evaluation of reactivity monitoring techniques …oa.upm.es/35262/1/vicente_becares_palacios.pdf ·...

EVALUATION OF REACTIVITY MONITORING TECHNIQUES AT THE YALINA-BOOSTER SUB-CRITICAL FACILITY

SEPTIEMBRE 2014

Vicente Bécares Palacios

DIRECTOR DE LA TESIS DOCTORAL:

David Villamarín Fernández Óscar L. Cabellos de Fco.

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UNIVERSIDAD POLITÉCNICA DE MADRID

ESCUELA TÉCNICA SUPERIOR DE INGENIEROS INDUSTRIALES

Evaluation of reactivity monitoring techniques at the Yalina – Booster subcritical facility

(Tesis Doctoral)


Licenciado en física

2014

DEPARTAMENTO DE INGENIERÍA NUCLEAR

ESCUELA TÉCNICA SUPERIOR DE INGENIEROS INDUSTRIALES

Evaluation of reactivity monitoring techniques at the Yalina – Booster subcritical facility

Autor:


Licenciado en Física

Director:

David Villamarín Fernández

Doctor en Física

Tutor:

Oscar Luis Cabellos de Francisco

Doctor en Ingeniería Nuclear

2014

Acknowledgements

The rst person I want to thank is my PhD director, David Villamarín. Apart from his guidance andvaluable advice, I would like to stress his patience and encouragement during all these years. I wish tothank as well my co-director at the Department of Nuclear Engineering at the Polytechnical Universityof Madrid, Óscar Luis Cabellos de Francisco, for his valuable comments and, particularly, for his helpwith all the bureaucratic work.

Second, I want to thank all contributors to the success of the EUROTRANS experiments at Yalina-Booster, in particular to Calle Berglöf from KTH, Manuel Fernández-Ordóñez from CIEMAT, and veryspecially, the Yalina team from the JIPNR-Sosny, particularly to Victor Bournos, Yurii Fokov, HannaKikavitskaya, Christina Routkovskaia, Sergei Mazanik and Ivan Seramovich. Apart from their profes-sionalism and hard work during the experiments, I would like to highlight their kindness in making ushaving a agreeable stay at Belarus. I wish to acknowledge as well the institutions that have providednancial support for the work carried out during this PhD thesis: the EU 6th Framework programthrough IP-EUROTRANS Contract No. FI6W-CT2005-516520 and the ENRESA-CIEMAT agreementfor the Transmutation Applied to High Level Radioactive Waste.

Third, I want to thank as well all the colleagues from the Nuclear Innovation Unit and NuclearFission Division at CIEMAT during all these years, for all that I have learned with them and all the helpthey have provided at many instances, which would be endless to detail. In particular, special thanksto those that have revised the manuscript: Manuel Fernández-Ordóñez, Francisco Álvarez-Velarde andJuan Blázquez. I am also indebted to the administrative and services sta at CIEMAT for their usuallyinvisible, albeit indispensable, assistance in many daily tasks. In particular, I would like to mention theDocumentation Service for their help in nding many of the bibliographic references of this thesis.

And nally, last but not least, I would like to acknowledge my family for all their support during allthese years.

1

Contents

Introduction 7

I Motivation 9

1 Radioactive waste and its management 11

1.1 Radioactive waste and its classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 HLW disposal through isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.1 Geological disposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Seabed, ice sheet and extraterrestrial disposal of HLW . . . . . . . . . . . . . . . . 18

1.3 Transmutation of HLW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.1 Charged-particle induced transmutation . . . . . . . . . . . . . . . . . . . . . . . . 191.3.2 Photon-induced transmutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.3 Neutron-induced transmutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.3.1 Accelerator-based neutron sources . . . . . . . . . . . . . . . . . . . . . . 221.3.3.2 Nuclear fusion devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.3.3 Nuclear ssion reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.4 Intercomparison of transmutation technologies and conclusions . . . . . . . . . . . 25

2 Transmutation in Accelerator Driven Systems 29

2.1 Limits of critical reactors and Accelerator Driven Systems . . . . . . . . . . . . . . . . . 292.2 ADS concepts and evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2.1 ADSs in the nuclear fuel cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.2 ADSs outside the nuclear fuel cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 Technical challenges of Accelerator Driven Systems . . . . . . . . . . . . . . . . . . . . . . 35

3 Experimental ADS facilities 39

3.1 Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.1 FEAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.2 MUSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1.3 TRADE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1.4 GUINEVERE / FREYA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1.5 MYRRHA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Former USSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.1 Energy plus Transmutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.2 Yalina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.3 SAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.4 Other proposed experimental ADS facilities . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.1 Kyoto University Critical Assembly (KUCA) . . . . . . . . . . . . . . . . . . . . . 453.3.2 Transmutation Experimental Facility (TEF) at J-PARC . . . . . . . . . . . . . . . 46

3.4 USA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.6 Summary of Accelerator Driven System experiments . . . . . . . . . . . . . . . . . . . . . 48

3

4 CONTENTS

II Theoretical introduction 51

4 Neutron transport theory 53

4.1 The Boltzmann transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Application to neutron transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2.1 The neutron transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.2 Prompt and delayed neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2.3 The diusion approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Solutions of the neutron transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3.1 Stationary case and criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.3.2 Variable separation and spectrum of the transport operator . . . . . . . . . . . . . 67

5 Perturbation theory and the point kinetics model 71

5.1 Adjoint neutron transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1.1 Adjoint operator and adjoint functions . . . . . . . . . . . . . . . . . . . . . . . . 715.1.2 Interpretations of the adjoint ux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3 The point kinetics model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 The Monte Carlo method 83

6.1 Montecarlo application to neutron transport . . . . . . . . . . . . . . . . . . . . . . . . . . 836.2 MCNPX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.2.1 Geometry description and nuclear data in MCNP/MCNPX . . . . . . . . . . . . . 856.2.2 Types of calculations (tallies) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2.3 Criticality calculations (KCODE mode) . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3 Calculation of the eective delayed neutron fraction . . . . . . . . . . . . . . . . . . . . . 886.3.1 Calculation methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.3.1.1 k-eigenvalue methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.3.1.2 Methods based on dierent interpretations of the adjoint ux . . . . . . . 91

6.3.2 Benchmark systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.3.3 Validation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.4 Calculation of the eective mean neutron generation time . . . . . . . . . . . . . . . . . . 97

7 Reactivity monitoring 103

7.1 The current-to-ux technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047.2 Pulsed Neutron Source (PNS) techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.2.1 Response to a single pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057.2.2 Response to a series of pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.3 Beam trip techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097.4 Correction of spatial and energy eects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.4.1 Multi-region kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.4.2 Neutron intergeneration time distribution . . . . . . . . . . . . . . . . . . . . . . . 1127.4.3 Correction factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.4.4 Generalized area-ratio and prompt neutron decay techniques . . . . . . . . . . . . 113

III The Yalina-Booster experiment 115

8 Experimental set-up 117

8.1 The Yalina-Booster facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.2 Data Acquisition System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

9 Dead time 131

9.1 Dead time theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319.2 Determination of dead times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329.3 Correction of dead times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

CONTENTS 5

10 Pulsed Neutron Source (PNS) experiments 14510.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14510.2 Reactivity determination with prompt neutron decay constant technique . . . . . . . . . . 15010.3 Reactivity determination with the area-ratio technique . . . . . . . . . . . . . . . . . . . . 15010.4 Generalized version of the methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

11 Continuous source experiments 16711.1 Steady-state reactivity monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

11.1.1 Prompt decay constant method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16811.1.2 Source-jerk method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

11.2 Reactivity monitoring during system perturbations . . . . . . . . . . . . . . . . . . . . . . 17211.2.1 Fast variation of the system reactivity . . . . . . . . . . . . . . . . . . . . . . . . . 17211.2.2 Fast variation of the neutron source . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Conclusions 179

6 CONTENTS

Introduction

High level radioactive waste produced in nuclear power plants remains highly radiotoxic for long periodsof time (up to millions of years) after reactor discharge. Therefore, to prevent damage to persons or theenvironment, it is necessary to guarantee its ecient isolation from the biosphere during these periods,which represents a considerable challenge. One of the possible alternatives to reduce the required isolationperiod of time consists in the transmutation of the long lived isotopes present in the high level radioactivewaste into shorter lived ones.

One of the possible technologies to achieve this goal is the transmutation in accelerator-driven sub-critical systems (ADS). These devices consist in a subcritical nuclear reactor coupled to an external,accelerator driven neutron source. The advantage of these systems over other technologies is that thesubcriticality condition allows to relax the stability requirements of a critical system, which allows theloading of larger amounts of problematic isotopes, and hence higher transmutation rates, while the sub-critical reactor can produce enough energy to make the whole system exoenergetic.

Nevertheless, the deployment of full scale ADSs still requires several technological developments. Inparticular, a technique to monitor the subcriticality level of the system is required in order to guaranteethat appropriate safety margins are kept. In this context, this thesis is concerned with the investigationof dierent techniques for the monitoring of the subcriticality level in an ADS and the experimentalvalidation of these techniques in a zero power subcritical assembly, namely the Yalina-Booster facility ofthe JIPNR-Sosny of the Belorussian Academy of Sciences.

This thesis is structured in three parts: (I) motivation, (II) theory and (III) application to the Yalina-Booster experiment.

Part I is divided in tree chapters. In chapter 1, an overview of the problem of long lived radioactivewaste is presented, as well as a survey of the state-of-the-art of the dierent alternatives for its man-agement (isolation from the biosphere and transmutation), with a focus on the last one. In particular,section 1.3 comprises a bibliographical survey of the dierent technologies and nuclear reactions that havebeen proposed historically for transmutation of long lived isotopes (accelerated particles, lasers, fusiondevices, etc).

Chapter 2 is focused on ADSs. The principles of ADS operation, as well as an outline of the historicalevolution of the ADS concept and a summary of the main challenges that are still to be addressed by thistechnology, are presented in this chapter. Finally, chapter 3 presents a summary of the main experimentsthat have been carried out up to date in the eld of ADS (FEAT, MUSE, E+T, RACE, Yalina, KUCA,GUINEVERE/FREYA), some projected but nally unrealized (TRADE, SAD), and some in the planningstage (TEF, MYRRHA).

Part II is devoted to the development of the reactor kinetic theory and the theoretical basis forreactivity monitoring techniques, beginning by the basic principles of neutron transport theory. Thus, inchapter 4, it is started from the Boltzmann transport equation and its application to neutron transport.Fundamental concepts are introduced, such as the eective multiplication factor keff (or alternatively,the reactivity ρ) and the source multiplicity ks.

In chapter 5, perturbation theory in neutron transport and the concept of adjoint ux are introduced.These concepts are used to derive the point-kinetics model, upon which the most common reactivitymonitoring techniques are based, and to dene the kinetic parameters: eective delayed neutron fraction(βeff ) and mean neutron generation time (Λeff ).

In chapter 6, the Monte Carlo method for neutron transport applications, and in particular, theMCNPX code, that is used in this thesis, are presented. In addition, it is described how the Monte Carlotechnique can be applied for determining the adjoint-weighted kinetic parameters βeff and Λeff . For thecase of βeff , an extensive validation of a several Monte Carlo calculation techniques has been performed.

Finally, in chapter 7, the main ADS reactivity monitoring techniques are presented. They include thecurrent-to-ux technique, PNS techniques (area-ratio and prompt neutron decay constant) and beam-

7

8 INTRODUCTION

trip techniques (source-jerk and again prompt neutron decay constant). Since both PNS and beam-triptechniques are based on the point kinetics model, they require corrections in order to take into accountthe spatial and energy eects present in actual systems. Existing techniques for this purpose (multi-region kinetics, use of the neutron intergeneration time and correction factors) are discussed, as well asa proposed generalization of the technique based on correction factors (section 7.4.4).

Part III is concerned with the Yalina-Booster experiment. In chapter 8 the Yalina-Booster facility andthe data acquisition system used for the experiments are described. Since the counting rates encounteredduring the measurements were high enough to make dead time eects relevant, chapter 9 is devoted todescribe the methodology that has been applied to determine the magnitude of the dead time and correctits eect.

Last two chapters are concerned with the analysis of the experimental data and the validation ofreactivity monitoring techniques. Two types of experiments were performed, both with a Pulsed NeutronSource (PNS) and a continuous beam with beam-trips. The experimental results of reactivity monitoringwith PNS experiments are presented in chapter 10, while the results of experiments with the quasi-continuous beam (current-to ux and beam trips techniques) are presented in chapter 11.

For the case of the PNS experiments, the area-ratio (Sjöstrand) and the prompt neutron decay tech-niques have been applied to the experimental results. The impact of spatial eects is investigated, andthe performance of the methodologies based on conventional correction factors and the generalized oneintroduced in section 7.4.4 is assessed. This last methodology is also applied to correct the spatial eectsand determine the reactivity of the system with the source-jerk and prompt neutron decay techniquesapplied to the beam trip experiments. Furthermore, the ability of these techniques to measure the re-activity during system transients has been also investigated, both during a fast variation of the systemreactivity (control rod insertion and extraction) and during a fast external source variation (long beamtrip).

Part I

Motivation

9

Chapter 1

Radioactive waste and its management

Abstract - The management of the long lived, high level radioactive waste produced as a result of

nucleoelectric generation (mostly consisting of irradiated fuel) poses a major challenge as its radiotox-

icity remains at unacceptable levels for long periods of time (up to millions of years). This makes

obviously desirable the development of technologies to reduce the time while this waste represents

a risk for the human health or the environment. This can be achieved by transmuting the most

hazardous long lived nuclei into stable or short lived ones. In this chapter, the existing bibliography

on dierent transmutation technologies has been revised. They have been classied according to the

nature of the nuclear reaction involved in the transmutation process (charged particles, photons or

neutrons). Final conclusion is that the most feasible way to reduce the long term radiotoxicity of

high level nuclear waste is the transmutation of the long lived actinides present in the irradiated fuel

into shorter lived ssion products.

Since the beginnings of the industrial revolution, world's energy supply relies on the burning of fossilfuels: coal, oil and natural gas. Thus, in year 2013, 86.7% of the world's primary energy supply camefrom these sources [1]. However, growing concerns about the exhaustion of the economically exploitableresources as well as about the environmental eects of this massive burning of fossil fuels are causing awidespread claim for a change in our energetic model. Among the possible alternatives is nuclear ssionenergy. Nuclear ssion energy was responsible of 4.4% of the world's primary energy production in 2013[1] and it is the main energy source used for electricity generation in some countries like France.

Nuclear ssion, which had remained virtually unsuspected until late 1938, is the last of the energysources presently known to the mankind to have been discovered. Actually, it was discovered after nuclearfusion, as the D-D fusion process had already been experimentally produced by M. L. Oliphant, P. Hartekand E. Rutherford in 1934 [251]. The discovery of nuclear ssion is credited to O. Hahn and F. Strassman,who noticed that the bombardment of uranium by neutrons gave rise to the appearance of isotopes ofbarium as a reaction product [136]. L. Meitner and O. R. Frisch were the rst to correctly interpret thiseect as the splitting of the uranium nucleus into two lighter nuclei in an article published in January1939 [217], where they named the process nuclear ssion. In this same article, they correctly predictedas well the large amounts of energy (about 200 MeV) released in the process. The actual occurrence ofssion was conrmed in other experiments during the following months, such as those of F. Joliot [167]and O. R. Frisch [119] himself.

The next crucial step in the development of nuclear ssion energy was the discovery of the emissionof secondary neutrons in the ssion process, which opened the possibility to achieve self-sustained ssionchain reactions. This discovery was published by H. von Halban, F. Joliot and L. Kowarski as early asof March 1939 [137]. They also were the rst to envisage the concept of a self-sustained ssion reactorcapable to produce energy, for which they led a patent [168]. The rst such self-sustained ssion reactor,the so-called Chicago Pile 1, was built by a team of the University of Chicago led by E. Fermi and L.Szilard which went critical for the rst time on 2 December 1942.

Nuclear ssion technology saw a very rapid development in the subsequent years as the result of wareorts aiming to develop nuclear weapons and after the war, civilian nuclear ssion energy programs werestarted in many countries. Thus, on 20 December 1951 the rst electricity of nuclear origin was producedat the EBR-I reactor at Idaho Falls (USA). This reactor was capable to produce 200 kW of electric power,which were used to light the reactor building. The next step, supplying nuclear-generated electricity tothe grid, was achieved by a 5 MWe experimental reactor at Obninsk (USSR) on 26 June 1954 [187]. Therst truly commercial power plant was the Calder Hall nuclear power plant at Sellaeld (UK), which was

11

12 CHAPTER 1. RADIOACTIVE WASTE AND ITS MANAGEMENT

Table 1.1: Principles of radioactive waste management [157].

No. Title Statement

1 Protection of human healthRadioactive waste shall be managed in such a way as to

secure an acceptable level of protection for humanhealth

2 Protection of the environmentRadioactive waste shall be managed in such a way as to

provide an acceptable level of protection of theenvironment

3Protection beyond national

borders

Radioactive waste shall be managed in such a way as toassure that possible eects on human health and the

environment beyond national borders will be taken intoaccount

4 Protection of future generations

Radioactive waste shall be managed in such a way thatpredicted impacts on the health of future generationswill not be greater than relevant levels of impact that

are acceptable today

5 Burdens on future generationsRadioactive waste shall be managed in such a way thatwill not impose undue burdens on future generations

6 National legal framework

Radioactive waste shall be managed within anappropriate national legal framework including clear

allocation of responsibilities and provision forindependent regulatory functions

7Control of radioactive waste

generationGeneration of radioactive waste shall be kept to the

minimum practicable

8Radioactive waste generation

and managementinterdependencies

Interdependencies among all steps in radioactive wastegeneration and management shall be appropriately

taken into account

9 Safety of facilitiesThe safety of facilities for radioactive waste

management shall be appropriately assured during theirlifetime

ocially opened on 17 October 1956. Since these beginnings, nuclear energy generation has expanded sothat, according to International Atomic Energy Agency [159], at the end of 2013 a total of 434 nuclearpower reactors were in operation worldwide, totaling over 370 GW of electric power.

The generation of nuclear ssion energy is not accomplished without drawbacks, however. Thoselikely causing the largest public concerns are the possible health consequences of radioactive releasesduring nuclear accidents and the radioactive waste generated as the result of nuclear energy production.This work is concerned with the last of these issues, the problem of nuclear waste. The production ofradioactive waste is inherent to the ssion process and since the beginnings of nuclear energy generation,large amounts of radioactive waste have been generated and accumulated. Although radioactive wasteis also produced in other activities, such as medical treatment or scientic research, nuclear energygeneration is by far the main source of radioactive waste. The radioactive nuclides in this waste maypose a risk to the human health and the environment if they are incorporated in enough large quantitiesto the biosphere, hence the need to manage them properly.

The International Atomic Energy Agency (IAEA) has dened the objective of the management ofradioactive waste as to deal with radioactive waste in a manner that protects human health and theenvironment now and in the future without imposing undue burdens on future generations [157]. TheIAEA has also established a set of fundamental principles of radioactive waste management to betterdetail this objective (table 1.1). Anyway, it must be taken into account that under radioactive wastemany dierent materials comprising a wide range of levels of activity and radiotoxicity are comprised,and consequently the options to meet this objective are dierent as well. For this reason, a survey of thedierent types of radioactive waste and their characteristics is presented in next section.

1.1. RADIOACTIVE WASTE AND ITS CLASSIFICATION 13

1.1 Radioactive waste and its classication

Several classications of radioactive waste have been issued by dierent organizations. The IAEA classiesnuclear waste into Exempt Waste (EW), Low and Intermediate Level Waste (LILW) and High Level Waste(HLW) [156]. The characteristics of each these categories are listed in table 1.2a. The EU commission hasrecommended a similar classication [109] that has been adopted by several countries including Spain.This classication comprises three categories of nuclear waste: Transition Radioactive Waste, Low andIntermediate Level Waste (LILW, again subdivided into long lived and short lived) and High Level Waste(HLW). Transition Radioactive Waste is dened as the type of radioactive waste (mainly from medicalorigin) which will decay within the period of temporary storage and may then be suitable for managementoutside of the regulatory control system subject to compliance with clearance levels. The denitions ofthe other categories are the same of that in the IAEA classication, the only dierence being that nodecay heat generation limit is set to distinguish LILW and HLW, but it is rather left as site-specic.

In the USA, the classication of radioactive waste is accomplished according to the Nuclear WastePolicy Act of 1982, which establishes four categories of nuclear waste: high level waste, spent nuclearfuel, transuranic waste and low-level radioactive waste (table 1.2b). The rst three of these categories areclassed as HLW category in the IAEA classication while low-level radioactive waste roughly correspondto LILW.

Most of the radioactive waste produced as a result of nuclear energy generation is classied as LILW.To illustrate the order of magnitude of the amounts of HLW and LILW into consideration, the case ofSpain is presented, according to the forecast of the Spanish 6th General Plan of Radioactive Waste [221].It was considered there an operational life of 40 years for all the eight operating LWRs in that date(7716 MWe altogether), no new nuclear plant built and complete dismantling after decommissioning.Furthermore, there were another two shutdown nuclear power plants in process of decommissioning, a160 MWe PWR (José Cabrera, operated between 1968 and 2006) and a 480 MWe gas-graphite reactor(Vandellós-I, operated between 1972 and 1989). Spanish nuclear power plants operate all in an openfuel cycle; only the fuel of Vandellós-I and some of the Garoña power plants have been reprocessedand vitried. With these premises, the total volume of radioactive waste generated will be 189,100 m3

including the small amounts generated in activities not related with nuclear power generation. LILWaccounted for over 93% of this volume (176,300 m3) while HLW represented less than 7% (12,800 m3,about the volume of ve Olympic-size swimming pools). The composition of these wastes is furtherdetailed in gure 1.1.

LILW is produced in nuclear facilities during operation and decommissioning, and comprises mainlyresiduals from the reactor water treatment system (resins, lters, sludge and evaporator concentrates) andfrom the laboratories (e.g. contaminated protective clothing and cleaning materials), as well as activatedstructural materials and contaminated tools and pieces of equipment. Other parts of the fuel cycle alsoproduce LILW. Radioactive waste from activities other than nuclear energy production is essentiallyLILW as well. Nevertheless, LIWL waste can be stored in surface or near surface facilities and mostcountries have already implemented schemes to manage LILW radioactive waste, so its management anddisposal is not considered to pose a major problem. In Spain, a LILW repository is operating since 2001at the El Cabril site in the province of Córdoba [2].

On the contrary, no universally accepted solution has been found yet for the management of HLW.HLW is generated almost exclusively in nuclear power generation and consists mostly of irradiated nuclearfuel. As an example, typical composition of irradiated PWR fuel is presented in gure 1.2. It can beobserved that over 90% of the mass of irradiated fuel is composed by U-238 and U-235 that remains non-ssioned. The other constituents of irradiated fuel are ssion products and other actinides originated byneutron capture in the U-235 and U-238 (U-236, Np, Pu, Am, Cm). These ssion products and actinidesare the responsible of the high radiotoxicity of irradiated fuel.

Fission products are the nuclei resulting from the ssion process and all the isotopes originated fromthem either by radioactive decay or neutron capture. Since the number of neutrons in stable nuclei risesfaster than the number of protons, ssion products tend to be neutron rich nuclei and thus many of themare unstable. However, most of them are short lived and thus ssion products are the main contributorsto the radiotoxicity of the irradiated fuel only during a few decades after fuel discharge from the reactor.In gure 1.3, it can be seen that the radiotoxicity due to ssion products falls below the level of naturaluranium about 300 years after discharge. The most relevant isotopes contributing to radiotoxicity duringthe rst decades after irradiation are Cs-137 (half-life: 30 years) and Sr-90 (29 years), while I-129 (16×106

years) and Tc-99 (2 × 105 years) are the main contributors to the ssion products radiotoxicity in thevery long term.


Table 1.2: IAEA and NWPA classications of nuclear waste

(a) Classication of nuclear waste, according to IAEA

Waste class Description Disposal option

Exempt Waste (EW)Activity levels so that dose to the publicshould not exceed 0.1 mSv in any case

No radiologicalrestrictions

Low andIntermediateLevel Waste(LILW)

Short Lived(LILW-SL)

Activity levelsexceeding those forEW and decay heatgeneration below

2 kW/m3

Restricted long livedradionuclidesconcentrations

Near surface orgeological disposal

Long Lived(LILW-LL)

Long livedradionuclideconcentrations

exceeding limitationsfor LILW-SL

Geological disposal

High Level Waste (HLW)Decay heat generation above 2 kW/m3 and

long lived radionuclide concentrationsexceeding limitations for LILW-SL

Geological disposal

(b) NWPA classes of nuclear waste

Waste class Denition

Spent nuclear fuelFuel that has been withdrawn from a nuclear reactor following irradi-ation, the constituent elements of which have not been separated byreprocessing

High level waste

(A) the highly radioactive material resulting from the reprocessing ofspent nuclear fuel, including liquid waste produced directly in reprocess-ing and any solid material derived from such liquid waste that containsssion products in sucient concentrations; and (B) other highly ra-dioactive material that the Commission, consistent with existing law,determines by rule requires permanent isolation

Transuranic wasteMaterial contaminated with elements having atomic numbers greaterthan uranium (92 protons) in concentrations greater than 100nanocuries/gram

Low level waste

Radioactive material that (A) is not high level radioactive waste, spentnuclear fuel, transuranic waste, or by-product material [...]; and (B) theCommission, consistent with existing law, classies as low-level radioac-tive waste


Figure 1.2: Typical composition of irradiated PWR fuel. Source: elaborated from data supplied by F.Álvarez Velarde (CIEMAT) calculated with the ORIGEN 2.2 code.

Figure 1.3: Evolution of the radiotoxicity due to actinides and ssion products in the case of the irradiatedLWR fuel presented in gure 1.2. The radiotoxiciy level of natural uranium is also plotted for comparison.Source: same as gure 1.2

1.2. HLW DISPOSAL THROUGH ISOLATION 17

is the most widely accepted alternative for the long term isolation of irradiated fuel, but other disposalalternatives, such as ocean disposal or ice sheet disposal have also been investigated. Extraterrestrialdisposal may represent an ultimate solution for the problem of HLW disposal, but huge costs and therisk of accidents make it unrealistic with present day technology. A comprehensive, although old, reviewof all these options is presented in [310].

The period of time during which HLW is required to be conned has been a matter of debate. In1995, the U. S. National Academy of Sciences stated in a study that peak risks might occur tens tohundreds of thousands of years or even farther into the future [236]. Based on these conclusions, the U.S. Environmental Protection Agency (EPA) established in 2008 a compliance period of 1 million yearsfor a HLW repository, extending its previous gure of 10,000 years [358].

1.2.1 Geological disposal

Geological disposal of high level radioactive waste consists in burying it in a repository drilled in a deepstable geological formation. This concept has been considered since the 1950s: a 1957 report of the U.S. Academy of Sciences [147] already asserted that radioactive waste could be disposed of safely in avariety of ways and at a large number of sites in the United States.

Present-day geological disposal is based in the concept of defense in depth, in which multiple barriersare placed between the HLW and the biosphere in order to minimize the risk for the radionuclides in thespent fuel to leak into the environment. The main barrier, which must last for the long isolation timesrequired until the radionuclides have decayed to a non hazardous level, is the natural geological barrier,i.e. the rock in which the HLW is buried in.

This barrier is enhanced by several engineered barriers. The rst barrier will be the waste form itself,which must be capable to retain its constituent radionuclides during long periods of time. The most usualwaste forms are a UO2 or PuO2 matrix, for the case of spent fuel, or borosilicate glass, for the case ofreprocessing waste. In any of these forms, the HLW will be then introduced in metallic canisters, thatwill constitute the second barrier. The third barrier will be an intermediate layer of a sealing materialbetween the canisters and the host rock. This material has to be able to accommodate the mechanicalstresses that may occur due to tectonic movements, as well as having low permeability and self sealingcapacity, to prevent ground water to leak into the HLW, and good thermal conductivity, to dissipate thedecay heat. The preferred material for this layer in most repository concepts is bentonite clay.

The host rock will be the nal barrier. Requested characteristics are long term mechanical andchemical stability, impermeability and sealing properties to prevent migration of the radionuclides in theHLW to the environment if the other barriers eventually fail. Several rock formations have been proposedas candidates to host a deep geological repository. They include granites and clay formations, salt domesand volcanic tus.

To test the performance of deep geological disposal, several underground research laboratories havebeen in operation in several countries [158]. The most relevant ones are the HADES UndergroundResearch Facility at Mol, Belgium; the Lac du Bonet Underground Research Laboratory in Canada;the Grimsel Test Site and the Mont Terri Rock Laboratory, both in Switzerland; the Äspö Hard Rocklaboratory, in Sweden and the Meuse/Haute Marne Underground Research Laboratory, in France. Othertwo underground research laboratories are being built in Horonobe and Mizuname, in Japan. Apart fromthese purpose built laboratories, a large number of additional test have been performed in other facilitiessuch as tunnels or mines.

The condence in deep geological disposal is backed by the existence of natural analogues [291]. Oneof the most cited ones is the so called Oklo natural reactor, located in Gabon. There it was discoveredthat the isotopic composition of some uranium deposits was lower in U-235, a fact that is explained ifthe deposits operated as natural ssion reactors some two billion of years ago, when the concentration ofU-235 was high enough to allow a sustained ssion chain reaction moderated by natural water. (noticethat the half life of U-235 is much shorter than the half life of U-238). The reaction products have sincebeen retained on site.

Despite of all these facts, a universal consensus about the safety of geological disposal has not beenreached (e. g. [379]). Critics with deep geological disposal of HLW claim that the behavior of a repositoryover the long isolation period required is still uncertain and that the safe connement of HLW is thereforenot guaranteed. The disposal of high level nuclear waste has also been criticized on the basis of ethicalgrounds as well (e.g. [319]), arguing that high level nuclear waste disposal is ethically unacceptable dueto the burden it can pose for future generations. Inadvertent human intrusion in a distant future whenknowledge about the existence of the repository may be lost has also been a matter of some concerns.


Furthermore, possible future uses of irradiated nuclear fuel (e.g. as fuel for breeder reactors) makequestionable the convenience of nal, irretrievable disposal.

Finally, repositories have to face a strong local and public contest. Hence, few countries have decidedto implement deep geological storage and, as of 2014, there is no operational deep geological repositoryfor HLW. Only the USA had one repository under construction, now frozen, and other two countries,Sweden and Finland, project to build deep geological repositories in the short term.

In the USA, the Nuclear Waste Policy Act of 1982 established deep geological disposal as the nalsolution for the US high level radioactive waste. In 1987 the Yucca Mountain site in the state of Nevadawas selected for a deep geological repository [107]. The project envisaged that the high level radioactivewaste was to be stored in steel canisters and placed in a tunnel complex ∼ 300 m below the surface. Oncelled, the repository would be monitored for at least 50 years and then sealed. Although the repositorywas approved in 2002, the project was heavily contested and funding for the project was eventuallysuspended in 2011.

In Finland, the Olkiluoto site was selected in 2001 to host a deep geological repository [4]. Anunderground rock characterization facility called ONKALO [12] began to be built at the site in 2004. InSweden, the Forsmark site was selected in 2009 to host the Swedish repository and a building licenceapplication was submitted to the Swedish Radiation Safety Autority in 2011 [5]. The design of boththe Swedish and Finish repositories is based on the KBS-3 concept [268], developed by the SwedishNuclear Fuel and Waste Management Company (SKB). This design contemplates the encapsulation ofthe irradiated fuel elements in a double capsule, the innermost made of iron and the outermost of cooper,to better resist corrosion; and the nal deposit inside a bentonite layer placed in a hole in a cave drilledat ∼ 500 m of depth in granite.

1.2.2 Seabed, ice sheet and extraterrestrial disposal of HLW

Disposal of high-level waste in the seabed has been proposed as an alternative to deep geological reposi-tories. Here, it is worthwhile to remark that although low and intermediate level waste has been dumpedinto the sea since as early as 1946 and until 1982 [76, 85], no HLW has ever been dumped into the sea. Itmust be noticed that while LILW was simply dumped in metal or concrete canisters in the hope that evenif they ultimately fail the large amounts of ocean water will guarantee the dilution of the radioactivity tolevels low enough not pose a risk to the human health or the environment, the much higher radiotoxicityof HLW make necessary more durable isolation strategies. Hence, sea disposal of HLW contemplates itsburial in some form under the seabed.

Seabed disposal has some disadvantages with respect to geological disposal that cause that less at-tention has been paid to it. First, waste retrievability would be very dicult should it be required at alater time. Second, the dumping of HLW into the sea is prohibited by several international conventions,including the UN Convention on the Law of Sea (UNCLOS), the Convention on the Prevention of MarinePollution by Dumping of Wastes and Other Matter (also referred as the London Dumping Convention,or LDC) and the Convention for the Protection of the Marine Environment of the North-East Atlantic(OSPAR) [209]. It may be argued, though, that these conventions do not apply to seabed disposalconcepts since they ban dumping HLW into the sea, not burying it under the seabed.

In spite of these facts, and the considerable public opposition it faces, several concepts for burying HLWinto the seabed have been investigated. One of the proposed concepts consists in the burial of radioactivewaste into stable clay formations in the seabed [149, 150]. A research program was initiated at the SandiaNational Laboratory of the USA in 1974 which lasted until 1986, when the U.S. Department of Energywithdrew funding to concentrate on geological disposal [228]. The Nuclear Energy Agency of the OECDalso supported a Seabed Working Group to coordinate international R&D activities in the eld between1976 and 1987 [182]. These studies contemplate either storing waste canisters in holes previously drilledin the seabed or dropping the waste from ships inside special canisters called penetrators, designed tobury themselves to depths of tens of meters into the seabed.

Other concept for disposal of HLW contemplates its burial into subduction faults [72]. The appealingidea behind this concept is that the subduction process would eventually send the waste deep into theEarth's mantle. Nevertheless, there are many uncertainties concerning the long term behavior of wastedisposed through this option, including the risk that they are released again into the biosphere by thevolcanoes that are associated with subduction zones. A similar concept consists in placing the high levelwaste canisters in high rate sedimentation rate areas, specically in the deltas around the mouths of largerivers, where they would become buried at increasing depths [310].

Ice sheet disposal in Greenland or Antarctica has also been proposed [269, 310]. Under this scheme,

1.3. TRANSMUTATION OF HLW 19

canisters would be left in the surface or in shallow holes in the ice. From there, because of the decay heat,they would melt the surrounding ice and become progressively buried at increasing depths. One variantof this method that allows for retrievability during a period of time after disposal consists in anchoringthe canisters to a surface plate. Retrievability would thus be possible until the anchor plates are nallycovered by the accumulating snow and buried themselves. Another variant that would also allow fortemporary retrievability considers simply leaving the canisters in a surface facility, which eventuallywould be also covered by snow. Anyway, apart from other objections, international law make ice sheetdisposal dicult; in the case of Antarctica, the disposal of radioactive waste is specically banned underthe terms of the article V of the 1959 Antarctic Treaty.

A nal alternative is extraterrestrial disposal. The U.S. NASA has sponsored a number of reportsexploring several possibilities [285, 350], both for the nal destination of the wastes (Earth orbit, solarorbit, the Sun or the extrasolar space) and the vehicle to be used (expendable launchers, space shuttlesor even railguns). However, high costs and the risk of accidents have cause that little attention has beenpaid to this alternative.

1.3 Transmutation of HLW

Given the concerns and public opposition against geological disposal of HLW described in the previoussection, the interest in transmutation technologies seems obvious. In the context of radioactive wastemanagement, transmutation refers to the the process of conversion of long lived radioactive nuclides intoshorter lived or stable ones, thus removing or reducing the requirements for long term storage.

Earliest concepts of transmuting systems trace back to the 1960s; comprehensive summaries of earlyHLW transmutation studies can be found in [97] and in [310]. Nevertheless, institutional reports of theseyears tended to be skeptical about the advantages of transmutation of HLW over geological disposal. Forinstance, two Oak Ridge National Laboratory reports [98, 116] concluded that although the transmutationof actinides and certain ssion products was likely to be feasible, no clear incentives were found takinginto account the economical costs and the short term radiological risks associated to the partitioning andtransmutation process, that could oset the long term benets of transmutation. Other studies of thoseyears reached similar conclusions (see [247], annex C).

However, costs, uncertainties and the little public acceptance of geological disposal of HLW has sincecaused an increased interest in transmutation technologies and several research programs have beenlaunched in dierent countries. The 1980s and especially the 1990s constitute a turning point in thiseld. Thus, for instance, already in 1988, the Atomic Energy Commission of Japan launched the OMEGAprogram (Options Making Extra Gains from Actinides and ssion products), in which several Japaneseinstitutions and research institutes took part and whose aim was to explore the requested technologiesfor the partition and transmutation of high level nuclear waste [215, 227]. In France, the 1991 lawabout the management of radioactive waste (law 91-1381, 30 December 1991) encouraged the researchon partition and transmutation of long lived radionuclides and hence the SPIN program (SeParation-INcineration) [37] was launched to accomplish this goal. The European Commission has also supportedseveral R&D programs on partition and transmutation (P&T) of HLW since the the Fourth FrameworkProgram, started in 1994 [185, 61]. Finally, although the USA have been less enthusiastic to P&T ofHLW, possibly because of their once-through fuel cycle policy and their commitment to built the YuccaMountain repository, concerns about reaching the maximum capacity of the repository and other reasonshave caused a rising interest in P&T of HLW since the 1990s. This interest has materialized in severalR&D programs including the Accelerator Transmutation of Waste (ATW) Project [54, 356] and theAdvanced Fuel Cycle Initiative (AFCI) [357].

The transmutation of the long lived radioactive nuclei of HLW into stable or shorter lived nuclei canbe achieved in principle through a number or nuclear reactions. According to the particle that originatesthe reaction, they can be classed into charged-particle induced, photon-induced or neutron-induced. Adiagram of these reactions and the technologies available to produce them is presented in gure 1.4. Theremaining part of this chapter is dedicated to a survey of these technologies.

1.3.1 Charged-particle induced transmutation

The rst possibility to achieve the transmutation of long lived nuclei are charged particles reactions. Thispossibility is not new, in fact, it traces its roots back to 1919, when Rutherford produced the rst articialtransmutation of a chemical element, using an α particle source to transmute nitrogen into oxygen. Thedevelopment of particle accelerators allowed for more intense particle sources and could be used for


Figure 1.4: Summary of the dierent nuclear reactions that have been proposed or can be used for HLWtransmutation purposes

transmutation and production of articial elements at larger scales. For instance, it was in cyclotronsat the University of California in Berkeley where the rst articial elements (technetium, neptunium,plutonium, etc) where produced in the 1930s and 40s. Today, cyclotrons and LINACs are widely usedto produce proton-rich isotopes for medical purposes. In a similar way, the usage of accelerated chargedparticles may be appealing at a rst glance for ssion product transmutation: since they tend to beneutron-rich isotopes, reactions with protons seem the most obvious way to transmute them into stableisotopes.

Nevertheless, the application of accelerators in this way for the transmutation of nuclear waste inindustrial scale is hampered by the energy requirements to accelerate the particles, that can oset thebenets of nuclear energy generation. A possibility to overcome at least partially this problem is the useof exothermic reactions, so that reaction energy can be recovered to compensate for the energy spentin accelerating the particles. With this consideration, the only charged-particle induced reactions thathas aroused a certain interest for HLW transmutation purposes are spallation reactions. These reactionsare originated by high energy particles (most commonly, protons) impinging over heavy nuclei. In theprocess, the initial nuclei is transformed into a lighter ones, with the emission of large numbers of lightparticles, neutrons among them. A brief survey on spallation reactions can be found in [58].

Spallation reactions were discovered in 1947 in the Berkeley 184-inch cyclotron in experiments thatinvolved the bombardment of heavy nuclei (e.g. arsenic, cooper, antimony, uranium) with deuterons andα particles. There, reactions were observed in which large numbers of light particles were ejected andthe product nuclei had considerable lower masses than the target nuclei [99, 151, 206, 220, 243, 313]. W.H. Sullivan named these reactions spallation reactions [312], a term which is borrowed from mechanicalengineering where it describes the ejection of small fragments from a material when it is impacted by aprojectile. In fact, spallation reactions occur naturally in the Earth when cosmic rays interact with thenuclei of the atmosphere, although a detailed characterization of these reactions was only possible whenintense particle beams from accelerators became available.

The currently accepted mechanism to describe the spallation process was proposed by R. Serberalready in 1947 [315]. It considers a two stage mechanism consisting of an intranuclear cascade followedby evaporation. The rst stage, the intranuclear cascade, consists of individual collisions of the incidentparticle with the nucleons, ejecting some of them if the transferred energy is high enough. This occursbecause the wavelength of the high energy incident particle is less than the radius of the nuclei. Thesecond stage, evaporation, consists in the de-excitation of the nucleus, left in an excited state after thecascade. The excess energy is distributed between the nucleons and some of the periphery may be ejected.

Some studies have been published assessing the potential of spallation reactions for the transmutationof several problematic nuclides present in HLW, both ssion products (Sr-90, Tc-99, Cs-137) and actinides(Am-241, Am-243, Cm-244) [172, 174, 191, 377]. The general conclusion that can be gathered fromthese studies is that the positive energy balance requirement is dicult to be met taking into account


conversion losses at dierent stages and consequently some external energy supply will always be required.Furthermore, the accelerator beam intensity requirements to achieve practical transmutation rates arechallenging. Hence, the direct use of spallation reactions for transmutation of HLW is presently littleconsidered.

1.3.2 Photon-induced transmutation

Nuclear reactions induced by high energy photons (γ-ray range) have also aroused some interest for thetransmutation of certain nuclides. Actinides can be transmuted into ssion products through photossionand long lived ssion products can be transmuted into stable or shorter lived nuclei trough (γ, n) or otherphotonuclear reactions.

The required γ photons to produce these reactions can be obtained by several techniques. A rstpossibility is through Compton backscattering. Some experiments concerning the transmutation of certainnuclides using Compton backscattering photons have been performed at the NewSUBARU facility inJapan [154, 205, 223]. They have consisted in transmutation rate and eciency measurements withtargets of stable Au-197 and I-127 targets. Similar experiments have been proposed as well for the SLEGSlaser Compton scattering facility at the Shanghai Synchrotron Research Facility (SSRF) in China [83, 84].Nevertheless, these techniques, as it was the case of transmutation with charged particles, require energyto generate the photons, a challenge that must be addressed for industrial transmutation of HLW.

Another possibility to obtain γ photons is through high intensity lasers. These techniques haveattracted considerable attention in recent years due to the development of very high intensity short-pulse lasers. Although these lasers operate in the optical to infrared range, they can be focused ontoa target of a high atomic number (tantalum or tungsten are commonly used) to generate a relativisticelectron plasma. When these electrons are stopped by the heavy atoms of the target, they in turn cangenerate Bremsstrahlung photons of energy high enough to induce photonuclear reactions. Comparedwith the Compton backscattering technique described above, short pulse laser based techniques have theadvantage that they allow obtaining neutrons with relatively small and cheap (tabletop) facilities. Theyhave the disadvantage, however, that Bremsstrahlung photons have broader energy spectra than Comptonbackscattering photons, hence worse matching the reaction resonances and providing lower transmutationeciencies. Furthermore, the application in an industrial scale of these techniques is severely limited bythe low energy eciency of laser systems. Anyway, experimental results on the photo-transmutation ofI-129 have been obtained both with the JeTi laser at the Jena University (Germany) and the VULCANlaser at the Rutherford Appleton Laboratory (UK) [202, 210] and theoretical works about the viabilityof similar transmutation of other isotopes such as Cs-135 [341], Zr-93 [294] or Sr-90 [293] have also beenpublished. Laser induced photo-ssion of actinides (U-238 and Th-232) has also been experimentallyobserved in several facilities [95, 201, 311]. Finally, laser induced transmutation is one of the activitiesproposed for the European ELI project [20].

As a nal remark, it is also possible to excite the nuclei to higher energy isomer states with γ-rayphotons of lower energies than the previous techniques, even with photons in the X-ray range. This processmight nd application for HLW transmutation purposes if long lived radioactive nuclei have isomer stateswith, for instance, shorter beta-decay lives than the initial state. The advantage of this technique isthat it requires less energetic photons that may be obtained from a wider range of sources, such assynchrotrons or X-ray free electron lasers (FELs). Nevertheless, the application of this technique fortransmutation is hampered by many factors (photon losses in other processes, too wide photon spectrato match resonances) [299] and, in any case, it is also unlike to become practical for industrial scaletransmutation once more because of the energy requirements of the photon source and because it isnot clear if the most hazardous radionuclides have an structure of isomer states adequate to apply thistechnique.

1.3.3 Neutron-induced transmutation

The third possibility for transmutation of radioactive isotopes is through neutron induced reactions.Neutron induced transmutation is well suited for the case of actinides because they have large thermal orfast ssion cross sections and hence they can be transmuted into ssion products through neutron inducedssion. Neutron induced transmutation of long lived ssion products is in principle more problematicsince they are neutron-rich nuclei and hence they are unlike to be transmutted into more stable nucleiby simple neutron capture, but there are other possibilities such as (n, xn) reactions or neutron captureto very short lived nuclei. For instance, Tc-99 can be transmuted into stable Ru-100 through the chain:


10n + 99

43Tc −→ 100

43Tc −−−−−−→

β− 15.8 s

10044Ru

Any transmutation scheme based on neutron reactions will be built around an intense neutron source,which can produce neutrons through several technologies. At present, intense neutron sources make useof proton or heavier particle reactions, nuclear fusion and nally, nuclear ssion itself. The applicationand the potential of these technologies for transmutation purposes is discussed next.

1.3.3.1 Accelerator-based neutron sources

There are several nuclear reactions that allow obtaining neutrons from accelerated particles (see forinstance [219]). Anyway, the ones providing the largest neutron yields and hence the most usually usedin high intensity sources are D-Be and D-Li stripping reactions, (γ, n) reactions and spallation.

A rst possibility to obtain neutrons from accelerated particles is from D-Be and D-Li strippingreactions. Stripping reactions are nuclear reactions in which the incident nucleus transfers a nucleon to atarget nucleus, while the remaining of the incident nucleus carries the balance of energy and momentum.In the D-Be and D-Li process, the deuteron loses the proton to the Be or Li nucleus leaving behind aneutron. The emergence of neutrons from deuteron bombardment of lithium and beryllium targets wasrst observed in 1933 at the Kellogg Radiation Laboratory using deuterons of energies up to 0.9 MeVfrom an electrostatic accelerator [96].

Since the required energy of the deuterons is not very high (typically it ranges from a few MeV to tensof MeV), D-Li and D-Be neutron sources can be built from small-size cyclotrons or LINACs and theyhave been proposed as a compact, inexpensive way to replace research reactors to produce neutrons thatcan be hosted in a university or hospital without requiring large scale facilities. At this energy range itis also possible to obtain neutrons from several reactions involving dierent incident particles (protons,deuterons, tritons) and target materials, but D-Li and D-Be provide the highest neutron yields. For thisreason, several facilities using D-Be or D-Li reactions for neutron production have been built in the lastyears or are being built currently. They include the LENS facility at the Indiana University (USA), usinga 13 MeV, 2.5 mA LINAC [41, 204], the SARAF facility at the Soreq Nuclear Research Center (Israel),using a 40 MeV, 2 mA LINAC [211] and the projected NFS facility at SPIRAL2 (France) with a 40 MeV,5 mA LINAC [203]. In addition, D-Li and D-Be reactions are of interest in the eld of fusion materialstesting, since the energy ranges of neutrons obtained in these reactions are similar to those of fusionreactions. The International Fusion Materials Irradiation Facility (IFMIF) [163, 264] is a proposed largescale facility for fusion materials testing based on D-Li reactions.

Other possibility to obtain neutrons from accelerated particles is from (γ, n) reactions with pho-toneutrons produced by the Bremsstrahlung photons obtained by stopping accelerated electron beamsin metallic targets. This process has already been mentioned in section 1.3.2. (γ, n) reactions have beenknown for a long time, having been discovered by J. Chadwick and M. Goldhaber in 1934 in the reaction21H (γ, n)

11H that they observed when they irradiated a tank of deuterium with γ-rays from a radioactive

source [81]. Neutron sources based on photoneutron reactions have been in operation from the 1950s,such as the 100 MeV betatron at the Knolls Atomic Power Laboratory [30, 387], coupled to a naturaluranium target, and the 45 MeV Harwell linear electron accelerator in the UK, which used a mercurytarget [207, 275]. At present, several intense neutron sources coupling electron LINACs to photoneutronproducing targets are in operation in the world since some decades ago. Their main role is neutron crosssections measurements. They include the GELINA facility at the European Commission's Joint ResearchCenter in Geel (Belgium); the ORELA facility at the Oak Ridge National Laboratory and the GaerttnerLINAC laboratory at the Rensselaer Polytechnic Institute (both in the USA) and the electron LINACat the Kyoto University Research Reactor Institute (Japan). Other two photoneutron sources have beenput into operation in the last years, namely the Pohang Neutron Facility (Korea) [179] in 1999 and theIntense Resonance Neutron Source (IREN) at the Frank Laboratory of Neutron Physics of the JINR inDubna (Russia) [21, 52] in 2009.

Finally, a third possibility to obtain neutrons from accelerated particle beams are spallation reactions,which were already described in section 1.3.1. There it was stated that spallation reactions are producedby high energy particles (100 MeV - 10 GeV range), usually protons, impinging on heavy nuclei. Theprocess is accompanied by the emission of a large number of light particles, neutrons among them, ina number that depends on the energy of the incident proton beam and the composition of the target[38, 219]. Concerning the dependence of the neutron yields with the incident proton energy, it hasbeen found that the number of neutrons per incident proton increases with increasing proton energy.Nevertheless, above about 1 GeV, additional particles begin to arise as the result of the nucleon-nucleon


Table 1.3: Intercomparison of neutron producing reactions with accelerated particles. Source: [219]. Note:spallation neutron have some dependence on target dimensions. Data presented here were measured withtargets of 10.2 cm dia × 61 cm (Sn, Pb, U) and 20.3 cm dia × 61 cm (Pb). See the reference for furtherdetails.

Reaction Incidentparticleenergy

Neutronsper incidentparticle

Neutrons perMeV of incidentparticle energy

9Be (D, n) 15 MeV 1.2× 10−2 8× 10−4

Photoneutrons (W target) 35 MeV 1.7× 10−2 4.9× 10−4

Spallation in Sn 1 GeV ∼ 10 0.01

Spallation in Pb 1 GeV ∼ 20 0.02

Spallation in 238U 1 GeV ∼ 40 0.04

collisions (pions). This fact limits the eciency of spallation reactions for neutron production and hencededicated spallation neutron sources use incident particles with energies of up to a few GeV.

The other aspect that aects the spallation neutron yield is the target composition. It has been foundthat the number of neutrons by incident proton increases with the atomic number of the target material.Hence, heavy materials are preferred for the target, although considerations concerning material damage,heat transfer and activation in the target have also to be taken into account. Hence, several materialsincluding Hg, W, Ta, Sn, Pb (both in solid and in molten states) and U have been used or proposed tobe used as spallation targets.

Spallation reactions are used as neutron sources in several facilities around the world since the 1980s.Early facilities were built making the most of already existing proton accelerators. This is the case ofthe n-TOF facility at CERN, the Pulsed Neutron Source of the Moscow Meson Factory (Russia), theSINQ neutron source at the Paul Scherrer Institute (Switzerland), the LANSCE neutron source at the LosAlamos National Laboratory (USA), and the ISIS neutron source at the Rutherford Appleton Laboratory(United Kingdom). This last has been the most intense pulsed neutron source of the world for manyyears. In the recent years, a second generation of purpose-designed spallation sources are being built.They include the Spallation Neutron Source (SNS) at the Oak Ridge National Laboratory (USA), thespallation neutron source at J-PARC (Japan) and the future European Spallation Source (ESS) at Lund(Sweden).

To assess the suitability of these reactions for HLW transmutation purposes in an industrial scale, themost relevant issue to be taken into account is that since a particle accelerator is an energy consumingdevice, using accelerators for transmutation of HLW requires an external supply of energy; if the requiredsupply is too large it can make transmutation economically unfeasible. The most relevant parameterthat aects the energy requirements of a neutron source is the number of neutrons produced in thereaction per unit of energy of the accelerated particle. An intercomparison of the neutron yield per unitof energy of the incident particle of the reactions described above can be found in table 1.3. It can beobserved that the neutron yield per unit of energy of the incident particle of spallation reactions is severalorders magnitude larger than for D-Be and photoneutron reactions. Furthermore, as it has been alreadysaid in section 1.3.1, spallation reactions are exothermic processes so the energy released can be used tocompensate, at least partly, for the energy required to accelerate the protons. Hence spallation is theonly reaction that has deserved certain interest for transmutation purposes. Nevertheless, some studies[172, 173, 174] indicate that transmutation of certain isotopes (Cs-137, Tc-99) by spallation neutronswould require large amounts of energy that would likely make it unpractical.

