Nuclear and Particle Physics PDF

 Nuclear and Particle Physics Nuclear and Particle Physics B. R. Martin # 2006 John Wiley & Sons, Ltd. ISBN: 0-470-01999-9 Nuclear and Particle Physics B. R. Martin Department of Physics and Astronomy, University College London Copyright # 2006 John W...

 Nuclear and Particle Physics Nuclear and Particle Physics B. R. Martin # 2006 John Wiley & Sons, Ltd. ISBN: 0-470-01999-9 Nuclear and Particle Physics B. R. Martin Department of Physics and Astronomy, University College London Copyright # 2006 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. 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Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop # 02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging-in-Publication Data Martin, B. R. (Brian Robert) Nuclear and particle physics/B. R. Martin. p. cm. ISBN-13: 978-0-470-01999-3 (HB) ISBN-10: 0-470-01999-9 (HB) ISBN-13: 978-0-470-02532-1 (pbk.) ISBN-10: 0-470-02532-8 (pbk.) 1. Nuclear physics–Textbooks. 2. Particle physics–Textbooks. I. Title. QC776.M34 2006 539.70 2–dc22 2005036437 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13 978-0 470 01999 9 (HB) ISBN-10 0 470 01999 9 (HB) 978-0 470 02532 8 (PB) 0 470 02532 8 (PB) Typeset in 10.5/12.5pt Times by Thomson Press (India) Limited, New Delhi Printed and bound in Great Britain by Antony Rowe Ltd., Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. To Claire Contents Preface xi Notes xiii Physical Constants and Conversion Factors xv 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 4 1.2 Relativity and antiparticles 7 1.3 Symmetries and conservation laws 9 1.3.1 Parity 10 1.3.2 Charge conjugation 12 1.4 Interactions and Feynman diagrams 13 1.4.1 Interactions 13 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 21 1.6.2 Cross-sections 23 1.6.3 Unstable states 27 1.7 Units: length, mass and energy 29 Problems 30 2 Nuclear Phenomenology 33 2.1 Mass spectroscopy and binding energies 33 2.2 Nuclear shapes and sizes 37 2.2.1 Charge distribution 37 2.2.2 Matter distribution 42 2.3 Nuclear instability 45 2.4 Radioactive decay 47 2.5 Semi-empirical mass formula: the liquid drop model 50 2.6 -decay phenomenology 55 2.6.1 Odd-mass nuclei 55 2.6.2 Even-mass nuclei 58 viii CONTENTS 2.7 Fission 59 2.8 -decays 62 2.9 Nuclear reactions 62 Problems 67 3 Particle Phenomenology 71 3.1 Leptons 71 3.1.1 Lepton multiplets and lepton numbers 71 3.1.2 Neutrinos 74 3.1.3 Neutrino mixing and oscillations 76 3.1.4 Neutrino masses 79 3.1.5 Universal lepton interactions – the number of neutrinos 84 3.2 Quarks 86 3.2.1 Evidence for quarks 86 3.2.2 Quark generations and quark numbers 89 3.3 Hadrons 92 3.3.1 Flavour independence and charge multiplets 92 3.3.2 Quark model spectroscopy 96 3.3.3 Hadron masses and magnetic moments 102 Problems 108 4 Experimental Methods 111 4.1 Overview 111 4.2 Accelerators and beams 113 4.2.1 DC accelerators 113 4.2.2 AC accelerators 115 4.2.3 Neutral and unstable particle beams 122 4.3 Particle interactions with matter 123 4.3.1 Short-range interactions with nuclei 123 4.3.2 Ionization energy losses 125 4.3.3 Radiation energy losses 128 4.3.4 Interactions of photons in matter 129 4.4 Particle detectors 131 4.4.1 Gas detectors 131 4.4.2 Scintillation counters 137 4.4.3 Semiconductor detectors 138 4.4.4 Particle identification 139 4.4.5 Calorimeters 142 4.5 Layered detectors 145 Problems 148 5 Quark Dynamics: the Strong Interaction 151 5.1 Colour 151 5.2 Quantum chromodynamics (QCD) 153 5.3 Heavy quark bound states 156 5.4 The strong coupling constant and asymptotic freedom 160 5.5 Jets and gluons 164 5.6 Colour counting 166  CONTENTS ix 5.7 Deep inelastic scattering and nucleon structure 168 Problems 177 6 Electroweak Interactions 181 6.1 Charged and neutral currents 181 6.2 Symmetries of the weak interaction 182 6.3 Spin structure of the weak interactions 186 6.3.1 Neutrinos 187 6.3.2 Particles with mass: chirality 189 6.4 W and Z0 bosons 192 6.5 Weak interactions of hadrons 194 6.5.1 Semileptonic decays 194 6.5.2 Neutrino scattering 198 6.6 Neutral meson decays 201 6.6.1 CP violation 202 6.6.2 Flavour oscillations 206 6.7 Neutral currents and the unified theory 208 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 223 7.3.1 Shell structure of atoms 223 7.3.2 Nuclear magic numbers 225 7.3.3 Spins, parities and magnetic dipole moments 228 7.3.4 Excited states 230 7.4 Non-spherical nuclei 232 7.4.1 Electric quadrupole moments 232 7.4.2 Collective model 236 7.5 Summary of nuclear structure models 236 7.6 -decay 238 7.7 -decay 242 7.7.1 Fermi theory 242 7.7.2 Electron momentum distribution 244 7.7.3 Kurie plots and the neutrino mass 246 7.8 -emission and internal conversion 248 7.8.1 Selection rules 248 7.8.2 Transition rates 250 Problems 252 8 Applications of Nuclear Physics 255 8.1 Fission 255 8.1.1 Induced fission – fissile materials 255 8.1.2 Fission chain reactions 258 8.1.3 Nuclear power reactors 260 8.2 Fusion 266 8.2.1 Coulomb barrier 266 x CONTENTS 8.2.2 Stellar fusion 267 8.2.3 Fusion reaction rates 270 8.2.4 Fusion reactors 273 8.3 Biomedical applications 278 8.3.1 Biological effects of radiation: radiation therapy 278 8.3.2 Medical imaging using radiation 282 8.3.3 Magnetic resonance imaging 290 Problems 294 9 Outstanding Questions and Future Prospects 297 9.1 Particle physics 297 9.1.1 The Higgs boson 297 9.1.2 Grand unification 300 9.1.3 Supersymmetry 304 9.1.4 Particle astrophysics 307 9.2 Nuclear physics 315 9.2.1 The structure of hadrons and nuclei 316 9.2.2 Quark–gluon plasma, astrophysics and cosmology 320 9.2.3 Symmetries and the standard model 323 9.2.4 Nuclear medicine 324 9.2.5 Power production and nuclear waste 326 Appendix A: Some Results in Quantum Mechanics 331 A.1 Barrier penetration 331 A.2 Density of states 333 A.3 Perturbation theory and the Second Golden Rule 335 Appendix B: Relativistic Kinematics 339 B.1 Lorentz transformations and four-vectors 339 B.2 Frames of reference 341 B.3 Invariants 344 Problems 345 Appendix C: Rutherford Scattering 349 C.1 Classical physics 349 C.2 Quantum mechanics 352 Problems 354 Appendix D: Solutions to Problems 355 References 393 Bibliography 397 Index 401 Preface It is common practice to teach nuclear physics and particle physics together in an introductory course and it is for such a course that this book has been written. The material is presented so 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 traditional 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 appendix B 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. Never- theless, space restrictions have still meant that it has been necessary to make a choice of topics covered 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 the 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 and Sons, 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, David Miller and Peter Hobson for comments on Chapter 4 and 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 Notes References References are referred to in the text in the form Ab95, where Ab is the start of the first author’s surname and 1995 is the year of publication. A list of references with full publication details is given at the end of the book. Data Data for particle physics may be obtained from the biannual publications of the Particle Data Group (PDG) and the 2004 edition of the PDG definitive Review of Particle Properties is given in Ei04. The PDG Review is also available at http://pdg.lbl.gov and this site contains links to other sites where compilations of particle data may be found. Nuclear physics data are available from a number of sources. Examples are: the combined Isotopes Project of the Lawrence Berkeley Laboratory, USA, and the Lund University Nuclear Data WWW Service, Sweden (http://ie.lbl.gov/toi.html), the National Nuclear Data Center (NNDC) based at Brookhaven National Laboratory, USA (http://www.nndc.bnl.gov), and the Nuclear Data Centre of the Japan Atomic Energy Research Institute (http://www.nndc.tokai.jaeri.go.jp). All three sites have extensive links to other data compilations. It is important that students have some familiarity with these data compilations. Problems Problems are provided for all chapters and appendices except Chapter 9 and Appendices A and D. They are an integral part of the text. The problems are mainly numerical and require values of physical constants that are given in a table following these notes. A few also require input data that may be found in the references given above. Solutions to all the problems are given in Appendix D. xiv NOTES Illustrations Some illustrations in the text have been adapted from, or are loosely based on, diagrams that have been published elsewhere. In a few cases they have been reproduced exactly as previously published. In all cases this is stated in the captions. I acknowledge, with thanks, permission to use such illustrations from the relevant copyright holders. Physical Constants and Conversion Factors Quantity Symbol Value Speed of light in vacuum c 2:998
