Foundations of nuclear and particle physics

6.1 Introduction to Chiral Symmetry6.2 Renormalization 6.3 Baryon Chiral Perturbation Theory 6.4 On to Higher Energy: Dispersion Relations 7 Introduction to Lepton Scattering 7.1 Unpolar

This textbook brings together nuclear and particle physics, presenting a balancedoverview of both fields as well as the interplay between the two The theoretical aswell as the experimental foundations are covered, providing students with a deepunderstanding of the subject In-chapter exercises ranging from basic experimental tosophisticated theoretical questions provide an important tool for students to solidifytheir knowledge Suitable for upper undergraduate courses in nuclear and particlephysics as well as more advanced courses, the book includes road maps guidinginstructors on tailoring the content to their course Online resources including colorfigures, tables, and a solutions manual complete the teaching package This textbookwill be essential for students preparing for further study or a career in the field whorequire a solid grasp of both nuclear and particle physics.

Key features

Contains up-to-date coverage of both nuclear and particle physics, particularly theareas where the two overlap, equipping students for the real-world occasionswhere aspects of both fields are required for study

Covers the theoretical as well as the experimental foundations, providing studentswith a deep understanding of the field

Exercises ranging from basic experimental to sophisticated theoretical questionsprovide an important tool for readers to consolidate their knowledge

THOMAS WILLIAM DONNELLY is a Senior Research Scientist at MIT He received hisPhD in Theoretical Nuclear Physics in 1967 from the University of British Columbia

JOSEPH ANGELO FORMAGGIO is an Associate Professor of Physics at MIT He received

his PhD in Physics at Columbia University in 2001 He has been a member on a number

of experiments including the Sudbury Neutrino Observatory and the KATRIN neutrinoexperiment

BARRY R HOLSTEIN is an Emeritus Professor Physics at the University of Massachusetts.

He received his PhD in Physics from Carnegie Mellon University in 1969 He is Editor

of Annual Reviews of Nuclear and Particle Physics, Consulting Editor of the American Journal of Physics, and Associate Editor of the Journal of Physics G.

RICHARD GERARD MILNER is a Professor of Physics at MIT He received his PhD from

the California Institute of Technology in 1985 He has proposed and led experiments atSLAC, DESY, MIT-Bates, and Jefferson Laboratory

BERND SURROW is a Professor of Physics at Temple University He gained his PhD in

Physics at the University of Hamburg in 1998 He has been a member of a number ofexperiments including the STAR experiment at BNL, the CMS and OPAL experiments at

CERN and the ZEUS experiment at DESY.

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Library of Congress Cataloguing in Publication Data

Names: Donnelly, T W (T William), 1943– author | Formaggio, Joseph A., 1974– author | Holstein, Barry R., 1943– author | Milner, Richard Gerard, 1956– author | Surrow, Bernd, 1998– author.

Title: Foundations of nuclear and particle physics / T William Donnelly (Massachusetts Institute

of Technology), Joseph A Formaggio (Massachusetts Institute of Technology), Barry R.

Holstein (University of Massachusetts, Amherst), Richard G Milner (Massachusetts Institute of Technology), Bernd Surrow (Temple University, Philadelphia).

Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, [2016]

| Includes index.

Identifiers: LCCN 2016026959| ISBN 9780521765114 (hardback) |

ISBN 0521765110 (hardback)

Subjects: LCSH: Nuclear physics–Textbooks | Particles (Nuclear physics)–Textbooks.

Classification: LCC QC776 D66 2016 | DDC 539.7–dc23 LC record available

at https://lccn.loc.gov/2016026959

ISBN 978-0-521-76511-4 Hardback

Additional resources for this publication at www.cambridge.org/9780521765114

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication, and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Joe ⇔ to Mike, Hamish, Janet, and John, for their unwavering wisdom; to Jaymi, Coby, and Joshua, for their unquestioning love

Barry ⇔ to Jeremy and Jesse

Richard ⇔ to Liam Milner for inspiration and to Eileen, Will, Sam, and David for love and support

Bernd ⇔ to Suzanne, Alec, Arianna, and Carl for their love and support

2.6 Discrete Symmetries: P, C, and T

3 Building Hadrons from Quarks

3.1 Light Mesons Built from u, d, and s Quarks

3.2 Baryons

3.3 Baryon Ground-State Properties

4 The Standard Model

4.1 Electroweak Interaction: The Weinberg–Salam Model4.2 The Higgs Mechanism

4.3 The Higgs Boson

6.1 Introduction to Chiral Symmetry

6.2 Renormalization

6.3 Baryon Chiral Perturbation Theory

6.4 On to Higher Energy: Dispersion Relations

7 Introduction to Lepton Scattering

7.1 Unpolarized Electron Scattering

7.2 Spin-Dependent Lepton–Nucleon Scattering

7.3 Electron–Nucleus Scattering

7.4 Electromagnetic Multipole Operators

7.5 Parity-Violating Lepton Scattering

8 Elastic Electron Scattering from the Nucleon

8.1 The Elastic Form Factors of the Nucleon

8.2 The Role of Mesons

8.3 Beyond Single-Photon Exchange

8.4 PV Electron Scattering and Strange-Quark Content in the Nucleon8.5 The Shape of the Proton

8.6 Electromagnetic Form Factors in QCD

9 Hadron Structure via Lepton–Nucleon Scattering

9.1 Deep Inelastic Scattering

9.2 The Parton Model

9.3 Evolution Equations

9.4 Hadronization/Fragmentation

9.5 The Spin Structure of the Nucleon: Lepton Scattering

9.6 Spin Structure Functions in QCD

9.7 Generalized Parton Distributions

9.8 The Role of Partons in Nuclei

10 High-Energy QCD

10.1 Introduction

10.2 Building the Tools

10.3 Spin Structure of the Nucleon: Polarized Proton Collider

10.4 Flavor Asymmetry of the Sea via the Drell–Yan Process

10.5 Low-x Physics

10.6 Jets, Bosons, and Top Quarks

10.7 The Path Forward

11 The Nucleon–Nucleon Interaction

11.7 Effective Field Theory: the NN Interaction

11.8 Nucleon–Nucleon Interaction from QCD

12 The Structure and Properties of Few-Body Nuclei

13 Overview of Many-Body Nuclei

13.1 Basic Properties of Finite Nuclei

13.2 Nuclear and Neutron Matter

13.3 Relativistic Modeling of Nuclear Matter

14 Models of Many-Body Nuclei

14.1 Hartree–Fock Approximation and the Nuclear Mean Field

14.2 Rotational Model of Deformed Nuclei

14.3 Vibrational Model

14.4 Single-Particle Transitions and Giant Resonances

15 Electron Scattering from Discrete States

15.1 Parity-Conserving Elastic Electron Scattering from Spin-0 Nuclei

15.2 Parity-Violating Elastic Electron Scattering from Spin-0 Nuclei

15.3 Elastic Scattering from Non-Spin-0 Nuclei: Elastic Magnetic Scattering15.4 Electroexcitation of Low-Lying Excited States

