Accelerator Driven Subcritical Reactors

ACCELERATOR DRIVEN SUBCRITICAL REACTORS Related Titles Nuclear Dynamics in the Nucleonic Regime D Durand, E Suraud an...

Author: H Nifenecker | O Meplan | S David

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ACCELERATOR DRIVEN SUBCRITICAL REACTORS

Related Titles Nuclear Dynamics in the Nucleonic Regime D Durand, E Suraud and B Tamain Statistical Models for Nuclear Decay A J Cole Neutrons, Nuclei and Matter J Byrne Basic Ideas and Concepts in Nuclear Physics (2nd Edition) K Heyde Non-accelerator Particle Physics H V Klapdor-Kleingrothaus and A Staudt Nuclear Physics: Energy and Matter J M Pearson Nuclear Decay Modes D N Poenaru Nuclear Particles in Cancer Treatment J F Fowler Linear Accelerators for Radiation Therapy (2nd Edition) D Greene and P C Williams Nuclear Methods in Science and Technology Y M Tsipenyuk

SERIES IN FUNDAMENTAL AND APPLIED NUCLEAR PHYSICS

Series Editors R R Betts and W Greiner

ACCELERATOR DRIVEN SUBCRITICAL REACTORS H Nifenecker Laboratoire de Physique Subatomique et de Cosmologie, Grenoble, France

O Meplan Laboratoire de Physique Subatomique et de Cosmologie, Grenoble, France and

S David Institut de Physique Nucleáire, Orsay, France

INSTITUTE OF PHYSICS PUBLISHING BRISTOL AND PHILADELPHIA

# IOP Publishing Ltd 2003 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK). British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0743 9 Library of Congress Cataloging-in-Publication Data are available

Commissioning Editor: John Navas Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing: Nicola Newey and Verity Cooke Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 929, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset by Academic þ Technical, Bristol Printed in the UK by MPG Books Ltd, Bodmin, Cornwall

Contents

1 Introduction

1

2 The energy issue 2.1 World energy perspectives 2.1.1 Energy consumptions 2.1.2 Fossil reserves 2.1.3 Greenhouse effect 2.2 Renewable energies 2.2.1 Solar energy 2.2.2 Biomass 2.2.3 Wind energy 2.2.4 Hydroelectricity 2.3 Nuclear energy 2.3.1 Standard reactors 2.3.2 Breeder reactors 2.3.3 Nuclear waste disposal options 2.3.4 Deployment of a breeder park 2.4 Costs 2.5 The possible role of accelerator driven subcritical reactors 2.5.1 Safety advantages of subcriticality 2.5.2 Use of additional neutrons

4 4 4 4 6 17 17 18 19 19 20 20 23 24 31 35 36 37 38

3 Elementary reactor theory 3.1 Interaction of neutrons with nuclei 3.1.1 Elementary processes 3.1.2 Properties of heavy nuclei 3.1.3 Neutron density, flux and reaction rates 3.2 Neutron propagation 3.2.1 Boltzmann equation 3.2.2 Integral form of the Boltzmann equation 3.2.3 Fick’s law

39 39 39 40 42 45 46 47 47 v

vi

Contents

3.3 3.4

3.5

3.6

3.7

3.2.4 Diffusion equation 3.2.5 Slowing down of neutrons Neutron multiplying assemblies Limiting values 3.4.1 Critical masses 3.4.2 Maximum flux Reactor control 3.5.1 Delayed neutrons 3.5.2 Temperature dependence of the reactivity 3.5.3 Critical trip 3.5.4 Residual heat extraction Fuel evolution 3.6.1 The Bateman equations 3.6.2 The long-term fuel evolutions Basics of waste transmutation 3.7.1 Radiotoxicities 3.7.2 Neutron balance for transmutation and incineration

4

ADSR principles 4.1 Properties of the multiplying medium 4.1.1 Energy gain 4.1.2 Neutron balance 4.1.3 Neutron importance

5

Practical simulation methods 5.1 Neutron reaction data files 5.2 Deterministic methods 5.3 Monte Carlo codes 5.3.1 Deterministic versus Monte Carlo simulation codes 5.3.2 MCNP, a well validated Monte Carlo code 5.4 Physics in MCNP 5.4.1 Precision and variance reduction 5.5 MCNP in practice 5.5.1 Introduction 5.5.2 Units 5.5.3 Input file structure 5.6 Examples 5.6.1 Reactivity calculation 5.6.2 Homogeneous versus heterogeneous cores 5.6.3 Subcritical core 5.6.4 Precision 5.7 Fuel evolution 5.7.1 Evolution constraint 5.7.2 Spatial flux

49 53 60 62 63 66 68 68 73 76 78 81 82 82 87 87 88 93 93 94 94 97 99 99 103 104 104 105 105 110 111 111 111 111 122 122 123 126 132 133 134 134

Contents 5.7.3 5.7.4

vii

Special cross-section data Time step between two MCNPs

134 135

6 The neutron source 6.1 Interaction of protons with matter 6.1.1 Electronic energy losses 6.1.2 Nuclear stopping 6.1.3 The nuclear cascade 6.1.4 Experimental tests of the INC models 6.1.5 The neutron source 6.1.6 State of the art of the simulation codes 6.2 Alternative primary neutron production 6.2.1 Deuteron induced neutron production 6.2.2 Muon catalysed fusion 6.2.3 Electron induced neutron production 6.3 Experimental determination of the energy gain 6.4 Two-stage neutron multipliers 6.5 High-intensity accelerators 6.5.1 State of the art of high-intensity accelerators 6.5.2 Requirements for ADSR accelerators 6.5.3 Perspectives for high-intensity accelerators for ADSRs 6.5.4 Examples of high-intensity accelerator concepts

138 138 138 139 140 142 148 154 155 155 158 159 160 161 164 165 166

7 ADSR kinetics

171

8. Reactivity evolutions 8.1 Long-term evolutions 8.2 Short-term reactivity excursions 8.2.1 Protactinium effect 8.2.2 Xenon effect 8.2.3 Temperature effect 8.2.4 Impact of reactivity excursions

177 177 177 179 181 183 184

9. Fuel reprocessing techniques 9.1 Basics of reprocessing 9.2 Wet processes 9.2.1 The purex process 9.3 Dry processes 9.3.1 Vaporization 9.3.2 Gas purge 9.3.3 Liquid–liquid extraction 9.3.4 Selective precipitation 9.3.5 Electrolysis

185 185 188 188 199 200 201 201 204 204

168 170

viii

Contents

10 Generic properties of ADSRs 10.1 The homogeneous spherical reactor 10.1.1 General solution of the diffusion equation 10.1.2 The three-zone reactor 10.1.3 Model calculations 10.2 Parametric study of heterogeneous systems

209 209 210 210 211 213

11 Roˆle of hybrid reactors in fuel cycles 11.1 The thorium–uranium cycle 11.1.1 Radiotoxicity 11.1.2 Breeding rates 11.1.3 Doubling time 11.1.4 Transition towards a 232 Th-based fuel from the PWR spent fuel, using a fast spectrum and solid fuel 11.1.5 Thorium cycle with thermal spectrum 11.2 Incineration 11.2.1 Plutonium incineration 11.2.2 Minor actinide incineration 11.2.3 Initial reactivity of MA fuels 11.2.4 Fuel evolution 11.2.5 Solid versus liquid fuels 11.2.6 The paradox of minor actinide fuels

215 215 215 217 219 222 225 229 229 231 232 234 238 239

12 Ground laying proposals 12.1 Solid fuel reactors 12.1.1 Lead cooled ADSR: the Rubbia proposal 12.2 Molten salt reactors 12.2.1 The Bowman proposal 12.2.2 The TIER concept 12.3 Cost estimates

242 242 242 246 246 247 249

13 Scenarios for the development of ADSRs 13.1 Experiments 13.1.1 The FEAT experiment 13.1.2 The MUSE experiment 13.2 Demonstrators 13.2.1 Japan 13.2.2 United States 13.2.3 Europe

252 253 253 253 255 255 255 256

Appendix I Deep underground disposal of nuclear waste I.1 Model of an underground disposal site I.1.2 Radioelement diffusion in geological layers I.1.3 Physical model of diffusion in the clay layer

263 263 264 265

Contents I.1.4

I.2

I.3

I.4

I.5 I.6 I.7

Simplified solution of the diffusion problem through the clay layer I.1.5 Solubility as a limiting factor of the flow of radioactive nuclei Determining the dose to the population I.2.1 Some dose determination examples I.2.2 Full computation example of the dose at the outlet Accidental intrusion I.3.1 Drilled samples I.3.2 Using the well to draw drinking water Heat production and sizing of the storage site I.4.1 Schematic determination of the temperature distribution I.4.2 Examples Geological hazard An underground laboratory. What for? Conclusion

ix 266 267 267 268 269 271 272 272 274 274 275 276 276 277

Appendix II The Chernobyl accident and the RMBK reactors II.1 The RBMK-1000 reactor II.2 Events leading to the accident II.3 The accident

279 279 281 283

Appendix III Basics of accelerator physics III.1 Linear accelerators III.1.1 The Wideroë linear accelerator III.1.2 The Alvarez or drift tube linac (DTL) III.1.3 Phase stability III.1.4 Beam focusing III.1.5 The radio frequency quadrupole (RFQ) III.2 Cyclotrons III.3 Superconductive solutions III.4 Space charge limitations

284 285 285 287 293 294 300 300 301 302

Bibliography

305

Index

313

Chapter 1 Introduction

Although, at the beginning of nuclear energy deployment, many different reactor systems were proposed and more or less tested, only a few of them became standard in the Western hemisphere: light-water reactors like the PWR (pressurized water reactor) or BWR (boiling water reactor), heavywater reactors of the CANDU (Canadian deuterium uranium) type and, finally, the sodium cooled fast-neutron reactors (LMR: liquid metal reactor). It was clear from the beginning that breeder reactors like the LMR were necessary for a sustainable development of nuclear power. However, the experience with the LMR is far from conclusive, especially since the universally used sodium coolant seems to lead to too strong safety constraints on the building and operation of the reactors. Furthermore, light-water reactors are plagued by the problem of radioactive waste which, to date, has not found a consensual solution. The waste problem added to the fear of catastrophic accidents (Chernobyl syndrome) explains the large societal opposition to nuclear power observed in many countries. To some extent one may think that nuclear energy will come to a dead end. This is an unfortunate situation when the rising concern about the greenhouse effect pleads for a severe reduction in the use of fossil fuels. The development of safer and less polluting means of producing energy from nuclear fission is, thus, of great relevance. In this context, in recent years, a great deal of interest has been displayed, worldwide, in accelerator driven subcritical nuclear reactors (ADSR or ADS), also called subcritical or hybrid reactors, to produce energy and transmute radioactive wastes in a, possibly, cleaner and safer way than at present. Pioneers in this revival have been Furukawa [1], Bowman [2] and Rubbia [3]. Similar ideas were first proposed more than 50 years ago [6–16]. At that time they were not carried through, not so much because of technical difficulties but for lack of economic incentive. It is true, also, that while building reliable GeV accelerators achieving intensities of several tens of milliamperes was by no means considered to be a trivial matter, efficient critical reactors were available, and were thus thought to be the simplest and most 1

2

Introduction

natural way to harness nuclear fission. It was, however, acknowledged that accelerator driven nuclear devices might offer interesting transmutation possibilities. For example, without a large enough concentration of 235 U in natural uranium, the only way to exploit fission energy would have been the use of such subcritical systems. High-energy accelerators appear to be a promising way to incinerate heavy actinides [17, 18]. It has also been acknowledged that a thoriumbased fuel cycle would considerably limit the amount of transuranic wastes produced. The implementation of such a cycle would be made easier with subcritical reactors, due to the improved neutron economy of such systems as compared with classical critical reactors [1–3]. Hybrid reactors appear to be a credible alternative to fast breeder and fusion reactors. In chapter 2 we examine the general question of world energy needs and possible supplies. In chapter 3 we give an elementary reminder of reactor theory. Chapter 5 is devoted to the description of commonly used reactor simulation codes. In chapter 6 we describe the basic physics of hybrid reactors, including an account of the experimental and computational state of the arts of spallation reactions. In chapter 7 we discuss safety questions specific to ADSRs. In chapter 8 we examine the evolution of hybrid reactor fuel as a function of time. This evolution determines the most important constraints on the neutronics of hybrid reactors. Chapter 9 describes the main fuel reprocessing techniques, since these condition the possibility of implementing a durable nuclear energy production. Chapter 10 shows how the existence of a spatially confined neutron source influences the size of the reactor. Chapter 11 examines the possible use of hybrid reactors in the nuclear waste issue. Chapter 12 gives some examples of ‘historic’ projects which used either solid or liquid fuels. Chapter 13 discusses the present state of thoughts for the possible development of ADSRs. In this book we have, deliberately, chosen to place the emphasis on simple, intuitive, treatments of the different aspects of the physics of hybrid reactors, in order to provide the reader with the possibility of gaining physical insight into these systems. However, the reader should be aware that realistic calculations require the use of complex codes, most frequently of the Monte Carlo type. Nevertheless, it is our experience that starting with simple models helps the understanding of the results of the ‘real’ calculations, and provides intuition of the most fruitful ways to improve systems or discover new ones. As far as possible this review is self-contained and does not require a priori knowledge of the field. It elaborates on reviews of the subject previously published in Progress in Particle and Nuclear Physics [4] and in Nuclear Instruments and Methods [5].

Using any kind of energy source, accelerators allow ample production of neutrons which might be used for synthesis of fissile nuclei starting from the fertiles 238 U and 232 Th.

Acknowledgments

3

In many cases it can be seen that critical reactors could do as well as ADSRs. We hope that this book will give the reader the elements needed to form an opinion on the possible usefulness of ADSRs, as well as to have a good basis for starting working on these systems. More generally, students and practitioners in nuclear reactor technology might find useful the more recent developments presented here.

Acknowledgments We thank the ISN group, and especially Professor J M Loiseaux, Dr D Heuer, A Nuttin and F Perdu, for contributing to many ideas and much of the material presented here. Professor C Rubbia was the originator of our interest in the subject of hybrid reactors and the main inspiration for our thoughts on the subject. We enjoyed many fruitful interactions with Professors J P Schapira, M Salvatores, M E Brendan and C D Bowmann. Professors D Hilscher, L Tassan-Got, W Mittig, B Lott and S Leray were kind enough to provide us with originals of their figures. Mrs E Huffer was kind enough to read our manuscript carefully and correct the English. The patience and support of Hele`ne Me´plan and Marguerite Nifenecker are gratefully acknowledged.

Chapter 2 The energy issue

A discussion of the possible future of nuclear energy requires some information on the general question of world energy perspectives.

2.1 2.1.1

World energy perspectives Energy consumptions

The relative contribution of the main energy sources to world needs is shown in table 2.1 [19]. The table shows that nuclear energy provides only a modest part of the total amount produced. This is partly related to the fact that nuclear energy is only used for electricity production, which only represents about 30% of global energy needs. However, table 2.2 [20] shows that this is not the whole story. Clearly, a number of countries with nuclear capabilities like Germany, the US and the UK resort to nuclear energy on a modest scale, especially when compared with a country like France. 2.1.2

Fossil reserves

Proven fossil and nuclear reserves in 1997 were as shown in table 2.3 [21]. Note that if nuclear energy were to rise to 40% of the total energy production, the reserves, which are estimated presently at 5 million tons of natural uranium at market rates, would decrease to 20 years in the PWR case and to 1000 years in the breeding case. In this last case it would, however, become cost effective to exploit very low grade ores, such as oceanic uranium, so that reserves would be much larger. It would also be

The market rate is assumed to be less than $150/kg. The OECD organization estimates that the total of assured and plausible uranium underground exploitable reserves amounts to 17 million metric tons.

4

World energy perspectives

5

Table 2.1. Shares of different types of energy in world consumption (1998). Energy types Oil Coal Gas Hydro Traditional2 Nuclear Renewables

33.0 21.3 19.4 6.8 11.6 5.8 1.9

Total 1 2

Gtoe1

%

3.4 2.2 2.0 0.7 1.2 0.6 0.2

100

10.3

Ton oil equivalent. Traditional: essentially biomass which is mostly wood.

Table 2.2. Consumptions in Mtoe for selected countries and different forms of energy. Values of specific consumptions (rows 2–5) are given for 1996. Those for the total consumption are from 1997 (row 6) and are not necessarily equal to the sum of rows 1–4. 1

2 Coal

3 Oil

4 Gas

5 Nuclear

6 Total

7 Nuclear (%)

8 Toe per capita

Germany China France UK Japan Russia USA

88.9 666.0 14.7 44.9 88.3 126.5 516.0

137.4 144.1 91.0 83.7 268.7 162.7 806.8

75.2 14.9 29.0 76.7 54.3 335.0 335.0

39.8 3.1 97.3 23.0 67.3 25.3 173.6

347.3 1098.9 247.5 228.0 514.9 592.0 2162.2

11 0.2 39 10 13 4 8

4.23 0.90 4.22 3.86 4.08 4.02 8.10

Table 2.3. World energy reserves.

Energy type

Reserves (Gtoe)

Solid fuels Oil Gas Non-conventional oil Methane hydrates Nuclear PWR Nuclear breeding

1032 141 133 Several hundred Gtoe More than 1000 Gtoe 50 7000

Annual production (Gtoe)

Number of years at present production rate

2.32 3.47 2.00

219 41 64 ? ? 125 20 000

0.4

6

The energy issue

possible to use oceanic uranium, which amounts to nearly 3 billion tons, in non-breeding reactors provided a 50% cost increase of the electricity produced were acceptable [22]. One should take the above reserve estimates with some caution since exploration efforts seem to level off when the estimated reserves amount to about 40 to 50 years. For example, in the case of oil, current reserves have constantly increased since, at least, 1940. However, a recent careful study [23] shows evidence for a decrease of the real estimated reserves starting around 1980. This study predicts a decrease in oil production by 2010. Taking into account the large reserves of coal, unconventional gas and oil, it seems that fossil fuels could be available at an adequate level during the 21st century. The main limitation to their use will rather, probably, be related to the greenhouse effect. 2.1.3

Greenhouse effect

The International Panel on Climate Change (IPCC) reviews periodically the evidence for climate change and, depending on the rate of greenhouse gas emissions, makes predictions on future evolutions. Because the future of nuclear energy cannot be discussed without reference to these aspects, we summarize the main conclusions reached by the IPCC as well as by organizations such as the World Energy Council (WEC) and the International Institute for Applied Systems Analysis (IIASA) who focus on the prediction of future energy consumption. Evidence for anthropically induced climate change Figure 2.1 shows how the concentration of three important greenhouse gases in the atmosphere follows the world population increase. The correlation between the four curves is striking with an especially fast increase of greenhouse gas concentrations after 1950. Although the greenhouse efficiency of methane and other gases is much larger than that of CO2 , its much larger abundance gives it the dominant contribution. Typically, CO2 accounts for 62% of the additional radiative forcing due to anthropic emissions, methane for 20% and other gases for the rest. Water has an amplifying role but is not a driving factor. Although the increase of greenhouse gas concentrations is unquestionable, its effects on temperature are more difficult to establish. Figure 2.2 shows a clear rise of the temperature during the last century. However, this increase is not attributable solely to increased concentration of greenhouse gases. Climatic models indicate that only the temperature increase observed between 1960 and the present can be attributed to greenhouse gas emissions.


7

Figure 2.1. Evolution of the atmospheric concentration of three important greenhouse gases as a function of time. The evolution of the world population is also shown (from IPCC [24]).

Atmospheric kinetics of CO2 The present annual rate of anthropic emission of CO2 amounts to 6 Gtons of carbon. It appears that about 3 Gtons are reabsorbed into the ocean. Therefore about 3 Gtons contribute to increasing the CO2 concentration in the

8

The energy issue

Figure 2.2. Evolution of the world average surface temperature with time. Climatic models attribute the rise between 1910 and 1940 to natural causes, and that between 1960 and the present to radiative forcing by anthropic greenhouse gas emissions (from IPCC [24]).

atmosphere. Climatic models [25, 26] show that atmospheric concentration stabilization of CO2 can be stabilized only if anthropic emissions are reduced below 3 Gtons. Figures 2.3 and 2.4 illustrate this. Figure 2.3 shows that emission rates much below the present value are required to obtain a stabilization of the CO2 concentration. Examination of figure 2.4 shows that this stabilization will take a long time to be established. For example, in the S450 case (stabilization of the concentration at 450 ppmv) the stabilization occurs only after year 2075, although the emissions decrease as soon as 2020. For the S750 case the corresponding dates are 2200 and 2070 respectively. Energy policies Anthropic CO2 emission amounted, in 2000, to 24 Gigatons. The regional emissions of polluting agents and greenhouse gases are summarized in table 2.4 [27]. The comparison between emissions from different countries, apparent in table 2.5, is instructive and gives a measure of the effort which these countries will have to achieve in order to improve the situation [28]. CO2 emissions are given either in CO2 weight or in weight of the carbon included with the correspondence: weight(C) ¼ weight(CO2 Þ 12 44.


9

Figure 2.3. Evolution of anthropic CO2 emission annual rates for different scenarios. The curves are labelled by the asymptotic value reached shown in figure 2.4. Solid lines and dashed lines correspond to two emission profiles which lead to the same final concentration [25].

Figure 2.4. Evolution of CO2 concentrations corresponding to the emission profiles shown in figure 2.3 [25].

10

The energy issue Table 2.4. Emission rates of main polluting agents in different regions.

North America Latin America Western Europe Central Europe CIS Arab countries Black Africa South Asia Pacific Total

Carbon (Gtons)

Sulfur (Mtons)

Nitrogen (Mtons)

1.55 0.26 1 0.25 1.08 0.22 0.11 0.2 1.27 5.94

12.1 3.2 10.4 3.9 12.4 2.2 1.9 3.4 15.1 64.6

5.5 1.4 3.7 1.0 4.0 1.0 0.7 1.1 5.7 24.1

Countries showing CO2 emission rates of more than 10 tons per capita rely heavily on fossil fuels for electricity production. Other developed countries, usually, have large hydroelectric resources. This is the case of the Nordic countries and of Switzerland. France is characterized by extensive use of nuclear power, while Japan uses a relatively very small proportion of coal among the fossil fuels. The small emission of China, and other big countries like India, reflect their degree of development. Note that a large part of greenhouse gas emissions is due to transportation systems.

Table 2.5. Per capita emissions of main polluting agents for different countries.

Norway Switzerland Sweden Netherlands France Canada Poland UK Germany USA CIS Japan China n.a. ¼ not available

CO2 (tons per capita)

SO2 (kg per capita)

NO2 (kg per capita)

7.5 7.1 7.0 12.5 7.0 17.0 14.0 11.0 13.0 20.0 12.5 8.0 2.5

12 10 20 14 20 140 70 68 70 85 n.a. n.a. n.a.

55 28 45 37 30 70 30 49 40 78 n.a. n.a. n.a.


11

Table 2.6. Comparison of pollutant emission rates for different technologies. Emission for 1 kWh

CO2 (kg)

SO2 (g)

NO2 (g)

Coal (1% S) Fuel (1% S) Gas Cogeneration coal Cogeneration fuel Cogeneration gas

0.95 0.80 0.57 0.57 0.46 0.34

7.5 5.0 – 4.4 2.9 –

2.80 1.80 1.30 1.17 0.99 0.70

CO2 is essentially a greenhouse gas. Although sulfur and nitrogen oxides are much more effective greenhouse gases than CO2 at the molecular level, their much smaller concentrations and shorter lifetimes in the atmosphere make them contribute relatively little to the overall greenhouse effect. They are the main cause of acid rains, as well as of atmospheric ozone. The three main different fossil fuels (coal, gas and oil) have different greenhouse gas emission rates, as shown in table 2.6 [28]. Table 2.6 also shows that, whenever possible, the co-utilization of electricity and heat allows significant gains on the emission rates. Aside from CO2 , methane also has a strong greenhouse effect, about one third of that of CO2 . Apart from leaks in the gas transportation system and releases from coal mining, methane is essentially produced in the agriculture sector and will not be considered further here. The same applies to CFC and other stable and complex gaseous molecules. Scenarios for energy production The WEC and IIASA [29] have considered different scenarios for energy production up to 2100. It seems useful to discuss them as examples of possible energy futures. These scenarios belong to three main types depending upon the Gross Domestic Product per capita reached in different geographical aggregates. Table 2.7 shows the parameters chosen by WEC for these scenarios in 2050. Scenario A corresponds to a fast growth of the GDP per capita in all regions. It assumes a significant reduction of inequality between them. The growth is especially fast in former Soviet Union countries. Scenario C has a rather slow average GDP per capita growth but is, clearly, of the egalitarian type. Table 2.8 shows the regional energy intensities typical of the three scenarios. For scenarios A and B the energy intensities decrease as a

In an evaluation of the contribution of natural gas to the greenhouse effect, any losses to the atmosphere at each stage, from production to final use, should be taken into account since methane is at least 20 times more efficient than CO2 for inducing a greenhouse effect.

12

The energy issue

Table 2.7. Regional parameters of the WEC–IIASA scenarios: population and gross domestic product per capita for eleven geographical aggregates in 2050. The 1990 values for the GDP per capita are given for reference. Scenarios Population (million, year 2050) North America Western Europe Pacific OECD Former Soviet Union Eastern Europe Latin America Middle East, North Africa Africa Centrally planned Asia Other Pacific Asia South Asia World

1990 (k$ per capita)

A (k$ per capita)

B (k$ per capita)

C (k$ per capita)

362.42 494.6 148.12 394.67 141.06 838.58 924.25

21.62 16.15 22.78 2.71 2.39 2.50 2.12

54.47 45.88 58.68 14.09 16.27 8.33 5.64

45.84 37.06 45.80 7.48 7.83 7.07 4.03

38.79 32.95 42.80 7.14 7.97 7.39 4.09

1 735.73 1 984.17

0.54 0.38

1.57 6.99

1.03 3.36

1.19 5.40

750.55 2 281.28 10 055.43

1.53 0.33 3.97

12.21 2.00 10.10

7.86 1.33 7.24

10.20 1.75 7.46

Table 2.8. Energy intensity toe/kilodollar (2050). Scenarios

North America Western Europe Pacific OECD Former Soviet Union Eastern Europe Latin America Middle East and North Africa Africa Centrally planned Asia Other Pacific Asia South Asia World

1990

A

B

C

0.360 0.208 0.165 1.786 1.137 0.560 0.608 1.085 1.994 0.646 1.178 0.430

0.179 0.105 0.088 0.565 0.258 0.288 0.383 0.630 0.323 0.219 0.480 0.245

0.184 0.105 0.090 0.654 0.406 0.308 0.450 0.780 0.533 0.276 0.591 0.272

0.096 0.080 0.060 0.476 0.290 0.224 0.369 0.632 0.240 0.162 0.416 0.190


13

Figure 2.5. Correlation between GDP per capita and energy intensity. The energy intensity is the ratio of energy consumption to the GDP.

consequence of the increase in the GDP per capita, as currently observed, and shown in figure 2.5. In particular the lower GDP per capita in former Soviet countries retained in scenario B leads to higher energy intensities for these countries. Scenario C assumes a voluntary decrease of energy intensities, especially in the most developed countries. Table 2.9 shows the contribution of electricity to the primary energy consumed. This table shows the same features as table 2.8: in scenarios A Table 2.9. Share of electricity (2050). Scenarios

North America Western Europe Pacific OECD Former Soviet Union Eastern Europe Latin America Middle East and North Africa Africa Centrally planned Asia Other Pacific Asia South Asia World

A2

B

C2

36.53 39.41 38.56 16.74 24.46 14.70 11.05 13.18 14.26 20.16 13.06 20.50

35.46 38.76 39.59 16.18 19.97 13.19 10.46 12.29 11.04 16.53 12.41 18.77

41.05 41.78 43.12 15.56 22.31 14.48 11.35 11.29 17.21 22.90 12.64 19.77

14

The energy issue

Table 2.10. Total primary energy Mtoe (1990 and 2050). Scenarios 1990 Coal Oil Nat. gas Nuclear Hydro Biomass (comm.) Biomass (nonc.) Solar Others Total CO2 (MtC)

2176.36 3063.84 1684.93 450.07 488.68 246.30 848.86 0.00 16.92 8975.96 5931.63

B

A1

A2

A3

C1

C2

4 135.69 4 040.48 4 498.93 2 737.99 916.73 1 122.31 859.98 432.41 1 086.83 19 831.35 9 571.72

3 786.28 7 900.83 4 698.98 2 903.96 992.79 1 124.21 717.29 1 858.43 852.15 24 834.92 11 618.61

7 827.21 4 780.77 5 459.40 1 092.24 1 104.31 2 207.35 747.24 420.19 1 200.80 24 839.51 14 667.51

2 240.51 4 329.44 7 913.04 2 823.81 1 061.70 2 906.17 743.48 1 636.45 1 006.87 24 661.47 9 293.69

1 504.26 2 668.24 3 919.01 521.43 1 031.03 1 480.96 822.11 1 552.25 746.73 14 246.02 5 343.35

1 472.41 2 615.77 3 343.51 1 770.91 962.22 1 357.13 824.28 1 377.41 526.29 14 249.93 5 114.21

and B the share of electricity increases gradually with the GDP per capita. Scenario C assumes a deliberately increased share of electricity. Note that in table 2.9 we refer to subscenarios labelled A2 and C2. Indeed, the WEC and IIASA subdivide their scenarios A and C into three and two subscenarios respectively. The main differences between the subscenarios are the energy mixes producing the primary energy. In subscenarios A1, A2 and A3 the relative shares of coal, oil and gas are different. In subscenarios C1 and C2 the relative shares of renewable and nuclear energies are different. These features are displayed in Table 2.10. Note, also, the relative importance of gas in the energy mix of subscenario C1. From table 2.10 it is seen that only scenarios C might lead to CO2 stabilization within this century. These scenarios require very strong limitations on energy consumptions which might be very difficult to implement. Scenario C assumes a voluntary decrease of energy intensities, especially in the most developed countries. Table 2.11 compares the expected fuel consumptions cumulated from 1990 to 2050 to the reserves estimated in 1990. It is not clear from the Table 2.11. Cumulative fuel consumptions from 1990 to 2050 compared with 1990 reserves (Gtoe). Scenarios

Coal þ lignite Oil Gas

A1

A2

A3

B

C1

C2

Reserves 1990

200 300 210

275 260 211

158 245 253

194 220 196

125 180 181

123 180 171

540 146 133

15


Table 2.12. Reduction factors used in the nuclear intensive scenarios (first four columns). In the last column the share of hydrogen in the transportation sector is given.

North America Western Europe Pacific OECD Former Soviet Union Eastern Europe Latin America Middle East and North Africa Africa Centrally planned Asia Other Pacific Asia South Asia

2030 reduction factor electricity

2050 reduction factor electricity

2050 reduction factor coal

2050 reduction factor gas

2050 H2 share

0 0 0 0.5 0.5 0.5 0.3

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0.3 0 0 0

0.8 0.8 0.8 0.4 0.6 0.6 0.3

1 0.3 0.3 0.3

0 0 0 0

0 0.3 0 0

0 0.3 0 0

0.3 0.3 0.3 0.2

table that scenarios A and B are compatible with oil and gas reserves. Even scenarios C might come close to exhausting these reserves. Nuclear intensive scenarios The IIASA scenarios foresee a rather modest contribution of nuclear energy to the global energy mix. It seems that the main reason for this shyness is more related to ‘political correctness’ than to economical or technological constraints. It is interesting to examine how much a deployment of nuclear energy limited only by these constraints might limit global warming and resource exhaustion. Such an approach has been followed in reference [30] where it was found that, with a nuclear production potential of 9000 GWe in 2050, the asymptotic temperature increase could be limited to 2 8C, even in the case of scenario A2. We now give the outline of such a treatment for the three scenarios A2, B and C2. We assume that, by 2030, the use of fossil fuels for electricity production will be drastically reduced. Table 2.12 displays the reduction factors used for several geographic aggregates. The same reduction was applied in the three scenarios. Fossil fuels were assumed to be saved by resorting to nuclear power, although renewable energies could equally well be used provided they become reasonably competitive. The choice we have made of nuclear power provides a kind of existence theorem for a solution to curb CO2 emissions, and, at the same time, tests the capabilities of nuclear power in terms of fuel availability, wastes produced and capital needs.

16

The energy issue Table 2.13. CO2 emission and nuclear primary energy for the different nuclear intensive scenarios compared with the reference IIASA. Scenario

CO2 emission (Gton C)

Nuclear production (Gtep)

1990 2030 2030 2030 2030 2030 2030 2050 2050 2050 2050 2050 2050 2050 2050 2050 2050 2050 2050

6 11.9 8.7 9.9 7.2 6.3 5.3 14.8 10.2 4.2 2.7 9.8 8.0 3.5 2.3 5.7 4.9 2.2 1.4

0.49 0.7 3.8 1.3 3.3 1.1 2.3 1.1 6.3 13.3 15.4 2.7 4.7 10.3 12.2 1.8 2.3 6.1 7.1

A2 A2E B BE C2 C2N A2 A2E A2N A2H B BE BN BH C2 C2E C2N C2H

Table 2.12 shows different types of reduction factor used for 2050. These reduction factors are used differently in three different scenarios with increasing use of nuclear power: 1. The ‘electric’ scenario, labelled E, assumes that fossil fuels are no longer used for electricity production. 2. The ‘nuclear’ scenario, labelled N, assumes that coal and gas are no longer used in industry, homes or offices (in particular for heating), except in former USSR and China. 3. The ‘hydrogen’ scenario, labelled H, assumes that hydrogen has largely replaced oil in the transportation sector. The share of hydrogen fuel is displayed in the last column of table 2.12. Table 2.13 summarizes the basic differences between the reference IIASA scenarios and the various more nuclear intensive scenarios considered here. It gives CO2 emission and nuclear power production for 1990, 2030 and 2050. It appears from table 2.13 that the substitution of nuclear for fossil fuels in electricity production, although it achieves a significant reduction of CO2 emissions, is not sufficient to reach the 3 Gt carbon emission target. In contrast, when fossil fuels are replaced by nuclear power both for electricity

Renewable energies

17

Table 2.14. Cumulative needs of natural uranium assuming nuclear production with lightwater reactors only. Scenario

Cumulated needs 2030 (Mtons)

Cumulated needs 2050 (Mtons)

A2H BH C2H

5.8 4.8 3.3

20 16 10

production and for other uses in industry, offices and homes, the target can be reached in 2050 in all the scenarios. The replacement of fossil fuels by electrolytic hydrogen in transportation systems allows further reduction. While we have noted that known reserves of natural uranium amount to approximately 125 years of production at the present rate, it is clear that a nuclear intensive scenario will put much more strain on uranium resources. This is apparent in table 2.14, where the cumulated needs of natural uranium for the three ‘hydrogen’ scenarios in 2030 and 2050 are displayed. Presently known reserves would only be sufficient up to 2030, while the higher estimate of 17 million tons given by the OECD might allow reaching 2050. The need for a timely development of breeder reactors is, thus, clearly demonstrated and will be discussed later.

2.2

Renewable energies [31]

One possible way to allow the production of energy essential to the development of developing countries, which account for the majority of humankind, without catastrophically increasing greenhouse gas emissions might be to rely increasingly on renewable energies. In order to assess their possibilities, we briefly review the main ones. 2.2.1

Solar energy

On the earth’s surface the solar constant measures the power received from the sun by a 1 m2 surface perpendicular to the sun’s rays. It amounts to 1 kW. Practically, in order to estimate the energy available at a specified location, one has to take into account the latitude and the average daily and yearly insolation. Typical annual sunshine ranges from 1000 kWh/m2 in northern Europe to 2500 kWh/m2 in deserts like the Sahara. In Spain values reaching 1600 kWh/m2 are obtained, while in California values as high as 2000 kWh/m2 are observed [32]. Taking, as an example, an insolation of 1800 kWh/m2 and an efficiency for transformation of the solar energy into electricity of 15%, one gets an annual electricity production of 270 kWh/m2 , a number approximately valid for photovoltaic as well as thermodynamical

18

The energy issue

systems. To obtain an annual production of 7 TWh, similar to that of a 1 GW nuclear plant, 26 km2 are needed. At current solar cell costs, such a facility would cost around $17 billion, more than ten times that for a nuclear plant of similar power. The corresponding electricity cost would reach about 50¢ per kWh, compared with the current costs of around 5¢ per kWh. Such high costs will restrict the use of photovoltaic systems to remote locations not connected to an electricity distribution network. Note, however, that the need to store the energy would increase its cost by at least a factor of two. Thermodynamic systems allow much lower costs down to 12¢ per kWh and may become competitive for significant energy production in such places as Africa, India and South America, provided the current output of the facility can be fed into a network able to cope with the essentially intermittent form of the solar energy. One should note, in this context, that day/night storage capacities are present in all thermal solar systems, in the form of oil, sodium or molten salt heat storage reservoirs. 2.2.2

Biomass [33]

In temperate regions the average annual production of dry ligneous material is 10 tons per hectare, with the maxima reaching 20 t/ha. This corresponds to a gross resource between 3.6 and 7.2 toe/ha, i.e. between 40 and 80 MWh. As compared with the average annual insolation of 1:5 104 MWh/ha this corresponds to an efficiency between 0:2 and 0:5%. Taking into account the thermodynamical efficiency for electricity production, the maximum efficiency for electricity production is 0:2%. A facility producing 7 TWh per year would require a cultivated area of about 2500 km2 , in the best cases. This surface could be halved in tropical conditions. It is generally considered that the use of biomass is neutral as far as CO2 emissions are concerned. However, this is only true if it is ensured that all incinerated biomass is compensated by adequate replantating. This is, presently, not the case in most developing countries where deforestation contributes to greenhouse gas emission. Today’s world biomass energy production amounts to approximately 70 GToe/year. Humans use about 4% of this production either for producing food (2 GToe) or for energy production (1 Gtoe). In the most biomass intensive scenarios given at the 1992 UN Rio Conference, biomass energy production would be as high as 5 GToe, and the total human use would amount to 13 GToe, i.e. around 20% of the available resources. Because of its large volume, biomass has to be transformed into high energy content material close to its production location. Aside from local uses, the transformation of biomass into gas (methane), alcohol (ethanol, ETBE) or vegetable oil ester is considered. At present, electricity production using biogas is only marginally competitive when the cost of the biomass is negligible. Otherwise the cost of biogas electricity is three times more than that which can be

Renewable energies

19

obtained with fossil fuels. The cost of biofuels is about three times that of fossil fuels. 2.2.3

Wind energy [34, 35]

It is usual to express the wind energy annual resource in terms of kWh/m2 of swept-through surface. The best sites are close to the sea coasts. As an example, we consider the case of France. There the best sites generate up to 5000 kWh/m2 per year. A 1000 m2 windmill, with a peak power of 1 MW, would yield about 5 GWh/year. Existing large windmills reach peak powers of a few MW. The average surface requirement is of the order of 8 ha/MW, i.e. a production of 60 kWh/m2 /year. This figure is noticeably less than that expected from photovoltaic facilities, which reach close to 300 kWh/m2 /year. For France, the production potential is estimated to be 66 TWh/year for ground-based facilities and 97 TWh/year for offshore facilities. This figure corresponds to approximately the production of 20 nuclear reactors (57 are in use at this time) and would require 100 000 high power windmills with an average density of 20 windmills per km of coast. It is clear that the economically competitive potential is much less than the technically feasible potential and would depend upon the selling price as well as the environmental constraints. Note that since wind energy is intermittent, it can only become competitive when the facility is connected to a network. Under such conditions competitiveness is only marginal for good sites. However, the unpredictable intermittency of wind energy will make network control rather difficult, should the share of wind energy become significant. Furthermore, the possibility of windless periods requires that backup electricity production systems be available. This means that wind energy is only able to save fuel but not investment. It is more adapted to fuel-intensive electricity production means, such as gas turbines, than to capital-intensive production means like nuclear reactors. It is also compatible with hydroelectricity. 2.2.4

Hydroelectricity [33]

The production potential of hydroelectricity is considerable. It amounts, theoretically, to 36 000 TWh, while the resources that can be harnessed in practice are estimated at 14 000 TWh, more than the present world electric energy production of 12 000 TWh. In 1990 hydroelectricity produced was only 2200 TWh. From these numbers, it would appear that hydroelectricity could be the main alternative to the use of fossil fuels for electricity production. However, several factors will limit this possibility: .

Most of the potential is located in Asia (27%), South America (24%) and the former USSR (24%). For the industrially developed countries of Europe, North America and Japan, the unused potential is small.

20

The energy issue

The local environmental impact of large hydroelectricity is, usually, quite significant. Not only are the local climate and ecosystem disturbed, but large populations have to be displaced. This is, certainly, the main limiting factor to the establishment of large hydroelectric dams. . The risks of catastrophic dam rupture. In the recent past, dam ruptures led to some of the most catastrophic technological events, certainly comparable with Chernobyl. Some of the most dreadful events were: .

Morvi (India 1979) 30 000 dead Vaiont (Italy, 1963) 2118 dead L’Oros (Brazil, 1960) 1000 dead St Francis (USA, 1928) 700 dead Gleno (Italy, 1923) 600 dead Logan (USA, 1972) 450 dead Malpasset (France, 1959) 423 dead. Each year, although with less catastrophic outcomes, several dam ruptures are observed in the world. These limitations led the World Energy Council [27] to anticipate a maximum share of 7% for hydroelectricity in the fulfilment of world energy needs. Hydroelectricity, after the large initial investment is repaid over a period of 15 to 30 years, is very cheap. In many cases the kWh cost is less than 2¢. Finally, note that if hydroelectricity is easily modulated according to need, it is dependent upon the pluviometric regime of the region where the dams are implanted. Long droughts may significantly affect its availability, as has been experienced recently in California.

2.3

Nuclear energy

2.3.1

Standard reactors

Most existing energy producing reactors are of the light-water cooled type, either pressurized water (PWR) or boiling water (BWR). Although other types of commercial reactor like the heavy-water CANDU have interesting characteristics, our discussion focuses on the light-water reactors. The power of commercial reactors ranges between 600 and 1500 electric MWatts (MWe), with thermodynamical efficiencies close to 33%. As an example, we consider a 1000 MWe reactor. Each fission produces approximately 200 MeV (185 MeV at the moment of fission and 15 MeV produced by subsequent radioactivity). Accordingly the fission of 1 kg of a fissile isotope typically produces 80 TJ (or

radioactivity produces energy in the form of fast electrons (betas) and gamma rays.

Nuclear energy

21

Table 2.15. Inventories at loading and discharge of a 1 GWe PWR. Nuclides 235

U U 238 U U total 239 Pu Pu total Minor actinides 90 Sr 137 Cs Long-lived FP Total FP Total mass

Initial load (kg) 954

236

26 328 27 282

27 282

Discharge inventory (kg) 280 111 25 655 26 047 156 266 20 13 30 63 946 27 279

1900 toe). A 3 GW (thermal gigawatts) reactor, yielding 1 GWe (electrical gigawatt), produces annually about 7 TWhe for an availability of 80%. It burns annually about 1 ton of fissile isotopes which is equivalent to two million tons oil equivalent (toe). More precise numbers are given in table 2.15, where material inventories at loading and discharge are given [36]. In table 2.15 a burn-up of 33 GWd/ton (gigawatt-day/metric ton) is assumed. The table shows the following interesting features: The amount of 235 U which has disappeared equals 674 kg. This accounts not only for the fission of this nuclide, but also for its neutron captures, at least the 111 kg of 236 U produced. . This means that at least 383 kg of the higher isotopes, mostly 239 Pu, have contributed to fission. This can also be considered as an indirect fission of the 238 U isotope, which lost 673 kg corresponding essentially to the production of plutonium. Of these 673 kg only 286 kg are found in the form of plutonium isotopes and minor actinides. . The mass balance between the initial and discharge inventories is not exact. This is due to the mass equivalence of the energy produced (about 1 kg) and to the neutrons captured in the structure elements and cooling water (2 kg corresponding to approximately 0.5 neutron per fission). .

