Empirical and Semi-Empirical Parameterization of Radiative Neutron Capture Cross Sections for Thirteen Actinides Across Three Energy Regimes

Abstract

We present a systematic, experiment-only parameterization of the radiative neutron capture cross section σ_n, γ(E) for thirteen actinides of importance to reactor physics, fuel-cycle modeling and transmutation (²³²Th, ²³³U, ²³⁵U, ²³⁸U, ²³⁷Np, ²³⁹Pu, ²⁴⁰Pu, ²⁴¹Pu, ²⁴²Pu, ²⁴¹Am, ²⁴³Am, ²⁴⁴Cm and ²⁴⁶Cm). A total of 90,299 EXFOR points were reduced to a per-isotope tabular form and analyzed in three regimes whose boundaries follow standard nuclear-data conventions: thermal (E < 0.5 eV), resonance (split into RRR, 0.5 eV–1 keV and URR, 1–100 keV) and fast (0.1–1.0 MeV). In every regime, an empirical and a semi-empirical description are fitted to the same data under a common point-uncertainty model; the effect of dataset-wide normalization offsets is assessed separately by refits with per-dataset log-normalization parameters. Models are compared by reduced χ², AIC/BIC and—in the URR—by dataset-level hold-out validation, and every derived quantity carries both a delta-method uncertainty and a dataset-level resampling 95% interval. In the thermal region the bound-level-plus-first-resonance form recovers first-resonance energies for ²³⁹Pu, ²⁴¹Pu, ²³⁷Np, ²⁴¹Am and ²⁴³Am to more than 3% from cross-section data alone; in the URR the Porter–Thomas-corrected statistical model is preferred on AIC for 7 of 12 nuclides, with fission competition included for the four fissile targets, but on dataset-level hold-out for only 3 of 5; and in the fast region the closed-form black-nucleus statistical model used here fails for ²³²Th, ²³⁸U and ²³⁷Np, whereas a default TALYS calculation reproduces the energy dependence for ²³²Th and ²³⁸U after a single normalization, attributing the failure to the approximations of the closed-form model rather than to statistical-model theory. Dataset-level resampling intervals are up to eighty times wider than delta-method uncertainties, demonstrating that inter-experiment discrepancy—not counting statistics—dominates the uncertainty of the pooled actinide capture record.

1. Introduction

Radiative neutron capture on the actinides governs the buildup of higher actinides in irradiated fuel, the neutron economy of thermal and fast systems, the design of minor-actinide transmutation schemes, and the interpretation of safeguards and burnup measurements. Evaluated libraries—ENDF/B-VIII.0 [1], JEFF-3.3 [2], JENDL-5 [3] and CENDL-3.2 [4]—provide recommended cross sections constrained by experiment, but the underlying record is heterogeneous: different laboratories, techniques, normalizations, resolutions and epochs, with strongly non-uniform energy coverage from nuclide to nuclide.

That record is compiled in EXFOR, maintained by the Nuclear Reaction Data Centres network [5] and distributed through a retrieval system preserving the identity of every original dataset [6]. Because EXFOR stores each measurement with its dataset identifier, the pooled record can be analyzed either as one large sample or as a collection of correlated experimental campaigns. The second view is the physically correct one: points from the same campaign share a flux normalization, sample mass, and detector efficiency, so they are not independent.

The modeling of actinide capture rests on a well-established hierarchy. Below and around the first resonances, the cross section follows multilevel R-matrix theory [7], in practice, through the Reich–Moore approximation [8] as implemented in SAMMY [9] and documented for evaluation work by Fröhner [10]. Where individual levels can no longer be resolved, the average cross section follows from compound-nucleus statistical theory [11], corrected for Porter–Thomas fluctuation of the reduced neutron widths [12] and, more completely, for the Moldauer elastic-enhancement effect [13]. Above the resonance region, a level-density prescription is required: Gilbert–Cameron [14] or the back-shifted Fermi gas of Dilg et al. [15]. Modern evaluations assemble these ingredients inside full reaction codes such as TALYS [16] or EMPIRE [17], with input parameters from RIPL-3 [18,19].

The energy-regime division used here is not arbitrary. Separating thermal from epithermal at the cadmium cut-off near 0.5 eV is the basis of the Westcott convention [20,21] and remains standard in activation-analysis nuclear data [22]. Dividing the resonance region into resolved and unresolved parts is prescribed by the evaluated-file formats, which place the actinide boundary near 1 keV and the upper URR edge near 100 keV [23]; the special numerical treatment the URR requires has been assessed by the OECD-NEA [24], and the same limits reappear as group boundaries in standard multigroup structures [25]. The regimes therefore coincide with boundaries the nuclear-data community already treats as physically meaningful.

Interest in these nuclides is driven by fuel-cycle and transmutation applications [26,27]. The OECD-NEA target-accuracy assessment identified minor-actinide capture—²³⁷Np, ²⁴¹Am, ²⁴³Am and the curium isotopes in particular—as among the largest contributors to design uncertainty in fast and accelerator-driven systems [28,29], and later reviews restated the need for improved differential data and consistent covariances [30,31]. That demand produced a sustained program of high-resolution time-of-flight and calorimetric measurements: ²⁴¹Am capture at DANCE [32] and n_TOF [33,34], a partial-wave resonance analysis of n + ²⁴¹Am [35], and the ²³⁷Np resonance-region measurement at n_TOF [36]. These form the modern, well-characterized part of the record analyzed here.

Empirical and data-driven descriptions are a complementary tradition: compact analytical forms interpolate smoothly, can be embedded in engineering codes, and can be fitted directly to measurements without a full model calculation. Machine-learning surrogates such as Bayesian neural networks have recently been applied to (n,γ) systematics [37] with the same motivation but far less interpretability. This work sits deliberately at the transparent end of that spectrum: every formula used has at most five parameters. The statistical machinery is standard but applied more strictly than is usual in cross-section systematics: models are ranked by the Akaike [38] and Bayesian [39] information criteria, and uncertainties are obtained both by linear propagation and by resampling in its clustered form [40,41,42,43], so that whole campaigns rather than individual points are resampled. Full evaluation frameworks, by contrast, combine model calculations and measurements through unified Monte Carlo [44] or generalized least-squares [45] updating. The transparent, low-parameter regression approach adopted here has previously been applied to isotopic concentrations in irradiated nuclear fuel [26,27] and to empirical cross-section formulae for light-target and cyclotron-production reactions [46,47].

The question asked is deliberately narrow: what can be extracted from the raw experimental record alone, using transparent low-parameter formulae, and where does simple physics stop working? All evaluated-library input is excluded from the fitting; evaluated and compiled quantities are used only a posteriori as benchmarks. Empirical formulae are chosen for functional convenience and provide a model-independent reference against which the added complexity of physics-based models can be assessed. Semi-empirical formulae are derived from resonance or statistical-model theory with a few nuclear parameters left free; if such a model is correct, its fitted parameters should reproduce independently tabulated nuclear data—a falsifiable test no purely empirical fit can offer. Thirteen nuclides were retained (Table 1). Section 2 describes the compilation, the uncertainty and normalization model, and the coverage that constrains what can be attempted; Section 3.1, Section 3.2 and Section 3.3 treat the three regimes; Section 3.4 examines variation across the actinide chain; Section 4 and Section 5 discuss and conclude.

Table 1. Nuclide selection grouped by fuel-cycle role.

Group	Nuclides
Thorium–uranium fuel cycle	232Th, 233U, 235U, 238U
Plutonium	239Pu, 240Pu, 241Pu, 242Pu
Minor actinides	237Np, 241Am, 243Am, 244Cm, 246Cm

The symmetry/asymmetry-related component of this study concerns nuclear composition: the neutron–proton asymmetry I = (N − Z)/A and the even/odd character of the target proton and neutron numbers. Section Neutron–Proton Asymmetry and the Rγ Benchmark connects these descriptors to the discrepancy between fitted and reference R_γ values, providing a composition-resolved diagnostic of the unresolved-resonance model. Even/odd classification is kept distinct from equality of neutron and proton numbers and from spatial symmetry. The comparison is descriptive; it neither establishes symmetry breaking nor replaces the multiple-comparison assessment of the cross-nuclide correlations.

2. Materials and Methods

2.1. Source, Compilation and Uncertainty Model

All (n,γ) data were retrieved from the IAEA EXFOR database through the Nuclear Data Services Data Explorer [48]. Each record retains the first author, publication year, EXFOR dataset identifier (DSID), incident energy E, energy uncertainty, cross-section σ, and its uncertainty where reported. The compilation used in the fits contains 90,299 points, of which 48,276 (53.5%) carry a reported uncertainty. Retention of the DSID is essential: it is what makes the hold-out validation of Section 3.2.5 and the dataset resampling of Section 3.1.3 possible, and what allows correlated normalization errors within a campaign to be treated as a block rather than as independent noise. An overview of the complete pipeline is given in Figure 1.

