Elementary introduction to nuclear reactor physics [First Edition] 1124145885, 9781124145884

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Elementary Introduction to

NUCLEAR REACTOR PHYSICS

Phoenix

GC

777

, L78

cop.2

+

1960 BY JOHN WILEY

COPYRIGHT

All

rights

thereof

& SONS , INC .

This book or any part not be reproduced in any form

reserved .

must

without the written permission of the publisher . LIBRARY

OF CONGRESS

CATALOG CARD NUMBER : 60-11725

PRINTED IN THE UNITED STATES OF AMERICA

SECOND PRINTING , SEPTEMBER , 1966

Preface In less than a generation nuclear reactor physics has developed from esoteric beginnings to a branch of knowledge that has found its way into the college curriculum . Although still taught primarily at the graduate level , there is already a significant number of colleges and institutions offering or planning to offer undergraduate degrees in nuclear science or engineering where courses in reactor physics are an integral part of the program . Un-

doubtedly

, this trend will persist and interest in nuclear technology at the undergraduate level will continue to grow through the added stimulus of large - scale financial support of the U. S. Atomic Energy Commission .

This book is intended to serve as a textbook for an undergraduate course in nuclear reactor physics . It has been my aim to give an elementary but coherent account of that branch of physics involved in the study and design of nuclear reactors at a standard of presentation judged to be suitable for advanced undergraduate students . The book is the outcome of a course which was originally developed at New York Maritime College as early as 1951. During the last seven years I have been giving this course to selected groups of junior and senior engineering students who have previously had one semester of atomic and nuclear physics and one semester of differential equations .

I

have attempted to follow a consistent and logical line of development of the subject matter , steering a middle course between a too detailed and rigorous mathematical treatment and a too shallow and purely descriptive exposition . The mathematical skill required by students using this book does not go beyond calculus and elementary differential equations and , when mathematical arguments are used to derive physically significant results which may not be immediately evident to the student

,

all necessary intermediate steps are shown in

Preface

vi

I

detail . have tried not to overemphasize and treat in excessive detail any one topic , so as to keep the book well -balanced and within the bounds of an undergraduate course .

Many of the concepts are introduced early in the book , and their is reserved for the latter part of the book where the mathematical technique for their use and application is explained . This procedure has considerable pedagogic value since it separates the conceptual difficulty or novelty from the purely technical difficulty involved in the learning of a new concept . As the student later on encounters again a concept which he has already met earlier in his course , the sense of complete newness and sometimes overwhelming strangeness will be absent and , instead , the feeling of relative familiarity will be of great help to him in learning its practical use and more detailed description

application

.

The first three chapters deal with some fundamental aspects of nuclear physics as far as they have a direct bearing on the physics of nuclear reactors . In this basic review course only those topics which are of immediate importance to the nuclear reactor physicist or engineer have been emphasized , whereas some others which may be indispensable to an over - all general understanding of nuclear phenomena have not been touched upon . Starting with the nucleus as a composite structure with inherent stability or the lack of it the logical line of development leads to a consideration of radioactivity , to the concept of

binding energy , and to an examination of the character of nuclear forces which are responsible for it . The liquid drop model of the nucleus is next introduced and nuclear reactions are explained in terms

of the formation of a compound nucleus . Neutron reactions representing the most important type of nuclear reactions for our purpose are then considered , which leads to the concept of neutron cross sections . The various neutron cross sections are subsequently examined and their energy dependence is described with some reference to neutron resonances and their relation to the compound nucleus . The neutron fission cross section leads to an examination of the physical aspects of nuclear fission and its explanation in terms of the previously described It has been found that the consistent and systematic use of a nuclear model such as this is of tremen-

liquid drop model of the nucleus .

dous help to students , especially engineering students , notwithstanding the shortcomings of such a model in some respects . The possibility of a chain - reacting system is subsequently presented and the necessary conditions for its satisfactory operation are examined . This material introduces the need for studying in some detail the interaction of neutrons with matter in bulk , the physics of thermal neutrons , and the

Preface

vii

thermalization of fission neutrons . This discussion is followed by an elementary exposition of neutron diffusion theory in a manner suitable for undergraduates , a consideration of the critical equation and of the spatial distribution of neutrons in finite reactor assemblies of simple . Some aspects of the nonstationary reactor are then presented in an elementary manner together with some of the causes that

geometries

lead to its nonstationary

character . The concluding chapters deal with nuclear radiations that are associated with the operation of a nuclear reactor , their detection and measurement , and , finally , the need for protection against them and some elementary aspects of health physics . It is hoped that this book will also be found helpful to graduate engineers , or scientists who want or need to familiarize themselves with some aspects of nuclear science as applied to reactors or allied fields , to those who require an intermediate textbook for their preparatory

reading before embarking on a more advanced and intensive study of the subject , and to those who wish to gain a maximum of insight into the physical principles with a minimum of mathematical technique . A large number of worked examples have been included which serve

to illustrate the ideas developed in the book and to demonstrate their use for obtaining numerical answers to physical problems . Readers who are using this book for self - instruction should find these worked examples throughout

the text especially helpful . wish to express my indebtedness and gratitude to Captain J. Barton Hoag , U.S.C.G. , Professor of Physics and Head of the Science Department of the U. S. Coast Guard Academy , for reviewing the entire manuscript , for his many helpful suggestions , constructive criticism , and encouraging comments , and to Dr. Meir H. Degani ,

In

conclusion

I

Professor of Physics and Chairman

of the Science Department at the University State of New York Maritime College , for his personal interest , valuable advice , and friendly encouragement throughout . also wish to thank my students for working out the solutions to the

I

problems at the end of each chapter and for their interest and enthusiasm which originally gave me the idea that it might be worth while to undertake the writing of a text such as is here presented . S. E.

May

1960

LIVERHANT

1

1

Contents

1

The Atomic Nucleus

1

1.4

Introduction 1 1 Nuclear Structure Distribution of Nuclides Nuclear Stability 3

1.5

Isobars

1.6

1.12

Natural and Induced Radioactivity 6 The Transuranium Elements 14 The Laws of Radioactive Decay 14 Activity 17 Average Lifetime 18 Half - Life 19 Radioactive Equilibrium and Serial Transformations

1.13

Neutron Activation and the Production of Isotopes

1.1 1.2 1.3

1.7 1.8 1.9 1.10 1.11

5

2 2.1 2.3

2.4 2.5 2.6 2.7 2.8 2.9

2

2.10 Nuclear Reactions and Q - Value 46 2.11 Nuclear Masses and Isotopic Masses 2.12

Threshold

30

Nuclear Forces and Nuclear Binding

Introduction 30 Short Range and Saturation of Nuclear Forces Nuclear Size 31 Nuclear Potential 32 Coulomb Barrier Height 33 Explanation of a - Particle Emission 39 Metastable and Bound Energy States 40 Binding Energy and Isotopic Mass 44 The Atomic Mass Unit 45 48

Energy for Nuclear Reactions ix

50

21

26

30

X

Contents

Binding Binding 2.14 2.15 Binding 2.16 Binding 2.13

Energy Energy Energy Energy

2.17 Semiempirical

and Nuclear Stability and Q - Value 56 of Mirror Nuclei per Nucleon 57

Interpretation

Binding Energy Curve

53

56

of the

58

3 The Compound Nucleus and Nuclear Reactions 3.1

3.2 3.3 3.4 3.5

3.6 3.7

Introduction 65 The Liquid Drop Model Analogy 65 Some Features of Nuclear Reactions 66 The Compound Nucleus 68 Energy Levels of Nuclei 70 Level Widths and De - excitation Isomers and Isomeric States

4 4.1 4.2 4.3 4.4

65

75

78

Neutron Reactions

Introduction 81 Slow Neutron Reactions 81 Nuclear Reaction Cross Section Neutron Cross Sections 83

4.5

Determination

4.6

Attenuation of Neutrons

81

83

of the Cross Section

84

88

Cross Section and Mean Free Path 90 Neutron Flux and Reaction Rate 93 96 4.9 Energy Dependence of Neutron Cross Sections 4.10 The Fission Cross Section 103

4.7 4.8

Macroscopic

5 5.1 5.2

5.3 5.4 5.5

5.6 5.7 5.8 5.9

109

Nuclear Fission

Introduction 109 Materials 109 Yields and Mass Distribution of Fission Products Energy Distribution of Fission Fragments 114 Energy Release from . Fission 115 Neutron Yield and Neutron Production Ratio Prompt and Delayed Neutrons 118 Energy Distribution of Fission Neutrons 119 Nuclear Fission and the Liquid Drop Model 122 Fissionable

110

xi

Contents 5.10

Spontaneous Fission and Potential Barrier

5.11

Deformation

6 6.1 6.2

6.3 6.4 6.5

6.6 6.7 6.8

of the Liquid Drop

124

127

Thermal Neutrons

134

Introduction 134 Energy Distribution of Thermal Neutrons 134 Effective Cross Section for Thermal Neutrons The Slowing Down of Reactor Neutrons 142 Scattering Angles in L and C.M. Systems 147 Angular and Energy Distribution 148 Forward Scattering in the L System 153 Transport Mean Free Path and

140

Scattering Cross Section 155 Average Logarithmic Energy Decrement 155 6.10 Slowing - Down Power and Moderating Ratio 159 6.11 Slowing - Down Density 161 164 6.12 Slowing - Down Time

6.9

6.13 6.14

Resonance Escape Probability 164 The Effective Resonance Integral 167

7 7.1 7.2 7.3

The Nuclear Chain Reaction

Introduction 170 Neutron Cycle and Multiplication Factor The Thermal Utilization Factor 173

7.8

Neutron Leakage and Critical Size 174 Nuclear Reactors and Their Classification Power Reactors 179 Reactor Control 194 Reactor Shielding 196

7.9

Research Reactors

7.4 7.5

7.6 7.7

170

170

176

196

7.10 Calculation of k for a Homogeneous Reactor 215 7.11 Heterogeneous Reactors 224 7.12 Effect of Heterogeneous Arrangement on p , , and e 7.13 Calculation of k for Heterogeneous Reactor 230

f

8 8.1

8.2 8.3 8.4

Neutron Diffusion

Introduction 245 Thermal Neutron Diffusion 246 The Diffusion Equation 248 The Thermal Diffusion Length 250

227

245

xii 8.5

8.6 8.7 8.8

Contents

The Exponential Pile 253 The Diffusion Length for a Fuel - Moderator Mixture 256 Fast Neutron Diffusion and Fermi Age Equation 258 Correction for Neutron Capture 264

9 9.1

9.2 9.3 9.4

9.5 9.6 9.7

9.9 9.10

10.2 10.3 10.4 10.5 10.6 10.7

Moderators

276

Critical Size and Geometrical Buckling Extrapolation Length Correction 286 Effect of Reflector 290

278

294

The Nonsteady Nuclear Reactor

Introduction 294 Thermal Lifetime and Generation Time 294 Time - Dependent Reactor Equation 296 Excess Reactivity and Reactor Period 297 Effect of Delayed Neutrons 299 Delayed Neutrons and Reactor Periods The Inhour Equation 303 11

302

Conditions Affecting the Reactivity

11.2 11.3

Introduction 311 Effect of Temperature Changes on the Reactivity Effect of Fission Products Accumulation 315

11.4

Fuel Depletion and Fuel Production

11.1

268

The Nonleakage Factors 272 Criticality of Large Thermal Reactors 275 The Critical Equation for Reactors with

10 10.1

267

Introduction 267 Diffusion Equation Applied to a Thermal Reactor Thermal Neutron Source as Obtained from the Fermi Age Equation 268 Critical Equation and Reactor Buckling , 270

Hydrogenous 9.8

The Critical Equation

12

311

312

324

Nuclear Radiations and Their Interactions 329

with Matter 12.1 12.2

Introduction 329 Absorption of Heavy Charged Particles in Matter

330

xiii

Contents 12.3 12.4 12.5

12.6 12.7

Absorption of Electrons in Matter 337 Interaction of Electromagnetic Radiation with Matter 343 Compton Scattering 347 Photoelectric Absorption 353 Pair Production 355 13

Radiation Detection and Measurement

13.1

Introduction

13.2 13.4

Lauritsen Electroscope 362 Ionization Chambers 363 Ionization in Gases 365

13.5

Gas Amplification

13.3

13.6 13.7 13.8 13.9

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10

2

369

380

Radiation Protection and Health Physics

Introduction 386 Radiation Damage 386 Internal Irradiation Damage Acute and Chronic Exposure Radiation Intensity 388

386

387 388

Radiation Units and Radiation Dosage 389 Relative Biological Effectiveness 391 Radiation from a Point Source 394 Neutron Irradiation Damage 395 Radiation Shielding and Build - Up Factor 396 15

A.1

362

Proportional Counters 371 Geiger - Müller Counters 373 Scintillation Counters 378 Cerenkov Radiation and Cerenkov Counters 14

362

405

Appendixes

Relativistic Expressions for Mass , Momentum , and Energy 405 The De Broglie Relations 406

The Heisenberg Uncertainty Principle 407 B.1 The Neutron Current Density and Fick's Law of Diffusion 408 2 Neutron Leakage Rate 410 3 Boundary Conditions and Extrapolation Length 3

410

xiv

Contents

4 5

Solution of Diffusion from a Point Source

413

Solution of Diffusion Equation for Thermal Neutrons from a Plane Source

C.1

Equation for Thermal Neutrons

414

One - Group Calculation for Reflected Reactor (a ) Infinite Slab Reactor with Reflector (b ) Spherical Reactor with Reflector 419 Physical Constants 423 Slow- Neutron Cross Sections of the Elements

416

424

Answers to Problems

427

Index of Tables

429

Index

431

chapter 1

The Atomic Nucleus

1.

Introduction

Nuclear reactor physics is a branch of nuclear physics that deals with the large - scale interaction of neutrons with fissionable materials and with such other materials as are used in the construction of nuclear reactors . Various types of nuclear reactors have been built for the purpose of studying nuclear fission and of phenomena associated with it , and for the purpose of utilizing the energy released in nuclear fission . The energy obtained from the combustion of traditional fuels which has of civilization can properly be called " atomic energy, " since it is liberated during chemical processes which are essentially atomic or electronic rearrangements that do not involve the nuclei of the atoms . been used by man since almost the beginning

In contrast with this process the energy released by nuclear fission is of an entirely different origin , since it arises as a result of a nuclear reaction in the course of which the constituents of a uranium or other fissionable nucleus directly affected , with radically different nuclear arrangements emerging after the reaction . The energy liberated in this type of reaction is properly termed " nuclear energy" because source are the nuclei the

of

its

are

.

atoms

of

to

as

.

a

of

,

a

a

in

of

large The practical importance nuclear fission lies the release amount of energy accompanying this process which occurs on scale of compared energy obtained about million times larger the amount from the combustion of an atom of traditional fuel

to

.

a

of

in

is of of

of

of

,

a

.

is

of

The study of reactors and reactor physics requires some familiarity with the principles of nuclear physics and the general properties atomic presented nuclei This information the first three chapters of this book beginning with general survey the kinds nuclear species and description nuclear structure some general features The concept nuclear stability introduced and applied the

2

Elementary Introduction to Nuclear Reactor Physics

classification of nuclei according to their type of nuclear stability . The unstable nuclei , commonly called radioactive nuclei , have been an important source of information about nuclei in general and , because of this , the laws of radioactivity are described in some detail and are then applied

to the important topic of isotope production 1.2

.

Nuclear Structure

are collectively known as nucleons , there are two kinds , protons and neutrons . Each proton carries a positive charge that is numerically equal to that of the electron , whereas the neutron is an electrically neutral particle .

The constituents of atomic nuclei

of which

The masses of the proton and neutron are very nearly equal , although that of the neutron is somewhat larger than that of the proton . The number of neutrons , N, inside an atomic nucleus together with the number of protons , Z , inside the same nucleus determine the atomic mass number ,

A , of the atom .

A= N

+ Z

The chemical properties of atoms are determined by the number of atomic electrons in the neutral atom , which must be equal to Z so as to result in an electrically neutral atom by exactly balancing the positive charge of an equal number of nuclear protons .

All atoms

having the same number of nuclear protons , and therefore the of the same element . Most elements occur in nature as a mixture of several isotopes (some even as many as 20 ) , i.e. , they all have the same characteristic Z , but differ in the number of their nuclear neutrons N. Consequently , their mass numbers A are different . For example , 8016 , 8017 , 8018 are three isotopes of the same element oxygen (Z = 8 ) with atomic mass numbers 16 , 17 , 18 , respectively , of which 8016 is the most abundant by far of the three isotopes . same Z , are isotopes

It

The term is ,

is often used interchangeably with the term isotope . a more general term and it denotes any species of atom

nuclide

however ,

without having reference to any particular nuclear property . 1.3

Distribution of Nuclides

Over 1300 different nuclei are presently known

,

of which only about

one - fifth are stable , the rest being unstable and decaying spontaneously

with the accompanying emission of particles or radiation . Unstable nuclides occur both naturally and can also be obtained artificially by bombarding stable nuclei with high - energy particles .

3

The Atomic Nucleus

If

we include artificially

elements , a total of 102 different known , with mass numbers ranging from 1 to about 260 , with only the mass number A = 5 missing in an otherwise continuous chain of mass numbers . elements from

Z

= 1 to

Z =

created

102 are

All nuclides beyond bismuth (Z = 83 ) are unstable and disintegrate spontaneously with the emission of y -rays or energetic particles . By far the largest number of stable nuclei ( ~ 60 % ) have an even number of protons and an even number of neutrons . These are the even -even nuclei . The odd - even and even - odd nuclei , i.e. , those having an odd number of protons and an even number of neutrons and vice versa , make up about

20 % each of the total number of stable nuclei . Only four stable nuclei fall into the odd -odd category , namely , 1 H² , Li6 , B10 , 7N14 . ( The odd - odd nuclei 19K40 , Lu176 , 23V50 , 57La138 occur in minute quantities in nature , but all are radioactive .) These facts seem to indicate that the even -even nuclei represent a more stable nuclear arrangement than do the odd - odd or even - odd arrangements . The reason for this preference is not yet fully understood , although it can be made plausible on the basis

The neutron in

of the

general

properties of nuclear forces .

the free state , when it is not bound inside a nucleus , is

itself an unstable particle which decays spontaneously into electron after an average lifetime of about 12 min . n

a

proton and an

→p + e

When contained inside a nucleus , however , the neutron behaves as an indivisible fundamental particle and should then not be looked upon as a particle having a composite structure . Certain nuclei having what is called a “ magic ” number of neutrons are especially stable . These numbers are 2 , 8 , 20 , 28 , 50 , 82 , and 126. The nuclei 36 Kr86 and 54Xe136 are examples of this stability . The magic numbers of neutrons or protons correspond to the formation of closed shells in nuclei and can be pictured to be somewhat similar to the electron shells in the atomic structure of the elements . The most stable of all nuclei are the

"doubly

magic " nuclei which have a magic number of protons as well as a magic number of neutrons . Familiar examples of doubly magic nuclei are 2He4 , 8016 , 20Ca40 , and 82Pb208 . ( For additional information see M. G. Mayer and J. H. D. Jensen , Elementary Theory of Nuclear Shell Structure ,

John Wiley & Sons ,

1955. )

1.4

Nuclear Stability

A nucleus can be considered stable if it does not undergo transformations spontaneously , i.e. , of its own accord and without the addition of outside.

4

Elementary Introduction to Nuclear Reactor Physics

energy , such as by bombardment with high - energy nuclear projectiles or irradiation with y- rays . It is instructive to plot the known stable nuclei graphically in such a way that the neutron number N = A >> Z appears along the y -axis and the

proton number Z along the x -axis ( Fig . 1.1 ) . Such a graph shows that for stable nuclei the number of neutrons and the number of protons tend to be equal in the light nuclei , with a gradual increase in the number of neutrons over that of protons appearing as the atomic number of the nuclides goes up . The graph also shows clearly that all stable nuclei fall within a narrow band which encloses the so - called line of stability . The ratio of N/Z starts with the value 1 for H2 and gradually increases to the maximum value of

for bismuth - 209 , the heaviest stable nuclide . plotted All unstable nuclei when on the same diagram will be located stability region outside the . Those lying above and to the left of the 1.52

stability zone are in the neutron excess region and will thus show a tendency to reduce their N/ Z ratio . This can be achieved either ( 1) by ẞ--decay or (2 ) by the much rarer event of neutron emission . its

1. A ẞ -decay is equivalent to the transformation of a neutron into a proton with the simultaneous creation of an electron and ejection from the nucleus

to

carbon

-

emission

14

one has the decay

--

ẞ

a

As an example of

of

.

pte

n

nitrogen-

6C14

₂N14

+

→

:

14

−1e0

is ,

.

on¹

the right of the stability zone They will tend ( 1 ) ẞ +

-emission

capture

).

K -

or

(

2 )

(

or

)

(

as

.

or

excess protons

Z

N / of

a

lie in

neutron deficiency increase their ratio by such spontaneous processes positron emission electron capture orbital region

to

are located below and

to

Kr³6

+

86

36

36

-87

Kr

All nuclei which

a

is

86 , a

a

in

-

87

a

of

an

.

,

a

A

2.

highly excited nucleus and neutron emission occurs only from example under ordinary circumstances very infrequent As neutron emission one has the transformation of Krypton into the preceding stable isotope Krypton reaction which associated with significant number of cases nuclear fission

.

a

,

a

e +

+

→

n

a

of

a

of

by the transformation

a

of

is

,

+

ẞ

-emission which

p

is

1.

positron from the emission nucleus proton into neutron inside the positive electron nucleus with the simultaneous creation

The

preceded

5

The Atomic Nucleus

An example of this process stable carbon - 12 :

is the transformation

N126C12

of nitrogen - 12 into

+ 1e0

of

a

to

.

a

.

its

2. The alternative possibility for reducing the excessive number of protons in a nucleus exists in the capture of an extra - nuclear electron by the nucleus , thereby bringing about a reduction in net positive charge equivalent proton process This would be the fusion and an electron to form neutron

P

+e to

.

K -

in

,

K -

,

is

,

,

to

The electrons that are most likely be captured are those closest the nucleus of the atom i.e. the two electrons the shell of the atom commonly called capture Because of this orbital electron capture

is

.

-

is

.

-

K -

.


0 , the reaction is exoergic , and the liberated energy will appear in the form of kinetic energy of the reaction products .

48

Elementary Introduction to Nuclear Reactor Physics

If

Q < O , the reaction is endoergic , and it can proceed only if sufficient kinetic energy is contributed by the initial particles . It follows from Eq . 2.25 that during a nuclear reaction some conversion of rest mass energy into an equivalent amount of kinetic energy , for Q > 0 , takes place , and a similar conversion of kinetic energy into rest mass energy occurs for Q < 0 , by Eq . 2.26.

2.11

In

Nuclear Masses and Isotopic Masses

numerical calculations

,

it must be remembered that the isotopic of nuclei but those of the neutral atoms .

masses listed in tables are not those

In

order to obtain the corresponding nuclear masses we would have to subtract the mass of the appropriate number of orbital electrons from the listed values .

This procedure , however , is

in nuclear reaction calculations since the electron masses are exactly balanced in these reactions . The number of electron masses that would have to be added , in order to unnecessary

convert the nuclear masses to atomic masses in Eq . 2.17 are Z₁ + Z₂2 on the left - hand side and Z3 + Z4 on the right - hand side of the equation . By the charge- conservation condition 2.18 these additions are equal , so that when we take the mass differences the electron masses would cancel out . Consider

,

as an example, the reaction

, N¹4

+ ₂He¹

= 80¹7

+ ₁H¹ + Q

Here , the number of orbital electrons involved on the left-hand side is given by the prefixes 7 and 2 and their sum is exactly equal to those on the right -hand side , namely , 8 and 1. The omission or inclusion of 9 electron masses on both sides of the equation does not affect the Q - value of the reaction . This justifies our working with the isotopic masses as listed instead of using nuclear masses which are not so conveniently tabulated . One exception to this general rule , however , must be mentioned here . It occurs when the reaction involves a positron emission when the positive charge is not associated with a nucleonic mass .

