knief chapter 2 final

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NUCLEARENGINEERING

Theory and Technologyof Commercial Nuclear Power

RONALD ALLEN KNIEFMechanicsburg, Pennsylvania

American Nuclear Society, Inc.

555 North Kensington Avenue

La Grange Park, Illinois 60526 USA


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Library of Congress Cataloging-in-Publication Data

Knief, Ronald Allen, 1944

Nuclear engineering : theory and technology of commercial nuclear power 0 Ronald AllenKnief. 2nd ed.

p. cm.

Includes bibliographical references and index.

ISBN 0-89448-458-3

1. Nuclear engineering. 2. Nuclear energy. I. Title.

TK9145.K62 2008

621.48dc22

2008029390

ISBN-10: 0-89448-458-3

ISBN-13: 978-0-89448-458-2

Library of Congress Catalogue Card Number: 2008029390

ANS Order Number: 350023

2008 American Nuclear Society, Inc.

555 North Kensington Avenue

La Grange Park, Illinois 60526 USA

All rights reserved. No part of this book may be reproduced in any formwithout the written permission of the publisher.

Printed in the United States of America


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CONTENTS

Foreword to the First EditionPreface xvPreface to the First Edition

Overview1 Introduction 3

Nuclear Fuel Cycles 4Nuclear Power Reactors 10Exercises 20Selected Bibliography 22

II Basic Theory2 Nuclear Physics 27

The Nucleus 28Radioactive Decay 31Nuclear Reactions 36Nuclear Fission 41Reaction Rates 46Exercise 63Selected Bibliography 64

xiiiXVll

3 Nuclear Radiation Environment 67Interaction Mechanisms 69Radiation Effects 72Dose Estimates 79

vii


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viii Contents

Radiation Standards 87Exercises 94Selected Bibliography 96

4 Reactor Physics 99Infinite Systems 100Finite Systems 108Computational Methods 113Exercises 131Selected Bibliography 133

5 Reactor Kinetics and Control 135Neutron Multiplication 136Feedbacks 145Control Applications 151Exercises 157Selected Bibliography 159

6 Fud Depletion and Related Effects 161Fuel Burnup 162Transmutation 163Fission Products 170Operational Impacts 176Exercises 181Selected Bibliography 182

7 Reactor Energy Removal 185Power Distributions 187Fuel-Pin Heat Transport 191Nuclear Limits 197Exercises 205Selected Bibliography 206III Nuclear Reactor Systems8 Power Reactors: Economics and Design Principles 211

Economics of Nuclear Power 212Reactor Design Principles 226Reactor Fundamentals 231Exercises 237Selected Bibliography 240

9 Reactor Fuel Design and Utilization 241Fuel-Assembly Design 242Utilization 254Exercises 259Selected Bibliography 260


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10 Light-Water ReactorsBoiling-Water ReactorsPressurized-Water ReactorsExercises 282Selected Bibliography

261262 268

284

Contents ix

11 Heavy-Water-Moderated and Graphite-ModeratedReactors 287Heavy-Water-Moderated Reactors 288Graphite-Moderated Reactors 296Exercises 308Selected Bibliography 310

12 Enhanced-Converter and Breeder Reactors 313Spectral-Shift Converter Reactors 315Thermal-Breeder Reactors 317Fast Reactors 321Exercises 332Selected Bibliography 333IV Reactor Safety

13 Reactor Safety Fundamentals 337Safety Approach 338Energy Sources 340Accident Consequences 343Exercises 355Selected Bibliography 356

14 Reactor Safety Systems and Accident Risk 359Engineered Safety Systems 360Quantitative Risk Assessment 384Advanced Reactors 404Exercises 410Selected Bibliography 41315 Reactor Operating Events, Accidents, and Their Lessons 417

Significant Events 419TMI-2 Accident 423Chernobyl Accident 450Common Accident Lessons 467Exercises 468Selected Bibliography 472

16 Regulation and Administrative Guidelines 475Legislation and Its Implementation 476Reactor Siting 480Reactor Licensing 487


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x Contents

Administrative Guidelines 495Exercises 500Selected Bibliography 503

V The Nuclear Fuel Cycle17 Fuel Cycle, Uranium Processing, and Enrichment 507Nuclear Fuel Cycle 508

Uranium 513Exercises 532Selected Bibliography 533

18 Fuel Fabrication and Handling 535Fabrication 536Fuel Recycle 541Spent Fuel 546Exercises 553Selected Bibliography 557

19 Reprocessing and Waste Management 559Reprocessing 560Fuel-Cycle Wastes 566Waste Management 573Exercises 593Selected Bibliography 596

20 Nuclear Material Safeguards 599Special Nuclear Materials 601Domestic Safeguards 604International Safeguards 618Fuel-Cycle Alternatives 625Exercises 628Selected Bibliography 630

VI Nuclear Fusion21 Controlled Fusion 635Fusion Overview 636

Magnetic Confinement 643Inertial Confinement 650Commercial Aspects 655Non-Thermonuclear Fusion 659Exercises 661Selected Bibliography 662


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Contents xi

AppendixesI Nomenclature 667

II Units and Conversion Factors 671III The Impending Energy Crisis: A Perspective on the Needfor Nuclear Power 677Energy Crisis 678Options 683Proposed Solutions 694

Exercises 698Selected Bibliography 702

IV Reference Reactor Characteristics 707Answers to Selected Exercises 719General Bibliography 721Index 747


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BASIC THEORY

Goals1. To introduce the basic theoretical concepts of nuclear physics, radiation pro-tection, reactor physics, reactor kinetics, fuel depletion, and energy removal2. To develop fundamental calculational skills that can aid in understanding

nuclear energy problems and solutions3. To identify the bases for some of the "uniquely nuclear" features of the

operations and systems described in the remaining parts of the bookChapters in Part II

Nuclear PhysicsNuclear Radiation EnvironmentReactor PhysicsReactor Kinetics and ControlFuel Depletion and Related EffectsReactor Energy Removal


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2NUCLEAR PHYSICS

ObjectivesAfter studying this chapter, the reader should be able to:

1. Describe the atom, including the constituent parts and their relationships.2. Write and perform calculations using balance equations for alpha, beta, and

gamma radioactive decay and for the time-dependent size of a sample ex-periencing decay.

3. Identify the principal nuclear reactions that involve neutrons and write theirbalance equations for specified reactants.

4. Define fissile, fissionable, and fertile. Identify the major nuclides in each ofthese three categories.

5. Describe the distribution ofenergy among the product particles and radiationsassociated with fission. Explain the basis for decay heat.

6. Describe the energy distribution of fission neutrons.7. Define microscopic cross section, macroscopic cross section, and mean free

path. Sketch the first three of the four tiers in the cross-section hierarchy.8. Define neutron flux, write the equation for neutron reaction rate, and use

the equation to perform calculations.9. Estimate radioactive decay rates and neutron reaction rates using data from

the "Chart of the Nuclides" and cross-section plots.10. Interpret the energy dependencies of neutron-reaction cross sections and

fission-particle emission to explain key features of and limitations on reactordesign and operation.

27


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28 Basic Theory

Prerequisite ConceptsEnergy FormsBasic ChemistryBasic SI Units Appendix II

The characteristics of the radioactive decay and nuclear reaction processes have beenthe driving forces behind nearly every unique design feature of nuclear energy systems.Thus, familiarity with some of the fundamental principles of nuclear physics is essentialto understanding nuclear technology. .