The nuclear wastes to be considered can be divided into three categories: 1. The plutonium and minor actinides, with very high radiotoxicities due to their dominant alpha-decay. They have long lifetimes, up to 25 000 years for 239 Pu and more than two million years for 237 Np. They would require either long-term underground disposal or transmutation. In the latter case they can only disappear by fission (this is usually called incineration). The fission of 280 kg of plutonium and minor actinides would produce

22

The energy issue

Table 2.16. Long-lived fission products with their half-lives and production rates. Nuclide

79

Se

90

Zr

99

Tc

107

Pd

126

Sn

T1=2 (years) 70 000 1:5 106 2:1 105 6:5 106 105 Production (kg/y) 0.11 15.5 17.7 4.4 0.44

129

I

135

Cs

1:57 107 2 106 3.9 7.7

about 2 TWh of electrical energy. This means that at least one incinerating reactor for four PWRs would be needed if one wants to completely incinerate the plutonium and the minor actinides. 2. The long-lived fission products, nuclides with lifetimes longer than 1000 years which decay by emission. The main fission fragments involved are shown in table 2.16, together with the amounts produced yearly by a 1 GWe reactor. 3. The medium-lived fission products, essentially 90 Sr and 137 Cs, which have very high activities at discharge and small neutron capture cross-sections. It does not seem realistic to transmute them and they would, then, set a minimum duration of around 300 years during which the wastes are radioactive and require supervised storage. The inefficient use of uranium in current thermal reactors has consequences on the amount of mining required, as well as on the level of resources. In the absence of recycling, each 1 GWe reactor requires annually about 100 tons of fresh natural uranium. Typically currently used uranium ores have grades around 0.25% [37]. This means that a 1 GWe reactor requires the extraction of 40 000 tons of ore, to be compared with the two million tons of oil which would be needed to produce the same amount of energy. The rather large amount of mill tailings is associated with radioactivity due to the descendents of uranium, especially to a continuous flow of radon during a long period (75 400 years as defined by the half-life of the parent 230 Th). This radon gas escapes more readily from the tailings than from the unmined uranium ore. The uranium reserves are estimated around 5 million tons at costs close to the present. The present world power production is about 350 GWe, requiring an annual 40 000 tons of natural uranium. Thus the present known reserves are estimated to last 125 years. Again, it is not a problem as long as the present small contribution of nuclear power to the overall energy production is maintained. However, as in the wastes case, should the nuclear share increase to a 30% level, the reserves would be reduced to approximately 40 years, no more than the oil reserves. One should note, however, that there is a very large reserve of uranium in sea water, amounting to about three billion tons [37], at a concentration of 3.2 parts per billion. It seems possible to extract this uranium at a cost ten times higher than the current cost, which would increase the cost of the produced electricity by 50%.

Nuclear energy 2.3.2

23

Breeder reactors

The use of breeding or converter reactors would change the picture considerably. Converters and breeders allow full use not only of the fissile 235 U isotope, but also of the fertile 238 U and 232 Th isotopes. Thus, in principle, a 1 GWe reactor requires only 1 ton of natural uranium or an equivalent amount of thorium. This means that, at the current market cost, assuming a production capacity of 9000 GWe, corresponding to the most nuclearintensive scenario of table 2.13, the reserves would amount to 2000 years for natural uranium and about four times more for thorium. In fact, the very efficient use of the uranium and thorium would allow the use of very low grade ores, including sea water uranium, which means that the resources would be practically unlimited. The mill tailings would also be considerably reduced by a factor more than 100. While the plutonium present in spent fuels has to be considered a waste, it is the fissile material for breeders and converters. Only long-lived fission products (LLFPs) and minor actinides (MAs)† can thus be considered as nuclear wastes. In the absence of specific transmutation of these wastes, their radiotoxicity, after a cool-down period of 300 years,‡ would be at least one order of magnitude smaller than that of the PWR spent fuels, for an equivalent energy production. Since fuel reprocessing is a prerequisite for any breeding or converting cycle, it is quite logical to consider the transmutating of LLFPs and MAs. We shall discuss this possibility in some detail below. It has been shown that the incineration of MAs and the transmutation of some of the most significant LLFPs are feasible. Nuclear wastes would then be reduced to the reprocessing losses. Modern reprocessing is claimed to have 99.9% efficiency in the recovery of plutonium and 99% in the recovery of MAs [38]. It would then be possible to reduce the total radiotoxicity of the wastes by several orders of magnitude after a few hundred years of cooling. With such a reduction, long-term disposal might not be necessary, or at least will be considerably reduced. While the reliability and safety of PWR reactors has been widely demonstrated in industrialized western countries,x the experience with breeder or

Breeder reactors produce more fissile material than they consume, while converter reactors produce as much fissile material as they consume. † Np, Am and Cm are produced in relatively small quantities in standard reactors and are thus called minor actinides. ‡ This cooling time is necessary to allow for the decay of 137 Cs and 90 Sr, whose transmutation would be very difficult and costly. Due to their short half-life these isotopes dominate the short-term waste radiotoxicity. x Even in the former USSR the reactors similar to the PWR, the VVER, are considered to be safe by international experts, while the RBMK reactors, such as those of Chernobyl, are unanimously considered unsafe.

24

The energy issue

converter reactors is limited and ambiguous. By far the best known breeders are of the liquid metal fast reactor type. Practically all of these liquid metal reactors have used sodium as their coolant, with the exception of several recent Russian submarine propulsion reactors which are cooled with a liquid lead–bismuth eutectic. While it seems that the records of the Russian sodium cooled reactors like BOR60, BN350 and BN600 are very good, the records of such western reactors are much more questionable. The small American reactor EBR2 worked satisfactorily until its final shutdown in 1995. Another American reactor, Enrico Fermi, could never run. The small French reactor RAPSODIE ran very nicely until its final shutdown. The 250 MWe Phenix reactor ran satisfactorily for ten years until unexplained reactivity fluctuations led to its shutdown. It has been started again recently at reduced power, pending safety improvements. The large, 1200 MWe Superphenix reactor was plagued by sodium leaks and administrative imbroglio until the decision was made to stop it indefinitely. The Japanese Monju reactor is, also, suffering from sodium leaks. It seems that the combination of increasingly stringent safety constraints and the use of the very reactive liquid sodium led to difficult running conditions. Furthermore, the investment costs for a reactor like Superphenix are about twice as large as those necessary for a standard PWR reactor with the same power. The cost of electricity which would have been produced by Superphenix in normal running conditions would have been twice that produced by a PWR. Of course, it can be argued that, for an industrial-scale series of reactors, the investment cost as well as the fuel cost would have decreased. In conclusion, while the interest of breeder reactors is clear in the hypothesis of an extension of nuclear power, one cannot consider the present type of sodium cooled reactor to be the only solution. In this context it is possible that hybrid reactors may help develop alternatives to sodium cooled reactors. They may also facilitate a switch to the thorium breeding cycle which would lead to a much reduced production of minor actinides. 2.3.3

Nuclear waste disposal options

The magnitude of the nuclear waste problem can be deduced from table 2.17 which shows the amount of spent fuels discharged in the OECD countries in 1992 [36]. We recall that nuclear power only accounts for 5.8% of the total world energy production. This small percentage will, however, lead to a spent fuel inventory of about 200 000 tons by the year 2020. The annual production of spent fuels amounts to about 8000 tons. This figure is to be compared with

Sodium has high heat conduction, low viscosity, similar to water, low melting point and low neutron absorption cross-section. The first sodium cooled reactor was designed to be airborne, and thus the low density of sodium was an important asset.

Nuclear energy

25

Table 2.17. Data concerning the end of cycle in OECD countries.

France Belgium Sweden Switzerland Spain Finland Germany Japan United Kingdom USA Canada Netherlands Total

Nuclear power (GWe)1

Share of nuclear power (%)2

Spent fuels3

58.5 5.5 10.0 3.0 7.1 2.3 22.7 38.9 11.7 98.8 15.8 0.5 274.8

76.4 55.8 51.1 36.8 35.0 29.5 29.3 27.2 25.8 22.0 19.1 4.9

11 770 1 400 3 240 1 300 1 775 975 6 315 8 600 7 0004 28 600 20 0005;6 150 91 125

1

1 January 1995 (AIEA). As compared with the total electric energy production. 3 Cumulated tons in 1995 (EU estimates). 4 Authors’ estimate. 5 Canada uses natural uranium reactors (CANDU), hence the large inventory. 6 Authors’ estimate for 1995. 2

the present spent fuel recycling capabilities of around 2000 tons per year, mostly by the COGEMA La Hague facility. At present, two different strategic approaches are proposed for highactivity nuclear waste disposal: 1. Direct spent fuel element disposal, without any reprocessing. Such an approach is favoured by, among others, the US, Sweden and Swizerland. 2. Spent fuel reprocessing with the aim of optimized extraction of transuranics and fission products and, possibly, their transmutation by nuclear reactions into less radiotoxic or short-lived species. This approach is followed, notably, by the UK, Japan, France and Belgium. In both cases, some sort of storage of radioactive wastes is needed. Two options are considered: 1. Deep underground storage with or without possible retrieval. 2. Surface or sub-surface storage. It is clear that these two last solutions can only be temporary, since the halflife of many of the wastes exceeds, by far, the life span of civilizations. The proponents of such solutions argue that technical progress may allow a better evaluation of the safety or feasibility of alternative solutions. One should note, however, that such progress requires experimenting with deep

26

The energy issue

underground storages in dedicated laboratories on the one hand, and separation and transmutation studies on the other hand. It may seem, therefore, paradoxical that the most vocal advocates of temporary storage oppose both underground laboratories and reprocessing. The paradox can be understood as an aspect of a strategy aimed at pulling out of nuclear power altogether. Only after a withdrawal is obtained will the question of existing wastes be seriously examined. In that case the only possible solution will be deep underground disposal, but it is untimely to acknowledge that fact while fighting the anti-nuclear struggle! In the future, the relevance of the two basic choices, direct storage or reprocessing, will depend on the development of nuclear power. 1. In the case of withdrawal from nuclear power in the near future, direct storage is the most natural choice. Reprocessing policies are consequences of investments made in the frame of the deployment of fast breeders. The phasing out or standing still of the breeder programmes raised the question of the future of the large reprocessing facilities like those of BNFL and COGEMA. It was found that using the separated plutonium as fuel in thermal neutron reactors had some advantages (decreased need for enriched uranium and reduced volume of the most active wastes) at a very modest cost [39]. An a posteriori policy of waste separation and transmutation followed. 2. At the present world level, nuclear power has only a marginal role in alleviating the waning of reserves and the environmental degradation problems associated with energy production. Long-term continuation of nuclear power would only be justified at a much higher level than at present. In that case, we have previously noted that breeding will be mandatory. Reprocessing will be necessary and the nuclear waste issue will be completely different. For example, using the values given above for scenario A2N in 2050, nuclear power would reach as much as 9000 GWe, more than 20 times more than at present. With the PWR technology, the annual discharge of spent fuel would rise to 260 000 tons. Should this be disposed of underground, four sites equivalent to the US Yucca Mountain would be needed each year. Using fast reactor technology, both plutonium and uranium should be recovered from the spent fuel and, aside from technological losses, the highly active wastes would be limited to fission products and minor actinides, i.e. about 9000 tons per year. Furthermore, given the existence of reprocessing facilities, it might be feasible to transmute minor actinides as well as some of the long-lived fission products. Underground disposal From the preceding, it seems probable that deep underground disposal will be necessary in all cases. It is, therefore, important to understand the

27

Nuclear energy Table 2.18. Evolution with time of the activity of 100 000 tons of irradiated fuel. Time (years)

1000

10 000

100 000

1 million

1 billion

Activity (Bq)

4 1018

2 1018

1017

1016

1015

nature and amplitude of the hazards which might be associated with such a site. An order of magnitude of the dangers associated with deep underground storage can be obtained by comparing the activity of the stored wastes with the radioactivity of the earth’s crust. Assuming that the storage site is 500 m deep, the comparison can be made with that of the first kilometre of crust. The mean activity of the crust is around 1500 Bq per kg. Considered sites have areas of order 1 km2 corresponding to a crust activity of the order of 3:5 1015 Bq, a little less than half being due to 40 K and more than half to thorium and uranium decay. The crust activity for the area of a country like France amounts to 1:7 1021 Bq. This activity has to be compared with that of the materials stored. We take the example of a storage of 100 000 tons of irradiated fuel, corresponding to 50 years of operation of the nuclear reactors of France, a very highly nuclearized country. The activity of the storage is shown as a function of time in table 2.18 and figure 2.6.

Figure 2.6. Evolution with time of the activity of 100 000 tons of irradiated fuel. The activity of 1 km2 of the first km of the crust is also shown.

28

The energy issue

Although this is a very rudimentary approach, the comparison of the activities of table 2.18 with the total activity of the crust of France shows that the average increase of radioactivity over France due to nuclear waste storage will remain very small at all times. Assuming that at least 100 000 years are needed for a complete diffusion of the wastes one sees that, even locally, the dose which might be delivered to the most exposed population will not exceed a few times that due to natural radioactivity. More precise diffusion calculations, such as those displayed in Appendix I, give the following results. At no time in the future, in a normal situation, will the dose delivered to the most exposed population exceed 0.25 mSv/year, i.e. a factor of ten below natural irradiation . The main contributor to the dose is 129 I. Almost all stored 129 I will be released within a time span of approximately 1 million years. Due to their small solubility and mobility, actinides have a very small contribution. . In case of an accidental situation, such as drilling a well through the repository and drinking the extracted contaminated water, the maximum dose to the most exposed population should not exceed a few mSv/year. In this case 129 I remains an important contributor but 226 Ra takes the lead for longer times. It is a descendent of 238 U. In the case of reprocessing, its influence will decrease considerably. . The amount of heating by the radioactive wastes at the storage site will have an essential roˆle in determining the surface, and thus the cost of the storage facility. After 100 years plutonium and minor actinides will play the dominant roˆle in heat production and the cost of their incineration will have to be evaluated in comparison with the ensuing cost saving of storage. .

In order to have an estimate of the cost of deep underground disposal, we take the example of the US Yucca Mountain site which has been accepted as a site for deep underground storage of nuclear wastes in the US. The site would cover about 6 km2 honeycombed with about 100 km of tunnels [37], while the maximum storage capacity should be 70 000 tons. The cost of the site would be more than 15 billion dollars, corresponding to an additional cost of nuclear electricity of about 1 mil/kWh. Waste transmutation The nuclear reactions available for nuclear waste processing are of two types: 1. Transmutation, which by neutron capture transforms a radioactive nucleus into a stable one. This method is suitable for fission products.

This may involve intermediary steps through short-lived isotopes.

Nuclear energy

29

As stable nuclei could be, simultaneously, transformed into radioactive ones, the method may require an initial separation of the isotopes to be transmuted. However 99 Tc and 129 I do not require such separation. 2. Incineration, which amounts to nuclear fission following neutron capture. This method is suitable for transuranic elements. It is always associated with energy and neutron production. It is already applied, on an industrial scale, to plutonium.

The plutonium case From the preceding, plutonium can be considered according to two different viewpoints. In the breeding strategy it is a nuclear fuel. In standard PWR reactors it appears to be a nuclear waste which is apt to be incinerated. Incineration is possible with thermal reactors like PWRs, but complete incineration will be difficult in this case. Indeed, it is associated with the production of transplutonic elements (americium and curium) which are difficult to incinerate in a PWR. For the thorium–uranium cycle, 233 U would have a role similar to plutonium in the uranium–plutonium cycle. However, in this case, the production of transuranic elements is greatly reduced. Different nuclear waste reprocessing strategies are possible, depending on the availability of existing reprocessing plants, on the experience of incineration in thermal reactors and on the prospect to use fast reactors. Incineration with fast reactors A reprocessing policy has taken shape, especially in France and Japan. Plutonium obtained after the reprocessing of standard PWR fuel is used to manufacture MOx fuel, a mix of plutonium and depleted uranium oxides. The MOx fuel elements are used as substitutes of normal enriched uranium fuel elements in PWR. Note that the presence of 238 U in the MOx fuel is needed for safety reasons: to preserve a negative temperature coefficient and partially breed the nuclear fuel so as to prevent too fast a decrease of the reactivity. While being irradiated the plutonium mixture is depleted of the fissile 239 isotope and enriched in the 240 isotope, which is not fissile by thermal neutrons but has a large capture cross-section (it is a neutron poison), as well as in transplutonic nuclei (minor actinides, especially americium) which are not or are poorly fissile. In the course of successive reprocessing cycles it is, therefore, necessary to increase the total concentration of plutonium with respect to that of uranium. Such an increase is, however, not possible ad infinitum, since while 240 Pu is a poison for thermal neutrons, it is not for fast neutrons. Thus a fuel too enriched in

Fission may happen after one or several radiative captures.

30

The energy issue

plutonium might lead to a criticality of the reactor for fast neutrons, and to a divergence in the case of partial or total loss of coolant. For the time being only one MOx irradiation is done and irradiated MOx fuels are not reprocessed. In principle further reprocessings and irradiations are possible. However, after two or three reprocessing cycles in PWR, a ‘dirty’ plutonium would be produced with an increased quantity of minor actinides. It has been suggested that this mix could be incinerated in fast, sodium cooled, reactors [38]. These would incinerate more plutonium than they would produce, in contrast to breeder reactors. However, due to safety considerations, the fuel would include a minimum amount of 238 U which would limit the net consumption of plutonium. Fast reactors should also be able to incinerate minor actinides efficiently. Incineration of dirty plutonium and minor actinides would require one fast reactor for four to five PWRs. Complete plutonium incineration in thermal reactors Rather than specializing about one-third of the PWRs into MOx-PWRs, it seems that replacing uniformly the traditional 3.5% 235 U enriched fuel elements by elements where about two-thirds of the fissile nuclei would be 235 U and the remaining one-third 239 Pu and 241 Pu would allow a stabilization of the plutonium inventory. A practical method to do so might be to mix UOx and MOx needles in a fuel element [40, 41]. Minor actinides should be extracted at each reprocessing, since these cannot be incinerated in thermal reactors. The minor actinides could then be incinerated in fast [42] or hybrid [43–46] reactors. Recently, it has been proposed [41, 47] to incorporate special annular fuel rods highly enriched in plutonium in standard PWR reactors. The PWR reactor could then consume 160 kg of plutonium per year instead of producing 200 kg as in present standard PWRs. Incinerating minor actinides in dedicated fuel elements also seems possible. Such a solution would be attractive, at least as long as uranium reserves do not command breeding reactors. Indeed, with minor modifications, the existing reactor system could be run with a stable plutonium and maybe minor actinide inventory. Another solution, proposed by Rubbia [45], among others, is to replace depleted uranium by thorium. Thorium has neutronic properties close to those of 238 U. The incineration of plutonium would be associated with the production of 233 U. The proposed system could burn, annually, around 1.2 tons of plutonium while producing 0.7 ton of 233 U. This nucleus could then either be a substitute of 235 U in standard PWR fuel, or be a part of a new fuel based on the mixture 232 Th–233 U. Such a fuel could be reprocessed as many times as desired in PWR reactors, at variance with the 238 U–239 Pu mixture. The main difficulty of such a scheme would be the fuel element

Minor actinides are weakly fissile by thermal neutrons, but easily fissile by fast neutrons.

Nuclear energy

31

fabrication: irradiation of 233 U produces a significant amount of 232 U by (n, 2n) reactions on 233 U, whose decay is accompanied by intense high-energy gamma activity which would require large biological shielding for fuel fabrication. The whole fuel cycle would have to be redesigned. A renewed interest has been displayed in gas cooled high-temperature reactors (HGTRs or HTRs). In these reactors the fuel is composed of tiny fissile–fertile particles (uranium, thorium, and plutonium carbides) coated with carbon and Si–C layers. Such fuel allows operation at very high temperatures (up to 900 8C). Even after a loss of coolant accident the fuel retains its integrity, its temperature being limited only by radiation cooling to an admissible value (more than 2000 8C). The high stability of the fuel would allow very high burn-ups of more than 200 MWd/ton. With such burn-ups a very efficient incineration of plutonium should be obtained, while reducing the amount of spent fuels or reprocessing wastes. However the question of the fate of the extremely active and minor-actinide-rich spent fuel remains open. 2.3.4

Deployment of a breeder park

We have seen above that a large increase of the share of nuclear power in meeting energy needs will most probably require the deployment of a large breeder park. How quickly such a park could be developed will depend on the amount of plutonium available from standard reactor fuel reprocessing, the initial inventory of the breeders, as well as on the doubling time of the breeder park. A relevant study has been made in [30] where the deployment of 3000 PWR reactors in 2030 and an additional 6000 breeders by 2050 was considered. Such a reactor park could stabilize the global temperature, while preserving the possibility of a strong increase of world energy consumption. The question addressed was whether such a deployment is compatible with uranium reserves and doubling times of the breeders. Two possible breeding cycles were considered: the U–Pu cycle using fast neutron reactors and the Th–U cycle using thermal neutron reactors. In both cases the initial loads are assumed to be mixtures of the fertile element (U or Th) with plutonium taken from the spent fuels of PWR and BWR reactors. It is important to make sure that the amounts of plutonium available would be sufficient to supply all the breeding reactors by 2050. The U–Pu cycle Experience with fast breeders shows that a typical [48] 1.2 GWe reactor requires an initial inventory of 5 tons of plutonium. A 1200 MWe reactor However, efficient enough radiation cooling limits the size and power of the reactor to less than about 1 GWth.

32

The energy issue

Figure 2.7. Size of installed nuclear power (in GWe) for the U–Pu cycle as a function of time. In the first stage, a PWR park is developed which produces plutonium used to start a fast reactor U–Pu breeder park.

produces around 0.25 tons of plutonium annually, corresponding to a doubling time of 20 years. However this value of the doubling time does not take into account the reprocessing stage. The longer the cooling time of the used fuel before reprocessing, the longer the effective doubling time. As an example, if the residence time of the plutonium in the reactor is 4 years, and the cooling time also 4 years, the plutonium inventory is doubled, and so is the doubling time. The transition from a PWR- (BWR-) based system to a fast reactor system was studied. It was assumed that a strong PWR programme starts in 2010, first breeders starting progressively in 2020. By 2030 no new PWRs are connected to the grid, leaving the field to fast reactors. Figure 2.7 shows the evolution of the reactor park corresponding to a plutonium production of 250 kg/GWe by the PWRs and 200 kg/GWe by the fast breeders. A cooling time of 1 year was assumed. The target of 9000 GWe by 2050 can be reached. For longer cooling times it is found that the target cannot be reached. Cooling times as short as 1 year are probably not possible with standard aqueous reprocessing and would require pyro-chemical reprocessing. After 2050 the PWRs would be phased out progressively and the doubling time of the FR could be adjusted to the desirable evolution of the reactor park. In figure 2.7 a 1.5% annual increase of the nuclear park was assumed. Figure 2.8 shows the plutonium stockpile outside the reactors. It displays three regimes. First the Pu inventory increases slowly until 2030 as

Nuclear energy

33

Figure 2.8. Evolution of the Pu inventory in the case of deployment of a fast reactor U–Pu breeder park.

a result of the production by the increasing PWR (BWR) park. The decrease between 2030 and 2050 reflects the sharp increase in the number of fast reactors. The increase after 2050 is due to the slower increase of the number of fast reactors. Instead of keeping the total plutonium stockpile at such a high value, it could be possible to use the excess neutrons for the transmutation of fission products, for example. Alternatively, reactor sizes could be reduced, giving more flexibility to the power system. In our scenario the last PWR reactors will be phased out in 2070. At that time the total amount of natural uranium used would reach 12 million tons, close to the presently estimated reserves. This means that the number and lifetime of the PWR park cannot be considered as an easily adjustable variable to achieve the strong increase of nuclear power between 2030 and 2050. This increase will be difficult to achieve and requires the early development of breeders, as well as the availability of as much reprocessed plutonium as possible. The generalization of MOx incineration has to be weighed against this requirement. Similarly, incinerating plutonium in HTR reactors may be counterproductive if spent fuel reprocessing is not possible. Until the development of breeder reactors the best use of reprocessing facilities might be the fabrication of Pu–Th fuels for PWRs, producing 233 U, which could be used as described in section 4.2.2. Of course, accepting a lower value for the target in 2050 would make things easier. For example, a target of 7000 GWe could be reached with a doubling time of 32.5 years. Another possibility would be to increase the share of more efficient plutonium-producing reactors such as the CANDUs.

34

The energy issue

Figure 2.9. Size of installed nuclear power (in GWe) for the Th–U cycle as a function of time. In the first stage, a PWR park is developed which produces plutonium used to start the molten salt Th breeder park.

The Th–U cycle The possibility of breeding 233 U from thorium was demonstrated by the MSRE experiment [49]. The MSBR [50] project has produced a rather detailed design for a large molten salt reactor, with interesting breeding possibilities. As an alternative to the solid fuel U–Pu breeders we have studied the potential of the Th–U cycle with MSR reactors. The first fissile loads of the 1 GWe MSR are made of industrial plutonium obtained from spent PWR fuel reprocessing. Due to the mediocre neutronic properties of this plutonium, our simulations show that 4 tons/GWe are needed to ensure criticality. The initial plutonium load is replaced by 233 U. Every year, all 5 year old available plutonium is used for new MSRs. This is not sufficient to ensure the required rate of increase. The complement is obtained from the excess 233 U produced in the operating MSRs, used to start new Th–233 U reactors. Only 1 ton/GWe of 233 U is needed to ensure criticality of an MSR. The doubling time is 25 years with a 10 day cycling time of the salt. The chemical treatment amounts to extracting fission products and protactinium. 233 U is re-injected into the salt after protactinium decay. Figure 2.9 is similar to figure 2.7 for the U–Pu cycle, and shows the evolution of the reactor park. We have distinguished Th–Pu and Th–233 U

The equality between this number and the inventory of the U–Pu breeders is fortuitous.

Costs

35

Figure 2.10. Evolution of the 233 U stockpile in the case of deployment of a Th–U molten salt breeder park.

reactors according to their initial loads. The lifetime of the reactors was assumed to be 40 years, which explains the decrease of the ‘Th–Pu’ reactors after 2070. Figure 2.10 shows the evolution of the 233 U stockpile outside the reactors. The plutonium stockpile is not displayed since all the plutonium produced is, after 5 years of cooling, used for new Th–Pu reactors. As for the U–Pu cycle, the amount of available 233 U measures the flexibility of the system which could be used for fission product transmutation and/or smaller production units. It is interesting to note that the final stockpile of 233 U is only 16 000 tons, to be compared with the much larger stockpile of 80 000 tons of Pu displayed in figure 2.4. However, due to the difference of inventories (1 ton versus 4 tons), the number of new reactors which could be fed with these stockpiles is the same, namely 16 000 GWe. This illustrates the fact that the value of (2.9 for Pu versus 2.3 for 233 U) is not the only relevant quantity to evaluate breeding potentials. In the case of MSRs, the ability to remove the fission products continuously is another determining factor.

2.4

Costs

If one wants to assess the future of nuclear energy, it is, of course, useful to compare its cost with that of other means of electricity production. We report in table 2.19 some cost estimates, given in US¢ per kWh. The table shows that, among the renewable energies, only wind energy has reached competitiveness with fossil fuels and nuclear power. However,

36

The energy issue Table 2.19. Cost estimates in US¢ per kWh for different energy production technologies. Fossil fuels: traditional Fossil fuels: combined cycles Nuclear Hydro-power Wind Geothermal Solar: thermal Solar: photovoltaic Biomass

5 [53] 3.7 [53] 3.3 (France)1 [53] 2.5 [52] 6 [52] 7 [52] 12 [52] 50 [52] 8 [52]

1 This figure includes investment cost for a new plant with 5% actualization rate. At present the cost of nuclear electricity is much lower (down to 2¢ per kWh) since the initial investment has been paid off [39].

this close competitiveness is only attained if the electricity produced by the windmills is used as input to the general network. Biomass may also be competitive in specific sites if no long-distance transportation of the bio-fuel is required. In the future, thermal solar energy might reach competitiveness in well-insolated sites, and if long-term energy storage is not needed. To be complete one should stress that wind energy, small biomass facilities, and solar devices might be very suitable in remote sites where no electricity network exists. Also, energy saving efforts should have a high priority: it is largely preferable to invest a given amount of money to save, say, 7 TWh annually, than to build an additional 1 GWe facility. The table shows that the only competitive and massive energy producing method which could be an alternative to fossil fuel facilities is nuclear power. However, the market trends favour combined gas fuelled facilities which reach very high efficiencies and involve investment costs only one-third of those required for a nuclear facility, even in a favourable institutional context like France. A ‘rebirth’ of nuclear power is only probable, in the not-too-distant future, if a strong policy to reduce greenhouse gas emissions is put into effect worldwide. This ‘rebirth’ will probably be possible only if, in public opinion, the nuclear waste problem is solved satisfactorily, and if a Chernobyl-type accident is demonstrated to be impossible.

2.5

The possible role of accelerator driven subcritical reactors

When compared with critical reactors, ADSRs have two specific characteristics.

The possible role of accelerator driven subcritical reactors

37

1. If well designed, they prevent any possible criticality accidents. This has led their most vocal advocates [45] to propose that they should replace critical reactors altogether. In a less radical but also more common approach it has been proposed [38, 46] to take advantage of this subcriticality in order to use certain types of fuel with poor neutronic properties: in particular, because of their very small delayed neutron fractions, homogeneous incineration of minor actinides appears to be feasible only with subcritical reactors. 2. Subcriticality provides additional neutrons which can be used for increased breeding of 233 U or 239 Pu. It is even possible to breed them in the absence of any fissile element. Another possible use of the additional neutrons is to transmute long-lived fission products. Let us elaborate on these points. 2.5.1

Safety advantages of subcriticality

In principle, criticality accidents such as that of Chernobyl should be impossible for an ADSR. However, this is true only as long as one can monitor the effective value of the reactivity. As shown in chapter 7, this monitoring cannot be done solely by relating the beam energy to the reactor output energy: an increase of the reactivity of the subcritical part can be accompanied by local poisoning of the spallation source in such a way that the output energy does not increase but may, on the contrary, decrease until a critical situation appears. It is thus necessary to devise elaborate ways to monitor the effective reactivity of the subcritical array. One aspect, which is seldom stressed, of ADSRs is that it requires more technical skill and good maintenance to keep them running than for critical reactors. Indeed, high-intensity accelerators are and will remain rather difficult to operate. Loss of expertise of the staff as well as poor maintenance will, inexorably, decrease the performance of the machine until it finally stops. In contrast, as the recent past shows, critical reactors are apt to run in rather bad shape and do not necessarily need the best team to be operated, with the dangers associated with such a situation. It can be argued, therefore, that ADSRs can offer safety against a societal disorder. Aside from criticality accidents, ADSRs are subject to risks similar to those of critical reactors, such as solid fuel core melt-down, radioactive leaks into the environment, etc. In addition, the coupling between the accelerator and the subcritical medium may be the origin of weaknesses with respect to safety, such as window breaking or propagation of radioactivity through the accelerator. For large subcriticality levels of more than a few per cent, the delayed neutron fraction has no influence on the safety of the reactor. This means that it becomes possible to use fuels with large minor actinide concentrations

38

The energy issue

or plutonium without compensating for the small delayed fraction by the presence of 238 U. Similarly, the sign of the temperature and void coefficients have a reduced influence. However, they should be limited so that subcriticality should be guaranteed at all power levels of the reactor. In particular, overly negative coefficients should be avoided to prevent criticality in the case when an accelerator trip leads to a sharp fall of the reactor power. The high tolerance level of ADSRs with respect to the fuel’s neutronic properties should make them excellent tools to study new reactor concepts by relaxing many safety conditions. For example, the same accelerator could feed different subcritical systems like molten salt, gas or lead cooled reactors. Such prototypes could allow studies of corrosion, radiation defects and fuel evolution in realistic conditions with less stringent criticalitycontrol-related constraints. 2.5.2

Use of additional neutrons

We show in section 4.1.2 that the additional number of neutrons provided by an ADSR is the number of initial spallation neutrons N0 whatever the value of the multiplication coefficient ks . It is then possible to estimate the additional transmutation capabilities of an accelerator. Typically an accelerator can provide 1:5 1025 neutrons/year/mA. Thus if all neutrons are captured in the nuclei to be transmuted, one gets an annual transmutation rate of 25 moles/year/mA. For example, the production rate of 233 U would be around 5 kg/year/mA. One sees that a 20 mA accelerator, as typically considered for ADSRs, would produce 100 kg/year of 233 U, in addition to the production of the subcritical system, which in the best case amounts to 50 kg/GWe/year. Thus ADSRs can increase quite significantly the 233 U breeding capabilities. For 239 Pu the gain is smaller since a fast breeder can produce up to 150 kg/GWe/year. Similarly around 2.5 kg of long-lived fission products could be transmuted each year per mA proton beam. Since a typical 1 GWe reactor produces each year 4 kg of 129 I, for example, a 20 mA accelerator could transmute annually the production of more than ten reactors.

Chapter 3 Elementary reactor theory

Before describing the physics specific to hybrid reactors, it is appropriate to review the basics of nuclear reactor theory. We first recall some elements of neutron physics which apply to both critical and subcritical systems.

3.1 3.1.1

Interaction of neutrons with nuclei Elementary processes

In a nuclear reactor neutrons are produced, slowed down and captured. Furthermore, energy is produced by the fission process and, to a lesser extent, by radioactive decay. The most important nuclear characteristics of a nucleus present in a reactor are therefore: The fission cross section F . The capture or (n,Þ cross section c . . The number of neutrons emitted following the capture of a neutron by a fissile nucleus. This quantity is crucial to the possibility of establishing a chain reaction. It can be split into two factors: the probability that an absorption leads to fission, F =ðF þ c Þ ¼ 1=ð1 þ Þ where ¼ c =F ; and the mean number of neutrons emitted per fission. Thus, ¼ =ð1 þ Þ. . The scattering cross sections, either elastic s or inelastic in , which control the propagation of neutrons in the medium. . The atomic mass of the nucleus A which controls the amount of slowing down of the neutron following an elastic scattering. After scattering at an angle in the centre of mass, the final laboratory energy of a neutron, whose initial energy is E0 , is given by E A1 2 A1 2 Ef ¼ 0 1 þ þ 1 cosðÞ : ð3:1Þ Aþ1 Aþ1 2 . .

The absorption cross-section a ¼ c þ F .

39

40

Elementary reactor theory If the scattering in the centre of mass is isotropic it follows that all final energies between E0 and E0 f½ðA 1Þ=ðA þ 1Þ2 g are equiprobable. Since, in this latter case, the neutron energy loss is proportional to its initial energy, it is convenient to measure energies in terms of lethargy u ¼ lnðE0 =EÞ where E0 is some arbitrary initial energy (usually the average energy of fission neutrons), and E is the actual neutron energy. We define A1 2 $¼ : ð3:2Þ Aþ1 It is convenient to define the average lethargy gain, or, equivalently, average logarithmic energy loss per collision ð $E0 E dE $ ln $ ð3:3Þ ¼ ¼1þ ln 0 E 1 $ E ð1 $Þ E0 0 which expressed as function of the mass A yields ¼ 1 þ ðA 1Þ2

1 A1 ln : 2A A þ 1

ð3:4Þ

For large A, ’ 2=ðA þ 23Þ. 3.1.2

Properties of heavy nuclei

In the context of fission reactors, the properties of nuclei heavier than thorium are of paramount importance. In particular a distinction is made between fissile and fertile nuclei. This distinction is based on the response of these nuclei to the absorption of a slow neutron: while fissile nuclei have a high probability of fissioning after such absorption, as shown in figure 3.1, fertile nuclei do not, although they have significant fission cross-sections for neutrons with energy in the MeV range, as shown in figure 3.2. Neutron capture by fertile nuclei eventually leads to the production of a fissile species, usually following beta decay. The best known examples of fissile nuclei are 233 U, 235 U and 239 Pu. Typical fissile nucleus production processes following neutron capture by fertile species are:

232

Th þ n ! 233 Th ! 233 Pa ! 233 U 22:3 min 26:97 days

238

U þ n ! 239 U ! 239 Np ! 239 Pu: 23:45 min 2:35 days

ð3:5Þ ð3:6Þ

Figure 3.3 shows that the capture cross-sections, above the resonance region, decrease sharply with energy. As a rule, heavy nuclei with an even number of neutrons are fertile while those with an odd number of neutrons are fissile. This is the result of the even/odd effect on neutron binding energies as well as of the

Interaction of neutrons with nuclei

41

Figure 3.1. Fission cross-sections of fissile nuclei.

fact that fission barrier heights lie between odd and even neutron binding energies. Aside from fission and capture cross-sections, the values of are very important in order to assess the potentialities of the nuclei to sustain a chain reaction. Variations of with neutron energy are shown for some nuclei in figure 3.4. The figure shows that 233 U has a particularly high value of at low neutron energies, while, at high energies, 239 Pu takes the lead. Indeed, only 233 U has allowed breeding in a thermal neutron reactor, the Molten Salt Reactor Experiment at ORNL [49]. Here the breeding rate was barely 5% per year and was only obtained with an online extraction of the neutron capturing 233 Pa. Breeding is obtained much more readily with fast neutron reactors using 239 Pu as fuel, a rate of 18% per year having been reached with Superphenix. Macroscopic cross-sections Nuclear reactors are macroscopic media where neutrons are propagated. It is, therefore, worthwhile to define macroscopic entities characteristic of the neutronic properties of the medium. We consider a homogeneous mixture of different nuclei i in number N. Let ni be the number of nuclei i ðÞ per unit volume (usually cm3 ). Let i be the cross-section of type ðÞ (for example fission, absorption, capture or scattering) of nucleus i. The

42

Elementary reactor theory

Figure 3.2. Fission cross-sections of fertile nuclei.

macroscopic cross-section is defined as: ðÞ ðcm1 Þ ¼ 1024

N X

ðÞ

ni i

ðbarnsÞ:

ð3:7Þ

i

The mean free path for reaction is simply ðÞ ¼

3.1.3

1 : ðÞ

ð3:8Þ

Neutron density, flux and reaction rates

In reactor physics it is customary [48] to define the number of neutrons per unit volume, per velocity bin and solid angle unit nðr; v; ; tÞ such that the number of neutrons in volume d3 r, at a position r; with a velocity between v and v þ dv pointing in direction (unit vector) within solid angle d2 is nðr; v; ; tÞ d3 r dv d2 : The flux of neutrons is defined as ðr; v; ; tÞ ¼ vnðr; v; ; tÞ:

ð3:9Þ

The number of neutrons per time unit with velocity v and direction which cross a planar unit surface at position r and unit normal vector u is:

Interaction of neutrons with nuclei

43

Figure 3.3. Capture cross-sections of 232 Th and 238 U.

ðr; v; ; tÞ u: The case where is isotropic is particularly interesting, since it is nearly satisfied in reactors. Then, if measuring angles with respect to the normal vector u; we obtain the total number of neutrons crossing the surface, regardless of their direction, by ð =2 4 ðr; v; ; tÞ cosðÞ sinðÞ d ¼ 2 ðr; v; ; tÞ: 0

We now consider a thin slab with unit surface and an atomic thickness of ns identical nuclei per unit surface. The nuclei have reaction cross-section : The number of reactions in the slab per unit time reads ð =2 ns cosðÞ sinðÞ d ¼ 4 ðr; v; ; tÞns : nreac ¼ 4 ðr; v; ; tÞ 0 cosðÞ The total flux is the directional flux integrated over angle, thus ’ðr; v; tÞ ¼ 4 ðr; v; ; tÞ: This quantity is usually called the neutron flux. In term of this quantity, the number of reactions per time unit is thus nreac ¼ ns ’ðr; v; tÞ

ð3:10Þ

44


Figure 3.4. Energy dependence of for the principal fissile nuclei.

while the number of neutrons crossing a unit surface plane per time unit is 1 2 ð’ðr; v; tÞÞ. Instead of computing the number of reactions per unit time, it is instructive to compute the total length travelled by all neutrons traversing a thin unit surface slab, which we assume to have thickness l, that is very small compared with the transverse dimensions of the slab. Then the total length travelled by the neutrons is ð =2 L ¼ 4 ðr; v; ; tÞl sinðÞ d ¼ ’ðr; v; tÞl 0

where the volume V of the slab is simply l, since it has unit surface. Thus we obtain an expression for the flux: ’ðr; v; tÞ ¼

L : V

ð3:11Þ

We have given this derivation of the neutron flux properties because the confusion is often made that it is the number of neutrons crossing the unit surface plane per time unit. In fact it is the number of neutrons crossing a sphere of unit surface cross-section. This is simply obtained if one remembers that each neutron crosses the sphere twice. Equivalently it is the number of neutrons crossing a unit surface circle kept perpendicular to the neutron direction.

Neutron propagation

45

The formula was demonstrated for an infinitely thin slab. However, for isotropic neutron fluxes, it can be generalized to any arbitrary volume. Indeed, any volume can be subdivided, at will, into n small, thin, parallelograms. For each of the elementary volumes we have the flux value ’i ¼ Li =Vi . The average flux over the volume is the average of the ’i weighted by the volume, P P V’ Li L ð3:12Þ ¼ h’i ¼ P i i ¼ V V Vi since the total length travelled by the neutrons is clearly the sum of the elementary lengths. This is an important formula since it is the one used in all Monte Carlo simulations. It can be shown that this formula also holds for anisotropic neutron fluxes. Note that the definition of the neutron flux is different from the usual definition of flux in other fields of physics. For example, in thermodynamics, a finite heat flux through a surface requires a temperature gradient across this surface. The analogy of the heat flux in neutronics is the neutron current defined as ð Jðr; v; tÞ ¼

ðr; v; ; tÞ d2 : ð3:13Þ ð4 Þ

This current is only different from 0 for anisotropic neutron fluxes. One-sided currents are also frequently used and defined by ð

ðr; v; ; tÞ d2 ð3:14Þ J þ ðr; v; tÞ ¼

N>0

J ðr; v; tÞ ¼

ð

ðr; v; ; tÞ d2

ð3:15Þ

N 0 and z < 0 have opposite directions. This means that only odd terms will be involved in the total neutron current, i.e. the term r cosðÞð@’=@zÞ0 . Terms from z < 0 and z > 0 are equal, so that it suffices to evaluate the contribution of the z < 0 region and double it. In the absence of external sources, neutrons crossing surface dAz come either from a scattering or a fission event. In a first approach we neglect the contribution of fission neutrons, assuming that s f . We also assume that the time delay between neutron scattering and the arrival of the neutron at the origin is negligible. The number of neutrons scattered in an elementary volume dV is s ’ðr; v; tÞ, and those heading towards the surface dAz , assuming isotropic laboratory scattering, is s ’ðr; v; tÞ cosðÞ dAz dV : 4 r2 On their way towards surface dAz , these neutrons may undergo a reaction which takes them out, thus the number of neutrons reaching the surface is eT r s ’ðr; v; tÞ cosðÞ dAz dV : 4 r2 Taking into account the remarks of the preceding paragraph we obtain the neutron current, ð 2

ð

ð1 s @’ 2 Jz ¼ d cos ðÞ sinðÞ d r eT r dr 2 @z 0 ¼ 0 ¼ =2 r¼0 and @’ Jz ¼ s2 : ð3:22Þ 3T @z 0 Similar relations hold for the other components of J; so that we get Fick’s law, equation (3.21), with D ¼ s2 : ð3:23Þ 3T Note that, for a s , D ¼ 1=3s .

This circumstance is responsible for the validity of Fick’s law up to second order.

