Table Of ContentRESONANCE
SELF-SHIELDING
CALCULATION
METHODS IN
NUCLEAR REACTORS
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Woodhead Publishing
Series in Energy
RESONANCE
SELF-SHIELDING
CALCULATION
METHODS IN
NUCLEAR REACTORS
LIANGZHI CAO
Professor, School of Nuclear Science and Technology,
Xi’an Jiaotong University, Peoples Republic of China
HONGCHUN WU
Professor, School of Nuclear Science and Technology,
Xi’an Jiaotong University, Peoples Republic of China
QIAN ZHANG
Distinguished Research Fellow, Department of Physics,
Zhejiang University, Peoples Republic of China
QINGMING HE
Associate Professor, School of Nuclear Science and
Technology, Xi’an Jiaotong University,
Peoples Republic of China
TIEJUN ZU
Professor, School of Nuclear Science and Technology,
Xi’an Jiaotong University, Peoples Republic of China
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Contents
Foreword vii
Preface ix
1 Introduction 1
1.1 Background 1
1.2 Someelementsforresonanceself-shielding calculation 3
1.3 Developmentofresonanceself-shieldingcalculation methods 5
1.4 Layoutofthisbook 12
References 13
2 Resonance cross-section processing 17
2.1 Evaluated nucleardatalibrary 17
2.2 Resonancereconstruction 20
2.3 Dopplerbroadening 30
2.4 Unresolvedresonancetreatment 35
2.5 Resonanceintegral table 40
2.6 Resonanceelastic scattering 42
2.7 Goldstein-Cohenfactor 59
References 61
3 Equivalence theory method 63
3.1 Approximatespectrumofhomogeneoussystems 63
3.2 Approximatespectrumofheterogeneoussystems 71
3.3 Theestablishment ofclassicalequivalence theory 75
3.4 Thecalculationoftheeffectivecross-section 85
3.5 Equivalence theoryoflattice systems 90
3.6 Discrepancy analysis 103
References 117
4 Subgroup method 121
4.1 Introduction 121
4.2 Methodologyofsubgroup method 122
4.3 Practicalapplication ofsubgroup method 145
4.4 Improvements ofsubgroup method 156
References 162
v
vi Contents
5 Ultrafine group method 165
5.1 Introduction 165
5.2 Neutronslowing-down equation 166
5.3 Collisionprobability calculationandacceleration 171
5.4 Couplingwithmethodofcharacteristic (MOC) 178
References 192
6 Wavelet expansion method 195
6.1 Introduction 195
6.2 Theoreticalmethod 197
6.3 Numericalresults 210
6.4 Acouplingmethodofsubgroup andwaveletexpansion 220
6.5 Numericalresultsforthecouplingmethod 224
6.6 Conclusions 229
References 230
7 Resonance treatment in Monte Carlo method 233
7.1 Overview 233
7.2 TheMonteCarlomethod 234
7.3 OverviewoftheMonteCarlocodes 239
7.4 Temperaturetreatmentoftheresolvedresonance 246
7.5 Temperaturetreatmentoftheunresolvedresonance 254
7.6 Treatmentoftheresonanceelastic scatteringeffect 254
References 257
8 High-fidelity resonance self-shielding calculation 261
8.1 Overview 261
8.2 Continuous-energy quasi-one-dimensionalslowing-down based
method 262
8.3 Pin-basedpointwiseenergyslowing-downmethod 267
8.4 Global-localself-shielding calculationscheme 275
References 327
9 Resonance treatment for double heterogeneity 331
9.1 Chordlengthsamplingforstochasticmedia 331
9.2 Theanalytical dancofffactorapproach 335
9.3 Thedisadvantage factorapproach 340
9.4 TheSanchez-Pomraningapproach 356
References 388
Index 391
Foreword
Piero Ravetto
Politecnico di Torino, Torino, Italy
The study of resonance effects is one of the classic topics in reactor
physics since the early stages of the history of nuclear energy. Resonance
absorption plays an important role in establishing the equilibrium and sta-
bility of the neutron field and, therefore, it constitutes a central point for
the design and assessment of multiplying systems in both steady-state and
transient conditions. The evaluation of resonance self-shielding is a key
pointinthegenerationofnuclearparametersforreactorsimulationsstarting
from basic nuclear data.
