*2.4. SDF and Evaluation Procedure*

The SDF can be created by the model used in the evaluation. The only way to find the SDF caused by the evaluation procedure is a comparison of the evaluations based on the same experimental data. The understanding of the influence of the selected data set is also important. The different sets of experimental data for 235U(nth,f) PFNS in fission induced by thermal neutrons were evaluated using the GMA (Gauss–Markov–Aitken approach) code. Because all measurements were undertaken relative to the 252Cf(sf) PFNS standard, the shape of the ratio 252Cf(sf) to 235U(nth,f) and Mannhart's standard for absolute 252Cf(sf) spontaneous PFNS with covariances was used in the evaluation. Using the shapeof-ratio experimental data in the fit excludes large components of uncertainties contributing to absolute measurements. The normalization constraint was applied in the fit with an uncertainty close to the uncertainty of the evaluated average number of prompt neutrons for 235U. The evaluated PFNS properly extrapolated to 0 and to 20 MeV on the energy of emitted neutrons was used for calculation of average neutron energy <E> of the spectrum. The data used in GMA evaluation were also used for <E> calculation with the scale method [7] fitting procedure. The 2% uncertainty at each point from GMA was applied in the SM analysis. It was applied for scaling of the PFNS shape. The results are presented in Table 2.

**Table 2.** The comparison of the <E> calculated with numerical integration of GMA evaluation in the energy range 0–20 MeV, and SM evaluation for the energy range 0.43–7.3 MeV.


We can reach the same conclusion as before. The inclusion of the experimental data as they are given in [11] increases the <E> outside the estimated uncertainties. Therefore, there is no evidence that the procedure of the evaluation itself contains the SDF.

#### *2.5. The Evidence of the SDF in LANL Double TOF Experiment*

The final experimental data obtained at LANL are still not available. The values for analysis in this paper were obtained by digitalizing the data from figures presented in [13,14]. The uncertainty of this digitalization is small. The data were analyzed with the SM. The experimental PFNS as a ratio to the SM functions is shown in Figures 7 and 8. Data were analyzed in the energy range of the emitted neutrons 0.25 < E < 7 MeV. The parameters of the SM fit were used for PFNS calculation in the energy range of emitted neutrons above 0.25 MeV. The data below 0.25 MeV were excluded from the analysis due to possible problems with neutron registration by 6Li-glass detectors near 0.244 MeV 6Li resonance. One should keep in mind that only three points are available in the range 7–10 MeV for the LANL experiment. The average energy for thermal point and LANL data are different. After the correction due to different <E> (ratio to SM fit, Figures 7 and 8), the shape of LANL PFNS is in reasonable agreement (inside uncertainties) with old data.

Parameters of the SM (the normalization and <E>) are given in Table 3. The same parameters with numerical integration of data are given for comparison.

**Table 3.** Parameters for the LANL result of PFNS measurements evaluated with SM for different incident neutron energies (Eo) in comparison with the results of numerical integration.


**Figure 7.** Ratio of experimental data for PFNS to the calculated SM functions. Starostov's data for thermal neutrons are shown for comparison.

**Figure 8.** Ratio of experimental data for PFNS to the calculated SM functions. Starostov's data for thermal neutrons are shown for comparison.

The analysis with SM demonstrated very interesting peculiarities:


One may assume that the origin of the SDF in double TOF LANL results can be some unaccounted background neutrons. These neutrons reached the chamber at the same time interval due to a longer flight path. These background neutrons cause fission at higher neutron energy, and as a consequence, they contribute to a higher average neutron energy of PFNS.

SDF connected with the time structure of the proton beam or other causes of SDFs are also possible.

One may assume that LANL results consistent with the results in [11] are accurate and that other measurements performed in the years 1983 to 2018 contain the SDF. This seems a doubtful assumption, at least concerning the results [11]. As discussed in Section 2.3, the results of the measurements [11] contain rather strong SDF, and this should be accounted for in the PFNS evaluation procedure.

**Figure 9.** Comparison of 235U PFNS average neutron energies estimated with the SM for new LANL-2020 double TOF measurements [13] (closed black circles, Table 3) with the results of DB and PB measurements. The open red triangles present the results of numerical integration when calculating the average energy. The thin blue line is a linear fit of LANL-2020 data, and the black line is a linear fit of the results of DB measurements. The data were taken from [7].

**Figure 10.** Ratios of the measured PFNS to the Maxwellian spectra (see details in the text).

#### **3. SDF in Fission cross Section Ratio Measurements**

A new experiment using the Time Projection Chamber (TPC) for registration of FF was implemented in Los Alamos National Laboratory (LANL) by a team from different universities and laboratories (NIFFTE collaboration). The results of the absolute ratio of 239Pu(n,f) to 235U(n,f) cross section measurements are shown in Figure 11. The data were taken from [16].

