Basic Nuclear Phys

7/23/2019 Basic Nuclear Phys

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Nuclear Physics Explained in Simple Terms

Ernest M. Henley

University of Washington

Alejandro Garca

University of Washington


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Preface

O God, I could be bounded in a nut shell and countmyself a king of infinite space...

Shakespeare,Hamlet, Act II, Scene 2.

Most of the things we know are made of molecules. These are made of atoms.At the core of each of these atoms is an atomic nucleus. The quote above seemsto express the awe we feel when we realize that the secrets to the behavior of theuniverse may lie in the tiny core of atoms. In this book we will try to describe someof these secrets. Although atomic nuclei occupy a tiny fraction of the volume ofatoms, understanding its properties is one of the biggest challenges that the humanmind has ever undertaken. Nuclear physics is a mature science, more than onehundred years old, but in recent years there have been shifts in paradigms on how

the nucleus can be understood. We wrote this book to address the curiosity of thelay person with respect to nuclear physics but trying to give answers coherent withthe contemporary view. We felt there was a need for simple explanations of someaspects of nuclear physics. It is written as a series of questions with answers. It isaddressed without a particular age group in mind: it should be good for anybodythat is curious about the workings of the most amazing phenomena that take placein a large variety of nuclei that Nature has provided us with.

Because nuclei play an important role in energy production, nuclear astrophysics,and the understanding of basic phenomena predicted by the theory of elementaryparticles, it is at the crossroads of almost every other subfield of physics so we hopethat our discussions will inspire neophytes that may have some interest in any othersubfield of physics as well.

In the quote above we carefully omitted the rest of Hamlets phrase ... were itnot that I have bad dreams... but this part may also be appropriate to describethe status of nuclear physics: although many problems have been solved, manymysteries remain to be understood and continue to give headaches to scientists. Wewill describe some of these problems and the present outlook toward solutions inthe last chapter.

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iv Preface


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Contents

Preface ii

1 What is a nucleus ? 1

2 How was the nucleus discovered? 32.1 Beginnings of nuclear physics . . . . . . . . . . . . . . . . . . . . . . 32.2 What are nuclear isotopes? . . . . . . . . . . . . . . . . . . . . . . . 4

3 How big and how heavy are nuclei? 5

4 Why do some nuclei decay quickly while others remain intact foreternity? 74.1 Why dont nuclei blow apart? . . . . . . . . . . . . . . . . . . . . . . 74.2 Why do nuclei not collapse? . . . . . . . . . . . . . . . . . . . . . . . 84.3 What Can One Learn from Nuclear Masses? . . . . . . . . . . . . . . 84.4 Why Are Some Nuclei More Stable Than Others? . . . . . . . . . . . 9

5 Can we turn lead into gold in the laboratory? 115.1 What are nuclear reactions? . . . . . . . . . . . . . . . . . . . . . . . 125.2 What are reaction cross sections? . . . . . . . . . . . . . . . . . . . . 125.3 Is it feasible to produce large amounts of chosen nuclei by nuclear

reactions between other nuclei? . . . . . . . . . . . . . . . . . . . . . 12

6 Is quantum mechanics important for understanding nuclei? 156.1 What are waves? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2 What are the uncertainty relationsfor waves? . . . . . . . . . . . . . 16

6.3 What is light? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.4 What is quantum mechanics? . . . . . . . . . . . . . . . . . . . . . . 206.5 What are the uncertainty principles of quantum mechanics? . . . . . 226.6 What are nuclear excited states? . . . . . . . . . . . . . . . . . . . 23

7 Why is nuclear energy important to our society? 257.1 Why is nuclear energy so much larger than chemical or atomic energy? 257.2 When and how can a single nucleus be split into pieces (fission) and

when and how can two nuclei fuse together (fusion)? . . . . . . . . . 26

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7.3 How do nuclear reactors work? . . . . . . . . . . . . . . . . . . . . . 277.4 How do nuclear weapons work? . . . . . . . . . . . . . . . . . . . . . 28

8 What is radioactivity and which kind is most dangerous? 318.1 What is radioactivity? What type of interaction drives it? . . . . . . 318.2 What type of radioactivity is the most hazardous for our health? . . 338.3 How can we minimize risks in case of emergencies? . . . . . . . . . . 33

9 Are there simple models useful as well as beautiful? 359.1 The Liquid Drop Model . . . . . . . . . . . . . . . . . . . . . . . . . 359.2 The Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.3 Collective Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10 How Are The Properties of Light Nuclei Investigated? 37

11 Are Nuclei Responsible for the Workings of the Sun and Stars? 4111.1 Nucleosynthesis in the Sun . . . . . . . . . . . . . . . . . . . . . . . 4211.2 Do stars get older? The evolution of stars . . . . . . . . . . . . . . . 4211.3 Explosive nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . 4211.4 N eutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12 What symmetries are applicable to nuclei? 4312.1 What are symmetries? . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.2 B asic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.3 Space-Time Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 44

13 What are the Microscopes for Examining Nuclei? 4713.1 I ntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

14 Can Light and/or Neutrinos Change Carbon to Gold? 4914.1 I ntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.2 Photo induced reactions . . . . . . . . . . . . . . . . . . . . . . . . . 4914.3 Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

15 Is nuclear physics useful other than for generating energy or un-derstanding the workings of the cosmos? 5115.1 Cancer treatment with protons . . . . . . . . . . . . . . . . . . . . . 5115.2 Use of radiation to detect contraband or explore pyramids . . . . . . 51

15.3 Neutrons as probes of small devices . . . . . . . . . . . . . . . . . . . 5115.4 Imaging with nuclear magnetic resonance . . . . . . . . . . . . . . . 5115.5 Imaging with positron-emission tomography . . . . . . . . . . . . . . 5115.6 Imaging with 99mTc . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

16 What are the problems that nuclear physicists are presently work-ing on? 5316.1 Nuclear structure far from stability . . . . . . . . . . . . . . . . . . . 5316.2 Nuclear structure from QCD . . . . . . . . . . . . . . . . . . . . . . 53


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CONTENTS vii

16.3 Relativistic collisions and quark-gluon plasma . . . . . . . . . . . . . 5316.4 Origin of heavy elements and explosive nucleo-synthesis . . . . . . . 5316.5 The structure of the proton and the neutron . . . . . . . . . . . . . . 5316.6 Using nuclei to find new physics . . . . . . . . . . . . . . . . . . . . . 53


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Chapter 1

What is a nucleus ?

Atoms are often described as the basic building blocks of matter. While this remainstrue in many respects, major discoveries in physics over the past 100 years haveshown that atoms have a simple yet rich internal structure. We know today, that atthe heart of each atom is a very small, dense and positively charged core called thenucleus surrounded by a diffuse cloud of relatively light negatively charged particlescalled electrons. The atom as a whole is electrically neutral, and it is in this sensethat we think of them as basic building blocks. However, that atom is a largelyempty, and this is illustrated in the figure.

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2 CHAPTER 1. WHAT IS A NUCLEUS ?


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Chapter 2

How was the nucleus

discovered?

2.1 Beginnings of nuclear physics

Nuclear physics actually had two beginnings. The first one was in 1896 with HenriBecquerel in France. He knew nothing about nuclei; they were unknown. Roentgenhad previously discovered that x rays , now known to be light of frequency muchhigher than visible, could penetrate materials that visible light could not, and leftmarks on photographic plates. X rays were shown to yield images of bones and

quickly became useful for medical purposes. Becquerel noticed that uranium saltshad similar effects on photographic plates. He initially thought that some mysteri-ous mechanism (via light from the Sun or heat) made the uranium salts generatex rays. But soon, in the rainy days of Paris, he discovered that the radiation fromthe uranium salts was not affected by temperature or light. It did not change withtime, nor with ultraviolet light or even X-rays. No cathode ray tube was needed.It emanated spontaneously from uranium and got to be known as Becquerel rays. These rays occurred naturally, and blackened photographic plates, even if thesample was wrapped in dark paper. He was followed by Marie Curie, who triedto isolate the radiations source. She realized that they came from some particularatoms. It took over a decade to identify the radiation , which is now known to be amixture of rays (ionized Helium atoms), rays (electrons and positrons(1)), andrays (light).

The second beginning was the real discovery of nuclei. Following Bohrs discov-ery of the quantum structure of hydrogen, Ernest Rutherford, in England, bom-barded thin gold foils with alpha particles, and thereby disproved the generallyaccepted plum pudding model of the atom. This model, put forward by J.J.

1Positrons are particles very similar to electrons but of opposite electric charge. They were

predicted by Diracs theory of electromagnetism in quantum mechanics and found later in ex-

periments. Positrons are the anti-particles of electrons and, in contact, these can annihilate into

light.

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4 CHAPTER 2. HOW WAS THE NUCLEUS DISCOVERED?

