CHAPTER 3 Nuclear Processes and Neutron Physics - Nuclear Processes and...We will also introduce some language specific to nuclear physics that will be useful throughout this book

1

©UNENE, all rights reserved. For educational use only, no assumed liability. Nuclear Physics – June 2015

CHAPTER 3

Nuclear Processes and Neutron Physicsprepared by

Dr. Guy MarleauÉcole Polytechnique de Montréal

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 .............................................................................................................................31.1 Overview......................................................................................................................... 31.2 Learning Outcomes......................................................................................................... 4

2 Structure of the Nucleus..........................................................................................................42.1 Fundamental Interactions and Elementary Particles...................................................... 42.2 Protons, Neutrons, and the Nuclear Force ..................................................................... 62.3 Nuclear Chart .................................................................................................................. 72.4 Nuclear Mass and the Liquid Drop Model.................................................................... 102.5 Excitation Energy and Advanced Nuclear Models ........................................................ 122.6 Nuclear Fission and Fusion ........................................................................................... 15

3 Nuclear Reactions ..................................................................................................................163.1 Radioactivity and Nuclear Decay .................................................................................. 163.2 Nuclear Reactions ......................................................................................................... 203.3 Interactions of Charged Particles with Matter.............................................................. 223.4 Interactions of Photons with Matter ............................................................................ 243.5 Interactions of Neutrons with Matter .......................................................................... 30

4 Fission and Nuclear Chain Reaction.......................................................................................344.1 Fission Cross Sections ................................................................................................... 354.2 Fission Products and the Fission Process...................................................................... 364.3 Nuclear Chain Reaction and Nuclear Fission Reactors ................................................. 404.4 Reactor Physics and the Transport Equation ................................................................ 42

5 Bibliography ...........................................................................................................................446 Summary of Relationship to Other Chapters.........................................................................457 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 numbers......................................................................................14Figure 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 atom.24Figure 7 Photoelectric effect..........................................................................................................26Figure 8 Compton effect. ...............................................................................................................28Figure 9 Photonuclear cross sections for deuterium, beryllium, and lead....................................29Figure 10 Cross sections for hydrogen (left) and deuterium (right) as functions of energy..........32

Figure 11 Cross sections for 23592 U (left) and23892 U (right) as functions of energy. ........................33

Figure 12 Effect of temperature on 238U absorption cross section around the 6.64-eVresonance...............................................................................................................................34

Figure 13 Capture and fission cross sections of 23592U (left) and23892U (right) as functions of

energy.....................................................................................................................................36

Figure 14 Relative fission yield for 23592U as a function of mass number........................................37

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 state...................14Table 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 nuclides........37Table 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 ,21D ,

and 126C in the energy range 0.1 eV < E

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 fuel,whether 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.

1.1 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 2.4) 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 models.We 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 neutrons.This is followed by a description of the neutron transport equation that can be used to charac-terize the behaviour of neutrons inside a reactor.

1.2 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

2.1 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 force,the so-called “colour” 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 “weak” charge. Table 1 provides a brief description of the general proper-ties of the four fundamental forces, where the s’ߙ 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 example,the 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 ݁= 1.6 × 10ିଵଽ C is the charge of an electron, ߝ = 8.854 × 10ିଵଶ F/m is the electric

permittivity of vacuum, ℏ = 1.0546 × 10ିଷସ J × s is the reduced Planck constant, andܿ= 2.9979 × 10଼ m/s 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 ≈ 1 ≈ ݎ

Electromagnetic Electric charge ߙ ≈ 1/137 ݎିଶ

Weak Weak charge ௪ߙ ≈ 10ି ହିݎ to ିݎ

Gravitational Mass ீߙ ≈ 10ିଷଽ ଶିݎ

In the standard model of particle physics [Halzen1984, Le Sech2010], these forces are repre-sented by 12 virtual particles of spin 1 called “gauge bosons”:

the massless photon that carries the electromagnetic interaction;

eight massless gluons that carry the strong (colour) force;

three massive bosons (W± and Z) that mediate the weak interaction.

These bosons are exchanged between the 12 “physical fermions” (spin 1/2 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 (GeV/c2) Charge Particle Mass (GeV/c2) Charge

݁ 0.000511 −1 ݑ ≈ 0.003 2/3

ߥ < 2.5 × 10ିଽ 0 ݀ ≈ 0.006 −1/3

ߤ 0.1057 −1 ܿ ≈ 1.2 2/3

ఓߥ < 0.000170 0 ݏ ≈ 0.1 −1/3

߬ 1.7777 −1 ݐ ≈ 173 2/3

ఛߥ < 0.018 0 ܾ ≈ 4.1 −1/3

Table 2 provides a description of the main properties of leptons and quarks with the massesgiven in GeV/c2, with 1 GeV/c2≈ 1.782661 × 10ିଶ kg, and the charges in terms of the chargeof one electron [Le Sech2010].

