Combined Use of Integral Experiments and Covariance Data

Available online at www.sciencedirect.com

Nuclear Data Sheets 118 (2014) 596–636 www.elsevier.com/locate/nds

Combined Use of Integral Experiments and Covariance Data G. Palmiotti,1, ∗ M. Salvatores,2, 1 G. Aliberti,3 M. Herman,4 S.D. Hoblit,4 R.D. McKnight,3 P. Obloˇzinsk´ y,4 P. Talou,5 G.M. Hale,5 H. Hiruta,1 T. Kawano,5 C.M. Mattoon,6 4 G.P.A. Nobre, A. Palumbo,4 M. Pigni,7 M.E. Rising,5 W.-S. Yang,3, 8 and A.C. Kahler5 1 Idaho National Laboratory, Idaho Falls, ID 83415, USA Commissariat a l’Energie Atomique, DRN Cadarache, France 3 Argonne National Laboratory, Argonne, IL 60439, USA 4 National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY 11973, USA 5 Los Alamos National Laboratory, Los Alamos, NM 87545, USA 6 Lawrence Livermore National Laboratory, Livermore, CA 94550, USA 7 Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA 8 Purdue University, West Lafayette, IN 47907, USA 2

In the frame of a US-DOE sponsored project, ANL, BNL, INL and LANL have performed a joint multidisciplinary research activity in order to explore the combined use of integral experiments and covariance data with the objective to both give quantitative indications on possible improvements of the ENDF evaluated data files and to reduce at the same time crucial reactor design parameter uncertainties. Methods that have been developed in the last four decades for the purposes indicated above have been improved by some new developments that benefited also by continuous exchanges with international groups working in similar areas. The major new developments that allowed significant progress are to be found in several specific domains: a) new science-based covariance data; b) integral experiment covariance data assessment and improved experiment analysis, e.g., of sample irradiation experiments; c) sensitivity analysis, where several improvements were necessary despite the generally good understanding of these techniques, e.g., to account for fission spectrum sensitivity; d) a critical approach to the analysis of statistical adjustments performance, both a priori and a posteriori; e) generalization of the assimilation method, now applied for the first time not only to multigroup cross sections data but also to nuclear model parameters (the “consistent” method). This article describes the major results obtained in each of these areas; a large scale nuclear data adjustment, based on the use of approximately one hundred high-accuracy integral experiments, will be reported along with a significant example of the application of the new “consistent” method of data assimilation. CONTENTS

I. INTRODUCTION: EARLY STATISTICAL ADJUSTMENT STUDIES AND THE PROBLEM OF UNCERTAINTIES II. THE NEA-WPEC SUBGROUP 26 AND TARGET ACCURACIES. THE DOE NUCLEAR DATA ADJUSTMENT PROJECT

597

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III. COVARIANCE DATA: FROM EDUCATED GUESS TO COMMARA-3.0 599 A. COMMARA Evaluation Methodology 601 1. Light Nuclei 601

∗

Corresponding author: [email protected]

http://dx.doi.org/10.1016/j.nds.2014.04.145 0090-3752/2014 Published by Elsevier B.V.

2. Structural Materials and Fission Products 3. Actinides - Cross Sections 4. Actinides - Nubars and Fission Spectra B. Validation of Covariance Data IV. INTEGRAL EXPERIMENT SELECTION AND ANALYSIS A. Integral Experiment Selection B. Integral Experiment Analysis 1. Integral Experiment Analysis: ZPR-6 Assembly 7 (ZPR-6/7) 2. Integral Experiment Analysis: Irradiation Experiments PROFIL-1 and -2 V. SENSITIVITY STUDY A. Multiplication Factor and Reactivity

602 603 603 605

606 606 608 608

609 612

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Coefficients B. Reaction Rate Ratios or Spectral Indices C. Isotope Buildup D. Results

612 612 613 613

sis. Early proposals were presented and discussed at the United Nations Conference on the Peaceful Uses of Nuclear Energy in Geneva in 1964 [1, 2]. Successively during the 1970’s and ’80’s, the development of more powerful computers allowed for a continuous improvement of the analytical tools and the reduction of approximations used in the solution of the Boltzmann equation and in the multigroup generation algorithms. In the process the major source of uncertainty in the assessment of the total neutron balance and of safety and operational parameters was found to be in nuclear cross sections data. At the same time a very large number of high accuracy integral experiments were performed in several critical facilities all over the world, providing evidence of potential significant uncertainties if extrapolation to design was to be attempted. Generally, two types of integral experiments can and have been used in the validation of simulation tools:

VI. COVARIANCES OF INTEGRAL PARAMETERS 614 A. Application to Criticality Experiments keff 616 B. Application to Reaction Rate Ratio (RRR) Experiments 616 C. Application to Irradiation Experiments 617 VII. A GLOBAL MULTIGROUP ADJUSTMENT A. Experiment and Parameters Down Selection B. Value of the Experiments and Adjustment Margins C. Analysis of Specific Cross Section Adjustments D. On the “A-posteriori” Correlations Between Parameters and Experiments

618 618 619

• Mock-up experiments (“global” validation): in this case there is the need for a very close experimental simulation of a reference configuration. Bias factors can be defined but cannot be extrapolated beyond the reference configuration, for example, [3–5] related to the CRBR mock-up experiments at the ZPPR facility.

622 624

VIII. A NEW FRONTIER: THE CONSISTENT METHOD 625 A. The Method 625 B. Consistent Method Implementation 626 C. Evaluation of Nuclear Physics Parameter Covariances 627 D. Evaluation of Sensitivity Coefficients for Integral Experiments 627 E. Direct Monte-Carlo Assimilation of Integral experiments 627 IX. APPLICATIONS A. Assimilation of 239 Pu 1. First Round of Assimilation 2. Second Round of Assimilation 3. PFNS B. Results of Direct Assimilation

• Use of “clean,” “representative” integral experiments (usually called the “bias factor and adjustment” method). This approach allows defining bias factors and uncertainties that can be used for a wide range of applications. It also allows defining “adjusted” application libraries or even “adjusted” data files [6–10]. In practice, it can be shown that by using a set of J integral experiments A characterized by a sensitivity matrix SA , besides a set of statistically adjusted cross-section ˜σ can be data, a new “a posteriori” covariance matrix B obtained [11–13]. As a result of the statistical adjustment procedure, one can then evaluate the resulting reduction of uncertainty matrix for the different design parameters Rq (q = 1, · · · ) as

628 628 628 630 630 631

X. THE FUTURE

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XI. CONCLUSIONS

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˜R = S T B ˜ B R σ SR =   (S T B S )2 = BR 1 − (S T Bσ SR )(SAT Bσσ SRA +BE +BM ) , R

References

634

A

(1)

where Bσ is the a priori nuclear data covariance matrix ˜σ is the new covariance matrix for the nuclear data, and B SR is the sensitivity matrix of the set of Q design parameters in the reference system to the K nuclear data. As before, SA is the sensitivity matrix of the set of J integral experiments. BE and BM are the covariance matrices associated to, respectively, the experimental values and to the calculated values, in order to account for method uncertainties [14]. The mock-up approach was mainly followed within the US, while more analytical experimental programs were performed in France, UK, Japan and Russia (former USSR). While in the mock-up approach one would

I. INTRODUCTION: EARLY STATISTICAL ADJUSTMENT STUDIES AND THE PROBLEM OF UNCERTAINTIES

Since the early 1960’s, physicists involved in the early design of fast reactors have proposed using integral experiments to improve multigroup cross sections. This approach was justified by the limited knowledge at the time of the very large number of nuclear data needed in a wide energy range both for design and for experiment analy597

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In this respect, in 2004 D.L. Smith reminded that: “Interest in nuclear data uncertainties is growing robustly after having languished for several years. Renewed attention to this topic is being motivated by the practical need for assuring that nuclear systems will be safe, reliable, and cost effective, according to the individual requirements of each specific nuclear technology... The availability of fast computers and the concurrent development of sophisticated software enable nuclear data uncertainty information to be used more effectively than ever before. For example, data uncertainties and associated methodologies play useful roles in advanced data measurement, analysis, and evaluation procedures. Unfortunately, the current inventory of requisite uncertainty information is rather limited when measured against these evolving demands. Consequently, there is a real need to generate more comprehensive and reasonable nuclear data uncertainty information, and to make this available relatively soon in suitable form for use in the computer codes employed for nuclear analyses and the development of advanced nuclear energy systems. This conference contribution discusses several conceptual and technical issues that need to be addressed in meeting this demand during the next few years” [22].

attempt to apply directly the observed calculation-toexperiment discrepancies to the calculation of the reference design configuration (with an appropriate use of the integral experiment uncertainties and an appropriate evaluation of possible calculation approximation differences between experimental and design configurations), the adjustment methodology was applied when a set of appropriately designed experiments was available. Major requirements for the integral experiments were, from one side, to be as “clean” (e.g., well documented and easy to analyze) as possible and, from another side, to provide complementary information on specific nuclear data (e.g., structural material capture, actinide fission cross sections, inelastic cross sections, etc.) in specific energy ranges (e.g., tailoring the neutron spectrum or the neutron leakage component of the core, . . .). A careful assessment of experiment uncertainties was systematically performed. The so-called “adjusted data sets” have been used directly in the design of fast reactors, e.g. PHENIX and SUPERPHENIX in France. As an example of the performance of adjusted data, the use of the CARNAVAL-IV adjusted library [15] did allow the prediction of the critical mass of SUPERPHENIX with accuracy of the order of 300 pcm. Further successful applications were made in the field of shielding and of fission products data [16, 17]. Most international groups had become interested in these techniques (e.g., see Proceedings of the NEACRP Specialists’ Meeting on Application of Critical Experiments and Operating Data to Core Design Via Formal Methods of cross-section data adjustment, Jackson Hole, USA, September 23-24 1988, NEACRP-L-307 (1988)). In the field of thermal reactor core analysis, similar techniques were applied to improve nuclear data. The methods were neither explicitly called “adjustments” nor were they based on the formal approaches described in Ref. [11], but they were rather of the type “tendency research” and eventually modified cross sections were imbedded in commercial data sets. Due to the difficulty experienced in the roughly last two decades in the field of fast reactor development, the drastic reduction of integral experiment programs, and significantly reduced activity in the adjustment method area, only a few new adjusted libraries have been derived (e.g., ERALIB1 in France and ADJ2000 in Japan [18, 19]). However, the growing interest for waste management studies, advanced fuel cycles and Partitioning & Transmutation technologies revived the interest if not at first for data adjustments, at least for sensitivity and uncertainty analyses. These new studies focused on the cross section needs and target accuracies for new innovative systems: Accelerator-Driven Assemblies (ADS) [20] and Generation-IV [21]. It was quickly recognized that the value and credibility of any uncertainty analysis was strictly dependent on the scientific quality of the data uncertainty (variance and correlation) used, as can be easily seen by inspection of Eq. (1).

II. THE NEA-WPEC SUBGROUP 26 AND TARGET ACCURACIES. THE DOE NUCLEAR DATA ADJUSTMENT PROJECT

Based on the results of the pioneer uncertainty studies quoted previously, the nuclear data user and evaluator communities suggested to the Working Party on Evaluation Cooperation (WPEC) of the OECD Nuclear Energy Agency Nuclear Science Committee to establish an International Expert Group to develop a systematic approach to define data needs for advanced reactor systems and to make a comprehensive study of such needs for Generation-IV (Gen-IV) reactors. That expert group (called Subgroup 26) was established at the end of 2005. The group performed and published [23] a comprehensive sensitivity and uncertainty study to evaluate the impact of neutron cross-section uncertainty on the most significant integral parameters related to the core and fuel cycle of a wide range of innovative systems, even beyond the Gen-IV range of systems. Results were obtained for the Advanced Breeder Test Reactor (ABTR), the Sodium-cooled Fast Reactor (SFR), the European Fast Reactor (EFR), the Gas-cooled Fast Reactor (GFR), the Lead-cooled Fast Reactor (LFR), the Accelerator Driven Minor Actinide Burner (ADMAB), the Very High Temperature Reactor (VHTR) and a Pressurized Water Reactor with extended burn up (PWR). These systems correspond to studies in the Generation-IV initiative and in the advanced fuel cycle and Partitioning & Transmutation studies in the USA, Japan and Europe. As far as uncertainties, it was found that there is a strong impact of correlation data (i.e. off-diagonal el598

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• Sensitivity analysis of selected configurations including reference design configurations,

ements) on the uncertainty assessment. Any credible uncertainty analysis should include the best available covariance data accounting for energy correlations (as in the present study) and possibly for cross correlations among reactions (a typical case would be the interrelation among total, elastic and inelastic cross sections) and even for cross correlation among isotopes, if needed for example to account for normalization issues. Moreover, it was concluded that the calculated integral parameter uncertainties, resulting from the uncertainties on nuclear data available at that time (see next paragraph), were probably acceptable in the early phases of design feasibility studies. In fact, the uncertainty on keff was found to be less than 2% for all systems (with the exception of Accelerator Driven Systems) and reactivity coefficient uncertainties were below 20%. Power distributions uncertainties were also relatively small, except, once more, in the case of ADS. However, it was pointed out that later conceptual and design optimization phases of selected reactor and fuel cycle concepts would need improved data and methods, in order to reduce margins, both for economical and safety reasons. For this purpose, a compilation of preliminary “Design Target Accuracies” was put together and a target accuracy assessment was performed to provide an indicative quantitative evaluation of nuclear data improvement requirements by isotope, nuclear reaction and energy range, in order to meet the design target accuracies. First priorities were formulated on the basis of common needs for fast reactors and, separately, thermal systems. The results of the nuclear data target accuracy assessment indicated at that time that a careful analysis was still needed in order to define the most appropriate and effective strategy for data uncertainty reduction. Successively, a project for nuclear data adjustments for Global Nuclear Energy Partnership (GNEP) and Gen-IV applications was launched by DOE in 2007 with motivations and objectives derived from the extensive sensitivity/uncertainty studies quoted previously. These studies pointed out that current uncertainties on the nuclear data should be significantly reduced, in order to get full benefit from the advanced modeling and simulation initiatives. Only a parallel effort in advanced simulation and in nuclear data improvement will provide designers with more general and well validated calculation tools that would be able to meet design target accuracies. Despite the preliminary nature of the covariance data used in the studies, it was possible to point out a number of important features and to outline a possible strategy for data improvement. The use of integral experiments had been essential in the past to ensure enhanced predictions for power fast reactor cores; in some cases, these integral experiments had been documented in an effective manner and associated uncertainties were well understood. The process for producing the adjusted data set involves several ingredients and steps:

• Use of science based covariance data for uncertainty evaluation and target accuracy assessment, • Analysis of experiments using the best methodologies available, • Use of calculational and experimental discrepancies in a statistical adjustment. The set of experiments should be selected based on their relevance to the reference design, available documentation, and experimental uncertainty assessment. The sensitivity analysis should be performed for a wide range of integral experiments including: critical mass, reaction rates, reactivity coefficients, and irradiation data. This will require appropriate and general enough sensitivity analysis tools. The uncertainty evaluation and target accuracy assessment will use the best available covariance data for isotopes of interest. The analysis of experiments will be performed using, whenever possible, independent sets of calculation tools both stochastic and deterministic in order to produce the best possible C/E. In this way, systematic errors due to the calculation methodology will be minimized and should not affect the adjustment process. Finally the adjustment procedure will be based on a statistical approach that allows using a number of experiments less than that of parameters (i.e. cross sections) to be adjusted, as indicated in Ref. [11].

III.

COVARIANCE DATA: FROM EDUCATED GUESS TO COMMARA-3.0

At first, the earliest comprehensive uncertainty analyses for transmutation systems (ADS) and GEN-IV [20, 21], were performed with covariance matrices developed at ANL through educated guesses based on nuclear data performance in the analysis of selected clean integral experiments. In establishing these very preliminary uncertainties and covariance data for uranium (U) and transuranics (TRU) multigroup nuclear data, integral experiment analysis was used and four classes of isotopes were defined as follows: 1. The major isotopes (i.e.,

235

U,

2. Other U and Pu isotopes (e.g., 240 Pu, 241 Pu and 242 Pu).

238

U and

234

U,

236

239

Pu).

U,

238

Pu,

3. Minor actinides up to 245 Cm (i.e., 237 Np, 241 Am, 242 Am, 243 Am, 242 Cm, 244 Cm and 245 Cm). 4. Higher-mass minor actinides, for which very few reliable integral experiments were available.

• Selection of a set of relevant experiments, 599

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Finally, only a few uncertainty evaluations were available for structural materials and crude estimations of uncertainties were mainly based on the inspection of discrepancies among available data files. Partial energy correlations were also introduced. The same correlations for all isotopes and reactions were used, under the form of full energy correlation in 5 energy bands. The idea was to single out energy regions of relevance, in particular for actinides:

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TABLE I. The COMMARA 33-energy group structure (eV). Group 1 2 3 4 5 6 7 8 9 10 11

1. The region above the threshold of fertile isotope fission cross-sections, and of many inelastic crosssections, up to 20 MeV. 2. The region of the continuum down to the upper unresolved resonance energy limit.

Upper Upper Upper Group Group Energy Energy Energy 1.96 × 107 12 6.74 × 104 23 3.04 × 102 7 4 1.00 × 10 13 4.09 × 10 24 1.49 × 102 6.07 × 106 14 2.48 × 104 25 9.17 × 101 3.68 × 106 15 1.50 × 104 26 6.79 × 101 6 3 2.23 × 10 16 9.12 × 10 27 4.02 × 101 1.35 × 106 17 5.53 × 103 28 2.26 × 101 5 3 8.21 × 10 18 3.35 × 10 29 1.37 × 101 5 3 4.98 × 10 19 2.03 × 10 30 8.32 × 100 3.02 × 105 20 1.23 × 103 31 4.00 × 100 5 2 1.83 × 10 21 7.49 × 10 32 5.40 × 10−1 1.11 × 105 22 4.54 × 102 33 1.00 × 10−1

3. The unresolved resonance energy region. At present, the 33 groups (see Table I for energy boundaries) cross section covariance library COMMARA “Covariance Multigroup Matrix for Advanced Reactor Applications” has been under development by BNL-LANL collaborative effort over the last three years. The project builds on two covariance libraries developed earlier, with considerable input from BNL and LANL. In 2006, as indicated above, an international effort under WPEC Subgroup 26 produced BOLNA covariance library by putting together data, often preliminary, from various sources for most important materials for nuclear reactor technology. This was followed in 2007 by collaborative effort of four US national laboratories to produce covariance data, often of modest quality – hence the name low-fidelity, for a virtually complete set of materials included in ENDF/BVII.0. At present the covariance data of 4-5 major reaction channels for 110 materials of importance for power reactors have been evaluated. In 2011, the COMMARA library version 2.0 was released [28]. It should be noted that COMMARA-2.0 covariance data refer to central values given in the 2006 release of the US neutron evaluated library ENDF/B-VII.0 [29]. The next release of the COMMARA library, version 3.0, will refer to the central values of ENDF/B-VII.1, released in 2011 [30]. This release of the ENDF/B library contained covariance data for 190 materials, many of which were based on covariance data in COMMARA2.0. COMMARA-3.0 will be the processed covariances found in ENDF/B-VII.1, increasing the number of materials from 110 to over 180. The covariances will be vetted by a procedure similar to that done for version 2.0, modifying those where the resulting uncertainties are found too small or large, the result of model extrapolation, or other inconsistencies, such as negative eigenvalues or unitarity violations, but keeping such modifications to a minimum. Another goal for version 3.0 will be the addition of as many mubar and nubar covariances as possible, and to include cross material covariances by using a modeling scheme based on EXFOR experimental interdependencies.