Thomson, had all the constituents of the atom spread evenly throughout the spaceoccupied by the atom. He thus expected the alpha particles to be scattered at smallangles. He asked one of his assistants, Hans Geiger, and a younger one, Ernest Mars-den, to look for alpha particles scattered backward. Imagine their surprise, whenthey actually found a few scattered all the way back. This is like hitting tissuepaper with a bowling ball and finding that the bowling ball sometimes turns all theway around. Rutherford could not believe these results and had to check them forhimself; he found it was true. He concluded that an atom had a tiny kernel at itscenter that contained most of its mass. The kernel was called the nucleus of theatom. Rutherford found that the size of the nucleus was less that 1/1000 that ofthe atom. Today we know that atoms consist mainly of empty space, with electronson the outside (up to 108 cm) and a small central nucleus (of about 1013 cm).

On the way to understanding the structure of the nucleus, which was assumedto contain the known particles, protons and electrons, there were many wrong turnsand puzzles. The electrons observed in radioactive decays were initially assumed toexist in the nucleus previous to the decay. Now we know that they are generatedin the radioactive decay process and that there are no electrons in the nucleus.Admitting that particles transform or get produced in decays beyond the photonsthat were observed in atomic transitions took time.

In 1932, the British physicist James Chadwick took a penetrating radiationfound by Walther Bothe and Wilhelm Becker in Germany and bombarded paraffinwith it. The protons inside the paraffin (hydrogen nuclei) showed that the incidentradiation had about the same mass as the protons. Chadwick concluded that thenew radiation was composed of new neutral particles of about the mass of theproton. They were called neutrons. It was then realized that nuclei consisted of

protons and neutrons. In some of the radioactive decays a neutron within a nucleusgets transmuted into a proton and an electron and, in others a proton decays intoa neutron and a positron. Thus, the electric charge remains constant. As we willsee later an elusive particle, the neutrino, is also emitted in radioactive decays aswell. Because neutrons and protons are very similar in every way except for theirelectric charge and they are the main constituents of nuclei, they are genericallycallednucleons.

2.2 What are nuclear isotopes?

Nuclei are identified by the number of protons and neutrons they contain. Becauseatoms are electrically neutral the number of protons is equal to the number of

electrons in the atom. Thus the number of protons is also frequently referred toas the atomic number and the letter Z is used to denote it. The atomic numberdetermines the chemical properties of an atom. The number of neutrons for a givenatomic number may vary. For example, carbon can be found as 12C but also as 13C.Both have six protons, but the former has six neutrons while the latter has seven.These two are called isotopesof carbon.


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Chapter 3

How big and how heavy are

nuclei?

Before we continue pondering questions about nuclei it is convenient to have anestimation of their sizes and weights.

Nuclei are about ten thousand times smaller than atoms. To get an idea one canconsider shrinking the Earth to the size of the thickness of a hair. In that processthe thickness of a hair would become as big as a typical nucleus. The unit tomeasure nuclear radii is the fermi, with 1 fermi = 1015 meters or 0.0000000000001centimeters.

Instead of quoting weights (that are really the forces exerted by gravity ondifferent objects) physicists quote the masses of objects. As example, the massof a person that weighs 150 pounds is about 68 kilograms. The mass of a persontimes the acceleration of gravity yields the persons weight. For macroscopic objectsthe units of weight are Newtons and one pound is about 4.45 Newtons. Mass canbe converted to energy and their relationship was discovered by Einstein, E =mc2, where c is the speed of light. Later in this book we will explain how thetransformation from mass to energy can take place. But here we need to be awareof that relationship because it has led to the common use of units of energy toquote masses. The reader may be thinking We started with weights, then movedto masses, and now we are in energies!. This is correct, but these three quantitiesare all simply related to each other by constants. The unit of energies used in

atoms and nuclei are electron-Volts, or eV: 1 eV is the kinetic energy that oneelectron gains in moving from the negative to the positive side of a battery of 1Volt. One eV is roughly equal to 1019 of the energy delivered by a 1-Watt lampduring one second. The excitation energies of atoms are measured in eV. As wewill see the excitation energies of nuclei are typically one million times bigger, ora few mega-electron-Volts, MeV. The mass of a single proton (or a single neutron)is approximately mpc2 1000 MeV. Because the masses of nuclei are are roughlyproportional to the number of protons plus neutrons in the nucleus and the massesof atoms are roughly equal to their nuclear masses (the masses of the electrons

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6 CHAPTER 3. HOW BIG AND HOW HEAVY ARE NUCLEI?

being much smaller) another unit called the Atomic Mass Unit, or AMU , is alsofrequently used: 1 AMU = 1/12 of the mass of the atom of 12C. This unit isalso useful because in grams it expresses the mass of approximately 6 1023 (theAvogadro number) atoms. Thus 1 gram of 1H contains approximately 6 1023hydrogen atoms and 56 grams of 56Fe contain the same number of atoms of iron.


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Chapter 4

Why do some nuclei decay

quickly while others remainintact for eternity?

4.1 Why dont nuclei blow apart?

The nucleus is composed of protons and neutrons. Protons have positive electriccharge and neutrons are electrically neutral. What then keeps the nucleus togetherwhen the Coulomb force between the protons wants to blow it apart? It is just

this Coulomb force between electrons and protons which holds atoms together. Butthere the force is attractive whereas in nuclei it is repulsive. There must be astronger attractive force which overcomes the repulsion between the protons. Butwhere does it come from and what mediates the interaction?

In 1935 the Japanese physicist Hideki Yukawa came up with an answer. Bythen Diracs theory of the electromagnetic interaction for quantum systems (knownas quantum electrodynamics or QED) was well established. According to it theelectromagnetic interaction is mediated by photons. Photons are massless particlesand this, according to Diracs theory, implies that the interaction falls off slowlywith distance as 1/r2. This behavior is described as the force having infinite rangebecause at very large distances there can still be an influence. The nuclear force, onthe other hand, has to have a much shorter range. Otherwise solids would collapse,

for example. Inspired by Diracs theory Yukawa showed that the radial dependenceof a force mediated by a particle of mass is exp [(c)/]r. When the radialdistance from the source is large, the force goes to zero exponentially. The rangeis given by /c. Requiring the range to be about the size of a nucleus, on theorder of a few times 1013 cm, Yukawa came up with a mass for the mediator ofabout c2 100 MeV, about a tenth of the mass of the proton or neutron, butmuch heavier than electrons. The particle was thus called a meson and Yukawapredicted that this particle should be produced in high-energy collisions, such asthose taking place when cosmic rays enter the atmosphere. The Yukawa particle

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8CHAPTER 4. WHY DO SOME NUCLEI DECAY QUICKLY WHILE OTHERS REMAIN INTACT FOR E

was sought in cosmic rays and, indeed, something of that mass turned up, but theparticle was weakly and not strongly interacting. Nowadays we call that particlea muon and we know that it is like an electron but about 200 times heavier. TheYukawa particle, now called a pion, was found by a British physicist, Cecil FranklinPowell, in cosmic rays in 1947. Pions can be either neutral (0) or charged ().Nowadays we know that there are many other mesons. Like photons, they are partof a class of particles referred to as bosons, particles with integer spin, as opposedto fermions, like the electron, proton and neutron, with half-integer spin. The spinis the angular momentum of the particle or nucleus.

4.2 Why do nuclei not collapse?

In view of the attractive nuclear force, what keeps nuclei from collapsing? Whydont the neutrons and protons all end up on top of each other? If so, the density ofnuclei would increase withA, whereAis the number of nucleons, whereas it is foundto be constant. To answer this question we have to think about a few issues. First,and perhaps foremost, there is the Pauli exclusion principle. It allows only oneproton or neutron in each quantum state. But there are also additional featureswhich prevent collapse. The nuclear forces are complex. There are exchangeforces, in which a proton becomes a neutron and vice-versa, or a proton of spin upturns into one with spin down. The first kind of exchange force occurs because thepions exchanged between a neutron and proton can be charged, so that a protonemits a+ and turns into a neutron. These temper the attraction between pairs ofnucleons because some of the exchange forces are repulsive. There is an additional

feature of nuclear forces that prevents collapse. Nuclear forces exhibit a repulsionat short distances, e.g. 0.5 Fermi. Where does this repulsion come from? In thecontext of the Yukawa meson theory it stems from the exchange of other heaviermesons. The meson, for example, is about 5 times heavier than the pion and itsexchange provides forces of shorter range. The combination of exchange forces andrepulsive core is quite enough to prevent the collapse and to yield a constant centraldensity. Thus, the nucleons are as close to each other as they can be. This propertyis known as saturation. In this sense nuclei are similar to liquids and surfacetension also plays a significant role. The volume of nuclei is then proportional toAand the radius to A1/3.

4.3 What Can One Learn from Nuclear Masses?

The masses of nuclei are the sums of the masses of N neutrons and Z protonsplus the (negative) binding energy. The binding energy is directly related to thenuclear forces. Fortunately, the dominant part of the nuclear forces are two-bodyinteractions. Nucleons interact primarily with their nearest neighbors. Forces thatinvolve three nucleons are much smaller and those that involve four nucleons aresmaller again. Measurements of the binding energies of nuclei with A 20 showan approximate proportionality to A. Thus, the binding energy per nucleon isapproximately constant.