16.1 Introduction

16.2 Quasielastic Electron Scattering and the Fermi Gas Model

16.3 Inclusive Electron Scattering and Scaling

16.4 Δ-Excitation in Nuclei

16.5 Nuclear Spectral Function and the Nucleon Momentum Distribution

17 Beta Decay

17.1 Introduction

17.2 Nuclear Beta Decay

17.3 The Nucleus as a Laboratory

18.5 Outstanding Questions in Neutrino Physics

19 The Physics of Relativistic Heavy Ions

20.1 Big Bang Nucleosynthesis

20.2 Nuclear Reaction Rates

21.3 BSM Physics: Theoretical Approaches21.4 Summary

Appendix A Useful Information

A.1 Notations and Identities

A.2 Decay Lifetimes and Cross SectionsA.3 Mathematics in d Dimensions

Appendix B Quantum Theory

B.1 Nonrelativistic Quantum MechanicsB.2 Relativistic Quantum Mechanics

B.3 Elastic Scattering Theory

B.4 Fermi–Watson Theorem

References

Subject Index

The first question one might ask about this book is: Why do we need another text on thesubject of nuclear and particle physics when excellent texts already exist in both of theseareas? Indeed, it is true that each sub-discipline has texts that range from elementary tovery advanced and cover specific topics in varying degrees of depth that can be used forthe appropriate types of courses For instance, there are fine books on quantum fieldtheory [Bjo64, Pes95, Wei05, Sch14], on the constituent quark model [Clo79], on high-energy physics [Gri08, Hal84], on hadron scattering [Col84], and on nuclear structure[Des74, Wal95, Won98, Pov08, Row10] However, there are relatively few textbooksthat cover several sub-disciplines in a coherent and balanced way, and those that doexist are either more elementary, e.g., Povh et al [Pov08] than the present book, or arecast at a more theoretical level and are too advanced for the goals we as authors haveset for ourselves Having a book that stresses the interconnections between the twoareas of subatomic physics is crucial, since increasingly one finds that the two fieldsoverlap and that it is essential for a graduate student conducting frontier research andpreparing for a career in the field to have an understanding of both An example of thisoverlap occurs, for instance, in modern neutrino physics wherein experiments utilizingseveral-GeV neutrinos as probes almost always involve targets/detectors constructedfrom nuclei and specifics of nuclear structure are unavoidably required to properlyinterpret such data.

One specific decision we have made in designing this book is to assume that thereader is familiar with the basics of quantum field theory More elementary textstypically do not make this assumption and thus much of the discussion, for instance, oflepton scattering from hadrons and nuclei, or of the foundations of chiral symmetry andeffective field theory is limited and not at the frontier of the field We realize that manystudents today do have at least an introductory course in quantum field theory, or aretaking one simultaneously with a course that this book covers, and thus we havefollowed a somewhat more advanced approach than has been customary We haveincluded in Appendix B an overview of the essential aspects of quantum mechanics andquantum field theory that are needed for the book Furthermore, the subject of many-body theory underlies much of nuclear physics and the presentation of this subject canalso be rather elementary, as is usually the case in texts that cover the two fields, or tooadvanced for our purposes, focusing on Green’s functions, diagrammatic techniques andnonperturbative approximations at a theoretical level We have chosen a middle course:

we have covered the basics of many-body theory, but also have introduced some of theimportant diagrammatic representations of the nonperturbative approximationsemployed very widely in quantum physics ranging from atomic and condensed matter

physics to the present context of nuclear and hadronic physics.

The book’s central focus is to describe the current understanding of the sub-atomicworld within the framework of the Standard Model The layout of the book issummarized as follows: In the first quarter of the book, the Standard Model isdeveloped The structure of the nucleon and few-body nuclei are discussed in the secondquarter In the third quarter, the structure and properties of atomic nuclei are described.Lepton scattering is the principal tool used in the central narrative of the book tounderstand hadrons In the final quarter of the book we present extensions of the earlierfocus on EM lepton scattering to include the weak interactions of leptons with nucleonsand nuclei This begins with a chapter on beta-decay and progresses to intermediate-to-high energy neutrino-induced reactions These two chapters are followed by two morethat build on what occurs earlier in the book, namely, on applications to nuclear andparticle astrophysics and to studies of the hot, dense phase of matter formed in heavy-ion collisions The book closes with a brief perspective on physics beyond the StandardModel

We should also emphasize that the use of word “foundations” in the title of the book

is intentional, indicating that this text is not an encyclopedia where one might findmaterial on all of the major topics in the field, albeit at a superficial level Rather, wehave consciously made choices in what and what not to present We have, for instance,not developed the topic of intermediate-energy hadron scattering, emphasizing leptonscattering instead and have not attempted to cover the lattice approach to the solution ofQCD While the important areas of nuclear structure and the high-energy frontier arecovered, we note that excellent, up-to-date, comprehensive textbooks on these importantareas are available Our intent has been to provide the reader with basic material uponwhich to build by subsequently employing the more advanced sources that exist when itbecomes necessary for a more in-depth understanding of specific subjects In thisregard, we have included references to review articles, so that the interested reader canpursue material to a more advanced level Just what to emphasize and what merely torefer to in passing is, of course, subjective; however, having five co-authors hasallowed us to debate the choices we have made

We view the approximately 120 exercises provided throughout the book and located

at the end of each chapter as an important tool for the reader to consolidate theirunderstanding of the material in the book There exists significant variety in theseexercises, ranging from basic experimental issues to sophisticated theoretical questions.Many owe their origins to other sources, but we have tried to tailor them to the materialdiscussed here

The authors have all taught courses of the type described above at various levels.Specifically, at MIT the book covers the scopes set out for the introductory first-yeargraduate course in nuclear and particle physics (8.701), together with the second-yeargraduate courses in nuclear (8.711) and particle (8.811) physics All graduate students

in experimental nuclear/particle physics at MIT are required to take the latter two, withthe former being a prerequisite Additionally, at MIT there is an advanced undergraduatecourse in nuclear/particle physics (8.276), as well as more advanced courses in many-body theory (8.361), nuclear theory (8.712) and electroweak interactions (8.841) – all

origins to Professor Frank Wilczek We acknowledge that Chapter 19 was shaped by thework of Professor Berndt Müller and his colleagues We thank the Super-KamiokandeCollaboration for permission to use their image on the cover.

