Determination of the parameters of a Skyrme type effective interaction using the simulated annealing approach

We implement for the first time the simulated annealing method to the problem of searching for the global minimum in the hypersurface of the ${\chi }^2$ function, which depends on the values of the parameters of a Skyrme-type effective nucleon-nucleon interaction. We undertake a realistic case of fitting the values of the Skyrme parameters to an extensive set of experimental data on the ground-state properties of many nuclei, ranging from normal to exotic ones. The set of experimental data used in our fitting procedure includes the radii for the valence $1d_{5/2}$ and $1f_{7/2}$ neutron orbits in the ${}^{17}\mathrm{O}$ and ${}^{41}\mathrm{Ca}$ nuclei, respectively, and the breathing-mode energies for several nuclei, in addition to the typically used data on binding energy, charge radii, and spin-orbit splitting. We also include in the fit the critical density ${\rho }_{\mathrm{cr}}$ and further constrain the values of the Skyrme parameters by requiring that (i) the quantity $P=3\rho (\mathit{dS}/d\rho )$, directly related to the slope of the symmetry energy S, must be positive for densities up to $3{\rho }_0$; (ii) the enhancement factor κ, associated with the isovector giant dipole resonance, should lie in the range of 0.1–0.5; and (iii) the Landau parameter $G_0^{\text{'}}$ is positive at $\rho ={\rho }_0$. We provide simple but consistent schemes to account for the center-of-mass corrections to the binding energy and charge radii.

Figure 1

Variation of the average value of $\langle {\chi }^2\rangle {}_T$ as a function of the inverse of the control parameter T for the KDE0 interaction for the two different choices of the starting parameters (see text for details).

Figure 2

Variation of the fluctuations $\Delta {\chi }_T^2$ in the value of ${\chi }^2$ as a function of $1/T$ for the KDE0 interaction for the two different choices of the starting parameters (see text for details).

Figure 3

Variation of the symmetry-energy coefficient $S(\rho )$ as a function of the nuclear matter density ρ.

Figure 4

Energy per particle for pure neutron matter $E^{(n)}/A$ as a function of density. Results for the two newly generated Skyrme interactions KDE0 and KDE are compared with those obtained for the SLy7 Skyrme force and the realistic UV14+UVII model of Wiringa, Fiks, and Fabrocini [53].