chapter 3 nuclear processes and neutron physics - nuclear...nuclear physics – june 2015 chapter 3...

1

copyUNENE all rights reserved For educational use only no assumed liability Nuclear Physics ndash June 2015

CHAPTER 3

Nuclear Processes and Neutron Physicsprepared by

Dr Guy MarleauEacutecole Polytechnique de Montreacuteal

Summary

In this section we first describe the nucleus including its composition and the fundamentalforces that affect its behaviour Then after introducing the concepts of radioactivity andnuclear decay we discuss the various interactions between radiation and matter that can takeplace in and around a nuclear reactor We finish with a presentation of neutron physics forfission reactors

Table of Contents

1 Introduction 311 Overview 312 Learning Outcomes 4

2 Structure of the Nucleus421 Fundamental Interactions and Elementary Particles 422 Protons Neutrons and the Nuclear Force 623 Nuclear Chart 724 Nuclear Mass and the Liquid Drop Model 1025 Excitation Energy and Advanced Nuclear Models 1226 Nuclear Fission and Fusion 15

3 Nuclear Reactions 1631 Radioactivity and Nuclear Decay 1632 Nuclear Reactions 2033 Interactions of Charged Particles with Matter 2234 Interactions of Photons with Matter 2435 Interactions of Neutrons with Matter 30

4 Fission and Nuclear Chain Reaction3441 Fission Cross Sections 3542 Fission Products and the Fission Process 3643 Nuclear Chain Reaction and Nuclear Fission Reactors 4044 Reactor Physics and the Transport Equation 42

5 Bibliography 446 Summary of Relationship to Other Chapters457 Problems 468 Acknowledgments 46

2 The Essential CANDU


List of Figures

Figure 1 Nuclear chart where N represents the number of neutrons and Z the number ofprotons in a nucleus The points in red and green represent respectively stable andunstable nuclei The nuclear stability curve is also illustrated 8

Figure 2 Average binding energy per nucleon (MeV) as a function of the mass number (numberof nucleons in the nucleus)9

Figure 3 Saxon-Wood potential seen by nucleons inside the uranium-235 nucleus 13Figure 4 Shell model and magic numbers14Figure 5 Decay chain for uranium-238 (fission excluded) The type of reaction (β or α-decay)

can be identified by the changes in N and Z between the initial and final nuclides 20Figure 6 Microscopic cross section of the interaction of a high-energy photon with a lead atom24Figure 7 Photoelectric effect26Figure 8 Compton effect 28Figure 9 Photonuclear cross sections for deuterium beryllium and lead29Figure 10 Cross sections for hydrogen (left) and deuterium (right) as functions of energy32

Figure 11 Cross sections for 23592 U (left) and 238

92 U (right) as functions of energy 33

Figure 12 Effect of temperature on 238U absorption cross section around the 664-eVresonance34

Figure 13 Capture and fission cross sections of 23592U (left) and 238

92U (right) as functions of

energy36

Figure 14 Relative fission yield for 23592U as a function of mass number37

Figure 15 Fission spectrum for 23592U 39

List of Tables

Table 1 Properties of the four fundamental forces 5Table 2 Properties of leptons and quarks 6Table 3 Main properties of protons and neutrons 6Table 4 Energy states in the shell model where g is the degeneracy level of a state14Table 5 Comparison of the energy U required for a fission reaction in a compound nucleus

with the energy Q available after a neutron collision with different isotopes 35Table 6 Average numbers of neutrons produced by the fission of common heavy nuclides37Table 7 Distribution of kinetic energy among the particles produced by a fission reaction 38

Table 8 Maximum values of the elastic scattering and absorption cross sections for 11H 2

1D

and 126C in the energy range 01 eV lt E lt01 MeV 41

Nuclear Processes and Neutron Physics 3


1 Introduction

Energy production in an operating nuclear reactor is mainly the result of fission reactionsinitiated by neutrons Following these reactions radiation is produced in the form of alphaparticles electrons (particles-ߚ) photons and neutrons as well as a large number of unstablenuclei (including actinides and fission products) that may decay thereby producing additionalradiation and energy The neutrons generated directly in the fission reaction as well as thoseresulting from fission product decay are a key factor in maintaining and controlling the chainreaction as discussed in Chapters 4 and 5 Radiation also has an impact on the properties ofthe various materials in the core as well as on living cells It is therefore important to under-stand how radiation interacts with matter if one needs for example to evaluate the radio-toxicity of burned fuel (Chapter 19) or to determine the kind of barrier that must be put inplace to protect the public the environment or the personnel in a nuclear power plant(Chapter 12) Moreover the energy produced by these decays contributes to heating the fuelwhether it is still in the reactor or in the spent fuel pools The aim of this chapter is to providethe reader with an overview of the nuclear processes that take place in and around a reactor

11 Overview

Several very important processes take place simultaneously in a nuclear reactor includingneutron slowdown as a result of collisions with nuclei nuclear fission and decay of radioactivenuclei with radiation emissions that subsequently lose energy or are absorbed

To understand how all these physical processes take place it is important to possess a generalknowledge of the physics of the nucleus including its composition and the fundamental forcesthat affect its behaviour Although one rarely uses a description of the nucleus in terms ofelementary particles and their interaction in power-reactor applications such a descriptionprovides a practical background for a study of nuclear structure In fact the nucleus is such acomplex object that it can be described only through simplified models that provide usefulinformation For example the liquid drop model (Section 24) gives an empirical description ofhow one can extract energy from a nucleus by breaking it into smaller components (fission ordecay) or by combining nuclei (fusion) These models are also used to predict the stability of anucleus under various decay processes the energy associated with its excited states and theassociated gamma-ray spectrum In Section 2 an overview of the structure of the nucleus isprovided starting from a fundamental description and progressing to more practical modelsWe will also introduce some language specific to nuclear physics that will be useful throughoutthis book

The majority of the fission reactions taking place in a reactor are initiated following theabsorption of a neutron by a heavy nucleus Radioprotection also involves the interactionbetween radiation in the form of alpha particles electrons photons and neutrons andmatter Two important points must then be addressed before describing the physics ofneutrons in a nuclear reactor the source of this radiation and its behaviour over time and theinteraction of the emitted particles with matter Therefore Section 3 is divided into threeparts it starts with a description of radioactivity and the decay of nuclei continues with adescription of the interaction of ionizing radiation with matter (everything but neutrons)which is a very important topic for radiation protection studies and ends with a brief discus-sion of the interaction of neutrons with matter



The last section of this chapter is dedicated to the nuclear chain reaction It starts by giving anextensive description of the fission reaction and discusses the need to slow down neutronsThis is followed by a description of the neutron transport equation that can be used to charac-terize the behaviour of neutrons inside a reactor

12 Learning Outcomes

The goal of this chapter is for the reader to

Understand the fundamental forces in nature and the structure of the nucleus

Learn how to compute the mass of a nucleus based on the liquid drop model and toevaluate the energy released by the fission and fusion processes

Understand how to evaluate the activity and radio-toxicity of spent fuel

Become familiar with nuclear cross-section databases for neutrons and ionization radiationand the computation of reaction rates between particles and matter

Recognize the need to slow down neutrons in thermal reactors and to evaluate the effec-tiveness of a moderator

2 Structure of the Nucleus

21 Fundamental Interactions and Elementary Particles

Four forces or interactions are currently sufficient to understand and explain nature fromparticle physics to astrophysics including chemical and biological processes These forces areclassified relative to their intensity at the nuclear level (distances of the order of 1 fm) asfollows

The strong force responsible for the cohesion of the proton as well as the nucleus Thisforce is always attractive

The electromagnetic force governs the physics of the atom and the associated chemicaland biological processes This force can be both attractive and repulsive

The weak force responsible for the existence of the neutron as well as many other radio-active nuclei This force is always attractive

The gravitational force governs the large-scale structure of the universe This forceensures the stability of the solar system as well as the fact that people can keep their feeton the ground This force is always attractive

From the point of view of physics these forces arise from the specific intrinsic properties ofthe particles on which they act For example the electromagnetic force arises between twoparticles that have electric charges while the gravitational force appears between two parti-cles with masses (or more generally between two particles with energy) For the strong forcethe so-called ldquocolourrdquo of the particle plays a role equivalent to that of electric charge inelectromagnetic interactions while the weak force is the result of the interaction of twoparticles having a ldquoweakrdquo charge Table 1 provides a brief description of the general proper-ties of the four fundamental forces where the srsquoߙ are dimensionless coupling constantsrelated to the intensity of the interaction and the spatial dependence of the force is expressedin powers of ݎ the distance between the two interacting particles [Halzen1984] For examplethe electromagnetic force between two particles of charge ݖ and is given by



2

2 20

| | ( ) 4 | |

e

e zZ zZF r c

rr

(1)

where = 16 times 10ଵଽ C is the charge of an electron ߝ = 8854 times 10ଵଶ Fm is the electricpermittivity of vacuum ℏ = 10546 times 10ଷସ J times s is the reduced Planck constant and= 29979 times 10 ms is the speed of light For the gravitational force the coupling constantis expressed in terms of proton masses One can immediately see that the range of thegravitational and electromagnetic forces is substantially larger than that of the weak force due

to theirଵ

మdependence Accordingly the weak force is important only when two particles

with weak charges are in close contact On the other hand the strong (colour) force remainssignificant irrespective of the distance between particles with colour charge This observationled to the concept of colour confinement meaning that it is impossible to isolate a particlehaving a net colour charge different from zero [Halzen1984]

Table 1 Properties of the four fundamental forces

Force Equivalent charge Coupling constant Spatial dependence

Strong Colour charge ௦ߙ asymp 1 asymp ݎ

Electromagnetic Electric charge ߙ asymp 1137 ଶݎ

Weak Weak charge ௪ߙ asymp 10 ହݎ to ݎ

Gravitational Mass ߙ asymp 10ଷଽ ଶݎ

In the standard model of particle physics [Halzen1984 Le Sech2010] these forces are repre-sented by 12 virtual particles of spin 1 called ldquogauge bosonsrdquo

the massless photon that carries the electromagnetic interaction

eight massless gluons that carry the strong (colour) force

three massive bosons (Wplusmn and Z) that mediate the weak interaction

These bosons are exchanged between the 12 ldquophysical fermionsrdquo (spin 12 particles) thatmake up all the matter in the universe

six light fermions or leptons that have a weak charge Three of these have no electriccharge (the neutrinos) while three are charged (electron muon and tau)

six heavy fermions or quarks that have colour weak and electric charges These fermionscannot exist freely in nature because of colour confinement and must be combined in trip-lets of quarks or quark-antiquark pairs to form respectively baryons (protons neutrons)and mesons (pion kaon) that have no net colour charge



Table 2 Properties of leptons and quarks

Leptons Quarks

Particle Mass (GeVc2) Charge Particle Mass (GeVc2) Charge

0000511 minus1 ݑ asymp 0003 23

ߥ lt 25 times 10ଽ 0 asymp 0006 minus13

ߤ 01057 minus1 asymp 12 23

ఓߥ lt 0000170 0 ݏ asymp 01 minus13

17777 minus1 ݐ asymp 173 23

ఛߥ lt 0018 0 asymp 41 minus13

Table 2 provides a description of the main properties of leptons and quarks with the massesgiven in GeVc2 with 1 GeVc2asymp 1782661 times 10ଶ kg and the charges in terms of the chargeof one electron [Le Sech2010]

22 Protons Neutrons and the Nuclear Force

The two lightest baryons that can be created out of a quark triplet are the positively chargedproton which is composed of two ݑ quarks and one quark and the neutral neutron madeup of two quarks and one ݑ quark These are present in the nucleus of all the elementspresent naturally in the universe The main properties of the proton and the neutron areprovided in Table 3 [Basdevant2005] One can immediately see that the masses of the protonand neutron are very large compared with those of the ݑ and quarks In fact most of theirmass is a result of the strong colour force (gluon interaction) One can also observe thatprotons are stable while neutrons are unstable and decay relatively rapidly into protonsthrough the weak interaction A second observation is that the magnetic moment which iscreated by the movement of charged particles is negative for the neutron This confirms atleast partially the theory that neutrons are made of quarks because even though the neutronis neutral the distribution of charge inside it is not uniform

Table 3 Main properties of protons and neutrons

Property Proton Neutron

Mass (GeVc2) = 0938272 = 0939565

Charge (electron charge) 1 0

Magnetic moment (Nߤ) ߤ = 2792847 ߤ = minus1913042

Spin 12 12

Mean-life (s) gt 10ଷ = 8815

Although the proton and neutron have a neutral colour they interact through the nuclearforce which is a residual of the strong colour force However instead of exchanging gluonsdirectly as do the quarks inside the proton or neutron they interact by exchanging of virtualpion As a result the nuclear force is much weaker than the strong force in the same way that



the Van der Waals force between neutral atoms is weaker than the electromagnetic forceMoreover in contrast to the strong colour force the magnitude of the nuclear force decreasesrapidly with distance the nuclear interaction being given approximately by the Yukawa poten-tial

VYukawa

(r) = -aNc

e-

mp cr

r (2)

where గ is the mass of the pion (140 MeVc2) and ேߙ asymp 145 is a constant that controls thestrength of the nuclear interactions between baryons (protons or neutrons) The minus sign

here indicates that the resulting force ܨ) = minusnablaሬሬ ) is always attractive This potential is identi-cal for proton-proton proton-neutron and neutron-neutron interactions

The strength of this force at short distances is so intense that it can compensate for therepulsive electromagnetic force between protons leading to the formation of bonded nucleias well as inhibiting neutron decay For example the helium nucleus which is composed oftwo protons and two neutrons is stable Similarly the stable deuteron is composed of oneproton and one neutron The absence in nature of a nucleus composed of two protons is not aresult of the fact that the electromagnetic force is greater than the strong force but ratherdue to the exact structure of the nuclear force In addition to having a spatial behaviourprovided by the Yukawa potential this potential is also spin-dependent and repulsive betweenparticles with anti-parallel spins (a nucleus with two protons having parallel spins is notpermitted because of Paulirsquos exclusion principle) This also explains why stable nuclei contain-ing only neutrons are not seen in nature

23 Nuclear Chart

As indicated in the previous section not all combinations of neutrons and protons can lead tobonded states Figure 1 provides a list of all bonded nuclei [Brookhaven2013 KAERI2013]The points in red represent stable nuclei while those in green indicate radioactive nuclei Thestraight black line indicates nuclei having the same number of neutrons and protons Notethat this chart is quite different from the periodic table where the classification of the ele-ments is based on the number of electrons surrounding the nucleus Because the number ofelectrons surrounding the nucleus is identical to the number of protons inside the nucleuseach element corresponds to a combination of the nuclei that are found on a horizontal line ofthe nuclear chart The nomenclature used to classify these nuclei is the following

The number of protons in the nucleus is used to identify the element For example allnuclei with = 8 are known as oxygen (atomic symbol O) independently of the numberof neutrons present in the nucleus

For a given element the specific isotope is identified by ܣ = + where ܣ is known asthe mass number and is the number of neutrons in the nucleus For example oxygen-17 denoted by ଵO corresponds to an oxygen atom where the nucleus has eight protonsand nine neutrons Instead of the notation ேାX with X the element associated with the notation

