ISBN 92-64-02302-X

NEA/WPEC–20

International Evaluation Co-operation

VOLUME 20
COVARIANCE MATRIX EVALUATION
AND PROCESSING IN THE RESOLVED/
UNRESOLVED RESONANCE REGIONS

A report by the Working Partyon International Evaluation Co-operationof the NEA Nuclear Science Committee

CO-ORDINATOR
T. Kawano
Kyushu University/
Los Alamos National Laboratory
JAPAN/USAMONITORR. Jacqmin
CEA Cadarache
FRANCE© OECD 2006
NEA No. 6198
NUCLEAR ENERGY AGENCY
ORGANISATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENTORGANISATION FOR ECONOMIC CO-OPERATION AND DEVELOPMENTThe OECD is a unique forum where the governments of 30 democracies work together to address
the economic, social and environmental challenges of globalisation. The OECD is also at the forefront ofefforts to understand and to help governments respond to new developments and concerns, such as corporate
governance, the information economy and the challenges of an ageing population. The Organisation provides
a setting where governments can compare policy experiences, seek answers to common problems, identify
good practice and work to co-ordinate domestic and international policies.
The OECD member countries are: Australia, Austria, Belgium, Canada, the Czech Republic,
Denmark, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Japan, Korea, Luxembourg,
Mexico, the Netherlands, New Zealand, Norway, Poland, Portugal, the Slovak Republic, Spain, Sweden,
Switzerland, Turkey, the United Kingdom and the United States. The Commission of the European
Communities takes part in the work of the OECD.
OECD Publishing disseminates widely the results of the Organisation’s statistics gathering andresearch on economic, social and environmental issues, as well as the conventions, guidelines and standards
agreed by its members.
***This work is published on the responsibility of the Secretary-General of the OECD. The opinions
expressed and arguments employed herein do not necessarily reflect the official views of the Organisation orof the governments of its member countries.

NUCLEAR ENERGY AGENCYThe OECD Nuclear Energy Agency (NEA) was established on 1st February 1958 under the nameof the OEEC European Nuclear Energy Agency. It received its present designation on 20th April 1972, when
Japan became its first non-European full member. NEA membership today consists of 28 OECD member
countries: Australia, Austria, Belgium, Canada, the Czech Republic, Denmark, Finland, France, Germany,
Greece, Hungary, Iceland, Ireland, Italy, Japan, Luxembourg, Mexico, the Netherlands, Norway, Portugal, theRepublic of Korea, the Slovak Republic, Spain, Sweden, Switzerland, Turkey, the United Kingdom and theUnited States. The Commission of the European Communities also takes part in the work of the Agency.
The mission of the NEA is:

to assist its member countries in maintaining and further developing, through international cooperation, the scientific, technological and legal bases required for a safe, environmentallyfriendly and economical use of nuclear energy for peaceful purposes, as well as
to provide authoritative assessments and to forge common understandings on key issues as
input to government decisions on nuclear energy policy and to broader OECD policy analyses
in areas such as energy and sustainable development.

Specific areas of competence of the NEA include safety and regulation of nuclear activities,
radioactive waste management, radiological protection, nuclear science, economic and technical analyses of
the nuclear fuel cycle, nuclear law and liability, and public information. The NEA Data Bank provides nuclear
data and computer program services for participating countries.
In these and related tasks, the NEA works in close collaboration with the International Atomic
Energy Agency in Vienna, with which it has a Co-operation Agreement, as well as with other internationalorganisations in the nuclear field.

© OECD 2006
No reproduction, copy, transmission or translation of this publication may be made without
written permission. Applications should be sent to OECD Publishing: rights@oecd.org or byfax (+33-1) 45 24 13 91. Permission to photocopy a portion of this work should be addressed to the Centre
Français d’exploitation du droit de Copie, 20 rue des Grands-Augustins, 75006 Paris, France(contact@cfcopies.com).

FOREWORD

A Working Party on International Evaluation Co-operation was establishedunder the sponsorship of the OECD/NEA Nuclear Science Committee (NSC) topromote the exchange of information on nuclear data evaluations, validation, andrelated topics. Its aim is also to provide a framework for co-operative activities
between members of the major nuclear data evaluation projects. This includes thepossible exchange of scientists in order to encourage co-operation. Requirements
for experimental data resulting from this activity are compiled. The working
party determines common criteria for evaluated nuclear data files with a view toassessing and improving the quality and completeness of evaluated data.
The parties to the project are: ENDF (United States), JEF/EFF (NEA Data
Bank member countries) and JENDL (Japan). Co-operation with evaluation
projects of non-OECD countries are organised through the Nuclear Data Sectionof the International Atomic Energy Agency (IAEA).
The following report was issued by Subgroup 20, whose task was toinvestigate methods for assessing the uncertainties associated with evaluatednuclear data in the resonance range. These uncertainty (covariance) data areessential for practical applications of evaluated nuclear data, for example for the
adjustment of group constants and the estimation of design accuracies. The
objectives of the subgroup were to:xidentify the needs of the user community;xdevelop a methodology for the production of covariance matrices;x

produce a few examples of covariance matrices and review the formats
used to store the data.

The opinions expressed in this report are those of the authors only and do not
represent the position of any member country or international organisation. This
report is published on the responsibility of the Secretary-General of the OECD.