108 ms1 Planck’s constant h 4:136
1024 GeV s h  h=2  6:582
1025 GeV s  hc 1:973
1016 GeV m hcÞ2 ð 3:894
1031 GeV2 m2 electron charge (magnitude) e 1:602
1019 C Avogadro’s number NA 6:022
1026 kg-mole1 Boltzmann’s constant kB 8:617
1011 MeV K1 electron mass me 0:511 MeV=c2 proton mass mp 0:9383 GeV=c2 neutron mass mn 0:9396 GeV=c2 W boson mass MW 80:43 GeV=c2 Z boson mass MZ 91:19 GeV=c2 atomic mass unit 1 u  ð12 mass12 C atomÞ 931:494 MeV=c2 Bohr magneton B  e h=2me 5:788
1011 MeV T1 Nuclear magneton N  e h=2mp 3:152
1014 MeV T1 gravitational constant GN hcðGeV=c2 Þ2 6:709
1039  fine structure constant   e2 =4"0  hc 7:297
103 ¼ 1=137:04 Fermi coupling constant hcÞ3 GF =ð 1:166
105 GeV2 strong coupling constant s ðMZ c2 Þ 0.119 1 eV ¼ 1:602
1019 J 1 eV=c2 ¼ 1:783
1036 kg 15 1 fermi ¼ 1 fm  10 m 1 barn ¼ 1 b  1028 m2 1 Tesla ¼ 1T ¼ 0:561
1030 MeV=c2 C1 s1 1 year ¼ 3:1536
107 s 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 Henri 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. In the years that followed, the phenomenon was extensively investigated, notably by the husband and wife team of Pierre and Marie Curie and by Ernest Rutherford and his collaborators,2 and it was established that there were three distinct types of radiation involved: these were named (by Rutherford) -, - and -rays. We know now that -rays are bound states of two protons and two neutrons (we will see later that they are the nuclei of helium atoms), -rays are electrons and -rays are photons, the quanta of electromagnetic radiation, but the historical names are still commonly used. 1 An interesting account of the early period, with descriptions of the personalities involved, is given in Se80. An overview of the later period is given in Chapter 1 of Gr87. 2 The 1903 Nobel Prize in Physics was awarded jointly to Becquerel for his discovery and to Pierre and Marie Curie for their subsequent research into radioactivity. 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 B. R. Martin # 2006 John Wiley & Sons, Ltd. ISBN: 0-470-01999-9 2 CH1 BASIC CONCEPTS 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 performed in 1911 at the suggestion of Rutherford by his collaborators, Geiger and Marsden. These consisted of scattering
-particles by very thin gold foils. In the Thomson model, most of the
-particles would pass through the foil, with only a few suffering deflections through small angles. Rutherford suggested they should look for large-angle scattering and to their surprise they found that some particles were indeed scattered through very large angles, even greater than 90. Rutherford showed that this behaviour was not due to multiple small-angle deflections, but could only be the result of the
-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 þe, where e is the magnitude of the charge on the electron4, 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 per cent smaller than the mass of the hydrogen atom. Examples are carbon and nitrogen, with masses of 12.0 and 14.0 in these units. However, it could not explain why not all atoms obeyed this rule. For example, chlorine has a mass of 35.5 in these 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 very long- standing puzzle, the solution to which is suggested by some as yet unproven, but widely believed, theories of particle physics that will be discussed briefly in Chapter 9.  HISTORY 3 units. At 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 The explanation of isotopes had to wait 20 years until a classic discovery by Chadwick in 1932. His work followed earlier experiments by Irène Curie (the daughter of Pierre and Marie Curie) and her husband Frédéric Joliot.6 They had observed that neutral radiation was emitted when
-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 experi- ments 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 under- standing 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,8 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-range9 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. 5 Frederick Soddy was awarded the 1921 Nobel Prize in Chemistry for his work on isotopes. 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 Werner Heisenberg received the 1932 Nobel Prize in Physics for his contributions to the creation of quantum mechanics and the idea of isospin symmetry, which we will discuss in Chapter 3. 9 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. 4 CH1 BASIC CONCEPTS 1.1.2 The emergence of particle physics: the standard model and hadrons By the early 1930s, the 19th 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 () and the neutrino (). The photon was postulated by Planck in 1900 to explain black-body radiation, where the classical description of electromagnetic radiation led to results incompatible with experi- ments.10 The neutrino was postulated by Fermi in 1930 to explain the apparent non-conservation of energy observed in the decay products of some unstable nuclei where -rays are emitted, the so-called -decays. Prior to Fermi’s suggestion, -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. Fermi’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 -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 Fermi’s hypothesis was confirmed by Reines and Cowan in a classic experiment in 1956 that detected free neutrinos from -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 devel- oped 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 Murray Gell-Mann and independently and simultaneously by George Zweig, who postulated that the new particles were bound states of three families of more fundamental physical particles. 10 X-rays had already been observed by Röntgen in 1895 (for which he received the first Nobel Prize in Physics in 1901) and -rays were seen by Villard in 1900, but it was 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. 11 However, in Section 3.1.4 we will discuss recent evidence that neutrinos have very small, but non-zero, masses. 12 A description of this experiment is given in Chapter 12 of Tr75. Frederick Reines shared the 1995 Nobel Prize in Physics for his work in neutrino physics and particularly for the detection of the electron neutrino.  HISTORY 5 Gell-Mann called these quarks (q).13 Because no free quarks were detected experimentally, there was initially considerable scepticism for this view. We now know that there is a fundamental reason why quarks cannot be observed as free particles (it will be discussed in Chapter 5), but at the time many physicists looked upon quarks as a convenient mathematical description, rather than physical particles.14 However, evidence for the existence of quarks as real particles came in the 1960s from a series of experiments analogous to those of Rutherford and his co-workers, where high-energy beams of electrons and neutrinos were scattered from nucleons. (These experiments will also be discussed in Chapter 5.) Analysis of the angular distributions of the scattered particles showed that the nucleons were themselves bound states of three point-like charged entities, with properties consistent with those hypothesized in the quark model. One of these properties was unexpected and unusual: quarks have fractional electric charges, in practice
13 e and þ 23 e. This is essentially the picture today, where elementary particles are now considered to be a small number of physical entities, including quarks, the electron, neutrinos, the photon and a few others we shall meet, but no longer nucleons. The best theory of elementary particles we have at present is called, rather prosaically, the standard model. This aims to explain all the phenomena of particle physics, except those due to gravity, in terms of the properties and interactions of a small number of elementary (or fundamental) particles, which are now defined as being point-like, without internal structure or excited states. Particle physics thus differs from nuclear physics in having a single theory to interpret its data. An elementary particle is characterized by, amongst other things, its mass, its electric charge and its spin. The latter is a permanent angular momentum possessed by all particles in quantum theory, even when they are at rest. Spin has no classical analogue and is not to be confused with the use of the same word in classical physics, where it usually refers to the (orbital) angular momentum of extended objects. The maximum value of the spin angular momentum about any axis is s