ேାX is also found in the literature

Two other terms are less frequently used isobars corresponding to nuclei having the samemass numbers but different values for and and isotones corresponding to nucleihaving the same number of neutrons but a different number of protons

8

copyUNENE all rights reserved For educational use only no assumed liability

Several conclusions can be reached

Isotopes containing a small number of protons (and are nearly equal If the differenceunstable For combination of protons and neutrons with large valuescan be formed

For nuclei with values of ranging from 20 to 82 the number of excess neutronrequired to create bonded nuclei increasescific value of ǡseveral stable nuclei can be produced with different numbers of neutrons

No stable nucleus exists with still be found in nature because they decay at a very slow rremains in nature

Approximately 3200 bonded nuclei have been identifiedstability of a nucleus is ensured by the presence of a sufficient number of neutrons to copensate for the effect of the Coulomb force between protonsonly at short distances while the Coulomb force has a longer range additional neutrons arerequired when the radius of the nucleus increasesons (protons and neutrons)

where is the average radius of a single nucleon

Figure 1 Nuclear chart where N represents the number of neutrons and Z the number ofprotons in a nucleus The points in red and green represent respectively stable and unstable

nuclei The nuclear stability curve is also illustrated

The Essential

copyUNENE all rights reserved For educational use only no assumed liability Nuclear Physics

Several conclusions can be reached from Figure 1

Isotopes containing a small number of protons ( Ͳʹ) will generally be stable only ifIf the difference ȁ െ ȁ is slightly too large the nucle

For combination of protons and neutrons with large values of | െ

ranging from 20 to 82 the number of excess neutronrequired to create bonded nuclei increases steadily One can also observe that for a sp

several stable nuclei can be produced with different numbers of neutrons

gt 83 although some radioactive nuclides withstill be found in nature because they decay at a very slow rate No isotope with

3200 bonded nuclei have been identified out of which 266 are stablenucleus is ensured by the presence of a sufficient number of neutrons to co

of the Coulomb force between protons Because the nuclear force actsonly at short distances while the Coulomb force has a longer range additional neutrons are

of the nucleus increases because of the presence of more nucl

130R A R

the average radius of a single nucleon

Nuclear chart where N represents the number of neutrons and Z the number ofprotons in a nucleus The points in red and green represent respectively stable and unstable


The Essential CANDU

Nuclear Physics ndash June 2015

) will generally be stable only if the nucleus becomes

| no nuclei

ranging from 20 to 82 the number of excess neutrons ( െ )e that for a spe-


although some radioactive nuclides with ͻ͵ canNo isotope with above 92

out of which 266 are stable Thenucleus is ensured by the presence of a sufficient number of neutrons to com-

the nuclear force actsonly at short distances while the Coulomb force has a longer range additional neutrons are

the presence of more nucle-

(3)


Nuclear Processes and Neutron Physics


The stability of a nucleus does not only depend on the number of nucleons it contains but alsoon how these protons and neutrons are paired

159 contain an even number of protons and of neutrons

53 have an even number of protons

50 have an odd number of protons and an even number of neutrons and

only 4 are made up of an odd number of protons and of neutrons

In addition for values of or equal to 8 20 50 82 and 126 the number of stable nuclei isvery high These numbers are known as ldquomagic numbersrdquoand protons just like electrons in atoms are more tightly bonded when they fill a quantumenergy shell (see Section 25)

The mass శೋX of a nucleus produced by combiningthe sum of the masses of its constituents

N Zm Z m Nm AB c

where ܤ Ͳ the average binding energy per nucleon is the result of the negative potentialthat binds the nucleons inside the nucleifunction of ܣ is provided for stable nucleireaching a maximum around ܣ ൌ 60the average nucleons are more tightly bonded inside nuclei having intermediate values of( Ͳʹ gtܣ 120) than for very low or very high values

Figure 2 Average binding energy per nucleon (MeV) as a function o(number of nucleons in the nucleus)

Information on the atomic mass of over 3000 isotopes is available inthat the atomic mass is different from the nuclear mass of an isotopethe mass and the binding energy of thesimple way to compute the atomic mass is using



leus does not only depend on the number of nucleons it contains but alsoon how these protons and neutrons are paired For example out of the 266 stable nuclei


53 have an even number of protons but an odd number of neutrons



equal to 8 20 50 82 and 126 the number of stable nuclei isThese numbers are known as ldquomagic numbersrdquo They suggest that the neutrons

and protons just like electrons in atoms are more tightly bonded when they fill a quantum

produced by combining protons and neutrons isconstituents

2

N Z p nX

m Z m Nm AB c

the average binding energy per nucleon is the result of the negative potentialthat binds the nucleons inside the nuclei As can be seen in Figure 2 where a plot of

is provided for stable nuclei ܤ first increases rapidly for low mass numbers60 and then decreases slowly with ܣ This means that on

the average nucleons are more tightly bonded inside nuclei having intermediate values of120) than for very low or very high values

Average binding energy per nucleon (MeV) as a function of the mass number(number of nucleons in the nucleus)

Information on the atomic mass of over 3000 isotopes is available in [Livermore2013]that the atomic mass is different from the nuclear mass of an isotope because it includes both

he binding energy of the electrons gravitating around the nucleussimple way to compute the atomic mass is using

9


leus does not only depend on the number of nucleons it contains but alsoFor example out of the 266 stable nuclei

equal to 8 20 50 82 and 126 the number of stable nuclei isThey suggest that the neutrons


neutrons is less than

(4)

the average binding energy per nucleon is the result of the negative potential where a plot of ܤ as a

first increases rapidly for low mass numbersis means that on

the average nucleons are more tightly bonded inside nuclei having intermediate values of ܣ

f the mass number

[Livermore2013] Noteit includes both

electrons gravitating around the nucleus One



1

2

N Z nX H

M Z M Nm AB c (5)

where the mass of the proton has been replaced by that of the hydrogen atom which includesthe electron mass as well as its binding energy The approximation sign comes from the factthat the binding energy of each of the electrons around a heavy nucleus is different fromthat of a single electron attached to a proton in the hydrogen atom

Finally the natural composition of a given element will include stable as well as some radioac-tive isotopes Assuming that the relative atomic abundance of isotope of atomic mass ܯ inthe natural element is ߛ then the mass of the natural element is given by

X i i

i

M M (6)

In reactor physics one often works with the mass (or weight) fraction for the abundance of anisotope defined as

i ii

X

Mw

M

(7)

For completeness one can compute the isotopic concentration X of an element (atomscm3)with atomic mass ܯ X (gmoles) and density ߩ (gcm3) using

X A

X

NM

(8)

where ℕA = 6022 times 10ଶଷ atomsmoles is the Avogadro number The concentration ofisotope is then given by

i i A i X

X

N NM

(9)

More information related to the isotopic contents of natural elements can be found in[KAERI2013 WebElements2013]

24 Nuclear Mass and the Liquid Drop Model

As discussed in Section 23 the mass of the nucleus is smaller than its classical mass (the massof its constituents) because the strong force linking the nucleons reduces the global energy ofthe nuclear system the binding energy being related to the missing mass Δ X according to

2Δ Xm c AB (10)

This missing mass is very difficult to compute based on theoretical models because the form ofthe nuclear potential is known only approximately and because quantum models involvingmore than two particles in interaction cannot be solved exactly For evaluation of the missingmass one generally relies on a semi-empirical model known as the liquid drop model Themain assumption used in this model is that the nuclear force is identical for protons andneutrons

The liquid drop model first assumes that each of the ܣ nucleons in the nucleus sees the samenuclear potential [Basdevant2005] Moreover this potential is independent of the number ofnucleons in the nucleus This second assumption can be justified by the fact that the range ofthe nuclear force is small and therefore each nucleon feels the effect of only the limited



number of nucleons that are close by Accordingly the contribution to the missing mass fromthe ܣ nucleons is given by

2Δ V Vm c a A (11)

where the constant asymp15753 MeV has been determined experimentally This contributionis known as the volume effect because the volume of a nucleus can be assumed to be propor-tional to the number of nucleons it contains This term would be sufficient if the nucleus hadan infinite size However this is not the case and the protons and neutrons that are locatednear the boundary of the nucleus will interact with a smaller number of nucleons If the

surface of the nucleus is given by ଶߨ4 prop ଶଷܣ one can assume that the ଶଷܣ nucleons nearthe outer surface of the nucleus see a reduced uniform potential This leads to a negativesurface contribution to the mass defect of the form

2 23Δ S Sm c a A (12)

where ௌ asymp17804 MeV

Until now only the nuclear force has been taken into account However the protons presentinside the nucleus repel each other through the Coulomb force thereby decreasing thebinding energy Because the potential associated with this force is long-range (() prop 1 prop

(ଵଷܣ1 each of the protons will interact with the remaining minus 1 protons leading to acontribution of the form

22

13 13

( 1)Δ C C C

Z Z Zm c a a

A A

(13)

where only the dominant term in has been preserved and asymp07103 MeV

It can be observed in the nuclear chart that stable nuclei with a low mass number generallycontain an equal number of protons and neutrons As the number of excess neutrons orprotons in a nucleus increases it becomes more and more unstable until no bonded state canbe produced This means that nuclear binding should decrease as a function of | minus | =minusܣ| 2| the form of the contribution being

22 ( 2 )

Δ A A

A Zm c a

A

(14)

where asymp2369 MeV is known as the asymmetry term The nuclear chart also shows thatfew stable nuclei with an odd number of neutrons and protons (odd-odd nuclei) can be foundin nature while the number of isotopes with an even number of protons and neutrons (even-even nuclei) dominates This means that the binding energy should be large for even-evenand small for odd-odd nuclei Assuming that the contribution to the missing mass presentedpreviously remains valid only for odd-even or even-odd nuclei ܣ) odd) an additional empiricalcorrection of the form

2

34

(1 1 )Δ ( 1)

2

A

ZP Pm c a

A

(15)

is required Here asymp336 MeV is known as the pairing term

The final formula for the missing mass is therefore



2 2 22 3

1 34

3

(1 1 )( 2 )Δ ( 1)

2

A

ZX V S C A P

Z A Zm c a A a A a a a

A AA

(16)

which is known as the Bethe-Weizsaumlcker or semi-empirical mass formula (SEMF) Even if thismechanistic model is relatively simple it provides a very good approximation to the mass ofthe nuclei found in nature It can also be used to determine the value of ܣ that minimizes themass of a nucleus for a specific value of thereby maximizing the probability that it is stableNeglecting the pairing term one then obtains

2

2

3

4Z A

8 2

A n p

A C

Aa A m m c

a a A

(17)

This equation which is also illustrated in Figure 1 (the nuclear stability curve) closely followsthe nuclear stability profile

The main weakness of the liquid drop model is its lack of predictive power For example itneither explains the presence of ldquomagic numbersrdquo nor the gamma-ray absorption spectrumnor the decay characteristics of different nuclei These explanations can be obtained only bylooking at the interactions between protons and neutrons using quantum theory as explainedin the next section

25 Excitation Energy and Advanced Nuclear Models

All chemists and physicists are familiar with atomic ldquomagic numbersrdquo even though the explicitterm is rarely used For example the number of electrons in noble gases could be consideredldquomagicrdquo because these gases are odourless colourless and have very low chemical reactivityFor such atoms all the states associated with principal quantum energy levels 1 to areoccupied by electrons (closed or filled shells) In addition the absorption (emission) spectrumof light by different atoms is the result of electrons being excited (de-excited) from one elec-tronic shell to another One can therefore expect nuclear magic numbers to have a similarnature namely that only specific energy levels are permitted for the protons and neutronswhen they are bonded inside a nucleus (which would explain also the photon emission andabsorption spectra) These energy levels are not uniformly distributed but occur in bandscalled shells The fact that all the quantum states in a shell (a neutron or proton shell) areoccupied by a nucleon increases the stability of the nucleus compared to nuclides with par-tially filled shells

To obtain the energy levels permitted for protons and neutrons inside the nucleus one needsto solve the quantum-mechanics Schroumldinger equation for ܣ strongly coupled nucleons[Griffiths2005] However when trying to solve this equation two problems arise

the Schroumldinger equation has no analytic solution for problems where three or moreparticles are coupled

the interaction potential between nucleons is not known exactly

As a result this many-body problem is generally simplified using the following assumptionsthat are inherent to the shell model [Basdevant2005]

each nucleon in the nucleus is assumed to move independently without being affected bythe displacement of other nucleons



a nucleon moves in a potentialincreases sharply near its outer surface namely when

Note that in this model there are no onemeans that instead of solving the Schroumldinger equation forproblem is reduced to the solution ofnucleon with an averaged potentialmodel is the Saxon-Wood potential (see

where asymp30 MeV is the potential depth and

Figure 3 Saxon-Wood potential seen by nucleons inside the uranium

The energy levels are then very similar to those associated with thenamely

where is the mass of a nucleon (proton or neutron) ൌ ͳǡʹ ǡڮ the principal quantum number

number that takes into account spinindependent of the magnetic quantum numberis important when determining the probability of interaction between a nucleus (spinother particles is a combination of its internal spin and



a nucleon moves in a potential (ݎ) that is relatively uniform inside the nucleus ander surface namely when ൎݎ (mean field approximation)

Note that in this model there are no one-to-one interactions between the nucleonsmeans that instead of solving the Schroumldinger equation for ܣ strongly coupled nucleons the

ced to the solution of ܣ independent Schroumldinger equations for a singlean averaged potential The conventional nuclear potential used in the shell

Wood potential (see Figure 3) which has the form [Basdevant2005]

0

1

1

Rr R

V r Ve

30 MeV is the potential depth and ן ଵȀଷܣ is the radius of the nucleus in

Wood potential seen by nucleons inside the uranium-235 nucleus

The energy levels are then very similar to those associated with the harmonic potential

02 3

2

n

N

VE N

R m

is the mass of a nucleon (proton or neutron) ൌ ሺʹ െ ሻʹൌ 0the principal quantum number and 012l the angular momentum quantum

number that takes into account spin-orbit coupling In general one assumes that the energy isindependent of the magnetic quantum number The spin of each of the nucleons whichis important when determining the probability of interaction between a nucleus (spinother particles is a combination of its internal spin and its angular momentum

13


that is relatively uniform inside the nucleus and(mean field approximation)

one interactions between the nucleons Thisstrongly coupled nucleons the

independent Schroumldinger equations for a singleThe conventional nuclear potential used in the shell

[Basdevant2005]

(18)

is the radius of the nucleus in fm

235 nucleus

harmonic potential

(19)

012 ⋯ withthe angular momentum quantum

In general one assumes that the energy isof each of the nucleons which

is important when determining the probability of interaction between a nucleus (spin (ܬ and

14


The energy states of the shell model are presented intween the magic numbers and these states is illustrated

The model is never used to evaluate the mass of nuclei (the SEMF modelpurpose) but it is useful for classifyi( ேାX)lowast) and for determining the magic numbers

