Generation of fission yield covariance matrices and its application in uncertainty analysis of decay heat

Abstract: The uncertainties and covariance matrices of fission yield are important in the uncertainty analysis of decay heat. At present, there are no covariance matrixes of fission yield given in the evaluated nuclear data library, although they have provided the uncertainties with good estimates. In this work, the generalized least squares (GLS) updating approach was adopted to evaluate the fission yield covariances with the constraints from basic physical conservation equation and chain yield data, using the nuclear data files from ENDF/B-VIII.0, JENDL-5 and JEFF-3.3. Based on these original and updated data, summation calculation was performed for fission pulse decay heat of thermal neutron-induced fission of $<sup>{235}$U. The uncertainties of decay heat were obtained through generalized perturbation theory, including the uncertainties propagated from fission yield, decay energy, decay constant and branching ratio. The original uncorrelated yield data contributes a $\sim 4 \%$ uncertainty at all times and dominates the decay heat uncertainty at cooling times longer than \SI{100}{s}. With the generated covariance matrixes, the uncertainty of calculated decay heat is strongly reduced and decay energy data makes a major contribution in general. The relative uncertainties at cooling time \SI{0.1}{\second} are $\sim$10$\%$ for ENDF/V-VIII.0 and JEFF-3.3 and $\sim$5$\%$ for JENDL-5 and those at cooling time 10$<sup>{5}$ s are about 1$\%$ for three libraries. The influence of the GLS updating procedure on the contributions of important fission products to decay heat and their sensitive coefficients was also discussed.

• The paper generates independent fission-yield covariance matrices for ENDF/B-VIII.0, JENDL-5, and JEFF-3.3 using generalized least squares constrained by conservation laws, charge symmetry, normalization, and chain yields.
• The paper shows that incorporating yield correlations reduces the roughly 4% fission-yield uncertainty floor, producing total decay-heat uncertainties of about 5–10% at 0.1 seconds and roughly 1% at 10⁵ seconds for thermal-neutron fission of ²³⁵U.
• The paper finds that decay-energy data become the main remaining uncertainty source, while library differences, assumed 100% uncertainties, and JENDL-5-specific sensitivity shifts limit cross-library comparisons and motivate further evaluation.

Motivation and scope

Decay heat from fission products is a central quantity in reactor safety analysis, spent-fuel management, and post-shutdown heat removal design. Summation calculations of fission pulse decay heat (FPDH) depend on independent fission yields (IFYs) and decay data, but the major general-purpose evaluated libraries—ENDF/B-VIII.0, JENDL-5, and JEFF-3.3—provide only best-estimate IFYs with standard deviations, without covariance matrices. Because uncorrelated yield uncertainties propagate additively into decay heat, the absence of correlations substantially inflates the estimated uncertainty. This work addresses that gap by generating IFY covariance matrices via a generalized least squares (GLS) updating procedure constrained by physical conservation laws and chain yield data, and then quantifying how these covariances alter decay heat uncertainties for thermal neutron-induced fission of $^{235}$U (2604.04350).

Methodology: GLS-constrained covariance generation

The authors treat the library IFYs as prior parameters $\boldsymbol{\theta}_a = \boldsymbol{Y}_I$ with a diagonal prior covariance, and impose five classes of constraints through an observation equation $$\boldsymbol{\eta} \sim \boldsymbol{S}^t\boldsymbol{\theta}_a$$:

• Mass and charge conservation: $\sum_i A_i Y_{I(i)} = A_{CN} - \bar{\nu}_p$ and $\sum_i Z_i Y_{I(i)} = Z_{CN}$, with light charged particles (LCPs) from ternary fission included for JENDL-5 and JEFF-3.3.
• Normalization of non-LCP IFYs to 2, and normalization of heavy-side mass yields to 1 about the midpoint mass $(A_{CN}-\bar{\nu}_p)/2$.
• Charge symmetry, applied following Mills to complementary charge pairs when both charge yields exceed 1%, with a 0.5% tolerance reflecting ternary fission probability.
• Chain yield constraints, relating IFYs to cumulative yields via the branching matrix, $$\boldsymbol{Y}_{Ch} = \boldsymbol{D}^t\boldsymbol{Y}_I$$ with $$\boldsymbol{D}^t = \boldsymbol{d}^t[\boldsymbol{E}-\boldsymbol{b}]^{-1}$$.

