Nuclear Data Uncertainties for Typical LWR Fuel Assemblies and a Simple Reactor Core

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Nuclear Data Sheets 139 (2017) 1–76 www.elsevier.com/locate/nds

Nuclear Data Uncertainties for Typical LWR Fuel Assemblies and a Simple Reactor Core D. Rochman,1, ∗ O. Leray,1 M. Hursin,1 H. Ferroukhi,1 A. Vasiliev,1 A. Aures,2 F. Bostelmann,2 W. Zwermann,2 O. Cabellos,3 C.J. Diez,3 J. Dyrda,3 N. Garcia-Herranz,4 E. Castro,4 S. van der Marck,5 H. Sj¨ostrand,6 A. Hernandez,6 M. Fleming,7 J.-Ch. Sublet,7 and L. Fiorito8 1

Reactor Physics and Systems Behaviour Laboratory, Paul Scherrer Institut (PSI), 5232 Villigen, Switzerland 2 Gesellschaft f¨ ur Anlagen- und Reaktorsicherheit (GRS), 85748 Garching, Germany 3 OECD Nuclear Energy Agency (NEA), Issy-les-Moulineaux, France 4 Universidad Polit´ecnica de Madrid (UPM), Madrid, Spain 5 Nuclear Research and Consultancy Group (NRG), Petten, The Netherlands 6 Uppsala University, Department of Physics and Astronomy, Uppsala, Sweden 7 United Kingdom Atomic Energy Authority (UKAEA), Abingdon, UK 8 Institute for Advanced Nuclear Systems, SCK•CEN, Belgium (Received 17 June 2016; revised received 27 October 2016; accepted 20 November 2016) The impact of the current nuclear data library covariances such as in ENDF/B-VII.1, JEFF3.2, JENDL-4.0, SCALE and TENDL, for relevant current reactors is presented in this work. The uncertainties due to nuclear data are calculated for existing PWR and BWR fuel assemblies (with burn-up up to 40 GWd/tHM, followed by 10 years of cooling time) and for a simplified PWR full core model (without burn-up) for quantities such as k∞ , macroscopic cross sections, pin power or isotope inventory. In this work, the method of propagation of uncertainties is based on random sampling of nuclear data, either from covariance files or directly from basic parameters. Additionally, possible biases on calculated quantities are investigated such as the self-shielding treatment. Different calculation schemes are used, based on CASMO, SCALE, DRAGON, MCNP or FISPACT-II, thus simulating real-life assignments for technical-support organizations. The outcome of such a study is a comparison of uncertainties with two consequences. One: although this study is not expected to lead to similar results between the involved calculation schemes, it provides an insight on what can happen when calculating uncertainties and allows to give some perspectives on the range of validity on these uncertainties. Two: it allows to dress a picture of the state of the knowledge as of today, using existing nuclear data library covariances and current methods.

CONTENTS

I. INTRODUCTION II. METHODS A. Uncertainty Propagation Methods 1. PSI-SHARKX 2. PSI-SHARKX Fission Yields 3. GRS-XSUSA 4. UPM-SCALE 5. NRG-Fast TMC 6. UU-TMC 7. UKAEA-FISPACT-II 8. SCK Fission Yields B. Considered Systems

∗

Corresponding author: [email protected]

http://dx.doi.org/10.1016/j.nds.2017.01.001 0090-3752/© 2017 Elsevier Inc. All rights reserved.

2 3 3 3 4 6 9 10 12 13 15 16

1. PWR UO2 Assembly 2. PWR UO2 +Gd Assembly 3. PWR MOX Assembly 4. BWR UO2 +Gd Assembly 5. BWR MOX+Gd Assembly 6. Martin-Hoogenboom Benchmark C. Calculated Quantities

16 18 18 19 20 21 22

III. NUCLEAR DATA 23 A. Libraries 26 1. ENDF-6 Format 26 2. JEFF-3.2 27 3. ENDF/B-VII.1 27 4. SCALE 27 5. JENDL-4.0 28 6. TENDL-2014 28 7. Comparison of Uncertainties for Criticality Benchmarks 29 B. Isotopes and Reactions 29

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C. Processing 1. PREPRO 2. CALENDF 3. NJOY for Covariances 4. AMPX D. Missing Data IV. RESULTS A. Assemblies k∞ B. Assemblies Number Densities C. Assemblies Two-group Cross Sections D. Assemblies Pin Power Distribution E. k∞ Partial Contributors F. Impact of Fission Yields 1. On k∞ 2. On Inventory 3. On Inventory (TMC Comparison) G. Convergence, Correlation and Skewness 1. Convergence 2. Correlation 3. Skewness H. Full Core - Power Distribution 1. Preliminary Tests 2. Tally F7 3. Tally F6 4. Comparison with the Fast GRS Method I. Other Sources of Uncertainties 1. Operating Conditions 2. Manufacturing Tolerances 3. Burnup-induced Technological Changes 4. Computational Biases 5. Example J. Self-shielding Treatment 1. Reactivity Coefficient UQ 2. Handling the Implicit Effect in SHARK-X 3. UQ Results 4. Conclusion for the Implicit Effect V. CONCLUSIONS AND OUTLOOK

31 31 31 31 32 33

References

55

INTRODUCTION

The simulation of the neutronic behavior of a reactor core is crucial for its safety. It intervenes at each stage of the life of a reactor: design, loading core licensing, cycle calculations, and dismantling. It also influences the fate of irradiated fuel: design, burn-up, transport, eventually reprocessing and finally storage. At the start of these simulations are the nuclear data (e.g. microscopic cross sections, angular and energy distributions for emitted particles, fission yields), and also their degree of knowledge (uncertainties, or covariances). One of the recurrent question is “how important are nowadays the nuclear data ?” In other terms: are they known enough, or do we (still) need to improve their precision with new measurements or new theoretical models. The answer is certainly not simple and depends on the field of application. For instance it is widely accepted that capture cross sections have an important impact in astrophysics (see for instance Ref. [1]). But the isotopes involved for such calculations may not exist in nuclear energy applications. Also, specific (p,n) cross sections are of high importance for the production of new positron emission tomography (PET) tracers [2]; these reactions are nevertheless not common in energy applications. As a final example outside nuclear energy applications, neutron-induced single event effects in modern microelectronics can have a large impact for planes, space shuttles and the International Space Station, but the neutron energy range for damage is often above 10 MeV [3]. In nuclear energy applications the question of the relevance of nuclear data needs to be answered in unequivocal terms for the safety of nuclear operations: yes the knowledge of nuclear data is enough for the usual operations of current nuclear reactors; and no it is not enough for their simulations outside the very restrictive limits related to their normal use. This apparent contradiction is based on the fact that the drive by experience is more reliable than the drive by fundamental understanding for thermal reactors. Additionally, the simulation tools have often been adjusted to achieve good performances in their validation limits, for instance by using adjusted nuclear data libraries. The consequence is that it becomes difficult to predict system behaviors with enough accuracy in the following areas:

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A. Additional Results on Uncertainties

I.

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Acknowledgments

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61

• Outside the usual reactor operations, • In typical fuel inventory, • In local power distribution in the core, • In dosimetry and neutron source predictions, • In ex-core calculations, • Outside neutronics code applicability (e.g. new reactor designs), • Improvement of safety margins for cost reduction, 2

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• Higher fuel burn-up.

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approach allows to simulate what happens in “real-life” assignments, where systems uncertainties are provided to another party. This is the second goal of this paper: providing a perspective on the relevance of uncertainties due to nuclear data, because the uncertainties due to nuclear data might not only be contained in its covariance files. Finally, for nuclear simulations at the assembly or core level, nuclear data are not the only source of uncertainties. Even if it is not the main subject of this paper, a few indications are provided in Sec. IV I and IV J for the other potential effects on calculated quantities. In a broader overview, the question of the uncertainties obtained from nuclear data with respect to specific requirements from the industry and safety authorities is a key issue. Such requirements might be different than the calculated uncertainties, as presented in Ref. [10]. The present paper does not touch on this subject, and further information can be obtained on the cited reference. In the case of current light water reactors, specific studies for the uncertainties on targeted quantities would be required and are not included here. The present paper also does not intend to present results to benchmark a specific nuclear data library, as performed in Ref. [11]. For this intention, only specific and well dedicated efforts are necessary as it was realized for JENDL-4.0 [11]. Finally, the question of the feedback to improve nuclear data libraries based on benchmark results and sensitivities is not discussed here.

For additional information, the following references can be read as examples of some simulations compared to experiments: [4–6]. The improvement of predictions in the above fields does not automatically imply a nuclear data improvement. If the propagation of nuclear data uncertainty certainly indicates the need of better basic nuclear data measurements and theories, two other aspects can be improved: (1) better handling of nuclear data with improved processing codes, and (2) improvement of simulations by using more exact transport and solver systems. If these two points are not the subject of this paper, they should still be mentioned for completeness: 1. So-called “processing” of nuclear data allows to provide ready-to-use sets of data for simulations. The fact that the vast majority (with the exception of the AMPX code [7]) of such processing systems rely on (parts of) a single code such as NJOY [8] does not follow the principle of redundancy, often mandatory in nuclear environments. There is a very little significance for funding agencies to better measure quantities that NJOY can not handle, even if they can improve simulations (e.g. fission neutron emission as a function of fragment mass). 2. Neutronics simulation codes (transport, depletion, activation) as used as reference in the nuclear industry are often designed to compensate existing nuclear data inaccuracy. Such codes can therefore poorly handle “more correct” nuclear data as they will then provide deteriorated C/E ratios (calculation over experiment ratios). But it should be noticed that Monte Carlo reference calculations can nowadays be used for the estimation of numerical biases.

II.

METHODS

The methods to calculate reactor quantities and their uncertainties differ because of the use of different transport codes, base libraries, covariance libraries and energy groups. A base library is a library which is used to calculate the nominal values, such as JEFF, ENDF/B, or JENDL [12–14]. It can differ from the sources of covariance information, for many practical reasons. Table I summarizes the main characteristics of each method used in this paper, depending on where the calculations are performed. As presented in this table, the methods to calculate the uncertainties differ between the participants. More details are given in the following subsections.

Once we are aware of the current limitations of the calculation capabilities, it is however important to assess to which degree of accuracy such important quantities as nuclear data can provide. This is the first goal of this paper: presenting in a single study calculated uncertainties due to nuclear data for relevant systems in nuclear energy production. To fulfill this goal, different institutes agreed to participate, with their own method and database (see Sec. II A for the different definitions of methods). The only common points in these studies are (a) the systems (see Sec. II B) and (b) the quantities to calculate (see Sec. II C). Such an approach is definitively different compared to the Expert Group on Uncertainty Analysis in Modelling (UAM [9]). Here, because the libraries (both for nominal values and for covariances) are different, the calculated uncertainties can not be the same (see Sec. IV for results). The use of recommended covariance libraries is very convenient for the comparison and understanding of the various results, but it also restricts the possible spread of calculated responses due to the differences in the various covariance data libraries available. This difference of

A.

Uncertainty Propagation Methods 1.

PSI-SHARKX

At PSI, a methodology for the propagation of nuclear data covariance information is developed since 2011 [15, 20–22]. Since then, it was applied to many different burn-up calculations, see for instance Refs. [23, 24]. The SHARK-X methodology is based on the covariance files as provided in existing nuclear data libraries, and on a modified version of the deterministic lattice code 3

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TABLE I. Summary of the methods used to calculate uncertainties on the different quantities. Institute Method name Code Base library Covariance library

Covariance FY

Covariance groups Processing

Samples Main reference

PSI SHARK-X CASMO-5 ENDF/B-VII.0 ENDF/B-VII.1

GRS XSUSA TRITON ENDF/B-VII.0 SCALE-6.1

UU UPM UKAEA NRG Monte Carlo SAMPLER Fast-TMC DRAGON-5 TRITON FISPACT-II MCNP6 JENDL-3.3 ENDF/B-VII.1 ENDF/B-VII.1 ENDF/B-VII.1 JENDL-4.0 SCALE-6.2beta5 TENDL-2015 TENDL-2014 (fast range )

SCALE-6.2beta4 JEFF-3.2 TENDL-2014 Many

-

-

44 56 for SCALE-6.2 NJOY2012

44

69

ENDF/B-VII.1 + SAMPLER 235 U correlation 56

AMPX

NJOY99

AMPX

500 [15]

1000 [16]

300 -

200 [17]

TENDL-2015

NJOY2012 PREPRO2015 CALENDF2010 500 [18]

TENDL-2014

pointwise NJOY99

500 [19]

self-shielding factors are not perturbed, even though by definition it should change given a new microscopic cross section. This effect is known as the “implicit effect” and has been shown to be small for k∞ uncertainty and two-group macroscopic cross section uncertainty [29]. In Ref. [29], the implicit effect is further analyzed in terms of the “direct effect” (directly due to the cross section perturbation) and the “indirect effect” (due to the induced change in the neutron flux). In the resonance range, it appears that the large increase (direct effect) is almost entirely compensated by a large decrease in the flux (indirect effect). So although this is a limitation of the current SHARK-X implementation, it is believed that the implicit effect is small, at least for responses which are not highly influenced by energy ranges including large resonances.

CASMO-5M. Modified CASMO-5 CASMO-5 is a 2-dimensional deterministic fuel assembly code developed by Studsviks Scanpower [25]. It is delivered with its own processed library in binary format, which include cross sections, self-shielding factors, resonance integrals and other quantities. In its later versions, this library is based on ENDF/B-VII versions [12, 26], but also include other nuclear data evaluations from other libraries such as TENDL [27]. The processing of the original nuclear data evaluations into the library used by CASMO-5 is performed with NJOY, together with additional proprietary modules for specific formatting. It is therefore not possible to conveniently produce a different library version for CASMO-5, as it was performed for instance for DRAGON [28]. As a consequence, the CASMO-5 version was modified to allow for the alteration of some nuclear data while they are being used. This modified version of CASMO-5 used in SHARKX is called CASMO-5MX. The layout of the modified CASMO-5 code flow is presented in Fig. 1. Self-shielding factors The new modules allowing for the modifications of the nuclear data are indicated in red. In Fig. 1, the calculation flow of CASMO-5 shows that first a resonance calculation is performed for a representative set of pins (e.g. average, burnable absorbers). The resonance calculation step, which produces a set of resonance self-shielded microscopic cross section data, is performed in the resonance range. In the entire energy range, the step called “Microscopic cross section” produces a set of infinitely dilute cross sections (interpolated at the right temperature). These infinitely dilute cross sections are later replaced by the “PERTXS” module. It means that the

2.

PSI-SHARKX Fission Yields

Fission yields can be randomly varied in the SHARK-X method as the other nuclear data quantities. The variations are also based on specific fission yield covariance matrices. The CASMO-5 neutron data libraries (ENDF/BVII.0, ENDF/B-VII.1 or JENDL 4.0) contain fission yield data for 27 fissioning species. The code does not contain all isotope decay chains (a limited set of 296 isotopes are considered in the version 5 for 235 U) and therefore some approximations are made to limit the number of variables to handle. The included fission yields in the CASMO-5 database are either independent, cumulative or a combination of both, depending on the mass and nuclear charge of the considered fission fragment. Nuclides that are on the southeast edge of the CASMO-5 depletion chains are 4

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CASMO-5 fission product chains, then the cumulative yield is given separately for each state. The amount of IT from the meta-state to the ground state must, however, be subtracted from the cumulative yield for the ground state before use in CASMO-5 (IT is included in the fission product chains).

Input File

Resonance calc. (incl. self-shielding)

ENDF/B-VII-based library

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PERTXS

Perturbation file

• If the ground and meta-states are combined to a single ident in the CASMO-5 fission product chains then the cumulative yield for the combined ident is obtained by summing the cumulative yields for the two states and subtracting the amount of IT from the meta to the ground state.

Microscopic cross section (inf. dilute)

The propagation of fission yield (FY) uncertainty through neutronics codes is non-trivial because fission yields constitute a very constrained system [30] and the following physical constraints must thus be respected. The following conservation equations can be considered, depending on the selected approach. The sum of independent FY from a given father (no ternary fission considered) is given by Eq. (1)  Y (A, Z) = 2, (1)

Macroscopic cross section Assembly

Microgroup spectral calculation

Z,A Collapse to 2D group structure

with Z, A the nuclear charge and the mass of the fission product, Y its independent fission yield. The mass conservation (no spontaneous fission considered) implies that  A × Y (A, Z) = Afather + 1 − νp , (2)

2D MOC transport calculation

Few group constant reaction rates calculation

Z,A

with Afather the mass of the target nucleus and νp the number of prompt emitted neutron per fission. The charge conservation implies that  Z × Y (A, Z) = Zfather, (3)

Output File

Z,A

with Zfather the charge of the target nucleus, and finally, for every nuclear charge of the fission products Z, there exists a Z-complementary fission product   Y (A, Z) = Y (Zfather − Z, A). (4)

FIG. 1. (Color online) Work flow of the modified CASMO5MX code. The “red” boxes represent the added modules to CASMO-5M in order to modify the nuclear data.

A

A

A consequence of these constraints is the difficulty to perturb FYs related to the evaluated fission yields: there are very simple covariance matrices available (only diagonal terms). These matrices do not contain correlation terms, neither for the independent nor cumulative fission yields. Additionally, the evaluated cumulative yields can be inconsistent with the independent yields, for both the nominal values and their uncertainties. Therefore, when performing stochastic sampling, different possibilities exist. In the current version of SHARK-X, four types of fission yield uncertainty propagation can be applied. In each case, all FYs are perturbed and an important number of code runs (e.g. 500-1000 samples) is therefore required in order to obtain trustworthy statistics. The four sampling types are:

represented with cumulative yield, whereas other nuclides are represented with an independent yield. The cumulative yield is approximately the sum of independent yields along the mass diagonal. Two possible cases exist for the cumulative yield if both ground and meta-states exist for a particular ident. The cumulative yield for the ground state in the evaluated fission product library includes isomeric transition (IT) from the meta-states. The amount of IT is then computed from the branching ratio given in the radioactive decay data library. • If both ground and one meta-state exist in the 5

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1. Sampling (according to the provided standard deviations) and perturbation without any constraints between the FYs (no constraints are respected). These direct perturbations can result in creating more fission products and inducing a bias on the output (e.g. reactivity, compositions),

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uncertainties are close to the reference data (ENDF/BVII.1) and a correlation matrix between all fission products can be obtained. Examples are presented in Ref. [33] and an example of the correlation matrix is reproduced here for the thermal neutron-induced fission of 235 U (Fig. 3). The fission yields are ordered by sets of fission products in increasing masses, and by complementary couples (the nuclear charges of the two fragments sum to the nuclear charge of the fissioning system). For instance in the case of 235 U, the plotted isotopes are all Ge, followed by Nd (presented in Fig. 3 by Ge/Nd), then As/Pr, Se/Ce, Br/La,... up to Tc/In. In total the correlation between about 250 fission yields are presented. This defines the different possibilities in SHARK-X, which will be used and presented in section IV F.

2. Sampling without any constraints but with a normalization to 2 done after the sampling to fulfill the first condition (Eq. (1)). Consequently, the standard deviations of the samples change with the normalization process. 3. Sampling of the FYs using a constraint matrix as correlation matrix. The building of this matrix is explained in the next subsection. An alternative approach is becoming nowadays possible with the help of theoretical codes such as GEF [31]. The GEF code allows to calculate among other quantities independent fission yields, which are globally in good agreements with current evaluations and selected experimental data. GEF is based on physical models and its parameters are adjusted to reproduce such data (one single set of parameters for all considered fissioning systems). The production of fission yield covariance files with GEF is based on the combination with a Bayesian Monte Carlo procedure (BMC), as presented in Ref. [32] for cross section evaluations and in Ref. [33] for fission yields. The succinct description is as follows and presented in Fig. 2:

3.

GRS-XSUSA

The XSUSA (“Cross Section Uncertainty and Sensitivity Analysis”) method is based on the random sampling GRS method (Gesellschaft f¨ ur Anlagen- und Reaktorsicherheit) implemented in the code package SUSA “Software for Uncertainty and Sensitivity Analysis”) [35]. The probability distributions of the uncertain input parameters are used to generate random variations of these input quantities. When applying this method with neutron cross section uncertainties, this means that many nuclear data libraries are generated, where all quantities with available uncertainties, namely inelastic and elastic scattering, (n,2n) and capture cross sections, in the case of fissionable nuclides additionally the fission cross section, the number of neutrons per fission, and the fission neutron spectrum, are varied at the same time for a large number of nuclides. As basis for generating the data variations, the SCALE 6.1 covariance data library is currently being used. This library contains uncertainties for relevant nuclides on the basis of various sources, including high-fidelity evaluations from ENDF/B-VII, ENDF/BVI, and JENDL-3.3, as well as approximate uncertainties obtained from a collaborative project performed by Brookhaven National Laboratory, Los Alamos National Laboratory, and Oak Ridge National Laboratory [36]. These covariance matrices are processed in a multi-group structure with 44 energy groups. When performing depletion calculations with the TRITON sequence from the SCALE 6.1 package [37], the default 94 nuclides in the fuel are taken into account. Thus, while the “traditional” SUSA method is predominantly being applied to problems with a limited number of parameters and only few correlations between them, the application to depletion calculations with nuclear data uncertainties leads to a large amount of uncertain parameters (94 nuclides, typically 2-8 reactions per nuclide, 44 energy groups per reaction), with a large amount of correlations between the energy group data of each nuclide/reaction combination, and also cross correlations between data of different reactions or even data for different nuclides.

1. Select models and parameter distributions from GEF, 2. Produce a set of random calculated quantities by sampling the parameter distributions, 3. Compare theoretical calculations and selected reference data, for instance with the χ2 quantity, 4. Use weights to update the parameter distributions, 5. Sample again to produce new calculated quantities based on the new weighted parameter distributions, 6. Repeat (3) to (5) until convergence of the parameter distributions. Some details for the above steps are described below. There is fundamentally no difference with Ref. [32], except in the step (3) for the choice of the reference data. The set of reference data is chosen here to be the evaluated fission yields as given in the ENDF/B-VII.1 library. A list of the 21 most-important parameters for the GEF code is considered in this work, as obtained from Ref. [34] (see Table 7 in this reference) and is presented in Table II. The adjustment of parameters and their updates is performed with a simplified χ2 goodness of fit-estimator. After n iterations, the calculated GEF fission yields and 6

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GEF model parameters distribution for position, widths of the fission channels, deformation, polarization, temperatures...

0.35

P A Width S2

0.3

Iteration 0

0.25 0.2 GEF random fission yields

0.15 0.1 0.05 0

11

12

13

14

15

16

17

18

17

18

parameter parameter updates i = 1...n

0.35

P A Width S2

0.3

Iteration 1

0.25 Comparison with EXFOR or evaluations (JEFF) exp(−χ2 /2)

0.2 0.15 0.1 0.05 0

11

12

13

14

15

16

parameter Updated parameter distributions (mean, standard deviation, skewness and correlation)

FIG. 2. (Color online) Representation of the Bayesian Monte Carlo in the case of the GEF code for fission yields. On the right are presented the probability distributions for one GEF parameter, along two iterations of the updating process (iteration 0 and 1). In practice, the number of iterations is larger than 1.

For depletion calculations, another simplification can be applied. For each burn-up step, the self-shielding calculation is performed with nominal isotopic inventories for this specific burn-up step, and not with the actual isotopic inventory originating from the varied nuclear data. This is justified by the fact that the shielded microscopic cross sections resulting from the self-shielding calculations are only little sensitive to the isotopic content of the fuel. When performing the fuel assembly transport calculation, of course macroscopic cross sections are generated with the actual material composition for each individual fuel pin. This proceeding again has the advantage that the self-shielding calculations have to be performed in advance only for one complete depletion calculation with nominal nuclear data, and can be saved for later use in all of the calculations with varied data, leading to a

In the current XSUSA implementation, the neutron cross sections are varied after the self-shielding calculations, i.e. so-called implicit effects are not taken into account. In other words, it is assumed that cross section perturbations are propagated linearly through the selfshielding calculation. This has the practical advantage that the self-shielding calculations have to be performed only for the calculation case with nominal nuclear data. To verify this simplification, many comparisons have been performed with TSUNAMI [38], which takes into account implicit effects, for multiplication factors, reaction rates, few-group macroscopic cross sections, and reactivity differences in criticality calculations. Good agreement between the uncertainties has been obtained in all cases; examples can be found in Refs. [16, 39]. Nevertheless, the inclusion of implicit effects is under development.

