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UNIVERSITY OF LJUBLJANAFACULTY OF MATHEMATICS AND PHYSICS

DEPARTMENT OF PHYSICS

Gasper Zerovnik

Use of covariance matrices for estimating uncertainties in reactorcalculations

DOCTORAL DISSERTATION

ADVISER: doc. dr. Andrej Trkov

Ljubljana, 2012

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UNIVERZA V LJUBLJANIFAKULTETA ZA MATEMATIKO IN FIZIKO

ODDELEK ZA FIZIKO

Gasper Zerovnik

Uporaba kovariancnih matrik za oceno negotovosti v reaktorskihpreracunih

DOKTORSKA DISERTACIJA

MENTOR: doc. dr. Andrej Trkov

Ljubljana, 2012

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Abstract

A novel method for the random sampling of correlated parameters has been proposed. The comparison to stan-dard random sampling methods shows significant improvements especially for inherently positive parameterswith large relative uncertainties. Self-shielding factors for standard monitor materials have been calculatedusing deterministic MATSSF and Monte Carlo MCNP code. The MATSSF code and its limitations have beenverified by comparison with the reference MCNP calculations. The calculated self-shielding factors for rho-dium foil of different thicknesses have been validated experimentally. Uncertainties in 55Mn capture resonanceintegral as a function of the level of self-shielding have been estimated starting from the randomly sampled re-sonance parameters. It was shown that for the 55Mn case the reduction of the resonance parameter covariancedata to 640-group cross section covariance matrix, commonly used in neutron dosimetry, does not produceequivalent estimation of the resonance integral uncertainty when the self-shielding effect is significant. Crosssections for the isotopes of tungsten have been adjusted by taking into account experimental isotopic andelemental microscopic cross section data using the generalized least squares method in the program systemGANDR. The new evaluation has been used for the analysis of the FNG-W experiment. The results agreewithin the two-sigma band. Experimental data for additional reactions on elemental tungsten and integral dataare recommended to be included to improve the evaluation.

Keywords: reactor calculations, nuclear data, cross section, resonance parameter, covariance matrix, determi-nistic methods, Monte Carlo methods, random sampling methods, self-shielding, resonance integral, nucleardata adjustment, MCNP, MATSSF, GANDR

PACS: 24.30.-v, 28.20.-v, 28.41.Ak

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Povzetek

Glavni prispevek doktorskega dela je nova metoda za nakljucno vzorcenje koreliranih parametrov, ki omogocakonsistentno vzorcenje poljubne kombinacije koreliranih normalno in log-normalno porazdeljenih parametrov.Primerjava s standardnimi metodami vzorcenja je pokazala bistveno izboljsavo predvsem za pozitivne parame-tre z veliko relativno negotovostjo. Opisana je tudi primerjava izracunov faktorjev samoscitenja za resonancniintegral s programoma: deterministicnim MATSSF in Monte Carlo MCNP. Program MATSSF in njegoveomejitve so bile verificirane z referencnimi izracuni z MCNP. Izracunani faktorji samoscitenja so bili validi-rani tudi eksperimentalno z obsevanjem rodijevih folij razlicnih dimenzij v reaktorju TRIGA in aktivacijskimimeritvami. Analiza primera reakcijskega preseka za zajetje nevtrona v 55Mn je pokazala, da redukcija kova-riancnih matrik za resonancne parametre na kovariancne matrike za reakcijski presek v 640-grupnem priblizkune more zadovoljivo opisati vpliva samoscitenja na resonancni integral. Metoda korekcije reakcijskih prese-kov z upostevanjem eksperimentalnih podatkov je bila demonstrirana za volfram v hitrem delu nevtronskegaspektra. Z uporabo programa GANDR so bili selektivno vkljuceni eksperimentalni podatki za totalni presek velementarni mesanici volframa. Posledicna vkljucitev korelacij med izotopi omogoca natancnejso oceno nego-tovosti integralnih referencnih testnih primerov. Tako je na primer analiza integralnega eksperimenta FNG-Wpokazala ujemanje znotraj dveh standardnih deviacij. Za nadaljnje izboljsanje evaluacij presekov za volfram bibilo potrebno vkljuciti se eksperimentalne podatke za druge reakcije na elementarnem volframu in integralneeksperimente z veliko obcutljivostjo na volfram.

Kljucne besede: reaktorski preracuni, jedrski podatki, reakcijski presek, resonancni parameter, kovariancnamatrika, deterministicne metode, metode Monte Carlo, metode nakljucnega vzorcenja, samoscitenje, reso-nancni integral, korekcije jedrskih podatkov, MCNP, MATSSF, GANDR

PACS: 24.30.-v, 28.20.-v, 28.41.Ak

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Po-wer

Zahvala

V prvi vrsti bi se zahvalil svojima mentorjema, doc. dr. Andreju Trkovu in zal pokojnemu prof. dr. MatjazuRavniku, za pomoc in usmerjanje pri raziskovalnemu delu. Pri raziskovalnem delu sta mi bila v veliko pomoctudi sodelavca Luka Snoj in Ivo Kodeli. Zahvalil bi se tudi vsem ostalim sodelavcem Odseka za reaktor-sko fiziko, se posebej Bojanu Zefranu za neprecenljivo tehnicno pomoc, Albertu Miloccu za pregled vmesneverzije doktorskega dela, Manci Podvratnik za pomoc pri razlagi antikorelacij za asimetricne porazdelitve,Vladimirju Radulovicu za pomoc pri izrazanju v angleskem jeziku in obema tajnicama, Urski Tursic in DarjiStich, za pomoc pri urejanju administrativnih zadev.

Zahvala gre tudi nekaterim zunanjim sodelavcem, in sicer Janezu Zerovniku za pomoc pri matematicnirazlagi metode koreliranega vzorcenja, Dusku Kancevu za sodelovanje pri clanku o uporabi log-normalne po-razdelitve v verjetnostnih varnostnih analizah, Radojku Jacimovicu za izvedbo eksperimenta z obsevanjemrodijevih folij, Robertu Capoteju za pomoc pri Metropolisovemu algoritmu, Dimitriju Rochmanu za posredo-vanje najnovejsih evaluacij in vzorcev iz knjiznice TENDL-2010, Dougu Muiru za pomoc pri uporabi pro-grama GANDR in Luizu Lealu za evaluaciji za mangan in torij.

Posebej bi se rad zahvalil tudi komisiji za pregled doktorskega dela, se posebej prof. dr. Alojzu Kodretuza zelo podroben pregled preliminarne verzije disertacije in stevilne koristne komentarje, ki so pripomogli kizboljsanju kvalitete koncne verzije.

Hvala tudi Maji Remic za lektoriranje.

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Abbreviations

A Nuclide to neutron mass ratio

1 − α Maximum relative neutron energy loss per collision

a∗ Bell factor

C Correlation matrix

corr Correlation coefficient

cov Covariance

f (Resonance) self-shielding factor for a single nuclide

G f (Resonance) self-shielding factor (general definition)

Γ Resonance width

ke f f Effective multiplication factor

L Mean chord length

λ Goldstein-Cohen parameter

µ Mean value

n Number density of the target (nuclides)

N Number density of the projectile particles (neutrons)

ϕ Neutron flux spectrum

ϕ0 External (source) spectrum

ϕg Group g flux

Φ Angular flux

p Probability distribution function (PDF) for a single variable

P Multivariate PDF

RI Resonance integral

s, s0 Neutron source term

S Sensitivity matrix

σ Standard deviation

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Σ(mi) Microscopic cross section

Σ(mi)0 , Σ0 Dilution cross section

Σ Macroscopic cross section

V Covariance matrix

x(m)i Parameter i sample m

ξ Random variable

Contents

1 Introduction 15

2 Nuclear data 182.1 Resonance parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Reaction cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Thermal energy range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Resolved resonance range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.3 Unresolved resonance range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.4 Fast energy range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Other important data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Covariance data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Analytical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Reactor calculations and uncertainty propagation 273.1 Deterministic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.1 Transport equation and its approximations . . . . . . . . . . . . . . . . . . . . . . . . 283.1.1.1 Group approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.1.2 Neutron slowing-down equation and Bondarenko method . . . . . . . . . . 303.1.1.3 Narrow resonance (NR) approximation . . . . . . . . . . . . . . . . . . . . 313.1.1.4 Wide and intermediate resonance (WR and IR) approximations . . . . . . . 32

3.1.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.2.1 First order approximation (sandwich formula) . . . . . . . . . . . . . . . . 333.1.2.2 Higher order approximations . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.1 Monte Carlo (neutron) transport calculations . . . . . . . . . . . . . . . . . . . . . . 34

3.2.1.1 Program MCNP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.2 Random sampling methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.2.1 I. Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.2.2 II. Metropolis algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.2.3 III. Correlated sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.2.4 Weighted sampling of log-normal variables . . . . . . . . . . . . . . . . . . 46

3.2.3 Total Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3 Combinations of deterministic and probabilistic approaches . . . . . . . . . . . . . . . . . . . 50

3.3.1 Combinations of random sampling methods and deterministic transport calculations . 503.3.2 Combinations of sensitivity analysis and Monte Carlo transport calculations . . . . . . 50

3.4 An analytical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Comparison of the random sampling methods . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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14 CONTENTS

4 Resonance integrals and self-shielding factors 614.1 Self-shielding factors for standard monitor materials . . . . . . . . . . . . . . . . . . . . . . 61

4.1.1 MATSSF code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.1.2 Generalization of the mean chord length . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.1.2.1 Isotropic neutron source . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.1.2.2 Cylindrical source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.3 MCNP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.1.4 Verification of the MATSSF mean chord length estimator . . . . . . . . . . . . . . . . 664.1.5 Self-shielding factors of Nickel alloy wire . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1.5.1 Mean chord length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.1.5.2 Multigroup approximation of resonance interference . . . . . . . . . . . . . 68

4.1.6 Resonance self-shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.1.7 Experimental validation of self-shielding factors for Rh foils . . . . . . . . . . . . . . 74

4.2 Example of resonance covariance analysis: 55Mn resonance integral . . . . . . . . . . . . . . 794.2.1 Random sampling procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.2 Resonance integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.3.1 Uncertainty in the resonance integral from resonance parameters . . . . . . 804.2.3.2 Uncertainties in the resonance integral – reactor dosimetry 640-group ap-

proximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Nuclear data adjustment 855.1 EXFOR database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Generalized least squares fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3 Global Assessment of Nuclear Data Requirements (GANDR) . . . . . . . . . . . . . . . . . . 875.4 Example: tungsten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4.1 Tungsten cross section adjustment and correlation treatment . . . . . . . . . . . . . . 885.4.1.1 Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.4.1.2 Full covariance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.4.2 Integral experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6 Conclusions 96

Chapter 1

Introduction

In the early days of nuclear technology the reactor designers were relying on experimental mockups to testnew concepts and optimize the geometry and material composition. With increasing costs of running versa-tile experimental facilities, together with stricter licensing requirements and increased computational powerof modern computers, much of the role of the experiments has been taken over by computational models.Since even the best and most accurate models can only be as good as the input data, there is an increasingdemand from nuclear research, industry, safety and regulatory bodies for best estimate predictions of systemperformance in terms of the design and operational parameters of nuclear reactors, which are to be providedwith their confidence bounds. Uncertainties propagated from evaluated microscopic neutron cross section dataare an important component needed to estimate the accuracy of predictions of such integral quantities. Theconcern about nuclear data uncertainties is also related to the need to ensure that nuclear power will continueto be safe, reliable and economically competitive with other alternative energy options [1].

Uncertainties and errors of reactor parameters (as well as any other physical parameters) can be dividedinto two components: statistical and systematic (in the literature also classified as type A and type B, respec-tively [2]). In principle, by repeating the experiment or simulation many times, the statistical uncertainty canbe reduced to an arbitrarily low value since statistical deviations from expected value are uncorrelated by defi-nition. On the other hand, repeating the experiment does not reduce the systematic errors (or biases). In orderto reduce biases, the measurement method and/or the experimental procedure has to be changed. Furthermore,the classic data treatment with uncertainties (e.g. in the form of standard deviations or variances) does not suf-fice; in general, all data are cross-correlated, therefore they have to be described by a vector of expected valuesand a covariance matrix (or a correlation matrix accompanied by a vector of variances or standard deviations),which contains all information about uncertainties and cross-correlations [3]. (Differential) covariance dataare the basis for (integral) parameter uncertainty estimations and can consistently be adjusted by adding newexperimental data [4].

In reactor calculations, sources of uncertainties may be attributed to two different categories: uncertaintiesin nuclear data and uncertainties due to (physical, geometric, and numerical) approximations in the calcula-tional models. By the use of the Monte Carlo method [5], which in principle enables exact geometry and(almost) exact physics, the latter is mainly reduced to the statistical component. For realistic systems however,there are geometric and material composition uncertainties, but they are NOT intrinsic to the Monte Carlomodel. And the statistical uncertainty is, because of the rapid computer development, becoming less and lesssignificant in reactor parameter calculations.

Therefore, the main sources of uncertainties and biases in (integral) reactor parameters are uncertaintiesand inaccuracies of the nuclear data. Introduction of covariance matrices in nuclear data began in late 1970s.However, due to insufficient computer power and a declining interest in nuclear energy after the Three MileIsland accident, this field of research soon came to a stagnation. The computational requirements for a fullcovariance treatment of nuclear data are immense since merely the incident neutron reaction cross sections(which represent a significant part of nuclear data) include:

• more than 100 isotopes that are highly important in reactor and fusion applications,

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16 CHAPTER 1. INTRODUCTION

• typically, up to a few dozens of reactions that are relevant for a given isotope,

• for many of the important isotopes, cross section definitions for some of the reactions require more than105 points, or equivalently a similar number of resonance parameters.

Because of such data volume, even with modern computers, the covariance treatment has to be limited accord-ing to the problem dealt with. Furthermore, information on experimental data uncertainties and consequentlyalso nuclear data evaluations are often incomplete and inconsistent. The program system GANDR (GlobalAssessment of Nuclear Data Requirements) [6] is the first attempt of a wholly general cross section covariancetreatment. It enables cross-covariance treatment between more than 100 different isotopes, each described byup to 25 custom defined reactions.

In collaboration with foreign research institutions, the co-workers of the Reactor Physics Division at theJozef Stefan Institute calculated correlations between different reactions and energies of chosen isotopes [7].Initial cross sections had been calculated from global nuclear models, and the corresponding covariance ma-trices from model parameter uncertainties by the Monte Carlo method. Introducing experimental data fordifferent reaction cross sections for separate isotopes in GANDR, new (corrected) cross sections and adjustedcovariance matrices were calculated. The main step forward compared to older evaluations is in the use ofmore accurate nuclear models, which include a set of parameters, common to a large number of elements.Older evaluations were mostly based upon experimental data and partial nuclear models, which were lessaccurate and incomplete regarding the consistency of the parameters between elements/isotopes.

Cross sections and covariance matrices, obtained by the above described method, serve as a basis forfurther research. Recently, the covariance treatment has been extended to multiple isotopes (of tungsten) byincluding experimental data for the natural element [8].

After the integral parameters are calculated, another step can be made: by comparing these calculationswith measurements we can adjust the cross sections in order to obtain the best possible agreement of theintegral parameters with a broad range of integral experiments. This procedure requires calculations of integralparameter sensitivity coefficients on cross sections; since the number of these coefficients is quite large, itwould be overly time consuming to calculate them by the Monte Carlo method. Deterministic programs,on the other hand, add biases due to geometrical and/or physical approximations, therefore one has to bevery careful when applying cross section adjustments. The inclusion of integral parameter measurements incross sections is in principle an extension of the inclusion of cross section measurements: when includingexperimental data for a single isotope, the sensitivity coefficients equal 1, and when including experimentaldata for an element, the sensitivity coefficients equal the isotopic abundances in the natural mixture.

Resonance self-shielding is a well-known phenomenon in reactor physics, and is very important e.g. inheterogeneous reactor cores, or in large absorbent samples, irradiated in a neutron field [9]. In essence, the self-shielding correction factor is a measure of internal disturbance of the external neutron field. Different methodsfor self-shielding factor calculations for standard monitor materials used in neutron activation analysis havebeen compared [9] and validated experimentally [10]. The computational model has further been extended,taking into account neutron field anisotropy [11, 12]. However, it has been shown experimentally that forsmall reactors, e.g. the TRIGA research reactor [13] at the Jozef Stefan Institute, the anisotropy effect isimmeasurably small with the existing experimental equipment [14, 15].

The energy grid in GANDR is much too coarse for covariance data to properly take into account the res-onance phenomena. There is typically a number of resonances between neighbouring points of the grid, andthere are large variations of relative uncertainty and correlations between resonance peaks and ’dips’. Due tothe self-shielding effect, the flux is very sensitive to the reaction cross sections, therefore the use of uncertain-ties, averaged over broad energy intervals, may lead to very distorted results. Therefore, it is more appropriateto start with basic data, i.e. the resonance parameters and corresponding covariance matrices. The effect ofuncertainties and correlations in resonance parameters on the resonance integral and indirectly self-shieldingfactors has been estimated [16]. Two different types of resonance-covariance evaluations have been compared,one based on a simple approach starting from published resonance parameters with uncertainties [17] (devel-oped at Nuclear Research and Consultancy Group – NRG) and the other based on a more rigorous resonanceanalysis with the SAMMY code [18] (evaluated at Oak Ridge National Laboratory – ORNL). The resonance

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integral uncertainties have been estimated starting with random sampling of the resonance parameters. Fromeach sampled set an ENDF file was produced. The resonance integrals have been calculated by the determin-istic PREPRO code [19], which uses narrow resonance approximation ([20], pp. 423-433). Different randomsampling methods of correlated parameters have been proposed and compared [21]. The new, so-called cor-related sampling method [22], in principle enables sampling of any combination of correlated normally andlog-normally distributed parameters with arbitrary precision.

The thesis is structured as follows. In Chapter 2, nuclear data, with emphasis on resonance parameters,cross sections and covariance data, are described. Next, reactor calculation and uncertainty propagation meth-ods, in both deterministic and probabilistic approach, are given in Chapter 3. The use of these methods isdemonstrated on self-shielding factor calculations for selected monitor materials and uncertainty estimationsof the manganese neutron capture resonance integral in Chapter 4. Finally, the methodology of nuclear dataadjustment by taking into account experimental data (along with the demonstration on the tungsten case) isexplained in Chapter 5.

Chapter 2

Nuclear data

In order to perform a reactor calculation, one needs the information about geometry and material composition,specific for each system, and, on the other hand, nuclear data, which are general, and should in principle bethe same in all universe. However, nuclear data are only exceptionally known extremely accurately (e.g. fora neutron, incident on a proton). On the contrary, large amounts of differential data include large errors andsignificant correlations, which should be given in the form of (co)variance data along with the expected valuesof the corresponding differential data. Unfortunately, in many evaluations, covariance data are either missingor inconsistent. Nevertheless, there is a tendency to include covariance data in all new evaluations to ensureconsistent uncertainty analyses. Nuclear data are generally based on different semi-empirical quantum orsemi-classical nuclear models, e.g. the nuclear optical model [23], which typically include several parameters,which are either free or have relatively loose constraints. In nuclear data evaluations, these parameters aredetermined in order to fit experimental data, both differential and integral ones. The process of adjustment ofreaction cross section data by taking experimental data into account is described in detail in Section 5.

Presently, nuclear data evaluations are most commonly stored in ENDF-6 (Evaluated Nuclear Data Format,version 6) format [24]. The nuclear data Files in ENDF-6 format are called ENDF files. The ENDF-6 formatis not to be confused with the ENDF/B (Evaluated Nuclear Data Files) nuclear data libraries, maintained atthe US National Nuclear Data Center (NNDC), with the newest evaluation denoted by ENDF/B-VII.1 [25].Nowadays, all major nuclear data libraries, e.g. the European JEFF3.1.1 [26] and the Japanese JENDL-4.0[27] libraries, use the ENDF-6 format.

ENDF-6 format includes all data which may be important in reactor calculations. An ENDF evaluation fora nuclide is divided into several logical blocks, which for historical reasons are called Files [24]. File 1 includesgeneral information about the nuclide, like its rest mass, excitation energy, temperature, number of neutronsper fission (if fissionable), delayed neutron and photon data, and several other parameters. File 2 includesresonance parameters (Section 2.1), and File 3 the cross sections as a function of energy (Section 2.2), whileFiles 30-35 and File 40 contain different types of covariance data (Section 2.4). Other Files include angularand/or energy distributions, thermal neutron scattering data, decay and fission product yields, etc.

The ENDF-6 format is also very general. The cross section may be represented as a sum of the resonancecontributions and the background cross section, or given on an arbitrary energy grid with different interpolationrules between energy points. For instance, if the cross section is given pointwise (at around 105 differentenergies), the file is called PENDF, which is a sub-format of ENDF-6. Covariance data may also be given indifferent forms, e.g. as full (absolute or relative) covariance matrix or as variances and non-zero correlationcoefficients.

2.1 Resonance parameters

Most nuclear reactions (excluding the elastic scattering which in some aspects may not be regarded as a ’real’reaction since the internal structure of all particles remains the same after the collision), especially with neu-trons with relatively low energies, occur via compound nucleus formation. The other type of reactions, moreprobable for higher incident particle energies, is the so-called direct reaction. All reactions via a compound

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2.1. RESONANCE PARAMETERS 19

nucleus exhibit a common feature – they are localized in incident particle energy which corresponds to the en-ergy of an excited state of the compound nucleus. This sudden local increase of the probability for compoundnucleus formation, which is measured in terms of reaction cross section, is called a resonance. The compoundnucleus is unstable and typically decays with a half-life which is several orders of magnitude larger comparedto the time of crossing of the energetic particle through the target nucleus, long enough for the compoundnucleus to ’forget’ all the pre-collision properties of the system incident particle-target (e.g. the impact param-eter) except the total energy, momentum or equivalently wave vector, and angular momentum which have to beconserved. Because of the compound nucleus instability the width of the resonance is finite and inversely pro-portional to the decay time, what follows from the Heisenberg uncertainty principle. In reality, the transitionto direct reactions, for which the interaction times are comparable to those of crossing of the particle throughthe nucleus, is continuous. The reactions that occur at intermediate time scales are called pre-equilibriumreactions ([28], pp. 11). An alternative classification can be given with the number of intranuclear collisions,which is one or two for direct reactions, a few for pre-equilibrium reactions, and many for compound nucleus(or equilibrium or resonance) reactions.

The theory of Breit and Wigner is based on the time-dependent quantum theory of perturbations [29].Although it is clear that the strong nuclear force cannot be realistically described by perturbations, the Breit-Wigner formula [24] for reaction cross sections has the correct form, i.e. energy dependence, and for isolatedresonances gives excellent fits to experimental cross section data at the peak. Contradictions may appear onlywhen attempts are made to interpret the values of the parameters obtained from such a fit. The reason forthe correctness of the form is that it only depends on the condition that the reaction proceeds through anisolated (i.e. long-lived) intermediate state ([30], pp. 259). If the resonances can be treated separately, thesingle-level Breit-Wigner (SLBW) formula is used. The multi-level Breit-Wigner (MLBW) formula takes intoaccount also interference between the neighbouring resonances. However, it does not take into account anyinterference between different reactions.

In recent years, there is a tendency to use the more general R-matrix theory [30] (for the so-called R-Matrix Limited (RML) format in ENDF-6) or its approximation, the Reich-Moore (RM) representation [31],to describe the neutron induced resonances in nuclear data evaluations. In R-matrix theory of nuclear reactions,the collision matrix U, from which the reaction cross sections can be derived, and which describes the relationbetween the incoming and outgoing wave functions, is expressed in terms of the channel matrix R, whichcontains all information about resonance parameters ([32], pp. 51):

Rcc′ =∑λ

γλcγλc′

Eλ − E. (2.1)

c and c′ denote reaction channels, i.e. the pairs of incoming/outgoing particles. In reactor physics, the incom-ing pair c is usually a neutron and the target nucleus. Eλ are the resonance energies, and γλc are the decayamplitudes, which are in close relation with resonance widths Γλc = γ

2λc.

In the full R-matrix treatment, the cross sections as functions of incident particle energy and possiblythe scattering angle depend on the parameters of all resonances (multi-level treatment) and all channels ofpossible reactions in a very complicated way (for details see [32], pp. 51-54, or [24], pp. 326-336). The onlyassumptions of the R-matrix theory are ([30], pp. 262):

• non-relativistic quantum mechanics is valid,

• all processes which lead to the formation of more than two outgoing particles, are neglected (rare butnot impossible three-body decays, like the ternary fission, may be approximately described as two sub-sequent two-body decays),

• all creation and destruction processes are neglected (this would e.g. prohibit photon production – the(n,γ) reaction, but this restriction can be removed with perturbation treatment)

• all polarizing (non-central) forces on the incoming/outgoing particles have finite range.

20 CHAPTER 2. NUCLEAR DATA

The Reich-Moore approximation, which is presently the most frequently used resonance parameter repre-sentation in the evaluated nuclear data libraries, is a limited subset of the full R-matrix formalism. It is limitedto one entrance channel (neutron) and up to four exit channels (neutron, gamma, and two fission channels),which is sufficient for most resonances in the resolved resonance range. The interference between reactionchannels as well as resonances is properly treated – as in the R-matrix format.

For the sake of completeness, let us mention also the rather archaic Adler-Adler (AA) formalism, whichcan be used in ENDF-6 File 2 for resonance parameters, but this method is slowly being abandoned.

2.2 Reaction cross section

For practical purposes it is most convenient to express neutron-nucleus interactions in the form of cross sectionwhich is a function of incident neutron energy E on a target at rest (zero temperature) or moving at thermalenergies. The microscopic cross section1 Σ(mi)(E) is the property of a nucleus and is proportional to theprobability for the reaction to occur. It corresponds to the effective cross section area of the nucleus, in whichthe reaction occurs if a neutron with kinetic energy E crosses the nucleus. The sum of the cross sections of allpartial reactions r is called the total cross section:

Σ(mi)t =

∑r

Σ(mi)r . (2.2)

Physically, the microscopic cross section depends on the motion of the neutron relative to the target. Inreality, any material is at non-zero temperature, therefore even for monoenergetic neutrons the probability for areaction varies. The measurable quantity is the effective cross section Σ(mi)(E, T ), which is the zero temperaturecross section Σ(mi)(E, 0), averaged over the distribution of all possible target velocities f (v, T ), which may wellbe assumed Maxwellian for each velocity component:

Σ(mi)(E, T ) =∫

f (v′, T )Σ(mi)(m(v − v′) · (v − v′)/2, 0)d3v′, (2.3)

where the simple relation E = mv2/2 was used. If the temperature of the material is known, only Σ(mi)(E,T ) isactually relevant. Consequently, nuclear data libraries for applications usually include tabulated effective crosssections for room temperature and possibly some other temperatures, so that the user does not need to evaluateEq. (2.3). The temperature effect is important due to the Doppler broadening of resonances (for more detailssee Section 2.2.2).

For convenience, the effective cross section Σ(mi)(E, T ) will simply be written as Σ(mi)(E) in subsequentsections of this text, while the temperature effects (default: room temperature) will be implicitly taken intoaccount when necessary.

2.2.1 Thermal energy range

For most nuclei, only reactions with neutral outgoing particles, i.e. elastic scattering (n,n) and radiative capture(n,γ), are energetically possible with thermal incident neutrons. For some light nuclei, some other absorptionreactions, e.g. the (n,α) reaction on 10B, are exothermic. On the other hand, some heavy nuclei can be fissionedwith thermal neutrons, which are essential for the existence of thermal multiplication systems.

All non-threshold reactions except the elastic scattering exhibit a 1/v (or 1/√

E) cross section dependencefar below the lowest energy resonance. Some strong resonance absorbers or neutron poisons, e.g. 113Cd or135Xe, have low-energy resonances, however in general resonances below 1 eV are extremely rare, which is thereason why the absorption cross section dependence of most nuclides is very close to 1/v at thermal energies(around 0.0253 eV). The 1/v relationship may simply be explained by assuming that the reaction probabilityis proportional to the length of time while the incident neutron is in contact with the target nucleus, which is

1In standard literature, the microscopic reaction cross section is most commonly denoted by σ. However, due to extensive use ofthe symbol σ for standard deviation and uncertainty in this work, a rather non-conventional notation Σ(mi) is used for the microscopiccross section.

2.2. REACTION CROSS SECTION 21

inversely proportional to the neutron speed v. Formally, 1/v relationship is derived from the general R-matrixtheory of nuclear reactions through compound nucleus formation or its approximations, e.g. the Breit-Wignerformula for absorption cross section assuming the lowest resonance energy is well above the observed range([33], pp. 65-66). Furthermore, since the reaction probability is proportional to the time of ’direct contact’between a neutron and a nucleus, the reaction rate is proportional to the density of neutrons in the medium,assuming a constant density of the nuclei. Consequently, the effective thermal absorption cross section istemperature independent.

The event of elastic scattering may occur via compound nucleus formation or via direct reaction of potentialscattering, i.e. solid ball elastic collision. The latter is practically independent of the incident neutron energy,and its contribution frequently dominates the contribution from resonance scattering at thermal energies, wherethe elastic scattering cross section is consequently energy independent. Trivially, the thermal elastic scatteringcross section is also temperature independent.

In the thermal energy range, neutrons may experience coherent scattering on the crystal lattice due totheir wave nature (see introduction of the Section 3). Consequently, interference can be measured on theoutgoing neutron waves. In other words, the scattering becomes highly anisotropic even in the centre-of-masscoordinate system of neutron and target nucleus – the differential scattering cross section is a complicatedfunction of the outgoing neutron angle. However, these data are separated from other cross section data in File7 of the ENDF-6 format (for details, see [24], pp. 144-152) and will not be used in this work.

2.2.2 Resolved resonance range

Above the thermal energy range, the lower bound of the resolved resonance range is defined by the resonancewith the lowest energy which is, in general, decreasing with nucleus size. In the resolved resonance range,where all resonances are clearly separable and measurable, the cross sections for reactions as a function ofneutron energy may be represented in two ways:

• as a sum of contributions from all resonances plus some background2, or

• as a tabulated function of energy Σ(mi)(E) on an extremely fine mesh.

In general, the correspondence between the resonance parameters and cross section is quite complicated.For illustration, in the most primitive SLBW representation, the absorption cross section for any partial reactionr, scattering and total cross sections around a single resonance equal ([33], pp. 62-63):

Σ(mi)r = πo2g

ΓnΓr

(E − E0)2 + Γ2/4, (2.4)

Σ(mi)s = πo2g

Γ2n

(E − E0)2 + Γ2/4+ 2

√πo2gΣ(mi)

pΓn(E − E0)

(E − E0)2 + Γ2/4+ Σ

(mi)p , (2.5)

and

Σ(mi)t =

∑i,n

πo2gΓnΓi

(E − E0)2 + Γ2/4+ πo2g

Γ2n

(E − E0)2 + Γ2/4+ 2

√πo2gΣ(mi)

pΓn(E − E0)

(E − E0)2 + Γ2/4+ Σ

(mi)p

= πo2gΓnΓ

(E − E0)2 + Γ2/4+ 2

√πo2gΣ(mi)

pΓn(E − E0)

(E − E0)2 + Γ2/4+ Σ

(mi)p , (2.6)

respectively. The parameters involved in Eqs. (2.4)-(2.6) are:

• o = ~/√

2µE, where µ is the reduced mass of the neutron-target system;

2Information about individual resonances is often incomplete, therefore an additional energy dependent term, also called ’thebackground cross section’, is sometimes added to the contributions from all resonances in order to obtain a consistent reaction crosssection as a function of energy.


• g = (2J +1)/((2s+1)(2l+1)), where s = 1/2, l and J are the neutron, the target nucleus, and compoundnucleus spins, respectively; for s-waves, J equals l ± 1/2, while for p-, d-, and higher order waves, theorbital angular momentum ℓ of the neutron-target system is reflected in additional possibilities for thequantum number J;

• partial resonance widths Γi, corresponding to different reaction channels;

• total resonance width Γ =∑

r

Γr, proportional to the decay constant of the compound nucleus;

• resonance peak energy E0;

• potential scattering cross section Σ(mi)p .

The second term in Eq. (2.5) represents the interference between the potential elastic scattering and elasticscattering via compound nucleus, resulting in an asymmetric elastic scattering cross section near the resonancepeak. Cross section for any reaction which occurs via compound nucleus only, is symmetric around theresonance peak. In the resolved resonance range, all reactions except elastic scattering exhibit such behaviour,as seen from Eq. (2.4).

An important property of the cross section in the resonance range is the Doppler broadening of resonancesdue to the thermal motion of the target nuclei. The exact mathematical expression is rather complex; anapproximation for the absorption cross section, for example, may be obtained by inserting Eq. (2.4) intoEq. (2.3) and assuming Maxwellian distribution for f . As a curiosity, let us mention that the area belowthe resonance, equal to the integral of the cross section over energy, is temperature independent, while inthe high temperature limit the resonance shape approaches Gaussian (rather than Lorenzian) distribution withasymptotic width proportional to

√T ([34], pp. 48-52). The temperature dependence is connected to the

self-shielding effects, discussed in Section 4.

2.2.3 Unresolved resonance range

In general, the product of the density of resonances and their average widths increases with increasing inci-dent neutron energy. The unresolved resonance range starts with neutron energies where the neighbouringresonances are so narrow and close together that they are hard to measure individually, but still sufficiently farapart that the cross section is a complex function of energy with many sharp peaks and dips between them.These oscillations of the cross section have a significant impact on certain integral parameters, therefore theyhave to be taken into account. Since the resonances are virtually immeasurable, the resonance energies andother resonance parameters are predicted statistically ([20], pp. 410-420).

The boundary between the resolved and unresolved resonance range may be very hard to define for cer-tain nuclides. Nuclear data evaluations may include some small unresolved resonances in the upper part ofthe resolved resonance range, which may be added to the background cross section, or they include ficti-tious resonances, placed randomly to fulfill the statistical expectation. However, the background cross sectionrepresentation cannot take into account the self-shielding effect (Section 4.1 and Ref. [9]), therefore the contri-bution of the unresolved resonances to cross section in resolved resonance range cannot be allowed to becomesignificant [24] in order to ensure accurate and reliable use of cross section data in all circumstances.

Furthermore, in general, the d-wave resonances (corresponding to orbital angular momentum ℓ = 2) arenarrower than p-wave (ℓ = 1), which are narrower than s-wave resonances (ℓ = 0), thus the unresolved regionfor d-wave resonances is expected to begin at lower energies compared to the p-wave and s-wave resonances.The current format does not permit different resolved-unresolved cutoff-points for different ℓ-values [24].

2.2.4 Fast energy range

When incident neutron energy increases further, the resonances are spaced so closely together that the oscil-lations in the cross section simply average out. Consequently, the resonances seem to disappear and the crosssection is again a smooth function of energy as e.g. the absorption cross section in thermal energy range.

2.3. OTHER IMPORTANT DATA 23

Therefore, the cross section in this range can be tabulated in a relatively coarse mesh while ensuring suffi-cient accuracy. The fast energy range usually corresponds to neutron energies above a few MeV, but againsignificantly varies with mass number.

The accuracy of the partial reaction cross section measurements in the fast energy range is compromiseddue to the number of threshold reactions (reaction channels of p-, α-, other single-, and multi-particle emissionsare endothermic for most target nuclei) which appear in MeV range and are sometimes hard to measure sepa-rately both by direct measurement, which is favourable for total cross sections, and neutron activation analysis(NAA). In NAA, the reaction rate is determined via measurements of the activation products. If a nuclide is anactivation product of several reactions and/or product of their decay chains, it is difficult or even impossible toseparate out single reactions. Activation production rates also depend on γ-emission probabilities, which aresometimes not known accurately enough.

Temperature effects are negligible in this energy range.

2.3 Other important data

Apart from cross sections, there are other nuclear data which may be important for neutron transport calcula-tions: the average number of neutrons per fission ν(E), fission spectrum χ(E), i.e. the energy distribution ofthe neutrons emitted after the fission event, differential angle/energy and double-differential cross sections etc.For other purposes, e.g. dose rate estimations, decay and fission product yields, photon production yields, etc.,are also included in the ENDF data libraries. However, these data will not be used in this work.

2.4 Covariance data

In reality, measured quantities are never known exactly. Their uncertainties may be expressed in terms ofthe standard deviations (second central moments) of the parameter distributions (while nominal or expectedvalues are the first moments of the distributions), or better as entire probability distributions. Even if derivedfrom nuclear models, cross section and other nuclear data include uncertainties since these models inevitablyinclude parameters, which are unknown or at least uncertain to some degree. For example, the microscopicscattering cross section of a neutron incident on a proton (1H nucleus) can be derived analytically from non-relativistic quantum mechanics, but the result includes parameters which do not follow directly from the theoryand need to be measured, such as proton/neutron mass and size. Therefore, in principle, all nuclear data includeuncertainties.