hð
h  h=2Þ, where h is Planck’s constant and s is the spin quantum number, or spin for short. It has a fixed value for particles of any given type (for example s ¼ 12 for electrons) and general quantum mechanical principles restrict the possible values of s to be 0, 12, 1, 32,.... Particles with half-integer spin are called fermions and those with integer spin are called bosons. There are three families of elementary particles in the standard model: two spin-12 families of fermions called leptons and quarks; and one family of spin-1 bosons. In addition, 13 Gell-Mann received the 1969 Nobel Prize in Physics for ‘contributions and discoveries concerning the classification of elementary particles and their interactions’. For the origin of the word ‘quark’, he cited the now famous quotation ‘Three quarks for Muster Mark’ from James Joyce’s book Finnegans Wake. Zweig had suggested the name ‘aces’, which with hindsight might have been more appropriate, as later experiments revealed that there were four and not three families of quarks. 14 This was history repeating itself. In the early days of the atomic model many very distinguished scientists were reluctant to accept that atoms existed, because they could not be ‘seen’ in a conventional sense. 6 CH1 BASIC CONCEPTS at least one other spin-0 particle, called the Higgs boson, is postulated to explain the origin of mass within the theory.15 The most familiar elementary particle is the electron, which we know is bound in atoms by the electromagnetic interaction, one of the four forces of nature.16 One test of the elementarity of the electron is the size of its magnetic moment. A charged particle with spin necessarily has an intrinsic magnetic moment l. It can be shown from the Dirac equation that a point-like spin-12 particle of charge q and mass m has a magnetic moment l ¼ ðq=mÞS, where S is its spin vector, and hence l has magnitude  ¼ q
h=2m. The magnetic moment of the electron very accurately obeys this relation, confirming that electrons are elementary. The electron is a member of the family of leptons. Another is the neutrino, which was mentioned earlier as a decay product in -decays. Strictly speaking, this particle should be called the electron neutrino, written e , because it is always produced in association with an electron (the reason for this is discussed in Section 3.1.1). The force responsible for -decay is an example of a second fundamental force, the weak interaction. Finally, there is the third force, the (fundamental) strong interaction, which, for example, binds quarks in nucleons. The strong nuclear force mentioned in Section 1.1.1 is not the same as this fundamental strong interaction, but is a consequence of it. The relation between the two will be discussed in more detail later. The standard model also specifies the origin of these three forces. In classical physics the electromagnetic interaction is propagated by electromagnetic waves, which are continuously emitted and absorbed. While this is an adequate descrip- tion at long distances, at short distances the quantum nature of the interaction must be taken into account. In quantum theory, the interaction is transmitted discon- tinuously by the exchange of photons, which are members of the family of fundamental spin-1 bosons of the standard model. Photons are referred to as the gauge bosons, or ‘force carriers’, of the electromagnetic interaction. The use of the word ‘gauge’ refers to the fact that the electromagnetic interaction possesses a fundamental symmetry called gauge invariance. For example, Maxwell’s equa- tions of classical electromagnetism are invariant under a specific phase transfor- mation of the electromagnetic fields – the gauge transformation.17 This property is common to all the three interactions of nature we will be discussing and has profound consequences, but we will not need its details in this book. The weak and strong interactions are also mediated by the exchange of spin-1 gauge bosons. For the weak interaction these are the W þ , W
and Z 0 bosons (again the superscripts denote the electric charges) with masses about 80–90 times the mass of the proton. 15 In the theory without the Higgs boson, all elementary particles are predicted to have zero mass, in obvious contradiction with experiment. A solution to this problem involving the Higgs boson will be discussed briefly in Chapter 9. 16 Gravity is so weak that it can be neglected in nuclear and particle physics at presently accessible energies. Because of this, we will often refer in practice to the three forces of nature. 17 See, for example, Appendix C.2 of Ma97.  RELATIVITY AND ANTIPARTICLES 7 For the strong interaction, the force carriers are called gluons. There are eight gluons, all of which have zero mass and are electrically neutral.18 In addition to the elementary particles of the standard model, there are other important particles we will be studying. These are the hadrons, the bound states of quarks. Nucleons are examples of hadrons,19 but there are several hundred more, not including nuclei, most of which are unstable and decay by one of the three interactions. It was the abundance of these states that drove the search for a simplifying theory that would give an explanation for their existence and led to the quark model in the 1960s. The most common unstable example of a hadron is the pion, which exists in three electrical charge states, written ðþ ; 0 ; 
Þ. Hadrons are important because free quarks are unobservable in nature and so to deduce their properties we are forced to study hadrons. An analogy would be if we had to deduce the properties of nucleons by exclusively studying the properties of nuclei. Since nucleons are bound states of quarks and nuclei are bound states of nucleons, the properties of nuclei should, in principle, be deducible from the properties of quarks and their interactions, i.e. from the standard model. In practice, however, this is far beyond present calculational techniques and some- times nuclear and particle physics are treated as two almost separate subjects. However, there are many connections between them and in introductory treatments it is still useful to present both subjects together. The remaining sections of this chapter are devoted to introducing some of the basic theoretical tools needed to describe the phenomena of both nuclear and particle physics, starting with a key concept: antiparticles. 1.2 Relativity and Antiparticles Elementary particle physics is also called high-energy physics. One reason for this is that if we wish to produce new particles in a collision between two other particles, then because of the relativistic mass–energy relation E ¼ mc2 , energies are needed at least as great as the rest masses of the particles produced. The second reason is that to explore the structure of a particle requires a probe whose wavelength is smaller than the structure to be explored. By the de Broglie relation ¼ h=p, this implies that the momentum p of the probing particle, and hence its energy, must be large. For example, to explore the internal structure of the proton using electrons requires wavelengths that are much smaller than the 18 Note that the word ‘electric’ has been used when talking about charge. This is because the weak and strong interactions also have associated ‘charges’ which determine the strengths of the interactions, just as the electric charge determines the strength of the electromagnetic interaction. This will be discussed in more detail in later chapters. 19 The magnetic moments of the proton and neutron do not obey the prediction of the Dirac equation and this is evidence that nucleons have structure and are not elementary. The proton magnetic moment was first measured by Otto Stern using a molecular beam method that he developed and for this he received the 1943 Nobel Prize in Physics. 8 CH1 BASIC CONCEPTS classical radius of the proton, which is roughly 10