Table 4 Energy states in the shell model where

0 1

1 1

0 1

2 6

orbital 1s 1p

Figure 4 Shell model and magic numbers

The second model that is often used is the collective model of the nucleuslooking at the motion of the individual nucleons one considers thevibration movement of the nucleusthe shape of an ellipsoid (only nuclei that contain magispherical) Distorted nuclei can then acquire rotational and vibrational motionsquantized energy levels can be associatedhaving a moment of inertia I and an angular momentumWong2004]

The Essential


1

2j l

l model are presented in Table 4 whereas the association btween the magic numbers and these states is illustrated in Figure 4

The model is never used to evaluate the mass of nuclei (the SEMF model is used) but it is useful for classifying the energy levels of excited nuclei (denoted

the magic numbers

Energy states in the shell model where g is the degeneracy level of a state

2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s

Shell model and magic numbers

The second model that is often used is the collective model of the nucleus Here instead oflooking at the motion of the individual nucleons one considers the overall rotation and

of the nucleus The justification for this model is that most nuclei takethe shape of an ellipsoid (only nuclei that contain magic numbers of protons and neutrons are

Distorted nuclei can then acquire rotational and vibrational motionsquantized energy levels can be associated In classical mechanics the energy of a rotating solid

and an angular momentum ܬ is given by [Basdevant2005

2

2

JE

I

The Essential CANDU


(20)

the association be-

is used for thisng the energy levels of excited nuclei (denoted as

is the degeneracy level of a state

Here instead ofrotation and

The justification for this model is that most nuclei takec numbers of protons and neutrons are

Distorted nuclei can then acquire rotational and vibrational motions to whichIn classical mechanics the energy of a rotating solid

[Basdevant2005

(21)



In quantum mechanics the angular momentum is quantized with the energy levels beinggiven by

EJ

= 2J J +1( )

2I (22)

with =ܬ 012 ⋯ For nucleus having an ellipsoid shape only even values of areܬ allowedand the energies satisfy the following relation

2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)

where ܯ and are respectively the mass and radius of the nucleus These energy levels aregenerally much smaller than those associated with the shell model

26 Nuclear Fission and Fusion

Let us look back at Figure 2 where the average binding energy per nucleon is provided as afunction of mass number As mentioned previously this curve indicates that the total bindingof one heavy nuclide containing ܣ gt 180 nucleons which is given by

AE AB A (25)

is larger than that of two identical nuclides containing 2ܣ protons and neutrons

2 22

A A

AE AB E

(26)

In physical terms this means that two lighter nuclides generally represent a lower energy statethan a heavier nucleus Accordingly a spontaneous reaction where the heavy nuclide ܪ breaks down into two smaller components ଵܮ) and (ଶܮ is permitted from an energy-conservation point of view with the energy released by the reaction given by భܧ + మܧ minus ுܧ

This spontaneous reaction called fission has been observed experimentally for long-livedactinides such as ଶଷଶTh ଶଷହU and ଶଷU even though this is not their preferential decaymode (seven in 109 decays of ଶଷହU follow this path) For example the spontaneous fissionreaction

92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)

which releases 116 MeV has been observed

A second observation is that even if a spontaneous fission reaction is very improbable thefission process can be initiated externally by an absorption collision between a projectile and aheavy nuclide For example the neutron-induced reaction

1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)

is possible even with very low-energy neutrons and produces the same amount of energy asthe spontaneous fission reaction above This reaction is facilitated by the fact that the main



interaction between the neutron and a nucleus is the attractive nuclear force The fact thatseveral neutrons are generally produced following such reactions finally leads to the conceptof a controlled chain reaction if sufficient neutrons are produced following a fission reactionto initiate another fission reaction The conditions required to achieve and maintain such achain reaction are discussed in Section 4

Figure 2 also indicates that the nucleons inside very light nuclides are loosely bonded com-pared to intermediate-mass nuclei Combining two such nuclides could therefore result in anucleus that is more strongly bonded again releasing energy This process is called nuclearfusion Examples of such exothermal fusion reactions are

2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)

which release respectively 176 and 183 MeV in the form of kinetic energy for the final fusionproducts Producing such reactions is much more difficult than neutron-induced fissionbecause the two initial nuclides have positive charges and must overcome the Coulomb forceto come into close contact This can be achieved by providing sufficient kinetic energy (plasmatemperature) for a sufficient long time (confinement) to ensure that the energy released byfusion is greater than the energy required to reach these conditions (the break-even point)The two main technologies that are currently able to attain the conditions necessary forcontrolled nuclear fusion are based respectively on confinement by magnetic fields (Tokamak)and inertial confinement (fusion by laser)

3 Nuclear Reactions

From the nuclear chart (Figure 1) it is clear that only 266 nuclides are stable This means thatmost of the bonded nuclei that can be created will decay This raises two questions

How do they decay

How are they created

Another observation is that if one succeeds in creating a nucleus (stable or not) it may notnecessarily be in its fundamental (ground) energy state (see Section 25) These are the mainquestions to answer in the first part of this section We start by discussing the decay processbefore continuing with a description of nuclear reactions The second topic involves theinteraction of particles with matter with most of these particles being produced during thedecay process

31 Radioactivity and Nuclear Decay

Nuclear decay the emission of radiation from an unstable nucleus or radioisotope is a sto-chastic process controlled by the laws of statistics and physics Therefore it cannot take placeunless several fundamental quantities are conserved [Basdevant2005 Smith2000]

total energy (including mass which is a form of energy according to special relativity)

linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge



The latter two are required because of the intrinsic properties of quarks and leptons Forcommon nuclear reactions these last two conservation relations can be translated to

conservation of electrons and neutrinos where the electron and the neutrino have aleptonic charge of +1 and the positron (anti-electron) and anti-neutrino have a leptoniccharge of -1 and

conservation of the total number of protons plus neutrons (mass number)

A decay process transforms through emission of particles a parent nucleus into a more stabledaughter nucleus (a nucleus with a lower mass where the nucleons are more strongly bondedor equivalently have a lower mass) In fact if one considers a nucleus at rest that decaysaccording to the following reaction

1

i

i

NCA B

Z W Vi

X Y b

(31)

then total energy conservation implies that

2 2 2

1

A B B C Ci iZ W W V Vi i

N

X Y Y b bi

m c m c T m c T

(32)

where is the kinetic energy of the different decay products This reaction can take placespontaneously only if

2 2 2

1 1

0B C A B Ci iW Z WV Vi i

N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)

that is when the energy balance is favourable (the last two equations are for mass numberand charge conservation) For a nucleus in an excited state the daughter nucleus will often bethe same as the parent nucleus but at a different energy level (ground or lower-lying excitedstate) the secondary particle being a photon (a ray-ߛ because the photon is produced follow-ing a nuclear reaction)

Nuclear decay is characterized by a constant definedߣ according to [Smith2000 Nikjoo2012]

Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)

where (ݐ) is the number of nuclei present at time ݐ Δ(ݐ) is the change (negative becauseof the decay process) in the number of nuclei after time Δݐ and ߣ is known as the decayconstant This definition leads to the Bateman equation [Bell1982]

( )

( )dN t

N t S tdt

(37)



where (ݐ) represents the net rate of production of nuclei (from the decay of other nuclei orfrom nuclear reactions) One can also define

A t N t (38)

the activity of a radionuclide that represents the number of decays per second taking place attime ݐ This activity is generally stated in Becquerels (Bq) where 1 Bq corresponds to onedecay per second For the case where (ݐ) = 0 the solution to the Bateman equation has theform

Δ0 0( Δ ) tN t t N t e (39)

where (ݐ) is the number of nuclei at time ݐ

Two other concepts related to decay constants are also commonly used The mean life defined as

1

(40)

represents the average time required for the final number of nuclei to reach a value of(ݐ)

00( )

N tN t

e (41)

Similarly the half-life ଵଶݐ

12

(

2)2

lnt ln

(42)

represents the average time required for the initial number of nuclei to decrease to a value of2(ݐ)

For some radionuclides several decay reactions (decay channels) may be observed each beingcharacterized by a specific relative production yield such that

1

1J

j

j

Y

(43)

The decay constant associated with each channel is given by =ߣ ߣ For example potas-sium-40 ( ଵଽ

ସK) which has a mean life of 1805 times 10ଽ years decays 8928 of the time intocalcium-40 (

మబరబCa = 08928) and 1072 of the time into argon-40 (

భఴరబAr = 01072)

Note that the decay constant ߣ associated with a reaction can be evaluated using Fermirsquossecond golden rule [Schiff1968]

22| | ( Ξ) Ξ

f

f H i f d

(44)

where |⟩ is the wave function associated with the initial state ⟨ |is the wave function associ-ated with a final state and ᇱܪ is the interaction potential associated with the reaction Theintegral is over phase space Ξ with the term )ߩ Ξ) being the density of state for the finalwave function

The most common decay reactions are the following



ray-ߛ emission for an excited nucleus( X)lowast rArr

X + ߛ (45)

Internal conversion for an excited nucleus( X)lowast rArr [

X]ା + (46)This process corresponds to the direct ejection of an orbital electron leaving the nu-cleus in an ionized state ([ ௭

X]ା)

ߚ decay (weak interaction) for a nucleus containing too many neutrons

X rArr [ ାଵ

Y]ା + + ҧߥ (47)

Neutron emission (nuclear interaction) for a nucleus having a very large neutron ex-cess

X rArr

ଵX + ଵ (48)

ାߚ decay (weak interaction) for a nucleus containing too many protons

X rArr [ ଵ

Y] + ା + ߥ (49)

Orbital electron capture (weak interaction) for a nucleus containing too many protons

X rArr ଵ

Y + ߥ (50)

Proton emission (nuclear interaction) for a nucleus having a very large protonexcess

X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)

This process corresponds to the ejection of an atom of hydrogen

emission-ߙ for heavy nuclei (nuclear interaction)

X rArr [ ଶ

ସY]ଶ + ଶସߙ = ଶ

ସY + ଶସHe (52)

This process corresponds to the ejection of an atom of helium

Spontaneous fission for heavy nuclei (nuclear interaction)

X rArr ܥ

ଵ+ F +

G (53)This process corresponds to the fragmentation of a heavy nucleus into two or moresmaller nuclei with the emission of several neutrons

For heavy or very unstable nuclei (see for example Figure 5 for uranium-238) several of thesereactions must often take place before a stable isotope is reached

The Janis software from OECD-NEA can be used to retrieve and process the decay chain for allisotopes [OECD-NEA2013]



Figure 5 Decay chain for uranium-238 (fission excluded) The type of reaction (β or α-decay) can be identified by the changes in N and Z between the initial and final nuclides

32 Nuclear Reactions

As we saw in the previous section several types of particles are produced through decay ofradioactive nuclides These particles have kinetic energy and as they travel through matterthey interact with orbital electrons and nuclei A nuclear reaction takes place if these particlesproduce after a collision one or more nuclei that are different from the original nucleus (anexcited state of the original nucleus or a new nucleus) Otherwise the collision gives rise to asimple scattering reaction where kinetic energy is conserved

For the case where charged particles are traveling through a material scattering collisions withthe orbital electrons or the nucleus are mainly observed (also called potential scattering)More rarely the charged particles are captured by the nucleus to form a compound nucleus (avery short-lived semi-bonded nucleus) that can decay through various channels

Similarly the electric and magnetic fields associated with the photons interact mostly with thenucleus or the orbital electrons They can also initiate nuclear reactions after being absorbedin the nucleus Finally neutrons rarely interact with electrons because they are neutralparticles and the electrons are not affected by the nuclear force (electromagnetic interactionwith the quarks inside the nucleon magnetic interactions with unpaired electrons and weakinteractions are still present) However they can be scattered (nuclear potential scattering) orabsorbed (nuclear reaction) by the nucleus Again this absorption can lead to the formationof a very unstable compound nucleus that will decay rapidly

The main nuclear reactions of photons electrons neutrons and charged heavy particles withnuclei are often the inverses of the decay reactions presented earlier



ray absorption

A AZ ZX X (54)

ray-ߛ scattering (elastic and inelastic)

A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)

proton capture1 11 1 A A

Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)

neutron capture1 10 A A

Z Zn X X (60)

neutron scattering (elastic and inelastic)

1 1

0 0 A AZ Zn X X n (61)

photon-induced fission10 A B A B C

Z Y Z YX C n F G (62)

neutron-induced fission1 1 10 0 A B A B C

Z Y Z Yn X C n F G (63)

These reactions must obey the same conservation laws as those described earlier for thedecay reaction For most reactions (except elastic scattering) the final nucleus will be in ahighly excited state and will decay rapidly by ray-ߛ emission (prompt photons)

The probability that any given reaction will take place between a nucleus and a particle isrepresented by the microscopic cross section ߪ Much like the decay constant the crosssections have a quantum mechanical interpretation and can be expressed in terms of the wavefunction |⟩ associated with the initial state (the product of the wave functions of the projec-tile and the nucleus) the wave function ⟨ |associated with the final state and ᇱܪ the interac-tion potential The differential cross section is then given by Fermirsquos first golden rule whichcan be written as

22| | ( )

Ξ i

d Vf H i f

d v

(64)



where is the volume of the system ݒ is the velocity of the projectile (assuming that theinitial nucleus is at rest) and )ߩ ) is again the density of states in the phase space Ξ for finalparticles The cross section which is then the integration over the final phase space of thedifferential cross section has units of surface The most common unit used for the micro-scopic cross section is the barn (b) with 1 b = 10ଶସ cmଶ = 10ଶ mଶ

These cross sections are very difficult to evaluate explicitly because of the complexity of thewave functions (the wave function for a nucleus is the product of the wave functions of all thenucleons in the nucleus bonded by the nuclear potential) and the interaction potential Theseare generally evaluated experimentally and the results are stored in evaluated nuclear datafiles A software package such as Janis can be used to retrieve and analyze these cross sec-tions [OECD-NEA2013] For computational reactor physics the NJOY program is generallyused to process the cross-section databases for neutron-induced reactions [MacFarlane2010]

33 Interactions of Charged Particles with Matter

As mentioned earlier charged particles traveling through matter interact mainly with atomicelectrons because they are much more numerous than nuclei ( electrons for a nucleus withatomic number ) and also because the wave functions associated with these electrons have avery large spatial extension (the size of an atom) compared to that associated with the nu-cleus

Heavy particles such as protons particles-ߙ and ions are scattered by the Coulomb potentialproduced by the electron with the cross section given by the Mott scattering formula[Leroy2009 Bjorken1964] As a result a heavy particle loses a very small quantity of energyand momentum to secondary electrons and travels mainly in a straight line as it slows downThe secondary electrons on the other hand can gain enough energy to be knocked loose fromthe atom leaving it in an ionized state

For electrons and positrons traveling through matter the problem is somewhat more complexThe cross section for electron-electron collisions is given by the Moumleller scattering formulawhile Bhabha scattering is used for positron-electron collisions [Leroy2009 Bjorken1964]Positrons lose on average half their energy in collision with atomic electrons and theirtrajectory is somewhat erratic Positron-electron annihilation reactions can also take placeproducing two rays-ߛ Finally the electrons and positrons that are accelerated in the strongelectromagnetic field of a heavy nucleus lose some of their energy by ray-ߛ emission due tosynchrotron radiation (also known as Bremsstrahlung) [Leroy2009 Bjorken1964]

