Covariances and parameter conﬁdence intervals from light-element R-matrix evaluations

. R-matrix parameter covariances for light elements with A ≤ 16 are calculated within the Wigner-Eisenbud multichannel unitary R-matrix theory. We review the theoretical foundation, numerical approach, and determination of the parameter covariances within this approach. We derive the relation between the parameter variance, as the diagonal elements of the covariance matrix, and the parameter conﬁdence interval based upon the chi-squared distribution. Cross section covariances computed from the parameter conﬁdence intervals are calculated for several compound systems and discussed.


Introduction
Nuclear reaction and scattering cross section information continues to be an indispensible component in design, modeling, simulation and refinement of complex systems in nuclear science, energy, and security. The ability to assess the validity of modeling and simulation capabilities, in particular, depends on the characterization and quantification of uncertainty associated with nuclear reaction and scattering observables. We should emphasize that we are not just referring to the energy-dependent, angle-integrated cross sections σ(E), but the full, complete set of observables that comprise those associated with a particlar compound system A Z that couples to all of the two-(and more) body partitions. For example, for the compund system of 5 Li, relevant two body partitions in order of increasing total partition mass, are 5 Li p ⊗ 4 He + d ⊗ 3 He · · · . (1) The observables for this system are the set of total, angleintegrated, and angular distributions of both unpolarized (spin-independent) and polarized or spin-dependent versions of these quantities. The advantage of being able to handle all of these types differential data has been discussed in Ref. [1]. These observables may be formed from the initial state spin density matrix elements ρ (i) and the transition T matrices for reactions and scattering (where the final partition composed of the same nuclides as the initial partition, but the masses may differ for inelastic scattering) as This spin-density matrix approach, which follows closely along the lines of Ref. [2], constitutes a general methodology for computing model representations and confers a * e-mail: mparis@lanl.gov * * e-mail: ghale@lanl.gov smooth (differentiable) representation of reaction and scattering data for general compound system. Our focus in this present work is on light A ≤ 20 systems; the R-matrix formalism is discussed in some detail in Ref. [1]. Observed data for all processes (total, elastic and reactions for both unpolarized and polarized data) is fit via least squares, the topic of the next section, Sec.2. The determination of the covariances from such a fit is the main focus of the present work. In section 3, we apply the method to the several compound systems. Conclusions are given in Sec.4.

Least squares fitting and covariance determination
Least squares fitting is accomplished via the R-matrix approach to reaction theory. The χ 2 function is formed as a sum over the included data for all experiments of processes that couple to a particular compound system. The general expression for the χ 2 function, which may be regarded as a log-likelihood function, is given as: The sum is over the measurement setups M or different experiments. Each experiment M is comprised of kinematical points and observable values denoted R i M , indexed by i M . Predictions X i M (p) are given generally as ratios of the elements of the spin-density matrix and their sums (such as the trace for the unpolarized angular distributions) and expressed as functions of the variational parameters p α , which are N p in number. Normalization factors n M are varied for either relative or absolute measurements. In the case of relative measurements, the experimental normalization is chosen to eliminate the second term in the above equation [eq. (3)]. Absolute measurement normalizations are allowed to vary in accordance with either the quoted experimental normalization uncertainty ∆S M or, if this uncertainty is underdetermined, by an assumed amount deemed to be reasonable, sometimes as large as a few 10's of percent but usually < 10%. The above form for χ 2 EDA (p, n M ) should be compared to a standard form [3], which takes into account data covariance through the elements of a matrix V i j as in a simplified notation that keeps the experimental setup index M implied. (We also drop explicit reference to the normalizations, n M , which may be considered to present.) Here absolute measurements are defined as where the observable R i M and the experimental normalization S M are defined above. The form χ 2 EDA (p) is recovered in the limit that the data covariance matrix has the form

