Prediction for ($p,n$) charge-exchange reactions with uncertainty quantification

Background: Charge-exchange reactions are a powerful tool for exploring nuclear structure and nuclear astrophysics; however, a robust charge-exchange reaction theory with quantified uncertainties is essential for extracting reliable physics.

Purpose: The goal of this work is to determine the uncertainties due to optical potentials used in the theory for charge-exchange reactions to isobaric analog states.

Method: We implement a two-body reaction model to study $(p,n)$ charge-exchange transitions and perform a Bayesian analysis. The $(p,n)$ reaction to the isobaric analog states of ${}^{14}\mathrm{C}$, ${}^{48}\mathrm{Ca}$, and ${}^{90}\mathrm{Zr}$ targets are studied over a range of beam energies. We compare predictions using standard phenomenological optical potentials with those obtained microscopically.

Results: Charge-exchange cross sections are reasonably reproduced by modern optical potentials. However, when uncertainties in the optical potentials are accounted for, the resulting predictions of charge-exchange cross sections have very large uncertainties.

Conclusions: The charge-exchange reaction cross section is strongly sensitive to the input interactions, making it a good candidate to further constrain nuclear forces and aspects of bulk nuclear matter. However, further constraints on the optical potentials are necessary for a robust connection between this tool and the underlying isovector properties of nuclei.

Figure 1

Angular distributions for ${}^{14}\mathrm{C}(p,n){}^{14}\mathrm{N}^{\text{IAS}}$, ${}^{48}\mathrm{Ca}(p,n){}^{48}\mathrm{Sc}^{\text{IAS}}$, and ${}^{90}\mathrm{Zr}(p,n){}^{90}\mathrm{Nb}^{\text{IAS}}$, at $E=25,35,45$ MeV. Insets show the same results in logarithmic scale. The curves represent predictions from the WLH (blue solid) and KD (green dash-dot) global optical potentials. Experimental data with error bars (black) are from [34, 35].

Figure 2

Histogram and correlation plots of optical potential parameters for $p+{}^{48}\mathrm{Ca}$ at $E=35$ MeV that characterize the entrance channel for the ${}^{48}\mathrm{Ca}(p,n){}^{48}\mathrm{Sc}^{\text{IAS}}$ reaction at a beam energy of $E=35$ MeV. Parameter distributions of the Bayesian posteriors are shown in green and distributions of the WLH potential are shown in blue. The black vertical lines indicate the mean value of the Bayesian prior which is taken from the Koning-Delaroche global optical potential.

Figure 3

Histogram and correlation plots of optical potential parameters for $n+{}^{48}\mathrm{Sc}$ at $E=28$ MeV that characterize the exit channel for the ${}^{48}\mathrm{Ca}(p,n){}^{48}\mathrm{Sc}^{\text{IAS}}$ reaction at a beam energy of $E=35$ MeV. Parameter distributions of the Bayesian posteriors are shown in green and distributions of the WLH potential are shown in blue. The black vertical lines indicate the mean value of the Bayesian prior which is taken from the Koning-Delaroche global optical potential.

Figure 4

Angular distributions for ${}^{14}\mathrm{C}(p,p)$ at $E=25,35,45$ MeV and ${}^{14}\mathrm{N}(n,n)$ at $E=22,32,42$ MeV. The Bayesian 95% confidence interval is shown in green and the 95% confidence interval of the WLH optical potential is shown in blue. Experimental data are from [38, 39]. Mock data are predictions of the Koning-Delaroche global optical potential.

Figure 5

Angular distributions for ${}^{48}\mathrm{Ca}(p,p)$ at $E=25,35,45$ MeV and ${}^{48}\mathrm{Sc}(n,n)$ at $E=18,28,38$ MeV. The Bayesian 95% confidence interval is shown in green and the 95% confidence interval of the WLH optical potential is shown in blue. Experimental data are from [40]. Mock data are predictions of the Koning-Delaroche global optical potential.

Figure 6

Angular distributions for ${}^{90}\mathrm{Zr}(p,p)$ at $E=25,35,45$ MeV and ${}^{90}\mathrm{Nb}(n,n)$ at $E=13,23,33$ MeV. The Bayesian 95% confidence interval is shown in green and the 95% confidence interval of the WLH optical potential is shown in blue. Experimental data are from [41]. Mock data are predictions of the Koning-Delaroche global optical potential.

Figure 7

Angular distributions for ${}^{14}\mathrm{C}(p,n){}^{14}\mathrm{N}^{\text{IAS}}$, ${}^{48}\mathrm{Ca}(p,n){}^{48}\mathrm{Sc}^{\text{IAS}}$, and ${}^{90}\mathrm{Zr}(p,n){}^{90}\mathrm{Nb}^{\text{IAS}}$ at $E=25,35,45$ MeV. The Bayesian 95% confidence interval propagated from elastic scattering is shown in green and the 95% confidence interval of the WLH optical potential is shown in blue. Experimental data with error bars shown in black are from [34, 35]. Note that the local minima of the charge-exchange cross sections cause the lower band to be jagged; these features have been smoothed out so that the uncertainty bands may be more clearly interpreted.

Figure 8

Range of the 95% confidence interval defined as the ratio of the upper limit to the lower limit as a function of the scattering angle. Results for charge exchange (solid), neutron elastic scattering (dash-dot), and proton elastic scattering (dashed) are shown for the Bayesian (green thick lines) and WLH (blue narrow lines) approaches.

Figure 9

Left: Charge-exchange cross sections for three cases taken from the Bayesian posterior for the ${}^{48}\mathrm{Ca}(p,n){}^{48}\mathrm{Sc}^{\text{IAS}}$ reaction at $E=35$ MeV. Right: The corresponding real (thick lines) and imaginary (narrow lines) isovector potentials.