A key methodological choice is the half-life cutoff $T_{cut}$ in the chain yield calculation. The authors show numerically that calculated chain yields converge at $T_{cut} = 1$ minute, with converged $\boldsymbol{\theta}_a = \boldsymbol{Y}_I$0 deviating from the IFY mass distribution by up to ~10% in the mass regions $\boldsymbol{\theta}_a = \boldsymbol{Y}_I$1–97 and 135–140, driven by crossing-mass-chain decays of nuclides with half-lives between 1 ms and 1 minute. This cutoff is consistent with the England–Rider evaluation. England–Rider chain yields are used as constraints for ENDF/B-VIII.0 and JENDL-5, and Nichols et al. for JEFF-3.3; branching ratio uncertainties are neglected in this constraint.

The GLS update simultaneously adjusts the central values and produces the posterior covariance. Validation against the constraints shows residual uncertainties reduced by orders of magnitude—for example, the mass conservation residual uncertainty drops from ±3.79 to ±7.48×10⁻³ for ENDF/B-VIII.0—and reduced $\boldsymbol{\theta}_a = \boldsymbol{Y}_I$2 for both charge symmetry and chain yield constraints. The one exception is charge symmetry for JEFF-3.3, whose IFYs were already adjusted to enforce complement charge equality more precisely than the present procedure; this is a benign limitation rather than a failure of the method.

The resulting correlation structure is physically interpretable: mass yields exhibit predominantly negative correlations between neighboring masses, with the strongest negative correlations ($\boldsymbol{\theta}_a = \boldsymbol{Y}_I$3 = 84–86, 87–88, 136–137) coinciding exactly with the mass regions where crossing-mass-chain decays separate $\boldsymbol{\theta}_a = \boldsymbol{Y}_I$4 from $\boldsymbol{\theta}_a = \boldsymbol{Y}_I$5—direct evidence that the chain yield constraint drives those correlations. Charge-yield correlations show both signs, indicating a more complex dependence structure.

Decay heat calculations and uncertainty propagation

FPDH is computed with the summation method, solving the Bateman equations numerically, and uncertainties are propagated via generalized perturbation theory using the sandwich formula $\boldsymbol{\theta}_a = \boldsymbol{Y}_I$6. Sensitivities to decay constants and decay energies are analytic; those to IFYs and branching ratios are numerical. Branching ratio covariances within each nuclide are derived analytically in an appendix under the exact normalization constraint $\boldsymbol{\theta}_a = \boldsymbol{Y}_I$7, yielding off-diagonal elements $\boldsymbol{\theta}_a = \boldsymbol{Y}_I$8. Where libraries omit decay energy or decay constant uncertainties, 100% values are assumed—an assumption that materially affects cross-library comparisons, as discussed below.

Two data sets are compared per library: Set-A (original yields, diagonal covariance) and Set-B (GLS-updated yields with full covariance). Decay data are unmodified.

Results

Central values. Calculated light-particle and electromagnetic decay heats agree well among the three libraries above 1000 s cooling time. Below 1000 s discrepancies grow, becoming significant below 1 s; JEFF-3.3 underestimates electromagnetic decay heat between 5 and 50 s. The GLS adjustment itself shifts decay heat only modestly—up to ~5% enhancement below 1 s and 1–2% at the curve peaks for ENDF/B-VIII.0 and JENDL-5, negligible for JEFF-3.3.

Uncertainties. The headline quantitative result concerns the role of yield correlations. With uncorrelated Set-A yields, fission yield contributes a roughly flat ~4% uncertainty at all cooling times and dominates the total beyond 100 s. With Set-B covariances, the yield contribution collapses, and decay energy becomes the dominant source in general. The resulting total relative uncertainties are approximately:

Cooling time	ENDF/B-VIII.0	JENDL-5	JEFF-3.3
0.1 s	~10%	~5%	~10%
10⁵ s	~1%	~1%	~1%

The markedly lower JENDL-5 uncertainty at short cooling times is attributed not to superior physics but to its broader coverage of decay energy uncertainties for short-lived radionuclides, reducing reliance on the assumed 100% values. This is an important caveat when comparing libraries: part of the spread reflects documentation completeness rather than underlying data quality.

