Advances in Nuclear Physics

ADVANCES IN NUCLEAR PHYSICS VOLUME 26

CONTRIBUTORS TO THIS VOLUME Paul Fallon

A. Z. Mekjian

Lawrence Berkeley National Laboratory Berkeley, California

Physics Department Rutgers University Piscataway, New Jersey

B. W. Filippone W. K. Kellogg Laboratory California Institute of Technology Pasadena, California

M. Garçon

M. B. Tsang National Superconducting Cyclotron Laboratory Michigan State University East Lansing, Michigan

DAPNIA/SPhN, CEA-Saclay Gif-sur-Yvette, France

David Ward

S. Das Gupta

Lawrence Berkeley National Laboratory Berkeley, California

Physics Department McGill University Montreal, Quebec, Canada

Xiangdong Ji Department of Physics University of Maryland College Park, Maryland

J. W. Van Orden Old Dominion University Norfolk, Virginia & Thomas Jefferson National Accelerator Facility Newport News, Virginia

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

ADVANCES IN NUCLEAR PHYSICS Edited by

J. W. Negele Center for Theoretical Physics Massachusetts Institute of Technology Cambridge, Massachusetts

E. W. Vogt Department of Physics University of British Columbia Vancouver, B.C., Canada

VOLUME 26

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

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ARTICLES PUBLISHED IN EARLIER VOLUMES

Volume 1 The Reorientation Effect • J. de Boer and J. Eicher The Nuclear Model • M. Harvey The Hartree-Fock Theory of Deformed Light Nuclei • G. Ripka The Statistical Theory of Nuclear Reactions • E. Vogt Three-Particle Scattering—A Review of Recent Work on the Nonrelativistic Theory • I. Duck

Volume 2 The Giant Dipole Resonance • B. M. Spicer Polarization Phenomena in Nuclear Reactions • C. Glashausser and J. Thirion The Pairing-Plus-Quadrupole Model • D. R. Bes and R. A. Sorensen The Nuclear Potential • P. Signell Muonic Atoms • S. Devons and I. Duerdoth

Volume 3 The Nuclear Three-Body Problem • A. N. Mitra The Interactions of Pions with Nuclei • D. S. Koltun Complex Spectroscopy • J. B. French, E. C. Halbert, J. B. McGrory, and S. S. M. Wong Single Nucleon Transfer in Deformed Nuclei • B. Elbeck and P. O. Tjøm Isoscalar Transition Rates in Nuclei from the Reaction • A. M. Bernstein

Volume 4 The Investigation of Hole States in Nuclei by Means of Knockout and Other Reactions • Daphne F. Jackson High-Energy Scattering from Nuclei • Wieslaw Czyz Nucleosynthesis by Charged-Particle Reactions • C. A. Barnes Nucleosynthesis and Neutron-Capture Cross Sections • B. J. Allen, J. H. Gibbons, and R. L. Macklin Nuclear Structure Studies in the Z = 50 Region • Elizabeth Urey Baranger An s-d Shell-Model Study for A = 18 – 22 • E. C. Halbert, J. B. McGrory, B. H. Wildenthal, and S. P. Pandy

Volume 5 Variational Techniques in the Nuclear Three-Body Problem • L. M. Delves Nuclear Matter Calculations • Donald W. L. Sprung Clustering in Light Nuclei • Akito Arima, Hisashi Horiuchi, Kunihara Kubodera, and Noburu Takigawa v

vi

Articles Published in Earlier Volumes

Volume 6 Nuclear Fission • A. Michaudon The Microscopic Theory of Nuclear Effective Interactions and Operators • Bruce R. Barrett and Michael W. Kirson Two-Neutron Transfer Reactions and the Pairing Model • Ricardo Broglia, Ole Hansen, and Claus Riedel

Volume 7 Nucleon-Nucleus Collisions and Intermediate Structure • Aram Mekjian Coulomb Mixing Effects in Nuclei: A Survey Based on Sum Rules • A. M. Lane and A. Z. Mekjian The Beta Strength Function • P. G. Hansen Gamma-Ray Strength Functions • G. A. Bartholemew, E. D. Earle, A. J. Ferguson, J. W. Knowles, and M. A. Lone

Volume 8 Strong Interaction in A-Hypernuclei • A. Gal Off-Shell Behavior of the Nucleon-Nucleon Interaction • M. K. Strivastava and D. W. L. Sprung Theoretical and Experimental Determination of Nuclear Charge Distributions • J. L. Friar and J. W. Negele

Volume 9 One- and Two-Nucleon Transfer Reactions with Heavy Ions • Sidney Kahana and A. J. Baltz Computational Methods for Shell-Model Calculations • R. R. Whitehead, A. Watt, B. J. Cole, and I. Morrison Radiative Pion Capture in Nuclei • Helmut W. Baer, Kenneth M. Crowe, and Peter Truöl

Volume 10 Phenomena in Fast Rotating Heavy Nuclei • R. M. Lieder and H. Ryde Valence and Doorway Mechanisms in Resonance Neutron Capture • B. J. Allen and A. R. de L. Musgrove Lifetime Measurements of Excited Nuclear Levels by Doppler-Shift Methods • T. K. Alexander and J. S. Forster

Volume 11 Clustering Phenomena and High-Energy Reactions • V. G Neudatchin, Yu. F. Smirnov, and N. F. Golovanova Pion Production in Proton-Nucleus Collisions • B. Holstad Fourteen Years of Self-Consistent Field Calculations: What Has Been Learned • J. P. Svenne Hartree-Fock-Bogoliubov Theory with Applications to Nuclei • Alan L. Goodman Hamiltonian Field Theory for Systems of Nucleons and Mesons • Mark Bolsterli

Articles Published in Earlier Volumes

Volume 12 Hypernetted-Chain Theory of Matter at Zero Temperature • J. G. Zabolitzky Nuclear Transition Density Determinations from Inelastic Electron Scattering • Jochen Heisenberg High-Energy Proton Scattering • Stephen J. Wallace

Volume 13 Chiral Symmetry and the Bag Model: A New Starting Point for Nuclear Physics • A. W. Thomas The Interacting Boson Model • A. Arima and F. Iachella High-Energy Nuclear Collisions • S. Nagamiya and M. Gyullasy

Volume 14 Single-Particle Properties of Nuclei Through (e, e'p) Reactions • Salvatore Frullani and Jean Mougey

Volume 15 Analytic Insights into Intermediate-Energy Hadron-Nucleus Scattering • R. D. Amado Recent Developments in Quasi-Free Nucleon Scattering • P. Kitching, W. J. McDonald, Th. A. J. Maris, and C. A. Z. Vasconcellos Energetic Particle Emission in Nuclear Reactions • David H. Boal

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

Volume 17 P-Matrix Methods in Hadronic Scattering • B. L. G. Bakker and P. J. Mulders Dibaryon Resonances • M. P. Locher, M. E. Saino, and A. Švarc Skrymions in Nuclear Physics • Ulf-G. Meissner and Ismail Zahed Microscopic Descriptions of Nucleus-Nucleus Collisions • Karlheinz Langanke and Harald Friedrich

Volume 18 Nuclear Magnetic Properties and Gamow-Teller Transitions • A. Arima, K. Shimizu, W. Bentz, and H. Hyuga Advances in Intermediate-Energy Physics with Polarized Deuterons • J. Arvieux and J. M. Cameron Interaction and the Quest for Baryonium • C. Amsler Radiative Muon Capture and the Weak Pseudoscalar Coupling in Nuclei • M. Gmitro and P. Truöl Introduction to the Weak and Hypoweak Interactions • T. Goldman

vii

viii

Articles Published in Earlier Volumes

Volume 19 Experimental Methods for Studying Nuclear Density Distributions • C. J. Batty, E. Friedman, H. J. Gils, and H. Rebel The Meson Theory of Nuclear Forces and Nuclear Structure • R. Machleidt

Volume 20 Single-Particle Motion in Nuclei • C. Mahaux and R. Sartor Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics • B. D. Keister and W. N. Polyzou

Volume 21 Multiquark Systems in Hadronic Physics • B. L. G. Bakker and I. M. Narodetskii The Third Generation of Nuclear Physics with the Microscopic Cluster Model • Karlheinz Langanke The Fermion Dynamical Symmetry Model • Cheng-Li Wu, Da Hsuan Feng, and Mike Guidry

Volume 22 Nucleon Models • Dan Olof Riska Aspects of Electromagnetic Nuclear Physics and Electroweak Interaction • T. W. Donnelly Color Transparency and Cross-Section Fluctuations in Hadronic Collisions • Gordon Baym Many-Body Methods at Finite Temperature • D. Vautherin Nucleosynthesis in the Big Bang and in the Stars • K. Langanke and C. A. Barnes

Volume 23 Light Front Quantization • Matthias Burkardt Nucleon Knockout by Intermediate Energy Electrons • James J. Kelly

Volume 24 Nuclear Charge-Exchange Reactions at Intermediate Energy • W. P. Alford and B. M. Spicer Mesonic Contributions to the Spin and Flavor Structure of the Nucleon • J. Speth and A. W. Thomas Muon Catalyzed Fusion: Interplay between Nuclear and Atomic Physics • K. Nagamine and M. Kamimura

Volume 25 Chiral Symmetry Restoration and Dileptons in Relativistic Heavy-Ion Collisions • R. Rapp and J. Wambach Fundamental Symmetry Violation in Nuclei • H. Feshbach, M. S. Hussein, A. K. Kerman, and O. K. Vorov Nucleon-Nucleus Scattering: A Microscopic Nonrelativistic Approach • K. Amos, P. J. Dortmans, H. V. von Geramb, S. Karataglidis, and J. Raynal

ARTICLES PLANNED FOR FUTURE VOLUMES

Fifty Years of the Nuclear Shell Model • Igal Talmi Chiral Perturbation Theory for Mesons and Nucleons • Stefan Scherer

PREFACE The four articles of the present volume address very different topics in nuclear physics and, indeed, encompass experiments at very different kinds of experimental facilities. The range of interest of the articles extends from the nature of the substructure of the nucleon and the deuteron to the general properties of the nucleus, including its phase transitions and its rich and unexpected quantal properties. The first article by Fillipone and Ji reviews the present experimental and theoretical situation pertaining to our knowledge of the origin of the spin of the nucleon. Until about 20 years ago the half-integral spin of the neutron and proton was regarded as their intrinsic property as Dirac particles which were the basic building blocks of atomic nuclei. Then, with the advent of the Standard Model and of quarks as the basic building blocks, the substructure of the nucleon became the subject of intense interest. Initial nonrelativistic quark models assigned the origin of nucleon spin to the fundamental half-integral spin of its three constituent quarks, leaving no room for contributions to the spin from the gluons associated with the interacting quarks or from the orbital angular momentum of either gluons or quarks. That naive understanding was shaken, about fifteen years ago, by experiments involving deep-inelastic scattering of electrons or muons from nucleons. These experiments suggested that less than half of the nucleon’s spin came from the intrinsic spin of the constituent up and down quarks. As a result, the search for the true origin of nucleon spin became one of the most important problems of subatomic physics. New generations of experiments were planned at many of the world’s largest facilities for nuclear and particle physics, involving the deep-inelastic scattering of polarized leptons from nucleons. Given the progress in recent years, it is now an especially appropriate time to assess the results of these new experiments and of the associated theoretical ideas. The article by Fillipone and Ji provides an excellent review of the subject given by one of the leading experimentalists and one of the leading theorists of the field. Understanding the phases of hadronic matter and the transitions between them is one of the key problems of contemporary nuclear science. Given the xi

xii

Preface

excitement associated with the exploration of the phase transition to a quarkgluon plasma - and to quark deconfinement and chiral symmetry restoration in relativistic heavy-ion collisions, it is possible to overlook other interesting phase transitions. Hence, we have chosen for the second review an article by Das Gupta, Mekjian and Tsang that describes our present understanding of the liquid-gas phase transition which occurs at much lower temperatures and densities. When two heavy nuclei collide at medium to high energy, they may undergo multifragmentation in which the hot system disassembles into a region of liquid-gas coexistence. This article reviews the present experimental and theoretical knowledge of such systems and discusses the wealth of physics ideas needed to describe the experiments. The third article, by Ward and Fallon describes the rich lode of experimental data and of theoretical models emerging now about very high spin states of nuclei. This field is pursued with the conventional accelerators of low energy nuclear physics - tandem accelerators or cyclotrons - augmented recently with very large gamma-detection arrays such as Gammasphere and Euroball. This is a new direction for nuclear spectroscopy which, decades ago, pursued the understanding of the multitude of low energy states of nuclei using ideas of mean fields and of collective motion. The study of high angular momentum states with the new detector arrays explores nuclei under truly unique and extreme conditions challenging all of the basic models. As the article describes, one not only finds extreme rotational bands – superbands – extending to very high angular momentum, but also observes unexpected and striking results. For example, the identical bands in different nuclei which have gamma ray transition energies differing by about one part in a thousand (a kilovolt difference in transitions of more than one MeV) imply that different pairs of nuclei have moments of inertia which differ by at most one part in a thousand. What powerful symmetry is involved? Such results clearly expose the inadequacy of our present microscopic understanding of nuclear structure. The field has grown very large and this fine review covers most of the important new results and ideas. The understanding of the atomic nucleus became a serious study with the discovery of the neutron in 1932, the same year in which the deuteron was discovered. Since that time no other composite particle – other than the nucleon itself – has contributed so much to the development of our field. On the one hand the deuteron is the lightest nucleus (other than the proton) and its two-body nature makes it a very useful and tractable object for the exploration of ideas and models pertaining to heavier nuclei: on the other hand it differs dramatically from all other nuclei because it is a very loosely bound system of a proton and a neutron and therefore provides a useful target of essentially free neutrons. Both aspects are important. In the last decade high-resolution, medium-energy

Preface

xiii

continuous electron beams have produced beautiful experimental results for the nucleon and the deuteron. This is especially so for the very high quality beams of the Jefferson laboratory which has emerged as a flagship of nuclear physics. Writing in a historical perspective Garçon and Van Orden describe what we have learned from the deuteron, especially from the electromagnetic form factors derived from recent electron-deuteron scattering experiments with observed polarizations. The new data is a feast for nuclear models. This article does an excellent job of sharing that feast with the whole community of subatomic physicists. Taken together, the four articles of the present volume display the breadth and rich interplay between theory and experiment which continues to characterize nuclear physics. J.W. NEGELE E.W. VOGT

CONTENTS

Chapter 1 THE SPIN STRUCTURE OF THE NUCLEON B.W. Filippone and Xiangdong Ji 1.

Introduction 1.1. A Simple Model for Proton Spin 1.2. Lepton Scattering as a Probe of Spin Structure 1.3. Theoretical Introduction

2.

Experimental Overview 2.1. SLAC Experiments 2.2. CERN Experiments 2.3. DESY Experiments 2.4. RHIC Spin Program

10 12 13 15 16

3.

Total Quark Helicity Distribution 3.1. Virtual Photon Asymmetries 3.2. Extraction of 3.3. Recent Results for 3.4. First Moments of 3.5. Next-to-Leading Order Evolution of

16 17 19 21 24 27

4.

Individual Quark Helicity Distributions 4.1. Semi-Inclusive Polarized Lepton Scattering High Energy Collisions 4.2.

30 32 36

5.

Gluon Helicity Distribution from QCD Scale Evolution 5.1.

37 38

xv

2 3 4 6

xvi

Contents

5.2. 5.3. 5.4. 5.5. 5.6. 5.7.

from Di-jet Production in Scattering from Hadron Production in Scatter-

38

ing

42

from Open-charm (Heavy-quark) Production in Scattering from Direct Photon Production in Collisions from Jet and Hadron Production in Collisions Experimental Measurements

43 45 48 48

6.

Transverse Spin Physics Structure Function of the Nucleon 6.1. The 6.2. Tranversity Distribution 6.3. Single-Spin Asymmetries From Strong Interactions

54 54 58 61

7.

Off-Forward Parton Distributions 7.1. Properties of the Off-Forward Parton Distributions 7.2. Deeply Virtual Exclusive Scattering

64 67 70

8.

Related Topics in Spin Structure 8.1. The Drell-Hearn Gerasimov Sum Rule and Its Generalizations 8.2. Spin-Dependent Fragmentation Conclusions Acknowledgements

73 73 77 79 79

References

80

Chapter 2 LIQUID-GAS PHASE TRANSITION IN NUCLEAR MULTIFRAGMENTATION S. Das Gupta, A.Z. Mekjian and M.B. Tsang 1.

Introduction

91

2.

Liquid Gas Phase Transition in Nuclear Meanfield Theory

92

3.

Experimental Overview

98

4.

Event Selection 4.1. Central Collisions

99 99

Contents

4.2.

xvii

Peripheral Collisions

102

5.

Evidence for Nuclear Expansion

102

6.

Space-Time Determination

105

7.

Temperature Measurements 7.1. Kinetic Temperatures 7.2. Excited State Temperature 7.3. Isotope Temperature 7.4. Effect of Sequential Decays 7.5. Cross-Comparisons Between Thermometers 7.6. Summary of Temperature Measurements

109 109 110 110 111 113 115

8.

Excitation Energy Determination

115

9.

Signals for Liquid-Gas Phase Transition Rise and Fall of IMF 9.1. 9.2. Critical Exponents 9.3. Nuclear Caloric Curve 9.4. Isospin Fractionation

118 118 119 120 123

10. A Class of Statistical Models

126

11. A Thermodynamic Model

128

12. Generalisation to a More Realistic Model

138

13. A Brief Review of the SMM Model

142

14. The Microcanonical Approach

142

15. The Percolation Model

144

16. The Lattice Gas Model(LGM)

146

17. Phase Transition in LGM

149

18. Isospin Dependent LGM Including Coulomb Interaction

152

19. Calculations With Isospin Dependent LGM

153

20. Fragment Yields from a Model of Nucleation

156

21. Isospin Fractionation in Meanfield Theory

157

22. Dynamical Models for Fragmentation

158

23. Outlook Acknowledgements

161 161

References

162

xviii

Contents

Chapter 3 HIGH SPIN PROPERTIES OF ATOMIC NUCLEI D. Ward and P. Fallon 1.

Introduction 1.1. Nuclear Structure at High Angular Momentum 1.2. Basic Ideas 1.3. Electromagnetic Decays 1.4. Organisation of the review

168 168 169 173 173

2.

The Development of Large Arrays for Gamma-Ray Spectroscopy 2.1. Early Ideas-Selectivity, Sensitivity and Population Mechanisms 2.2. Second and Third Generation Arrays 2.3. Close-packed Composite-Detectors 2.4. Segmented Detectors 2.5. Ancillary Detectors

175

3.

Rotations, Particle Alignments and the Nuclear Shape 3.1. Introduction 3.2. Triaxial Shapes 3.3. High-K States 3.4. Rotational Bands Built on Vibrational Intrinsic States

194 194 197 205 208

4.

Superdeformation 4.1. The Existence and Stability of Superdeformed Nuclei 4.2. Rotations of Superdeformed Nuclei

218 219 229

5.

Limits to Nuclear Rotations 5.1. Termination of Rotational Bands 5.2. Rotations and Pairing Correlations 5.3. Rotations in N = Z Nuclei 5.4. Shears Bands

242 242 244 249 252

6.

Studies of the Quasi-Continuous Radiation 6.1. Introduction 6.2. The Nuclear Inertia at the Highest Angular Momentum Acknowledgements

256 256 274 282

References

282

175 180 186 189 190

Contents

xix

Chapter 4 THE DEUTERON: STRUCTURE AND FORM FACTORS M. Garçon and J.W. Van Orden 1.

A Historical Introduction 1.1. Discovery of the Deuteron 1.2. Early Theories 1.3. Spin 1.4. Connection with OPE

294 294 295 296 297

2.

The Non-Relativistic Two-Nucleon Bound State 2.1. The Potential Model of the Deuteron 2.2. The Deuteron Wave Function

297 297 298

3.

Static 3.1. 3.2. 3.3.

301 302 304 305

4.

Elastic Electron-Deuteron Scattering 4.1. Deuteron electromagnetic form factors 4.2. Observables data 4.3. Review of elastic 4.4. Empirical features of form factors

5.

Theoretical Issues 319 5.1. Deuteron Elastic Form Factors in the Simple Potential Model 319 322 5.2. Limitations of the simple potential model 324 5.3. Construction of Relativistic Models expansions 327 5.4. 330 5.5. Relativistic Constraint Dynamics 332 Field Theoretical Models 5.6. 348 5.7. Deuteron models with nucleon isobar contributions 350 5.8. Quarks and gluons 353 5.9. Further Comparison Between Models and Data

6.

The Nucleon Momentum Distribution in the Deuteron 6.1. Measurements at High Missing Momenta measurements and y-scaling 6.2.

and Low Energy Properties Deuteron Static Properties (Experiment) Low Energy Scattering Parameters Static Properties and the NN Potential

305 305 307 311 316

358 358 359

xx

Contents

6.3.

Hadronic deuteron break-up at high energy

360

7.

The Deuteron as a Source of “Free” Neutrons

361

8.

Prospects for the Future Acknowledgements

361 364

Appendix A.1. Beyond One Photon Exchange at High A.2. Polarized Deuteron Targets — Polarimeters A.3. Nucleon Electromagnetic Form Factors

365 365 365 369

References

372

Index

379

Chapter 1

THE SPIN STRUCTURE OF THE NUCLEON B.W. Filippone W.K. Kellogg Laboratory California Institute of Technology Pasadena, CA 91125, USA and

Xiangdong Ji Department of Physics University of Maryland College Park, MD 20742, USA

1.

Introduction

2.

Experimental Overview

10

3.

Total Quark Helicity Distribution

16

4.

Individual Quark Helicity Distributions

30

5.

Gluon Helicity Distribution

37

6.

Transverse Spin Physics

54

7.

Off-Forward Parton Distributions

64

8.

Related Topics in Spin Structure

73

References

80

2

1

2

B.W. Filippone and Xiangdong Ji

1. INTRODUCTION Attempting to understand the origin of the intrinsic spin of the proton and neutron has been an active area of both experimental and theoretical research for the past twenty years. With the confirmation that the proton and neutron were not elementary particles, physicists were challenged with the task of explaining the nucleon’s spin in terms of its constituents. In a simple constituent picture one can decompose the nucleon’s spin as

where and represent the intrinsic and orbital angular momentum respectively for quarks and gluons. A simple non-relativistic quark model (as described below) gives directly and all the other components = 0. Because the structure of the nucleon is governed by the strong interaction, the components of the nucleon’s spin must in principle be calculable from the fundamental theory: Quantum Chromodynamics (QCD). However, since the spin is a low energy property, direct calculations with non-perturbative QCD are only possible at present with primitive lattice simulations. The fact that the nucleon spin composition can be measured directly from experiments has created an important frontier in hadron physics phenomenology and has had crucial impact on our basic knowledge of the internal structure of the nucleon. This paper summarizes the status of our experimental and theoretical understanding of the nucleon’s spin structure. We begin with a simplified discussion of nucleon spin structure and how it can be accessed through polarized deep-inelastic scattering (DIS). This is followed by a theoretical overview of spin structure in terms of QCD. The experimental program is then reviewed where we discuss the vastly different techniques being applied in order to limit possible systematic errors in the measurements. We then address the variety of spin distributions associated with the nucleon: the total quark helicity distribution extracted from inclusive scattering, the individual quark helicity distributions (flavor separation) determined by semi-inclusive scattering, and the gluon helicity distribution accessed by a variety of probes. We also discuss some additional distributions that have recently been discussed theoretically but are only just being accessed experimentally: the transversity distribution and the off-forward distributions. Lastly we review a few topics closely related to the spin structure of the nucleon. A number of reviews of nucleon spin structure have been published. Following the pioneering review of the field by Hughes and Kuti [177] which set the stage for the very rapid development over the last fifteen years, a number of reviews have summarized the recent developments [77, 43, 216, 176, 105].

The Spin Structure of the Nucleon

3

Also Ref. [91] presents a detailed review of the potential contribution of the Relativistic Heavy Ion Collider (RHIC) to field of nucleon spin structure.

1.1.

A Simple Model for Proton Spin

A simple non-relativistic wave function for the proton comprising only the valence up and down quarks can be written as

where we have suppressed the color indices and permutations for simplicity but enforced the normalization. Here the up and down quarks give all of the proton’s spin. The contribution of the and quarks to the proton’s spin can be determined by the use of the following matrix element and projection operator:

where the matrix element gives the number of up quarks polarized along the direction of the proton’s polarization. With the above matrix element and a similar one for the down quarks, the quark spin contributions can be defined as

Thus the fraction of the proton’s spin carried by quarks in this simple model is

and all of the spin is carried by the quarks. Note however that this simple model overestimates another property of the nucleon, namely the axial-vector weak coupling constant In fact this model gives

compared to the experimentally measured value of The difference between the simple non-relativistic model and the data is often attributed to relativistic effects. This “quenching” factor of ~ 0.75 can be

4

B.W. Filippone and Xiangdong Ji

applied to the spin carried by quarks to give the following “relativistic” constituent quark model predictions:

1.2. Lepton Scattering as a Probe of Spin Structure Deep-inelastic scattering (DIS) with charged lepton beams has been the key tool for probing the structure of the nucleon. With polarized beams and targets the spin structure of the nucleon becomes accessible. Information from neutral lepton scattering (neutrinos) is complementary to that from charged leptons but is generally of lower statistical quality. The access to nucleon structure through lepton scattering can best be seen within the Quark-Parton Model (QPM). An example of a deep-inelastic scattering process is shown in Fig. 1.1. In this picture a virtual photon of fourmomentum (with energy and four-momentum transfer ) strikes an asymptotically free quark in the nucleon. We are interested in the deepinelastic (Bjorken) limit in which and are large, while the Bjorken scaling variable is kept fixed (M is the nucleon mass). For unpolarized scattering the quark “momentum” distributions — — are probed in this reaction, where is the quark’s momentum fraction. From the cross-section for this process, the structure function can be extracted. In the quark-parton model this structure function is related to the unpolarized quark distributions via

where the sum is over both quark and anti-quark flavours. With polarized beams and targets the quark spin distributions can be probed. This sensitivity results from the requirement that the quark’s spin be anti-parallel to the virtual photon’s spin in order for the quark to absorb the virtual photon. With the assumption of nearly massless and collinear quarks, angular momentum would not be conserved if the quark absorbs a photon when its spin is parallel to the photon’s spin. Thus measurements of the spin-dependent cross-section allow the extraction of the spin-dependent structure function Again in the quark-parton model this structure function is related to the quark spin distributions via

The Spin Structure of the Nucleon

5

The structure function is extracted from the measured asymmetries of the scattering cross-section as the beam or target spin is reversed. These asymmetries are measured with longitudinally polarized beams and longitudinally and transversely polarized targets (see Sect. 3.). Beyond the QPM, QCD introduces a momentum scale dependence into the structure functions The calculation of this dependence is based on the Operator Product Expansion (OPE) and the renormalization group equations (see e.g., Ref. [261, 180, 43]). We will not discuss this in detail, but we will use some elements of the expansion. In particular, the expansion can be written in terms of “twist” which is the difference between the dimension and the spin of the operators that form the basis for the expansion. The matrix elements of these operators cannot be calculated in perturbative QCD, but the corresponding coefficients are calculable. The lowest order coefficients (twist-two) remain finite as while the higher-twist coefficients vanish as (due to their de-

6

B.W. Filippone and Xiangdong Ji

pendence). Therefore, the full dependence includes both QCD radiative corrections (calculated to next-to-leading-order (NLO) at present) and highertwist corrections. The NLO corrections will be discussed in Sect. 3.5..

1.3.

Theoretical Introduction

To go beyond the simple picture of nucleon spin structure discussed above we must address the spin structure within the context of QCD. We discuss several of these issues in the following Sections.

1.3.1. Quark Helicity Distributions and In polarized DIS, the antisymmetric part of the nucleon tensor is measured,

where is the ground state of the nucleon with four-momentum and polarization and is the electromagnetic current. The antisymmetric part can be expressed in terms of two invariant structure functions,

In the Bjorken limit, we obtain two scaling functions,

which are non-vanishing. If the QCD radiative corrections are neglected, is related to the polarized quark distributions as shown in Eq. (1.11). In QCD, the distribution can be expressed as the Fourier transform of a quark light-cone correlation,

where is a light-cone vector and is a renormalization scale. is a path-ordered gauge link making the operator gauge invariant. When QCD radiative corrections

The Spin Structure of the Nucleon

7

are taken into account, the relation between and is more complicated (see Sect. 3.5.). When is not too large one must take into account the higher-twist contributions to which appear as power corrections. Some initial theoretical estimates of these power corrections have been performed [62, 198]. Integrating the polarized quark distributions over yields the fraction of the nucleon spin carried by quarks,

The individual quark contribution is also called the axial charge because it is related to the matrix element of the axial current in the nucleon state. is the singlet axial charge. Because of the axial anomaly, it is a scaledependent quantity.

1.3.2. The Nucleon Spin Sum Rule To understand the spin structure of the nucleon in the framework QCD, we can write the QCD angular momentum operator in a gauge-invariant form [186]

where

(The angular momentum operator in a gauge-variant form has also motivated a lot of theoretical work, but is unattractive both theoretically and experimentally [253].) The quark and gluon components of the angular momentum are generated from the quark and gluon momentum densities and respectively, is the Dirac spin matrix and the corresponding term is the quark spin contribution. is the covariant derivative and the associated term is the gauge-invariant quark orbital angular momentum contribution. Using the above expression, one can easily construct a sum rule for the spin of the nucleon. Consider a nucleon moving in the z direction, and polarized

8

B.W. Filippone and Xiangdong Ji

in the helicity eigenstate The total helicity can be evaluated as an expectation value of in the nucleon state,

where the three terms denote the matrix elements of three parts of the angular momentum operator in Eq. 1.18. The physical significance of each term is obvious, modulo the momentum transfer scale and scheme dependence (see Sect. 3.5.) indicated by µ. There have been attempts to remove the scale dependence in by subtracting a gluon contribution [37]. Unfortunately, such a subtraction is by no means unique. Here we adopt the standard definition of as the matrix element of the multiplicatively renormalized quark spin operator. As has been discussed above, can be measured from polarized deep-inelastic scattering and the measurement of the other terms will be d iscussed in later Sections. Note that the individual terms in the above equation are independent of the nucleon velocity [188]. In particular, the equation applies when the nucleon is traveling with the speed of light (the infinite momentum frame). The scale dependence of the quark and gluon contributions can be calculated in perturbative QCD. By studying renormalization of the nonlocal operators, one can show [186, 253]

As

there exists a fixed-point solution

Thus as the nucleon is probed at an infinitely small distance scale, approximately one-half of the spin is carried by gluons. A similar result has been obtained by Gross and Wilczek in 1974 for the quark and gluon contributions to the momentum of the nucleon [164]. Strictly speaking, these results reveal little about the nonperturbative structure of bound states. However, experimentally it is found that about half of the nucleon momentum is carried by gluons even at relatively low energy scales (see e.g., [214]). Thus the gluon degrees of freedom not only play a key role in perturbative QCD, but also are a major component of nonperturbative states as expected. An interesting question is then, how much of the nucleon spin is carried by the gluons at low energy

The Spin Structure of the Nucleon

9

scales? A solid answer from the fundamental theory is not yet available. Balitsky and Ji have made an estimate using the QCD sum rule approach [61]:

which yields approximately 0.25. Based on this calculation, the spin structure of the nucleon would look approximately like

Lattice [226] and quark model [65] calculations of have yielded similar results. While has a simple parton interpretation, the gauge-invariant orbital angular momentum clearly does not. Since we are addressing the structure of the nucleon, it is not required that a physical quantity have a simple parton interpretation. The nucleon mass, magnetic moment, and charge radius do not have simple parton model explanations. The quark orbital angular momentum is related to the transverse momentum of the partons in the nucleon. It is well known that transverse momentum effects are beyond the naive parton picture. As will be discussed later, however, the orbital angular momentum does have a more subtle parton interpretation (see Sect. 7.). In the literature, there are suggestions that be considered the orbital angular momentum [253]. This quantity is clearly not gauge invariant and does not correspond to the velocity in classical mechanics [143]. Under scale evolution, this operator mixes with an infinite number of other operators in light-cone gauge [175]. More importantly, there is no known way to measure such “orbital angular momentum”.

1.3.3. Gluon Helicity Distribution In a longitudinally-polarized nucleon the polarized gluon distribution contributes to spin-dependent scattering processes and hence various experimental spin asymmetries. In QCD, using the infrared factorization of hard process, with can be expressed as

where Because of the charge conjugation property of the operator, the gluon distribution is symmetric in

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B.W. Filippone and Xiangdong Ji

The even moments of are directly related to matrix elements of charge conjugation even local operators. Defining

we have

The renormalization scale dependence is directly connected to renormalization of the local operators. Because Eq. (1.24) involves directly the time variable, it is difficult to evaluate the distribution on a lattice. However, the matrix elements of local operators are routinely calculated in lattice QCD, hence the moments of are, in principle, calculable. From the above equations, it is clear that the first-moment of does not correspond to a gauge-invariant local operator. In the axial gauge the first moment of the non-local operator can be reduced to a local one, which can be interpreted as the gluon spin density operator. As a result, the first moment of represents the gluon spin contribution to the nucleon spin in the axial gauge. In any other gauge, however, it cannot be interpreted as such. Thus one can formally write in the axial gauge, where is then the gluon orbital contribution the nucleon spin. There is no known way to measure directly from experiment other than defining it as the difference between and

2. EXPERIMENTAL OVERVIEW A wide variety of experimental approaches have been applied to the investigation of the nucleon’s spin structure. The experiments complement each other in their kinematic coverage and in their sensitivity to possible systematic errors associated with the measured quantities. A summary of the spin structure measurements is shown in Table 2.1 where the beams, targets, and typical energies are listed for each experiment. The kinematic coverage of each experiment is indicated in the table by its average four-momentum transfer and Bjorken range Also given are the average or typical beam and target polarizations as quoted by each experimental group in their respective publications (or in their proposals for the experiments that are under way). The column labeled lists the dilution factor, which is the fraction of scattered events that result from the polarized atoms of interest, and the column labelled is an estimate of the total nucleon luminosity in units of for each experiment.

The Spin Structure of the Nucleon

11

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B.W. Filippone and Xiangdong Ji

In an effort to eliminate possible sources of unknown systematic error in the measurements the experiments have been performed with significantly different experimental techniques. Examples of the large range of experimental parameters for the measurements include variations in the beam polarization of 40-80%, in the target polarization of 30-90% and in the correction for dilution of the experimental asymmetry due to unpolarized material of 0.1-1. We now present an overview of the individual experimental techniques with an emphasis on the different approaches taken by the various experiments.

2.1.

SLAC Experiments

The SLAC program has focused on high statistics measurements of the inclusive asymmetry. The first pioneering experiments on the proton spin structure were performed at SLAC in experiments E80 [35] and E130 [66]. These experiments are typical of the experimental approach of the SLAC spin program. Polarized electrons are injected into the SLAC linac, accelerated to the full beam energy and impinge on fixed targets in End Station A. The polarization of the electrons is measured at low energies at the injector using Mott scattering and at high energies in the End Station using Moller scattering. Target polarization is typically measured using NMR techniques. The scattered electrons are detected with magnetic spectrometers where electron identification is usually done with Cerenkov detectors and Pb-Glass calorimeters. For E80 and El30, electrons were produced by photoionization of produced in an atomic beam source. Electron polarization is produced by SternGerlach separation in an inhomogeneous magnetic field. Polarized protons were produced by dynamic polarization of solid-state butanol doped with a paramagnetic substance. Depolarization effects in the target limited the average beam currents to ~10 nA. In these experiments a considerable amount of unpolarized material is present in the target resulting in a dilution of the physics asymmetry. For E80 and E130 this dilution reduced the asymmetry by a factor of ~0.15. Over the last ten years a second generation of high precision measurements have been performed at SLAC. Information on the neutron spin structure has been obtained using polarized in experiments E142 [45,46] and E154 [6]. Here the polarized behaves approximately as a polarized neutron due to the almost complete pairing off of the proton spins. The nuclear correction to the neutron asymmetry is estimated to be ~5-10%. Beam currents were typically .5-2 µA and the polarization was significantly improved for the E154 experiment using new developments in strained gallium-arsenide photocathodes [225]. A schematic diagram of the spectrometers used for E142 is shown in Fig. 2.1.

The Spin Structure of the Nucleon

13

Additional data on the neutron and more precise data on the proton has come from E143 [2, 3, 9] and E155 [48, 49] where both and H polarized targets using polarized ammonia and were employed. The main difference between these two experiments was again an increase in beam energy from 26-48 GeV and an increase in polarization from 40% to 80%.

2.2. CERN Experiments Following the early measurements at SLAC, the EMC (European Muon Collaboration) experiment [57,58] performed the first measurements at Polarized muon beams were produced by pion decay yielding beam intensities of The small energy loss rate of the muons allowed the use of very thick targets (~1 m) of butanol and methane. The spin structure measurements by EMC came at the end of a series of measurements of unpolarized nucleon and nuclear structure functions, but the impact of the EMC spin measurements was significant. Their low measurements, accessible due to the high energy of the muons, suggested the breakdown of the naive parton picture that quarks

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provide essentially all of the spin of the nucleon. The SMC (Spin Muon Collaboration) experiment [19, 15, 13, 21, 16] began as a dedicated follow-on experiment to the EMC spin measurements using an upgraded apparatus. An extensive program of measurements with polarized and targets was undertaken over a period of ten years. Improvements in target and beam performance provided high precision data on inclusive spindependent structure functions. The large acceptance of the SMC spectrometer in the forward direction (see Fig. 2.2) allowed them to present the first measurements of spin structure using semi-inclusive hadron production. As with EMC, the high energy of the muon beam provided access to the low regime A new experiment is underway at CERN whose goal is to provide direct information on the gluon polarization. The COMPASS [111] (COmmon Muon Proton Apparatus for Structure and Spectroscopy) experiment will use a large acceptance spectrometer with full particle identification to generate a high statistics sample of charmed particles. Using targets similar to those used in SMC and an intense muon beam improved measurements of other semi-inclusive asymmetries will also be possible.

The Spin Structure of the Nucleon

2.3.

15

DESY Experiments

Using very thin gaseous targets of pure atoms and very high currents (~40 mA) of stored, circulating positrons or electrons HERMES (HERa MEasurement of Spin) has been taking data at DESY since 1995. HERMES is a fixed target experiment that uses the stored beam of the HERA collider. The polarization of the beam is achieved through the Sokolov-Ternov effect [266], whereby the beam becomes transversely polarized due to a smallspin dependence in the synchrotron radiation emission. The transverse polarization is rotated to the longitudinal direction by a spin rotator – a sequence of horizontal and vertical bending magnets that takes advantage of the precession of the The beam polarization is measured with Compton polarimeters [64]. HERMES has focused its efforts on measurements of semi-inclusive asymmetries, where the scattered is detected in coincidence with a forward hadron. This was achieved with a large acceptance magnetic spectrometer [11] as shown in Fig. 2.3. Initial measurements allowed some limited pion identification with a gas threshold Cerenkov detector and a Pb-glass calorimeter. Since 1998 a Ring Imaging Cerenkov (RICH) detector has been in operation allowing full hadron identification over most of the momentum acceptance of the spectrometer.

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Up to the present, HERMES has taken data only with a longitudinally polarized target. Future runs will focus on high statistics measurements with a transversely polarized target to access, e.g., transversity (see Sect. 6.1) and (see Sect. 6.2). Promising future spin physics options also exist at DESY if polarized protons can be injected and accelerated in the HERA ring. The [205] program would use the stored 820 GeV proton beam and a fixed target of gaseous polarized nucleons. This would allow measurements of quark and gluon polarizations at complimenting the higher energy measurements possible in the RHIC spin program. A stored polarized proton beam in HERA would also allow collider measurements [120] with the existing H1 and ZEUS detectors. Inclusive polarized DIS could be measured to much higher and lower than existing measurements. This would allow improved extraction of the gluon polarization via the scaling violations of the spin-dependent cross-section. Heavy quark and jet production as well as charged-current vs. neutral-current scattering would also allow improved measurements of both quark and gluon polarizations.

2.4.

RHIC Spin Program

The Relativistic Heavy-Ion Collider (RHIC) [254] at the Brookhaven National Laboratory recently began operations. This collider was designed to produce high luminosity collisions of high-energy heavy ions as a means to search for a new state of matter known at the quark-gluon plasma. The design of the accelerator also allows the acceleration and collision of high energy beams of polarized protons and a fraction of accelerator operations will be devoted to spin physics with colliding Beam polarizations of 70% and center-ofmass energies of are expected. Two large collider detectors, PHENIX [235] and STAR [167], along with several smaller experiments, BRAHMS [276], PHOBOS [273] and PP2PP, will participate in the RHIC spin program. As an example a schematic diagram of the STAR detector is shown in Fig. 2.4. Longitudinal beam polarization will be available for the PHENIX and STAR detectors enabling measurements of quark and gluon spin distributions (see Sect. 4.2 and 5.7.4).