1.3.3.2 Nuclear fusion devices

The application of fusion devices to transmute both actinides and ssion products has been in consider-ation since the 1970s [256, 310, 380]. These isotopes are usually considered to be introduced as a blanketin magnetic or inertial connement devices.

Fusion devices have some appealing advantages for the transmutation of ssion products. First, thehigh neutron uxes achievable allow for high transmutation rates. Second, at fusion neutron energies (14MeV for the D-T reaction), transmutation of ssion products is possible through several reactions, such


as (n, γ), (n, 2n), (n, 3n), (n, p), (n, d) or (n, α), while for lower energy neutrons, such as ssion neutrons,the only relevant process is (n, γ). Recent reviews of the potential of transmutation of certain ssionproducts (Cs-137, I-129, Sn-126, Zr-93) in fusion devices are [296, 297, 298, 337, 343, 344].

Actinides can also be transmuted in fusion devices. As these elements are transmuted by ssion, asstated above, the introduction of an actinide blanket in a fusion reactor has the eect of turning it into ahybrid ssion-fusion reactor, sometimes also referred as fusion-driven systems (FDS). This fact constitutesindeed a motivation for the study of transmutation in fusion devices because energy gain requirements offusion-driven transmutation devices are less challenging than pure energy-producing devices and hencethey may constitute an intermediate step towards pure fusion power plants. The main disadvantagesof fusion driven transmutation are the high complexity and present low degree of development of fusiontechnology, even considering these lower energy gain requirements, that causes that no fusion-basedtransmutation system has surpassed the conceptual stage.

Several proposals of such fusion-ssion hybrid systems have been issued in the last years. Among them,a team of the Georgia Institute of Technology headed by W. M. Stacey has developed some concepts oftokamaks based on variants of the ITER design with liquid metal or gas cooled ssile blankets [332, 333].Another similar tokamak-based proposal has been issued by a team of the University of Texas at Austin[188], this time based on their Compact Fusion Neutron Source (CFNS) tokamak design. The Institute ofPlasma Physics of the Chinese Academy of Sciences has been also pursuing a series of conceptual hybriddesigns intended for transmutation purposes including the FDS-I tokamak and the FDS-ST sphericaltokamak [381].

Other magnetic connement devices than tokamaks have also been proposed for transmutation, suchas the ARIES compact stellarator concept, developed by the University of California at San Diego [338];Z-pinch devices, by the Sandia National Laboratory [91, 216] or Gas Dynamics Traps, by the BudkerInstitute of Nuclear Physics of Novosibirsk [242]. Hybrid designs based on inertial connement fusiondevices have also been proposed, such as the Prometheus [385, 386] and the LIFE [295] conceptualreactors. A brief summary including many of these and some other proposals for hybrid fusion-ssionsystems for transmutation is presented in [11].

Another more exotic proposed alternative for transmutation of ssion products are muon catalyzedfusion devices. Muons are produced through the acceleration of protons, deuterons or tritium ions againstlow atomic mass targets to produce pions, which rapidly decay into muons. Muons are then directed tothe reaction chamber by proper magnetic eld. The resulting whole system is then comparable to thoseaccelerator based techniques in terms of energy balance. Once again, some more or less optimistic studiesabout the possibilities of this technology [139, 173, 174, 372] conclude that the energy balance criterionis dicult to met also with this technology if solely used for ssion product transmutation, although theenergy balance can be largely improved if actinides are transmuted simultaneously.

As a nal comment, the potential of thermonuclear explosions as neutron sources for transmutationhas also been investigated [310]. Needless to say, this possibility has raised much opposition and henceonly little attention has been paid to it.

1.3.3.3 Nuclear ssion reactors

Finally, the most obvious option to achieve high neutron uxes is a ssion reactor. Furthermore, ssionreactors are the sole existing option today to produce neutrons without an external supply of energy orat least, with an energy gain in the process, opposite to accelerator driven neutron sources, which requirean external supply of energy.

Actinide and ssion product transmutation in nuclear reactors has been in consideration since at leastthe 1960s. Some early studies include those of H. C. Claiborne [92], about the transmutation capabilitiesof LWRs, and of S. L. Beaman and E. A. Aitken [44], about the transmutation capabilities of LMFBRs.Fission reactors are well suited for the transmutation of actinides, because as actinides are transmutedby ssion they can constitute themselves the fuel of the nuclear reactor. The potential of transmutationof ssion products in ssion reactors is however much more dubious. Some ssion products can betransmuted by neutron capture, at the expense of neutron economy [304, 382]. Nevertheless, neutroninduced transmutation of ssion products by neutron capture is dicult because of the fact that ssionproducts tend to be neutron rich isotopes, and hence many of them have very low neutron capture crosssections to make neutron induced transmutation ecient.

In principle, any reactor type is capable to reduce the amount of transuranic elements in the fuelproviding that large enough irradiation periods are allowed so that a large fraction of the actinides endsby undergoing ssion. Nevertheless, to reduce the amount of actinide waste without the need of very longburnup cycles and without the buildup of heavier actinides, which often pose a higher radiological risk


than the initial ones, ssion must occur faster than neutron capture, and hence ssion cross sections mustbe larger than capture cross sections. Hence, the simplest parameter to intercompare the transmutationperformance of dierent reactor systems is the neutron ssion to capture ratio for every nuclide of interest[363]. A good transmutation system should have a neutron capture to ssion ratio as low as possible.In table 1.4, the integral neutron capture and ssion cross sections together with their ratios for severalactinides weighted with both a thermal and a fast (ssion) spectrum are presented. It can be observedthat, with a thermal spectrum, ssile nuclei have capture to ssion ratios lower than one but non-ssilenuclei have capture to ssion ratios much larger than one. With a fast spectrum, on the contrary, bothssile and non-ssile nuclei have capture to ssion ratios lower than one. This suggest that fast reactors arebetter suited for transmutation than thermal reactors. For thermal reactors to be useful as transmuters,high neutron uxes, long irradiation periods or many irradiation cycles are required to overcome thelow ssion to capture ratios. This brings additional issues such as reactivity penalty, material damageor accumulation of high mass number actinides during the irradiation, with their associated radiologicalrisk, that make transmutation in thermal reactors unlike to be feasible. For example, Claiborne's earlywork considered 15 recycling cycles to achieve up to a 200-fold reduction of toxicity one million yearsafter fuel discharge (with 99.9% of eciency in actinide separation). With the associated losses, costsand radiological risks during reprocessing, 15 cycles make this approach unpractical.

Anyway, today some countries already extract plutonium from irradiated fuel and recycle it in LWRs.A comprehensive study on the issues of plutonium recycling in dierent types of reactors has beenpublished by the OECD/NEA [246]. The capacity of this recycling is limited, however, since the low ssionto capture ratio achievable with the thermal spectrum of an LWR causes a decrease in the concentrationof the ssile Pu-239 and the accumulation of high mass number plutonium isotopes alongside with thehighly radiotoxic americium and curium. For these reasons, as far as I am aware, plutonium is recycledonly once (also called plutonium mono-recycling, MOX fuel is not reprocessed again); to achieve anysignicant reduction in the irradiated fuel radiotoxicity several cycles would be necessary (plutoniummulti-recycling). Hence, although plutonium recycling in LWRs is advantageous in terms of better fueleconomy, it has little advantage in terms of radiotoxicity reduction of the irradiated fuel.

Some more detailed studies on the intercomparison of the HLW transmutation potential of dierentreactor technologies have been published [70, 300, 301, 302, 305]. These studies also agree in the advan-tages of fast reactor technology and hence, most concepts of transmutation reactors are based on fastsystems. Furthermore, many other studies focused on specic designs, either thermal and fast, have alsobeen published. Without trying to be exhaustive, some of these studies are dedicated to sodium-cooledfast reactors [17, 114]; lead or lead-bismuth eutectic (LBE) cooled fast reactors [208]; light water reactors[224]; heavy water reactors [122] or even more unusual systems such as molten salt reactors [348, 349] orgaseous fuel reactors [375]. A summary on the transmutation capacities of Generation IV designs can befound in [340].

Finally, it is important to remark that the ability of nuclear reactors to transmute minor actinideshas not only be assessed theoretically but also several irradiation experiments of fuel samples containingminor actinides in nuclear reactors have been performed. The most extensive tests have been performedin the Phénix sodium-cooled fast reactor (Marcoule, France). The rst of these experiments was theSUPERFACT set of experiments [240], carried out in the 1980s. These experiments were targeted tostudy the viability of transmutation of minor actinides, specically Np-237 and Am-241 in a sodiumfast reactor. Additional irradiation experiments have been carried out at the Phénix reactor in the2000s [135, 376]. They include the ECRIX and the CAMIX-COCHIX experiments (both concerning thetransmutation of Am), the METAPHIX experiment (Am, Np, Cm) and the FUTURIX experiment (Pu,Am). After the shutdown of the Phénix reactor in 2009, France, the USA and Japan have launchedan international project called GACID (Global Actinide Cycle International Demonstration) [234] tocontinue the irradiation experiments at the Joyo sodium fast reactor in Japan.

Irradiation experiments of minor actinides in thermal reactors have also been performed, they includethe ACTINEAU experiment (Np, Am) at the OSIRIS reactor (Saclay, France) [133] and the EFTTRA-T4experiment (Am) carried out at the HFR reactor (Petten, The Netherlands) [186]. Finally, there have alsobeen some experiences concerning the transmutation of ssion products such as the RAS-1 experimentat the HFR [184] (Tc-99 and I-129) and the ANTICORP-1 at Phénix (Tc-99) [135].

1.3.4 Intercomparison of transmutation technologies and conclusions

With the considerations presented in this section, the most promising way to reduce the long termradiotoxicity of the irradiated nuclear fuel is the transmutation of actinides (essentially plutonium and


Table 1.4: Integral ssion and neutron capture cross sections with a thermal and a fast for the mostrelevant actinides. Results obtained with the JANIS program [244] and the ENDF/B-VII.0 evaluatednuclear data library (Parameters chosen: (1) PWR spectrum: Emax,th = 0.108 eV, θth = 0.054 eV,Emax,epi = 2.1 MeV, θepi = 1.4 MeV; (2) Fission spectrum: θ = 1.0 MeV).

PWR spectrum Fission neutron spectrum

〈σf 〉〈σc〉〈σc〉 / 〈σf 〉〈σf 〉〈σc〉〈σc〉 / 〈σf 〉

U-234 0.2091 40.4875 193.6730 1.0656 0.1156 0.1085

U-235 53.3005 14.3890 0.2700 1.2276 0.1200 0.0977

U-236 0.3137 18.6231 59.3614 0.4631 0.1221 0.2636

U-237 0.4601 47.3858 102.9853 0.5607 0.0885 0.1579

U-238 0.0367 14.9103 405.7665 0.2217 0.0868 0.3916

U-239 2.5202 15.5655 6.1763 0.4016 0.1097 0.2732

Np-237 0.2200 57.5638 261.6715 1.2003 0.2611 0.2176

Np-238 199.1731 43.1364 0.2166 1.4634 0.1409 0.0963

Np-239 0.2210 29.9454 135.5070 1.3622 0.2714 0.1993

Pu-238 2.3646 38.3855 16.2334 1.8497 0.1852 0.1001

Pu-239 143.2365 83.3579 0.5820 1.7595 0.0574 0.0326

Pu-240 0.3383 480.4125 1419.9062 1.2073 0.1006 0.0833

Pu-241 139.3128 51.5759 0.3702 1.6255 0.1130 0.0695

Pu-242 0.1635 69.4410 424.7468 0.9996 0.0931 0.0931

Pu-243 42.0372 20.6615 0.4915 0.9783 0.0584 0.0597

Am-241 1.216 176.2773 143.1288 1.1607 0.3743 0.3225

Am-242 257.9438 31.0388 0.1203 2.0309 0.05826 0.0287

Am-242m 695.0043 131.9188 0.1898 1.9862 0.09036 0.0455

Am-243 0.2572 103.7250 403.2495 0.8944 0.2492 0.2786

Am-244 251.2461 63.9520 0.2545 1.7932 0.4314 0.2406

Am-244m 228.6466 57.4755 0.2514 1.7933 0.4414 0.2461

Cm-242 0.3552 7.0896 19.9593 0.7702 0.0337 0.0437

Cm-243 129.6335 19.8947 0.1535 1.8432 0.1070 0.0581

Cm-244 0.6021 36.3369 60.3458 1.4000 0.2380 0.1700

Cm-245 155.9664 23.2442 0.1490 1.7533 0.1132 0.0645


americium) into shorter lived ssion products. Long lived ssion products may also be transmutedinto shorter lived or stable isotopes through several reactions, but none of these reactions is either self-sustained or has a positive enough energy balance, which makes unlike their applicability in industrialscale. Anyway, due to the much lower contribution of ssion products to the long-term radiotoxicity ofirradiated fuel than actinides, its transmutation constitutes a secondary goal.

The technology most likely to be developed in the shorter term to achieve this goal is ssion reactortechnology. Actinide transmutation in ssion reactors has two important advantages for its industrialscale application. First, since nuclear ssion is an exoenergetic process, the recovery of the released energycan make the transmutation of minor actinides positive in terms of energy balance; it can become, infact, a source of useable energy. Second, since the ssion process allows for self sustained neutron chainreactions no external supply of energy is required. For these reasons, the industrial scale transmutationof actinides in ssion reactors is widely regarded as feasible. Fusion devices can also provide intense,exoenergetic and self-sustained neutron sources well suited for actinide or long lived ssion productstransmutation, but they are in a much earlier stage of development than ssion reactors. Finally, otherpossible technologies for transmutation, such as proton spallation or photossion have the disadvantageof requiring an external supply of energy, which make them unlike to be practical for industrial scaletransmutation.

Chapter 2

Transmutation in Accelerator DrivenSystems

Abstract - For reactor stability considerations, the amount of plutonium and americium that can

be introduced in a conventional, critical reactor is limited. For this reason, the concept of accelerator

driven subcritical systems (ADS) has received considerable attention in the last years, as these systems

are free from critical stability issues and hence, they allow for larger loads of these minor actinides. In

this chapter, a brief historical outline of the evolution of the concept of ADS is presented. In the end,

a brief survey of the main technological developments required for the development of industrial-sized

ADSs is presented.

2.1 Limits of critical reactors and Accelerator Driven Systems

As it was described in the last chapter, transmutation of HLW is mainly concerned with two elements,plutonium and americium. Nevertheless, the introduction of these elements in a critical reactor bringssome adverse eects regarding its stability. These include lower delayed neutron fractions, lower controlelements worth, reduced Doppler feedback eect and positive void coecient.

The usage of plutonium fuels, as it is the case of MOX fuel in LWRs described in section 1.3.3.3,already brings some adverse eects (see, e. g. [331]) to the reactor stability. First, plutonium isotopeshave large thermal neutron capture cross sections (gure 2.1), which causes the worth of control elementsand burnable poisons, which are based on thermal neutron absorbers such as boron or gadolinium, todecrease. Second, the delayed neutron fraction of Pu-239 is about one third of U-235, thus reducing thesafety margin to reach prompt criticality (gure 2.2). These eects, in fact, limit the amount of MOXfuel that can be loaded into LWRs, although some recent designs such as the EPR and the ABWR arecapable to run on full MOX fuel loads, according to the vendors.

The introduction of americium in a fast reactor brings additional adverse eects. First, both Am-241and Am-243 have large capture cross sections in the 100 keV to 1 MeV range (gure 2.3). This increasesneutron capture before reaching the resonance region where the nuclear Doppler eect takes place andhence, the introduction of americium in a nuclear reactor will reduce the ability of the nuclear Dopplereect to stabilize the reactor [373, 374, 378]. Second, the large rise of the ssion cross sections of bothAm-241 and Am-243 in the 100 keV to 1 MeV range (gure 2.4) has the eect of causing positive voidcoecients, since the formation of voids in the coolant will drive to a harder neutron spectrum and hence,to an increase of the reaction rate and the power produced.

Some more detailed papers and studies have been published where these eects are quantied forseveral designs with various fuels and coolants, including sodium-cooled fast reactors, lead and lead-bismuth fast reactors and gas fast reactors [351, 374, 378, 390]. The nal conclusion of these studies isthat for stability issues, critical reactors can be loaded with only a limited amount of minor actinides inthe fuel. There has also been studies considering particular designs for critical cores with high Pu andMA content and acceptable safety features [142], but they have been paid little attention.

For these reasons, Accelerator Driven Subcritical systems (ADSs), also referred sometimes as HybridSystems or Electronuclear Systems, have been proposed to act as dedicated minor actinide burners. Thesesystems consist of a subcritical assembly (values of keff between 0.95 and 0.98 are usually considered)maintained in a steady power level by an external neutron source driven by an accelerator. With the

29

30 CHAPTER 2. TRANSMUTATION IN ACCELERATOR DRIVEN SYSTEMS

Figure 2.1: Radiative capture (n,γ) cross section for the most relevant isotopes of Pu, compared withsame cross section of U-238 (ENDF/B-VII.0 library).

Figure 2.2: Delayed neutron fractions of Pu-239, Am-241 and Am-243, compared with the delayed neutronfraction of U-235 (ENDF/B-VII.0 library).

2.1. LIMITS OF CRITICAL REACTORS AND ACCELERATOR DRIVEN SYSTEMS 31

Figure 2.3: Radiative capture (n,γ) cross section for the most relevant isotopes of Am, compared withsame cross section of U-238 (ENDF/B-VII.0 library).

Figure 2.4: Fission cross sections of Am-241 and Am-243, compared with ssion cross section of U-235and Pu-239 (ENDF/B-VII.0 library).


considerations made in section 1.3.3.1, spallation reactions are the preferred choice for the external sourceas they produce higher neutron yields per unit of energy of the accelerated particle. The subcriticalassembly produces ssion energy, part of which is used to keep the accelerator running and a the restcould be supplied to the grid, thus achieving a system with a positive energy balance.

Although obviously more complex and expensive than conventional critical reactors, ADSs oer someadvantages. First, it is obvious that stability and negative feedback eects (although always desirable)are not strictly necessary requirements to operate such subcritical systems. In addition to this, otheradvantages of ADSs over critical reactors include more exibility in core loading patterns and less relevanceof fuel depletion with increasing burnup. This allows for a higher burnup than in critical reactors, althoughwith a penalty in core multiplication and hence, in energy balance. For further information regardingADSs features and performance, a large number of publications about the matter have appeared sincethe 1990s. Some of the most exhaustive ones are [241, 247, 248]. A useful summary can be found in [36].

It must be remarked that ADSs and critical reactors are not necessarily mutually exclusive. Forinstance, under the double strata concept, originally proposed by JAERI, a fuel cycle is consideredincluding both fast reactors and ADSs, alongside with an existing LWR eet. Under this concept, fastreactors were to be used for plutonium recycling while ADSs were to be used solely for MA burning. Inthis way it was intended to maximize the eciency and minimize the cost of the fuel cycle.

To better illustrate the working of an ADS, let us consider a system where the external neutron sourceintroduces S neutrons per unit of time in the system. The total number of neutrons (N) produced in thesubcritical system will be given by:

N = S

∞∑n=1

kns = Sks

1− ks

where ks is the source multiplicity dened in section 4.3.1. And hence the number of ssions per unitof time produced in the system by the introduction of S neutrons are given by

Nfis =S

ν

ks1− ks

where ν is the average number of neutrons per ssion. To evaluate the energy gain of an ADS itis required to take into account the energy produced in a ssion and the energy required to produce aspallation. One nds out that:

εgain =S

ν

ks1− ks

E (MeV/ssion)

E (MeV/proton)

Considering typical values of 200 MeV/ssion, 2.5 neutrons per ssion and 20 spallation neutrons per1000 MeV proton, one nds out that the energy gain of such a system is 30.4 for ks = 0.95 and 78.4for ks = 0.98. However, to compute the feasibility of such an ADS, the thermal eciency (εthermal)and the accelerator eciency (εacc) must be taken into account. The condition for having energy gain isobviously:

εgainεthermalεacc > 1

Considering for instance εthermal = 0.3 and εacc = 0.5, the previously considered system will stillproduce 4.56 times more electric power than it consumes, for ks = 0.95, and 11.76 times, for ks = 0.98,high enough to guarantee the positive energy balance of the system.

2.2 ADS concepts and evolution

Although the concept of coupling an accelerator to a subcritical assembly has received much attentionsince the 1990s for actinide waste transmutation purposes, this concept is in fact much older and traces itsroots back to beginnings of the nuclear age. Some information about early ADS concepts can be found in[131, 170, 171]. Most of the ADS concepts since proposed has been as part of nuclear fuel cycles, intendedfor nuclear fuel breeding or nuclear waste transmutation, but the advantages of the ADS concept havealso attracted interest for other applications, such as increasing the yield of neutron sources or isotopeproduction.

2.2. ADS CONCEPTS AND EVOLUTION 33

2.2.1 ADSs in the nuclear fuel cycle

The earliest concepts involving the coupling of an accelerator to a subcritical assembly appeared in the1950s. Main motivation behind these early designs was to act as fuel breeders as an alternative to criticalfast breeder reactors. The main advantage of an accelerator driven subcritical breeder over a critical fastbreeder reactor is that the larger neutron excess of the rst ones allows for a larger ssile fuel productionfor a given reactor power. This is specially relevant for the case of the thorium fuel cycle, since theaverage neutron yield of U-233 ssion in a fast spectrum is lower than the neutron yield of Pu-239 ssion.

The rst of these accelerator driven breeder concepts that was attempted to take into practice wasthe so-called Materials Testing Accelerator (MTA) project at Berkeley [141], leaded by O. H. Lawrence.In spite of its name, the project was intended for plutonium production for military purposes. Theproject consisted on a linear accelerator capable to produce a 500 mA beam of 350 MeV deuterons, whichwas to be focused on a thorium target surrounded by a breeding blanket of depleted uranium from thedepleted tails produced in uranium enrichment processes. The device was expected to produce about500 kg of plutonium per year. The project ran for a few years between 1950 and 1952, when it wascanceled because the discovery of large uranium deposits in the USA made disappear the fear of shortterm exhaustion of the ssile material sources. Only a deuteron linear accelerator providing 50 mA of30 MeV deuterons, which nevertheless constituted a considerable technological achievement for the time,was built and operated during 1952-1953 for demonstration purposes.

At about the same time, AECL of Canada began developing a similar concept, but this time intendedto produce U-233 from Th-232 to fuel CANDU reactors in the thorium cycle [35]. The motivation behindthis system was that CANDU reactors operating in the U-233/Th-232 were expected to have enoughneutron excess to breed the same amount of fuel they consume but still some supply of external fuel wasrequired, at least to achieve initial criticality. To produce this fuel, an accelerator breeder was proposed,consisting of a high intensity (300 mA) linear accelerator delivering a beam of 1 GeV particles (protonsor deuterons) to a spallation target surrounded by a blanket of fertile material. To enhance the energyproduced in the target, and hence the system eciency, some amount of ssile material can be mixedinto the target. Interestingly enough, although never surpassed the conceptual design phase, this projectwas the basis for a proposal of the so-called Intense Neutron Generator [355] project, a precedent of thepresent day neutron spallation sources.

The Oak Ridge National laboratory of the USA proposed another similar thorium breeder conceptin 1979, the so-called TMF-ENFP (Ternary Metal Fueled - ElectroNuclear Fuel Producer) [73]. Thisconcept consisted of a sodium cooled blanket containing all three isotopes U-238, Pu-239 and Th-232(hence the designation Ternary Metal Fueled) surrounding a sodium column that acted as neutronproducing target. The Pu-239 was produced from the U-238 and was used to achieve a higher neutronexcess to breed U-233. The system was to be coupled to a 300 mA beam of 1 GeV protons. It wasintended for U-233 fuel breeding to be used in LWRs or CANDU reactors. A later modication of thisdesign considered a proton beam spread over the entire reactor that acted itself as the neutron producingtarget [166].

The Brookhaven National Laboratory in the USA also worked on another accelerator breeder conceptduring the late 1970s and early 80s [128, 129]. In this case, they considered the uranium cycle insteadof the thorium cycle and the main motivation was to develop proliferation resistant fuel cycles. Theproposed concept consisted of a 300 mA, 1.5 GeV proton accelerator coupled to a target of liquid lead-bismuth, in the shape of falling liquid columns placed inside a vacuum vessel. A breeding blanketwill be placed around this target consisting of conventional LWR fuel elements in pressure tubes. Fastssions in these fuel elements, together with the energy generated in the target itself would guaranteethe energetic self-suciency of the system. The purpose of this concept of accelerator breeder was toincrease the ssile material content of spent LWR fuel (by breeding Pu-239 from U-238) to allow itsreuse in additional cycles. This could be achieved without reprocessing in between, thus increasing theproliferation resistance of the cycle. Alternatively, instead of using spent fuel, fresh fuel could be loadedto increase its ssile material content. Hence the system could become an alternative or complement toconventional enrichment techniques.

The rst concept of ADS intended for transmutation of long lived radioactive isotopes, more specif-ically minor actinides, was the PHOENIX concept [361, 362], developed in the Brookhaven NationalLaboratory in the 1980s. Under this concept, a high intensity LINAC, capable to produce a 104 mAbeam of 1.6 GeV protons, was to be coupled to some modules, each consisting of a spallation target anda subcritical assembly. The design of these facilities was based on the core of the Fast Flux Test Facility(FFTF) reactor. These facilities were to be sodium cooled 450 MWth subcritical assemblies with a minoractinide fuel which would serve as well as spallation target. Up to eight of these modules can be driven


to a single accelerator, to build up a 3600 MWth system capable to supply 850 MWe of electric powerto the grid. It was foreseen that a single 3600 MWth system would be capable to get rid of all the longlived nuclides produced in 75 LWRs.

In 1992, another ADS concept was issued by a team of the Los Alamos National Laboratory (USA)[67]. This concept envisaged a very high intensity proton LINAC coupled to a liquid lead spallationsource and a D2O moderated molten-salt subcritical assembly. The very high neutron ux (1016n/cm2s)expected to be attained in the device were to allow for ecient transmutation of minor actinides in spiteof the lower ssion-to-capture ratios of a thermal neutron spectrum. Although this concept of molten saltsubcritical system was not further developed, it inspired the later Accelerator Transmutation of Waste(ATW) program [54, 356], which envisaged a transmutation strategy based on 840 MWth LBE-cooledfast ADS modules.

In Europe, a CERN's team led by Carlo Rubbia proposed an ADS concept named Energy Amplierin 1993 [77]. Initially, the concept, a thorium thermal breeder driven by a spallation source, was verysimilar to the AECL's concepts described above, with the dierence that it was intended not only toproduce U-233 but also to burn the U-233 itself, operating in a equilibrium state between the ssioned U-233 and the U-233 bred from the Th-232. Two target options were considered: either the fuel/moderatoritself or a dedicated target. For the accelerator, two options were considered to provide a beam withsuitable energy and intensity: a LINAC or a isochronus cyclotron. Several options for the fuel, moderatorand coolant were studied. The primary role envisaged for the device was the production of energy usingthe abundant thorium fuel resources, with the large criticality safety margins that a subcritical systemprovides and, according to the authors, with a low proliferation risk due to the use of the thorium fuelcycle.

A more detailed proposal was presented two years later [289]. The concept was still based in thethorium fuel cycle but it has evolved into a lead cooled fast system. Now, the concept envisaged asystem formed by 1500 MWth modules cooled by molten lead circulating by natural convention. Thelead also served as spallation target. A compact cyclotron system, delivering a proton beam of about 10mA and 1 GeV, was designed for the Energy Amplier. Still, the system was not intended primarily fortransmutation but for energy production.

The CERN's Energy Amplier has strongly inuenced later European ADS designs. The last ADSconcept issued in Europe is the so-called EFIT (Experimental Facility for Industrial Transmutation)[34], developed under the 6th European Framework Program. This concept envisages a 400 MWth lead-bismuth cooled pool-type subcritical core coupled to a spallation target driven by a 800 MeV protonaccelerator. Contrary to the Energy Amplier concept, EFIT is optimized for the MA-burning role. It isexpected to be fueled with Pu-minor actinide oxide, U-free fuel and it is expected to be able to transmuteabout 40 kg of minor actinides per produced TWh(th) with no net production of plutonium. This gurecan be compared with the data of the composition of the irradiated LWR fuel presented in gure 1.2.There it was stated that less than 2 kg of MAs were produced out of one ton of fuel at 50 GWd burnup,or, equivalently, 1.2 TWh(th). With these gures, for getting rid of the whole MA actinide production ofa LWR park EFIT-like facilities should represent less than the 5% of the installed nuclear thermal power.

In Japan, considerable attention has also been paid to accelerator driven systems, specially afterthe OMEGA program mentioned in section 1.3 was launched in 1988. Several concepts were proposed[197, 226]. The earliest concept (1989) was a liquid metal cooled system, either sodium or lead-bismuth,of a simple parallelepiped shape, with minor actinide fuel acting as well as the spallation target. Later,several concepts were issued, included a sodium cooled ADS, a molten-salt ADS, a molten-alloy ADSand a lead-bismuth cooled ADS. Finally a design consisting on a 800 MWth LBE subcritical systemcoupled to a 1.5 GeV superconducting LINAC was chosen as the reference JAERI's (later JAEA's)design [345, 353, 354]. Similarly to the EFIT design, it is expected to be fueled with a Pu-minor actinide,U-free fuel. According to the authors, this design is estimated to be able to transmute about 250 kg ofminor actinides per year without net plutonium production.

The Korean Atomic Energy Research Institute (KAERI) has also being developing a concept of aLBE cooled ADS, called HYPER, since the 1990s [257, 258, 323]. It consists of a 1000MWth systemcoupled to a proton beam of about 1 GeV energy. Unlike EFIT and JAEA's concepts, it is to be fuelednot with U-free but with U-TRU fuel. The design is calculated to be able to burn 282 kg of TRU peryear. Its ability to incinerate also long lived ssion products (Tc-99, I-129) has also been analyzed.

The Indian Atomic Energy commission has also showed interest in the development of ADSs, bothfor the MA burning role and for the thorium breeding role. With this purpose, a so called Co-ordinationCommittee for Accelerator-Driven Sub-Critical Reactor Systems was constituted in December 1999. Aroadmap was prepared in 2001 that should end up with the development of a 1500 MW(th) LBE or lead

2.3. TECHNICAL CHALLENGES OF ACCELERATOR DRIVEN SYSTEMS 35

cooled ADS [103].Finally, to remark that several gas-cooled ADS concepts have also been proposed. General Atomics

has developed a concept of gas-cooled ADS based on their existing GT-MHR design [40]. Variants of thissystem with a thermal and a fast spectrum were proposed. The thermal variant is expected to be ecientas transmuter thanks to the extremely long burnup cycles achievable with the GT-MHR fuel, while thefast variant can achieve harder neutron spectra than any other reactor technology, thus achieving thehighest possible ssion to capture ratios. Both the thermal and the fast variant could be combined tooptimize the eciency of the whole fuel cycle. Similarly, an accelerator-driven subcritical variant of thepebble bed modular reactor was also developed by the Spanish company LAESA [8].

As a nal comment, a small size (200-400 MWth) compact helium-cooled ADS designed as Low EnergyElectronuclear Power Plant (LEEPP) has also been proposed by a Russian team [339]. It was designedto be compact enough to t into a 10m diameter× 25m length container, including the accelerator. Themotivation of this design was to provide steam or energy in isolated areas with the safety advantage of asubcritical facility.

2.2.2 ADSs outside the nuclear fuel cycle

Apart from all these applications of ADSs within the nuclear fuel cycle, the concept of coupling anneutron source to a multiplicative subcritical blanket has also found application for other purposes, suchas increasing the yield of neutron sources and the production of radioisotopes.

Subcritical blankets have been used in a few occasions to improve the neutron yield of neutron sourcesused for research. In fact, they represent the only practical application up to now of the ADS concept.This is is the case of the booster target coupled to the Harwell electron linear accelerator and the IBR-30reactor of the Joint Institute of Nuclear Research (JINR) at Dubna, both coupled to photoneutron sources.The rst of them was built at Harwell during the late 50s and consisted of a 235U subcritical assemblycoupled to a photoneutron target driven by a 45 MeV electron LINAC. The IBR-30 reactor was a pulsedfast reactor with a mean power of about 25-30 kW used as a neutron source for neutron spectroscopyexperiments that operated between 1969 and 2001 [14, 219]. It was loaded with metallic plutonium fueland was cooled by air. The IBR-30 was a developmental step between the IBR-1 (1 kW mean power,commissioned in 1960) and the larger, sodium cooled IBR-2 (4 MW mean power, commissioned in 1978)pulsed reactors. Pulsed operation of these reactors was achieved by means of a rotating disk containingan amount of 235U at some point. When, as a result of the disk rotation, the part containing the 235U wasintroduced in the core, the reactor became supercritical for a short time, producing neutron bursts. Inaddition to this mode of operation, the IBR-30 was operated also as a booster for a photoneutron target(made of plutonium rst, later of tungsten) coupled to a 40 MeV electron LINAC. A photoneutron targetdriven by a 30 MeV electron LINAC was also designed for the IBR-2 [22, 219], but, to my knowledge,it was never operated in this way. As a nal remark, the IREN neutron source project of the JINRalready mentioned in section 1.3.3.1 envisages adding a metallic plutonium subcritical assembly to thephotoneutron source already in operation. In fact, the IREN facility has been built in the same place ofthe decommissioned IBR-30 reactor and its associated accelerator.

Finally, concerning the application of ADSs for radioisotope production, the Belgian company IBApursued for some time a cyclotron driven subcritical assembly to serve for medical Mo-99 production asan alternative to nuclear reactors [169]. Nevertheless, this concept, called ADONIS, did not surpassedthe design stage.

2.3 Technical challenges of Accelerator Driven Systems

It does not seem to be any physical or major technical obstacle for the development of ADSs, but someelds where technological development is required in order to design and operate an industrial scale ADShave been identied. Apart from the issues that are dependent on the reactor technology chosen (fastreactor, MSR, etc), such as reduced stability or coolant and materials or fuel issues, specic issues ofADSs include:

Accelerator intensity Using the relationship for εgain given in section 2.1, it can be obtained that tohave a 3000 MWth ADS, it is required to have a beam of 1000 MeV protons of a power of 37.5 MW forkeff = 0.98 or 93.75 MW times for keff = 0.95, or, respectively, 37.5 mA and 93.75 mA of intensity.This is more than one order of magnitude larger than the values achieved by present day highest intensityaccelerators in this energy range. For comparison, the present highest power accelerators of the world,


namely the LINACS used for the spallation sources at Oak Ridge and J-PARC have beam powers of 1.4MW and 1 MW, respectively, although work is underway at present to increase their power up to 5 MW.The ESS is also intended to operate initially at 5 MW. For the case of cyclotrons, the SINQ cyclotronat PSI has similarly a power of about 1 MW. The feasibility of accelerators in the power level of tens ofMW required for transmutation has still to be demonstrated.

Accelerator reliability An industrial scale facility needs to operate in a stable way with a limitednumber of unexpected shutdowns. Apart from the obvious inconvenience of unexpected shutdowns in ahigh power facility connected to the grid, it is feared the eect of a large numbers of accelerator beam tripsin terms of thermal stress in the reactor components [342]. In fact, the accelerator reliability requirementsfor ADS development is one of the motivations for many of the research carried out over the last yearsin the eld of accelerator reliability [140, 225, 270].

Spallation target stress The stress on the spallation target caused by impinging protons is consider-able. This can limit the operational life of the spallation target or it can require its periodical replacement,with the implications it has in terms of costs and system availability. Cooling the spallation target is alsoan issue, since the amount of power dissipated (expected to reach the tens of MW range) is expected tobe large. Both issues can be solved with a uid target, such as the circulating mercury targets used in theSNS or the liquid lead-bismuth eutectic target tested at SINQ in 2006 under the MEGAPIE project [39].This last was specically intended for demonstrating the performance of such a target for high powerADS applications.

Coupling between the accelerator and the spallation target The coupling of the accelerator tothe spallation target is one of the most challenging issues for the development of industrial scale ADSs(e. g. [94]). The proton beam must penetrate the pressure vessel of the reactor and thus a window mustbe present that is transparent to the neutrons but at the same time it must keep the integrity of thepressure vessel barrier. The damage caused by the proton beam to this window and the possibility forit to fail has also to be taken into account. Windowless target designs, where no mechanical window isplaced between the accelerator tube and the reactor and the coolant does not invade the accelerator tubeby means of an adequate hydrodynamical design are also receiving considerable attention, but remain tobe proven, and they arise concerns about the integrity of the pressure vessel and the eects of an invasionof the accelerator tube in case of failure.

Spallation target activation The buildup of spallation products in the spallation target during theoperation of an ADS has raised some concerns as it may oset the benets of the transmutation of othernuclides in an ADS [335, 352]. Nevertheless, it has been found that although this may be the case ifthe transmutation of certain ssion products is considered, the radiotoxicity increase due to spallationproducts buildup is much less signicant than the radiotoxicity decrease as the result of minor actinidestransmutation. A proposal to reduce the target activation is the use of low atomic mass targets, such astin, in spite of the lower neutron yield of spallation in these materials. A further advantage of using tinspallation targets is that the ssion product Sn-126 can be introduced in them, thus contributing to thetransmutation of this isotope [138, 336].

Reactor dynamical behavior Reactor dynamical behavior of subcritical systems is dierent fromthe well known behavior of the critical systems. New source or reactivity transients dierent than thosepresent in critical systems may arise. In particular, at increasing subcriticality levels the point-kineticsmodel looses validity and space-dependent dynamics plays a much more relevant role.

Reactivity monitoring A nal issue that has been identied is the need to develop a reactivitymonitoring system that guarantees the maintenance of an enough large reactivity margin at any moment.Instrumentation and control systems of present critical reactors based on neutron ux and power levelmeasurements are of limited application in an ADS since in an ADS ux and power are dependent bothon the neutron source and on the reactivity and hence changes in the neutron source can mask changesin the reactivity. This fact is more relevant since it is desirable that an ADS operates with the highestpossible value of the reactivity in order to reduce accelerator power requirements. In these circumstancesthe possibility to inadvertently reach criticality constitutes a risk, particularly taking into account thatADSs intended for transmutation are to be designed to operate with large plutonium and americiumcontents and hence with poor self-stabilizing performance.

2.3. TECHNICAL CHALLENGES OF ACCELERATOR DRIVEN SYSTEMS 37

This PhD thesis is concerned with the last two points and in particular with the last one, and hencethey will be treated with much more detail later. But before entering in details, and to describe the currentstatus of the art of the research in these elds, next chapter is a summary of the experiments performedworldwide, concerning the analysis of the dynamical behavior of subcritical systems, the validation ofexperimental reactivity measurement techniques and the development of a reactivity monitoring system.

Chapter 3

Experimental ADS facilities

Abstract - Since the 1990s, in order to support the development of accelerator driven systems, several

experiments with small size facilities have been performed around the world. They include the FEAT,

MUSE, GUINEVERE/FREYA, E+T, Yalina, KUCA, RACE and VENUS experiments. This chapter

presents a brief description of all these experiments, alongside with a number of projects that were

planned but were not realized in the end (TRADE, SAD), and other projects (TEF at JPARC,

MYRRHA) that are still at the planning stage. They have been classied by regions (Europe, former

USSR, Japan, USA and China), and for each of these regions, they are described in chronological

order.

Main goals of experiments carried out in experimental ADS facilities include to demonstrate theenergy gain of an ADS, to validate neutron transport codes, to study the kinetics of these systems andto develop reactivity monitoring techniques. A timeline of the main experiments that have been carriedout up to date is given in gure 3.1.

3.1 Europe

As far as 2014, it is in Europe where ADS research has reached its highest degree of development. Researchinto ADS physics was started in Europe in the early 1990s both at CERN, in connection with the EnergyAmplier project (FEAT experiment), and at the French CEA, as a result of the 1991 law concerningthe management of nuclear waste mentioned in section 1.3 (MUSE series of experiments). Later, ADSresearch in Europe have been coordinated under dierent Euratom's Framework Programs, beginningwith the last experiment of the MUSE series (MUSE-4, 5th FP), the GUINEVERE experiment andthe unrealized TRADE project (6th FP) and the FREYA experiment (7th FP). Under these FrameworkProjects, European participation in other ADS experiments (RACE, Yalina, SAD) has also been enabled.Finally, the most ambitious ADS project considered up to date, with a power of 50-100 MWth, is theMYRRHA project which aims to be operative by 2025.

3.1.1 FEAT

The FEAT experiment (First Energy Amplier Test) [10], which took place in 1993, consisted in thecoupling of a subcritical system to the CERN proton synchrotron (PS). This experiment was intended todemonstrate the feasibility of the Energy Amplier concept mentioned in section 2.2. More specically,the aim was to measure the energy gain of such devices, in addition to measurements of the ux andpower distribution within the assembly.

The subcritical system used was a small training subcritical assembly that belonged to the PolytechnicUniversity of Madrid. It consisted of an hexagonal array of natural uranium fuel rods with aluminumcladding. This array was contained in a cylindrical stainless steel tank of demineralized water that actedas moderator and reector. Dimensions of the tank were 152.4 cm height and 122.0 cm diameter. Withthis conguration a keff of about 0.91-092 was achieved. For the FEAT experiment, a cylindrical depleteduranium spallation target was introduced horizontally along the mid plane of the device, which causedthe keff to decrease to 0.895 ± 0.010. The CERN Proton Synchrotron provided proton pulses with anintensity of about 109 protons per pulse and a repetition rates of about one pulse every 10 s. In theseconditions, the assembly power was about 1 W. The energy of the proton beam ranged between 600 MeVand 2.75 GeV, which allowed investigating the energy gain with protons of dierent energies.

39

keff

2

410

3.1. EUROPE 41

their lower part. This conguration allows for an homogeneous core when fully retracted. The pilot rodis composed of aluminum and enriched uranium surrounded by polyethylene.

The MUSE-1 experiment took place in December 1995 [303]. The external neutron source usedwas a 252Cf source placed in the central channel of the core. It provided 7.55 × 107n/s and could belocated at three dierent positions along the channel. The experiments performed consisted in reactivitydetermination using the Modied Source Multiplication (MSM) method; axial and radial neutron uxdistributions using 235U ssion chambers, and source importance (ϕ∗) measurements.

The MUSE-2 experiment [325] took for two months in 1996 and used a core with a similar com-position than that of MUSE-1. The source was again a 252Cf source, but this time the activity was6.11× 107n/s. In MUSE-2, the target was surrounded by buers of stainless steel and sodium to modifythe source importance. The experimental program was similar to that of MUSE-1 and included reac-tivity measurements with the MSM method; ssion rate distribution measurements with 235U ssionchambers; assembly power determination with a calibrated 235U ssion chamber and source importancemeasurements.

The MUSE-3 [108] experiment took place between February and April 1998. During this experiment,a commercial D-T neutron generator (SODERN Génie 26) was coupled to the facility, with the tritiumtarget located at the mid plane. It provided an intensity of 2×108n/s. Three dierent congurations of thecore with dierent levels of subcriticality were obtained by removing a dierent number of fuel elementsfrom the critical reference conguration. Furthermore, smaller reactivity variations were obtained bychanging the control rods position. Contrary to the 252Cf used in the previous experiments, this neutrongenerator was capable of operating both in pulsed and in continuous mode. In continuous mode, similarexperiments than in MUSE-1 and -2 were carried out, that included reactivity measurements with theMSM technique and the study of the modication of the source importance after surrounding the sourcewith sodium and lead. With the generator operating in pulsed mode it was also possible to analyze thekinetic response of the system to fast neutron pulses and to apply PNS reactivity measurement techniques(see section 7.2).

Nevertheless, presence of light materials in the neutron generator (used for high voltage insulation)cause some thermalization of the neutron spectrum that considerably perturbed the PNS experiments.For this reason, a new experiment was designed using a purpose-built neutron generator without lightmaterials. This new experiment, designed as MUSE-4 [218, 368], began in November 2001 and lasted until2004. It is important to notice that since September 2000, participation of several European institutionsin the experiment was granted within the 5th EURATOM framework program. These include SCK/CEN(Belgium), FZK and FZJ (Germany), ENEA (Italy), Delft University of Technology and NRG (TheNetherlands), AGH (Poland), CIEMAT (Spain), KTH and Chalmers University of Technology (Sweden)and BNFL (United Kingdom).

The neutron source used in the MUSE-4 experiment was a D-D or D-T fusion source driven by ahigh intensity deuteron accelerator. This accelerator was designed by the LPSC-CNRS at Grenobleand was designated as GENEPI (GÉneratéur de NEutrons Pulsés Intenses - Intense neutron pulsesgenerator [101]), and as its name indicates, it was capable to operate only in pulsed mode. GENEPI is anelectrostatic accelerator capable to provide a beam of 240 keV deuterons (this energy was chosen becauseit maximizes the cross section of the D-T fusion reaction) in short and sharp pulses of < 1 µs and peakintensities larger than 50 mA, with repetition rates of up to 4 kHz. The neutron producing target wasplaced in the center of the MASURCA assembly. It had the shape of a copper disk of 50 mm thicknesswith a 30 mm titanium depot with embedded deuterium or tritium. The target was surrounded by alead block in order to better simulate a spallation source, which is more likely to be used in an industrialADS. Neutron production was calibrated to be of about 3× 104 n/pulse with a D-D source and 3.3× 106

n/pulse with a D-T source.During the MUSE-4 experiment, three basic subcritical congurations of the reactor were investigated,

with keff being successively about 0.994, 0.97 and 0.95, in addition to a critical reference conguration.They were obtained by removing an increasing number of fuel elements from the core periphery. A fourthconguration, with part of the sodium replaced by lead, was also investigated. In addition to changingthe fuel load, changing the control and pilot rods positions allowed to extend the reactivity range downto a keff of about 0.85.

The MUSE-4 experiment has been the largest experiment in the eld of ADS kinetics carried out up todate and has been instrumental for the denition of later experiments, in particular, the EUROTRANSand FREYA projects of the 6th and 7th EU Framework Programmes. Extensive research was carriedout, mainly focused on the experimental validation of subcritical core characterization techniques andreactivity monitoring techniques. In particular, special attention was devoted to Pulsed Neutron Source

42 CHAPTER 3. EXPERIMENTAL ADS FACILITIES

(PNS) techniques, and correction methodologies to take into account the spatial and spectral eects thataect them. On the other hand, the main shortcoming of the experiment was the impossibility of theGENEPI neutron source to operate in continuous mode with short beam trips. These techniques havebeen tested in later experiments like Yalina-Booster and GUINEVERE/FREYA, the second one usingan upgraded version of the GENEPI neutron source capable of operating in the beam-trip mode.

3.1.3 TRADE

The successor of MUSE in the 6th EURATOM framework should have been the TRADE (TRiga Ac-celerator Driven Experiment) project, to be carried out at the ENEA facilities at Cassacia, near Rome(Italy) [146, 155, 286, 287, 290]. The experiment consisted in coupling an existing TRIGA reactor to anewly built spallation source.

The reactor (ocial designation ENEA RC-1), of the TRIGA Mark II type, started operation in 1960and upgraded to a maximum power of 1 MWth in 1967. Fuel consists of a uranium-ZrH alloy, withan enrichment of 20%. For the TRADE experiment, the spallation target will be placed in the centralthimble of the reactor and dierent number of fuel elements will be added or removed to achieve dierentcore congurations with keff ranging from approximately 0.90 and 0.99.

The spallation target envisaged for the TRADE experiment was made of solid tantalum and had theshape of a conical surface with the basis facing the proton beam. It was to be coupled to a cyclotronproviding a 140 MeV proton beam; with this energy, neutron yield was estimated to be about 0.8 neu-trons per incident proton. The accelerator was designed for a maximum beam intensity of 500µA, butactual intensity was limited by target and core coolability issues. The target was to be cooled by forcedconvention while the core was to be cooled by natural convention. Maximum power levels achievable inthis way were to be about 20-40 kW in the target and 200-400 kW in the core.

The main objective of the TRADE experiment was to study the dynamical behavior of an ADS withreactivity feedback eects. The envisaged power levels were expected to make these eects noticeable,something that did not happen in the MUSE experiment due to the low power at which this experimentwas performed. Unfortunately, the TRADE experiment was canceled due to nancial problems after onlysome core characterization experiments were performed in 2003-2004. Nevertheless, after this cancellationand in support of EUROTRANS involvement in the RACE project, an experimental campaign wasperformed in the reactor in 2004-2006, under the name of RACE-T [288], centered in the validationof dierent subcriticality determination techniques and the development of instrumentation and a dataacquisition system.

3.1.4 GUINEVERE / FREYA

After the cancellation of the TRADE experiment, part of the resources were diverted to the less ambitiousGUINEVERE experiment [27, 63, 130]. This project couples the existing VENUS reactor, located atthe SCK-CEN facilities in Mol (Belgium) with the GENEPI accelerator already used in the MUSEexperiments. The interest of this experiment with respect to the MUSE experiment is that it will have afull lead core instead of a sodium core and that it will be allowed to operate with a continuous neutronsource, in addition to the pulsed source. However, unlike the TRADE experiment, the power level willnot be enough to measure reactivity feedback eects. The extension of the GUINEVERE experimentinto the 7th EURATOM framework has been named FREYA. Although it was initially envisaged to startthe experiments in 2009, the program suered several delays and it did not started until 2011.

The VENUS reactor is a zero-power open-pool light water moderated experimental reactor that hasbeen in operation since 1964 and has been used mainly for training purposes. For the GUINEVEREexperiment it has been modied into a fast reactor by designing a new core of uranium fuel within amatrix of lead blocks. A lead reector has also been added. The reactor with these modications hasbeen designed as VENUS-F. The core is now formed by fuel elements each of them containing nine fuelrods arranged in a 3× 3 square lattice with lead blocks between them and surrounded by additional leadplates, to simulate a lead coolant. Fuel is 30% enriched metallic uranium.

Each fuel rod is formed by three fuel rodlets of about 20 cm length piled up, for an overall fuel elementheight of about 60 cm. The width of these fuel elements is about 8 cm and they are arranged within a12×12 square grid to form an approximately cylindrical core. Not all positions of this grid are lled withthese fuel elements; the remaining outermost positions of the grid that are not lled with fuel elementsare lled with elements similar in shape but containing only fuel blocks. This grid is surrounded by axialreectors (top and bottom, 40 cm thickness) and radial lead reectors that ll all the remaining spacewithin the existing VENUS vessel, so that the shape of the whole reactor is a cylinder of about 140

3.2. FORMER USSR 43

cm height and 160 cm diameter. Dierent number of fuel elements can be used to have dierent corecongurations with dierent subcriticality levels.

A new reactor shutdown system has been developed for the modied VENUS-F reactor. It consists ofsix safety rods that are composed by an absorbing element (B4C with natural boron) and a fuel follower.During normal operation, electromagnets keep the fuel part introduced in the core in order to minimizethe perturbation in the ux shape; for shutdown, the electromagnets are de-energized and the fuel partis replaced by the absorbing part, thus the reactivity decrease is achieved both by fuel removal from thecore and by its replacement by absorbing material. In addition to these six safety rods, two control rodsof absorbing material are provided to allow for ne reactivity adjustments.

The GENEPI accelerator has been also modied into the GENEPI-3C (Continu - continuous) version[42, 43]. In addition to the pulsed mode of operation, which similar performance than in the MUSEexperiments (section 3.1.2), for GUINEVERE it will be also capable to operate in continuous mode, withsustained beam intensities of up to 1 mA, with maximum neutron production rates of 5× 1010 n/s. Theaccelerator beam enters the reactor from above; both D-D and D-T targets will also be available andthey will be placed in the center of the assembly, replacing the four central fuel elements.

3.1.5 MYRRHA

The next step of the European roadmap for the development of ADS is the MYRRHA reactor currentlyin the design phase.

The MYRRHA project traces its roots to the working group created in 1998 by the governments ofFrance, Italy and Spain to perform the conceptual design of an experimental ADS facility to act as aprototype for larger, industrial scale devices. Later, this activity was incorporated in the 5th EURATOMframework program under the PDS-XADS (Preliminary Design Studies of eXperimental ADS) project,carried out between 2001 - 2004. As a result of these activities, three dierent designs emerged [90].The rst of them was proposed by a team led by Ansaldo Nucleare and was an 80 MWth lead-bismuthcooled subcritical reactor coupled to a 600 MeV, 6 mA proton linear accelerator [89, 145]. The secondwas proposed by Framatome ANP and consisted of a 80 MWth helium cooled subcritical reactor coupledto a similar accelerator that the Ansaldo design [123]. Finally, the third design, named MYRRHA(Multipurpose hYbrid Research Reactor for High-end Applications), was proposed by a partnership ofthe SCK-CEN and IBA and was a 50 MWth lead-bismuth cooled subcritical reactor coupled to a 350MeV, 5 mA proton cyclotron [13].