Neutron propagation

49

Although our derivation of Fick’s law assumed isotropic scattering in the laboratory frame, it is in fact possible to extend its validity to the case of moderately anisotropic scattering. In particular if a s , D ¼ 1=3s ð1 Þ where is the average value of the cosine of the scattering angle. The derivation also assumed an infinite, homogeneous medium. It is in fact valid when applied in regions several mean free paths away from the medium’s boundary. It is even valid at the boundary between two media, provided the absorption cross-section is small. Similarly Fick’s law is valid in the presence of external sources, in regions sufficiently far from the sources (several mean free paths). One of the most serious limitations of Fick’s law, in its simple, monogroup, form, is that it assumes no velocity modification after scattering. This is true in thermal reactors where neutron spectra can be considered to have reached an equilibrium in which up-scattering is as probable as down-scattering. It may also be true for low lethargy fast flux reactors. In both cases Fick’s law can be used to simplify the Boltzmann equation to a diffusion equation. 3.2.4

Diffusion equation

The first term of the right-hand side of equation (3.18) reads: divðJðr; v; tÞÞ ¼ Dr2 ’ðr; v; tÞ. The diffusion equation is obtained from the Boltzmann equation when neutrons are assumed to be monoenergetic, or, in other words, to belong to a single group. The integration† over velocities in equation (3.18) can be dropped, giving X X ð jÞ @’ðr; tÞ ðiÞ 2 ¼ Dr ’ðr; tÞ þ ’ðr; tÞ i f ðrÞ a ðrÞ þ Sðr; tÞ: v @t j i ð3:24Þ Note that we have replaced T in equation (3.18) by a , since diffusion has no effect on the flux in one-group formalism. Using relation (3.75), equation (3.24) can be written as X ð jÞ @’ðr; tÞ ¼ Dr2 ’ðr; tÞ þ ’ðr; tÞ a ðrÞðk1 1Þ þ Sðr; tÞ: ð3:25Þ v @t j Equation (3.25) leads to a few interesting remarks. In an infinite and homogeneous medium, with an evenly distributed neutron source, the Lethargy is the logarithm of the neutron energy. Low lethargy media are such that the energy (lethargy) change after a collision is small. This is true for high mass nuclei. † We suppose that the slowing down distance for the neutrons is negligible. A better, but still simple, handling can be obtained using the Fermi age theory as described in standard books like that of Lamarsh [55] and Bussac and Reuss [48].

50


equation should not include derivatives of ’ðr; tÞ, since ’ðr; tÞ should be independent of r. Thus equation (3.25) simplifies to X ð jÞ @’ðtÞ ¼ ’ðtÞ a ðk1 1Þ þ SðtÞ: v @t j

ð3:26Þ

Consider first the case that, for t > 0, SðtÞ ¼ 0 and ’ð0Þ is finite. Then equation (3.26) has the solution X ð jÞ ’ðtÞ ¼ ’ð0Þ exp vðk1 1Þt a ð3:27Þ j

which shows that if k1 > 1 the flux diverges, while it decreases to 0 for k1 < 1. It is time independent only if k1 ¼ 1: This condition can never be met in reality. Rather, in critical reactors, a time-dependence of the absorption cross-sections is used, so that k1 fluctuates about 1. Equations (3.26) and (3.27) have a simple interpretation if one defines the neutron lifetime as the mean time separating creation and absorption of the neutron,

¼

a 1 ¼ X ð jÞ v v a

ð3:28Þ

j

then, equation (3.26) reads @’ðtÞ ðk 1Þ ¼ ’ðtÞ 1 þ SðtÞ @t

ð3:29Þ

omitting the source term. This equation simply expresses that every time a neutron disappears, k1 neutrons are re-emitted, the average time between two neutron absorption events being the time : With the same notations equation (3.27) reads ’ðtÞ ¼ ’ð0Þ eðk1 1Þt=

ð3:30Þ

which shows that the characteristic evolution time of the neutron flux is

=jk1 1j: Other quantities like neutron density nðtÞ, fission rates, specific power WðtÞ, obey the same type of evolution. Consider, now the case where k1 < 1, and SðtÞ ¼ S0 is time independent, but positive. The solution of equation (3.26) for stationary states reads ’¼

S0 X ð jÞ : ð1 k1 Þ a j

ð3:31Þ

Neutron propagation

51

The number of absorption reactions per second is then nreac ¼

X j

ð jÞ

a

S0 S0 X ðiÞ ¼ ð1 k1 Þ ð1 k1 Þ a

ð3:32Þ

j

which agrees with equation (3.71). Boundary conditions The treatment of finite systems requires that boundary conditions be defined. In this simplified discussion we consider the case of a homogeneous medium surrounded by a vacuum. At the boundary of the medium, there is only an outgoing one-sided flux Jþ while J ¼ 0: This means that the current J ¼ Jþ J > 0 and thus that, since gradð’Þ > 0, ’ decreases from the inner to the outer region. Extrapolating ’ linearly in the vacuum region, where the diffusion equation is not valid, the extrapolated value should vanish at some distance dextra . By comparison with exact calculations one finds that ’ is a good approximation of the true solution for dextra ’ 2D: This is usually very small compared with the multiplying medium size so that a simple, but sufficient, approximation of ’, at least for qualitative discussions, is obtained by requiring it to vanish at the boundaries of the medium. To illustrate this, we solve the diffusion equation for a semi-infinite homogeneous reactor bounded by two parallel planes [55]. One-dimensional time-dependent diffusion equation Slab reactor. The diffusion equation reduces to a one-dimensional equation d’ðx; tÞ d2 ¼ D 2 ’ðx; tÞ þ ’ðx; tÞa ðk1 1Þ þ SðxÞ v @t dx

ð3:33Þ

where we have used a single absorption cross-section a , independent of x, and a plane neutron source at position x ¼ 0. At the boundaries x ¼ a=2, we require ’ðx ¼ a=2; tÞ ¼ 0. It is, therefore, convenient to use a Fourier development of ’ and , ’ðx; tÞ ¼

X n odd

An ðtÞ cos Bn x;

ðxÞ ¼

2X cos Bn x; a n odd

with Bn ¼ n =a ðn ¼ 1; 3; . . .Þ. The coefficients An ðtÞ are obtained by solving the equations dAn ðtÞ S ¼ ðDB2n þ a ðk1 1ÞÞAn ðtÞ þ 2 : v dt a

ð3:34Þ

52


If S ¼ 0; the solution is D An ðtÞ ¼ An ð0Þ exp k1 1 B2n a vt: a For k1 < 1 þ B21 ðD=a Þ ¼ 1 þ ð 2 D=a2 a Þ all terms vanish exponentially. For k1 > 1 þ ð 2 D=a2 a Þ, the first term, and possibly some other low order ones, increases exponentially. The reactor becomes critical for k1 ¼ 1 þ ð 2 D=a2 a Þ; in this case A1 ðtÞ becomes time independent, while higher order terms decrease exponentially. Therefore, the neutron flux distribution becomes time independent and is a solution of the time-independent diffusion equation D

d2 ’ðx; tÞ þ ’ðx; tÞa ðk1 1Þ ¼ 0 dx2 d2

2 ’ðx; tÞ þ ’ðx; tÞ ¼ 0 dx2 a2

ð3:35Þ

which has the form ’ðxÞ ¼ A1 cos

x : a

Simple solutions are also obtained for spherical and cylindrical reactors. Spherical reactor. For spherically symmetric systems the time-independent diffusion equation reads 1 d 2 d’

2 r þ 2 ’ðrÞ ¼ 0; 2 @r R r dr

ð3:36Þ

R being the radius of the reactor. The solution satisfying the boundary conditions is

’ðrÞ ¼ A

sinð r=RÞ r

ð3:37Þ

2 D : R2 a

ð3:38Þ

and the critical equation reads k1 ¼ 1 þ

Cylindrical reactor. Similarly, for infinite cylindrical systems 1 d d’ r þ B2 ’ðrÞ ¼ 0 r dr dr

ð3:39Þ

Neutron propagation

53

Table 3.1. Fission and capture cross-sections (barns) averaged over a PWR neutron spectrum [56]. PWR spectrum Nuclide

Fission

Capture

235

40.62 0.107 101.02 0.44 109.17 0.28 0.462 0.092 0.62

11.39 1.03 42.23 109.39 37.89 57.55 11.51 72.257 29.261

U U 239 Pu 240 Pu 241 Pu 242 Pu 243 Pu 243 Am 244 Cm 238

whose solution is ’ðrÞ ¼ AJ0 ðBrÞ; with J0 the ordinary Bessel function of order 0. J0 ðBRÞ ¼ 0. This condition is fulfilled for a number of values, the smallest being BR ¼ 2:405. One-group cross-sections The solution of the diffusion equation (3.24) requires that the one-group cross-sections be known. Some of the most important fission and capture cross-sections for heavy nuclei are given in tables 3.1 and 3.2. Table 3.1 gives the cross-sections for a PWR spectrum while table 3.2 gives the cross-sections for a fast reactor of the Superphenix type. 3.2.5

Slowing down of neutrons

Slowing down of neutrons is, in general, a complicated problem which does not allow analytical formulation. However, for hydrogen and very heavy scatterers simple treatments are possible, and we give two examples in the following. Energy spectra We give a very simple treatment of neutron slowing down under the simplifying assumptions: . .

No absorption No direct contribution of the neutron source at energy E

54


Table 3.2. Fission and capture cross-sections (barns) averaged over an LMFBR neutron spectrum [56]. Superphenix spectrum

Superphenix spectrum

Nuclide

Fission

Capture

Nuclide

Fission

Capture

232

0.0137 0 2.742 0.51 2.03 0.116 1.82 0.0428 0.36 3.1 0.36 1.38 1.85 0.354 2.49

0.444 0.8 0.257 0.45 0.566 0.663 0.41 0.296 0.765 0.36 0.828 0.211 0.503 0.415 0.432

242

0.278 2.03 0.28 0.463 1.83 0.237 3.25 0.42 0.32 0.412 2.45 0.3 2.15 0.293

0.342 0.568 0.34 0.300 0.403 0.555 0.210 0.38 0.4 0.373 0.4 0.302 0.362 0.306

Th Pa 233 U 234 U 235 U 236 U 237 U 238 U 237 Np 238 Np 239 Np 238 Pu 239 Pu 240 Pu 241 Pu 233

Pu Pu 244 Pu 241 Am 242 Am 243 Am 241 Cm 242 Cm 243 Cm 244 Cm 245 Cm 246 Cm 247 Cm 248 Cm 243

Isotropic elastic scattering No inelastic scattering . Monatomic medium . .

We consider an infinite medium with a source of N0 neutrons/s. The number of neutrons which are scattered each second from an energy larger than E to an energy smaller than E is evidently equal to N0 . A neutron of energy E 0 can be scattered equiprobably to reach energies between E 0 and $E 0 , $ being defined in equation (3.2). It follows that the number of collisions falling below energy E is ð E=$ E $E 0 N0 ¼ s ðE 0 Þ 0 nðE 0 Þ dE 0 ð3:40Þ E $E 0 E ÐÐÐ where nðEÞ dE ¼ dE ’ðr; EÞ d3 r is the integral of the neutron flux over the whole system with energy E within dE. Assuming that s ðEÞ is independent of E, we see that ð d E=$ E $ ð3:41Þ nðE 0 Þ dE 0 0 dE E E 0 $E 0 1 $ which reads ð E=$ E

1 nðE 0 Þ dE 0 nðEÞ 0 E 0 ð1 $Þ

ð3:42Þ

Neutron propagation

55

which, after differentiation, gives d nðE=$Þ nðEÞ 1 1 E 1 nðEÞ ¼ ¼ n nðEÞ ð3:43Þ dE Eð1 $Þ Eð1 $Þ ð1 $Þ E $ E which is realized if nðEÞ ¼

C : E

ð3:44Þ

To determine C we use ð E=$

E $E 0 nðE 0 Þ dE 0 E 0 $E 0

ð3:45Þ

s C ð1 $ þ $ ln $Þ: ð1 $Þ

ð3:46Þ

N0 ¼ s

E

and get N0 ¼ Thus, nðEÞ ¼

N0 ð1 $Þ N0 ¼ s ð1 $ þ $ ln $ÞE s E

ð3:47Þ

where has been defined in equation (3.3). One observes the so-called 1=E slowing down spectrum, and that the neutron flux is approximately proportional to the mass of the scatterer. This allows us to consider the use of a heavy medium for fission product transmutation in the resonance region, as tested in the TARC experiment [57]. Similarly, the case of an absorbing medium can be treated for these two extremes. We treat the case of the heavy scatterer which will present itself later. We give a simple derivation of the evolution of the neutron distributions profiles as a function of energy and distance to the source, based on the random walk process. The random walk process. Suppose that a neutron is created at x, y, z, t ¼ 0. It suffers collisions with mean free path . The collisions are isotropic so that one can write, after n collisions, xn ¼

n X i¼0

xi ;

yn ¼

n X i¼0

yi ;

zn ¼

n X

zi :

i¼0

Since the signs of the are positive or negative with equal probability one has hxn i ¼ h yn i ¼ hzn i ¼ 0: The average distance travelled by a neutron between two successive collisions is : The average of the square of the distance is 22 . Thus h2x þ 2y þ 2z i ¼ 22 so that hx2n i ¼ 23n2 ; with similar expressions for the other coordinates. It follows that, after n collisions, the probability

56


distributions are given by a normal distribution (we only consider the x distribution): 1 3x2 Pðn; xÞ ¼ qffiffiffiffiffiffiffi exp : ð3:48Þ 4n2 43 n If one considers the distance r to the source, one gets, for the density distribution, 1 3r2 exp : ð3:49Þ Pðn; rÞ ¼ 4n2 3 ð43 nÞ3=2 For very heavy scatterers it is a reasonable approximation to assume that the energy is decreased by a fixed relative amount, following each collision. This is given by E 1 $ 2m ¼’ ’ : E 2 M

ð3:50Þ

Since E=E ¼ ln E, the number of collisions required for the neutron to decrease its energy from E0 to E is 1 E n ¼ ln 0 : E Thus the spatial distribution reads: PðE; rÞ ¼

1

2

0

ð3:51Þ

3r2

13

C7 : 6expB 2 E0 A5 2 1 E0 3=2 4 @ 4 log log E 3 E

ð3:52Þ

3

The root-mean square radius reads sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 E0 ln ðr : EÞ ¼ 3 E and the r probability distribution thus reads more simply as 1 r2 PðE; rÞ ¼ : exp 2 2 ðr : EÞ ð2 Þ3=2 3 ðr : EÞ

ð3:53Þ

ð3:54Þ

This result is, essentially, similar to that of the Fermi age theory, which reads [55] 2

qðr; Þ ¼

er =4 ð4

Þ3=2

ð3:55Þ

where is the Fermi age; qðr; Þ is the slowing down density, i.e. the number of neutrons at position r and age which cross the energy E. The Fermi age is

Neutron propagation

57

defined as

¼

ð E0 E

DðEÞ dE s ðEÞ E

with DðEÞ ¼ 1=½3s ðEÞð1 Þ and ¼ 23 =A. will be neglected for large A, such as in the case of lead. Thus, for D and s independent of E, 22 ðr : EÞ ¼

42 E0 4 E ln ¼ ln 0 ¼ 4 2 3 E 3s E

ð3:56Þ

which shows the identity of the age distribution with that obtained from the random walk calculation. Since the age equation accepts a dependence of D and s on energy, it is more general than what we have just deduced, so we shall keep it. The relation between and n is obtained for constant D and s :

¼

2 E0 n2 ln ¼ : 3 E 3

Inclusion of absorption in the model. Every time a collision takes place the neutron may be absorbed; it survives with probability ðEÞ ¼

s ðEÞ 1 ¼ : a ðEÞ þ s ðEÞ 1 þ a ðEÞ=s ðEÞ

ð3:57Þ

It follows that the probability distribution after n collisions becomes Qðr : nÞ ¼ Pðr : nÞ

n Y

ðEi Þ:

ð3:58Þ

1

In the limit when the macroscopic absorption cross-section remains always small, compared with a constant scattering cross-section, one obtains X n a ðEi Þ Qðr : nÞ ¼ Pðr : nÞ exp : ð3:59Þ s ðEÞ 1 Since the average energy interval is Ei the discrete sum may be approximated by an integral as ð En n X a ðEi Þ a ðEÞ ¼ dE: ð3:60Þ E ðE Þ E s i s ðEÞ 0 1 Thus, we obtain the age distribution, including weak absorption, 2 ð Eð Þ r a ðEÞ dE exp exp 4 E0 Es ðEÞ qðr; Þ ¼ : ð4

Þ3=2 A similar expression is obtained for hydrogen scattering medium.

ð3:61Þ

58


Absorption by a strong resonance The potential of transmutation in a resonance region [57] comes from two effects: 1. The presence of very strong resonances in the nucleus to be transmuted. 2. The fact that the change of energy in very heavy scatterers like lead is very progressive, allowing several chances for the slowing down neutron to interact with a nucleus to be transmuted. We give a schematic handling of the effect of a very strong absorption resonance characterized by its width ; its energy ER and a cross-section abs . We want to compute the number of captures in the absorbing nucleus per incident neutron, as a function of the concentration of the absorber. The Breit and Wigner formula reads ðEÞ ¼

0 0 ¼ 2 1 2 1 þ ½ðE ER Þ =ð2 Þ 1 þ x2

ð3:62Þ

with x ¼ ðE ER Þ=2. A neutron at energy E enters the medium that has na absorbing nuclei and a macroscopic scattering cross-section of ð0Þ s . We define a ¼ na 0 and ðEÞ ¼ na ðEÞ. The probability that the neutron survives an interaction, i.e. that it is not captured, is Psurv ¼ s ðEÞ=½s ðEÞ þ ðEÞ. After n interactions the survival probability is Psurv ¼

Y i ¼ 1;n

Y s ðEi Þ 1 ¼ s ðEi Þ þ ðEi Þ i ¼ 1;n 1 þ ½ðEi Þ=s ðEi Þ

thus ln Psurv ¼ ¼

X

X

ðEi Þ ln 1 þ s ðEi Þ

lnð1 þ

x2i Þ

X

ln 1 þ

ð0Þ

x2i

a þ s

the interval between two successive values of Ei is ER and that between two xi is, accordingly, 2ER =. Using the integral approximation of the sum, and integrating from 0 to 1, we get ln Psurv

¼ 2ER

ð1

ð0Þ

a lnð1 þ x Þ ln 1 þ x þ s 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0Þ

a 1þ 1 ¼ ER s 2

2

dx

ð3:63Þ

Neutron propagation

59

and Psurv

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 01 : ¼ exp ER s

ð3:64Þ

Slowing down spectrometer When the value of is small, the number of collisions needed to slow down a neutron from 2 MeV, for example to thermal energies, becomes large. For lead, for example, 760 collisions are needed. Since the mean free path in lead is around 3 cm, the neutron travels as much as 23 m before reaching thermal energies. It is clear that this process takes a long time, hence the possibility to use this slowing down time to characterize the neutron. We give a schematic derivation of the relation between the neutron velocity and energy and the slowing down time [58]. The average velocity loss at each collision is v=v ¼ 12 . The velocity after n collisions is simply vn ¼ v0 ð1 12 Þn :

ð3:65Þ

Assuming a constant scattering cross-section and mean free path ; the moment at which collision n occurs is therefore, on the average, 1 1 1 þ þ þ tn ¼ 1þ v0 1 12 ð1 12 Þ2 ð1 12 Þn 1 ¼

1 ð1 12 Þn þ 1 1 1 n 2 ð1 2 Þ

ð3:66Þ nþ1

tn ¼

1 ð1 12 Þ : v0 12 ð1 12 Þn

ð3:67Þ

Using equations (3.67) and (3.65) and dropping the index n one gets

v¼

v0 ð1 12 Þ ð12 v0 =Þt þ ð1 12 Þ

ð3:68Þ

and E¼

E0 ð1 12 Þ2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðð12 =Þ ð2E0 =mÞ t þ ð1 12 ÞÞ2

which is commonly written as E¼

K ðt þ t0 Þ2

ð3:69Þ

60


with K¼

2m2 ð1 12 Þ2 ; 2

t0 ¼ ð1 12 Þ

2

rffiffiffiffiffiffiffiffi m : 2E0

For a lead scatterer, the TARC experiment [57] provided EkeV ¼

172 ðtms þ 0:3Þ2

ð3:70Þ

in good agreement with Slovacek et al. [58] who calculated K ¼ 165.

3.3

Neutron multiplying assemblies

In nuclear reactors the fission of a nucleus results from the absorption of a neutron. This fission is accompanied by the emission of neutrons, with between 2.2 and 3, depending on the fissioning species. These neutrons, in turn, may induce fissions, and thus produce new neutrons. However, each neutron does not produce a fission. It may be absorbed either in a non-fissile or in a fissile nucleus without fission of the said (fission probability after neutron absorption by a fissile nucleus is never 100%). A neutron created in a medium (which we first consider infinite) containing fissile nuclei will give birth to k1 second-generation neutrons. The number of neutrons of the third generation will be k21 and that of generation n, kn1 1 . Each neutron generation is the result of a neutron-producing nuclear reaction which can be a fission or, more rarely, an (n; xn) reaction. The total number of neutrons following the apparition of a neutron in the multiplying medium will be nchain ¼ 1 þ k1 þ k21 þ þ kn1 þ ¼

1 : 1 k1

ð3:71Þ

The total number of neutrons created in the medium per source neutron is simply k1 nchain . One defines a neutronic ‘gain’ as the ratio of the total number of neutrons (source þ created) to the number of source neutrons. This gain is then 1=ð1 k1 Þ. Since all neutrons are ultimately absorbed, the number of absorption reactions is thus nreac ¼ nchain : For finite media one has to replace k1 by an effective value of keff † which is less than k1 due to neutrons escaping from the system. One should also consider local values, ks , dependent on the specific location of the apparition of the initial neutron. If keff is larger than unity the reaction diverges, i.e. from †

For k1 < 1. See section 4.3.3 for a more systematic discussion of the different multiplication factors.

Neutron multiplying assemblies

61

one initial neutron the final number of neutrons goes to infinity. A controlled divergence allows one to start a reactor. When uncontrolled it leads to a criticality accident as in Chernobyl. Of course, in the case of nuclear weapons, the divergence is sought. When keff is kept equal to unity one obtains a critical reactor. The possibility to keep precisely the condition keff ¼ 1 is due to the presence of a small fraction of delayed neutrons which allow time to compensate for deviation of the criticality coefficient keff from unity. If keff is less than unity an incident neutron gives birth to a finite number of secondary neutrons. The medium is said to be multiplying. The multiplication factor is 1=ð1 keff Þ. Expression of k1 We derive an expression for k1 : For simplicity we assume that the only possible reactions are scattering, capture and fission, neglecting such reactions as (n; xnyp). Since the number k1 is the number of secondary neutrons produced, on the average, following absorption of the primary neutron one can write k1 ¼ hi

probability for fission after absorption probability for absorption

ð3:72Þ

where hi is the average number of neutrons emitted per fission. One should note that this expression is of interest only if k1 remains constant with time during the multiplication process, i.e. if the neutron spectrum itself remains time invariant. In particular this requires that the neutron of the first generation have a spectrum similar to that of fission neutrons. If this is not the case, a correction has to be made. A quantitative expression for k1 can be obtained as follows. One considers that, at a given time, the medium is immersed in a neutron flux ’ðE; rÞ; where we indicate a spatial dependence of the flux to take into account any possible inhomogeneities of the medium. Equivalent to equation (3.72) we can write k1 ¼ hi

number of fissions after absorption : number of absorptions

ð3:73Þ

In this form we can obtain the expression in terms of cross-sections: ðððð f ðE; rÞ’ðE; rÞ dE d3 r k1 ¼ hi ðððð : ð3:74Þ 3 a ðE; rÞ’ðE; rÞ dE d r

A fraction of less than 1% of neutrons is emitted by fission fragments with a delay of up to a few seconds after fission.

62


If we consider a medium involving n nuclei, and use cross-sections averaged over r and E, as in equation (3.74), we can write X ðiÞ i f i k1 ¼ X

ðiÞ

ð3:75Þ

:

a

i

Consider the simple case where the medium involves only three types of nucleus, one fissile, one fertile and one absorbing. Then ðfisÞ

k1 ¼

ðfisÞ

f ðfisÞ

a

ðfertÞ

þ a

ðabsÞ

þ a

¼

a ðfisÞ

a

ðfertÞ

þ a

ðabsÞ

þ a

ð3:76Þ

where we have used the relation ¼ ðf =a Þ ¼ ðf =a Þ, since it is clearly valid when there is only one fissile species. It follows that ðfisÞ

k1 <

a ðfisÞ

a

ðfertÞ

þ a

:

The number of fissile nuclei per unit volume disappearing per unit time is ðfisÞ a , while the number of such nuclei created following neutron capture ðfertÞ ðfertÞ ðfisÞ by fertile nuclei is a : Thus the breeding condition is that a > a . It follows that breeding is only possible if > 2k1 , and in particular, for critical systems, > 2. It is often useful and quite common to write k1 as a product of four factors, k1 ¼ "pf

ð3:77Þ

where " is the enhancement factor due to fertile nuclei fissions occurring by fast neutrons, f the probability that the neutron absorption occurs in the fuel, p the probability for a neutron absorbed in the fuel to be specifically absorbed by a fissile nucleus, and the mean number of neutrons emitted following an absorption in a fissile nucleus. While these definitions are valid for fast reactors, they are different for thermal reactors: " becomes the enhancement factor due to fissions of fertile and fissile nuclei by fast neutrons, p the probability that the neutron escapes capture during the slowing down process (especially in the large resonances of the fertile nuclei), f the fraction of thermal neutrons absorbed in the fuel, and the number of neutrons emitted after absorption in one of the fuel nuclei (both fertile and fissile).

3.4

Limiting values

Criticality can be obtained only if the probability of neutron escape is small enough. This condition leads to the concept of a critical minimum mass of

Limiting values

63

fuel needed to sustain the chain reaction, in the absence of an external neutron source. 3.4.1

Critical masses

We give a schematic determination of critical sizes and masses of two model homogeneous reactors: a lead cooled fast neutron reactor and a heavy-water moderated thermal neutron reactor. Fast reactor In the one-group diffusion formalism, the critical size of a spherical homogeneous reactor is given by equation (3.38) k1 ¼ 1 þ

2 L2c R2

ð3:78Þ

with L2c ¼

D s ¼ a 3a 2T

ð3:79Þ

which leads to the minimum size of the sphere, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Rc ¼ Lc : ðk1 1Þ

ð3:80Þ

The physical characteristics of the medium components are given in table 3.3. The relative atomic concentrations are also given in table 3.3. The relative concentration of 232 Th and 233 U are in the proportion required for regeneration: U nth Th c ¼ nu a :

ð3:81Þ

The number of lead atoms is taken as set to that of the fuel atoms. Table 3.4 gives the macroscopic cross-sections. Table 3.3. Physical properties of the elements of the model fast reactor. Cross-sections are in barns. n is the relative proportion of the element.

232

Th U Pb

233

s

a

f

n

11.72 18.95 11.35

10 10 10

0.458 2.999 0.01

0.014 2.742 0

0.435 0.065 0.5

64

Elementary reactor theory Table 3.4. Macroscopic cross-sections (cm) for the model fast reactor. s

a

f

0.327

1:28 102

5:94 103

With a value of ¼ 2:53 one gets k1 ¼

f ¼ 1: 17 a

ð3:82Þ

and sffiffiffiffiffiffiffiffiffiffiffiffiffiffi s Lc ¼ ¼ 8:6 cm 3a 2T

ð3:83Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Rc ¼ Lc ¼ 65 cm: ðk1 1Þ

ð3:84Þ

and

The mass of the fuel is around 7: 2 metric tons. Thermal neutron reactor In the thermal reactor neutrons should be slowed down before they are captured. The survival probability was expressed in equation (3.61). The survival integral reads ðE a ðEÞ Na I p ¼ exp dE ¼ exp ð3:85Þ s E0 Es ðEÞ with I¼

ðE

a ðEÞ dE: E E0

ð3:86Þ

This integral is only valid for small absorption. For E lying between fission neutron energy and thermal energy a parametric expression of the survival probability p can be used [55] a 0:033cabs 1 c p ¼ exp ð3:87Þ s with the values of a and c for 238 U and 232 Th (which are the main resonance absorbers) shown in table 3.5. cabs is the atomic concentration of the absorber nuclei.

65

Limiting values Table 3.5. Values of the parameter a and c of the effective integral parametrization [55]. 232

238

Th

8.33 0.253

a c

U

2.73 0.486

For heavy water ¼ 0:509 while s ¼ 0:452 cm1 . Taking the case of Th and requiring that p should be equal to 0:95; we get the concentration of 232 Th atoms. 232

cabs ¼ 6 103 :

ð3:88Þ

The macroscopic absorption cross-section by the fertile ðThÞ a

Th follows:

1

ð3:89Þ

ð233 UÞ

¼ 1:47 103 cm1

ð3:90Þ

ð233 UÞ

¼ 1:35 103 cm1

ð3:91Þ

¼ 1: 47 10

3

232

cm

and, at equilibrium a f and the value of k1 k1 ¼ 1:15:

ð3:92Þ

The diffusion length has to take into account the slowing down stage. This amounts to adding half the age of the neutron given by equation (3.56) to the diffusion length Lc 2 ðr : EÞ ¼

2 E ln 0 ¼ 2 : 2 E 3s

ð3:93Þ

For thermal final energies the slowing down term reads, in the case studied, 2 ðr : Eth Þ ¼ 116 cm2

ð3:94Þ

L2c ¼ 245 cm2

ð3:95Þ

while

so that the critical radius reads sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Rc ¼ L ¼ 154 cm ð3:96Þ ðk1 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with L ¼ L2c þ 2 ðr : Eth Þ ¼ 19 cm. The critical mass of fuel is around 1: 2 tons. The lower critical mass for thermal systems is, obviously, a consequence of the higher capture and fission cross-sections.

66


The above calculations of critical masses neglected absorption in the structural elements, and, most important, by fission products. 3.4.2

Maximum flux

Because of structural constraints, the flux in critical reactors cannot be too arbitrarily high. We first discuss the maximum neutron flux which could be achieved in either a fast or a thermal reactor. This is an important quantity since the lifetime of a nucleus in a neutron flux is ’a , independent of the nucleus concentration. The maximum heat density which can be extracted provides the maximum value of the product f ’. At present, values of 500 W/cm3 are design values for liquid metal cooled reactors. This leads to a value of f ’ ¼ 500="f ¼ 1:5 1013 fissions/cm3 /s. Very high values of the flux can be obtained if f is very small. However, f cannot be arbitrarily small since the reactor has to be critical, thus k1 ¼ ff =½f ð1 þ Þ þ c g > 1 where c is the macroscopic capture cross-section of components other than the fissile part. Thus one should have f =c > 1=ð 1 Þ ’ 1. As an example of a thermal system, we consider pipes containing a molten salt immersed in a heavy water tank. The lower limit of c is given by that of heavy water, c ¼ 0:000 044: With a fission cross-section of 500 barns this corresponds to a fissile nucleus density of 0:8 1017 /cm3 . The maximum maximorum of the neutron flux in a thermal reactor is thus 3:4 1017 . Of course, due to the components of the salt and of the pipes, such a value will probably never be reached. However, fluxes ten times smaller have been considered, for example, by Bowman [2]. For such high flux the lifetime of fissile nuclei would be extremely short: 5 hours for a 1017 n/cm2 /s flux. The inventory in fissile material of the reactor would also be extremely small: for a density of fissile nuclei of 3:0 1017 =cm3 , corresponding to a flux of 1017 n/cm2 /s, the total fissile mass necessary to produce 3 GW would be only 700 g! The total volume of the reactor would be 6 m3 : For fast reactors we consider a fissile species diluted in molten lead with c ¼ 3 104 cm1 and thus a maximum maximorum flux of 5:0 1016 n=cm2 =s: For such a flux the lifetime of the fissile species would be around 3 hours. The minimum inventory for a 3 GW reactor would be 350 kg. This shows that thermal reactors have a higher potential for small inventories and fast burn-up. Of course the actual implementation of these potentialities could be very difficult. In fact, the more or less fissile nature of the fuel for thermal neutrons has a deep influence on the incineration rate achievable. This can be seen in the following more quantitative, although schematic, analysis. We consider a homogeneous infinite reactor with only two components:

Limiting values

67

1. the fuel, characterized by its atomic density nfuel , absorption cross-section ðfuelÞ a , and its neutron multiplication coefficient kfuel > 1, and 2. the coolant, characterized by its atomic density ncool , and its absorption ðcoolÞ cross-section a . The aggregate reactor is characterized by its atomic density nreac ¼ nfuel þ ncool , its absorption cross-section n n ðreacÞ ðfuelÞ ðcoolÞ a ¼ fuel a þ cool a ; nreac nreac and an effective multiplication coefficient ðfuelÞ

kreac ¼ kfuel

nfuel a ðfuelÞ

nfuel a

ðcoolÞ

þ ncool a

:

We define the atomic fraction of the fuel x ¼ nfuel =nreac . The criticality condition kreac ¼ 1 allows us to write x as ðcoolÞ

x¼

a ðfuelÞ

a

ðcoolÞ

ðkfuel 1Þ þ a

:

ð3:97Þ

Aside from the criticality condition it seems reasonable to assume that the fission density is limited to a specific value w: Thus ðfuelÞ

w¼x

a n ’ ð1 þ Þ reac

ð3:98Þ

with ðfuelÞ

¼

a

ðfuelÞ

f ðfuelÞ

:

f

Thus, the incineration rate reads inc ¼

ðfuelÞ ðfuelÞ a w w a ðkfuel 1Þ ’¼ ¼ 1þ : ðcoolÞ xnreac nreac ð1 þ Þ a

ð3:99Þ

In the case of fissile mixtures (kfuel > 1), it appears that the main difference ðfuelÞ ðcoolÞ between fast and thermal reactors lies in the ratio a =a . There is a clear advantage to using coolants with small absorption cross-sections. As ðcoolÞ examples, for heavy water ncool a ¼ 4 105 for thermal reactors, and ðcoolÞ 4 ncool a ¼ 3 10 for lead and fast spectra. Fuel absorption crosssections for thermal neutrons exceed 500 barns while they lie around 2 barns only for fast neutrons. It follows that, for fissile mixtures, incineration rates with thermal neutrons could, in principle, be three orders of magnitude larger than those with fast neutrons. The situation is different for non-fissile (minor actinides, for example) mixtures. In this case the major difference between incineration of thermal

68


and fast neutrons is that of the corresponding fuel multiplication factors. The subcritical nature of the MA fuel with thermal neutrons implies the use of an ADSR to perform the incineration. Dilution of the fuel would thus be counterproductive, since it would decrease the reactor multiplication coefficient kreac below kfuel , and thus require higher accelerator current to keep the neutron flux constant. The incineration rate reduces to the first term of the right-hand side of equation (3.99), i.e. w inc ¼ ð3:100Þ nreac which means that it depends, essentially, on the fission density. Indeed, because of the condensed nature of the components of all practical reactor designs, it is not possible to play very much on the value of nreac . For example, for water, the atomic density is 1023 =cm3 , while for lead it is 0:3 1023 =cm3 and for uranium 0:6 1023 =cm3 .

3.5

Reactor control

In order to evaluate some of the possible advantages of hybrid reactors over critical reactors, we think it useful to review how these are controlled. It is essentially through the motion of neutron-absorbing control rods that the value of the criticality coefficient keff is modified. In discussions of the evolution of a reactor it is usual to use the reactivity ¼ ðkeff 1Þ=keff . The time constant associated with the motion of control rods is, typically, measured in seconds. The time delay nf between two neutron generations is much smaller, typically 107 s for fast reactors and 104 s for thermal reactors [48]. Such numbers would imply a very fast evolution of the reactor, even for very small positive reactivities. As we saw in section 3.2.4, the reactor power increases exponentially with time t: t WðtÞ ¼ W0 exp : ð3:101Þ ð1 Þ nf For ¼ 0:01 the power is multiplied by 2 after 70 neutron generations, i.e. less than 10 ms for fast reactors and less than 10 ms for thermal reactors! With such a fast rise in the reactor power, one might think that reactor control by control rods would be hopeless. In fact, the presence of a small fraction of delayed neutrons makes things tractable. 3.5.1

Delayed neutrons

Delayed neutrons are associated with the beta decay of fission fragments. Indeed, after prompt fission neutron emission the residual fragments are still neutron rich. They undergo a beta decay chain. The more neutron rich

Reactor control

69

Figure 3.5. Illustration of the delayed neutron emission process. On the left the precursor nucleus (A, Z), in its ground state, beta decays to excited states of the possible neutronemitting nucleus (A, Z þ 1). The most excited levels of this nucleus may be above the neutron binding energy, and thus may emit neutrons, leaving a residual nucleus (A 1, Z þ 1).

the fragment, the more energetic and faster the beta decay. In some cases the available energy in the beta decay is high enough to leave the residual nucleus in such a highly excited state that neutron emission instead of gamma emission occurs. This process is illustrated in figure 3.5. The eventually emitted neutron is said to be delayed (with respect to the fission). The delay is determined by the beta decay time constant. Delays vary between fractions of a second and several tens of seconds. Probabilities for delayed neutron emission are of the order of or less than 1% per fission, or per prompt fission neutron. Beta-delayed neutron emission is highest when the emitted neutron binding energy is small. This is true when the neutron emitter has an odd number of neutrons just above a neutron shell closure. In particular beta decaying nuclei with neutron numbers equal to 52 (N ¼ 50 closed shell) and 84 (N ¼ 82 closed shell) are good delayed neutron emitter precursors. Examples are 87 Br and 137 I. Beta-delayed neutrons are characterized by their yields i , relative to the total neutron number per fission, and their decay constants

i : The total P number of delayed neutrons yieldPper fission is ¼ i . One may, also, define a mean decay time Td ¼ ð i Ti Þ=. Thus, in first approximation, the time which determines the time constant of the reactor is

nf ð1 Þ þ Td rather than nf : Table 3.6 shows the values of , Td and

d ¼ Td for a number of nuclei. The data are for fast neutron fission. We have also given the values of N=A for these nuclear species, since the more neutron rich fissioning nuclei generally lead to higher values of , but often to smaller values of Td . From table 3.6 and using equation (3.101), we see that the doubling time will range between 0.1 s and 1 s. The smaller the value of d , the more difficult

70

Elementary reactor theory Table 3.6. Properties of delayed neutrons.

232

Th U 235 U 238 U 239 Pu 241 Pu 241 Am 243 Am 242 Cm 233

Td (s)

d (s)

N=A

0.020 3 0.002 6 0.006 40 0.014 8 0.002 0.005 4 0.001 3 0.002 4 0.000 4

6.98 12.40 8.82 5.32 7.81 10 10 10 10

0.141 0.032 0.056 0.079 0.020 0.054 0.013 0.024 0.004

0.612 0.605 0.608 0.613 0.607 0.609 0.606 0.609 0.603

Values estimated by the authors.

will reactor control be. In particular reactors fuelled exclusively with minor actinides would have low values of d . Because of the important influence of the delayed neutron fraction on reactor safety it is customary to express reactivity in $ units: a positive reactivity of 1 $ is a reactivity equal to , corresponding to a multiplication coefficient keff ’ 1 þ : Of course reactivities can also be expressed in fractions of unity. The importance of the delayed neutrons in reactor safety can be seen better examining a modified neutron kinetic equation that includes the effect of delayed neutrons. Delayed neutron kinetic equation The variation of the neutron population can be written as dnðtÞ absorptions escapes þ prompt neutrons ¼ +delayed neutrons+source dt

ð3:102Þ

where nðtÞ is the total number of neutrons present in the reactor. The number Ð of neutrons with velocity u is nðtÞ ðuÞ, with 01 ðuÞ du ¼ 1. The absorption rate reads ð ð nðtÞ nðtÞ u ðuÞa ðuÞ du ¼ ha inðtÞ u ðuÞ du ¼ nðtÞha ihui ¼ ð3:103Þ

a where a ¼ 1=ha ihui is the partial lifetime of neutrons for absorption. Similarly, the escape term can be written as nðtÞ ¼ nðtÞhPihui

P

ð3:104Þ

71

Reactor control

where P is the is the partial lifetime of neutrons for escape. hPi is defined in such a way that hPi=ðhPi þ ha iÞ is the relative probability that a neutron disappears from the reactor by escape rather than by absorption. The prompt neutron production rate is ð1 Þnf ðtÞ

ð3:105Þ

where is the delayed neutron fraction. Let Xi ðtÞ be the population of delayed neutron precursor nuclei, i their decay constant, and "i their relative production of neutrons at decay time. Then the delayed neutron production rate reads X i "i Xi ðtÞ: i

We define the total neutron lifetime 1 1 1 ¼ þ

P a so that equation (3.102) takes the form: X dnðtÞ nðtÞ ¼ þ ð1 Þnf ðtÞ þ i "i Xi ðtÞ þ SðtÞ: dt

i

ð3:106Þ

The evolution equations for the Xi ðtÞ are dXi ðtÞ ¼ decays+formation by fission dt dXi ðtÞ ¼ i Xi ðtÞ þ ci nf ðtÞ dt

ð3:107Þ ð3:108Þ

where ci is the yield of fission fragment i. Given a fission event, the partial delayed neutrons yield is clearly i ¼ " i ci so that equation (3.106) becomes X dnðtÞ nðtÞ X ðtÞ ¼ þ ð1 Þnf ðtÞ þ i i i þ SðtÞ dt

ci i which can be further simplified by writing X dnðtÞ nðtÞ ¼ þ ð1 Þnf ðtÞ þ i i Yi ðtÞ þ SðtÞ dt

i Yi ðtÞ ¼

Xi ðtÞ ci

dYi ðtÞ ¼ i Yi ðtÞ þ nf ðtÞ: dt

ð3:109Þ

ð3:110Þ ð3:111Þ ð3:112Þ

72


Finally we express nf ðtÞ,

ð

nf ðtÞ ¼ nðtÞ u ðuÞf ðuÞ du ¼ nðtÞhuihf i:

ð3:113Þ

Note that keff ¼

hf i ha i þ hPi

ð3:114Þ

so that huihf i ¼ keff huiðha i þ hPiÞ ¼ nf ðtÞ ¼ nðtÞ

keff

keff

and we finally get the neutron kinetics equation system X dnðtÞ nðtÞ ¼ ð1 ð1 Þkeff Þ þ i i Yi ðtÞ þ SðtÞ dt

i dYi ðtÞ nðtÞ keff ¼ i Yi ðtÞ þ : dt

ð3:115Þ ð3:116Þ

ð3:117Þ ð3:118Þ

With Ci ¼ i Yi ¼ i

Xi ðtÞ ¼ "i Xi ðtÞ ci

one gets X dnðtÞ nðtÞ ¼ ð1 ð1 Þkeff Þ þ i Ci ðtÞ þ SðtÞ dt

i dCi ðtÞ nðtÞ ¼ i Ci ðtÞ þ i k : dt

eff

ð3:119Þ ð3:120Þ

It is instructive to consider the time-independent solutions of the system (3.119): n ð3:121Þ Ci ¼ i 0 keff i n0 ð1 keff Þ ¼ S0 ð3:122Þ

which shows that the concentration of delayed neutron precursors is proportional to keff . The number of delayed neutrons produced by unit time is ðn0 = Þkeff . This proportionality reflects the fact that delayed neutrons are only emitted by fission, the number of neutrons originating from fission in the system being n0 keff S0 ¼ S0 :

1 keff

ð3:123Þ

Reactor control

73

Equation (3.119) accepts exponential solutions. Suppose that nðtÞ ¼ nð0Þ e!t

ð3:124Þ

Ci ðtÞ ¼ Ci ð0Þ e!t

ð3:125Þ

it follows that, for SðtÞ ¼ 0, exponential solutions exist if ! 1X

i ¼ 1þ ð! þ 1Þ

i ð! i þ 1Þ i

ð3:126Þ

and X d

i ¼ þ ð! þ 1Þ i d! ð! þ 1Þ ð! i þ 1Þ i

1 !2 i : ð! þ 1Þð! i þ 1Þ

ð3:127Þ

The ð!Þ function is always positive for ! > 0: The trivial solution ¼ ! ¼ 0 corresponds to the constant neutron flux of a critical reactor. The limiting value ¼ 1 (keff ! 1Þ corresponds to infinitely fast exponential increase with ! ! 1: Figure 3.6 shows how depends upon the exponential divergence slope. It shows that the approximate calculation using average values of the delayed neutron yield and lifetime is very close to the exact calculation, except for very small values of : Both converge with the prompt curve for larger than a few $. Figure 3.7 compares the critical divergence with and without delayed neutrons for a PWR reactor. In the first case, it is seen that, for a reactivity below 1$, the evolution of the reactor is slow, with characteristic times above 1 s. For > 1$, in contrast, the evolution of the reactor becomes very fast and is comparable with that of the case without delayed neutrons. For ¼ 2$, for example, the characteristic time is less than 0.02 s. 3.5.2

Temperature dependence of the reactivity

A reactor’s safety is crucially dependent on the influence that temperature increase has on the reactivity. Indeed, if a temperature increase leads to a reactivity increase the reactor may become unstable and uncontrollable. This type of instability may be observed in certain states of the RBMK reactors, such as those of Chernobyl, discussed in some detail in Appendix 2. Several effects determine the influence of temperature on the reactivity. The most important is the variation of the number of captures in the region of the resonances. Doppler effect Fertile nuclei are characterized by large capture resonances. In heterogeneous reactors, fissile and fertile nuclei are concentrated in fuel rods.