3.

TOTAL QUARK HELICITY DISTRIBUTION

A large body of data has been accumulated over the past ten years on inclusive polarized lepton scattering from polarized targets. These data allow the extraction of the spin structure functions and the nearly model-

The Spin Structure of the Nucleon

17

independent determination of the total quark contribution to the nucleon spin Inclusive data combined with assumptions about flavor symmetry, and results from beta decay provide some model-dependent information on the individual flavour contributions to the nucleon spin. Studies of the dependence of allow a first estimate of the gluon spin contribution albeit with fairly large uncertainties. These results are discussed in the following Sections.

3.1.

Virtual Photon Asymmetries

Virtual photon asymmetries can be defined in terms of a helicity decomposition of the virtual photon-nucleon scattering cross-sections. For a transversely polarized virtual photon (e.g., with helicity ±1) incident on a longitudinally polarized nucleon there are two helicity cross-sections and and the longitudinal asymmetry is given by

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B.W. Filippone and Xiangdong Ji

is a virtual photon asymmetry that results from an interference between transverse and longitudinal virtual photon-nucleon amplitudes:

These virtual photon asymmetries, in general a function of lated to the nucleon spin structure functions and

and

are revia

where These virtual photon asymmetries can be related to measured lepton asymmetries through polarization and kinematic factors. The experimental longitudinal and transverse lepton asymmetries are defined as

where is the cross-section for the lepton and nucleon spins aligned (anti-aligned) longitudinally, while is the cross-section for longitudinally polarized lepton and transversely polarized nucleon. The lepton asymmetries are then given in terms of the virtual photon asymmetries through

The virtual photon (de)polarization factor D is approximately equal to (where is the energy of the virtual photon and E is the lepton energy), but is given explicitly as

where is the magnitude of the virtual photon’s transverse polarization

and

The Spin Structure of the Nucleon

19

is the ratio of longitudinal to transverse virtual photon cross-sections. The other factors are given by

3.2.

Extraction of

The nucleon structure function is extracted from measurements of the lepton-nucleon longitudinal asymmetry (with longitudinally polarized beam and target)

where represents the cross-section when the electron and nucleon spins are aligned (anti-aligned). These cross-sections can also be expressed in terms of spin-independent and spin-dependent cross-sections

In the limit of stable beam currents, target densities and polarizations, the experimentally measured asymmetry is usually expressed in terms of the measured count rates N and the number of incident electrons

is then determined via

where and are the beam and target polarizations respectively, is a dilution factor due to scattering from unpolarized material and accounts for QED radiative effects [34]. If however there is a time variation of the beam or target polarization or luminosity, the asymmetry should be determined using

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B.W. Filippone and Xiangdong Ji

since in this case the measured count rates can be written in terms of

and

where now represents the product of beam current and target areal density – the luminosity. In Eq. 3.18 we have ignored a factor accounting for the acceptance and solid angle of the apparatus which is assumed to be independent of time. The spin structure function can then be determined from the longitudinal asymmetry

where is the unpolarized structure function. The unpolarized structure function is usually determined from measurements of the unpolarized structure function and R using

To use the above equation we need an estimate for is constrained to be less than but can also be determined from measurements (see Sect. 6.1.) with a longitudinally polarized lepton beam and a transversely polarized nucleon target (when combined with the longitudinal asymmetry). As a guide to the relative importance of various kinematic terms in the above equations we present examples of the magnitude of these terms in Table 3.1 typical for the SMC and HERMES experiments. For extraction of the neutron structure function from nuclear targets, e.g., and additional corrections must be applied. For the deuteron, the largest contribution is due to the polarized proton in the polarized deuteron which must be subtracted. In addition a D-state admixture into the wave function will reduce the deuteron spin structure function due to the opposite alignment of the spin system in this orbital state; thus

where

is the D-state probability of the deuteron. Typically a value of is used for this correction. For polarized a wavefunction correction for the neutron and proton polarizations is applied using

The Spin Structure of the Nucleon

21

where and as taken from a number of calculations [146, 106]. Additional corrections due to the neutron binding energy and Fermi motion have also been investigated [75, 106, 260] and shown to be relatively small.

3.3.

Recent Results for

Most of the experiments listed in Table 2.1 have contributed high precision data on the spin structure function Where there is overlap (in and ), the agreement between the experiments is extremely good. This can be seen in Fig. 3.1 where the ratio of the polarized to unpolarized proton structure function is shown. Analysis of the dependence of this ratio [4] has shown that it is consistent experimentally with being independent of within the range of existing experiments, although this behaviour is not expected to

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B.W. Filippone and Xiangdong Ji

persist for all A comparison of the spin structure functions are shown in Fig. 3.2. Some residual dependence is visible in the comparison of the SMC data with the other experiments. The general dependence of will be discussed in Sect. 3.5.

The Spin Structure of the Nucleon

23

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B.W. Filippone and Xiangdong Ji

While the results shown in Fig. 3.1 and Fig. 3.2 correspond to data also exists at lower because of the large kinematic acceptance in many of the experiments. Much of this data [4, 25, 238], when expressed as appears to be largely independent of

3.4.

First Moments of

The initial interest in measurements of was in comparing the measurements to several predicted sum rules, specifically the Ellis-Jaffe and Bjorken sum rules. These sum rules relate integrals over the measured structure functions to measurements of neutron and hyperon beta-decay. The Ellis-Jaffe sum rule [137] starts with the leading-order QPM result for the integral of

where the sum is over for three active quark flavours and the dependence has been suppressed as it is absent in the simple QPM. Introducing the nucleon axial charges:

where the Ellis-Jaffe sum rule then assumes that the strange quark and sea polarizations are zero Then for the proton and neutron integrals the Ellis-Jaffe sum rule gives:

To evaluate the integrals it is assumed that which is true if Then is determined from the ratio of axial-vector to vector coupling constants in neutron decay A value for can be estimated with the additional assumption of SU(3) flavour symmetry which allows one to express for hyperon beta decays in terms of and (see Table 3.2), giving Nucleon and hyperon beta

The Spin Structure of the Nucleon

25

decay is sometimes parameterized in terms of the F and D coefficients. These coefficients are related to the axial charges and with

The assumptions implicit in the Ellis-Jaffe sum rule, e.g., 0 and symmetry, may be significantly violated. On the contrary, the Bjorken sum rule [74]

requires only current algebra and isospin symmetry in its derivation. Note that both the Ellis-Jaffe and Bjorken sum rules must be corrected for QCD radiative corrections. For example, these corrections have been evaluated up to order and amount to ~10% correction for the Ellis-Jaffe sum rule and ~15% correction for the Bjorken sum rule at Comparison of these predictions with experiment requires forming the integrals of the measurements of over the full range from at a fixed Thus extrapolations are necessary in order to include regions of unmeasured both at high and low For the large region this is straightforward: since is proportional to a difference of quark distributions it must approach zero as as this is the observed behaviour of the unpolarized distributions. However, the low region is problematic, as there is no clear dependence expected. In the first analyses simple extrapolations based on Regge parameterizations [170, 135] were used. Thus was assumed to be nearly constant for Later, Next-to-Leading-Order (NLO) QCD calculations [136] (see Sect. 3.5.) suggested that these parameterizations likely underestimated the low contributions. The NLO calculations cannot predict the actual dependence of the structure function, but can only take a given

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dependence and predict its dependence on Thus by using the Regge parameterizations for low they can give the low x behaviour at the of the experiments, e.g., Evaluating the experimental integrals at a fixed requires an extrapolation of the measured structure function. In general, for each experiment, the experimental acceptance imposes a correlation between and preventing a single experiment from measuring the full range in at a constant value of Thus the data must be QCD-evolved to a fixed value of This has often been done by exploiting the observed independence of (see Fig. 3.1). In this case most of the dependenceof results from the dependence of the unpolarized structure function which is well measured in other experiments. Alternatively, NLO QCD fits (as described in the next Section) can be used to evolve the data sets to a common The E155 collaboration has recently reported [49] a global analysis of spin structure function integrals. They have evolved the world data set on and and have extrapolated to low and to high using a NLO fit to the data. Their results are compared in Table 3.3 with the predictions for the Ellis-Jaffe and Bjorken sum rules (Eqs. 3.27,3.29) including QCD radiative corrections for up to order using the calculations of Ref. [217] and world-average for [241].

As seen in Table 3.3 the Bjorken sum rule is well verified. In fact some analyses [136] have assumed the validity of the Bjorken sum rule and used the dependence of to extract a useful value for In contrast there is a strong violation of the Ellis-Jaffe sum rules. Many early analyses of these results interpreted the violation in terms of a non-zero value for (in which case using only the leading order QPM. However, modern analyses have demonstrated that a full NLO analysis is necessary in order to interpret the results. This analysis will be described in the next Section. Here, for completeness, we give the leading order QPM result. Within the leading order QPM, and can be determined by using Eqs. 3.27 with the experimental values from Table 3.3. Dropping the assumption of but retaining the assumption to

The Spin Structure of the Nucleon

determine

27

one finds:

after applying the relevant QCD radiative corrections to the terms in Eq. 3.27 (corresponding to a factor of 0.859 multiplying the triplet and octet charges and a factor of 0.87 8 multiply ing the singlet charge for This then gives a very small value for the total quark contribution to the nucleon’s spin, Note that the quoted uncertainties reflect only the uncertainty in the measured value of and not possible systematic effects due to the assumption of symmetry and NLO effects. Studies of the effect of symmetry violations have been estimated [9] to have little effect on the uncertainty in and but can increase the uncertainty on by a factor of two to three. NLO effects are the subject of the next Section.

3.5.

Next-to-Leading Order Evolution of

As discussed above the spin structure functions possess a significant dependence due to QCD radiative effects. It is important to understand these effects for a number of reasons, including comparison of different experiments, forming structure function integrals, parameterizing the data and obtaining sensitivity to the gluon spin distribution. As the experiments are taken at different accelerator facilities with differing beam energies the data span a range of In addition, because of the extensive data set that has been accumulated and the recently computed higher-order QCD corrections, it is possible to produce parameterizations of the data based on Next-to-Leading-Order (NLO) QCD fits to the data. This provides important input to future experiments utilizing polarized beams (e.g., the RHIC spin program). These fits have also yielded some initial information on the gluon spin distribution, because of the radiative effects that couple the quark and gluon spin distributions at NLO. At NLO the QPM expression for the spin structure function becomes

where for three active quark flavors the sum is again over quarks and antiquarks: and are Wilson coefficients and correspond to the polarized photon-quark and photon-gluon hard scattering cross-section respectively. The convolution is defined as

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B.W. Filippone and Xiangdong Ji

The explicit dependence of the nucleon spin structure function on the gluon spin distribution is apparent in Eq. 3.31. At Leading Order (LO) and and the usual dependence (Eq. 3.23) of the spin structure function on the quark spin distributions emerges. At NLO however, the factorization between the quark spin distributions and coefficient functions shown in Eq. 3.31 cannot be defined unambiguously. This is known as factorization scheme dependence and results from an ambiguity in how the perturbative physics is divided between the definition of the quark/gluon spin distributions and the coefficient functions. There are also ambiguities associated with the definition of the matrix in dimensions [272] and in how to include the axial anomaly. This has lead to a variety of factorization schemes that deal with these ambiguities by different means. We can classify the factorization schemes in terms of their treatment of the higher order terms in the expansion of the coefficient functions. The dependence of this expansion can be written as:

In the so-called Modified-Minimal-Subtraction scheme [231, 277] the first moment of the NLO correction to vanishes such that does not contribute to the first moment of In the AdlerBardeen [63, 38] scheme (AB) the treatment of the axial anomaly causes the first moment of to be non-zero, leading to a dependence of on This then leads to a difference in the singlet quark distribution in the two schemes:

A third scheme, sometimes called the JET scheme [101, 218] or chirally invariant (CI) scheme [104], is also used. This scheme attempts to include all perturbative anomaly effects into Of course any physical observables (eg. are independent of the choice of scheme. There are also straightforward transformations [38, 237, 219] that relate the schemes and their results to one another. Once a choice of scheme is made the dependence of can be calculated using the Dokahitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) [163] equations. These equations characterize the evolution of the spin distributions

The Spin Structure of the Nucleon

in terms of

29

splitting functions

where the non-singlet quark distributions are defined with

for three quark flavors

The splitting functions can be expanded in a form similar to that for the coefficientfunctions in Eq. 3.33 and have been recently evaluated [231, 277] in NLO. The remaining ingredients in providing a fit to the data are the choice of starting momentum scale and the form of the parton distributions at this The momentum scale is usually chosen to be so that the quark spin distributions are dominated by the valence quarks and the gluon spin distribution is likely to be small. Also, as discussed above, at lower momentum transfer some models for the dependence of the distributions (e.g., Reggetype models for the low region) are more reliable. The form of the polarized parton distributions at the starting momentum scale are parameterized by a variety of dependences with various powers. This parameterization is the source of some of the largest uncertainties as the dependence at low values of is largely unconstrained by the measurements. As an example, Ref. [38] assumes for one of its fits that the polarized parton distributions can be parameterized by

With such a large number of parameters it is usually required to place additional constraints on some of the parameters. Often symmetry is used to constrain the parameters, or the positivity of the distributions is enforced (note that this positivity is strictly valid only when all orders are included; see Ref. [39]). Thus in other fits, the polarized distributions are taken to be proportional to the unpolarized distributions as in e.g., Ref. [49]:

A large number of NLO fits have recently been published [156,151,63,38, 24, 8, 86, 162, 219, 220, 221, 161, 49, 115]. These fits include a wide variety

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B.W. Filippone and Xiangdong Ji

of assumptions for the forms of the polarized parton distributions, differences in factorization scheme and what data sets they include in the fit (only the most recent fits [161] include all the published inclusive data). Some fits [115] have even performed a NLO analysis including information from semi-inclusive scattering (see Sect. 4.1.). A comparison of the results from some of these recent fits is shown in Table 4.1. Note that in the JET and AB schemes includes a contribution from Thus the overriding result of these fits is that the quark spin distribution is constrained between 0.05-0.30 but that the gluon distribution and its first moment are largely unconstrained. The extracted value for is typically positive but the corresponding uncertainty is often 50-100% of the value. Note that the uncertainties listed in Table 4.1 are dependent on the assumptions used in the fits. Estimates of the contribution from higher twist effects [62, 198] corrections) suggest that the effects are relatively small at the present experimental This is further supported by the generally good fits that the NLO QCD calculations can achieve without including possible higher-twist effects. Lattice QCD calculations of the first moments and second moments of the polarized spin distributions are under way [150, 125, 158, 165]. Agreement with NLO fits to the data is reasonable for the quark contribution, although the Lattice calculations are not yet able to calculate the gluon contribution.

4.

INDIVIDUAL QUARK HELICITY DISTRIBUTIONS

As shown in the last Section, the inclusive lepton asymmetries generally provide spin structure information only for the sum over quark flavours. Access to the individual flavour contributions to the nucleon spin requires assumptions including symmetry in the weak decay of the octet baryons (nucleons and strange hyperons). Potentially more direct information on the individual contributions of and quarks as well as the separate contributions of valence and sea quarks is possible via semi-inclusive scattering. Here one or more hadrons in coincidence with the scattered lepton are detected. The charge of the hadron and its valence quark composition provide sensitivity to the flavor of the struck quark within the Quark-Parton Model (QPM). Semi-inclusive asymmetries also allow access to the third leading-order quark distribution called transversity. Because of the chiral odd structure of this distribution function it is not measurable in inclusive DIS. Transversity will be discussed in Sect. 6.2. Additionally, semi-inclusive asymmetries

The Spin Structure of the Nucleon

31

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B.W. Filippone and Xiangdong Ji

can provide a degree of selectivity for different reaction mechanisms that are sensitive to the gluon polarization. The sensitivity of semi-inclusive asymmetries to the gluon polarization will be discussed in Sect. 5. The flavour decomposition of the nucleon spin using semi-inclusive scattering will be discussed in the next two Sections.

4.1.

Semi-Inclusive Polarized Lepton Scattering

Within the QPM, the cross-section for lepto-production of a hadron (semiinclusive scattering) can be expressed as

where

is the inclusive DIS cross-section, the fragmentation function, is the probability that the hadron originated from the struck quark of flavour is the hadron momentum fraction and the sums are over quark and antiquark flavours To maximize the sensitivity to the struck “current” quark, kinematic cuts are imposed on the data in order to suppress effects from target fragmentation. These cuts typically correspond to and In general the fragmentation functions depend on both the quark flavour and the hadron type. In particular for a given hadron This effect can be understood in terms of the QPM: if the struck quark is a valence quark for a particular hadron, it is more likely to fragment into that hadron A flavour sensitivity is therefore obtained as is a sensitivity to the antiquarks Equation 4.1 displays a factorization of the cross-section into separate and dependent terms. This is an assumption of the QPM and must be experimentally tested. Measurements of unpolarized hadron lepto-production [53] have shown good agreement with the factorization hypothesis. Data from hadrons can also be used to extract fragmentation functions [209]. Both the and dependence of the fragmentation functions have been parameterized within string models of fragmentation [263] that are in reasonable agreement with the measurements. Recently the dependence of the fragmentation functions have been calculated to NLO [73]. Assuming factorization of the cross-section as given in Eq. 4.1, we can write the asymmetry for lepto-production of a hadron as

The Spin Structure of the Nucleon

33

Due to parity conservation the fragmentation functions contain no spin dependence as long as the final-state polarization of the hadron is not measured (spin-dependent fragmentation can be accessed through the self-analyzing decay of - see Sect. 8.2.). By making measurements with H, D and targets for different final-state hadrons and assuming isospin symmetry of the quark distributions and fragmentation functions a system of linear equations can be constructed:

and solved for the In these equations, the unpolarized quark distributions are taken from a variety of parameterizations (e.g., Ref. [157, 214]) and the fragmentation functions are taken from measurements [53, 209] or parameterizations [263]. EMC, SMC and HERMES have made measurements of semi-inclusive asymmetries. A comparison of the measurements from SMC and HERMES is shown in Fig. 4.1. As the HERMES data are taken at and the SMC data at these data suggest that the semiinclusive asymmetries are also approximately independent of It is important to note, especially for the lower data of HERMES, that Eq. 4.2 must be modified if parameterizations of the unpolarized quark distributions are used. In some parameterizations it is assumed that the unpolarized structure functions are related by the Callen-Gross approximation rather than by the complete expression Thus some experimental groups will present Eq. 4.2 with an extra factor of included. Up to now results have only been reported for positively and negatively charged hadrons (summing over and K) because of the lack of sufficient particle identification in the experiments. This reduces the sensitivity to some quark flavours (e.g., strangeness) and requires additional assumptions about the flavour dependence of the sea-quark and anti-quark distributions. Two assumptions have been used to extract information on the flavour and sea dependence of the quark polarizations, namely

or

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B.W. Filippone and Xiangdong Ji

Here and represent the and sea-quark spin distributions. A comparison of the extracted valence and sea-quark distributions from HERMES and SMC is shown in Fig. 4.2. The valence distributions are defined using Typical systematic errors are also shown in Fig. 4.2 and include the difference due to the two assumptions for the sea distributions given by Eqs. 4.4-4.5. The solid lines are positivity limits corresponding to The dashed lines are parameterizations from Gehrmann and Stirling (Gluon A-LO) [151].

The Spin Structure of the Nucleon

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B.W. Filippone and Xiangdong Ji

36

Values for the integrals over the spin distributions from SMC and HERMES are compared in Table 4.2. The dominant sensitivity to , within the quark sea is due to the factor of two larger charge compared to and

While the experimental results presented in Table 4.2 have been extracted through a Leading-Order QCD analysis, NLO analyses are possible [114] and several such analyses have recently been published [115]. Future measurements from HERMES and COMPASS will include full particle identification providing greater sensitivity to the flavour separation of the quark spin distributions. In particular, due to the presence of strange quarks in the K valence quark distribution, K identification is expected to give significant sensitity to

4.2.

High Energy

Collisions

The production of weak bosons in high energy collisions at RHIC provides unique sensitivity to the quark and antiquark spin distributions. The maximal parity violation in the interaction and the dependence of the production on the weak charge of the quarks can be used in principle to select specific flavour and charge for the quarks. Thus the single spin longitudinal asymmetry for production can be written [85].

where and refer to the value of the quark and antiquark participating in the interaction (see for example Fig. 4.3). Making the replacement gives the asymmetry for production. In the experiments the are detected through their decay to a charged lepton in PHENIX and in STAR) and the values are determined from the angles and energies of those detected leptons. Thus for production with the valence quarks are selected for and while for valence quarks are selected for and Detection

The Spin Structure of the Nucleon

37

of then gives and An example of the expected sensitivity of the PHENIX experiment after about four years of data taking is shown in Fig. 4.4.

5.

GLUON HELICITY DISTRIBUTION

As remarked in the Introduction, the gluon contribution to the spin of the nucleon can be separated into spin and orbital parts. As with its unpolarized counterpart, the polarized gluon distribution is difficult to access experimentally. There exists no theoretically clean and, at the same time, experimentally straightforward hard scattering process to directly measure the distribution. In the last decade, many interesting ideas have been proposed and some have led to useful initial results from the present generation of experiments; others will be tested soon at various facilities around the world. In the following Subsections, we discuss a few representative hardscattering processes in which the gluon spin distribution can be measured.

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B.W. Filippone and Xiangdong Ji

from QCD Scale Evolution

As discussed in Sect. 3.5, the polarized gluon distribution enters in the factorization formula for spin-dependent inclusive deep-inelastic scattering. Since the structure function involves both the singlet quark and gluon distributions as shown in Eq. 3.31, only the dependence of the data can be exploited to separate them. The dependence results from two different sources: the running coupling in the coefficient functions and the scale evolution of the parton distributions. As the gluon contribution has its own characteristic behaviour, it can be isolated in principle from data taken over a wide range of Because the currently available experimental data have rather limited coverage, there presently is a large uncertainty in extracting the polarized gluon distribution. As described in Sect. 3.5., a number of NLO fits to the world data have been performed to extract the polarized parton densities. While the results for the polarized quark densities are relatively stable, the extracted polarized gluon distribution depends strongly on the assumptions made about the dependence of the initial parameterization. Different fits produce results at a fixed differing by an order of magnitude and even the sign is not well constrained. Several sets of polarized gluon distributions have been used widely in the literature for the purpose of estimating outcomes for future experiments. An example from Ref. [151] of the range of possible distributions is shown in Fig. 5.1. Of course the actual gluon distribution could be very different from any of these.

5.2.

from Di-jet Production in

Scattering

In lepton-nucleon deep-inelastic scattering, the virtual photon can produce two jets with large transverse momenta from the nucleon target. To leadingorder in the underlying hard scattering subprocesses are Photon-Gluon Fusion (PGF) and QCD Compton Scattering (QCDC) as shown in Fig. 5.2. If the initial photon has momentum and the parton from the nucleon (with momentum P) has momentum the invariant mass of the di-jet is the at which the parton densities are probed is

where is the Bjorken variable. Therefore the di-jet invariant mass fixes the parton momentum fraction. Depending on the relative sizes of and can be an order of magnitude larger than

The Spin Structure of the Nucleon

39

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The Spin Structure of the Nucleon

41

If the contribution from the quark initiated subprocess is small or the quark distribution is known, the two-jet production is a useful process to measure the gluon distribution. The di-jet invariant mass provides direct control over the fraction of the nucleon momentum carried by the gluon Indeed, di-jet data from HERA have been used by the H1 and ZEUS collaborations to extract the unpolarized gluon distribution [29, 255]. With a polarized beam and target, the process is ideal for probing the polarized gluon distribution. The unpolarized di-jet cross-section for photon-nucleon collisions can be written as [118]

where and are the gluon and quark densities, respectively, and A and B are the hard scattering cross-sections calculable in perturbative QCD (pQCD). Similarly, the polarized cross-section can be written as

where the first and second ± refer to the helicities of the photon and nucleon, respectively. The double spin asymmetry for di-jet production is then

The experimental asymmetry by

in DIS is related to the photon asymmetry

where and are the electron and nucleon polarizations, respectively, and D is the depolarization factor of the photon. At low the gluon density dominates over the quark density, and thus the photon-gluon fusion subprocess dominates. There we simply have

which provides a direct measurement of the gluon polarization. Because of the helicity selection rule, the photon and gluon must have opposite helicities to produce a quark and antiquark pair and hence Therefore, if is positive, the spin asymmetry must be negative. Leading-order calculations in Refs. [118, 141, 245, 119] show that the asymmetry is large and is strongly sensitivitive to the gluon polarization.

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At NLO, the one-loop corrections for the PGF and QCDC subprocesses must be taken into account. In addition, three-jet events with two of the jets too close to be resolved must be treated as two-jet production. The sum of the virtual ( processes with one loop) and real ( leading-order processes) corrections are independent of the infrared divergence. However, the two-jet cross-section now depends on the scheme in which the jets are defined. NLO calculations carried out in Refs. [233, 234, 246], show that the strong sensitivity of the cross-section to the polarized gluon distribution survives. In terms of the spin asymmetry, the NLO effects do not significantly change the result. Since the invariant mass of the di-jet is itself a large mass scale, two-jet production can also be used to measure even when the virtuality of the photon is small or zero (real photon). A great advantage of using nearlyreal photons is that the cross-section is large due to the infrared enhancement, and hence the statistics are high. An important disadvantage, however, is that there is now a contribution from the resolved photons. Because the photon is nearly on-shell, it has a complicated hadronic structure of its own. The structure can be described by quark and gluon distributions which have not yet been well determined experimentally. Some models of the spin-dependent parton distributions in the photon are discussed in Ref. [155]. Leading-order calculations [270, 99] show that there are kinematic regions in which the resolved photon contribution is small and the experimental di-jet asymmetry can be used favorably to constrain the polarized gluon distribution.

5.3.

from

Hadron Production in

Scattering

For scattering at moderate center-of-mass energies, such as in fixed target experiments, jets are hard to identify because of their large angular spread and the low hadron multiplicity. However one still expects that the leading hadrons in the final state reflect to a certain degree the original parton directions and flavors (discounting of course the transverse momentum, of order from the parton intrinsic motion in hadrons and from their fragmentation). If so, one can try to use the leading hadrons to tag the partons produced in the hard subprocesses considered in the previous Subsection. Bravar et al. [87] have proposed to use hadrons to gain access to To enhance the relative contribution from the photon-gluon fusion subprocess and hence the sensitivity of physical observables to the gluon distribution they propose a number of selection criteria for analysis of the data and then test these “cut” criteria in a Monte Carlo simulation of the COMPASS experiment. These simulations show that these cuts are effective in selecting the

The Spin Structure of the Nucleon

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gluon-induced subprocess. Moreover, the spin asymmetry for the two-hadron production is large (10-20%) and is strongly sensitive to the gluon polarization. Because of the large invariant mass of the hadron pairs, the underlying sub-processes can still be described in perturbative QCD even if the virtuality of the photon is small or zero [144]. This enhances the data sample but introduces additional sub-processes to the hadron production. The contribution from resolved photons, e.g., from fluctuations, appears not to overwhelm the PGF contribution. Photons can also fluctuate into mesons with scattering yielding hadron pairs. Experimental information on this process can be used to subtract its contribution. After taking into account these contributions, it appears that the low-virtuality photons can be used as an effective probe of the gluon distribution to complement the data from DIS lepton scattering.

5.4.

from Open-charm (Heavy-quark) Production in Scattering

Heavy quarks can be produced in scattering through photon-gluon fusion and can be calculated in pQCD (see Fig. 5.3). In the deep-inelastic scattering region, the charm quark contribution to the structure function is known [154],

where

and

with This result assumes that, because of the large charm quark mass, the direct charm contribution (e.g., through ) is small and the light-quark fragmentation production of charm mesons is suppressed. The dependence of the structure function, if measured, can be deconvoluted to give the polarized gluon distribution. The renormalization scale µ can be taken to be twice the charm quark mass Following Ref. [111], the open charm electro-production cross-section is large when is small or vanishes and can be written

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where the virtual photon flux is

E and are the lepton and photon energies and For a fixed the flux is inversely proportional to The second factor in Eq. 5.9 is the photonucleon cross-section. The cross-section asymmetry is the simplest at the real-photon point The total parton cross-section for photon-gluon fusion is

where is the center-of-mass velocity of the charm quark, and is the invariant mass of the photon-gluon system. On the other hand, the spin-dependent cross-section is

The photon-nucleon asymmetry for open charm production can be obtained by convoluting the above cross-sections with the gluon distribution, giving

The Spin Structure of the Nucleon

45

where is the gluon momentum fraction. Ignoring the dependence, the spin asymmetry is related to the photon-nucleon spin asymmetry by where D is the depolarization factor introduced before. The NLO corrections have recently been calculated by Bojak and Stratmann [79] and Contogouris et al. [112]. The scale uncertainty is considerably reduced in NLO, but the dependence on the precise value of the charm quark mass is sizable at fixed target energies. Besides the total charm cross-section, one can study the distributions of the cross-section in the transverse momentum or rapidity of the charm quark. The benefit of doing this is that one can avoid the region of small where the asymmetry is very small [270]. Open charm production can be measured experimentally by detecting mesons from charm quark fragmentation. On average, a charm quark has about 60% probability of fragmenting into a The meson can be reconstructed through its two-body decay mode the branching ratio is about 4%, Additional background reduction can be achieved by tagging through detection of the additional production is, in principle, also sensitive to the gluon densities. However, because of ambiguities in the production mechanisms [184], any information on is likely to be highly model-dependent.

5.5.

from Direct Photon Production in

Collisions

can be measured through direct (prompt) photon production in proton-proton or proton-antiproton scattering [70]. At tree level, the direct photon can be produced through two underlying sub-processes: Compton scattering and quark-antiquark annihilation as shown in Fig. 5.4. In proton-proton scattering, because the antiquark distribution is small, direct photon production is dominated by the Compton process and hence can be used to extract the gluon distribution directly. Consider the collision of hadron A and B with momenta and respectively. The invariant mass of the initial state is Assume parton from the hadron A (B) carries longitudinal momentum The Mandelstam variables for the parton sub-process are

where we have neglected the hadron mass. The parton-model cross-section for

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inclusive direct-photon production is then

For the polarized cross-section the parton distributions are replaced by polarized distributions and the parton cross-sections are replaced by the spin-dependent cross-section The tree-level parton scattering cross-section is

where the gration over, say,

reduces the parton momentum integration into one intewith range and

For the polarized case, we have the same expression as in Eq. (5.16) but with

The Spin Structure of the Nucleon

47

In the energy region where the Compton subprocess is dominant, we can write the proton-proton cross-section in terms of the deep-inelastic structure functions and and the gluon distributions G and [70],

Here the factorization scale µ is usually taken as the photon transverse momentum Unfortunately, the above simple picture of direct photon production is complicated by high-order QCD corrections. Starting at next-to-leading order the inclusive direct-photon production cross-section is no longer well defined because of the infrared divergence arising when the photon momentum is collinear with one of the final state partons. To absorb this divergence, an additional term must be added to Eq. (5.15) which represents the production of jets and their subsequent fragmentation into photons. Therefore, the total photon production cross-section depends also on these unknown parton-to-photon fragmentation functions. Moreover, the separation into direct photon and jet-fragmented photon is scheme-dependent as the parton cross-section depends on the methods of infrared subtraction [147]. To minimize the influence of the fragmentation contribution, one can impose an isolation cut on the experimental data [139]. Of course the parton cross-section entering Eq. (5.15) must be calculated in accordance with the cut criteria. An isolation cut has the additional benefit of excluding photons from or decay. When a high-energy decays, occasionally the two photons cannot be resolved in a detector or one of the photons may escape detection. These backgrounds usually reside in the cone of a jet and are largely excluded when an isolation cut is imposed. The NLO parton cross-sections in direct photon production have been calculated for both polarized and unpolarized scattering [147]. Comparison between the experimental data and theory for the latter case is still controversial. While the collider data at large are described very well by the NLO QCD calculation [1], the fixed-target data and collider data at are under-predicted by theory. Phenomenologically, this problem can be solved by introducing a broadening of the parton transverse momentum in the initial state [51]. Theoretical ideas attempting to resolve the discrepancy involve a resummation of threshold corrections [215] as well as a resummation of double logarithms involving the parton transverse momentum [212, 102]. Recently, it

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has been shown that a combination of both effects can reduce the discrepancy considerably [213].

5.6.

from Jet and Hadron Production in

Collisions

Jets are produced copiously in high-energy hadron colliders. The study of jets is now at a mature stage as the comparison between experimental data from Tevatron and other facilities and the NLO QCD calculations are in excellent agreement. Therefore, single and/or di-jet production in polarized colliders can be an excellent tool to measure the polarized parton distributions, particularly the gluon helicity distribution [84]. There are many underlying subprocesses contributing to leading-order jet production: Summing over all pairs of initial partons and subprocess channels and folding in the parton distributions etc., in the initial hadrons A and B, the net two-jet cross-section is:

For jets with large transverse momentum, it is clear that the valence quarks dominate the production. However, for intermediate and small transverse momentum, jet production is overwhelmed by gluon-initiated sub-processes. Studies of the NLO corrections are important in jet production because the QCD structure of the jets starts at this order. For polarized scattering, this has been investigated in a Monte Carlo simulation recently [113]. The main result of the study shows that the scale dependence is greatly reduced. Even though the jet asymmetry is small, because of the large abundance of jets, the statistical error is actually very small. Besides jets, one can also look for leading hadron production, just as in electroproduction considered previously. This is useful particularly when jet construction is difficult due to the limited geometrical coverage of the detectors. One generally expects that the hadron-production asymmetry has the same level of sensitivity to the gluon density as the jet asymmetry.

5.7. Experimental Measurements The first information on has come from NLO fits to inclusive deepinelastic scattering data as discussed in Sect. 3.5.. Also recent semi-inclusive data from the HERMES experiment indicates a positive gluon polarization at a moderate Future measurements from COMPASS at CERN, polarized RHIC, and polarized HERA promise to provide much more accurate data.

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49

5.7.1. Inclusive DIS Scattering As discussed in Sect. 3.5., the spin-dependent structure function is sensitive to the gluon distribution at NLO. However, to extract the gluon distribution, which appears as an additive term, one relies on the different dependence of the quark and gluon contributions. The biggest uncertainty in the procedure of the NLO fits is the parametric form of the gluon distribution at It is known that by taking different parameterizations, one can get quite different results.

5.7.2. HERMES Semi-inclusive Scattering The HERMES experiment has been described in Sect. 2.3.. In a recent publication [32], the HERMES collaboration reported a first measurement of the longitudinal spin asymmetry in the photoproduction of pairs of hadrons with high transverse momentum which translate into a at an average Following the proposal of Ref. [87], the data sample contains hadron pairs with opposite electric charge. The momentum of the hadron is required to be above 4.5 GeV/c with a transverse component above 0.5 GeV/c. The minimum value of the invariant mass of the two hadrons, in the case of two pions, is A nonzero asymmetry is observed if the pairs with 1.5 GeV/c and are selected. The measured asymmetry is shown in Fig. 5.5 with an average of If is not enforced the asymmetry is consistent with zero. The measured asymmetry was interpreted in terms of the following processes: lowest-order deep-inelastic scattering, vector-dominance of the photon, resolved photon, and hard QCD processes – Photon Gluon Fusion and QCD Compton effects. The PYTHIA [263] Monte Carlo generator was used to provide a model for the data. In the region of phase space where a negative asymmetry is observed, the simulated cross-section is dominated by photon gluon fusion. The sensitivity of the measured asymmetry to the polarized gluon distribution is also shown Fig. 5.5. Note that the analysis does not include NLO contributions which could be important. The HERMES collaboration will have more data on this process in the near future.

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The Spin Structure of the Nucleon

51

5.7.3. COMPASS Experiment The COMPASS expriment at CERN will use a high-energy (up to 200 GeV) muon beam to perform deep-inelastic scattering on nucleon targets, detecting final state hadron production [111]. The main goal of the experiment is to measure the cross-section asymmetry for open charm production to extract the gluon polarization For the charm production process, COMPASS estimates a charm production cross-section of 200 to 350 nb. With a luminosity of they predict about 82,000 charm events in this kinematic region per day. Taking into account branching ratios, the geometrical acceptance and target rescattering, etc., 900 of these events can be reconstructed per day. The number of background events is on the order of 3000 per day. Therefore the total statistical error on the spin asymmetry will be about Shown in Fig. 5.6 are the predicted asymmetries and for open charm production as a function of The curves correspond to three different models for From the results at different one hopes to get some information about the variation of as a function of Measurements with high hadrons will also be used to complement the information from charm production.

5.7.4.

from RHIC Spin Experiments

One of the primary goals of the RHIC spin experiments is to determine the polarized gluon distribution. This can be done with direct photon, jet, and heavy quark production.

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Direct photon production is unique at RHIC. This can either be done on inclusive direct photon events (PHENIX) or photon-plus-jet events (STAR). Estimates of the background from annihilation show a small effect. Shown in Fig. 5.7 is the sensitivity of STAR measurements of in the channel The solid line is the input distribution and the data points represent the reconstructed For inclusive direct photon events, simulations show very different spin asymmetries from different spin-dependent gluon densities. Jet and heavy flavour productions are also favorable channels to measure polarized gluons at RHIC. The interested reader can consult the recent review in Ref. [91].

The Spin Structure of the Nucleon

5.7.5.

53

from Polarized HERA

The idea of a polarized HERA collider has been described in Sect. 2.4. Here we highlight a few experiments which can provide a good measurement of the polarized gluon distribution [120], First of all, polarized HERA will provide access to very large and low regions compared with fixed-target experiments. At large can be as large as Thus, one can probe the gluon distribution through the variation of the structure function. An estimate from an NLO pQCD analysis shows that the polarized HERA data on can reduce the uncertainty on the total gluon helicity to ±0.2(exp)±0.3(theory). Polarized HERA can also measure the polarized gluon distribution through di-jet production. Assuming a luminosity and with the event selection criteria the expected error bars on the extracted are shown in Fig. 5.8. The measured region covers can also be measured at polarized HERA through hadrons and jet production with real photons.

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6. TRANSVERSE SPIN PHYSICS 6.1.

The

Structure Function of the Nucleon

As discussed in Sect. 3.2. the structure function can be measured with a longitudinally polarized lepton beam incident on a transversely polarized nucleon target. For many years, theorists have searched for a physical interpretation of in terms of a generalization of Feynman’s parton model [142, 210], as most of the known high-energy processes can be understood in terms of incoherent scattering of massless, on-shell and collinear partons [142]. It turns out, however, that is an example of a higher-twist structure function. Higher-twist processes cannot be understood in terms of the simple parton model [138]. Instead, one has to consider parton correlations initially present in the participating hadrons. Higher-twist processes can be described in terms of coherent parton scattering in the sense that more than one parton from a particular hadron takes part in the scattering. Higher-twist observables are interesting because they represent the quark and gluon correlations in the nucleon which cannot otherwise be studied. Why does contain information about quark and gluon correlations? According to the optical theorem, is the imaginary part of the spin-dependent Compton amplitude for the process

where and N represent the virtual photon and nucleon, respectively, and the labels in the brackets are helicities. Thus Compton scattering involves a helicity exchange. When the process is factorized in terms of parton subprocesses, the intermediate partons must carry this helicity exchange. Because of the vector coupling, massless quarks cannot undergo helicity flip in perturbative processes. Nonetheless, the required helicity exchange is fulfilled in two ways in QCD: first, through single quark scattering in which the quark carries one unit of orbital angular momentum through its transverse momentum; second, through quark scattering with an additional transverselypolarized gluon from the nucleon target. These two mechanisms are combined in such a way to yield a gauge-invariant result. To leading order in can be expressed in terms of a simple parton distribution [172],

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55

where

and is the transverse polarization vector and is the component along the same direction. Although it allows a simple estimate of in the models [180], the above expression is deceptive in its physical content. It has led to incorrect identifications of twist-three operators [26, 258] and incorrect next-to-leading order coefficient functions [204]. When the leading-logarithmic corrections were studied, it was found that mixes with other distributions under scale evolution [261]. In fact, is a special moment of more general parton distributions involving two light-cone variables

where the

are defined as

where Under a scale transformation, the general distributions evolve autonomously while the do not [90, 131]. The first result for the leading logarithmic evolution of the twist-three distributions (and operators) [90] has now been confirmed by many studies [252, 60]. Thus an all-order factorization formula is much more subtle than is indicated by the leading-order result. It involves the generalized two-variable distributions,

where are the coefficient functions with summing over different quark flavours and over gluons. Accordingly, a perturbative calculation of in terms of quark and gluon external states must be interpreted carefully [228]. Recently, the complete one-loop radiative corrections to the singlet and nonsinglet have been published [192]. The result is represented as the orderterm in and is one of the necessary ingredients for a NLO analysis of data. Note that the Burkhardt-Cottingham sum rule,

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survives the radiative corrections provided the order of integrations can be exchanged [96]. As an example of the interesting physics associated with we consider its second moment in

where

is the second moment of the structure function. Here is the matrix element of a twist-three operator,

where and the different brackets – (. . .) and [. . .] – denote symmetrization and antisymmetrization of indices, respectively. The structure of this twist-three operator suggests that it measures a quark and a gluon amplitude in the initial nucleon wave function. To better understand the significance of we consider a polarized nucleon in its rest frame and consider how the gluon field inside of the nucleon responds to the polarization. Intuitively, because of parity conservation, the color magnetic field can be induced along the nucleon polarization and the color electric field in the plane perpendicular to the polarization. Introducing the color-singlet operators and we define the gluon-field polarizabilities and in the rest frame of the nucleon,

Then it is easy to show

Thus measures the response of the color electric and magnetic fields to the polarization of the nucleon. The experimental measurements of the structure function started with the SMC [14] and E142 [45, 46] collaborations. Subsequently, the E143 [5], E154 [7], and E155 [47, 83] collaborations have also measured and published their data. The combined E143 and E155 data for proton and deuteron are shown in Fig. 6.1. The solid line shows the twist-two contribution to only [278]. The dashed and dash-dotted lines are the bag model calculations by Song [267] and Stratmann [269].