-

For example, 8015

=

N15 + 1e⁰

+Q

(nuclear masses )

If

This equality is correct if we use nuclear masses . we were to use the atomic masses instead , this would increase the left - hand side by 8 electron masses and that of the right - hand side by only 7. To put matters right , if we want to use isotopic masses instead of the original nuclear masses , we

49

Nuclear Forces and Nuclear Binding must add an extra electron mass on the right - hand side

of the equation .

Thus , 8015 = „7N¹5 + ₁eº + Q +

=

N15 + 2m

(atomic

me

masses )

+ Q

Example 2.6 . Calculate the Q - values for the fusion reactions : (a) H²(d , n) He³ ; (b) H²(d , p ) H³ ; (c ) H³(d , n ) He¹ . H2- mass = 2.014741 He³-mass = 3.016977 (a) Therefore Σm ; = 2 x 2.014741 mn = 1.008987 = 4.029482

е

therefore

Σmi =

(b)

therefore Σm , = 4.025964 Σm = 0.003518 amu = 3.27 Mev

-

= Σmi

,

H¹ - mass = 1.008145 H³- mass = 3.016997

4.029482 as (a )

,

therefore Σm = 4.025142 4.03 Mev Σm, = 0.004340 amu

е =3 Σmi

therefore

(c)

He¹ - mass = 4.003879 mn = 1.008987

H³-mass

= 3.016997 H2 - mass = 2.014741

therefore Σm ; = 5.031738

Q

therefore

therefore Σm

,

= 5.012866

- Σm , = 0.018872 amu

= Σm ;

= 17.5 Mev

Example 2.7 .

of this

U238 emits a- particles of 4.159 Mev energy . Find the Q -value nuclear transformation .

In the

decay process U238

=

He + Q

Th234 +

the a- particle will carry off most of the available disintegration energy , although not all of it , since some of it will appear as recoil energy of the Th234 nucleus . Q will , therefore , be somewhat greater than the kinetic energy of the x - particle . We apply the conditions of conservation of energy and momentum to the motion of the product nuclei , using the suffix r for the recoil nucleus , Th234 . = 0 conservation of momentum Pa + Pr

Er

-Q

(pr)2 E, = , 2m

Hence

E, Therefore

-

2

= Px² 2ma

е Q = E

+

-4.159

Ma xx - EEx mr ma

=

mr

Ex ; E mr

(

1

+

ma

-EΕπ(1+ ) =

mr

)

also

Ea +

244

By Eq . 2.26 ,

= 4.267 Mey

50 50

Elementary Introduction to Nuclear Reactor Physics

Threshold Energy for Nuclear Reactions

2.12

by

to

Q.

,

+

,

0 ,




-

Z

Therefore

be

1M4

By adding electron masses into isotopic masses and the

zM4

2M4-4

possible

+

> z

zM¹

to

decay

--

a ẞ

,

for

zM¹

to

Qa

z

zMA

so that

x -

>

,

,

follows

Similarly

be

,

its

,

Q -

shown the

It

the parent The difference mass we have value for the decay process therefore that for an process be possible we must have

less than that

=

is

which

mass

by

spontaneous nature must an exoergic process the course parent nucleus decays into daughter products whose combined

of

of

virtue of

a

It

Be8 →→→

Nuclear Forces and Nuclear Binding

55

(

convert

2e

)

2.39

2m¸

(

–

)

(

-1₁Mª

-

zM¹

atomic masses

z

+

=

QB

> z -

₁Mª

—

zM¹ Therefore

+

,

order

to

)

to

Z

to

Again adding electron masses both sides isotopic masses this condition becomes ,

.

,

nuclear masses in

+

1M4

+ e

> -z

zM¹

,

be

to

+

a ẞ

Z -

its

This shows that a ẞ-- decay is energetically possible if the isotopic mass of an atom is greater than that of isobar with the next higher number -decay energetically possible we must have For

,

at

be

to

,

is ,

nuclear masses

of

the atomic binding energy

)

(

be

,

EK

+

+

1M4

in

to

K -

in

K -

,

of

-

Ek

-

)

2.40

(

—

₁M¹

)

(

atomic masses

z -

zMA

K Ek

at

Z

1

if

a

is

K -

capture energetically possible the isotopic mass by atom exceeds the mass of its isobar of atomic number

EK

Example 2.9

.

of

Examine the possibilities Be and Li² The isotopic masses of these nuclei are

isobaric transitions

between the

Li

7.01822 amu

Be7

7.01916 amu possible

This

is

0.874 Mev

.

0.00094 amu

of

The mass difference for the two nuclei

is

.

--

decay

is

?

.

no

ẞ

Be

',


z

1

Z

-

to

K -

of

,

is

the electron the shell provided detach order the energy the parent atom which from the shell the parent atom Adding electron masses both sides this inequality we get

it of

where

E

zMA

energetically possible the condition

> z -

to

apture

e

a

For

K -c

.

1

Z

-

at

a ẞ +

It

-decay energetically follows from this relation that for possible the isotopic mass of the nuclide must exceed that of its isobar of by atomic number least two electron masses i.e. by least 1.022 Mev

apture

→ Li

?

53 days

.

53 .

of

a

K -c

-

is

to

4Be7

K -

.

is

of

.

is

,

K -

is

,

so ,

ẞ +

-

,

in

is

1.022 Mev two electron masses which less than the energy equivalent -decay possible Hence no The atomic binding energy the electron considerably less than the available energy difference the Be¹ atom however capture energetically possible of 0.874 Mev that days This transition known occur with half life

Elementary Introduction to Nuclear Reactor Physics

56

2.14

Binding Energy and Q -Value

The Q -value of a reaction can also be obtained in terms of the B.E. of the interacting nuclei . It follows from Eqs . 2.15 and 2.16 that , expressing all masses in energy equivalents , and applying it to the reaction 2.27 m₁ = W — ( B.E. )

= Z₁m

m2

therefore

= Zam ,

+ ( A₁ − Z₁ ) m „ + Z₁m¸ − ( B.E. ) , +

(A₂

-

mz = Zgm , + (A

-

m₁ =

—

Zm

+ (А

Q = (m₁ + m²)

-

Z½) m „ + Zâm¸e − ( B.E. ) ½

-

Z ) m „ + Zâm¸ −

( B.E. ) 3

Z¸) m „ + Zâm¸ − ( B.E. ) Ą

-

− ( m² + m₁ ) = [ ( B.E. ) 3 + ( B.E. ) , ] —

[ (B.E. ) , + (B.E. ) 2] =

Σ (B.E . ) , —- Σ ( B.E .) ;

=

−A( B.E .)

( 2.41 )

Example 2.10 . From the Q - values for the reactions H2 ( d , n ) He³ and H²(d, p) H³ , as calculated in Example 2.6 , find the B.E. of He³ and H³ , respectively . How can the difference in the B.E. be explained ? For the H²(d , n) He³ reaction

ΣB₁ For the H²(d, p )H³

= Q + ΣB; = 3.27 +2 x 2.23 = 7.73 Mev

reaction

ΣΒ,

= Q + ΣΒ; = 4.03 + 2 x 2.23 = 8.49 Mev

-

The difference in the B.E. is 8.49 7.73 0.76 Mev . The smaller B.E. of the He³ nucleus must be attributed to the Coulomb potential between the two protons in the He³ nucleus . To decide this point we can calculate the Coulomb energy directly

:

Ec

-R e2

=

(4.8

× 10-10 )2

erg 1.3 × 10-13 × 31½ -20 23 x 10-2 1.9

× 10

13

× 1.6 × 10-6

Mev

= 0.77

Mev

This is in good agreement with the B.E. difference as calculated before .

2.15

H³, because

Binding Energy of Mirror Nuclei

He³ and Li' , Be are examples of mirror nuclei pairs , so - called one nucleus of each pair can be transformed into the other by

Nuclear Forces and Nuclear Binding

57

merely interchanging the number of protons and neutrons in the nucleus . Further examples of mirror nuclei are : ( C¹¹ , B¹¹ ) , ( N13 , C13 ) , ( O15 , N15) , (F17 , O¹7 ) , (Si29 , P29 ) , (Sc41 , Ca41 ) . No mirror nuclei above the mass number 41 have been observed . The difference in the B.E. of mirror nuclei can be explained as due solely to the difference in their Coulomb energy as was done for the H3 , He³ pair in the last example . With the exception of this pair and the Li7 , Be7 pair (compare Example 2.9) , all mirror nuclei of the higher atomic number decay into the image nucleus of lower atomic number by a ẞ+-decay process .

zM4

2.16

-

Z 1 MA

++ eº

Binding Energy per Nucleon

The B.E. per nucleon , B.E./A , sometimes called the binding fraction , for a given nuclide is obtained by dividing the total B.E. by the number of nucleons contained in that nucleus . With the exception of the very light elements , this quantity varies in a regular manner as shown in Fig . 2.7 . The B.E. per nucleon

varies very slowly as we pass from A = 12 ( carbon ) = 238 (uranium ) and it remains within a range of ~7.5 Mev for carbon , its lowest value , and ~ 8.8 Mev for iron , its highest value . It is larger for the nuclei between A = 40 and A = 100 than for those outside

to A

this interval , which indicates that these intermediate nuclei are more stable than those outside this region , if we apply as our criterion of stability that a greater B.E. is evidence of greater nuclear stability . The most stable nuclide on this basis will be the one for which B.E./A is a maximum . This maximum occurs in the vicinity of A = 58. Those nuclides that are situated in the region of small mass numbers are able to increase the B.E./A ratio by undergoing fusion processes (compare Example 2.6) in which several light nuclei combine to form a heavier nucleus with a higher degree of stability resulting from this transformation . In contrast with this tendency of the light nuclei , those in the region of

A will show a readiness to move into the region of greater stability the center of the curve by some process such as the emission of αparticles ( natural radioactivity ) or by splitting into heavier fragments (fission ) . In both the fission and fusion processes the B.E. per nucleon , B.E./A , is reduced (algebraically ) and an equivalent amount of energy is liberated ,

large

in

which , in the former case , is utilized in nuclear reactors and , in the latter in the H - bomb , if it be permissible to apply the term “ utilization " to

case ,

58

Elementary Introduction to Nuclear Reactor Physics

an , as yet , purely destructive process . Fusion processes are also thought to occur on a large scale on the sun and other similar stars . Fusion reactions could explain the continued production of vast amounts of energy that the sun constantly pours out into space without apparent diminution . * 9

Rulo Sn 119

016

8c12

190 205 TI

235

LU238-

B11Be 9 B10

6

H3 He3

80

60

100

140

120

160

The B.E. per nucleon shown

Semiempirical

as a

2.7

.

.

FIG

Interpretation

180

200

220

260

240

A

Mass number

function

the mass number

.

40

of

1

D2

20

2.17

155

N14

He

7

Binding energy per nucleon BE 4 3 ~

-Mev

)A/ (

S34

Gd

Cr52 Kr84

of the Binding Energy Curve

been described

of

of

of

which have already

,

nuclear forces some

.

from the general properties

of

a

in

A

of

fairly satisfactory explanation the general shape the B.E. curve contributing factors which arise can be given terms of small number

is

is

.

is to )

to

is

of

is

in

M. ).

(

,

Phys 29 547 1957 .,

.

1939

),

434

Rev. Mod

(

55

,

,

Rev.

",

Stars

or ,

.

.

,

Phys

in

"

Carbon Cycle the Elements

of

,

Bethe

nuclear interactions

"

,

,

of the

Synthesis

“

example

al .

For

Burbridge

et

*

-

short range nature

E.

a

its

.

a

to a

of

,

of

.

(

of

the graph Fig 2.7 the essential constancy This behavior be expected from and consistent with the saturation character the nuclear forces which gives binding energy potential due the for each nucleon which rise limited interaction with only few adjacent nucleons This interaction between nucleon and immediate neighbors not influenced by the the presence of other more remote nucleons the nucleus because

The dominant feature of the B.E. per nucleon

Nuclear Forces and Nuclear Binding

59

If the binding

were the result of a simultaneous interaction between all of the nucleons present , one would expect a binding that increased with increasing the number of nucleons . Thus , the binding of a nucleon in a large nucleus where there are many nucleons present should be far larger than the binding of a nucleon in a light nucleus with only a few nucleons

This , however , is not the case . The B.E. per nucleon for all nuclei to a first approximation . For a nucleus of mass number A ,

present .

B.E.

A

=

B.E. =

Therefore

This is the saturation contribution

is the same

constant = a

aA to the B.E.

This preliminary and still rather crude estimate must be modified by considering several significant and obvious corrections , namely ; ( 1) a surface energy correction ; ( 2 ) a Coulomb energy correction ; ( 3 ) an asymmetry correction ; and (4 ) an odd - even (" pairing " ) correction . 1.

The nucleons which are situated at or near the surface of the nucleus

as tightly as are the nucleons well inside the nucleus , they completely since are not surrounded by other nucleons . This effect is quite analogous to the surface tension in liquids and , for this reason , is sometimes referred to as the surface tension contribution . Owing to the

will not be bound

in binding of the surface nucleons , their contribution to the total B.E. will not be equal to the full saturation binding that had originally been assumed to be the same for all nucleons . deficiency

The reduction in the B.E. will be proportional to the number of nucleons that are situated in the nuclear surface region . This number is proportional to the surface area of the nucleus . Since the nuclear radius is proportional to A , the surface area is proportional to A and , hence , the B.E. deficiency will also be proportional to 4 % . The correction to be made is therefore a negative term proportional to

A

, i.e. ,

-bas

In a relative sense , the surface tension effect will be most pronounced for light nuclei . For heavier nuclei it becomes less noticeable , since the ratio of surface nucleons to total number of nucleons decreases with increasing A. For this

reason , one must expect the B.E. for the light elements to

fall

well below the saturation value , as is indeed borne out by Fig . 2.7 . 2. A second contributory cause for a decrease in the B.E. is to be found in the Coulomb interaction between the intranuclear protons . The electrostatic repulsion between the protons must lead to a decrease in the

Elementary Introduction to Nuclear Reactor Physics

60

B.E. of the nucleus since it opposes and , in a sense , counteracts the nuclear binding . we assume a uniform distribution of the protons throughout the nucleus , we can set the electrostatic potential energy proportional to

If

/

/

Z2 R, i.e. , to Z A%. The negative factor can then be written as

-c

contribution to the B.E. due to this Z2

A

This term becomes progressively more important as Z increases . We should therefore get a noticeable falling off from the saturation value as we get to the heavier nuclei . This is , in fact , shown clearly by the B.E. graph

(Fig .

2.7) . The symmetry correction is necessary to take account of the fact that of all possible arrangements of neutrons and protons for a given A , the most stable configuration will be that for which the proton and neutron numbers are equal . Any deviation from this symmetry between protons and neutrons in the nucleus will reduce the stability and , thus , the B.E. of the nucleus . The connection between proton - neutron symmetry and 3.

maximum stability is suggested by experimental evidence that indicates a stronger n -p interaction in a nucleus than either the p -p or n -n interaction . The reduction in the B.E. owing to the asymmetry between neutrons and protons is allowed for by including the term

-d

(N ― Z )² A

-

-

term the pairing term for even -even or odd - odd nuclei only is also required further to improve the agreement with the experimental values of nuclear B.E.'s . This term is 4.

A

purely

empirical

±

(+ for even -even

e

-for odd - odd

nuclei ,

nuclei )

If we include

all contributions that have just been described , the total B.E. for a nucleus ( Z , A) appears finally in this form B.E. = aA

Division by the

bAs -

――

mass number

B.E.

A

= a

-

b

A

с

Z2

A

gives

― c

Z2

- (NZ A d

)2

±

1/ e

(2.42 )

for the B.E. per nucleon

-

d

-

( N − Z) ² A2

±

e

Α

( 2.43 )

Nuclear Forces and Nuclear Binding

61

and the result

represented

;

graphically

volume energy

.

)

(

Saturation

14 13 12 11

( )a

)

()-b c(

4

)

(a

5

9

B.E./A

Mev

) (

saturation energy-

∞

9

Binding energy

=

10

energy contribution

3

Coulomb

1

(b)

2

Surface energy contribution

energy contribution

)

(c

Symmetry

80

100

120

Energy contributions

A

140

160

180

200

220

240

the B.E. per nucleon curve

.

2.8

60

to

FIG

.

40

.

20

260

in

,

Table 2.2

.

in

are tabulated 2.8

Fig

is

:

a

;

d =

c =

;

b

e

:

;

in

The B.E. and the B.E./A are given Mev for the following numerical values of the constants = 13.0 a = 14.0 0.585 19.3 = 33. The numerical values of the corrective terms for few elements

62

Elementary Introduction to Nuclear Reactor Physics

TABLE

2.2

B.E.

(c)

(b)

(a) A127

4.33

1.22

0.03

0

Cu63

3.26

1.96

0.03

Mo98

2.68

2.28

0.39

Pt195

2.22

3.14

0.77

0 -0.03 0

U238

2.09

3.36

0.99

-0.00

† The numbers nucleon .

in parentheses

are the experimentally

A

(d)

8.42 (8.33 ) † 8.75 (8.75) 8.62 (8.63) 7.87 (7.92 ) 7.56 (7.58 )

found values for the B.E. per

PROBLEMS (1 ) By means of Eq .2.1 estimate the density of nuclear matter (2 ) Derive expression 2.10 for G starting with Eq . 2.8 .

in a C¹² nucleus .

(3) Calculate the barrier height for an a - particle inside a 84P0210 nucleus , and compare it with the energy of a - particles emitted by this nucleus in spontaneous α - decay .

(4)

Calculate the distance of closest approach for a 5.298 Mev a- particle to a in terms of the nuclear radius of this

(5 ) nuclei

Calculate the difference in the total binding energy of the two mirror C¹¹ and B¹¹ .

Pb210 nucleus and express this distance nucleus .

(6) (7)

Calculate the separation energy for a neutron in He¹ and in O¹6 .

84P0212 emits x - particles of 8.776 Mev energy . Calculate the disintegration energy that corresponds to it, and compare the x- particle energy inside the Po212 nucleus to the barrier height for the a - particle .

( 8 ) Measurements made on the products of the reaction Li² ( d, α) , He³ have led to an isotopic mass of 5.0137 for the hypothetical nuclide He³ . Show that this nuclear configuration cannot be stable by considering the reaction 2He5

(9) Examine are as given : ( 10 )

system

of the following nuclides

the stability Li5 = 5.0136 ;

Li8 = 8.02502 ;

5B8

, whose isotopic

masses

= 8.0264 .

E of two colliding particles in the center of mass is also called the energy of relative motion . Show that

The kinetic energy

of coordinates

(a) E = (b) EEo

¿He¹ + n .

=

uv2 , where µ =

μv²

+

(m

+

mM/ (m +

M) V² .

M)

is the reduced mass , and that

What is the physical interpretation of the last term in (b) ?

Nuclear Forces and Nuclear Binding

(11 ) Calculate the Q - value 9F19 (n , p ) ¸O¹⁹ , using the isotopic

63

and the threshold energy for the reaction masses for F19 = 19.0044 and O19 = 19.0091 .

( 12 ) Calculate the Q -value for the ẞ +-decay of kinetic energy of the emitted positron .

)

energy for a uniform proton distribution in a

(13 ) The Coulomb potential 3 Z (Z = 1) e² X

nucleus is

N13 and the (maximum

By means of this expression , calculate the difference R 5 in the B.E. of two mirror nuclei and compare it with the result obtained by using the semiempirical formula for the B.E. ·

96

).

.

.

)

(

,

its

isotopic

(

mass of 42Mo⁹6 is 95.9349 . Using the known masses of proton , neutron , and electron , calculate the total B.E. of this isotope . Similarly , find the total B.E's . for two isobars 40Zr96 95.9385 and 44Ru96 95.9379 Compare these values for the B.E's with those obtained from the empirical B.E. relation

(14 ) The

of of

B -d

,

,

of ( B A , 16

of isobaric transformations for the isobaric the binding energy equation 2.42 estimate the Mn53 .

)

( 15

Examine the possibilities

pair Mn5³ Cr5³ and by means ecay decay energy for the

in

a

to (b )

is

?

-

-

ẞ -

.

is

A

if

( )a a

A

if

a

Z ,

as

.

)

By treating Eq 2.42 quadratic show that for constant value single parabola the B.E. equation represents odd and two identical parabolas separated along the energy axis by an amount equal twice the pairing term even What conclusions can be drawn from this result about the respective stability of odd mass number and even mass number nuclei ( Z to

,

( b )

U235

,

)

( a

added

to

is

a

neutron

Pu239

.

( c )

the nearest

.

=

)

A

a

=

) )

(

, 18

Calculate the energy liberated when

U238 and

find the value

Z to

17

Z

(

By differentiating the B.E. equation 2.42 with respect Zo which makes the B.E. maximum and evaluate integer for 27

of

BIBLIOGRAPHY .

1955

No.

9 (S

, "

, 5,

,

,

:

of

.:

,

Elementary Nuclear Theory Wiley Morrison Fundamentals Nuclear Physics Nucleonics

P.

,

and

U

E.

,

Condon

"

H. A.

Bethe

eptember

).

1947

.

.

,

.

,

:

L.

,

,

.

,

-

,

.:

P

C.

,

, G. ,

,

,

.:

E

,

.:

D

The Atomic Nucleus McGraw Hill 1955 Fermi Nuclear Physics University of Chicago Press 1950 Principles of Modern Physics Wiley 1958 French A. Gamov and Critchfield Theory of the Atomic Nucleus and Nuclear Energy Sources Clarendon Press 1949 ,

Evans R.

.

).

(

.

,

,

.

,

,

.

-W

,

,

-

,

:

,

I

,

,

.: D .: E.

,

,

S S. , .:

Green A. E. Nuclear Physics McGraw Hill 1955 Green A. and N. A. Engler Phys Rev. 91 40 1953 Halliday Introductory Nuclear Physics Wiley 1955 Kaplan esley Nuclear Physics Addison 1955

64

Elementary Introduction to Nuclear Reactor Physics

Lapp , R. E. , and H. L. Andrews : Nuclear Radiation Physics, Prentice - Hall , 1954 . Mansfield , W. K .: Elementary Nuclear Physics , Temple Press , 1958 . Perlmann , I. , and F. Asaro : " Alpha Radioactivity , ” Ann . Rev. Nucl . Sci . , 4 ( 1954) . Richtmyer , F. K. , E. H. Kennard , and T. Lauritsen : Introduction to Modern Physics , McGraw - Hill , 1955.

H.: Introduction to Atomic and Nuclear Physics , Rinehart , 1954 . Strominger , D. , J. M. Hollander , and G. T. Seaborg : " Table of Isotopes , " Rev. Mod. Phys . , 30 , No. 2 , Part 2 (April 1958) . Semat ,

chapter

3

The Compound Nucleus and Nuclear Reactions

Introduction

3.1

The theory of nuclear binding presented so far has made use of some of the familiar properties of a liquid drop . The "liquid drop model " of the nucleus is expanded still further in this chapter and is applied to the theory of nuclear reactions which can be explained in terms of the compound nucleus picture . The energy levels of nuclei and the nuclear resonances of the compound nucleus are described , and the relation between level width and half-life is derived .

3.2

The Liquid Drop Model Analogy

The analogy between the behavior of the molecules of a liquid and that of the nucleons inside a nucleus can be extended to lead to a clearer picture

of nuclear

reactions in general

.