The constant bias at about 2% is clearly visible in the energy range 0.2–15 MeV and has an even larger spread above 15 MeV. The authors of [16] provided a very detailed and deep analysis of the modeling of their experiment (experimental details, data reduction procedure, uncertainties of different parameters and so on) but at present could not explain the existing bias, which can be treated as SDF. From our point of view, it is premature to assign the SDF to this measurement, especially taking into account that the reaction rate ratios measured in clean benchmarks show better consistency with the NIFFTE results.

The authors of [16] came to the conclusion that the difference in 2% absolute normalization with the ENDF/B-VIII.0 [20] evaluation based on its turn on neutron standards evaluation [2] is too large to be ignored. The large non-uniformity and mass value of the 239Pu sample, which can degrade with time, are the largest concern. Although the TPC results with multi-parametric data allow an estimate of many sources of systematic uncertainties (or potential SDF by other words), the present decision is to repeat the measurements with a newly prepared 239Pu sample. At the same time, we should admit that the results of the reaction rate measurements obtained as benchmarks with fast neutron spectrum and Mannhart's evaluations [21] of 252Cf PFNS-averaged cross sections endorse the data obtained with the TPC.

**Figure 11.** 239Pu(n,f)/235U(n,f) cross section ratio measured with a fission TPC [16] in comparison with the data [17–19] and the results of the GMA fit (solid line) with fission TPC data added to the standard (old) database.

#### **4. SDF in Maxwellian Averaged cross Sections (MACS) for Astrophysical Application**

MACS for neutron capture are used in astrophysics to model the stellar nucleosynthesis of elements. The range of needed neutron temperatures (kT) is varied from a few keV to 100 keV. The novel method of direct MACS measurement for kT = 25–30 keV was proposed by Beer and Kaeppeler [22]. It is based on the kinematics of 7Li(p,n) reaction at a proton energy of 1912 keV. As it was shown experimentally and through modeling, the neutrons in this case are emitted in the forward cone with the spectrum integrated on the angles close to the Maxwellian spectrum with kT between 25 and 30 keV.

Ratinsky and Kaeppeler recorded the accurate activation MACS measurements [23] for 197Au(n,γ) (582 ± 9 mb at kT = 30 keV, stellar definition), which were used as the standard for measurements of other nuclides by this method. Slightly renormalized results of Macklin's 197Au(n,γ) microscopic cross section, which provided this MACS value, were used in calculations for extrapolation of calculated MACS to lower and higher kT. These values were inconsistent with MACS calculated for the evaluation of 197Au(n,γ) standards [22] based on a combined fit of 62 measurements of captured cross sections and their ratios to other standard reactions. MACS calculated for standard cross section evaluation in the energy range 5 keV–2.8 MeV, supplemented by the ENDF/B-VII.0 evaluation below and above this energy range, was 614 mb at kT = 30 keV. The ENDF/B-VII.0 evaluation had missed resonances at the upper end of the resolved resonance range. The correction at the missed resonances increases the MACS value to 619 mb [24].

This controversy and new results of nTOF [25] and GELINA [26] measurements of microscopic capture cross sections, consistent with the standards evaluation, led to a new cycle of measurements and analysis of the MACS. To resolve the discrepancy, PINO [27] and SimLiT [28] Monte Carlo codes were developed for modeling the neutron source. GEANT4 code was used for modeling the neutron transport. There have been a number of publications, but the latest review of the Spectrum Averaged Cross Section (SACS) measurements and re-evaluation is published in [29]. Two major sources that may cause bias in the MACS values were discussed [29]: attenuation of the neutron flux at the copper

backing of the 7Li target, and the difference between the measured neutron spectrum and "true" neutron spectrum incident at the gold sample following a reduction in the measured value to the MACS value at kT = 30 keV.

Simulation of the neutron source had shown [29] that backing of the 7Li target with copper of 1 mm thickness requires the introduction of neutron scattering correction between 6.4 and 7.1% depending on the nuclear data library used in simulation calculations. The comparison with experimental data obtained for different thicknesses of backings and size of deposited 7Li targets shows [29] that the large part of the discrepancy with MACS obtained with the standards evaluation and calculated with the latest experimental data [25,26] may be explained by an improper account of backing in the activation measurements. The authors of [29] were unable to introduce corrections in the results [23] at the base of their simulation because of the inconsistencies between experimental data [23] after introducing corrections based at the simulation.

