Quantum and classical analyses of intertwined phase transitions in odd-mass Nb isotopes

TL;DR

This work demonstrates intertwined quantum phase transitions in odd-mass Nb isotopes using the Interacting Boson-Fermion Model with Configuration Mixing (IBFM-CM). A quantum analysis reveals a Type II QPT across normal (A) and intruder (B) configurations near $N=60$, superimposed on a Type I QPT within the intruder sector as deformation evolves, yielding a rich, configuration-moultured spectrum. A classical analysis with matrix coherent states produces Landau-like eigen-potentials $E_{-,K}(\beta)$ that reproduce the deformation trends and the A↔B mixing topology observed quantum mechanically. The results extend the previously known Zr-based intertwined QPTs to Nb isotopes, providing a unified framework for understanding shape coexistence and single-particle motion in deformed cores. Overall, the study highlights how weak to strong mixing and configuration crossing drive complex phase-change behavior in odd-mass nuclei.

Abstract

Quantum phase transitions (QPTs) in odd-mass Nb isotopes are investigated in the framework of the interacting boson-fermion model with configuration mixing. A quantum analysis reveals a Type I QPT (gradual shape-evolution within the intruder configuration) superimposed on a Type II QPT (abrupt crossing of normal and intruder states), thus demonstrating the occurrence of intertwined QPTs. A classical analysis highlights the implications for the single particle motion in the deformed field generated by the even-even Zr cores.

Figures (9)

• Figure 1: Schematic representation of the two coexisting shell-model configurations ($A$ and $B$) for $^{99}_{41}$Nb$_{58}$. The corresponding numbers of proton bosons ($N_{\pi}$) and neutron bosons ($N_{\nu}$), are listed for each configuration and $N=N_{\pi}+N_{\nu}$.
• Figure 2: Comparison between (a) experimental and (b) calculated lowest-energy positive-parity levels in Nb isotopes. Empty (filled) symbols indicate a state dominated by the normal A (intruder B) configuration. In particular, the $9/2^+_1$ state is in the A (B) configuration for neutron number 52--58 (60--64) and the $5/2^+_1$ state is in the A (B) configuration for 52--54 (56--64).
• Figure 3: Evolution of spectral properties along the Nb chain. Symbols (solid lines) denote experimental data (calculated results). (a) Percentage of the intruder (B) component [the $b^2$ probability in Eq. (\ref{['Prob-ab-cm']})], in the ground state ($J^+_{gs}$) and first-excited state ($7/2^+_1$). Values of $J^+_{gs}$ are indicated at the top. (b) $B(E2; 7/2^+_1\to J^{+}_{gs})$ in W.u. (c) Quadrupole moments of $J^+_{gs}$ in $eb$. (d) Magnetic moments of $J^+_{gs}$ in $\mu_N$. For the data, see Fig. 2 of gavleviac22.
• Figure 4: Evolution of spectral properties along the Zr chain. (a) Order parameters: $\braket{\hat{n}_d}_{0^+_1}$ (solid line) and $\braket{\hat{n}_d}_{A},\,\braket{\hat{n}_d}_{B}$ (dotted lines), normalized by the respective boson numbers. (b) B(E2) values for $2^+\!\to\!0^+$ transitions in Weisskopf units (W.u.). Dotted lines denote calculated $E2$ rates for transitions within a configuration. Solid line denotes calculated $2^{+}_1\to 0^{+}_1$ rates. For the data, see Fig. 17 of Gavrielov2022.
• Figure 5: Experimental (left) and calculated (right) energy levels in MeV, and $E2$ (solid arrows) and $M1$ (dashed arrows) transition rates in W.u., for $^{93}$Nb and $^{92}$Zr. Lines connect $L$-levels in $^{92}$Zr to sets of $J$-levels in $^{93}$Nb, indicating the weak coupling $(L\otimes \tfrac{9}{2})J$. For the data, see Fig. 3 of gavleviac22. Note that the observed $4^+_{\rm 1;A}$ state in $^{92}$Zr is outside the $N\!=\!1$ model space.