ADVANCES INNUCLEAR PHYSICSVOLUME 27

CONTRIBUTORS TO THIS VOLUME

Stefan SchererInstitut fr KernphysikJohannes Gutenberg-Universitt MainzMainz, Germany

A Continuation Order Plan is available for this series. A continuation order will bringdelivery of each new volume immediately upon publication. Volumes are billed onlyupon actual shipment. For further information please contact the publisher.

Igal TalmiThe Weizmann Institute of ScienceRehovot, Israel

ADVANCES INNUCLEAR PHYSICS

E. W. VogtDepartment of PhysicsUniversity of British ColumbiaVancouver, B.C., Canada

VOLUME 27

Edited byJ. W. NegeleCenter for Theoretical PhysicsMassachusetts Institute of TechnologyCambridge, Massachusetts

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ARTICLES PUBLISHED IN EARLIER VOLUMES

Volume 1The Reorientation Effect J. de Boer and J. EicherThe Nuclear Model M. HarveyThe Hartree-Fock Theory of Deformed Light Nuclei G. RipkaThe Statistical Theory of Nuclear Reactions E. VogtThree-Particle ScatteringA Review of Recent Work on the Nonrelativistic Theory

I. Duck

Volume 2The Giant Dipole Resonance B. M. SpicerPolarization Phenomena in Nuclear Reactions C. Glashausser and J. ThirionThe Pairing-Plus-Quadrupole Model D. R. Bes and R. A. SorensenThe Nuclear Potential P. SignellMuonic Atoms S. Devons and I. Duerdoth

Volume 3The Nuclear Three-Body Problem A . N. MitraThe Interactions of Pions with Nuclei D. S. KoltunComplex Spectroscopy J. B. French, E. C. Halbert, J. B. McGrory, and S. S. M. WongSingle Nucleon Transfer in Deformed Nuclei B. Elbeck and P. O. TjmIsoscalar Transition Rates in Nuclei from the Reaction A. M. Bernstein

Volume 4The Investigation of Hole States in Nuclei by Means of Knockout and Other Reactions

Daphne F. JacksonHigh-Energy Scattering from Nuclei Wieslaw CzyzNucleosynthesis by Charged-Particle Reactions C. A. BarnesNucleosynthesis and Neutron-Capture Cross Sections B. J. Allen, J. H. Gibbons, and

R. L. MacklinNuclear Structure Studies in the Z = 50 Region Elizabeth Urey BarangerAn s-d Shell-Model Study for A = 18 22 E. C. Halbert, J. B. McGrory, B. H. Wildenthal,

and S. P. Pandy

Volume 5Variational Techniques in the Nuclear Three-Body Problem L. M. DelvesNuclear Matter Calculations Donald W. L. SprungClustering in Light Nuclei Akito Arima, Hisashi Horiuchi, Kunihara Kubodera, and

Noburu Takigawa

v

Articles Published in Earlier Volumes

Nuclear Fission A. MichaudonThe Microscopic Theory of Nuclear Effective Interactions and Operators Bruce R. Barrett

and Michael W. KirsonTwo-Neutron Transfer Reactions and the Pairing Model Ricardo Broglia, Ole Hansen, and

Claus Riedel

Volume 11Clustering Phenomena and High-Energy Reactions V. G Neudatchin, Yu. F. Smirnov, and

N. F. GolovanovaPion Production in Proton-Nucleus Collisions B. HolstadFourteen Years of Self-Consistent Field Calculations: What Has Been Learned J. P. SvenneHartree-Fock-Bogoliubov Theory with Applications to Nuclei Alan L. GoodmanHamiltonian Field Theory for Systems of Nucleons and Mesons Mark Bolsterli

Volume 7Nucleon-Nucleus Collisions and Intermediate Structure Aram MekjianCoulomb Mixing Effects in Nuclei: A Survey Based on Sum Rules A. M. Lane and

A. Z. MekjianThe Beta Strength Function P. G. HansenGamma-Ray Strength Functions G. A. Bartholemew, E. D. Earle, A. J. Ferguson,

J. W. Knowles, and M. A. Lone

Volume 8Strong Interaction in A-Hypernuclei A . GalOff-Shell Behavior of the Nucleon-Nucleon Interaction M. K. Strivastava and

D. W. L. SprungTheoretical and Experimental Determination of Nuclear Charge Distributions J. L. Friar and

J. W. Negele

Volume 9One- and Two-Nucleon Transfer Reactions with Heavy Ions Sidney Kahana and A. J. BaltzComputational Methods for Shell-Model Calculations R. R. Whitehead, A. Watt, B. J. Cole,

and I. MorrisonRadiative Pion Capture in Nuclei Helmut W. Baer, Kenneth M. Crowe, and Peter Trul

Volume 10Phenomena in Fast Rotating Heavy Nuclei R. M. Lieder and H. RydeValence and Doorway Mechanisms in Resonance Neutron Capture B. J. Allen and

A. R. de L. MusgroveLifetime Measurements of Excited Nuclear Levels by Doppler-Shift Methods

T. K. Alexander and J. S. Forster

vi

Volume 6

Volume 12Hypernetted-Chain Theory of Matter at Zero Temperature J. G. ZabolitzkyNuclear Transition Density Determinations from Inelastic Electron Scattering

Jochen HeisenbergHigh-Energy Proton Scattering Stephen J. Wallace

Volume 13Chiral Symmetry and the Bag Model: A New Starting Point for Nuclear Physics

A. W. ThomasThe Interacting Boson Model A. Arima and F. IachellaHigh-Energy Nuclear Collisions S. Nagamiya and M. Gyullasy

Volume 14Single-Particle Properties of Nuclei Through (e, ) Reactions Salvatore Frullani and

Jean Mougey

Volume 15Analytic Insights into Intermediate-Energy Hadron-Nucleus Scattering R. D. AmadoRecent Developments in Quasi-Free Nucleon Scattering P. Kitching, W. J. McDonald,

Th. A. J. Maris, and C. A. Z. VasconcellosEnergetic Particle Emission in Nuclear Reactions David H. Boal

Volume 16The Relativistic Nuclear Many-Body Problem Brian Serot and John Dirk Walecka

Volume 17P-Matrix Methods in Hadronic Scattering B. L. G. Bakker and P. J. MuldersDibaryon Resonances M. P. Locher, M. E. Saino, and A. varcSkrymions in Nuclear Physics Ulf-G. Meissner and Ismail ZahedMicroscopic Descriptions of Nucleus-Nucleus Collisions Karlheinz Langanke and

Harald Friedrich

Volume 18Nuclear Magnetic Properties and Gamow-Teller Transitions A. Arima, K. Shimizu,

W. Bentz, and H. HyugaAdvances in Intermediate-Energy Physics with Polarized Deuterons J. Arvieux and

J. M. CameronInteraction and the Quest for Baryonium C. Amsler

Radiative Muon Capture and the Weak Pseudoscalar Coupling in Nuclei M. Gmitro andP. Trul

Introduction to the Weak and Hypoweak Interactions T. Goldman

Articles Published in Earlier Volumes vii

Articles Published in Earlier Volumes

Volume 25Chiral Symmetry Restoration and Dileptons in Relativistic Heavy-Ion Collisions R. Rapp

and J. WambachFundamental Symmetry Violation in Nuclei H. Feshbach, M. S. Hussein, A. K. Kerman, and

O. K. VorovNucleon-Nucleus Scattering: A Microscopic Nonrelativistic Approach K. Amos,

P. J. Dortmans, H. V. von Geramb, S. Karataglidis, and J. Raynal

viii

Volume 24Nuclear Charge-Exchange Reactions at Intermediate Energy W. P. Alford and B. M. SpicerMesonic Contributions to the Spin and Flavor Structure of the Nucleon J. Speth and

A. W. ThomasMuon Catalyzed Fusion: Interplay between Nuclear and Atomic Physics

K. Nagamine and M. Kamimura

Volume 23Light Front Quantization Matthias BurkardtNucleon Knockout by Intermediate Energy Electrons James J. Kelly

Volume 22Nucleon Models Dan Olof RiskaAspects of Electromagnetic Nuclear Physics and Electroweak Interaction T. W. DonnellyColor Transparency and Cross-Section Fluctuations in Hadronic Collisions Gordon BaymMany-Body Methods at Finite Temperature D. VautherinNucleosynthesis in the Big Bang and in the Stars K. Langanke and C. A. Barnes

Volume 21Multiquark Systems in Hadronic Physics B. L. G. Bakker and I. M. NarodetskiiThe Third Generation of Nuclear Physics with the Microscopic Cluster Model

Karlheinz LangankeThe Fermion Dynamical Symmetry Model Cheng-Li Wu, Da Hsuan Feng,

and Mike Guidry

Volume 20Single-Particle Motion in Nuclei C. Mahaux and R. SartorRelativistic Hamiltonian Dynamics in Nuclear and Particle Physics B. D. Keister and

W. N. Polyzou

Volume 19Experimental Methods for Studying Nuclear Density Distributions C. J. Batty, E. Friedman,

H. J. Gils, and H. RebelThe Meson Theory of Nuclear Forces and Nuclear Structure R. Machleidt

Volume 26The Spin Structure of the Nucleon B. W. Filippone and Xiangdong JiLiquid-Gas Phase Transition in Nuclear Multifragmentation S. Das Gupta, A. Z. Mekjian,

and M. B. TsangHigh Spin Properties of Atomic Nuclei D. Ward and P. FallonThe Deuteron: Structure and Form Factors M. Garon and J. W. Van Orden

Articles Published in Earlier Volumes ix

This volume contains two major articles, one providing a historical retrospec-tive of one of the great triumphs of nuclear physics in the twentieth centuryand the other providing a didactic introduction to one of the quantitative toolsfor understanding strong interactions in the twenty-first century.