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TABLE II. List of the 21 most relevant parameters with their uncertainties considered in this work for the 235,238 U and 239,241 Pu targets. The original table can be found in Ref. [34], Table 7. For more details, see Ref. [34]. Parameters

GEF name

Position of the shell for the S1 channel Position of the shell for the S2 channel Position of the shell for the S3 channel Position of the shell at Z ≈ 42 Shell effect for the S1 channel Shell effect for the S2 channel Shell effect for the S3 channel Shell at Z ≈ 42 Rectangular contribution to the width of S2 channel Shell effect at mass symmetry Curvature of shell for the S1 channel Curvature of shell for the S2 channel Curvature of shell for the S3 channel Curvature of shell at Z ≈ 42 Weakening of the S1 shell with 82/50 − N/rmCN /ZCN (ω)eff for tunneling of the S1 channel (ω)eff for tunneling of the S2 channel (ω)eff for tunneling of the S3 channel (ω)eff for tunneling at Z ≈ 42 Width of the fragment distribution in N/Z Charge polarization

P DZ Mean S1 P DZ Mean S2 P DZ Mean S3 P DZ Mean S4 P Shell S1 P Shell S2 P Shell S3 P Shell S4 P A Width S2 Delta S0 P Z Curv S1 P Z Curv S2 P Z Curv S3 P Z Curv S4 T low SL T low S1 T low S2 T low S3 T low S4 HOMPOL POLARadd

short name in this paper P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15 P16 P17 P18 P19 P20 P21

Uncertainty (1σ) 0.1 Z units 0.13 Z units 0.1 Z units 0.12 Z units 0.2 MeV 0.37 MeV 0.57 MeV 0.12 MeV 0.05 mass units 0.1 MeV 5% 5% 5% 5% 10 % 3% 3% 3% 3% 10 % 0.1 Z units

nuclear data uncertainty codes, as described, e.g., in Refs. [21, 41–44]. With the energy-dependent neutron cross sections, for estimating the main contributions to the result uncertainties it is convenient to determine group sensitivities, where typically the nuclear multigroup data for a certain reaction of a certain nuclide are treated as one group. The group sensitivity analysis is performed by determining the squared multiple correlation coefficient R2 as uncertainty importance indicator to quantify uncertainty importance of a group of input variables (X1 ,. . .,Xk ) with respect to an output variable Y. R2 is usually defined as the maximum (squared) simple correlation coefficient between the output variable Y and any linear combination of input variables from the group. It can be computed by the following formula

FIG. 3. (Color online) Updated correlation matrices for the fission fragments for 235 U with the BMC method. Only half of the symmetric matrices are presented. Correlations are presented in percent (from -100 % to +100 %).

⎡

⎤ ρ(Y, X1 ) ⎢ ⎥ .. −1 R2 = [ρ(Y, X1 ), . . . , ρ(Y, Xk )] × CX ×⎣ ⎦. . 1 ρ(Y, Xk )

significant performance gain. XSUSA can also handle fission yield and decay data uncertainties which may be relevant for the uncertainties of nuclide densities in depletion calculations. Such results can be found in Ref. [40]. It was also shown there that fission yield correlations can have a significant influence on the magnitude of the uncertainties of fission product densities, leading to substantial reduction in many cases. However, the methods to generate fission yield correlations are not unique and currently under discussion in the nuclear uncertainty community. Therefore, it was decided to completely omit fission yield uncertainties in XSUSA depletion calculations until generally accepted values for the correlations are available. Within XSUSA, special emphasis is put on performing sensitivity analyzes for the calculation results; this so far is not yet standard with other sampling based

(5)

Here, ρ(Y, Xi ) is the correlation coefficient between the output variable Y and the input variable Xi , (i=1,. . .,k). −1 CX is the inverse of the (k×k)- correlation matrix CX1 1 of the group X1 of input variables X1 , . . . , Xk , i.e. the inverse of the matrix of correlation coefficients ρi,j = ρ(Xi , Xj ) between all the input variables Xi and Xj , (i, j = 1, . . . , k) from this group. The squared multiple correlation coefficient can be interpreted as the relative amount of output uncertainty coming from the uncertainty of the respective group (i.e. the particular nuclide reaction). 8

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the independent yields covariance matrix. The yield uncertainties are taken from ENDF/B-VII.1, which are given by fissionable nuclide and for three energies: 0.025 eV, 0.5 MeV and 14 MeV. Only correlations in yields from 235 U are included; correlations in yields from other fissionable nuclides are not available. During Sampler execution, the perturbation factors are read for each data sample, and a complete set of perturbed independent yields for all fissionable nuclides and energies is computed. Those values will be read by ORIGEN.

UPM-SCALE

This section describes the uncertainty analysis methodology applied at UPM. This analysis is based on Monte Carlo stochastic sampling using the Sampler module along with TRITON depletion sequence from the SCALE6.2 code system [17]. Calculation methodology for uncertainty propagation: SAMPLER sequence SCALE6.2 includes Sampler super-sequence to assess the uncertainty of a particular response (due to the uncertain knowledge of input data) computed by any SCALE sequence. The workflow can be summarized as follows:

Calculation methodology for depletion: TRITON sequence TRITON is the SCALE sequence to perform the coupled cross-section processing, transport and depletion calculations. It couples the 2-D discrete-ordinates transport code NEWT and the point depletion and decay code ORIGEN-S using a predictor-corrector approach. In this approach, cross-section processing and transport calculations are performed at the mid-point of each depletion interval specified by the user, using the isotope concentration computed by predictor depletion steps. Transport results at mid-points (cross sections and flux distribution) are used to perform the ORIGEN-S corrector depletion calculations over the full interval, providing isotopics at the end of the interval. Predictor steps use cross-sections and flux at the mid-point of the previous interval to lead to the transport calculation at the mid-point of the actual depletion interval which will produce updated transport values. Using TRITON sequence, it is not possible to get both the nuclide composition and k∞ at the same depletion time, as requested in the benchmark. Consequently, k∞ at the end of the time interval is obtained by linear interpolation of the computed values at the mid-points. Details of the different calculation steps are given as follows. Cross-section library All calculations were performed using the broadgroup cross section library in 56 energy groups from ENDF/B-VII.1 distributed with SCALE6.2 and stored in AMPX master-format. This library, developed for light water reactor physics calculations, was preferred to the fine-group library in 252 energy groups also available in SCALE6.2 based on considerations of computational time and accuracy for current reactors applications. Cross-section processing Each perturbed multigroup library containing the problem-independent cross-sections was processed into a problem-dependent library by performing resonance self-shielding. The CENTRM method was used for this task as it is the most rigorous model in both energy and geometry since it performs a continuous energy 1-D transport calculation. The required data for the self-shielding calculations (self-shielding factors and continuous energy cross sections) were also perturbed by using the same perturbation factors computed for

• First, Sampler produces random samples of the input data, • Then, a large number of calculations of the SCALE sequence of interest are performed, propagating the perturbed data to the specified response, • Finally, the uncertainty and correlations in the response are obtained by statistical analysis of the output distributions. Different input data can be perturbed (nuclear data, nuclide concentrations, geometrical dimensions and other model parameters) provided that uncertainties and correlations are known. Concerning nuclear data, SCALE6.2 includes covariances for multigroup neutron cross-sections, fission product yields and radioactive decay data, so all those parameters can be perturbed, individual or simultaneously. In order to produce the random samples of nuclear data, Sampler uses pre-computed perturbation factors generated by XSUSA code [45] by sampling covariance information. A master file containing perturbation factors for 1000 samples is stored and become available during the Sampler execution. That is, nuclear data parameters are not sampled “on the fly”. In this work, the uncertainty analysis is performed using 200 samples where both multigroup cross-sections and fission yields are perturbed together. • With respect to cross-sections, perturbation factors for all groups and reactions of all materials were pre-generated by sampling the SCALE6.2 56group covariance library. During Sampler execution, those perturbation factors are read for each sample and a complete set of perturbed problemindependent (infinitely-dilute) multigroup crosssections is computed as well as consistent perturbed parameters required to perform the self-shielding. Those values will be used by TRITON to perform each coupled cross-section processing, transport and depletion calculation. • With respect to fission product yields, perturbation factors were pre-generated by sampling 9

Nuclear Data Uncertainties . . .

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D. Rochman et al.

having different compositions or relative positions in the assembly, to prevent TRITON from averaging the neutron flux in regions with different flux distributions that can impact isotopics. The following number of materials was found adequate for an accurate depletion prediction for the five considered assemblies: 8 for the PWR UO2 assembly, 48 for the PWR UO2 +Gd, 3 for the PWR MOX, 68 for the BWR UO2 +Gd and 50 for the BWR MOX+Gd (those numbers include the 5 different materials considered in the 5 equal-area radial rings per Gd-bearing absorber rod). A simplification was applied for depletion calculations with a large number of fuel materials to be depleted. In order to decrease calculation run-time in cross-section processing, some materials were grouped together so that a common set of microscopic cross sections were used for all of them, although each material was tracked independently in the depletion calculation. Normalization was performed using total full assembly power for non-gadolinium rods, and flux for Gd rods. The length of depletion intervals was set to match the burn-up steps requested in the benchmark. Cross-section updates are required more often when properties change rapidly, that is, during poisoning and Gd depletion. Consequently, finer steps were set at the beginning of the burnup for all assembly types and maintained for larger burnups for assemblies containing Gd rods. Decay calculations were performed by independent ORIGEN-S calculations after the assembly burnup using the requested benchmark steps.

the infinitely-dilute multigroup cross sections; that way, “implicit effects” are treated in a consistent manner, since all data are based on the same fundamental ENDF/B information. Different lattice cell calculations were specified for: i) different U or Pu-enriched rods and ii) rods with the same enrichment but different relative positions in the assembly (e.g. different positions with respect to the water channel and channel box) in order to permit a more accurate prediction of the fuel depletion. Moreover, for the BWR assemblies, accurate Dancoff factors were also calculated using de SCALE/MCDancoff module. The following number of cell calculations was performed for the five considered assemblies: 4 for the PWR UO2 assembly, 5 for the PWR UO2 +Gd, 3 for the PWR MOX, 17 for the BWR UO2 +Gd and 12 for the BWR MOX+Gd. Note the large number of cell calculations required e.g. for the BWR UO2 +Gd assembly. Only 8 different rod types are found, but since rods with the same composition have different positions with respect to the water channel and the channel box, the corresponding Dancoff factors differ significantly and up to 17 lattice cells were to be specified to accurately catch the effect of the flux distribution on the isotopics and pin-powers along depletion. In particular, for UO2 and MOX fuel rods, the LATTICECELL treatment in CENTRM was used. For Gadolinium-bearing fuel rods, a MULTIREGION treatment with five equal-area rings to capture radial depletion of gadolinium was used. In Ref. [46] the impact of the subdivision of U+Gd rods was assessed, concluding that while the impact on the isotopic is minimal, on k∞ is very high in the period when the gadolinium is burning out, being essential the subdivision of the U+Gd fuel for criticality calculations. Transport calculations In the NEWT transport calculations the order of SN was set to 10; a fine mesh of 4 × 4 was used for the square-pitched units and all fuel mixtures and structural materials used P1 scattering whereas all moderator mixtures used P3 scattering. Coarse-mesh finite-difference acceleration option was employed on the global grid to activate a low order solution for homogenized cells in the coarse spatial grid to substantially reduce the computational time to reach convergence. Depletion and decay calculations Isotope transmutation and decay calculations were performed with the ORIGEN-S code, which uses a 1-group cross section library. During depletion calculations, only cross sections for isotopes included in the transport calculation are updated on the 1-group library. In order to account for all nuclides having a significant impact on flux, a total of 388 nuclides were added in the transport calculations to update cross sections for ORIGEN-S. When performing the depletion calculation, materials in the fuel assembly were depleted individually when

In summary, to propagate uncertainties in nuclear data along depletion, the applied calculation scheme used random pre-computed perturbation factors to generate, for each sample, a perturbed infinitely-dilute multigroup cross-section library, perturbed self-shielding factors and a perturbed yield library for TRITON sequence. Those data were used in an individual coupled cross-section processing, transport and depletion calculation to obtain a sample of the output quantities of interest along burnup. A total of 200 samples were obtained, and it is worth noting that all samples were independent, being the set of perturbed data propagated to downstream codes independently on other runs.

5.

NRG-Fast TMC

The Fast TMC method can be applied in the case of Monte Carlo transport calculations [19]. It can be explained by a careful examination of how one should combine results of several Monte Carlo runs. This method, although not new, is detailed in the next subsection, to introduce the notation, and to prepare the ground for an explanation of the Fast TMC method. After that, in Subsection II A 5 b, it will be argued that a clever bookkeeping of the individual run results can yield an estimate of the variance due to the differences between these indi10

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for the combination of M runs of N random realizations is therefore

vidual runs. a. Combination of Monte Carlo runs. A Monte Carlo program generates random realizations of a quantity of interest. If the quantity of interest is x, the program produces N random realizations xi . Using these xi , the program does some internal bookkeeping along the way. The program keeps track of the average of the xi , and of the average of x2i S1 =

N 1  xi , N i=1

S2 =

N 1  2 x . N i=1 i

x¯ =

(7)

The expectation value for σ 2 is the variance of the variable x divided by N V (x) . N

So far, we have assumed that the M Monte Carlo runs were statistically independent, but otherwise equal. In that case the expectation value for the variance of x ¯j is given by Eq. (8), i.e. V (¯ xj ) = V (x)/N for all j. This is the classical Monte Carlo convergence rate. The situation changes when the M runs are not identical. When one changes, from one run to the next, not only the random number seed, but also one of the input parameters, the variance of x¯ will increase. The variance will reflect the statistical variance as well as the variance due to input parameters variations. A Monte Carlo estimator for this variance, including both statistical and input parameter effects, is ⎛ ⎞2 M M 1  2 ⎝1  ⎠ 2 σtot = x ¯ − x ¯j . (13) M j=1 j M j=1

(8)

It should be noted that internally the program works with the three variables N, S1 , S2 , but on output it writes the three variables N, x ¯, σ. It is, however, trivial to reconstruct the internal variables, based on the output variables S1 = x ¯, S2 = (N − 1)σ 2 + x¯2 .

(9)

Now suppose one has performed M statistically independent runs of the program, each giving output variables N, x ¯j , σj , with j = 1, . . . , M . For the sake of simplicity, it is assumed here that all runs were performed with an equal number (N ) of independent realizations of x. The generalization of Eq. (9) then reads S1 =

M 1  x ¯j , M j=1

S2 =

M M N −1 2 1  2 σj + x¯ . M j=1 M j=1 j

(11)

Here we have assumed N >>1 and M >>1. The result (11) is a special case (viz. the same N for all runs) of the more general situation described in Ref. [47]. Eq. (11) tells us that it is possible, based on run results x ¯j and σj , to construct the correct Monte Carlo estimator for the total number (N × M ) of random realizations. b. The variance of the run results. There is, however, more that can be done with the set of M Monte Carlo runs. The expression between curly brackets in the equation for σ 2 in Eq. (11) is in fact the definition of the variance of the M Monte Carlo estimators x ¯j (j = 1, . . . , M ) denoted as V (¯ x) ⎛ ⎛ ⎞2 ⎞ M M   1 ⎜ 1 ⎟ E⎝ x ¯2 − ⎝ x ¯j⎠ ⎠ = V (¯ x). (12) M j=1 j M j=1

At the end of the run, the program uses the standard Monte Carlo estimators for the average of x, which is denoted by x ¯ here, and for the standard deviation σ that goes with it, as defined by

E(σ 2 ) = V (¯ x2 ) =

M 1  x ¯j , M j=1

⎧ ⎛ ⎞2 ⎫ ⎪ ⎪ M M M ⎬ ⎨    1 1 1 1 2 2 2 ⎝ ⎠ . σ = 2 σj + x¯j − x ¯j ⎪ M j=1 N M⎪ M j=1 ⎭ ⎩ M j=1

(6)

x ¯ = S1 ,

1 S2 − S12 . σ2 = N −1

D. Rochman et al.

This estimator can be calculated based on the output parameters of the M individual Monte Carlo runs. Moreover, one can also combine the individual Monte Carlo results according to Eq. (11). The final question is then which part of this total variance is due to statistics, and which part is due to the input parameter variation? This question can be answered by estimating the statistical variance. Eq. (11) is no longer a good estimator for that, because the input parameter variation also influences its results. Instead one can look at the variance of the individual runs, σj2 . For each individual run, the input parameters are of course fixed, and therefore the variance is due to statistics only. This variance is estimated by the usual Monte Carlo estimator σj2 .

(10)

In other words, if one would have performed one large Monte Carlo run with N × M independent realizations of x, the program would have calculated exactly the same S1 and S2 . For this one run, the program would have N M done it as S1 = (1/N M ) i=1 xi , and similarly for S2 . Having reconstructed the variables S1 and S2 , one can use Eq. (7) once more. The final Monte Carlo estimator 11

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NUCLEAR DATA SHEETS

many parameters that are needed for simulations, such as cross section for various reaction channels, energies and angular distributions of outgoing particles, etc. Therefore this method is applied here to the calculation of local power densities in a full core reactor simulation: with the amount of computer time needed to calculate local power densities in the Martin-Hoogenboom benchmark, it is possible to simultaneously calculate the uncertainties in these local power densities due to the uncertainties in nuclear data.

A proposal for an estimator of the statistical variance of the combined set of Monte Carlo is then 2 σst =

M 1  2 σ . M j=1 j

(14)

The expectation value of this estimator is equal to the expectation value of each of the σj2 2 V (x) = V (¯ x), = E σst N

(15)

and is therefore suitable as an estimator of the statistical part of the total variance. Another way to calculate the statistical variance is to repeat the set of M Monte Carlo calculations, this time without input parameter variation. This method doubles the amount of necessary calculation time, though, and one would only like to do this when really necessary. In general, however, it is difficult to know whether this is necessary without performing these calculations explicitly. One could of course use very high numbers of his2 tories (high N ), to get low values for σst . If these values 2 2 are much lower than σtot , than the precise estimate of σst 2 is less important for the end result σip . On the other hand, there is always the problem with Monte Carlo eigenvalue calculations, that the Monte Carlo estimator for the standard deviation is not entirely correct: the estimator is based on the assumption that there is no correlation between successive neutron batches (’generations’), but in actual fact such correlations do exist. Therefore, the method to repeat the set of Monte Carlo calculations with statistical variation only will be used here as a check on Eq. (15). Further details are given in Subsection IV H 1 b. Finally, having calculated the total variance using Eq. (13), and the statistical variance using Eq. (15), one can calculate the variance due to the input parameter variation. Here it is necessary to assume that the variance due to statistics and the one due to input parameter variation are independent. This assumption was checked implicitly in the work reported in Ref. [19], through a comparison of several methods, for both high and low values of M . The variance due to input parameter variation is then 2 2 2 σip = σtot − σst .

D. Rochman et al.

6.

UU-TMC

This section describes the method used at the Uppsala University with DRAGON. The deterministic modeling of the transport of neutrons in a fuel assembly utilizes as material parameters the so-called multi-group microscopic cross-sections as presented earlier. The term multigroup refers to the discretization of the energy variable of such cross-sections and therefore, they are presented as being averaged along different energy bins. One of the most utilized code to compute multi-group microscopic cross-sections is NJOY. It will not only process point-wise information from the different nuclear databases that exist around the world in order to obtain energy averaged parameters, but it also has the capability of arranging the information in different multi-group library formats that can be read by different lattice codes (e.g. WIMS, DRAGON, APOLLO, etc). The NJOY code, developed at Los Alamos National Laboratory, is modular and sequential. The module that averages the cross-sections in energy and represents them in ENDF format is known as “GROUPR”. The ENDF format allows understanding in an easy way the different multi-group information required for an assembly lattice calculation (i.e. microscopic cross-sections, different Legendre orders of the scattering matrices, energy spectrum, released neutrons per fission, etc.). Afterwards, other NJOY modules, such as WIMSR, processes the multigroup ENDF file and arrange it into an ASCII library that some lattice codes understand and may utilize for transport calculations. This final library format is known as WIMS library, and codes such as DRAGON and the ones belonging to the WIMS-family are able to run with the WIMS library format. As uncertainty analysis has become nowadays part of any physical modeling and code development endeavor, methodologies to propagate (in this particular study) cross-sections uncertainties through lattice codes are needed to be developed in accordance to the library format that a certain code utilizes. Due to the fact that the ENDF multi-group format given by GROUPR is easy to read, understand and manipulate, a new methodology was developed to statistically perturb the GROUPR ASCII file in order to create posterior perturbed WIMS libraries. Thus, as many different WIMS libraries are created, as many times the lattice code will be run in order

(16)

This idea can be applied to any and all input parameters in a Monte Carlo simulation. One can vary one parameter at a time, or many parameters at the same time. The beauty of Monte Carlo is that many parameters can be varied at the same time, while preserving the same convergence rate: in this case, the variance converges as √ 1/M (the standard deviation as 1/ M ), irrespective of the number of input parameters varied. Since the amount of nuclear data used in a reactor simulation is very large, the Fast TMC method is well suited for this situation: each nuclear data file contains 12

Nuclear Data Uncertainties . . .

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D. Rochman et al.

2. MF=6, MT=2 and 221 should be statistically perturbed in order to change the elastic scattering matrices and also the total cross-section. Since only the P0 covariances are given, then only the P0 matrix can be perturbed.

to create a deterministic output into a stochastic one. This is exemplified in Fig. 4. The module WIMSR reads and manipulates only certain information contained on the ENDF file. Therefore, such information is the one that is required to be statistically perturbed. The information that WIMSR reads from the output of GROUPR is as follows:

3. For the inelastic scattering term, the perturbation was done only at the MT=91 data. This is good enough since WIMSR adds up all the MT=51-91 terms and thus, the propagation of the perturbation is carried out correctly.

1. From the MF=3 data, it reads only the fission (MT=18) and the capture (MT=102) crosssections. These will be used to form the total crosssection of the final WIMS library.

4. The Nu-fission term (MF=6, MT=18) was modified in a way that only the ν uncertainty was propagated, since the fission perturbation is carried out at the MF=3, MT=18 data.

2. From the MF=6 data, it would read only the scattering matrix given by MT=2 correspondent to the fast energy spectrum region. For the resonance and thermal regions, scattering matrices are given by MT=221 (which is computed by the NJOY free-gas model for thermal scattering). It should be noted that the WIMS library format only utilizes P1 and P0 terms for moderator nuclides. For other nuclides such as actinides, etc., only the P0 matrix is considered. The energy-integration of both MT=2 and MT=221 will be used to compute the elastic scattering part of the total cross-section.

5. For the WIMS library, even if MF=5 data could be perturbed in order to change the fission spectrum, no effect whatsoever would have on the spectrum of the final WIMS library. This is the only term that cannot be perturbed in the ENDF file and thus, such perturbation should be carried out at the final WIMS library.

7.

3. From the MF=6 and MT=50-91, WIMSR will add them per energy group in order to form inelastic scattering and thus, after energy integration, update the total cross-section.

UKAEA-FISPACT-II

Provided with the reactor burn-up simulation data from the lattice code CASMO-5, FISPACT-II [18] was modified to also accommodate the fine 586-group neutron spectrum data with newly ENDF/B-VII.1 processed data and CALENDF-generated [48] probability tables for self-shielding corrections. FISPACT-II possesses a new POWER keyword which causes re-normalisation of the total flux to match a specified W/cc volumetric power to a defined set of fissile isotopes. The normalisation uses isotopic KERMA values provided by collapsing of the neutron spectrum with KERMA data from the nuclear data files, which can be specified using any appropriate mt values (for example the total mt=301, non-elastic mt=303, inelastic mt=304, fission mt=318, disappearance mt=401, photons mt=442, total kinematic mt=443 or any combination thereof). To best match the self-shielding factors (SSF) for fission and 238 U capture from CASMO-5, the probability tables were applied as energy-dependent SSF for each group based on total cross-section dilution self-shielding (rather than using macro-partial SSF, a more sophisticated approach available to FISPACT-II). As shown in Fig. 5, the energy-dependent reaction rates for BOL SF97 require virtually no self-shielding factors for energies below 10 eV (due to the fine group resolution). Above 10 eV, the 235 U(n,f) has some SSF correction which is not significant for the overall reaction rate. The 238 U capture is quite different, with significant resonances contributing to the reaction rate up to the keV energy region. Group SSF range down to 0.02, and average to an effective SSF around 0.1-0.2, which evolves

4. From MF=6 and MT=18, the Nu-Fission crosssection is taken. 5. Even though the MF=5 data contained in the ENDF file gives the fission spectrum, the WIMS library format fixes the fission spectrum and thus, does not take into account these info from the ENDF file. Now, the uncertainty analysis would perturb ALL terms of the parameters that are required for the transport calculation. This is how the perturbation methodology works in order to conserve any further transport calculation: 1. Covariance data is given to the total energy integrated terms of the multi-group cross-sections. From this information (which can be obtained from the ERRORJ module of NJOY) the fission and capture cross-sections can be perturbed. It was realized that if the fission cross-section (MF=3, MT=18) is perturbed in the ENDF ASCII file, the module WIMSR will change not only the total cross-section but also Nu-fission. This is a good thing since perturbations to the fission cross-section are being conserved by WIMSR. Also, any perturbation to the capture cross-section will immediately update the total cross-section as well. 13

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D. Rochman et al.

