Calculate the nuclear binding energy of 5525mn in joules.

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To Claire

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Contents

Preface to the First Edition xiii

Preface to the Second Edition xv

Notes xvii

1 Basic Concepts 1 1.1 History 1

1.1.1 The Origins of Nuclear Physics 1 1.1.2 The Emergence of Particle Physics: the Standard Model and

Hadrons 3 1.2 Relativity and Antiparticles 6 1.3 Space-Time Symmetries and Conservation Laws 8

1.3.1 Parity 9 1.3.2 Charge Conjugation 10 1.3.3 Time Reversal 12

1.4 Interactions and Feynman Diagrams 14 1.4.1 Interactions 14 1.4.2 Feynman Diagrams 15

1.5 Particle Exchange: Forces and Potentials 17 1.5.1 Range of Forces 17 1.5.2 The Yukawa Potential 19

1.6 Observable Quantities: Cross-sections and Decay Rates 20 1.6.1 Amplitudes 20 1.6.2 Cross-sections 22 1.6.3 Unstable States 26

1.7 Units: Length, Mass and Energy 28 Problems 29

2 Nuclear Phenomenology 31 2.1 Mass Spectroscopy 31

2.1.1 Deflection Spectrometers 32 2.1.2 Kinematic Analysis 33 2.1.3 Penning Trap Measurements 34

2.2 Nuclear Shapes and Sizes 38 2.2.1 Charge Distribution 39 2.2.2 Matter Distribution 43

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2.3 Semi-Empirical Mass Formula: the Liquid Drop Model 45 2.3.1 Binding Energies 45 2.3.2 Semi-empirical Mass Formula 47

2.4 Nuclear Instability 52 2.5 Radioactive Decay 53 2.6 β–Decay Phenomenology 56

2.6.1 Odd-mass Nuclei 56 2.6.2 Even-mass Nuclei 58

2.7 Fission 59 2.8 γ Decays 62 2.9 Nuclear Reactions 63

Problems 67

3 Particle Phenomenology 71 3.1 Leptons 71

3.1.1 Lepton Multiplets and Lepton Numbers 71 3.1.2 Universal Lepton Interactions: the Number of Neutrinos 74 3.1.3 Neutrinos 76 3.1.4 Neutrino Mixing and Oscillations 77 3.1.5 Oscillation Experiments and Neutrino Masses 80 3.1.6 Lepton Numbers Revisited 86

3.2 Quarks 87 3.2.1 Evidence for Quarks 87 3.2.2 Quark Generations and Quark Numbers 90

3.3 Hadrons 92 3.3.1 Flavour Independence and Charge Multiplets 92 3.3.2 Quark Model Spectroscopy 96 3.3.3 Hadron Magnetic Moments and Masses 101 Problems 107

4 Experimental Methods 109 4.1 Overview 109 4.2 Accelerators and Beams 111

4.2.1 DC Accelerators 111 4.2.2 AC Accelerators 112 4.2.3 Neutral and Unstable Particle Beams 119

4.3 Particle Interactions with Matter 120 4.3.1 Short-range Interactions with Nuclei 120 4.3.2 Ionization Energy Losses 122 4.3.3 Radiation Energy Losses 124 4.3.4 Interactions of Photons in Matter 125

4.4 Particle Detectors 127 4.4.1 Gas Detectors 128 4.4.2 Scintillation Counters 132 4.4.3 Semiconductor Detectors 133

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4.4.4 Čerenkov Counters 134 4.4.5 Calorimeters 135

4.5 Multi-Component Detector Systems 138 Problems 143

5 Quark Dynamics: The Strong Interaction 147 5.1 Colour 147 5.2 Quantum Chromodynamics (QCD) 149 5.3 Heavy Quark Bound States 151 5.4 The Strong Coupling Constant and Asymptotic Freedom 156 5.5 Quark-Gluon Plasma 160 5.6 Jets and Gluons 161 5.7 Colour Counting 163 5.8 Deep Inelastic Scattering and Nucleon Structure 165

