subatomic lecture notes oxford

8/13/2019 Subatomic Lecture notes Oxford

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B2.IV Nuclear and Particle Physics

A.J. Barr

January 22, 2014


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Contents

1 Introduction 1

2 Nuclear 32.1 Structure of matter and energy scales . . . . . . . . . . . . . . . . 32.2 Binding Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Semi-empirical mass formula . . . . . . . . . . . . . . . . . 42.3 Decays and reactions . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Alpha Decays . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Beta decays . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Nuclear Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.1 Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 Resonances and the Breit-Wigner formula . . . . . . . . . . 182.4.3 Nuclear scattering and form factors . . . . . . . . . . . . . . 21

2.5 Key points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Appendices 242.A Natural units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.B Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.B.1 Decays and the Fermi Golden Rule . . . . . . . . . . . . . . 252.B.2 Density of states . . . . . . . . . . . . . . . . . . . . . . . . 252.B.3 Fermi G.R. example . . . . . . . . . . . . . . . . . . . . . . 262.B.4 Lifetimes and decays . . . . . . . . . . . . . . . . . . . . . 262.B.5 The ux factor . . . . . . . . . . . . . . . . . . . . . . . . 272.B.6 Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.C Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.D Gamma decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Hadrons 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Pions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.2 Baryon number conservation . . . . . . . . . . . . . . . . . 323.1.3 Delta baryons . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Linear Accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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3.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.1 Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.2 Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.3 Quark ow diagrams . . . . . . . . . . . . . . . . . . . . . 363.3.4 Strangeness . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.5 Pseudoscalar octet . . . . . . . . . . . . . . . . . . . . . . 383.3.6 Baryon octet . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Colour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.5 Heavier quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.6 Charmonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7 Hadron decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Appendices 46

3.A Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.B Discovery of the Omega . . . . . . . . . . . . . . . . . . . . . . . 484 Scattering theory 51

4.1 Scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.1.1 Scattering amplitudes . . . . . . . . . . . . . . . . . . . . . 524.1.2 The Born approximation . . . . . . . . . . . . . . . . . . . 54

4.2 Virtual Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 The Yukawa Potential . . . . . . . . . . . . . . . . . . . . . . . . . 55

Appendices 594.A Beyond Born . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 Feynman diagrams 615.1 Aim of the game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2.1 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2.2 Anti-particles . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.3 Distinct diagrams . . . . . . . . . . . . . . . . . . . . . . . 635.2.4 Relativistic propagators . . . . . . . . . . . . . . . . . . . . 645.2.5 Trees and loops . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6 The Standard Model 716.1 Matter particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.1.1 Lepton avour number . . . . . . . . . . . . . . . . . . . . 726.2 Force particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.3 Strong force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3.1 Deep inelastic scattering . . . . . . . . . . . . . . . . . . . 79

6.3.2 (e+ + e hadrons) (e+ + e + ) . . . . . . . . . . . . . . . . . . . . . . . 80

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6.4 W and Z bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.4.1 The Z 0 particle . . . . . . . . . . . . . . . . . . . . . . . . 846.4.2 Production and decay . . . . . . . . . . . . . . . . . . . . . 84

6.4.3 Parity violation . . . . . . . . . . . . . . . . . . . . . . . . 886.5 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.5.1 Solar neutrinos . . . . . . . . . . . . . . . . . . . . . . . . 926.5.2 Atmospheric neutrinos . . . . . . . . . . . . . . . . . . . . 92

6.6 The Higgs eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.7 Beyond the Standard Model . . . . . . . . . . . . . . . . . . . . . . 96

6.7.1 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.7.2 A theory of avour . . . . . . . . . . . . . . . . . . . . . . 976.7.3 Matter / antimatter asymmetry . . . . . . . . . . . . . . . 976.7.4 The dark side of the universe . . . . . . . . . . . . . . . . . 97

6.7.5 The hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . 986.7.6 Strings and things . . . . . . . . . . . . . . . . . . . . . . . 98

Appendices 996.A Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Applications 1017.1 Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.1.1 Energy and barriers . . . . . . . . . . . . . . . . . . . . . . 1017.1.2 Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . 1037.1.3 Chain reactions . . . . . . . . . . . . . . . . . . . . . . . . 1037.1.4 Fission reactor principles . . . . . . . . . . . . . . . . . . . 105

7.2 Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.3 Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.3.1 The pp-II and pp-III chains . . . . . . . . . . . . . . . . . . 1117.3.2 The CNO cycles . . . . . . . . . . . . . . . . . . . . . . . . 1117.3.3 Solar neutrinos . . . . . . . . . . . . . . . . . . . . . . . . 1137.3.4 Heavier elements . . . . . . . . . . . . . . . . . . . . . . . 113

8 Accelerators and detectors 1178.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.2 Accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.2.1 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.3 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.4 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8.4.1 Photon interactions . . . . . . . . . . . . . . . . . . . . . . 1218.4.2 Very high-energy electrons and photons . . . . . . . . . . . 1238.4.3 Very high-energy, strongly interacting particles . . . . . . . . 1248.4.4 Detecting neutrons . . . . . . . . . . . . . . . . . . . . . . 1258.4.5 Detecting neutrinos . . . . . . . . . . . . . . . . . . . . . . 125

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8.4.6 Measuring properties . . . . . . . . . . . . . . . . . . . . . 125

Appendices 1278.A Linacs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278.B Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1308.C Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8.C.1 Semiconductor detectors . . . . . . . . . . . . . . . . . . . 1308.C.2 Gas and liquid ionization detectors . . . . . . . . . . . . . . 1318.C.3 Scintillator detectors . . . . . . . . . . . . . . . . . . . . . 131

9 Examples 1359.1 Problems 1: Radioactivity and nuclear stability . . . . . . . . . . . 1369.2 Problems 2: Quarks and deep inelastic scattering . . . . . . . . . . 1429.3 Problems 3: Scattering & reactions . . . . . . . . . . . . . . . . . . 1489.4 Problems 4: The Standard Model . . . . . . . . . . . . . . . . . . . 1549.5 Problems 5: Energy production, stars . . . . . . . . . . . . . . . . . 158

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Chapter 1

Introduction

When the great American physicist and bongo-drums player, Richard Feynman, wasasked to think of a single sentence that would convey the most important scienticknowledge, he answered simply: Everything is made of atoms.

Our understanding of matter at the atomic scale has made possible much of mod-ern life, with its the mobile phones, computers and communications technology.Technologies from semiconductors, lasers, displays, and materials developments allrequire knowledge of how atoms behave and interact. The fundamentals of chem-istry, drug development and biochemistry all rely on that fundamental insight. Butwe also know that atoms are not fundamental, and not indivisible; smaller structuresexist inside.

The discovery of the atomic nucleus within the atom had profound consequences.The implications were not initially obvious; indeed the pioneering New Zealandnuclear physicist, Ernst Rutherford, reportedly said that the idea of getting practicalenergy out of the atomic nucleus was moonshine.1 Later in the 20th century,theory and experiment on nuclear structure allowed us to understand the energysource of stars (including the sun), the most violent supernovae, the heating of theearth, and even the method of formation of the chemical elements around us. Moreimmediately, ssion power stations continue to provide the greatest contribution tolow-carbon electricity in the UK, generating 18% of electricity in the UK and asmuch as 75% in France.

Like ssion, the physics of nuclear fusion (combining nuclei) has been understoodfor decades. The nuclear physics is not an area of current research in the eld of nuclear physics, but future developments in controlling the plasma in which fusionoccurs will be needed if we are to unlock the potential of this almost inexhaustiblesource of energy. Large-scale experimental facilities at the Culham Laboratory inOxfordshire and the ITER fusion prototype plant in France are investigating waysto control high-energy plasmas for long times.

Today the physics questions about the fundamental make-up of forces and matter1He turned out to be right, in a way, since the moonlight has since been understood to originate

from fusion of hydrogen to helium inside the sun. Its perhaps unlikely that this is what he intended.

Richard Feynman havingfun.

The plasma inside theMAST tokamak reactorruns at temperatures of upto 3,000,000K.

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CHAPTER 1. INTRODUCTION

The rst web server. When TimBerners Lee wrote his proposalfor the World Wide Web it was

annotated by his managervague but interesting.

