ENDF/B-VIII.1 neutron covariances in the SCALE 56-group structure: two processing routes, NJOY and AMPX, and their effect on criticality uncertainties

3.1 The group collapse

Both codes collapse the evaluated covariances on a working grid that contains every group boundary and every evaluator bin boundary (ERRORR's union grid, PUFF-IV's super grid), so that each working interval s lies in one group and one evaluator bin. With reaction rates R_x,s = ∫_s σ_x(E) φ(E) dE as weights, the relative covariance of the group cross sections of reactions x and y is

where C_xy(s, t) is the evaluated relative covariance of the bins containing s and t. Resonance-parameter covariances V_p are propagated as D_x V_p D_y^T, with D_x,Ik = ∂σ_x,I/∂p_k, and added to any MF33 contribution. The two codes differ in how they obtain the reaction rates on the working grid and the derivatives D.

3.2 NJOY route

NJOY2016.78 [4] was run through SANDY 1.2 [7] (RECONR, BROADR, GROUPR, ERRORR), with SANDY's input writers extended for the weighting, cross-material blocks and MF35 options. ERRORR assumes that cross section and flux are flat inside each group of the GENDF file it receives, so a 56-group GENDF would lose the shape of the reaction rates inside each group. Each nuclide was therefore processed twice. A probe run reads the union grid that ERRORR prints; GROUPR then produces a GENDF on exactly that grid, snapped to the ENDF energies, and the final ERRORR run uses it, with a check that the union grid did not change. For U-235 this reproduces ERRORR working from the point-wise file alone to within 10⁻³ percentage points. Resonance-parameter covariances were processed with irespr=1: ERRORR reports using its ERRORJ-derived method for the Breit–Wigner and Reich–Moore formats and a SAMMY-based method for the R-matrix-limited format (LRF=7). Cross-material blocks used iread=2. MF35 was processed on a 56-group GENDF, because the union-grid method does not apply to it.

3.3 AMPX route

AMPX was built from the SCALE-Public source [9] and run per nuclide through the chain of Table 2, which follows the methods of the AMPX-6 manual [6]. PUFF-IV integrates the point-wise cross sections and weighting function directly on its super grid and computes analytic resonance-parameter derivatives with routines from SAMMY [10]. Four adjustments to the chain were needed. TGEL had to run twice, because its totals leave out reactions it creates in the same run (for Be-7 a single run gave a total of 43 b instead of 1.4 × 10⁵ b). Its MT3 follows an older sum rule that omits absorption, so MT3 was rebuilt as total minus elastic. Y12 cannot read the new LAW=−5 multiplicity data of ENDF/B-VIII.1 (MF6 MT18), so it received a copy of each evaluation without that section, which affects only the fission-spectrum weights. Finally, X10 reads the group structure from a unit that the SCALE driver forces to big-endian byte order. All 266 nuclides completed, in 4.3 CPU-hours.

3.4 Changes to the evaluated data

Every change to an evaluation was recorded with its reason. Table 3 summarizes them; the NJOY route made 134 changes in 40 files.

3.5 Checks of the products

• COVERX format. The writer reproduces SCALE's own 56- and 252-group libraries byte for byte. A first version packed the band records by rows instead of columns, transposing every asymmetric block; a self-consistent round trip cannot detect this. It was found through the comparison with AMPX and confirmed on SCALE's library: read by columns, all 1836 of its asymmetric blocks have |ρ| ≤ 1; read by rows, 934 do not.
• Display. Every NJOY output file was loaded with the import and plotting code of JANIS 4.0 [11], 408 cases in all. All matrices display; the cross-material blocks need their partner materials in the same file. JANIS imported nothing from the W tapes that contained NaN, which is how that problem was found.
• Regression tests. A tiered test suite, from format unit tests up to full NJOY, AMPX and JANIS runs of selected nuclides compared with stored reference records, accompanies the code [14].
• Sandwich code. The sandwich results of Section 5 were repeated with the independent reader and sandwich function of SAUNA [12]: 0.6661, 0.7191 and 0.7553 % Δk/k against 0.6661, 0.7191 and 0.7554 % Δk/k.