The book’s evolution profited from its use in draft form as a resource for the MITcourse 8.711 taught by one of us (RGM) and Dr Stephen Steadman in the springsemesters of 2014, 2015, and 2016 We acknowledge the constructive feedback from theMIT graduate students in those classes Further, we acknowledge careful and criticalreading of drafts by Dr Jan Bernauer, Charles Epstein, Dr Douglas Hasell, Dr RebeccaRussell, Dr Axel Schmidt, Dr Stephen Steadman, Reynier Cruz Torres and ConstantinWeisser at MIT, Professor James Napolitano, Dr Matt Posik, Devika Gunarathne,Amani Kraishan and Daniel Olvitt at Temple University, Rosi Reed at Lehigh Universityand Rosi Esha at UCLA We are grateful to Dr Brian Henderson for a careful reading ofall of the exercises We thank Connor Dorothy-Pachuta for his considerable expertise increating many of the figures in the book There are, of course, many others to thank who,over the years, have been our collaborators – we cannot list them all, but they will findtheir efforts reflected in many of our choices for what to present We do, however, wish

to acknowledge three who directly played roles in developing some of the figures in

Chapters 16 and 18, namely, Professors Maria Barbaro and Juan Caballero, andGuillermo Megias

In addition to being an integrated text, there are other aspects of this presentation that

we feel are important Specifically, we have attempted to make strong connections withcontemporary experiments and have tried, whenever possible, to help the reader becomeaware of the relevant frontier experimental facilities available and planned worldwide.Doing so is, of course, time dependent; but we have tried to be as up to date as possible

We have also made liberal use of the Particle Data Group website [PDG14] as aresource with which we encourage all students to become familiar Finally, in Appendix

A we have collected information that we believe will be useful to readers

Introduction

The past one hundred years has witnessed enormous advances in human understanding

of the physical universe in which we have evolved For the past fifty years or so, theStandard Model of the subatomic world has been systematically developed to providethe quantum mechanical description of electricity and magnetism, the weak interaction,and the strong force Symmetry principles, expressed mathematically via group theory,serve as the backbone of the Standard Model At this time, the Standard Model haspassed all tests in the laboratory Notwithstanding this success, most of the matteravailable to experimental physicists is in the form of atomic nuclei The most successfuldescription of nuclei is in terms of the observable protons, neutrons, and other hadronicconstituents, and not the fundamental quarks and gluons of the Standard Model Thus, theprofessional particle or nuclear physicist should be comfortable in applying thehadronic description of nuclei to understanding the structure and properties of nuclei.Experimentally, lepton scattering has proved to be the cleanest and most effective toolfor unraveling the complicated structure of hadrons Its application over differentenergies and kinematics to the nucleon, few-body nuclei, and medium- and heavy-massnuclei has provided the solid body of precise experimental data on which the StandardModel is built

In addition, the current understanding of the microcosm described in this bookprovides answers to many basic questions: How does the Sun shine? What is the origin

of the elements? How old is the Earth? Further, it underscores many aspects of modernhuman civilization, e.g., MRI imaging uses the spin of the proton, nuclear isotopes areessential medical tools, nuclear reactions have powered the Voyager spacecraft since

1977 into interstellar space

The purpose of the book is to allow the graduate student to understand thefoundations and structure of the Standard Model, to apply the Standard Model tounderstanding the physical world with particular emphasis on nuclei, and to establishthe frontiers of current research There are many outstanding questions that the StandardModel cannot answer In particular, astrophysical observation strongly supports theexistence of dark matter, whose direct detection has thus far remained elusive

Essential to making progress in understanding the subatomic world are thesophisticated accelerators that deliver beams of particles to experiments Existinglepton scattering facilities include Jefferson Laboratory in the US, muon beams atCERN, and University of Mainz and University of Bonn in Germany Intense photon

beams are used at the HIγ S facility at Duke University in the U.S., and in Japan at LEPS

at SPring-8, and at Elphs at Tohoku University Hadrons beams are used at the TRIUMF

Research (JINR), Dubna, Russia Neutron beams are used for subatomic physicsresearch at the Institut Laue-Langevin (ILL), Grenoble, France, at both the Los AlamosNeutron Science Center (LANSCE) and the Spallation Neutron Source (SNS) in the US,and at the future European Spallation Source (ESS) in Sweden The hot, dense matterpresent in the early universe is studied using heavy-ion beams at the Relativistic HeavyIon Collider (RHIC) in the US and at the Large Hadron Collider (LHC) at CERN Ofcourse, searches for new physics beyond the Standard Model are underway at the high-energy frontier of 13 TeV at CERN Understanding the structure of nuclei, withparticular emphasis on the limits of stability, is a major worldwide endeavor The mostpowerful facility at present is the Rare Isotope Beam Facility (RIBF) at RIKEN inJapan In the US, the frontier experiments at present are carried out at the NationalSuperconducting Cyclotron Laboratory at Michigan State University (MSU) and at theATLAS facility at Argonne National Laboratory A future Facility for Rare IsotopeBeams (FRIB) is under construction at MSU and is expected to have world-leadingcapabilities by 2022, as is a facility in South Korea, the Rare Isotope Science Project(RAON) Hadron beams for research are available at Los Alamos and the SpallationNeutron Source in the US, GSI in Germany, J-PARC in Japan, and NICA at Dubna,Russia A major new facility FAIR is planned at GSI Neutrino beams are generated atFermilab, CERN, and J-PARC and directed at detectors located both at the Earth’ssurface and deep underground A major new Deep Underground Neutrino Experiment(DUNE) is planned in the US using the Fermilab beam and the Sanford Underground