74


Figure 3.6. Variations of as a function of the slope of the exponential increase of the neutron flux !. The variations are shown for the exact calculation defined by the values of i and i (l), the average calculation where the delayed neutron fraction is described by the average values and (T) and the prompt response (g).

Neutrons having an energy within a resonance are strongly absorbed within a short distance. As a result, not all fertile nuclei participate in the capture process. The apparent width of the resonances, as seen by a neutron moving in the medium, is increased by the thermal motion of the nuclei. Higher temperatures lead to larger apparent widths, and thus to a broader energy region where all neutrons are captured. A larger number of neutrons captured results in a smaller number of neutrons available to induce fission. Thus the Doppler effect leads to a negative value of the temperature coefficient, at least in heteregeneous reactors. dk1 < 0: dT ðDopÞ We have given a plausibility argument on the sign of the Doppler effect. More elaborate and quantitative treatment can be found in specialized books [55, 48].

Reactor control

75

Figure 3.7. Variation of the divergence slope as a function of the reactivity for the prompt (l) and the exact (g) calculations.

Effect of the temperature on the neutron energy spectrum For thermal reactors a temperature increase leads to a hardening of the neutron spectrum. In turn this leads, in general, to a decrease of the capture and fission cross-sections. The effects of these reductions depend on the specific properties of the fissile and fertile nuclei in the thermal region. For example, a standard enriched uranium fuel has a negative spectrum temperature coefficient which becomes positive with increase of the plutonium content of the fuel.

76


Dilatation effects A temperature increase tends to decrease the density of the materials. This effect is especially important for liquid coolants whose dilatation transfers matter into the expansion vessels. Thus, a temperature increase decreases the relative concentration of a liquid coolant. This may have very different effects for different systems. In PWR reactors the slowing down of the neutrons tends be less efficient because of the drop in water density. This leads to a drop of the fission probability while the capture rate in water is also decreased. These two effects are opposite but their net result is a decrease of the reactivity. In RBMK reactors the neutrons are slowed down by the graphite, while water ensures the cooling and captures a fraction of the neutrons. A temperature increase leads to a decrease of the number of captures in water which is not counterbalanced by a decrease in the fission rates; thus, the temperature dilatation effect tends to increase the reactivity. In liquid sodium cooled fast reactors the decrease of the sodium density leads to a hardening of the spectrum and, therefore, increases the fission rate, at least for large reactors. For lead cooled reactors, because of the smaller slowing down power of lead, this effect is very small. Void effect In the case of a large temperature increase, in an accidental configuration, vapour bubbles may appear in the coolant. For PWR reactors this has a negative effect on the reactivity, because of the dominant influence of the spectrum hardening. For RBMK reactors the effect is strongly positive because of the dominant effect of the decrease of the capture rate in the coolant. In sodium cooled reactors the effect on reactivity is positive, but it occurs at a much higher temperature than for PWR reactors. In lead cooled reactors the void coefficient is negative [45]. 3.5.3

Critical trip

Finally, reactors are designed so that a power increase leads to a reactivity decrease. This has the interesting consequence that the reactor is autostable. Placing the control rods in a predetermined position ensures that the reactor power will converge to a given value. Slow reactivity insertion As an example we consider the case of a PWR. The average normal temperature of the coolant is around 300 8C. For fresh fuel the reactivity change In small reactors the decrease of the coolant density leads to an increase of neutron escapes, which may decrease the reactivity.

Reactor control

77

between zero and nominal power is close to 0:016 [48]. The nominal power is taken to be 3 GWth. We make the simplifying assumptions that the temperature is proportional to the reactor’s power and that the power rise is slow enough for the equilibrium temperature to be reached at each power level. We neglect the contribution of radioactive processes to the power. The initial power is assumed to be 1 MWth. The initial value of the reactivity is taken to be ¼ 0:016.† The evolution of the power [see equation (3.101)] is given by the set of equations dW ¼ ðTðWÞÞW= D dt W Wð0Þ Wnom Wð0Þ TðWÞ TðWð0ÞÞ ðTðWÞÞ ¼ ðTðWð0ÞÞ 1 TðWnom Þ TðWð0ÞÞ

TðWÞ ¼ TðWð0ÞÞ þ ðTðWnom Þ TðWð0ÞÞÞ

where Wð0Þ is the initial power, which we chose to be Wð0Þ ¼ 1 MW. Wnom is the nominal thermal power of the reactor, which we chose to be 3000 MW. The temperature at nominal power is TðWnom Þ which we take to be 300 8C while the initial temperature is 30 8C. This set of equations reduces to dW ðTðWð0ÞÞÞ Wð0Þ W2 ¼ W 1þ dt

D ðWnom Wð0ÞÞ Wnom Wð0Þ whose solution is found to be WðtÞ ¼

a b þ eat ða bÞ

with a¼

ðTðWð0ÞÞÞ Wð0Þ 1þ

D ðWnom Wð0ÞÞ

ð3:128Þ

ðTðWð0ÞÞÞ :

D ðWnom Wð0ÞÞ

ð3:129Þ

and b¼

The result obtained with this approximate treatment for a typical PWR reactor is shown in figure 3.8. We see in the figure that power stabilization occurs within around 50 s.

Some finite value of the power, or neutron flux, is necessary to ensure divergence. This reactivity exceeds 1$, and in principle the slow power rise approximation is not justified. The treatment given here is therefore only indicative. Inserting the additional reactivity by smaller steps would be more justified but would give very similar results. †

78


Figure 3.8. Evolution of the power of a reactor starting in a supercritical state at very small energy (1 MeV). The reactivity decreases due to the negative temperature coefficient. Criticality is reached at the nominal power.

Fast reactivity insertion Consider, now, that a large, sudden reactivity is inserted into the reactor while it is running at nominal power. In this case the time to be considered between neutron generations is the prompt time, i.e. for a PWR,

nf ¼ 0:1 ms. The power increase is too rapid for the cooling system to be efficient. The cooling system is no longer efficient and melt-down of the fuel elements may occur. Boiling of the water coolant leads to a strong decrease of the reactivity and to a limitation of the power surge. In other types of reactor like RBMK and fast reactors the decrease of the reactivity only occurs with fuel dissemination. In conclusion, we see that the safety of critical neutron reactors depends strongly on the delayed neutron fraction of the fuel as well as on the temperature coefficients, especially on the Doppler temperature coefficient. These strong dependences severely constrain the type of fuel which can be used in critical reactors in safe conditions. We shall see that hybrid reactors might be very helpful in alleviating these constraints. 3.5.4

Residual heat extraction

Even if catastrophic criticality excursions are prevented by a judicious choice of the different reactivity coefficients, combined with efficient active measures, possible serious accidents, such as that of Three Mile Island, may be caused by a defective extraction of the residual heat produced in the fuel by the radioactivity of the fission fragments after reactor shutdown. Immediately after shut-down the residual heat amounts to 7% of

Reactor control

79

the heat produced at full power. This means that a 1 GWe reactor (3 GWth) produces 200 MWth of residual heat after shut-down. This value drops to 16 MWth after 1 day and 9 MWth after 5 days [37]. In principle, if the coolant is still present and the circulating system active, this residual heat is easily disposed of. However, both loss of coolant (LOCA) and a cooling fluid circulation system failure are possible and their probabilities depend on the type of reactor. Since subcritical assemblies of hybrid reactors are not different, in this respect, from critical assemblies, we discuss the properties of the most representative reactor types, as far as heat extraction is concerned. PWR and BWR reactors [37] The maximum operating temperature of the water in PWR (resp. BWR) is 325 8C (resp. 288 8C) and the pressure 155 bar (resp. 72 bar). Because of the high pressure a pipe or vessel rupture may occur and lead to partial or total loss of coolant. In such a case, emergency core-cooling systems would have to come into play. If these fail, the core will melt partially or completely. This is called a core melt accident. The probability of a core melt accident has been calculated to be around 5 105 per reactor-year for PWRs, and 4 106 per reactor-year for BWRs [59]. Continuous safety improvements have been made, taking advantage of experience, and recent safety evaluations yield a core melt probability of 105 per reactoryear for PWRs [60]. Although core melting induces a large release of radioactivity, the reactor containment structure should prevent significant release to the external atmosphere, as was indeed demonstrated in the Three Mile Island accident. The probability that, despite the containment,† significant radioactivity would be released to the exterior is one to two orders of magnitude lower than that of core melting. The risk for an individual living in the vicinity of the reactor to die from a cancer induced by accidental radioactivity release is estimated to be around 108 for today’s PWRs. Although the above numbers appear to be small or very small, some prominent experts such as Weinberg [61] have argued that, should the use of nuclear power expand again, a core melt probability of 104 (which would lead to one core melt every other year for a 5000 nuclear reactor fleet) would be socially unacceptable. It was, therefore, important to

This value was given in the Rasmussen report. Following the Three Mile Island accident improvements were made both on the equipment and on the operational procedures so that a value of 105 is claimed to be more representative. † One possibility for containment disruption is a secondary, explosive, hydrogen reaction with air. Hydrogen is produced from the thermal decomposition of water in contact with the hot zircalloy casting of the fuel elements. More efficient hydrogen extraction from within the reactor building is one of the major safety improvements considered in future reactor design.

80


design deterministically safe reactors. Such is the PIUS [62] design. The PWR reactor is immersed in a huge pool of borated water, and special passive locks ensure that the cooling water and the borated water do not normally mix. If the pressure of the cooling water becomes too high the locks open automatically and the reactor is flooded by the borated water. This would, first, make the chain reaction impossible, and second, ensure residual heat evacuation via natural convection. However PIUS would have a small thermodynamical efficiency and make inspection and maintenance difficult. Liquid metal reactors [37] The most documented type of liquid metal reactor is the sodium cooled reactor. Here the coolant is liquid sodium at a temperature of 500 8C, while the boiling temperature of sodium at atmospheric pressure is 883 8C. The pressure within the vessel is slightly above 1 atm of argon gas, in order to prevent air intrusion. The most modern sodium cooled reactors like Phenix and Superphenix [63] are of the swimming pool type. These characteristics practically prevent loss of coolant accidents. The negative temperature reactivity coefficients have been shown to lead to a break-off of the chain reaction and to a moderate temperature increase within minutes, for the small EBR2 (20 MWe) reactor. For Phenix (250 MWe) natural convection was demonstrated to evacuate the residual heat. In general it appears that sodium cooled reactors are much safer than water cooled reactors with respect to core melting. The risk for an individual living in the vicinity of the reactor to die from a cancer induced by accidental radioactivity release is estimated to be around 1010 for these reactors. The main safety concern with sodium cooled reactors is, precisely, with the use of sodium which burns spontaneously when in contact with air and dissociates water, with the consequent risk of a hydrogen explosion. The risk associated with violent sodium–water reactions explains the choice made by the former USSR to use a molten bismuth–lead eutectic as coolant for their most modern nuclear submarine reactors. It also prompted the recent proposal of the CERN group [76]. In addition lead has a boiling temperature of 1749 8C, which makes coolant boiling completely impossible, due to radiation cooling. A swimming pool type lead cooled reactor would have a very high safety level. High-temperature gas reactors [37] The largest coolant temperatures are limited to 350 8C by pressure in the case of water cooled reactors and to 600 8C by corrosion in the case of liquid metal cooled reactors. Higher temperatures would allow higher

Fuel evolution

81

efficiencies for electricity conversion, using combined cycles, as well as heat cogeneration. They might also have interesting chemical applications like thermal decomposition of water to produce hydrogen. High temperatures can be reached with a gas coolant, especially helium. These considerations were at the origin of the studies on high-temperature gas reactors (HTGRs). These reactors also have, potentially, interesting safety properties, although they use graphite as their neutron moderator like the British Windscale or the Chernobyl RBMK reactors. The high operating temperature would prevent the Wigner effect which led to the Windscale reactor accident. Using helium rather than water as coolant would ensure strong negative temperature coefficients, in contrast to the case of the water cooled RBMK reactors. The strong negative temperature coefficient ensures a breaking off of the chain reaction in the case of a loss of cooling. After reactor shutdown, the fuel element temperatures will rise until radiation cooling takes over. This is possible because fuel elements are designed to be able to sustain very high temperatures. The fuel is made of microspheres (TRISO spheres) of fissile and fertile nuclei surrounded by several layers of carbon, which ensure that no fission products can escape from the spheres. The microspheres are, themselves, embedded in carbonaceous materials which constitute the fuel rods. These are placed in graphite blocks, through which holes allow cooling gas circulation. Extensive tests were carried out in Germany, on the AVR reactor, to evaluate the behaviour of the fuel with temperature. The operating temperature is around 1000 8C. The fuel was tested at 1600 8C for several hundred hours and very small fission product release was observed. For moderate power reactors with around 150 MWe, calculations show that, in the absence of cooling, a maximum temperature of 1600 8C can be reached for a few tens of hours. The temperature is limited by radiation cooling. This is efficient because not only the total power but also the specific power of the reactor are kept small. The specific power is limited to 6 kW/l, to be compared to the 100 kW/l for PWRs. The probability of significant radiation release has been estimated to be 108 per year, i.e. three orders of magnitude less than for PWRs. The main safety concern for the HTGR is that air intrusion in the vessel would cause the graphite to burn.

3.6

Fuel evolution

During irradiation the nuclear fuel evolves due to several processes, the most significant being: . .

fission of heavy nuclides or decay

82


transformation of fertile nuclei into fissile nuclei due to neutron captures followed by radioactive decay . generation of fission products which may act as neutronic poisons. .

3.6.1

The Bateman equations

In general, the evolution of the nuclear fuel is followed by solving the Bateman equations, which read X X c dni ðtÞ ¼ Ti ’ þ i; j ni þ ðj;i ’ þ j;i Þnj dt j j 6¼ i where ni is the number of nuclei of type i per unit volume, i; j is the decay constant of nucleus i to nucleus j, ci; j is the capture cross-section of nucleus i giving nucleus j, Ti , the total cross-section of nucleus i, is the sum of the capture and fission cross-sections, ’ is the neutron flux. These equations are summarized in the vector–matrix form. dn ¼ An: dt 3.6.2

The long-term fuel evolutions

In order to stress the main trends of the evolution of the nuclear fuel we consider a model where only three types of nucleus are present: 1. the fertile nuclei (cap); 2. the fissile nuclei (fis); 3. the fission products (fp). The fuel is replenished in fertile nuclei at a rate SðtÞ. Absorption crosssections are denoted ðaÞ , and fission cross-section ðfÞ . The evolution of the nuclei is given by the system dncap ðaÞ ¼ ncap cap ’ þ SðtÞ dt

ð3:130Þ

dnfis ðaÞ ðaÞ ¼ ncap cap ’ nfis fis ’ dt

ð3:131Þ

dnfp ðfÞ ¼ nfis fis ’ dt

ð3:132Þ

where nfp is the number of fission products’ pairs. In order to discuss the dominant features of the fuel evolution we shall make the simplifying assumption that the number of fertile nuclei is kept constant. This assumption is approximately valid as long as the characteristic

83

Fuel evolution

evolution time of the fissile part is much shorter than that of the fertile part, ðaÞ ðaÞ i.e. fis cap . Then dncap ¼0 dt and the number of fissile nuclei obtained is† nfis ðtÞ ¼

1 ðaÞ

fis

ðaÞ

ðaÞ

ðaÞ

ðaÞ

fncap cap ½1 expðfis ’tÞ þ nfis ð0Þfis expðfis ’tÞg: ð3:133Þ

ðaÞ The term ncap cap ½1

ðaÞ expðfis ’tÞ expresses the rise of nfis ðtÞ due to the ðaÞ ðaÞ nuclei into fissile ones. The term nfis ð0Þfis expðfis ’tÞ

conversion of fertile corresponds to the disappearance, by fission, of the fissile nuclei present at the initial time. It appears that nfis ðtÞ tends towards an equilibrium value ðequÞ ðaÞ ðaÞ ðequÞ nfis ¼ ncap cap =fis at large times. If nfis ð0Þ < nfis the number of fissile nuclei will increase with time, so that the reactor is of the breeder type. ðequÞ Inversely, if nfis ð0Þ > nfis the reactor will be an incinerator. It is important to note that a hybrid reactor can always be a breeder, in contrast to critical reactors. The evolution of the number of fission products is given by ðaÞ

ðaÞ

nfp ðtÞ ¼ tncap cap ’ þ ncap

cap ðaÞ fis

ðaÞ

½expðfis ’tÞ 1

ðaÞ

þ nfis ð0Þ½1 expðfis ’tÞ:

ð3:134Þ

Here the first term corresponds to the linear consumption of fertile nuclei, the second term to the building up of the fissile nuclei from the fertile ones, and the last term to the disappearance of the initial load of fissile nuclei. For large times, the first term dominates. Knowing the evolution of the concentrations one gets the evolution of the multiplication factor‡ ðaÞ

k1 ðtÞ ¼

nfis ðtÞ fis ðaÞ

ðaÞ

ðaÞ

ncap cap þ nfis ðtÞfis þ npf ðtÞpf þ PðtÞ

ð3:135Þ

This condition is satisfied for both fast and thermal reactors. However if a reactor could be made to operate in the resonance region, it might be incorrect. In this case, the fertile part would, progressively, disappear in favour of the fissile part. To our knowledge, no system based on this property has yet been proposed. It would lead to a very high breeding ratio. † Here, we neglect the effect of radiative captures in fissile nuclei except for the difference between ðfÞ ðaÞ ðfÞ ðaÞ ¼ fis =fis and . Thus, in the evolution equation fis ¼ fis . ‡ In the following considerations the value of k1 may be significantly larger than unity. In such cases additional neutron absorbers or leakage will ensure subcriticality.

84


where PðtÞ is the number of neutrons lost in structural materials, control rods or escaping the reactor. In critical reactors the condition k ¼ 1 is kept via modulation of PðtÞ. For hybrid reactors the value of k is allowed to evolve within prescribed limits around a nominal value provided it remains sufficiently smaller than unity. This may be obtained by periodical regeneration of the fuel as well as by defining working conditions between two regeneration events that minimize the variations of k1 . These conditions are implemented differently in systems using liquid fuels and in those using solid fuels. Liquid fuel systems In this case fission products are, generally, extracted from the fuel soon after they are produced. It is also possible to keep the concentration of fertile elements constant by continuous feeding. The relative proportion of fissile and fertile nuclei evolves towards an equilibrium: dnfis ðaÞ ðaÞ ¼ ncap cap ’ nfis fis ’ ¼ 0 dt

ð3:136Þ

that is, ðeqÞ

nfis

ðeqÞ

ncap

ðaÞ

¼

cap ðaÞ

fis

:

ð3:137Þ

In the case of simple fissile nuclei regeneration, equations (3.135) and (3.137) show that the maximum value of k1 is equal to =2. So far, proposed liquid fuels have been molten salts. A reactor using a mixture of uranium, thorium, beryllium and lithium fluorides has run successfully for several years in Oak Ridge National Laboratory [49]. This experience led to the molten salt breeding reactor project. This reactor was supposed to use the 232 Th–233 U cycle described in section 3.5. The capture of neutrons by 233 Pa tends to decrease the reactivity of the reactor. This is why, in the MSBR project, online fuel processing was assumed. This processing aimed at extracting both the fission products and the protactinium. After protactinium decay, the resulting 233 U was reinjected into the reactor. This procedure predicts breeding of the order of 5% per year. Liquid fuels have also been considered for fast reactors. In this case chlorides rather than fluorides have been proposed [64]. Solid fuels In systems using solid fuels as small a variation of k1 as possible between two refuelling events is sought. From equations (3.133), (3.134) and (3.135) it is In the case of the U–Pu cycle effects due to captures in 239 Np, analogous to those due to 233 Pa in the Th–U cycle, are ten times smaller, because of the much shorter half-life of 239 Np.

Fuel evolution

85

Figure 3.9. Model fuel evolution in a Th–U hybrid system. The fast neutron flux is 4 1015 n/cm2 /s. The evolution of the concentrations of 233 U and fission fragments (F.F.) with respect to 232 Th are shown in (a), the evolution of k1 in (b).

seen that the value of k1 ðtÞ depends on the initial concentration of the fissile element. An initially breeding value of this concentration induces an increase of k1 ðtÞ with time. This increasing trend may be more or less exactly compensated by the decrease of k1 caused by the increase of the concentration of fission products. Rubbia et al. [76] have shown that such a compensation was possible over long periods of time. To illustrate the mechanism of this compensation, we use our simple three-component model where we choose representative values of the cross-sections for a fast reactor using the thorium cycle. Thus, referring to table 3.2, the capture cross-section of the fertile nucleus is taken to be 0.45 barns and the fission cross-section of the fissile nucleus to be 2.75 barns. The average capture cross-section for fission products was taken to be 0.15 barns, according to recent calculation results. Starting from a state without any fissile component, figure 3.9 shows the evolution of the fissile part, of the fission product part (a), and that of the multiplication factor k1 (b). The evolution of k1 shows a maximum after about 7 years, starting from zero concentration of 233 U. After 3 years, the concentration of 233 U is close to 0:135. Starting with this concentration the value of k1 is reasonably found to be constant for at least 5 years, as shown in figure 3.10(b). The maximum value of k1 shows that the neutron economy for a critical reactor would be difficult since only 6% of the neutrons are available for sterile captures and leakage. This point will be discussed later, in more realistic terms. In figure 3.10(a) we show the evolution of k1 when the fissile component initial load noticeably exceeds the equilibrium value. Here there is a fast and continuous decrease of the reactivity. This means that solid fuel hybrid reactors would not be a good choice for incinerating without regenerating a highly fissionable nucleus like 239 Pu, for example.

D. Heuer, private communication.

86


Figure 3.10. Evolution of the model Th–U fuel with an initial concentration of 0.5 (a) and 0.135 (b) of 233 U with respect to thorium.

Figures 3.11(a) and (b) are equivalent to figures 3.9(b) and 3.10(b), but for a thermal reactor with the same specific power corresponding to a flux of 4 1014 n/cm2 /s. One sees that, if the neutron economy is slightly improved (higher value of k1 at maximum), k1 is stable only for a very short time, less than 1 year. This difference between fast and thermal systems was stressed by Rubbia et al. [76]. Figure 3.11(a) also shows that the electro-breeding† of 233 U is much faster for thermal reactors than for fast reactors. This is a reflection of the fact that the equilibrium concentration of 233 U is seven times smaller for thermal reactors.

Figure 3.11. Variation of k1 for a thermal system using the Th–U cycle. (a) Starting with no 233 U present in the system at time 0. (b) Starting with an initial concentration of 233 U slightly below the equilibrium value.

We shall see in the next section that such a high thermal flux may be difficult to accept, because of the ‘protactinium effect’. † Electro-breeding denotes the process by which the fertile to fissile conversion is achieved by the spallation neutrons produced by the accelerator.

Basics of waste transmutation

3.7

87


Nuclear energy production is accompanied by the production of various radioactive wastes: fission products activation products due to neutron captures by nuclei belonging to the structure of the reactor, such as, for example, cobalt 60 . transuranic nuclei due to neutron captures by the nuclear fuel. . .

Nuclear wastes are characterized by their radiotoxicity and their halflife. Only wastes with half-lives exceeding ten years are associated with significant storage problems. These are essentially some fission products (LLFP) and transuranic elements. Their noxiousness is, traditionally, measured by their ingestion radiotoxicity. 3.7.1

Radiotoxicities

The ingestion radiotoxicity of an element is a measure of the biological consequences of its ingestion. The radiotoxicity is defined as RðSvÞ ¼ Fd ðSv=BqÞAðBqÞ

ð3:138Þ

where RðSvÞ is the radiotoxicity in Sievert per mass unit, Fd ðSv=BqÞ is the dose factor in Sievert per Becquerel activity and AðBqÞ is the activity. For 1 kg mass AðBq=kgÞ ¼

1:32 1019 T1=2 ðyearsÞ A

ð3:139Þ

where A is the atomic mass of the element. The International Commission on Radiation Protection (ICRP) has evaluated the dose factors [65], some of which are given in table 3.7 [66]. Fission products decay by radiation, while transuranic elements decay essentially through radiation. For the same disintegration rate, emitters are much more radiotoxic than emitters, as can be seen from table 3.7, with the exception of 129 I which has very peculiar biological properties, with a very high affinity for the thyroid gland. The use of ingestion radiotoxicity as a measure of noxiousness is subject to question. For example, in the case of underground storage, the probability for the radioactive species to enter the biosphere is of paramount importance. Plutonium and, generally, other actinides, have very low mobility, especially in clay, so that they contribute little to the radiotoxicity released to the biosphere. In contrast, elements like niobium, technetium and iodine are very mobile and are, potentially, the chief contributors to radiotoxological release from deep underground storage. Calculations such as those given

88


Table 3.7. Radiotoxological data for the most important long-lived fission products and actinides. Nucleus

Half-life (years)

Dose factor (Sv/Bq)

Activity (Bq/kg)

Radiotoxicity (Sv/kg)

99

2:111 105 0:157 108 0:230 107 0:214 107 0:159 106 0:877 102 0:241 105 0:656 104 0:143 102 0:373 106 0:433 103 0:737 104 0:291 102 0:181 102 0:850 104

0:78 109 0:11 106 0:20 108 0:11 106 0:25 106 0:23 106 0:25 106 0:25 106 0:47 108 0:24 106 0:20 106 0:20 106 0:20 106 0:16 106 0:30 106

6:3 1011 6:5 109 4:2 1010 2:6 1010 3:6 1011 6:3 1014 2:3 1012 8:3 1012 3:8 1015 1:5 1011 1:3 1014 7:4 1012 1:9 1015 3:0 1015 6:3 1012

4:9 102 0:7 103 0:8 102 0:3 104 0:9 105 1:4 108 0:6 106 2:1 106 1:8 107 0:4 105 0:3 108 1:5 106 0:4 109 0:5 109 1:9 106

Tc I 135 Cs 237 Np 233 U 238 Pu 239 Pu 240 Pu 241 Pu 242 Pu 241 Am 243 Am 243 Cm 244 Cm 245 Cm 129

in the Appendix show that, indeed, the dominant contribution to the future possible exposition of populations comes from 129 I. 3.7.2

Neutron balance for transmutation and incineration

The possibility of transmuting and incinerating nuclei depends on the neutron cost of these reactions. The simplest case is that of fission products. Fission product transmutation The transmutation of fission products requires, obviously, at least one neutron per nucleus. The production rate of the most important LLFPs, 99 Tc and 129 I, are given in table 3.8. From this table it appears that at least 0.07 neutron per fission would be required to achieve transmutation of these two nuclei. Ideally the most efficient way to transmute fission products is to use neutrons which Table 3.8. Yields of technetium 99 and iodine 129 per fission of three important nuclei. Fissioning species

233

99

4:9 102 1:8 102

Tc I

129

U

235

U

6:1 102 7:8 103

239

Pu

6:2 102 1:4 102


89

would be lost to captures in the structural elements or which would escape the reactor. This is why it has been proposed to capture neutrons in the resonances of fission fragments, whenever these display strong resonances [57]. In this way, it is hoped that neutrons are captured by the fission products before they reach thermal energies where captures in structure materials are significant. We discuss these ideas in the case of a fast reactor using a lead reflector, such as was proposed by Rubbia et al. [76]. 99 Tc is characterized by the existence of a strong resonance at ER ¼ 5584 meV, with ¼ 149:2 meV and 0 ¼ 104 barns. We apply equation (3.64): sffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 01 Psurv ¼ exp ð3:140Þ ER s which we write, after numerical evaluation (s ¼ 10 barns for lead)

149:2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ffi ð 1 þ x 10 1Þ Psurv ðxÞ ¼ exp ð3:141Þ 55:84 where x is the concentration of 99 Tc nuclei with respect to lead. We find that 90% of the neutrons are captured for a 99 Tc concentration of 6 104 : In the energy amplifier original design [76], about 6% of the neutrons were captured in the lead. About half of these are captured below 5 eV and could, thus, be captured in the diluted technetium. Since each fission produces 2.5 neutrons, it follows that 7.5 neutrons could be absorbed in technetium per 100 fissions. The volume of lead to consider is that where the neutron flux is high enough, rather than the full volume of the lead pool described in the energy amplifier proposal. The transport length in lead is around 1 m. It is found that the total weight of lead irradiated by a high neutron flux is around 600 tons. The amount of 99 Tc which should be dissolved in order to capture 90% of the available neutrons would then be around 180 kg. The number of neutrons captured per year in the 99 Tc would be ðcapÞ

NTc

¼ 8:4 1025

ð3:142Þ

assuming a 10 MW beam and a value of ks ¼ 0:98: These captures correspond to a transmuted mass of 14 kg. The half-life of the 99 Tc in the neutron flux would be 7.5 years. These data can be compared with those obtained with critical reactors. Calculations have been made both for fast and PWR reactors [66]. In the case of fast reactors the best results are obtained using moderated assemblies where 99 Tc is mixed with a hydrogeneous material like CaH2 . In the case of fast reactors, the shortest half-life is 15 years, while it is 21 years in the case of PWR. Thus, it appears that capture by adiabatic resonance crossing, like that discussed above, might be advantageous. For transmutation, the most important parameter is the neutron flux, since the effective lifetime of a nucleus in a neutron flux is inversely

90


proportional to the flux value. As an example, a set of nuclei with a crosssection of 1 barn, typical of some fission products, needs 200 years in a 1014 neutron/cm2 /s flux to be reduced by a factor of 2. Such numbers explain, partly, why projects such as that of Bowman et al. [2] aimed at a thermal neutron flux as high as 1016 /cm2 /s. Actinide incineration The only practical way to dispose of actinides is to induce their fission. Fission is accompanied both by energy and neutron production. However, several neutron captures may be necessary before fission occurs, so that the net neutron number necessary for actinide incineration will be Ncap þ ð1 Þ where Ncap is the number of captures before fission and the number of fission neutrons. The number of neutrons required depends on the neutron flux magnitude as well as on how hard it is. Let us consider one nucleus of species jðZj ; Aj Þ. It can suffer fission with average crossðfÞ ðcÞ section j , capture a neutron with average cross-section jk (here nucleus k is Zk ¼ Zj , Ak ¼ Aj þ 1) or decay to several possible other nuclei k, with partial decay rates jk (here nucleus k is (Zk ¼ Zj þ 1, Ak ¼ Aj ), (Zk ¼ Zj 1, Ak ¼ Aj ), (Zk ¼ Zj 2, Ak ¼ Aj 4) depending on the type of radioactivity involved). The fission probability reads ðfÞ

ðFÞ

Pj

¼

’j ðfÞ

ðcÞ

’j þ ’j þ

P

: jk

k

The production of nucleus k from nucleus j can be defined as ðcÞ

jk þ ’jk ðFÞ Ajk ¼ ð1 Pj Þ P : ðcÞ ðjl þ ’jl Þ

ð3:143Þ

l

Starting with one nucleus i; the number of nuclei j which are ultimately produced is given by the system X Akj yk þ ij : ð3:144Þ yj ¼ k

The Kronecker symbol ij expresses the fact that, initially, there was one nucleus i. Knowing the yj , it is possible to compute the number of neutrons necessary to incinerate nucleus i: X ðÞ Di ¼ R Pj yj ð3:145Þ j;

where the set f yj g is the solution of system (3.144), R is the neutron balance ðÞ for reaction (fission, capture or decay) and Pj is the reduced transition


91

Table 3.9. Values of the neutron balance for different types of reaction.

R

Capture

Fission

Decay

1

1

0

rate for reaction and nucleus j: The values of R are given in table 3.9. The expression of D was first given in a slightly different form by Salvatores and Zaetta [38], and generalized to mixtures of nuclei. Table 3.10 gives values of D for important nuclei, as well as for spent fuel mixtures. Table 3.10 shows [38] that incineration by fast neutrons is always a net neutron producer. This is due to the fact that fission cross-sections of fertile nuclei, which are very small or vanishing for thermal neutrons, are large for fast neutrons. The table also shows under which conditions breeding can be obtained from 232 Th and 238 U. The protoactinium effect is clearly visible for high thermal fluxes where its extraction is, clearly, mandatory. While, with a moderate flux, breeding can be obtained for 232 Th for both a thermal and a fast flux, only a fast flux allows breeding for 238 U. Table 3.10. Values of neutron consumptions per fission for the incineration of selected nuclei in three representative fuel mixtures: transuranium isotopes at discharge of a PWR, transplutonium isotopes and neptunium extracted at discharge of a PWR, and plutonium isotopes at discharge of a PWR [38].

Isotope or fuel 232

Th (with Pa extraction) Th (without Pa extraction) 238 U 238 Pu 239 Pu 240 Pu 241 Pu 242 Pu 237 Np 241 Am 243 Am 244 Cm 245 Cm DTru (PWR) DTPu þ Np (PWR) DPu (PWR) 232

Fast spectrum ð1015 n/cm2 /s)

PWR ð1014 n/cm2 =s)

0.39 0.38 0.62 1.36 1.46 0.96 1.24 0.44 0.59 0.62 0.60 1.39 2.51 1.17 0.7 1.1

0.24 0.20 0.07 0.17 0.67 0.44 0.56 1.76 1.12 1.12 0.82 0.15 1.48 0.05 1.1 0.2

92


Neutron balance is not the only parameter that should be considered for the choice of a particular system with the aim of waste incineration. The half-life of the waste in the neutron flux is also very important since it determines the inventory needed to reach a specified transmutation rate and the time it takes to get rid of the waste.

Chapter 4 ADSR principles

We have seen that, if the neutron multiplication factor of a reactor assembly keff < 1, the chain reaction cannot be sustained. However, if a source of neutrons is introduced inside the multiplying medium, the initial neutron number is multiplied by a factor which can be very large. Since neutrons are associated with an equivalent number of fissions, a large energy could be produced with a subcritical system, provided a neutron source is available. If N0 is the number of primary neutrons following, for example, the interaction of a proton with a target surrounded by a multiplying medium, chacterized by a multiplication factor k, the total number of created neutrons, after multiplication, is N0 : ð1 kÞ

ð4:1Þ

The number of secondary neutrons (produced after at least one multiplication) is kN0 : ð1 kÞ

4.1

ð4:2Þ

Properties of the multiplying medium

The purpose of hybrid reactors is to produce energy as well as a neutron excess which could be used for nuclear waste transmutation. As a consequence, it is important to evaluate to what extent the energy produced by fission in the multiplying medium exceeds the energy of the primary particle beam. As an example of expected values of k; we discuss the

In principle k depends on the properties of the source neutron (energy, spatial distribution) and on the geometry of the reactor and is usually noted ks (‘k source’). For simplicity we keep the notation k, at this stage.

93

94

ADSR principles

experimental measurement of the energy produced in a subcritical system as was done in the FEAT experiment at CERN [122]. 4.1.1

Energy gain

As seen from equation (6.11), the number of secondary neutrons is kN0 =ð1 kÞ. Each of these neutrons is produced by a fission (we disregard ðn; xnÞ reactions), which itself produces neutrons. Thus, the number of secondary fissions in the system is kN0 =ð1 kÞ. Since each fission releases about 0.18 GeV energy, the thermal energy produced in the medium will be 0.18kN0 =ð1 kÞ. This energy has to be compared with the energy of the incident protons Ep to define an energy gain of the system: G¼

0:18kN0 G k ¼ 0 : ð1 kÞEp 1 k

ð4:3Þ

The CERN FEAT experiment [122] gave a constant value of G0 k ¼ 3, for incident proton energies larger than 1 GeV and for a uranium target. The proton beam is produced with a finite efficiency which is the product of the thermodynamic efficiency for producing electricity from heat (typically 40% in foreseen reactors† ) by the acceleration efficiency. For high intensity accelerators, most of the power is used for the high-frequency cavities, at variance with low-intensity accelerators where most of the power is spent in the magnetic devices. High intensities, therefore, are expected to allow 40% efficiencies (see Appendix III) [2, 76]. Finally, the total efficiency for proton acceleration is expected to be in the vicinity of 0.16. This leads to a minimum value of the multiplication factor for obtaining a positive energy production (ignition), km ¼ 0:68: For a value k ¼ 0:98 a net energy gain of 16 is achieved. 4.1.2

Neutron balance

Aside from energy production, it is important to evaluate the potential of hybrid reactors for transmutation, i.e. to what extent they produce excess neutrons. A standard reactor can be viewed as a device producing energy and neutrons. Both energy and neutrons are primarily produced by fission. Fission releases about 200 MeV and 2.5 neutrons. It follows that one may

This energy corresponds to the kinetic energy of the fission fragments and their prompt deexcitation (neutrons and photons). In the total energy balance one should also include the beta radioactivity energy which amounts to approximately 20 MeV. However, in the FEAT [122] experiment, which is the only existing direct measurement of k, only the fission kinetic energy was measured. † The thermodynamic efficiency of present PWRs is around 0.33, while that of gas combined cycle turbines reaches 0.5. Lead (and sodium) cooled reactors can reach efficiencies of 0.4. High temperature gas reactors can reach even higher efficiencies of 0.5.

Properties of the multiplying medium

95

say that 80 MeV are needed to produce one neutron. The spallation process requires only 30 MeV to produce one neutron. Should the fission 200 MeV be available for proton acceleration one would, then, get more than nine neutrons per fission (2.5 þ 6.6)! True enough, no usable energy would be produced. In fact, assuming a thermodynamic efficiency of 40% and an accelerating efficiency of 40%, one finds that about 6 GeV are needed to accelerate a proton to 1 GeV. It would then be possible to obtain about 3.5 neutrons per fission, still without producing usable energy. For more realistic scenarios one sees that an accelerator allows an increase of the number of neutrons available for transmutation at the expense of usable energy. It is interesting to see if, as far as neutron availability is concerned, hybrid reactors are more or less efficient than the association of a critical reactor and an accelerator. The number of neutrons produced in the hybrid reactor is N0 ; 1k

ð4:4Þ

N0 k : ð1 kÞ

ð4:5Þ

N¼ while the number of fissions is NF ¼

On average a fission is produced by ðF þ c Þ=F neutrons. The total number of neutrons needed to produce NF fissions is Nnf ¼ NF

F þ c N0 k ¼ NF ð1 þ Þ ¼ ð1 kÞ F

ð4:6Þ

where is the number of neutrons produced following the capture of an initial neutron by a fissile nucleus. The total number of neutrons available for transmutation is therefore N0 k NDhyb ¼ N Nnf ¼ : ð4:7Þ 1 1k We now consider a critical reactor coupled to an accelerator. NDr is the number of neutrons available when using a reactor producing NF fissions, in addition to the N0 spallation neutrons. The number of neutrons necessary per fission is F þ c ¼1þ F

ð4:8Þ

while the number of neutrons produced per fission is . It follows that the number of neutrons available per fission is 1 . The total number of neutrons available in the reactor is then NDf ¼ NF ð 1 Þ ¼

N0 k ð 1 Þ ð1 kÞ

ð4:9Þ

96

ADSR principles

and the total number of neutrons available for the reactor þ accelerator system is k N0 k ð 1 Þ ¼ NDr ¼ N0 1 þ : ð4:10Þ 1 ð1 kÞ 1k Thus NDhyb ¼ NDr :

ð4:11Þ

It follows that the choice of a specific value of k is irrelevant as far as the transmutation capabilities are concerned. Whatever the method of coupling between the fission reactor and the accelerator, the number of available neutrons is ND ¼ N0 þ NF ð 1 Þ:

ð4:12Þ

From the preceding, it is seen that using 10% of the available energy allows us to obtain about 0.1 additional neutrons per fission. Although small, this number has to be compared with the number of neutrons which are effectively available in reactors. We know that the maximum number of available neutrons per fission amounts to 1 . In practice the real number is smaller than this value due to captures in structural materials and to transmutations of fertile nuclei. Let the number of such capture neutrons be c : The number of available neutrons is then 1 c . Captures in structural materials cannot be much less than 0.2 neutron per fission, particularly as reactivity changes are counterbalanced by the presence of consumable neutronic poisons. For each fissioning nucleus fissile nuclei suffer neutron capture leading, in general, to a fertile nucleus. If one requires regeneration of the nuclear fuel, one sees that c ¼ 0:2 þ 1 þ at least. The number of available neutrons amounts to 2ð1 þ Þ 0:2. We consider four cases. 1. The thermal 238 U–239 Pu system. Then, ¼ 2:871, ¼ 0:36. The number of available neutrons is 2:871 ð2 1:36Þ 0:2 ¼ 0:05. Regeneration is not possible and no neutrons are available for transmutation. The 0.1 additional neutrons made available by the use of an accelerator would allow regeneration. 2. The thermal 232 Th–233 U system. In this case ¼ 2:492, ¼ 0:09. The number of available neutrons is 2:492 ð2 1:09Þ 0:2 ¼ 0:11. Regeneration is possible and 0.1 neutrons are available for transmutation. The additional number of neutrons that an accelerator would bring is significant. 3. The fast 238 U–239 Pu system. In this case ¼ 2:98, ¼ 0:14. The number of available neutrons is 2:98 ð2 1:14Þ 0:2 ¼ 0:5. Regeneration is easy. The advantage of an accelerator is not compelling. 4. The fast 232 Th–233 U system. In this case ¼ 2:492, ¼ 0:093. The number of available neutrons becomes 2:492 ð2 1:093Þ 0:2 ¼ 0:10. Regeneration is possible. The additional number of neutrons that an accelerator would bring is significant.

Properties of the multiplying medium 4.1.3

97

Neutron importance

We have already mentioned the difference between k1 and keff . The definition of k1 was unambiguous: it was the number of neutrons produced in a homogeneous, infinite, medium following the absorption of a neutron. It is, therefore, a property of the medium. For finite reactors, some neutrons escape the medium, so that the effective multiplication coefficient keff is less than k1 . For example, in section 3.2.4 we discussed the simplest slab reactor and found that the criticality condition became k1 ¼ 1 þ 2 D=a2 a , rather than k1 ¼ 1: Thus, close to criticality, keff ¼ k1

2 D a2 a

ð4:13Þ

the criticality being reached when keff ¼ 1. In a subcritical finite system it seems evident that the progeny of a neutron created at the centre will not be the same as that of a neutron created near the boundary. This leads to the concept of a source multiplication factor ks ðr; EÞ, depending on the initial position and energy of the neutron, and such that its neutronic progeny will be ks =ð1 ks Þ. This progeny number is called the importance of the neutron † which is also the adjoint flux † ¼

ks : 1 ks

ð4:14Þ

There is, a priori, no reason why the number of first-generation neutrons would be ks . Rather, ks can be viewed as defined by ks ¼ k1 þ k1 k2 þ þ k1 k2 ki þ : 1 ks The number of first-generation neutrons k1 also depends upon the position and energy of the neutron. It is also called (unfortunately!) the adjoint flux ’† ¼ k1 : The adjoint flux obeys an integral equation, which is particularly instructive in the simple case of the one-group theory. Let a neutron be produced at r. Its first collision will occur at r 0 with probability T jr r 0 jÞ expð T ðr 0 Þ jr r 0 j2 T is the average of the total cross-section between r and r 0 . If the where interaction is an absorption f ðr 0 Þ † 0 ðr Þ T ðr 0 Þ

98

ADSR principles

neutrons will be produced, by definition of the adjoint function. Similarly, if the collision is a diffusion the number of neutrons produced will be s ðr 0 Þ † 0 ðr Þ: T ðr 0 Þ Finally, one gets ððð T jr r 0 jÞ † 0 expð d3 r ðr Þðs ðr 0 Þ þ f ðr 0 ÞÞ: † ðrÞ ¼ jr r 0 j2

ð4:15Þ

This equation has exactly the form of the one-group version of equation (3.19) for the normal flux. Thus the flux and the adjoint flux, or importance, are proportional in the one-group theory. This need not be the case in multigroup theories. Since the kernel of the integral of equation (4.15) is regular in three dimensions, it follows that if † ðrÞ becomes infinite somewhere, it is infinite everywhere. A local criticality ðks ¼ 1Þ leads to a global criticality ðkeff ¼ 1Þ: For subcritical systems it is quite possible that ks > keff , however the series of ki converges towards keff when i ! 1. Similarly the i generation neutron density ni ðrÞ, although decreasing in magnitude like kieff , converges towards the critical distribution nðrÞ. Consider a neutron created in the medium with a probability following the asymptotic density distribution. If absorbed, it produces k1 new neutrons. However, in a finite system it only produces keff neutrons, with some neutrons escaping the system. Thus keff ¼ Pcap k1 . The escape probability is Pesc ¼ 1 keff =k1 . If we consider a system with N0 injected neutrons and multiplication keff , the number of escaping neutrons will be N0 k1 keff : 1 keff k1 In most cases, in this book, we have not distinguished between ks and keff .