2.2 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 (GeV/c2) ݉ = 0.938272 ݉ = 0.939565

Charge (electron charge) 1 0

Magnetic moment (Nߤ) ߤ = 2.792847 ߤ = −1.913042

Spin 1/2 1/2

Mean-life (s) ߬ > 10ଷ

߬ = 881.5

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 force.Moreover, 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 MeV/c2) and ேߙ ≈ 14.5 is a constant that controls the

strength of the nuclear interactions between baryons (protons or neutrons). The minus sign

here indicates that the resulting force ⃗ܨ) = −∇ሬሬܸ⃗ ) 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 Pauli’s exclusion principle). This also explains why stable nuclei contain-ing only neutrons are not seen in nature.

2.3 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 nucleus,each 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

©UNENE, 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

©UNENE, 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 ȁܰ െ ܼ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.

> 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

1/30,R 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

nuclei. The nuclear stability curve is also illustrated.

The Essential CANDU

Nuclear Physics – 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-

several stable nuclei can be produced with different numbers of neutrons.

, 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 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

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 “magic numbers”and protons, just like electrons in atoms, are more tightly bonded when they fill a quantumenergy shell (see Section 2.5).

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 nuclei,reaching a maximum around ܣ ൌ 60the average, nucleons are more tightly bonded inside nuclei having intermediate values of( Ͳʹ ൏ >�ܣ 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

159 contain an even number of protons and of neutrons,

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

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.

equal to 8, 20, 50, 82, and 126, the number of stable nuclei isThese numbers are known as “magic numbers”. 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 nXm 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 numbers,60, 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

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

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 numbers,is 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 HM 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 ii

M M (6)

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

.i iiX

Mw

M

(7)

For completeness, one can compute the isotopic concentration ܰX of an element (atoms/cm3)

with atomic mass ܯ X (g/moles) and density ߩ (g/cm3) using:

,X AX

NM

(8)

where ℕA = 6.022 × 10ଶଷ atoms/moles is the Avogadro number. The concentration of

isotope i݅s then given by:

.i i A i XX

N NM

(9)

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

2.4 Nuclear Mass and the Liquid Drop Model

As discussed in Section 2.3, 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 Δ݉ 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 ܽ ≈15.753 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 ଶܴߨ4 ∝ ,ଶ/ଷܣ one can assume that the ଶ/ଷܣ 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 2/3Δ ,S Sm c a A (12)

where ௌܽ ≈17.804 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 (ܸ(ܴ) ∝ 1/ܴ ∝

,(ଵ/ଷܣ/1 each of the ܼ protons will interact with the remaining ܼ− 1 protons, leading to acontribution of the form:

22

1/3 1/3

( 1)Δ ,C C C

Z Z Zm c a a

A A

(13)

where only the dominant term in ܼ has been preserved, and ܽ ≈0.7103 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 |ܰ − |ܼ =−ܣ| 2ܼ|, the form of the contribution being

22 ( 2 )Δ ,A A

A Zm c a

A

(14)

where ܽ ≈23.69 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

23/4

(1 1 )Δ ( 1)

2

A

ZP Pm c a

A

(15)

is required. Here, ܽ ≈33.6 MeV is known as the pairing term.

The final formula for the missing mass is therefore



2 2 22 31 3/4

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-Weizsäcker 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 stable.Neglecting 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 “magic numbers”, nor the gamma-ray absorption spectrum,nor 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.

2.5 Excitation Energy and Advanced Nuclear Models

All chemists and physicists are familiar with atomic “magic numbers”, even though the explicitterm is rarely used. For example, the number of electrons in noble gases could be considered“magic” because these gases are odourless, colourless, and have very low chemical reactivity.For 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 Schrödinger equation for ܣ strongly coupled nucleons[Griffiths2005]. However, when trying to solve this equation, two problems arise:

the Schrödinger 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 Schrödinger equation forproblem is reduced to the solution ofnucleon with an averaged potentialmodel is the Saxon-Wood potential (see

where ܸ ≈30 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 Schrödinger equation for ܣ strongly coupled nucleons, the

ced to the solution of ܣ independent Schrödinger 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 0,1,2,l 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 Schrödinger 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)

0,1,2, ⋯ 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)∗) 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 0,1,2, ⋯. For nucleus having an ellipsoid shape, only even values of areܬ allowed,and 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.

2.6 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 > 180 nucleons, which is given by

,AE AB A (25)

is larger than that of two identical nuclides containing 2/ܣ 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 − ுܧ .

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

92

235U Þ 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 17.6 and 18.3 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 2.5). 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.

3.1 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

,ii

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

0,B 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 Δܰ(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 (ݐܵ) 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 (ݐܵ) = 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, ,ଵ/ଶݐ

1/2(

2)

2 ,ln

t ln

(42)

represents the average time required for the initial number of nuclei to decrease to a value of.)/2ݐ)ܰ

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

1

1.J

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 1.805 × 10ଽ years, decays 89.28% of the time intocalcium-40 (ܻ

మబరబCa = 0.8928) and 10.72% of the time into argon-40 (ܻభఴరబAr = 0.1072).