Research Laboratory in South Dakota Belle II, an experiment at the high luminosity e+e−

collider SuperKEKB in Japan, will come online within the next several years andprovide new stringent tests of flavor physics Annihilation of electrons and positrons is

used to probe the Standard Model at both the Double Annular ϕ Factory for Nice

Experiments (DAFNE) collider in Frascati, Italy as well as the Beijing ElectronPositron Collider (BEPC) in China Finally, a high luminosity electron–ion collider hasbeen widely identified by as the next machine to study the fundamental quark and gluonstructure of nuclei and machine designs are under development in the US, Europe, andChina

To begin, let us remind the reader of the particles that comprise the Standard Model(see Fig 1.1) As will be discussed in due course, the Standard Model starts withmassless particles and then, through spontaneous symmetry breaking, these interactingparticles acquire masses in almost all cases The measured spectrum of masses is still amystery; indeed, in the case of the neutrinos, intense effort is going into determining theactual pattern of masses in Nature Note that at this microscopic level, but also at thehadronic/nuclear level, when one says that particles interact with one another what ismeant is that some particle is exchanged between two other particles, thereby mediatingthe interaction For instance, an electron can exchange a photon with a quark whereby

the photon mediates the e − q interaction Or two nucleons (protons and neutrons) can exchange a pion and one has the long-range part of the NN interaction.

Fig 1.1 The particles of the Standard Model.

The organizational principle for this book centers on building from the underlyingfundamental particles (leptons, quarks, and gauge bosons) to hadrons (mesons and

baryons) built from q q and qqq, respectively, and on to many-body nuclei or

hypernuclei built from these hadronic constituents At very low energies and momentathe last are the relevant effective degrees of freedom, since, using the HeisenbergUncertainty Principle, such kinematics translate into large distance scales where themicroscopic ingredients are packaged into the macroscopic hadronic degrees offreedom Then, as the energy/momentum is increased, more and more of the sub-structure becomes relevant, until at very high energy/momentum scales the QCD degrees

of freedom must be used to represent what is observed

Naturally, there can be a blending between the different degrees of freedom and,where they overlap, it may be possible to use one language or the other And in somecases it turns out to be important to address both the “fundamental” physics issues andthe larger-scale nuclear structure issues at the same time This book attempts to presentthe foundations of the general field of nuclear/particle physics – sometimes calledsubatomic physics – in a single volume, trying to maintain a balance between the verymicroscopic QCD picture and the hadronic/nuclear picture

The outline of the book is the following After this introductory chapter, in Chapter 2

the basic ideas of symmetries are introduced In general discussions of quantum physics

it is often advantageous to exploit the exact (or at least approximate) symmetries in theproblem, for then selection rules emerge where, for instance, matrix elements betweenspecific initial and final states of certain operators can only take on a limited set of

operators (see Chapter 7) where conservation of angular momentum leads to a small set

of allowed values for matrix elements of such operators taken between states that haveknown spins Another example of an important (approximate) symmetry is provided byinvariance under spatial inversion, namely, parity: to the extent that parity is a goodsymmetry again only specific transitions can occur Other symmetries discussed in

Chapter 2 include charge conjugation and time reversal, as well as discrete unitaryflavor symmetries, the latter being important for classifying the hadrons built fromconstituent quarks, namely, the subject of Chapter 3

After these introductory discussions the book proceeds to build up from particles tohadrons to many-body nuclei, starting in Chapter 4 with the Standard Model of particlephysics In this one begins with massless leptons, quarks, and gauge bosons togetherwith the Higgs and then through spontaneous symmetry breaking generates the basicfamiliar building blocks with their measured masses The recent successful discovery ofthe Higgs boson at the Large Hadron Collider (LHC) is summarized

The Standard Model has proven to be extremely successful and, at the time ofwriting, there is as yet no clear evidence that effects beyond the Standard Model (BSM)are needed; in the final chapter of the book, Chapter 21 we return to summarize some ofthese BSM issues For the present, following the path of increasing complexity, in

Chapters 5 and 6 the ideas and models employed in descriptions of low-Q2, strongcoupling QCD are discussed in some detail, including what is not typically covered in abook at this level, namely, chiral symmetry

Chapters 7 through 10 form a distinct section where the aim is to visualize thestructure of the proton, neutron, and nuclei in terms of the fundamental quarks and gluons

of QCD At low and medium energies, this is carried out using lepton scattering whereintense beams of high quality are available Thus, snapshots of the nucleon charge andmagnetism and quark momentum and spin distributions are directly obtainable in theform of structure functions and form factor distributions Chapter 7 provides anintroduction to lepton scattering, including both parity-conserving and parity-violatingscattering Since lepton scattering is being used as a common theme in much of the rest

of the book, Chapter 7 is the first stop along the way where the multipole decomposition

of the electromagnetic current is developed in some detail This is followed in Chapter

8 by a discussion of elastic scattering from the nucleon At this time, a direct connectionbetween elastic scattering and QCD remains elusive and the most successful theoreticaldescription is in terms of hadrons Chapter 9 describes the current understanding of thestructure of hadrons in terms of high-energy lepton scattering and this is directlyinterpretable in terms of perturbative QCD Further, the gluon momentum and spindistributions are indirectly determined via the QCD evolution equations The partondistributions are snapshots of nucleon structure over different spatial resolutions andwith different shutter speeds Lepton scattering constitutes a theme of the book at bothhigh- and low-energy scales and with the full electroweak interaction Due to the lack ofsuitable lepton beams, QCD is at present probed at the highest energies using hadronbeams This is the focus of Chapter 10 and the measurements extend and complement

those with lepton beams in the previous chapters For example, direct experimentalinformation on the contribution of gluons to the spin of the proton has become possibleonly through polarized proton–proton collisions.