Figure 1. End-to-end analysis pipeline: EXFOR retrieval and reduction into (E, σ, δσ) triplets with dataset identifiers retained, an energy split at 0.5 eV, 1 keV and 100 keV into three regimes, empirical and semi-empirical fits under a common point-uncertainty model, a separate dataset-normalization robustness check, and joint uncertainty quantification by delta-method propagation and cluster-bootstrap resampling of whole EXFOR datasets (B = 400).

Point Uncertainty and the 10% Surrogate: Where an experimental uncertainty δσ is reported, it is used directly; where absent, the assigned point uncertainty is s_i = 0.10σ_i. This fixed relative scale allows measurements without reported uncertainties to enter the weighted fit without assigning them zero uncertainty or excluding them solely because uncertainty information is missing. The same rule is applied to the empirical and semi-empirical models. As a descriptive comparison, the source EXFOR export used for the uncertainty comparison contains 82,855 records with positive finite energy, cross section, and reported uncertainty; their point-weighted median relative uncertainty, 100δσ/σ, is 9.07%, with an interquartile range of 4.61–41.04% (Supplementary Table S10). Thus, 10% is close to the overall source-export median and is retained as a rounded common working scale. This comparison is descriptive, not evidence that each missing uncertainty equals 10% or that the choice is statistically optimal. The surrogate applies to the 42,023 points (46.5%) of the compilation that lack reported uncertainties. It is distinct from the dataset-wide normalization term below: assigning a pointwise relative scale does not itself model a correlated shift in an entire experimental campaign. Because this comparison cannot establish the size of the missing uncertainties, every fit that uses point uncertainties was repeated with the surrogate set to 5%, 20%, and the regime-specific reported medians (Section 4.4; Supplementary Table S11).

A closely related precedent is provided by Alhassan et al. [49], who used reported experimental uncertainties and assigned a 10% relative uncertainty to selected EXFOR datasets with unreported uncertainties, including cases where they were the only measurements available in the energy range or had previously been assessed as being of reasonable quality. This provides a precedent for an explicit provisional assignment, rather than establishing 10% as a universal or statistically optimal experimental uncertainty.

Per-Dataset Normalization: Measurements within one experimental dataset may share an overall scale error arising from flux, sample mass, detector efficiency, or the reference standard. Each dataset d is therefore assigned a log-normalization parameter n_d, and the prediction for observation i is exp(n_d(i)) · m(E_i;θ). Here, y_i is the measured cross section, E_i is the incident energy, θ is the vector of model parameters, d(i) is the dataset containing observation i, and m is the unnormalized model prediction. A Gaussian prior n_d ~ N(0, σ_n²), with σ_n = 0.05, centers each multiplicative scale exp(n_d) on unity. The fitted residual vector is

$$\mathbf{r}(\theta ,\mathbf{n})=\left[{\displaystyle \frac{y_i-\exp \left(n_{d\left(i\right)}\right)\hspace{0.17em}m(E_i;\theta )}{s_i}};{\displaystyle \frac{n_d}{{\sigma }_n}}\right].$$

(1)

The first block of Equation (1) measures each observation–prediction difference in units of its assigned point uncertainty s_i. The second block penalizes dataset normalization shifts relative to the prior scale σ_n. Minimizing the squared residual norm therefore balances agreement with individual measurements against excessive rescaling of whole datasets. The two blocks should not be interpreted as interchangeable uncertainty sources. The fits reported in Section 3 use the point uncertainties only; the normalization terms of Equation (1) are activated in the robustness checks of Section 3.1.4 and Section 3.3.3 and in the fitted-normalization diagnostics of Section 3.3.3.

The 5% baseline sits near the lower edge of the credible range for absolute-flux budgets in modern time-of-flight campaigns (n_TOF ²⁴¹Am 4–6%, n_TOF ²³⁷Np ≈ 5%, DANCE ²⁴¹Am ≈ 6%). Older activation and lead-slowing-down measurements are typically quoted at 5–15% [50]; a single common prior therefore restrains the nuisance parameters rather than licensing them, and is conservative in the direction that matters.

2.2. Exclusion of ²⁴⁵Cm

²⁴⁵Cm was excluded on data-availability grounds despite its acknowledged fuel-cycle importance. Its entire EXFOR (n,γ) record comprises only two experimental points, drawn from two distinct datasets, both lying at thermal energy. No regime can therefore be fitted, hold-out validation is undefined with two datasets, and a single thermal point would enter the systematics of Section 3.4 as a leverage point with no independent confirmation. Because the three modern evaluations for this nuclide are not mutually independent, their spread understates the true uncertainty, and no ²⁴⁵Cm value is quoted from them; Section 4.2 treats the nuclide as a priority measurement gap.

2.3. Coverage per Regime

Here, data coverage means the availability and distribution of experimental measurements across nuclides and energy intervals. Table 2 summarizes coverage by counting the experimental points in each interval; it does not report the statistical coverage probability of a confidence interval. Coverage is extremely uneven and largely determines which nuclides can be analyzed in which regime. The regime limits are 0.5 eV for the thermal/epithermal boundary [20,22], and 1 keV and 100 keV for the resolved/unresolved and upper URR boundaries [23,24].

Table 2. Number of experimental EXFOR points per nuclide in each energy regime; the fast-region column covers the entire range above 100 keV.

Nuclide	Thermal (<0.5 eV)	RRR (0.5 eV–1 keV)	URR (1–100 keV)	Fast (≥100 keV)
232Th	73	35	182	250
233U	7	4148	363	7
235U	4	2680	46	17
238U	10	3252	1864	367
239Pu	415	2805	287	35
240Pu	2	0	428	0
241Pu	264	1883	772	0
242Pu	10	331	271	0
237Np	11,448	12,001	6209	342
241Am	2303	5552	4024	91
243Am	6308	10,289	214	7
244Cm	2	6437	311	0
246Cm	2	3950	1	0

Two entries require comment. ²³³U has 4148 RRR points but only 7 below 0.5 eV, and ²³⁵U shows the same pattern (2680 against 4). Both are fissile, and experimental effort has overwhelmingly targeted the fission cross section and the capture-to-fission ratio in the resolved resonance region. Below 0.5 eV, the thermal capture cross section of a fissile nuclide is conventionally established once, to high precision, by a small number of integral and standards measurements. Three structural features drive the whole analysis: the thermal region is well covered for only seven nuclides; the RRR is well covered for almost all, but with point distributions that no smooth formula can fit point-by-point; the fast region is essentially empty for eight of thirteen nuclides.

2.4. Model Selection, Fit Indicators and Resampling

Information Criteria: The Akaike information criterion (AIC) [38] and Bayesian information criterion (BIC) [39] compare goodness of fit while penalizing additional model parameters. Lower values indicate the preferred model within a comparison using the same observations and uncertainty model. The expressions used for the tabulated comparisons are

AIC = χ² + 2k,  BIC = χ² + klnn.

(2)

Here, n is the number of observations entering the fit, or the number of group-averaged observations when fitting resonance-region groups. In the tabulated model-complexity penalties, k counts the model parameters: the URR empirical power law has two (A, b), whereas the statistical model has three (S₀, S₁, R_γ); the thermal models have two and five, respectively. Thus, “two against three” refers to parameter counts, not to numbers of datasets or energy regimes. The dataset-normalization structure is held the same between the compared models. An identical additional parameter-count penalty for those shared nuisance parameters would cancel in within-pair AIC/BIC differences; this does not constitute a calibration of their effective degrees of freedom. The two criteria give the same preferred model in the reported comparisons; BIC tabulations are in Supplementary Tables S2–S4 and S8.

A hierarchy of evidence is adopted and applied consistently: (i) dataset-level hold-out validation is primary wherever the data permit, being the only test used here that is out-of-sample with respect to the correlated systematics that limit this corpus; (ii) information criteria are secondary; (iii) agreement of fitted parameters with independently tabulated nuclear data is a separate falsification test that no purely empirical model can pass or fail.

Fit-quality Indicators: Four indicators are reported because no single number is sufficient when the number of points per nuclide spans four orders of magnitude and the cross section spans six. Reduced χ² is χ² per degree of freedom; the fitting objective of the fits reported in Section 3 is the weighted sum of squares of the first block of Equation (1); the full extended residuals are used in the normalization robustness checks. The root-mean-square error (RMSE) is expressed in barns, while the relative RMSE (rRMSE) measures errors relative to the measured cross sections. R² is evaluated on the natural logarithm of the cross section as a descriptive log-space indicator, not as the fitted objective. A negative R² means that the model performs worse, on that scale, than a constant prediction equal to the mean of the logarithmic observations.