The measured neutron spectrum induced by 1912 keV protons and integrated on the angles is similar to the 25.3 keV Maxwellian neutron spectrum [23] but shows a clear lack of neutrons above 80 keV. The neutron temperature, which should be best assigned to the measured MACS values, has rather large uncertainty. The comparison of the measured spectrum [23] with Maxwellian spectra at kT = 25.3 (best fitted to the experimental simulated spectrum [23]) and kT = 28.5 keV (obtained from calculated mean energy of the experimental simulated spectrum [24]) is shown in Figure 12. It was estimated [24] that, depending on the procedure of the reduction to the true Maxwellian spectrum, the difference between the measured value [23] at the best assigned temperature and the calculated value may reach 2.5%. A similar difference (1.7%) is shown in [29] between recommended values obtained from the latest measurements of SACS and SACS calculated for the true Maxwellian spectrum at the same temperatures.

**Figure 12.** Comparison of experimental neutron spectrum with a true Maxwellian neutron spectrum for two temperatures kT = 25.3 and 28.5 keV. All spectra have free normalization.

A series of new measurements of MACS with a 7Li(p,n) reaction for different thicknesses of lithium target and spectrum of protons incident on the target supported by modeling of the experiment had allowed obtaining the best simulation of the Maxwellianlike spectra for different temperatures [29]. A new version of the neutron cross section standards [2] took into account the results of the new measurements [29] of 197Au(n,γ) cross sections. MACS at kT = 30 keV calculated for the standard evaluation in the energy range 3 keV–2.8 MeV embedded in the ENDF/B-VIII.0 evaluation provides the new recommended value of 611.4 ± 4.2 mb with the uncertainty increasing to 11.2 mb when USU is accounted for. This value has excellent consistency with the new value 612 ± 6 mb [29] recommended for use by the astrophysical community.

We may conclude that the modeling of the neutron source, including the energy angular correlation and attenuation of neutrons in lithium target backing excludes the bias (~5%) caused by SDF from the results of SACS measurements. The proof of this is the consistency between results derived from simulated SACS measurements and calculated from the evaluated cross sections obtained in the independent microscopic cross section measurements.

## **5. Unrecognized Sources of Uncertainty (USU) in the Data Evaluation**

#### *5.1. Small Uncertainties Problem in Neutron Cross Section Standards Evaluation and USU*

In 1991, the Cross Section Evaluation Working Group (CSEWG) concluded that the uncertainties of the evaluated neutron cross section standards [30] are strongly underestimated. The standards were obtained in the combined model-independent statistical fit of about 400 data sets for 10 reactions and their combinations. The relative uncertainties obtained from variances of the covariance matrix of the evaluated standards were two to three times lower than the spread of the experimental data estimated for the same broad energy groups.

The spread of the experimental data can be best characterized by the variances of the evaluation obtained with the use of the sample statistics. For this, the model-independent least square fit of the data in the energy groups can be performed without consideration of the uncertainties assigned to the data. The uncertainty of integral data calculated with an account of the evaluated covariance matrices (such as SACS for 197Au(n,γ) reaction discussed above) was also considered as too small. This can be partly explained by an incomplete budget of uncertainty sources for some measurements and absence or not a full account of cross-correlations between the same components of uncertainties in different measurements, which use the same sample or detector, or even method. This conclusion remains generally true with the revision of the outlaying experimental data uncertainties.

The difference in the data values obtained in different experiments, which cannot be explained by uncertainties assigned to them, indicates the presence of the USU [31]. In the case when better consistency cannot be achieved through the revision or introduction of the corrections based on the Monte Carlo modeling of the experiments (SDF removing), the additional uncertainty can be introduced in the evaluated covariance matrix, making the evaluated uncertainties more realistic.

This approach can be applied to the neutron cross section standards evaluation [2]. The GMA code [32] for model-independent evaluation of the standards uses the GLSQ method for a simultaneous fit of the cross sections and integral parameters with an iterative approach. Starting from the second iteration, a posterior evaluation for data values with an uninformative covariance matrix is used as a new prior. Usually, three iterations are needed to obtain full convergence when the last posterior data evaluation is practically indistinguishable from the last prior evaluation. All experimental data are reduced in the model-independent fit to a common grid of energy nodes.

The covariance matrix for each experimental data set is constructed from statistical, fully correlated systematical and medium energy range correlated to systematical components of uncertainties. The correlations between the same components of the uncertainties in different measurements can be accounted for.

## *5.2. Sample Method for Determination of USU Covariance Matrices*

A sample method can be adapted for construction of the USU component of the evaluated covariance matrix using the biases between the evaluated and experimental data with an account of recognized (known) uncertainties. A sample method for the uncertainty evaluation in the measurements is usually formulated for a set of repeated measurements of multivariate (vector) data [33].