We know from the kinetic theory of liquids that the molecules participate in a random thermal motion . The exact counterpart of this behavior is found in the movement of the nucleons inside a nucleus . We have seen that these nucleons have a kinetic energy of about 20 to 25 Mev and their kinetic energy is analogous to the thermal energy of the molecules of a liquid . The principle of equipartition of energy applies within the nucleus and any excess energy contributed by a particle which has just entered the nucleus from the outside is shared by all the nucleons . The rapid sharing of added energy can readily be understood on the basis of the strong short - range forces that are acting between the nucleons . 65

Elementary Introduction to Nuclear Reactor Physics

66

They will

effect a quick distribution

of any excitation energy which has amongst been introduced into the nucleus all the constituents of the nucleus . By way of analogy , this rapid sharing of energy by a large number of nucleons is referred to as " heating of the nucleus . ” The sharing of excitation

energy is a random process and there is always probability a definite that after some time sufficient energy may be concentrated on a single nucleon , which would then be able to escape from the nucleus . The minimum energy required for this to happen would

be equal to the B.E. of this particular nucleon in the parent nucleus , i.e. , somewhere between 2 to 8 Mev , depending on the particular nucleus . This process , again in analogy with the corresponding event in liquids , is referred to as " boiling off " of a nucleon . The liquid drop picture of the nucleus offers a very satisfactory physical description of nuclear properties and it is the basis of a general theory put

forward by Bohr to explain some important general features of nuclear reactions . This theory is known as the compound or intermediate nucleus theory .

An explanation of the fission process in terms of the liquid drop model of the nucleus can be given and is presented in outline in a later chapter . 3.3

Some Features of Nuclear Reactions

When a specific target material is bombarded with a single type of particles , more than just one type of nuclear reaction may be observed to occur . For example , the bombardment of 11Na23 with a - particles might lead to any one or several of these transformations :

11Na23

+ He¹

13A127

+y

13A126

+ n¹

12Mg26

+ ₁H¹

12Mg25

+ 1H²

11Na23

+

He

The same end products might also be obtained by bombarding 12Mg25 with deuterons , or by bombarding12Mg26 with protons . We can summarize these facts schematically as is shown on page 67 .

,

.

-

in

of

all

This shows that any one of the interactions in the left - hand column might lead to any or the right hand column the reactions appearing Of course not all the possible transmutations will occur with equal

The Compound Nucleus and Nuclear Reactions

13A127

+γ

12Mg26

+ 1H¹

12Mg25

+ 1H²

11Na23

+ ₂He¹

67

13A127

+Y

13A126

+ on¹ + ₁H¹

12Mg26

12Mg25 + 1H²

+ 2He

11Na23

probability or frequency . The yield of the reaction , which is a measure of the number of transmutations that have taken place during the bombardment , will depend on the energy of the bombarding particles . The yield for each possible reaction will show an energy dependence that is characteristic

for the particular reaction and that transformations

is different

for the various possible

.

600 Sb116 500

arbitrary units

)

400

Sb 118 300

200

100

10

5

Cross section

(

Sb 117

15

25

20

35

30

of

)

(

Sb¹¹8

b )

a -p

.

: ( a )

In¹¹5

( α, n )

(

)

3n )

) ( x,

);

Sb¹19

G. M.

)

(

of

the competition

).

1949

consequence

the compound nucleus

.

of

is a

of

the curves

de excitation -

of (

424

(c

(

(

( x,

76 ,

,

.

.

Phys Rev.

modes

reactions

articles with The three curves

Sb116.

the cross section

the different

Temmer

)b 3n ) ( x,

x, n ) ;

The maximum between

In¹¹5

yield curves for the interaction the compound nucleus 5Sb119

in

2n Sb¹¹7 )

( α,

In¹¹5

( to )a (

Excitation functions the formation ( c )

3.1

.

.

FIG

In¹¹5 leading represent the

of or a 2n -

particle energy Mev

slowly .

(

Fig

). as

,

a

is

at

to

,

in

as

,

In

.

is

it

A

particular process often predominates different energy regions and possible for several nuclear reactions occur simultaneously particle energies general energy certain suitable the increased the yield from one process will go through maximum and then will decrease 3.1

another transmutation

reaction

becomes

more prominent

68

Elementary Introduction to Nuclear Reactor Physics

It follows , therefore , that there must be some competition between the various possible modes of transformations . Increase in one mode must imply a consequent decrease in another or several other modes . Further-

emitted particles

more , each mode of transformation has its characteristic threshold energy , with the yield for the particular mode becoming measurable only when the energy of the bombarding particles exceeds this threshold energy .

Resonances and resonance energies

Yield

of

E

nuclear

reaction

yield curve

.

resonances

a

of

representation

in

schematic

bombarding nuclear particles

of

3.2

.

.

FIG

A

Energy

3.4

of

is

.

which the various resonances appear

).

.

(

,

called resonance and the energies are the resonance energies Fig 3.2

at

is is

of

,

or

in

of

is

of

Another important feature nuclear reactions the appearance distinct peaks maxima the yield the reaction for certain discrete values of the bombarding energy which observed when the experiment projectile energies This phenomenon repeated over an extended range

The Compound Nucleus

,

.

b .

B

of

an intermediate

or

+ b

B

the formation

of

+

a

A stage consists

in

The first

a

A

being bombarded with stream of particles Consider the target nuclei particles and this resulting the production and in

a,

in

a as

of

(1 )

(

)

2

:

a

,

.

be

The various experimental results just described can understood and explained along lines that have been suggested by Bohr According to taking place these nuclear reaction should be looked upon two compound nucleus distinct and independent stages formation disruption of the compound nucleus and

compound

The Compound Nucleus

69

and Nuclear Reactions

nucleus C through the capture and assimilation of the bombarding particle a by the target nucleus A.

A+

C

a

stage , completely independent of the first , is the disintegration compound of the nucleus C into the residual nucleus B and the ejected particle b.

The second

Bb

C

stages are assumed to be independent of each other , the disintegration of the compound nucleus will be completely independent of the particular reaction mode that may have led to its formation . The compound nucleus , once having been formed , can decay in several ways . we look again at the reactions shown diagrammatically at the beginning of section 3.3 , we see that we can describe the separate stages

As the two

If

( 1) and (2) now 1.

Formation

as

of the

compound nucleus

+y

13A127 12Mg26

and 2. Disintegration

+ H¹

12Mg25

+ 1H²

11Na23

+ 2He¹

of the compound

13A127

13A127 *

*

.

13A127

nucleus 13A127 *

13A127

+ y

13A126

+

on¹

12Mg26 + 1H¹ 12Mg25 + 11Na23

H2

+ ₂He¹

The asterisk used in the symbol for the compound nucleus is intended

to remind

us that this nucleus has absorbed more energy than it would normally have . It is , therefore , in an excited state , which accounts for its instability and subsequent decay . The excess energy , or excitation energy , is very nearly equal to the sum of the kinetic energy of the incident particle

whose absorption brought about the formation of the compound nucleus and its B.E. in the compound nucleus .

Elementary Introduction to Nuclear Reactor Physics

70

Example

3.1 .

When

F19 is bombarded

a (p , n ) reaction

with protons

with

subsequent a - emission occurs .

H¹ +

9F19

10Ne¹⁹ + on¹

10Ne20 *

One of several resonances for this reaction occurs with a proton energy of 4.99 Mev . Calculate the excitation energy of the compound nucleus that corresponds to this resonance . Since we are interested only in the internal energy gain of the compound nucleus the calculation will be simplest if we work in the center - of -mass system . Thus , by Eq . 2.34 , the convertible energy of the protons , or the energy of relative motion is

E=

Eo 1 + m/

M

=

4.99 1 + 1/19

= 4.74 Mev

To this contribution , by virtue of the kinetic energy of the incident proton , must be added the B.E. contribution of the absorbed proton in the compound nucleus . This can be obtained from the isotopic masses 10Ne20

= 19.004444 ₁H¹ = 1.008146

= 19.998772 = Σm,

-

Ση ;

9F19

therefore Σm ; = 20.012590

Σm, = 0.013818 amu =

12.88

Mev

The excitation of the compound nucleus therefore is Eexc

= 4.74

3.5

Energy Levels of Nuclei

+ 12.88 = 17.62 Mev

its

The idea of the intermediate compound nucleus was originally put forward in order to explain the phenomenon of resonances in nuclear reactions . Such resonances in the yield curve of a nuclear transmutation will occur whenever the energy of the incident particle together with in to

to

is

)

~ 8

(

tationary state

is

to

.

of

state

is

the nucleus the lowest energy level or the nucleus that corresponds this state the B.E. as calculated from the mass defect of the nucleus For excitation .

The most stable energy ground state The energy

of

.

quasi

-s

a

be

in

is a

of

).

.

(

is

of in

equal the compound nucleus and coincides with an energy that nucleus Fig 3.3 An energy level that corresponds an excitation energy greater than the B.E. Mev the particle the compound nucleus virtual energy level and the nucleus then said to B.E.

level

,

less than the B.E. 3.5

).

Fig

.

energy levels

(

energies

the corresponding energy levels are bound

The Compound Nucleus

71

and Nuclear Reactions

A nucleus can exist in a higher bound energy state for a certain length of time , but it will remain in this excited state for only a short period before it returns to its ground state . It has a definite lifetime in the excited state before making a transition back to its ground state . This it can do by emitting the excess energy in the form of a y - ray . Evidence for the existence of discrete energy levels in nuclei comes from the emission of discrete energy groups of a - particles and from y - rays emitted by radioactive substances . In general , these yield information about the low - lying or bound energy levels .

.

Ec

Compound nucleus energy level Ec

13

0

Binding energy particle compound nucleus B.E.

Energy incident particle

of

of

of

the kinetic energy

to

is

E,

EE

.

+

,

in

its

compound nucleus equal the sum compound nucleus and B.E. B.E

E,

.

.

FIG 3.3 Energy level of bombarding particle

in

in

) a -

to

-

,

(

,

C '

C '

of

,

,

particles The very rare and exceptionally long ranged i.e. high energy that are emitted for example by Ra and Th have been shown be

.

of

a

of

.

in

,

x -

.

of

,

energies are observed

,

( x, p )

in

certain reactions proton groups different group being with the energies for the protons the same for all particles that group Discrete energy values are particles observed amongst the ejected protons although monoenergetic are being used the bombardment of the target ,

For example

a

in

of

of

of

.

emitted from the excited energy levels these nuclei Further evidence for the existence excitation energy levels comes from the appearance distinct energy groups amongst the product nuclei when monoenergetic particles are used given target the bombardment

of .

to

of

in

.

in

is

a

of

,

if

be

These experimental results can understood one attributes each corresponding energy level their origin the nuclei associated with the formation the group energies Thus the largest the residual nucleus its lowest or ground state The smaller energy values energy group

72

Elementary Introduction to Nuclear Reactor Physics

of the groups are associated with residual nuclei which have been left in a higher , excited energy state . Consequently , the energy differences between the various groups can be related to the respective energy level spacings in the residual nucleus . The energy level spacings are in fact equal to the difference in the Q - values for

the respective proton groups . Highest energy group of

Eo

emitted particles ; no accompanying Y- rays .

E1 E2 E3

Outgoing particle groups of lower energies with excess energy emitted as y- rays .

Y2

Compound nucleus

FIG .

Residual nucleus

3.4 . The diagram illustrates the relation between the energy levels of the residual and the energies of the observed outgoing particle groups .

nucleus

This conclusion is supported by the fact that the lower energy groups are accompanied by simultaneous y - ray emission , whereas the highest energy group is not accompanied by any y -emission from the residual nucleus.

( Fig . 3.4 ) . The experimental evidence here presented can be summed up by saying that ( 1 ) the observed energy groups which result from the bombardment of a target with monoenergetic projectiles yield information about the energy levels in the residual or product nucleus , whereas ( 2 ) the resonances which appear in the yield curves when the energy of the projectile is varied give us information about the virtual energy levels of the compound nucleus .

The relationship

of

a

.

a

to

of

its

between the various energy levels and energy regions schematically is shown in Fig . 3.5 for a typical nucleus . ground configuration In state all levels the nucleus are occupied by up the nucleons definite level Above this level there exist series

of

.

of

is

higher levels which can become occupied when the nucleus excited by the energy addition from outside Above the region excited energy states

73

and Nuclear Reactions

the virtual

quasi stationary levels which are no longer energy levels that can be -

or

the region

of

is

The Compound Nucleus

This region contains those

.

bound energy states

.

Virtual levels and adjacent continuum

,

of

to

is of

.

as

a a

in

an interoccupied during nuclear reaction while the nucleus exists compound nucleus Since each virtual energy level mediate state corresponds the excitation energy one of many possible distributions very large amongst the nucleons the number virtual levels

Separation energy of nucleon

Bound levels

filled levels Unfilled or excited levels

S

nucleus

.

Energy level regions

a

3.5

.

.

FIG

of

Region

of

10

.

†

as

/

1

of

)

(

A

~

in

of

The spacings between successive energy levels the case intermediate nuclei between ~ 100 and 150 are the order of ~ Mev near the ground level and gradually decrease with increasing energy we move up from the ground state to

a

evaporated

"

emitted

or "

a

.

be

or -y

as

dozen neutrons can

.

as a

,

many nucleus even from the nucleus

,

in

a

de -

,

-y

of

,

-

,

De excitation of bound energy levels i.e. return the ground state rays from excited bound energy levels can take place only by emission variety of whereas excitation from virtual energy level can occur ways either by particle emission very highly excited emission For

rule

,

"

.

"

,

is

of

general

,

a

†

the level spacing decreases with increasing atomic mass number the exception the magic number nuclei where the level spacing very wide and very much like that for the light nuclei

As

of the nuclei with

74

Elementary Introduction to Nuclear Reactor Physics

The radiative capture of slow neutrons is an example of the formation of a compound nucleus with subsequent de - excitation by y -emission . This capture of slow neutrons plays an important part in nuclear reactor processes . It occurs mainly for neutron energies below 1000 ev and the following reaction is a typical example . resonance

n¹ +

U239*

U238

U239

+7

Example 3.2 . In the B¹º( α , p ) C¹³ reaction with 4.77 Mev a - particles the two most energetic proton groups which were observed to be emitted at an angle of 90 ° with the direction of the incident a - particle beam had energies of 6.84 Mev and 3.98 Mev , respectively . What information can be gained from these results about the energy levels of the residual C¹³ nucleus ? We first calculate the Q -values for the two proton groups of the given reaction 2He¹ + 5B10

1H¹ + 6C13

If

we use suffixes 0 , 1 , 2 to denote the kinetic energy and momentum of αparticle , proton , and C¹³ nucleus , respectively , we can apply the laws of conservation diagram

of energy and in Fig . 3.6 .

to the reaction that is represented by the

momentum

P1

Po

P2 FIG .

P₁ = P₂ sin 0

Po =P₂ cos 0 Therefore

P²²

=

Therefore

2m¿½

=

P₁²

+

2m₁₁

Po² + 2m Eo

mi E2 = E₁ + mo Eo

Therefore

Therefore

3.6 .

m2

Q

=

ΣE,

-

ΣΕ; ΣE

m2

-

= 13Е₁

+ 13E0

= E₁ +

E₂

− E Eo =

13E1-13E0

This shows that the Q - value is different for each proton group energy E₁ .

The Compound Nucleus

75

and Nuclear Reactions

The difference in the Q -value for the two proton groups is the excitation energy nucleus and it represents the energy difference between the ground

of the C¹³

state and the first excited state

Q

=

(13E₁

of the

– 13Eŋ ) E , =6.84 -−

== 13 (6.84

Hence

(13E ) — 13E9 ) E₁ =3.98

× 2.86

— 3.98 ) = 1

Thus , the first excited state of the

3.6

C13 nucleus .

= 3.08 Mev

C13 nucleus is 3.08

Mev above the ground state .

Level Widths and De - excitation

A nucleus

left in an excited state is unstable and so will remain in this only state for a finite time interval . The excited nucleus has a definite mean life before it decays by one of the possible modes of de - excitation . The relation between the probability per second of its de - excitation and the mean lifetime is the same as was found for radioactive decay, namely , the mean lifetime 7 is the reciprocal of the decay probability per second

T=

1

( 3.1 )

2

A

new quantity , г , which is called the level width of the excited energy level , is defined in terms of 2 by

h г= 2.

( 3.2 )

2π

so that one also has

h 2πτ

=г

(3.3 )

This relationship between the level width П and the mean lifetime is a method of expressing the Heisenberg uncertainty relationship in a suitable manner so that it can be applied to the de - excitation process . we equate the uncertainty in the time - measurement At with the mean lifetime of the excited state , we obtain for the uncertainty in the energy of the excited state AE,

If

ΔΕ = This shows that , what we

h 2π Δι

=

h 2πτ

=г

(3.4 )

have called the level width of the excited state , is simply a small spread in the energy of the excited level which appears because of the uncertainty in the exact energy of the level . Since the level width T is proportional to the decay probability and inversely proportional to the mean lifetime 7 , it is clear that a long lifetime

76

Elementary Introduction to Nuclear Reactor Physics

means a very fine and narrow energy level , whereas a short lifetime will be associated with a broad or diffuse energy level of larger level width ( Fig . 3.7 ) . For each individual mode of decay we can , similarly , define a partial

I

of the

I'

=

each particular mode

for

mean lifetime

h

of

(3.5 )

2πT ' Energy

level width ' in terms de - excitation

Narrow

Broad level

( level

Cross section

(a )

(b )

Some and the corresponding they might appear from cross section measurements are shown level widths reflects the difference the lifetimes of the nuclear energy .

and level widths

in ( )b .

in

The difference

in

level widths

-

as

( a )

in

.

.

FIG 3.7 Schematic representation of nuclear energy levels of the energy levels of the compound nucleus are shown

the sum of the

)

3.6

(

"

+

+

2

+

+

2 "

2 '

+

'

T "

,

de -

of

λ

individual

excitation must be equal partial decay probabilities therefore

=

or

-

The over all probability

to

.

levels

+

(

)

··

3.7

·

г ″

=

T

Hence

of

.

of

to

of

a

is

in

.

,

..

г ”

,

,

г "

г '

is

I

equal which shows that the total width the sum the partial widths length and measure the average time that the nucleus remains an excited state before decaying

The Compound Nucleus The total level width

77

and Nuclear Reactions

I

can be obtained experimentally . Experimental compound nucleus are a representative

of the average lifetime of found to be ~ 10-14 sec . values

If

this time interval is compared to the natural nuclear time , which is the time required for a nuclear projectile to traverse a target nucleus , it is found that the mean life of the compound nucleus is considerably longer than the natural time . To get an estimate , if we take a nuclear diameter of ~ 10-12 cm and a velocity of ~ 109 cm per sec for the absorbed projectile , we obtain a natural nuclear time of ~ 10-12 / 109 sec , i.e. , ~ 10-21 sec . Comparing this time with the experimental value for the mean lifetime we see that it is ~ 10-14 / 10-21 = 107 times as long as the natural nuclear time . The lifetime of the compound nucleus is , therefore , long enough to allow

of

is

.

of

a

of

.

its

numerous collisions between the incident particle and the nucleons inside the nucleus . A kind of statistical equilibrium can readily be established during the lifetime of the compound nucleus so that by the time it has reached the end of its natural life span , it has " forgotten " the mode of its formation and the original cause of formation again energy The breakup occurs when sufficient amount concentrated on one single nuclear subparticle Each one the several ,

is

a

in

,

a

.

to

of

a

breakup will happen with modes definite probability and these individual probabilities are the same for all modes formation leading the same compound nucleus As rule emission of radiation nuclear interaction much less a

,

so

a

of

.

a

-ry

in

,

is ,

of

a

.

,

-ry

,

in a

is ,

-y a to

of

likely than the formation new product nuclei that reaction leading the emergence nuclear particles on the whole more probable than ay Emission of reaction which would result the emission of ray other less favored nuclear event than the formation therefore ay reaction products nuclear reactions In fact the probability for emission becomes exceedingly small whenever the expulsion of nuclear

of is

is

,

a

as

,

of

.

particle from the compound nucleus becomes energetically possible However when the excitation energy the compound nucleus only about the same order the separation energy for nucleon which -y

de -

is

,

a

-

)y

(n

is

-

.

,

8

~

5

to

~

ray emission somewhere from Mev Mev excitation by becomes the favored mode since only very little kinetic energy would be left available for the emitted nucleon This circumstance can be used to explain why radiative capture which favored reaction an is

or to

(

of

,

),

of

in

a

of

.

-

reaction with low energy neutrons Assuming that we have energy available which certain amount be carried away by the emitted particle the form kinetic energy radiation the relative probability emission from the compound nucleus .

,

-ry

,

,

a -

,

,

-

or

the level widths would be greatest for neutron emission followed by ay emission particle emission and last proton emission then

Elementary Introduction to Nuclear Reactor Physics

78

Example 3.3 . The B¹º (x , p ) C¹³ reaction shows among others a resonance for an excitation energy of the compound nucleus of 13.23 Mev . The width of this level , as found experimentally , is 130 kev . Calculate the mean life of the nucleus for this excitation . The given reaction leads to the formation of the compound nucleus 5B10 +

· ( N14 ) * . →→→

He¹

( N¹4 ) * .

C¹³ + 1H¹

r = 2πh 1τ therefore T =

h 1 2π

I

=

x

6.62

10-27 erg sec

x

6.28 × 130,000 ev

1.6

x

10-12 erg /ev

== 5 x

10-21 sec

Example 3.4 . The neutron capture of slow neutrons by U235 shows a resonance for an energy of excitation of 0.29 ev . The compound nucleus can become de -excited either by a y -emission or by a fission into larger nuclear fragments . The mean lifetime of the compound nucleus was found to be 4.7 × 10-15 sec = 34 × 10 ³ ev . and the partial width for y -emission г,, Calculate the partial fission width The total width is found from the given lifetime ,

I

r=

h

-

г.

=

2пт

6.62 × 10-27 6.28

x

4.7 × 10-15

erg X

1

1.6 × 10-12 erg /ev

= 140 × 10-3 ev Since therefore

г = =

"

-

= 140 × 10-3 ev

=

3.7

106

x

10

- ³ ev

-

34

x

10-3 ev

Isomers and Isomeric States

The mean lifetime of an excited nuclear energy state before de - excitation by y -emission is generally < 10-13 sec , which is well below the present limits of experimental time measurements . There are , however , a number of known cases where a nucleus can exist in an excited state for a length of time which is sufficiently long to be measurable . Nuclei which can exist in these measurably long states of excitation are called isomers and the excited energy states of long enough lifetimes to be measurable are isomeric states of the nucleus . Isomeric nuclei are otherwise identical nuclei which differ only in the amount of their excitation . The existence of isomeric states was first noticed in the case of bromine , where different nuclear reactions were found to lead to the same radioactive isotope Br80 with , however , different half-lives . The discrepancy in

The Compound Nucleus and Nuclear Reactions

79

the half- lives of Br80 isotopes was explained as owing to the formation of the isotope in its ground state in one reaction , and its formation in an excited state in the other reaction ( Fig . 3.8 ) . Metastable state

Br80 Br80

Ground state

B

β

Kr80 FIG .

3.8 .

state to

Nuclear isomerism . The nucleus ( Br ) can decay directly from metastable or by preliminary transition from metastable state to ground state with

Krº ,

subsequent

decay to

Kr

.

