Issue 
EPJ Nuclear Sci. Technol.
Volume 4, 2018
Special Issue on 4th International Workshop on Nuclear Data Covariances, October 2–6, 2017, Aix en Provence, France – CW2017



Article Number  46  
Number of page(s)  8  
Section  Applied Covariances  
DOI  https://doi.org/10.1051/epjn/2018027  
Published online  14 November 2018 
https://doi.org/10.1051/epjn/2018027
Regular Article
Comments on the status of modern covariance data based on different fission and fusion reactor studies
Jožef Stefan Institute,
Jamova 39,
Ljubljana, Slovenia
^{*} email: ivan.kodeli@ijs.si
Received:
29
September
2017
Received in final form:
6
February
2018
Accepted:
14
May
2018
Published online: 14 November 2018
Both the availability and the quality of covariance data improved over the last years and many recent crosssection evaluations, such as JENDL4.0, ENDF/BVII.1, JEFF3.3, etc. include new covariance data compilations. However, several gaps and inconsistencies still persist. Although most modern nuclear data evaluations are based on similar (or even same) sets of experimental data, and the agreement in the results obtained using different crosssections is reasonably good, larger discrepancies were observed among the corresponding covariance data. This suggests that the differences in the covariance matrix evaluations reflect more the differences in the (mathematical) approaches used and possibly in the interpretations of the experimental data, rather than the different nuclear experimental data used. Furthermore, “tuning” and adjustments are often used in the process of nuclear data evaluations. In principle, if adjustments or “tunings” are used in the evaluation of crosssection then the covariance matrices should reflect the crosscorrelations introduced in this process. However, the presently available crosssection covariance matrices include practically no crossmaterial correlation terms, although some evidence indicate that tuning is present. Experience in using covariance matrices of different origin (such as JEFF, JENDL, ENDF, TENDL, SCALE, etc.) in sensitivity and uncertainty analysis of vast list of cases ranging from fission to fusion and from criticality, kinetics and shielding to adjustment applications are presented. The status of the available covariance and future needs in the areas including secondary angular and energy distributions is addressed.
© I. Kodeli, published by EDP Sciences, 2018
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The performance of the new crosssection evaluations, if judged by the agreement with the large set of integral experiments, greatly improved over the last decades. Indeed, using the recent nuclear data evaluations, the calculationtoexperiment (C/E) ratios for the large series of critical integral benchmarks are indeed excellent. For example, the comparison presented in [1] reveals that almost 50% of the calculated k_{eff} values (about 900 out of over 2000 critical benchmarks analysed using ENDF/BVII.1, JENDL4.0 and JEFF3.1.1) lie within one standard deviation (1σ) of the experimental plus MCNP statistical uncertainty. However, such good agreement of C/E is difficult to understand from the mathematical (statistical) point of view. Indeed, much larger dispersion of results is to be expected from the statistical point of view taking into account in addition also the calculational uncertainties due to nuclear data, unless (1) the later are very small (highly unlikely), or (2) are correlated with the integral results, suggesting some adjustment or tuning procedure was used in the evaluation process. Manifestly, these “tunings” are not reflected in the crosssection covariance matrices, which include practically no crossmaterial correlation terms, with the exception of crosscorrelations between (n,f) reactions of U and Pu isotopes in the JENDL evaluations (−3.3 and on) [2]. The total uncertainty to cover 68% of the 2000 analysed C/E cases is around 1.8σ of the experimental uncertainty, which would correspond to the average 1σ computational uncertainty of only around 500 pcm, i.e. of a similar order of magnitude as the measurement uncertainties and lower than the typically calculated values.
2 SUSD3D and XSUN2017 computer code package
The SUSD3D [3] code was developed in the 1990s to allow 1, 2, and 3dimensional crosssection sensitivity and uncertainty calculations. In the past few decades the code was applied to waste range of different nuclear applications, including neutron and gamma ray shielding, criticality, and kinetics. The latest version of SUSD3D is part of the XSUN2017 [4] code package.
An important factor limiting the use of S/U analysis is the availability and the quality of crosssection covariance data. Several tools and nuclear data libraries were developed to facilitate the access and allow the validation of the data. This will be presented in Section 3.
2.1 Examples of applications
The SUSD3D code was used since early 1990s for very various applications, such as:

reactor pressure vessel surveillance dosimetry [3]: uncertainty in predicted dosimeter reaction rates and pressure vessel exposition, determination of realistic safety margins and consequently the reactor lifetime predictions;

fission shielding benchmarks [3]: sensitivity and uncertainty in the measured reaction rates were calculated for the several benchmarks from the SINBAD database, such as the ASPIS Iron, ASPIS Iron88 and VENUS3 pressure vessel dosimetry benchmark;