The subsequent EUROTRANS project of the 6th EURATOM framework program [110], started in2005, comprised among its activities the detailed design of a facility for the short-term eXperimentaldemonstration of Transmutation in an ADS (XT-ADS). Lead-bismuth cooling was preferred over heliumcooling and the MYRRHA design was selected as starting point. The design work continues in the the7th framework program under the project CDT under the name MHYRRA-FASTEF (FAst SpectrumTransmutation Experimental Facility) [28].

The facility is to be built at the SCK-CEN site in Mol (Belgium) and (as of 2014) it is expected tobe commissioned by 2025. As of 2013 [26, 334] the thermal power of the MYRRHA facility was expectedto be between 65 and 100 MWth and it was expected to be capable to operate both in subcritical andcritical modes. The fuel used will be of the MOX type. The original accelerator requirement of 350 MeVand 5 mA has been changed to 600 MeV and 4 mA. Furthermore, a linear accelerator has been selectedinstead a cyclotron.

Apart from serving as a prototype of an industrial scale ADS, MYRRHA is intended to serve as amultirole facility whose roles will include material irradiation, radioisotope production and fundamentalphysics research, such as the ISOL@MYRRHA project [360], that aims to make use of part of theaccelerator beam to produce radioactive beams.

3.2 Former USSR

Several experimental ADSs have also been built or planned in various countries of the former USSR.Many of them make use of existing reactors, reactor fuel or accelerators. As of 2014, only three subcriticalfacilities have been successfully built: the Energy plus transmutation project at the JINR in Dubna(Russia) and the Yalina-Thermal and Yalina-Booster experiments at the JIPNR in Minsk (Belarus).Another project, the so-called SAD (Subcritical Assembly at Dubna) project in Dubna, also reached aconsiderable degree of development, although it was abandoned in the end. Other projects that havebeen abandoned after a certain degree of development are the XADS at ITEP and other facilities at the


Moscow Meson Factory. Finally, there is an ongoing project to build a neuron source at the KharkovInstitute of Physics and Technology (KIPT) in Kharkov (Ukraine) that is based on an accelerator drivensubcritical assembly.

3.2.1 Energy plus Transmutation

Under the Energy plus Transmutation project carried out at the Joint Institute for Nuclear Research ofDubna (Russia), a lead spallation target has been surrounded by a natural uranium subcritical assembly[192, 193, 371, 388]. The lead target has the shape of a cylinder of 520 mm length and 84 mm diameter.The subcritical assembly is made up of several consecutive sections separated by berglass panels, eachof them consisting of 30 natural uranium rods in aluminum cladding arranged in an hexagonal lattice.The fuel rods are 104 mm length and 36 mm diameter, hence four of these sections are required tocover the whole length of the lead target. The assembly was driven by proton beams produced in theSynchrophasotron or the Nuclotron synchrotrons, which are capable to provide beams of protons andheavier ions in the multi-GeV range.

Experiments started in 1999, and they have been mainly focused into the characterization of neutronuxes and spectrum inside both the spallation target and the surrounding blanket and the energy gainof the device using proton beams of dierent energies, ranging between 0.7 and 2.0 GeV. In a secondphase, similar experiments have been performed with a 2.52 GeV deuteron beam [194]. Furthermore,some irradiation experiments have been performed to measure ssion cross sections and transmutationrates of certain minor actinides and ssion products.

3.2.2 Yalina

The Joint Institute for Power and Nuclear Research (JIPNR) of the National Academy of Sciences ofBelarus, with nancial support of the ISTC and the U. S. Department of Energy, has built two small, zeropower subcritical assemblies, namely Yalina-Thermal and Yalina-Booster [66, 88, 126, 180, 314]. Theyhave been mainly used for kinetic studies of ADSs.

The rst of them, which began operation in 2000, consisted of a square lattice of EK-10 type fuelrods in polyethylene blocks with a graphite reector. This type of fuel rod has been used by manySoviet designed research reactors and consist of an alloy of UO2 and MgO with 10% enriched uraniumand aluminum alloy cladding. Its successor, the larger Yalina-Booster, began operation in 2005, andits most noticeable feature is the presence of a booster region in the centermost part of the assembly,surrounding the neutron source. This region is composed by rods of highly enriched uranium in leadblocks, and its aim is to increase the source importance. This booster is surrounded by a thermal zonecomposed by EK-10 fuel rods in polyethylene blocks, itself surrounded by a graphite reector. Originally,the booster was made up by 90% enriched metallic uranium fuel rods and 36% enriched UO2 fuel rods,but its enrichment has been since them reduced, in a rst step the 90% enriched fuel rods were replacedby 36% enriched UO2 fuel rods and in a second step, they were replaced by 21% enriched UO2. Furtherdetails of the Yalina-Booster facility are given in section 8.1.

The main neutron source available is a high intensity deuteron generator (NG-12-1), capable to pro-duce a 12 mA current of 250 keV deuterons and that can be coupled both to deuterium or tritium targets.Cf-252 and Am-Be sources has been also used.

3.2.3 SAD

The project SAD (Subcritical Assembly at Dubna) traces its roots to a series of projects raised in thelate 1990s that envisage the coupling of a subcritical core to one of the beam ports of the Phasotronsynchrocyclotron of the Joint Institute of Nuclear Research (JINR) of Dubna (Russia). The Phasotronsynchrocyclotron is operating since 1949 and accelerates protons to 660 MeV.

Earliest proposals envisage to couple the Phasotron to the core of the plutonium reactor IBR-30[31, 33]. When coupled to the Phasotron, this photoneutron source would have been replaced by auranium-molybdenum spallation target. Initially, it was envisaged to keep the metallic plutonium fuel ofthe IBR-30, but it was later decided to utilize MOX fuel elements of the type used in the BN-600 sodiumcooled fast reactor [23, 32, 271]. The SAD project itself, which involved several foreign institutions,including CEA (France), KTH (Sweden), FZK (Germany) and CIEMAT (Spain), was launched in 2001.By 2006 [320], the planned subcritical assembly was to consist of up to 141 fuel elements of same typeused in the BN-600. Each of these elements consisted of 18 fuel rods arranged in an hexagonal array.The fuel was of the MOX type (UO2 + PuO2, with 29.5% in weight of Pu). The spallation target was

3.3. JAPAN 45

made of lead and it was water-cooled. The keff achieved in this way was to range between approximately0.95 and 0.97, which allowed to reach a power level in the range of 20-30 kW using a beam power of 1kW. An upgrade consisting in the replacement of part of the lead of the spallation target by tungsten orberyllium, which allowed a 2 kW proton beam without melting, was also investigated [272]. This upgradewould have allowed to increase the assembly power to 100 kW.

The design phase of the SAD project was completed by 2006 along with some supporting experiments[132]. However, due to nancial reasons and a re in April 2005 that damaged the accelerator, the projectwas indenitely postponed. The possibility to use a tungsten photoneutron target driven by a 200 MeVelectron LINAC instead of the Phasotron was also proposed [273]. Beam power with this congurationwas to be 10 kW, but the lower neutron yields of photoneutron reactions than those of spallation causedthe ssion power of the assembly to be only of about another 10 kW. Anyway, this proposal did neithersuccess.

3.2.4 Other proposed experimental ADS facilities

The Institute of Theoretical and Experimental Physics (ITEP) proposed building an experimental ADSby refurbishing an already decommissioned research reactor and coupling it to an existing proton linearaccelerator [189, 190]. The accelerator was to provide a 36 MeV proton beam with an average current of0.5 mA. This beam was directed to a beryllium target placed in the center of the assembly. The targetwas designed in the shape of a conical surface with a thickness of 6 mm with the proton beam impingingfor the basis side. Neutron production was expected to be about 3× 1014 n/s. The assembly itself was toconsist of 90% enriched uranium fuel elements arranged in an hexagonal lattice immersed in D2O whichserved both as moderator and as coolant for the assembly and the target. Finally, the whole assemblywas to be surrounded by a graphite reector. The keff of the assembly was expected to be 0.95, enoughto achieve 100 kW of power. The facility was intended not only for ADS studies, but also for severalother usages using both the accelerator beam and the neutron ux. These include isotope production,material irradiation testing, activation analysis, silicon doping and neutron capture therapy.

Another project aimed to build a subcritical facility at the IN-06 neutron source of the Moscow MesonFactory belonging to the Institute for Nuclear Research of the Russian Academy of Sciences [274, 282].Presently this facility consists of a pulsed tungsten spallation source coupled to a 209 MeV proton LINAC[292]. Building the subcritical assembly would have required upgrading the accelerator energy to 500-600MeV and additional licensing [15].

A nal project is to build an ADS at the Kharkov Institute of Physics and Technology of Ukraine [115,124, 125, 389]. This project has been being pursued in collaboration with Argonne National Laboratoryof the USA. The subcritical assembly uses low-enriched (19.7 %) uranium fuel and is coupled to aphotoneutron source driven by a 100 kW beam of 100 MeV electrons. The system has been designed asa multi-role facility and its functions include medical isotope production, basic physics, material researchand personnel training. As of May of 2013, the facility was envisaged to start operations in 2014.

3.3 Japan

In Japan, there have been two independent projects to build experimental ADSs. The rst one is theNeutron Factory Project at the Kyoto University Research Reactor Institute (KURRI) and the other isthe Transmutation Experimental Facility that is to be built as a part of the J-PARC research complex,carried out by the KEK and the Japan Atomic Energy Agency.

3.3.1 Kyoto University Critical Assembly (KUCA)

In 1996, the Kyoto University Research Reactor Institute (KURRI) proposed the so called NeutronFactory Project, a plan to build an intense pulsed neutron source to complement or replace the existingKyoto University Reactor, a 5 MW pool type research reactor that had been in operation since 1964. Sincea spallation source of enough intensity was considered too expensive to be built, boosting a less intensespallation source with a subcritical assembly was considered a good option. Hence, it was proposed touse another existing facility of the KURRI, the Kyoto University Critical Assembly (KUCA), and turnit into an ADS [175, 318] by coupling it to a spallation target.

KUCA [222] has been operative since 1974 and it has been mainly used for neutronic studies andstudent training. The facility consists in fact of three dierent assemblies inside the same building, two ofthem (designated A and B cores) have solid moderator (dierent materials can be used, e.g. polyethylene


or graphite) and the third (the C core) is light-water moderated. Nevertheless, the control system of thefacility only allows for a single core to reach criticality at a given time. The A core is the one that hasbeen coupled to an accelerator within the Neutron Factory Project. Its maximum power output is 100W(usually 10W or less), although it can attain 1 kW in short term operation.

The proton accelerator complex is still in the development phase. Presently, its main component is a150 MeV Fixed Field Alternating Gradient (FFAG) synchrotron ring. It is coupled to a two stage injectorconsisting of two FFAG rings providing pulses of up to 6.0 × 108 protons at a repetition rate of 30 Hz,which corresponds to an average intensity of about 3 nA [249]. Two upgrades are being considered forthe accelerator system. The rst is to replace the current injector by a proton LINAC capable to providepulses of 3.12 × 1012 protons at repetition rates of up to 200 Hz, which will bring the beam averageintensity to the multi-µA range. The second upgrade is to add an additional FFAG synchrotron ring toincrease the proton energy to 700 MeV, which will allow to increase the neutron yield by a factor of 30for a given beam intensity. The spallation target is made of tungsten and, due to safety regulations, isplaced not in the center but outside the assembly, which has forced to install a neutron guide and a beamduct through the reector of the assembly to conduct the maximum possible number of neutrons to theassembly.

First injection of a beam of 100 MeV protons from the FFAG synchrotron into the assembly wasachieved in March 2009 [162, 280]. Experiments were performed with three dierent values of keff : 0.91,096 and 0.99. Due to beam losses in the main ring beam intensity was limited to 10 pA, which allowed fora production of 106n/s in the spallation target. During these rst experiments, the assembly was fueledwith 93% enriched uranium in the shape of metallic plates and polyethylene moderator and reector. Itis worth mentioning than before these experiments with the FFAG synchrotron, some characterizationexperiments were performed with a (D,T) fusion target driven by a 300 keV Cockcroft - Walton typeaccelerator. This accelerator operated in pulsed mode and allowed neutron production rates of up 108n/s.[277, 278, 279, 281].

3.3.2 Transmutation Experimental Facility (TEF) at J-PARC

As already commented in sections 1.3 and 2.2, in 1988, the Atomic Energy Commission of Japan launchedthe OMEGA partition and transmutation program. Their aims included the development of the tech-nologies required for high intensity accelerators and neutron spallation targets required for ADSs [215].It was soon realized that these technologies had a much broader eld of application than merely nucleartransmutation, which drove to the proposal by the Japan Atomic Energy Research Institute (JAERI)of its Neutron Science Project, that pretended to built a high intensity proton accelerator for severalapplications, among them, it was to be coupled to a subcritical assembly [226, 227].

In 2001, this project merged with KEK's Japan Hadron Facility (JHF) project into the so calledJ-PARC (Japan Proton Accelerator Research Complex) project. This large scale facility will consist ofcomplex of proton accelerators reaching energies of 50 GeV intended for a large number of applications,including a spallation neutron source, muon and neutrino production facilities and a hadron physicsresearch facility [229, 254, 255]. Among them, the Japan Atomic Energy Agency (successor of the JAERI)plans to build the so-called Transmutation Experimental Facility (TEF). The facility, however, has beenpostponed for the second phase of the J-PARC development and as of 2010 there were no scheduled datefor construction start.

The TEF at J-PARC will consist in fact of two facilities, both driven by a 600 MeV proton LINAC.They are Transmutation Physics Experimental Facility (TEF-P) and the ADS Target Test Facility (TEF-T) [307, 308, 309]. Of them, only TEF-P is a subcritical assembly; TEF-T, is an materials irradiationfacility using 200kW of the 600 MeV proton beam and intended to test a Pb-Bi spallation target.

TEF-T will be a very low (< 1kW) critical facility coupled to a lead spallation target and will take10W of the 600 MeV proton beam, corresponding to a neutron intensity of 1.5×1012 n/s in the spallationtarget. The subcritical assembly will be built using the fuel and components of the existing JAERI's FastCritical Assembly (FCA). It will have the shape of a vertical stainless steel grid containing the fuel, in theshape of plates of enriched uranium and plutonium, plus several other materials to simulate the dierentcomponents of an industrial ADS: lead or sodium to simulate the coolant, AlN to simulate nitride fuel,etc. The spallation target will be made of lead, but another materials can be tested at a later stage(Pb-Bi, W, etc). keff will be kept lower than 0.98. Although the main experiences are intended to studyADS physics, at later stages it under consideration as well to replace part of the fuel by minor-actinidenitride fuel. This last experience will require the development of novel fuel remote handling techniques.

3.4. USA 47

3.4 USA

ADS research in the USA was performed under the auspices of the Department of Energy Advanced FuelCycle Initiative (AFCI) . Within the frame of this program, the RACE (Reactor Accelerator CouplingExperiment) project was initiated in July 2003. This project envisaged coupling an electron LINACdriven photoneutron source to several subcritical cores [55, 56]. However, almost all research into ADSswas terminated with the cancellation of the AFCI in 2006 in favor of the development of the GenerationIV program.

Originally, the RACE project was to be carried out in three phases. In phase I, to be performedat the Idaho Accelerator Center of the Idaho State University (ISU-IAC), the subcritical core was acompact, zero power subcritical assembly specically built for the experiment. During the second phase,the external neutron source will be coupled to a TRIGA reactor of the University of Texas at Austin, andduring the third phase, it will be coupled to the TRIGA reactor of the Texas A&M University. Later, therst phase was renamed as RACE-LP (low power) and the other two were renamed as RACE-HP (highpower). During this second phase, the power level was expected to be large enough to observe thermalfeedback eects in the reactivity. In the end, before the program was terminated, only the rst phasewas realized during 2005 and 2006, in addition to some preliminary tests with an electron LINAC drivenphotoneutron target coupled to the TRIGA reactor at the University of Texas at Austin [250].

The neutron source used during the RACE experiments at the ISU-IAC consisted of an electron linearaccelerator operated at energies between 20 and 25 MeV coupled to a neutron producing target made ofan alloy of tungsten and copper. The accelerator provides pulses of neutrons of a few microseconds length(up to 5µs) at a frequency of up to 200 Hz. In this way it was possible to produce about 2×10−3 neutronsper incident neutron, or equivalently, about 1010 neutrons per µA of electron beam. Although maximumpower was about 1 kW, activation issues forced to limit beam power to the tens of Watt range, whichcorresponds to a total neutron yield in the target of about 1010n/s. The advantage of this photoneutronsource over the D-D or D-T sources used in similar experiments is that it produces a small contribution ofhigh energy neutrons (in the tens of MeV range) that better resembles the spectrum of spallation sources.

The subcritical assembly using during this phase consisted of a modular core made up of 150 fuelelements in the shape of plates of an alloy of 20% enriched uranium and aluminum in an aluminumcladding. These plates were placed in six boxes, each containing between 20 and 30 of them, andseparated by aluminum plates. This core was surrounded by a graphite reector and was placed into awater tank, so that the water did not only lled the space between the reector and the tank walls, butalso the spacing between fuel plates. The maximum keff achieved with this conguration was about 0.92and the experiments were carried out with values of keff between 0.88 and 0.92.

Experimental activities during the RACE experiments were mostly devoted to the development ofinstrumentation and techniques for reactivity measurements [57, 165]. It must be remarked that Euro-pean participation in this project was enabled within the EUROTRANS program and a series of jointexperiments were conducted at the Idaho Accelerator Center in 2006. The experiments at the Universityof Texas at Austin were also devoted to the study of reactivity measurement techniques.

3.5 China

Research into ADSs has also been performed at the Chinese Institute of Atomic Energy (CIAE) since1995. A research program was established in 2000 that resulted in the building of the subcritical assemblyVenus-I [316, 317, 384], which is operating since 2005. It is intended for ADS neutronic studies, but alsofor measuring minor actinide and ssion products transmutation rates.

This facility consists of a core with two coupled regions: a fast neutron zone in the center, surroundingthe neutron source, formed by natural uranium fuel within an aluminum matrix (aluminum was chosento simulate the spectrum of a sodium fast reactor) and a thermal zone surrounding it formed by 3%enriched UO2 in a polyethylene matrix. The reason to choose this conguration is to combine of a fastspectrum zone suitable to measure transmutation rates and a thermalized spectrum zone for to generateenough power. This conguration is similar to the Yalina-Booster facility, but in Venus-1 no valve zoneis present between the two zones. This core is surrounded by a polyethylene reector and shield and thecomplete structure is placed in a stainless steel cylindrical container. The assembly is designed to operatewith values of keff between 0.9 and 0.98. The main sources used are D-D and D-T ssion sources drivenby the CIAE Pulsed Neutron Generator (CPNG), a 600 keV, 15 mA Cockcroft-Walton type acceleratorAm-Be and Cf-252 sources are also employed.


3.6 Summary of Accelerator Driven System experiments

In tables 3.1 and 3.2 the main features and purposes of the dierent ADS experiments described in thischapter are presented. The earliest experiments (FEAT, Energy plus Transmutation) were dedicatedto experimentally demonstrate the concept of an ADS. Later experiments were zero power assembliesdedicated to the study the kinetic behavior of subcritical assemblies and the validation of reactivitymeasurement techniques, rst with a pulsed neutron source only (MUSE, RACE-LP) and later with acontinuous beam as well (Yalina, Guinevere), closer to a commercial facility. The next step was to bea facility of enough power level for reactivity temperature feedback eects to be present, and with aspallation source to be closer to an industrial facility, but the cancellation of TRADE, RACE-HP andSAD has caused this step to remain untested as of today. Hence, the MYRRHA reactor will be the rstone, but MYRRHA is a prototype reactor of relatively high power level that represents a considerableleap from previous experiments.

3.6. SUMMARY OF ACCELERATOR DRIVEN SYSTEM EXPERIMENTS 49

Table3.1:

Summaryof

ADSexperimentsperformed

orplannedup

todate.

Subcriticalassembly

Externalneutronsource

Experiment

Pow

erFuel

Moderator

Coolant

Reactiontype

Accelerator

type

Beam

energy

Beam

intensityor

power

FEAT

Verylow

Unat

H2O

H2O

Spallation

inU

Synchrotron,

p0.6-2.75

GeV

Verylow

MUSE

Verylow

MOX(25%

Pu)

Air

D-T,D-D

Electrostatic,D

250keV

50mA(pulse

peak)

TRADE

200-400

kWU(20%

)-ZrH

H2O

H2O

Spallation

inTa

Cyclotron,p

140MeV

500µA

Guinevere/F

REYA

Verylow

Umet

(30%

)

Air

D-T,D-D

Electrostatic,D

250keV

50mA(pulse

peak)

1mA(continuous)

MYRRHA

57MWth

MOX

Pb-Bi

Spallation

inPb

LINAC,p

600MeV

2.33

mA

E+T

Verylow

Unat

Air

Spallation

inPb

Synchrotron,

p,D

0.7-2.0GeV

(p)

2.52

GeV

(D)

Verylow

Yalina-T

Verylow

UO

2(10%

)-MgO

Polyethylene

Air

D-T,D-D

Electrostatic,D

250keV

12mA(pulse

peak)

1.5mA(continuous)

Yalina-B

Verylow

Umet

(36%

,90%)

UO

2(10%

)-MgO

Polyethylene

Air

D-T,D-D

Electrostatic,D

250keV

12mA(pulse

peak)

1.5mA(continuous)

SAD

20-30

kWMOX(29.5%

Pu)

Air

Spallation

inPb

Synchrocyclotron,

p660MeV

1kW

KUCA

Verylow

Umet

(93%

)Polyethylene

Air

Spallation

inW

FFA

Gsynchrotron,

p150MeV

∼µA

TEF-P

Verylow

U,Pu

Air

Spallation

inPb

LINAC,p

600MeV

10W

RACE-LP

Verylow

U(20%

)-Al

H2O

H2O

Photoneutrons

LINAC,e−

20-25MeV

∼10

W

RACE-HP

?U-ZrH

H2O

H2O

Photoneutrons

LINAC,e−

20-25MeV

?

Venus-1

Verylow

Unat-LEU(3%)

Polyethylene

Air

D-T,D-D

Electrostatic,D

upto

600keV

15mA(pulse

peak

?)


Table

3.2:Activities

carriedout

orplanned

indierent

ADSexperim

entsperform

edor

plannedup

todate.

Exp erim

entEnergy

gainMAs&

LLFPsfuel,

transmuta-

tionrates

Subcriticalreactorphysics

Reactivity

monitoring

with

pulsedsource

Reactivity

monitoringwith

continuoussource

Reactivity

temperature

feedbackeects

Spallationsource

Prototype

forindustrialsystem

FEAT

YES

NO

NO

NO

NO

NO

YES

NO

MUSE

YES

NO

YES

YES

NO

NO

NO

NO

TRADE

YES

NO

YES

YES

YES

YES

YES

NO

Guinev ere

YES

NO

YES

YES

YES

NO

NO

NO

MYRRHA

YES

YES

YES

YES

YES

YES

YES

YES

E+T

YES

YES

NO

NO

NO

NO

YES

NO

Yalina-T

YES

YES

YES

YES

YES

NO

NO

NO

Yalina-B

YES

YES

YES

YES

YES

NO

NO

NO

SAD

YES

NO

YES

YES

NO

YES

YES

NO

KUCA

YES

NO

YES

YES

NO

NO

YES

NO

TEF-P

YES

YES

YES

??

NO

YES

NO

RACE-LP

YES

NO

YES

YES

NO

NO

NO

NO

RACE-HP

YES

NO

YES

YES

NO

YES

NO

NO

Venus

YES

YES

YES

YES

?NO

NO

NO

Part II

Theoretical introduction

51

Chapter 4

Neutron transport theory

Abstract - In this chapter, neutron transport theory is developed, starting from the general derivation

of the Boltzmann transport equation and then applying it to the specic case of neutron transport

in multiplicative media. After that, the solutions of the neutron transport equation are discussed. In

particular, the concepts of eective multiplicative constant (keff ) and source multiplication (ks) are

introduced and calculated for the case of a spherical reactor (for which an analytical solution exists)

in order to show numerically the dierence between these two concepts.

The basic problem of nuclear reactor design and analysis is the knowledge of the space distribution andtime evolution of the neutron population within the reactor. As this is essentially a transport problem itcan be studied with the Boltzmann transport equation. This equation was originally derived by LudwigBoltzmann (1844-1906) within the frame of the classical kinetic theory of gases, but it has since foundapplication to study transport phenomena in many other elds such as condensed matter physics, plasmaphysics or, in the matter in hand, nuclear reactor physics. Thus, in the rst section of this chapter,the Boltzmann transport equation is derived in its general form. This derivation is based on my ownclass notes from the course on Advanced Statistical Physics imparted by Prof. J. M. Mateos Roco atthe University of Salamanca in 2007. Other references are [152, 284]. Then, in the second section,this equation is applied to the case of neutron transport to nally obtain the fundamental equation ofneutron transport theory. Main references used for this section are [53, 252], other references used include[100, 106, 144, 199] and the class notes from the course on Nuclear Technology given by Prof. O. Cabellosat the Polytechnical University of Madrid in 2007.

4.1 The Boltzmann transport equation

As stated above, the Boltzmann transport equation was originally derived within the frame of the classicalkinetic theory of gases. In classical kinetic theory of gases, the system under study is a dilute gas formedby particles that can be considered to be point-like (that is to say, the average distance among theparticles within the gas is much larger than their dimensions) and that are well described by its positionand momentum. These particles are supposed to follow the laws of classical mechanics. The particlesinteract among themselves through short range interactions whose eect is a change in the energy andthe direction of movement of the particles. Furthermore, the particles can also be aected by externalforce elds, such as gravitational or electromagnetic forces.

Let us begin by dening a phase space formed by the three spatial components and the three velocitycomponents of the particles. Let us dene the distribution (or density) function f (~r,~v, t) so that thenumber of particles in the phase space element d~rd~v around the point (~r,~v) in the instant of time t is givenby f (~r,~v, t) d~rd~v. In the absence of collisions and by eect of their own movement, in an innitesimallylater instant of time t′ = t + dt all the particles in this volume element will lay in the volume elementd~r′d~v′ around the point (~r′, ~v′) (see gure 4.1). Hence:

f (~r,~v, t) d~rd~v = f (~r′, ~v′, t′) d~r′d~v′

where ~r′ and ~v′ are given by:

~r′ = ~r + ~rdt = ~r + ~vdt

53

54 CHAPTER 4. NEUTRON TRANSPORT THEORY

Figure 4.1: Geometry to derive the Boltzmann transport equation in a simplied phase space consistingonly of one spatial component and one velocity component.

~v′ = ~v + ~vdt = ~v +~F

mdt

Where ~F denotes the force due to any external force eld, should it be applied. For the shake ofsimplicity, let us suppose that it depends only on the spatial components ~r and not on the velocitycomponents ~v, that is, ~F = ~F (~r).

The volume element d~rd~v is related to d~r′d~v′ by the Jacobian of the transformation, that is:

d~r′d~v′ = |J |d~rd~v

where the Jacobian |J | is given by:

J =∂ (x′, y′, z′, vx′ , vy′ , vz′)

∂ (x, y, z, vx, vy, vz)=

∣∣∣∣∣∣∣∣∣∣∣∣∣

1 0 0 dt 0 00 1 0 0 dt 00 0 1 0 0 dt

1m∂Fx∂x dt

1m∂Fx∂y dt

1m∂Fx∂z dt 1 0 0

1m∂Fy∂x dt

1m∂Fy∂y dt

1m∂Fy∂z dt 0 1 0

1m∂Fz∂x dt

1m∂Fz∂y dt

1m∂Fz∂z dt 0 0 1

∣∣∣∣∣∣∣∣∣∣∣∣∣=

= 1− 1

m

(∂Fx∂x

+∂Fy∂y

+∂Fz∂z

)dt2

If only a linear transport theory is considered, neglecting non linear eects, we can limit to rst orderapproximation in J , which turns out to be a simply one:

d~r′d~v′ = d~rd~v

so we end up with:

f (~r,~v, t) d~rd~v = f (~r′, ~v′, t) d~rd~v ⇒ f (~r,~v, t) = f (~r′, ~v′, t)⇒ f (~r,~v, t) = f

(~r + ~vdt,~v +

~F

mdt, t

)

Again, if a linear transport theory is considered, a Taylor series expansion to the rst order can beperformed at the right-hand side of the previous equation and obtain that:

4.2. APPLICATION TO NEUTRON TRANSPORT 55

f

(~r + ~vdt,~v +

~F

mdt, t

)= f +

∂f

∂tdt+ ~v∇~rfdt+

~F

m∇~vfdt

where the dependence of f on ~r, ~v and t has been dropped for simplicity. With this result, we havearrived to the conclusion that in absence of collisions the distribution function f satises the equation:

f

(~r + ~vdt,~v +

~F

mdt, t

)− f (~r,~v, t) = 0⇒ ∂f

∂tdt+ ~v∇~rfdt+

~F

m∇~vfdt = 0⇒

⇒ ∂f

∂t+ ~v∇~rf +

~F

m∇~vf = 0 (4.1)

Equation 4.1 is known as the collisionless Boltzmann transport equation. The generalization to thecase with collisions is straightforward, just by adding a term that takes into account the eect of collisions:

∂f

∂t+ ~v∇~rf +

~F

m∇~vf =

(∂f

∂t

)coll

(4.2)

This last expression constitutes the Boltzmann transport equation. To clarify the meaning of thisequation, let us write it in the way:

∂f

∂t= −~v∇~rf −

~F

m∇~vf +

(∂f

∂t

)coll

And let us integrate it over a volume V in the phase space:∫V

∂f

∂td~rd~v = −

∫V

~v∇~rfd~rd~v −∫V

~F

m∇~vfd~rd~v +

∫V

(∂f

∂t

)coll

d~rd~v

It is immediate to see that the term on the left-hand side represents the net variation of the number ofparticles in the phase space volume V . On the right-hand side, ~v can be introduced into the ∇~r operatorin the rst term using that in the phase space formalism the velocity coordinates ~v are independent ofthe space coordinates ~r and then apply the divergence theorem to nd that:∫

V

~v∇~rfd~rd~v =

∫V

∇~r (~v·f) d~rd~v =

∫∂V

f~vd~s~rd~v

where ∂V denotes the boundary of V . This term represents the net number of particles leaving V byeect of their own movement, in absence of collisions. Thus this is often referred as the streaming term.

The second term in the right-hand side can be worked similarly and it represents the net number ofparticles leaving V due to the eect of the external force eld. Finally, the last term on the right-handside represents the net change in the number of particles in the phase space volume due to the collisions.Hence, we have arrived to the common interpretation of the Boltzmann transport equation as a balanceequation of the number of particles within a phase space volume: the net variation of the total number ofparticles within this volume is equal to net variation of the number of particles due to their own movementin absence of collisions (streaming) and due to the eect of external force elds plus the net variation inthe number of particles due to collision processes. This interpretation as balance equation is in fact usedin many basic texts as the starting point to directly derive the Boltzmann transport equation.

4.2 Application to neutron transport

4.2.1 The neutron transport equation

To assess if the Boltzmann transport equation derived in the previous section is also suitable for describingthe neutron transport in a medium, it is necessary to determine whether the assumptions made forgases (that they are formed by point-like particles that follow the laws of classical mechanics and thatinteractions are of short range) do apply for the case of neutrons. These assumptions are discussed inthe following points:


Neutrons can be considered point-like particles. For a thermal neutron with an energy of about0.025 eV, its reduced wavelength, given by the De Broglie formula:

λ =~p

=~√

2mE' 2.9× 10−11m

is about one order of magnitude smaller than typical interatomic distances in a solid, which areof the order of several Å or ' 10−10 m. Since λ is inversely proportional to

√E, fast neutrons

have shorter wavelengths. Consequently, as a rst approximation, neutrons can be considered tobe point like particles in fast and thermal systems, although this approximation is near the limitof validity for the case of thermal systems and it is not valid in systems with a large componentof lower energy neutrons. The wave-like behavior of low energy neutrons is relevant enough forthermal scattering eects to be included in nuclear data libraries and it can be taken into accountby computer codes like MCNP/MCNPX (chapter 6). This wave-like behaviour constitutes, in fact,the basis of neutron diraction and interferometry techniques that make use of low energy (coldand ultracold) neutrons.

Neutrons behave like classical particles. Typical neutron energies can rise up to a few MeV, stillseveral orders of magnitude below its mass (roughly 1 GeV), thus rendering relativistic eectsnegligible.

Neutron interactions are short-ranged. Since neutrons are chargeless particles, electromagneticforces do not aect neutron transport, and therefore neutrons interact among themselves or withthe nuclei in the medium through short range nuclear forces. This fact implies as well that the term~Fm∇~vf can be dropped from the Botzmann's transport equation when applied to neutrons, sincethe other force eld also present, the gravitational, is usually too weak to aect them1.

With these considerations, it seems that the Boltzmann's transport equation can be applied to neutrontransport with enough accuracy in most cases of interest in nuclear reactor physics. Following the notationof Bell & Glasstone [53] and Duderstat & Hamilton [106], instead of the distribution functions of thekinetic theory of gases, let us write it in terms of an equivalent quantity which is the neutron angular

density (also referred as neutron density in the phase space in [144]) that will be denoted by N(~r,E, ~Ω, t

).

In a completely analogous way to the distribution function, it is dened so that the number of neutronsN(~r,E, ~Ω, t

)in the space element d~r around the point ~r with energies in the element of energy dE around

E and with directions lying in the element d~Ω around ~Ω at time t is given by N(~r,E, ~Ω, t

)d~rdEd~Ω2.

Another dierence with the kinetic theory of gases, where it is considered that the gas particles onlyinteract among themselves, is that neutrons do not only interact among themselves but also with thenuclei present in the medium. Therefore, is necessary to write a transport equation for each one ofthe isotope species present in the medium. Hence, let us dene an anglar density function for the i-thisotope N ′i

(~r,E, ~Ω, t

)in an analogous manner as the neutron angular density N

(~r,E, ~Ω, t

). With these

denitions, providing that there are n dierent isotopes present in the medium, it is possible to write aset of n+ 1 transport equations for the neutrons and the n isotopes as:

∂N

∂t+ ~v∇~rN =

(∂N

∂t

)neutronneutron

(N,N) +

(∂N

∂t

)neutronnucleus

(N,N ′1...N′n) (4.3)

∂N ′i∂t

+ ~v∇~rN ′i =

(∂N ′i∂t

)nucleusnucleus

(N ′1...N′n, N

′1...N

′n) +

(∂N ′i∂t

)nucleusneutron

(N,N ′1...N′n) (4.4)

1Incidentally, the eect of gravity on very low-energy (ultracold) neutrons has been experimentally observed [9].2Notice that f (~r, ~v, t) can be also expressed as f

(~r, v, ~Ω, t

), just by splitting the dependence on ~v in dependence on the

modulus v and the direction ~Ω. Furthermore, as v is v = v (E) as E = 12mnv2 (in the non-relativistic case, that, as stated

above, holds for neutrons), we can nally write f(~r, E, ~Ω, t

). Notice that although ~Ω has three components only two of

them are independent, as their value is linked by the normalization condition√

Ω2x + Ω2

y + Ω2z = 1.


where the collisions terms have been explicitly separated in two terms each due to neutron-neutron,neutron-nucleus, nucleus-nucleus and nucleus-neutron interactions. The term with ~F has been droppedbecause, as stated above, no external force eld can act upon the neutrons.

These equations can be simplied taking into account that neutron-neutron interactions are usuallynegligible with respect to neutron-nucleus interactions. This is because neutron densities are several ordersof magnitude lower than typical atomic densities of materials even in the highest ux reactors3. For thesame reason, neutron-nucleus collisions can be considered to have no signicant impact in modifying thenuclei distribution of the media (at least in the short term), which eectively decouples equation 4.4 fromequation 4.3. Hence, if the nuclei distributions N ′i are considered to be in equilibrium (i. e., to be timeindependent) and to be known, the neutron transport equation can be written in the shape:

∂N

∂t+ ~v∇~rN =

(∂N

∂t

)coll

(4.5)

where the term(∂N∂t

)neutronnucleus

(N,N ′1...N′n) has been rewritten as

(∂N∂t

)coll

, that takes into account

only the interactions of the neutrons with the nuclei.The task is now to obtain an expression for this collision term. For this, all the processes that can

modify the number of neutrons in a phase space element must be considered. The processes that aregoing to be considered in this work are:

1. Neutron absorption. This process always causes a loss of neutrons in the phase space elementd~rd~ΩdE .

2. Scattering. This process causes the transference of neutrons from the element with E and ~Ω intoother elements with a dierent E′ and ~Ω′. This process can cause either a gain in the neutronnumber in the phase volume element due to neutrons coming from other phase volume elements(in-scattering) or a lost due to neutrons being scattered to other phase volume elements (out-scattering). Within this process we are including both elastic and inelastic scattering.

3. Fission. This process has two eects: on the one hand, it causes the loss of the neutrons thatproduce the ssion, in a similar way than absorption. On the other hand, as new neutrons emergefrom ssion it causes a gain of neutrons.

4. (n, xn) reactions. in the same way as ssion, this process has two eects: the loss of the neutronsthat produce the reaction and the production of additional neutrons in the process.

5. External sources. By external sources we refer to neutron sources that are independent of theneutron density. Thus ssion and (n, xn) are not considered as external sources.

With these considerations the collision term can be split in ve terms in the way:

∂N

∂t+ ~v∇~rN = −

(∂N

∂t

)abs

−(∂N

∂t

)outscat

+

(∂N

∂t

)inscat

+

(∂N

∂t

)fis

+

(∂N

∂t

)sour

(4.6)

To nd an expression for these terms, it is necessary to introduce the concepts of reaction rate andcross section. If we have a medium formed by nuclei and neutrons interacting, the reaction rate of anX reaction will be proportional to both neutrons and nuclei densities and to the relative speed of theneutrons respect to the nuclei. That is, considering a reference system where the nuclei are at rest:(

∂N

∂t

)X

(~r;E, ~Ω; t

)= σX

(~r;E, ~Ω

)N(~r;E, ~Ω; t

)ρ (~r) v (4.7)

where N(~r;E, ~Ω; t

)is the neutron angular density dened before, ρ (~r) is the nuclei density and v

is the modulus of the neutron velocity which in this reference system correspond with the velocity ofthe neutrons with respect to the nuclei. The proportionality factor is denoted by σX and will depend ingeneral on the composition (and thus on ~r) and on the energy and direction of the incident neutrons. If

3A ux of 1015n/cm3s of thermal neutrons of 0.025 eV, equivalent to 2.2 × 105cm/s, corresponds to a neutron densityof 4.5× 109n/cm3, while typical atomic densities of materials are of the order of 1022at/cm3


the reaction is isotropic, the dependence on ~Ω will drop. It is easy to see that the dimensions of σX arethose of an area and thus it is referred as cross section.

In case the reaction causes a change in the energy or the direction of the particles, equation 4.7 turnsinto:

(∂N

∂t

)X

(~r;E, ~Ω→ E′, ~Ω′; t

)= σX

(~r;E, ~Ω

)fX

(~r;E, ~Ω→ E′, ~Ω′; t

)N(~r;E, ~Ω

)ρ(~r)v (4.8)

Where fX(~r;E, ~Ω→ E′, ~Ω′

)is named the transfer function and denotes the fraction of neutrons that

are transferred from an element d~ΩdE around the energy E and the direction ~Ω into an element aroundthe energy E′ and the direction ~Ω′ . The other terms have the same meaning than in equation 4.7.

Before going further, let us make two usual denitions that will allow for simplications. First, it isusual to dene macroscopic cross section as the product of the cross section by the nuclei density:

ΣX

(~r;E, ~Ω

)= σX

(~r;E, ~Ω

)ρ (~r)

The dimensions of this new quantity are those of an inverse length. It can be shown that the physicalmeaning of this quantity is the inverse of the mean free path that a neutron has to travel since it isgenerated until it undergoes the reaction X. To distinguish from this new quantities, the cross sectionsσX are usually referred as microscopic cross sections.

Another useful denition is the neutron scalar ux. It is dened as the product of the neutron densityby the velocity v :

Φ(~r;E, ~Ω, t

)= vN

(~r;E, ~Ω, t

)The dimensions of this new dened quantity are those of inverse area by inverse time and its physicalmeaning is the number of neutrons with energy E and direction ~Ω that go through the element of areaat the position ~r at time t. In fact, it is the neutron scalar ux rather than the neutron angular densitythe most commonly used quantity to describe the neutron population within a reactor.

In terms of these new dened quantities, the Boltzmann transport equation can be written as:

1

v

∂Φ

∂t+ ~Ω∇~rΦ =

(∂N

∂t

)coll

(4.9)

And the above reaction rates:(∂N

∂t

)X

(~r;E, ~Ω; t

)= ΣX

(~r;E, ~Ω

)Φ(~r;E, ~Ω; t

)(∂N

∂t

)X

(~r;E, ~Ω→ E′, ~Ω′; t

)= ΣX

(~r;E, ~Ω

)fX

(~r;E, ~Ω→ E′, ~Ω′

)Φ(~r;E′, ~Ω′; t

)With all these denitions, we can give expressions for each of the terms of equation 4.6:

1. Absorption. The absorption term can be straightforwardly written as:(∂N

∂t

)a

(~r;E, ~Ω; t

)= Σa (~r;E) Φ

(~r;E, ~Ω; t

)2. Scattering. The eect of the neutrons coming from other phase volume elements (in-scattering) can

be described with a term of the shape:(∂N

∂t

)inscat

(~r;E, ~Ω; t

)=

∫ ∫Σs (~r;E′) f

(~r;E′, ~Ω′ → E, ~Ω

)Φ(~r;E′, ~Ω′; t

)dE′d~Ω′

The eect of the neutrons being scattered to other phase volume elements (out-scattering) can bedescribed in a simpler way, as in this case the integral of the transfer function over the entire rangeof energies and angles must be one:(

∂N

∂t

)outscat

(~r;E, ~Ω; t

)=

∫ ∫Σs (~r;E) f

(~r;E, ~Ω→ E′, ~Ω′

)Φ(~r;E, ~Ω; t

)dE′d~Ω′ =

= Σs (~r;E) Φ(~r;E, ~Ω; t

)Notice that with these expressions we are taking into account both elastic and inelastic scattering.


3. Fission. The losses due to ssion can be described by a term similar to the absorption or out-scattering ones: (

∂N

∂t

)fis,−

(~r;E, ~Ω; t

)= Σfis (~r;E) Φ

(~r;E, ~Ω; t

)The neutron production in ssion processes can be described by:(∂N

∂t

)fis,+

(~r;E, ~Ω; t

)=

1

4π

∫ t

0

∫ ∫ν (E′) Σfis (~r;E′)χ (E′, t′ → E, t) Φ

(~r;E′, ~Ω′; t′

)dE′d~Ω′dt′

Where ν (E′) denotes the average number of neutrons produced in a ssion induced by an incomingneutron of energy E′ and χ (E′, t′ → E, t) is a function that describes the spectrum of energy Eof the neutrons produced in the ssion and the time t at which these neutrons appear in a ssioninduced at time t′ by an incoming neutron of energy E′. It is necessary to consider the time at whichssion neutrons appear because, as it will be described later with greater detail, ssion neutronscan appear in the same instant of the ssion (prompt neutrons) or a certain time after it as resultof the decay of the ssion fragments (delayed neutrons). The term 4π arises from considering thatthe neutrons emerging from ssion have an isotropic distribution.

4. (n, xn) reactions. This process can be described in a similar way to ssion, being the number ofneutrons lost due to this process given by:(

∂N

∂t

)(n,xn),−

(~r;E, ~Ω; t

)=∑x

Σ(n,xn) (~r;E) Φ(~r;E, ~Ω; t

)And the neutron production by: (

∂N

∂t

)(n,xn),+

(~r;E, ~Ω; t

)=

=∑x

1

4π

∫ ∫xΣ(n,xn) (~r;E′) ξ (E′ → E) Φ

(~r;E′, ~Ω′; t

)dE′d~Ω′

Where ξ (E′ → E) is a function that describes the spectrum of energy E of the neutrons producedin a (n, xn) reaction produced by an incoming neutron of energy E′. In this case, and contrary tossion, the new neutrons appear at the same instant of time that the reaction occurs. Again, theterm 4π arises from considering that the neutrons produced have an isotropic distribution.

5. External source. We will denote any external neutron source by:(∂N

∂t

)sour

(~r;E, ~Ω; t

)= Sext

(~r;E, ~Ω; t

)To simplify, let us write all the previous eects that cause lost of neutrons in one term together in theway (

∂N

∂t

)tot

(~r;E, ~Ω; t

)=

(∂N

∂t

)abs

(~r;E, ~Ω; t

)+

(∂N

∂t

)outscat

(~r;E, ~Ω; t

)+

+

(∂N

∂t

)fis,−

(~r;E, ~Ω; t

)+

(∂N

∂t

)(n,xn),−

(~r;E, ~Ω; t

)=

=

Σa (~r;E) + Σs (~r;E) + Σfis (~r;E) +

∑x

Σ(n,xn) (~r;E)

Φ(~r;E, ~Ω; t

)= Σtot (~r;E) Φ

(~r;E, ~Ω; t

)So we can nally write the Boltzmann equation for neutron transport:

1

v

∂Φ

∂t+ ~Ω∇~rΦ = −Σtot (~r;E) Φ

(~r;E, ~Ω; t

)+

+

∫ ∫Σs (~r;E′) f

(~r;E′, ~Ω′ → E, ~Ω

)Φ(~r;E′, ~Ω′; t

)dE′d~Ω′+


+1

4π

∫ t

0

∫ ∫ν (E′) Σfis (~r;E′)χ (E′, t′ → E, t) Φ

(~r;E′, ~Ω′; t′

)dE′d~Ω′dt′+

+∑x

1

4π

∫ ∫xΣ(n,xn) (~r;E′) ξ (E′ → E) Φ

(~r;E′, ~Ω′; t

)dE′d~Ω′ + Sext

(~r;E′, ~Ω′; t

)(4.10)

Taking into account the complexity of this expression, it is useful to dene the following operators:

Migration and losses operator, M . This operator takes into account the neutrons that are lost eitherby absorption and ssion, as well as the net losses either by streaming or scattering.

MΦ = ~Ω∇~rΦ(~r;E, ~Ω; t

)+ Σtot (~r;E) Φ

(~r;E, ~Ω; t

)−

−∫ ∫

Σs (~r;E′) f(~r;E′, ~Ω′ → E, ~Ω

)Φ(~r;E′, ~Ω′; t

)dE′d~Ω′

Creation operator, F . This term takes into account the neutrons that are created in ssions and in(n, xn) reactions.

FΦ =1

4π

∫ t

0

∫ ∫ν (E′) Σfis (~r;E′)χ (E′, t′ → E, t) Φ

(~r;E′, ~Ω′; t′

)dE′d~Ω′dt′+

+∑x

1

4π

∫ ∫xΣ(n,xn) (~r;E′) ξ (E′ → E) Φ

(~r;E′, ~Ω′; t

)dE′d~Ω′

In terms of these operators, the neutron transport equation can be written as:

1

v

∂Φ

∂t= FΦ− MΦ + Sext (4.11)

where the dependencies of the terms have been omitted for simplicity.It is also interesting to introduce the transport operator L dened as:

L = F − M

so that the neutron transport equation can take the simple form:

1

v

∂Φ

∂t= LΦ + Sext (4.12)

4.2.2 Prompt and delayed neutrons

As stated above, not all neutrons produced in the ssion process are emitted in the same instant of timethat ssion, but a fraction of them appears some time later as the result of the decay of certain ssionproducts called the precursors. These neutrons are called delayed neutrons to be distinguished from theones produced instantaneously that are referred as prompt neutrons.

Following the notation of Bell & Glasstone (chapter 9) [53], let us denote by βj the fraction of thewhole number of neutrons emitted in the ssion process due to the decay of a j precursor with decayconstant λj . This fraction is dependent on the composition of the medium (as it depends on the ssilematerial present) and the energy of the incident neutron, hence it holds that βj = βj (~r,E′). Thus, thefraction of delayed neutrons that appear per unit time with an energy E as the result of the decay of thej-th precursor produced by a ssion induced by a neutron of energy E′ will be given by:

χj (E)βj (~r,E′)λje−λjt

where the χj (E) is the normalized energy spectrum of the delayed neutrons emitted by the j-th precursor.This new parameter is introduced because each precursor has a particular spectrum of delayed neutronswhich is dierent from the spectrum of prompt neutrons.


Let us now denote by β (~r,E′) the total fraction of neutrons that is emitted as the result of decayingssion fragments, that is to say, the sum of the fractions of all precursors:

β (~r,E′) =∑j

βj (~r,E′)

The fraction of prompt neutrons is then obviously 1− β (~r,E′). Hence, the neutron fraction that emergeas prompt neutrons with an energy E after a ssion induced by a neutron of energy E′ will be given by:

χp (E′ → E) (1− β (~r,E′)) δ (t− t′)

where the Dirac delta function accounts for the fact that the prompt neutrons are produced in thesame instant as the ssion and χp (E′ → E) is the normalized energy spectrum of the prompt ssionneutrons induced by an incoming neutron of energy E′. So we arrive to the conclusion that the functionχ (E′, t′ → E, t) takes the shape:

χ (E′, t′ → E, t) = χp (E′ → E) (1− β (~r,E′)) δ (t− t′) +∑j

χj (E)βj (~r,E′)λje−λj(t−t′) (4.13)

With all these considerations, we can write the ssion term above in the way:(∂N

∂t

)fis,+

(~r;E, ~Ω; t

)=

1

4π

∫ t

0

∫ ∫ν (E′) Σfis (~r;E′) χp (E′ → E) (1− β (~r,E′)) δ (t− t′) +

+∑j

χj (E)βj (~r,E′)λje−λj(t−t′)

Φ(~r;E′, ~Ω′; t′

)dE′d~Ω′dt′ (4.14)

It is usually more convenient to write the above expression as:(∂N

∂t

)fis,+

(~r;E, ~Ω; t

)=

1

4π

∫ ∫ν (E′) Σfis (~r;E′)χp (E′ → E) (1− β (~r,E′)) Φ

(~r;E′, ~Ω′; t

)dE′d~Ω′+

+∑j

χj (E)

4πλjCj (~r;E; t) (4.15)

where the quantity Cj (~r;E; t) is given by:

Cj (~r;E; t) =

∫ t

0

∫ ∫βj (~r,E′) e−λj(t−t

′)ν (E′) Σfis (~r;E′) Φ(~r;E′, ~Ω′; t

)dE′d~Ω′dt′ (4.16)

and is just the concentration of the j precursor. It is usual to give an integro dierential equationfor this quantity, which is obtained just by applying the derivation under the integral sign in the lastexpression:

∂Cj (~r;E; t)

∂t= −λjCj (~r;E; t) +

∫ ∫βj (~r,E′) ν (E′) Σfis (~r;E′) Φ

(~r;E′, ~Ω′; t

)dE′d~Ω′ (4.17)

that states just that the net variation in the concentration of the j-th precursor is just the dierencebetween the production rate in ssions and the destruction rate by decay.

In a similar way it has been done above, it is possible to dene a creation operator that takes intoaccount only prompt ssion neutron production as:

Fpφ =1

4π


(~r;E′, ~Ω′; t

)dE′d~Ω′+

+∑x

1

4π

∫ ∫xΣ (~r;E′) ξ (E′ → E) Φ

(~r;E′, ~Ω′; t

)dE′d~Ω′ (4.18)

and a decay neutron source as:

Sd (~r;E; t) =∑j

χj (E)

4πλjCj (~r;E; t)


Table 4.1: Summary of neutron densities and uxes and their relationships.

Quantity Notation Denition

Neutron angular density N(~r, t; ~Ω, E)

Neutron density n(~r, t;E) n(~r, t;E) =∫

4πN(~r, t; ~Ω, E)d~Ω

Neutron angular ux Φ(~r, t; ~Ω, E) Φ(~r, t; ~Ω, E) = v (E)N(~r, t; ~Ω, E)

Neutron scalar ux φ(~r, t; , E) φ(~r, t;E) = v (E)n(~r, t;E)

Neutron current ~J(~r, t;E) ~J(~r, t;E) =∫

4π~ΩΦ(~r, t; ~Ω, E)d~Ω

In terms of these two new dened quantities we can write the neutron transport equation with delayedneutrons as:

1

v

∂Φ

∂t= FpΦ− MΦ + Sd + Sext (4.19)

Notice that this equation can also be expressed as:

1

v

∂Φ

∂t=(F − Fd

)Φ− MΦ + Sd + Sext (4.20)

Where the delayed neutron operator is just:

FdΦ =∑j

FdjΦ =∑j

χj (E)

4π

∫ t

0

∫ ∫βj (~r,E′) ν (E′) Σfis (~r;E′) Φ

(~r;E′, ~Ω′; t′

)dE′d~Ω′dt′ (4.21)

Notice that equation 4.20 may appear to have little sense since Sd = FdΦ. However, this expression willbe used later.

4.2.3 The diusion approximation

A common approximation to the transport equation is the so-called diusion approximation. In fact,many basic reactor physics books limit themselves to this approximation, instead of fully developingthe transport theory based on the Boltzmann transport equation. The motivation to introduce thisapproximation here is because it will be used in section 4.3.1 to obtain simple solutions of the neutrontransport equation.