FIG. 4. (Color online) Processing steps for the Uppsala methodology to propagate DRAGON libraries.

reaction channels [49–51], the inventory is handled in full. The use of cross section uncertainties, as provided by covariance files, requires identification of each combination of parents, reactions, decays and loops which result in the production of every nuclide of interest. TMC analyses can be performed without these combinatorially complex tasks, providing uncertainties on all nuclides simultaneously as shown in Fig. 7. Note the exceptionally low uncertainties for nuclides within the fission yield peaks, which are not realistic for many. This feature is a direct consequence of the effect of limited range of values generated by default parameter variation within GEF, which must be modified to generate experimentally-faithful yield variation. Nuclides with relatively small yields, approximately those below 1 per 105 fissions, have an increased uncertainty which is an artifact due to statistical variation inherent to the MonteCarlo GEF method. FISPACT-II employs a sophisticated reaction rate uncertainty quantification method using available cross section covariances and deep reaction/decay pathway analysis [52]. All independent fission yields, open reaction channels and decays are considered, following from hundreds to a few thousand nuclides. FISPACT-II can process ENDF-6 NI-type covariance data with LB=1, 5 or 6, using a projection operation to match covariance and cross-section energy grids, which are stored in multigroup format (586-group for CASMO-5 based simulation). Once individual reaction rate uncertainties are calculated for a given incident particle spectrum, the reaction/decay production pathway for each nuclide must be determined. In the Takahama case a typical production route for 137 Xe, a precursor to 137 Cs, the path is broken

through burn-up. The plutonium production rate is extremely sensitive to these SSFs, which dictate the fission 235 U/239 Pu ratio and composition throughout the fuel lifetime. For this analysis, SSFs are updated for each change in spectrum, as reaction rates are re-calculated for each of these steps. For this uncertainty quantification exercise, the TotalMonte Carlo fission yield uncertainties on the thermal 235 U, 239 Pu and 241 Pu, as well as 400 keV 238 U are quantified through use of random fission yield files. 500 neutron-induced fission yield files, which are generated by input parameter variation using the GEF-4.2 fission simulation code (as presented in section II A 2), were used for multiple simulations. Simulations for each case are performed in parallel and code outputs are statistically collapsed to determine moments for nuclide inventories, heat outputs, activity, etc. The tremendous advantage of this approach is that all covariances are effectively treated through sampling of physically-consistent input physics, without the need for complex covariance methods or pathway analyzes. An example of these simulations is shown in Fig. 6. A correlated uncertainty has also been calculated, using the variation in collapsed covariance uncertainty with variation in nFY files. This coupled contribution is typically less than 10 % of the reaction rate collapsed-covariance uncertainties. Uncertainty in an integral quantity such as decay heat hides a tremendous amount of information, since dozens or hundreds of nuclides each contribute similar fractions of the total quantity for most cooling times - particularly for times not long after irradiation. Since FISPACTII also handles all independent fission yields, decays and

14

0.9 0.8

1 0.9 0.8

100

10-1

-2

101

10

102 Energy (eV)

103

104

0.7

Self-shielding factor U238 (n,γ) Spectrum in CASMO-586 Unshielded 586 RR U238 (n,γ)

Neutrons/RR per unit lethargy (arb. norm.)

100

10-3 10-2 10-1 100 101 102 103 104 105 106 107 Energy (eV)

1.1

0.7

Self-shielding factor U238 (n,γ) Spectrum in CASMO-586 Unshielded 586 RR U238 (n,γ)

1.2

100

10-1

100

101

102 Energy (eV)

103

10 104

Self-shielding factor (σd / σinf)

1

1.3

SS factor [PT 00 MT 301] U235 (n,f) Spectrum in CASMO-586 Unshielded 586 RR U235 (n,f)

Neutrons/RR per unit lethargy (arb. norm.)

1.2 1.1

10-3 10-2 10-1 100 101 102 103 104 105 106 107 Energy (eV) Neutrons/RR per unit lethargy (arb. norm.)

1.3 Self-shielding factor (σd / σinf)

Self-shielding factor U235 (n,f) Spectrum in CASMO-586 Unshielded 586 RR U235 (n,f)

D. Rochman et al.

Self-shielding factor (σd / σinf)

NUCLEAR DATA SHEETS

Self-shielding factor (σd / σinf)

Neutrons/RR per unit lethargy (arb. norm.)

Nuclear Data Uncertainties . . .

-2

FIG. 5. (Color online) FISPACT-II collapsed energy-dependent unshielded reaction rates for 235 U(n,f) [top] and 238 U(n,γ) [bottom], with probability table self-shielding factors superimposed. The ultra-fine energy resolution with the CASMO-5 586 group eliminates the need for self-shielding below 10 eV, but for 238 U capture the factors above 10 eV are significant, resulting in an effective SSF around 0.1-0.2.

down through fission of 29 % 235 U, 46 % 239 Pu, 14 % 241 Pu and 8 % 238 U, with varying contributions through 137 137 I, Te, isomeric and direct production. Some minor contributions from other fission and paths including reactions are also analyzed in FISPACT-II. Uncertainties for each reaction are then calculated and weighted to determine a final nuclide inventory uncertainty. For fission scenarios, this can be extremely complex due to combinatorial growth, requiring intelligent path ‘pruning’. This employs minimum path/loop values for inclusion and maximum path depth values, which have been set to defaults except where proven insufficient. Both the fission yield and reaction rate uncertainties are calculated together, so that the uncertainties are propagated together. For each simulation with varied fission yield files, the reaction rate uncertainty is also calculated. While the uncertainty in the effective cross section for each reaction channel in each time step does not vary in each of these simulations, the fraction of each pathway does. Difference uncertainties between, say, the 235 U or 241 Pu fission rates and yields are coupled, so that the reaction rate uncertainty will vary as the ratio is changed. Exam-

ples are shown in Fig. 8, with varied dominant nuclide sets, TMC-based nFY uncertainty, covariance reaction rate uncertainty and their correlated contributions. Note that the uncertainty files mf=33 for the major nuclear data libraries do not contain uncertainties within the resolved resonances or below, which completely discards any uncertainty quantification for reactions where those energies are significant. This is precisely the case for neutron capture and non-threshold fission, which plays a central role in thermal reactor physics. Without these data all uncertainties are guaranteed to be underestimates.

8.

SCK Fission Yields

SANDY [53, 54] is a new NJOY-wrapping numerical code developed at SCK-CEN to perform the sensitivity analysis (SA) and uncertainty propagation (UQ) of nuclear data. Amongst its capabilities, such code generates covariance matrices for independent fission yields using a generalized least-square (GLS) approach. After collecting 15

Nuclear Data Uncertainties . . .

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0

10-1 1

10-2

10-3 10-2

100

102 104 Time after irradiation (s)

106

108

Relative uncertainty (%)

10

Sampled mean heat after irradiation TMC nFY ENDF/B-VII covariances Coupled

10 Heat output after irradiation (kW/kg)

D. Rochman et al.

0.1

FIG. 6. (Color online) Example decay heat from Takahama SF97-1 after 45 MWd/TU, using ENDF/B-VII.1 nuclear data file covariances for reaction rate uncertainty and TMC thermal neutron sampling for yield uncertainty. The coupled combination includes both, with an additional correlation due to variation in the collapsed covariance uncertainty with nFY variation. The uncertainty bands have been multiplied by 10 to aid visualization on log-log axes.

best-estimate and uncertainty values from the ENDF-6 files, such data are constrained to comply with the conservation equations of a fission event e.g. conservation of mass, charge and number of fission fragments, symmetry, correlation between independent and chain fission yields. Hence, covariance matrices are produced and independent fission yields are upgraded. Such an update tackles the need for more consistency of fission yield uncertainties in the major general purpose libraries (JEFF, ENDF/B) [12, 13, 26], which only loosely abide by the conservation laws at the current stage. Then, given an upgraded array of best-estimate independent fission yields Y and a covariance matrix C, random samples are drawn from a multivariate normal probability density function (pdf) N (Y, C) using a standard Monte Carlo sampling procedure. The new random sets of independent fission yields replace the original values in “perturbed” ENDF-6 files. In each “perturbed” file, cumulative fission yields are also recalculated using the random independent yields and the Q-matrix equation which involves the use of radioactive branching ratios [55]. In this work, we created “perturbed” files for the U and Pu fissioning systems of ENDF/B-VII.1. Additionally, the perturbation of the Q-matrix is performed and an extra set of files with perturbed “fission yields and branching ratios” is made. The fission yields from SANDY are used in this work with CASMO-5 as an option in the SHARK-X approach.

B.

Considered Systems

In the following, details for the different assembly systems are given and are summarized in Table III. Then, the Martin-Hoogenboom simplified full core is presented.

1.

PWR UO2 Assembly

The 2-dimensional assembly model for the PWR with UO2 fuel is the Three Mile Island Unit 1 (TMI-1) type. The specifications for such assembly model can be found in Refs. [56, 60]. The general characteristics are reproduced here and in Table III. The TMI-1 assembly is a UO2 fuel assembly with a 15×15 lattice of 212 fuel rods and is represented in Fig. 9. The number of guide tubes is 16 (in light blue in Fig. 9 left), with one instrumental tube at the center of the assembly. Four fuel rods contain a small amount of gadolinium (2 w%), which is believed to have a small impact on the assembly characteristics. The number of UO2 fuel rod (without Gd) is 204, with an enrichment in 235 U of 4.12 wt %. Some specific geometrical data are given in Table IV. For the presented burn-up calculations, the limit in the burn-up value is 40 GWd/tHM.

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FIG. 7. (Color online) FISPACT-II uncertainties of dominant 30 nuclides for Takahama SF97 at 45 MWd/tn burn-up, with cooling times of 100 s, 104 s, 106 s and 108 s. TMC fission yield uncertainty, covariance reaction rate and nFY-RR correlated uncertainties are shown in red, blue and green, respectively.

17

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Relative uncertainty (%)

100 Takahama SF97 45 GWd/t 1.0E+02s cooling

NUCLEAR DATA SHEETS

TMC thermal nFY ENDF/B-VII.1 covariances Variation in coupled RR-nFY

TABLE III. Assembly specifications for the five considered cases. All concentrations (Gd and actinides) are given in wt %.

10

Assembly PWR UO2 UO2 + Gd Type TMI Takahama-3 UAM Definition Yes Yes Lattice size 15×15 17×17 Nbr. fuel rods 212 264 Nbr. fuel rods with Gd 4 16 2b 6 Conc. Gd2 O3 Fuel Temp. (K) 900 900 Moderator Temp. (K) 565 575 Cladding Material Zr-4 Zr-4 Power density 33.75 36.7 (W/gU or W/gHM) Boron (ppm) 900 630 Fuel pellet outer  (cm) 0.929 0.805 Burn-up EOC 40 40 (GWd/tHM) 235 U concentration 4.12 2.6-4.1 235 U ave. conc. 4.12 4.03 Pu total conc. Pu fissile conc. Pu fissile ave. conc. Reference [56] [57] Assembly BWR UO2 + Gd MOX + Gd Type ATRIUM-10 UAM Definition No No Lattice size 10×10 10×10 Nbr. fuel rods 91 91 Nbr. fuel rods with Gd 11 11 3 3 Conc. Gd2 O3 Fuel Temp. (K) 900 900 Moderator Temp. (K) 559 559 Cladding Material Zr-4 Zr-4 Power density 27.6 27.6 (W/gU or W/gHM) Boron (ppm) Fuel pellet outer  (cm) 0.884 0.884 Burn-up EOC 40 40 (GWd/tHM) 235 U concentration 2.2-4.8 0.2 235 U ave. conc. 3.99 0.43 Pu total conc. 2.7-10.6 1.6-6.3 Pu fissile conc. Pu fissile ave. conc. 3.9 Reference [59] [59]

1

Np239 Xe138 Cs138 I134 La140 La142 I135 Tc104 Y95 Y94 Mo101 U239 Tc105 Sr93 Kr88 Xe137 Ba141 I132 Xe133 Nb97 I133 Zr95 Tc102 Rb90 Rb89 Ba142 Cs139 Te134 I136 Cs140

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Ba137m Cs134 Kr85 Ce144 Eu154 Pr144 Pu238 Cm244 Rh106 Cs137 Am241 Y90 Sb125 Pu241 Eu155 Pm147 Sr90 Cm243 Cm242 Te125m Pu240 Pu239 Eu152 Ru106 Am243 Pr144m Am242m Am242 Nb95 Zr95

0.1

FIG. 8. (Color online) FISPACT-II uncertainties of dominant 30 nuclides for Takahama SF97 at 45 MWd/tn burn-up, with cooling times of 1 sec, 100 sec and 108 sec.

PWR UO2 +Gd Assembly

2.

The 2-dimensional assembly model for the PWR with UO2 fuel and Gd is the Takahama-3 type as used in the UAM benchmarks. The specifications for such assembly model can be found as previously in Refs. [56, 60]. The general characteristics are reproduced here and in Table III. The Takahama assembly is a UO2 + Gd fuel assembly with a 17×17 lattice of 264 fuel rods and is represented in Fig. 10. The number of guide tubes is 25 (in light blue in Fig. 10), and 16 fuel rods contain an amount of gadolinium (6.0 w%) (in red in Fig. 10 left). The total number of UO2 fuel rod (with and without Gd) is 264, with an enrichment in 235 U of 4.11 wt % (without Gd) and 2.61 wt % (with Gd). Some specific geometrical data are given in Table V.

a b

3.

D. Rochman et al.

MOX Beznaua No 14×14 179 0 788 575 Zr-4 27.7 600 0.929 40

0.22 3.4-5.5 2.5-4.0 3.65 [58]

These data come from the ARIANE program. The amount of Gd2 O3 in this assembly is not significantly changing the assembly behavior compared to an assembly without gadolinium.

PWR MOX Assembly

This 2-dimensional assembly model for the PWR with MOX fuel corresponds to the Swiss reactor Beznau type as used for the ARIANE program and the BM5 sample. The specifications for such assembly model can be found in Ref. [61, 62]. The general characteristics are reproduced here and in Table III.

This Beznau assembly is a MOX fuel assembly with a 14×14 lattice of 179 fuel rods and is represented in Fig. 11. It corresponds to the assembly M308. Different Pu content zones are included in this assembly: low, medium and high. In weight percent (w/o) of (Pu+Am)/(U+Pu+Am), these different zones corre18

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FIG. 9. (Color online) Layout of the PWR UO2 assembly for the TMI-1 model. The colors represent the different rods and guide tubes (see text). On the left is the representation with CASMO-5 (full model) and on the right with TRITON (1/4 model).

FIG. 10. (Color online) Layout of the PWR UO2 + Gd assembly for the Takahama-3 model. The colors represent the different rods and guide tubes for the CASMO-5 model (see text). On the left is the representation with CASMO-5 (full model) and on the right with TRITON (1/4 model).

sponds to 3.36, 4.28 and 5.50 %, respectively. In Fig. 11, each Pu content is differentiated by colors: gray for low Pu content, red for medium and green for the high Pu content. The number of guide tubes is 16 (in light blue in Fig. 11), with one instrumental tube at the center zone of the assembly. Some specific geometrical data are given in Table VI. For the presented burn-up calculations, the limit in the burn-up value is 40 GWd/tHM.

4.

BWR UO2 +Gd Assembly

This 2-dimensional assembly model for the BWR with UO2 fuel corresponds to the ATRIUM-10 type (10-9Q) with symmetrical water gaps, as in the case of the model of the BWR with MOX fuel. The specifications for such assembly model can be found in Refs. [63]. The general characteristics are reproduced here and in Table III. This ATRIUM-10 assembly is a UO2 fuel assembly with a 10×10 lattice of 91 fuel rods and is represented

19

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FIG. 11. (Color online) Layout of the PWR MOX assembly M308 for the Beznau model. The colors represent the different rods, Pu contents and guide tubes (see text for details): gray (low Pu content), red (medium Pu content) and green (high Pu content). On the left is the representation with CASMO-5 and on the right with TRITON.

TABLE IV. Fuel assembly geometry for the PWR UO2 case (TMI-1). Geometry parameters Rod array Number of fuel rods Number of fuel rods with Gd Fuel rod outer diameter Number of guide tubes Guide tube outer diameter Guide tube inner diameter Instrumentation rod outer diameter Instrumentation rod inner diameter Fuel rod pitch Fuel assembly pitch Cladding outer diameter Cladding inner diameter Fuel density

TABLE V. Fuel assembly geometry for the PWR UO2 + Gd case (Takahama-3).

value 15×15 208 4 10.922 mm 16 13.462 mm 12.650 mm 12.522 mm 11.202 mm 14.427 mm 218.11 mm 1.0922 cm 0.958 cm 10.28 g/cm3

Geometry parameters Rod array Number of fuel rods Number of fuel rods with Gd Fuel rod outer diameter Number of guide tubes Guide tube outer diameter Guide tube inner diameter Fuel rod pitch Fuel assembly pitch Cladding outer diameter Cladding inner diameter Fuel density

value 17×17 264 14 8.05 mm 25 6.12 mm 5.71 mm 12.59 mm 214 mm 0.95 cm 0.822 cm 95 %

(large square at the center of Fig. 12). Some specific geometrical data are given in Table VII. For the presented burn-up calculations, the limit in the burn-up value is 40 GWd/tHM.

in Fig. 12. Eight different U content zones are used in this assembly, as presented with different colors in Fig. 12. Only one fuel type contains Gd2 O3 : in dark blue in Fig. 12 (11 rods) 4.3 wt% UO2 and 3.00 wt% Gd2 O3 . All other fuel rods do not contain gadolinium. In Fig. 12, each U content is differentiated by colors: 2.20 wt% (light green), 2.80 wt% (orange), 3.40 wt% (dark green), 4.30 wt% with Gd(dark blue), 3.80 wt% (pink), 4.30 wt% without Gd (blue), 4.80 wt% (brown) and 4.10 wt% (gray). The unique water tube is equivalent to the space of 9 rods

5.

BWR MOX+Gd Assembly

This 2-dimensional assembly model for the BWR with MOX fuel also corresponds to the ATRIUM-10 type (109Q) with symmetrical water gaps. The specifications for such assembly model can be found in Ref. [59]. The general characteristics are reproduced here and in Table III. 20

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FIG. 12. (Color online) Layout of the BWR UO2 assembly for the ATRIUM-10 model. The colors represent the different rods, UO2 contents and the guide tube (see text for details). On the left is the representation with CASMO-5 and on the right with TRITON.

TABLE VI. Fuel assembly geometry for the PWR MOX case (Beznau M308 assembly). Geometry parameters Rod array Number of fuel rods Number of fuel rods high Pu content (5.50 %) Number of fuel rods medium Pu content (4.28 %) Number of fuel rods low Pu content (3.36 %) Fuel rod outer diameter Number of guide tubes Guide tube outer diameter Guide tube inner diameter Instrumentation rod outer diameter Instrumentation rod inner diameter Fuel rod pitch Fuel assembly pitch Cladding outer diameter Cladding inner diameter Fuel density

TABLE VII. Fuel assembly geometry for the BWR UO2 case (ATRIUM-10 assembly)

value 14×14 179 115

Geometry parameters Rod array Number of fuel rods Number of fuel rods with Gd Fuel rod outer diameter Number of guide tubes Guide tube outer diameter Guide tube inner diameter Fuel rod pitch Fuel assembly pitch Cladding outer diameter

52 12 9.29 mm 16 13.36 mm 12.5 mm 9.48 mm 10.72 mm 14.12 mm 198.2 mm 1.072 cm 1.25 cm 10.3 g/cm3

value 10×10 91 11 8.84 mm 1 (3×3) 34.89 mm 33.48 mm 12.95 mm 305 mm 1.005 cm

do not contain gadolinium and UO2 . In Fig. 13, each Pu content is differentiated by colors: For the presented burn-up calculations, the limit in the burn-up value is 40 GWd/tHM.

6.

As for the BWR UO2 case, this ATRIUM-10 assembly consists of a 10×10 lattice of 91 fuel rods and is represented in Fig. 13. Six different Pu content zones are used in this assembly, as presented with different colors in Fig. 13. Only one fuel type contains Gd2 O3 : in brown in Fig. 13 (6 rods) 3.95 wt% UO2 and 1.5 wt% Gd2 O3 . All other fuel rods

Martin-Hoogenboom Benchmark

This benchmark differs from the other geometries as it is a full core and the calculations do not include burnup. It is included in this study to show the impact of nuclear data beyond the assembly level, but is still a simplification compared to a core design with realistic fuel loading. Additionally, this case indicates what can be achieved nowadays in terms of uncertainty propaga21

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FIG. 13. (Color online) Layout of the BWR MOX assembly for the ATRIUM-10 model. The colors represent the different rods, MOX contents and the guide tube (see text for details). On the left is the representation with CASMO-5 and on the right with TRITON.

The use of F6 in this study is not complete. The heating function used by F6 was calculated by NJOY [8] under the assumption that photon energies would be tallied separately. In the runs performed here, only neutrons were simulated, and therefore the energy included in F6 was not complete. Also, the energy due to fission product decay is not taken into account in this study (for F6 nor F7). Still, despite its shortcomings, it was deemed interesting to produce results for the deposited energy. The number of zones in which the F7 tally is non-zero is 6, 362, 400, because there are 241 assemblies with 264 fuel pins, divided in 100 axial zones. For F6 there are 12, 744, 900 zones with non-zero results, because part of the power is deposited in the guide tube zones and also outside the active core. The total number of non-zero tally results is therefore over 19 million. For each of these 19 million values, its uncertainty due to nuclear data uncertainty was calculated.

tion with a computational power commonly available in a working environment, without accessing large computer clusters. For systems with less simplications, such as full core studies with irradiated assemblies coming from a number of reactor cycles, the combination of Monte Carlo transport simulation and Monte Carlo uncertainty propagation might not be adequate. As presented later, two uncertainty propagation methods will be applied to this model, using similar nuclear data. The Martin-Hoogenboom benchmark consists of a reactor core with 241 identical fuel assemblies, each consisting of 17 × 17 regions, 264 of which are filled with fuel pins and the remaining ones are for guide tubes. A full description of the benchmark can be found in Refs. [64, 65]. Here, the calculations for the benchmark were performed with MCNP6 [66]. In the MCNP model, a tally grid was defined with 357 × 357 × 100 identical regions, so that the active core and some space around it are covered. The size of each region was 1.26 × 1.26 × 3.66 cm3 . The tallies used were the standard F6 and F7 tallies, while the grid option was the same as described in Ref. [67]. The F7 tally calculates the power generated in each fuel zone, which is the quantity required for the benchmark. The F6 tally calculates the power deposited in each zone, and is not necessary for the benchmark. Nevertheless, the F6 tally is used here, because one can judge whether the uncertainties are equal for generated and deposited energy. Also, the deposited power has a tail extending into the reflector, where the generated power is zero. By including F6 in the present study, it can be seen to which depth in the reflector one can calculate uncertainties in this way.

C.

Calculated Quantities

The uncertainties on relevant quantities are provided in this paper, as a function of the assembly burn-up, up to 40 MWd/kgU. • k∞ : as the considered assemblies are surrounded by reflective boundaries (no leakage), the multiplication factor provided here is the k∞ , see section IV A. • Number densities: during burn-up and also after shutdown, the uncertainties on a number of atom concentrations (or number densities) are provided 22

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measurement level. This step is of course costly in time and efforts. Up to now, the measured data are not always directly available to the evaluator community, but rather some derived data such as cross sections. There is nevertheless a recent effort to make the original data available with the maximum amount of information to allow new analysis [68]. This step can vary in time and take from one to several years.

for important isotopes such as major and minor actinides and important fission products. • The uncertainties on other quantities important for reactor physics: pin power and macroscopic cross sections. Parameters such as the peak power, Doppler and moderator temperature coefficients were also calculated but are not presented here. • The impact of neglecting the implicit effect in some calculations is also quantified in the case of the CASMO-5 calculations: see section IV J.

2. After the measurements and the analysis of the results, a publication takes place with different information and the final results. Reports can also appear with details. Based on these publications, the compilation of experimental data can take place. Compilers of experimental data enter the published data (as well as specific information provided by the authors) into a database called EXFOR [68]. This database does not contain evaluated data (unless mentioned) and is the source of experimental data for later evaluation. Including the publication and compilation, the needed time is at least of one year. It is still possible to find data in EXFOR compiled decades after their publications.

