Particle-number conservation in odd mass proton-rich nuclei in the isovector pairing case

An expression of a wave function which describes odd-even systems in the isovector pairing case is proposed within the BCS approach. It is shown that it correctly generalizes the one used in the pairing between like-particles case. It is then projected on the good proton and neutron numbers using the Sharp-BCS (SBCS) method. The expressions of the expectation values of the particle-number operator and its square, as well as the energy, are deduced in both approaches. The formalism is applied to study the isovector pairing effect and the number projection one on the ground state energy of odd mass N approximate to Z nuclei using the single-particle energies of a deformed Woods-Saxon mean-field. It is shown that both effects on energy do not exceed 2%, however, the absolute deviations may reach several MeV. Moreover, the np pairing effect rapidly diminishes as a function of (N - Z). The deformation effect is also studied. It is shown that the np pairing effect, either before or after the projection, as well as the projection effect, when including or not the isovector pairing, depends upon the deformation. However, it seems that the predicted ground state deformation will remain the same in the four approaches.