15 m. This in turn requires electron energies that are greater than 103 times the rest energy of the electron, implying electron velocities very close to the speed of light. Hence any explanation of the phenomena of elementary particle physics must take account of the requirements of the theory of special relativity, in addition to those of quantum theory. There are very few places in particle physics where a non-relativistic treatment is adequate, whereas the need for a relativistic treatment is far less in nuclear physics. Constructing a quantum theory that is consistent with special relativity leads to the conclusion that for every particle of nature, there must exist an associated particle, called an antiparticle, with the same mass as the corresponding particle. This important theoretical prediction was first made by Dirac and follows from the solutions of the equation he first wrote down to describe relativistic electrons.20 The Dirac equation is of the form @Cðx; tÞ i
h ¼ Hðx; p ^ÞCðx; tÞ; ð1:1Þ @t ^ ¼
i
hr is the usual quantum mechanical momentum operator and the where p Hamiltonian was postulated by Dirac to be ^ þ mc2 : H ¼ ca p ð1:2Þ The coefficients a and  are determined by the requirement that the solutions of Equation (1.1) are also solutions of the Klein–Gordon equation 21 @ 2 Cðx; tÞ

h2 ¼

h2 c2 r2 Cðx; tÞ þ m2 c4 Cðx; tÞ: ð1:3Þ @t2 This leads to the conclusion that a and  cannot be simple numbers; their simplest forms are 4 4 matrices. Thus the solutions of the Dirac equation are four- component wavefunctions (called spinors) with the form22 0 1 C1 ðx; tÞ B C2 ðx; tÞ C Cðx; tÞ ¼ B C @ C3 ðx; tÞ A: ð1:4Þ C4 ðx; tÞ 20 Paul Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger. The somewhat cryptic citation stated ‘for the discovery of new productive forms of atomic theory’. 21 This is a relativistic equation, which is ‘derived’ by starting from the relativistic mass–energy relation E2 ¼ p2 c2 þ m2 c4 and using the usual quantum mechanical operator substitutions, p ^ ¼
i
hr and E ¼ i
h@=@t. 22 The details may be found in most quantum mechanics books, for example, pp. 475–477 of Sc68.  SYMMETRIES AND CONSERVATION LAWS 9 The interpretation of Equation (1.4) is that the four components describe the two spin states of a negatively charged electron with positive energy and the two spin states of a corresponding particle having the same mass but with negative energy. Two spin states arise because in quantum mechanics the projection in any direction of the spin vector of a spin-12 particle can only result in one of the two values 12, called ‘spin up’ and ‘spin down’, respectively. The two energy solutions arise from pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the two solutions of the relativistic mass–energy relation E ¼ p2 c2 þ m2 c4. The latter states can be shown to behave in all respects as positively charged electrons (called positrons), but with positive energy. The positron is referred to as the antiparticle of the electron. The discovery of the positron by Anderson in 1933, with all the predicted properties, was a spectacular verification of the Dirac prediction. Although Dirac originally made his prediction for electrons, the result is general and is true whether the particle is an elementary particle or a hadron. If we denote a particle by P, then the antiparticle is in general written with a bar over it, i.e. P
. 23 For example, the antiparticle of the proton is the antiproton
p, with negative electric charge; and associated with every quark, q, is an antiquark,
q. However, for some very common particles the bar is usually omitted. Thus, for example, in the case of the positron eþ , the superscript denoting the charge makes explicit the fact that the antiparticle has the opposite electric charge to that of its associated particle. Electric charge is just one example of a quantum number (spin is another) that characterizes a particle, whether it is elementary or composite (i.e. a hadron). Many quantum numbers differ in sign for particle and antiparticle, and electric charge is an example of this. We will meet others later. When brought together, particle–antiparticle pairs, each of mass m, can annihilate, releasing their com- bined rest energy 2mc2 as photons or other particles. Finally, we note that there is symmetry between particles and antiparticles, and it is a convention to call the electron the particle and the positron its antiparticle. This reflects the fact that the normal matter contains electrons rather than positrons. 1.3 Symmetries and Conservation Laws Symmetries and the invariance properties of the underlying interactions play an important role in physics. Some lead to conservation laws that are universal. Familiar examples are translational invariance, leading to the conservation of linear momentum; and rotational invariance, leading to conservation of angular momentum. The latter plays an important role in nuclear and particle physics as it leads to a scheme for the classification of states based, among other quantum 23 Carl Anderson shared the 1936 Nobel Prize in Physics for the discovery of the positron. The 1958 Prize was awarded to Emilio Segrè and Owen Chamberlain for their discovery of the antiproton. 10 CH1 BASIC CONCEPTS numbers, on their spins.24 Another very important invariance that we have briefly mentioned is gauge invariance. This fundamental property of all three interactions restricts the forms of the interactions in a profound way that initially is contra- dicted by experiment. This is the prediction of zero masses for all elementary particles, mentioned earlier. There are theoretical solutions to this problem whose experimental verification (or otherwise) is probably the most eagerly awaited result in particle physics today. In nuclear and particle physics we need to consider additional symmetries of the Hamiltonian and the conservation laws that follow and in the remainder of this section we discuss two of the most important of these that we will need later – parity and charge conjugation. 1.3.1 Parity Parity was first introduced in the context of atomic physics by Eugene Wigner in 1927.25 It refers to the behaviour of a state under a spatial reflection, i.e. x !
x. If we consider a single-particle state, represented for simplicity by a non- relativistic wavefunction ðx; tÞ, then under the parity operator, P ^, ^ ðx; tÞ  P ð
x; tÞ: P ð1:5Þ Applying the operator again, gives ^ 2 ðx; tÞ ¼ PP P ^ ð
x; tÞ ¼ P2 ðx; tÞ; ð1:6Þ implying P ¼ 1. If the particle is an eigenfunction of linear momentum p, i.e. ðx; tÞ  p ðx; tÞ ¼ exp½iðp x
EtÞ; ð1:7Þ then ^ P p ðx; tÞ ¼P p ð
x; tÞ ¼P
p ðx; tÞ ð1:8Þ and so a particle at rest, with p ¼ 0, is an eigenstate of parity. The eigenvalue P ¼ 1 is called the intrinsic parity, or just the parity, of the state. By considering a multiparticle state with a wavefunction that is the product of single-particle wavefunctions, it is clear that parity is a multiplicative quantum number. The strong and electromagnetic interactions, but not the weak interactions, are invariant under parity, i.e. the Hamiltonian of the system remains unchanged under 24 These points are explored in more detail in, for example, Chapter 4 of Ma97. 25 Eugene Wigner shared the 1963 Nobel Prize in Physics, principally for his work on symmetries.  SYMMETRIES AND CONSERVATION LAWS 11 a parity transformation on the position vectors of all particles in the system. Parity is therefore conserved, by which we mean that the total parity quantum number remains unchanged in the interaction. Compelling evidence for parity conservation in the strong and electromagnetic interactions comes from the absence of transitions between nuclear states and atomic states, respectively, that would violate parity conservation. The evidence for non-conservation of parity in the weak interaction will be discussed in detail in Chapter 6. There is also a contribution to the total parity if the particle has an orbital angular momentum l. In this case its wave function is a product of a radial part Rnl and an angular part Ylm ð ; Þ: lmn ðxÞ ¼ Rnl Ylm ð ; Þ; ð1:9Þ where n and m are the principal and magnetic quantum numbers and Ylm ð ; Þ is a spherical harmonic. It is straightforward to show from the relations between Cartesian ðx; y; zÞ and spherical polar co-ordinates ðr; ; Þ, i.e. x ¼ r sin cos ; y ¼ r sin sin ; z ¼ r cos ; ð1:10Þ that the parity transformation x !
x implies r ! r; !