Dataset resampling. The procedure is a non-parametric cluster bootstrap in the sense of Field and Welsh [43], with the EXFOR dataset as the cluster. For a nuclide with D distinct DSIDs in a regime, one replicate draws D labels uniformly with replacement, concatenates all points of each drawn label, and refits from the original converged parameter vector. Derived quantities are recomputed per replicate, and the reported interval is the 2.5%/97.5% percentile interval over B = 400 replicates. Fits with fewer than three distinct DSIDs are flagged (“deg.”) rather than quoted.

3. Results

3.1. Regime 1—The Thermal Region (E < 0.5 eV)

3.1.1. Empirical and Semi-Empirical Descriptions

The standard empirical description of thermal capture is the power law σ_γ(E) = A·E^−b, with ideal 1/v behavior at b = 1/2. For a Maxwellian flux φ(E)dE ∝ E·exp(−E/E₀) dE with E₀ = kT = 0.0253 eV, the Westcott factor evaluates analytically to g_W = (2/√π)·Γ(2 − b), with δg_W = |g_W · ψ(2 − b)| · δb, and returns exactly 1 at b = 1/2. The closed form is a physical Westcott factor only for ²³²Th and ²³³U; for the five resonance-bearing nuclides it is reported as an effective g_W^eff.

The semi-empirical form follows from Breit–Wigner theory. Two contributions matter below 0.5 eV: a bound (negative-energy) level whose tail is the physical origin of near-1/v behavior, and the first positive-energy resonance where it lies low enough to intrude:

$${\sigma }_{\gamma }(E)={\displaystyle \frac{A_b}{\sqrt{E}(E+E_b{)}^2}}+{\displaystyle \frac{A_r}{\sqrt{E}\left[(E-E_r{)}^2+{\mathsf{\Gamma }}_r^2/4\right]}}.$$

(3)

In Equation (3), A_b sets the amplitude of the bound-level contribution and E_b its positive energy-distance parameter; A_r sets the first-resonance amplitude, E_r its position, and Γ_r its fitted width. The bound-level term describes the low-energy background, while the second term allows curvature associated with a nearby resonance. These five parameters (A_b, E_b, A_r, E_r, Γ_r) are fitted without supplying tabulated resonance parameters. Agreement with independently tabulated values is therefore an external test of the model. The objective is non-convex in E_r; fits used fifteen starts per nuclide on a grid of E_r ∈ {0.26, 0.30, 0.42, 0.51, 1.0} eV, and Γ_r ∈ {0.02, 0.05, 0.10} eV, retaining the lowest-χ² converged solution. Attainment of the global minimum was verified by χ² profiling and 200 random restarts.

3.1.2. Results and Recovery of Resonance Parameters

Seven nuclides have four or more distinct thermal energies and can be fitted (Table 3, Figure 2). BIC selects the same model as AIC for all seven (Supplementary Table S2).

Table 3. Thermal-region (E < 0.5 eV) fits comparing the empirical power law and the bound-level-plus-first-resonance model; asterisks mark the AIC-preferred model. The column headed Γ_r (eV) reports the fitted resonance width in Equation (3), not an uncertainty in the resonance energy. A dash indicates that no resonance width is quoted for the bound-level-dominated fit. The previously reported width values are retained.

Nuclide	n	b	χ2ν (emp)	AIC (emp)	Er (eV)	Fitted Width Γr (eV)	χ2ν (semi)	AIC (semi)	Preferred
232Th	73	0.5382 ± 0.0090	0.324	27.0 *	(none)	—	0.329	32.4	empirical
233U	7	0.5731 ± 0.0205	0.335	5.7 *	(none)	—	0.574	11.1	empirical
239Pu	415	0.0291 ± 0.0061	19.80	8181	0.2987 ± 0.0005	0.0995	0.044 *	28.2 *	semi-empirical
241Pu	264	0.5188 ± 0.0054	17.33	4545	0.2602 ± 0.0008	0.1082	0.762 *	207 *	semi-empirical
237Np	11,448	0.4598 ± 0.0006	30.17	345,380	0.4884 ± 0.0001	0.0443	0.813 *	9318 *	semi-empirical
241Am	2303	0.4013 ± 0.0016	27.95	64,307	0.3092 ± 0.0001	0.0544	3.288 *	7566 *	semi-empirical
243Am	6308	0.3921 ± 0.0020	1.965	12,393	0.4243 ± 0.0008	0.1121	0.818 *	5164 *	semi-empirical

Figure 2. Thermal-region fits for the seven fittable nuclides: gray points are EXFOR data, solid line the empirical power law A·E^−b, dotted line the bound-level term alone, and dashed line the full bound-level-plus-first-resonance form. The vertical dotted line marks 0.0253 eV.

For the two nuclides whose lowest resonance lies far above the thermal range—²³²Th (first resonance at 21.8 eV) and ²³³U—the extra three parameters do not improve model selection, and the power law is preferred. For the five nuclides with a resonance below or near 0.5 eV, the semi-empirical form is strongly preferred by the reported information criteria, reducing χ²ν by factors from 2.4 (²⁴³Am) to 450 (²³⁹Pu). The first-resonance energies recovered by the semi-empirical form are compared with their tabulated values in Table 4.

Table 4. Recovery of first-resonance energies from thermal data alone: fitted E_r versus the tabulated JENDL-5 value [3].

Nuclide	Er Fitted (eV)	Er Tabulated (eV) [3]	Deviation
239Pu	0.2987	0.296	+0.9%
241Pu	0.2602	0.2603	−0.04%
237Np	0.4884	0.490	−0.3%
241Am	0.3092	0.3076	+0.5%
243Am	0.4243	0.4155	+2.1%

Agreement is more than 3% throughout and better than 1% for four of five. The fitted widths (44–112 meV) are of the correct magnitude for actinide s-wave resonances. For ²³²Th and ²³³U, the multi-start fit correctly collapses to the bound-level term, making A_r consistent with zero once more.

3.1.3. Thermal Cross Sections, Westcott Factors and Resonance Integrals

Cross sections at 0.0253 eV were obtained from the data for all thirteen nuclides and compared with the IAEA safeguards reference compilation [51,52], itself based on Mughabghab’s Atlas of Neutron Resonances [53] (Table 5).

Table 5. Thermal cross-section σ₀ at 0.0253 eV, Westcott factors g_W and partial resonance integrals versus the IAEA reference table [51]; asterisks mark agreement within 3% and † marks effective (non-pure-Maxwellian) g_W values. Note that RI (0.5 eV–1 keV) is a partial integral of this work and is not the IAEA infinite-dilution RI∞.

Nuclide	σ0 This Work (b)	σ0 IAEA (b)	Δ	gW This Work	gW IAEA	RI (0.5 eV–1 keV) (b)	RI∞ IAEA (b)
232Th	7.375	7.35 ± 0.03	+0.3% *	0.9993	0.9982	1.8	83.3 ± 1.5
233U	49.295	45.5 ± 0.7	+8.3%	0.9998	1.0495	122.1	138 ± 6
235U	99.866	98.8 ± 0.8	+1.1% *	—	0.9956	146.9	146 ± 6
238U	2.636	2.683 ± 0.012	−1.8% *	—	1.0009	83.5	277 ± 3
239Pu	276.503	269.3 ± 2.9	+2.7% *	1.1149 †	1.1369	278.6	180 ± 20
240Pu	289.322	289.5 ± 1.4	−0.1% *	—	1.0264	—	8452 ± 200
241Pu	370.816	362.1 ± 5.1	+2.4% *	0.9995 †	1.038	242.5	162 ± 8
242Pu	18.545	18.5 ± 0.5	+0.2% *	—	1.0096	991.3	1115 ± 40
237Np	175.917	175.9 ± 2.9	+0.01% *	1.0026 †	0.982	595.2	652 ± 24
241Am	699.775	587 ± 12	+19.2%	1.0085 †	1.051	1481.1	1425 ± 100
243Am	86.931	75.1 ± 1.8	+15.8%	1.0090 †	1.014	2003.1	1820 ± 70
244Cm	15.284	15.2 ± 1.2	+0.6% *	—	0.999	789.6	655 ± 30
246Cm	1.162	not tabulated	—	—	—	192.8	not tabulated

Nine of the twelve comparable values agree within 3%, and five agree within 1%—a strong validation of the reduction pipeline. Three deviations deserve comment. ²⁴¹Am (+19%): the IAEA value of 587 b is old; a modern OECD-NEA-referenced re-evaluation yields 747.8 b [54], between which the present 699.8 b lies, and Section 3.1.4 shows the discrepancy is not statistically significant once inter-experiment scatter is accounted for. ²⁴³Am (+16%): a normalization discrepancy issue between modern time-of-flight and older activation campaigns; the discrepancy survives dataset-level resampling. ²³³U (+8%): only seven thermal points exist.

