Background: While diproton decay was first theorized in 1960 and first measured in 2002, it was first observed only in 2012.The measurement of 14 Be in coincidence with two neutrons suggests that 16 Be does decay through the simultaneous emission of two strongly correlated neutrons.Purpose: In this work, we construct a full three-body model of 16 Be (as 14 Be + n + n) in order to investigate its configuration in the continuum and in particular the structure of its ground state.Method: In order to describe the three-body system, effective n-14 Be potentials were constructed, constrained by the experimental information on 15 Be. The hyperspherical R-matrix method was used to solve the three-body scattering problem, and the resonance energy of 16 Be was extracted from a phase shift analysis.Results: In order to reproduce the experimental resonance energy of 16 Be within this three-body model, a three-body interaction was needed. For extracting the width of the ground state of 16 Be, we use the full width at half maximum of the derivative of the three-body phase shifts and the width of the three-body elastic scattering cross section. Conclusions:Our results confirm a dineutron structure for 16 Be, dependent on the internal structure of the subsystem 15 Be.

Recently a variety of studies have shown the importance of including non-locality in the description of reactions. The goal of this work is to revisit the phenomenological approach to determining non-local optical potentials from elastic scattering. We perform a χ 2 analysis of neutron elastic scattering data off 40 Ca, 90 Zr and 208 Pb at energies E ≈ 5 − 40 MeV, assuming a Perey and Buck [1] or Tian, Pang, and Ma [2] non-local form for the optical potential. We introduce energy and asymmetry dependencies in the imaginary part of the potential and refit the data to obtain a global parameterization. Independently of the starting point in the minimization procedure, an energy dependence in the imaginary depth is required for a good description of the data across the included energy range. We present two parameterizations, both of which represent an improvement over the original potentials for the fitted nuclei as well as for other nuclei not included in our fit. Our results show that, even when including the standard Gaussian non-locality in optical potentials, a significant energy dependence is required to describe elastic-scattering data.

Until recently, uncertainty quantification in low energy nuclear theory was typically performed using frequentist approaches. However in the last few years, the field has shifted toward Bayesian statistics for evaluating confidence intervals. Although there are statistical arguments to prefer the Bayesian approach, no direct comparison is available. In this work, we compare, directly and systematically, the frequentist and Bayesian approaches to quantifying uncertainties in direct nuclear reactions. Starting from identical initial assumptions, we determine confidence intervals associated with the elastic and the transfer process for both methods, which are evaluated against data via a comparison of the empirical coverage probabilities. Expectedly, the frequentist approach is not as flexible as the Bayesian approach in exploring parameter space and often ends up in a different minimum. We also show that the two methods produce significantly different correlations. In the end, the frequentist approach produces significantly narrower uncertainties on the considered observables than the Bayesian. Our study demonstrates that the uncertainties on the reaction observables considered here within the Bayesian approach represent reality more accurately than the much narrower uncertainties obtained using the standard frequentist approach.

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 of 10 − 25 MeV/u on targets with mass A = 48 − 208. The case of 48 Ca(d,p) 49 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 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 distorted-wave Born approximation.

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 C is discussed in detail: the elastic-scattering fit and the prediction of 12 C(d,p) 13 C transfer angular distributions, using both uncorrelated and correlated χ 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 χ 2 functions (in comparison to the uncorrelated χ 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.

Background: Uncertainty quantification for nuclear theories has gained a more prominent role in the field, with more and more groups attempting to understand the uncertainties on their calculations. However, recent studies have shown that the uncertainties on the optical potentials are too large for the theory to be useful.Purpose: The purpose of this work is to explore possible experimental conditions that may reduced the uncertainties on elastic scattering and single-nucleon transfer cross sections that come from the fitting of the optical model parameters to experimental data.Method: Using Bayesian methods, we explore the effect of the uncertainties of optical model parameters on the angular grid of the differential cross section, including cross section data at nearby energies, and changes in the experimental error bars. We also study the effect on the resulting uncertainty when other observables are included in the fitting procedure, particularly the total (reaction) cross sections.Results: We study proton and neutron elastic scattering on 48 Ca and 208 Pb. We explore the parameter space with Markov-Chain Monte Carlo, produce posterior distributions for the optical model parameters, and construct the corresponding 95% confidence intervals on the elastic-scattering cross sections. We also propagate the uncertainties on the optical potentials to the 48 Ca(d,p) 49 Ca(g.s.) and 208 Pb(d,p) 209 Pb(g.s.) cross sections.Conclusions: We find little sensitivity to the angular grid and an improvement of up to a factor of 2 on the uncertainties by including data at a nearby energy. Although reducing the error bars on the data does reduce the uncertainty, the gain is often considerably smaller than one would naively expect. We also find that the inclusion of total reaction cross section can improve the uncertainty although the magnitude of the effect depends strongly on the cases considered.

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