;  !  þ ; ð1:11Þ and from this it can be shown that Ylm ð ; Þ ! Ylm ð
;  þ Þ ¼ ð
Þl Ylm ð ; Þ: ð1:12Þ Equation (1.12) may easily be verified directly for specific cases; for example, for the first three spherical harmonics, 1 1 1 1 2 3 2 3 2 Y00 ¼ ; Y10 ¼ cos ; 1 Y1 ¼ sin e i : ð1:13Þ 4 4 8 Hence ^ P lmn ðxÞ ¼P lmn ð
xÞ ¼ Pð
Þl lmn ðxÞ; ð1:14Þ i.e. lmn ðxÞ is an eigenstate of parity with eigenvalue Pð
1Þl. An analysis of the Dirac Equation (1.1) for relativistic electrons, shows that it is invariant under a parity transformation only if Pðeþ e
Þ ¼
1. This is a general result for all fermion–antifermion pairs, so it is a convention to assign P ¼ þ1 to all leptons and P ¼
1 to their antiparticles. We will see in Chapter 3 that in strong interactions quarks can only be created as part of a quark–antiquark pair, so the intrinsic parity of a single quark cannot be measured. For this reason, it is also 12 CH1 BASIC CONCEPTS a convention to assign P ¼ þ1 to quarks. Since quarks are fermions, it follows from the Dirac result that P ¼
1 for antiquarks. The intrinsic parities of hadrons then follow from their structure in terms of quarks and the orbital angular momentum between the constituent quarks, using Equation (1.14). This will be explored in Chapter 3 as part of the discussion of the quark model. 1.3.2 Charge conjugation Charge conjugation is the operation of changing a particle into its antiparticle. Like parity, it gives rise to a multiplicative quantum number that is conserved in strong and electromagnetic interactions, but violated in the weak interaction. In discussing charge conjugation, we will need to distinguish between states such as the photon  and the neutral pion 0 that do not have distinct antiparticles and those such as the þ and the neutron, which do. Particles in the former class we will collectively denote by a and those of the latter type will be denoted by b. It is also convenient at this point to extend our notation for states. Thus we will represent a state of type a having a wavefunction a by ja; a i and similarly for a state of type b. Then under the charge conjugation operator, C ^, ^ ja; C ai ¼ Ca ja; a i; ð1:15aÞ and ^ jb; C bi ¼ j
b;
b i; ð1:15bÞ where Ca is a phase factor analogous to the phase factor in Equation (1.5).26 Applying the operator twice, in the same way as for parity, leads to Ca ¼ 1. From ^ with eigenvalues Equation (1.15a), we see that states of type a are eigenstates of C 1, called their C-parities. States with distinct antiparticles can only form ^ as linear combinations. eigenstates of C As an example of the latter, consider a þ 
pair with orbital angular momentum L between them. In this case ^ jþ 
; Li ¼ ð
1ÞL jþ 
; Li; C ð1:16Þ because interchanging the pions reverses their relative positions in the spatial wavefunction. The same factor occurs for spin-12 fermion pairs f
f , but in addition there are two other factors. The first is ð
1ÞSþ1 , where S is the total spin of the pair. 26 A phase factor could have been inserted in Equation (1.15b), but it is straightforward to show that the relative phase of the two states b and
b cannot be measured and so a phase introduced in this way would have no physical consequences.  INTERACTIONS AND FEYNMAN DIAGRAMS 13 This follows directly from the structure of the spin wavefunctions: 9 "1 "2 Sz ¼ 1 = p1ffiffi ð" # þ # " Þ Sz ¼ 0 S¼1 ð1:17aÞ 2 1 2 1 2 ; #1 #2 Sz ¼
1 and p1ffiffi ð" #
#1 "2 Þ Sz ¼ 0 S ¼ 0; ð1:17bÞ 2 1 2 where "i ð#i Þ represents particle i having spin ‘up’ (‘down’) in the z-direction. A second factor ð
1Þ arises whenever fermions and antifermions are interchanged. This has its origins in quantum field theory.27 Combining these factors, finally we have ^ j f
f ; J; L; Si ¼ ð
1ÞLþS j f
f ; J; L; Si; C ð1:18Þ for fermion–antifermion pairs having total, orbital and spin angular momentum quantum numbers J, L and S, respectively. 1.4 Interactions and Feynman Diagrams We now turn to a discussion of particle interactions and how they can be described by the very useful pictorial methods of Feynman diagrams. 1.4.1 Interactions Interactions involving elementary particles and/or hadrons are conveniently summarized by ‘equations’ by analogy with chemical reactions, in which the different particles are represented by symbols which usually – but not always – have a superscript to denote their electric charge. In the interaction e þ n ! e
þ p; ð1:19Þ for example, an electron neutrino e collides with a neutron n to produce an electron e
and a proton p, whereas the equation e
þ p ! e
þ p ð1:20Þ 27 See, for example, pp. 249–250 of Go86. 14 CH1 BASIC CONCEPTS represents an electron and proton interacting to give the same particles in the final state, but in general travelling in different directions. In such equations, conserved quantum numbers must have the same total values in initial and final states. Particles may be transferred from initial to final states and vice versa, when they become antiparticles. Thus the process 
þ p ! 
þ p; ð1:21aÞ also implies the reaction p þ
p ! þ þ 
; ð1:22Þ which is obtained by taking the proton from the final state to an antiproton in the initial state and the negatively charged pion in the initial state to a positively charged pion in the final state. The interactions in Equations (1.20) and (1.21a), in which the particles remain unchanged, are examples of elastic scattering, in contrast to the reactions in Equations (1.19) and (1.22), where the final-state particles differ from those in the initial state. Collisions between a given pair of initial particles do not always lead to the same final state, but can lead to different final states with different probabilities. For example, the collision of a negatively charged pion and a proton can give rise to elastic scattering (Equation (1.21a)) and a variety of other reactions, such as 
þ p ! n þ 0 and 
þ p ! p þ 
þ 
þ þ ; ð1:21bÞ depending on the initial energy. In particle physics it is common to refer (rather imprecisely) to such interactions as ‘inelastic’ scattering. Similar considerations apply to nuclear physics, but the term inelastic scattering is reserved for the case where the final state is an excited state of the parent nucleus A, that subsequently decays, for example via photon emission, i.e. a þ A ! a þ A ; A ! A þ ; ð1:23Þ where a is a projectile and A is an excited state of A. A useful shorthand notation used in nuclear physics for the general reaction a þ A ! b þ B is Aða; bÞB. It is usual in nuclear physics to further subdivide types of interactions according to the underlying mechanism that produced them. We will return to this in Section 2.9, as part of a more general discussion of nuclear reactions. Finally, many particles are unstable and spontaneously decay to other, lighter (i.e. having less mass) particles. An example of this is the free neutron (i.e. one not bound in a nucleus), which decays by the -decay reaction n ! p þ e
þ 
e ; ð1:24Þ  INTERACTIONS AND FEYNMAN DIAGRAMS 15 with a mean lifetime of about 900 s.28 The same notation can also be used in nuclear physics. For example, many nuclei decay via the -decay reaction. Thus, denoting a nucleus with Z protons and N nucleons as (Z, N ), we have ðZ; NÞ ! ðZ
1; NÞ þ eþ þ e : ð1:25Þ This is also a weak interaction. This reaction is effectively the decay of a proton bound in a nucleus. Although a free proton cannot decay by the -decay p ! n þ eþ þ e because it violates energy conservation (the final-state particles have greater total mass than the proton), a proton bound in a nucleus can decay because of its binding energy. This will be explained in Chapter 2. 1.4.2 Feynman diagrams The forces producing all the above interactions are due to the exchange of particles and a convenient way of illustrating this is to use Feynman diagrams. There are mathematical rules and techniques associated with these that enable them to be used to calculate the quantum mechanical probabilities for given reactions to occur, but in this book Feynman diagrams will only be used as a convenient very useful pictorial description of reaction mechanisms. We first illustrate them at the level of elementary particles for the case of electromagnetic interactions, which arise from the emission and/or absorption of photons. For example, the dominant interaction between two electrons is due to the exchange of a single photon, which is emitted by one electron and absorbed by the other. This mechanism, which gives rise to the familiar Coulomb interaction at large distances, is illustrated in the Feynman diagram Figure 1.1(a). In such diagrams, we will use the convention that particles in the initial state are shown on the left and particles in the final state are shown on the right. (Some authors take time to run along the y-axis.) Spin-12 fermions (such as the electron) Figure 1.1 One-photon exchange in (a) e