The above constitutes the first part of the book after which the building-up processmoves from hadrons to nuclei The next step is to deal with the simplest system that is

not a single baryon, namely, the system of two nucleons, discussing NN scattering and

the properties of the only bound state with baryons number two, the deuteron in Chapter

11 For the latter the EM form factors and electrodisintegration are treated in somedetail After this, in Chapter 12 the so-called few-body nuclei, those with A = 3 and 4,

constitute the focus

For nuclei heavier than the A = 2, 3, and 4 cases, treating the many-body problem

forms the basic issue, and accordingly in Chapter 13 an overview of the general nuclear

“landscape” is presented, showing the typical characteristics of nuclei, including theregions where nuclei are stable (the “valley of stability”) out to where they are justunstable (the “drip lines”), and their regions of especially tight binding (the “magicnumbers”) Also in this chapter the concept of infinite nuclear matter and neutron matter

is introduced and treated in some detail This is followed in Chapter 14 by a discussion

of a selection of typical nuclear models As mentioned earlier, this book is not intended

to be a theoretical text on nuclear many-body theory That said, this chapter hassufficient detail that the basic issues in this area can be appreciated Importantly, thetools used in this part of the field must be capable of dealing with nonperturbativeinteracting systems and accordingly this provides a theme in this chapter wherediscussions of the so-called Hartree–Fock (HF) and Random Phase Approximations(RPA) are provided together with an introduction to diagrammatic representations of theapproximations Also typical collective models are discussed as examples of how onemay start with some classical oscillation or vibration of the nuclear fluid, makeharmonic approximations to those movements, and then quantize the latter to arrive atsemi-classical descriptions of nuclear excitations (“surfons,” “rotons,” etc.), as is done

in many areas of physics where similar techniques are employed

The above discussions are then followed by two chapters focused on electronscattering from nuclei, Chapter 15 where elastic scattering is treated in some detail,together with some applications of the models introduced in Chapter 14 for low-lyingexcited states Chapter 16 continues this by treating higher-lying excitations wheredifferent modeling is required Specifically, the Relativistic Fermi Gas (RFG) model isderived and used as a prototype for more sophisticated approaches It is also the startingpoint for similar discussions of neutrino scattering from nuclei to follow in Chapter 18.Before those are presented, in Chapter 17 the weak interaction provides the focus and

we see how precision beta-decay experiments can be used as a probe for beyondStandard Model physics Chapter 18 deals with the subject of neutrinos and the fact thatone flavor can oscillate into another, since neutrinos are known to have mass At thetime of writing, the detailed nature of the mass spectrum, whether or not CP violation ispresent in the leptonic sector and whether neutrinos are Dirac or Majorana particles arestill under investigation and intensive efforts are being undertaken worldwide to shedlight on these interesting questions

modeling is somewhat different from that discussed in most of the rest of the book withstatistical mechanics being called into play together with fluid dynamics An informedpractitioner in the general field of nuclear/particle physics should be familiar with thissubject as well.

The book concludes with Chapter 20 on nuclear and particle astrophysics using many

of the concepts treated in the rest of the book, and with Chapter 21 where the types ofsignatures of effects beyond the Standard Model are summarized, together with twoappendices where some useful material is gathered

While we strongly advocate using the book to explore both nuclear and particlephysics in a coherent, balanced way, nevertheless it might be that it will also be used in

a course that emphasizes one subfield or the other Accordingly, we suggest thefollowing “road maps” to help the reader negotiate the text for those purposes When theemphasis is placed on particle physics we suggest paying the closest attention to

Chapters 2 to 10 and 21, with some parts of Chapters 17, 18, and perhaps 19, and whenthe emphasis is on nuclear physics Chapters 2, 7, 11 to 18, 20 and perhaps 19

We strongly recommend the following online resources as important tools forenhancing the material presented in this book

1 The Review of Particle Physics, Particle Data Group

http://pdg.lbl.gov includes a compilation and evaluation of measurements of theproperties of the elementary particles There is an extensive number of reviewarticles on particle physics, experimental methods, and material properties as well

as a summary of searches for new particles beyond the SM

2 National Nuclear Data Center

http://www.nndc.bnl.gov is a source of detailed information on the structure,properties, reactions, and decays of known nuclei It contains an interactive chart ofthe nuclides as well as a listing of the properties for ground and isomeric states ofall known nuclides

We conclude this introductory chapter with some exercises designed to introducesome of the concepts which we hope our particle/nuclear students will be able toaddress

Exercises

1.1 US Energy Production

In 2011, the United States of America required 3,856 billion kW-hours ofelectricity About 20% of this power was generated by ∼100 nuclear fissionreactors About 67% was produced by the burning of fossil fuels, which accountedfor about one-third of all greenhouse gas emissions in the US The remaining 13%was generated using other renewable energy resources Consider the scenariowhere all the fossil fuel power stations are replaced by new 1-GW nuclear fission

reactors How many such reactors would be needed?

1.2 Geothermal Heating

It is estimated that 20 TW of heating in the Earth is due to radioactive decay: 8 TWfrom 238U decay, 8 TW from 232Th decay, and 4 TW from 40K decay Estimate thetotal amount of 238U, 232Th, and 40K present in the Earth in order to produce suchheating

1.3 Radioactive Thermoelectric Generators

A useful form of power for space missions which travel far from the Sun is aradioactive thermoelectric generator (RTG) Such devices were first suggested bythe science fiction writer Arthur C Clarke in 1945 An RTG uses a thermocouple

to convert the heat released by the decay of a radioactive material into electricity

by the Seebeck effect The two Voyager spacecraft have been powered since 1977

by RTGs using 238Pu Assuming a mass of 5 kg of 238Pu, estimate the heat producedand the electrical power delivered (Do not forget to include the ∼ 5%thermocouple efficiency.)

1.4 Fission versus Fusion

Energy can be produced by either nuclear fission or nuclear fusion

a) Consider the fission of 235U into 117Sn and 118Sn, respectively Using the massinformation from a table of isotopes, calculate (i) the energy released perfission and (ii) the energy released per atomic mass of fuel

b) Consider the deuteron–triton fusion reaction

Using the mass information from the periodic table of the isotopes, calculate(i) the energy released per fusion and (ii) the energy released per atomic massunit of fuel

1.5 Absorption Lengths

A flux of particles is incident upon a thick layer of absorbing material Find theabsorption length, the distance after which the particle intensity is reduced by a

factor of 1/e ∼ 37% (the absorption length) for each of the following cases:

a) When the particles are thermal neutrons (i.e., neutrons having thermalenergies), the absorber is cadmium, and the cross section is 24,500 barns.b) When the particles are 2MeV photons, the absorber is lead, and the crosssection is 15.7 barns per atom

c) When the particles are anti-neutrinos from a reactor, the absorber is the Earth,and the cross section is 10−19 barns per atomic electron

are several important examples that are believed to be absolute symmetries and henceexact conservation laws Some of these specific examples are discussed in more detail

in what follows

Table 2.1 Exact conservation laws

Furthermore, there are symmetries that are not completely respected in Nature,although characterizing the states used in terms of eigenstates of these approximatesymmetries often proves fruitful; some examples are given in Table 2.2 We shall beusing all of these concepts throughout the book Next let us turn to a brief discussion ofsome of the basics needed when treating symmetries using group theory