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4.4. WHY ARE SOME NUCLEI MORE STABLE THAN OTHERS? 9

Because of the (approximate) regularity of nuclear masses and binding energies,physicists thought that it must be possible to find a mass formula for nuclei. Indeed,Carl von Weiszacker did just that in 1935. It is based on a liquid drop analogy.In an infinite large nucleus, since each nucleon interacts on average with the samenumber of neighbors, the binding energy is proportional to A. A smaller effect isgiven by a surface term, since nucleons on the outside are not totally surroundedby other nucleons; this term is proportional to the surface area or A2/3. Thereis, of course, the Coulomb energy between protons which depends inversely on thenucleus radius and thus should be proportional to A1/3. Thus, the main terms are

BE= aVA + aSA2/3 + aCZ(Z 1)A1/3. (4.1)

Here aV 16 MeV, aS 19 MeV, aC 0.7 MeV are constants that wereoriginally determined experimentally (using the known binding energies of somenuclei) but can nowadays be calculated from models as well. There are additionalterms, e.g. a symmetry energy because nuclei like to have N Z, if the Coulombeffects are neglected. Figure 4.1 shows a plot of the binding energy per nucleonplotted versus the number of nucleons in nuclei. This figure shows that for most

Figure 4.1: Plot of the binding energy per nucleon versus the number of nucleonsin nuclei.

nuclei the rough binding energy per nucleon is about 8 MeV. It also shows that the

largest binding corresponds to nuclei close to iron (A 56). Lighter elements haveless binding energy per nucleon and heavier ones too.

4.4 Why Are Some Nuclei More Stable Than Oth-ers?

There are smaller terms in the formula for nuclear masses, that we neglected towrite in the formula of the previous section. They also make a difference in the


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10CHAPTER 4. WHY DO SOME NUCLEI DECAY QUICKLY WHILE OTHERS REMAIN INTACT FOR E

stability of nuclei. For instance, the symmetry energy, mentioned briefly above,tends to make nuclei withN=Zmost stable. The figure below shows that, indeed,for light nuclei, the stability line is around N Z. However, as nuclei get larger,the effect of the Coulomb repulsion between protons becomes significant, so thatnuclei with N Zare more stable. For instance 208Pb, the most common isotopeof lead, has 82 protons and 126 neutrons.

Figure 4.2: The chart of nuclei. Nuclei are represented by squares. The numberof protons increases vertically and the number of neutrons horizontally. The blacksquares represent the stable nuclei. Courtesy of National Nuclear Data Center,

Brookhaven National Laboratory.

What makes some nuclei unstable? Given that mass and energy are relatedby E = m c2, if the mass of a nucleus is larger than that of a lighter nucleusplus the mass of the remaining neutrons(s) and protons(s) when they are free frominteractions between them, then the nucleus can decay to that lighter nucleus, e.g.,8Be +. Alternatively, the nucleus may be beta unstable, in that it emitsan electron while simultaneously a neutron gets converted into a proton (or emitsa positron while simultaneously a proton gets converted into a neutron). As canbe seen on Fig. 4.2 nuclei group into regions of stability where nuclei are stable.On the border of the region there are nuclei that are almost stable, such that theirdecay lifetimes are large. These nuclei live long enough that they can be studied

experimentally. The methods of investigating unstable or radioactive nuclei haveimproved to such an extent that nuclei with lifetimes of a few milliseconds can nowbe studied. An example is a nucleus with one or several neutrons that are quasi-freeand reside at much larger distances from the center than the mean radius of thenucleus; they are called halo nuclei. In these nuclei neutrons extend their orbitsseveral fermis beyond the usual sizes. An example is 11Li, which has four or fiveneutrons more than 7Li and 6Li, which are the stable isotopes of lithium.


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Chapter 5

Can we turn lead into gold in

the laboratory?

In the literature of years previous to the Renaissance one finds intriguing tales aboutmysterious characters that seek the secret for the transformation (or transmutation)of common elements into gold. Since lead is much cheaper than gold this may seemhighly profitable. Science revealed the secret in the XX century: since atoms are

Figure 5.1: Photograph of Allegory of Alchemy on the walls of Notre Dame inParis. This image refers to secret encounters of obscure characters that were thoughtto posses the secrets of transmutation.

made of electrons and nuclei and the latter are made of protons and neutrons onecan, at least in principle, produce any atom by simply recombining the particlesfrom some atoms into others. In this chapter we will try to explain how feasiblethis is.

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12CHAPTER 5. CAN WE TURN LEAD INTO GOLD IN THE LABORATORY?

5.1 What are nuclear reactions?Nuclear reactions are the transformations that occur when a nucleus exchangesparticles or is excited in a collision with another nucleus or by some other source ofenergy, like light. At energies small compared to the mass of the nucleons (E 1000MeV) the reactions consist of rearrangements of neutrons and protons from somenuclei into others. We typically think of them as two-body processes. For example,the combination of a proton and 13C can yield a nucleus of 14N and a photon.Also, vice-versa, an incoming photon can separate 14N into 13C and a proton. Thenotation commonly used specifies first the target nucleus, which is considered atrest, and, in parenthesis, the incoming and outgoing particles. For our first example:13C(p, )14N, where represents the photon.

5.2 What are reaction cross sections?

Reaction cross sections give the area of an imaginary region over which an incomingparticle would generate a reaction in a nucleus. In the typical laboratory experimentsome nucleus (perhaps protons, alpha particles, or some other heavier nucleus) isaccelerated onto a target (which may consist of a foil made of the target nuclei)that will have a certain number of target nuclei per unit area, Ntrgt/A. Thebeam will consist of a certain number of incoming particles per unit time that wewill call Nin/t. If we could measure how many reactions are taking place perunit time, which we will call Nout/t we should observe a proportionality withthe incoming beam intensity and the target density:

Noutt

= NtrgtA

Nint

(5.1)

where the constant of proportionality has the units of area and is called the reactioncross section. Once this quantity is known one can calculate how many reactionswill take place per unit time under an infinite number of different conditions, like amore intense incoming beam, or a less dense target, etc...

5.3 Is it feasible to produce large amounts of cho-sen nuclei by nuclear reactions between othernuclei?

How about producing 197Au (gold, Z= 79) starting with 208Pb (lead, Z= 82)? Itseems we only need to throw away three protons and eight neutrons, repeat, makea few pounds, and we are done. If this were as simple as it sounds here the priceof gold would not be so much higher than that of lead, but it is about 10000 timesmore expensive. What is the difficulty?

First we will need to excite the 208Pb to liberate some of the protons and neu-trons. Roughly speaking, as can be seen in Fig. 4.1 we will need about 8 MeV pereach nucleon that we would like to come out. So we will need an accelerator to send


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5.3. IS IT FEASIBLE TO PRODUCE LARGE AMOUNTS OF CHOSEN NUCLEI BY NUCLEAR REACTIO

some particle to deliver about 100 MeV. This can be done. There are acceleratorsthat could deliver particles and transfer this amount of energy. But one shouldbear in mind that the typical maximum intensities are of order of 1012 particlesper second. Next we need to get a rough idea of the order of magnitudes of thereaction cross sections. The nucleus 208Pb has a radius of approximately 2081/3

fermis or approximately 6 1013 cm. Thus the nuclear cross section is roughly41025 cm2. The reaction cross section does not need to be equal to the geometriccross section, but to get a rough estimate we will assume it is. This may be off bya couple of orders of magnitude, but, as we shall see, it will still be useful. Nextwe can imagine that our Pb target has the normal density of Pb and a thicknessof 1 millimeter. This leads to about 3 1021 Pb nuclei per cm2. A target muchthicker than this will lead to too much energy loss for the incoming particles. Withthese numbers and the equation we wrote in the previous question we can estimatethat we could make approximately 109 reactions per second. Thus, even if all re-actions would generate one atom of gold (a gross over-estimate because there aremany other possible final destinations), we would need a long time to producemacroscopic amounts. To produce a few hundred grams of gold we would need toproduce about the Avogadro number, 6 1023, which implies about 1014 secondsor about 3 million years! This shows that while in principle it may be possible tomake gold from Pb, it is not practical.


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14CHAPTER 5. CAN WE TURN LEAD INTO GOLD IN THE LABORATORY?


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Chapter 6

Is quantum mechanics

important for understandingnuclei?

The simple answer to the question above is yes. But in what follows we explainwhy this is so and give answers to several questions. We will need these answerslater to answer other questions.