Chapter 5 Practical simulation methods

Practical computer simulations of reactors use either deterministic or Monte Carlo codes. The deterministic codes are the most used for critical reactor simulations, while Monte Carlo methods are almost exclusively used for hybrid reactors. Since our emphasis is on the latter, we only give a short reminder of the deterministic methods. Both methods require extensive neutron reaction data which are obtained from various data bases. The quality of the calculations is often limited by that of the data. Prior to discussing calculation methods, it is fitting to discuss data files and their evaluation.

5.1

Neutron reaction data files

The simulation of the neutronic behaviour (safety features, fuel evolution, radioprotection, etc.) of a nuclear reactor requires precise knowledge of the reaction rates, mainly elastic and inelastic scattering, fission, radiative capture, (n; ) and (n; xn) reactions, fission product yields, angular and energy distribution of the produced neutrons, and, of course, the decay time of the radioactive nuclei. The neutron energies in a reactor range from 0.01 eV to a few MeV. The ADSR concept extends the neutron energy up to a few hundred MeV, and requires a precise knowledge of the interactions of charged particles with the nuclei of the spallation target (neutron emission, angular distribution, production of residues, etc.). These specific codes and data are presented in chapter 6. At present, most of the codes used to simulate the neutronics of a reactor core use data libraries describing neutron-induced reactions from 10 meV to 20 MeV. The deterministic method and the Monte Carlo approach do not use the data bases in the same way. Both methods require, however, a huge number of data in order to perform precise calculations. The neutronic libraries provide evaluated values of the cross-sections at a given temperature. 99

100

Practical simulation methods

There is at present no nuclear model able to predict exactly the neutron cross-sections in the energy range covered by reactor physics. Thus, most of the data available in the neutronic libraries are evaluated from experimental measurements. Different effects have to be taken into account, for instance background, impurities in the sample, etc. In general, the evaluation of neutronic data requires different types of measurement, which do not necessarily correspond to the same energy range. Thus, a compilation step is required, to combine different types of experimental data together, and produce a list of evaluated parameters of the cross-sections. This very important step consists of fitting the data with a nuclear model. One of the approaches that can be considered is the Bayesian one, which allows a compilation of different data sets. This statistical method is based on the equation pðAjBÞ pðBjAÞpðAÞ;

ð5:1Þ

where A is the quantity to be determined (a resonance parameter for example) and B is the set of data. pðA=BÞ gives the probability distribution of the evaluated parameter. pðB=AÞ is the probability that quantity B will be observed if the parameter is A (likelihood function). This quantity is determined by the nuclear model chosen. Finally the last term pðAÞ represents what is known about A before the experimental measurement. This equation shows how the information contained in a new measurement can be added to the initial data base corresponding to pðAÞ, in order to update the knowledge of the parameter A. A simplified example of this approach is detailed below. Depending on the neutron energy, different models ( pðB=AÞ) are used to extract the cross-section parameters from the measurements. The R-matrix theory [67, 68] is generally applied in the energy domain where the resonances can be resolved experimentally. This range of energy depends on the nucleus studied and on the quality of the measurements. It corresponds in general to thermal energies, up to a few tens of keV. This formalism considers only binary collisions: an ingoing wave function describes the two incident particles (the neutron and the nucleus); an outgoing wave function describes the emerging reaction products (the neutron and the excited nucleus, or fission fragments, etc.). The space is divided into, on one hand, the external area, where nuclear forces are negligible, and, on the other hand, the internal region, where nuclear forces dominate. The external region is handled by pure Coulombic wave functions of the particles. Different approximations can be used to describe the excited nucleus, the Breit and Wigner approach being preferred for an isolated resonance. For the different absorption reactions i, the cross-section is a Lorentzian: n i ðE E0 Þ2 þ ð12 Þ2 P where n is the neutronic width and ¼ i i the total width. i ðEÞ ¼ 2 gJ

ð5:2Þ

Neutron reaction data files

101

For elastic scattering, we have to add the symmetric resonance term (i ¼ n in the previous equation), the term corresponding to the potential scattering cross-section (p ) and a term corresponding to the interferences between these two phenomena: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n n ðE E0 Þ s ðEÞ ¼ 2 gJ þ 2 2 gJ p þ p : 2 2 1 ðE E0 Þ2 þ ð12 Þ2 ðE E0 Þ þ ð2 Þ ð5:3Þ In these equations, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ¼ h= 2ECM where is the reduced mass and ECM the energy of the particle in the centre of mass system. . gJ ¼ ð2J þ 1Þ=2ð2I þ 1Þ where I is the spin of the target nucleus, and J the spin of the compound nucleus level. . E0 is the energy of the resonance being studied. More precise models calculate the resonance profile in more complex cases. However, the different parameters (width, mean energy) have to be fitted on experimental measurements. We give here a simplified example of Bayes’ method, using the least-squares approximation. Let us consider a resonance of a given nucleus. We will assume that the partial widths of this resonance are perfectly known, and that we are trying to determine only its position in energy. We consider that a first experimental measurement has been performed, and that we have prior knowledge of the resonance position, in the form of an estimated value "1 and an associated error 1 . The prior distribution of E0 , given "1 and 1 , is then ðE " Þ2 pðE0 j"1 ; 1 Þ dE0 exp 0 2 1 ð5:4Þ dE0 : 21 We suppose now that we perform a new experiment concerning the position of this resonance (for example a fission experiment), and that we obtain a data set f yi g. We choose the Breit–Wigner description which leads to a theoretical observable fyth i BW ðEi Þg. Thus, the likelihood of obtaining the values f yi g, given the theoretical f yth i g; is X 2 ðyi yth i Þ th pðyi jyi Þ dfyi g exp ð5:5Þ dfyi g 222 i 2 where 2 ¼ hð yi yth i Þ i and df yi g ¼ dy1 dy2 dyN . Finally we obtain a new distribution of the energy position, given by the product of the likelihood function by the prior distribution: 2 ðE0 "1 Þ2 X ð yi yth i Þ pðE0 Þ dE0 ¼ exp ð5:6Þ dE0 : 221 222 i

102


This evaluation contains the knowledge taken from the two successive experimental data sets. The correlation between prior and new data has been neglected here, but could be taken into account. This approach can be generalized to a vector of parameters X ¼ fX1 ; X2 ; . . . ; XN g (for example position, total and partial widths of a resonance). Prior knowledge of this vector is in the form of an estimated vector E and an associated covariance matrix A (with A ¼ hðE XÞ† ðE XÞi), which describes the uncertainties and correlations of these parameters. The probability distribution of X can be written as pðXjE; AÞ expð 12 ðE XÞ† A1 ðE XÞÞ dX

ð5:7Þ

where dX is the volume element of the N-dimensional parameter space. This evaluation step requires sophisticated computer programs. For example, the SAMMY [69] code has been developed at the Oak Ridge National Laboratory (ORNL), and allows one to parametrize the experimental data (essentially neutron time-of-flight data), using Bayes’ method, together with the multilevel R-matrix description. This code gives the best fit of the experimental data and the associated covariance matrix. Physics effects, such as the Doppler effect, are taken into account. For heavy nuclei, the resonances can be separated for energies below a few tens of keV. Above this energy, the resonances are too close to be experimentally separated. In this domain, the evaluated cross-sections are built with resonance parameters that are chosen randomly. Individual partial widths are chosen following the Porter–Thomas [70] distribution: 1n 1 1 n nx 2 Pn ðxÞ ¼ e2nx 1 2ð2 nÞ 2 with x ¼ ðÞ =hðÞ i.† The average values hðÞ i are extrapolated from the region of well-separated resonances or from nuclear model estimates. In the Porter–Thomas distribution, n is the number of degrees of freedom. For neutron elastic scattering widths, there is only one final state, so that n ¼ 1. For gamma rays, there are many available levels for the primary gamma decays, n ’ 30–40. For fission, the relevant degrees of freedom are the Bohr and Wheeler transition states, and one finds typically n ’ 3–4. Note that large values of n correspond to small fluctuations around the average. Resonance energies are chosen according to the Wigner interval distribution [71] between next-neighbour 2 1 levels with same spin and parity which reads PðSÞ ¼ 12 S e4S with S ¼ D=hDi and D the distance between two nearest neighbours.‡

This law is also known as a chi-square distribution with n degrees of freedom. Note that widths ðÞ should not be confused with the function. ‡ The Wigner law shows that levels with same spin and parity repulse each other. They do not follow the random Poisson distribution, and reflect quantum chaos. †

Deterministic methods

103

Families of resonances with different spins and/or parities are handled independently. At higher energies, the resonance widths are larger than the spacing between each of them, and the cross-section varies smoothly with energy. In this energy domain, the optical model is generally used to evaluate the experimental measurements. The evaluated cross-sections are usually found in nuclear data evaluated files like ENDF-B6, JEF 2.2, JENDL or BROND. These files contain the parameters of each resonance and sometimes the correlation matrix of these parameters, which allow sensitivity calculations. These files, as well as the experimental files (*.EXFOR files in the CSIRS library), can be found on the National Nuclear Data Center (NNDC) site at Brookhaven National Laboratory. In general, an additional step is required and consists of decoding the data files, and producing a new one at the chosen temperature, which will be consistent with the code and the system studied. For deterministic codes, the reaction cross-section must be calculated and averaged on different energy groups, while Monte Carlo codes require, in general, continuous cross-section values obtained by a list of points and an interpolation procedure. For example, the program NJOY reads an ENDF format file, and writes a specific file for a Monte Carlo code such as MCNP, taking into account the Doppler broadening of the resonances. The spallation source of the ADSR implies neutron energies of a few hundred MeV. Present nuclear data libraries contain neutronic cross-sections up to 20 MeV. Different experimental programmes aim to widen the neutronic cross-section available in the libraries up to 200 MeV.

5.2

Deterministic methods

These methods consist of more or less elaborate approximations of the Boltzmann equation. The most widely used approximation is the multigroup diffusion theory which we outline here, as an example. The different groups correspond to energy bands Ei < E < Ei þ 1 . The set of multigroup equations reads X X Di ’i ðrÞ t;i ’i ðrÞ þ r; j ! i ’j ðrÞ þ i f; j ’j ðrÞ ð5:8Þ j

j

where ið jÞ denotes the ið jÞth group. r; jP ! i is the cross-section for a jump from group j to group i. t;i ¼ a;i þ j r; j ! i is the cross-section for removing neutrons from group i. f; j is the fission cross-section in group j. http://www.nndc.bnl.gov; another important site is that of the Nuclear Energy Agency, http://www.nea.fr.

104


i is the fraction of the fission neutrons which have energies within group i. The cross-sections should be computed as averages over the group energy domain by ð Ei þ 1 i ðEÞ’i ðEÞ dE Ei i ¼ ð Ei þ 1 ð5:9Þ ’i ðEÞ dE Ei

which means that equation (5.8) is, in fact, a set of complicated integrodifferential equations. In particular, in the resonance regions, the flux has a complicated structure due to its depletion at energies in the vicinity of a resonance energy. Thus, approximations are made on the calculation of the group cross-sections. In particular, in heterogeneous reactors the crosssections for the cells are first computed, with a large number of groups, with simplifying assumptions on the shape of the flux, and, possibly, correction factors. In a second step the flux on the cell network is computed. In practice, experiments are needed to validate the group cross-sections for each type of reactor.

5.3

Monte Carlo codes

Monte Carlo calculations follow the history of individual neutrons. The most used codes are MORSE [72] and MCNP [74]. The CERN group has written its own code, MC2 [76], which is, however, not in the public domain. The physics involved is basically the same in all these codes. In the Monte Carlo scheme, there is no space (or time) discretization. One does not solve any differential equations and there is no need to write such an equation. MC methods supply information only about specific quantities, requested, a priori, by the user. In that sense, MC codes solve the integral transport equations. The principle is to follow individual particle histories (as many as possible). Then, the particle average behaviour is inferred, using the central limit theorem, from the simulated particles. Individual probabilistic events (such as interactions of particles with materials) are simulated sequentially. The probability distributions governing these events are statistically sampled to describe the total phenomenon. This is very similar to a real physics experiment: you plan to measure some quantities (with specific detectors). By recording the result for many particles, the experiment supplies information on the physics of your system. 5.3.1

Deterministic versus Monte Carlo simulation codes

Deterministic codes need an a priori knowledge of the solution in order to obtain the convergence of the iterative process described previously (for

Physics in MCNP

105

example, the calculation of mean cross-sections is a very delicate step which is generally system dependent). They have been developed in the past years, when computers were too slow. The predictions of such codes are limited but they can be useful to study perturbations to a given system. The main drawback is that the discretization of space, time and energy implies approximations. Moreover, they are very memory and computer time consuming if a realistic system is to be described, and, in practice, it is not possible to have a reasonably good description of a real 3D system. Monte Carlo codes are very well adapted to the description of complex 3D systems. There are no approximations due to discretization. These codes allow very detailed representations of all physical data. With increasing computer speed, very precise results can be obtained for a system within a few hours. The same codes could be used to describe experimental set-ups and give precise predictions to which experimental results can be compared, improving confidence in the code, in the case of good agreement. Code reliability lies in the validity of cross-sections (which are directly taken from evaluations); if they are not correct, the results will also be wrong. Of course, this is also true of deterministic codes. 5.3.2

MCNP, a well validated Monte Carlo code

The Monte Carlo code MCNP (general Monte Carlo N-particle transport code) is one of the best known and used Monte Carlo codes. It can be obtained from the NEA. This code can transport photons, electrons and neutrons. In the following, we will only consider neutron transport. The statistical sampling process is based on the selection of random numbers, analogous to throwing dice in the Monte Carlo gambling casino.

5.4

Physics in MCNP

As already said, particle transport looks like a theoretical experiment: a particle is followed from its birth (the source), throughout its life, to its death (absorption, escape). Probability distributions are randomly sampled using transport data to determine the outcome at each step of its life (figure 5.1). Specific techniques have to be implemented for critical problems, where the neutron chain length can reach infinity. In MCNP, critical calculations are known as KCODE calculations. In all cases one has to know: if the neutron interacts or not in a medium if yes, on which nuclei of the medium . what kind of reaction occurs . what are the ‘secondary’ particles emitted . .

Let us see how these steps are handled in MCNP.

106


Figure 5.1. Random walk of a neutron.

Interaction: yes or no? If one considers a neutron in a material, this neutron can escape or interact in the material. The probability for a collision to occur between l and l þ dl is pðlÞ dl ¼ expðT lÞT dl

ð5:10Þ

where T is the macroscopic total cross-section. One has to sample l according to this exponential probability law. Let be a random number in ½0; 1½ uniformly distributed. One can write ðl ¼ pðlÞ dl ¼ 1 eT l ð5:11Þ 0

that is to say l ¼ ð1=T Þ lnð1 Þ which can be replaced by l ¼ ð1=T Þ ln because 1 and have the same distribution. The probability distribution of l is obtained by estimating the length of the interval corresponding to the interval dl, d ¼ ¼ expðT lÞT dl dl

ð5:12Þ

which verifies that pðlÞ obeys the distribution given in equation (5.10). If l is greater than the distance to the edge of the material, the neutron escapes; the neutron is then placed on the surface separating the medium being exited and the test for the medium being entered is done again. . Otherwise, an interaction occurs at distance l. .

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107

What is the interaction? Depending on the interaction, MCNP answers the following: 1. What is the velocity of the target nucleus? 2. On which nucleus does the collision happen? 3. How many photons are emitted? (This is optionally done if MCNP follows neutrons and photons; but here, we will not discuss that process.) 4. Is the neutron still alive or is it captured? 5. Is it an elastic scattering or an inelastic reaction? 6. What are the energies and directions of the new outgoing particles (if any)? In the following, we show the main ways to answer these questions. will denote a random number in ½0; 1½ uniformly distributed. (1) Velocity of the target nucleus A collision between a neutron and an atom is affected by the thermal motion of the atom, and in most cases the collision is also affected by the presence of other atoms nearby. Depending on the compound (essentially, moderators) and on the neutron energy, two treatments are done. For slow neutrons (En < 4 eV) and for carbon, water, etc., the crystalline structure as well as effects of chemical binding have to be taken into account via an Sð; Þ treatment (these Sð; Þ tables are special sets of ENDF files). For higher energies or other compounds, a free gas thermal treatment is applied: the medium is assumed to be a free gas, and, in the range of atomic weight and neutron energy where thermal effects are significant, the elastic scattering crosssection at zero temperature is assumed to be independent of neutron energy and the reaction cross-sections are independent of temperature; the elastic scattering cross-section at temperature T is then given by kT ðTÞ ¼ ðT ¼ 0Þ 1 þ ð5:13Þ 2AEn where A is the atomic weight and En the neutron energy.† Then the collision is sampled in the target-at-rest frame and the target laboratory velocity is chosen according to the Maxwell law PðVÞ / V expðV 2 =kTÞ. However, if the elastic scattering cross-section has been processed (say, by NJOY) at the desired temperature, this treatment is not applied. The temperature entered in MCNP is used only to modify (if

S is known as the scattering function of , the reduced momentum transfer, and , the reduced energy transfer. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi † In fact, this is an approximation for AEn =kT > 2; a more general formula is used when this number is smaller than 2.

108


Figure 5.2. Selection of nucleus k from N.

needed) elastic scattering, and thus total, cross-sections; no other Doppler effect is taken into account via this temperature. (2) Selection of nucleus k among N If the material is composed of N different nuclei, the interaction occurs on nucleus k if k 1 X i¼1

iT <

N X

iT

i¼1

k X

iT :

ð5:14Þ

i¼1

Figure 5.2 shows an illustration of this selection method. (3) Capture There are two ways to handle capture: Analogue capture, which is a ‘true’ capture: the neutron is absorbed with probability a =T where absorption denotes all reactions like (ðn; Þ; ðn; Þ; . . .) except ðn; n0 ; . . .Þ. . Implicit capture: the neutron has a weight Wn which is decreased to Wn0 ¼ ð1 ða =T ÞÞWn by the capture. This way of handling captures is useful to obtain better statistics (the neutron is still alive after a capture, but its contribution is smaller). .

(4) Elastic scattering or inelastic reactions Inelastic reactions (ðn; n0 Þ, ðn; f Þ, ðn; npÞ, . . . ) are assumed to be independent of the temperature, whereas the elastic scattering cross-section (and thus the total cross-sections) is adjusted according to the free gas model previously described. The probability that the reaction is elastic scattering is el el ¼ : inel þ el T a

ð5:15Þ

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109

The probability that the reaction is an inelastic one is inel : T a

ð5:16Þ

If the inelastic reaction ‘wins’, the jth reaction is chosen among M according to j1 X

i <

i¼1

M X i¼1

i

j X

i :

ð5:17Þ

i¼1

Sampling angular distribution For both elastic and inelastic scattering, the direction of outgoing particles is determined by sampling angular distribution tables (in fact it is the cosine angle which is sampled) from the cross-section files. If the distribution is isotropic for a reaction, the cosine angle is chosen uniformly in ½1; 1½: . Otherwise, 32 groups of cosine angles are used depending on nuclear data. This group discretization could be a limitation of MCNP (mainly sensitive for high energies). .

Energy of outgoing particles (non-fission inelastic scattering) Various probability laws (67) are used for the Np outgoing particles, depending on nuclear data (equiprobable, evaporation, Maxwell or Watt spectrum, etc.). Fission inelastic reactions If the reaction is a fission, Np neutrons are emitted according to the value of ðEn Þ, the mean number of neutrons per fission (given in cross-section files). The Np neutrons are chosen as Np ¼ I þ 1 Np ¼ I

if ðEn Þ I if > ðEn Þ I

where I is the largest integer smaller than ðEn Þ: The energies of outgoing neutrons are chosen using a Maxwell fission spectrum, or an energydependent Watt spectrum, or an evaporation spectrum. However, such treatment is not totally correct, because the NP obey a Poissonian distribution; for example, if ¼ 2:5, MCNP gives two or three neutrons (in order to have 2.5 on average), but it never gives bigger and smaller neutron number (with the same average).

110 5.4.1

Practical simulation methods Precision and variance reduction

Monte Carlo results represent an average of contributions from many histories sampled during the run. A statistical error (or uncertainty) is associated with the result. This number is of course very important but it cannot be taken into account alone. It is also very important to know how this uncertainty evolves with the number of histories. Indeed, this behaviour can reflect whether the result is statistically well behaved; if this is not the case, the uncertainty will not reflect the true confidence interval of the result which thus could be completely erroneous. MCNP is certainly one of the best codes that provide very detailed and efficient methods to determine the quality of the confidence interval, as well as methods to improve the precision. Here, we just want to give an insight as an introduction of all the methods that can be used. Precision and accuracy There is a very big difference between precision (statistical error or uncertainty) and accuracy (systematic error). Accuracy will depend on the code (bugs, nuclear data, specific treatment, etc.) and on the user (wrong hypothesis, bad use of particular method, etc.). The precision will depend on the number of events in the region concerned and on the statistical method used. What is a good uncertainty? If the result of a run is with N histories contributing to this quantity, the pffiffiffiffi uncertainty is / 1= N ; thus, to divide this error by 2, N has to be multiplied by 4. But the computer time is proportional to N, so the duration of the run is also multiplied by 4. In general, an MCNP result with an uncertainty less than 10% is sufficient, but only if the entire volume has been visited by particles. In order to check the confidence in a result, MCNP gives the evolution of a number, the figure of merit (FOM) which is defined by FOM ¼ 1=ð2 TÞ where T is the computerptime. This ffiffiffiffi number must be approximately constant (T / N and / 1= N ) for the result to be credible. Analogue and non-analogue Monte Carlo Analogue Monte Carlo. One particle is followed event by event. This is the standard way to transport particles as shown in figure 5.1. This method is correct if the number of histories is large enough. Non-analogue Monte Carlo. Only interesting particles are followed: each time a particle is an interesting one, it is multiplied into Q particles with a weight

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111

1=Q. This method decreases the statistical error, but could introduce a bias in the results. The variance reduction techniques in MCNP Truncation. MCNP can operate energy or time cut-offs; the particle is killed below an energy threshold. This is good because it saves time, but it is dangerous because low-energy particles can produce highly energetic ones (fissions) and thus lead to erroneous results. Similar time cut-offs exist when the particle time exceeds a given cut-off. Population control methods use particle splitting and Russian roulette to control the number of particles in various regions. Each MCNP cell is given an importance I. For example, if a neutron of weight W passes from a cell of importance 2 to one of importance 8, it is split into 8=2 ¼ 4 identical neutrons each with a weight W=4. Conversely, if the neutron passes from a cell of importance 8 to one of importance 2, a Russian roulette is played and the neutron is killed with the probability 1 28 ¼ 75% or, worded differently, followed with a 25% probability and a weight W 4. This type of population control can be applied to the energy range.

5.5 5.5.1

MCNP in practice Introduction

The aim of this part is to present some reactor simulation examples; it cannot replace the MCNP reference manual. When MCNP is running, two special files are needed; one is, of course, the problem description (geometry, materials, neutron source, etc.). The other one, named xsdir, is a special file that contains some nuclear data and the path of the ENDF library files for each material. This file can be placed in a default directory, but MCNP first looks in the ‘current’ directory if such a file exists. 5.5.2

Units

The specific units used in MCNP are summarized in table 5.1. 5.5.3

Input file structure

The problem description is in one file , the input file. This file contains geometry description (cells defined with surfaces), materials, neutron source, calculation

Note that the neutron source may be defined in a separate file.

112

Practical simulation methods Table 5.1. Main units used in MCNP. Magnitude

Units

Lengths Energies Times Temperature Atomic densities Mass densities Cross-sections

cm MeV Shakes (108 s) MeV (kT) atoms/barn-cm g/cm3 barn

type (critical or standard), etc. This file has a special format. It contains delimiters, data entries (or cards) and comments. Each line is limited to 80 characters; but a single data entry may lie on more than one line if each of these lines (except the last one) ends with an & (ampersand). MCNP is case insensitive. The general structure of the input file is: Title line Cell cards Blank line delimiter Surface cards Blank line delimiter Data cards Blank line terminator Comments can be inserted on separate lines: the line must begin with a ‘c’ followed by at least one blank. They can also be placed at the end of a data entry: after a $ (dollar sign). The surfaces The geometry of a problem is described in the Cell cards; cells are volumes delimited by surfaces; these volumes may or may not be filled with materials. But before seeing how to define cells, let us see how Surface cards are defined. MCNP has a wide choice of surface types to describe cells; however, here we will just present the surface types that we will use for our reactor examples. A Surface card consists in a surface number, a special keyword indicating the type of the surface, and parameters for that keyword. Table 5.2 lists these surfaces and their parameters.

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Table 5.2. Some surface cards. Surface type

Keyword Parameters

Equations

General plane Plane perpendicular to X-axis Plane perpendicular to Y-axis Plane perpendicular to Z-axis Cylinder parallel to X-axis Cylinder parallel to Y-axis Cylinder parallel to Z-axis Cylinder along X-axis Cylinder along Y-axis Cylinder along Z-axis General sphere

P PX PY PZ C/X C/Y C/Z CX CY CZ S

ABCD D D D y0 z0 R x0 z0 R x0 y0 R R R R x0 y0 z0 R

Sphere centred at origin

SO

R

Ax þ By þ Cz D ¼ 0 xD¼0 yD¼0 zD¼0 ð y y0 Þ2 þ ðz z0 Þ2 R2 ¼ 0 ðx x0 Þ2 þ ðz z0 Þ2 R2 ¼ 0 ðx x0 Þ2 þ ð y y0 Þ2 R2 ¼ 0 y2 þ z2 R2 ¼ 0 x2 þ z 2 R 2 ¼ 0 x2 þ y2 R2 ¼ 0 ðx x0 Þ2 þ ð y y0 Þ2 þ ðz z0 Þ2 R2 ¼ 0 2 x þ y2 þ z2 R2 ¼ 0

The cells In order to describe a cell, one can use intersections, unions or exclusions of surfaces (and also exclusion of other cells). Thus, a cell will be defined with a cell number, identifying the cell either a material number with its density or a ‘0’ if the cell is void, . a list of surfaces (or cells) with operators (intersection: a blank; union: a colon ‘:’; exclusion ‘#’). . .

First, let us see how cells are built from surfaces; for the sake of simplicity we will deal with a two-dimensional empty cell (see figure 5.3). 1 0 -1 2 3 -4

Figure 5.3. A simple geometry (a) and a somewhat more difficult one (b). Circled numbered are the cell numbers, plain numbers are surface numbers.

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In this example, the ‘–’ sign before surface 1 means that one considers the region below surface 1 (i.e. y d1 0, if surface 1 is given by y d1 ¼ 0). Similarly, the region on the left of surface 4 is considered (x d4 0) whereas the ‘positive’ surfaces 2 and 3 mean that the region being considered is to the right and above them respectively. Cell 1 is thus the intersection of these four ‘signed’ surfaces. The ‘exterior’, i.e. cell 2, can be defined as 2 0 1:-2:-3:4 that is, as the union of the outside region defined by the surfaces; the exclusion operator can also be used: 2 0 #1 which means that cell 2 is the whole space except cell 1. Consider now the more difficult case of figure 5.3(b). First, forget cell 3 and surface 6. Because cell 1 has a concave corner the cell has to be defined also with unions, namely: 1 0 -1 2 3 (-4:-5)

ðFig. 5.3(b) without surface 6Þ

Now, taking into account surface 6 and cell 3, 1 0 -1 2 3 (-4:-5) 6 3 0 -6

ðFig. 5.3(b)Þ

where ‘–6’ means the inside of circular surface 6. The outside of cell 1 could be defined as 2 0 1:-2:-3:(4 5) or, with the exclusion operator, 2 0 #1 #3 Don’t forget to exclude cell 3 also! Indeed, #1 means ‘the whole space except cell 1’ and cell 3, although it is inside cell 1, does not belong to that cell. To illustrate a real input geometry with a short example, let us consider the geometry of figure 5.4. First simple geometry c c Cell cards c 1 0 -1 $ the inner sphere 2 0 -2 3 -4 1 $ the cylinder without the sphere 3 0 #2 #1 $ exterior c c Surface cards

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Figure 5.4. Simple geometry: a sphere with R ¼ 5 cm inside a cylinder centred on the Z axis with R ¼ 20 cm and height ¼ 40 cm.

c 1 2 3 4

SO CZ PZ PZ

5 20 -20 20

$ $ $ $

centred sphere with R=5 cm infinite cylinder with R=20 cm bottom plane intersecting the cyl. top plane intersecting the cyl.

c end of the file Suppose that this geometry is written in a file named ‘mygeom’. In order to see this geometry, simply type on the command line: mcnp ip i=mygeom A prompt ‘plot>’ allows you to view a section perpendicular to the z axis at the value z0 by the command pz z0 Other views can be seen with ‘px’ and ‘py’. This shows a 2D view of the geometry. If there are errors in the geometry the ‘bad’ surfaces are drawn with a dotted red line. Note We have already mentioned that cells may have different importances (see section 5.4.1). These importances are defined with the IMP card. The aim

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of increasing cell importance is to increase the number of histories, i.e. to reduce the statistical error. Going from one cell with importance 1 to a cell with importance 2 will multiply tracks (i.e. the number of neutrons) by 2, halving the weight of each track. Conversely, going from a cell of importance 2 to a cell of importance 1 will halve the number of tracks, doubling the weight of each particle. A zero importance cell means that no transport is done. By default, cell importances are zero. Thus, before doing a calculation, you have to specify the importance of each cell. One way is to do it along with the cell definition; in the previous example, the cell part is just: c c c 1 2 3

Cell cards 0 -1 imp:n=1 $ the inner sphere 0 -2 3 -4 1 imp:n=1 $ the cylinder without the sphere 0 #2 #1 imp:n=0 $ exterior

In cells 1 and 2, neutron importance (‘:n’) is 1 whereas the exterior (cell 3) where neutrons are not followed has a null importance. Material Now, we know how to define empty geometries. Materials that fill the cells are entered in the Data cards. A material is defined with an ‘M’ followed by a number (without blank), a cross-section reference and a proportion. Let us see an example on our simple geometry test: First simple geometry c c Cell cards c 1 1 -18.75 -1 $ the inner sphere 2 2 -1.0 -2 3 -4 1 $ the cylinder without the sphere 3 0 #2 #1 $ exterior c c Surface cards c 1 SO 5 $ centred sphere with R=5 cm 2 CZ 20 $ infinite cylinder with R=20 cm 3 PZ -20 $ bottom plane intersecting the cylinder. 4 PZ 20 $ top plane intersecting the cylinder. c Materials M1 92235.60c 1 $ 235 U M2 1001.60c 2 8016.60c 1 $ H2 O

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We have placed in cell 1 (the sphere) material 1 of density 18.75 g/cm3 (the ‘–’ sign means that it is a mass density in g/cm3 ). Material 1 is pure 235 U (note that the cross-section code is ZZAAA.id for a nucleus of atomic number ZZ and mass number AAA; id refers to the version of cross-section used and the type (continuous or discrete). The complete list of cross-section codes is found in appendix G of the MCNP reference manual). The cylinder is filled with water (density 1 g/cm3 ); the water is composed of two nuclei of hydrogen (code 1001.60c) and one of oxygen (8016.60c). If you try to view the geometry as previously explained, you will see that non-empty cells are now in colour. In association with the Material data cards, it is possible to use so-called MT cards for some materials (moderators) in order to specify that an Sð; Þ treatment must be supplied for low-energy neutrons (En < 4 eV). This treatment replaces the free-gas treatment below 4 eV (the most significant effects are below 2 eV). For example for material 2 (water) in the previous example one can write: M2 1001.60c 2 8016.60c 1 $ H2 O MT2 LWTR.07 This treatment does not exist for all materials. Source MCNP has a lot of possibilities for the definition of neutron sources and it is beyond the scope of this book to describe them. Source description is done in the Data cards section of the input file. Here we just want to give two simple source examples. The first one (the simpler) is a punctual isotropic and mono-energetic source, SDEF POS 0 0 0 ERG=2.5 which defines a neutron source at position (0, 0, 0) of energy 2.5 MeV. This kind of source is very useful for starting a KCODE. When dealing with an ADSR, the spallation source has to be defined; because this involves high-energy particles, MCNP alone cannot process that kind of source; nevertheless, if one can create a file accounting for neutrons below 20 MeV from such a source, MCNP can read that file (the position (x; y; z), direction (cosine), energy, time and weight of each neutron have to be specified). Such a source file is read with the FORTRAN subroutine ‘source.f ’ included in the MCNP distribution. The source can be built by any high-energy transport code like FLUKA, HETC, or directly by MCNPX, which is a version of MCNP coupled to LAHET. In order to give a more realistic ADSR description in the following section, we give here an MCNP source to represent, more or less, the

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spallation source. It is a cylindrical source: SDEF POS 0 0 0 ERG=D1 RAD=D2 EXT=D3 AXS 0 0 -1 $ Dn=to be described later SP1 -5 a $ (related to D1) SI2 r1 r2 $ Radial extension (ring of radii r1 and r2 ) (related to D2) SI3 zmin zmax $ Z extension (from source position) (related to D3) Its energy is described by the Source Probability card (SP1); the first parameter (–5) means that the source energy probability is pðEÞ / E expðE=aÞ (evaporation spectrum) and the second parameter is the value of a in MeV. The axis of the cylinder passes through point POS (0; 0; 0) in the direction AXS (0; 0; 1) ¼ z axis. Neutron positions are sampled uniformly (in volume) within a ring of inner radius r1 and outer radius r2 (Source Information, SI2 card). The ring lies in a plane perpendicular to AXS at a distance from POS sampled by EXT (here defined on the SI3 card from zmin to zmax ). Note A source neutron, when it is born from a punctual source or an external source (like a spallation one) cannot be on a surface. If a neutron is born on a surface, it has to be pushed a little (let us say, by 1 mm). Basic tallies Tallies are quantities that are stored during an MCNP run. There are several tally types depending on what one wants to store (current, flux, . . . ). Here, we will present only one type of tally and associated ‘modifier’: this tally is the one which calculates the neutron flux in a cell (or, when used with modifiers, counting rate, mean cross-section and so on). They are always normalized by the number of source neutrons and the volume (or surface for surface tallies) of the cell. Tallies are defined in the Data cards block. The keyword for tally declarations is ‘F’ followed by a number (three digits maximum). The last digit corresponds to the tally basic type. For example, the neutron fluence in cell 1 is written F4:n 1

This last digit ranges from 1 (number of neutrons integrated over a surface) to 7 (fission energy deposition) for neutrons. For our purpose, tally basic type 4 (fluence over a cell) is the most useful; then, one can choose for this tally type F4, F14, . . . , F994, i.e. a total of 100 flux tallies in the same MCNP run.

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where ‘:n’ indicates that neutrons are concerned. It is possible to calculate a given quantity in more than one cell; for example, the neutron flux in cell 1 and 2 will be F14:n 1 2 The result of this tally will consist of four numbers, the flux value and its statistical error in each of the two cells, whereas the line F24:n (1 2) is the mean flux in cells 1 and 2 (thus two numbers, the flux and its error). Energy and/or time bin The previous examples show integrated flux over a cell, independent of energy and time. It is, however, useful to know the energy and/or time dependence of such quantities. Below are three examples related to the three previous tallies: E4 0.01 T14 0.1 E24 0.1 T24 10.

0.1 1. 10. 0.2 0.3 0.4 1. 10. 99I 1000.

The first line defines, for tally 4, an energy binning where the upper bounds of each bin are 0.01, 0.1, 1 and 10 MeV; tally 14 give the flux in cell 1 and in cell 2 as a function of time (from 0.1 to 0.4 shakes). Tally 24 will give the mean flux of cells 1 and 2 as a function of energy (three bins) and of time (101 linear bins from 10 to 1000 shakes). . .

The first bin is always from 0 to the first specified upper bound. MCNP gives also the sum of all bins (1D binning), or for 2D binning the sum of all energy bins as a function of time and the sum of all time bins as a function of energy.

Tally multiplier It is possible to calculate quantities of the form ð C ’ðEÞRðEÞ dE where ’ðEÞ is the fluence and RðEÞ is an additive and/or multiplicative response function from the MCNP cross-section library. It is very useful to find the reaction rates, the averaged cross-sections (over the flux) and so Note that the sum made by MCNP takes into account correlations between histories, therefore the arithmetic summing of all bins is not equal to the sum bin given by MCNP.

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Table 5.3. Some reaction codes for neutrons. Reaction codes

Meaning

1 1 3 2 6 7 2 102 16

total cross-section (without thermal adjustment, T ¼ 0 K) total cross-section (adjusted for temperature dependence) elastic cross-section (without thermal adjustment, T ¼ 0 K) elastic cross-section (adjusted for temperature dependence) total fission cross-section average number of fission neutron absorption cross-section (n; ) cross-section (n; 2n) cross-section

on. Such quantities are obtained with FM cards (related to F (tally) cards); an FM card is written as FMn C m1 R1 or FMn (C m1 R1 )(C m2 R2 ) where n is the tally number, C is the multiplicative constant, mi is the material and Ri is a reaction code (or a list of reactions); table 5.3 gives some useful reaction codes. But let us see some useful examples of FM cards. Suppose that cell 1, of volume V, is made of material 2 with a given density. F4:n 1 F14:n 1 FM14 (1 2 102) (-1 2 102) (-1 2 -6:102) (-1 2 -6 -7) Tally 4 just gives the flux in cell 1. Tally 14 gives: Ð . ð1=VÞ ’ðEÞn; ðEÞ dE for material 2. Ð . ð1=VÞ ’ðEÞn; ðEÞ dE for material 2; the density of cell 1 is used because the multiplicative constant is negative to calculate the number of atoms. Ð . ð1=VÞ ’ðEÞðn; ðEÞ þ fis ðEÞÞ dE for material 2; the ‘:’ means that reactions Ð –6 and 102 are summed. . ð1=VÞ ’ðEÞðEÞfis ðEÞ dE for material 2; the ‘ ’ means that reaction –6 is multiplied by ‘reaction’ –7. To calculate

ð hn; i ¼

’ðEÞn; ðEÞ dE ð ’ðEÞ dE

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one just has to take the ratio of the first value of tally 14 by the value of tally 4 (for complex cells, MCNP is not able to calculate the cell volume and one has to specify it with an SD or VOL card). It is also possible to calculate these quantities for a material which is not directly present in the geometry; it is just necessary to define this material. This is particularly interesting if one wants to see the contribution of individual nuclei to a mixture (for example the role of hydrogen and of oxygen in the water). Note that, in this case, the multiplicative constant C must be positive, because the density of this perturbing material is not given. Other useful cards NPS and CTME cards An NPS card specifies the number of particle histories: NPS n means that the Monte Carlo calculation stops after n source particles. . A CTME card specifies the computer time limit (in minutes): if a run duration is greater, the Monte Carlo calculation stops. .

TotNu and NoNu cards NoNu means that fission is treated as a simple capture; this card may be useful to see primary fission neutrons. . TotNu has two meanings, depending on the type of calculation: (1) In KCODE mode (critical calculation), if TOTNU is not present or is present with no argument, the total neutron number (including delayed neutrons) is used; if TOTNU is followed by NO, the prompt is used. (2) In non-KCODE mode, if TOTNU is absent or present with NO, the prompt is used, whereas if TOTNU is present with no argument, the total is used. .

PHYS card This card is used as PHYS:n Emax Elim Emax is the maximum neutron energy and Elim is the boundary separating implicit and analogue capture: below Elim, capture is analogue (i.e. a true capture) and above it is implicit (the weight of neutrons is reduced). In the latter case, the CUT card is useful to specify a minimum neutron weight before killing them. PRDMP card When an MCNP calculation is run, a lot of information and the tally results are written in an ‘o’ file, which is easy to read for a human being, but less so for a computer (lots of text). The PRDMP card is useful to save tally results in an ‘m’ file, which has a pretty standard format and is easier to read with a

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computer; in addition, MCNP can read the ‘m’ file and plot some results. To produce such a file, the following example is pretty good: PRDMP 2j -1 (2j means jump the first two entries, and –1 is to write the ‘m’ file).

