Quantum phase transition in the spherical mean-field plus quadrupole-quadrupole and pairing model in a single-j shell

The quantum-phase-transitional behavior of the spherical shell-model mean field plus the geometric quadrupole-quadrupole and standard pairing model within a single- j shell is analyzed in detail. Various quantities, such as low-lying energy levels, some typical energy ratios, the overlaps of the excited states with those of the corresponding limiting cases, B(E2) values and electric quadrupole moments of some low-lying states and their ratios, as functions of the control parameter of the model in a j = 15/2 shell are calculated as an example, in which only a crossover occurs due to the Pauli exclusion. The results show that there are noticeable changes within the crossover region of the rotation-like to the pair-excitation (superconducting) phase transition, especially in the half-filling case. As an application to realistic nuclear systems, a chain of isotones Rn-212 -Fr-213 -Ra-214 -Ac-215 is chosen to be described by the model with valence protons in the 1h(9/2) shell. As far as the low-lying energy levels, the experimentally observed B(E2) values, and the electric quadrupole moment within the yrast band are concerned, these nuclei seem fitted reasonably well. The results indicate that these nuclei are all within the rotation-like to the pair-excitation phase transition near the crossover point.