The article by Igal Talmi on Fifty Years of the Shell Model the Questfor the Effective Interaction, pertains to a model that has dominated nuclearphysics since its infancy and that developed with astonishing results over thenext five decades. Talmi is uniquely positioned to trace the history of the ShellModel. He was active in developing the ideas at the shell models inception,he has been central in most of the subsequent initiatives which expanded, clar-ified and applied the shell model and he has remained active in the field tothe present time. Wisely, he has chosen to restrict his review to the dominat-ing issue: the choice of the effective interactions among valence nucleons thatdetermine the properties of low lying nuclear energy levels.

The treatment of the subject is both bold and novel for our series. Theideas pertaining to the effective interaction for the shell model are elucidatedin a historical sequence. In a massive article, which will be valued both forits completeness and its sound judgment of the various contributions to thesubject, Talmi succeeds without use of a single figure comparing model re-sults with each other or with experimental data. Instead, the compelling flowof ideas and results for the Shell Model are described in a manner that makesclear why this magnificent edifice has had such great resilience. Many of theideas of the Shell Model are very elegant. They converge on a picture of nu-clear states that is so all-encompassing that there has been very little room, sofar, for the intrusion of subnucleon physics. Talmi takes us right up to the mostrecent decade in which very large computers have been able to verify and ex-pand upon earlier views of the effective interaction by directly diagonalizingthe matrices pertaining to the myriad of nuclear states that arise from severalvalence nucleons beyond closed shells. This review is then a celebration of amost successful model of atomic nuclei and of its underlying ideas.

xi

PREFACE

The second article of this volume is also massive and pertains to a subjectthat has become very important for the field. Chiral Perturbation Theory hasbeen one of the few quantitative tools for coping with the theory of the stronginteractions, QCD, which is both remarkably simple to express as a fundamen-tal Lagrangian and notoriously intractable to solve. Chiral Perturbation Theoryis particularly useful for the nucleon-nucleon and the nucleon-meson interac-tions. Stefan Scherer has worked on the theory for more than a decade and hastaught a course on it at the University of Mainz. He has chosen in his present re-view to write a didactic article which should be especially helpful for nuclearphysicists who are entering the field. The subject has had major critics whohave worried about the convergence of the theory and also about its relation toeffective interactions. These subjects are dealt with clearly in this review.

Our series has now produced twenty-seven volumes and the technologyproducing it, which began thirty five years ago with an article set in the hotlead of linotype machines, has now evolved to electronic formatting and pro-duction. Having recently concentrated our effort in the successful transitionto this new technology, we are now positioned to devote renewed effort to thecommissioning of review articles. With the ever increasing breadth on contem-porary nuclear physics and related fields, the need for insightful, pedagogicalreviews has never been greater.

J.W. NEGELEE.W. VOGT

Prefacexii

CONTENTS

1.2.

3.

4.

Chapter 1FIFTY YEARS OF THE SHELL MODEL

THE QUEST FOR THE EFFECTIVEINTERACTION

Igal Talmi2

1112192626374453

58626873858991

102109

xiii

IntroductionThe (Re)Emergence of the Shell Model2.1.2.2.

Single Nucleon OrbitsThe Supermultiplet Scheme

Early Calculations3.1.3.2.3.3.3.4.3.5.

Energy Levels of Simple ConfigurationsNuclear Magnetic MomentsElectromagnetic Moments and Transitions. Beta DecaySeniority and theApplications of the to Even-Even and Odd-OddNuclei

3.6.3.7.3.8.3.9.

Simple Potential Interactions. Configuration MixingThe Beta Decay ofSimple Potential Interactions. Ignoring the Hard CoreThe Pb Region I

Effective Interactions from Experimental Nuclear Energies4.1.4.2.4.3.

Simple ConfigurationsProtons and Neutrons in Different Orbits.The Zr Region I

5.

6.

7.

8.

Chapter 2INTRODUCTION TO CHIRAL

PERTURBATION THEORYStefan Scherer

xiv Contents

4.4.4.5.4.6.4.7.4.8.4.9.

115121133140144168175179181181

193202

215215220

232232241249253265

The ShellMixed Configurations in the ShellThe Nickel IsotopesThe Pb Region IIThe Zr region IIThe ShellThe Complete and BeyondPseudonium Nuclei

Some Schematic Interactions and Applications5.1.5.2.

5.3.

The Pairing InteractionThe Surface Delta Interaction, the Quasi-Spin Scheme andExtensionsThe SU (3) Scheme

Seniority and Generalized Seniority in Semi-Magic NucleiThe Seniority Scheme and Applications6.1.

6.2. Generalized SeniorityLarge Scale Shell Model Calculations7.1.7.2.7.3.

The and BeyondTheNuclei in which N = 20 Loses Its Magicity

What have we learned? Where do we stand?

References

1.

2.

Introduction 278278

279290290

1.1.1.2.

2.1.QCD and Chiral Symmetry

Scope and Aim of the ReviewIntroduction to Chiral Symmetry and Its Application to Mesonsand Single Baryons

Some Remarks on SU(3)

4.10.4.11.

Index

3.

4.

5.

6.

xvContents

2.2.2.3.2.4.

The QCD LagrangianAccidental, Global Symmetries ofGreen Functions and Chiral Ward Identities

293297310323324

Spontaneous Symmetry Breaking and the Goldstone Theorem3.1.3.2.

3.3.3.4.

4.1.4.2.4.3.4.4.4.5.4.6.4.7.4.8.4.9.4.10.

5.1.5.2.5.3.5.4.5.5.5.6.

A.1.A.2.A.3.A.4.

330335338

339340347352360365370380384388405417418423427440449483496501501507512522529

539

Chiral Perturbation Theory for Mesons

Degenerate Ground StatesSpontaneous Breakdown of a Global, Continuous, Non-AbelianSymmetryGoldstones TheoremExplicit Symmetry Breaking: A First Look

Spontaneous Symmetry Breaking in QCDTransformation Properties of the Goldstone BosonsThe Lowest-Order Effective LagrangianEffective Lagrangians and Weinbergs Power Counting SchemeConstruction of the Effective LagrangianApplications at Lowest OrderThe Chiral Lagrangian at OrderThe Effective Wess-Zumino-Witten ActionApplications at OrderChiral Perturbation Theory at

Chiral Perturbation Theory for BaryonsTransformation Properties of the FieldsLowest-Order Effective Baryonic LagrangianApplications at Tree LevelExamples of Loop DiagramsThe Heavy-Baryon FormulationThe Method of Infrared Regularization

Summary and Concluding RemarksAppendix

Green Functions and Ward IdentitiesDimensional Regularization: BasicsLoop IntegralsDifferent Forms of in SU(2) SU(2)

References

Chapter 1

FIFTY YEARS OF THE SHELL MODEL THE QUEST FOR THE EFFECTIVEINTERACTION

Igal TalmiThe Weizmann Institute of ScienceRehovot, Israel

1.

2.

3.

4.

5.

6.

7.

8.

Introduction

The (Re)Emergence of the Shell ModelEarly Calculations

Effective Interactions from Experimental Nuclear Energies

Some Schematic Interactions and Applications

Seniority and Generalized Seniority in Semi-Magic Nuclei

Large Scale Shell Model Calculations

What have we learned? Where do we stand?

References

2

11

26

89

181

215

232

253

265

1

1. INTRODUCTION

In 1999 we celebrated 50 years of the modern version of the shell modelfor nuclei. In this model it is assumed that the nuclear constituents - protonsand neutrons - move independently in a central potential well. This well shouldbe due to the average interaction between these constituents, the nucleons. Toobtain the observed spacings between single nucleon orbits, the potential wellshould include a rather strong interaction between the spin and orbital angularmomentum of each nucleon. The strong spin-orbit interaction turned out to bean essential ingredient. It was introduced by Maria G. Mayer (1949) and in-dependently by Haxel, Jensen and Suess (1949) and led to the observed singlenucleon orbits and the correct magic numbers for which major shells are com-pletely filled. The shell model revolutionized the theory of nuclear physics.The field mostly affected was nuclear structure theory which deals with energylevels of nuclei, their wave functions and electromagnetic and beta transitionsbetween them. Another wide field, of various reactions between nuclei and be-tween nucleons and nuclei, was also strongly affected. The shell model wasimposed on nuclear theorists by a large variety of experimental data summa-rized by M.G. Mayer (1948). Theoretical developments inspired experimental-ists and, more frequently, novel experiments presented challenges to theorists.To review the impact of the shell model on nuclear physics in the first 50 years,a real encyclopaedia would be necessary. This review has a rather modest aim.It deals with the rather limited goal of calculation of certain energies and wavefunctions within the shell model.