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Neglecting the contributions from and and the dependence, the E155 collaboration [47, 83] has integrated their data to get and The results are consistent with the Burkardt-Cottingham sum rules within the relatively large errors. The second moments allow an extraction of the matrix elements. E155 found and at an average of A combined analysis of the E142, E143, E154, and E155 data yields and These numbers are generally consistent with bag model [267, 269, 198, 193] and chiral quark model [279] estimates, and are 1 to away from QCD sum rule calculations [268, 62, 134]. The error bars on the present lattice calculation are still relatively large [159]. According to the simple quark model, the matrix element in the neutron should be much smaller than that in the proton because of SU(6) spin-flavor symmetry. While the proton has been constrained with reasonable precision, the neutron has a much larger error bar. In the near future, JLab experiments with a polarized target [59] can improve the present error on the neutron and hence test the quark model predictions.

6.2. Tranversity Distribution Along with the unpolarized and polarized quark distributions – and – discussed above, a third quark distribution exists at the same order (twist two) as the other two distributions. Note that no corresponding transverse spin distribution exists for gluons (due to helicity conservation). This transversity distribution, can be described in the Quark Parton Model as the difference in the distribution of quarks with spin aligned along the nucleon spin vs. anti-aligned for a nucleon polarized transverse to its momentum. The structure function related to transversity is given by

The first moment of the transversity distributions also leads to an interesting observable – the nucleon’s tensor charge

In terms of nucleon matrix elements, this tensor charge is defined [168] as:

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59

Recent calculations have made estimates of the tensor charges using QCD Sum Rules [169, 199], Lattice QCD [50], and within the Chiral Quark Model [202]. In a non-relativistic model the transversity is equal to the longitudinal spin distribution because the distribution would be invariant under the combination of a rotation and a Lorentz boost. Relativistically, this is not the case and could be significantly different from The challenge to gaining experimental information on lies in its chiral structure. In the helicity basis [181, 182] represents a quark helicity flip, which cannot occur in any hard process for massless quarks within QED or QCD. This chiral-odd property of transversity makes it unobservable in inclusive DIS. In order to observe a second non-perturbative process that is also chiral-odd must take place. This was first discussed by Ralston and Soffer [251] in connection with Drell-Yan production of di-muons in polarized collisions. Here the transversity distribution of both protons results in a chiral-even interaction. Several calculations have suggested that the transversity distribution may be accessible in semi-inclusive lepton-nucleon scattering [108, 183, 206, 236, 41]. In this process a chiral-odd fragmentation function, leading to a leptoproduced hadron, offsets the chiral-odd transversity distribution. Many of these calculations take advantage of an inequality

discovered by Soffer [264] to limit the possible magnitude of Calculations [206, 236, 207] have also detailed a set of spin distribution and fragmentation functions that are accessible from leading and next-to-leading twist processes. In fact in some cases the next-to-leading twist processes can dominate, especially at low and with longitudinally polarized targets. Potentially relevant experimental information has recently come from the HERMES collaboration and SMC collaboration. HERMES has measured the single-spin azimuthal asymmetry for pions produced in deep-inelastic scattering of unpolarized positrons from a longitudinally polarized hydrogen target [33] (see Fig. 6.2). A related measurement has been reported by SMC [88] using a transversely polarized target. The HERMES asymmetry is consistent with a sin distribution, where is the angle between the lepton scattering plane and the plane formed by the virtual photon and pion momenta. While the dependence of the asymmetry is relatively weak except for the smallest point, the dependence shows a rapid rise up to The average asymmetry averaged over the full acceptance is 0.022±0.005±0.003 while the asymmetry for production is consistent with zero. Some models [133] suggest that the product of the transversity distribution times the chiral-odd fragmentation function can account for the observed asymmetry.

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Transversity may also play a role in observed single-spin asymmetries in collisions. These possibilities are discussed in the next Section.

The Spin Structure of the Nucleon

6.3.

61

Single-Spin Asymmetries From Strong Interactions

As we have discussed in Sect. 4.2., single-spin asymmetries can arise from processes involving parity-violating interactions, such as W and Z boson production in collisions at RHIC. In this Subsection, we discuss a different class of single-spin asymmetries which are generated entirely from strong interaction effects. While we do not have enough space here to make a thorough examination of the subject, we briefly discuss the phenomena, a few leading theoretical ideas, and some other related topics. A recent review of the subject can be found in Ref. [222]. For more than two decades, it has been known that in hadron-hadron scattering with one beam transversely polarized, the single-particle inclusive yield at non-zero has an azimuthal dependence in a coordinate system where is chosen to lie along the direction of the polarized beam, and x along the beam polarization [203]. It is easy to see that the angular dependence is allowed by strong interaction dynamics. If the momentum of the polarized beam is and that of the observed particle the angular distribution reflects the existence of a triple correlation, where is the beam polarization. The correlation conserves parity and hence is not forbidden in strong interactions. Although it is nominally time-reversal odd, the minus sign can be canceled, under the time-reversal transformation, by a factor of from an interference of two amplitudes with different phase factors. The angular correlation is usually characterized by the spin asymmetry

where are the cross-sections with reversed polarizations, and is the transverse momentum of the produced particle. is the Feynman variable, where is the longitudinal momentum of the produced hadron and is the maximum allowed longitudinal momentum. An example of a single spin asymmetry for production is shown in Fig. 6.3. After examining the existing data, one finds the following interesting systematic effects [222]: is significant only in the fragmentation region of the polarized beam. It increases almost linearly with when the target is unpolarized. is large only for moderate transverse momentum and its sign show a strong dependence on the type of polarized beam and produced particles That

is strikingly large is the most impressive aspect of the phenomenon.

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63

The simplest theory explaining is one that assumes an underlying parton process: partons from the parent hadrons scatter and fragment to produce the observed particle. To get the single spin asymmetry, one requires, for instance, that quarks change their helicity during hard scattering. However, chiral symmetry then dictates that the asymmetry is proportional to the quark mass which is vanishingly small for light quarks. Thus the simple parton model for cannot yield the magnitude of the observed symmetry [200]. For the moment, the leading theoretical ideas in the literature are still based on the parton degrees of freedom. However, the spin-flip is introduced through more complicated mechanisms: Either the initial and final partons are assumed to have novel nonpertrubative distribution and fragmentation functions, respectively, or the parton hard scattering involves coherent processes. In the latter case, the asymmetry can arise from the coupling of chiral even (odd) twist-two (twist-three) parton correlations in the polarized nucleon and chiral even (odd) twist-three (twist-two) fragmentation functions of the scattered partons [132, 243, 185, 244]. The required phase difference is generated from the interference of the hard scattering amplitudes in which one of the hard propagators is on-shell. The predicted asymmetry is of order in the large limit, which is a characteristic twist-three effect. For moderate can be a slowly decreasing function of [244]. The comparison between the experimental data and the phenomenological prediction seems to yield good agreement [244]. It is not clear, however, that the available fixed target data can be fully described by perturbative parton scattering. One needs more data at higher energy to test the scaling property inherent in a perturbative description. The alternative is to consider nonperturbative mechanisms to generate the phase difference. This can be done by introducing transverse-momentum dependent parton distributions [262] and fragmentation functions [108]. In a transversely polarized nucleon, the transverse momentum distribution may not be rotationally invariant. It may depend on the relative orientation of the spin and momentum vectors. Likewise, when a transversely polarized quark fragments, the amplitude for hadron production can depend on the relative orientation between the hadron momentum and the quark spin. Both mechanisms have been shown to produce large single spin asymmetries [41, 76, 78, 42]. Here again the applicability of the model for the existing data is not clear. In particular, the fitted fragmentation functions and parton distributions must be tested in different kinematic regions. Moreover, the new distributions do not possess colour gauge invariance. A phenomenological model for parton scattering with formation of largehadrons was proposed by Boros, Liang and Meng [81]. Although not derived from field theory, the model has a very intuitive physical picture and suc-

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cessfully describes the data. It would be interesting to test the predictive power of the model in future experiments. The RHIC spin facility can test many of these theoretical ideas with a variety of experimental probes including [254] polarized Drell-Yan, dimeson production, etc. A subject closely related to the single-spin asymmetry is the polarization of hyperons, such as produced in unpolarized hadron collisions [171]. The observed polarization is perpendicular to the plane formed by the beam and hyperon momenta. Many theoretical models have been invented to explain the polarization [140]. Most models are closely related to those devised to explain the single spin asymmetry.

7.

OFF-FORWARD PARTON DISTRIBUTIONS

In this Section, we discuss some of the recent theoretical developments on generalized (off-forward) parton distributions (OFPD) and their relation to the angular momentum distributions in the nucleon. We will also consider possible experimental processes, such as deeply virtual Compton scattering (DVCS) and meson production, to measure these novel distributions. OFPD’s were first introduced in Ref. [123] and discovered independently in Ref. [186] in studying the spin structure of the nucleon. Radyushkin and others have introduced slightly different versions of the distributions, but the physical content is the same [247, 248, 249, 110]. The other names for these functions range from off-diagonal, non-forward and skewed to generalized parton distributions. Here we follow the discussion in Ref. [189]. One of the most important sources of information about the nucleon structure is the form factors of the electroweak currents. It is well known that the vector current yields two form factors

where and and are the Dirac and Pauli form factors, respectively. gives the anomalous magnetic moment of the nucleon, The charge radius of the nucleon is defined by

with factors,

The axial vector current also defines two form

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65

The axial form factor is related to the fraction of the nucleon spin carried by the spin of the quarks, and can be measured from polarized deepinelastic scattering as discussed in previous Sections. The pseudoscalar charge, can be measured in muon capture. A generalization of the electroweak currents can be made through the following sets of twist-two operators,

where all indices are symmetric and traceless as indicated by (...) in the superscripts. These operators form the totally symmetric representation of the Lorentz group. One can also introduce gluon currents through the operators:

For the above operators are not conserved currents from any global symmetry. Consequently, their matrix elements depend on the momentumtransfer scale µ at which they are probed. For the same reason, there is no low-energy probe that couples to these currents. One can then define the generalized charges from the forward matrix elements of these currents

The moments of the Feynman parton distribution charges

are related to these

where is defined in the range For is just the density of quarks which carry the fraction of the parent nucleon momentum. The density of antiquarks is customarily denoted as which in the above notation is for One can also define the form factors and of these currents using constraints from charge conjugation, parity, time-reversal and Lorentz symmetries

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where

and U(P) are Dirac spinors, and Mod is 1 when is even and 0 when is odd. Thus is present only when is even. We suppress the renormalization scale dependence for simplicity. In high energy experiments, it is difficult to isolate the individual form factors. Instead it is useful to consolidate them into generalized distributions — the off-forward parton distributions (OFPD’s). To accomplish this a light-light vector is chosen such that

Then,

where OFPD

and are polynomials in of degree or The coefficients of the polynomials are form factors. The and are then defined as:

Since all form factors are real, the new distributions are also real. Moreover, because of time-reversal and hermiticity, they are even functions of The OFPD’s are more complicated than the Feynman parton distributions because of their dependence on the momentum transfer As such, they contain two more scalar variables besides the x variable. The variable is the usual invariant which is always present in a form factor. The variable is

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67

a natural product of marrying the concepts of the parton distribution and the form factor: The former requires the presence of a prefered momentum along which the partons are predominantly moving, and the latter requires a four-momentum transfer is just a scalar product of these two momenta.

7.1.

Properties of the Off-Forward Parton Distributions

The physical interpretation of parton distributions is transparent only in light-cone coordinates and light-cone gauge. To see this, we sum up all the local twist-two operators into a light-cone bilocal operator and express the parton distributions in terms of the latter,

The light-cone bilocal operator (or light-ray operator) arises frequently in hard scattering processes in which partons propagate along the light-cone. In the light-cone gauge A = 0, the gauge link between the quark fields can be ignored. Using the light-cone coordinate system

we can expand the Dirac field

where and The quark (antiquark) creation and annihilation operators, and obey the usual commutation relation. Substituting the above into Eq. (7.12), we have [189]

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where V is a volume factor. The distribution has different physical interpretations in the three different regions. In the region it is the amplitude for taking a quark of momentum out of the nucleon, changing its momentum to and inserting it back to form a recoiled nucleon. In the region it is the amplitude for taking out a quark and antiquark pair with momentum Finally, in the region we have the same situation as in the first, except the quark is replaced by an antiquark. The first and third regions are similar to those present in ordinary parton distributions, while the middle region is similar to that in a meson amplitude. By recalling the definition of in terms of the QCD energymomentum tensor

it is clear that they can be extracted from the form factors of the quark and gluon parts of the Specializing Eq. (7.8) to

Taking the forward limit of the µ = 0 component and integrating over threespace, one finds that the give the momentum fractions of the nucleon carried by quarks and gluons On the other hand, substituting the above into the nucleon matrix element of Eq. (7.16), one finds [186]

Therefore, the matrix elements of the energy-momentum tensor provide the fractions of the nucleon spin carried by quarks and gluons. There is an analogy for this. If one knows the Dirac and Pauli form factors of the electromagnetic

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69

current, and the magnetic moment of the nucleon, defined as the matrix element of is Since the quark and gluon energy-momentum tensors are just the twisttwo, spin-two, parton helicity-independent operators, we immediately have the following sum rule from the off-forward distributions;

where the dependence, or contamination, drops out. Extrapolating the sum rule to the total quark contribution to the nucleon spin is obtained. When combined with measurements of the quark spin contribution via polarized DIS measurements, the quark orbital contribution to the nucleon spin can be extracted. A similar sum rule exists for gluons. Thus a deep understanding of the spin structure of the nucleon may be achieved by measuring OFPD’s in high energy experiments. A few rigorous results about OFPD’s are known. First of all, in the limit and they reduce to the ordinary parton distributions. For instance,

where and are the unpolarized and polarized quark densities. Similar equations hold for gluon distributions. For practical purposes, in the kinematic region where an off-forward distribution may be approximated by the corresponding forward one. The first condition, is crucial—otherwise there is a significant form-factor suppression which cannot be neglected at any and For a given is restricted to

Therefore, when is small, is automatically limited and there is in fact a large region of where the forward approximation holds. The first moments of the off-forward distributions are constrained by the form factors of the electromagnetic and axial currents. Indeed, by integrating over we have [186]

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where and are the Dirac, Pauli, axial, and pseudo-scalar elastic form factors, respectively. The dependence of the form factors are characterized by hadron mass scales. Therefore, it is reasonable to speculate that similar mass scales control the dependence of the off-forward distributions. The first calculation of the OFPD has been done in the MIT bag model [194]. The parameters are adjusted so that the electromagnetic form factors and the Feynman parton distributions are well reproduced. The shapes of the distributions as a function of are rather similar at different and The dependence of the energy-momentum form factors is controlled by a mass parameter between 0.5 and 1 The same distributions were also studied in the chiral quark-soliton model by Petrov et al. [242], In contrast to the bag model results, the chiral soliton model yields a rather strong dependence. The model also predicts qualitatively different behaviours in the regions in line with the physical interpretation of the distributions. In the case of the distribution, the pion pole contribution is important [149]. The OFPD’s have also been modeled directly without a theory of the structure of the nucleon. In Ref. [274], the distributions are assumed to be a product of the usual parton distributions and some form factors, independent of the variable In Ref. [249, 250], the so-called double distributions are modeled in a similar ansatz from which a strong dependence is generated. Scale evolution of the OFPD’s has received wide attention and is now completely solved up to two loops. In the operator form, the evolution has been studied at the leading logarithmic approximation long before [227]. In terms of the actual distributions, the evolution equations at the leading-log can be found in Refs. [123, 187, 247, 248, 249, 103] in different cases and forms. In a series of interesting papers, Belitsky and Müller have calculated the evolution of the off-forward distributions at two loops [69]. The key observation is that perturbative QCD is approximately conformally invariant. The breaking of the conformal symmetry can be studied through conformal Ward identities, which allows one to obtain the two-loop anomalous dimension by calculating just the one-loop conformal anomaly.

7.2.

Deeply Virtual Exclusive Scattering

Of course the eventual utility of the OFPD’s depends on whether they can actually be measured in any experiment. The simplest, and possibly the most promising, type of experiments is deep-inelastic exclusive production of photons, mesons, and perhaps even lepton pairs. Here we consider two experiments that have been studied extensively in the literature: deeply virtual Compton scattering (DVCS) in which a real photon is produced, and diffractive meson production. There are practical advantages and disadvantages from

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both processes. Real photon production is, in a sense, cleaner but the crosssection is reduced by an additional power of The Bethe-Heitler contribution can be important but can actually be used to extract the DVCS amplitude. Meson production may be easier to detect, however, it has a twist suppression of In addition, the theoretical cross-section depends on the unknown light-cone meson wave function. Deeply virtual Compton scattering was first proposed in Ref. [186, 187] as a practical way to measure the off-forward distributions. Consider virtual photon scattering in which the momenta of the incoming (outgoing) photon and nucleon are and respectively. The Compton amplitude is defined as

where In the Bjorken limit, and their ratio remains finite, the scattering is dominated by the single quark process in which a quark absorbs the virtual photon, immediately radiates a real one, and falls back to form the recoiling nucleon. In the process, the initial and final photon helicities remain the same. The leading-order Compton amplitude is then

where and are the conjugate light-cone vectors defined according to the collinear direction of and and is the metric tensor in transverse space, is related to the Bjorken variable by Much theoretical work has been devoted to DVCS in the last few years. The one-loop corrections to DVCS have been studied by Ji and Osborne [195]. An all-order proof of the DVCS factorization has been given in Ref. [247, 248, 249, 195, 109]. Suggestions have also been made to test the DVCS scattering mechanism [122]. Asymmetries for polarized DVCS have been considered in [187] and reconsidered in [145, 68]. DVCS with double photon helicity flips have been investigated in Ref. [174, 67]. The estimates for cross-sections have been made in Ref. [274, 275]. Development on the experimental front is also promising. Recently, both ZEUS and H1 collaborations have announced the first evidence for a DVCS

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signature [98], and the HERMES collaboration has made a first measurement of the DVCS single-spin asymmetry [40]. More experiments are planned for COMPASS, JLAB and other future facilities [121]. Heavy quarkonium production was first studied by Ryskin as a way to measure the unpolarized gluon distribution at small [256]. In the leading-order diagram, the virtual photon fluctuates into a pair which subsequently scatters off the nucleon target through two-gluon exchange. In the process, the pair transfers a certain amount of its longitudinal momentum and reduces its invariant mass to that of a The cross-section is:

where M is the mass, and is the decay width into the lepton pair. The equation was derived in the kinematic limit and the Fermi motion of the quarks in the meson was neglected. Two other important approximations were used in the derivation. First, the contribution from the real part of the amplitude is neglected, which may be justifiable at small Second, the off-forward distributions are identified with the forward ones. The above result was extended to the case of light vector-meson production by Brodsky et al., who considered the effects of meson structure in perturbative QCD [89]. They found a similar cross-section,

where the dependence on the meson structure is in the parameter

and is the leading-twist light-cone wave function. Evidently, the above formula reduces to Ryskin’s in the heavy-quark limit The amplitude for hard diffractive electroproduction can be calculated in terms of off-forward gluon distributions [247]. With the virtual photon and vector meson both polarized longitudinally (i.e., determined using a Rosenbluth separation, with the vector meson polarization measured via its decay products), one finds

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where again The above formula is valid for any and smaller than typical hadron mass scales, Hoodbhoy has also studied the effects of the off-forward distributions in the case of production [173]. He found that Ryskin’s result needs to be modified in a similar way once the off-forward effects become important. More detailed theoretical studies of meson production have been done in Refs. [224, 160, 149]. Longitudinal production data has been collected by the E665 and the HERMES collaborations [54] and the comparison with model calculations is encouraging [274, 148].

8.

RELATED TOPICS IN SPIN STRUCTURE

In this Section, we review two interesting topics related to the nucleon spin. First, we consider the Drell-Hearn-Gerasimov sum rule and its generalization to finite Then we briefly review polarized production from fragmentation of polarized partons where the polarization can be measured through its weak non-leptonic decay.

8.1. The Drell-Hearn Gerasimov Sum Rule and Its Generalizations The Drell-Hearn-Gerasimov (DHG) sum rule [126] involves the spindependent photo-nucleon production cross-section. Consider a polarized real photon of energy scattering from a longitudinally polarized nucleon and producing arbitrary hadronic final states. The total cross-sections are denoted as where the subscripts 3/2 and 1/2 correspond to the helicity of the photon being parallel or antiparallel to the spin of the nucleon. The sum rule relates the integral of the spin-dependent cross-section from the inelastic threshold to infinity to the anomalous magnetic moment of the nucleon

For the proton and neutron, the sum rule is

and

respectively.

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There has been much interest in recent years in testing the above sum rule by determining the integral on the left-hand side. Direct experimental data on the spin-dependent photoproduction cross-section has become available recently [27] (see Fig. 8.1) and more data at higher energy are coming soon [28]. However, many of the published “tests” in the literature rely on theoretical models for the photoproduction helicity amplitudes which are only partially constrained by unpolarized photoproduction data [52, 166]. Because of the weighting, the low energy amplitudes play a dominant role in the DHG integral [128]. In fact, one can show that in the large limit, the integral is entirely dominated by the resonance contribution [107]. We will not discuss in detail how the phenomenological estimates of the DHG integral are done in the literature [201, 281, 94, 257]. The interested reader can consult a recent review on the subject [127]. The main conclusion from these calculations is that the isoscalar part of the sum rule is approximately satisfied, whereas a large discrepancy remains for the isovector part Typically, the proton integral is estimated to be in the range of to A more up-to-date analysis [129] including the recent data from MAMI and the extrapolation of DIS data gives a result of for the proton, but disagrees with the expected neutron sum by What do we learn about nucleon spin physics by testing the sum rule? Moreover, the DHG sum rule is the analogue of the Bjorken sum rule at 0 [44] (here, we discuss the Bjorken sum rule in the generalized sense that the first moment of is related to nucleon axial charges in the asymptotic limit). If both sum rules are important to study, how do we extend these sum rules away from the kinematic limits Finally, how is the DHG sum rule evolved to the Bjorken sum rule and what can we learn from the evolution? In recent years, there has been much discussion in the literature about the generalized DHG integrals and their dependence [93, 94, 71, 265, 130, 259]. A summary of different definitions of the generalized DHG integrals can be found in Ref. [240, 129]. As pointed out in [197], the key to addressing the above questions is the dispersion relation for the spindependent Compton amplitude The virtual-photon forward scattering tensor defines the spin-dependent amplitude

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75

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B.W. Filippone and Xiangdong Ji

where is the electromagnetic current. From general principles, such as causality and unitarity as well as assumptions about the behaviour of one can write down a dispersion relation

where is the spin-dependent structure function discussed in Sect. 1.3.1. Whenever is known, in theory or experiment, the above relation yields a dispersive sum rule. For instance, the Bjorken and DHG sum rules are obtained from theoretical predictions for respectively [74, 223]. What do we learn by testing these dispersive sum rules? First, we learn about the assumptions required for the derivation of the relation; in particular, the high-energy behaviour of the Compton amplitude [170]. Second, we learn about the scattering mechanisms in the virtual-Compton process. For the Bjorken sum rule, it is perturbative QCD and asymptotic freedom; for the DHG sum rule, it is nucleon-pole dominance and gauge symmetry [223]. Finally, if the sum rules are reliable, we have a new way to measure nucleon observables. In earlier Sections, we discussed how to extract and (the fraction of the nucleon spin carried by quark spin) from polarized DIS data. Assuming the validity of the DHG sum rule, we obtain the magnetic moment of the nucleon from inclusive photoproduction. How do we extend these sum rules to other kinematic regions? According to Eq. 8.3, the virtual Compton amplitude is the key. As discussed in Sect. 1.3., at large but finite perturbative QCD introduces two types of corrections. The first are the radiative corrections: gluons are radiated and absorbed by active quarks, etc. The second are the higher twist corrections in which more than one parton from the target participates in the scattering. With these corrections, we can extend Bjorken’s result for the Compton amplitude from to finite [198, 193]. Since the scale that controls the twist expansion is on the order of the perturbative QCD prediction for is valid down to Combined with Eq. 8.3, it yields a generalized Bjorken sum rule. It is the generalized Bjorken sum rule that is commonly tested experimentally. At small but finite chiral perturbation theory provides a sound theoretical method to calculate corrections to the low-energy theorem [71, 197]. Recently, a fourth-order chiral perturbation theory calculation for the inelastic part of yielded [191]

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The result shows a rapid dependence near which is qualitatively, though not quantitatively, consistent with a recent phenomenological analysis [129]. For a quantitative test, one needs polarized electron scattering data soon available from JLab [95]. How does the DHG sum rule at evolve to the Bjorken sum rule at The physically most interesting quantity which connects both sum rules is

where and are the elastic nucleon form factors. It is the elastic contribution which dominates at low starts at from the proton (neutron) at and rapidly decreases to about 0.2 at and remains essentially flat as The interpretation for the variation is as follows [197]. The forward Compton amplitude is an amplitude for the photon to scatter from a nucleon target and remain in the forward direction. This is very much like a diffraction process and is the “brightness” of the diffraction center. For low photons, scattering from the different parts of the proton is coherent, and the scattered photons produce a large diffraction peak at the center. As becomes larger, the photon sees some large scale fluctuations in the nucleon; the scattering becomes less coherent. The large scale fluctuations can largely be understood in terms of the dissociation of the nucleon into virtual hadrons. When the photons see parton fluctuations at the scale of the photons see individual quarks inside the nucleon and the scattering is completely incoherent. The diffraction peak is just the sum of diffractions from individual quarks. In short, the variation of the sum rules reflects the change of the diffraction intensity of the virtual photon as its mass is varied. A clear theoretical understanding of the virtual photon diffraction at is not yet available, but there are two distinct possibilities. First, there is a gap in which neither parton nor hadron language describes the scattering well. In this case, an interesting theoretical question is how the transition from low to high happens. Second, some extensions of the twist expansion and chiral perturbation theory may overlap in the intermediate region. If so, we have parton-hadron duality at a new level. In any case, a lattice calculation of may shed important light on this [190].

8.2. the

Spin-Dependent

Fragmentation

In the constituent quark model, the spin structure of the baryon is simple: quark pair couples to give zero angular momentum and isospin, and the

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spin of the is entirely carried by the spin of the remaining quark. From our present knowledge of the spin structure of the nucleon, we expect that this naive picture will fail to explain the actual spin structure of the In fact, if SU(3) flavour symmetry is valid, we can deduce from the beta decay data and polarized deep-inelastic scattering on the nucleon that (in leading order) and Unfortunately the spin structure of the cannot be measured because of the lack of a stable target. However, the spin-dependent fragmentation of partons to the baryon can be studied experimentally because the polarization can be measured through the self-analyzing decay The fragmentation functions are difficult to calculate in QCD, even in principle. We have little experience in modeling the fragmentation functions compared with the internal structure of the nucleon. Nevetherless, one hopes that the spin physics in the fragmentation process corroborates what we learn about the spin structure. Moreover, if a or is exclusively produced from the fragmentation of a strange or antistrange quark, respectively, the measurement of the polarization is a way to access the strange quark polarization in the nucleon. A relatively simple process from which the spin-dependent fragmentations to A can be studied is annihilation with one of the beams (say, electron) polarized. Considering only the intermediate photon state, the asymmetry in polarized production is

where is a spin-dependent fragmentation function and are the fragmentations of the quarks with helicities ±1/2 to a of helicity +1/2. At the peak, the parity violating coupling induces polarizations in the quark-antiquark pairs produced. Hence even without beam polarization, the particles produced through fragmentations are polarized [92]. Recently, several collaborations at LEP have extracted the polarization from quark fragmentation at the peak [97]. A number of models for spin-dependent quark fragmentation functions have been proposed to explain the results [208, 116, 80, 229], and data are consistent with very different scenarios about the flavour structure of fragmentation. The polarized fragmentation functions can also be measured in deepinelastic scattering, in which the polarized beam produces a polarized quark from an unpolarized target, which then fragments [178]. Within the QPM, the measured polarization from a lepton beam with polarization is,

The Spin Structure of the Nucleon

where is the depolarization factor. A process-independent can be defined from

79

polarization

The A polarization from DIS scattering was first measured by the E665 Collaboration with a polarized muon beam [18]. The data sample was taken at _ with with and The A polarization was found to be and at The polarization was 0.26±0.6 and 1.1 ±0.8 for the two bins, respectively. The comparisons with different fragmentation models can be found in Refs. [56, 230]. Recently, HERMES has also reported a measurement of the polarization from polarized deep-inelastic positron scattering from an unpolarized proton target. The result is at an average [31]. The result seems to be consistent with the assumption of the naive quark model that the polarization is entirely carried by the valence quark [116]. In Ref. [117], predictions for production from collisions at RHIC and with a single beam polarization was studied. Spin asymmetry measurements as a function of the rapidity provide a way to discriminate various models of the spin-dependent fragmentation. The main theoretical uncertainties, such as the NLO corrections and the unknown polarized parton distributions, have no major impact on the asymmetry. In Ref. [82], it is argued that the hyperfine interaction responsible for the mass splitting induces a sizable fragmentation of polarized up and down quarks into a which leads to large positive polarizations at large rapidity.

CONCLUSIONS Since the EMC publication of the measurement on the fraction of the nucleon spin carried by quarks, understanding the spin structure of the nucleon has become an important subfield in hadron physics. In this review, we have tried to highlight some of the important developments over the last ten years and discuss some of the future prospects in this field.

ACKNOWLEDGEMENTS The authors would like to thank J.W. Martin for a careful reading of the manuscript.

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240. R. Pantförder, hep-ph/9805434. 241. Particle Data Group, Eur. Phys. J. C15 (2000) 1. 242. V. Petrov et al., Phys. Rev. D57 (1998) 4325; M.V. Polyakov and C. Weiss, Phys. Rev. D60 (1999) 114017. 243. J.W. Qiu and G. Sterman, Phys. Rev. Lett.67 (1991) 2264. 244. J.W. Qiu and G. Sterman, Phys. Rev. D59 (1999) 014004. 245. G. Rädel, A. De Roeck and M. Maul, hep-ph/9711373. 246. G. Rädel and A. De Roeck, hep-ph/9909403. 247. A.V. Radyushkin, Phys. Lett. B385 (196) 333. 248. A.V. Radyushkin, Phys. Rev. D56 (1997) 5524. 249. A.V. Radyushkin, Phys. Rev. D59 (1999) 014030. 250. A.V. Radyushkin, Phys. Lett. B449 (1999) 81; I.V. Musatov and A.V. Radyushkin, Phys. Rev. D61 (2000) 074027. 251. J. Ralston and D.E. Soper, Nucl. Phys. B152 (1979) 109. 252. P.G. Ratcliffe, Nucl. Phys. B264 (1986) 493. 253. P.G. Ratcliffe, Phys. Lett. B192 (1987) 180; R.L. Jaffe and A. Manohar, Nucl. Phys. B267 (1990) 509; X. Ji, J. Tang and P. Hoodbhoy, Phys. Rev. Lett.76 (1996) 740; P. Hagler and A. Schafer, Phys. Lett. B430 (1998) 179; S.V. Bashinsky and R.L. Jaffe, Nucl. Phys. B536 (1998) 303; A. Harinderanath and R. Kundu, Phys. Rev. D59 (1999) 116013; X. Chen and F. Wang, hep-ph/9802346; P. Hoodbhoy, X. Ji and W. Lu, Phys. Rev. D60 (1999) 114042. 254. (RHIC) See www.rhic.bnl.gov. 255. J. Respond, proceedings of workshop ‘Deep-Inelastic Scattering and Related Phenomena’, Rome (1996), eds. G. D’Agostini and A. Nigro, World Scientific, Singapore (1997) 439. 256. M.G. Ryskin, Z. Phys. C37 (1993) 89. 257. A.M. Sandorfi, C.S. Whisnant and M. Khandaker, Phys. Rev. D50 (1994) R6681. 258. K. Sasaki, Prog. Theor. Phys.54 (1975) 1816. 259. O. Scholten and A. Yu. Korchin, Eur. Phys. J. A6 (1999) 211. 260. R.W. Shulze and P.U. Sauer, Phys. Rev. C48 (1993) 38. 261. E.V. Shuryak and A.I. Vainshtein, Nucl. Phys. B199 (1982) 451; Nucl. Phys. B201 (1982) 141. 262. D, Sivers, Phys. Rev. D41 (1990) 83; Phys. Rev. D43 (1991) 261. 263. T. Sjostrand, CPC82 (1994) 74. 264. J. Sorter, Phys. Rev. Lett.74 (1995) 1292. 265. J. Soffer and O. Teryaev, Phys. Rev. Lett.70 (1993) 3373. 266. A.A. Sokolov and I.M. Ternov, Sov. Phys. Doklady8 (1964) 1203. 267. X. Song, Phys. Rev. D54 (1996) 1955.

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Chapter 2

LIQUID-GAS PHASE TRANSITION IN NUCLEAR MULTIFRAGMENTATION S. Das Gupta Physics Dept., McGill University 3600 University Street Montreal, QC H3A 2T8, Canada

A.Z. Mekjian Physics Dept., Rutgers University Piscataway, NJ 08854, USA and

M.B. Tsang National Superconducting Cyclotron Laboratory Michigan State University East Lansing, MI 48824, USA

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Introduction

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3.

Experimental Overview

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Event Selection

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Evidence for Nuclear Expansion

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Space-Time Determination

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7.

Temperature Measurements

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Excitation Energy Determination

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9.

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10. A Class of Statistical Models

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11. A Thermodynamic Model

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12. Generalisation to a More Realistic Model

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13. A Brief Review of the SMM Model

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14. The Microcanonical Approach

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15. The Percolation Model

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16. The Lattice Gas Model(LGM)

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17. Phase Transition in LGM

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18. Isospin Dependent LGM Including Coulomb Interaction

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19. Calculations With Isospin Dependent LGM

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20. Fragment Yields from a Model of Nucleation

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21. Isospin Fractionation in Meanfield Theory

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22. Dynamical Models for Fragmentation

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23. Outlook

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References

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INTRODUCTION

Heavy-ion collisions allow one to pump energy into a nuclear system. In central collisions of equal size nuclei one can also create a significant amount of compression using high energy nuclear beams. The possibility of studying nuclei far from normal conditions raises the question: can we study phase transitions in nuclei similar, for instance, to the way, one can study phase transition in water? This is the subject of the present article. Two important phase transitions are being studied using heavy-ion collisions from medium to very high energies. One phase transition occurs at densities that are subnormal and at temperatures of a few MeV Nuclei at normal density and zero temperature behave like Fermi liquids so that this transition is a liquid to gas phase transition. The second phase transition of current interest is expected at a much higher temperature and at a much higher density (several times normal density) and will be the subject of intense experimental investigation at the Relativistic Heavy Ion Collider (RHIC) and at CERN in the coming decade. There one expects to see transition from hadronic matter to a quark-gluon plasma. In very high energy collisions many new particles are created. This is a domain very much beyond the limits of non-relativistic quantum mechanics with conservation of particles. Thus, we will not treat this phenomenon at all. Instead at an energy scale of tens of MeV, we should be able to stretch or compress pieces of nuclear matter and we expect to see Van der Waals type of behaviour. As a Van der Waals gas is considered to be a classic example of a liquid-gas phase transition, we have a situation similar to that in condensed matter physics. Unfortunately, the experimental conditions in the nuclear physics case are quite severe. The collisions which produce different phases of nuclear matter are over in seconds. Thus, we can not keep matter in an “abnormal” state long enough to study the properties. Furthermore, the detectors measure only the products of these collisions where all the final products are in normal states. We have to extrapolate from the end products to what happened during disassembly. This is a difficult task which complicates confirmation of theoretical predictions. This article is written so that it is suitable for nuclear physicists not specialised in the area of heavy-ion collisions. We hope it is also accessible to non-nuclear physicists since the ideas are quite general and well-known from statistical physics. We hope practising heavy-ion collision physicists will also find this a useful reference. The plan of the article is as follows. Section 2 deals with early theoretical discussions which showed that well established models predict that during disassembly after heavy-ion collisions bulk matter will enter liquid-gas co-existence region provided the beam energies are suitably chosen.

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In Sect. 3 an experimental overview is provided. Sections 4 to 9 bring us in contact with some experimental results. Here we show, for example, how estimates of temperature or freeze-out density are extracted from experiments. Sections 10 to 22 are primarily theoretical. We introduce and develop some models; some we simply sketch without providing all the details, as that would make the article extremely long.

2.

LIQUID GAS PHASE TRANSITION IN NUCLEAR MEANFIELD THEORY

Nuclear matter is an idealised system of equal number of neutrons N and protons Z. The system is vary large and the Coulomb interaction between protons is switched off. For light nuclei the Coulomb interaction has a very small effect and N=Z nuclei have the highest binding energy. As nuclei get bigger the Coulomb energy shifts the highest binding energy towards nuclei with N>Z. This brings into play the symmetry energy which is repulsive and is proportional to Stable systems are scarce after mass number A=N+Z>260. Thus no known nuclei approach the limit of nuclear matter. However, extrapolation from known nuclei leads one to deduce that nuclear matter has density and binding energy We will choose an Equation of State (EOS) of this idealised nuclear matter to examine if a liquid-gas phase transition can be expected and at what temperature and density. The following parametrisation, called the Skyrme parametrisation for the interaction potential energy, has been demonstrated [1] to be a good approximation for Hartree-Fock calculations. We take the potential energy density arising from nuclear forces to be

Our unit of length is and unit of energy is MeV. In the above are in MeV, is attractive, repulsive and is a parameter. The constants should be chosen such that in nuclear matter the minimum energy is obtained at with energy E/A=-16 MeV. This fixes two of the three parameters and the third can be obtained by the compressibility coefficient (nuclear physics has its own unique definition of compressibility coefficient at ). The Skyrme parametrisation is simple enough that we will write down all the relevant formulae. From Eq. (2.1) the energy per particle as a function of

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at zero temperature is given by

In Eq. (2.2) the last term on the right hand side is the zero-temperature Fermigas value for kinetic energy. The pressure due to the interaction at zero or any temperature is

The condition that E /A minimise at

The condition that E /A is -16 MeV at

gives

gives

Lastly, compressibility is given by

The EOS for the Skyrme parametrisation with a=-356.8 MeV, b=303.9 MeV and (this gives ) is shown in Fig. 2.1. In the figure isotherms are drawn for various temperatures (10, 12, 14, 15, 15.64 and 17 MeV). The pressure contributed by kinetic energy was calculated in the finite temperature Fermi-gas model. The similarity with Van der Waals EOS is obvious; for a more quantitative comparison we refer to Jaquaman et al. [2]. With the parameters chosen here the critical temperature is 15.64 MeV. The spinodal region can be seen clearly. The coexistence curve which is shown in the figure is obtained using a Maxwell construction [3]. We now describe how in heavy-ion collisions one will sweep across the plane. In heavy-ion collisions one distinguishes between spectators and participants. Imagine two equal ions colliding at zero impact parameter. Some highly excited nucleons are emitted first. The other nucleons are called participants because each nucleon will collide with at least one nucleon in its path if all nucleons are assumed to move in straight line paths. In peripheral collisions where the impact parameters are non-zero, nucleons outside the overlapping zone would not have collided with nucleons from the other nucleus.