Let us begin by considering equation 4.10. For simplicity, let us not consider either delayed neutronsor (n, xn) reactions. Upon integration over all possible directions ~Ω, it is found that:

1

v

∂φ

∂t+∇~r ~J = −Σtot (~r;E)φ (~r, t;E) +

∫Σs (~r;E′) f (~r;E′ → E)φ (~r, t;E′) dE′+

+

∫ν (E′) Σfis (~r;E′)χ (E′ → E)φ (~r, t;E′) dE′ + Sext (~r, t;E′) (4.22)

where φ(~r, t;E) is called neutron scalar ux, dened as:

φ(~r, t;E) =1

4π

∫Ω

Φ(~r, t; ~Ω, E)d3~Ω (4.23)

that represent the number of neutrons that go through the unit surface per unit time in any direction,and ~J(~r, t;E) is called neutron current, dened as:

~J(~r, t;E) =

∫Ω

~ΩΦ(~r, t; ~Ω, E)d3~Ω (4.24)

The physical signicance of the neutron current can be easily understood by applying the divergencetheorem: ∫

V

∇~r ~Jd3~r =

∫∂V

~Jd~s


So the neutron current integrated over a given surface is equal to the net variation of the neutronpopulation in the volume enclosed by this surface.

Equation 4.22 written in terms of these two quantities is much simpler than 4.10, but it has thedisadvantage of being in terms of two dierent quantities, namely, the neutron scalar ux φ and theneutron current ~J . Hence, to go further, a relationship between these two quantities must be obtained.Here is where the diusion approximation is performed, considering that the current is proportional tothe negative gradient of the neutron scalar ux, namely:

~J = −D ~∇φ (4.25)

This expression is known as Fick's law and the constant D is referred as diusion constant. This lawwas stated in the XIX century for the solute diusion in a solution (where it is usually expressed in terms ofconcentrations rather than uxes). For a detailed discussion of the validity of the diusion approximation,see for instance [252]. Using this expression, the neutron transport equation in the diusion approximationcan be written as:

1

v

∂φ

∂t−D∇2φ = −Σtot (~r;E)φ (~r, t;E) +

∫Σs (~r;E′) f (~r;E′ → E)φ (~r, t;E′) dE′+

+

∫ν (E′) Σfis (~r;E′)χ (E′ → E)φ (~r, t;E′) dE′ + Sext (~r, t;E′) (4.26)

It can be shown (see [252] for instance) that the diusion constant D can be expressed as:

D =1

3Σtr(4.27)

where Σtr is referred as the transport cross section and is in turn given by:

Σtr = Σtot − µsΣs (4.28)

where µs is the average cosine of elastic scattering, which can be found to be µs = 23A , with A equal

to the mass number of the medium.An additional simplication can be achieved if a single energy group is considered. In this way, the

following simple equation is obtained:

1

v

∂φ

∂t= D∇2φ− Σtot (~r, t)φ (~r, t) + νΣfis (~r, t)φ (~r, t) + Sext (~r, t) (4.29)

This expression constitutes the one group diusion equation for the total ux. Σtot and νΣfis arerespectively dened as:

Σtot (~r, t) =

∫Σtot (~r;E)φ (~r, t;E) dE∫

φ (~r, t;E) dE

νΣfis (~r, t) =

∫ ∫ν (E′) Σfis (~r;E′)χ (E′ → E)φ (~r, t;E′) dEdE′∫

φ (~r, t;E) dE

The interpretation of equation 4.29 is straightforward. It states that the the net variation of theneutron population in a given volume element is equal to the net number of neutrons that enter or leftthe volume element, plus the number of neutrons created in ssions within this volume element minus thenumber of of neutrons lost by capture within this volume element, plus the number of neutrons createdby the external source. With these approximations, the operators M and F take the simple form:

F φ = νΣfis (~r, t)φ (~r, t)

Mφ = Σtot (~r, t)φ (~r, t)−D∇2φ (~r, t)


4.3 Solutions of the neutron transport equation

The neutron transport equation derived in the previous sections, either in its exact form or in the diusionapproximation, is a partial derivative equation that involves derivatives in all three spatial componentsand in time. There is no general analytical solution for this equation and even numerical solutions canbe extremely complicated to nd. A priori, it is not even guaranteed the existence or uniqueness of thesolutions in the most general case, although the existence and uniqueness of solutions has been provedfor some cases with certain restrictions [80, 213, 367].

Hence, there is an obvious interest in nding approximations of the neutron transport equation forwhich simple analytical or numerical solutions can be found. The rst and most simple case is thestationary case in which we consider the scalar ux independent of time. The second case that will beanalyzed here is the case in which variable separation in space and time in the neutron ux is possible.The analysis of these two cases will allow the introduction of important concepts of reactor theory suchas the multiplicative constant, keff .

4.3.1 Stationary case and criticality

Let us begin by considering the stationary case of the neutron transport equation, that is to say, the casein which Φ = Φ

(~r;E′, ~Ω′

). Let us consider rst the case with no external neutron source in the system.

Then equation 4.11 reduces to: (M − F

)Φ = 0 (4.30)

It is immediate to see that this equation always has the trivial solution Φ = 0. In order to have additionalsolutions, the operator M− F is required to be singular. This implies that there must be solutions for theux such that MΦ = FΦ. According to the meaning of these operators, this is to say that the numberof neutrons created in ssions must be the same as the number of neutron lost either by leakage orabsorption. In this case, the system is said to be critical and the neutron chain reaction is self-sustainedin time.

In case the system is not critical, there must happen either that the number of neutrons createdis larger than the number of neutrons lost or that the number of neutrons created is smaller than thenumber of neutrons lost. In the rst case, the neutron population will increase in time and the system issaid to be supercritical. In the second case, the neutron population will decrease in time and the systemis said to be subcritical.

In neither case there is a stationary solution and obtaining a solution for the ux requires solving thetime dependent transport equation. However, it is desirable to be able to give an estimation of how farfrom criticality the system is. In the case of a subcritical system, a steady state solution can be foundby introducing an external neutron source in the system, in the way:

FΦ− MΦ + Sext = 0 (4.31)

Although this equation gives physical solutions for the ux and it actually describes the case of a sub-critical system coupled to an external neutron source as it is the case of ADS, this procedure is not validfor the case of a supercritical system, where the introduction of an external source is incompatible witha stationary system. Thus, the usual procedure is to introduce an eigenvalue in equation 4.30 in order toobtain the eigenvalue equation:

MΦλ = λFΦλ (4.32)

Notice that the physical meaning of this eigenvalue is the ratio between the number of neutrons lostin the system and the neutrons produced by ssions in the system that is in the state Φλ. Hence, it iscustomary to work with not with λ but with its inverse, which is denoted as the eective multiplicationconstant, keff . In terms of keff , equation 4.32 can be rewritten as:

MΦλ =1

keffFΦλ (4.33)

It must be pointed out, however, that unless keff is equal to one, that is, the reactor is critical, thesolutions obtained by this method do not correspond in fact with the solutions of the neutron transportequation. It has been found however, as it will be analyzed later in this section with a numerical example,that they can constitute a good approximation providing that the departure from the critical case is not

4.3. SOLUTIONS OF THE NEUTRON TRANSPORT EQUATION 65

too large (see [253] for further details on this topic). The eigenfunctions of equation 4.33, Φλ, are referredas the lambda mode and this is why we use the subscript λ.

Notice as well that, in principle, equation 4.33 can have no solution for λ and Φλ, or, on the contrary,which is the usual case, it can have more than one solution. In this case, the valid solution can bedetermined by physical considerations (e. g., by requesting the niteness and positiveness of Φλ in allthe space). The corresponding Φλ is accordingly known as the fundamental λ-mode.

Let us now introduce the concept of reactivity as an alternative way to describe the departure fromcriticality of a nuclear system. The reactivity of a system is dened as:

ρ = 1− λ = 1− 1

keff=keff − 1

keff(4.34)

This magnitude is zero for a critical system, negative for a subcritical system and positive for asupercritical system. Because keff has been dened in a system in a stationary solution, the lambdamode mentioned earlier, this reactivity is sometimes referred as static reactivity as opposed to the dynamicreactivity that can be dened for systems in a non stationary state and that will be introduced in section5.3. In this work we are not going to consider these two denitions of reactivity and we are going todesignate by reactivity the one dened by equation 4.34 (the static reactivity). Notice that with thisdenition, the reactivity, as well as the keff , are parameters that depend only on the geometry andcomposition of the system and are independent of the neutron population in the system or its timeevolution.

Another concept that often appears in subcritical systems dynamics is that of source multiplicity orks, which is dened as:

ks =FΦ

MΦ=

FΦ

FΦ + Sext(4.35)

That is, ks is the multiplication of a neutron considering the physical ux within the system (i. e.,the solution of the transport equation) instead of the λ-mode solution of the ux. It must be noticedthat since ks depends on the source, it is not a characteristic parameter of the system, as keff is. Noticeas well that both parameters are the same for a critical system, but their dierence increases with thelevel of subcriticality.

To better understand the dierence between keff and ks and to analyze the eect of approximating Φby Φλ let us consider a numerical example. For this, let us consider the simple case of a bare, homogeneoussubcritical sphere kept in a stationary state by a point source in its center. Let us consider the diusionapproximation to study this problem. In this approximation, and considering neutron scalar uxes insteadof neutron angular uxes, equation 4.31 takes the shape:

νΣfisφ (~r) = Σtotφ (~r)−D∇2φ (~r) (4.36)

Making use of the expression of the Laplace operator in spherical coordinates and rearranging theterms in equation 4.36 we arrive to the second order equation:

d2

dr2φ (r) +

2

r

d

drφ (r) +

νΣfis − ΣtotD

φ (r) = 0 (4.37)

It is important to notice that, since the dependence of time has been removed and the geometry of theproblem allows to be left with only one independent variable (r), equation 4.37 is an ordinary dierentialequation, and hence the existence and uniqueness of solutions is guaranteed for given boundary conditions.The solution of this equation takes the shape:

φ (r) =C1sin (Br)

r+C2cos (Br)

r(4.38)

where B =

√νΣfis−Σtot

D and where C1 and C2 are arbitrary constants that can be determined withtwo proper boundary conditions. The rst boundary condition is that of a point source in the center:

limrs → 0

[4πr2

sJ (rs)]

= Sp

Which in terms of the ux φ translates into:


limrs → 0

[4πr2

sDd

drφ (rs)

]= Sp ⇒

⇒ limrs → 0

[4πr2

sD

(C1Bcos (Brs)

rs− C1sin (Brs)

r2s

− C2Bsin (Brs)

rs− C1cos (Brs)

r2s

)]= Sp ⇒

⇒ limrs → 0

[rsC1BDcos (Brs)− C1Dsin (Brs)− rsC2BDsin (Brs)− C2Dcos (Brs)] = Sp ⇒

⇒ limrs → 0

[C2Dcos (Brs)] = Sp ⇒ C2 =SpD

The second boundary condition is that of vanishing neutron ux at the extrapolated distance R′

(the extrapolated distance is usually dened in diusion theory as R′ = R + 23

1Σtr

, where R denotes thephysical boundary of the reactor and Σtr is the transport cross section dened in equation 4.28):

φ (r = R′) = 0⇒ C1sin (BR′)

R′+SpD

cos (BR′)

R′= 0⇒ C1 = −Sp

D

cos (BR′)

sin (BR′)

So it is nally obtained that:

φ (r) =SpD

(− cos (BR′)

sin (BR′)

sin (Br)

r+cos (Br)

r

)The ks of the assembly will be given by replacing this solution of φ (r) in equation 4.35.Now, let us turn to determine the λ mode of this system. In this case, equation 4.32 can be written

as:

λνΣfisφλ (~r) = Σtotφλ (~r)−D∇2φλ (~r)

Making use of the expression of the Laplace operator in spherical coordinates and rearranging theterms we arrive to the second order equation:

d2

dr2φλ (r) +

2

r

d

drφλ (r) +

λνΣfis − ΣtotD

φλ (r) = 0

The solution of this equation takes again the shape:

φλ (r) =Cλ1sin (Bλr)

r+Cλ2cos (Bλr)

r

where B =

√λνΣfis−Σtot

D and where Cλ1 and Cλ2 are arbitrary constants that can be determined withboundary conditions. The boundary conditions that are considered in this case are that φλ vanishes atthe extrapolated distance R′ and that the neutron ux is nite in all the volume. The second conditionimposes that C2 = 0. The rst condition imposes:

C1sin (BR′)

R′= 0

The physical requirement of the ux to be positive in all the volume imposes a condition for B, thatis:

BR′ = π

which in turns gives a condition for the eigenvalue λ and the criticality constant keff :

λ =π2

R′2D + Σtot

νΣfis⇒ keff =

νΣfisπ2

R′2D + Σtot

An undened constant C1 still remains undetermined. Several criteria can be applied to x it, suchas the total reactor power.


In gure 4.2 it is plotted the dierence between the values of r2φ (r) and r2φλ (r) values for dierentvalues of keff for the case of a homogeneous spherical reactor with a point source in the center (r2φ (r)and r2φλ (r) are plotted instead of simply φ (r) and φλ (r) to avoid the singularity due to the presenceof the point source in the center). Notice how for values of keff very close to criticality, φ (r) and φλ (r)are very similar, and the dierence between them increases with increasing subcriticality level. In gure4.3, the corresponding value of ks is represented for dierent values of keff for the same system. Noticehow both parameters tend to one for a critical reactor and its dierence increases with increasing level ofsubcriticality. In this case, ks is larger than keff since the external neutron source is placed in the centerof the sphere, that is, in the position that minimizes the number of losses, and hence, the multiplicationof the source neutrons is the largest.

4.3.2 Variable separation and spectrum of the transport operator

The second case in which analytical solutions of the neutron transport equation can be found is the casein which the scalar ux can be factorized into a part dependent only on time and a part dependent onspace and energy:

Φ(~r,E, ~Ω, t

)= P (t) Ψ

(~r,E, ~Ω

)(4.39)

In this case, considering the homogeneous equation (for simplicity, we neglect the time dependence in Ldue to the the delayed neutrons):

1

v

∂P

∂tΨ = PLΨ⇒ 1

P

∂P

∂t=

v

ΨLΨ (4.40)

Since the left-hand side of this equation depends only on the time and the right-hand side of the equationdepends only on the spatial variables, the only possible solution for this equation is that both terms areequal to a constant:

1

P

∂P

∂t=

v

ΨLΨ = α (4.41)

So we end up with a separable ordinary dierential equation for the time dependence of the scalar ux:

1

P

∂P

∂t= α⇒ ∂P

∂t= αP ⇒ P (t) = P (0) eαt (4.42)

And an eigenvalue equation for the spatial dependence:

LΨ(~r,E, ~Ω

)=α

vΨ(~r,E, ~Ω

)(4.43)

Hence, the nal solution for the scalar ux has the shape:

Φ(~r,E, ~Ω, t

)= eαtΨα

(~r,E, ~Ω

)(4.44)

Where we have included the constant P (0) in the spatial part. The set of eigenvalues α is known as thespectrum of the transport operator L. Notice that in principle, and opposite to other operators such asthe Hamiltonian operator in quantum mechanics, the transport operator is not self adjoint, and thus theireigenfunctions do not form a basis and an expansion of any solution in eigenfunctions of the transportoperator is not always possible.

Among the eigenvalues of the transport operator, the one with the lowest eigenvalue is called thefundamental eigenvalue. The corresponding eigenfunction Ψα is called the alpha mode of the reactor. Asthe time dependence of the ux with this eigenvalue is in an exponential, this will be the only one thatwill be relevant after a certain time, while all the higher ones will die o. In this case, the system is saidto be in an asymptotic state.


(a)keff

=0.9

95

(b)keff

=0.9

5

(c)keff

=0.9

0(d)keff

=0.8

5

Figure

4.2:r

2φ(r)

vs.r

2φλ

(r)for

dierentvalues

ofkefffor

asphericalhom

ogeneousreactor

with

apoint

sourcein

thecenter.

Dierent

valuesofsubcriticality

areachieved

byvarying

theradius

ofthe

sphere.Cross

sectionsand

constantshave

beentaken

tocorrespond

tothose

fora93%

enricheduranium

sphereusing

theone-group

valueslisted

in[198],

table6.1,

p.222.

Both

φ(r)

arenorm

alizedto

haveφλ

(r)the

sameintegral

value.


Figure 4.3: ks for dierent values of keff for the subcritical spherical reactor with a point source in thecenter described in gure 4.2.

Chapter 5

Perturbation theory and the pointkinetics model

Abstract - In this chapter, the well known point kinetics model of reactor kinetics is derived, upon

which the reactivity monitoring techniques that will be used later in this PhD are based. This is done

in three steps. First, the concept of adjoint neutron ux is introduced, and its usual interpretation as

neutron importance is discussed. Second, once this concept has been introduced, perturbation theory

is derived. Finally, perturbation theory is used to obtain the point kinetics model equations. This

approach allows the denition of a number of important parameters, including the reactivity ρ, the

eective delayed neutron fraction βeff and the eective mean neutron generation time Λeff .

The point kinetics model is a simple model that often constitutes the starting point for many studiesof reactor dynamics. The interest in introducing the point kinetics model now is that it will be usedin chapter 7 to derive the formulae of some commonly used reactivity monitoring techniques. Althoughthe point kinetics model can be introduced in a simple, intuitive way (see for instance [148, 253]), theprocedure that we will be followed here is a more rigorous one, based on perturbation theory, that willallow relating it with the neutron transport equation derived in chapter 4.

Perturbation theory is applied not only in reactor theory but also in many other elds, possibly thebest known example is in quantum mechanics. In all cases, the objective of perturbation theory is tocalculate approximate solutions of certain problems for which an exact solution is dicult or impossible tond but the problem is close enough (i.e. it is a perturbation of) other problem for which an exact solutiondoes exist. Applications of perturbation theory in the eld of reactor physics include the calculation ofthe eective kinetic parameters (see sections 6.3 and 6.4) or the theoretical of the control rod calibrationcurve [199].

In a similar way that perturbation theory in quantum mechanics requires the introduction of anadjoint wave function and an adjoint wave equation, the application of perturbation theory in reactorphysics requires the denition of an adjoint neutron ux, which in turn requires the denition of anadjoint transport equation. Furthermore, it will be shown that in addition of being necessary for the thederivation of the perturbation theory, the concept of adjoint ux plays an important role because, as itwill be shown in section 5.1.2, it can be interpreted as the neutron importance, that is, the ability of acertain neutron to generate additional ssions within the system.

This chapter is mainly based on the works of Bell and Glasstone [53] and Ott and Neuhold [253].Other references are [144, 148, 176, 199, 276].

5.1 Adjoint neutron transport equation

5.1.1 Adjoint operator and adjoint functions

Let us begin by dening the inner or scalar product of two functions Ψ (ξ) and Φ (ξ) as:

(Ψ,Φ) =

∫Ψ (ξ) Φ (ξ) dξ (5.1)

where the integral extends over the allowed range of values of the variable ξ. With this denition ofinner product, the adjoint operator L† to the transport operator L is dened as an operator that fullls

71

72 CHAPTER 5. PERTURBATION THEORY AND THE POINT KINETICS MODEL

the condition: (Ψ†, LΦ

)=(

Φ, L†Ψ†)

(5.2)

for any two suitable Φ functions Ψ†. The set of functions Ψ† on which the adjoint operator L† acts arenamed adjoint functions or adjoint uxes. Here it is important to remark an important dierence withthe well-known case of quantum mechanics: the transport operator is not self-adjoint, i.e. L 6= L†. Forthis reason, the adjoint uxes Ψ† are in principle a dierent set of functions that the set of functions (oruxes) Φ on which the transport operator L acts (i.e. they may have to satisfy dierent properties thatthe neutron uxes Φ). Therefore, the task now is to determine the shape of L† and the properties of theadjoint uxes Ψ†. The physical signicance of the adjoint ux will be discussed in section 5.1.2.

However, before going further, let us prove a useful property of the eigenfunctions of L and L† thatwill be used later. This property is that two eigenfunctions of L and L† corresponding to dierenteigenvalues are orthogonal with the inner product previously dened. For that, let us consider Φ to be aneigenfunction of L corresponding to the eigenvalue λ and Ψ† to be an eigenfunction of L† correspondingto the eigenvalue η. If we perform the following inner products:

LΦ = λΦ

L†Ψ† = ηΨ†

⇒

(Ψ†, LΦ

)= λ

(Ψ†,Φ

)(Φ, L†Ψ†

)= η

(Φ,Ψ†

)

From the denition of adjoint operator, we have that(

Ψ†, LΦ)

=(

Φ, L†Ψ†). Hence:

(λ− η)(Ψ†,Φ

)= 0

And because we have imposed that λ and η are dierent eigenvalues, then(Ψ†,Φ

)= 0

Let us try now to determine the form of the adjoint transport equation and the properties of the adjointfunctions. Let us begin by writing down again the neutron transport equation in the shape:

1

v

∂Φ

∂t+ ~Ω∇~rΦ = −Σtot (~r;E) Φ

(~r;E, ~Ω; t

)+

+

∫ ∫Σs (~r;E′) f

(~r;E′, ~Ω′ → E, ~Ω

)Φ(~r;E′, ~Ω′; t

)dE′d~Ω′+

+1

4π


(~r;E′, ~Ω′; t

)dE′d~Ω′+

+∑x

1

4π

∫ ∫xΣ(n,xn) (~r;E′) ξ (E′ → E) Φ

(~r;E′, ~Ω′; t

)dE′d~Ω′ + Sd (~r;E; t) + Sext

(~r;E, ~Ω; t

)(5.3)

Using the linearity of the scalar product dened above we can search for the adjoint operator termby term. Let us begin by the rst term on the left-hand side, which contains a time derivative. Uponintegrating by parts:(

Ψ†,∂

∂tΦ

)=

∫Ψ† (t)

∂

∂tΦ (t) dt =

∫Ψ† (t) dΦ (t) = Ψ† (t) Φ (t)

∣∣∞0−∫

Φ (t) dΨ† (t) =

= Ψ† (t) Φ (t)∣∣∞0−∫

Φ (t)∂

∂tΨ† (t) dt (5.4)

The integration has to be carried out over the whole range of the variables, which in the case of thetime it is from 0 to ∞. From equation 5.2 it is obvious that it is necessary to get rid of the rst termon the right-hand side in order to get a simple form for the adjoint operator. This can be achieved bymeans of imposing appropriate boundary conditions to the adjoint functions. Hence, the condition thatmust be imposed is that Ψ† (t) Φ (t)

∣∣∞0

= 0 for any ux Φ. If we limit our study to subcritical systemsonly, it must hold the condition that Φ (t)→ 0 for t→∞, so the condition Ψ† (t) Φ (t)

∣∣∞0

simplies into

5.1. ADJOINT NEUTRON TRANSPORT EQUATION 73

Ψ† (t = 0) = 0. With this, we can conclude that the adjoint operator for the time derivative term is thederivative itself but with a minus sign: (

∂

∂t

)†= − ∂

∂t

The same reasoning is also valid for the case of the second term on the left-hand side of equation 5.3 thatinvolves spatial derivatives. In this case, if the boundary condition we have for Φ is that of no incomingux, that is ~Ω·Φ = 0 for all incoming directions, the boundary condition that Ψ† must satisfy so that therst term on the right-hand side in equation 5.4 vanishes is that of ~Ω·Ψ† = 0 for all outgoing directions(no outgoing ux).

Now, let us turn to the integral terms on the right-hand side of equation 5.3. For the rst term it isstraightforward to see that it has the same shape that its adjoint. For the second term we have that:∫ ∫

Ψ(~r;E, ~Ω; t

)Σs (~r;E′) f

(~r;E′, ~Ω′ → E, ~Ω

)Φ(~r;E′, ~Ω′; t

)dE′d~Ω′

d~rdEd~Ωdt =

=

∫ ∫Ψ(~r;E′, ~Ω′; t

)Σs (~r;E) f

(~r;E, ~Ω→ E′, ~Ω′

)Φ(~r;E, ~Ω; t

)dEd~Ω

d~rdE′d~Ω′dt =

=

∫ ∫Φ(~r;E, ~Ω; t

)∫ ∫Σs (~r;E) f

(~r;E, ~Ω→ E′, ~Ω′

)Ψ†(~r;E′, ~Ω′; t

)dE′d~Ω′

d~rdEd~Ωdt (5.5)

where the name of the bounded variables E, ~Ω and E′, ~Ω′ has been interchanged and the integralreorganized. This result also imply that the adjoint uxes Ψ† must satisfy the same boundary conditionson E and ~Ω that the direct uxes Φ. So it can be concluded that the adjoint of this term is:

Σs (~r;E)

∫ ∫f(~r;E, ~Ω→ E′, ~Ω′

)Ψ†(~r;E′, ~Ω′; t

)dE′d~Ω′

where the factor Σs (~r;E) has been taken outside the integral sign as it does not depends on the variablesE′ or ~Ω′. A similar reasoning can be applied to the third and fourth terms. So nally, the followingequation for the adjoint ux is found:

−1

v

∂Ψ†

∂t− ~Ω∇~rΨ† = −Σtot (~r;E) Ψ†

(~r;E, ~Ω; t

)+

+Σs (~r;E)

∫ ∫f(~r;E, ~Ω→ E′, ~Ω′

)Ψ†(~r;E′, ~Ω′; t

)dE′d~Ω′+

+1

4πν (E) Σfis (~r;E) (1− β (~r,E))

∫ ∫χp (E → E′) Ψ†

(~r;E′, ~Ω′; t

)dE′d~Ω′+

+∑x

1

4πxΣ(n,xn) (~r;E)

∫ ∫ξ (E → E′) Ψ†

(~r;E′, ~Ω′; t

)dE′d~Ω′ + S†d (~r;E; t) + S†ext

(~r;E, ~Ω; t

)(5.6)

Notice that this expression has an ambiguity in the sense that it gives no expression for the adjoint ofthe external source. This fact allows dening several adjoint problems depending on the denition of thisterm. Dierent denitions of the adjoint of the external source will give dierent adjoint uxes. Notice,however, that as the adjoint ux is taken to be dimensionless, the dimensions of this adjoint source mustbe those of inverse length, that is, of a macroscopic cross section. This fact will be further discussed innext section where an interpretation of the adjoint ux is given.

In terms of operators, equation 5.6 can be rewritten as:

− 1

v

∂Ψ†

∂t= F †Ψ† − M†Ψ† + S†ext (5.7)

or even in a more compact form:

− 1

v

∂Ψ†

∂t= L†Ψ† + S†ext (5.8)

Where the adjoint creation, losses and transport operators are dened respectively as:


F †Ψ† =1

4πν (E) Σfis (~r;E) (1− β (~r,E))

∫ ∫χp (E → E′) Ψ†

(~r;E′, ~Ω′; t

)dE′d~Ω′+

+∑x

1

4πxΣ(n,xn) (~r;E)

∫ ∫ξ (E → E′) Ψ†

(~r;E′, ~Ω′; t

)dE′d~Ω′ + S†d (~r;E; t) (5.9)

M†Ψ† = −~Ω∇~rΨ† + Σtot (~r;E) Ψ†(~r;E, ~Ω; t

)−

− Σs (~r;E)

∫ ∫f(~r;E, ~Ω→ E′, ~Ω′

)Ψ†(~r;E′, ~Ω′; t

)dE′d~Ω′ (5.10)

L† = F † − M† (5.11)

5.1.2 Interpretations of the adjoint ux

In certain cases, it is possible to obtain a physical interpretation of the adjoint ux. Let us consider thecase of a subcritical reactor maintained in a steady state by an external neutron source. The neutrontransport equation applied for this case takes the form:

LΦ = −S ⇒ FΦ− MΦ = −Sext (5.12)

As stated in the previous section, to dene the adjoint of this equation there is an ambiguity in thedenition of the external source. It was also remarked there that the adjoint source has the dimensionsof inverse length or equivalently, of macroscopic cross section. Hence, let us consider it to be the ssioncross section of the system, that is:

L†Ψ† = −Σf ⇒ F †Ψ† − M†Ψ† = −Σf (5.13)

Then, subtracting equation 5.12 multiplied by Φ and equation 5.13 multiplied by Ψ† and integratingthe resulting expression over the entire range of the variables taking into account the properties of theadjoint operators, we end up with the following expression:∫

Sext

(~r, ~Ω, E

)Ψ†(~r, ~Ω, E

)d~rd~ΩdE =

∫Σf

(~r, ~Ω, E

)Φ(~r, ~Ω, E

)d~rd~ΩdE (5.14)

The right-hand side of the last equation is just of the response of the system to the external source,that is to say, the number of ssions per unit time that takes place in the whole system in the presenceof the external source. We can further clarify this if we consider the external source to have the shape of:

Sext

(~r, ~Ω, E

)= δ

(~r0, ~Ω0, E0

)(5.15)

Then we nd out that:

Ψ†(~r0, ~Ω0, E0

)=

∫Σf

(~r, ~Ω, E

)Φ(~r, ~Ω, E

)d~rd~ΩdE (5.16)

where Φ(~r, ~Ω, E

)is the solution of the equation Lφ

(~r, ~Ω, E

)= −δ

(~r0, ~Ω0, E0

). That is to say, the

value of the adjoint ux in a point of the phase space is the response of the system to a unit sourceintroduced in this point, or, in other words, the number of ssions produced in the whole system becauseof the introduction of a single neutron in this point. Hence, adjoint functions are usually interpreted asthe source importance.

To clarify the meaning of the concept of source importance, consider that a neutron source is intro-duced in a certain position of a subcritical system. Depending on the energy, direction and position ofthis source, the multiplication in the system will be dierent, i.e., the number of ssions caused by thesource is not the same if the source is introduced in the center or a highly multiplicative region of thesystem or if it is introduced in the outermost part of it or in a low multiplicative region of the system.The relevance of the source importance was already mentioned in section 4.3.1 where it was remarkedthat for a sphere reactor with an external source placed in the center ks is larger than keff because ofthe higher importance of the neutrons in the center than in the periphery of the system.

Notice that the interpretation of the adjoint as source importance is somewhat arbitrary as it isdetermined by the choice of Σf as the adjoint source. Other choices for this neutron source will give

5.2. PERTURBATION THEORY 75

rise to other interpretations of the adjoint neutron ux. For instance, we could have chosen a detectorcross section Σd instead of Σf , to obtain in this case that the adjoint ux measures the response of thedetector to the introduction of an external source in this point.

Notice as well that this interpretation of the adjoint ux is consistent with one of the conditionsimposed to the adjoint functions in section 5.1.1, specically that of no outcoming ux, since a neutronleaving the system will have no chance of producing any more ssions.

Other developments that drive to similar interpretations of the adjoint ux to this one are available inthe literature. In [144], for instance, the same interpretation of adjoint functions as source importance isreached in another way, considering the introduction of an external source in a critical system instead thatin a subcritical one. Other equivalent interpretation of the adjoint ux based on the concept of iteratedssion probability (i.e., the expected number of ssions produced by a given neutron in the system) ispresented in [153].

5.2 Perturbation theory

The interest of perturbation theory, either in the eld of nuclear reactor theory or in any other eld,is to allow the approximate calculation of the change in the values of certain parameters after smallchanges, or perturbations, in a system that initially is in a state in which these parameters are known,without having to determine the exact nal state of the system. The main interest for us to introducethis technique is because perturbation theory will be used in section 5.3 to derive the well known pointkinetics model. Nevertheless, perturbation theory has another applications in the eld of reactor kinetics;for instance, in section 6.4 a technique derived from perturbation theory will be applied to calculate theeective neutron generation time of the Yalina-Booster subcritical facility.

A common application of perturbation theory in the eld of nuclear reactor dynamics is to calculatethe eects of perturbations in the criticality constant, keff . For this, perturbation theory is applied toeigenvalue functions and stationary states such as those described in section 4.3.1. To apply perturbationtheory, it must be considered a system in an initial state characterized by a neutron production operatorF0 and neutron loss operator M0. The λ mode equation (equation 4.32) of this system is:

M0Φ0 = λ0F0Φ0 (5.17)

And its correspondent adjoint equation is 1:

M†0 Φ†0 = λ0F†0 Φ†0 (5.18)

Where λ0 is considered to be the fundamental mode. Now, let us consider a system in anotherstate characterized by a neutron production operator F and neutron loss operator M . In this state,the solutions for the ux and the eigenvalue are respectively Φ and λ, both of which we consider to beunknown. Ideally, for perturbation theory to be practical, this system should be similar to the previousone, that is, it should be a perturbation of the previous system. With these considerations, the system inthis state will be described by the equation:

MΦ = λFΦ (5.19)

Remember, as it was explained in section 4.3.1, that the eigenvalue λ is the inverse of the criticalityconstant keff . Performing the scalar product by Φ†0 on each side of equation 5.19 and by Φ on each sideof equation 5.18 one gets the equations:(

Φ†0, MΦ)

= λ(

Φ†0, FΦ)

(5.20)

(Φ, M†0 Φ†0

)= λ0

(Φ, F †0 Φ†0

)(5.21)

1It can be easily proven that the adjoint eigenvalue is the same that the direct eigenvalue in the same way it was provenin section 5.1.1 that eigenfunctions corresponding to dierent eigenvalues are orthogonal. For this, considering that theseeigenvalues can be dierent we have that:

M0Φ0 = λ0F0Φ0

M†0Φ†0 = λ†0F†0 Φ†0

⇒

(Φ†0, M0Φ0

)= λ0

(Φ†0, F0Φ0

)(Φ0, M

†0Φ†0

)= λ†0

(Φ0, F

†0 Φ†0

) ⇒ λ†0 = λ0

unless(

Φ†0, M0Φ0

)and

(Φ†0, F0Φ0

)are both 0 (see [144], p. 348).


Upon application of the properties of the adjoint operator, last equation turns into:(Φ†0, M0Φ

)= λ0

(Φ†0, F0Φ

)(5.22)

Let us make now the following denitions:

∆λ = λ− λ0

∆Φ = Φ− Φ0

∆F = F − F0

∆M = M − M0

With these denitions, equation 5.22 can be expressed as:(Φ†0, M0Φ

)= λ0

(Φ†0, FΦ

)− λ0

(Φ†0,∆FΦ

)(5.23)

Now, subtracting equations 5.20 and 5.23 we arrive to:(Φ†0, MΦ

)−(

Φ†0, M0Φ)

= λ(

Φ†0, FΦ)− λ0

(Φ†0, FΦ

)+ λ0

(Φ†0,∆FΦ

)⇒

⇒(

Φ†0,∆MΦ)

= ∆λ(

Φ†0, FΦ)

+ λ0

(Φ†0,∆FΦ

)⇒

⇒ ∆λ =

(Φ†0,

[∆M − λ0∆F

]Φ)

(Φ†0, FΦ

) (5.24)

The last equation is the expression for ∆λ in perturbation theory. This expression can be rewrittenstraightforwardly in terms of reactivities using the denitions given in section 4.3.1:

∆λ = λ− λ0 =1

keff− 1

keff,0= ρ0 − ρ = −∆ρ⇒ ∆ρ =

(Φ†0,

[λ0∆F −∆M

]Φ)

(Φ†0, FΦ

) (5.25)

Notice that both 5.24 and 5.25 are exact, as no approximation has been done so far. For this reason,5.25 is known as the exact perturbation formula. However, it is of little use as it contains the perturbedux Φ on the right-hand side, which, as it has been stated before, is in principle unknown. The way toproceed now is to write the perturbed ux Φ as the original unperturbed ux Φ0 plus a perturbation,that is:

Φ = Φ0 + ∆Φ (5.26)

Hence, upon substituting in equation 5.26 we nd that:

∆ρ =

(Φ†0,

[λ0∆F −∆M

]Φ)

(Φ†0, FΦ

) ⇒

⇒ ∆ρ(

Φ†0,[F0 + ∆F

][Φ0 + ∆Φ]

)=(

Φ†0,[λ0∆F −∆M

][Φ0 + ∆Φ]

)⇒

⇒ ∆ρ(

Φ†0, F0Φ0

)+ ∆ρ

(Φ†0, F0∆Φ

)+ ∆ρ

(Φ†0,∆FΦ0

)+ ∆ρ

(Φ†0,∆F∆Φ

)=

=(

Φ†0,[λ0∆F −∆M

]Φ0

)+(

Φ†0,[λ0∆F −∆M

]∆Φ)

5.3. THE POINT KINETICS MODEL 77

Working out the ∆ρ in the rst term in the left-hand side:

∆ρ =

(Φ†0,

[λ0∆F −∆M

]Φ0

)(

Φ†0, F0Φ0

) +

(Φ†0,

[λ0∆F −∆M

]∆Φ)

(Φ†0, F0Φ0

) −∆ρ(

Φ†0, F0∆Φ)

(Φ†0, F0Φ0

) −

−∆ρ(

Φ†0,∆FΦ0

)(

Φ†0, F0Φ0

) −∆ρ(

Φ†0,∆F∆Φ)

(Φ†0, F0Φ0

)And substituting all ∆ρs in the right-hand side by equation 5.25 we end up with:

∆ρ =

(Φ†0,

[λ0∆F −∆M

]Φ0

)(

Φ†0, F0Φ0

) +

+

(Φ†0,

[λ0∆F −∆M

]∆Φ)

(Φ†0, F0Φ0

) −

(Φ†0,

[λ0∆F −∆M

]Φ)

(Φ†0, FΦ

)(

Φ†0, F0∆Φ)

(Φ†0, F0Φ0

) −(

Φ†0,[λ0∆F −∆M

]Φ)

(Φ†0, FΦ

)(

Φ†0,∆FΦ0

)(

Φ†0, F0Φ0

) −

−

(Φ†0,

[λ0∆F −∆M

]Φ)

(Φ†0, FΦ

)(

Φ†0,∆F∆Φ)

(Φ†0, F0Φ0

) (5.27)

The rst term on the right-hand side of equation 5.27 is of rst order in the perturbation; the second, thirdand fourth are of the second order in the perturbation, and the fth is of third order in the perturbation.Hence, if the perturbation is small enough, we can neglect all terms of order larger than one and we areleft-hand then with the rst order perturbation formula:

∆ρ '

(Φ†0,

[λ0∆F −∆M

]Φ0

)(

Φ†0, F0Φ0

) (5.28)

This equation is an approximation, but it has the advantage that the right-hand side depends onlyon the initial unperturbed state and the perturbation in the operators F and M , it has no dependenceon the perturbed uxes Φ and Φ†. Notice the similitude of this result with the result of rst ordertime-independent perturbation theory of quantum mechanics. There the energy of a perturbed state ofa system was obtained from the perturbed Hamiltonian and the unperturbed initial state. Here it isthe reactivity of the perturbed state which is obtained from the perturbed transport operator and theunperturbed initial state.

The accuracy of the results of perturbation theory depend of course of the similarity of the perturbedstate to the initial state, and their validity dismisses as both states depart from each other. This eectmust be taken into account when performing perturbative calculations and will be discussed in nextsection, when applied to derive the point kinetics model and in section 6.4, to determine the meanneutron generation time of a system.

5.3 The point kinetics model

Now the perturbation theory derived in the previous section will be applied to obtain the equations ofthe point kinetics model. As the initial unperturbed state let us choose the fundamental lambda modedened in section 4.3.1, that is, the solution of equation:(

Mλ − λFλ)

Φλ = 0⇒(M†λ − λF

†λ

)Φ†λ = 0 (5.29)

The perturbed state will be given by equation:

1

v

∂Φ

∂t=(F − Fd

)Φ− MΦ + Sd + Sext (5.30)


where the prompt and delayed neutrons parts are written separated for later convenience. Performingthe scalar product on both sides of equation 5.30 by Φ†λ one obtains that:

∂

∂t

(Φ†λ,

1

vΦ

)=(

Φ†λ,[F − M

]Φ)−(

Φ†λ, FdΦ)

+(

Φ†λ, Sd

)+(

Φ†λ, Sext

)(5.31)

where it has been used the fact that Φ†λ, as well as Φλ, are time independent and thus they can beintroduced inside the time derivatives. In a similar way, performing the scalar product on both sides ofequation 5.29 by Φ one obtains that: (

Φ†λ,[Mλ − λFλ

]Φ)

= 0 (5.32)

Combining equations 5.31 and 5.32 one nds that:

∂

∂t

(Φ†λ,

1

vΦ

)=(

Φ†λ,[∆F −∆M

]Φ)−(

Φ†λ, FdΦ)

+(

Φ†λ, Sd

)+(

Φ†λ, Sext

)(5.33)

Where ∆F and ∆M denote, respectively:

∆F = F − λFλ

∆M = M − Mλ

Now, let us separate the ux in two parts:

Φ(~r;E, ~Ω; t

)= n (t) Ψ

(~r;E, ~Ω; t

)with the condition that:

∂

∂t

(Φ†λ,

1

vΦ

)=∂n (t)

∂t

(Φ†λ,

1

vΨ

)(5.34)

This condition means that the variation of the adjoint-weighted ux integrated over the entire reactoris due only to the variation of n (t). The function n (t) that only depends on time is normally referred

as the amplitude factor and the function Ψ(~r;E, ~Ω; t

)is referred as the shape factor or shape function.

This condition is equivalent to:

∂

∂t

(Φ†λ,

1

vΨ

)= 0⇒ ∂

∂t

(Φ†λ,

1

vΨ

)= 0 (5.35)

Or, in other words, the adjoint-weighted shape factor integrated over the entire reactor remainsconstant in time. Notice that as far as now no simplication has been made because time dependenceappears on both amplitude and shape factors, that is to say, we are not making variable separation. Withthis separation, equation 5.33 turns into:

∂n (t)

∂t

(Φ†λ,

1

vΨ

)+ n (t)

∂

∂t

(Φ†λ,

1

vΨ

)= n (t)

(Φ†λ,

[∆F −∆M

]Ψ)− n (t)

(Φ†λ, FdΨ

)+

+(

Φ†λ, Sd

)+(

Φ†λ, Sext

)(5.36)

But because of equation 5.35, the second term in the left-hand side is zero and it is found that:

∂n (t)

∂t

(Φ†λ,

1

vΨ

)= n (t)

(Φ†λ,

[∆F −∆M

]Ψ)− n (t)

(Φ†λ, FdΨ

)+(

Φ†λ, Sd

)+(

Φ†λ, Sext

)(5.37)

This equation can be rewritten as:

∂n (t)

∂t=

(Φ†λ,

[∆F −∆M

]Ψ)−(

Φ†λ, FdΨ)

(Φ†λ,

1vΨ) n (t) +

(Φ†λ, Sd

)(

Φ†λ,1vΨ) +

(Φ†λ, Sext

)(

Φ†λ,1vΨ) (5.38)


In order to be able to give a physical interpretation for each of the terms that appear on the right-handside of this equation, let us multiply and divide every one of them by

(Φ†λ, FΨ

)to obtain:

∂n (t)

∂t=

(Φ†λ, FΨ

)(

Φ†λ,1vΨ)(

Φ†λ,[∆F −∆M

]Ψ)

(Φ†λ, FΨ

) −

(Φ†λ, FdΨ

)(

Φ†λ, FΨ)n (t) +

+

(Φ†λ, FΨ

)(Φ†λ, Sd

)(

Φ†λ,1vΨ)(

Φ†λ,1vΨ) +

(Φ†λ, FΨ

)(Φ†λ, Sext

)(

Φ†λ,1vΨ)(

Φ†λ,1vΨ) (5.39)

Let us now analyze the physical meaning of each of these terms. Let us begin by the factor thatappears multiplying all terms. It is better to consider the inverse of this factor, which can be written as:(

Φ†λ,1vΨ)

(Φ†λ, FΨ

) =

∫ ∫Φ†λ

(~r;E, ~Ω

) ∫ ∫1vΨ(~r;E′, ~Ω; t

)dE′d~Ω′dEd~Ω∫ ∫

Φ†λ

(~r;E, ~Ω

)χ(E)4π

∫ ∫ν (E′) Σfis (~r;E′) Ψ

(~r;E′, ~Ω; t

)dE′d~Ω′dEd~Ω

(5.40)

The numerator of this factor is the adjoint-weighted neutron population of the system and the denom-inator is the adjoint-weighted neutron production rate in ssion reactions. The inverse is therefore thession rate per neutron in the system. Hence, this factor is interpreted as the inverse of the mean neutrongeneration time, that is, the mean time between two successive generations of neutrons. This signicancewill be further discussed in section 6.4. In the general case we will have that this term is time dependentas Ψ is time dependent, so let us denote it by by Λ (t). Hence,

Λ (t) ≡

(Φ†λ,

1vΨ)

(Φ†λ, FΨ

) (5.41)

The second term on the right-hand side of equation 5.39 is easily interpreted as the adjoint-weightedratio of delayed neutrons emitted in ssion neutrons versus the adjoint-weighted total number of neutronsemitted in the ssion. Once more this is a time dependent quantity. Let us denote it by β (t).

β (t) ≡

(Φ†λ, FdΨ

)(

Φ†λ, FΨ) (5.42)

The rst term on the right-hand side of equation 5.39 is called dynamic reactivity by some authors(see for instance [253]). Let us denote it by ρ (t) that can be dened in either of the tree following ways:

ρ (t) ≡

(Φ†λ,

[∆F −∆M

]Ψ)

(Φ†λ, FΨ

) =

(Φ†λ,

[F − M

]Ψ)

(Φ†λ, FΨ

) = 1−

(Φ†λ, MΨ

)(

Φ†λ, FΨ) (5.43)

To see the equivalence between these three denitions, notice that this term can be expanded in themanner: (

Φ†λ,[∆F −∆M

]Ψ)

(Φ†λ, FΨ

) =

(Φ†λ,

[F − λFλ − M + Mλ

]Ψ)

(Φ†λ, FΨ

) =

=

(Φ†λ,

[F − M

]Ψ)

(Φ†λ, FΨ

) −

(Φ†λ,

[λFλ − Mλ

]Ψ)

(Φ†λ, FΨ

) =

= 1−

(Φ†λ, MΨ

)(

Φ†λ, FΨ) −

(Φ†λ,

[λFλ − Mλ

]Ψ)

(Φ†λ, FΨ

) (5.44)

The second term in these two last expressions is 0 because(

Φ†λ,[λFλ − Mλ

]Ψ)

=([λF †λ − M

†λ

]Φ†λ,Ψ

)=([

λF †λ − M†λ

]Φ†λ,Ψ

)= 0.


Before giving an interpretation for the last two terms, let us nd out an equation for the delayedneutron source. It is possible to write that:

(Φ†λ, Sd

)=

Φ†λ,∑j

χj (E)

4πCj (~r, t)λj

=∑j

λj

(Φ†λ,

χj (E)

4πCj (~r, t)

)(5.45)

From equation 4.17, using the denition of delayed neutron operator (equation 4.21) and the aboveseparation of the neutron ux in amplitude and phase factors it is found that:

∂

∂t

(Φ†λ,

χj (E)

4πCj (~r, t)

)= −λj

(Φ†λ,

χj (E)

4πCj (~r, t)

)+(

Φ†λ, FdjΨ)n (t) (5.46)

Dividing both sides of this equation by the time independent quantity (see equation 5.24):

1

Λ (t)(

Φ†λ, FΨ) =

(Φ†λ, FΨ

)(

Φ†λ,1vΨ)(

Φ†λ, FΨ) =

1(Φ†λ,

1vΨ) (5.47)

equation 5.46 turns into:∂

∂tcj (t) = −λjcj (t) +

1

Λ (t)βj (t)n (t) (5.48)

where

cj (t) =1

Λ (t)(

Φ†λ, FΨ) (Φ†λ,

χj (E)

4πCj (~r, t)

)is the importance weighted concentration of the j-th precursor, normalized by the time-independentquantity Λ (t)

(Φ†λ, FΨ

), and

βj (t) =

(Φ†λ, FdjΨ

)(

Φ†λ, FΨ)

is the fraction of the neutrons that are due to the decay of the j-th precursor. In the same way, theexternal source can be redened as:

Sext (t) =

(Φ†λ, FΨ

)(

Φ†λ,1vΨ)(

Φ†λ, S)

(Φ†λ, FΨ

) =

(Φ†λ, S

)Λ (t)

(Φ†λ, FΨ

) (5.49)

which is now the adjoint-weighted with the same normalization that the precursor concentration. Sonally, the following equations are found:

∂n (t)

∂t=ρ (t)− β (t)

Λ (t)n (t) +

∑j

λjcj (t) + Sext (t) (5.50)

∂


1

Λ (t)βj (t)n (t) (5.51)

These constitute the exact equations of the point kinetics model. These equations are exact in thesense that as far as now no approximation has been made, but they are of little use since time-dependentparameters appear on the two sides of the equation. The time dependence of these parameters is dueto the time dependence of the shape factor Ψ. To nd a simplied model, we apply the principles ofperturbation theory are applied and it is considered that Ψ is approximately equal to the reference state,i. e., the time-independent Φλ. If we consider as well that the operators F and M are approximatelyequal to the operators Fλ and Mλ, the dynamic reactivity will be:

ρ (t) = 1−

(Φ†λ, MΨ

)(

Φ†λ, FΨ) = 1−

(M†Φ†λ,Ψ

)(F †Φ†λ,Ψ

) = 1−

(ˆM†λΦ†λ,Ψ

)(

ˆF †λΦ†λ,Ψ

) = 1−λ = 1− 1

keff=keff − 1

keff= ρ (5.52)


So it coincides with the reactivity dened in section 4.3.1. Similarly, with these hypothesis β (t) andΛ (t) become time independent and are denoted as βeff and Λeff , e meaning eective to indicatethat both these parameters are weighted by the adjoint ux. Hence, the common equations of thepoint-kinetics model are nally found:

∂n (t)

∂t=ρ− βeff

Λeffn (t) +

∑j

λjcj (t) + Sext (t) (5.53)

∂


1

Λeffβeff,jn (t) (5.54)

It must be remarked that the election of the initial state for applying perturbation theory is critical forthe validity of the point kinetics approximation and for giving a physical signicance to the parametersinvolved in the equations. The importance of this election implies that point kinetics will be a goodapproximation for small departures from this state, but they will lose validity as the departure from itincreases. In the derivation of the previous equations the fundamental λ-mode has been chosen, but manyother elections for the initial state can be made. Choosing the fundamental λ-mode as initial state hasthe advantage of allowing relating the parameters in the equation with the keff dened in section 4.3.1and nding relatively simple physical interpretations of all the other terms involved in terms of βeff andΛeff . This election has been also made in the books of Hetrick [148] and Keepin [176].

Derivations of point kinetics from initial states others than the λ-mode can be found in the literature.For example, in the books of Ott and Neuhold [253], Bell and Glasstone [53] or Henry [143] the initialstate is considered to be a critical state and hence the weighting function is given by the solution of

the equation ˆF †0 Φ†0 =

ˆM†0 Φ†0. This election coincides with the former and is the obvious one for small

departures from criticality, but its convenience is more questionable for deeper subcritical states. J.Dorning [105] proposed to use the fundamental α-mode instead (which he called the ω-mode). Manyother elections can be considered, we can think for instance in the adjoint uxes dened in section 5.1.2,that is F †Φ†f − M†Φ

†f = −Σf or F †Φ†det − M†Φ

†det = −Σdet. Nevertheless, with any of these options

there is the issue of relating the results with the denition of keff of section 4.3.1 and it will also aectthe denition of the parameters β and Λ.

Chapter 6

The Monte Carlo method

Abstract - The purpose of this chapter is to give an overview of the MCNPX Monte Carlo neutron

transport code, which has been used in this thesis. Sections 6.1 and 6.2 are an introduction to the

Monte Carlo techniques applied to neutron transport and to the Monte Carlo code MCNPX in partic-

ular. One feature that it is not standard of the MCNPX code is the calculation of kinetic parameters,

namely the eective delayed neutron fraction βeff and the eective mean neutron generation time

Λeff . Nevertheless, a number of methodologies have been proposed for this aim. In particular, for

the calculation of βeff , several methodologies have been analyzed and benchmarked. The results of

this study have been published in [51].

As it was stated in section 4.3, the neutron transport equation, even in the diusion approximation,is a partial derivative equation for which analytical solutions are restricted to very few simple cases.Therefore, numerical solutions are usually required. Numerical methods to solve the neutron transportproblem are classied into deterministic and stochastic (or Montecarlo) methods.

Deterministic methods are variants of the nite dierence method, the nite volume method or thenite element method, which are widely used to solve PDEs in many other elds of physics or mathe-matics. These methods start with the neutron transport equation and attempt to solve it through thediscretization of the space and angular directions in a number of nite domains. The energy domain isalso divided into a nite number of discrete intervals (multigroup theory). The discretization of the prob-lem always implies approximations and therefore deterministic methods can only be applied to relativelysimple problems, for which the discretization can be performed easily.