Fig. 14 presents the different calculated k∞ for the five considered systems. The effect of the gadolinium, at the beginning of cycle, can be easily observed. The calculated k∞ are obtained in this case with CASMO-5 and the ENDF/B-VII.1 library. As in the case of k∞ , the number densities for important isotopes are presented in Figs. 52 to 54 in Appendix A, average over all the pincells for each assembly. These quantities were also calculated with CASMO5 and the ENDF/B-VII.1 library. The major differences in trends come from the fuel composition at the beginning of irradiation: UO2 or MOX fuel. As presented the influence of the type of assembly (PWR or BWR) is of secondary effect. III.

D. Rochman et al.

3. Based on EXFOR and possibly other data not included in this database, the evaluation step begins. It is the production of “recommended” values based on the expert judgment of the evaluators. Modern evaluations make use of EXFOR and theoretical models coming from nuclear reaction codes such as TALYS [27] or EMPIRE [69]. In the resonance range, the analysis of experimental data can be performed with a Bayesian approach, as performed with the SAMMY, REFIT or CONRAD codes [70– 72]. The final product is then formatted into a specific format, usually the ENDF-6 format [73]. Groups of evaluators usually meet once or twice a year to discuss about such evaluations. The evaluation process can also take up to several years.

NUCLEAR DATA

As the nuclear data are the heart of this study, some specific details on their life and maturity are presented here. It helps to understand the origin of these values and the developments happening “behind the scene”. The nuclear data community is in full effervescence. Many activities are driving such state: new library releases, needs of covariance information, standardization of the evaluation format, elimination of error compensations, clear processing and finally stronger links between the three aspects of the nuclear data: measurements, evaluation and applications. Such an activity might not be visible for an outside bystander, but many of these changes are modernizing the full chain of the production, using automation when possible and leaving the time of scientists for improving the quality of their work. Quality and assurance rules directly applied to nuclear data contributes to a better assessment of risks using covariances. If this does not solve all issues related to nuclear safety, it certainly make the neutronics calculations under a higher control, for instance compared to thermohydraulic calculations or divers accident scenario outside the “neutron influence sphere”. The life cycle of a nuclear data quantity such as a cross section is summarized in Fig. 15 and can be helpful to understand how a nuclear data library (with its covariance files) is being created. A short summary is given below:

4. Once the evaluations are performed, they are included in a so-called library, such as JEFF, ENDF/B, JENDL or TENDL. Depending on the library, it is released every year (TENDL), or when needed. After the creation of a library, it is processed in files and formats which can be used by different codes such as transport codes. This step is not trivial and often defines the performances of transport codes. Different organizations are responsible for the distributions of the nuclear data libraries: the International Atomic Energy Agency (IAEA), the Nuclear Energy Agency (NEA) or the National Nuclear Data Center (NNDC). 5. The last steps are the testing of the library. This is called “benchmarking”. It can be done based on a series of benchmarks from the open literature (such as the ICSBEP, IRPhE or SFCOMPO databases [74–76]), or from proprietary benchmarks

1. As presented, the start of the nuclear data reactions relevant to energy production happens at the 23

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fuel measurements.

(often the case for Post Irradiation Examination, or PIE benchmarks, or reactor experiments such as ZPPR or EOLE). This step usually is spread out in time, depending on the benchmarks used for the validation. A first set of tests is usually performed by the evaluation community (often together with other specialists with experience in reactor simulation). It can be done before the library release, and/or continued shortly after. A second set of benchmarking and validation is later performed by the users community, often using other types of proprietary data coming from the online reactor core measurements, follow-up simulations and irradiated

6. Finally, feedback loops can be used for improving the evaluations after the results of benchmarking. The first ones (in red in Fig. 15) consist in correcting or adjusting quantities which were pointed out as incorrect by the benchmarking. This type of correction, if performed, needs to be done with care, since integral benchmarks are not designed to unequivocally be used for nuclear data corrections. Processing, code simplifications or compensation can play a role and give incorrect indications. But if there is a general trend, using different 24

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Measurements

Differential, integral, with covariances

Publication Compilation

Articles, reports, eventually in EXFOR

Reaction Theory

TALYS, SAMMY

Evaluation in data libraries

JEFF, ENDF/B, JENDL, TENDL

Processing & Distribution

NJOY, PREPRO, CALENDF, NEA, IAEA, NNDC

Inclusion in neutronics codes

Deterministic, Monte Carlo depletion, activation

Benchmarking validation 1 (evaluation community)

ICSBEP IRPhE SINBAD SFCOMPO

Inclusion in production, commercial codes

SIMULATE, TRACE, APOLLO. . .

Benchmarking validation 2 (industry community)

PIE, reactor data

D. Rochman et al.

FIG. 15. (Color online) Life cycle for the nuclear data: from the measurements to the industrial validation, and back. Acronyms are presented in the text.

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JENDL-4.0 [14], or the TENDL library [27]. These libraries are not the ones directly used by simulation codes, but the processing step is necessary in order to format the existing data and to possibly interpret such data. Important comparisons for the major libraries were performed as part of the working group on nuclear data and covariance adjustment and can be found in Ref. [79].

benchmarks, different simulation tools and different processing codes, the modifications of specific nuclear data can be justified. The drawback of such adjustment is the limitation of the adjusted data to specific applications and a library can lose its characteristics of being general purpose. The other alternative is that the results of different benchmarks are used as indication to “re-visit” a specific quantity at the experimental and differential level (such as the measurement of pointwise cross sections). This is the second loop (in black in Fig. 15). This use of benchmarking is more appropriate and more general as the information is verified again at a more fundamental level.

1.

ENDF-6 Format

Complete detail on the ENDF-6 format can be found in Ref. [78]. As all the nuclear data (and their covariances) used in this work are derived from the ENDF-6 format, a short description of the major sections included in an ENDF-6 evaluation is presented here for neutron-induced reactions. For all nuclear data except covariance information, the evaluated nuclear data files include the following sections:

In general, this approach to nuclear data is also valid for uncertainties and covariances. One difference is that there are no benchmarks to rigorously test the accuracy of covariances. One can nevertheless assess their quality to some extent with application of “common sense”, user experience and comparative analysis with existing data. For instance, in development of covariance data in evaluations, inter-comparisons to other libraries may be performed. New efforts in comparing covariance information are now realized at the NEA databank with the NDast computer system. Some details can be found in Ref. [77]. This might be done for instance against the same types of integral experiments used for benchmarking of the cross section values themselves - ICSBEP, IRPhE and so on. An exercise of this nature can aid evaluators in better understanding their consistency against files produced by peers using similar data and techniques; or, by highlighting questionable or incorrect data, which might be a result of either the evaluation itself or the processing thereof. It is also possible therefore to follow the same loops as shown in Fig. 15 for covariance data feedback. This being said, as uncertainties and correlations are only reflecting a specific method of analysis, it is of prime importance to describe and take into account how these analysis was performed. In the following, general information about the libraries used in this work is presented. For more details, the readers are referred to specific publications.

MF1 Description of the file, average number of neutrons per fission (total, prompt, delayed), energy release in fission, β-delayed photon spectra, MF2 Resonance parameters, MF3 Reaction cross sections, MF4 Angular distributions of emitted particles, MF5 Energy distributions of emitted particles, MF6 Energy-angle distributions of emitted particles, MF7 Thermal scattering data, MF8 Fission yields and decay data, MF9 Production cross sections for radioactive nuclei (fraction of MF3), MF10 Production cross sections for radioactive nuclei (absolute values), MF12 Photon production multiplicities and branching ratios. For covariance information, the most important sections are MF31 Average number of neutrons per fission,

A.

Libraries

MF32 Resonance parameters,

In this section, the different nuclear data libraries used in this work are presented. A given library concerns both the reactions themselves (cross sections, angular distributions, emitted particles and spectra, etc), but also their covariances. Details about the different libraries can be found in the presented cited papers, and some descriptions related to their available covariances will be given below. The basic nuclear data are usually described in a specific format, called ENDF-6 [78], followed by all the existing libraries, such as the US ENDF/B-VII.1 [12, 26], the OECD NEA JEFF-3.2 and JEFF-3.3 [13], the Japanese

MF33 Reaction cross sections, MF34 Angular distributions, MF35 Energy distributions, MF40 Activation cross sections. Based on the above sections, all libraries following the ENDF-6 format present similar structures, allowing similar processing and data extractions for use in different simulation codes. Existing tools such as from the IAEA, the NNDC or the NEA are extremely useful to visualize such data (see for instance JANIS [80]). 26

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quantities are above acceptable values (more than 5 % for k∞ ), as presented in Fig. 24. Results for the JEFF-3.2 library will therefore not be presented for other quantities, as the uncertainties are not complete. At the date of the writing of this paper, a new release of JEFF, version 3.3, is under preparation. This version is supposed to contain a larger number of covariance files, and especially full covariance information for the main actinides.

JEFF-3.2

The JEFF library and its latest release 3.2 is provided by the OECD Nuclear Energy Agency [13]. It contains a large number of neutron evaluations, but is rather limited in terms of covariance information. The JEFF library is also not available in the form of processed library to be used with CASMO-5 or the SCALE package. In the case of CASMO-5, it can be obtained from the CASMO-5 developers but for an additional charge. Regarding the number of covariances, the version 3.2 does not contain covariances for 235 U; in the case of 239 Pu, covariances are provided in the resonance region. An example for 238 U is presented in Fig. 16, where the covariance information extends from the thermal range up to 100 keV. Above this energy range, no uncertainties are provided.

3.

ENDF/B-VII.1

The American library ENDF/B-VII, release 1 [12] contains a large number of covariances for important reactions related to reactor applications. The first release of ENDF/B-VII was distributed with a set of covariances based on the BOLNA database, used for the assessment of uncertainties in innovative systems [10]. This latest release contains covariances for 190 isotopes. Apart from the TENDL library, it is one of the most complete in terms of covariances. Another advantage of the ENDF/B-VII.1 library is its extensive testing of covariances. The impacts of the covariances for the main actinides were assessed by different institutes, providing confidence in the results. Additionally, this library is available in many formats (for CASMO-5 or SCALE), which makes the use of the covariance files consistent with the cross sections. This issue was reported in Ref. [81]. An example for the 235 U(n,f) covariance is presented in Fig. 17. Additionally, as there is a large difference between the prompt and the total nubar uncertainties (they were independently evaluated), the prompt nubar covariances are not used in this work. Only the total nubar covariances are used, as it was the case in previous work (see for instance Ref. [82]).

4.

SCALE

The SCALE package [37] contains a large amount of covariance information. It is an application-oriented covariance library, created using a variety of sources and approximations for all isotopes in the SCALE cross section libraries. It contains uncertainties and correlations from ENDF/B-VII (released versions and pre-released versions), ENDF/B-VI.8, JENDL (for 240,241 Pu), LANL specific covariances and BOLNA covariances [10]. An example for the 239 Pu covariance total nubar is presented in Fig. 18. Significant differences with ENDF/B-VII.1 can be observed. In the present paper, covariance files from the SCALE-6.1 and 6.2 libraries are used. Participants used different beta version of this library as the final version was not available at the time the calculations. For TRITON calculations from UPM, version beta5 is used, whereas in the case of SHARK-X (CASMO-5 with ENDF/B-VII.1 cross sections), the covariances of the version beta4 are used.

FIG. 16. (Color online) Covariances for 238 U(n,γ) from JEFF3.2.

The processing of this covariance file (in the format MF32) is not easy because of the complexity and size of the MF32. Additionally, because it is not complete in terms of covariances, the JEFF-3.2 library is not a suitable candidate for uncertainty propagation. It will be used in this work in the SHARK-X method, applied as relative covariances to the ENDF/B-VII.1 cross sections, as provided in CASMO-5 . Examples will be presented for k∞ only, as the obtained uncertainties on the reactor 27

Nuclear Data Uncertainties . . .

FIG. 17. (Color online) Covariances for ENDF/B-VII.1.

5.

239

NUCLEAR DATA SHEETS

D. Rochman et al.

Pu(n,f) from

239

FIG. 18. (Color online) Covariances for 6.0.

tainties on the capture cross section are also very different above 500 eV.

JENDL-4.0

The Japanese library JENDL-4.0 [14] also contains a large amount of isotopes with covariance files, including the main actinides. Important differences with ENDF/BVII.1 covariances exist and were analyzed in Ref. [83]. Noticeable differences between JENDL-4.0 and ENDF/BVII.1 exist for the main actinides. The most important of them are presented here: •

•

Pu from SCALE-

•

239 Pu. For the fission cross section in the unresolved resonance range (from 2.5 to 30 keV), JENDL-4.0 indicates 2.5 % and ENDF/B-VII.1 1 %. The uncertainties for JENDL-4.0 are likely to be over-estimated with regards to the lower part of the resonance range. In the case of the capture cross section (also in the unresolved resonance range), the situation is inverted where JENDL-4.0 proposes about 5 % and ENDF/B-VII.1 16 %.

238

U. For the capture cross section, uncertainties are rather similar, with some artificial differences in the resonance range due to format conversion. Both evaluated data are based on the same measurement. The inelastic covariance files also present important differences uncertainties can be of 5 or 15 % for the same energy range. These cross sections are important enough to impact calculated uncertainties.

6.

TENDL-2014

The TENDL library is produced since 2008 and is based on TALYS calculations in the fast neutron range. In the resonance neutron range, it is using either experimental data, compiled data or resonance parameters from other libraries. If no information is available, systematics are used. Every isotope in the TENDL library is produced with its own set of covariance files [27], from thermal energy up to 20 MeV (see for instance Refs. [84– 86]). The applied method is the TMC approach, where all relevant nuclear data parameters are randomly varied to produce different cross sections and other nuclear

235

U. In the case of the fission cross section, whereas the uncertainties between JENDL-4.0 and ENDF/B-VII.1 are similar below 500 eV and above 100 keV, large differences can be observed in between (about 5 % for JENDL-4.0 and 1 % for ENDF/B-VII.1, with a peak at 2 keV). The uncer28

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TABLE VIII. Results of keff uncertainties due to cross section data for three typical types of plutonium ICSBEP benchmarks, propagated by linear expansion (sandwich rule; U = C · S · CT pmf1 pci1 pst-034-001 ENDF/B-VII.1 0.78 % 0.55 % 0.70 % 0.54 % 0.55 % 0.70 % JENDL-4.0

FIG. 19. (Color online) Covariances for JENDL-4.0.

235

shown in Fig. 20. Comparative testing of these initial files was performed using the OECD NEA software tool NDaST [89] for a large series of ICSBEP cases, whose nuclear data sensitivities are stored in a freely accessible online database. The resulting keff uncertainty propagated by simple linear expansion (sandwich rule; U = C ·S ·C T ) may then be found quickly and with minimal user effort. The complete series as calculated with JENDL-4.0 is shown in Fig. 21, whilst selected example cases are compared in Table VIII. Fast spectra cases of metallic composition, as represented by case PMF001 (Jezebel) are found to show some significant variation between the three libraries. An intermediate spectrum lattice benchmark, PCI001 (HECTOR) also exhibits important variations. In contrast, thermal solution case results, as exemplified by PST034-001 are in much closer overall agreement. Taking account of these types of results and their origin in reaction and energy can aid the nuclear data project with feedback for the validation of this important 239 Pu nuclide evaluation for use in safety applications.

U(n,f) from

data. These random cross sections can be used to produce standard deviations and correlations (for covariance files), but also can be formatted into so-called random nuclear data files. The advantage of using the random data files is that there is no loss of information, compared to the covariance files. Depending on the application and the simulation codes, both methods can be used. In the case of simulation codes for which the processing steps are well-known, such as SERPENT [87], MCNP [66] or DRAGON [88], the random ENDF-6 files can be processed and used in a Monte Carlo procedure (repeating many times the same calculation with different nuclear data libraries).

7.

B.

Isotopes and Reactions

The list of isotopes and reactions taken into account in the following calculations may vary from one library to another. But the most important isotopes and reactions are nowadays included in ENDF/B-VII.1 and SCALE for the covariance information. Table IX presents the isotopes provided with covariance information which are used in this work. If an isotope is not used, it means that it does not contain any covariance files. As presented, some libraries contain only partial data. For the fission products, the JEFF-3.2 library contains covariances for 22 isotopes (from Rb to I), whereas ENDF/B-VII.1 provides covariances for 33 isotopes. For each isotopes, the following covariances are considered when available (see possible restrictions in section II for each method):

Comparison of Uncertainties for Criticality Benchmarks

The differences of covariance files for the main actinides will be at the source of discrepancies for the reactor quantities presented later. At the evaluation level, efforts are needed to understand and explain such differences for the cross section uncertainties. As discussed previously, the comparative results of these main differences may be analyzed for common ICSBEP criticality (keff ) benchmarks. As a demonstration, the 239 Pu covariance data for fission reaction are

1. Cross sections from the thermal range up to 20 MeV: all provided channels (elastic, capture, fission, inelastic...), 2. ν: as mentioned earlier, as there can be inconsistencies between the total number of emitted neutrons per fission and the number of prompt neutrons, only 29

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FIG. 20. (Color online) Inter-comparison of the relative standard deviations as a function of energy, taken from the covariances for three evaluated files: JEFF-3.3T1 (not official library), ENDF/B-VII.1 and JENDL-4.0.

239

Pu(n,f)

FIG. 21. (Color online) Results of keff values with uncertainties propagated over a wide series of ICSBEP benchmarks. C/E values from MCNP6 and JENDL-4.0 are shown in red, uncertainties due to nuclear data is represented by the green bars, and experimental uncertainties by the blue bars.

30

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be given for the processing with NJOY, except regarding the treatment of covariances.

TABLE IX. List of the main isotopes which contain covariance information (with a “x”). The covariances for these isotopes are used for the uncertainty propagation. “x” means that covariances are provided from 0 to 20 MeV, “-” means that no covariances are provided, and “/” means that partial covariances are provided.“M.A.” means minor actinides, “F.P.” fission products and “T.S.L.” thermal scattering data for H in H2 O.

1.

U 238 U 239 Pu 1 H 16 O M.A. F.P. T.S.L.

x x x x x x x -

x x x x x x x -

/ / / -

x x x x x -

x x x x x x x x

the covariances for the total ν is considered, 3. χ MF5: values for the prompt fission neutron spectra are considered (see section III C 3 on CVX format files for specificities). 4. Fission yields MF8, see section IV F for more details.

2.

CALENDF

CALENDF [48, 91] is an important R-matrix analysis and alternative processing code with unique capabilities. As for PREPRO-2015, it is exclusively used in this work for data processing for the FISPACT-II inventory code. It also is extensively used for the TENDL library to generate statistical resonances in evaluations when no experimental information is available [92]. It focuses on producing probability tables and performing mathematical operations on these tables. Effective cross sections can then be extracted for the resolved and unresolved resonance range of each evaluation. As for the PREPRO package, an overlap of capabilities exists with NJOY in producing probability tables, but different methods are used towards different energy ranges and applications [93].

5. Other data, such as angular distributions, doubledifferential data, thermal scattering data are usually not considered in processed covariance files. In the TMC approach used for the full core calculations, these quantities are nevertheless considered.

C.

PREPRO

PREPRO is a powerful processing code for ENDF-6 file, originally developed at Lawrence Livermore National Laboratory [90]. It is a collection of 18 computer codes, designed to process the original data into formats to be used in specific applications. In this paper, PREPRO is used to process the ENDF-6 files for the FISPACTII code, in combination with CALENDF and NJOY. Each of the modules, or computer codes, is designed to fulfill a specific task and are used in a given sequence. The recent release consists of an important modernization of all codes, following the latest necessities linked to the ENDF6 format (portability, digit precision, latest FORTRAN). It also comes with a set of “best parameters”. Although PREPRO can perform similar tasks as NJOY, there are differences in the way the nuclear data are treated, leading to different processed quantities. It is therefore not a trivial step to correctly process basic nuclear data into usable formatted files.

ENDF/B-VII.1 SCALE JEFF-3.2 JENDL-4.0 TENDL 235

D. Rochman et al.

Processing

The processing of the basic nuclear data libraries (in the ENDF-6) format is a necessary and important step. It is often not a single step, but is performed using one or more processing codes, with specific input options. One of the most used formats is the ACE format for Monte Carlo transport codes. This format is created with specific NJOY modules. Even if the ACE format is not used in this work for the assembly calculations, the NJOY processing code is still used and some of its modules are also used here. Some specificities on the use of NJOY were already presented (see section II A 6), but other codes are used such as PREPRO for FISPACT-II or AMPX for SCALE [37]. The processing of ENDF-6 files can be crucial for the performances of deterministic codes, as microscopic cross sections are used to produce macroscopic cross sections in specific energy groups and different dilutions. These steps are often not in the public domain as for the CASMO-5 code. Therefore no details can be provided, except that it is based on different versions of NJOY (with proprietary modifications). In the following, no further details will

3.

NJOY for Covariances

In addition to the processing of cross sections, NJOY can be used to process covariances and provide them in a format for the SHARK-X system. As indicated in Fig. 1, SHARK-X modifies the cross section in some of the CASMO-5 modules. The covariance information used for the perturbation of the CASMO-5 library is obtained from existing covariance files as provided by the international nuclear data libraries such as JEFF, ENDF/BVII, JENDL or TENDL. The original format of such covariance files is the ENDF-6 format [78], and can be represented following different formalisms in the thermal and resonance range, or in the fast neutron range. In 31

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a simplified manner, two formats exists: (1) covariances for the quantities of interest (cross sections, angular distributions, energy distributions) and (2) covariances for parameters used in the reconstruction of cross sections. Both types of data are processed differently, and the outcome of such processing is given in a unique format, ready to be used by SHARK-X to generate random data. These original covariance files are processed with NJOY [8] to produce the so-called COVERX format. Different modules are used and the most important ones are RECONR for cross section reconstruction, followed by BROADR for Doppler broadening and GROUPR to produce group cross sections. Then, depending on the initial covariance format, the module ERRORR is used with different options. Possibilities for the input parameters of NJOY are given in Ref. [8]. The final step consists in producing the covariance information in the COVERX format with the NJOYCOVX program [94]. For details on the COVERX format, see Ref. [95]. Once the covariance files are translated into the COVERX format, they are expanded in a database for convenient handling. Not all covariance information are used by CASMO-5 and only some reactions are converted into a local database, as defined in the ENDF-6 format by their MT number:

D. Rochman et al. 4.

AMPX

The AMPX code is a modular code system, developed by the Reactor and Nuclear Systems Division (RNSD) at Oak Ridge National Laboratory (ORNL), for processing ENDF-6 format [78] evaluations into multi-group (MG) and continuous energy (CE) cross sections data ready to be used within transport and depletion codes provided in the SCALE tool suite [7, 37]. Production of MG cross section data was the first aim of AMPX, latter adding the generation of CE cross sections. For neutron transport calculations, CE cross section data consists in up to three different parts: • Point cross section data, which represent cross section data as pairs of points (σ,E) at incident particle energy. As long as cross section data are provided in different representation, such as resonances and interpolation laws, here they are transformed into a set of pairs (σ,E) through linearization, keeping the former representation with a given accuracy. • Probability tables, when Unresolved Resonance data are provided, which represent stochastic behavior of cross section in that region due to the fact that resonances are so closely spaced in energy that it is either impossible or impractical to resolve each resonance. Thus, cross section value probability distributions are provided in order to reproduce such mentioned behavior.

• Fission cross section (MT=18), • Average neutron per fission (MT=452), • Neutron absorption cross sections σ(n,γ) (MT=102), σ(n,p) (MT=103), σ(n,d) (MT=104), σ(n,t) (MT=105), σ(n,3 He) (MT=106) and σ(n,α) (MT=107),

• Scattering kinematics, which store differential cross section data: outgoing energy particles and angles as a function of incident particle energy. These parts can be obtained by running different modules, as shown in Fig. 22, and then, assembly them into one single file per isotope. For generating point cross section data, the following modules shall be used in given order:

• Total scattering cross sections for elastic and inelastic (MT=2 and MT=4), • (n,2n) cross section (MT=16), • Average fission spectrum χ,

1. Polident creates point-wise cross section at 0 K from evaluated ENDF-6 files, reconstructing cross section data from resonance parameters, if provided, and combining them with background cross sections.

• Decay data: decay constants (MT=504, 505 and 514), energy per decay (MT=500 to 503 and 510 to 513),

2. Broaden performs Doppler-broadening of pointwise cross section provided by Polident.

• Fission yields (MT=454 and 457). One should notice that some limitations exist in the use of the COVERX format related to the quantity χ, as noticed in Ref. [44]. In the existing implementation of this format, the covariance information for the prompt fission neutron spectrum is independent of the neutron incident energy. In practice, only one covariance matrix is then processed into COVERX. This means that the user needs to make an arbitrary choice as the original covariance information can be given for different incident neutron energy. In the case of thermal systems, the choice of the thermal energy might be preferred.

