Mass predictions of the relativistic mean-field model with the radial basis function approach

The radial basis function (RBF) is a powerful tool to improve mass predictions of nuclear models. By combining the RBF approach with the relativistic mean-field (RMF) model, the systematic deviations between mass predictions of the RMF model and the experimental data are eliminated to a large extent and the resulting rms deviation is reduced from $2.217$ to $0.488$ MeV. Furthermore, it is found that the RBF approach has a relatively reliable extrapolative power along the distance from the $\beta$-stability line except for a large uncertainty around the region at magic number. From the deduced neutron separation energies, we found that the description of the nuclear shell structure and shape transition is also significantly improved by the RBF approach, thus improving agreement with the solar $r$-process abundances before $A=130$ and speeding up the $r$-matter flow. Therefore, a shorter irradiation time is enough to reproduce the solar $r$-process abundance distribution for the improved RMF mass model, which is closer to the irradiation time for those sophisticated mass models.

Figure 1

(a) Mass differences between the experimental data and the predictions of the RMF model. (b) Reconstructed functions $S(N,Z)$ of the RMF model. Boundary of nuclei with known masses in AME12 are shown by the black contours. (c) Mass differences between the experimental data and the predictions of the RMF + RBF model. Dotted lines denote magic numbers.

Figure 2

(a) Reconstructed functions $S(N,Z)$ based on the known masses of nuclei with a distance from the $\beta$-stability line $\varepsilon \le 3$. Outer and inner black contours indicate all nuclei with known masses and known nuclei with $\varepsilon \le 3$, respectively. (b) Differences in the reconstructed functions $S(N,Z)$ between those based on measured masses with $\varepsilon \le 3$ and those based on all measured masses.

Figure 3

The rms deviations of nuclear masses between experimental data and theoretical predictions for the RMF (squares) and RMF + RBF (circles) models as a function of the distance from the $\beta$-stability line $\varepsilon$. To determine the RBF weights, only nuclei with $\varepsilon \le 3$ are included, which are shown by the shaded area.

Figure 4

Two neutron separation energies $S_{2n}$ for nuclei with $50\le Z\le 90$. For clarity, only nuclei with even $Z$ are plotted. Black lines denote those predicted by the RMF model (a) and the RMF + RBF model (b). Experimental data are indicated by points.

Figure 5

Best fits to the solar $r$-process abundances (circles) for the RMF (dotted line) and RMF + RBF (solid line) models. For comparison, the $r$-process simulation using the mass predictions of the RMF model and the astrophysical condition $({\omega }_2,{\tau }_2)$ from the best fit of the RMF + RBF model is represented by the dashed line. In all calculations, nuclear $\beta$-decay properties are taken from NUBASE2012 [58] if available; otherwise, predictions from the FRDM plus quasiparticle random phase approximation approach [26] are employed.

Figure 6

(a) Superposition weight $\omega \left(n_n\right)$ and (b) neutron irradiation time $\tau \left(n_n\right)$ of 16 $r$-process components determined by the best fits to the solar $r$-process abundances. Open circles and filled circles denote the corresponding $\omega \left(n_n\right)$ and $\tau \left(n_n\right)$ for the RMF and RMF + RBF models, respectively.

Figure 7

The $r$-process paths for the RMF (open squares) and RMF + RBF (crosses) models. Filled squares represent stable nuclei.