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These “non-interacting” nucleons are defined as spectators. For the same beam energy much more energy is pumped into the participating zone. There will even be compression in this zone if the excitation is very high. Spectators are only mildly excited. They are excited for many reasons: highly non-spherical shapes, unfavourable N/Z ratios, migration from participants etc.. In general, spectators should have little compression. Both central and peripheral collisions have been studied to find signals of phase transitions. Imagine then, as a result of heavy-ion collisions, nuclear matter has been excited to a high temperature, with or without compression. Looking at Fig. 2.1 we see that the pressure will be positive and matter will begin to expand [4, 5, 6]. The exit path is hard to guess but the simplest expectation supported by transport models is that it is approximately isentropic in the beginning part of the expansion. However, if the system reaches the spinodal region, the meanfield description is inappropriate and the system is expected to break up into chunks. In Fig. 2.1 we have nonetheless followed the isentropic trajectory. Expansion continues till it reaches a freeze-out volume, a theoretical idealisation. Once the freeze-out volume is reached there is no exchange of matter between different fragments. Since the fragments are still hot, they will get rid of their excitation by evaporation (sequential two body decays [7, 8]) before they reach the detector. The freeze-out density is significantly lower than the normal density. It is probably not as low as one-tenth the normal density because interactions between fragments (except for Coulomb forces) will cease well before that. The freeze-out density is often a parameter in the theory adjusted to get the best fit and is model dependent. The ‘best’ choice seems to be always less than half the normal density. In Fig. 2.1 we have shown this arbitrarily to be that is, one-quarter of normal nuclear density. Normally, the EOS is drawn with isotherms but some additional insight can be gained by looking at isentrops [6]. For this we refer to Fig. 2.2 where we have drawn, for the same Skyrme interaction, and but now for constant entropy instead of constant temperature. Imagine then an excited spectator is formed at normal density indicated by the vertical dashed line. Let us focus on two isentrops, S /A=1.86 and S /A=0.76. In the first case the system starts with and positive pressure. It will begin to expand; the value of “thermal” E /A as it expands along the isentrop, drops. For conservation of energy it must then develop a collective flow. This collective flow will take it beyond the minimum of E/A and drives it to the spinodal region. For the isentrop with value 0.76, even though it starts with positive pressure, it does not gain enough collective energy to drive it to the spinodal region. It will therefore oscillate around zero pressure and has to deexcite by other means (two-body decay [7]). The intermediate case with S /A= 1.39 just makes it to the spinodal region.

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It is also clear both from Fig. 2.1 and Fig. 2.2 that if the starting point is too high (i.e., too much excitation energy) the system will entirely miss the liquid region and probe only the gas region at the time of dissociation. This was the case at Bevalac [9] (incident energy in GeV) where the goal of studying heavyion reactions was quite different. While meanfield theory, as described above, easily leads to a liquid-gas phase transition picture, one clearly needs to go much beyond. There is hardly any observable that one can calculate using meanfield theory alone. Most common experimental observables are the clusters, their compositions, their excitations, velocities etc. Meanfield theory does not give these values although it suggests that the system must break up because of spinodal instability. However, with Maxwell construction obtained from the meanfield EOS one can draw a coexistence curve (Fig. 2.1) and this has experimental relevance. As we will describe in much greater detail later, one may measure the caloric curve [10] defined as T vs. E* / A in heavy ion experiments. The experiment gives a measure of the specific heat, in the vicinity of temperature 5 MeV. Indeed many models (to be described in later Sections) produce a peak in the specific heat at about this temperature. The peak is reminiscent of the crossing of the coexistence curve [11]. These models are also able to calculate many other observables with reasonable success. Let us refer to Fig. 2.1 to see where we would expect to see the peak in meanfield theory. If we consider the freeze-out density to be the intersection of the line and the coexistence curve suggests a temperature of about 15 MeV. This “boiling” temperature will come down if a lower freeze-out density is used but even at one tenth the normal density the boiling temperature is still 12 MeV. Since we have used the nuclear matter theory, the temperature is expected to be lower due to finite particle number and Coulomb interaction. In reference [2] and later in [12], the effect of finite particle number was estimated to be significant. In addition we must remember that meanfield theories normally overestimate the critical temperature. For example, in the Ising model, this overestimation is about 50 per cent [13]. In meanfield Thomas-Fermi theory that includes the Coulomb interaction De et al. [14] find the peak in specific heat at 10 MeV for Without the Coulomb interaction, in bulk matter with the same isospin asymmetry as the peak is located at 13 MeV. As will be described in greater detail later, both experimental data and more realistic models point to much lower temperature. Thus in meanfield theory interesting things seem to happen at too high a temperature.

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EXPERIMENTAL OVERVIEW

Experimentally the following features are well known. At excitation energy successive emissions of particles by evaporation of the compound nucleus or its fission are the basic deexcitation mechanisms. The picture can be justified by saying that there is enough time between successive emissions so that the nucleus can relax to a new equilibrium state where R is the radius of the compound nucleus and is the velocity of sound. At the time interval between successive emissions is comparable with At excitation energy comparable to binding energy the very existence of a long-lived compound nucleus is unlikely which leads to the scenario of an explosion-like process involving the whole nucleus. This will lead to multiple emission of nuclear fragments of different masses. This is what is called “multifragmentation” where ‘multi’ is more than two. Associated with multifragmentation is a term Intermediate Mass Fragment (IMF) that we will use often. This refers to particles with charge Z between 3 and 20 to 30. The lower charge limit is set to 3 because of exceptional binding of the alpha particle. The upper limit is set not to include fission like fragments; if the nucleus broke up into several chunks in the spinodal region, we could expect some of them to be IMF’s. In the meanfield scenario described earlier multifragmentation is associated with the co-existence region. Thus, it is considered to be the most promising experimental observable to study the liquid-gas phase transition in nuclear matter. However, while phase transition signals will always be weakened by finite particle number effects, multifragmentation is usually found at the appropriate energy and occurs in nuclear collisions even when the thermodynamic limit is not reached. Thus multifragmentation is a more general process than phase transition. In the co-existence region, light particles such as neutrons, hydrogen isotopes (p, d, t) and helium isotopes are considered as gas while the IMF’s are treated as droplet forms of the liquid. In collisions where larger residues remain, they are the liquid remnants from the original colliding nuclei. Since nuclei are two-component systems consisting of neutrons and protons, the isotopic contents of the gas and the liquid phase will be different. This is specially so when bound nuclei of smaller sizes are usually found along the valley of stability and have nearly equal number of protons and neutrons. Thus if the initial collisions consist of heavy nuclei which have more neutrons than protons, one would expect the excess neutrons to diffuse out to the gas region resulting in a neutron enriched nucleon gas. This has already been seen in experiments and will be discussed later. Thus preliminary glimpse of the phase transition in nuclei suggests a much richer structure than what has been implied by nuclear matter alone.

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Multifragmentation was seen in high-energy proton-nucleus collisions [15, 16, 17, 18] before systematic studies were undertaken in nucleus-nucleus collisions. For a proton incident on a nucleus the picture is as follows. Shortly after the collision between the proton and the target nucleus, several prompt nucleons leave the system and carry off much of the energy of the collision. At low incident proton energies only remnants near the mass of the target are produced. For incident proton energies around 0.5 GeV, the system undergoes fission leaving two large fragments. When the incident proton energies are between 1.0 GeV and 10 GeV, the cross-section for multifragmentation rises by an order of magnitude. At energies above 20 GeV the cross-section becomes independent of energy, reaching the limiting fragmentation region. Systematic studies of multifragmentation have been undertaken using heavyion beams since the mid eighties, when these beams became routinely available and large detection arrays were built. The production of fragments from central collisions reaches a maximum around 100A MeV. In the following Section we will examine various aspects of the multifragmentation process, which may be employed to signal the liquid-gas phase transition.

4.

EVENT SELECTION

Most early multifragmentation experiments are inclusive measurements [15, 16, 17, 18, 19, 20], i.e., particles are identified with no requirements that other particles from the same event should be detected in coincidence. These types of experiments do not provide information about the collision dynamics or properties of the emission sources from the nuclear reaction. Since multifragmentation of spectators, produced in peripheral collisions has different characteristics from fragments emitted from the participant zone formed in central collisions, it is important that the emission sources be identified. There are both advantages and disadvantages of using central or peripheral collisions to find the signals of phase transition. In this Section, the methods used to select central and peripheral collisions will be discussed.

4.1. Central Collisions In central collisions, the excitation energy pumped into the participant zone is higher and the source characteristics, i.e. selection of a single source, can be accomplished easily with large detector arrays which provide nearly angular coverage. Intuitively, one expects more particles to be produced in violent or central collisions than peripheral collisions. Thus the number of emitted particles can be related to the collision geometry and the simplest ob-

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servable is the number of charged particles detected, There are variations of the observable such as the hydrogen multiplicity, or light charge particle multiplicity, All these observables work reasonably well in distinguishing central collisions with small impact parameters from peripheral collisions [21]. However, the neutron multiplicity, [22] and the IMF multiplicity, [23] do not work as well. Aside from multiplicities, there are other observables such as the mid-rapidity charge, [24] and the total transverse kinetic energy, of the identified particles [25], which can be used as an impact parameter filter. is the summed charge of particles with rapidity between that of the target and projectile. This quantity reflects properties of the participant zone. The total transverse energy is defined as where and denote the kinetic energy, momentum and emission angle of particle with respect to the beam axis. The most common way to relate an experimental observable to the impact parameter is to assume a monotonic relationship between the observable and the impact parameter [26, 21]. In general, a reduced impact-parameter scale, which ranges between 0 (head-on collisions) to 1 (glancing collisions), is defined as

where is the normalized probability distribution for the measured quantity X, and is the maximum impact parameter for which particles were detected in the near detection array. For illustration of the impact parameter determination, the top panel of Fig. 4.1 shows the charged particle multiplicity distribution of the induced reaction on at 35 MeV per nucleon incident energy [27]. The bottom panel shows the relationship between the reduced impact parameter, and with a lower cut of applied in the analysis. While is the most simple observable to measure the impact parameters, it is not very precise due to fluctuations and geometry efficiencies of the detection device. In cases where the single source from the central collision needs to be better defined or determined, additional constraints are applied. In head-on collisions, the angular momentum transfer is zero and all the emitted particles are emitted isotropically in the azimuthal angle [28]. Thus additional constraints on central collisions can be placed by requiring the detected particles to have isotropic emission pattern. Other constraints include requiring the total charge detected to be a substantial fraction of that of the initial system [29], the ratio of total transverse momentum to longitudinal kinetic energy

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or by requiring the velocity of the emitted particles to be about half of the center of mass velocity [24, 30]. Obviously, each additional constraint reduces the number of events available for analysis. Too many constraints may reduce the data to the extreme tails of the distributions where large fluctuations of the observable become a problem.

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During the compression stage, pressure in the central region causes the participant zone to expand [31]. Thus part of the available energy in central collision is converted to collective energy such as radial flow which expands outward, or transverse flow caused by the spectators being pushed by the participant region to the side, and the squeeze-out of nucleons from the participant region perpendicular to the reaction plane due to blocking by the spectators [32]. Clearly, all these collective motions strongly affect the signals of the phase-transition observed in central collisions and reduce the amount of excitation energy available for heating up the system. They must be understood and taken into account in the study of phase transition.

4.2. Peripheral Collisions Theoretically, spectators should be less affected by the effects of collective motion than the participants. The collision kinematics focus the emitted fragments from the projectile to the forward direction in the laboratory. These fragments are generally detected with spectrometers or detectors placed at forward angles and the charges, velocities etc. are identified. The decay of a projectile spectator is easier to study experimentally than the target spectator which is emitted backward with very low energy in the laboratory frame. Unlike central collisions, the impact parameter is strongly correlated with the size of the source in peripheral collisions. Thus the size of the projectilelike residue such as the charge, provides some indication of the impact parameter [33]. In the case where most of the projectiles fragment into many small pieces, the quantity defined as the sum of atomic numbers of all fragments with has been found to be a good measure for impact parameter [34]. It represents the charge of the spectator system reduced by the number of hydrogen isotopes emitted during its decay and thus, it is the complement of the hydrogen multiplicity, In an experiment where both was measured by the forward spectrometer and was measured by a array in the reaction of Au+Au at E /A=400 MeV, the two observables are anti-correlated [35]. Thus, like and other observables discussed in the previous Section, can be used in Eq. (4.1) to provide a quantitative measure of the impact parameter.

5.

EVIDENCE FOR NUCLEAR EXPANSION

Around incident energy of 50A MeV, fragment multiplicities increase with the size of the emission source and excitation energy. In examining reactions of Xe on various targets, and even though

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the targets span a range of N/Z from 1.0 to 1.5, a near-universal correlation has been observed between the average number of emitted IMFs, and the charge-particle multiplicity, for non- central collisions [36]. Figure 5.1 shows the mean number of IMF detected in the collision of at 50A MeV as a function of the detected charge particle multiplicity, [37]. In the most central collision, the mean number of is 7 but up to 14 IMF fragments have been observed. The large fragment multiplicities cannot be reproduced by the break-up of the hot system at normal nuclear matter density with either the dynamical or statistical models. (Predictions of various statistical models are lower than the data as shown by the dashed lines, open circles and crosses.) Calculations requiring expansion to less than 1/3 of the normal nuclear matter density is needed to explain the large increase in as shown by the solid lines.

If the hot nuclear system expands, the “radial” component of the velocity should be evidenced in the particle energy spectra. Without the influence of radial expansion, the energy spectra resulting from the collision of a target and

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projectile at intermediate energy are composed of three isotropically emitting thermal sources corresponding to the projectile-like and target-like spectators in addition to the participant region formed by the overlap of the projectile and target. Instead, the IMF and light particle energy spectra from the central collisions of Au+Au reaction show a shoulder like shape [38, 39]. To fit the energy spectra, large radial expansion velocities are required in addition to the three sources [38, 39]. Similarly, the mean kinetic and transverse energy of emitted fragments also provide measure of the radial collective velocities when compared to the predictions of thermal models [40, 41, 42]. Figure 5.2 shows a nearly linear relationship between the radial velocities with the incident energy [40]. The plot suggests that 30% to 60% of the available energy is used in the radial expansion. This energy is thus not available for thermal heating of the nuclear matter. Evidence for the “nuclear expansion” of the hot nuclear systems is a necessary but not sufficient condition for the occurrence of a liquid-gas phase transition.

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SPACE-TIME DETERMINATION

The average radial velocity plotted in Fig. 5.2 indicates that the nuclear expansion occurs in a rather short time As a result of the fast expansion, the density of the reaction zone is below normal nuclear matter density. Information about the space-time evolution of the reaction zone can be obtained via intensity interferometry. The principle behind such experiments is similar to the intensity interferometry [43] employed to determine the radius of stars, where both singles and coincident yields of photons from the same source (star) are measured. Intuitively, one expects the correlation to be small if the source size is large and a large correlation from a small source. In nuclear physics, particles are detected instead of photons. A correlation function constructed from these yields is defined as, where is the laboratory momenta of particle At large relative momenta where the final interaction is negligible, should be zero. Unlike astronomy where the space-time evolution of stars is slow, the time scale involved in nuclear physics is very short. Thus there are ambiguities in determining the size and time-scale of nuclear reactions using intensity interferometry, because a small source emitted over a long period of time behaves like a large source emitted over a short period of time [44]. The space-time information of the emitting source can be obtained by measuring the correlation function. An example of the fragment-fragment correlation from the Ar+Au reaction at E/A=50 MeV is shown in Fig. 6.1 [45] as a function of the reduced velocity, where is the relative velocity between fragment 1 and 2. The use of allows summing over different combinations of fragment-fragment correlations. Basic features of the correlation functions for different particle pairs depend on details of the final state interaction between the two particles. For intermediate massfragments, the most important interaction is the Coulomb interactions between the particles as shown by the suppression of the correlation functions at small However, if the fragments are emitted in the vicinity of a heavy reaction residue, the Coulomb interactions with the residue may not be neglected [46]. This space-time ambiguity is illustrated by the calculations shown as lines in Fig. 6.1 [45]. The calculations are Monte Carlo simulations of many body Coulomb trajectory calculations of fragments emitted from a spherical source of radius and lifetime The data are equally well described by calculations using four different combinations of and as shown in the figure. Even with this ambiguity, the “valley” exhibited at low provides some measure of the spacetime extent of the source. As the energy increases, the width of this “valley”

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increases suggesting emission from a smaller and may be a faster source. In order to get more definite results about the emission time, information about the source size must be obtained independently. Such information is most commonly extracted by comparing predictions with data and thus is model dependent. For example, assuming that the source sizes can be obtained from the linear momentum transferred to the system, one can obtain the emission time from fragment-fragment correlation functions. The left panel of Fig. 6.2 shows the dependence of mean emission time as a function of incident energy for the system Kr+Nb [47]. Above 55 MeV per nucleon, multifragmentation seems to occur in a time scale that saturates at The result is consistent with breakup of a fragmenting source at low density including those driven by Coulomb instabilities as in the Au+Au reaction at E/A=35 MeV [42]. In the latter experiment, the source size was obtained by comparing various experimental observables to the prediction of the statistical multifragmentation model. Recent analysis of the IMF correlation functions from high energy hadron induced multifragmentation suggests the saturation time occurs at much shorter time scale (<100 fm/c) as shown in the right panel of Fig. 6.2 [48]. Consid-

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ering the space-time ambiguity and model dependence in extracting the time information, the correlation analysis is probably not reliable in extracting a time scale less than 50 fm/c.

Without precise time information, the quantitative measurements of freezeout densities have been difficult to obtain since the density is quite sensitive to the emission time and volume of the source. Assuming zero lifetime, the density or source sizes can be obtained from light charged particle correlation measurements [49, 50, 51]. The left panel of Fig. 6.3 shows the radii extracted for different reactions using the p-p correlation as a function of the proton velocity normalized by the beam velocity. The middle data set with lots of data points are experimental results from the and induced reaction on Au. The solid diamonds and solid circles are radii extracted from the induced reaction on Au and induced reaction on Ag, respectively. The dot-dashed and the dash lines are scaled from the solid lines by the radii of the projectile. At high velocity where the protons originate from the projectile, the scaled predictions agree with the data very well, suggesting that the method of using p-p correlations to extract source size information is consistent within the same

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method.

The right panel of Fig. 6.3 shows the radii deduced from the p-p [solid squares], [open stars], [open squares] correlations as a function of for the projectile decay of Au+Au collisions at E/A=1 GeV. The radii obtained from the p-p correlation are smaller than the radii obtained from correlations which are in turn smaller than the radii obtained from correlations. There is no logical explanation for such an observation. The inconsistencies of the measurements, regarding the different radii values obtained from different particle correlations, illustrate the present experimental difficulties in extracting the precise values of the freeze-out densities since the analysis is highly model dependent. Other methods, such as comparing IMF multiplicities or mean kinetic energies with statistical models have been employed to determine the source sizes [42]. Independent of different analysis methods, densities

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lower than 1/3 of the normal nuclear matter density provide the best agreement with the data. Consistent with such a conjecture, models that assume normal nuclear matter density underpredict the fragment multiplicities [37, 52, 53, 54]. Considering the complexities of the issue, the available experimental data suggest a time scale for the multifragmentation process of the order of 100 fm/c and density values of less than a third of the normal nuclear matter density. Obviously, it is desirable to have more exact values. Further experimental work and better understanding of the reaction mechanisms in the coming years will allow more precise measurements in this area.

7. TEMPERATURE MEASUREMENTS 7.1.

Kinetic Temperatures

How valid is the concept of temperature in heavy ion reactions? Long before intermediate energy collisions which are best described by multifragmenation mechanisms, the concept of temperature was being used routinely to describe heavy ion collisions at Bevalac (see for example [16]). In cascade [55] or transport calculations [56] one can follow in microscopic models how the original ordered motion of the beam gets dispersed into Maxwell-Boltzmann distribution through two-body collisions. In the Purdue experiment of proton on Xe [18], the high energy tails of the kinetic energy spectra provide evidence that the fragments originate from a common remnant system somewhat lighter than the target which disassembles simultaneously into a multibody final system. Theoretically, the slopes of the particle kinetic energy spectra assuming a Maxwellian distribution, should be sensitive to the initial temperature.

where Here is a normalization constant; E and are the energy and mass of the emitted particle; is the detection angle relative to the incident beam; is the slope or kinetic temperature; and corrects for the Coulomb repulsion from the target residue. As discussed earlier, collective motions complicate the energy spectra. Furthermore, fluctuations in Coulomb barriers, sequential feedings from higher-lying states [57], Fermi motion [58] and pre-equilibrium emissions all contribute to the complications associated with extracting emission temperatures from the energy spectra.

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7.2. Excited State Temperature To circumvent some of these problems, other thermometers, which are less sensitive to the collective motion, have been sought. Thermometers based upon the relative populations of excited states of emitted light particles have been used quite extensively in extracting the temperatures of the hot nuclear systems.

Here is the excitation energy, is the measured yield and is the spin of the state To minimize the influence of sequential decays, nuclei with levels that are widely separated are often chosen [59]. One of the most commonly used nuclei is the unstable isotope. The ground state decays to p and particles while the excited state decays into d and particles. Statistical models incorporating the effect of sequential decays suggest that temperatures up to 6 MeV should be obtainable with this nucleus based on the excited states population [60]. Other nuclei include particles, and which all have relatively widely separated states. Even though the ground state and first excited state of the alpha particle are separated by 20.1 MeV, a substantial part of the measured ground state alpha yields can be attributed to sequential decays from heavy nuclei due to the unusual binding energy of the alpha particles. The consistency of the method is normally checked by measuring the temperatures of several nuclei.

7.3. Isotope Temperature Another thermometer, which utilizes the yield ratios of two pairs of isotopes have been under intense study in the past few years [61, 62]. If chemical and thermal equilibrium are achieved, in the limit of the Grand Canonical Ensemble, one can obtain the isotope temperature information from a double isotope ratio defined by

where are the yields of one isotope pair and is another isotope pair. To cancel the nucleon chemical potential terms, the mass number differences of isotope pair (1,2) must be the same as the mass number differences of isotope pair (3,4); B is the binding energy difference, a contains the statistical weighting factor. This equation assumes that the sequential decay corrections to the yields are negligible.

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This assumption is rather problematic as the experimental measured yields are “cold” fragments containing contributions from the decays of many excited nuclei. In experiments where a large number of isotope yields are measured such as the proton induced reaction on Xe [18], thousands of values can be extracted [63] using Eq. (7.3). If Eq. (7.3) is correct, all the values of thus obtained should be the same. However, the experimental values fluctuate over a large range of values, including negative numbers. These fluctuations arise from sequential decays and can be minimized by selecting double ratios with large binding energy differences (B>10 MeV). However, such requirements select mainly isotope pairs that involve proton rich isotopes such as or which are not well bound. Thus instead of having many independent thermometers, there are in general two classes of isotopes thermometers, those involving the pair and those using pair. The fact that and to a lesser extent have been found to exhibit anomalous energy spectra may invalidate the simple relationship of Eq. (7.3). The difference in shape of the energy spectra between and means that the isotope temperature depends on experimental energy thresholds. It has been shown that the temperature depends strongly on the energy gates used [64, 65, 66, 67]. This dependence has been exploited to examine the evaporative cooling of the Xe on Cu collisions at E/A=30 MeV [65]. Since energy thresholds are often employed to minimize the contributions of the pre-equilibrium emissions [66, 67], this directly affects the temperature values measured.

7.4. Effect of Sequential Decays In recent years, many models have been developed to describe the emission of particles from the multifragmentation process successfully. However, to compare with experimental data, these models must take into account the effects of sequential decays. Inclusions of nuclear spectral information into the calculations to simulate the effects of secondary decay has not been fully successful because the task is not only computationally difficult, but it is hampered by the lack of complete information about the resonances in many nuclei. Figure 7.1 shows the effect of sequential feedings on the temperature determination using two assumptions about the unstable states [68]. The horizontal axis is the emission temperature used in the statistical calculations [60] while the vertical axis is the apparent temperatures obtained using different classes of thermometers. In the left panel, sequential decay calculations including only known bound states and resonances are shown for (He-Li) and denoted by the solid and dashed lines respectively. The apparent isotope temperatures increase monotonically with the input temperature. With inclusion of

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continuum states (right panel), both temperatures flatten out at an asymptotic value of about 6 MeV. Thus, inclusion of sequential decay contributions from the continuum enhances decays to low-lying states and renders temperatures involving alpha particles insensitive to the emission temperature at high excitation energy. For comparison, the calculated excited state temperature of (dot-dashed line) is plotted in the right panel: continues to increase monotonically beyond 6 MeV emission temperature. The rate of increase becomes much less only after 9 MeV temperature.

Of course, the dependence of the apparent temperature on the emission temperature is model dependent. When the sequential decay effect is large such as at high temperature, reaction models with accurate description of the sequential decay processes are needed to relate the measured temperature to the

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emission temperature. Efforts have been made to include structural information to describe secondary decays. However, even with the best code available, the disagreement between measured and predicted isotope yields could be a factor of 10 or more especially for the neutron rich or proton-rich isotopes [69].

7.5. Cross-Comparisons Between Thermometers Cross-comparisons between different thermometers exist. In the case of kinetic temperature measurement, the slope of the energy spectra measured at backward angles and at low incident energy give reliable temperature information for systems with low excitation energy. Under these conditions, the collective flow effect and pre-equilibrium contributions are minimized. At higher energy, the kinetic temperatures are not reliable but one can cross-compare the isotope ratio and excited state temperatures [64, 70, 71, 72, 73]. Careful measurements of these temperatures in various systems suggest that below E/A=35 MeV, there are good agreements between and [70]. However, the disagreement increases with incident energy [71, 73] as shown in Fig. 7.2. For the system of Kr+Nb, temperatures obtained from excited state populations and isotope yields have been measured as a function of the incident energy [73]. The open symbols represent the temperatures extracted from the excited state populations of and respectively. Within experimental uncertainties, they are the same. The consistencies of the experimental results from different nuclei render credence to this thermometer. The closed symbols represent temperatures extracted from isotope yield ratios; (closed squares) rely on the double ratio while (closed circles) use the double ratio Values for vary little with incident energy, similar to the trends exhibited by the excited states temperatures of and In contrast, values of (closed squares) increase monotonically with incident or excitation energy. Similar discrepancies between and have been observed in Au + Au central collision from E/A=50 to 200 MeV [71], Ar+Sc reaction from E/A=50 to 150 MeV [75] and in Ar+Ni system [72] at E/A=95 MeV. Independent of models describing sequential decays, thermometers using alpha particles should be affected by sequential decays in the same manner and should give the same experimental temperature. However, current data [60, 71, 72] show that there are substantial differences between these two thermometers when the excitation energy or incident energy increases. This may indicate that different reaction mechanisms may be involved in the production of primary and particles. In such case, isotope yield temperatures constructed from Eq. (7.3) are problematic.

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7.6. Summary of Temperature Measurements Figure 7.3 shows an overall picture provided by the present data using different thermometers. The kinetic temperatures extracted from fitting the charged particle energy spectra with an intermediate rapidity source exhibit a smooth trend over a wide range of incident energies from a few MeV to nearly 1 GeV per nucleons [74]. The open diamond points shown in Fig. 7.3 are the proton kinetic temperatures extracted from [74] over a narrow range of incident energies for comparison purposes. The dashed lines are drawn to guide the eye. The temperature values depend slightly on the particle types. However, the other light charged particles, d and t, exhibit similar trends, namely, the kinetic energy temperature increases rapidly with the incident energy. A collection of the over a range of incident energies from 30 to 200A MeV are plotted as solid points in Fig. 7.3 [59, 71, 73]. Contrary to the kinetic temperature, there is only a slight increase from 3 to 5.5 MeV, in the excited state temperature as a function of the incident energy spanning over one order of magnitude. The open circles in Fig. 7.3 represent the most commonly used isotope ratio temperature, [71, 73, 75] as a function of incident energy from 35 to 200A MeV. The increase from 4 to 10 MeV as a function of incident energy is much less than but the increase is larger than the nearly constant value of are plotted as open diamonds in Fig. 7.3. This latter isotope thermometer does not agree with Instead, remain relatively constant over the incident energy studied. They behave more like the excited state temperature. Experimentally, temperatures extracted from excited states or yield ratios involving carbon isotopes are around 4-5 MeV [77, 59, 73, 69, 63]. Around 4 MeV emission temperatures, the secondary decay effects are small and can be corrected with current models incorporating sequential decays. The near constant temperature may signal enhanced specific heat. However, if the low temperature of 4 MeV is caused by the limiting temperature due to sequential decays, it becomes difficult to deduce the freeze-out temperature from the measured quantities.

8.

EXCITATION ENERGY DETERMINATION

In nuclear physics experiments, the collision conditions are reconstructed from the particles detected. Even if all the emitted particles can be measured experimentally, it is still difficult to disentangle the contributions from various emitting sources arising from the spectator and participant zones. Before multifragmentation occurs, the hot systems first deexcite by emitting neu-

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trons and light charged particles (including very light IMF’s). These particles are normally emitted in the forward directions and in a very short time scale (<30 fm/c) before equilibrium is established. It is important that these preequilibrium particles are not included in the determination of excitation energy characterizing the fragmenting source. Experimentally the contamination of the observables used in characterizing the emitting source is very difficult to assess without the use of model assumptions. Cascade and transport calculations can be used to estimate the number of particles emitted and the energy lost in the “prompt” or early stage of the reaction. Such calculations may suggest some optimum ways of estimating the number such as imposing energy thresholds on the data to minimize the “preequilibrium” contributions [48, 78]. However, the prompt contributions cannot be completely eliminated from the data using the energy threshold gates. This increases the uncertainties and fluctuations in the excitation energy determined. The model calculations may also indicate the size, mass, N/Z ratio and energy of the residues which undergo “multifragmentation”. Assume we can detect

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and identify all the particles emitted from this excited source, conservation of energy suggests that

Where is the excitation energy, is the kinetic energy of the charged particles, is the energy of the neutrons, is the energy of the gamma rays emitted during the deexcitation and Q is the mass difference between the parent nucleus and all the emitted particles. Gamma energies are relatively small and contribute little to the total excitation energy compared to the other terms. In reality, most experimental apparatus does not have a complete coverage. Furthermore, thresholds in energy and geometry exist in the detection arrays. Thus there are uncertainties in determining the terms and Q. More importantly, neutrons are often not measured. Neutrons do not interact with matter as much as the charged particles so they are more difficult to detect. Very often, is estimated using the average number of neutrons emitted and the mean neutron energy, Conservation of particles imposes some constraints on the value of neutron multiplicity As the neutron data is difficult to obtain, the mean neutron energy values are usually adopted from other experiments or assumed to be the same as the proton mean energy. Therefore in general, poses the largest uncertainty to Eq. (8.1) and determination of excitation energy of the fragmenting system becomes quite a difficult task. Recently, intense effort has been placed in extracting the excitation energy of heavy nuclei induced by high-energy hadron beams such as protons, pions and anti-protons at high energy [79]. In these reactions, the collective excitation and existence of multiple sources are minimal. Even with such “simplified” systems, it is difficult to extract precise excitation energy without the use of model assumptions. For heavy ion reactions, the task is much more daunting. In addition, all collective motions strongly affect the signals of the phase-transition and must be understood before exact values of the excitation energy can be assigned. With so many uncertainties associated with extracting the excitation energy, any experimental observables that utilize the fluctuations of excitation energy such as measuring the heat capacities are subjected to the same problems [80]. The results must be viewed cautiously. While more work is needed to determine the excitation energy accurately, one consensus is that increasing incident energy corresponds to increasing excitation energy. It is quite common to extract the excitation energy using projectile fragmentation. Assuming that the projectile is only modestly excited, only one source, the projectile-like residue, exists. Then the excitation energy of the projectile-like residue should be inversely proportional to impact parameter.

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Thus it has been argued that rather directly, reflects the energy transfer to the excited spectator system [10]. Larger energy transfers, then, correspond to smaller value of and vice versa. However, geometrical arguments suggest that the source size also varies with impact parameter [81]. Furthermore, the energy spectra of light charged particles are inconsistent with one source but require multiple sources to fit the spectra [73]. It is thus incorrect to assume that there are no contributions from pre-equilibrium emissions. As discussed previously, collective flow also plays a role in projectile fragmentation. Besides, all the uncertainties associated with Eq. (8.1) as described above apply to these reactions. Currently determination of the excitation energy presents the biggest challenge to the experimenters. With care and the increasing availability of large neutron detection devices, the excitation energy measurements will be improved in the coming years. A firm grip of this parameter is very important in our understanding of the liquid-gas phase-transition.

9.

SIGNALS FOR LIQUID-GAS PHASE TRANSITION

Over the years, many experimental observables involving IMF have been used to study the nuclear liquid-gas phase transition. Discussions of all the proposed experimental signatures will require too much time and space. Instead, we will focus our discussions on four experimental signatures, which have attracted most attention in the past years.

9.1.

Rise and Fall of IMF

Copious emission of intermediate mass fragments is one predicted consequence of the liquid-gas phase transition of nuclear matter, both by statistical models and transport models. At low excitation energy, few fragments are “evaporated” from the liquid while at very high excitation energy, the liquid “vaporizes” to produce nucleon gas. The “rise and fall” of IMF multiplicities has been observed in both central and peripheral collisions. For central collisions, maximum fragment productions occur around incident energy of 100A MeV as shown in the left panel of Fig. 9.1 for the Kr+Au reaction [82]. At incident energy above 400A MeV, production of IMF shift from central to more peripheral collisions [83, 84]. The right panel of Fig. 9.1 shows the impact parameter dependence (obtained by measuring for the fragmentation of Au projectiles at incident energy from 400 MeV to 1 GeV [83]. In both panels of Fig. 9.1, fragment multiplicities increase to a maximum with increasing excitation energy. The fragment production then declines and “vaporizes”

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completely into nucleons and light particles.

9.2. Critical Exponents The observation by the Purdue group [17] that the yields of the fragments produced in p+Xe and p+Kr obeyed a power law led to a conjecture that the fragmenting target was near the critical point of liquid-gas phase transition. The origin of this conjecture is the Fisher model [85] which predicts that at the critical point the yields of droplets will be given by a power law. The power law has since then been established very firmly in collisions between heavy ions [83] with the value of the exponent being close to 2. But the power law is no longer taken as the ‘proof ’ of criticality. There are many systems that exhibit this sort of power law: mass distributions of asteroids in the solar system, debris from the crushing of basalt pellets [86] and the fragmentation of frozen potatoes [87]. In fact, the lattice gas model which has been used a great deal for calculations of phase transitions and multifragmentations in nuclei [88, 89] gives a power law at the critical point, at the coexistence curve (that is a first order phase transition provided the freeze-out density is less than the critical density) and also along a line in the plane away from the coexistence curve. Nonetheless, the occurrence of a power law is an experimental fact and it is therefore desirable that models which aim to describe multifragmentation produce a power law, phase transition or not. Even if we expect to see phase transition in nuclear collisions it is unlikely that the system dissociates at the critical point. Much of the literature in intermediate energy heavy-ion collisions assumes that when one is seeing

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phase transition one is actually seeing critical phenomena [90, 91]. To reach the critical point one has to hit the right temperature and the right density. While one may be able to hit the right temperature by varying the beam energy, one has no control over the freeze-out density. Thus it is unlikely that the dissociation takes place at the critical point. We think that this strong emphasis on critical phenomenon rather than first order phase transition in nuclear multifragmentation came about for several reasons. Firstly, the experimentally observed power law was interpreted in terms of critical phenomena. Secondly, a bond percolation model [92, 93, 94, 95, 96] was among the first to be applied to multifragmentation in nuclear collisions. This model has only continuous phase transition. The bond percolation model can be demonstrated to be a special case of a lattice gas model [97] which is more versatile and has both first order phase transition and critical phenomena. An excellent review of the early history of this topic exists [98]. This covered the period to the end of 1984. More recently, the study of the liquid-gas phase transition in nuclear matter focuses more on measuring the thermodynamical properties, such as the temperature and densities, of the disassembling system.

9.3.

Nuclear Caloric Curve

Experimentally, production of particles in multifragmentation appears to be dominated by their phase space [42, 54, 99]. Thus, one should be able to measure temperature and densities, basic quantities in statistical physics. If the liquid-gas phase transition is of first order, one would expect to see enhanced specific heat corresponding to a plateau region in the caloric curve defined as temperature, T vs. heat or excitation energy E*/A. Aside from the experimental difficulties associated with measuring both quantities as discussed in Sect. 7 and 8, the simple caloric curve of temperature vs. excitation energy with a plateau in the temperature assumes that the pressure is constant [100]. There is no experimental evidence that such a condition is met in nuclear collisions. Thus even without introducing the isospin degree of freedom, the caloric curves depend on three variables, pressure, volume and temperature. Such complicated, three-dimensional nuclear caloric curves have been recently calculated [101]. Different shapes of the caloric curves have been obtained depending on the condition of the experiments and analysis. Therefore, one-dimensional caloric curves are useful only if the exact conditions can be determined or modeled. By themselves, these curves can be misleading and definitely do not constitute a signature for the liquid-gas phase transition even though the idea is very attractive.

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One of the purposes of this review article is to examine the experimental efforts in extracting the liquid-gas phase transition signals. Since many caloric curves have been measured since 1995, we will discuss the experimentally obtained curves keeping the above “warnings” in mind. If the incident energy is assumed to be related to the excitation energy, (this is particularly true for central collisions), then Fig. 7.3, which is a plot of temperature versus incident energy from E/A=30 to 200 MeV, is one form of caloric curve. It shows that the trend highly depends on the specific thermometers chosen to measure the temperature; the kinetic temperature increases rapidly (from 12 to 30 MeV); the excited states temperatures are nearly constant (from 3 to 7 MeV); the isotope temperatures involving He isotopes increase moderately (from 4 to 10 MeV) but isotope temperatures involving stay nearly constant at 4 MeV. Figure 7.3 sums up the most serious experimental problems we are faced with i.e., the discrepancies in the temperature measurements. However, this figure also shows that the excitation functions of the temperature measurement exhibit a smooth behaviour within each class of thermometer. Moreover, the trends are consistent from experiment to experiment since the data shown in Fig. 7.3 come from many different sources. The more traditional caloric curves of plotting temperatures versus the extracted excitation energy are shown in Fig. 9.2. In the left hand panel, all the curves are plotted on the same scale. The temperatures obtained have been extracted using the isotope yield ratios; are denoted by circles and are represented by the squares. To avoid confusions, all the temperatures plotted are the experimental apparent temperatures since sequential decay corrections are highly model dependent. Sequential decays account for part of the differences between and and there are empirical correction factors to reduce such differences. However, in this plot, the differences between the curves constructed with or are much larger than the correction factors. Thus, to keep the discussion simple, only the reported raw data are shown. In order to view each successive curve better, they are re-plotted in the right panel with a scale compressed by a factor of 2. Each curve is offset from from its predecessor by 2 MeV and the corresponding reactions are labelled close to the curves. The most interesting curve is the one labeled “Au+Au” plotted at the bottom of both the right and left panels. It was obtained from the spectator decays of the Au+Au reaction at E/A=600 MeV [10]. remains relatively constant as a function of deduced excitation energy, E*/A, between 3 and 10 MeV but increases rapidly at E*/A greater than 10 MeV. The resemblance to the first order phase transition of liquid raises a lot of excitement in the field. It also resembles the prediction from the statistical model [102].

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Unlike the predicted caloric curves from realistic models which will be discussed in more details in Sect. 12 or the temperature excitation functions of Fig. 7.3, the experimental caloric curves depend strongly on the reaction systems and analysis. (Due to the effect of sequential decay, caloric curves determined from the experimentally extracted isotope yields may not resemble the curves of the deduced primary fragments [67].) On the other hand, if one ignores the highest excitation data point in the Au+Au system, all caloric curves exhibit a smooth increase of temperature with excitation energy. This trend is very similar to the increase of as a function of incident energy as shown in Fig. 7.3. However in that case, the excitation function of the is nearly independent of reaction systems, Au+Au, Kr+Nb and Ar+Sc reactions, measured by different experimental groups. The differences in the curves shown in Fig. 9.2 again point to the uncertainties associated with the experimental procedures in extracting temperature and excitation energy. All the caloric curves measured so far suffer from the same uncertainties in determining the excitation energy. Some data may have better handle on the excitation energy because of better detector coverage or simpler reaction mechanisms. For example, the caloric curves obtained from the projectile fragmentation of Au, La and Kr on C, [67] have been extracted with a time-projection

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chamber (EOS-TPC) where a complete reconstruction of the projectile charge can be accomplished. The curves obtained from Au+C and Kr+C overlap very nicely with each other even though the La+C system shows lower temperatures measured. (The experimenters of Ref. [67] claim that the discrepancies observed in the temperatures (~15%) are within the experimental uncertainties.) It is also encouraging to see that these two curves overlap with the and data which used similar procedures in determining the temperatures and excitation energy [103]. Many of the extracted caloric curves do not agree with each other. Part of the differences can be attributed to the energy thresholds applied to extract the isotope yields. The high energy thresholds used in the systems to isolate the prompt component [104] probably account for the highest temperatures obtained in all the curves. For the Au+C, La+C, Kr+C, [67] and [103] systems, assumptions have been made regarding the preequilibrium contributions and the missing neutrons. Energy thresholds are used to eliminate pre-equilibrium emissions. This might account for the relatively high temperatures measured as compared to the temperatures extracted from Au+Au reaction. In the latter case, the pre-equilibrium contributions were minimized using other methods and attempts were made to extrapolate yields to zero energy thresholds. In the past year, some of the caloric curves have been revised [67] and others including the original “caloric curve” data are being reanalyzed [10, 105]. With more attention paid to the experimental problems associated with determining these curves, some of the discrepancies might be resolved in the near future. Future studies might extract other underlining physics from these data. Without further understanding of the reaction dynamics and experimental limitations, one should be extremely cautious in interpreting these curves as experimental signatures for the liquid-gas phase transitions of nuclear matter.