PROBLEMS (1) How long will it take a 1 Mev neutron to cross a U238 nucleus ? (2 ) Enumerate the possible modes of decay of the compound nucleus 10Ne20 . (3 ) Calculate the kinetic energy in the C.M. and L systems of the x - particles which are formed in the (p , a) reaction of 2 Mev protons with Li7. (4) Show that the excitation energy Eexe imparted to a compound nucleus is given by Eexc = (M/M + m) E + B.E. , where m , E, are mass and energy in laboratory system of the incident particle , M the mass of the target nucleus , and B.E. the binding energy of incident particle in the compound nucleus . (5 ) Identify the possible residual nuclei for the compound nucleus Zn65 * . If this compound nucleus is obtained by bombarding Cu63 with 6 Mev deuterons , calculate the excitation

of the

compound

(6) In the Be⁹ (x , p )B12 reaction

nucleus .

Mev a - particles , proton groups of 6.96 Mev , 6.08 Mev , and 5.45 Mev were observed at right angles to the incident x- particles . Calculate the Q -values for these groups and the corresponding energy levels . To what nucleus do these energy levels refer ? , using 21.7

(7) When Co59 is irradiated with neutrons , the radioisotope Co60 is produced . This isotope decays by ẞ - emission of maximum energy 0.31 Mev and two successive y - rays of energies 1.1715 Mev and 1.3316 Mev . What information about the excited energy levels of a certain nucleus can be derived from these experimental data ? Identify the nucleus to which these excited energy levels refer .

(8) The first excited energy level of O¹7 is at 0.87 Mev above ground level . The mean life of this excited state is 2.5 x 10-10 sec . Calculate the width of this energy level .

80

Elementary Introduction to Nuclear Reactor Physics

BIBLIOGRAPHY Devons , S .: Excited States of Nuclei , Cambridge University Press , 1949 . Evans , D. R .: The Atomic Nucleus , McGraw - Hill , 1955 . French , A. P.: Principles of Modern Physics , Wiley , 1958 .

Glasstone , S. , and M. C. Edlund : The Elements of Nuclear Reactor Theory, van Nostrand , 1952. Green , A. E. S.: "Nomogram for Estimating Nuclear Reaction Energies , " Nucleonics , 13 , No. 2 ( February 1955 ) . Green , A. E. S .: Nuclear Physics , McGraw - Hill , 1955 . Halliday , D.: Introductory Nuclear Physics , Wiley , 1955 . Kaplan , I .: Nuclear Physics , Addison - Wesley , 1955 . Mansfield , W. K .: Elementary Nuclear Physics , Temple Press , 1958. Segrè , E. ( Ed .) : Experimental Nuclear Physics , vol . II , part VI , Wiley , 1953 . Semat , H.: Introduction to Atomic and Nuclear Physics , Rinehart , 1954 . Weinberg , A. M. , and E. P. Wigner : The Physical Theory of Neutron Chain Reactors , University of Chicago Press , 1958.

chapter

4 Neutron Reactions

Introduction

4.1

Because of their special importance for reactor physics nuclear reactions initiated by neutrons are more fully described in this chapter . The concept of reaction cross section is introduced and is applied to the calculation of the macroscopic cross section , mean free path , and neutron reaction rates . The energy dependence of the various cross sections is then outlined and the Breit - Wigner formula is explained for simple cases . Finally , the fission cross section is introduced and some general properties of fissionable materials are summarized .

4.2

Slow Neutron Reactions

The most important reactions in reactor physics and in nuclear engineering are nuclear reactions with slow neutrons as these are essential as initiators and perpetuators of nuclear chain reactions in the most common type

of reactors . It

is , therefore , necessary to gain some understanding

of

neutron interactions with other nuclei , in particular with fissionable nuclei , and with matter in bulk . Consider the general type of nuclear reaction

a+

X

b +

Y

where the projectiles a are now assumed to be neutrons with kinetic energy of not more than a few Mev .

If

81

in

to

a

is

it

is

,

)y

(n

1

+

its

1. b is a y -ray this process is a ( n , y ) reaction and it represents the radiative capture of a neutron by the target nucleus with the subsequent emission of a y - ray and the conversion of the target nucleus X4 into isotope X4 The reaction an important factor be considered the design of nuclear reactors in that cause of neutron loss to the neutron

Elementary Introduction to Nuclear Reactor Physics

82

of a reactor and also requires protective measures to shield the experimenter or operator from the injurious effects of the emitted y-radiation . A well - known example of a ( n , y ) reaction is the radiative capture of a economy

neutron by Cd¹¹³ .

n¹ +

→ Cd114 *

Cd113

Cd114

-

If a and

+y

-

b are identical particles in the present case neutrons we process , which may be either elastic or inelastic . scattering the have a target nucleus is raised into an excited state by retaining some of the 2.

If

kinetic energy of the incident neutron , the scattering is inelastic ; otherwise , it is elastic .

If b is not an elementary

particle such as a neutron , proton , deuteron , particle , or a the result of the reaction will be two nuclei of intermediate mass numbers ( assuming of course that X is not an elementary particle ) . This type of nuclear reaction is called a fission process . 3.

In the case of the readily fissionable isotope U235 all of the preceding processes can occur and compete with each other , although the probability for their occurrence is very different . a sample of that isotope is subjected

If

frequency of occurrence for ( 1) a fission process , ( 2) a radiative capture process , and ( 3 ) a scattering process is roughly in the proportion of 60 : 10 : 1. As a rule , several reactions can occur between a neutron and a given nucleus . The probability with which any particular one of the various possible reactions will actually take place depends , however , very strongly on the neutron energy. The probability of a neutron capture as compared to other possible reactions is greatest with slow neutrons , where it is the

to

a slow neutron bombardment , the relative

most common process by far . The radiative capture of neutrons shows pronounced resonances at certain definite energies , and in the neighborhood of resonances г , г, for the heavy nuclei . ( In a few exceptional cases such as Pd108 , Sm152 , W186

In

г N,

r≈ 7

ev , and

0.1

For the light

for Mn and Co , г

>

>

г.)

,, so that neutron scattering is the predominating result in this general energy region . Notable exceptions , however , are the light nuclei He³ , Li , Be7 , B¹º , N14 , where the (n , p ) or

(n , «)

elements ,

n

reactions have the highest probability by far . Thus on¹ + ₂He³

071 + Li6 on¹ +

Be

on¹ + ¿ B¹0

0¹

+

N14

,

→ ₁H¹ + ¸H³ + 0.74 Mev H³ + ₂He¹ + ¿ H¹

4.785

Mev

+ 3¿ Li² + 1.65 Mev

´¿He¹ + 3Li² + 2.791

Mev

H¹ + C¹4 +0.626 Mev

Neutron Reactions

83

Nuclear Reaction Cross Section

4.3

The probability of occurrence of a particular nuclear reaction is described by the effective cross section for that process . Using this terminology , we can state that for slow neutron reactions with U235 , the fission cross section is about six times as great as the radiative capture cross section and about

sixty times as great as the scattering cross section . The probability that a given reaction will occur between one neutron and one nucleus is usually called the microscopic cross section . Specifying the cross section is an alternative method of describing the yield of a reaction . Whereas the latter gives the number of nuclear transformations that take place per specified number of particles that are shot into a thick target , the effective cross section measures the circular area that a target nucleus must have so that each collision within this area produces a reaction . The reaction will certainly occur if the particle passes through the cross - sectional area ; otherwise , no reaction will take place .

We

particular

of nuclear Thus of cross section σ , when referring to a nuclear scattering process , an absorption cross section σ, when dealing with absorption processes , or a fission cross section σ, when considering nuclear collisions that lead to the fissioning of the target nucleus . associate

interaction

a cross section with , we speak

.

each

type

a scattering

Although the reaction cross section bears no direct or simple relationship

to the " geometrical " nuclear cross section # R² ( R being the nuclear radius ) , nevertheless , for most nuclear interactions it agrees with it as far as orders go , and it generally falls within 10-23 and 10-27 cm² . Nuclear reaction cross sections are measured in units called barns , where

of magnitude

1

barn

=

10-24 cm²

The cross section for a nuclear reaction depends not only on the target nucleus as one would expect if the cross section were identical with the geometric " cross section of the target nucleus but also , and in many primarily , energy cases on the neutron .

-

"

4.4

If a neutron

Neutron Cross Sections

beam is allowed to pass through a slab

of target material , intensity owing variety it will to a of processes that during through place passage will have taken the material such scattering absorption with accompanying absorption emission fission

.

subsequent

The various possibilities

as

or

,

-y

,

to

leading

,

its

emerge with reduced

are represented

84

Elementary Introduction to Nuclear Reactor Physics

schematically in Fig . 4.1 . The attenuation of a neutron beam or its loss in intensity due to the combined effect of all causes is described in terms of the total cross section σ . It is equal to the sum of the individual cross sections for all processes involved σt =

. σs

(4.1)

+ σa

In

cases where some of the neutron absorptions can lead to a fission reaction , the absorption cross section can be further divided into a fission Target material

A

B 1B Incident neutron beam

D

detector

-

-

-

FIG . 4.1 . Three types of neutron interactions with matter are illustrated . A represents radiative neutron capture ; B represents elastic scattering collision ; C represents neutron induced fission process . The attenuated direct neutron beam passes through the material to the detector at D. cross section

σ,, and

a nonfission or ( radiative ) capture cross section

σa =

σ.

(4.2)

σcс + σ₁ of

The scattering cross section can also be subdivided into an elastic scattering part and an inelastic scattering part .

σ =

σelinel (scattering)

4.5

Therefore + of +

Oc

(4.3 )

(absorption )

Determination of the Cross Section

We now proceed to find

a relation between the cross section and the experimentally available and directly measurable observables , such as the

85

Neutron Reactions incident beam intensity , and the number of observed transmutations .

interactions or

I

The beam intensity is measured in terms of the neutron flux density referred to in an abbreviated form as " neutron flux " ) , i.e. , the number of neutrons crossing unit area perpendicular to the beam in 1 sec .

(commonly

If all

the neutrons in the beam move with the same uniform velocity v of neutrons crossing 1 cm² per sec is equal to the number of neutrons contained in a parallelepiped of base area 1 cm²

(Fig .

4.2 ) , then the number

B

A

FIG .

4.2 . Flux density and neutron

A per second

equals the number 1 cm² and length v cm .

density . The number

of neutrons

contained

of neutrons

crossing

in a parallelepiped

surface

of base area

and of length v cm . Neutrons which cross face A at a given instant will have reached face B 1 sec later , where the distance separating A and B is v cm . All neutrons crossing face A during this time interval will be contained within the volume bounded by A and B.

If

n is the neutron density volume , then

, i.e. ,

the number of neutrons in

1

cm³ of the

I = n(

neutrons / cm³) v ( cm / sec )

=

nv (neutrons /cm² sec )

(4.4)

If a homogeneous

beam of neutrons is allowed to pass through a thin of target material of area A , thickness t , and having N, nuclei per cm³ , the effective nuclear target area which is available for nuclear reactions to occur is given by the product of cross section per target nucleus and total number of target nuclei contained in the target sheet . sheet

Cross section per nucleus = σ

Total number of nuclei Therefore

Available nuclear target area =

NoAt

(N,At)o

).

.

(

to

its

We are assuming that the target is thin enough so that no overlapping of nuclei in successive layers occurs that might cause a screening of some nuclei by those in the preceding layers . Thus , every nucleus presents entire effective cross section the incident neutron beam Fig 4.3

86

Elementary Introduction to Nuclear Reactor Physics

The probability or the chance for one incoming neutron to hit a nuclear of this area to the total area A presented to

target area is equal to the ratio

the incident neutron . Hence Probability of interaction per neutron

=

nuclear target area total target area

= No Ato = Noto

A

The number of nuclear reactions per second , r ,

is obtained

(4.5)

from this by

Direction of neutron beam

FIG . 4.3 . Neutrons in the impinging neutron beam have an unobstructed view of all target nuclei , if the target is thin enough so that no nucleus lies in the " shadow" of any other nucleus . multiplying the probability per neutron ( Eq . 4.5 ) by the total number of neutrons incident on the target per second . This number is given by the product of beam intensity (flux) and the area of incidence A. Hence

I

Number of incident neutrons per second =

The number of reactions per second

=

IA

(4.6)

(probability of interaction per neutron ) x ( number of incident neutrons per second )

= (4.5) × (4.6 ) Therefore

r = (Noto )(IA) =

(Noo ) IV

(4.7)

87

Neutron Reactions where V is the volume of the target

σ=

At . Hence

,

r

r

=

NOIAt

(4.8)

NOIV

Expression 4.7 for the reaction rate contains the product of o, the cross section per nucleus or the microscopic cross section , and N, the number of nuclei per cm³ . This product is called the macroscopic cross section and is denoted by Σ.

Σ=

(4.9 )

Νοσ

From expression 4.7 we get for the interaction rate interactions per second ) per unit volume , r

rr =

At

number of

( i.e. ,

= ΣΙ

(4.10)

showing that the interaction rate per unit volume is equal to the macroscopic cross section multiplied by the flux density . It also follows from 4.10 that

Σ = Tr

I

(4.11 )

n

its

showing that the macroscopic cross section can also be interpreted as the rate of interactions per unit volume per unit neutron flux . Very often the target material is specified by areal nuclear density where

total number of target nuclei

NA

total target area Not

be . )

nAl

= TA

is

4.13

(

A

r/

σ=

of

"

"

"

to

(

as

"

.

to

between the volume density No and the areal density by 4.12 The term density used here refers number nuclei per unit volume and not their mass per unit volume given by expression 4.8 can now also The cross section written this form expressed

in

The connection

)

4.12

nal

)

A

(

= NoAt =

.

,

,

if

as

to

.

is

A

r/

A

= where the rate of nuclear reactions per unit target area Expression 4.13 corresponds 4.11 and they both give the nuclear cross section the reaction rate per nuclear density per neutron flux we match per corresponding density the reaction rate unit area with the areal and the reaction rate per unit volume with the corresponding volume density of target nuclei

88

Elementary Introduction to Nuclear Reactor Physics

If

the density and the atomic weight of the target material are known , the volume density of the nuclei in the target , No , can be derived in terms of the density p, the atomic weight M , and the Avogadro number N. The mass per atom can be obtained either as the ratio of p /N, or the ratio of

M/ N .

Hence p / N =

M /N

and

PN

No =

(4.14)

M

Example 4.1 . A thin sheet of Co59 , 0.03 cm thick , is irradiated with a neutron beam of flux density 1012 neutrons per cm² sec for a period of 2 hr . the cross section for neutron capture by Co59 is 30 barns , how many nuclei of the isotope Co6⁰ will have been produced at the end of the irradiation period per cm² and activity of the sample ? The half-life of Co⁰ is 5.2 what will be the initial years , and the density of Co59 is 8.9 grams per cm³ .

If

x

30

10-24

therefore

,

is ,

hr

2

1013 reactions cm²

/

59

x

3600

=

period

of

/

× 2

x

=

1010

a

during

of

R

1012

1010 reactions cm² sec

The number of transmutations 8.19

0.03

x

(

59

×

×

=

= 8.19

8.9

×

1023

by

6.03

x

tle

(

-N )

=

×

= Notla

4.14 )

by (4.13 ) by (4.12 )

ra = nalo

4.6

1013

.

decays sec

/

×

59

3.15

x

×

×

of

to

2.5 3.7

1013

107

106 decays sec

1010

/

×

5.2

x

0.693

= 2.5

=

59

× ×

=

×

=

2

-

of

is

,

of

to

is

of

as

equal The number Co60 nuclei produced the number neutron captures negligible the fraction Co60 that decay during the irradiation process because their long half life compared the length the irradiation period The initial activity per cm²

106

curies

= 67 microcuries

Attenuation

of Neutrons a

or

to

,

in

of

a

We have seen that neutron beam when passing through matter suffers intensity through collision processes with the nuclei reduction the intervening matter which lead either scattering absorption of the

Neutron Reactions

89

neutrons (Fig . 4.1 ) . The emerging beam will be attenuated or weakened by an amount which can be readily calculated . In order to do this , let us start with a homogeneous neutron beam that passes through a slab of material of 1 cm² cross-sectional area ( Fig . 4.4) .

If the

I, I dI

incident flux is

it will

become change in the flux ,

and the flux after penetrating a distance x is

I.

after a further penetration of distance dx . The dl, between the penetration distances x and x + dx

Lo

-

dx

I

I

beam , is reduced to intensity after penetrating a thickness the material . A further penetration of distance de causes a further change in the beam intensity of amount -dl .

FIG . 4.4 . The neutron

x in

is caused by the various collision events that have taken place within this distance . We can ; therefore , equate and the number of collisions that have occurred within dx .† By Eq . 4.7 the number of collisions in the volume V = A dx is

-dl

r = σNIA dx = σNol dx Therefore

— dI

or

By integrating both

since

A =

1

cm²

= σNol dx

dI

dx = Σ dr

I

σNo

sides

of the last equation

(4.15 ) between

the limits x = 0

↑ Strictly speaking , this conclusion is only true for absorption collisions , since for scattering collisions the neutrons are not entirely removed from the beam but may be scattered in such a way as to reappear in the beam unless special experimental arrangements are employed . (See page 345. )

Elementary Introduction to Nuclear Reactor Physics

90

and x = x we find the intensity of the neutron beam after penetrating a distance x in the target material .

dI = -log

Therefore and

I Io I

dx

= Σχ

(4.15a)

= Io exp

(-2x)

(4.16 )

This result shows that Σ corresponds to a linear absorption coefficient and has the dimensions of a reciprocal length . Penetration through a distance equal to 1/2 reduces the beam intensity by a factor

of e.

10-24

cm

-1

20800

×

.

1023

x

6.03

.

by Eq 4.14

962 cm

x

Therefore

=

4.60

=

2.30

x

2

log

=

logio 100

10

100

)

=

= log

by Eq 4.15a .

lo

= (

Σx = log

-1

=

-

113

P No

M

×

8.67

=

×

Νοσ =

=

Σ

its

Example 4.2 . The absorption cross section of Cd¹¹3 for certain neutrons is 20,800 barns . Taking the density of this material to be 8.67 grams /cm³ , calculate the macroscopic cross section and the thickness of Cd¹¹³ required to original value reduce the intensity of the neutron beam to 1 % of

4.60

= 0.0048 cm

962

Macroscopic Cross Section and Mean Free Path

4.7

a

is

.

a

a

or

1 e .

to

.

)

4.17

(

)

−Σx

(

exp

the neutron density

as

be is

no

=

n

.

so

),

.

of

finding an average quantity proportional We recall that the beam intensity q 4.4 that Eq 4.16 can also written

standard method

(E

2

/

of

is a

in

it

a

as

be

=

λ

The penetration distance 1/2 can also defined the path length over which the neutrons can travel without suffering collision with target nucleus with probability The distance called the mean free path because also the average mean distance that neutron can travel the material without making collision We can prove this by evaluating the average path length for the neutrons according to the

Neutron Reactions

91

which gives the number of neutrons per cm³ that can penetrate to a distance x a without making a nuclear collision of any kind ( Fig . 4.4 ) . Upon penetrating a further distance dx , of the n remaining neutrons , a number dn will undergo a collision and drop out where dn is obtained by differentiating Eq . 4.17 and is dn

= −n ,

exp

( −2 )

(4.18 )

The combined path length of the group of neutrons that travel a distance x and then suffer a collision within the short distance between x and x + dx is , therefore , x dn .

If we

consider the neutron beam to consist of a collection of many groups of neutrons that stick together for a distance x and then similar drop out together , we can find the total combined path length for all the neutrons in the beam by integrating x dn over all possible values of x from O to co . Therefore 00

Total combined path length =

S x dn

The average path length is obtained from the total path length by dividing it by the total number of neutrons , no . Hence

Average path length =

Therefore

x= =

0

x dn

= x

no

(4.19 )

∞

exp S -Ex (-2x ) dx

=

by Eq . 4,18

2

(4.20 )

It

can be shown that this result applies not only to a uniform neutron beam but that it is quite general . The average path length between collisions for neutrons moving in a medium of macroscopic cross section is equal to 1/2 . No distinction was made in our derivation between a scattering collision and an absorption collision as they both contribute to the attenuation of the neutron flux . The macroscopic cross section to which we have referred successive

is therefore the total macroscopic cross section . that

Σ, = Not Νοσ , = =

It

follows from Eq . 4.1

Νοσ + Νοσα

Σ +Σ

(4.21 )

92

Elementary Introduction to Nuclear Reactor Physics

In analogy with Eq . 4.20 we can define a mean free path for scattering collisions only , and one for absorption collisions only . 1

4.22

Σα

)

=

(

Σ

λ

and

follows from Eq 4.21 that

Σα

it to

as

)

be .

(

4.23

28

2a

λa

1

/

1

+

or 2,

λ¸

/

1

Clearly

77

=

+ .

Σ ,

+

-Σ =

=

1

.

It

2.5

=

,

2

particular type

of

.

.

a

A

a

collision are greater than the chance evading any collision smaller chance smaller mean free path

of evading all types of collisions means also

of

evading

of

chances

,

or

is

,

λ,

one would expect Since possibilities the total cross section includes more collision than either the scattering cross section the absorption cross section separately the smaller than either

=

x

λs

=

λa

= 1.88 cm 1.88

3.2

4.8

×

=

x

x

2.25

S

x

1023

;

M σs

)

(

Np

12

6.03

a =

,

1

.

/

=

Noos

/

1

=

σ

:

,

=

p

Σ .

=

1

;

x

in

.

Example 4.3 Calculate and compare the collision and absorption mean free = 4.8 barns paths for neutrons graphite using these values 3.2 103 barns density 2.25 grams cm³

x 4.8

x

10-3

10-

cm

cm

2820 cm

it a

,

of

.

in

so

,

.

a

in

as

,

is

to

The absorption mean free path seen be considerably greater than the scattering mean free path that on the average neutron can travel over 1500 times far before being absorbed than can travel before making scattering collision graphite graphite This result has important implications for the usefulness nuclear reactor design Example 4.4

σ0 = 4.2 barns σH = 38.0 barns

.

O ,

,

H

in

a

of

.

Calculate the macroscopic scattering cross section for neutrons certain energy H₂O assuming the following values for the microscopic scattering cross sections of respectively and

Neutron Reactions

93

The macroscopic cross section for a compound or a mixture of several components is equal to the sum of the individual macroscopic cross sections

Σ = Σ + Σ2 + Σ = ΣΗ + Σο

Therefore

(4.24 )

23 +

= Νησι + Νοσο The number of water molecules per cm³ is No =

6.03 × 1023 18

The number of O nuclei per cm³ H nuclei is twice this number . No( 0 ) ==0.0335

Hence ΣΗ ,

= 0.067 x = 2.54

1024

x

is equal to this number , whereas the number

× 1024 ;

38

= 0.0335 x 1024

x

No ( H)

==0.067

of

× 1024

+ 0.0335 x 1024 × 4.2 × 10-24

10-24

+ 0.14

-

= 2.68 cm 1

4.8

Neutron Flux and Reaction Rate

It is convenient and physically reasonable , when working with large numbers of neutrons , to consider them as constituents of a neutron gas and to describe their movements and random motion in a manner analogous to and familiar to us from the molecular theory of gases . the neutrons that make up such a gas move with a speed u , which we shall assume at present

If

same for all the neutrons , and have a mean free path 2 , the time interval t between two successive collisions for a given neutron , will be

to be the

1

=

7

(4.25 )

‫ט‬

The reciprocal of this , v /2 , will give the number of collisions per second made by a neutron . For an assembly of n neutrons per cm³ , the total number of collisions made each second is then Number of collisions /cm³ sec = (number of collisions per neutron / sec ) × (number of neutrons / cm³) that is ,

ry =

λ

(4.26)

n

This is the same result as was found for a beam of neutrons with all neutrons traveling in a given direction ( Eq . 4.10) .