5.6

Examples

5.6.1

Reactivity calculation

As already mentioned, in a Monte Carlo calculation, a particle is followed from its birth to its death. The code ends with the end of the last history. But in an (over-)critical system, neutron chains are infinite and such a calculation will not end. A special technique has been developed to handle this case. It allows us to obtain keff and tallies for any criticality. Starting with Ns source neutrons, the calculation develops over cycles (or neutron generations); within a cycle, a neutron is followed from its birth to its death but, this time, fission is considered as a cycle termination (like a capture). Within a cycle, at each collision point, the number of fission neutrons stored n is randomly sampled in order to have a mean value hni ¼

fis W t keff

(W is the neutron weight, fis=t is the microscopic fission/total cross-section and keff is estimated from the previous cycle, or Pis a user-given value if this is the first cycle). For the next cycle, M ¼ n particles (M Ns Þ are emitted on the corresponding collision site. The effective multiplication factor can be obtained as Nifis Nifis ¼ lim i ! 1 N fis i ! 1 N abs i i1

keff ¼ lim

where Nifis and Niabs are respectively the number of fissions and of absorptions in generation i. In MCNP, three methods (based on cross-section calculations) are used to obtain the effective multiplication factor keff and the final value is a weighted average of the three factors. To perform a critical calculation, one has to use the KCODE card (in the Data block card); a ‘starting’ source has to be defined to initiate the first cycle. The syntax is KCODE Ns kexpt NI NT where Ns is the number of ‘source’ neutrons per cycle, kexpt is the expected value of the effective multiplication factor, NT is the total number of cycles (100 is usually sufficient) and NI is the number of initial cycles that are

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excluded from the keff (and tallies) calculation, in order to give enough time for the fission source to be established. Going back to the example of the uranium sphere in a light water cylinder, the input file could be First simple geometry c c Cell cards c 1 1 -18.75 -1 $ the inner sphere 2 2 -1.0 -2 3 -4 1 $ the cylinder without the sphere 3 0 #2 #1 $ exterior c c Surface cards c 1 SO 5 $ centred sphere with R=5 cm 2 CZ 20 $ infinite cylinder with R=20 cm 3 PZ -20 $ bottom plane intersecting the cylinder. 4 PZ 20 $ top plane intersecting the cylinder. c Materials M1 92235.60c 1 $ 235 U M2 1001.60c 2 8016.60c 1 $ H2 O c Source : kcode sdef pos 0 0 0 erg 2.5 kcode 1000 1 10 80 totnu Suppose that this input file is named 1stgeo. MCNP can be run by entering the command mcnp n=1stgeo At the end of the run, the keff value is displayed on the screen and is stored in the 1stgeo file (after the short table summarizing the number of source neutrons, captures, escapes, . . .). The result is keff ¼ 0:840 0:003 (with more cycles, the precision is improved). If the sphere radius is changed from 5 cm to 6.312 cm, keff ¼ 0:999 0:003 which corresponds to the critical case. Then, using the Sð; Þ treatment (by inserting an MT2 lwtr.07 card after material M2), we see that keff is reduced to keff ¼ 0:974 0:003; this demonstrates the importance of this treatment for thermal neutrons. 5.6.2

Homogeneous versus heterogeneous cores

After these very simple examples, we are able to ‘build’ a more realistic reactor. As a starting point, we want to study a light water reactor (critical)

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with a lead reflector, loaded with UO2 fuel. Suppose that the water volume is about 4 times more than the fuel volume. The core is a cylinder with a diameter equal to its height. The core radius is 1 m, the reflector thickness is 50 cm and the iron tank containing the two is 5 cm thick. Homogeneous core In a first step, suppose that we consider a homogeneous core of (H2 O þ UO2 ) with 1.35% (compared to 238 U) 235 U enrichment. The condition VH2 O =VUO2 ¼ 4 implies a fuel density of fuel ¼ 2:8 g/cm3 (taking UO2 10 g/cm3 Þ and the atomic composition of the fuel is 6 moles of H2 O for 1 mole of UO2 . Now, we are able to write the MCNP input file: Homogeneous core c c Exterior c 1 0 1:-2:3 imp:n=0 c c Iron tank c 2 1 -7.87 -1 2 -3 (4:-5:6) imp:n=1 c c Lead reflector c 3 2 -10.34 -4 5 -6 (10:-11:12) imp:n=1 c c Core c 4 3 -2.8 -10 11 -12 imp:n=1 c tank/reflector surfaces 1 cz 155 2 pz -155 3 pz 155 4 cz 150 5 pz -150 6 pz 150 c reflector/core surfaces 10 cz 100 11 pz -100 12 pz 100 c Material

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$ Iron of the tank $ Lead of the reflector 0.0135 92238.60c 0.987 & $ 235 U (1.35%) + 238 U 12. 8016.60c 8. $ 6 H2 O + 1 O2 (of the fuel) 0 erg 2.5 10 150

m1 26000.55c 1 m2 82000.50c 1 m3 92235.60c 1001.60c sdef pos 0 0 kcode 1000 1 totnu

The keff of the reactor with that geometry is keff ¼ 1:001 0:001. Heterogeneous core The previous reactor is very simple; a more realistic configuration consists of replacing the homogeneous core by a core with fuel rods in a moderator. Each rod has a radius of 0.5 cm. The core will then be filled by a square lattice. In order to satisfy the moderator/fuel volume ratio, the side of each square is 1.98 cm. Of course, it is not possible to describe each small cell. But it is possible to define a lattice (hexagonal or hexahedra). One has to define a mesh (the hexahedra or the hexagon) and two universes: a universe is either a lattice or a collection of cells; it defines different ‘geometry levels’ (somewhat like Russian dolls). Heterogeneous core c c Exterior c 1 0 1:-2:3 imp:n=0 c c Iron tank c 2 1 -7.87 -1 2 -3 (4:-5:6) imp:n=1 c c Lead reflector c 3 2 -10.34 -4 5 -6 (10:-11:12) imp:n=1 c c Core c 4 0 -10 11 -12 imp:n=1 fill=1 5 0 21 -22 23 -24 imp:n=1 u=1 fill=2 lat=1 6 3 -1. 30 imp:n=1 u=2 7 4 -10. -30 imp:n=1 u=2

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Practical simulation methods c tank/reflector surfaces 1 cz 155 2 pz -155 3 pz 155 4 cz 150 5 pz -150 6 pz 150 c reflector/core surfaces 10 cz 100 11 pz -100 12 pz 100 c square mesh 21 px -0.99 22 px 0.99 23 py -0.99 24 py 0.99 c 30 cz 0.5 c Material m1 26000.55c 1 $ Iron of the tank m2 82000.50c 1 $ Lead of the reflector m4 92235.60c 0.0135 92238.60c 0.9865 & 8016.60c 2. $ 235 U(1.35%)+238 U(98.65%)+1 O2 m3 1001.60c 2. 8016.60c 1. $ 6 H2 O + 1 O2 (of the fuel) sdef pos 0 0 0 erg 2.5 kcode 1000 1 10 150 totnu

The geometry is shown in figure 5.5. In this example, the core is a cylinder filled with universe 1 (cell 4, the fill card ¼ 1). This universe is defined in cell 5 (u ¼ 1). Cell 5 is filled with universe 2 (fill ¼ 2) with a hexahedron lattice (lat ¼ 1); the hexahedral mesh is defined by the planes 21 to 24. Universe 2 is defined as cells 6 and 7 (u ¼ 2). The keff of the heterogeneous reactor is keff ¼ 1:050 0:001, which is higher because the flux is slightly more thermal (self-shielding of uranium capture) than in the homogeneous case. 5.6.3

Subcritical core

Let us see now the case of a subcritical ADSR. Suppose it is filled with Th/233 U and cooled by lead; the target is a lead spallation target. In this example, the core is filled with a hexagonal lattice. As already said in the previous example, the lattice that we use is a nice tool but some fuel 232

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Figure 5.5. Z-Slice of the critical heterogeneous reactor: (a) complete view, (b) loop around core/reflector. One can see that some fuel rods are partially cut.

rods are cut. In order to have a better description, we present here the way to overcome that problem; the resulting MCNP file is much longer; we will just illustrate the method on our design with non-realistic sizes (large hexagons, thick fuel rods, small reactor). The spallation source cannot be described with MCNP alone (one can use MCNPX of course); we will take a cylindrical source as described in section 5.5.3 to ‘represent’ the spallation source. The following MCNP input file (named ads) describes the reactor: Heterogeneous core c c Exterior c 1 0 1:-2:3 imp:n=0 c c Iron tank c 2 1 -7.87 -1 2 -3 (4:-5:6) (50:-6) imp:n=1 $cyl.-(reflector hole(middle))-(beam hole(top)) c c Lead reflector c 3 2 -10.34 -4 5 -6 (10:-11:12) (50:-12) imp:n=1 $cyl -(core hole(middle))-(beam hole(top)) c c Core c 4 0 -10 11 -12 40 imp:n=1 fill=2 5 0 21 -22 23 -24 25 -26 imp:n=1 u=2 lat=2 fill=-10:10 -10:10 0:0 &

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Practical simulation methods 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 & 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 & 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 & 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 1 1 1 1 & 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 1 1 1 & 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 1 1 1 & 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 1 1 1 & 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 & 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 & 1 1 1 1 3 3 3 3 3 3 1 1 3 3 3 3 3 3 1 1 1 & 1 1 1 1 3 3 3 3 3 1 1 1 3 3 3 3 3 1 1 1 1 & 1 1 1 3 3 3 3 3 3 1 1 3 3 3 3 3 3 1 1 1 1 & 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 & 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 & 1 1 1 3 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 & 1 1 1 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 & 1 1 1 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1 1 & 1 1 1 1 3 3 3 3 3 3 1 1 1 1 1 1 1 1 1 1 1 & 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 & 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 & 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 3 -10.34 30 imp:n=1 u=3 $ 7 4 -10. -30 imp:n=1 u=3 $ fuel rod surround by lead 8 2 -10.34 31 -32 imp:n=1 u=1 $ c c Target and beam pipe c 20 2 -10.34 -40 41 11 -12 imp:n=1 $ lead from core to the target vessel surface 21 1 -7.87 -41 43 -44 (42:-45:46) (51:-46) imp:n=1 $ iron vessel for the target 22 3 -10.34 -42 45 -46 imp:n=1 $ lead target 23 2 -10.34 -41 11 -43 imp:n=1 $ lead under the target vessel 24 2 -10.34 -41 50 44 -12 imp:n=1 $ lead above target vessel to pipe c beam pipe 30 0 -51 46 -44 imp:n=1 $ hole for the beam in the target vessel 31 1 -7.87 -50 51 44 -12 imp:n=1 $ pipe in the core 32 0 -51 44 -12 imp:n=1 $ vacuum in pipe 33 1 -7.87 -50 51 12 -6 imp:n=1 $ pipe in the lead reflector 34 0 -51 12 -6 imp:n=1 $ vacuum in pipe 35 1 -7.87 -50 51 6 -3 imp:n=1 $ pipe in the tank 36 0 -51 6 -3 imp:n=1 $ vacuum in pipe c Tank/Reflector surfaces 1 cz 155 2 pz -155 3 pz 155 4 cz 150 5 pz -150

Examples 6 pz 150 c Reflector/Core surfaces 10 cz 100 11 pz -100 12 pz 100 c Hexagonal Mesh surfaces 21 py -6 22 py 6 23 p 8.66025e-01 5.00000e-01 24 p 8.66025e-01 5.00000e-01 25 p 8.66025e-01 -5.00000e-01 26 p 8.66025e-01 -5.00000e-01 c Fuel Rod surfaces 30 cz 2 31 pz -101 32 pz 101 c Target surfaces 40 cz 13 41 cz 10 42 cz 9.5 43 pz -30 44 pz 10 45 pz -29.5 46 pz 9.5 c Beam Pipe surfaces 50 cz 2 51 cz 1.5

0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00

129

-6 6 -6 6

c Material m1 26000.55c 1 $ Iron of the tank m2 82000.50c 1 $ Pb of the reflector m3 82000.50c 1 $ Pb of target and core m4 92233.60c 0.15 90232.60c 0.85 & $ 233 U(15%) and 232 Th 8016.60c 4. $ 16 O of ThO2 and UO2 c Neutron Source SDEF POS 0 0 0 ERG=D1 RAD=D2 EXT=D3 AXS 0 0 -1 SP1 -5 1.3 $ pðEÞ / E expðE=1:3Þ SI2 0. 9. $ Radial extension (disk of radius 9 cm) SI3 -9 29 $ Z extension (from source position) TOTNU PRDMP 2J -1

As can be seen, the core is a cylinder (cell 4) filled with universe 2, the hexagonal lattice (lat ¼ 2). But this time, the position of each mesh of the lattice is specified: the lattice is made of 10 ð10Þ þ 1 ¼ 21 hexagons in x, the same in y and there is no z lattice. Each mesh is filled with a specific universe: 1 for lead hexagons and 3 for fuel þ lead hexagons (we have organized the line in order to visualize a schematic view of the hexagonal

This is not necessary, but lines must not stop beyond column 80.

130


Figure 5.6. ADSR geometry. (a) Z-slice for z ¼ 0; (b) X-slice for x ¼ 0.

lattice: the ‘fuel’ part of the core is more or less cylindrical and the ‘hole’ in its centre is for the target). The geometry is displayed in figures 5.6 and 5.7. Flux as a function of energy To obtain the flux as a function of the energy in the fuel (cell 7), and in the lead of the core (cell 6), we just have to add the following to the MCNP input file (ads): F4:N 7 6 SD4 392070.76 3498808.17 $ Vol of cell 7 and cell 6 in cm3 E4 1.00e-08 1.58e-08 2.51e-08 3.98e-08 6.31e-08 1.00e-07 1.58e-07 & 2.51e-07 3.98e-07 6.31e-07 1.00e-06 1.58e-06 2.51e-06 3.98e-06 & 6.31e-06 1.00e-05 1.58e-05 2.51e-05 3.98e-05 6.31e-05 1.00e-04 & 1.58e-04 2.51e-04 3.98e-04 6.31e-04 1.00e-03 1.58e-03 2.51e-03 & 3.98e-03 6.31e-03 1.00e-02 1.58e-02 2.51e-02 3.98e-02 6.31e-02 & 1.00e-01 1.58e-01 2.51e-01 3.98e-01 6.31e-01 1.00e+00 1.58e+00 & 2.51e+00 3.98e+00 6.31e+00 1.00e+01 1.58e+01

The first line is the tally definition; the second line (SD4) is necessary because, for these complex cells, MCNP is not able to calculate the cell volumes. An SD card is a means to specify the volume of each cell of the tally bin (the first entry is the volume of one fuel rod times the number of rods (156) and the second entry is (the volume of a hexagon minus the volume of a fuel rod) times the number of hexagons). Then we have chosen a logarithmic energy binning. Of course, the energy dependence could be much more precise by specifying more bins. In

One can also use the VOL card where all cell volumes have to be given.

Examples

131

Figure 5.7. ADSR geometry: the target.

order to see the result of this calculation, one can use MCNP with the command line mcnp z then, one has to read the ‘m’ file (here it is adsm) by rmctal adsm Tally 4 for cell 7 is plotted by default (first tally, first cell bin) with linear x axis. To obtain a log–log representation, loglog By default, the tallies are ‘normalized’ by the bin width, i.e. in our case d=dE is plotted; but, because the chosen binning is logarithmic, the physical quantity to be plotted is d=d logðEÞ; MCNP can only plot quantities such as d=dE or d; because we have chosen a constant logarithmic step for the binning, one can plot d / d=d logðEÞ in this case; the normalization is suppressed by the nonorm command.

132

Practical simulation methods One can plot tally 4 for cell 6 (lead moderator) with the command tally 4 fixed f 2

which means to plot the cell bin 2 (f 2) for tally 4. To see the flux in cell 7 and cell 6 on the same plot one can use tally 4 fixed f 1 coplot tally 4 fixed f 2

Flux as a function of the radius It is possible to obtain quantities such as neutron fluence as a function of distance (radial or axial). For example, to obtain ðrÞ in the core, one has to divide the core cylinder into slices (like an onion). We illustrate that point for the ads file. Some new cylinders have to be added: 13 14 15 16

cz cz cz cz

80 60 40 20

Then, cell 4 is modified and new cells are added: 4 0 104 105 106 107

-10 13 11 -12 40 imp:n=1 fill=2 0 -13 14 11 -12 40 imp:n=1 fill=2 0 -14 15 11 -12 40 imp:n=1 fill=2 0 -15 16 11 -12 40 imp:n=1 fill=2 0 -16 11 -12 40 imp:n=1 fill=2

and, of course, a tally card F4:N 4 104 105 106 107 5.6.4

Precision

The influence of very long multiplication chains on the accuracy of Monte Carlo simulations has been discussed by the CERN group [80]. M ¼ 1=ð1 kÞ being the total number of neutrons originating from one initial neutron, these authors give the number N of cascades that have to be generated to obtain a relative error " on M: M N ¼ 2:56 2 2:55 " with the number of neutrons per fission. Equivalently, the precision for N cascades is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M " ¼ 1:6 : N 2:55

Fuel evolution

5.7

133

Fuel evolution

The aim of this section is to discuss fuel evolution by coupling MCNP to an external programme that solves Bateman equations, thus permitting a correct treatment of the time evolution of the flux, of the composition, and the average reaction cross-sections during the reactor operation. The results of such rather involved calculations are a significant improvement over the schematic treatment of section 3.6. A study of 232 Th/233 U, lead cool, fast ADS has been carried out by Brandan et al. [77]; here, we just present the method and some results of this study. The Bateman equations are solved by fourth-order Runge–Kutta (sufficient for typical evolution times between a few weeks and a few years). One only uses the mean cross-section for a given nuclear reaction, defined as, ð ð ðEÞðE; rÞ dE d3 r Eð r ð ¼ ; ðE; rÞ dE d3 r E

r

where ðEÞ is the cross-section for the reaction and ðE; rÞ is the differential neutron flux at energy E and position r within the volume V. The integral flux is defined as ð ð 1 ðE; rÞ dE d3 r: ¼ V E r After an initial MCNP calculation: one extracts from each cell the values of the average reaction rate per target nucleus, of the average flux normalized to one source neutron, and the average cross-sections (read from the ‘m’ file), . the linear system of equations is solved by Runge–Kutta, . one writes a new MCNP file with the new values of the materials composition, . and continues the calculation loop one more step until the predetermined cycle duration time is reached. .

However, one has to pay attention to some other points.

One can find at NEA (http://www.nea.fr) the MCB (Monte Carlo Continuous Energy Burnup) code which is a general-purpose code that can be used for calculation of nuclide density evolution with burn-up or decay, including keff calculations of critical and subcritical systems and neutron transport calculation together with all necessary reaction rates and energy deposition (at six different temperature). MCB is compatible with MCNP-4C and complete burn-up calculations can be done in a single run that requires preparation of a single input file with a very few more data lines compared with a regular MCNP input.

134 5.7.1

Practical simulation methods Evolution constraint

One has to decide which are the relevant parameters for controlling the evolution (fixed ks=eff , fixed power, . . .). For example, in the study by Brandan et al. [77], the average reactor power density (per fuel volume unit) was fixed at 330 W/cm3 , which permits local values of the order of 600 W/cm3 at the reactor centre, an acceptable value from a thermodynamical point of view. The proton beam intensity (in this study a true spallation target was generated by FLUKA) is adjusted during the reactor evolution to compensate the reactivity fluctuations from one cycle to the next. This restriction imposes a constant number of fissions per unit time, so that it is possible to associate a fixed burn-up to all cycles (defined at fixed operation periods). In turn, this allows the direct comparison among inventories after different cycles for all types of reactor fuel. 5.7.2

Spatial flux

Since the neutron flux depends on the spatial position within the reactor, the core volume is divided into evolution cells (cell dimensions are typically z 5 cm and r 5 cm). The evolution of each type of nucleus within the cell takes into account nucleus disappearance by neutron-induced reactions and by natural decay, and nucleus production by neutron reactions on a parent nucleus and by natural decay of a parent. The description of these four processes constitutes a system of Bateman’s coupled equations whose solution is a vector that contains the number of nuclei of each type. 5.7.3

Special cross-section data

Fission product Since not all the relevant cross-sections for the production and disappearance of fission products are included in the MCNP data base (ENDF-VI), one has to use a kind of ‘mean fission product’; the cross-sections for all fission fragments have to be analysed, together with the mass distribution of fission fragments for each of the fissile nuclei relevant to the study to evaluate the mean fission product and its cross-section (a linear combination of the data available in ENDF-VI is used to reproduce the mean values). A time evolution study of these average cross-sections for the fissile nuclei has to be done to modify, if necessary, the linear combination after exposure to the neutron flux. In fast spectra, these cross-sections are generally more or

For example, in the study of Ref. [77], it was found that the average neutron capture crosssection for the nuclei produced by the fission of 233 U was equal to 0.15 barn.

Fuel evolution

135

less stable, but in a thermal case, the variation of these cross-sections being much faster, one has to modify the linear combination. Americium The production of the metastable 242m Am (excitation energy 48.6 keV, halflife 141 years) has to be taken into account, assuming that 10% of the 241 Am neutron capture produces this particular state [78]. 5.7.4

Time step between two MCNPs

The uncertainties associated with the coupling between successive Monte Carlo (MC) steps and the integration of the (non-linear) differential equations were carefully studied by Brandan et al. [77]. They found that an integration step equal to one fifth of the shortest nuclide half-life was sufficient to reach convergence of the solution to the differential equations. They anticipate two types of error: statistical errors in the average crosssection determination after one MC step, and systematic errors due to the evolution of these cross-sections during the time of a single MC calculation. The CPU time devoted to each MC step was fixed, thus fixing the total number of neutrons being followed and the statistical errors at all cycles, independent of the system multiplication. For instance, a CPU equal to 15 min (in a DEC Alpha 500/500 workstation) permits one to follow some 20 000 neutrons, which correspond to 400 source neutrons for a typical ks ¼ 0:98. This way, the statistical errors in the calculated cross-sections were typically less than 10% in each cell. However, the total error after a series of MC steps may depend dramatically on the particular nucleus being studied. Tests were performed simulating the evolution of a 233 U reactor over 5 years. In a first case, the differential equations were solved every 3 months of operation, i.e. the complete evolution included a total of 20 MC steps. These results were compared with a second case, with only one MC step for the 5-year solution. Figure 5.8 illustrates the resulting time evolution of the number of 239 Pu 244 and Cm nuclei during the last year at the end of the evolution, for the two cases. The data points and thick curve show the results from the 20-step calculation. The thin solid curves show independent one-step results and the dashed curve represents their average. The dispersion of the one-step results reflects statistical fluctuations in the initial average cross-sections. These fluctuations, indicated in figure 5.8 by the (F) arrows, reach 1% of the average for the production of 239 Pu and 10% for the production of 244 Cm. Arrows (S) in figure 5.8 show the (systematic) discrepancies between the two procedures. The systematic error that could be made when treating the 5 year evolution with only one intermediate integration is of the order of 1% for 239 Pu and of 20% for 244 Cm. The difference

136


Figure 5.8. Evolution of the 239 Pu (a) and 244 Cm (b) abundances at the end of a 5 year cycle for different numbers of Monte Carlo steps. Solid symbols show results for Monte Carlo calculations every 3 months. Thin curves are independent 5 year calculations and the thick dashed curve is their average. Arrows (F) indicate the fluctuations in the 5 year evolutions. Arrows (S) show the systematic difference between the two methods.

between these nuclei is due to the time evolution of their mean cross-sections, caused by the neutron spectral evolution during the operation. As figure 5.8 has shown, this variation cannot be properly taken into account by one-step calculations. All the results reported by Brandan et al. [77] have been obtained with 15 minute CPU Monte Carlo simulations per step and

Fuel evolution

137

20 MC steps per 5 year calculation; test calculations have shown that the systematic errors associated with this choice are negligible. Examples of particular cases where this detailed approach is absolutely required have been given by David et al. [79].

Chapter 6 The neutron source

In most hybrid reactor concepts, the external neutrons are provided by the interaction of accelerated charged particles with matter. The most widely proposed systems use high-energy protons. A few other propositions resort to electrons or deuterons as well as muons as originators of neutronproducing reactions. Because of the importance of high-energy protons we discuss this case more thoroughly.

6.1

Interaction of protons with matter

Energetic protons and nuclei interact with matter mostly by collisions with electrons. These lead to progressive energy loss. 6.1.1

Electronic energy losses

The energy loss due to electronic collisions is given by Bethe’s formula which reads, to a very good approximation, dE DZ Zp 2 2me c2 2 2 2 ¼ ln ð6:1Þ MeV=cm dx A IðZÞ with A, Z and respectively the mass number, the charge and the mass density of the target nucleus and Zp , and E respectively the charge, velocity and energy of the projectile. The constant D ¼ 0:3071 MeV cm2 /g; I is the average ionization potential of target atoms, with approximate value IðZÞ ¼ 16Z 0:9 eV. Also me c2 ¼ 0:511 MeV. Bethe’s formula does not let us obtain an analytic expression of the projectile range. A common approximation, which allows reasonable proton range estimates, is 0:75

dE Z Zp Ap ¼ 496 ¼ cR E 0:75 MeV=cm dx A E 0:75 138

ð6:2Þ

139

Interaction of protons with matter where Z cR ¼ 496 Zp A0:75 p : A

ð6:3Þ

The range is obtained by integration of dx E 0:75 ¼ dE cR

ð6:4Þ

which gives Rel ðEÞ ¼

E 1:75 : 1:75cR

ð6:5Þ

Note that the stopping power per atom is proportional to Z, and independent of A and : For a 1 GeV proton one gets an electronic range Rel ðE ¼ 1 GeVÞ ¼ 205

A : Z

ð6:6Þ

Taking the examples of beryllium and lead we get, for a 1 GeV proton, . .

for beryllium: Rel ðE ¼ 1 GeVÞ ¼ 250 cm for lead Rel ðE ¼ 1 GeVÞ ¼ 45 cm.

6.1.2

Nuclear stopping

While being slowed down, protons may undergo nuclear reactions. For proton energies larger than, typically, 100 MeV, the most violent reactions are called spallation. These account for most of the neutrons produced. In a crude, black nucleus model, the reaction cross-section reads Vc ðEÞ ¼ 0 1 ð6:7Þ E with the geometrical cross-section ð1:3A1=3 þ 1Þ2 barns: 0 ¼ 100

ð6:8Þ

Vc is the coulomb barrier: Vc ¼

1:44Z A þ A : A 1:3A1=3 þ 1

ð6:9Þ

With these expressions it is possible to derive a nuclear range. For highenergy protons, the cross-section reduces to the black nucleus value, and thus the nuclear range reads approximately Rnuc ¼

A 31A1=3 : ¼ 0:60

140

The neutron source

Thus, for . .

beryllium Rnuc ¼ 35 cm lead Rnuc ¼ 16 cm.

The probability that 1 GeV protons suffer nuclear reactions is very high both for beryllium and for lead. The nuclear range is smaller, relative to the electronic range, for light nuclei. On the other hand, the energy deposited in the target nucleus following a nuclear encounter is larger for heavy targets. In a simple forward scattering picture (a` la Glauber) one expects that the number of target nucleons hit, and hence the energy deposited in the target nucleus, is proportional to the target thickness, i.e. to A1=3 . It follows that the ratio of nuclear energy loss to electronic energy loss scales like ðA=ZÞE 0:75 . The over-simplistic considerations we have just made are only intended to give a feeling of the physics of the interaction of high-energy protons with nuclei. It showed that the proton energy should be chosen high enough that nuclear energy losses exceed electronic energy losses. A more detailed treatment requires nuclear cascade simulations. 6.1.3

The nuclear cascade

Most existing codes used for high-energy proton–nucleus reactions are based on the intranuclear cascade (INC) model of Bertini [82] for the first stage of the reaction, the final stages being described by an evaporation (EVAP) model like those by Dostrovski et al. [83] or Dresner [84]. The philosophies of the INC and EVAP models are very different: the INC calculations follow the history of individual nucleons in a classical or semi-classical manner, while the EVAP calculations follow the de-excitation of the whole nucleus while it decays from one nuclear level to a lower one. The connection between the two approaches is one of the delicate points of high (or intermediate) energy simulations of proton–nucleus reactions. In principle the single-particle approach of INC is justified as long as the wavelength of the incident nucleon is smaller than the nucleon radius, i.e. < r=2 Fermi and E > 160 MeV. On the other hand, the evaporation approach is valid as long as the energy of the nucleon does not too much exceed the nuclear potential depth, i.e. about 40 MeV. Thus, the transition energy between the INC and EVAP calculations cannot be specified rigorously. For that reason several codes have added an intermediate step whose domain of validity is expected to overlap the INC and EVAP domains. This step is the pre-equilibrium (PE) step [85].

‘Smaller’ is rather ambiguous. The black nucleus cross-section for n–n interactions is ð2r þ ðcm =2ÞÞ2 and comes close to the classical limit for, say, cm =2 ¼ 0:5r or ¼ r=2.


141

The intranuclear cascade In the INC code of Bertini the incident proton collides with one or several nucleons of the target nucleus. These, in turn, collide with the unperturbed nucleons. A cascade develops. The INC calculation for a specific nucleon stops whenever its energy falls below a specified value, related to the depth of the nuclear potential well. Collisions between cascade nucleons are not allowed. This limitation is lifted in more recent calculations, like those of Yariv and Fraenkel [86, 87], Iljinov et al. [88] and Cugnon and co-workers [89] where cascade–cascade collisions are allowed, at the cost of calculation time. The initial code of Bertini, which is still the most widely used, was poorly documented and is difficult to improve. In particular, better nucleon–nucleon and pion–nucleon cross-sections are now available. Efforts are being made to use newer codes like the code ISABEL of Yariv et al. incorporated as an option in the Los Alamos code LAHET [90], the code of Cugnon incorporated in the CERN GEANT system [94], or a code written by the authors of the FLUKA system [95]. Classical intranuclear cascade calculations, like those just referred to, do not treat, in general, the emission of clusters of nucleons during the process. Work is being done to account for such emissions in the frame of the QMD model [96]. However, such calculations are very time consuming and will, probably, be restricted to specific calculations like those necessary for estimating the production of helium in structural materials. Note that both the evaporation and pre-equilibrium model account for the emission of light nuclei, in the low-energy regime. The pre-equilibrium step The INC model lacks justification for nucleon energies (inside the nucleus) below around 100–150 MeV. Pre-equilibrium models [85, 97–99] have been used for some time in nuclear physics in this energy domain. These models follow a population of quasi-particle excitations of the nuclear Fermi gas by means of a master equation. Quasi-particle states are characterized by their particle escape and damping widths. Angular distributions are associated with the escaping particles. In a sense, pre-equilibrium models allow an easier phenomenological adjustment of angular distributions than does the intranuclear cascade. There are many versions of pre-equilibrium models, but unhappily no clear criteria to choose among them, except their ability to reproduce experimental data. The evaporation step Evaporation calculations are based on the Weisskopf [100] approach, with a few using the Hauser–Feshbach [81] one. Most actual calculations are based

142

The neutron source

on the code developed by Dresner [84]. The most important ingredients of the codes are the level densities. It is important to account for the influence of shell effects on the level density parameters and of their washing out with nuclear temperature. In this respect important improvements have been made with respect to the original Dresner code. They often resort to the level density formula derived by Ignatyuk et al. [101]. Shell effects also have to be treated carefully in their influence on fission barriers. Most INC calculations use the fission model of Atchison [102]. Atchison gives a different treatment for the fission of heavy nuclei with Z > 88 and for that of nuclei with 70 < Z < 89. In the first case a fixed fission barrier, Bf ¼ 6 MeV, is used and ÿn =ÿf is assumed to depend only on the charge of the fissioning nucleus, not on its energy. These approximations are based on the experimental data reviewed by Vandenbosch and Huizenga [103]. For lighter nuclei a statistical model calculation is carried out, using fissility dependent parametrizations of Bf and of af =an : Above a few MeV the results of the calculations are in reasonable agreement with experiments. However, it might be timely to improve the present treatment of fission by including such recently discovered features as the time delay to fission [104], which decreases the fission probability at high excitation energies, the temperature dependence of the fission barriers and symmetry and surface dependence of the level density parameters. 6.1.4

Experimental tests of the INC models

The OECD has recently organized benchmarks [105, 106] comparing available INC calculations. These benchmarks have shown that smallangle neutron spectra were particularly sensitive tests of the code. Small-angle neutron spectra The SATURNE group studied these observables before the close-down of the Saclay (France) synchrotron SATURNE. Figure 6.1 shows the smallangle neutron spectra for a number of targets and 1 GeV incident protons. The data are compared with a calculation using the Bertini INC and with one using the Cugnon INC. Although still not perfect, it is clear that the Cugnon INC is a significant improvement over the Bertini INC. The figure shows the importance of the delta charge exchange peak, 300 MeV lower than the high-energy quasi-elastic peak. Both peaks correspond to a process by which the proton is changed into a neutron. Figure 6.2 shows similar results and comparisons made by the Los Alamos group [90]. Here again, the improvement obtained with the more modern ISABEL [86] INC calculation is clear. The delta peak involves real charged pion exchange, the quasi-elastic peak the excitation of the target into an isobar analogue state. The delta peak is a preferential doorway for the energy deposition of


143

Figure 6.1. Energy spectra of neutrons emitted at different angles, following interaction of 1.2 GeV protons with Pb target [107]. The histogram represents TIERCE simulations [108] using the Bertini (solid line) or Cugnon (dotted line) cascade model.

the high-energy proton (or neutron) into the target nucleus. Indeed, for heavy nuclei the average energy deposition is close to 300 MeV. Fission probabilities Fission probabilities have important implications both on the neutron production of the p–nucleus reaction and on the production of nuclear waste. Figure 6.3 compares experimental fission cross-sections for the highly fissile 235 U to calculations using the Bertini code with or without

144

The neutron source

Figure 6.2. Comparison of experimental neutron energy spectra following the reaction p þ 7 Li with spectra calculated using the Bertini and ISABEL INC codes [90]. Light dots: Bertini; solid line: ISABEL; thick dots: data.

pre-equilibrium treatment. It is seen that, in this case, the pre-equilibrium treatment does not influence the results very much. This is a consequence of the very high fission probability. For neutron energies below 100 MeV, all INC calculations become unsatisfactory. This reflects a deficiency of the INC codes in reproducing reaction cross-sections below that energy. Figure 6.4 compares experimental fission cross-sections for the poorly fissile 208 Pb with calculations using the Bertini and the ISABEL INC codes


145

Figure 6.3. Comparison of experimental neutron induced fission cross-sections of 235 U with calculations based on the Bertini INC with or without pre-equilibrium [90]. Solid line: standard pre-equilibrium MPM calculation [109]. Dashed line: no pre-equilibrium step. Dotted line: hybrid MPM. See Prael [90] for details.

with or without pre-equilibrium treatment, and for different level densities. Here the interest of the pre-equilibrium treatment is far from obvious. The standard level density is that of Igniatyuk [101] and gives the best results. The Ju¨lich level density [84] includes the effects of shells on level densities but not the washing out of these effects with temperature. It clearly underestimates the level density. In any case, as stated earlier, it seems clear that progress has to be made in the fission treatment. Neutron yields The most important experimental observable with respect to hybrid reactors is the number of neutrons produced per proton–nucleus reaction. Recent measurements of neutron multiplicities have been made by Hilscher et al. [110] for thin and thick targets of Pb and U between 1 and 5 GeV. Thin target measurements can be compared with those obtained by different INC calculations done in the frame of the OECD workshop [106]. The measurement for thin targets was carried out at 1.22 GeV, while the benchmark calculations were carried out at 0.8 and 1.6 GeV. In table 6.1 we give a sample of the results of the benchmark and a linear interpolation to 1.22 GeV together with the experimental result. Also present in the last column of the table is a systematics established by Pearlstein [105]. Table 6.1 shows a scatter of close to 30% both on the calculated and measured values.

146

The neutron source

Figure 6.4. Comparison of experimental 208 Pb neutron induced fission cross-sections with calculations based on the Bertini (left) and ISABEL (right) INC models, with and without pre-equilibrium and using different level densities [90]. Bold dots: data by Vonach et al. [91]. Upper left: Bertini INC, RAL fission, and default level density [101]; solid line: standard MPM; dashed line: no MPM; dotted line: hybrid MPM. Lower left: Bertini INC, RAL fission, and standard MPM: solid line: default level density; dashed line: Ju¨lich level density; dotted line: HETC level density. Upper right: ISABEL INC, RAL fission, and default level density; solid line: standard MPM; dashed line: no MPM. Lower right: ISABEL INC, RAL fission, and standard MPM; solid line: default level density; dashed line: Ju¨lich level density; dotted line: HETC level density.


147

Table 6.1. Experimental and computed neutron multiplicities for GeV protons on thin lead targets. Energy

Experiment

PSI1

LANL2

Dubna3

Jaeri4

BNL5

0.80 GeV 1.22 GeV 1.60 GeV

14.5

13.6 16.27 18.7

14.7 18.00 21.0

12.1 14.9 17.5

11.46 13.89 16.10

13.9 17.31 20.4

1

The PSI calculation was made by F. Atchison and H. U. Wenger. It used the Bertini INC code with the Dresner Evaporation code. 2 The calculation was made by E. Prael with LAHET using the Bertini ICNC, precompound emission and the Dresner EVAP4 evaporation code, with the fission model of Atchison. 3 The calculation was made by Mashnik using the code CEM92M [105] which includes INC þ PE þ EVAP modules. 4 The calculation was made by T. Nishida et al. and used the Bertini INC þ Dresner EVAP. 5 Systematics established by S. Pearlstein.

Residue mass and charge distributions Another very important characteristic of spallation reactions is the nature of the heavy residues. This reflects the light particle emission rates and determines the radiotoxicity of the wastes of the spallation reaction. In particular, for heavy target nuclei like lead, the amount of 194 Hg created is a key quantity since 194 Hg has a long half-life of 520 years and decays to 194 Au which has a short lifetime of 38 hours and a large decay energy of 2.5 MeV. Thus the decay of 194 Hg is both long lived and energetic. Up until recently determination of residue mass and charge distributions used standard radiochemical techniques, mostly gamma ray counting of the reaction products. The most extensive study has been that by Michel et al. [111]. This technique is based on the observation of gamma decays following beta decays, and thus misses stable residues as well as very long-lived ones whose activity is too weak to be visible. Recently, the very beautiful inverse kinematic technique promoted in GSI, for fission studies and neutron-rich nuclei synthesis, has been used to tackle the residue question. This technique, essentially, allows determination of all yields of heavy residues and fission fragments. These yields are measured immediately after the reaction and thus cannot be compared directly with those obtained from radiochemical measurements. However, for the so-called shielded isotopes which are directly produced by the reaction mechanism, a direct comparison between the two techniques can be done, and is found to be satisfactory [112]. Figure 6.5 shows a comparison between the experimental data and a few calculations for the reaction p þ Au (0.8 GeV/A). All calculations reported use the Dresner evaporation code. The figure shows a significant improvement when using modern INC calculations for the region of spallation products. However, fission appears to be under-evaluated. Figure 6.6 shows that an

148

The neutron source

Figure 6.5. Comparison of the experimental data and INC calculations for the reaction Auð800 MeV=AÞ þ p [112].

even better agreement is obtained if the Dresner evaporation code is replaced by a modern evaporation code elaborated at GSI [113]. 6.1.5

The neutron source

While the preceding dealt with thin targets, hybrid reactors require thick targets with, at least, complete stopping of the proton beam. All particles produced in the first nuclear interaction should be followed until they escape in the vacuum or are absorbed. They may be the origin of new particles and so forth. A particle cascade is thus generated. These cascades are simulated by transport codes, which are all built on the same scheme. Nuclear reactions induced by particles whose energy is larger than a specified value (at the moment 20 MeV for neutrons) are treated by INC modules. Neutrons below the cut-off are then followed with specific neutron Monte Carlo transport codes like MCNP [74], MORSE [72] or MC2 [75]. In principle the calculation can proceed until all neutrons have been absorbed or have escaped the medium, whatever its properties. However, in the case of a neutron multiplying medium, it is much more efficient to distinguish between source neutrons and secondary neutrons produced by the As said in the previous subsection these include a high-energy part like the Bertini cascade, an evaporation part like the Dresner EVAP, and, possibly, a pre-equilibrium part.


149

Figure 6.6. Comparison [112] of the experimental data and a calculation using the INC code of Cugnon et al. [89] and the GSI evaporation code [113] for the reaction Auð800 MeV=AÞ þ p.

multiplication. Then if N0 is the number of primary neutrons following (see chapter 4), for example, the interaction of a proton with a target surrounded by a multiplying medium, characterized by a multiplication factor k, the total number of created neutrons, after multiplication, is N0 : ð1 kÞ

ð6:10Þ

The number of secondary neutrons (produced after at least one multiplication) is kN0 : ð1 kÞ

ð6:11Þ

The distinction between source and secondary neutrons is by no means trivial. It is relatively easy if the source and multiplying media are distinct. In this case one could define the source neutrons as those coming out of the source medium and penetrating into the multiplying medium. However, even for this simple case, neutrons can originate in the multiplying medium, penetrate the source medium and be scattered back into the

In principle k depends on the properties of the source neutron (energy, spatial distribution) and on the geometry of the reactor and is usually denoted ks (‘k source’). For simplicity we keep the notation k, at this stage.

150

The neutron source

multiplying medium. Furthermore, high-energy neutrons penetrating the multiplying medium have different multiplication properties from ‘average secondary neutrons’: for example, they can produce more (n; xn) reactions, and, in the case of fission, lead to a fission with higher than average neutron multiplicity. In practice, it is fortunate that spallation (evaporation) neutrons have energy spectra which are close to fission neutron spectra. Thus, in most cases, neutrons are not followed if they are born with an energy below a cut-off value Ecut , considered high with respect to fission and evaporation neutrons (typically 5 to 10 MeV). Such neutrons with energy E < Ecut are then considered as the source neutrons and kept as inputs for calculations with the neutron propagation codes. This procedure allows a very convenient possibility of disconnecting the study of the multiplying medium from that of the neutron source. However, this ‘operational’ definition of source neutrons makes the comparison with experiment difficult. Indeed, experimentally [110], one usually measures the number of neutrons coming out from a thick target. Thick target calculations As explained above, for thick target measurements, the connection between the high-energy INC codes and the low energy is usually done at neutron energies of 20 MeV. However, at such low energies, the INC codes are not expected to perform well. Work is in progress to extend the low-energy codes to much higher energies, typically 150 MeV. To that end, the MCNPX code of Los Alamos [114] uses pre-equilibrium models to generate data files specific to the different nuclear species. The approach of the Bruyeres le Chatel group is, rather, to use an optical model calculation [115]. Figure 6.7, prepared by Koning, compares an experimental neutron spectrum, as observed at JAERI [116], to the traditional calculation and to a calculation incorporating the Bruyeres le Chatel approach. The improvement is striking. Calculations using the new library prepared by Los Alamos give an agreement with experiment similar to that obtained with the Bruyeres le Chatel method. In particular one can see that the strong discontinuity at 20 MeV displayed by the traditional calculation, which is due to the inaccuracy of the INC model, is completely removed in the new calculations. Thick-target neutron multiplicities Thick-target neutron yields are, evidently, of great interest for hybrid reactor designs. As mentioned above, most measurements resort to a measure of the number of neutrons exiting from a large piece of material. The recent measurements of Hilscher et al. [110] use a large, gadolinium loaded, liquid scintillator. Such detectors allow an event by event measurement of the neutron multiplicity, with a very high detection efficiency. However, the


151

Figure 6.7. 68 MeV neutrons on 40 cm of iron, detector at 40 cm: comparison of experimental data (circles) with two calculations, the best one using MCNPX up to neutron energies of 150 MeV. Figure communicated by A J Koning.

neutron energy is not measured and the detection efficiency does depend on the neutron energies, especially above 10 MeV. In the Hilscher work an average neutron detection efficiency of 0.85 was assumed. Independently of the difficulties mentioned above for making a direct comparison between these measurements and the results of a calculation, it is not trivial to define an optimal size for the target. If the target is too small, the initial cascade may not be contained. This is exemplified in figure 6.8 which shows that the multiplicity measured depends on the length and diameter of the target. It is possible to find a target thickness and diameter such that the multiplicity saturates. However, this does not guarantee that the measured multiplicity will be exactly the number of neutrons emitted per incident particle. This is due to two opposite effects: (a) neutrons may be absorbed in the target, even after being reflected from the detector, and (b) neutron multiplication may already be effective in the target. Lead targets should be largely immune from both effects, although Hilscher et al. find that, using a very pure lead target, as compared with From here on we call neutron multiplicity the number of neutrons per incident particle Nn =p rather than the average multiplicity of the neutron distribution hMn i. The difference between the two numbers in the work by Hilscher et al. comes from cases where the incident particle did not suffer any nuclear reaction. In practice, for targets thick enough, the two measurements converge.

152

The neutron source

Figure 6.8. From top to bottom: survival probability, mean number of neutrons per proton, mean energy escaping from the target and detected in the prompt pulse, and mean neutron multiplicities and most probable neutron multiplicity as measured on the multiplicity distribution of neutrons following a nuclear reaction. From Hilscher et al. [110].


153

Figure 6.9. Mean neutron multiplicity per incident proton on lead, as a function of the proton energy. Solid circles: data of Hischer et al. [110]. Open circles: moderator measurement of Vassilkov et al. [117].

their standard one, the neutron multiplicity is increased by approximately 2.5%. Figure 6.9 shows the ‘asymptotic’ results obtained by Hilscher et al. Uranium, on the other hand, is a multiplying medium whose k1 can be estimated from equation (3.75) and table 3.2, with a value of ¼ 2:3: kUnat ¼ 0:29 corresponding to a total multiplication in an infinite medium 1 of 1=ð1 0:29Þ ¼ 1: 4: This effect is probably responsible for the much higher neutron multiplicities observed with uranium as compared with lead, as can be seen by comparing figures 6.9 and 6.10. Another difficulty in interpreting the data, in view of an application to accelerator driven systems, is that the energy of the neutrons and thus their ‘importance’ is not known. Therefore, the results of direct thick-target multiplicity measurements should not be taken as source data to be input into a neutron propagation code, but rather as benchmarks for selecting valid INC calculations. Such calculations have been done by Hilscher et al.† who find that, as a very good agreement between their measured value for their lead target and a HERMES calculation is obtained, the same calculation yields 30.5 n=p reaction/GeV for a target 100 cm long and 150 cm in diameter. In-medium measurements An alternative way to measure neutron multiplicities has been used by the TARC collaboration [57]. The goal of the TARC experiment was to evaluate the possibility to transmute long-lived fission products like 99 Tc by adiabatic resonance crossing, as discussed in section 3.2.5. To this aim a large 330 ton

Qualitatively, the importance of a neutron, in a multiplying medium, measures the number of its descendents, relative to that of an ‘average’ fission neutron. † Private communication.