4.1 Contents

The ten nuclides with significant |ρ| > 1 in the AMPX library are also among the nineteen in the NJOY library. They come from the evaluations: O-16, whose evaluated elastic covariance (1584 bins) contains 18 isolated elements of magnitude 7–112 among neighbours of order 10⁻⁸; six platinum isotopes (Pt-191 to 194, 196 and 198) and Ta-181, whose inelastic × capture blocks exceed the two reactions' own variances; Mn-55 (elastic × (n,2n)); and V-51, with blocks that are non-zero where the (n,α) variance is zero. The other nine NJOY cases are discussed in Section 4.3. All of these were left as processed.

4.2 Agreement

For the reactions that dominate thermal systems the two routes give the same uncertainties (Fig. 1): the largest difference in relative standard deviation over the significant groups of U-235 fission and capture, U-238 capture, Pu-239 fission, H-1 capture and O-16 elastic scattering is 0.0004 percentage points. For U-235 the correlations agree within 0.0005. The ν̄ group values of the two codes differ by up to 0.45 % (U-235, 1.13–5 eV), because they weight ν̄ slightly differently inside a group, but the relative covariances agree within 0.001 percentage points.

Across the whole library, Fig. 2a compares the 57 870 significant group standard deviations of reactions present in both libraries. 90.7 % agree within 1 % of their value and 94.3 % within 10 %. Per nuclide (Fig. 2b), the largest difference in standard deviation is at most 0.1 percentage points for 140 of the 266 nuclides and at most 1 percentage point for 162; 104 nuclides differ by more. Counting correlations as well (within 0.01 and 0.1), 153 of the 295 SCALE IDs agree, 10 differ slightly and 132 substantially. For 105 of the 130 nuclides that differ by at least 1 percentage point or 0.1 in correlation, the largest difference lies inside the resolved-resonance range of MF32, for 14 in the group that straddles its upper end, for 9 only in correlations between reactions, and 2 have no MF32 data.

4.3 Where the routes disagree

R-matrix resonance parameters. For Cl-35, Ca-40, Cu-63, Cu-65, Rh-103 and W-182, 183, 184 and 186, all evaluated with R-matrix-limited (LRF=7) resonance parameters, NJOY gives correlations between elastic scattering and capture far outside ±1. Each reaction's own block is positive semi-definite, but the joint matrix is not (Rh-103: eigenvalues from −24.6 to +24.6). For Rh-103, 301 of the 1512 significant group pairs of the elastic × capture block have |ρ| > 1, up to 1098; in the AMPX library none do (Fig. 3a, b). Repeating the NJOY run without MF32 removes these values, and AMPX processes the same evaluations without them, so they come from ERRORR's processing of LRF=7 parameters rather than from the evaluations. The same step wrote NaN for W-182 to 184 above the resolved range. Ca-40, whose MF32 contribution in the NJOY library even contains negative variances, is also the source of the very large standard deviations at thermal energies in Fig. 2a.

Isolated one-group values. ERRORR also produces single groups out of line with both their neighbours. Of 111 such one-group outliers between the libraries, the NJOY value is the odd one in 85, 67 of them in the narrow group from 6.875 to 7 eV, which is 1.8 % wide; the AMPX value is the odd one in 26. For Ti-46 elastic scattering, NJOY gives 0.40 % in that group, between neighbours of 17.35 % and 17.34 %, where AMPX gives 17.33 % (Fig. 3c). ERRORR working from the point-wise file alone gives the same values, so they are not caused by the union-grid GENDF.

Cross-reaction blocks of O-16. The NJOY library contains 45 O-16 matrices (including empty ones) and the AMPX library 14. In particular the AMPX library has no elastic × inelastic block, which NJOY derives from the evaluation. This block decides the small differences in Section 5; why PUFF-IV does not produce it was not investigated.