It

is shown here to be

94

Elementary Introduction to Nuclear Reactor Physics

valid also for an assembly of neutrons with the neutrons traveling different and arbitrary directions .

If we interpret

in

by a neutron in 1 sec , then the product nu represents the total distance traveled in 1 sec by all the neutrons which are contained in 1 cm³ . By dividing the total distance by the average distance between successive collisions , we arrive at the number of collisions per second per cm³ , i.e. , ry, confirming Eq . 4.26 . The product nu is called the neutron flux (i.e. , flux density) and commonly denoted by . v as the distance traveled

$= In

nv

(4.27)

terms of the neutron flux 4 , the reaction rate / cm³ can now be written

ry = ΦΣ The reaction

rate

for

a medium

(4.28 )

of volume V cm³ ,

R = ryV = VΣ

is , therefore ,

(4.28a)

Previously , we had defined the flux density as the number of neutrons falling on unit area each second . That definition is satisfactory and equivalent to the new definition for a parallel beam of neutrons , all traveling in the same direction . It is , however , not general enough to be applicable to an assembly of neutrons that move in all directions at random like the molecules of a gas . The neutrons in a nuclear reactor are more realistically described in terms of the random motion of gas molecules than as a parallel beam of particles incident on a plane surface . The reaction rate should depend only on the product nu , i.e. , the flux of the neutrons and not on the direction of approach of the neutrons . The number of neutrons falling on 1 cm² per sec would , however , be less for random motion of the neutrons than if they were all moving in a direction perpendicular to that area , so that the number

of

reactions would also be

correspondingly smaller .

As with

gas molecules , the neutrons in a reactor do not all have the same

speed , but instead manifest a considerable spread in the range of their velocities . It is therefore necessary and desirable to generalize the definition

of the neutron

flux ( Eq . 4.27 ) . This can be done by defining a flux element do for a small range of neutron velocities between v and v + dv , so that n(v ) dy is the number of, neutrons per cm³ having velocities within this range ( Fig . 4.5) . dp = n(v)v dv Therefore

[

= 4=

®n (v) v

dv

(4.29 )

Neutron Reactions

95

Alternately , we can divide the neutrons into groups of small energy ranges , denoting the number of neutrons with energies between E and E + dE by n(E) dE. In terms of this decomposition the neutron flux can be defined as

$=

[

®n

(E)v dE

(4.30 )

Flux per unit velocity interval

n(v)v

nu dv

dv

of

+

v

The shaded area represents the flux

neutrons

with velocities lying between

of

defining the flux which we shall find useful later on

the

:

third method following

is

.

4.5

+ .

and

A

v

.

FIG

Velocity

dv

v

v

do

∞

$ ( E )

dE

)

(

4.31

+

and

E

the small energy range between

E

the flux

in

E )

dE

dE

).

4.6

.

(

Fig

(

where

is

0

=

,

a

/

/

,

in

"

"

in

3.15

VNox 3.15

107

)

reactions sec

107

x

=

x /

x

,

= VE

of fission

107 sec year

×

3.15

/

number

(

x

/

Number of fission reactions year

=

.

is

in

of

given by the number The number U235 nuclei used up one year U235 nuclei that undergo fission reactions during that time Hence

of

.

to

of

σ,

.

of

%

of

is

A

of

.

Example 4.5 typical natural uranium reactor assembly contains about 50,000 kg natural uranium which 0.7 the readily fissionable isotope U235 Assuming an average neutron flux 1012 neutrons cm² sec and fission = 580 barns for the readily fissionable isotope U235 estimate cross section of the fraction this isotope burnt up by fission one year assuming the reactor be continuous operation and neglecting other U235 consuming reactions

96

Elementary Introduction to Nuclear Reactor Physics

Hence , the fractional burn - up is this number , divided by the original number NOV. Therefore

Fractional burn- up = 3.15

x

x 0, x

10

&

= 3.15 x 107 x 580 × 10-24 × 1012

= 1.83

x 10-2

Therefore , the annual burn - up through fission is somewhat less than 2 % .

.

'

$ ( E )

=

EdE

of

The shaded area represents the flux

4.6

and

E

E dE neutrons

Energy

with energies lying between

.

E

.

FIG

dE

+

E

$

-d

Neutron flux per unit energy interval

$ (E)

of Neutron Cross Sections

Energy Dependence

4.9

In our

to

is

It

It

.

will

to

.

distinguish four principal regions

.

Mev

divide the neutron

these general trends

nergy region which comprises neutron energies between 0.1 Mev Neutrons within this range will be called fast

~

high

to -e

10 1.

~

A

be sufficient for our purposes

take account

of

to

the various cross sections and

energies into several regions

to

general behavior

of

in

to

.

it

of

preliminary discussion neutron reactions was noted briefly only depend that the neutron cross sections not on the nature of the target interacting energy nucleus but also on the of the neutron convenient classify neutrons that are involved nuclear reactions according the

An

intermediate energy region for energies between

.

.

1000 ev neutrons

Neutrons with energies

in

~

2.

.

neutrons

0.1 Mev and this range will be termed intermediate

97

Neutron Reactions 3.

A region

for

energies between

energies lying in this range 4. A region for energies

~ 1000

ev and

~ 1 ev .

Neutrons with

will be called epithermal neutrons . of ~1 ev and less . Neutrons with

energies

this region will be referred to as thermal neutrons or slow neutrons

in

.

Using this classification , we can survey the neutron cross sections in the given order . 1. FAST NEUTRONS . The most probable interaction between neutrons and nuclei in this region is the (n , n ) reaction , i.e. , scattering , so that the absorption cross section will be very much smaller than the scattering cross section , σ < σ,. The total cross section σ , is almost entirely due to

scattering and it approaches a value of σt = σ = 2πR²

For medium and heavy nuclei , o , can σι

be expressed

(4.32 ) fairly well by

= 0.125A%

(4.33 )

2. INTERMEDIATE NEUTRONS . In this region the ( n , n ) process is still the most favored reaction for intermediate and heavy nuclei , with scattering making the chief contribution to the total cross section . σ, = σ , is of the order of 1 barn , whereas the cross section σ, for the ( n , y ) or radiative capture process is of the order of 1 millibarn . For light nuclei ( A < 25 ) this region contains well separated and distinct

for the medium and heavy nuclei the overlap and appear smoothed out . The nature of these is described in more detail in what follows .

resonances , whereas

resonances resonances

In this energy region which is also known as region , the resonance the neutron cross sections of most elements show many distinct and high maxima in the total cross section . The peaks in the nuclear cross section point out the existence of resonance levels in the 3. EPITHERMAL NEUTRONS .

compound nucleus of the kind discussed in an earlier chapter , and they are , therefore , generally known as resonances . They appear to be superimposed on a background that varies as E- (or 1/v) . The number of absorption peaks and their mutual separations vary considerably

and the general nucleus is shown

appearance

in Fig . 4.7.

of

the energy - dependence

for different nuclei , of σ , for a typical

The maxima are resonances in the capture cross section σ , that are superimposed on- a background , which is practically entirely a scattering cross section . The scattering cross section for energies lying between the individual resonance energies is of the order of a barn , with the capture cross section o, of the order of a millibarn .

98

Elementary Introduction to Nuclear Reactor Physics

The total background cross section σ ,

=

σt

between

is given by

resonances

4πR2

(4.34)

which is twice that of the corresponding cross section for fast neutrons

(Eq . 4.32 ) .

σ

Cross section

Resonances

-region

.

a

region

the resonance

)

(

ev

typical nucleus is

dependence

in

The energy

neutron cross section with energy for

of σ c,

Variation

4.7

.

.

FIG

of

Energy

given by the

igner formula

Г,

Г,

²

2 )

T /

)²

+ (

(

4.35

the called single level formula which holds for separated resonances that are well and do not overlap that the collision responsible single reaction which for this maximum can be ascribed compound resonance level of the nucleus In this formula the de Broglie wavelength the incident neutron

,

of

of

is

,

=

is 2

.

a

is

to

so

,

-

is

4.35

so -

Equation

E

4π

)

22

—

=

(

Oc

E

-W

.

Breit

at

a

,

,

n, )y

and that for the

(

T

/

is

à „

(

r

reaction

)

4.36

=

г,

n, )n

(

г

.

/

г

+

,

.

so

г₂n that the probability for the reaction ,

of

of

in

).

,

.

(

to

,

is

г,

г,

).

(

,

( n,

n )

)y

.

I

,

h /

( of n

λ

mv and the width the resonance line half the maximum n and value the cross section Fig 4.8 are the partial level widths for the and reactions respectively and each measure of the probability for the corresponding reaction compare Eqs 3.2 place take only decay processes and 3.7 Since these two are the modes the compound nucleus this energy region that need be considered we must have by Eq 3.7 that

Neutron Reactions

99

In most cases , when E is small , the probability for the (n , y) process is much higher than for the (n , n ) process ( see section 4.2 ) , so that гY, > гn and , consequently , г = y' For most nuclei г , is ~ 0.1 ev , whereas г is usually two or three orders of magnitude smaller .

1200

г

Cross section

г.

E

Eo

г.

-

its

in

of

the form

(

4.37

)

-

—

1+

E )²

1

E

E

=

half width

=

E

the maximum value

of

where σο

is

( г / 2 ) ²

σc

written

terms

in

be

expressed

igner formula can also 0

-W

The Breit

resonance

(

a

.

.

FIG 4.8 The width

is

of

Neutron energy

the resonance

capture cross section for

Eo-

–

E

>

r

,

,

a

Eo

constant

(

4.38

υ

)

n, α

in

the capture cross section for slow the cross section with boron

of

/v 1

is

.

),

.

(

,

-

is

dependence the well known neutrons which exhibited clearly Fig 4.10 for example

(

This

)

=

,

σc

0

to

.

a

of

a

).

of

.

(

-

Eo ,

a

is

of

-

A

well known example neutron absorption resonance that of cadmium with maximum value 7200 barns for neutron energy of 0.18 ev Fig 4.9 Eq 4.37 For the case broad resonance i.e. when simplifies

100

Elementary Introduction to Nuclear Reactor Physics

barns

) (

Absorption cross section

σ

8000

6000

4000

2000

0.08

0.24

0.40

ev

,

strong resonance peak

.

,

of

The slow neutron cross section cadmium showing maximum of about 7200 barns

0.18 ev with

a

.

at

4.9

.

FIG

a

Neutron energy

10,000

1000

barns

) (a

100

1 10

-

10

1

10-3

1

1

10

100 1000 104 105 106

as 1 v

)

Mev

.

5

fairly constant and has several resonances between 0.5 and

it

ev

.

/

cross section boron The cross section varies remains and 0.1 Mev For energies between 100

ev .

.

.

FIG 4.10 Neutron absorption for neutron energies below 100

of

(

Energy ev

Neutron Reactions

101

For light

nuclei the resonances are very broad and widely separated from > E – E, is well satisfied and the 1/v dependence is quite generally obeyed . 4. THERMAL NEUTRONS . In the thermal energy region the scattering cross section σ8, rises steadily as we go from the lighter to the heavier each other , so that the condition

barn to a value below 10 barns . The capture cross law , and the Breit -Wigner formula may be applied to the (n , y ) cross section , even though there are no resonances present in the thermal energy region . For E, we use the value of the nearest resonance in the low - energy region . A schematic representation of the variation of neutron cross section with energy for a typical nucleus is given in Fig . 4.7 , and the thermal elements from about section σ follows the

1

/

1

neutron absorption cross sections for some representative listed according to orders of magnitude in Table 4.1 .

nuclides are

Example 4.6 . Assuming a resonance cross section of 55,000 barns for Cd113 at a resonance energy of 0.176 ev , estimate the relative probability of a neutron emission as compared to a resonance capture if the total width is 0.113 ev and Tn.

Ty

Setting

E

= E

in Eq . 4.35 , we get

= With

λ ==h / mv = h [ ( 2mE

22

- TIYIn

г,г

π

22

1'2

Therefore

= 2πmЕoo

y

) ½ , this gives for =

In

since

h² гn 2πmЕ Ty

6.28 × 1.67 × 10-24 × 0.176 × 1.6 × 10-12 × 55,000 × 10-24

h2

(6.63 × 10-27 )2 = = 3.7 × 10-3

The probability for a radiative capture is , therefore , nearly the probability for neutron scattering .

300 times as great as

Example 4.7 . The cross section for the ( n , a ) reaction with boron follows the law . If the cross section for 50 ev neutrons is 16.8 barns , calculate the cross section for 0.025 ev neutrons . Using Eq . 4.38 , we get

/

1v

σ = 0

=

E½

16.8

E

50½ 1/40

= 336 × 5½ = 753 barns

Elementary Introduction to Nuclear Reactor Physics

102

TABLE Nuclide

4.1

ба

Thermal Neutron Absorption Cross Sections of Some Nuclides ( in barns ) Nuclide

σα

Nuclide

σα

He¹

0

N14

1.75 (n , p)

H2

0.00046

K39

1.94

V50

$32

0.002 (n, α)

K41

1.24

Kr83

220

Be⁹

Ti48

8.3

Rh103

156

0.010 Fe56

2.7

Te123

410

Ni58

4.4

Xe131

120

Zr91

1.58

Nd143

324

Mo 96

1.2

Eu153

450

[127

7.0

Hf177

380

W184

2.0

Pt194

fission factor .

The number of

fast neutrons has now increased

1,

called the fast

to

пое

is being reduced steadily by collisions assembly with the other nuclei in the until they eventually enter the epithermal energy region with strong U238 absorption resonances Fig 7.1 Some of the neutrons will be absorbed by U238 event whereas most of them will escape resonance absorption The number neutrons that will pass through this region without being absorbed can obtained

.

2 ),

be

.

of

(

).

(

its

The energy of the fast neutrons

neutrons surviving thus far

is

Hence the number

of

).

6.67

enter the region by the resonance escape

,

p (

by multiplying the number that

probability

noεp

in

or

4 )

3

(

5 ).

to

of

is

,

shall obtain the number the fuel which

,

of

the previous number thermal neutrons we thermal neutrons that are actually absorbed by

ƒ

If

.

f

utilization factor we now multiply by

is

in

as

of

(

in

,

These neutrons will next reach thermal energies where their impending absorption fate may be either absorption U235 events and materials other than U235 event compared The fraction thermal neutrons absorbed by the fuel all thermal neutron absorptions the assembly called the thermal

noεpf

of

a

,

.

as

is

,

no

of

is n

a

This number of thermal neutron absorptions will yield number fast large Hence having started with an fission neutrons that times initial number fast neutrons we have now obtained new generation of fast neutrons whose total number поєpfn ,

Hence

Final number of fast neutrons = noεpfn Initial number of fast neutrons no )

7.1

(

=

= εpfn

The Nuclear Chain Reaction

173

)

(

as

of

.

it

is

the number

as

A

schematic picture

the ratio thermal neutrons thermal neutrons destroyed per second the neutron cycle we have just traced shown

of

to

can also be interpreted

7.1

created per second

in

leading fissions Neutrons 235

Thermal fissioning of U235

Fast fissioning 238 of U

Fast

Fast neutrons

Neutrons nonfission captures

Fast neutrons

to

U2

neutrons

.

Fig 7.2

Fastneutrons producedfrom fissioning 235and 238

U

of

Neutrons absorbedin uraniumfuel

down slowing Neutrons

U

.

in

.

-

known as the four factor formula Expression

of

is

=

...

This ratio is called the reproduction or multiplication factor and is denoted by k The relation εpfn 7.2 k∞

(

Neutrons

)

,

poisonsabsorbed nonproductively etc.

Neutrons

Neutrons absorbedin reactor materials

U

entering 238 resonance region

.

.

FIG 7.2 Neutron

escaping region Neutrons reaching resonance and energies thermal

Neutrons in absorbed resonance region

cycle

.

Neutrons reaching thermal energies

escaping as Neutrons reactor from neutrons thermal

7.3

Neutrons during escaping slowing down

The Thermal Utilization Factor

.

n

= 1.34

.

to

.

η

,

,

f

as

the we conventionally consider the uranium mixture When defining fuel although the thermal neutrons can produce fissions with the U235 component only We must then also use the numerical value for which applies the uranium mixture For the natural mixture this would be

Elementary Introduction to Nuclear Reactor Physics

174

238

)

(

7.3

+

Noii

)

238 ) ( ( 238 )238 ) (

No

No + (

)

(

No

235 235 ) ) ( + )235

No (

=

(

)

(

nat

235

f

Thus , if o stands for the thermal absorption cross sections and the suffix i referring to all nonuranium materials and impurities and employing an obvious notation , we can write for a natural uranium fuel mixture that

,

.

)

(

Of nat

235

)

(

7.4

238 )

(

(

238 )

/(

235

)

235

)

+

235

(

No

235

(a

) 235 )

No

No

)

(

a (

)

(

σa nat

)

(V

)

nat

=

n (

.

,

in

,

,

,

Here the numerator represents the thermal neutron absorptions by the two uranium isotopes only and the denominator represents similarly the total number of thermal neutron absorptions the assembly Eq by Also 5.3

U235

portion

+

(

7.5

)

Noii

(

235 238 ) ) ( a ( 235 ) 238 )

235 )

No

of

considering only the

,

Alternatively

(

,

(

(

No

+

)

(

)

No

=

235 )

nat

f

nat

235 ) a (

n (

.

,

Hence by multiplying Eqs 7.3 and 7.4

the natural uranium

)

( +

238 )

238 (a ) 235 a ) (

)

(

)

+

ƒ(

(

) 238 )

+

No

235

Noii

235

)

(

7.8

)

(

ƒ

235 ) (

.

.

by Eq 7.5 =

nat )

235

238 ) ) ( a (

235 )

(

235

)

No

)

235 ) σa ( (

)n nat )

f(

(

(

+

7.7

)

(

a

(

nat

f

7.6

Noiơi

235

nat

No

Therefore

No

235

= =

(V

)

)

235

(

)

235

(

Therefore

235 )

(

= Oƒ 235 (

N (

235

235 ) a (

(

No

)235

No

=

)

)

(

235

n (

f

,

mixture as the fuel we have

.

-

,

as a

as

is

or

it

immaterial whether we consider the U235 isotope alone the uranium mixture whole the fuel provided we match the correct with the adopted utilization factor fin the four factor formula which shows that

In

7.4 Neutron Leakage and Critical Size

6 )

is

.

(

of

of

the preceding derivation the multiplication factor we omitted leakage from the assembly event completely the possibility and tacitly assumed zero leakage during the neutron cycle This omission

The Nuclear Chain Reaction

175

tantamount to the assumption of an infinite size for the assembly, because only then can there be no neutron leakage from the system . For that reason , in order to remind us of the implied assumption of an assembly of infinite extent , the suffix co was added to k . For an assembly of finite dimensions , the effective reproduction constant kett will be less than k by a factor L , ( L < 1) , which is determined by the neutron leakage from the system

.

Keff

=

kL

(7.9)

We conveniently separate the total leakage effect into two components , a fast neutron nonleakage factor 1,, and a thermal neutron nonleakage factor Ith This separation is suggested by diffusion theory , which treats the diffusion of fast neutrons and that of thermal neutrons separately . We , therefore , set

L=

(7.10 )

14th

The two nonleakage factors are a measure of the fraction of neutrons that do not escape from an assembly of finite size . Hence , Keff

= Kollth

(7.11 )

kett , for a given reactor system , is the ratio of the number of neutrons (at corresponding stages of the neutron cycle ) in successive generations . This self- multiplication of neutrons is the essential feature of a nuclear chain reaction . The magnitude of kert determines the speed with which the number of neutrons builds up and the rate at which nuclear fissions occur in the assembly . In a nuclear bomb type of assembly , this build -up must

take place very rapidly , whereas in industrial and research reactors this self- multiplication must be slow enough to allow the fission rate to remain always under the control of the operator . keff > 1 , the assembly continues to produce more neutrons than it consumes and is then said to be supercritical .

If

kett 1 , fewer neutrons are produced than are consumed . Such assembly is said to be subcritical . an , For keff = 1 the rate of neutron production is exactly balanced by the

For

rate of neutron consumption and , in this case , the assembly is called a critical one . we start with an assembly for which keff > 1 , we can decrease kett by

If

progressively reducing the reactor size , thereby increasing the neutron loss through leakage from the assembly . this reduction in the dimensions of the assembly is continued until keft = 1 , the reactor size of the assembly at that point is called critical size Thus the critical size an assembly .

to

in

to

is

that size for which the rate neutron loss due all causes equal the rate of neutron production the assembly

is

of

,

of

.

its

If

exactly

Elementary Introduction to Nuclear Reactor Physics

176

The fundamental problem in the design of a nuclear reactor is to obtain an assembly with keff > 1. As a first step to this end k must be calculated and then , by introducing the finite size and geometry of the reactor , kett can be calculated from the layout and the dimensions of the reactor . Although this plan of action is very straightforward , the actual mathematical work involved can be very complex .

Of the four factors

in formula 7.2 , η ʼn is a constant of nature for a given nuclear fuelt which must be obtained from experimental measurements . The numerical values of this constant for the important nuclear fuels have been listed in Table 5.2 . The other three factors allow the nuclear engineer

,

in -

,

as

.

,

as

its

and designer some leeway , as they depend on the physical properties of the geometry fuel arrangement fuel as well as on the size of the reactor , incorporated moderator well on other materials the reactor assembly

p

to

nf

)

7.12

(

=

k∞

-

1

;

, &

of

as

is

If

part no U238 used the reactor fuel and differ only very slightly from the numerical value thus the four factor formula reduces

generally not welcomed by the nuclear neutron leakage important realize that the critical size requirement for consequence chain reaction become possible neutron leakage against spontaneous chain reactions with small safeguard acts

It

.

as a

is a

of

a

to

to

is

,

it

is

Although engineer

of

.

to

,

or

to

,

of

quantities U233 U235 Pu239 by reducing the number neutrons from spontaneous fissions below the minimum number required sustain continued fission reactions For these three fuels the number of neutrons conse-

all the

.

);

(

is

in

,

quently

if

per neutron absorbed well above unity Table 5.2 all three materials chain reactions would occur generated neutrons were retained inside the fissionable material generated

7.5

Nuclear Reactors and Their Classification of

at

a

to

)

;

3

)

(

;

( 2

)

; ( 5

as

,

(

;

6 )

)

(4

of

;

of

)

( 1

,

of

variety characteristic type average neutron energy such fuel used which the greater part all fissions occur moderator material used arrangement and spatial disposition purpose fuel and moderator heat removal methods and other features such of the reactor and as :

We can classify nuclear reactors according

features

.

coolants employed

,

of

be

,

,

.

,

be

is

n η

in a

†

of

By changing the composition enrichment the fuel e.g. increasing the degree higher value for could achieved but this would no longer the same given fuel being used here the sense this term

The Nuclear Chain Reaction

177

Let us consider and compare

features in the given order :

of these

some

1. The fuels that can be used are uranium containing U235 in its natural concentration of 0.715 % or in an enriched proportion , U233 or Pu239 . The fraction of delayed fission neutrons is greatest with U235 , which is important from the point of view of reactor control . U233 is potentially the most abundant fuel , as the fertile material from which it can be derived , namely , thorium , is about four times as abundant as uranium . Pu239 is

favored as a fuel in fast breeding reactors (i.e. , reactors that convert fertile materials into fissionable materials and create more fuel than they consume) because of its high breeding gain . U233 is the only one of the fuels cited that can sustain a breeding reaction with either thermal or fast neutrons . 2. A fast reactor employs high - energy neutrons to sustain the chain reaction that makes the use of a moderator unnecessary . Enriched uranium and Pu239 are suitable fuels for this type of reactor . Natural uranium cannot be used as a fast reactor fuel because it is not possible to achieve a chain reaction with it unless a suitable moderator is also used .