Radioactive decay and nuclear reactions are unique in that theyI. provide clear evidence that mass and energy can be interconverted2. involve a variety of particles and radiations, which often have discrete, or quantized,energies3. require descriptive formulations based on laws of probability

Such characteristics have prompted the development of many new experimental andanalytical methods.

THE NUCLEUSThe atom is the basic unit of matter. As first modeled by Niels Bohr in 1913, itconsists of a heavy central nucleus sU'rrounded by orbital electrons. The nucleus, inturn, consists of two types of particles, namely protons and neutrons. Table 2-1compares the atom and its constituents in terms of electric charge and mass. t

The proton and electron are of exactly opposite charge. A complete atom hasthe same number of protons and electrons, each given by the atomic number Z. Theelectrostatic [Coulomb] attractive forces between the oppositely charged particles is

t Unless otherwise noted, data here and in the remainder of the book are based on the GeneralElectric Chart of the Nuclides (GE/Chart. 1989). The definition for the "amu" is provided at the end ofthis section.

TABLE 2-1Characteristics of Atomic and Nuclear Constituents

ConstituentElectronProtonNeutronNucleusAtom

t e = 1.6022 x 10- 19 C.+amu = 1.6606 x 10- 27 kg.*Z = atomic number. = atomic mass number.

Charge(e)t

-I+1o+z*o

Mass(amu)+5.5 X 10--11.0072761.008665- -A

Radius(m )

- 10 II>- 10- 11


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Nuclear Physics 29

the basis for the electron orbits (in a manner similar to the way solar system orbitsresult from gravitational forces).

Although the atom is electrically neutral, the number and resulting configurationof the orbital electrons uniquely determine the chemical properties of the atom, andhence its identity as an element. It may be recalled that the atomic number is the basisfor alignment in the periodic table of elements.StructureVery strong, short-range forces override the Coulomb repulsive forces to bind theposit ively charged protons (along with the uncharged neutrons) into the compact nu-cleus. The protons and neutrons in a nucleus are collect ively referred to as nucleons.Since each nucleon has roughly the same mass, the nucleus itself has a mass that isnearly proportional to the atomic mass number A, defined as the total number ofnucleons. The electrons are very light compared to the particles in the nucleus, so themass of the atom is nearly that of the nucleus.

The characteristic dimensions of the nucleus and atom are listed in Table 2-1.The latter value is based on the effective radius of the outer electron orbits in arraysof atoms or molecular combinations.

A useful shorthand notation for nuclear species or nuclides is 1X, where X isthe chemical symbol, Z is the atomic number, and A is the atomic mass number. (Analternative formulation, zXA, is also found, although current practice favors retentionof the upper right-hand location for charge-state information.) The subscript Z isactually redundant once the chemical element has been identified; its use is discre-tionary.Different nuclides of a single chemical element are called isotopes. For example,uranium isotopes U, U, and U were mentioned in the last chapter. Each has92 protons and electrons with 141, 143, and 146 neutrons, respectively. Anotherimportant isotope group is the hydrogen family-iH, rH, and 1H. The latter two arethe only isotopes often given separate names and symbols-deuterium lTD] and tritium[TTL respectively.Binding EnergyOne of the most startling observations of nuclear physics is that the mass of an atomis less than the sum of the masses of the individual constituents. When all parts areassembled, the product atom has "missing" mass, or a mass defect given by

(2-1)where the masses mp ' me, and mn of the proton, electron, and neutron, respectively,are multiplied by the number present in the atom of mass Malam'

The defect mass is converted into energy at the time the nucleus is formed. tThe conversion is described by the expression

E = mc 2 (2-2)t Mass changes also occur with chemical binding (e.g., electrons with a nucleus to form an atom

and atoms with each o ther to form molecules), but they are so small as to defy measurement.


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30 Basic Theory

9> 8)z0 7w-I(J:::> 6za:w 5Q.>- 4a:wzw 3 . . . - 3He?Z 2Q 2zco 2H10----1"""""--..------------------------....o 20 40 60 80 100 120 140 160 180 200 220 240

MASS NUMBER AFIGURE 2-1Binding energy per nucleon as a function of mass number.

for energy E, mass m, and proportionality constant c2 , where c is the speed of lightin a vacuum. This simple-appearing equation (one of the world's most famous!) wasdeveloped by Albert Einstein with the "theory of relativity. "

The energy associated with the mass defect is called the binding energy. It issaid to put the atom into a "negative energy state" since positive energy from anexternal source would have to be supplied to disassemble the constituents. (This iscomparable to the earth-moon system, which could be separated only through anaddition of outside energy.) The binding energy [BE] for a given nucleus may beexpressed as

(2-3)As the number of particles in a nucleus increases, the BE also increases. The

rate of increase, however, is not uniform. In Fig. 2-1, the BE in MeV* per nucleonis plotted as a funct ion of atomic mass number. The nuclides in the center of the rangeare more tightly bound on the average than those at either very low or very highmasses.

The existence of the fission process is one ramification of the behavior shownin Fig. 2-1. Compared to nuclei of half its mass , the 235U nucleus is bound relativelylightly. Energy must be released to split the loosely bound 235U into two tightly bound

*The MeV is a convenient unit for energy, which is defined shortly.


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Nuclear Physics 31

fragments. A reasonably good estimate for the energy released in fission can be obtainedby using data from the curve in Fig. 2-1.

Energy production in our sun and other stars is based on the fusion process,which combines two very light nuclei into a single heavier nucleus. As shown by Fig.2-1, two deuterium nuclei [TH or rD], for example, could release a substantial amountof energy if combined to form the much more tightly bound helium [iHe]. A numberof fusion reactions are considered as potential terrestrial nuclear energy sources inChap. 21.Mass and Energy ScalesThe masses and energies associated with nuclear particles and their interactions areextremely small compared to the conventional macroscopic scales. Thus, special unitsare found to be very useful.

The atomic mass unit [amu] is defined as 1/12 of the mass of the carbon-12 atom. t The masses in Table 2-1 are based on this scale.

When an electron moves through an electrical potential difference of I volt [V],it acquires a kinetic energy of I electron volt leV]. This unit (equal to 1.602 x 10- 19J, as noted in App. II), along with its multiples keV for a thousand and MeV for amillion, is very convenient for nuclear systems.:j:

Mass and kinetic energy, as noted previoUSly, may be considered equivalentthrough the expression E = mc2 . Thus, for example, it is not uncommon to expressmass differences in MeV or binding energies in amu based on the conversion I amu= 931.5 MeV. (Other useful factors are contained in App. II.)

RADIOACTIVE DECAYThe interactions among the particles in a nucleus are extremely complex. Some com-binations of proton and neutron numbers result in very tightly bound nuclei, whileothers yield more loosely bound nuclei (or do not form them at all).