4.4 Fission spectrum

An MF35 matrix gives the covariances of the probabilities of emission into outgoing-energy bins. Collapsing it to groups needs the shape of χ inside each bin: a group lying wholly inside one bin must carry that bin's relative uncertainty. An independent exact collapse was written as a reference, using the tabulated MF5 spectrum at the thermal incident energy integrated with its own interpolation laws. Against it, the NJOY χ group values are exact, but ERRORR splits the bins with χ taken as flat inside each 56-group, so the standard deviations are wrong where χ changes shape inside a group. For U-235 (Fig. 4) the NJOY values above 3.74 keV are within 0.61 percentage points of the reference, and those of AMPX within 0.65. Below 3.74 keV, where the reference is 14.7–22.9 %, NJOY gives 0.04–57 % and AMPX 0.28–94 %; this part of the spectrum holds only about 10⁻⁴ of the fission neutrons. Across the 67 files with MF35, the NJOY standard deviations above 3.74 keV are off by up to 0.5–1.7 percentage points for most actinides. AMPX takes its weights from the X10 master library, whose χ is an average over all incident energies weighted by the fission source, 1–3 % different from the thermal χ in the fast groups. In the benchmarks of Section 5 the effect on k_eff is at most 0.0001 % Δk/k; for fast systems it may be larger.

A related property of the evaluations was found on the way. Because the bin probabilities sum to one, every row of an MF35 matrix should sum to zero. In the thermal blocks this holds within a few parts per million for 64 of the 67 files. For Pu-238 and Pu-239 it does not: their matrices imply uncertainties of 0.91 % and 0.23 % in a total probability that must be exactly one, and Pu-240 misses marginally. Such a matrix adds variance without physical basis whenever the χ sensitivities do not sum to zero. The matrices were processed as given.

5.1 Method and cases

For a response with relative sensitivities S_x,I to the group cross sections, the relative variance due to nuclear data is

where the cross terms (x ≠ y) can be negative. Each reaction pair's term is reported once, as the signed square root of 2 S_x C_xy S_y^T. The sensitivities come from SCALE 6.2 TSUNAMI-3D continuous-energy calculations [8], made with SCALE's ENDF/B-VII continuous-energy library (ce_v7), of three cases of the ICSBEP Handbook [13]: LEU-COMP-THERM-093 case 1 (LCT-093-1), HEU-SOL-THERM-046 case 1 (HST-046-1) and PU-SOL-THERM-039 case 1 (PST-039-1). They were computed in SCALE's 252 groups, which nest the 56 groups, and were collapsed exactly by summing. Three covariance libraries were applied: the two of this work and the 56-group library of ENDF/B-VII.1 [1] covariances distributed with SAUNA [12].

5.2 Results

The two routes. The totals of the two ENDF/B-VIII.1 libraries differ by 0.0005, 0.0019 and 0.0030 % Δk/k, at most 3 pcm. In each case almost the whole difference is the O-16 elastic × inelastic term, present in the NJOY library and absent from the AMPX library (Section 4.3): −0.025, +0.051 and −0.068 % Δk/k in signed square root. Every other leading term agrees within 0.0005 % Δk/k (Fig. 5), because the reactions that matter in these thermal systems are the ones that agree in Fig. 1; the large differences of Section 4.3 concern nuclides with small sensitivities here. Setting the corrupted O-16 elastic elements to zero, as a test, changes the totals by at most 0.007 % Δk/k.

ENDF/B-VIII.1 against ENDF/B-VII.1. With the new covariances the totals are 26 %, 36 % and 29 % lower. The largest reductions come from U-235 ν̄ (0.42 against 0.66 % Δk/k in LCT-093-1, 0.45 against 0.70 in HST-046-1), U-238 capture (0.26 against 0.43), the U-235 fission spectrum (0.076 against 0.468 in HST-046-1) and Pu-239 fission (0.20 against 0.54 in PST-039-1). The Pu-239 fission × capture term changes sign, from +0.33 to −0.16 % Δk/k, and ENDF/B-VIII.1 adds an anticorrelation between H-1 elastic scattering and capture that reduces the variance (−0.06 to −0.15 % Δk/k) and that the ENDF/B-VII.1 library does not have. Some terms grow: Pu-239 ν̄ (0.44 against 0.17), Pu-240 capture (0.18 against 0.04), U-235 fission (0.20 against 0.15) and H-2 (n,2n) (0.29 against 0.25).

The ENDF/B-VIII.1 libraries have no covariances for a few reactions to which the cases are sensitive, which therefore contribute nothing: N-14 (n,p) and elastic scattering in all three cases (integral sensitivity up to 0.020 for N-14 (n,p) in PST-039-1), Al-27 (n,p) and O-17 (n,α) in LCT-093-1, and S-32 and Cd-113 capture in HST-046-1.