5.8.1 Scaling 165 5.8.2 Quark-Parton Model 167 5.8.3 Scaling Violations and Structure Functions 170 Problems 173

6 Weak Interactions and Electroweak Unification 177 6.1 Charged and Neutral Currents 177 6.2 Symmetries of the Weak Interaction 178 6.3 Spin Structure of the Weak Interactions 182

6.3.1 Neutrinos 182 6.3.2 Particles with Mass: Chirality 184

6.4 W ± and Z0 Bosons 187 6.5 Weak Interactions of Hadrons: Charged Currents 188

6.5.1 Semileptonic Decays 189 6.5.2 Selection Rules 192 6.5.3 Neutrino Scattering 195

6.6 Meson Decays and CP Violation 197 6.6.1 CP Invariance 197 6.6.2 CP Violation in K 0L Decay 199 6.6.3 CP Violation in B Decays 201 6.6.4 Flavour Oscillations 203 6.6.5 CP Violation and the Standard Model 205

6.7 Neutral Currents and the Unified Theory 207 6.7.1 Electroweak Unification 207 6.7.2 The Z0 Vertices and Electroweak Reactions 210 Problems 213

7 Models and Theories of Nuclear Physics 217 7.1 The Nucleon-Nucleon Potential 217 7.2 Fermi Gas Model 220 7.3 Shell Model 222

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7.3.1 Shell Structure of Atoms 222 7.3.2 Nuclear Magic Numbers 224 7.3.3 Spins, Parities and Magnetic Dipole Moments 227 7.3.4 Excited States 229

7.4 Non-Spherical Nuclei 231 7.4.1 Electric Quadrupole Moments 231 7.4.2 Collective Model 234

7.5 Summary of Nuclear Structure Models 234 7.6 α Decay 235 7.7 β Decay 238

7.7.1 Fermi Theory 239 7.7.2 Electron and Positron Momentum Distributions 240 7.7.3 Selection Rules 242 7.7.4 Applications of Fermi Theory 243

7.8 γ Emission and Internal Conversion 247 7.8.1 Selection Rules 247 7.8.2 Transition Rates 248 Problems 250

8 Applications of Nuclear Physics 253 8.1 Fission 253

8.1.1 Induced Fission and Chain Reactions 253 8.1.2 Fission Reactors 257

8.2 Fusion 262 8.2.1 Coulomb Barrier 262 8.2.2 Fusion Reaction Rates 264 8.2.3 Stellar Fusion 266 8.2.4 Fusion Reactors 268

8.3 Nuclear Weapons 271 8.3.1 Fission Devices 273 8.3.2 Fission/Fusion Devices 275

8.4 Biomedical Applications 278 8.4.1 Radiation and Living Matter 278 8.4.2 Medical Imaging Using Ionizing Radiation 283 8.4.3 Magnetic Resonance Imaging 289 Problems 294

9 Outstanding Questions and Future Prospects 297 9.1 Overview 297 9.2 Hadrons and Nuclei 298

9.2.1 Hadron Structure and the Nuclear Environment 298 9.2.2 Nuclear Structure 300 9.2.3 Nuclear Synthesis 302 9.2.4 Symmetries and the Standard Model 303

9.3 The Origin of Mass: the Higgs Boson 305 9.3.1 Theoretical Background 305

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9.3.2 Experimental Searches 307 9.4 The Nature of the Neutrino 311

9.4.1 Dirac or Majorana? 311 9.4.2 Neutrinoless Double β Decay 312

9.5 Beyond the Standard Model: Unification Schemes 315 9.5.1 Grand Unification 315 9.5.2 Supersymmetry 318 9.5.3 Strings and Things 321

9.6 Particle Astrophysics 322 9.6.1 Neutrino Astrophysics 323 9.6.2 The Early Universe: Dark Matter and Neutrino Masses 327 9.6.3 Matter-Antimatter Asymmetry 330

9.7 Nuclear Medicine 331 9.8 Power Production and Nuclear Waste 333

Appendix A Some Results in Quantum Mchanics 339 A.1 Barrier Penetration 339 A.2 Density of States 341 A.3 Perturbation Theory and the Second Golden Rule 343 A.4 Isospin Formalism 345