þ e
! e
þ e
and (b) eþ þ eþ ! eþ þ eþ 28 The reason that this decay involves an antineutrino rather than a neutrino will become clear in Chapter 3. 16 CH1 BASIC CONCEPTS are drawn as solid lines and photons are drawn as wiggly lines. Arrowheads pointing to the right indicate that the solid lines represent electrons. In the case of photon exchange between two positrons, which is shown in Figure 1.1(b), the arrowheads on the antiparticle (positron) lines are conventionally shown as pointing to the left. In interpreting these diagrams, it is important to remember: (1) that the direction of the arrows on fermion lines does not indicate the particle’s direction of motion, but merely whether the fermions are particles or antiparticles, and (2) that particles in the initial state are always to the left and particles in the final state are always to the right. A feature of the above diagrams is that they are constructed from combinations of simple three-line vertices. This is characteristic of electromagnetic processes. Each vertex has a line corresponding to a single photon being emitted or absorbed, while one fermion line has the arrow pointing toward the vertex and the other away from the vertex, guaranteeing charge conservation at the vertex, which is one of the rules of Feynman diagrams.29 For example, a vertex like Figure 1.2 would correspond to a process in which an electron emitted a photon and turned into a positron. This would violate charge conservation and is therefore forbidden. γ e– e+ Figure 1.2 The forbidden vertex e
! eþ þ  Feynman diagrams can also be used to describe the fundamental weak and strong interactions. This is illustrated by Figure 1.3(a), which shows contributions to the elastic weak scattering reaction e
þ e ! e
þ e due to the exchange of a Z 0 , and by Figure 1.3(b), which shows the exchange of a gluon g (represented by a coiled line) between two quarks, which is a strong interaction. Feynman diagrams can also be drawn at the level of hadrons. As an illustration, Figure 1.4 shows the exchange of a charged pion (shown as a dashed line) between a proton and a neutron. We shall see later that this mechanism is a major contribution to the strong nuclear force between a proton and a neutron. We turn now to consider in more detail the relation between exchanged particles and forces. 29 Compare Kirchhoff’s laws in electromagnetism.  PARTICLE EXCHANGE: FORCES AND POTENTIALS 17 Figure 1.3 (a) Contributions of (a) Z 0 exchange to the elastic weak scattering reaction e
þ e ! e
þ e , and (b) gluon exchange contribution to the strong interaction q þ q ! q þ q Figure 1.4 Single-pion exchange in the reaction p þ n ! n þ p 1.5 Particle Exchange: Forces and Potentials This section starts with a discussion of the important relationship between forces and particle exchanges and then relates this to potentials. Although the idea of a potential has its greatest use in non-relativistic physics, nevertheless it is useful to illustrate concepts and is used in later sections as an intermediate step in relating theoretical Feynman diagrams to measurable quantities. The results can easily be extended to more general situations. 1.5.1 Range of forces At each vertex of a Feynman diagram, charge is conserved by construction. We will see later that, depending on the nature of the interaction (strong, weak or electromagnetic), other quantum numbers are also conserved. However, it is easy to show that energy and momentum cannot be conserved simultaneously. Consider the general case of a reaction A þ B ! A þ B mediated by the exchange of a particle X, as shown in Figure 1.5. In the rest frame of the incident 18 CH1 BASIC CONCEPTS Figure 1.5 Exchange of a particle X in the reaction A þ B ! A þ B particle A, the lower vertex represents the virtual process (‘virtual’ because X does not appear as a real particle in the final state), AðMA c2 ; 0Þ ! AðEA ; pA cÞ þ XðEX ;
pA cÞ; ð1:26Þ where EA is the total energy of particle A and pA is its three-momentum.30 Thus, if we denote by PA the four-vector for particle A, PA ¼ ðEA =c; pA Þ ð1:27Þ and P2A ¼ EA2 =c2
p2A ¼ MA2 c2 : ð1:28Þ Applying this to the diagram and imposing momentum conservation, gives EA ¼ ðp2 c2 þ MA2 c4 Þ1=2 ; EX ¼ ðp2 c2 þ MX2 c4 Þ1=2 ; ð1:29Þ where p ¼ jpj. The energy difference between the final and initial states is given by E ¼ EX þ EA
MA c2 ! 2pc; p!1 ! MX c2 ; p!0 ð1:30Þ and thus E  MX c2 for all p, i.e. energy is not conserved. However, by the Heisenberg uncertainty principle, such an energy violation is allowed, but only for a time  
h=E, so we immediately obtain r  R  h
=MX c ð1:31Þ 30 A resumé of relativistic kinematics is given in Appendix B.  PARTICLE EXCHANGE: FORCES AND POTENTIALS 19 as the maximum distance over which X can propagate before being absorbed by particle B. This maximum distance is called the range of the interaction and this was the sense of the word used in Section 1.1.1. The electromagnetic interaction has an infinite range, because the exchanged particle is a massless photon. In contrast, the weak interaction is associated with the exchange of very heavy particles – the W and Z bosons. These lead to ranges that from Equation (1.31) are of the order of RW;Z  2 10
18 m. The funda- mental strong interaction has infinite range because, like the photon, gluons have zero mass. On the other hand, the strong nuclear force, as exemplified by Figure 1.4 for example, has a much shorter range of approximately (1–2) 10
15 m. We will comment briefly on the relation between these two different manifestations of the strong interaction in Section 7.1. 1.5.2 The Yukawa potential In the limit that MA becomes large, we can regard B as being scattered by a static potential of which A is the source. This potential will in general be spin dependent, but its main features can be obtained by neglecting spin and considering X to be a spin-0 boson, in which case it will obey the Klein–Gordon equation @ 2 ðx; tÞ

h2 ¼

h2 c2 r2 ðx; tÞ þ MX2 c4 ðx; tÞ: ð1:32Þ @t2 The static solution of this equation satisfies MX2 c4 r2 ðxÞ ¼ ðxÞ; ð1:33Þ
h2 where ðxÞ is interpreted as a static potential. For MX ¼ 0 this equation is the same as that obeyed by the electrostatic potential, and for a point charge
e interacting with a point charge þe at the origin, the appropriate solution is the Coulomb potential e2 1 VðrÞ ¼
eðrÞ ¼
; ð1:34Þ 4"0 r where r ¼ jxj and "0 is the dielectric constant. The corresponding solution in the case where MX2 6¼ 0 is easily verified by substitution to be g2 e
r=R VðrÞ ¼
; ð1:35Þ 4 r where R is the range defined earlier and g, the so-called coupling constant, is a parameter associated with each vertex of a Feynman diagram and represents the 20 CH1 BASIC CONCEPTS basic strength of the interaction.31 For simplicity, we have assumed equal strengths for the couplings of particle X to the particles A and B. The form of VðrÞ in Equation (1.35) is called a Yukawa potential, after the physicist who first introduced the idea of forces due to massive particle exchange in 1935.32 As MX ! 0, R ! 1 and the Coulomb potential is recovered from the Yukawa potential, while for very large masses the interaction is approximately point-like (zero range). It is conventional to introduce a dimensionless parameter
X by g2
X ¼ ; ð1:36Þ 4
hc that characterizes the strength of the interaction at short distances r  R. For the electromagnetic interaction this is the fine structure constant
 e2 =4"0 h
c  1=137 ð1:37Þ that governs the splittings of atom energy levels.33 The forces between hadrons are also generated by the exchange of particles. Thus, in addition to the electromagnetic interaction between charged hadrons, all hadrons, whether charged or neutral, experience a strong short-range interaction, which in the case of two nucleons, for example, has a range of about 10