Whenever a nucleus can attain a more stable (i.e., more tightly bound) config-uration by emitting radiation, a spontaneous disintegration process known as radio-active decay may occur. (In practice this "radiation" may be actual electromagneticradiation or may be a particle.) Examples of such processes are delayed briefly toallow for an examination of important basic principles in the following paragraphs.Conservation PrinciplesDetailed studies of radioactive decay and nuclear reaction processes have led to theformulation of very useful conservation principles. For example, electric charge isconserved in a decay; the total amount is the same before and after the reaction even

tTwo earlier mass scales defined the amu as 1116 the mass of elemental oxygen and of oxygen-16,respectively. Each of these differs from the current standard, so caution is advised when using data frommultiple sources [e.g., Kaplan (1962) is based on the 160 scale, while the Chart of the Nuclides (GE/Chart,1989) employs the 12C scale].:f:ey is usually pronounced as "ee-vee," keY as "kay-ee-vee," etc.


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32 Basic Theory

though it may be distributed differently among entirely different nuclides and/or par-ticles. The four principles of most interest here are conservation ofI. charge2. mass number or number of nucleons3. total energy4. linear and angular momentum

Conservation of electric charge implies that charges are neither created nordestroyed. Single positive and negative charges may, of course, neutralize each other.Conversely, it is also possible for a neutral particle to produce one charge of eachsign.

Conservation of mass number does not allow a net change in the number ofnucleons. However, the conversion of a proton to a neutron is allowed. Electronsfollow a separate particle conservation law (which is beyond the scope of this dis-cussion). By convention, a mass number of zero is assigned to electrons.

The total of the kinetic energy and the energy equivalent of the mass in a systemmust be conserved in all decays and reactions. This principle determines which out-comes are and are not possible.

Conservation of linear momentum is responsible for the distribution of the avail-able kinetic energy among product nuclei, particles, and/or radiations. Angular mo-mentum considerations for particles that make up the nucleus play a major role indetermining the likelihood of occurrence of the outcomes that are energetically possible.(This latter consideration is substantially beyond the scope of this book.)Natural RadioactivityA wide range of radioactive nuclides, or radionuclides, exist in nature (and did sobefore the advent of the nuclear age). Artificial radionuclides produced by nuclearreactions are considered separately.

The naturally occurring radioactive decay processes may produce any of threeradiations. Dating back to the time of their discovery and identification, the arbitrarynames aLpha. beta, and gamma are still employed.Alpha RadiationAlpha radiation is a helium nucleus, which may be represented as either or An important alpha-decay process with 2W is written in the form

where 26Th [thorium-231 ] is the decay or daughter product of the U parent nucleus.It may be noted that the reaction equation demonstrates conservation of mass numberA and charge (or equivalently, atomic number Z) on both sides of the equation. Thus,once any two of the three constituents are known, the third may be determined readily.

Most alpha-emitting species have been observed to generate several discretekinetic energies. Thus, to conserve total energy, the product nuclei must have cor-respondingly different masses. The discrete or quantum differences in energy are relatedto a complex energy LeveL structure within the nucleus. (Explanation of such phenomena


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Nuclear Physics 33

is delegated to the field of quantum mechanics, which again is outside the scope ofthis book.)Beta RadiationBeta radiation is an electron of nuclear, rather than orbital, origin. Since, as notedearlier, the electron has a negative charge equal in magnitude to that of the protonand has a mass number of zero, it is represented as _?e or _? {3. A beta-decay reaction,which is important to production of plutonium (e.g., in a breeder reactor), is

where [neptunium-239] is the parent of the and 8v* is an uncharged,massless "particle" called an antineutrino.t As required by conservation principles,the algebraic sums of both charge and mass number on each side of the equation areequal.The nuclear basis for beta decay is

where the uncharged neutron emits an electron and an antineutrino while leaving aproton (and a net additional positive charge) in the nucleus. The slight mass differencebetween the neutron and proton may be noted from Table 2-1 to be sufficient to allowfor electron emission as well as a small amount of kinetic energy.

The nature of the antineutrino defies the human senses. Because it has neithercharge nor mass, the antineutrino does not interact significantly with other materialsand is not readily detected. However, it does carry a portion of the kinetic energy thatwould otherwise belong to the beta particle. In any given decay, the antineutrino maytake anywhere from 0 to 100 percent of the energy, with an average of about two-thirds for many cases.

Analogously to alpha decay, the beta process in a given radionuclide may produceseveral discrete transition energies based on the energy levels in the nucleus. Here,however, a range of beta energies from the transition energy down to zero are actuallyobserved, due to the sharing with antineutrinos.Gamma RadiationGamma radiation is a high-energy electromagnetic radiation that ongmates in thenucleus. It is emitted in the form of photons, discrete bundles of energy that haveboth wave and particle properties.

Gamma radiation is emitted by excited or metastable nuclei, i .e ., those with aslight mass excess (which may, for example, result from a previous alpha or betatransition of less than maximum energy). In one gamma-decay reaction

tThere is also a neutrino 81', which is associated with positron and electron capture decay processes.Because neutrinos are not significant in the context of commercial nuclear energy, they are not consideredhere. The interested reader may consult a textbook on nuclear physics.


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34 Basic Theorv

an excited nucleus is transformed into one that is more "stable" (although, in thiscase, still radioactive). Such processes increase the binding energy but do not affecteither the charge or the mass number of the nucleus.

The gamma-ray energies represent transitions between the discrete energy levelsin a nucleus. As a practical matter, nuclides can often be readily identified or differ-entiated from each other on the basis of their distinctive gamma energies.

Decay ProbabilityThe precise time at which any single nucleus will decay cannot be determined. How-ever, the average behavior of a very large sample can be predicted accurately by usingstatistical methods.

An average time dependence for a given nuclide is quantified in terms of a decayconstant A-the probability per unit time that a decay will occur. The activity of asample is the average number of disintegrations per unit time. For a large sample, theactivity is the product of the decay constant and the number of atoms present, or

Activity = An(t) (2-4)where n(t) is the concentration, which changes as a function of the time t. BecauseA is a constant, the activity and the concentration are always proportional and may beused interchangeably to describe any given radionuclide population.

It has been typical to quote activities in units of the Curie [Ci], defined as 3 .7x \0'0 disintegrations per second, which is roughly the decay rate of I g of radium(the material studied by Marie Curie in her pioneering studies of radioactivity). Thecurrently favored SI unit is the Becquerel, which is I disintegration per second.

The rate at which a given radionuclide sample decays is, of course equal to therate of decrease of its concentration, or

Activity = rate of decreaseMathematically, this is equivalent to

An(t) dn(t)dt

By rearranging terms,dn(t)/n(t)

dt (2-5a)

dn(t)or net) -A dt (2-5b)


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Nuclear Physics 35

where Eq. 2-5a shows that the decay constant A is the fractional change in nuclideconcentration per unit time.

The solution to Eq. 2-5b isnet) = nCO) e- Ar (2-6)

where nCO) is the radionuclide concentration at time t O. As a consequence of theexponential decay, two useful t imes can be identified:I. The mean lifetime 7-the average [statistical mean) time a nucleus exists before

undergoing radioactive decay. Because it may be shown thatI

7= Athis lifetime is also the amount of time required for the sample to decrease by afactor of e (see Eq. 2-6).

2. The half-life TIi2- the average amount of time required for sample size or activityto decrease to one-half of its initial amount.The half-life, mean lifetime, and decay constant are found to be related by

In2In 2T = Aor equivalently

0.693T I /2 = 0.6937 = --AThe basic features of decay of a radionuclide sample are shown by the graph in

Fig. 2-2. Assuming an initial concentration nCO), the population may be noted todecrease by one-half of this value in a time of one half-life. Additional decreases occursuch that whenever one half-life elapses, the concentration drops to one-halfof whateverits value was at the beginning of that time interval.