The Monte Carlo technique, on the contrary, does not consider the neutron transport equation butattempts to simulate the actual physical transport process and record some parameters (tallies) of them.Monte Carlo methods have the advantage over deterministic methods that they do not require the dis-cretization of the phase space and therefore they allow the detailed consideration of complex geometriesand the continuous treatment of the energy. For this reason, extensive usage of Monte Carlo simulationshas been used for a series of calculations required for this work (neutron ux distribution, keff , kineticparameters). The purpose of this chapter is to describe the way these calculations have been performedwith the MCNPX code. For more details about the Monte Carlo method, the reader is referred to thebibliography [79, 327].

6.1 Montecarlo application to neutron transport

Although Monte Carlo techniques are widely used today in many elds of physics and mathematics, oneof the elds where Monte Carlo techniques are most widely used is in particle transport problems. Infact, the Monte Carlo technique was originally developed for this purpose, as it was rst developed in the1940s by J. von Neumann, S. Ulam and N. Metropolis working for the Manhattan project. Since then,Montecarlo techniques have found application in many particle transport problems: radiation shielding,reactor calculations, detector design, dosimetry, radiotherapy...

Monte Carlo neutron transport codes are used in nuclear engineering for two main types of calcula-tions. The rst of them is for xed-source calculations, that is, neutron ux or ssion power distributionsin a subcritical system with a known source. These calculations appear for instance in detector design,reactor shielding and irradiation or dosimetry calculations. The second main application is for critical-ity/eigenvalue calculations in multiplicative systems. Several production Montecarlo codes are publicly

83

84 CHAPTER 6. THE MONTE CARLO METHOD

Table 6.1: Summary of several modern Monte Carlo neutron transport codes.

Name Institution Comments

FLUKA

INFN(Italy)CERN(Europe)

Originally intended for high energy applications. It simulates thepropagation and interaction with matter of about 60 dierent par-ticles, including photons and electrons from 1 keV to thousands ofTeV, hadrons of energies up to 20 TeV (neutrons down to thermalenergies) and heavy ions.

GEANTCERN(Europe)

Originally intended for the simulation of detectors used in high en-ergy applications. It incorporates some routines of FLUKA.

MAVRICKENO

ORNL(USA)

Both are part of the SCALE code system and use the same geomerypackage. MAVRIC is intended for shielding calculations. KENO isintended for reactor calculations, including criticality constants andux and power distributions.

MCBENDMONK

AMEC plc(UK)

Originally developed by the UKAEA. Both are part of the AN-SWERS code package and share a common geometry modellingpackage. MCBEND is intended for shielding, dosimetry and radi-ation transport in subcritical congurations. It can be applicableto neutron, gamma and electron transport. MONK is intended forreactor phyiscs and criticality calculations.

MCNPMCNPX

LANL(USA)

Multipurpose: reactor physics, shielding, dosimetry. It can trans-port neutrons (up to 20 MeV), photons (up to 1 GeV) and electrons(up to 1 GeV). MCNPX is an extension of MCNP to more particlesand more energy ranges, including neutrons up to 150 MeV. It usessome of the FLUKA routines.

MCUKurchatovInstitute(Russia)

A family of codes for dierent purposes. It can transport neutrons,photons, electrons and positrons.

MVPGMVP

JAERI(Japan)

For xed source, criticality and burnup calculations. Capable oftransporting neutrons and photons. MVP uses the continuous en-ergy model and GMVP uses the multigroup energy model. Vector-ized.

TRIPOLICEA

(France)

Multipurpose: reactor calculations, shielding, instrumentation. Itcan transport neutrons (up to 150 MeV), photons (up to 100 MeV),electrons and positrons (up to a few GeV).

available for these purposes. They are listed in table 6.1. Codes intended for xed source calculationsinclude FLUKA [113], GEANT [7], MAVRIC [261] and MCBEND [6]. Codes intended for criticalitycalculations are KENO [127] and MONK [6]. Finally, multipurpose codes capable of types of calculationsare MCNP/MCNPX [383, 260], TRIPOLI [104], MCU [3] and MVP/GMVP [230].

Although as it has been previously said Montecarlo methods oer some advantages over deterministicmethods, they also have some disadvantages. The main disadvantage of Monte Carlo methods over deter-ministic methods is their slower convergence that causes them to require the usage of large computationalresources. For this reason, the spread of Monte Carlo techniques has been linked to the development ofhigh performance computing. The precision of the results of a Monte Carlo calculation is proportionalto the inverse of the square root of the number of particle stories (as it is the standard deviation of themean), which implies, e.g., that to get a ten-fold increase in the precision of the results it is necessarya 100-fold increase in the number of stories, and hence in the computational requirements. Consideringthe application or not of techniques to accelerate the convergence, Monte Carlo methods are classiedinto analog and non-analog Montecarlo methods.

Analog Monte Carlo methods are named in this way because they attempt to simulate the physicaltransport process as truly as possible and therefore they use a fully random sampling of transport events.This causes analog Montecarlo methods to have poor eciency for solving certain types of problems.Think, for instance a case in which we are interested in calculating the counting rate in a detector that isonly a small fraction of the volume of a nuclear system so that only a small number of particles end upin the detector. In this case, having to simulate all the particles will imply a large use of computational

6.2. MCNPX 85

resources. For this reason, Montecarlo techniques usually apply variance reduction techniques, thatconsists in alternating the random sampling in an attempt to favor those particle stories that have thehighest contribution to the required calculation. Monte Carlo methods that apply variance reductiontechniques are known as non-analog Monte Carlo methods, as they do not exactly simulate the physicalprocess. Quite obviously, the diculty in the application of these techniques lays in that it must be donein a way so that no systematic errors are introduced in the result.

Another disadvantage of Monte Carlo methods with respect to deterministic methods lays in thediculty of calculating adjoint uxes with Montecarlo techniques. In deterministic codes, the adjointequation can be solved in a very similar way than the direct (or forward) transport equation, but thebackwards transport of neutrons in Montecarlo codes is not so obvious and it is still a matter of research[71]. For this reason, kinetic parameters βeff and Λeff , which are adjoint weighted quantities, are alsodicult to compute with Monte Carlo calculations and they are usually computed with deterministiccodes. Nevertheless some Monte Carlo techniques are available for this purpose and they will be describedlater in sections 6.3 and 6.4.

For the case of βeff , several methodologies have been analyzed and benchmarked against experimentalmeasurements or deterministic calculations, in order to analyze their performance and determine whatto choose. On the contrary, for the case Λeff , only one Monte Carlo methodology has been proposed toour knowledge and it has been adopted without benchmarking.

6.2 MCNPX

MCNP (Monte Carlo N-Particle) is a particle (neutrons, photons and electrons) Montecarlo transportcode developed at the Los Alamos National Laboratory. It rst version was released in 1977, although it isan evolution of previous Montecarlo transport codes whose development had started more than a decadebefore. MCNP is capable to perform both xed-source and criticality calculations (keff eigenvalues).Latest available version is MCNP 6.1, released in 2013.

MCNPX (from MCNP eXtended) is an extension of the MCNP to allow a wider range of particleenergies and a wider range of particles (protons, pions, muons, heavy ions, etc). It has incorporated codefrom FLUKA and LAHET (a Montecarlo transport code for nucleons and pions in the middle energyrange, also developed at LANL and that uses the same geometry descriptions that MCNP). First versionpublicly released was the 2.1 version in 1997. Last available version is 2.7, released in 2010.

A brief summary of the features of MCNP/MCNPX follows. Main references used are the manuals ofMCNP 4C [69], MCNP 5 [383], MCNPX 2.5 [259] and MCNPX 2.6 [260].

6.2.1 Geometry description and nuclear data in MCNP/MCNPX

The problem geometry is divided in MCNP/MCNPX in a number of domains or cells with a xedcomposition and density. The cells are dened as unions and intersections of dierent types of surfaces.Surfaces that can be considered in MCNP/MCNPX are planes, spheres, cylinders, cones, quadric surfacesand circular or elliptical torus. MCNP/MCNPX also has the capacity to deal with repeated structuresand lattices.

Main source of nuclear data for MCNP/MCNPX are continuous energy nuclear data libraries, butdiscrete reaction and multigroup data can also be used. MCNP/MCNPX can deal as well with thermalneutron dispersion libraries. Dosimetry cross section data can also be supplied to MCNP/MCNPX.Although these data are not used for particle transport, they are used by MCNP/MCNPX to convertparticle uxes into dosimetry data. It is important to remark that MCNP/MCNPX can not use evaluateddata itself, but data should be provided in a specic format (called ACE). Therefore, a previous processingof the data with a code such as NJOY is required. Doppler resonance broadening and other eects in thenuclear data that are not considered by MCNP/MCNPX must also be considered during this previousdata processing.

6.2.2 Types of calculations (tallies)

MCNP/MCNPX can perform seven standard types of calculations or tallies, apart from the denitionof additional user-dened tallies. They are listed in table 6.2. The result of these basic tallies can bemultiplied by cross sections to determine reaction rates instead of neutron uxes or by dose-conversionfactors to obtain dose rates. Furthermore, MCNP/MCNPX also allows the usage of mesh tallies with auser-dened mesh.


Up to 13 variance reduction techniques are available in MCNP/MCNPX ([69], p. 2-132). Thesevariance reduction techniques are classied in truncation methods, population control methods, modiedsampling methods and partially deterministic methods. Truncation methods consist in curtailing theproblem space or the particle energy so that only regions of interest are sampled. Truncation methodsavailable in MCNP/MCNPX are energy and time cuto. Population control methods attempt to samplemore in detail the regions of higher importance. To prevent biasing the results, they assign a weight wi toeach particle so that in high importance regions many low-weight stories are run and in low importanceregions a few high-weight stories are run. Geometry and energy splitting and Russian roulette belong tothis class. Modied sampling methods use modied random number distributions to send more particlesinto the regions of interest, also assigning dierent weights wi to the particles not to bias the results.Finally, under partially deterministic methods are included dierent techniques that alter the statisticalnature of the Monte Carlo method. The reader is referred to the MCNP manual for a detailed descriptionof all these methods.

Since the Montecarlo technique uses discrete random variables, to determine continuous variables suchas neutron uxes or reaction rates, the usage of statistical estimators is required. The simplest estimatoris the collision estimator. In chapter 4, the reaction rate was related with the neutron ux through theconcept of macroscopic cross section. Therefore, the time integrated average neutron ux in a volume ofconstant composition can be estimated as:

1

V

∫V

∫t

∫E

φ (~r, t, E) d3~rdtdE =1

V

∫V

∫t

∫E

(∂N∂t

)coll

Σt (~r,E)d3~rdtdE =

1

V

∑i

wiΣt (~r,E)

where the index i ranges over all collisions in the volume in the given volume. Reaction rates for anx reaction can be estimated in a similar fashion, by simply scoring the parameter wiΣx (~r,E) /Σt (~r,E).Similarly, other estimators can be dened for specic reactions by simply replacing Σt (~r,E) by themacroscopic cross section for the given reaction. Thus an absorption estimator, a capture estimator, etccan be dened.

These collision estimators have the disadvantage that may drive to large statistical errors in areaswith low collisions rates. To solve this problem, another type of estimators named track length estimatorsare dened which is the preferred estimator in MCNP/MCNPX. Track length estimators calculate theaverage reaction rate of a reaction x in a volume by scoring the quantity wi

∫t.l.

Σx (~r,E) ds for everyparticle going through the volume, where the integral extends to the track of the particle within thevolume. Quite obviously, the average neutron ux within the volume can be estimated as wi × t.l..Track length estimators allow for better statistics since they apply to all particles entering a cell, notonly to particles undergoing reactions in the cell. The reader is referred to the bibliography [327] for ajustication of this estimator.

Both collision and track length estimators pose problems when applied to very small volumes. Forthese cases another type of estimator has been developed which is known as next-event estimator whichis also implemented in MCNP/MCNPX (point-detector tally, [69], p. 2-85). This estimator calculatesthe ux at a point in space from the probability of a particle at every collision or source point of beingscattered and reach that point, that is, the scored quantity is:

wip (µ) /2πR2e−∫R0

∑t(s)ds

where p (µ) /2πR2 is the probability of being scattered in the angle between the incident particledirection and the detector point (assuming azimuthal symmetry of the scattering angle) and e−

∫R0

Σt(s)ds

gives the probability of reaching the nal point (i. e., of not suering a collision in between).

6.2.3 Criticality calculations (KCODE mode)

MCNP/MCNPX has a mode (KCODE) to perform criticality calculations. To perform a criticalitycalculation, MCNP/MCNPX requires the number of KCODE cycles, the number of particle histories byKCODE cycle and an initial estimate for the keff . An initial source distribution needs to be provided aswell.

In the rst KCODE cycle, a user-specied number of particles N is started isotropically from a set ofM source points. The source points can either be dened in a user-supplied le, or they can be speciedby the user in the input le (KSRC card) or they are distributed uniformly over the volume (SDEF card).The energy distribution of the particles can be also specied in the external le or in the KSRC card;otherwise a Watt ssion spectrum is considered. A good specication of the initial source distribution

6.2. MCNPX 87

Table 6.2: Summary of MCNP/MCNPX tallies.

Tally type Denition Particles

Track lenghtestimate of cell ux

1V

∫V

∫t

∫Eφ (~r,E, t) dEdtdV = WTl/V

where W represents the particle weight and V the cell volume.

Neutrons,photons,electrons

Track lengthestimate of energy

deposition

1Vρaρg

∫V

∫t

∫EH (E)φ (~r,E, t) dEdtdV

where ρa is the atom density, ρg is the gram density and H (E)is the heating response integrated over all nuclides in the mate-rial. H (E) is dened dierently for neutrons, photons or otherparticles.

Neutrons,photons

Track lengthestimate of ssionenergy deposition

Similar to the previous tally, but it only considers ssion reactions.Unlike the previous tally, it also takes into account the energydeposition of ssion gammas, which causes that the result of thistally may be larger than the previous one.

Neutrons

Surface current∫S

∫µ

∫t

∫EJS (~r,E, t, µ) dEdtdµdS1


Surface ux

1S

∫S

∫t

∫Eφ (~r,E, t) dEdtdS

This tally can be understood as the limit of the track length esti-mate of the cell ux when the cell thickness tends to zero.


Flux at a point orring detector

The point detector tally estimates the ux at a point in space (apoint detector) using a next event estimator. A ring detector tallyis similar, but the detector is not a xed point but it is sampledwithin a ring.

Neutrons,photons

Pulse height

Distribution of the energy pulses deposited in the cell by individualparticles. It intends to simulate physical detectors. It is not atrack lenght cell tally, but the energy deposited is computed fromthe dierence in the kinetic energies between the particles enteringand exiting in the cell.


1 The MCNP/MCNPX denition of surface current should not be confused with the denition of neutron current given insection 4.2.3. MCNP/MCNPX denes surface current as the total (not net) number of particles crossing a surface in aparticular direction. It is related with the neutron scalar ux by JS (~r, t;E;µ) = |µ| ·S·φ (~r, t;E). Notice that with thisdenition the surface current is always a positive quantity.


can help to accelerate the convergence of the calculation. The number of initial points M is computed asN/keff , with the initial estimation for the keff . If the number of points in the le or in the KSRC cardis dierent, some points will be either replicated or ignored.

These particles are then transported following a random walk process until a capture or ssion eventtakes place. These ssion points are taken as initial source points for a new KCODE cycle. The numberof KCODE cycles is specied by the user. Usually, some initial cycles (inactive cycles) are skipped beforestarting the accumulation of keff values, this is done to allow the ux to stabilize before begin to calculatekeff . The number of inactive cycles is also specied by the user.

MCNP provides three dierent estimators for the criticality constant keff ([69], p. 2-164 and subse-quent). The rst of them, the collision estimator, is dened as:

kCeff =1

N

∑i

Wi

[∑k fkνkσf,k∑k fkσT,k

]where the sum index i ranges over all collisions in a cycle where ssion is possible, the sum index k

ranges over all materials involved in the i-th collision collision, σT,k and σf,k are respectively the totaland the ssion microscopic cross sections, νk is the average number of neutrons produced per ssion atthe incident energy, fk is the atomic fraction for the nuclide k, Wi is the weight of the particle enteringcollision and N is the nominal source size for cycle.

The second estimator for keff is the absorption estimator. This estimator is dened in two dierentways depending whether implicit or analog capture is considered. For analog capture, which keeps thehistory weight constant, the absorption estimator is dened as:

kAeff =1

N

∑i

Wiνk

[σf,k

σC,k + σf,k

]where σC,k is the total capture cross section. For implicit capture, the absorption estimator is dened

as:

kAeff =1

N

∑i

W ′i νk

[σf,k

σC,k + σf,k

]where W ′i is dened as W ′i = Wi(σC,k + σf,k)/σT,k. With this denition, the dierence with the

collision estimator is that it only considers the isotope involved in the collision rather than all the nuclidespresent in the cell.

Finally, the third estimator for keff is the track length estimator, which is dened as:

kTLeff =1

N

∑i

Wiρd∑k

fkνkσf,k

where the sum index i ranges over all neutron trajectories, ρ is the atomic density in the cell and dis the trajectory track length for the last event. Notice that

∑k ρfkσf,k is the macroscopic ssion cross

section of the material and the mean neutron track is the inverse of the total macroscopic capture crosssection of the material.

The nal result of keff provided by MCNP/MCNPX is a combination of all these three estimators.The computation and the errors of this combination is rather complex and the reader is referred to tothe manual for details. Apart from this nal combination, the results for every estimator alone and forcombinations of pairs of estimators are also provided.

6.3 Calculation of the eective delayed neutron fraction

As stated in section 6.1, one of the disadvantages of Monte Carlo methods versus deterministic methodsis the diculty of performing adjoint-weighted calculations and hence the diculty in calculating thekinetic parameters. For instance, kinetic parameters calculation is not standard feature of MCNPX1.

A large number of publications have appeared over the last years considering dierent techniquesfor the calculation of βeff with Monte Carlo codes. Trying to group them, they can be classied intotwo categories. The rst one comprises techniques based on k-eigenvalue calculations; the second one

1It must be remarked that the latest versions of the MCNP code (version 6.1) can also provide values for adjoint-weightedparameters. The calculation methodology is based on an interpretation of the adjoint ux as iterated ssion probability([177, 178]).

6.3. CALCULATION OF THE EFFECTIVE DELAYED NEUTRON FRACTION 89

comprises techniques based on dierent interpretations of the adjoint weighting, such as those basedon interpreting the adjoint weighting as the next ssion probability or the iterated ssion probability.In addition, a third category of techniques can be considered to include those based on perturbativemethods, such as these derived in [232, 233], but they have not been considered in this work.

The rst of these categories, techniques based on k-eigenvalue calculations, includes techniques basedon the denition and calculation of certain parameters by analogy to the eective multiplication constant(keff ), such as those of [68] and [330]. Techniques based on the interpretation of the next-ssion prob-ability as the adjoint weighting have been analyzed by [237], [181] and [231]. Techniques based on theinterpretation of the iterated ssion probability as the adjoint weighting can be seen as an improvementof the previous ones, and they have been proposed by [238], [283], [87] and [161]. In section 6.3.1 itwill be provided some discussion on the derivation and the physical meaning of all these techniques.In sections 6.3.2 and 6.3.3 the results of the application of the above mentioned techniques against anumber of critical and subcritical benchmark systems will be presented. These systems are considered tobe representative of a wide range of nuclear systems. For this, the Monte Carlo code MCNPX has beenused, with three dierent nuclear data libraries (ENDF/B-VII.0, JEFF-3.1 and JENDL-3.3). The use ofseveral nuclear data libraries allows setting a lower limit to the uncertainty on βeff estimators, due toboth the accuracy of the dierent techniques and the uncertainties of the basic nuclear data.

6.3.1 Calculation methodologies

The eective delayed neutron fraction, βeff , has been dened in section 5.3 (equation 5.42) as:

βeff =

(Φ†λ, FdΦλ

)(

Φ†λ, FΦλ

)where F is the creation operator, that takes into account all neutrons (prompt and delayed) created

in the phase space by ssion, and Fd is the delayed neutron creation operator, that takes into accountonly delayed neutrons. In this work, it is considered "eective" only the delayed neutron fraction denedin Eq. 5.42 with the λ-mode neutron ux Φλ dened in section 4.3.1 and its adjoint counterpart Φ†λ, thatis, the fundamental mode solutions of the eigenvalue equations2:

MΦλ =1

keffFΦλ (6.1)

M†Φ†λ =1

keffF †Φ†λ (6.2)

being M the migration and losses operator, that takes into account the net number of neutrons leavingthe phase space element by capture, out-scattering or streaming, and F is the creation operator, alreadydened. M† and F † denote their corresponding adjoints.

Several other delayed neutron fractions β can be dened considering uxes other than Φλ or Φ†λ butthey will not be the "eective" values anymore. For instance, considering the adjoint ux to be constantover the whole phase space, a non-adjoint weighted delayed neutron fraction, β0, can be dened, thatcan be expressed as:

β0 =

(FdΦλ

)(FΦλ

) (6.3)

The determination of β0 with Monte Carlo codes poses no major diculty and can be performed bysimply counting the number of total and delayed neutrons produced in ssion processes. On the contrary,the determination of adjoint-weighted parameters requires the development of specic methodologies.

2Notice that the λ-mode ux, Φλ, obtained as solution of Eq. 6.1 only corresponds to the physical ux for a criticalsystem. As the system departs from criticality, the physical ux also begins to dierentiate from Φλ. Hence, the conceptof eective delayed neutron fraction losses signicance for systems far away from critical.


6.3.1.1 k-eigenvalue methods

Some of these methodologies can be classied as k-eigenvalue methods because they are based in deningand solving eigenvalue equations similar to 6.1 and 6.2. A rst method is applied by [68] and it has beennamed the prompt method by [181] and the prompt k-ratio method by [233]. It is obtained by deningthe following eigenvalue equations:

MΦp =1

kpFpΦp (6.4)

M†Φ†p =1

kpF †pΦ†p (6.5)

where Fp is the prompt neutron creation operator. Assuming that Φp ' Φλ, which in principle seemsa good approximation since over 99% of the neutrons produced in ssion are prompt neutrons, it isobtained that:

βeff =

(Φ†λ, FdΦλ

)(

Φ†λ, FΦλ

) = 1−

(Φ†λ, FpΦλ

)(

Φ†λ, FΦλ

) ' 1−

(Φ†λ, kpMΦλ

)(

Φ†λ, keffMΦλ

) = 1−kp

(Φ†λ, MΦλ

)keff

(Φ†λ, MΦλ

) =

= 1− kpkeff

(6.6)

The eigenvalue kp can be easily evaluated in MCNPX without requiring any modication of the code,making the most of the ability of this code to switch o the delayed neutron transport (this is a standardfeature of MCNPX and MCNP since version 4C, and it is performed by overriding the KCODE card witha TOTNU NO card). To perform this calculation it is obviously required that nuclear data librariescontain information about delayed neutron spectra, which is included in the latest versions of the mostcommon ones.

A problem of the prompt method lies in understanding how it takes into account adjoint weighting.In fact, the calculation of an adjoint-weighted parameter like βeff should not be possible using expres-sions that only contain k-eigenvalues, i.e., that are not adjoint-weighted parameters. Notice that botheigenvalues keff and kp can be determined (Eqs. 6.1 and 6.4) with no need to dene any adjoint ux.For instance, [68] considers no adjoint uxes for deriving the method.

Notice as well that, in principle, any weighting function (e.g. Φ†p as dened by Eq. 6.5) can be usedinstead of Φ†λ in Eq. 6.6, leading to the same numerical value for βeff . In particular, a constant weightingfunction over the whole phase space can be considered, and therefore the method must provide valuesfor β0 instead of βeff . In our opinion, it is not fully understood how the appropriate adjoint weightingis taken into account when applying the prompt method, as dened by Eq. 6.6, and, in particular, whenit is applied with a Monte Carlo code. These issues have not been addressed when the prompt methodis discussed by authors like [181] or [233].

In [181], it is also remarked that the prompt method is always an approximate method, because ofthe approximation Φp ' Φλ. For this reason, it has been considered the possibility of removing theapproximation by dening a new parameter k′p as:

k′p ≡

(FpΦλ

)(MΦλ

) (6.7)

Notice that Eq. 6.7 is not an eigenvalue equation. With this parameter, a new delayed neutronfraction β′ can be dened in an analogous manner to Eq. 6.6:

β′ = 1−k′pkeff

(6.8)

Although with this denition of k′p it is expected that β′ will be equivalent to the non-adjoint weighted

neutron fraction β0, as dened by Eq. 6.3, results obtained with this parameter are also included in section6.3.3, to help clarifying how the adjoint weighting inuences the prompt method. The prompt methodusing k′p will be denoted as prompt method with the total eigenfunction, as opposite to the previousmethod, which will be denoted as prompt method with the prompt eigenfunction.


k′p can be calculated with MCNP performing a single KCODE cycle switching o the delayed neutrontransport and taking as initial ssion source distribution a previously calculated one with a ux thatincludes both prompt and delayed neutrons. An alternative to improve statistics is to perform many ofthese KCODE switching o delayed neutrons and upgrading the initial ssion source in another KCODEcycle run in parallel, this one considering both prompt and delayed neutrons.

6.3.1.2 Methods based on dierent interpretations of the adjoint ux

A second class of methods for the calculation of βeff is based on the interpretation of the physicalmeaning of the adjoint weighting. In section 5.1.2, it has been obtained, considering the adjoint sourceto be the macroscopic cross section of the sytem, that the adjoint ux can be interpreted as the sourceimportance, i.e., the value of the adjoint ux in a point of the phase space is the response of the systemto a unit source introduced in this point, meaning by response of the system the total number of ssionreactions caused by the introduction of a source neutron in the point of the phase space

(~r,E′, ~Ω′

). In

mathematical terms (equation 5.16):

Ψ†(~r0, ~Ω0, E0

)=

∫Σf

(~r, ~Ω, E

)Φ(~r, ~Ω, E

)d~rd~ΩdE

being Φ(~r, ~Ω, E

)the solution of the equation Lφ

(~r, ~Ω, E

)= −δ

(~r0, ~Ω0, E0

)3.

To implement the interpretation of the adjoint weighting as neutron source importance to calculateβeff it is necessary to determine the dierent importances of prompt and delayed neutrons in generatingnew ssions. For this, rst notice that, as it is well known, the total progeny of a given neutron is given byk+k2+k3+... in case the multiplication constant k remains constant between ssion generations; otherwiseit will be given by k0 + k0k1 + k0k1k2 + .... Therefore, it is necessary to determine the multiplicationconstant of delayed neutrons.

To obtain these values, a modication of the MCNPX code has been implemented by D. Villamarín(CIEMAT) to track the delayed neutron creation. In this way, if we start from an already convergedssion source distribution, we can now obtain the subset of the ssion source corresponding to delayedneutrons. Once this delayed neutron source is known, a value of kd0 can be calculated (in MCNPXthis is performed with a single KCODE cycle, taking this delayed neutron source as initial ssion sourcedistribution). Notice that, with this denition, kd0 is not an eigenvalue, as it is calculated from thedelayed component of a converged source.

This rst KCODE cycle will also provide a second ssion source distribution that can be used as theinitial ssion source distribution for another KCODE cycle, in order to obtain a second multiplicativeconstant kd1 and a third ssion source distribution. In turn, this third source distribution can be used toobtain a third multiplicative constant, kd2, and so on. Notice that after several cycles the value of kdi willtend to the value of keff , as the neutron sources used to calculate them tend to reach the initial ssionsource distribution (considering both prompt and delayed neutrons), and therefore the results obtainedwith this technique will converge.

Once the values kd0, kd1, etc, have been determined, the value of βeff can be calculated as:

βI.F.P. =Equilibrium number of delayed neutronsEquilibrium number of all neutrons

× Average multiplication of delayed neutronsAverage multiplication of all neutrons

=

= β0 limn→∞

kd0 + kd0kd1 + kd0kd1kd2 + ...+∏n−1i=0 kdi

keff + k2eff + k3

eff + ...+ kneff(6.9)

This technique to calculate βeff will be denoted as integrated ssion probability (I.F.P). This newdenition is introduced to dierentiate it from the concept of iterated ssion probability introduced in[153]. In Hurwitz's derivation, which is done for a critical system, the iterated ssion probability is denedas the number of ssions produced in the n-th generation after a neutron is introduced in the system,and therefore it is usually interpreted as kd0kd1kd2...kdn. Nevertheless, although this interpretation isappropriate for a critical system, it seems less appropriate for a subcritical system, because it tends to 0after a large number of generations. Therefore, in this case a better estimate of the neutron importanceshould be the total (or integrated) number of ssions produced by a given neutron, i.e. kd0 + kd0kd1 +

3Take into account that this is equivalent to the track length estimator of k. Hence, this estimator will be used for allcalculations.


Table 6.3: βeff and β0 results (in pcm) with the ENDF/B-VII.0 library. Statistical errors are given whenavailable. βP.E.: prompt method with the prompt eigenfunction; βT.E.: prompt method with the totaleigenfunction; βN.F.P : next ssion probability; βI.F.P : integrated ssion probability.

ENDF/B-VII.0Benchmark βeff reference β0 βP.E. βT.E. βN.F.P. βI.F.P.

Godiva 659 ± 10 640 651 ± 6 637 ± 3 643.05 ± 0.09 642 ± 1Jezebel 194 ± 10 204 176 ± 7 210 ± 3 183.35 ± 0.04 178.5 ± 0.3Topsy 665 ± 13 812 698 ± 6 816 ± 3 620.77 ± 0.13 684 ± 1Popsy 290 ± 10 536 282 ± 6 546 ± 3 256.73 ± 0.08 273.7 ± 0.5Big Ten 720 ± 7 904 724 ± 4 906 ± 3 697.72 ± 0.09 712 ± 2CORAL 663 ± 17 822 679 ± 6 811 ± 8 620.91 ± 0.12 681 ± 1MUSE-4 331 ± 5 352 313 ± 6 354 ± 3 313.99 ± 0.08 313.6 ± 0.6TCA 771 ± 17 702 763 ± 8 701 ± 6 763.32 ± 0.21 763 ± 2

Yalina-B (902 f. e.) 761 663 716 ± 8 669 ± 6 724.82 ± 0.24 731 ± 1Yalina-B (1141 f. e.) 753 662 723 ± 7 665 ± 4 715.69 ± 0.23 719 ± 1

kd0kd1kd2 + ..., which tends to converge to a nite value. Thus, this is the interpretation adopted inthis work. It must be remarked that this same interpretation has already been applied by [111, 112] tocalculate neutron importance functions and importance-weighted neutron generation times4.

In practice, it is only needed to calculate a limited number of values of kdi, providing that thisnumber is enough to reach convergence. If we are left with a single cycle we will be considering onlythe multiplicity of the delayed neutrons but not of their progeny, which is dened as the next ssionprobability in [181]:

βN.F.P. = β0kd0

keff(6.10)

In [181] it is also discussed the validity of this approximation, arguing that the exact knowledge ofthe adjoint ux is not critical since the value of βeff is largely determined by the value of β0 and hencethe accuracy in the denition of the adjoint weighting function has little impact. However, the validity ofthis approach has been questioned by other authors, for instance [161] and [86]. Therefore, the validityof this approximation will be discussed in section 6.3.3.

6.3.2 Benchmark systems

The techniques described in the previous section have been implemented with a modied version of theMonte Carlo code MCNPX 2.7 and validated against a set of benchmark experiments for which measuredor deterministic data for βeff are available. These benchmarks have been chosen to be representative ofwide range of systems. Hence, they include two homogeneous bare fast systems (Godiva and Jezebel), fourreected fast systems (Topsy, Popsy, Big-Ten and CORAL-I), a thermal system (TCA), two congurationsof a subcritical system (Yalina-Booster) and three large fast systems (MUSE-4, ESFR and MYRRHA).

MCNP inputs have been taken from the [239] (the reference number is given), unless indicated oth-erwise. A brief description of these system follows. The same βeff reference values considered by [181]have been used when available; otherwise, the source is referenced5.

Godiva (HEU-MET-FAST-001). A bare, highly enriched (94 w/o) uranium spherical core containing52.42 kg of uranium. The proposed MCNP model consists of ve shells with slightly varying

4In fact, for a critical system both interpretations should be equivalent. In a critical system, the expressions kd0 +kd0kd1 + kd0kd1kd2 + ... in the numerator and keff + k2eff + k3eff + ... in the denominator both diverge. Therefore, we can

neglect the rst m terms (which take a nite value) in both the numerator and the denominator, and take out the commonfactors, to be left with:

βI.F.P. = β0 limn,m→∞

kd0...kdm

(1 + kd(m+1) + ...+

∏n−1i=m+1 kdi

)km+1eff

(1 + keff + ...+ kn−m−1

eff

)If after the m-th generation kdi has converged to the value of keff , then the terms between brackets in the numerator

and in the denominator must cancel out, and thus we are left with the iterated ssion interpretation of the adjoint ux.5MCNP inputs and references for the cases of ESFR and MYRRHA were supplied, respectively, by S. Pérez-Martín and

M. Vázquez-Antolín (CIEMAT)


Table 6.4: βeff and β0 results (in pcm) with the JENDL-3.3 library.

JENDL-3.3Benchmark βeff reference β0 βP.E. βT.E. βN.F.P. βI.F.P.



Table 6.5: βeff and β0 results (in pcm) with the JEFF-3.1 library (JEFF-3.1.1 for the ESFR andMYRRHA).

JEFF-3.1Benchmark βeff reference β0 βP.E. βT.E. βN.F.P. βI.F.P.



ESFR (1) 389.5 491 398 ± 7 465 ± 21 391.62 ± 0.08 388.7 ± 0.7ESFR (2) 360.3 454 362 ± 6 404 ± 35 360.33 ± 0.07 358.1 ± 0.7ESFR (3) 363 477 364 ± 7 475 ± 27 362.22 ± 0.08 357.3 ± 0.8ESFR (4) 338.7 436 339 ± 7 393 ± 31 337.59 ± 0.07 335.3 ± 0.9

MYRRHA (1) 321 363 331 ± 10 347 ± 13 321.27 ± 0.06 321.3 ± 0.7MYRRHA (2) 332 377 331 ± 9 399 ± 12 332.68 ± 0.09 330.5 ± 0.6


composition and an additional sixth shell added to compensate for reections in the supportingelements and other factors.

Jezebel (PU-MET-FAST-001). A bare plutonium sphere (4.5 w/o of Pu-240) containing 17.020 kgof a plutonium and gallium alloy (1.02 w/o of gallium).

Topsy (HEU-MET-FAST-028). A sphere of 93 w/o enriched uranium (17.84 kg) surrounded by athick (19 inches) U-238 reector.

Popsy (PU-MET-FAST-006). A plutonium/gallium sphere (6.06 kg) surrounded by a uraniumreector.

Big Ten (IEU-MET-FAST-007). The Big Ten core contained uranium with three dierent enrich-ments (93%, 10% and natural). It was surrounded by a depleted uranium reector. Total uraniummass in the reactor was 10 metric tons.

CORAL-I. A reected, highly enriched uranium cylinder (containing 26 kg of 90% enriched uranium)surrounded by a reector of natural uranium ([366]). It was in operation at CIEMAT (formerlyJEN) between 1968 and 1988. MCNP models for CORAL-I are available ([369]). Experimentalβeff data were measured and are presented in [102].

TCA (LEU-COMP-THERM-006). The TCA (Tank Critical Assembly) consists of a tank containingan array of fuel assemblies of 2.6 w/o UO2 and water that acts as moderator. The number of fuelrods and the lattice geometry and pitch can be changed. The reactor is made critical by adjustingthe water level.

Yalina-Booster. The Yalina-Booster subcritical facility is located at the JIPNR-Sosny of the Na-tional Academy of Sciences of Belarus. The facility consists of a fast and a thermal zones partiallydecoupled by a thermal neutron absorbing layer between them. The fast zone is formed by HEU(90% and 36% enrichment) in a lead matrix and the thermal zone is formed by LEU (10% enrich-ment). The whole assembly is surrounded by a graphite reector. Experimental βeff data are notavailable, but deterministic calculations with the ERANOS code have been reported by [16] fortwo dierent congurations of the system with a dierent number of fuel rods in the thermal zone(902 and 1141, respectively). The reference value considered here is the average of several libraries.No estimation of the error is given. MCNP inputs were supplied by JIPNR-Sosny and have beenmodied to t these two congurations.

MUSE-4. The MUSE-4 experiment was carried out at the MASURCA reactor at the CEA-Cadarache facility (France) between 2001 and 2004 and comprised several critical and subcriticalcongurations. The reference critical conguration consisted of a core formed by a mixture of MOXfuel and sodium rodlets surrounded by a reector made up of stainless steel and sodium rodlets.The MCNP input has been made using the specications given in [324]. βeff data have been takenfrom [200]. It must be remarked that the values of βeff were measured in a later conguration thanthose of the MCNP input le used, with 1115 fuel elements instead of 1112. These three elementswere added to compensate for the decay of some of the Pu-241 in the MOX fuel. Nevertheless thedierence in the measured value of βeff is not expected to be signicant.

ESFR (European Sodium Fast Reactor). The ESFR is a conceptual design for a large scale sodium-cooled fast reactor. The conguration considered in this work consists of a 3600 MWth core,composed of two zones with dierent fuel compositions, surrounded by a stainless steel reector.The MCNP input was developed within the participation of CIEMAT in the CP-ESFR project[265]. The reference results for βeff have been taken from [214] and were calculated by PSI with thedeterministic code ERANOS. Four dierent core congurations have been considered to calculatethe value of βeff :

1. (U,Pu)O2 MOX fuel (averaged Pu content 15% in weight) at beginning-of-cycle.

2. Same fuel after 1230 equivalent full power days.

3. (U,Pu,MA)O2 fuel (MA content 4% in weight) at beginning-of-cycle.

4. Same fuel after 1230 equivalent full power days.


MYRRHA. The MYRRHA facility has been already described in section 3.1.5. MCNP referenceinputs were developed by SCK-CEN within the CDT FASTEF project. Reference results for βeffwere calculated by ENEA using the ERANOS code [306]. Fuel depletion calculations for the end-of-cycle core have been performed with the EVOLCODE system [19]. The eective delayed neutronfraction has been calculated in two congurations:

1. 100 MW(th) critical core at beginning-of-cycle (fresh fuel Pu content 34% in weight).

2. 94 MW subcritical core at end-of-cycle (fresh fuel Pu content 30% in weight).

In all the cases except the ESFR and MYRRHA, the βeff results with three dierent nuclear datalibraries (ENDF-VII.0, JEFF-3.1 and JENDL-3.3) processed at room temperature are presented. For theESFR and MYRRHA, only results for the JEFF-3.1.1 library processed at operating temperatures arepresented.

6.3.3 Validation results

The results of the validation are presented in Tables 6.3, 6.4 and 6.5. They include results of βeff withthe techniques described in section 6.3.1 as well as results for β0. The ratios between the values of βeffcalculated with the dierent techniques and the reference values are presented graphically in Fig. 6.1.The prompt method (both with the prompt and the total eigenfunction) has been applied consideringthe simple average of the three keff estimators provided by MCNPX (collisions, absorptions and tracklength estimators)6. In the case of the I.F.P and N.F.P methods only the track length estimator has beenconsidered.

Results show that the prompt method, with the prompt eigenfunction, ts the reference experimen-tal or deterministic results with good accuracy in all the cases. These results are consistent with theconclusions of previous benchmarks (e.g. [181]), in spite that, as it was commented in section 6.3.1, thequestion of how adjoint weighting is taken into account with this method, is not, to our understanding,fully explained.

On the other hand, the prompt method with the total eigenfunction has been found to provide valuesof β0 rather than values of βeff , as expected. Although in homogeneous systems (Godiva, Jezebel) bothvalues are similar, in heterogeneous (reected) systems (Topsy, Popsy, Big Ten, CORAL), where β0 andβeff are very dierent, the prompt method with the total eigenfunction overestimates the value of βeffby a factor of up to two. For thermal systems (TCA and Yalina-B), however, it has been found tounderestimate the values by about a 10%.

The integrated ssion probability methodology has also been found to provide accurate results forβeff in all cases for the number of KCODE cycles considered (50). If only a single KCODE cycle isperformed (next ssion probability approximation) the values of βeff obtained can be up to about 10%lower than the reference values for the case of reected systems (Popsy, Topsy, Big-Ten, CORAL).

In these cases, the number of KCODE cycles required in order to reach convergence in the value ofβeff has been analyzed. The results of this analysis are presented in Fig. 6.2. Notice how the value ofβeff increases rapidly with the rst cycles, and after about 15 cycles, a relatively constant level is reached.A very accurate determination of keff is required for this level to remain constant with the number ofcycles; otherwise, the bias in keff will accumulate with the increasing number of cycles and it will resultin an increasing systematic deviation in the value of βeff with the number of cycles. Therefore, a largernumber of KCODE cycles require a higher precision in the determination of keff (more statistics).

To better understand the evolution of the βeff results with the number of KCODE cycles, the radialdistributions of the initial delayed neutron ssion source have been obtained, and the ssion sources aftera dierent number of cycles for the cases of the four spherical reactors (Fig. 6.3). It can be observed how,for the cases of the non-reected systems (Godiva and Jezebel), the ssion source distribution does notexperience major changes with the number of cycles. For these systems there was no noticeable dierencebetween the βeff results considering the next ssion and the integrated ssion probability interpretationsof the adjoint weighting. In the reected systems (Topsy and Popsy), however, there is a considerabledierence among the initial delayed neutron ssion source, the neutron source obtained from the rstKCODE cycle, and subsequent ssion sources (from the second one, it is apparent that the ssion sourcehas reached a distribution that is stable with the number of cycles). These noticeable dierences amongssion sources can explain the dierences in the βeff results obtained with the N.F.P. and the I.F.P.techniques.

6It has to be noticed that the βeff estimator must keep the correlation among the two calculations (with and withoutdelayed neutrons), which, in general, is lost using the combined average keff estimator.


0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Go

div

a

Jeze

bel

To

psy

Po

psy

Big

Ten

CO

RA

L

MU

SE

-4

TC

A

Y-B

(1)

Y-B

(2)

β x/β ef

f,re

f

Prompt method, prompt eigen.

Prompt method, total eigen.

Next fission probability

Integrated fission probability

(a) ENDF/B-VII.0

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Go

div

a

Jeze

bel

To

psy

Po

psy

Big

Ten

CO

RA

L

MU

SE

-4

TC

A

Y-B

(1)

Y-B

(2)

β x/β ef

f,re

f





(b) JENDL-3.3

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Go

div

a

Jeze

bel

To

psy

Po

psy

Big

Ten

CO

RA

L

MU

SE

-4

TC

A

Y-B

(1)

Y-B

(2)

ES

FR

(1)

ES

FR

(2)

ES

FR

(3)

ES

FR

(4)

MY

R. (

1)

MY

R. (

2)

β x/β ef

f,re

f





(c) JEFF-3.1

Figure 6.1: Graphical depiction of the ratios of the βeff calculated with dierent techniques to thereference values.

6.4. CALCULATION OF THE EFFECTIVE MEAN NEUTRON GENERATION TIME 97

Finally, concerning the uncertainty in the results due to the nuclear data library used, it has beenfound to be of the same magnitude as the dierence between the prompt method (with the prompteigenfunction) and the integrated ssion probability method. With both methods, changing the nucleardata library can drive to changes in the βeff results of up to the order of 10 pcm. Therefore, it cannotbe concluded whether any library is better suited than the others for βeff calculations for a particulartype of systems.

6.4 Calculation of the eective mean neutron generation time

To obtain the eective mean neutron generation time Λeff from the MCNPX simulations, the perturba-tion method also applied in [365], which has been known since the 1950s [143], has been used. It consistsin calculating the variation of the system reactivity when it is perturbed by the introduction of ctitiousisotope with a capture cross section of the shape 1/v. This technique has the advantage of requiring noevaluation of the adjoint ux and hence it can be tackled with Monte Carlo codes.

Before describing the procedure to obtain Λeff from Monte Carlo simulations, it is convenient tospend some time in discussing the concept of mean neutron generation time and distinguish it from othersimilar parameters such as neutron lifespans and lifetimes that are commonly used [329]. As it wasdened in section 5.3, the mean neutron generation time is the mean time between two ssion events andit is dened as:

Λ =

(Ψ†, 1

vΦ)(

Ψ†, FΦ) (6.11)

where ψ† is an arbitrary weighting function. When this weighting function is the λ-mode adjoint ux,Φλ, we have the eective mean neutron generation time, Λeff , as dened in section 5.3. In a similarway, some neutron lifetimes can be dened for dierent processes, again by dividing the adjoint-weightedneutron population of the system by the reaction rate of this process. For instance, a removal lifetimeτr, an absorption lifetime τa and a leakage lifetime τl can be dened in the following way:

τr =

(Ψ†, 1

vΦ)(

Ψ†, MΦ) =

∫ ∫Ψ† 1

vΦd~rdEd~Ω∫ ∫ Ψ†ΣtotΦ−DΨ†∇2Φ

d~rdEd~Ω

(6.12)

τa =

∫ ∫Ψ† 1

vΦd~rdEd~Ω∫ ∫Ψ†ΣtotΦd~rdEd~Ω

(6.13)

τl =

∫ ∫Ψ† 1

vΦd~rdEd~Ω

−∫ ∫

DΨ†∇2Φd~rdEd~Ω(6.14)

Other lifetimes can be dened for other process in a similar way, e. g. a ssion lifetime or a capturelifetime. The lifetimes of a certain process is interpreted as the mean time between two events of thisprocess, or, equivalently, as the inverse of reaction rates, normalized to the weighted neutron population.It is straightforward to prove that the following addition rule hold for lifetimes:

1

τr=

1

τa+

1

τl(6.15)

Neutron lifespan by a process x (x can be ssion, capture or leakage) is dened as the mean timebetween a neutron is born in the system and it is terminated by this process [74]. The dierence betweenlifespan and lifetime may be confusing; the key point to notice is that a neutron lifespan is a time frombirth to event while a neutron lifetime is a time from event to event. Hence:

tx =1

N

N∑i

ti (6.16)

Where N is the total number of neutrons and ti is the time from birth to event for every single neutronin the system.

To better understand the dierence between these concepts and the relationship between them it isinteresting to consider the following numerical example, proposed in [74]. Let us consider a subcriticalsystem with keff = 0.9 a mean number of neutrons emitted by ssion ν = 2.5 coupled to a neutron


580

600

620

640

660

680

700

020

4060

80100

No

. of K

CO

DE

cycles

βeff (p.c.m.)

EN

DF

/B-V

II.0JE

FF

-3.1JE

ND

L-3.3

(a)Topsy

230

240

250

260

270

280

290

300

310

020

4060

80100

No

. of K

CO

DE

cycles

βeff (p.c.m.)

EN

DF

/B-V

II.0JE

FF

-3.1JE

ND

L-3.3

(b)Popsy

680

690

700

710

720

730

740

020

4060

80100

No

. of K

CO

DE

cycles

βeff (p.c.m.)

EN

DF

/B-V

II.0JE

FF

-3.1JE

ND

L-3.3

(c)Big-Ten

580

600

620

640

660

680

700

020

4060

80100

No

. of K

CO

DE

cycles

βeff (p.c.m.)E

ND

F/B

-VII.0

JEF

F-3.1

JEN

DL

-3.3

(d)CORAL

Figure

6.2:βeffresults

applyingthe

integratedssion

probabilityconsidering

adierent

number

ofcycles.

The

hatchedarea

representsthe

referencevalue

with

itsuncertainty.


02

46

810

12

Del

ayed

neu

tro

ns

100

KC

OD

E c

ycle

s

1 K

CO

DE

cyc

le

2 K

CO

DE

cyc

les

Co

reE

xter

nal

s r (c

m)

φF.S.(r) (a.u.)

(a)Godiva

02

46

810

12

Del

ayed

neu

tro

ns

100

KC

OD

E c

ycle

s

1 K

CO

DE

cyc

le

2 K

CO

DE

cyc

les

Co

reE

xter

nal

s

r (c

m)

φF.S.(r) (a.u.)

(b)Jezebel

05

1015

2025

30

Del

ayed

neu

tro

ns

100

KC

OD

E c

ycle

s

1 K

CO

DE

cyc

le

2 K

CO

DE

cyc

les

Co

reR

efle

cto

rE

xter

nal

s

r (c

m)

φF.S.(r) (a.u.)

(c)Topsy

05

1015

2025

30

Del

ayed

neu

tro

ns

100

KC

OD

E c

ycle

s

1 K

CO

DE

cyc

le

2 K

CO

DE

cyc

les

Co

reR

efle

cto

rE

xter

nal

s

r (c

m)

φF.S.(r) (a.u.)

(d)Popsy

Figure6.3:

Distributionof

delayedandtotalneutronsource

forfour

sphericalreactors,norm

alized

tothenumberof

source

points.


source. Within the system, a neutron can disappear because ssion, capture (e.g. (n, γ)) or leakage.Let us consider that the average time from birth to ssion (tf ) is 100µs, the average time from birthto capture (tc) is 90µs and the average time from birth to leakage (tl) is 50µs. A neutron has a 20%of probability to undergo capture, a 36% of probability to undergo ssion and a 44% of probability toundergo leakage. Hence, the removal lifetime of the system, or the inverse rate a with as neutron isremoved from the system by any of these processes, will be given by:

τr = Pctc + Pf tf + Pltl = 0.2× 90µs+ 0.36× 100µs+ 0.44× 50µs = 76µs

Considering a reactor population of normalized to the unity, this removal lifetime correspond to aremoval rate of 13158 n/s. If 20% are lost to capture, this corresponds to 2632 n/s or τc = 378µs; if36% are lost to ssion, this corresponds to 4737 n/s or τf = 211µs; and if 44% are lost to leakage, thiscorresponds to 5790 n/s or τl = 173µs. And considering ν = 2.5 this implies the creation of 11842 n/s inthe system, which correspond to a mean neutron generation time Λ = 84µs.

KCODE calculations in MCNP/MCNPX provide collision, absorption and track length estimators forall these lifetimes and lifespans but the values provided by MCNP/MCNPX are not adjoint weighted.Hence, it is necessary to implement a methodology to calculate the eective (adjoint-weighted) meanneutron generation time with MCNP/MCNPX. To derive this technique, let us start from the exactperturbation formula for reactivity increments, as obtained in section 5.2:

∆ρ =

(Φ†0, [λ0∆F −∆M ] Φ

)(

Φ†0, F0Φ0

) (6.17)

As perturbation, let us consider the introduction in the system of an uniform density of a neutronabsorber of cross section c/v. In this case, we have that:

∆M = Σc =c

v

∆F = 0

Notice that for Σc to have dimensions of inverse length, c must have dimensions of inverse time. Withthis choice for the perturbation, equation 6.17 becomes:

∆ρ = −

(Φ†0,

cvΦ)

(Φ†0, FΦ

) = −c

(Φ†0,

1vΦ)

(Φ†0, FΦ

) = −cΛeff (6.18)

And using the denition for Λeff given in equation 5.41, we obtain that:

∆ρ = −cΛeff ⇒ Λeff = −∆ρ

c(6.19)

This method is implemented in MCNP/MCNPX by introducing at every point of the assembly thesame density of a ctitious isotope with cross section of the shape c/v. This was achieved by creatingan ENDF le containing an isotope with such a cross section. In fact, since Σc = N (c)σc, beingN (c) the atomic density of the isotope, σc is introduced in the shape A/v. A is chosen so that thedensities N (c) remain low enough, since it is preferable, in order not to alter the composition of theMCNP/MCNPX model of the assembly, to introduce a small density of an isotope with a large capturecross section than a high density of an isotope with a small cross section. The value of A chosen isA = 1012barn×cm/s = 10−12cm3

/s. With this value, the resulting cross section is several orders of magnitudelarger than other typical large neutron cross sections (gure 6.4). It must be nally remarked that sincein the ENDF format energies must be introduced in eV and cross sections in barns, in order to have thisvalue of A, the pairs energy-cross section were introduced into the ENDF format le with the formula:

σc(b) =A

v=

A′√E(eV )

where A′ = 0.7229826248× 106barn×√

eV in order to have A = 1012barn×cm/s.


10-1

1

10

10 2

10 3

10 4

10 5

10 6

10 7

10 8

10-5

10-4

10-3

10-2

10-1

1 10 102

103

104

105

106

E (eV)

σ (b

)

235U MT=18: (n,fission) cross section

10B MT=107: (n,α) cross section

3He MT=103: (n,p) cross section

Fictitious isotope neutron capture cross section

Figure 6.4: Fictitious isotope neutron capture cross section compared with other typical large neutroncross sections (cross section data from the ENDF/B-VII.0 library)

Chapter 7

Reactivity monitoring

Abstract - In this chapter, the basic techniques available for reactivity monitoring are derived. They

are the current-to-ux technique, Pulsed Neutron Source (PNS) techniques (area-ratio (Sjöstrand)

and prompt neutron decay constant techniques) and beam-trip techniques (source-jerk and prompt

neutron decay constant techniques). Furthermore, the limitations of these techniques due to the

presence of spatial and spectral eects in actual systems will be discussed. An original methodology

for correcting these eects will be also presented. This methodology consists in a generalization of

the use of correction factors, and it will be validated against the experimental results measured at

the Yalina-Booster facility in chapters 10 and 11.