3

Members of Subgroup 20

CHIBA, GoJapan Atomic Energy Agency
Oarai, Japan

LEAL, Luiz C.
Oak Ridge National Laboratory
Oak Ridge, USA

COURCELLE, Arnaud
CEA Cadarache
St. Paul les Durance, France

MACFARLANE, Robert E.
Los Alamos National Laboratory
Los Alamos, USA

ISHIKAWA, Makoto
Japan Atomic Energy Agency
Oarai, JapanNABEREJNEV, Dimitri G.
Argonne National Laboratory
Argonne, USA

JACQMIN, Robert
CEA Cadarache
St. Paul les Durance, France

SHIBATA, Keiichi
Japan Atomic Energy Agency
Tokai-Mura, JapanKAWANO, Toshihiko
Los Alamos National Laboratory
Los Alamos, USA

TAGESEN, SiegfriedInst. f. Radiumforschung und
Kernphysik der Univ. Wien
Vienna, Austria

LARSON, Nancy M.
Oak Ridge National Laboratory
Oak Ridge, USA

TALOU, Patrick
Los Alamos National Laboratory
Los Alamos, USA

4

TABLE OF CONTENTS

Summary ........................................................................................................... 7
1. Introduction............................................................................................... 9
2. Summary of issues discussed in the subgroup........................................ 10
2.1 Covariance for resonance parameters................................................ 10
2.2 Data size problem ............................................................................... 10
2.3 The retroactive technique ................................................................... 11
2.4 Processing problem ............................................................................ 11
3. Procedure to evaluate resonance parameter covariances ..................... 12
3.1 Retroactive method ............................................................................. 12
3.1.1 Example 1, simple case ........................................................... 13
3.1.2 Example 2, using the SAMMY retroactive option .................. 15
3.2 Compact format .................................................................................. 15
3.3 Processing........................................................................................... 16
3.4 Unresolved resonance region ............................................................. 17
4. Results and discussions............................................................................. 17
5. Concluding remarks ................................................................................. 18
References ......................................................................................................... 21
Figures............................................................................................................... 25

5

SUMMARY

This document serves as a summary of the work of Subgroup 20 (SG20) on
covariance matrix evaluation and processing in the resolved/unresolved resonanceregions, organised under the auspices of the Nuclear Energy Agency’s Nuclear
Science Committee Working Party on International Evaluation Co-operation(WPEC).
The work described in this report focuses on:xsummarising the issues related to covariance evaluation in the resonanceregion;x

discussing the retroactive method used in the SAMMY code [1];xdescribing the compact format for storing huge covariance matrices in
ENDF-6 files;x

recent developments and upgrades of processing codes to generate amulti-group covariance matrix from resonance parameter covariance
data.

7

COVARIANCE MATRIX EVALUATION AND PROCESSING
IN THE RESOLVED/UNRESOLVED RESONANCE REGIONS

1.

Introduction

Requirements for covariance data for advanced technologies of nuclear
energy applications are steadily growing. The evaluation of covariance data is,however, difficult and normally requires more effort than the cross-section
evaluation itself. To provide the covariance data for nuclear data users is to givea qualitative assessment of the data library. Major customers of the covariancedata are the reactor core calculations (estimation of uncertainty in the keff,
criticality safety study, adjustment of nuclear data libraries [2,3]) and radiation
shielding designs.
In WPEC/Subgroup 2 (SG2) [4], methods to evaluate the covariance of 56Fe
were investigated. After the completion of SG2, extensive progress was madeby the JENDL group, and the JENDL-3.2 covariance file [5] for some importantnuclides for fast-reactor applications was released. In practice this covariancefile would be used for a few limited purposes. However, a long-standing problem
remains, concerning the resonance region. For the JENDL-3.2 covariance file in
the resonance region, a simple method was adopted to estimate the covariancematrices of the 235,238U and 239Pu resolved-resonance parameters [6]. The
technique was not applicable to the 235U resonance parameters in JENDL-3.3,
because there are more than 3 000 resonances in one energy region.
As many nuclear data libraries – ENDF, JENDL and JEFF, for example –adopted the same 235U resonance parameters obtained by Leal, Derrien, Larson
and Wright [7], the issue to be resolved is of general interest among the nuclear
data communities in the world.
Our main objective is to evaluate and report the resonance parameter
covariance matrix in such a way that processing codes can make use of it. Boththe method and some of the tools for the evaluation are described in this report.
To demonstrate our achievements, we first apply our method to the gadolinium
resonance parameters as a simple example, and then extend the technique into
the actinide region.

9

2.

Summary of issues discussed in the subgroup

2.1 Covariance for resonance parameters
An alternative to providing resonance parameter covariance matrices (RPCM)
would be to evaluate uncertainties in multi-group-averaged cross-sections [8].
In this method, nuclear data users would find the covariance data in MF=33 of
ENDF-6 files, not in MF=32. Although this may be practical for estimating theuncertainties in the resonance region for one energy group structure, the RPCM
method is preferable for the following reasons:xUse of the RPCM method makes it possible to calculate uncertainties
for a self-shielding factor.

x

The cross-section covariance matrix (CSCM) depends on the energy
group structure used. It is easy to construct a covariance of groupedcross-sections in any kind of structure if the RPCM is available.

x

Correlations between different reaction channels, such as fission andcapture, can be generated using the RPCM.

