Mean-field proton-neutron pairing correlations with the Gogny D1S energy density functional

Abstract

We study proton-neutron pairing correlations within the Hartree-Fock-Bogoliubov (HFB) framework using Gogny-type energy density functionals. By allowing for proton-neutron mixing in the quasi-particle transformation, both isovector ($T=1$) and isoscalar ($T=0$) pairing channels are explicitly included at the mean-field level. The \texttt{TAURUS} code has been extended to treat density-dependent Gogny interactions in this generalized HFB scheme. We examine the numerical behavior of the widely used Gogny D1S functional and compare it with calculations performed using the Hamiltonian-based Brink-Boecker B1 interaction supplemented by a zero-range spin-orbit term. When proton-neutron mixing is included and large single-particle spaces are employed, instabilities are observed for Gogny D1S due to the zero-range density-dependent term contribution to the proton-neutron pairing field, whereas stable solutions are obtained with the B1 interaction. Constrained HFB calculations performed in reduced configuration spaces allow us to explore total energy curves as functions of proton-neutron pairing collective coordinates in selected $sd$-shell nuclei. In all cases studied, the self-consistent minima correspond to vanishing proton-neutron pairing, with energy increasing rapidly as proton-neutron pairing correlations are introduced. These results provide insight into the behavior of Gogny functionals under generalized HFB conditions and offer useful guidance for future developments.

Figures (6)

• Figure 1: HFB energy as a function of the number of oscillator shells included in the working basis for the nucleus $^{20}$Mg, calculated with the Brink-Boecker B1 (red symbols) and Gogny D1S (blue symbols) interactions. Open symbols correspond to wave functions without $pn$ mixing, while filled symbols correspond to those allowing for $pn$ mixing. Energies are given relatively to the value obtained with the largest $N_{\mathrm{s.h.o.}}$ used in each case.
• Figure 2: (a) HFB energy and (b) gradient as functions of the number of iterations of a particular computational run for the nucleus $^{20}$Mg, calculated with the Brink-Boecker B1 (red symbols) and Gogny D1S (blue symbols) interactions. Open symbols correspond to wave functions without $pn$ mixing, while filled symbols correspond to those allowing for $pn$ mixing. Energies are given relatively to the value obtained with the largest $N_{\mathrm{s.h.o.}}$ used in each case, and those corresponding to the Gogny D1S interaction are shifted by 3 MeV for clarity of presentation.
• Figure 3: (a) HFB energy as a function of (a) the axial deformation $\beta_{2}$ and, (b) pairing parameters $\delta^{T}_{\tau\tau'}$ calculated with Gogny D1S using 5 major oscillator shells. Continuous, dashed, dotted and dash-dotted lines corresponds to $\delta^{T=1}_{pp}$, $\delta^{T=1}_{nn}$, $\delta^{T=1}_{pn}$ and $\delta^{T=0}_{pn}$ parameters, respectively.
• Figure 4: HFB energy (top panel) and pairing energies (bottom panel) as a function of the pairing parameters $\delta^{T}_{\tau\tau'}$ for even-even Mg isotopes calculated with Gogny D1S using 5 major oscillator shells. The line style has the same meaning as in Fig. \ref{['Fig3']}. Blue, green, and red colors are used for $E^{pp}_{\mathrm{pair}}$, $E^{nn}_{\mathrm{pair}}$, and $E^{pn}_{\mathrm{pair}}$, respectively.
• Figure 5: Same as Fig. \ref{['Fig4']} but for $N=Z$$sd$-shell nuclei.