9.4. Isospin Fractionation Since nuclei are composed of neutrons and protons, isospin effects may be very important for the nuclear liquid-gas phase transition [106]. As the asymmetry between neutron and proton densities becomes a local property in the system, calculations predict neutrons and protons to be inhomogenously distributed within the system resulting in a relatively neutron-rich gas and relatively neutron-poor liquid [107, 108, 109]. The critical temperature may also be reduced with increasing neutron excess reflecting the fact that a pure neutron liquid probably does not exist [107]. While, recent calculations suggest that the rather narrow range of isospin values available in the laboratory might not allow us to observe the decrease in critical temperature [110], efforts are

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under way to study the fractionation of the isospin in the co-existence region. As the isospin effects are not large, the influence of sequential decays becomes important and may obscure the isospin fractionation effect one wishes to study. To minimize such problems, isobar pairs, such as which have the same number of internal excited states have been used. Some indications for isospin fractionation are provided by the sensitivity of distributions to the overall N/Z ratio of the system [33]. The ratios of also have been observed to decrease with incident energies, in qualitative agreement with the predictions from the isospin dependent lattice gas model [109, 111, 112]. Light isobars such as pair may suffer from contamination of pre-equilibrium processes. Attempts have been made to use additional mirror isobar pairs such as and Figure 9.3 shows the isobaric yield ratios of 3 pairs of mirror nuclei as a function of the binding energy difference, for two reactions, (open points) and (solid points) at E/A=50 MeV. If the sequential decay and the Coulomb effects are small, the dependence of the isobaric yield ratios on the binding energy difference should be exponential, i.e., of the form where and are the neutron and proton densities. The experimental data fluctuate around such a relationship. Extrapolation to using best fit lines (dashed and solid lines) allows one to obtain values for of 2.5 for the system (top line) and 5.5 for the system (bottom line). Both of these numbers are larger than the initial N/Z values of the two system, 1.24 and 1.48 for and respectively. The change in the N/Z values of the two systems is about 20%. However the changes between any of the mirror nuclei ratios are of the order of 200%, much larger than what one expects if the extracted neutrons introduced into the neutron rich systems are homogenously distributed. This observation suggests that the free neutron density needed to determine the light particle yields emitted from multifragmentation is much enhanced in the neutron rich system. To bypass the sequential decay problems completely and to avoid using only selected ratios, an observable employing ratios of all measured isotope yields are used. This method relies on extracting the relative neutron and proton density from two similar reactions, which mainly differ in isospin. Adopting the approximation of a dilute gas in the Grand-Canonical Ensemble limit with thermal and chemical equilibrium, the production of isotopes with neutron number N and proton number Z are governed by the nucleon densities, the temperature T plus the individual binding energies of the various isotopes B(N,Z).

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The factor F(N, Z, T) includes information about the secondary decay from both particle stable and particle unstable states to the final ground state yields as well as the volume terms. (Some readers may notice the similarity of Eq. (9.1) to the Saha equation used to describe the nucleation of a neutron and proton gas in astrophysics. In that case the prefactor F(N,Z,T) is dominated by the entropy term.) If one constructs the ratio of Y(N, Z) from two different related reactions, the observable called the relative isotope ratio, (N, Z) has a simple dependence on the relative neutron density and proton density of the free nucleon gas.

In the study of the central collisions of four Sn systems at incident energy of 50 MeV per nucleon [30], the relative neutron and proton densities have been measured for the (N/Z=1.36), (N/Z=1.36), (N/Z= 1.48)with respect to the (N/Z= 1.24) system.

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More than 20 isotope ratios are measured and they follow the dependence of Eq. (9.2) very well [30, 113]. The extracted and ratios are shown in Fig. 9.4; increases while decreases with the N/Z ratio of the total system. The increase of is consistent with neutron enrichment in the gas phase while the decrease of suggests proton depletion, a consequence of n-enrichment in the nucleon gas. The experimental trend (data points with the solid line drawn to guide the eyes) is much stronger than the trend expected if neutrons and protons were homogeneously mixed (dashed lines) in the breakup configuration. Adopting an equilibrium breakup model, results of Figs. 9.3 and 9.4 are consistent with isospin fractionation, a signal predicted in the liquid-gas phase transition. However, as with other signatures for phase transition observed so far, since the isospin fractionation is governed by the symmetry energy of the neutron and protons, “isospin fractionation” is a more general characteristic of heavy-ion reaction than the liquid-gas phase phenomenon. In fact, dynamical models also give predictions of isospin amplication, in qualitative agreement with the data [114].

10.

A CLASS OF STATISTICAL MODELS

A class of statistical models has been very successful in explaining multifragmentation processes in heavy-ion collisions. These models assume the following scenario. One defines a freeze-out volume. At this volume an equilibrium statistical mechanics calculation is done. However, these statistical calculations do not start from a fundamental two-body interaction or even a simplified schematic two-body interaction. Instead the inputs are the properties of the composites (which appear as bound objects because of the fundamental two-body interaction); their binding energies and the excited states. Their populations are solely dictated by phase-space. This is very similar to chemical equilibrium between perfect gases as, for example, discussed in [3]. The only interaction between composites is that they can not overlap with each other in the configuration space. Coulomb interaction between composites can be taken into account in different stages of approximation. These models have the virtue that they can be used to calculate data for many experiments whether these experiments relate to phase transition or not. The Copenhagen model, a statistical multifragmentation model abbreviated SMM (also referred to as SMFM), has become, de facto, the “shell-model” code for intermediate energy heavy ion data. An excellent review of this model exists [102]. The Berlin Model, a microcanonical multifragmentation model, usually abbreviated MMMC, has also been used to fit experimental data [115]. Some other references for microcanonical simulation of similar physics are [116, 117]. While there have been

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tremendous improvements in techniques and details, the roots of such models for heavy-ion collisions go back more than 20 years [118]. With some simplifications, the model of composites within the freeze-out volume at a given temperature can be exactly solved. In order to distinguish this model from SMM and MMMC (which are much harder to implement) we will coin a name. We will call this the thermodynamic model. As phase transition aspects are easily studied in the model, we treat this in detail.

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A THERMODYNAMIC MODEL

If there are A identical particles of only one kind in an enclosure, the partition function of the system can be written as

Here is a one-particle partition function of the particle. For a spinless particle this is is the mass of the particle; V is the available volume within which each particle moves; A! corrects for Gibb’s paradox. One might argue that this is not a rigorous way of treating symmetry or antisymmetry of particles but a recent paper [119] demonstrates that for nuclear disassembly this correction is very adequate. If there is only one species, Eq. (11.1) is trivially calculated. If there are many species, the generalisation is

Here is the partition function of a composite which has nucleons. We are at the moment ignoring the distinction between a neutron and a proton and thus our composites are bound states of nucleons. For a dimer, for a trimer, etc. In a more realistic version we will introduce the distinction between neutrons and protons but this model of one type of nucleon is highly illustrative, so we will continue with this for a while. Equation (11.2) is no longer trivial to calculate. The trouble is with the sum in the right hand side of Eq. (11.2). The sum is restrictive. We need to consider only those partitions of the number A which satisfy This restriction is hard to implement in an actual calculation and yet for A of the order of 100, the number of partitions which satisfies the sum is enormous. We can call a given allowed partition to be a channel. The probability of the occurrence of a given channel is

The average number of composite of nucleons is easily seen from Eq. (11.3) to be

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one readily arrives at a recursion relation [120]

For one kind of particle, above is easily evaluated on a computer for A as large as 3000 in matter of seconds. Thus in this model we can explore the thermodynamic limits. It is this recursion relation that makes computation so easy in the model. In the realistic model with two kinds of particles which we will introduce later, systems as large as 400 particles are easily done. It is important to realise that millions of channels possible in the partitions (Eq. (11.3)) are exactly taken into account although numerically. No Monte-Carlo sampling of the channels is required. We now need an expression for which can mimic the nuclear physics situation. We take

where the first part arises from the centre of mass motion of the composite which has nucleons and is the internal partition function. For 1 and for it is taken to be

Here, as in [102], MeV is the volume energy term, is a temperature dependent surface tension term and the last term arises from summing over excited states of the composite in the Fermi-gas model. For high temperatures, this will overestimate the contribution of the excited states and a modified expression is sometimes used to correct for this [116]. The explicit expression for used here is with 18 MeV and The value of is taken to be 16 MeV. The energy carried by one composite is given by Of these, the first term comes from cm motion and the rest from In [121], the term was neglected. It is included here but makes little difference. In the nuclear case one might be tempted to interpret the V of Eq. (11.6) as simply the freezeout volume but it is clearly less than that; V is the volume available to the particles for the centre of mass motion. In the Van der Waals spirit we take where is taken here to be constant and equal to The precise value of is inconsequential so long it is taken to be constant. Calculations employ V; the value of enters only if

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results are plotted against density. The energy of the system is

where

is the freeze-out the pressure is These can be deduced from Eqs. (11.2)

and (11.4). The surface tension term is crucial for phase transition in this model. At a given temperature the free energy F = E – TS has to be minimised. It costs in the energy term E to break up a system. A nucleus of A nucleons has less surface than the total surface of two nuclei each of A/2 nucleons (the volume energy term has no preference between the two alternatives). Therefore at low temperature one will see a large chunk. The –TS term favours break up into smaller objects. The competition between these two effects leads to the following features seen in experiments. At low temperature (low beam energy) each event will have one large composite (the fission channel is not included in these models) and few small composites. This leads to the inclusive crosssection being U-shaped (Fig. 11.1). (For illustration, starting from this figure and and upto and including Fig. 11.5, all calculations employ V of Eq. (11.6) ). As the temperature increases, the peak in the large side begins to decrease, finally disappearing entirely. When this happens, one has crossed from the coexistence zone to the gas phase. A graph of the yield against a is shown in Fig. 11.5. At temperature 6.2 MeV one sees both a large residue and smaller clusters, at 6.7 MeV the large cluster just disappears and at 7.2 MeV one has only the gas phase. It is instructive to plot for the same nucleus the specific heat per nucleon labelled by the total multiplicity and the number of intermediate mass fragments as a function of temperature (Fig. 11.2). One sees the specific heat maximising at the same temperature as the one at which, in Fig. 11.1, the peak in the high side of the yield function just disappears. One should remember that fragments produced in this model appear at non-zero temperature. They will further decay by sequential emissions. Thus the total multiplicity plotted here is lower than actual value to be expected finally. After the sequential decays, the yields of very light elements such as monomers, dimers etc. will increase substantially as the heavy composites decay by emitting these. In in Fig. 11.2 we have included With sequential decays included will go down from the values shown in Fig. 11.2.

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Keeping these reservations in mind we see in Fig. 11.2 that the sudden increase of the multiplicity and imply coexistence. At higher temperature, the system is in the gas phase. The cross-section for a large residue is very small (Fig. 11.1). In order to understand the nature of phase transition we now go to much larger systems so that one can feel confident about extrapolation to the thermodynamic limit. With this in mind we have plotted in Fig. 11.3, for a system of 1400 and 2800 particles, the free energy per particle as a function of temperature (The free energy is simply A brake appears to develop in the first derivative of F/A which signifies first order phase transition. We follow this up in Fig. 11.4 by calculations of specific heat per particle for 200, 1400 and 2800 particles. As the number of particles increases, the maximum in the per particle becomes sharper and the height increases. In Fig. 11.5 we have tried to understand the origin of this singularity in greater detail. Let us denote by the ensemble average of the mass number of the heaviest composite (the technique for this calculation is given in [121]). This should scale like A where A is the number of particles in the disintegrating system. In Fig. 11.5 we have plotted as a function of where is the temperature at which the specific heat maximises. As A becomes large, the drop in the value of at becomes sharp. The sudden disappearence of this large blob of size causes this behaviour of In Fig. 11.6 we have drawn a diagram for a system of 200 particles at various temperatures. We have also drawn a line that is labelled co-existence which passes through the points where the specific heat attains the highest values. For plotting this graph we have used We stop below At higher density the approximation of non-interacting clusters (even after including Van der Waals type correction for finite volumes of the composites) would be very questionable. One approximation in the above calculation is the assumption of constant excluded volume. The excluded volume (as can be verified in Monte-Carlo simulation) is a function of the total multiplicity . It is also a function of the freeze-out volume inside which the particles are constrained to move. For 200 particles, the effect of this variability of the excluded volume on the diagram was investigated in [123]. The difference is not large. However, this has not been studied in the thermodynamic limit. It will be very interesting to investigate what effect it will have on the nature of the phase transition in the very large A limit. The thermodynamic properties of this model have been further studied in [124, 125].

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GENERALISATION TO A MORE REALISTIC MODEL

For comparisons with actual data the model must be made more realistic. Towards that goal, a composite is now labelled by two indices: where the the first index in the subscript refers to the number of protons and the second, to the number of neutrons in the composite. The partition function for a system with Z protons and N neutrons is given by

There are two constraints: and These lead to two recursion relations any one of which can be used. For example,

where

Here is the internal partition function. These could be taken from experimental binding energies, excited states and some model for the continuum or from the liquid drop model or a combination of both. The versatility of the model lies in being able to accommodate any choices for A choice of from a combination of the liquid-drop model for binding energies and the Fermi-gas model for excited states that has been used is

where and One can recognise in the parametrisation above the volume term, the surface tension term, the Coulomb energy term, the symmetry energy term and contributions from excited states. The Coulomb interaction is long range; some effects of the Coulomb interaction between different composites can be included in an approximation called the Wigner-Seitz approximation. We assume, as usual, that the break up into different composites occurs at a radius which is greater than normal

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radius Considering this as a process in which a uniform dilute charge distribution within radius collapses successively into denser blobs of proper radius we write the Coulomb energy [126] as

It is seen that the expression is exact in two extreme limits: very large freezeout volume or if the freeze-out volume is the normal nuclear volume so that one has just one nucleus with the proper radius. For the thermodynamic model that we have been pursuing, the constant term in the above equation is of no significance since the freeze-out volume is assumed to be constant. In a meanfield sense then one would just replace the Coulomb term in Eq.(12.4) by Calculations with the thermodynamic model with two kinds of particles and realistic were done in [127, 128]. A plateau in the caloric curve is found around 5 MeV which is in accordance with experimental finding. An interesting point in the calculation is the following observation. Without the Coulomb, the height in the peak of the specific heat increases with A (see previous Section). With Coulomb the height is reduced and the dependence on A nearly disappears. The growth in size is compensated by the growth in Coulomb repulsion. This means the caloric curve is approximately universal, i.e., does not depend strongly on the specific system which is disintegrating. We show the caloric curves computed for three disintegrating systems in Fig. 12.1. This is taken from [127]. Of course, the model can also be used to calculate other observables, not necessarily related with any phase transition aspect. However, for many purposes an “afterburner” calculation is required. The composites obtained in the calculations are “hot”. They will subsequently decay by particle emissions. In [129] these subsequent decays were included in an approximate manner so that one can compare with experimental yields of boron, carbon and nitrogen isotopes. This comparison is shown in Fig. 12.2.

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A BRIEF REVIEW OF THE SMM MODEL

A very comprehensive review of this model exists [102] so we only give a brief resumé here. It is relevant to mention that a peak in the specific heat at about 5 MeV temperature was predicted in the model much before experiments were done [131]. In the SMM each break-up channel is treated separately. A given channel is specified by the set of V and temperature T. The volume V which is the free volume available for the motion of the cms of the composites is taken to be where is multiplicity dependent

In the above is taken to be 1.4 fm; is the normal radius of a nucleus of A nucleons. The concept of temperature is used but its primary use is to make the energy in each channel the same value and to correspond to the experimental situation. It is therefore a microcanonical calculation in spirit. The energy in a given channel is given by

where gives the intrinsic energies at temperature T (includes binding energies, contribution from excited states etc.; see the discussion in Sect. 12) and completes the Wigner-Seitz estimation of Coulomb energy. The crucial thing here is the choice of channels. It is impossible to include all channels. For A=200 the number of possible multifragment partitions is so a Monte-Carlo sampling which is geared to include the most important channels at a given excitation energy is needed. This is a very elaborate story in itself and all we can do here is to provide references. An important paper which elaborates on the procedure is [132]. The review article on the model [102] gives a more complete list of references. As in all models of this type, sequential decays of the hot fragments need to be included to compare with most experimental data. The fit with data on IMF multiplicity, mean energy etc. is normally quite good.

14.

THE MICROCANONICAL APPROACH

Pioneers of this approach were the Berlin group [133] and Randrup and collaborators [116]. In the Berlin approach, the clusters, which have finite sizes, are all totally inside the freeze-out volume. This freeze-out volume is the

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same for all channels. Randrup et al., demand that centres of all the clusters should be within the freeze-out volume. There are other practical differences between the two formulations. Let us sketch the general procedures that any microcanonical calculation will have to accomplish. Suppose that we are interested only in calculating the average number of clusters of a composite which has protons and neutrons, i.e., when the total energy is E and the disintegrating system has Z protons and N neutrons. How do we proceed? For simplicity we will regard that the only interaction between the clusters is that they can not overlap. We will also assume that the composites have only ground state. There are many break-up channels that are possible. All divisions (that satisfy and where is the number of cluster which has i protons and neutrons) are allowed and equally likely to occur. The phase space available to each channel, however, will be, in general, different and will strongly affect the final result. The phase space associated with each break-up channel is

Here M is the multiplicity in the channel the binding energy of the cluster and is the potential energy between the clusters and In our case it is either 0 (when they are separated) or (when they overlap). The momentum integral is analytic:

Once the momentum integral is done we still need to do the configuration space integral. This is by no means trivial but one can estimate it in a MonteCarlo procedure. The first particle is placed at a random position inside the freeze-out volume. Having placed the first one we try to place the second one, again at a random position in the freeze-out volume. We may succeed but we may fail also if, by chance, the second chosen position was such that the new particle overlaps with the particle already in. A successful run occurs if we are able to put M particles in without failing once. An unsuccessful run happens if anytime in the chain we failed to put a particle. Then the volume integral is where is the number of successful runs and

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is the number of unsuccessful runs. The quantity just calculated is where V is the volume of the thermodynamic model. Provided all this is done for each channel and we have calculated the phase space for each channel, the average number of particles of a composite with protons and neutrons will be given by where is the number of composites with proton number neutron number in the channel labelled by and is the phase space integral associated with this channel. The sum is over all In practice, this procedure is impossible to carry out as the number of channels is inordinately large, An “importance sampling” [134] of the phase space is necessary. This is usually done with the Metropolis method [134, 135]. We will have occasion to use this technique later also. One attempts elementary moves by which one migrates from one channel (here with multiplicity M) to a neighbouring channel (multiplicity M + 1 or M – 1). These moves are “fission” (take a composite and arbitrarily break it into two pieces) and “fusion” (join two composites). Let us call the phase space integral before the move. One also calculates the phase space integral after the attempted move. Let us call this the move is accepted. If the probability of switching is given by the ratio (see [134] why these give the correct transition probabilities). An event is accepted every N attempted moves (some successful and some not) and averages will be calculated with many such events; N should be sufficiently large to avoid event-to-event correlation and of course one should have sufficient number of events to reduce statistical errors. There are a great many details which need to be worked out before such a program can be instituted for practical calculations. The interested reader should consult the original literature. The techniques for microcanonical calculations were developed in the mid eighties spanning several papers, each one an improvement over the previous one. What we have outlined here are the general principles.

15.

THE PERCOLATION MODEL

This model is strikingly different from the models described above. The model has been extensively studied in condensed matter physics. A delightful monograph exists [96] which has all the material needed to follow the application in nuclear physics. The applications in nuclear physics were made by Campi and collaborators [92, 94] and by Bauer and collaborators [93, 95]. There are two types of percolation models: site percolation and bond percolation. For applications to nuclear physics, bond percolation was used. In bond

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percolation there are N lattice sites. One uses a three-dimensional cubic lattice, thus etc. The number of nucleons is also N. Each lattice site contains one nucleon. We do not distinguish between neutrons and protons. The crucial parameter is the bonding probability whose value can vary between 0 and 1. The probability that two nearest neighbour nucleons will be part of a cluster is given by the value of If (high excitation energy) all the nucleons will emerge as singles. If the nucleons stay together as one nucleus (low excitation energy, not enough to break up the nucleus). For the values of between the two extremes Monte-Carlo sampling is needed to generate events and determine in each event the occurrence of clusters of different sizes. There is a phase transition in this model. One can define a percolating cluster; this is a cluster, which, if it exists, spans the walls, i.e., connects opposite walls through an unbroken cluster. For N very large, this appears at the value of This value of will be labelled by The order parameter in this model is the probability that an arbitrary site (equivalently an arbitrary nucleon) is part of this percolating cluster. Below this is zero since there is no percolating cluster. It starts from zero at and continuously moves towards the value 1 as the value of is increased. The phase transition in this model is continuous and not a first-order phase transition. This aspect had a very important and strong influence in the history of search for phase transition in heavy-ion collisions. Near critical points, One can define critical exponents and try to evaluate them from experiment. We will see later that even though we regard now the phase transition in nuclear heavy ion collisions to be first order, it is meaningful to try to measure certain exponents. In the lattice gas model (to be described below) these retain significance even when one is far from a critical point and is in the vicinity of a first-order phase transition. We give the values of the more common exponents in the thermodynamic limit (i.e., One of these exponents we have already encountered many times. Near the critical point the yield of mass is given by

At the critical point the yield is a power law. The value of in the percolation model is 2.18. The value of in the above equation is 0.45. Let us denote the mass of the largest cluster by The second moment is defined by

Here the summation excludes the and is the number of nucleons. The second moment diverges: where the value of is 1.80. In finite systems will not become infinity but will go through a

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maximum as is traversed. Above the percolation point the order parameter is given by where In spite of its simplicity, the percolation model was an aid in understanding various phenomena. It has now been replaced by a lattice gas model which is more realistic, more versatile and indeed contains the percolation model as a special case.

16. THE LATTICE GAS MODEL(LGM) The advantage of the percolation model is that clusters are easily obtained. This break-up can be compared with experiment. But there is no equation of state in the usual sense. The equation of state requires two variables: and V, then T is automatically known from the EOS: or and T then EOS gives V etc. There is only one parameter in the percolation model, the bond probability. There is no obvious way this model can be linked to finite temperature Hartree-Fock theory or the thermodynamic model or SMM or the microcanonical model. There is no Hamiltonian. In [88, 89] LGM was introduced so that one has an EOS as in Hartree-Fock theory but also has the capability of predicting clusters as in the percolation model. The EOS of LGM in meanfield theory in a grand canonical ensemble is done in textbooks [13]. To obtain clusters in the model an extension of the well known model is necessary. Although LGM today is more complete with the inclusion of isospin dependence and Coulomb interaction, we introduce first the simplest version. This will be very easily generalised later. As in percolation, we have lattice sites but now, in general, is greater than A, the number of nucleons that need to be put in these sites. When A the nucleus has normal density. We are not allowed to put more than one nucleon in a site. Thus the model is limited to normal volume or larger. Because cluster formation presumably takes place in a volume significantly larger than normal volume, this restriction is not debilitating. The nearest neighbours have a bond which is negative. Only nearest neighbours interact. The exclusion of the possibility of two nucleons occupying the same site mimics a short range repulsion. The attractive nearest neighbour interaction simulates the attractive interaction which is also short range (but longer than the short range repulsive interaction). Let be the number of nn bonds in a specific lattice configuration. The energy carried by these bonds is Thus the partition function is

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Here is the number of configurations which have nearest neighbour bonds and which can be formed from A nucleons in lattice sites. This is not analytically solvable. Hence calculation of observables where configurations have the above weighting requires Monte-Carlo simulations. The simulations are usually done in the Metropolis algorithm. Starting from an initial configuration chosen suitably [109], one attempts a switch between an unoccupied site and an occupied site. If the resulting change of energy is negative, the switch is accepted. If is positive, it is accepted with a probability After a large number N of attempted switches (some successful and some unsuccessful) an event is accepted. N should be large enough to avoid event to event correlation. The grand canonical ensemble of the LGM (sum over all possible A’s) can be mapped onto a three dimensional Ising model [13, 136]. The latter has been extensively studied and indeed serves as a model for liquid-gas phase transition. Many of the known results of the Ising model can be directly applied. For example in the large A limit the critical temperature will be and the critical density We consider now an extension of the model so that clusters can be computed. Suppose we have generated a configuration. At finite temperature, the nucleons will not be frozen at the lattice sites. They will have momenta. In this configuration each of the A nucleons can be given a momentum by MonteCarlo sampling of a Maxwell-Boltzmann distribution at the given temperature. Thus in an event we have nucleons at definite lattice sites with definite momenta. There may be some isolated nucleons which have no nearest neighbours. These clearly are monomers. The next case is when there is a cluster of two nucleons which are nearest neighbours of each other. They will form a bound cluster of two if the kinetic energy of relative motion is insufficient to overcome the attraction between the two nucleons, i.e., Here and of one nucleon). It turns out that this prescription which is rigorously correct for a cluster of two also works statistically for larger clusters. That is, we can formulate a rule that independent of other neighbours, two nearest neighbours form part of the same cluster if the relative kinetic energy of the two is insufficient to overcome their attraction. It is obvious that this recipe, introduced in [88, 89], reduces the many body problem of recognising a cluster of many nucleons into a sum of independent two body problems. For brevity we refer to this as PD recipe. To see why this works statistically even if not individually let us specifically consider a three body cluster [109]. For three particle clusters, nearest neighbours are either linear or L shaped. In either case there is only one particle which has two bonds (label this by particle 2) and two others (label them 1 and 3) which have one bond each. Accord-

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ing to the PD recipe this will form a three body cluster if and To check if particle 3 is part of a three body cluster (similar arguments will be needed for particles 1 and 2) we should instead verify if Here (12,3) is the relative momentum between the centre of mass of (1+2) and 3; is the reduced mass for this relative motion. Thus there may be cases where the PD recipe gives a three body cluster whereas in reality the third one will separate. But there will also be cases where the PD recipe will deem that the third one will separate whereas in reality it stays attached. Statistically overestimation will cancel out underestimation because for a Maxwell-Boltzmann distribution all relative motions are also Maxwellian at the same temperature. That is, in Monte-Carlo simulation, will be as many times below the will be. The same argument applies to particle 1. For particle 2, it can be verified that if and then is always satisfied. Campi and Krivine have used a different approach and come to the conclusion [137] that the PD recipe gives the correct number of particle stable clusters. Recognition of clusters in a many body system is a complicated issue. The PD recipe was tested in [138] in numerical simulations and found reliable. In the PD recipe once the configuration of A nucleons and their momenta are given, the cluster decomposition is immediate. One may however, starting from this initial condition, switch to a different model. One may propagate particles using molecular dynamics. At asymptotic times clusters are easily identified as different clusters will separate from each other. Of course the result will depend upon the interaction potential used for molecular dynamics propagation. To test the PD recipe one must use an interaction consistent with the assumptions of the lattice gas model. Let be the length of each side of the elementary cubic lattice. The interaction between particles must become repulsive when the distance between them gets to be less than a. At distance it is deepest at -5.33 MeV. At distance beyond it must go to zero rapidly. Given the same initialisation and such interaction, molecular dynamics produced very similar results as the PD recipe. It follows that with this recipe of calculating composites, we do not need to worry about subsequent evaporation as one needs to in many other models; thermodynamic, SMM and microcanonical. This is a tremendous advantage. Evaporation was already taken into account when we applied the PD recipe. One does not take the size of the cluster to be given by just the number of nucleons which are connected to each other through the nearest interaction [139]. Some of these will fly away. The rest that remain and are counted, are particle

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stable. With a prescription for obtaining clusters, the LGM will show many features in common with the percolation model. This leads to interesting properties.

17. PHASE TRANSITION IN LGM Whether one later ascribes momenta to nucleons and calculates clusters or not, there is phase transition in the traditional LGM. The thermodynamics of the system does not depend upon the definition of clusters. The co-existence curve can be drawn. This diagram is simply transferable from studies in the three dimensional Ising model. We show this in Fig. 17.1. The thermal critical point is shown in the diagram as C.P. With a rule for calculating clusters it will be very interesting if the thermal critical point also coincides with the onset of a percolating cluster. This aspect was studied by Coniglio and Klein [140]. They propose that the probability that two nearest bonds have an active bond between them be given by

With this definition, these authors, using renormalisation group techniques, proved that at percolation just sets in. However, percolation sets in not just at the thermodynamic critical point but rather along a continuous line in the plane (the dotted line in Fig.24). This was studied in [141] and the line is called Kertész line. Thus the critical exponents and are meaningful not only at the critical point but along an entire line. It turns out the PD recipe which is a natural choice for calculation of clusters in the case of nuclear disintegration is very close to the Coniglio-Klein (CK) formulation. The PD formula for p (using the fact that in a MaxwellBoltzmann distribution the relative motion is also Maxwell-Boltzmann) is

A comparison of according to the above formula to the Coniglio-Klein formula is shown in Fig. 17.2. They are very close. As far as we know, all cluster calculations in nuclear physics use the PD recipe.

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With the aid of Fig. 17.1 we can now discuss phase transition in nuclear disintegration according to LGM. The freeze-out volume that fits the data best [89] is bigger than twice the normal nuclear volume. In that case as the temperature of the disintegrating system is raised from a low value to a high value (either by changing the beam energy or by gating on appropriate impact parameter) the system will cross the coexistence curve on the low density side of the critical point (to the left of C.P. in Fig. 17.1). Thus we will have firstorder phase transition [142]. As the line is crossed one will see a discontinuity in specific heat, a peak in and other features.

18. ISOSPIN DEPENDENT LGM INCLUDING COULOMB INTERACTION For many practical applications, it becomes necessary to distinguish between like particle interactions (bond between proton and proton or/and between neutron and neutron) and unlike particle interaction (bond between neutron and proton); between like particles must be repulsive or zero otherwise we can obtain dineutron or diproton bound states. The bond between unlike particles will be attractive. At zero temperature in nuclear matter, energy considerations imply that sites will be alternately populated by neutrons and protons. Thus the only nearest neighbour interactions will be those between unlike particles. Nuclear matter binding energy then dictates that MeV. This however does not fix the value of or It is clear that the Monte-Carlo technique of generating events for finite nuclear systems can also be used when the interaction between like and unlike particles are different. The Coulomb interaction between protons can also be included. When this is done at zero temperature we obtain the ground state energies. This was done in [109] for a range of nuclei for The binding energies of these nuclei thus computed were then fitted to a simple liquid-drop mass formula:

Here I = (N – Z)/A is the neutron-proton asymmetry of the nucleus. The fit of LGM binding energies to this four parameter formula is quite good. We compare the four parameters deduced from LGM to liquid-drop model values [143] in the table. Considering the simplicity of the model, the agreement is gratifying. We also notice that the asymmetry parameter is larger than the liquid-drop value. Since is fixed from nuclear matter binding energy, the only way we can bring down is to make and go negative. As

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explained already this is not permissible. We are therefore led to the conclusion that and are the best choices for isospin dependent LGM.

19. CALCULATIONS WITH ISOSPIN DEPENDENT LGM We mentioned earlier that the isotopic content of the gas phase can be different from that of the liquid phase. This comes out nicely in LGM. Figure 19.1, taken from [109] demonstrates this. Here one considers breakup of at different temperatures. The average value of the charge of the largest residue at each temperature is denoted by This will drop in value as the temperature is increased. Also calculated is the average value of neutron content of the largest cluster. We may regard the largest cluster as the liquid phase and the rest of the nucleons as primarily belonging to the gas phase. The disintegrating system has N/Z = 1.49. With an isospin dependent LGM, the is much closer to 1. This means the gas phase has a higher N/Z ratio, higher than that of the parent system. The reason for this behaviour is that the symmetry energy drives the N/Z ratio of the largest cluster towards the value unity. The Coulomb effect will offset this as shown in the figure. The simplest version of the LGM which had no isospin dependence and no Coulomb term will keep the at the value pertaining to the disintegrating system. This is also shown in the figure. This contradicts experiment. Au we show several quantities as a function of temperature. In experiments one extracts the quantity where the yield Y(Z) as a function of Z is fitted to The power law comes out quite well in LGM. We notice that the maximum in the maximum in and the minimum in all bunch around T=4.2 MeV which we then associate with the crossing of the coexistence curve. The maximum in is at a slightly higher temperature.

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The EOS for isospin dependent LGM in meanfield theory using the BraggWilliams and the Bethe-Peierls approximation can be found in [144]. Phase transition aspects, much more than what we have covered here, can be found in [108, 145, 146, 147]. The model has been used successfully to obtain several experimental results, for example, ratios as the isotopic content of the parent system changes [108, 109]. Some applications were made in [110]. The shortcomings of the model for detailed fittings to experimental data are obvious. The model has cubical symmetry rather than spherical symmetry. The shell effects are missing although the smooth part of binding energy is approximately reproduced. The excitation spectrum of the composites is incorrect. But it has many nice features not present in other models. Here composites are formed directly out of fluctuations. One starts with interactions between two nucleons. The inclusion of Coulomb effects is precise, though numerical. Best of all, it includes interactions between composites.

20.

FRAGMENT YIELDS FROM A MODEL OF NUCLEATION

A phenomenological droplet model based on homogeneous nucleation theory has also been used to describe mass yields provided in heavy-ion collisions [148]. The nucleation model is an extension of the Fisher droplet model which was originally used to describe such yields. The extension allows for the possibility that supersaturated systems are produced during the brief encounter of two colliding nuclei. Homogeneous nucleation occurs in supersaturated systems when chance collisions of particles in the gas phase yield to local density inhomogeneties. These inhomogenities are droplets of particles of the liquid phase that will grow in size if the systems lived for a long time. Specifically, in the supersaturated phase, a critical size droplet exists which is determined by the surface tension and the difference of liquid and gas chemical potentials. Droplets larger than the critical size will grow by accumulating nucleons in order to lower the free energy of the system while droplets of smaller size will evaporate nucleons also lowering the free energy of the system. This behaviour in growth and evaporation reflects itself in a yield distribution which is U-shaped with an initial decrease with A until the critical size is reached and then an increase in yield with A above In this nucleation description, the probability of formation of droplets is determined by calculating the change in Gibbs free energy with and without a drop at constant temperature and pressure. For example, if a droplet of size A is surrounded by B droplets of the gas phase, then and Here and are the liquid and gas chemical potentials, is the radius of the drop (with and is the surface

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free energy such that MeV at T=0. The term is a term introduced by Fisher to account for the power law fall-off of yield distributions at a critical point with a critical exponent. The probability of forming a drop of A nucleons is where This gives a cluster distribution N(A), to a constant C,

On the coexistence curve of a liquid-gas phase transition and N(A) is a monotonically decreasing function of A. For a supersaturated system and N(A) has a minimum value at a critical size droplet Neglecting the term, is given by The model fits many yields of heavy-ion collisions.

21. ISOSPIN FRACTIONATION IN MEANFIELD THEORY Early studies of the liquid-gas phase transition were carried out using a Skyrme interaction and focussed mostly on a one component system made of nucleons even though expressions were developed for two component system of protons and neutrons [12]. The one component aspects are given in Sect. 2 of this review. The two component aspects will now be discussed in a meanfield approach with this Section based mostly on the work of Muller and Serot [107]. Extensions of the results of [12] are now being carried out in [149] using a Skyrme interaction while the Muller-Serot analysis is based on a relativistic meanfield model. Initial results in [149] are qualitatively similar to those of [107]. Both approaches allow a complete calculation various thermodynamic properties, such as the pressure and proton and neutron chemical potentials. In one component systems, the Skyrme interaction and the relativistic meanfield model lead to an S-shaped behaviour of pressure versus volume or density at fixed temperature with stable, unstable, supersaturated and supercooled regions. The standard Maxwell construction describes the liquid to gas phase transition with both phases having the same pressure and the chemical potential The equality of chemical potentials of the liquid and gas phases in phase equilibrium is equivalent to equal areas of the regions above and below the Maxwell or vapour pressure line in the S-shaped loop in vs. V. For two component systems, phase equilibrium becomes more complicated since the proton to neutron ratio can be different in the two phases because of the symmetry energy which favours N = Z. Since the symmetry energy will be large in the denser liquid phase, the proton-neutron asymmetry

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will be bigger in the gas phase than in the liquid phase. In two component systems, the phase equilibrium conditions consist of setting, at fixed pressure, the proton chemical potentials in the two phases equal to each other and similarly, the neutron chemical potentials are equal in the liquid and gas phases. The properties of the phase separation boundaries (binodals) and instability boundaries (spinodals) are studied as a function of a quantity labelled which is the proton fraction, with the neutron fraction given as and For two component systems, the binodals are now a surface in plots in space. By contrast, for one component systems, the binodal is a line for the vapour or Maxwell pressure vs. T which terminates at the critical temperature The line is at in the space of two component systems. The two dimensional binodal surface of the phase coexistence boundary now contains a line of critical points for different values of and a line of maximal asymmetry. In a plot of the binodal surface, the intersection of a fixed T plane with the surface gives a loop of vs. The maximal asymmetry point is at and physically corresponds to the smallest proton ratio or the largest neutron ratio on the binodal surface at each T. The critical point is at on this loop. The loop degenerates to a point at the critical temperature of a symmetric system (see Fig. 6.2 in [107]). If T and are both fixed, the binodal surface has two values of and corresponding to different values of the proton fraction in the liquid and gas phase. These different values arise because of the difference in symmetry energy in the liquid and the gas phase. One of the interesting conclusions of the meanfield two component model of the liquid-gas phase transition is that the first-order transition of a one component system becomes a second-order transition. Because of the greater dimensionality in the physical situation, the phase transition is continuous. The role of dimensionality due to the number of components of the system on the order of the phase transition was also pointed out by Glendenning [150].

22.

DYNAMICAL MODELS FOR FRAGMENTATION

The one common characteristic of all the theoretical models considered so far is that they are static, i.e., they all assume that equilibrium is achieved and hence laws of equilibrium statistical mechanics apply. A more fundamental calculation would use a transport equation. Here two nuclei approach each other in their ground state. By solving a time dependent equation we see what the final outcomes are. The BUU model does start with two nuclei, in their ground states, boosted towards each other. One does not have to introduce a temperature. The model

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has a meanfield as well as hard collisions and has indeed proven to be highly successful in predicting sideward flow, squeeze-out etc., [56]. But BUU is good for predicting expectation values of one-body operators, it does not have fluctuations. Thus it will not produce clusters. A great deal of effort went into introducing fluctuations in BUU type formalism. A very short list of references are [151, 152, 153, 154, 155]. Unfortunately, practical calculations are extremely time consuming. In rare cases calculations have been done to compare with experimental data [156] but the method has not been pushed to see, for example, the rise and fall of IMF production, an accurate estimation of the parameter etc. Maybe, in future such calculations will be done. While exact calculations for heavy-ion collisions based on quantum mechanics are impossible to carry out, classical calculations for ion-on-ion collisions for particles are entirely feasible with present day computers. This requires solving for each particle and Here is the two body interaction. One starts at time with initial values of and numerically integrates out time. In a set of three papers [157, 158, 159] Pandharipande and coworkers studied disassembly of hot classical drops as well as collisions between cold charged argon balls containing and particles. The chosen values of were (108,108), (200,16) and (65,65) [157]. The interaction between atoms was a truncated (12,6) potential and a Coulomb interaction was added. Although this is thirteen years later, the reader will find reference [157] still very relevant and revealing. As a function of time, the evolution of temperature and density of the central region is plotted so that one can see under what initial conditions the central region reaches spinodal instability and what the final products are in such cases. They point out that the mass yields calculated in the disassembly of hot equilibrated drops having 216 particles and density somewhat less than normal density is very similar to mass yield of 108+108 collisions. Thus the assumption of statistical equilibrium is valid. The authors stress that the classical argon balls used in the study are not intended to be mock nuclei, but instead to provide simple systems whose time evolution can be studied exactly by solving Newton's equations of motion. Direct comparisons with nuclear data are difficult. Nonetheless, there are many similarities. A power law for fragment yields followed from these calculations, the minimum value being about 1.7. Another remarkable feature is that the apparent temperature deduced from the fragment kinetic energies is much larger than that of the system. The large kinetic energies of fragments in the simulation appear to come from collective motion of expansion and Coulomb repulsion. In a later paper, the Illinois group devised a simple nucleon-nucleon potential for classical calculations of nuclear heavy ion collisions [160]. They did not use this potential to study liquid-gas phase transition but instead used this

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at Bevalac energies to investigate flow angles and transverse momenta. Later, disassembly of a collection of nucleons which start with initial temperature and density of interest in this article and interact via this potential was considered by other groups [161, 162, 163, 164]. Pratt et al. [165] used a truncated (8,4) potential to study similar dissociation. It has not been demonstrated that such simulations apply to nuclear cases very well since actual nuclear data for specific cases have not been compared. The Frankfurt group proposed a simulation which they called quantum molecular dynamics(QMD). This was used for Bevalac energies initially. Relativistic versions exist and are in frequent use. This has also been used for energy region of interest here. Detailed exposition of the model exists [166]. Here each nucleon is represented by a Gaussian in coordinate and momentum space with widths consistent with uncertainty principle. The centroids of the Gaussian move but the widths are kept fixed. The centroids move under the influence of meanfield except when the centroids move very close to each other Then they scatter as in two-body scattering. Pauli blocking is taken into account for scattering. Because each particle is represented by a centroid in phase space with fixed widths in momentum and coordinate space, there is fluctuation built into the system and in the final stage one can recognise clusters. In calculations reported in reference [84] fragment multiplicities were underpredicted in the energy region of interest here. The Copenhagen group has done simulations using a prescription which they dubbed nuclear molecular dynamics [167]. This is quite similar to QMD. All such simulations which provide clusters at the end are quite computer intensive.. The clusters are easily recognised if each event is run until “asymptotic” times so that different clusters are well separated from each other. But that requires considerable computer time. Quite sophisticated algorithms have been introduced for early recognition of clusters [168]. This is of practical importance. Many other models are on the market which will simulate ion-on-ion collisions. They were not necessarily introduced to study liquid-gas phase transition. Some references are [169, 170, 171]. Phase transition aspects were discussed in reference [172]. A very attractive model, expanding emitting source (EES) model was proposed by Friedman [173]. This model assumes that initially the hot system evaporates as well as expands. For low initial temperature the system will cease expanding and will revert towards normal density. But beyond a certain temperature at the end of this slow expansion the system will explode. The relationship of this model to liquid-gas phase transition will be interesting to explore.