Furthermore, nuclear data are always inter-correlated to a certain degree, for instance cross section betweendifferent energies, different reactions, and even different materials. The correlations are induced by using thesame or similar instruments and/or methods for direct measurements of different nuclear data, by nuclearmodel parameter fitting, and finally when adjusting nuclear data by taking integral experiments into account.To ensure consistent reactor calculations, it is of utmost importance to correctly estimate uncertainties andcorrelations of nuclear data.

Nuclear data uncertainties are usually expressed in the form of covariance matrices. Generally, a covari-ance matrix V corresponds to a vector of parameters x with components xi when

Vi j = cov(xi, x j) =⟨(xi − x0i)(x j − x0 j)

⟩= σiσ jCi j, (2.7)

where x0i and σi are the expected value and standard deviation of the parameter xi, respectively, Ci j is thecoefficient of correlation between parameters xi and x j, and ⟨⟩ denotes the averaging operator. Since thecorrelation matrix C is symmetric, so is the covariance matrix by definition (2.7).

Even though in recent years there has been a tendency to include covariances to all new nuclear dataevaluations, a significant fraction of the data still either lack covariances or include inconsistent ones. The lattermay be misleading and therefore even worse than no data in some cases. Especially cross-material covariancesare very rare even in new evaluations, which is understandable regarding the amount of information it wouldrequire.


For the purposes of this work, the most important covariance data are: the covariances of resonance pa-rameters (which are included in File 32 according to the ENDF-6 format), and cross section covariances (File33). One of the important findings (for details see Section 4.2) is that these two (File 32 and File 33) represen-tations are not always equivalent. To roughly illustrate this phenomenon, the transformation of the uncertaintyof the resonance parameters to uncertainty of the cross section as a function of energy is shown on a simplifiedexample below.

2.4.1 Analytical example

In the SLBW formalism without Doppler broadening, radiative capture cross section in the vicinity of anisolated resonance can be expressed as

Σ(mi)γ (E) = K

ΓnΓγ

(E − E0)2 + Γ2/4, (2.8)

where K = πo2g has a weak energy dependence (1/E), which can be neglected in the vicinity of a resonancepeak, where |E − E0| ≪ E.

For instance, let us assume both the neutron width Γn and the capture width Γγ ’by chance’ equal one halfof the total resonance width Γ with 100% correlation:

Γ = Γ0 ± ∆Γ, (2.9)

Γn = Γγ =12

(Γ0 ± ∆Γ) , (2.10)

where ∆Γ ≪ Γ0, while the relative uncertainty of the resonance energy E0 is negligible. Then

Σ(mi)γ (E) = K

(Γ0 ± ∆Γ)2/4(E − E0)2 + (Γ0 ± ∆Γ)2/4

≃ KΓ2

0/4(1 ± 2∆Γ/Γ0)

(E − E0)2 + Γ20/4(1 ± 2∆Γ/Γ0)

= KΓ2

0/4

(E − E0)2 + Γ20/4

1 ± 2∆Γ/Γ0

1 ± Γ0∆Γ/[2(E − E0)2 + Γ20/2]

≃ (Σ(mi)γ )0(E)

1 ± ∆Γ[4(E − E0)2 + Γ20] + Γ2

0∆Γ

Γ0[2(E − E0)2 + Γ20/2]

, (2.11)

where

(Σ(mi)γ )0(E) = K

Γ20/4

(E − E0)2 + Γ20/4

. (2.12)

Absolute and relative cross section uncertainty equal

∆Σ(mi)γ (E) =

∆Γ

Γ0

4(E − E0)2 + 2Γ20

2(E − E0)2 + Γ20/2

(Σ(mi)γ )0(E) =

∆Γ

Γ0K fA(E) (2.13)

and∆Σ

(mi)γ (E)

(Σ(mi)γ )0(E)

=∆Γ

Γ0

4(E − E0)2 + 2Γ20

2(E − E0)2 + Γ20/2=∆Γ

Γ0fR(E), (2.14)

respectively, where fA(E) and fR(E) denote the ’form factors’ of the absolute and relative radiative capturecross section uncertainties as functions of energy. These functions are plotted in Fig. 2.1. Both relative andabsolute cross section uncertainties are largest in the resonance peak, but far away from the peak, relativeuncertainty converges to a certain finite value while the absolute uncertainty converges to 0, together withcontribution of the resonance to the cross section.

2.4. COVARIANCE DATA 25

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

f

(E-E0)/ 0

fR fA

Figure 2.1: Shape of the relative and absolute uncertainty of the capture cross section ( fR and fA) due touncertainty in the resonance width in vicinity of the resonance peak.

From the example, described by Eqs. (2.13) and (2.14) and Fig. 2.1, it can easily be inferred that if theresonances and their covariances are to be tabulated as functions of energy, several points per resonance haveto be used in order to ensure sufficient accuracy of either the relative or the absolute covariance matrix. Itssize has to be extended by a large factor when transforming from covariance matrix of resonance parametersto covariance matrix of cross sections.

On the other hand, in reality, the total resonance width is often known more precisely than individual chan-nel widths. Consequently, anti-correlations between channels may be induced. If in Eq. (2.8) the uncertaintyof the total width Γ is neglected, the relative total cross section uncertainty is constant with energy. In thesecase, one may argue that the size of the covariance matrix can stay the same or even be reduced when trans-forming from resonance parameter representation to cross section representation. A counterargument to thelast statement is the ORNL evaluation of 55Mn (for extensive explanation see Section 4.2).

Let us further extend this case to two separate resonances, both with widths, uncertainties, and correlationsbetween widths of different reaction channels equal to the previous single resonance case. The two resonancesare referred to as ’equal’ due to equal widths, even though they are shifted – located at different energies.Fig. 2.2 shows the shape of the relative uncertainty of the capture cross section between the two resonances,which are separated by 2 total widths. If the resonances are fully correlated (solid curve), the variation of therelative cross section uncertainty is slightly decreased. If they are fully anti-correlated, even the relative uncer-tainty reaches zero in the dip between the resonances. Clearly, the behaviour of the cross section uncertaintygreatly depends on the amount of correlation between the resonances.

Considering the cross section of a medium or heavy nuclide with a large number of resonances with a widerange of widths, uncertainties and correlations, the uncertainty of integral parameters is very hard to predictanalytically, and the final numerical results may often seem counter-intuitive.


0.0 0.5 1.0 1.5 2.00

1

2

3

4

f R

(E-E0)/ 0

100% correlation 100% anti-correlation

Figure 2.2: Shape of the relative uncertainty of the capture cross section ( fR) due to uncertainty in the resonancewidths between two equal resonances, separated by 2 total widths, for different amount of correlation betweenboth resonances. E0 denotes the resonance energy of the lower resonance.

Chapter 3

Reactor calculations and uncertaintypropagation

In reactor calculations, sources of uncertainties may be attributed to two different categories: uncertainties innuclear data and errors due to physical, geometry, and numerical approximations in the calculational models.

The estimation of (integral) parameter uncertainties may be performed in two steps. Integral parametersare calculated from input parameters, either by deterministic (Section 3.1.1) or Monte Carlo (Section 3.2.1)methods. Additionally, the uncertainty propagation from input data to the final parameters can also be treatedeither by deterministic (Section 3.1.2) or probabilistic (Section 3.2.2) methods. The deterministic uncertaintypropagation methods are based on sensitivity analysis, while the probabilistic uncertainty propagation methodsrely on random sampling of input parameters combined with either deterministic or probabilistic calculationsof final parameters for each sample.

To accurately describe a reactor core (e.g. to calculate multiplication factor, neutron reaction rates, poweror flux distribution) it is sufficient to take into account the transport of neutrons only. Other particles, likephotons, have negligible effect on neutron multiplication. However, their effect is not negligible when calcu-lating the dose rates, for example. When performing a neutron transport calculation in a medium (material),interaction between the pairs of neutrons, neutron decay as well as relativistic effects and wave nature of theneutrons (except for thermal neutron scattering on crystal lattices) can be neglected:

• Typically, the neutron density is lower than 1014 m−3, which is several orders of magnitude less than anymaterial density. Neutron-neutron interactions are thus very rare and may be neglected.

• Free neutrons decay with a half-life of about 10.3 minutes. Since mean neutron lifetimes in criticalsystems are typically well below the millisecond range, free neutrons do not have enough time to decaybefore being absorbed.

• The majority of neutrons (with rest energy mc2 = 938 MeV) in a reactor have the kinetic energy E <

2 MeV, which corresponds to the relative speed

β =vc=

√2Emc2 < 0.065 (3.1)

which is only a small fraction of the speed of light c.

• De Broglie neutron wavelength equals:

λB =h

mv=

hc

mc2√

2E/(mc2)< 0.18 nm, (3.2)

where hc = 1240 eVnm and E > Eth = 0.0253 eV. Thermal neutron wavelength is comparable to atypical crystal lattice unit cell constant, which implies interference in scattering. The wavelength offaster neutrons is smaller, therefore their wave nature can be neglected.

27

28 CHAPTER 3. REACTOR CALCULATIONS AND UNCERTAINTY PROPAGATION

The physical equations, which describe neutron transport, are very general, and can be applied to anyneutron multiplication system, either close to critical (e.g. reactor core), deeply subcritical (e.g. so-calledAccelerator Driven Systems), or even system with external neutron source(s) but no internal multiplication.Roughly speaking, there are two major concepts:

• Deterministic Boltzmann transport equation (Section 3.1.1) along with its approximations, e.g. theslowing-down equation.

• Direct simulation of individual particles (neutrons) using the Monte Carlo method (Section 3.2.1).

In principle, in addition to the uncertainties in nuclear data and tolerances in geometry and material data, MonteCarlo method produces statistical uncertainty only, whereas deterministic methods essentially introduce biasesdue to model approximations.

Most of the methods for uncertainty propagation, described in this section, are general and may be appliedto almost any data with uncertainties or probability distributions, even though in this work all applications arerestricted to nuclear data. In nuclear data evaluations including uncertainties, cross sections, resonance andother parameters are most frequently given in the form of expected values and corresponding covariance ma-trices. The representation with covariance matrices is very well suited for deterministic sensitivity analysis. Ifrandom sampling is chosen, on the other hand, one has to apply multivariate probability distributions for inputparameters, which are to be constructed from expected values and (co)variances if no additional informationis present. Usually, normal or log-normal (for inherently positive parameters with large relative uncertainties)distribution is assumed, though other distributions, such as uniform, can sometimes be used instead.

3.1 Deterministic approach

3.1.1 Transport equation and its approximations

Taking into account the well-founded assumptions of non-relativistic kinematics, no neutron-neutron interac-tions and point neutron-nucleus interactions, the Boltzmann neutron transport equation ([34], pp. 103-117)describes the transport of neutrons in any system with or without neutron multiplication. The Boltzmannequation is a continuity equation. The neutron field is usually described by the neutron flux Φ. In general, theflux is a function of the position, neutron kinetic energy and direction (i.e. velocity vector), and in the case oftransients, also a function of time:

Φ = Φ(r, E, Ω, t). (3.3)

The flux is usually defined so that

Φ(r, E, Ω, t)v

d3rdEdΩ = N (r, E, Ω, t)d3rdEdΩ (3.4)

equals the number of neutrons in differential volume d3r around r, with kinetic energy between E and E + dE,and direction within dΩ around Ω, at time instant t.

The flux is not a measurable physical quantity. It is a product of two physical quantities: number density Nand particle speed v. However, the flux is a very handy mathematical tool which, multiplied by the macroscopiccross section Σi for reaction i:

Σi =∑

j

n jΣ(mi)i, j , (3.5)

where n j is the number density of the nuclide j, represents the reaction rate density, a measurable integralparameter:

Ri(r, t) =∫ ∞

0

∫4πΣi(E)Φ(r, E, Ω, t)dEdΩ. (3.6)

3.1. DETERMINISTIC APPROACH 29

Transport equation is frequently written in the form of a continuity equation for an observation point,which travels along with neutrons of a chosen energy and direction:

1v∂Φ

∂t+ Ω · ∇Φ = (3.7)

= −Σt (r, E)Φ +∫ ∞

0

∫4πΣ

(d)s (r, E′→E, Ω′→Ω)Φ(r, E′, Ω′, t)dE′dΩ′ + s(r, E, Ω, t)

with initial condition Φ(r, E, Ω, 0) = Φ0(r, E, Ω) and the boundary condition Φ(rS , E, Ω, t) = 0 for all rS onthe boundary surface S of the observed system and Ω · n < 0, where n is a vector, normal to surface S in thepoint rS . This form of the boundary condition implies that any neutron crossing the surface S does not returnto the system, described by Eq. (3.7).

The terms on the left-hand side (LHS) of Eq. (3.7) form the substantial or material derivative of the conti-nuity equation, while on the right-hand side (RHS) neutron sources and sinks are taken into account. The firstterm on RHS describes the decrease of the number of neutrons because of all interactions with matter. Thesecond term on RHS represents the increase of the number of neutrons with chosen direction Ω and energyE because of scattering of neutrons from other energies E′ and directions Ω′. The last term on RHS is theneutron source:

s(r, E, Ω, t) =χ(E)4π

∫ ∞

0

∫4πν(E′)Σ f (r, E′)Φ(r, E′, Ω′, t)dE′dΩ′ + s0(r, E, Ω, t), (3.8)

where χ(E) is the fission spectrum, ν(E) average number of neutrons per fission induced by incident neutronof energy E, Σ f the fission cross section, and s0 the external neutron source. The external neutron sourceis a source of neutrons, independent of the neutron multiplication in the system. Usually, these neutrons areproduced at constant rate either from a combination of α-decay (e.g. of 241Am) and (α,n) reaction (e.g. on9Be) or by spontaneous fission (e.g. of 252Cf). In power reactors, for instance, the external source s0 is usedfor reactor start-up and is negligible near full power while the fission source is non-zero in fuel regions only.

Since the interaction between pairs of neutrons is negligible, the transport equation (3.7) is linear. Inspite of that, its solutions are far from trivial having 7 independent variables in general. Furthermore, themacroscopic reaction cross sections are very complicated functions of energy and discontinuous functions ofthe position. Therefore, in almost all realistic cases where analytical solution does not exist, numerical and/orphysical approximations have to be used. In reactor physics, there are several physical approximations of thetransport equation, including the diffusion equation, point kinetics equations, energy group approximations,PN and S N approximations.

3.1.1.1 Group approximations

With discretization of the energy variable, the integro-differential transport equation (3.7) is reduced to asystem of linear differential equations for group fluxes Φg. The effective group cross sections are defined insuch a way that the reaction rate is conserved:

ΣgΦg =

∫ Eg+1

Eg

ΣΦdE, Φg =

∫ Eg+1

Eg

ΦdE, (3.9)

where Eg and Eg+1 are the lower and upper energy of the group interval g, respectively.The effective group cross sections have to be pre-calculated, therefore for serious calculations at least two

approximations are combined. In the beginning, the shape of the neutron spectrum, i.e. the energy dependenceof the flux, within the group intervals is unknown, therefore it has to be assumed e.g. from theory. Thedenser the energy grid, the more accurate the approximation. Therefore typically, in the first phase the neutronspectrum is accurately calculated on a fine energy grid and a coarse spatial grid and used for estimation ofgroup cross sections, which are in the second phase used for a more accurate fine-grid spatial flux distributioncalculation in a few- or even only one-energy group approximation.


3.1.1.2 Neutron slowing-down equation and Bondarenko method

One of the approximations of the neutron transport equation is the neutron slowing down equation. For infinitehomogeneous isotropic non-multiplying medium in a steady state Eq. (3.7) reduces to

0 = −Σt(E)ϕ(E) +∫ ∞

0Σ

(d)s (E′→E)ϕ(E′)dE′ + s(E), (3.10)

where ϕ(E) is the neutron spectrum. Eq. (3.10) is a form of neutron slowing-down equation for infinitemedium. The flux depends on neutron energy only, and is softer, i.e. the average neutron energy is lower,than the source spectrum in s(E) unless the source spectrum is already in thermal equilibrium. The spectrumis softer compared to the source since above thermal energies there is no neutron scattering to higher energies(so-called upscattering) and the kinetic energy of neutrons is dissipated to the surrounding nuclei, hence thename slowing-down equation.

If upscattering is neglected and isotropic neutron scattering in the laboratory system is assumed, the scat-tering term in Eq. (3.10) can be written as ([34], pp. 34-45):∫ ∞

0Σ

(d)s (E′→E)ϕ(E′)dE′ =

∑k

∫ E/αk

E

Σs,k(E′)(1 − αk)E′

ϕ(E′)dE′ +∫ E/αm

E

Σm

(1 − αm)E′ϕ(E′)dE′, (3.11)

where αk = [(Ak − 1)/(Ak + 1)]2 is the maximal possible fraction of neutron kinetic energy loss per collisionand index k defines isotope. In Eq. (3.11), the scattering cross section Σs(E) is divided into:

Σs(E) =∑

k

Σs,k(E) + Σm, (3.12)

where Σs,k is the scattering cross section of the nuclide k, and Σm is the isolated constant contribution tothe scattering cross section. Historically, it is called the moderator cross section since in realistic systems itoften describes the scattering contribution of a moderator, which is homogeneously mixed with resonance ab-sorber(s). However, if there is no moderator present, Σm simply represents the potential scattering componentof the absorber, which is always present since there is no ideal absorber.

In the resonance range, between around 0.5 eV and 2 MeV, the external (reactor) spectrum is often takenas ϕ0(E) = 1/E, which is the homogeneous solution of the slowing-down Eq. (3.10), corresponding to asystem with no source (or more realistically, for energies below the fission source), for energies above thermalequilibrium so that the upscattering can be neglected, further assuming non-absorbent medium with constantscattering cross section (Σs(E) = Σm = Σt =⇒ Σs,k = 0), which is a relatively good approximation for reactormoderators, e.g. light water and graphite in TRIGA:∫ ∞

0Σ

(d)s (E′→E)ϕ0(E′)dE′ =

∫ E/αm

E

Σm

(1 − αm)E′dE′

E′=Σm

E≃ Σt

E= Σtϕ0(E). (3.13)

In a near-1/E spectrum, the influence of resonance absorption of the moderator contribution in Eq. (3.12)on the scattering rate can be neglected and the scattering term (Eq. (3.11)) can be simplified further, as inEq. (3.13): ∫ E/αm

E

Σm

(1 − αm)E′ϕ(E′)dE′ ≃

∫ E/αm

E

Σm

(1 − αm)E′ϕ0(E′)dE′ = Σmϕ0(E). (3.14)

This approximation is justified by the fact that the integral over energy averages out local fluctuations of thespectrum ϕ(E) due to the resonance absorption. Introducing this approximation into Eq. (3.10) and writingΣt′(E) = Σt(E) − Σm,

[Σt′(E) + Σm] ϕ(E) = Σmϕ0(E) +∫ ∞

0Σ

(d)s (E′→E)ϕ(E′)dE′ (3.15)

is obtained. Also, for energies below the range where direct contribution of the fission source is significant,the source term s0 is neglected.


For a medium of finite dimensions, an additional term, the escape cross section Σe, is added to the moder-ator cross section based on the principle of equivalence of geometrical and resonance self-shielding, formingthe so-called dilution cross section:

Σ0 = Σm + Σe, (3.16)

The escape cross section Σe is connected to the probability of neutron escape out of the medium Pe; in Wignerrational approximation ([35], vol. 3, pp. 253-257), the following dependence is used:

Pe =Σe

Σt′ + Σe. (3.17)

The escape cross section equals

Σe =a∗

L. (3.18)

All geometrical properties of the system are lumped into just one parameter – the sample mean chord lengthL, which in isotropic neutron field represents the mean no-collision path of a neutron through the sample.The constant a∗ is the Bell factor, which slightly depends on the sample shape and ratio between absorptionand scattering rates, and represents a zero-order correction to the Wigner rational approximation. Commonlyadopted value of the Bell factor is a∗ = 1.16; this value was determined empirically and gives satisfactoryresults for most samples of cylindrical shape [9].

Taking all these considerations into account the slowing-down equation for finite medium transforms into:Σa(E) +∑

k

Σs,k(E) + Σ0

ϕ(E) = Σ0ϕ0(E) +∑

k

∫ E/αk

E

Σs,k(E′)(1 − αk)E′

ϕ(E′)dE′, (3.19)

noting that in the resonance range the total cross section is the sum of absorption and scattering cross section:Σt = Σa + Σs. The amount of resonance peak contribution to the perturbations of the neutron spectrum ϕ(E)from ϕ0(E) depends on the value of the dilution cross section: the smaller the dilution cross section, the greaterthe influence of the resonance on the flux. On the other hand, at ’infinite dilution’ Σ0 → ∞, which correspondsto the case with a very small amount of resonance absorber, the neutron field is left virtually unperturbed. Thisapproach is called the Bondarenko method [36].

Eq. (3.19) approximately describes the average spectrum in a homogeneous object with no internal neutronmultiplication, placed into an external neutron field ϕ0, typically a sample, located in an irradiation facility of areactor. Eq. (3.19) can be solved numerically to any desired degree of accuracy, if the pointwise cross sectionsare known. However, the procedure from first principles is rather cumbersome for routine application becauseit requires handling of pointwise cross section libraries, which may require more than 105 points in some casesto accurately represent a single reaction cross section. The slowing-down equation can also be solved in groupform, which is useful for approximate routine spectrum and reaction rate calculations of samples irradiatedby an external neutron source [9]. Furthermore, some analytical expressions, such as the narrow (NR), wide(WR), or intermediate resonance (IR) approximations can be derived.

3.1.1.3 Narrow resonance (NR) approximation

In the narrow resonance approximation (NR) the energy loss per collision with the resonance absorber atomis assumed to be sufficient so that any scattering event removes the neutron from the resonance [9]. The NRapproximation for a resonance j is valid if the resonance width Γ satisfies the condition ([20], pp. 423-427)

Γ ≪ (1 − α)E j

√Σs + Σ0

Σ0. (3.20)

In that case, the self-scattering contribution of the resonance can be neglected. Approximating the scatteringcross section of the absorber(s) with a constant potential scattering Σp, Eq. (3.19) becomes:[

Σ0 + Σp + Σa(E)]ϕ(E) =

[Σ0 + Σp

]ϕ0(E) ; ϕ(E) =

Σ0 + Σp

Σ0 + Σp + Σa(E)ϕ0(E) (3.21)

in the vicinity of the resonance. The approximation is valid for resonances at sufficiently high energies inrelatively light moderators, e.g. 55Mn [16].


3.1.1.4 Wide and intermediate resonance (WR and IR) approximations

In some cases where the NR approximation fails for the integral in Eq. (3.19), the wide resonance (WR)approximation can be used [9, 20]. Assuming αk → 1, the above mentioned integral is reduced to Σs(E)ϕ(E),giving

ϕ(E) =Σ0

Σ0 + Σa(E)ϕ0(E) (3.22)

in the vicinity of the resonance. The WR approximation is valid for some low-energy resonances of heavynuclides.

For resonances which are neither narrow nor wide, the so-called intermediate resonance absorption treat-ment appears to be quite successful. The flux is estimated by the formula

ϕ(E) =Σ0 + λΣp

Σ0 + λΣp + Σa(E)ϕ0(E), (3.23)

where the Goldstein-Cohen parameter 0 < λ < 1 is adjusted so that the resonance absorption of ϕ is equal toabsorption of iterated ϕ using Eq. (3.19). It is a generalization of the above described approximations, sinceobviously setting λ→ 1 gives the NR and λ→ 0 gives the WR approximation.

3.1.2 Sensitivity analysis

In the domain of deterministic methods, complementary to the calculation of the expected values of the reactorparameters from the neutron transport equation or its approximations, the uncertainties of these parameters areestimated by sensitivity analysis. As opposed to the transport equation, the sensitivity analysis is completelygeneral and can be applied to any set of parameters. Compared to the random sampling methods, the mainadvantage of the deterministic sensitivity analysis is its numerical efficiency. Usually, a first-order approxima-tion, based on the Taylor series expansion of the integral parameters as functions of the input parameters, issufficient. This approximation, the so-called sandwich formula, is exact when integral parameters are linearfunctions of the input parameters. Moreover, it works very well for most smooth non-linear functions, whenuncertainties are relatively small. However, when dealing with data with large relative uncertainties, determin-istic uncertainty propagation may be used only if higher terms of the Taylor expansion are taken into account.Of course, if the functions are discontinuous, the Taylor expansion is invalid, but in reactor physics, examplesof such functions are very rare and rather artificial.

Let x = (x1, . . . , xp) be a vector of arbitrary parameters xi which, in general, are not independent. Thecorrelations between the parameters may be expressed in the form of covariances

cov(xi, xi′) = ⟨xixi′⟩ − ⟨xi⟩⟨xi′⟩, (3.24)

or directly in the form of correlation coefficients

corr(xi, xi′) =cov(xi, xi′)√

cov(xi, xi)cov(xi′ , xi′)=

cov(xi, xi′)σiσi′

(3.25)

accompanied by the vector of corresponding parameter uncertainties σ.Let fk be an arbitrary analytic function of the parameters x. Then fk may be expanded about x0 as [37]:

fk(x0 + σ) =∞∑j=0

1j!

p∑i=1

σi∂

∂xi

j

fk(x0). (3.26)

The exact expression for the covariances between the functions fk follows straightforward from Eqs. (3.24)


and (3.26):

cov( fk, fk′) = ⟨ fk fk′⟩ − ⟨ fk⟩⟨ fk′⟩

=

⟨ ∞∑j=0

1j!

p∑i=1

(xi − x0i)∂

∂xi

j

fk(x0)∞∑

j′=0

1j′!

p∑i′=1

(xi′ − x0i′)∂

∂xi′

j′

fk′(x0)⟩− fk(x0) fk′(x0)

=

∞∑j, j′=0

1j! j′!

⟨ p∑i1=1

· · ·p∑

i j=1

[(xi1 − x0i1)

∂

∂xi1· · · (xi j − x0i j)

∂

∂xi j

]fk(x0)·

·p∑

i′1=1

· · ·p∑

i′j′=1

(xi′1 − x0i′1)∂

∂xi′1

· · · (xi′j′− x0i′j′

)∂

∂xi′j′

f ′k (x0)⟩− fk(x0) fk′(x0)

=

∞∑j, j′=0

1j! j′!

p∑i1=1

· · ·p∑

i j=1

p∑i′1=1

· · ·p∑

i′j′=1

(∂

∂xi1· · · ∂

∂xi j

)fk(x0)

∂

∂xi′1

· · · ∂

∂xi′j′

fk(x0)·

·⟨(xi1 − x0i1) · · · (xi j − x0i j)(xi′1 − x0i′1) · · · (xi′j′

− x0i′j′)⟩− fk(x0) fk′(x0) (3.27)

Thus, the covariances are expressed explicitly in terms of mixed higher order moments of the x probabilitydistribution: ⟨

(xi1 − x0i1) · · · (xi j − x0i j)(xi′1 − x0i′1) · · · (xi′j′− x0i′j′

)⟩, (3.28)

and higher order derivatives, i.e. sensitivity coefficients:(∂

∂xi1· · · ∂

∂xi j

)fk(x0). (3.29)

3.1.2.1 First order approximation (sandwich formula)

The most commonly used first-order approximation for covariances is obtained directly from Eq. (3.27) bytruncating the series (3.27) at j, j′ = 1:

cov( fk, fk′) ≃ fk(x0) fk′(x0) + fk(x0)p∑

i′=1

∂ fk′(x0)∂xi′

⟨xi′ − x0i′⟩ + fk′(x0)p∑

i=1

∂ fk(x0)∂xi

⟨xi − x0i⟩

+

p∑i,i′=1

∂ fk(x0)∂xi

∂ fk′(x0)∂xi′

⟨(xi − x0i)(xi′ − x0i′)

⟩ − fk(x0) fk′(x0)

=

p∑i,i′=1

∂ fk(x0)∂xi

cov(xi, xi′)∂ fk′(x0)∂xi′

(3.30)

since x0i = ⟨xi⟩ by definition.If we introduce the covariance

V xii′ = cov(xi, xi′) (3.31)

and sensitivity matrices

S ki =∂ fk(x0)∂xi

, (3.32)

Eq. (3.30) expresses in the following simple form:

V f = S V xS T (3.33)

also called the ’sandwich formula’.


Case. Let R =∑

g ϕgΣg be the total reaction rate density, calculated from group fluxes ϕg and effective groupmacroscopic cross sections Σg. The first-order approximation for the relative variance of R can be expressedas:

σ2R

R2 =cov(R,R)

R2 ≃ 1R2

∑g,h

(∂R∂ϕg

cov(ϕg, ϕh)∂R∂ϕh+∂R∂Σg

cov(Σg,Σh)∂R∂Σh+ 2

∂R∂ϕg

cov(ϕg,Σh)∂R∂Σh

)

=∑g,h

(Σg

Rϕg

cov(ϕg, ϕh)ϕgϕh

ϕhΣh

R+ϕg

RΣg

cov(Σg,Σh)ΣgΣh

Σhϕh

R+ 2Σg

Rϕg

cov(ϕg,Σh)ϕgΣh

Σhϕh

R

)

=∑g,h

(ϕgΣg

R

[cov(ϕg, ϕh)

ϕgϕh+

cov(Σg,Σh)ΣgΣh

+ 2cov(ϕg,Σh)

ϕgΣh

]ϕhΣh

R

)=

∑g,h

(Rrel

g

[(Vrelϕ

)gh+

(VrelΣ

)gh+ 2

(Vrelϕ,Σ

)gh

]Rrel

h

), (3.34)

i.e. the relative reaction rate variance is the product of the relative reaction rate vectors (which correspondto the relative sensitivity coefficients) with the sum of the relative covariance matrix of the cross section, thespectrum, and the relative spectrum-cross section cross-covariances.

3.1.2.2 Higher order approximations

In some cases, especially when the functions fk exhibit strong non-linearity within the range of the uncertaintyof the input parameters xi, higher orders of the expansion (3.26) have to be taken into account. However, ana-lytical expression for the f covariance matrix quickly becomes very complex when adding higher order terms.For example, under the assumptions of independent parameters (any set of parameters may be transformed intolinear combinations which are mutually independent, as described in Section 3.2.2) and symmetric distribution(e.g. uniform or normal), second-order covariance matrix can be expressed as [38]:

V fkk′ =

p∑i=1

∂ fk(x0)∂xi

∂ fk′(x0)∂xi

V xii +

14

p∑i=1

∂2 fk(x0)∂x2

i

∂2 fk′(x0)∂x2

i

((µ4)i − (V x

ii)2)+

12

p∑i,i′,i

∂2 fk′(x0)∂xi∂xi′

∂2 fk(x0)∂xi∂xi′

V xiiV

xi′i′ ,

(3.35)where (µ4)i denotes the 4th central moment of the parameter xi probability distribution. A simple analyticalexample showing where second-order approximation has to be taken into account is presented in Section 3.4.

3.2 Monte Carlo method

3.2.1 Monte Carlo (neutron) transport calculations

As already mentioned, the density of neutrons in the majority of realistic multiplication systems is so smallthat the neutron-neutron interactions can be neglected and consequently the transport equation is linear. Fur-thermore, the absence of neutron-neutron interactions gives rise to the Monte Carlo simulations of individualneutrons and enables straightforward implementation of parallel processing. One of the most commonly usedMonte Carlo codes is the program MCNP [5] (Monte Carlo N-Particle Transport Code). MCNP simulatestransport of individual neutrons, photons, and/or electrons through a medium of arbitrary geometry and mate-rial composition. As far as reactor calculations are concerned, the transport of neutrons is sufficient to describethe reactor core, since the photons and electrons have a negligible effect on neutron multiplication.

3.2.1.1 Program MCNP

Most of the concepts described in this section are common to all Monte Carlo neutron transport codes. How-ever, some of the practical solutions, e.g. the definition of neutron source and the normalization of integralparameters, are program-specific. Since these specific practical solutions should not compromise the compre-hension of the general method, the description below refers entirely to the program MCNP.

3.2. MONTE CARLO METHOD 35

The simulated system-dependent MCNP input data are the geometry model and the material composition.The geometry model is defined with surfaces (first-order, second-order and tori, which is sufficient for mostpractical applications), which are combined into areas and cells using Boolean operators. Within each cell, thematerial composition is homogeneous. The correct temperature of the system is taken into account by usingcontinuous energy microscopic cross sections from an evaluated nuclear data library, Doppler broadened tothe appropriate temperature, and data for thermal scattering on crystal lattices.

The program tracks every neutron through its entire lifetime. A neutron is ’born’ from fission or someother neutron multiplication reaction, or in an external source at a randomly generated position (in a cell), andwith a random velocity vector. Then, from a uniformly distributed random number ξ, the distance L of theneutron before next interaction with matter is calculated as

L = − 1Σt

ln ξ, (3.36)

which is a consequence of the exponential distribution of the interaction distance:

χ ≡ 1 − ξ =∫ L

0P(s)ds =

∫ L

0Σt exp(−Σt s)ds = 1 − exp(−ΣtL), χ ∈ [0, 1) and ξ ∈ (0, 1]. (3.37)

If the collision location is inside the same cell, the reaction type is sampled randomly according to corre-sponding partial cross section ratios. Absorption means ’death’ of the neutron. New, next generation neutronsmay be produced e.g. by fission or (n,2n) reaction, which are again randomly generated according to thecorresponding neutron emission spectrum and angular distribution. Scattering, on the other hand, does notterminate the neutron – the scattering angle and energy are determined and the neutron is being tracked further(for example see Fig. 3.1). If the collision location is outside the initial cell, the neutron is carried over tothe surface of the cell while conserving the velocity vector. The path in the initial cell is taken into account,therefore its path in the next cell is suitably shorter than if it was born at the surface crossing between the cells.

Neutron source. The neutrons, simulated in MCNP, are obtained either from an external neutron source(with arbitrary spectrum, spatial distribution, and time dependence if needed) with optional additional internalmultiplication or by internal neutron multiplication. If the internal multiplication option is used, only the initialsource has to be defined, which eventually converges to the real fission source distribution. The external sourceoption converges for subcritical systems only, whereas the internal multiplication option may be used for bothsub- and super-critical systems.

Flux and reaction rate calculations. MCNP enables calculation of flux, averaged over an arbitrary volumeV , e.g. the volume V j of a cell j. The flux in cell j is calculated according to [5]:

φ j = φ(r j) =∫

V j

∫E

∫4π

∫tΦ(r, Ω, E, t)dtdΩdE

dVV j, (3.38)

where the volume integration is performed over the cell, the angular integration is performed over the entiresolid angle, whereas the time and energy intervals are chosen arbitrarily. If the total flux is sought, the integra-tion is performed over the entire energy range E ∈ (0,∞). If the energy spectrum is sought instead, the energyintervals (groups) may be arbitrarily dispersed.

For most applications, the system is in a steady state, thus the integration may be performed over theentire time interval. Under this assumption, the quantity φ j is proportional to the neutron flux in cell j. φ j

corresponds to the physical quantity of fluence – time integrated flux by definition.To better understand the MCNP flux calculator, let us rewrite Eq. (3.38) in a different way:

φ j =

∫V j

∫E

∫4π

∫tN (r, Ω, E, t)vdtdΩdE

dVV j

=

∫E

∫4π

∫V j

∫s

N (r′ + sΩ, Ω, E, t)dsdV ′

V jdΩdE, (3.39)


Figure 3.1: Example of paths (moderation and absorption in the end) of three random neutrons in graphite withinitial kinetic energy 2 MeV, generated by MCNP. Projections were plotted by the program VOXLER [39].

where N denotes the number density (of neutrons), and a new variable s = vt has been introduced. The timedependence in N (r′ + sΩ, Ω, E, t) tracks the paths of neutrons with initial position r′, direction Ω and energyE, therefore the integration over volume dV ′ should be performed over all possible positions of neutrons at thebeginning of the time interval, and depends on energy and direction of travel of the neutrons. Integration overthe path s is performed within the chosen cell and the integration limits are functions of energy, direction oftravel, and initial position.

From the discrete Monte Carlo treatment of neutrons

N (r, Ω, E, t) =I∑

i=1

δ(E − Ei)δ(Ω − Ωi)δ3(r − (r′i + vitΩi)) (3.40)


it follows

φ j =

∫E

∫4π

∫V j

∫s

I∑i=1

δ(E − Ei)δ(Ω − Ωi)δ3(r′ + sΩ − (r′i + vitΩi))dsdV ′

V jdΩdE

=

I∑i=1

∫V j

∫sδ3(r′ − r′i )ds

dV ′

V j=

1V j

I∑i=1

Li, (3.41)

where Li is the path of i-th neutron in cell j, and I is the total number of neutrons passing the chosen cell inthe time interval of observation.

Eq. (3.40) holds for neutrons that are travelling at constant velocity. The integral in Eq. (3.38) can bebroken into arbitrarily small intervals, where Eq. (3.40) is exact, consequently Eq. (3.41) holds generally,since it does not include any time dependence.