3.1.4. Robustness and Uncertainty Quantification

Repeating the thermal fits with the per-dataset normalization parameters activated shows them to be robust to the systematic-error model: every σ₀ moves by less than 2.1% and every resonance energy by less than 0.1%. Two complementary uncertainties are computed for every derived quantity—the delta method using the full parameter covariance, and dataset resampling (Section 2.4; Table 6 and Figure 3; Supplementary Table S5). The gap between the two is itself a result: where the resampling interval is far wider, uncertainty is dominated by disagreement between laboratories rather than by counting statistics.

Table 6. σ₀ with delta-method uncertainty and dataset-resampling 95% confidence intervals, against IAEA A-5 [51]: “pow” is the empirical power law, “br” the bound-level + first-resonance model; “n.d.” flags fits unidentifiable over the available energy span and “deg.” flags resampling intervals with fewer than three independent datasets.

Nuclide	Datasets	σ0 Pow (b)	Delta±	Resampling 95% CI (pow)	σ0 br (b)	Delta±	Resampling 95% CI (br)	IAEA A-5 (b)	Consistent?
232Th	13	7.344	0.076	7.248–7.421	7.316	0.073	7.258–7.404	7.35 ± 0.03	yes
233U	3	49.49	1.76	40.91–51.70	49.51	1.76	48.36–200.7	45.5 ± 0.7	yes (wide)
235U	4	99.37	2.4 × 107	n.d.	—	—	—	98.8 ± 0.8	n.d.
238U	10	2.404	4.6 × 104	n.d.	—	—	—	2.683 ± 0.012	n.d.
239Pu	2	276.7	2.97	deg.	270.3	3.15	deg.	269.3 ± 2.9	—
241Pu	2	366.1	2.89	deg.	352.9	3.06	deg.	362.1 ± 5.1	—
242Pu	6	18.44	0.55	n.d.	—	—	—	18.5 ± 0.5	—
237Np	16	181.50	0.088	145.7–183.5	178.09	0.089	176.5–178.9	175.9 ± 2.9	yes
241Am	9	686.7	1.11	212.1–696.1	701.2	1.14	526.2–702.4	587 ± 12	yes
243Am	8	88.17	0.171	85.19–88.97	87.90	0.169	84.26–88.60	75.1 ± 1.8	no

Figure 3. Uncertainty of the derived thermal quantities: (a) σ₀ normalized to the IAEA A-5 reference [51] with the long teal bar showing the dataset-level resampling 95% interval and the short rust bar the delta-method interval; (b) partial-resonance integrals normalized to their own point estimate, restricted to nuclides with at least three independent datasets.

Four findings follow. (1) The ²⁴¹Am discrepancy is not significant: the resampling interval 526–702 b contains 587 ± 12 b, whereas the delta-method uncertainty is a factor of eighty narrower. (2) The ²⁴³Am discrepancy is real: the interval 84.3–88.6 b excludes 75.1 ± 1.8 b by a wide margin. (3) The ²³⁵U and ²³⁸U thermal fits are unidentifiable over the narrow available energy span (delta-method uncertainties of order 10⁴–10⁷ b). (4) For ²³⁹Pu and ²⁴¹Pu, the resampling degenerates to only two datasets; the delta-method uncertainties should be read as a lower bound.

3.2. Regime 2—The Resonance Region

3.2.1. Group Averaging and the RRR/URR Split

Between 0.5 eV and 100 keV, the cross section contains many resonances, so a smooth few-parameter formula cannot describe individual measurements. To summarize the energy dependence on a coarser scale, the data were binned into equal-lethargy groups, meaning intervals of equal width in ln E, at 25 bins per decade and averaged with a 1/E weight:

$$⟨\sigma {⟩}_g={\displaystyle \frac{{\int }_{E_g}^{E_{g+1}}\sigma (E)E^{-1}\hspace{0.17em}\mathit{dE}}{{\int }_{E_g}^{E_{g+1}}{E}^{-1}\hspace{0.17em}\mathit{dE}}}.$$

(4)

In Equation (4), E_g and E_g+1 are the lower and upper boundaries of group g. The denominator normalizes the 1/E weight, so the result is an average cross section rather than an integral. Partial resonance integrals then follow as RI = Σ_g ⟨σ⟩_g · ln(E_g+1/E_g). The resonance regime is split into Regime 2a (RRR, 0.5 eV–1 keV, empirical description only) and Regime 2b (URR, 1–100 keV, both descriptions applicable); the boundary follows the actinide convention of the ENDF-6 formats [23].

3.2.2. Regime 2a—Resolved Resonance Region

Group averages were fitted with ⟨σ⟩_g = A·E^−b (Table 7).

Table 7. Empirical power-law fits to 1/E-weighted group averages in the resolved-resonance region (0.5 eV—1 keV).

Nuclide	Groups	Points	brrr	R2	RIpartial (b)
232Th	24	35	0.666 ± 0.023	0.937	1.8
233U	83	4148	0.245 ± 0.004	0.204	122.1
235U	76	2680	0.166 ± 0.006	−0.155	146.9
238U	76	3252	0.565 ± 0.017	−6.530	83.5
239Pu	75	2805	0.108 ± 0.007	−0.769	278.6
241Pu	83	1883	0.143 ± 0.007	−0.136	242.5
242Pu	70	331	0.394 ± 0.022	−0.874	991.3
237Np	83	12,001	0.182 ± 0.002	−0.611	595.2
241Am	83	5552	0.127 ± 0.003	−0.267	1481.1
243Am	82	10,289	0.126 ± 0.004	−0.306	2003.1
244Cm	81	6437	0.156 ± 0.006	−0.007	789.6
246Cm	81	3950	−0.161 ± 0.004	−0.646	192.8

A negative R² for ten of twelve nuclides establishes an important diagnostic negative result: after 1/E averaging at 25 bins per decade, the RRR admits no power-law description, and RRR cross sections are better represented by tabulated group values than by a single smooth parameterization. The b_rrr values should be read only as a coarse slope indicator (as used in Section 3.4), never as a predictive model. The partial resonance integrals, with their group and point counts, are also listed in Supplementary Table S6.

3.2.3. Regime 2b—Unresolved Resonance Region

The empirical description again fits ⟨σ⟩_g = A·E^−b. The semi-empirical description is the statistical average over many overlapping resonances, retaining s- and p-waves, with the reduced wavelength in barns π ƛ² = (6.507 × 10⁵/E[eV]) · (A/(A + 1))², the s-wave strength function ⟨Γ_n⁰⟩/⟨D⟩ = S₀·√E, and a p-wave term. Reduced neutron widths follow a Porter–Thomas distribution, whereas radiation widths are nearly constant; Jensen’s inequality implies that the width-fluctuation average is less than or equal to the no-fluctuation expression, and the correction is applied throughout as the primary URR model:

$$F(r)={\mathbb{E}}_{x\sim {\chi }_1^2}\left[{\displaystyle \frac{\mathit{xr}}{\mathit{xr}+1}}\right], W(r)={\displaystyle \frac{F\left(r\right)}{r/(1+r)}}\le 1.$$

(5)

$$⟨{\sigma }_{\gamma }⟩=2{\pi }^2{\bar{\lambda }}^{\hspace{0.17em}2}{\sum }_{\mathcal{l}=0,1}(2\mathcal{l}+1)R_{\gamma }F(R_{\mathcal{l}}/R_{\gamma }).$$

(6)