As it has been stated in chapter 2.3, this work is concerned with the reactivity monitoring in ADSs.Several techniques are available for this purpose. They can be classied in static techniques, dynamictechniques and noise (stochastic) techniques (e. g. [253]).

Static techniques measure the reactivity with the system in an steady state. For this reason, they arethe most likely to be applied to continuously monitor the reactivity of a power ADS, that will obviouslyoperate in a steady state. Static techniques have the disadvantage, however, that they are only capable ofmeasuring changes of reactivity and hence they must be complemented by other techniques that provideabsolute values of the reactivity in a reference state. The static technique that it is applicable to systemsin a subcritical state is the current-to-ux technique. The current-to-ux technique is based on theModied Source Method (MSM), which is commonly used in the calibration of control rods [62], adaptedto ADS characteristics. This technique will be described in section 7.1.

Dynamic reactivity monitoring techniques are based on the analysis of reactor transients. For thisreason, they are less useful than static techniques for continuously monitoring the reactivity of a powerADS. Nevertheless, they have the advantage of being capable to provide absolute reactivity values thatcan be used to calibrate static techniques. In this work, dynamic techniques based on two types oftransient are considered: pulsed neutron source (PNS) experiments and beam-trips experiments, as theyare the experiments performed at the Yalina-Booster facility that will be presented in part III.

In a PNS experiment (section 7.2) the response of the subcritical system to the introduction of veryshort pulses from the external neutron source is analyzed. Two main methods are available to determinethe reactivity from the results of PNS experiments: the area-ratio (or Sjöstrand) method and the promptneutron decay constant method. For its part, in a beam-trip experiment (section 7.3), the neutron sourceis operated in a steady state with small beam interruptions (beam trips). Again, two main methodsare available to determine the reactivity from beam-trips experiments: one is the prompt neutron decayconstant method that is also applied to the PNS experiments and the other one is the so-called source-jerkmethod, which is equivalent to the area-ratio technique of PNS experiments.

Finally, the third group of techniques available for reactivity measurement are the noise techniques[60]. They are based on analyzing the behavior of individual ssion chains and are only applicable tovery low power assemblies where it is possible to dierentiate neutrons belonging to the same or dierentssion chains. Therefore, they are not useful for reactivity monitoring purposes in a power ADS andhence they will not be treated in this work.

It must be noticed, however, that these techniques are all based in the point-kinetics model. However,as it was already noticed in section 5.3, the validity of the point kinetics model is determined by theproximity of the state of the system to a reference state that is usually considered to be the λ-mode inorder to identify the reactivity that appears in the point kinetics model with the reactivity dened insection 4.3.1. As it was described in section 4.3.1, in a critical system the λ-mode coincides with the

103

104 CHAPTER 7. REACTIVITY MONITORING

actual state of the system but as the subcriticality level increases the dierence between the λ-mode andthe actual state of the system also increases. This brings diculties concerning the applicability of thesereactivity measurement techniques and the interpretation as the reactivity of the measured quantities.A full discussion of these issues, centered in the case of the Yalina-Booster facility, will be presented inpart III.

7.1 The current-to-ux technique

In section 4.3.1 the concept of source multiplicity, ks, was introduced as:

ks =FΦ

FΦ + S

This equation can be rearranged in the shape:

FΦ + S =S

1− ks(7.1)

The term F φ + S is the number of neutrons produced in the system after the introduction of Sneutrons from the external source. If we denote this term by N , we can rewrite equation 7.1 in thefashion:

N = S1

1− ks(7.2)

and ks can be rewritten as well as:

ks = 1− S

N(7.3)

By analogy with the MSM method, routinely used to evaluate rod worth in critical reactors, equation7.2 is normally rewritten in terms of keff introducing the concept of source eciency, dened as1:

ϕ∗ ≡ 1− keff1− ks

(7.4)

resulting in:

keff = 1− ϕ∗ SN

(7.5)

From this equation it is possible to evaluate the keff of the system by measuring N and S or theratio between them. However, this measurement requires that ϕ∗ is calibrated at specic congurations.Furthermore, neither N nor S will be experimentally measured but some related measurable magnitudesM and R (e.g. detector counting rates or current intensities). M and R are respectively proportionalto N and S and are related with them through the corresponding eciencies, namely M = εDN andR = εSS. Hence, in terms of M and R, equation 7.5 becomes:

keff = 1− ϕ∗ εDεS

R

M(7.6)

The neutron source is expected to be deduced from a measurement of the intensity of the chargedparticle beam (typically protons) generating the neutrons in the spallation target. Consequently, thevalue of εS and ϕ∗ depend on the beam position, energy, shape and density of the spallation targetand probably other factors. εD will also be aected by the detector nature, position and other factors.Some of these parameters can be measured easily but others are very dicult to monitor, and hence arecalibration of the entire ϕ∗ εDεS factor, with a technique other than the current-to-ux, is needed fromtime to time.

In any case, these factors usually evolve slowly and they can be considered constant during a periodof time. Therefore, the current-to-ux method is useful for online monitoring of the reactivity. Thedierence of reactivity between two congurations of a system will be given by:

1Other authors [121] dene the source importance alternatively as ϕ∗ ≡ 1−keff1−ks

kskeff

. In this work the denition given

by equation 7.4 is always used.

7.2. PULSED NEUTRON SOURCE (PNS) TECHNIQUES 105

∆keff = keff,2 − keff,1 = 1− ϕ∗2εD2

εS2

R2

M2−(

1− ϕ∗1εD1

εS1

R1

M1

)= ϕ∗1

εD1

εS1

R1

M1− ϕ∗2

εD2

εS2

R2

M2=

= ϕ∗1εD1

εS1

R1

M1

(1−

ϕ∗2εD2

εS2

R2

M2

ϕ∗1εD1

εS1

R1

M1

)= (1− keff,1)

(1−

ϕ∗2εD2

εS2

ϕ∗1εD1

εS1

M1/R1

M2/R2

)(7.7)

If both congurations are similar enough as to assume that εD1 = εD2, εS1 = εS2 and ϕ∗1 = ϕ∗2, thenequation 7.7 can be approximated by:

∆keff ' (1− keff,1)

(1− M1

R1

R2

M2

)(7.8)

which is equivalent to the ASM method (e.g. [62]).

7.2 Pulsed Neutron Source (PNS) techniques

As stated in the introduction of the chapter, in PNS experiments it is studied the response of the systemto a very short neutron pulse (ideally Dirac-δ shaped). The response of a subcritical system to such aDirac-δ pulse is studied in section 7.2.1. The knowledge of this response is relevant not only to analyzePNS experiments, but also because it represents the Green's function of the system (also called pulse-response), from which the response of the system to any source of an arbitrary shape can be obtained.

In practice, PNS techniques are not applied to single pulses but to series of pulses repeated with acertain period and superposed. In this way the statistics can be largely improved. For this reason, theresponse of a subcritical system to a series of pulses will be also analyzed in section 7.2.2.

7.2.1 Response to a single pulse

The equations of the point kinetics derived in the previous section applied to this case, considering onlyone family of delayed neutrons, take the form:

dn

dt=ρ− βeff

Λeffn+ λc+ n0δ (t) (7.9)

dc

dt=βeffΛeff

n− λc (7.10)

where we are considering the introduction of n0 source neutrons in t = 0 . c denotes the precursordensity λ is the decay constant of the precursors.

To obtain the solution of this system of equations, the Laplace transform technique will be applied.Hence, taking Laplace transforms on both sides of equations 7.9 and 7.10 they turn into:

sN − n (0) =ρ− βeff

ΛeffN + λC + n0 (7.11)

sC − c (0) =βeffΛeff

N − λC (7.12)

where N and C denote, respectively, the Laplace transforms of n (t) and c (t) and n (0) and c (0) arethe initial condition. Over isolating C in the second equation and substituting into the rst, we arriveto:

N (s) =n0 + n (0) + λ

s+λc (0)

s− ρ−βeffΛeff

− λ(s+λ)

βeffΛeff

=(n0 + n (0)) (s+ λ) + λc (0)(s− ρ−βeff

Λeff

)(s+ λ)− λ βeffΛeff

=

=(n0 + n (0)) s+ (n0 + n (0) + c (0))λ(

s− ρ−βeffΛeff

)(s+ λ)− λ βeffΛeff

(7.13)

The denominator of this expression is a second order polynomial in s, whose roots are:


(s− ρ− βeff

Λeff

)(s+ λ)− βeffλ

Λeff= 0⇒ s2 +

(−ρ− βeff

Λeff+ λ

)s− ρλ

Λeff= 0⇒

⇒ s =

−(−ρ+βeff+λΛeff )+

√(−ρ+βeff+λΛeff )2+4Λeffρλ

2Λeff= ω1

−(−ρ+βeff+λΛeff )−√

(−ρ+βeff+λΛeff )2+4Λeffρλ

2Λeff= ω2

As these two roots will be generally dierent, the following relationships can be readily applied to getthe solutions in the time domain:

F (s) =1

(s− a) (s− b)with a 6= b⇒ f (t) =

1

(b− a)

ebt − eat

(7.14)

F (s) =s

(s− a) (s− b)with a 6= b⇒ f (t) =

1

(b− a)

bebt − aeat

(7.15)

However, before going further, I am going to make some simplications considering the dierent ordersof magnitude of the parameters that appear in those equations. The rst term in the right-hand side ofthe equation 7.9 is related to the inverse of prompt neutron period while the second is related with theinverse of the delayed neutron period. Thus the rst term is expected to be much larger (in absolutevalue) than the second, that is to say:∣∣∣∣ρ− βeffΛeff

∣∣∣∣ >> λ⇒ ρ− βeff >> λΛeff

This conclusion can be also reached by comparing the magnitude of the dierent terms in this equation.According to [148], the mean neutron generation time Λeff may range from about 10−3 seconds (largethermal reactor) to about 10−8 seconds (unreected fast metal assembly). The numerical value of βeff/λ,calculated as βeff/λ =

∑i βi/λi with i representing the dierent families of delayed neutrons, is found to

be always larger than 10−2s, considering several isotopes (U-235, Pu-239, U-233) and dierent energies.Therefore, taking into account than in a subcritical system ρ is negative and hence |ρ− βeff | > βeff

Λeff <<βeffλ

<

∣∣∣∣ρ− βeffΛeff

∣∣∣∣⇒ ∣∣∣∣ρ− βeffΛeff

∣∣∣∣ >> λ

With this result equation 7.13 can be simplied. Let us begin with the second root, ω2, which can besimplied more easily, just by neglecting all terms of rst order in λΛeff :

ω2 =− (−ρ+ βeff + λΛeff )−

√(−ρ+ βeff + λΛeff )

2+ 4Λeffρλ

2Λeff'

'− (−ρ+ βeff )−

√(−ρ+ βeff )

2

2Λeff= −−ρ+ βeff

Λeff(7.16)

If we try to do the same with ω1 we will nd that the 0th order approximation is just 0, so we haveto keep up to the rst order in λΛeff :

ω1 =− (−ρ+ βeff + λΛeff ) +

√(−ρ+ βeff + λΛeff )

2+ 4Λeffρλ

2Λeff=

=− (−ρ+ βeff + λΛeff ) +

√(−ρ+ βeff )

2+ (λΛeff )

2+ 2 (ρ+ βeff ) Λeffλ

2Λeff'

'− (−ρ+ βeff + λΛeff ) +

√(−ρ+ βeff )


2Λeff(7.17)

where I have neglected the second order terms in λΛeff in the radicand. Now, performing a Taylorseries expansion of the square root term and neglecting again all terms of order greater than one in λΛeff :

7.2. PULSED NEUTRON SOURCE (PNS) TECHNIQUES 107

ω1 =− (−ρ+ βeff + λΛeff ) +

√(−ρ+ βeff )


2Λeff=

=− (−ρ+ βeff + λΛeff ) + (−ρ+ βeff )

√1 + 2

(ρ+βeff )

(−ρ+βeff )2 Λeffλ

2Λeff=

=− (−ρ+ βeff + λΛeff ) + (−ρ+ βeff )

(1 +

(ρ+βeff )

(−ρ+βeff )2 Λeffλ)

2Λeff=

=− (−ρ+ βeff + λΛeff ) + (−ρ+ βeff ) +

(ρ+βeff )(−ρ+βeff )Λeffλ

2Λeff=

ρλ

(−ρ+ βeff )(7.18)

With these simplications, and considering n (0) and c (0) to be 0, equation 7.13 simplies into:

N (s) =n0 (s+ λ)(

s− ρλ(−ρ+βeff )

)(s+

−ρ+βeffΛeff

) (7.19)

which nally drives us to the following solution in the time space [25]:

n (t) = n0

(e−−ρ+βeff

Λefft

+λβeffΛeff

(−ρ+ βeff )2 e

ρλ−ρ+βeff

t

)(7.20)

From this expression, we can deduce two dierent ways of determining the value of ρ from the ex-perimental data. The rst of them is by simply tting the decay curve of the prompt neutrons to anexponential. Then we will nd that:

α =ρ− βeff

Λeff⇒ ρ

βeff=

αβeff/Λeff

+ 1 (7.21)

where α is the exponential decay constant obtained from the tting. Hence if the value of βeff/Λeffis known the value of ρ (in $) can be calculated. This method is referred as the prompt decay constantmethod and was rst proposed by B. E. Simmons and J.S. King in 1958 [321].

The second method consists in calculating the ratios of the areas under the curves of the prompt anddelayed neutrons. If we denote them by Ap and Ad, it is found that:

Ap =∫∞

0n0e−−ρ+βeff

Λefftdt = n0

Λeff−ρ+βeff

Ad =∫∞

0n0λβeffΛeff(−ρ+βeff )2 e

ρλ−ρ+βeff

tdt = −n0

Λeffβeffρ(−ρ+βeff )

⇒ ρ

βeff= −Ap

Ad(7.22)

which allows us to nd out the value of ρ (again in $) with no need of other parameters than theexperimental ones. This method is referred as the areas method, the area ratio method or the Sjöstrandmethod, as it was rst proposed by N.G. Sjöstrand in 1956 [322].

One important point to take into account when applying the area ratio method is the presence ofan intrinsic source, which must be subtracted in order to correctly estimate the delayed area. This isrelevant in the case of plutonium fueled reactors, but it is negligible in the case of uranium fueled reactors.

Actually, there is a simpler way to obtain this formula from the point kinetics equations withouthaving to solve them. If we consider the former equations for a single source writing separately thecontribution of the fast and the delayed neutrons:

dnpdt

=ρ− βeff

Λeffnp + n0δ (t)

dnddt

=ρ− βeff

Λeffnd + λc

dcidt

=βeffΛeff

(np + nd)− λc

Integrating these equations from 0 to ∞:


0 =ρ− βeff

ΛeffAp + n0

0 =ρ− βeff

ΛeffAd + λAc

0 =βeffΛeff

(Ap +Ad)− λAc

Where I have denoted:

Ac =

∫ ∞0

c (t) dt

and Ap and Ad are already dened. With a simple algebraic manipulation of these equations, eq. 7.22can be obtained. See [148] for further information.

7.2.2 Response to a series of pulses

Let us analyze now the case in which we do not have a single pulse but a series of pulses. Let us considernow this case, in which we have a series of pulses of the same amplitude n0 separated by a time intervalτ . In this case, the point kinetics equations take the form:

dn

dt=ρ− βeff

Λeffn+ λc+

∑i, iτ<t

n0δ (t− iτ) (7.23)

dc

dt=βeffΛeff

n− λc (7.24)

The Laplace transforms of the equations above are:

sN − n (0) =ρ− βeff

ΛeffN + λC +

∑i, iτ<t

n0e−iτs (7.25)

sC − c (0) =βeffΛeff

N − λC (7.26)

With the same approximations that in the previous section, we end up with:

N (s) =∑i, iτ<t

n0 (s+ λ) eiτs(s− ρλ

(−ρ+βeff )

)(s+

−ρ+βeffΛeff

) (7.27)

And performing the inverse Laplace transform:

n (t) =∑i, iτ<t

n0

(e−−ρ+βeff

Λeff(t−iτ)

+λβeffΛeff

(−ρ+ βeff )2 e

ρλ−ρ+βeff

(t−iτ)

)(7.28)

Let us now rewrite this equation in terms of a new index j dened as:

t− t′ = (i+ j) τ with t′ < τ

That is, j = 0 correspond to the last neutron pulse before t, j = 1 correspond to the second to thelast, etc, while t′ is time elapsed at time t since the last pulse. In terms of this new index, equation 7.29becomes:

n (t) =∑j, jτ<t

n0

(e−−ρ+βeff

Λeff(jτ+t′)

+λβeffΛeff

(−ρ+ βeff )2 e

ρλ−ρ+βeff (jτ+t′)

)(7.29)

Now, let us assume that τ >> Λ and τ << 1/λ, that is, we are assuming that the superposition of theprompt components is negligible and thus there is superposition of the delayed components only. Hence,it is immediate to see that in the rst exponential the terms corresponding to j's others than 0 can beneglected. In the second exponential we can apply the well known formula of the sum of a geometricseries to nd:

7.3. BEAM TRIP TECHNIQUES 109

0 2 4 6 8 10 12 14 16 18

~ eαt

Time (ms)

Neu

tro

n f

lux

(a.u

.)

Ad

Ap

Figure 7.1: Example of application of the prompt decay constant and the area ratio method to a typicalPNS experiment. The gure represents many pulses superposed.

∑i

n0λβeffΛeff

(−ρ+ βeff )2 e

ρλ−ρ+βeff (jτ+t′)

=λβeffΛeff

(−ρ+ βeff )2

n0

1− eρλ

−ρ+βeffτe

ρλ−ρ+βeff

t′ '

' λβeffΛeff

(−ρ+ βeff )2

n0

1−(

1 + ρλ−ρ+βeff

)e ρλ−ρ+βeff

t′

= n0βeffΛeff

−ρ (−ρ+ βeff ) τe

ρλ−ρ+βeff

t′ ' n0βeffΛeff

−ρ (−ρ+ βeff ) τ

(7.30)In the last step we have used that t′ < τ << 1/λ. So we end up with:

n (t) = n0

(e−−ρ+βeff

Λefft′

+βeffΛeff

−ρ (−ρ+ βeff ) τ

)(7.31)

It is immediate to see that equation 7.31 and hence the prompt decay constant method are still validin the case of having series of pulses instead of single pulses. And integrating over the period τ , we obtainthe same expressions as before for the prompt and delayed components:

Ap = n0Λeff−ρ+βeff

Ad = −n0Λeffβeffρ(−ρ+βeff )

⇒ ρ

βeff= −Ap

Ad(7.32)

So the area ratio method is also valid in the case of having series of pulses instead of single pulses, ifwe consider the areas integrated over one period τ . An example of the application of both the promptdecay shape and the area ratio techniques is presented in gure 7.1.

7.3 Beam trip techniques

Let us consider now the case of a subcritical system coupled to an external neutron source operatingin a steady state that suers a sudden removal of the neutron source (gure 7.2). The evolution of theneutron ux in the assembly after this beam trip can be analyzed by performing the convolution of theimpulse response determined in section 7.2 with a constant function. Considering an external source thatintroduces ns neutrons/s in the system:

n (t) =

∫ T

−∞ns

(e−−ρ+βeff

Λeff(t−τ)

+λβeffΛeff

(−ρ+ βeff )2 e

ρλ−ρ+βeff

(t−τ)

)dτ (7.33)


0 25 50 75 100Time (ms)

Neu

tro

n f

lux

(a.u

.)

Time (ms)

Beam

inten

sity (a.u.)

Figure 7.2: Scheme of the point kinetics response of the ux (thick line) during a trip of the beam current(thin line).

0 5 10 15 20Time (ms)

Neu

tro

n f

lux

(a.u

.)

n(t) = n1 + (n0 - n1) eαt

n0 - n1

n1

n0

Figure 7.3: Example of application of the prompt decay shape technique and the source-jerk techniqueto a beam trip.

7.4. CORRECTION OF SPATIAL AND ENERGY EFFECTS 111

where we are considering that the beam trip occurs at time T . The solution of this integral is:

n (t) = nsΛeff

βeff − ρ

(e−−ρ+βeff

Λeff(t−T ) − βeff

ρe

ρλ−ρ+βeff

(t−T ))

(7.34)

From this formula we can infer two techniques to determine the reactivity from the results of a beamtrip experiment. First, it is obvious that the prompt decay constant technique, as it is described inprevious sections, can be applied as well to a beam trip experiment. The second technique is obtainedevaluating the ux before and after the beam trip. From equation 7.34, the ux before the beam trip is:

n0 = nsΛeff

βeff − ρ

(1− βeff

ρ

)(7.35)

After the beam trip, for a time t such that t >> −Λeff/(−ρ + βeff ) and t << 1/λ, the promptneutrons have died away and the delayed neutrons reach a quasi-stationary level equal to:

n1 = −nsΛeff

βeff − ρβeffρ

(7.36)

Combining equations 7.35 and 7.36, it is immediate to nd that:

ρ

βeff= −n0 − n1

n1(7.37)

Equation 7.37 constitutes the basis of the source-jerk technique for reactivity determination. Anexample of the application of these both techniques to a typical beam trip experiment is presented ingure 7.3.

7.4 Correction of spatial and energy eects

The previous derivation of both PNS and beam-trip techniques have been based on the point-kineticsmodel. As explained in section 5.3, the point-kinetics model is based on the assumption that the neutronux is fully separable into a time dependent and a space dependent part. However, in actual systems,this variable separation is not always appropriate. In these cases, the kinetics of the system will bedominated by complex space-time modes that will result into non-exponential prompt neutron decays thatfurthermore will be dependent on the detector position. In addition to these spatial eects, energy eectsmay also be relevant. The consequence of all these eects is that the relationship between the measuredparameters (prompt neutron decay slope α and area-ratio) and the reactivity of the system will not begiven by the simple formulae 7.21 and 7.22 any more. For this reason, correction methods are neededto obtain unbiased values of the reactivity. Several approaches have been proposed to deal with theseeects, such as multi-region kinetic models, correction factors or using the neutron intergeneration timedistribution. All these techniques will be described next, alongside with a newly developed methodologythat has been referred as generalized versions of the prompt decay constant and the area-ratio techniques.

7.4.1 Multi-region kinetics

In multi-region kinetic models [59, 93, 164, 328, 359, 368], the system is divided into a number of regions(e.g. the core and the reector), each of them characterized by certain parameters ρj , βj and Λj , denedby analogy to the point-kinetics ones, but specic for each one of the regions instead that for the wholereactor. These regions are related by a set of coupling coecients fik that describe the probability ofa neutron to leak from the kth-region to the ith-region. With these assumptions, the equations of thepoint kinetics model can be expanded into a set of equations of the like:

∂ni (t)

∂t=ρi − βi

Λini (t) +

∑j

λjcij (t) +∑k

fikΛk

nk + Sext (t) (7.38)

∂

∂tcij (t) = −λjcij (t) +

1

Λiβijn (t) (7.39)

where the indexes ni denote the neutron ux in the ith-region. Notice that in addition of spatialregions, this formalism can be extended to consider as well energy groups. If several of the dened regionscontain ssile material, this methodology is usually referred as theory of coupled reactors [24, 29, 65].


Although this methodology may seem appealing, some objections can be made. First, the methodologyconsists indeed in a discretization of the spatial variables. There is no better mathematical justicationfor this discretization than for the variable separation of the point kinetics model. Furthermore, theconsequence of increasing the degree of the system of dierential equations of the model is to introduceadditional exponentials in the solution, an additional exponential for every additional region or energygroup. Quite obviously, with an enough large number of exponentials it is possible to describe any complexbehavior of the neutron ux in the system, which make multi-region models somewhat contrived, moreso considering the diculty in measuring or interpreting many of the parameters that are introduced inthese models.

7.4.2 Neutron intergeneration time distribution

Another methodology to determine the reactivity from the results of a PNS experiment in presence ofspatial and energy eects is based on the concept of neutron intergeneration time distribution [262, 263].This methodology is also referred as kp method [64]. In this methodology, the neutron intergenerationtime distribution P (τ) (i.e. the distribution of times between two consecutive generations of neutronsin a ssion chain) is computed (e.g. with Monte Carlo simulations). Once P (τ) is known, the neutronpopulation of the i-th neutron generation ni can be determined as a time convolution of P (τ) with theprevious neutron generation:

ni (t) = kp

∫ ∞0

ni−1 (t− τ)P (τ) dτ = kpP ∗ ni−1 = k2pP ∗ P ∗ ni−2 = ... (7.40)

where kp denotes the prompt neutron multiplicative constant (which can be related with keff if thevalue of /betaeff is known). The neutron population n(t) at a given time t will be then given by:

n (t) =∑i

ni (t) = n0 (t) + kpP ∗ n0 + k2pP ∗ P ∗ n0 + ... (7.41)

where n0 denotes the initial distribution of neutrons. From this n(t), a time-dependent exponentialprompt neutron decay parameter Ω(t) (which replaces the constant value of α of the point kinetics model)can be determined as the logarithmic derivative of n(t).

The procedure to determine the kp of the system consists in calculating dierent shapes of n(t) (or,equivalently, Ω(t)) for dierent values of kp and compare them with the experimental measurements.According to the proposers of this method, the shape of P (τ) is little sensitive to changes in the reactorgeometry, the nuclear data or the criticality level. Therefore, the values of n(t) will be largely determinedby the value of kp.

Disadvantages of this method are its large mathematical complexity and the diculty of experimen-tally validating the calculated shapes of P (τ) upon which the method relies. It has been neverthelessapplied to the results of the MUSE-4 experiment [370].

7.4.3 Correction factors

A simpler approach to correct spatial eects involves the use of correction factors. This methodology ismost commonly applied to the area-ratio technique [59, 78, 120, 165, 212, 346, 347]. This method consistsin modifying equation 7.22 by the introduction of a factor Cdet, which is specic of the detector positionand type:

ρ = CdetApAd

(7.42)

These correction factors Cdet are usually determined with neutron transport codes. For instance, withthe MCNP code, a criticality calculation can be performed to obtain ρ, and two xed-source calculations,with and without delayed neutrons, can be performed to determine the values of Ap + Ad and Ap for agiven detector position and type. From these three results, the value of Cdet can be easily obtained.

A similar methodology can also be applied to the results of the prompt decay constant method[47, 195, 196]. In this case, equation 7.21 is replaced by a general linear relationship between ρ and α:

ρ = aα+ b (7.43)

The parameters a and b can be determined with neutron transport code in the same way than for thecase of the area-ratio technique. In this case, as there are two parameters that need to be determined,

7.4. CORRECTION OF SPATIAL AND ENERGY EFFECTS 113

the calculation of two pairs of values (ρ, α) is required, instead of the single pair of values (ρ,Ap/Ad) thatis required for the area-ratio technique. The number of required pairs (ρ, α) can be reduced to one is, forinstance, to x b = βeff ; in this way, the value of a can be determined from a single pair of values (ρ, α)obtained with the neutron transport code.

7.4.4 Generalized area-ratio and prompt neutron decay techniques

The methodology based on correction factors is based on considering the equations of the prompt decayconstant (eq. 7.21) and area-ratio (eq. 7.22) techniques and replacing the parameters of these equationsby a new set of parameters calculated with neutron transport codes. However, the linear relationshipbetween ρ and the measured quantities given by the point kinetic model is still considered.

Hence, a further generalization of the prompt decay constant and the area ratio methods has beendeveloped. This generalization is based in relaxing even more the conditions of equations 7.21 and 7.22,considering the more general relationships:

ρ = ρ1 (α) (7.44)

and

ρ = ρ2

(ApAd

)(7.45)

These relationships can be determined with neutron transport codes. For this reason, a numberof calculations of the system in a set of dierent congurations is performed, and for each of thesecongurations, pairs of values (ρ, α) and (ρ,Ap/Ad) are calculated. For the t of the sets of pairs of valuesthus calculated to a given model (e.g. a linear one) the functional form of the relationships ρ1 and ρ2 canbe determined.

These relationships can be dierent for each detector position and type due to spatial and energyeects, but for being useful for reactivity determination purposes the only requirement is that they mustbe univocal for a given detector position and type. By univocal it is meant that the same function relatesthe reactivity ρ and the measurable parameters α or Ap/Ad for any variation of the reactor congurationor of some physical constants. Such a strong hypothesis is not expected to be true for the whole rangeof congurations and the whole range of reactivities. Nevertheless, it seems reasonable to make a lessrestrictive hypothesis, and search whether such univocal relationship exist for perturbations made arounda given conguration with independence of the perturbed parameter.

This methodology has two advantages over the use of conventional correction factors. First, it canextend the range of application of the area-ratio and prompt decay constant techniques to regions wherethe linearity assumed in equations 7.42 and 7.43 is not maintained. Second, the generalized methodologyproposed in this section also provides a way to estimate the systematic uncertainty that aects theobtained value of ρ. Systematic errors are due to the uncertainties in the knowledge of parameters of thesubcritical system (geometry, composition, cross sections), that cause that the calculations are performedwith a model that never corresponds exactly to the system. Another source of systematic uncertaintyis the change of system parameters during the operation (e.g. changes in the moderator density as achanges in the temperature or changes in the fuel composition due to burn-up). With the generalizedmethodology, it is possible to estimate the impact of these systematic uncertainties in the obtained valueof ρ from the dispersion of the pairs of values (ρ, α) and (ρ,Ap/Ad). Alternatively, ranges may be foundfor which the methodology is not applicable, because the lack of a univocal relationship between thereactivity and the measured parameters.

The results of the application of this generalized methodology to the Yalina-Booster subcritical as-sembly, alongside with the direct application of the prompt decay constant and area-ratio techniques andthe use of conventional correction factors, will be presented in part III.

Part III

The Yalina-Booster experiment

115

118 CHAPTER 8. EXPERIMENTAL SET-UP

Figure 8.2: Photograph of the Yalina Booster subcritical assembly with part of the biological shieldingremoved to show the dierent regions. (Photograph taken by D. Villamarín).

8.1 The Yalina-Booster facility

The Yalina Booster subcritical facility belongs to the Joint Institute for Power and Nuclear Research(JIPNR) of the National Academy of Sciences of Belarus. It should not be confused with its predecessor,Yalina Thermal, also owned by the JIPNR, which was not used in the experiments presented here.

The Yalina Booster subcritical assembly began to operate in June 2005. Its most notable featureis the presence of two well dierentiated regions in the core: a highly multiplicative zone in the centeraround the neutron source known as booster surrounded by a larger, less multiplicative zone, where theessential of the power of the reactor is produced. The interest of this kind of congurations is that thanksto its high multiplicative power on the source neutrons the booster can increase the source importance andconsequently it allow to reduce the accelerator intensity required for a given power level in the reactor.Such assemblies can be considered as a system of two coupled reactors, the booster being one and thezone around it the other.

The booster is formed by highly enriched uranium fuel rods inserted in lead blocks. The amplicationof the neutrons coming from the source is achieved through the ssion reactions in the uranium andthe (n, xn) reactions in the lead. With this composition is obvious that the booster is a fast spectrumzone. This is what allows for a large multiplicative eect on the source neutrons, as the losses due toabsorptions will be very low. However, it must be pointed out that the value of the keff of the boosterconsidered alone would be very low due to large geometrical losses. These losses are desirable, though,as these are precisely the neutrons feeding the thermal zone, which, as stated above, is surrounding it.

The booster is subdivided in two regions: the inner booster, with a spacing between consecutivefuel rods of 11,43 mm and the outer booster with a spacing between consecutive fuel rods of 16 mm.Originally, two dierent enrichments were used within the booster: the fuel in the innermost part (calledinner booster) was metallic uranium enriched to the 90% and the fuel in the outermost part (calledouter booster) was UO2 enriched to the 36%. Later, the 90% enriched uranium fuel was removedand replaced by 36% enriched UO2 fuel, leaving a single enrichment in the booster. The experimentspresented in this work were all performed with this single enrichment booster.

8.1. THE YALINA-BOOSTER FACILITY 119

Figure 8.3: Scheme of the Yalina Booster facility in the SC3a conguration showing the dierent zonesand the location of the experimental channels (image courtesy of C. Berglöf (KTH)).

The booster is formed by a pile of lead blocks each measuring 80 x 80 x 648 mm for total boosterdimensions of 480 x 480 x 648 mm. The fuel rods have a 0.2 mm thick stainless steel cladding for anexternal diameter of 7 mm. The spacing between fuel rods is of 11.43 mm in the inner booster and of 16mm in the outer booster. The central part contains no uranium and its rear half is empty to hold theneutron source.

The zone surrounding the booster is composed of fuel rods of a mixture of UO2 (10% enrichment)and Mg in polyethylene blocks which acts as moderator. This zone will then have a thermal spectrum,and so it is referred as the thermal zone. As stated above, this zone produces most of the power of thereactor, which is favored by the thermal spectrum.

The thermal zone is square shaped with 960 mm of side and 576 mm depth and is formed by pilingup blocks of polyethylene of 80 x 80 x 48 mm (12 blocks in depth). The fuel rods have a diameter of 10mm with an aluminum alloy cladding of 1.5 mm and have an active length of 500 mm.

A key point in the design of these booster assemblies is the need to prevent, or at least minimize,thermal neutrons from the thermal zone to reect back into the booster, which will limit largely thecritical size of the reactor because of the eect of the multiplications of these thermal neutrons in thehighly enriched core. This must be achieved, however, without preventing fast neutrons from the boosterto leak into the thermal zone. Thus, a so called valve zone formed by thermal neutron absorbers is placedbetween the booster and the thermal zone. This valve zone has two layers: the innermost one is formedby natural uranium rods and the outermost one is formed by B4C absorbing rods.

One of the interesting points of this facility is the great exibility in core loading patterns with dierentlevels of subcriticality, by adding or removing control rods in the dierent zones of the assembly. Thereactivity can also be changed by moving a bank of three control rods of B4C that is also provided.

Finally the whole system is radially surrounded by a 250 mm thick graphite reector and axially andradially by a 150 mm thick borated polyethylene biological shielding. Several experimental channels areavailable at dierent locations throughout the facility. The location of these channels is depicted in gure


Table 8.1: Summary table of the Yalina-Booster characteristics. Source: [180]

Booster

No. of fuel pins, inner booster 132

Fuel pin spacing, inner booster 11.43 mm

No. of fuel pins, outer booster 575

Fuel pin spacing, outer booster 16 mm

Fuel composition UO2, 36% enrichment

Fuel density 9.8 g/cm3

Average 235U weight per pin 49.5 g

Fuel pin outer diameter 7 mm

Cladding thickness 0.2 mm

Cladding material Stainless steel (X18H10T)

Valve zone

No. of pins, innermost layer 108

Material, innermost layer UNAT

Material density 18.7 g/cm3

No. of pins, outermost layer 116

Material, outermost layer B4C (20% 10B)

Material density 1.2 - 1.3 g/cm3

Thermal zone

Max. no. of fuel pins 1180

Fuel pin spacing 20 mm

Fuel composition UO2 + Mg, 10% enrichment

Fuel density 5.172 g/cm3

Average 235U weight per pin 7.76 g

Fuel pin outer diameter 10 mm

Cladding thickness 1.5 mm

Cladding material Aluminum alloy (CAB)

Reector and shielding

Material, radial reector Graphite / borated polyethylene

Thickness, radial reector 250/50 mm

Material, axial reector Borated polyethylene

Thickness, axial reector 100 mm


Table 8.2: Number of fuel rods in the dierent congurations

Internal booster External booster Thermal zone

SC3a 132 563 1077

SC3b 0 563 1090

SC6 132 563 726

Table 8.3: Summary table of the NG-12-1 neutron generator characteristics. Source: [180]

Accelerator

Max. deuteron energy 250 keV

Max. current, pulse mode 12 mA

Pulse length 0.5 -100 µs

Pulse repetition frequency 1 Hz- 10 kHz

Max. current, continuous mode 2 mA

Tritium target

Matrix material Titanium

Diameter 45 mm

Rotation speed 560 r.p.m

Cooling Water

Max. neutron yield ∼ 1011 n/s


8.3. The experiments presented here were taken with detectors at the locations EC1B, EC2B and EC3Bin the booster; EC5T and EC6T in the thermal zone and MC2 and MC3 in the reector. A photographand a schematics of Yalina-Booster are presented in gures 8.2 and 8.3 and in table 8.1 a summary oftheir characteristics is presented.

The measurements have been performed with three dierent core congurations, named SC3a (16th 27th June 2008), SC3b (27th October 10th November 2008) and SC6 (deep subcritical) (12th 13thNovember 2008). Congurations SC3a and SC3b were intended to have the same reactivity (keff w 0.95)but in SC3b the fuel rods in the inner booster were removed and additional fuel rods were added in thethermal zone to compensate them and thus having the same reactivity. The interest of having these twodierent congurations is to study the eect of having removed the inner booster, which is the part ofthe reactor closest to the source, on the neutron population evolution. In the last conguration, SC6, adeep subcriticality (keff w 0.85) is achieved by the removal of a large number of fuel rods in the thermalzone. The interest of this conguration is to simulate shutdown or refueling procedures, as this levelof subcriticality is not envisaged for commercial ADSs. The composition of the core in each of thesecongurations is described in table 8.2 and gure 8.4.

Several external neutron sources are available, including Cf-252, (D,D) and (D,T). The experimentspresented here were performed with a (D,T) source, consisting of a tritium target coupled to a deuteronaccelerator. The deuteron accelerator (NG-12-1) is of the electrostatic type and accelerates the deuteronsto 250 keV. This energy has been chosen to maximize neutron production, taking into account both theneutron penetration in the target material (the deeper the tritium is, the more energy is required) and thereaction cross sections. The target has the tritium embedded in titanium, has a diameter of 45 mm andis water cooled. It is placed in the geometrical center of the assembly to maximize the source importance.In the pulsed mode, it provides ~ 1011 neutrons per second. It must be pointed out that as the tritiumtarget depletes some deuterons from the accelerator beam became inserted in the target. This, alongsidewith the depletion in T of the target itself, causes that the (D-D) reactions began to play a signicantrole. The characteristics of the accelerator and the neutron source are summarized in table 8.3.

The accelerator can be operated both in pulsed and in continuous mode. In pulsed mode, it is possibleto modulate the pulse widths, the pulse repetition rate and the peak current. In continuous mode, it isalso possible to modulate the current. In pulsed mode, the current is limited by the accelerator optics;in the continuous mode, the current is limited by the coolability of the target. There is a possibility toovercome this limit by using a larger target capable to dissipate more heat, but it was not used in theexperiments presented here.

During the PNS experiments presented here, the accelerator was operated at repetition rates of 50 Hz(in the SC3a conguration), 57 Hz (in the SC3b conguration) and 166 Hz (in the SC6 conguration), withpulses of 5 µs of duration and 6 mA of D peak current. The higher repetition rate in the SC6 congurationwas possible because the higher subcriticality level made prompt neutron decay faster, and hence a highernumber of pulses per second was possible without superposition of the prompt neutron populations aftereach single pulse. In the beam-trip experiments, beam trips were produced at a frequency of 50 Hz.

During the beam trips experiments, maximum intensity was 1.5 mA. An important remark is that duea malfunction, source intensity was not constant but was aected by a 50 Hz oscillation. The implicationsof this oscillation for the application of beam monitoring techniques will be later discussed.

Dierent simulations have been performed with MCNPX 2.7 within the frame of the analysis ofthe Yalina-Booster experiments. These include the simulation of the evolution of the counting rates atdierent detector positions, criticality calculations and calculations of the kinetic parameters βeff andΛeff . While the both counting rate and criticality calculations are straightforward, the calculation ofkinetic parameters is more cumbersome since, as stated above, it requires adjoint weighting calculations.The results are presented in table 8.4, 8.5 and 8.6. The geometry description of the Yalina-Boosterassembly was provided by the JIPNR-Sosny. Nuclear data libraries used are ENDF/B-VII.0, JEFF-3.1and JENDL-3.3, processed at 293K with the NJOY-99.161-NEA12 code.

The values of βeff were obtained with the k-prompt method with the same source described in section6.3. Notice that the values of βeff obtained with the JEFF-3.1 library are about 15 pcm larger than thoseobtained with the ENDF/B-VII.0 and the JENDL-3.3 libraries. This is eect was already explained insection 6.3 as due to an overestimation of the value of β0 with this library. Apart from this eect thatcauses dierence between libraries, for a given library the values of βeff have been found not to varysignicantly between congurations.

The calculation of Λeff has been performed with the perturbative method described in section 6.4.An important issue to take into account is choosing an adequate atom density for the ctitious isotopeintroduced in the system. If the density is too low the variation in the reactivity will be small and hence,


(a) SC3a conguration.

(b) SC3b conguration.

(c) SC6 conguration.

Figure 8.4: The Yalina-Booster core congurations in the experiments presented in this work (imagescourtesy of C. Berglöf (KTH)).


Table 8.4: Calculated values of the keff reactivity with dierent nuclear data libraries for each of thecongurations with the control rods extracted and inserted.

Conguration C. R. ENDF-VII.0 JEFF-3.1 JENDL-3.3

SC3aOut 0.94873 ± 0.00011 0.94905 ± 0.00004 0.94908 ± 0.00011

In 0.94588 ± 0.00011 0.94600 ± 0.00011 0.94593 ± 0.00011

SC3bOut 0.94851 ± 0.00011 0.94909 ± 0.00011 0.94891 ± 0.00011

In 0.94544 ± 0.00012 0.94584 ± 0.00012 0.94601 ± 0.00011

SC6 Out 0.85070 ± 0.00012 0.85133 ± 0.00012 0.85180 ± 0.00010

Table 8.5: Results for βeff (in p.c.m.) with dierent nuclear data libraries for the dierent congurationsof Yalina-Booster measured during the EUROTRANS experimental campaign.

ENDF/B-VII.0 JEFF-3.1 JENDL-3.3

SC3a 729 ± 5 747 ± 6 734 ± 5

SC3b 728 ± 6 742 ± 6 741 ± 6

SC6 757 ± 7 772 ± 7 754 ± 7

Table 8.6: Comparison of dierent estimates of the mean neutron generation time obtained with dier-ent libraries and for dierent congurations. First column contains their values of the adjoint-weighedmean neutron generation time calculated as explained in section 6.4. Second and third columns contain,respectively, the prompt removal lifetime divided by keff and the ssion lifespan tf , all these parameterstaken from the MCNPX outputs. All values are given in µs.

Λeff lp/keff tf

ENDF/B-VII.0 60.8 ± 0.4 181.9 59.7

SC3a JEFF-3.1 60.8 ± 0.4 181.6 59.7

JENDL-3.3 60.9 ± 0.4 180.6 59.6

ENDF/B-VII.0 61.6 ± 0.4 183.5 60.2

SC3b JEFF-3.1 60.4 ± 0.3 183.4 60.2

JENDL-3.3 63.4 ± 0.4 185.4 60.1

ENDF/B-VII.0 67.0 ± 0.5 192.3 59.7

SC6 JEFF-3.1 68.1 ± 0.5 191.9 59.7

JENDL-3.3 69.9 ± 0.5 190.4 59.6


(a) ENDF/B-VII.0

(b) JEFF-3.1

(c) JENDL-3.3

Figure 8.5: Antireactivity introduced in various congurations of the Yalina-Booster subcritical assemblycalculated with dierent nuclear data libraries for dierent values of the perturbation (i. e., dierent atomdensities of the ctitious absorber).


large statistical errors will appear. On the other hand, if the density is too high, systematic errors as theresult of a too large perturbation will arise. Therefore, the calculation has been performed using dierentdensities of the ctitious isotope in order to guarantee that these densities are the appropriate ones.The results are shown in gure 8.5, where it is represented the value of the antireactivity introducedin the system when dierent densities of the ctitious absorber are present. Notice that the dierentpoints are always very close to a straight line, even those corresponding to the highest densities, whichindicates that for this range of densities the perturbation is small enough to be treated as a rst orderperturbation. Λeff is in fact the slope of these lines and the values calculated in this way are presentedin table 8.6. The values of the prompt removal lifetime divided by keff and the ssion lifespan tf arealso shown. These parameters are provided by MCNPX itself, and, as it has been discussed in section6.4, are sometimes taken as estimators of the mean neutron generation time. Notice that both SC3a andSC3b congurations have very similar values of Λeff and somewhat dierent than the SC6 conguration.

8.2 Data Acquisition System

During the EUROTRANS experiments at the Yalina-Booster subcritical assembly data were taken si-multaneously with detectors operating in pulse and in current mode. In pulse mode operation individualdetector events (or, more specically, certain properties of individual events such as the deposited energyor the time of occurrence) are registered independently. In current mode operation individual eventsare not registered but the average current intensity between the terminals of the detector is measuredinstead. The advantage of pulse mode operation is that it allows to retrieve useful information aboutthe individual events. However, it has the disadvantage that it is only applicable in case of low countingrates, where dead time eects due to signal pile-up are negligible or can be corrected. In this case, whichis likely to be the case of an industrial high power ADS, the current mode is preferred instead. In thecase of the EUROTRANS measurements at Yalina-Booster, detector counting rates at certain detectorpositions are high enough to have signicative dead time eects and therefore both modes of operationmade sense and they can complement each other. For this reason, two data acquisition systems wereused, one for each mode of operation.

This work is only concerned with the data taken in pulse mode. The pulse mode data acquisition sys-tem was already tested in previous experiments (mainly MUSE) and hence is considered as the referenceone for reactor kinetic studies or reactivity measurement techniques. The current mode data acquisitionsystem was tested by the rst time at Yalina-Booster and is aected by a series of issues concerningelectronic noise or detector saturation and therefore it was considered for electronics testing purposesrather than for reactor kinetic studies. Hence, it will not be considered in this work.

A schematics of the pulse mode data acquisition system including the detectors and the electronicchains is presented in gure 8.6. Four detectors were used simultaneously in pulse mode: two ssionchambers, a He-3 proportional counter and one liquid scintillator. The neutron ux within the assemblywas measured with U-235 ssion chambers. Two models were available: a larger one (KNT-31), containing500 mg of active layer and a smaller one (KNT-5), containing 1 mg of active layer. The larger one wasgenerally used in the booster and the reector, the zones with the lower thermal uxes, and the smallerone was used in the thermal zone, to prevent saturation due to the higher thermal ux in this zone.

Apart from these ssion chambers, other two detectors were operated in pulse mode. The thirddetector was a CANBERRA 0.5NH1/1K He-3 proportional counter, which was placed in the experimentalroom but outside the assembly and was intended to be used as a reference detector for normalizationpurposes. Finally, the fourth detector was a BC-501A liquid scintillator coupled to a Hamamatsu R877-01photomultiplier tube used to monitor the intensity of the D-T neutron source. This is a typical detectorused with fast neutrons and it is noted for its fast time response that allows the discrimination of neutronsand γ signals through digital pulse shape analysis techniques [134]. The application of these techniquesrequires a parallel data acquisition system capable to register the shape of the signals as well. For thisreason, the BC-501A detector was also coupled to an additional data acquisition system based on anAcquiris DC271 8 bit digitizer capable of sampling rates of up to 1 GS/s per channel (only for some ofthe experiments in the SC3a conguration).

CANBERRA 7820 ADS pulse ampliers and CANBERRA 7821 HT high voltage power supplies [82]were used with the ssion chambers and the He-3 detector and the signals from these ampliers werefed to a digital data acquisition system consisting of two National Instruments PCI-6602 counter/timercards to register the times at which they occurred. The signals from the BC-501A, for its part, passedthrough a constant fraction discriminator, to remove low energy neutron signals from the assembly andallow only high energy neutrons from the D-T reactions in the source (14 MeV) to be detected. Then,

8.2. DATA ACQUISITION SYSTEM 127

Figure 8.6: Scheme of the data acquisition system with detectors in pulse mode used during the Yalina-Booster subcritical experiment.


Figure 8.7: Operation principle of the NI PCI-6602.

they passed through a NIM/TTL adapter to be nally fed into the PCI-6602 cards. No amplier wasrequired for this detector.

The NI PCI-6602 counter/timer cards were intended to register the times of individual detector signals.Recording the times of individual detector signals rather than counting rates has the advantage that inthis way a complete reconstruction of the experiment at a later time is allowed. The operation principleof the NI PCI-6602 cards is schematically presented in gure 8.7. Each of these cards have an internal80 MHz clock that increments the card's counter one unit every clock pulse. When a detector signalarrives to any of the card's channels, the value of the counter at this instant is stored in an internal databuer. Therefore, the 80 MHz clock allows to register the times of the detectors signals with a precisionof 12.5 ns. It must be remarked that since the cards are 32-bit cards, the counter restarts to 0 every232 clock pulses, corresponding to approximately 53 seconds, and hence this time must be added to allsubsequent registered signals. The cards acquire data until the data buer is full (maximum buer size is32 MB/channel) or a time limit given to the data acquisition system as an input parameter is exceeded.At this moment, the system stops and a le with the vectors containing the arrival times of the signalsat the dierent channels is written into the hard disk in binary format. It was found when analyzing theexperimental data that the recorded times of some counts were out of order. The reason is unknown,nevertheless, this problem was easily solved by applying a quick sort algorithm. A header including aseries of parameters of the experiment (initial time at which the trigger signal started and number ofchannels used) are also stored in the le in order to help in the subsequent data processing.

Each NI PCI-6602 card allows for simultaneous measurement in up to eight channels, although onlythree can be used simultaneously at maximum transfer rates of about 5 MS/s in the DMA (Direct MemoryAccess) mode. The other ve channels use IRQ (Interruption Request Queue) and thus they are muchslower (up to 1 kS/s). In our case, the counting rates were high enough to require data transfer by DMA,and thus two cards were necessary for the four detectors and a synchronization line between them wasrequired. IRQ channels were used for lower frequency signals, such as trigger signals from the accelerator(∼ 100 Hz).

The way to present the experimental data is dierent in the case of the PNS and the beam tripexperiments. For the case of PNS experiments, the data from the detectors were registered alongside witha trigger signal from the accelerator indicating the times of the source pulses. This allows superimposingthe detector signals after a large number of accelerator pulses simply by subtracting the time of each countto the time of the last neutron pulse. The histograms produced in this way are referred as histogramof reduced times. The need to superimpose the results after many accelerator pulses is more evidentwith a practical example taken from the Yalina-Booster experiments: the 1 mg ssion chamber in theexperimental channel EC5T in the thermal zone registered 209132 counts during 115239 neutron pulses(with a repetition frequency of 50 Hz) that is, less than two detector counts per accelerator pulse onaverage.

A nal remark is that the eect of the unplanned beam losses must be taken into account since itcan aect the reactivity monitoring techniques, the area-ratio technique (or source-jerk technique) inparticular, because of the time the delayed neutrons require to stabilize after a change in the neutronux. For the case of the beam-trip experiments, as les with the counting rate during a long intervalof time are produced, it is possible to analyze the evolution of the delayed neutron level and hence

8.2. DATA ACQUISITION SYSTEM 129

wait for stabilization before applying the source-jerk technique. For the case of the PNS experiments,however, where one has histograms of reduced times and this analysis is not possible, the number ofcounts exceeding the trigger period is computed and recorded.

For the case of the experiments with a continuous neutron source with beam-trips, histograms ofreduced times are not used, but instead les containing the counting rate during the experiment aregenerated from the experimental les. The binning of these les has been chosen to be 100µs, corre-sponding to 8000 bins of 12.5 ns. Nevertheless, trigger signals from the accelerator at the times of thebeam trips were also recorded, thus making possible the superposition of dierent beam trips to achievebetter statistics.

Chapter 9

Dead time

Abstract - This chapter describes the procedures used to determine and correct the dead-time of

the detectors used during the experiments at the Yalina-Booster subcritical assembly. The chapter

is divided in three sections. First, some generalities about dead times are presented. Next, dierent

possibilities to determine the dead time of a detector system are discussed. Finally, the procedure

used to correct the dead time of the detectors used in the Yalina-Booster experiment is described.

Some of the information presented in this chapter has been published in [45].

All types of radiation detectors are aected by a dead time. The dead time of a detector is theminimum time interval that two consecutive counts must be separated in order to be recorded as twodierent events. The eect of the presence of a dead time is that the measured counting rate will belower than the real rate of particles impinging on the detector, as the counts that occur during the deadtime of a previous one are not recorded. This eect is the more relevant the higher is the counting rate,as the higher is the counting rate the more counts will be lost for having occurred in the dead time of aprevious one. At the counting rates measured in the Yalina-Booster experiment the eect of dead timeis deemed to be relevant. Thus, it is necessary to apply correction methods to the experimental data inorder to get rid of this eect and to recover the real counting rates.

9.1 Dead time theory

The rst point to take into account when dealing with dead times is the fact that there are two types ofdead time, referred as paralyzable and non paralyzable (also called extendable and non extendable,respectively). In a detector aected by paralyzable dead time, events that occur during the dead timeof a previous one, and consequently not recorded, also produce a dead time. In other words, the eectof this last event can be regarded as extending the dead time of the previous one. On the contrary, in anon paralyzable detector, events occurred during the dead time of a previous one are neither detectednor they cause an additional dead time.

