Uncertainty analysis of the nuclear liquid drop model

The present work investigates the uncertainties in a nuclear liquid-drop model. The model parameters, their uncertainties and correlations, are determined through standard regression analysis that includes a statistical treatment of the errors of the model. The adjustment of the model is based on experimental binding energies corrected by the Thomas-Fermi shell corrections. The uncertainties are then propagated onto observables to reveal the reliability of the predictions and insight is provided to guide further research. Standard regression analysis is further extended to encompass experimental uncertainties as well as correlations. Strong support for such an extension can be found in Bayesian inference. The uncertainties and the correlations regarding microscopic corrections are not currently known; however, when available, these contributions may be included through this extended formalism.

Figure 1

Uncertainties in the predicted binding energies (blue dots) as a function of proton number $Z$ along with the mean of the uncertainties (red line).

Figure 2

Correlations between the theoretical binding energy of ${}^{208}\mathrm{Pb}$ and the 2315 binding energies. Notice, the extreme correlations in the region of ${}^{208}\mathrm{Pb}$ and their downfall away from this region. The position of ${}^{208}\mathrm{Pb}$ is indicated by the red dashed lines.

Figure 3

Correlations between the theoretical $Q_{\alpha }$ value of ${}^{208}\mathrm{Pb}$ and the 2315 $Q_{\alpha }$ values. Notice, the extreme correlations in the region of ${}^{208}\mathrm{Pb}$ and their downfall away from this region, as well as the complex patterns due to the involvement of a greater number of nuclei. The position of ${}^{208}\mathrm{Pb}$ is indicated by the red dashed lines.

Figure 4

Histogram (blue bars) formed of 50 bins restricted to $\widehat{\mathbf{\epsilon }}\in$ $[-2.1$;2.1] MeV and representing the distribution of the errors ($\widehat{\mathbf{\epsilon }}={\mathbf{B}}_{\mathbf{\text{Exp}}}-\mathbf{F}\widehat{\mathbf{p}}$). The normal distribution constructed from the mean (2 keV) and the standard deviation ($\widehat{\sigma }=602$ keV) of the errors is also provided (red line).

Figure 5

Quantile-quantile plot (blue dots) for the distribution of the errors restricted to $\widehat{\mathbf{\epsilon }}\in$ $[-2.1$;2.1] MeV and the expected theoretical straight line (dashed blue) in the case where normality is perfectly satisfied.

Figure 6

The errors ($\widehat{\mathbf{\epsilon }}={\mathbf{B}}_{\mathbf{\text{Exp}}}-\mathbf{F}\widehat{\mathbf{p}}$) restricted to $\widehat{\mathbf{\epsilon }}\in$ $[-2.1$;2.1] MeV are shown (blue dots) as a function of proton number $Z$. The mean of the errors (red line) and the standard deviation of the errors (red band), i.e., the root-mean-square deviation, are also displayed.

Figure 7

The 1088 nuclei involved in the study were divided into two equal groups each containing 544 nuclei. The first group (blue) includes those having a binding energy uncertainty below 2.6 keV while the second (red) contains those having a binding energy uncertainty above 2.6 keV. The second group is composed of more exotic nuclei than the first group.

Figure 8

Histogram formed of 25 bins, representing distribution of the binding energies uncertainties, restricted to the range [0;25] keV and including 1029 nuclei. A fourth of the 1088 nuclei have uncertainties below 1 keV and half below 2.6 keV.