A systematic approach to Covariance matrix formulation in charged particle activation experiments

Abstract: This work presents a detailed covariance and correlation matrix analysis for experimentally measured cross sections obtained using the activation technique. Both statistical and systematic contributions to the covariance matrix were explicitly calculated using sensitivity coefficients. The detector efficiency was determined by refitting standard source data with an exponential function, and the associated covariance matrix of the fitted parameters was propagated to estimate the uncertainty in efficiency at the relevant $γ$-ray energy. The cross sections and the corresponding experimental parameters, such as beam flux, target thickness, $γ$-ray intensity, and decay corrections, were taken from previously published measurements and are used here for the purpose of illustrating the covariance formalism. The resulting covariance and correlation matrices provide a comprehensive representation of uncertainties and their interdependencies. This formalism demonstrates the importance of including correlated uncertainties for reliable interpretation and comparison of experimental cross section data.

• The paper provides a Jacobian-based formalism to explicitly incorporate both correlated and independent uncertainty sources in activation cross section measurements.
• It applies the methodology to (p,γ) data, quantifying statistical and systematic contributions with notable off-diagonal correlations of up to 3.3%.
• This approach enhances nuclear data evaluation, model validation, and uncertainty quantification in charged particle activation experiments.

Systematic Covariance Matrix Formulation in Charged Particle Activation Experiments

Introduction

This work presents an explicit and systematic formalism for constructing covariance and correlation matrices in activation cross section measurements with charged particle projectiles (2603.29628). The analysis confronts the long-standing issue that conventional uncertainty reporting—often via straightforward quadratic summation of statistical and systematic components—fails to account for inter-measurement correlations, particularly those originating from normalization parameters (e.g., detector efficiency, beam intensity, decay data). The author offers a comprehensive Jacobian-based propagation approach which formalizes the inclusion of all correlations, clarifies common versus independent uncertainty sources, and provides analytic expressions for all relevant contributions.

Methodological Framework

The author defines the activation cross section determination process in detail, starting from the general relationship between measured activity and nuclear reaction parameters. The cross section is modeled as a function $\sigma_i = \xi(x_k)$ of experimental variables, including counts, detector efficiency, beam flux, target thickness, decay data, and timing corrections. Each contributing parameter is classified by both its statistical/systematic nature and its correlation structure (uncorrelated, fully correlated, partially correlated), with special attention to the classification of the target areal density.

Propagation of uncertainties is performed via sensitivity coefficients (i.e., partial derivatives) forming the Jacobian matrix. The analysis provides explicit analytic forms for all derivatives involved. For the standard set of parameters, the coefficients are constant; for decay constant and timing parameters, scenario-dependent expressions are derived, reflecting their nonlinear inclusion in decay corrections. The author also addresses the practical evaluation of detector efficiency and its covariance, recognizing that the fit to multi-gamma calibration data induces parameter correlations that must be propagated to efficiency uncertainty at the relevant $\gamma$ energy.

Covariance Matrix Formulation and Implementation

The total covariance is expressed as the sum of statistical and systematic components. Systematic uncertainties—dominated by normalization factors—propagate as $\sigma_i \sigma_j (\Delta x_p / x_p)^2$ for fully correlated parameters. Statistical uncertainties, dominated by counting statistics, carry no off-diagonal elements. The formal covariance propagation is presented in the compact matrix form $V_\sigma = J\eta J^T$, where $\eta$ is the parameter covariance matrix.

The correlation matrix is subsequently constructed by normalizing covariance elements with variances, yielding a quantification of the degree of correlation between all measured cross sections. The paper demonstrates that proper treatment of correlated uncertainties is essential for the reliable weighting and comparison of cross-section data, especially in the context of nuclear data evaluation and astrophysical modeling.

Numerical Application

The author applies the formalism to experimental $(p,\gamma)$ cross section data on $^{144}$Sm at low energies. All input parameters are supplied with granular uncertainty estimates. Detector efficiency is refitted, and the covariance of the fit parameters is propagated. The resulting covariance and correlation matrices illustrate that statistical uncertainties dominate the diagonal terms, whereas systematic uncertainties (especially those from efficiency and normalization) yield significant off-diagonal correlations. For the studied data (129.2, 30.0, 3.11, and 0.24 $\mu$b at energies 4.11, 3.68, 3.17, and 2.59 MeV, respectively), variance contributions dominate, but off-diagonal correlations of 3–3.3% are observed, quantifying the non-negligible systematic coupling across energy points.

A strong numerical result is that for these measurements, the detector efficiency uncertainty at $E_\gamma = 893.73$ keV is reported as 3.59%, and the dominant off-diagonal correlations, entirely absent from naïve uncertainty budgets, are fully specified.

Implications and Future Directions

The presented formalism addresses a central challenge in nuclear data science: correlations among measurements fundamentally impact the reliability of data evaluations, model validation, and reaction rate computations. Systematic uncertainty propagation using covariance matrices is explicitly required in all major evaluated data libraries, and the methodology here establishes a transparent standard reproducible across future activation datasets.

Beyond immediate applications to nuclear astrophysics and isotope production, rigorous covariance formalism has direct implications for uncertainty quantification in simulation-driven engineering, model-based inference, and Bayesian statistical fits. It enables optimal statistical weighting, avoids bias from mischaracterized error bars, and underpins robust uncertainty propagation in complex models. The extension of this methodology to surrogate reaction approaches, complex decay chains, and correlated target preparation parameters is anticipated as the next stage of research.

Conclusion

This work delivers an exhaustive, reproducible methodology for constructing covariance and correlation matrices in charged particle activation cross section experiments, rigorously accounting for both statistical and systematic contributions and their interdependencies. The analytic Jacobian-based approach, combined with practical guidance for uncertainty identification and sensitivity evaluation, ensures that full covariance information is accessible for experimental datasets—supporting reliable model comparisons, global fits, and propagation to downstream applications. This formalism constitutes a necessary foundation for future precise nuclear measurements and data evaluation efforts.