The reaction rates are obtained by multiplying the flux by a cross section and integrating over energy. Ifintegrated over entire energy interval, the quantity

Rr, j =

∫V j

∫E

∫4π

∫tΣr (r, E)Φ(r, Ω, E, t)dtdΩdE

dVV j

(3.42)

is proportional to the reaction rate density as defined by Eq. (3.6). The index r refers to the type of reaction.Like in many other probabilistic methods, the statistical uncertainty of the Monte Carlo method is pro-

portional to the inverse square root of the number of samples, in this case neutron histories. Therefore, thenumerical efficiency depends very much on the local properties of the system, but less on the characteristicsof the system as a whole. In other words, the relative uncertainty of the calculated flux or reaction rate in acell is very sensitive to the flux distribution, i.e. is proportional to the inverse square root of the fraction of theneutrons passing the cell, but almost insensitive to the system complexity.

3.2.2 Random sampling methods

The first step in probabilistic uncertainty propagation is to produce random representative samples of the inputdata, from which (integral) parameters can be calculated. Three different random sampling methods withseveral variations are proposed in this work:

I. Diagonalization,

I. Metropolis algorithm,

III. Correlated sampling.

The last one, the so-called correlated sampling method, is completely new, while the other two are variationsof well-known methods. In the field of reactor physics, these methods are mostly used for sampling of nucleardata, but they may be used for any data with uncertainties. The random sampling methods are described inthe following text and their performance is compared in Section 3.5. In this work, the practical examples ofrandom sampling are restricted to resonance parameters.

Random sampling of single parameters is straightforward. Let p be the normalized probability distributionfunction (PDF) of the parameter x, and ξ a random variable, uniformly distributed between 0 and 1. Then, arandom sample of the parameter x can be derived by inverting

ξ =

∫ x

−∞p(x′)dx′. (3.43)

If the indefinite integral of p is an elementary function, Eq. (3.43) can be inverted analytically; otherwise,numerical methods have to be employed, for example the rejection sampling method [40]. Furthermore, anynumber of independent parameters, represented by a vector x, may be sampled separately since the corre-sponding multivariate PDF is a product of single parameters’ PDF’s:

P(x) =∏

i

pi(xi). (3.44)

On the other hand, if parameters are correlated, they cannot be sampled separately since P(x) is coupled.


3.2.2.1 I. Diagonalization

Though in principle the entire multivariate distribution P(x) may be given, the parameters are in most casesrepresented by a vector of expected values ⟨x⟩ and the corresponding covariance matrix V only. Since V ispositive semidefinite, it is diagonalizable:

V = QT DQ, (3.45)

where Q is an orthogonal matrix, and D = diag(d1, d2, . . . ) a diagonal matrix with all di > 0. The diagonalelements di, that correspond to the variances of the linear combinations of the initial parameters

y = Qx, (3.46)

are uncorrelated, with expected value ⟨y⟩ = Q⟨x⟩ and standard deviation√

di. Consequently, they may beseparately sampled according to (3.43). Samples of the initial set of parameters x are then obtained by inverselinear transformation:

x = QT y. (3.47)

The main limitation of the diagonalization method is that the linear transformation (3.46) preserves thedistribution shape only for multivariate normal distribution. Therefore, if the shape of the distribution is farfrom normal, the linear transformation generates distortions of the sampled parameters distributions from theidealized distributions, preserving only the first two distribution moments (for instance see Section 3.5).

Different probability distributions. When the mean value ⟨x⟩ and the standard deviation σx (which corre-spond to the best estimate and uncertainty for parameters with small relative uncertainty [42]) provide the onlyavailable information about a physical quantity, normal distribution with mean value and standard deviationas its parameters:

p(x) =1√

2πσ2x

exp(− (x − ⟨x⟩)2

2σ2x

)(3.48)

is the best possible assumption for the parameter PDF according to the Maximum Entropy Principle [43].Some physical quantities, like the resonance widths, are inherently positive. Therefore, when large relative

standard deviations are present (above ∼ 0.5), a significant fraction of negative samples of inherently positiveparameters is produced if sampled according to the normal distribution, which is unphysical. Samples withnegative values may be ignored, but this affects the mean and the standard deviation of the parameter. Ifuniform instead of normal distribution is used:

p(x) = (2

√3σx)−1, x ∈

[⟨x⟩ −

√3σx, ⟨x⟩ +

√3σx

]0, otherwise

, (3.49)

this reduces the number of cases with negative parameters and may also speed up the convergence of the inte-gral parameter uncertainties, calculated from the samples1, but on the other hand, when the relative standarddeviation exceeds 1/

√3 ≃ 0.58, negative values are not excluded. Furthermore, when the relative standard

deviation exceeds ≃ 0.666 (see Fig. 3.2), the fraction of negative samples becomes larger than for the normaldistribution. Therefore, the applicability of the uniform sampling is limited to a narrow interval of relativeuncertainties. Also, since the uniform distribution has no tails, sampling may give very different results inintegral calculation, if the observable happens to be highly sensitive to parameter values sampled from the tailsof the distribution.

For inherently positive parameters with known mean value and standard deviation, log-normal instead ofnormal distribution should in principle be employed [41] since the Maximum Entropy Principle is applied inthe logarithmic rather than the linear space. The form of the PDF is:

p(x) =1

√2πσ2x2

exp(− (ln x − µ)2

2σ2

), x > 0 (3.50)

1Dimitri Rochman, personal communication, September 2010.


0.00 0.25 0.50 0.75 1.00 1.25 1.500.00

0.05

0.10

0.15

0.20

0.25

0.30

fract

ion

of n

egat

ive

valu

es

x/ x

normal distribution uniform distribution

Figure 3.2: Fraction of negative samples as a function of relative standard deviation for normal and uniformdistribution.

where

σ =

√ln

(1 +

σ2x

⟨x⟩2

), (3.51)

µ = ln⟨x⟩ − σ2

2, (3.52)

if the mean value ⟨x⟩ and standard deviation σx of the PDF are to be preserved. The log-normal distributionhas the desirable properties:

• it gives only positive parameter values (Fig. 3.3, left), and

• it resembles normal distribution for small relative uncertainties (Fig. 3.3, right).

This ensures positive values of the directly sampled parameters yi, implicit in the transformation (3.45), how-ever, when transformed back via (3.47), negative parameter values may re-appear, although their occurrenceis less likely than if using normal distribution (a practical example is analyzed in Section 3.5).

Negative values may be avoided completely by the logarithmic transformation. In this method, mean valuesof the parameters are logarithmized whereas the corresponding absolute covariance matrices are converted intorelative covariance matrices. Sampling of the parameters in log-space according to normal distribution withrelative covariance matrix is approximately equivalent to sampling in the original parameter space accordingto log-normal distribution with absolute covariance matrices. Since this method is analogous to the first-order Taylor expansion, significant biases in mean values and standard deviations of the sampled parametersarise when operating with large relative uncertainties. Even though one may expect that this method willfail to accurately reproduce the distribution moments for parameters with large relative uncertainties, it is notobvious that the first-order approximation will also fail in the case of parameters with relative uncertainties inthe ’intermediate’ range, where the negative value fraction is small but not negligible.


0

0

Figure 3.3: Comparison of normal (solid) and log-normal (dashed) distribution for large (σx/⟨x⟩ = 0.5, left)and small (σx/⟨x⟩ = 0.1, right) relative uncertainty.

Negative values may also be avoided in a less-elegant, but very simple way: by zero cutoff. Obviously,this method increases the mean value and decreases the standard deviation of the sampled parameters (seeFig. 3.4). Here, negative values are set to zero rather than ignored because this measure slightly reduces biasesin the mean and standard deviation of the sampled parameters. The mean value biases may be significantlyreduced (or completely compensated if the PDF is symmetric) by employing symmetric cutoff at both ends ofthe PDF (below 0 and above 2⟨x⟩), however at the cost of further reducing the originally intended standarddeviation.

3.2.2.2 II. Metropolis algorithm

In recent decades, usage of the Markov chain Monte Carlo methods to simulate complex multivariate distri-butions is increasing steadily [45]. The Metropolis-Hastings method, first published in [46] and subsequentlygeneralized by Hastings [47], produces a random sequence following any PDF P(x). The Metropolis algo-rithm generates a Markov chain, where each state is randomly generated and depends on the previous stateonly, therefore successive states are correlated and this correlation is transferred throughout the chain.

One has to choose a starting point x(0) and perform a sequence of randomized steps within the parameterspace in order to generate a sequence of states. An ’intermediate’ state

x(n+1)′ = F(x(n)). (3.53)

is generated by a randomized transition function F, while the change of state is always conditional: the in-termediate state is accepted if P(x(n+1)) ≥ P(x(n)) or accepted with probability P(x(n+1))/P(x(n)) if P(x(n+1)) <P(x(n)). If the transition is accepted, x(n+1) = x(n+1)′ , otherwise, x(n+1) = x(n).

In general, the transition function F is quite arbitrary, but it has to be suitably defined in order to ensureergodicity and reversibility of the generated Markov chain. By definition, the Markov chain is ergodic if thesequence x(n) covers the entire phase space of x, and reversible if for each pair of states x and y condition:

P(x)T (x→ y) = P(y)T (y→ x) (3.54)

is satisfied. T (x → y) is the probability of transition from x to y, which for Metropolis-algorithm-generatedsequence equals the product of probability density for y = F(x) (denoted by ρ(x → y)) and the conditionalprobability P(y) of acceptance of this transition. It immediately follows that

ρ(x→ y) = ρ(y→ x) (3.55)

is equivalent to Eq. (3.54). When both the ergodicity and reversibility conditions are fulfilled, the generatedsequence is representative of the theoretical multivariate PDF [48].


0.00 0.25 0.50 0.75 1.00 1.25 1.50-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

rela

tive

bias

x/ x

zero cutoff - mean bias symm. cutoff - mean bias zero cutoff - st. dev. bias symm. cutoff - st. dev. bias

Figure 3.4: Relative bias in the mean value and standard deviation as a function of the relative standarddeviation for zero and symmetric cutoff. Normally distributed parameter.

The choice of the transition function F determines the acceptance ratio acc and the amount of correlationbetween consequent states. The acceptance ratio is defined as the ratio between the number of accepted stepsand the total number of steps in the limit of infinite number of steps.

From the generated Markov chain, random samples of the PDF can be extracted. However, for someapplications (almost) uncorrelated samples are required. In this case, a certain number of Metropolis statesbetween each pair of samples has to be disregarded. A reasonable correlation limit, below which the samplesare allowed to be correlated, has to be chosen in order to achieve an acceptable trade-off between the amountof correlation between the samples and sample-generation computer time. In this study (results are presentedin Section 3.5), correlations below 0.4 will be allowed.2 Reflecting the above, the correlation length l of aMarkov chain is defined as the average distance, measured in terms of Metropolis states, between the stateswith 0.4 correlation. The distance between the subsequent samples is proportional to the correlation length l,therefore the choice of the transition function is vital for the efficiency of the Metropolis algorithm.

Sampling in original parameter space. For the purposes of this study (results are presented in Section 3.5),a well-tested transition function, which satisfies all required conditions (see [44]), has been chosen:

x(n+1)′ = x(n) + c√

diag(V) · (2ξ − 1), (3.56)

where diag(V) is the diagonal matrix of the x variances, ξ is a vector of completely uncorrelated, between 0and 1 uniformly distributed random variables, and c is a predefined constant, which may be altered to tune theacceptance ratio acc of the transition between the successive states.

The main advantage of this transition function is that all operations are performed in the original parameterspace, so there is no need for any matrix transformation or diagonalization.

2Personal communication with Roberto Capote, August 2010.


Sampling in diagonal space. If the number of correlated parameters within x is increasing, the constant chas to be significantly decreased in order to conserve non-negligible acc. Consequently, the correlation lengthl increases, reducing the efficiency of the method. This effect may become a serious problem when the numberof correlated parameters exceeds about 100.

This problem may be easily understood by imagining a picture of the multivariate PDF. When correlationsbetween parameters are strong, the regions with large probabilities are narrowed down to the direction alongthe correlations, and uncorrelated sampling of the parameters frequently results in very low values of the PDF,making the Metropolis step almost unacceptable.

On the other hand, if the sampling is performed in the direction of the correlations, i.e. in the orthogonalparameter space where the covariance matrix is diagonal, the problem is reduced to uncorrelated sampling,where the PDF value is kept within acceptable limits:

y(n+1)′ = y(n) + c√

D · (2ξ − 1). (3.57)

Transforming back (multiplying by QT , see Eq. (3.47)) into the original parameters space, we obtain:

x(n+1)′ = x(n) + cQT√

D · (2ξ − 1). (3.58)

Of course, for each PDF there is a need for additional diagonalization of the covariance matrix, whichmight be numerically unstable. Luckily, inherent stability of the Metropolis algorithm, which is ensured bythe fact that ’inappropriate’ transitions are automatically rejected, assures numerical stability of this samplingprocedure at the cost of lower computational efficiency compared to the direct diagonalization method.

3.2.2.3 III. Correlated sampling

As already mentioned before, inherently positive parameters with a known estimate of the expected value anduncertainty are best described by the log-normal distribution [43].

Since multivariate log-normal distribution is used in many other applications, from radiation protection[49] to probabilistic safety assessment [50] and astrophysics [51], from hazard modelling [52] to pattern recog-nition [53], the method described below reaches beyond nuclear engineering.

The above-described diagonalization and Metropolis methods have problems when dealing with corre-lated log-normally distributed parameters. The diagonalization method is limited by the fact that only normaldistribution is preserved when performing linear transformations thus producing distorted original parame-ter distributions and negative values, while the Metropolis algorithm would in principle allow inclusion ofmultivariate coupled log-normal probability distribution function [54]

P(x) = det(2πV)−1/2

n∏i=1

x−1i

exp[−1

2(ln x − µ)T V−1(ln x − µ)

], (3.59)

where extended definition of the logarithm (ln x

)i = ln xi, ∀i (3.60)

is used, µ is the vector of expected values, and V is the covariance matrix for the vector of parameters inlog-space (ln x). Therefore, µ and V are not the true vector of expected values and covariance matrix of the setof parameters x, but merely the parameters of the multivariate log-normal distribution. The relation to the realexpected values and covariance matrix of x is no longer as simple as in Eqs. (3.51) and (3.52). Since withinthe scope of the present investigation it was not possible to find nor derive any analytical relations, this methodwill be left for further studies.

Alternatively, a novel method is proposed, based on correlated sampling of parameters, which in principleallows the use of arbitrary probability distribution functions for parameters. The advantage of the method isthat it exactly preserves the entire multivariate PDF’s of the sampled parameters. In this work, only normal andlog-normal distributions are considered. The main limitation of the method is the need to solve a system of n2


quadratic equations, where n is the number of correlated parameters, before applying the parameter sampling.Luckily, the system of quadratic equations is special so that it may be written in a form of a matrix equationwith one solution obtainable by performing the matrix square root operation.

The basic idea of the method is very simple: to produce samples of n parameters distributed according totheir distribution functions from n independent random variables ξi, e.g. uniformly or normally distributed. Asample x(m) would then be calculated as a known function F of the random variables, which would depend onthe parameter distributions and correlations:

x(m) = F(ξ(m)). (3.61)

In practice, the function F has to be derived for every distribution or family of distributions, which is in generalfar from trivial.

Normal distribution. Probably the most simple example is the normal distribution since any linear com-bination of normal variables is again a normal variable. If in ξ(m) all components are normally distributedvariables, the samples could be produced by

x(m) = A · ξ(m) + µ, (3.62)

where matrix A and vector µ are defined such that mean values, standard deviations and correlations betweenthe components of x are preserved. Without limiting the generality, all ξ can be assumed to follow the so-calledstandard normal distribution, i.e. the Gaussian distribution with zero mean and unit standard deviation.

Following straightforwardly from this assumption, µ represents the vector of expected values of x. Thematrix A has to satisfy

Vi j = limM→∞

1M − 1

M∑m=1

(x(m)

i − µi) (

x(m)j − µ j

)= lim

M→∞

1M − 1

M∑m=1

n∑k,l=1

Aikξ(m)k A jlξ

(m)l

=

n∑k,l=1

AikA jl limM→∞

1M − 1

M∑m=1

ξ(m)k ξ(m)

l =

n∑k,l=1

AikA jlδkl =

n∑k=1

AikA jk, (3.63)

where Vi j are the absolute covariances of the parameters xi and x j. For a diagonal element of the correlationmatrix Eq. (3.63) simplifies to:

Vii =

n∑k=1

A2ik. (3.64)

Eq. (3.63) defines n2 quadratic equations for n2 unknowns, the elements of the matrix A. When the systemof Eq. (3.63) is solved, an unlimited number of normally distributed parameter set x samples can be produced,taking into account all correlations between individual parameters. Note that any solution for A is sufficientfor the purpose of random sampling since though each solution produces different random samples x(m), thelatter all obey the same distribution. Probably the easiest way to find a solution A is to first write Eq. (3.63) inmatrix form:

Vi j =

n∑k=1

AikA jk =

n∑k=1

Aik(AT

)k j=

(AAT

)i j=⇒ V = AAT . (3.65)

Since covariance matrix V is symmetric and diagonalizable with non-negative eigenvalues, there exists at leastone solution, where A is symmetric. Therefore, Eq. (3.65) simplifies to

V = A2. (3.66)

This matrix equation has several solutions, one of them being the matrix square root of V:

A = V1/2 (3.67)


which may be obtained via diagonalization:

A = QD1/2QT = Q(QT VQ

)1/2QT (3.68)

where D is the diagonal matrix of the V eigenvalues, and Q is an orthogonal matrix of the V eigenvectors.Here, we take advantage of the fact that square root of a diagonal matrix is simply a matrix with square rootsof diagonal elements.

Log-normal distribution. As already mentioned above, natural logarithms of log-normally distributed vari-ables are distributed normally. In the following discussion, this property will be used several times.

Let us start with independent log-normally distributed random variables ξ(m)i , i = 1, . . . , n. Then variables

ln ξ(m)i are normally distributed and so are all of their linear combinations:

ln x(m)i =

n∑k=1

Aik ln ξ(m)k + µi, (3.69)

for practical reasons denoted by ln x(m)i . Obviously,

x(m)i = exp

n∑k=1

Aik ln ξ(m)k + µi

= eµi

n∏k=1

exp(Aik ln ξ(m)

k

)= eµi

n∏k=1

(ξ(m)

k

)Aik (3.70)

are again log-normal variables. Analogously to normally distributed case, we may assume all ln ξi have zeromeans and unit standard deviations (corresponding to µ = 0 and σ = 1 of the log-normal distribution functionin Eq. (3.50), respectively) without limiting the generality.

Coefficients Ai j have to be chosen in such a way that all covariances Vi j are taken into account:

Vi j = limM→∞

1M − 1

M∑m=1

(x(m)

i − ⟨xi⟩) (

x(m)j − ⟨x j⟩

)= lim

M→∞

1M − 1

M∑m=1

x(m)i x(m)

j − ⟨x j⟩M∑

m=1

x(m)i − ⟨xi⟩

M∑m=1

x(m)j + ⟨xi⟩⟨x j⟩

, (3.71)

where p(x) is the standard log-normal PDF (from Eq. (3.50) with µ = 0 and σ = 1). Furthermore,

⟨xi⟩ = limM→∞

1M − 1

M∑m=1

x(m)i = lim

M→∞

1M − 1

M∑m=1

eµi

n∏k=1

(ξ(m)

k

)Aik= eµi

n∏k=1

∫ ∞

0ξAik

k p(ξk)dξk

= eµi

n∏k=1

exp

A2ik

2

(3.72)

and

⟨xix j⟩ = limM→∞

1M − 1

M∑m=1

x(m)i x(m)

j = limM→∞

1M − 1

M∑m=1

n∏k=1

eµieµ j(ξ(m)

k

)Aik(ξ(m)

k

)A jk

= eµieµ j

n∏k=1

∫ ∞

0ξ

Aik+A jk

k p(ξk)dξk = eµieµ j

n∏k=1

exp [Aik + A jk]2

2

. (3.73)

Inserting Eqs. (3.72) and (3.73) into Eq. (3.71) we obtain

Vi j = eµieµ j

n∏k=1

exp [Aik + A jk]2

2

− ⟨xi⟩⟨x j⟩

= eµieµ j

n∏k=1

exp

A2ik

2

exp

A2jk

2

exp(AikA jk

)− ⟨xi⟩⟨x j⟩

= ⟨xi⟩⟨x j⟩ exp

n∑k=1

AikA jk

− ⟨xi⟩⟨x j⟩. (3.74)


Finallyn∑

k=1

AikA jk = ln(

Vi j

⟨xi⟩⟨x j⟩+ 1

)= Vi j, (3.75)

where a new matrix V , which in fact is the absolute covariance matrix for the set of parameters ln x, is defined.Analogously to the normal distribution case (see Eq. (3.65)), we may again write a matrix equation

V = AAT . (3.76)

The coefficients of matrix A can be calculated in exactly the same way as for the normal distribution, i.e.by calculating the square root of a matrix, for multivariate log-normal distribution V instead of the originalcovariance matrix V . Note that V is still symmetric and diagonalizable.

From Eq. (3.75) we can see that the relative covariances for log-normal distribution always have to exceed-1. The reason is that the log-normal distribution is non-zero for positive parameter values only. Also, an inter-esting property of the correlation matrix for log-normal distribution can be derived [55]. If the distributions oftwo observed parameters are the same, correspondence between normal (Ci j) and log-normal (Ci j) correlationcoefficient may be written as

Ci jσ2 = Vi j = ln

Ci jσ2x

⟨x⟩2 + 1⇔ Ci j =

⟨x⟩2

σ2x

(eσ

2Ci j − 1)=

eσ2Ci j − 1

eσ2 − 1, (3.77)

using Eqs. (3.51) and (3.52) in the last step. If normally distributed parameters ln xi and ln x j are perfectlycorrelated (Ci j = 1), so are the log-normally distributed parameters xi and x j (Table 3.1). This is intuitiveand may be extended to any function of the initial parameters as long as the same function is applied toboth parameters in question. Furthermore, if ln xi and ln x j are independent (Ci j = 0), so are xi and x j, aswell as any of their functions which in this case may be different for each parameter. On the other hand, ifln xi and ln x j perfectly anti-correlate (Ci j = −1), xi and x j are still anti-correlated, but not perfectly. Forthe special case of standard log-normal distribution (µ = 0, σ = 1) the figure is given in Table 3.1. Atfirst this may be counter-intuitive, but it is a consequence of the asymmetry of the log-normal distribution.In fact, the largest anti-correlation is obtained when one parameter’s cumulative distribution function (CDF)equals one minus the other parameter’s CDF: C(xi) = 1−C(x j). This may be understood as if one parameter’sdistribution is in some way inverted relative to the other parameter. If the distribution is symmetric, the invertedfunction has exactly the same shape as the original function, yielding the largest possible absolute value of theintegral, corresponding to -1 correlation. Otherwise, perfect anti-correlation is impossible. A more asymmetricdistribution means a higher lower bound for the correlation coefficient. For any distribution, this lower boundhas to be within the interval [−1, 0]. Log-normal distribution converges to the symmetric normal distributionin the case of small uncertainties, i.e. in the limit σ→ 0, Eq. (3.77) expectedly reduces to

Ci j ≥e−σ

2 − 1eσ2 − 1

&1 − σ2 − 11 + σ2 − 1

& −1. (3.78)

In the opposite limit σ→ ∞, where asymmetry is large, Eq. (3.77) reduces to

Ci j ≥e−σ

2 − 1eσ2 − 1

& e−2σ2& 0. (3.79)

Table 3.1: Relation between correlation coefficients for standard normal (Ci j) and standard log-normal (Ci j)distribution.

Ci j 1 0 -1Ci j 1 0 (1 − e)/(e2 − e) ≃ −0.368

In general, the influence of the distribution asymmetry on the lower bound for correlation coefficient isimportant for experimenters an well as (nuclear) data evaluators. If they are measuring/evaluating inherentlypositive parameters with large relative uncertainties, they have to be very careful to define/use consistentcorrelation coefficients for anti-correlated parameters [55].


Combinations of normal and log-normal distribution. In reality, cases may occur where some of thecorrelated parameters are normally while others are log-normally distributed. For example, regarding nucleardata: the resonance energies are normally distributed while the resonance widths are log-normally distributedand often have large relative uncertainties.

On any fixed confidence interval, the log-normal distribution converges to the normal distribution in thelimit of small relative uncertainties. Therefore, a multivariate log-normal distribution can be assigned to theparameter vector

x ′ = x + X, (3.80)

where X has to be chosen to satisfy the conditions: Xi + xi ≫√⟨x2

i ⟩ for normally distributed xi and Xi = 0for log-normally distributed xi. The samples of x ′ are then produced by employing Eq. (3.69). From these,final samples x can trivially be obtained. Arbitrary precision can obviously be achieved by increasing Xi fornormally distributed parameters. Note that the covariance matrices of x and x ′ are exactly the same.

3.2.2.4 Weighted sampling of log-normal variables

In all above descriptions of the random sampling methods, the discussion was limited to the average behaviourof the generated samples, when the number of samples approaches infinity or is at least large enough for allthe observed integral parameters to converge within acceptable limits. However, in reality this condition isdirectly connected to the required computer time and memory, which are often the limiting factors.

For log-normal distribution, the speed of convergence of the distribution moments as a function of thenumber of random samples n strongly depends on the ratio σx/⟨x⟩. If the relative uncertainty is increased, theshape of the distribution approaches log-uniform (1/x) on a progressively broader interval around the meanvalue. Due to the high tail of the log-uniform distribution, the convergence of the mean value (Fig. 3.5) andespecially the standard deviation (Fig. 3.6) of the sampled parameters becomes very slow. If the relative un-certainty is below 100%, 1000 samples suffice to achieve an accuracy within a few percent. In some realisticcases, relative uncertainties are much larger. For example, relative uncertainties of gamma widths of some ofthe less important resonances in the 232Th ENDF/B-VII.1 evaluation [1] reach even 5000% [56]! Due to theslower convergence of the random sampling of log-normal variables with larger relative uncertainties, the re-quired number of samples soon increases by several orders of magnitude. For example, for relative uncertaintyof 1000% the standard deviation is still far from the converged one even after 105 generated samples.

The slow convergence of the higher moments of the randomly sampled log-normal parameters is inherentto the distribution. Therefore, the only way to accelerate the convergence is to introduce a deterministiccomponent into the sampling procedure.

The idea behind the weighted sampling is to pre-define intervals with different cumulative probabilitiesand sample the same number of samples from each interval. Of course, in order to be consistent the sampleshave to be weighted by a factor, proportional to the cumulative probability of the corresponding intervals.

Following intuition, the following interval boundaries have been chosen:

b(i) = exp ikσx

√2

c

, (3.81)

where k and c are constants, and i runs from 0 to I = ⌈cσx/⟨x⟩⌉. The last interval is [b(I),∞). Optimalk ≃ 5⟨x⟩/σx has been estimated experimentally. Therefore, in the following discussion

b(i) = exp5i⟨x⟩

√2

c

(3.82)

is used.Figs. 3.7 and 3.8 show the convergence of the mean value and standard deviation, respectively, for a log-

normally distributed parameter with relative uncertainty of σx/⟨x⟩ = 50 for different values of the constantc, which is proportional to the total number of intervals. The convergence is faster with a larger number of


10 100 1000 10000 100000

0.6

0.8

1.0

1.2

1.4

1.6

1.8

<x>(n

) /<x>

n

x=0.1 x=1 x=3 x=10

Figure 3.5: Convergence of the mean value of the sampled parameters for different ratios σx/⟨x⟩ of the log-normal distribution.

intervals, however regardless of the value of c it is incomparably faster than non-weighted sampling (Figs. 3.5and 3.6).

The weighted sampling appears to be very successful for single log-normally distributed parameters evenwith huge relative uncertainties. But the main limitation of the weighted sampling becomes apparent when thenumber of correlated parameters is increased – namely, the number of intervals increases exponentially withthe total number of correlated parameters. Therefore, the weighted sampling in the above described form isuseful for only up to a few correlated parameters. A possible solution for multiple correlated parameters isto re-randomize the choice of the interval i but then the convergence is slowed down significantly for a singleparameter (Fig. 3.9) and gets much worse with increasing number of sampled parameters.


10 100 1000 10000 1000000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8x(n

) /x

n

x=0.1 x=1 x=3 x=10

Figure 3.6: Convergence of the standard deviation of the sampled parameters for different ratios σx/⟨x⟩ of thelog-normal distribution.

100 1000 10000

0.97

0.98

0.99

1.00

1.01

1.02

<x>(n

) /<x>

n

c=1 c=0.75 c=0.5 c=0.25

Figure 3.7: Convergence of the mean value of the sampled parameters for values of the constant c of thelog-normal distribution with σx/⟨x⟩ = 50.


100 1000 100000.94

0.96

0.98

1.00

1.02

1.04

1.06x(n

) /x

n

c=1 c=0.75 c=0.5 c=0.25

Figure 3.8: Convergence of the standard deviation of the sampled parameters for values of the constant c ofthe log-normal distribution with σx/⟨x⟩ = 50.

100 1000 100000.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

<x>(n

) /<x>

n100 1000 10000

0.80

0.85

0.90

0.95

1.00

1.05

1.10

x(n) /

x

n

Figure 3.9: Convergence of the mean value (left) and standard deviation (right) of the sampled parameters fordeterministic (solid curve) and randomized (dotted curve) choice of the sampling interval i. Constant c = 1,log-normal distribution with σx/⟨x⟩ = 50.


3.2.3 Total Monte Carlo

In recent years, there have been attempts to combine the random sampling of the input nuclear data with theMonte Carlo (neutron) transport simulations. The combined method is called Total Monte Carlo [57]. MonteCarlo transport simulations are repeated for a large number of random nuclear data samples. All data containedin these samples (usually in ENDF-6 format) should be randomized taking into account all uncertainties andcorrelations. In each result, corresponding to a nuclear data sample, statistical uncertainty is involved whereasthe distribution of the results is a convolution of integral parameter distribution due to uncertainties in nucleardata and statistical uncertainty distribution. If the statistical uncertainty is insignificant, the standard deviationof the results approaches the integral parameter uncertainty due to uncertainties in nuclear data. Following thisprocedure, the best estimate for the uncertainty in integral parameters due to uncertainties in nuclear data canbe obtained.

The main advantage of the method is the accuracy when the number of samples approaches infinity assum-ing the data and models are consistent. The main disadvantage lies in the huge numerical demands (large bothin computer time and memory requirements), but this problem is constantly diminishing with the continuousdevelopment of fast computer systems.

3.3 Combinations of deterministic and probabilistic approaches

3.3.1 Combinations of random sampling methods and deterministic transport calculations

If the random sampling methods are combined with deterministic neutron transport calculations, a consid-erable amount of computer time can be saved compared to the Total Monte Carlo method. This option isespecially useful when correlated parameters with large relative uncertainties, typically nuclear data, influencethe uncertainty of integral parameters and the user is more interested in the uncertainties rather than the meanvalues of the integral parameters. Random sampling methods ensure consistent uncertainty propagation in thenon-linear regime where first-order deterministic sensitivity analysis fails, while the deterministic transportcodes enable fast calculation of integral parameters.

In principle, the procedure is very simple. For each random sample of input data a deterministic codeis used to calculate the integral parameter(s) in question. The expected values of integral parameter(s), theiruncertainties and correlations (if more than 1 integral parameter is observed) are then determined by simplestatistical analysis of the results corresponding to the series of calculations using sampled data files. Forexample, this combination of methods will be used for the resonance integral uncertainty calculation due tothe uncertainties and correlations in the resonance parameters, as described in Section 4.2.

3.3.2 Combinations of sensitivity analysis and Monte Carlo transport calculations

The opposite option, the use of deterministic sensitivity analysis in combination with Monte Carlo transportcalculations, is typically employed when a large number of mostly uncorrelated parameters with small relativeuncertainties influence the uncertainty of some integral parameters. By using Monte Carlo transport calcula-tions accurate mean values of the integral parameters are obtained, while the (usually first-order) sensitivityanalysis is sufficient for accurate estimation of their standard uncertainties. This combination of methods is fre-quently used to estimate the uncertainty of integral parameters for benchmark experiments due to experimentaluncertainties in geometry and material data.

For example, in our contribution to the International Handbook of Evaluated Critical Safety BenchmarkExperiments [58], we analyzed the experimental uncertainty of multiplication factor ke f f of a critical assembly,fueled by highly enriched uranium and moderated by polyethylene [59]. The total experimental uncertainty inke f f was estimated via first-order sensitivity coefficients:

(∆ke f f

)2=

∑i, j

S iS jσiσ jCi j, (3.83)

3.4. AN ANALYTICAL EXAMPLE 51

where σi is the uncertainty/tolerance of the geometry/material parameter xi. Correlation coefficients Ci j wereusually either 0 or ±1. Each sensitivity coefficient S i was determined from two separate MCNP [5] transportcalculations: a reference calculation with expected values of all input parameters and a calculation where asingle parameter xi was perturbed:

S i =ke f f (xi) − kre f

e f f

xi − x0i. (3.84)

Each Monte Carlo transport calculation was continued until the statistical uncertainty was negligible comparedto the total uncertainty ∆ke f f .

3.4 An analytical example

As explained before, first-order sensitivity analysis for estimating uncertainties of integral parameters is accu-rate when the relationship between input and integral parameters is close to linear. In practice, this methodis typically sufficient when input parameters have small uncertainties. The limitation of the first-order ap-proximation is exposed by a comparison of the first- and second-order approximations for the estimation ofthe variance of a product of two parameters with arbitrary uncertainties and correlation. In reactor physics, arealistic example of such functional dependence would be the reaction rate, which equals the product of thetotal flux and the effective one-group reaction cross section.

Case. Uncertainty of the ’integral’ parameter f (x, y) = xy, where x and y are normalized (mean valuesx0 = y0 = 1) and normally distributed ’input’ parameters with standard deviations σx, σy and correlationcoefficient c.

Sandwich formula. It follows straightforward from Eq. (3.30) that

σ2f = y2

0σ2x + x2

0σ2y + 2x0y0σxσyc = σ2

x + σ2y + 2σxσyc (3.85)

Second order approximation. Since f is a quadratic function, all derivatives higher than the order of 2 arezero, and second-order Taylor expansion is exact. In general x and y are correlated, therefore if Eq. (3.35)is to be applied, they have to be transformed into new, independent variables. Since normal distribution isconserved when performing linear operations, linear combinations u and v (of x and y) are independent if andonly if cov(u, v) = 0. Such linear combinations are:

u =xσx− yσy

v =xσx+

yσy

(3.86)

with variances

σu = 2(1 − c)

σv = 2(1 + c). (3.87)

In terms of u and v, function f transforms into:

f (x(u, v), y(u, v)) =σxσy

4(v2 − u2). (3.88)

After several straightforward steps taking into account (µ4)i = 3σ4i for normal distribution, Eq. (3.35) reduces

to:

σ2f =

σ2xσ

2y

2

[u2

0(1 − c) + v20(1 + c) + (1 − c)2 + (1 + c)2

]= σ2

x + σ2y + σ

2xσ

2y + σxσyc + σ2

xσ2yc2. (3.89)


Parametric study and comparison of the methods. In the limit of small relative errors of initial parameters(σx, σy → 0), first and second-order approximations (Eqs. (3.85) and (3.89)) are identical (Fig. 3.10). Asexpected, the second-order correction becomes significant when relative errors approach 1 regardless of thecorrelation between the parameters (Fig. 3.10). In this case ( f (x, y) = xy), second-order correction is exactand thus it gives the same results as the random sampling method in the limit of infinite number of randomsamples. In general, even though second-order approximation is always better than first-order, it may notgive satisfactory results in cases with extremely large relative errors. On the other hand, for any distribution,random sampling in principle always gives unbiased results regardless of the functional dependence, and isthus safer to use.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

2 f

x

y=0.1, Eq. (3.85) y=0.1, Eq. (3.89) y=0.3, Eq. (3.85) y=0.3, Eq. (3.89) y=0.6, Eq. (3.85) y=0.6, Eq. (3.89) y=1, Eq. (3.85) y=1, Eq. (3.89)

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

6

2 f

c

x= y=0.1, Eq. (3.85) x= y=0.1, Eq. (3.89) x= y=0.5, Eq. (3.85) x= y=0.5, Eq. (3.89) x=0.5, y=1, Eq. (3.85) x=0.5, y=1, Eq. (3.89) x= y=1, Eq. (3.85) x= y=1, Eq. (3.89)

Figure 3.10: Comparison of the first- (Eq. (3.85)) and second-order (Eq. (3.89)) approximations of the vari-ance of function f . The left graph indicates the dependence of the variance on the relative error of the inputparameters for zero correlation while the right graph shows the dependence of the same quantity on the corre-lation coefficient c. Since f is quadratic, the second-order approximation gives exact results, equivalent to therandom sampling in the limit of infinite number of samples.

3.5. COMPARISON OF THE RANDOM SAMPLING METHODS 53

3.5 Comparison of the random sampling methods

All three random sampling methods with several variations, described in Section 3.2.2, have been tested andcompared on realistic data: evaluated resonance parameters for 55Mn.