154

The neutron source

Figure 6.10. Mean neutron multiplicity per incident proton on uranium, as a function of the proton energy. Solid circles: data of Hilscher et al. [110]. Open circles: moderator measurement of Fraser et al. [118].

block of pure lead was used as a target for the CERN PS protons at various energies. A number of holes were drilled through the block, allowing the placement of different types of neutron detector inside it. In particular, 3 He gas proportional counters and silicon detectors viewing 6 Li or 233 U targets allowed the measurement of the neutron fluxes. In the range between 1 eV and 1 keV, neutron energies were determined by their slowing down time as discussed in section 3.2.5. A detailed mapping of the neutron flux could, therefore, be obtained. This mapping was reproduced by a Monte Carlo simulation, as shown in figure 6.11. The figure shows excellent agreement between the different types of measurement and the calculated values. The simulation found a multiplicity of 31/p/GeV for neutrons falling below the threshold of 20 MeV, in agreement with the HERMES calculation of Hilscher et al. [110]. 6.1.6

State of the art of the simulation codes

From the above discussion it appears that most simulation codes account reasonably well for neutron multiplicities. However, the traditional approach which associates the Bertini INC and the Dresner evaporation for the highenergy part of the cascade, and the MCNP or MORSE codes for neutrons below 20 MeV, has serious failures, especially for the prediction of the neutron energy spectra and the residual nucleus mass and charge distributions. Significant improvements are obtained when either the ISABEL or the Cugnon codes are used, especially in combination with the relatively new GSI evaporation code. The extension of the MCNP type calculations up to 150 MeV, which is being carried out at Los Alamos and Bruyeres le Chatel, is also a very significant improvement. In this respect a large amount of work, both experimental and calculational, has to be done for

Alternative primary neutron production

155

Figure 6.11. Comparison of the neutron fluxes as a function of neutron energy and position in the lead block measured with different types of detector with Monte Carlo simulated values. Figure from the TARC collaboration. The measurement was carried out at a proton momentum of 3.5 GeV/c.

the completion of the evaluated data files for neutrons and protons between 20 and 150 MeV.

6.2


While most proposals for the neutron source of hybrid reactors resort to spallation reactions with high-energy protons, some other possibilities have been proposed, which we discuss briefly here. 6.2.1

Deuteron-induced neutron production

Deuterons are weakly bound nuclei which easily undergo break-up reactions. This is well known at low energies, around 10 MeV, where deuteron beams

156

The neutron source

Figure 6.12. Ratio of neutron production with deuteron projectiles relative to that of proton projectiles, for uranium, beryllium and iron targets, as a function of beam energy [120].

have been used to produce neutrons very efficiently. Light targets, like beryllium, are used in this context to obtain a peaked forward neutron beam, with a most probable energy close to half the deuteron beam energy. The mechanism at work is that, in the nuclear field of the target, the extended deuteron may break up into its proton and neutron constituents. The neutron then escapes the target easily. On the other hand, proton beams are not expected to produce many forward peaked neutrons, since charge exchange reactions at low energies are not probable. At high energies one expects that, because of the high cross-section for charge exchange, neutrons and protons behave similarly. Thus, after break-up, the deuteron would become equivalent to two protons with half the deuteron energy. Therefore, one would not expect any significant gain by using a highenergy deuteron beam as a source of spallation neutrons. Ridikas and Mittig [120] have done a comparison between protons and deuterons. They simulated the interactions of deuterons and protons with thick targets, using the LAHET þ MCNP system. Figure 6.12 shows the variations of the ratio of total neutron multiplicities with deuteron beams divided by those with proton beams as a function of the beam energy. Even at 1 GeV there seems to be an advantage for the deuteron beam, except for the uranium target. This may be related to a larger cross-section of the A recent experimental measurement [121] of neutron multiplicities for deuterons and protons of 200 MeV confirms that, for uranium, the two projectiles are equally neutron prolific.


157

Figure 6.13. Energy gain calculated with different projectiles and targets [120]. Also shown are the results of the FEAT experiment at CERN [122]. The targets were either Be or U surrounded by natural uranium (fuel).

deuteron. The striking feature is the very large enhancement of neutron multiplicities for a deuteron beam impinging on a beryllium target, even at rather high energies. The calculations shows that the excess neutrons are strongly forward peaked. It is true that the neutron multiplicities of the d þ Be reaction are less than those of the p þ U reaction, for energies higher than 200 MeV. However, the authors suggest using a hybrid target where deuterons impinge on a thick beryllium target surrounded by a neutron multiplying medium. They compare this arrangement with a similar one where the beryllium is replaced by uranium. The result is shown in figure 6.13, in terms of energy gain (see below) of the set-up. It appears that, at 1 GeV, the advantage of the beryllium target is not striking. The neutron yield decreases less rapidly with energy with deuterons than with protons, which might ease the requirement on the accelerator. However, deuteron beams are known for activating the accelerator very strongly, so that the final advantage of employing deuteron beams is not obvious. One interesting result shown in figure 6.13 is that, even for protons, the light target is almost as prolific as the heavy one. This is true although, as can be seen in figure 6.14, the neutron multiplicity obtained from a beryllium thick target is nearly three times less than that obtained with a uranium target. This surprising apparent contradiction means that the neutrons produced in the beryllium are more prolific than those born in the uranium: they are more energetic. This is a clear illustration of the difficulty of defining source neutrons.

158

The neutron source

Figure 6.14. Comparison of neutron multiplicities obtained with proton and deuteron beams of different energies as functions of the target atomic masses. Figure from Ridikas and Mittig [120].

6.2.2

Muon catalysed fusion

Muon catalysed d–t fusion has been suggested as a possible means for producing high yields of 14 MeV neutrons. As an example we mention the proposal by Petitjean et al. [123]. In the muon catalysed fusion process, negative muons are captured on the lower Bohr orbit of deuterium or tritium atoms. The muon’s orbit radius is around 2.5 fermis, close to the nuclear radius, so that the Coulomb field of the deuteron (or triton) is almost cancelled. The probability of fusion of the muon accompanied deuteron or triton with another t or d nucleus becomes large. After fusion,


159

most often, the muon is shaken out and becomes available for another cycle until it decays or is captured by a heavy nucleus. It has been found that up to 150 fusions per muon can be obtained. According to Petitjean et al. the optimum beam–target combination for negative pion, and thus negative muon, production is a beam of 1.5 GeV deuterons impinging on a carbon target. The HETC [92] simulation gives a maximum negative pion yield of 0.16. It follows that the maximum possible number of produced 14 MeV neutrons per GeV of deuteron is around 15. These neutrons could be further multiplied by (n, 2n), and even (n, 3n), reactions. A multiplication factor of 2 seems a maximum. Finally we see that no more than 30 neutrons per GeV-deuteron can be produced. Since not all muons will be captured by the heavy hydrogen atoms, a maximum number of 15 is more likely. This is a factor of 2 below the neutron yield from protons on uranium. Note the advantage that the pion production target can be completely disconnected from the neutron source. However, the (d, t) cell requires high pressures, and a high magnetic field is necessary to trap and focus the muons. It is doubtful that this technique could be used competitively. 6.2.3

Electron induced neutron production

Electron beams have been used extensively to produce neutron pulses. It has been proposed, for example by Abalin et al. [124], to use an electron accelerator to generate the neutron source of hybrid reactors. Electrons with energies above a few tens of MeV are slowed down essentially by Bremsstrahlung photon emission. Above 10 MeV the dominant process for energy loss of the photons is pair creation. In that manner an electromagnetic shower develops, consisting of a population of electrons, positrons and photons. Independently of their electromagnetic interactions these particles may also interact with nuclei. The photonuclear cross-section is dominated by the giant dipole resonance, with an energy which depends weakly on the nuclear mass and is around 10 MeV. Thus, if one wants to optimize the photonuclear rate of interactions within the electromagnetic shower, one should maximize the number of photons (real or virtual) with energies around 10 MeV. The optimum initial electron energy is then found to be between 100 and 200 MeV. For example, the Euratom Geel linear accelerator provides electrons with a maximum energy of 140 MeV, and operates, in practice, at 100 MeV. At this energy only 0.1 neutron is produced per incident electron. This corresponds to a neutron production yield of 1/GeV, to be compared with the typical 30 neutrons per GeV which can be obtained with protons. To overcome this small neutron production efficiency, Abalin et al. [124] suggest using multiplying media with ks very close to one. We shall examine their proposal in more detail below.

160

6.3

The neutron source

Experimental determination of the energy gain

As said in chapter 4, the CERN FEAT experiment [122] gave a value of G0 k ¼ 3, for incident proton energies larger than 1 GeV and for a uranium target. The experiment consisted of mapping out the number of fissions produced in a multiplying array surrounding a uranium target bombarded by the CERN PS proton beam. The multiplying array consisted of natural uranium bars immersed in a light-water swimming pool. The fission density within the uranium bars was obtained from the measurements, after careful corrections for the flux depression within the bars and influence of the surroundings of the detectors. The value of k was deduced from a measurement using a known 252 Cf source. From the value of G0 it is possible to deduce a value of N0 ¼ G0 Ep =0:18 ¼ 41.5 neutrons per GeV proton, to be compared with a value of 32 which can be read on figure 6.10. The ratio of neutron multiplicities for uranium and lead amount to 1:35 for the CERN experiments and 1:50 for the measurements of Hilscher et al., to be compared with the value of 1:4 corresponding to the multiplication in uranium. Another important result of the FEAT experiment was that the neutron multiplicity per GeV saturated for proton energies above 0.8 GeV. This behaviour is illustrated in figure 6.15. For lead and 1 GeV protons, the value of G0 should be between 2:5 and 1:8 according to the value retained for the neutron multiplicity. The value of G0 ¼ 2 was retained by the CERN group for its calculations of the energy amplifier [76].

Figure 6.15. Energy gain measured in the FEAT experiment as a function of the proton energy [122]. Also shown is the result of the CERN Monte Carlo calculation using the high-energy code FLUKA [95] and the MC2 [75] neutron transport code written by the CERN group.

161

Two-stage neutron multipliers

6.4


We show in chapter 8 that, if control rods are to be avoided, the multiplication factor keff should be limited to about 0.98 for the standard fast-neutron hybrid reactors and 0.95 for the slow-neutron ones. This limitation on keff also limits the energy gain G0 =ð1 keff Þ accordingly. An interesting suggestion was made as early as 1958 by Avery [131], in order to increase the energy gain of a hybrid system; it was reactivated by Abalin et al. [124] and Daniel and Petrov [132]. It consists of coupling two multiplying systems in such a way that neutrons produced in the first one can penetrate the second while those produced in the second cannot penetrate the first. We quantify the possible gain which can be obtained in this way. Let one neutron be created in a multiplying medium. If absorbed it produces k1 new neutrons. However, in a finite system it only produces keff neutrons. Since keff ¼ Pcap k1 the escape probability is Pesc ¼ 1

keff : k1

ð6:12Þ

If we consider a system with N0 injected neutrons and multiplication keff , the number of escaping neutrons will be N0 k1 keff : 1 keff k1 Now, we consider two multiplying media which communicate. Let k1 !21 K ¼ !12 k2 be the matrix of efficiencies: a neutron born in medium 1 gives k1 progeny in ð1Þ ð2Þ medium 1, and !12 in medium 2. Let ni and ni be the number of generation i neutrons in media 1 and 2. The number of neutrons of the next generation in each medium is ð1Þ

ð1Þ

ð2Þ

ð2Þ

ð2Þ

ð1Þ

ni þ 1 ¼ k1 ni þ !21 ni ni þ 1 ¼ k2 ni þ !12 ni that is, ni þ 1 ¼ K ni

and the final number of neutrons as a function of the initial one: nðIniÞ nðFÞ ¼ 1 K This is only true on the average for a neutron chosen randomly according to the flux distribution of the adjoint reactor.

162

The neutron source

which yields, for nðIniÞ ¼ ðFÞ

n1 ¼

N0

¼ N0

ðFÞ

N0 0

ð6:13Þ

1 k2 ð1 k1 Þð1 k2 Þ !21 !12

!21 !12 1 k2 !12 !12 ¼ N0 ¼ N0 ! ! ð1 k1 Þð1 k2 Þ !21 !12 1 k1 21 12 ð1 k2 Þ 1 k2 1 k1

n2

while, if one neutron is created in medium 2, the final numbers are n1 ¼

!21 ð1 k1 Þð1 k2 Þ !12 !21

n2 ¼

1 ! ! ð1 k2 Þ 12 21 1 k1

:

ð6:14Þ ð6:15Þ

If one could define a system where !12 6¼ 0 and !21 ¼ 0, one would get ðFÞ

n2 ¼

!12 N0 ð1 k1 Þð1 k2 Þ

ð6:16Þ

N0 : 1 k1

ð6:17Þ

ðFÞ

n1 ¼

Abalin et al. [124] propose that the first medium could be a fast-neutron multiplier but a strong thermal-neutron absorber while the second medium would be both a slowing down and thermal-neutron multiplier. The fast neutrons created in medium 1 could, eventually, reach medium 2 and be slowed down and multiplied. In contrast, slow neutrons from medium 2 could not reach medium 1 without being immediately absorbed in the strong thermal-neutron absorber (for example a gadolinium nucleus). As suggested by equation (6.12), an estimate of !12 is !12 ¼

k11 k1 k11

ð6:18Þ

which shows that it is interesting to maximize k11 , and thus to use as pure fissile material as possible. !12 will, in general, be of the order of unity. It is mandatory that ð1 k1 Þð1 k2 Þ > !21 !12 ’ !21 . In order for the system to be of interest as compared with standard ones, k1 and k2 should be close to 0.95. This means that the coupling term, !21 ; should be less than 2 103 . A serious safety problem might arise from an unwanted decrease of the amount of absorber in medium 1.


163

Rather than using a difference between the neutronic properties of medium 1 and 2, it is possible to play on the relative geometrical arrangement of the two media. For example, consider that the first medium is a sphere with radius R1 surrounded by a spherical shell between R1 and R2 comprising medium 2. A neutron exiting medium 1 has, evidently, a unit probability of entering medium 2 so that !12 is given by equation (6.18). A emitted pneutron ffi from the inner surface of the shell has probability 1 1 ðR21 =R22 Þ of entering medium 1: In the absence of medium 1; neutrons lost by medium 2 exit by the external surface of the shell. If the shell is not too thick, the number of neutrons crossing the inner surface of the shell should be approximately equal to this last quantity. It follows that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi k21 k2 R1 !21 ’ 1 1 : ð6:19Þ k21 R22 The minimization of !21 implies a minimization of ðk21 k2 Þ=k21 in agreement with other constraints. Typically, for breeder reactors, ðk21 k2 Þ=k21 is close to 0:1: It follows that the condition on the product !12 !21 implies that R1 =R2 < 0:2: With R1 =R2 ¼ 0:1, !12 !21 ’ 5 104 and ðFÞ

n2 ¼ 500N0

ð6:20Þ

for k1 ¼ k2 ¼ 0:95 and !12 ¼ 1: In these conditions, if a neutron is created in medium 2; the final number of neutrons will be n1 ¼ 0:25 and n2 ¼ 25, so n1 þ n2 ¼ 25:25. The large amplification difference when neutrons are created in medium 1 and when they are created in medium 2 shows that very high neutron multiplication can be obtained in the case being discussed, the system remaining, however, safely far from criticality. This could give the possibility of reducing by almost one order of magnitude the power requirement for the accelerator. Figure 6.16 shows the result of a very simple Monte Carlo calculation which illustrates the preceding discussion, and shows how a very high multiplication can be obtained, while staying very far from criticality. The model reactor is made of a central plutonium sphere with a radius of 4.62 cm surrounded by a plutonium shell with an inner radius of 10 m and a thickness of 1.54 cm. Each single component is characterized by keff ¼ 0:95. The very high values of ki for small i reflects the high value of for pure plutonium. The sharp decrease is due to the large escape probability of neutrons created in the inner sphere. After 20 generations, the multiplication process takes place essentially in the outer shell. The simulated value of ks ¼ 0:997 is to be compared with the analytically calculated value of ks ¼ 0:9964: Figure 6.16 also illustrates the consideration of section 4.1.3. Finally Daniel and Petrov [132] have proposed the use of the difference in fissile concentration in zone 1 (booster) and zone 2 to obtain a high value of ks while keeping a reasonably small value of keff : They did a one-group

164

The neutron source

Figure 6.16. Evolution of the multiplication factor as a function of the neutron generation number for a model reactor made of a central plutonium sphere with a radius of 4.62 cm surrounded by a plutonium shell with an inner radius of 10 m and a thickness of 1.54 cm. Each single component is characterized by keff ¼ 0:95.

diffusion calculation for a two-zone fast reactor with a subcriticality 2 ¼ 1 keff 2 ¼ 0:03, for the external zone and k11 ¼ 1:2, corresponding to keff 1 ¼ 0:98. For the spherical geometry, with a volume ratio between the two zones of 103 ; they obtained a booster gain of 3.6, allowing a corresponding decrease in the beam power.

6.5

High-intensity accelerators

In chapter 4 we have seen that, without elaborate source enhancement as described in section 6.4, an energy gain of the subcritical array on the order of 100 can be anticipated. With a desired total thermal power of the ADSR around 1 GWth, one sees that beam powers around 10 MW are needed. This is an order of magnitude larger than beam powers delivered by present accelerators. It is, therefore, important to examine the feasibility of such high-intensity accelerators. Since this discussion requires some knowledge of accelerator physics we give an introduction to this subject in Appendix III. It is shown there that, in practice, linear accelerators and cyclotrons are the best choices available for obtaining high average intensities that are almost continuous. It is thus instructive to review the characteristics of the existing high-intensity Linacs and cyclotrons.

High-intensity accelerators 6.5.1

165

State of the art of high-intensity accelerators

Linear accelerators The highest-power operational proton linear accelerator is that of Los Alamos Meson Factory (LAMPF). The LAMPF accelerator. The LAMPF accelerator accelerates protons up to 800 MeV. It is 800 m long. It provides 1 ms bunches of protons with an intensity of 15 mA and with a repetition rate of 120 Hz. Correspondingly its duty cycle is 12%, and its average beam power 1.4 MW. The accelerator comprises three parts: 1. The source and injector. The source can provide either Hþ or H ions. The largest intensity of 30 mA is obtained with the Hþ ions. The ions are accelerated at 750 kV by a Cockroft–Walton electrostatic accelerator. The continuous beam is then bunched at 201.25 MHz. 2. The 0.75 MeV protons are injected into a drift tube Linac (DTL or Alvarez). The DTL is 16.7 m long. It accelerates protons up to 100 MeV. It works at a 201.25 MHz frequency. The Q value of the accelerating structures reaches 5 104 . The drift tubes are enclosed in three tanks. Each tank shelters 165 drift tubes. Each tank is fed with an RF unit with 2.7 MW peak power. Beam focalization is achieved with 135 quadrupoles arranged in the focusing–defocusing–focusing–defocusing (FDFD) mode. 3. The 100 MeV protons are further accelerated up to 800 MeV in a sidecoupled Linac (SCDTL). This is 726.9 m long. It works at a 805 MHz frequency. Only one out of four micro bunches is filled with particles. The Q value of the accelerating structures reaches 2:4 104 : The 5000 cavities are enclosed in 104 tanks. The RF power is provided by 44 units with 1.25 MW peak power each. Beam focalization is achieved with 204 quadrupoles arranged in doublets. The main characteristics of LAMPF are given in table 6.2. From the value of the shunt resistance and the phase angle of 648, using equation (16.22) of Appendix III, P¼

Vm2 2sh

ð6:21Þ

one obtains a thermal RF power of approximately 10 MW, to be compared with a beam power of 12 MW, and thus an RF-to-beam efficiency of 55%. The total power of the RF power units amounts to 65 MW, which gives a line-to-RF efficiency of 34% and a line-to-beam efficiency of 19%.

Here one should consider the power during the macro-pulse rather than the average power.

166

The neutron source Table 6.2. Characteristics of the LAMPF Linac. Accelerator Length Energy Pulse intensity Repetition rate Macro pulse length Average intensity

800 m 800 MeV 15 mA 120 Hz 1 ms 1.2 mA

Injector Energy Intensity

0.75 MeV 30 mA

DTL Energy Length Shunt impedance Frequency Axial field RF power rating Normalized acceptance

100 MeV 100 m 42 M /m 201.25 MHz 1.6–2.4 MV/m 10.8 MW 7 mm-mrad

Side-coupled DTL Energy Length Shunt impedance Frequency Axial field RF power rating Normalized acceptance

800 MeV 700 m 30–42 M /m 805 MHz 1.1 MV/m 55 MW 17 mm-mrad

Cyclotrons The highest-power cyclotron system presently in operation is the cyclotron ensemble of the Paul Scherrer Institute (PSI) at Zu¨rich, Switzerland. The PSI cyclotrons. The characteristics of the PSI cyclotrons are given in Table 6.3. From the table we obtain a 27% RF-to-beam efficiency. The line-to-RF efficiency amounts to 66%, while the total line-to-beam efficiency reaches 18%. Compared with LAMPF, one notes a better efficiency of the RF power units, but a lower RF-to-beam transfer efficiency. 6.5.2

Requirements for ADSR accelerators

The requirements for a high-proton-intensity accelerator connected to an ADSR can be summarized as follows:

High-intensity accelerators Table 6.3. Characteristics cyclotron.

of

the

167

SIN

Accelerator Energy Average intensity Beam power

590 MeV 1.8 mA 1 MW

Pre-injector Energy Intensity

0.8 MeV 12 mA

Injector cyclotron Energy Intensity Frequency Injection radius Extraction radius Magnet power RF power dissipation Extraction efficiency Total RF power

72 MeV 2 mA 50.6 MHz 40.6 cm 350 cm 20 kW 0.4 MW 99.97% 2 0:18 MW

Sector cyclotron Energy Intensity Frequency Injection radius Extraction radius Magnetic field Energy gain/turn Magnet power RF power dissipation Total RF power

590 MeV 1.8 mA 50.63 MHz 210 cm 445 cm 2.09 T 2.46 MeV 1 MW 1.3 MW 2.1 MW

Radio frequency Beam power Total RF power Total line-to-RF losses

1 MW 2.61 MW 1.35 MW

Proton energy should be larger than 600 MeV, in order to optimize the number of neutrons produced per MeV of incident energy. . Beam power should exceed 10 MW, possibly more. This means that for 1 GeV protons currents larger than 10 mA are needed. . Beam losses should be made very small in order to minimize accelerator structure activation. . High beam availability is required. This is obvious for long-lasting shutdowns due to equipment breakdown: this requirement is analogous to .

168

The neutron source

those for power reactors. Another event specific to accelerators is the occurrence of short-duration trips where the beam is lost. If the duration of a trip exceeds the time it takes for the subcritical system to reach thermal equilibrium after an input power variation (temperature relaxation time) the ADSR structures will be submitted to thermal stress and, thus, increased fatigue. The reduction of the frequency and duration of trips is therefore an important request for high-intensity accelerators. . The energy efficiency of the accelerator complex should be reasonably high. 6.5.3

Perspectives for high-intensity accelerators for ADSRs

Intensities Intensities may be limited by several factors: The intensity which can be provided by the ion source. ECR plasma sources are able to provide very high intensities with good emittance and very high reliability. . The pre-injection accelerator. Until recently pre-acceleration was achieved with electrostatic accelerators like Cockroft–Waltons or SAMES. The need to have the source at a potential close to 1 MV leads to complexity and is a cause of breakdowns. RFQs are nowadays able to accelerate reliably several tens of mA at several MeV [126, 127]. . Space charge increases with the maximum accelerated current. The critical region is at low energy. As shown in Appendix III, a way to decrease space charge effects is to reduce the distance between focusing devices. Here again RFQs are helpful since their wavelengths are of the order of a few mm. It seems that Linacs are more promising than cyclotrons as far as space charge limitations are concerned. In cyclotrons, space charge limitation decreases when the separation between turns increases [125] and thus when the energy gain per turn increases.† However, increasing the energy gain per turn also increases the high voltage on the cavity and, as discussed below, the risks of trips. In practice, cyclotrons’ intensities seem to be limited to approximately 10 mA. Intensities as high as 100 mA should be feasible with Linacs and have been demonstrated for the low-energy injection part [126, 127]. . High transmission of the accelerators is required in order to minimize beam losses. In principle, Linacs seem to have an advantage here. Indeed extraction of the beam from cyclotrons is a delicate point. However, transmissions as high as 99.98% have been reported for PSI [125]. The .

Possible designs of accelerators for ADSRs have been described in several workshops of the NEA on ‘Utilization and Reliability of High Power Proton Accelerators’. † Maximum intensity has been shown by the SIN physicists to increase as Eg3 where Eg is the energy gain per turn.

High-intensity accelerators

169

operational experience from PSI shows that high-intensity accelerators can be run with very limited irradiation risks for operators. . Beam availability already reaches 85 to 90% today both at LAMPF and PSI. Such figures could be improved for industrial accelerators by using components far enough from their design values and through redundancy of the critical pieces of equipment. Regular and more frequent maintenance would also be efficient. With these improvements, it is estimated that availabilities exceeding 95% should be possible. . Short-duration trips cause sharp decreases of the beam intensity. These might induce fatigue in the structure elements of the ADSR so that they have to be reduced in number as much as possible. Many of them are due to the failure of individual elements such as RF power units, cavity windows and focusing devices. The number of trips at LAMPF and PSI amount to over 10 000 per year [127]. This number should be drastically reduced by several orders of magnitude. Several dispositions seem able to improve the situation considerably. (i) Minimize the number of single items whose failure leads to a beam loss, e.g. RF cavities and power units. These should be such that when one of them breaks down the energy decrease is small enough that the beam is not defocused away from the target. An example is provided by the ATW project [127] which plans to have an individual power unit for each SC cavity. The failure of one unit leads to a decrease of the beam energy by 5.5 MeV to be compared with a total energy of 1000 MeV. This 0.55% decrease is not sufficient for a beam loss at the target position. (ii) Use devices which rapidly compensate the energy loss consecutive to the breakdown of a unit [127] by increasing the gain of the nearby units. . It is shown, on general grounds, in Appendix III, that energy efficiency improves for higher intensities. Table 6.4 is an example of this gain. It compares the energy efficiencies of the present PSI cyclotrons and of a possible 10 MW accelerator based on an extrapolation of the PSI concept [128]. Table 6.4. Power needed to drive the SIN facility and a proposed 10 MW facility.

Energy Intensity Beam power RF power Magnet power Line-to-RF conversion Total power Energy efficiency

PSI

10 MW

590 MeV 1.8 mA 0.9 MW 2.6 MW 1.6 MW 1.35 MW 5.55 MW 0.18

1000 MeV 10 mA 10 MW 14 MW 3 MW 7 MW 24 MW 0.41

170 6.5.4

The neutron source Examples of high-intensity accelerator concepts

Cyclotron As an example we take the proposal of a high-intensity cyclotron system put forward by the PSI physicists [128]. This proposal is very similar to that made by the CERN group for the Energy Amplifier [3]. It consists of an injector cyclotron providing protons at 120 MeV. The second stage is a cyclotron with 12 sectors and eight RF cavities. This provides an energy gap per turn greater than that of PSI and a smaller number of turns. The space charge effect is then reduced to the point that 10 mA beams might be extracted. Linac As an example we take the ATW proposal [127]. It is similar to other proposals like the TRASCO one [126]. The proposal includes an ECR source providing up to 100 mA, a 350 Mcs RFQ accelerating protons up to 6.7 MeV, a normal conducting SCDTL accelerator up to 10 MeV working at 350 Mcs, and super-conducting sections each optimized with respect to the average proton velocity, operating at 350 Mcs up to 210 MeV energy and 700 Mcs from 210 to 1000 MeV. The final energy is 1 GeV for a 45 mA intensity. The most reliable version of the proposal assigns one RF power unit for each cavity with an energy gain of 5.5 MeV.

Chapter 7 ADSR kinetics

In section 3.5 we discussed some aspects of the control and safety of critical reactors. We discuss in what respect these are modified by subcriticality. As mentioned in section 3.5.3 a reactivity insertion of more than 1 $ in a critical reactor leads to a fast exponential excursion, equation (3.101): ðpromptÞ t WðtÞ ¼ W0 exp : ð7:1Þ n For lead cooled fast reactors n ¼ 3 108 s [7]. It follows that the power would be multiplied by 100 after 14 108 =ðpromptÞ seconds. Even for ðpromptÞ ¼ 0:001, i.e. a total reactivity insertion of 0:004 (for a 233 U fuelled reactor) the power is multiplied by 100 after 0.14 ms! Consider now the case of a hybrid system with k ¼ 0:98, and a reactivity increase of 0:4% equal to that just considered for the critical system. The energy gain is proportional to 1=ð1 kÞ and increases by 25% only! The preceding considerations are very schematic. An example of a realistic calculation [76] is displayed in figure 7.1, where the behaviour of a critical system is compared with that of a subcritical system. In the figure, the total reactivity inserted is as much as 2.55 $. Temperature reactivity dependence is taken into account. The advantage of hybrid reactors, even with a moderate amount of subcriticality, is quite striking. A critical system is designed so as to make a prompt criticality almost impossible. In a subcritical system the reactivity margins can be increased and new types of fuel can be considered, which would be very difficult to use in a critical system. Note that the power increase due to a reactivity insertion in a subcritical core does not depend on the proportion of delayed neutrons. Thus, this safety parameter is no longer restrictive. The comparison becomes much more complex in the case of an incident leading to an automatic stopping of the critical system, for instance a temperature increase or the loss of heat extraction. A critical reactor is dimensioned so that such an incident leads immediately to a decrease of the reactivity. Let us take the example of a diminution of the coolant mass 171

172

ADSR kinetics

Figure 7.1. Comparison of the power increase of a critical reactor and of different subcritical systems after insertion of an additional reactivity. The additional reactivity amounts to an increase rate of 170 $/s for 15 ms, after which the reactivity remains constant. Note that hybrid reactors are not supposed to be less than 6 $ subcritical. Figure from Rubbia et al. [76].

in the core of a PWR. Several aspects play an important role: on one hand, the capture rates on hydrogen and boron decrease in the core. This effect has a positive impact on the reactivity. On the other hand, the neutron slowing down decreases and the neutronic spectrum becomes harder, leading to a decrease of the reactivity, as shown in figure 7.2, which represents the infinite multiplication factor as a function of the energy of the incident neutron: the neutronic captures become much more frequent (compared with fission) when the energy of the neutrons increases. This modification of the spectrum shape has, thus, a negative effect on the reactivity. Finally, a diminution of the heat extraction leads to an increase of the fuel temperature, which has two main consequences: a shift of the thermal spectrum towards higher energies, and the Doppler effect, which increases the reaction rates in the resonances. As already mentioned, a hardening of the spectrum leads to a decrease of the reactivity. In a PWR, the Doppler effect leads to an increase of neutron captures in the resonance of 238 U at 4.6 eV, and makes the reactivity decrease. The characteristics (geometry, fuel composition, coolant proportion, boron concentration, etc.) are determined so that the global effect leads to an immediate decrease of reactivity, without any external intervention. The power is reduced to the residual heat, essentially

ADSR kinetics

173

Figure 7.2. Variations of k1 of a PWR fuel (UO2 /H2 O) as a function of the energy of the incident neutron.

due to the radioactivity of fission products, which represents, a few seconds after the end of the chain reaction, a few per cent of the nominal power. If the same incident occurs in a subcritical core, the reactivity loss () due to passive effects explained before (void, temperature, etc.) makes the power decrease in a few seconds and converge towards P ¼ P0 1 þ : 0 For a typical of the order of a few tens of ppm (part per million), and of the order of 300 ppm, the power decrease is of the order of a few per cent only. An additional step is thus required, which consists of stopping the external neutron source. This task is not difficult to carry out in itself, but requires a quick and precise detection of the problem, which could be not so crucial for a critical system. Some designs propose passive systems to manage this kind of incident, as the TIER concept proposed by Bowman [133] where fuse-wire put inside the fuel would immediately stop the beam after a temperature increase. In the Energy Amplifier design proposed by Rubbia et al. [3], the safety is provided by an overflow of the molten lead into the beam pipe, thus stopping the beam outside the subcritical assembly. In the case of an incident which could not be managed by the self-responding behaviour of the fuel and which would require detection and human or automatic intervention, the fall of the control rods of a critical system can be compared with beam stopping. Note that this can be immediately

174

ADSR kinetics

achieved, a few fractions of a second after incident detection, whatever the core geometry, while the fall of control rods requires a few seconds and can be prevented by core deformation in the case of a major accident. In any case, an accelerator driven subcritical reactor requires precise measurement of the reactivity, during operation as well as during rest periods. Different methods are considered to measure the reactivity. They are the object of present experimental campaigns, for example the MUSE experiment [134]. One of the aims of this experimental program is to determine a method to calculate the reactivity of a subcritical system, based on the measurement of the time evolution of the neutronic flux in response to a pulsed neutron source. The theoretical time response of a subcritical system characterized by its prompt effective multiplication factor kpeff to a delta excitation is, in the approximation of the one-group point kinetics: NðtÞ ¼ N0 ðtÞ et where is the decrease rate defined by ¼

1 kpeff l

where l is the average time between a fission and the previous one in the chain reaction. In the one-group point kinetics approach, l is constant and equal to the mean lifetime of a neutron. l could be calculated by simulation and kpeff could thus be determined by the measurement of the time evolution of the subcritical core, following a neutron source pulse. A recent simulation development [135] shows that this simplified approach is not sufficient to determine the subcriticality level with enough accuracy and proposes a more detailed formulation which takes into account, on the one hand, the differences of the neutron source and the subsequent fission source (energy and spatial distributions), and, on the other hand, the reflector effect, which strongly modifies the time response of the subcritical core. Monte Carlo simulations, performed on a simplified system, show that the decrease rate is clearly not constant with time, as is shown in figure 7.3. This strongly modifies the shape of the population decrease and forbids its description as an exponential decrease. The decrease rate ðtÞ converges towards an asymptotic value 1 , which is different from the decrease rate defined by the one-group point kinetics approach. The time required to converge towards 1 corresponds to several hundreds of generations. This very slow evolution is due to the presence of the reflector where the absorption cross-sections are low, and where the neutrons may spend a lot of time before coming back to the fissile zone. Furthermore, the importance of these long-lived neutrons increases with the level of subcriticality, for the same reasons that a population with a small birth rate is mainly old.

ADSR kinetics

175

Figure 7.3. Decrease rate ðtÞ, its asymptotic value 1 and the point kinetic value ð1 kpeff Þ=l: The decrease rate was calculated for a spherical fast reactor (90 cm diameter core of 27% Pu MOx, 20 cm thick sodium/stainless steel reflector), with an effective multiplication factor kpeff ¼ 0:972. In the Monte Carlo simulation, ðtÞ decreases and tends towards the asymptotic value 1 .

The method proposed by Perdu and coworkers [135] is to calculate, by detailed Monte Carlo simulation, the intergeneration time distribution, denoted PðÞ, which satisfies ð PðÞ d ¼ kpeff : It can be shown that this distribution is essentially determined by the characteristics of the reflector and does not depend strongly on the composition of the fuel, nor on the value of kpeff . Once the function PðÞ is calculated, the fission rate at time t is a solution of ð1 Nf ðtÞ ¼ Nf ðt ÞPðÞ d 0

which leads, for a Dirac pulse, to the expression of Nf ðtÞ Nf ¼ P þ P P þ P P P þ

176

ADSR kinetics

where the star denotes convolution. Since the distribution PðÞ does not depend on the value of kpeff , we can write Nf ¼ kpeff P0 þ ðkpeff Þ2 P0 P0 þ ðkpeff Þ3 P0 P0 P0 where P0 is the normalized intergeneration time distribution P0 ðÞ ¼ ð 1

PðÞ

:

PðuÞ du 0

Then, the decrease rate can be calculated by kp ðtÞ ¼ eff

1 dNf ðtÞ : Nf ðtÞ dt

This means that we are able to calculate theoretically the decrease rate kp ðtÞ eff for any given value of kpeff . An experimental measurement of the response of a subcritical core to a neutron pulse provides an experimental decrease rate exp ðtÞ. It is possible to determine the value of kpeff which gives the best agreement between exp ðtÞ and kpeff ðtÞ. This method could provide a determination of the subcriticality level, which takes into account complex effects, like the energy spectrum of the source neutrons, or the effects of the reflector, where some neutrons spend a lot of time before coming back to the core. As far as residual heat extraction is concerned, hybrid reactors have essentially the same properties as a critical reactor using the same technology as the subcritical reactor: hence, following the considerations of section 3.4.4, the potential interest of lead cooled and high-temperature gas reactors.

Chapter 8 Reactivity evolutions

Since hybrid reactors should not require control rods, it is of course very important to check that, in time, the reactor cannot become critical. We address this question in the present chapter, having in mind, especially, the possible evolution of the fuel.

8.1

Long-term evolutions

More complete calculations than those presented in section 3.6 are needed in order to characterize the behaviour of specific fuels which might be used in hybrid reactors. As examples we show the results of three such calculations [129] in figure 8.1 where we show the evolution of k1 for a plutonium mixture originating from PWR spent fuel, for the 232 Th–233 U system, and for a fuel made of a mixture of minor actinides. Figure 8.1 confirms that hybrid reactors with solid fuels are not fit for plutonium incineration. On the other hand, minor actinide fuels behave like a mixture of fissile and fertile nuclei. Figure 8.2 shows the evolution of the fission rates due to the various nuclei as a function of time. It appears that the stabilization of the variation of k1 is chiefly due to the formation of 238 Pu, which has a high fast-neutron fission probability. It is formed by neutron capture by 237 Np, which behaves like a fertile species. To a lesser extent the rise of 244 Cm fissions counteracts the decrease of the 241 Am and 243 Am fissions. Figure 8.2 also gives an idea of the long times required for a significant decrease of the fission rate of the fuel and, correspondingly, of the total number of transplutonium nuclei.

8.2

Short-term reactivity excursions

Short-term fuel evolution as well as temperature changes may lead to reactivity changes. For thermal reactors the very large capture cross-section of 135 Xe 177

178

Reactivity evolutions

Figure 8.1. Variations of k1 for (1) (K1 Th þ U) a mixture of 232 Th (90%)–233 U (10%); (2) (K1 Pu) a mixture of 238 Pu (2.5%), 239 Pu (60.8%), 240 Pu (24.9%), 241 Pu (11.7%), as considered by Rubbia et al. [45]; (3) (K1 A.M.) a mixture of minor actinides of 237 Np (33.3%), 241 Am (21.6%), 243 Am (40%), 242 Cm (2.1%), 243 Cm (0.032%), 244 Cm (1.4%), 245 Cm (0.9%) [45]. The flux assumed in the calculations was 4 1015 n/cm2 /s.

Figure 8.2. Evolution of the fission rates for the minor actinide fuel, as a function of time, and according to the fissioning nucleus. The neutron flux assumed was 4 1015 n/cm2 /s.


179

and 149 Sm lead to such effects. In the case of the Th–U cycle, a specific effect arises, both for thermal and fast reactors, due to the 27 day half-life of 233 Pa. We first examine this effect. 8.2.1

Protactinium effect [76]

As we have seen in section 3.5, followed by two beta decays: 232

233

U is formed by neutron capture by

Th þ n ! 233 Th ! 233 Pa ! 233 U: 22:3 min 26:97 days

232

Th

ð8:1Þ

The presence of protactinium imposes limits on the admissible neutron flux when using solid fuels. This limitation is due to two detrimental effects of protactinium: 1. Protactinium captures neutrons, and thus decreases the reactivity of the reactor. 2. After a reactor stop, the 233 Pa inventory decays to 233 U, which leads to an increase of the reactivity and of k. This increase may lead to reactor criticality. The characteristic time for such an evolution is of the order of the half-life of 233 Pa, i.e. about one month. Corrective actions could easily be taken by inserting a negative reactivity. However, the advantage that passive safety of hybrid systems represents would be lost. It is, thus, interesting to keep the system subcritical in all instances. The evolution equations of the Th–Pa–U system read dnTh ðaÞ ¼ nTh Th ’ þ SðtÞ dt

ð8:2Þ

dnPa ðaÞ ðaÞ ¼ nTh Th ’ nPa nPa Pa ’ dt dnU ðaÞ ¼ nPa nU U ’: dt

ð8:3Þ ð8:4Þ

Thus, at equilibrium, ðaÞ

Th ’ nPa ¼ nTh þ ðaÞ ’ Pa

ð8:5Þ

and ðaÞ

Th nU n ¼ : ¼ Pa nTh nTh ðaÞ ’ ðaÞ ð þ ðaÞ ’Þ U U Pa ðaÞ

ð8:6Þ ðaÞ

For thermal neutrons Pa ¼ 43 barns and for fast neutrons Pa ¼ 1:12 barns. The lifetime of Pa in the neutron flux is only significantly shortened ðaÞ if ’ > =Pa , i.e. ’ > 7 1015 n/cm2 /s for thermal reactors and

180


’ > 2:7 1017 n/cm2 /s for fast reactors. Except for the very high-flux molten salt reactor which has been proposed by Bowman [2], such fluxes are ðaÞ never reached, so that Pa ’ can be neglected with respect to : Thus, at equilibrium, ðaÞ

ðaÞ

nPa Th ’ ; ¼ nTh

nU ¼ Th : nTh ðaÞ U

It is seen that the amount of protactinium is a measure of the neutron flux. It is also proportional to the specific power of the reactor, itself proportional to the density of fission c , with ðaÞ

ðfÞ

c ¼ n U U ’ ¼

nU U ’ : 1þ

Since the specific power is the limiting factor of reactor designs rather than the neutron flux we express the modifications to the reactivity due to Pa in terms of the number of captures in uranium. The multiplication coefficient reads ðaÞ

k1 ¼

nU U ðaÞ

ðaÞ

ð8:7Þ

ðaÞ

nU U þ nTh Th þ nPa Pa þ P

and 1

k1 ¼ 2þ ðaÞ

ðaÞ Pa c ð1

þ Þ ðaÞ

nTh Th

þ

ð8:8Þ

P ðaÞ

nTh Th

ðaÞ

where we used nTh Th ¼ nU U . During neutron irradiation, it follows that k1 is decreased by ðaÞ

ð1 þ Þ k1 k1 : ¼ Pa c ðaÞ k1 nTh

ð8:9Þ

Th

ðaÞ

ðaÞ

ðaÞ

ðaÞ

For thermal reactors the ratio Pa =Th ¼ 74 while Pa =Th ¼ 24 for fast reactors. It follows that, for a given decrease of k1 ; fast reactors allow a specific power 3 times larger than thermal reactors and, hence, three times more compact cores. After a reactor stop, the protactinium will decay into 233 U, leading to an increase of k1 : Asymptotically, the final value of k1 will be ðaÞ

1þ kðasÞ 1 ¼

nPa U nTh ðaÞ Th ðaÞ

n P 2 þ Pa U þ nTh ðaÞ nTh ðaÞ Th Th

:

ð8:10Þ


181

Since the perturbation on the reactivity is small, we can write that the relative change with respect to the unperturbed value of k1 is ðaÞ

k1 k1

ðaÞ

nPa U nPa U ðaÞ nTh ðaÞ nTh ðaÞ ð1 þ Þ U Th Th : ¼ ’ 0:5 c ðaÞ P 1 nTh Th 2þ ðaÞ nTh Th

ð8:11Þ

In order to estimate the true reactivity excursion, the decrease of the reactivity during irradiation has to be taken into account so that the total, maximum excursion is k1 c ð1 þ Þ k1 ðaÞ ðaÞ ¼ þ 0:5 : ð8:12Þ U ðaÞ k1 Pa nTh Th For fast reactors we get k1 ð1 þ Þ ¼ 1:4 107 c nTh k1 and, for thermal reactors, k1 ð1 þ Þ ¼ 1:6 108 c : nTh k1 One sees that the limit on the specific power is ten times more stringent for thermal systems than for fast ones. For k1 =k1 ¼ 2 102 the corresponding capture densities are on the order of 3 1013 for fast systems and 2:5 1012 for thermal ones. The corresponding fluxes are then 4 1015 for fast reactors and 4 1013 for thermal reactors. In conclusion, it appears that the protactinium effect greatly favours fast reactors if solid fuels and the Th–U cycle are to be used. This is not true for the U–Pu cycle where 239 Np, which plays a role analogous to 233 Pa, has a much shorter half-life of 2.35 days. 8.2.2

Xenon effect [55]

Some fission products like 135 Xe and 149 Sm have very large absorption crosssections for thermal neutrons. They are not produced directly by fission but by beta decay of precursor fission fragments. The decay chains by which they are produced as 135

I ! 135 Xe ! 135 Cs 6! 135 Ba ðstableÞ 6:7 h 9:2 h 2:6 10 yr 149

Nd ! 149 Pm ! 149 Sm ðstableÞ: 2h 54 h

ð8:13Þ ð8:14Þ

182


135

Xe has an absorption cross-section of 2:7 106 barns for thermal (0.025 eV) neutrons, while that of 149 Sm is 40 800 barns. Because of the dominant effect of 135 Xe we give the derivation of the xenon effect. The evolution equations read dnI ¼ yI f ’ I nI dt

ð8:15Þ

dnXe ¼ I nI Xe nXe nXe Xe ’ dt

ð8:16Þ

where we have neglected captures by iodine. yI ’ 0:06 is the yield of mass 135 in fission. At equilibrium yI f ’ I yI f ’ ¼ Xe þ Xe ’

nI ¼ nXe

ð8:17Þ ð8:18Þ

for a flux of 4 1013 n/cm2 /s, Xe ’ ¼ 104 to be compared with Xe ¼ 2 105 : Thus nXe ’

yI f : Xe

ð8:19Þ

It follows that k1 y ’ I ¼ 0:03: k1 2

ð8:20Þ

After a reactor shut-down, the evolution of xenon is given by dnI ¼ I nI dt

ð8:21Þ

dnXe ¼ I nI Xe nXe dt

ð8:22Þ

which yields nXe ðtÞ ¼ yI f ’ expðXe tÞ

1 Xe ’

þ

1 expððI Xe ÞtÞ : I Xe

ð8:23Þ

Figure 8.3 shows the evolution of the xenon-induced reactivity decrease after shut-down of a thermal reactor at two flux levels: 4 1013 and 2 1014 n/cm2 /s. It is remarkable that, as apparent from equation (8.19), the initial xenon concentration is independent of the neutron flux, at least for fluxes that are not too small. In critical reactors the reactivity decrease prevents restarting of the reactor if a large enough positive reactivity reserve is not available. Hybrid systems can be restarted at any time, although the gain will be smaller if the xenon concentration is high. However, if the reactor is


183

Figure 8.3. Variation of the xenon-induced reactivity decrease after reactor shut-down. The thermal-neutron flux was (a) 4 1013 n/cm2 /s, (b) 2 1014 n/cm2 /s.

stopped long enough, the xenon concentration vanishes and thus the reactivity is larger by 0.0035 than the reactivity during operation. This is true for any thermal reactor, and has to be added to the protactiniumrelated reactivity increase, in the case of thorium reactors. For fast reactors the xenon effect is negligible. 8.2.3

Temperature effect

The reactivity of any reactor is generally temperature dependent. Critical reactors have, for obvious safety reasons which we have discussed in chapter 3, a negative reactivity temperature coefficient. For example, PWRs have a coefficient between 5 105 and 104 per 8C [48]. This means that a PWR reactor has a reactivity at zero power between 0.03 and 0.015 higher than at nominal power. The temperature coefficient of fast

184


reactors is, usually, smaller than that of thermal rectors. For sodium cooled fast reactors it is around 105 per 8C [48]. A similar value has been calculated by Rubbia et al. [76] for their Fast Energy Amplifier. 8.2.4

Impact of reactivity excursions

From the above considerations it is apparent that fast hybrid reactors have more favourable neutronic characteristics than do thermal reactors. Practically it seems difficult to design a hybrid thermal reactor with keff larger than 0.95 for the U–Pu cycle and 0.92 for the Th–U cycle. The corresponding values for fast reactors would be 0.99 and 0.98 respectively.