Table 2.2 Approximate conservation laws

particle–antiparticle interchange charge conjugation, C

temporal inversion time-reversal invariance, T

transformations in isospace isospin, I (or T)

transformations in flavor space flavor

Representations

By an n-dimensional representation of a group G one means a mapping

(2.1)(2.2)

which assigns to every element g a linear operator A(g) in some n-dimensional complex vector space, the so-called carrier space of the representation GL(n), such that the image of the identity e is the unit operator I and that group operations are preserved

(2.3)

Throughout the book we shall frequently encounter infinite-dimensional continuousgroups (Lie groups) whose elements are labeled uniquely by a set of parameters whichcan change continuously (see [Rom64] for an introductory discussion) An example isprovided by the rotation group, that is, the group of continuous rotations For the Liegroups that are encountered frequently in this book it is sufficient to study the mapping

from the Lie algebra into GL(n),

(2.4)

where the {T α} preserve the Lie-algebra commutation relations If a subspace of the

carrier space of some representation is left unchanged by all operators T α, it is called aninvariant subspace and the representation is reducible; otherwise it is irreducible If thecorrespondence

When discussing the implications of symmetries in particle and nuclear physics one

frequently encounters the special unitary groups in N dimensions, SU(N), which can be represented using N × N matrices U satisfying

We shall see several examples of physical states labeled using various symmetries,

specifically by spin and by isospin (SU(2)), by flavor and by color (SU(3)), or by higher groups, e.g., SU(6) for spin-flavor Within the context of SU(N), a representation

is reducible if it is possible to choose a basis in which the matrices T α take the blockform

(2.10)

where A, B, C, are lower-dimensional irreducible sub-matrices when the original matrix T α is fully reduced Given an irreducible representation {T α}, the only linearoperators O which commute with every T α are multiples of the identity and also theconverse:

(2.11)Any unitary matrix can be written as

(2.12)

where H is a traceless Hermitian matrix For a Lie group the elements of the group are characterized by a finite number of real parameters {a α } and for SU(N) one finds that there are n = N2 − 1 such parameters Accordingly, one can write

(2.13)

where the {L α } form a basis for the N × N Hermitian matrices known as the generators

of the group SU(N) To study the representations, it is sufficient to study the generators

and their commutation relations,

(2.14)where the latter are characterized by the antisymmetric structure constants

2.2 Angular Momentum and SU(2)

Let us begin by discussing the representations of SU(2) in a systematic way The basis

space is three-dimensional and is spanned by S = (S1, S2, S3), that satisfy thecommutation relations [Edm74]

(2.15)

where ϵ ijk is the antisymmetric tensor, +1 if ijk is an even permutation of 123, −1 if an

odd permutation and zero otherwise In the carrier space a Hermitian scalar productexists:

(2.16)

Next we need to label the states in the carrier space using the Cartan subalgebra,

namely, the maximal set of mutually commuting operators that span the space For SU(2) the subalgebra only contains a single operator, usually chosen to be S z , where the z-axis

is chosen by convention to point in some convenient direction; later in Section 2.4 we

shall see that for SU(N) with N ≥ 3 the situation is more complicated The importance of

devising such a mutually commuting set is well-known from quantum mechanics: it isthen possible to diagonalize all of the matrices in the set simultaneously and to label thestates with the corresponding eigenvalues From this set of generators there are specialoperators that can be constructed which commute with all generators of the group,

namely, the so-called Casimir operators Again for SU(2) there is only one such operator (although more for SU(N) with N ≥ 3) namely the quadratic Casimir operator

(2.17)

As discussed above, such operators commute with all generators of the group,

(2.18)and hence must be proportional to the unit matrix, i.e., their eigenvalues may be used to

eigenvalues of the Casimir operator, and with quantum numbers m, the eigenvalues

belonging to the operators in the Cartan subalgebra,

(2.19)(2.20)

Since S2 and S z are Hermitian, λ and m are both real, and moreover, λ is positive and

may be chosen by convention to be

(2.21)

where j then labels the representation Correspondingly, we now have

(2.22)(2.23)

and, being eigenstates of Hermitian matrices, the states |j, m are orthogonal and can be

normalized Defining raising and lowering operators

(2.24)

it is straightforward to show that

(2.25)(2.26)

Next using Eq (2.25) one proves that, after operating on the states |j, m with the raising or lowering operators to form new states, S± |j, m, the latter are also

(2.29)and using the fact that

(2.30)(2.31)(2.32)one then has that

(2.33)Since is made up from quadratic Hermitian operators, one has that

raising and lowering operators acting on states |j, m yield real c-numbers times states with m ± 1:

(2.36)

Next let us focus on spin SU(2), taking j → S with m → S z and discuss the dimensional representations in somewhat more detail The simplest is the one-

lowest-dimensional, singlet representation (S = 0) with basis state |0, 0 and having S z |0, 0 =

S+ |0, 0 = S− |0, 0 = 0 The first nontrivial representation is the so-called fundamental

one, which for SU(2) is two-dimensional (S = 1/2) with basis states |S = 1/2, S z =

±1/2

Letting S± and S z act on the basis states, it is straightforward to obtain explicitexpressions for the representation matrices:

(2.38)

or equivalently

(2.39)

Conventionally one writes S i ≡ σ i /2, thereby defining the Pauli matrices σ i , with i = 1, 2,

3, corresponding to x, y, z, respectively; we use the two types of notation interchangably.

A more complicated case is the one for S = 1 (dimension three) with basis states labeled

This is the so-called adjoint or regular representation This is an example of an N2 − 1

dimensional representation of SU(N) given by the mapping in Eqs (2.1) and (2.2) withstructure constants (see Eq (2.14))

(2.43)

Later when building hadrons in Chapter 3 we shall find it convenient to use weightdiagrams Since the generator in the Cartan subalgebra can be used to label states of arepresentation, the corresponding eigenvalues can be plotted in a diagram of this type,

which here for SU(2) amounts to drawing a line with dots to indicate where the

eigenvalues occur, as shown in Fig 2.1 Below we shall see that in SU(N) with N ≥ 3

one has patterns in (N − 1)-dimensional space.

Fig 2.1 Weight diagrams for SU(2) for spins S =1/2, 1, and 3/2.