In Equation (6), R_ℓ = ⟨Γ_n^ℓ⟩/⟨D⟩ and R_γ = ⟨Γ_γ⟩/⟨D⟩ are average-width-to-level-spacing ratios, and ℓ = 0, 1 labels the retained s- and p-wave contributions. In Equation (5), x is the fluctuating reduced neutron width normalized to its mean, F averages the capture factor over that fluctuation, and W compares the result with the no-fluctuation expression. These equations provide an average description of the unresolved region rather than a reconstruction of each individual resonance. Equation (6) contains no fission width: compound-nucleus decay is shared between the neutron and γ channels only. For the fissile targets ²³³U, ²³⁵U, ²³⁹Pu, and ²⁴¹Pu, the ENDF/B-VIII.0 unresolved-resonance parameters give s-wave ⟨Γf⟩/⟨Γγ⟩ ratios of about 6–25, so fission competes strongly with capture; for ²³⁷Np, ²⁴¹Am, and ²⁴³Am, the ratio is below 0.01, and it is zero for ²³²Th and ²³⁸U. To quantify the effect, the URR fits of the four fissile nuclides were repeated with a fission channel for each spin J, using the ENDF/B-VIII.0 ⟨Γf⟩/⟨Γγ⟩ ratios and numbers of fission degrees of freedom, with Porter–Thomas fluctuations of both the neutron and the fission widths; S₀, S₁, and Rγ remain the only fitted parameters (Supplementary Table S12). Relative to the fits without fission, the fitted Rγ increases only by factors of 1.1–1.9 and remains far below the RIPL-3 values, whereas S₀ rises to 1.4–3.1 times the RIPL-3 values because a fission-to-radiation width ratio held fixed is largely compensated by a rescaling of the neutron strength functions. Because fission competition cannot be neglected for these targets, the fission-inclusive fits are adopted as the primary statistical-model results for ²³³U, ²³⁵U, ²³⁹Pu, and ²⁴¹Pu in Table 8; the fits without fission are retained only in Table 9, where they isolate the effect of the width-fluctuation correction. With fission included, the AIC preference changes only for ²³³U, for which the empirical form becomes preferred (ΔAIC = 4.6); for ²³⁵U, the empirical form remains preferred by a reduced margin (ΔAIC = 0.7), and for ²³⁹Pu and ²⁴¹Pu, the statistical model remains preferred (ΔAIC = 10.9 and 5.6, respectively).

Table 8. URR (1–100 keV) fits: empirical power law versus width-fluctuation-corrected statistical model; asterisks mark the AIC-preferred model. † the fissile targets ²³³U, ²³⁵U, ²³⁹Pu, and ²⁴¹Pu; the statistical model includes fission competition, with ⟨Γf⟩/⟨Γγ⟩ and the number of fission degrees of freedom fixed at the ENDF/B-VIII.0 values (Section 3.2.3; Supplementary Table S12); for the remaining nuclides, Equation (6) is used.

Nuclide	grps	bemp	R2emp	AIC (emp)	S0 (10−4)	S1 (10−4)	Rγ (10−3)	R2semi	AIC (semi)	Preferred
232Th	49	0.416 ± 0.011	0.957	78.9	0.261 ± 0.017	0.76	3.1	0.965	66.2 *	semi
233U †	10	0.318 ± 0.030	0.920	22.9 *	1.231 ± 0.241	20.00	3.2	0.908	27.5	emp.
235U †	33	0.433 ± 0.019	0.958	24.8 *	1.720 ± 0.220	11.61	4.7	0.962	25.4	emp.
238U	50	0.504 ± 0.004	0.964	786.1	0.296 ± 0.019	1.30	2.0	0.969	277.2 *	semi
239Pu †	43	0.740 ± 0.012	0.974	75.2	3.700 ± 1.169	1.23	5.0	0.981	64.3 *	semi
240Pu	12	0.537 ± 0.087	0.845	10.5 *	0.005 (at bound)	5.04	2.4	0.878	11.5	emp.
241Pu †	37	0.530 ± 0.017	0.937	71.5	3.416 ± 1.380	6.91	3.5	0.946	65.9 *	semi
242Pu	39	0.423 ± 0.010	0.652	534.1	0.035 ± 0.022	3.69	2.6	0.805	257.1 *	semi
237Np	50	0.470 ± 0.002	0.986	720.2 *	0.926 ± 0.015	0.14	6.7	0.991	1185.0	emp.
241Am	50	0.452 ± 0.001	0.981	1298.0	1.537 ± 0.015	2.75	18.8	0.996	451.5 *	semi
243Am	48	0.386 ± 0.004	0.909	480.1	1.086 ± 0.023	2.14	20.4	0.973	215.9 *	semi
244Cm	25	0.924 ± 0.032	0.665	313.1 *	30.0 (at bound)	0.64	3.0	0.668	313.2	emp.
246Cm	—	no URR data

Table 9. Effect of the width-fluctuation correction (WFC) on the fitted s-wave strength function and comparison with RIPL-3 [18]; asterisks mark the better AIC and strength-function agreement. For ²³³U, ²³⁵U, ²³⁹Pu, and ²⁴¹Pu, both fits omit fission and serve only to isolate the effect of the WFC; the primary, fission-inclusive values for these targets are given in Table 8.

Nuclide	S0 No WFC	S0 with WFC	S0 RIPL-3	Ratio No WFC	Ratio WFC	Rγ WFC	Rγ RIPL-3	⟨W⟩s	AIC No WFC	AIC WFC
232Th	0.229	0.261 ± 0.017	0.84 ± 0.07	0.27	0.31	3.1 × 10−3	1.46 × 10−3	0.708	65.2 *	66.2
233U	0.016	0.005 (at bound)	0.90 ± 0.05	0.02	0.01	1.7 × 10−3	7.69 × 10−2	0.965	15.8	14.9 *
235U	0.394	0.467 ± 0.056	0.98 ± 0.07	0.40	0.48	4.0 × 10−3	8.44 × 10−2	0.701	28.0	26.2 *
238U	0.263	0.296 ± 0.019	1.03 ± 0.08	0.26	0.29	2.0 × 10−3	1.16 × 10−3	0.695	198.0 *	277.2
239Pu	30.0 (at bound)	1.395 ± 0.283 *	1.20 ± 0.10	25.0	1.16 *	4.6 × 10−3	1.95 × 10−2	0.699	120.9	65.5 *
240Pu	0.005 (at bound)	0.005 (at bound)	1.07 ± 0.10	0.005	0.005	2.4 × 10−3	2.39 × 10−3	0.953	11.0 *	11.5
241Pu	0.727	0.784 ± 0.256	1.18 ± 0.09	0.62	0.66	2.7 × 10−3	4.39 × 10−2	0.689	65.8	65.7 *
242Pu	0.079	0.035 ± 0.022	0.98 ± 0.08	0.08	0.04	2.6 × 10−3	1.70 × 10−3	0.832	236.5 *	257.1
237Np	0.442	0.926 ± 0.015 *	0.97 ± 0.07	0.46	0.96 *	6.7 × 10−3	7.16 × 10−2	0.695	666.3 *	1185.0
241Am	1.329	1.537 ± 0.015	0.88 ± 0.06	1.51	1.75	1.88 × 10−2	7.93 × 10−2	0.710	493.7	451.5 *
243Am	0.987	1.086 ± 0.023	0.98 ± 0.06	1.01	1.11 *	2.04 × 10−2	5.34 × 10−2	0.731	185.9 *	215.9
244Cm	30.0 (at bound)	30.0 (at bound)	1.00 ± 0.20	30.0	30.0	3.0 × 10−3	3.05 × 10−3	0.891	311.0 *	313.2

The expectation in Equation (5) was evaluated with an 80-node Gauss–Hermite quadrature, which converged to better than one part in 10⁵ (Supplementary Text S1). No free parameters are added; S₀, S₁, and R_γ remain the only fitted quantities. Averaged over 1–100 keV, the correction factor W ranges from 0.689 (²⁴¹Pu) to 0.965 (²³³U).

The statistical model is preferred for 7 of 12 nuclides on both AIC and BIC (Table 8 and Figure 4; Supplementary Tables S3 and S12) despite having one additional parameter. Because the semi-empirical parameters are bounded whereas A and b are not, the effective complexity of the statistical model is lower than its nominal parameter count; therefore, the criteria, if anything, over-penalize the physics model, so this bias works against the conclusion drawn rather than in its favor.

Figure 4. Regime 2 for the eight best-covered nuclides: 1/E-weighted group averages at 25 bins per decade, gray squares in the RRR and blue circles in the URR, with the empirical power law (solid) and statistical model (dashed) fitted to URR groups only.

3.2.4. Physical Validity of the Fitted Parameters and the RIPL-3 Benchmark

Reference S₀, ⟨D₀⟩ and ⟨Γ_γ⟩ are taken from the RIPL-3 recommended average s-wave resonance parameter file [18]. To isolate the effect of the correction, the URR data were also refitted with W ≡ 1 on the same group averages (Table 9 and Figure 5; Supplementary Table S4).

Figure 5. The Porter–Thomas width-fluctuation correction: (a) W(r) with its minimum of 0.680 at r = 1.64; (b) fitted s-wave strength function divided by the RIPL-3 value, without and with the correction; shaded band is the mean RIPL-3 uncertainty.