3. Tgel reconstructs total, capture, inelastic and absorption cross sections from their partials in order to keep the consistency of such reaction cross sections. For generating probability tables, these modules shall be run: 1. Purm generates the probability tables from the Unresolved Resonance parameters, if provided, for each desired temperature. 32

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FIG. 22. (Color online) Flowchart of AMPX modules for processing ENDF-6 format files into SCALE format.

both outcomes, point-wise cross section data and thermal scattering data.

2. Purm up corrects the probability tables generated by Purm depending on the evaluation content. For generating kinematics data, these modules shall be called in the following order:

D.

1. Y12, is used to generate kinematics data for neutron scattering and gamma-production scattering, producing double-differential data in tabulated form.

Missing Data

As presented, a large part of the basic nuclear data are interpreted by the processing codes and therefore can be also used by the transport and depletion codes. Some data are nevertheless not considered for the uncertainty propagation for at least one of the two following reasons: (1) the data do not have covariance information which can be processed, or (2) the data are hardcoded in the code of interest and can therefore not be changed. Examples of such data are thermal scattering data (for instance for H in H2 O), delayed neutron fractions, and to some extent fission yields. For the fission yields, practical solutions exist (see section II A 2 or Refs. [28, 96]) but these solutions (repeating many times the same calculation, changing each time the fission yields data) are not easily applicable by users with limited experience on nuclear data. Additionally, the lack of official full covariance information for fission yields leads to a variety of solutions. In the case of thermal scattering data, if a solution also lies in the TMC method [97, 98], random external thermal scattering data files are also not easily handled by users. Finally, in the case of delayed neutron groups and yields, they are often located within the transport codes (as in the case

2. Jamaican converts the double differential pointwise distributions into marginal probability distributions in angle and conditional probability distributions in exit energy. All above generated data shall be merged into one single file per isotope, so Platinum is called to compose the final CE library. Additional modules/codes are used in between which take care of selecting e.g. cross section to broaden or selecting energy range to apply up-scattering (See AMPX manual for further detail). Thermal scattering data are provided for some moderators found in compound states, like Hydrogen in water (H2 O). Their processing, to obtain Thermal Scattering Libraries (TSL), is the same as for scattering kinematics data in CE libraries, using Y12 and Jamaican. New broadened pointwise data should be obtained for the corresponding temperatures for which thermal scattering data are provided, using Polident and Broaden again. Platinum combines 33

Nuclear Data Uncertainties . . .

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of CASMO-5 ), and often repeated at different places, which makes their modification difficult. Also, the decay data are not considered here as a source of uncertainties. It was already presented in many publications that the decay data uncertainties generally have a limited impact on the quantities of interest (with the exception of the 241 Am capture branching ratio). The lack of correlations in the evaluated decay data files, as for the fission yields, can also lead to different interpretations. These hidden data are known to have an impact on reactor calculations. In the case of fission yields (see later for the impact of the lack of evaluated correlations), many studies can be found, but their impacts greatly vary depending on the source of covariance. In the case of thermal scattering data, also less studies exist (see for instance Refs. [97, 99]), there is nowadays a study group at the OECD/NEA dedicated to updating these data and their covariance information [100].

A.

Assemblies k∞

One of the first parameters to study is the k∞ of the different assemblies. The results are presented in Fig. 23 for the five different considered systems. Not all the participants were able to calculate the uncertainties for all the systems, but at least three methods were applied for each system. As mentioned before, the presented uncertainties are obtained using different methods, different libraries for nominal values and different libraries for the nuclear data covariances. It is therefore not expected to obtain similar results. These results are a reflection of what would happen if each institute was asked to calculate the uncertainties for a specific system, using its own method almost independent of each other. The following observations can be made: • A first observation is that although the calculated uncertainties are all within the same range (less than 1 %), they can vary by a factor 2. These differences mainly originate from the use of different nuclear data libraries and their covariance files. At the beginning of irradiation, the uncertainties due to the prompt ν for 235 (MT=452) are sensibly different between ENDF/B-VII.1 and SCALE-6.2: 0.61 % and 0.34 %, respectively.

As presented in the previous sections, the comparison of results at a macroscopic level will be influenced by many criteria: the nuclear data library, with different nominal values and covariance information, but also the way these libraries are interpreted and processed. The uncertainty propagation methods themselves are also influencing the final quantities. Therefore it will not be straightforward to unequivocally identify the origin of the observed differences at the level of global parameters such as k∞ , macroscopic cross sections or isotope inventories. The full calculation path from basic nuclear physics to applications needs to be under control and understood.

IV.

D. Rochman et al.

• A second observation is that the trend as a function of burn-up can also differ (from decreasing to increasing, with some constant behavior for the PWR UO2 case). • In the case of the assemblies with Gd, the shape of the uncertainties can be affected at about 10 MWd/kgU (PWR and BWR UO2 ), but the effect is not strong for the BWR MOX case.

RESULTS

Different results for specific assembly and full core quantities are presented in this section. A large amount of results can be obtained from the calculations, and it is not realistic to present all of them. The following quantities are then presented and discussed for the assemblies: k∞ , number densities for actinides, group cross sections and pin power distributions. The uncertainties for the number densities of fission products are discussed in a separated section, due to the large differences in methods. Again, it is not specifically expected to obtain the same uncertainties between the participants, as many calculational parameters can be different, even if the definitions of the assemblies are the same. In the case of the uncertainties on the power distribution for a full core, the results are also separately presented (see section IV H). Other studies performed by the authors of the present papers were presented in other publications, using similar methods as the ones presented here, see for instance Refs. [101, 102]. Also, the present study can be compared to some extent to the ongoing studies on the benchmarks for uncertainty analysis in modeling (UAM) [56].

It should also be noticed that the JEFF-3.2 library can not be used for reliable uncertainty results, as shown in Fig. 24, for the main reason that it does not contain enough covariance information. With the JEFF-3.2 library, the uncertainties on k∞ are too high (about 8 % for all systems). These large uncertainties for JEFF-3.2 are most likely due to 90 Zr which contains an unrealistically large thermal capture uncertainty. It should be nevertheless noted that at the time of the production of JEFF-3.2, the available NJOY version for the processing of the covariance information contain an incorrect treatment of the specific format used for 90 Zr. This malfunction was discovered in 2016 [103]. It was indeed verified that the corrected version of NJOY provides a processed uncertainty closer than the expected value (between 20 and 30 % for the thermal capture uncertainty). This points out once more the importance of the processing step and the danger linked to the use of a single processing tool. In a sake of fairness towards the JEFF library, it can be noticed that the JEFF-3.2 library contains relevant covariance information for the resonance range of 238 U 34

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0.9

0.9 PWR UO2 + Gd

PWR UO2 0.8

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Δk∞ (%)

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D. Rochman et al.

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CASMO + ENDF/B-VII.1 (PSI) CASMO + SCALE-6.2 (PSI) DRAGON + JENDL-4.0 (UU) DRAGON + ENDF/B-VII.1 (UU) TRITON + SCALE-6.1 (GRS) TRITON + SCALE-6.2 (UPM)

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0.8 0.7 0.6 0.5 0.4 0

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Burn-up (MWd/kgU) FIG. 23. (Color online) Uncertainties in percent for k∞ for the 5 different considered assemblies, obtained with different codes and libraries.

and 239 Pu in the context of PWR and BWR calculations. Such covariance files are necessary to obtain a full assessment of uncertainties.

B.

the actinides, the effect of the fission yields is secondary compared to the variations of the actinide cross sections. Results for some actinides are presented in Figs. 60 to 62 in Appendix A.

Assemblies Number Densities

• Apart from the uncertainties on 234 U, the calculated uncertainties for specific actinides follow the same trends among the different methods. For this isotope, the major source of differences between the results comes from the use of different nuclear data libraries and their covariance files. For instance,

In this section, only the inventory for actinides (number densities) will be presented. Concerning the fission products, there uncertainties can be strongly affected by the variations of the fission yields, which are not always considered by the participants. In the case of 35

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PWR UO2

8.0 4.0 2.0

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Burn-up (MWd/kgU) FIG. 24. (Color online) Uncertainties in percent for k∞ for the 5 different considered assemblies, obtained with CASMO-5 and different libraries.

• As shown in Fig. 61, SHARK-X tends to overpredict the concentration uncertainty of 239 Pu at the end of cycle: 2.2 % vs. 1.7 % obtained with the GRS and UPM methods for the UO2 assembly. The overprediction of SHARK-X with respect to the other codes is explained by neglecting the implicit effect, i.e. the perturbation of self-shielding (see section IV J), which is especially important for 238 U(n,γ) and consequently for the 239 Pu build-up. Considering the implicit effect, SHARK-X predictions for the 239 Pu uncertainty are in line with the other solutions as shown in Sec. IV J.

ENDF/B-VII.1 leads to an uncertainties of 1.7 % in the case of the PWR UO2 at 40 MWd/kgU, whereas the JENDL-4.0 library leads to 5.9 %. • Important variations of amplitudes exist (see 236 U, 237 Np or 244 Cm, and to some extent 239 Pu), which are most likely related to the different covariance libraries. • As in the case of the k∞ uncertainties, the spread of the uncertainties for specific isotopes can be larger than the calculated uncertainties themselves. 36

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NUCLEAR DATA SHEETS

and different libraries are used. The origin of the differences for the impact of 235 U lies in the covariance library. The effect of the Gd can be seen especially for 238 U, with a change of slope around 10 MWd/kgU. The sensitivity to 238 U is showing different shape, depending on the library: decreasing then increasing for ENDF/B-VII.1 and decreasing for JENDL-4.0. This was already observed a few times. In Refs. [22, 85, 96], the change in slope for 238 U was observed and related to the effect of the capture cross section. Alternatively, in Ref. [104], the change in slope was not observed, as a different nuclear data library was used.

Assemblies Two-group Cross Sections

The assembly group cross sections are presented in Figs. 63 and 64 in Appendix A. Similar to the k∞ and number densities, important variations (factor 2) can appear between the results, but the amplitudes are still within the same range. Again, these differences are expected due to (at least) the use of different covariance libraries.

D.

Assemblies Pin Power Distribution

The uncertainties on the assembly pin power distributions are presented in Fig. 65 in Appendix A at the end of irradiation (40 MWd/kgU), where the impact of nuclear data is supposed to be high. Still in this case, all uncertainties are relatively small (less than 0.7 %) and the agreement among the participants is rather good. There is a noticeable difference between the BWR and PWR assemblies: the power distributions for the BWR assemblies present higher uncertainties than for PWR assemblies. Although a complete sensitivity study is outside the scope of this paper, a simple comparison of results is showing that the major source of uncertainties for the power distributions is the capture cross section on 238 U. This cross section is at the origin of the differences between the PWR and BWR results.

E.

D. Rochman et al.

F.

Impact of Fission Yields

The fission yields, like any other nuclear data, can have impacts on assembly quantities. Different types of fission yields are usually given in the evaluated files (independent and cumulative), and depending on the depletion codes, one type or more are used during the assembly burn-up calculations. In the case of the Monte Carlo code SERPENT or in XSUSA, only the independent yields are used. In the case of CASMO-5, the majority of the fission yields are independent, although some are cumulative. Additionally, there are no agreed methods to sample fission yields, independently of the type of yields, as presented in Ref. [105]. Therefore some participants, such as GRS, decided to not sample fission yields. As presented in Ref. [82], the methods of sampling the fission yields have large impacts on the calculated uncertainties. The methods used in SHARK-X are presented in section II A 2, but other methods exist:

k∞ Partial Contributors

With the Monte Carlo method applied to propagate the uncertainties, it is relatively straightforward to obtain some estimations of the sensitivities to specific nuclear data. A simple method consists in repeating the same calculations, but varying only the isotope or reaction of interest. This approach can be performed if the calculation time does not exceed some practical limits. Other methods, using correlation coefficients and similarity indexes can be found in the literature. Examples for the calculations of sensitivity vectors, as well as providing feedback to nuclear data, can be found in Ref. [83]. In the present case, as the assembly calculations can be performed in acceptable calculation time, the Monte Carlo method is chosen; it gives access to the partial contributions of specific nuclear data for all calculated quantities. Such results can represent a large amount of data and only a selection of results are presented here for k∞ : see Fig. 66 in Appendix A. It is important to realize that the Monte Carlo methods (for uncertainty propagation) do no provide sensitivity vectors as in the case of the perturbation methods. When randomly modifying the nuclear data as inputs, the whole cross sections are changing, following correlations between energy groups. Therefore the sensitivity vectors as used in the so-called “sandwich rule” can not be obtained. In the comparison of Fig. 66, different codes

• CEA method: At the CEA Cadarache, a method is under development to combine the Brosa model or the Wahl’s systematics for the distribution of fission fragments before neutron emission with the saw-tooth curve, representing the number of emitted neutrons as a function of fragment masses. Normalization to evaluated fission yields is performed during the process. More details can be found in Ref. [106]. • SCK/UPM method: At the SCK Mol, Belgium and the UPM University, Spain, a method is developed coupling the GEF code with a Bayesian/general least-squares method to mathematically include experimental (or evaluated) information, see Ref. [107] and section II A 8. • ORNL methods: At the Oak Ridge National Laboratory (ORNL), two methods are considered: (1) the Wahl’s systematics and its parameters are used to generate random yields (based on random parameters) and covariances are extracted; (2) a Bayesian approach based on constraints from the mass yields, the conservation of the mass and charge number [108]. 37

Nuclear Data Uncertainties . . .

NUCLEAR DATA SHEETS

• Normalization 1: as defined in section II A 2, different normalizations are applied to the random fission yields with Eqs. (1) to (4). More details can be found in Ref. [82].

• PSI methods: see section II A 2 • Additional efforts are focusing on better theoretical description of the fission process. Many observables (such as fission yields) can be extracted from such models, see for instance the effort at the Uppsala University to combine the GEF and TALYS codes [109], and the FIFRELIN development at the CEA [110, 111]. Once these efforts lead to reasonable fission yields, Monte Carlo and data assimilation methods could be used to derive fission yield covariances.

• Normalization 2: the correlations between fission yields are directly extracted from GEF calculations, i.e. the uncertainties come from ENDF/B-VII.1, but the correlations from GEF. • Normalization 3: as presented in Ref. [33], the GEF results are updated using a Bayesian method to reproduce the ENDF/B-VII.1 fission yield uncertainties. In this case, both (updated) uncertainties and (updated) correlations come from GEF.

As long as the nuclear data community is not recommending a set of covariance files for fission yields (or a set of random fission yields), the user community is applying “ad-hoc” solutions, leading to a spread of calculated uncertainties.

1.

• Normalization 4: a simple normalization on the random yields to a sum of 2 is applied. The uncertainties come from ENDF/B-VII.1, and the sampling is performed without correlations. • Normalization 5: the method of the SANDY code is used with the fission yield uncertainties from from ENDF/B-VII.1 (see section II A 8).

On k∞

The impact of the fission yields on k∞ was already presented in the previously cited references. Two examples for the PWR MOX and UO2 assemblies are presented in Figs. 25 and 26. These uncertainties are obtained using

For consistency, all the CASMO-5 calculations are performed at 19 groups, which is the default number of groups. It was checked that in the case of nominal calculations, similar results were obtained between 19, 56, 95 and 586 groups (586 groups is the maximum number of groups given that the nuclear data libraries are provided in 586 groups). Additionally, it was checked that for cross section uncertainty propagation, 19 groups provided a similar answer than 586 groups. In the case of fission yields, there is also no effect on the calculated uncertainties due to the number of groups for transport, as presented in Fig. 26, where the uncertainty propagation for the fission yields of the four major contributors are performed with CASMO-5 for three different groups: 19, 95 and 586 (the bias induced by the group collapsing is less than 50 pcm at 10 MWd/kgU). The CASMO-5 calculations are also compared to SERPENT calculations for the PWR UO2 assembly. SERPENT is using pointwise data and can be considered as a reference in this case. As observed, the results using 19 groups are similar to the reference case. The normalization 5 provides higher uncertainties for the UO2 fuel compared to the MOX case, related to higher fission yield uncertainties in the case of 235 U.

0.5 PWR MOX (FY only)

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Burn-up (MWd/kgU) ENDF/B-VII.1 ENDF/B-VII.1 + ENDF/B-VII.1 + ENDF/B-VII.1 + ENDF/B-VII.1 +

D. Rochman et al.

GEF normalization 1 normalization 2 normalization 3 normalization 4 normalization 5

FIG. 25. (Color online) Uncertainty on k∞ for the PWR MOX assembly due to the fission yields of 239,241 Pu and 235,238 U for different types of correlations and normalizations (see text). The number of groups for the CASMO-5 transport is 19.

2.

On Inventory

The fission yields have an impact on the inventory of isotopes during burn-up and cooling down time. This effect is minor on actinides, but is of prime importance for fission products. Because of the multiple methods as presented in the previous section, large differences for the uncertainties on fission products were observed among participants. An example for the PWR UO2

CASMO-5 and the SHARK-X method. Still, for a given library (being the source of uncertainties only as there are no correlation between yields), the uncertainties for k∞ can differ in amplitude and in trend for different normalization and correlations. In this figure, the different normalization are: 38

Nuclear Data Uncertainties . . .

PWR UO2 (FY only) ENDF/B-VII.1

0.2

Δk∞ (%)

NUCLEAR DATA SHEETS

TABLE X. Comparison for the fission product inventories between the FISPACT-II calculations (UKAEA with CASMO-5 fluence) and SHARK-X (CASMO-5 PSI). The uncertainties are due to random fission yields for 235,238 U and 239,241 Pu from the BMC method for both calculation codes. The considered system is the BWR MOX assembly and results are presented at 20 MWd/kgU.

0.1 normalization 3 normalization 5

Isotope Nominal uncertainty skewness atom/cc ratio PSI PSI PSI/UKAEA ×1017 157 Gd 1.8 0.6 1.25 0.124 ± 28.1 % 155 Gd 1.5 0.7 0.84 0.071 ± 20.8 % 156 Eu 1.5 0.6 0.89 0.526 ± 21.3 % 152 Sm 1.0 0.8 0.41 37.0 ± 13.2 % 150 Sm 1.0 0.9 0.21 59.5 ± 8.2 % 149 Nd 1.0 0.9 0.23 0.01 ± 8.4 % 139 Nd 1.1 0.7 0.23 263 ± 3.0 % 137 Cs 1.0 0.9 0.23 291 ± 2.5 % 131 Xe 1.0 1.0 -0.04 131 ± 7.1 % 127 Te 0.8 1.1 0.78 12.7 ± 25.1 % 101 Ru 1.0 0.8 -0.17 267 ± 2.5 % 91 Y 1.0 0.8 0.08 12.3 ± 4.7 % 89 Sr 1.1 0.9 0.17 7.50 ± 7.4 %

0.0 0

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Burn-up (MWd/kgU) PWR UO2 (FY only) ENDF/B-VII.1 + normalization 3

Δk∞ (%)

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19 groups 95 groups 586 groups SERPENT pointwise

0.1 0.0 0

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D. Rochman et al.

40

Burn-up (MWd/kgU) FIG. 26. (Color online) Uncertainty on k∞ for the PWR UO2 assembly due to the fission yields of 239,241 Pu and 235,238 U for number of energy groups. The fission yields are from ENDF/B-VII.1. Top: uncertainties obtained with two different methods. Bottom: uncertainties obtained with CASMO in different energy group structures.

section II A 7, the UKAEA method makes uses of the neutron fluence calculated by CASMO-5, these neutron fluences being the same as in the PSI calculations. Therefore the comparison between the calculated uncertainties can directly provide insights on how these fission yields are treated by the two codes. The comparisons for some fission product inventories are presented in Table X, where the nominal values are compared as well as the calculated uncertainties and skewness. The results concern the BWR case with a MOX fuel at 20 MWd/kgU, and the random fission yields are taken from the TENDL library, using the BMC method. The number of random calculations is 300 in both cases. From Table X, one can see that the nominal number densities for the selected fission products are in general in good agreement. Some differences exist and should be later investigated in a dedicated study, but the agreement is rather good for the most produced fission products. Concerning the calculated uncertainties, there seems to be a systematic difference (the SHARKX uncertainties being larger than the ones from FISPACT-II), but the differences are also relatively small for fission products which are predominantly produced. One should realize that this type of comparison is not straightforward, as it was shown in Refs. [44, 112] for nuclear data uncertainties on k∞ and requires a extensive control on all the processing steps. The detailed comparison of these two methods will be performed in a dedicated study.

case is presented in Fig. 27 for three important fission products. It is therefore important to establish recommended method for fission yield sampling, otherwise the calculated uncertainties on fission products largely vary, undermining the trust in these values. As presented in Fig. 27, small uncertainties were obtained by not considering fission yield uncertainties, which lead to unrealistically small uncertainties on inventory.

3.

On Inventory (TMC Comparison)

Among the different methods of calculations and the participants, two methods are based on the TMC/BMC approach and are using similar “random” fission yield files. The SHARK-X method for the variations of fission yields can use the random fission yields as produced in the Bayesian Monte Carlo method (see section II A 2). Alternatively, FISPACT-II can use the same random fission yields (see section II A 7). Additionally, FISPACT-II is not a neutron transport code and can not solve transport equations, but is a robust inventory tool. As explained in 39

95

Mo

1.00 0.10 0.01 0

10

20

30

Burn-up (MWd/kgU)

2

10

PWR UO2

99

D. Rochman et al.

Tc

Uncertainty (%)

PWR UO2

NUCLEAR DATA SHEETS

Uncertainty (%)

Uncertainty (%)

Nuclear Data Uncertainties . . .

1.00 0.10 0.01 0

Cooling (years)

10

20

30

2

Burn-up (MWd/kgU)

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Cooling (years)

FIG. 27. (Color online) Uncertainty on the inventory for 3 fission products for the PWR UO2 assembly from 3 participants. The PSI method uses the normalization method 3.

G.

Convergence, Correlation and Skewness

TABLE XI. Indications of the standard errors for the different moments of a distribution, as a function of the number of samples.

As in many Monte Carlo processes, it is crucial to extract information from probability density function showing signs of convergence as a function of the number of iterations. Other quantities are also of interest, such as the correlation between random calculated quantities. Therefore the calculated uncertainties in terms of standard deviations, main subject of this paper, are only one aspect of the total information obtained during the process.

1.

Nbr samples 10 50 100 200 300 500 1000 2000 5000

Convergence

Fig. 28 presents an example of a calculated distribution for k∞ in the case of the PWR UO2 assembly. Only the 238 U(n,γ) cross section is changed at 10 keV from one sample to the other. The k∞ is calculated with CASMO5 and the ENDF/B-VII.1 library. A large number of samples are taken in order to observe the variations of the different moments and their convergence. In practical cases, the number of samples considered varies between 300 and 1000, depending on how long a single calculation is. Apart from the latest developed codes, many of the reactor codes are not easily run in a parallel mode to take advantage of computers or clusters with many cpus. But in the case of the uncertainty propagation, some degree of parallelization is relatively easy since all the calculations are independent of each other. Then sets of calculation with sampled inputs can be run on different cpus and the outputs are simply collected at the end. Still, in practice such method can not be faster than a single calculation on a single cpu. Therefore no more than a thousand set of calculations are usually used. In the present example, the burn-up value was intentionally taken low to shorten the calculation time. In Fig. 28, the three first moments of the k∞ distribution are presented, and the gray bands represent the standard errors on the plotted quantities. As the convergence of the average and standard deviation is usually considered, it is also of interest to consider other moments, such as the skewness and the kurtosis. Table XI provides indications on the standard errors as a function of the number

Standard Error Average/σ skewness kurtosis 31 % 0.7 1.3 14 % 0.3 0.7 10 % 0.25 0.5 7% 0.17 0.35 6% 0.14 0.28 4% 0.10 0.19 3% 0.08 0.15 2% 0.05 0.11 1.4 % 0.04 0.07

of samples for these moments [113]. It is then worth to realize that for about 500 samples, a skewness of less than 0.1 does not necessarily indicate a skewed distribution. 2.

Correlation

The calculated quantities such as k∞ , macroscopic cross sections, or inventories present a certain amount of correlations due to the sampled inputs and to the physics included in the simulation tools. This was presented in many references (see for instance Refs. [85, 114] for burnup calculations). As presented in Fig. 29, strong correlations can exist. Such correlation can be quantified by the Pearson correlation ρ defined as follows for two quantities x and y n 

ρ=

i=1

(xi − x)(yi − y) (n − 1)sx sy

(17)

with xi and yi the random realization of x and y, x and y the calculated average of x and y and sx and sy the two standard deviations. Examples of such correlation factors in the case of the TMC method can be found in 40

Nuclear Data Uncertainties . . .

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D. Rochman et al.

Average

1.2125 0.2 Average

1.2174

0.1

Scattered data

counts/bin (a.u.)

