Abstract. The usual neutron-proton BCS wave function is simultaneously projected on both the good neutron and proton numbers using a discrete projection operator. The projected energy of the system is deduced as a limit of rapidly convergent sequence. It is numerically studied for the N = Z nuclei of which "experimental" pairing gaps may be deduced from the experimental odd-even mass differences. It then appears that the particle-number fluctuation effect is even more important than in the case of pairing between like-particles.PACS. 21.60.-n Nuclear structure models and methods -21.30.Fe Forces in hadronic systems and effective interactions
An expression of the number-projected electric quadrupole moment Q2 has been established in the isovector pairing case using the SBCS discrete projection before variation method. It has been verified that this expression reduces to the pairing between like-particles one at the limit when the np pairing gap parameter Δ np goes to zero. The convergence of the projection method has been numerically tested and a fast convergence has been observed. The electric quadrupole moment has been numerically calculated for some even–even proton-rich nuclei such as 16 ≤ Z ≤ 56 and 0 ≤ (N-Z) ≤ 4. The single-particle energies and eigen-states used are those of a Woods–Saxon mean-field. The np pairing effect on Q2 has been studied either before and after the projection; it seems that it is somewhat small since the relative discrepancies do not exceed 12%. Moreover, the np pairing effect is roughly the same in both situations. However, it has been shown that this effect diminishes with increasing values of (N-Z). The projection effect on Q2 has also been studied when including, or not, the np pairing correlations. It appears that this effect is slightly less important in the np pairing case than when only the pairing between like-particles is considered.
Expressions of temperature-dependent perpendicular (ℑ⊥) and parallel (ℑ‖) moments of inertia, including isovector pairing effects, have been established using the cranking method. They are derived from recently proposed temperature-dependent gap equations. The obtained expressions generalize the conventional finite-temperature BCS (FTBCS) ones. Numerical calculations have been carried out within the framework of the schematic Richardson model as well as for nuclei such as N = Z, using the single-particle energies and eigenstates of a deformed Woods–Saxon mean-field. ℑ⊥ and ℑ‖ have been studied as a function of the temperature. It has been shown that the isovector pairing effect on both the perpendicular and parallel moments of inertia is non-negligible at finite temperature. These correlations must thus be taking into account in studies of warm rotating nuclei in the N ≃ Z region.
The effect of the particle-number symmetry restoration on the root mean square (rms) proton and neutron radii of neutron-deficient nuclei is studied in the isovector pairing case. As a first step, an expression of the nuclear radii which includes the neutron-proton pairing effects and which strictly conserves the particle-number has been established using the SBCS (Sharp BCS) method. It is shown that this expression generalizes the one obtained in the pairing between like-particles case. As a second step, the proton and neutron rms radii are numerically evaluated for even-even nuclei such as 16 Z 56 and 0 (N − Z) 4 using the single-particle energies of a Woods-Saxon mean-field. The results are compared with experimental data when available and with the results obtained when one considers only the pairing between like-particles.Finally, the projection effect, when including or not the isovector pairing effect, has been studied by evaluating the relative discrepancies: δr n-proj and δr n-proj-np . This effect is roughly the same in both cases since the average values of these latter are respectively 0.70% and 0.74%. These values are also very close to that of the proton system.However, one notices that, in the neutron case, the np pairing effect seems to be more important than that of the projection. The particle-fluctuation effects in the BCS states being, by definition, nonphysical it is difficult to explain why Figs. 1 and 2 show very different trends.
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