3.5.1 Datasets

Two evaluated 55Mn nuclear data files (both in ENDF-6 format [24]) have been chosen for comparison in thisthesis. In total, the resolved resonance parameters in the NRG library [17] extend up to 100 keV and include172 resonances, while in the ORNL file3 the resolved resonance range extends up to 125 keV and includes 187resonances. In the ORNL file, all resonance energies, neutron and gamma widths have non-zero uncertaintiesassigned, whereas in the NRG library, only neutron and gamma widths have non-zero uncertainties.

In each set, every resonance parameter with non-zero uncertainty is assigned a number Np. The resonancesare first sorted by increasing energy, and secondly, by parameter type: resonance energy first (ORNL only),then neutron width, and finally gamma width. Furthermore, resonances N are labeled in increasing energyorder.

In the NRG data, some resonance parameters have large relative uncertainties (Fig. 3.11) but the corre-lations are limited within the parameters of a single resonance (Fig. 3.12). In the ORNL data, the resonanceparameters have much smaller relative uncertainties (Fig. 3.11) but are strongly correlated throughout the res-onance energy range (Figs. 3.12 and 3.13). Certain contributions to the overall uncertainty, such as directreaction contribution, normalization, etc., are not included in the ORNL resonance parameter covariances.They are given as background cross section contribution (File 33 in ENDF terminology).

0 50 100 150 200 250 3000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

p/p0

Np

0 50 100 150 200 250 300 350 400 450 500 5500.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

p/p0

Np

Figure 3.11: Relative uncertainties σp/p0 of resonance parameters Np for NRG (left) and ORNL (right) data.Black color corresponds to neutron widths, and red to gamma widths, respectively. Resonance energy un-certainties (green) are either non-existing (NRG) or small and almost invisible in full linear scale (ORNL).Furthermore, particularly note the difference in scales between both graphs – the scale in the left figure (NRG)is about 8 times larger than the one in the right figure (ORNL).

3Luiz Leal, personal communication, October 2010.


1 20 40 60

60

40

20

1

Np

Np

-1

1

1 20 40 60

60

40

20

1

Np

Np

-1

1

Figure 3.12: Correlation matrix for the first 60 resonance parameters (corresponding to resonances with lowestenergies) for NRG (left) and ORNL (right) data.

1 50 100 150 187

187

150

100

50

1

N

N

-1

1

Figure 3.13: Correlation matrix for gamma widths of resonances throughout the resolved resonance range forthe ORNL data.


3.5.2 Results

First, the sampling methods are compared graphically. In order for the methods to be equivalent, it is necessaryfor the sample distributions to follow similar patterns. One has to be aware, however, that this is of courseinsufficient for the equivalence. Mean value, standard deviation, and other quantitative measures provide morereliable information about the distributions.

XY-plots of samples of correlated parameter pairs are used for a rough illustration of eventual differencesin sampling methods and/or distributions. Two arbitrary strongly anti-correlated resonance parameters of theNRG dataset are shown in Fig. 3.14 comparing samples obtained by different variations of the diagonalizationmethod: normal-, uniform-, zero cutoff with normal-distribution, and sampling in log-space. Since one ofthe parameters has large uncertainty (50%), normal distribution clearly generates negative parameter values.Employing uniform distribution reduces the negative value fraction but also distorts the pattern. Zero cutoffdoes not affect the pattern of the sampled parameter in the positive value range but generates additional 0 valueswhich slightly bias the distribution moments. On the other hand, logarithmizing and sampling in log-spaceaffects the distribution very much and should be avoided unless backed by physical or other arguments.

Sampling with different methods is qualitatively compared in Fig. 3.15. Diagonalization method andMetropolis with sampling both in the original parameter space and in the diagonal space are considered. Cor-related sampling method is omitted in this and the following figures since for normal distribution it producessamples, equivalent to the diagonalization method. Although the absolute value of the correlation between thetwo plotted parameters (-35%) is smaller than on Fig. 3.14, one has to recall that all resonance parameters ofthe ORNL dataset are correlated which demands a reduction of the Metropolis step parameter c when samplingin the original parameter space (Eq. (3.56)) and consequently significant increase of the correlation length. If100 Metropolis states are skipped between successive samples, diagonalization and Metropolis sampling indiagonal space (Eq. (3.58)) seem equivalent according to Fig. 3.15. On the other hand, in the original-spaceMetropolis samples are more ’densely’ distributed, i.e. they are visibly correlated. In this case, a much largernumber of Metropolis steps has to be skipped between the samples. Furthermore, a closer look (Fig. 3.16) re-veals that even successive diagonal-space Metropolis samples are partially correlated since on average they aremuch closer together than samples, generated by the diagonalization method. However, with a large numberof samples included, this small correlations seem to cancel out.

Fig. 3.17 reveals an example of centred and normalized parameter sequence for different Metropolis stepsand diagonalization method as a reference. While the diagonalization method essentially produces white noisewhich is perfect for this purpose, consequent Metropolis states are more or less correlated depending on thechosen type of transition function between the states.

The amount of correlation can be measured by the autocorrelation function:

Rn =

∑i, j(x(i+n)

j − ⟨x j⟩)(x(i)j − ⟨x j⟩)/σ2

x j∑i, j(x(i)

j − ⟨x j⟩)(x(i)j − ⟨x j⟩)/σ2

x j

(3.90)

with properties: R0 = 1 and |Rn| ≤ 1. Naturally, one seeks to minimize the correlations between the sampleswhich are to be used for further calculations. As evident from Fig. 3.18, correlation generally converges to0 with increasing distance between the Metropolis steps. As already mentioned, in this discussion the limitfor correlations is set at 0.4 and the correlation length l is defined accordingly.4 In order to be able to neglectthe correlations between the samples, at least l Metropolis states have to be omitted between the successivesamples.

Correlation length greatly depends on the dataset and on the transition function used by the Metropolisalgorithm (Fig. 3.18), especially on the value of the parameter c (see also Fig. 3.19). For data with a largenumber of correlated parameters (ORNL), performing steps in diagonal space (according to Eq. (3.58)) mayreduce the correlation length by as much as an order of magnitude (Fig. 3.18).

In order to ensure consistent results for any integral parameter, the sampling method has to conservethe mean value and standard deviation for all parameters which are varied. Furthermore, negative values ofinherently positive physical quantities have to be avoided.

4Personal communication with Roberto Capote, August 2010.


300 400 500 600 700 800 900-0.5

0.0

0.5

1.0

1.5

2.0

2.5

normal uniform log-space zero cutoff

p 215[eV]

p214[eV]

Figure 3.14: Gamma (p213) versus neutron width (p212) for 71st lowest energy resonance of the NRG 55Mndataset (-93% correlation). Different variations of the diagonalization method. 1000 samples are shown foreach distribution.

Comparison of the normalized mean value biases ((p − p0)/σp), normalized standard deviations (σ p−p0σp

),

and negative value fractions, obtained by different sampling methods and distributions, averaged over allgamma widths of the NRG dataset, are summarized in Table 3.2. Different distributions used by the diago-nalization method produce very consistent means and standard deviations of the sampled parameters, whereasMetropolis and the cutoff methods cause slight biases in mean and standard deviation. For most purposes, thisbiases are negligible, although one has to be aware of potential discrepancies, especially when dealing withlarge uncertainties (for the cutoff) and/or large number of strongly correlated parameters (for the Metropolis).On the other hand, if the data are sampled in the logarithmic space and then transformed back to the originalparameter space, large biases are produced. Even though negative values are completely avoided, this first-order approximation method is too crude to be safely used for dealing with large uncertainties. The correlatedsampling using normal distribution is equivalent to the diagonalization method with the same distribution func-tion. However, contrary to the diagonalization method, the correlated sampling method preserves the shape ofthe log-normal distribution and therefore completely avoids negative parameter values while ensuring consis-tent mean values and standard deviations. For this dataset, the correlated sampling method with log-normallydistributed parameters is the best performer of all the proposed methods.

A similar comparison of the mean value biases and standard deviation was done with the ORNL dataset(Table 3.3). Due to a much larger correlation length for the Metropolis, performing Metropolis steps in originalparameter space causes a significant reduction of the sampled parameters’ standard deviation. Again, if a suf-ficient Metropolis step is chosen and enough states between the samples are omitted, both methods give meansand standard deviations, equivalent within the statistical standard deviation. Since the relative uncertaintiesare small in this case, the correlated sampling method is virtually equivalent to the diagonalization method.


340.77 340.78 340.79 340.8023.660

23.665

23.670

23.675

23.680

23.685

23.690

23.695 diagonalization Metropolis orig. Metropolis diag.

p 11[e

V]

p10[eV]

Figure 3.15: Neutron width (p11) versus resonance energy (p10) for 4th lowest energy resonance of the ORNL55Mn dataset (-35% correlation). Different sampling methods. Consecutive Metropolis samples are separatedby 100 steps. 1000 samples are shown for each method.

Table 3.2: Negative values and biases for different sampling methods and distributions. Distance betweensuccessive Metropolis samples is 100 steps. NRG data, gamma widths.

Method/distribution p < 0 [%] ⟨ p−p0σp⟩ σ p−p0

σp

diag./normal 1.41 0.0044 ± 0.0024 1.0008 ± 0.0034uniform 0.20 0.0011 ± 0.0024 0.9999 ± 0.0023log-normal 0.32 −0.0024 ± 0.0024 0.9988 ± 0.0053normal - zero cutoff 0 0.0098 ± 0.0024 0.9882 ± 0.0032normal - symmetric cutoff 0 −0.0015 ± 0.0024 0.9757 ± 0.0030log space 0 0.1817 ± 0.0027 1.1279 ± 0.0085Metropolis (orig.) (c = 0.25, acc = 11%) 1.39 0.0126 ± 0.0025 0.9884 ± 0.0036zero cutoff 0 0.0177 ± 0.0024 0.9710 ± 0.0034symmetric cutoff 0 0.0124 ± 0.0024 0.9585 ± 0.0031log-normal (c = 0.2, acc = 20%) 0 0.1853 ± 0.0028 1.1080 ± 0.0089corr. sampl./log-normal 0 0.0000 ± 0.0024 0.9958 ± 0.0054


340.77 340.78 340.79 340.80

23.665

23.670

23.675

23.680

23.685

23.690

diagonalization Metropolis diag.

p 11[e

V]

p10[eV]

Figure 3.16: Neutron width (p11) versus resonance energy (p10) for 4th lowest energy resonance of the ORNL55Mn dataset (-35% correlation), comparing sampling methods. Consecutive Metropolis samples are separatedby 100 steps. 200 samples are shown for each method.

Table 3.3: Relative mean value and standard deviation biases for different sampling methods and distributions.Distance between Metropolis samples is 1000 steps. Correlations between the parameters within a samplehave been neglected since the absolute value of the average correlation coefficient is 0.14%, only. ORNL data,gamma widths.

Method ⟨ p−p0σp⟩ σ p−p0

σp

diag. −0.0016 ± 0.0023 1.0014 ± 0.0033Metropolis (orig.) (c = 0.03, acc = 16%) −0.0044 ± 0.0021 0.9075 ± 0.0033Metropolis (diag.) (c = 0.2, acc = 14%) −0.0022 ± 0.0023 1.0009 ± 0.0033corr. sampl. 0.0004 ± 0.0023 0.9988 ± 0.0033


0 500 1000 1500 2000 2500 3000-3

-2

-1

0

1

2

3(p

-p0)/

p

n

diagonalization Metropolis (c=0.03) Metropolis diag. (c=0.2)

Figure 3.17: Centred and normalized sequence for a chosen resonance parameter (p− p0)/σp from NRG 55Mndataset as a function of step n. Different sampling methods. Sample resolution (smallest ∆n shown on graph)is 10 steps.

0 500 1000 1500 2000 25000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Rn

n

diagonalization Metropolis (c=0.25) Metropolis (c=0.1) Metropolis (c=0.03)

0 500 1000 1500 2000 25000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Rn

n

diagonalization Metropolis (c=0.03) Metropolis diag. (c=0.2)

Figure 3.18: Autocorrelation function R as a function of the distance between the Metropolis states n fordifferent Metropolis steps. NRG (left) and ORNL (right) data. Autocorrelation function of the diagonalizedsampling is shown as a reference.


0.0 0.1 0.2 0.3 0.4 0.50

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

l

c

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

l

acc

Figure 3.19: Correlation length l of the Metropolis algorithm as a function of the step parameter c fromEq. (3.56) (left) and acceptance ratio acc (right). NRG data.

Chapter 4

Resonance integrals and self-shieldingfactors

4.1 Self-shielding factors for standard monitor materials

When an object is placed in an external neutron field, it interacts with neutrons. If these interactions arenon-negligible, the object alters the neutron field, which influences the rate of interactions. This is called theself-shielding effect. The effect is especially pronounced in the resonance energy range, where the high reso-nance peaks absorb practically all neutrons with corresponding energies creating dips in the neutron spectrumaround the resonance energy and spectrum peaks below the resonance energy due to down-scattering from theresonance, and consequently usually lowering the reaction rate. The absorption reaction rate increase is pos-sible if a strong absorption resonance is placed directly below a strong scattering resonance. For instance, theself-shielding effect is important when dealing with heterogeneous systems, e.g. reactor cores [60], or whendetermining activation of relatively large samples of strong absorbers, irradiated by neutrons [9].

Neutron activation analysis (NAA) is a versatile multi-elemental analytical technique that relies on thecharacteristic properties of nuclei formed by the radiative capture reaction, whereby the constituent elementsof irradiated samples can be identified. Sample irradiation is often performed in irradiation facilities that havea significant fraction of the neutron flux at epithermal energies – in the resonance range. Activation equations[61] allow for the contribution of epithermal neutrons, but if resonant absorbers are present in the sample, acorrection for resonance self-shielding has to be made.

Resonance capture is a rather complex process and is well known in reactor physics. In the past, the ten-dency in neutron activation analysis was to select the samples by size, form, or chemical pre-treatment so thatresonance effects were negligible. However, this is not always possible. Recently, Chilian et. al. [62] proposeda simple empirical approach to address the problem of self-shielding. An in-depth analysis of the self-shieldingin neutron activation analysis was made by Monte Carlo simulations [9]. The aim of the work was to comparethe results of the method of Chilian with a variety of more rigorous approaches from resonance theory, thesolution of the integral slowing-down equation and the detailed Monte Carlo calculations for a number of real-istic test cases. The final objective was to develop a simple procedure for calculating the self-shielding factorsthat could be used routinely in neutron activation and similar applications. To this purpose, the MATSSF codewas developed within the scope of a Coordinated Research Project of the International Atomic Energy Agencyon Neutron Activation Analysis and is available from the project web site http://www-nds.iaea.org/naa/matssf/.

4.1.1 MATSSF code

The resonance or epithermal self-shielding factor f for single resonance absorbers (nuclide i) is defined by thereaction rate ratio with the perturbed and the unperturbed flux:

f =

∫ E3

ECdΣk(E)ϕ∗(E)dE∫ E3

ECdΣk(E)ϕ0(E)dE

. (4.1)

61

62 CHAPTER 4. RESONANCE INTEGRALS AND SELF-SHIELDING FACTORS

A library for the MATSSF code containing f factors was generated from the ENDF/B-VII.0 evaluated nucleardata library [63] using the NJOY data processing system [64] with the flux calculator option in the resolvedresonance range that solves Eq. (3.19) for the perturbed flux and a pure 1/E spectrum for the unperturbed flux.Above the resolved resonance range the NR approximation was used. The integration limits are 0.55 eV to2 MeV. The lower bound is also called the cadmium cutoff energy ECd and corresponds to the effective energylimit for neutrons, absorbed in (below limit) or transmitted through (above limit) 1 mm thick layer of cadmium.Above E3 = 2 MeV, the contribution to the reaction rate for non-threshold reactions is usually negligible intypical reactor spectra. For threshold reactions, the resonance self-shielding effect is not important, thereforeit is not included in MATSSF. The self-shielding factors are tabulated as a function of the microscopic dilutioncross section Σ(mi)

0,k = Σ0/nk, where nk is the number density of the absorber k. The MATSSF code calculatesthe dilution cross section according to the sample composition and geometry and obtains the self-shieldingfactor for the single resonance absorber by interpolation from the tables.

There are cases when other resonance absorber nuclides admixed with the absorber nuclides under investi-gation cause interference. The solution of Eq. (3.19) using pointwise cross sections might be too cumbersomefor routine application. As a compromise, a multigroup library of cross sections in 640 energy groups was gen-erated. The SAND-II group structure [65] is used, same as in the IRDF-2002 dosimetry library [66]. Insteadof solving Eq. (3.19) directly, its multigroup equivalent is derived and used in MATSSF.

The effective epithermal self-shielding factor of reaction rates G f is given by

(G f

)k=

∫Σk(E)ϕ(E)dE∫Σk(E)ϕ0(E)dE

=

∫Σk(E)ϕ(E)dE∫Σk(E)ϕ∗(E)dE

∫Σk(E)ϕ∗(E)dE∫Σk(E)ϕ0(E)dE

=


f , (4.2)

where ϕ∗ is the flux, perturbed by the absorber k, whereas ϕ is the flux, perturbed by all constituents of thesample material. The integration limits ECd and E3 are intentionally dropped. In MATSSF, the unperturbedflux is taken as ϕ0 = 1/E, the perturbed fluxes ϕ∗ and ϕ are calculated by directly solving the slowing-downequation using 640-group approximation, while the f factor is taken from the MATSSF library of self-shieldingfactors for single absorbers, generated with the NJOY code system. The epithermal self-shielding factors G f

can be approximated by f when the cross section library for MATSSF is not available.In multigroup form (

G f)k=

∑g Σk,gϕg∑g Σk,gϕ

∗g

f . (4.3)

The macroscopic resonance cross section for nuclide k in group g is defined by

Σk,g = nkΣ(mi)k,g = nk

∫ Eg+1

EgΣ

(mi)k (E)ϕ0(E)dE∫ Eg+1

Egϕ0(E)dE

, (4.4)

where E1 < E2 < · · · < E641.The single-nuclide perturbed flux ϕ∗ is calculated by first integrating Eq. (3.19) over a chosen energy group

g: ∫ Eg+1

Eg

(Σ0 + Σt(E)) ϕ∗(E)dE =(Σ0 + Σt,g

)ϕ∗g = Σ0 ϕ0,g +

∫ Eg+1

Eg

dE∫ E/α

E

Σs(E′)(1 − α)E′

ϕ∗(E′)dE′. (4.5)

In Eq. (4.5), the inner integral cannot be evaluated easily because of its variable integration limits. Still,we can write ∫ Eg+1

Eg

dE∫ E/α

E

Σs(E′)(1 − α)E′

ϕ∗(E′)dE′ =∫ Eg+1

Eg

G(g)∑h=g

Υs,h(E) dE, (4.6)

4.1. SELF-SHIELDING FACTORS FOR STANDARD MONITOR MATERIALS 63

where G(g) is such that EG(g) < 2 MeV and EG(g) < Eg+1/α < EG(g)+1 if allowed by the first condition.Following from (4.6),

Υs,h(E) =∫ E2h

E1h

Σs(E′)(1 − α)E′

ϕ∗(E′)dE′, (4.7)

where

E1h =

E, h = gEh, h > g

(4.8)

and

E2h =

Eh+1, Eh+1 < E/αmaxE/α, Eh, Eh+1 ≥ E/α.

(4.9)

We further writeΥs,h(E) = Υ0

s,hXh(E), (4.10)

where Υ0s,h is a group constant, independent of the outer integral variable in (4.6) and Xh(E) < 1 is dimension-

less:

Υ0s,h =

∫ Eh+1

Eh

Σs(E′)(1 − α)E′

ϕ∗(E′)dE′. (4.11)

Assuming constant cross section Σs(E) = Σs,h and ϕ∗(E) = 1/E within the group h we obtain:

Υ0s,h =

Σs,h

1 − α

∫ Eh+1

Eh

1(E′)2 dE′ =

Σs,h

1 − α (E−1h − E−1

h+1) =Σs,h

1 − αE−1

h − E−1h+1

ln(Eh+1/Eh)ϕ∗h. (4.12)

The above equation can be used to re-write the RHS of Eq. (4.6) as

G(g)∑h=g

Υ0s,h

∫ Eg+1

Eg

Xh(E) dE =G(g)∑h=g

Σs,h

1 − αE−1

h − E−1h+1

ln(Eh+1/Eh)ϕ∗h

∫ Eg+1

Eg

Xh(E) dE. (4.13)

On the other hand, the RHS of Eq. (4.6) can be written as

G(g)∑h=g

Σs,hg ϕ∗h, (4.14)

where Σs,hg is the physical scattering cross section from group h into group g. Thus

Σs,hg =Σs,h

1 − αE−1

h − E−1h+1

ln(Eh+1/Eh)

∫ Eg+1

Eg

Xh(E) dE. (4.15)

Furthermore, assuming again Σs(E) = Σs,h and ϕ∗(E) = 1/E, Xh(E) is simplified:

Xh(E) =Υs,h(E)Υ0

s,h

=

∫ E2h

E1h

1(E′)2 dE′∫ Eh+1

Eh

1(E′)2 dE′

. (4.16)

Inserting Eqs. (4.8) and (4.9) into (4.16) and together into (4.15), we obtain a rather complex expression forthe effective scattering cross section:

Σs,hg =Σs,h

1 − αYh,g − Zh,g

ln(Eh+1/Eh), (4.17)

where Y and Z are defined as

Yh,g =

ln Eg+1Eg, h = g

Eg+1−EgEh

, h > g(4.18)


and

Zh,g =

Eg+1−EgEh+1

, Eh < Eh+1 <Egα <

Eg+1α

α ln αEh+1Eg+

Eg+1Eh+1− α, Eh <

Egα < Eh+1 <

Eg+1α

Eg+1Eh+1− Eg

Eh+ α ln Eh+1

Eh,

Egα < Eh < Eh+1 <

Eg+1α

α ln Eg+1Eg, Eh <

Egα <

Eg+1α < Eh+1

α − EgEh+ α ln Eg+1

αEh,

Egα < Eh <

Eg+1α < Eh+1.

(4.19)

After clarifying Σs,hg we can introduce the RHS of (4.6) into Eq. (4.5) to get

[Σ0 + Σt,g

]ϕ∗g = Σ0 ϕ0,g +

G(g)∑h=g

Σs,hg ϕ∗h. (4.20)

Considering all groups g of the 640-energy group system that fall within the range of integration, Eqs. (4.20)form a system of linear equations for the perturbed group flux ϕ∗g. Written down in explicit form

ϕ∗g =Σ0 ϕ0,g +

∑G(g)h=g+1 Σs,hg ϕ

∗h

Σ0 + Σt,g − Σs,gg, (4.21)

the system can easily be solved recursively for all ϕ∗g starting with the highest energy group.The multi-nuclide perturbed flux ϕ is calculated in the same fashion as ϕ∗, but taking several integrals for

different absorber nuclides in the Eq. (3.19).The G f factor takes into account the flux depression (or an increase because of scattering resonances) due to

other isotopes present in the sample. Note that the approximate multigroup treatment of nuclide interference isvery crude and does not treat interfering resonances, which are much narrower than the energy-group structure.

4.1.2 Generalization of the mean chord length

The latest version of MATSSF uses a generalized expression for the parameter L used in Eq. (3.16). Let usdefine L as the average length of the sample, as ’seen’ by the neutrons

L =

∫VΦ(Ω)dΩ∫

S⊥(Ω)Φ(Ω)dΩ, (4.22)

where Φ(Ω) is the angular flux, and S⊥(Ω) component of the surface of the sample, normal to the direction Ω.As will be explained in further discussion, the mean chord length is just an example of the length L accordingto the above definition.

Eq. (4.22) is valid for arbitrary (neutron) source and sample. The only requirement is that the sample hasa convex shape. However, Eq. (4.22) implicitly assumes that the angular flux at the position of the sample isnot space dependent. In other words, the sample is small enough to neglect the effects of finite dimensions ofthe neutron source.

In the MATSSF code, the sample body is restricted to cylinders, which is sufficient for most realistic cases.Thus, in Eq. (4.22),

V = πr2d (4.23)

andS⊥ = 2rd| cosα| + πr2| sinα|, (4.24)

where α is the angle relative to the normal plane of the axis of the cylinder (same as the polar coordinate inspherical coordinate system).


4.1.2.1 Isotropic neutron source

For isotropic neutron source, which can be simulated by a large isotropic spherical source centred at theposition of the sample, the angular flux is constant:

Φ(Ω) = 1. (4.25)

In spherical coordinates (if we take the sample cylinder axis perpendicular to the azimuthal coordinate φ),

S⊥ = 2rd| cosϑ| + πr2| sinϑ|, (4.26)

where ϑ is the polar coordinate. Taking the symmetries of the cylinder and the sphere into account, andinserting Eqs. (4.25), (4.23), and (4.26) in Eq. (4.22) yields

L =

∫ π/20

∫ π/20 πr2d cosϑdϑdφ∫ π/2

0

∫ π/20 (2rd cosϑ + πr2 sinϑ) cosϑdϑdφ

=2rd

r + d=

4VS. (4.27)

Thus, in the case of isotropic neutron source, the generalized mean sample length L leads to the well-knownexpression 4V/S for the mean chord length of a convex sample. A conventional derivation of this expressioncan be found e.g. in [20], pp. 115-122.

4.1.2.2 Cylindrical source

A void irradiation channel passing through the reactor can be approximated by an isotropic cylindrical sourcewith radius R and height H.

For a uniform infinite source of cylindrical shape, the angular flux in the axis of the cylinder is proportionalto:

Φ(Ω) =1

cosα, (4.28)

where α is the angle relative to the normal plane of the axis of the cylinder.Angular flux in the middle of the channel, at the expected position of the sample, is proportional to:

Φ(Ω) = 1/ cosα, α ∈

− arctan H

2R , arctan H2R

0, elsewhere.

(4.29)

Furthermore, we considered two different examples of sample orientation.First, let us consider the sample cylinder (foil of wire) with its axis parallel to the source cylinder. If we

define spherical coordinate system where the equatorial plane is normal to the cylinder axis, then in Eq. (4.24)we just have to replace the angle α with the polar coordinate ϑ. Obviously, the system source-sample is axiallysymmetric. Regarding all other symmetries, the parameter L in Eq. (4.22) reduces to

L =

∫ ϑmax

0 πr2d 1cosϑ cosϑdϑ∫ ϑmax

0 (2rd cosϑ + πr2 sinϑ) 1cosϑ cosϑdϑ

=πr2d ϑmax

(2rd sinϑmax + πr2(1 − cosϑmax))

=πrd

√1 + H2

4R2 arctan H2R

d HR + πr

(√1 + H2

4R2 − 1) . (4.30)

Second, if the sample cylinder (usually a wire) is lying in the irradiation channel so that the axes of thesource and the sample cylinders are perpendicular, Eq. (4.24) requires a transformation. The result in sphericalcoordinates is not very simple nor intuitive:

S⊥ = 2rd√

cos2 ϑ cos2 φ + sin2 ϑ + πr2| cosϑ sinφ|. (4.31)


The integration of Eq. (4.22) yields

L =π2r2d arctan H

2R

2

2rd g(arctan H

2R

)+ πr2H

2R√

1+ H2

4R2

, (4.32)

where g(x) is a non-elementary uniformly continuous real function, defined for x ∈ [0, π/2]:

g(x) =∫ π/2

0dφ

∫ x

0

√cos2 ϑ cos2 φ + sin2 ϑ dϑ. (4.33)

For our purposes, approximation by 8th degree polynomial in the given range

g(x) ≃ 0.999105x + 0.029183x2 + 0.427788x3 − 0.456087x4 +

+ 0.339182x5 − 0.185970x6 + 0.059748x7 − 0.008226x8 (4.34)

is sufficient, since the relative error of the above approximation is within 0.1% on the domain of g(x).Expressions (4.30) and (4.32) are incorporated as an option in the MATSSF code. The need and the

applicability of the anisotropic source model was tested with the Monte Carlo approach and experimentally.

4.1.3 MCNP model

The most rigorous treatment of self-shielding that takes the energy-dependence of cross sections and geometryinto account is to directly apply the Monte Carlo technique. The MCNP5 code (Ref. [5]) and the associatedlibrary based on ENDF/B-VII.0 data (Ref. [25]) were used in the calculations. The irradiation facility wasmodelled by a cylindrical or spherical isotropic surface source, emitting neutrons with a pure 1/E distributionin epithermal range (from ECd = 0.55 eV to E2 = 2 MeV). The obtained reaction rates were proven to beinsensitive (within the reasonably small statistical uncertainty of less than 1%) to small deviations from the1/E spectrum, e.g. spectrum of light water reactors. The radius of the source surface was equal to the radiusof the irradiation channel in the TRIGA Mark II reactor [13]. The height of the surface source was takenequal to the effective core height. The monitor material samples were modelled explicitly. Two separate runswere made for each sample, one with full geometry including the sample material, and the other with voidin the sample material cell (zero density). Thus, in the latter case, the macroscopic total cross section of themoderator in the flux calculation was zero. The true self-shielding factor was calculated as the ratio of theperturbed and the unperturbed (zero density) reaction rates. The number of particle histories was sufficientlylarge so that the statistical uncertainty in the ratio was well below 1%. If R±∆R and R0±∆R0 are the calculatedreaction rates with and without the sample in place, respectively, the self-shielding factor equates

G f ± ∆G f =RR0

1 ±√(∆RR

)2

+

(∆R0

R0

)2 . (4.35)

The results obtained by MCNP are always represented along with their statistical uncertainty of one standarddeviation.

4.1.4 Verification of the MATSSF mean chord length estimator

The MATSSF mean chord length estimators (4.30) and (4.32) were verified by comparison with Monte Carlomean-free-path-in-cell calculation for realistic source-sample configurations. In Fig. 4.1, a cylindrical sourceof height H = 30 and varying radius R was analyzed for different sample sizes and orientations. The defaultunit is cm but in this discussion units are intentionally omitted due to the dimensionlessness of the problem.Examples shown in Fig. 4.1 are: thick foil (r = 0.4, d = 0.025), thin foil (r = 0.4, d = 0.002), thick wire(r = 0.1, d = 1), and thin wire (r = 0.01, d = 1), for two orientations: along the source cylinder axis, and


Figure 4.1: Generalized mean chord length L of a cylindrical sample as a function of the cylindrical sourceradius R for source height H = 30, and different sample radius r and height d. Red dashed curves andcorresponding data points with error bars represent L according to Eq. (4.32) and Monte Carlo calculation, re-spectively, both for perpendicular orientation of the cylinders. Green dotted curve indicates parallel orientationof the sample with respect to the source. Grey line shows the mean chord length 4V/S for isotropic source.


perpendicular to the axis. The source and sample are always centred in the same point. Generally, the resultsagree within the statistical uncertainty of the Monte Carlo calculation.

In the above discussion it was assumed that an isotropic source appears in a long, voided irradiation chan-nel. The assumption was tested on a TRIGA and SLOWPOKE reactor experimentally [14]. The measurementsshowed no dependence on the sample orientation within the experimental uncertainty. Apparently, the assump-tion of isotropic neutron field on the edge of the irradiation channel fails. This is confirmed by the fact thatthe measurements could be explained by a refined Monte Carlo calculational model, in which the source wasmoved into the moderator surrounding the irradiation channel [15]. The conclusion from the analysis is thatfor small reactors flux anisotropy in irradiation channels is negligible and sample orientation does not have tobe taken into account. However, it might be important for void irradiation channels in larger reactors.

4.1.5 Self-shielding factors of Nickel alloy wire

A detailed analysis was performed for a realistic sample of a 1 mm thick wire, about 3.8 mm in length, madeof nickel-based alloy with the following (mass percent) composition

Ni 80.93%Mo 15.16%W 2.76%Mn 0.41%Au 0.29%

The remainder (0.45%) was arbitrarily assigned to Fe. The density of the material was 9.21 g/cm3. Thesample is one of the standard monitor materials used for neutron activation analysis.

For comparison of computational methods only, generic sample(s) with much smaller precision of thematerial and geometric data could be chosen. However, use of the standard monitor material poses no disad-vantage for the purposes of the verification of computational methods, while the presented results are usefulfor self-shielding corrections of experimental reaction rates for NAA.

The source was assumed to have a 1/E energy dependence, placed on a cylindrical surface of radiusR = 1.2 cm and effective height H = 30 cm, representative of an irradiation channel in TRIGA reactor [13],or placed on a spherical surface around the sample. In this work, only the results for spherical source arepresented in most cases, since it was shown to produce a more representative angular flux at the position of thesample in TRIGA irradiation channels compared to the bare cylindrical source-sample configuration [14, 15].

4.1.5.1 Mean chord length

First, the mean chord length as calculated in MATSSF and MCNP for all three source-sample configurationswas compared (Table 4.1). The MCNP calculated values agree very well with the corresponding MATSSFvalues again verifying the expressions (4.30) and (4.32), which differ significantly for different sample config-urations, justifying the use of MATSSF mean chord length estimator.

Table 4.1: Calculated mean chord in void sample for different orientations and source distributions.

MATSSF [cm] MCNP [cm]along channel axis 0.0988 0.09875 ± 0.00003

sample flat 0.0853 0.08511 ± 0.00001isotropic source 0.0884 0.08833 ± 0.00001

4.1.5.2 Multigroup approximation of resonance interference

To validate the multigroup approximation of the treatment of resonance interference the fine-group spectrumcalculated with MCNP for the isotropic source case was compared to the 640-group MATSSF calculation (the


slowing-down approximation – Eq. (4.21)) and the narrow resonance approximation. The results are shown inFig. 4.2, where the lethargy spectra

ψ(E) = Eϕ(E) (4.36)

are plotted to compensate for the general 1/E behaviour of the spectra ϕ(E) in the resonance range. Themultigroup calculation reproduces well the general features of the spectrum, even though the amplitude ofthe perturbations does not always agree with the reference MCNP calculated spectrum. For comparison only,the spectrum in NR approximation is also shown in Fig. 4.2. This approximation may work well enoughfor narrow resonances, but is totally inadequate for the purposes discussed herein, especially when strongscattering resonances are present, which give rise to the peaks in the spectra.

Figure 4.2: Comparison of the neutron energy spectrum obtained by different methods.

To verify the magnitude of the direct resonance interference effects, the epithermal spectrum was calculatedwith the so-called flux calculator option of NJOY for each constituent nuclide at its appropriate dilution. Inthe second step these calculated spectra were used to calculate the self-shielding factors of the chosen nuclide.The case of 58Fe is a clean case because its implied concentration in the sample is so small that by itself itapproximates well the infinite dilution. The interference with other nuclides increases the self-shielding effect,reducing the self-shielding factor by more than 3%. The results are given in Table 4.2 for the case of thecylindrical source with the wire oriented along the channel axis. This case was chosen for demonstration dueto the most pronounced self-shielding effect among the three source-sample configurations – a consequence ofthe largest mean chord length (Table 4.1).

Table 4.2 indicates that interference does not arise from the most abundant nuclides, but mainly from 95Mo,100Mo and 55Mn, which are present in relatively small amounts. The cumulative resonance integral of 58Fe inthe spectrum with individual interfering nuclides is defined as

RIC(E) =∫ E

E1

Σ(mi)(E′) ϕ∗(E′) dE′. (4.37)

The cumulative resonance integral is shown in Fig. 4.3, with the ratio to the integral at infinite dilution in thelower part of the figure.


Figure 4.3: Cumulative resonance integral of 58Fe in spectra perturbed by different components in the alloy asa function of the upper integration energy boundary.


Table 4.2: Contributions to resonance interference of 58Fe from other sample constituents, expressed as f − 1.The dilution cross section Σ(mi)

0 of the constituents is also given.

nuclide f − 1 Σ(mi)0 [b]

58Ni -0.07% 25560Ni -0.03% 67261Ni 0.00% 1555062Ni 0.00% 487464Ni 0.00% 19150

92Mo -0.11% 1042094Mo 0.00% 1672095Mo -2.42% 971496Mo -0.10% 927197Mo -0.11% 1620098Mo -0.10% 6407

100Mo 0.94% 16060182W -0.13% 61450183W -0.14% 113800184W -0.11% 53150186W -0.11% 5728055Mn -0.72% 32760197Au -0.25% 166100Total -3.45%

As evident from Fig. 4.3, the largest contribution to the resonance integral of 58Fe comes from the 359 eVresonance. This is also the energy where large differences between different spectra occur. The cross sectionsof 95Mo, 100Mo and 55Mn around this energy are shown in Fig. 4.4.

Fig. 4.5 shows the spectra per unit lethargy in the presence of interfering nuclides; 55Mn causes a broaddip below the 58Fe resonance. The dips due to the resonances of 95Mo and 100Mo are narrower, but closer tothe 58Fe resonance. Since the solution of the slowing-down equation takes resonance scattering into account,there are bumps in the spectrum below the resonance. In the case of 100Mo this bump partly compensatesthe dip above the 58Fe resonance and hence the interference due to 100Mo is smaller compared to 95Mo, eventhough the dip at the resonance energy is stronger.