6.1 What are waves?Before we discuss light and quantum mechanics we need to consider some basicproperties of waves . One simple example of a wave is the perturbation that prop-agates on a guitar string after we pulse it. A remarkable phenomenon is that thetone that we hear doesnt depend on how we pulse the string: it only depends onthe length and tension of the string. Although the initial perturbation may be adisplacement at the point where we pulse the string, shortly (a fraction of a second)after the string is vibrating all along its length. Each point of the string may beconsidered as oscillating back and forth. The displacement from equilibrium is afunction of position along the string, x, and of time,Y(x, t). On the left of Fig. 6.1we show an example of an oscillation at the center of the string versus time and onthe right we show a snapshot of what we may see at a given time. The time for one

oscillation,T, is called the periodof the wave. We will find convenient to refer alsoto the number of oscillations per unit time, or frequency, f:

f= 1

T.

The period is measured in seconds and the frequency in oscillations per secondorHertz. The typical tuning pitch for musicians, for example, is the A above middleC and has a frequency of 440 Hz. Because the string is anchored on its two ends

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16CHAPTER 6. IS QUANTUM MECHANICS IMPORTANT FOR UNDERSTANDING NUCLEI?

Figure 6.1: Sketch of a wave on a guitar string. Left: oscillation at the center ofthe string versus time. Right: snapshot of possible modes: fundamental (in red)and first harmonic (in blue).

a snapshot of the wave at any time will show zero amplitude at those points. Weshow two sketches of possible waves. The distance after which the wave repeatsitself is called the wavelength, . The wave in red (called the fundamental) has awavelength twice as long as that for the wave in blue (called the first harmonic).The fundamental dominates the pitch of the sound we hear from a guitar.

Waves are common in all realms of physical phenomena. We discussed abovea wave on a string. The vibrations on the anchoring points produce oscillationson the body of the guitar and this makes oscillations in the air pressure that aretransmitted and perceived by our ears. Organ pipes have similar properties. Thewaves on the surface of the water form another example. As we will see below lightis also described by waves.

6.2 What are theuncertainty relationsfor waves?

The waves sketched in Fig. 6.1 are the simplest versions of waves and can be de-scribed by sines and cosines functions of time. However, in order to understandsome features of quantum mechanics we need to consider other wave shapes. Tofix ideas we will concentrate for now on the variation versus time and assume thatwe are looking at the wave at a fixed point in space. Although waves in generaldont have a sinusoidal form they can practically always be expressed as the sumof many sinusoidal functions with different frequencies. Fig. 6.2 shows an example.On top we show a periodic wave looking like a negative step followed by a positivestep. In the middle graphs we show a composition of 3 sine waves that have been

given appropriate amplitudes so as to do the best in reproducing the pulse on thetop graph. If we call the frequency of the top pulse f0 the frequencies of the 3sine waves are f0, 2 f0, 3 f0. On the middle right we see the amplitudes of the 3sine waves plotted versus the frequency. This plot is called the frequency spectrum.In the bottom we show a similar pair of plots corresponding to 10 sine waves. Asmore components are included the spectrum gets richer and the shape of the sumresembles the top wave.

These waves are periodic. How about a pulse? One can think of a pulse asa periodic wave that has a very long period. As the period goes to infinity the


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6.2. WHAT ARE THEUNCERTAINTY RELATIONS FOR WAVES? 17

Figure 6.2: Description of periodic waves with sines and cosines. Top: periodicwave looking like a negative step followed by a positive step. Middle graphs: sum(dark blue) of 3 sine waves with frequencies equal (black), double (red) and triple(green) of that for the top pulse. On the right we see the amplitudes of the 3 sinewaves plotted versus the frequency. Bottom: same as middle but now with 10 sinewaves. As more components are included the shape of the sum resembles the topwave.

fundamental frequency will go to zero and the multiples will fill the spectrum ina dense fashion. Fig. 6.3 shows, on the top left, the same pulse that we used forFig. 6.2 but now it is not a periodic wave. The spectrum, on the top right, is nowa continuum. The bottom plots are the equivalent plots for a pulse that is twiceas wide in time. Here there is something very important to notice: the width ofthe corresponding frequency spectrum is half of that of the original pulse. This isactually a general property of the frequency spectra corresponding to pulses and it isvery important for communications and for many applications of wave phenomena.If we call the width of the pulse in time tand the width of its frequency spectrum


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Figure 6.3: Top: a non-periodic pulse versus time on the left and its frequency spec-trum on the right. For a non-periodic pulse the frequency spectrum is continuous.Bottom: doubling the width of the pulse results in a spectrum that is half as wide.

f then:

tf 1.

To summarize: as one tries to define wave pulses in time (t 0) the frequencyrange gets very large (f ). And vice-versa, if one tries to define the frequencyvery well, the wave will look like a pure sinusoidal function and will spread overtime from minus to plus infinity.

All the discussion above took place thinking about the wave versus time at aposition in space. But we could repeat all the arguments thinking about the waveversus position at a given time. We would replace time by position and frequency,which is the inverse of the period, by the inverse of the wavelength and we get the

uncertainty relation for position and inverse wavelength(also called wave number )k= 1/:

xk 1.

Theseuncertainty relationsare properties of waves. Notice that they are not relatedto quantum mechanics so far in our discussions, although we will show below theconnection.


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6.3. WHAT IS LIGHT? 19

6.3 What is light?One important clue as to what light is came towards the end of the nineteenthcentury when Maxwell came up with a way of understanding electric and magneticphenomena in a unified way. Maxwell concluded that light is radiation emittedwhen electric charges are accelerated. For example, the visible light that comesfrom the Sun is generated by agitated electrons and the x rays used by dentists aregenerated by electrons that are accelerated and then quickly stopped in the tubethat is place near our mouths. In Maxwells theory light is made of waves of electric

Figure 6.4: Light can be considered as made of waves of electric and magnetic fields.On the left we show an example of how the electric field may look versus time atone point in space. On the right we show a time snapshot of the electric field asa function of distance from the source.

and magnetic fields that propagate in vacuum. An example of a wave is shownin Fig. 6.4. At a given point in space the electric field oscillates as a function of

time and our eyes perceive these oscillations when we look at something. But ata given time the electric field also shows oscillations as a function of distance fromthe source. The period Tand the wavelength are, respectively, the interval oftime and distance over which the wave repeats itself. The two are related by thespeed of lightc:

c=

T

which is about 3 108 meters per second. The difference between the differentkinds of light is in the wavelength or frequency. For visible light the difference isin the color. The wavelength of visible light is in the approximate range 400 700nanometers (1 nm = 109 m) while that of x rays is about 100 times smaller. The

corresponding frequencies are 4 7 1014

Hz (oscillations per second) for visiblelight and 4 7 1016 Hz for x rays.One important phenomenon that is also explained by Maxwells theory is that of

interference: since light is made of electric fields it can cancel itself. Two electricfields of equal magnitude oriented in opposite directions cancel each other. Sincelight is made of electric fields this can also happen with light so that light plus lightcan yield darkness. The apparent distortion of images that can be observed whenlight is passed through apertures of sizes comparable to their wavelength is partlycaused by this effect.


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Another important consequence of Maxwells theory is that light carries momen-tum as well as energy. For an ob ject moving at speeds small compared to the speedof light momentum is the product of mass times speed, p =mv. The variation ofmomentum per unit time is the force exerted on an object. Maxwell showed thatelectromagnetic waves (including light) carry momentum as well as energy and theyare related by:

E= p c.

Thus, light exerts a force on objects that absorb it. This force is too faint for usto feel it when standing on the rays of the Sun, but has to be taken into accountwhen calculating orbits of spacecrafts moving outside the Earths atmosphere, forexample. Fig. 6.5 shows an artists rendition of the spacecraft Ikaros, which makes

use of the radiation force to sail in space. Another example of the use of the

Figure 6.5: Since light carries momentum it generates a pressure which is used bythe Ikaros spacecraft for sailing in space.

momentum carried by radiation is the trapping of atoms by laser beams. This hasallowed for manipulation of atoms to a degree that had been previous unconceivable.

But more was in store about the nature of light which came out during the earlytwentieth century, which is addressed in the answer to the next question.

6.4 What is quantum mechanics?

Quantum mechanics describes the behavior of nature at the microscopic level. Thefirst ideas were developed by Einstein, Plank, Bohr, de Broglie and others in theearly twentieth century to explain several observations that could not be understoodwith the classical theory.

One such observation was the photoelectric effect, which is the emission of elec-trons from metals when illuminated by light. Today this effect is used in photocellsto start and stop circuits according to the presence of light and it has many practicalapplications. Einstein showed that the energies of the emitted electrons under light


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6.4. WHAT IS QUANTUM MECHANICS? 21

could only be explained if one assumed that light of a given frequency transfersenergy in quanta, called photons, each with energy Erelated to the frequency by:

E = h f

whereh 6.6 1034 kg m2/s is Planks constant. A consequence of this and therelation that Maxwell had found between energy and momentum for light is thatthe momentum of each photon is related to the light wavelength by:

p = h

.