One can approximate the energy loss of a particle of charge andݖ mass inside a material ofdensity ߩ atomic number and atomic mass ܯ using the continuous slowing-down approxi-mation (CSDA) where one assumes that the particle is interacting constantly with an electrongas of density given by

An ZM

(65)

The energy loss per unit distance traveled by the particle inside the material (ݔܧminus)known as the stopping power is then given by the Bethe-Block formula [Smith2000Leroy2009]

2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)



where ݒ is the speed of the particle and (ݒ)ܮ the stopping number which can be written as

22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)

where isܫ the mean excitation potential for the atomic electrons in this material The correc-tion term (ݒ)ܨ takes into account the fact that the atomic electrons are bonded as well asother quantum effects For electrons one must also consider the energy lost by synchrotronradiation which is given approximately by [Leroy2009]

3 2

e

radiation e

dEZn

dx m

(68)

where is the mass of the electron Here it is assumed that the kinetic energy of theelectrons ≪

ଶ

For heavy particles (protons (particles-ߙ nuclear slowing-down nuclear(ݔܧminus) becomesimportant when they reach a low energy

The CSDA range of a particle is defined as the distance (path of flight) that it must travel tolose all its kinetic energy It is computed as

0

1

T

total

R dEdE

dx

(69)

where is the initial kinetic energy of the particle and

total collision radiation nuclear

dE dE dE dE

dx dx dx dx

(70)

These CSDA expressions are helpful to understand the general behaviour of particles slowingdown in a material However for practical applications tabulated values for ݔܧminus and are more useful This type of information for electrons protons and particles-ߙ can be foundon-line for various elements and different material compositions on the National Institute ofStandard and Technology Web sites [Berger2013a] The databases available are

ESTAR for electrons

PSTAR for protons

ASTAR for particles-ߙ

For heavy particles this site also provides information on nuclear stopping power Also notethat the stopping power and range are defined in a somewhat different way from the notationabove The following expressions are used instead

2 1 (MeV cm g ) (MeV cm)dE dx dE dx

(71)

and

2(g cm ) (cm)R R (72)

24


because these are the conventional units used in radiation shielding studiesnumber of electronion pairs produced per unit distance in a material is given approximatelyby

where is the average energy required to ionize an atom in this material

34 Interactions of Photons with Matter

The behaviour of photons as they travel in a material is somewhat diffecharged particles which lose their kinetic energyelectrons and nuclei the Coulomb field being a longparticle that carries the energy ܧ =of frequency ߥ and wavelength ߣlength the photon behaves as a wave (diffraction and interference for lowas a particle (photoelectric effect for highHere we will consider only the interactions with matter of the relatively highܧ) ͳ) found in nuclear reactorsfirst absorbed by the atomic electrons or the nucleusparticles can be produced including photons (scattering reactions) electrons and positrons(pair creation) and neutrons (photonuclear reaction)

Figure 6 Microscopic cross section of the interaction of a high

The Essential


these are the conventional units used in radiation shielding studies Note that thenumber of electronion pairs produced per unit distance in a material is given approximately

1

pairs

total

dn dE

dx W dx

is the average energy required to ionize an atom in this material

Photons with Matter

The behaviour of photons as they travel in a material is somewhat different fromlose their kinetic energy mainly by continuously interacting withoulomb field being a long-range force The photon is the quantum

= ߥ and momentum Ԧൌ ሬ Ȁߣof an electromagnetic fieldൌ Ȁߥ travelling in direction ሬ Depending on its wav

length the photon behaves as a wave (diffraction and interference for low-energy photons) ort for high-energy photons) when it interacts with matter

Here we will consider only the interactions with matter of the relatively high-energy photons) found in nuclear reactors In this case the photon behaves as a particle that is

sorbed by the atomic electrons or the nucleus Following this absorption secondaryincluding photons (scattering reactions) electrons and positrons

and neutrons (photonuclear reaction)

Microscopic cross section of the interaction of a high-energy photon with a leadatom

The Essential CANDU


Note that thenumber of electronion pairs produced per unit distance in a material is given approximately

(73)

rent from that ofby continuously interacting with

The photon is the quantumof an electromagnetic field

Depending on its wave-energy photons) or

energy photons) when it interacts with matterenergy photons

In this case the photon behaves as a particle that isFollowing this absorption secondary

including photons (scattering reactions) electrons and positrons

energy photon with a lead



The five main reactions taking place between high-energy photons and matter that are ofinterest in nuclear reactor-related studies are

Photoelectric effect

Rayleigh scattering

Compton scattering

Pair production and

Photonuclear reactions

With each such reaction is associated a macroscopic cross section Σ ௫(ܧ) representing theprobability per unit distance travelled by a photon of energy ܧ that an interaction of type ݔhas taken place in material The macroscopic cross section takes into account the probabili-ties of interaction both between single photons and between particles and the fact that thematerial contains particles of type per unit volume

( ) ( ) m x m i i xE N E (74)

where (ܧ)௫ߪ is the microscopic cross section of the interaction of a photon of energy ܧ witha particle of type through reaction ݔ (see Section 32) The total microscopic photon crosssection can be expressed as

( ) ( ) ( ) ( ) ( ) ( )Photoelectric Rayleigh Compton Pair production PhotonuclearE E E E E E (75)

Examples of microscopic cross sections for the interaction of photons with a lead atom arepresented in Figure 6

Let us now describe in detail the five reactions contributing to (ܧ)ߪ and discuss their respec-tive dependences on the energy of the incident photon

341 Photoelectric effect

This effect is illustrated in Figure 7 A photon with an energy ܧ = ℎߥ that is greater than thebinding energy of atomic electrons around the nucleus is absorbed Following this absorp-tion the electron becomes free leaving behind an ion The energy balance for this reaction is

e NT T E W (76)

where is the kinetic energy of the final electron and ே that of the recoil nucleus (generallyvery small compared with )

26


Figure

From quantum mechanics the binding energy of electrons aroundbe approximated using [Griffiths2005]

where ൌ ͳ for K-shell electrons 2 for Lof the photon decreases the electrons in the more tightly bonded shells (K Lare successively excluded producing a large drop in the photoelectric crossenergy falls below (see Figuredifficult and only approximate relations are available[Leroy2009]

sPhotoelectric

which is valid within 2 for photon with energies ranging from 100 keVtotal photoelectric cross section (the[Leroy2009]

Photoelectric Photoelectric KE E Z Z

Clearly this cross section decreases rapidly with photon energytons of energies less than a few tenths of

Several processes can take place after this reactioninitially in the valence band the ionized atom is in its ground stateemission is observed When electrons coming from more tightly bonded energy levelejected the ionized atom is left in an excited stateground state by two different meansvalence electron) can emit a photon and fill the vacant energy levelcome from a slightly higher energy level leaving the ion in a less excited state that will furtherdecay before reaching its ground state

The Essential


Figure 7 Photoelectric effect

From quantum mechanics the binding energy of electrons around a nucleus of charge[Griffiths2005]

2

2

1361 n

ZW eV

n

shell electrons 2 for L-shell electrons and so on Therefore as the energyof the photon decreases the electrons in the more tightly bonded shells (K L and M shells)are successively excluded producing a large drop in the photoelectric cross section when the

Figure 6) The evaluation of this cross section is therefore veryand only approximate relations are available For a K-shell electron one can use

c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2

which is valid within 2 for photon with energies ranging from 100 keV to 200 MeVthe sum over all energy shells) is given approximately by

2 3

1 001481 ln 0000788 ln Photoelectric Photoelectric KE E Z Z

section decreases rapidly with photon energy It is dominant only for phtons of energies less than a few tenths of an MeV

Several processes can take place after this reaction If the electron ejected from the atom wasinitially in the valence band the ionized atom is in its ground state and no further radiation

When electrons coming from more tightly bonded energy levelejected the ionized atom is left in an excited state This excited ion can then decay to its

two different means First an electron in a much higher energy state (ievalence electron) can emit a photon and fill the vacant energy level The electron can alsocome from a slightly higher energy level leaving the ion in a less excited state that will further

reaching its ground state When the photons emitted in these successive decays

The Essential CANDU


a nucleus of charge can

(77)

as the energyand M shells)

section when thesection is therefore very

hell electron one can use

(78)

to 200 MeV Thesum over all energy shells) is given approximately by

2 31 001481 ln 0000788 ln E E Z Z (79)

It is dominant only for pho-

If the electron ejected from the atom wasand no further radiation

When electrons coming from more tightly bonded energy levels areon can then decay to its

First an electron in a much higher energy state (ie aThe electron can also

come from a slightly higher energy level leaving the ion in a less excited state that will furtherWhen the photons emitted in these successive decays



are absorbed by less-bonded electrons so-called soft Auger electrons are ejected from theatom producing an Auger electron cascade

342 Rayleigh scattering

This elastic scattering reaction takes place when a photon after being absorbed by a boundatomic electron is re-emitted with the same energy but in a different direction by the elec-tron returning the atom to its original state [Leroy2009 Jauch1976] The cross section for thisreaction is very difficult to evaluate because it depends strongly on the wave function of thebonded electrons surrounding the nucleus It is generally written as an integral of the form

2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)

where (ߠܧ)ܨ is the atomic form factor that represents the effect of the electrons sur-rounding the nucleus on the scattering of a photon at an angle ߠ with respect to the initialphoton direction Several theoretical expressions for (ߠܧ)ܨ are available in the literaturefor different values of and various energy ranges For example when the photon energy isgreater than 100 keV the main contributors to Rayleigh scattering are the electrons in the K-shell and the relativistic Bethe-Levinger form factor is used [Hubbell1975]

F Eq Z( ) =sin(2gatan(Q))

gQ 1+ Q2( )g

(81)

with

Q Eq Z( ) =2E sin(q 2)

aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)

Because Rayleigh scattering is never the dominant photon reaction (see Figure 6) for thephoton energy range considered in nuclear reactors its contribution to the total cross sectionis often neglected

343 Compton scattering

The Compton effect is the incoherent inelastic scattering of a photon by an electron that is

weakly bonded to a nucleus (see Figure 8) Here a photon with momentum ሬ (หሬห= ℎߥ=

ܧ ) is scattered at an angle withߠ respect to its original direction with the final momentum

of the photon being ሬprime The electron is then ejected with a kinetic energy obtained byconservation of momentum and energy

2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)

where the binding energy of the electron is assumed to be small compared to the photonenergy and the recoil energy carried by the nucleus is neglected For a nearly free electronthe cross section for this reaction is given by the Klein-Nishina formula which can be writtenas

28


2 2e

Compton

e

cE ln

m c

with 2 ( )eE m c This cross section decreases with energy

the photoelectric effect It dominates the photoelectric reaction for energies above approxmately 05 MeV

Figure

344 Pair production

A photon with energy ܧ ൌ ߥ 2charge (or an electron) can be converted into an electronthe nucleus carries a small part of the kinetic energy of the photonenergy being transferred in the form of mass and kinetic energy to the pairenergies in the range found in nuclear reactors the crossapproximately by [Leroy2009]

Pair production eE Z

with 2 ( )eE m c This cross section vanishes for

Compton cross section above a few MeV

345 Photonuclear reactions

Photonuclear reactions are the result of photons being absorbed by a nucleus to form anexcited nucleus which gets rid of its energy through different decay processes namely theemission of a secondary photon the ejection of one or more nucleons (protonthe ejection of particles-ߙ or throughabove a threshold which lies between 7 and 9 MeV for heavy nuclides butlower for very light isotopes (1666 MeV and 2226 Merium) The absorption cross sections for these reactions (seedifficult to evaluate and only approximate expressions are available

The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c

section decreases with energy but much more slowly tha

It dominates the photoelectric reaction for energies above approx

Figure 8 Compton effect

2 ଶ traveling through the Coulomb field of a nucleus of

can be converted into an electron-positron pair After the reactionthe nucleus carries a small part of the kinetic energy of the photon with the bulk of the

ansferred in the form of mass and kinetic energy to the pairenergies in the range found in nuclear reactors the cross section for this reaction is given

2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c

section vanishes for ܧ ͳǤͲʹ MeV and is larger than the

above a few MeV

Photonuclear reactions are the result of photons being absorbed by a nucleus to form anexcited nucleus which gets rid of its energy through different decay processes namely theemission of a secondary photon the ejection of one or more nucleons (protons and neutrons)

or through a fission reaction Photonuclear reactions occur onlybetween 7 and 9 MeV for heavy nuclides but is considerably

lower for very light isotopes (1666 MeV and 2226 MeV respectively for beryllium and deutions for these reactions (see Figure 9) are generally very

and only approximate expressions are available

The Essential CANDU


21 2 1

2 1 2

(85)

but much more slowly than for

It dominates the photoelectric reaction for energies above approxi-

traveling through the Coulomb field of a nucleus ofAfter the reaction

the bulk of the For photon

section for this reaction is given

(86)

MeV and is larger than the

Photonuclear reactions are the result of photons being absorbed by a nucleus to form anexcited nucleus which gets rid of its energy through different decay processes namely the

s and neutrons)Photonuclear reactions occur only

considerablyV respectively for beryllium and deute-

) are generally very



Figure 9 Photonuclear cross sections for deuterium beryllium and lead

For deuterium which is important to CANDU reactorheavy water the expression [Leroy2009]

s 2 DPh

is often used where ǡtheܧ photon energy

00624 and (ܧ)Dߪ is in barns The end result of such a photonuclear reaction is a hydrogenatom and a fast neutron that can initiate

Tabulated values for Rayleigh scattering (coherscattering (incoherent scattering) and pair production crossmaterials and for energies ranging from 1 keV to 100 GeV are available onof the National Institute of Standards and Technology (NIST)reactions the energy-dependent crossOECD-NEA Janis software [OECD-NEA2013]

For each reaction presented above the photons first transfer all

a nucleus following a collision As a result starting with

direction Ωሬሬ incident at point Ԧonݎ a material of uniform macroscopic cross

number of initial photons ሺܧǡݎԦ+given by [Basdevant2005]

n E r s n E r e



Photonuclear cross sections for deuterium beryllium and lead

For deuterium which is important to CANDU reactors because they use large amounts of[Leroy2009]

hotonuclear(E) raquo C 2 D

E - 2226( )3

E 3

the photon energy is given in MeV ܥ Dమ varies between 0061 and

The end result of such a photonuclear reaction is a hydrogenatom and a fast neutron that can initiate a fission reaction (see Section 26)

Tabulated values for Rayleigh scattering (coherent scattering) photoelectric effect Comptonand pair production cross sections with different atoms or

materials and for energies ranging from 1 keV to 100 GeV are available on-line on theof the National Institute of Standards and Technology (NIST) [Berger2013b] For photonuclear

dependent cross sections for various isotopes can be obtainedNEA2013]

esented above the photons first transfer all their energy to an electron or

As a result starting with ሺܧǡݎԦǡȳሬሬ) photons of energy

on a material of uniform macroscopic cross section

+ ȳሬሬݏ Ωሬሬ) still present after a distance hasݏ been crossed is

Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)

varies between 0061 and

The end result of such a photonuclear reaction is a hydrogen

ent scattering) photoelectric effect Comptonsections with different atoms or

line on the Web siteFor photonuclear

obtained using the

energy to an electron or

photons of energy ܧ and

on ሻܧሺߑ the

has been crossed is

(88)