A.4.1 Isospin Operators and Quark States 345 A.4.2 Hadron States 347

Appendix B Relativistic Kinematics 351 B.1 Lorentz Transformations and Four-Vectors 351 B.2 Frames of Reference 353 B.3 Invariants 355

Problems 358

Appendix C Rutherford Scattering 361 C.1 Classical Physics 361 C.2 Quantum Mechanics 364

Problems 365

Appendix D Gauge Theories 367 D.1 Gauge Invariance and the Standard Model 367

D.1.1 Electromagnetism and the Gauge Principle 368 D.1.2 The Standard Model 370

D.2 Particle Masses and the Higgs Field 372

Appendix E Data 377 E.1 Physical Constants and Conversion Factors 377 E.2 Tables of Particle Properties 378

E.2.1 Gauge Bosons 378 E.2.2 Leptons 379 E.2.3 Quarks 379

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E.2.4 Low-Lying Baryons 380 E.2.5 Low-Lying Mesons 382

E.3 Tables of Nuclear Properties 384 E.3.1 Properties of Naturally Occurring Isotopes 384 E.3.2 The Periodic Table 392

Appendix F Solutions to Problems 393

References 437

Bibliography 441

Index 443

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Preface to the First Edition

It is common practice to teach nuclear physics and particle physics together in an intro- ductory course and it is for such a course that this book has been written. The material presented is such that different selections can be made for a short course of about 25–30 lectures depending on the lecturer’s preferences and the students’ backgrounds. On the latter, students should have taken a first course in quantum physics, covering the tradi- tional topics in non-relativistic quantum mechanics and atomic physics. A few lectures on relativistic kinematics would also be useful, but this is not essential, as the necessary background is given in an appendix and is only used in a few places in the book. I have not tried to be rigorous, or present proofs of all the statements in the text. Rather, I have taken the view that it is more important that students see an overview of the subject, which for many, possibly the majority, will be the only time they study nuclear and particle physics. For future specialists, the details will form part of more advanced courses. Nevertheless, space restrictions have still meant that it has been necessarily to make a choice of topics and doubtless other, equally valid, choices could have been made. This is particularly true in Chapter 8, which deals with applications of nuclear physics, where I have chosen just three major areas to discuss. Nuclear and particle physics have been, and still are, very important parts of the entire subject of physics and its practitioners have won an impressive number of Nobel Prizes. For historical interest, I have noted in the footnotes many of these awards for work related to the field.

Some parts of the book dealing with particle physics owe much to a previous book, Particle Physics, written with Graham Shaw of Manchester University, and I am grateful to him and the publisher, John Wiley & Sons, Ltd, for permission to adapt some of that material for use here. I also thank Colin Wilkin for comments on all the chapters of the book; to David Miller and Peter Hobson for comments on Chapter 4; and to Bob Speller for comments on the medical physics section of Chapter 8. If errors or misunderstandings still remain (and any such are of course due to me alone) I would be grateful to hear about them. I have set up a website (www.hep.ucl.ac.uk/∼brm/npbook.html) where I will post any corrections and comments.

Brian R Martin January 2006

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Preface to the Second Edition

The structure of this edition follows closely that of the first edition. Changes include the rearrangement of some sections and the rewriting and/or expansion of others where, on reflection, I think more explanation is required, or where the clarity could be improved; the inclusion of a number of entirely new sections and two new appendices; modifications to the notation in places to improve consistency of style through the book; the inclusion of additional problems; and updating the text, where appropriate. I have also taken the opportunity to correct misprints and errors that were in the original printing of the first edition, most of which have already been corrected in later reprints of that edition. I would like to thank those correspondents who have brought these to my attention, par- ticularly Roelof Bijker of the Universidad Nacional Autonoma de Mexico, Hans Fynbo of the University of Aarhus, Denmark and Michael Marx of the Stony Brook campus of the State University of New York. I will continue to maintain the book’s website, (www.hep.ucl.ac.uk/∼brm/npbook.html) where any future comments and corrections will be posted.