For independent random vectors *Xi (i =* 1, ... , *I*) of dimension n and with a zero mean, the sample covariance matrix (without discloser of the nodes indices) is:

$$CovX\_I = \frac{1}{I} \sum\_{i=1}^{I} X\_i \otimes X\_i \tag{1}$$

The closeness of the sample covariance matrix to the actual covariance matrix depending on vector dimension *(n)* and number of samples *(I)* was studied in [34]. Sample covariances for the USU component can be constructed in the framework of an ad hoc procedure based on the biases between evaluated and experimental data reduced by known (and accounted for in the evaluation) systematic uncertainties.

The sample vector *δ<sup>i</sup>* for the USU component can be written as:

$$\begin{array}{l} \delta\_{i} = (y\_{i} - \mu) - u\_{i\prime} \text{ if } y\_{i} > \mu \text{ and } \delta\_{i} > 0;\\ \delta\_{i} = (y\_{i} - \mu) + u\_{i\prime} \text{ if } y\_{i} < \mu \text{ and } \delta\_{i} < 0;\\ \delta\_{i} = 0 \text{ in all other cases,} \end{array}$$

where *yi* is a vector of *i*-the experimental data set, *μ* is a vector of evaluated data (best approximation to the true value), *(yi* − *μ)* is a vector of biases between experimental and evaluated data, *ui = ε*2 *<sup>i</sup>* + *<sup>η</sup>*<sup>2</sup> *<sup>i</sup>* is a vector of total uncertainty of experimental data, *ε<sup>i</sup>* is a vector of the statistical component of the uncertainty, and *η<sup>i</sup>* is a vector of the systematic component of the uncertainty, which consists of two components: assigned to the analysis of experimental uncertainty *ηi,exp* and assigned to the outlaying data *ηi,out*:

$$
\eta\_i = \sqrt{\eta\_{i,exp}^2 + \eta\_{i,out}^2}
$$

There is an established procedure for obtaining the evaluated values with GMA. It includes:


This work with outlaying data allows avoiding big local discrepancies and reducing the general chi-square per degree of freedom for standard evaluation from the initial 3.4 to a value close to 1 without strong local discrepancies.

The main differences to the classic sampling method are the following:


Covariances for the USU component can be written with the discloser of the node indices *m* and *n* for vectors as

$$\text{Cov}\delta^{mn} = \frac{1}{K\_{mn}}\sum\_{i=1}^{I} \delta\_i^m \delta\_i^n \,\, \, \, \tag{2}$$

In our case, not all data sets contribute at the energy node *m* or *n*, and *Kmn* is a number of non-zero terms in the sum. The total covariances of evaluated data (*Covδmn tot* ) can be written as a sum of covariances obtained in the GMA fit with the ad hoc increase in the uncertainties for the outlaying data (*Covδmn GMA*) and covariance estimated for the USU component (*Covδmn USU*).

$$\text{Cov}\delta\_{\text{tot}}^{\text{mm}} = \text{Cov}\delta\_{\text{GMA}}^{\text{mm}} + \text{Cov}\delta\_{\text{LISII}}^{\text{mm}} \tag{3}$$

The covariance matrix for the USU component (2) may turn out to be semi-positive definite, and because of this, the total covariances (3) may lose its semi-positive definite. This introduction of USU covariances can be considered as a rather crude approach to the estimation of the realistic uncertainties of the evaluated data, but it is definitely a better approach than expert estimation [30]. If we increase the uncertainties of the outlaying data using stricter ad hoc requirements to the data consistency, we will reduce the USU covariances, or even exclude them. Then we have a strong connection between the treatment of the outliers and USU uncertainties, with a clear distinction that uncertainties for outliers are introduced into the experimental data iteratively in the fit, and USU uncertainties as additional components to the evaluated data.

The work with the GMA database of experimental data has shown that outliers are often the "poor" data with large uncertainties. Increasing the uncertainty of these outliers with the procedure described above reduces the chi-square per degree of freedom, changes the evaluated (mean) values, and to a lower extent, changes the covariances. The "smallness" of the evaluated uncertainties is determined mainly by the "good" experimental data with small uncertainties.

#### *5.3. Numerical Example of USU Covariance Matrix Construction*

This approach for determination of USU was applied for testing of 238U(n,f) to the 235U(n,f) cross section ratio evaluated with the GMA for the model case of 11 cross section ratio measurements taken as absolute in 11 nodes (Figure 13). The outliers were determined, and their uncertainties were increased.

**Figure 13.** Experimental data used in the model case of the USU determination for the 238U(n,f)/235U(n,f) cross section ratio are shown by different symbols. GMA evaluation with an uncertainty band is shown by lines.

The results shown in Figure 14 demonstrate an increase in total percent uncertainty of up to 4–5 times in a few nodes where the spread of the data was large. The covariance matrix for the USU component has rather large positive correlations, which shows that most experimental data have normalization problems.

**Figure 14.** Contribution of the USU in the total uncertainty of evaluated data for the model case of the 238U(n,f)/235U(n,f) cross section ratio.