4. The resolved resonance region. 5. The thermal range. No correlation among different reactions or different isotopes had been considered at that stage, even if it was suggested that this assumption may lead to nonconservative results in some cases. In the frame of the Subgroup 26 activity, the integral parameter uncertainties previously calculated using the ANL covariance matrices were revised by the use of covariance data developed in a joint effort of several laboratories (BNL, ORNL, LANL, NRG, ANL) contributing to Subgroup 26. This new set of covariance matrices was referred to as BOLNA. The BOLNA covariance matrices were at the time unique in terms of completeness and applied methodology. However, it was understood that although BOLNA was the state of the art, it was termed preliminary and conclusions about achieved accuracies were conditioned by the partly untested quality of that set of covariance matrices. A major step forward in the production of sciencebased covariance data was performed within the DOE “adjustment” project. At first the covariance data were produced using the existing BNL-LANL evaluation capabilities and tools, such as nuclear reaction model code empire [24], Atlas of Neutron Resonances [25], Bayesian code kalman [26] and experimental database EXFOR [27], improving the methodology for evaluation of neutron cross section covariance data both in the low energy and fast neutron energy range. Covariance evaluations for major actinides such as 235,238 U and 239 Pu by ORNL-LANL were also used. The required amount of data was very large, 112 materials, including actinides, structural materials, light and heavy nuclei as well as fission products. To keep the scope of the effort practical often simplified estimates of covariance data were made. This did mean that although experimental data would have been consulted, they were not at first fully incorporated in the evaluation process. 600

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expressed on the third line in terms of the cross section standard uncertainty Δσ and correlation coefficient ρ. EDA R-matrix analyses were performed for several light systems, including n+1 H, n+4 He, n+6 Li, n+10 B and n+16 O. In all cases, the covariances were calculated from the R-matrix parameters, covariances, and derivatives, using Eq. (3). Only selected systems and reactions are presented here for illustration. The evaluated n-p cross section uncertainty is shown in Fig. 1. Very low uncertainties (less than 1%) are obtained from an excellent fit to more than 5,000 experimental data points.

COMMARA Evaluation Methodology

A description of BNL and LANL covariance methodology can be found in several papers and reports published in 2008. Refs. [31, 32] provide summaries of LANL methodologies for light nuclei and actinides, while a summary of BNL methodologies developed primarily for structural materials and fission products can be found in [26, 33]. In 2009-2010, further advances were made by both BNL and LANL. Notably, BNL developed a new procedure for the resonance region based on the kernel approximation and LANL improved methods for covariance data of prompt fission neutron spectra [34]. 1.

G. Palmiotti et al.

Light Nuclei

Light nuclei are defined here as materials from 1 H to F. For these materials, the covariance methodology is based on two distinct methods: (1) R-matrix evaluations that also provide cross-section covariance matrices; (2) simple estimates when no R-matrix evaluation is available. The EDA code [35] has been used at Los Alamos to perform R-matrix evaluations [31] for light nuclei. The cross sections and covariances from these analyses were used as input for the two previous versions of the lightelement standard cross section evaluation, and more recently in some of the low-fidelity covariances provided for the Nuclear Criticality Safety program. The multichannel R-matrix calculations are compared with the measurements using   nXi (p) − Ri 2  nS − 1 2 2 χEDA = + (2) ΔRi ΔS/S i

19

FIG. 1. Standard deviation of the n-p scattering cross sections for energies up to 20 MeV.

in which for a given scattering observable, Xi (p) are the values calculated from R-matrix parameters p, Ri , ΔRi are the measured relative values and their standard uncertainties, respectively; S and ΔS are the measured scale with its standard uncertainty, and n is the associated adjustable normalization parameter. This expression differs from the usual one in which the deviations are weighted by the inverse of the full variance/covariance matrix for the measurements Mi = Ri S. However, if the relative and scale parts of the measurement are assumed to be independent, as in Eq. (2), the usual expression closely approaches the EDA one near a solution, when all the parameters (including normalizations) are adjusted to minimize χ2 . Near a solution, chi-square assumes a quadratic form, from which an iterative minimization procedure leads to the parameter covariance matrix C0 . Crosssection covariances are then obtained by first-order error propagation as

The R-matrix analysis of the 11 B system extends only to neutron energies up to 1 MeV, to encompass the standards region. A very good fit was obtained in this region to the neutron total cross section, n+10 B elastic scattering cross sections and polarizations, differential cross sections for the 10 B (n, α0 ) and 10 B(n, α1 ) reactions, integrated cross sections for the two α-producing reactions, as well as measurements of the branching ratio between them. Below 1 MeV, the R-matrix analysis provided the correlations for the various cross sections, and then extend to 20 MeV using a simple form that decays exponentially from the diagonal as the square of the energy difference. The correlation matrix for 10 B(n, n) is shown in Fig. 2. The covariances for the n+16 O cross sections are based on a recent R-matrix analysis of reactions in the 17 O system that included all reactions possible between the channels n+16 O and α+13 C at incident neutron energies up to 7 MeV. This resulted in a fit to more than 5700 data points with a chi-square per degree of freedom of 2.59. The cross-section covariances calculated from firstorder error propagation reflect the significant amount of

cov[σi (E)σj (E  )] =

 = [∇p σi (E)]T C0 [∇p σj (E  )]p=p

= Δσi (E)Δσj (E  )ρij (E, E  ),

0

(3) 601

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mation is available. Another issue concerns the use of different approaches for the evaluation of related quantities. For instance, the total (n, α) cross section of 10 B is defined as the sum of (n, α0 ), (n, α1 ), (n, α2 ), etc., and for each level excitation cross sections are evaluated differently: while the ground-state production (n, α0 ) is based on experimental data, other excited cross-sections are obtained from model calculations. No consistent and automated uncertainty quantification techniques can deal with such inconsistent evaluations, and therefore ad-hoc simplified evaluations of covariance matrices had to be used.

2.

For the purposes of the present project this category includes materials from 23 Na to 209 Bi. The BNL methodology covers the whole energy range using one set of approaches for the resonance region and another one for the fast neutron region. In the resonance region two methods were broadly used:

FIG. 2. Correlation matrix of the 10 B(n, n) cross section with energy up to 2 MeV. The axis labels are in eV. Light areas denote high correlation, while dark areas denote low or negative correlations.

resonance structure in the cross sections at energies below 7 MeV. This is also true of the μ-bar covariances that were calculated for the first Legendre moment of the neutron elastic scattering angular distribution. Cross-section uncertainties vary from tenths of a percent to tens of percent, depending on whether the energy is in the peak or valley of a resonance, as shown in Fig. 3 for elastic scattering.

1. A new method based on the idea of kernel approximation [36] has been used. Major strengths of the kernel approximation are its transparency, its capability to treat level-level correlations and to incorporate uncertainties related to potential elastic scattering. The kernel approximation was used for the most important structural materials, for examples see [37, 38].

Relative Standard Deviation (%)

100 16

2. The integral approximation. This simple, yet reasonable approach was originally developed for the purposes of low-fidelity covariance project [39]. The method approximates covariance data by two uncorrelated uncertainties, one in the thermal region and another in the low energy part of the resonance region. These uncertainties are supplied by thermal cross sections and by resonance integrals, which works reasonably well for capture. For elastic scattering use was made of uncertainties of thermal cross sections and those of scattering radii. For the present project the method was mostly used for materials of lower priority, primarily fission products.

O (n,n)

10

1

0.1

0.01 0.1

1 Incident Neutron Energy (MeV)

Structural Materials and Fission Products

10

O(n, n) standard deviation (%) at energies up to 7

In the fast neutron region the following methods were used:

While the R-matrix theory gives the most reliable evaluation and uncertainty quantification, it is often limited to a certain energy range and for few elements for which experimental data exist. We have complemented Rmatrix results with less sophisticated approaches, ranging from least-square fits or simpler analyses of experimental data, statistical model calculations when appropriate, or even just simple extrapolations when very limited infor-

1. The empire-kalman method. This approach, described in detail Ref. [26] published in 2008, uses the well-known reaction model code empire coupled to the RIPL library for a priori model parameters and produces a posterior parameters by including experimental data with the Bayesian code kalman [40]. The method has been recently improved by incorporation of model parameter uncertainties and accounting for model uncertainties

FIG. 3. MeV.

16

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• Limited evaluation. Evaluation performed by BNL with empire-kalman using parameterization reproducing ENDF/B-VII.0 central values.

through the marginalization of the so-called ‘nuisance’ parameters. Concept on marginalization of the nuisance parameters is critical for accounting for model uncertainties, and systematic uncertainties in the experimental data. This technique makes use of the model parameters, which scale certain parts of the reaction calculations. These parameters are not used in the analysis of the experimental data but their “a priori” assigned uncertainty is used in the final propagation of parameter uncertainties to the cross section covariances.

• Simple estimates. Estimates based on dispersion analysis performed for BNL by V. Maslov [42].

4.

Actinides - Nubars and Fission Spectra

LANL has developed a code package aimed at analyzing, predicting and evaluating the average prompt fission neutron spectrum (PFNS) and multiplicity (PFNM) for a wide range of actinides, through model calculations as well as experimental data. The main calculation tool is centered around an extended version of the Los Alamos model [43], whose input parameters can be tuned to fit available experimental spectrum data. Other experimental data such as the average outgoing neutron energy Eout , and the average neutron multiplicity ν can also be used to adjust the model parameters. By default, the first-order linear Kalman filter is used to combine model calculations and experimental data and infer a best value for the spectrum as well as its associated covariance matrix. Sensitivity studies of the calculated spectrum to the choice of model input parameters are performed and folded into the Kalman filter equations with the experimental PFNS to infer a final spectrum and set of optimal model input parameters. The model parameter covariance matrix is easily transformed into a spectrum covariance matrix following a linear approximation. As a probability distribution, the PFNS is normalized to unity. This mathematical constraint implies a zerosum rule to be verified by the PFNS covariance matrix, as explained in the official ENDF-102 manual, Section 35.3 [44]. This additional constraint is properly taken into account in LANL’s evaluation procedure. Experimental PFNS data come in many various forms: ratio to 252 Cf (sf) PFNS, ratio to Maxwellian at different temperatures, absolute, etc. The different types of PFNS are considered automatically, and “raw” data from the EXFOR database can be used directly. Any PFNS measurement is limited to an outgoing neutron energy range [Emin , Emax ], specific to each measurement. Therefore, to be incorporated in the analysis, each experiment has to be re-normalized. We follow the approach described in Appendix B of Ref. [43], and use the Los Alamos model to perform this re-normalization. An ENDF toolkit has also been developed to read and write PFNS and PFNM related quantities in ENDF format, including covariance matrices. Such a toolkit greatly simplifies the final steps of an evaluation, as well as comparison with existing evaluations. This code package has been used to perform PFNS evaluations and uncertainty quantifications for n (0.5 MeV)+239 Pu [34], as well as n (0.5 MeV)+235,238 U. While the PFNS calculated in this work are close to the ones

2. Dispersion analysis. This approach, suitable for materials where independent evaluations are available, compares central values in major evaluated nuclear data libraries and infers estimate of cross section uncertainties from dispersion between these cross sections. 3. The propagation of model parameter uncertainties. This method uses code empire, accounts for assessment of model parameter uncertainties such as given in RIPL-3 library, and propagates them into cross section covariance [41]. The method was used for the mass production of covariances included in the low-fidelity covariance library. 3.

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Actinides - Cross Sections

Covariance data for actinides consist of three types, reaction cross sections, prompt fission neutron multiplicity and prompt fission neutron spectra, each of them being evaluated independently. Covariance methodology for reaction cross sections used approaches of different complexity depending on the material priority. Major actinides – full scale evaluation for ENDF/BVII.0. Central values for major actinides were evaluated already for ENDF/B-VII.0, but covariances were completed after the ENDF/B-VII.0 release. Thus, covariances for major actinides can be viewed as stemming from simultaneous evaluations combining LANL fast region, with resonances supplied by ORNL. In the fast region, fission is obtained from detailed analysis of experimental data, while other reaction channels are evaluated mostly by the nuclear reaction model code GNASH with experimental data included through the Bayesian code KALMAN. In the resonance region ORNL evaluations are based either on full scale SAMMY analysis or retroactive SAMMY analysis. Minor actinides – three methods of different complexity were used depending on the material priority and the laboratory supplying covariances: • Full scale evaluation retrofitted to ENDF/B-VII.0. Simultaneous evaluations performed for ENDF/BVII.1 library by LANL and ORNL, with covariances retrofitted to account for discrepancies between ENDF/B-VII.1 and ENDF/B-VII.0 central values. 603

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present in the ENDF/B-VII.0 evaluated library, they are not exactly identical. To remain consistent with this library, the PFNS covariance matrices were retrofitted to the evaluated PFNS, keeping intact the calculated correlations in outgoing neutron energy. The evaluated average PFNS for the 0.5 MeV neutroninduced fission on 239 Pu is shown in Fig. 4, and compared to selected experimental data sets and evaluated libraries. More details on the evaluation procedure can be found in [34].

210

Energy Release (MeV)

205

Ratio to Maxwellian (T=1.42 MeV)

1.4

G. Palmiotti et al.

Prior Posterior Madland [16]

200 195 190 185 180 175

n(0.5 MeV)+239Pu PFNS

n+ 170 0.768

1.2 1

0.4 0.2 0.01

Knitter, 1975 (0.215 MeV) Staples, 1995 (0.5 MeV) Lajtai, 1985 (thermal) Bojcov, 1983 (thermal) Kalman filter result ENDF/B-VII.0 JENDL-4.0 0.1

1

U

0.772 0.774 0.776 Fissility Parameter

0.778

0.780

FIG. 5. The average energy release Er  is plotted as a function of the fissility parameter (see Eq. 4 and text for details) for a suite of uranium isotopes. Because of the functional form for Er  used to evaluate the PFNS across the suite, cross-isotope correlations are produced in the final covariance matrix evaluation. For more details, please see Ref. [45].

0.8 0.6

0.770

238

10

Outgoing Neutron Energy (MeV)

FIG. 4. The prompt fission neutron spectrum for 0.5 MeV neutron-induced fission of 239 Pu is plotted as a ratio to a Maxwellian at temperature T=1.42 MeV, along with selected experimental data and evaluated libraries. The raw result of our Bayesian evaluation is shown in dotted black with a one-sigma band shown in grey. The final PFNS covariance matrix was retrofitted to ENDF/B-VII.0 mean values for the COMMARA-2.0 library.

through a statistical analysis of the available experimental data. This is true at least for the major actinides for which sufficient data exist. In fact, the Los Alamos model cannot predict accurately the average PFNM without a precise knowledge of the total prompt gamma-ray energy released in a fission event. The evaluated average neutron multiplicity is plotted in Fig. 6 as a function of the incident neutron energy, and compared to experimental data available in this energy range. Note that the ENDF/B-VII.0 PFNM values for 239 Pu have been slightly adjusted (lowered) to better predict the neutron multiplication factor keff of the JEZEBEL plutonium critical assembly. Such an adjustment is now being reassessed in view of newer data and better modeling. Note that the recent development of Monte Carlo Weisskopf and Hauser-Feshbach techniques [47, 48] offer the promise of better, more predictive, evaluations that would not only contain the average spectrum but also the average multiplicity. Recent calculations for n+239 Pu have shown that ν(Einc ) can be predicted quite precisely, within the uncertainty bands obtained from a statistical analysis of the experimental data, at least up to incident neutron energies of 5 MeV. Many other observables, e.g., P(ν), n-n correlations, etc., can be computed through such Monte Carlo simulations, but are not necessarily for direct usage for nuclear energy applications. However, they do help place stringent constraints on the modeling of the fission process, in turn providing much better predictive capabilities and more realistic quantification of uncertainties.

A more systematic approach to the evaluation and quantification of uncertainties of PFNS for the suites of uranium and plutonium isotopes has also been performed recently [45]. In this work, a functional form [46] of the Los Alamos model input parameters is used across a suite of isotopes, giving rise to cross-isotope correlations in the final PFNS covariance matrix. Fig. 5 shows the average energy release parameter as a function of the fissility parameter, defined as x(Z, A) =

Z 2 /A  −Z 2  , 50.883 1 − 1.7826 NA

(4)

across the suite of uranium isotopes studied in this work. An important advantage of this approach is that PFNS can be estimated more reliably for isotopes for which no experimental data exist. Such covariance matrices have yet to be incorporated in the COMMARA library. While the evaluation of average PFNS is strongly influenced by model calculations, in particular due to the limited range in outgoing energy of the measured spectra, the evaluation of average PFNM as a function of the incident neutron energy is more commonly treated 604

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3.2

Neutron Multiplicity (n/f)

3.15 3.1 3.05

ENDF/B-VII.0/1 JENDL-4.0 JEFF-3.1 Gwin, 1986 Boldeman, 1971 Savin, 1970 Hopkins, 1963

239

TABLE II. Contributions to the uncertainty in keff (pcm) for EFR system using COMMARA-2.0.

Pu PFNM

Isotope 238 U 240 Pu 241 Pu 239 Pu 238 Pu 58 Ni 56 Fe 242 Pu 16 O 23 Na 241 Am 245 Cm 244 Cm Total

3 2.95 2.9 2.85 2.8 0.001

0.01 0.1 Incident Neutron Energy (MeV)

1

FIG. 6. The average prompt fission neutron multiplicity ν p is plotted as a function of the incident neutron energy in the case of neutron-induced fission of 239 Pu, and compared to selected experimental data and evaluated libraries.

B.

G. Palmiotti et al.

σcap 229 157 34 228 35 25 79 47 6 4 18 1 28 376

σfiss 28 134 19 175 39 0 0 18 0 0 5 56 19 235

ν 111 252 12 55 34 0 0 16 0 0 8 7 16 284

σel 150 16 0 24 1 5 25 1 79 44 1 0 0 180

σinel 761 73 11 99 8 23 118 4 2 93 4 0 1 786

χ 0 100 0 154 0 0 0 0 0 0 0 0 0 184

P1el 0 0 0 0 0 0 15 0 0 54 0 0 0 56

Total 817 349 42 347 63 34 145 53 80 117 20 57 37 982

TABLE III. Contributions to the uncertainty in keff (pcm) for EFR system using COMMARA-3.0β.

Validation of Covariance Data

Isotope 238 U 240 Pu 241 Pu 239 Pu 238 Pu 58 Ni 56 Fe 242 Pu 16 O 23 Na 241 Am 245 Cm 244 Cm Total

The problem of how to validate covariance matrices is twofold: a) validation of the mathematical features of the matrices and validation of the internal consistency of the data and b) validation of variance and correlations numerical values. As for the second point a possible procedure to detect potential anomalies is to calculate a set of uncertainties on integral parameters and to compare, e.g., the evolution between successive versions of a specific covariance data set. The results of this exercise as applied to the impact of COMMARA-2.0 and a preliminary release of the next version of the library, COMMARA-3.0β, on standard fast reactor configurations (critical or sub-critical, with different core size and fuel types and compositions) already used in Refs. [21, 23] for uncertainty analysis, are given in Tables II - VII. The EFR (European Fast Reactor) is a large size ( 1500 MWe) Na-cooled core with standard mixed oxide (MOX) fuel. The SFR and ADMAB systems are loaded with significant amounts of minor actinides and they were chosen to point out the covariance matrix data for those elements and isotopes. SFR is a Na-cooled, metal fueled fast reactor loaded with TRU fuel of a composition as discharged by standard PWRs (i.e. Pu/MA ratio 10) and ADMAB, which is a Pb-cooled nitride fueled fast spectrum ADS loaded with a Pu/MA ratio close to one. These results allow us to point out the impact and order of magnitude of the impacts of updated covariance data, in the present case mostly due to revised covariance data for higher Pu isotopes and, in general, for minor actinide data. In general, smaller uncertainties are observed with preliminary COMMARA-3.0β. More validation is expected in the future.