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23. OUTLOOK The possible links of many experimental observables to expected liquidgas phase transition in finite nuclear systems continue to be a fascinating story. Much has been learnt and much remains to be learnt. The topic has forced nuclear physicists to delve into realms that were not familiar to them. Necessity has prompted us to do interesting theoretical work, for example, [145] in finitesize scaling and in general, about phase transitions in “small” systems [174]. We are exploring ideas which are of relevance in other fields. There is a broadening of the horizon which is refreshing. We foresee substantial effort in several directions, for example, in dynamical models. Just fifteen years ago, multifragmentation was barely mentioned in the literature. However, in the past decade tremendous progress has been made in understanding the multifragmentation process and its relationship to the liquidgas phase transition in nuclear matter. Even though we have not found one definitive experimental signature pinpointing the phenomenon, we know that a mixed phase can be created in the heavy-ion reactions, that multifragmentation occurs within 50-200 fm/c after the initial collisions with a freeze-out density of less than 1/3 of the normal nuclear density, and that the freeze-out temperature is probably in the range of 4-6 MeV. In the near future, experiments will be designed to measure excitation energy, reaction time and freeze-out densities and other observables more precisely. Nonetheless, the availability of comprehensive experimental data has stimulated intense interest in the theoretical front leading to better understanding of the statistical and dynamical nature of nuclear collisions. More exciting development will be awaiting in the exploration of the isospin degree of freedom in the liquid-gas phase transition with the availability of high to moderate intensity radioactive beams. In writing this article we have also realised that vast amount of work has been done in this field by groups widely spread geographically. The literature on this subject is colossal. Our reporting had to be necessarily selective, influenced largely by our own involvement and experience. We are particularly aware that in writing an article about a subject whose scope is this large we must have left out a significant amount of interesting work. For example, several review articles can be written on the subject matter of the last Section alone. We finish therefore by apologising for all the omissions that occurred.

ACKNOWLEDGEMENTS Subal Das Gupta thanks J. Lopez for an advanced copy of a monograph on heavy-ion collision reactions written by Lopez and Dorso [175]. Research

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support was provided by Natural Sciences and Engineering Research Council, le Fonds pour la Formation de Chercheurs et l’aide à la Recherche du Québec, the National Science Foundation Grant No. PHY-95-28844 and the U.S. Department of Energy Grant No. DE FG02-96ER40987.

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Chapter 3

HIGH SPIN PROPERTIES OF ATOMIC NUCLEI David Ward and Paul Fallon Lawrence Berkeley National Laboratory Berkeley, CA 94720, USA

1.

Introduction

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2.

The Development of Large Arrays for Gamma-Ray Spectroscopy

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3.

Rotations, Particle Alignments and the Nuclear Shape

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4.

Superdeformation

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5.

Limits to Nuclear Rotations

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6.

Studies of the Quasi-Continuous Radiation

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References

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1. INTRODUCTION 1.1. Nuclear Structure at High Angular Momentum The nucleus is one of nature’s most interesting quantal many-body systems. It concentrates into a single system many types of behaviour found individually in other systems but which, in nuclei, interact with one another. These interactions lead to new phenomena which may have no parallel in other systems. Because the nucleus is a finite quantum system, its behaviour at a phase change is very different from that of ordinary matter. The finiteness of the nucleus is also manifest in the influence that just one nucleon may have on determining the nuclear properties, particularly the nuclear shape. A main direction of nuclear physics has thus become understanding this quantal many-body system and its relationship to other such systems. On one hand, many nuclear properties can be determined largely by the motion of a single nucleon, described within the framework of a shell model: the nucleons are assumed to move independently in their average potential, in close analogy to the atomic shell model. On the other hand, the nucleons, especially in heavier nuclei, often behave collectively, and there are close analogies to solid-state physics; this behavioural dichotomy is the central theme in studies of nuclei at high angular momentum. Much of the current emphasis involves the study of nuclei stressed so that they are different from ordinary stable nuclei. The stress of high angular momentum and temperature in the nucleus adds further dimensions; new coordinate axes upon which to study nuclear properties. The study of emergent properties, i.e., properties of the system that do not follow in any simple way from the fundamental interactions of the constituents, have proven fruitful in nuclei, and particularly at high angular momentum. The most obvious manifestations of an emergent property in nuclei are rotational excitations which give rise to band structure in the spectra of gammarays de-exciting from high angular momentum. Although only a model, it is productive to think of the total angular momentum of a nucleus as being separable into the angular momenta of individual particles, collective vibrations and rotations of the nucleus as a whole. With increasing angular momentum, inertial forces modify nuclear structure considerably; they weaken the pairing correlations, induce shape changes, and cause the transfer of angular momentum from collective modes to energetically more favoured individual particle modes. The interleaving of these effects leads to a rich variety of phenomena at high spin. At the ground-state in an even-N, even-Z nucleus, the individual nucleons are packed in pairs: each pair shares a common orbit but moves in opposite

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(time-reversed) directions so that the combined angular momentum of the pair is zero. In this way, the nucleons gain maximum advantage of the binding offered by the short-range nucleon-nucleon force. Because of the pairing field, it is usual to speak of quasiparticles rather than particles. Levels within an energy of the Fermi-level have a mixed particle-hole character. When increasing angular momentum is impressed upon a nucleus, the structure of its quantum states at the lowest temperature (yrast states) evolve. The liquid-drop aspects of nuclear dynamics suggest that the system should respond in the same way as would a rotating star; that is, it should become increasingly oblate: but the finiteness and quantum nature of the nucleus gives rise to new phenomena. For instance, even at low rotational frequency, the Coriolis force acting on nucleons occupying time-reversed orbits can be sufficiently strong to overcome their pairing energy and align both their angular momenta to the rotation axis. Depending on the quantum states of this pair, the nuclear shape may be polarized into a prolate or oblate shape or remain unchanged. The aligned angular momentum is a significant fraction of the total; therefore, after such an alignment event the nucleus will rotate more slowly, even though the total angular momentum is larger. The Coriolis stress on the nucleus is relieved temporarily by this process, but with further increases in angular momentum the stress will build up again until a second pair break and align to the rotation axis. The breaking of several pairs eventually causes the main part of the pairing to collapse.

1.2.

Basic Ideas

Many nuclei have a permanent quadrupole deformation in their ground states. Their low energy spectrum is dominated by rotational bands characterised by regular level spacings, and greatly enhanced probabilities for electric quadrupole (E2) decay between band members.* Rotations about an axis perpendicular to the symmetry axis of the nuclear shape, (labelled x below) are described by a rotational frequency, a moment of inertia, and an angular momentum In terms of the total angular momentum, I, and its projection on the symmetry axis, K, the component and frequency are given by:

*

Alternative models of nuclear structure have been developed to exploit dynamical symmetries. These include the Interacting Boson Model of F. Iachello and A. Arima, “The Interacting Boson Model” Cambridge Univ. Press, Cambridge (1987), and the Fermion Dynamical Symmetry Model of Guidry et al., Adv. Nucl. Phys. 21 (1994) 227.

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where E(I) refers to the energy of the state of spin I. At high spin Eq. 1.2 above reduces to

then

i.e., numerically, the rotational frequency is one half of the corresponding gamma-ray transition energy, and reduces to I. We define a kinematical moment of inertia by: The dynamical moment of inertia is defined by:

For rigid rotation, and will be equal; however this is seldom the case because of pairing and alignment gains. At low spin, nuclear moments of inertia are typically one half of the inertia of a corresponding rigid sphere. This is an effect of pairing that effectively causes the nucleus to behave as a superfluid. In a typical ground-state rotational band, initially shows a steady rise with increasing spin, but around spin 12 to it often rises sharply and may go through a series of zig-zags. The initial rise is mainly an effect induced by Coriolis stress. At the microscopic level, the nucleus generates collective spin by making small adjustments to the orbits of many particles, aligning a small part of each of their spins to the rotation axis, and in so doing, the pairing field is weakened, allowing the moment of inertia to rise. This phenomenon, known as Coriolis-anti-pairing is analogous to the quenching of superconductivity by a magnetic field, where rotational frequency is analogous to field strength. Because the nucleus is a finite system, the phase transition towards the normal state is gradual, rather than sharp. The zig-zags or backbends in the nuclear moment of inertia are due to band crossings. At the first such backbend, the ground-state rotational band is crossed by a band containing a broken pair whose spins are largely aligned with the rotation axis. For example, if the pair were in orbitals, as much as units of spin could be transferred from the collective rotation to the single-particle degrees of freedom at the backbend. Alignment events in rotating nuclei produce a wide range of effects. Nuclear shapes are determined in a delicate and complex balance among the pairing field, the shell structure, and the inertial forces of rotation. A single alignment event influences each of these factors and can therefore have dramatic effects on the nuclear shape. By the same token, the experimental determination of rotational frequencies at which alignments occur gives a very sensitive measure of these primary nuclear properties.

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1.2.1. Cranked Shell Model The development of the cranked shell model was an important mile-post in our progress to understanding nuclear structure. This model brings a new unity to individual nucleon and collective aspects of nuclear behaviour by the equal treatment of those quasiparticles aligned to the rotation axis, and those aligned to the mean nuclear field. This is achieved by effectively solving a shell model in a deformed potential, where the quasiparticle energies are allowed to respond to the rotation of the mean field by adding the cranking term The realization that rotational frequency rather than angular momentum was the key variable brought new insights. It is, for example, obvious in the cranked shell model that the excited two-quasiparticle rotational bands in neighbouring even nuclei will experience alignment events at a common rotational frequency, as is observed experimentally, rather than at a common spin or excitation energy. The transformation to the rotating frame, in which the quasiparticle trajectories are called Routhians, introduces a simplification, since in that frame, the energies of multi-quasiparticle configurations are the sums of their component single-quasiparticle structures (with possible small residual interactions). This additivity property of Routhians allows empirical treatments that predict the excitation energies of complex rotational structures in much the same way that empirical treatments of the spherical shell model relate excited configurations near closed shells. In Fig. 1.1, we show typical results of a cranked shell model calculation. Plotted are quasiparticle energies (Routhians, ) in the rotating frame as a function of rotational frequency. With cranking, the time-reversal degeneracy of the solutions to the deformed shell model (e.g., these could be Nilsson wavefunctions) are broken, and each level spilts into two components characterised by the quantum number usually designated signature (the eigenvalue of the operator ). Promoting a pair of protons, say from the levels -e and -f to the levels e and f, excites the two-quasiparticle band designated ef. This band has rotationally aligned components in it and its energy rapidly falls with increasing rotational frequency. It crosses the quasiparticle vacuum at as seen in Fig. 1.1. This corresponds to the first alignment event, or band crossing in The aligned spin of the configuration is To attain self-consistency in the cranked shell model, it is necessary to compute the pairing gap and nuclear shape at each rotational frequency. Various prescriptions have been devised to parameterise the nuclear shape. In the course of this article we will use the parameters and all of which are related to shape parameterisations, and which are inter-related. This allows us to retain a close connection to the original work and concepts.

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The inclusion of the macroscopic potential into the cranked shell model, a procedure known as Strutinski averaging, has proven to be remarkably successful. For example, the occurrence of superdeformed shapes along the yrast line in nuclei with certain Z and A values was predicted more than a decade before it was observed experimentally.

1.3. Electromagnetic Decays For convenience, we have collected some of the most useful relationships for calculating reduced transition probabilities from experimentally determined lifetimes, or more accurately, partial lifetimes, for gamma-decay. The electromagnetic properties of the nucleus are expressed by the following equations. The photon transition strength, T, defined as the decay probability per unit time by a given multipolarity is related to the partial mean-life by:

The reduced transition probabilities, B(M1) and B(E2) are obtained from:

where B(M1) is given in nuclear magnetons squared, and B(E2) is given in In a rotor model, the reduced transition probabilities are given as:

where is the electric quadrupole moment. The quantities and are gyromagnetic factors referring to the intrinsic configuration, and to the collective rotation respectively. The angular momentum coupling matrix element, or Clebsch-Gordan coefficient is denoted C.G.

1.4.

Organisation of the review

The development of the field of nuclear structure at high angular momentum closely follows the development of techniques and instrumentation in gamma-ray spectroscopy. The most rapid advances occurred in the mid1980’s with the development of large gamma-detector arrays, culminating in todays Gammasphere and Euroball arrays. Perhaps the development of a large

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gamma-ray tracking array will mark the next major advances. Section 2 of the review is dedicated to a history of the development of large arrays. To make the length of our review manageable, we have tried to concentrate on nuclear structure physics at the very highest angular momentum. In doing so (and not always consistently at that) we have either omitted or have shortchanged many interesting and important aspects of nuclear structure. The shapes of nuclei are central to any understanding, and this is discussed in Sect. 3. There is mounting evidence that some nuclei have rather stable triaxial deformation at high spin, but the best evidence for static triaxial deformation, namely rotational bands exploiting the “wobbling” degree of freedom, have not been firmly identified. The phenomenon of chiral symmetry, whilst not strictly pertaining to high spin, is sufficiently new and out of the ordinary that we have included it in Sect. 3. Also in this Section we discuss the behaviour of rotational bands built upon vibrational states: as the rotational frequency is increased, what happens to the phonons? Can rotational bands built on octupole-phonon states make the transition to static octupole shapes at sufficiently high spin? The study of superdeformation has been the most active subfield of high spin studies over the last fifteen years. This work has produced a lot of experimental data which we have not attempted to review in detail, rather we have concentrated on phenomena common to most superdeformed nuclei. Despite the tremendous experimental progress, there are still some conceptual problems, for example, is it more useful to think of superdeformation as special to the integer axis ratios, 2:1 of the deformed shape; or should we accept all axis ratios greater than the “normal” deformed value of 3:2, as manifestations of the same phenomena? This is discussed in Sect. 4. Many results coming out of the experimental effort of the last decade are even now only partially understood, for example identical bands, and the so-called staggering effect. In Sect. 5, we have reviewed topics concerning the limits to rotational behaviour; these include the collapse of static pairing, spin exhaustion or band termination, and shears bands, sometimes called magnetic rotors. These subjects retain some mysteries – for example why do shears bands, which are essentially multi-particle structures, exhibit such beautifully regular spectra characteristic of collective rotation? The last Section addresses nuclei at the highest spins sustainable as probed by studies of the quasi-continuous gamma radiation. In these areas, progress has been made despite the many difficulties. There are strong theoretical grounds to expect a transition to the extremely elongated, triaxial shapes first described by Jacobi. For all but the heaviest nuclei, this transition should occur at lower angular momentum than fission, and should be observable; yet it remains elusive. There is overwhelming evidence that the nuclear moment of inertia in-

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creases rapidly at very high spins, but whether this is due to a transfer of spin from the single-particle degrees of freedom to the collective rotation, i.e., an alignment gain, or whether it represents an increasing deformation has yet to be determined.

2.

2.1.

THE DEVELOPMENT OF LARGE ARRAYS FOR GAMMARAY SPECTROSCOPY Early Ideas-Selectivity, Sensitivity and Population Mechanisms

Between 1978 (when the field was last reviewed in Advances in Nuclear Physics [1]) and today, there have been enormous gains in the selectivity and sensitivity of gamma-ray experiments in studies of high angular momentum states. The best way to populate high spin states for studies by gamma-ray spectroscopy remains the heavy-ion fusion-evaporation reaction, but to extend studies to nuclei not accessible by fusion-evaporation, other mechanisms have been explored. These include: (a) deep-inelastic transfer, e.g., Ref. [2] and multi-nucleon transfer reactions, e.g., Ref. [3]; (b) Coulomb excitation, e.g., Ref. [4]; (c) relativistic fragmentation [5]; and (d) incomplete fusionevaporation, e.g., Ref. [6]. Generally speaking, these mechanisms allow exploration of a more neutron-rich species than can be reached by fusion-evaporation. A problem common to all these alternative mechanisms is that they do not populate the highest spins, and with the exception of Coulomb excitation, they populate many different final nuclei. Although techniques for product selection have improved greatly over the last decade, as we describe below, nevertheless, the cleaner the reaction at the outset, the greater will be the sensitivity of the experiment. Even with the cleanest fusion-evaporation reaction there are unavoidable problems at high spin in that the reaction cross-sections fragment into at least three, and generally many more, channels. This follows from the energy initially tied up in rotation – typically 40 MeV or more which must be made available to the nucleus simply to power the rotation resulting from large impact parameter (high angular momentum) collisions; this energy will ultimately appear as gamma radiation. But in small impact parameter (low angular momentum) collisions, the energy is thermal energy, and will be used to evaporate particles in the cooling process. Considering that the energy lost per evaporated particle (binding plus kinetic) generally lies in the range 8-15 MeV, the highest rotational energies will be separated from the lowest by typically three evaporated particles. Therefore, we need to select a particular residual nucleus of the reaction, and to select the interesting decays from high angular momen-

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tum states. It fallows from this discussion that the highest angular momentum is correlated to reaction channels with the fewest evaporated particles, and vice versa. After the evaporation process has finished, de-excitation of the nucleus is continued by a succession of gamma-decays; since each transition generally removes no more than two units of angular momentum, the number of gammarays in cascade, M, is closely related to the initial spin, and strongly correlated to the summed gamma-ray energy, H, carried by the cascade. Restricting ourselves to purely gamma-ray techniques, although there are many methods involving ancillary detectors discussed later in this Section, product and angular momentum selection are obtained by placing restrictions on the summed energy of coincident gamma radiation in an event, H, and on the number of coincident gamma-rays (or multiplicity), M. In an array of detectors, the M-coincident gamma-rays in an event will produce K-hits on the array, where K is generally denoted the “fold”. In an ideal array which covers all the solid angle with many individual detectors, the quantities K and M will be nearly equal. Deviations from this will be caused by scattering between detectors, by multiple hits where two or more gamma-rays hit one detector, and by failure to detect a gamma-ray which passes through the detector array without any interaction. Varying degrees of product selection can also be obtained by demanding the coincident detection of a characteristic X-ray, or a gammaray known to be emitted by the desired nucleus, or a characteristic isomeric decay. All of these methods are successful only if they can be executed with high efficiency. Summed gamma-ray energy was first measured by placing large (typically two 12”×12”) NaI(Tl) detectors close to the target in such a way as to intercept most of the gamma radiation whilst allowing some high resolution germanium detectors an uninterrupted view of the target. In the same era, states of high angular momentum were studied with multiplicity filtering, whereby only events triggering a higher than average number of hits, K, in a (small) array of NaI(Tl) detectors were accepted into the data stream. Multiplicity filtering suppresses events of very low multiplicity such as room background, induced beta-decays and Coulomb excitation of the target by the beam. Since the total solid angle covered by the multiplicity array was much less than the sphere, these techniques could not assign a multiplicity to a given event. Not until arrays covering most of the spherical solid angle, were developed was it possible to measure K on an event-by-event basis. A useful analysis of the mathematics of tiling the sphere with a few shapes, each subtending equal solid angle, is provided in Ref. [7]. The first large array to be completed (circa 1980) was the “Spin Spectrometer” comprising 72 NaI(Tl) detectors in a spherical geometry [8], c.f., Fig. 2.1. It was soon followed by

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the Darmstadt-Heidelberg “Crystal-Ball” comprising 162 NaI(Tl) detectors in a spherical geometry [9]. For the first time it was then possible to study the nuclear de-excitation cascade as a function of initial spin and temperature, and to measure the entry distribution as illustrated in Fig. 2.2 taken from Ref. [10]. By replacing elements of the ball with high-resolution germanium detectors, it was possible to take advantage of the selectivity provided by H and K measured event-by-event in the ball. Leading up to this period (late 1970’s) the advantages of Compton suppression of high resolution germanium detectors began to be considered. Techniques for doing this had been used much earlier and in a variety of applications, but the important realisation was how very poorly naked germanium detectors performed in coincidence experiments. For example, if the photopeak-to-total ratio were typically 0.15 in a single detector for a 1 MeV gammaray, then in a coincidence experiment the useful photopeak data would be only 2.25% of events recorded; nearly 98% of the recorded data is then not only worthless, but constitutes an unavoidable background! Compton suppression techniques typically raise the photopeak-to-total ratio to the range 0.5-0.65, making the photopeak efficiency in a coincidence experiment 25-40%. But how to realise Compton suppression? The NaI(Tl) suppressors of the late 70’s were large, and shielding necessary to prevent their direct view of the target tended to push the germanium detectors well back from the target. With only two or three suppressed germanium modules, the coincidence rates were so poor as to make experiments discouragingly difficult. Also in that era, the very large liquid- nitrogen Dewars on the germanium detectors (a relic from the days of Li-drifted detectors†) precluded efficient close packing, even if one laboratory would have had more than two or three such modules at that time. The time was also one of vigorous debate concerning how best to move forward with the field. One school argued that discrete-line spectroscopy had reached its limit and to push to higher spins it would be necessary to study the quasi-continuous radiation. Another school thought that with sufficiently powerful instrumentation, the quasi-continuum would to some extent be resolved into discrete lines. These differing philosophies ideally required differing instrumentation; for example the attainment of high-resolution is not an issue for quasi-continuum studies. Both schools agreed that the way forward must

†

Since a warm-up to room temperature was fatal to Li-drifted Ge detectors, they tended to be provided with large Dewars. HPGe detectors survive many thermal recycles, and can without risk be provided with small Dewars.

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combine the sensitivity afforded by Compton suppression with the selectivity provided by a system. If one were to marry a low-resolution inner-array with a Compton suppressed high-resolution outer array it was clear that NaI(Tl) would not work for the inner array either, since the thickness required for a useful shell of NaI(Tl) pushed the outer array too far back from the target. Operating in June 1980, the TESSA 1 array, which comprised five NaI(Tl)suppressed germanium detectors, was one of the first systems to obtain coincidence data with Compton suppression showing that large improvements in sensitivity could be obtained in the discrete-line spectrum. For the moment, the community agreed that the way forward was with high-resolution Comptonsuppressed arrays. It was clear that in the long-term, NaI(Tl) scintillator would simply not do the job because its relatively low density meant that efficient suppressors were much too large to pack around the target, c.f., Fig. 2.3. For-

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tunately, just at this time, a new scintillator, Bismuth Germanate (BGO), was becoming available. Although BGO has only about 1/10 of the light output of NaI(Tl), its higher density, and higher average Z gives one inch of BGO the same detection efficiency as roughly two to three inches of NaI(Tl). The TESSA 2 array operational in 1983 comprised six NaI(Tl) suppressed HPGe detectors, and an inner castle of 62 small BGO scintillation detectors. The Hera array at LBNL, completed in 1985, was the first array to operate with suppressors fabricated of BGO.

2.2. Second and Third Generation Arrays The technical developments needed to construct powerful arrays were: 1. Bismuth germanate crystals (BGO) large enough to allow construction of Compton shields. 2. Efficient high-purity germanium detectors (HPGe) with small, all-orientation Dewars.

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These developments guaranteed that several Compton-suppressed HPGe detectors could be close-packed around a target, and it all came together in the mid 1980’s when several 2nd generation arrays were built at laboratories around the world. These arrays had typically 16-20 HPGe detectors with 25% efficiency relative to NaI(Tl) [11]. Compton suppression was by BGO shields, and the high-resolution array was packed around an inner BGO ball or castle that provided sum-energy and multiplicity spectroscopy on an event-by-event basis. About a dozen 2nd generation arrays have been built around the world, and most are still operating in 2000. A view of the 8PI Spectrometer, from which many of our examples will be taken is shown in Fig. 2.4. The BGO inner arrays of the 2nd generation instruments provided very useful channel selection as illustrated in Fig. 2.5 taken from Ref. [12]. In favourable cases, the improvement in the signal-to-background ratio for the lines of interest are as much as a factor of three. Data acquisition with the 2nd generation arrays, although it presented a challenge, was solved employing techniques developed for the much larger detectors of particle physics. A good example is provided by the 8PI Spectrometer acquisition described in Ref. [13]. Data analysis was another matter. The greatly improved sensitivity now meant that level schemes could be studied in much more detail than before. The information in a matrix, typically containing the coincidence relationships between 300–500 transitions, exceeded the ability of the human brain to organize into a level scheme. This remains an incompletely resolved problem; however the development of computer-assisted level-scheme analysis has proved invaluable. These codes effectively increase the sensitivity of the instrumentation by making it possible to extract more information than would be otherwise possible, c.f., Ref. [14, 15]. An example of how well this works is shown in Fig. 2.6 taken from Ref. [15]. The discovery of superdeformed rotational bands [16] gave a boost to the field and stimulated ideas to go beyond the 2nd generation arrays. The Ga.Sp spectrometer at Legnaro [17] may be considered as the last and most developed of the 2nd generation arrays. It has a spherical inner ball of BGO detectors , and a suppressed array of forty, large volume HPGe detectors. We define 3rd generation arrays by their very high coverage of the sphere with HPGe detectors. To date, two such arrays have been built, namely, Gammasphere [18] and Euroball [19]. The Gammasphere array is depicted in Fig. 2.7, and more information may be found at the Gammasphere website [20].

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Fig. 2.7. General view of the gammasphere array.

Whereas the 2nd generation arrays had typically 5% or less coverage of the sphere with HPGe detectors, Gammasphere has ~45% coverage in HPGe. These arrays have taken advantage of developments in HPGe detector fabrication that made 75% efficiency (relative to NaI(Tl), c.f., Ref. [11]) routinely attainable. The result is that Gammasphere typically operates in four-fold coincidences in the suppressed HPGe array at rates comparable to two-fold coincidences in 2nd generation arrays. The net gain of the third generation arrays is difficult to quantify as it depends on the experiment and on the scientific goals. For most purposes, the sensitivity of an array may be expressed as the intensity of the weakest cascade of coincident gamma-rays that can be isolated and identified in an experiment, and here we follow the analysis given by Radford, Ref. [21]. Not including additional selection techniques, it depends on two factors: 1) the inherent resolving power; and 2) the statistical accuracy achieved. The inherent resolving power, R, increases with coincidence fold, f, as an exponent:

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is a characteristic both of the experiment and of the instrument:

where P is the peak-to -total ratio and is the ratio of the typical separation of gamma-ray energies between adjacent members of a cascade, and R is the resolution of the high-resolution array. It should be noted that in most applications where the residues emit gamma-rays whilst they are in motion, the resolution is not the 2 keV typical of large HPGe detectors, but is dominated by Doppler broadening effects which, depending on the recoil velocity and gamma-ray energy, may be more typically 5 or even 10 keV. If we consider a superdeformed band in a nucleus with then and the term in equation will be typically 4 or 5. The resolving power rapidly increases with increasing fold, but, as remarked earlier, the final sensitivity also depends on the statistical accuracy and this is falling with increasing fold. Detailed calculations for Gammasphere indicate that in most experiments the maximum sensitivity will be in fold four. If additional selection techniques are invoked, then the result should be scaled up by the gain in signal-to-background, and scaled down by the decreased statistical sensitivity. An illustration of how the resolving power increases with coincidence fold is shown in Fig. 2.8.

2.3. Close-packed Composite-Detectors Without gamma-ray tracking, which is discussed in the next Section, an array made solely of close-packed germanium detectors performs very poorly on account of crystal-to-crystal scattering and multiple hits. However, a few germanium detectors close-packed into small sub-assemblies are proving useful, provided the sub-assemblies are separated by Compton shields in the larger assembly. Such units offer the advantages of mechanical convenience, and a high photo-peak efficiency in add-back modes (where the energies of adjacent hits are added to reconstruct the photo-peak energy), together with excellent sensitivity to gamma-ray linear polarization. Two such systems have been developed, namely the CLOVER and the CLUSTER detectors. The CLOVER detector comprises four, tapered, HPGe detectors in a very close-packed arrangement, mounted in a common cryostat and surrounded by a single BGO Compton suppression shield was discussed in Ref. [22]. The CLOVER detectors have proven to be very flexible and

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several laboratories have acquired a few of them to construct small‡ spectrometers which can be used without specialised and rigid mechanical structures. Figure 2.9 shows a CLOVER detector at Oak Ridge National Laboratory. These small systems will be very effective in situations where the gamma-ray multiplicity is low, and where there is an externally derived trigger on which to select the interesting events. The CLUSTER detector comprises seven very close-packed HPGe detectors in a common cryostat surrounded by a single BGO suppression shield. The crystals are individually encapsulated, which offers some convenience for annealing and maintenance, c.f., Fig. 2.10 after Ref. [23]. Both the CLOVER and the CLUSTER detectors are employed in the Euroball array.

‡

These arrays are small only in comparison with Gammasphere, or with Euroball. In fact the close distance to the target gives them a total photo-peak efficiency comparable to, or even larger than the bigger instruments. This is bought at the price of poorer segmentation which makes these instruments unsuited to studies at the highest multiplicities.

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Segmented Detectors

A recent development in HPGe fabrication has been the ability to subdivide the outer surface electrode. Each zone, or segment, may be coupled to its own high-resolution pre-amplifier. Two-thirds of the Gammasphere detectors are segmented into two zones, a left, and a right sector, each with separate read-out, and the high-resolution signal is taken from a common inner contact. On replay, these outputs are compared and algorithms have been developed to ascribe the first interaction to be located either, a) in the left half, b) in the right half, or c) in the middle. This sub-division allows a finer Doppler correction to be made (than simply taking the average detector angle with respect to the beam) which improves the resolution for detectors positioned near 90° [24]. With technical improvements, it is now possible to sub-divide a detector’s surface into many zones both in depth and in azimuth. For example the

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CLOVER detectors in Euroball, described above, were originally conceived without segmentation but were eventually manufactured with the read-out on each detector segmented into four zones in azimuth; these zones are marked externally on the ORNL detector shown in Fig. 2.9. Such an arrangement has improved the sensitivity to linear polarization, as well as provided a much finer Doppler correction [25]. The success of electronic segmentation has stimulated the concept of gammaray tracking spectroscopy whereby the hit coordinates of each gamma-ray interaction in the crystal are deduced on an event-by-event basis through signal analysis of the outputs from each of the segments. Parallel developments with planar [26], and with co-axial detectors [27] are being pursued. Prototype co-axial detectors with as many as 36-zones have been manufactured and tested [28]. The goal of these developments is to build a array entirely of segmented HPGe crystals containing no BGO elements. With tracking, there is no need for Compton suppression; events in which a gamma-ray interacts and then escapes one crystal to strike a neighbour are tracked to the neighbouring crystal and the photopeak energy reconstructed [29]. Since the tracking array works by Compton acceptance, rather than by Compton rejection, its photopeak efficiency is approximately doubled over a conventional array. Furthermore, since the array would contain only HPGe and no BGO, there is a further factor of near two gain in the solid angle compared to, say, the Gammasphere array. These gains in efficiency translate to order-of-magnitude improvements in sensitivity for studies where the multiplicity is high, so that full advantage of the high detected coincidence fold can be taken, or in cases where the recoil velocity is very high, where a much finer Doppler correction will be achieved.

2.5. Ancillary Detectors

2.5.1. Light ion detectors Arrays optimised to detect proton and with near geometry have greatly improved the selectivity of experiments in which the reaction cross-section fragments into multiple proton and alpha evaporation channels. These arrays typically have a granularity of 40-90 detectors. Silicon diodes, and CsI(Tl) scintillators [30, 31] have been used in this application. Pulseshape discrimination between protons and with CsI(Tl) has proven very effective. The detection efficiency needs to be high, so that ideally there is sufficient experimental leverage to distinguish between, for example, a threeand a four-proton evaporation. The detectors must not present any significant absorption to the gammarays of interest, nor significantly degrade the peak-to-total response. In addi-

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tion to selecting a channel by directly gating on the appropriate numbers of identified charged particles, these detectors allow a fine tuning of the Doppler correction derived from the residue’s velocity vector [32]. This can be a large gain for a light residue, say where the evaporation of one or more causes a major perturbation to the recoil vector. An additional quantity provided by the light-ion array is the sum-energy of all evaporated charged-particle kinetic energies. This quantity compared with the gamma-ray sum-energy, c.f., Fig. 2.11, gives another measure of the reaction channel and provides a discriminant upon which to reject events where evaporated particles were missed by the array [33]. Further information on light-ion detector arrays may be found linked to the Gammasphere website [20].

2.5.2.

Neutron detector arrays

One method of selecting the most neutron-deficient residues in a fusion reaction is to take coincidences with evaporated neutrons. To do this efficiently requires a large array of neutron detectors placed at forward angles in order to take advantage of the kinematic focusing. Selection of one neutron in the array is straightforward, but selection of two or more coincident neutrons is more problematic since it is difficult to distinguish between two-neutron events and scattered one-neutron events, particularly if the real two-neutron events are orders of magnitude fewer. Experience has shown that better results are obtained by removing neutrons from the compound system in the form of rather than by direct assault. Generally speaking, as the proton drip-line is approached, the cross-section for the evaporation of to a given residue is much higher than any reaction involving the evaporation of neutrons to the same residue. Nevertheless, physics may dictate that the residue of interest cannot be reached without evaporation of neutrons. In such cases a neutron array may offer the best hope. Large arrays have recently been commissioned at both Gammasphere [20], and at Euroball [34]. The neutron array designed to operate with Gammasphere comprises thirty liquid-scintillation detectors whose hexagonal, tapered shape closely matches Gammasphere modules; the neutron detectors then replace the thirty Gammasphere modules situated in the first five rings at forward angles. Pulse-shape analysis, and time-of-flight techniques give good neutron-gamma discrimination. Because the array was engineered to match the Gammasphere modules, it is close-packed, and has a relatively high efficiency for detecting 2-neutron events. For example, in a typical reaction with kinematic focussing, the efficiency for detecting a single neutron is approximately 30%, and for two neutrons 9%. Further details may be found linked to the Gammasphere website [20].

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Heavy ion detectors

In the gamma-ray spectroscopy of binary reactions with heavy projectile and target nuclei, there are large Doppler shifts. When applying a Doppler correction, there can be ambiguities concerning which gamma-rays originate from which partner. Heavy ion detectors have been used for many years to address these problems, and to enhance signal-to-background ratios. In Coulomb excitation studies, for example, systems as simple as an annular silicon surface barrier detector to as complex as large position-sensitive parallel-plate avalanche detector have been used [35]. One of the most developed systems in present use is the CHICO detector built specifically to operate inside of the Gammasphere array [36]. It comprises a position-sensitive system with twenty parallel-plate avalanche detectors. The scattering angles are measured to approximately one degree in polar angle, and nine degrees in azimuth; in conjunction with 500 psec timing resolution, the mass resolution is typically

2.5.4.

Heavy residue separators and analysers

A number of residue separators capable of being used with gammaarrays have been developed. These may be vacuum instruments (e,g., the FMA at Argonne National Laboratory [37] and the RMS at Oak Ridge National Laboratory [38]), or gas-filled such as the RITU [39] separator at Jyvaskyla, and the BGS [40] (Berkeley Gas-filled Separator). The vacuum instruments provide unit mass resolution of approximately 1/200, and may truly be called analysers. For light residues at high energies, detectors at the focal plane may also provide some Z identification. The gas-filled instruments simply collect residues without any useful mass resolution beyond separating fusion residues from the beam (and projectile-like fragments) and from fission fragments: they have a much higher efficiency (approaching 100% in some cases), than the vacuum instruments which typically operate in the range 5-10%. The technique of “recoil tagging” may be applied to any residue separator or analyser. For those residues that have a delayed alpha or delayed proton in their decay chain, recoil tagging extends the sensitivity for detection of gamma-rays down to very low limits. This technique uses high pixelation detectors in the focal plane to track the arrival of the residue, and its subsequent decay. The prompt, in-beam gamma-rays associated with a particular residue are tagged by a characteristic proton or alpha decay which may occur many milli-seconds later. The potential true-to-random problem is solved by demanding a tight position-correlation between the recoil implantation signal, and the subsequent decay signal ( proton, or fission-fragment). A

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recent review of the technique may be found in Ref. [41]. These techniques have had considerable success, perhaps most spectacularly in the observation of a rotational band in [42]. Evidently this nucleus can be excited to spin without fissioning, and provides a remarkable example of the importance of shell correction energies to the macroscopic liquid drop model.

3. 3.1.

ROTATIONS, PARTICLE ALIGNMENTS AND THE NUCLEAR SHAPE Introduction

Nuclear shapes are determined in a delicate and complex balance between the pairing field, the shell structure and the inertial forces of rotation. In the Lund convention (c.f., Fig. 3.1), the various shapes of order that a nucleus can adopt, and their relationship to the rotational axis, are characterised by the parameters and In this Section, we are concerned with reflection symmetric shapes, but nuclei can also adopt shapes that violate this symmetry, e.g., octupole shapes, and these will be discussed later in this Section. The evolution of nuclear shapes with rotational frequency can be represented in a plane described by the coodinates and as shown in Fig 3.1. The semi-axes of the corresponding ellipsoidal figure, for small are given by:

k = l , 2, 3. A sector spanning describes all shapes with whereby various orientations can be obtained by a suitable choice of the axes. For that are integral multiples of 60°, we obtain axially symmetric shapes. In Fig. 3.1, and -120° yield prolate shapes; yield oblate shapes. All other correspond to triaxial shapes. In a quantum mechanical treatment, there can be no collective rotation about an axis of symmetry, and these are labelled “single-particle” in Fig. 3.1 corresponding to and -120°.

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In Fig. 3.1 as we proceed clockwise from

we encounter:

1. Non-collective oblate shapes, in which the total spin is generated by the alignment of quasiparticle spins to the oblate symmetry axis. An example is provided by the terminating bands discussed in Sect. 5. The role that non-collective structures might play at very high spins is discussed in Sect. 6.2. 2. Prolate collective rotors, which are the most prevalent type of collective rotation at low spins. 3. Triaxial collective rotors, to be discussed in this Section. 4. Oblate collective rotors. A few examples of this type of behaviour are known, but none at very high spin[43, 44, 45]. At a typical deformation, say 3:2 axis ratio, the rigid-body moment of inertia for oblate collective rotation is much smaller than for prolate collective or for noncollective oblate rotation. Without some very strong compensating shellenergy, these structures can not compete at high spin. Evidence for collective oblate rotation to spin in the nucleus has been published recently[46]. 5. Non-collective prolate shapes, in which the total spin is generated by the alignment of quasiparticle spins to the prolate symmetry axis. Examples are provided by the sequences of high-K isomeric states seen in the mass A ~180 region and discussed later in this Section. For rigid rotation, the non-collective prolate shape has the smallest moment of inertia in the Lund diagram, and it is remarkable that these states can lie so close to, or on the yrast line for spins up to Again this illustrates the vital importance of the shell energy.

We then find in nuclei every possible mode of carrying angular momentum depicted here, and in some cases, several modes may be found to be co-existing in the same energy and spin domain within the same nucleus; a phenomonen known as “shape co-existence”. Particles with large orbital angular momentum alignable to the rotation axis, for example, prefer particular nuclear shapes where their energy is minimised. As the first particles are placed in a shell, they tend to drive the nucleus towards more positive whereas, at the end of the shell they drive towards more negative Depending on the softness of the core, these effects may induce a sizeable triaxial deformation, or may help to stabilise a particular shape in an otherwise weakly defined potential. The nuclear system can be remarkably sensitive to polarising effects; the outcome may be decided by just one nucleon.

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3.2.1.