sensitivity/uncertainty pre and postanalysis of the fusion shielding benchmarks performed at the Frascatti Neutron Generator (FNG) at ENEA Frascatti (sensitivity/uncertainty of the measured fast/thermal activation rates and the tritium production in FNGBulk Shield benchmark, FNGStreaming, FNGSiC, FNGTungsten [5], FNG HCPB and FNGHCLL tritium breeding modules [6,7] and FNG Copper [8,9] benchmarks);

criticality benchmarks (sensitivity to k_{eff} and β_{eff}): many benchmarks from IRPhE and ICSBEP (KRITZ2 [10], SNEAK7A and −7B [11], VENUS2, etc.), MYRRHA reactor [12], etc.;

oil well logging: sensitivity and uncertainty of the carbontooxygen gammaray ratio [13].
3 Types of covariance matrices
Different data formats for crosssection covariances are available in the evaluated files according to the type of nuclear data:

MF = 31: covariance of average number of neutrons per fission ( − MT = 452, 455, 456);

MF = 32: shape and area of individual resonances;

MF = 33: covariance of neutron crosssection;

MF = 34: covariance of angular distribution of secondary neutron (SAD). NJOY processing is available for the reaction MT = 251/P_{1} only;

MF = 35: covariance of energy distribution of secondary neutron (SED). NJOY processing is available for the reaction MT = 18 only;

MF = 30: covariances obtained from parameter covariances and sensitivities (no NJOY processing available yet);

MF = 40: covariances for production of radioactive nuclei.
Several nuclear data processing codes and multigroup covariance data libraries are available from the OECD/NEA Data Bank, such as:

NJOY99/2012/2016 (ERRORR, COVR) [14]: processing of files MF = 31–35 (COVFILS format);

PUFFIV: code system to generate multigroup covariance matrices from ENDF/BVI uncertainty files (COVERX Format);

SUNJOY/ERRORR34 (part of SUSD3D package) [3]: secondary angular distributions (SAD) covariance (MF = 4 and 34) processing code (COVFILS format);

ANGELOLAMBDA [15]: utility programs for interpolation and mathematical verification of the matrices (COVERX and BOXER format input data, COVFILS output format);