σcap 225 102 34 229 18 32 79 12 5 90 18 7 18 362

σfiss 28 60 26 182 5 0 0 18 0 0 5 8 4 196

ν 111 61 12 55 13 0 0 5 0 0 8 11 13 141

σel 151 15 0 24 1 6 25 1 79 42 1 0 0 180

σinel 762 64 11 99 2 24 118 4 3 93 4 0 1 786

χ 169 61 0 155 7 0 0 12 0 0 0 4 1 238

P1el 0 0 0 0 0 0 13 0 0 14 0 0 0 19

Total 834 160 46 350 24 40 144 20 80 137 20 15 22 947

TABLE IV. Contributions to the uncertainty in keff (pcm) for SFR system using COMMARA-2.0. Isotope 240 Pu 56 Fe 245 Cm 238 U 244 Cm 238 Pu 242 Pu 239 Pu 241 Pu 243 Am 23 Na Total

605

σcap 206 195 8 50 147 117 176 131 95 55 8 423

σfiss 241 0 444 9 141 119 93 136 45 58 0 567

ν 439 0 50 33 111 154 81 41 34 13 0 492

σel 39 320 0 51 2 8 7 27 2 2 21 329

σinel 113 321 3 230 5 32 15 78 38 18 182 459

χ 169 0 0 0 0 28 0 100 0 0 0 198

P1el 0 88 0 0 0 0 0 0 0 0 145 170

Total 580 501 447 244 232 231 216 233 117 83 234 1063

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TABLE V. Contributions to the uncertainty in keff (pcm) for SFR system using COMMARA-3.0β. Isotope 240 Pu 56 Fe 245 Cm 238 U 244 Cm 238 Pu 242 Pu 239 Pu 241 Pu 243 Am 23 Na Total

σcap 153 194 39 48 94 63 40 132 95 55 102 345

σfiss 94 0 46 9 31 26 41 141 70 58 0 206

ν 119 0 67 33 90 56 26 41 34 13 0 186

σel 36 320 0 52 1 5 6 27 2 2 17 327

σinel 106 320 2 231 3 8 14 78 38 18 182 456

χ 95 0 21 45 3 30 55 100 0 0 0 160

P1el 0 80 0 0 0 0 1 0 0 0 35 87

TABLE VII. Contributions to the uncertainty in keff (pcm) for ADMAB system using COMMARA-3.0β.

Total 261 499 93 248 133 93 85 236 128 83 212 738

Isotope 241 Am 241 Pu 245 Cm 244 Cm 243 Am 237 Np 209 Bi 240 Pu 15 N 239 Pu 242m Am 238 Pu Total

TABLE VI. Contributions to the uncertainty in keff (pcm) for ADMAB system using COMMARA-2.0. Isotope 241 Am 241 Pu 245 Cm 244 Cm 243 Am 237 Np 209 Bi 240 Pu 15 N 239 Pu 242m Am 238 Pu Total

IV.

σcap 231 73 13 505 300 183 64 56 1 91 11 32 674

σfiss 122 42 1063 651 449 171 0 88 0 126 142 56 1360

ν 172 33 115 516 95 65 0 151 0 35 10 59 593

σel 11 0 0 1 3 1 66 8 216 15 0 2 227

σinel 116 57 10 37 229 97 166 64 13 118 4 19 355

χ 0 0 0 0 0 0 0 82 0 129 0 3 153

P1el 0 0 0 0 0 0 0 0 0 0 0 0 0

σcap 230 73 57 311 300 314 63 44 1 91 0 17 602

σfiss 122 74 112 148 449 171 0 34 0 131 0 10 551

ν 172 33 149 416 95 65 0 37 0 35 0 21 492

σel 11 0 0 1 3 1 66 8 216 15 0 1 227

σinel 116 57 6 25 229 64 166 60 13 118 0 5 346

χ 0 0 69 45 0 0 0 60 0 129 0 14 165

P1el 0 0 0 1 0 0 0 0 0 0 0 0 1

Total 333 123 207 543 594 369 190 109 216 240 0 33 1052

VIII the list of the pre-selected experiments to be used in the nuclear data adjustment is provided. The selection has been based on relevance of the experiment with respect to typical fast reactor concept(s) (e.g., 1000 MWt ABR metallic and/or oxide fuel cores [49]), quality of the experimental data, and availability of data, documentation, and calculation models (Monte Carlo and/or deterministic) as indicated in Table VIII. The relevance has been determined with physical considerations as well as to sensitivity, uncertainty, and the so-called “representativity” analyses. In order to plan for specific experiments able to reduce uncertainties on selected design parameters, a formal approach, initially proposed in [52] has been applied extensively in [53] and further developed in [12]. This approach is summarized as follows. In the case of a reference parameter R, once the sensitivity coefficient matrix SR and the nuclear data covariance matrix Bσ are available, the uncertainty on the reference integral parameter can be evaluated by the equation

Total 333 107 1070 973 594 277 190 211 216 237 143 89 1690

INTEGRAL EXPERIMENT SELECTION AND ANALYSIS A.

G. Palmiotti et al.

Integral Experiment Selection T ΔR2 = SR Bσ SR .

The selected integral experiments should meet a series of requirements: (a) low and well documented experimental uncertainties; (b) enabling to separate effects (e.g., capture and fission); and (c) allowing validating global energy and space dependent effects. As for the point (b) above, irradiation experiments with separate isotope samples allow us one to obtain significant information on capture data, while fission rate experiments in well characterized spectra provide high accuracy information on fission data. As for the point (c), one should approach validation using “representative” experiments as much as practical, according to the definition given below, while specific spatial effects (as reflector effects in the ABR cores, see below) should be singled out with appropriate experiments (e.g., experiments with or without blankets, to underline possible specific effects due to the presence of a steel reflector). In Table

(5)

We consider an integral experiment conceived in order to reduce the uncertainty ΔR2 . Let us indicate by SE the sensitivity matrix associated with this experiment. If we define the “representativity factor” by the following expression T Bσ SE SR rRE = , T B S )(S T B S ) (SR σ R E σ E

(6)

it can be shown [52] that the uncertainty on the reference integral parameter R is reduced by 2

2 ). ΔR = ΔR2 (1 − rRE

(7)

If more than one experiment is available, then Eq. (6) can be generalized [12]. 606

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TABLE VIII. List of pre-selected integral experiments. The assembly name, availability of documentation, analyzed quantity, and calculation model are given. Experiments analyzed Calculation Model Document Critical React. React. Irrad. Availability Monte Carlo Deterministic mass Rates Coeff. Exp. GODIVA [50] Yes Yes Yes Yes [50] Yes Yes Yes Yes JEZEBEL-239 [50] Yes Yes Yes Yes JEZEBEL-240 [50] Yes Yes Yes Np Sphere [50] Yes Yes Yes BIGTEN [50] Yes Yes Yes FLATTOP [50] Yes Yes Yes Yes Yes ZPR-6/6-7 [50] Yes Yes Yes Yes ZPR-3/53-54 [51] Yes Yes ZPPR-2 [51] Yes Yes Yes ZPPR-9 [51] Yes Yes Yes Yes ZPPR-10 [51] Yes Yes Yes ZPPR-15 NEA Yes Yes Yes Yes MUSE/COSMO CEA Yes Yes Yes PROFIL CEA Yes Yes Yes TRAPU CEA Yes Yes PROFIL CEA Yes Yes TRAPU Assembly

Table IX shows some selected results of a representative study for the keff . The ABR Metal and Oxide cores, chosen as reference cores, are flexible fast reactor designs developed at ANL [49]. As a reminder, a representativity factor equal to 1 indicates a perfect mock up experiment, i.e. able to reduce to a minimum the uncertainties on the reference core design, at the condition that the experimental and calculation uncertainties are kept to very low levels. In the following we provide a brief description of the selected experiments and the physical considerations on why they were selected. Most of the information is coming from Refs. [50, 51]. GODIVA, JEZEBEL (239 Pu, and 240 Pu), FLATTOP, and Np Sphere are all critical sphere experiments, while BIGTEN is a cylindrical criticality experiment. Each was performed at LANL and used as reference benchmarks for validation of nuclear data libraries, in particular ENDF/B data. The inclusion of these critical masses and associated central spectral indices will ensure for the adjustment phase consistency with high energy data that enjoy a long and well documented validation. Furthermore several ZPR (Zero Power Reactor) and ZPPR (Zero Power Physics Reactors) assemblies were selected :

TABLE IX. Representativity factors determined with diagonal only values and full covariance BOLNA matrix, for the keff of selected experiments with respect to ABR metal and oxide fuel design. Representativity ABR-Metal ABR-Oxide Experiment diagonal full diagonal full matrix matrix matrix matrix ZPPR-15 0.816 0.814 0.781 0.738 0.748 0.780 0.753 0.740 ZPPR-2 0.788 0.796 0.760 0.723 ZPPR-9 0.309 0.435 0.360 0.434 ZPR-3/53 0.051 0.065 0.114 0.115 ZPR-3/54 0.305 0.190 0.310 0.175 ZPR-6/6 0.770 0.792 0.765 0.739 ZPR-6/7

53 and 54 were plutonium-fueled assemblies with simple geometries and material compositions. Although the critical masses of the assemblies were small, their neutron spectra were generally similar to those expected in large fast power reactors. The interest for the adjustment here it is related to the reflector effect. The main difference between the two assemblies is that the blanket surrounding the core zone of assembly 53 was replaced by an iron reflector in assembly 54.

• ZPR-6 Assemblies 6 and 7 – these were large clean, cylindrical, single zone (uranium for assembly 6 and plutonium for assembly 7) oxide fuel benchmark assemblies. The interest for the adjustment resides in the cleanness of the configurations, the many high accuracy measurements performed, and the comparison between the uranium and plutonium fuel in two very close configurations.

• ZPPR-2 – This assembly was a clean, two-zone cylindrical core with approximate size, geometry, neutron spectra, and composition of the Demonstration Reactor, a large size fast reactor proposed in the 1970s. Interest here is for a classical two enrichment zone core for a large size reactor.

• ZPR-3 Assembly 53 and 54 – the ZPR-3 Assembly 607

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• ZPPR-9 – This assembly was a clean, cylindrical two-zone physics benchmark for a reactor of approximately 700 MWe. It is a larger version of ZPPR-2. The measurements were directed at evaluating the effects of the larger core size on all the important physics parameters.

adjustment, etc. The 87 corresponding C/E will be provided in the adjustment section when compared against the “a posterior” C/E values. In general, whenever possible, Monte Carlo (in particular the MCNP5 code) was used in the experimental analysis in order to have the most detailed description of the geometry and to allow us to use continuous-energy cross sections. The ENDF/B-VII.0 library was adopted as starting point. In the following we provide two typical examples of experimental analysis: ZPR-6 Assembly 7, and the irradiation experiments PROFIL-1 and -2.

• ZPPR-10 – This assembly was an evolution of ZPPR-9. It contained control rods with variable control rod positions, and the shape of core at core midplane was made hexagonal. • ZPPR-15 – This was a clean, two-zone core of about 330 MWe size with metallic fuel. The focus of the program was to provide experimental support for metal fuel designs. Tests included radial and axial expansion effects and the use of a B4C radial shield.

1.

Integral Experiment Analysis: ZPR-6 Assembly 7 (ZPR-6/7)

ZPR-6/7 is a large cylindrical assembly surrounded by a thick depleted-uranium reflector based on mixed Pu-U oxide fuels. There were two principal core configuration established for the ZPR-6/7 program. Those were the uniform core loading and high 240 Pu-zone core loading [50]. The former had a relatively uniform core composition. A central zone of 61 matrix locations in each half of the assembly was defined as the exact core. This exact core region had the same unit cell and the same average composition as the outer core, but the plates used in the exact core were those for which knowledge of material properties was most precise. The latter configuration was a variant of the uniform core. The plutonium in the standard Pu-U-Mo fuel plates used in the uniform core contains 11% 240 Pu. The analyses of these assemblies were performed using detailed geometry models (see Fig. 7), MCNP5 code and ENDF/B-VII.0 cross sections. The corresponding C/E for the critical masses were respectively: 1.00043 for the reference configuration and 0.99937 for the high 240 Pu core content. These results indicate a very good performance by the adopted cross section library when experimental uncertainties (typically of the order of 200 pcm) and calculation ones (typically 10 pcm) are taken into account. The cell averaged spectral indices were initially calculated based on the volume weighted approach over each fuel plate at the central drawer (Fig. 8). However, some of spectral indices showed large discrepancies. This indicated that the volume weighted approach was inappropriate. Therefore, a more accurate atomic density weighted approach was used. The formula used is given by

• MUSE/COSMO – This was a benchmark from experiments performed at the French experimental facility of MASURCA in Cadarache (France). This was a clean cylindrical type core, with oxide fuel of about 25% content in plutonium, made in preparation of the MUSE experimental campaign in support of ADS design. Of particular interest is the presence of a reflector (no blanket), and an extensive set of minor actinides reaction rates measurements. • PROFIL – The PROFIL-1 and -2 irradiation experiments had special pins containing samples of pure isotopes, including fission products, major and minor actinides (Uranium, Plutonium, and Americium isotopes) and were irradiated in the French PHENIX fast reactor. The major aim was to gain information on capture, and where possible (n,2n), cross sections of the irradiated samples. These were very good quality experiments with low uncertainties. • TRAPU – The TRAPU experiment consisted of a six-cycle irradiation in the PHENIX reactor of mixed-oxide pins that contained plutonium of different isotopic compositions but heavily charged in the higher isotopes (240 Pu, 241 Pu and 242 Pu) compared to typical PHENIX fuel. The aim was to gain knowledge on transmutation rates, including that of minor actinides.

B.

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Integral Experiment Analysis

Reaction rate of isotope n =

An initial set of 148 integral experimental quantities was analyzed (using the best calculation tool available) in order to provide C/E and associated calculation and experimental uncertainties and correlations. The initial set was then reduced to 87 experimental values based on several considerations, such as duplications, some covariance data not available, experiments reserved for Fe

i=1,3

RRin Nin

i=1,3

Nin

,

(8)

where RRin and Nin refer to reaction rate and atomic density of Isotope n at fuel plate i, respectively. Calculated results are compared in Table X. As can be seen, more accurate solutions have been obtained by the atomic density weighted approach. 608

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TABLE X. Comparison of spectral indices averaged over the central 2x2x2 inch box in ZPR-6/7. Spectral Index F49/F25 F28/F25 C28/F25

C/E (Volume C/E (Atomic weighted) density weighted) 0.9435 ± 0.021 0.9690 ± 0.025 0.9640 ± 0.025 0.0223 ± 0.030 0.9148 ± 0.035 1.0045 ± 0.035 0.1323 ± 0.024 1.0227 ± 0.026 1.0098 ± 0.026 Experiment

Depleted U3O8 RR1n, N1n

Sodium Can NA Iron Oxide

2 in.

Pu-U-Mo DOW RR2n, N2n Iron Oxide Sodium Can NA Depleted U3O8 RR n, N n 3 3

FIG. 8. Configuration of measurement of cell-averaged reaction rates in the first 2 inches of the central drawer of ZPR-6/7.

transport burn-up calculations. The first major task for such calculations is to build accurate reactor core models. Figs. 9, 10 show the radial cross sectional views of MCNP5 PHENIX reactor models for PROFIL-1 and PROFIL-2 experiments, respectively. In each configuration, only the assembly containing irradiated samples was modeled with detailed descriptions in order to capture pin-by-pin spatial self-shielding effects. Such an assembly corresponds to the central assembly for PROFIL-1, containing one irradiated pin with 46 samples, and the assembly in the second ring of the core for PROFIL-2 containing two irradiated pins, each containing 42 samples. Each irradiated fuel pin for both experiments is located in the third ring of this assembly. The full detailed descriptions of irradiated samples and their containers are available in [54, 55]. Calculated values of keff for the initial PROFIL configurations are shown in Table XI. These quantities indicate that the models are very close to the initial condition that is designed to be critical. Additionally, it was observed that calculated and experimental axial flux profiles over the irradiated samples show a good agreement. These results validate accuracy of our models. In order to perform three-dimensional burn-up Monte Carlo calculations, it is important to obtain accurate and statistically reliable one-group cross sections for each ir-

FIG. 7. Radial cross sectional view of ZPR-6/7 with the high 240 Pu zone model. The upper view shows the entire ZPR matrix loading for this assembly; the lower view indicates the heterogeneous structure of the plate loadings in each core matrix location.

2.