Four conclusions follow. (1) The correction removes the largest parameter-bound inconsistency: for ²³⁹Pu the uncorrected fit reaches the imposed upper bound for S₀ while the corrected fit without fission gives a finite S₀ = (1.40 ± 0.28) × 10⁻⁴; with fission competition included (Table 8), S₀ = (3.70 ± 1.17) × 10⁻⁴, about three times the RIPL-3 value of (1.20 ± 0.10) × 10⁻⁴, so the correction removes the bound problem but does not by itself bring ²³⁹Pu into agreement with RIPL-3. (2) For ²³⁷Np, it moves S₀ from a factor-of-two deficit to 5% agreement while worsening the in-sample fit, a signature of model misspecification, most likely a missing d-wave term or spin-statistical weight. (3) The correction does not resolve the discrepancies for ²³²Th or ²³⁸U: S₀ rises by only 13–14%, which is insufficient to close a factor-of-four gap. (4) A systematic even–odd signature appears in R_γ: for the five even–even targets, the fitted-to-reference R_γ ratios range from approximately 0.98 to 2.12, whereas the odd-A targets give ratios of 0.042–0.382 with the fission-inclusive fits of Table 8 (0.022–0.382 without fission). This pattern is consistent with the single-J approximation being an important remaining model limitation; it does not establish that approximation as the unique cause of the discrepancy. Including fission competition for the fissile targets leaves this pattern intact (Section 3.2.3; Table 8): their Rγ ratios are 0.042–0.258, still far below the even–even values, so the even–odd signature is not an artifact of the omitted fission width.

3.2.5. Dataset-Level Hold-Out Validation

Information criteria reward in-sample fit; a model can be preferred by AIC yet generalize poorly when systematic errors are correlated within a campaign. Entire EXFOR datasets (about 20% per fold, up to 12 folds) were therefore withheld from URR fits, and the root-mean-square logarithmic error (RMSLE) was evaluated on the held-out data (Table 10; Supplementary Table S7).

Table 10. Dataset-level hold-out validation in the URR (1–100 keV): RMSLE on held-out data for empirical and semi-empirical models across up to 12 folds per nuclide.

Nuclide	Datasets	Withheld per Fold	Folds	RMSLE Empirical	RMSLE Semi-Empirical	Winner
232Th	22	4	9	0.296	0.263	semi-empirical
238U	46	9	12	0.322	0.390	empirical
237Np	7	1	10	0.710	0.728	empirical
243Am	14	3	12	0.152	0.097	semi-empirical
239Pu	6	1	2	0.399	0.396	semi-empirical

With only five nuclides, the 3-of-5 split carries no inferential weight. The exercise does, however, establish one qualitative negative result: the ranking produced by the information criteria is not reproduced out of sample. ²³⁸U is the decisive case: the semi-empirical model is strongly preferred by AIC and BIC, yet its hold-out RMSLE is 21% worse than that of the empirical model, which is the signature of a model whose parameters have absorbed dataset-specific normalization structure.

3.2.6. ²³²Th and ²³⁸U Strength Functions and Dataset Normalizations

The s-wave strength functions fitted for ²³²Th and ²³⁸U lie a factor of three to four below the RIPL-3 recommended values, whereas for ²³⁷Np the corrected fit agrees with RIPL-3 to within about 5%; for ²³⁹Pu the 16% agreement obtained without fission does not survive the inclusion of fission competition, which places S₀ about three times above RIPL-3 (Table 8). The URR group fits do not include dataset normalizations, but a factor of three to four far exceeds credible normalization errors of 5–15% [50], so the residual gap reflects a genuine limitation of the s + p statistical kernel. This gap therefore indicates where a full Hauser–Feshbach treatment with optical-model transmission coefficients and explicit fission competition, with fitted rather than fixed fission widths, becomes indispensable.

3.3. Regime 3—The Fast Region (0.1–1.0 MeV)

3.3.1. Scope, Screening and Empirical Descriptions

The fast region is restricted to 0.1–1.0 MeV, which for most of these nuclides lies below the fast-fission threshold. Only five nuclides have usable data: ²³²Th, ²³⁸U, ²³⁷Np, ²³⁹Pu and ²⁴¹Am. Outlier screening within quarter-decade lethargy windows rejected three points in total. Two empirical forms were fitted: a log-quadratic log₁₀ σ_γ = a + b·log₁₀ E + c·(log₁₀ E)² and an exponential σ_γ = A·E^−α·exp(−β·E).

Even the empirical fits have limited descriptive accuracy (R² of 0.31–0.72) because the fast-region record is dominated by scatter between campaigns, not by structure in the cross section. A variance decomposition on the residual ln σ_γ, splitting the residual into between-dataset and within-dataset components with intraclass correlation ρ_I = σ_B²/(σ_B² + σ_W²), gives ρ_I ≈ 0.5: roughly half the unexplained variance is disagreement between campaigns. No smooth function of energy can materially exceed the observed R² on pooled data without fitting the disagreement itself.

3.3.2. Failure of the Closed-Form Hauser–Feshbach Model

A closed-form statistical calculation was attempted with the compound-nucleus formation cross section evaluated in the black-nucleus approximation σ_CN = π(R + ƛ)², a back-shifted Fermi-gas level density, a standard Lorentzian E1 γ-strength function (E_G = 14 MeV, Γ_G = 5 MeV), and σ_γ = σ_CN · T_γ/(T_γ + C·T_n) with the level-density parameter a and neutron-competition scale C free. The scope of this test should be stated precisely: it concerns a closed-form model with a black-nucleus formation cross section, not Hauser–Feshbach theory as implemented in modern reaction codes.

For ²³²Th, ²³⁸U, and ²³⁷Np the R² is negative (−1.14, −1.04, −0.41), and for all five nuclides the level-density parameter reaches its imposed lower bound with a/A ≈ 0.034 against the established actinide value near 0.11 MeV⁻¹ [19]. The closed-form model used here omits optical-model transmission coefficients, fission competition, direct/semi-direct capture, and the width-fluctuation correction, and it does not describe the measured capture of ²³²Th, ²³⁸U, and ²³⁷Np over 0.1–1.0 MeV; this result applies to that model and should not be read as a limitation of statistical models in general. To identify the missing physics, the same screened data were compared with a complete Hauser–Feshbach calculation, TALYS-2.24 [16] with its default global parameters and no adjustment (Figure 6; Supplementary Table S13). Without any fitted parameters, TALYS gives R² = −0.25 for ²³²Th (closed-form model −1.14) and reproduces the energy dependence of ²³²Th, ²³⁸U, and ²³⁹Pu: after a single overall normalization, its R² is 0.32, 0.32, and 0.57, comparable to the empirical fits. Its absolute magnitude is low by factors of 1.19 (²³²Th), 1.64 (²³⁸U), and 2.08 (²³⁹Pu), a known consequence of unadjusted global γ-strength and level-density parameters. The wrong slope of the closed-form model therefore arises mainly from its omitted physics, whereas the magnitude requires the parameter adjustment that is routine in evaluations. Neither calculation reproduces the ²³⁷Np data (TALYS R² = −0.05), whose structure near 0.15–0.25 MeV is not captured by either model.

Figure 6. Regime 3: fast-region (0.1–1.0 MeV) fits for the five usable nuclides showing the two empirical parameterizations and the closed-form Hauser–Feshbach curve, which has the wrong slope for ²³²Th, ²³⁸U, and ²³⁷Np (negative R²), together with a complete TALYS-2.24 calculation with default parameters (dotted; no fitted parameters). Closed-form curves are drawn with the fitted parameters of Table 11.

Figure 6. Regime 3: fast-region (0.1–1.0 MeV) fits for the five usable nuclides showing the two empirical parameterizations and the closed-form Hauser–Feshbach curve, which has the wrong slope for ²³²Th, ²³⁸U, and ²³⁷Np (negative R²), together with a complete TALYS-2.24 calculation with default parameters (dotted; no fitted parameters). Closed-form curves are drawn with the fitted parameters of Table 11.

Table 11. Fast-region (0.1–1.0 MeV) fits: log-quadratic and exponential empirical forms versus the closed-form Hauser–Feshbach model (HF) of Section 3.3.2.

Nuclide	Points	Sets	R2 log-quad	AIC log-quad	R2 Exp	AIC Exp	R2 HF	AIC HF	a/A Fitted (MeV−1)
232Th	190	24	0.309	1111	0.279	1146	−1.143	4436	0.034
238U	265	34	0.426	1369	0.420	1439	−1.041	6192	0.034
237Np	333	11	0.716	1438	0.617	1811	−0.409	4822	0.057
239Pu	31	4	0.514	258.7	0.515	258.9	0.506	255.4	0.034
241Am	91	1	0.394	44.0	0.393	44.1	0.361	43.6	0.033

Table 11. Fast-region (0.1–1.0 MeV) fits: log-quadratic and exponential empirical forms versus the closed-form Hauser–Feshbach model (HF) of Section 3.3.2.

