Uncertainty quantification for optical model parameters

Background: Although uncertainty quantification has been making its way into nuclear theory, these methods have yet to be explored in the context of reaction theory. For example, it is well known that different parameterizations of the optical potential can result in different cross sections, but these differences have not been systematically studied and quantified.

Purpose: The purpose of this work is to investigate the uncertainties in nuclear reactions that result from fitting a given model to elastic-scattering data, as well as to study how these uncertainties propagate to the inelastic and transfer channels.

Method: We use statistical methods to determine a best fit and create corresponding 95% confidence bands. A simple model of the process is fit to elastic-scattering data and used to predict either inelastic or transfer cross sections. In this initial work, we assume that our model is correct, and the only uncertainties come from the variation of the fit parameters.

Results: We study a number of reactions involving neutron and deuteron projectiles with energies in the range of 5–25 MeV/u, on targets with mass $A=12–208$. We investigate the correlations between the parameters in the fit. The case of deuterons on ${}^{12}\mathrm{C}$ is discussed in detail: the elastic-scattering fit and the prediction of ${}^{12}\mathrm{C}$($d,p$)${}^{13}\mathrm{C}$ transfer angular distributions, using both uncorrelated and correlated ${\chi }^2$ minimization functions. The general features for all cases are compiled in a systematic manner to identify trends.

Conclusions: Our work shows that, in many cases, the correlated ${\chi }^2$ functions (in comparison to the uncorrelated ${\chi }^2$ functions) provide a more natural parameterization of the process. These correlated functions do, however, produce broader confidence bands. Further optimization may require improvement in the models themselves and/or more information included in the fit.

Figure 1

Pairwise two-dimensional ${\chi }_{UC}^2$ contour plots for the best-fit parameterization of Table 2. Black stars show the best-fit parameters.

Figure 2

95% confidence bands constructed from the uncorrelated fitting of $d+{}^{12}\mathrm{C}$ elastic-scattering data, for elastic-scattering angular distribution (a) and the ${}^{12}\mathrm{C}$($d,p$)${}^{13}\mathrm{C}$(g.s.) transfer angular distribution (b, predicted) both at 11.8-MeV deuteron energy. The cross-section calculations from the best-fit parameterization are shown in red (solid), and the 95% confidence bands in brown (hatched); the black circles show the elastic-scattering and transfer data.

Figure 3

Visual representation of model correlation for the elastic-scattering data set for ${}^{12}\mathrm{C}$($d,d$)${}^{12}\mathrm{C}$, for select angles. The blue histograms show the frequency around the average value of the cross-section values at the given angle. The scatter plots show the correlation between each pair of angles.

Figure 4

Same as Fig. 1 for the correlated best-fit parameterization of Table 2.

Figure 5

Same as Fig. 2 for the correlated fit.

Figure 6

Comparison of uncorrelated (black solid) and correlated (red dashed) cross-section calculations using the best-fit parameterization when elastic cross-section data were fit to predict transfer cross sections for ${}^{12}\mathrm{C}$($d,d$)${}^{12}\mathrm{C}$. Panel (a) shows the elastic-scattering calculations, and panel (b) shows the predicted transfer calculations.

Figure 7

Comparison of correlated and uncorrelated cross-section calculations using the best-fit parameterization when elastic-scattering data were fit to predict inelastic cross sections, for ${}^{54}\mathrm{Fe}-n$ scattering. Panel (a) shows the elastic-scattering fits, and panel (b) shows the inelastic-scattering predictions. Shown are uncorrelated (black solid) and correlated (red dashed) fits and predictions.