An example of an important application of radioactive decay is in the managementof radioactive wastes (a subject considered in detail in Chap. 19). The fission products85Kr [krypton-85) and 87Kr, which have half-lives of roughly II years and 76 min,respectively, are generally both present in LWR cooling water. In 10 half-lives, eachwould be reduced in population and activity to about 0.1 percent (actually 1/1024).Thus , 87Kr would essentially disappear of its own accord in a little over one-half day.The 85Kr, on the other hand, would be of concern for on the order of a hundred years.(The latter, for example, was a problem following the accident at the Three MileIsland reactor, discussed in Chap. 15.)


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36 Basic Theory

1.0 - n(O)

-;;\0 0.9c:c:z 0.80I - 0.7l:II :I -z 0.6wuz0 0.5u0w 0.4N...J


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Nuclear Physics 37

(Cl* x X A U 0- 0------ ------- V NUCLEUSPROJECTILE TARGET COMPOUND

PARTICLE NUCLEUS NUCLEUS PARTICLE

FIGURE 2-3Generic nuclear reaction.

Compound NucleusThe compound nucleus temporarily contains all of the charge and mass involved inthe reaction. However, it is so unstable in an energy sense that it only exists for onthe order of 10- 14 S (a time so short as to be insignificant, and undetectable, on ascale of human awareness). Because of its instability, a compound nucleus shouldnever be considered equivalent to a nuclide that may have the same number of protonsand neutrons.Nuclear reactions are subject to the same conservation principles that govern

radioactive decay. Based on conservation of charge and mass number alone, a verywide range of reactions can be postulated. Total energy considerations determine whichreactions are feasible. Then, angular momentum (and other) characteristics fix therelative likelihood of occurrence of each possible reaction. Equations for a number ofimportant reactions are considered at the end of this section. Reaction probabilitiesare the subject of the last section of this chapter.

Conservation of total energy implies a balance, including both kinetic energyand mass. The simple reaction in Fig. 2-3 must obey the balance equation

(2-7)where E; and M;c2 , respectively, are the kinetic and mass-equivalent energies of theith part icipant in the reaction. Rearranging the terms of Eq. 2-7 shows that

where the left-hand bracket is the Q-value for the reaction.When Q > 0, the kinetic energy of the products is greater than that of the initial

reactants. This implies that mass has been converted to energy (a fact that may beverified by examining the right-hand side of Eq. 2-8). Such a reaction is said to beexothermal or exoergic because it produces more energy than that required to initiateit.For cases with Q < 0, the reaction reduces the kinetic energy of the system and

is said to be endothermal or endoergic. These reactions have a minimum thresholdenergy, which must be added to the system to make it feasible (i.e., to allow the massincrease required by Eq. 2-8).


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38 Basic Theorv

A different type of energy threshold exists in reactions between charged particlesof like sign (e.g., an alpha particle and a nucleus) due to the repulsive Coulomb forces.In this case, however, the reaction may still be exoergic as long as Q > 0 or,equivalently, as long as the product kinetic energy exceeds the energy required tooverride the electrostatic forces. The fusion reactions considered in Chap. 21 are animportant example of threshold reactions of this latter type.

Reaction TypesA very wide range of nuclear reactions have been observed experimentally. Of these,the reactions of most interest to the study of nuclear reactors are the ones that involveneutrons.

When a neutron strikes for example, a compound nucleus is formed,as sketched in Fig. 2-4. The compound nucleus then divides in one of several possibleways. These reactions and several others are discussed in the remainder of this section.(It should be noted that charge Z and mass number A are conserved in each case.)ScatteringA scattering event is said to have occurred when the compound nucleus emits a singleneutron. Despite the fact that the initial and final neutrons do not need to be (andlikely are not) the same, the net effect of the reaction is as if the projectile neutronhad merely "bounced off," or scattered from, the nucleus.

The scattering is elastic when the kinetic energy of the system is unchanged bythe reaction. Although the equation for elastic scattering, 235U(n, n), or

looks particularly uninteresting, the fact that the neutron generally changes both its235U + 1n ELASTIC92 0 SCATTERING

235* 1 INELASTIC92 U + On SCATTERING

235u + 1 (236u) * 236U + 0")"2 On .. 92 ... RADIATIVE92 0 CAPTURE

234U + 2 'n MULTIPLE92 0 NEUTRON

L..-__ FISSIONFIGURE 2-4Possible outcomes from neutron irradiation of U.


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Nuclear Physics 39

kinetic energy and its direction is significant. The former change, in particular, maysubstantially alter the probability for further reactions.

If the kinetic energy of the system decreases, the scattering is inelastic. Theequation 2W(n, n'), or

represents the fact that kinetic energy is not conserved and that, thus, the productnucleus is left in an excited state. This extra loss of kinetic energy makes inelasticscattering significant for certain applications to neutron slowing down. The excited23;U nucleus quickly decays to its more stable form by emitting gamma radiation.Radiative CaptureThe reaction U(n, 1) or

is known as radiative capture or simply n, 1. The capture gamma in this case has anenergy of about 6 MeV (corresponding roughly to the binding energy per nucleon onFig. 2-1 for this extra neutron).

The same reaction occurring in materials other than the fuel is often calledactivation. When sodium coolant in an LMFBR gains a neutron in the reaction

HNa + An (fiNa)* TiNa + 81the TiNa is radioactive (i.e., the sodium has been activated). The desire to eliminatethe possibility of radioactive sodium contacting water is responsible for the introductioninto the LMFBR steam cycle (Fig. 1-5) of an intermediate loop of "clean" sodium.

On the more positive side, the entire field of activation analysis is based onproducing artificial radionuclides by neutron bombardment of an unknown sample.Then, by using gamma-ray information to determine the identity and quantity of eachspecies, the composition of the initial material can be determined, often to a highdegree of accuracy.

Multiple NeutronThe compound nucleus may also deexcite by emitting more than one neutron. Thereaction (n, 2n) or

evolves a pair of neutrons. Reactions that produce three or more neutrons are alsopossible.

The multiple neutron reactions are generally endoergic. For example, above aneutron threshold energy of about 6 MeV, the reaction


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40 Basic Theorv

is possible. The relatively short half-life of 232U can have a significant impact onoperations in mU-thorium fuel cycles (described in Chap. 6).

FissionThe typical fission reaction f) or

yields two fission-fragment nuclei plus several gammas and neutrons. As consideredin the next section, fission produces many different fragment pairs.Charged ParticlesAlthough not common for 235U, there are many neutron-initiated reactions that producelight charged particles. An important example is the a) reaction

where boron-IO is converted to lithium-7 plus an alpha particle. On the basis of thisreaction, boron is often used as a "poison" for removing neutrons when it is desiredto shut down the fission chain reaction.

Other neutron-induced reactions yieid protons or deuterons. Multiple charged-particle emissions, possibly accompanied by neutron(s) and/or gamma(s), have alsobeen observed.Neutron ProductionTwo other types of reactions are of interest because their product is a neutron. Thereaction a, n) or

can occur when an alpha emitter is intimately mixed with beryllium. Plutonium-beryllium [Pu-Be) and radium-beryllium [Ra-Be) sources both employ this reactionto produce neutrons. Either may be used to initiate a fission chain reaction for startupof a nuclear reactor.