Integral Experiment Analysis: Irradiation Experiments PROFIL-1 and -2

The PROFIL-1 and PROFIL-2 are the highly accurate set of irradiation experiments performed at the PHENIX reactor. These two experiments irradiated a total of 130 small separate samples which contain almost pure isotopes. Thus, these experimental results are a powerful source of information for the nuclear data adjustment for major and minor actinides and several fission products. In order to perform the analysis of these irradiation experiments, one needs to carry out full three-dimensional 609

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TABLE XI. Calculated keff based on initial PROFIL configurations. PROFIL-1 PROFIL-2 keff 1.000006(±8 pcm) 1.00487(±6 pcm)

FIG. 10. Radial cross sectional view of the PROFIL-2 as modeled by MCNP5. The upper plot shows the radial configuration and the lower plot a subassembly containing irradiated samples.

calculation. If the number of recorded source particles is small, then it is not feasible to obtain statistically reliable solutions even with variance reduction techniques. In order to resolve this problem, a program that duplicates the recorded information of source particles was written. However, the number of duplications is limited because adding additional source particles enlarges the size of the source file. In order to address this issue, several fixed source calculations were performed by skipping random numbers corresponding to the number of histories for each run. After finishing all fixed source calculations, the solutions were collected, and then the batch statistics were taken. In the case of PROFIL-1 the “experimental” axial flux distribution has been derived by measurements of reaction rates at different places and times. Because of the differences between the experimental and calculated axial flux distributions, a comparison of the neodymium production for the samples of 235 U was performed using the two distributions. There were 6 samples of 235 U located along the axis of the core. The results indicated that the experimental distribution provides a consistent (almost constant) set of C/E’s, while the calculated one shows a drift in the bottom part. Based on this observation it was decided that the experimental axial distribution

FIG. 9. Radial cross sectional view of the PROFIL-1 as modeled by MCNP5. The upper plot shows the radial configuration and the lower plot a subassembly containing irradiated samples.

radiated sample that are to be used for solving Bateman equations. However, it is extremely difficult to obtain statistically reliable tally results since the size of each sample is very small (∼ 0.06 cm3 ), and some of reactions, e.g., (n,2n), are caused by very fast neutrons (∼ 5 MeV) which are usually not well populated. Moreover, it is not straightforward to perform variance reduction in criticality calculations. Thus, a calculation procedure that uses MCNP’s surface source capability (Fig. 11) was adopted. In this approach, first, a full-core criticality calculation is performed with the surface source write (SSW) card in order to generate the binary source file containing the surface and fission volume sources around and in the irradiated fuel samples, respectively. This source file is used to perform the fixed source calculation with the reduced geometry modeling only the irradiated fuel pin. The fixed source calculation can be performed with only the recorded particles from the full-core criticality 610

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FIG. 11. Procedure for calculating one-group cross sections by MCNP.

respectively, as

should be used in the analysis. The likely reason for the observed discrepancy has to be attributed to the lack of information on the control rod movement during the cycles, and the actual flux calculations were performed with a fixed average control rod position. The next step was to correctly normalize the isotope buildup results to the actual values of the fluence (and hence eliminate the uncertainty in the irradiation history). For this purpose the neodymium production in the 235 U samples were calculated and compared with the corresponding experimental values. The corrective factor, by which the experimental fluxes are divided, has been subsequently derived. The normalization value used in the successive analysis is 1.047. Depletion calculations have been carried out using the NUTS [56] code in order to evaluate the isotope buildup. The one group cross sections from the MCNP5 calculations and normalized fluxes were provided as inputs for the depletion calculations. The information that can be gathered from the PROFIL-1 post-irradiation analysis is related to the evaluation of the reaction rates (mainly capture and (n,2n) rates) for a given isotope. In particular, the analysis of the experiment is based on the relation existing between the burn-up dependent variations of the atom number densities and the microscopic cross sections. For isotopes for which the neutron capture product is stable or has a long decay half-life, the most accurate experimental technique for obtaining information on the integral capture cross section is to determine the variation in composition that results from high-flux irradiation of a pure sample. Capture and (n,2n) reaction rates for an isotope of mass A, which has received a total fluence of τ , can be evaluated by the measurement of ratios of concentrations using Eqs. (9) and (10),

σc,A τ f (τ ) ∼ = ΔNA+1 /NA = NA+1 (τ ) NA+1 (0) − , NA (τ ) NA (0)

(9)

and σ(n,2n),A τ f (τ ) ∼ = ΔNA−1 /NA = NA−1 (τ ) NA−1 (0) − , NA (τ ) NA (0)

(10)

where f (τ ) is a correction factor which takes into account the physical phenomena difference from capture (or (n,2n) reactions) that the considered isotope A can experience during the irradiation. Because of its definition, f (τ ) is a measure of the fertile or fissile properties of a given isotope, being lower and higher than unity for fertile and fissile isotopes respectively. It can be evaluated by a time dependent calculation   (τ ) (0) NA+1 NA+1 1 − (0) × . (11) f (τ ) = (τ ) σ · c τcalc N N A

A

Here, σc and τcalc represent the one-group capture cross section for the isotope A and the calculated fluence, respectively. This approach works very well when we are considering a reaction rate that is dominated by the formation of the resulting measured isotope (i.e., capture cross section) but it will attribute the same C/E also when the reaction rate is not dominant like in the case of an (n,2n) cross section. In order to avoid this problem a slightly different approach was adopted. We correct the experimental density variation by a calculated quantity that takes out 611

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the variation due to all the phenomena other than the reaction rate that we are considering,

obtained results are also presented.

corr /NA = τ · σc,A ∼ = ΔNA+1 exp (τ ) ΔNA+1

−

calc (ΔNA+1

A.

−

(0) NA e−στ )

NA

,

Multiplication Factor and Reactivity Coefficients

(12) The Boltzmann transport equation or its diffusion theory approximation can be written in operator form as

exp (τ ) is the experimental measured density where ΔNA+1 calc variation, and ΔNA+1 is the calculated one. In the end Eq. (7) was used to derive an initial guess for the unknown experimental cross section and then the latter is computed by changing its value until the final measured experimental densities were matched. Using this new approach C/E’s were calculated for the sample isotope buildup. Table XII shows some typical results, in particular for the case relative to the 241 Am samples. The analysis of neutron irradiation on a sample of 241 Am provides information on several reactions, as indicated in Fig. 12. In fact, the build-up of 242m Am gives direct information on the 241 Am capture cross section, while the measured values of the buildup of 238 Pu, 242 Pu and 242 Cm provide information on the overall decay scheme and in particular the branching ratios. Table XII summarizes the C/E obtained from the irradiation of two separate 241 Am samples and shows the consistency of the observed C/E when specific values of the branching ratios (as shown in Fig. 12) are used.

V.

G. Palmiotti et al.

BΦ = (A − λF )Φ = 0,

(13)

where A is the loss operator, F is the fission production operator, and Φ is the neutron flux. Using conventional perturbation theory [65], the sensitivity coefficients for the multiplication factor, k = λ−1 = F Φ/AΦ, can be obtained as     ∂A 1 ∂F kσ ∗ Sk = − ∗ Φ , − Φ , (14) Φ , F Φ ∂σ k ∂σ where ,  denotes the integration over space, angle and energy, and the fundamental mode adjoint flux Φ∗ is determined from the following adjoint equation B ∗ Φ∗ = (A∗ − λF ∗ )Φ = 0. ∗

(15)

∗

Here, A and F are the adjoint operators of A and F , respectively. Reactivity coefficients such as the coolant void worth are generally defined as the reactivity change of a perturbed system from an unperturbed, reference system and can be represented as the eigenvalue difference of the two systems, as Δρ = λ−λ = 1/k−1/k  , with the primed variables for the perturbed system. Thus, the sensitivity coefficients for a reactivity change can be determined as [66]   1 ∂k  σ 1 ∂k σ ∂Δρ = − 2 SΔρ = Δρ ∂σ Δρ k  2 ∂σ k ∂σ   Sk  1 Sk = . (16) − Δρ k  k

SENSITIVITY STUDY

For the sensitivity analysis of the “static” experiments, sensitivities were calculated in transport S4P1 approximation with the BISTRO code [57] of ERANOS [58], using 33-group cross sections prepared with ECCO [59] and based on ENDF/B-VII.0 data [29]. Results were obtained with respect to elastic scattering (σel ), inelastic scattering (σinel ), capture (σc ), P1 anisotropic elastic scattering (P1 σel ), fission (σf ), average number of fission neutrons (ν) and prompt fission spectrum (χp ). Transport theory was used for the sensitivity calculations of all static experiments. However, sensitivities obtained in diffusion theory would generally show a quite good agreement with the transport solution, except in the case of small size systems with large leakage effects [60]. For the sensitivity study of the irradiation experiments, cross sections were generated in 33-energy groups using the MC2-3 code [61] and the ENDF/B-VII.0 nuclear data. Depletion calculations were performed using the fast reactor fuel cycle analysis code REBUS-3 [62]. The flux calculations used the finite difference diffusion theory option of the DIF3D code [63] and sensitivities were obtained with the DPT code [64]. Results were determined with respect to elastic scattering, inelastic scattering, capture, fission, average number of fission neutrons and (n,2n). In the following, a brief overview of the equations used for the sensitivity analysis of the different parameters investigated in the present study is given. Some of the

B.

Reaction Rate Ratios or Spectral Indices

For a reaction rate ratio, RR = Σn , Φ/Σd , Φ, defined by macroscopic cross sections Σn and Σd , the sensitivity coefficients can be calculated using generalized perturbation theory [67–69] as σ dRR = SR R = RR dσ      ∂RR ∂A 1 ∂F σ ∗ − Γ , − Φ , (17) RR ∂σ ∂σ k ∂σ where Γ∗ is the generalized adjoint flux, which is the solution of the following generalized adjoint equation ∂RR (A∗ − λF ∗ )Γ∗ = =  ∂Φ  Σn Σn , Φ Σd − . (18) Σd , Φ Σn , Φ Σd , Φ 612

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FIG. 12. Decay scheme of

241

TABLE XII. C/E values for isotope buildup relative to Samples 11 44 a b c

238

Pu 0.949 0.949

Calculated from Calculated from Calculated from

242

Pu 0.965 0.988

242

Am 0.972 1.001

Am.

241

Am samples in PROFIL-1.

Am σc b 0.965 0.988

241

Am σc c 0.944 0.945

241

242 Am

buildup. buildup. 238 Pu buildup (see text). 242 Pu

limited to the finite difference diffusion theory option in RZ geometry, and the burn cycle is modeled with only a single time interval. Sensitivities S for the generic integral parameters R are calculated by DPT according to   I  ti+1 α ∂R  ∂M S = + N N ∗ dt, R ∂α i=1 ∂α ti     I I   ∂Bi∗ ∗ ∗ ∂Bi + Φi + Φ Γi , Γi , ∂α ∂α i i=1 i=1    I  ∂kσf ∂kσc ∗ − + , Ni Φ i Pi , (19) ∂α ∂α i=1

This equation has solutions since it is a singular inhomogeneous equation with a source term orthogonal to the flux Φ. Its unique solution is obtained by imposing the bi-orthogonal condition Γ∗ , F Φ = 0. The first term in the rightmost side of Eq. (17) is called the direct effect and accounts for the variation of the response RR directly due to the changes in the cross sections explicitly present in the formula of RR. The second term is called the indirect effect and represents the variation of RR due to the flux change subsequent to the cross section variation. In the case where Σn and Σd are microscopic cross sections at a given point, Eq. (17) can be also used for the sensitivity analysis of spectral indices (i.e. reaction rate ratios).

C.

Am σc a 0.969 1.001

241

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where i is the time node, N ∗ is the adjoint number density, Γ∗ is the generalized adjoint flux, Γ is the generalized flux, and P ∗ is the adjoint power. As it can be seen in Eq. (19), DPT has computational capabilities to decompose the sensitivities of nuclide density evolution during burnup into the separate effects of direct, number density, flux, adjoint flux, and power terms. Specifically, the nuclide density term represents the response parameter increase through the changes in nuclide transmutation rates caused by the increase in a cross section of interest, the flux terms represent the response increase through the changes in the neutron flux distribution in space and energy for a fixed flux level, and the power terms represent the response increase through the changes in the neutron flux level for a given power level.

Isotope Buildup

The sensitivity calculations of final number densities were performed using the depletion perturbation theory code DPT. The depletion perturbation method implemented in DPT is based on the non-equilibrium and equilibrium fuel cycle analysis methodologies of REBUS3. The code calculates the sensitivity coefficients of a given response functional with respect to cross sections and initial nuclide densities. The adjoint nuclide density equations are solved using iteration methods similar to those used for solving the nuclide density equations, and the generalized adjoint flux equations are solved using DIF3D. DPT currently generates sensitivity coefficients for five responses: the beginning and end of cycle keff values, the burnup reactivity swing, the power fraction in any core region, and the end of cycle mass of any isotope in any core region. Currently, the flux solution option is

D.

Results

For selected integral parameters, Table XIII shows the reactions of larger sensitivity contributions. In the case 613

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TABLE XIII. Most relevant reaction sensitivities for selected integral parameters. Reaction U σc U σf 235 U ν¯ 238 U σc 238 U σf 238 U ν¯ 238 U σel 238 U σinel 238 U χ 239 Pu σc 239 Pu σf 239 Pu ν¯ 239 Pu σel 239 Pu χ 239 Pu P1 σel 240 Pu σc 240 Pu ν¯ 241 Pu σf 241 Pu ν¯ 242 Pu σc 243 Am σc 0 C σel 16 O σel 23 Na σc 23 Na σel 23 Na σinel 56 Fe σc 56 Fe σel 56 Fe σinel Total 235 235

a

ZPPR-9 ZPPR-9 keff Void STEP-3 -a -0.272 0.092 0.148 -0.061 0.149 -0.063 0.588 0.810 0.810 2.267

0.695 0.095 0.267 -0.086 -0.437 0.347 0.459 -0.821 -1.240 -1.320 0.054 -0.058 -0.551 0.077 0.282 0.356 0.061 -0.111 -0.106 -2.254

JEZEBEL PROFIL-1 COSMO 244 239 Pu Cm Buildup F28/F25 keff in 242 Pu Sample 0.232 -0.996 -1.548 -1.577 -0.170 -0.422 0.983 -0.142 0.125 -0.300 0.181 -0.272 0.722 0.286 -0.126 0.961 1.430 0.061 0.962 -0.098 0.057 0.052 0.985 0.951 -0.204 -0.107 -0.072 -0.080 -0.102 2.736 -0.515 -0.096

Sensitivity coef. null or smaller than the values listed for the same integral parameter.

of the ZPPR-9 multiplication factor, it can be seen that most of the sensitivity effects are due to the 239 Pu and 283 U data. Concerning the reactivity coefficients, nonnegligible sensitivities are generally also due to structural materials as 23 Na, 56 Fe, 16 O especially with the elastic and/or the inelastic scattering. In the case of small size assemblies with large leakage effects such as JEZEBEL 239 Pu, the Legendre coefficient P1 anisotropic elastic scattering generally produce quite relevant sensitivity effects. Regarding the spectral indices, the major sensitivities are mostly due to the cross sections contributing to the direct terms (cross sections at the numerator and denominator of the rate ratio). However, indirect effects may also lead to non-negligible sensitivities (see e.g. 238 U inelastic, 239 Pu fission, 16 O elastic and 238 U capture in the case of the 238 U fission to 235 U fission spectral index in the COSMO experiment). Finally, in the case of the isotope buildup as, e.g., the 244 Cm buildup in the 242 Pu sample of PROFIL-1, it is noticed that the spectral effects due to 238 U capture, 238 U elastic

and inelastic produce relevant sensitivities even if smaller than the density term contributions (242 Pu capture and 243 Am capture in the considered case). Similarly, the flux and power terms through 235 U and 239 Pu fission and ν¯ show non negligible sensitivities as well. Figs. 13 to 18 show the energy profile of the sensitivity associated with selected reactions among those listed in Table XIII.

VI.

COVARIANCES OF INTEGRAL PARAMETERS

The experimental value of an integral parameter can only be useful for validation if its uncertainty is known. It may be slightly less obvious, but equally true, that if a series of experimental values are to be useful for validation, then their uncertainties and correlations must also be known. For example, analysis of a series of N criticality measurements performed on a particular lattice 614

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FIG. 15. Energy distribution of JEZEBEL 239 Pu keff sensitivity coefficients for 239 Pu fission neutron spectrum.

FIG. 13. Energy distribution of ZPPR-9 keff sensitivity to 238 U inelastic and 56 Fe P1 anisotropic elastic scattering.

FIG. 14. Energy distribution of ZPPR-9 void STEP-3 sensitivity coefficients for 23 Na inelastic.

FIG. 16. Energy distribution of COSMO F28/F25 sensitivity coefficients for 238 U fission.

apparatus (all using the same fuel, moderator, etc.) may provide little more validation information than analysis of a single measurement from the series. On the other hand, when experimental values are used in a data adjustment study, the requirement for the integral experiment covariance data is quite obvious, as noted by the matrix term BE in Eq. (1) in Sect. I. Generally, experimental uncertainties of integral parameters are reported by experimenters along with the experimental values. Often these uncertainties are given in the form of components. However, the correlations between different integral parameters are scarcely reported. Therefore, an estimate from available experimental information must be performed as a separate task. A typical procedure for evaluating integral covariance data is described in [70, 71]. In the latter reference, Ishikawa (JAEA) introduced the terms “independent” and “common” for the standardly used terms “random” and “systematic.” The following procedure, described in [71], has been adopted in the present work. Step 1. Analyze the components of experimental uncertainties for data sets A and B quantitatively and

separate them into 2 groups: the random uncertainties (independent uncertainties, with correlation factor ρA,B = 0 between A and B) and the fully systematic uncertainties (common uncertainties, with correlation factor ρA,B = 1). Step 2. Sum up the experimental uncertainties of data sets A and B for random (uran ) and systematic (usyst ) grouped separately. Step 3. Calculate the total uncertainty and correlation factor. The total uncertainty of set A is given by utot,A =

u2ran,A + u2syst,A

(20)

and similar for data set B. Then, the correlation between sets A and B is given by ρA,B =

615

usyst,A · usyst,B . utot,A · utot,B

(21)

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TABLE XIV. Sources of experimental uncertainties for keff in ZPR-9 Assembly 1 (IEU-MET-FAST-013). Source of uncertainty uncertainty % Measurement Technique Data Fitting 0.0002 Inhours to Δk 0.0093 Temperature 0.0037 Geometry Matrix Interface Gap < 0.0001 Nominal Plate Dimensions < 0.0001 Matrix Tube Pitch 0.0450 Room Return 0.0520 Composition Enriched Uranium 0.0732 Depleted Uranium 0.0036 Protective Paint on Fuel Plates 0.0008 Aluminum 0.0124 Humidity 0.0001 Total 0.1018

FIG. 17. Energy distribution of 244 Cm buildup coefficients for 242 Pu sample of PROFIL-1 to selected reactions.

bined in a single uncertainty matrix, there were 62 unique (i.e., independent) uncertainty components contributing to keff uncertainties for these assemblies. Table XV identifies the most significant of these uncertainties. The overall covariance matrix for the 33 ANL critical assembly multiplication factors keff is given in Table XVI. TABLE XV. Major uncertainty components for the ANL ZPR/ZPPR experimental keff values. Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14

FIG. 18. Energy distribution of 106 Pd buildup coefficients for 105 Pd sample of PROFIL-1 to 105 Pd capture.

A.

Application to Criticality Experiments keff

A wealth of integral experiments have now been evaluated in the International Handbook of Evaluated Criticality Safety Benchmark Experiments (ICSBEP Handbook) and the International Handbook of Evaluated Reactor Physics Benchmark Experiments (IRPhEP Handbook) [50, 72]. In the evaluation process for these benchmark experiments, it is expected that the component experimental uncertainties are identified and quantified. Uncertainty data for 33 of the ANL ZPR/ZPPR critical assemblies included in the ICSBEP Handbook were used to estimate covariance data for these experiments as illustrated below. As an example, the experimental uncertainties of keff in ZPR-9 Assembly 1 (IEU-MET-FAST-013) are shown in Table XIV. Experimental uncertainties for keff generally fall into three categories: measurement techniques, geometry and composition. Uncertainty data similar to that in Table XIV were obtained for each of the 33 different ZPPR assemblies. When all of these component uncertainties were com-

B.

Major Uncertainty Component Excess Reactivity Inhours to Δk Room Return Matrix Interface Gap Matrix Tube Pitch Aluminum in Matrix Tubes Steel in Matrix Tubes Steel in the Cans Enriched Uranium Depleted Uranium in Core Depleted Uranium in Blanket Depleted Uranium Oxide Plates Plutonium Depleted Uranium in Pu-U-Mo Plates

Application to Reaction Rate Ratio (RRR) Experiments

For determination of covariance information between reaction rate measurements, components of these uncertainties must include: uncertainties in measurement of foil mapping activities, detector calibration, composition, etc. These component uncertainties are combined as described above. Table XVII presents the uncertainty matrix for the foil activation method for the RRR mea616

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TABLE XVI. Covariance matrix for experimental integral parameter keff for 33 ANL ZPR/ZPPR benchmarks. Included are 8 assemblies of ZPR-3, 5 ZPR-6, 10 ZPR-9 and 10 assemblies of ZPPR.

surement in the ZPPR facility shown for the case of the F49/F25 and C28/F25 reaction rate ratios [71]. In ZPPR-9, the reaction rates were measured in the same run, and at the same foil place. Consequently, common uncertainties of three reaction rate ratios (e.g., F28/F25, F49/F25 & C28/F25) come from the uncertainty in the common reaction rate (F25).

C.