Not dark matter.

have moved on. We observe the most basic building blocks of nature to be point-like constituents . the quarks, leptons, making up matter, and with gauge bosonsas the force-carrying particles. The experimental observations are described toamazing accuracy by the Standard Model of particle physics, a theoretical triumphof relativistic quantum eld theory that correctly predicts the gyromagnetic ratio of the electron to an accuracy of one part in a trillion.

Laboratory measurements, together with a mathematical formalism, let us enquireinto the nature of the vacuum, and into the dense and hot conditions of the universea fraction of a second after its birth. They are also providing insights into thereasons why matter dominates over anti-matter, and the origins of mass. Performingexperiments length scales and higher energies than ever before requires the inventionof new technologies. The technologies of the future will, no doubt, grow out of ourcurrent areas of research. At the same time, spin-offs are already affecting thewider world. Perhaps the most remarkable invention of our era, the World WideWeb, was developed by Queens college graduate Tim Berners Lee when working at

CERN in order to help physicists collaborate on designing and building the LHC.The technologies developed for current particle physics experiments have been usedin medical imaging, climate forecasting, decoding the human genome, nuclear anti-proliferation, cancer treatment, information analysis and drug development.

Recently the LHC has opened up a new eld by discovering a completely new typeof particle. Observations suggest it is remarkably similar to the Higgs boson of the Standard Model. This apparently fundamental spin-0 particles, and is only juststarting to be investigated and understood. It is almost certainly the manifestationof entirely new fundamental force of nature, different from all of the others observeduntil now. Much about that force has not yet been investigated. To understand theproperties of that force will require a long programme at a high-energy LHC andmost likely new facilities and new ideas.

Many other experimental and theoretical questions remain open. The properties of neutrinos, are only now starting to be probed with precision. Soon we may knowwhether or not neutrinos are their own anti-particles. Other crucial differencesbetween matter and anti-matter, differences essential to our existence, are beingstudied with ever greater detail in the decays of hadrons.

The Dark Matter particle, believed to be responsible for the missing 80% of thematter in the universe, is being hunted by astroparticle and underground directdetection experiments. And it is hoped that future theories or experiments maythrow light onto the enormous difference in strength between the forces.

New theories exist which can solve these problems. All predict the existence of newparticles or phenomena, often within reach of either operating or proposed facilities.The close interplay of theory with experiment at the cutting edge of knowledge willbe required if these new phenomena are to be predicted, measured and added tothe canon of human knowledge.

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Chapter 2

Nuclear physics and decays

2.1 Structure of matter and energy scales

Subatomic physics deals with objects of the size of the atomic nucleus and smaller.We cannot see subatomic particles directly, but we may obtain knowledge of theirstructures by observing the effect of projectiles that are scattered from them. Theresolution any such probe is limited to of order the de Broglie wavelength,

= h p

(2.1)

where h is Plancks constant, and p is the momentum of the projectile. If wewish to resolve small distances, smaller than the size of the atom, we will need todo so with probes with high momenta. Smaller objects also tend to have largerbinding energies holding them together, so require larger energies to excite theirinternal components. Some typical sizes of objects are given below, together withthe momementum of the projectile required to resolve their size, and typical bindingenergies in electron-volt (eV) units.

Object Size p = h Binding energyAtom 1010 m 10keV/c eV

Nucleus 1015 m 1GeV/c MeVQuark < 1019 m > TeV/c > TeV

We can see that small objects also tend to have high binding energies, and henceprobes of large energy will be required in order to excite them or break them up. Themomenta are indicated in units of eV/c where c is the speed of light. These unitsmake it easy to compare the momentum of the projetile to its corresponding energyE = pc for the case of a massless probe such as a photon. The most convenientunit for describing the size of nuclei is the femtometer 1015 m.1 No sub-structurehas yet been found for quarks even when using very high energy (TeV) probes.

1The unit of 10 15 m or femtometer is sometimes called the fermi reecting the many seminalcontributions of the Italian physicist Enrico Fermi to the eld of nuclear physics.

keV 103 eVMeV 106 eVGeV 109 eVTeV 1012 eV

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2.2. BINDING ENERGY CHAPTER 2. NUCLEAR

Isotopes Same Z Isotones Same N

Isobars Same A

Notation for related nuclei

2.2 The Nuclear Periodic Table and Binding Energy

Nuclei are found to be made out of two constituents: protons and neutrons. We

label nuclei by their atomic number Z which is the number of protons they contain,by their neutron number N , and by their mass number A = Z + N .

The symbol used to identify a nucleus is

AZXN

where X is the name of the chemical element. For example the Carbon-14 nucleus,which contains 8 neutrons and 6 protons is denoted 146C8 . Since the elementsname species the number of electrons, and hence the atomic number Z , and sinceA = N + Z , we can fully specify the nucleus by just the symbol for the chemicaland the mass number,

A X e.g. 14C.

Most nuclei are spherical in shape. The nuclear radius r can be measured in scat-tering experiments, and follows the general rule

r = r0A1/ 3 (2.2)

where the constant r 0 is the characteristic nuclear size and is about 1.2 1015 m.The fact that r is proportional to A1/ 3 indicates that the volume of the nucleusV r 3 is proportional to the mass number A. Each proton or neutron is thereforemaking an equal contribution to the overall nuclear volume.

2.2.1 Binding energy and the semi-empirical mass formula

The mass m(A, Z ) of the nucleus containing Z protons and A Z neutrons shouldbe given by the mass of its consituents, less the mass associated with the bindingenergy. The mass-energy is therefore

m(A, Z )c2 = Zm pc2 + ( A Z )mn c2 B (A, Z ), (2.3)where m p 938.3 MeV/c 2 and mn 939.6 MeV are the masses of the proton andneutron respectively. In nuclear physics it is convenient to measure energies in unitsof MeV and masses in units of MeV/c 2 . Using these units makes it easy for us to

convert from mass to mass-energy and vice versa. By assuming such units, we canomit the factors of c2 in what follows. 2

We can build up a functional form for the binding energy B(A, Z ) by consideringthe forces between the nuclear constituents. To nd the full quantum mechanicalground state for all of the protons and neutrons would be a very difficult problem.However we can understand a great deal about nuclear behaviour by building up amodel of the mass which encapsulates its key features. This we will do over the restof the section, building up towards the semi-empirical mass formula of equation(2.5). The semi-empirical means that the model is built partly partly by demandingagreement with data, and partly from our understanding of the underlying theory.

2For more on natural units see appendix 2.A.

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CHAPTER 2. NUCLEAR 2.2. BINDING ENERGY

Figure 2.1: Binding energy per nucleon (B/A ) as a function of A for some commonnuclei. Data taken from [5]. Plot from [source].

Firstly, we will need an attractive force in order to hold the nucleus together againstthe mutual electrostatic replusion of its constituent protons. That force must bevery strong, since the Coulomb electrostatic repulsion between a pair of protons,each of charge e and separated by distance d 1 fm, is

F = e2

4 0 d2

230Nwhich is macroscopic comparable to the weight of a child.

What form should that nucleon-nucleon attractive force take? We can get cluesabout the force by looking at the binding energy per nucleon B/A is shown for somecommon nuclei, shown in Figure 2.1. For nuclei this binding energy is typically of order 8 MeV per nucleon. It can be seen that the most stable nuclei are foundaround 56 Fe. Different behaviours can be seen in different regions. There is a broadattish plateau for the central region 30 < A < 200 for which B/A 8MeV.For A below about 30 the binding energy per nucleon is smaller than the plateauvalue and is spiky. There is a systematic drop in B/A for large A, particularly forA > 200.

To obtain a value of B/A that is rather at, we cannot permit strong attractionsbetween each of the constituent nucleons and every one of the others. If everynucleon felt an attraction to each of the others, then the binding energy would beexpected to grow as approximately B A(A 1)A2 , and hence B/A would beapproximately proportional to A. Such a linear growth in B/A is ruled out by thedata (Figure 2.1).

To obtain the at B/A found in nature we must assume that the strongly attractiveforce acts only between nearest neighbour nucleons. In this way, each nucleonbinds to the same number of nearest neighbours, independently of the size of thenucleus, and hence the binding energy per nucleon is the same regardless of the

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2.2. BINDING ENERGY CHAPTER 2. NUCLEAR

NP

F

E

g N Eg Z E

The density of states g(E ) forprotons and neutrons as a

function of energy E .

nuclear size,B A

where is a constant with units of energy. The use of nearest-neighbour interactions

indicates that the force must either be short-range, or screened from long-rangeinteractions by the effects of the nucleons in between.