High-energy gamma rays may interact with certain nuclei to product photoneu-trons. One such reaction with deuterium, TO( y, n) or

To + (TO)* IH + bnoccurs in any system employing heavy water. A similar reaction occurs with inresearch reactors that have beryllium components. Both reactions have energy thres-holds based on overcoming the "binding energy of the last neutron" (i.e., the bindingenergy difference between the isotopes with A - Z neutrons and A - Z - I neutrons,respectively).


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Nuclear Physics 41

NUCLEAR FISSIONThe fission reaction is the basis for current commercial application of nuclear energy.Several of the important features of the process are discussed in this section.Nature of FissionAccording to a very simple qualitative model, a nucleus may be considered as a liquiddrop that reacts to the forces upon and within it. The nucleus, then, assumes a sphericalshape when the forces are in equilibrium. When energy is added, the nucleus is causedto oscillate from its initially spherical shape. If the shape becomes sufficiently elon-gated, it may neck down in the middle and then split into two or more fragments. Alarge amount of energy is released in the form of radiations and fragment kinetic energy(e.g., see Fig. 1-1).

Almost any nucleus can be fissioned if a sufficient amount of excitation energyis available. In the elements with Z < 90, however, the requirements tend to beprohibitively large. Fission is most readily achieved in the heavy nuclei where thethreshold energies are 4-6 MeV or lower for a number of important nuclides.

Certain heavy nuclides exhibit the property of spontaneous fission wherein anexternal energy addition is not required. In californium-252 for example, thisprocess occurs as a form of radioactive decay with a half-life of about 2.6 years. Even235U and 239U fission spontaneously, but with half-lives of roughly 10 17 years and 10 16years, respectively; these values are at least 107 times greater than the ex-decay half-lives.

Charged particles, gamma rays, and neutrons are all capable of inducing fission.The first two are of essentially no significance in the present context. As indicated inthe previous chapter, neutron-induced fission chain reactions are the basis for com-mercial nuclear power.Neutron-Induced FissionWhen a neutron enters a nucleus, its mere presence is equivalent to an addition ofenergy because of the binding energy considerations discussed earlier. The bindingenergy change mayor may not be sufficient by itself to cause fission.

Afissile nuclide is one for which fission is possible with neutrons of any energy.Especially significant is the ability of these nuclides to be fissioned by thermal neutrons,which bring essentially no kinetic energy to the reaction. The important fissile nuclidesare the uranium isotopes 2W and and the plutonium isotopes and It has been noted that 235U is the only naturally occurring member of the group.

A nuclide is fissionable if it can be fissioned by neutrons. All fissile nuclides,of course, must fall in this category. However, nuclides that can be fissioned only byhigh-energy, "above-threshold" neutrons are also included. This latter category in-cludes 26Th, and all of which require neutron energies in excess ofI MeV.

The fissile nuclides that do not exist in nature can only be produced by nuclearreactions. The target nuclei for such reactions are said to be fertile. Figure 2-5 tracesthe mechanisms by which the three major fertile nuclides, 26Th, and produce and respectively. The first two are each based on radiative


26/49

42 Basic Theory

rr 22.3 min

rr 27 d

(al

rr 23.5 min

rr 56 h

(b)

240p94 u

(e)FIGURE 2-5Chains for conversion of fertile nuclides to fissile nuclides: (a) "


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Nuclear Physics 43

10

0...JLU;;:Z0Ul 10- 1UlU.I -ZLUUa:LU 10 - 2

r If\I\. )

80 100 120 140 160 180MASS NUMBER A

FIGURE 2-6Fission yield as a function of mass number for thermal-neutron fission of 235U. (Adapted from NuclearChemical Engineering by M. Benedict and T. H. Pigford, 1957 by McGraw-Hill Book Company,Inc. Used by permission of McGraw-Hill Book Company.)

The fission fragments tend to be neutron-rich with respect to stable nuclides ofthe same mass number. The related energy imbalance is generally rectified by suc-cessive beta emissions, each of which converts a neutron to a proton. Two beta-decay"chains" (from different fissions) are shown in Fig. 2-7. The gamma rays are emittedwhenever a beta decay leaves the nucleus in an excited state. The antineutrinos thataccompany the beta decays are not shown because they have no direct effect on nuclearenergy systems.

The chain in Fig. 2-7, which contains strontium-90 mSr], is especially trou-blesome both in reactor accidents (Chaps. 13-15) and waste management (Chap. 19)because strontium is relatively volatile and this isotope has a long 29-year half-lifecoupled to the high fission yield shown in Fig. 2-6. Similar considerations apply tocesium-137 which has a 30-year half-life.

Another problem associated with the fission fragments is the presence withinthe decay chains of nuclides, which capture neutrons that would otherwise be availableto sustain the chain reaction or to convert fertile material. Two especially importantneutron "poisons" are xenon-135 and samarium-149 Each poses aslightly different problem for reactor operation (as discussed in Chap. 6).

Neutron ProductionThermal-neutron fission of 235U produces (an average of) 2.5 neutrons per reaction.The majority of these are prompt neutrons emitted at the time of fission. A smallfraction are delayed neutrons, which appear from seconds to minutes later.

The number of neutrons from fission depends on both the identity of the fis-


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44 Basic Theorv

0.96 s '. 7 s 12 s 14 min

fJ-

33 h

fJ-

13.6 d STABLEFIGURE 2-7Two representative fission-product decay chains (from dijferelll fissions). (Adapted courtesy of U.S.Department of Energy.)

sionable nuclide and the energy of the incident neutron. The parameter v (referred tosimply as "nu") is the average number of neutrons emitted per fission.

The energy distribution, or spectrum, for neutrons emitted by fission X(E) isrelatively independent of the energy of the neutron causing the fission. For manypurposes, the expression

X(E) = 0.453e-L036E sinh \ /2.29E (2-9)provides an adequate approximation to the neutron spectrum for 235U shown by Fig.2-8. The most likely neutron energy of about 0.7 MeV occurs where X(E) is amaximum.

The neutron spectrum X(E) is actually defined as a probability density, or theprobability per unit energy that a neutron will be emitted within increment dE aboutenergy E. Thus, neutron fractions can only be obtained by integration over finiteenergy intervals. As a probability density, it is required that

L'" dE X(E) = I(where for actual data the infinite upper limit would be replaced by the maximumobserved energy). The average or mean energy (E) of the distribution is then

(E) = L'" dE E X(E)which for Eq. 2-9 evaluates to very nearly 2.0 MeV.


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r 1\ I I I I II \ x(E) = 0.453 e-1.036E sinh ,j2.29E1\\ I\.

'" r--- '--

0.300

,> 0.200)...::UJX

0.100

oo 2 3 4 5 6 7 8

Nuclear Physics 45

ENERGY E. MeVFIGURE 2-8Fission-neutron energy spectrum X(E) for the thermal fission of 235U approximated by an empiricalexpression.

Almost all fission neutrons have energies between 0.1 MeV and 10 MeV. Thus,reactor concepts that are based on using the very-low-energy thermal neutrons (E V> 100V>oc:U..J 10:?

, ltwlJf ' .""

10

NEUTRON ENERGY, eV104

FIGURE 2-11Total microscopic cross section for 238U as a function of incident neutron energy. (Data from Hughes/BNL-325, 1955.)