The calculation of nuclear energies is a very difficult problem. The spe-cial stability of nuclei with magic proton and neutron numbers follows directlyfrom their energy levels in the potential well. Still, the average potential is dueto two-nucleon interactions and, hence, a better approximation to the potentialenergy could be obtained by calculating them using shell model wave func-tions. This, however, is very difficult for several important reasons. First, theinteraction between free protons and neutrons is not sufficiently well known.Results of scattering experiments and properties of the deuteron have been fit-ted by several, rather phenomenological, realistic interactions. Still, in theabsence of an established theory for these interactions, no definite prescriptionis available. The internal structure of protons and neutrons in terms of quarksand gluons adds much to the complications. This problem, however, is onlypart of the difficulty of calculating nuclear energies from first principles.The solution of the many body problem is usually extremely complicated. Thestrong and singular interaction between free nucleons makes ordinary pertur-bation methods useless. Motivated by the apparent success of the shell model,nuclear many body theorists have tried to derive a renormalized (effective) in-

2 Igal Talmi

teraction which could be used with shell model wave functions to calculatenuclear energies. Much extensive and intensive theoretical research has beencarried out in this direction. Most results were obtained for 3 and 4 nucleonsystems or for nuclear matter. Properties of the latter have been extrapolatedfrom data on heavy nuclei since this species is not available for experiment.The results are encouraging but not definitive. It is still not clear what roleis played by effective 3 body interactions and those involving more nucleons.Such effective interactions are not excluded by many body theory even if theydo not appear in the original interaction between nucleons.

Starting from the shell model, it is possible to aim at a simpler goal. If weconsider a nucleus in which there are several nucleons outside closed shells(valence nucleons), energies of states can be divided into three parts. The firstis the binding energy of the closed shells, the second part is the sum of singlenucleon energies, which includes their kinetic energies and their interactionswith the core nucleons, and the third is the mutual interaction of the valencenucleons. Calculating binding energies of closed shells is the most difficulttask. It amounts to solving the nuclear many-body problem. Single nucleonenergies are probably easier to calculate since they are much smaller. Still, thisinvolves the kinetic energy of the valence nucleons and the effective interactionbetween them and core nucleons (those in closed shells). The easiest part tocalculate is the interaction between valence nucleons. If they occupy a singleorbit, only the matrix elements of the effective interaction between nucleonsin this orbit are required. The energy of this part is rather small and hence, thematrix elements need not be very accurate. If valence nucleons occupy severalorbits, differences between single nucleon energies are also needed. These canbe usually taken from experiment.

Since early days of the shell model, theorists tried to calculate energies dueto the interaction between valence nucleons in the various states allowed by thePauli principle. Such calculations are necessary not only to obtain energies ofnuclear states but also to determine wave functions of these states. Adopting acentral potential well, states of closed proton and neutron shells, or states withone nucleon outside or missing from closed shells, have well determined wavefunctions. For these nuclei, the predictions of the shell model, at this stage, areonly qualitative. Determination of the central potential well yields magic num-bers at shell closures, the order and spacings of single nucleon orbits as wellas radial functions. This determination, however, involves reliable solution ofthe nuclear many body problem.

If there are several valence nucleons (outside closed shells), there are usu-ally several possible states allowed by the Pauli principle. Eigenstates of theshell model Hamiltonian can be determined only by taking into account themutual interaction of the valence nucleons. The calculated eigenstates should

Fifty Years of the Shell Model 3

have energy eigenvalues and other observables like moments and transitionrates which agree with experiment. Shell model theorists have tried for severaldecades to obtain by quantitatively significant calculations, these eigenvaluesand wave functions.

The calculation of nuclear energies, which also yields the shell model wavefunctions, is greatly facilitated by use of symmetries. The isospin symmetry,which will be described in the following, is rather strictly observed in nuclei inspite of their great complexity. With certain conditions on the nature and rangeof the two-body interaction, other symmetries have been found. The supermul-tiplet (SU(4)) symmetry was introduced by Wigner and by Hund, in the earlierversion of the shell model. The system of nucleons moving independently ina potential well turned out to be fertile ground for other, more specific sym-metries. The mathematical expression of symmetries is by introducing groupsof transformations which act on the Hamiltonian and on wave functions. Theinteracting boson model, which can be viewed as an approximation to the shellmodel, has several symmetries corresponding to simplified physical situations.That model, in which extensive use is made of group theory, will not be de-scribed in this review. Some of these groups, however, may occur also in theshell model and they will be briefly described, without going into details.

The main difficulty in shell model calculations was the choice of the effec-tive interaction between valence nucleons to be used. Over the years differentapproaches to this problem were developed which will be described in the fol-lowing pages. The evolution of ideas about the determination and use of effec-tive interactions is the subject of this review. Its aim is to show how physicistshave been grappling with this problem, the ways they developed to solve it andsome results of these efforts.

An attempt was made to review all relevant papers. It is difficult to be-lieve that this has been achieved. Some prejudice may have been applied in thechoice of papers, but most of those which are not included, are simply victimsof oversight. Their authors are asked to accept a profound and sincere apology.The results of most papers included, are presented without comments or criti-cism. It is easy to be critical in retrospect but the papers should be judged bytaking into account the time and circumstances.

Even this rather limited subject is sufficiently wide so that some interestingand important developments will not be included. For example, very little willbe mentioned of effective interactions calculated in many body theory fromthe interaction between free nucleons. Other important topics which will notbe considered are listed below.

4 Igal Talmi

The Collective Model

A very important development in nuclear structure physics was the intro-duction of the collective or unified model. This model is very successful in de-scribing nuclei which exhibit collective spectra like the well known rotationalbands. Nevertheless, this model will not be considered in this review for severalreasons. The very large quadrupole moments and very strong electromagneticE2 transition probabilities observed in certain nuclei, attracted the attention ofAage Bohr (1950). He thought that such effects cannot be simply describedby the motion of a few nucleons and they must be due to collective motion ofmost nucleons in the nucleus. Bohr and Mottelson (1953) developed the the-ory of the collective model. The relevant degrees of freedom were taken to bethe shape and orientation of the nuclear surface. They limited the deformationfrom the spherical shape to be given by the spherical harmonic. A differ-ential equation which describes the dynamics of this quadrupole deformationwas written. It is expressed in terms of the Euler angles specifying the directionin space and derivatives with respect to them. In the intrinsic frame of refer-ence two more variables are required to obtain 5 variables which completelycharacterize the system. These are the angles and Also deformations de-fined by higher values, and also have been later considered.Spins of individual nucleons could be coupled to the nuclear surface. In their1953 paper, Bohr and Mottelson considered various possibilities of collectivemotion, one of which was the case of strong, axially symmetric, deformationof the surface. Such a shape could rotate around an axis perpendicular to theaxis of symmetry giving rise to states of the nucleus with energies proportionalto J(J+1). In the same year, such rotational bands were discovered by Hey-denburg and Temmer (1953) and the collective model scored a major success.

In order to incorporate motion of individual nucleons into the collectivemodel, it was extended and further developed (the unified model). It startswith a potential well which may be deformed. The motion of nucleons insidethe well is considered to be fast while changes of the nuclear shape, by eitherrotations or vibrations, are considered to be adiabatic. Thus, for every valueof the deformation parameter(s) of the potential, the single nucleon orbits canbe determined. The equilibrium deformation is found by minimizing the sumof energies of occupied single nucleon orbits. This procedure which was intro-duced by Nilsson and Mottelson (1955), leads for magic numbers to a sphericalequilibrium shape. Other nuclei turn out to be deformed. An anisotropic har-monic oscillator, with axial symmetry, was used by Nilsson (1955) to calculatesingle nucleon orbits which are named after him. Whereas the shell model wasmotivated by atomic structure, the unified model resembles to some extent,the theory of molecular structure. Unlike the situation in nuclei, in molecules

Fifty Years of the Shell Model 5

there are large energy separations between rotations, vibrations and electronicexcitations.

A better way to calculate equilibrium deformations is the use of Hartree-Fock calculations without the constraint of a spherical self-consistent field.Whenever there are valence nucleons outside closed shells, the self-consistentfield does not have spherical symmetry. The eigenstates of a rotationally in-variant Hamiltonian have definite values of J. Projecting such states from theHartree-Fock wave function would certainly result in better wave functions. Ifthe deformation is small, the projected states may correspond to shell modelstates and no rotational band is expected to emerge. On the other hand, if thedeformation is large and there are many valence nucleons, the situation maybe different. Wave functions with even slightly different orientations of thedeformation axis will be close to orthogonal. In such cases, projected stateswith definite angular momenta may form a rotational band whose energies areclose to being degenerate. In actual cases it would be too complicated to carryout exact projection of states with definite J and approximation methods havebeen developed.

Still, the notion of states of valence nucleons determined by their effectiveinteractions is confined to the shell model whose central potential well is spher-ically symmetric. In this case, the degeneracies or near degeneracies of singlenucleon energies are removed by introduction of the mutual effective interac-tion. The latter is usually taken as a perturbation and its diagonalization yieldswave functions of the various states. In the collective model, nucleons are as-sumed to move independently in a deformed potential well. Hence, the portionof the inter-nucleon interaction included in the common potential well is largerthan in the spherical case. Even without considering further interactions, wavefunctions with definite angular momenta are determined by projecting fromstates of nucleons in the deformed potential. As pointed out above, the prob-lem of projection is usually very complicated and is carried out by using someapproximation. For these reasons, not much effort was aimed at looking for thecorrect effective interaction.

