Statistical tools for a better optical model

Background: Modern statistical tools provide the ability to compare the information content of observables and provide a path to explore which experiments would be most useful to give insight into and constrain theoretical models.

Purpose: In this work we study three such tools in the context of nuclear reactions with the goal of constraining the optical potential.

Method: The three statistical tools examined are (i) the principal component analysis, (ii) the sensitivity analysis based on derivatives, and (iii) the Bayesian evidence. We first apply these tools to a toy-model case, comparing the form of the imaginary part of the optical potential. Then we consider two different reaction observables, elastic angular distributions and polarization data for reactions on ${}^{48}\mathrm{Ca}$ and ${}^{208}\mathrm{Pb}$ at two different beam energies.

Results: For the toy-model case, we find significant discrimination power in the sensitivities and the Bayesian evidence, showing clearly that the volume imaginary term is more useful to describe scattering at higher energies. When comparing between elastic cross sections and polarization data using realistic optical models, sensitivity studies indicate that both observables are roughly equally sensitive but the variability of the optical model parameters is strongly angle dependent. The Bayesian evidence shows some variability between the two observables, but the Bayes factor obtained is not sufficient to discriminate between angular distributions and polarization.

Conclusions: From the cases considered, we conclude that, in general, elastic scattering angular distributions have similar impact in constraining the optical potential parameters compared with the polarization data. The angular ranges for the optimum experimental constraints can vary significantly with the observable considered.

Figure 1

${}^{48}\mathrm{Ca}(p,p)$ at 9 MeV parameter posterior distributions: surface model (orange) and volume model (blue).

Figure 2

${}^{48}\mathrm{Ca}(p,p)$ at 65 MeV parameter posterior distributions: surface model (orange) and volume model (blue).

Figure 3

Angular distribution 95% confidence intervals for $\frac{d\sigma }{d\mathrm{\Omega }}$ for ${}^{48}\mathrm{Ca}(p,p)$ at (a) 9 MeV and (b) 65 MeV: surface model (orange) and volume model (blue). Mock data generated with Ref. [8] (black circles).

Figure 4

Principal component analysis for ${}^{48}\mathrm{Ca}(p,p)$ at 65 MeV using $\frac{d\sigma }{d\mathrm{\Omega }}$ data. Shown are the weights $w_a$ obtained for three different scattering angles (a) for the surface model and (b) for the volume model.

Figure 5

Parameter sensitivities for ${}^{48}\mathrm{Ca}(p,p)$: (a) surface model at 9 MeV, (b) surface model at 65 MeV, (c) volume model at 9 MeV, and (d) volume model at 65 MeV.

Figure 6

95% confidence intervals calculated using $\frac{d\sigma }{d\mathrm{\Omega }}$ data (orange intervals) or $i{T}_{11}$ data (blue intervals) in the optimization procedure: (a) $d\sigma /d\mathrm{\Omega }$ for ${}^{48}\mathrm{Ca}(p,p)$ at 12 MeV; (b) $d\sigma /d\mathrm{\Omega }$ for ${}^{48}\mathrm{Ca}(p,p)$ at 21 MeV; (c) $i{T}_{11}$ for ${}^{48}\mathrm{Ca}(p,p)$ at 12 MeV; (d) $i{T}_{11}$ for ${}^{48}\mathrm{Ca}(p,p)$ at 21 MeV.

Figure 7

Sensitivity matrix for $\frac{d\sigma }{d\mathrm{\Omega }}$ data and $i{T}_{11}$ data: (a) ${}^{48}\mathrm{Ca}(n,n)$ at 12 MeV; (b) ${}^{48}\mathrm{Ca}(p,p)$ at 12 MeV, and (c) ${}^{48}\mathrm{Ca}(p,p)$ at 21 MeV. More details can be found in the text.

Figure 8

${}^{48}\mathrm{Ca}$(p,p) at 14 MeV 95% confidence intervals calculated using $\frac{d\sigma }{d\mathrm{\Omega }}$ data (orange bands) or $i{T}_{11}$ data (blue bands) in the optimization procedure: (a) $d\sigma /d\mathrm{\Omega }$ and (b) $i{T}_{11}$. Data from Ref. [23].