The overall decrease of the self-shielding factor of 58Fe amounts to more than 3%.

4.1.6 Resonance self-shielding

Resonance self-shielding factors of the nickel alloy were calculated for all nuclides of interest for NAA andare given in Tables 4.3 and 4.4.

First, a single-isotope analysis was performed, assuming that the number density of the nuclide underinvestigation was the same as in the real material, but the number densities of all other components werezero, thus ignoring the resonance interference between different isotopes. The results are shown in Table 4.3.The values obtained by MATSSF are in column 3 ( f ) and MCNP in column 4 ( f0). The values in column 2( f ∗) were calculated according to the method described by Chilian et. al. [62], which considers neither theinter-nuclide interference nor the differences in source-sample configuration. Also, for some nuclides column2 is left blank because the data for the calculation of self-shielding factors according to Chilian et. al. arenot available. Because of the explicit geometry and energy dependence treatment in the Monte Carlo method,the MCNP values (column 4) were taken as the reference single nuclide epithermal self-shielding factors f0.Columns 5 and 6 show deviation of the Chilian (∆ f ∗ = f ∗ − f0) and MATSSF (∆ f = f − f0) calculated self-


Incident Energy (eV)

Cro

ss S

ectio

n (

barn

s)

300 350 40010-2

10-1

1

10

102

103

ENDF/B-VII.0: MN-55(N,G)MN-56ENDF/B-VII.0: FE-58(N,G)FE-59ENDF/B-VII.0: MO-95(N,G)MO-96ENDF/B-VII.0: MO-100(N,G)MO-101

Figure 4.4: Cross sections of interfering nuclides around 359 eV [25].

Figure 4.5: Spectra per unit lethargy in the presence of interfering nuclides.


shielding factors in comparison with the reference MCNP value. Since the importance of the self-shieldingeffect is directly measured by the difference of the self-shielding factor from unity (1− f ), the deviation relativeto this difference, i.e. ∆ f /(1− f0), is a good estimator for the comparison of different methods. However, fromthe practical point of view, only the relative error of the self-shielding factor is important. Furthermore, ifthe self-shielding effect is small (. 1%), the statistical uncertainty in MCNP represents a significant fractionof the ∆ f /(1 − f0). Therefore, the relative deviation ∆ f / f0 is shown in the tables. The agreement of theself-shielding factor is generally very good for both methods (Chilian and MATSSF) for all nuclides andsource-sample configurations.

Table 4.3: Comparison of resonance self-shielding factors for the constituents of the nickel-alloy wire.Isotropic source. Separate isotopic analysis.

nuclide Chilian MATSSF MCNP ∆ f ∗

f0

∆ ff0( f ∗) ( f ) ( f0)

55Mn 0.990 0.9940 0.996 ± 0.000 -0.6% -0.2%56Fe 1.000 0.9998 1.000 ± 0.001 0.0% -0.0%58Fe 1.000 1.0000 1.002 ± 0.002 -0.2% -0.2%58Ni 0.9918 0.995 ± 0.000 -0.3%64Ni 0.995 0.9997 1.000 ± 0.000 -0.5% -0.0%

92Mo 0.9928 0.996 ± 0.002 -0.3%98Mo 0.956 0.9349 0.950 ± 0.002 0.6% -1.6%100Mo 0.965 0.9396 0.953 ± 0.002 1.3% -1.4%184W 0.9664 0.976 ± 0.001 -1.0%186W 0.797 0.8011 0.814 ± 0.002 -2.1% -1.6%197Au 0.923 0.9236 0.942 ± 0.001 -2.0% -2.0%

Due to the interference of resonances of different isotopes, the separate isotopic analysis does not giverealistic results when several resonant nuclides are present in the sample. Therefore, an additional full sampleanalysis (Table 4.4) has been performed for the observed Ni-alloy sample. Since the method by Chilian et.al. does not allow any treatment of inter-nuclide interference effects, the values in the full sample analysis(column 2, f ∗) are exactly equal to the corresponding values of the separate isotopic analysis. The MATSSFcalculated single nuclide resonance self-shielding factors (column 3, f ) are similar to corresponding valuesin Table 4.3, except for minor correction of the background cross section Σ0 which, in contrast to the caseof separate isotopic analysis, now includes the moderator cross section Σm of other nuclides. Column 4 inTable 4.4 shows the self-shielding factors G f calculated with the flux obtained with the MATSSF 640-groupslowing-down equation solver, expression (4.21), taking into account all isotopes present in the sample. Thecalculated full sample MCNP values (column 5, G f0) are again taken as reference. Columns 6-8 show relativedeviation of the Chilian (∆ f ∗ = f ∗ −G f0) and both MATSSF (∆ f = f −G f0 and ∆G f = G f −G f0) calculatedself-shielding factors from the reference G f0 .

Since MATSSF includes the treatment of interference of different isotopes, it is expected that the G f

values (column 4) should agree better with the reference MCNP calculation, compared to columns 2 and 3.Indeed, for Ni-alloy sample, the MATSSF approximations (column 4) works quite well for isotropic sourcesand reproduces to some extent the amplification of the self-shielding effect in 58Fe in spite of the coarsenessof the multigroup interference treatment. Note that the isotropic source case is representative of the problemdiscussed in Section 4.1.5.2 and that the Monte Carlo results nicely reproduce those in Table 4.2. However,for this example, the improvement due to multi-nuclide treatment is not as significant as for some samples,analyzed in [9].

Original results for a wire lying flat in the irradiation channel were less favourable [9], therefore addi-tional analysis was performed. The equivalence principle takes into account only first-order interactions withneutrons. When the wire is lying flat in the channel, most of the neutrons intersect the sample along the short


Table 4.4: Comparison of resonance self-shielding factors for the constituents of the nickel-alloy wire.Isotropic source. Full sample analysis.

nuclide Chilian MATSSF MATSSF MCNP ∆ f ∗

G f0

∆ fG f0

∆G f

G f0( f ∗) ( f ) (G f ) (G f0)55Mn 0.990 0.9941 0.9902 0.993 ± 0.000 -0.3% 0.1% -0.3%56Fe 1.000 0.9998 0.9930 0.994 ± 0.001 0.6% -0.6% -0.1%58Fe 1.000 1.0000 0.9904 0.986 ± 0.002 1.4% 1.4% 0.4%58Ni 0.9919 0.9863 0.991 ± 0.000 0.1% -0.5%64Ni 0.995 0.9997 0.9934 0.995 ± 0.000 0.0% 0.5% -0.2%

92Mo 0.9929 0.9880 0.990 ± 0.001 0.3% -0.2%98Mo 0.956 0.9359 0.9347 0.952 ± 0.001 0.4% -1.7% -1.8%100Mo 0.965 0.9407 0.9369 0.952 ± 0.002 1.4% -1.2% -1.6%184W 0.9669 0.9651 0.977 ± 0.001 -1.0% -1.2%186W 0.797 0.8033 0.7860 0.817 ± 0.001 -2.4% -1.7% -3.8%197Au 0.923 0.9247 0.9213 0.940 ± 0.001 -1.8% -1.6% -2.0%

dimension and therefore the generalized mean chord length is relatively short. If the wire contains a significantamount of material with strong scattering resonances (such as Ni), many neutrons are deflected along the wireaxis and consequently ’see’ a longer flight-path, what increases the reaction rate and reduces the self-shielding.This can be accounted for by a suitable first-order adjustment of the Bell factor. The values of the Bell factoradopted for different sample-source configurations are given in Table 4.5. Note that the described Bell factoradjustment is valid for long (H ≫ R), bare irradiation channels only. It is not applicable to small reactors, e.g.the TRIGA reactor.

Table 4.5: Bell factor for different source-sample configurations. Σs and Σt are full sample one-group scatteringand total cross sections, respectively.

sample-source configuration a∗

Wire along channel axis 1.16Wire lying flat in the channel 1.30 + 0.5Σs/Σt

Isotropic 1.16

The nickel-alloy wire serves as an interesting example for the analysis of the self-shielding factors due tothe number of interfering non-threshold neutron activation reactions. In [9], self-shielding factors have beencalculated and compared for 11 further different samples of standard monitor materials. In general, conclusionsanalogous to the nickel-alloy wire can be drawn for MATSSF and MCNP, while the inability of the method ofChilian to describe inter-nuclide resonance interference is exposed.

4.1.7 Experimental validation of self-shielding factors for Rh foils

Rhodium foils of about 5 mm diameter, 0.006 mm (labelled Rh1) and 0.112 mm (Rh2) thick, were irradiated[10] in the core and in the reflector of the TRIGA reactor at Jozef Stefan Institute [13]. Another series ofmeasurements (samples Rh01, Rh02, Rh03) was performed1 using samples with diluted rhodium solution incellulose matrix. Geometrical and material specifications of all samples are shown in Table 4.6.

Two irradiations were performed for each foil: one for the bare foil and the other for the foil enclosed ina small cadmium box of 1 mm thickness. The measured 104Rh activation product specific activities Asp, and

1Personal communication with Radojko Jacimovic, 2011.


Table 4.6: List of irradiated rhodium samples.

label material ρ [g/cm3] thickness (d) [mm] diameter (2r) [mm] Rh mass [mg]Rh1 Rh 12.41 0.0057 5.02 1.40Rh2 Rh 12.41 0.112 5.04 27.7

Rh01 Rh in C6H10O50.85 2.0 5.0

0.016Rh02 Rh in C6H10O5 0.014Rh03 Rh in C6H10O5 0.019

cadmium ratios RCd for irradiation channel IC40 in the TRIGA reflector are summarized in Table 4.7. Thespecific activity Asp is proportional to the activation reaction rate [67], and the cadmium ratio RCd is defined asthe ratio of specific activity between bare and cadmium-covered identical samples, irradiated simultaneously.

Table 4.7: Measured 104Rh specific activities Asp and cadmium ratios RCd for irradiated rhodium samples. Theexperimental uncertainties consist of statistical counting uncertainties of the γ detector only.

label Cd cover Asp [min−1g−1] RCd

Rh1yes 5.44 · 108(1 ± 0.005)

5.55(1 ± 0.013)no 3.02 · 109(1 ± 0.012)

Rh2yes 2.48 · 108(1 ± 0.005)

9.33(1 ± 0.007)no 2.32 · 109(1 ± 0.005)

Rh01yes 1.91 · 106(1 ± 0.016)

5.01(1 ± 0.017)no 9.59 · 106(1 ± 0.006)

Rh02yes 1.86 · 106(1 ± 0.013)

5.21(1 ± 0.014)no 9.69 · 106(1 ± 0.006)

Rh03yes 1.85 · 106(1 ± 0.0144)

5.15(1 ± 0.016)no 9.54 · 106(1 ± 0.0062)

The resonance self-shielding factors G f and so-called thermal self-shielding factors

Gth =

∫ ECd

0 Σ(E)ϕ(E)dE∫ ECd

0 Σ(E)ϕ0(E)dE, (4.38)

where ECd = 0.55 eV is the cadmium cutoff energy, were calculated by MATSSF and MCNP and are shownin Table 4.8. Again, note that ϕ0(E) is the spectrum in the void irradiation channel and ϕ(E) is the spectrum,perturbed by the irradiated sample. The resonance self-shielding corrections for the 0.006 mm and 0.112 mmthick samples amount to about 10% and 60%, respectively. For this comparison, the extended MCNP modelof the irradiation channel [14, 15] has been used in order to more realistically describe the experiment: the ir-radiation channel and the surrounding moderator have been modelled explicitly, the cylindrical neutron sourcehas been pulled back into the moderator, the spectrum and axial distribution have been calculated using thefull MCNP model of the TRIGA reactor. In this more realistic case, ϕ0(E) is the local reactor spectrum ratherthan the pure 1/E spectrum. The MCNP results for 1/E spectrum are still included in the Table 4.8 sincethey are directly comparable to the MATSSF results, due to the same external spectrum and source/sampleconfiguration. In this case the systematic errors for the MATSSF calculated resonance self-shielding factorof the thick foil due to physical approximations and realism of the model (irradiation channel geometry andsource configuration), affect the self-shielding factor in opposite directions.

Since self-shielding factors cannot be measured directly, a different quantity, which is proportional to the


Table 4.8: Calculated self-shielding factors and corresponding dilution cross sections Σ(mi)0 for irradiated

rhodium samples.

label code Gth G f Σ(mi)0 [b]

Rh1MCNP 0.982 ± 0.001 0.857 ± 0.007

1.8 · 104MCNP (1/E) 0.983 ± 0.001 0.873 ± 0.004MATSSF 0.9833 0.8777

Rh2MCNP 0.828 ± 0.001 0.378 ± 0.005

910MCNP (1/E) 0.824 ± 0.001 0.389 ± 0.001MATSSF 0.8339 0.3734

Rh01* MATSSF 0.9979 0.9983 2.2 · 106

Rh02* MATSSF 0.9979 0.9984 2.5 · 106

Rh03* MATSSF 0.9978 0.9981 1.8 · 106

*Due to small corrections, the self-shielding factors for diluted samples were not calculated in MCNP.

thermal to fast reaction rate ratio, has been observed:

frel = (RCdFCd − 1)G f

Gth. (4.39)

FCd is the cadmium transmission factor for rhodium, calculated from ENDF/B-VII.1 library [25]:

FCd =

∫ ∞0 t(E)Σ(E)ϕ(E)dE∫ E3

ECdΣ(E)ϕ(E)dE

= 0.936, (4.40)

where t(E) is the cadmium filter transmission function [67], and E3 = 2 MeV. The cadmium transmissionfactor FCd is not as sensitive to the spectrum as the self-shielding factors, therefore it is assumed constant.

It can be shown straightforwardly that the factor frel is by definition almost independent of the self-


shielding, which is connected to the perturbed flux ϕ(E):

frel = (RCdFCd − 1)G f

Gth

=

∫ ∞

0 Σ(E)ϕ(E)dE∫ ∞0 t(E)Σ(E)ϕ(E)dE

∫ ∞0 t(E)Σ(E)ϕ(E)dE∫ E3

ECdΣ(E)ϕ(E)dE

− 1

G f

Gth

=

∫ ∞

0 Σ(E)ϕ(E)dE∫ E3

ECdΣ(E)ϕ(E)dE

− 1

G f

Gth

=

∫ ∞0 Σ(E)ϕ(E)dE −

∫ E3

ECdΣ(E)ϕ(E)dE∫ E3

ECdΣ(E)ϕ(E)dE

G f

Gth

=

∫ ECd

0 Σ(E)ϕ(E)dE +∫ ∞

E3Σ(E)ϕ(E)dE∫ E3

ECdΣ(E)ϕ(E)dE

G f

Gth

≃∫ ECd


ECdΣ(E)ϕ(E)dE

G f

Gth

=

∫ ECd


ECdΣ(E)ϕ(E)dE

∫ E3

ECdΣ(E)ϕ(E)dE∫ E3

ECdΣ(E)ϕ0(E)dE

∫ ECd

0 Σ(E)ϕ0(E)dE∫ ECd

0 Σ(E)ϕ(E)dE

=

∫ ECd

0 Σ(E)ϕ0(E)dE∫ E3

ECdΣ(E)ϕ0(E)dE

. (4.41)

The only assumption in the above derivation is that the contribution to the reaction rate above E3 = 2 MeV isnegligible, which is a well-founded assumption for non-threshold reactions in thermal reactor spectra.

In the formulation (4.39), the factor frel is parameterized as a function of measured (RCd) and calculated(FCd, G f and Gth) quantities. Strictly speaking, independence of frel on the absorber content/sample size isessential but not sufficient for the consistency of the calculations and the experiment. In spite of that, factorfrel can be considered a good indicator of the agreement between calculations and experiment.

The factors frel for both rhodium foils (Rh1 and Rh2) have been compared to the reference sample (Rh0)with diluted rhodium. The results (Table 4.9 and Fig. 4.6) show that the self-shielding factors, calculatedwith MCNP and MATSSF agree within 10% even for the foil with significant self-shielding. Even thougha significant part of the discrepancy of the factor frel of the thick foil from the reference can be accountedas systematic error, this 10% relative difference corresponds to less than 4% absolute difference in the self-shielding factor, which is satisfactory considering the complex resonance interference phenomena. Throughthis, a step towards a successful validation of the calculated self-shielding factors has been made [10], althoughfor a completely reliable validation additional measurements would have to be performed.

Table 4.9: Comparison of the factor frel for rhodium wires of different thicknesses. Irradiation channel IC40of the TRIGA reactor. Rh0 corresponds to the weighted average of all three ’infinitely’ diluted samples.

sample/code Rh01 Rh02 Rh03 Rh0 Rh1 Rh2MCNP / / / / 3.66 ± 0.05 3.53 ± 0.03

MATSSF 3.69 ± 0.05 3.87 ± 0.05 3.82 ± 0.06 3.80 ± 0.03 3.74 ± 0.02 3.46 ± 0.02


103 104 105 106

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4.0

f rel

(mi)0 [b]

MATSSF MCNP

Figure 4.6: Factor frel versus dilution cross section Σ(mi)0 for self-shielding factors, calculated by MATSSF and

MCNP. The uncertainty band (within the dotted lines) corresponds to the ±1σ interval of the weighted averageof the three samples with diluted rhodium.

4.2. EXAMPLE OF RESONANCE COVARIANCE ANALYSIS: 55MN RESONANCE INTEGRAL 79

4.2 Example of resonance covariance analysis: 55Mn resonance integral

The slowing-down of neutrons introduces a component with a 1/E shape into the neutron spectrum downto thermal energies, where the shape turns into approximately Maxwellian distribution when neutrons comeinto thermal equilibrium with the nuclei of the medium. Deviations from the idealized shape of the spectrumdepend primarily on the absorption of the medium. In many cases critical systems are designed to enhancethermalization. In such systems, the resonance absorption above the thermal energy range, where the neutronspectrum follows closely the 1/E behaviour, may represent a significant contribution to the absorption reactionrate of many nuclides.

The tendency in modern evaluated nuclear data files is to include cross section covariance information sothat uncertainties in integral observables like resonance integral (RI) can be calculated from the uncertaintiesin cross section. In the resonance range the uncertainties in the cross section are often given in terms ofthe covariance matrix of the resonance parameters (ENDF-6 File 32). For nuclides with a large number ofresonances the covariance matrix can become very large and expensive to process in terms of the computationtime. By converting the covariance matrix of resonance parameters into a covariance matrix of cross sectionuncertainties (ENDF-6 File 33) valid over specified coarse energy bins, the amount of information can beconsiderably reduced. The question is how important is the information that is discarded in the process.Differences in covariance formatting are expected to show up in the Doppler-broadened and self-shielded crosssections. In the case of cross section covariances in ENDF-6 File 33, the relative uncertainties remain almostthe same at all temperatures and levels of self-shielding. The uncertainties of integral quantities calculateddirectly from the covariances of the resonance parameters have a more complex relationship.

The main objective of the investigation [16] was to study the derived uncertainties in the neutron captureresonance integral at room temperature for various levels of self-shielding on the ENDF-6 format of covari-ances in the resonance region. The resonance integrals were calculated from the reconstructed cross sectionsby the narrow resonance approximation (NR) with the GROUPIE code of the PREPRO series [19]. The crosssections were generated by the random sampling of the resonance parameters, taking parameter correlationsinto account. The 55Mn capture was chosen as an example, for which two resonance-covariance evaluationsare available, developed at NRG [17] and ORNL2, respectively. The uncertainties and correlations of theresonance parameters for these two evaluations are shown in Section 3.5.1.

For comparison, the resonance integral uncertainty is also calculated with the ORNL resonance covariancematrix converted into the SAND-II 640-group structured [65] cross section covariance matrix using NJOY pro-cessing system [64] as customary in reactor dosimetry applications. The results are discussed in Section 4.2.3.

4.2.1 Random sampling procedure

A random sampling method (the diagonalization method, described in Section 3.2.2.1) has been used to cal-culate resonance cross sections and their uncertainties taking into account the full covariance matrices. Thenominal resonance parameter values and their covariance matrix are read from the selected ENDF file. Byrandom sampling around the nominal values and considering correlations according to the information in thecovariance matrix, a number of perturbed cross section files in ENDF-6 format is obtained. Each of the per-turbed ENDF files is then processed to investigate the influence of the uncertainties in resonance parameterson integral observables, in particular on the resonance integral (RI). For calculations in this section, 1000perturbed ENDF files have been produced.

4.2.2 Resonance integral

The following resonance integral definition is used:

RI =∫ E3

ECd

Σ(mi)(E)ϕ(E)dE, (4.42)

2Luiz Leal, private communication, October 2010.


where ECd = 0.55 eV, E3 = 2 MeV, ϕ(E) is neutron spectrum, and Σ(mi)(E) is the microscopic cross section,calculated from the sampled resonance parameters. In the case presented below, Σ(mi)(E) represents the 55Mnneutron capture microscopic cross section. The neutron flux has been calculated with the GROUPIE code ofthe PREPRO series [19] using the NR approximation (Section 3.1.1):

ϕ(E) =Σ

(mi)0 + Σ

(mi)p

Σ(mi)0 + Σ

(mi)t (E)

ϕ0(E) =Σ

(mi)0 + Σ

(mi)p

Σ(mi)0 + Σ

(mi)t (E)

1E, (4.43)

where Σ(mi)t is the total microscopic cross section, Σ(mi)

p is microscopic potential scattering cross section, Σ(mi)0

is the dilution cross section, which is inversely proportional to the sample size, Eq. (3.16), and ϕ0(E) = 1/E isproportional to the infinite dilution neutron slowing-down spectrum in a medium with negligible absorption.

However, for finite Σ(mi)0 the weighting spectrum φ(E) in GROUPIE is renormalized φ(E) = kϕ(E), where

the scaling factor k is such that the integral of ϕ0(E) is preserved∫ E3

ECd

φ(E)dE =∫ E3

ECd

kϕ(E)dE = lnE3

ECd. (4.44)

The renormalized resonance integral RI′ equals:

RI′ =∫ E3

ECd

Σ(mi)(E)φ(E)dE = k∫ E3

ECd

Σ(mi)(E)ϕ(E)dE = kRI (4.45)

where the scaling factor k = ln E3ECd

/ϕtot is derived from Eq. (4.44) and ϕtot is the integral of ϕ(E).This normalization merely affects the absolute value of the resonance integral, but more importantly, it

does not alter the relative uncertainty of RI.

4.2.3 Results

4.2.3.1 Uncertainty in the resonance integral from resonance parameters

The main result of this study is obtained from the comparison of the resonance integral and its relative un-certainty as a function of the dilution cross section, calculated with the ORNL and NRG resonance parameterdata.

For each sampled parameter set, the resonance integral has been calculated following Eqs. (4.42) and(4.43). The resonance integral uncertainty has been calculated as simply the statistical standard deviation ofthe resonance integral from 1000 random ENDF samples. The convergence of RI and its uncertainty has beenchecked against the number of samples, and for 1000 samples the error is at least an order of magnitude lowerthan the calculated parameter. Resonance integral RI′ and its relative uncertainty as a function of the dilutioncross section Σ(mi)

0 are shown in Fig. 4.7. Mean fully-shielded resonance integrals from the NRG and ORNLevaluations are quite similar, but at infinite dilution the discrepancy is as high as 15%. This is in contradictionwith the uncertainty estimated from the covariance data. This is caused by the inconsistency of the evaluatedresonance parameters, which is not the subject of the present investigation. The resonance contribution to theuncertainty for the ORNL data is of the order of 0.05% for infinite dilution, and the total uncertainty is 3.7%,including background cross section uncertainty. The uncertainty for the NRG data amounts to around 0.5%, ispractically independent of the dilution and is given by the uncertainties of the resonance parameters alone.

The influence of self-shielding on the relative resonance integral uncertainty is emphasized in Fig. 4.8.The NRG resonance data at high dilutions give a relatively high uncertainty, which is a direct consequenceof larger uncertainties assigned to the resonance parameters. Due to lack of correlations between resonances,the uncertainty remains approximately constant with increasing level of self-shielding. On the contrary, theresonance contribution in the ORNL data to the overall uncertainty is very small. The total uncertainty isdominated by the background, which is assumed to be independent of the self-shielding level. As the levelof self-shielding increases, the uncertainty also strongly increases for the strongly correlated ORNL data (seecomparison of the relative uncertainties with and without background on Fig. 4.8). At infinite dilution, the


1 10 100 1000 10000

6

8

10

12

14

RI'

[b]

(mi)0 [b]

ORNL data NRG data

Figure 4.7: Comparison of resonance integral RI′ and its uncertainty as a function of the dilution cross sectionΣ

(mi)0 calculated from NRG and ORNL resonance data.

resonance contribution to the RI uncertainty is dominated by the uncertainties of the strongest resonances. Athigh levels of self-shielding the importance of the highest resonance peaks is diminished. If resonances arecorrelated, the relative uncertainties in the dips between resonances increase, giving rise to an overall increasein the resonance integral uncertainty.

When self-shielding becomes stronger, the dips between the resonances, where the cross section variationsare absolutely small but relatively still significant, gain a larger influence on the resonance integral, whileon the other hand, the magnitude of the peaks does not affect the resonance integral much since the flux isalready strongly depressed. In the case of uncoupled resonances the uncertainty contributions from separateresonances tend to much better cancel out (i.e. as a sum of the squares) than in the case with full resonanceparameter covariance matrix, where the total uncertainty increases linearly with the number of resonances.

For the NR approximation this effect can easily be shown. Inserting Eq. (4.43) in Eq. (4.42) we get:

RI =∫ E3

ECd

Σ(mi)(E)(Σ(mi)0 + Σ

(mi)p )

Σ(mi)0 + Σ

(mi)t (E)

dEE. (4.46)

Near resonance peaks and/or for low dilution cross section Σ(mi)t (E) ≫ Σ(mi)

0 holds, and the contribution tothe resonance integral reduces to:

dRI ≃ (Σ(mi)0 + Σ

(mi)p )Σ(mi)(E)

Σ(mi)t (E)

dEE. (4.47)

Since the small-scale variations (i.e. resonances) in Σ(mi)(E) are frequently proportional to the variations inΣ

(mi)t (E), they apparently have only negligible influence on RI.


1 10 100 1000 10000

0.1

1

10RI/R

I [%

]

(mi)0 [b]

ORNL data ORNL (resonance only) NRG data

Figure 4.8: Relative uncertainty σRI/RI of the resonance integral as a function of the dilution cross sectionΣ

(mi)0 of sparse covariance matrix and large uncertainties (NRG data) to full covariance matrix but smaller

uncertainties (ORNL data) of the resonance parameters.

On the other hand, in dips between resonances and/or for high dilution cross section Σ(mi)t (E) ≪ Σ(mi)

0holds. It follows:

dRI ≃ Σ(mi)(E)dEE. (4.48)

In this case, contribution to the RI is proportional to Σ(mi)(E) and since the latter is small by assumption, itscontribution to RI, both relative and absolute, is very small.

Hence the important differences in RI occur due to Σ(mi)t (E) variations in the range comparable to Σ(mi)

0 . Dueto the shape of the resonances, the gradient of the cross section dΣ(mi)/dE decreases proportionally to (E−E0)−3

far away from the resonance peak, what follows e.g. from Eq. (2.8) by assuming K is constant. Consequently,the energy interval around the resonance peak, where the contribution of the cross section uncertainty on RI issignificant, is greatly broadened when self-shielding effect is strong. Since the cross section Σ(mi)

t (E) ∼ Σ(mi)0

far away from resonance peaks where relative variation of the cross section is expected to be similar to thevariation around the resonance peak (Section 2.4.1), the increase of both relative and absolute uncertainty ofthe resonance integral is expected when the self-shielding is amplified.

4.2.3.2 Uncertainties in the resonance integral – reactor dosimetry 640-group approximation

It has been shown for the 55Mn case that taking strong self-shielding effect into account, the relative uncer-tainty in the resonance integral may increase by several orders of magnitude if calculated directly from theuncertainties in the resonance parameters, i.e. based on the original ENDF-6 File 32. In this work we inves-tigate if the commonly used reactor dosimetry SAND-II 640-group approximation is sufficient to accuratelyquantitatively describe this increase in the relative uncertainty of the resonance integral.

The uncertainty in the resonance integral has been calculated in the NR approximation with the ORNLevaluation for 55Mn. The 640-group covariance matrices have been generated from the resonance parameter


covariances using the NJOY processing system [64], including the uncertainty of the background.The uncertainty in the resonance integral has been calculated for infinite dilution, corresponding to the

unperturbed 1/E spectrum, and for Σ(mi)0 = 1 b, where only the uncertainty in the cross section has been taken

into account. The relative uncertainty equals 3.8% and 3.2%, for Σ(mi)0 = ∞ and Σ(mi)

0 = 1 b, respectively.As shown in the following paragraph the total uncertainty for Σ(mi)

0 = 1 b is less than twice the uncertaintydue to cross sections only, i.e. < 6.4%, which is much less than when calculated directly from the resonanceparameters (16%, recall Fig. 4.8). This demonstrates that the 640-group approximation is not sufficientlyaccurate to describe the increase in the relative uncertainty of the resonance integral for increased self-shieldingdue to correlations of the resonance parameters.

Contributions of the cross section and flux uncertainties to the total resonance integral uncertainty. Ingeneral, the relative variance of the resonance integral may be expressed as (see Eq. (3.34)):(

σRI

RI

)2=

(RIrel

)T(VrelΣ + Vrel

ϕ + 2VrelΣ,ϕ)RIrel, (4.49)

where

• RIrel is the vector of relative group resonance integrals RIg/RI,

• VrelΣ

is the relative covariance matrix of the cross sections,

• Vrelϕ is the relative covariance matrix of the spectrum, and

• VrelΣ,ϕ is the matrix of relative covariances between cross section and spectrum.

Thus VrelΣ

, Vrelϕ , and Vrel

Σ,ϕ are blocks of the total cross section-spectrum relative covariance matrix:

Vrel =

VrelΣ

VrelΣ,ϕ

VrelΣ,ϕ Vrel

ϕ

(4.50)

In NR approximation,

(Vrelϕ

)gh=

∑i, j

∂ϕg

∂Σ(mi)t,i

Σ(mi)t,i

ϕg

(VrelΣt

)i j

∂ϕh

∂Σ(mi)t, j

Σ(mi)t, j

ϕh=

Σ(mi)t,g

Σ(mi)0 + Σ

(mi)t,g

(VrelΣt

)gh

Σ(mi)t,h

Σ(mi)0 + Σ

(mi)t,h

,∣∣∣∣(Vrelϕ

)gh

∣∣∣∣ ≤ ∣∣∣∣(VrelΣt

)gh

∣∣∣∣ , (4.51)

the equality being valid in the limit of infinite self-shielding (Σ(mi)0 = 0). Furthermore,∣∣∣∣(Vrel

Σt

)gh

∣∣∣∣ < ∣∣∣∣(VrelΣ

)gh

∣∣∣∣ (4.52)

is practically always true, since the measurements of the total cross section are much more precise comparedto the measurements of any partial cross section. Moreover,∣∣∣∣(Vrel

Σ,ϕ

)gh

∣∣∣∣ < ∣∣∣∣(VrelΣ

)gh

∣∣∣∣ (4.53)

since the relative spectrum uncertainties do not exceed the relative cross section uncertainties and the correla-tion is at most 100%.

Since negative spectrum covariances are larger (less negative) than corresponding cross section covari-ances, the contribution to the overall RI uncertainty could in principle be larger. However, since the correlationmatrices for spectrum and cross section are equal, the dominance of the diagonal elements of the covariance


matrix ensures that the increase due to anti-correlation is always smaller compared to the decrease due toreduced variances. Finally, we obtain: (

σRI

RI

)2< 4

(RIrel

)TVrelΣ RIrel. (4.54)

Clearly, the total relative resonance integral uncertainty is at most twice the uncertainty due to cross sectionsonly, the only assumption being that the relative uncertainty in total cross section does not exceed the uncer-tainty in the partial (usually (n,γ)) cross section. This result confirms that the correlated uncertainty in thespectrum due to the uncertainty in nuclear data cannot contribute sufficiently to the total uncertainty to accountfor the large difference compared to the uncertainty, calculated from the resonance covariances.

Chapter 5

Nuclear data adjustment

Nuclear data are generally based on different semi-empirical quantum or semi-classical nuclear models, e.g.the nuclear optical model [23], which typically include several parameters which are either free or with rel-atively loose constraints. In nuclear data evaluations, these parameters are determined to fit the differentialexperimental data. The data may be adjusted further by taking into account integral experiments as well.

When performing nuclear data adjustment, the evaluator usually starts with nuclear models, which includea number of parameters with uncertainties. From these, cross sections and/or other nuclear data with priorcovariance matrices are generated. Prior probability distribution or simply prior of an uncertain quantity is thebest guess of its probability distribution before additional data is taken into account. In the case of nuclear dataadjustment, the prior includes the expected value and standard deviation of observed quantities, i.e. the tabu-lated cross section and the corresponding covariance matrix. The cross sections along with covariance data arethen adjusted by taking into account both differential (various cross sections, fission spectrum, etc.) measure-ments (e.g. from EXFOR database, Section 5.1) and integral experiments (e.g. from the ICSBEP handbook[58]) using the generalized method of least squares (Section 5.2) . If nuclear models and experimental dataare consistent, the uncertainties of the final evaluation should on average decrease relative to the prior uncer-tainties, however especially integral experiments generally increase the cross-correlations between differentnuclear data. In reality, for various reasons, experimental data are not always consistent, for example due tounnoticed or underestimated correlations between different experimental data, underestimated uncertainties,or simply erroneous measurements. Therefore, nuclear data evaluator often has to filter experimental dataor manually increase their uncertainties and correlations to artificially reduce their impact on the adjustment.Unfortunately, this filtering procedure is highly subjective and therefore, as a rule, two different evaluators willproduce two different evaluated nuclear data files even with exactly the same data at their disposal! However,on the plus side, two experienced evaluators will produce similar evaluated nuclear data files.

5.1 EXFOR database

The Experimental Nuclear Reaction Data (EXFOR) library contains an extensive compilation of experimentalnuclear reaction data [68]. Neutron reactions have been compiled systematically since the discovery of theneutron, while charged particle and photon reactions have been covered less extensively. EXFOR is an ex-change format designed to allow transmission of nuclear reaction data between the members of the NuclearReaction Data Centers Network (NRDC) which has been organized under the auspices of the InternationalAtomic Energy Agency to coordinate the collection, compilation, and dissemination of nuclear data on aninternational scale. The EXFOR format has been designed for flexibility rather than optimization of data pro-cessing in order to meet the diverse needs of the nuclear reaction data centers [68]. The EXFOR format iscontinuously refined and expanded to include new types of data as the need arises. A series of keywords andcodes have been designed to implement this; these keywords and codes are defined in the EXFOR Dictionar-ies. For example, a typical series of experimental data for a particular neutron induced cross section reactionwith a nuclide include: author and year, internal EXFOR identification number, common (systematic) errorfor all data of the series, and information for single data points: cross sections with uncertainties at different

85

86 CHAPTER 5. NUCLEAR DATA ADJUSTMENT

energies which should also have estimated uncertainties.

5.2 Generalized least squares fitting

The generalized least squares method assumes normal distribution of all variables in question [69, 70]. Let usstart with a vector of parameters x, with expected values x0 and covariance matrix V x, which represent the onlyand best available information about x at a certain instant of time. The parameter vector x may, for example,correspond to tabulated cross section as a function of incident neutron energy or resonance parameters. Then,our best guess about the probability distribution p(x) is the multivariate normal distribution with parametersx0 and V x [3], which is proportional to:

p(x) ∝ exp[−1

2(x − x0)T

(V x

)−1(x − x0)

]. (5.1)

Afterwards at some point, new measurements of known functions y(x), arbitrary integral or differentialobservables, y0 with covariance matrix V y and, in general, cross-covariance matrix with initial data V xy aswell, become available. An extended parameter vector, consisting of both x and y, is constructed:

z(x) =(

xy(x)

). (5.2)

It follows

V z =

(V x V xy

V xy V y

). (5.3)

The probability distribution p(z) of the new parameter vector is again assumed to be normal:

p(z) ∝ exp[−1

2(z − z0)T

(V z

)−1(z − z0)

]. (5.4)

If written in terms of initial parameter vector, the expression

(z − z0)T(V z

)−1(z − z0) = Q(x) = Q(x0,ad j) + (x − x0,ad j)T

(V x,ad j

)−1(x − x0,ad j) (5.5)

can be manipulated to show a new, adjusted probability distribution for initial data x, where the index ad jstands for adjusted (expected values, covariances, etc.). The relation between the adjusted and initial covari-ance matrices can be obtained by differentiating Q(x):

12∇x∇x

T Q(x) = ∇x

(∂z∂x

)T (V z

)−1(z − z0)

=

(∂z∂x

)T (V z

)−1 ∂z∂x

= S T(V z

)−1S (5.6)

from LHS of Eq. (5.5), and

12∇x∇x

T Q(x) = ∇x

(V x,ad j

)−1(x − x0,ad j)

=(V x,ad j

)−1(5.7)

from RHS of Eq. (5.5), where ∇x is the gradient operator in the parameter space of vector x and S = ∂z∂x is the

matrix of sensitivity coefficients. Combining Eqs. (5.6) and (5.7), the adjusted covariance matrix is expressedin terms of initial (prior) covariance matrix V z and sensitivity matrix S , which are both known:

V x,ad j =

[S T

(V z

)−1S]−1= S −1V z(S T )−1. (5.8)

5.3. GLOBAL ASSESSMENT OF NUCLEAR DATA REQUIREMENTS (GANDR) 87

The adjusted parameters x0,ad j can be obtained from Eqs. (5.6) and (5.7) by setting x = 0 and z = 0:

x0,ad j = V x,ad jS T(V z

)−1z0. (5.9)

The generalized least squares method is a first-order approximation, analogous to the first-order Taylorexpansion based ’sandwich formula’ for uncertainty propagation (Section 3.1.2), with additional assumptionof normal distribution of all relevant variables. The limitations of the method are clear: it has problems withnon-linearities (a very illustrative example can be found in Section XII. of [3]) which are usually encounteredwhen dealing with large relative uncertainties, and obviously with non-Gaussian distributions which in realisticapplications are a minor difficulty compared to the non-linearities.