The relationship of the real counting rates with the measured counting rates is well known for thesetwo idealized models (see [183] for instance). If we denote by n the real counting rate, by m the measuredcounting rate and by τ the dead time and considering that the physical process of detection is Poissonian1,the measured counting rates are given by:

m =n

1 + nτ(9.1)

for the case of a non-paralyzable dead time, and by

m = ne−nτ (9.2)

for the case of a paralyzable dead time.In practice, however, these two models of dead time are ideal models, and actual detectors systems

have neither a paralyzable nor a non paralyzable dead time, but a combination of both. Hence,1Nuclear ssion is not a Poissonian process, since a after a ssion has occurred, and because of the secondary neutrons

emitted in the ssion process, it is more likely that another ssion will occur. In fact, the non-Poissonian nature of thession process constitutes the basis of the neutron noise techniques mentioned at the beginning of chapter 7. Nevertheless,for dead time correction purposes the deviation from the Poisson distribution is small enough and the formulae derived forthe Poisson process can be applied for ssion.

131

132 CHAPTER 9. DEAD TIME

it is necessary to study the behavior of series arrangements of dead times. Quite obviously, a seriesarrangement of dead times is only relevant when the rst dead time is shorter than the following ones,because otherwise only the rst dead time is to be taken into account. From now on, let us restrict to thecase of a series arrangement of two dead times, namely τ1 and τ2. First, let us denote the ratio betweenthese two dead times by:

α = τ1/τ2

It can be obtained that the measured counting rate can be expressed by:

m = nT1T2

where T1 and T2 are respectively two transmission factors related to each dead time. It can be obtainedthat:

T2 =

1/ (1 + nτ2) for τ2non-paralyzable

e−nτ2 for τ1paralyzable

while T1 can take four dierent shapes, one for any possible combination of the two types of deadtimes. Analytical forms can be obtained only in the cases that the rst time is non-paralyzable and thesecond is paralyzable and vice-versa. In the case the rst dead time is of the non-paralyzable type andthe second is of the paralyzable type:

T1 =eαx

1 + αx(9.3)

And in case the rst dead time is paralyzable and the second is not:

T1 =1 + x

(1− α)x+ eαx(9.4)

Another important issue is the distribution of the time intervals between consecutive counts. It iswell known that for a Poissonian process the distribution of time intervals between consecutive countstakes the shape of a simple exponential. However, the presence of a dead time alters this distribution.Analytical formulae for this distribution have only be obtained for the cases of a simple paralyzable ornon-paralyzable dead time. For the non-paralyzable case this formula reads:

f (t) = ρe−ρ(t−τ) for t > τ (9.5)

While for the paralyzable case:

f (t) = ρJ∑j=1

1

(j − 1)![−ρ (t− jτ)]

j−1e−jρτ (9.6)

where J is the largest integer below t/τ . For the case of a series arrangement of dead times, noanalytical formulae has been determined, to our knowledge. Hence, they have been determined withMonte Carlo simulations. In gure 9.1 the known cases of the time distributions of a purely paralyzableand non-paralyzable cases are presented, as well for the cases of series arrangements of two dead times:paralyzable - non-paralyzable and non-paralyzable - paralyzable.

9.2 Determination of dead times

In our case, dead time eects were relevant only in the case of the ssion chamber in the booster. Threesources of dead time are possible: the ssion chamber itself, the pulse amplier and the NI PCI-6602cards. Fission chambers introduce a paralyzable dead time in the system, because during the timethe gas is ionized after a count additional events can take place in the gas, thus extending the deadtime. The dead time of the ssion chambers was known to be about 80 ns. The electronic chain, forhis part, is also a source of dead time. In our case, the Canberra 7820 amplier has a resolving time,according to specications, of 50 ns [82]. During this time additional signals cannot be processed, butnew signals received during this time do not add an additional dead time. Hence, the amplier adds a non- paralyzable dead time to the system. Since this dead time is shorter than this of the ssion chambersit should not aect the measurement.

9.2. DETERMINATION OF DEAD TIMES 133

0

0.02

0.04

0.06

0.08

0.1

x 10-2

0 1 2 3 4 5 6

Poisson process (without dead time)

Non-paralyzable dead time

Paralyzable dead time

t/τ

Fre

quen

cy o

f tim

e in

terv

als

(per

sou

rce

coun

t)

(a) Unperturbed distribution of time intervals and distributionof time intervals with paralyzable and non-paralyzable dead time(400 ns in any case)

0

0.02

0.04

0.06

0.08

0.1

x 10-2

0 1 2 3 4 5 6t/τ

Fre

quen

cy o

f tim

e in

terv

als

(per

sou

rce

coun

t)

(b) Distribution of time intervals for a series arrangement of aparalyzable and a non-paralyzable dead time (200 ns paralyzableand 400 ns non-paralyzable)

0

0.02

0.04

0.06

0.08

0.1

x 10-2

0 1 2 3 4 5 6t/τ

Fre

quen

cy o

f tim

e in

terv

als

(per

sou

rce

coun

t)

(c) Distribution of time intervals for a series arrangement of a non-paralyzable and a paralyzable dead time (200 ns non-paralyzableand 400 ns paralyzable)

Figure 9.1: Distribution of time intervals for dierent dead time models (source counting rate 106

counts/s, 109 counts in the simulation)


The NI PCI-6602 cards may also introduce a dead time if the counting rates are high enough tosaturate the data transfer capacity. According to specications, maximum transfer rates is over 1 MHzfor a single data stream or 500 kHz for continuous buered acquisitions, but these gures depend onthe computer utilized. Since during the experiments at Yalina-Booster there were short peaks of above2 MS/s and sustained counting rates of about 1 MS/s, there were an obvious concern that counts werebeing lost at the PCI-6602. To determine if this was indeed happening and to better characterize themaximum counting rates that were achievable with the NI PCI-6602 without loses, a series of tests with anarbitrary waveform generator was performed. In gure 9.2 the real counting rate fed into the card is shownalongside with the measured counting rate in the card. It can be noticed how the measured countingrate follows the real counting rate for counting rates of up to about 5 MS/s. An additional analysis thatwas carried out was the structure of the time dierences between consecutive counts at dierent countingrates (gure 9.3). Notice how at 1 MS/s the time dierence between consecutive counts is always itscorresponding 1 µs, while at 6 MS/s there begin to appear counts with time dierences in between notof the corresponding 167 ns but of 333 ns, indicating that a count has been lost in between. At aneven larger counting rate of 10 MS/s a larger fraction of counts are lost, appearing counts with timesdierences in between of 100, 200, 300 and even 500 ns. Spurious counts also begin to appear, some ofthem with very short time dierences between them. The conclusion of these measurements is that it isnot expected that the NI cards have caused counts to be lost during the Yalina-Booster experiments.

Apart from the specications of the dierent components, several standardized techniques are availablefor measuring the dead time of a detector system. Some references are [160, 183, 235]. However, it wasnot possible to apply them for the EUROTRANS experimental campaign at the Yalina-Booster facility,so it is necessary instead to make the most of the measured data to determine the dead time of thedetector system.

First of all, it is in principle possible to determine the dead time of a detector system by simplyanalyzing the distribution of the dead time intervals between consecutive counts and comparing it withthe shape of the distributions derived in section 9.1. Notice that as the data acquisition system stores thetimes of the individual events it is possible to obtain the experimental time interval density. It must beremarked, however, that the counting rates for the experiments at Yalina-Booster were not constant. Thisis obvious for the case of the PNS experiments but also for the case of the continuous source experimentswith beam trips, because of the presence of the large 50 Hz oscillation already mentioned. This factcomplicates the comparison of the experimental dead time distributions with the theoretical ones derivedin section 9.1. Nevertheless, continuous source experiments are in principle more adequate to apply thistechnique since the counting rate is more close to constant than in the case of the PNS experiments.

The structure of the distribution of time dierences between consecutive counts for a some beam tripsexperiments in the SC3a conguration with dierent counting rates is shown in gure 9.4. In this gureit is clearly shown the presence of a dead time of about 6 - 7 bins of the NI card of 12.5 ns each, whichcorrespond to a dead time of 75 - 87.5 ns, which is consistent with the expected value. This last value hasbeen taken. The plots of the structure of the time intervals are plotted alongside with the t of the decaycurve to an exponential decay (gray line). It can be noticed that the for the les with the lowest countingrates this exponential ts quite well the experimental results but for higher counting rates the goodnessof the t to an exponential decreases. The causes of this behavior are dicult to determine, rst of allbecause of the 50 Hz oscillation that causes that the counting rate is not constant. Notice that with it isalso possible in principle the determination of the type of the dead time (paralyzable or non-paralyzable)in this way, just comparing the experimental time interval distribution with the calculated distributionsof gure 9.1. Nevertheless, the presence of some ringing eects, which are visible in gure 9.4, made thisdetermination almost impossible.

Another possible way to determine the dead times is through the analysis of the relationship betweenthe real counting rate n versus the measured counting rate m in the detector. Some experiments wereperformed in the SC3a conguration with varying source intensities ranging between 0.1 and 1.5 mA,which make possible to attempt this technique. The real counting rate is obviously an unknown quantity,but some other quantities proportional to it and that they are not aected by dead time are known. Firstwe have the counting rates in several detectors with lower counting rates and hence not aected by deadtime. They are the ssion chamber with the lower counting rate, the He-3 detector outside the assemblyand the BC501A used for neutron source monitoring. Furthermore, we have also have the lower level inthe beam trips (n2 in equation 7.37), which, according to equation 7.37, is proportional to the higherlevel, being the proportionality constant related with the reactivity of the system.

Notice that both equations 9.1 and 9.2, up to the second order in n, can be expanded as:

9.2. DETERMINATION OF DEAD TIMES 135

0

1

2

3

4

5

6

7

8

0 2 4 6 8 10

n (MS/s)

m (

MS

/s)

m=n

Figure 9.2: Real (n) versus measured (m) counting rates in the NI PCI-6602 cards with increasingintensities.

10-3

10-2

10-1

1

0 0.2 0.4 0.6 0.8 1 1.2∆t (µs)

Fra

ctio

n o

f to

tal n

o. c

ou

nts

1 MS/s

6 MS/s

10 MS/s

Figure 9.3: Structure of time dierences between consecutive counts in data measured with the NI PCI-6602 cards with dierent intensities.


0

0.02

0.04

0.06

0.08

0.1

x 10-20

0.51

1.52

2.53

3.54

t (µs)

Relative frequency of time intervals

(a)0.1

mA

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x 10-20

0.51

1.52

2.53

3.54

t (µs)


(b)0.5

mA

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

00.5

11.5

22.5

33.5

4t (µs)


(c)1.0

mA

0

0.002

0.004

0.006

0.008

0.01

00.5

11.5

22.5

33.5

4t (µs)


(d)1.5

mA

Figure

9.4:Experim

entaltim

einterval

distributionfor

experiments

with

dierentdeuteron

beamintensities.

9.3. CORRECTION OF DEAD TIMES 137

m = n− τn2 +O(τ2n3

)(9.7)

As we do not know n, but we know some quantities that are proportional to them, say n′. If wedenote the proportionality factor as χ so that n = χn′,we can rewrite equation 9.7 as:

m w χn′ − τχ2n′2 +O(τ2χ2n′3

)(9.8)

n′ can be a ssion chamber with a lower counting rate, the He-3 detector outside the assembly, theBC501A used for neutron source monitoring or the lower level during the beam trip. The results of theapplication of this technique to the experimental results is shown in gure 9.5. Nevertheless, althoughthe results may appear appealing at rst glance, numerical results for the dead time showed a largedispersion depending on the reference detector used and the only conclusion that could be reached withthis technique is that the dead time was of the order of 100 ns.

Finally, the possible presence of higher frequencies in the oscillation has also been analyzed by meansof a publicly available fast Fourier transform routine [118, 117]. The presence of such frequencies higherthan the 10 kHz used can aect the dead time correction since it implies dierent corrections and beamlosses in the peaks and in the valleys of the oscillation. The results of the analysis for a certain le arepresented in gure 9.6. No frequency higher than 200 Hz has been found, and thus the dead time eectis not expected to be aected by higher frequencies.

9.3 Correction of dead times

First of all, as it has been previously said, apart from the dead time, there is some ringing present due tothe bounces of the detector signal in the electric chain. This ringing also alters the measured intensities.To get rid of them, a second additional dead time is introduced via software. This dead time is imposedto be non-paralyzable. To determine how long this dead time must be, we have imposed and correctedthe experimental data introducing dierent dead times and we have analyzed the convergence of thecorrected data to a constant value, independent of the imposed dead time. The results obtained withthis method of correction for dierent values of the imposed non-paralyzable dead time are shown ingure 9.7 and 9.8. Notice how as the imposed non-paralyzable dead time becomes larger, the correctedcounting rates converge, which is a necessary condition for this method to be consistent. Thus, for theresults presented in this work, a good choice for this paralyzable dead time has been considered to be212.5 ns (corresponding to 17 bins of the counter/timer cards).

Let us now detail how the dead time correction was implemented. In section 9.1, it was obtainedthat in the case of a series arrangement of a paralyzable and a non-paralyzable dead time, the measuredcounting rate m can be expressed in terms of the real counting rate n as:

m = n

(1− τpτn )nτn+eτpn(9.9)

where τp is the paralyzable dead time and τnp is the non-paralyzable dead time. In order to givean analytical expression for n, we perform a expansion to the second order of the exponential in thedenominator and simply work out n in the resulting second order equation:

m = n(1− τp

τnp

)nτnp+eτpn

= n(1− τp

τnp

)nτnp+(1+nτp+ 1

2n2τ2p+O(n3τ3

p))= n

1+nτnp+ 12n

2τ2p+O(n3τ3

p)⇒

⇒ n =

1−mτnp −√

(1−mτnp)2 − 2m2τ2p

m2τ2p

+O(m3τ3

p

)m (9.10)

where it has been taken into account thatm and n are of the same order of magnitude to rst order, soO(n3τ3

p

)= O

(m3τ3

p

). Since m ' 106c/s at the maximum peaks and τp ' 10−7s, O

(m3τ3

p

)is expected

to be about 10−3.The way this correction is implemented in the computer is dierent for the PNS and for the continuous

source experiments. For the case of the PNS experiments, what we have is a le consisting in a list ofcounts per bin with the results of a large number of pulses superimposed. Let us denote by ni and mi

respectively the true counting rate (true counts per bin) and the recorded counting rate (recorded countsper bin) in the ith bin. These are normalized by the total number of pulses that are superimposed. Notice


0 10 20 30 40 50 60 70 80 90

01

23

45

6n

1 , EC

1B(co

un

ts/s × 104)

n0, EC1B (counts/s × 104)

(a)n0vs.n1(upper

andlow

erlev

elsin

thebeam

tripsexperim

ents)

inEC1B

0 10 20 30 40 50 60 70 80 90

00.05

0.10.15

0.20.25

0.3M

C3 (co

un

ts/s × 104)

EC1B (counts/s × 104)

(b)EC1Bvs.

MC3

0 10 20 30 40 50 60 70 80 90

00.5

11.5

22.5

33.5

44.5

BC

501A (co

un

ts/s × 104)


(c)EC1Bvs.

BC501A

0 10 20 30 40 50 60 70 80 90

00.25

0.50.75

11.25

1.51.75

22.25

2.5H

e-3 in Y

.T. (co

un

ts/s × 104)


(d)EC1Bvs.

He-3

inYalin

a-Therm

al

Figure

9.5:Measured

versusreal

countingrate

forthe

ssioncham

berin

theexperim

entalchannel

EC1B

inthe

boosterfor

severaldierent

beamintensities

(SC3a

conguration).


(a)12.5nstimebin

(b)100µstimebin

Figure9.6:

Frequencycontentof

aselected

le.


that this approach is valid only in the case that the pulse intensity and hence the counting rates aftereach pulse are approximately the same after each pulse. Otherwise it would be necessary to perform thecorrection pulse by pulse. Then it is found that the value of ni is given by the former formula where:

mτnp =

j=nnp∑j=1

mi−j

mτp =

j=np∑j=1

mi−j

where nnp and np are the duration of the non-paralyzable and paralyzable dead times expressed innumber of bins. In fact, the eect of the last bin is also considered in the correction, so the formulaeactually used are:

mτnp =

j=nnp∑j=1

mi−j +1

2mi

mτp =

j=np∑j=1

mi−j +1

2mi

The results of this correction method applied to the PNS experiments are shown in table 9.1. Thereit is presented the total number of counts in the histogram of reduced times as well as in the bin with thepeak counting rate. It can be observed that the dead time correction is only relevant for the detectorsin the booster, which have high peaks in the counting rate. In the thermal zone and in the reector theeect is much smaller; the result of the correction may be even a decrease in the number of counts withrespect to the experimental results. This can be understood as an eect of removing the ringing eectsas the result of imposing the additional 212.5 ns of dead time via software. This eect particularly highin the case of the detector in the position MC3 (only is SC3a), where the peak counting rate is reducedup to about one third. Nevertheless, this eect is concentrated in just one or two bins and taking intoaccount the low counting rate in this detector (the 1 mg ssion chamber was used) it turns out that it iscaused in fact by only a small change in the number of counts ∼ 10 in this bin so it is considered not tobe very signicative.

The correction method applied to the PNS results, however, cannot be applied to the beam tripexperiments, since in this case we do not have a histogram of reduced times but a histogram containingthe number of counts during the experiment with a certain bin size. Since the bin sizes considered arelarger than the dead time, the dead time correction for the beam trip experiments is performed by simplyapplying equation 9.10 to the results of each bin.

The application of this correction technique has been tested against the experimental results of the realcounting rate n versus the measured counting ratem for dierent beam intensities that were mentioned insection 9.2. The results are shown in gure 9.9. Specically, the lower level in the beam trips experimentshas been considered to estimate the real counting rate n. It can be observed how the corrected resultsare closer to a straight line than the measured data, but they are still below the linear extrapolationof the rst term of the t of the measured counting rate to a second degree polynomial, which is thevalue to which in principle the corrected data should tend to converge. Nevertheless, the uncertainty inthe determination of the slope of this line is large and hence this dierence does not seem to be verysignicant.

Finally, another way to determine and correct the dead time in this case where there are a set valuesfor the true counting rate n and the measured counting rate m for several dierent values of n is simplytrying dierent values of (paralyzable) dead time until the relationship between m and n becomes linear,without imposing any additional dead time. The application of this method is shown in gure 9.10.Nevertheless, the value of dead time which was required to obtain a linear relationship is about 200 ns,which is not consistent with the observed value of the dead time obtained from the histogram of timedierences between consecutive counts. Furthermore, since the linearity of the relationship between theaccelerator current and the assembly power is the basis of the current to ux technique whose validity isto be analyzed, it does not seem adequate to impose a linear relationship between these two magnitudesbeforehand.


Table 9.1: Eect of the dead time correction in the PNS experiments in Yalina-Booster. Values areexpressed in counts per second.

SC3a control rods outDetectorposition

Total no. of counts,corrected

Total no. of counts,experimental

Max. no. of counts,corrected

Max. no. of counts,experimental

EC1B 26.0 24.1 2.55× 106 2.01× 106

EC5T 1.81 1.81 3.61× 103 3.61× 103

MC2 92.5 92.8 3.79× 104 3.81× 104

MC3 7.93× 10−2 7.95× 10−2 1.22× 102 1.83× 102

SC3a control rods inDetectorposition


Total no. of counts,non-corrected


Max. no. of counts,non-corrected

EC1B 24.5 22.5 2.60× 106 2.03× 106

EC2B 25.7 24.1 2.34× 106 1.88× 106

EC3B 79.2 78.2 1.85× 106 1.55× 106

EC5T 2.23 2.23 4.12× 103 4.12× 103

EC6T 2.80 2.80 2.45× 103 2.45× 103

MC2 76.9 77.1 3.37× 104 3.39× 104

MC3 7.88× 10−2 7.90× 10−2 1.00× 102 1.62× 102

SC3b control rods outDetectorposition





EC1B 16.8 16.2 1.38× 106 1.21× 106

EC2B 16.3 15.8 1.23× 106 1.09× 106

EC3B 88.4 87.7 0.96× 106 0.88× 106

EC5T 2.99 2.99 0.56× 104 0.56× 104

EC6T 1.32 1.32 0.11× 104 0.11× 104

MC2 49.1 49.2 2.01× 104 2.02× 104

SC3b control rods inDetectorposition





EC1B 15.9 15.3 1.40× 106 1.23× 106

EC2B 15.8 15.3 1.26× 106 1.12× 106

EC3B 78.6 78.0 0.94× 106 0.86× 106

EC5T 2.57 2.57 0.52× 104 0.52× 104

EC6T 1.40 1.40 0.13× 104 0.13× 104

MC2 47.9 48.0 2.08× 104 2.09× 104

SC6 control rods outDetectorposition





EC2B 12.9 12.2 1.54× 106 1.33× 106

EC5T 0.93 0.93 0.29× 104 0.29× 104


0

20

40

60

80

100

0 50 100 150 200 250 300 350 400

0.1 mA

0.5 mA

1.0 mA

1.5 mA

Imposed non-paralyzable dead time (ns)

Co

un

ts/s

× 1

04

Figure 9.7: Eect on the counting rate of imposing and correcting by an additional non-paralyzable deadtime for dierent values of the accelerator beam intensity.

0

20

40

60

80

100

120

140

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

No n. p. dead time

212.5 ns n. p. dead time

Time (s)

Co

un

ts/s

× 1

04

Figure 9.8: Eect of imposing and correcting by an additional non-paralyzable dead time on a typicalbeam trip experiment (SC3a conguration).


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6

Corrected values

Measured values

m = χn + τχ2n2

m = χn

Linear fit to the corrected data

n1, EC1B(counts/s × 104)

n0,

EC

1B (

cou

nts

/s ×

104 )

Figure 9.9: Correction of dead time imposing 212.5 ns of additional non-paralyzable dead time (n0 andn1 are the upper and the lower levels of the beam trips experiments).

20

40

60

80

100

1 2 3 4 5

Corrected values, 400 ns non-par. d.t.Corrected values, 200 ns non-par. d.t.Measured values

n1 (counts/s × 104)

n0

(co

un

ts/s

× 1

04 )

Figure 9.10: Correction of dead time without imposing any additional dead time (n0 and n1 are theupper and the lower levels of the beam trips experiments).

Chapter 10

Pulsed Neutron Source (PNS)experiments

Abstract - In this chapter, the results of the Pulsed Neutron Source (PNS) experiments carried out

at the Yalina-Booster facility within the frame of the EUROTRANS project are analized. Both the

prompt decay constant method and the area-ratio (Sjöstrand) method to determine the reactivity

are applied to these experiments. The methodologies based on traditional and generalized correction

factors (section 7.4.4) are applied to correct the spatial eects present in the assembly and the results

are compared and discussed. Part of the contents of this chapter have been published in [48], [47]

and [59].

As explained in section 7.2, PNS experiments investigate the evolution of the neutron populationin the reactor after the injection of very short neutron pulses (ideally instantaneous injection) from anexternal source. Although this is not intended to be the normal mode of operation of an industrialADS, which will rather operate in a continuous or quasi-continuous way, PNS experiments are of twofoldinterest. First, the response of the system to such short neutron pulses provides the Green's function(also known as impulse response) of the system, from which the response of the system to any neutronpulse with arbitrary time dependence can be derived 1. Second, PNS experimental techniques havealready been extensively validated in previous experiments such as MUSE-4 [326, 368], RACE-LP [165],Yalina-Thermal [267] or Yalina-Booster itself [266, 346, 347] and thus constitute a reliable starting pointfor further analysis of reactor kinetics.

These experiments have shown that PNS techniques can provide very precise measurements of theprompt neutron decay constant or the prompt to delayed neutron area ratio. However, those experimentshave also shown that the deviation from point kinetics behavior due to local or spectral eects requiresadditional corrections to obtain unbiased values of the reactivity of the system. Furthermore, they havealso shown the dependence of both methods on the knowledge of the eective kinetic parameters of thesystem to extract the reactivity from the experimental data. However, in most cases, it is very dicultto determine these kinetic parameters from experiments, specially if the system can not be made critical,so they are obtained from detailed computer simulations of the system.

In this chapter, the correction methodologies based on correction factors and the generalized method-ology presented in section 7.4.4 are applied to experimental results Yalina-Booster. The Monte Carlocode MCNPX has been applied to obtain the kinetic parameters of the system and the correction factors,as well as to determine the functional relationships between the reactivity and the prompt neutron decayslopes or the area-ratio.

10.1 Experimental results

In gures 10.1, 10.2 and 10.3 the measured evolution of the neutron ux at dierent positions within thereactor after an external neutron pulse is presented for congurations SC3a, SC3b and SC6, respectively.In these gures it can be observed that a certain time after the pulse injection the ux decays exponentiallyin all the core channels. Furthermore, the decay slopes in all channels in the core, both in the boosterand in the thermal zone, are essentially the same. This means that the core has reached its fundamental

1Assuming that there are neither thermal feedbacks or other elements in the reactor behavior that could limit thelinearity of the transport equation.

145

146 CHAPTER 10. PULSED NEUTRON SOURCE (PNS) EXPERIMENTS

10-5

10-4

10-3

10-2

10-1

1

0 2 4 6 8 10 12 14 16 18

EC1BEC2BEC3B

EC5TEC6TMC2

Time (ms)

Co

un

ts/µ

s/p

uls

e

Figure 10.1: Evolution of the counting rate in dierent detectors after a source pulse in the SC3a congu-ration. EC1B, EC2B, EC3B and MC2 are 500 mg FCs and EC5T and EC6T are 1 mg FCs. To comparethe results, the counting rate in the 1 mg FCs have been multiplied by a factor of 500. Repetition rateof the pulses was 50 Hz, source pulse duration was 5 µs and pulse intensity was 6 mA.

10-5

10-4

10-3

10-2

10-1

1

0 2 4 6 8 10 12 14 16

EC1BEC2BEC3B

EC5TEC6TMC2

Time (ms)

Co

un

ts/µ

s/p

uls

e

Figure 10.2: Evolution of the counting rate in dierent detectors after a source pulse in the SC3b congu-ration. EC1B, EC2B, EC3B and MC2 are 500 mg FCs and EC5T and EC6T are 1 mg FCs. To comparethe results, the counting rate in the 1 mg FCs have been multiplied by a factor of 500. Repetition rateof the pulses was 57 Hz, source pulse duration was 5 µs and pulse intensity was 6mA.

10.1. EXPERIMENTAL RESULTS 147

10-5

10-4

10-3

10-2

10-1

1

0 1 2 3 4 5

EC2BEC5TMC2

Time (ms)

Co

un

ts/µ

s/p

uls

e

Figure 10.3: Evolution of the counting rate in dierent detectors after a source pulse in the SC6 cong-uration. EC2B and MC2 are 500 mg FCs and EC5T is a 1 mg FC. To compare the results, the countingrate in the 1 mg FC has been multiplied by a factor of 500. Repetition rate of the pulses was 166 Hz,source pulse duration was 5 µs and pulse intensity was 6mA.

decay mode and that all higher order modes have die o. These modes are visible at the beginning of thehistograms. This is a very important nding because this means that the neutron ux can be eectivelyfactorized into a time-dependent part and a space-dependent part, which, as it was described in section5.3, it was a necessary condition for the applicability of the point kinetics model. The detectors in thereector, however, do not reach this fundamental mode (this is specially clear in gure 10.3), which istelling us that the reector is introducing lower decaying modes. Notice as well the stability of the delayedneutrons level in all the channels.

Notice that the shape of the higher order modes in the detectors closest to the source (all the boosterchannels and EC5T) is a peak at the beginning. This is due to the source neutrons and to the initialneutrons produced in the booster. However, in the detectors located the farthest from the source (EC6Tand MC2) there is no peak or is much smaller. This is because very few neutrons from those produced inthe source or in the booster reach these detectors, as most of them are either absorbed or cause anotherssions before reaching these points. Furthermore, the larger distance from the source causes that theneutrons require some time to reach these detectors, and that is way the maximum in this channel isobserved some time later than in the other channels. This eect is specially noticeable in the channel inthe reector, MC2.

The fact that the decay slope of the detectors in the core and in the thermal zone is the same is tellingus that in spite of the valve zone, the core and the thermal zone are coupled and that the fundamentaldecay mode is driven by the thermal zone. This is because, as explained in section 8.1, the essential ofthe power of Yalina B is produced in the thermal zone and, in spite of the presence of the valve zone,some thermal neutrons leak into the booster and there they suer a high multiplicative eect by the highenriched uranium.

In gure 10.4, 10.5 and 10.6 the evolution of the counting rate after one pulse in some detectors withcontrol rods inserted and extracted are presented for the case of the SC3b conguration. It can be clearlynoticed how the control rod position changes the slopes of the prompt decay constant.

In gure 10.7 it is shown the counting rate during the rst 200 µs after the injection of a neutronpulse in the 500 mg FC placed in the experimental channel EC1B for the SC3a and SC3b congurationswith the control rods inserted, alongside with the counting rate in the BC501A detector used as sourcemonitor. In this channel the eect of having removed the inner booster should have the largest eect inall the channels as it is the closest channel to it.

First of all notice the small peak that appears in the source monitor at around 30 µs in the SC3bconguration. This was due to an accelerator malfunction that was sending a small additional neutron


10-4

10-3

10-2

10-1

0 1 2 3 4 5 6 7 8 9 10

EC2B with control rods inEC2B with control rods outEC3B with control rods inEC3B with control rods out

Time (ms)

Co

un

ts/µ

s/p

uls

e

Figure 10.4: Evolution of the counting rate in the booster detectors after a source pulse in the SC3bconguration with control rods inserted and extracted.

10-2

10-1

1

10

0 1 2 3 4 5 6 7 8 9 10

EC5T with control rods in

EC5T with control rods outEC6T with control rods inEC6T with control rods out

Time (ms)

Co

un

ts/µ

s/p

uls

e

Figure 10.5: Evolution of the counting rate in the thermal zone detectors after a source pulse in the SC3bconguration with control rods inserted and extracted.

10.1. EXPERIMENTAL RESULTS 149

10-4

10-3

10-2

0 1 2 3 4 5 6 7 8 9 10

MC2 with control rods in

MC2 with control rods out

Time (ms)

Co

un

ts/µ

s/p

uls

e

Figure 10.6: Evolution of the counting rate in the reector detectors after a source pulse in the SC3bconguration with control rods inserted and extracted.

10-3

10-2

10-1

1

0 25 50 75 100 125 150 175 200

SC3a EC2B

SC3b EC2BSC3a source monitor

SC3b source monitor

Time (µs)

Co

un

ts/µ

s/p

uls

e

Figure 10.7: Evolution of the counting rate in the EC1B 500 mg FC and in the source monitor duringthe rst 200 µs after the pulse for the SC3a and SC3b congurations with the control rods inserted.


pulse after the initial one. Notice that although it is hard to notice this small pulse was also detected inthe detector in channel EC1B, in the shape of the small increase in the counting rate during this smallpulse and a short time after it.

The eect of having removed the inner booster in the SC3b conguration is noticed in the lower heightof the peak after the neutron pulse, which is an obvious consequence of the lower neutron multiplicationin the booster.

10.2 Reactivity determination with prompt neutron decay con-stant technique

In section 7.2, it was obtained that the relationship between the prompt decay constant α and thereactivity ρ in the point kinetics model is given by equation 7.21:

ρ

βeff=

αβeff/Λeff

+ 1

Hence, it is possible to perform an exponential t to the results of each detector in the time regionwhere an exponential decay is observed and use the kinetic parameters βeff and Λeff to obtain thereactivity.

The prompt decay constants α and the estimated keff with the ENDF/B-VII.0, JEFF-3.1 andJENDL-3.3 nuclear databases are shown in tables 10.1, 10.2 and 10.3. The results show that the promptneutron decay constant, and hence the reactivity values, obtained for dierent detector positions in thefuel region are compatible among each other.

In the reector region, prompt decay constants show a clear trend to provide lower values than in thefuel region. This indicates that the point kinetics hypothesis is not valid for all detector positions andcorrections are required.

It must be also noted that the dierence in reactivities due to the control rods insertion, estimated byMCNPX in 305 ± 15 pcm, is clearly noticed and consistent with the calculated values. This shows thehigh sensitivity of the method and its reliability to estimate reactivity changes of this order of magnitude.

Compared with the simulations, it can be observed that the average experimental keff are slightlylower (about 300 pcm) than the values calculated with MCNPX (tables 8.5 and 8.6). This eect hasbeen analyzed and found to be consistent with the simulations providing lower absolute values for theexponential decay constants, as it can be noticed in gures 10.8a and 10.8b, where the experimentaland simulated evolution of the counting rates at a U-235 detector after a D-T pulse in each of the threeregions of the assembly (the booster, the thermal zone and the reactor) are plotted alongside. In all cases,the constant level due to delayed neutrons has been subtracted and the experimental results have beencorrected to take into account the dead time of the detector system, as described in chapter 9.

In addition to the direct application of the prompt decay constant method presented in this section,the correction factors proposed in section 7.4.3 has also been applied to the experimental data. As statedin this section, this extension, that is denoted as the prompt decay constant method with correctionfactors, consists in replacing equation 7.21 by the more general linear relationship given by equation 7.43:

ρ = aα+ b

where parameters a and b are not necessarily the kinetic parameters Λeff or βeff . Parameters a andb have been determined with detailed MCNPX simulations of the system where, for simplicity, it hasbeen xed b = βeff . In this way, the value of a can be determined from a pair of values (ρ, α) obtainedfrom the simulations. The results obtained with the prompt decay constant method with correction factorswith the ENDF/B-VII.0, JEFF-3.1 and JENDL-3.3 databases are presented in tables 10.1, 10.2 and 10.3.Note the similitude of the values of a obtained in this way with the values of Λeff for all the channels inthe core, where the point kinetics is known to be largely valid. The value of a obtained for the reector islarger, however, and when it is used, the experimental values obtained for keff with this detector becomesimilar to those obtained with the detectors in the core.

10.3 Reactivity determination with the area-ratio technique

The second technique to determine the reactivity from the PNS experiments is the area-ratio technique.As explained in chapter 7, equation 7.22 can be used to determine the reactivity of the system from the

10.3. REACTIVITY DETERMINATION WITH THE AREA-RATIO TECHNIQUE 151

(a) EC1B

(b) EC5T

(c) MC2

Figure 10.8: Comparison of the evolution of the measured and simulated counting rates after a D-T neu-tron pulse in a U-235 ssion chamber (SC3a conguration with the control rods extracted). Simulationsare made with the ENDF/B-VII.0 library.


Table 10.1: Experimental keff calculated with the direct prompt decay constant method and with theprompt decay constant method with correction factors in SC3a, SC3b and SC6 congurations with theENDF/B-VII.0 library. With direct prompt decay constant method it is meant the direct application ofthe method, using equation 7.21 with the experimental values of α and the calculated kinetic parameters(tables 8.5 and 8.6). With prompt decay constant method with correction factors it is meant the applicationusing equation 7.43 with the values of a calculated using MCNPX (also given in the table) and the valuesof βeff in table 8.5.

(a) SC3a conguration

Det.a

(µs)Control rods out Control rods in ∆keff (pcm)

α (s−1) keff,direct keff,corr α (s−1) keff,direct keff,corr direct corr

EC1B 61.0± 0.2

-1057± 3

0.94609± 0.00042

0.94595± 0.00028

-1128± 4

0.94223± 0.00045

0.94207± 0.00030

387± 61

387± 41

EC2B 61.0± 0.2

-1124± 3

0.94249± 0.00043

0.94231± 0.00027

EC3B 61.2± 0.2

-1097± 1

0.94393± 0.00040

0.94349± 0.00018

EC5T 61.1± 0.2

-1094± 8

0.94409± 0.00060

0.94381± 0.00046

-1134± 6

0.94193± 0.00051

0.94164± 0.00035

216± 79

217± 58

EC6T 61.2± 0.2

-1098± 5

0.94385± 0.00048

0.94338± 0.00031

MC2 76.2± 0.2

-869± 4

0.95643± 0.00038

0.94437± 0.00028

-921± 6

0.95354± 0.00047

0.94084± 0.00042

289± 60

353± 51

(b) SC3b conguration

Det.a



EC1B 61.6± 0.3

-1057± 7

0.94531± 0.00053

0.94530± 0.00046

-1143± 7

0.94062± 0.00057

0.94061± 0.00050

469± 78

469± 68

EC2B 61.7± 0.3

-1048± 3

0.94581± 0.00041

0.94568± 0.00032

-1111± 3

0.94236± 0.00043

0.94222± 0.00033

345± 59

346± 46

EC3B 61.9± 0.2

-1043± 1

0.94609± 0.00039

0.94584± 0.00021

-1085± 1

0.94377± 0.00040

0.94351± 0.00021

233± 55

234± 30

EC5T 61.9± 0.2

-1073± 6

0.94443± 0.00051

0.94414± 0.00039

-1125± 6

0.94161± 0.00054

0.94130± 0.00040

282± 74

283± 56

EC6T 62.0± 0.2

-1038± 7

0.94639± 0.00054

0.94599± 0.00043

-1091± 7

0.94347± 0.00055

0.94304± 0.00042

292± 77

294± 61

MC2 77.2± 0.2

-856± 4

0.95653± 0.00038

0.94445± 0.00030

-913± 4

0.95334± 0.00041

0.94054± 0.00033

319± 56

390± 45

(c) SC6 conguration

Det. a (µs)Control rods out

α (s−1) keff,direct keff,corr

EC2B 69.9± 0.3

-2624± 16

0.85054± 0.00124

0.85043± 0.00097

EC5T 69.5± 0.4

-2659± 22

0.84883± 0.00143

0.84943± 0.00132

10.3. REACTIVITY DETERMINATION WITH THE AREA-RATIO TECHNIQUE 153

Table 10.2: Experimental keff calculated with the direct prompt decay constant method and with theprompt decay constant method with correction factors in SC3a, SC3b and SC6 congurations with theJEFF-3.1 library.


Det.a



EC1B 61.1± 0.2

-1057± 3

0.94625± 0.00042

0.94600± 0.00026

-1128± 4

0.94239± 0.00045

0.94212± 0.00028

387± 61

388± 38

EC2B 61.1± 0.2

-1124± 3

0.94265± 0.00043

0.94232± 0.00025

EC3B 61.3± 0.1

-1097± 1

0.94409± 0.00040

0.94363± 0.00015

EC5T 61.0± 0.1

-1094± 8

0.94425± 0.00060

0.94402± 0.00045

-1134± 6

0.94209± 0.00051

0.94186± 0.00033

216± 79

217± 56

EC6T 61.3± 0.1

-1098± 5

0.94401± 0.00048

0.94355± 0.00030

MC2 76.2± 0.1

-869± 4

0.95660± 0.00038

0.94449± 0.00026

-921± 6

0.95370± 0.00047

0.94096± 0.00041

289± 60

353± 49


Det.a



EC1B 61.4± 0.2

-1057± 7

0.94657± 0.00046

0.94559± 0.00044

-1143± 7

0.94196± 0.00050

0.94090± 0.00048

461± 68

468± 65

EC2B 61.7± 0.2

-1048± 3

0.94707± 0.00032

0.94587± 0.00028

-1111± 3

0.94367± 0.00034

0.94241± 0.00030

345± 59

340± 47

EC3B 61.5± 0.2

-1043± 1

0.94734± 0.00030

0.94631± 0.00019

-1085± 1

0.94505± 0.00031

0.94399± 0.00020

233± 55

229± 43

EC5T 61.6± 0.2

-1073± 6

0.94570± 0.00044

0.94458± 0.00038

-1125± 6

0.94293± 0.00046

0.94176± 0.00040

282± 74

277± 64

EC6T 61.8± 0.2

-1038± 7

0.94764± 0.00048

0.94632± 0.00043

-1091± 7

0.94476± 0.00048

0.94339± 0.00042

292± 77

287± 68

MC2 76.7± 0.2

-856± 4

0.95760± 0.00032

0.94498± 0.00030

-913± 4

0.95446± 0.00035

0.94110± 0.00033

319± 56

314± 47

(c) SC6 conguration



EC2B 69.8± 0.2

-2624± 16

0.84846± 0.00124

0.84673± 0.00144

EC5T 69.3± 0.3

-2659± 22

0.85002± 0.00121

0.85002± 0.00121


Table 10.3: Experimental keff calculated with the direct prompt decay constant method and with theprompt decay constant method with correction factors in SC3a, SC3b and SC6 congurations with theJENDL-3.3 library.


Det.a



EC1B 60.9± 0.2

-1057± 3

0.94604± 0.00042

0.94606± 0.00028

-1128± 4

0.94217± 0.00045

0.94219± 0.00030

387± 61

387± 41

EC2B 60.8± 0.2

-1124± 3

0.94243± 0.00043

0.94250± 0.00028

EC3B 61.1± 0.2

-1097± 1

0.94388± 0.00040

0.94366± 0.00018

EC5T 60.9± 0.2

-1094± 8

0.94403± 0.00060

0.94407± 0.00046

-1134± 6

0.94187± 0.00051

0.94191± 0.00035

216± 79

216± 58

EC6T 61.3± 0.2

-1098± 5

0.94379± 0.00048

0.94342± 0.00031

MC2 75.8± 0.2

-869± 4

0.95640± 0.00038

0.94472± 0.00028

-921± 6

0.95350± 0.00047

0.94120± 0.00042

290± 60

351± 51


Det.a



EC1B 61.7± 0.3

-1057± 7

0.94373± 0.00053

0.94537± 0.00046

-1143± 7

0.93892± 0.00058

0.94068± 0.00050

481± 79

470± 68

EC2B 61.7± 0.3

-1048± 3

0.94424± 0.00041

0.94587± 0.00032

-1111± 3

0.94070± 0.00043

0.94242± 0.00034

354± 59

346± 47

EC3B 61.7± 0.2

-1043± 1

0.94453± 0.00038

0.94608± 0.00020

-1085± 1

0.94215± 0.00040

0.94374± 0.00021

239± 55

233± 29

EC5T 61.6± 0.2

-1073± 6

0.94282± 0.00052

0.94451± 0.00038

-1125± 6

0.93993± 0.00054

0.94169± 0.00040

289± 75

282± 55

EC6T 62.1± 0.2

-1038± 7

0.94484± 0.00055

0.94600± 0.00043

-1091± 7

0.94184± 0.00054

0.94306± 0.00042

300± 78

295± 61

MC2 76.8± 0.2

-856± 4

0.95524± 0.00038

0.94493± 0.00030

-913± 4

0.95197± 0.00041

0.94104± 0.00033

328± 57

388± 45

(c) SC6 conguration



EC2B 69.4± 0.3

-2624± 16

0.84827± 0.00124

0.85138± 0.00097

EC5T 68.9± 0.4

-2659± 22

0.84653± 0.00144

0.85063± 0.00130

10.4. GENERALIZED VERSION OF THE METHODS 155

prompt and delayed counting rate areas. In Yalina-Booster both areas can be easily calculated since thefrequency of the neutron pulses is high enough for the delayed neutrons to be considered constant butlow enough for the prompt neutrons to decay to negligible values before the following pulse, as shown insection 10.1.

The area ratio measured in congurations SC3a, SC3b and SC6 for dierent detectors, as well as thekeff values obtained with the direct application of equation 7.22 are shown in tables 10.4, 10.5 and 10.6.In can be observed that the dierences in the values among dierent detectors due to spectral or spatialeects are large, even a factor of more than two. These dierences result in a large spread of the keffresults obtained with detectors at dierent positions. Therefore, the direct application of the area-ratiomethod is of little use for the case of Yalina-Booster.

Correction factors have been applied to overcome this problem, as explained in section 7.4.3. It mustbe noticed that correction factors used here have not been calculated as in equation 7.42, but instead inthe manner:

Cdet =(Ap/Ad)MC

|ρMC |=

(Ap/Ad)MC

|keff,MC−1keff,MC

|(10.1)

And the experimental keff is thus obtained as:

keff,exp =1

1 +(Ap/Ad)exp

Cdet

(10.2)

Note that with this implementation the eective delayed neutron fraction is also embedded in thecorrection factors, in addition to the spatial eects. It is also worth mentioning that in this way keffvalues are directly obtained, instead of values of ρ. The MCNPX code was used to determine thesecorrection factors for the case of Yalina-Booster, but other simulations tools can also be used to investigatespatial and energy eects for the area-ratio method, as shown by [78]. Results obtained in this way withthree dierent nuclear data libraries (ENDF/B-VII.0, JEFF-3.1 and JENDL-3.3) are presented in tables10.4, 10.5 and 10.6are presented in tables. It can be noticed that the large spread of keff estimators isreduced and compatible values are found for all detectors after corrections.

10.4 Generalized version of the methods

In section 7.4.4, it was presented a further generalization of both the prompt decay contant methodand the area ratio method. This generalization consisted in replacing the linear relationships given byequations 7.21 and 7.22, respectively, by the more general relationships 7.44 and 7.45:

ρ = ρ1 (α)

ρ = ρ2

(ApAd

)As it is discussed in section 7.4.4, these univocal relationships are specic for every subcritical system

and every detector position. The existence of these relationships has been investigated for the case ofYalina-Booster. In this case, the search of these relationships has been performed with a series of simula-tions of the system using MCNPX. Starting from the most accurate description of congurations, SC3a,SC3b and SC6, small variations of dierent parameters have been performed, namely the polyethylenedensity, the fuel enrichment and the geometry (height to width ratio). For every variation of these pa-rameters, values of ρ, α and Ap

Adare calculated. Notice that both α and Ap

Adare detector dependent. The

value of α has been obtained as the decay constant resulting from the t to an exponential function ofthe simulated detector counting rate after a neutron pulse for a xed time interval.

The choice of the parameters to be perturbed is not trivial, since the number of possibilities is verylarge. Although a sensitivity analysis can assist in this choice, a conventional sensitivity analysis is notenough because it implies assuming a linear dependency between the perturbed parameters and both ρand the measured parameters (α and Ap

Ad), which, in turn, implies assuming a linear relationship between

ρ and the measured parameters. As stated above, with the generalized version proposed we do not wantto limit ourselves to linear relationships (which in fact, it is the case of considering conventional correctionfactors). Even if non-linear relationships are considered, it still would be necessary to assume a givendependency among the perturbed parameters, the measured parameters and the reactivity, when, in fact,


Table 10.4: Experimental values of the prompt to delay area ratio in Yalina-Booster experiments.keff,direct denotes the values calculated using equation 7.22 and the ENDF/B-VII.0 values of βeff (table8.5). keff,corr denotes the values calculated using equation 7.42 and correction factors (Cdet) calculatedfrom simulations using the same library. Dierent correction factors have been applied for the cases withthe control rods inserted and extracted. They are also given in the table.


Det.Control rods out Control rods in ∆keff (pcm)

Ap/Ad Cdet keff,direct keff,corr Ap/Ad Cdet keff,direct keff,corr direct corr

EC1B15.31± 0.03

269.58± 2.19

0.89960± 0.00064

0.94626± 0.00045

17.64± 0.04

292.83± 2.67

0.88606± 0.00073

0.94318± 0.00053

1354± 97

308± 70

EC2B 244.70± 1.83

15.63± 0.03

252.94± 2.03

0.89771± 0.00065

0.94180± 0.00048

EC3B 166.73± 1.55

10.20± 0.01

165.11± 1.59

0.93079± 0.00045

0.94182± 0.00056

EC5T8.70± 0.06

150.26± 1.95

0.94036± 0.00055

0.94527± 0.00080

9.44± 0.04

152.91± 2.11

0.93561± 0.00049

0.94185± 0.00084

475± 73

342± 116

EC6T 129.41± 1.48

7.48± 0.03

128.86± 1.54

0.94829± 0.00039

0.94514± 0.00069

MC27.23± 0.01

125.26± 1.39

0.94993± 0.00033

0.94543± 0.00061

7.85± 0.01

129.60± 1.55

0.94587± 0.00036

0.94289± 0.00069

406± 49

254± 92

MC37.24± 0.18

126.90± 1.41

0.94987± 0.00123

0.94578± 0.00148

7.88± 0.21

125.71± 1.57

0.94568± 0.00141

0.94102± 0.00174

419± 187

477± 228




EC1B15.17± 0.03

286.89± 3.39

0.90055± 0.00076

0.94978± 0.00064

17.48± 0.09

310.69± 4.76

0.88711± 0.00097

0.94673± 0.00086

1343± 123

304± 107

EC2B13.92± 0.02

259.96± 2.43

0.90799± 0.00070

0.94918± 0.00048

15.28± 0.03

270.41± 2.77

0.89990± 0.00076

0.94652± 0.00056

809± 103

266± 74

EC3B9.64± 0.01

168.81± 1.42

0.93442± 0.00051

0.94598± 0.00046

10.21± 0.01

167.03± 1.47

0.93081± 0.00053

0.94240± 0.00051

361± 74

358± 69

EC5T9.26± 0.04

151.53± 1.85

0.93684± 0.00055

0.94241± 0.00075

10.07± 0.05

153.67± 2.00

0.93170± 0.00061

0.93850± 0.00086

515± 82

391± 114

EC6T7.42± 0.04

128.17± 1.26

0.94875± 0.00048

0.94528± 0.00061

7.60± 0.04

127.74± 1.32

0.94757± 0.00049

0.94385± 0.00065

118± 68

143± 89

MC27.31± 0.01

128.38± 1.26

0.94947± 0.00040

0.94613± 0.00053

7.97± 0.01

130.52± 1.37

0.94516± 0.00043

0.94245± 0.00061

431± 59

368± 81

(c) SC6 conguration

Det.Control rods out

Ap/Ad Cdet keff,direct keff,corr

EC2B43.63± 0.07

268.15± 7.74

0.75172± 0.00175

0.86006± 0.00404

EC5T23.96± 0.85

138.88± 6.34

0.84647± 0.00476

0.85287± 0.00674


Table 10.5: Experimental values of the prompt to delay area ratio in Yalina-Booster experiments andvalues of keff,direct, correction factors (Cdet) and keff,corr obtained with the JEFF-3.1.




EC1B15.31± 0.03

265.81± 1.64

0.89737± 0.00076

0.94554± 0.00035

17.64± 0.04

287.56± 2.10

0.88357± 0.00086

0.94220± 0.00044

1380± 115

334± 56

EC2B 238.40± 1.33

15.63± 0.03

248.28± 1.60

0.89545± 0.00077

0.94078± 0.00040

EC3B 161.29± 1.25

10.20± 0.01

159.60± 1.34

0.92920± 0.00053

0.93993± 0.00051

EC5T8.70± 0.06

143.63± 1.51

0.93898± 0.00061

0.94289± 0.00072

9.44± 0.04

146.55± 1.68

0.93413± 0.00056

0.93948± 0.00074

485± 82

341± 103

EC6T 124.67± 1.06

7.48± 0.03

122.45± 1.20

0.94708± 0.00045

0.94243± 0.00061

MC27.23± 0.01

125.09± 1.07

0.94876± 0.00040

0.94536± 0.00047

7.85± 0.01

127.00± 1.29

0.94461± 0.00043

0.94179± 0.00059

415± 58

357± 76

MC37.24± 0.18

122.43± 1.12

0.94869± 0.00127

0.94484± 0.00150

7.88± 0.21

121.77± 1.19

0.94441± 0.00146

0.93922± 0.00173

428± 194

562± 229




EC1B15.17± 0.03

297.39± 3.11

0.89883± 0.00076

0.95147± 0.00055

17.48± 0.09

319.99± 4.37

0.88519± 0.00097

0.94820± 0.00076

1364± 123

326± 94

EC2B13.92± 0.02

268.69± 2.57

0.90638± 0.00070

0.95074± 0.00074

15.28± 0.03

279.53± 2.91

0.89817± 0.00076

0.94817± 0.00055

822± 103

257± 73

EC3B9.64± 0.01

168.81± 1.42

0.93325± 0.00051

0.94647± 0.00046

10.21± 0.01

168.53± 1.48

0.92958± 0.00053

0.94288± 0.00050

367± 74

359± 68

EC5T9.26± 0.04

150.62± 1.82

0.93571± 0.00055

0.94208± 0.00074

10.07± 0.05

152.36± 1.95

0.93048± 0.00061

0.93800± 0.00085

523± 83

408± 113

EC6T7.42± 0.04

129.73± 1.28

0.94782± 0.00048

0.94590± 0.00061

7.60± 0.04

127.78± 1.30

0.94662± 0.00049

0.94386± 0.00064

120± 68

204± 88

MC27.31± 0.01

127.43± 1.24

0.94855± 0.00040

0.94575± 0.00053

7.97± 0.01

130.32± 1.35

0.94416± 0.00043

0.94237± 0.00060

439± 59

338± 80

(c) SC6 conguration



EC2B43.63± 0.07

258.62± 7.17

0.74804± 0.00174

0.85565± 0.00401

EC5T23.96± 0.85

136.30± 6.09

0.84390± 0.00482

0.85049± 0.00670


Table 10.6: Experimental values of the prompt to delay area ratio in Yalina-Booster experiments andvalues of keff,direct, correction factors (Cdet) and keff,corr obtained with the JENDL-3.3.




