Unexpected transitional paths in the prolate to oblate shape phase transitions for Bose-Fermi systems

The quantum shape phase transitions in odd-even nuclei are investigated within the intrinsic frame approach to the interacting boson-fermion model. In this work, the case of a single-j fermion coupled to a bosonic core that performs a transition from prolate to oblate shapes is considered. The focus is on the effect of the coupled fermion on the whole system along the transitional path from prolate to oblate shapes, passing through the gamma -unstable shape. One could expect that all the magnetic substates of the coupled single-fermion with j=9/2 would be driven by the shape of the bosonic core. However, the present work shows that the odd-fermion follows some unexpected and unique paths. The five-components of the j=9/2 orbital do show quite interesting and diverse behaviour. Two of them move slowly from the prolate to the oblate shape by venturing into the triaxial region and also show gamma -softness around their slow shape-changing region. The other three odd-states show sudden jumps from prolate side to oblate side and shape coexistence appears, although one of them is fairly close to gamma -instability. These unexpected situations make this shape phase transition worth of investigation and discussion.