A very interesting question which has occupied the interest of many the-orists, is the relation between the collective model and the (spherical) shellmodel. In principle, the answer is straightforward since shell model states forma complete set of states. Wave functions of the unified model, as any wave func-tion of A bound nucleons, can be expanded as a linear combination of them.Such expansions, however, are extremely complicated. The situation is actu-ally somewhat simpler but still far from simple. Even in the collective model,closed shell nuclei are considered to be spherical and so are also nuclei nearthem. There is no simple way to describe nuclei in the transition region byusing the collective model. Strongly deformed nuclei, however, have a simple

6 Igal Talmi

description in that model. But even in most of them, it is the orbits of valencenucleons which are considerably affected by the deformation of the potentialwell. The effect of deformation on nucleons in closed shells may then be ap-proximated by renormalization of the effective charges of the valence nucleons.Thus, it should be possible, in principle, to express states of rotational bands bycertain linear combinations of states of valence nucleons. These should be cal-culated from the effective interaction between valence nucleons. The numberof the latter in relevant cases is much too big and approximate methods shouldbe developed to handle this problem. This point is important for understandingthe relation between the shell model and the collective model. For actual calcu-lations, the collective model may be considered as an approximation methodwhich is very efficient for certain regions of nuclei. It is indeed remarkablethat very complicated states of the spherical shell model may be successfullyapproximated by wave functions of nucleons moving independently in a de-formed potential well.

Hartree-Fock Calculations

The very foundation of the shell model is, in principle, the Hartree-Fock(HF) approach. If nucleon wave functions describe independent motion in acentral potential well, there is a simple prescription for the state which mini-mizes the total energy. This occurs when the potential well acting on a nucleonin any given orbit, is equal to its average interaction with the other nucleonsin their various orbits. In other words, the minimum energy is obtained whenthe potential well is self-consistent. The HF ground state is the best varia-tional wave function of all states which are antisymmetrized products of singlenucleon wave functions (Slater determinants). The HF wave functions can de-scribe precisely ground states of nuclei with closed orbits or those with one nu-cleon outside (or missing from) closed orbits. In all other cases, a single deter-minant cannot be an eigenstate of the shell model Hamiltonian since it does nothave a definite angular momentum. It is a linear combination of several stateswith various angular momenta. If the deformed potential well is determinedby variation of some parameters, a more accurate procedure is to project stateswith good angular momentum and then vary the parameters (variation afterprojection). This procedure, however, is very complicated in actual cases. Asmentioned above, Hartree-Fock calculations may yield in certain cases selfconsistent fields which are deformed, as assumed in the collective model. Incertain cases, where the deformation of the HF potential well is large, suchstates may form a rotational band. In such cases, variation after projectionis expected to yield the same deformation parameters for all members of theband.

Fifty Years of the Shell Model 7

In such cases, it is possible to restore the rotational symmetry of the sys-tem, remove the degeneracy and obtain a J(J+1) dependence of energy levelsby cranking. A component of spin perpendicular to the symmetry axis ofthe potential well is added to the Hamiltonian multiplied by a Lagrange mul-tiplier This combination is then minimized by using perturbation theory.Eigenvalues of the Hamiltonian turn out to depend quadratically on Thus,

and the rotational frequency, is related to the angular momentum of a state Jby where I is the resulting moment of inertia of the band. Clearly,this approximation may work only for strongly deformed nuclei which exhibitrotational bands. In other cases, the Hartree-Fock theory cannot be directly ap-plied to nuclear states. Nevertheless, there were many attempts to apply theHartree-Fock method to nuclei. These calculations have not played an impor-tant role in the quest for the effective interaction between valence nucleons.They are based on some phenomenological interaction which yields the bind-ing energies and nuclear radii of doubly magic nuclei. Closed shell nuclei turnout to be spherical in HF calculations. The presence of one or more valence nu-cleons leads to self consistent potentials which are deformed. In certain cases,described above, this deformation may have physical significance, as in nu-clei exhibiting collective rotational spectra. In other cases, it just exhibits theinadequacy of the HF approach. In most shell model calculations, it is sim-ply assumed that nucleons occupy certain orbits in a central and spherical po-tential well. Interactions used in Hartree-Fock calculations are usually inade-quate to reproduce eigenstates due to the interaction between valence nucleons.Hartree-Fock calculations attempt to calculate global properties of nuclei. Theinteractions used in them have a sufficient number of parameters to fit someproperties of doubly magic nuclei. The fact that these interactions are unsuitedfor calculating nuclear level spacings raises doubts about their physical mean-ing.

Relativistic Mean Field Theory

Another development in nuclear structure physics, which will not be dis-cussed, is the relativistic mean field theory. The usual shell model and thecollective model are non-relativistic. The velocities of nucleons in nuclei aresmall compared to the speed of light and hence, it is assumed that relativis-tic effects may introduce only small corrections. Possible such effects wereidentified and discussed. An apparently different approach was suggested byWalecka (1974). He introduced a relativistic mean field theory (RMF) which

8 Igal Talmi

is based on certain meson exchange interactions. In the past, exchange of pi-ons, the lightest mesons, was considered as the most important contribution tothe interaction of free nucleons at larger distances. In the RMF theory, pionexchange interaction was not considered since this interaction essentially av-erages to zero in nuclear matter because of its strong spin and isospin depen-dence. Instead, exchanges of sigma and omega mesons were intro-duced. The neutral scalar meson cannot be found in tables of mesons. It wasnot discovered experimentally and there are no reliable theoretical estimatesof its mass. It is widely used, however, in the description of the interaction be-tween free nucleons in terms of meson exchanges. It gives rise to an attractive,Yukawa type interaction between any pair of free nucleons. Its mass is assumedto be smaller than the mass and hence, the range of the potential due to ex-change is larger than the range of the repulsive potential due to exchange ofThe combined interaction resembles the free nucleon interaction with its hardcore and strong attraction at longer range. Still, if a non-relativistic mean fieldcalculation is used, ignoring two-body correlations between nucleons, no sta-ble nuclei result. If the attraction is stronger, the nucleus shrinks to a point andif the repulsion is dominant, the nucleus has zero density. The reason is thatthe source of the (static) fields of both mesons is the same nucleon density.

This is no longer the case in the relativistic treatment of nucleons. There,the scalar mesons are coupled to the scalar density of the Dirac nucleonswhereas the vector mesons are mainly coupled to the 4th component of thenucleon current vector. The finite binding energy emerges in such calculationsfrom the slight difference between these two sources. In this way, the inter-actions which bind the nucleus are completely separated from interactions be-tween valence nucleons. In particular, it is acknowledged that some propertiesof nuclei (such as the spectrum of two-nucleon states near the Fermi surface)may depend on finer details like the one-pion-exchange tail but, as quotedabove, it essentially averages to zero in nuclear matter. To obtain sufficientbinding, the average scalar potential S and the average vector potential V mustbe rather large. This leads to an attractive feature of the model. The spin-orbitinteraction which arises naturally in the relativistic model, is determined essen-tially by the of S and V. The latter is very large and the resultingspin-orbit interaction is of the order of magnitude needed. On the other hand,the single nucleon wave functions are determined essentially by the of Sand V which is rather small, like the non-relativistic potentials in common use.The question which is relevant here is whether this RMF model is in contra-diction with the non-relalivistic shell model which has been successfully usedsince 1949. It turned out that this is not the case. In application to nuclei, cal-culated lower components of the Dirac wave functions of single nucleons turnout to be rather small compared with the upper ones, as is seen in the paper of

Fifty Years of the Shell Model 9

Lalazissis et al., (1998). Hence, non-relativistic wave functions are good ap-proximations of the single nucleon wave functions in the RMF theory. Thus,the RMF leads to a central potential well, which determines single nucleonenergies and wave functions, which is to a very good approximation, a poten-tial well used in the non-relativistic model. Another attractive feature of therelativistic model is a bunching of certain single nucleon energies. This bunch-ing, experimentally observed, was shown by Ginocchio (1997) to be due to acertain symmetry of the Dirac Hamiltonian which is an exact symmetry whenS+V = 0.

Coulomb Energy of the Protons

A very important symmetry of the interaction between nucleons is chargeindependence. Apart from the Coulomb interaction between protons, the in-teraction between two nucleons depends entirely on their space and spin state.Any state of two protons and of two neutrons is antisymmetric in the space andspin variables. A proton and a neutron when in the same antisymmetric statehave the same interaction as a proton pair or a neutron pair. A proton and a neu-tron may be also in space and spin states which are symmetric. In such states,the nuclear interaction may well be different from the one in antisymmetricstates. This symmetry of the Hamiltonian leads to a certain classification ofstates according to their symmetry properties under exchanges of proton andneutron variables.

Charge independence has a mathematical description in terms of isospinquantum numbers. Proton and neutron are taken to be the same particle in twodifferent charge states (the small mass difference is ignored). In analogy tospin, they are assigned an isospin 1/2 with projections +1/2 and 1/2 respec-tively. Each nucleon is assigned an isospin vector t in an abstract space. Thevalue of tt is equal to with Isospin vectors of several nucleonsmay be combined to form the total isospin vector T. The space-spin antisym-metric states are assigned total isospin T = 1 with projection for aproton-neutron pair. The space-spin symmetric state of a proton neutron pairis assigned isospin T = 0 so that protons and neutrons obey a generalizedPauli principle. Their wave functions are fully antisymmetric in space, spinand isospin (charge) variables. Charge independence of the nuclear Hamilto-nian implies that its eigenstates may depend on the total isospin T but not on its3-projection That is, the eigenstates are independent of the charge of thestates which is determined by the number of protons among the A nucleons.