TABLE XVII. Covariance data for measurement of Reaction Rate Ratio (RRR) in ZPPR-9. Reaction Rate Ratio F28/F25 F49/F25 C28/F25 Total Uncert. 2.7% 2.0% 1.9% F28/F25 1.0 F49/F25 0.23 1.0 C28/F25 0.23 0.32 1.0

Application to Irradiation Experiments

The experimental uncertainty UE is max(ue , us ) where ue is the measured uncertainty of the final density (in 617

Combined Use of Integral . . . most cases  0.5%), and us is given by   n   1 2 us =  [(C/E)i − av] , n(n − 1) i=1

NUCLEAR DATA SHEETS

(and therefore 56 Fe) have been excluded (ZPR-3 53 and 54 and their associate reaction rate distributions). This is a long standing issue that will be the focus of future work in nuclear uncertainty reduction. In order to limit the number of cross sections to be adjusted, the multigroup cross sections have been preselected based on their contribution to the total uncertainty of each measured integral parameter taken into account in the adjustment. In practice the following formula is used

(22)

where n is the number of samples and av is the average C/E value 1 (C/E)i . n i=1 n

av =

T Bσ SRp | ≤ unc2 , (ΔRp )2 = |SR p

The calculational uncertainty UC has 3 components combined statistically: flux ( 2%, 1% coming from Nd normalization dispersion, and 1% from axial distribution because rod movement is not known), time dependence of cross sections ( 1%), and statistical cross section uncertainty (MCNP uncertainty doubled and weighted by sensitivity coefficients). The common (systematic) uncertainty is the one associated with the flux. In general Eq. (21) should be applied to both experimental and calculational correlations. In practice, for the 20 experiments used in the OECD/NEA Subgroup 33 exercise [71] the correlations proposed by M. Ishikawa have been adopted, while for the irradiation experiments (PROFIL-1, PROFIL-2, and TRAPU) and COSMO experiments Eq. (21) has been used in the present work. VII. A.

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(23)

where (ΔRp )i is the uncertainty for the measured integral parameter p induced by the uncertainty on cross section i, SRp is the sensitivity array, Bσ the covariance matrix, and  a user input parameter that is used as relative contribution to the total uncertainty (i.e., if  = 0.001, all cross sections that contribute more than one thousandth of the total uncertainty are taken into consideration for adjustment). In particular for keff integral parameters  = 0.0001 has been used; for spectral indices, reactivity coefficients (sodium void, control rod worth), and isotope buildup an  = 0.001 has been adopted. The list of the 34 isotopes included in the adjustment is the following: 10 B, 16 O, 23 Na, 52 Cr, 56 Fe, 58 Ni, 95 Mo, 97 Mo, 101 Ru, 105 Pd, 106 Pd, 133 Cs, 143 Nd, 145 Nd, 149 Sm, 151 Sm, 153 Eu, 234 U, 235 U, 236 U, 238 U, 237 Np, 238 Pu, 239 Pu, 240 Pu, 241 Pu, 242 Pu, 241 Am, 242m Am, 243 Am, 242 Cm, 243 Cm, 244 Cm, and 245 Cm. The list of the 8 reactions and other quantities considered is the following: capture, fission, nu-bar, elastic scattering, inelastic scattering, (n,2n), fission spectrum, and P1 elastic anisotropic scattering. This list is consistent with that available in the COMMARA-2.0 covariance matrix, even though very few isotopes have covariance data ¯ elastic for fission spectrum (only 235,238 U,239 Pu) and μ scattering (only 23 Na and 56 Fe). As indicated in the previous section, when the selection criteria in Eq. (23) were applied to the selected 87 experiments, the resulting number of cross sections to be adjusted amounts to 1,126 against a theoretical initial number of 8,976 (this does not take into account zero cross sections of threshold reactions). The selected cross sections present 8,133 nonzero elements in their distinct covariance matrix. The COMMARA-2.0 library mostly contains energy correlation terms with few inter-reaction terms, and practically no inter-isotope terms. There were 14 correlations used among experiment uncertainties and 357 among calculational uncertainties on the integral parameters. Adjustment was run with and without these two sets of correlations. The impact is relatively negligible as the most visible effect was a slight increase of the normalized χ2 (going from 1.43 to 1.63). The difference in the two χ2 is essentially due to one experiment: the buildup of 239 Pu in the sample of 238 Pu. The results that are presented in the following are those that take into account the correlations.

A GLOBAL MULTIGROUP ADJUSTMENT Experiment and Parameters Down Selection

In this section we present the multigroup adjustment performed using the experiment analysis presented in Sect. IV, the sensitivity analysis of Sect. V, the experiment covariance of Sect. VI, and the covariance data COMMARA-2.0 described in Sect. III. As said before, the original number of analyzed experiments is 148; however, the actual number used in the adjustment is 87. This reduction is due to several reasons. First, experiments with very similar sensitivity coefficients with respect to the same cross sections were eliminated, specifically the samples in the PROFIL2 irradiation experiment that were already measured in PROFIL-1, and the TRAPU-1 and TRAPU-3 isotope build-up (only the TRAPU-2 data were retained). The duplicate results will be used as cross-check for the adjusted data. Second, for some isotopes no covariance data were available. That has been the case for the 144 Nd and 147 Sm samples of the PROFIL-2 irradiation experiments. Next, for some configurations (ZPPR-15 sodium void and central control rod) no reliable sensitivity coefficient calculation model was available. Also, for certain integral parameters (control rod rings of ZPPR-10) generalized perturbation three-dimensional transport capability is necessary and at present not available. Finally, all the configurations relevant to reflector effects 618

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Table XVIII shows the list of the experiments and associated Q, UC/E , UN D , AM , χ, and I values. Concerning the adjustment margins 13 experiments carry negative values: the fission spectral indices of 239 Pu in JEZEBEL, BIGTEN and ZPR-6/7, 238 U in GODIVA, 238 Pu and 241 Am in COSMO, the reactivity worth of the central control rod in ZPPR-10, the buildups of 239 Pu in the PROFIL-1 238 Pu sample, of 240 Pu in the PROFIL-1 239 Pu sample, of 238 Pu in the PROFIL-1 239 Pu sample, of 102 Ru in the PROFIL-1 101 Ru sample, of 106 Pd in the PROFIL-1 105 Pd sample, of 134 Cs in the PROFIL-1 133 Cs sample, and of 243 Cm in the TRAPU-2 experimental pin. Therefore, one should expect some difficulties in the performing the adjustment for these experiments in the one sigma range. The more negative the value, the greater the negative impact on the total χ2 value. However, the more stringent χ value, which considers the three sigma range, shows only one value greater than three. This happens for the buildup of 238 Pu in the PROFIL-1 239 Pu sample. We will see later, how, actually, this experiment should have been eliminated from the adjustment. Concerning the I quantities, one can observe that there are no values less than 0.1; therefore, based on this criterion, all considered experiments should see a significant uncertainty reduction after adjustment. Table XIX shows the results in terms of previously observed C/E (calculation results over experimental one) and the associated uncertainties. In general the agreement is satisfactory as the new C/E after adjustments stay close to 1 within two σ of the uncertainties with some notable exceptions, among others: 239 Pu fission spectral index in ZPR-6/7 and ZPPR-9, STEP-2 sodium void and central rod worth in ZPPR-10, 238 Pu and 241 Am fission spectral indices in COSMO, 239 Pu buildup in the 238 U sample of PROFIL-1, 238 Pu buildup in 241 Am sample of PROFIL-1, and 238 Pu buildup in TRAPU-2. In Table XX experiments are ordered following the magnitude of their contribution to χ2 . This is a negative ranking as a large contribution indicates some problem with the experiment that produces it. An ideal adjustment would have a normalized χ2 of 1 or less. As it can be seen, the largest contributor, the buildup of 239 Pu in the sample of 238 Pu, gives almost all the difference with respect to χ2 = 1. Other notable contributors, but in a lesser magnitude, are the 241 Am fission spectral index in COSMO, 106 Pd buildup in 105 Pd sample of PROFIL-1, the 238 U fission spectral index in Godiva, the 238 Pu fission spectral index in COSMO, the central control rod worth in ZPPR-10, the 239 Pu fission spectral index in JEZEBEL and BIGTEN, the 243 Cm buildup in TRAPU2, and 240 Pu buildup in 239 Pu sample of PROFIL-1. As anticipated, this is practically the same list that was indicated for the AM negative values. For these experiments we have taken a look at both the sensitivity coefficients and the major contributors to the change of C/E. The first indication, if we trust the integral experimental data and their associated uncertainties, is that the current estimate for uncertainties of the follow-

Value of the Experiments and Adjustment Margins

Before proceeding to the adjustment results analysis, let us investigate the relation between the C/E’s and related experimental, calculational and nuclear data uncertainties. This analysis helps foresee what can be expected from the adjustment, especially in terms of reduced uncertainties and improved C/E’s. In order to carry out this analysis let us define some helpful quantities. If we call Q the (E − C)/C quantity that identifies the discrepancy between the experimental result E and the associated calculation result C, UC/E the uncertainty related to Q (which is equal to the statistical combination of UE and UC , respectively the experimental and calculation uncertainty), and UN D the uncertainty coming from nuclear data for a specific integral parameter, we can express the adjustment margin AM as AM = UN D + UC/E − |Q|.

(24)

The AM quantity [71] establishes if in the adjustment there is enough room provided by the nuclear data uncertainty to accommodate the C/E discrepancy. Of course, the C/E discrepancy has to take into account its associated uncertainty. If the AM values are negative, this implies that there is not enough uncertainty for the adjustment in the one sigma range. This will be reflected, afterwards, in the χ2 values. One could interpret the appearance of negative values as a signal of some inconsistency in the covariance matrix (usually due to too low uncertainties associated to specific cross sections). The AM quantity is similar to the χ quantity used by JAEA [11]. The quantity χ is defined as χ=

|Q| 2 UN D

2 + UC/E

.

(25)

This χ is used with a three sigma range criterion for deciding whether to eliminate an experiment from the adjustment. Finally, let us define the quantity I, suggested by M. Ishikawa [71], as I=

UN D . UC/E

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(26)

I can help to assess a posteriori uncertainty of the adjustment. If I  1 the uncertainty associated to nuclear data after adjustment will be close to UC/E , and, therefore, improved if we use good quality experiments and calculation tools with very reduced uncertainties. If I  1, then there is no gain as the a posteriori uncertainty associated to the adjusted nuclear data will remain about the same as the initial one (i.e. the experiments have associated uncertainties larger than those attached to nuclear data). In the special case of I = 1, the resulting a posteriori uncertainty will be ∼ UN D /2. 619

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TABLE XVIII. List of integral experiments and associated Q, UC/E , UN D , AM , χ, and I values in %. Experiment Q UC/E UN D AM 1 JEZEBEL 239 Pu keff 0.014 0.201 0.636 0.823 239 238 235 2 JEZEBEL Pu U/ U 2.354 1.421 3.696 2.763 239 237 235 3 JEZEBEL Pu Np/ U 1.317 1.432 3.615 3.73 4 JEZEBEL 239 Pu 239 Pu/235 U 2.533 0.949 0.823 -0.761 5 JEZEBEL 240 Pu keff 0.019 0.201 0.656 0.838 6 FLATTOP keff -0.097 0.302 0.765 0.970 7 FLATTOP 238 U/235 U 1.812 1.86 3.093 3.141 237 235 8 FLATTOP Np/ U 0.442 1.432 3.446 4.436 9 ZPR-6/7 keff -0.043 0.230 0.971 1.158 10 ZPR-6/7 F8/F5 -0.448 3.499 6.396 9.447 11 ZPR-6/7 F9/F5 3.756 2.524 0.836 -0.396 12 ZPR-6/7 C8/F5 -0.970 2.683 1.512 3.225 13 ZPR-6/7 240 Pu keff 0.063 0.221 0.974 1.132 14 ZPPR-9 keff 0.078 0.117 1.193 1.232 15 ZPPR-9 F8/F5 2.987 2.915 7.897 7.826 16 ZPPR-9 F9/F5 1.958 2.119 0.87 1.031 17 ZPPR-9 C8/F5 -0.921 1.992 1.545 2.616 18 ZPPR-9 STEP-3 -1.884 7.737 7.564 13.417 19 ZPPR-9 STEP-5 2.754 7.543 9.740 14.529 20 JOYO keff 0.255 0.181 0.885 0.811 21 GODIVA keff 0.017 0.201 0.891 1.075 22 GODIVA F28/F25 4.712 1.342 2.231 -1.139 23 GODIVA F49/F25 1.420 1.844 0.716 1.140 24 GODIVA F37/F25 0.908 1.649 3.273 4.014 25 BIGTEN keff -0.002 0.071 2.345 2.414 26 BIGTEN F28/F25 5.597 0.922 13.079 8.404 27 BIGTEN F49/F25 2.669 0.922 1.014 -0.733 28 BIGTEN F37/F25 3.413 1.345 7.626 5.559 29 NP SPHERE keff 0.562 0.365 0.885 0.688 30 ZPR-6/6A keff 0.124 0.096 1.57 1.542 31 ZPPR-15 keff 0.127 0.006 0.966 0.845 32 ZPPR-10 keff -0.015 0.111 1.109 1.205 33 ZPPR-10 STEP-2 -13.717 9.688 6.931 2.901 34 ZPPR-10 STEP-3 -5.338 5.675 6.994 7.331 35 ZPPR-10 STEP-6 -3.535 4.535 7.844 8.844 36 ZPPR-10 STEP-9 -0.819 5.457 8.908 13.546 37 ZPPR-10 center rod -6.279 2.195 1.557 -2.527 38 COSMO F28/F25 1.626 1.803 5.913 6.090 39 COSMO F37/F25 -0.498 1.581 5.089 6.173 40 COSMO F48/F25 -6.760 2.532 3.139 -1.089 41 COSMO F49/F25 0.918 1.295 0.921 1.298 42 COSMO F40/F25 -4.853 2.297 4.377 1.821 43 COSMO F41/F25 -0.369 2.032 0.573 2.237 44 COSMO F42/F25 -1.778 2.309 6.001 6.532 45 COSMO F51/F25 -8.189 2.305 5.022 -0.863 46 COSMO F53/F25 -1.000 2.319 10.03 11.349 47 PROFIL-1 236 U in 235 U sample 5.374 2.309 14.822 11.757 48 PROFIL-1 239 Pu in 238 U sample 2.881 2.500 2.183 1.802 49 PROFIL-1 239 Pu in 238 Pu sample -27.378 2.433 5.797 -19.148 50 PROFIL-1 240 Pu in 239 Pu sample 10.375 2.474 4.657 -3.244 51 PROFIL-1 238 Pu in 239 Pu sample 32.802 8.773 16.428 -7.601 52 PROFIL-1 241 Pu in 240 Pu sample 4.167 2.508 5.581 3.922 53 PROFIL-1 239 Pu in 240 Pu sample 29.199 14.405 27.461 12.667 242 241 54 PROFIL-1 Pu in Pu sample 4.384 2.319 8.190 6.125 55 PROFIL-1 243 Am in 242 Pu sample -5.660 3.466 17.748 15.553 56 PROFIL-1 242 Am in 241 Am sample 1.317 3.089 2.817 4.589 57 PROFIL-1 244 Cm in 242 Pu sample 11.732 6.054 18.878 13.200 96 95 58 PROFIL-1 Mo in Mo sample -3.195 4.522 5.125 6.453 59 PROFIL-1 98 Mo in 97 Mo sample 1.112 4.264 5.034 8.186 60 PROFIL-1 102 Ru in 101 Ru sample -9.42 2.474 5.407 -1.539 61 PROFIL-1 106 Pd in 105 Pd sample 17.925 2.751 5.962 -9.211 62 PROFIL-1 134 Cs in 133 Cs sample 13.766 2.997 8.302 -2.467 63 PROFIL-1 146 Nd in 145 Nd sample 5.042 2.879 4.152 1.989 2.669 2.452 11.069 10.851 64 PROFIL-1 150 Sm in 149 Sm sample 65 PROFIL-1 242 Pu in 241 Am sample 2.354 3.36 2.611 3.617 66 PROFIL-1 238 Pu in 241 Am sample 5.374 3.483 2.706 0.815 67 PROFIL-2 245 Cm in 244 Cm sample -8.759 2.5 33.424 27.165 68 PROFIL-2 154 Eu in 153 Eu sample 9.769 2.377 11.507 4.114 69 PROFIL-2 144 Nd in 143 Nd sample 2.041 2.807 3.452 4.218 70 PROFIL-2 107 Pd in 106 Pd sample 6.724 2.846 14.365 10.487 152 151 71 PROFIL-2 Sm in Sm sample -9.991 2.404 13.259 5.672 72 PROFIL-2 238 Pu in 237 Np sample 6.724 3.114 4.199 0.59 73 TRAPU-2 234 U buildup -2.248 3.27 2.013 3.034 235 74 TRAPU-2 U buildup -1.961 2.408 1.786 2.234 75 TRAPU-2 236 U buildup 0.503 2.433 13.87 15.801 76 TRAPU-2 237 Np buildup 3.842 10.53 10.042 16.730 77 TRAPU-2 238 Pu buildup 1.010 2.433 1.990 3.413 78 TRAPU-2 239 Pu buildup -1.186 2.717 0.550 2.081 79 TRAPU-2 240 Pu buildup 1.626 2.617 1.468 2.459 80 TRAPU-2 241 Pu buildup 0.806 3.015 1.757 3.966 242 81 TRAPU-2 Pu buildup -0.99 2.433 3.353 4.796 82 TRAPU-2 241 Am buildup 1.420 4.327 0.859 3.766 83 TRAPU-2 242 Am buildup -3.754 4.665 3.704 4.615 243 84 TRAPU-2 Am buildup 4.275 4.826 18.224 18.775 85 TRAPU-2 242 Cm buildup -1.672 3.748 2.938 5.015 86 TRAPU-2 243 Cm buildup 107.039 4.036 49.186 -53.817 244 87 TRAPU-2 Cm buildup 5.708 3.64 19.397 17.329 #

620

χ

I

0.021 3.167 0.595 2.600 0.339 2.525 2.016 0.868 0.028 3.267 0.118 2.532 0.502 1.663 0.118 2.407 0.043 4.214 0.061 1.828 1.413 0.331 0.315 0.563 0.063 4.405 0.065 10.183 0.355 2.709 0.855 0.411 0.365 0.775 0.174 0.978 0.224 1.291 0.282 4.892 0.019 4.437 1.810 1.663 0.718 0.388 0.248 1.985 0.001 32.849 0.427 14.186 1.948 1.100 0.441 5.668 0.587 2.425 0.079 16.336 0.132 160.799 0.013 9.976 1.152 0.715 0.593 1.232 0.390 1.729 0.078 1.632 2.333 0.709 0.263 3.280 0.093 3.219 1.676 1.24 0.578 0.711 0.982 1.906 0.175 0.282 0.276 2.599 1.482 2.179 0.097 4.326 0.358 6.420 0.868 0.873 4.355 2.383 1.968 1.882 1.761 1.873 0.681 2.225 0.942 1.906 0.515 3.531 0.313 5.121 0.315 0.912 0.592 3.118 0.467 1.133 0.169 1.181 1.584 2.186 2.730 2.167 1.560 2.770 0.998 1.442 0.235 4.515 0.553 0.777 1.218 0.777 0.261 13.37 0.831 4.841 0.459 1.230 0.459 5.047 0.741 5.515 1.286 1.348 0.586 0.616 0.654 0.742 0.036 5.701 0.264 0.954 0.321 0.818 0.428 0.202 0.542 0.561 0.231 0.583 0.239 1.378 0.322 0.199 0.630 0.794 0.227 3.776 0.351 0.784 2.169 12.187 0.289 5.329

Combined Use of Integral . . .

NUCLEAR DATA SHEETS

TABLE XX. List of 87 integral experiments ordered according to contribution to χ2 (χ2 = 1.6315).