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Triaxial Shapes

Low spin

The shape of a nucleus in its intrinsic frame can not be determined by experiment, but it may be inferred from experimental data with varying degrees of model-dependence. In a series of experiments over many years, the Rochester group and collaborators have analysed E2 matrix elements by the model-independent sum-rule method (a recent example and futher references are provided in Ref. [47]). These techniques are amongst the most direct ways of determining intrinsic shapes, and, for example in principal, they can distinguish between static and Reviewing this, and other information, we conclude that there are regions of the periodic table where nuclei if not actually triaxial, are at least soft to deformation in the parameter, and this is evinced by low lying collective and rotational bands built upon such states. Nevertheless, static has not been convincingly identified at low spins and if it occurs at all, it is a high spin phenomenon where the macroscopic shape-defining potential has been softened, and where quasiparticles aligned to the rotation axis can exert their influence. The effects of aligned quasiparticles[48] are summarised in Fig 3.2. The shape-driving force is largest where the Routhians have their greatest slope, and as seen in Fig 3.2, that is towards positive at the beginning of the shell, and towards negative at the end. These plots demonstrate that the one-quasiparticle signature splitting for a particular orbital tends to be largest whenever the nucleus adopts the shape most prefered by that orbital. Considerable effort has gone into the analysis of signature splitting effects as a probe of triaxiality. An early result is shown in Fig 3.3, taken from Ref. [49]. The series through represent decreasing softness in deformation. The disappearance of signature splitting in the [523] band in after the AB-crossing indicates that the core has been polarised to a less favourable for that orbital. This would be expected, since the mid-shell [523] orbital prefers negative whereas the AB crossing involves orbitals at the beginning of the shell, which drive the core to positive

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By varying whilst holding the other parameters fixed, cranked shellmodel, or particle-rotor calculations§ are capable of reproducing virtually any observed signature splitting, however, the signature splitting may have some dependence on these other parameters which can not be separated from the effects of Other indicators of triaxiality must be invoked to obtain a consistent picture. Experimental indicators in addition to signature splitting are: a) crossing frequencies, and interaction strengths; b) B(M1) and B(E2)values between signature partner bands; and c) relative B(E2)-values between the odd and adjacent even nuclei. Such effects have been examined by several authors, for example Ref. [51, 52, 53, 54]. In for example[51], when the [523]7/2 orbital is occupied, the nucleus adopts a triaxial shape consistent with ~ –20°. This is in agreement with the signature splitting and with the measured electromagnetic properties, however, these indicators can not distinguish between static and For example, according to Ref. [55], for the ratio of the transition moments between signature partner bands, averaged over both signatures is related to the by:

where is an effective K-value for the quasiparticle.The results are summarised in Fig 3.4 and Fig. 3.5. The relationship between the signature splitting of the Routhians and the staggering in B(M1)-values between signature partner bands is an indicator of triaxiality[55]. If A denotes the signature averaged B(M1)-oscillation and B denotes the factor:

then in a cranking model it can be shown that A and B are equal for axial symmetry. In a case like where the B-term is much bigger than A-term, then triaxiality must be invoked with for and for (the situation in ). This same kind of analysis has also been applied to and and The analysis of suggests that the nucleus may have two distinct triaxial deformations according to the signature. This might be expected from Fig. 3.2 which shows that the shape-driving effects are very different between the signatures. The B(E2)-values as a function of spin within a rotationally aligned band are sensitive to triaxiality in the core[54]. Furthermore, the relative B(E2)values of the yrast aligned band, and the adjacent even core, compared at the §

A particle-rotor model was first discussed by Bohr and Mottelson[50]

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same core angular momentum, have a very strong dependence on triaxiality. For example, for an orbital, the case measured in Ref. [54], one must compare with The data for show an effect indicating that However, a difficulty with this method is that the analysis assumes that the aligned quasiparticle is a passive spectator to the triaxiality of the core; in reality the quasiparticle might polarise the core thereby invalidating the analysis.

3.2.2. High spin Calculations predicting triaxial shapes with large deformation [56, 57, 58] stimulated searches for these exotic shapes in nuclei. It should be noted that these shapes are caused by shell structure, and have a very different origin from the Jacobi shapes discussed in Sect. 6.2.2. Jacobi shapes are also highly deformed, and triaxial, but they arise purely in the liquid drop properties of nuclei, and would exist without the benefit of shell structure.

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Superdeformed triaxial bands have been claimed to have been seen in and triaxial bands with enhanced deformation have been reported in the mass A~ 130 region, and More recently, evidence for stable triaxial bands was found in These bands have small and nearly constant dynamical moments of inertia, typically over a wide range of rotational frequency, say 0.3 to 0.9 MeV, (corresponding to spins in the range 20-40 ). The small moments of inertia are characteristic of triaxial shapes, but not uniquely so. However, in these nuclei the calculated potential energy surfaces (PES) Fig. 3.6, show very deep well-defined minima for For example, the PES for a fixed configuration in is 2 MeV lower than the energy for the axially symmetric shape, and at least in the calculation, the triaxial minimum is stable over the whole spin range for which the bands are observed experimentally.

For large, static theory predicts the so-called “wobbling mode” which gives rise to multiple bands with identical moments of inertia, as first discussed by Bohr and Mottelson[64], and more recently by Shimizu and Matsuzaki[65]. The multiple bands observed in may be the best evidence to date for this long-standing prediction.

3.2.3. Chiral Doubling The possibility for chirality in nuclei occurs if there were situations where three distinguishable angular momentum vectors form an orthogonal system: we can then assign left-handed and right-handed couplings. Such a situation

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could occur in a nucleus having a static triaxial deformation with From the viewpoint of tilted axis cranking[67], the triaxiality allows ‘aplanar’ solutions where the total angular momentum, J, does not lie in any principal plane (defined by any two of the principal axes) [68]. In Ref. [68] a simple system involving an proton particle and neutron hole coupled to a triaxially deformed core was studied. The contribution of particles to the total angular momentum lies in the principal plane formed by the short and intermediate axes (giving the maximal overlap between their toroidal density distribution and the core). Whereas the hole contribution lies in the principal plane defined by the long and intermediate axes (the holes have a dumbbell shaped density distribution). The angular momentum from core rotation lies perpendicular to both the particle and to the hole contributions along the principal axis where the moment of inertia is largest; this will be the intermediate axis for irrotational-like flow[69]. This situation is illustrated in Fig. 3.7. At the bandhead, where the contribution from core-rotation is small, J will lie in the principal plane defined by the 1–3 axes. With increasing core rotation J will be gradually tilted towards the intermediate axis, 2. Detailed calculations have been performed by Dimitrov et al.[70].

The existence of this aplanar solution gives rise to a pair of identical bands with opposite intrinsic chirality (handedness). This is a new kind of

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intrinsic symmetry breaking in nuclei.¶ The occurrence of the two identical bands has been given the name ‘chiral doubling’ in analogy to ‘parity doubling’ for reflection asymmetric (octupole) shapes. An experimental signature for such exotic behaviour would be a pair of regular bands with the same parity which gradually merged into the degenerate chiral doublet with increasing spin; on further increase in spin the bands should pull apart again. Any interband transitions must disappear after aplanar geometry is established, since the two degenerate bands will correspond to very different orientations of the angular momentum which can not be connected by low multi-polarity operators (a phenomenon similar in origin to K– forbiddeness). A possible case has been seen in [60] where two positive parity bands follow the general pattern described above. However, the bands do not form a good identical pair which may reflect the fact that the triaxial core has insufficient deformation or is too soft to maintain a stable shape. A negative result has also been reported in [73]. More recently, studies of bands built on configurations in the isotones of i.e., in Cs, La, and Pm (N = 75) have observed new partners of the yrast bands that might be candidates for chiral symmetry-breaking doublets [74]. These doublets are not degenerate, and the authors interpret this to indicate the presence of soft chiral vibrations, rather than chiral rotation.

3.3.

High-K States

The quantum number K is defined to be the projection of the total angular momentum on the shape symmetry axis of the nucleus. In the simplest description, a resultant K arises from the individual nucleons by summing components of angular momentum, along the shape symmetry axis, In certain regions of the periodic table, it is economic to build-up angular momentum by breaking up nucleon pairs in orbits and maximally aligning their spins to the shape symmetry axis. This corresponds to the situation in Fig. 3.1. Neutrons and protons generally participate equally in nuclear excitations, therefore generating a lot of angular momentum by this mechanism requires that both protons and neutrons occupy favoured orbitals. These are the top orbitals of a high-j shell where for prolate shapes, the are high. Nuclei near A~180 are in one such region where neutrons are closingout the shell, and protons the In this region, high-K states with up to ten quasiparticles are known, and multi-quasiparticle states lie close to the yrast line, or are yrast themselves. ¶

Bohr and Mottelson considered the case where the nucleon-nucleon interactions themselves violate time-reversal symmetry (which is related to chiral symmetry)[71]

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High-K states are frequently isomeric. The K-selection rule on gammadecay, namely where is the multi-polarity, hinders the decay as discussed below. Also, a high-K level may be so favoured that there are no levels below it with a spin within such a level is called a “spin trap”. Clas[75], and the sic examples are the (5.5 hr) two-quasiparticle level in (31 yr) four-quasiparticle level in [76]. A more recent example is [77] for which bandheads up to are proposed. A semi-empirical method of predicting the configurations, energies, spins and parities of multiquasiparticle states has been developed by Jain et al.[78]. A summary of the A ~180 region is shown in Fig. 3.8[79]. High-K states are important probes of nuclear structure. The collective moment of inertia of a rotational band built on a high-K bandhead gives a measure of the pairing strength. By examining many such bands, one can build up a picture of how the pairing strenth collapses with increasing numbers of broken pairs, or one might examine the effect of specific orbitals on the pairing strength[80]. Precise analysis is not possible because first, the collective moment of inertia can not be derived unambiguously from experiment on account of Coriolis effects and second, the relationship of the moment of inertia to the pairing gaps and is generally taken from the Migdal formula[81], which is only an approximation. In the example nucleus the rigid rotor value for the moment of inertia is never reached, even after breaking three neutron and two proton pairs to make the ten-quasiparticle state. In effect, the moment of inertia appears to saturate at a value well short of the rigid body value. The authors of Ref. [77] suggest that their observed value is the one appropriate to zero static pairing, and that either dynamical pairing effects are reducing the moment of inertia, or alternatively, one can argue[82] that the rigid value does not have to be satisfied in any particular nucleus, but is only satisfied on average. In fact, in Ref. [82], it is shown that a more exact calculation of the moment of inertia with no pairing gives values greater than rigid at the beginning of the shell, and less than rigid at the end. Another way in which a high-K state probes nuclear structure derives from comparing its magnetic moment with the magnetic dipole, (M1), transition strengths in the rotational bands built on top. This allows one to extract the collective g-factor, and the single-particle g-factor, as separate quantities[83]. As pointed out in Ref. [83], the collective g-factor is just the fractional contribution of protons to the total collective angular momentum, (protons plus neutrons), and as such is sensitive to the collective moments of inertia:

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which will in turn depend on the pairing strengths. The one-quasiparticle gfactors summarised in Ref. [83] show a clear effect due to the blocking of the pairing. The average in odd-proton nuclei (protons blocked) is 0.395, versus the average in odd-neutron nuclei (neutrons blocked), 0.260. There are not enough data to do a study of multi-particle effects so as to dissect the collapse of the pairing field by breaking an increasing number of pairs. In the nucleus the isomer is believed to be a pure two-quasiproton configuration, and measurements give [84]. In the nucleus a similar analysis gives for the state, which is believed to be a pure three-quasineutron state[85]. These values appear to be in the right general area, but much more data needs to be taken.

3.3.1.

The decay of High-K states, and the question of K-purity

A further topic of interest in the study of high-K states is the question of K-purity, or lack of it. The nuclei and exhibit very strong K-hindrance in their decay; for example a factor ~20-30 per degree of K-forbiddeness, where is the multipolarity of the transition. These data are summarised in Ref. [79]. But there are also cases where the K-hindrance is less than a factor of two per degree of K-forbiddeness, as for example in and in [79, 86]. The lack of K-hindrance in these Kforbidden decays has been associated with Coriolis-induced K-mixing of lowK components into the high-K band[87, 88, 89]; alternatively, it has been attributed to tunneling through the potential-barrier in [86, 90, 91]. In tilted-axis cranking theory[67], there can be high-K components in the lowK band. In the mass A ~180 region, bands with which comprise the so-called tilted bands (or t-bands), can be mixed into the usual rotational-aligned (s-bands) with as discussed in Ref. [92]. It remains to be seen which of the proposed mechanisms best characterises the decay of the high-K bands.

3.4. Rotational Bands Built on Vibrational Intrinsic States

3.4.1. Classifications The low-order shape vibrations of the nuclear surface are classified as:

1. Beta Vibrations. Parity positive: K=0, in which the nuclear shape vibrates in the deformation coordinate, whilst maintaining axial symmetry. 2. Gamma Vibrations. Parity positive: K=2, in which the nuclear shape vibrates in the deformation coordinate, thereby departing from axial

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symmetry. 3. Octupole Vibrations. Parity negative: K=0,1,2 and 3, in which the nu-

clear shape vibrates in thereby deviating from a reflectionsymmetric shape in the intrinsic frame. In all these cases, the equilibrium deformation may be zero or have some finite value. The energy and collectivity of the lowest vibrational state is a measure of the softness of the potential against deformations in that coordinate. Furthermore, if these states are to have a well-defined meaning as a shape vibration, then they must lie significantly lower than twice the pairing-gap energy, where two-quasiparticle states appear. The identification of rotational bands built on vibrational states is generally based on the measurement of gamma-ray branching ratios at low spin. In lowest order, these should follow the squares of appropriate Clebsch-Gordan coefficients, often called the Alaga rules. There are large differences in the branching ratios expected for transitions from a level of say spin in an excited band to the levels of spin and in the ground-state rotational band, according to whether the excited state has the character of a beta- or gamma-vibration. Similarly, the various K-components of the octupole bands can be distinguished by their branching ratios to the ground-state band. In the case of a beta-vibration, the presence of enhanced E0 components seen in the conversion electron spectrum for the spin J to J transitions is an unmistakeable signature.

3.4.2. Mixing of Vibrational Modes The simplicity of this picture is marred by Coriolis mixing between levels of the same spin in the beta and ground-bands, between the gamma and ground-bands, and between the K-components of the octupole multiplet. These interactions are very strong, and the unperturbed levels need not lie close by one another in order to experience the perturbation. By way of contrast, the interaction between the beta and gamma bands is not particularly strong, but levels of the same spin in these bands frequently lie very close to one another, and may mix strongly. Finally, at higher excitation, there will be mixing with two-quasiparticle, and particle-hole states. In lowest order, the two-band mixings, beta-ground, and gamma-ground can be described in a formalism due to Mikhailov[93]. There are numerous examples where this treatment has proven to be remarkably accurate. Coriolis mixing of the K-components of the octupole vibrator are well understood at lower spins. In many examples, one sees that the mixing of the components corresponds with a state in which the phonon spin of aligns to the rotation

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axis, a result discussed by Vogel[94].

3.4.3. Alignment of the Phonon Spin to the Rotation Axis Although there are interesting aspects of vibrational states at low angular momentum, we wish to concentrate on high angular momentum properties. A question one might ask in this context is: “what happens to phonons when the intrinsic frame is rotated at high frequency?” The cranked random phase approximation (RPA theory) is an ideal tool for this discussion, since the effects of rotation on the quasiparticle components of the phonon are treated selfconsistently. Some recent references for the theory may be found in Ref. [65] for beta and gamma vibrations, and in Ref. [95] for octupole vibrations. For the octupole case, in the lowest band (usually ), the nucleus aligns of angular momentum to the rotation axis by tilting the orbits of easilly aligned orbitals involved in the phonon. Since many such orbitals are invoved, a resultant can be attained by many small adjustments. This may be termed a phonon-alignment process. The aligment of octupole phonons to the rotation axis is very clear in experimental data; examples are provided by uranium and plutonium isotopes shown in Fig. 3.9 from Ref. [96], and Fig. 3.10 from Ref. [97]. The details of the alignment process, such as the rate of increase

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of the aligned spin and its final value, vary from case to case. There is a dependence on the Fermi level, since that effects the mix of available orbitals. Also, the K = 0 band need not be lowest; for example, in the K = 1 band is calculated to lie lowest[98]. In nuclei with neutron number N=90, c.f., Fig. 3.11 from Ref. [99], alignment of the octupole phonon happens at near zero rotational frequency, and the full value is reached immediately. Quadrupole phonons cannot align to the rotation axis. One may think of this in the following way: octupole phonons have components with K=0,1,2, and 3 and can therefore point in any direction governed by the relative amplitudes of the various K-values: quadrupole phonons have only the K-component, K=0 (beta phonons), and K=2 (gamma phonons), and can therefore not be realigned. The effect of the Coriolis interaction acting in 2nd order on quadrupole phonons is to renormalise the moment of inertia of rotational bands built on them. Of course, it is possible to express the “alignment” of beta or gamma bands to the rotation axis as shown in Fig. 3.11 for the N=90 nuclei, and in Fig. 3.12 for the gamma-vibrational bands in [100] and in [101]. The near-constant difference in the moments of inertia of the gamma- and ground-bands in and in (top panels of Fig. 3.12) translates into a constantly increasing “alignment gain”(bottom panels of Fig. 3.12), which contrasts with the behaviour of the octupole bands which asymptote to a constant alignment gain.

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In the positive signature of the gamma-band has a strong alignment gain due to a crossing with the lowest aligned two-quasiparticle state as discussed later.

3.4.4.

Breakdown of Phonons at High Rotational Frequencies

Octupole phonons: The system can be cranked at progressively higher frequency until the Routhian for the lowest negative-parity two-quasiparticle band crosses that for the octupole phonon, cf. Fig. 3.13, taken from Ref. [96]. One can view this as the frequency for which a particular pair of nucleons coupled to spin (where they have taken advantage of the octupole correlation energy) breakup and align most of their angular momentum to the rotation axis. It is interesting to note that the rotational alignment of the phonon raises the crossing frequency considerably, as can be seen in Fig. 3.13. An example for a realistic case is shown in Fig. 3.9. In the figure, the size of the dots is meant to convey a measure of the octupole collectivity. The lowest signature of the octupole band is seen to be suddenly broken down at and becomes an aligned two-quasiparticle state.

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At normal deformations, for example in the region, the alignable orbital is always an intruder, such as or and the loss of this critical orbital from the octupole correlation substantially weakens the octupole collectivity. This process whereby the collectivity of the phonon is progressively weakened by crossings with two-quasiparticle levels, which release high-j orbitals from the octupole correlation, has been termed phonon- breakdown by Nakatsukasa[96]. In the context of superdeformation, we note that for a 2:1 axis ratio, the balance of positive and negative parity orbitals is roughly equal, whereas at normal deformation, opposite parity within a shell is provided only by the intruder orbitals. This has consequences for the octupole bands. Firstly, the presence of roughly equal numbers of orbitals with either parity means that there are more pairs participating in the octupole correlations, making them stronger. Secondly, the superdeformed octupole bands should be more robust against phonon breakdown at high spin in the sense that there are pairs in the octupole correlation that do not involve an intruder orbital, and are therefore not so easilly aligned. These considerations explain why the octupole vibrational bands in the superdeformed well of some Hg isotopes lie at very low excitation energy, and why they persist to high angular momentum, as discussed in Sect. 4.2.1[173]. Quadrupole phonons The breakdown of quadrupole phonons happens for the same reasons as discussed above for the octupole case. An example taken from is shown in Fig. 3.14 Ref. [101] which gives calculated Routhians for the collective vibrational bands with signature and the unperturbed two-quasiparticle Routhians are also depicted. The beta-band interacts strongly with both the lowest neutron and proton two-quasiparticle bands, whereas interactions with the gamma-band are small. In the calculation, after the crossing at the beta-band loses its collectivity rapidly by successive crossings. In contrast, both signatures of the gamma-band retain there character to high frequency with little alignment. These predictions agree well with the observed properties of the gammaband shown in Fig. 3.12, which shows that there are no large signature splitting effects. By way of contrast, for the gamma-band in the same RPA theory predicts that only the positive signature should be crossed at whereas the negative signature is calculated to extend beyond without any crossing. This is in very good agreement with what is observed, c.f., Fig. 3.12.

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3.4.5. Reduced Transition Probabilities Octupole bands The most accessible reduced transition probabilities come from measurements of gamma-ray branching ratios, particularly if we assume that the in-band moments are given by the rigid rotor model. The ratios:

can be derived directly from experiment without any model assumptions; here is in the octupole bands,(K=0 unless otherwise stated), and and are in the ground-state band. For large J, the ratio of the Clebsch-Gordan coefficients asymptotes to unity and hence so does the expression. Coriolis mixing of the K-components in the octupole multiplet modifies these matrix elements (c.f., Ref. [93]):

where is understood to be J(J+1), and is the effective transition dipole moment. The parameter, q, is zero in the event of no mixing. In fact Shimizu and Nakatsakasa have shown how results of cranked-RPA theory can be cast into the form of the generalised intensity relationships so that the parameter, q, can be given a microscopic interpretation [102]. There are very few cases where these B(E1)-values, or ratios of B(E1)’s have been measured to high spin. In the cases of [101] and [98] the measured values are close to unity, indicating little mixing. By contrast, in [100], the values for are up to ~50 × stronger than Such large effects would appear to be outside the range of validity of the generalised intensity relationships, which are derived in first-order perturbation theory, nevertheless, they do explain the data shown in Fig. 3.15. In nuclei absolute from K=0 octupole bands to the ground-band generally increase with spin much more rapidly than can be explained by cranked RPA theory. For example, in the nuclei [101], [103] and [100], these increase typically by a factor of 5 to 10 times between spin and spin In the plutonium isotopes there is a similar rise between spin [97]. The behaviour of these nuclei makes a strong contrast to with the nuclei reviewed by Butler and Nazerewicz[104], namely and which uniformly show constant B(E1)-values over the spin range to

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The existence of large B(E1)-values between octupole bands and the ground state band has been taken to be evidence for strong octupole correlations ever since it was first discussed in 1957 by Bohr and Mottelson[105]. Recent summaries of these effects may be found in Ref. [104], and in Ref. [106]. It has also been suggested that large B(E1)-values may be a signature of static octupole deformation [104, 106]. That being the case, it is tempting to speculate, as do Wiedinhover et al.[97], that an increasing B(E1)-value with spin indicates a shifting from octupole vibration to static octupole deformation. Static versus vibrational octupoles It is important to understand whether or not in nuclei the minimum in the the potential versus octupole deformation is always centred at or can be centred at a finite value,with a barrier between ± Certainly in the case of molecules, reflection-asymmetric shapes are commonplace, and the rotational spectra of such molecules comprises alternating levels of positive and negative parity, e.g., Ref. [106]. The wavefunction of the rotating octupole state is a linear combination of the two signs of the deformation (in order to have good parity) with a relative phase that depends on Jolos and von Brentano[107] have suggested that the height of the potential barrier between the two octupole shapes will increase with increasing angular momentum, thereby decreasing the probabilty of tunneling, so that the system will stabilise with a static deformation. Numerous attempts to develop a signature for the onset of static octupole deformation have been explored. For example the rotational-like interleaving of positive and negative parity states[104]. More recently, Cocks et al.[108], have argued that the observation of zero aligned spin between the positive and negative parity bands may indicate a static deformation. By their reckoning, the radon isotopes would be vibrational with approximately alignment, whereas and could have a permanent octupole deformation since the alignment is smaller than and close to zero for

4.

SUPERDEFORMATION

First observed in [109], through the discovery of fission isomerism and then understood a few years later as a second minimum in the fission barrier, superdeformation is now established in several nuclear mass regions ranging from A ~ 40 through A ~240. Following the observation of fission isomers, there was a gap of about 25 years before the first high spin superdeformed nucleus was discovered[16]: a wait determined by the need for multi

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detector high-resolution gamma-ray detector arrays (c.f., Sect. 2). The rapid experimental progress on high spin superdeformation has been presented in several reviews[110, 111], and reference to the theoretical developments can be found in[112, 113]. A summary of experimental and theoretical developments of fission isomers is given in[114]. Here, and in the following, our aim is not to provide a full review of superdeformation, but to concentrate on the phenomena that arise from characteristics common to superdeformed bands. Before discussing their rotational properties it will be instructive to consider the existence of superdeformed nuclei in terms of their shell structure. By doing so, we will be able to draw important distinctions between superdeformed and normal deformed nuclei.

4.1. The Existence and Stability of Superdeformed Nuclei Superdeformed nuclei can be considered to be the deformed analog of spherical magic nuclei. Superdeformation is therefore a phenomenon distinct from normal deformations, and provides an opportunity to study rotating nuclei in regions of low level density where the deformation induces a significant change in the quantum numbers of the valence states. Its occurrence is a clear illustration of the role of single-particle (microscopic) effects on macroscopic properties. In the following Sections we will consider the separate microscopic and macroscopic contributions to the total energy, and the manner in which they determine the superdeformed shape.

4.1.1.

Shell Structure

The existence of shell gaps imparts an extra binding to nuclei with completely filled shells. Although this energy is only a few percent of the total binding energy, the occurrence of non-uniform level spacings|| has a profound effect on the low-lying excitation spectrum and relative energies of various nuclear shapes. The spectrum of single-particle states derived from an axially-symmetric harmonic oscillator potential, and shown as a function of the deformation parameter in Fig. 4.1, provides the simplest realization of superdeformed magic shell gaps. Here, the degeneracies in the single-particle spectrum, which give rise to regions of high and low level densities, occur when the deformation corresponds to an ellipsoid with integer values for the major-to-minor axis ratios; e.g., spherical (1:1 axis ratio) or superdeformed (2:1). Shell gaps arising from ||

The exact details of the shell structure, such as the number of nucleons at closed shells, will depend on the form and deformation of the nuclear potential. A detailed discussion on the effects of nuclear shell structure in a deformed system is given by Bohr and Mottelson[115].

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shape symmetries occur at the same deformation independent of nuclear mass; for the harmonic oscillator superdeformed nuclei will have a prolate 2:1 axis ratio that corresponds to a quadrupole deformation of The harmonic oscillator is an idealized potential. Nevertheless, in a realistic potential, regions of low level density still occur[116], albeit less pronounced. The main differences are due to the spin-orbit coupling term. These terms lower the energy of the highest state of a given oscillator shell (N) to such an extent that it is shifted down into next shell (N-1). Such levels are called intruders and they have a different parity than the “normal” states of the N-l shell. High-j intruder states have characteristic properties; they are not easily mixed with the normal parity states, they exhibit the largest quadrupole moments, and they carry large amounts of angular momentum. These characteristics cause the intruder states to play an important role in generating high angular momenta both at normal and at superdeformations. The single-particle level energies as a function of deformation for a realistic potential (Woods-Saxon including spin-orbit coupling) are shown in Fig. 4.2. The deformed shell gaps are not centered at a constant deformation, but are skewed to lower deformations in a manner determined by the splitting of the substates of the intruder states. It was shown[117] that this feature closely resembles that given by a pseudo-oscillator potential that has the intruder states added back in the appropriate way. The pseudo-oscillator spectrum is constructed from the harmonic oscillator by removing those states that, after introducing the term, make up the intruder levels discussed above. It is a general feature, common to all potentials, that the lighter the nucleus the more dispersed are the regions of low level density as a function of deformation. In the lightest nuclei the single-particle level density is so low that there are no distinct regions of high and low level density, even for the harmonic oscillator potential. Let us now consider the deformations of normal deformed nuclei. Bohr and Mottelson provide[118] a simple estimate for the nuclear deformation in the absence of shell corrections, i.e., for a more-or-less uniform level-energy spectrum. They show that the deformation is proportional to which can be thought of as arising from the competing effects of the core nucleons (volume effect A) and valence nucleons (surface effect ) that provide the spherical and deformation driving tendencies, respectively. In Fig. 4.3 we plot the normalized quadrupole moment, which is proportional to the quadrupole deformation obtained from B(E2) values, for nuclei in the mass range The deformation, expressed in terms of axis ratios, is given to the right side. There is a tendency for the ground-state deformations to exhibit a mass dependence that follows a dependence (full line), but with strong deviations in the vicinity of closed spherical shells. The dependence was

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scaled to coincide with the deformations of mid-shell nuclei.

The experimentally derived deformations obtained from normalized quadrupole moments for nuclei classified as superdeformed in the literature are also shown in Fig. 4.3. For nuclei A ~150, 190 and 240 there are large deviations from the dependence arising from the superdeformed shell gaps. These deviations are comparable to the effects of the spherical shell gaps at Z=82, N=126 but in the opposite direction. The superdeformed

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nuclei in these three regions are therefore clearly separated from their normal deformed counterparts. They are restricted to distinct mass regions with deformation values that are rather close to each other; thus supporting the concept of deformed magic shell closures that occur because of nuclear shape symmetries. Below A ~150 two effects combine to blur the distinction between superdeformed and normal deformed nuclei. First, in very light nuclei even “normal” deformations driven by the polarizing tendencies of the valence nucleons can be large with magnitudes approaching the superdeformed limit. Second, measured deformations deviate significantly from the ideal 2:1 shape and in some cases form continuous chains of deformations spanning the range from normal deformed to superdeformed.

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Although the picture of deformed magic shell gaps (i.e., a series of gaps occurring at the same deformation; e.g., 2:1) presented by the harmonic oscillator is over simplified, the notion of superdeformation as a phenomenon closely connected with the occurrence of deformed (magic) shell gaps imparts a useful basis within which to examine and characterize their properties. Superdeformation is a phenomenon whose origin goes beyond the deformationdriving tendencies of the valence nucleons and is one that provides a beautiful illustration of the importance of the interplay between the microscopic and macroscopic degrees of freedom discussed below.

4.1.2. The Role of the Fission Barrier To determine the total energy as a function of deformation, the shell correction energy is superimposed[121] onto the macroscopic liquid-drop energy. If the macroscopic energy is sufficiently flat as a function of deformation (due to either the Coulomb and/or rotational energies), then the shell correction can produce a “pocket” or second minimum in the potential energy surface (Fig. 4.4). The superdeformed (second) minimum is then isolated from the normal-deformed first minimum by an inner barrier and from fission by an outer barrier. In other words, the formation of a superdeformed minimum will depend on the shape and size of the fission barrier. Too large a fission barrier, which is steeply rising as a function of deformation, and the relatively small shell-correction terms will not be able to produce a superdeformed minimum that is sufficiently well separated from the normal-deformed states; on the other hand, the superdeformed minimum will not be isolated from fission if the macroscopic energy surface were tilted too far the other way in favour of fission. Thus, the existence of superdeformed nuclei depends on a delicate interplay between the microscopic and macroscopic properties of nuclei. In very heavy nuclei the Coulomb energy alone has the desired effect on the fission barrier; i.e., to flatten the energy surface. The resulting region of superdeformation is centered around Z~94 and A~240 and comprises the fission isomers discovered in the early 1960’s. However, the majority of nuclei have too few protons to produce a large Coulomb energy and rotation is required to reduce the fission barrier. In many cases, and particularly for very high angular momentum is needed with values often approaching the fission limit; which may not be too surprising given that a “flat” potential energy surface is not so far from fission. The superdeformed nuclei around A~190 make use of both a large Coulomb energy and high angular momentum to provide a suitable macroscopic energy surface. The idea that large nuclear deformations are favoured by rotation was presented in[122] (some 10 years prior to the

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observation of high spin superdeformation) and that they survive rapid rotation was discussed in[123, 124, 125, 126, 127]. Cohen, Plasil, and Swiatecki[122] pointed out that even if a nucleus is considered to be a purely classical object where shell structure is not included, then fast rotations may lead to extremely deformed shapes with axis ratios greater than 2:1; the formation of which is connected to the Jacobi instability in analogy to the behaviour of some astronomical objects, such as rotating stars. Swiatecki and Myers have recently revisited this topic, which is discussed further in Sect. 6.

4.1.3. Regions of Superdeformation The extent of the regions of superdeformation is illustrated in Fig. 4.3. With each region discovered different aspects of the physics of very deformed rotating nuclei have been emphasized; e.g., around A ~150 the extreme singleparticle properties were readily identified providing insight into the symmetries of superdeformed nuclei, while in A ~190 the pairing correlations were seen to have a significant influence on the rotation of the superdeformed nucleus, promoting a deeper understanding of the importance of higher order pairing terms. Prior to the large gamma-ray arrays, Gammasphere and Euroball, no superdeformed structures were observed below A ~130. The increased experimental sensitivity provided by these detectors quickly led to the discovery of superdeformed shapes in increasingly lighter nuclei – starting at A ~80[128, 129], proceeding through mass 60[130], and culminating more recently in [131]. These new regions have expanded the physics of superdeformation to include, for example, the termination of rotational bands and the role of interactions. In the lightest nucleus, the properties of the superdeformed band have been used to study the microscopic origin of deformations and rotational collectivity through a comparison with large scale spherical shell models. The superdeformed nuclei around A ~100, e.g., [132] and [133], reported during the preparation of this manuscript, exhibit very large deformations with axis ratios 2:1 or greater. Recent calculations[134] predict that around Z ~48 and A ~110 nuclear shapes corresponding to axis ratios of 2.3:1 become yrast at I ~60 to It is interesting to note that nuclei near A ~110 are expected[122] to posses macroscopic properties which favour the formation of very large deformations prior to fission. Being able to experimentally access the spin regime between the onset of Jacobi-like liquid drop shapes and the fission limit may well play a decisive role in populating even larger deformations by heavy-ion fusion reactions; e.g., hyperdeformed nuclei with 3:1 axis ratios.

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4.1.4. Hyperdeformation and Cluster Structure

Shell-gaps for hyperdeformed shapes appear in the simple harmonic oscillator level diagram (c.f., Fig. 4.1) at exactly integer 3:1 axis ratio: they survive in a realistic potential (although not in the same places), for quadrupole deformations in the range 0.7-0.8 (c.f., Fig. 4.2). Evidence for a 3rd minimum in the fission barrier for Th isotopes (e.g., [135]) at low spin has been known for many years in (n,f) and (d,pf) reactions. Evidence for hyperdeformed states at high spin has been elusive, and to some extent, contradictory. A ridge structure with the right spacing was reported for by Galindo-Uribarri et al.[136], (c.f., also[137]). This structure has been confirmed in experiments at Ga.Sp[138, 139], at Eurogam2, and at Euroball3[140], although an assigment to is preferred by these experiments. Todate, attempts to observe a rotational sequence of discrete lines that could be associated with hyperdeformation at high spin either failed, or remain unconfirmed. Aberg has suggested that in the hyperdeformed minimum, rotational decays proceed along parallel bands above the yrast line, and no individual band has sufficient intensity to be detected [141]. Very deformed nuclear shapes can be regarded as having an underlying cluster structure; for example a hyperdeformed state of could be regarded as a linear “molecule” of three This topic has been reviewed by Fulton[142], and recent applications may be found in Ref.[143]. In fact there is a deep connection between the shell-energy correction approach and the cluster approach. The magic numbers for superdeformation and hyperdeformation can be generated by combining clusters made up of nuclei with spherical magic numbers[144]. In doing so, one can see how the sequence alternates between reflection symmetric and reflection assymmetric (octupole) shapes. In reality, these properties, which apply strictly for the harmonic oscillator potential are modified by the real nuclear potential. Nevertheless, in microscopic calculations with a realistic potential, strong octupole correlations are found in superdeformed and in hyperdeformed nuclei near the particle numbers suggested by the underlying cluster structure. The search for hyperdeformed rotational structure at high angular momentum remains an important goal for the future. It may take a better understanding of where to look, and of how to bring in, and to hold the greatest possible angular momentum. Whether the present generation of detectors are sensitive enough to detect the extremely weak signal remains to be seen.

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4.1.5. Population and Decay An important aspect of high spin superdeformed studies has been to understand their population and decay. Here we briefly mention a few of the main points, the reader is referred to the reviews[110, 111] and references therein for a more complete picture. Superdeformed bands exhibit a characteristic gamma-ray intensity profile, marked by a relatively short region of rapid feeding at high spins followed by a long sequence of transitions with constant intensity and an abrupt depopulation, often over only two or three states. In almost all cases the high spin limit to the population is determined by the competition with fission. Early work aimed at understanding the population of superdeformed nuclei was presented in[145]. Since then, there have been many experimental investigations that have attempted to quantify the effects of different target-beam combinations. Several works in the A ~150 region[146, 147, 148] have indicated that symmetric reactions lead to a greater population of superdeformed states in a given nucleus compared with asymmetric reactions. However, this does not appear to be the case for A ~190 superdeformed nuclei[149]. The physics describing the population of superdeformed nuclei is linked to the physics of compound fusion reactions*. Precisely which aspects of the dynamics of the fusion process are visible in the feeding of superdeformed states is an interesting question; e.g., it has been suggested[150] that the enhanced populated may indicate an increase in the fusion time for the mass-symmetric reactions due to dissipative effects. The decay of a superdeformed band involves the dynamics of quantum tunneling through the potential barrier separating the two minima and is intimately connected with the coupling of “cold” ordered states in the superdeformed minimum with “hot” highly mixed states within the normal deformed minimum. It is generally accepted that pairing correlations can play an important role in determining the angular momentum at which the bands decay, their rate of decay, and the shape of the quasi continuous gamma-ray spectrum associated with the decay. (Barrier penetration models are discussed, for example, in [145, 151, 152], and recently the decay process has been connected with the idea of chaos assisted tunneling [153].) The search for discrete gamma-ray decays linking the superdeformed bands to the known normal deformed states is an important part of the study of superdeformation in all mass regions and is the only way to firmly establish the spins, excitation energies, and parities of superdeformed bands. The highly fragmented nature of the decay has made it extremely difficult to firmly link the superdeformed bands and only in a few cases has this been possible, although where it has been possible the re* This is discussed further in Sect. 6.

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sults have proven extremely important (see the discussion on identical bands in 4.2.2, for example). Experimental data on superdeformed decays is reviewed in[154] for the A ~60 region, in[155] for the A ~130 region, and in[156] for the A ~150 and 190 regions.

4.2. Rotations of Superdeformed Nuclei We can, broadly speaking, isolate two characteristics of superdeformed bands that emphasize different aspects of their rotational properties compared with normal deformed nuclei. These characteristics are the large deformations and large shell gaps (low level densities). Large deformations cause the levels of a given to be split by an amount approximately proportional to giving rise to reduced Coriolis interactions. Furthermore, deformation distorts the mean-field potential and favours nucleon states that have a high wavefunction density in the plane containing the symmetry axis (the equatorial plane); i.e., states with large but small (component of along the symmetry axis). The large superdeformed shell gaps give rise to a low density of states near the Fermi surface, which will have the effect of reducing the pair correlation strength (more precisely the static pairing strength). The shell gaps also provide stability against large changes in deformation as a function of spin, and justify the concept of an “inert” superdeformed core. The latter two points provide an environment where the motion of the individual single-particles exhibits the properties of an extreme shell model – e.g., the observed additivity of such properties as single-particle alignments and orbital quadrupole moments (see Sect. 4.2.1). The above characteristics, derived from the large deformations and shell gaps, give superdeformed bands their distinct experimental signature of highly regular, often long, cascades of transitions. The transition energy spacing is both much smaller and far more constant compared with normal deformed rotational structures in the same mass region. Rotational bands based on superdeformed shapes in various nuclei span the whole range of angular momentum space from depending on the specific mass region. to Pair correlations will therefore be of varying importance, depending on the spin or frequency regime spanned by the superdeformed band. The rotational frequency range covered by superdeformed bands is illustrated in Fig. 4.5. The moment of inertia, scaled to remove the gross mass dependence is plotted for a superdeformed band in each of the mass regions. This type of plot has been used, e.g., [128], to illustrate the common features of superdeformed bands. Nevertheless, there are differences between the different mass regions that result from the spread in deformations, the varying role of pairing, and reduced valence configuration space (i.e., spin content) for the lightest nuclei.

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Valence Particle Configurations and Observed Properties

In the absence of strong pairing correlations, primarily at high angular momenta, the way in which the dynamic moments of inertia of superdeformed bands vary with rotational frequency is sensitive to the occupation of specific intruder orbitals[157]. The superdeformed “core” gives a fairly constant contribution to and because the non-intruder orbitals generally have lower values and a larger component they do not contribute as much to By comparing the observed with those calculated assuming different intruder occupations, one has a method to classify superdeformed bands according to the number and type of intruder orbitals occupied. This procedure was used extensively in the A ~150 region to characterize and understand the properties of these superdeformed bands Fig. 4.6. Discrepancies between calculated and measured dynamic moments of inertia were attributed to non-negligible pairing effects [158, 159].

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At lower angular momenta pairing plays a more visible role in determining the form of The superdeformed bands in the A ~190 region, which extend from I ~10 to are good examples. In contrast to the A ~150 region, most A ~190 superdeformed bands have very similar smoothly rising values as a function of rotational frequency. The common cause of the smooth increase in is the consecutive alignment of pairs of intruder protons and neutrons in the presence of pair correlations[162, 163]. In other words, these are quasiparticle bandcrossings. An odd particle occupying an intruder state will block the quasiparticle crossing leading to a far more constant contribution to the moment of inertia as a function of frequency. Several such blocked configuration have been identified[164]. At the highest angular momenta the value of the moment of inertia is seen to decrease[160] in agreement with the quasiparticle bandcrossing interpretation. The observed increase in the A ~190 moments of inertia as a function of frequency is remarkably similar to that seen in the normal deformed rotational bands near indeed, the intruder orbitals responsible for the quasiparticle bandcrossing, i.e., protons and neutrons, are common to both regions. The important relationship between particle number, deformation, and the relevant intruder orbitals is illustrated[161] in Fig. 4.7.