Another important observation was the fact that gases under electrical dis-charges produce emission of light with well defined frequencies. So when observing

this light through a prism, for example, instead of a rainbow-like structure withsmooth intensities versus color, one observes sharp lines at some particular colors.If one considers atoms as made of a central nucleus that has positive electric chargeand electrons going around it with negative electric charge one can calculate thepossible orbits of electrons, just as one calculates the orbits of planets around theSun. The classical theory, however, predicted that electrons could have any amountof energy and should loose energy continuously and there was no way to understandthe emissions with well-defined frequencies from gases or even the stability of atoms.What was eventually understood is that at the microscopic level the electron hasto be described by a wave with wavelength and frequency given by the same rela-tionships as Einstein proposed to understand the photoelectric effect. Thus, if weconsider an electron moving in a circular orbit, for example, the circumference has

to be an integer number of wavelengths:2r = n

where n can be 1, 2, 3, etc. If this condition is not fulfilled the wave will can-cel itself as many turns over the circumference are considered. Fig. 6.6 shows anexample with only two turns around the circle but one can visualize how the wavefunction of turn number 2 (in blue) cancels part of the wave function of turn num-ber 1 (in red). As more and more turns are considered the whole wave tends todisappear.

This phenomenon is common with waves: sound waves in organ pipes of well-defined lengths have well-defined pitches; radio waves propagating in electric circuitswith fixed capacitors and coils have well-defined frequencies; etc. Here, assuming

that electrons in atoms are described by waves leads to a quantization of the possibleorbits radii. And this in turn leads to a quantization of the energy of the orbits. Ifthe electrons can only have certain discrete energies, then emissions correspondingto de-excitations or absorptions will have sharply defined frequencies.

The above model forces the electron to be in a circular orbit but there really isno reason for this to be so. Heisenberg and Schrodinger found a way of solving theproblem without assumptions. Schrodinger found a way of establishing an equationthat allows for calculating the wave function if the potential energy versus position isknown. Once the wave function is known one can calculate any observable quantity.


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Figure 6.6: Cancellation effect on a wave on a circular path. The wave amplitudeis shown versus the distance along the circle from an arbitrary starting point. Ifthe condition 2r = n is not met the wave cancels itself. The wave in blue isthe continuation of the wave in red. These two only partially cancel, but it can beseen that as many more turns are added into the picture the remaining wave getssmaller.

In particular, the square of the wave function yields the probability distribution ofthe particle in space. These methods explained the initial ideas of Bohr and othersin a more general context.

6.5 What are the uncertainty principles of quan-tum mechanics?

Once one accepts the idea that particles are represented by waves with frequencyand wavelength given by the Einstein relations

f=E/h

= h/p,

one has to accept as a consequence that there will be uncertainty relationsbetweenenergy and time as well as between momentum and position:

Et hpx h

as can be verified by simply multiplying the wave uncertainty relationsof Sect. 6.2by Planks constant, h.

These relations are usually referred to as the uncertainty principles: the ultimateaccuracy for determining the energy of a particle is limited by the length of timeavailable for the observation. But equivalently there is a necessary relation between


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6.6. WHAT ARE NUCLEAR EXCITED STATES? 23

the width of a quantum state (a minimal spread in energy) and its lifetime: theshorter the lifetime, the bigger the width. Similarly for momentum and position:the momentum and velocity of a particle at a well-defined point in space can notbe a single value but is rather given by a distribution of values.

6.6 What are nuclear excited states?

A nucleus, like an atom, has quantum stateswith different excitation energies abovethe ground state. Due to the relation between the lifetime of the states and theirwidths, mentioned above, the states that decay faster have larger widths. Thelowest excited states can usually decay only by emission of photons, via the elec-tromagnetic interaction. These are called bound states. Above a certain threshold,

which is different for each nucleus, the decay with emission of neutrons, protons,alpha particles, or other light nuclei becomes possible. Because these are drivenby the strong interaction the lifetimes for these decays are shorter and the widthsof the states are much larger. These are called unbound states. Some nuclei areunbound even in their ground state. This is the case for 8Be, for example, whichdecays into two particles. Fig. 6.7 shows some of the states of the nucleus 18Ne.

Figure 6.7: Example of nuclear excited states in 18Ne showing some states with their

excitation energies and widths. The states below the particle emission threshold aremuch narrower than those above it. The dotted lines indicate many other statesnot shown here.


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Chapter 7

Why is nuclear energy

important to our society?

7.1 Why is nuclear energy so much larger thanchemical or atomic energy?

Consider a ball rolling back and forth in the valley between two hills. If we want toconfine the ball to a smaller space we can make the valley steeper. If we consider aball attached to a spring we would need a stifferspring in order to confine the ball tovery small space. The valley steepness or the stiffness of the spring are proportionalto the force they provide and the energy of the oscillation. Thus we see that moreenergy is required to confine objects to smaller spaces. This is also true for theatom and nucleus. Electrons are confined into atoms by the electrostatic attractionand the energies are on the order of eVs (1) but in order to confine neutrons andprotons to the much smaller nucleus the energies needed are of the order of millionsof eVs, or MeVs. One can check in Sect. 4.3 that the binding nuclear energiespernucleon indeed are several MeVs.

In the macroscopic world of our everyday life we can alternatively let the balloscillate with smaller and smaller initial amplitude and we get the feeling we can getinfinitely small oscillations simply by having infinitely small energies. This alter-native way of confining turns out to have a limit according to quantum mechanics,which dictates the behavior of the tiny world of atoms and nuclei. For a given steep-

ness or stiffness there is an absolute minimal energy. Quantum mechanics tells usthat particles are described by waves whose wavelength is inversely proportionalto their speed. The minimal energy corresponds to the situation in which the os-cillation spreads over the orbit. This gives the situation for an orbit that doesnthave the cancellation effect shown in Fig. 6.6. The electrons in the atoms or theneutrons and protons in nuclei are close to this limit so the mechanism described atthe beginning, which requires more energy for more confinement, is the only viable

1See Chapter 3 for a definition this unit of energy

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26CHAPTER 7. WHY IS NUCLEAR ENERGY IMPORTANT TO OUR SOCIETY?

way.In summary, given that nuclei are much smaller than atoms the nuclear binding

energies are about one million times bigger than the atomic binding energies. Theplot of the binding energies per nucleon, shown in Fig. 4.1 shows that combining twovery light nuclei into a heavier one (like two deuterium nuclei into a 4He nucleus)leads to more binding energy per nucleon. Thus, since the total energy is conserved,there is a net excess energy that is liberated in the process in the form of kineticenergy. This can then be used to move engines. The process described above iscalled nuclear fusion. On the other hand Fig. 4.1 also shows that splitting a veryheavy nucleus, like Uranium, also leads to more average binding energy per nucleon.That process is called nuclear fission. The next sections describe more details aboutthese reactions.

7.2 When and how can a single nucleus be splitinto pieces (fission) and when and how can twonuclei fuse together (fusion)?

In 1938, Otto Hahn and Fritz Strassman found that the addition of a neutron tosome heavy nuclei, like uranium, can produce lighter fragments, like barium. Forinstance a neutron can combine with 235U yielding 142Ba: n+235U 142 Ba+91Kr+3n. In general, the two fragments are not of the same mass, but there tends to be aheavier one and a lighter one. Lise Meitner and Otto Frisch were the first to producea theoretical interpretation of these findings. This nuclear reaction, called fission ,

occurs more frequently if the incoming neutrons are slow, with energies of the orderof a few thousandths of eVs. At these very small incident energies protons or othercharged particles can not penetrate the repulsive electrostatic energy generated bythe other protons in the nucleus, so this phenomenon is unique to neutrons. Notethat there is a single neutron to start the reaction, but three neutrons are emitted. Ifthe outgoing neutrons can be used to start similar reactions the number of neutronswould increase very rapidly, as would the energy produced in the process of splittingthe nuclei. A so-called self-sustaining process can occur. It is used all over the worldin nuclear reactors. The process has also been used in so-called atomic bombswith a tremendous energy release in a very short time.

Fission reactors necessarily generate long-lived isotopes (radioactivity) with half-lives that can be of thousands of years. This radioactivity is difficult to contain over

such long periods of times and can drift into rivers and contaminate large regions ofspace producing environmental hazards. As mentioned above the process of fusingtwo very light nuclei into a heavier one should also liberate energy. In fact, the Sunproduces energy by this process which has the great advantage of not producing anylong-lived isotope. Much effort has been dedicated to trying to achieve controlledfusion in the laboratory, for example, via the reaction 2H+3H 4He. The problemis difficult because in order to produce the fusion the electrostatic repulsion needsto be overcome. Magnetic confinement of a plasma has been sought for manyyears without success. The Sun works naturally making use of the huge amount


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7.3. HOW DO NUCLEAR REACTORS WORK? 27

Figure 7.1: National Ignition Facility at Lawrence Livermore National Laboratory.Left: the pellet that contains the light nuclei (deuterium and tritium, for example).Right: workers inside of the chamber. At its center the laser beams will deliver apeak power of hundreds of petaWatts onto the pellet inducing fusion.

of hydrogen atoms that attract each other by gravity and liberate energy in aslow fashion (see more details in Sect. 11.1). A big facility, called the NationalIgnition Facility at Lawrence Livermore National Laboratory, seems to be closeto demonstrating a possible solution in the laboratory. It makes use of a systemof lasers delivering a peak power of hundreds of petaWatts (1 peta Watt = 1015

Watts) onto a target containing the light nuclei. It is expected that high enoughcompression will occur to achieve controlled fusion.