Finally, a word about footnotes: readers have always had strong views about these, (‘Notes are often necessary, but they are necessary evils’ – Samuel Johnson), so in this book they are designed to provide ‘non-essential’ information only. Thus, for those readers who prefer not to have the flow disrupted, ignoring the footnotes should not detract from understanding the text.

Brian R. Martin November 2008

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Notes

References

References are referred to in the text in the form of a name and date, for example Jones (1997), with a list of references with full publication details given at the end of the book.

Data

It is common practice for books on nuclear and particle physics to include tables of data (masses, decay modes, lifetimes etc.) and such a collection is given in Appendix E. Among other things, they will be useful in solving the problems provided for most chapters. However, I have kept the tables to a minimum, because very extensive tabulations are now readily available at the ‘click of a mouse’ from a number of sites and it is educationally useful for students to get some familiarity with such sources of data.

For particle physics, a comprehensive compilation of data, plus brief critical reviews of a number of current topics, may be found in the bi-annual publications of the Particle Data Group (PDG). The 2008 edition of their definitive Review of Particle Properties is referred to as Amsler et al. (2008) in the references. The PDG Review is available online at http://pdg.lbl.gov and this site also contains links to other sites where compilations of particle data may be found.

Data for nuclear physics are available from a number of sources. Examples are: the Berkeley Laboratory Isotopes Project (http://ie.lbl.gov/education/isotopes.htm); the Na- tional Nuclear Data Center (NNDC), based at Brookhaven National Laboratory, USA (http://www.nndc.bnl.gov); the Nuclear Data Centre of the Japan Atomic Energy Research Institute (http://wwwndc.tokai-sc.jaea.go.jp/NuC); and the Nuclear Data Evaluation Lab- oratory of the Korea Atomic Energy Research Institute (http://atom.kaeri.re.kr). All four sites have links to other data compilations.

Problems

Problems are provided for Chapters 1–8 and some Appendices; their solutions are given in Appendix F. The problems are an integral part of the text. They are mainly numerical and require values of physical constants that are given in Appendix E. Some also require data that may be found in the other tables in Appendix E and in the sites listed above.

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xviii Notes

Illustrations

Some illustrations in the text have been adapted from, or are based on, diagrams that have been published elsewhere. In a few cases they have been reproduced exactly as previously published. I acknowledge, with thanks, permission to use such illustrations from the relevant copyright holders, as stated in the captions. Full bibliographic details of sources are given in the list of references on page 437.

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1 Basic Concepts

1.1 History

Although this book will not follow a strictly historical development, to ‘set the scene’ this first chapter will start with a brief review of the most important discoveries that led to the separation of nuclear physics from atomic physics as a subject in its own right and later work that in its turn led to the emergence of particle physics from nuclear physics.1

1.1.1 The Origins of Nuclear Physics

Nuclear physics as a subject distinct from atomic physics could be said to date from 1896, the year that Becquerel observed that photographic plates were being fogged by an unknown radiation emanating from uranium ores. He had accidentally discovered radioactivity: the fact that some nuclei are unstable and spontaneously decay. The name was coined by Marie Curie two years later to distinguish this phenomenon from induced forms of radiation. In the years that followed, radioactivity was extensively investigated, notably by the husband and wife team of Pierre and Marie Curie, and by Rutherford and his collaborators,2 and it was established that there were two distinct types of radiation involved, named by Rutherford α and β rays. We know now that α rays are bound states of two protons and two neutrons (we will see later that they are the nuclei of helium atoms) and β rays are electrons. In 1900 a third type of decay was discovered by Villard that involved the emission of photons, the quanta of electromagnetic radiation, referred to in this context as γ rays. These historical names are still commonly used.

1 An interesting account of the early period, with descriptions of the personalities involved, is given in Segrè (1980). An overview of the later period is given in Chapter 1 of Griffiths (1987). 2 The 1903 Nobel Prize in Physics was awarded jointly to Henri Becquerel for his discovery and to Pierre and Marie Curie for their subsequent research into radioactivity. Ernest Rutherford had to wait until 1908, when he was awarded the Nobel Prize in Chemistry for his ‘investigations into the disintegration of the elements and the chemistry of radioactive substances’.