Nuclide	Points	Sets	R2 log-quad	AIC log-quad	R2 Exp	AIC Exp	R2 HF	AIC HF	a/A Fitted (MeV−1)
232Th	190	24	0.309	1111	0.279	1146	−1.143	4436	0.034
238U	265	34	0.426	1369	0.420	1439	−1.041	6192	0.034
237Np	333	11	0.716	1438	0.617	1811	−0.409	4822	0.057
239Pu	31	4	0.514	258.7	0.515	258.9	0.506	255.4	0.034
241Am	91	1	0.394	44.0	0.393	44.1	0.361	43.6	0.033

3.3.3. Normalization Treatment and Its Diagnostic Value

Fitted per-dataset normalizations in the fast region spread over a factor of two (18/24 ²³²Th and 24/34 ²³⁸U datasets with fitted scale shifts exceeding 5%, many of them reaching 0.5–0.6; Figure 7), far outside any credible normalization uncertainty for capture measurements [50]; the aggregate fast-region normalization diagnostics are given in Supplementary Table S9. The nuisance parameters are therefore absorbing model error rather than experimental systematics, and the conclusion that the closed-form model fails is strengthened, not weakened, by allowing normalization freedom (χ²ν remains at 98 and 144 for ²³²Th and ²³⁸U).

Figure 7. Fitted per-dataset normalization factors exp(n_d) for (a) the thermal region (Regime 1), (b) the URR group averages (Regime 2b) and (c) the fast region (Regime 3); each point is one EXFOR dataset, the solid line marks unity and the gray band marks the ±5% Gaussian prior.

3.4. Systematics Across the Actinide Chain

Fitted parameters were correlated with mass number A, neutron excess (N − Z)/A, compound-nucleus separation energy S_n [55], and target odd/even character, drawing on the thermal cross sections of Section 3.1, the RRR slopes of Section 3.2.2 and the URR strength functions of Section 3.2.3 (Figure 8). The nominally significant relationships are as follows:

Figure 8. The Three Principal Candidate Systematics: (a) thermal σ₀ versus compound-nucleus neutron separation energy, coded by even–even, odd-Z, and odd-N characters; (b) fitted URR s-wave strength function versus A; (c) RRR slope b_rrr versus A with odd-A targets highlighted. Dashed lines are least-squares trends that do not survive multiple-comparison correction.

• log₁₀ σ₀ = −2.009 + (0.658 ± 0.320)·S_n, r = 0.526, p = 0.065, n = 13;
• σ₀ versus odd/even character: Spearman ρ = 0.660, p = 0.014;
• b_rrr versus A: r = −0.610, p = 0.035; versus (N − Z)/A: r = 0.579, p = 0.048;
• RI_partial versus A: ρ = 0.648, p = 0.023.

Multiple-Comparison Caveat: Thirty-six tests were performed, for which ≈1.8 false positives are expected at α = 0.05; none survive the Bonferroni correction (α = 0.0014). This section is therefore a screening exercise producing four candidate relationships, each with a physical rationale and each requiring a larger independent sample. No other result in this paper depends on it.

Neutron–Proton Asymmetry and the Rγ Benchmark

Figure 9 combines the target neutron–proton asymmetry I = (N − Z)/A with a comparison of the fitted and reference R_γ values in Table 9. Here A = N + Z, and the even/odd classes refer to the target nucleus, not the compound nucleus. All thirteen targets have I > 0; the figure therefore compares degrees of neutron excess rather than dividing the sample into N = Z and N ≠ Z nuclei. The even/odd classification describes the proton and neutron counts and is not a classification of spatial symmetry.

Figure 9. Neutron–proton asymmetry and the R_γ benchmark comparison. (a) I = (N − Z)/A for the thirteen target nuclides, with color and symbol distinguishing even Z/even N, even Z/odd N and odd Z/even N. (b) R_γ from the width-fluctuation-corrected fit divided by the RIPL-3 reference value, plotted against target I on a logarithmic vertical scale. Ratios use the rounded entries in Table 9; the dashed line marks unity. ²⁴⁶Cm is omitted from panel (b) because no paired entry is tabulated. No uncertainty bars are shown because Table 9 does not report R_γ uncertainties. In panel (b), the dagger (†) marks fits with S₀ at an imposed parameter bound. The comparison is descriptive and does not establish symmetry breaking.

Among the twelve targets with paired R_γ entries in Table 9, the five even–even targets have fitted-to-reference ratios of approximately 0.98–2.12; the seven odd-A targets have ratios of 0.022–0.382. With the fission-inclusive fits of Table 8, the four fissile targets give 0.042–0.258, so the odd-A range becomes 0.042–0.382, and the separation is unchanged. The plot shows this separation by target class across overlapping ranges of I. This is a descriptive, composition-resolved model diagnostic, not a demonstration that neutron excess causes the discrepancy or that R_γ follows a universal monotonic dependence on I. The pattern is consistent with the model limitations discussed in Section 3.2.4, but does not uniquely identify their origin. No new significance test is claimed; R_γ uncertainties are not tabulated in Table 9, and three displayed fits have S₀ at an imposed bound.

3.5. Fit Quality Across the Three Regimes

The following comparison summarizes fit quality across the three regimes, as discussed in Section 3.1, Section 3.2 and Section 3.3 and illustrated in Figure 10. In the thermal region, adding the bound level and first positive-energy resonance reduces RMSE by factors of 5 to 53 for ²³⁹Pu, ²⁴¹Pu, ²³⁷Np, and ²⁴¹Am and turns a negative R² into 0.885–0.999. ²⁴³Am is the only unsatisfactory case because its thermal record is dominated by mutually inconsistent normalizations.

Figure 10. Fit Quality Across the Three Regimes: (a) R² on ln σ_γ in the thermal region for empirical (gray) and semi-empirical (teal) models; (b) R² in the URR for empirical and width-fluctuation-corrected statistical models; (c) distribution of relative RMSE for the preferred model in each regime on a logarithmic scale.

The resonance region contains the central negative result of this paper. In the RRR, the relative RMSE is 107–677% for eleven of twelve nuclides, and R² is negative for ten (Table 7). The URR behaves quite differently, with relative RMSE of 6–41% and R² > 0.90 for ten of twelve, and the width-fluctuation correction lowers RMSE for eight nuclides.

In the fast region, the two empirical forms are practically indistinguishable, whereas the closed-form Hauser–Feshbach model is worse by 25–102% for ²³²Th, ²³⁸U, and ²³⁷Np with negative R² for all three. Relative RMSE of 7–41% also makes the fast region the least well constrained of the three.

4. Discussion

4.1. Where Semi-Empirical Models Improve the Description

What works is the thermal region. The semi-empirical model is decisively superior wherever a low-lying resonance exists, recovering its energy to more than 3% from cross-section data alone; where none exists, it correctly collapses to the bound-level term, and the simpler power law is preferred.

What partly works is the URR. The semi-empirical model is preferred by AIC for 7 of 12 nuclides but by dataset-level hold-out validation for only 3 of 5. The difference between these two selection rates is itself informative: it indicates how much of the AIC advantage reflects physics and how much reflects the absorption of dataset systematics.

What does not work is the RRR and the fast region. The systematically negative R² of Section 3.2.2 provides a diagnostic result that is easy to overlook: RRR cross sections are better represented by tabulated group values than by a single smooth parameterization. In the fast region, the closed-form model does not describe the data even when 24–34 free normalizations are allowed, showing that this specific closed-form black-nucleus approximation is inadequate for actinides at these energies; a default TALYS calculation reproduces the energy dependence for ²³²Th, ²³⁸U, and ²³⁹Pu after a single normalization (Supplementary Table S13), so the inadequacy lies in the approximation rather than in statistical-model theory. Independently, roughly half the unexplained fast-region variance is inter-campaign disagreement.

4.2. ²⁴⁵Cm as a Measurement Priority, and the Leading URR Defect

²⁴⁵Cm was excluded because only two thermal EXFOR points exist. This is a serious gap: ²⁴⁵Cm has a high thermal fission cross section, dominates buildup of higher curium isotopes, and its capture-to-fission ratio directly affects transmutation-efficiency estimates for minor-actinide burning [28,29]. A modern time-of-flight capture measurement covering the resonance region would be of substantial value.