In modelling a nearest-neighbour force we ought to make a correction for the factthat those nucleons on the surface have fewer neighbours. To correct for the reducednumber of binding opportunities on the surface we reduce the binding energy by anamount proportional to the surface area, leading to the corrected formula

B A A23 . (2.4)

The new contribution is negative since it reduces the binding energy. We have madeuse of the observation (2.2) that since the volume of the nucleus scales as r 3 A,the surface area scales as r 2 A

2/ 3 .

These two terms ( 2.4) in this rst approximation to the binding energy are knownas the volume term and the surface term respectively. Together they form what isknown as the liquid drop model , since a similar result would be found for a drop of uid with nearest neighbour interactions and a surface tension parameterised by .The liquid drop model is consistent with the observation that each nucleon requiresthe same volume of space, in agreement with equation (2.2).

So far, so good. However there is nothing in this liquid drop model to prevent thegrowth of arbitrarily large nuclei. Such large nuclei are not observed in nature, sowe must be missing something. The obvious candidate is the Coulomb repulsion ,which interacts over long distances, and so will tend to push larger nuclei apart.This electrostatic repulsion between protons will reduce the binding energy by anamount proportional to Z (Z 1) Z 2 because every proton feels the repulsionfrom all of the other protons (not just nearest neighbours). The binding energy willbe reduced by the electrostatic binding energy which can be parameterised by

Z 2

A13

.

Here is a another constant with dimensions of energy, which we will calculate avalue for in the examples. The Coulomb repulsion energy is inversely proportional tothe radius of the nucleus, and hence to A

13 , since the potential energy of a uniform

sphere of charge Q is proportional to Q2 /r .

Two further terms are required to give a good match between our model and thedata. Both of them are quantum mechanical in origin.

Firstly there is an asymmetry term . The origin of this term is as follows. Sinceprotons are identical fermions, the Pauli exclusion principal states that no two of them may exist in the same state. Nor may any neutron occupy the same state asany other neutron. However it is possible for a proton and a neutron to exist in thesame state since the two particles are not identical. The allowed states are thereforedistinct, and are separately lled for the protons compared to the neutrons.

We can work out the size of the asymmetry effect by calculating the number of states available. Neutrons and protons are both fermions, and so obey Fermi-Dirac

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CHAPTER 2. NUCLEAR 2.2. BINDING ENERGY

777

7.25

7.5

7.5

7.5

7.75

7.75

8

8

8

8.25

8.258.5

8.5

8.5

50 100 150N

20

40

60

80

100

Z

(a)

3

4 5 6

7

7.5

8

8.5

50 100 150N

20

40

60

80

100

Z

(a)

Figure 2.2: Diagram showing binding energies as a function of proton and neutronnumber for (a) data [5] and (b) the Semi-Empirical Mass Formula.

statistics. The temperatures we are interested in are small compared to the chemicalpotential ( kB T ). Under these circumstances the Fermi-Dirac distribution tendstowards a step function all levels are lled up to some energy level, known as theFermi Energy F , with all states with energy above F left vacant.

At large mass number A the Coloumb repulsion term would tend to favour largerN and smaller Z , since neutrons do not suffer from the Coulomb repulsion asprotons do. However this energic advantage of neutrons over protons will be partiallycancelled out by the fact that the additional neutrons must (on average) be placedin higher energy levels than additional protons, since all of the lower-energy neutronstates will already be lled.

The density of available states is found to be proportional to E

12

. In the exampleswe show that this leads to an energy equation of the form

(N Z )2

A .

This asymmetry term reduces the binding energy, doing so most when the differ-ence between the number of protons and of neutrons is largest.

Finally there is a pairing term which accounts for the observation that nuclei witheither even numbers of protons (Z even) or with even numbers of neutrons (N even) tend to be more stable than those with odd nuclei. The pairing term is zero

for odd-A nuclei. Even A nuclei have two possibilities. If both Z and N are eventhen the nucleus is more tightly bound and have an extra binding contribution, soB is increased by . If both Z and N are odd then the nucleus is less tightly boundand so B is decreased by .

Putting all ve terms together we obtain a formula for the binding energy,

B (A, Z ) = A A23

(A 2Z )2A

Z 2

A13

+ (A, Z ),

having eliminated N in favour of A. Substituting this into the formula dening the

p(E i ) = 1

e( E i ) /k B T + 1

The Fermi-Dirac function givesthe probability p(E i ) of lling a

state with energy E i for asystem at temperature T andwith chemical potential . kB

is the Boltzmann constant.

NP

F

E

g N Eg Z E

The density of states g(E ) forprotons and neutrons as a

function of energy E .

N Z pairing termeven even even odd 0odd even 0odd odd

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2.3. DECAYS AND REACTIONS CHAPTER 2. NUCLEAR

Volume Surface Asymmetry Coulomb Pairing

15.835 18.33 23.2 0.71 11.2/ AFigure 2.3: Typical values of the SEMF parameters (in MeV). From Bowler.

Sketch of the shape of thevalley of stability.

, ECZ

NThe changes in (Z, A ) inducedby various decays.

binding energy (2.3) we obtain the semi-empirical mass formula (SEMF)

M (A, Z ) = Zm p + ( A Z )mn A + A23 +

(N Z )2A

+Z 2

A13 (N, Z ).

(2.5)

Other than for A < 30, where our approximations are less valid, the SEMF gives arather good description of the binding energies of the observed nuclei (Figure 2.2).

In particular the SEMF correctly predicts the shape of the curved valley of stabilityin the Z, N plane within which the stable nuclei are found. The relative numbersof protons and neutrons along this valley reects a trade-off between the Coulomband asymmetry terms. At low A the asymmetry term favours N = Z . At larger Athe Coulomb term starts to compete with the asymmetry term, reducing the ratioof protons to neutrons.

It is energetically favourable for nuclei far from that valley to migrate towards it bynuclear decay, in the ways we describe in the following section.

2.3 Decays and reactions

A table of the nuclides can be found in Figure 2.4. The stable long-lived nuclideslie along the valley of stability where the binding energy per nucleon is largest. Thevalley lies along N Z for light nuclei but has N > Z for heavier nuclei. Nuclei farfrom that valley, and very heavy nuclei, tend to be unstable against nuclear decay.While unstable nuclei will decay spontaneously, other reactions can be initiated byring projectiles at a nucleus. Reactions are said to be elastic if the nal statecontains the same set of particles e.g. the elastic scattering of a photon from anucleus, via an excited intermediate state:

X + X X + .Reactions are inelastic if there is a change in particle content during the reactione.g. radiative capture of a neutron

A X + n A+1 X A+1 X + .

For all nuclear decays and reactions we dene the Q value to be amount of energyreleased by the decay,

Q = M i M f

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CHAPTER 2. NUCLEAR 2.3. DECAYS AND REACTIONS

F i g u r e

2 . 4 :

T a b

l e o f t h e n u c l

i d e s a s a

f u n c t

i o n o

f t h e n u m

b e r o f n e u t r o n s

( N ,

o n t h e x - a x

i s ) a n

d t h e n u m

b e r o f p r o t o n s

( Z o n t h e y - a x

i s ) .

D a r

k e r c o

l o u r s r e p r e s e n t l o n g e r - l

i v e d n u c l

i d e s ,

w h i c h c a n

b e

f o u n

d i n t h e

v a l

l e y o

f s t a b

i l i t y . T h e M a g

i c N u m

b e r s

i n d i c a t e p a r t i c u

l a r l y s t a b

l e

n u c l e i a n

d a r e

d e s c r

i b e d

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l m o

d e l

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d i x 2

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. F r o m

h t t p : /

/ w w w . n

n d c .

b n

l . g o v / n u

d a t

2 / .

9
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Figure 2.5: Decay modes of the nuclei. From [5].

The rst sum is over the masses of the initial particles in the decay (includingtheir binding energies), while the second sum is over the masses of the nal-stateparticles (including their binding energies). A positive Q value shows that a reactionis energetically favourable.

2.3.1 Alpha Decays

Alpha decays occur when (usually heavy) nuclei eject an particle, that is a heliumnucleus containing two protons and two neutrons. The decay is

AZX A4Z2Y +

42He.