15 m, corresponding to the exchange of a particle with an effective mass of about 17th the mass of the proton. The dominant contribution to this force is the exchange of a single pion, as shown in Figure 1.4. This nuclear strong interaction is a complicated effect that has its origins in the fundamental strong interactions between the quark distributions within the two hadrons. Two neutral atoms also experience an electromagnetic interaction (the van der Waals force), which has its origins in the fundamental Coulomb forces, but is of much shorter range. Although an analogous mechanism is not in fact responsible for the nuclear strong interaction, it is a useful reminder that the force between two distributions of particles can be much more complicated than the forces between their components. We will return to this point when we discuss the nature of the nuclear potential in more detail in Section 7.1. 1.6 Observable Quantities: Cross Sections and Decay Rates We have mentioned earlier that Feynman diagrams can be turned into probabilities for a process by a set of mathematical rules (the Feynman Rules) that can be 31 Although we call g a (point) coupling constant, in general it will have a dependence on the momentum carried by the exchanged particle. We ignore this in what follows. 32 For this insight, Hideki Yukawa received the 1949 Nobel Prize in Physics. 33 Like g, the coupling
X will in general have a dependence on the momentum carried by particle X. In the case of the electromagnetic interaction, this dependence is relatively weak.  OBSERVABLE QUANTITIES: CROSS SECTIONS AND DECAY RATES 21 derived from the quantum theory of the underlying interaction.34 We will not pursue this in detail in this book, but rather will show in principle their relation to observables, i.e. things that can be measured, concentrating on the cases of two- body scattering reactions and decays of unstable states. 1.6.1 Amplitudes The intermediate step is the amplitude f, the modulus squared of which is directly related to the probability of the process occurring. It is also called the invariant amplitude because, as we shall show, it is directly related to observable quantities and these have to be the same in all inertial frames of reference. To get some qualitative idea of the structure of f, we will use non-relativistic quantum mechanics and assume that the coupling constant g2 is small compared with 4
hc, so that the interaction is a small perturbation on the free particle solution, which will be taken as plane waves. If we expand the amplitude in a power series in g2 and keep just the first term (i.e. lowest-order perturbation theory), then the amplitude for a particle in an initial state with momentum qi to be scattered to a final state with momentum qf by a potential VðxÞ is proportional to ð f ðq2 Þ ¼ d3 xVðxÞ exp½iq x=
h; ð1:38Þ i.e. the Fourier transform of the potential, where q  qi
qf is the momentum transfer.35 The integration may be done using polar co-ordinates. Taking q in the x- direction, gives q x ¼ jqjr cos ð1:39Þ and d3 x ¼ r2 sin d dr d; ð1:40Þ where r ¼ jxj. For the Yukawa potential, the integral in Equation (1.38) gives h2
g2
f ðq2 Þ ¼ : ð1:41Þ q2 þ MX2 c2 34 In the case of the electromagnetic interaction, the theory is called Quantum Electrodynamics (QED) and is spectacularly successful in explaining experimental results. Richard Feynman shared the 1965 Nobel Prize in Physics with Sin-Itiro Tomonoga and Julian Schwinger for their work on formulating quantum electro- dynamics. The Feynman Rules are discussed in an accessible way in Gr87. 35 See, for example, Chapter 11 of Ma92. 22 CH1 BASIC CONCEPTS This amplitude corresponds to the exchange of a single particle, as shown for example in Figures 1.3 and 1.4. The structure of the amplitude, which is quite general, is a numerator proportional to the product of the couplings at the two vertices (or equivalently
X in this case), and a denominator that depends on the mass of the exchanged particle and its momentum transfer squared. The denomi- nator is called the propagator for particle X. In a relativistic calculation, the term q2 becomes q2 , where q is the four-momentum transfer. Returning to the zero-range approximation, one area where it is used extensively is in weak interactions, particularly applied to nuclear decay processes. In these situations, MX ¼ MW;Z and f !
GF , where GF , the so-called Fermi coupling constant, is given from Equation (1.36) by GF 4
W 3 ¼ ¼ 1:166 10
5 GeV
2 : ð1:42Þ ð
hcÞ ðMW c2 Þ2 The numerical value is obtained from analyses of decay processes, including that of the neutron and a heavier version of the electron called the muon, whose properties will be discussed in Chapter 3. All the above is for the exchange of a single particle. It is also possible to draw more complicated Feynman diagrams that correspond to the exchange of more than one particle. An example of such a diagram for elastic e
e
scattering, where two photons are exchanged, is shown in Figure 1.6. Figure 1.6 Two-photon exchange in the reaction e
þ e
! e
þ e
The number of vertices in any diagram is called the order n, and when the amplitude associated p with ffiffiffiffi n any given Feynman diagram is calculated, it always contains a factor of ð
Þ. Since the probability is proportional to the square of the modulus of the amplitude, the former will contain a factor
n. The probability associated with the single-photon exchange diagrams of Figure 1.1 thus contain a factor of
2 and the contribution from two-photon exchange is of the order of
4.  OBSERVABLE QUANTITIES: CROSS SECTIONS AND DECAY RATES 23 As
 1=137, the latter is usually very small compared with the contribution from single-photon exchange. This is a general feature of electromagnetic interactions: because the fine structure constant is very small, in most cases only the lowest-order diagrams that contribute to a given process need be taken into account, and more complicated higher-order diagrams with more vertices can (to a good approximation) be ignored in many applications. 1.6.2 Cross-sections The next step is to relate the amplitude to measurables. For scattering reactions, the appropriate observable is the cross-section. In a typical scattering experiment, a beam of particles is allowed to hit a target and the rates of production of various particles in the final state are counted.36 It is clear that the rates will be proportional to: (a) the number N of particles in the target illuminated by the beam, and (b) the rate per unit area at which beam particles cross a small surface placed in the beam at rest with respect to the target and perpendicular to the beam direction. The latter is called the flux and is given by J ¼ nb vi ; ð1:43Þ where nb is the number density of particles in the beam and vi their velocity in the rest frame of the target. Hence the rate Wr at which a specific reaction r occurs in a particular experiment can be written in the form Wr ¼ JNr ; ð1:44aÞ where r , the constant of proportionality, is called the cross-section for reaction r. If the beam has a cross-sectional area S, its intensity is I ¼ JS and so an alternative expression for the rate is Wr ¼ Nr I=S ¼ Ir nt t; ð1:44bÞ where nt is the number of target particles per unit volume and t is the thickness of the target. If the target consists of an isotopic species of atomic mass MA (in atomic mass units-defined in Section 1.7 below), then nt ¼ NA =MA , where  is the density of the target and NA is Avogadro’s constant. Thus, Equation (1.44b) may be written Wr ¼ Ir ðtÞNA =MA ; ð1:44cÞ 36 The practical aspects of experiments are discussed in Chapter 4. 24 CH1 BASIC CONCEPTS where ðtÞ is a measure of the amount of material in the target, expressed in units of mass per unit area. The form of Equation (1.44c) is particularly useful for the case of thin targets, commonly used in experiments. In the above, the product JN is called the luminosity L, i.e. L  JN ð1:45Þ and contains all the dependencies on the densities and geometries of the beam and target. The cross-section is independent of these factors. It can be seen from the above equations that the cross-section has the dimensions of an area; the rate per target particle Jr at which the reaction occurs is equal to the rate at which beam particles would hit a surface of area r , placed in the beam at rest with respect to the target and perpendicular to the beam direction. Since the area of such a surface is unchanged by a Lorentz transformation in the beam direction, the cross-section is the same in all inertial frames of reference, i.e. it is a Lorentz invariant. The quantity r is better named the partial cross-section, because it is the cross- section for a particular reaction r. The total cross-section  is defined by X  r : ð1:46Þ r Another useful quantity is the differential cross-section, dr ð ; Þ=d, which is defined by dr ð ; Þ dWr  JN d; ð1:47Þ d where dWr is the measured rate for the particles to be emitted into an element of solid angle d ¼ d cos d in the direction ð ; Þ, as shown in Figure 1.7. The total cross-section is obtained by integrating the partial cross-section over all angles, i.e. ð 2 ð1 dr ð ; Þ r ¼ d d cos : ð1:48Þ 0