Note that the decay constant, ,ߣ associated with a reaction can be evaluated using Fermi’ssecond golden rule [Schiff1968]:

2'2 | | ( ,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)∗ ⇒

X + .ߛ (45)

Internal conversion for an excited nucleus:( X)∗ ⇒ [

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 ⇒ [ ାଵ

Y]ା + ݁ି + .ҧߥ (47)

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

X ⇒

ିଵX + ଵ .݊ (48)

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

X ⇒ [ ିଵ

Y]ି + ା݁ + .ߥ (49)

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

X ⇒ ିଵ

Y + .ߥ (50)

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

X ⇒ [ ି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 ⇒ [ ିଶ

ିସY]ଶି + ଶସߙ = ିଶ

ିସY + ଶସHe. (52)

This process corresponds to the ejection of an atom of helium.

Spontaneous fission for heavy nuclei (nuclear interaction):

X ⇒ ܥ

ଵ݊+ 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.

3.2 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 matter,they 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 AZ ZX X (55)

electron capture:

1 γ.A AZ Ze X Y

(56)

positron capture:

1 A AZ Ze X Y

(57)

proton capture:1 11 1 .

A AZ Zp X Y

(58)

ߙ capture:24 4

2 2 .A AZ ZX Y

(59)

neutron capture:1 10 .

A AZ Zn X X

(60)

neutron scattering (elastic and inelastic):

*1 1

0 0 .A AZ Zn X X n (61)

photon-induced fission:10 .

A B A B CZ Y Z YX C n F G

(62)

neutron-induced fission:1 1 10 0 .

A B A B CZ 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 Fermi’s first golden rule, whichcan be written as:

2'2 | | ( ),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 = 10ିଶସ cmଶ = 10ିଶ଼ 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].

3.3 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 down.The 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 complex.The cross section for electron-electron collisions is given by the Möeller scattering formula,while 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 place,producing 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 ,(ݔ݀/ܧ݀−)known as the stopping power, is then given by the Bethe-Block formula [Smith2000,Leroy2009]:

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(ݔ݀/ܧ݀−) 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 ݔ݀/ܧ݀− 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 electron/ion pairs produced per unit distance in a material is given approximatelyby

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

3.4 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 electron/ion 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 ሬ݄⃗݊ Ȁߣof an electromagnetic fieldൌ Ȁܿߥ travelling in direction ሬ⃗݊. 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 electron/ion 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 matter.energy 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 3.2). 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.

3.4.1 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

13.61 ,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( ) »

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 0.01481 ln 0.000788 ln .Photoelectric Photoelectric KE E Z Z section decreases rapidly with photon energy. It is dominant only for ph

tons 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 (i.e.valence 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 0.01481 ln 0.000788 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 (i.e., 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.

3.4.2 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 ,eRayleighe

cE F E Z d

m c

(80)

where (ܼ,ߠ,ܧ)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 (ܼ,ߠ,ܧ)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 E,q ,Z( ) = sin(2gatan(Q))gQ 1+ Q2( )

g, (81)

with

Q E,q ,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.

3.4.3 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 ሬ⃗݇ (ห݇ሬ⃗ห= ℎߥ/ܿ=

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

of the photon being ሬ⃗݇′. 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 electron,the 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 0.5 MeV.

Figure

3.4.4 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 photon,energy 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.

3.4.5 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 (1.666 MeV and 2.226 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.

2݉ ܿଶ traveling through the Coulomb field of a nucleus of

can be converted into an electron-positron pair. After the reaction,the 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 pair.energies 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 (1.666 MeV and 2.226 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 D,Ph

is often used, where ǡtheܧ photon energy

0.0624, 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 Ωሬሬ⃗ 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) » C 2 D

E - 2.226( )3

E 3

the photon energy, is given in MeV, ܥ Dమ varies between 0.061 and

The end result of such a photonuclear reaction is a hydrogenatom and a fast neutron that can initiate a fission reaction (see Section 2.6).

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 ݊ሺܧǡݎԦǡȳሬሬ⃗) photons of energy

on a material of uniform macroscopic cross section

+ ,ȳሬሬ⃗ݏ Ωሬሬ⃗) still present after a distance hasݏ been crossed is

, Ω , .,Ω , Ω E sn E r s n E r e

29


Photonuclear cross sections for deuterium, beryllium, and lead.

large amounts of

(87)

varies between 0.061 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 Ωሬሬ⃗ following the absorption of photons of energy ′ܧ and direction Ωሬሬ⃗′ will also con-

tribute to the total population of photons at point +Ԧݎ .Ωሬሬ⃗ݏ 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 Σ௦൫ܧᇱ→ ,ܧ Ωሬሬ⃗ᇱ→ Ωሬሬ⃗൯ is the differential scattering cross section representing the prob-

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

energy ܧ and direction Ωሬሬ⃗.

3.5 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

Documents

CHAPTER 3 Nuclear Processes and Neutron Physics - Nuclear Processes and...We will also introduce some language specific to nuclear physics that will be useful throughout this book