7.3 How do nuclear reactors work?A typical reactor contains some fissible material, like 235U. The fission products areallowed to transfer the kinetic energy to a medium, like water in a pool, which canthen be used to move an engine. However, in describing the process of fission abovewe indicated that in order to produce useful amounts of energy it is crucial thatthe neutrons produced by one nuclear reaction can be used in subsequent reactions.Here is an important problem: the outgoing neutrons from a typical reaction arefast, with energies of the order of millions of eVs (or MeVs). At these energiesthe probability of capturing neutrons in subsequent reactions is small. So neutronreactors are designed to moderate (i.e. slow down) the neutrons from the nuclearreactions. Since neutrons are electrically neutral this has to be achieved by having

the neutrons collide with other nuclei in elastic collisions. If the target nuclei areheavy the neutrons bounce back without loosing energy so what is needed for themoderator are lighter nuclei. The hydrogen in water can be used for this purpose buthydrogen can also absorb neutrons. For that reason some reactors use deuterated(called heavy) water because deuterium is still light enough to take energy fromthe neutrons in elastic collisions and the cross section for absorption is much smallerthan that on hydrogen. The first neutron reactor produced by Enrico Fermi and hiscollaborators used high-purity graphite as a moderator. The high-purity was neededto minimize neutron absorption. Fermi distributed the uranium in small pellets


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Figure 7.2: Sketch of a pressurized-water reactor. The water absorbs the kineticenergy of the nuclear reaction products and then exchanges heat at the steamgenerator. The steam is subsequently used to move an electricity generator.

that were embedded in the graphite so the neutrons produced in one pellet couldmove through the graphite, get moderated, and then produce nuclear reactions onanother pellet. Of course, if the system is small, the fraction of neutrons that escape

through the surface is too large and the sustaining reaction can not take place. Asthe system becomes bigger the ratio of volume to surface (and consequently theratio of neutrons that react versus those that escape) becomes higher. Fermi andcollaborators figured that there should be a critical size for a reactor. Because theydidnt know exactly how many neutrons were absorbed in the medium they carefullymonitored the number of neutrons produced as they made the system bigger. Inmodern reactors the pellets are still distributed within a moderator and some sheetsof material that absorb neutrons efficiently (for example indium) are used to controlthe reaction rate.

7.4 How do nuclear weapons work?

Nuclear reactors are designed to slowly consume the fuel so that they can lastfor years delivering energy at a rate of approximately a GigaWatt (109 Watts). Innuclear weapons the main interest is to yield as much energy as possible very quicklyso the moderation and efficiency common to nuclear reactors cant be achieved.Nevertheless, even when the probability of capturing a given neutron is smaller, ifsufficient density of235U is present, the system can be brought to itscritical pointwhere large neutron multiplication and liberation of energy occurs. It turns out thatnatural uranium is mostly composed of 238U with only less than 1% of 235U. In


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7.4. HOW DO NUCLEAR WEAPONS WORK? 29

order for the system to reach the multiplication point, a much higher concentration(larger than 80%) of 235U is required. Uranium with these concentrations of the235U is known as weapons grade uranium. Achieving these concentrations wasone of the main obstacles that the scientists working toward the production of thefirst nuclear weapon, in the Manhattan project, had to overcome. For this reasoncountries that show plans to enrich uranium, or to obtain enriched uranium, areconsidered to be aiming at producing nuclear weapons.


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Chapter 8

What is radioactivity and

which kind is mostdangerous?

8.1 What is radioactivity? What type of interac-tion drives it?

Radioactivity is the emission of radiation from matter with long half-lives. As we

described in Chapter 2 it was accidentally discovered by Becquerel. Soon after thediscovery the emitted radiation was classified as alpha, beta, and gamma radiationsaccording to the thickness of materials needed to stop them. Alpha radiation couldbe stopped with a sheet of paper, beta radiation could penetrate a few millimetersor more of aluminum, and gamma radiation could go through centimeters of lead.Today we know that alpha radiation consists of ionized helium atoms, beta radia-tion consists of electrons and their anti-particles, positrons, and gamma radiationconsists of photons, i.e. light. These radiations are emitted by a variety of nuclei,and the half-lives can range from microseconds or less to millions of years or more.

The three processes are actually driven by different interactions and we describeeach one here. Notice that in all these transitions the total number of nucleons andthe electric charge is conserved. Also electrons (positrons) are never produced alone

but always accompanied by anti-neutrinos (neutrinos) so that the total numberof light particles (lepton number) is conserved. The light anti-particles (like thepositron and the anti-neutrino) are counted as having negative lepton numbers,while the electron and neutrino are counted as having positive lepton number.

In some heavy nuclei two protons and two neutrons can get together into analpha particle and, if allowed by energy conservation, can be ejected leaving adifferent nucleus. For example 237Np can decay into +233 Pa. The interactionthat drives this transition is the strong interaction (the same interaction that holdsnuclei together) but the half-life for the transition is extremely long, about two

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32CHAPTER 8. WHAT IS RADIOACTIVITY AND WHICH KIND IS MOST DANGEROUS?

r

V

E

r(E)R

Figure 8.1: Sketch of potential nuclear energy versus distance for alpha particles.The heavy nucleus radius R determines the region of strong attraction. If the alpha

particle finds itself at the point marked r (E) it will leave the nucleus.

million years! The reason for this is that in order for the process to occur the alphaparticle has to penetrate an energy barrier. Fig. 8.1 shows a sketch of the potentialenergy that the alpha particle perceives. According to classical mechanics the alphaparticle should be confined and not be able to ever escape, since its kinetic energyis smaller than the potential energy at the top of the barrier. Quantum mechanics,however, allows for this kind of process, but the probability of the penetrationdecreases exponentially as with the difference between the barrier potential and thekinetic energy of the particle. This implies very long half-lives.

The beta radiations are driven by the weak interaction. The basic transitionsthat can occur in nuclei are:

n p + e + ep n + e+ + e

wheren and p indicate a neutron or proton, respectively; e indicate the electronor positron; and e is the neutrino of the electron and e its corresponding anti-particle. Nuclei that have an excess of neutrons will undergo the first process whileproton-rich nuclei will follow the second process as they move towards stable nuclei.For example, 14C decays emitting an electron: 14C 14N +e + e; while 14Odecays by emitting a positron: 14O 14N + e+ + e. The weak interaction is muchweaker than the strong interaction but the half-lives are determined by many otherfactors, like the amount of energy available for the decay, and the amount of orbital

angular momentum transfered between the particles.The gamma radiation is produced by the electromagnetic interaction. In generalthese decays have shorter lifetimes than the weak decays, but many nuclei thatundergo a beta transition feed an excited state of the daughter nucleus, whichsubsequently decays emitting gamma radiation. As example, 60Co decays by a betatransition with a half-life of approximately five years. But the beta decay is followedby emission of a photon within a few pico-seconds (1012 seconds).

All these radiations can ionize elements in our cells and can thus create a hazard.We are constantly bombarded by radiation so clearly our bodies can function with-


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8.2. WHAT TYPE OF RADIOACTIVITY IS THE MOST HAZARDOUS FOR OUR HEALTH?33

out problems under certain levels of radiation. But under very intense doses theprobability of generation of cancer cells increases. For this reason the radioactivewaste produced by reactors is considered a big problem.

8.2 What type of radioactivity is the most haz-ardous for our health?

8.3 How can we minimize risks in case of emer-gencies?


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34CHAPTER 8. WHAT IS RADIOACTIVITY AND WHICH KIND IS MOST DANGEROUS?


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Chapter 9

Are there simple models

useful as well as beautiful?

Nuclei are complex objects composed of protons and neutrons. Each nucleus has alowest (ground) state and excited (energies larger than the ground state) states. Inatoms, the Coulomb force determines the energies of the various states. What aboutnuclei? Here it is the nuclear forces that determine the energies, and the nuclearforce comes from QCD. In principle, we can calculate all these states and theirproperties from QCD, but it is immensely difficult and the task has not yet beenaccomplished. Instead, physicists work with models which reproduce the energiesand properties. Some are prettier than others and some are simpler. Not all

models are equally beautiful, but all are useful.