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52 Basic Theory

more time near the nucleus and experience the nuclear forces for a longer time. Theabsorption probability, then, tends to vary inversely as the neutron speed and, thus,give rise to the IIv-behavior. When an absorption cross section 0"0 is known for energy0, values for other energies in the I/v-region may be readily calculated from

O"a() Vo0"0 -;-

O"a() YEoor O"0y'E (2-14)

because the kinetic energy in terms of speed isjust = fmv 2 . The "thermal" energy = 0.025 eY (or v = 2200 rn/s) is generally used as the reference whenever it lieswithin the range.Almost all fissionable and other nuclides have regions that are roughly IIv, manyto higher energies than shown for 238U. In the important neutron poison lOB, thisregular behavior actually continues rather precisely up to energies in the keY range.

The very high, narrow resonance peaks in the center region of Fig. 2-11 are aresult of the nucleus's affinity for neutrons whose energies closely match its discrete,quantum energy levels. It may be noted for the lowest-energy resonance that the totalcross section changes by a factor of nearly 1000 from the peak energy to those justslightly lower. All fissionable materials exhibit similar resonance behavior.

Scattering, the other contributor to the total cross section, tends to take twoforms. In potential scattering, the cross section is essentially constant (i .e., independentof energy) at a value somewhat near the effective cross-sectional area of the nucleus.Resonance scattering, like its absorption counterpart, is based on the energy-levelstructure of the nucleus.The distinction between fissile and nonfissile isotopes is readily observed fromthe plot of the fission cross sections for 235U and 238U in Fig. 2-12. As neutron energy

NEUTRON ENERGY, eVFIGURE 2-12Microscopic fission cross section for fissile mU and fissionable mU, (Data from Hughes/BNL-325.1955.)


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Nuclear Physics 53

increases, the 235U curve is characterized by large (near-lIv) values, resonance be-havior, and slightly irregular low values. The nonfissile 238U shows a fission thresholdnear 1 MeV followed by a maximum cross section of one bam. The fission crosssection in 235U clearly exceeds that for 238U at all neutron energies of interest in reactorsystems.Figure 2-12 also shows that for 235U thermal-neutron fission is more probablethan fast-neutron (E p 0.1 MeV) fission by over three orders of magnitude. This isone of the important factors that has favored thermal reactors over fast reactors.Interaction RatesThe cross section quantifies the relative probability that a nucleus will experience aneutron reaction. The overall rate of reaction in a system, however, also depends onthe characteristics of the material and of the neutron population. The following sim-plified derivation identifies the important features of interaction-rate calculations.

Considering first a parallel beam of monoenergetic neutrons with a speed v anda density of N per unit volume, it is necessary to determine the rate at which theypass through a sample like that of Fig. 2-9. If, as before, the disk has an area dA anda thickness dx, then:I. When the beam is perpendicular to dA, only neutrons directly in front of and within

a certain distance l can reach the surface in a given time dt .2. The distance l traveled by the neutrons is equal to their speed v multiplied by thetime dt , or l = v dt.

3. As shown by Fig. 2-13, only the neutrons within the cylinder of length l and areadA can enter the sample in time dt (all others will either not reach the target orwill pass outside of its boundaries).

4. The number of neutrons in the cylinder is the density N times the volume dV [=l dA, or v dt dA from (2)],

Number of neutrons passingdA per unit time

NdVdt

N v dt dAdt Nv dA (2-15)

The reaction rate must be equal to the product of the rate of entry of neutronsand the probability that a neutron will interact with a nucleus. Thus, combining theresults of Eqs. 2-15 and 2-11,

Reaction rate = (Nv dA)(nCT dx)

J) [)dA FIGURE 2-13Basic geometry for developing the concept of nuclear reaction rate in terms of macroscopic cross sectionand neutron flux: I = v dt; V = IdA = v dt dA.


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54 Basic Theory

for neutron density N, neutron speed v, nuclide density n, cross section CT, and diskarea and thickness dA and dx, respectively. Rearranging tenns and noting that dA dxis just the disk volume dV,

Reaction rate = (nCT)(Nv) dV (2-16)This result separates the effects of nuclide characteristics, neutron population, andsample volume.

The macroscopic cross section L is defined asL = nCT (2-17)

With n the number of atoms per unit volume and CT the area per atom, the macroscopiccross section is "per unit distance" or generally cm -I. It is the probability per unitdistance of travel that a neutron will interact in a sample characterized by atom densityn and microscopic cross section CT.

It is, perhaps, unfortunate that both CT and L bear the title "cross section"because they have different units. The microscopic cross section is an "effective area"used to characterize a single nucleus. The macroscopic cross section is the probabilitythat a neutron will interact in traveling a unit distance through a (macroscopic) sampleof material. Use of the common nicknames "micro" and "macro" without attachingthe words "cross section" may provide a partial solution to the problem.

The neutron flux is defined by = Nv (2-18)

for neutron density N and speed v. The density N is in tenns of neutrons per unitvolume, and the speed v is distance per unit time, thus, neutron flux is neutronsper unit area per unit time, or generally neutrons/cm 2s. Although the earlier derivationwas based on a monodirectional, monoenergetic beam, the definition itself is com-pletely general.

The reaction rate in Eq. 2-16 may be rewritten asReaction rate = L dV (2-19)

for macroscopic cross section L and neutron flux as defined by Eqs. 2-17 and2-18, respectively, and for sample volume dV. An alternative fonn is

Reaction rate per unit volume = L (2-20)where now the volume dV is unspecified.

The macroscopic cross section L is constructed from microscopic cross sections,and thus varies with nuclide, interaCtion type, and neutron energy. In the formn j = it represents the probability per unit distance of travel that aneutron of energy E will interact by mechanism r with nuclide j. These interactionprobabilities for a given reaction and given nucleus are independent of other reactions


39/49

Nuclear Physics 55

and the other nuclei present, so they may be summed to describe any desired com-bination. The total macroscopic cross section for any given mixture is

L;niX = L L all j all r (2-21 )where the summations are over all nuclides j in the mixture and all reactions r. Insumming the reactions, care is advised in assuring that each is independent (e.g., Lamay not be added to Lf or L, because La = Lf + L", as shown by Fig. 2-10).

Based on the inherent limitations of the earlier development of the concept of amicroscopic cross section, it must be recognized that Eq. 2-11 and others based on itare valid only for "small" samples where the nuclei do not shadow or obscure eachother from the neutron beam. "Thick" samples have the effect of removing neutronsand, therefore, changing the neutron flux seen by the more internal nuclei. If a beamof neutrons passes into a material sample, it is said to experience attenuation or adecrease of intensity.

I f a beam of parallel monoenergetic neutrons crosses the surface of a sample,one neutron will be removed each time a reaction occurs. Thus, the rate of decreaseof neutron flux


40/49

56 Basic Theory

4

1.00 0.9 0.8X 0.7X 0.6::)...Jl! - 0,50UJ 0.4N...J 0,3


41/49

Nuclear Physics 57

use for overview purposes dealing with a limited range of nuclides, reactions, and/orenergies.

The computer-based ENDF is the workhorse for nearly all large-scale calcula-tions. The ENDF/B set contains complete, evaluated sets of nuclear data for about 80nuclides and for all significant reactions over the full energy range of interest. Revisedversions are developed, tested, and issued periodically.