Multigroup covariance data libraries: ZZVITAMINJ/COVA, ZZSCALE5.1/COVA and ZZSCALE6/COVA44G (44group crosssection covariance matrix library extracted from SCALE6.0 [16]).
3.1 Uncertainties in prompt and delayed Nubar (/) (MF31)
The uncertainties in prompt nubar directly influence the uncertainty in k_{eff} and are therefore often among its major contributors. Large differences can be observed among different evaluations, the standard deviations ranging from as low as ∼0.1% (most probably unrealistically optimistic) up to ∼1%. Inconsistencies between the prompt and total neutron multiplicities were also found in ENDF/BVII.1 [17]. This results in very different uncertainty estimations (see an example in Tab. 1).
Examples of covariances of ^{239}Pu are shown in Figure 1. The standard deviations passed from ∼1% in older SCALE5.1 and −6.0 m libraries to ∼0.1% in the recent ENDF/BVII.1 and JENDL4.0 evaluations, most probably to accommodate a better C/E agreement for a large series of integral benchmarks, rather than reflecting the uncertainties in differential measurements. Whereas this approach may provide relatively realistic uncertainties in k_{eff} for classes of problems covered by the integral benchmarks, it is likely to lead to biased results of adjustment analysis since preventing any modifications of and thus freezing the values including the tunings introduced during the evaluations. Furthermore, no crossisotope correlations are included in the available evaluations. Due to their importance for burnup calculations the fission yield correlation matrices were evaluated in [18].
Similarly, the uncertainties in delayed nubar were found important for reactor kinetics calculations, such as the uncertainties in effective delayed neutron fraction −β_{eff}. Only JENDL4.0 [2] evaluation includes the corresponding covariance matrices (Fig. 1), therefore most β_{eff} S/U analyses were based on these data [11]. However, here again no correlations are proposed between delayed nubar values of different isotopes even if it is evident that such correlations exist because of the use of similar measurement techniques and theoretical computational model. Missing correlation in evaluated files are likely to result in misleading uncertainty calculations. Uncertainties calculated assuming no and fullcorrelations between the of different isotopes strongly vary on the cases studied (see few examples in Tab. 1).
An attempt to estimate the correlations among the values of different actinides is described in [19]. The GEF code [20] was used to calculate the variance–covariance of the delayed fission yield data for ^{235}U, ^{238}U and ^{239}Pu actinides as a function of input model parameters and the corresponding uncertainties. Typical values of the correlations coefficients as high as around 0.8 between ^{235}U and ^{239}Pu, and around 0.3 between ^{238}U and ^{239}Pu were observed. Is was concluded that this is likely to have considerable impact on the uncertainty propagation calculations, such as those of the effective delayed neutron fraction and the burnup evolution [21].
Uncertainties in k_{eff} and β_{eff} calculated using the SUSD3D code. The two values for the β_{eff} uncertainty correspond to the assumption of no/full correlation among the uncertainties of different actinides. SAD/SED uncertainties are not included.
Fig. 1 Covariance matrices of ^{239}Pu for from the SCALE6.0 m, ENDF/BVII.1 and JENDL4.0 evaluations. 
3.2 MF33 covariance matrices
The covariance information of the type MF33 is most widely evaluated and used, also because the processing is in general well established. The main concerns represent the lack of correlations between different isotopes and rather large differences among evaluations in some cases. An example of the use of different copper and iron covariance evaluations is shown in Tables 2 and 3, respectively. More details can be found in references [8] and [9].
FNGCu benchmark: uncertainty due to transport crosssections of different origin compared to the C/E values.
ASPIS IRON88 benchmark: computational (ΔC) vs. experimental (ΔE) uncertainties.
3.3 SAD uncertainties (MF34)
The importance of the uncertainties in the SAD was demonstrated in several fast neutron applications such as fusion [22], fast reactors, etc. and the processing of these data and te S/U methodology is available and tested since decades. In the EFF2 evaluations in the 1990s, the covariance matrices for angular distribution of secondary particles became available for the elastic crosssections for _{56}Fe, _{52}Cr, _{58}Ni and ^{60}Ni [23,24]. The matrices were prepared in the file MF = 34 ENDF/B6 format in terms of covariances among Legendre coefficients, and energydependent correlation was included as well. The evaluations included the terms from P_{1} to P_{6}.
In the scope of the European Fusion File project in 1995 a procedure was developed to process the SAD covariance matrices into a multigroup form to be used subsequently by the SUSD3D S/U code [25]. The processing code, called ERRORR34, now part of the SUSD3D [3] code package, can process the ENDF/B6 format files MF = 4 and 5 (SAD/SED crosssections), and MF = 34 (SAD covariances). Groupcollapse strategy similar to the one used in NJOY [14] was adopted, therefore many NJOY91.91 (ERRORR) subroutines were used. As in the ERRORR module, union groups are first formed as an union of the user's and ENDF/B grids. The SAD partial crosssections, weighting flux and covariance matrices are defined to produce multigroup values on this grid. The covariance matrices in the user defined energy structure are then calculated from: (1) where g refers to the union group, and G to the user defined energy group, represent the lth Legendre polynomial coefficients of the SAD partial crosssections, in energy group G, is the SAD relative covariance in union group structure, Φ_{G} is the weighting flux in group G. Finally the relative covariance in the new energy grid is obtained from: (2)Modifications were subsequently needed also in the SUSD3D code, in order to take into account the full covariance matrix information provided by ERRORR34.
Among the recent nuclear data evaluations, the JENDL4.0 [2] includes the SAD (MF34) covariances relative to the reaction type MT251 (average scattering cosine) for several important isotopes (Fe, U, Pu, etc.). The recent versions of NJOY (NJOY99, −2012 and −2016) can also process these data in the multigroup form. Note however that these data (and the NJOY processing) is of course limited to the P_{1} Legendre term. MT34/MF251 covariances for few isotopes (^{56}Fe) are likewise included in the ENDF/BVII.1 [26] evaluation. Even more SAD covariances are available in the TENDL [27] libraries for elastic and some inelastic reactions. The evaluations include also higher than P_{1} Legendre terms, however only P_{1} can be processed using the recent NJOY (−99 and above) codes. The ERRORR34 code sequence can not be used in these cases since it is based on the older NJOY91 version and needs to be updated (e.g. to the NJOY2016 version) for this purpose.