Nevertheless, the CSCM method may be used for those cases where
high-quality covariance data are not required. One example is resonanceparameters for fission products with weak absorption, because few people use
this kind of covariance data at present. For such cases, the simple solution is toestimate uncertainties in the averaged capture or total cross-sections in theresonance region, and give them in MF=33.
2.2 Data size problem
In the current ENDF-6 format, the RPCM is stored in MF=32, for which thenumber of elements is NJNp u (NJNp + 1)/2, where NJ is the number ofresonances and Np is the number of parameters per resonance. If there are 100
resonances with three parameters (E, *n, *J), which might be a common case formany fission-product and structural materials, there are 45 150 elements (= 7 525
lines), and the file size is about 600 kB. It is still possible to store everything ina single file.
However, when the number of resonances is large, this format is notpractical. In the case of 235U, there are more than 3 000 resonances, and eachresonance has five parameters. The covariance file for the resonance parameters
may exceed one gigabyte.

10The compact format proposed by Larson [9] permits storage of the fullcovariance matrix within a reasonable file size by dropping information that is
not so important. This new format is essential to store the huge covarianceof resonance parameters of actinides. The proposal was approved by the
Cross-section Evaluation Working Group (CSEWG) in 2004, and now it is apart of the official ENDF-6 format.
2.3 The retroactive technique
For new resonance analyses, uncertainty information is obtained as aby-product of the fitting procedure used to determine the resonance parameters.
Resonance analysis codes such as SAMMY [1] first generate the RPCM andthen write it into an appropriate format, i.e. either the original ENDF-6 or the
new compact format. The resonance parameter set and the RPCM data are thenincluded in a single ENDF file.
However, it is not always possible or practical to perform completely new
evaluations. Instead, RPCM must be found for resonance parameters in theexisting nuclear data libraries. To generate these RPCM, it is necessary tocollect information regarding how the resonance analysis was carried out in thepast, and mimic the analysis procedure to obtain uncertainty information for theparameters.
The retroactive technique is essentially a simulation of previous resonanceanalyses. The reliability of the RPCM obtained with the retroactive method
depends on the accuracy with which the information used for the resonanceanalysis can be reconstructed. This information includes experimental data,
resolution functions of experiments and other systematic error sources in the data.
In the retroactive technique, experimental data are recreated from the given
resonance parameters. Uncertainties in these simulated data are estimated fromexperimental conditions, etc., and should be realistic. The RPCM is then found by
propagating all the errors from the simulated data to the resonance parameters,
i.e. by using the R-matrix code to fit the simulated data by varying the resonanceparameters. If the resulting fitted parameter values are very nearly the same as
the original resonance parameter values, then the associated RPCM is expected
to be a good approximation of the RPCM for the original resonance parameters.
2.4 Processing problem
The current release of the NJOY [10] nuclear data processing system canprocess neither the Reich-Moore [11] resonance parameter covariance matrix
11nor the compact format. However, the ERRORJ code [12,13] can process bothof them. This code can be used as a module of NJOY to generate the covariancematrix for multi-group cross-sections. We have encouraged the development of
ERRORJ throughout the subgroup activity.
The ERRORJ code [12] was initially written by Kosako and Yamano of
Sumitomo Atomic Energy Industries as a means of processing the JENDL-3.2
covariance file [5]; the code was then taken over by Chiba and Ishikawa [5] ofJapan Nuclear Cycle Development Institute (JNC) [present affiliation, Japan
Atomic Energy Agency (JAEA)]. They developed the code in order to make it
usable with the other libraries, and implemented new features. The code has
been distributed to many institutes, including JAEA, Toshiba, ORNL, ANL,
LANL, NEA Databank, IAEA and IPPE.
Recently, ORNL established a new capability for processing covariancedata. The PUFF code [14,15] can now process resonance parameter covariancematrices in both the original and the new compact format.
Although SAMMY is not a processing code, it also has the capability ofgenerating the multi-group averaged cross-sections and the associated covariancematrix in the resonance region. SAMMY has therefore been used to check themulti-group cross-section covariances generated by ERRORJ and PUFF.
3.

Procedure to evaluate resonance parameter covariances

3.1 Retroactive method
The retroactive method is a simulation of the resonance analysis. Instead of
performing a full resonance analysis, it is simulated with reconstructed fictitious
experimental data. Figure 1 shows a usual procedure of resonance analysis.
Point-wise cross-sections are generated once from initial parameters taking into
account the experimental resolution function, Doppler broadening, self-shielding
effects and so forth, and the calculated cross-sections are compared with theexperimental data. With a Bayesian update technique, a set of resonanceparameters and the associated covariance matrix are obtained, which give agood fit to the experimental data within the accuracy of the data.
The retroactive method (Figure 2) starts with the existing resonanceparameters. We assume that resonance parameters in the nuclear data libraries
are the best estimates based on the experimental data available for the data
analysis. Point-wise cross-section data (or transmission data) are reconstructedfrom the resonance parameters with the appropriate resolution function and the12target temperature. A typical resolution function can be found at several
time-of-flight (TOF) facilities, such as ORELA (ORNL) and GELINA (IRMM).
The data generated are used as the simulated experimental data, and theparameter fitting to the data is performed. The uncertainties in the simulated
data are also simulated, using typical statistical and systematic errors found in
the literature. The parameter fitting gives F2 very close to zero, and there isessentially no change in the parameters, because the generated experimental andcalculated cross-sections are identical.
Larson has implemented the retroactive covariance generation option into
the SAMMY code, and we recommend that this capability be used for covarianceevaluations. The JENDL-3.2 resonance covariance data [6] adopted a similar
but simplified method. Table 1 shows a comparison of methods used in
JENDL-3.2 and those used in SAMMY.
Table 1. Comparison of JENDL-3.2/3.3 and SAMMY retroactive methods