Looking at levels of light to medium nuclei with the same isospin T, theirspacings are indeed fairly independent of or charge. Ground state energies(binding energies) depend on and their differences are fairly consistent

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with those calculated by using the Coulomb energy of the protons as a per-turbation. Both experimental tests and theoretical estimates indicate that theisospin T is a very good quantum number. Admixture of different isospinsis extremely small and has usually a negligible effect on the calculation ofnuclear energies. Such slight admixtures have a measurable effect in variousweak transitions. The differences in binding energies have an important effecton the stability of heavier nuclei. Although weaker than the nuclear interaction,the Coulomb energy is roughly proportional to the square of the proton numberZ. For higher Z values it becomes comparable to the nuclear interaction sincethe latter has saturation properties which make it roughly proportional to thenumber of nucleons A. As a result, in heavier nuclei which are stable enoughto be studied, protons and neutrons occupy different major shells in groundstates and low lying levels. In such states, the isospin T does not help in clas-sification of states, even if it is a good quantum number. All such states havethe same isospin T = (N Z)/2 and its use does not introduce any simplifi-cation. In such nuclei, protons and neutrons can be considered independentlywhich avoids the complications due to isospin mixing expected in such nuclei.Hence, this review will not deal with Coulomb energies of nuclei even thoughcomparison of calculated and experimental Coulomb energies is important forour understanding of certain aspects of nuclear structure.

2. THE (RE)EMERGENCE OF THE SHELL MODEL

In 1999 we celebrated 50 years of the nuclear shell model. Actually it wasproposed much earlier than 1949. Nuclear structure physics started with thediscovery of the neutron and the historical paper by Heisenberg (1932), inwhich he suggested that protons and neutrons can be considered as two statesof the same particle and that they are the building blocks of atomic nuclei.Physicists who tried to understand this system had in front of their eyes thesuccessful model of electron shells in atoms. In the shell model of the atom,electrons move independently in a central potential well in orbits characterizedby a radial quantum number (number of nodes of the radial part of the wavefunction in the finite-range of ) and orbital angular momentum The centralpotential is due to the positive charge of the nucleus and the average repul-sive interaction of the electrons. In certain atoms in which electrons occupyall states, allowed by the Pauli principle, in orbits which are close in energy, amajor shell is closed. Such atoms have high excitation energies and high ion-ization energies. This is the reason why such noble gas atoms barely participatein chemical reactions.

Fifty Years of the Shell Model 11

2.1. Single Nucleon Orbits

In spite of the clear differences between atoms and nuclei, physicists lookedfor similarities and managed to find them. On the basis of the few binding en-ergies which were measured at that time, Bartlett (1932), in the same year ofHeisenbergs paper suggested that, in analogy with electron shells in the atom,in the orbit is completely filled with two protons and two neutrons while

is obtained by adding on a closed with six protons and six neu-trons. Elsasser (1933) started a serious study of proton and neutron shells innuclei. In addition to 2 and 8 he identified the magic numbers, for which a ma-jor shell is fully occupied, 50, 82 and even 126. He also suggested some orderand bunching of single nucleon which reproduce these numbers. It wasrecognized that the order of single nucleon orbits in nuclei must be differentfrom the one in atoms. There, it is roughly derived from a Coulomb potential,due to the central nuclear charge and modified by the repulsion of the elec-trons. In nuclei, however, the possible order of levels was similar to that ina (3-dimensional) harmonic oscillator. In it, major shells are characterized by

and contain degenerate orbits with or0. Orbits will be denoted by and are replaced, for historicalreasons by Harmonic oscillator levels start with the orbit whichcloses at proton or neutron number 2. The next is the orbit, yielding themagic number 8 when it is fully occupied. The next shell, with degenerateand orbits, closes at magic number 20. To reproduce shell closures at 50, 82or 126, drastic shifts of single nucleon orbits were required.

It is not difficult to understand why the shell model for nuclei seemed hardto understand. In the atom, there is a massive nucleus whose large electriccharge supplies a strong central potential acting on the light mass electrons.Although the potential is modified by the mutual interaction of the electrons,the approximation of a central field seems to be very good. It is the basis ofapproximation methods like the Hartree-Fock approach. In nuclei, nothing re-sembles this simple picture and the mutual interaction of protons and neutronsis rather strong, as seen from the large binding energies. Adopting the shellmodel implies that in nuclei, protons and neutrons move independently in acentral potential well. A justified question is how could such motion take placein view of strong interactions between the consistent nucleons. In a series of3 papers published in Reviews of Modern Physics, Bethe and Bacher (1936)reviewed the status of nuclear physics at that time. They present a fair and bal-anced view of the shell model, its successes and short comings. They argue thatalthough the order of single nucleon orbits proposed by Elsasser and others re-produces the magic numbers, their model lacks a theoretical foundation. Adeeper argument which they present, concerns the effect of nucleon-nucleon

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interactions on the single nucleon picture of the shell model. It is fair to admitthat this problem has remained with us until now. It is in this aspect that theshell model still lacks theoretical foundation.

A more serious argument against the shell model for nuclei was raised byNiels Bohr. In a paper published in Nature, Bohr (1936) wrote: It is, at anyrate, clear that the nuclear models hitherto treated in detail are unsuited to ac-count for the typical properties of nuclei for which, as we have seen, energyexchanges between the individual nuclear particles is a decisive factor. In fact,in these models it is, for the sake of simplicity, assumed that the state of motionof each particle in the nucleus can, in the first approximation, be treated as tak-ing place in a conservative field of force, and can therefore be characterized byquantum numbers in a similar way to the motion of an electron in an ordinaryatom. In the shell model, a nucleon moves in an orbit which is essentiallyundisturbed by the interaction with other nucleons. The mean free path of anucleon in the nucleus, as estimated from nuclear reactions, is much smallerthan nuclear dimensions. Bohr continues with a devastating conclusion: In theatom and in the nucleus we have indeed to do with two extreme cases of me-chanical many-body problems for which a procedure of approximation restingon a combination of one-body problems, so effective in the former case, losesany validity in the latter....

In their review article, Bethe and Bacher just argued that the interaction be-tween nucleons should be introduced into the shell model. Niels Bohr arguedthat in view of the strong interaction, observed in nuclear reactions, the shellmodel cannot be used even as a first approximation. His criticism had a pro-found influence on many young physicists (e.g., Racah). Only those who weredeeply involved, like Wigner and Hund, were not discouraged and continuedtheir work using the shell model. Their work, however, was mostly limitedto nuclei lighter than for which the order of single nucleon orbits couldbe simply understood. The SU(4) scheme or the supermultiplet theory whichthey developed independently, could be applied to nuclei also without adoptingexplicitly the shell model. Some of its predictions agreed with experiment butothers could not be made with sufficient accuracy without deeper understand-ing of nuclear states.

In 1948, Maria G. Mayer published a paper on closed shells in nuclei. Shepresented there the experimental facts...to show that nuclei with 20, 50, 82or 126 neutrons or protons are particularly stable. She quotes the 1934 pa-per of Elsasser who suggested that special numbers of neutrons or protonsin the nucleus form a particularly stable configuration. Then she writes, Thecomplete evidence for this has never been summarized, nor is it generally rec-ognized how convincing this evidence is. The experimental evidence in thepaper concerns isotopic abundances, number of isotones, edges of the stability

Fifty Years of the Shell Model 13

curve, delayed neutron emission, neutron absorption cross sections and also,tentatively, asymmetric fission. She mentions the fact that shell closure at 20is understood but does not speculate about the other numbers. It was her paperwhich forced the shell model on the physics community. The realization thatthere are magic numbers was in sharp contrast with a statistical approach tothe nucleus or to the liquid drop model. In those models, there could be no dis-tinction between nucleon numbers 50 and 52 or between 82 and 84. The solidexperimental facts which Mayer presented became a challenge to physicists.

Maria Mayers paper was noticed by Nordheim and by Feenberg and Ham-mack who at the end of the same year submitted their versions of the shellmodel. Feenberg and Hammack (1949) followed essentially the level schemeof Elsasser. In addition to shell closures they discuss spins and magnetic mo-ments, as well as islands of isomerism - concentration of electromagneticisomeric transitions just before shell closure. Nordheim (1949) has anotherscheme which also agrees with observed shell closures and with observed spinsand magnetic moments. He makes explicit use of the results of Schmidt for nu-clear magnetic moments. The model of Lande (1934), in which particleonly, one proton or one neutron, is responsible for the total spin and magneticproperties of the whole nucleus, was taken up by Schmidt (1937) who hadby then, more experimental data. Both level schemes of Nordheim and Feen-berg and Hammack, list the various orbits by their orbital angular momentum

and characterize the states by the total L and total intrinsic spin S like inRussell-Saunders coupling of atomic electrons (LS-coupling).

Maria Mayer was trying to understand the origin of the shell closures whichshe so convincingly demonstrated. She was also aware of the experimental datawhich Nordheim and Feenberg presented in favour of their schemes. She musthave discussed the situation with some colleagues since Enrico Fermi, listeningto her, asked Is there any indication of spin-orbit coupling?. In her paper, inwhich she introduced the shell model, she thanked him for this remark, whichwas the origin of this paper. In some other place she said that she thought fora moment and answered Yes and it explains everything. In that paper, sheconsidered the effect of a strong spin-orbit interaction on order and spacing ofsingle nucleon orbits. She shows how in her scheme the observed shell clo-sures are reproduced as well as most observed nuclear spins as presented byNordheim and Feenberg. She tentatively suggests shell closure at 28 and alsomentions how, in her scheme the prevalence of isomerism towards the end ofthe shell, noticed by Feenberg and Nordheim, is easily understood.