Chart of the NuclidesThe Chart of the Nuclides is very useful for performing preliminary or scoping cal-culations related to reactors. A portion containing important fissile and fertile nuclidesis sketched in Fig. 2-15. Because of the vast amount of information, the figure showsthe basic structure plus detail for only the 235U, 238U, and 239pU nuclides. The basicgrid has the elements in horizontal rows with vertical columns representing neutronnumbers. The block at the left end of each row gives the chemical symbol and, asappropriate, the thermal [0.025 eV] absorption and fission cross sections for thenaturally occurring isotopic composition.

Basic data for individual nuclides includes (as appropriate): nuclide ID natural isotopic composition in atom percent [at %] half-life type of radiation(s) emitted and energies in decreasing order (" ... " implies ad-

ditional lower energy values not included) thermal neutron cross sections and resonance integrals, both in barns isotopic (atomic) massBecause the isotopic compositions are given in atom percent [at %], while fuel cyclemass flows are usually in weight percent [wt %], there is occasional confusion (e.g.,the enrichment of natural uranium is 0.72 at % or 0.711 wt % 235U). The resonanceintegral mentioned above is intended as a measure of an "effective" cross section thatfission neutrons would see in slowing down through the resonance-energy region (seealso Chap. 4).

The masses used in the Chart of the Nuclides are for the entire atom rather thanfor the nucleus. Thus, they include the mass of the electrons less the contribution ofelectron binding energy. The electron binding energies are almost always too smallto be significant in comparison with nuclear binding energies. Further, because nuclearmass differences (as opposed to the masses themselves) are of most interest, atomicmasses can be used consistently. (I t may be noted that charge conservation guaranteesthat electrons are handled appropriately. In beta decay, for example, the new electroncan enter an orbit to balance the extra proton in the nucleus.)

The Chart of the Nuclides also contains a wealth of useful information not shownin Fig. 2-15. One interesting feature is the use of color codes for both half-life andcross-section ranges. A recent addition has been fission-product production data fortypical power reactors. Fortunately, the introduction and guide to use of the Chart ofthe Nuclides are well written and easily understood. Currently, it is available in bothwall-chart and booklet form.


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OTOPICOMIC)ASS

CLiDE10ALFL1FEsecond;minute;hour;day;ear)

p 'M .unML . I . ; t v -U 235 TOPIC ABUNDANCE U 238 Pu 239 ..... N(ATOM %)I 0.7200 99.2745 2.411 X 104 a _HRADIATION TYPES 4.468 x 109 a a 5.155, 5.143, 5.105, (s7.04 X 108 a AND ENERGIES - l a 4.196,4.15, ... 'Y .0496 .. .'Y .0516, .0301- IT

a 4.401, 4.365, ... 'Y .18572 (MeV) l SF weak 1.057 SF ay 271 h..., SF ay 99,14 X 10 1, ay 2.68,277 20 X 101 at 750, dat 585,-275 t 0.025eV CROSS- at 51'b, 1.3 mb 30 x 101 a235.043924 SECTIONS, acx 11'b 239.052157RESONANCE 2380W/ ISINTEGRALS (,II(BARNS) M

Pu Pu235 Pu236 Pu237 Pu238 Pu239'1 Pu240 Pu241 Pu242 Pu243 Pu24494 - - - I'. . /Np Np228 Np233 Np234 Np236 Np238 Np239 Np240 Np24193 --- r---U U226 U227 U232 U233 U234 U235 U236 U2J7 U2J8 ' U239 U24Q

92 --- !'- ./ /Pa224 Pa225 Pa226 Pa2Jl Pa232 Pa233 Pa234 Pa235 Pa236 Pa237 Pa238---Th22J Th224 Th225 Th230 Th231 Th2J2 Th233 Th2J4 Th2J5 Th236 150- - -Ac222 Ac223 Af224 Ac229 Ac230 Ac231 Ac232 148---Ra221 Ra222 Ra223 Ra228 Ra229 Ra230- - -

134 140 142 144 146FIGURE 2-15Structure and data presentation in the Chart of the Nuclides for 235U, 238U, and 239Pu. (Adapted from theChart of the Nuclides, courtesy of Knolls Atomic Power Laboratory, Schenectady, New York. Operatedby the General Electric Company for the United States Department of Energy Naval Reactors Branch.)


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Nuclear Physics 59

SummaryMost of the important nuclides and many of the concepts of this chapter are directlyor indirectly summarized in Fig. 2-16. Isotopes are in rows with arrows representingradiative capture or (n, ,,) reactions. Beta decays are shown by downward-pointingarrows. Thermal-neutron fission is represented by diagonal dashed arrows. Decay half-lives and thermal cross sections are from the Chart of the Nuclides.

As a review, the reader should identify on Fig. 2-16: the four major fissile nuclides three chains for converting fertile nuclides to fissile nuclides an important (n, 2n) reaction nonfission capture events in fissile nuclidesOverall, the information on this figure should be of substantial value in a number ofthe later chapters.

KEY

_ (n, 2n)-REACTION IE ;;, 6 MeV)_ In, 1')-REACTION,:';x FISSION Il-DECAYXX THERMAL CROSS-SECTION IBARNS)YY HALF-LIFECROSS-SECTIONS AND HALF-LIVESFROM CHART OF THE NUCLIDES (1989)

a-DECAY HALF-LIVES ARE ALL ;;, 10 Y

FIGURE 2-16Neutron irradiation chains for heavy elements of interest for nuclear reactors.


44/49

(Fig. 2-5)

bn mass mass

60 Basic Theory

Sample CalculationsThe data in Figs. 2-15 and 2-16 plus the equations developed in this chapter can beused to determine a number of important reactor and fuel cycle characteristics. Thefollowing are representative examples:1. The reaction and decay equations and total mass change for conversion of fertile

to fissile + bn + goy} + _? f3 + _? f3

Noting that charge and mass number are conserved,238.050785 amu (Fig. 2-15)

+ 1.008665 amu (Table 2-1)239.059450 amu

mass -239.052157 amu (Fig. 2-15)Total change 10.007293 amul

In terms of kinetic energy,0.007293 amu x 931.5 MeV/amu = 16.79 MeV 1

2. The nuclide density n28 of 238U. The definition of SI units (see App. II) providesthat one mole [mol] of any element contains the same number of atoms as 0.012kg [12 g] of 12C. In the same manner that a mole of 12C has a mass equivalent toits mass number (A = 12.000, as used to define the atomic mass unit [amu]) ingrams, a mole of each other element consists of its own A-value in grams. A moleof substance, in turn, contains Avogadro's number Ao of atoms (-6.02 x 10 23at/mol as noted in App. II). Thus, nuclide density n j may be expressed as

. Ao (at/mol) Aon J = x p (g/cm3) = - p (at/cm3)A (g/mol) Afor density p . Uranium-238 with nominal density p28 = 19.1 g/cm3 would thenhave

6.022 X 10 23 at/mol I In28 = X 19.1 g/cm3 = 4.83 x 10 22 at/cm3238.05 g/mol(Note that use of the integer value 238 compared to the actual atomic mass of238.05 (Fig. 2-15) produces the same result.) [Nuclide density calculations for


