q-deformed pairing-vibrations
  Boson creation operators constructed from linear combinations of q- deformed zero coupled nucleon pair operators acting on the nucleus (A,0), are used to derive pp-RPA equations. The solutions of these equations are the pairing vibrations in (A${\underline{+}}$2) nuclei. For the $0^+_1$ and $0^+_2$ states of the nucleus $^{208}$Pb, the variations of relative energies and transfer cross-sections for populating these states via (t,p) reaction, with deformation parameter $\tau$ have been analysed. For $\tau=0.405$ the experimental excitation energy of 4.87MeV and the ratio $\frac{\sigma(0^+_2) }{\sigma(0^+_1)}=0.45$ are well reproduced. The critical value of pairing interaction strength for which phase transition takes place, is seen to be lower for deformed zero-coupled nucleon pair condensate with $\tau$ real, supporting our earlier conclusion that the real deformation simulates the two-body residual interaction. For $\tau$ purely imaginary a stronger pairing interaction is required to bring about the phase transition. The effect of imaginary deformation is akin to that of an antipairing type repulsive interaction. Using deformed zero coupled quasi-particle pairs, a deformed version of Quasi-boson approximation for $0^+$ states in superconducting nuclei is developed. For the test model of 20 particles in two shells, the results of q-deformed boson and quasi-boson approximations have been compared with exact results. It is found that the deformation effectively takes into account the anharmonicities and may be taken as a quantitative measure of the correlations not being accounted for in a certain approximate treatment. 
S. Shelly Sharma, and N.K. Sharma