The change of mass number of the heavy nucleus is A = 4 and the change inits atomic number is 2.The process of decay of heavy nuclei is often via sequential chains involving bothalpha and beta decays. Since A = 4 for decays and A = 0 for and decays we see that for any nucleus starting with mass number A (and if it onlydecays via alpha, beta or gamma decay processes) all other nuclei in that chainmust have some other set of mass numbers A = A 4m, where m is an integerindicating the number of alpha decays that have occured.There are therefore four non-overlapping decay chains for the heavy elements, cor-responding to A = n , A = n + 1 , A = n + 2 , and A = n + 3 respectively, wheren is an integer. So for example a chain initiated by 238 U will decay via nuclideswith mass numbers separated by four units A = {238, 234, 230, . . . , 4n + 2 , . . .}.

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This occurs via the series of decays:

238 U 234 Th

234 Pa

234 U

230 Th

226 Th etc .

The other three types of chain can, for example, be initiated from decays of thefollowing isotopes: 232 Th (leading to nuclides in the A = 4n series); 237 Np (leadingto nuclides in the A = 4n + 1 series); and 239 Pu (leading to nuclides in the A =4n + 3 series).

Rate calculation for decays

We can model the decay as a process in which proto particles are pre-formedinside the nucleus. Each is assumed to have a large number of collisions with theedge of the nucleus, but a small probability on each collision of tunnelling throughthe Coulomb barrier and escaping.

If the Q value of the decay is positive, then the decay is energetically favourable,but it may still be suppressed by a large tunnelling factor. Let us try to modelthe probability of tunnelling through the barrier. We will assume that the largeexponential in the quantum tunnelling factor will dominate the calculation of therate of decay, so we will neglect differences in the probability of formation of theproto-alpha particle, and its rates of hitting the barrier.

The time independent Schr odinger equation denes the energy eigenstate | ,E | =

p2

2m + V | .

where E is the energy of the alpha particle, p is the momentum operator, m is itsmass, and V is the potential in which it moves.

For simplicity, we will ignore the spherical geometry and treat the problem as one-dimensional in the radial direction r so that for a state with energy Q,

Q r | = 12m

2

r 2 + V (r ) r | , (2.6)

where we use natural units such that = h2 = c = 1 (see appendix 2.A). In theDirac notation r | represents the wave function that is the amplitude to ndthe alpha particle located between r and r + dr . Without losing any generality wecan write the wave function as the exponential of some other function (r ),

r | = exp[ (r )]. (2.7)After inserting (2.7) into (2.6) and dividing by exp() we nd

Q = 12m

+ ( )2 + V (r ),

where the primes indicate derivatives by r. We can model the potential V (r ) feltby any particle by the function

V (r ) =const r < R azZ EM

r r > R a

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Re[ ]

VQ

R b

R a

1 2 3 4 5 6 7r

1

1

2

3

Real part of r | (ascalculated in the WKBJapproximation) and V (r ) for athin Coulomb potential barrier.

where inside the nucleus V is large and negative, and outside the nucleus it is givenby the Coulomb potential and hence characterised by the charges z and Z of the-particle and the daughter nucleus respectively. The constant EM in the Coulombpotential is the dimensionless electromagnetic ne structure constant

EM = e2

4 0 c 1137

.

Within the barrier the potential is smoothly varying, so should be a smoothlyvarying function of (r ). We then expect ( )2 , and we can safely neglect the term compared to the ( )2 .3

The tunnelling probability can be found from the ratio of the mod-squared ampli-tudes:

P = | Rb| |2

|Ra

|

|2 = e

2G .

where G (> 0) is given by

G = (Rb) (Ra ) = 2m R bR a dr V (r ) Q.The minus sign before the radical ensures that we select the exponentially fallingsolution. The inner limit of the integration is the radius of the nucleus

Ra r 0A13 ,

and the outer limitRb =

Z 1Z 2 EMQ

,

the radius for which Q > V , i.e. where the particle enters the classically allowedregion.

2.3.2 Beta decays, electron capture

There are three related nuclear decay processes which are all mediated by the weaknuclear interaction . Neutron-rich isotopes can decay through the emission of anelectron e and an anti-neutrino e in the beta decay a process:

AZX

A

Z+1 Y + e + e . (2.8)

The effect is to increase the atomic number by one, but to leave the mass numberunchanged. At the level of the individual nucleons the reaction is

n p + e + e . (2.9)

The emitted electron can be observed and its energy measured. The associatedanti-neutrino has a very small interaction probability, and so is expected to escape

3This is known as the WKBJ approximation. It is a good approximation if many wave-lengths(or in the classically forbidden regions, as here, many factors of 1/e ) of the wave-function occurbefore the potential changes signicantly.

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unobserved. Long before neutrinos were observed, Wolfgang Pauli realised that anadditional, invisible, massless particle was required in order to conserve energy andmomentum in the decay (2.8). His arguments ran as follows. The emitted electronsare observed to have a variety of different kinetic energies, up to Q. Meanwhilethe mass difference between the parent and daughter nucleus is xed to a singlevalue. The energy given to the recoiling daughter nucleus is small and is xed bymomentum conservation, so it cant be responsible for the decit observed whenthe electron has energy less than Q. Energy conservation is then only possible if the total energy Q can be shared between the electron and some other unobservedpartice the (anti-)neutrino.

Pauli also argued that without the neutrino the reaction ( 2.9) would violate angularmomentum conservation. Adding the angular momenta of just the two observednal state spin-half particles the electron and the proton according to the rulesof quantum mechanical angular momentum addition we would nd

12

12 = 0 or 1.

Neither of the possibilities of total angular momentum s = 0 or s = 1 match thespin of the initial neutron, which has s = 12 . However by adding a third spin-half particle to the nal state the s = 12 anti-neutrino we can reconstruct a statewhich has total angular momentum equal to that of the proton ( s = 12 ) since

12

12

12

= 12

or 32

.

Isotopes which have a surplus of protons can proceed via one of two processes. Therst is the emission of a positively charge anti-electron. This is known as positronemission or + decay A

ZX AZ1Z + e+ + e (2.10)The positron is the anti-particle of the electron. It has the same mass as theelectron, but positive charge.

The second method of decay of proton-rich nuclei is by the nucleus removing oneof the atomic electrons, the electron capture process:

AZX + e AZ1Z + e (2.11)

These two processes ((2.10) and (2.11)) result in the same change to the nucleus,and so compete with one another to reduce the Z number of proton-rich nuclei.

When considering whether electron capture or + decay will dominate we note that

The Q value for positron emission is 2 m e c2 smaller than that for thecorresponding electron capture. Electron capture relies on there being a substantial overlap of an electronwave-function with the nucleus.

When viewed at the level of the nuclear consitutents, all three of the interactionsabove decay (2.8), + decay (2.10) and electron capture ( 2.11) involve the

EO OE

Z

EE

OO

Z

Mass as a function of Z fornuclides of the same A , forodd- A nuclei (above) andeven-A nuclei (below). Theeven-A case has two curvesseparated by 2.

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interaction of four particles: a proton, a neutron, an (anti-)electron and an (anti-)neutrino.

n p + e + e (2.12) p n + e+ + e , (2.13) p + e n + e . (2.14)

We note that all of the reactions ( 2.12)(2.14) are assumed to be occurring insidethe complex environment of the nucleus. Of these three reactions, only neutrondecay (2.12) can occur in isolation, since it is the only one with Q > 0, (the neutronbeing about 1.3 MeV/c 2 heavier than the proton). The other two reactions ( 2.13)(2.14) occur only within a nucleus, when the energy released from the rearrangementof the nuclear constituents is sufficient to compensate for the endothermic natureof the reaction at the level of the individual nucleon.

We note that all of three transitions , + , and e capture leave the mass

number A unchanged. The parent and daughter nuclei are isobars. The decayprocesses (2.12)(2.14) allow transitions between isobars, and mean that for odd- Anuclei for any value of A there is usually only one stable isobar, that for which themass of the system is minimum.

Even-A nuclei may have one stable isobar, but can also have two or very occasionallythree. Multiple stable states are possible for A even because the binding curves for(even-Z , even-N ) and (odd-Z odd-N ) are separated by 2 , where is the pairingenergy in the SEMF. For an even-even nucleus, since all of the reactions (2.12)(2.14) change both |Z | and |N | by one they result in an odd-odd nucleus, and so atransition to the higher of the two curves. None of the reactions permit a changein Z of two units, and the probability of two such reactions happening at once isextremely small, so an even-even nucleus with some Z can be stable provided thatboth the neighbouring nuclei with Z + 1 and Z 1 have larger mass.

