple particle-hole configurations represents a major challenge in nuclear structure physics. The odd-mass niobium isotopes ($Z = 41$), located near the $N = 60$ region, are of particular interest due to shape coexistence and quantum phase transitions. This work investigates the structure of the $^{93-103}$Nb isotopes using the intrinsic-frame formalism of the interacting boson-fermion model with configuration mixing (IBFM-CM), aiming to determine nuclear shapes and explore shape coexistence, configuration crossing, and quantum phase transitions. We employ the intrinsic formalism of the IBFM-CM, including both 0p-0h (regular) and 2p-2h (intruder) configurations interacting with the unpaired nucleon, providing a self-consistent framework to study energy surfaces, shape coexistence, and intruder bands for both positive- and negative-parity states. A realistic Hamiltonian for niobium, determined in previous studies, is adopted. The formalism is applied to the $^{93-103}$Nb isotopes for both positive- and negative-parity bands. A detailed analysis of the mean-field energy surfaces has been performed, including axial energy curves, triaxial energy surfaces in the $β-γ$ plane, and the corresponding equilibrium deformation parameters. The results reveal clear evidence of configuration coexistence and crossing along the isotopic chain. The existence of crossing configurations is demonstrated around $N = 60$, corresponding to a quantum phase transition previously identified in the Sr and Zr isotopic chains. Furthermore, the presence of an unpaired nucleon in Nb influences the abruptness of the quantum phase transition, underscoring the sensitivity of the structural evolution to single-particle degrees of freedom.

Figures (9)

• Figure 1: Equilibrium deformation values, excitation energies, and regular content of the wavefunction for the positive parity states. (a) equilibrium values of $\beta$ for the unmixed case ($V_{mix}=0$), full lines for the regular states and dashed for the intruder ones, thick dashed yellow line for the IBM-CM result; (b) equilibrium value of $\beta$ for $V_{mix}\neq 0$, full lines for the first five states and dashed for the last five ones; (c) equilibrium value of $\gamma$ for $V_{mix}=0$; (d) equilibrium value of $\gamma$ for $V_{mix}\neq 0$; (e) excitation energies for $V_{mix}=0$; (f) excitation energies for $V_{mix}\neq 0$; (g) regular content of the wave function for $V_{mix}\neq 0$.
• Figure 2: Axial energy curves as a function of $K^\pi$ for positive-parity states with $V_{mix}\neq 0$. For reference, the unmixed regular and intruder energy surfaces are also included.
• Figure 3: Ground-state energy surfaces in the $\beta–\gamma$ plane for positive-parity states along the isotopic chain. White dots indicate the minimum of the energy surfaces. The energy scale varies across panels, with blue representing the minimum and red the maximum, divided into $40$ contour levels.
• Figure 4: Same as Fig. \ref{['fig-energies-betas-gammas-pos']} but for negative-parity states.
• Figure 5: Same as Fig. \ref{['fig-2D-energies-pos']} but for negative-parity states.