TABLE XIX. A Prior and a posteriori C/E with associated uncertainties. #

Experiment

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87

JEZEBEL 239 Pu keff JEZEBEL 239 Pu 238 U/235 U JEZEBEL 239 Pu 237 Np/235 U JEZEBEL 239 Pu 239 Pu/235 U JEZEBEL 240 Pu keff FLATTOP keff FLATTOP 238 U/235 U FLATTOP 237 Np/235 U ZPR-6/7 keff ZPR-6/7 F8/F5 ZPR-6/7 F9/F5 ZPR-6/7 C8/F5 ZPR-6/7 240 Pu keff ZPPR-9 keff ZPPR-9 F8/F5 ZPPR-9 F9/F5 ZPPR-9 C8/F5 ZPPR-9 STEP-3 ZPPR-9 STEP-5 JOYO keff GODIVA keff GODIVA F28/F25 GODIVA F49/F25 GODIVA F37/F25 BIGTEN keff BIGTEN F28/F25 BIGTEN F49/F25 BIGTEN F37/F25 NP SPHERE keff ZPR-6/6A keff ZPPR-15 keff ZPPR-10 keff ZPPR-10 STEP-2 ZPPR-10 STEP-3 ZPPR-10 STEP-6 ZPPR-10 STEP-9 ZPPR-10 CENTER ROD COSMO F28/F25 COSMO F37/F25 COSMO F48/F25 COSMO F49/F25 COSMO F40/F25 COSMO F41/F25 COSMO F42/F25 COSMO F51/F25 COSMO F53/F25 PROFIL-1 236 U in 235 U sample PROFIL-1 239 Pu in 238 U sample PROFIL-1 239 Pu in 238 Pu sample PROFIL-1 240 Pu in 239 Pu sample PROFIL-1 238 Pu in 239 Pu sample PROFIL-1 241 Pu in 240 Pu sample PROFIL-1 239 Pu in 240 Pu sample PROFIL-1 242 Pu in 241 Pu sample PROFIL-1 243 Am in 242 Pu sample PROFIL-1 242 Am in 241 Am sample PROFIL-1 244 Cm in 242 Pu sample PROFIL-1 96 Mo in 95 Mo sample PROFIL-1 98 Mo in 97 Mo sample PROFIL-1 102 Ru in 101 Ru sample PROFIL-1 106 Pd in 105 Pd sample PROFIL-1 134 Cs in 133 Cs sample PROFIL-1 146 Nd in 145 Nd sample PROFIL-1 150 Sm in 149 Sm sample PROFIL-1 242 Pu in 241 Am sample PROFIL-1 238 Pu in 241 Am sample PROFIL-2 245 Cm in 244 Cm sample PROFIL-2 154 Eu in 153 Eu sample PROFIL-2 144 Nd in 143 Nd sample PROFIL-2 107 Pd in 106 Pd sample PROFIL-2 152 Sm in 151 Sm sample PROFIL-2 238 Pu in 237 Np sample TRAPU-2 234 U buildup TRAPU-2 235 U buildup TRAPU-2 236 U buildup TRAPU-2 237 Np buildup TRAPU-2 238 Pu buildup TRAPU-2 239 Pu buildup TRAPU-2 240 Pu buildup TRAPU-2 241 Pu buildup TRAPU-2 242 Pu buildup TRAPU-2 241 Am buildup TRAPU-2 242 Am buildup TRAPU-2 243 Am buildup TRAPU-2 242 Cm buildup TRAPU-2 243 Cm buildup TRAPU-2 244 Cm buildup

A priori C/E Value Unc. (%) 0.99986 0.97700 0.98700 0.97530 0.99981 1.00097 0.98220 0.99560 1.00043 1.00450 0.96380 1.00980 0.99937 0.99922 0.97100 0.98080 1.00930 1.01920 0.97320 0.99746 0.99983 0.95500 0.98600 0.99100 1.00002 0.94700 0.97400 0.96700 0.99441 0.99876 0.99873 1.00015 1.15898 1.05639 1.03665 1.00826 1.06700 0.98400 1.00500 1.07250 0.99090 1.05100 1.00370 1.01810 1.08920 1.01010 0.94900 0.97200 1.37700 0.90600 0.75300 0.96000 0.77400 0.95800 1.06000 0.98700 0.89500 1.03300 0.98900 1.10400 0.84800 0.87900 0.95200 0.97400 0.97700 0.94900 1.09600 0.91100 0.98000 0.93700 1.11100 0.93700 1.02300 1.02000 0.99500 0.96300 0.99000 1.01200 0.98400 0.99200 1.01000 0.98600 1.03900 0.95900 1.01700 0.48300 0.94600

0.20 1.42 1.43 0.95 0.20 0.30 1.86 1.43 0.23 3.50 2.52 2.68 0.22 0.12 2.92 2.12 1.99 7.74 7.54 0.18 0.20 1.34 1.84 1.65 0.07 0.92 0.92 1.35 0.36 0.10 0.01 0.11 9.69 5.68 4.54 5.46 2.20 1.80 1.58 2.53 1.30 2.30 2.03 2.31 2.30 2.32 2.31 2.50 2.43 2.47 8.77 2.51 14.40 2.32 3.47 3.09 6.05 4.52 4.26 2.47 2.75 3.00 2.88 2.45 3.36 3.48 2.50 2.38 2.81 2.85 2.40 3.11 3.27 2.41 2.43 10.53 2.43 2.72 2.62 3.01 2.43 4.33 4.66 4.83 3.75 4.04 3.64

A posteriori C/E Value Unc. (%) 1.00083 0.99065 0.99426 0.98341 0.99975 1.00048 0.99554 1.00385 1.00091 1.01843 0.96731 1.00953 0.99974 0.99962 0.99096 0.98502 1.00812 0.99810 0.95484 1.00029 0.99770 0.97720 0.99573 1.00635 1.00012 0.99422 0.98455 0.99466 0.99738 0.99981 1.00000 1.00070 1.14126 1.04072 1.02213 0.99518 1.04314 0.99004 1.00320 1.05624 0.99508 1.02060 1.00473 1.00370 1.06561 1.00170 0.97608 0.96082 1.01171 0.97259 0.92935 0.97669 0.93717 0.97964 1.04031 0.98020 0.93942 0.99810 0.98171 0.98563 0.96948 0.97384 0.97345 0.97916 0.97310 0.95218 0.98493 0.98701 0.98438 0.98635 0.98658 0.97713 1.01736 1.02121 1.02314 0.94227 1.06889 1.00272 1.00300 1.00053 1.01980 0.99027 1.02799 0.94249 0.98851 1.00618 1.00297

G. Palmiotti et al.

Experiment (E-C)/C (%) Contrib. to χ2 1 PROFIL-1 239 Pu in 238 Pu sample -27.38 0.480 2 COSMO F51/F25 -8.19 0.107 3 PROFIL-1 106 Pd in 105 Pd sample 17.92 0.093 4 GODIVA F28/F25 4.71 0.072 5 COSMO F48/F25 -6.76 0.063 6 ZPPR-10 center rod -6.28 0.061 7 BIGTEN F49/F25 2.67 0.057 8 TRAPU-2 243 Cm buildup 107.04 0.057 9 JEZEBEL 239 Pu 239 Pu/235 U 2.53 0.054 10 PROFIL-1 240 Pu in 239 Pu sample 10.38 0.051 11 PROFIL-1 243 Am in 242 Pu sample -5.66 0.048 12 BIGTEN F28/F25 5.60 0.046 13 PROFIL-1 102 Ru in 101 Ru sample -9.42 0.041 238 239 14 PROFIL-1 Pu in Pu sample 32.80 0.038 15 ZPR-6/7 F9/F5 3.76 0.032 16 TRAPU-2 238 Pu buildup 1.01 -0.031 134 133 17 PROFIL-1 Cs in Cssample 13.77 0.029 18 PROFIL-1 239 Pu in 238 U sample 2.88 0.028 19 PROFIL-1 238 Pu in 241 Am sample 5.37 0.023 20 ZPPR-10 STEP-2 -13.72 0.020 21 TRAPU-2 243 Am buildup 4.28 0.020 22 PROFIL-1 236 U in 235 U sample 5.37 0.020 23 COSMO F40/F25 -4.85 0.020 24 PROFIL-1 244 Cm in 242 Pu sample 11.73 0.019 238 237 25 PROFIL-2 Pu in Np sample 6.72 0.015 26 NP SPHERE keff 0.56 0.013 27 BIGTEN F37/F25 3.41 0.012 28 TRAPU-2 244 Cm buildup 5.71 0.011 29 TRAPU-2 240 Pu buildup 1.63 0.011 30 PROFIL-1 239 Pu in 240 Pu sample 29.20 0.010 31 PROFIL-1 146 Nd in 145 Nd sample 5.04 0.010 32 ZPPR-9 F9/F5 1.96 0.010 33 COSMO F28/F25 1.63 0.009 34 PROFIL-2 154 Eu in 153 Eu sample 9.77 0.008 35 ZPPR-10 STEP-3 -5.34 0.007 36 TRAPU-2 239 Pu buildup -1.19 -0.007 152 151 37 PROFIL-2 Sm in Sm sample -9.99 0.007 38 TRAPU-2 242 Cm buildup -1.67 -0.006 39 ZPPR-15 keff 0.13 0.006 40 PROFIL-1 241 Pu in 240 Pu sample 4.17 0.005 41 JEZEBEL 239 Pu 238 U/235 U 2.35 0.004 42 PROFIL-1 96 Mo in 95 Mo sample -3.19 0.004 43 ZPPR-10 STEP-6 -3.54 0.004 44 ZPPR-9 C8/F5 -0.92 0.004 45 COSMO F49/F25 0.92 0.004 46 TRAPU-2 241 Pu buildup 0.81 0.004 47 ZPPR-9 keff 0.08 0.004 48 FLATTOP 238 U/235 U 1.81 0.004 49 TRAPU-2 235 U buildup -1.96 0.003 50 TRAPU-2 237 Np buildup 3.84 0.003 51 ZPPR-9 STEP-5 2.75 0.003 52 ZPPR-9 F8/F5 2.99 0.003 53 ZPR-6/7 C8/F5 -0.97 0.003 54 TRAPU-2 241 Am buildup 1.42 0.003 55 ZPR-6/6A keff 0.12 0.003 56 PROFIL-2 107 Pd in 106 Pd sample 6.72 0.003 57 JOYO keff 0.25 -0.003 58 PROFIL-1 242 Pu in 241 Am sample 2.35 0.003 59 GODIVA F37/F25 0.91 -0.002 60 TRAPU-2 242 Am buildup -3.75 0.002 61 GODIVA F49/F25 1.42 0.002 62 PROFIL-2 144 Nd in 143 Nd sample 2.04 0.002 63 TRAPU-2 236 U buildup 0.50 -0.002 64 TRAPU-2 234 U buildup -2.25 -0.002 65 ZPR-6/7 F8/F5 -0.45 0.001 66 FLATTOP 237 Np/235 U 0.44 -0.001 67 ZPR-6/7 keff -0.04 0.001 68 GODIVA keff 0.02 0.001 69 ZPPR-9 STEP-3 -1.88 0.001 70 JEZEBEL 239 Pu 237 Np/235 U 1.32 -0.001 71 ZPPR-10 keff -0.01 0.001 72 PROFIL-1 242 Pu in 241 Pu sample 4.38 0.001 73 TRAPU-2 242 Pu buildup -0.99 0.001 74 COSMO F42/F25 -1.78 0.001 75 FLATTOP keff -0.1 0.001 76 COSMO F37/F25 -0.50 0.000 77 PROFIL-2 245 Cm in 244 Cm sample -8.76 0.000 78 COSMO F41/F25 -0.37 0.000 79 PROFIL-1 150 Sm in 149 Sm sample 2.67 0.000 239 80 JEZEBEL Pu keff 0.01 0.000 81 PROFIL-1 98 Mo in 97 Mo sample 1.11 0.000 82 PROFIL-1 242 Am in 241 Am sample 1.32 0.000 240 83 ZPR-6/7 Pu keff 0.06 0.000 84 ZPPR-10 STEP-9 -0.82 0.000 85 JEZEBEL 240 Pu keff 0.02 0.000 86 BIGTEN keff 0.00 0.000 87 COSMO F53/F25 -1.00 0.000 #

0.16 1.03 0.86 0.44 0.18 0.18 0.87 0.80 0.08 1.07 0.33 0.60 0.08 0.07 1.03 0.33 0.60 2.28 2.67 0.14 0.16 0.93 0.35 0.89 0.07 0.84 0.38 1.03 0.29 0.09 0.01 0.07 2.11 2.14 2.22 2.49 0.73 0.89 0.89 1.75 0.33 1.62 0.39 2.03 1.27 2.24 1.01 0.59 1.47 1.36 7.59 1.52 12.67 1.32 2.21 1.40 2.58 3.18 3.04 1.58 1.81 2.26 1.94 1.60 1.21 1.24 2.16 2.03 2.08 2.55 2.07 2.26 1.55 0.19 1.00 7.12 0.67 0.21 0.41 0.46 0.54 0.43 2.26 2.44 1.51 3.63 2.50

621

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NUCLEAR DATA SHEETS

mostly for cases with no or small changes from ENDF/B-VII.0 to ENDF/B-VII.1.

ing reactions in specific energy ranges (consistent with the spectrum of the experiment), are likely underestimated: 238

Pu 105 Pd 238 Pu 242 Cm

capture, capture, fission, capture,

241

Am 238 U 239 Pu 239 Pu

This is the case of 239 Pu, 242 Cm, 105 Pd, 151 Sm, and Nd. Figs. 22, 23, and 24 show the comparison for 239 Pu, 242 Cm, and 105 Pd capture cross sections. A special case is 238 Pu: the adjustments suggest a significant reduction of the capture cross section that for some energy interval can reach unphysical values. The trend for reduction is probably justified, but the present COMMARA-2.0 uncertainties are not consistent with the indications coming from the integral experiment (PROFIL-1 irradiation) that gives most information on that cross section (see previous discussion on this subject).

fission, fission, fission, (n,2n).

145

In other words, the existing covariance data do not provide enough room to perform a robust adjustment. One specific cross section deserves particular attention: 238 Pu capture. In fact, for this parameter the adjustment produced unphysical solutions (i. e. negative cross sections), confirming what was anticipated by the associated χ value that indicated the elimination of the buildup of 239 Pu in the sample of 238 Pu PROFIL-1 experiment, to which the 238 Pu capture carry most of the sensitivity. As pointed out before, eliminating this experiment would generate a very reasonable χ2 value close to 1. In the previous list of cross sections, and using the same approach (sensitivity coefficients and contributions to the adjusted C/E) applied to the experiments previously indicated in which the new C/E are outside the 2σ margins, one can add the following reactions, where uncertainties are likely underestimated: 238

U U 23 Na 16 O 238

C.

capture, inelastic, inelastic, elastic,

239

Pu Na 56 Fe 239 Pu 23

G. Palmiotti et al.

capture, elastic, elastic, fission spectrum.

FIG. 19. Comparison of a priori and a posteriori values for 239 Pu inelastic cross sections.

Analysis of Specific Cross Section Adjustments

A more complex task is to discuss the adjustment impact on the individual cross sections given the large amount of data used (1126 cross sections). The wide range of cross section adjustments can be analyzed according to different criteria. In what follows we have selected, rather arbitrarily, four categories: • Inelastic cross section adjustments for the BIG THREE actinides (235 U, 238 U, and 239 Pu). The inelastic cross sections for these materials are all reduced in the range ∼ 1 − 5 MeV. Their adjustments can be coupled to significant uncertainty reduction. Fig. 19 shows the 239 Pu inelastic cross section for ENDF/B-VII.0 (same as ENDF/B-VII.1) compared against the a posteriori values (after adjustment). Fig. 20 shows the associated uncertainty bands. One can appreciate the considerable reduction induced by the adjustment. In Fig. 21 the adjusted 238 U inelastic cross section is shown. The main differences (of the order of ∼ −5%) with respect to the ENDF/B-VII.0 values ooccur in the plateau region.

FIG. 20. Comparison of a priori and a posteriori values for 239 Pu inelastic cross sections uncertainty band.

• Adjustments that help to discriminate between ENDF/B VII.0 and VII.1. – Adjustments closer to ENDF/B-VII.0. This is the case of, e.g., 241 Am capture cross section  5 keV; 238 Pu fission cross section; 242 Pu fission cross section; 95 Mo capture cross section. Figs. 25, 26, and 27 show the comparison for 241 Am capture, 238 Pu fission, and 95 Mo capture cross sections.

• Capture cross section adjustments both for TRU (TRans-Uranics) and FP (Fission Products), 622

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G. Palmiotti et al.

FIG. 21. Comparison of a priori and a posteriori values for 238 U inelastic cross sections.

FIG. 24. Comparison of a priori and a posteriori values for 105 Pd capture cross sections.

FIG. 22. Comparison of a priori and a posteriori values for 239 Pu capture cross sections.

FIG. 25. Comparison of a priori and a posteriori values for 241 Am capture cross sections.

– Adjustments closer to ENDF/B-VII.1. This is the case of, e.g., 243 Am capture cross section and 240 Pu(n,2n). Fig. 28 shows the comparison for the 240 Pu(n,2n) cross sections.

This is the case of 235 U capture cross section where nominal values adjustments are coupled to a very

significant reduction in the uncertainty. Other cases include: 238 U(n,2n), 242m Am and 243 Am fission, 244 Cm capture, 237 Np capture, and most FP capture cross sections. In Figs. 29, 30, and 31 a priori and a posteriori uncertainty bands are plotted for 235 U (reduction over all energy range), 244 Cm (above ∼ 3 keV), and 149 Sm (reduction over all energy range) capture cross sections.

FIG. 23. Comparison of a priori and a posteriori values for 242 Cm capture cross sections.

FIG. 26. Comparison of a priori and a posteriori values for 238 Pu fission cross sections.

• Small or even near zero adjustments associated with strong uncertainty reductions.

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FIG. 27. Comparison of a priori and a posteriori values for 95 Mo capture cross sections.

FIG. 30. Comparison of a priori and a posteriori values for 244 Cm capture cross section uncertainty band.

FIG. 31. Comparison of a priori and a posteriori values for 149 Sm capture cross section uncertainty band.

FIG. 28. Comparison of a priori and a posteriori values for 240 Pu (n,2n) cross sections.

D.

where Bσ is the “a-priori” correlation matrix of the parameters and BE is the “a-priori” correlation matrix of the experiments. That form corresponds to the hypothesis that for “a-priori” there is no correlation between parameters and experiments. The “a-posteriori” correlation matrix (i.e. after adjustment) is given by

On the “A-posteriori” Correlations Between Parameters and Experiments

The global “a-priori” correlation matrix By has the form  B 0 By =  σ 0 BE

  , 

G. Palmiotti et al.

By˜ = (I − By AT G−1 A)By ,

(27)

(28)

where A is the sensitivity matrix with dimension (K + J) × J, K is the total number of parameters (e.g. cross sections) and J is the total number of experiments. A is given by    A11 A12 · · · A1r     A21 A22 · · · A2r    (29) A= . .. ..  .  .. . .    Aq1 Aq2 · · · Aqr  The matrix A can be rewritten as a vector with two components, each being a matrix A = |Aσ , AE | = |S, 1|,

(30)

where S (dimension K × J) is the sensitivity matrix of the integral experiments with respect to the parameters

FIG. 29. Comparison of a priori and a posteriori values for 235 U capture cross section uncertainty band.

