The quon algebra, which interpolates between the Bose and Fermi algebras and depends on a free parameter q , is used to generate deformed Marumori boson expansions for the pair and quadrupole operators. Then we obtain a pairing plus quadrupole-quadrupole Hamiltonian, for a single j -shell, in terms of these deformed bosonic operators. The Hamiltonian is diagonalized and its eigenvalues are compared with the ones obtained from the counterpart fermionic Hamiltonian. The deformation parameter helps in achieving the correct energy levels, which cannot be encountered in practice with the usual non-deformed boson expansion.
We consider the pairing-plus-quadrupole model Hamiltonian within the framework of a many-fermion shell model. The Hamiltonian exhibits a transition from a superconducting to a rotational phase with variation of a parameter. The model has analytical solutions in its two limits due to the presence of dynamical symmetries. However, the symmetries are basically incompatible with one another which means that no simple solution exists in intermediate situations. Exact (numerical) solutions are nevertheless possible on a computer and enable one to study the behavior of competing but incompatible symmetries and the phase transitions that result in a semirealistic situation. This mixed-symmetry dynamics generates a rich structure which gives new insight for the study of phase transitions.
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 $^{212}\mathrm{Rn}\text{\ensuremath{-}}^{213}\mathrm{Fr}\text{\ensuremath{-}}^{214}\mathrm{Ra}\text{\ensuremath{-}}^{215}\mathrm{Ac}$ is chosen to be described by the model with valence protons in the
1h9/2 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.