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1.2122 1.2100

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0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 0 150

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400 1000

400 1000

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FIG. 28. (Color online) Example of the convergence of the three first moments of the k∞ distribution for the at 4 MWd/kgU PWR UO2 , changing the 238 U(n,γ) at 10 keV nuclear data from ENDF/B-VII.1

culation - core calculation” at once, without approximation in between (the TMC approach).

Refs. [97, 115] for the variations of the copper isotopes and of the thermal scattering data. In the case of the BWR MOX assembly, the correlations between fission products, due to the variations of fission yields in the SHARKX method are presented in Fig. 30 at 20 MWd/kgU and at 7 years after shutdown. It can be seen that strong correlations exist between fission products, due to the variations of the fission yields. These correlations appear at the beginning of irradiation and stay more or less constant until the shutdown time. Even during cooling, the remaining isotopes (which have half-life long enough to be observed) exhibit the same correlation. This has an important impact in case of propagation of uncertainties to the next “step”: to a core simulator. The calculated quantities at the assembly level can be used as inputs for core simulators such as SIMULATE or VERA [116, 117]. For nuclear data uncertainty propagation, a two-step approach can be foreseen: first uncertainty propagation as in this paper, collect the random calculated quantities and simplify them by averages, standard deviations and skewnesses. The second step would consist in sampling again the calculated quantities and perform the core calculations with these inputs. This happens in practice since the assembly and core simulators are two distinct codes, sometimes used for different purposes. In this case, neglecting the parameter correlations will lead to a different results compared to the one-step method: loop over “nuclear data - assembly cal-

3.

Skewness

The skewness is the indication of a non-symmetric distribution. A value of 0 indicates a symmetry, whereas a positive or negative value quantifies an asymmetry (towards values higher or lower than the average, respectively). One of the first observation of skewed distributions for k∞ was presented in Ref. [41]. In principle, a positive skewness for a quantity such as k∞ indicates a higher probability for high k∞ , compared to the usual understanding of the 1σ probability, which is related to a Gaussian characteristics. So the skewness is an important quantity with safety implications, which should not be ignored. With a Monte Carlo uncertainty propagation scheme (as performed in this paper), this quantity can always be obtained. It certainly depends on the input parameters and their covariances. Figure 31 is presenting the skewness of the number density for a quantity called “Pseudo fission products” in CASMO-5, separately varying two types of nuclear data: the 238 U(n,γ) cross section at 10 keV, and the 235 U fission yields. As observed, different skewnesses are obtained, showing the importance of the input data in the output distribution. In the case of a non-zero skewness and non-zero 41

Nuclear Data Uncertainties . . .

NUCLEAR DATA SHEETS

239

Pu (g/cm3 )

ρ = −0.99

1.1· 10−4

1.0· 10

Cm (g/cm3 )

U-239 Pu

−4

2.056· 10−2

3.8· 10−6

244

were performed with random nuclear data sets for 235 U, U, 239 Pu, and for the H in H2 O thermal scattering. For each of these 508 runs, a different random number seed was used in MCNP. For all the runs it was checked whether the MCNP6 fission source entropy indicator gave a warning message. In 146 out of 508 runs, this was the case. Therefore, only the remaining 362 runs were accepted for further analysis. It was checked explicitly that no bias was introduced by this procedure. b. Estimates of standard deviation The calculation of standard deviations in a Monte Carlo eigenvalue calculation is a difficult matter. It is well known effect that correlations between generations of neutrons lead to incorrect calculations of the standard deviations (see e.g. Refs. [119, 120]). Several efforts have been undertaken to develop improved algorithms [121, 122], but none of these methods has so far been accepted as a new standard. Recently, a systematic study has been performed to show the effects of improper source convergence, and of cycle to cycle correlations [123]. In the present work, the estimate of standard deviation plays an important role, through Eqs. (14) and (16). Therefore, 389 extra runs were performed with constant nuclear data for all the runs, but with varying random number seed (the 389 seeds used were identical to the ones for the first 389 of the 508 runs with nuclear data variation). Therefore the only difference between these runs is statistical, and the true standard deviation can be calculated based on the output of these runs. Also for these runs it was checked whether the source was converged before tally data were accumulated. Based on this same criterion as used in the previous section, 274 runs were accepted for further analysis. The average of the MCNP standard deviation in keff of the 362 runs was 3.42 ± 0.02 pcm. On the other hand, the real standard deviation observed in the 274 keff values was 3.77 pcm. The MCNP calculated standard deviation therefore underestimated the real standard deviation by 9%. It was not possible to establish the statistical accuracy of this overestimation, given the limited statistics. For the standard deviation of local power results, the results are shown for tally F7 in Fig. 32, along a horizontal line through the reactor core. In the center of the reactor core, the standard deviation of each of the 362 runs was of the order of 5%. Away from the center the standard deviation was higher, increasing to roughly 18% near the edge of the core. The ratio of the MCNP calculated standard deviation to the real standard deviation is close to one, and varies mostly between 0.95 and 1.1. The average ratio along this line through the reactor core is 1.019 ± 0.040. It can be concluded that, in this case, there is no significant bias in the MCNP calculated standard deviation for the local power results. If anything there is some indication that the MCNP standard deviation is a slight overestimation. Because the differences between the MCNP standard 238

1.2· 10−4

238

2.058· 10−2 2.060· 10−2 U (g/cm3 )

238

ρ = 0.96

3.7· 10−6 3.6· 10−6 3.5· 10−6 243

3.4· 10−6 6.8· 10−6

Am-244 Cm

7.0· 10−6 7.2· 10−6 243 Am (g/cm3 )

FIG. 29. (Color online) Examples of correlation between number densities, changing the nuclear data for 238 U for the PWR UO2 at high burn-up. ρ represents the Pearson correlation.

correlation between quantities, the correct sampling becomes difficult and uses methods which are usually not applied in the nuclear field (see Ref. [118] for details). Such methods should still be considered for a general capability on random sampling.

H.

Full Core - Power Distribution 1.

D. Rochman et al.

Preliminary Tests

a. Source convergence In Section II A 5 it was explained that the Fast TMC method is very efficient. For eigenvalue calculations, however, there is the issue of source convergence. When one splits a big run into M small runs, one has to ensure that each of the M runs has a converged source, before tally data are accumulated. Since in general this cannot be predicted on beforehand, source convergence was appraised a posteriori, using the MCNP6 built-in indicator, the fission source entropy. The model was run for some time with fixed nuclear data, to obtain a reasonably converged fission source with which all subsequent simulations could be started. All the subsequent simulations were done with 10 inactive cycles and 90 active cycles of 4 × 106 histories. In total 508 runs 42

Ce

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1

NUCLEAR DATA SHEETS

103

Nuclear Data Uncertainties . . .

7 years cooling

FIG. 30. (Color online) Correlation matrix between fission products in the BWR MOX assembly due to the variations of fission yields. Left: at 20 MWd/kgU, Right: at 7 years after shutdown.

0.2

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U(n,γ) 235 U FY

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FIG. 31. (Color online) Examples of two different skewness for the same quantity, the number density of the pseudo fission product, changing the 238 U(n,γ) cross section at 10 keV in one case (red) and the 235 U fission yields in the other case (blue).

-100

0 Horizontal position [cm]

100

0 200

FIG. 32. (Color online) The ratio of the MCNP standard deviation of F7 for a single run, and the real standard deviation, on a horizontal line through the reactor core.

deviation and the real standard deviation are small in this case, the results reported in this paper are all based on the estimates using the MCNP calculated standard deviations. Nevertheless it was checked explicitly that all results using the real standard deviations did not deviate more than 1σ from the ones reported here. c. Output Convergence It was stated in Section II A 5 that the Fast TMC method is highly efficient, and can yield uncertainty results without increasing the amount of CPU time needed. Later, in Section IV H 1 a, it turned out that this statement does not hold true for eigenvalue calculations, because one has to ensure that the source is converged for all M short runs, before the tally data can be accumulated. The efficiency of the method is therefore less than 100% for eigenvalue calculations, and the amount of time spent in source convergence will be roughly proportional to M , the number of small runs. The interesting question then is, how fast do the uncer-

tainty results converge as a function of M ? For the keff values of the short runs, the uncertainty due to nuclear data was estimated as 740 pcm. The statistical uncertainty associated with this estimate can be estimated as 28 pcm, by using the ’variance of the variance’ as explained in Ref. [124]. This result was obtained on the basis of 362 runs. The result can also be calculated as a function of the number of runs, as shown in Fig. 33, compared to the Fast GRS method [125] (see section IV H 4 for details).The dashed lines in this figure are drawn as 740 ± (28 × 362/M). The behavior of the √ observed variance in keff values is in line with the 1/ M convergence one would expect. The convergence can be quantified for the local power results in the same way. Because of the relatively high statistical uncertainty in these results, the results between x = −50 cm and x = 50 cm were averaged. This 43

Nuclear Data Uncertainties . . .

1000

2.

Tally F7

500

0 1

10

100

Number of runs

Std. dev. [%]

FIG. 33. The calculated uncertainty in keff as a function of M , the number of runs (for the Fast TMC method the number of runs is M , for the Fast GRS method it is 2M ). 3%

F7 Uncertainty [pcm]

D. Rochman et al.

total, therefore, the run time was increased by some 40%, compared to a single, long Monte Carlo run that would yield the same F6 and F7 results (but without nuclear data uncertainties).

Fast TMC Fast GRS

Power density [a.u.]

keff Uncertainty [pcm]

1500

NUCLEAR DATA SHEETS

4 3 2 1

F7 local power density

1% MCNP6 standard deviation 0.5%

0% -200 -150 -100

2%

-50

0 x [cm]

50

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FIG. 35. The combined F7 result of 362 runs, shown along a horizontal line, at center line core, through the reactor core. 1%

The F7 results of 362 runs were combined, making use of Eq. (11), into one final result shown in Fig. 35. In the figure, the results for all the guide tubes, in which the F7 results are zero, are left out. Together with the results are shown, in the lower half of the figure, the combined results for the relative standard deviation. In the center of the core, the relative standard deviation is roughly 0.25%. This is much lower than the 5% in √ Fig. 32, because of the combination of 362 results: 5%/ 362 ≈ 0.25%. Away from the center, the relative standard deviation increases, up to just below 1% in the outermost fuel pins. However, a similar horizontal line through the core near the top or bottom of the reactor, yields relative standard deviations up to 3%. It is clear therefore that also the combined result of all 362 runs does not satisfy the benchmark criterion, even though the total number of histories in this combination is high. The reason for this is that the power distribution in the core is highly peaked. The power density in the top of a peripheral fuel pin is roughly 100× lower than in the center of the core. In the normal way of running a Monte Carlo eigenvalue calculation, there will be far fewer neutrons reaching the peripheral pin, and therefore the standard deviation will be higher. If the number of neutrons in the peripheral pins is 100× lower, then the relative standard deviation can be expected to be 10× higher. This is indeed was is observed here, with standard deviation of 0.25% in the center, and 3% in the outer edges of the core. In Ref. [126], where the benchmark criterion was fully satisfied, several techniques were applied to flatten the

0% 1

10 100 Number of runs (M)

FIG. 34. The calculated uncertainty in F7, averaged along a line through the reactor core from x = −50 cm to x = 50 cm, as a function of the number of runs.

average is shown as a function of M in Fig. 34. Also here, the dashed √ lines indicate the 1σ margin with the theoretical 1/ M convergence rate. The behavior of the observed variance in the average F7 values is in line with this theoretical convergence rate. All in all the evidence suggests that the results presented here are well converged. In the horizontal plane, the calculated uncertainty values inside the active core agree well with a quadratic curve. The efficiency of the method can be loosely quantified as follows. At first, a preparation run was performed for source convergence. This run would also be needed the normal way of running Monte Carlo, except that in that case it would be the so-called inactive part of a big run. For the Fast TMC method, more is needed for source convergence. The CPU time needed for source convergence in each of the M runs was 10% of the run time. Moreover, 30% of the runs were considered to be ’not well converged’, and were excluded from the analysis. In 44

Nuclear Data Uncertainties . . .

NUCLEAR DATA SHEETS

distribution of neutrons, in order to solve this efficiency problem. This is not done in the present work, because it was not the primary objective to meet the specifications of the benchmark. Rather, the objective was to prove that with a given number of histories used for a Monte Carlo calculation, one can extract nuclear data uncertainties as well, provided one chops the one run into many smaller runs.

3.

Uncertainty [%]

6% 5% 4% 3% 2% 1% -150

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0 x [cm]

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150

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FIG. 36. Uncertainty in F7 results due to nuclear data uncertainty. Also shown is a fit to the data (see text).

The results for the uncertainty due to nuclear data uncertainty are shown in Figs. 36 and 37. Some of the results were zero or negative, and these are all shown in the figure as zeroes. Not shown are the results for the guide tubes, because the F7 tally values and their uncertainties are zero there anyway. There is of course still a sizable statistical uncertainty in these results, but the general trend is quite clear. In the middle of the core, uncertainties due to nuclear data are lowest, of the order of 1%. This is in line with results for infinitely reflected pin cell or assembly level studies [19]. Toward the edge of the core along this horizontal line through the core, uncertainties due to nuclear data are somewhat larger, up to roughly 3%. The results, including the zeroes in Fig. 36, but excluding the guide tubes, can be fitted to a simple polynomial, as a function of the position along the horizontal line through the core f (x) = a0 + a2 x2 + a4 x4 .

Tally F6

The F6 results of the 362 runs were also combined, in the same way as those for F7. Results, along the same horizontal line through the core, are shown in Fig. 38, this time on a logarithmic scale on the y-axis. In this figure, the results for the guide tubes are also shown. These results are, as one would expect, much lower than in the fuel pins. In the figure it would seem almost as if there were two lines drawn: the upper line for the fuel pins, and the lower one for the guide tubes. The F6 results extend beyond the boundary of the active core. The results outside the active core almost connect to the results for the guide tubes, and then decline steeply when going away from the core. The standard deviation for F6 is similar to the one for F7, when looking at the fuel pins. For the guide tubes, the standard deviation is lower. Also for the standard deviation it looks as if there are two lines drawn: one for the fuel pins (the higher line) and one for the guide tubes. Near the edge of the active core, around ±182 cm, the standard deviation in the fuel pins is around 1%, roughly the same as for F7, but lower for the guide tubes. Beyond the active core boundary the standard deviation increases from well below 1% to more than 8%. The latter value belongs to an F6 power density value that is four order of magnitude lower than in the middle of the core. The results for the uncertainty in F6 due to nuclear data uncertainty shown in Fig. 39. The upper half of the figure shows the range of uncertainties until 100%, while the lower half zooms in on the range between 0% and 7%. Overall the values in the center of the core are similar to those for F7, around 1%, but in the case of F6 a little above it. Near the edge of the core, and outside the core, the values increase very steeply. The value of 100% is reached around x = ±220 cm (i.e. roughly 40 cm outside the active core), where the standard deviation in Fig. 38 is roughly 7%. The tally data have been calculated up to x = ±224 cm, where the standard deviation is roughly 8%, but the calculated uncertainty due to nuclear data is 150%. It was checked that the values around 100% uncertainty were well converged. These values were calculated based on M runs, with M = 32, 64, 128, 256, and362. Starting from M = 64, all values at x = ±224 cm were within 10% of one another. It should be stressed, though, that this uncertainty value was calculated for the part of the power that is deposited by neutrons. At these peripheral locations, it can be expected that most of the power deposition is due to photon interactions, but photon transport was not simulated in the runs reported here. The uncertainties in F6 can be fitted to a polynomial as well. In this case it makes sense to fit only to the data inside the active core, i.e. with position −182 < x < 182 cm. The results of a second order fit were that a0 = (1.2 ± 0.1)%, and a2 = (4.7 ± 0.7) × 10−5 %/cm2 . As in the case of F7, the results of a fourth order fit were consistent, within 1σ, with the values of the second order

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All terms with an odd power of x, such as a1 x and a3 x3 have been left out, because the reactor is symmetrical (all fuel assemblies are identical in this benchmark). The results of a second order fit were that a0 = (0.91 ± 0.16)%, and a2 = (5.3 ± 0.9) × 10−5 %/cm2 . A fit with an extra fourth order term included did not give significantly different results: the values for a0 and a2 were within 1σ from the above ones, and the value for a4 was consistent with zero. 45

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FIG. 37. (Color online) Same as Fig. 36 (generated power) through an horizontal plan at the center of the core.

method was in fact the first highly efficient Monte Carlo method for calculating uncertainties [125]. It calls for two sets of M Monte Carlo runs, the first set with a constant random number seed r1 , and the second set with a different, but also constant random number seed r2 . For each of the sets one uses ’random’ nuclear data, in such a way that the nuclear data used for the first run of set 1 are equal to the nuclear data used for the first run of set 2. For any quantity x one thus obtains two series of results, x ¯j,1 and x ¯j,2 , for j = 1, . . . , M . The variance in this quantity can then be calculated as the covariance of x ¯j,1 and x¯j,2 . Based on this prescription, 2×328 runs were performed

fit. The uncertainties for F6 in the guide tubes are slightly higher than for F6 in the fuel zones. When the guide tubes are excluded from the fit, the values for a0 and a2 are roughly equal to the one for F7. For the guide tubes only, the values are a0 = (2.0 ± 0.1)%, and a2 = (1.9 ± 0.4) × 10−5 %/cm2 .

4.

Comparison with the Fast GRS Method

The results obtained in this paper can be checked by also calculating the results with another method. The method that will be referred to here as the Fast GRS 46

Standard deviation [%]

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be compared between the Fast TMC and the Fast GRS method. The data along a horizontal line through the reactor core are shown in Fig. 40, together with the data from the Fast TMC method (for M = 362). Also shown in the figure are the quadratic fits to these data. The values for both methods are close to one another near the center of the core, but differ somewhat more near the edge of the active core. This difference is not significant, which can be judged by looking at the uncertainty in the fit parameters, and the convergence as a function of M .

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The convergence of the Fast GRS method can be quantified as well, based on the fit results. The values for a0 and a2 are shown in Fig. 41 as a function of M , together with the results from the Fast TMC method. It can be seen that the results of the Fast GRS method for both a0 and a2 show a slight trend as a function of the number of runs, whereas the results of the Fast TMC method are more flat. This different behavior is not understood at present. For a0 , the results for both methods are in reasonable agreement with one another, starting from 128 runs. For a2 this is the case roughly from 256 runs onwards. It can be concluded that the uncertainty results of the Fast TMC method are confirmed by those of the Fast GRS method. In Fig. 41, there are also three solid points, at M = 84, 64, and 58. These points represent the results of the Fast GRS method based on the runs that were considered ‘not well converged’. The point at M = 58 is based on the cases where the runs for both random number seeds were not well converged, while the points at M = 84, 64 are based on the cases where runs with random number seed 1 were not well converged (M = 84), or the runs with random number seed 2. In all cases, the results do not differ significantly from those based on well converged runs. So also in this case, the results appear to be less sensitive to source convergence than the fission source entropy, which was used as the indicator for convergence.

FIG. 39. Uncertainty in F6 results due to nuclear data uncertainty. Also shown is a fit to the data inside the active core, i.e. with −182 < x < 182 cm (see text).

for the Martin-Hoogenboom benchmark, of which 2×122 runs were accepted for further analysis, based on the same criterion as was used for the Fast TMC method in Subsection IV H 1 a. The acceptance rate is lower in this case, because the method requires two runs for each set of (random) nuclear data. Both of these runs should be converged, which was the case for 2×122 runs. For 148 sets of runs, one of the runs was converged, but not the other. In 58 cases, both runs were considered ’not well converged’. For keff , the result for the uncertainty was 707 pcm. Although the method does not automatically give an estimate for the statistical uncertainty, this value is judged to be consistent with the 740 ± 28 pcm reported in Sec. IV H 1 c, see also Fig. 33. Also for the files that were considered ‘not well converged’, the uncertainty estimate using the Fast GRS method was calculated, which gave 712 pcm. Again, this result appears to be less sensitive to source convergence than the fission source entropy, which was used as the indicator for convergence. Also the uncertainties for local power densities can 47

Nuclear Data Uncertainties . . .

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Fit parameter a0

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2%

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D. Rochman et al.

TABLE XII. Uncertainties and probability density functions in the literature for a selection of operating conditions considered in this work (U means uniform and N normal probability density function. ρ(T) is the correlation the fuel or the moderator temperature T ).

1.5% 1%

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Ref.[129] Ref. [130] Ref. [131] Fuel T 2%U 2%U Moderature T Reactor pressure Boron concen. 2%U Power ρ(T)=1 Moderator dens. 2 % U Irradiation hist. 1 % U

1e-4 5e-5 Fit parameter a2 0

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FIG. 41. Comparison of the Fast GRS method and the Fast TMC method for the fit parameters a0 and a2 (see text), as a function of the number of runs (for the Fast TMC method the number of runs is M , for the Fast GRS method it is 2M ). For an explanation of the full circle points, see text.

I.

1.5 % U 1%N 1.5 % N -

2%N -

This work 2%U 2%U 1%U 2%U ρ(T)=1 ρ(T)=1 1%U

in terms of uncertainties for calculated quantities with any simulation codes. In the following, we will present some of these quantities and their published uncertainties. We will also present an example for the impact of such quantities in terms of uncertainties on k∞ for a simple PWR UO2 assembly.

Other Sources of Uncertainties

Although this paper is focusing on the impact of the nuclear data on specific quantities, other sources of uncertainties can play an important role during the burnup calculation of a specific assembly. It is important to have a general overview of the impact of different parameters on calculated quantities, relatively to the impact of nuclear data, in order to recognize the limits in the predictive power of assembly burn-up simulations. Previous work performed by national and international organizations already extensively covers this subject, even if it is still a field under development. For more dedicated studies on this matter, the reader is invited to visit Refs. [127, 128]. In a relatively simplistic manner, the following different types of input quantities can be considered as sources of uncertainties:

1.

Operating Conditions

The operating conditions can concern different quantities such as the fuel and moderator temperatures, the reactor power and pressure, the boron concentration, the moderator density and of course the irradiation history. A convenient method to assess their impact on calculated quantities is again to apply a Monte Carlo variation of these parameters, in a similar way as for the nuclear data. Concerning their pdf, different assumptions can be made, and Table XII presents some selected published values. Some of these quantities can be repeated many times during a burn-up simulation for different burn-up steps. For instance, the reactor pressure is generally defined only once at the beginning of the calculation and can therefore be changed once for the complete burn-up scheme. Other quantities, such as the fuel and moderature temperatures, the boron concentration can be defined for many different groups of burn-up steps and can therefore be changed many times for one calculation (until the end of cooling time). Some other parameters can be correlated with other quantities. Such correlations can be calculated by the burn-up code (like the moderator density, correlated with the moderator temperature), or imposed during the random variation of parameters.

1. Nuclear data, 2. Reactor operation conditions, 3. Manufacturing tolerances, 4. Burnup induced technological changes, 5. Computational biases. It is not the goal of this paper to fully describe the sources and impact of such quantities. But in order to draw a non too partial view on the sources of uncertainties, the above conditions need to be mentioned and described in a minimal manner. Historically, nuclear data and their uncertainties received a large attention and as presented in the previous sections, covariance files can be found in the literature, together with different methods of uncertainty propagation. The reactor operating conditions and manufacturing tolerances are nevertheless also important to quantify

2.

Manufacturing Tolerances

Similar to the operating conditions, the manufacturing tolerances for the characteristics of the pin cells and the 48

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D. Rochman et al.

TABLE XIV. Uncertainties considered for the burnup induced effects.

TABLE XIII. Uncertainties and probability density functions in the literature for five manufacturing conditions considered in this work (“U” means uniform and “N” normal probability density function).

Fuel Pin Moderator Pin position pin position radius shift shift Ref.[136] 15-50μm, 28 GWt/t 70 μm, 50 GWt/t Ref.[137] Ref.[138] 80 μm, 30 GWt/t Ref.[139] 160 μm, 60 GWt/t Ref.[140] 150 μm, 60 GWt/t This work 0.2mm per 0.05mm per 150 μm, 55 GWt/t burn-up burn-up groups groups

235 pin Fuel pin Guide Fuel U radius position thimble density enrichment shift shift Ref.[129] 0.8 % N 0.5 % N 1%N Ref.[130] 0.1 % U 0.3 % N Ref.[132] Ref.[133] 0.2 % 1% Ref.[60] 0.1 % N 0.5 % N 0.02 % N Ref.[134] 0.1 % Ref.[135] 0.1 % 1.65 % 0.05 % Adopted in 0.5 % U 1 mm 0.2 mm 1.5 % U 0.2 % U this work

3.