The even–odd signature in R_γ (Section 3.2.4) points to the collapse of two J channels into one effective ⟨D⟩ as the principal remaining model deficiency, ahead of the missing d-wave term and the omitted elastic-enhancement correction. Because RIPL-3 tabulates ⟨D₀⟩ for all thirteen nuclides, R_γ can be constrained, breaking the S₀–R_γ degeneracy and making S₀ individually meaningful.

4.3. What the Uncertainty Analysis Implies for the Corpus

The factor-of-eighty gap between delta-method and dataset-resampling intervals is more than a technical detail: it shows that for actinide capture the uncertainty of the pooled EXFOR record is dominated by systematic disagreement between independent measurement campaigns rather than by the counting statistics of any individual measurement. Two operational consequences follow: derived quantities should carry a dataset-level resampling interval, and nuclides resting on fewer than three independent datasets (²³⁹Pu and ²⁴¹Pu thermally; ²⁴¹Pu and ²⁴²Pu for RI_partial) should be flagged as having no meaningful empirical uncertainty. Every table entry reports its contributing dataset count.

4.4. Limitations

Several limitations should be noted. Experimental provenance is not explicitly weighted: all datasets share the same normalization prior, so modern time-of-flight campaigns are not favored over older activation measurements. Within the model, the single-ℓ approximation biases the URR by up to a factor of 4–5 and is the leading known model defect, while the URR d-wave term remains approximate and fission competition is included for the fissile targets (Table 8) only with ⟨Γf⟩/⟨Γγ⟩ fixed at the ENDF/B-VIII.0 values, so their S₀ and S₁ remain correlated with that assumed ratio, and the Moldauer elastic-enhancement term is omitted. Normalization parameters can also absorb model error: fitted values of 0.5–0.6 in the fast region are not credible as experimental normalizations. Uncertainty quantification is uneven: dataset resampling requires at least three independent datasets, so for ²³⁹Pu, ²⁴¹Pu, ²⁴²Pu, and several RRR cases, only the delta method of uncertainty is available. Finally, thirteen nuclides cannot establish systematics surviving multiple-comparison correction.

The source-export comparison in Supplementary Table S10 shows substantial regime dependence: the reported median is 35.22% in the RRR but approximately 5–6% in the thermal, URR, and fitted fast-energy intervals. The 10% surrogate is therefore not a regime-specific estimate or a conservative bound for all observations. Missing uncertainties may have a different distribution from reported uncertainties, and densely sampled datasets contribute more strongly to point-weighted summaries. To test the effect of the 10% choice, every fit that uses point uncertainties—the thermal fits (Table 3, Table 4, Table 5 and Table 6) and the fast-region fits (Table 11)—was repeated with the surrogate set to 5%, 20%, and the regime-specific medians (5.54% thermal, 5.49% fast), with all other settings unchanged (Supplementary Table S11). The RRR and URR group fits, and the hold-out test, do not use point uncertainties: each group carries the standard error of its points, with a 15% fallback for single-point groups; varying that fallback between 7.5% and 30% leaves R² negative for ten of twelve RRR nuclides and changes the URR AIC count by, at most, one nuclide (²³⁵U). In the thermal region σ₀ changes by at most 0.3% for six of seven nuclides and by up to 6.6% for ²⁴¹Am, whose power-law fit is poor (χ²ν ≈ 28); first-resonance energies change by less than 1%, the AIC preference is unchanged for all seven nuclides, and every σ₀ remains inside its baseline resampling interval except the bound-plus-resonance ²³⁷Np value under the regime-median surrogate (176.2 b against 176.6–178.9 b). In the fast region, the closed-form Hauser–Feshbach model gives negative R² for ²³²Th, ²³⁸U, and ²³⁷Np under every assumption (−0.29 to −1.26), and the log-quadratic form remains preferred for these three nuclides; only the ²⁴¹Am preference, decided at baseline by ΔAIC = 0.3, changes. The conclusions are therefore insensitive to the surrogate, although missing-error scales were not estimated jointly with the fits. The normalization robustness check in Section 3.1.4 concerns a different component of the uncertainty model.

No point-by-point comparison with evaluated libraries was performed. Fits are benchmarked against compiled integral quantities and RIPL-3 parameters rather than pointwise against ENDF/B-VIII.0, JEFF-3.3 or JENDL-5. The omission is deliberate: below 1 keV an evaluated file is a reconstruction from resolved-resonance parameters, so the comparison would measure smooth curve against resonance structure; over 1–100 keV the evaluated curve is itself a statistical-model calculation of the same family. Such a comparison would be informative mainly in the fast region, where this study has already concluded that a full reaction-code calculation is required.

4.5. Methodological Extensions

A useful future extension is to estimate uncertainty scales for observations with unreported uncertainties jointly with the physical parameters, potentially within a hierarchical model that also represents dataset-wide normalization differences. These are separate components: estimating normalization scales alone does not determine the missing pointwise uncertainties. Such an extension is not implemented in the present revision; the common 10% surrogate is retained. A crossed variance-component analysis by epoch and technique (activation, total-absorption calorimetry, total-energy detection) could additionally attribute between-dataset variance to its sources.

Three further extensions target the model and the benchmarking. Re-integration of the evaluated libraries over 0.5 eV–1 keV would reconstruct pointwise cross sections onto a common grid, with MF33 covariances supplying the deferred weighted comparison. Explicit spin weighting with J-dependent level spacings, constrained by RIPL-3, would break the degeneracy between level density and strength function for the seven odd-A targets—the highest-value physics extension requiring no new data. A d-wave extension over a wider energy window is a natural next step. Beyond these, a full reaction-code calculation for the fast region using TALYS [16] or EMPIRE [17] is warranted. Finally, additional independent experimental campaigns are needed to strengthen the hold-out validation.

5. Conclusions

This analysis establishes a consistent three-regime empirical and semi-empirical parameterization of (n,γ) cross sections for thirteen actinides, based on 90,299 EXFOR data points and fitted solely to experimental data. As an external validation, the derived thermal cross sections agree with the IAEA reference compilation to within 3% for nine of the twelve nuclides with available reference values.

In the thermal region, the semi-empirical bound-level-plus-resonance model recovers the first-resonance energies for ²³⁹Pu, ²⁴¹Pu, ²³⁷Np, ²⁴¹Am, and ²⁴³Am to more than 3% from cross-section data alone, reducing χ²ν by factors of 2.4 to 450 relative to the power-law baseline. A closed-form Westcott factor g_W = (2/√π)·Γ(2 − b) follows analytically from the empirical power law.

The resolved resonance region admits no such power-law description: R² is negative for ten of twelve nuclides even after 1/E group averaging. In the URR, the width-fluctuation-corrected two-partial-wave statistical model, with fission competition included for the fissile targets, is preferred on AIC for seven of twelve nuclides but on dataset-level hold-out for only three of five; this gap indicates the extent to which information criteria can overstate the case for physics-based models when applied to heterogeneous experimental data. The Porter–Thomas correction frees the bound-limited ²³⁹Pu strength function (AIC 120.9 → 65.5) and brings ²³⁷Np to 5% agreement with RIPL-3, but it raises S₀ for ²³²Th and ²³⁸U by only 13–14% and cannot explain their factor-of-four deficit. A systematic even–odd pattern in the fitted R_γ, which persists when fission competition is included for the fissile targets, identifies the single-J approximation as the principal remaining model deficiency; with fission competition included, as adopted for the fissile targets in Table 8, their S₀ values lie at 1.4–3.1 times RIPL-3, so close agreement for ²³⁹Pu is obtained only when fission is omitted.

For the fast region, the closed-form black-nucleus model used here proves inadequate: the black-nucleus approximation distorts the level-density parameter by a factor of two to three, with χ²ν remaining at 98 and 144 for ²³²Th and ²³⁸U. A complete TALYS calculation with default parameters reproduces the energy dependence for ²³²Th, ²³⁸U, and ²³⁹Pu but underestimates the magnitude by factors of 1.2–2.1, so the failure is attributed to the approximations of the closed-form model, while absolute predictions still require adjusted parameters. Roughly half of the unexplained fast-region variance stems from inter-campaign disagreement.

The reported systematics across the actinide chain do not survive multiple-comparison correction and are therefore hypothesis-generating only. The central quantitative result emerges from the uncertainty comparison: dataset-level resampling intervals are up to eighty times wider than delta-method uncertainties, demonstrating that uncertainty in the actinide capture corpus is overwhelmingly dominated by systematic disagreement between independent measurement campaigns rather than by counting statistics. The main practical recommendation is therefore that no derived actinide capture quantity extracted from pooled EXFOR data should be published without an accompanying dataset-level resampling interval.