Coupling of Angular Momentum

By taking the direct product of two representations, we find a new representation which

in general is reducible For instance, as an example in SU(2) let us consider the direct product of two S = 1/2 (two-dimensional) representations (see Eq (2.37)), written in

the following way |S(1) = 1/2, S3(1) = ±1/2; S(2) = 1/2, S3(2) = ±1/2, now for brevitysimply indicated |±±, yielding four states

(2.44)with

it is straightforward to check that the matrices in Eq (2.46) are now reduced to blockform with a 3×3 block and a 1×1 block along the diagonal, the former being therepresentation found above for spin 1 (triplet) and the latter being for spin 0 (singlet).

Using the quantum number S to label a representation one can write

(2.48)

or, equivalently, using the dimensions of the representations

(2.49)

both types of notation will be used and will be generalized for SU(N) with N ≥ 3.

Equations (2.48) and (2.49) are the most basic versions of what is called theClebsch– Gordan series: the product of two states with given angular momenta can berewritten (recoupled) and expressed as a sum of states with good angular momentum

quantum numbers For SU(2), which is involved in recoupling angular momenta, only

one state of each angular momentum occurs,

(2.50)

where jmin = |j1 − j2| and jmax = j1 + j2 Later we shall see that the situation for SU(N) with N ≥ 3 is more complicated The Clebsch–Gordan or vector coupling coefficients,

j1m1j2m2|j1j2jm, enter when states in a coupled scheme |( j1j2)jm are written in terms

of states in an uncoupled scheme |j1m1; j2m2 :

(2.51)

here the other quantum numbers characterizing the states are suppressed These

coefficients are related to the 3 -j symbols which are generally more convenient to use

(we employ the conventions of [Edm74] throughout the book):

special 3 -j symbol that occurs frequently is the so-called parity coefficient,

(2.57)and inverse

(2.58)

The tensors satisfy the following commutation relations with the angular momentumoperators

(2.59)(2.60)

where From this a very useful result can be obtained (see[Edm74]), namely, the Wigner–Eckart theorem:

(2.61)

fact proportional to a single number, the reduced matrix element, weighted by

well-known coefficients, the 3 -j symbols.

2.3 SU(2) of Isospin

Because of their closeness in mass and the very similar roles they play, as will bediscussed later in Chapter 8, one may consider the proton (mass m p = 938.27 MeV) and

neutron (mass m n = 939.57 MeV) to be two states of a common particle, the nucleon N.

In analogy to a system with spin S = 1/2 and spin projections S3 = ±1/2 (see Section 2.2)

one may assign a new quantum number to the nucleon, namely, isospin T = 1/2 The convention typically used in particle physics is to assign an isospin projection T3 = +1/2

to the proton and −1/2 to the neutron, although in nuclear physics the oppositeconvention is sometimes employed In this book we use the former convention and so

one has the two basic states of the nucleon, written |T = 1/2, T3 = ±1/2, respectively:

(2.62)(2.63)The analog of Eq (2.13) is

(2.64)

As with spin, for isospin one has raising and lowering operators T± = T1 ± iT2 thatsatisfy the analogs of Eqs (2.25,2.26) and of Eq (2.36)

(2.65)

which will be employed in Chapter 3 when moving across the isospin subspace of the

baryon multiplets We also have T i ≡ τ i/2, yielding the analogs of the Pauli matrices (seeEqs (2.39))

(2.66)

that obey the relations

(2.67)

We shall also see in Chapter 3 that isospin will be used in building hadrons from

quarks In this case the fundamental representation 2 (at the level of SU(2)) will be the

doublet {|u, | d}, up and down, corresponding to isospin 1/2 with projections ±1/2,

If we change to the basis then the representations in Eqs (2.66) and (2.71)

become identical Note that for SU(2) the conjugate representation is equivalent to the fundamental representation – we shall see that this is not the case for SU(N) with N ≥ 3.

As with ordinary spin, it is possible when considering many-particle systems to

vector-couple the isospins For example, when treating NN scattering in Chapter 11, onehas an isosinglet (the analog of the spin singlet discussed above)

(2.72)and the three states of the isotriplet

(2.73)(2.74)(2.75)

In strong-interaction physics the isospin appears to be a good or nearly good symmetryand so, for example, the nuclear force is (approximately) invariant under isospin

transformations Note, however, that it is broken by the electromagnetic interaction – pp interactions differ from nn and pn interactions, for instance.

In analogy to the developments in the previous section for the case of SU(2) with up and

down quarks an extended basis of three states, up, down, and strange, occurs whendiscussing flavor for low-lying mesons and baryons (see Chapter 3) and the special

unitary group SU(3) becomes relevant:

(2.76)

As in Eq (2.64) one has

(2.77)

now with eight transformation matrices rather than three for SU(2) making up the

representation (the “eight-fold way”) When acting on the basis states in Eqs (2.76)these matrices induce transformations within the basis, just as the Pauli matrices induce

transformations within the doublet space of SU(2) in Eqs (2.70) Defining the

Gell-Mann λ-matrices by λ a ≡ 2F a these are conventionally given by

where the ds are symmetric and the f s are antisymmetric under interchange of any two indices One has the following nonzero SU(3) structure constants

of SU(N) using the elegant constructions called Young tableaux Following [Clo79] westate the rules without proof and give a few examples (more examples are found in the

Exercises) The fundamental representation N in SU(N) of dimension N is denoted by a box □, and the conjugate representation N by a column of N − 1 boxes As noted above,

in SU(2) the fundamental 2 and conjugate 2 representations are the same, namely, a

single box, whereas for SU(N) with N ≥ 3 this is not the case; for instance, for SU(3) the

fundamental 3 and conjugate 3 representations are those shown in Fig 2.2

Fig 2.2 The fundamental and conjugate representations of SU(3).

The next set of rules is the following: Any row of boxes is totally symmetric under

singlet For example, two boxes in a column in SU(2) for spin denotes the singlet spin-0 state (likewise for the isosinglet state) and three boxes in a column in SU(3) flavor

denotes the totally antisymmetric flavor singlet state As in the above discussions ofcoupling of angular momenta, one can take the direct product of two or morerepresentations and decompose this into a direct sum of representations For instance,suppose that two fundamental representations are multiplied together, then one obtainstwo-box tableaux of the type shown in Fig 2.3

Fig 2.3 Multiplication of two fundamental representations.