EC1B15.31± 0.03

273.45± 2.23

0.89898± 0.00064

0.94698± 0.00044

17.64± 0.04

294.34± 2.19

0.88537± 0.00073

0.94220± 0.00044

1361± 97

334± 56

EC2B 246.13± 1.84

15.63± 0.03

255.78± 1.69

0.89708± 0.00065

0.94241± 0.00040

EC3B 164.46± 1.50

10.20± 0.01

163.27± 1.23

0.93035± 0.00045

0.94120± 0.00045

EC5T8.70± 0.06

148.18± 1.89

0.93998± 0.00055

0.94454± 0.00080

9.44± 0.04

151.08± 1.78

0.93520± 0.00049

0.94119± 0.00074

477± 73

335± 109

EC6T 126.10± 1.40

7.48± 0.03

122.45± 1.20

0.94795± 0.00039

0.94243± 0.00061

MC27.23± 0.01

125.01± 1.38

0.94961± 0.00033

0.94533± 0.00061

7.85± 0.01

127.81± 1.30

0.94552± 0.00036

0.94214± 0.00059

409± 49

319± 85

MC37.24± 0.18

124.02± 1.36

0.94954± 0.00124

0.94416± 0.00148

7.88± 0.21

121.64± 1.19

0.94532± 0.00142

0.93916± 0.00173

422± 188

500± 227




EC1B15.17± 0.03

286.83± 3.35

0.89895± 0.00076

0.94977± 0.00063

17.48± 0.09

305.33± 4.57

0.88533± 0.00097

0.94585± 0.00086

1362± 123

392± 106

EC2B13.92± 0.02

254.83± 2.32

0.90650± 0.00070

0.94820± 0.00048

15.28± 0.03

266.17± 2.67

0.89829± 0.00076

0.94571± 0.00055

821± 103

249± 73

EC3B9.64± 0.01

165.42± 1.36

0.93333± 0.00051

0.94493± 0.00046

10.21± 0.01

164.04± 1.42

0.92967± 0.00053

0.94141± 0.00051

366± 74

353± 68

EC5T9.26± 0.04

147.80± 1.75

0.93579± 0.00055

0.94104± 0.00074

10.07± 0.05

148.05± 1.86

0.93056± 0.00061

0.93632± 0.00086

523± 83

473± 114

EC6T7.42± 0.04

125.63± 1.20

0.94788± 0.00048

0.94423± 0.00061

7.60± 0.04

125.12± 1.26

0.94669± 0.00049

0.94273± 0.00065

120± 68

150± 89

MC27.31± 0.01

125.81± 1.20

0.94862± 0.00040

0.94613± 0.00053

7.97± 0.01

128.08± 1.32

0.94424± 0.00043

0.94142± 0.00061

438± 59

367± 81

(c) SC6 conguration



EC2B43.63± 0.07

262.55± 7.36

0.75246± 0.00175

0.85565± 0.00401

EC5T23.96± 0.85

136.30± 6.09

0.84699± 0.00475

0.85049± 0.00670


this relationship is what is being sought for. Therefore, a brute force strategy is followed, and a completecalculation of is performed for every perturbed conguration.

Thus, the perturbed parameters are chosen among those that are expected to have an enough sig-nicant impact in ρ, α or Ap

Adto be able to apply the generalized technique. Furthermore, the three

parameters chosen are related, each one of them, with a dierent aspect of a nuclear system: geometry(height to width ratio), composition (fuel enrichment) and material densities (polyethylene density). Fi-nally, although this is not the case of a zero-power facility such as Yalina-Booster, these parameters arelikely to change during system transients in a hypothetical high power ADS. In eect, moderator densityand system geometry can vary as the result of temperature changes in the reactor, while fuel enrichmentmay vary as the result of the burn-up. In this way, the existence of a single analytical relationship be-tween ρ that holds for a whole transient can be investigated. This is of particular interest as subcriticalitymonitoring is specially critical during reactor transients.

Of course, the selection of the perturbed parameters is still rather arbitrary and relevant parametersmay have been omitted. Nevertheless, it must be noted that the same may occur in a conventionalsensitivity analysis.

In gures 10.10, 10.11 and 10.12 the pairs (∆α,∆ρ) and(

∆ApAd,∆ρ

)obtained in this way for several

experimental channels are shown for the SC3a, SC3b and SC6 congurations. The origin of coordinatescorresponds to the non-perturbed conguration using the ENDF/B-VII.0 database. As we can observe,for small variations in the three parameters, there exists a relationship between ∆ρ and either ∆α or∆ApAd

that is independent of the varied parameter.

To remark that the univocal relationship between ∆α and ∆ApAd

with ∆ρ is only locally true, largevariations of the polythene density have been performed. As it can be observed in the gures, this leads toa multi-valued function for the case of the detectors in the core (EC1B, EC2B and EC5T), while it remainsa well dened linear relationship for the detector in the reector (MC2). Hence, a way to determine therange of validity of the generalized method at dierent detector positions has been obtained.

Furthermore, it can be observed in the gures that the relationship between ∆ρ and the measuredparameters keeps a good linearity up to about ∆ρ = 1000 pcm. This linearity allows obtaining thereactivity from the experimental measurements with the simple equations:

ρexp = ρ0 + Λ∗ (∆α)exp (10.3)

for the case of the generalized version of the prompt decay constant method and

ρexp = ρ0 + β∗(

∆ApAd

)exp

(10.4)

for the case of the generalized version of the prompt to delayed area ratio method. ρ0 denotes thereactivity of the reference non-perturbed system conguration and the parameters Λ∗ and β∗ are obtainedby the linear tting of the pairs (∆α,∆ρ) and

(∆ApAd,∆ρ

), respectively. They have been denoted in this

way by analogy with the parameters Λeff and βeff in equations 7.21 and 7.22. The results for Λ∗ and β∗

obtained with this method are presented in table 10.7. In principle, the comparison of these parameterswith Λeff and βeff provides a way to estimate the validity of the point kinetics approximation.

In the case of the Λ∗ parameter, two dierent behaviors can be observed in Yalina-Booster. First,in the fuel region, where the calculated value of Λeff is about 15% lower than Λ∗ for SC3a and SC3bcongurations and 30% in the case of SC6 conguration. This eect is consistent with the fact that pointkinetics provides worse results as reactivity lowers. Second, in the reector region, where the value ofΛ∗ is very dierent from Λeff even for SC3a and SC3b congurations. In this case, the dierence isproduced by the thermalization in graphite as well as the low absorption cross section.

For its part, the obtained values of β∗ vary largely with the detector position. For the case of Yalina-Booster, it is remarkable, however, its nearly positive monotonic tendency from the centermost to theoutermost detector position and the tendency to take much lower values than βeff in the booster.

Finally, it is important to remark that the generalized method proposed in this section also provides away to estimate the systematic uncertainty introduced in the value of keff . Here, it has been adopted thevalue given by the maximum dierence between the mean value of Λ∗ and β∗ and any of the values of Λ∗

and β∗ that would be obtained by considering the variation of any of the system parameters (polyethylenedensity, fuel enrichment, height to width ratio) alone (gure 10.9). The uncertainties calculated in thisway are presented in table 10.7. It can be observed that the systematic errors obtained in this waymay range from less to 10% to about 50% for the dierent congurations and detector positions. This


-6 -4 -2 0

0

0.005

0.01

0.015

0.02

0.025∆ρ

∆(Ap/Ad)

Varying polyethylene density

Varying uranium enrichment

Varying height to width

∆ρ=β*•∆(Ap/Ad) (polyethylene density)

Systematic error

∆ρ=β*•∆(Ap/Ad) (all three series)

L.P.D.C.

Figure 10.9: Example of determination of the systematic uncertainty in the parameter β∗. The valueof β∗ (slope of the tted line) obtained considering only the points corresponding to perturbations inthe polyethylene density has the largest dierence with the slope β∗ obtained considering a t to allperturbed points. Notice that the points corresponding to large polyethylene density changes are notconsidered for calculating the slopes.

variation is to be considered as well to determine the best position for placing a detector for determiningthe reactivity with the generalized method.

The generalized method provides values for the reactivity in absolute units, rather than values in unitsof dollars, without the need of knowing the value of βeff . Therefore, values for keff can be obtaineddirectly. In table 10.7 the results for keff obtained with the generalized versions of both the promptdecay constant and the area ratio techniques are presented. It can be observed that the generalized arearatio technique signicantly reduces the dispersion of the direct area ratio results. An important remarkis that regardless of the keff absolute values, for a given detector position both methods are capable toclearly noticing the dierence in keff due to the control rods movement. Moreover, when both methodsare compared, it can also be observed that they are compatible.

Figure 10.13 shows the results obtained with the generalized method compared with the results ofthe prompt decay constant and the area ratio techniques (sections 10.2 and 10.3) for the SC3a and SC3bcongurations with the control rods inserted and extracted. It is found that the generalized methods,presented in this section, provides very similar keff results for the prompt decay constant method and thearea ratio method using the correction factors presented in sections 10.2 and 10.3. Nevertheless, it canbe also observed that the generalized prompt decay constant method tends to slightly shift downwardsthe results of keff with respect to the results obtained with the previous versions of the prompt decayconstant method (equation 7.43), which may be due to the better account of the spatial and spectraleects that it is in principle allowed with the generalized method.


(a) EC1B, prompt decay constant method (b) EC1B, area-ratio method

(c) EC5T, prompt decay constant method (d) EC5T, area-ratio method

(e) MC2, prompt decay constant method (f) MC2, area-ratio method

Figure 10.10: (∆α,∆ρ) and (∆(Ap/Ad),∆ρ) values obtained by changing dierent parameters in the SC3aconguration obtained with the ENDF/B-VII.0 library. The ranges in the variation of the parameters thathave been studied are up to ±40% in the polyethylene density, up to ±7.5% in the uranium enrichmentand up to ±15% in the height to width ratio. The values corresponding to large polyethylene densitychanges (L.P.D.C.) are highlighted in the cases where they considerably depart from linearity. The crosssections have also been changed, by considering dierent nuclear data libraries, but the range of variationis very small in this scale. The dotted lines are the result of tting these pairs of points to a straightline (in the range where they can be considered to follow a linear relationship). The pairs of pointsresulting from the application of this generalized method to the experimental values of (∆α,∆ρ) and(∆(Ap/Ad),∆ρ) with the control rods inserted an extracted are also plotted.




(e) MC2, prompt decay constant method (f) MC2, area-ratio method

Figure 10.11: Id. for the SC3b conguration.


Table 10.7: keff calculated with the generalized method. Upper values have been calculated with thegeneralized prompt decay constant method and lower values have been calculated with the generalizedarea-ratio method.

(a) SC3a

Detector positionΛ∗(µs)

keff , control rods out keff , control rods in ∆keff (pcm)β∗ (pcm)

EC1B69.1 ± 9.1 0.94558 ± 0.00051 0.94119 ± 0.00103 438 ± 115299 ± 60 0.94674 ± 0.00047 0.94053 ± 0.00165 621 ± 172

EC2B68.6 ± 9.5 0.94151 ± 0.00103 338 ± 77 0.94147 ± 0.00166

EC3B69.1 ± 10.3 0.94282 ± 0.00089 552 ± 73 0.94285 ± 0.00083

EC5T69.2 ± 5.9 0.94316 ± 0.00072 0.94071 ± 0.00077 245 ± 106653 ± 68 0.94534 ± 0.00066 0.94104 ± 0.00094 430 ± 114

EC6T69.8 ± 8.4 0.94264 ± 0.00080 787 ± 37 0.94529 ± 0.00056

MC295.5 ± 12.3 0.94326 ± 0.00077 0.93885 ± 0.00135 441 ± 156860 ± 5 0.94518 ± 0.00041 0.94044 ± 0.00052 474 ± 66

MC3Not enough statistics

839 ± 50 0.94561 ± 0.00142 0.94083 ± 0.00168 478 ± 220

(b) SC3b


keff , control rods out keff , control rods in ∆keff (pcm)β∗ (pcm)

EC1B72.1 ± 5.6 0.94476 ± 0.00059 0.93928 ± 0.00090 548 ± 107276 ± 55 0.94954 ± 0.00040 0.94382 ± 0.00102 572 ± 109

EC2B71.6 ± 5.8 0.94523 ± 0.00043 0.94123 ± 0.00067 401 ± 80317 ± 66 0.94908 ± 0.00030 0.94521 ± 0.00074 386 ± 80

EC3B71.2 ± 5.2 0.94544 ± 0.00030 0.94276 ± 0.00047 268 ± 55560 ± 81 0.94611 ± 0.00044 0.94326 ± 0.00080 285 ± 92

EC5T70.4 ± 5.1 0.94354 ± 0.00055 0.94032 ± 0.00073 322 ± 92666 ± 67 0.94236 ± 0.00078 0.93759 ± 0.00120 477 ± 43

EC6T71.1 ± 7.5 0.94562 ± 0.00057 0.94225 ± 0.00081 337 ± 99854 ± 32 0.94497 ± 0.00049 0.94360 ± 0.00050 137 ± 70

MC297.1 ± 14.7 0.94340 ± 0.00085 0.93851 ± 0.00155 489 ± 176809 ± 19 0.94604 ± 0.00035 0.94129 ± 0.00038 475 ± 52

(c) SC6


keff , control rods outβ∗ (pcm)

EC2B112 ± 30 0.85028 ± 0.00156360 ± 124 0.85972 ± 0.00404

EC5T110 ± 29 0.84868 ± 0.002141010 ± 259 0.84876 ± 0.00798




Figure 10.12: Id. for the SC6 conguration.


(a)SC3a,promptdecay

technique

(b)SC3a,arearatiotechnique

(c)SC3b,promptdecay

technique

(d)SC3b,arearatiotechnique

Figure10.13:

Com

parisonof

thereactivity

resultsobtained

withthedierent

versions

presentedin

thispaperof

both

theprom

ptdecayconstant

andthearea

ratiotechniques

fortheSC

3aandSC

3bcongurations(allresultshave

been

obtained

withtheENDF/B

-VII.0library).

Chapter 11

Continuous source experiments

Abstract - This chapter completes the analysis of the results of PNS experiments at the Yalina-

Booster facility presented in chapter 10 with the analysis of the results of the experiments with the

system in a steady state with short beam trips. For the rst time, to my knowledge, the reactivity of

a subcritical system has been monitorized during operation with three dierent techniques: current-

to-ux, source-jerk and prompt neutron decay constant techniques. The essential of the contents of

this chapter have been published in [49].

PNS techniques presented in chapter 10 can not be applied during the normal operation of an industrialADS, since an stable and continuous power level is required. Hence, it is necessary to use other techniquescompatible with a continuous or quasi-continuous operation of the accelerator.

In the nal conclusions of the MUSE-4 experiments carried out during the 5th European FrameworkProgramme [218] it was proposed to combine two independent techniques for continuously monitoringthe reactivity of an ADS. The rst of these techniques is the current-to-ux technique. However, as itis stated in chapter 7, this method has the drawback that only provides relative measurements of thereactivity, and therefore, it must be complemented with another technique capable of providing absolutevalues for the reactivities. One possibility consists in performing very short interruptions of the beamcurrent (beam trips) and applying slightly modied PNS methodologies, also explained in chapter.

This chapter is divided in two sections. First, in section 11.1, they will be shown the results obtainedin experiments performed when the reactor was in steady state, that is, the reactivity and neutron sourceproduction remained constant or variations were slow compared with the delayed neutrons scale of time.Results are compared with those obtained using PNS techniques presented in chapter 10. Second, insection 11.2, the results obtained during fast variations of the reactivity of the assembly and the neutronsource intensity will be presented.

11.1 Steady-state reactivity monitoring

As mentioned above and in section 7.1, the current-to-ux technique is suitable only for relative reactivitymeasurements. Therefore, steady-state experiments have been only applied to validate the absolute reac-tivity determination techniques with beam trip interruptions. Steady-state experiments were performedin the two main congurations SC3a and SC3b and the two sub-congurations obtained by the insertionor extraction of the control rods. In addition, for the SC3a conguration, dierent intensities were ex-plored to investigate possible eects of the neutron source intensity in the reactivity determination usingbeam trips. Beam trips were forced at a rate of one per second with a trip duration of about 40 ms.

It is important to notice that the accelerator beam during the experiments was not perfectly constant.As it can be observed in gure 11.1, a large 50 Hz oscillation in the neutron source could not be avoided.This oscillation was due to an oscillation in the deuteron beam impinging position. Nevertheless, sincethe period of this oscillation is much larger than the characteristic decay time of the prompt neutrons(about 1 ms) but still an order of magnitude shorter than the shortest of the decay periods of any ofthe delayed neutron families (between 0.23 and 55.72 s in a six group model [148]), the oscillation is notexpected to signicantly aect the reactivity determination when it is properly time averaged.

As a nal comment, it may be worth investigating the possibility to use the oscillating nature of thesource to obtain the reactivity. However, this is not an expected operating condition of an industrial-sizeADS and it has not been included in this work.

167

168 CHAPTER 11. CONTINUOUS SOURCE EXPERIMENTS

Figure 11.1: External neutron source (pale gray) and 235U detector response (dark gray) before and duringa beam trip. Both signals have been averaged over periods of 0.1 ms to reduce statistical uctuations.Notice that the source has a strong 50 Hz oscillation, which drives the 235U detector between beam trips.

11.1.1 Prompt decay constant method

First of all, it must be remarked that the accurate determination of the prompt neutron decay constants,that are used in the prompt decay constant method, requires large statistics. Hence, the histograms ofcounts per unit time after a large number of beam trips (∼ 1000) have been superimposed in order tohave enough statistics. The prompt decay slopes have been obtained simply by tting the slope of thedecay of the prompt neutrons after a beam trip to an exponential plus a constant. An adequate range forthe t must be chosen to get rid of the non point-kinetics eects present at the very rst microsecondsafter the beam interruption. The 50 Hz oscillation described above is not expected to aect the results ofthe prompt decay constant method, which is only function of the neutron population just a few promptneutron decay periods before the beam trip.

The results of the application of the prompt decay constant method are shown in Table 11.1, alongsidewith the values of the prompt neutron decay constants obtained in complementary PNS experiments inthe same conguration of the system (Table 10.1). It must be remarked that the slopes have been foundto be largely compatible, within errors, for dierent detector positions. There are only two exceptionsthat deviate considerably from the others (position EC1B with 1.2 mA beam intensity and position EC5Twith 0.5 mA intensity, both in the SC3a conguration with the control rods inserted) and that can beconsidered spurious results.

The similarity of the prompt neutron decay slopes among dierent detector positions is a remarkablending since it implies that, at least for the case of Yalina-Booster and for the detector positions inves-tigated, the same values for the reactivity will be obtained at these detector positions, in spite of spatialeects (notice, however, that the similitude of the results at dierent detector positions remarked abovedoes not necessarily imply that spatial or spectral eects are not present). Furthermore, the results arealso compatible with the results obtained in PNS experiments.

From the prompt neutron decay slopes it is possible to compute the reactivity using equation 7.21,provided that Λeff and βeff are known. However, in this work, the generalized methodology proposedin section 7.4.4 that has been already applied to the PNS experiments in Yalina-Booster in section 10.4,will be used instead. The values of keff thus obtained are shown in table 11.2, alongside with thevalues obtained in the PNS experiments, for comparison. As the values of α obtained in beam-tripsexperiments were largely compatible with the ones obtained in PNS experiments, the results of keffare also compatible. It can also be observed in table 11.2 that, with the exception of the two casesabove mentioned, the results for keff are about 300-400 p.c.m. lower than the MCNPX results, with anuncertainty of the order of 100 p.c.m., taking into account both statistical errors and the estimated range

11.1. STEADY-STATE REACTIVITY MONITORING 169

Table 11.1: Prompt decay constant results for SC3a and SC3b congurations, compared with the resultsof the PNS experiments


Detector Beam inten- Control rods extracted Control rods insertedposition sity (mA) α (s−1) αPNS(s−1) α (s−1) αPNS(s−1)

1.2 mA -1059 ± 7 -1053 ± 9EC1B 1.0 mA -1042 ± 11 -1057 ± 3 -1121 ± 14 -1128 ± 4

0.5 mA -1054 ± 10 -1126 ± 121.2 mA -1066 ± 7 -1094 ± 8

EC2B 1.0 mA -1062 ± 9 -1105 ± 10 -1124 ± 30.5 mA -1055 ± 14 -1118 ± 151.2 mA -1062 ± 18 -1105 ± 21

EC5T 1.0 mA -1033 ± 23 -1094 ± 8 -1116 ± 27 -1134 ± 60.5 mA -1050 ± 41 -1054 ± 10


Detector Beam inten- Control rods extracted Control rods insertedposition sity (mA) α (s−1) αPNS(s−1) α (s−1) αPNS(s−1)EC2B 1.0 mA -1035 ± 5 -1048 ± 3 -1085 ± 5 -1111 ± 3EC5T 1.0 mA -1034 ± 12 -1073 ± 6 -1073 ± 14 -1125 ± 6

of systematic uncertainties in Λ∗. Furthermore, the dierence in keff due to the change in the controlrods position is of the order of magnitude of the expected value (about 300 p.c.m).

Finally, when observing in table 11.2 the variation of the ∆keff values with the beam intensity, adecreasing tendency may be apparent (for the EC2B and EC5T positions in the SC3a conguration).Nevertheless, the range of the uncertainties is too large to be conclusive.

11.1.2 Source-jerk method

The source-jerk method (equation 7.37) has been applied to the Yalina-Booster experiments. It is impor-tant to remark that thanks to the large statistics available for this method it was possible to determinethe reactivity at every single beam trip with a precision better than 2$. An example of the distribution ofthe source-jerk results every beam trip for a given experiment is presented in gure 11.2. This might bean advantage with respect to the prompt decay constant method, that needs much higher system powerto allow single beam trip determination of the reactivity with the same precision.

The 50 Hz oscillation in the beam already mentioned should not aect the results of the source-jerktechnique, since the half lives of the dierent families of delayed neutrons are much longer than theperiod of the oscillation. Thus the delayed neutron level n1 is determined by the average neutron ux.Consequently, the prompt neutron level n0 will be determined by the average neutron ux over one orseveral oscillation cycles before the beam trip.

Other aspect to take into account when applying the source-jerk technique is that beam interruptionsreduce the eective neutron source. This causes in turn a reduction of the delayed neutron level, n1,while not altering the prompt neutron level, n0 − n1. In the case of the experiments at Yalina-Boosterpresented here, there was a beam trip of about 40 ms every second, that is, the source is down 4% ofthe time. To take this eect into account, a duty cycle factor ξ is dened as the fraction of time theaccelerator is working at average intensity. Equation 7.37 then becomes:

ρ ($) = −n0 − n1n1

ξ

= −ξ n0 − n1

n1(11.1)

with an estimated value, from the previous argument, of ξ ' 0.96.In table 11.3, the experimental results obtained with the source-jerk technique are presented. It also

includes, when available, the results obtained with the PNS area-ratio technique for comparison purposes.As it was found for the case of the PNS area-ratio results, it is also found that the dierences betweensource-jerk results at dierent detector positions can reach a factor of up to two. This was to be expectedsince the source-jerk technique is equivalent to the area ratio technique applied to a source interruption


Table 11.2: keff estimations from the prompt decay constant method for SC3a and SC3b congurationscompared with the results of the PNS experiments.


Detector Beam inten- keff c.r. extracted keff c.r. inserted ∆keff (p.c.m.)position sity (mA) BT PNS BT PNS BT PNS

1.2 mA 0.94545± 0.00063

EC1B 1.0 mA 0.94651± 0.00078

0.94558± 0.00058

0.94163± 0.00130

0.94119± 0.00103

488 ± 152 438 ± 115

0.5 mA 0.94579± 0.00078

0.94134± 0.00124

445 ± 146

1.2 mA 0.94501± 0.00070

0.94330± 0.00092

171 ± 116

EC2B 1.0 mA 0.94529± 0.00076

0.94262± 0.00107

0.94151± 0.00103

267 ± 131

0.5 mA 0.94571± 0.00097

0.94185± 0.00132

386 ± 164

1.2 mA 0.94513± 0.00117

0.94249± 0.00139

264 ± 182

EC5T 1.0 mA 0.94695± 0.00143

0.94316± 0.00072

0.94181± 0.00176

0.94071± 0.00077

514 ± 227 245 ± 106

0.5 mA 0.94587± 0.00258


Detector Beam inten- keff c.r. extracted keff c.r. inserted ∆keff (p.c.m.)position sity (mA) BT PNS BT PNS BT PNS

EC2B 1.0 mA 0.94605± 0.00049

0.94523± 0.00043

0.94288± 0.00068

0.94123± 0.00067

317 ± 83 401 ± 80

EC5T 1.0 mA 0.94599± 0.00080

0.94354± 0.00055

0.94353± 0.00099

0.94032± 0.00073

246 ± 127 322 ± 92

Table 11.3: Source-jerk raw parameter (ξ n0−n1

n1) results for SC3a and SC3b congurations, compared

with the results of the area-ratio raw parameter (ApAd ) results of the PNS experiments


Detector Beam inten- Control rods extracted Control rods insertedposition sity (mA) Source-jerk Area-ratio (PNS) Source-jerk Area-ratio (PNS)

1.2 mA 15.17 ± 0.03 17.99 ± 0.03EC1B 1.0 mA 15.63 ± 0.03 15.31 ± 0.03 18.26 ± 0.04 17.64 ± 0.04

0.5 mA 16.12 ± 0.04 18.65 ± 0.051.2 mA 13.75 ± 0.02 15.37 ± 0.03

EC2B 1.0 mA 14.02 ± 0.03 15.66 ± 0.04 15.63 ± 0.030.5 mA 14.61 ± 0.04 16.31 ± 0.041.2 mA 8.89 ± 0.06 9.68 ± 0.07

EC5T 1.0 mA 9.01 ± 0.08 8.70 ± 0.06 9.64 ± 0.08 9.44 ± 0.040.5 mA 9.07 ± 0.09 9.99 ± 0.09


Detector Beam inten- Control rods extracted Control rods insertedposition sity (mA) Source-jerk Area-ratio (PNS) Source-jerk Area-ratio (PNS)EC2B 1.0 mA 13.84 ± 0.02 13.92 ± 0.02 15.45 ± 0.02 15.28 ± 0.03EC5T 1.0 mA 9.39 ± 0.04 9.26 ± 0.04 10.22 ± 0.04 10.07 ± 0.05

11.1. STEADY-STATE REACTIVITY MONITORING 171

Table 11.4: keff estimations from the source-jerk method for SC3a and SC3b congurations, comparedwith the area-ratio results of the PNS experiments.


Detector Beam inten- keff c.r. extracted keff c.r. inserted ∆keff (p.c.m.)position sity (mA) SJ PNS SJ PNS SJ PNS

1.2 mA 0.94710±0.00041

0.93961±0.00183

749 ± 188

EC1B 1.0 mA 0.94587±0.00063

0.94674±0.00056

0.93890±0.00197

0.94053±0.00199

697 ± 207 621 ± 172

0.5 mA 0.94457±0.00087

0.93786±0.00217

671 ±234

1.2 mA 0.94713±0.00044

0.94225±0.00149

488 ± 155

EC2B 1.0 mA 0.94631±0.00060

0.94139±0.00168

0.94147±0.00199

492 ± 179

0.5 mA 0.94452±0.00099

0.93944±0.00212

508 ± 234

1.2 mA 0.94422±0.00072

0.93964±0.00111

459 ± 132

EC5T 1.0 mA 0.94353±0.00082

0.94534±0.00078

0.93991±0.00112

0.94104±0.00112

362 ±138 430 ± 114

0.5 mA 0.94319±0.00089

0.93789±0.00129

530 ± 158


Detector Beam inten- keff c.r. extracted keff c.r. inserted ∆keff (p.c.m.)position sity (mA) SJ PNS SJ PNS SJ PNS

EC2B 1.0 mA 0.94930±0.00032

0.94908±0.00035

0.94474±0.00083

0.94521±0.00088

456 ± 89 386 ± 80

EC5T 1.0 mA 0.94160±0.00083

0.94236±0.00093

0.93671±0.00127

0.93759±0.00146

489 ± 152 477 ± 143

Figure 11.2: Distribution of source-jerk results every beam trip for a given experiment (SC3a congurationwith the control rods extracted; EC2B position). The t to a lognormal distribution is plotted alongside.


Table 11.5: Estimates of the keff changes due to control rod movement (p.c.m.) observed by dierenttechniques (SC3b conguration; detector position EC2B (booster zone)).

Control rod Control rodextraction insertion

Current-to-ux 194 ± 8 -194 ± 8Source-jerk 416 ± 81 -399 ± 80

Prompt decay constant 808 ± 182 -492 ± 182MCNPX (ENDF/B-VII.0) 307 ± 16

instead of a source pulse, and therefore, it should be equally aected by spatial eects. For this reason, thesame correction methods should be applicable; in particular, it will applied the generalized methodologyapplied in section 10.3 to the PNS area-ratio results. The results of the application of this method arepresented in table 11.4. It can be observed that after this correction the dispersion in the keff results isconsiderably reduced.

The keff values obtained are largely compatible with the PNS area-ratio method. Still, the resultsobtained with the source-jerk technique show systematically a larger variation between detector positionsthan the results of the prompt decay constant method.

The dierence in reactivity between the results with the control rods inserted and extracted for acertain detector (about 400-500 p.c.m in most cases) is somewhat larger than the expected value from theMCNPX simulations (300 p.c.m) and from the prompt decay constant method. However, the uncertaintyin the results is too large for being conclusive (considering both statistical errors and uncertainties in β∗).

It is interesting as well to remark that a dependence of the reactivity with the source intensity can benoticed, particularly in the detector at the EC1B position, which was the one with the highest countingrate. Although this dependence is rather small (less than one dollar), it is always in the same sense,showing lower values of the reactivity as the source intensity increases, making it unlikely to be onlystatistical uctuations. Several explanations for this behavior are being evaluated. In principle, themost likely explanation seems to be an incomplete correction of the dead time eects, but there areother possible causes, such as spectral eects due to dierent relevance of the D-D reactions with thebeam intensity (due to dierent distribution of the tritium or the implanted deuterons in the target, forinstance). The lack of information on the neutron generator target components does not allow to conrmor reject any of these explanations, but it can be worth mentioning it in order to assist in the analysis offuture experiments.

11.2 Reactivity monitoring during system perturbations

11.2.1 Fast variation of the system reactivity

To determine the capability to measure the reactivity and detect changes during a fast variation in thereactivity of the system, a series of experiments have been performed in the SC3b conguration insertingand extracting the control rods. During these experiments, a beam trip per second was still produced.Hence, it was possible to apply both the current-to-ux and the beam trip techniques (source-jerk andprompt decay constant) to determine the reactivity during control rods movement. For the cases of thesource-jerk and the prompt decay constant methods, the same generalized methodologies used in section11.1 have been applied to translate the experimental results into criticality constant and the uncertaintiesgiven include statistical as well as an estimate of the error due to systematic eects. The current-to-uxtechnique, for its part, has been applied using the approximate equation 7.8 (i.e., considering that thefactor εDϕ

∗

εSdoes not change as a result of the change of conguration caused by the movement of the

control rods) and using as reference keff for calibration the source-jerk results before the control rodinsertion.

The results are shown in Figures 11.3 and 11.4. With the current-to-ux and the beam trip techniquesthe reactivity has been monitored with a resolution of 1 second, corresponding with the beam tripfrequency, while with the prompt decay constant method data of ve consecutive beam trips have beenaccumulated to reduce statistical uctuations. All three techniques allow detecting the change of keffbefore and after the control rods movement and allow having estimates of ∆keff (table 11.5) compatiblewith the values measured in steady state (tables 11.2 and 11.4) and of the order of magnitude of theMCNPX estimate.

11.2. REACTIVITY MONITORING DURING SYSTEM PERTURBATIONS 173

(a) Current-to-ux, control rod extraction (b) Current-to-ux, control rod insertion

(c) Source-jerk, control rod extraction (d) Source-jerk, control rod insertion

(e) Prompt neutron decay constant, control rod extraction (f) Prompt neutron decay constant, control rod insertion

Figure 11.3: Reactivity values measured using the current-to-ux, source-jerk and prompt neutron decayconstant techniques during the fast movement of the control rods in SC3b conguration. The detector islocated in the EC2B position in the booster region.


(a) Current-to-ux, control rod extraction (b) Current-to-ux, control rod insertion

(c) Source-jerk, control rod extraction (d) Source-jerk, control rod insertion

(e) Prompt neutron decay constant, control rod extraction (f) Prompt neutron decay constant, control rod insertion

Figure 11.4: Reactivity values measured using the current-to-ux, source-jerk and prompt neutron decayconstant techniques during the fast movement of the control rods in SC3b conguration. The detector islocated in the EC6T position in the booster region.


The current-to-ux technique is less aected by statistical uctuations than the other two techniques,which allows higher monitoring frequencies. This is due to the large statistics that can be accumulatedwith both the ux and source monitoring detectors. In the present case of Yalina-Booster, the current-to-ux technique allows following the control rods during their movement with good statistical uncertainty,as it is evident in gures 11.3a and 11.3b. In a power ADS with larger statistics higher monitoringfrequencies may be possible. The measured keff change between the two control rods positions has beenfound to be about 200 p.c.m., lower than the MCNPX result (300 p.c.m) and the changes measured withthe source-jerk and the prompt decay constant technique (table 11.5). It must be taken into account thatin addition to the statistical errors, systematic errors due to local eects are also present. These localeects have been neglected by assuming that the factor εDϕ

∗

εSremains constant during the control rod

movement.Actually, it is possible to estimate the variation of the factor εDϕ

∗

εSwith the control rod position using

MCNPX. If we consider that εS does not depend on the control rod position (which is normally true),equation 7.7 becomes:

∆keff = (1− keff,1)

(1− εD2ϕ

∗2

εD1ϕ∗1

M1/R1

M2/R2

)(11.2)

The factor εD2ϕ∗2

εD1ϕ∗1can be determined with MCNPX using equation 7.6. If MMCNPX is the number of

neutron counts in the detector per source neutron, as calculated by MCNPX, then:(εD2ϕ

∗2

εD1ϕ∗1

)MCNPX

=(1− keff,2)

(1− keff,1)

MMCNPX2

MMCNPX1

(11.3)

Using equation 11.2 it has been observed that the variation in ∆keff measured with the current-to-ux method increases in 60 p.c.m, which is enough to explain most of the dierence with the expectedvalue of about 300 p.c.m.

The source-jerk technique also allows obtaining an estimate of the reactivity every trip, although thelow statistics in the determination of the delayed neutron level n1 causes larger uncertainties than thecurrent-to-ux technique. Another important remark is that the source-jerk technique requires some timefor the delayed neutrons level to stabilize after a reactivity change. From the gure it can be observedthat while the control rod extraction takes about 6 seconds to complete, the source jerk estimation ofkeff takes more than 50 seconds to fully adapt to the new value, due to the time required by delayedneutrons to stabilize to their new level. The continuous line represents the theoretical evolution of thesource-jerk results obtained solving the point kinetics equation, where the initial and nal value of thekeff have been xed to the experimental values. The experimental data follow closely the theoreticalmodel. Hence, it is necessary to stress that the source-jerk technique will underestimate the keff forsome time after a keff increase and it will overestimate it for some time after a decrease, which has tobe taken into account for safety analysis.

Finally, the prompt decay constant technique has much larger statistical errors than the source jerktechnique, even averaged over 5 seconds time intervals, and it is dicult to distinguish the eect of thecontrol rod position due to statistical uctuations. This limits the maximum frequency of reactivitymonitoring, although this may not be an issue in a power ADS, where much larger statistics will beavailable. However, as it is based on prompt neutrons only, the prompt decay constant method shouldnot be aected by the slow adaptation to a new value of keff after a reactivity change as it was the caseof the source-jerk technique and therefore it should be capable of providing a fast monitoring capacity.Nevertheless, the statistics of the Yalina-Booster experiments do not permit to conrm it.

11.2.2 Fast variation of the neutron source

In addition to the fast movement of the control rods, during the Yalina-Booster experiments it was alsopossible to determine the capabilities of the reactivity monitoring techniques to measure the reactivityafter a long (several seconds) beam trip interruption. This is equivalent to a fast variation of the beampower.

In Figure 11.5 it is shown the reactivity monitoring during more than 1000 seconds of the Yalina-Booster reactor in the SC3a conguration. In the experiment, the beam was lost for approximately30 seconds and recovered afterward. It can be observed that both the current-to-ux and the promptneutron decay constant techniques (in the last, beam trips every 10 seconds have been accumulated toincrease statistics) continue providing the same estimation of the keff after the beam recovery followingthe interruption. On the other hand, as it was already explained in section 11.2.1, the delayed neutrons


require about 50s to stabilize after a modication of the reactor conditions or the power level and hencethe source-jerk method requires this time to provide correct values for the reactivity. This eect mustbe taken into account if it is intended to apply the source-jerk technique for monitoring the reactivity innon-steady conditions.

Conclusions

The aim of this PhD thesis was to investigate the feasibility of measuring the reactivity of a subcriticalreactor, with special focus on ADSs. For this purpose, reactivity monitoring techniques and methodologiesto correct spatial and energy eects have been validated against experimental data measured at the Yalina-Booster subcritical assembly of the JIPNR-Sosny (Minsk, Belarus). Furthermore, in addition to existingtechniques, a generalization of the correction technique based on correction factors is proposed, and alsovalidated against experimental results.

Generalized methodology

The generalized methodology consists in removing the linearity requirement between the reactivity andthe measured parameters, such as the area-ratio or the prompt neutron decay constant. This linearity isobtained in the point-kinetics model and is retained when traditional correction factors are used. Withthis generalized methodology, more general relationships of the shape ρ = ρ1 (α) and ρ = ρ2 (Ap/Ad) areconsidered. Quite obviously, these relationships will be specic for a given system and, due to spatial andenergy eects, they will be specic for a given position and detector type. Nevertheless, for reactivitymonitoring purposes, it is only required that they are univocal for a given detector type at a given positionwithin the system, i.e., the same value of ρ it is obtained for dierent congurations with the same valueof α or Ap/Ad.

Although it is not reasonably to expect to nd such univocal relationships that hold for all possiblerange of subcriticalities and congurations of the subcritcal assembly in study, it is reasonably to hopeto nd relationships that hold for a range of perturbations made around a given conguration. In fact,this is the way how these relationships can be determined with the help of neutron transport codes, bymaking perturbations of certain parameters of the system (geometry, materials, nuclear data) around agiven conguration.

The main advantage of this generalized methodology with respect to the use of traditional correctionfactors, apart from removing the linearity requirement, is that, as several congurations of the systemare considered, it also provides a way to estimate the systematic uncertainties in the reactivity results asa consequence of the uncertainties in the modelization of the system (geometry, cross sections, materialproperties), either because a poor determination, or because they change during reactor operation. Thisis not possible with traditional correction factors where a single simulation of the system in a singleconguration is considered. Furthermore, from the limits of the univocity of the relationships found, itis also possible to determine the validity range of the relationship itself.

Finally, it must be remarked that the basic hypothesis of this method is to nd one or more param-eters of the subcritical system that present a univocal dependence with the reactivity independently ofthe conguration, but this parameter need not necessarily be α or Ap/Ad, which greatly expands theapplicability of the method. Once such a parameter is determined, a neutron transport code can be usedto characterize its dependence with the reactivity.

Correction of dead time

The experimental data measured at the Yalina-Booster subcritical assembly have been used to validate thegeneralized methodology mentioned above, alogside with the correction techniques based on traditionalcorrection factors. A problem that has been encountered when analyzing the raw data is the need ofcorrecting dead time eects, that are relevant because of the high counting rates measured during theexperiments. It must be remarked that these eects are dicult to avoid for reactivity measurementpurposes, since the techniques involved require measuring neutron uxes that may extend over a range

179

180 CONCLUSIONS

of several orders of magnitude. This causes that while in the upper part of this range dead time eectsmay appear, in the lower part of the range counting statistics may be still too poor. In the case ofthe experiments presented here, the determination of the dead time has been accomplished making themost of the ability of the data acquisition system to register the times of the individual detector signals,and therefore, the ability to determine the distribution of time intervals between consecutive counts. Inprinciple, this technique can allow as well for the determination of type of dead time (paralyzable or non-paralyzable), even in the case of complex arrangements of dead times, but this has been found not to bethe possible for the case of the experimental results of Yalina-Booster. In order to remove as well possiblespurious counts due to the presence of ringing eects, a second dead time has been imposed via software,and experimental data have been corrected considering a model consisting of a series arrangement of twodead times: the (paralyzable) dead time due to the detectors and the second (non-paralyzable) dead timeimposed via software.

PNS experiments

As stated above, two types of experiments to apply reactivity determination techniques are analyzed inthis work: pulsed neutron source (PNS) experiments and beam-trip experiments.

Upon simple inspection of PNS results, some observations can be made. First, PNS experiments inYalina-Booster have shown that the neutron ux in the core reaches an asymptotic state driven by theneutron ux in the thermal zone a certain time after a neutron pulse is injected in the system. This factis inferred because the counting rate in all the detectors in the booster and in the thermal zone decayexponentially with the same exponential decay constant a certain time after the pulse injection. Thisnding has important consequences since this implies that the neutron ux can be factorized into anamplitude and a shape factor, and therefore it can be expected that the point kinetics model works wellin the core. In the reector, however, low decaying modes appear and this asymptotic state is not reachedany more. The eect of having removed the inner booster between the SC3a and SC3b congurationswas reected the counting rate in the very few instants after the neutron pulse injection, that was nearlytwofold larger in SC3a than in SC3b. Otherwise, no other eect has been found. The small variationof reactivity caused by the insertion and extraction of the control rods was also clearly noticeable as avariation of the decay slope in the dierent detectors.

Kinetic parameters calculation

Two basic reactivity determination techniques have been applied to the PNS experiments: the promptdecay constant method and the area ratio (or Sjöstrand) method. Both these techniques require theknowledge of the kinetic parameters of the system to provide absolute values of the reactivity (or, equiva-lently, the keff ) of the system. The prompt decay constant method requires the knowledge of the eectivedelayed neutron fraction βeff and the mean neutron generation time Λeff , while the area-ratio techniqueonly requires the knowledge of βeff . As they were no experimental measurements for these parame-ters, they have been obtained through Monte Carlo simulations with the MCNPX code. Since these areadjoint-weighted parameters, its calculation with Monte Carlo codes is not straightforward. The valueof Λeff has been obtained using an unique method based on perturbation techniques, but for the caseof βeff , several proposed techniques have been analyzed and benchmarked. These techniques are twoversions of the so called prompt method, both with the prompt eigenfunction and the total eigenfunction,and the techniques based on the interpretation of the next ssion probability (N.F.P.) and the integratedssion probability (I.F.P.) as the adjoint weighting.

Results of the benchmark show that the prompt method with the prompt eigenfunction and thetechnique based on I.F.P. provide the most accurate values for βeff in spite of its weak theoreticaljustication (it is necessary to explain how adjoint weighting is approximated with it). The usage of thetotal eigenfunction instead of the prompt eigenfunction has also been investigated and has been found toprovide values for β0 instead of βeff . The technique based on I.F.P. provides similar values for βeff , butit has been noticed that as the number of calculation (KCODE) cycles becomes larger, the uncertaintiesin the determination of keff and kdi cause the uncertainty in the βeff results obtained with this methodto increase. The technique based on N.F.P. can be considered the rst cycle of the technique basedon I.F.P. and has been found to work reasonably well for most of the systems considered but it causesnoticeable systematic errors in reected heterogeneous systems.

CONCLUSIONS 181

Direct application of the area-ratio and prompt decay constantmethods

For the case of the area ratio method, it has been found that, in the case of Yalina-Booster, strong spatialeects are present that cause large dierences (up to a factor of two) between the reactivity estimationsobtained with detectors in dierent positions, being especially relevant for those closest to the neutronsource. Thus, in order for the area ratio method to be useful for determining the reactivity of the system,a correction method is needed in order to get rid of these spatial eects.

In the case of the prompt decay constant method, on the other hand, the presence of a single exponen-tial decay mode in the core of Yalina-Booster allows the direct application of the prompt decay constantmethod, with little inuence of spatial eects, and hence providing very similar results for the dierentdetectors positions in the booster and in the thermal zone, on condition that the limits to t the decayslopes to exponentials are properly chosen in order to avoid the eects of the higher order modes thatappear in the initial instants of time after the pulse and decay very fast. Furthermore, as the interval forthe t excludes the initial instants after the neutron pulse where the counting rates are the highest, theprompt decay constant method is little aected by dead time eects. The applicability of the promptdecay constant method in the reector, however, is limited by the appearance of slower decaying modes,and again corrections are required.

Correction factors

The rst correction method that has been applied to both the area-ratio and the prompt decay constanttechnique is the use of traditional correction factors. This correction methodology consists in allowingfor more general relationships between the reactivity and either the prompt decay slope or the the promptto delayed area ratio, but keeping the linearity of the equations, that is, considering equations of the shapeρ = aα+ b and ρ = c (Ap/Ad). The parameters of these equations are determined from the knowledge ofpairs of values (ρ, α) or (ρ,Ap/Ad). More precisely, the determination of the parameters a and b for theprompt decay constant method requires the knowledge of a couple of values (ρ, α), but, to simplify, ithas been considered b = βeff for similitude with the point kinetics model and hence only a single point(ρ, α) is also required for the prompt decay constant method. The Monte Carlo code MCNPX has beenused to obtain these pairs of values. The results of the application of this correction technique to theexperimental results show that, in the case of the area ratio technique, the relative dispersion of the keffestimates obtained the dierent channels is largely reduced and compatible results are obtained.

For the case of the prompt decay constant technique, the values obtained were compatible with thoseobtained with the direct application of the point kinetics and it can be also applied in the reector.Furthermore, it must be also remarked that, even if the reactivity determination by both the area ratioand the prompt decay constant methods may not give accurate enough estimates for the reactivity, it hasbeen found that they are still able to detect small variations of the reactivity, as it was the case whenthe control rods were inserted or extracted (∼ 300 p.c.m.), that is very clearly observed in the resultsobtained with both methods. This is an interesting conclusion for a power ADS, as it can allow to designa reactor protection system that shuts down the reactor in case of a reactivity jump larger than a certainmagnitude is observed.

Application of the generalized methodology

Regarding the generalized version of the correction factor methodology, the existence and univocityof relationships of the type ρ = ρ1 (α) and ρ = ρ2 (Ap/Ad) for the case of Yalina-Booster have beeninvestigated with the Monte Carlo code MCNPX. As stated above, this has been done by performing smallvariations in the geometry and composition of the system (polyethylene density, uranium enrichment,height to width ratio). In this way, univocal relationships have been found to exist over a range ofreactivities around the base case of ∆ρ =∼ 1000 pcm. Furthermore, within this range, these relationshipshave been found to be largely linear, and thus relationships of the shape ρexp = ρ0 + Λ∗ (∆α)exp and

ρexp = ρ0 + β∗(

∆ApAd

)exp

have been considered, where ρ0 denotes the reactivity of the reference non-

perturbed system conguration and the parameters Λ∗ and β∗ are obtained by the linear tting of thepairs (∆α,∆ρ) and

(∆ApAd,∆ρ

)obtained from the simulations.

182 CONCLUSIONS

The keff estimates thus obtained have been found to be very similar to the results obtained with theprompt decay constant method and the area ratio method using traditional correction factors. Never-theless, the generalized prompt decay constant method shows a trend to slightly underestimate (in theorder of 100 p.c.m.) the results of keff obtained with the previous versions of the prompt decay constantmethod. Furthermore, the systematic uncertainty in the values of keff has also been evaluated by con-sidering the values obtained when perturbing system parameters (polyethylene density, fuel enrichment,height to width ratio) individually. The largest dierence with the average value has been chosen as anestimator of the systematic uncertainty. It has been found that the the value of this systematic uncer-tainty is in the order of 100 p.c.m, of the same order than the statistical uncertainty obtained with thetraditional methods. Therefore, both statistical and systematic uncertainties contribute similarly to thetotal uncertainty in keff .

Beam trip experiments

The other type of experiments that were carried out at the Yalina-Booster subcritical facility in orderto validate reactivity measurement techniques used a continuous deuteron beam with short beam trips.Two dierent situations have been analyzed: when the system is in an steady state and during systemperturbations.

Three dierent techniques for reactivity determination have been applied: the current-to-ux tech-nique, the prompt decay constant technique and the source-jerk technique. The current-to-ux techniqueis the most likely to be used for continuous reactivity monitoring in a large-scale ADS since it only re-quires the system operating in a steady state. However, this technique can not provide absolute reactivitymeasurements and it is largely dependent on having a stable external neutron source. Furthermore, tomeasure the reactivity dierence between two dierent congurations, it requires that either the param-eter εDϕ

∗

εSis conserved between the two congurations or that it is possible to determine its variation

between these two congurations (e.g. through detailed Monte Carlo simulations). Hence, this tech-nique has been applied only to monitor the change in the reactivity during system perturbations, morespecically, during the fast movement of the control rods and fast variations of the neutron source. Thecurrent-to-ux technique has been found to be able to precisely track the control rod movement (esti-mated with MCNP to be ∼ 300 p.c.m.) within one-second intervals and should work even at higherrates.

The other two techniques, namely the prompt decay constant technique and the source-jerk technique,are both capable to provide absolute reactivity measurements and hence they are able to complementthe relative reactivity measurements with the current-to-ux technique. Hence, in addition to monitorthe reactivity during system perturbations, they have been applied to obtain absolute reactivity valueswith the system in a steady state.

The prompt decay constant technique and the source-jerk technique are based in analyzing the systemresponse to short beam interruptions (beam trips or induced interruptions). Both techniques are basedon the point kinetic model and are, respectively, equivalent to the prompt neutron decay constant andthe area ratio and techniques applied in PNS experiments. Because both techniques are based on thepoint kinetic model, they will be aected by spatial and spectral eects in a practical system. The samecorrection techniques based on detailed Monte Carlo simulations already validated in PNS measurementshave been used to take into account these eects. Once corrected, both the prompt decay constant andthe source-jerk techniques provide largely compatible results for the reactivity and compatible with thevalues obtained with the PNS experiments (the dierence for a given conguration is always below 1$ or∼ 700 p.c.m.).

In comparing the source-jerk and the prompt decay constant techniques, it has been found that forthe neutron ux of level of Yalina-Booster the source-jerk technique provides more precise data than theprompt-decay constant technique and thus allows monitoring the reactivity during fast variations of thereactivity or the assembly power. In the case of Yalina-Booster, the source-jerk method is precise enoughto monitor the reactivity every second. However, for power ADS with higher ux levels both the source-jerk and the prompt decay constant techniques could reach similar levels of precision. The drawback of thesource-jerk technique is that it requires some time for delayed neutrons to stabilize after fast variations ofthe reactivity or the assembly power and hence it requires some time to provide correct reactivity valuesafter a system perturbation while the current-to-ux and the prompt neutron decay constant methodcan be applied during and immediately after the perturbation. Hence, it is recommended to use both thesource-jerk and the prompt decay constant methods for the interim calibration of the reactivity, and usethe current-to-ux technique for the continuous monitoring of the reactivity.

CONCLUSIONS 183

Future work

Future work will be centered in analyzing the performance of the reactivity monitoring techniques tothe experimental data of the current FREYA project currently ongoing at the VENUS-F reactor of theBelgian SCK-CEN. This system is very dierent to Yalina-Booster as it is a fast reactor lacking anythermal zone, which represents an important advantage in order to extend the validation range of thereactivity monitoring techniques. Experimental data from previous experiments (MUSE, RACE) mayalso be reanalyzed in light of the ndings presented in this thesis.

At present, the generalized methodology presented in this thesis is being applied to the experimen-tal PNS results of the FREYA experiment [46]. As it has been done for Yalina-Booster, generalizedrelationships between the area-ratio and the reactivity are being searched varying a set of parameters,over a range down to 0.85. Preliminary results have found that it is possible to accurately describe therelationship between the area-ratio and the reactivity by means of two linear relationships, providingfurther validation of the method.

Work will continue with the analysis of the applicability of the generalized techniques to the resultsof the beam trip experiments. Finally, it is expected to pursue the validation eorts as new experimentaldata from new core congurations measured within this experiment are available.

Finally, in a longer term, it is expected that the generalized methodology will be applicable to theMYRRHA project and other ADS designs that are presently under consideration (e.g. EFIT). In particu-lar, the ability of the Monte Carlo code MCNP to reproduce the area-ratio and prompt decay techniquesresults, as it is shown in this thesis, will made possible to simulate the application of the reactivitymonitoring techniques during reactor transients. We expect that the generalized method proposed inthis thesis will be specially useful for this purpose, as it is potentially capable to provide a single (or afew number of) relationship between the reactivity and the measured parameters (area-ratio and promptdecay constant) that holds for the whole transient. This work can be done with the help of other toolsdeveloped at CIEMAT, in particular, the computer programs for coupling MCNP to fuel burnup andthermal-hydraulics codes [18, 364].

184 CONCLUSIONS

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