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Nuclei in the $Z\!\approx\!40,N\!\approx\!60$ region have one of the mostcomplicated structural evolution across the nuclear chart, with coexistingshapes arising from different mixed configurations. In such a region, it isdifficult to investigate odd-mass nuclei. In this paper a new algebraicframework is introduced, the interacting boson-fermion model with configurationmixing. Using this framework, with a boson core and a proton in the$1f_{5/2},2p_{3/2},2p_{1/2},1g_{9/2}$ orbits, a calculation is carried out tounderstand the structural evolution of the odd-mass niobium isotopes $(Z=41)$with neutron number 52-62. The calculated results are compared to energylevels, two neutron separation energies, $E2$ and $M1$ transition rates, and toquadrupole and magnetic moments. The detailed analysis discloses the effects ofan abrupt crossing of states between normal and intruder configurations (TypeII QPT), which is accompanied by a gradual evolution from spherical- todeformed-core shapes within the intruder configuration (Type I QPT), where bothtypes of QPTs occur around the critical point of neutron number 60. Theidentification of both types of QPTs in the same chain of isotopes provides anempirical manifestation of intertwined quantum phase transitions (IQPTs) inodd-mass nuclei and the relevance of IQPTs to the niobium chain.

Configuration mixing and intertwined quantum phase transitions in odd-mass niobium isotopes