Gogny force with a finite-range density dependence

In the present work, we have investigated an extension of the effective nucleon-nucleon Gogny interaction in which the zero-range density-dependent term has been replaced with a finite-range term. The parameters have been adjusted on both nuclear-matter properties and a few observables of stable nuclei. The traditional and unified fitting procedure of the Gogny force used in Bruyères-le-Châtel ensures common basic properties between the extended analytical form of the Gogny interaction, called $D2$, and the original one. In particular, symmetric infinite nuclear-matter and neutron-matter properties as well as pairing correlations have been investigated. A few static properties obtained in finite nuclei using the Hartree-Fock-Bogoliubov approach and the $D2$ parametrization of the Gogny interaction are analyzed and compared to the results obtained with $D1$-type parametrizations and experiment when it is possible. The $D2$ parametrization makes it possible to reproduce nuclear structure properties with improved accuracy.

Figure 1

Potential energy in $ST$ channels for the $D1S$ (empty squares) and $D2$ (empty circles) parametrizations of the Gogny interaction. The solid triangles and the solid squares correspond to Bethe-Brueckner-Goldstone calculations without and with 3-body interaction (BBG and BBG+3B respectively).

Figure 2

Equation of state of the neutronic matter for the $D1S$ (empty squares) and $D2$ (empty circles) parametrizations of the Gogny interaction. A comparison with the predictions of the variational Monte-Carlo approach (FP) by Friedman and Pandharipande (solid line) is done.

Figure 3

Neutron (solid line) and proton (dashed line) effective mass splitting as a function of the asymmetry $\beta$ for the $D1S$ (empty squares) and $D2$ (empty circles) parametrizations of the Gogny interaction. The BBG microscopic predictions are indicated.

Figure 4

Landau parameters $F_0^{ST}$ represented according the density, in each $\mathit{ST}$ channels. The results have been obtained, in symmetric nuclear matter, with the $D1S$ (empty squares) and $D2$ (empty circles) parametrizations of the Gogny interaction.

Figure 5

Contributions in the $S=0,T=1$ channel of various $D2$-type parametrizations of the Gogny force which differ only by the pairing attractivity.

Figure 6

Spatial shape of the pairing interaction for the $D1S$ (black open circles) and $D2$ parametrizations. On each curve, the crosses indicate the position of the minima.

Figure 7

Pairing gap in $ST=01$ channel, in symmetric nuclear matter, according to the Fermi momentum $k_F$. The results are shown for the $D1S$ (empty squares) and $D2$ (empty circles) parametrizations. The curve associated with the realistic Paris potential is also presented (solid line).

Figure 8

The simplest loop structure to calculate $s$ (see text).

Figure 9

Graphical representation of the quantity $s$ (see text).

Figure 10

Graphical representation of the quantity $s$ (see text).

Figure 11

Structuring of the loops that optimizes the execution time of the quantity $s$ (see text).

Figure 12

Spin, parity, and excitation energies $\Delta \left(E\right)$ (in MeV) of the ground and the first excited states in 11 odd Sn isotopes, with a number of neutron ranging from 61 to 81. Experimental results as well as the predictions of the $D1S$ and $D2$ parametrizations are indicated. The $D2A$ parametrization is similar to the $D2$ parametrization but differs by a stronger surface pairing.

Figure 13

Spatial distribution of the neutron density in the ${}^{116}\mathrm{Sn}$ isotope calculated with the $D1S$ (empty squares) and the $D2$ (empty circles) parametrizations.

Figure 14

Spatial distribution of the correlated neutron pairs ${\kappa }_n\left(r\right)$ in the ${}^{116}\mathrm{Sn}$ isotope, calculated with the $D1S$ (empty squares) and the $D2$ (empty circles) parametrizations.

Figure 15

Odd-even mass difference ${\Delta }^{\left(3\right)}$ according to the number of nucleons $A$ in the Sn isotopic chain. The results are shown for the $D1S$ (empty squares) and $D2$ (empty circles) parametrizations and compared to the experimental data (EXP) (solid squares).

Figure 16

Neutron pairing energy $E_{app}$ (in MeV) of a few Sn isotopes, calculated with the $D1S$ (empty squares) and $D2$ (empty circles) parametrizations.

Figure 17

Spatial distribution of the neutron density in the ${}^{208}\mathrm{Pb}$ isotope calculated with the $D1S$ (empty squares) and the $D2$ (empty circles) parametrizations.

Figure 18

Isotopic variation of charge radii $\Delta r_c^2$ for the Pb isotopes according to the mass $A$. The theoretical prediction obtained at the HFB approximation for the $D1S,D1M$, and $D2$ parametrizations are compared to the experimental ones (EXP).

Figure 19

(Top) Pairing energy calculated at the HFB approximation. (Middle) Moments of inertia. (Bottom) Axial deformation $\beta$. Calculations have been done for a few nuclei of the rare-earth region with the $D1S$ (blue) and $D2$ (red) parametrizations. Experimental data are indicated with black points.

Figure 20

Same as Fig. 19 for a few actinides.

Figure 21

Potential energy evolution $E_{\mathrm{HFB}}$ according to the axial quadrupole deformation parameter $\beta$ for a few actinides (see text). The calculations have been done for the $D1S$ (empty squares) and $D2$ (empty circles) parametrizations. The curves corresponding to the $D2$ parametrization have been translated in order that their minimum correspond to the ones calculated with the $D1S$ parametrization.