Parameter uncertainty scaling
The determination of uncertainties in the parameter values is given by the usual analysis [4] at a local (assumed to be global) minimum of the function χ 2 EDA (p) given by parameter values p =p. About this minimum, the function χ 2 EDA (p) assumes the quadratic form where δp α ≡ p α − p 0,α and C is the parameter covariance matrix (the summation convention on repeated indices is implied). The standard analysis, given in Arndt    3 He for deuteron incident laboratory energy 324 keV of the unpolarized differential cross section in the center of mass as a function of the center of mass angle θ cm . The evaluation is shown by the solid curve. The gray band is the 90% confidence interval determined as described in the text.
and MacGregor [4], at this local minimum, which considers the marginalization of a single parameter p α , gives the uncertainty in terms of the change of the χ 2 function and the diagonal covariance element (the variance) C αα as Choosing δχ 2 in the usual manner, δχ 2 ≡ 1 however, leads to the following scaling behavior with N p , the number of fit parameters: as can readily be seen in fig. 1 867 σ(E), σ(θ), A y , P y Total: 6963 Table 1. Channel configuration (top) and data summary (bottom) for the 5 Li system analysis. The column labeled "Observables" indicates the following data types: σ(E), integrated cross section; σ(θ), unpolarized angular distributions (energy-dependence suppressed); A initial-state analyzing power; P final-state polarization; C spin correlation coefficients; K polarization transfer coefficients. (We have suppressed the indices i, j, . . . which take on values x, y, z for spins/polarization directions in configuration space.) All polarization and spin distributions are angular distributions, which depend on the angle of the outgoing particle. Chi-squared per degree of freedom for the analysis is χ 2 /dof 2.7 over 7,178 data points, 215 of which were discarded by eliminating individual data points which contribute to χ 2 > 40.  face that depends on the number of parameters N p . Following Ref. [5], we recognize that, given the fact that the χ 2 -function is a statistic distributed according to the χ 2distribution. A detailed analysis [5] supports this observation and we arrive at a description of the parameter uncertainty in terms of the confidence interval, chosen as the upper limit of the cumulative χ 2 distribution function P(δχ 2 ; k) for k degrees of freedom as: which is shown in fig. 2. Given that the mode and average of the χ 2 distribution are k − 2 and k, respectively, we estimate using eq. (9), the scaling of the uncertainty of p α to be The above analysis has been applied to the 5 Li compound system. The R-matrix configuration for this system is shown in the  Figure 5. The angular distribution for 3 He(d, p) 4 He for deuteron incident laboratory energy 455 keV of the unpolarized differential cross section in the center of mass as a function of the center of mass angle θ cm . The evaluation is shown by the solid curve. The gray band is the 90% confidence interval determined as described in the text.

Application to 5 Li compound system
We have applied the uncertainty quantification in terms of parameter confidence intervals described above to the 5 Li he3(d,p)he4 d /d E= 6.000 MeV CP2020 90% confidence range CP2011 (no data) diff cross section 3he(d,p)4he ed= 6.00 mev klinger 1971 i(theta) 3he(d,p)4he at ed = 6.0 mev gruebler et al Figure 6. The angular distribution for 3 He(d, p) 4 He for deuteron incident laboratory energy 6.0 MeV of the unpolarized differential cross section in the center of mass as a function of the center of mass angle θ cm . The evaluation is shown by the solid curve. The gray band is the 90% confidence interval determined as described in the text. At the solution, set by an upper limit on the size of the gradient of 1 part in 10 6 , we determine the parameter uncertainties as described in the previous section and propagate their error according to standard error propagation via the "sandwich" rule; see Ref. [1]. A few examples (out of hundreds generated by fitting the ∼ 7k data points for the 5 Li system) are shown in figs. 3 to 8. Figures 3 and 4 show the elastic scattering of d from 3 He for a deuteron laboratory energy of 324 keV. Curves are shown in fig. 3 for a more recent evaluation (CP2020, which will be included in the ENDF/B-VIII.1 release in 2024) and the older evaluation (CP2011, which is equivalent to ENDF/B-VIII.0 [6]) plotted against the data by Ref. [7]. Figure 4 shows data similar to the previous figure, plotted against the data from Ref. [8]. The (d, p) reaction data is shown in figs. 5 and 6 and finally the proton elastic data is displayed in figs. 7 and 8.

Conclusions
Confidence intervals for R-matrix evaluations have been determined from the χ 2 distribution. We have motivated an improvement in the scaling of the parameter uncertainties with the number of parameters N p in the fit. We applied this methodolgy to the 5 Li compound system in an evaluation of ∼ 7k differential and integral data points. The improved scaling of the parameter uncertainties resolves some long standing issues associated with R-matrix evaluations yielding cross section uncertainties that are too small [9].