Inference of parameters for the back-shifted Fermi gas model using a feedforward neural network

The back-shifted Fermi gas model is widely employed for calculating nuclear level density (NLD) as it can effectively reproduce experimental data by adjusting parameters. However, selecting parameters for nuclei lacking experimental data poses a challenge. In this study, a feedforward neural network (FNN) was utilized to learn the level density parameters at neutron separation energy $a\left(S_n\right)$ and the energy shift $\Delta$ for 289 nuclei. Simultaneously, parameters for nearly 3000 nuclei are provided through the FNN. Using these parameters, calculations were performed for neutron resonance spacing in $s$ and $p$ waves, cumulative number of levels, and NLD. The FNN results were also compared with the calculated outcomes of the parameters from fitting experimental data (local parameters) and those obtained from systematic studies (global parameters), as well as the experimental data. The results indicate that parameters from the FNN achieve performance comparable to local parameters in reproducing experimental data. Moreover, for extrapolated nuclei, parameters from the FNN still offer a robust description of experimental data.

Figure 1

Level density parameters $a$ at the neutron separation energy $S_n$. The parameters obtained from the FNN (red circles with error bars) and global parameters (blue crosses) were compared with the local parameters (black crosses).

Figure 2

The energy shift $\mathrm{\Delta }$ for even-even nuclei, odd-odd nuclei, and odd-$A$ nuclei. The parameters obtained from the FNN (red spheres with error bars) were compared with the local values (black spheres).

Figure 3

The ratio of the $s$-wave neutron resonance spacings $D_0$ predicted by the BFM for even-even, odd-odd, and odd-$A$ nuclei to the experimental data from RIPL-2 as a function of mass number $A$. The parameters obtained from the FNN (red circles) and global parameters (blue triangles) were compared with the local parameters (black squares).

Figure 4

The $D_0$ values calculated using parameters from FNN (a) and their ratios to the results obtained from global parameters (b) for 3171 nuclei, including 289 nuclei within the dataset (represented by blue boxes) and 2882 nuclei outside the dataset.

Figure 5

The ratio of the $D_0$ values predicted by BFM for 12 nuclei outside the dataset to the experimental data in RIPL-3. The results obtained from FNN parameters (red spheres) were compared with the results from global parameters (blue triangles).

Figure 6

The ratio of the $p$-wave neutron resonance spacings $D_1$ predicted by the BFM for even-even, odd-odd, and odd-$A$ nuclei to the experimental data from RIPL-3 as a function of mass number $A$. The FNN parameters (red circles) and global parameters (blue triangles) were compared with the local parameters (black squares).

Figure 7

Cumulative number of levels for ${}^{58\text{–}66}\mathrm{Cu}$. Comparison was made between the results calculated from FNN parameters (red line), global parameters (blue dotted-dashed line), and local parameters (black dashed line), and the experimental data (green dot line).

Figure 8

The cumulative number of levels calculated with FNN parameters, local parameters, and global parameters are compared with the experimental data, for ${}^{155\text{–}166}\mathrm{Dy}$.

Figure 9

The cumulative numbers of levels calculated with FNN parameters, local parameters, and global parameters are compared with the experimental data, for ${}^{236\text{–}244}\mathrm{Pu}$.

Figure 10

Comparison among the total NLDs calculated using FNN parameters (red line), local parameters (black dashed line), and global parameters (blue dotted-dashed line) in the BFM with experimental data (green dots with error bars).