Data point

JENDL
Averaged over several
resonances

Resolution

N/A

Uncertainties Inferred from realexperiments, but rounded
into a single value – 5% forexampleCorrection
N/A

Sensitivity

Sensitivity to the V

SAMMY retroactive
Reconstructed (simulated)from original resonanceparameters
Taken from typicalexperiments
Estimated from realexperiments, can be rounded
into a single value
Doppler and resolution
broadening, multiple
scattering, normalisation,and background, etc.
At each data point

3.1.1 Example 1, simple case
The covariance matrix of resonance parameters obtained in this manner
reproduces the accuracy of the simulated data, so the generated data uncertainties
should be realistic. As a simple example, we ignore all the data broadening anddistortion effects, and just calculate cross-sections with the given resonanceparameters. Figure 3 shows the change in the capture cross-sections of 156Gd
when the capture width *J of the 33 eV resonance was increased by 50%. This13

results in a 30% increase in the calculated capture cross-section at 10 eV, and a
relative sensitivity of 30/50 = 0.6. If we assume that the uncertainty in the capture
cross-section at 10 eV is 5%, *J should have an uncertainty of 5/0.6 = 8.3%.
We generate the “simulated experimental data” using the 156Gd resonanceparameters in ENDF/B-VI, which were evaluated using the Multi-level
Breit-Wigner formula. First, we consider the capture data only. Important
quantities are the number of data points included (relevant to the TOF data
energy interval) and the energy resolution. Here we consider simple Gaussianbroadening due to the experimental energy resolution. In the measurement of
the 99Tc capture cross-section [16] at the KURRI TOF facility, there are about
100 energy points in the TOF data. Sources of uncertainty in the neutron capture
measurement are statistical errors (3-10%), normalisation errors (6%) and othercorrections (about 2%). We assumed that the simulated neutron capture data areof a similar quality as the KURRI 99Tc data, so 10 uncertainties in the capture
cross-sections with 50% correlation that comes from normalisation are given to
the simulated data. The correlation coefficients tend to be large because of theuncertainties in the data normalisation. The energy resolution (FWHM) of 1%
was also taken from the real experiment.
It should be emphasised that the data uncertainties are not copied from the
KURRI experiments. Rather, the uncertainty information is estimated by referring
to a typical experimental set-up, assuming that accuracies of measurements ateach experimental facility are not very different.
The KALMAN code [17] was used to estimate the RPCM. The CSCM
was reconstructed from the RPCM and the sensitivity matrices; calculated
uncertainties in the capture cross-section are shown in Figure 4. The average
uncertainty in the capture cross-sections in the energy range 0-1 keV is 5.4%.
The total cross-section has larger uncertainties, because transmission data were
not included in this evaluation.
The neutron transmission data for gadolinium isotopes were analysed by
Mughabghab and Chrien [18]. Since no experimental transmission data are
available in EXFOR, we simulated the data and included realistic estimates ofexperimental conditions. The simulated data are based on the following
assumptions: transmission data energy interval of GE ~ 0.01En (obtained fromthe literature), energy resolution of 1% and 10% total uncertainties with 50%
correlation. We did not include the data in the off-resonance region, because
they are not informative. The calculated uncertainties in the total cross-sections
are shown in Figure 5. The average uncertainty is 2%. Since capture data were
not used in this calculation, the uncertainties in the capture cross-section
became unrealistic.

14

The combined result, which includes simulated total and capture data, isshown in Figure 6. The average uncertainty in the total cross-section is 1.7%,
and in capture is 3.2%.
3.1.2 Example 2, using the SAMMY retroactive option
Experimental data in the resonance region, as discussed before, often
reveal a strong correlation due to the data normalisation. If the data are strongly
correlated, statistical errors are not so important because they disappear whenmany data points are included, and only systematic (normalisation) error
remains. In this situation the retroactive analysis of the resonance covariancewith SAMMY becomes fairly efficient.
The normalisation error is nn important quantity in the SAMMY retroactive
method. We again employ the gadolinium isotope as an example. It is assumed
that the capture normalisation error is 5% and the total normalisation error is
3%. Those values were chosen so as: (1) to reproduce the evaluated uncertainties
in the thermal cross-sections reported by Mughabghab [19]; and (2) to bereasonably consistent with realistic situations of measurements.
Figure 7 shows generated neutron capture cross-sections for 155Gd withSAMMY. The data were made for 12 370 energy points in the energy range
0-183.3 eV, which covers the entire resolved-resonance region. The total
cross-section data were also generated on the same energy grid. The statisticaluncertainty in the capture reaction was assumed to be 20%, and in the total was
2%. Those values do not have a large impact on the final result. Note that those
uncertainties are just statistical, and they are not correlated.
SAMMY generates the RPCM by fitting the resonance parameters to thesimulated experimental data. The resonance parameters do not change during
the fitting process, because the calculated and experimental cross-sections are
the same. The RPCM is then written into the ENDF-6 format (standard ENDF-6
or compact format); this RPCM is shown in the standard format in Figure 8.
3.2 Compact format
The compact covariance matrix format proposed by Larson drasticallyreduces the file size of resonance parameter covariance. The format was approved
by CSEWG, and the ENDF-6 data format manual (Revision 2005) [9] explains
this format in detail. In this format, the covariance of resonance parameters isdecomposed into the correlation matrix and uncertainties. The correlation15