5.3 Global Assessment of Nuclear Data Requirements (GANDR)

Program system GANDR (Global Assessment of Nuclear Data Requirements) [6] is the first attempt of awholly general cross section covariance treatment. It enables cross-covariance treatment between more than100 different isotopes, each described by up to 25 custom defined reactions.

Due to the vast number of reactions and isotopes, the cross section energy grid for each reaction is limitedto 74 energy points or nodes E(n) in order not to run out of computer memory. Linear interpolation for crosssection in between nodes is used ([6], Vol. 1, pp. 24-27). The linear interpolation is implemented via theso-called ’hat’ functions, analogously to the finite element method for solving partial differential equations:

H(n, E) =

(E − E(n − 1))/(E(n) − E(n − 1)), E(n − 1) ≤ E < E(n)(E(n + 1) − E)/(E(n + 1) − E(n)), E(n) ≤ E < E(n + 1)0, elsewhere

(5.10)

where trivially74∑

n=1

H(n, E) = 1. (5.11)

A correction function is defined

Aad j(E) =74∑

n=1

aad j(n)H(n, E) (5.12)

where aad j(n) is the correction factor at energy E(n). Obviously, between energy nodes n the correctionfunction is linear. The adjusted cross section is calculated by simply multiplying the initial (prior) crosssections by the correction function:

Σ(mi)ad j,r,k(E) = Aad j,r,k(E)Σ(mi)

r,k (E) =74∑

n=1

aad j,r,k(n)H(n, E)Σ(mi)r,k (E) (5.13)

where indices r and k define reaction and isotope, respectively. Input data for GANDR are prior cross sectionsΣ

(mi)r,k (E) and covariance matrices, defined on energy grid n, which are then adjusted by including experimental

(EXFOR) data through generalized least squares method (Refs. [69, 70], Section 5.2) which is linear – it workswell for small uncertainties only.

GANDR adjustments alter cross sections on a coarse scale, whereas the fine shape of energy dependence isleft intact. The relatively coarse energy grid is problematic mainly in the resonance part of the spectrum, wherethe mesh is much too coarse to correctly describe the resonance behaviour of the cross sections, and near thethreshold for threshold reactions. In thermal and fast energy ranges, the density of the mesh is sufficient formost applications. In some cases, the coarse energy mesh may even be an advantage since it prevents smallscale oscillations of the cross sections due to numerical instabilities.


5.4 Example: tungsten

Modern evaluations of nuclear reaction data rely heavily on nuclear model codes, which are also used todefine the prior for the cross section covariance matrix, including cross-reaction correlations. Systematicuncertainties due to the deficiencies in nuclear models can be treated by a suitable sampling scheme of scalingfactors for individual reaction channels. By treating the total cross section as redundant, the constraint that thesum of the partial cross sections of a material equals the total cross section is guaranteed automatically evenfor the uncertainties, as long as the covariance matrices of all constituting reactions including cross-reactioncorrelations are included. Cross-material correlations exist whenever the same set of nuclear model parametersis applied to a group of nuclides, such as the isotopes of the same element or neighbouring nuclides in termsof atomic and mass numbers. This feature has not been explored in any detail yet. Experimental data arealso correlated due to systematic errors and uncertainties in a single set of measurements, as well as betweendifferent sets of measurements that use the same standards, measuring technique, etc. Information on suchuncertainties and correlations is often lacking, but is crucial for obtaining realistic uncertainty estimates. This isprobably an area of greatest subjectivity in the evaluation of covariances. Measurements on elemental samplesare more abundant because enriched isotopic samples are expensive. There are indications that omission ofcross-isotope correlations may lead to an underestimation of uncertainties by a factor of two or more (anexample is presented in Table 5.2). The study of such correlations requires a detailed investigation. Each ofthe items listed above has to be investigated carefully for its impact on the evaluated cross sections and theassociated uncertainties, so as not to overlook important effects when reducing the complexity in structure andvolume of the archived information. In the covariance analysis [8] of the tungsten (W) cross section in the fastenergy range the following were considered:

• The FNG-W experiment [71] performed at the Frascatti Neutron Generator [72] with a deuterium-tritium(D-T) neutron source (14.1 MeV) in front of a large tungsten block was analysed as an example.

• The reference case was the calculation with tungsten isotopic data evaluated in the previous work [7].

• The experimental cross section data used in the analysis were extended by including the total crosssection measurements on natural element. The impact on the cross sections and the uncertainties wasstudied.

• The covariance data were extracted from the GANDR system. In addition to the covariances of individ-ual isotopes, cross-material covariances were extracted for all major isotopes of tungsten.

• The evaluated nuclear reaction data files were processed with the NJOY [64] and PREPRO [19] systems.

• Activation rates and uncertainties for monitor reactions in the FNG-W benchmark experiment werecalculated and compared to previous calculations and measurements.

The work was a contribution to the continuing effort for developing the methodology of generating covarianceinformation that is validated in practical applications. In the process, the GANDR system [6] was upgradedby the author to Version 4; a major upgrade of the local covariance processing and formatting tools was alsorequired.

5.4.1 Tungsten cross section adjustment and correlation treatment

Previously, cross section adjustments, uncertainties and covariances of four different tungsten isotopes (182W,183W, 184W, and 186W) [7] were determined by the GANDR system [6]. The IAEA evaluation version labelled’ib21g’ was taken for the initial isotopic pointwise cross sections in ENDF format (PENDF). Prior covariancematrices were calculated by the Monte Carlo method: nuclear model input parameters were randomly sampledwithin their uncertainty bands [1]. Model calculations were done with the EMPIRE-3 code. The prior was fedinto the GANDR system. A selection of experimental data from the EXFOR database was made, correctingknown errors and renormalizing to newer standards, when needed. Quoted uncertainties were supplemented

5.4. EXAMPLE: TUNGSTEN 89

with estimated systematic uncertainties, as described in [7]. By applying the experimental data for differentreactions in GANDR, cross section adjustment factors and within-material covariance matrices were calcu-lated. Cross section correlation treatment between energy nodes and reactions was enabled by constructing alarge covariance matrix, including all reactions. The final cross section equal the product of the prior crosssection and the adjustment function. The original analysis provided covariances between different cross sec-tions energy ranges for individual materials. However, some results indicate that there may be a need for fullcross-material covariance treatment between the tungsten isotopes.

5.4.1.1 Experimental data

Since the isotope-enriched samples are expensive, cross section measurements on high-purity elemental sam-ples are more convenient. Consequently, there is a larger number of experimental data available for elementaltungsten. Experimental datasets of the total cross section of elemental tungsten for the purpose of covarianceanalysis were carefully chosen from the EXFOR database in order to ensure a consistent evaluation of the crosssection uncertainties and correlations. The full list of experimental datasets taken into consideration is givenin Table 5.1. Individual data sets can be identified by the EXFOR entry number. Full reference to the data canbe found in the EXFOR database. Since the covariances in the resonance range are determined separately bythe resonance analysis, experimental data in the resonance range and below were excluded. Inconsistent dataand data, for which uncertainty information was incomplete, were rejected. Also, a systematic error was addedartificially to all measurements. Due to insufficient uncertainty information about some of the experiments,assignment of systematic uncertainties may be subjective, but it is necessary in order to avoid underestimationof the overall uncertainties. If one is conservative enough, the difference in results due to the subjectiveness ofthe method should be consistent within the calculated uncertainty.


Table 5.1: Natural W total cross section data from EXFOR.Author (year) EXFOR No. AcceptedJones (71) X10194 9 -Murthy (75) X10403 9 -Jones (68) X11206 6 -Parker (70) X13026 9 -Knopf (87) X22045 7 -Mezentseva (06) X41491 21 -Schmunk (60) X11634 7 -Havens JR (47) X12184 2 -Kim (03) X31581 2 -Schwartz (56) X12152 2 -Selove (51) X12168 2 -Chrien (58) X12153 2 -Harvey (58) X12154 4 -Egelstaff (55) X21037 9 -Grigoriev (06) X41505 7 -Berlev (07) X41510 7 -Egelstaff (57) X21038 10 -Dilg (72) X20583 27 -Fields (47) X11260 21 -Filippov (68) X40082 19 -Adair (50) X11636 6 -Whalen (66) X11540 8 -Divadeenam (68) X10523 14 yesGrigoriev (87) X40989 3 -Harvey (80) X13770 6 -Whalen (64) X12176 2 -Miller (52) X11712 21 -Foster JR (71) X10047 78 yesNereson (54) X11308 22 -Walt (55) X11215 44 -Abfalterer (01) X13753 25 yesKao (67) X30050 3 -Bratenahl (58) X11155 27 -Goodman (52) X11057 19 -Coon (52) X11056 49 -Bessheyko (92) X41250 5 yesTutubalin (77) X40559 3 -Peterson (60) X11108 35 yesHildebrand (50) X11039 29 -De Juren (50) X11176 9 -Dzhelepov (55) X41261 11 -


5.4.1.2 Full covariance analysis

In order to construct a full W covariance matrix, EXFOR data for the total cross sections of natural tungstenwere entered in a GANDR calculation. The covariance matrix prior was the one resulting from the previousisotopic (182W, 183W, 184W, and 186W) GANDR calculations [7]. Elemental tungsten PENDF cross sectionswere treated as redundant information and were generated from the previous PENDF isotopic evaluationsusing the MIXER module of the program package PREPRO [19]. Using EXFOR data from Table 5.1 ina GANDR calculation, new cross section adjustment factors for all isotopes and reactions and a full cross-material covariance matrix have been obtained.

Cross section adjustment. Initial cross sections from IAEA evaluation labelled ’ib21g’ have been adjustedpreviously [7] by applying experimental data for different reactions on 182W, 183W, 184W, and 186W in GANDRsystem [6]. These adjusted cross sections, along with the corresponding covariance data, have been taken as aprior. By including experimental data for total cross section of natural tungsten, new cross section adjustmentshave been calculated. The new evaluation is labelled ’ib21i’. Due to the prior in-material correlations, inclu-sion of the data for natural W very differently affects cross sections of different isotopes, reactions, and incidentneutron energies. For example, the adjustments are very small (within ±0.5%) below 1 MeV (Fig. 5.1). Thecapture cross section above 1 MeV slightly increases for 182W, decreases significantly for 183W, changes inboth directions for 184W, and remains almost unchanged (relative differences within 1%) for 186W (Fig. 5.1).Significant changes in the cross section adjustment factors occur at energies where the cross section is smalland the inherent uncertainty is very large. The combined effect on the elemental capture cross section is almostnegligible in the cross section as well as in the associated uncertainty (Fig. 5.3). The effect of this adjustmenton the elemental total cross section (Fig. 5.2) is also very small. Non-negligible differences occur only in theextremely high energy region well above 10 MeV, where the new ’ib21i’ almost exactly follows the includedexperimental data of Abfalterer (2001), while the ’ib21g’ does not include these data.

Cross section uncertainties and full covariance matrix. It must be emphasized that the preparation ofthe covariance data is focused on the energy region above the resonance range, since the resonance analysisis performed separately. It is based on capture and transmission time-of-flight measurements, which do notgive the cross sections directly, and are not suitable for inclusion in the present analysis. Therefore, thecross-material covariances are given mainly above the resonance range. At the moment it is not possible tostore cross-material covariances of resonance parameters in an ENDF formatted file. The covariance dataof individual isotopes did not change significantly from the original version [7]. The strongest correlationswere observed in the elastic channel between different isotopes, which is reasonable, since at low energies thischannel represents the dominant contribution to the total cross section. The correlation factors just above 100keV range from about -0.2 to -0.4, while the elastic-to-capture and capture-to-capture correlation factors aregenerally below 0.1 and differ in sign. An example of the covariance matrix between the elastic cross sectionsof 182W and 184W are shown in Fig. 5.4.


100000 1000000 1E70.80

0.85

0.90

0.95

1.00

1.05

A adj

E [eV]

182W 183W 184W 186W

Figure 5.1: Relative (n,γ) cross section adjustment factors after inclusion of the EXFOR data for the elementaltotal cross section [8], relative to the isotopic evaluation [7], as a function of incident neutron energy fordifferent tungsten isotopes.


Figure 5.2: Comparison of the elemental tungsten total cross section between the original ’ib21g’, the new’ib21i’, the ENDF/B-VII evaluations (’e70’) and experimental data, which are taken into account in the new’ib21i’.

Figure 5.3: Comparison of the elemental tungsten capture cross section between the original ’ib21g’, the new’ib21i’, the ENDF/B-VII evaluations (’e70’) and experimental data. Note: elemental capture cross sectionexperimental data were not taken into account in any of the presented evaluations.


∆σ/σ vs. E

for 182W(n,el.)

10-2

10-1

100

101

102

103

104

105

106

107

10-1

100

101

102

∆σ/σ vs. E for 184W(n,el.)

10-2 10-1 100 101 102 103 104 105 106 10710-1

100

101Ordinate scale is %

relative standard deviation.

Abscissa scales are energy (eV).

Correlation Matrix

0.00.20.40.60.81.0

0.0-0.2-0.4-0.6-0.8-1.0

Figure 5.4: Cross-material covariance matrix for the elastic cross sections of 182W and 184W. Covariances atlow energies would come from resonance analysis and are not available.


5.4.2 Integral experiment

Measurements of integral parameters are important to take into account when evaluating nuclear data librariessince they are complementary to the direct, differential, cross section measurements. In general, an integralparameter adjusts cross sections over the entire energy range (spectrum) for all reactions of all materialspresent in the experiment, whereby essentially introducing (cross-energy, cross-reaction, cross-isotope andcross-material) correlations between all these differential data. In [8], however, the integral experiment wasnot used to adjust the cross sections and covariance matrices, but merely to asses the difference induced by thecross-isotope correlations in tungsten relative to the cross-reaction correlations, only.

The FNG-W benchmark experiment with a D-T source in front of a large tungsten block was taken as anexample. Activities of different monitor samples were measured at different penetration depths. The impact ofthe reaction cross section covariance data on the predicted activities of monitor foils inside the block and theassociated uncertainties were studied. Table 5.2 shows the relative uncertainties in the predicted activities ofthe gold monitor sample at two positions, 15 cm (Pos. 2) and 35 cm (Pos. 4) deep in the tungsten block. Inboth cases the predicted activities were just outside the lower limit of the uncertainty interval in the measureddata [7]. Small adjustments to the cross sections in the present evaluation affected the results less than thestatistical uncertainty in the calculations.

Table 5.2: Uncertainties in the predicted activity of gold monitor at 15 cm (Pos. 2) and 35 cm (Pos. 4) in thetungsten block of the FNG-W benchmark experiment.

Uncertainty [%] Uncertainty [%]Pos. 2 Pos. 4

182W 0.8 1.4183W 0.9 1.6184W 0.9 2.2186W 0.7 1.1

Total – ’ib21g’ 1.68 3.26Total – ’ib21i’ 1.54 3.15

Total – full correlations 3.34 6.29

The results in Table 5.2 show the relative contributions of different isotopes to the total uncertainty usingthe original covariances that do not include any cross-material covariances. The last line in Table 5.2 shows theuncertainty for the same case if, hypothetically, all cross sections between different isotopes of tungsten werefully correlated. Note the increase in the total uncertainty of nearly a factor of two. In the new evaluation theoverall uncertainty is slightly lower due to the application of the elemental total cross section data. A separatecalculation has shown that the influence of the correlations in this particular case is practically negligible.This is understandable, knowing that only the correlations between the elastic and capture cross sections wereprocessed and that the sensitivities to these cross sections are low. Additional work, which is in progress, willinclude cross-material covariances for more reactions and elemental data of at least the capture and the (n,2n)reaction will be included in the analysis. Elemental capture data will inevitably introduce stronger correlationsbetween the isotopic capture data. Introduction of the (n,2n) data is also likely to have an effect due to the highsensitivity of the results to the cross sections of this reaction channel. Therefore, one should be careful aboutjumping to conclusions based on this study alone.

Chapter 6

Conclusions

In this analysis, the use of covariance matrices in reactor calculations has been widely discussed. Differentmethods for random sampling have been described, compared, and demonstrated for resonance parameters.The methods are general and their applicability is not restricted to nuclear data. With combinations of sam-pling methods and deterministic resonance integral calculations, different covariance representations have beencompared. Furthermore, self-shielding factors, which are important for neutron activation analysis, have beencalculated using deterministic and probabilistic methods for a number of realistic samples of monitor mate-rials. They were experimentally validated for rhodium foils of different dimensions. In contrast to the useof covariance matrices of nuclear data for estimating uncertainties of reactor parameters, measurements ofthese quantities can be used to inversely adjust the nuclear data and corresponding covariances. This methodhas been used for inclusion of isotopic and elemental experimental data for the evaluation of tungsten crosssections.

Sensitivity coefficients of integral observables on input parameters (e.g. cross sections) are usually ob-tained from deterministic calculations of the adjoint transport equation. Combined with the covariance ma-trices of the input parameters, the propagation of uncertainties due to parameters on integral observables canbe calculated. Alternatively, by random sampling of the correlated parameters each sample gives a set ofperturbed parameters. Using each set in a direct calculation, the uncertainty in the integral observable can bedetermined from the statistical analysis of the calculations with the perturbed parameters.

Three Monte Carlo based sampling methods have been proposed in this work: the diagonalization method,the Metropolis algorithm, and the correlated sampling method. In favourable conditions, i.e. small uncertain-ties and weak correlations, all sampling methods produce equivalent results. Differences occur when dealingwith large uncertainties or strong correlations between a large number of parameters.

The main advantages of the diagonalization method are simplicity, numerical efficiency, and completelynon-correlated samples. For parameters with a well-conditioned covariance matrix and small relative uncer-tainties (below ∼ 30%), this is the method to choose. If the uncertainties are large, the method experiencesproblems with negative parameter values, which can be effectively reduced by using different probability dis-tributions or cutoffs, but cannot be completely avoided.

The main advantage of the Metropolis algorithm with random sampling in the original space is the factthat it completely avoids matrix diagonalization which may be numerically unstable. Because of the correla-tions between successive samples (which is inherent to the Metropolis scheme), the method is well suited forparameters with sparse covariance matrices, e.g. for resonance parameters if correlations between differentparameters of a single resonance are given, but there are no correlations between different resonances. Withsimple implementation of the log-normal distribution, negative values of inherently positive parameters canbe completely avoided, at the expense of significant biases in average values and standard deviations of thesampled parameters. This method is favourable when large uncertainties but weak correlations between theparameters are present.

The advantage of the Metropolis algorithm with random sampling in the diagonal space compared to thesampling in the original space is the reduction of the correlation distance, which is substantial (several orders ofmagnitude) when correlations are strong. It requires initial diagonalization of the covariance matrix; however,

96

97

the eventual numerical instability is not as critical as in the diagonalization method, since the Metropolis algo-rithm inherently reduces instabilities with the step rejection/acceptance principle. The method of Metropolisalgorithm with random sampling in the diagonal space is the most sophisticated of all methods described inthis work, and is the only acceptable method in cases of a badly-conditioned covariance matrix with strongcorrelations.

The main advantage of the correlated sampling method is that it enables consistent sampling of inherentlypositive parameters with large relative uncertainties using log-normal distribution without producing any bi-ases. Since the correlated sampling method is numerically analogous to the diagonalization method, it alsoefficiently produces completely non-correlated samples. Though in practice it might be numerically unstableif the covariance matrix is badly conditioned, in contrast to all other random sampling methods the correlatedsampling method in principle enables sampling of any combination of normally distributed and log-normallydistributed correlated parameters with arbitrary precision and accuracy.

The random sampling methods were applied to the resonance parameters. However, before elaborating onthe parameters, the propagation of the uncertainties in resonance parameters on self-shielded cross sectionswas investigated.

Different methods of calculating resonance self-shielding factors of the resonance integrals were intercom-pared on a set of realistic cases of monitor sample materials. The following conclusions can be drawn:

• The simple empirical approach of Chilian et. al. to calculate epithermal self-shielding factors producesreasonable results in many cases. The disadvantage of the method is that it has little physical justifi-cation. The limitations of the method are difficult to assess, so it should be used with caution, unlessverified for specific materials and on specific irradiation facilities.

• The simple method of the MATSSF code is based on the resonance theory and the equivalence prin-ciple, which are well-established in reactor physics. Application of the method by interpolation ofpre-computed self-shielding factors is quick and simple. It was verified by comparison with MonteCarlo calculations involving a single resonance absorber in the sample.

• None of the above-mentioned methods addresses interference between constituent nuclides. A more so-phisticated method was tested, whereby the spectral effects were considered using average cross sectionson an energy-grid in the 640-group SAND-II structure. The method is useful for thicker samples wherestrong absorbers produce a significant distortion of the local flux, but not in cases where only specific(narrow) resonances from different nuclides overlap. When several resonance absorbers are present, theproposed approximation works well for wires and foils with their axis along the irradiation channels andfor isotropic sources. The scattering effects in wires lying flat in a channel (perpendicular to the channelaxis) change the effective flight path through the sample, thus effectively increasing the neutron flux andreducing the self-shielding. The effect is compensated by suitably adjusting the Bell factor. The methodis now included in the MATSSF code.

• Interference due to overlapping resonances from pairs of nuclides was studied by solving the integralslowing-down equation for a constituent nuclide at appropriate dilution and applying the calculatedspectrum to generate epithermal self-shielding factors for selected nuclides. In this way the interferencesbetween different nuclides were treated explicitly to reveal the nature of the phenomena.

• The results were compared against direct Monte Carlo calculations for a full model of the sample, in-cluding geometry and material composition. The analysis helped to understand qualitatively and quan-titatively the nature of the resonance self-shielding in activation monitor material samples.

All in all, the simple method of MATSSF (and possibly the method of Chilian et. al.) can be used to makefirst-order corrections for epithermal self-shielding. The more advanced method of MATSSF that uses the640-group cross section library greatly increases the accuracy of the resonance self-shielding factors whenresonance interference is present, without significantly increasing the computational effort. If still higheraccuracy is needed, it is best to calculate the self-shielding factors directly by the Monte Carlo technique,

98 CHAPTER 6. CONCLUSIONS

but special care is required about the statistics, since the self-shielding factors are the ratios of reaction rates,which are numerically of comparable magnitude.

The self-shielding factor calculations have been validated by a neutron activation experiment performed inthe TRIGA reactor at the Jozef Stefan Institute. Different rhodium sample foils that span from about 10% to60% correction due to the resonance self-shielding, have been irradiated in two irradiation channels with dif-ferent spectral characteristics. With accurately calculated self-shielding factors, reasonably good consistencyof results is obtained, validating both MCNP and MATSSF self-shielding factor calculations.

The impact of self-shielding on the resonance integral uncertainty due to uncertainties and correlations ofthe resonance parameters has been estimated using narrow resonance approximation. If all resonance param-eters or at least a significant part of them are correlated, the resonance integral uncertainty greatly depends onthe dilution cross section. Consequently, the relative uncertainty in the self-shielded cross sections of large anddense material samples can be much larger than that of smaller samples. The effect of converting the covari-ances of the resonance parameters into the cross section covariance matrix was investigated. The importantconclusion is that in general it is not appropriate to replace resonance parameter covariances with cross sectioncovariances . The two are not equivalent even when the energy mesh for the cross section covariance matrix isrelatively dense. It is shown that commonly used reactor dosimetry SAND-II representation of 640 groups isnot adequate to derive the uncertainty of the resonance integral for strongly-shielded samples.

For covariance analysis in the fast energy range, where the self-shielding effect is negligible, a different ap-proach is used. The principle of nuclear data adjustment by taking into account new measurements of arbitrary(differential or integral) observables has been described focusing on the linear least squares fitting method.The methodology was illustrated on the example of tungsten cross section, where the evaluation of isotopicreaction cross section covariance data was extended to include experimentally measured data on elementalsamples. The GANDR code and the data formatting utilities were used to produce consistent evaluated datafiles in ENDF-6 format. At present, the total cross section data of elemental tungsten were included in theanalysis and the cross-material correlations were limited to the elastic and capture reactions. Since the crosssections in the original evaluations were determined well by the relatively abundant experimental data for theisotopes, the inclusion of the additional experimental data had little effect on the overall uncertainties. Cross-material correlations appeared mainly between the cross sections for the elastic channels of the major isotopes.The predicted activities of the monitor samples in the tungsten block of the FNG-W experiment remained prac-tically unchanged when the new data were used for the calculations. The uncertainties were slightly reduced,mainly due to the smaller uncertainties in the isotopic cross sections and not from the cross-material correla-tions. One should not jump to conclusions about the importance of cross-material covariances based on thepresent analysis alone. The present benchmark is known to have a high sensitivity to the (n,2n) reaction andcapture. The main value of the present analysis is the demonstration of data consistency. Work is in progressto include cross-material covariances of the (n,2n) reaction and to include experimentally measured captureand (n,2n) cross section data for elemental tungsten samples.

The use of covariance matrices for estimating uncertainties is still a rarity in reactor calculations. Becauseof the rapid increase in computer efficiency, more such analyses can be expected in the future. This workincludes a thorough description of the general covariance treatment from input data to integral parameters,with the emphasis on the random sampling methods, one of which is completely new, and of the generalmethod of cross section and corresponding covariance matrix adjustment by inclusion of experimental data.The use of these methods is illustrated by individual, original examples. The most important practical resultof the analysis is that in general it is not appropriate to replace resonance parameter covariances with the crosssection covariances. The two may be equivalent at infinite dilution, but not for high levels of self-shieldingwhere the effects are most important, unless the energy mesh for the cross section covariance matrix is denseenough to take the resonance structure into account explicitly, but then the objective of a more compact datarepresentation is lost.

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[32] F.H. Frohner, ”Evaluation and Analysis of Nuclear Resonance Data,” JEFF Report 18, Nuclear EnergyAgency (NEA), Organisation for Economic Co-operation and Development (OECD).

[33] P. Reuss, ”Neutron Physics,” EDP Sciences, 2008.

[34] J.J. Duderstadt, L.J. Hamilton, ”Nuclear Reactor Analysis,” John Wiley & Sons, Inc., 1976.

[35] D.G. Cacuci, Ed., ”Handbook of nuclear engineering,” Springer, 2010.

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[36] I.I. Bondarenko, Ed., ”Group Constants for Nuclear Reactor Calculations,” Consultants Bureau, NewYork, USA, 1964.

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[38] P. Dossantos-Uzarralde, H.P. Jacquet, G. Dejonghe, I. Kodeli, ”Methodology Investigations on Uncer-tainties Propagation in Nuclear Data Evaluation,” International Conference Nuclear Energy for New Eu-rope 2010, Portoroz, Slovenia, September 6-9. Proceedings. Ljubljana: Nuclear Society of Slovenia,2010.

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[41] D.L. Smith, D.G. Naberejnev, L.A. Van Wormerb, ”Large errors and severe conditions,” Nucl. Instr. Meth.A, 488 (2002) 342-361.

[42] D.L. Smith, D.G. Naberejnev, ”Confidence intervals for the lognormal probability distribution,” Nucl.Instr. Meth. A, 518 (2004) 754-763.

[43] E.T. Jaynes, ”Prior Probabilities,” IEEE T. Syst. Sci. Cyb., 4 (1968) 227-241.

[44] R. Capote, D.L. Smith, ”An Investigation of the Performance of the Unified Monte Carlo Method ofNeutron Cross Section Data Evaluation,” Nucl. Data Sheets (N.Y. N.Y.), 109 (2008) 2768-2773.

[45] S. Chib, E. Greenberg, ”Understanding the Metropolis-Hastings Algorithm,” Am. Stat., 49 (1995) 327-335.

[46] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, ”Equations of state calculations byfast computing machine,” J. Chem. Phys., 21 (1953) 1087-1091.

[47] W.K. Hastings, ”Monte Carlo Sampling Methods Using Markov Chains and Their Applications,”Biometrika, 57 (1970) 97-109.

[48] L. Tierney, ”Markov Chains for Exploring Posterior Distributions,” Ann. Stat., 22 (1994) 1701-1762.

[49] G. Miller, D. Melo, H. Martz, L. Bertelli, ”An Empirical Multivariate Log-normal Distribution Repre-senting Uncertainty of Biokinetic Parameters for 137Cs,” Radiat. Prot. Dosim., 131 (2008) 198-211.

[50] D. Kancev, G. Zerovnik, M. Cepin, ”Uncertainty Analysis: Analytical Unavailability Modelling Incor-porating Ageing of Safety Components,” to be published.

[51] R. Stanek, E. Rasia, A.E. Evrard, F. Pearce, L. Gazzola, ”Massive Halos in Millenium Gas Simulations:Multivariate Scaling Relations,” Astrophys. J., 715 (2010) 1508-1523.

[52] S. Kim, D.K. Dey, ”Modeling multilevel survival data using frailty models,” Commun. Stat. TheoryMeth., 37 (2008) 1734-1741.

[53] B. Sahiner, H.-P. Chan, L.M. Hadjilski, ”Performance analysis of three-class classifiers: Properties of a3-D ROC surface and the normalized volume under the surface for the ideal observer,” IEEE Trans. Med.Imaging, 27 (2008) 215-227.

[54] S.J. Fletcher, M. Zupanski, ”A hybrid multivariate Normal and lognormal distribution for data assimila-tion,” Atmos. Sci. Let., 7 (2006) 43-46.

[55] G. Zerovnik, A. Trkov, D.L. Smith, ”Transformation of correlation coefficients between normal and log-normal distribution,” technical note, in preparation.

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[59] L. Snoj, G. Zerovnik, M. Ravnik, I. Lengar, R. Sanchez, ”1 × 1 array of highly enriched uranium, mod-erated and reflected by polyethylene,” HEU-MET-THERM-032, International handbook of evaluatedcriticality safety Benchmark experiments, NEA/NSC/DOC(95)03, September 2010.

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[72] M. Martone, M. Angelone, M. Pillon, ”The 14 MeV Frascati Neutron Generator,” J. Nucl. Mater., 212-215 (1994) 1661-1664;

0

Uporaba kovariancnih matrik za oceno negotovosti v reaktorskihpreracunih

(Razsirjen povzetek v slovenskem jeziku)

1 Uvod

Negotovosti in napake v reaktorskih (enako kot katerih koli drugih fizikalnih) parametrih lahko razdelimo vdve glavni skupini: statisticne in sistematske. V principu lahko z zadostnim ponavljanjem eksperimenta alisimulacije statisticno negotovost poljubno zmanjsamo, saj so statisticna odstopanja od povprecja po definicijimed seboj nekorelirana. Po drugi strani pa ponavljanje meritev ne zmanjsa sistematske napake. Za dosegotega je potrebno spremeniti metodo in/ali postopek izvajanja meritve. Za pravilen opis sistematskih napakpotrebujemo vektor pricakovanih vrednosti parametrov s kovariancno matriko, ki vsebuje vse informacije onegotovostih in korelacijah [3]. (Diferencialni) kovariancni podatki so osnova za oceno negotovosti (integral-nih) parametrov in jih lahko konsistentno prilagajamo z dodajanjem novih eksperimentalnih podatkov [4].

V reaktorskih preracunih negotovosti glede na njihov izvor razdelimo v dve veliki skupini: negotovostiv jedrskih podatkih in negotovosti zaradi (fizikalnih, geometrijskih in numericnih) priblizkov v racunskihmodelih. Z uporabo metode Monte Carlo [5], ki v principu omogoca uporabo eksaktne geometrije in (skoraj)eksaktne fizike, od slednje preostane predvsem statisticna komponenta, ki pa z razvojem strojne opreme postajavse manj motec dejavnik.

Zato so v danasnjem casu glavni vir negotovosti v (integralnih) reaktoskih parametrih negotovosti innetocnosti v jedrskih podatkih. Z vpeljavo kovariancnih matrik v reaktorske preracune so priceli ze v poznih70-tih letih prejsnjega stoletja, vendar so bili prvi poskusi relativno neuspesni zaradi pomanjkanja racunskemoci in potem tudi zaradi zmanjsanega interesa po nesreci na Otoku treh milj. Racunalniske zahteve za po-polno kovariancno obravnavo jedrskih podatkov so ogromne, saj samo preseki za reakcije z nevtroni (ki resdapredstavljajo znaten del jedrskih podatkov, relevantnih za reaktorske preracune) obsegajo:

• vec kot 100 izotopov, ki so pomembni za reaktorske in fuzijske aplikacije,

• tipicno je relevantnih nekaj deset reakcij za posamezen izotop,

• veliko pomembnih izotopov zahteva definicijo presekov za nekatere reakcije v vec kot 105 energijskihtockah ali v obliki primerljivega stevila resonancnih parametrov.

Zaradi toliksnega obsega podatkov moramo tudi v modernem casu kovariancno obravnavo omejiti glede napotrebe problema, ki ga raziskujemo. Poleg tega so informacije o eksperimentalnih negotovostih, kar seposledicno odraza tudi v evaluacijah jedrskih podatkov, nepopolne in nekonsistentne. Programski sistemGANDR (Global Assessment of Nuclear Data Requirements) [6] je prvi poskus popolnoma splosne kova-riancne obravnave reakcijskih presekov. Omogoca sklopljeno obravnavo vec kot 100 izotopov, ki jih opisujedo 25 poljubno definiranih reakcij.

V sodelovanju s tujimi raziskovalnimi ustanovami so sodelavci Odseka za reaktorsko fiziko Instituta JozefStefan izracunali korelacije med razlicnimi energijami in reakcijami izbranega izotopa z upostevanjem ekspe-rimentalnih presekov za ta izotop [7]. Zacetni preseki so bili izracunani iz globalnih jedrskih modelov, zacetnekovariancne matrike pa iz negotovosti parametrov modela s pomocjo metode Monte Carlo. Z upostevanjem ek-sperimentalnih podatkov za razlicne reakcijske preseke za posamezne izotope v GANDR so izracunali nove,popravljene preseke in kovariancne matrike. Bistvo napredka glede na starejse evaluacije je v uporabi na-tancnejsih jedrskih modelov, ki vsebujejo nabor parametrov, ki so skupni vecim elementom. Starejse evaluacijetemeljijo predvsem na delnih jedrskih modelih, ki imajo probleme predvsem s konsistenco med razlicnimi ele-menti/izotopi. Zgoraj opisani preseki sluzijo kot osnova za nadaljnje delo. Razsirjena obravnava kovariancnih

II

podatkov z uporabo eksperimentalnih podatkov za elementarni volfram je bila testirana na evaluacijah prese-kov za izotope volframa [8].

Po izracunu integralnih parametrov je v principu mozen se en korak naprej: s primerjavo teh izracunov zmeritvami lahko spreminjamo reakcijske preseke, tako da se izracuni integralnih parametrov cim bolje ujemajos sirokim spektrom integralnih eksperimentov. V ta namen potrebujemo izracune obcutljivosti integralnih para-metrov na preseke; ker je stevilo obcutljivostnih koeficientov tipicno relativno veliko, bi jih bilo zelo zamudnoracunati z metodo Monte Carlo. Po drugi strani pa deterministicni programi za seboj povlecejo sistematskenapake zaradi geometrijskih in/ali fizikalnih priblizkov. Vkljucitev integralnih eksperimentov v evaluacijepresekov je v principu posplositev vkljucitve meritev presekov: ce upostevamo meritve za posamezen izo-top, so ustrezni obcutljivostni koeficienti enaki 1, ce pa upostevamo eksperimentalne podatke za element, soobcutljivostni koeficienti enaki pojavnosti izotopa v naravni elementarni mesanici.