As we shall see in Chapter 3, it is important to be able to determine the dimension of

a specific tableau and hence the number of particles that fit into related multiplets Therules for computing the dimension may be found in [Clo79]; these yield a ratio of two

numbers which is the dimension of the representation Working in SU(N), for the numerator one starts in the upper left-hand corner and inserts the number N down the diagonal of the tableau, then N + 1 for the next box to the right of the left-hand corner and down the diagonal lying above the main diagonal, N + 2 for the next box to the right and down its diagonal, and so on, and then one does the same with N −1 starting with the

box below the upper left-hand corner and its diagonal, and so on The structure obtained

is made clearer with an example – see Fig 2.4

Fig 2.4 Labeling of Young tableaux to determine the dimension of the representation.

Note that only tableaux of this type, namely concave downwards and to the right, areallowed Then the numerator of the ratio is given by the product of all of these numbers

For the denominator one uses the following rule: one draws a line entering the tableaufrom the right-hand side for each row, for each line one terminates the line in allpossible ways, i.e., ending in all possible boxes it encounters, and for each choice theline turns downwards exiting the tableau via the particular column being considered Allsuch constructions, which Close [Clo79] calls “hooks,” are made, the number of boxesencountered enumerated and then finally the denominator is the product of all of thosenumbers The detailed proofs of how to determine the dimension of a specific tableauare given in [Ham62] Again an example should help to make this rule clear Consider

two boxes in a row in SU(N) as in Fig 2.5 The numerator is the product N(N + 1) and the denominator is 2, yielding dimension N(N + 1)/2 For two boxes in a column in SU(N) as in Fig 2.6, one finds a numerator of N(N − 1) and a denominator of 2, yielding dimension N(N − 1)/2 For SU(2) these two dimensions are 3 and 1, respectively, while for SU(3) they are 6 and 3, namely one has found that (see also Eq (2.49))

Fig 2.5 Illustration of the hook rule for two boxes in a row.

Next let us take the direct product of three fundamental representations Starting fromthe two results above with either two boxes in a row or two in a column, the procedure

is to add a third box in all posible ways that yield diagrams that are concavedownwards and to the right, resulting in the tableaux in Fig 2.7 Using the rules stated

above the dimensions are immediately found to be N(N +1)(N +2)/6 and (N −1)N(N +1)/3 for the upper left-and right-hand tableaux, respectively, and N(N − 1)(N − 2)/6 and (N − 1)N(N + 1)/3 for the lower left- and right-hand tableaux, respectively For SU(2) the tableau with a column of three boxes cannot occur, whereas it can for SU(3),

and so we arrive at the answers

namely, a quartet and two doublets for SU(2) and a decuplet, two octets, and a singlet for SU(3) In Chapter 12 few-body nuclei will be discussed and there one has both spin

and isospin as SU(2) properties of the three-body states obtained for 3He and 3H, and in

Chapter 3 we shall see how to build low-lying baryons from triplets of u, d, and s 1/2) quarks requiring both the SU(2) characterization of the spin content together with the SU(3) characterizations of their flavor and color The tableau with three boxes in the

(spin-same row is completely symmetric, the one with three in a column is completelyantisymmetric, whereas the others with both rows and columns is of mixed symmetry,i.e., for particles in the first row (column) it is symmetric (antisymmetric), but has nogood permutation symmetry between the upper right-hand and lower boxes

Fig 2.7 Direct products of three fundamental representations (see Fig 2.3 ).

Finally, let us consider the direct product of the fundamental representation N (a single box) with the conjugate representation N, namely, a column of N − 1 boxes as

shown in Fig 2.8; here SU(2) is uninteresting, since a single box represents both,

although for SU(N) with N ≥ 3 important new results are found One has N ⊗ N = 1 ⊕ (N2−1), namely, a singlet and a representation of dimension N2 − 1 For instance, in

SU(3) this yields the result in Fig 2.9 and using the above rules one finds that 3 ⊗ 3 = 1

⊕ 8, that is, one has a singlet and an octet These will be used in Chapter 3 when

discussing the flavor structure of the low-lying mesons built from qq pairs of u, d, and s

quarks

Fig 2.8 Direct product of the fundamental and conjugate representations.

Fig 2.9 Direct product of the fundamental and conjugate representations in SU(3).

2.6 Discrete Symmetries: P, C, and T

A finite group is one which contains only a finite number of elements In particle andnuclear physics we encounter very simple discrete groups with just two elements,

namely the identity e and an element g satisfying g2 = e For such groups, the element g

This means that if the system is an eigenstate of U then transitions can only occur

between eigenstates with the same eigenvalue ±1 As mentioned at the beginning of thischapter, even when the symmetry is not exact it may still be useful to characterize thetrue states in terms of eigenstates of the symmetry

A first important example of such a bi-modal symmetry is spatial inversion

(2.86)

with associated parity eigenvalues P = ±1 (even or odd, respectively) A few things to

know about parity, some of which will be the subject of later discussions in the book,are the following: (1) Parity is a multiplicative quantum number; (2) Parity is believed

to be conserved in strong and electromagnetic interactions, but is certainly violated bythe weak interaction; (3) Fermion states have opposite parities for particles andantiparticles, whereas bosons have the same parities for particles and antiparticles; and

(4) Fermions and bosons may be catalogued according to their spin-parity, J P, integer for the former and integer for the latter Bosons are denoted 0+ (scalar), 0−(pseudoscalar), 1− (vector), 1+ (axial-vector), and so on For example, the pion is a 0−pseudoscalar meson

half-One example of using parity properties when discussing quantum systems is provided

by the case of solutions in a spherically symmetric potential There one has H(r) = H(−r) = H(r), so that [P, H] = 0 and bound states in the potential have definite parity A

simple example involves a system moving in such a potential having no other internaldegrees of freedom where the wavefunctions have the form

(2.87)

where is a spherical harmonic [Edm74] It can be shown that making the

transformation r → −r, which implies that θ → π − θ and ϕ → π + ϕ yields

and consequently such wavefunctions have parity P = (−) L Anotherexample involves electromagnetic transitions between nuclear states which will bediscussed in more detail in Chapter 7 There it will be shown that electric transitions of

multipolarity J entail natural parity changes, ΔP = (−) J, whereas magnetic transitions are

the opposite, having non-natural parity changes, ΔP = (−) J+1 Thus, given good parityquantum numbers for two nuclear states, only transitions of a given multipolarity of onetype or the other can occur For instance, if an excited state with spin-parity 1− decays