Fermi theory of beta decays

To understand the lifetimes of the nuclei, we wish to calculate the expected ratesfor decays. We follow the method and approximations of Enrico Fermi.

If we put the initial state particles on the left hand side of the diagram, and the nalstate particles on the right hand side, then we obtain the following three diagramsfor the reactions (2.12)(2.14).

n

p

e

e

p

n

e+

e

p

e

n

e

We shall assume that each of these four-particle interactions will happen at a singlepoint in space. The amplitude for each reaction is given by the same constant

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a four-body coupling constant which tells us the amplitude for each interaction atthat point in space. For each of the three diagrams that coupling is the Fermiconstant ,

GF

1.17

105 GeV2

which has units of inverse energy squared.

In using a single, constant factor, we implicitly make the simplifying assumptionthat the four-body interaction does not depend on the spins of the incoming oroutgoing particles. To simplfy calculations we will also follow Fermi in assumingthat the wave-functions of the electron and the anti-neutrino can be represented byplane waves. This ignores the effect of the Coulomb attraction between the electronand the nucleus, and is a good approximation provided that the electron energy issufficiently high.

We recall that in quantum mechanics, the rate of some a transition from an initialstate to a nal state characterised by a continuum of energy levels is given by theFermi Golden Rule 4

= 2

|Afi |2 dN dE f

(2.15)

Here the transition rate is given in terms of the amplitude Afi connecting theinitial and the nal states, and the degeneracy dN dE f of states at the nal energy. Itis the A fi and dN dE f that we shall have to calculate.

To be concrete, let us consider the beta decay reaction ( 2.12). We denote the initialnuclear wave-function by x |i , the nal nuclear wave-function by x |f . Theelectron and anti-neutrino wave-functions are approximated as plane waves

x |e e = exp ( i p e x ) (2.16)x | = exp ( i p x ). (2.17)

We can now write down the initial state |i , which is just that of the parent nucleus

|i = |i ,and nal state |f , which is the product of the daughter nucleus state |f , theelectron state |e and the anti-neutrino state | e

|f = |f |e | e .The matrix element

Afi controls the transition from the initial to the nal state

Af i = f |A|iIt can be obtained by working in the position representation and recognising that theamplitude GF associated with the point-like interaction (2.12) should be integratedover the volume of the nucleus,

Afi = d3x GF e f i .4For a refresher, see appendix 2.B.1

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The and terms are the position representations (wave functions) of the fourparticles, and in the nal state are found in complex conjugate form. The inte-gral sums over the amplitudes for the point-like reaction to occur anywhere in thenucleus, since the reaction could have occurred anywhere within.

To perform the integral we rst Taylor expand the exponentials in the plane wavefunctions (2.16)(2.17) for the electron and the neutrino. The expansion is usefulbecause the exponents p x are small.5 The product of e and can therefore bewritten

ei (p e + p )x 1 i(p e + p ) x + . . . (2.18)Provided that the rst term in this expression does not vanish when performing theintegral, it can be expected to dominate, and the whole integral can be approximatedby

Afi = GF d3x f iGF M nucl

where in the lower line M nucl denotes the overlap integral between the neutron inthe parent nucleus and the proton in the daughter nucleus. The size of the quantityM nucl depends on the participating nuclei, and is known as the nuclear matrixelement .

In some particularly simple cases M nucl can be calculated analytically. In particular,if the initial-state neutron, and the nal-state proton happen to inhabit the samestate within a nucleus, the overlap integral is maximal, i.e. for those nuclei

|M nucl | = 1 .An example of a maximum overlap integral is found for the simplest case of theisolated neutron decay. 6

To complete the job of calculating we need to nd the density of states factordN dE f . The density of states for the outgoing electron can be calculated from thedensity of states inside a box7,

dN = d3p(2)3

.

Assuming spherical symmetry the angular integrals yield 4 so

dN = 4 p2dp

(2)3 .

A similar result holds for the neutrino. The states allowed by the daughter nucleusare xed by total momentum conservation, so provide no further contribution tothe density of states. The recoil energy of the heavy daughter nucleus is negligible,so conservation of energy gives

E e + E = Q,5The size of x is of order the typical nuclear size, i.e. 10 fm, which in natural units is10 fm/ (197 MeV fm)10 1 MeV 1 . The typical momenta of the out-going particles are of orderMeV, so the dot products in the exponents are of order 10 1 .6Such decays are called super-allowed.7See appendix 2.B.2 for the source of this term.

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CHAPTER 2. NUCLEAR 2.4. NUCLEAR SCATTERING

where E e is the kinetic energy of the electron. Hence the rate of decays that yieldelectrons with momenta between pe and pe + dpe is

d( pe ) = G2F |M nucl

|2 (Q E e )2

23 p2e dpe .

In the relativistic limit where E e me we can perform the integral and obtain thesimple result Q5 .

i.e. the rate depends on the fth power of the available energy.

Forbidden decays (Non examinable )

We have assumed that the rst term in ( 2.18) will dominate, but it canvanish due to selection rules. For example, if the nuclear matrix element hasodd parity the rst term vanishes, since then we are integrating the productof an odd and an even function. In that case, the next term in the seriesis required, and the reaction rate is suppressed. Such decays are said tobe rst forbidden. In general the larger the change in angular momentumrequired in the nuclear transition, the further along the series one will needto go to nd a non-zero term, and the slower will be the decay.

2.4 Nuclear Scattering

The structure of the nucleus can be probed by scattering projectiles from it. Thoseprojectiles might be protons, electrons, muons, or indeed other nuclei.

2.4.1 Cross sections

Many experiments take the form of scattering a beam of projectiles into a target.

Provided that the target is sufficiently thin that the ux is approximately constantwithin that target, the rate of any reaction W i will be proportional to the ux of incoming projectiles J (number per unit time) the number density of scatteringcentres n in the target (number per unit volume), and the width x of the target

W i = i n J x. (2.19)

The constant of proportionality i has dimensions of area. It is known as the crosssection for process i and is dened by

i = W in J x

(2.20)

We can get some feeling for why this is a useful quantity if we rewrite (2.19) as

W i = ( nAx )

N targetJ

A

P scatt

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2.4. NUCLEAR SCATTERING CHAPTER 2. NUCLEAR

We will later nd that Beta decays are mediated at very small lengthscales ( 10

18 m) by charged spin-1 force-carrying particles known asW bosons.

W d

u

e

e

W +u

d

e

e +

W +

u

e

d

e

The Feynman diagrams above show the W bosons responsible for decay, + decay, and electron capture. For probes with wavelength c,W , where

c,W =

mW cis the W boson Compton wavelength, or equivalently for probes with mo-mentum p mW , the small-distance behaviour of the interaction is notapparent. We do not resolve the W boson and instead we get what appearsto be a single four-body interaction.

where A is the area of the target. Here N target is the total number of targetsilluminated by the projectile, and the cross section can be interpreted as the effec-tive area presented to the beam per target for which a particular reaction can beexpected to occur.

The total rate of loss of beam is given by W = W i , and the corresponding totalcross section is therefore

=i

i .

We could choose to quote cross sections in units of e.g. fm2 or in natural units of GeV2 , however the most common unit used in nuclear and particle physics is theso-called barn (b) where

1 barn = 1028 m2

We can convert the barn to natural units of MeV 2 using the c conversion constantas follows

1 barn = 1028 m2

= 100 fm2 / (197 MeV fm)2

= 0 .00257 MeV2 .

The differential cross section dd is the cross section per unit solid angle of scat-tered particle. It is dened to be the rate of scattering per target per unit incomingux density per unit solid angle (d) of deected particle.

2.4.2 Resonances and the Breit-Wigner formula

Sometimes the projectile can be absorbed by the nucleus to form a compoundstate, then later reabsorbed. The time-energy uncertainty relationship of quantum

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As well as scattering experiments, the size of various nuclei can be deter-mined by other methods, including:

Shifts in the energy-levels of atomic electrons from the change of their Coulomb potential caused by the nite size of the nucleus.