The number of photons therefore decreases exponentially with distance This behaviour issimilar to what is observed when light is attenuated by passing through a material with alinear attenuation coefficient (ܧ)ߤ = (ܧ)ߑ Secondary photons produced with energy ܧ and

direction Ωሬሬ following the absorption of photons of energy primeܧ and direction Ωሬሬprime will also con-

tribute to the total population of photons at point +Ԧݎ Ωሬሬݏ The equation that describes thephoton population in a material assuming a constant neutron incident population is the staticphoton transport equation which takes the form [Pomraning1991]

Ω Ω Ω Ω Ω Ω Ω sn E r E n E r dE d E E n E r

(89)

where Σ௦൫ܧᇱrarr ܧ Ωሬሬᇱrarr Ωሬሬ൯ is the differential scattering cross section representing the prob-

ability that the absorption of a photon of energy ᇱܧ and direction Ωሬሬᇱ produces a photon of

energy ܧ and direction Ωሬሬ

35 Interactions of Neutrons with Matter

Neutron interaction with matter is the result of the nuclear force although weak and electro-magnetic interactions the latter being the result of anomalous magnetic moment are alsopossible As a result only neutron-nucleus interactions are generally considered including

elastic potential scattering with cross section σĞůĂƐƟĐ

inelastic scattering with cross section σŝŶĞůĂƐƟĐ

neutron absorption with cross section σĂďƐŽƌƉƟŽŶ and

spallation reactions σƐƉĂůůĂƟŽŶ

In the first three reactions the neutron interacts with the nucleus as a whole In spallationreactions the neutrons interact directly with the nucleons in the nucleus The latter reactioncan take place only when the energy of the incident neutron ܧ is sufficiently high (thresholdreactions with ltܧ 01 MeV) and therefore σspallation is either neglected because it is too

small or simply included in the absorption cross section The probability that the neutroninteracts with a nucleus is then given by the total cross section σ where

σ( ) σ ( ) σ ( )n i absorption n i scattering n iE E E (90)

σ σ σ scattering n i elastic n i inelastic n iE E E (91)

Elastic potential scattering is an interaction in which kinetic energy is conserved and theneutron does not penetrate the nucleus As a result the neutron is slowed down transferringpart of its energy to the nucleus For a collision between a neutron with an initial kineticenergy andܧ a nucleus of mass ܯ X at rest the final neutron kinetic energy ܧ (momentum

and energy conservation) will be given by [Bell1982]

2

2 1 2 cos 1

n in f

EE A A

A

(92)

where ܣ = ܯ X and ߴ the scattering angle for the final neutron in the of mass system isrelated to the scattering angle in the laboratory system ߠ by

2

cos 1cos

1 2 cos

A

A A

(93)



On average the neutron will lose an energy Δܧ

1

2n iE E

(94)

2

2

1α

1

A

A

(95)

after each collision Because Δܧ decreases as ܣ increases neutron slowing-down (also calledmoderation) is more efficient when scattering collisions with light rather than heavy nuclei areinvolved

For low-energy neutrons the potential scattering cross section in the centre of mass system isnearly constant and has the form [Bell1982]

s = 4p R2 (96)

where is the nuclear radius for this material At higher energy one can use Fermirsquos goldenrules to obtain an approximation for (ܧ)ߪ of the form

s (E) = 4p2 sin2 d

2mnE

(97)

where ߜ represents the phase shift of the neutron wave function in isotropic scattering

Inelastic scattering and absorption reactions generally proceed by creation of a compoundnucleus

1 A A A

Z Z Zn X X n X (98)

1 A A

Z Zn X X other decay channels (99)

The creation of the compound nucleus is a resonant reaction because the resulting weaklybonded nucleus can exist only if it corresponds to an excitation state of X

ାଵ The lifetime ofthis excited nucleus is generally very short (asymp 10ଵ s) and with each decay channel including inelastic scattering is associated a resonance width given by [Bell1982]

Γ k

k

(100)

with being the mean life for decay through this channel The total resonance width for thecreation of the excited nucleus Γ is then given by

Γ Γ k (101)

For light nuclei Γinelastic ≫ Γabsorption and the inelastic scattering process dominates For

heavy nuclei the reverse is true (absorption dominates) The resonance widths for inelasticscattering also increase with energy the explicit dependence being

0 inelastic nE E E (102)

where Γ is a constant that depends on the nucleus considered

The probability (cross section) for the creation of a compound nucleus with resonant energyܧ is given by

32


1AZ X

E l

where is the orbital-angular-momentum quantum numbthrough channel the cross section is

Note that at low energy ܧ) ا (ܧ Γ

k

and the cross section is a function of

Figure 10 Cross sections for hydrogen (l

Examples of cross sections for light (hydrogen and deuterium) and heavy nuclei (Uଽଶ

ଶଷ ) are presented in Figure 10clearly dominates One can also observe that the total crosstimes larger than that of deuterium (better scattering nucleus for neutron moderation)absorption representing 10 of the total crosswill lead to neutron capture For deuterium the ratio of neutron absorptions to collisions iscloser to 1 making this isotope a better neutron moderatorabsorption reaction is the most prevalent and exhibits a very large number of resonancesvery high energy these resonances do not disappear but are so closely packed that theyproduce a nearly continuous cross section

The Essential


2

22

0

E2 1

2

2

nE lmE

E E

momentum quantum number For a nucleus that decayssection is

1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E

section is a function of ͳȀݒ with beingݒ the incoming neutron velocity

Cross sections for hydrogen (left) and deuterium (right) as functions of energy

sections for light (hydrogen and deuterium) and heavy nuclei (and Figure 11 For light nuclides the scattering process

One can also observe that the total cross section of hydrogen is about tenthat of deuterium (better scattering nucleus for neutron moderation)

bsorption representing 10 of the total cross section of hydrogen one out of ten collisionsFor deuterium the ratio of neutron absorptions to collisions is

closer to 1 making this isotope a better neutron moderator overall For heavy nuclides theprevalent and exhibits a very large number of resonances

resonances do not disappear but are so closely packed that theysection

The Essential CANDU


(103)

For a nucleus that decays

(104)

(105)

being the incoming neutron velocity

of energy

sections for light (hydrogen and deuterium) and heavy nuclei ( Uଽଶଶଷହ and

or light nuclides the scattering processsection of hydrogen is about ten

that of deuterium (better scattering nucleus for neutron moderation) Withone out of ten collisions

For deuterium the ratio of neutron absorptions to collisions isFor heavy nuclides the

prevalent and exhibits a very large number of resonances Atresonances do not disappear but are so closely packed that they



Figure 11 Cross sections for

The derivations above are valid only if one assumes that the nucleus is at restnot the case for material having temperatureskinetic energy that is proportional toon the cross sections can be taken into account for a material at temperature[Bell1982]

2

k n X k n X X XE T v v v v p E dE

where Ԧݒ and ԦXݒ are the velocity of the neutron and the nucleusis the Maxwell-Boltzmann distributionwith energy ܧ in the range Xܧ ܧ

X X X Xp E dE E e dE

where is the Boltzmann constantsections exhibiting ͳȀݒ behaviour (ieAt low neutron energies the scattering crossbehaviour and is not affected by this transformation although the energy distribution of thefinal neutrons resulting from these collisions will be such that they end up in thermal eqrium with the nuclides in the medium

The thermal motion of atoms also changes the form of resonant crossheight of the resonance while incresonance is known as the Doppler effect



Cross sections for 23592 U (left) and 238

92 U (right) as functions of energy

The derivations above are valid only if one assumes that the nucleus is at rest This is clearlynot the case for material having temperatures Ͳ because atoms acquire an averagekinetic energy that is proportional to the temperature of their medium The resulting effect

sections can be taken into account for a material at temperature

1

2


mE

are the velocity of the neutron and the nucleus respectively HereBoltzmann distribution which represents the probability of finding

Xܧ Xܧ due to thermal motion

( )

32

2XE k T

X X X Xp E dE E e dEk T

B

B

is the Boltzmann constant This thermal motion of nuclei has no impact on crossbehaviour (ie absorption cross sections of hydrogen at low energy)

ies the scattering cross section for light nuclides also exhibitsbehaviour and is not affected by this transformation although the energy distribution of thefinal neutrons resulting from these collisions will be such that they end up in thermal eqrium with the nuclides in the medium [Bell1982]

The thermal motion of atoms also changes the form of resonant cross sections by reducing theheight of the resonance while increasing its width (see Figure 12) This widening of theresonance is known as the Doppler effect

33


(right) as functions of energy

This is clearlyacquire an averageThe resulting effect

sections can be taken into account for a material at temperature using

(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)

This thermal motion of nuclei has no impact on crossof hydrogen at low energy)

exhibits ͳȀݒbehaviour and is not affected by this transformation although the energy distribution of thefinal neutrons resulting from these collisions will be such that they end up in thermal equilib-

sections by reducing theThis widening of the



Figure 12 Effect of temperature on 238U absorption cross section around the 664-eV reso-nance

The neutron-nucleus interaction cross sections are very complex and in most cases are evalu-ated using a combination of theory and experiment They are generally tabulated in evaluatednuclear data files that can be read by nuclear calculation software (NJOY for example)[MacFarlane2010] They can also be extracted on-line from nuclear databases or off-line usingthe Janis software [KAERI2013 Livermore2013 OECD-NEA2013]

Once the energy-dependent cross sections are known one can easily evaluate the number ofreactions of type ݔ per second (the reaction rate ௫(ݒ)) taking place inside a region ofvolume V containing nuclei per cm3 of a material (cross section σ ௫(ݒ)) with a popula-

tion of (ݒ) neutrons per cm3 having a velocity =ݒ ඥ2ܧ

m m x

m x m x

N v n vR v V V v v

v

(108)

where (ݒ) = ݒ(ݒ) is the neutron flux and Σ ௫(ݒ) = σ ௫(ݒ) the macroscopic crosssection for a reaction of type ݔ in this material

4 Fission and Nuclear Chain Reaction

The concept of controlled energy production in a nuclear reactor is based on three fundamen-tal ideas that were discussed in Sections 26 31 and 32 namely

energy is released if a heavy nucleus is fragmented into lighter nuclei the fragmentationprocess being called nuclear fission

the spontaneous nuclear fission process which has a very low yield for several heavynuclei generates neutrons in addition to the light nuclei as end products

the nuclear fission process can be initiated by neutrons and photons

Accordingly a neutron of kinetic energy travelling through a medium containing heavynuclei at rest could initiate a fission reaction

1 1 1 A B A B Co Z o Y Z Yn X C n F G

(109)

The kinetic energy ோ released by this reaction is then

1

2 2 2 2(1 ) A B A B CZ Y Z Y

R n n X F GT T C m c m c m c m c

(110)



with part of the energy carried by the neutrons and the lighter nuclei These lighter nucleirapidly lose their energy in matter (see Section 33) thereby producing heat that is removedby the coolant and transformed to electricity

Not all the neutrons produced after this reaction initiate a fission reaction A number of themare absorbed by material without producing a fission reaction Others simply leave the system(with or without collisions) Provided that on average a single neutron per fission reaction caninitiate a new fission a controlled exothermal chain reaction can be established Establishingsuch a chain reaction looks relatively simple however this is far from the case as shown inthe following subsections

41 Fission Cross Sections

The probability of fission is given by the value of the fission cross sections associated with anucleus and depends on the energy of the neutron that initiates the reaction All isotopeswith atomic number gt 90 are fissionable (have a non-zero probability of undergoing fissionfollowing the capture of a neutron) Table 5 compares the average energy required toinitiate a fission reaction in different nuclei with the energy available in the compoundnucleus after a neutron with a vanishing kinetic energy has been captured For nuclei contain-ing an even number of protons and an odd number of neutrons gt Such isotopes arecalled fissile isotopes because the internal energy of the compound nucleus resulting fromneutron capture is sufficient to surmount the potential barrier for fission even when theincident neutrons are very slow ( asymp 0) For isotopes with lt the fission barrier can beovercome only using neutrons with relatively high kinetic energy Fission for these nuclei is athreshold reaction

Table 5 Comparison of the energy U required for a fission reaction in a compound nucleuswith the energy Q available after a neutron collision with different isotopes

Isotope (MeV) (MeV)

Thଽଶଷଶ 51 59

Uଽଶଶଷଷ 66 55

Uଽଶଶଷହ 64 58

Uଽଶଶଷ 49 59

Puଽସଶଷଽ 64 55

As shown in Figure 13 the energy dependence of the fission cross sections for Uଽଶଶଷହ and Uଽଶ

ଶଷ

reflects the behaviour described above The probability that a neutron of energy ܧ lt 10

MeV will collide with Uଽଶଶଷ and result in a fission reaction is very low For Uଽଶ

ଶଷହ there is nolower limit on the neutron energy and the fission probability increases as the energy of theneutron decreases The neutron capture cross section (neutron absorption followed by

photon emission) for Uଽଶଶଷ dominates at low energy while for Uଽଶ

ଶଷହ it remains relatively smallcompared to the fission cross section (approximately 10) over the full energy range As a

result most neutron absorption in Uଽଶଶଷହ will lead to fission followed by production of new

neutrons (neutrons are regenerated) For Uଽଶଶଷ a collision with a neutron having energy below


the fission threshold will lead to a net neutron loss As a result the natural uranium fuels used

in CANDU reactors (072 atomic of Uଽଶଶଷହ ) will be much less reactive (prone to fission) than

the enriched fuels (enrichment factors ranging from 25 to 5 atomic of Uଽଶଶଷହ ) used in light

water reactors (LWR)

42

Theorcal

ass

14funnu




Fission Products and the Fission Process

fission process generally produces two light nuclei called fission products although threemore nuclei can also be generated The relative rate of production of a given nuclide ܫled the fission yield is denoted by ூand satisfies

2I

I

Y (111)

uming that only two nuclei are produced by each fission reaction As one can see in Figure

where the fission yields of all the nuclei resulting from the fission of Uଽଶଶଷହ are plotted as a

ction of their mass number the fission process favours the production of relatively lightclides (90 lt Alt105) combined with heavier isotopes (130 lt Alt145)

energy



Figure 14 Relative fission yield for

The average number of neutrons emitted bysmall compared to the neutron excessvalue of for a given reaction depends on the nature of the fissile nuclides the fission proucts released and the energy of the neutronsneutron-rich and unstable They will reach stabilityand neutron emission All the transformations that take place withinreaction are generally included in the soprompt electrons and prompt photons)milliseconds to minutes) due to decay of isotopes with longdelayed

Table 6 Average numbers of neutrons produced by the fissio

Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ

The fission reaction (see Section 26) also releases a large amount of kinetic energy (on theorder of 200 MeV) that is distributed between fission products neutronsTable 7) Clearly the fission products carry most of the energy and because of their very shortrange in matter this energy heats the nuclear fuelparticles) also lose most of their energy in the fueldepositing their energy in the reactor structures while the weakly interacting neutrinos willgenerally be lost to the environment