Nuclear and Particle Physics: An Introduction, Second Edition Brian R. Martin C© 2009 John Wiley & Sons, Ltd

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2 Nuclear and Particle Physics

At about the same time as Becquerel’s discovery, J.J. Thomson was extending the work of Perrin and others on the radiation that had been observed to occur when an electric field was established between electrodes in an evacuated glass tube and in 1897 he was the first to definitively establish the nature of these ‘cathode rays’. We now know the emanation consists of free electrons, (the name ‘electron’ had been coined in 1894 by Stoney) denoted e− (the superscript denotes the electric charge) and Thomson measured their mass and charge.3 The view of the atom at that time was that it consisted of two components, with positive and negative electric charges, the latter now being the electrons. Thomson suggested a model where the electrons were embedded and free to move in a region of positive charge filling the entire volume of the atom – the so-called ‘plum pudding model’.

This model could account for the stability of atoms, but could not account for the discrete wavelengths observed in the spectra of light emitted from excited atoms. Neither could it explain the results of a classic series of experiments started in 1911 by Rutherford and continued by his collaborators, Geiger and Marsden. These consisted of scattering α particles by very thin gold foils. In the Thomson model, most of the α particles would pass through the foil, with only a few suffering deflections through small angles. Rutherford suggested they look for large-angle scattering and indeed they found that some particles were scattered through very large angles, even greater than 90 degrees. Rutherford showed that this behaviour was not due to multiple small-angle deflections, but could only be the result of the α particles encountering a very small positively charged central nucleus. (The reason for these two different behaviours is discussed in Appendix C.)

To explain the results of these experiments Rutherford formulated a ‘planetary’ model, where the atom was likened to a planetary system, with the electrons (the ‘planets’) occupying discrete orbits about a central positively charged nucleus (the ‘Sun’). Because photons of a definite energy would be emitted when electrons moved from one orbit to another, this model could explain the discrete nature of the observed electromagnetic spectra when excited atoms decayed. In the simplest case of hydrogen, the nucleus is a single proton (p) with electric charge +e, where e is the magnitude of the charge on the electron,4 orbited by a single electron. Heavier atoms were considered to have nuclei consisting of several protons. This view persisted for a long time and was supported by the fact that the masses of many naturally occurring elements are integer multiples of a unit that is about 1 % smaller than the mass of the hydrogen atom. Examples are carbon and nitrogen, with masses of 12.0 and 14.0 in these units. But it could not explain why not all atoms obeyed this rule. For example, chlorine has a mass of 35.5 in these units. However, about the same time, the concept of isotopism (a name coined by Soddy) was conceived. Isotopes are atoms whose nuclei have different masses, but the same charge. Naturally occurring elements were postulated to consist of a mixture of different isotopes, giving rise to the observed masses.5

3 J.J. Thomson received the 1906 Nobel Prize in Physics for his discovery. A year earlier, Philipp von Lenard had received the Physics Prize for his work on cathode rays. 4 Why the charge on the proton should have exactly the same magnitude as that on the electron is a puzzle of very long-standing, the solution to which is suggested by some as yet unproven, but widely believed, theories of particle physics that will be briefly discussed in Section 9.5.1. 5 Frederick Soddy was awarded the 1921 Nobel Prize in Chemistry for his work on isotopes.