8.1 What SCALE's library holds, and where it came from

SCALE's own ENDF/B-VII.1-based 56-group covariance library has 452 SCALE IDs. The v1.0 libraries of this work had 233 of them, and 62 IDs that SCALE's library does not have; 219 of SCALE's IDs were missing, nearly all because ENDF/B-VIII.1 has no covariance data for those nuclides, or no evaluation of them. The SCALE manual [15] names the source of each of SCALE's entries. For the 219 missing IDs these sources, and what releases v1.1 and v1.2 did with each, are in Table 6.

The BLO data come from the low-fidelity covariance project of Brookhaven, Los Alamos and Oak Ridge (2008) [16]. Oak Ridge estimated the uncertainties of capture, fission and elastic scattering below 0.5 eV and from 0.5 eV to 5 keV from the uncertainties of measured thermal cross sections and resonance integrals in Mughabghab's atlas [18], each range fully correlated; Brookhaven used optical-model calculations with estimated parameter uncertainties above 5 keV for about 300 structural, fission-product and heavy nuclides; Los Alamos provided actinides and light nuclides. The NNDC publishes the result [17] as MF33 and MF31 files for 387 ENDF/B-VII.0 materials, without cross sections. They are not approved by the Cross Section Evaluation Working Group and are estimates, not evaluations; SCALE's manual calls them approximate.

8.2 Older ENDF/B evaluations

Release v1.1 searched ENDF/B-VIII.0, VII.1, VII.0 and VI.8 for each missing nuclide and used the most recent evaluation whose covariances cover every reaction SCALE's entry has, processing the whole evaluation with its own cross sections: Fe-57 from ENDF/B-VII.1 and Fe-58, Ni-61, Ni-62 and Ni-64 from ENDF/B-VI.8. Release v1.2 refined the search in two ways. It counts the total cross section, and total ν̄, as covered when the partial reactions are, because PUFF-IV forms them from the partials (as it does for SCALE's own library), and it also searches the superseded versions on the ENDF/B-VI release tapes. This adds Pm-147 and Eu-155 from ENDF/B-VIII.0, whose covariances cover elastic, inelastic, (n,2n) and capture; Nb-93 from ENDF/B-VI.8, a complete evaluation, where the newer ENDF/B-VII.0 gives only a total covariance; and Sc-45, whose only covariance data are a capture covariance in the ENDF/B-VI.0 version. That version is a partial evaluation without total or elastic cross sections, so its covariance section was added to the ENDF/B-VIII.1 evaluation instead.

8.3 The BLO entries

Each BLO file was added to the ENDF/B-VIII.1 evaluation of the same nuclide, which supplies the cross sections, under that evaluation's material number. Sections for reactions without an MF3 cross section were left out, except the summed reactions (MT4, MT103–107), which both codes form from their partials. SCALE's manual [15] describes the changes SCALE made in building its library; Table 7 lists them and how each was applied. The resulting 192 nuclides, including the SG-26 Am-242, were processed by the unchanged NJOY and AMPX routes, and the multigroup steps were then applied to both libraries.

8.4 Agreement with SCALE's entries

The BLO entries were compared with SCALE's own, group by group, in groups 2–56 (below 6.434 MeV) where the cross section is significant, as in Table 4; group 1 is considered separately below. As an integral measure, the relative uncertainty of a one-group reaction rate,

with the flux of Eq. (1), is the uncertainty that a sensitivity profile of that shape would get. Only differences between the libraries are reported. To separate the processing from the cross sections, the BLO data were also combined with the ENDF/B-VII.1 cross sections, which SCALE's library used (this variant is not released).

The remaining differences have four causes.