9.1 The Liquid Drop Model

9.2 The Shell Model

In atoms, there is a central force which attracts the electrons to the nucleus. Innuclei, by contrast, there are forces between the nucleons, but no central force. Inatoms the shells are around the center (nucleus) and fill up according to the Pauliexclusion principle. No more than 2 electrons can be in any one state (one of spin upand one of spin down). It came as a great surprise that there appeared to be nuclei

with Z or N= 2, 8, 20, 28, 50, 82 and 126 that were particularly stable, just asclosed atomic shells. How do we know? These nuclei had more isotopes and excitedstates lie higher in energy than for neighboring nuclei. Except for the first two shellclosures, no one could understand the others until Maria Goeppert-Mayer and J.H.D. Jensen came along. They realized that spin-orbit forces (forces which couple thespin of a nucleon to its orbital angular momentum) is the feature that must be takeninto account. The simplest shell model, the independent particle model, assumesthat each nucleon moves in an average potential due to all the others. Why this isthe case took quite a bit longer to understand. The evidence for the shell model

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36CHAPTER 9. ARE THERE SIMPLE MODELS USEFUL AS WELL AS BEAUTIFUL?

stems in large part from the above magic numbers, corresponding to closed shells.One piece of evidence comes the separation energy of the last nucleon. In atoms, itis the ionization energy of the last electron. If the last electron just closes a shell,it will be much harder to remove than other ones. By contrast, the first electronoutside a closed shell has a smaller removal (ionization) energy; hence these atomstend to be good conductors. Similar considerations hold in nuclei for nucleons.There is an additional effect in nuclei that must be considered, namely a pairing energy for two nucleons in a total angular momentum =0 state. Closed shells tendto have a total angular momentum of zero and be spherically symmetric, as well asparticular stability. An example is 208Pb with 82 protons and 126 neutrons. Thefirst excited states of these nuclei tend to be higher than for other ones.

We can use a harmonic oscillator (V = 12

kr2) to obtain the closed shells if weadd a large spin-orbit potential. This tends to lower the energies of the total angularmomentumj = + 1/2 state, which is needed to get the correct shell closures. Apotential somewhere between a harmonic oscillator and a square well does evenbetter.

We have concentrated on ground state energies, above. What about excitedstates? Clearly some of these come from shifting a neutron or proton into highershell model states, but there are other effects as well. For instance, there areresidual interactions. An example is the pairing force mentioned earlier betweentwo like nucleons. Extended shell model calculations can now be carried out, thanksto computers. They take all these effects into account.

Recently, a lot of attention has been focused on unstable nuclei or so-calledhalo nuclei. These nuclei tend to be far from the center of the valley of stability.They have Z/Nthat are either large or small compared to the more stable nuclei,

and have very small binding energies. An example is 11Li. The last particle(s)tend to be far from the nuclear center. These nuclei can now be studied because ofspecial accelerators and detectors

9.3 Collective Model

The Shell Model has a number of failures. For instance, the quadrupole moments ofmany nuclei are much larger than expected. This suggests that the central core ofnucleons are ellipsoidal rather than spherical. These nuclei have collective motions.

Whereas closed shell nuclei are spherical, nuclei far removed from them havesurface oscillations, like surface waves of a liquid drop. These nuclei can acquirepermanent deformations; the entire deformed nucleus then can rotate or vibrate.

The deformed nucleus may also serve as a deformed potential for single particlemotions. The rotations lead to rotational bands of energy levels, which for even Zand even N nuclei may have angular momenta of 0, 2, 4, 6, etc. Vibrations mayalso have angular momentum of 2with equally spaced energies.

There are of course, variants of this and other models, but these two modelsserve as the basis for many others.


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Chapter 10

How Are The Properties of

Light Nuclei Investigated?

In order to determine the properties of light nuclei, one needs to know the forcesinvolved. Consider a nucleus with more than two nucleons. One may ask whetherthe forces are simply the addition of all the two-body forces present, or whether thepresence of a third nucleon affects the way the pairs interact. What is the role ofthree-body and four-body forces, i.e., forces that act between 3 or 4 bodies at once?Do we know enough about these and two-body forces? These are the questions weaddress in this Chapter.

We can test our knowledge of two-body forces by examining properties of thedeuteron, the nucleus of deuterium, which is composed of just one proton and oneneutron. There are then no direct Coulomb forces. Do we know enough to be ableto calculate its static properties, the binding energy, the magnetic and quadrupolemoments? The answer, unfortunately, is no, even though the subject is almost 100years old! So, what do we know? We know that the long range force is due to theexchange of a pion, a particle of mass 140MeV/c2, which comes in 3 varieties,+,, and0. The proton can emit a +, which can be absorbed by the neutron.The neutron then becomes a proton and the proton becomes a neutron. The twoparticles have traded places in what is called an exchange force. The proton canalso emit a 0 and remain a proton and the neutron can absorb the pion. How dothese forces compare? The charge effects are small compared to the nuclear force

and can be neglected (to lowest order) such that the forces are the same. The factthat the forces are (almost) independent of charge was first recognized by Heisenbergas early as 1932. This independence has been developed and is now called isospinsymmetry. With the neglect of the Coulomb force, this means that thep p,n n,and n p forces are the same in the same state. The latter imperative is addedbecause two protons and two neutrons are restricted to certain states, but the n ppair is not. For an even state of angular momentum, the two particles must be in asinglet (spin =0 and odd under interchange of the two nucleons) spin. The tripletspin (spin=1 and even under interchange) is forbidden for two like particles because

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38CHAPTER 10. HOW ARE THE PROPERTIES OF LIGHT NUCLEI INVESTIGATED?

n

p

n

p

Figure 10.1: Sketch of pion exchange between two nucleons. The nuclear interaction

can be modeled in terms of meson exchanges.

the state must be antisymmetric under the exchange of the two particles. Thecorresponding potential is written as f2 e

(mr/c)/(4r), where f is called thepion-nucleon coupling constant and the force is called a Yukawa force after HidekiYukawa, who first suggested this force. The range of this force is about 1.4 1013cm or 1.4 fm. The one pion exchange force gives rise to a spin-dependent forcethat is called a tensor force. This force exists, but it cannot be the entire story.Neither the binding energy, nor the quadrupole and magnetic dipole moments ofthe deuteron can be understood from just the one pion exchange. So, what else isthere? What about the exchange of two or three pions? These are also important,but less so; their range is smaller than /mc, e.g. /(2mc). We see that the

calculation is not as simple as hoped for! There are other terms in the force.First of all, it is known that the force at short distances is not due to one pion

exchange. From scattering experiments we know that there is a short distancerepulsion between nucleons, with a range of 0.3f m. This could come about fromthe exchange of more massive mesons, such as the , of mass = 782 MeV/c2, which isa vector meson of spin 1. There are further heavy mesons, which must be considered.Can we treat them all in perturbation theory? Do we know their couplings to thenucleons well enough? Despite considerable effort over many decades, we still donot know exactly how to explain the two-body force in terms of meson exchange.This lack of knowledge clearly makes it difficult to explain the properties of thedeuteron or heavier nuclei and to explain nucleon-nucleon scattering.

In recent years a different approach has taken hold. It makes use of the pion

exchange force that we do know and gets around the lack of knowledge of theshorter distance physics by expanding in terms of the relative momentum of thetwo nucleons times some effective radius. This is called an effective field theory(EFT) and is exact, but with unknown constants. The theory is based on QCD,the theory of strong interactions and makes use of chiral dynamics and chiralsymmetry.

A calculation of nuclear forces with QCD directly is extremely difficult, becausethe theory is highy nonlinear. It can only be carried out numerically, on a lattice ofpoints about 2 fm (?) apart, The theory is called an effective field theory, as outlined


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39

by S.Weinberg. The chiral symmetry arises because the up and down quarks havevery small masses ( 5MeV/c2). To the extent that these masses can be neglected,the chiral symmetry is exact, and left-handed quarks separate from (do not mixwith) right-handed ones The masses of the quarks involve both left-handed andright-handed quarks, and couples them, thus breaking the chiral symmetry. Thisgives rise to Goldstone bosons, which are the pions of low mass. Here low meanssmall compared to the mass of heavy mesons ( 800MeV/c2) or of the so-calledchiral symmetry scale ( 1GeV/c2)). In an effective field teory, one expands interms of momenta/Mc and (m/M), where M 1GeV/c2.

Although the quarks and pion masses are small compared to M, the nucleonmass is not. You can treat the nucleon as a static source, such that pion exchangedoes not cause any motion. The short distance scale is separated from the long

one in that the pion mass is small and all other meson masses are of the orderof the chiral symmetry scale. In EFT, these mesons lead to a contact interactionbetween nucleons and derivatives of momenta. The expansion is similar to thelow energy effective range expansion of nuclear forces, which can be written ask tan() =1/a+ (1/2)kr20+..., where is the low energy phase shift, k is therelative momentum of the nucleons, a is called the scattering length, r0the effectiverange, and the ellipses involve higher powers of momenta.

There are problems in the EFT perturbative approach because the deuteron isbound and the spin zero lowest state is almost bound. Various methods have beensuggested for getting around these problems. The method has many advantages.Two and many-body (e.g. 3) forces are developed on the same footing and showthat the strength of N-body (N 3) forces are smaller than two-body ones, sincethey only occur as higher order terms in the expansion.

