Constraining transfer cross sections using Bayes' theorem

Background: Being able to rigorously quantify the uncertainties in reaction models is crucial to moving this field forward. Even though Bayesian methods are becoming increasingly popular in nuclear theory, they have yet to be implemented and applied in reaction theory problems.

Purpose: The purpose of this work is to investigate, using Bayesian methods, the uncertainties in the optical potentials generated from fits to elastic-scattering data and the subsequent uncertainties in the transfer predictions. We also study the differences in two reaction models where the parameters are constrained in a similar manner, as well as the impact of reducing the experimental error bars on the data used to constrain the parameters.

Method: We use Bayes' theorem combined with a Markov chain Monte Carlo to determine posterior distributions for the parameters of the optical model, constrained by neutron-, proton-, and/or deuteron-target elastic scattering. These potentials are then used to predict transfer cross sections within the adiabatic wave approximation or the distorted-wave Born approximation.

Results: We study a number of reactions involving deuteron projectiles with energies in the range 10–25 MeV/nucleon on targets with mass $A=48$–208. The case of ${}^{48}\mathrm{Ca}(d,p){}^{49}\mathrm{Ca}$ transfer to the ground state is described in detail. A comparative study of the effect of the size of experimental errors is also performed. Five transfer reactions are studied, and their results are compiled in order to systematically identify trends.

Conclusions: Uncertainties in transfer cross sections can vary significantly (25–100 %) depending on the reaction. While these uncertainties are reduced when smaller experimental error bars are used to constrain the potentials, this reduction is not trivially related to the error reduction. We also find smaller uncertainties when using the adiabatic formulation than when using the distorted-wave Born approximation.

Figure 1

Representation of the MCMC samplings for ${}^{90}\mathrm{Zr}(n,n){}^{90}\mathrm{Zr}$ at 24.0 MeV. Different shades of gray (different colors online) represent different processors.

Figure 2

Comparison of the posterior distributions (histograms) resulting from various prior distributions (corresponding solid lines) for a wide Gaussian (WG), medium Gaussian (MG), and narrow Gaussian (NG) as defined in Table 2 for ${}^{90}\mathrm{Zr}(n,n){}^{90}\mathrm{Zr}$ at 24.0 MeV.

Figure 3

Comparison of the (a) ${}^{90}\mathrm{Zr}(n,n){}^{90}\mathrm{Zr}$ elastic scattering and (b) ${}^{90}\mathrm{Zr}(d,p){}^{91}\mathrm{Zr}(\mathrm{g}.\mathrm{s}.)$ transfer at 24.0 MeV for the posterior distributions shown in Fig. 2.

Figure 4

Same as Fig. 2 for a fixed prior shape—wide Gaussian—but varying $\epsilon$, as given in the legend in (j).

Figure 5

Systematic study of the mean of the posterior distribution as a function of the width of the prior distribution for neutron elastic scattering on ${}^{90}\mathrm{Zr}$ at 24.0 MeV: the prior width along the $x$ axis is given as the percentage of the original parameter value from [38], the posterior mean is given as a percentage of the prior center (solid circles), and the error bars show the width of the posterior mean as a percentage of the width of the prior.

Figure 6

Posterior distributions for the optical model parameters conditional on ${}^{48}\mathrm{Ca}(n,n)$ elastic scattering at 12.0 MeV. Gray (light blue) histograms show the posterior assuming 10% (5%) error on the experimental data. Overlaid solid line shows the Gaussian prior distribution (magnitude is arbitrary).

Figure 7

Same as Fig. 6 for ${}^{48}\mathrm{Ca}(p,p)$ elastic scattering at 14.03 MeV.

Figure 8

Same as Fig. 6 for ${}^{48}\mathrm{Ca}(p,p)$ elastic scattering at 25.0 MeV.

Figure 9

The 95% confidence intervals for the elastic scattering of (a) ${}^{48}\mathrm{Ca}(n,n)$ at 12.0 MeV, (b) ${}^{48}\mathrm{Ca}(p,p)$ at 14.08 MeV, and (c) ${}^{48}\mathrm{Ca}(p,p)$ at 25.0 MeV. Gray solid (light blue dashed) lines outline the 95% intervals when 10% (5%) experimental errors were used.

Figure 10

The 95% confidence interval for ${}^{48}\mathrm{Ca}(d,p){}^{49}\mathrm{Ca}(\mathrm{g}.\mathrm{s}.)$ at 24.0 MeV, compared to data at 19.3 MeV (extracted from [43]). Gray solid (light blue dashed) regions show the intervals when 10% (5%) experimental errors are used for the ADWA calculation.

Figure 11

Same as Fig. 9 for ${}^{48}\mathrm{Ca}(d,d){}^{48}\mathrm{Ca}$ at 23.2 MeV.

Figure 12

Same as Fig. 10 for the DWBA calculation.

Figure 13

Comparison of angular distributions between ADWA with 10% errors (gray solid lines), ADWA with 5% errors (light blue dashed), DWBA with 10% errors (green dash dotted), and DWBA with 5% errors (pink dotted) for ${}^{48}\mathrm{Ca}(d,p){}^{49}\mathrm{Ca}(\mathrm{g}.\mathrm{s}.)$ at 24.0 MeV (compared to data at 19.3 MeV).

Figure 14

Comparison of the posterior distributions (histograms) resulting from various prior distributions (corresponding solid lines) for a wide linear (WL), medium linear (ML), and narrow linear (NL) as defined in Table 2 for ${}^{90}\mathrm{Zr}(n,n){}^{90}\mathrm{Zr}$ at 24.0 MeV.

Figure 15

95% confidence intervals of the elastic-scattering angular distributions for ${}^{90}\mathrm{Zr}(n,n){}^{90}\mathrm{Zr}$ at 24.0 MeV using the posterior distributions from Fig. 14.