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˜σ is a matrix of the type As shown above, B

and AE is a square identity matrix (dimension J × J). Finally G = ABy AT

˜σ = Bσ (1 − A), B

(31)

(32)

The form of G−1 means that the “a-posteriori” correlation matrices for both parameters and experiments will contain terms depending on both Bσ and BE . We can write the more general expression for By˜ as    Bσ˜ B ˜  σ ˜ E .  By˜ =  (33) BE˜ σ˜ BE˜ 

T ˜ T ˜R = SR Bσ (1 − A)SR . Bσ SR = SR B

VIII.

If we multiply both sides  of the previous equation once 1 0 by   and once by   we obtain, respectively, 0 1     1  B  By˜ ×   =  σ˜  (35) B˜ 0    E σ˜    Bσ   Bσ   Bσ S  −1       G |S − 1| ×  =  − 0   −BE  0       Bσ   Bσ S  −1    G SBσ  − =  0   −BE       B   B SG−1 SBσ  . =  σ  −  σ 0 −BE G−1 SBσ 

Analogously we obtain     B ˜   Bσ SG−1 BE  σ˜ E  =   B ˜   BE − BE G−1 BE E

  . 

  . 

(36)

(37)

From these equations, one can get the explicit expression of the expected “a-posteriori” matrices for both the parameters (σ) and the experiments Bσ˜ = Bσ − Bσ SG−1 SBσ

(38)

BE˜ = BE − BE G−1 BE .

(39)

(41)

At present, a limited use is made of these “a-posteriori” correlations and further studies are needed to prove their relevance from a physics point of view, in order to extract from them further information on the performed adjustment and some insight on their use for e.g. assessing the extrapolation of an adjustment to a wider set of integral parameters in different systems.

To find the explicit form of the matrix elements, we can use the following procedure    S  −1   G |S − 1| × By (34) By˜ = By − By ×  −1     B S  = By −  σ  G−1 |S − 1| × By . −BE

Finally we have     Bσ˜   Bσ − Bσ SG−1 SBσ     B˜  =  BE G−1 SBσ Eσ ˜

(40)

˜σ is used to where A is in principle a full matrix. When B evaluate the new uncertainty on any integral parameter, the terms off-diagonal will give significant or even dominant contributions to the a posteriori uncertainty reduction of an integral parameter R if most of the aij terms are positive (positive correlations), see e.g. Ref. [73]

with dimension J × J . The inverse of G, G−1 , is given by G−1 = (S T Bσ S + BE )−1 .

G. Palmiotti et al.

and

The off-diagonal terms, although they do not enter explicitly into the adjustment, represent “a-posteriori” correlations between experiments and parameters. 625

A NEW FRONTIER: THE CONSISTENT METHOD A.

The Method

The major drawback of the classical adjustment method is the potential limitation of the domain of application of the adjusted cross section data since adjustments are made on multigroup data, and the multigroup structure, the neutron spectrum used as weighting function and the code used to process the basic data file are significant constraints. A new approach has been developed in order to adjust physical parameters and not multigroup nuclear data, the objective being now to correlate the uncertainties of some basic parameters that characterize the neutron cross section description, to the discrepancy between calculation and experimental value for a large number of clean, high accuracy integral experiments. This new approach is the first attempt to build up a link between the wealth of precise integral experiments and basic theory of nuclear reactions. A large amount of exceptionally precise integral measurements have been accumulated over the last 50 years. These experiments were driven by the necessities of nuclear applications but were never fully exploited for improving the predictive power of nuclear reaction theory. Recent advances in nuclear reaction modeling and neutron transport calculations, combined with sensitivity analyses methods offer a reasonable possibility of de-convoluting results of the integral experiments in a way to obtain feedback on parameters entering nuclear reaction models. Essential ingredients of such a procedure will be covariance data for model parameters and sensitivity matrices. The latter will provide direct link between reaction theory and integral experiments. By using integral reactor physics

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NUCLEAR DATA SHEETS

experiments (at the meter scale), information is propagated back to the nuclear level (at the femtometer scale) covering a range of more than 13 orders of magnitude. The assimilation procedure results in more accurate and more reliable evaluated data files that will be of universal validity rather than tailored to a particular application. These files will naturally come with cross section covariance data incorporating both microscopic and integral measurements as well as constraints imposed by the physics of nuclear reactions. Thus, these covariance data will encompass the entire relevant knowledge available at the time of evaluation. On the physics side, the assimilation improves knowledge of model parameters, increasing the predictive power of nuclear reaction theory and bringing a new quality to nuclear data evaluation as well as refinements in nuclear reaction theory. The classical “statistical adjustment” techniques as described in Sect. I provide adjusted multigroup nuclear data for applications, together with new, improved covariance data and reduced uncertainties for the required design parameters, in order to meet target accuracies. One should, however, set up a strategy to cope with the drawbacks of the methodology, which are related to the energy group structure and energy weighting functions adopted in the adjustment. In fact, the classical statistical adjustment method can be improved by “adjusting” reaction model parameters rather than multigroup nuclear data. The objective is to associate uncertainties of certain model parameters (such as those determining neutron resonances, optical model potentials, level densities, strength functions, etc.) and the uncertainties of theoretical nuclear reaction models themselves (such as optical model, compound nucleus, pre-equilibrium and fission models) with observed discrepancies between calculations and experimental values for a large number of integral experiments. The experiments should be clean (i.e., well documented with high QA standards) and high accuracy (i.e., with as low as possible experimental uncertainties and systematic errors), and carefully selected to provide complementary information on different features and phenomena, e.g., different average neutron spectrum energy, different adjoint flux shapes, different leakage components in the neutron balance, different isotopic mixtures and structural materials, etc. In the past, a few attempts were made [74–76] to apply a consistent approach for improving basic nuclear data, in particular to inelastic discrete levels and evaporation temperatures data of 56 Fe for shielding applications, and to resolved resonance parameters of actinides (e.g., G and total and partial widths, peak positions, etc.). More recently [77], the method was applied to 23 Na. This effort indicated the validity of the approach but also challenges to be overcome for its practical application. This was mainly related to the way of getting the sensitivity coefficients and to the need of reliable covariance information.

B.

G. Palmiotti et al.

Consistent Method Implementation

The Consistent Data Assimilation methodology allows one to overcome both difficulties, using the approach that involves the following steps: • Selection of the appropriate reaction mechanisms along with the respective model parameters to reproduce adopted microscopic cross section measurements with the empire [24] code calculations. Use of coupled channels, quantum-mechanical pre-equilibrium theories, and advanced statistical model accounting for width fluctuations and full gamma cascade ensure state of the art modelling of all relevant reaction mechanisms. • Determination of covariances matrices for the set of nuclear reaction model parameters obtained in the previous step. This is achieved by combining initial estimates of parameter uncertainties, with uncertainties/covariances for the adopted experimental data through the kalman [26] code. This way, the resulting parameter covariances will contain constraints imposed by nuclear reaction theory and microscopic experiments. Several parameters have been considered, including resonance parameters for a few dominating resonances, optical model parameters for neutrons, level density parameters for all nuclei involved in the reaction, parameters entering pre-equilibrium models, and parameters determining gamma-strength functions. • Sensitivity of cross sections to the perturbation of the above mentioned reaction model parameters are calculated with the empire code. • Use of the adjoint technique to evaluate sensitivity coefficients of integral reactor parameters to the cross section variations, as described in the previous step. To perform this task, the eranos code system [58] that computes sensitivity coefficients based on generalized perturbation theory is employed. • Performing analysis of selected experiments using the best calculation tools available (in general Monte Carlo codes like MCNP). • Performing consistent data assimilation on basic nuclear parameters using integral experiment analysis with best methodology available to provide discrepancies between calculation and measured quantities. After the C/E’s are available, they are used together with the sensitivity coefficients coming from the previous step in a data assimilation code. • Constructing new ENDF/B type data files based on modified reaction theory parameters for use by neutronic designers. 626

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the isotope under consideration is generated and a subsequent run of njoy on this file generates multigroup cross sections in the same energy structure used for the computation of the reactor physics integral parameters. The multigroup cross section variations associated with the individual fundamental parameter that have been varied in the corresponding empire calculation are readily computed by difference with the reference njoy calculation for the isotope under consideration. In parallel, the cross section sensitivity coefficients to integral parameter R, ΔR/Δσj , are provided using the standard Generalized Perturbation Theory in the eranos code system [58]. Folding the two contributions (from empire and eranos) one obtains the sensitivity coefficients of the nuclear physics parameters to the integral measured parameters.

Evaluation of Nuclear Physics Parameter Covariances

As indicated in the outline of the methodology, the first step is to provide estimated range of variation of nuclear physics parameters, including their covariance data. To this end the code empire coupled to the kalman code is employed. The kalman code is an implementation of the Kalman filter technique based on minimum variance estimation. It naturally combines covariances of model parameters, of experimental data and of cross sections. This universality is a major advantage of the method. kalman uses measurements along with their uncertainties to constrain covariances of the model parameters via the model parameter sensitivity matrix. Then, the final cross section covariances can be calculated from the updated covariances for model parameters. This procedure consistently accounts for the experimental uncertainties and the uncertainties of the nuclear physics parameters. We emphasize that under the term ‘reaction model’ we mean also the resonance region described by models such as the Multi-Level Breit-Wigner formalism.

D.

E.

Direct Monte-Carlo Assimilation of Integral experiments

An alternative method was developed at BNL. Here, rather than compute the sensitivities of the multigroup cross sections σj for each empire parameter pk , the variations in the pk are propagated through empire to individual ENDF files. These are then processed with NJOY to produce ACE cross section files, which then replace the standard ENDF/B-VII.1 ACE file in MCNP Monte-Carlo simulations of an integral experiment. In this way we can approximate the sensitivity of the integral experiment observable(s) (for example, keff ) to the empire parameters pk . These sensitivities are then used in a kalman fit, where there may be only one experimental point - the observable in an integral experiment. In this case, the sensitivity matrix is reduced to a simple vector of sensitivities to each empire parameter. In an attempt to modify the empire parameters in a way that preserves agreement with the differential data, the output covariance matrix for the empire parameters from the differential kalman fit is used as the input covariance matrix when fitting the integral data. The resulting covariance matrix contains information concerning correlations from both differential and integral experiments. These parameter covariances can then be used to generate single and cross-reaction covariances, and, once the method is extended to multiple materials, cross-material covariances as well, as constrained by the model and experimental data. Of course, the applicability of this sensitivity-matrix approach rests on the assumption that the dependence of the integral experiment observables will be fairly linear with respect to varied empire parameters, at least within the range they are allowed to vary. To check this, we calculate the non-linear correction for each varied parameter using the central value along with the results from each ±Δpk . If the non-linear corrections are large, as is the case for some parameters, particularly the optical model and prompt-fission neutron spectra parameters, then ei-

Evaluation of Sensitivity Coefficients for Integral Experiments

In order to evaluate the sensitivity coefficients of the nuclear parameters to the integral parameters measured in a reactor physics experiment, a folding procedure is applied, where the sensitivity calculated by empire, are folded with those calculated by eranos (i.e. multigroup cross section sensitivity coefficient to integral parameters). Following this procedure, the sensitivities of integral experiments to nuclear parameters pk are defined as  ΔR ΔR Δσj = × , Δpk Δσ Δp j k j

G. Palmiotti et al.

(42)

where R is an integral reactor physics parameter (e.g., keff , reaction rates, reactivity coefficient, etc.), and σj a multigroup cross section (the “j” index accounts for isotope, cross section type and energy group). In general to compute σj one can use a) empire with an appropriate set of parameters pk to generate first b) an ENDF/B file for that specific isotope and successively, c) njoy, to obtain multi-group cross sections. As specified in the previous section, one can compute the variation of the cross sections Δσj resulting from a variation of each parameter pk . Specifically, the procedure would consist in the generation of the Δσj corresponding to fixed, well chosen variations of each pk taken separately and generating the Δσj /Δpk as a two-point approximation to the first derivative of the σj with respect to each pk , dσj /dpk . Following each empire calculation, an ENDF/B file for 627

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ther the size of the step Δpk must be reduced, or a nonlinear fitting algorithm implemented. While the effects of non-linearities was not a significant issue when fitting a single integral observable such as keff , they will likely become more important as more integral observables are fit across multiple materials.

IX.

G. Palmiotti et al. A.

Assimilation of

239

Pu

239 Pu is a major actinide produced in light water reactors through neutron capture on 238 U followed by two successive β − decays. Fast reactor sensitivity studies have shown the need for decreasing the uncertainties associated with 239 Pu fission cross section in the fast region. It was chosen for the present assimilation exercise due to its paramount importance and availability of a clean and a relatively simple to model integral experiment (JEZEBEL). As in 235 U, two rounds of assimilation have been scheduled for 239 Pu. The first was completed in 2011 and reported in the INL Report [79]. The second round has advanced to the point of producing an improved prior, model parameter covariances, and group-wise sensitivity matrices, but the final assimilation is not yet complete.

APPLICATIONS

The mechanics of assimilation as described in Sect. VIII D were applied to a number of materials. A set of prior empire parameters, the resulting cross sections, the sensitivities of the cross sections to the parameters and the parameter covariances were prepared by the NNDC at BNL. The application of these priors to integral experiments was carried out at INL, which completes a single cycle of assimilation. The first assimilations performed at BNL and INL were for the structural materials 23 Na and 56 Fe, as described in INL report INL/EXT-10-20094 [78]. While much was learned from the assimilation of these materials, the resulting evaluations were not considered an improvement over the the existing ENDF/B-VII.0 and were not used in ENDF/B-VII.1. Another set of assimilations were performed for 235 U and 239 Pu as described in INL report INL/EXT-11-23501 [79]. For both of these materials a first round of assimilation was completed, with limited success, and a second round of empire calculations has been completed using an improved version of empire. In addition, the testing of a ‘direct’ method of assimilation of the empire calculations to integral experiments using Monte-Carlo methods was tested at BNL for 239 Pu with encouraging results. See the following section on 239 Pu for further details. A first round of assimilation was also performed for 242 Pu and 105 Pd, described in INL report INL/EXT-12-27127 [80]. The integral experiment PROFIL-1 was used for 105 Pd and was found to be sensitive to only a few empire parameters. In spite of this relatively simple adjustment, a good assimilation was only possible if the empire uncertainties were adjusted ad hoc, perhaps resulting from a discrepancy between the differential and integral data sets, especially capture. 242 Pu was assimilated using a number of irradiation experiments, with limited success, where some parameters required variations greater than 1σ to obtain reasonable values for C/E, again suggesting conflict between integral and differential data. Finally, 238 U, not originally considered for assimilation, was added due to its role as an important actinide. Here, the calculations were performed using a new version of empire employing new potentials with good results. The cross sections, sensitivities and covariances of parameters were prepared but have not yet been assimilated with integral data.

1.

First Round of Assimilation

The version of empire used in these calculations was revision 1978. The nuclear reaction models and major options used to prepare the prior are summarized below: • Coupled Channels (ECIS code) used for direct inelastic scattering and absorption calculations; • Spherical optical model transmission coefficients used in compound nucleus decay; • Optical model parameters used: – direct inelastic scattering RIPL-3 no. 2408 [81, 82]; – neutrons RIPL-3 no. 2408; – protons RIPL-3 no. 5408 [81, 82]; • Exciton model used for pre equilibrium emission of neutrons, protons and gammas; • Iwamoto-Harada model used for cluster emission; • HRTW width fluctuation correction was used for compound nucleus decay for incident neutrons energies below 3.00 MeV; • Hauser-Feshbach model with full γ-cascade used for compound nucleus decay; • Optical model for fission used for fission calculations; • Internal empire library used for fission barrier parameterization; • Discrete levels above fission barriers taken into account; • empire-specific EGSM level densities used in Hauser-Fashbach model and fission calculations; 628

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• E1 γ-strength function set to modified Lorentzian (RIPL-3 MLO1 option).

G. Palmiotti et al.

improvement was obtained on the discrepancies of keff , as mentioned above, and the fission spectral index of 239 Pu, while that of the fission spectral index of 238 U stays essential the same after adjustment. The remaining two fission spectral indices were already in good agreement and do not change significantly. The two improved integral parameters are directly related to the 239 Pu fission cross sections, and, therefore, one should expect such amelioration. For the 238 U spectral index it is likely that an improvement would be obtained if we take into account the dependence from the 238 U fission cross section.

The default empire input was adjusted by modifying several parameters, especially level densities, fission barriers, and transitional states above fission barriers, to reproduce experimental data in all reaction channels. In doing so we used the ENDF/B-VII.0 evaluation as a guide. Some energy dependent tuning of the parameters was invoked to bring calculations closer to the ENDF/B-VII.0 cross sections. In this initial attempt PFNS and ν¯ were taken over from the ENDF/B-VII.0 evaluation and were not subject to assimilation. Also the resonance parameters were imported from ENDF/B-VII.0. BNL provided the multigroup cross sections, covariance matrix for the model parameters and the sensitivities in terms of the multigroup cross sections to INL. The assimilation was performed at INL using the JEZEBEL integral experiment. Detailed description of this work was reported in Ref. [79] and we only summarize its final results. The integral parameters considered were keff , and the fission spectral indices. The prior cross sections from empire calculations resulted in a keff of 0.9857 ± 8 pcm. The results from the assimilated cross sections resulted in keff = 0.99980 ± 8 pcm. For comparison, the ENDF/B-VII.0 yields keff of 0.99986 ± 9 pcm. Fig. 32 shows the assimilated fission cross section compared to the prior and ENDF/B-VII.0, which are equal to ENDF/B-VII.1. The assimilation increased fission of 239 Pu below 6 MeV bringing it closer to ENDF/B-VII.0 between 1 and 6 MeV and worsening the agreement below 1 MeV. The agreement with ENDF/B-VII.1 has slightly improved but overall ENDF/B-VII.1 is still in much better agreement with the differential data than our postassimilation evaluation. This happens in spite of the fact that both files show perfectly equivalent performance with JEZEBEL keff .

TABLE XXI. (C/E) values before and after adjustment to JEZEBEL experiments in the first round of the 239 Pu assimilation. Experiment keff F28/F25 F49/F25 F37/F25 F23/F25

prior C/E 0.9857 ± 0.002 0.9561 ± 0.009 0.9708 ± 0.020 0.9988 ± 0.017 1.0003 ± 0.017

post C/E 0.9998 ± 0.002 0.9598 ± 0.002 0.9917 ± 0.003 1.0010 ± 0.001 1.0002 ± 0.001

TABLE XXII. empire parameters varied during the assimilation of 239 Pu with JEZEBEL. Each parameter varied is listed along with the % variation from the assimilation and the initial and final % uncertainties. All parameters using energy units are given in MeV. % Initial % Final % Variation Std. Dev. Std. Dev. VA000a -0.141 0.134 0.121 0.432 0.951 0.612 FUSRED000b 0.299 0.705 0.692 LDSHIF010c -0.120 0.671 0.668 DELTAF000d -0.076 0.965 0.958 ATILNO010e -0.079 0.480 0.479 VB000f 0.128 1.240 1.239 ATLATF000g 0.918 0.815 TOTRED000h -0.0831 HA000i -0.155 0.474 0.471 Parameter

a b c d e f g h i

FIG. 32. Comparison of the post-assimilation fission cross sections for 239 Pu compared with the respective prior, ENDF/BVII.1 (equal to ENDF/B-VII.0) and selected experimental data.

Height of first fission barrier hump in 240 Pu. Factor multiplying reaction (fusion, absorption, compound nucleus formation) cross sections. Shift (LDSHIFT-1) of the level densities in target at the point of discrete levels. Pairing energy used in the level densities at the saddle point in 240 Pu. Factor multiplying asymptotic level density parameter in the target. Height of the second fission barrier hump in 240 Pu. Factor multiplying asymptotic level density parameter at the saddle point in 240 Pu. Factor multiplying total cross section. Width of the first fission barrier hump in 239 Pu.