Muonic equivalents of the above. Muons are about 200 times heavierthan electrons, so their Bohr raduis is about 200 times smaller. Oneobserves the series of x-rays from the atomic (muonic) transitions

The decay width can be generalised to a particle which has many different decaymodes. The rate of decay into mode i is given i . The total rate of decay isgiven by the sum over all possible decay modes

= Xi =1 ...n i .The fraction of particles that decay into nal state i , is known as the branchingratio

B = i

.

The quantity i is known as the partial width to nal state i , whereas the sumof all partial widths is known as the total width .

mechanics tells us that if a state has only a nite lifetime (of order t), then it hasan uncertaintity on its energy E given by

E t .

Using the fact that the decay rate = 1/ , and using natural units to set = 1 ,

we nd that E . (2.21)

In these units, the uncertainty in the rest-energy of a particle is equal to the rateof its decay. This means that if we take a set of identical unstable particles, andmeasure the mass of each, we will expect to get a range of values with width of order .

Long-lived intermediate states have small and hence well-dened energies. Wetend to think of these reasonably long-lived intermediate states as particles. Theneutron is an example of an unstable state that lives long enough for the wordparticle to be meaningfully applied to it.

Short-lived intermediate states have large widths and less well dened energies.When the intermediate state is so short-lived that its width is similar to its mass,then the decay is so rapid that it is no longer useful to think of it as a particle itsreally some transition through which the state happens to be momentarily passing.

We can develop these ideas more quantitatively by considering the general process

A + B O C + D. (2.22)The initial particles A and B collide to form an unstable intermediate O, which thendecays to the nal state C and D (which may or may not have the same particle

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E 0 E

The Breit-Wigner line shape.

content as the initial state). An example of a familiar process is the absorption andthen emission of a photon by an atom, with an intermediate excited atomic state,

A +

A

A + .

Alternatively the reaction could represent an inelastic nuclear interaction, for exam-ple the nuclear absorption of a neutron to form a heavier isotope followed by itsde-excitation

25 Mg + n 26 Mg 26 Mg + .Other reactions can create and annihilate other types of particle.

The reaction will proceed most rapidly when the energies of the incoming particlesare correctly tuned to the mass of the intermediate. The reaction rate will be fastestwhen the energy of A + B is equal to the rest-mass energy E O of the intermediatestate. The energy need only match E O to within the uncertainty in the energyof the intermediate.

Under the condition that the transition from A + B to C + D proceeds exclusivelyvia the intermediate state 0, of mass m0 and that the width of the intermediateis not too large ( m0), the probability for the scattering process, as a functionof total energy E takes the familiar Lorentzian shape

p(E ) 1

(E E 0)2 + 2 / 4. (2.23)

This is the same peaked shape as seen in resonances in situations involving oscil-lators, and so the excited intermediate state is often called a resonance , and theprocess is known as resonant scattering.

Taking into account density of states and ux factors, and the possibilities of decayinto multiple different nal states, the overall cross-section for the process (2.22) isgiven by the famous Breit-Wigner formula

i0f = k2

i f (E E 0)2 + 2 / 4

. (2.24)

Since excited states are very common, this is an important result not just in nuclearand particle physics, but also in any process where excitations are found. The termsin this equation are as follows:

i is the partial width of the resonance to decay to the initial state A + B

f is the partial width of the resonance to decay to the nal state C + D is the full width of the resonance at half-maximum (and equal to the sum

j j over all possible decay modes)

E is the centre-of-mass energy of the system E 0 is the characteristic rest mass energy of the resonance k is the wave-number of the incoming projectile in the centre-of-mass framewhich is equal to its momentum in natural units.

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The cross-section is non-zero at any energy, but has a sharp peak at energies E closeto the rest-mass-energy E 0 of the intermediate particle. Longer lived intermediateparticles have smaller and hence sharper peaks.

Resonant scattering experiments can tell us about the excited states of nuclei,and hence provide further information about nuclear structure and interactions.All sorts of particles which are too short-lived to travel macroscopic distances cannevertheless be created as intermediate states and studied from the properties of their Breit-Wigner peaks.

2.4.3 Nuclear scattering and form factors

When a target only slightly perturbs the wave-function of the projectile, the resultingscattering behaives rather like optical diffraction.

In optics, the properties of a microscopic aperture can be understood from thepattern obtained when light, of wavelength similar to the size of the aperture, isdiffracted by that apperture. Far from the aperture, the optical pattern observed isthe two dimensional Fourier transformation of the aperture function. This is trueeven if the optical aperture is too small to observe directly.

Now consider scattering a wave from a three-dimensional projectile. Again, theobserved diffraction pattern comes from a Fourier transform of the object, but nowthe aperture function is replaced with the potential V (x ).

In the Born approximation, which is valid for weak potentials, the amplitude f ( k )for scattering a projectile such that its change in momentum is k , is proportionalto the 3D Fourier transform of the scattering potential V ,

f ( k ) = A d3x V (x ) ei k x (2.25)where A is a normalising constant. The probability to scatter into some small angled is then proportional to |f ( k )|2 .Let us consider the scattering of a projectile of charge z from a nucleus with chargeZ and spherically symmetric local charge density (r ), centred at the origin. Thepotential at some point x is given by summing over the Coulomb potentials fromdistributed charges at all other locations x ,

V (x ) = ze2

4 0 d3x

(x )

|x x |= z d3x (x )|x x |

where in the second step we again use the relation (valid in natural units) that theelectromagnetic ne structure constant = e

2

4 0 .

Substituting this form of the potential into the Born relation ( 2.25) we nd

f ( k ) = zA d3x d3x ei ( k )x (x )|x x |.

Diffractive scattering from a slitand from an object.

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Sketch of a nuclear form factordiffraction pattern.

We may simplify this expression by dening a new variableX = x x . This changeof variables allows us to factorize the two integrals, giving the result

f ( k ) =1

Z d3x (x )eik x

Form Factor ZzA

d3X

eik X

|X | Rutherford. (2.26)

The scattering amplitude from a distributed charge is therefore equal to the productof two terms. The second term can be recognised as the Rutherford scatteringamplitude the amplitude that would be obtained from scattering from a pointcharge density (x ) = Z (x ). The second term therefore tells us nothing aboutthe internal structure of the nucleus. All of the interesting information about thenuclear structure is encapsulated in the rst term,

F nucl ( k ) =

d3x N (x ) eik x

which is known as the nuclear form factor . The form factor is the three-dimensionalFourier transform of the normalised charge density N (x ) = (x )/Z . All of the inter-esting information about the size and structure of the nucleus is found in F nucl ( k ).We will nd interesting scattering that is interesting diffraction patterns if the exponent is of order unity. For this to be true the de Broglie wave-length of the projectile must be of the same order as the nuclear size, as was noted in theintroduction to this chapter.

The mod-squared of the optical amplitude gives the intensity. Similarly it is |F |2which is important when considering the ux of scattered projectile particles. Therate at which particles are scattered into unit solid angle is given by

dN d

= |F nucl (|k |)|2dN d Rutherford

. (2.27)

This equation is more often written in terms of the differential cross section forscattering

dd

= |F nucl (|k |)|2dd Rutherford

.

By examining the form factor for particles scattered with various changes in mo-mentum | k | we can infer information about N (x ) and hence about the size andshape of the nuclear potential V (x ).

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CHAPTER 2. NUCLEAR 2.5. KEY POINTS

2.5 Key points

In natural units (appendix 2.A), = c = 1 and[Mass] = [Energy] = [Momentum] = [Time] 1 = [Distance] 1

A useful conversion constant is

c 197 MeV fm

The nuclear mass is well described by the semi-empirical mass formula

M (A, Z ) = Zm p + ( A Z )mn A + A23 +

(A 2Z )2A

+Z 2

A13 (A, Z ).

The binding energy leads to a valley of stability in the (A, Z ) plane wherethe stable nuclei lie In a reaction or decay, the Q-value is the energy released in a decay

Q = M i M f If Q > 0 the reaction is exothermic it gives out energy, whereas if Q < 0the reaction is endothermic , and energy must be supplied for it to proceed.

Alpha decay rates are dominated by quantum tunnelling through the Coulombbarrier.