Relative fission yield for 23592U as a function of mass number

of neutrons emitted by the fission process (see Table 6) is relativelythe neutron excess െ ͶͲ required for heavy isotopes to exist

for a given reaction depends on the nature of the fissile nuclides the fission proand the energy of the neutrons Most fission products are therefore very

They will reach stability through two main processesAll the transformations that take place within 10ଵସs of the fission

reaction are generally included in the so-called prompt contribution (prompt neutronsand prompt photons) Particles produced on a longer time scale (from

milliseconds to minutes) due to decay of isotopes with longer mean lifetime

Average numbers of neutrons produced by the fission of common heavy nuclides

1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31

The fission reaction (see Section 26) also releases a large amount of kinetic energy (on theorder of 200 MeV) that is distributed between fission products neutrons rays-ߛ etc

the fission products carry most of the energy and because of their very shortrange in matter this energy heats the nuclear fuel Similarly the charged electron

their energy in the fuel The highly penetrating rays-ߛ will end upreactor structures while the weakly interacting neutrinos will

generally be lost to the environment Finally for LWR and CANDU reactors only a very sm

37


as a function of mass number

) is relativelyrequired for heavy isotopes to exist The

for a given reaction depends on the nature of the fissile nuclides the fission prod-Most fission products are therefore very

through two main processes ߚ decays of the fission

called prompt contribution (prompt neutronsParticles produced on a longer time scale (from

mean lifetimes are called

n of common heavy nuclides

The fission reaction (see Section 26) also releases a large amount of kinetic energy (on therays etc (see

the fission products carry most of the energy and because of their very shortSimilarly the charged electrons ߚ)

rays will end upreactor structures while the weakly interacting neutrinos will

Finally for LWR and CANDU reactors only a very small



part of the neutron energy will be transferred to the fuel in the form of heat with most of itbeing lost to the environment (mostly through slowing-down in the moderator)

Table 7 Distribution of kinetic energy among the particles produced by a fission reaction

Particles Kinetic energy(MeV)

Fission products 168

Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207

Note that the data presented in Table 7 represent average values meaning that the neutronsproduced in the fission process have an energy distribution The probability that a neutron ofenergy ܧ is produced in the fission process which is known as the fission spectrum (ܧ) isoften approximated by the Watt relationship [Watt1952]

sinh naEn nE Ce bE (112)

where and are constants associated with the isotope undergoing fission and dependweakly on the incident neutron energy The normalization factor ܥ is defined so that

0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)

This fission spectrum for Uଽଶଶଷହ is illustrated in Figure 15 Clearly most of the neutrons are

released with energies greater than 1 MeV and can initiate fission reactions with fissionablematerial However the fission cross section for this energy range is relatively low (see Figure13) and fissions at such energies are highly improbable Nevertheless a controlled chainreaction that relies mainly on fast fission can be achieved in so-called ldquofast neutron reactorsrdquowhere Puଽସ

ଶଷଽ is often used because of its large fission cross section for high-energy neutronsThe alternative to fast neutron reactors is moderated reactors in which the fast neutrons areslowed down to energies where the fission cross section is significantly larger (below 1 eV)This is the concept used in ldquothermal neutron reactorsrdquo which are discussed in the next sec-tion



Now let us look at the properties of fission products These nuclides have intermediatemasses and vanishing fission cross sections However they can have very large neutron-capture cross sections and thereby can have a considerable negative impact on the neutronpopulation in the fuel where they are produced For this reason they are often called neutron

poisons Examples of poisons that have a large impact on nuclear-reactor operation are Xeହସଵଷହ

ߪ) = 31 times 10 b) and Smଶଵସଽ ߪ) = 61 times 10ସ b) A second observation is that most of these

nuclides are unstable and decay naturally through one of the reactions described in Section31 These reactions produce both particles including delayed neutrons and energy that isoften deposited in the fuel These delayed neutrons play an important role in the control ofnuclear reactors as will be explained in Chapter 5 Moreover energy will still be produced inthe fuel for a long time after the controlled fission reaction has been stopped due to thedecay of fission products This is the main source of the heat produced in neutron-irradiatednuclear fuels (some energy is also produced through photonuclear fission) This residual heatis removed from shut-down reactors and irradiated fuel pools by continuously circulating thecoolant

Figure 15 Fission spectrum for 23592U

Finally we must address the problem of the changes in the composition of a material exposedto a neutron flux In Section 31 we introduced the Bateman equation which takes the form[Smith2000 Nikjoo2012 Magill2005]

ௗே(௧)

ௗ௧= ூߣminus ூ(ݐ) minus (ݐ)ூܮ + ூ(ݐ) (115)

where ூ(ݐ) is the concentration of isotope inܫ the material and the net source (ݐ) is nowdivided into a production ூ(ݐ) and a loss (ݐ)ூܮ term The loss term from sources other thandecay is the result of a neutron being absorbed (captured) by isotope ܫ

0

σ ( ) I I I absorptionL t N t E E dE

(116)



where (ܧ) is the neutron flux The production term has two contributions decay from othernuclei and transformation of a different nucleus into ܫ following a neutron-induced nuclearreaction

0

σ d xI J I J J J I J J x

J J x

P t Y N t Y N t E E dE

(117)

where ூௗ is the yield for the production of isotope fromܫ the decay of isotope ܬ while ூ

௫ is

the isotope productionܫ yield from a reaction of type ݔ (fission capture etc) between aneutron and isotope ܬ

43 Nuclear Chain Reaction and Nuclear Fission Reactors

The previous sections have introduced all the concepts required to design a conceptualnuclear fission reactor

The nuclear fuel contains heavy isotopes that are broken into lighter nuclides (fissionproducts) following a neutron-induced fission reaction Such a reaction also generates en-ergy and secondary neutrons that can be used to maintain a chain reaction

Most of the energy released as a result of the fission process is carried away by the fissionproducts and deposited in the fuel because of the short range of these nuclei

The secondary neutrons produced in the fission process are fast neutrons

These neutrons can lose their energy through scattering collisions with the various materi-als present in our conceptual reactor Neutron slowing-down by collisions with light iso-topes is more efficient than by collision with heavy isotopes (see Section 35)

This energy deposition increases the temperature of the fuel which must then be cooledto make use of the heat and avoid damage to the reactor For a power reactor this heat iscollected in a coolant (generally gas or liquids circulating around the fuel) and used to gen-erate electricity

Now let us look at the fuel and coolant materials that are the most readily available

Three heavy fissionable isotopes are present in nature Thଽଶଷଶ Uଽଶ

ଶଷ and Uଽଶଶଷହ Of these only

Uଽଶଶଷହ is fissile by neutrons of all energies Because the density of thorium is 11000 kgm3 andit has a microscopic fission cross section of 01 barn for 1 MeV neutrons the mean free path ofa neutron for fission in this material is 35 m The mean free path for neutron capture inthorium is very similar in magnitude Therefore nearly half the 22 neutrons emitted byfission in thorium will be lost to unproductive capture For Uଽଶ

ଶଷ which represents 9927 ofall the atoms found in natural uranium (density of 18000 kgm3) the odds are slightly betterwith mean free paths for fission and capture respectively of 50 and 300 cm and fewer neu-trons emitted per fission are lost to capture

The main advantage of Uଽଶଶଷହ even though it represents only 072 of natural uranium compo-

sition is that its fission cross section is quite high (1000 barns at 001 eV) giving a mean free

path for fission of 2 cm The capture cross section of Uଽଶଶଷହ at this energy is also six times

smaller than the fission cross section It is possible to decrease the mean free path for fission

with Uଽଶଶଷହ by increasing its relative concentration in uranium-based fuel (fuel enrichment)

The ratio of capture to fission in uranium-enriched fuels remains small even if the fuel containsa large amount of Uଽଶ

ଶଷ (capture cross section less than 1 barn)



The heat produced in the fuel can then easily be extracted by conventional means (by circulat-ing a fluid such as water) The main problem is to slow down the secondary fission neutronsto low energies (lt 1 eV) while ensuring that few are lost by capture inside other materials orby leaving the reactor A neutron slowing-down material the moderator must therefore beintroduced into the reactor to serve this purpose


1D

and 126C in the energy range 01 eV lt E lt01 MeV

Isotope ƐĐĂƩĞƌŝŶŐߪ (b) ĂďƐŽƌƉƟŽŶߪ (b)

Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016

For this moderator two questions are of prime importance what should be its compositionand where should it be located The first question is easily answered by combining the kinet-ics of the neutron-nucleus collision and a survey of the scattering and absorption cross sec-tions of the most promising candidates In Section 35 we concluded that the slowing-downefficiency of a nucleus increases as its mass number decreases which points to the use of lightnuclides as moderators The most promising candidates are the isotopes of hydrogen andcarbon that are readily available in water and graphite Table 8 provides the maximum valuesof the elastic scattering and absorption cross sections for Hଵ

ଵ Dଵଶ and C

ଵଶ in the energy range01 eV lt ܧ lt 01 MeV As one can see Hଵ

ଵ is the best moderator if one does not take intoaccount its absorption cross section However up to one in ten collisions of a neutron with Hଵ

ଵ

could lead to absorption (for 1eV neutrons) On the other hand even if the number of colli-sions required to slow down neutrons using Dଵ

ଶ and Cଵଶ is significantly higher (the distance the

neutron will travel between collisions is longer) less than 0007 of the collisions with Dଵଶ and

003 of the collisions with Cଵଶ (1 eV neutron) lead to capture Accordingly if one wants to

build a core that is compact a Hଵଵ -based moderator is a requirement (light water for example

in LWR) otherwise the best candidate is a moderator that contains large concentrations of Dଵଶ

(heavy water in CANDU)

Now let us try to answer the question of the spatial distribution of this moderator Two verydifferent options can be considered a homogeneous mixture of fuel and moderator and smallisolated fuel elements placed strategically in the moderator Again the answer to this ques-tion can be inferred from Figure 13 The neutrons in a homogeneous mixture of fuel andmoderator materials have a large chance of colliding with fuel isotopes as they are slowingdown This means that neutrons with energies in the ldquoresonancerdquo range (4 eV lt ܧ lt 10 keV)will be in direct contact with fuel nuclides Because their capture cross sections at resonanceenergies increase often by several orders of magnitude with respect to the background valueneutron capture will have a detrimental effect on the number of neutrons that can reachenergies below 1 eV On the other hand by separating different fuel elements by a sufficientamount of moderating materials to ensure that a maximum number of neutrons escape theresonance energy range one decreases neutron loss by capture and increases the probabilityof fission



These are the general physics considerations that led to current reactor designs including thepressurized water (PWR) and CANDU reactors In the next section we conclude this chapterby introducing the neutron transport equation which provides a means to evaluate numeri-cally the neutron population in the reactor and to determine whether a given reactor conceptcan lead to a sustained chain reaction This equation also provides the neutron flux that isrequired to follow the evolution of the fuel and the fission products in the reactor using theBateman equation

44 Reactor Physics and the Transport Equation

When designing a nuclear power reactor physicists have two main objectives

controlling the nuclear chain reaction

extracting the energy produced inside the fuel to avoid damage to the core and to use thisenergy to produce electricity

Because most of the energy produced in a nuclear reactor is the result of the fission processthe power produced in the core can be expressed by

3 2

0 4

( ) ( Ω) Ω

J f J

JV

P t d r dE r t E r t E d

(118)

where is the reactor volume ܧݐԦݎ) Ωሬሬ) is the flux of neutrons with energy ܧ and direction

Ωሬሬ at a point Ԧinݎ space and a time ݐ and Σ(ݎԦܧݐ) is the macroscopic fission cross section

for isotope givenܬ by

f J J f Jr t E N r t E

(119)

where (ݎԦݐ) is the time-dependent concentration of ܬ the solution of the flux-dependent

Bateman equations (see Section 42) Finally ߢ represents the average energy produced by

the fission of isotope ܬ which we assume to be energy-independent

Now let us consider the flux which is a measure of the neutron population in the reactor(Section 35) For a given time interval ݐ the change of the neutron population

ܧݐԦݎ) Ωሬሬ) = ܧݐԦݎ) Ωሬሬ)ݒ due to loss of neutrons by collisions is [Bell1982 Duder-stadt1979 Heacutebert2009 Lewis1993]

Ω Ω CL r t E dt r t E r t E dt

(120)

where Σ is the total cross section and ܧݐԦݎ)ܮ Ωሬሬ) the rate of collision per unit volume of

neutrons with energy ܧ and direction Ωሬሬ at point Ԧandݎ time ݐ This term includes both ab-sorbed and scattered neutrons Some neutrons will be lost to migration when the net flow ofneutrons reaching a region in space of volume ଷݎ is different from that leaving this sameregion This rate of neutron loss due to migration is given by

LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)

meaning that neutrons which can be considered to form a gas tend to move towards regions

of low neutron population (the kinetic theory of gas) Neutron with energy ܧ and direction Ωሬሬ

can also be produced in this region of space due to scattering of neutrons with differentdirections and energies



2

0 4

Ω Ω Ω Ω Ω S sS r t E dt dE d r t E E r t E dt

(122)

where Σ௦(ݎԦܧݐᇱrarr ܧ Ωሬሬprime rarr Ωሬሬ) is the probability (also known as the double differential

scattering cross section) that a neutron of energy ᇱܧ moving in direction Ωሬሬprime is scattered in

direction Ωሬሬwith energy ܧ For the case where fission takes place the neutron production rateis

2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj

S r t E dt E dE r t E d r t E dt

(123)

where (ܧ) is the fission spectrum and ߥ the average number of neutrons produced by

fission of isotope ܬ Neutron productions from other sources ா൫ݎԦܧݐ Ωሬሬ൯can also be added

to the balance for example sources for other types of nuclear reactions (photonuclear fissionand photoneutron production in Dଵ

ଶ ) or flux-independent contributions from the decay ofradioactive isotopes

The global neutron population will then satisfy the following balance equation

1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)

which is known as the Boltzmann transport equation If one assumes that the cross sectionsare nearly constant over time (the change in the composition of the materials in the reactorvaries substantially only over long time periods) and that the core is at static equilibrium (theflux remains constant) one obtains the following time-independent neutron transport equa-tion [Lewis1993]

2 2

0 4 0 4

Ω Ω Ω

1 Ω Ω Ω Ω Ω Ω s j j f J

j

r E r E r E

dE d r E E r E E dE r E d r Ek

(125)

where is known as the multiplication factor It is inserted in the transport equation tomodify the fission rate in such a way as to reach this static equilibrium When = 1 equilib-rium is achieved and the reactor is critical For lt 1 the reactor is sub-critical and neutrongeneration is insufficient to compensate for losses whereas for gt 1 the reactor is super-critical and neutron production exceeds losses

For reactor calculations where fission reactions take place only inside the core boundaryconditions of the form

fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)

are generally imposed on the transport equation Here the point Ԧݎ is on the outer boundary

of the reactor with ሬሬbeing a unit vector normal to this boundary and directed outwards Thiscondition simply means that no neutrons will enter the reactor from the outside

The static transport equation is an eigenvalue equation with being the inverse of theeigenvalue Therefore the neutron flux is the eigenvector for this equation and is only defined



to within a normalization constant Therefore if ൫ݎԦܧ Ωሬሬ൯ is a solution so is ൫ݎԦܧ Ωሬሬ൯

with an arbitrary constant For reactor calculations the normalization constant is directlyrelated to the power produced in the core