Table XXII shows the empire parameter variations and standard deviations obtained by the data assimilation (using a prior obtained with an older version of empire). For this assimilation one parameter variation, that of VA000 (the height of the first fission barrier hump),

The results of the assimilation for the full set of integral parameters are summarized in Table XXI. A significant 629

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slightly exceeded the initial uncertainty, while the other variations lie within range. It is interesting to note that the new post-assimilation standard deviations of Table XXII generate a reduction of the evaluated uncertainty of the JEZEBEL keff by 18.7%, mostly coming from the fission cross section contribution. This is already an indication of the potential gain, in terms of uncertainty reduction, that the data assimilation can produce. One should expect more reductions when other integral experiments are included in the data assimilation process.

2.

G. Palmiotti et al.

rier in the compound nucleus (CN) (FISVF100), (ii) the height of the second hump in the fission barrier in CN (FISVF2000), (iii) CN asymptotic level density parameter (ATILNO000), (iv) the width of the first hump in the fission barrier in CN (FISHO100). The global parameters TOTRED and FUSRED that scale total and fusion cross sections, respectively, have a constant impact on the fission cross section over the whole energy range, although a small kink is observed at the threshold for second-chance fission. Once the first chance fission opens the situation becomes more complicated with more parameters affecting the cross sections. The most important added parameters are (i) the level density parameter in the target (ATILNO0100), level density shift in the target (LDSHIF0100), and level density parameter at the second hump of the CN fission barrier. (FISAT20000). Thus, while fission barrier parameters are determining the under-barrier fission, target level densities add to this list above the threshold for the first chance fission. It is worth noting, that another factor that strongly influences fission above the barrier are level densities above the second hump (not the first hump!) of the fission barrier in the CN. When the second chance fission opens many additional parameters enter the game and the picture becomes quite complicated but it is still possible to identify parameters that are most important; there are just many more of them.

Second Round of Assimilation

The second round of 239 Pu assimilation was undertaken to take advantage of the new version of the empire code (revision 2893) which allows for a much better description of the differential data without or with minimal use of the energy-dependent scaling of model parameters. In addition, the second round of assimilation also takes into account PFNS and ν¯. Extending the methodology by including these two new quantities along with a better prior should lead to more sound evaluation that is consistent with the differential and integral experiments. It should also shed light on the correlations between cross sections and PFNS and possible cancellation effects in the previous evaluations. The empire modeling of 239 Pu employed coupledchannels calculations for the inelastic scattering to the first five levels in 239 Pu using the optical model potential by Capote et al. (RIPL-3 no. 2408 [81, 82]). The fission channel was calculated within the simplified, full damping, approach although discrete transitional states above the fission barrier were taken into account. The level densities at the saddle points were provided by the low-K approximation EGSM model. The experimental fission barriers of RIPL-3 were taken as a starting point. The default input was manually modified to improve agreement with experimental data. In particular, additional discrete transition states above fission barriers were added. Then, repeated iterations with kalman were performed to fine tune model parameters to the experimental data in all reaction channels and produce covariances for the model parameters. Finally, empire calculations were completed with the 239 Pu resonance region taken from the ENDF/B-VII.1 file. The resolved resonance region extends from 0 up to 2.5 keV and is followed by the unresolved range that ends at 30 keV. Taking into account that integral experiment JEZEBEL is not sensitive to the resonance region, this range was not subject to adjustment during the assimilation. The 239 Pu fission cross sections in the low-energy range below threshold for the first-chance fission were most sensitive to the following nuclear reaction parameters (in order): (i) the height of the first hump in the fission bar-

3.

PFNS

Prompt fission neutron spectra (PFNS) were fit to Pu thermal data sets of Boytsov [83] and Starostov [84]. Fig. 33 shows empire calculations for 239 Pu PFNS using both the Los Alamos and Kornilov models. The Kornilov data was normalizied with a Maxwellian at temperature 1.32 MeV while the Los Alamos model was normalized at a temperature 1.42 MeV. For both plots the initial empire calculation is shown in green. kalman was then used to fit the default Kornilov parameters. The fitted adjustments for the both models are listed in Table XXIII. For the Kornilov model this fit (blue curve) resulted in better agreement with the data at lower energy. No signficant improvement was achieved with the the Los Alamos model. At an incident energy of 2.5 MeV, the average PFNS total energy for the Los Alamos model is 2.163 MeV and for the Kornilov model is 2.154 MeV. Table XXIII shows the final kalman parameters for both models. One sees the strong sensitivity of the Kornilov model to the kinetic energy of the fragment due to neutron emission. In contrast, the Los Alamos model parameter values showed a markedly decreased variation. For both models, ENDF/B-VII.1 fails to reproduce the data at the lowest energy range (up to about 500 keV). 239

630

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ENDF/B-VII.1 Empire Ei2.53E-2 KALMAN 94-Pu-239(N,F),DE Ei2.53E-2

Parameter PFNALP PFNRAT PFNERE PFNTKE

1

239Pu(n,f)

3

10-1

1

10

Ratio to Maxwellian (T = 1.42)

ENDF/B-VII.1 Empire Ei2.53E-2 94-Pu-239(N,F),DE Ei2.53E-2

2.5

Pu(n,f)

1.5

1 0.5

Smith (57) Kalini (58) Smirenkin (62) Gayther (75) Szabo (76) Weston (84) Shcherbakov (01) Tovesson (10)

Pre-assimilation Post-assimilation

0.1

1

Incident Neutron Energy (MeV)

10

FIG. 34. The pre-assimilation fit to differential fission data for 239 Pu data shown in solid black with the post-assimilation shown in dashed blue. A sample of the experimental data fitted with empire are shown for comparison. Note the small difference between the curves relative to the uncertainties and scatter of the differential data. These small differences from the pre- to post-assimilation cross sections are enough to bring keff into much better agreement with experiment.

800

239Pu(n,f)

239

2

0

1000

600

Scale Factor Kornilov Los Alamos 0.926 1.000 0.999 0.990 0.990 1.002 0.975 0.999

PFNS

Outgoing Neutron Energy (MeV)

1200

G. Palmiotti et al.

TABLE XXIII. Variation of the default empire parameters of the Kornilov & Los Alamos PFNS models in case of 239 Pu obtained after kalman adjustment to experimental data.

Fission Cross Section (barns)

Ratio to Maxwellian (T = 1.32)

Combined Use of Integral . . .

PFNS

400 10-2

10-1

1

not required. The changes to the empire parameters were minor, as shown in Table IX B, and well within the uncertainty of the parameters when fitted to differential data, and the resulting differences in the differential cross sections calculated by empire are small compared to the uncertainties in the differential cross sections, as shown for fission in Fig. 34, where the changes in the empire calculations pre- and post-assimilation are shown with a small sample of available differential data. The change required for assimilation is very small in comparison to the uncertainties of the experimental data sets. Integral experiments serve as global constraints on the cross sections which can help guide the fitting of the empire parameters, and these constraints are sensitive to minor modifications to the differential cross sections which are far smaller than the uncertainties currently available for most differential data sets. The fitting of integral experiments is an important additional tool to help determine a consistent set of values for the differential cross sections. Fig. 34 shows the pre- and post-assimilation empire calculations for fission for 239 Pu as an example of the relatively minor modifications required of the cross sections to bring the resulting MCNP simulation of keff into

10

Outgoing Neutron Energy (MeV)

FIG. 33. The Empire calculations for PFNS of 239 Pu using the Kornilov (top) and Los Alamos (bottom) models as fit to experimental data shown in blue. For comparison, the ENDF/B-VII.1 evaluation is shown in red.

B.

Results of Direct Assimilation

The ‘direct’ assimilation method described in Sect. VIII E was tested for 239 Pu. For this material the integral experiment modeled was JEZEBEL, a solid sphere of 239 Pu [85]. The simulation also required ACE cross section tables for trace amounts of 240,241 Pu and gallium, which were not varied and taken from ENDF/B-VII.1. Using the ACE file for 239 Pu (using the central values from the kalman fit to differential data), generated a value of keff = 1.00516 ± 0.00008. After fitting keff with the kalman code and then running empire with the fitted parameters (to account for any non-linearities) we obtained a final value of keff = 0.99959 ± 0.00008. As this value was within the quoted uncertainty for keff another iteration of assimilation was 631

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channel. There are three more off-diagonal correlation values greater than 50% in Table XXV. One of them is between the parameters indexed as 12 and 16, displaying the obvious anticorrelation between the parameters controlling the heights of the first and second fission barriers of the compound nucleus. Another strong correlation is observed between the fission level density at the saddle point for the compound nucleus, with index 37, and the level-density parameter for the target nucleus, indexed with the number 2. Finally, the last pair of parameters with strong correlation (50%) is formed by the fissionbarrier-height parameter for the compound nucleus (index 16) the parameter that shifts the excitation energy in level densities in the target nucleus (index 47). This first round of assimilation for 239 Pu has shown the potential of the method to improve the integral performance of a file and reduce associated uncertainties. We note, however, that this improvement was obtained with a file that is visibly inferior to ENDF/B-VII.0 when compared to differential data. This illustrates a long standing issue of error compensation when “good agreement is obtained for bad reasons.”

TABLE XXIV. Results of direct assimilation of 239 Pu. empire parameters varied are listed with values before and after assimilation of integral experiment JEZABEL. Parameters which had the default value of 1.0 and were not varied during assimilation are not listed. Parameter Pre-assimilation Post-assimilation ATILNO-000 1.083 1.0851 ATILNO-001 0.907 0.9034 ATILNO-020 0.938 0.9380 ATILNO-030 0.988 0.9880 TUNEFI-010 0.833 0.8327 TUNE-000 2.228 2.2230 FUSRED-000 0.970 0.9700 RESNOR-000 1.320 1.3200 FISVF1-000 1.000 0.9995 FISVF1-010 1.000 1.0005 FISVF2-000 1.000 1.0042 FISVE1-000 1.000 0.9985 FISVE2-000 1.000 0.9995 FISHO1-000 1.000 0.9992 FISHO2-000 1.000 0.9992 FISAT1-000 0.917 0.9157 FISAT2-000 0.971 0.9717 FISAT2-010 0.981 0.9810 FISDL1-000 1.000 0.9999 FISDL2-000 1.000 0.9999 LDSHIF-000 1.100 1.0990 LDSHIF-010 1.063 1.0647 LDSHIF-020 0.917 0.9170 PFNALP-000 0.963 0.9613 PFNRAT-000 0.928 0.9279 PFNERE-000 0.999 1.0002 PFNTKE-000 0.984 0.9853

X.

THE FUTURE

The research activity documented in the present paper has not only produced significant improvements of nuclear data and of the related uncertainties but, together with international activities performed in parallel [11, 23], has succeeded in providing a deeper understanding of nuclear data adjustment methods and of their application. In fact, as for international activities, the findings and conclusions of the NEA WPEC Subgroup 33 on “Methods and issues for the combined use of integral experiments and covariance data” have pointed out, as detailed in a companion paper [73] that the statistical adjustments methodologies in use worldwide for different reactor analysis and design purposes are essentially equivalent and that they can provide a powerful tool for nuclear data improvement if used in appropriate manner. In fact, it has been indicated that the associated sensitivity analysis requires careful use of existing methods and that the choice of specific integral experiments of different types (critical masses but also reaction rates, reactivity coefficients and irradiation experiments) and sensitive to different energy neutron spectra, is of high relevance to avoid as much as possible compensating effects in the adjustments. Finally, it has been pointed out the crucial role of the covariance data used, both those associated to the nuclear data and those associated to the integral experiments. As a result, the role for cross section adjustment is more and more perceived as that of providing useful feedback to evaluators and differential measurement experimentalists in order to improve the knowledge of neutron cross sections to be used in a wider range of applications. This new role for cross section adjustment requires tackling and solving a new series of issues: definition of criteria to assess the

agreement with the experimentally measured value. A sample of a few of the more recent data sets are shown to give an indication of the uncertainty and spread among the measurements, and give a perspective to the effects of the keff assimilation. Other channels showed similar or smaller changes. Table XXV shows the posterior (post-assimilation) correlation matrix for the 29 parameters, out of the total of 53 parameters varied, which have a correlation of 10% or higher with any other parameter. It may be seen that there are seven pairs of parameters with strong correlations. Three of those pairs correspond to PFNS parameters (indices 50 through 53). PFNS parameters are very strongly correlated among themselves, but weakly with cross section parameters. Excluding the PFNS correlations, the one with highest value (anticorrelation of 98%) is the one connecting the parameters indexed 9 and 10, which correspond to scaling parameters of the total and reaction cross sections. This is expected since implementation of these parameters in empire is such that changing total or reaction cross sections by a certain amount results in changing the other cross section by the same amount, thus preserving value of the elastic 632

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TABLE XXV. Correlations among reaction model parameters (in %) resulting from the direct assimilation for 239 Pu. See text for discussion of the values presented below. Columns and rows with all off-diagonal correlations below 10% were omitted. Parameter

1 2 3 5 6 7 9 10 11 12 13 14 16 17 18 20 23 24 26 34 37 38 46 47 48 50 51 52 53

a b c d e f g h i j k

1 2 3 5 6 7 9 10 11 12 13 14 16 17 18 20 23 24 26 34 37 38 46 47 48 50 51 52 53 ATILNO-000a 100 ATILNO-010a 4 100 ATILNO-020a -2 0 100 TUNEFI-010b 0 1 4 100 TUNEFI-000b -1 2 -1 0 100 c TUNE-000 -19 -2 1 0 -1 100 TOTRED-000d 0 0 0 0 0 0 100 FUSRED-000d 0 0 0 0 0 0 -98 100 e RESNOR-000 -5 15 -7 1 0 2 1 0 100 FISVF1-000f -3 47 -12 2 8 -3 0 0 17 100 FISVF1-010f -2 -13 22 -2 0 1 0 0 -47 -7 100 f FISVF1-020 2 6 -21 -1 0 -1 0 0 0 0 -5 100 FISVF2-000f -13 -38 17 -3 12 4 0 0 -19 -67 6 3 100 FISVF2-010f -2 -5 -21 19 -1 0 0 0 -2 -16 -26 2 22 100 FISVF2-020f 0 3 -24 -1 0 0 0 0 0 -2 1 -29 4 6 100 g FISVE1-000 -1 -2 0 0 -1 -1 0 0 0 17 0 0 9 0 0 100 FISVE2-000g -2 7 -2 0 -1 -1 0 0 0 0 0 0 18 -2 0 -1 100 FISVE2-010g 0 0 5 -2 0 0 0 0 0 2 -1 0 -3 12 0 0 0 100 FISHO1-000h 4 3 1 0 2 1 -1 0 6 34 0 -2 3 0 0 1 2 0 100 i FISAT1-000 -1 10 -3 1 -1 -1 0 0 1 3 -3 1 20 -4 0 -1 -2 0 -1 100 i FISAT2-000 -2 67 21 -3 -2 0 0 0 -4 -2 10 7 20 20 8 0 -4 -3 3 -3 100 FISAT2-010i 2 -1 37 -3 0 -1 0 0 4 7 -12 12 -10 17 17 0 1 -3 -1 2 -14 100 LDSHIF-000j 21 0 0 0 0 4 0 0 2 3 0 -1 2 0 0 0 1 0 -4 0 1 0 100 j LDSHIF-010 -9 -18 5 -1 -7 -1 1 0 -17 -13 -15 7 50 7 3 -10 -6 -1 11 -6 8 -2 2 100 j LDSHIF-020 0 1 1 -6 0 0 0 0 1 3 -5 -4 -3 30 -8 0 0 -3 0 0 -2 -1 0 -1 100 PFNALP-000k 0 -1 0 0 0 0 0 0 2 -1 1 -1 3 0 0 0 0 0 0 0 0 -1 0 0 0 100 PFNRAT-000k 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 65 100 k PFNERE-000 -1 2 -1 0 0 1 0 0 -2 1 -1 1 -4 0 0 0 0 0 0 0 0 1 0 -1 0 -65 -1 100 k PFNTKE-000 -1 1 0 0 0 0 0 0 -2 1 -1 1 -4 0 0 0 0 0 0 0 0 1 0 -1 0 -24 44 88 100

Level density parameters Fission decay width parameters Equilibrium decay width parameter Scaling parameters Response function parameter Heights of fission barriers Vibrational enhancements of fission level density at saddle point Width of fission barriers Fission level densities at saddle point Shifts of excitation energy Prompt fission neutron spectra parameters

be most important. Specific feedback between the adjustment work and cielo can evolve as the two activities progress.

reliability and robustness of an adjustment; requisites to assure the quantitative validity of the covariance data; criteria to alert for inconsistency between differential and integral data; definition of consistent approaches to use both adjusted data and a-posteriori covariance data to improve quantitatively nuclear data files; provide methods and define conditions to generalize the results of an adjustment in order to evaluate the extrapolability of the results of an adjustment to a different range of applications (e.g., different reactor systems) for which the adjustment was not initially intended; suggest guidelines to enlarge the experimental data base in order to meet needs that were identified by the cross section adjustment. A future development should then focus on a research activity to provide criteria and practical approaches to use effectively the results of sensitivity analyses and cross section adjustments for feedback to evaluators and differential measurement experimentalists in order to improve the knowledge of neutron cross sections, uncertainties, and correlations to be used in a wide range of applications. This activity is of particular relevance to the foreseen objective to improve future data files using synergies from different nuclear data projects. In particular, a good coordination with the new cielo initiative [86] with testing and feedback on the foreseen new cielo evaluations will

XI.

CONCLUSIONS

For four decades, statistical data adjustment methods have been successfully applied to the design of reactors, and recently to fast and innovative (e.g. Generation-IV) reactors. However, only recently these methods have been recognized as powerful tools to be used also for the improvement of nuclear data files, within a very wide range of applications. The establishment of science-based covariance data, motivated by the need to understand and to quantify uncertainties on reactor design parameters, has been one of the most important developments in order to achieve both goals, i.e. improvements of evaluated data files and reduction of design parameters uncertainties. However, other important achievements have been necessary to pursue those objectives: integral experiment analysis and experimental uncertainties assessments, sensitivity methods, critical analysis of the adjustment features, both a priori and a posteriori. The examples given in the paper show the robustness and significance of the adjustment methods as they 633

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are applied at present, but they also indicated further R&D needs in order to better exploit e.g. a-posteriori covariance data or to widen the application domain of adjusted parameters. In this respect, the new “consistent” method of adjustment applied to nuclear model parameters shows a very important potential. Finally, the use of well documented integral experiments, in particular in terms of experimental uncertainties and correlations, the further expected improvements in covariance data and the use of appropriate sensitivity analysis, will help to plan and define fewer but more focused experiments (both integral and differential) for future needs of reactor and advanced fuel cycles.

Acknowledgements: The authors would like to thank the following: P. Finck who has been at the origin of the research project and has shown continuous interest in it; A. Hill for his crucial support; P.G. Young and M.B. Chadwick (LANL) for technical contributions and useful discussions and advice. Work at INL supported by the U.S. Department of Energy, Office of Nuclear Energy and Office of Science, under DOE Idaho Operations Office Contract DE-AC0705ID14517. The work at BNL was sponsored by the Office of Nuclear Physics, Office of Science of the U.S. Department of Energy under Contract No. DE-AC0298CH10886 with Brookhaven Science Associates, LLC.

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