Pojav resonancnega samoscitenja je dobro poznan v reaktorski fiziki, in je zelo pomemben npr. v hete-rogenih reaktorskih sredicah ali v relativno velikih vzorcih, ki vsebujejo mocne absorberje, v nevtronskempolju [9]. Korekcijski faktor samoscitenja je v bistvu merilo za (notranji) vpliv na (zunanje) nevtronsko po-lje. Na osnovi negotovosti v reakcijskih presekih smo ocenili negotovost v izracunu faktorjev samoscitenja[9, 10, 11, 12]. Energijska mreza v sistemu GANDR je veliko preredka, da bi lahko pravilno upostevali re-sonancne pojave. Pogosto je med sosednjima tockama mreze vec resonanc, kjer obstajajo velike variacijev relativni negotovosti in korelacijah med resonancnimi vrhovi in ’dolinami’ med sosednjimi resonancami.Zaradi pojava samoscitenja je nevtronski fluks zelo obcutljiv na reakcijske preseke, zato bi lahko obravnavanegotovosti, povprecena po sirokih energijskih intervalih, vodila do zelo izkrivljenih rezultatov. Zato je boljpravilno izhajati iz osnovnih podatkov, to je resonancnih parametrov in pripadajocih kovariancnih matrik.Ocenili smo vpliv negotovosti in korelacij resonancnih parametrov na resonancni intergral in posledicno nafaktorje samoscitenja [16]. Obravnavali smo dve razlicni resonacno-kovariancni evaluaciji, prva bazira naenostavem pristopu iz objavljenih resonancnih parametrov z negotovostmi [17] (narejena je bila na NuclearResearch and Consultancy Group – NRG), druga pa na bolj rigorozni resonancni analizi s kodo SAMMY [18](narejena na Oak Ridge National Laboratory – ORNL). Resonancne integrale smo izracunali z deterministicnokodo PREPRO [19] v priblizku ozkih resonanc ([20], str. 423-433), negotovosti v resonancnih integralih paso bile ocenjene s pomocjo vzorcenja resonancnih parametrov z metodo Monte Carlo. Raziskali in med sebojprimerjali smo razlicne metode vzorcenja koreliranih parametrov [21, 22].

Disertacija je strukturirana sledece. V poglavju 2 so opisani jedrskih podatki s poudarkom na resonancnihparametrih, reakcijskih presekih in kovariancnih podatkih. Transportni izracuni in analiza negotovosti z opi-som tako deterministicnih kot probabilisticnih metod so glavna tema poglavja 3. Uporaba teh metod je demon-strirana v poglavju 4 na izracunih faktorjev samoscitenja za izbrane vzorce (monitorje) in analizi negotovostiresonancnega integrala za zajetje nevtrona v manganu. Metodologija korekcije jedrskih podatkov na osnovieksperimentalnih podatkov je razlozena in demonstrirana na primeru volframa v poglavju 5.

III

2 Jedrski podatki

Medtem ko je geometrija in materialna sestava specificna za vsak sistem, so na drugi strani jedrski podatkiuniverzalni. V realnosti so le redko poznani eksaktno; obicajno bazirajo na semi-empiricnih kvantnih ali semi-klasicnih jedrskih modelih, npr. na opticnem modelu [23], ki tipicno vsebujejo vec (delno) prostih parametrov,ki jih dolocimo tako, da se izracuni cim bolje ujemajo z eksperimentalnimi podatki.

V danasnjem casu je vecina evaluacij jedrskih podatkov shranjenih v ENDF-6 formatu [24], ki med dru-gim zajema vse podatke, ki so lahko pomembni za reaktorske preracune. Evaluacije za posamezne nuklideso razdeljene na t.i. ’File’. File 1 vsebuje splosne informacije o nuklidu, File 2 resonancne parametre (po-glavje 2.1), File 3 reakcijske preseke v odvisnosti od enegije vpadnega nevtrona (poglavje 2.2), File-i 30-35ter 40 pa pokrivajo razlicne tipe kovariancnih podatkov.

2.1 Resonancni parametri

Vecina jedrskih reakcij (razen elasticnega sipanja) pri relativno nizkih energijah (pod nekaj MeV) potece prekovmesnega jedra. Te reakcije imajo skupno lastnost – resonancno strukturo preseka v odvisnosti od energijevpadnega delca (nevtrona).

Breit-Wignerjeva teorija je osnovana na casovno odvisni kvantni teoriji perturbacij [29]. Ceprav je jasno,da mocne jedrske sile ne moremo natancno opisati s perturbacijami, ima Breit-Wignerjeva formula za izoliraneresonance v okolici vrhov pravilno obliko (odvisnost od energije). To je posledica dejstva, da na obliko vplivale pogoj, da reakcija poteka preko vmesnega stanja ([30], str. 259). V zadnjih letih bolj splosna teorija’R-matrix’ [30] oz. njen Reich-Moorov priblizek [31] vedno bolj izpodriva enostavnejsi Breit-Wignerjevpristop. V teoriji ’R-matrix’ jedrske reakcije opisujemo z matriko trkov U, iz katere lahko potem izracunamopreseke, in matriko reakcijskih kanalov R, ki opisuje valovne funkcije zacetnih in koncnih delcev in vsebujevse informacije o resonancnih parametrih ([32], str. 51):

Rcc′ =∑λ

γλcγλc′

Eλ − E, (2.1)

kjer c in c′ oznacujeta pare zacetnih in koncnih delcev (v reaktorski fiziki je c obicajno par nevtron-jedro), Eλ soresonancne energije, γλc pa razpadne amplitude, ki so v tesni zvezi z resonancnimi sirinami: Γλc = γ

2λc. Teorija

’R-matrix’ v splosnem sklopljeno obravnava vse resonance in reakcijske kanale. Reich-Moorov formalizem, kise trenutno v praksi najpogosteje uporablja, je omejen na en vhodni (nevtron) in stiri izhodne kanale (nevtron,gama in dva fisijska kanala).

2.2 Reakcijski preseki

Iz prakticnih razlogov interakcije nevtron-jedro obicajno zapisemo v obliki presekov v odvisnosti od ener-gije vpadnega nevtrona na mirujoco tarco. Mikroskopski presek Σ(mi)(E) je lastnost jedra in je sorazmerenverjetnosti za potek reakcije.

V realnosti tarca (jedro) ne miruje, temvec se giblje z neko termicno hitrostjo, zato je dejanska merljivakolicina efektivni presek Σ(mi)(E,T ), ki ga dobimo s povprecenjem preseka pri nicelni temperaturi po vseh

IV

moznih hitrostih tarce:

Σ(mi)(E,T ) =∫

f (v′, T )Σ(mi)(m(v − v′) · (v − v′)/2, 0)d3v′. (2.2)

V nadaljevanju bomo Σ(mi)(E, T ) poenostavljeno pisali kot Σ(mi)(E), temperaturni efekti (pri sobni temperaturi)pa bodo ze implicitno upostevani.

2.2.1 Obmocje termicnih energij

Preseki za vse nepragovne reakcije razen elasticnega sipanja imajo pri nizkih energijah (pod najnizjo reso-nanco) odvisnost 1/v. Ker so resonance pod 1 eV izredno redke (npr. 113Cd ali 135Xe, ki sta izredno mocnatermicna absorberja), je odvisnost preseka za absorpcijo pri termicnih energijah obicajno zelo blizu 1/v. Ela-sticno sipanje je sestavljeno iz potencialnega sipanja (analog trku trdih kroglic), za katerega je verjetnost neod-visna od energije, in sipanja preko vmesnega jedra, ki ima odvisnost 1/v. Vsi preseki pri termicnih energijahso neodvisni od temperature.

Pri termicnih energijah je mozno koherentno sipanje nevtronov na kristalni mrezi, kar povzroci mocno ani-zotropijo elasticnega sipanja. V disertaciji kotne odvisnosti diferencialnih sipalnih presekov ne bomo obrav-navali.

2.2.2 Obmocje locljivih resonanc

Spodnja meja obmocja nelocljivih resonanc je definirana z resonanco pri najnizji energiji. Kot pove ze ime,so v tem obmocju resonance jasno merljive in razlocljive, preseke pa se lahko izraza direktno v odvisnosti odenergije ali pa kot vsoto prispevkov vseh resonanc in ozadja. Zveza med resonancnimi parametri in preseki jev splosnem zelo zapletena.

Pomembna lastnost presekov v resonancnem obmocju je Dopplerjeva razsiritev resonanc zaradi termicnegagibanja jeder v tarci. Ceprav se integral preseka po energiji ohranja, pa povecanje temperature preko samoscitenja(poglavje 4) vpliva na nevtronski spekter, kar prinese porast reakcijske hitrosti.

2.2.3 Obmocje nelocljivih resonanc

V splosnem gostota resonanc z energijo vpadnega nevtrona narasca relativno glede na njihovo sirino. Obmocjenelocljivih resonanc se zacne neposredno nad obmocjem locljivih resonanc – je energijsko obmocje, kjer sososednje resonance tako blizu skupaj, da jih z razpolozljivimi napravami ne moremo nedvoumno lociti, hkratipa dovolj narazen, da je presek zapletena funkcija energije z veliko oscilacijami. Te oscilacije zaradi resonancobravnavamo statisticno.

2.2.4 Obmocje visokih energij

Pri se visjih energijah so resonance tako blizu skupaj, da se med seboj pokrivajo in energijska odvisnost presekase izpovpreci. Pri energijah reda velikosti MeV se zacnejo pojavljati t.i. pragovne reakcije. Temperaturni efektiso v tem energijskem obmocju zanemarljivi.

2.3 Ostali pomembni jedrski podatki

Poleg reakcijskih presekov so za nevtronske transportne preracune pomembni tudi: povprecno stevilo nevtro-nov na fisijo ν(E), fisijski spekter χ(E), diferencialni kotni/energijski, dvojni diferencialni preseki, itd.

V

2.4 Kovariancni podatki

Negotovosti v jedrskih podatkih ponavadi izrazamo v obliki kovariancnih matrik. Kovariancna matrika Vpripada vektorju parametrov x, ce velja

Vi j = cov(xi, x j) =⟨(xi − x0i)(x j − x0 j)

⟩= σiσ jCi j, (2.3)

kjer sta x0i in σi pricakovana vrednost in standardna deviacija parametra xi, Ci j pa je korelacijski koeficientmed parametroma xi in x j.

V disertaciji bomo uporabljali predvsem kovariancne podatke o resonancnih parametrih (File 32 v ENDF-6 formatu) in presekih (File 33). Na primeru bomo pokazali (poglavje 4.2), da oba nacina zapisa nista nujnoekvivalentna. Ta pojav je spodaj ilustriran na poenostavljenem primeru.

2.4.1 Analiticen primer

V Breit-Wignerjevem formalizmu lahko presek za zajetje v okolici izolirane resonance izrazimo kot

Σ(mi)γ (E) = K

ΓnΓγ

(E − E0)2 + Γ2/4, (2.4)

kjer ima K = πo2g sibko energijsko odvisnost (1/E), ki je zanemarljiva v okolici vrha resonance E0.Ce privzamemo, da sta ’slucajno’ nevtronska Γn in gama sirini Γγ obe enaki polovici sirine resonance Γ pri

100% korelaciji, za absolutno in relativno negotovost preseka dobimo:

∆Σ(mi)γ (E) =

∆Γ

Γ0

4(E − E0)2 + 2Γ20

2(E − E0)2 + Γ20/2

(Σ(mi)γ )0(E) =

∆Γ

Γ0K fA(E) (2.5)

in∆Σ

(mi)γ (E)

(Σ(mi)γ )0(E)

=∆Γ

Γ0

4(E − E0)2 + 2Γ20

2(E − E0)2 + Γ20/2=∆Γ

Γ0fR(E), (2.6)

kjer fA(E) in fR(E) ponazarjata odvisnost negotovosti od energije (Slika 2.1). Ce bi hoteli tabelirati negotovostpreseka v odvisnosti od energije, bi v tem primeru za eno resonanco potrebovali vec tock. Po drugi strani paje v realnosti pogosto resonancna sirina znana precej bolj natancno kot sirine posameznih reakcijskih kanalov,kar ustreza (ce zanemarimo negotovost Γ v enacbi (2.4)) konstantni relativni negotovosti preseka, torej bibila velikost kovariancne matrike presekov lahko enaka ali celo manjsa od kovariancne matrike resonancnihparametrov. Protiprimer zadnji trditvi prestavlja evaluacija ORNL za 55Mn (poglavje 4.2).

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

f

(E-E0)/ 0

fR fA

Slika 2.1: Oblika relativne in absolutne negotovosti v preseku zaradi negotovosti v resonancnih sirinah.

VI

3 Reaktorski preracuni in propagacijanegotovosti

Ocene negotovosti (integralnih) parametrov izvajamo v dveh korakih. Integralne parametre izracunamo iz vho-dnih podatkov z deterministicnimi (poglavje 3.1.1) ali probabilisticnimi (poglavje 3.2.1) metodami. Za analizonegotovosti lahko spet uporabimo deterministicne (obcutljivostna analiza, poglavje 3.1.2) ali probabilisticne(nakljucno vzorcenje vhodnih podatkov, poglavje 3.2.2) metode.

Za natancen opis delovanja reaktorske sredice zadosca obravnava transporta nevtronov. Pri tem lahko za-nemarimo interakcije med pari nevtronov, razpad nevtronov, relativisticne efekte in valovno naravo nevtronov(razen pri sipanju termicnih nevtronov na kristalih). Za izracun transporta nevtornov poznamo dve glavni me-todi: deterministicno Boltzmannovo transportno enacbo in simulacijo gibanja posameznih nevtronov z metodoMonte Carlo.

Metode za nakljucno vzorcenje in propagacijo negotovosti so splosne in jih lahko uporabimo na skorajkaterih koli podatkih z negotovostmi, ceprav se v disertaciji omejimo na jedrske podatke. Podajanje podatkovv obliki momentov porazdelitve (obicajno pricakovanih vrednosti in kovariancnih matrik) je zelo primerno zadeterministicno obcutljivostno analizo. Pri nakljucnem vzorcenju potrebujemo celotno porazdelitev, ki jo v tehprimerih sestavimo iz poznanih momentov porazdelitve. Obicajno privzamemo normalno ali log-normalno (zapozitivne parametre) porazdelitev.

3.1 Deterministicen pristop

3.1.1 Transportna enacba in njeni priblizki

Boltzmannova enacba ([34], str. 103-117) skoraj eksaktno opisuje transport nevtornov v poljubnem sistemu.Nevtronsko polje ponavadi opisujemo s fluksom Φ, kjer

Φ(r, E, Ω, t)v

d3rdEdΩ (3.1)

predstavlja stevilo nevtronov v diferencialnem elementu okrog pozicije r, energije E in smeri Ω v nekemtrenutku t. Merljiva fizikalna kolicina je gostota reakcijske hitrosti

Rr (r, t) =∫ ∞

0

∫4πΣr(E)Φ(r, E, Ω, t)dEdΩ, (3.2)

ki je enaka integralu produkta fluksa in makroskopskega preseka Σr za reakcijo r.Transportno enacbo pogosto zapisemo v obliki kontinuitetne enacbe za opazovalno tocko, ki se premika

skupaj z nevtroni izbrane energije in smeri:

1v∂Φ

∂t+ Ω · ∇Φ = (3.3)

= −Σt (r, E)Φ +∫ ∞

0

∫4πΣ

(d)s (r, E′→E, Ω′→Ω)Φ(r, E′, Ω′, t)dE′dΩ′ + s(r, E, Ω, t)

VII

z zacetnim Φ(r, E, Ω, 0) = Φ0(r, E, Ω) in robnim pogojem Φ(rS , E, Ω, t) = 0 za vse rS na mejni povrsini Sopazovanega sistema in Ω · n < 0, kjer je n vektor, pravokoten na povrsino S .

Zadnji clen na desni strani enacbe (3.3) predstavlja izvor:

s(r, E, Ω, t) =χ(E)4π

∫ ∞

0

∫4πν(E′)Σ f (r, E′)Φ(r, E′, Ω′, t)dE′dΩ′ + s0(r, E, Ω, t), (3.4)

kjer je χ(E) fisijski spekter, ν(E) povprecno stevilo nevtronov, ki nastanejo pri cepitvi jedra z nevtronom skineticno energijo E, in s0 zunanji izvor nevtronov.

Zaradi zapletenosti transportne enacbe so se v reaktorski fiziki uveljavili stevilni priblizki, kot so difuzijskaenacba, enacbe tockovne kinetike, PN in S N aproksimacije, grupne aproksimacije, itd.

3.1.1.1 Grupne aproksimacije

Z diskretizacijo energijske spremenljivke integro-diferencialno enacbo (3.3) prevedemo na sistem linearnihdiferencialnih enacb za grupne flukseΦg. Efektivne grupne preseke definiramo tako, da se ohranjajo reakcijskehitrosti:

ΣgΦg =

∫ Eg+1

Eg

ΣΦdE, Φg =

∫ Eg+1

Eg

ΦdE. (3.5)

Podobna diskretizacija je mozna tudi s kotno spremenljivko Ω.

3.1.1.2 Enacba upocasnjevanja nevtronov in princip ekvivalence

V neskoncnem homogenem nepomnozevalnem mediju brez sipanja k visjim energijam se enacba (3.3) poeno-stavi v:

0 = −Σt(E)ϕ(E) +∑

k

∫ E/αk

E

Σs,k(E′)(1 − αk)E′

ϕ(E′)dE′ + s0(E), (3.6)

kjer je ϕ(E) nevtronski spekter, αk = [(Ak − 1)/(Ak + 1)]2 pa maksimalna relativna izguba energije nevtona prielasticnem trku z jedrom k.

Za koncen medij vpeljemo zunanji spekter ϕ0(E) in razredcitveni presek Σ0 po principu ekvivalence geo-metrijskega in resonancnega samoscitenja:

Σ0 =a∗

L, (3.7)

kjer L oznacuje povprecno dolzino tetive vzorca in a∗ Bellov faktor, ki je rahlo odvisen od oblike vzorca inrazmerja med absorpcijskim in sipalnim presekom [9].

Enacba upocasnjevanja za medij koncnih razseznosti se zapise v obliki:

[Σ0 + Σt(E)] ϕ(E) = Σ0ϕ0(E) +∑

k

∫ E/αk

E

Σs,k(E′)(1 − αk)E′

ϕ(E′)dE′. (3.8)

Enacba (3.8) je se posebej uporabna za izracun spektrov, reakcijskih hitrosti in faktorjev samoscitenja zavzorce, obsevane v zunanjem nevtronskem polju ϕ0(E) [9]. Lahko jo resujemo v grupnem priblizku, ali pauporabimo kaksnega od analiticnih priblizkov, npr. priblizek ozkih resonanc.

3.1.1.3 Priblizek ozkih resonanc

V priblizku ozkih resonanc predpostavimo, da je sirina resonanc veliko manjsa od povprecne izgube energijenevtrona pri trkih. Ce sipalni presek aproksimiramo s konstantnim prispevkom potencialnega sipanja Σp, seenacba (3.8) v okolici resonance poenostavi v:[

Σ0 + Σp + Σa(E)]ϕ(E) =

[Σ0 + Σp

]ϕ0(E) ; ϕ(E) =

Σ0 + Σp

Σ0 + Σp + Σa(E)ϕ0(E). (3.9)

Priblizek ozkih resonanc je veljaven pri dovolj visokih energijah v lahkih moderatorjih, kot je 55Mn [16].

VIII

3.1.2 Obcutljivostna analiza

Deterministicna obcutljivostna analiza temelji na Taylorjevem razvoju (integralnih) parametrov kot funkcijevhodnih parametrov z negotovostmi.

3.1.2.1 Priblizek prvega reda

Najpogosteje se uporablja priblizek prvega reda:

cov( fk, fk′) =p∑

i,i′=1

∂ fk(x0)∂xi

cov(xi, xi′)∂ fk′(x0)∂xi′

, (3.10)

kjer so xi vhodni, fk pa integralni parametri. Z vpeljavo kovariancnih in obcutljivostne matrike

S ki =∂ fk(x0)∂xi

, (3.11)

lahko enacbo (3.10) zapisemo v skrceni obliki:

V f = S V xS T . (3.12)

Primer. Naj bo R =∑

g ϕgΣg gostota reakcijske hitrosti, izrazena v grupnem priblizku. Priblizek prvega redaza varianco R znasa:

σ2R

R2 =∑g,h

(Rrel

g

[(Vrelϕ

)gh+

(VrelΣ

)gh+ 2

(Vrelϕ,Σ

)gh

]Rrel

h

), (3.13)

torej relativna varianca gostote reakcijske hitrosti je vsota produktov relativnih grupnih gostot reakcijskihhitrosti in relativnih kovariancnih matrik.

3.1.2.2 Priblizki visjih redov

V dolocenih primerih, se posebej v primeru mocne nelinearnosti funkcij fk znotraj obmocja negotovosti xi, jeza dosego zelene natancnosti potrebno upostevati popravke visjih redov, katerih kompleksnost izredno hitronarasca. Na primer, ob predpostavki neodvisnih vhodnih parametrov in simetricnih porazdelitev priblizekdrugega reda zapisemo kot [38]:

V fkk′ =

p∑i=1

∂ fk(x0)∂xi

∂ fk′(x0)∂xi

V xii +

14

p∑i=1

∂2 fk(x0)∂x2

i

∂2 fk′(x0)∂x2

i

((µ4)i − (V x

ii)2)+

12

p∑i,i′,i

∂2 fk′(x0)∂xi∂xi′

∂2 fk(x0)∂xi∂xi′

V xiiV

xi′i′ ,

(3.14)kjer (µ4)i oznacuje 4. centralni moment porazdelitve parametra xi.

3.2 Metoda Monte Carlo

3.2.1 Monte Carlo (nevtronski) transportni preracuni

Ker so interakcije med pari nevtronov zanemarljive, lahko z metodo Monte Carlo simuliramo gibanje po-sameznih nevtronov v snovi. Eden najbolj razsirjenih Monte Carlo programov je MCNP [5] (Monte CarloN-Particle Transport Code), ki omogoca simulacijo gibanja posameznih nevtronov, fotonov in/ali elektronovna podlagi verjetnostnih porazdelitev za sipanje, absorpcijo in ostale reakcije v sistemu (skoraj) poljubne ge-ometrije in materialne sestave. Pri reaktorskih preracunih ponavadi zadosca, da simuliramo samo transportnevtronov, saj imajo ostali delci zanemarljiv vpliv na pomnozevanje nevtronov. S povprecenjem sledi velikegastevila nevtronov lahko dolocimo makroskopske kolicine, kot so pomnozevalni faktor ter porazdelitve fluksain reakcijskih hitrosti. Z metodo Monte Carlo so mozni najnatancnejsi izracuni, saj v principu omogoca po-ljubno natancno geometrijo ter zvezno energijsko in kotno obravnavo transporta nevtronov. Statisticna napakav splosnem pada s kvadratnim korenom iz stevila dogodkov.

IX

3.2.2 Metode nakljucnega vzorcenja

Probabilistice metode lahko uporabimo tudi za generiranje reprezentativnih vzorcev vhodnih parametrov. Izvsakega vzorca potem izracunamo (integralne) parametre, pricakovane vrednosti in negotovosti pa dolocimona podlagi enostavne statisticne analize.

Metode za nakljucno vzorcenje posameznih ali neodvisnih parametrov po poljubnih verjetnostnih poraz-delitvah p(x) so dobro znane in relativno enostavne. Vzorcenje koreliranih parametrov je bolj zapleteno; vdolocenih primerih lahko problem prevedemo na vzorcenje posameznih parametrov, kadar pa to ne daje zado-voljivih rezultatov, se je potrebno posluziti drugacnih prijemov.

Jedrski podatki so obicajno podani v obliki pricakovanih vrednosti in kovariancnih matrik. Pripadajoci po-razdelitvi sta normalna ali log-normalna (za pozitivne parametre) [43]. Pri spremenljivkah z majhno relativnonegotovostjo (. 0.3) lahko log-normalno porazdelitev z zadovoljivo natancnostjo nadomestimo z normalnoporazdelitvijo. V primeru vecjih negotovosti pa naletimo na probleme z negativnimi vrednostmi pozitivnihparametrov.

3.2.2.1 Diagonalizacija

Problem vzorcenja vecjega stevila med seboj koreliranih parametrov lahko prevedemo na vzorcenje neodvisnihparametrov z diagonalizacijo kovariancne matrike. Pri tem moramo biti zelo pazljivi, ker se pri diagonalizacijispremeni funkcijska odvisnost vseh porazdelitev razen normalne (ki ima edina lastnost, da ohranja obliko prilinearnih transformacijah).

Najvecjo tezavo predstavlja vzorcenje pozitivnih parametrov z velikimi relativnimi negotovostmi. Ce upo-rabimo normalno porazdelitev, naletimo na nefizikalne negativne vrednosti parametrov, ki jih lahko za siloumetno resimo, ce jih postavimo na 0. Po drugi strani lahko za neodvisne linearne kombinacije parametrov vdiagonalnem sistemu vzorcimo po log-normalni porazdelitvi, zaradi nelinearnosti log-normalne porazdelitvepa pri transformaciji nazaj v originalni sistem parametrov spet (sicer v manjsi meri) generiramo negativnevrednosti.

3.2.2.2 Metropolisov algoritem

Metropolisova metoda [46] omogoca generiranje nakljucnega zaporedja stanj po poljubni multivariatni po-razdelitvi P(x). Potrebno je izbrati zacetno stanje x(0) in definirati korak, ki mora zagotoviti ergodicnost inreverzibilnost. Ce je verjetnosna gostota v novi tocki koraka vecja kot v stari tocki, se korak izvede, v na-sprotnem primeru pa se izvede z neko verjetnostjo. Ker so zaporedna stanja tipicno mocno korelirana, je pristatisticni analizi rezultatov pogosto potrebno izpuscati veliko stevilo stanj.

Vzorcenje v originalnem sistemu parametrov. Za naso analizo smo izbrali preizkusen korak [44]:

x(n+1)′ = x(n) + c√

diag(V) · (2ξ − 1), (3.15)

kjer je diag(V) diagonalna matrika varianc x, ξ ∈ [0, 1) vektor neodvisnih nakljucnih enakomerno porazdelje-nih spremenljivk, in c konstanta.

Vzorcenje v diagonalnem sistemu. V primeru velikega stevila koreliranih parametrov je za dovolj velikoizvedljivost koraka potrebno mocno zmanjsati konstanto c, kar poveca korelacijsko dolzino vzorcev in zmanjsaucinkovitost metode. Problem lahko resimo, ce namesto v originalnem sistemu parametrov raje vzorcimo vsistemu linearnih kombinacij originalnih parametrov, ki so neodvisne.

3.2.2.3 Korelirano vzorcenje

Problema vzorcenja koreliranih log-normalno porazdeljenih parametrov z velikimi relativnimi negotovostmi zmetodo diagonalizacije ali Metropolisovim algoritmom ne moremo zadovoljivo resiti. Metropolis v principu

X

omogoca vzorcenje po multivariatni log-normalni porazdelitvi [54]

P(x) = det(2πV)−1/2

n∏i=1

x−1i

exp[−1

2(ln x − µ)T V−1(ln x − µ)

], (3.16)

kjer pa µ in V ne moremo enostavno izraziti s prvima momentoma porazdelitve. V prvi aproksimaciji seproblem lahko resi z vzorcenjem po normalni porazdelitvi v prostoru logaritmiranih parametrov, vendar pritem pride do znatnih odmikov momentov porazdelitve od pricakovanih vrednosti (poglavje 3.4).

Alternativo predstavlja metoda, ki temelji na hkratnem vzorcenju vseh koreliranih parametrov. V prin-cipu je metoda zelo enostavna; vzorce koreliranih parametrov izracunavamo iz enakega stevila neodvisnihnakljucnih stevil ξ(m) na podlagi vnaprej dolocene funkcije F:

x(m) = F(ξ(m)), (3.17)

kjer x(m) predstavlja m-ti vzorec vektorja parametrov. Funkcija F je seveda odvisna od porazdelitev parametrovin nakljucnih spremenljivk, in je v splosnem zelo netrivialna.

Normalna porazdelitev. Zaradi linearnosti najenostavnejsi primer predstavlja normalna porazdelitev vek-torja parametrov x. Ce predpostavimo standardno normalno porazdelitev nakljucnih spremenljivk ξi, lahkoreprezentativne vzorce generiramo po

x(m) = A · ξ(m) + µ, (3.18)

kjer je matrika A definirana tako, da ohranja srednje vrednosti, standardne deviacije in korelacije med parame-tri. µ predstavlja vektor pricakovanih vrednosti x. Izkaze se, da mora A zadoscati V = AAT , pri cemer je enaod resitev matricni kvadratni koren iz kovariancne matrike:

A = V1/2. (3.19)

Log-normalna porazdelitev. Pri log-normalno porazdeljenih spremenljivkah ξi izhajamo iz dejstva, da sonjihovi logaritmi normalno porazdeljeni, torej je njihova oblika invariantna na linearne transformacije:

ln x(m)i =

n∑j=1

Ai j ln ξ(m)j + µi, (3.20)

kjer je

µi = ln⟨xi⟩ −n∑

k=1

A2ik/2. (3.21)

Analogno normalni porazdelitviA = V1/2, (3.22)

vendar

Vi j = ln(

Vi j

⟨xi⟩⟨x j⟩+ 1

). (3.23)

Kombinacije normalne in log-normalne porazdelitve. V realnosti lahko naletimo na primer, ko so neka-teri od koreliranih paremetrov porazdeljeni normalno (npr. resonancne energije), drugi pa log-normalno (npr.resonancne sirine). Ker na vsakem omejenem intervalu zaupanja log-normalna porazdelitev konvergira k nor-malni v limiti majhnih relativnih negotovosti, lahko log-normalno porazdelitev priredimo vektorju parametrov

x ′ = x + X, (3.24)

kjer je Xi + xi ≫√⟨x2

i ⟩ za normalno porazdeljene xi in Xi = 0 za log-normalno porazdeljene xi. Z vecanjemXi lahko dosezemo poljubno natancnost.

XI

3.2.3 ’Total Monte Carlo’

’Total Monte Carlo’ (TMC) [57] oznacuje metodo, ki zdruzuje nakljucno vzorcenje jedrskih podatkov s tran-sportnimi izracuni z metodo Monte Carlo. Glavna prednost je natancnost metode, pomanjkljivost pa racunskazahtevnost, saj je potrebno ponoviti identicen izracun za vsak posamezen vzorec jedrskih podatkov.

3.3 Kombiniran deterministicno-probabilisticni pristop

Glede na potrebe in zmoznosti se pogosto kombinira deterministicne metodami s probabilisticnimi:

• Kombinacija nakljucnega vzorcenja z deterministicnimi transportnimi izracuni je uporabna npr. za hitrooceno negotovosti integralnih parametrov zaradi jedrskih podatkov.

• Kombinacija deterministicne obcutljivostne analize z Monte Carlo metodo za transport nevtronov jeuporabna npr. za oceno negotovosti integralnih parametrov kompleksnih sistemov.

3.4 Primerjava Monte Carlo metod vzorcenja

3.4.1 Vhodni podatki

Za primerjavo metod nakljucnega vzorcenja (resonancnih parametrov) smo izbrali dve razlicni evaluaciji jedr-skih podatkov za 55Mn:

• Evaluacija NRG [17] v obmocju locljivih resonanc (do 100 keV) vsebuje 172 resonanc. Korelacije samomed parametri posamezne resonance, velike relativne negotovosti (do 50%).

• Evaluacija ORNL1 v obmocju locljivih resonanc (do 125 keV) vsebuje 187 resonanc. Polna korelacijskamatrika, majhne relativne negotovosti (do 10%).

3.4.2 Rezultati

Metode vzorcenja smo najprej primerjali graficno, za bolj zanesljivo verifikacijo porazdelitev pa sledi se pri-merjava kvantitativnih parametrov: relativnega odmika povprecne vrednosti od pricakovane ((p − p0)/σp),relativne standardne deviacije (σ p−p0

σp) in deleza negativnih vrednosti pozitivnih parametrov (p > 0).

Glavna pomanjkljivost Metropolisove metode – korelacije med bliznjimi vzorci – je razvidna iz Slike 3.1,kjer je prikazana mnozica nakljucnih vzorcev dvojice antikoreliranih parametrov z razlicnimi metodami.Ceprav je med posameznimi vzorci na levem grafu Slike 3.1 izpuscenih po 100 korakov, je pri Metropoli-sovem algoritmu v originalnem sistemu opazna korelacija med vzorci, ki se odraza v zozani porazdelitvi. Privzorcenju v diagonalnem sistemu je korelacijska dolzina krajsa, in vzorci so vsaj na videz reprezentativni. Izdesnega grafa Slike 3.1 je razvidno, da je tudi pri vzorcenju v diagonalnem sistemu vcasih za reprezentativnevzorce potrebno izpustiti celo vec kot 1000 zaporednih korakov.

V Tabeli 3.1 so zbrana povprecna odstopanja prvih dveh momentov porazdelitev ter delez negativnih vre-dnosti za gama sirine iz evaluacije NRG z uporabo razlicnih metod vzorcenja. Uporaba razlicnih porazdelitev(normalne ali log-normalne) pri metodi diagonalizacije vpliva na delez negativnih vrednosti, medtem ko stamomenta porazdelitve zelo konsistentna. Negativnih vrednosti se lahko znebimo, ce jih preprosto postavimona 0 (odrez pri nicli), pri tem pa rahlo vplivamo na momente porazdelitve. Ce zelimo ohranjati simetrijoporazdelitev (simetricen odrez), so pricakovane vrednosti parametrov konsistentne, medtem ko je standardnadeviacija manjsa od realne. Drug nacin za odstranitev negativnih vrednosti je, da parametre logaritmiramo,vzorcimo po normalni porazdelitvi in na koncu pretvorimo nazaj v originalni sistem, vendar pa zaradi upo-rabe priblizka prvega reda pri transformaciji kovariancne matrike produciramo velika odstopanja momentovporazdelitev od pricakovanih. Podobnih metod se lahko posluzimo tudi pri Metropolisovem algoritmu, kjer je

1Luiz Leal, osebna komunikacija, oktober 2010.

XII

340.77 340.78 340.79 340.8023.660

23.665

23.670

23.675

23.680

23.685

23.690

23.695 diagonalizacija Metropolis orig. Metropolis diag.

p 11[e

V]

p10[eV]0 500 1000 1500 2000 2500 3000

-3

-2

-1

0

1

2

3

(p-p

0)/p

n

diagonalizacija Metropolis (c=0.03) Metropolis diag. (c=0.2)

Slika 3.1: Levo: Nevtronska sirina (p11) proti resonancni energiji (p10) za resonanco s 4. najnizjo energijoORNL evaluacije 55Mn (-35% korelacija). Razlicne metode vzorcenja, po 1000 vzorcev. Zaporedni vzorciMetropolisove metode so oddaljeni 100 korakov. Desno: Centrirano normalizirano zaporedje ((p − p0)/σp)za izbran resonancni parameter NRG evaluacije 55Mn kot funkcija koraka n. Razlicne metode vzorcenja,resolucija 10 korakov.

glavna razlika ze omenjena korealcija med vzorci, kar se odraza v vecji racunski zahtevnosti in/ali zmanjsanipovprecni standardni deviaciji vzorcev. Metoda koreliranega vzorcenja pricakovano edina producira povsemreprezentativne vzorce. Pri evaluaciji ORNL vse metode vzorcenja zaradi manjsih negotovosti dajejo zadovo-ljive rezultate.

Tabela 3.1: Negativne vrednosti in odmik od pricakovanih vrednosti prvih dveh momentov porazdelitev. Raz-dalja med sosednjimi vzorci pri Metropolisovi metodi je 100 korakov. Evaluacija NRG, gama sirine.

Metoda/porazdelitev p < 0 [%] ⟨ p−p0σp⟩ σ p−p0

σp

diag. 1.41 0.0044 ± 0.0024 1.0008 ± 0.0034log-normalna 0.32 −0.0024 ± 0.0024 0.9988 ± 0.0053odrez pri nicli 0 0.0098 ± 0.0024 0.9882 ± 0.0032simetricen odrez 0 −0.0015 ± 0.0024 0.9757 ± 0.0030logaritmiranje 0 0.1817 ± 0.0027 1.1279 ± 0.0085Metropolis (orig.) (c = 0.25, acc = 11%) 1.39 0.0126 ± 0.0025 0.9884 ± 0.0036odrez pri nicli 0 0.0177 ± 0.0024 0.9710 ± 0.0034simetricen odrez 0 0.0124 ± 0.0024 0.9585 ± 0.0031log-normalna (c = 0.2, acc = 20%) 0 0.1853 ± 0.0028 1.1080 ± 0.0089kor. vzorcenje/log-normalna 0 0.0000 ± 0.0024 0.9958 ± 0.0054

XIII

4 Resonacni integral in faktorjisamoscitenja

4.1 Faktorji samoscitenja za standardne monitorje

Vsak objekt, ki ga postavimo v nevtronsko polje, interagira z nevtroni, kar povratno vpliva na nevtronsko polje.Ce ta vpliv ni zanemarljiv, se posledicno spremeni tudi reakcijska hitrost. Ta pojav imenujemo samoscitenje,in je se posebej poudarjen v obmocju resonancnih energij. Tipicno samoscitenje pride do izraza v hetero-genih sistemih (npr. mreza gorivo-moderator) in pri obsevanju relativno velikih vzorcev mocnih nevtron-skih absorberjev. Slednjemu so se v preteklosti, ce je le bilo mozno, izogibali. V nasprotnem primeru jepotrebno upostevati resonancno samoscitenje, kar je dobro poznan, a netrivialen problem. Avtorji clanka[62] so kot resitev predlagali enostaven empiricen pristop (t.i. metodo Chilian). V clanku [9] omenjenometodo na seriji realisticnih vzorcev primerjamo z bolj eksaktnimi metodami: metodo Monte Carlo (pro-gram MCNP [5]) in deterministicnim pristopom resevanja enacbe (3.8) upocasnjevanja nevtronov (programMATSSF: http://www-nds.iaea.org/naa/matssf/).