coefficients, which range from -1.0 to 1.0, are scaled by a factor of 100, andexpressed by signed integer numbers, from -99 to 99.
The following procedure to compact the correlation coefficients is definedin the ENDF-6 manual [9]:1. Drop (set to zero) all values of correlation between -0.02 and +0.02.
2. Multiply the remaining coefficients by 100.
3. Map all positive values grater than K and less than or equal to K + 1 tothe integer K.
4. Map all negative values less than -K and greater than or equal to -K – 1
to the integer K.
As an example, the resonance parameter covariance for 157Gd in thecompact format is shown in Figure 9.
The compact covariance matrix format has a great advantage in storing theresonance parameter covariances of actinides. For many of the major actinides
such as 233,235U, only the compact format can hold the complete covariance datain the nuclear data library. Leal, et al. [20] performed the R-matrix resonanceanalysis of 233U, and the full covariance matrix of 769 resonance parameters
were obtained with SAMMY. When the data are stored using ENDF-6 format,
the file size becomes 94 MB; with the compact format, the size is a mere 2 MB.
3.3 Processing
The ERRORR module of NJOY [10] is able to process the RPCM if
it is given by the Multi-level Breit Wigner (MLBW) formula. A long-standing
problem regarding the processing code, however, involves processing thecovariances of the ENDF LRF=3 format (the so-called Reich-Moore format) [11].
The problem was resolved by the JENDL project. The ERRORJ code [12],an improved version of ERRORR, was developed to process the JENDL-3.2
covariance file [5]. The code was later extensively improved by Chiba andIshikawa [13], and Chiba reported on the current status of the ERRORJ code at
the workshop of “Perspectives on Nuclear Data for the Next Decade” [21].
To generate the covariance matrix of multi-group cross-sections, theprocessing code calculates error propagation from the resonance parameter
covariance to the cross-section, for which it requires the sensitivity coefficients
wVi/wpj, where Vi is the i-th group cross-section, and pj is the resonance16

parameter. ERRORJ calculates the sensitivity coefficients numerically. Thesenumerical derivatives have been tested through comparison with SAMMY’s
analytic calculations of wVi/wpj.
Recently Wiarda made improvements in the PUFF code [15] at ORNL.
PUFF now has the capability to process a LRF=3 resonance parameter covariancematrix, using the same analytical derivatives as SAMMY.
3.4 Unresolved resonance region
Issues to be discussed regarding the unresolved resonance region mainly
concern the format, comprising two parts: (1) uncertainty on the nuclear radius,and (2) energy-dependent unresolved resonance parameters.
The current ENDF-6 format does not include provision for an uncertainty
on the nuclear radius. Due to this limitation, the calculated uncertainties in the
averaged cross-sections in the unresolved resonance region tend to be small, as
discussed by Kawano and Shibata [22]. However, the nuclear radius is usually
held fixed while performing a resonance analysis and the uncertainty in the radiusparameter should be reflected (absorbed) by the uncertainties in the resonanceparameters themselves.
The covariance of unresolved resonance parameters must be in one energy
region, even if energy-dependent unresolved resonance parameters are given in
MF=2. This problem would be resolved by applying the retroactive evaluation
technique to the unresolved resonance parameters. The energy-independentunresolved resonance parameters are fitted to the simulated cross-sections
generated from the energy-dependent parameters.
Those two issues require modifications to the current ENDF-6 format.
However, motivation for the new format proposal is not that great, as the current
format is still able to accommodate the covariance data in the unresolved region
as described above.
4.

Results and discussions

Resonance parameter covariance matrices for the gadolinium isotopes for
ENDF/BVII were evaluated by Leal [23]. They were compiled into the ENDF-6
format files, and tested by processing with ERRORJ.