An example of the EFF2.4 covariance matrices for ^{56}Fe (processed by the code ERRORR34) is presented in Fig. 2), compared to the recent evaluation available in the JENDL4.0, ENDF/BVII.1 and TENDL2015 evaluations and processed using NJOY99. Note that contrary to the recent evaluations the EFF2.4 data include the terms P_{1} to P_{6}.
An example of the SAD uncertainties for the ASPISIRON88 benchmark calculated using the SUSD3D code and the above ^{56}Fe covariance matrices is given in Table 3. Considerable spread of results can be observed, however all suggesting that SAD uncertainties cannot be neglected for highenergy reactions.
Fig. 2 SAD covariance matrices of ^{56}Fe elastic crosssections from the EFF2.4, JENDL4.0, ENDF/BVII.1 and TENDL2015 evaluations. Warning: apparent similarity between the JENDL4.0 and ENDF/BVII.1 covariances is only an artifact of log/log scale which is for some reason used in the recent versions of NJOY. Switching back to the old (much more informative) lin/log presentation is strongly recommended (e.g. by redefining the “yrtest” parameter in the COVR module). 
3.4 SED uncertainties (MF35)
Uncertainties in the Secondary Energy Distributions are at present only available for the prompt fission neutron spectra (PFNS) and relatively complete data are included in recent evaluations such as JENDL4.0, ENDF/BVII.1 and JEFF3.3. However, the correletions among the covariances for different incident neutron energies are missing. The conservative assumption of total correlation is in this conditions probably the most appropriate.
Several sensitivity methods were studied in the scope of the WPEC26, concluding with recommending the constrained sensitivity method [28].
However, covariance information for other reactions is still missing. A simple method for evaluating covariances for delayed fission spectra, which are important for the calculation of β_{eff} uncertainty, was proposed in [11]. An approximate “twoblock” covariance matrices were constructed based on a simple common sense assumption of an energyuniform standard deviation of 15% and a complete anticorrelation between the energies above and below the mean delayed neutron energy for each of the 6 delayed groups. Conservative assumption of the complete correlation between the 6 individual groups was adopted.
To test the validity of this method a similar procedure, except assuming a uniform 4% standard deviation instead of 15%, was applied to the PFNS, where comparison with detailed covariance matrices available in some nuclear data evaluation (JENDL4.0, SCALE6, etc.) was possible. Table 4 compares the uncertainties in k_{eff} and β_{eff} calculated using the above “twoblock” PFNS covariances with those based on the PFNS covariances from JENDL4.0 and SCALE6.0. In spite of its simplicity the procedure is shown to predict similar uncertainties, both for k_{eff} and β_{eff} uncertainties, as the more sophisticated methods used in the JENDL4.0 and SCALE6.0 covariance data evaluations. This good agreement can be explained by the relatively narrowenergy sensitivity of the k_{eff} and β_{eff} to the fission spectra.
A similar procedure could be temporary applied to evaluate the SED uncertainties for other reactions such as (in)elastic scattering, until more sophisticated evaluations become available.
Fission spectra uncertainties in k_{eff} and β_{eff} calculated using the approximate “twoblock” prompt fission neutron spectra covariances (i.e. assuming flat anticorrelated 4% standard deviation) compared to those based on covariances from JENDL4.0 and SCALE6.0.
4 Conclusions
The availability of the covariance data improved over the last decades. Experience in using covariance matrices of different origin (such as JEFF, JENDL, ENDF, TENDL, SCALE, etc.) any types (MF31, MF33, MF34 and MF35) in sensitivity and uncertainty analysis of vast list of cases ranging from fission to fusion and from criticality, kinetics and shielding to adjustment applications is presented. The status of the available covariance and future needs in the areas including secondary angular and energy distributions is addressed. Of particular concern is the lack of correlation among different isotopes and reactions, the differences among the recent covariance matrices although the crosssection evaluations are mostly based on similar experimental data, and the lack of covariance information for some more specific reactions and reaction types (such as e.g. SAD/SED).
Acknowledgments
The authors acknowledge the financial support from the Slovenian Research Agency (research core funding No. PR07382). Part of the work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 under grant agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
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Cite this article as: Ivan Kodeli, Comments on the status of modern covariance data based on different fission and fusion reactor studies, EPJ Nuclear Sci. Technol. 4, 46 (2018)
All Tables
Uncertainties in k_{eff} and β_{eff} calculated using the SUSD3D code. The two values for the β_{eff} uncertainty correspond to the assumption of no/full correlation among the uncertainties of different actinides. SAD/SED uncertainties are not included.
FNGCu benchmark: uncertainty due to transport crosssections of different origin compared to the C/E values.
Fission spectra uncertainties in k_{eff} and β_{eff} calculated using the approximate “twoblock” prompt fission neutron spectra covariances (i.e. assuming flat anticorrelated 4% standard deviation) compared to those based on covariances from JENDL4.0 and SCALE6.0.
All Figures
Fig. 1 Covariance matrices of ^{239}Pu for from the SCALE6.0 m, ENDF/BVII.1 and JENDL4.0 evaluations. 

In the text 
Fig. 2 SAD covariance matrices of ^{56}Fe elastic crosssections from the EFF2.4, JENDL4.0, ENDF/BVII.1 and TENDL2015 evaluations. Warning: apparent similarity between the JENDL4.0 and ENDF/BVII.1 covariances is only an artifact of log/log scale which is for some reason used in the recent versions of NJOY. Switching back to the old (much more informative) lin/log presentation is strongly recommended (e.g. by redefining the “yrtest” parameter in the COVR module). 

In the text 
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