• The top group. In group 1 (6.434–20 MeV) the median difference is 0.69 percentage points, up to 15.89 for (n,2n), whose threshold lies in that group. The weighting SCALE used there is not documented; neither a 1/E spectrum nor fission spectra with temperatures from 0.9 to 3 MeV reproduce it, which suggests a different collapse path rather than an analytic spectrum. This work keeps the weighting of Eq. (1). The same effect reaches groups 2–4 for the Los Alamos actinide fission covariances, and changes the fission-rate uncertainty of Pa-231, Pa-233 and U-240, whose fission lies almost entirely above 1 MeV.
• The cross sections. The covariances are relative, but derived totals and groups straddling a covariance boundary depend on the cross-section shape. With ENDF/B-VII.1 cross sections, 21 IDs that differ on ENDF/B-VIII.1 come close or agree (the six Hf isotopes, Ba-134, Ba-137, Rh-105, Sm-153, Ag-107 and others); only one moves the other way, Co-58 in the NJOY route. On ENDF/B-VIII.1 these differences are a property of the newer evaluations.
• Thresholds. Inelastic scattering differs in one or two groups just above its threshold for some nuclides (Sn-112 to Sn-118, Sm-144, Kr-83, Nd-142), with little effect on the rate uncertainty.
• Versions of the BLO files. For H-3, He-3, Nd-144 elastic scattering and single capture groups of Xe-136 and Ba-138, SCALE's entries imply input data different from the files the NNDC publishes; for Nd-144, SCALE's derived total shows that its elastic variance was zero below 3.7 keV, where the published file gives 18 %.

The older-ENDF/B entries (AMPX route) reproduce SCALE's closely where SCALE used the same data. For Sc-45 the capture covariance is identical and the derived total agrees within 0.01 percentage points. For Nb-93 total, inelastic, (n,2n) and capture agree within 0.15 percentage points and elastic scattering, which ENDF/B-VI.8 derives from the others, within 1.24 (rate −3.7 %). Pm-147 and Eu-155 from ENDF/B-VIII.0 agree within 0.16 and 0.22 percentage points apart from one inelastic group of Pm-147 (2.95); their ENDF/B-VII.1 versions, which SCALE's manual names, differ by up to 20 percentage points, because SCALE corrected them and the corrections went into ENDF/B-VIII.0 [15]. For Re-185 and Re-187, SCALE added an elastic uncertainty “from Mughabghab” to the ENDF/B-VI capture covariance; the BLO elastic data for rhenium differ from it substantially, so the two isotopes were not included.

8.5 Libraries v1.2 and the two routes

For the 198 IDs added in v1.2 (196 entries and 2 duplicates), the two routes agree as in Section 4.2 for 165 IDs, differ slightly for 15 and substantially for 18, mostly in inelastic scattering just above its threshold (50–200 keV) and in a few derived totals. The ERRORR tapes of five BLO nuclides have within-material |ρ| > 1: four only where the cross sections are negligible, and Xe-136 capture, up to 1.42 between 3.74 keV and 1.85 MeV, where the BLO matrix itself is inconsistent. The tapes are kept as processed; in the COVERX and text libraries SCALE's rule sets these correlations to 1. All 619 files of release v1.2 load and display in JANIS 4.0 [11].

The build also exposed a pitfall of caching. The NJOY route kept its point-wise cross sections in a cache named after the evaluation file; when a nuclide was staged again from a different evaluation under the same name, the old cross sections were reused (In-115 and Nb-93, processed earlier from other ENDF/B versions). The cache now records a checksum of its input and is rebuilt when the input changes, and the affected nuclides were reprocessed; the comparison of the release libraries with the study libraries found the discrepancy.

8.6 Effect on the benchmarks

The added nuclides have small sensitivities in these thermal benchmarks, so the totals change by at most 0.001 % Δk/k. The reactions that Section 5.2 lists as missing for N-14, O-17 and Cd-113 are now covered. Al-27 (n,p) and S-32 capture are still missing, because ENDF/B-VIII.1 has covariances for those nuclides, but not for those reactions. Each new term agrees with the same term computed with SCALE's library to within 0.00004 % Δk/k, in both routes.

8.7 Conclusions of the revision

The entries of SCALE's library that ENDF/B-VIII.1 cannot supply were rebuilt from the same published sources and SCALE's documented treatment, with the processing of this work, for 196 of the 219 missing IDs; two more carry this work's data under SCALE's duplicate-ID convention. With the cross sections SCALE used, 97–98 % of the group values agree within 0.1 percentage points and 803–811 of 832 reaction-rate uncertainties within 1 %; with the ENDF/B-VIII.1 cross sections, as released, 95 % and 782–786. The remaining differences come from the undocumented top-group weighting, the newer cross sections, thresholds and differing versions of the low-fidelity files. In the three benchmarks the contribution of every added nuclide matches SCALE's library. Fifteen IDs could not be rebuilt because their source data are not published. The low-fidelity entries are estimates; the libraries mark their source nuclide by nuclide, so that users can treat them accordingly.