17

The relative uncertainties in the total and elastic cross-sections are shownin Figure 10. The uncertainties in the capture cross-section are very similar to
the total cross-section, and omitted from this plot. Figure 11 shows the correlation
matrix of 155Gd multi-group total cross-section in the resonance region (up to
100 eV), generated with ERRORJ. Regions with negative correlation coefficientare shown by the solid-line boxes. If the covariance in the resonance region isevaluated based on the averaged cross-section data, the correlation matrix may
have positive elements only, due to systematic (normalisation) errors in theexperimental data used. Negative correlations occur because of the introductionof the physics of resonances into the data analysis.
The first three resonances of 155Gd are at 0.0268 eV, 2.008 eV and 2.568 eV,shown by arrows in Figure 12. Below the first resonance the cross-section has asimple 1/v shape, so that those cross-sections should be almost fully correlated.
The resonance analysis tells us that the cross-sections between the first andsecond resonances must be anti-correlated to the first resonance region.
The correlation matrices for the elastic scattering and capture cross-sections
are shown in Figures 13 and 14. The correlation matrix of capture is similar to
the total cross-section in Figure 11. The cross-correlation between different
reaction channels can be given by the RPCM. Figure 15 shows the correlation
matrix between the elastic and capture channels generated by ERRORJ. Suchcross-correlation is especially important at thermal energy. For fissile materials
like 235U and 239Pu, we expect a strong correlation between the fission and thetotal cross-sections at thermal energy, as the fraction of fission in the total
absorption is very large. For the fertile and non-fissioning materials, the total
and capture cross-sections may be strongly correlated. When using the RPCM,
such a physical property is automatically taken into account.
Leal, et al. [24] presented a method to generate covariance data for 233U
resonance parameters using SAMMY. The 233U covariance data were processedwith ERRORJ to produce COVERX formatted data, and benchmark calculations
were performed with TSUNAMI. It was shown that the contribution of 233U
resonance parameter uncertainties to keff is about 0.5% for the 233U thermal
system (u233-sol-therm015).
5.

Concluding remarks

We have discussed several important issues concerning the covarianceevaluation in the resolved and unresolved resonance regions, including dataprocessing. For existing resonance parameters, we showed that the retroactive
technique is a powerful method for generating resonance parameter covariance18matrices (RPCM). The retroactive method has been implemented in the SAMMY
code, and examples were shown for the covariance data of gadolinium isotopes.
Resonance parameter covariance matrices tend to be large; it was shownthat the compact format drastically reduces the file size. This new format wasapproved by the CSEWG in 2004, and it is now a part of the official ENDF-6
format. The compact format enables storage of the full RPCM in the evaluatednuclear data library within a reasonable file size.
To process the Reich-Moore resonance parameter covariance to produce
multi-group cross-section covariance data, two processing codes are currently
available – ERRORJ and PUFF. Both codes are also able to read the compact
format. We presented the multi-group cross-section covariance of 155Gd.

19

REFERENCES[1]

Larson, N.M., Updated Users’ Guide for SAMMY: Multilevel R-matrix
Fits to Neutron Data Using Bayese’ Equation, ORNL/TM-9179/R6, Oak
Ridge National Laboratory (2003).

[2]

Ishikawa, M., “Development of a Unified Cross-section Set ADJ2000
Based on Adjustment Technique for Fast Reactor Analysis”, J. Nucl. Sci.
Technol., Suppl.2, 1073 (2002).

[3]

Ishikawa, M., “Recent Application of Nuclear Data to Fast Reactor Core
Analysis and Design in Japan”, Proc. Int. Conf. Nuclear Data for Scienceand Technology, Santa Fe, New Mexico, USA, 26 Sept.-1 Oct. 2004,
p. 1405 (2005).

[4]

WPEC/Subgroup 2 (Co-ordinator H. Vonach), Generation of Covariance
Files for Iron-56 and Natural Iron, Nuclear Energy Agency, OECD
(1996).

[5]

Shibata, K., A. Hasegawa, O. Iwamoto, S. Chiba, M. Sugimoto,
N. Odano, T. Kawano, Y. Nakajima, T. Murata, H. Matsunobu, S.Y. Oh,
K. Yokoyama, K. Sugino, M. Ishikawa, K. Kosako, N. Yamano, Y. Kanda,“JENDL-3.2 Covariance File”, J. Nucl. Sci. Technol., Suppl. 2, 40 (2002).

[6]

Kawano, T., K. Shibata, J. Nucl. Sci. Technol., 39, 807 (2002).

[7]

Leal, L.C., H. Derrien, N.M. Larson, R.Q. Wright, “R-matrix Analysis of235
U Neutron Transmission and Cross-sections in the Energy Range 0 to
2.25 keV”, ORNL/TM-13516, Oak Ridge National Laboratory (1997);
Nucl. Sci. Eng., 131, 230 (1999).

[8]

Shibata, K., J. Nucl. Sci. Technol., 42, 130 (2005).

[9]

Larson, N.M., “Compact Covariance Matrix”, in ENDF-102 Data
Formats and Procedures for the Evaluated Nuclear Data File ENDF-6,
BNL-NCS4494505-Rev., Cross-section Evaluation Working Group,
Brookhaven National Laboratory, NY, USA (2005).