Solving the transport problem is not trivial even for very simple cases (static one velocityneutrons in one dimension) In Chapters 4 (static) and 5 (time-dependent) approximation toand simplification of the transport equation commonly used in reactor simulations will bediscussed One should then be able to extract from the solutions of the resulting simplifiedequations information useful for the design operation and control of nuclear reactors

5 Bibliography

[Basdevant2005] J-L Basdevant R James S Michel Fundamentals in Nuclear PhysicsSpringer New York 2005

[Bell1982] G I Bell S Glasstone Nuclear Reactor Theory Robert E Krieger Malabar 1982

[Berger2013a] M J Berger J S Coursey M A Zucker J Chang J ESTAR PSTAR and ASTARComputer Programs for Calculating Stopping-Power and Range Tables for ElectronsProtons and Helium Ions httpphysicsnistgovStar visited April 2013

[Berger2013b] M J Berger J H Hubbel S M Seltzer J Chang J S Coursey R Sukumar D SZucker K Olsen XCOM Photon Cross Section Databasehttpwwwnistgovpmldataxcom visited April 2013

[Bjorken1964] J D Bjorken S D Drell Relativistic Quantum Mechanics McGraw-Hill NewYork 1964

[Brookhaven2013] National Nuclear Data Center at Brookhaven National Laboratoryhttpwwwnndcbnlgovchart visited April 2013

[Griffiths2005] D J Griffiths Introduction to Quantum Mechanics Pearson Canada Toronto2005

[Duderstadt1979] J J Duderstadt W R Martin Transport Theory Wiley New York 1979

[Halzen1984] F Halzen A Martin Quarks and Leptons An Introductory Course in ModernParticle Physics Wiley New York 1984

[Heacutebert2009] A Heacutebert Applied Reactor Physics Presses Internationales PolytechniqueMontreacuteal QC 2009

[Hubbell1975] J H Hubbell W J Veigele E A Briggs R T Brown D T Cromer and R JHowerton ldquoAtomic form factors incoherent scattering functions and photon scatter-ing cross sectionsrdquo J Phys Chem Ref Data 4 471-538 1975

[Jauch1976] J M Jauch F Rohrlich The Theory of Photons and Electrons Springer-VerlagNew York 1976

[KAERI2013] Nuclear Data Center at Korea Atomic Energy Research Institutehttpatomkaerirekrton visited April 2013

[Leroy2009] C Leroy P-G Rancoita Principle of Radiation Interaction in Matter and Detection2nd edition World Scientific Singapore 2009

[Le Sech2010] C Le Sech C Ngocirc Physique nucleacuteaire Des quarks aux applications DunodParis 2010



[Lewis1993] E E Lewis and W F Miller Jr Computational Methods of Neutron Transport 2ndedition Wiley New York 1993

[Livermore2013] Isotope Evaluation at Lawrence Livermore National Laboratoryhttpielblgovtoi2003MassSearchasp visited April 2013

[MacFarlane2010] R E MacFarlane A C Kahler ldquoMethods for Processing ENDFB-VII withNJOYrdquo Nuclear Data Sheets 111 2739-2890 2010

[Magill2005] J Magill and J Galy Radioactivity Radionuclides Radiation Springer Heidelberg2005

[Nikjoo2012] H Nikjoo S Uehara D Emfietzoglou Interaction of Radiation with Matter CRCPress Boca Raton FL 2012

[OECD-NEA2013] OECD-NEA Janis 30 Java-Based Nuclear Data Display Programhttpswwwoecd-neaorgjanis visited April 2013

[Pomraning1991] G C Pomraning Linear Kinetic Theory and Particle Transport in StochasticMixtures World Scientific London 1991

[Schiff1968] L I Schiff Quantum Mechanics McGraw-Hill New York 1968

[Smith2000] F A Smith A Primer of Applied Radiation Physics World Scientific Singapore2000

[Watt1952] B E Watt ldquoEnergy spectrum of neutrons from thermal fission of Uଶଷହ rdquo Phys Rev87 1037-1041 1952

[WebElements2013] WebElements The Periodic Table on the Webhttpwwwwebelementscomisotopeshtml visited April 2013

[Wong2004] S S M Wong Introductory Nuclear Physics Wiley-VCH Weinheim 2004

6 Summary of Relationship to Other Chapters

This chapter explains the structure of the nucleus and introduces the various interactions ofcharged particles photons and neutrons with matter It also defines the basic quantities andconcepts needed to study the fission chain reaction and to approach the analysis of reactorsThe Bateman equations are presented and used in other chapters to evaluate the rate ofevolution of fuel composition The Boltzmann transport equation which provides a way toevaluate the neutron population in a nuclear reactor is also introduced This is the fundamen-tal equation from which the neutron diffusion equation used in Chapters 4 and 5 is derived

Chapter 4 starts with the static transport equation to derive the time-independent neutron-diffusion equation for the analysis of steady-state (time-independent) neutron distributionseither in the presence of external sources or in fission reactors It also discusses the evolutionof the fuel composition in an operating reactor based on the coupled solution of the transportand Bateman equations using a quasi-static approximation

Chapter 5 covers the time-dependent neutron-diffusion equations and studies the phenomenaof fast-neutron kinetics fission-product poisoning reactivity coefficients etc in reactors

Chapter 12 deals with radioprotection and uses the concept defined in the present chapter forthe interactions of charged particles photons and neutrons with matter with the goal ofreducing the damage to living tissues



Chapter 19 examines the behaviour of fuel after irradiation in a nuclear reactor particularly itscomposition and the residual energy that it produces

7 Problems1 Evaluate the intensity of the electromagnetic force acting between two protons in

an atom of helium2 Determine the atomic mass of natural chlorine if it is composed of 7577 atomic

Clଵଷହ and 2423 atomic Clଵ

ଷ

3 What is the atomic concentration of Uଽଶଶଷହ in a sample of uranium oxide (UOଶ) of

density 104=ߩ g cmଷ enriched at a level of 35 weight

4 Using the semi-empirical mass formula evaluate the atomic mass of Feହ andcompare with the exact value

5 What is the ground state energy level of a neutron in the Saxon-Wood potential forUଽଶ

ଶଷ 6 Complete the following nuclear reactions and determine the net energy in MeV re-

leased by these reactionsa) ଵܪ

ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ

ଶସଵ = + ଷݎܭ

ଽ + 3 ଵ

7 What is the activity of one gram of potassium-408 Assuming that the activity of a sample of water due to tritium is 100 Bq determine

the mass of tritium in this sample What will be the activity of this sample afterthree years

9 Using the information available on the ESTAR web site plot the CSDA range in cmfor an electron in dry air at sea level as a function of its energy for 100 keV ltE lt 10MeV

10 Draw the deuterium cross section for photonuclear reactions as a function of en-ergy in the range 223 MeV lt E lt 10MeV (here assume ܥ మ = 0062) Compare

with the deuterium cross section for neutron radiative capture at 300 K in thesame energy range

11 How many collisions are required on average to slow down a neutron to half itsspeed (14 of its energy) through elastic collisions with deuterium nuclei

12 Evaluate the potential scattering cross section for oxygen and hydrogen at low en-ergy

13 A CANDU bundle contains 195 kg of natural uranium (enrichment of 0711

weight) Compute the fission rate in a bundle due to Uଽଶଶଷହ if it is exposed to a flux

of 10ଵସ neutronscm2s of energy 001=ܧ eV14 Show that the mean free path of 1 MeV neutrons in thorium is 35 m

8 Acknowledgments

Reviewers Marv Gold and Esam Hussein are gratefully acknowledged for their hard work andexcellent comments during the development of this Chapter Their feedback has much im-proved it Of course the responsibility for any errors or omissions lies entirely with the author

Thanks are also extended to Diana Bouchard for expertly editing and assembling the finalcopy

1Introduction

11Overview

12Learning Outcomes

2Structure of the Nucleus

21Fundamental Interactions and Elementary Particles

22Protons Neutrons and the Nuclear Force

23Nuclear Chart

24Nuclear Mass and the Liquid Drop Model

25Excitation Energy and Advanced Nuclear Models

26Nuclear Fission and Fusion

3Nuclear Reactions

31Radioactivity and Nuclear Decay

32Nuclear Reactions

33Interactions of Charged Particles with Matter

34Interactions of Photons with Matter

341Photoelectric effect

342Rayleigh scattering

343Compton scattering

344Pair production

345Photonuclear reactions

35Interactions of Neutrons with Matter

4Fission and Nuclear Chain Reaction

41Fission Cross Sections


43Nuclear Chain Reaction and Nuclear Fission Reactors

44Reactor Physics and the Transport Equation

5Bibliography

6Summary of Relationship to Other Chapters

7Problems

8Acknowledgments



List of Figures

Figure 1 Nuclear chart where N represents the number of neutrons and Z the number ofprotons in a nucleus The points in red and green represent respectively stable andunstable nuclei The nuclear stability curve is also illustrated 8

Figure 2 Average binding energy per nucleon (MeV) as a function of the mass number (numberof nucleons in the nucleus)9

Figure 3 Saxon-Wood potential seen by nucleons inside the uranium-235 nucleus 13Figure 4 Shell model and magic numbers14Figure 5 Decay chain for uranium-238 (fission excluded) The type of reaction (β or α-decay)

can be identified by the changes in N and Z between the initial and final nuclides 20Figure 6 Microscopic cross section of the interaction of a high-energy photon with a lead atom24Figure 7 Photoelectric effect26Figure 8 Compton effect 28Figure 9 Photonuclear cross sections for deuterium beryllium and lead29Figure 10 Cross sections for hydrogen (left) and deuterium (right) as functions of energy32

Figure 11 Cross sections for 23592 U (left) and 238

92 U (right) as functions of energy 33

Figure 12 Effect of temperature on 238U absorption cross section around the 664-eVresonance34



energy36

Figure 14 Relative fission yield for 23592U as a function of mass number37

Figure 15 Fission spectrum for 23592U 39

List of Tables

Table 1 Properties of the four fundamental forces 5Table 2 Properties of leptons and quarks 6Table 3 Main properties of protons and neutrons 6Table 4 Energy states in the shell model where g is the degeneracy level of a state14Table 5 Comparison of the energy U required for a fission reaction in a compound nucleus

with the energy Q available after a neutron collision with different isotopes 35Table 6 Average numbers of neutrons produced by the fission of common heavy nuclides37Table 7 Distribution of kinetic energy among the particles produced by a fission reaction 38


1D

and 126C in the energy range 01 eV lt E lt01 MeV 41



1 Introduction


11 Overview
























2

2 20

| | ( ) 4 | |

e

e zZ zZF r c

rr

(1)


to theirଵ



















Leptons Quarks


0000511 minus1 ݑ asymp 0003 23











Mass (GeVc2) = 0938272 = 0939565



Spin 12 12






VYukawa

(r) = -aNc

e-

mp cr

r (2)




23 Nuclear Chart






8










The Essential











130R A R




The Essential CANDU



| no nuclei







(3)











N Zm Z m Nm AB c














2

N Z p nX

m Z m Nm AB c







9






(4)




f the mass number





1

2

N Z nX H

M Z M Nm AB c (5)



X i i

i

M M (6)


i ii

X

Mw

M

(7)


X A

X

NM

(8)


i i A i X

X

N NM

(9)




2Δ Xm c AB (10)






2Δ V Vm c a A (11)



2 23Δ S Sm c a A (12)




22

13 13

( 1)Δ C C C

Z Z Zm c a a

A A

(13)



22 ( 2 )

Δ A A

A Zm c a

A

(14)


2

34

(1 1 )Δ ( 1)

2

A

ZP Pm c a

A

(15)





2 2 22 3

1 34

3

(1 1 )( 2 )Δ ( 1)

2

A

ZX V S C A P


A AA

(16)


2

2

3

4Z A

8 2

A n p

A C

Aa A m m c

a a A

(17)

























0

1

1

Rr R

V r Ve




02 3

2

n

N

VE N

R m



13





[Basdevant2005]

(18)


235 nucleus

harmonic potential

(19)




14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments



1 Introduction


11 Overview
























2

2 20

| | ( ) 4 | |

e

e zZ zZF r c

rr

(1)


to theirଵ



















Leptons Quarks


0000511 minus1 ݑ asymp 0003 23











Mass (GeVc2) = 0938272 = 0939565



Spin 12 12






VYukawa

(r) = -aNc

e-

mp cr

r (2)




23 Nuclear Chart






8










The Essential











130R A R




The Essential CANDU



| no nuclei







(3)











N Zm Z m Nm AB c














2

N Z p nX

m Z m Nm AB c







9






(4)




f the mass number





1

2

N Z nX H

M Z M Nm AB c (5)



X i i

i

M M (6)


i ii

X

Mw

M

(7)


X A

X

NM

(8)


i i A i X

X

N NM

(9)




2Δ Xm c AB (10)






2Δ V Vm c a A (11)



2 23Δ S Sm c a A (12)




22

13 13

( 1)Δ C C C

Z Z Zm c a a

A A

(13)



22 ( 2 )

Δ A A

A Zm c a

A

(14)


2

34

(1 1 )Δ ( 1)

2

A

ZP Pm c a

A

(15)





2 2 22 3

1 34

3

(1 1 )( 2 )Δ ( 1)

2

A

ZX V S C A P


A AA

(16)


2

2

3

4Z A

8 2

A n p

A C

Aa A m m c

a a A

(17)

























0

1

1

Rr R

V r Ve




02 3

2

n

N

VE N

R m



13





[Basdevant2005]

(18)


235 nucleus

harmonic potential

(19)




14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments





















2

2 20

| | ( ) 4 | |

e

e zZ zZF r c

rr

(1)


to theirଵ



















Leptons Quarks


0000511 minus1 ݑ asymp 0003 23











Mass (GeVc2) = 0938272 = 0939565



Spin 12 12






VYukawa

(r) = -aNc

e-

mp cr

r (2)




23 Nuclear Chart






8










The Essential











130R A R




The Essential CANDU



| no nuclei







(3)











N Zm Z m Nm AB c














2

N Z p nX

m Z m Nm AB c







9






(4)




f the mass number





1

2

N Z nX H

M Z M Nm AB c (5)



X i i

i

M M (6)


i ii

X

Mw

M

(7)


X A

X

NM

(8)


i i A i X

X

N NM

(9)




2Δ Xm c AB (10)






2Δ V Vm c a A (11)



2 23Δ S Sm c a A (12)




22

13 13

( 1)Δ C C C

Z Z Zm c a a

A A

(13)



22 ( 2 )

Δ A A

A Zm c a

A

(14)


2

34

(1 1 )Δ ( 1)

2

A

ZP Pm c a

A

(15)





2 2 22 3

1 34

3

(1 1 )( 2 )Δ ( 1)

2

A

ZX V S C A P


A AA

(16)


2

2

3

4Z A

8 2

A n p

A C

Aa A m m c

a a A

(17)

























0

1

1

Rr R

V r Ve




02 3

2

n

N

VE N

R m



13





[Basdevant2005]

(18)


235 nucleus

harmonic potential

(19)