This paper, Mayer (1949), sent as a letter to the editor of the Physi-cal Review, was delayed by the editor who asked Feenberg, Hammack andNordheim (1949) to write a letter in which they compared their schemes andMayers scheme. These two letters were published in the same issue. Mayers

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paper is On Closed Shells in Nuclei and in the title of the other one the nameNuclear Shell Models appears explicitly, perhaps for the first time. Beforeboth letters were published, a letter submitted two months later was publishedin the Physical Review by Haxel, Jensen and Suess (1949) On the MagicNumbers in Nuclear Structure. This letter was preceded by a short paper byJensen, Suess and Haxel (1949). They independently discovered that shell clo-sures at the numbers observed experimentally are reproduced if one assumesthat a strong spin-orbit coupling...leads to a strong splitting of a term withangular momentum into two distinct terms A more detaileddiscussion of spins of various nuclei according to their scheme, was publishedin three short papers in Naturwissenschaften (1949) in which the authors arelisted according to cyclic permutations of their names and in a longer paperin Z. f. Physik 128 (1950) 1295. In one of those they state explicitly that in-stead of LS-coupling the scheme should be used in nuclei. In thisscheme, orbits are denoted by and is replaced by a letter as explainedabove.

In the model suggested by Mayer and Jensen et al., single nucleon stateswith and have different energies. It was knownthat in nuclei some spin-orbit interaction was observed and its sign is oppositeto the sign in atoms. Hence, the single nucleon level with wasexpected to be the lower of the two. If there is strong splitting between thetwo states, the energy of the orbit with the highest value in anoscillator shell is lowered down to the lower oscillator shell. The orbit islowered into the oscillator shell, yielding shell closure at 28. Similarly,the magic number 50 is obtained by the orbit joining the orbits left in the

shell. Magic numbers 82 and 126 are due to the lowering of theand respectively. The splitting should be sufficiently large so that twonucleon interactions will not seriously mix states of nucleons withthose of ones.

In atoms, the prevalent coupling scheme is Russell-Saunders coupling orLS-coupling. In its eigenstates, the orbital angular momenta of the particles arecoupled to a definite total orbital angular momentum L. The spins are coupledto a definite total spin S which is coupled to L to yield a definite total angu-lar momentum J. If the interactions can be expressed by space coordinates,all possible states obtained from coupling those S and L are degenerate. Thedegeneracy of the various J states can be removed by certain spin interactionslike the spin-orbit interaction. In the 1949 version of the shell model the spin-orbit interaction is the dominant one and should be diagonalized first. Thus,the individual values of the nucleons are good quantum numbers. They arecombined to form the total angular momentum J of eigenstates. This schemeis the scheme which occurs also in some cases of atomic spectra.

Fifty Years of the Shell Model 15

The two body interactions are then treated as a perturbation.The definitive paper on the new version of the shell model was published

by Maria Mayer (1950I). The model is explained in detail and its predictionsof spins, isomeric states and even beta decay are presented, compared withthe experimental data. In that paper she states explicitly certain assumptionswhich were tacitly made in her papers and those of Jensen et al. They concernground states of several nucleons in a An even number of identicalnucleons...always couple to give a spin zero. An odd number of identicalnucleons in a state will couple to give a total spin There were only twoexceptions to this rule. The J = 3/2 spin of which should be equal tothat of a nucleon and the J = 5/2 of where the orbit is Sheexplained that three protons in a state can couple to give a spin 3/2,although as a rule the energy for the state with spin 5/2 is lower. A similarexplanation is given for the other case and the calculated magnetic moments ofthese exceptional states agree with the measured values better than the Schmidtmoments of and protons.

The title of that paper was Nuclear Configurations in the Spin-Orbit Cou-pling Model I. Empirical Evidence. In continuation of it, Mayer (1950II) pub-lished a second paper, II. Theoretical Considerations. The rules of couplingwhich she used to reproduce the observed ground state spins are stated and anattempt is made to derive them from a mutual interaction between nucleons.These rules are:

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1.

2.

3.

The lowest state of a configuration ( nucleons in the ) ofidentical nucleons is J = 0 if is even.

It is if is odd.

Of two in the same major shell, the one with the larger has alarger pairing energy.

The last rule was needed to explain why ground states with large in oddnuclei are rather rare, she suggested that higher tend to fill in pairs.The object of this paper is to investigate if there are theoretical reasons forthese empirical rules.

A given ordered set of single nucleon orbits determines the magic numbers.They are the maximum numbers of protons or neutrons, allowed by the Pauliprinciple, which may occupy the various orbits in a major shell. Nuclei withclosed major shells are particularly stable as summarized by Mayer (1948).States of one nucleon outside closed shells are also uniquely determined bythe central potential well. The single nucleon may occupy any of the states inthe higher shell(s). States of a nucleus with one nucleon missing from closed

shells, one hole states, are similarly determined. The central potential well ofthe shell model, should somehow emerge from the solution of the nuclear manybody problem. Fifty years ago, this solution was not available, nor is it now.Thus, these predictions of the shell model, as unexpected and strikingly suc-cessful as they turned out to be, are only qualitative. Quantitative results wereexpected for calculations of the mutual interaction (residual interaction) be-tween valence nucleons.

All states which several valence nucleons can form, have the same energyin the central potential well. This degeneracy may be removed by consideringthe mutual interaction of the valence nucleons. Eigenvalues and eigenstates areobtained by diagonalization of the sub-matrix of the Hamiltonian constructedfrom all states of the configuration (occupation of the various orbits). For ex-perimental reasons, the low lying states, and particularly the ground state, areof special interest. The Mayer rules specify the spins of ground states of suchconfigurations.

The rules of coupling that Mayer suggested are very simple. In atoms,where LS-coupling prevails, Hunds rule of coupling is that in the con-figuration, the lowest state has the maximum value of spin S. Such states havemaximum symmetry in electron spins and hence, minimum spatial symme-try and therefore, minimum repulsion of the electrons. Among all states withmaximum S, the lowest has the highest value of the total orbital angular mo-mentum L. In nuclei, the interaction is mostly attractive and relevant couplingrules for were not known. If the shell model is adopted, spins ofnuclei with closed shells are J = 0. If there is one valence nucleon, or onehole, in the outside closed shells, the spin of that state is Theenergy of a single nucleon is equal to the eigenvalue of the Hamiltonian, with acentral potential and a one-body spin-orbit interaction, for the In prin-ciple, it should be equal to the expectation value of the kinetic energy of thenucleon in the and its interaction with all nucleons in closed shells.

The configuration of several valence identical ornon-identical, has several states allowed by the Pauli principle. They are alldegenerate in the central potential, they all have the same energy equal totimes the single nucleon energy of the given In such cases, the groundstate and order of other levels are determined by the mutual interaction of the

It is considered as a perturbation and thus, calculated by diago-nalizing its matrix constructed from wave functions in the configuration.The two nucleon interaction is assumed to be rotationally invariant and hence,eigenstates of this interaction matrix have definite values of the total spin J.Apart from the Coulomb interaction between the protons, the interaction is as-sumed to be charge independent and hence, eigenstates are also characterizedby definite values of the total isospin T.

Fifty Years of the Shell Model 17

In pre-1949 papers, some authors tried to use two body interactions to cal-culate binding energies of nuclei. Mayers approach was much more modest.Mayer assumed that there is an attractive interaction between identical nucle-ons due to a potential which for reasons of definite and easy evaluation itwas assumed to have the shape of a . . Her method of calculation isindeed straightforward but inelegant. She calculated energies of states ofconfigurations up to where there is only one state with J = 0 orand thus, the sum method could be used. Mayer carried out a straightforwardbut tedious calculation of the interaction and obtained detailed results. Never-theless, she found that these states are indeed the lowest and found empiricallyan elegant expression for their energies. It is the correct one also for highervalues (for seniorities and see Sect. 6). These expressions are

where I is an integral of multiplied by radial parts of thewave function. However, no general proof, for all values, of the simpleresult...is given. She pointed out that these expressions reproduce the odd-even variation in binding energies. If the integral I does not strongly dependon then the pairing energy is indeed proportional to

This was the first time that two body interactions were included in the shellmodel with strong spin-orbit interaction and Maria Mayer waswell aware that using a zero range potential is only a crude approximation of ashort range interaction. Yet, nevertheless, it might be interesting to note thatthis assumption can explain qualitatively the empirical rules. In fact, this pa-per introduced the into nuclear structure physics to such an extentthat sometimes people would take it as an integral part of the shell model. Insome pre-1949 papers, the opposite approximation was used. The range of thetwo body interaction was taken to be larger than nuclear dimensions and thepotential was approximated by a constant in the region where nucleon wavefunctions are large. If this long range approximation is adopted and the in-teraction has a large component of space exchange (Majorana interaction) theresults for configurations are drastically different. Racah (1950) calculatedthe energies in such cases and found them equal to an attractive constant mul-tiplied by

Hence, the ground states have lowest possible values of J consistent with thePauli principle. For odd that value of J is equal to if (or ),

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i.e., a single nucleon (or hole). For (or ) it is equal to 3/2 andfor any other odd it is equal to 1/2. Racah took it as an argument in favour ofthe LS coupling against the coupling model. Like others, it took him sometime to accept the fact that is the prevalent scheme for nuclei. Inthese calculations and in many to be performed later, first-order perturbationtheory was used. Matrix elements of the two-body interaction were calculatedbetween shell model wave functions. Disagreement with experiment was oftenblamed on effects due to admixtures of higher configurations.