Contributors and sensitivities. At the 10 s peak of the light-particle heat curve, 40 dominant nuclides account for over 80% of $\boldsymbol{\theta}_a = \boldsymbol{Y}_I$9 (over 75% of $$\boldsymbol{\eta} \sim \boldsymbol{S}^t\boldsymbol{\theta}_a$$0). Notable inter-library discrepancies exist at the nuclide level even where aggregate heat agrees: JENDL-5 gives contributions for $$\boldsymbol{\eta} \sim \boldsymbol{S}^t\boldsymbol{\theta}_a$$1Nb, $$\boldsymbol{\eta} \sim \boldsymbol{S}^t\boldsymbol{\theta}_a$$2Y, and $$\boldsymbol{\eta} \sim \boldsymbol{S}^t\boldsymbol{\theta}_a$$3La less than half of the other libraries' values regardless of data set. For electromagnetic heat, $$\boldsymbol{\eta} \sim \boldsymbol{S}^t\boldsymbol{\theta}_a$$4Zr contributes 0.8% with ENDF/B-VIII.0 but nothing with JENDL-5 or JEFF-3.3, because the latter assign $$\boldsymbol{\eta} \sim \boldsymbol{S}^t\boldsymbol{\theta}_a$$5 versus 0.449 MeV—a concrete example of library-level decay data inconsistency propagating into component analyses.

Relative sensitivity coefficients $$\boldsymbol{\eta} \sim \boldsymbol{S}^t\boldsymbol{\theta}_a$$6 are largest for $$\boldsymbol{\eta} \sim \boldsymbol{S}^t\boldsymbol{\theta}_a$$7Zr (light particle) and $$\boldsymbol{\eta} \sim \boldsymbol{S}^t\boldsymbol{\theta}_a$$8Rb (electromagnetic). Yield adjustment changes most coefficients by less than 20%, but dramatic exceptions occur for JENDL-5 electromagnetic heat: the coefficient of $$\boldsymbol{\eta} \sim \boldsymbol{S}^t\boldsymbol{\theta}_a$$9Y increases roughly fivefold after GLS updating, while those of $\sum_i A_i Y_{I(i)} = A_{CN} - \bar{\nu}_p$0Nb and $\sum_i A_i Y_{I(i)} = A_{CN} - \bar{\nu}_p$1La fall by ~95% and ~65%. These large sensitivity redistributions for JENDL-5, absent for the other two libraries, indicate that its IFY adjustments interact nontrivially with the decay network and merit further scrutiny.

Limitations and open questions

Several assumptions bound the applicability of the results. The study covers only $\sum_i A_i Y_{I(i)} = A_{CN} - \bar{\nu}_p$2U($\sum_i A_i Y_{I(i)} = A_{CN} - \bar{\nu}_p$3); extension to other actinides and neutron spectra is not demonstrated here. Branching ratio uncertainties are excluded from the chain yield constraint, and decay data are never adjusted, so the reported decay-energy-dominated residuals reflect the libraries as given—including the 100%-uncertainty assumption where evaluations are silent. The midpoint-mass normalization constraint is explicitly approximate because mass number is discrete. Finally, the strong negative correlations induced by the chain yield constraint depend on the chosen $\sum_i A_i Y_{I(i)} = A_{CN} - \bar{\nu}_p$4 min; whether this cutoff remains appropriate for other fissioning systems, and how sensitive the generated covariances are to it, is left open. The pronounced JENDL-5-specific sensitivity changes also pose an unresolved question about the stability of importance rankings under yield re-evaluation.

Conclusion

This paper provides a practical, constraint-based route to IFY covariance matrices for three major evaluated libraries and demonstrates their value quantitatively: incorporating the generated correlations reduces the fission yield contribution to decay heat uncertainty from a dominant ~4% flat floor to a subdominant role, leaving total uncertainties of ~10% (ENDF/B-VIII.0, JEFF-3.3) or ~5% (JENDL-5) at 0.1 s and ~1% at 10⁵ s, now limited primarily by decay energy data. The updated yields and covariances are directly usable in nuclear design and safety analysis, and the residual dominance of decay energy uncertainties identifies decay data evaluation—not fission yields—as the next limiting factor for short-cooling-time decay heat predictions.