Beta decay rates and electron capture are governed by the Fermi couplingconstantGF 1.17 105 GeV2

The cross section is dened by:

i = W in J x

(2.28)

The differential cross section is the cross section per unit solid angle

di

d

Cross sections for sub-atomic physics are often expressed in the unit of barns .1 barn = 1028 m2

The Breit Wigner formula for resonant scattering is

i0f = k2

i f (E E 0)2 + 2/ 4

.

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2.A. NATURAL UNITS CHAPTER 2. NUCLEAR

In elastic nuclear scattering the form factor

F (| k|) =

d3x

(x)Z

ei k x ,

is the 3D Fourier transform of the normalised charge density, and is relatedto the Rutherford scattering differential cross section by

dd

= |F nucl (|k |)|2dd Rutherford

.

2.A Natural units

In the S.I. system of units, times are measured in seconds and distances in meters.In those units the speed of light takes the value close to 3 108 ms1 .We could instead have chosen to use unit of time such that c = 1 . For examplewe could have used units in which time is measured in seconds and distance inlight-seconds. In those units the speed of light is one (one light-second per second).Using units in which c = 1 allows us to leave c out of our equations (providedwe are careful to remember the units we are working in). Such units are useful inrelativistic systems, since now the relativistic energy-momentum-mass relations aresimplied to

E = m

p = mvE 2 p2 = m2 .

So for a relativistic system setting c = 1 means that energy, mass and momentumall have the same dimensions.

Since we are interested in quantum systems, we can go further and look for units inwhich is also equal to one. In such units the energy E of a photon will be equalto its angular frequency

E = = .

Setting = 1 therefore means that the units of energy are the same as the units of inverse time. Units with = 1 imply that time (and via c = 1 distance too) musthave the same dimensions as inverse energy, E 1 .

So in our system natural units with = c = 1 we have have that all of the followingdimensions are the same:

[Mass] = [Energy] = [Momentum] = [Time] 1 = [Distance] 1

We are still free to choose a convenient unit for all of these quantities. In subatomicphysics it is common to use units of energy (or inverse energy). The nuclear energylevels have typical energies of the order of 106 electron-volts, so we shall measure

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CHAPTER 2. NUCLEAR 2.B. TOOLS

energies, momenta and masses in MeV, and lengths and times in MeV1 . At theend of a calculation we might wish to recover, for example, a real length fromone measured in our MeV1 units. To do so we can make use of the conversionfactor

c 197 MeV fmwhich tells us that one of our MeV1 length units corresponds to 197 fm where1 fm = 1015 m.

2.B Tools for cross-section calculations

2.B.1 Decays and the Fermi Golden Rule

In subatomic physics we are interested in the decays of unstable particles, suchas radioactive nuclei, muons from the atmosphere, or Higgs bosons. Using time-dependent perturbation theory in quantum mechanics it is possible to show thatthat the transition rate of an unstable state into a continuum of other states isgiven by the Fermi Golden Rule :

= 2

|V fi |2 dN dE f

, (2.29)

where

is the rate of the decay

V fi is the matrix element of the Hamiltonian coupling the initial and the nalstates dN dE f is the density of nal states.

2.B.2 Density of states

The density of states for a single particle within a cubic box with sides length a canbe calculated as follows. The plane wave solution is of form

x | exp( ik x ) .If we require periodic boundary conditions, with period a equal to the side of thebox, then the values of the wavenumber kx are constrained to kx = 2n/a forinteger n . Similar conditions hold for ky and kz . The number of momentum stateswithin some range of momentum d3p = d3k (for = 1) is therefore given by

dN = d3p(2)3 V

where V = a3 is the volume of the box.

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2.B. TOOLS CHAPTER 2. NUCLEAR

2.B.3 Fermi G.R. example

Consider the isotropic decay of a neutral spin-0 particle A into two massless daugh-

ters, B and C A B + C.The Fermi G.R. gives the decay rate (in natural units) of A as

= 2 |V fi |2 dN dE f

.

The density of nal states can be found from the allowed momenta p B of particleB.

dN = d3p B(2)3 V

When p B is xed there is no further freedom for p C since the sum of the momenta

of the two nal state particles is xed by total momentum conservation. Thisconstraint means that for the two body nal state there is no additional term in thedensity of states for p C .8 Since all decay angles are equally probable, the integralsover the angles contribute 4, leading to

= 2 |V f i |2 4 p2B(2)3

dpBdE f V .

The relativistic decay products each have momentum |p B | = E f / 2 so dp BdE f = 12 .Normalising to one unstable particle in our unit volume gives V = 1, and results ina decay rate

= 1

2 |V fi

|2 p2B

= 1

8 |V fi |2 m2A .

2.B.4 Lifetimes and decays

The number of particles remaining at time t is governed by the decay law9

dN dt

= N,where the constant is the decay rate per nucleus. The equation is easily integratedto give

N (t) = N 0 exp(t) .We can calculate the particles average proper lifetime , using the probability thatthey decay between time t and t + t

p(t) t = 1N 0

dN dt

t = exp ( t) t.8For a three-body nal state there would be terms in dN of the form d

3 p(2 ) 3 for two of the three

particles, the third again being xed by momentum conservation.9The decay law was discovered experimentally by Frederick Soddy (1877-1956). Soddy, who

had been a scholar at Merton, was also rst person to understand that radioactivity led to thetransmutation of the elements in effect making him the rst true alchemist.

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CHAPTER 2. NUCLEAR 2.B. TOOLS

The mean lifetime is then

= t=

0 t p(t) dt

0 p(t) dt= 1The decay law can be justied from precise experimental verication. In essenceit represents a statement that the decay rate is independant of the history of thenucleus, its method of preparation and its environment. These are often excellentapproximations, provided that the nucleus lives long enough that has mass m where is its decay width, and provided it it not bombarded with disruptive probes,such as high-energy strongly interacting particles.

2.B.5 The ux factor

When calculating a cross section from a rate , we need to take into accountthat for scattering from a single xed target

= W

J where J is the ux density of incoming particles. The ux density is itself given by

J = n pvwhere n p is the number density of projectiles and v is their speed. If we normalise toone incoming particle per unit volume, then n p = 1 and the cross section is simplyrelated to the rate by

= W

v

2.B.6 Luminosity

In a collider a machine which collides opposing beams of particles the rateof any particular reaction will be proportional to the cross section for that reactionand on various other parameters which depend on the machine set-up. Thoseparameters will include the number of particles in each collding bunch, their spatialdistributions, and their frequency of bunch crossings.

We can dene a parameter called the luminosity L which encapsulates all therelevant machine parameters. It is related to the rate W and the cross section by

L= W

.

For any collider the luminosity tells us the instantaneous rate of reaction for anycross section. The product of the time-integrated luminosity and the cross sectiontell us the expected count of the events of that type

N events , i = i Ldt .

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2.C. SHELL MODEL CHAPTER 2. NUCLEAR

For a machine colliding trains of counter-rotating bunches containing N 1 and N 2particles respectively at a bunch-crossing rate f , we can show that the luminosityis

L= N 1N 2f

A ,

where A is the cross-sectional area of each bunch (perpendicular to the beam di-rection).

We have assumed above that the distributions of particles within each bunch isuniform. If that is not the case (e.g. in most real experiments the beams haveapproximately Gaussian proles) then we will have to calculate the effective overlaparea A of the bunches by performing an appropriate integral.

2.C Shell Model

Non examinable

The SEMF provides a reasonable description of the binding energies of the nucleifor A > 30 but only the overall structure, not the ner details.

Differences at small A (e.g. the tightly bound isotopes 42He and 168O) are alreadyobvious in Figure 2.1. Figure 2.6 shows in more detail the difference betweenthe measured binding energy (per nucleon) and the SEMF prediction. Islands of particularly high stability that is with anomalously large B/A are clearly visiblenear some special values of N or Z :

{2, 8, 20, 28, 50, 82, 126}.These are known as the magic numbers . They correspond to congurations of nuclear shells that are precisely lled with either protons or neutrons. Evidencefor this shell structure can be found in the binding energies, excitation energies,abundances, spins, and magnetic moments. Some nuclei, such the Helium nucleus42He have magic numbers both for N and for Z . This observation goes some way toexplaining why it is that Helium nuclei are emitted by heavy particles in the processof alpha decay. The shell model gives further insight into a variety of nuclearproperties, but is beyond the scope of this course.

2.D Gamma decaysNon examinable

Gamma decays are electromagnetic transitions, and are found when excited nuclearstates relax to their ground states.