4.1.1 Program MATSSF

Epitermicni faktor samoscitenja f za posamezen resonancni absorber je definiran kot razmerje reakcijskihhitrosti s perturbiranim in neperturbiranim fluksom:

f =

∫ E3

ECdΣk(E)ϕ∗(E)dE∫ E3

ECdΣk(E)ϕ0(E)dE

. (4.1)

Tabelirana knjiznica f faktorjev v MATSSF za posamezne izotope v odvisnosti od razredcitvenega preseka σ0je bila vnaprej izracunana s pomocjo sistema NJOY [64] iz neperturbiranega (cistega 1/E) in perturbiranega(dobljenega z resevanjem enacbe upocasnjevanja nevtronov) fluksa v mejah med 0.55 eV in 2 MeV.

Vsebnost vec mocnih resonancnih absorberjev v istem vzorcu povzroci interferenco med nuklidi razlicnihizotopov. Ker je uporaba tockovnih presekov za rutinske izracune prepocasna, MATSSF uporablja 640-grupnipriblizek enacbe (3.8). Ta priblizek ne uposteva pravilno prekrivanja resonanc, ki so ozje od grupne strukture.Na sreco ozke resonance ponavadi ne prispevajo bistveno k reakcijski hitrosti, tako da je v praksi priblizekpogosto zadovoljivo natancen.

Efektivni epitermicni faktor samoscitenja G f za reakcijo na nuklidu k dobimo iz:

⟨G f ⟩k =∫Σk(E)ϕ(E)dE∫Σk(E)ϕ0(E)dE

=


∫Σk(E)ϕ∗(E)dE∫Σk(E)ϕ0(E)dE

=


f , (4.2)

kjer je ϕ∗ fluks, perturbiran z nuklidom k, ϕ pa fluks, perturbiran z vsemi materiali v vzorcu. V grupni oblikiG f zapisemo:

⟨G f ⟩k =∑

g Σk,gϕg∑g Σk,gϕ

∗g

f . (4.3)

XIV

Enacba (3.8) se v grupnem priblizku prevede na sistem linernih enacb. Od tod lahko grupne flukse ϕ∗g izrazimoeksplicitno:

ϕ∗g =Σ0 ϕ0,g +

∑G(g)h=g+1 Σs,hg ϕ

∗h

Σ0 + Σt,g − Σs,gg, (4.4)

sistem enacb za ϕ∗g pa resimo rekurzivno zacensi z grupo, ki ustreza najvecjim energijam. Na podoben nacindobimo tudi ϕg za mesanico izotopov, z upostevanjem vecih integralov v enacbi (3.8).

4.1.2 Posplositev povprecne dolzine tetive

Zadnja verzija MATSSF uporablja posploseno definicijo parametra L (povprecne dolzine tetive v vzorcu) izenacbe (3.7), in sicer predstavlja povprecno dolzino iz perspektive nevtronov:

L =

∫VΦ(Ω)dΩ∫

S⊥(Ω)Φ(Ω)dΩ, (4.5)

kjer je Φ(Ω) kotni fluks in S⊥(Ω) presek vzorca v smeri, pravokotni na Ω. Enacba (4.5) implicitno privzame,da kotni fluks na mestu vzorca nima krajevne odvisnosti. Enacba (4.5) velja za vzorce konveksne oblike.

V MATSSF je edina mozna oblika vzorca cilindricna, kar zadosca za vecino realnih primerov:

V = πr2d (4.6)

inS⊥ = 2rd| cosα| + πr2| sinα|, (4.7)

kjer je α kot med izbrano smerjo in osjo cilindra.

4.1.2.1 Izotropen izvor

Z upostevanjem izotropije fluksa (Φ(Ω) = 1) dobimo:

L =

∫ π/20

∫ π/20 πr2d cosϑdϑdφ∫ π/2

0

∫ π/20 (2rd cosϑ + πr2 sinϑ) cosϑdϑdφ

=2rd

r + d=

4VS, (4.8)

kjer je ϑ polarna koordinata.

4.1.2.2 Cilindricen izvor

Prazen obsevalni kanal v reaktorju aproksimiramo z izotropnim izvorom na plascu cilindra s polmerom R invisino H. Kotni fluks na sredini kanala (kjer se nahaja vzorec) je sorazmeren:

Φ(Ω) = 1/ cosα, α ∈

− arctan H

2R , arctan H2R

0, sicer.

(4.9)

V aksialno simetricnem primeru, ko sta osi vzorca (tipicno folije ali zice) in obsevalnega kanala vzporedni,se parameter L izraza kot:

L =πrd

√1 + H2

4R2 arctan H2R

d HR + πr

(√1 + H2

4R2 − 1) . (4.10)

V nasprotnem primeru, ko cilindricni vzorec (obicajno zica) ’lezi’ v obsevalnem kanalu, tako da sta osipravokotni, rezultat ni tako enostaven:

L =π2r2d arctan H

2R

2

2rd g(arctan H

2R

)+ πr2H

2R√

1+ H2

4R2

, (4.11)

XV

kjer je g(x) neelementarna enakomerno zvezna funkcija, definirana za x ∈ [0, π/2]:

g(x) =∫ π/2

0dφ

∫ x

0

√cos2 ϑ cos2 φ + sin2 ϑ dϑ. (4.12)

Izraza (4.10) in (4.11) sta opcijsko vkljucena v MATSSF, g(x) pa je aproksimirana s polinomom 8. stopnje.

4.1.3 Model MCNP

Metoda Monte Carlo omogoca rigorozno obravnavo pojava samoscitenja, zato je vzeta za referenco pri pri-merjavi razlicnih metod. Obsevalna naprava je bila simulirana z izotropnim cistim 1/E ploskovnim izvoromna sferi ali plascu cilindra z radijem in efektivno visino, ki ustreza obsevalnemu kanalu reaktorja TRIGA [13].Vzorci so bili eksplicitno modelirani, faktorji samoscitenja pa izracunani direktno po definiciji: z razmerjemreakcijskih hitrosti v vzorcu pri dejanskem in pri neperturbiranem fluksu.

4.1.4 Faktorji samoscitenja za zico iz nikljeve zlitine

Za realisticen vzorec (zica debeline 1 mm in dolzine 3.8 mm) iz nikljeve zlitine (materialna sestava: Ni80.93%, Mo 15.16%, W 2.76%, Mn 0.41%, Au 0.29%, Fe 0.45%) z gostoto 9.21 g/cm3 je bila opra-vljena detajlna analiza. V tem delu prikazani zgolj rezultati, ki ustrezajo sfericnemu izvoru, ker se je zareaktor TRIGA izkazal za bolj reprezentativnega [14, 15]. Primerjava epitermicnih faktorjev samoscitenja,izracunanih z razlicnimi metodami, je prikazana v Tabeli 4.1. V modelih MATSSF (G f ) in MCNP (4. in 5.stolpec tabele) upostevamo interference med resonancami razlicnih izotopov, medtem ko je z metodo Chilian(2. stolpec) mozna le neodvisna obravnava posameznih izotopov. Za direktno primerjavo smo dodali locenoizotopsko obravnavo z MATSSF ( f ), 3. stolpec.

Tabela 4.1: Primerjava resonancnih faktorjev samoscitenja za reakcije (n,γ) na izotopih v zici iz nikljevezlitine.

nuklid Chilian MATSSF MATSSF MCNP ∆ f ∗

G f0

∆ fG f0

∆G f

G f0( f ∗) ( f ) (G f ) (G f0)55Mn 0.990 0.9941 0.9902 0.993 ± 0.000 -0.3% 0.1% -0.3%56Fe 1.000 0.9998 0.9930 0.994 ± 0.001 0.6% -0.6% -0.1%58Fe 1.000 1.0000 0.9904 0.986 ± 0.002 1.4% 1.4% 0.4%58Ni 0.9919 0.9863 0.991 ± 0.000 0.1% -0.5%64Ni 0.995 0.9997 0.9934 0.995 ± 0.000 0.0% 0.5% -0.2%

92Mo 0.9929 0.9880 0.990 ± 0.001 0.3% -0.2%98Mo 0.956 0.9359 0.9347 0.952 ± 0.001 0.4% -1.7% -1.8%

100Mo 0.965 0.9407 0.9369 0.952 ± 0.002 1.4% -1.2% -1.6%184W 0.9669 0.9651 0.977 ± 0.001 -1.0% -1.2%186W 0.797 0.8033 0.7860 0.817 ± 0.001 -2.4% -1.7% -3.8%197Au 0.923 0.9247 0.9213 0.940 ± 0.001 -1.8% -1.6% -2.0%

Ker je MATSSF zmozen skupne obravnave vec izotopov, je faktor G f pricakovano v povprecju blizjereferencnemu kot metoda Chilian in MATSSF faktor f (ceprav razlika ni tako izrazita kot pri nekaterih drugihvzorcih, obravnavanih v [9]), kljub relativno grobi (640-)grupni strukturi.

4.1.5 Eksperimentalna validacija faktorjev samoscitenja za rodijeve folije

Rodijeve folije s premerov 5 mm in debelino 0.006 mm oz. 0.112 m so bile obsevane v sredici in reflektorjureaktorja TRIGA na IJS [13], ustrezni popravki zaradi samoscitenja (1− f ) z MATSSF in MCNP so okoli 10%oz. 60%. Ujemanje z eksperimentom je znotraj 1% za tanjso in 5% za debelejso folijo [10], kar lahko stejemoza dodatno potrditev izracunanih faktorjev samoscitenja.

XVI

4.2 Primer resonancne kovariacne analize: resonancni integral 55Mn

V novejsih evaluacijah jedrskih podatkov so obicajno vsebovane tudi kovariance, na podlagi katerih lahkoocenimo negotovost integralnih parametrov kot npr. resonancni integral

RI =∫ E2

E1

Σ(mi)(E)ϕ(E)dE, (4.13)

zaradi negotovosti v presekih. V resonancnem obmocju so negotovosti v presekih pogosto podane posrednopreko negotovosti v resonancnih parametrih (ENDF-6 File 32). Ker za nuklide z velikim stevilom resonancnihparametrov kovariancna matrika postane zelo obsezna, se negotovosti v resonancnih parametrih pogosto pre-vede na negotovosti v presekih (ENDF-6 File 33), na relativno grobi energijski mrezi. Vprasanje je, kakopomembna je informacija, ki se pri reduciranju podatkov izgubi.

Glavni namen raziskave [16] je bil odgovoriti na zgornje vprasanje. Za dve evaluaciji presekov za 55Mn(NRG in ORNL, podrobneje opisani v poglavju 3.4.1) smo opazovali negotovost resonancnega integrala zajetja(v priblizku ozkih resonanc s kodo GROUPIE serije PREPRO [19]) v odvisnosti od stopnje samoscitenja,merjene z razredcitvenim presekom Σ(mi)

0 . Za vzorcenje resonancnih parametrov je bila uporabljena metodadiagonalizacije (poglavje 3.2.2.2), pri vsekem primeru je bilo generiranih 1000 vzorcev, kar zadostuje zazadovoljivo konvergenco RI.

Za primerjavo smo negotovost RI izracunali se iz kovarianc presekov v 640-grupni SAND-II strukturi [65]z deterministicno obcutljivostno analizo prvega reda.

4.2.1 Rezultati

4.2.1.1 Negotovost resonancnega integrala iz resonancnih parametrov

Relativna negotovost RI v odvisnosti od razredcitvenega preseka Σ(mi)0 je prikazana na Sliki 4.1. Prispevek

resonanc k negotovosti za podatke ORNL znasa od reda 0.05% pri neskoncnem razredcenju do vec kot 10%pri mocnem samoscitenju, medtem ko je prispevek ozadja dobre 3%. Za podatke NRG je skupna negotovostokrog 0.5%, kar je direktno posledica v povprecju vecjih relativnih negotovosti resonancnih parametrov, inje prakticno neodvisna od stopnje samoscitenja. Pri vecjih stopnjah samoscitenja pridejo do izraza korelacijemed resonancami, kar pri evaluaciji ORNL vpliva na povecanje relativne negotovosti, medtem ko pri evaluacijikorelacij med resonancami ni, zato se relativna negotovost RI prakticno ne spreminja.

4.2.1.2 Negotovost resonancnega integrala – 640-grupni priblizek

Relativna negotovost RI zaradi negotovosti v resonancnih parametrih lahko pri mocnem samoscitenju narasteza vec velikostnih redov (Slika 4.1). Zanima nas, ce lahko to povecanje negotovosti opisemo s pretvorbokovariancnih matrik resonancnih parametrov v kovariacne matrike presekov v 640-grupni SAND-II strukturi,ki jih pristejemo kovariancnim matrikam preseka ozadja, in uporabo deterministicne obcutljivostne analize 1.reda.

Na ta nacin izracunana relativna negotovost RI za podatke ORNL pri neskoncnem razredcenju znasa 3.8%,pri mocnem samoscitenju (Σ(mi)

0 = 1 b) pa 3.2%, pri cemer slednja stevilka predstavlja le direkten prispevekpreseka k negotovosti RI. Ker je fluks izracunan iz preseka v priblizku ozkih resonanc, presek k negotovostiRI prispeva tudi posredno preko negotovosti v fluksu, vendar se izkaze, da ta prispevek ne more biti vecji odprispevka preseka. Zato je skupna relativna negotovost RI v 640-grupnem priblizku < 6.4%, kar je bistvenomanj kot negotovost, izracunana direktno iz negotovosti resonancnih parametrov (16%). V primeru 55Mntorej 640-grupni priblizek ocitno ne zadosca za natancno oceno negotovosti resonancnega integrala za zajetjenevtrona [16].

XVII

1 10 100 1000 10000

0.1

1

10

RI/R

I [%

]

(mi)0 [b]

podatki ORNL ORNL (brez ozadja) podatki NRG

Slika 4.1: Relativna negotovost σRI/RI resonancnega integrala v odvisnosti od razredcitvenega preseka Σ(mi)0

za redko (angl. sparse) kovariancno matriko z velikimi negotovostmi (podatki NRG) ter polno kovariancnomatriko z manjsimi negotovostmi (podatki ORNL) resonancih parametrov.

XVIII

5 Korekcije jedrskih podatkov

Jedrski podatki so vecinoma osnovani na jedrskih modelih, ki tipicno vsebujejo vec parametrov, ki so prostiali imajo relativno blage omejitve. V evaluacijah jedrskih podatkov so ti parametri doloceni tako, da cim boljeopisejo eksperimentalne podatke, tako diferencialne (npr. iz baze EXFOR, poglavje 5.1) kot integralne (npr.iz zbirke ICSBEP [58]). Korekcije jedrskih podatkov se lahko izvaja npr. s posploseno metodo najmanjsihkvadratov (poglavje 5.2). V realnosti eksperimentalni podatki niso vedno konsistentni, zato jih je potrebnoustrezno filtrirati ali prilagoditi. S tem v proces evaluacije jedrskih podatkov vpeljemo subjektivno kompo-nento: po pravilu dva razlicna evaluatorja naredita dve razlicni evaluaciji tudi ce imata na razplago natankoiste eksperimentalne podatke. Kljub vsemu pa, ce sta dovolj izkusena, bosta njuni evaluaciji podobni oziromamed seboj konsistentni.

5.1 Baza eksperimentalnih podatkov EXFOR

Baza EXFOR (EXchange FORmat database) [68] sluzi za izmenjamo eksperimentalnih podatkov, in vsebujeobsezno zbirko eksperimentalnih podatkov o jedrskih reakcijah, ponavadi v obliki preseka za doloceno reakcijopri doloceni energiji, lahko pa tudi v drugih oblikah, npr. kot razmerja preseka med razlicnimi reakcijami, jedri,ali reakcijske hitrosti v tocno definiranem spektru. Vecina podatkov v bazi EXFOR je za reakcije z nevtroni,vsebuje pa tudi podatke za reakcije s fotoni in nabitimi delci.

5.2 ’Fitanje’ po posploseni metodi najmanjsih kvadratov

Posplosena metoda najmanjsih kvadratov predpostavlja normalno porazdelitev vseh parametrov [69, 70].Zacnemo z vektorjem parametrov x s pricakovanimi vrednostmi x0 in kovariancno matriko V x, ki predsta-vljata vse znane informacije o parametrih. V naslednjem koraku bi radi v oceno prvih dveh momentov xvkljucili (eksperimentalne) podatke y0 za neke znane funkcije y(x). Izkaze se, da je ob predpostavki normalneporazdelitve najboljsa ocena za popravljen vektor pricakovanih vrednosti in kovariancno matriko:

V x,ad j =

[S T

(V z

)−1S]−1= S −1V z(S T )−1,

x0,ad j = V x,ad jS T(V z

)−1z0, (5.1)

kjer je

z0 =

(x0y0

),

V z =

(V x V xy

V xy V y

).

S =∂z∂x

(5.2)

Posplosena metoda najmanjsih kvadratov je analogna priblizku 1. reda v razvoju kovarianc v Taylorjevo vrsto(poglavje 3.1.2). Omejitve metode so jasne: problem so nelinearnosti y(x), ki ponavadi pridejo do izraza privelikih relativnih negotovostih, in odstopanja realnih porazdelitev od normalne, kar pa se v praksi izkaze zamanjso tezavo od linearnosti.

XIX

5.3 ’Global Assessment of Nuclear Data Requirements’ (GANDR)

Program GANDR (Global Assessment of Nuclear Data Requirements) [6] je prvi poskus povsem splosnekovariancne obravnave reakcijskih presekov. Omogoca opis kovarianc za vec kot 100 izotopov in do 25 reakcij.Zaradi velikega stevila izotopov in reakcij je energijska mreza razdeljena na 74 tock E(n), med katerimi jeuporabljena linearna interpolacija:

H(n, E) =

(E − E(n − 1))/(E(n) − E(n − 1)), E(n − 1) ≤ E < E(n)(E(n + 1) − E)/(E(n + 1) − E(n)), E(n) ≤ E < E(n + 1)0, sicer

(5.3)

kjer velja74∑

n=1

H(n, E) = 1. (5.4)

Korekcijska funkcija je definirana kot

Aad j(E) =74∑

n=1

aad j(n)H(n, E), (5.5)

kjer je aad j(n) korekcijski faktor pri energiji E(n). Popravljen reakcijski presek izracunamo tako, da enostavnopomnozimo zacetni presek s korekcijsko funkcijo:

Σ(mi)ad j,r,k(E) = Aad j,r,k(E)Σ(mi)

r,k (E) =74∑

n=1

aad j,r,k(n)H(n, E)Σ(mi)r,k (5.6)

kjer se indeksa r in k nanasata na reakcijo in izotop. Vhodni podatki za GANDR so zacetni preseki Σ(mi)r,k (E)

in ustrezne kovariancne matrike, definirane na energijski mrezi n, ki so potem popravljeni z vkljucitvijo eks-perimentalnih podatkov po posploseni metodi najmanjsih kvadratov. Zaradi relativno redke energijske mrezeGANDR ne more vplivati na fino strukturo preseka, zato ni primeren za uporabo v resonancnem podrocju,medtem ko gostota mreze zadosca za aplikacije v obmocju visokih energij vpadnega nevtrona.

5.4 Primer: volfram

Ker vecina modernih evaluacij jedrskih podatkov bazira na jedrskih modelih, ki vsebujejo iste parametre zavec izotopov in celo elementov, so v principu prisotne korelacije med izotopi, ceprav v evaluacijah obicajnoniso podane. Dodatne korelacije med izotopi vpeljemo z vkljucitvijo eksprimentalnih podatkov za mesanicoizotopov. Pravilna ocena korelacij med izotopi je bistvena za zanesljivo oceno negotovosti integralnih ekspe-rimentov. Obstajajo indikacije, da neupostevanje korelacij med izotopi lahko pripelje tudi do vec kot dvakratpodcenjene negotovosti integralnih parametrov. Prispevek [8] obravnava primer preseka za volfram (W) vobmocju visokih energij:

• Izhajali smo iz referencnega primera iz [7], kjer so ze loceno upostevani eksperimentalni podatki zaposamezne volframove izotope.

• Evaluacijo presekov smo nadgradili z upostevanjem eksperimentalnih podatkov za totalni presek zanaravno mesanico volframa.

• Vpliv vkljucitve podatkov za naravno mesanico volframa in posledicne vpeljave korelacij med izotopismo testirali na integralnem eksperimentu FNG-W [71].

XX

5.4.1 Korekcija volframovega preseka in vpeljava korelacij

Izhajamo iz vmesne evaluacije, kjer so ze upostevani eksperimentalni podatki za stiri volframove izotope(182W, 183W, 184W, and 186W) [7], ki tako vsebuje oceno korelacij med razlicnimi energijami in reakcijamiposameznega izotopa. Nekateri rezultati [7] so pokazali potrebo po vpeljavi korelacij tudi med razlicnimi izo-topi, kar lahko dosezemo npr. z vkljucitvijo eksperimentalnih podatkov za reakcijske preseke za elementarnomesanico.

5.4.1.1 Eksperimentalni podatki

Ker so obogateni vzorci izredno dragi, so meritve preseka na elementarnih vzorcih z majhnim delezem pri-mesi bolj prirocne. Zato obstaja veliko eksperimentalnih podatkov za naravno mesanico volframa. Zaradiredke energijske mreze v GANDR smo vkljucili samo podatke nad resonancnim obmocjem. Nekonsistentni innekompletni podatki so bili zavrnjeni. Preostalim podatkom, ki so bili izbrani za korekcijo preseka, smo ume-tno pristeli sistematsko napako, ki v bazi EXFOR ni podana, jo je pa nujno potrebno upostevati, da ne pridedo podcenitve skupne negotovosti in korelacij v preseku. Ce smo pri tem dovolj konservativni, bo dobljenaevaluacija konsistentna znotraj negotovosti.

5.4.1.2 Kompletna kovariancna analiza

Korekcije preseka. Slika 5.1 prikazuje korekcijsko funkcijo Aad j(E) za presek za zajetje za posamezneizotope po vkljucitvi eksperimentalnih podatkov za totalni presek elementarne mesanice. Ker so bile korelacijeznotraj posameznih izotopov prisotne ze prej, vkljucitev podatkov za element zelo razlicno vpliva na razlicneizotope, reakcije in energije vpadnega nevtrona. Popravki so zelo majhni pri energijah pod 1 MeV, medtemko so pri visjih energijah znatni in v obe smeri (Slika 5.1). Zaradi splosnega padanja preseka z energijo sopopravki pomembni samo v sistemih s poudarjeno komponento nevtronskega spektra nad 10 MeV, npr. vsistemih s fuzijskim spektrom.

Negotovosti v preseku in polna kovariancna matrika. Zacetna evaluacija iz [7] ze vsebuje korelacije medrazlicnimi energijami in reakcijami posameznih izotopov. Elementi kovariancne matrike za posamezne iz-otope so se po vkljucitvi podatkov za elementarno mesanico v povprecju pricakovano nekoliko zmanjsali,korelacijski koeficienti pa se vecinoma niso znatno spremenili. Na novo smo vpeljali kovariance med presekiza razlicne izotope. Za ilustracijo, Slika 5.2 prikazuje kovariancno matriko za presek za elasticno sipanje na182W in 184W. Vecinoma je presek med izotopoma pricakovano antikoreliran, saj z vkljucitvijo podatkov zaelementarno mesanico dosezemo, da je njuna vsota oz. natancneje linearna kombinacija s pozitivnimi koefici-enti poznana bolj natancno kot preseka sama.

XXI

100000 1000000 1E70.80

0.85

0.90

0.95

1.00

1.05

A adj

E [eV]

182W 183W 184W 186W

Slika 5.1: Korekcijska funkcija Aad j po selektivni vkljucitvi podatkov iz baze EXFOR za totalni presek naelementarnem volframu [8], relativno glede na izotopsko evaluacijo [7], v odvisnosti od energije vpadneganevtrona za razlicne izotope volframa.

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∆σ/σ vs. E

for 182W(n,el.)

10-2

10-1

100

101

102

103

104

105

106

107

10-1

100

101

102

∆σ/σ vs. E for 184W(n,el.)

10-2 10-1 100 101 102 103 104 105 106 10710-1

100

101Ordinate scale is %

relative standard deviation.

Abscissa scales are energy (eV).

Correlation Matrix

0.00.20.40.60.81.0

0.0-0.2-0.4-0.6-0.8-1.0

Slika 5.2: Kovariancna matrika za presek za elasticno sipanje na 182W in 184W. Kovarianc pri nizkih ener-gijah ni, saj jih ne moremo natancno oceniti s programom GANDR; dobimo jih lahko z loceno resonancnoobravnavo.

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5.4.2 Integralni eksperiment

Evaluacija preseka za volframove izotope je bila preverjena na integralnem eksperimentu FNG-W. Eksperi-ment simulira transport fuzijskih nevtronov iz izvora devterij-tritij skozi velik kvader iz volframa. V Tabeli 5.1so podane relativne negotovosti aktivnosti zlatih folij, ki so bile obsevane 15 cm (Poz. 2) in 35 cm (Poz. 4)globoko v volframovem kvadru glede na izvor. Z upostevanjem korelacij med izotopi se ocenjena negotovostne spremeni bistveno, izracunana aktivnost pa ostaja nekoliko vec kot 1σ stran od meritev. V primeru, da pred-postavimo 100% korelacijo med izotopi, pa se ocena negotovosti skoraj podvoji (zadnja vrstica Tabele 5.1) inizracun postane konsistenten z meritvami aktivnosti. Za boljso oceno negotovosti integralnega eksperimentaFNG-W bo potrebno evaluacijo preseka za volfram razsiriti se z upostevanjem drugih reakcij na elementarnemvolframu, vsaj z zajetjem nevtrona in reakcijo (n,2n), na katero so rezultati tega integralnega eksperimenta zeloobcutljivi.

Tabela 5.1: Relativne negotovosti v napovedih aktivnosti zlatih folij, na globini 15 cm (Poz. 2) in 35 cm (Poz.4) v volframovem kvadru eksperimentalne konfiguracije FNG-W.

Negotovost [%] Negotovost [%]Poz. 2 Poz. 4

182W 0.8 1.4183W 0.9 1.6184W 0.9 2.2186W 0.7 1.1

Skupna – korelacije znotraj izotopov 1.68 3.26Skupna – korelacije med izotopi 1.54 3.15

Skupna – polna korelacija (C = 1) 3.34 6.29

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6 Zakljucki

Doktorsko delo obravnava uporabo kovariancnih matrik v reaktorskih preracunih. V njem so opisane in medseboj primerjane razlicne metode nakljucnega vzrocenja, ki so uporabljene na primeru resonancnih parametrov55Mn. Metode so splosne, njihova uporabnost ni omejena na jedrske podatke. Primerjava razlicnih reprezen-tacij kovariancnih matrik je bila narejena s kombiniranjem nakljucnega vzorcenja z deterministicnimi izracuniresonancnega integrala. Faktorji samoscitenja so bili izracunani in med seboj primerjani s probabilisticnimiin deterministicnimi metodami za realne vzorce standardnih monitorjev, ter eksperimentalno validirani zarodijeve folije razlicnih dimenzij. V nasprotju z uporabo kovariancnih matrik jedrskih podatkov za ocenonegotovosti reaktorskih parametrov lahko uporabimo eksperimentalne podatke slednjih za korekcijo jedrskihpodatkov in pripadajocih kovariancnih matrik.

Obcutljivostne koeficiente integralnih spremenljivk na vhodne parametre (npr. preseke) obicajno dobimoz deterministicnimi izracuni adjungirane transportne enacbe. V kombinaciji s kovariancnimi matrikami vho-dnih parametrov lahko ocenimo negotovost integralnih parametrov. Drug nacin je, da nakljucno vzorcimokorelirane vhodne parametre, tako da generiramo vzorce perturbiranih vektorjev parametrov. Za vsak vzorecpotem z direktnim izracunom in statisticno analizo rezultatov dolocimo pricakovane vrednosti in negotovostiintegralnih parametrov.

Obravnavane so tri razlicne metode vzorcenja: metoda diagonalizacije, Metropolisov algoritem in metodakoreliranega vzorcenja. V ugodnih okoliscinah, kadar operiramo s parametri z majhnimi negotovostmi insibkimi korelacijami, so vse metode ekvivalentne. Razlike se pojavijo pri velikih negotovostih in mocnihkorelacijah velikega stevila parametrov.

Glavne prednosti metode diagonalizacije so njena preprostost, numericna ucinkovitost in popolna neodvi-snost zaporednih vzorcev. Uporaba te metode je priporocljiva za normalno porazdeljene parametre in pozitivneparametre z majhno relativno negotovostjo (pod ∼ 30%). V primeru vecjih negotovosti metoda generira znatendelez negativnih vrednosti, kar je sicer mozno delno odpraviti, vendar se pri tem pojavijo nove tezave.

Glavna prednost Metropolisovega algoritma z vzorcenjem v originalnem parametricnem prostoru je, dase s tem popolnoma izognemo diagonaliziranju kovariancne matrike, ki je lahko numericno nestabilno. Za-radi korelacij med zaporednimi vzorci je metoda primerna za parametre z redko kovariancno matriko ali zavzorcenje relativno majhnega stevila parametrov. Pri velikih relativnih negotovostih pozitivnih parametrov sepojavijo podobne tezave kot pri metodi diagonalizacije.

Prednosti Metropolisovega algoritma z vzorcenjem v diagonalnem parametricnem prostoru v primerjavi zvzorcenjem v originalnem parametricnem prostoru je zmanjsanje korelacijske dolzine v zaporedju generiranihstanj Metropolisove metode. Razlika je ocitna pri mocnih korelacijah velika stevila parametrov. Ceprav me-toda na zacetku zahteva diagonalizacijo kovariancne matrike, se morebitne numericne nestabilnosti izravnajos principom sprejetja/zavrnitve koraka Metropolisovega algoritma. Je edina sprejemljiva od opisanih metod vprimeru slabo pogojene kovariancne matrike z mocnimi korelacijami.

Glavna prednost metode koreliranega vzorcenja je, da omogoca konsistentno vzorcenje pozitivnih para-metrov ne glede na relativno negotovost z uporabo log-normalne porazdelitve. Prav tako kot metoda diago-nalizacije tudi metoda koreliranega vzorcenja ucinkovito generira popolnoma nekorelirana zaporedja vzorcev.Metoda omogoca tudi vzorcenje poljubne kombinacije normalno in log-normalno porazdeljenih parametrov,edina slabost pa je, da je v praksi metoda lahko podvrzena numericnim nestabilnostim zaradi slabe pogojenostikovariancne matrike.

Metode nakljucnega vzorcenja so bile aplicirane na resonancne parametre. Se pred propagacijo negotovosti

XXV

iz resonancnih parametrov na resonancni integral pa je bila narejena analiza izracuna faktorjev samoscitenjaza resonancni integral.

Na realisticnih primerih so primerjane razlicne metode izracuna resonancnih faktorjev samoscitenja. Ugo-tovitve so sledece

• Enostavna empiricna metoda Chilian et. al. za izracun resonancnih faktorjev samoscitenja producirav mnogih primerih zadovoljive rezultate. Slabost metode je, da je na trhlih fizikalnih temeljih, zato jetezko oceniti njene omejitve. Uporaba metode je priporocljiva za ze preverjene materiale, obsevane vpreverjenih obsevalnih napravah.

• Enostavna metoda v programu MATSSF bazira na resonancni teoriji in principu ekvivalence. Aplikacijaz interpolacijo vnaprej tabeliranih vrednosti za posamezne izotope je hitra in ucinkovita. Daje dobrerezultate za vecino vzorcev, kjer ni znatne interference med resonancami razlicnih izotopov.

• Nobena izmed zgornjih metod ne uposteva resonancne interference med izotopi. Zato je bila razvitabolj sofisticirana metoda, ki vsakic posebej resuje enacbo upocasnjevanja nevtronov v 640-grupnempriblizku in je vkljucena v program MATSSF. Je nekoliko pocasnejsa od zgoraj navedenih metod, vendarse vedno izredno hitra v primerjavi z metodo Monte Carlo. Se posebej je primerna za rutinske izracunefaktorjev samoscitenja v vzorcih, ki vsebujejo vecje stevilo mocnih resonancnih absorberjev.

• Rezultati ostalih metod so bili primerjani z referencnimi izracuni z metodo Monte Carlo, s polnim geo-metrijskim modelom vzorca in obsevalnega kanala. Metoda je najnatancnejsa od vseh opisanih, vendartudi numericno najbolj zahtevna.

Izracuni faktorjev samoscitenja so bili validirani z nevtronskim aktivacijskim eksperimentom, ki je bilizveden na reaktorju TRIGA na Institutu ’Jozef Stefan’. Rodijeve folije razlicnih dimenzij so bile obsevanev dveh obsevalnih kanalih z razlicnimi spektri. S faktorji samoscitenja, izracunanimi z MCNP in MATSSF,dobimo dobro konsistentnost z eksperimentom.

Vpliv samoscitenja na negotovost resonancnega integrala zaradi negotovosti v resonancnih parametrih jebil ocenjen v priblizku ozkih resonanc. Kadar so vsi ali vsaj znaten delez resonancnih parametrov korelirani,je relativna negotovost resonancnega integrala mocno odvisna od razredcitvenega preseka. Tega pojava nemoremo opisati, ce kovariancne matrike resonancnih parametrov prej pretvorimo v 640-grupne kovariancnematrike preseka. Pomemben izsledek te raziskave je, da v splosnem ne moremo nadomestiti kovarianc reso-nancnih parametrov s kovariancami preseka tudi pri relativno gosti energijski mrezi.

Za kovariancno analizo v obmocju visokih energij, kjer je pojav samoscitenja zanemarljiv, uporabimodrugacen pristop. Opisan je princip korekcije jedrskih podatkov z upostevanjem novih eksperimentalnih po-datkov za poljubne (diferencialne ali integralne) opazljivke. Pri tem je uporabljen linearen priblizek - metoda’fitanja’ po metodi najmanjsih kvadratov. Metodologija je demonstrirana na primeru evaluacije preseka zavolfram, kjer so bili na novo vkljuceni eksperimentalni podatki za totalni presek na elementarni mesanici vol-frama, evaluacija pa je bila testirana na integralnem eksperimentu FNG-W. Ker je bilo pred tem v evaluacijovkljucenih veliko eksperimentalnih podatkov za posamezne izotope, vkljucitev podatkov za totalni presek zaelementarni volfram ni imela velikega vpliva niti na presek niti na integralne parametre. Ker pa ima omenjeniintegralni eksperiment veliko obcutljivost na reakcijo (n,2n), bi bilo za zanesljivejso oceno potrebno v evalu-acijo vkljuciti se eksperimentalne podatke za preseke na elementarni mesanici vsaj se za reakcijo (n,2n) in zazajetje nevtrona.

Uporaba kovariancnih matrik za oceno negotovosti v reaktorskih preracunih je se vedno dokaj redka. Za-radi velikega napredka v ucinkovistosti racunalniskih sistemov pa lahko upraviceno pricakujemo porast tovr-stnih analiz v prihodnosti. To delo vsebuje temeljit opis splosne kovariancne obravnave od vhodnih podatkov(reakcijskih presekov) do integralnih parametrov, kot tudi splosno metodo korekcije presekov in pripadajocihkovariancnih matrik z upostevanjem eksperimentalnih podatkov. Uporaba teh metod je demonstrirana na po-sameznih originalnih primerih. Najpomembnejsi rezultat analize je, da v splosnem ni mozno nadomestitikovarianc resonancnih parametrov s kovariancami preseka ozadja. Oba pristopa ne dajeta ekvivalentnih rezul-tatov, razen ce kovariancna matrika preseka ozadja ni tako gosta, da zajema podrobno resonancno strukturo,vendar v tem primeru ne dosezemo primarnega cilja – bolj kompaktnega zapisa podatkov.

XXVI

a

IZJAVA

Podpisani Gasper Zerovnik, rojen 9.11.1984 v Kranju, izjavljam, da sem avtor pricujocega dela.

Ljubljana, 30.1.2012

Gasper Zerovnik

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