$\beta$-delayed one-neutron emission probabilities within a neural network model

$\beta$-delayed neutron emission is one of the key ingredients for astrophysical $r$-process nucleosynthesis, and theoretical model predictions have still large uncertainties. In this work, we apply a novel feed-forward neural network model to calculate accurately $\beta$-delayed one-neutron emission probabilities. A model is trained with a set of input data of known physical quantities; one-neutron emission $Q$ value, the $Q$-value difference between the one- and two-neutron emissions, $\beta$-decay half-life, the distance from the least neutron-rich nucleus with $Q_{\beta 1n}>0$ in each isotope, and the exponential form of the ratio of $Q$-value $\exp (-Q_{\beta 2n}/Q_{\beta 1n})$. The results give improvements for predictions of medium heavy isotopes and provide reasonable results in $r$-process nuclei, especially in the waiting point nuclei for neutron magic numbers $N=50$ and 82, in comparison with other microscopic models.

Figure 1

Shape of the activation functions, $\tanh \left(x\right)$ and $\mathcal{F}\left(x\right)$. $\mathcal{F}\left(x\right)=2(1-\frac{1}{1+\text{exp}(-{\left(ax\right)}^b)})\text{tanh}\left(cx\right)$; the values $a,b$, and $c$ are taken as 0.05, 6, and 0.4, respectively. See the text for details.

Figure 2

The nuclei for training set (cyan squares) and the testing set (white squares). The upper panel shows the nuclei with only one-neutron emission, while the lower panel shows the nuclei both with and without multineutron emission. See the text for details.

Figure 3

Differences of $P_{1n}$ between theoretical results and the experimental data for the nuclei with only one-neutron emission. The data of $\mathrm{FRDM}12+\left(\mathrm{Q}\right)\mathrm{RPA}+\mathrm{HF}$ are taken from Ref. [17], and the data of $\mathrm{RHB}+\mathrm{RQRPA}$ are taken from Ref. [16]. See the text for details.

Figure 4

The same as Fig. 3, but for the nuclei with and without the multineutron emission. See the text for details.

Figure 5

Comparisons of $P_{1n}$ for the $r$-process nuclei around $N=82$ between the experimental data (black squares) [8, 10] and the theoretical results. The experimental data of ${}_{43}{}^{109\text{–}114}\mathrm{Tc}$ are taken from Ref. [8], while the others are taken from Ref. [10]. Three different theoretical models are ML-FNN (red diamonds), $\mathrm{FRDM}12+\left(\mathrm{Q}\right)\mathrm{RPA}+\mathrm{HF}$ (blue circles) [12], and $\mathrm{RHB}+\mathrm{RQRPA}$ (green triangles) [16]. See the text for details.