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Basic Concepts 3

The explanation of isotopes had to wait twenty years until a classic discovery by Chadwick in 1932. His work followed earlier experiments by Irène Curie (the daugh- ter of Pierre and Marie Curie) and her husband Frédéric Joliot.6 They had observed that neutral radiation was emitted when α particles bombarded beryllium and later work had studied the energy of protons emitted when paraffin was exposed to this neutral radiation. Chadwick refined and extended these experiments and demonstrated that they implied the existence of an electrically neutral particle of approximately the same mass as the proton. He had discovered the neutron (n) and in so doing had produced almost the final ingredient for understanding nuclei.7

There remained the problem of reconciling the planetary model with the observation of stable atoms. In classical physics, the electrons in the planetary model would be constantly accelerating and would therefore lose energy by radiation, leading to the collapse of the atom. This problem was solved by Bohr in 1913. He applied the newly emerging quantum theory and the result was the now well-known Bohr model of the atom. Refined modern versions of this model, including relativistic effects described by the Dirac equation (the relativistic analogue of the Schrödinger equation that applies to electrons), are capable of explaining the phenomena of atomic physics. Later workers, including Heisenberg, another of the founders of quantum theory, applied quantum mechanics to the nucleus, now viewed as a collection of neutrons and protons, collectively called nucleons. In this case however, the force binding the nucleus is not the electromagnetic force that holds electrons in their orbits, but is a short-range8 force whose magnitude is independent of the type of nucleon, proton or neutron (i.e. charge-independent). This binding interaction is called the strong nuclear force.

These ideas still form the essential framework of our understanding of the nucleus today, where nuclei are bound states of nucleons held together by a strong charge-independent short-range force. Nevertheless, there is still no single theory that is capable of explaining all the data of nuclear physics and we shall see that different models are used to interpret different classes of phenomena.

1.1.2 The Emergence of Particle Physics: the Standard Model and Hadrons

By the early 1930s, the nineteenth-century view of atoms as indivisible elementary particles had been replaced and a larger group of physically smaller entities now enjoyed this status: electrons, protons and neutrons. To these we must add two electrically neutral particles: the photon (γ ) and the neutrino (ν). The photon had been postulated by Planck in 1900 to explain black-body radiation, where the classical description of electromagnetic radiation led to results incompatible with experiments.9 The neutrino was postulated by Pauli in 193010 to explain the apparent nonconservation of energy observed in the decay products

6 Irène Curie and Frédéric Joliot received the 1935 Nobel Prize in Chemistry for ‘synthesizing new radioactive elements’. 7 James Chadwick received the 1935 Nobel Prize in Physics for his discovery of the neutron. 8 The concept of range will be discussed in more detail in Section 1.5.1, but for the present it may be taken as the effective distance beyond which the force is insignificant. 9 X-rays had already been observed by Röntgen in 1895 (for which he received the first Nobel Prize in Physics in 1901) and γ -rays were seen by Villard in 1900, but it was Max Planck who first made the startling suggestion that electromagnetic energy was quantized. For this he was awarded the 1918 Nobel Prize in Physics. Many years later, he said that his hypothesis was an ‘act of desperation’ as he had exhausted all other possibilities. 10 The name was later given by Fermi and means ‘little neutron’.

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4 Nuclear and Particle Physics

of some unstable nuclei where β rays are emitted, the so-called β decays. Prior to Pauli’s suggestion, β decay had been viewed as a parent nucleus decaying to a daughter nucleus and an electron. As this would be a two-body decay, it would imply that the electron would have a unique momentum, whereas experiments showed that the electron actually had a momentum spectrum. Pauli’s hypothesis of a third particle (the neutrino) in the final state solved this problem, as well as a problem with angular momentum conservation, which was apparently also violated if the decay was two-body. The β-decay data implied that the neutrino mass was very small and was compatible with the neutrino being massless.11 It took more than 25 years before Pauli’s hypothesis was confirmed by Reines and Cowan in a classic experiment in 1956 that detected free neutrinos from β decay.12

The 1950s also saw technological developments that enabled high-energy beams of particles to be produced in laboratories. As a consequence, a wide range of controlled scattering experiments could be performed and the greater use of computers meant that sophisticated analysis techniques could be developed to handle the huge quantities of data that were being produced. By the 1960s this had resulted in the discovery of a very large number of unstable particles with very short lifetimes and there was an urgent need for a theory that could make sense of all these states. This emerged in the mid 1960s in the form of the so-called quark model, first suggested by Gell-Mann, and independently and simultaneously by Zweig, who postulated that the new particles were bound states of three families of more fundamental physical particles.