Although there are unknown constants, the method can be called a theory in thatit is exact, in contrast to meson models, which can be said to be phenomenological.


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40CHAPTER 10. HOW ARE THE PROPERTIES OF LIGHT NUCLEI INVESTIGATED?


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Chapter 11

Are Nuclei Responsible for

the Workings of the Sun andStars?

Although nuclei are not involved directly in the formation of the sun and stars,they are responsible for the heat and light given off by these stars. Nuclei contain avery large amount of energy. Whereas atomic energies are measured in eV (electron-volts) nuclear energies tend to be a factor of a million times higher and are measuredin millions of electron-volts or MeV. Thus, the energy that binds a nucleon to the

nucleus is generally of the order of 8 MeV; in 4He the binding energy is 28 MeV.The mechanism for producing energy is called fusion because it brings several

light elements together to form a heavier nucleus of larger binding energy. Thedifference in energies is then released and available. There are several problemsthat must be overcome to have this happen. First of all, the lighter nuclei must bebrought in contact long enough to initiate the reaction; the Coulomb repulsion tendsto keep them apart! Second they must be contained at a very high temperature.The suns temperature is 6000 K; it is difficult to find a container that withstandsthat temperature. In the case of the sun, it is a plasma of almost 109 m in radius.

There are two cycles that have been proposed. known as the CNO cycle; herea C nucleus and 4 protons are transformed into an alpha particle (the nucleus of4He). The gist of the reactions are that 12C + 4p 12 C +4 He+2e+ + 2+photons.The

12

C acts as a catalyst and the reactions occur in 6 steps. Since 4 protons giverise to 4He, the gain in energy is about 27 MeV; the neutrinos carry away about10% of the energy. The CNO cycle is not the most important one in the sun, butit is in hot stars. In the sun, it is the so-called pp cycle that dominates. Thereactions in the sun begin with p +p d+e+ +, but the overall process is also4p converting to 4He, so that the energy release is the same as for the CNO cycle.Since neutrinos are involved in some of the reactions, the overall rates are quitesmall. Our confidence in the mechanism is quite high because the neutrinos fromthe sun have been observed on earth. On earth, we are trying to copy the fusion

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42CHAPTER 11. ARE NUCLEI RESPONSIBLE FOR THE WORKINGS OF THE SUN AND STARS?

reaction, since it is a cheap alternative to produce energy. It is cheap in thatsea water can be used for the process, namely 2H +3 H He4 +n, and there is alot of sea water available. To get the two hydrogen nuclei close enough together,both lasers and magnetic focusing have been attempted. A large tokamac is beingbuilt to carry out the focusing.

In any star (including the sun) there are two opposing pressures. Gravity tendsto contract the star, while the pressure due to the hot plasma tends to expand it.In the sun, the pressure at the center is about 1015P aor N/m2. The two opposingpressures balance each other as long as there is fuel left.

11.1 Nucleosynthesis in the Sun

4He is produced in the sun and in stars. How are the heavier elements formed?If two He nuclei collide they may form 8Be, which is unstable and decays back totwo alpha particles. If the density is sufficiently high at equilibrium, there is achance that He nuclei collide with the 8Be to form 12C. and this can continue with4He +12 C16 O + . This process can continue, but we may also have carbonburning, where 12C +12 C20 Ne + and other similar reactions. This type ofmechanism works until iron is formed. Before then, the binding energy per nucleonincreases, but it decreases again beyond iron. New mechanisms are needed here.One is called a slow (s) process and the other one a rapid (r) one. The slow processis a capture mechanism of neutrons by a nucleus; the resultant nucleus beta decaysand then another neutron capture can occur. The rapid process is also a neutroncapture but where many neutrons can be captured beyond stability with subsequent

decay. The r-process can produce heavy elements like uranium. Proton capturecan also occur, but in proton-rich environments. Although all these processes areunderstood qualitatively, much remains to be learned for a quantitative study.

11.2 Do stars get older? The evolution of stars

11.3 Explosive nucleosynthesis

11.4 Neutron stars


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Chapter 12

What symmetries are

applicable to nuclei?

12.1 What are symmetries?

Symmetries are not only applicable to nuclei, but to all of physics. Newtons equa-tions of motion , for instance, are time symmetric or reversible; they do not changeif the time t goes to -t. A ball is thrown from A to B; the motion is reversible; ifB throws the ball to A, then in midflight you cannot tell which direction the ballis going. If A and B are out of the picture, you cannot tell who threw the ball to

whom. This is an example of the application of a symmetry.Because some symmetries hold for some, but not all forces, we need to firstdiscuss the forces of nature. As far as we know, there only four forces and no more.They are called the strong or nuclear forces (QCD), the electromagnetic force, theweak force, and gravity.

12.2 Basic Forces

The best known force is the weakest one, namely gravity. Gravity is what makesthe earth go round the sun, the moon around the earth, and keeps us on the surfaceof the earth. It affects our climate. The strongest force is the nuclear force. We cansafely ignore gravity when it comes to nuclei. If we arbitrarily call the strong or

nuclear force as having unit strength, then the electromagnetic one has a strengthof about 1/100, the weak force about 1/1,000,000 (106), and the gravitational oneof the order of 1040. The comparison is made difficult by the different ranges ofthe forces. The gravitational and electromagnetic (e.g., Coulomb) forces stretchto infinity, they fall off as the square of the distance from the source. The strongone has a range of only approximately 1015 m, and the weak one about 1018m .The strength of the forces are compared at 1015 m. The nuclear and weak forcesfall off exponentially beyond their ranges. At 1 cm, the nuclear force is abysmallysmall This is what makes a comparison difficult. What binds atoms together is

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44 CHAPTER 12. WHAT SYMMETRIES ARE APPLICABLE TO NUCLEI?

the electrical (Coulomb) force between the electrons and the nucleus. This force ismuch, much stronger than the gravitational force.

The fourth force is the so-called weak force. At nuclear distances (1015 m)its strength is about 106 of the strong force. Hence the name weak. It is theforce responsible for beta-decay and it is the only force which acts on neutrinos.The range of the weak force is only about 1018 m. At that distance its strength iscomparable to the electric force. Indeed, the two forces have been unified into theelectroweak force. The force is weak at nuclear distances because of the exponentialfall-off.

In modern physics, forces are due to the exchange of particles between theobjects that feel the force. The range of the force thus depends on the mass (m) ofthe exchanged particle; it is /mc. The Coulomb (electric) force is carried by theexchange of a photon, the mass of which is zero. The range is inversely proportionalto the mass, and is thus infinite for the electric force. The same holds for gravity,although thegravitonhas not yet been found. The nuclear force effectively is due tothe exchange of mesons, like the pion (mass 140 MeV/c2); thus the range is about1015m. Actually, the basic force is QCD and the particle exchanged is a gluon ofzero mass. But gluons, like quarks are confined and we can talk of an effective masswhich is about 500 MeV - 1 GeV. The weak force is carried by heavy bosons, theW of mass close to 80 GeV/c2and the Z0 of mass near 90 GeV/c2. Thus theirranges are very short.

12.3 Space-Time Symmetries

Is there a difference between left and right? Can you tell left (L) from right (R)?The answer is no for nuclear forces, and even for electromagnetic forces. It is onlywhen you get to the weak forces that you can observe a difference, and this is aneffect on nuclear forces that is about 1 part in a million. In fact, until 1956, it wasnot believed that you can tell the difference between L and R. In that year T.D.Lee and C.N. Yang showed that no experiment had been done that would allow oneto observe a difference. They proposed several tests, among them beta-decay frompolarized nuclei. If more electrons are emitted antiparallel to the spin (polarization)direction than parallel to it, then so-called parity (handedness) is not conservedin the weak interaction and the emitted electrons are said to be left-handed. (Sinceneutrinos are involved, it is known that the interaction is weak.) C.S. Wu and hercollaborators showed that this was indeed the case and this was the beginning of a

quiet revolution in physics. C.S. Wu and collaborators took 60

Co and polarized itby means of a magnetic field. They observed the decay 60Co 60 Ni + e + andcounted the electrons emitted forward (parallel to the spin) and backwards; theyfound more electrons backwards than forwards. They should not have observed anydifference if parity were conserved.

What about other spatial or time symmetries? Can you tell the difference be-tween time going forward or backward? The time reversal symmetry appeared tohold , even for the weak interactions until 1964. A telling case for a time reversalsymmetry violation would be the measurement of an electric dipole moment for the


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12.3. SPACE-TIME SYMMETRIES 45

neutron. The neutron has an advantage over the proton in that it has no charge andconsequent Coulomb interaction. The argument for the lack of a dipole momentdepends on both parity conservation as well as time reversal symmetry. [ Outlinethe argument here with pictures]. Electric dipole moments have been sought forneutrons, electons, and atoms. The best test so far may be that in Hg atoms, wheredE 1027? e-cm. Since parity is known to be violated in the weak interactions,the null result means that the time reversal invariance hol

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