The re-emergence of the shell model raised again old arguments against theindependent nucleon picture of the nucleus. The question asked was whethersuch a picture could hold in view of the short-range strong interaction betweennucleons. At that time it was not known how singular this interaction really is.The new version of the model raised also a new problem - the origin of thestrong spin-orbit interaction. Several attempts have been made to explain thiseffect but even now, 50 years later, the definitive answer is still missing. It is in-teresting to note that most objections were directed towards the description ofnuclear states by The independent nucleon picture was somehowaccepted in spite of its difficulties. A strong spin-orbit interaction could not becompatible with a very elegant theory developed by Wigner (1937) and inde-pendently by Hund (1937). It is based on nucleon-nucleon interactions whichare due to a potential (Wigner force) or a potential multiplied by a spaceexchange operator (Majorana force). Eigenstates of Hamiltonians with suchinteractions could be classified according to irreducible representations of theSU(4) group. This theory was used in pre-1949 years and could account forsome important properties of nuclei, like symmetry energy. A particularly at-tractive feature of this theory was the clear distinction between favoured (al-lowed) and unfavoured (super-allowed) beta decays. It may be interesting tohave a look at the SU(4) theory and at some of its predictions. This can serveas a brief introduction to applications of group theory to nuclear spectroscopy,as well as describe some of the challenges facing the initial introduction of

2.2. The Supermultiplet Scheme

The SU(4) scheme is the first extensive application of group theory to nu-clear physics. It is the first of a long list of groups which have been applied overthe years to the theory of nuclear structure physics. The use of group theory isbased on symmetry properties of the nuclear Hamiltonian. Any such symmetryhas direct and important consequences for the eigenstates. If the Hamiltonianhas a certain symmetry, it is invariant under certain operations which define thesymmetry. Acting twice with such an operation on the Hamiltonian it remains

Fifty Years of the Shell Model 19

unchanged and hence, the composition of two such operations is another sym-metry operation. Thus, the symmetry operations form a group and the mathe-matical theory of groups can be applied to the eigenstates and eigenvalues ofthe Hamiltonian. A symmetry operation, which leaves the Hamiltonian invari-ant, may be applied to an eigenstate. The result is an eigenstate with the sameeigenvalue of the Hamiltonian. If P is the symmetry operation applied to astate of the system, it can be applied to the eigenvalue equation

Using the invariance of H, i.e., the result is

Thus, is also an eigenstate of H with the same eigenvalue.To demonstrate the implication of symmetry, the best known example of a

rotationally invariant non-relativistic Hamiltonian H may be used. The sym-metry operations in this case are 3-dimensional rotations which form the ro-tation group O(3). Eigenstates of H are also eigenstates of the square of theangular momentum with eigenvalues J(J + 1). They may be expressed aslinear combinations of 2J+1 independent states, characterized for instance, bythe M. Under such rotations, eigenstates transform among them-selves. These transformations are irreducible - no subsets can be formed whichtransform among themselves under all rotations. The matrix of the Hamilto-nian constructed from such eigenstates has a special form. Its non-vanishingelements form submatrices along the main diagonal, each submatrix is propor-tional to the unit matrix and is characterized (usually, not uniquely) by the totalangular momentum J. The matrices of rotations expressed in this basis havenon-vanishing matrix elements only in sub-matrices along the diagonal andeach submatrix is uniquely characterized by J. The submatrices with given Jof all rotations form an irreducible representation of the rotation group O(3).The eigenstates which define a given submatrix of the Hamiltonian form a basisof the given irreducible representation. In general, eigenstates of a Hamiltonianwith a given symmetry belong to certain bases of irreducible representations ofthe symmetry group. The labels of these irreducible representations can serveas quantum numbers which characterize eigenstates, like J in the example de-scribed above.

Wigner, in his papers, considered the SU(4) theory only as a first approx-imation. He assumed that the mutual interaction between nucleons dependsonly on the space coordinates of the nucleons. Such interactions include be-sides an ordinary potential (Wigner force) also the space exchange Majoranainteraction. The spin and isospin operators need not appear in the Hamiltonian

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which is then symmetric under any exchange of space coordinates of the nucle-ons. As a result, any function of space coordinates which is an eigenfunctionof the Hamiltonian, is transformed under a permutation of these coordinatesinto an eigenfunction with the same eigenvalue. If the transformed functionis different, another permutation may be applied to it etc. All such functionsmay be expressed as linear combinations of an independent set of functions.Even if there are several such independent functions they do not correspondto independent states of the system. They are all needed for construction ofeigenstates which are fully antisymmetric in space, spin and isospin variablesof the nucleons.

An eigenfunction fully symmetric in space coordinates should be simplymultiplied by a fully antisymmetric state of spin and isospin variables. In othercases, linear combinations should be formed of spatial eigenfunctions whichtransform among themselves under permutations multiplied by spin-isospinstates with the dual (opposite) symmetry. Thus, the space symmetry whichis allowed by the Pauli principle is limited by the available states of spin andisospin. These latter states transform among themselves under permutationsof spin-isospin variables of the nucleons. Hence, they form a basis of an irre-ducible representation of the group of permutations. A set of orthogonal statescan be obtained from them and any permutation can be expressed as a ma-trix in this basis. These sets of spin-isospin states are also bases of irreduciblerepresentations of the unitary group of transformations in the 4-dimensionalspace of spin and isospin states, constructed from single nucleon states with

Transformations of this unitary group U(4), mix states with different spinsand isospins but the Hamiltonian which does not contain spin and isospin op-erators, is trivially invariant under them. The eigenvalues depend only on thespatial parts of the eigenstates and, in particular, on their symmetry proper-ties. Due to the Pauli principle, the latter are determined by the symmetry ofthe spin-isospin functions. This way, eigenvalues of interactions which do nothave any dynamic dependence on spin or isospin, depend on quantum numberswhich characterize the symmetry of spin-isospin states. In the case of identi-cal particles, like atomic electrons, and interactions which are independent ofspins, eigenvalues are determined by the spatial parts of the wave functions.The latter are characterized by their orbital momentum L, and other quantumnumbers if needed, while their symmetry properties are determined by the to-tal spin value S. Energies of states with given L and S are then degenerateand do not depend on the total angular momentum J. They form multiplets inwhich eigenvalues are split by spin dependent forces, like the spin-orbit inter-action. In the U(4) scheme, all states with given spin-isospin symmetry forma It may contain states with different spins S and isospins

Fifty Years of the Shell Model 21

T. Since there are nuclei in which only some of the states ofa given supermultiplet may appear.

The irreducible representations of the U(4) group of spin-isospin states arecharacterized by 4 numbers. These can serve as quantum numbers to character-ize the eigenstates of the Hamiltonians considered here. One of them is simplythe total number of nucleons which is shared by all states of a given nucleus.The U(4) irreducible representations are also irreducible representations of itsspecial subgroup, the SU(4) group, with zero traces of its generators (infinites-imal elements of the group like and in the case of O(3)). The latterrepresentations are characterized by 3 numbers which together with deter-mine the irreducible representations of U(4). The 3 numbers include S andT in a special way. The 3 numbers are integers or half integers, usually de-noted by P, and which satisfy the inequality as wellas In a given supermultiplet, P is the largest value which Tor S can assume in any of its states. In states for which T = P (or S = P),

is the largest value which S (or T) can assume. In the SU(4) scheme, Tand S play equivalent roles in spite of their different physical meanings. Tosee which supermultiplets are predicted to be the lowest, Wigner introduceda drastic approximation. To obtain qualitative estimates he assumed that therange of the interaction is large compared with nuclear dimensions and incalculating matrix elements, it could be replaced by a constant. The constantwas taken as the value of the potential at which was reasonable for theGaussian potential used in those years. In this approximation, all states of agiven supermultiplet, those with the same spin-isospin symmetry, are degener-ate.

We saw above how this approximation was applied by Racah to a Majo-rana interaction in the case of identical nucleons. Due to the Pauli principle,the space exchange operator may be replaced by the spin exchange operatormultiplied by 1. In this case, the Majorana interaction may be replaced by

22 Igal Talmi

If the potential is replaced by a constant, the eigenvalue is a simple function ofthe spin S. It is a negative constant multiplied by

An ordinary potential interaction is simply proportional to in thisapproximation. Thus, in this approximation, the eigenvalues of the interactionwhich is independent of spins, are given by the eigenvalues of the spin vec-tor squared. The operator is the quadratic Casimir operator of the SU(2)group of the spin space (an operator constructed from the generators of the

Fifty Years of the Shell Model 23

group, here and which commutes with all elements of the group. Itseigenvalues characterize the irreducible representations of the group).