.

,

;

to

is

of

A

less than 25 % of fissionable material content relatively (i.e. , U233 , U235 , Pu239 ) is required for successful operation large amount fissionable material needed reach criticality however the critical core size may be only about one foot across because of the

its

An enriched fuel with not

a

at

.

is

.

,

of

as

is

in

.

to

at

.

elimination of the moderator This small critical size presents difficult high operating technical problems with respect the heat removal power levels present directed towards The main interest fast reactor design greater their application breeder reactors Fast reactors require greatly reduced amount fuel but parasitic neutron absorption The .

is

σ σ,

/

ratio for the fuel also greater for the fast reactor intermediate reactor uses neutron energies between thermal and several thousand yet generally but little information about this type available energies that In thermal reactor the fissions generally occur a

as

at

.

†

is

ev ,

An

.

to

is

,

considerably greater than any

,

-

and

documents

;

the backbone of of

.

. 3

,

Atomic Energy Commission

).

U.

(

Reactors

S.

Naval Reactors

be assembled

date reports have recently joined the declassified list Physics Handbook vol and LAMS 2288 Physics

to -

-

†

Two up

It

The thermal reactor was the first type reactor ever of the existing reactors are of this type still

most

is to

of

.

of

is

bypass the U238 cross section U235 for thermal neutrons of the competing cross sections assist the neutrons

Natural uranium can

employed suitable moderator resonance traps because the capture a

reactor

if

of

this type to

to

fuel

the temperature of the reactor core in

be used

as

correspond

of Intermediate

178

Elementary Introduction to Nuclear Reactor Physics

reactor engineering and construction and most of the available information

is concerned with this type .

3. Commonly used moderators are graphite , light and heavy water , and beryllium or its oxide . The requirements for a good moderator are , as previously noted , light mass of moderator nuclei , small absorption cross section , and large slowing - down power .

Of all moderators , H₂O has the greatest sdp but an appreciable neutron absorption cross section . Heavy water has a very small absorption cross section and the highest moderating ratio of all moderators . Graphite has a very small absorption cross section and the second highest moderating ratio of all moderators . Beryllium , too , has a small absorption cross section and the best sdp of all metals ( Table 6.2 ) . Hydrogenous organic compounds have proved to be suitable as moderators and have found some application . When high concentrations of fissionable materials are used as fuels , moderators are often employed not to improve the neutron balance but to ease the control operations of the nuclear reactor by lengthening the time interval between successive neutron generations . 4. Nuclear reactors are either of the homogeneous type or the heterogeneous type , depending on the fuel moderator arrangement

In

.

a homogeneous reactor the fissionable material is intimately and

uniformly mixed with the moderator , either as a solid mixture , as a slurry , as a liquid solution of a uranium salt in the moderator , or as a solution of uranium in a liquid metal . reactor the fissionable material is concentrated in or hollow cylinders , which are distributed in a regular array according to some geometric pattern or lattice throughout the moderator . The theory of moderation as outlined earlier assumed a uniform

In

a heterogeneous

plates , rods ,

.

distribution of all materials . It requires some modification in the case of a arrangement which will be introduced later . heterogeneous Homogeneous thermal reactors employing natural uranium can reach a multiplication factor greater than one only with heavy water as moderator . However , if the lattice or matrix arrangement (i.e. , a heterogeneous system ) is employed , a chain reaction becomes possible with natural uranium and graphite as moderator . The increase in the multiplication factor with the heterogeneous arrangement is due mainly to a reduction in the resonance absorption of the neutrons by the U238 which results in an increased value for p . The chief purposes for which nuclear reactors have been built are either for research , for thermal or electric power production , for breeding , for propulsion , or for a combination of these with a variety of subdivisions possible . 5.

The Nuclear Chain Reaction

179

In

order to remove the heat that is being generated in the reactor core of the fissioning taking place , materials are employed that are known as coolants , which are circulated through the core for the purpose of abstracting the heat and transferring it to the outside of the core where 6.

as a result

be utilized for various purposes . Possible coolants are ordinary water circulated under pressure , heavy water , liquid metal coolants such as sodium or sodium - potassium alloy , and mercury . In some reactor designs , air is circulated at subatmospheric pressure ( open cycle) , in others , high-

it can

N₂ are used . In still others , the heat is removed by pumping the fuel solution through an external heat exchanger and then returning it to the reactor core (closed cycle ) . It is obvious from this multiplicity of choice , that numerous possible combinations of the various components described can be selected for the pressure gas coolants such as He , CO2 , or

design and planning of a nuclear reactor . Considering the great number of variables that are at the disposal of the nuclear engineer , it should cause little surprise that well over a hundred designs for power reactors alone have been proposed . To these must be added a variety of research reactors and experimental units which have already been built and which still make

up the majority of the reactors which

have progressed from the planning to the operating stage . In Tables 7.1 to 7.10 are listed some of the better known units of various types to which frequent reference is made in the literature on nuclear reactors .

7.6

Power Reactors

To give a short description of some of them , we shall start with power reactors , and of these we select four main types : ( 1) the pressurized water reactor ( PWR ) ; ( 2 ) the boiling water reactor (BWR ) ; ( 3 ) the gas - cooled , natural uranium , graphite - moderated reactor ; and (4 ) the homogeneous reactor . The primary purpose of a power reactor is the utilization of the fission energy which is being produced in the reactor core and to convert it into useful power . The main task of the nuclear engineer consists in devising means and developing designs that will permit the efficient and economical exploitation of this energy source . Some ways by which this conversion can be achieved are indicated in Figs . 7.3 , 7.4 , and 7.7 .†

.

The Pressurized Water Reactor

(Table 7.1 ) Figure 7.3 is a schematic diagram of a pressurized water reactor ( PWR ) . This is a heterogeneous reactor that uses slightly enriched uranium ( 1.4 % † For a summary survey of power reactor developments , see " Atomic Energy Facts ," published by the U. S. Atomic Energy Commission , Chapter 5 and Table 24 .

,

. , ,

).

,

(

,

,)

2

-

) ( ) (1- (

;1

,

%

4

; enriched UO

pellets

; 2.8 tons

in

%

)

,

zircNatural uranium oxide pellets aloy tubes ~ tons of uranium oxide 120 rods of 19 tubes each

4.5

enriched sintered UO 330 kg U235

;

,

, H₂O 100 atm

,0D H₂O and graphite

1013 thermal flux

50,000

75,000-100,000

Core 5.1 5.5

Core tank 10 high diam 12

high

cylinder diam

,

,. :

8

,

,

5

5

high

Core size ft diam

ft ft

;

fuel enriched uranium concentric cooling channel

D.0

43,000

74,000

18,000

high base radius

.

550 kg of tubes tubes

H₂O

H₂O

H₂O

H₂O

H₂O

H₂O

~

~

Cylinder

:

) ) (3- ( (1-

atomic power station APS Russia

- -

ft ft

nuclear power NPDR demonstration reactor Canada

, ; ;

sheathing

10,000 1850 H₂O 1200 psi

:

Belgium

%

Enriched uranium zirconium first major use of Zr

H₂O

585,000 160,000

47

Mol

%

)el (

,.

BR

3 H₂O 1500 psi

ft ft

N. S. Savannah

%

(

Flat plate sandwich type elements highly enriched UO2 SS cladding

H₂O

400,000 130,000

H₂O 2000 psi

,

)el

subSTR STR SIW marine thermal reactor world's first mobile unit

in

Tho₂ 1.1 ton enriched UO over all enrichment 19.2 tons rod shaped fuel elements

H₂O

,

(

90

25 tons rods SS tube to form

cylinder diam high

ft ft

package power

enriched UO pellets assembled fuel rod

Core 6.8 5.9

Size

,.

Army APPR reactor

~

230,000 60,000

,

) el

Consolidated Edison CETR thorium reactor Indian Point

%

Mass

H₂O 2000 psi

(

Rowe

H₂O

Power kw Thermal

)el

Yankee

in

; , ; ; ; , , , - -U 4 ; , ;

alloy plate enriched Zr 93 pellets elements natural UO 14 tons natural UO2 75 kg tubes U235

Coolant

) ) ((

Penn

PWR

Moderator and Reflector

Type

Reactors

:

Shippingport

Arrangement

Fuel and Fuel

Water

Power

) (

Name Location Comments

7.1

Pressurized

TABLE

180 Physics Elementary Introduction to Nuclear Reactor

ft

5

)

×

5

(

%

in

,

,

Name Location Comments

,

)"

("1 ) -

, (,

"

"

2

-

3

-

(

,)

)

(, of

( 1

;

-

,

). , ) () - , (

) 30

%

) %

,

-

(

Slightly enriched uranium

66 ton UO2 1.5 Zr clad rods

kg UO2

;

enriched

enriched fuel plates

D₂O

(

H₂O

, ,0

el 3000

180,000

(

20,000

650,000

7.5

)

H₂O

H₂O

,0D

H₂O

30,000

high

cylinder diam

Steel tank diam 10 high

Core

Similar to BER

, :.

ft ft

;-

-

;

high

1012 Core size cylinder diam

ft

H₂O

H₂O

106

1013

4

40 plate type elements 91 % enriched 14 kg U235 Al Ni cladding

H₂O 1000 psi

D

H₂O

H₂O

H₂O normal pressure

1013

X

high

: ,.

ALPR Argonne low power reactor

,. -

-,

,.

90

6

,

power reactor

54 H₂O 15 in

)el

Norwegian Halden

;

ft

power station

, H₂O

5

highly 168 kg U235

20,000 5000

(

MTR type plates enriched uranium

,

H₂O 600 psi

H₂O 14

H₂O

12,000

3

Sandwich type plates assembled into 112 fuel elements of plates each over all enrichment 1.44 % 75 kg U235 4535 kg U238

H₂O

H₂O in

H₂O

diam

1013 Similar to BER

13

44

Ill

30

-

ft ft

Dresden Morris

-2

.

Vallecitos

ft

x

:,

VBWR

-

-

SPERT special power excursion reactor test Arco Idaho

1

1

experimental Argonne NL

cm

x

MTR type fuel elements core rad 65 cm high

3.5

Tank

-

EBWR BWR

x

power

7

H₂O

, . , . , ft

H₂O in

1013

Size

1

Arco BWR plant prototype

,.

H₂O

6400

1200

H₂O

ft

BER

H₂O 16 in

Flux Thermal

4

Similar to BER

H₂O

Coolant

Power kw Thermal

(

ft

as above

;

type fuel elements fuel 2ft 2ft test

Reflector

BWR

) ( ( )

:

.,

BER

MTR core

Reactors

)

BER Borax boiling reactor experiment Arco Idaho destroyed in runaway test

Moderator

Type

Power

Water

7.2

) (

Arrangement

Fuel and Fuel

Boiling

TABLE

The Nuclear Chain Reaction 181

22

x

Elementary Introduction to Nuclear Reactor Physics

182

Control rods

Steam out

Pressure vessel

Heat exchanger Fuel core

-Water

in

Coolant and moderator loop

(a) FIG .

7.3a . Schematic

diagram

of a

pressurized

water reactor .

2 % of U235 ) as fuel and light water as moderator and coolant . The core is contained in a pressure vessel under a pressure of 1000 to 2000 psi , with water being circulated by means of a pump through the core and an external heat exchanger . The purpose of the heat exchanger is the abstraction of heat energy from the coolant in order to make it available for conversion into other forms of energy in a ( for the engineer ) conventional manner . The main-

to

of a high pressure in the core and coolant loop is essential in order to raise the boiling point of the coolant to a high enough temperature ( 300 to 400° C ) which will permit a satisfactory thermodynamic efficiency to be achieved for the heat transfer cycle . Examples of this reactor type are the Shippingport , Pennsylvania , tenance

installation and the reactor unit on the U.S.S. Nautilus ( Table 7.1 ) . An interesting variant of this reactor type is the nuclear power demonstration reactor ( NPDR ) being built in Canada which is designed to employ D2O as moderator - coolant instead of the more traditional light water (Table 7.1 ) .

(Table 7.2 ) Figure 7.4a shows a boiling water reactor diagram which also uses light water as moderator - coolant . In contrast to the PWR , however , steam is The Boiling Water Reactor

† With light

water as moderator a minimum enrichment

of ~ 1 %

is required .

The Nuclear Chain Reaction

183

Control drive mechanism housing -Fuel port

0000 00

199

Locking assembly

Natural uranium assembly

Control rod-

(blanket)

Enriched uranium assembly

Core cage

(seed)

Spring

Thermal shields

Bottom plate

Flow baffle

(b) FIG . 7.36 . The Shippingport PWR fuel

core assembly

and pressure vessel .

184

Elementary Introduction to Nuclear Reactor Physics Control rods

Steam out to turbine , generator , etc.

FIG .

(a)

7.4 .

Boiling water reactor . diagram .

Schematic

Fuel core

Water in

in the reactor core itself with this unit , which is then passed directly to the turbines without having to go first through an intermediate generated

heat exchanger . Enriched uranium is used to fuel this reactor with a possible degree of enrichment that can vary within rather wide limits .

Working

examples of this type are the BORAX reactors , one of which was supply light and power to Arco , Idaho , for about an hour in 1955 . used to experimental boiling water reactor ( EBWR) at Argonne National The Laboratory , which develops a power of 5000 kw (electrical ) , was the first of the reactors to be completed under the AEC's nuclear power develop-

ment program (Fig . 7.4b) . The first BWR which uses D2O as moderator is the Halden (Norway) reactor which has a designed thermal power of 20,000 kw . This reactor is the only D₂O moderated BWR that has been constructed so far . An important advantage of the D₂O moderated BWR is the high conversion ratio , i.e. , the number of new fissionable or fuel nuclei produced per fuel nucleus consumed in the reactor . With natural uranium a ratio of nearly unity can be reached in this reactor . Extensive tests have shown that the BWR is a comparatively safe reactor type and that a properly designed BWR is largely self- regulatory . A sudden power surge , should it occur , will induce the formation of steam bubbles or steam voids in the liquid moderator , thus reducing the thermalization of neutrons while increasing the neutron leakage rate . As a consequence , the fissioning rate and , hence , the power production will

fall .

There is , however , some evidence that under certain conditions of operation power oscillations with increasing amplitudes can occur so that the claim of inherent stability of these reactors should be regarded with

185

The Nuclear Chain Reaction 84" I.D.

Thermal shock shield

Control rod guide Control rod

W.L.

Pressure vessel

Water level

Stainless steel cladding

Shroud

54 "

Fuel assembly Feedwater distribution

Dummy fuel assembly

ring 00 -Thermal shield 48 " fuel

Fuel

Lower grid Forced circulation , discharge

Core support Baffle Forced circulation inlet

Control rod extension Control rod thimble

FIG .

7.4b . Core

of EBWR .

some reservation until sufficient information about the origin and cause of these power oscillations becomes available ( see J. A. Thie , Nucleonics , 16 ,

No. 3 , 102 ( 1958 )) . Many of the designed power reactors which are fueled with enriched uranium are of the heterogeneous type to ensure a maximum contact area between fuel and coolant for the effective transfer of heat from the fuel to the circulating coolant . FUEL ELEMENTS . The fuel elements design are

of two

attachment ,

that have been used in reactor

main types , the cylindrical rod type with or without a fin and the plate or sandwich type ( Fig . 7.5 ) . For reactors

Elementary Introduction to Nuclear Reactor Physics

186

34

1/16 Zirconium - uranium - niobium alloy 0.018

3%

334

0.208

-Zircaloy 2 Stainless steel bottom tube

77 %

Cross section

CHISED

54

Fuel plate Spring Stainless steel top fitting

(a)

(b)

(c )

--

FIG . 7.5. Typical fuel element shapes . (a) EBWR fuel element . (b , c) Fuel elements for the Shippingport PWR . b Uranium oxide pellets in tubing of zircaloy with coolant able to flow through tubes . c U235 used as fuel with zircaloy cladding , with coolant flowing through channels . (d) Cross sectional view of sandwich - type fuel element as used in MTR . (e) Cross section through a fuel element of the Halden (Norway ) heavy water reactor .

The Nuclear Chain Reaction

187

(d)

f

()

(e)

(g) FIG . 7.5f. New , enriched fuel elements of the Brookhaven National Laboratory uranium -graphite reactor . (Courtesy of Brookhaven National Laboratory ) . (g ) Fuel element cross sections of the Canadian NRX - reactor (rod elements ) and NRU -reactor (plate - type elements ) . ( Courtesy of Atomic Energy of Canada Limited .)

Elementary Introduction to Nuclear Reactor Physics

188

operating at fairly high power levels the former are less suitable because of their relatively small contact area with the coolant , which is not sufficient for the required rate of heat removal from the core ; the sandwich type is generally preferred . The fuel elements must be " canned , " i.e. , they must be completely by cladding materials and sealed in to prevent corrosion of the fuel material by contact with the coolants , and also to retain the radioactive fission products and prevent their escape into the exposed parts of the unit . Aluminum and , to an increasing extent , zirconium are being widely used as cladding material . enclosed

The fuel UO2. The

elements can be either pure uranium metal or uranium oxide , advantages of the oxide are its longer fuel life and its com-

of

very likely

,

so

.

reactors the type far described fuel ratio and enriched fuel

to of

-

Water cooled and -moderated reactors employ lattices of relatively low moderator

is

.

†

of

to

large power reactors the preceding class

of

in

a

as

,

in

to

,

UO 2 fuel the further development

Nevertheless the use increase

of

as

its

patibility with hot water . The pure unalloyed metal undergoes violent chemical reactions when it comes in contact with hot water . On the other hand , the main drawback in the employment of the oxide fuel element is compared rather low thermal conductivity that the pure metal

a

,

(

)

,

(

.

to

of

,

to

.

is

required for their successful operation All existing designs necessitate the use of pressure vessels contain the core but their size can be of reasonable diameter for relatively high power output because the high fuel moderator ratio used They have generally relatively high power i.e. power per unit volume and high specific powers i.e. power densities ).

per unit mass of fuel

)

(

,

-

to

or

,

A

-

,

The Gas Cooled Natural Uranium Graphite Reactor Table 7.3 third type of power reactor which together with the previous two can ,

-

a

It

).

.

(

-

k to

.

a

a

of

.

or

,

be

,

a

of

.

or

it

-

is

have become more less standardized the gas cooled special natural uranium graphite moderated reactor Fig 7.6 holds attraction because can be constructed and operated without the use of fuel moderator that requires isotope separation The graphite moderator high degree purity containing must be of minimum amount of parasitic absorbers and the fuel must uranium metal As coolant either nitrogen carbon dioxide can be used The fuel moderator ratio

be considered

.

at

.

in

to

must be chosen close the value giving maximum for the reactor geometry selected fuel channels that The fuel elements are inserted regularly repeating intervals The pass through the graphite moderator

).

11 ,

,

17 ,

O ,

(

†

.p

156

to

Y,

It

has been found that the addition of small amounts of or Nb₂O the uranium oxide greatly improves its thermal conductivity see Nucleonics No.

England

, ) , , ,( ) - , /

type

optimized

1

-G (

Marcoule

15

(2 - ,3 -G -G to

Similar

-

natural

1

140 tons

G

to

uranium

32

Heterogeneous

Heterogeneous

-G

Similar

1

Graphite

Graphite

Graphite

graphite reflector

under

ft

29

)

,1

-

under pressure

CO2

pressure

CO₂

pressure

CO2 under

pressure

atm

Air

at

pressure

150

flux

neutron

thermal

5.5

40

160

180

150

30

el

1012

300

30

) (

France

, ,

France

Heterogeneous

long

ft of

Graphite

at atm

Air

Mw

Thermal

)el (

Marcoule

Pu production

France

diam

10

core

cm long

;

cylindrical

100 ton

Mg clad fuel

reactor

2.6 cm diam

uranium

to BNL

elements

natural

Similar

graphite reflector

Graphite

2

Marcoule

Heterogeneous

clad

finned

graphite reflector

under

pressure

CO2

Power

×

kg year

France

fuel

Al

235 tons

elements

clad

high

alloy

Graphite

,

Pu production

cylinder

31

BNL

Mg

finned

1.15

Coolant

,

for Pu production

ft

to BEPO

diam

long

21

Similar

3.5

size

elements

130 tons

,.; , , ft , -

in

diam

fuel

uranium

,. - , ; , - ., , ft, , ;. ft , ,.

cylindrical

Moderator

3

,

EDF

, ft

Heterogeneous

Arrangement

,

core

Fuel

8

England

Natural

and

×

Windscale

Heterogeneous

Fuel

Reactors

(

Hall

Arrangement

Uranium

Natural

Reactors

Power

)) (

Calder

Location

, ,

Comments

Moderated

7.3

,

Name

Graphite

TABLE

The Nuclear Chain Reaction 189

190

Elementary Introduction to Nuclear Reactor Physics

-

-

FIG . 7.6. Fuel loading face of G - 1 Marcoule reactor natural uranium graphitemoderated . (Courtesy Commissariat à l'Energy Atomique , through French Embassy Press and Information Division .)

The Nuclear Chain Reaction

191

gas coolant passes through the fuel channels and carries away the heat generated in the fuel elements .

The British Calder Hall reactor with a power of 182,000 kw belongs to this category of power reactors . Another important function of this reactor , besides the generation of power , is the production of Pu239 . The Oak Ridge National Laboratory reactor of this type with a power

of 700,000 kw

2 fuel elements uses partly enriched fuel in the form of UO₂ and He - gas as coolant . The French reactors at Marcoule , the G1 , G2 , and G3 , are dualpurpose reactors designed for plutonium production and for the production

of electrical

power .

The Homogeneous Reactor

(Table 7.4 ) reactor is shown schematically in Fig . 7.7 . The active solution of uranylsulfate ( or nitrate ) in light or heavy water circulates directly through the heat - exchanger and back to the active core which is contained in a stainless steel sphere . The size of this sphere is large enough to contain a sufficient amount of material for a chain reaction to be possible

A homogeneous

in the core itself

( i.e. ,

critical mass ) but nowhere else in the system . The to enable the liquid to reach an

system as a whole is kept under pressure

operating temperature which is well above the normal boiling point of the solvent .

An attractive feature of this type of reactor is the continuous operation that is possible without the need for periodic shutdowns to replenish the burnt - up fuel or to remove the Pu239 produced . A portion of the solution can be withdrawn for reprocessing and additional fuel can be introduced to compensate for fuel consumed without interfering with the continued operation of the reactor . It is essentially self- regulating and stable during operation , which obviates the need for elaborate control mechanisms . The operating power level is determined by the rate of heat removal from the reactor , and if this remains steady the reactor maintains a steady operating temperature also . † The use of a highly enriched fuel solution makes possible the attainment

of a for

high neutron flux and of a power level of several thousand kilowatts a relatively small reactor size . Its small critical size with its high power

density and specific power are distinct advantages of this class of reactors . By surrounding the active core with a blanket of fertile material this type of reactor can also be used as a breeder reactor of relatively high breeding ratio . † The corrosive

effect of the solution on the reactor components presents a problem and gold or platinum cladding must be used for their protection . In addition , the high activity of the circulating fuel solution ( ~ 109 curies ) requires effective shielding of the primary cooling system , which is a major difficulty .