The most referenced classical monographies on the subject are already
more than 40years old. The authors of this book have carried out a
praiseworthyjobtocollectinasinglepieceofliteraturetheenormouswork
that has been carried out over several decades on resonance self-shielding
evaluations, all the way to the most recent advancements. It is particu-
larlyusefulforareaderinterestedinthistopictohavethepossibilitytogain
a broad general view on what has been done in the past and to perceive
how the developments have taken place over the years going through the
successive nine chapters of this book. While the classic approaches are
properly reviewed and well-illustrated, the most recent developments,
consistent with the enormous advances in computing power and capabil-
itiesexperiencedinthelatesttimes,arealtogetherpresentedinanextensive
way.
The recent developments of nuclear reactors and the design of new
advanced systems are posing new challenges to the generation of nuclear
parameters. The authors are particularly careful to deepen the various as-
pects of resonance self-shielding for multiplying systems featuring different
neutron spectra and geometrical characteristics providing a useful physical
insight into the methodologies.
The book includes the relevant contributions in the development of
dedicated numerical methods for resonance self-shielding evaluations given
by the researchers of Xi’an Jaotong University. Of particular interest is the
presentation of the innovative wavelet method, of the treatment of
vii
viii Foreword
resonance effects within the stochastic Monte Carlo approach, of the
techniques for high-fidelity simulations and of double heterogeneity
problems.
Inthisbookthereadercanalsofindausefulpresentationandasynthetic
description of the most popular and widely used codes for neutronic sim-
ulations throughout the world. Many illuminating paradigmatic numerical
examples in various chapters allow the reader to well appreciate the quality
of the performance of the methodologies presented. The bibliographies
supplementing each chapter are really impressive for accuracy and
completeness, always including many classic references together with the
great deal of papers published in journals and in the proceedings of the
numerousspecialists’conferencesandmeetingsheldaroundtheworld.The
style of the presentation is concise and refreshing and it makes the reading
easy and relaxing. The classic material is always presented in a clear and
didactic fashion.
For the characteristics of presenting both the standard information as
wellastherecentadvancements,thisbookshouldconstituteaninteresting,
enjoyable,andinformativereadingforbothstudentsdealingwithadvanced
topics in nuclear engineering education, and young scientists and engineers
who are keen to deepen their knowledge of a crucial theme for the
neutronic simulations of nuclear reactors such as resonance physics and its
modern numerical simulation. However, I must say that thereading of this
book has been quite enjoyable and stimulating also for an old person like
me. I am rather sure that the whole international reactor physics com-
munity will welcome it and it will certainly be well appreciated for many
years to come.
Preface
Driven by the increasing energy demand and carbon footprint worldwide,
nuclear energy is playing a more significant role in the sustainable devel-
opmentofmankind.Alotofinnovativenuclearreactorconceptualdesigns
havebeenproposedforvariousapplicationsinthepastdecades.Thosenew
designs are always featured with complex geometry, multinuclide
compositionandspecialneutronspectrum,whichpresentsmanychallenges
to reactor physics computational methods, especially resonance self-
shielding calculation.
The history of research on the resonance self-shielding calculation dates
back to 1940s when Wigner developed the equivalence theory and then
Carlvik and Roman improved it with the rational approximation method.
After that, various methods have been proposed to deal with different
resonance problems in different applications. Those methods can be
roughly classified into three categories: equivalence theory method, sub-
group (multiband) method, and continuous-energy method. The literature
on those methods distribute in different journal papers and book chapters.
However, there were only two books focusing on this topic. One was
authored by Lawrence Dresner (Resonance Absorption in Nuclear Reactors,
1960781483222) and the other by A.A. Lukyanov (Moderation and Ab-
sorption of Resonance Neutrons, 1974). In recent years, the classical resonance
self-shielding methods, including equivalence theory, subgroup, and
ultrafine-group, have been extended with higher precision and efficiency.
Much work has been done on new topics like high-fidelity resonance
treatment, resonance interference effect, multigroup equivalence, and so
forth.Therefore,wethinkitisgoodtimingtopublishabookthatincludes
those recent progresses in resonance self-shielding computational methods.
Since 2004, the Nuclear Engineering Computational Physics (NECP)
lab at Xi’an Jiaotong University started to work on various numerical
methods for resonance self-shielding. Some traditional methods (equiva-
lencetheorymethod,subgroupmethod,etc.)wererevisitedandimproved.
Some advanced methods were programmed and implemented to reactor
physics codes. Some new methods were proposed and evaluated. Based on
those works, more than ten PhD and master students graduated from the
lab. Each of them made their own contribution to progresses of the reso-
nance self-shielding calculation. The primary motivation of this book is to
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