21

[10] MacFarlane, R.E., D.W. Muir, The NJOY Nuclear Data Processing
System Version 91, LA-12740-M (1994); The NJOY Nuclear Data
Processing System Version 99 (1999).
[11] Reich, C.W., M.S. Moore, Phys. Rev., 111, 929 (1968).
[12] Kosako, K., “Covariance Data Processing Code: ERRORJ”, Proc. of theSpecialists Meeting on Reactor Group Constants, J. Katakura, Ed., JAERI,
Tokai, Japan, 22-23 February 2001, JEARI-Conf 2001-009, p. 30, Japan
Atomic Energy Research Institute (2001); K. Kosako and N. Yamano,
Preparation of a Covariance Processing System for the Evaluated Nuclear
Data File, JENDL (III), JNC TJ9440 99-003, Japan Nuclear Cycle
Development Institute (1999) [in Japanese].
[13] Chiba, G., M. Ishikawa, “Revision and Application of the Covariance Data
Processing Code, ERRORJ”, Proc. Int. Conf. Nuclear Data for Scienceand Technology, Santa Fe, New Mexico, USA, 26 Sept.-1 Oct. 2004,
p. 468 (2005).
[14] Dunn, M.E., PUFF-III: A Code for Processing ENDF Uncertainty Data
Into Multigroup Covariance Matrices, Oak Ridge National Laboratory,
NUREG/CR-6650 ORNL/TM-1999/235 (2000).
[15] Wiarda, D., M.E. Dunn, N.M. Greene, N.M. Larson, L.C. Leal, “NewCapabilities for Processing Covariance Data in the Resonance Region”,
PHYSOR 2006 [to be published].
[16] Kobayashi, K., S. Lee, S. Yamamoto, T. Kawano, Nucl. Sci. Eng., 146,
209 (2004).
[17] Kawano, T., K. Shibata, Covariance Evaluation System, JAERI-Data/Code97-037, Japan Atomic Energy Research Institute (1997) [in Japanese].
[18] Mughabghab, S.F., R.E. Chrien, Phys. Rev., 180, 1131 (1969).
[19] Mughabghab, S.F., “Atlas of Neutron Resonances”, Academic Press(2006).
[20] Leal, L.C., H. Derrien, J.A. Harvey, K.H. Guber, N.M. Larson,
R.R. Spencer, R-matrix Resonance Analysis and Statistical Properties of
the Resonance Parameters of 233U in the Neutron Energy Range from
Thermal to 600 eV, ORNL/TM-2000/372, Oak Ridge National Laboratory(2001).

22[21] Chiba, G., “Resonance Self-shielding in Covariance Processing”, Proc.
Workshop of Perspectives on Nuclear Data for the Next Decade,
CEA/BRC, France, 26-28 Sept. 2005 [to be published].
[22] Kawano, T., K. Shibata, “Evaluation of Covariance Matrices for Resolved
and Unresolved Resonance Regions”, Proc. of the 11th Int. Symp. on
Reactor Dosimetry, Brussels, Belgium, 18-23 Aug. 2002, J. Wagemans,
H.A. Abderrahim, P. D’hondt, C. De Raedt, Eds.,World Scienti¿FV
p. 631 (2003).
[23] Leal, L. (2006) [private communication].
[24] Leal, L., G. Arbanas, N. Larson, H. Derrien, D. Wiarda, Nuclear Data
Covariance Generation in the Resonance Region, Talk at CSEWG &USNDP Meeting, Brookhaven National Laboratory, USA, 8-11 Nov. 2005.

23

FIGURES

25

Figure 1. Procedure to obtain resonance parameters
Resonance parameter

UpdateResolution function
Doppler broadening

Experimental data
Point-wise cross-sec.
and covariance Compare

Resonance parameter

Parameter covariance

Figure 2. Retroactive method
Resonance parameter(existing)

Resolution function
Doppler broadening

Generated dataand covariance

Point-wise cross-sec.
Identical

Parameter covariance27

Figure 3. Sensitivity of the capture width of the firstresonance to the capture cross-sections for 156Gd
1000

Capture Cross Section [b]Gd-156 Capture
100

Gamma-width, 50% increase

1030% increase
1

0.1

0.01
1

10
Neutron Energy [eV]100

Figure 4. Calculated uncertainties in the 156Gd
capture cross-sections, assuming the simulated capture
data have 10% uncertainties with 50% correlation
The arrows show the location of resonances

Uncertainties [%]30

Total Cross Section
Capture Cross Section

20
Resonance

10

Average = 5.4%
0
0

50

100

150

200

Neutron Energy [eV]

28

250

300

Figure 5. Calculated uncertainties in the 156Gd total
cross-sections, assuming the simulated total cross-section
data have 10% uncertainties with 50% correlation

Uncertainties [%]30

Total Cross Section
Capture Cross Section

20

10

Average=2%
0
0

50

100
150
200
Neutron Energy [eV]

250

300

Figure 6. Calculated uncertainties in the156
Gd capture and total cross-sections

Uncertainties [%]30

Total Cross Section
Capture Cross Section

20
Average (Capture) = 3.2%

Average (total) = 1.7%
10

0
0

50

100
150
200
Neutron Energy [eV]

29

250

300

Figure 7. Generated 155Gd capture cross-sections using SAMMY
The dotted and dashed curves are the 20% uncertainty band

Cross Section [b]

10000

1000

100

Capture
+ 20%
- 20%
10
0

0.5

1

1.5

2

2.5

3

Neutron Energy [eV]

Figure 8. ENDF-6 formatted 155Gd resonanceparameter covariance generated by SAMMY
64155.0
1.535920+2
64155.0
1.000000+0
1.000000-5 1.833000+2
1.57.900000-1
1.535920+2

0643432151
0643432151
1643432151
0643432151
0643432151
92643432151
643432151
643432151
643432151
643432151

1
2
3
4
5
6
7
8
9
10

175.5999999
2.0
2.080000-3 1.098000-1
643432151
178.0000000
1.0
9.733333-3 1.098000-1
643432151
180.3999999
1.0
1.466667-2 1.098000-1
643432151
1.875531-7 .3376958-9-2.264001-7 .7678523-9 .0042410-9 2.186969-8643432151.0196131-9-.0722565-9 1.476097-8 1.785823-9-.0036186-9-5.272415-9643432151-.0217329-9-.0853346-9 1.741534-9-.0046285-9-.0457875-9 .4604478-9643432151.0003844-9-.0071501-9-.8064558-9 .0359892-9-.0248011-9-.5288035-9643432151-.0759183-9-.0364101-9-.4262018-9-.0668140-9-.1325805-9-.5289678-9643432151......