192

Elementary Introduction to Nuclear Reactor Physics Heat exchanger

Heat exchanger

Reactor core

Fertile blanket

(a)

CORE ACCESS

FUEL PRESSURIZER

FUEL BLANKET

BLANKET

←

PRESSURIZER

EXPANSION

JOINT

BLAST SHIELD (74 in. I.D., 304 STAINLESS STEEL , 1-1/ 2 in. THICK)

CORE VESSEL (32 in. 1.D., ZIRCALOY- 2 , 5/16 in. THICK )

COOLING COILS

DIFFUSER

PRESSURE VESSEL (60 in. I.D., 347 STAINLESS STEEL CLAD, 4.4 in. THICK )

VESSEL ASSEMBLY

HRT REACTOR

(b) FUEL BLANKET

FIG . 7.7 . Homogeneous reactor . (a) Schematic diagram . (b) Reactor of the HRE -2 . (Courtesy of Oak Ridge National Laboratory .)

vessel assembly

(1-

(

(2-

1

-( ).

kg U235

in psi

-2

, U235

Pt

or

H₂O

Graphite 11 in

H₂O

and

in

Steel

circulation

exchanger

of fuel solution

Natural

heat

through

solution

forced

Fuel

10,000

5000 to

1000

2000

300

1.2

1.7

1013

1013

steel

sphere

Stainless

Cylinder

sphere

diam

diam

16 in

high

diam

cylinder 15 in

Core

high

15 in

core

diam

ft

: 2,. .

of fuel solution

necessary to withstand corrosive effect

-cladding

2.8

kg

gold

As above

4000

, ; )(

core

H₂O

;

Alamos

4.2

uranylsolution

,

Los

enriched

As above

x

LAPRE

%

.. : ,.

nical failure

U

phosphate

.)

3 (

tor exp dismantled in 1957 after mecha-

235

90

14 in

D0

,.

Los

reac-

,)

)el (

power

%

, psi

D₂O

1013

. .

Alamos

,4

2000

kg

×

LAPRE

)

enriched

kg to

uranylsulfate

) (

90

1.9

23

NL

exchanger

140

1000

el

HRT

1000 psi

pumped external

through heat

Size

ft

Ridge

pressurized

Fuel

solution

10 in

Flux Thermal

13

Oak

1954

U235

solution

H₂O

kw

Thermal

ft

HRE

,.)

dismantled

NL

enriched

sulfate

90

%

, ;3

uranium

,,0 . D

Ridge

,

Over

is

Oak

Coolant

Reflector

Power

(

exper

Moderator

Type

)

homogeneous

Fuel

Reactors Uranium

(

reactor

and

Enriched

,

Arrangement

Fuel

Power

)

HRE

,

Location

7.4

) (

Comments

Name

Homogeneous

TABLE

The Nuclear Chain Reaction

193

x

.

;

Elementary Introduction to Nuclear Reactor Physics

194

The HRE (homogeneous reactor experiment ) at Oak Ridge National is the prototype of a homogeneous power reactor , and its forerunner , the HRE - 1 ( later dismantled ) , was one of the earliest power reactors to produce electrical power from nuclear fission energy . † That reactor operated at a power of 1000 kw using a 93 % enriched uranium sulfate solution contained in a stainless steel sphere of 18 in . diameter . ORGANIC MODERATORS (Table 7.5 ) . The employment of water as moderator -coolant has certain drawbacks such as the need of high pressurization for reactors which must operate at high temperatures in order to Laboratory

prevent the water from boiling , as well as the need to use special materials to contain the water because of the highly corrosive property of pure hot water . Some of these drawbacks can be circumvented by the employment

of

organic moderator - coolants such as polyphenyls or their derivatives . These substances have high enough boiling points to permit their use at fairly high steam temperatures without the need for pressurization . The prototype of this arrangement is the OMRE ( organic moderated reactor experiment ) which is designed to operate at power levels of 5000 to 16,000

An

kw .

additional

advantage

of some significance which ought to be

mentioned in connection with organic moderator - coolants is the induced radioactivity in the pure materials . +

7.7

low

Reactor Control

Reactor control provisions are included in all reactor designs , the most common being control by means of movable neutron absorbing control rods which can be inserted in the core or the reflector (see page 267 ) of the reactor installation . Control by means of control rods is normally of three

of effectiveness : ( 1) a fine control by ( 2) a coarse or " shim " control ; and ( 3 ) an degrees

means

of " regulating rods " ; or " scramming "

emergency

control , which shuts down the reactor immediately . Other methods of control are available in the form of fuel control or configuration control . In the former method fuel is either added or withdrawn ; in the latter method the neutron multiplication of the system is altered by changing the reflector configuration or that of other movable reactor parts , thus affecting the neutron leakage loss ..

Control by the insertion or the withdrawal of strong neutron absorbing

† The first reactor to produce electricity from nuclear energy was the Argonne National Laboratory's EBR - 1 in Idaho . See C. A. Trilling, " OMRE Operating Experience , " Nucleonics , 17 , No. 11 , November 1959 .

(

). ,

(

,

(

plates

U2

type

cladding

terphenyl

300 psi

terphenyl

flux

x

1013

neutron

thermal

16,000 and

loops

)

, ;

)

25.5

steel 235

sandwich

,

stainless

Na

Diphenyl

and

,

Arco

Diphenyl and

) ,. ~ ( -

steel fuel

90

loop

Thorium

fuel

10,000

flux

1013

Core

in

.

3

kg

high

by 24

pressure 4.5 ft

high vessel

diam

by 36

. in ft : ;: . square

22

high

diam

core

Size

Cylinder

:

Idaho

stainless

enriched

around

%

UO2

Highly

blanket

circu-

(

experiment

organic moderated reactor

design

Bi solution

lating transferring heat to intermediate

U

neutron

thermal

3.5

6000

20,000

)

OMRE

,

)

BNL

Graphite

,

prototype

..

U233 to be used

Later

uranium

in Bi

enriched

,

type

%

dissolved

90

stainless

Sodium

-

homogeneous

metal

;

liquid

in

tubes

, Graphite

((

×

reactor

slugs

U238

kg

; 2190

kw

Thermal

)el

steel jacket

uranium

Coolant

Power

ft ft 66

fuel

kg U235

enriched

62.5

2.8

%

Calif

Reflector

and

Moderator

Reaction

:, .

LMFR

reactor

,

sodium

Fuel

)

experiment

and

Arrangements

Fuel

Types

Reactors

(

SRE

,

Location

Power

Miscellaneous

7.5

) (

Comments

Name

TABLE

The Nuclear Chain 195 95

196

Elementary Introduction to Nuclear Reactor Physics

materials in the shape of rods or strips is particularly suited to the control of thermal reactors because of the high thermal neutron cross sections of the materials that are incorporated in the control rods , such as boron , cadmium , or hafnium . The fuel control method is best suited for homogeneous or liquid fuel reactors , where this method presents no difficulty . The addition of poisons , " such as a boron salt solution , is difficult to reverse and is normally employed only as a shutdown measure .

"

7.8

Reactor Shielding

All

nuclear reactors , except those operating near zero power level , are of intense neutron and y -radiation and represent , therefore , a serious health hazard to the operating personnel and research workers . Provisions for their health protection must therefore be made by surroundsources

ing the reactor core and active components with a radiation shield . This consists usually of about 6 to 8 ft of high density concrete or of an equivalent thickness of other suitable materials . Because of this shield's primary function of health protection it is generally known as the biological shield . Through the absorption of neutrons and y -radiation , that part of the shield which is in immediate contact with the core can heat up considerably and may require special cooling facilities in order to prevent it from cracking or suffering other heat damage . For this reason , the inner portion of the shield generally known as the thermal shield constructed of steel plates .

7.9

Research

,

is usually

Reactors

The design of research reactors is aimed at providing relatively high neutron flux densities for experimental work and making the neutrons accessible to the experimenter . In contrast with power reactors , the power produced here in the form of heat is an undesirable by - product which should be kept to a minimum in order to eliminate the need for elaborate cooling arrangements . By starting with Eq . 4.28 , it follows easily that the average thermal flux and the reactor power P for U235 fuel are related by the expression

φ=

P (watts ) m (grams )

x 4x

1010 n

/cm² sec

which indicates that the flux is determined by the specific power P/m of the reactor . Since m is proportional to the volume of the reactor core , it is

The Nuclear Chain Reaction

197

4 , the necessary power can be reduced by making the core volume smaller . With a good moderator like D₂O and enriched fuel a very compact size ( linear dimensions of the order of 1 ft) is possible without falling below the critical size requirement . seen that , to achieve a given flux

A somewhat more realistic estimate of the neutron flux , instead of the optimum expression given earlier , is about one - third less , so that one can take the average thermal flux to be given approximately by Pth

= 2.6 x

1010 ×

P (watts ) m ( grams )

where m is the critical mass of the reactor fuel

U235 .

Experimental facilities are provided by openings that lead into the reactor core (“ glory hole " ) or into the lattice where the neutron flux will be composed of the entire reactor spectrum , with fast neutron flux and openings are provided thermal neutron flux in about equal proportions . leading into the moderator region where no fuel is present , the neutron flux will be preponderantly thermal , with some admixture of fast neutrons ,

If

however , since the fission neutron flux decreases distance from the fuel .

exponentially

with

If well - thermalized

neutrons are required , use is made of a thermal of the moderator against a portion of one side of the reactor from which the reactor shielding has been removed . We shall here describe four main types of research reactors : ( 1) the water boiler (Table 7.6) ; ( 2 ) the swimming pool ( Table 7.7 ) ; ( 3 ) the column , which is an extension

tank - type reactor ( Tables 7.8 and 7.9 ) ; and (4) the graphite - moderated reactor (Table 7.10 ) .

(Table 7.6) The water boiler ( Fig . 7.8 ) is usually a homogeneous mixture of a highly enriched uranium salt dissolved in ordinary water and contained in a small stainless steel vessel surrounded by a reflector and shield . Unless the The Water Boiler

reactor is run at a very low power ( a few watts ) , a stainless steel cooling coil must be provided inside the core . Furthermore , the vessel must also be provided with a gas circulation and recombination system because considerable amounts of H2 and O2 are dissociated during the operation of the reactor due to the intense ionizing radiation in the core . A good example of this type of reactor is the SUPO at Los Alamos which has been in operation since 1951. It is run at about 50 kw and provides a neutron flux of ~ 1012. The fuel solution normally contains about 1 kg of highly enriched uranium salt in a total volume of 15 liters .

This reactor type

is not to be confused with the

BWR

described

earlier ,

198

Elementary Introduction to Nuclear Reactor Physics

FIG . 7.8 . The Armour

Research Founda-

tion water boiler reactor . (a) Stainless steel reactor core with coolant circulating coils ..

Drivemotor

Reactor core

shielding Concrete

Controlrods exposure Neutron facility

Graphite reflector

-Movable concrete door

(b) Cutaway

view of the assembled reactor facility .

although the similarity in denotation may be misleading . The reason for the term " water boiler " is because of the fact that a sudden increase in power will cause the formation of steam bubbles in the solution , which in turn will quickly shut down the reactor . This behavior is an important safety feature of this reactor type . Under normal operating conditions boiling does not occur , since the temperature of the fuel solution is generally kept below 80 ° C by a water coolant circulating through the coils inside the core vessel .

The Nuclear Chain Reaction

2 in . diam . pneumatic tube

199 1½ in. diam . central exposure tube 3 in. diam . beam tube

1½ in . diam . curved pneumatic tube

6 in. square access ports Bismuth shield , aluminum clad

4 in . diam . beam tube

4 in. x 4 in. removable graphite stringers 6 in. to 4 in. 6 in. diam . access port

diam . beam tube 9 in.x9 in.x18in. dense -concreteshieldblocks

Rolling door

Boral liner Void space

4 in . diam . vertical beam tubes

-Control

- safety rod drive assembly

Core tank assembly

Reflector

Secondary enclosure envelope

Graphite thermal column Location of ion and fission chamber assemblies

Lead shield

FIG . 7.8c . Cross sections through the installation . Foundation .)

( Courtesy

of Armour

Research

,

Name Location Comments

,

,

( , ,) (

,)

,

(

()

. ) , ,

. 2-

,

,( ) ( ). 1 %

%

,) ) . ., ,

() (

.

salt in light water

H₂O

H₂O

107

50 H₂O

0.001

50 H₂O

-

1011

x

graphite

BeO and

Graphite

1012

×

9

Plutonium solution

enriched uranyl sulfate 90 1750 grams U235

Graphite

Stainless steel

As above

cylinder 103 diam high

spherical core

15 liter

stainless 14 steel sphere

. sphere

diam stainless steel

14

(

Saclay France

%

ft

Proserpine

%

.

KEWB kinetic exper WB Calif Atomics Internl

, H₂O

.

enriched uranyl sulfate

( x2

88

1011 average 1011

x

1010

109

As above

)

ft

Armour research ARR reactor Chicago

1000

10

108

As above

As above

11

,.

through heat exch

solution circulat

X 0.005

0.1-0.5

1.7

35-45

1012

1011

diameter One stainless steel sphere 15 liter

Size

ft

90 enriched uranyl sulfate under 1000 psi pressure

. , D.0

H₂O

Graphite

H₂O

90 enriched UO₂SO solution 848 grams U235 liters 14

,

5

H₂O

H₂O

Graphite

Graphite

H₂O

H₂O

H₂O

Graphite

H₂O

UO SO solution 694 grams U235 2950 grams U238

-,

ft

HRE homogen reactor exper Oak Ridge NL replaced by HRĒ

%

enriched uranyl nitrate 90 solution 700 grams U235

,

,

Carolina State College first university

in

×

reactor reactor

13

-

2

North

U

Cal

nitrate liters

,

Livermore

%

2

Water boiler

enriched

5

0.8 kg

H₂O

X

water boiler neutron WBNS Calif source NAA

U graphite

BeO and

X

90

H₂O

Thermal Flux

X 3

SUPO super power WB Los Alamos replaced HYPO

, , ; -

nitrate solution

;

,

897 grams U235 5341 grams U238

10-5

Power kw Thermal

ft

Enriched

H₂O

BeO and

H₂O graphite

Coolant

Reflector

Type

Moderator

Homogeneous

Reactors

,

HYPO high power WB Los Alamos replaced LOPO

uranyl sulfate 580 15 liters grams U235 3380 grams U238

Enriched solution

Research

) ((

low power water world's first water Los Alamos

Boiler

,

Arrangement

Uranium

Water

)

LOPO boiler boiler

Enriched

7.6

Fuel and Fuel

TABLE

200

Elementary Introduction to Nuclear Reactor Physics

5

The Nuclear Chain Reaction

201

The Swimming Pool Reactor (Table 7.7) This reactor (Fig . 7.9 ) contains a large amount of highly purified water ( 100 to 200 meters³ ) in a concrete tank into which is immersed the fuel assembly attached to a steel framework and suspended from a bridge which spans the width of the swimming pool . The fuel core consists of a rectangular assembly of plate type fuel elements made up of highly enriched uranium alloy and clad with aluminum which is immersed from 5 to 7 meters below the water surface . The total amount of fuel is about 3 kg of uranium . The water in the pool plays the triple role of moderator , coolant , and shield . Convective cooling is sufficient for power levels up to 100 kw . Beyond that , resort must be had to forced cooling by means of pumps to a downward flow of the water . This is necessary because of the p ) N16 reaction which is of considerable practical importance with water - cooled reactors in general . The product nucleus of this reaction , N16 , is short - lived , having a half- life of 7.35 sec , and emits a very penetrating cause

O¹6 (n ,

y -radiation of 6.2 Mev energy in

82 %

of the disintegrations

The downradioactive N16 from .

ward draft of the forced circulation the reaching the surface and contaminating the air. At a power level of 100 kw the neutron flux available for experimentation is 1012. The first reactor of this type at Oak Ridge was designed with the prevents

view of allowing radiation effects on bulky materials to be observed by immersing them in the water surrounding the core . The mobility of the reactor core is very convenient for the preparation and rearrangement of experimental set - ups . The BSF (bulk shielding facility ) at Oak Ridge and the NRLR ( Naval Research Laboratory reactor ) are large open pool reactors of this type . The Tank -Type Reactor

(Tables

7.8 and 7.9 ) very 7.10 ) is similar to the swimming pool as regards , the reactor core but the size of the pool has been reduced considerably to that of a tank . With proper heat removal equipment the operating power of these reactors can be pushed high enough to provide a neutron flux of 1014. The highest experimental flux attained so far is made available in the MTR (materials testing reactor) at Arco , Idaho , which uses 4 kg of highly

This reactor ( Fig .

enriched uranium assembled

in standard fuel elements

with aluminum

cladding . It is a water - cooled and moderated tank - type reactor , with an available average neutron flux of 2 × 1014 at a reactor power of 30,000 kw . A number of tank - type research reactors employ DO instead of light water as moderator - coolant as , for example , the CP - 5 at Argonne National Laboratory

,

which uses about

and assembled

with aluminum

1 .

to

2

At

kg of 90 % enriched uranium , alloyed an operating power of 1000 kw this

Thermal column

FIG

7.9a

Schematic

Control rod

202 side view

swimming

. pool reactor

-

of

Fuel element core

Experimental holes

Graphite reflector

Concrete shield

Irradiation facility

University

of

. .

University

Ford Michigan

compartments

Michigan

and

Reactor

is

)c( (

type that has two

The

a

core

swimming movable

a

Courtesy

pool

. ). of

203

Horizontal

experimental holes

7.9b

Two

compartment

Corporation

. .

FIG

swimming pool

reactor

Courtesy

Bendix

Thermal column

Fuel

)d( (

Dr.

during

J.

of

of

(.

).

Engineering

Shapiro

reactor

L. Nuclear

core

full

power University

,

Dept.

Courtesy

of

Aviation

Core bridge

Michigan

operation

., of

.)

of

,

,

)

L.

( (

.) ,

(,

.

.

.

.) (

(

, , ,

Fontenay

Fontenay

( ( ( 1

Minerve

,

France

France

uranium uranium

Enriched

Enriched

1013 1013

1000 1000

,

-U

%

Al alloy Alenriched 20 canned 3.5 kg of U235

) )

)

-

Italy

uranium

Enriched

)

France

1012

100

enriched type elements 3.3 kg U235 MTR

Al alloy 46

-U

Grenoble

-

,

-

Melusine

)

England

;

RS

( %

Triton

.

1013 1000-5000

H₂O H₂O H₂O H₂O

H₂O H₂O H₂O H₂O

H₂O H₂O H₂O H₂O

1011 Very low

H₂O H₂O

H₂O

28

diam 10 21 deep

ft

Harwell

30 ft

x 15

27 ft

21 ft

ft 24

,.

LIDO

1011 10-100 H₂O

H₂O

H₂O

X

type fuel assemblies

1012

27

32

10

ft

23 MTR

0.7 100 H₂O

H₂O or graphite

%

90

H₂O

1013 1000

H₂O

H₂O or graphite

ft

Geneva

-

type elements MTR enriched

; ; H₂O

3

kg U235

1013

ft

x

Swiss reactor exhibit

, -

BSF fuel elements

X 1.4

281 49

ft

reactor

. 1000

H₂O

H₂O or graphite

H₂O

1012

ft

x

1013 20

40

x

20

ft

x

State Univ

- -

x

type elements 2.8 kg MTR U235

100-1000

1-2

ft

x

x

Penn

ft

; 1 x.

x H₂O

100-1000

Size

ft

ft

Univ of Mich Ford nuclear reactor

% H₂O or graphite

H₂O

X

Battelle research reactor

x H₂O

BeO or H₂O

Thermal Flux

Type

ft

type fuel elements fuel MTR 24 in 21 in core 21 in

H₂O

Coolant

Pool

Power kw Thermal

Swimming

x

Naval research lab NRLR reactor Washington D.C.

. ;.

MTR sandwich type fuel enrichment elements over 90 Al cladding 3.5 kg of U235 2ft 1ft Fuel core

Reflector

Reactors

Reactors

) ) ((

BSR The original swimming pool reactor BSF bulk shielding facility Oak Ridge N.

Uranium

Research

Moderator

Enriched

,

Arrangement

Fuel and Fuel

Moderated

7.7

-

Name Location Comments

Water

-

Light

TABLE

204 Elementary Introduction to Nuclear Reactor Physics

x

ft

8

x ft

,

The Nuclear Chain Reaction

205

reactor provides an average thermal neutron flux of 1013 for experimental purposes . Other examples are the Canadian NRX and NRU reactors shown in Figs . 7.11 to 7.13 . The Graphite -Moderated Natural Uranium Reactor (Table 7.10 ) This reactor was the prototype of all reactors , starting with the CP - 1 Chicago pile ) built under Fermi's direction in Chicago . Similar assemblies , (

Instrument hole Experimental hole

Thermal column

Experimental hole

FIG . 7.10 . Tank - type

water - reflected reactor. (Courtesy Bendix Aviation Corporation .)

like the GLEEP (graphite low energy experimental pile ) and BEPO ( British experimental pile ) at Harwell , England , and the X - 10 Clinton pile at Oak Ridge and one at Brookhaven ( Fig . 7.14 ) are well -known examples as they have been the " work horses " of reactor technology and research . Their importance for research has lessened since the advent of the other types of research

reactors which were described earlier .

Primary interest in this

,

Name Location Comments

);

( ,

. ,

.-

-

,

;) . -

( . ( )(

) %

, ,3

-

)

(

H₂O

175,000

x

1014

22

18

Core in stainless steel tank 25 diam

Core tank 5.3 diam 14.8 high immersed in swimming pool

: 8ft

ft

x

( ,

)

., ..

32 in diam 36 in high

) 3(

U%

-

-4

49

average

40

.

graphite

Be and

1013

1014

10

26

.

MTR type fuel elements kg Al alloy more than 90 enriched

1000-5000

20,000

2

HO

H₂O

8

H.O

H₂O

26

: ,ft

H₂O

H.O

30 MTR type over 90 enriched kg of U235

H₂O

1013

average

X

Overall 32 34 reactor tank 56 diam 30 high fuel assembly 16 in 28 in 24 in

ft

plates per element

0.005-0.1

1014

x

MTR type 3-4 kg U235 51 fuel elements 19

H₂O

3000

X

maximum 2.7 X 10¹4

Size

ft

x

Heterogeneous

in H₂O

H₂O

)

MTR type fuel elements

Be

5.5

( (

,

ETR engineering test reactor Arco Idaho

-

3 x 3

x

ft

Heterogeneous

U

. .

ft

Heterogeneous

19

Elementary Introduction to Nuclear Reactor Physics ft ft ft

x

ORR Oak Ridge research reactor

>

,

x

Heterogeneous

,, -

in

., ; : . . ft .x .

RMF reactivity measuring facility Arco

%

)

H₂O

( 40,000

Flux Thermal

Neutron

(

)

MTR type 3.4 kg of U235

H₂O

Coolant

Cooled

Power kw Thermal

and Water

)

graphite

Be and

Reflector

Moderated

ft

:x

Heterogeneous

Reactors

x:

ft

LITR low inten sity test reactor Oak Ridge Natl Lab

H.O

Moderator

, -

Sandwich type 90 enriched fuel elements of Al alloy curved vertical fuel plates contained in x 24 in box About 4.5 kg U235 in Al tank 54 diam with side extension to form water well

Water

Research

) (

Heterogeneous

Fuel and Fuel Arrangement

Uranium

7.8

ft

OWR Omega West reactor Los Alamos

,

Arrangement

TABLE Enriched

-

Type

-

MTR materials testing reactor Arco Idaho

Tank

206

7.8

Continued

(

Name Location Comments

TABLE

,

(

,)

(

, ).

)

,

()

,,

%

2

-

-

4

,) %

10

Similar

RFT

,

Heterogeneous

H₂O

H₂O

H₂O

H₂O

300

2000

25,000-50,000

1011

1014

1014

1012

X

X

X

8

,,

enriched fuel elements 3.5 kg U235 32 units of 24 subunits 16 fuel elements per subunit

H₂O and Be

20,000

10

Cylindrical core 40 cm diam 50 cm high

Al tank diam

1

(

) (

) (

USSR

%

TRR

%

; H₂O and Be

3.4 X 109 1012 improved version

Core 28 in diam 36 high

ft

Heterogeneous

;

Over 90 enrichment type elements MTR kg U235

H₂O

H₂O

30

0.1

1014

Size

. :

USSR

in

4

RFT

x

-

Heterogeneous

in

-

Mol Belgium

, ,.

