Ground-state properties of even-even $N=Z$ nuclei within the Hartree-Fock-BCS and higher Tamm–Dancoff approaches

We calculate the ground-state properties of well deformed, even-even $N=Z$ nuclei in the region between ${}^{56}\mathrm{Ni}$ and ${}^{100}\mathrm{Sn}$ within two different approaches, focusing on the binding energy and deformation and pairing properties. First, we employ the Hartree-Fock-BCS (HFBCS) approximation with the Skyrme effective nucleon-nucleon interaction and discuss how the results depend on the parametrization of the interaction and on the pairing force parameters adjusted in various schemes to reproduce the experimental odd-even mass differences. Then, within the Higher Tamm-Dancoff Approximation (HTDA), which explicitly conserves the particle number, we calculate the same properties starting from the HFBCS solutions. The HTDA treatment of the ground-state correlations is converged within a $n$-particle-$n$-hole expansion using up to $n=4$ particle-hole excitations of the pair type (in the sense of Cooper pairs). We compare the ground-state properties calculated in these two descriptions of pairing correlations and deduce the importance of the particle-number conservation in weak pairing regimes. Finally, we extend the HTDA calculations so as to include the proton-neutron residual interaction and investigate the role of proton-neutron pairing on the above ground-state properties.

Figure 1

Convergence of the total correlation energy for ${}^{72}\mathrm{Kr}$ as a function of the pairing strength $V_0$. The percentage difference between correlation energies obtained in many-body spaces that differ by 2p-2h are shown.

Figure 2

Convergence of the occupation probability $v_i^2$ with enlarging of the particle-hole excitations space for neutrons and protons in ${}^{76}\mathrm{Sr}$. The cases of three different intensities $V_0$ of pairing interaction are shown.

Figure 3

Correlation energy (in MeV) normalized to the solution without $T=0$ pair correlations ($x=0$) as a function of $x$ for all considered nuclei. The case (a), respectively (b), corresponds to calculations with the pairing strength adjusted through the three-point and five-point formulas, respectively, in the $T=1$ channel.