1 d The final step is to write these formulae in terms of the scattering amplitude f ðq2 Þ appropriate for describing the scattering of a non-relativistic spinless particle from a potential. To do this it is convenient to consider a single beam particle interacting with a single target particle and to confine the whole system in an arbitrary volume V (which cancels in the final result). The incident flux is then given by J ¼ nb vi ¼ vi =V ð1:49Þ  OBSERVABLE QUANTITIES: CROSS SECTIONS AND DECAY RATES 25 Figure 1.7 Geometry of the differential cross-section: a beam of particles is incident along the z-axis and collides with a stationary target at the origin; the differential cross-section is proportional to the rate for particles to be scattered into a small solid angle d in the direction ð ; Þ and since the number of target particles is N ¼ 1, the differential rate is vi dr ð ; Þ dWr ¼ d: ð1:50Þ V d In quantum mechanics, provided the interaction is not too strong, the transition rate for any process is given in perturbation theory by the Born approximation37 ð 2 2  3    dWr ¼ dx f VðxÞ i  ðEf Þ: ð1:51Þ
h  The term ðEf Þ is the density-of-states factor (see below) and we take the initial and final state wavefunctions to be plane waves: 1 1 i ¼ pffiffiffiffi exp½iqi x=
h; f ¼ pffiffiffiffi exp½iqf x=
h; ð1:52Þ V V 37 This equation is a form of the Second Golden Rule. It is discussed in Appendix A. 26 CH1 BASIC CONCEPTS where the final momentum qf lies within a small solid angle d located in the direction ð ; Þ (see Figure 1.7.). Then, by direct integration, 2 dWr ¼ j f ðq2 Þj2 ðEf Þ; ð1:53Þ
hV 2 where f ðq2 Þ is the scattering amplitude defined in Equation (1.38). The density of states ðEf Þ that appears in Equation (1.51) is the number of possible final states with energy lying between Ef and Ef þ dEf and is given by38 V q2f ðEf Þ ¼ d: ð1:54Þ ð2
hÞ3 vf If we use this and Equation (1.53) in Equation (1.50), we have d 1 q2 ¼ 2 4 f j f ðq2 Þj2 : ð1:55Þ d 4
h vi vf Although this result has been derived in the laboratory system, because we have taken a massive target it is also valid in the centre-of-mass system. For a finite mass target it would be necessary to make a Lorentz transformation on Equation (1.55). The expression is also true for the general two-body relativistic scattering process a þ b ! c þ d. All the above is for spinless particles, so finally we have to generalize Equation (1.55) to include the effects of spin. Suppose the initial-state particles a and b, have spins sa and sb and the final-state particles c and d have spins sc and sd. The total numbers of spin substates available to the initial and final states are gi and gf , respectively, given by gi ¼ ð2sa þ 1Þð2sb þ 1Þ and gf ¼ ð2sc þ 1Þ ð2sd þ 1Þ: ð1:56Þ If the initial particles are unpolarized (which is the most common case in practice), then we must average over all possible initial spin configurations (because each is equally likely) and sum over the final configurations. Thus, Equation (1.55) becomes d gf q2 ¼ 2 4 f jMfi j2 ; ð1:57Þ d 4
h vi vf where jMfi j2  j f ðq2 Þj2 ð1:58Þ and the bar over the amplitude denotes a spin-average of the squared matrix element. 38 The derivation is given in detail in Appendix A.  OBSERVABLE QUANTITIES: CROSS SECTIONS AND DECAY RATES 27 1.6.3 Unstable states In the case of an unstable state, the observable of interest is its lifetime at rest , or equivalently its natural decay width, given by  ¼
h=, which is a measure of the rate of the decay reaction. In general, an initial unstable state will decay to several final states and in this case we define f as the partial width for channel f and X ¼ f ð1:59Þ f as the total decay width, while Bf  f = ð1:60Þ is defined as the branching ratio for decay to channel f. The energy distribution of an unstable state to a final state f has the Breit–Wigner form f Nf ðWÞ / ; ð1:61Þ ðW
MÞ2 c4 þ 2 =4 where M is the mass of the decaying state and W is the invariant mass of the decay products.39 The Breit–Wigner formula is shown in Figure 1.8 and is the same formula that describes the widths of atomic and nuclear spectral lines. (The overall factor depends on the spins of the particles involved.) It is a symmetrical bell- shaped curve with a maximum at W ¼ M and a full width  at half the maximum height of the curve. It is proportional to the number of events with invariant mass W. If an unstable state is produced in a scattering reaction, then the cross section for that reaction will show an enhancement described by the same Breit–Wigner formula. In this case we say we have produced a resonance state. In the vicinity of a resonance of mass M, and width , the cross-section for the reaction i ! f has the form i f fi / ; ð1:62Þ ðE
Mc2 Þ2 þ 2 =4 where E is the total energy of the system. Again, the form of the overall constant will depend on the spins of the particles involved. Thus, for example, if the resonance particle has spin j and the spins of the initial particles are s1 and s2 , then 
h2 2j þ 1 i f fi ¼ : ð1:63Þ qi ð2s1 þ 1Þð2s2 þ 1Þ ðE
Mc2 Þ2 þ 2 =4 2 39 This form arises from a state that decays exponentially with time, although a proper proof of this is quite lengthy. See, for example, Appendix B of Ma97. 28 CH1 BASIC CONCEPTS Figure 1.8 The Breit--Wigner formula (Equation (1.61)) In practice there will also be kinematical and angular momentum effects that will distort this formula from its perfectly symmetric shape. An example of resonance formation in 
p interactions is given in Figure 1.9, which shows the 
p total cross-section in the centre-of-mass energy range Figure 1.9 Total cross-sections for 
p interactions (data from Ca68)  UNITS: LENGTH, MASS AND ENERGY 29 1.2–2.4 GeV. (The units used in the plots will become clear after the next section.) Two enhancements can be seen that are of the approximate Breit–Wigner resonance form and there are two other maxima at higher energies. In principle, the mass and width of a resonance may be obtained by using a Breit–Wigner formula and varying M and  to fit the cross-section in the region of the enhancement. In practice, more sophisticated methods are used that fit a wide range of data, including differential cross-sections, simultaneously and also take account of non-resonant contributions to the scattering. The widths obtained from such analyses are of the order of 100 MeV, with corresponding interaction times of order 10
23 s, which is consistent with the time taken for a relativistic pion to transit the dimension of a proton. Resonances are also a feature of interactions in nuclear physics and we will return to this in Section 2.9 when we discuss nuclear reaction mechanisms. 1.7 Units: Length, Mass and Energy Most branches of science introduce special units that are convenient for their own purposes. Nuclear and particle physics are no exceptions. Distances tend to be measured in femtometres (fm) or, equivalently fermis, with 1 fm  10
15 m. In these units, the radius of the proton is about 0.8 fm. The range of the strong nuclear force between protons and neutrons is of the order of 1–2 fm, while the range of the weak force is of the order of 10
3 fm. For comparison, the radii of atoms are of the order of 105 fm. A common unit for area is the barn (b) defined by 1 b ¼ 10
28 m2. For example, the total cross-section for pp scattering (a strong interaction) is a few tens of millibarns (mb) (compare also the 
p total cross-section in Figure 1.9), whereas the same quantity for p scattering (a weak interaction) is a few tens of femtobarns (fb), depending on the energies involved. Nuclear cross-sections are very much larger and increase approxi- mately like A2=3 , where A is the total number of nucleons in the nucleus. Energies are invariably specified in terms of the electron volt (eV) defined as the energy required to raise the electric potential of an electron or proton by 1 V. In SI units, 1 eV ¼ 1:6 10
19 J. The units 1 keV ¼ 103 eV, 1 MeV ¼ 106 eV, 1 GeV ¼ 109 eV and 1 TeV ¼ 1012 eV are also in general use. In terms of these units, atomic ionization energies are typically a few eV, the energies needed to bind nucleons in heavy nuclei are typically 7–8 MeV per particle, and the highest particle energies produced in the laboratory are of the order of 1 TeV for protons. Momenta are specified in eV=c, MeV=c, etc.. In order to create a new part