×

4

BR

H₂O

H.O

ft

90 enriched MTR type fuel elements 4.2 kg U235

Graphite and H₂O

H₂O

,

enriched U3O Al 20 fuel plates kg U235

Graphite 24

H₂O

)( (

Heterogeneous

-U .2

in

5

HFR high flux reactor Petten Netherlands

Al

Heterogeneous

alloy disks stacked slug tubes 24 2.8 kg U235

20,000

H₂O

Neutron Flux Thermal

)) (

Argonne N.L. University training reactor

60

Schenectady

Heterogeneous

-

TTR thermal test reactor Knolls Lab

H₂O

H.O

(

Coolant

,. : . in.

ARGONAUT

, )

MTR type fuel assemblies

Reflector

(

Heterogeneous

Moderator

)

WTR Westinghouse test reactor Pittsburgh

) Arrangement

Power kw Thermal

) (

Arrangement

Fuel and Fuel

The Nuclear Chain Reaction 207

:, .

2

to

,

,

()

-3

)- ) ('3 (5

-

;3 5-

-

,

,

in

) ) ( ( ). . (

,

(

Norway

in

1

, Graphite 70 cm

D₂O

100-350

200,000

3

1014

1012

X

1013

cylinder

10

high diam

Steel tank diam high

Al vessel diam high 12

.

11

,0D

)

Tank meters diam

ft ft

(

Kjeller

in

,.

JEEP

,,0 D

D₂O tons

ft

,

fuel rods 65-76 natural 35.5 kg per rod Al tubes

D.0

.

Chalk NRU Canada

2 H₂O

9

fuel rods

6.8

×

:

U

30,000-40,000

108

ft ft 8

18 tons

, H₂O air and circulation of D₂O

ft in of graphite

0.01-0.03

ft diam

high

high

Physics Elementary Introduction to Nuclear Reactor

,. , ,. : . ft

200 natural

,

River

%

ft

NRX Canada

D7 Graphite 24

,,0

176 natural U rods Al clad 10.5 tons natural U

2

68

River

ft

D₂O 10 tons

2

diam slugs 1.285 Al jackets long slugs 148 rods holding

4

Nat

ft

in

Chalk

,. . ; , - . 9, ; ,,, U. -U -U . 6 6

22

Chalk River ZEEP Zero Energy Can exper pile

2

.

graphite

2000-4000

83

Core

: ft

enriched sandwich type fuel plates 1.7 kg of U235

D₂O

1013

Al tank diam

6

D₂O of and

X

1012

ft

tons

X

5 3

300

ft

Al alloy over 90

ft

D₂O

Al tank diam high

Size

,.

ANL

tons

2 Graphite

average 1011

1012

x

D₂O

300

Flux

Thermal

)

Al alloy fuel rods highly enriched 4.2 kg U235

ft

,,0 D6 tons

Neutron

ft ft 68

:

CP

ton

D₂O

Power kw Thermal

Reactors

(

long

Graphite

Type

,. :

improved ANL CP version of CP replaced by CP

in

, 3; -, ft ,6. U . Al cląd

Reflector

Moderator

, Coolant

Tank

( ) ) ( (

rods 120-136 diam 1.1 of natural U

D

Arrangement

Moderated

-O, )

Argonne N.L. first CP heavy water reactor

Heterogeneous

,

Fuel and Fuel

7.9

-

Name Location Comments

TABLE

208

:

2

, 7

U

,

)

7.9

Continued

(1

-R

(

,

)

)(

)

,

(-2

P

,

(

-U , ; -,9

%

)

,,

)(

( (( -E 2- 3-

(

))

, ,,)

)

Slightly enriched

U

20 enriched 2.5 kg U235 alloy

D₂O

D.0

ft

Graphite

5000

1014

1014

15 D₂O

Graphite

1013

2500 D₂O

1014

10,000

Graphite

D₂O

1014 10,000

x

108 maximum

1012 maximum

1012 maximum

8.7 1011

0.3 maximum

,D0 D₂O

,

U

, 2

Natural

in

2.5 kg of

Graphite ft

DO

D₂O

ft

1500-2000

N₂ or CO₂ passing between concentric Al fuel cylinders

Graphite

2

Similar to DIDO U235

D.0 10 tons

D₂O 15 tons

Graphite 90 cm some

150

D₂O

-

3

1

Italy

,

90 enriched Al alloy plates Al clad arranged boxes plates per box 2.5 kg U235

to DIDO

, ,

,

Saclay France

7 3

,

EL

ft

,

ISPRA

in

or similar

D.0 6.3 tons

, ,0D

Variable below

1.1 in diam concentric Al tons of U

,. . , 4 ,3

rods long cylinders

U

Graphite and some D₂O

,

Size

Al tank 2.5 meters high meters diam

meters high

Al vessel 1.8 meters diam 2.35

meters high

Reflectal tank 1.85 meters diam 2.54

Al tank 10 high diam

,. meters high meters diam

Al tank

Al tank ft in diam

6

:

Saclay France

,

) U

7

7

EL

"( in

×

8

:..

443 Harwell England

)"

2

ft ft

England

.,

U

-

:

,.

Harwell

in

.

DIDO

%

: , : .

DIMPLE deuteriummoderated pile low England energy Harwell

, . ( ,

136

D.O

,0D

Al tubes UO tablets Replaced by rods 1.9 tons of

,,0 D ,

-

Saclay France

U X

France

300-600

Flux

Thermal

Neutron

3

recirculation and air

(

Coolant

,.

Chatillon

Graphite 90 cm

Reflector

:

ZOE

Moderator

Power kw Thermal

( )

Stockholm Sweden SLEEP Swedish low energy experimental pile

Arrangement

Fuel and Fuel

) ( )

126 natural rods 2.9 cm diam canned very pure Al alloyed with Mg reflectal

) , ) , ( ,

Name Location Comments

TABLE

The Nuclear Chain Reaction 209

22

,0D ,0 D

,

-U

Al

%

210

Elementary Introduction to Nuclear Reactor Physics

STEEL SHIELD COOLING WATER -IN

CONCRETE SHIELDING

CONCRETE

HOTOPE SELF SERVE -SM MECHAN

STREL ELDS GRAPHITE HEUTRON BEAM HOLE

STER

PORAGE

HEAVY WATER COOLER

PUMP

FIG . 7.11 . Model of the NRX , which is a heavy water -moderated , light water - cooled tank type research reactor . ( Courtesy Atomic Energy of Canada Limited .) reactor type has now shifted to their adaptation and development as power reactors .

They all use natural uranium as fuel ( 20 to 50 tons) in the shape of cylindrical rods , or variations on this shape , clad usually in aluminum . Several hundred tons of graphite are used as moderator and reflector , with hundred cylindrical channels passing right through the mass of graphite to allow the introduction and positioning of the fuel elements . Cooling gas ( air or CO2 ) is made to pass over the elements and through the channels , usually at low pressure . 4 x 1010 for the The power and neutron flux range from 100 kw and several

2

The Nuclear Chain Reaction

211

FIG . 7.12 . The NRX -reactor and neutron research facilities at the left , radioisotope producing equipment at right . (Courtesy Atomic Energy of Canada Limited .)

GLEEP to

~ 4 x 1012 for the Brookhaven reactor . The partly for the production of Pu239 and for the preparation

30,000 kw and

X- 10 is also used of radioisotopes

A

.

heavy water - moderated reactor can be made much smaller than a graphite -moderated reactor of the same power , because the amount of D₂O required to moderate the fission neutrons is much smaller than that of

212

Elementary Introduction to Nuclear Reactor Physics

ISOTOPE FLASK larger Similar but flask is forfudrod removal used

DECK PLATE STEEL

CONCRETE SHIELDING REMOVABLE CONCRETE BLOCKS SHIELDING NATURAL URANIUM RODS FUEL GRAPHITE THERMAL COLUMN NEUTRON BEAM HOLES

Л

WATER -FILLED STEEL THERMAL SHIELDS 40TO87TONS EACH IONCHAMBERS TANK ALUMINUM

STEEL CONCRETE -FILLED BOTTOM RING 294 TONS

NEUTRON BEAM HOLES WARM HEAVY WATER

WATER -FILLED STEEL THERMAL SHIELD

HATCHWAYS TOHEAT EXCHANGERS HEAVY WATER COOLED

STEEL &MASONITE THERMAL SHIELD

HEAT EXCHANGER 17TONS

REACTOR COMPONENT SERVICE TROLLEY HEAVY WATER PUMP

WARMED RIVER WATER

PUMP MOTOR A.C. STANDBY D.C. MOTOR

COOL RIVER WATER

5 TON DOOR

FIG . 7.13 . Cutaway of NRU reactor . It is similar to NRX , but it uses heavywater as moderator - coolant . In addition to its research facilities and isotope - producing function it is also a plutonium - producing reactor . (Courtesy Atomic Energy of Canada Limited .) Since the neutron flux is determined by the power density , D₂O - moderated reactors , with their smaller volumes , can thus achieve a high neutron density , which is one of the advantages of that type of reactor . This fact explains the extensive use the D2O - moderated reactor has found as a research tool , as a test reactor , and as a Pu - production reactor , to a

is

as

,

the United States Atomic Energy

S.

,

-

,

or

,

,

-

.

†

.

to -

For an up date compilation of chain reacting assemblies see Nuclear Reactors Built Building Planned TID 8200 Commission †

very general

.

Table 7.11

yet limited

A

Norway

U. in

is

the Halden

in in

power producing reactor

reactor given research reactors

,

of

classification

a

application

single prototype

as

although

its

graphite .

,

,

,

(1

-

;

;

-)2

,2

-

. -,1 .

)

-X (

( )

, (. ( ). ( ,)

;)

.-

in

-

,

)

Graphite

Graphite

4000

.( .

Cubic

18-20

Cylinder 20 diam 20 long

25 ft cube

Cylinder diam 17 long

cube

See Second Geneva Conference

1012

4000-6000 1012

2

Fig 7.7

Air

Air

1010

30,000 X 1012

x

.

, . ,. 1 -( * /

is

reactor being reloaded with enriched uranium 685.

24 tons

ft

Air subatmospheric pressure

100 3.7

20

19 ft

,

The new fuel elements are shown

uranium

in

Natural

in

uranium

,. , . , ,7.

3

Belgium

in

Graphite

4

,

Heterogeneous

Natural

*

Heterogeneous

in

Uranium slugs 0.9 diam 12 long spacing Al clad

in

Heterogeneous

ft

Graphite

., ). ,8 ( . ,. 4 -

experimental England

,

,

20 ft

12 ft

Report

ft

The BNL 2366 Vol

12

at

6

Mol

in

4

BR

in

ft ft

305

in

,. :

Hanford

in

slugs 100 tons 1.1 long spacing

2

Heterogeneous

,

National

Air subatmospheric pressure

at

Uranium diam Al clad

ft

Graphite

1000-3800 ~ 1012

X

British Harwell

in

9

BEPO pile

49

ft ft

Brookhaven reactor

in

,. :

BNL Lab

to

2

Uranium slugs 0.97 diam long spacing Al clad

Air

,

Heterogeneous

10

ft

energy

1

ft

,

Graphite

18 ft

× 108

0.2-2

9ft

Size

x

GLEEP Graphite low Harwell exper pile England

in

3

Uranium slugs tons 1.1 diam long spacing diamond shaped fuel channels Al clad

1

ft

,

Graphite

9

Heterogeneous

2

. , , ,. , ). . . . ) ) 8 ,1 , ( 8, . 7, 6 ( . (. .4, - . -

spacing

,

Similar CP tons uranium metal 42 tons uranium oxide

x

x

pile ORNL Oak Ridge oldest of all operating reactors

Heterogeneous

ft

0.2

Thermal

Power kw Thermal Neutron Flux

x

CP Argonne Natl Lab modified version of CP dismantled

ft

Coolant

Reactors

( (

Graphite

Moderator Reflector

Uranium

)

Uranium tons uranium oxide 40 tons spherical lumps in diam at cubical

Arrangement

Reactors

Natural

(

Heterogeneous

Testing

Fuel and Fuel

and

Moderated

Research

Graphite

) ;)

CP Chicago pile Fermi's original pile Chicago dismantled and used as basis for CP

Arrangement

7.10

-

Name Location Comments

TABLE

The Nuclear Chain Reaction 213

in

)

p

.

12

P

Elementary Introduction to Nuclear Reactor Physics

214

FIG . 7.14 . Model of the Brookhaven graphite reactor core with the 5 ft heavy concrete shielding removed . At the right is the loading face showing large number of cylindrical holes into which uranium slugs or materials to be irradiated can be inserted . The round ports at the left are exit holes for neutron beams to be used for experiments . (Courtesy of Brookhaven National Laboratory .)

TABLE

7.11

Research Reactors Reactor

Type

Thermal

Fuel

Natural Natural uranium- uranium

Moderator Graphite

Reflector

Thermal Neutron Flux

Coolant Power (kw ) 4000-30,000

5

x

1011-5

x

1012

D₂O H₂O

300-40,000

5

x

1011-6

x

1013

H₂O

H₂O

100-5000

H₂O

H₂O

3000-30,000

1013-2

D20

D ,0

3000-20,000

1013-1014

Graphite

Gas

graphite

Natural uraniumD20

Natural

Swimming

20 %-90 % enriched uranium

H₂O

Light water tank

20 %-90 % enriched uranium

H₂O

Heavy water tank

20 %-90 % enriched uranium

D,0

Homo-

U235 in solution

H₂O

pool

geneous

uranium

D20

D₂O Graphite

Graphite

Graphite Beryllium Graphite Graphite

H₂O

10-50

1012-5 × 1013

1011-2

X

X

1014

1012

The Nuclear Chain 7.10

215

Reaction

Calculation of

The Fast Fission Factor

k

, for a Homogeneous

Reactor

&

For a homogeneous system & is practically unity because the fast fission neutrons , upon creation , almost immediately collide with the atoms of the moderator and suffer a reduction in their energy . This will reduce the neutron energy to below the threshold energy for U238 fission before they have had a significant chance to make a collision with a U238 nucleus .

f

The Thermal Utilization Factor For a homogeneous assembly that uses either natural or enriched uranium uniformly mixed with a moderator , can be obtained immediately from its definition

ƒ

f=

Σα (uranium) Za (uranium ) +

a(moderator)

(7.13 )

+ Za (other)

This expression gives the fraction of thermal neutrons absorbed per cm³ by the uranium (numerator ) as compared to all thermal neutron absorptions (denominator ) . The last term in the denominator takes account of the thermal neutron absorptions by foreign bodies in the assembly , such as impurities (" poisons " ) and structural components which , although not directly contributing to the chain reaction , are nevertheless essential for circulating the coolants , which in turn are required for removing the heat generated

inside the core of the reactor . 7.13 in the form

By writing Eq .

f=

1

1

+

Σα (moderator) La(uranium )

+

(7.14 )

Σα(other) Za(uranium )

ƒ

becomes evident that , in order to achieve as large an as possible , the reactor should be designed so as to keep the last term in the denominator as small as possible and to make the uranium /moderator ratio as large as possible .

it

-

The Resonance Escape Probability p For a given resonance absorber we are here primarily interested in U238 —the effective resonance integral ( Eq . 6.78 ) is seen to depend on Σs No

/

of This is not the same as σ (absorber ) because Σ 8, is the scattering cross section of the fuel - moderator mixture as a whole , Σ 8, = Σ , ( fuel ) + which represents

the scattering

cross section

of the mixture

per atom

absorber .

Σ , (moderator ) . In general

,

the scattering contribution of the uranium can

Elementary Introduction to Nuclear Reactor Physics

216

be neglected as will become apparent from the numerical examples that will be given presently . The 1/E spectrum of the neutron flux (6.61 ) will be considerably depressed in the neighborhood of an absorption resonance and the flux will be depleted after having crossed the resonance energy by an amount which depends on Σ / N (Fig . 7.15 ) . (E)

Neutron flux

,

absorber

Σ .

Flux in absence of

No

in

Flux

large presence of

presence

No

medium

E-

/ 1

small 11

,

Σ,

considerable amount of absorber

No

of

Flux

in

,

absorber

Σ,

moderate amount of

E

Eo Neutron energy

σα

U238

)

resonance absorber

(

Absorption cross section

Resonance line of

E

Eo FIG

its

Neutron energy

.

σ

as

is a

of of

a

in

is

is

of

a

.

/

7.15

.

.

dependence on Neutron flux near resonance line absorber and depressed The flux the neighborhood an absorption resonance and has minimum for an energy which very nearly the same that for which maximum No

it

/

Σ

is

,

as a

,

its

dependence on The value the effective resonance integral and No must be determined experimentally and result of such experiments essentially independent was found that the effective resonance integral

to

.

/

its

,

a

.

so

,

of the mass of the moderator atom that the dependence can be taken be the same for all commonly used moderators For given fuel value depends only on the fuel moderator ratio

The Nuclear Chain Reaction

For

217

the empirical relation between the effective resonance integral and No that was found to be very satisfactory for Σ /No0 1000 barns is as follows : U238

415

7.16

for

U238

and

)

(

7.15

Th232

.

Fig

.

(

)

3.85

+

in

Σ / N

shown

of

(

is

The integral

E

=

function

E

dE

Ja

( a)

as a

Eo

Graden eff

/

282 barns

U238 80

Limiting value

infinite dilution for 70 barns

-

thorium

Th232

20

40

8

Resonance integral barns

infinite dilution for uranium

-

Limiting value

200

at

) (

at

400

20

2000 4000 barns Σε Macroscopic scattering cross section per absorber atom

400

800

)

200

(

8

80

40

No

K

or in

°

at

,

,

). )

,

at

,

it

of

.

is

.

p

a

of

,

is

U238 the

of

limiting value 282 barns infinite dilution and this value used when calculating for highly diluted fuels For the pure U238 metal the value the integral 9.25 barns The numerical value the resonance escape probability for these two limits can be obtained directly from 6.75 from which follows that for

For

integral approaches

(

., 1,

.

.

,

(

.

or

.

.

.

FIG 7.16 The effective resonance absorption integrals of U238 and Th235 300 for various moderators The curves summarize experimental results with the absorber light water the form of pure metal oxide and diluted with graphite sucrose heavy water After Dresner Nucl Sci and Engin 68 1956

→

1 .

p

so

,

to

the requirement we had found be Thus the

.

of

,

,

which will lead

a

assembly

to

ƒ

an

.

/

a

p

a

,

therefore be chosen when designing maximum value the product pf

lead conoptimum ratio must

are seen

An

to

increase

conditions for maximum and maximum tradictory demands on the fuel moderator ratio

to

f.

the thermal utilization factor

,

order

of

just the opposite to

which

necessary

in

,p

tion

is

,

,

is

.

,

of

,

0 O ,

,p

,

/

2,

infinite dilution No → the exponent approaches zero that For the pure U238 metal course approaches zero The important conclusion the requirement of increasing dilution i.e. decreasing fuel content for increasing the value of the resonance absorp-

Elementary Introduction to Nuclear Reactor Physics

218

For a natural uranium - moderator assembly of the homogeneous type the four - factor formula reduces to the following condition for a chain reaction to be possible . k∞ = εnpf =

pf >

Therefore

1 × 1

1.34

pf ≥1

1.34 ×

= 0.746

( 7.16 )

In Table 7.12 are listed two of several other possible empirical relations for the calculation of the effective resonance integral that have been proposed ( see L. Dresner , Nucl . Sci . and Engin . , 1 , 68 ( 1956 ) ) .

TABLE

7.12

Effective Resonance

Material

Integral

Range

E /N )0.475

Uranium

3.04(

Thorium

3.20 (E

of Validity

/

800 barns

/No Νο
1) , B2 must be greater

,

than B 2. Similarly

,

a reduction

in

which makes the reactor subcritical without causing a similar change in B 2, so that we then than B „². We can then summarize the critical conditions for a thermal reactor for

(keff
1 , the power -level build - up after this initial period is determined by the stable reactor period alone .

This is the reactor period that was previously introduced by 10.12 and The transient periods for Ak > 0 are negative and have no direct

10.13 .

physical significance . A quantity which enters

, however , significantly in the calculation of the reactor period and which is important in nuclear reactor physics in general is the reactivity , usually denoted by p . It is defined as the excess multiplication Ak divided by the multiplication factor kett .

Р =

Ak kett

Keff

-

1

kett

= 1

It is apparent from this to criticality

1

+

L2B2

-B't ke -B

( 10.15 )

definition of p that , for reactors operating close

,

p

= Ak

( 10.16 )

When a reactor is operated so that it is critical on the prompt neutron contribution alone , it is said to be prompt critical and it can be shown that in this case p is numerically equal to the fraction of delayed neutrons . we

If

to signify this fraction , then

use

B P= В

( 10.17 )

In

the case of U235 fuel , ẞ = 0.0064 , so that a reactor using this fuel prompt critical for ρ = 0.0064 , or, by 10.16 , for Ak = 0.0064 . Ak > 0.0064 , the delayed neutrons will not be able to make their decisive contribution to the lengthening of the reactor period , because the neutron flux will rise at a rate which is determined by the prompt neutrons becomes

If

alone . Since these multiply at a much faster rate than the delayed neutrons †

See , e.g. , S. Glasstone , Principles

of Nuclear

Reactor

Engineering , Chapter 4 .

The Nonsteady Nuclear Reactor

303

the initial rate of increase in the neutron flux will also be much greater for a prompt critical reactor ( Figure 10.2 ) .

If,

however , 0 < Ak < 0.0064 , the rate of rise of neutron flux and reactor power will be much slower because now it is dependent upon the contribution from the delayed neutrons . The reactor , when operating in this manner , is said to be delayed critical . The flux multiplication for various values of Ak is shown in Fig . 10.1 . For safe operation when the power is to be increased , the reactor should be kept in the delayed critical operating condition because the reactor period is then sufficiently long to allow adequate control of the reactor to be exercised by the operator or by mechanical means . Figure 10.2 shows , qualitatively , the variation of the relative neutron flux / with time upon the introduction of a small sudden step change , , Ak > 0 in the multiplication factor of the reactor . Initially , within a very short time after the original disturbance , the flux increase will be determined by the prompt neutron lifetime . However , after a very short time interval has elapsed , the delayed neutrons will become effective and will cause the build - up rate of neutron flux to diminish considerably . The nature of a step - function and its effect on k is illustrated in Fig . 10.3 . For a negative step change , Ak < 0 , the flux will show an initial decrease and , within the very short time interval after the disturbance before the delayed neutron contribution becomes effective , the flux

a

.

a

,

.

,

,

its

behavior will be the mirror image of behavior for positive Ak After very short time however the flux change will flatten out As the precursors of the shorter lived delayed neutron groups rapidly disappear the flux will

.

to

is

in

,

.

is

is

.

in a

eventually fall off exponentially manner determined by the longest lived delayed neutron group This essentially what happens when the reactor shut down As the control rods are inserted into the core keff reduced below unity This exponential steep reactivity negative change decrease of causes an initial , .

p

.

10.18

(

TT

NiTi

+ Ti

,

t

Σ

+

P

Tkett

is

as

=

T ,

relation between the reactivity and the reactor period equation known the inhour stated here for reference

general is

which

The Inhour Equation

)

10.7

The

of

of

to

a

(

a

.

)

After short time interval of the order of minutes the period neutron flux and the power level decrease with 80.4 sec corresponding the average lifetime the longest delayed neutron group neutron flux

5

0

/

> Ak

1

=

0

0