Similarly, in the SU(4) case, the Majorana interaction can be replaced bya potential multiplied by the negative of the exchange operator in spin-isospinspace. The latter is the product of spin exchange and isospin exchange opera-tors

This operator can be evaluated by using the (quadratic) Casimir operator of theSU(4) group whose eigenvalues are simple functions of the numbers P,The resulting eigenvalues of the Majorana interaction in the Wigner approxi-mation are given by a negative (attractive) constant multiplied by

From this expression it follows that the lowest supermultiplets are thosewith minimum values of P, and The isospin of the ground state of agiven nucleus is usually equal to |N Z|/2 and cannot be smaller. Hence, thelowest states of such a nucleus are expected to have P = T = |N Z|/2and the supermultiplet characterized by (T, S, ). In even-even nuclei, T isan integer and the lowest value of S is 0. Also vanishes in this case and thesupermultiplet is specified by (T,0,0). In odd-even nuclei, with half integralT, the lowest value of S is 1/2 and the value of is either 1/2 (for oddproton nuclei) or 1/2 (for odd neutron nuclei). The interaction energy in thisapproximation, for a Wigner interaction with coefficient and a Majoranaforce with coefficient is thus given by the expression

multiplied by a negative constant which is the value of the (Gaussian) potentialat the origin.

The expression of the interaction energy contains the term T(T + 4) whichaccounts for the symmetry energy term in various mass formulae. It includesa linear term in T which is sometimes called the Wigner term. Such a termis indeed necessary for reproducing nuclear binding energies. These resultsare very general and completely independent of any assumption on the spa-tial parts of the eigenfunctions. Other observables of the lowest states in thisapproximation, like nuclear magnetic moments calculated by Margenau andWigner (1940), can also be evaluated. To describe this calculation, it is conve-nient to express explicitly the operators whose eigenvalues are P, and in

24 Igal Talmi

the state with highest values of and If P is the eigenvalue of andis the eigenvalue of then is the eigenvalue ofThe magnetic moment of the total spin S may be expressed as (the proton

state has and the neutron )

where is the gyromagnetic ratio of the proton spin, 22.79 n.m. (nuclearmagnetons) and is that of the neutron spin, 21.93 n.m. Taking the ex-pectation value of the component of the operator in the state withand inserting the values of S and Y mentioned above, we obtain the followingvalues for the magnetic moment of the spin

The spin S should be coupled to the total orbital angular momentum L,which has a definite value in any eigenstate of SU(4) Hamiltonians, to formthe total spin J. It follows that nuclear magnetic moments fall into 4 sets, oddproton nuclei with J = L + 1/2 and J = L 1/2 and odd neutron nucleiwith J = L + 1/2 and J L 1/2. The spin are the same as in theSchmidt-Lande model. The slopes of the Margenau-Wigner lines, however,depend on and may be different. Under the assumption that protons andneutrons contribute equally to L, Margenau and Wigner took to be equal toZ/A. In 1940 there were not many measured magnetic moments and no definiteconclusions could be drawn. Later it became clear that the slopes of measuredmagnetic moments follow the Schmidt lines rather than those of Margenauand Wigner. It is interesting to note that in a shell model without spin-orbitinteraction, magnetic moments in the SU(4) scheme could be identical to theSchmidt-Lande moments. If there are identical nucleons in a interact-ing by a short range attractive potential, ground states have L = 0 for evenand for odd. A small spin-orbit interaction would make the states with

lower for up to and the states with lower forthe rest of the orbit.

A very nice feature of the SU(4) scheme is the clear distinction betweenfavoured and unfavoured allowed beta decays. The favoured decays have ratherlarge matrix elements of the operator between initial and final states. Theunfavoured decays are also allowed ones, but the matrix elements are much

smaller. As an example, we can look at beta transitions in A = 43 nuclei. Thetransition from the J = 7/2 ground state of to the ground state ofhas logft=3.5 (the ft value is inversely proportional to the square of the matrixelement). On the other hand, the transition from the J = 7/2 ground state of

to the ground state of which has the same spin and parity, is muchless favoured, with logft=5.0. The decay proceeds also by a Fermi transi-tion but even when this contribution is subtracted, the Gamow-Teller transitionis still much stronger than the decay. The operator whose matrix elementsdetermine the allowed Gamow-Teller transitions is given by

This operator is independent of space coordinates and hence has vanishingmatrix elements between eigenstates in different supermultiplets. Spatial func-tions with different symmetries are orthogonal. Allowed Fermi beta transitions,due to the operators or can take place only between states in the samesupermultiplet which have the same isospin. Hence, beta decays between statesin different supermultiplets may take place only if there are some admixturesof other supermultiplets. If the SU(4) scheme gives a good description, suchadmixtures are expected to be small.

Thus, favoured transitions are those between states in the same supermul-tiplet and all other allowed decays are expected to be unfavoured. This resultof SU(4) symmetry agrees in most cases with experiment. There is no similardistinction between favoured and unfavoured beta decays in theshell model. Some systematic deviations from extreme can yieldsuch a distinction but not in such an elegant way. It is worth while to mentionthat there is one case of a highly attenuated Gamow-Teller beta decay betweentwo states which should be in the same supermultiplet. This is the decay of theJ = 0 ground state of to the J = 1 ground state of with log ft=9.(The attenuative of this decay created a major problem for nuclear physics butwas also important in archaelogy as the basis of carbon dating.) Both statesshould belong to the (1,0,0) supermultiplet and within it, the lowest state is ex-pected to have L = 0. In that state should have S = 0, T = 1 and inS = 1, T = 0. Attempts were made to overcome this glaring discrepancy byassigning the ground state L = 2( J = 1) rather than L = 0 but this wouldbe rather artificial and in contrast to the assumptions made for SU(4) symme-try. In case of exact SU(4) symmetry and the approximation made above, theL = 0 and L = 2 states should be degenerate in both nuclei. Use of a morerealistic potential would lead to equal splittings of these levels in andThe lowest J = 2 level in lies at 7 MeV above the J = 0 ground state.The assumption that the order of these levels is reversed in is in clear dis-agreement with the assumptions of the SU(4) scheme. In any case, there are

Fifty Years of the Shell Model 25

stronger arguments against applicability of the SU(4) scheme to nuclei.The most important assumption of the SU(4) scheme is that interactions

which depend on the spin operators are absent from the Hamiltonian or atleast that they are much smaller than other interactions. This refers to truespin dependent terms which cannot be eliminated, unlike the Majorana spaceexchange interaction considered above. In the SU(4) scheme, both S and Lhave definite values. The strong spin-orbit interaction, which determines theorder of single nucleon orbits and magic numbers, breaks SU(4) symmetry ina major way. Apart from the case of a single nucleon or hole, neither S norL have definite values and instead, the total spins of individual nucleons aregood quantum numbers. In view of the elegance and mathematical beauty ofthe SU(4) scheme, as well as its great past contributions to nuclear structuretheory, it is not difficult to imagine the emotional impact on its creator, Eu-gene Wigner, when it had to be abandoned. This was one case in which Naturedid not follow the trail which he blazed. After that, Wigner never turned activeattention to nuclear structure physics. He followed some attempts to attributeshell closures to tensor forces which are present in the nucleon-nucleon in-teraction. They failed, however, to explain even the meagre experimental dataavailable then. Yet, even until recent years, there were several attempts to res-urrect the SU(4) scheme and apply it to nuclei. Some physicists have realizedits merits and tried to find some evidence for its validity in nuclear physics.

3. EARLY CALCULATIONS

3.1. Energy Levels of Simple Configurations

The first calculations of nuclear energies due to two body interactions weremotivated by the results of Mayers calculations. In her paper she suggestedthat the ground state spins which do not follow her simple rules, are due tostates of the predicted configurations but with A possible reason forthis cross over of levels could have been that the assumption of zero rangeof the nuclear potential was too drastic. Calculation of energy levels in the

and configurations of identical nucleons were carried out byher student Dieter Kurath (1950) who used a Gaussian potential and harmonicoscillator wave functions for the nucleons. The use of a Gaussian potentialwas a time honored tradition, starting with Heisenberg and Weizsacker andcontinuing with Wigner and Feenberg. It was supposed to reproduce resultsof proton-proton and proton-neutron scattering and the binding energy of thedeuteron. Kurath found for longer ranges of the potential, that in theconfiguration the ground state spin was indeed J = 3/2 instead of J = 5/2.

26 Igal Talmi

This, he pointed out, is in agreement with the argument of Racah, describedabove. In the configuration the situation was less clear. At the ratherlong range of the potential where the J = 5/2 becomes the lowest state, theJ = 3/2 is very close and for slightly larger range it becomes the ground state.Similar calculations of the configuration of identical nucleons werecarried out by Talmi (1951a) who used a Yukawa type potential and hydro-gen like wave functions. The results indicated that a cross over between theJ = 5/2 and J = 3/2 levels occurs only at a rather very long range of the po-tential. This was attributed to the fact that even at very long range, the Yukawapotential goes over to a Coulomb potential rather than to a constant. It waspretty clear that these calculations could not explain the exceptional groundstate spins of and

In addition to experimental information and theoretical discussion ofground states, excited states were considered. In the seminal paper of Gold-haber and Sunyar (1951) on classification of nuclear isomers, they stated therule that the first excited state of even-even nuclei has usually spin 2 and evenparity. The experimental information was considered by Horie, Umezawa,Yamaguchi, and Yoshida (1951) who found evidence for several cases in whichthe first and second excited states of even nuclei had spins 2 and 4. They alsocalculated, but gave no details, levels of some configurations of identical nu-cleons and found that the short range attractive (they write repulsive) interac-tion of Mayer yields this order of levels. They also noticed that the spacings ofthese levels were not well reproduced by the The systematic studyof experimentally m