14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments




Leptons Quarks


0000511 minus1 ݑ asymp 0003 23











Mass (GeVc2) = 0938272 = 0939565



Spin 12 12






VYukawa

(r) = -aNc

e-

mp cr

r (2)




23 Nuclear Chart






8










The Essential











130R A R




The Essential CANDU



| no nuclei







(3)











N Zm Z m Nm AB c














2

N Z p nX

m Z m Nm AB c







9






(4)




f the mass number





1

2

N Z nX H

M Z M Nm AB c (5)



X i i

i

M M (6)


i ii

X

Mw

M

(7)


X A

X

NM

(8)


i i A i X

X

N NM

(9)




2Δ Xm c AB (10)






2Δ V Vm c a A (11)



2 23Δ S Sm c a A (12)




22

13 13

( 1)Δ C C C

Z Z Zm c a a

A A

(13)



22 ( 2 )

Δ A A

A Zm c a

A

(14)


2

34

(1 1 )Δ ( 1)

2

A

ZP Pm c a

A

(15)





2 2 22 3

1 34

3

(1 1 )( 2 )Δ ( 1)

2

A

ZX V S C A P


A AA

(16)


2

2

3

4Z A

8 2

A n p

A C

Aa A m m c

a a A

(17)

























0

1

1

Rr R

V r Ve




02 3

2

n

N

VE N

R m



13





[Basdevant2005]

(18)


235 nucleus

harmonic potential

(19)




14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments




VYukawa

(r) = -aNc

e-

mp cr

r (2)




23 Nuclear Chart






8










The Essential











130R A R




The Essential CANDU



| no nuclei







(3)











N Zm Z m Nm AB c














2

N Z p nX

m Z m Nm AB c







9






(4)




f the mass number





1

2

N Z nX H

M Z M Nm AB c (5)



X i i

i

M M (6)


i ii

X

Mw

M

(7)


X A

X

NM

(8)


i i A i X

X

N NM

(9)




2Δ Xm c AB (10)






2Δ V Vm c a A (11)



2 23Δ S Sm c a A (12)




22

13 13

( 1)Δ C C C

Z Z Zm c a a

A A

(13)



22 ( 2 )

Δ A A

A Zm c a

A

(14)


2

34

(1 1 )Δ ( 1)

2

A

ZP Pm c a

A

(15)





2 2 22 3

1 34

3

(1 1 )( 2 )Δ ( 1)

2

A

ZX V S C A P


A AA

(16)


2

2

3

4Z A

8 2

A n p

A C

Aa A m m c

a a A

(17)

























0

1

1

Rr R

V r Ve




02 3

2

n

N

VE N

R m



13





[Basdevant2005]

(18)


235 nucleus

harmonic potential

(19)




14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments

8










The Essential











130R A R




The Essential CANDU



| no nuclei







(3)











N Zm Z m Nm AB c














2

N Z p nX

m Z m Nm AB c







9






(4)




f the mass number





1

2

N Z nX H

M Z M Nm AB c (5)



X i i

i

M M (6)


i ii

X

Mw

M

(7)


X A

X

NM

(8)


i i A i X

X

N NM

(9)




2Δ Xm c AB (10)






2Δ V Vm c a A (11)



2 23Δ S Sm c a A (12)




22

13 13

( 1)Δ C C C

Z Z Zm c a a

A A

(13)



22 ( 2 )

Δ A A

A Zm c a

A

(14)


2

34

(1 1 )Δ ( 1)

2

A

ZP Pm c a

A

(15)





2 2 22 3

1 34

3

(1 1 )( 2 )Δ ( 1)

2

A

ZX V S C A P


A AA

(16)


2

2

3

4Z A

8 2

A n p

A C

Aa A m m c

a a A

(17)

























0

1

1

Rr R

V r Ve




02 3

2

n

N

VE N

R m



13





[Basdevant2005]

(18)


235 nucleus

harmonic potential

(19)




14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments










N Zm Z m Nm AB c














2

N Z p nX

m Z m Nm AB c







9






(4)




f the mass number





1

2

N Z nX H

M Z M Nm AB c (5)



X i i

i

M M (6)


i ii

X

Mw

M

(7)


X A

X

NM

(8)


i i A i X

X

N NM

(9)




2Δ Xm c AB (10)






2Δ V Vm c a A (11)



2 23Δ S Sm c a A (12)




22

13 13

( 1)Δ C C C

Z Z Zm c a a

A A

(13)



22 ( 2 )

Δ A A

A Zm c a

A

(14)


2

34

(1 1 )Δ ( 1)

2

A

ZP Pm c a

A

(15)





2 2 22 3

1 34

3

(1 1 )( 2 )Δ ( 1)

2

A

ZX V S C A P


A AA

(16)


2

2

3

4Z A

8 2

A n p

A C

Aa A m m c

a a A

(17)

























0

1

1

Rr R

V r Ve




02 3

2

n

N

VE N

R m



13





[Basdevant2005]

(18)


235 nucleus

harmonic potential

(19)




14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments



1

2

N Z nX H

M Z M Nm AB c (5)



X i i

i

M M (6)


i ii

X

Mw

M

(7)


X A

X

NM

(8)


i i A i X

X

N NM

(9)




2Δ Xm c AB (10)






2Δ V Vm c a A (11)



2 23Δ S Sm c a A (12)




22

13 13

( 1)Δ C C C

Z Z Zm c a a

A A

(13)



22 ( 2 )

Δ A A

A Zm c a

A

(14)


2

34

(1 1 )Δ ( 1)

2

A

ZP Pm c a

A

(15)





2 2 22 3

1 34

3

(1 1 )( 2 )Δ ( 1)

2

A

ZX V S C A P


A AA

(16)


2

2

3

4Z A

8 2

A n p

A C

Aa A m m c

a a A

(17)

























0

1

1

Rr R

V r Ve




02 3

2

n

N

VE N

R m



13





[Basdevant2005]

(18)


235 nucleus

harmonic potential

(19)




14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments




2Δ V Vm c a A (11)



2 23Δ S Sm c a A (12)




22

13 13

( 1)Δ C C C

Z Z Zm c a a

A A

(13)



22 ( 2 )

Δ A A

A Zm c a

A

(14)


2

34

(1 1 )Δ ( 1)

2

A

ZP Pm c a

A

(15)





2 2 22 3

1 34

3

(1 1 )( 2 )Δ ( 1)

2

A

ZX V S C A P


A AA

(16)


2

2

3

4Z A

8 2

A n p

A C

Aa A m m c

a a A

(17)

























0

1

1

Rr R

V r Ve




02 3

2

n

N

VE N

R m



13





[Basdevant2005]

(18)


235 nucleus

harmonic potential

(19)




14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments



2 2 22 3

1 34

3

(1 1 )( 2 )Δ ( 1)

2

A

ZX V S C A P


A AA

(16)


2

2

3

4Z A

8 2

A n p

A C

Aa A m m c

a a A

(17)

























0

1

1

Rr R

V r Ve




02 3

2

n

N

VE N

R m



13





[Basdevant2005]

(18)


235 nucleus

harmonic potential

(19)




14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments
















0

1

1

Rr R

V r Ve




02 3

2

n

N

VE N

R m



13





[Basdevant2005]

(18)


235 nucleus

harmonic potential

(19)




14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments

14





0 1

1 1

0 1

2 6

orbital 1s 1p



The Essential


1

2j l



the magic numbers


2 2 3 3 4 4 4

1 2 1 2 1 2 3

2 0 3 1 4 2 0

10 2 14 6 16 10 2

1d 2s 1f 2p 1g 2d 3s






2

2

JE

I

The Essential CANDU


(20)

the association be-






[Basdevant2005

(21)




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments




EJ

= 2J J +1( )

2I (22)


2 4 6 8

3 1 1

10 17 1

2E E E E (23)

with

E2

=152

2MR2 (24)




AE AB A (25)


2 22

A A

AE AB E

(26)



92

235U THORN 20

1n +50

133 Sn +42

100 Mo (27)



1 235 1 133 1000 92 0 50 42 3 n U n Sn Mo (28)






2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments





2 3 1 41 1 0 2 H H n He (29)

2 3 1 41 2 1 2 H He H He (30)


3 Nuclear Reactions


How do they decay






linear momentum

angular momentum

electric charge

leptonic charge and

baryonic charge







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments







1

i

i

NCA B

Z W Vi

X Y b

(31)


2 2 2

1


N

X Y Y b bi

m c m c T m c T

(32)


2 2 2

1 1


N N

Y X Yb bi i

T T m c m c m c

(33)

1

N

ii

Z W V

(34)

1

N

ii

A B C

(35)



Δ 0

Δ ( )( )

lim Δt

N tN t

t

(36)


( )

( )dN t

N t S tdt

(37)




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments




A t N t (38)


Δ0 0( Δ ) tN t t N t e (39)



1

(40)


00( )

N tN t

e (41)


12

(

2)2

lnt ln

(42)



1

1J

j

j

Y

(43)






22| | ( Ξ) Ξ

f

f H i f d

(44)






X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments




X + ߛ (45)



X]ା)


X rArr [ ାଵ

Y]ା + + ҧߥ (47)


X rArr

ଵX + ଵ (48)


X rArr [ ଵ

Y] + ା + ߥ (49)


X rArr ଵ

Y + ߥ (50)


X rArr [ ଵ

ଵY] + ଵଵ= ଵ

ଵY + Hଵଵ (51)



X rArr [ ଶ





X rArr ܥ

ଵ+ F +














ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments











ray absorption

A AZ ZX X (54)


A A

Z ZX X (55)

electron capture

1 γA AZ Ze X Y

(56)

positron capture

1 A AZ Ze X Y

(57)


Z Zp X Y

(58)

ߙ capture24 4

2 2 A AZ ZX Y

(59)


Z Zn X X (60)


1 1

0 0 A AZ Zn X X n (61)







22| | ( )

Ξ i

d Vf H i f

d v

(64)










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments










An ZM

(65)


2

24 ( )

e

collision

z cdEn L v F v

dx mv

(66)




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments




22

2

2( )

(1 ( ) )

mvL ln v c

I v c(67)


3 2

e

radiation e

dEZn

dx m

(68)


ଶ



0

1

T

total

R dEdE

dx

(69)



dE dE dE dE

dx dx dx dx

(70)


ESTAR for electrons

PSTAR for protons




(71)

and

2(g cm ) (cm)R R (72)

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments

24







The Essential



1

pairs

total

dn dE

dx W dx


Photons with Matter








The Essential CANDU



(73)












Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments





Rayleigh scattering

Compton scattering

Pair production and



( ) ( ) m x m i i xE N E (74)







e NT T E W (76)


26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments

26


Figure



sPhotoelectric





The Essential




2

2

1361 n

ZW eV

n



c KE( ) raquo

32 2pZ 5ae

6 mec2( )

3

2

3 E( )7

2

c( )2


2 3







The Essential CANDU



(77)




(78)


2 31 001481 ln 0000788 ln E E Z Z (79)











2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments






2

22

2

0

1 cos sin eRayleigh

e

cE F E Z d

m c

(80)



gQ 1+ Q2( )g

(81)

with


aeZm

ec2

(82)

g Z( ) = 1+ aeZ( )

2

(83)







2

2

(1 cos )

(1 cos )

e

e

T Em c E

(84)


28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments

28


2 2e

Compton

e

cE ln

m c



Figure

344 Pair production







The Essential


2

22 2

2 1 1 4 11 2 1

2 2 1 2

cE ln

m c







2

2

2

28 218 129ln

9 7 02

1 2e

Pair production e

e

cE Z

m c


above a few MeV





The Essential CANDU


21 2 1

2 1 2

(85)






(86)










s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments





s 2 DPh








n E r s n E r e






E - 2226( )3

E 3










Ω Ω ΩE s

n E r s n E r e

29



large amounts of

(87)





obtained using the



on ሻܧሺߑ the

has been crossed is

(88)







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments







(89)
















2

2 1 2 cos 1

n in f

EE A A

A

(92)


2

cos 1cos

1 2 cos

A

A A

(93)




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments




1

2n iE E

(94)

2

2

1α

1

A

A

(95)



s = 4p R2 (96)


s (E) = 4p2 sin2 d

2mnE

(97)



1 A A A

Z Z Zn X X n X (98)

1 A A




Γ k

k

(100)


Γ Γ k (101)






32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments

32


1AZ X

E l



k





The Essential


2

22

0

E2 1

2

2

nE lmE

E E


1 AZ

kk X

E E

Γ = Γradicܧ

2

0 02 1 2

kk E l

vm E








The Essential CANDU


(103)


(104)

(105)


of energy










2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments





2












1

2


mE



( )

32

2XE k T


B

B




33





(106)

Here Xܧ(Xܧ)

ing a nucleus

(107)










m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments







m m x

m x m x

N v n vR v V V v v

v

(108)









(109)


1

2 2 2 2(1 ) A B A B CZ Y Z Y


(110)








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments








Isotope (MeV) (MeV)




Uଽଶଶଷ 49 59



ଶଷ













42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments






42

Theorcal

ass

14funnu






2I

I

Y (111)




energy






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments






Isotope

Thଽଶଷଶ

Uଽଶଶଷଷ

Uଽଶଶଷହ

Uଽଶଶଷ

Puଽସଶଷଽ











1gt) eV) 1lt) MeV)

0 22

250 26

243 26

0 26

289 31





37



















Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments







Prompt rays-ߛ 7

Delayed rays-ߛ 7

Electrons 8

Neutrinos 12

Neutrons 7

Total 207




0

1n nE dE

(113)

and has the value

C =2a23e-b4a

pb (114)












ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments









ௗே(௧)



0


(116)




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments




0


J J x


(117)


௫ is

























1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments





1D



Hଵଵ 18 017

Dଵଶ 34 000025

Cଵଶ 46 00016


ଵ Dଵଶ and C



ଵ

















3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments









3 2

0 4

( ) ( Ω) Ω

J f J

JV


(118)





(119)







(120)



LM

r tE

Ω( ) dt = -

Ω timesNtildef

r tE

Ω( ) dt (121)






2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments



2

0 4


(122)




2

0 4

Ω ( ) ( ) Ω ( Ω) F j j f Jj


(123)






1

v

dfr t E

Ω( )

dt= L

C

r tE

Ω( ) + L

M

r tE

Ω( ) + S

S

r t E

Ω( ) + S

F

r t E

Ω( ) + S

E

r tE

Ω( ) (124)


2 2

0 4 0 4

Ω Ω Ω


j

r E r E r E


(125)



fr

B EΩ( ) = 0 for

Ω times

N lt 0 (126)









5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments






5 Bibliography









































ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments


























ଷ







ଶ + ଵܪଷ =

+ ଵ

b) ଵ + ଽܥ


ଽ + 3 ଵ











8 Acknowledgments



1Introduction

11Overview

12Learning Outcomes




23Nuclear Chart




3Nuclear Reactions


32Nuclear Reactions






344Pair production








5Bibliography


7Problems

8Acknowledgments

chapter 3 nuclear processes and neutron physics - nuclear...nuclear physics – june 2015 chapter 3...

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