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The influence of the isovector neutron–proton (np) pairing effect on nuclear statistical quantities is studied in N ≈ Z even–even systems. Expressions of the energy, the entropy, and the heat capacity are established using a recently proposed temperature-dependent isovector pairing gap equations. They generalize the conventional finite temperature BCS (FTBCS) ones. The model is first numerically tested using the schematic one-level model. As a second step, realistic cases are considered using the single-particle energies of a deformed Woods–Saxon mean-field. It is shown that: (i) the gap parameter Δ_np(T) behaves like Δ_tt(T), t = n, p, in the conventional FTBCS model and the critical temperature value T_cnp is such as T_cnp<T_cp<T_cn; (ii) the behavior of Δ_tt(T), t = n, p in the present model is different from that of the FTBCS one. This fact leads to a systematic discrepancy between the predictions of both models in the T_cnp<T<T_cn region for all studied statistical quantities; and (iii) in the 0≤T≤T_cnp region, the np pairing effect on the energy is a lowering of about 1%, on average, for all considered nuclei. Dealing with the entropy and the heat capacity, the np pairing effect appears only if the T_cnp value is sufficiently important.

The one-proton and two-proton separation energies are studied for "ordinary" and rare-earth proton-rich nuclei by including the isovector neutron–proton (np) pairing correlations using the BCS approximation. Even–even as well as odd nuclei are considered. In the latter case, the wave function is defined using the blocked-level technique. The single-particle energies used are those of a deformed Woods–Saxon mean field. It is shown that the np isovector pairing effects on the one-proton and two-proton separation energies are non-negligible. However, the only isovector BCS approximation seems to be inadequate for a good description of these quantities when including the np pairing effects: either a particle-number projection or the inclusion of the isoscalar pairing effect seems to be necessary. Another possible improvement would be a more realistic choice of the pairing strengths.

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.

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 ≈ 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.

Until now, the Sharp-Bardeen–Cooper–Schrieffer (SBCS) particle-number projection method, in the isovector neutron–proton pairing case, has been developed in the particle representation. However, this formalism is sometimes complicated and cumbersome. In this work, the formalism is developed in the quasiparticle representation. An expression of the projected ground state wave function is proposed. Expressions of the energy as well as the expectation values of the total particle-number operator and its square are deduced. It is shown that these expressions are formally similar to their homologues in the pairing between like-particles case. They are easier to handle than the ones obtained using the particle representation and are more adapted to numerical calculations.

The method is then numerically tested within the schematic one-level model, which allows comparisons with exact results, as well as in the case of even–even nuclei within the Woods–Saxon model. In each case, it is shown that the particle-number fluctuations that are inherent to the BCS method are completely eliminated by the projection. In the framework of the one-level model, the values of the projected energy are clearly closer to the exact values than the BCS ones. In realistic cases, the relative discrepancies between projected and unprojected values of the energy are small. However, the absolute deviations may reach several MeV.

The charge radii, the proton and neutron systems radii, as well as the matter radii of $N\approx Z$ odd-mass nuclei are studied using a particle-number projection approach in the neutron–proton (np) isovector ($T=1$) pairing case. Expressions of the proton and neutron systems quadratic radii are first established within a projection after variation method (i.e., of PBCS type) using a recently proposed wave function for odd-mass nuclei. It is checked that they reduce to the ones obtained in the like-particles pairing case. The new expressions are then used to study numerically the various radii of nuclei such as $16\le Z\le 48$ and $(N-Z)=1,3$, using the single-particle energies of a Woods–Saxon mean-field. It is shown that the few available experimental data are satisfactorily described by means of the present work approach. Furthermore, it appears that the np pairing and projection effects on the various radii are small on average in the case of odd-mass nuclei. However, the relative discrepancies with the values when only the pairing between like-particles is taken into account or the values obtained before the projection may reach up to 3% or 4% for some nuclei.

The eigensolutions of the collective Hamiltonian with different potentials suggested for description of the isovector pair correlations are obtained, analyzed and compared with the experimental energies. It is shown that the isovector pair correlations in nuclei around ${}^{56}$Ni can be described as anharmonic pairing vibrations. The results obtained indicate the presence of the $\alpha$-particle type correlations in these nuclei and the existence of the interaction different from isovector pairing which also influences on the isospin dependence of the energies.

The collective Hamiltonian including isovector pairing and $\alpha$-particle-type correlation degrees of freedom is constructed. The Hamiltonian is applied to description of the relative energies of the ground states of even–even nuclei around ${}^{56}$Ni. A satisfactory description of the experimental data is obtained. A significant improvement of the agreement with the experimental data compared to our previous calculations is explained by inclusion in the Hamiltonian of the dynamical variables describing $\alpha$-particle-type correlations.