Chapter 9 Fuel reprocessing techniques

9.1

Basics of reprocessing

Fuel reprocessing was born at the same time as nuclear energy. It was developed within the Manhattan project in order to recover the plutonium needed for the fabrication of the Nagasaki atomic bomb. An excellent account of the pioneering techniques used by Seaborg and his collaborators can be found in reference [138]. Here, a wealth of different methods which have been used for recovering uranium and plutonium can also be found. Aside from the military needs to recover plutonium, with the associated proliferation aspects, the attractive potential offered by fast neutron breeders led to the building of industrial plants to recover both plutonium and uranium from spent nuclear fuels. The largest of these plants that are operational today are La Hague and Sellafield. Because the breeding programmes have been stopped, these large plants have been converted to MOx fuel production for commercial light water reactors, a high recovery efficiency of plutonium and uranium being required. These plants use organic solvents for highly efficient recovery of these two elements. The reference molecule which has become a standard is tributyl phosphate (TBP) from which the so-called Purex (plutonium extraction) process was developed. The high efficiency allows quasi-complete plutonium extraction (99.9%), so that this element could be practically absent from the wastes. Of course, this requires complete incineration of plutonium in standard or dedicated reactors. Logically the minimization of the waste radiotoxicity requires that americium and curium should also be separated and incinerated. This led to the development of enhanced separation methods, based on the same principle as Purex, such as Diamex and Truex. Although the wet process, Purex (which implies dissolving the fuel elements in an aqueous acid solution), is by far the most used and the only one which has reached industrial status, other processes which do not require aqueous dissolution have been explored. This has been done in two main instances: 185

186

Fuel reprocessing techniques

1. The molten salt reactor programme initiated at Oak Ridge National Laboratory [49, 50]. Here the problem was, essentially, to recover thorium and uranium from a mixture of lithium, beryllium and fission product fluorides. 2. The fast breeder programme. The interest in the anhydric recovery of plutonium and uranium stems from the high plutonium enrichment of fast breeder fuels which leads to increased risks of criticality, especially with hydric processes, and from the interest of metallic fuels which lead to more energetic neutron spectra, with higher breeding capability. Indeed, the metallic fuels lend themselves very easily to fluorization. The law of mass action [139, 140] In general the behaviour of reactants in a solution or in a diphasic system, as often met in reprocessing, is governed by the law of mass action which we find useful to summarize. Consider a general binary chemical reaction A A þ B B ! C C þ D D: ð9:1Þ P This can also be written as i i Ai ¼ 0. By tradition the values of the lefthand side of the reaction equation, such as equation (9.1), are taken to be positive. At chemical equilibrium, the thermodynamical potential should P be minimum, which leads to the condition i i i ¼ 0, where i is the chemical potential of molecule Ai . Defining the concentrations of the molecules cA ¼ nA =N where nA is the number of molecules A, and N the total number of molecules, including those of the solvent(s), the condition of chemical equilibrium leads to the law of mass action, which is exact for perfect gases and dilute solutions: Y ci i ¼ KðP; T; f i gÞ ð9:2Þ i

where i is related to the chemical potential of molecule i and P and T are the pressure and temperature respectively. For example in a dilute solution i ¼ i T ln ci . For dilute solutions the equilibrium constant reads simply P KðP; T; f i gÞ ¼ exp i i i : ð9:3Þ T Here the P dependence disappears. KðP; TÞ is, more frequently, defined in terms of the free formation energies of the molecules. For reaction (9.1), for example, GA ðTÞ þ B GB ðTÞ C GC ðTÞ D GD ðTÞ KðP; TÞ ¼ exp A RT ð9:4Þ

Basics of reprocessing

187

where the temperature dependence of GðTÞ can be expressed as GðTÞ ¼ H TS

ð9:5Þ

where H and S are the formation enthalpy and entropy of the molecules. In the following, since we deal with condensed matter, we shall drop the dependence of KðP; TÞ upon P. Chemical activity In practice the conditions are more complex than for gases or very dilute solutions. Reaction rates depend upon the environment of the molecules which can form complexes with the solvent or other molecules. Such complex behaviour has led chemists to use the concept of activities ai instead of concentrations in the mass action relation, which then reads Y ai i ¼ KðP; T; f i gÞ: ð9:6Þ i

The activity coefficients are defined as the ratios of activities to concentrations: a i ¼ i : ð9:7Þ ci By convention it is assumed that, for a pure solvent where only molecules of the solvent interact, ¼ 1. For very dilute solutions, since the dissolved molecules interact essentially with the solvent, it is expected that the activity coefficients are independent of the concentrations so that ai ¼ i0 ci :

ð9:8Þ

In intermediate cases the activity coefficients vary with the concentrations. However, when these concentrations do not vary strongly during a reaction it is justified to consider that the activity coefficients are constant so that one can write a modified law of mass action in terms of concentrations rather than activities Y KðP; T; f i gÞ Q i ci i ¼ ¼ K ðP; T; f i gÞ: ð9:9Þ i i i An example of the dependence of the activity coefficients on the concentrations is given by the regular model which states that i ðci Þ ¼ exp½ð1 ci Þ2 :

ð9:10Þ

As expected, equation (9.10) gives the limiting values i ¼ 1 for ci ¼ 1 and i ¼ e for ci 0. With > 0 the dilute phase is more active than the concentrated one while the reverse is true for < 0. The two types of behaviour are shown in figure 9.1 for ¼ 1 and ¼ 1.

188


Figure 9.1. Examples of the variations of activity as a function of the concentration. The values of were chosen to be 1 and 1; respectively.

9.2

Wet processes

In these processes the spent fuels are first decladded by shearing and sawing. A dissolution of the oxide fuel in hot nitric acid follows. At present, on an industrial scale, uranium and plutonium are extracted by the Purex process. 9.2.1

The Purex process

Solvent properties The Purex process uses an organic phase consisting of tributyl phosphate (TBP) soluted in a hydrocarbon diluent as extractant. The formula of TBP is ðC4 H9 Þ3 PO4 . A possible structure for TBP is: H j C j C j C j C j O j HCCCCOPCCCCH j O

Wet processes

189

It is thought that TBP forms bonds via the electron of unsaturated oxygen. This molecule forms a complex with uranyl nitrate UO2 ðNO3 Þ2 which is soluble in the hydrocarbon diluent but not in water. Similarly it can form a complex with plutonium nitrate PuðNO3 Þ4 . The main complex forming equilibrium reactions for U and Pu read [138] : UO2þ 2 ðaqÞ þ 2NO3 ðaqÞ þ 2TBPðoÞ ! UO2 ðNO3 Þ2 2TBPðoÞ

ð9:11Þ

: Pu4þ ðaqÞ þ 4NO 3 ðaqÞ þ 2TBPðoÞ ! PuðNO3 Þ4 2TBPðoÞ

ð9:12Þ

which occur at the interface between the aqueous phase (aq) and the organic one (o). The equilibrium concentrations are correlated by the law of mass action: KU ¼

½UO2 ðNO3 Þ2 :2TBPðoÞ 2: 2 : ½UO2þ 2 ðaqÞ ½NO3 ðaqÞ ½TBPðoÞ

ð9:13Þ

KPu ¼

½PuðNO3 Þ4 :2TBPðoÞ : 4: 2 ½Pu ðaqÞ:½NO 3 ðaqÞ ½TBPðoÞ

ð9:14Þ

4þ

Distribution coefficients measure the ratio between the molar concentrations in the organic phase and those in the aqueous phase. For example, for uranium in the TBP–aqueous system: DU ¼

½UO2 ðNO3 Þ2 :2TBPðoÞ ½UO2þ 2 ðaqÞ

ð9:15Þ

which, using equation (9.13), reads 2: 2 DU ¼ KU ½NO 3 ðaqÞ ½TBPðoÞ :

ð9:16Þ

From equation (9.16) it appears that the concentration of uranium in the organic phase increases with the concentration of TBP in the organic phase as well as with the concentration of NO 3 in the aqueous phase. It is important to note that here we are dealing with free TBP concentration. In particular, a larger uranium concentration in the aqueous phase leads to a smaller free TBP concentration in the organic phase (for fixed total TBP concentration) since more TBP is used in the formation of the complex. The concentration of NO 3 in the aqueous phase can be adjusted by adding more or less nitric acid, which reacts with TBP according to : Hþ ðaqÞ þ NO 3 ðaqÞ þ TBPðoÞ ! HNO3 TBPðoÞ

ð9:17Þ

190

Fuel reprocessing techniques Table 9.1. Distribution coefficients for uranyl nitrate between aqueous nitric acid and 40% (volume) TBP in kerosene. ½UO2 ðNO3 Þ2 ðaqÞðmoles=lÞ

HNO3 ðaqÞðmoles=lÞ

DU

0.042 0.210 1.68

0.6 0.6 0.6

3.3 1.98 0.41

0.042 0.210 1.68

1.5 1.5 1.5

6.4 2.2 0.4

0.042 0.210 1.68

2.0 2.0 2.0

7.0 2.4 0.4

0.042 0.210

3.0 3.0

7.1 2.5

which leads to the equilibrium equation KH ¼

½HNO3 :TBPðoÞ : ½Hþ ðaqÞ:½NO 3 ðaqÞ ½TBPðoÞ

ð9:18Þ

and a distribution coefficient for H: : DH ¼ KH ½NO 3 ðaqÞ ½TBPðoÞ:

ð9:19Þ

Finally DU can be expressed as a function of the aqueous concentrations 2þ ½Hþ ðaqÞ; ½NO 3 ðaqÞ, UO2 ðaqÞ and the total concentration of TBP in the organic phase. Typical values of KU and KH are 5.5 and 0.145 respectively. Table 9.1 from reference [138] illustrates these considerations. It shows that the concentration of the uranyl ions strongly influences the value of DU . The influence of the nitric acid concentration is less dramatic and becomes small for a concentration larger than 1.5 moles/l. Figure 9.2 shows that TBP has much larger distribution coefficients for actinides than for most fission products. The separation properties can be controlled by playing on the concentration of nitric acid in the aqueous phase and on the uranium concentration in the organic phase. Properties of actinides in solution Figure 9.2 shows that the valence state of neptunium has a strong influence on the distribution coefficient. This is a general behaviour and elements with different valence states behave like different elements. In this context, it is useful to know the valence states of actinides in acid solutions.

Wet processes

191

Figure 9.2. Effect of nitric acid concentration on distribution coefficients for 30% (volume) TBP 80% saturated with uranium at 25 8C (from [141]). The valence states of Pu and Np are given.

Thorium. Thorium nitrate ThðNO3 Þ4 is very soluble in water, thorium being in a tetravalent state. It can, rather easily, form complexes with TBP. Uranium. In aqueous solution uranium can be found in trivalent U3þ , quadrivalent U4þ , pentavalent UV Oþ , and hexavalent UVI O2þ states. However the trivalent and pentavalent states are unstable so that, practically,

192


only the tetravalent and hexavalent states are important. The hexavalent state in the form of uranyl nitrate is highly soluble in TBP. Neptunium. In acid aqueous solutions neptunium can be found in trivalent Np3þ , quadrivalent Np4þ , pentavalent NpV Oþ , and hexavalent NpVI O2þ states. Np4þ and NpVI O2þ are the more soluble states in TBP. Plutonium. In aqueous solution plutonium can be found in trivalent Pu3þ , quadrivalent Pu4þ , pentavalent PuV Oþ , hexavalent PuVI O2þ and heptavalent PuVII O2þ states. The heptavalent and pentavalent states are unstable. The tetravalent and hexavalent states can form complexes with TBP, while the trivalent state cannot and is not soluble in organic solvents. Americium. In aqueous solution americium can be found in trivalent Am3þ , pentavalent AmV Oþ and hexavalent AmVI O2þ states. The tetravalent state is highly unstable. In the presence of oxidizable species, only Am3þ is of practical importance. It is moderately soluble in TBP. However, in the Purex process Am is not coextracted with uranium and plutonium and follows the fission products. Curium. Curium is only stable in the trivalent state and less soluble in TBP than americium. Single-step extraction The basic step for organic uranium and/or plutonium extraction involves first a thorough mixing of the aqueous and organic phases in order to maximize the interphase area, and second a reseparation of the phases. Figure 9.3 shows a schematic view of this process. Flux conservation implies that IF ðx0 x1 Þ ¼ IE ð y0 y1 Þ:

ð9:20Þ

Further, at equilibrium the concentrations are related by y0 ¼ Dx1 :

ð9:21Þ

In a single-step extraction y1 ¼ 0. In this case, substituting equation (9.21) into equation (9.20) yields x1 ¼

IF x ðIE D þ IF Þ 0

ð9:22Þ

which gives the fractional recovery ¼

IE y0 x0 x1 1 : ¼ ¼ ð1 þ ðIF =DIE ÞÞ IF x0 x0

ð9:23Þ

Multistep extraction Equation (9.23) shows that full extraction can only be obtained for an infinitely large organic solvent current. Improved extraction can be obtained

193

Wet processes

Organic Extract IE y0

Aqueous feed IF x0

Mixer

Extracter

Organic Feed IE y1

Aqueous Extract IF x1

Figure 9.3. Schematic representation of the extraction process. x0 and x1 are the extractable component (U or Pu) initial and final concentrations in the aqueous phase respectively. y1 and y0 are the extractable component initial and final concentrations in the organic phase respectively. IF and IE are the aqueous and organic feed currents respectively. Taken from Benedict et al. [138].

with a cascade arrangement, such as that shown in figure 9.4. For the twostage system shown in figure 9.4 the value of the fractional recovery becomes, for y2 ¼ 0, IF DIE þ ¼ ¼ IF IF IF2 I I I2 1þ þ 2 2 1þ 1þ F 1þ F þ F DIE DIE D IE DIE DIE DIE 1þ

IF DIE

1

ð9:24Þ which is larger than the value given by equation (9.23). Generalizing to N stages, assuming that yN ¼ 0 and defining ¼ DIE =IF we obtain the equation for y0 : y0 ¼ DxN

N 1 X

i :

ð9:25Þ

i¼0

Mass conservation implies IF x0 ¼ IF xN þ IE y0

ð9:26Þ

N 1 N þ 1 1

ð9:27Þ

which allows us to write y0 ¼ Dx0

194


Figure 9.4. Cascade arrangement of extraction modules. Note that, for simplicity, we have switched the geometry of the modules compared with that of figure 9.3. Taken from Benedict et al. [138].

while the fractional recovery reads ¼

I E y0 N 1 ¼ N þ1 : IF x0 1

ð9:28Þ

One sees that for large N and > 1 the efficiency tends towards one. For < 1; ! for N ! 1. The preceding calculation assumes that yN ¼ 0. This condition can easily be dropped and one gets an overall recovery of the extractable component N 1 1 yN N þ1 : ð9:29Þ þ ¼ N þ1 N þ 1 1 1 Dx0 Note that, because of the definition of ¼ IE y0 =IF x0 , the term depending on yN is partly trivial since it gives a finite contribution even when there is no transfer from the aqueous to the organic phase ( ¼ 0). If it is the extraction efficiency that is of interest, it might be more informative to use I E yN y ¼ N IF x0 Dx0 N 1 y ¼ N þ1 1 N : Dx0 1

reff ¼

ð9:30Þ ð9:31Þ

For yN ¼ 0; ¼ reff . When yN 6¼ 0, the extraction efficiency decreases, which seems natural since the final value xN ¼ yN 1 =D has a finite minimum value yN =D.

Wet processes Feed(aq)

195

Extraction Module

Extract(o)

Residue(aq)

Stripping Module

Stripping(aq)

Extract(aq) Figure 9.5. Extraction of a component from an aqueous phase into an organic solvent followed by redissolution of the component into an aqueous phase (stripping).

We have also assumed that the value of D does not depend on the stage number. This is only true for rather dilute solutions, as we have seen above. When this is not the case we refer to the discussion in reference [138]. It is important that the treatment is completely symmetric with respect to the nature of the phases. Thus it is possible to transfer a component from an organic phase to an aqueous phase. This makes recovering the expensive organic solvent possible, as shown in figure 9.5. Multicomponent extraction In the preceding we have discussed the extraction of a single component from the aqueous feed. The selection is possible because of the large differences between the values of D for this component and other species like fission products. In general, with TBP, uranium and plutonium have high distribution Note that our notations are slightly different from those of reference [138]. In particular we count stages starting from stage 0 as the aqueous feed input.

196


coefficient values and are co-extracted. Further separation between plutonium and uranium is thus required. In general this is done first by redissolving uranium and plutonium from the organic phase into an aqueous phase with low nitric acid concentration. Then plutonium is reduced from the tetravalent to the trivalent state with a mild reductant such as U4þ which does not reduce the hexavalent uranium. The trivalent plutonium cannot be dissolved back into an organic solvent, so that a new dissolution leaves the plutonium in the aqueous phase while uranium dissolves in the organic phase. In general, in order to increase the purity of the recovered uranium and plutonium, additional scrubbing is necessary. These different steps, as typically implemented in the Purex process, are shown in figure 9.6. Scrubbing solution

Pu stripping solution

FP(aq)

Solvent

U Scrubbing Module

U+Pu Extraction Module

U,Pu,FP

Pu(U)

U+Pu (FP)

FP

Feed(aq)

Pu Stripping Module

FP Scrubbing Module

U+Pu

Pu

Water

U Stripping Module

Solvent

U Extract(aq)

Figure 9.6. Diagram of the Purex process.

U

Wet processes

197

Thorium separation: the Thorex process Thorium has a distribution coefficient in TBP smaller by an order of magnitude than uranium. This makes thorium extraction by TBP more difficult than that of uranium. In particular, much larger fluxes of TBP are required for the same extracted quantity. Such an extraction, called the Thorex process, has been achieved at a pre-industrial stage in two instances [138]: At Hanford more than 200 metric tons of low-burn-up aluminium-clad thorium fuel were processed [142]. . In the frame of the THTR (thorium high-temperature reactor), laboratory extraction of thorium and uranium was carried out at Ju¨lich on highly irradiated fuels [143]. .

Aside from a low distribution coefficient, thorium extraction is made difficult by the existence of an additional organic phase which appears for a rather modest thorium nitrate concentration in the aqueous phase. This concentration decreases as a function of the concentration of nitric acid. Furthermore it was found that at moderate nitric acid concentrations, needed for good separation of thorium from fission products (around 1.5 mol/l), fission products, if present in large quantities, might precipitate. This has led [143, 144] to a rather complicated extraction process where an initial pre-decontamination of thorium and uranium with low thorium (1.15 mol/l) and medium nitric acid (1 mol/l) concentrations, was followed by an extraction stage with lower nitric acid concentration (0.15 mol/l) and, finally, separation and purification of thorium and uranium. The Purex process is unable to separate thorium from protactinium with good efficiency. An improved separation can be obtained by adding phosphoric acid H3 PO4 which complexes readily with protactinium. An interesting aspect of the Ju¨lich experiment is that it involved the treatment of fuel particles embedded in graphite and silicon carbide spheres. The graphite was burned in oxygen to CO2 . Because of the presence of 14 C this has to be sequestered in the form of CaCO3 , for example. Thus the treatment of thorium fuels by the Purex process has been demonstrated but leads to much larger radioactive effluents than does the Purex process for uranium and plutonium extraction. Separation of minor actinides The Purex process was designed to recover plutonium for military applications or for energy production with fast neutron reactors. In the latter case, recovered uranium was also useful. At present, plutonium is recovered for the fabrication of MOx fuels for PWRs and BWRs. The advocates of MOx fuel use insist that it not only is cost effective since it decreases the needs for enriched uranium, but that it can also help reduce the radiotoxicity

198


of the wastes. However, while plutonium is indeed the main long-term source of radiotoxicity for UOx spent fuels, minor actinide contributions become dominant in spent MOx fuels. Hence the idea to extract also minor actinides in order to transmute them. A number of processes have been proposed lately with that view. Actinides are extractable by the Purex process when they are in an even state of oxidation. For example uranium is readily extractable in its VI oxidation state as the uranyl ion UVI O2þ . Plutonium is also easily extracted as PuIV , but not as PuIII , which allows separation of plutonium from uranium. Neptunium is present in nitric acid solutions in states NpV and NpVI . This coexistence of odd and even oxidation states of neptunium allows us to consider its extraction by a rather simple modification of the classical Purex process, as shown in reference [145]. The cases of americium and curium are more difficult since they essentially appear in the odd III oxidation state in nitric acid solutions. Active research and development in several countries is pursued in order to find efficient processes, compatible with Purex, for their separation [145]. They test different complex organic molecules with high selectivity for americium and curium. One of the main difficulties is that rare earths are also present in the nitric solution in oxidation state III, and are co-extracted with americium and curium. Facing this challenge two strategies have been adopted: 1. Use selective stripping with complexants to separate americium and curium from rare earths. Pertaining to this approach are the so-called Talspeak [146], DIDPA [147] and Truex [148] processes. 2. Use a two-cycle separation process with two different solvents where the first step separates americium, curium and lanthanides from the other fission products, and the second step separates americium and curium from the lanthanides. Examples of such processes are TRPO [149] and Diamex [150]. As examples of the two approaches we discuss a little more precisely the Truex and Diamex processes. The Truex process The Truex process uses a neutral organophosphorus bidendate extractant: noctyl-phenyl-di-isobutyl-carbamoylmethyl-phosphine-oxide (CMPO). CMPO displays both high and low affinities for actinides(III) and lanthanides(III) at high and low nitric acid concentrations respectively. It is, then, relatively easy to obtain an An–Ln (actinide–lanthanide) mixture by the succession of extracting and stripping steps. The separation of An from Ln can then be done by selective complexing agents like DTPA (diethylenetriaminopentaacetic acid) which seems to make more stable complexes with Ans than with Lns.

Dry processes

199

The Diamex process The Diamex process uses malonamide extractants, the only one practically tested being di-methyl-dibutyltetradecylmalonamide (DMDBTDMA) which has very attractive properties as actinide extractant, with comparatively small technological wastes. The Diamex process has to be followed by an additional step to separate Ans from Lns.

9.3

Dry processes

Organic solvents suffer significantly from radiolysis. Therefore, before reprocessing they require sizeable cooling times of the spent fuels. Typical cooling times, such as those used at the COGEMA The Hague plant, are of the order of 5 years although cooling times as short as 1 year have been considered [138]. Shorter cooling times lead to a shorter lifetime of the expensive solvent, higher costs and larger amounts of wastes. When it is very important to minimize the cooling time, the fragility of organic solvents becomes a very serious drawback. This is especially the case in two occurrences: fuel reprocessing of breeders and molten salt reactors. In the first case the cooling time reflects directly on the doubling time. In the second case, online reprocessing involves treatment of an extremely active fuel. Dry inorganic processes appear to be much less sensitive to radiation effects and have been proposed and developed for the two aforementioned cases. In the breeder case, reprocessing of fast reactor fuels by chlorination was proposed, for example in the frame of the ‘integral fast reactor’ by the Argonne National Laboratory [151]. For molten salt reactors the reference is the fuel which was proposed in the MSBR project [50], which consisted of a mixture of fluorides. Because of the subject of this book we shall restrict our considerations to the processing of fluoride salts rather than chloride salts. However, with the exception of vaporization of fluorides the possibilities offered by fluorination and chlorination are rather similar. Elemental separation processes from a mixture of fluoride salts use one of the following techniques: vaporization of volatile fluorides gas purge . liquid–liquid exchange . selective precipitation . electrolysis. . .

The first molten salt reactor studies were done in the context of the airborne nuclear reactor project. It was based on the fluoride mixture NaF : ZrF4 : UF4 using uranium highly enriched in 235 U. The operation temperature was around 500 8C.

200


Efficiencies of these processes depend strongly on the relative amounts of fluor ions in the salt mixture. Large amounts of fluor ions lead to oxidizing conditions and shift the valence states of metals to high values. Oxidizing conditions often accelerate reaction rates and may be sought. However, the composition of the salts is often determined by other considerations such as fusion temperatures or corrosion reduction. Optimization with respect to such properties led the proponents of the molten salt breeder reactor to choose a mixture of the 7 LiF, BeF2 , ThF4 and UF4 fluorides in proportions (71.7 :16 : 12 :0.3 mol%) [154]. This mixture serves as a reference for our discussion. During irradiation 232 Th is transmuted into 233 Pa which can capture neutrons or decay to 233 U. It is efficient to minimize the sterile captures in 233 Pa, and therefore to extract it from the salt, and let it decay into 233 U outside the neutron flux. Subsequently, the 233 U can be re-injected into the salt or stored for future use. Similarly the quantity of fission products in the salt should be limited as much as possible. This is especially true for rare earths which have large neutron capture cross-sections. Reprocessing of the MSBR fuel, thus, consisted of extraction of 233 Pa and of as many fission products as possible, using the different techniques mentioned above. 9.3.1

Vaporization

It is well known that uranium hexafluoride is used in its gaseous state in isotopic separation plants. This property can be used in the vaporization separation of fluorides. The melting and boiling temperatures of stable actinide fluorides are given in table 9.2. Table 9.2. Melting and boiling temperatures of stable actinide fluorides [138]. Fluorides

Melts (8C)

Boils (8C) (1 atm)

ThF4 UF3 UF4 UF6 NpF3 NpF4 NpF6 PuF3 PuF4 PuF6 AmF3 AmF4 CmF3 CmF4

1110 1430 1036 64.05

1782 1457 56.541 (sublimation)

55.7 1426 1027 51.59

62.16

Dry processes

201

It is seen that UF6 , NpF6 and PuF6 are stable and gaseous at low temperature. They can be produced by fluorination of the tetrafluorides. Gas (helium) purging can then allow their extraction from the molten salt mixture. However, PuF6 is thermodynamically unstable and dissociates readily to PuF4 þ F2 at the high temperature of the molten salts. In practice, the extraction by fluorination of UF6 has been demonstrated and that of NpF6 is thought to be possible. The selective extraction of UF6 from other heavy-metal fluorides (Np, Pu, Zr) uses the fact that these form a stable complex with NaF at 400 8C while UF6 does not. Inversely, the separation of UF6 from lighter fluorides uses adsorption of UF6 on NaF at 100 8C. Finally, UF6 is desorbed from NaF at 400 8C. 9.3.2

Gas purge

A helium flow through the salt can be used to remove rare gases and, partially, noble metals which form small aggregates within the salt. 9.3.3

Liquid–liquid extraction

In principle, liquid–liquid extraction of components from the salt mixture resembles the solvent extraction of section 9.2. A contact is established between the molten salt phase and a liquid metallic phase. The most abundant component of the metallic phase is a metal with very weak interaction with the fluoride salts and a low melting point. Cadmium and bismuth have the required chemical and physical properties. However, for a reactor, bismuth is preferred because of its low neutron cross-section. The extraction proper is done via reduction by a reductive component added to the liquid bismuth. In the MSBR project the reductive component was chosen to be metallic lithium. This choice is advantageous since lithium is automatically in equilibrium with its own salt: LiF þ Li () Li þ LiF:

ð9:32Þ

The basic reduction reaction thus reads: MF þ Li () LiF þ M

ð9:33Þ

where M is the metallic element to be extracted from the molten salt. The distribution coefficient of element M is defined as the ratio of the molar concentration of M in the metallic phase to that in the salt phase: DM ¼

½M ½MF

Here we follow closely the treatment given in the thesis of Lemort [155].

ð9:34Þ

202


while DLi ¼

½Li : ½LiF

ð9:35Þ

When activity coefficients are reasonably constant, the law of mass action reads KM ðTÞ ¼

½M ½LiF ½MF ½Li

ð9:36Þ

and, therefore, one gets DM ¼ KM ðTÞDLi

ð9:37Þ

log DM ¼ log DLi þ log KM ðTÞ

ð9:38Þ

and where we recall that KM ðTÞ ¼ KM ðTÞ

ðTÞMF ðTÞLi : ðTÞM ðTÞLiF

ð9:39Þ

Using the expression (9.4) of KðP; TÞ, we obtain for KM GLiF GMF KM ðTÞ ¼ exp : RT

ð9:40Þ

Some examples of values of the formation enthalpies and entropies are shown in table 9.3. Values of logðKM ðT ¼ 600 8CÞÞ obtained from equation (9.40) are also shown in the table. Large values of KM generally correspond to large values of the distribution coefficients in the metallic phase.

Table 9.3. Values of the formation free enthalpies and entropies for representative fluorides. The value of logðKðTÞÞ for T ¼ 600 8C as deduced from these values are given in the fourth column of the table. Molecule

H (J/mol)

S (J/mol/K)

log(K)

LiF [156] ZrF4 [157] BeF2 [156] LaF3 [160] PdF2 [156] ThF4 [160] PaF4 [158] UF4 [158] PuF4 [157]

594 000 1 911 000 1 029 000 1 698 000 478 000 2 062 000 1 956 000 1 897 000 1 759 000

77 319 151 257 149 272 261 275 294

28 9 6 42 17 22 27 36

Dry processes

203

Figure 9.7. Variations of logðDM Þ with logðDLi Þ for representative metals at 600 8C.

Although, in practice, KM ðTÞ differs, sometimes significantly, from KM ðTÞ, assuming that the two values coincide (activity coefficients equal to unity) and using equation (9.38) allows us to obtain, at least, a qualitative understanding of the conditions for an efficient liquid separation. With this assumption, figure 9.7 shows the variations of the distribution coefficient for selected metals as a function of the concentration of lithium in the metallic phase. Positive values of logðKM ðTÞÞ correspond to possible extraction from the salt into the metal while the reverse is true for negative values. It is seen that noble metals (palladium) are easily extracted. Plutonium, uranium and protactinium are extracted with increasing difficulty, but still significantly less than rare earths (lanthanum) and thorium. Uranium and zirconium should be very difficult to separate. In practice, the situation is much more complex than that displayed in figure 9.7. This complexity is not only due to various values of the activity coefficients which differ from unity, but also to the presence of several states of oxidation of the actinides and rare earths. While the selective liquid–liquid extraction of uranium and protactinium appears to be easy, by playing with the content of lithium in bismuth, the extraction of rare earths in the presence of thorium appears to be a difficult challenge. It has been proposed in the frame of the MSBR project [50] to strip rare earths

204


Figure 9.8. Schematic representation of a galvanic cell.

from the liquid bismuth by a counter-current of an LiCl salt. This salt appears to have very different properties for thorium and rare earths. 9.3.4

Selective precipitation

For weakly soluble components like noble metals, it may be efficient to precipitate them in a low-temperature section of the molten salt loop. This will help prevent clogging of the pipes. 9.3.5

Electrolysis

Before examining the application of electrolysis to the separation of metals in a fluoride bath, we think it useful to give a short derivation of the main relations used in electrochemistry. The electric cell Let us begin with the consideration of an electric cell. Figure 9.8 is a schematic representation of a cell. A half cell is composed of a metallic electrode immersed in a salt solution. Here, for simplicity, we have assumed that the salt cation is the singly ionized metal of the electrode. Thus, on the electrodes, electron capture and loss reactions can take place such as: Mþ 1 þ e ! M1 M2 ! M þ 2 þe

More precisely this is a Daniel cell.

½G1

ð9:41Þ

½G2

ð9:42Þ

Dry processes

205

We give a positive value to the free energy release G1;2 of the electron capture by an ion, since this a rather natural choice corresponding to the elementary atomic reactions. If a conductor connects the two electrodes, an electron transfer between the two half cells can occur which results in the possible reactions þ Mþ 1 þ M 2 ) M2 þ M1

½G ¼ G1 G2

ð9:43Þ

þ Mþ 1 þ M 2 ( M2 þ M1

½G ¼ G2 G1 :

ð9:44Þ

The direction of the electron transfer depends upon the sign of G ¼ G1 G2 . If G > 0, electrons will flow from right to left, and therefore a positive current will flow from left to right. A positive potential V will be established between the left and the right electrodes. Assume that a mole of M1 is produced. The number of electrons transferred is thus equal to the Avogadro’s number, and the amount of charge transferred is a Faraday, i.e. 1 Faraday ¼ ð1:6 1019 Þ ð6 1023 Þ ¼ 96 000 Coulomb:

ð9:45Þ

The work done by this transfer is therefore W ¼ 96 000 V:

ð9:46Þ

Neglecting resistive losses, this work has to be provided by the molecular free energy change G, so that G ¼ 96 000V

ð9:47Þ

V ¼ 1:04 105 G:

ð9:48Þ

The generalization of these equations to the case when the M1 ions are the n1 charged Mnþ 1 species is straightforward and yields M1n1 þ þ n2 M2 ) M2n2 þ þ n1 M1 G ¼ 96 000n1 V V ¼ 1:04 105

½G

ð9:49Þ ð9:50Þ

G : n1

ð9:51Þ

In practice, electro-chemists use a standard reference for each element which is the cell of the element associated with the hydrogen cell H2 /Hþ with the reaction 2Hþ þ 2e ) H2 with one molar electrolyte concentration and normal temperature and pressure conditions. By convention the value of G for the hydrogen cell under these conditions is assumed to be 0. Each ionic state of elements is, therefore, characterized by the standard potential

Here we assume that the concentrations are kept constant by an external input or with a salt bridge.

206


of the cell it forms with the standard hydrogen half-cell, as well as by its free energy relative to that of Hþ : G0 ¼ 96 500nV0

ð9:52Þ

G0 : ð9:53Þ n Metals easily lose their electrons to hydrogen, and are therefore characterized by a negative value of both V0 and G0 . When the standard conditions are not fulfilled, equation (9.4) allows us to get V0 ¼ 1:04 105

G ¼ G0 þ RT lnðKÞ

ð9:54Þ

which reads, for reaction (II), G ¼ G0 þ RT ln

½M1 n1 ½M2n2 þ ½M1n1 þ ½M2 n2

ð9:55Þ

with R ¼ 8:314 510 J mol1 K1 . Since, on the electrodes, the molar concentrations are unity, n2 þ ½M2 G ¼ G0 þ RT ln : ð9:56Þ ½M1n1 þ The electrolysis cell While processes taking place in galvanic cells are spontaneous and follow the principle of free energy minimization, electrolysis involves the separation of the constituents of a molecule with a consequent free energy increase obtained from electric power. Figure 9.9 is a schematic drawing of an electrolysis cell. It describes the electrolysis of an MA salt. The reactions taking place at the electrodes are Mþ þ e ) M GM

at the cathode

Figure 9.9. Schematic representation of an electrolysis cell.

ð9:57Þ

Dry processes A ) A þ e GA

at the anode

MA ) M þ A ðGM þ GA Þ:

207 ð9:58Þ ð9:59Þ

The voltage applied between the electrodes has to exceed a threshold VTh for the separation between A and M to take place. This threshold is obtained when the work done in transporting an electron mole from the cathode to the anode equals the molar free energy of the separation reaction: 96 500Vt ¼ GM þ GA ¼ G:

ð9:60Þ

For multiply charged nþ negative ions one gets: 96 500nVt ¼ GM þ GA ¼ G:

ð9:61Þ

Electrolysis of fluorides This is not the place to provide a thorough and general discussion of electrolysis. We focus on the electrolysis of fluorides, and more specifically on the separation of lanthanide fluorides from thorium fluoride. The simplest method for separating various elements is to apply decreasing voltages to a batch mixture of the salts. Other, more elaborate, methods make use of the time variation of the electrolysis current when varying voltages are applied. However, all methods need a batch treatment. The electrolysis process can be compared with the reduction process by Li which was described in section 9.3.3. Consider the metallic fluoride salt Mnþ Fn . The reduction reaction reads: MFn þ nLi ¼) M þ nLiF

½Qred

ð9:62Þ

and the electrolysis reaction Mnþ þ ne ¼) M

½Qel :

ð9:63Þ

Reaction (9.62) can be written as Mnþ þ ne þ nF ne þ nLi ¼) M þ nLiþ þ ne þ nF ne

½Qred :

ð9:64Þ

Using equation (9.63), equation (9.64) can be written as nLi ¼) nðLiþ þ e Þ

½Qred Qel :

ð9:65Þ

It is seen that the difference between the reduction by Li and electrolysis reaction energies is expressed as a function of the Li ionization energy and of the ionization state of the metal: Qred Qel ¼ nQ½Li ¼) Liþ þ e :

ð9:66Þ

It follows that if some elements with different oxidation states are difficult to separate with the Li reduction technique, the addition of an additional

208


electrolysis step may be helpful. An example is the separation of Th and La. The most stable oxidation state of Th is 4þ while that of La and other lanthanides is 3þ. It follows that Qel ½Th Qel ½La ¼ Qel ½Th Qel ½La Q½Li ¼) Liþ þ e :

ð9:67Þ

Chapter 10 Generic properties of ADSRs

While the size of critical reactors may be arbitrary, the presence of a localized primary neutron source constrains the size and total power of hybrid reactors. In this chapter we shall examine this issue, first using a simple, intuitive spherical reactor model, and then taking the example of the optimization of the size of a possible demonstration set-up.

10.1 The homogeneous spherical reactor Because the neutron source is localized in character, one expects the size of hybrid reactors to be limited. Optimization of the reactor has to be done with respect to several key quantities: 1. The value of the source multiplication factor ks , which relates the beam power to the total power of the reactor, and thus to the energy gain. The possibility of innovative designs, in this respect, is discussed in section 6.4. 2. The value of the effective multiplication factor keff which determines the safety of the reactor. Because of the general positive correlation between keff and ks , the highest values of keff , compatible with safety, are sought. In section 3.6 we have seen that for fast systems a limit of keff ¼ 0:98 seems reasonable. 3. A specific power maximum value is set by the heat removal system. Practically, maximum specific powers of the order of 500 W/cm3 are possible with standard liquid metal cooling. 4. It is important to minimize the fuel volume, and at the same time to minimize the range of specific powers within the system. In order to give the reader a feeling for the size of hybrid reactors we study a simple spherical reactor model. The reactor is made of three concentric zones: 209

210

Generic properties of ADSRs

The central zone (1), where the spallation reaction takes place and where we neglect neutron absorptions. This spherical zone has radius R1 . . The fuel zone (2) between radius R1 and radius R2 . . A reflector zone (3) between R2 and infinity. .

Our treatment is based on the solution of the one-group diffusion equation, which appears to be a reasonable approximation for fast reactors. 10.1.1

General solution of the diffusion equation

The diffusion equation in a multiplying medium reads r2 ’

1 k1 ’¼0 L2c

ð10:1Þ

where L2c ¼

D ; a

D¼

s : 32T

The solution of the diffusion equation is uðrÞ ðA er þ B er Þ ¼ where ¼ ’ðrÞ ¼ r r

ð10:2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k1 : L2c

ð10:3Þ

The neutron current reads

r d’ðrÞ e 1 er 1 ¼D A þ þB : JðrÞ ¼ D dr r r r r

ð10:4Þ

We now apply the method to media 2 and 3. 10.1.2

The three-zone reactor

Let 2 and 3 be the attenuation factors in zones 2 and 3. In zone 3 the flux should remain finite at infinity. It must have the form u3 ðrÞ ¼ a3 e3 r :

ð10:5Þ

At radius R2 , one writes the continuity of neutron flux and current, and obtains a relation between the coefficients a2 and b2 which appear in u2 ðrÞ ¼ a2 e2 r þ b2 e2 r :

ð10:6Þ

We define R2 JðR2 Þ D3 ¼ ð1 þ 3 R2 Þ D2 ’ðR2 Þ D2

ð10:7Þ

2 R2 þ 1 < D2 ð2 R2 þ 1Þ D3 ð1 þ 3 R2 Þ ¼ : 2 R2 1 þ < D2 ð2 R2 1Þ þ D3 ð1 þ 3 R2 Þ

ð10:8Þ

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