Similarly to the beta decay case, one can work out the rate using the Fermi goldenrule. If one represents the initial nuclear wave-function by a and the nal nuclear

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CHAPTER 2. NUCLEAR 2.D. GAMMA DECAYS

40 60 80 100 120 140 N

0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

B SEMF A

0 5 10 15 20 25

4

5

6

7

8 B A

N

(a)

40 60 80 100 Z0.00

0.05

0.10

0.15

0.20

B SEMF A

0 5 10 15 20 253

4

5

6

7

8

9

B A

Z

(b)

Figure 2.6: Difference between the measured binding energy (per nucleon) andthe SEMF prediction. (a) The x-axis shows the number of neutrons in the nu-cleus; curves show isotopes (same Z ). (b) The x-axis shows the number of protons in the nucleus; curves show isotones (same N ). In both cases the insetshows the binding energy per nucleon for the low-A nuclei. The magic numbers

{2, 8, 20, 28, 50, 82, 126} are marked with dashed lines.

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2.D. GAMMA DECAYS CHAPTER 2. NUCLEAR

wave-function by b, then the appropriate matrix element is found to be

f |M |i = d3x b (A J ) eik x awhere A represents the electromagnetic 4-potential and J = q P/m is the electric 4-current operator. The electromagnetic selection rules and transitions are analogousto those of atomic physics.

Further Reading

An Introduction to Nuclear Physics, W. N. Cottingham and D. A. Green-wood, 2001 for the basics

Nuclear Physics, M.G. Bowler, Pergamon press, 1973

Introductory Nuclear Physics, P.E. Hodgeson, E. Gadioli and E. GadioliErba, OUP, 2003 The BNL table of the nuclides provides good reference datahttp://www.nndc.bnl.gov/nudat2/.

Bolwer and Hodgeson et. al. are good books which go well beyond this course.

30
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Chapter 3

Hadrons

3.1 Symmetry, patterns and substructure

The proton and the neutron have rather similar masses. They are distinguished fromone another by at least their different electromagnetic interactions, since the protonis charged, while the neutron is electrically neutral, but they have identical propertiesunder the strong interaction. This provokes the question as to whether the protonand neutron might have some sort of common substructure. The substructurehypothesis can be investigated by searching for other similar patterns of multipletsof particles.

There exists a zoo of other strongly-interacting particles. Exotic particles are ob-served coming from the upper atmosphere in cosmic rays. They can also be createdin the labortatory, provided that we can create beams of sufficient energy. TheQuark Model allows us to apply a classication to those many strongly interactingstates, and to understand the constituents from which they are made.

3.1.1 Pions

The lightest strongly interacting particles are the pions (). These can be producedby ring protons with energies of order GeV into material. Different pion creationinteractions are observed to occur, such as

p + p p + p + 0 p + p p + p + + + p + n p + n + 0 + + + .

There are three different pions with charges, +1 , 0 and 1 (+ , 0 and respec-tively). In each of these pion production interactions electric charge is conserved.However some of the energy of the incident particle(s) is turned into creation of new pion particles.

m p = 938.3 MeV/c 2

mn = 939.6 MeV/c 2

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3.1. INTRODUCTION CHAPTER 3. HADRONS

The three pions have masses

m + = m = 139 .6 MeV/c 2

m 0 = 135.0 MeV/c 2 .

Again we see an interesting pattern all three pions have similar masses, in thiscase that mass is about one seventh of that of the proton or neutron.

In fact the two charged pions have exactly the same mass. This is because the +and are anti-particles of one another. Anti-particles share the same mass, buthave opposite charges. The 0 has no charge, and is its own anti-particle.

Collisions also produce negatively charged anti-protons, p.

p + p p + p + p + p.

There is also an anti-neutron n , with the same mass mn as the neutron, and whichwhich can also be produced in collisions e.g.

p + p p + p + n + n.Though the neutron has no charge it is not its own anti-particle. We can tell thetwo are different because the anti-neutron decays differently from the neutron:

n p + e + n p + e+ + .

Another piece of evidence that neutrons are not the same as anti-neutrons is thatthey do not annihilate against one another inside nuclei.

3.1.2 Baryon number conservation

In all of the reactions above, we observe that the total number of protons andneutrons less anti-protons and anti-neutrons

N ( p) + N (n) N ( p) N (n)is conserved. This rule is a special case of the conservation of baryon number ,which is a quantum number carried by protons and neutrons, but not by pions.Protons and neutrons each have baryon number +1, while their anti-particles have

baryon number -1.Baryon number conservation keeps the proton stable, since it forbids the decay of the proton to e.g. a 0 and a + each of which have baryon number of zero.Experimental lower bound on the lifetime of the proton can be made by closeobservation of large tanks of water underground, yielding

p > 1.6 1025 years.This is very much longer than the lifetime of the universe ( 1.4 1010 years)so we would expect protons created in the early universe still to be around today.Thankfully they are as you can easily verify experimentally.

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CHAPTER 3. HADRONS 3.1. INTRODUCTION

Magnetic moments Non examinable

Hints about proton and neutron substructure can also be found in their magneticdipole moments. It is a prediction of the Dirac theory that any fundamentalspin-half fermion with charge Q and mass m should have a magnetic moment

= Q 2m

.

This would predict that if the proton was a fundamental particle it would havemagnetic moment equal to the nuclear magneton

N = e 2m p

However the proton has a magnetic moment of 2.79 N , in disagreement withthe Dirac prediction for a fundamental particle. The neutron, which would haveno magnetic moment in the Dirac theory, has magnetic moment equal to -1.91N . These observations suggest that protons and neutrons are not fundamentalparticles, but are made of something smaller.

3.1.3 Delta baryons

Other groups of strongly interacting particles are also observed. Charged pions livelong enough to be made into beams, and so we can study their reactions withprotons and neutrons. Examples of reactions observed include the production anddecay of a the multiplet of particles, which are observed as resonances in thecross-sections for processes such as

+ n + n + p 0 0 + n+ + n + 0 + p+ + p ++ + + p

The four short-lived delta particles have different charges (+2, +1, 0, -1), includ-ing a double-positively charged particle, ++ . All have rest-mass-energy close to1232MeV. All are produced in charge-conserving reactions. All have spin quantumnumber s = 3 / 2. They decay in a very short time of order 1022 s so can-not be observed as propagating particles. Instead they are observed as resonances.From the width of the resonance we can infer the lifetime of the correspondingparticle.

From the reactions above we can see that all four deltas must have baryon number+1, in order to conserve baryon number throughout each reaction these particles are baryons. Conservation of baryon number implies that none of the particles can be anti-particles of one another they must have separate anti-particles, which would be created in reactions with anti-protons or anti-neutrons.

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3.2. LINEAR ACCELERATORS CHAPTER 3. HADRONS

dpdt

= Q[E + v B ]

Reminder of the Lorentzforce law.

The linear accelerator injectorto the CERN protonsynchrotron. c CERN

3.2 Accelerating protons linear accelerators

We needed protons with kinetic energy of order GeV to perform these experiments.Unless we are willing to wait for the occasional high-energy cosmic ray coming fromspace, well need to accelerate them. Since the magnetic eld changes only thedirection of p , it is the electrical eld which is used to increase their energy of theparticles.

The problem we encounter if we try to use a constant electric eld to do ouracceleration is that to get these very high energies (of order GeV) we need to passthem through an enormous potential difference of order 109 volts. Van der Graaff generators can reach potentials of order ten million volts, but then tend to breakdown because of electrical discharge (sparking) to nearby objects. For the particlecreation reactions above, were looking for about two orders of magnitude moreenergy than this.

We can get around the limitations of a static potential difference by realizing thatonly that only the local E eld needs to be aligned along v , and only during theperiod in which the particle is in that particular part of space.

We can use then use time-varying electric elds. In the margin is a picture of a linearaccelerator or linac . In this device we have a series of cylindrical electrodes withholes through the middle of each which allow the beam to pass through. Electrodesare attached alternately to either pole of an alternating potential. The particles areaccelerated in bunches. As each bunch travels along we reverse the potential whilethe bunch is inside electrode (where this is no eld). We can then ensure that whenthe bunch is in the gap between the electrodes the eld is always in the correct

direction to keep it accelerating.

The oscillating potential on t

subatomic lecture notes oxford

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