96
97
98
99
100
101
102
103

0.026799999
2.007999999
2.568000000
3.616000000......

2.0
1.0
2.0
1.0

0
0
0
0
1
3
0
1
0
0
3
0
1.040000-4 1.080000-1
3.706666-4 1.100000-1
1.744000-3 1.110000-1
4.400000-5 1.300000-1

1
1
0
0
1
38778

8.837309-9 1.220777-8-7.495823-7 1.732927-8-1.512111-7-2.818956-9643432151 6467
3.366244-9-2.203100-8 3.179969-5 3.093391-7 8.139690-8-4.179741-6643432151 6468
6.019313-6-3.344716-8 6.208129-7 3.055970-8-2.288672-7 1.823992-5643432151 6469

30

Figure 9. The ENDF-6 resonance parameter
covariance of 157Gd in the compact format
6.415700+4 1.555760+2006.415700+4 1.000000+0001.000000-5 3.066000+2
13
1.5
8.000000-1
1
26.415700+4300.030874750
2.0
4.633883-4 1.072601-1
0.004709259
4.046326-5 1.809663-3
2.825350651
2.0
3.445999-4 9.710919-2
0.070088626
8.111083-6 1.940976-3
16.24198188
1.0
3.992890-4 9.102763-2......

1
2
0
1
720

300.9014716
1.0
5.518277-2 1.016247-1
0.007512823
8.754985-4 1.692468-3
306.4155768
2.0
2.815115-3 1.000446-1
0.013196548
5.442585-5 1.975326-3
0.0
0.0
0
180
2
1 99
3
1 -99 0
5
4 52
8
7 11
9
7
2
10
7 -1 4 1
11
1 -1 -1 -1
1 -2
-9
12
3 -3
5 -2 -1
-64
13
12
1
14
1 -1 -1 -1
1 4 1 -2
15
3 -2
-1
-3 -9 -5-72
16
13
2 1 2
17
11
3 1 -2 -1-12-15
18
13 -2 -3-11-12 1
19
15
1
1
20
11
3
3 -3
2 -1 -1......

314

0644032151
0644032151
1644032151
4644032151
60644032151
644032151
644032151
644032151
644032151
644032151

1
2
3
4
5
6
7
8
9
10

644032151
644032151
644032151
644032151
314644032151
644032151
644032151
644032151
644032151
644032151
644032151
644032151
644032151
644032151
644032151
644032151
644032151
644032151
644032151
644032151
644032151

122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142

Figure 10. Uncertainties in the multi-groupedcross-sections of 155Gd generated by ERRORJ
The solid line is for the total and dashed line is for the elastic scattering cross-sections
3
Total
Elastic

Uncertainty [%]

2

1

0
0.001

0.01

0.1

1
Energy [eV]

31

10

100

Figure 11. Correlation matrix of 155Gd total cross-sectionsin the resonance region, generated by ERRORJ
The regions surrounded by the solid-line boxes have negative correlations
100
1
10

Neutron Energy [eV]

0.5
0

1

-0.5
0.1

-1

0.01

0.001
0.001

0.01

0.1
1
Neutron Energy [eV]

10

100

Figure 12. Zoomed plot of Figure 11, in the energy range 0.01-5 eV
Locations of three resonances of 155Gd are shown by the arrows

1
0.5

Neutron Energy [eV]

1

0
-0.5
-1

0.1

0.01
0.01

0.1
1
Neutron Energy [eV]0.0268 eV

2.008 eV32

2.568 eV

Figure 13. Correlation matrix of 155Gd elastic
scattering cross-sections in the resonance region
The regions surrounded by the solid-line boxes have negative correlations
100
1
10
Neutron Energy [eV]

0.5
0

1

-0.5
0.1

-1

0.01

0.001
0.001

0.01

0.1

1

10

100

Neutron Energy [eV]

Figure 14. Correlation matrix of 155Gd
capture cross-sections in the resonance region
The regions surrounded by the solid-line boxes have negative correlations
100
1
10
Neutron Energy [eV]

0.5
0

1

-0.5
0.1

-1

0.01

0.001
0.001

0.01

0.1
1
Neutron Energy [eV]

33

10

100

Figure 15. Correlation matrix of 155Gd between
elastic and capture reactions in the resonance region
The horizontal axis is the energy of elastic scattering, and the vertical axis is the capture energy.
The regions surrounded by the solid-line boxes have negative correlations.

Neutron Energy (capture channel) [eV]

100
1
10
0.5
0

1

-0.5
0.1

-1

0.01

0.001
0.001

0.01
0.1
1
10
Neutron Energy (elastic channel) [eV]

34

100

OECD PUBLICATIONS, 2 rue André-Pascal, 75775 PARIS CEDEX 16
Printed in France.