Burnup-induced Technological Changes

During the burn-up of the assemblies in a reactor, the mechanical material and structural characteristics of the fuel rod pellet/gap/cladding will evolve due to several phenomena including e.g. densification, swelling, fission gas release, clad creep-down/creep out, oxidation, hydrogen uptake etc. All these phenomena will thus introduce sources of uncertainties in the burn-up calculations that should be accounted for. Because the quantification of these uncertainty sources is perhaps one of the most challenging tasks, only a limited number of uncertainties in relation to geometrical characteristics are considered here and shown in Table XIV. The displacements of the rods inside an assembly can be simulated at different burn-up steps, with a shift in the x and y directions from the center of the pin cell. By randomly repeating these keywords at different burn-up steps, the vibrations of the rods are therefore simulated. There is no available experimental information on how much a pin vibrates and assumptions have to be made. In the case of the manufacturing conditions, a maximum displacement is considered to be 1 mm for the fuel pin and 0.2 mm for a moderator pin (having a much larger diameter). Therefore, it can for instance be considered that the fuel pins can vibrate by a maximum of 0.2mm and that the moderator pins by 0.05mm. Additionally, one can consider the bowing effect of assemblies. The bowing effect (or channel deformation) has been widely observed in many (or all) commercial PWRs. It is an effect which affects the rod insertion and bring handling difficulties. Measurements of the bow indicated effects up to almost 10 mm in Ref. [141] and 20 mm in Ref. [142]. However, recent information has indicated displacements in the range of 35 mm. It is affecting the calculated quantities such as the number densities and needs to be taken into account. The bowing effect is still difficult to simulate with existing deterministic transport code at the assembly level. Some preliminary work is performed with Monte Carlo transport codes such as SERPENT [143], but more efforts are necessary.

assembly can be randomly modified following given pdfs. From the open source references, the quantities and uncertainties presented in Table XIII can be considered. For the (outer) pin radius, the variations of this quantity are presented in many references, from 0.1 % to 0.8 %, using either normal or uniform probability distributions. As a fuel pin can be defined with 3 radii: (1) for the fuel, (2) for the gap and (3) for the cladding, the three radii can be all varied together with a correlation of 1 as a simplified first approach. Considering the fuel and moderator pin position shifts (relatively to their central positions), random positionings of all pins can be realized at the beginning of the simulations and kept constant for complete burn-up calculations. This is certainly an approximation, which can be improved in a dedicated study. An example is presented in Fig. 42 for a few pin cells in a given PWR assembly.

FIG. 42. (Color online) Original positions of the fuel pins and moderator channels, dark yellow for fuel pins and blue for moderator channels (left). All pins are aligned. Right: same with random positioning of the fuel pins and moderator channels.

49

Nuclear Data Uncertainties . . . 4.

NUCLEAR DATA SHEETS

SHARK-X methods have shown that discrepancies can appear by neglecting the perturbation of the resonance self-shielding treatment, the so-called “implicit effect”. Such approximation has been shown to have some effect on k∞ UQ [146, 147] and the discrepancies might increase when the considered response is highly sensitive to capture in fertile isotopes like the Doppler reactivity coefficients [40]. A method to handle the perturbation of self-shielding in CASMO-5 was then developed recently at PSI and implemented in SHARK-X. The objectives of the work presented in this section are two folds. The first goal is to assess the consequences of neglecting the implicit effect on the uncertainty estimates of various responses considered (k∞ , Doppler coefficient, and nuclide densities). The linearity of k∞ , Doppler coefficient, nuclide densities to perturbation of the 238 U capture multi group cross-sections is the second motivation for the analysis presented here. The model considered is the PWR UO2 assembly as presented in section II B 1 (TMI-1 assembly). The Doppler Coefficient is computed by reducing the nominal fuel temperature (900 K) to 560 K. The nuclear data uncertainty information is the SCALE-6.2 relative variancecovariance matrices (VCMs) in a 56-energy group format. For all the calculations performed in this section, only the 238 U capture is perturbed. The large importance of 238 U capture with respect to the overall UQ estimates has been illustrated in section IV. The evolution of the k∞ uncertainty with exposure and the impact of the implicit effect are analyzed up to 40 GWd/THM. The uncertainties of nuclide densities are assessed at 40 GWd/THM for a selection of minor actinides produced mainly by capture in 238 U. The Doppler coefficient UQ is assessed at the beginning of Cycle (BOC=0 GWd/THM) and End of Cycle (EOC=40 GWd/THM).

Computational Biases

Sources of bias are distinct from sources of uncertainties as they do not occur because of limited knowledge on some input parameters. They are linked to the code and its methods to solve the transport and depletion problems. To evaluate in a broad manner the bias coming from a certain code, the best approach is the use of a different code, independent of the first one. For instance, the use of (1) a Monte Carlo transport code for a 2D geometry and (2) a deterministic 2D nodal code allows for a good estimation of the bias of the deterministic code.

5.

Example

Taking into account all the above sources of uncertainties, an example for the number densities of 234 U for a PWR UO2 assembly is presented in Fig. 43 (see Ref. [24] for details). As seen in this figure, the uncertainty due

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FIG. 43. (Color online) Example of the uncertainties on 234 U due to nuclear data and other sources, from Ref. [24]. The considered assembly is a PWR UO2 , similar to the one used in this study.

1.

Reactivity Coefficient UQ

The reactivity coefficient is defined in Eq. (19) and includes a reference and a perturbed state,

to other sources can be relatively important. It depends on the type of quantity, and in the present case, almost half of the uncertainties comes from other sources than nuclear data.

J.

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Δρ = ρpert − ρref 1 1 = ref − pert k∞ k∞ = λref − λpert .

Self-shielding Treatment

(19)

The SHARK-X scripts provide three major approaches for calculating the reactivity coefficient uncertainty. The first two rely on the calculation of sensitivity coefficients. Those coefficients are then folded with variance covariance matrix (VCM) of input parameters through the use of the sandwich rule to compute the uncertainty. The third method relies on Stochastic Sampling (SS). For reactivity change UQ, each input sample is used in the two transport calculations needed to compute the reactivity coefficient as expressed in Eq. (20). The uncertainty propagation through depletion and reactivity calculations

During a fast transient event and particularly the class of Reactivity Initiated Accident (RIA), the Doppler coefficient plays an important role [144, 145]. For this reason, an accurate quantification of the Doppler coefficient and its uncertainty is an essential prerequisite for a safe reactor design. As presented here, the SHARK-X methodology has recently been extended to perform reactivity UQ. The approach to do so is described briefly in the following. Moreover, previous verification work on the 50

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is quantified using the stochastic with a sample size of 1 000. For the calculation of the reactivity change sensitivities, the first method relies on the Direct Perturbation (DP) while the second uses the formulas of Equivalent Generalized Perturbation Theory (EGPT) [148] to compute the sensitivity of the reactivity changes using the sensitivity coefficients of the reference and perturbed state eigenvalues as shown in Eq. (20)   ∂λpert α ∂λref SΔρ,α = − Δρ ∂α ∂α   1 pert ref λpert Sk,α = . (20) − λref Sk,α Δρ

D. Rochman et al. 3.

UQ Results

First, the effect of the new self-shielding perturbation method is illustrated in terms of the k∞ UQ and its evolution with exposure. The results of the SS UQ method of SHARK-X is shown in Fig. 44 with and without perturbation of the self-shielding. 0.50

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The verification of the correct implementation of the methods has been done against TSUNAMI [17] for various pincell models and is published in Ref. [149].

0.30 0.25 0.20 0.15 PWR UO2

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Handling the Implicit Effect in SHARK-X

0.05 0.00

SHARK-X relies on CASMO-5 to perform the neutron transport in a given model. CASMO-5 uses the equivalence theory and the Intermediate Resonance (IR) approximation to compute its effective cross-section in the higher energy part of the resolved resonance range (10 eV to 10 keV). Below 10 eV, the energy mesh of the CASMO5 library is fine enough to avoid the need of self-shielding calculation within a particular group. A method has been recently developed and implemented in SHARKX to properly take into account the perturbation of the 238 U capture in the resonance self-shielding calculation of CASMO-5 . It is based on the predetermination of effective perturbation factors with reference and perturbed NJOY calculations using pointwise cross section data. The effective perturbation factors are parametrized with respect to perturbation magnitude, the dilution cross section and energy group index. Given the initial cross section library for a set of dilutions and temperatures, as well as a perturbation of the pointwise cross section α, the perturbed XS including the perturbation of the energy self-shielding is replaced by XS  (σ0 ) = γXS(σ0 ),

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FIG. 44. (Color online) Evolution of k∞ relative uncertainty with exposure for cross section uncertainty (SCALE-6.2 uncertainty data).

The effect of the self-shielding perturbation appears to be large initially, around 0.1 % but decreasing with exposure. The perturbation of the self-shielding is not negligible. It should be noted however, that the effect in terms of overall uncertainty may be smaller since other nuclide reaction pairs could play a large role (235 U and 239 Pu average number of neutron produced per fission for example). Using the DP method of SHARK-X, it is possible to analyze the evolution of the sensitivity coefficient of k∞ to the 238 U capture cross section during the cycle. It is shown with and without perturbation of the self-shielding at BOC in Fig. 45 top and EOC in Fig. 45 bottom. The standard deviation of the 238 U capture cross section is also plotted for the sake of completeness. As expected, the sensitivity of k∞ to the 238 U capture cross section is negative at BOC, and the proper handling of the self-shielding perturbation reduces the magnitude of the sensitivity in the range of energy where CASMO-5 perform self-shielding calculation (10 eV to 10 keV). Outside of this energy range, both implicit and explicit sensitivity coefficients are equal. At EOC, the sensitivity of k∞ to the 238 U capture cross section is now positive due to the breeding of 239 Pu. The proper handling of the implicit effect tends to reduce the magnitude of the sensitivity. The DP results suggest that the proper handling of the implicit effect will reduce the magnitude of the uncertainty and are consistent with the SS results.

(21)

where γ is determined by an additional NJOY calculation using point wise cross sections perturbed according to Eq. (22) and tabulated a priori. When perturbing a cross-section σx by a factor α, the cross section is modified in NJOY as σ  (E) = ασ(E),

10

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with E the energy of the incident neutron. The interpolation with respect to dilution and temperature is the usual one, i.e. square root. The interpolation for the perturbation magnitude is merely piecewise linear for now. The treatment of the implicit effect in SHARK-X is limited to the 238 U captures.

The effect of self-shielding perturbation on Doppler coefficient UQ is investigated similarly to the work reported 51

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45

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FIG. 45. (Color online) Comparison of k∞ sensitivity profiles (%/%/unit lethargy) for the 238 U capture cross section with and without considering the implicit effect at BOC (top) and EOC (bottom).

FIG. 46. (Color online) Comparison of Doppler coefficient sensitivity profiles (%/%/unit lethargy) for 238 U Capture for SHARK-X with and without considering the implicit effect at BOC (top) and EOC (bottom).

TABLE XV. Comparison of the Doppler UQ methods in SHARK-X at BOC and EOC.

UQ results for Doppler coefficient calculated with SS, EGPT and DP are shown in Table XV. SS, EGPT and DP UQ results of SHARK-X are in agreement considering the 95/95 confidence interval of the SS solution, supporting the claim that there is no visible linearity issue. Global UQ approach (SS method) and local ones (DP and EGPT) return consistent estimates. Looking more carefully at Table XV, even though both DP and EGPT rely on direct perturbation for the determination of the reactivity sensitivity coefficients, they do not return exactly the same results at BOC. Since the input uncertainty is the same, those discrepancies are due to difference in sensitivity coefficients. The Doppler coefficient sensitivity coefficients predicted SHARK-X DP and EGPT are shown in Fig. 47. Discrepancies are observed between DP and EGPT that do not lie with the “rounding uncertainty”, especially in the eV range, where EGPT tends to under predict the sensitivity of 238 U capture with respect to the DP sensitivities. The differences in sensitivity coefficients can be traced back to determination of perturbation magnitude in the DP method as described in Ref. [22]. The discrepancies introduced in SHARKX DP and EGPT UQ methods by the determination of the perturbation magnitude is a clear drawback of the SHARK-X methods. It should be noted however that such drawbacks do not exist with the SS UQ approach.

Exposure Method Δρ (pcm/K) Δρ std (%) SS 1.67 (1.60 / 1.74) BOC EGPT 2.16 1.64 DP 1.65 SS 1.41 (1.34/1.48) EOC EGPT 3.42 1.39 DP 1.39

on k∞ in the previous paragraph. The Doppler coefficient sensitivity profiles of SHARK-X generated with the EGPT method are compared for the 238 U captures at BOC and EOC. Both are shown in Fig. 46, with and without perturbing the self-shielding. Finally, the Doppler coefficient uncertainty does not vary much with exposure since the amount of 238 U in the fuel remains fairly constant. The spectral hardening due to the increased content of 239 Pu in the fuel is visible. Even though the relative standard deviation decreases with exposure, it is due to the larger magnitude of the Doppler coefficient at EOC. The absolute standard deviation of the Doppler coefficient has increased at EOC. 52

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1.27

1.275

Perturbed k-inf

100

50

0 1.255

1.26

1.265

1.27

1.275

1.28

1.285

Doppler Coefficient

100

50

0 2

2.05

2.1

2.15

2.2

2.25

2.3

0.925

0.93

0.935

0.925

0.93

0.935

0.94

0.945

Doppler Coefficient

3.3

3.4

3.5

3.6

Std. Dev. = 0.33% Skewness = 0.11 Lilliefors: Normal Jarque-Bera: Normal Anderson-Darling: Normal Std. Dev. = 1.41% Skewness = -0.12 Lilliefors: Normal Jarque-Bera: Normal Anderson-Darling: Normal

FIG. 49. Probability Distribution Function of k∞ for the reference and perturbed fuel temperature conditions, as well as the resulting Doppler Coefficients at EOC. The results of various normality tests are presented.

In order to confirm the normality of k∞ and Doppler coefficients probability distributions, various normality tests are applied to the 1000 samples of the SS results at BOC and EOC: the Lilliefors, Jarque-Bera and AndersonDarling tests are considered. Their results are shown in Figs. 48 and 49 as well as the mean, the standard deviation and the skewness of each distribution. Reference k-inf

0.92

Perturbed k-inf

50

FIG. 47. (Color online) Comparison of Sensitivity Profile (%/%/unit lethargy) from the DP and EGPT methods SHARK-X for the Doppler Coefficient (the width of the area around the sensitivity coefficient represents the effect of 1 pcm uncertainty in the assembly transport calculation on the sensitivity coefficient).

100

0.915

Std. Dev. = 0.34% Skewness = 0.11 Lilliefors: Normal Jarque-Bera: Normal Anderson-Darling: Normal

in 238 U with and without considering the implicit effect. Similarly to the k∞ and Doppler coefficient UQ, neglecting the self-shielding perturbation leads to a potentially large overestimation of nuclide density uncertainty, especially for 239 Pu, for which the uncertainty is reduced from 2.17 % to 1.73 % when considering the implicit effect. The observed linearity of the nuclides composition

Std. Dev. = 0.35% Skewness = 0.11 Lilliefors: Normal Jarque-Bera: Normal Anderson-Darling: Normal Std. Dev. = 0.34% Skewness = 0.11 Lilliefors: Normal Jarque-Bera: Normal Anderson-Darling: Normal Std. Dev. = 1.67% Skewness = -0.12 Lilliefors: Normal Jarque-Bera: Normal Anderson-Darling: Normal

FIG. 50. (Color online) Nuclide density UQ for various Minor Actinides at EOC.

FIG. 48. Probability Distribution Function of k∞ for the reference and perturbed fuel temperature conditions, as well as the resulting Doppler Coefficients at BOC. The results of various normality tests are presented.

with respect to the 238 U capture cross sections is investigated. The Lilliefors, Jarque-Bera and Anderson-Darling normality tests are applied to the probability distribution function of 239 Pu and 241 Pu concentrations at EOC; their results are shown in Fig. 51 as well as the mean, the standard deviation and the skewness of each distribution.

Finally, the effect of self-shielding perturbation on nuclide composition uncertainties is investigated using the SS UQ method of SHARK-X. The uncertainties of nuclide densities are shown in Fig. 50 at 40 GWd/THM for a selection of minor actinides produced mainly by capture 53

Nuclear Data Uncertainties . . .

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Nuclide 94239 Density

80

80

D. Rochman et al.

Nuclide 94241 Density

Mean = 4.10e-05 atom/b/cm2 70 Std. Dev. = 1.73% 60 Skewness = -0.16 Kurtosis = 2.86

70

60

Mean = 1.02e-05 atom/b/cm2 Std. Dev. = 1.04% Skewness = -0.19 Kurtosis = 2.88

50 50 40

Lilliefors: Not normal 30 Jarque-Bera: Not normal Anderson-Darling: Not normal

40

30

Lilliefors: Not normal Jarque-Bera: Not normal Anderson-Darling: Not normal

20 20

10

10

0 3.8

3.9

4

4.1

4.2

4.3

4.4

0 0.98

1

1.02

×10 -5

1.04

1.06 ×10 -5

FIG. 51. Probability Distribution Function of various Pu isotopes concentrations at EOC (239 Pu left and results of various normality test is displayed.

4.

Conclusion for the Implicit Effect

V.

241

Pu right). The

CONCLUSIONS AND OUTLOOK

The purpose of this work was (1) to obtain a general view of the impact of current nuclear data library uncertainties on relevant quantities for reactor applications and (2) to appreciate the differences obtained from different methods including simulation codes and nuclear data libraries. To fulfill these tasks, different participants have used their preferred codes and libraries (see Table I), on a series of predefined assemblies and a simplified full core (see Table III and section II B 6). This approach differs from the UAM method and it was not expected to lead to the same uncertainties from the different participants. The main results are presented in Figs. 23, 60 to 65, 26 and 37, and can be summarized as follows:

The consequences of neglecting the implicit effect on the uncertainty estimates of various responses considered (k∞ , Doppler coefficient, and nuclide densities) is found to be large in terms of sensitivities and also potentially in terms of uncertainties when the 238 U capture cross section is the primary contributor to the uncertainty, i.e. for Doppler coefficient. For now, the treatment of the implicit effect in SHARK-X is limited to the 238 U captures but could be extended to other nuclides in the future. Looking specifically to Doppler coefficient UQ, small discrepancies are introduced in SHARK-X DP and EGPT UQ methods by the determination of the perturbation magnitude. It is a clear drawback of SHARK-X and could be addressed by the development of an adjoint-based sensitivity coefficients. It should be stressed also that the UQ estimates obtained with the SHARK-X SS method do not suffer from such issue provided that the change in reactivity are larger than a few tens of pcm. Finally, the linearity of the various responses considered with respect to the 238 U capture cross section is investigated by looking at the consistency of the SHARKX UQ estimates and considering various normality tests on the SS results. Both k∞ and Doppler coefficients are found to be linear in the limit of perturbations within the SCALE-6.2 input uncertainty. Due to the coupling of the Boltzmann and Bateman equations, the nuclide densities show some non-linearity. Local UQ estimates based on the calculation of sensitivity coefficients may not be accurate.

• For the systems with results from different participants (all assemblies), the calculated uncertainties on k∞ and macroscopic cross sections can differ by a factor 2. Such differences can be expected as different covariance libraries are used, which are possibly the main source of discrepancies. • For k∞ , there seems to be a general consensus that the uncertainties are lower than ≈ 800 pcm. No such general agreement exist for other quantities. • For pin power distributions, the impact of nuclear data is relatively modest, with uncertainties less than 0.7 %. There seems to be higher uncertainties in the case of BWR assemblies compared to PWR. • For the uncertainties on number densities, the general agreement is less clear, especially due to the impact of fission yields. • In the case of the full core Monte Carlo calculations, the impact of nuclear data on the local generated 54

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tron emission (fraction and spectra). These nuclear data can affect specific quantities and should therefore be considered to obtain a more global assessment. Additionally, as shown in section IV I, other sources of uncertainties can have an important impact.

and deposited power distribution is not negligible (higher than 1 %). Based on these results, it can be recognized that different simulation tools (codes and libraries) lead to different results, which are often more spread than the calculated quantities themselves. This should help to appreciate the relative aspect of calculated uncertainties: as for experiments, three independent sources are better than a single one, allowing for a possible selection of only two results. Based on the experiences acquired during this work, some recommendations can be proposed to the different communities involved in nuclear simulations:

• The feedback to the nuclear data community can be achieved with the help of sensitivity studies, or at least with an ordering of the most sensitive nuclear data using importance factors. Such a study, not presented in this paper can be the subject of a dedicated analysis.

• Some nuclear data libraries contain partial covariance information, which can affect the results. This is the case for the JEFF-3.2 library, for which the NJOY processing code was not able to correctly process some covariance files.

Some important questions were not touched on in this paper, especially concerning the schemes and procedures to improve the quality of some nuclear data evaluations with the feedback from studies as presented here. For this important type of work, specific efforts are needed as performed at the international level, organized by the NEA. The present work is therefore part of the drive towards better nuclear data and possibly smaller uncertainties, and can be seen as one of the multiple steps in the general work for library improvements. In a close future, it is likely that the JEFF-3.2 library will be updated with a full set of covariances. Also, there is a growing recognition for the necessity of the testing of covariance files before their release. With the increasing automation of the nuclear data production, it is foreseen that the uncertainties as calculated in this paper will “converge” to acceptable values, thus decreasing the observed spread. Will this convergence of results suggests a better knowledge of calculated quantities ? This will be for the future to comment on.

• Because of the general use of a unique processing code, many (but not all) results depend on the NJOY processing. This does not follow the general recommendation of redundancy in the nuclear simulations. There is therefore a need for more cross checking among processing codes, and possibly for the developments of new processing capabilities. • Regarding the usual habit to present the uncertainties in terms of standard deviation, it seems to be important to take into account other moments of the probability distributions. With the modern computer power, it is not out of reach to calculate with a acceptable degree of confidence quantities such as the skewness. For safety assessment, this quantity should be considered as important as the standard deviation, as it quantifies events far from the mean, not specifically captured by the standard deviation.

ACKNOWLEDGMENTS

We would like to acknowledge the different funding agencies which made this work possible, as well as the UAM community for initiating the research work on uncertainty propagation and the development of methods. The GRS contribution is supported by the German Federal Ministry for Economic Affairs and Energy.

• There is a need for fission yield covariances. As shown in this study, this lack of information leads users to either not considering them, or to apply different methods leading to a wide spread of calculations. The covariance information needs to be provided with the nuclear data library, as for other quantities such as cross sections. • Some nuclear data are not usually considered, such as the thermal scattering data, or the delayed neu-

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Appendix A: Additional Results on Uncertainties

In the following pages, different figures are shown supporting the results presented in Secs. II C, IV B, IV C, IV D and IV E. Figures 52 to 59 present the nominal values for different quantities such as the power, and some number densisities. Figures 60 to 66 present the uncertainties for these quantities as well as for k∞ .

61

Nuclear Data Uncertainties . . . 234

10

U

8 6

PWR UO2

8 6 4

10

20

30

40

Burn-up (MWd/kgU) PWR UO2 + Gd

234

10

U

8 6 4

10

20

30

20

30

PWR UO2 + Gd

235

U

6 4

234

10

U

8 6 4

10

20

30

20

30

BWR UO2

235

U

6 4 0

Burn-up (MWd/kgU) 4

234

6

U

3 2

10

20

30

PWR MOX

235

20

30

3

234

10

12 10 8 6

10

20

30

10

20

30

Burn-up (MWd/kgU)

40

U

2 10

20

30

40

PWR MOX

236

U

5 4 3 2

40

0

BWR MOX

235

U

8 6

0

10

20

30

Burn-up (MWd/kgU)

10

20

30

40

Burn-up (MWd/kgU)

4 0

236

5

Burn-up (MWd/kgU)

U

40

1 0

atom/cm3 × 1019

atom/cm3 × 1017

BWR MOX

30

8

6

U

4

Burn-up (MWd/kgU) 14

20

Burn-up (MWd/kgU)

5

40

10

BWR UO2

11

0

atom/cm3 × 1018

10

U

4

40

2 0

236

7

Burn-up (MWd/kgU) atom/cm3 × 1019

PWR MOX

40

Burn-up (MWd/kgU)

8

40

30

PWR UO2 + Gd

0

atom/cm3 × 1018

10

20

10

40

2 0

13

Burn-up (MWd/kgU) atom/cm3 × 1020

BWR UO2

10

1 0

Burn-up (MWd/kgU) 10

4

Burn-up (MWd/kgU)

8

40

U

7

0

atom/cm3 × 1019

10

236

10

40

2 0

PWR UO2

Burn-up (MWd/kgU) atom/cm3 × 1020

atom/cm3 × 1018

10

13

1 0

atom/cm3 × 1019

0

atom/cm3 × 1018

U

2

4

atom/cm3 × 1017

235

D. Rochman et al.

atom/cm3 × 1019

PWR UO2

atom/cm3 × 1018

atom/cm3 × 1018

10

NUCLEAR DATA SHEETS

40

11

BWR MOX

236

U

8 5 2 0

10

20

30

40

Burn-up (MWd/kgU)

CASMO

FIG. 52. Number densities in atoms/cm3 for 3 uranium isotopes and the 5 different considered assemblies, using CASMO-5.