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Nuclear Physics 61

mixtures are somewhat more complex because compositions are typically expressedas percentages or fractions by molar composition (or, equivalently, atom number),weight, or volume. Accordingly, the corresponding A- and p-values must be weightedaverages of those for the constituent species.]3. The activity of I g of 238U:

Activity = ,\28n28(t) = 4.47 X 109Y (Eq. 2-4)(Fig. 2-15),\28 = In2 = 0.693 9 = 1.55 X 10- 10 years- 1T1/2 4.47 X 10 y

Ao 6.022 X 1023 atlmoln28 = - = = 2.53 X 1021 atlgA28 238.05 glmolActivity = 1.55 x 10- 10 y - I x 2.53 X 10 21Activity = 3.92 x 1011 y - I X I Y3.15 X 107 s

I CiActivity = 1. 24 X 104 s - 1 x 3. 7 X 10 10 S - I

4. Time required for 238U to decay by 1 percent:

I Bqx --I S - I= I 1. 24 X 104 Bq l

3.37 X 10- 7 Ci= 10.335 ]LCil

(Eq. 2-6)n(t) = 0.99n(0),\28 = 1.55 X 1O- lO y-1n(t) = 0.99 = e-(1.55 X IO - 10y-')tn(O)In(0.99) = - ( 1 . 5 5 x 1O- IO y - l ) tt = 6.48 X 107 Y = 164,800,000 y I

5. Average neutron density corresponding to a typical LWR thermal flux = 5 x10 13 cm -2 S - I , assuming an effective speed v = 2200 mls = 2. 2 x 105 cm/s:

= NvN= -v

(Eq. 2-18)5 x 10 13 c m - 2s - 12. 2 x 105 cm S - I


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62 Basic Theory

6. Power produced by 1 g of 235U fission in the LWR thermal flux in (5):Fission rate == dV == n25 a 25 dV. f (Eq. 2-19)

n' ==n 25 dV == n' == number of atoms in the I-g sample

6.022 X 1023 at = 2.56 x 10 21 at/g235.04 g10- 24 cm 2a 25 == 585 b x == 585 X 10- 24 cm 2lib

Fission rate == 7.49 x 10 13 fissionslslWPower == 7.49 x 10 13 fissions/s x 3.1 x 10 10 fissions/s 2.42 X 10 3 W

Power == 12.42 kW 1 per 1 g of 235U7. For equal nuclide densities of 235U and 239pU in a given reactor, find (a) fraction

of fissions for each and (b) absorption mean free path for each nucl ide and for themixture. Assume each has an atom densi ty of 10 21 at/cm3 a. Fiss ion fract ion

.1:Lmix .f

nJa J a J____1 .fn 25 a-}5 + n49 0:;9 fl25 = fl49 0:}5 + 0:;9

585b 585 b + 750 b ==

750 b585 b + 750 b

b. Mean free paths

in 235U

n 25 ==10- 24 cm 210 21 at/cm 3 x == 10- 3 at/bcm t1 b

10- 3 at/bcm (99 b + 585 b) == 0.684 cm- I10- 3 at/bcm (271 b + 750 b) == 1.021 cm- 1

for 235U

t The unit at/bcm is convenient and often used for atom densities employed in construction ofmacroscopic cross sections.


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Nuclear Physics 63

= - I = 10.979 cm I for mpuA;;'ix = (L;;,ix)-I = 10 .587 cml for both

EXERCISESQuestions2-1. Identify the constituent parts and describe the basic structure of an atom.2-2. Define the following terms: isotope, half-life, and mean free path.2-3. Define fissile, fissionable, and fertile. Identify the major nuclides in each of

these three categories.2-4. Describe the dis tribution of energy among the product particles and radiations

associated with fission. Explain the basis for decay heat.2-5. Sketch the relationship among the following reactor cross sections:a. total interaction

b. scatteringc. absorptiond. fissione. capture

2-6. Differentiate between microscopic and macroscopic cross sections.Numerical Problems2-7. Consider a thermal-neutron fission reaction in 235U that produces two neutrons.

a. Write balanced reaction equations for the two fission events correspondingto the fragments in Fig. 2-7.

b. Using the binding-energy-per-nucleon curve (Fig. 2-1), estimate the totalbinding energy for 235U and each of the fragments considered in (a).

c. Estimate the energy released by each of the two fissions and compare theresults to the accepted average value.

2-8. The best candidate for controlled nuclear fusion is the reaction between deu-terium and tritium. The reaction is also used to produce high energy neutrons.a. Write the reaction equation for this D-T reaction.b. Using nuclear mass values in Table 2-4, calculate the energy release for the

reaction.c. Calculate the reaction rate required to produce a power of I W, based on

the result in (b).d. Compare the D-T energy release to that for fission on per-reaction and per-

reactant-mass bases.2-9. Gamma rays interacting with or rH produce "photoneutrons." Write

the reaction equation for each and calculate the threshold gamma energy. (Usemass data from Table 2-4).

2-10. The nuclide emits either an alpha particle or a beta particle with a half-life of 3.10 min.a. Write equations for each reaction.b. Calculate the decay constant A and mean lifetime T.c. Determine the number of atoms in a sample that has an activity of 100 j.tCi.d. Calculate the activity after I, 2, and 2.5 half-lives.


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64 Basic Theory

TABLE 2-4Nuclear Mass Values for Selected NuclidesNuclide Nuclear mass (amu)bn 1.008665iH 1.007825iH 2.014102TH 3.016047 3.0160291He 4.002603 8.005305 9.012182

2-11. Natural boron has a density of 0.128 x 10 24 at/cm3 and cross section (J"a =764 band (J"s = 4 b at an energy of E = 0.025 eV.a. Calculate the macroscopic cross sections at 0.025 eV for absorption, scat-

tering, and total interaction.b. What fractional attenuation will a 0.025-eV neutron beam experience when

traveling through I mm of the boron? I cm?c. Assuming the absorption cross section is "one-over-v" in energy, calculate

the macroscopic cross sections for boron for neutrons ofO.0025-eV and 100-eV energies.d. What thickness of boron is required to absorb 50 percent of a 100-eV neutronbeam?

2-12. Estimate the fraction of thermal neutron absorptions in natural uranium thatcauses fission. Estimate the fraction of fast-neutron (2-MeV) fissions in naturaluranium that occurs in 238U. (Use data from tables and figures in this chapter.)

2-13. Recent measurements of particle fluxes from supernova place an upper limit onthe mass of a neutrino (and antineutrino) as I I eV. What fraction is this of thef3 particle or electron mass?

SELECTED BIBLIOGRAPHytNuclear Physics

Burcham, 1963Evans, 1955Hunt, 1987Hyde, 1964Kaplan, 1963Kramer, 1980Rhodes, 1986Turner, 1986

t Full citations are contained in the General Bibliography at the back of the book.


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Nuclear Data SourcesGE/Chart, 1989HoneckiENDF, 1966Hughes/BNL-325, 1955Lederer & Shirley, 1978

Other Sources with Appropriate Sections or ChaptersBenedict, 1981Cohen, 1974Connolly, 1978Duderstadt & Hamilton, 1976Etherington, 1958Foster & Wright, 1983G1asstone & Sesonske, 1981Henry, 1975Lamarsh, 1966, 1983Marshall, 1983aMurray, 1988Rydin, 1977WASH-1250,1973

Nuclear Physics 65

knief chapter 2 final

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