62

12 10 8 6 4 0

10

20

30

6 5 4 3 2 1 0

40

PWR UO2

239

Pu

12 10 8 6 4 0

10

20

30

6 5 4 3 2 1 0

40

239

10 8 6 4 0

10

20

30

6 5 4 3 2 1 0

40

239

40

0

240

3.5

Pu

20

30

Pu

10

20

30

4

240

10

20

PWR MOX

BWR MOX

239

6 5 4 3

30

20

30

Burn-up (MWd/kgU)

40

241

Pu

1 0

10

20

30

40

BWR UO2

241

Pu

2 1

40

0

240

10

20

30

40

Burn-up (MWd/kgU)

Pu

PWR MOX

241

Pu

1.5 1.0 0.5

10

20

30

40

0

BWR MOX

240

10

20

30

40

Burn-up (MWd/kgU) 2.5

Pu

4.5 4.0 3.5 3.0

10

40

2

3

Pu

2.0

5.0

7

0

PWR UO2 + Gd

Burn-up (MWd/kgU)

Pu

30

Burn-up (MWd/kgU)

2.5

0

atom/cm3 × 1020

8

20

3

40

3.0

40

10

Burn-up (MWd/kgU)

atom/cm3 × 1020

10

1

Burn-up (MWd/kgU)

Burn-up (MWd/kgU) atom/cm3 × 1020

30

1.5 0

2

0 0

atom/cm3 × 1020

atom/cm3 × 1020

PWR MOX

20

BWR UO2

Burn-up (MWd/kgU) 8 7 6 5 4 3

Pu

3

Burn-up (MWd/kgU)

Pu

241

0 0

atom/cm3 × 1019

atom/cm3 × 1019

BWR UO2

10

PWR UO2 + Gd

Burn-up (MWd/kgU) 12

PWR UO2

Burn-up (MWd/kgU) atom/cm3 × 1019

atom/cm3 × 1019

PWR UO2 + Gd

4

Pu

0 0

Burn-up (MWd/kgU) 14

240

atom/cm3 × 1019

Pu

atom/cm3 × 1019

239

atom/cm3 × 1019

PWR UO2

D. Rochman et al.

atom/cm3 × 1020

14

NUCLEAR DATA SHEETS

atom/cm3 × 1019

atom/cm3 × 1019

Nuclear Data Uncertainties . . .

BWR MOX

241

Pu

2.0 1.5 1.0

0

10

20

30

Burn-up (MWd/kgU)

40

0

10

20

30

40

Burn-up (MWd/kgU)

CASMO

FIG. 53. Number densities in atoms/cm3 for 3 plutonium isotopes and the 5 different considered assemblies, using CASMO-5.

63

Nuclear Data Uncertainties . . . 237

Np

1.0 0.5

PWR UO2

1.0

0.5

10

20

30

40

237

Np

1.0 0.5

20

30

0

PWR UO2 + Gd

1.0

241

10

20

30

Am

0.5

40

237

Np

1.0 0.5 0.0

10

20

30

20

30

BWR UO2

1.0

241

Am

237

1.5

Np

3 2 1 0

10

20

30

10

20

30

PWR MOX

237

2.0

Np

3 2 1 0

241

0 12

Am

10

20

30

10

20

30

Burn-up (MWd/kgU)

40

10

20

30

40

PWR MOX

244

Cm

9 6 3

40

0

BWR MOX

241

10

20

30

40

Burn-up (MWd/kgU) 15

Am

1.5 1.0 0.5

BWR MOX

244

Cm

12 9 6 3 0

0.0 0

Cm

3

Burn-up (MWd/kgU) atom/cm3 × 1019

BWR MOX

244

0 0

Burn-up (MWd/kgU) 4

40

Burn-up (MWd/kgU)

1.0

40

30

6

40

0.5 0

20

BWR UO2

9

Burn-up (MWd/kgU) atom/cm3 × 1019

PWR MOX

10

0 0

Burn-up (MWd/kgU) 4

Cm

Burn-up (MWd/kgU)

0.5

40

244

3

0

atom/cm3 × 1018

10

40

6

40

0.0 0

30

PWR UO2 + Gd

9

Burn-up (MWd/kgU) atom/cm3 × 1018

BWR UO2

20

0 0

Burn-up (MWd/kgU) 1.5

10

Burn-up (MWd/kgU)

atom/cm3 × 1017

0

Cm

3

40

0.0

0.0

atom/cm3 × 1019

10

atom/cm3 × 1018

atom/cm3 × 1019

PWR UO2 + Gd

244

6

Burn-up (MWd/kgU) atom/cm3 × 1018

Burn-up (MWd/kgU) 1.5

PWR UO2

9

0 0

atom/cm3 × 1017

0

atom/cm3 × 1018

Am

0.0

0.0

atom/cm3 × 1018

241

D. Rochman et al.

atom/cm3 × 1017

PWR UO2

atom/cm3 × 1018

atom/cm3 × 1019

1.5

NUCLEAR DATA SHEETS

0

10

20

30

Burn-up (MWd/kgU)

40

0

10

20

30

40

Burn-up (MWd/kgU)

CASMO

FIG. 54. Number densities in atoms/cm3 for 3 minor actinides of importance and the 5 different considered assemblies, using CASMO-5.

64

Nuclear Data Uncertainties . . .

NUCLEAR DATA SHEETS

D. Rochman et al.

Relative power distribution

1.2 1 0.8 0.6

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

0.4

235 U Weight percent

5 1 0.2 0.04

239 Pu Weight percent

5 1 0.2 0.04

240 Pu Weight percent

5 1 0.2 0.04

241 Pu Weight percent

5 1 0.2 0.04

242 Pu Weight percent

5 1 0.2 0.04

FIG. 55. (Color online) Pin distribution for the relative power and number densities for the PWR UO2 case, obtained with CASMO-5.

65

Nuclear Data Uncertainties . . .

NUCLEAR DATA SHEETS

D. Rochman et al.

Relative power distribution

1.2 1 0.8 0.6

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

0.4

235 U Weight percent

5 1 0.2 0.04

239 Pu Weight percent

5 1 0.2 0.04

240 Pu Weight percent

5 1 0.2 0.04

241 Pu Weight percent

5 1 0.2 0.04

242 Pu Weight percent

5 1 0.2 0.04

FIG. 56. (Color online) Pin distribution for the relative power and number densities for the PWR UO2 + Gd case, obtained with CASMO-5.

66

Nuclear Data Uncertainties . . .

NUCLEAR DATA SHEETS

D. Rochman et al.

Relative power distribution

1.2 1 0.8 0.6

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

0.4

235 U Weight percent

5 1 0.2 0.04

239 Pu Weight percent

5 1 0.2 0.04

240 Pu Weight percent

5 1 0.2 0.04

241 Pu Weight percent

5 1 0.2 0.04

242 Pu Weight percent

5 1 0.2 0.04

FIG. 57. (Color online) Pin distribution for the relative power and number densities for the PWR MOX case, obtained with CASMO-5.

67

Nuclear Data Uncertainties . . .

NUCLEAR DATA SHEETS

D. Rochman et al.

Relative power distribution

1.2 1 0.8 0.6

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

0.4

235 U Weight percent

5 1 0.2 0.04

239 Pu Weight percent

5 1 0.2 0.04

240 Pu Weight percent

5 1 0.2 0.04

241 Pu Weight percent

5 1 0.2 0.04

242 Pu Weight percent

5 1 0.2 0.04

FIG. 58. (Color online) Pin distribution for the relative power and number densities for the BWR UO2 case, obtained with CASMO-5.

68

Nuclear Data Uncertainties . . .

NUCLEAR DATA SHEETS

D. Rochman et al.

Relative power distribution

1.2 1 0.8 0.6

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

1 MWd/kgU

20 MWd/kgU

40 MWd/kgU

0.4

235 U Weight percent

5 1 0.2 0.04

239 Pu Weight percent

5 1 0.2 0.04

240 Pu Weight percent

5 1 0.2 0.04

241 Pu Weight percent

5 1 0.2 0.04

242 Pu Weight percent

5 1 0.2 0.04

FIG. 59. (Color online) Pin distribution for the relative power and number densities for the BWR MOX case, obtained with CASMO-5.

69

234

2

U

1.0

20

30

2

Cooling (years) 234

PWR UO2 + Gd

2 1

10

20

30

2

Burn-up (MWd/kgU)

2

U

PWR UO2

236

U

1.5 1.0 0.5

0

10

0

Cooling (years) 235

PWR UO2 + Gd

3

U

1

10

20

30

2

Burn-up (MWd/kgU)

10

Cooling (years) 236

PWR UO2 + Gd

U

2

1

0 10

20

2

30

Burn-up (MWd/kgU)

10.0

BWR UO2

10

0

Cooling (years)

234

2

U

1.0

10

20

2

30

Burn-up (MWd/kgU)

Uncertainty (%)

0

0.1

BWR UO2

10

0

Cooling (years)

235

3

U

1

10

20

2

30

Burn-up (MWd/kgU)

Uncertainty (%)

0

BWR UO2

10

Cooling (years)

236

U

2

1

0 20

30

2

Burn-up (MWd/kgU)

3

PWR MOX

10

0

Cooling (years) 234

2 1

20

30

2

Burn-up (MWd/kgU)

1.5

U

10

0

PWR MOX

10

235

3

U

1.0 0.5

20

30

2

Burn-up (MWd/kgU)

3

BWR MOX

10

Cooling (years) 234

U

UPM PSI GRS UU1

2 1 0

10

20

30

2

10

20

Burn-up (MWd/kgU)

30

2

10

Cooling (years)

30

2

PWR MOX

10

Cooling (years) 236

U

2 1

Burn-up (MWd/kgU)

BWR MOX

10

0

Cooling (years) 235

4

U

0.6 0.3

10

20

30

2

Burn-up (MWd/kgU)

0.0 0

20

0 0

Uncertainty (%)

10

10

Burn-up (MWd/kgU)

0.0 0

0

Cooling (years)

Uncertainty (%)

10

Uncertainty (%)

0

Uncertainty (%)

Uncertainty (%)

3

Uncertainty (%)

1

10

Uncertainty (%)

10

Burn-up (MWd/kgU)

Uncertainty (%)

2.0

U

0 0

Uncertainty (%)

235

Uncertainty (%)

0.1

PWR UO2

D. Rochman et al.

Uncertainty (%)

10.0

PWR UO2

NUCLEAR DATA SHEETS

Uncertainty (%)

Uncertainty (%)

Nuclear Data Uncertainties . . .

BWR MOX

10

Cooling (years) 236

U

3 2 1

0

10

20

Burn-up (MWd/kgU)

30

2

10

Cooling (years)

0

10

20

Burn-up (MWd/kgU)

30

2

10

Cooling (years)

FIG. 60. (Color online) Uncertainties in percent on 234,235,236 U number densities for the 5 different considered assemblies. “UU1” refers to the Uppsala calculations performed with JENDL-4.0.

70

Nuclear Data Uncertainties . . .

2

1 0

10

20

30

2

Burn-up (MWd/kgU) 239

0 3

2 1

30

2

2

30

240

PWR UO2 + Gd

1

3

BWR UO2

2 1

10

20

2

30

Burn-up (MWd/kgU)

3

Pu

20

30

2

Burn-up (MWd/kgU)

3

PWR MOX

BWR UO2

240

2 1

10

20

30

2

Burn-up (MWd/kgU)

0 20

30

2

Burn-up (MWd/kgU)

2

BWR MOX

PWR MOX

UPM PSI GRS UU1

1

0 10

20

Burn-up (MWd/kgU)

30

2

10

20

2

30

10

Cooling (years)

241

Pu

2 1

0

240

3

Pu

10

20

30

2

Burn-up (MWd/kgU)

10

20

30

2

PWR MOX

10

Cooling (years) 241

Pu

2 1

Burn-up (MWd/kgU)

BWR MOX

10

0

Cooling (years) 240

2

Pu

1.0 0.5

10

20

30

2

Burn-up (MWd/kgU)

0.0 0

Pu

0 0

1.5

Pu

10

BWR UO2

Cooling (years)

1

Cooling (years) 239

0

10

2

10

Uncertainty (%)

10

241

Burn-up (MWd/kgU)

0 0

10

0 0

3

Pu

2

Cooling (years)

1

3

Pu

1

Cooling (years) 239

30

PWR UO2 + Gd

Cooling (years)

2

10

Uncertainty (%)

10

20

2

10

0 0

10

Burn-up (MWd/kgU)

Uncertainty (%)

0

0

0 0

Cooling (years)

239

Pu

1

3

Pu

241

2

10

2

10

PWR UO2

Cooling (years)

Uncertainty (%)

20

Uncertainty (%)

10

Burn-up (MWd/kgU)

Uncertainty (%)

20

0 0

Uncertainty (%)

10

Burn-up (MWd/kgU)

0

Uncertainty (%)

1

10

Pu

3

Pu

2

Cooling (years)

PWR UO2 + Gd

240

Uncertainty (%)

Uncertainty (%)

3

PWR UO2

Uncertainty (%)

3

Pu

D. Rochman et al.

Uncertainty (%)

239

Uncertainty (%)

PWR UO2

Uncertainty (%)

Uncertainty (%)

3

NUCLEAR DATA SHEETS

BWR MOX

10

Cooling (years) 241

Pu

1

0 0

Cooling (years)

FIG. 61. (Color online) Uncertainties in percent on

10

20

Burn-up (MWd/kgU) 239,240,241

30

2

10

Cooling (years)

0

10

20

Burn-up (MWd/kgU)

30

2

10

Cooling (years)

Pu number densities for the 5 different considered assemblies.

71

Nuclear Data Uncertainties . . . 237

6

Np

15 10 5 0

4 2

30

2

10

10

Cooling (years) 237

PWR UO2 + Gd

15 10 5

20

30

2

Burn-up (MWd/kgU)

9

Np

0 2

30

241

PWR UO2 + Gd

Burn-up (MWd/kgU)

20

BWR UO2

3

10

15 10 5

20

2

30

Burn-up (MWd/kgU)

9

Np

20

30

2

Burn-up (MWd/kgU)

20

PWR MOX

BWR UO2

241

15 10 5

10

20

30

2

Burn-up (MWd/kgU)

0 20

30

2

Burn-up (MWd/kgU)

20

BWR MOX

PWR MOX

15 10 UPM PSI GRS UU1

5 0 0

20

Burn-up (MWd/kgU)

30

2

10

20

2

30

BWR UO2

10

Cooling (years)

244

Cm

15 10 5 0

241

20

Am

10

20

30

2

Burn-up (MWd/kgU)

10

20

30

2

PWR MOX

10

Cooling (years) 244

Cm

15 10 5

Burn-up (MWd/kgU)

BWR MOX

10

0

Cooling (years) 241

10

Am

3 2 1

10

20

30

2

Burn-up (MWd/kgU)

BWR MOX

10

Cooling (years) 244

Cm

8 6 4 2 0

0

10

10

0 0

4

Np

0

Cooling (years)

3

Cooling (years) 237

Cm

5

10

6

10

Uncertainty (%)

10

244

Burn-up (MWd/kgU)

0 0

10

0 0

9

Np

2

Cooling (years)

10

20

Am

3

Cooling (years) 237

30

PWR UO2 + Gd

Cooling (years)

6

10

20

15

10

Uncertainty (%)

10

Uncertainty (%)

0

10

Burn-up (MWd/kgU)

0

0

0

0 0

Cooling (years)

237

5

20

Am

6

10

Cm

10

Cooling (years)

Uncertainty (%)

20

Uncertainty (%)

10

244

15

10

0 0

PWR UO2

0 0

Uncertainty (%)

Uncertainty (%)

20

Uncertainty (%)

20

Am

Uncertainty (%)

20

Uncertainty (%)

10

Burn-up (MWd/kgU)

Uncertainty (%)

241

0 0

Uncertainty (%)

PWR UO2

D. Rochman et al.

Uncertainty (%)

PWR UO2

Uncertainty (%)

Uncertainty (%)

20

NUCLEAR DATA SHEETS

0

10

Cooling (years)

FIG. 62. (Color online) Uncertainties in percent on assemblies.

20

30

2

Burn-up (MWd/kgU) 237

Np,

241

Am and

72

10

Cooling (years) 244

0

10

20

Burn-up (MWd/kgU)

30

2

10

Cooling (years)

Cm number densities for the 5 different considered

Nuclear Data Uncertainties . . .

1.0 0.8

PWR UO2 Σabsorption thermal

1.5 1.0 0.5

10

20

30

40

10

Burn-up (MWd/kgU)

1.2

2.0 Uncertainty (%)

0.8

20

30

20

PWR UO2 + Gd Σabsorption thermal

1.5 1.0 0.5

30

40 2.0 Uncertainty (%)

2.0 1.5 1.0

2.0

10

20

30

30

BWR UO2 Σabsorption thermal

1.5 1.0 0.5

40

0.8

2.0 Uncertainty (%)

0.6

10

30

10

20

30

1.5 1.0 0.5

40

1.0 0.8 UPM PSI UU 1 UU 2 GRS

0.4 0

10

Burn-up (MWd/kgU)

BWR UO2 νΣfission fast

1.0 0.5

40

0 2.0

10

20

30

10

30

40

20

30

40

PWR MOX νΣfission fast

1.5 1.0 0.5

40

0

10

1.0

20

30

40

Burn-up (MWd/kgU)

BWR MOX Σabsorption thermal

0.5

2.0

0.0 20

20

1.5

Burn-up (MWd/kgU)

BWR MOX Σabsorption fast

0.6

10

0.0 0

Uncertainty (%)

1.2

0

Burn-up (MWd/kgU)

PWR MOX Σabsorption thermal

Burn-up (MWd/kgU)

Uncertainty (%)

20

0.0 0

40

0.5

2.0

Uncertainty (%)

Uncertainty (%)

1.0

30

1.0

Burn-up (MWd/kgU)

PWR MOX Σabsorption fast

40

0.0 0

Burn-up (MWd/kgU)

1.2

30

Burn-up (MWd/kgU)

Uncertainty (%)

20

20

PWR UO2 + Gd νΣfission fast

1.5

40

0.0 10

10

Burn-up (MWd/kgU)

BWR UO2 Σabsorption fast

0

0

0.0 0

Burn-up (MWd/kgU)

2.5

0.5

Burn-up (MWd/kgU)

Uncertainty (%)

10

1.0

40

0.0

0.6 0

1.5

Burn-up (MWd/kgU)

PWR UO2 + Gd Σabsorption fast

1.0

PWR UO2 νΣfission fast

0.0 0

Uncertainty (%)

0

Uncertainty (%)

2.0

0.0

0.6

Uncertainty (%)

D. Rochman et al.

Uncertainty (%)

2.0

PWR UO2 Σabsorption fast

Uncertainty (%)

Uncertainty (%)

1.2

NUCLEAR DATA SHEETS

BWR MOX νΣfission fast

1.5 1.0 0.5 0.0

0

10

20 Burn-up (MWd/kgU)

30

40

0

10

20

30

40

Burn-up (MWd/kgU)

FIG. 63. (Color online) Uncertainties in percent for two-group cross sections for the 5 different considered assemblies, obtained with different codes and similar libraries (ENDF/B-VII.0 or ENDF/B-VII.1). In the legend, “UU1” means DRAGON + JENDL-4.0 and “UU2” DRAGON + ENDF/B-VII.1.

73

NUCLEAR DATA SHEETS

1.2 0.8 0.4

1.5 1.0 0.5

20

30

40

Uncertainty (%)

10

30

0.8 0.4

40

1.0 0.5

30

40

0.8 0.4

1.5 Uncertainty (%)

1.2

10

20

30

30

0.5

0.4

1.5 Uncertainty (%)

0.8

10

20

30

20

30

0.8 0.4

10

20

30

10

20 Burn-up (MWd/kgU)

10

30

40

20

40

PWR MOX Σs,2→1

1.0 0.5

40

0

10

1.5

1.5

BWR MOX Σs,1→1

1.0

20

30

40

Burn-up (MWd/kgU)

0.5 0.0

0

0

0.0 0

Uncertainty (%)

1.2

30

0.5

Burn-up (MWd/kgU)

UPM PSI UU1 UU2 GRS

40

BWR UO2 Σs,2→1

1.5

0.5

Burn-up (MWd/kgU)

1.6 BWR MOX νΣfission thermal

30

Burn-up (MWd/kgU)

PWR MOX Σs,1→1

1.0

40

20

1.0

40

0.0 10

10

Burn-up (MWd/kgU)

1.2

0

0

0.0 0

Burn-up (MWd/kgU)

1.6 PWR MOX νΣfission thermal

40

0.5

1.5

BWR UO2 Σs,1→1

1.0

40

30

Burn-up (MWd/kgU)

Uncertainty (%)

20

1.0

40

0.0 10

20

PWR UO2 + Gd Σs,2→1

Burn-up (MWd/kgU)

1.6 BWR UO2 νΣfission thermal

0

10

0.0 0

Uncertainty (%)

20

0 1.5

0.0 10

0.5

Burn-up (MWd/kgU)

PWR UO2 + Gd Σs,1→1

Burn-up (MWd/kgU)

Uncertainty (%)

20

Uncertainty (%)

Uncertainty (%)

1.5

1.2

0

1.0

Burn-up (MWd/kgU)

PWR UO2 + Gd νΣfission thermal

1.6

PWR UO2 Σs,2→1

0.0 0

Uncertainty (%)

10

Burn-up (MWd/kgU)

Uncertainty (%)

1.5

PWR UO2 Σs,1→1

0.0 0

Uncertainty (%)

D. Rochman et al.

Uncertainty (%)

1.6 PWR UO2 νΣfission thermal

Uncertainty (%)

Uncertainty (%)

Nuclear Data Uncertainties . . .

BWR MOX Σs,2→1

1.0 0.5 0.0

0

10

20 Burn-up (MWd/kgU)

30

40

0

10

20

30

40

Burn-up (MWd/kgU)

FIG. 64. (Color online) Uncertainties in percent for two-group cross sections for the 5 different considered assemblies, obtained with different codes and similar libraries (ENDF/B-VII.0 or ENDF/B-VII.1). In the legend, “UU1” means DRAGON + JENDL-4.0 and “UU2” DRAGON + ENDF/B-VII.1.

74

Power distribution uncertainty (%)

Power distribution uncertainty (%)

Power distribution uncertainty (%)

Power distribution uncertainty (%)

Power distribution uncertainty (%)

Nuclear Data Uncertainties . . .

NUCLEAR DATA SHEETS

D. Rochman et al.

0.64 0.16 0.04

PWR UO2

0.01

PSI

UPM

GRS

0.64 0.16 0.04 0.01

PWR UO2 + Gd PSI

UPM

0.64 0.16 0.04 0.01

BWR UO2 PSI

UPM

0.64 0.16 0.04 0.01

PWR MOX PSI

UPM

0.64 0.16 0.04 0.01

BWR MOX PSI

UPM

GRS

FIG. 65. (Color online) Uncertainties in percent for the pin power distributions at 40 MWd/kgU. Same assemblies are presented horizontally.

75

Nuclear Data Uncertainties . . .

PWR UO2

235

U

CASMO + ENDF/B-VII.1 (PSI) DRAGON + JENDL-4.0 (UU)

0.6 0.4 0.2

U

0.6 0.4

10

20

30

40

0

Burn-up (MWd/kgU) 0.5 PWR UO2

238

10

20

30

40

Burn-up (MWd/kgU) 0.5 PWR UO2 + Gd

U

238

U

0.4

Δk∞ (%)

0.4

Δk∞ (%)

235

0.2 0

0.3 0.2 0.1

0.3 0.2 0.1

0.0

0.0 0

10

20

30

40

0

Burn-up (MWd/kgU) 0.4 PWR UO2

239

10

20

30

40

Burn-up (MWd/kgU) 0.4

Pu

PWR UO2 + Gd

0.3

Δk∞ (%)

Δk∞ (%)

D. Rochman et al.

PWR UO2 + Gd

0.8

Δk∞ (%)

Δk∞ (%)

0.8

NUCLEAR DATA SHEETS

0.2 0.1 0.0

239

Pu

0.3 0.2 0.1 0.0

0

10

20

30

40

0

Burn-up (MWd/kgU)

10

20

30

40

Burn-up (MWd/kgU)

FIG. 66. (Color online) k∞ sensitivities for the PWR UO2 assemblies (with and without Gd) to

76

235

U,

238

U and

239

Pu.