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When non-intruder orbitals are occupied the effect on the moments of inertia are more subtle, but they can still be observed, and with sufficiently high quality data they are generally distinct enough to enable the contributions from different orbitals to be separated. This effect is most sensitive when pairing can be neglected and when the level retains its individual single-particle identity. Proceeding one step further, it is possible to use the concept of additivity to describe the relative properties of superdeformed bands. This idea has been exploited with good success in the A ~150 region to obtain[165, 166] the relative spins of superdeformed states, which is is important because very few superdeformed bands have had their spins fixed by experiment, and none in the A ~150 region. Similar arguments have been made regarding the addition of single-particle quadrupole moments to that of the core[167, 168] (see discussion in Sect. 1.3.3). The success of additivity, when comparing the relative properties of superdeformed nuclei, relies on a common (stable) core. It breaks down if varying the nucleon number causes the nucleus to “jump” to a neighboring shell closure with a significantly different equilibrium deformation. A technique to establish the single-particle properties of nuclear states is to measure their g-factors, which can be inferred, for example, by measuring the strength of the magnetic dipole (M1) decay from that state. This has been possible in a few A ~190 high-K signature partner superdeformed bands; e.g., [169] and [170], and has providing confirmation of the quantum numbers of high-K neutron as well as intruder proton orbitals near to the Fermi surface. The direct, but more difficult measurement of magnetic moments by precession in a magnetic field has to-date only succeeded in ons case, and that in [171]. The majority of superdeformed nuclei possess multiple excited superdeformed bands and the majority of these can be understood in terms of relatively simple single-particle excitations within the superdeformed minimum. Nevertheless, there are a few excited superdeformed bands which are best understood in terms of collective oscillations about the mean superdeformed shape. Bands based on collective excitations may have large transition matrix elements connecting them with the yrast superdeformed band. A signature for collective modes would then be the observation of decays from the higher lying excited band to the yrast band. This is in contrast to the M1 decays between close-lying signature partners discussed above. Probably the best example for collective excitations in a superdeformed nucleus occurs in [172], where the observation of direct decays from an excited band to the yrast superdeformed band provides strong evidence for the presence of octupole correlations. The excited band can be interpreted as an octupole phonon based on the yrast superdeformed bands, as discussed within the framework of RPA calculations[173]. Additional support for octupole vibrations in the A ~190 region is found in

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[174] and [175]. The presence of orbitals near the Fermi surface with opposite parity and suggests that octupole vibrations may play a significant role in A ~150 and 190 superdeformed nuclei. However, apart from the few examples mentioned already for A ~190 superdeformed nuclei, collective excitations have been invoked for only a small number of bands in the A ~150 region[176, 177, 178]. It appears, therefore, that excitations within the superdeformed minimum are predominately single-particle.

4.2.2. Identical Bands The spectra in Fig. 4.8 illustrate one of the most surprising and unexpected observations to emerge from the study of superdeformed nuclei. It shows that over a long sequence of transitions the gamma-ray energies of superdeformed bands in different nuclei can be the same to within a few keV; an observation contrary to the general expectation that nuclei posses unique gamma-ray spectra.

The first identical bands were observed in the A ~150 region, which included the and pair[179, 180]. Shortly after, examples were also reported around A ~190[181, 182], and in the following few years many other

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cases of identical band pairs were reported throughout the regions of superdeformation as well as at normal deformations. An extensive review of identical bands in both superdeformed and normal deformed bands is presented in Ref.[183]. The peculiarity of the identical band phenomenon can be put succinctly: why do different nuclei choose to rotate with the same sequence of frequencies? For this to occur the two nuclei must posses the same dynamic moment of inertia and have the same or “quantized”† alignments over a large number of transitions. This latter condition follows from the fact that the level spins are themselves quantized. The requirement that the moments of inertia are the same constrains the alignment to be a constant, but not necessarily quantized (i.e., the gamma-rays to be “in-step”, but not necessarily identical). Early insight into the identical superdeformed band phenomenon was obtained by considering[180] the strong coupling limit of the particle-plus-rotor model[50], where a valence particle is coupled to an axially deformed rotating core. In this limit, the alignment of the odd particle is related to the decoupling parameter, by: where and which arises due the Coriolis interaction; note, for and no Coriolis mixing then and An example of coupling is given by the two excited superdeformed bands in which are assigned to the neutron [514]9/2 state and form identical bands with the superdeformed core. Here, the core gamma-ray energy is equal to the average of the gamma-ray energies in the two signature bands; In order for the rotational spectra of the even mass core and that of the core+particle to be exactly the same, the particle should occupy a state with i.e., of course the particle’s presence must not affect the rotational properties of the core The identical bands are an excellent illustration of the limit; their transition energies are equal to within 12 keV over practically the full spin range of the bands (close to ) which implies and hence The odd proton in the identical band is predicted to occupy the [301]1/2 Nilsson state and calculations[184] using the full Nilsson wavefunction gave a value of very close to experiment. Around A ~190, the even-even superdeformed nucleus acts as the appropriate core for several neighboring identical bands, e.g., the signature †

When both bands occur in even mass nuclei or both are in odd mass nuclei, then quantized alignment means or 1, or 2 ... etc. Comparing a band in an even mass nucleus with one in an odd mass nucleus requires half-integer alignment to give transitions of the same energy.

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degenerate excited bands in plays an analogous role for odd proton nuclei, e.g., In almost all cases in the A ~190 region the transition energies in the identical band follow the limit relative to the core, but the alignment is one and not zero as in the A ~150 region. To explain the observed unit alignment in this region it was suggested[181, 182] that the two nucleons occupy a pseudospin‡ doublet within which the pseudospins, of two valence particles align with the collective rotation to give the extra one unit of spin, At the time the pseudospin explanation was proposed, no superdeformed level spins had been measured in the A ~190 region and an extrapolation (spin-fit) procedure[191, 192] was used to determine the spins of the superdeformed states; consequently the occurrence of unit alignment was controversial. The gamma-ray decays from superdeformed to low-lying normal deformed states have subsequently been observed in (1,3)[193, 194] and (1)[195, 196] enabling the level spins in these SD nuclei to be unambiguously assigned and confirming the occurrence of unit alignment. However, these same data on the decay of the excited superdeformed bands led to the suggestion that the identical bands in bands have negative parity unlike their identical band partner, which most likely has positive parity. Pseudospin partners must have the same parity, therefore the observed parity difference between identical bands appears to rule out pseudospin alignment as the mechanism responsible for generating the observed unit alignment, but not the 1/2 unit alignment seen, for example, in [179] [190]. The idea that the pseudospin basis provides a useful framework in which to describe the properties of some superdeformed bands, and identical bands in particular, led to a revived interest in the underlying origin of pseudospin symmetry in nuclei. Exact pseudospin symmetry is equivalent to saying that the ratio of the and terms, which appear in the modified harmonic oscilla[197], For medium-heavy mass tor potential, has the value nuclei, µ is remarkably close to this limit. Within relativistic mean-field theories, the desired strength of the spin-orbit interaction is well reproduced and µ, turns out be ~0.6. The fact that relativistic mean-field theories give a value of µ close to the pseudospin limit has been related[199] to the near equality of the magnitudes of the attractive scalar and repulsive vector potentials present in the relativistic mean field. It is thus an intriguing thought that by studying nuclei at high spin one may be seeing the effects of fundamental symmetries ‡

In the pseudospin scheme[185, 186, 187] the pseudospin-orbit spitting and thus the Coriolis interaction can now readily decouple the pseudospin, from the pseudo-orbital angular momentum, The half-integer alignments discussed for the A ~150 identical bands can also be understood in terms of decoupled pseudospin in this case involving a pseudospin singlet[180].

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at the level of the nucleon interaction[198]. The particle-plus-rotor model, and the extension to include decoupled pseudospins provides a general mechanism for quantized alignments of the valence particles, but in its simplest form it relies on the fact that the core properties are not affected by the odd particle or hole. Within a mean-field approach, however, the observed alignments could have their origin in a number of effects; i.e., where is the contribution to the alignment from the valence particle(s), is due to the “quenching” of pairing, and and are the effects due to changes in deformation and mass respectively. (The decomposition into various contributions is somewhat arbitrary and it is recognized that the terms given above are not necessarily independent.) In such models the similarities in the rotational spectra are accompanied by cancellation effects which may be more or less systematic depending on the potential. In the mass 150 region, where pairing can often be neglected due to the high angular momenta, several studies using cranked single-particle models have shown that orbitals with small values have little effect on the moment of inertia, and changes due to mass, deformation, and alignment of the odd particle tend to cancel[201, 202, 203, 204]. Relativistic mean-field calculations have also had some success in reproducing identical moments of inertia in the region, and attribute the outcome to cancellation effects[205]. Returning to the mass 190 region, where pairing correlations are known to have a large influence on the properties of superdeformed bands, the differences in pairing strengths at low spins and subsequent differences in the response of the pair field to increasing rotational frequencies will generate different moments of inertia and hence non-zero alignments. Generally, the odd particle reduces the number of states available to the pairing interaction and thus reduces the pairing strength. Early studies[206, 207] have shown that after one neutron or proton orbital is blocked, the already reduced static pair gap is practically quenched leaving only the so-called dynamical pairing. Therefore, the addition of further neutrons(protons) does not greatly alter the strength of the pair correlations. More recently, a large scale comparison of both experimental and calculated alignments was carried out for A ~190 superdeformed bands[208]. It was concluded in this study that both the valence quasiparticles and the “core” (through the action of pair blocking) contribute to the observed alignments in these bands, and hence within a mean field model the peculiar unit alignments do not have a simple unique source. To conclude this Section we note that it has been ten years since the first identical superdeformed bands were observed. The physics behind this remarkable phenomenon has stimulated much discussion. Simply stated the question has been: do identical bands occur because of new physics, or, as others have suggested, do they arise due to a series of “accidental cancellations”? Despite

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a great deal of experimental data and many diverse theoretical approaches, a fully satisfactory answer to this question does not yet exist.

4.2.3. Lifetime Measurements and Deformation The measurement of lifetimes of excited nuclear states can give the reduced transition probabilities, B(E2)-values, describing the decay between the initial and final states. These quantities in turn are related to the intrinsic quadrupole moment§, in a rigid rotor model, from which one can infer the quadrupole deformation. Deformations inferred from other spectroscopic properties, such as moments of inertia, are very model dependent and are more susceptible to error; for example, a high moment of inertia does not necessarily imply a large deformation. The quadrupole moments carry important information on the underlying structure of the nucleus – they can be used to distinguish singleparticle configurations and are a stringent test of theory. High spin superdeformed states typically have lifetimes in the picosecond to femtosecond range and can be measured by Doppler shift techniques (see[209] for a review of these methods). The basic features of the method are as follows: (i) the shift in the gamma-ray transition energy gives the mean time at which the state depopulates as it slows down in the backing; (ii) the shift of the gamma-ray decays from the preceding state gives the mean time at which the state of interest is populated; and (iii) the stopping power determines the basic time scale. With the current sensitivity of large gamma-ray detector arrays the error on the Doppler shifted transition energy can be small. Provided we restrict our analysis to the region where the population intensity is constant with spin, the uncertainty in the so-called side feeding population is small and the uncertainty in the derived quadrupole moment is then dominated by systematic errors in the stopping powers, which are of the order of 5-15% depending on the stopping medium and on the effective velocity range of the measurement. A 5-15% uncertainty is sufficient to extract the gross deformation trends, as expected from Fig. 4.2 and illustrated in Fig. 4.3, but is not, in general, sufficient to make meaningful comparisons of deformations between neighboring bands. The systematic error introduced by the stopping powers can be largely eliminated if several superdeformed bands are measured in one reaction. The opportunity to perform simultaneous lifetime measurements for §

What is actually derived from a lifetimes measurement is the transition probability, B(E2) (and thus transition quadrupole moment ), which, under the assumption of a rigid axially symmetric deformed nucleus, can be expressed in terms of the intrinsic quadrupole moment, Throughout this Section we will use this relationship and refer only to the intrinsic moment. The assumption of axial symmetry is expected to be valid for the majority of superdeformed nuclei.

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superdeformed bands in the same nucleus as well as for bands in neighboring nuclei came with the enhanced sensitivity of the new gamma-ray detector arrays. First experiments of this type were performed on bands in [210], [211], and in and [212]. Since then, there have been similar studies in, A ~190[213, 214], A ~150[215, 216], A ~130-140[168], A ~80[217], and A ~60[130, 218, 219] superdeformed regions. The individual single-particle contributions to the total were investigated in Ref. [167] through a theoretical analysis of quadrupole (and hexadecapole) deformations of A ~150 superdeformed bands using a cranked Skyrme-Hartree-Fock model. In this work it was shown that the calculated quadrupole moments can be expressed as a sum of contributions from singleparticle/hole states around the double magic core; The contribution from a given state is independent of that from other valence states and the relative quadrupole moments are generally in agreement with the experimental values given in[212, 215]. Superdeformed nuclei around have been presented as systems exhibiting extreme single-particle motion where a simple shell model relationship (additivity of single-particle properties) provides a good description of the observed properties. In general, the results from simultaneous lifetimes measurements have shown a characteristic dependence of the deformation on the intruder configuration (consistent with expectation), and, so far, the evidence is that identicalband pairs have very similar¶ quadrupole moments (a fact consistent with their having the same high-j intruder configurations). In most cases this has led to the claim that they also posses very similar deformations, but if we compare systems with different mass then the macroscopic dependence of creates an ambiguity. There has been speculation[212] that, for example, the identical observed in band 5 and band 1 differ in deformation by 5%, and thus the identical gamma-ray energies requires a compensating difference in alignment (as suggested in Ref.[202]). However, deformations are not an observable and are model dependent, and it is not necessarily true that the scaling, valid on average, will hold over a limited range in particle number. Thus, it is not possible to unambiguously distinguish between solutions where both deformation and alignments are the same, or where they differ by small amounts such that their combined contribution to the moment of inertia compensates. ¶

A possible exception is the data on [214] where, although all five superdeformed bands in have very similar values, band 1 as measured in this experiment had Doppler shifts consistent with a higher recoil velocity and thus a larger intrinsic quadrupole moment; however, at this time other causes, such as differences in sidefeeding timescales, cannot be ruled out.

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4.2.4. Bifurcations Axially-symmetric prolate deformed nuclei possess a symmetry that leaves the system invariant following rotations through about an axis perpendicular to the symmetry axis. Evidence for higher symmetry terms came with the observation of a bifurcation, or staggering, in the yrast SD band of [220]. Here, the gamma-ray transition energies from every second superdeformed level are shifted by approximately 100 eV relative to the other levels. This perturbation that occurs every 4 units of angular momentum has been interpreted[220, 221, 222, 223] as evidence for a invariant term in the Hamiltonian. It is important to stress that this new term is a small perturbation on the main quadrupole shape. A oscillation can arise from a rotational Hamiltonian of the form:

I is the angular momentum vector, refers to its three projections onto the body fixed coordinates, and A, B are parameters (for regular oscillations ). To date, there is no clear microscopic justification for such a term; for example, scenarios based on a hexadecapole nuclear deformation do not appear to be likely[224, 225, 226]. Empirically, and with some theoretical justification, it can been argued[220] that the occurrence of staggering may be connected with the occupation of high-j intruder orbitals from the N=6 and N=7 oscillator shells and that the intrinsic hexadecapole moments of specific single particle states may play a role in generating the observed staggering[227]. The experimental evidence for bifurcation remains scarce and it is clear that this phenomenon is not a general property of superdeformed bands; indeed, several proposed cases for bifurcation in superdeformed bands have since proven false. Nevertheless, in a few examples (including the original case) the oscillations have been confirmed through two or more experiments and these are not considered to be statistical fluctuations. The reader is referred to Ref. [228] for a summary of the experimental status in A ~150 superdeformed bands. It is a further intriguing feature that bifurcation is also observed in superdeformed bands in and which have identical moments of inertia to band 1. These data were used to show that the staggering amplitudes, as a function of angular momentum, follow a cosine dependence, which constrain[228] the form of the invariant Hamiltonian presented in[221]. Further experimental evidence and theoretical study is needed to clarify the phenomenon of bifurcations.

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4.2.5. Superdeformation in Light Nuclei The properties of superdeformed bands in light nuclei (e.g., mass A ~60 and lower) may differ from their heavier counterparts because of the reduced number of valence particles and the fact that both neutrons and protons can take advantage of the same deformed shell gap (i.e., the neutron number is the same as, or very similar to, the proton number). The limited spin content available due to the small number of valence nucleons has noticeable effects. It establishes a maximum spin for a given configuration (see discussion in Sect. 5 on band terminations) and in light nuclei the spin of the terminating state is often within experimental reach. When the initial deformation is sufficiently large, complete termination is not expected; with intershell mixing, configurations from higher lying shells can contribute to the generation of angular momentum. In the harmonic oscillator marks the transition to the non-terminating regime. The A ~60 superdeformed bands[130, 218, 230] with are observed to within a few units of their calculated terminating spin (as defined by their unmixed configuration at low angular momentum), and the approach of the terminating state is signaled by a reduction in the moment of inertia, which reflects the higher cost associated with generating angular momentum. The large deformation of these bands is close to the critical deformation for termination and, for example, the superdeformed nucleus which is observed to is calculated to become triaxial rather than oblate and to extend beyond the terminating spin calculated without shell mixing. In the superdeformed band[131] has a deformation and is seen to its termination spin, In contrast to the case, the superdeformed band is expected to terminate at the maximally aligned configuration with an oblate deformation. The low number of valence nucleons means that fewer particle-hole (or quasiparticle) configuration changes are needed during the decay from the superdeformed band to the normal deformed structures. In A ~ 150 and 190 this process involves the rearrangement of ~ 10 or more nucleons, while for and the number is more like 4. Fewer particle-hole configuration changes may, depending on the level density, provide a higher probability for the decay flux to be less fragmented and thus a higher probability for the associated gamma-ray transitions to be detected. The chance to observe discrete gammaray decays is further enhanced when the bands extend to lower spins because the phase space open to the decay is reduced (this occurs, for example, in and ). The reduced Coulomb energy, in light nuclei ( say), means that it is possible to study nuclei that contain the same number of neutrons and

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protons. For N = Z the neutron and proton superdeformed shell gaps coincide; i.e., they occur at the same deformation, which is clearly the most favourable condition to provide a deep second minimum in the potential energy surface. The and bands are examples of N = Z superdeformed nuclei, both extend to low spins (I = 8 and respectively) and both bands carry a large fraction of the decay flux (50% and 40%, respectively) following production at high angular momentum. These features indicate a particularly favoured superdeformed configuration. The large population intensity (50%) for the superdeformed band should be contrasted to other A ~60 superdeformed bands, which only carry ~1% of full population intensity of the particular nucleus. Light mass superdeformed nuclei provide an opportunity to study the role of the interaction as well as the microscopic origin of collective rotations in highly deformed nuclei using large-scale spherical shell model calculations. These recent advances in superdeformation studies will be considered in Sect. 5 within the more general context of rotational properties of nuclei.

5.

LIMITS TO NUCLEAR ROTATIONS

5.1. Termination of Rotational Bands The collective angular momentum of rotation has to be generated from special motions of the individual nucleons within the nucleus; therefore, the rotational angular momentum cannot exceed the internal spin-content of the nucleus. The internal spin most directly available to the collective motion is provided by the valence nucleons. One immediate limit to the nuclear spin is then obtained by aligning the angular momenta of the valence nucleons to the maximum value permitted by the Pauli principle. If we apply this concept to say then we can say that the maximum collective spin generated from the valence nucleons (outside the doubly closed spherical-shell of ) is Similarly, for nuclei outside the doubly-closed spherical shell-gap at Z=N=50, ( is a good example) the valence spins limit the angular momentum of the rotation to a maximum in the range depending on the nucleus and on the configuration of the valence nucleons. This phenomenon is called “Band Termination” and it is the subject of this Section. In their book, Bohr and Mottleson[232] give estimates for the spin content of the valence nucleons and suggest some terminating spins as a function of mass. Already by A=160, the terminating spin is and higher than this

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is allowed by other factors, such as the stability against fission. We note that the spin-content of the valence space cannot be an absolute limit, since the nucleus can, and frequently does, increase the number of valence nucleons by breaking the core, and this process could presumably extend the available spin indefinitely. However, breaking the core entails a large energy cost, so that rotational bands formed by progressively breaking valence particles out of the core must lie at progressively higher excitation energy. Terminating bands are seen most clearly in the mass region around A=110. These nuclei contain just the right number of nucleons outside the core – too many valence nucleons make the terminating spin too large to be observed experimentally – too few will prevent a rotational structure from developing in the first place. Band terminations have been found in: 1) very light nuclei ( provided the first recognised case[233]); 2) the the and regions; 3) the region; and finally in 4) the region. In the region, the nucleus exhibits a band termination at spin 16, which exausts the spin content of the shell containing all the particles outside the doubly closed shell e.g., [234]. The region follows an interesting parallel with the region where the doubly closed shell plays the same role as the double closure e.g., [235]. The first band termination identified at high spin was in the nucleus and several examples are known in the region. These cases are different from those observed in the lighter mass region in the following sense. In the case of first observed by Janzen et al.[237], and interpreted in terms of band termination by Ragnarsson et al.[238], four terminating rotational bands are observed,the longest sequence running from spin 35/2 to spin 83/2[239]. The terminating states of these bands represent the physical end of the rotational sequence, and there are no higher-spin rotational states of that configuration. By contrast, the terminating states observed in the region belong to rotational bands that are either not observed at all below termination, or at best, only a few states are seen. The band that is observed to lower spin and that appears to have terminated has in fact not reached termination, rather its population has been cut-off to a level below the instrumental sensitivity for detection by the band that has terminated. All band termination phenomena contain the same physics, however the region is perhaps more instructive since a much wider range of the terminating configuration may be observed experimentally, and this is the region we will discuss to illustrate the physics. Results summarised by Afanesjev et al.[240], show some twenty or so nuclei in the In to in which terminating bands have been observed. Band-head spins typically lie in the range and the highest spins ob-

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served are in the range many of these bands can be followed over a wide spin range before terminating. A characteristic of the bands is their decreasing moments of inertia (both and ) as termination is approached: this can be seen directly in the gamma spectra as a rapid increase in the transition energies. Examples of this effect are shown in Fig. 5.1 and in the corresponding level scheme Fig. 5.2 as given in Ref. [240]. It reflects the energy cost to generate the next two units of angular momentum which in these bands can increase rapidly as termination is approached. A rigid rotor has a constant incremental increase in the cost of generating the next rotational state. Calculations by Afanasjev and Ragnarsson[241] show that the final stages before termination involve a rapid increase in the describing the nuclear deformation up until the terminating state itself lying at that is to say the state has an exactly oblate shape. The evolution of the nuclear shape from near prolate through maximum triaxiality to exactly oblate is characterised by a gradual reduction in the transition rates (e.g., B(E2)-values) describing the decay of the rotational band. In other words, the collectivity of the rotation is reduced and actually vanishes at termination where the angular momentum lies along a symmetry-axis of the nuclear shape. This effect has been confirmed in a few cases by measuring the lifetimes at high spins. These data are summarised in Ref. [240]: the nucleus provides a good example as shown in Fig. 5.3, and Fig. 5.4. Not all terminations show the high cost of extracting the last measure of angular momentum from the nucleus; and not al terminations move across the plane as described for the region. For example, in the energy cost of raising the spin from to which is just is actually smaller than Similarly in the nucleus the states preceding and including termination have a relatively constant energy cost per spin increment. In calculations by Afanasjev and Ragnarsson[241], which reproduce the main features of terminating bands, the reason for these different behaviours goes back to the relative costs of extracting the last parcel of angular momentum by aligning the nucleonic orbits, compared with the energy gain from corresponding changes to the nuclear shape.The balance of these competing effects is quite different between such nuclei as and

5.2.

Rotations and Pairing Correlations

At low spin, the moment of inertia of a rotational band is approximately half of the value expected for a system of independent nucleons moving in a deformed potential. The reduced moment of inertia can be understood as an

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effect of pairing correlations that favour the fully paired state, which consists of a condensate of J = 0 time-reversed pairs. This is the nuclear superfluid or vacuum state[242, 243]. The basic excitations of the vacuum are quasiparticle excitations[244] and in an even-even nucleus the lowest quasiparticle excitation is the two-quasiparticle state corresponding to a broken-pair. At zero rotational frequency, the energy of the two-quasiparticle state, relative to the vacuum, is approximately twice the static pair gap defined within the BCS equations[245]. It was suggested by Mottelson and Valatin[246] that rotation of the nucleus would have a destructive effect on the correlated motion of the nucleon pairs, and at a critical value of the rotational frequency there would be a transition from a superfluid phase to a normal (“aligned”) phase of nuclear matter. The rotation induced phase change is analogous to the quenching of superconductivity by an external magnetic field. However, nuclei are finite and this gives rise to dynamic fluctuations[247] which smooth out the transition from the paired to the unpaired regime. Conceptually one may still talk about a loss of static pairing associated with a condensate, but dynamic pairing correlations (also referred to as pairing vibrations) are still present, even at very high frequencies. For finite systems it would therefore seem more appropriate to associate the Mottelson-Valatin transition with the point when static pairing correlations have vanished and only dynamic pairing fluctuations remain. This is the approach we follow in remaining discussion. Much of the evidence for the demise of static pairing at high spins comes from the study of rotational bandcrossings, or backbends as reviewed for example in Ref. [248]. With increasing rotational frequency the rotational band based on the excited two- quasiparticle configuration becomes increasingly favoured with respect to the band based on the vacuum state (the groundstate band), and at some critical frequency these bands cross and the twoquasiparticle configuration becomes yrast[249]. The angular momentum of the broken pair is now free to add to the collective rotational angular momentum, R, and is aligned in the direction of R. At each quasiparticle alignment, the pairing strength is reduced, and in many cases only two such alignments are calculated to quench the static gap altogether[247]. At this point the nucleus is said to be “unpaired”, that is pair correlations are too weak to support a condensate and an experimental signature of a quenched static pair gap would then be the observation of band-crossings that can only be explained as excitations within the spectrum of single-particle states rather than quasiparticles. One form of an unpaired bandcrossing discussed in[250], involves the crossing of one single-particle state with another of the same parity and signature. However, it is also possible to generate unpaired crossings in which two nucleons are exchanged between pairs of single-particle states and at a rotational fre-

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quency where the combined energy of the single-particle Routhians of the first pair equals that of second pair. The first example of this latter type of unpaired crossing was observed in the band in [251], which is believed to be based on the exchange of a pair of neutrons between the states with (–, – l/2)(– l, 1/2)at low frequency and the states with (+, – l/2)(+, 1/2) at higher frequency. Both pairs have the same combined parity and signature, and thus the total parity and signature of the decay sequence is unchanged. These and similar[252] data at high spins have shown a lack of the systematic bandcrossings expected within a quasiparticle picture and provides evidence for the transition to the unpaired regime.

5.3.

Rotations in N = Z Nuclei

At N = Z, valence neutrons and protons occupy the same orbitals and their influence on nuclear properties is reinforced. With only small changes in nucleon number rapidly changing deformations are possible due to the enhanced shape polarizing effects of the valence particles and the occurrence of deformed shell gaps. Along the N = Z line between the spherical shell closures at and for example, both large prolate and oblate shell gaps occur, and an inspection of the single-particle level diagram suggests that these shell gaps and the down-sloping single-particle levels should favour welldeformed oblate and prolate shapes. Most deformed nuclei are known to have prolate deformations and large ground state oblate deformations are rare. The bias towards prolate shapes is subtle and difficult to quantify. For example, higher order terms in the liquid drop energy[253], effects of K = 1/2 level mixing, and particle residual interactions[254] all tend to favour a prolate shape. In A ~80 N ~Z nuclei, calculations [16, 56, 255] predict that prolate and oblate structures lie very close in energy giving rise to the phenomena of shape coexistence, and in particular cases, e.g., (N = Z = 34), they predict a substantial oblate ground state deformation. Early indications of shape coexistence in this region were found in light selenium isotopes[256], where near-spherical structures coexist with well-deformed prolate shapes, the latter becoming yrast at high spins. It was only recently, from a study of at high spin, that experimental evidence[46] was obtained to support the prediction of a well-deformed oblate ground state. For heavier N = Z nuclei, N, Z = 36,38,40, the combined effect of neutron and proton prolate-deformed shell gaps leads to well-deformed prolate ground states, In [257, 258, 259, 260], [261, 259], and [262, 259] the yrast sequence is characterized by a rotational band with properties consistent with that of a large prolate deformed rotor. A striking

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feature of all three nuclei is the increase in the rotational frequency at which the valence quasiparticle alignments occur, compared with those in neighbouring nuclei. In the isotopes, for example, and exhibit a band-crossing at a frequency of while in the crossing does not appear until The data on and although not observed to such high angular momenta as also indicate a delayed alignment compared with the N = Z + 2 neighbours. Comparisons of alignments in superdeformed nuclei[230] around also indicate a marked difference in the frequencies at which the intruder orbitals ( in this case) align. Delayed alignments at N = Z have been discussed in terms of pair correlations. Such effects are most likely to be of importance when neutrons and protons fill the same orbitals. The pairs can couple to spin J = 0 (T = 1 or isovector pairing) and in this coupling scheme the Coriolis force will tend to “break” the pair in the same way it affects the more usual and pairing. At N = Z the three types of T = 1 pairs, and are expected to coexist, due to the charge independence of the nuclear force. An pair can also couple to J > 0, T = 0 because it is not restricted by the Pauli exclusion principle. In this case the largest interaction occurs for pairs coupled to J = 1 and where corresponds to the parallel coupling of the nucleon angular momenta. The Coriolis force has a disruptive effect on the T = 0 J = 1 pairs, which have near anti-parallel angular momenta, but the pairs are already “aligned”, and pair correlations based on this coupling scheme will remain unaffected by rotation. It has been suggested that the presence of both T = 1[263] and T = 0[67] pairing can cause a delayed rotational alignment, compared with the situation where only and pairs contribute to the pair field. While pairing is a plausible explanation for delayed alignments, the evidence is not conclusive. Nevertheless, it is an intriguing possibility that familiar ideas of rotational behaviour, developed for nuclei near to the region of beta stability, may need to be modified in order to describe the generation of angular momentum in N = Z nuclei. High spin studies[264, 265, 266, 267] of T = 1/2, mirror nuclei have investigated the isospin symmetry breaking effects of the Coulomb interaction. The mirror pairs and show similar rotational band structures, yet there are small differences in excitation energy between states of the same spin that can be interpreted as Coulomb effects, and variations in the resulting Coulomb energy differences (CED) are sensitive indicators of macroscopic and microscopic nuclear structure effects. The CED is the difference in the excitation energies of states in the mirror pairs, it is zero for the ground state. As the mirror nuclei rotate the valence particles align. In for example, a pair of protons align at approximately spin I = 17/2, whereas

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in this same alignment involves a pair of neutrons. As the protons align the Coulomb self energy is reduced because the spatial overlap of the proton pair is reduced. Alignment of neutrons causes no change in the Coulomb energy and thus there is a net CED. At band termination, after the valence neutrons and protons have fully aligned, the CED returns to zero. During the alignment process the mirror nucleus undergoes a change in shape from deformed prolate in its ground state to near-spherical at termination. This change in the macroscopic charge distribution also has an effect on the CED. It should be noted that both the microscopic and macroscopic contributions to the CED are of the order of several tens of keV. These high spin data provide a stringent test of shell model calculations and it is a remarkable success of such calculations that the relatively small energy differences are well reproduced. Around A ~40 nuclei, the valence model space is large enough for collective rotation to develop, but sufficiently small for exact shell model diagonalization and the study of high spin states in this region allows a microscopic investigation of the structure of collective rotations. The rotational band in the deformed nucleus has been extensively studied, both experimentally[234] and theoretically[268]. Rotors such as and the nuclei and involve only one active major shell, whereas rotations in heavier nuclei involve protons and neutrons from two major shells. However, the superdeformed rotational band in the N = Z nucleus [131] for which the spins, parities, excitation energies and in-band as well as out-of-band B(E2) values have been measured, has a configuration that involves two active major shells. The comprehensive spectroscopic information for these collective states has enabled a detailed comparison with both the cranked shell model and large scale shell model calculations. The study of rapidly rotating nuclei with equal, or very nearly equal, numbers of neutrons and protons has advanced considerably over the past several years. Improved experimental techniques, which combine the increased gamma-ray detection efficiency of the new generation of gamma-ray arrays with the highly selective reaction channel tagging capabilities of recoil mass spectrometers as well as charged particle and neutron detector arrays, have been a main reason for these advances. The population of high spin states in N = Z nuclei by heavy ion fusion evaporations reactions is limited by the availability of suitable stable ions and by the extremely small production crosssections for nuclei far from beta stability. Advances in present experimental detection techniques will continue to enable progress in this area, but we may also expect other techniques, such as production by projectile fragmentation and re-accelerated radioactive beams, to play an increasingly important role in studying N = Z nuclei.

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5.4. Shears Bands The existence of a rotational spectrum is one of the defining characteristics of a deformed nucleus. The deviation from a spherical shape breaks rotational invariance and enables an orientation of the system to be specified in the intrinsic frame. Collective rotations involving many nucleons coherently aligning their spins are then possible. The nuclei described in this Section exhibit rotational-like spectra but have very small quadrupole deformations; the observation of these spectra has led to the identification of a new excitation mode in nuclei, namely, the shears mechanism. The essential shears mechanism involves a weakly deformed nucleus with the angular momentum generated largely by the behaviour of valence proton and neutron configurations, with associated angular momentum vectors and That this particular coupling of valence nucleon spins in a nearly spherical nucleus can result in a rotational-like spectrum was completely unexpected, and must surely rank as one of the most striking discoveries in nuclear structure in this last decade. The coupling of the shears blades’ angular momentum is shown in Fig. 5.5. The gamma-ray spectrum of the shears band is shown in Fig. 5.6 together with that of the superdeformed band. The regular spacing of gamma-ray transition energies typical of a deformed rotor is evident, and the rotational-like behaviour of the shears band is illustrated further in the energy versus angular momentum plot in Fig. 5.6. The excitation energy of this shears band follows an dependence, a feature common to most shears bands and we will find that it is the perpendicular coupling, which gives the bandhead spin, The rotational-like bands of magnetic dipole transitions later known as shears bands, were first observed in light Pb nuclei[269, 270, 271, 272]. To date, approximately fifty such bands have been identified in A ~200 nuclei, and as well as in several isotopes of Cd and Sn near A ~110. They are characterized by strong M1 transitions, but weak E2 transitions, and thus large B(M1)/B(E2) ratios. The M1 transition probabilities are a consequence of the large perpendicular component of the magnetic moment, since

The small quadrupole deformations have been confirmed by lifetime measurements[273] that gave quadrupole deformations of the order of 0.1.

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The shears mechanism, proposed by Frauendorf and based on the results of the tilted axis cranking (TAC) model[67], involves the generation of angular momentum through the alignment of a few specific proton and neutron states. In particular, for Pb isotopes, the occupation of high-K and/or proton states results in a large angular momentum vector, parallel to the symmetry axis, while the occupation of low-K, neutron-hole states gives rise to an angular momentum vector, aligned perpendicular to the symmetry axis. The total angular momentum is then tilted away from a principal axis, e.g., the 3-axis, by an angle Fig. 5.5. Thus aligned, the protons and neutron-holes have the maximum overlap with the core, hence the minumum energy; this perpendicular configuration coincides with the band-head configuration with energy and angular momentum Within the band, states of higher angular momentum arise from the alignment of the angular momentum vectors of the proton, and neutron-hole configurations in the direction of the total angular momentum, I. This process resembles the closing of a pair of shears where the vectors constitute the blades of the shears. Each blade is configured from several nucleonic orbitals, and within a blade the magnitude of (or ) remains fixed; in other words, the blades are considered to be rigid. The relevant degree of freedom is then the shears angle, which decreases with increasing angular momentum. An inevitable consequence of the shears mechanism is a reduction in the B(M1) values with increasing angular momentum. This follows because as the angle between the and closes to generate more angular momentum, the magnitude of is reduced. This characteristic feature of the shears mechanism, and summarised in Fig. 5.7, was observed in[274, 275, 276], providing the experimental confirmation of this excitation mode. The magnetic character of the transitions was unambiguously determined through electron conversion and linear polarization measurements[273]. A review of the experimental properties of shears bands is presented in[273]. High-K rotational bands with substantial prolate deformation, and exhibiting large B(M1)-values between the signature-partners have been known for many years prior to the discovery of shears bands. The semi-classical result of Donau and Frauendorf describes their M 1 transition strength by||

|| More generally, one can write an expression in terms of deformation-aligned (contributing

both signatures to the band), and rotational-aligned (contributing only the lowest signature) orbitals. The form given here is specialised to the lead region where, in this model, neutrons are rotationally-aligned, contributing one signature to the band.

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As with the shears bands, the B(M1)-values are explained by the large perpendicular component of the magnetic moment (the high K-value), and by the effect that the rotational aligned neutron adds coherently with the deformation aligned proton. What sets the shears bands apart from well-deformed bands with large B(M1)-values is their behaviour with increasing spin. The latter exhibit rather constant B(M1)-values with spin, whereas the shears bands show strongly decreasing trends with increasing spin, (c.f., Fig. 5.7). The observed B(M1)values for shears bands could be reproduced in a Donau and Frauendorf description by taking K as a free parameter, and allowing it to decrease with increasing spin. In the lead region, this allows the proton angular momentum to align in the direction of the rotation axis, which is essentially a shears mechanism.

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That the shears mechanism should lead to rotational-like behaviour is not obvious. In an attempt to explain this, and to retain the concept of rotational motion implied by the observed spectra, Frauendorf suggested[277] that it is the large magnetic dipole moment, that breaks spherical symmetry and allows an orientation to be specified. The magnetic dipole rotates about the total spin vector emitting M1 radiation and leads to a rotational band; this motion has been termed “magnetic rotation”. (Shears bands are sometimes referred to as magnetic rotors.) The idea that internal anisotropies, such as magnetic dipole vectors or nucleon current loops, can lead to spontaneous symmetry breaking and nuclear rotations is reviewed in Ref. [278]. In a series of papers Macchiavelli et al.[279] studied the shears mechanism with a semi-classical description that emphasized the role of the shears angle between and and the interaction between the proton and neutron blades. They concluded that the rotational character of the bands can be understood as an effective interaction of the form It was suggested that the interaction may have its origin in particle-vibration coupling to quadrupole phonons. These authors also noted that the term resembles the quadrupole-quadrupole force of the pairing-plus-quadrupole Hamiltonian. An analysis in terms of effective interactions does not require a deformation, nevertheless the appearance of a -like term, which is used to describe a deformed nucleus, can be considered a natural link between this interpretation and the deformed mean-field TAC model. Intuitively, we must expect that there is a continuous evolution from the situation of a well-deformed nucleus coupled to deformation- and rotationalaligned quasiparticles, to the situation of a weakly-deformed nucleus coupled to the shear’s blades. This question has been considered by Macchiavelli et al.[280], but much remains to be done. In a similar search for generality, Ragnarsson (c.f., e.g.,[281]) has pointed to the parallels between the shears mechanism, and the process of band termination dicussed earlier.

6. 6.1.

STUDIES OF THE QUASI-CONTINUOUS RADIATION Introduction

Although the study of the discrete peaks of the gamma-spectrum is technically simpler, there are interesting problems for which only study of the quasicontinuous radiation can provide answers. As its name suggests, the quasicontinuum is not a true continuum in that the number of paths is finite, gammaray energy is not a continuous variable and what is designated as continuum versus discrete is purely a matter of instrumental performance. We have arbi-

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trarily divided this topic into: a) Rotational Damping; and b) Spectroscopy. The study of rotational damping is directed towards understanding rotational phenomena in an environment where the average spacing between levels of the same spin and parity is comparable to, or smaller than the typical interaction matrix element; that is, a regime where the levels are begining to be strongly mixed so that there are no identifiable configurations. There are a number of interesting consequences and phenomena predicted to occur; including, and perhaps most importantly, the order-to-chaos transition in a manybody, quantal system. The spectroscopy of quasi-continuous radiation as we describe it here is directed mainly to the study of nuclear properties at the most extreme angular momentum sustainable. The chapter closes with a discussion of angular momentum effects in fusion.

6.1.1. Rotational Damping In the initial de-excitation from states of very high angular momentum, many if not most of the decay paths are far removed from the yrast-line, and the gamma-ray flux is diffused. Superdeformed bands are an exception to the rule, but generally speaking not until relatively low spins are reached does the flux concentrate to sufficiently few paths that one can recognise discrete peaks. This happens when the intensity of a given path exceeds some level determined by the sensitivity of the experiment. Decays down the very large number of pathways for which discrete peaks are not resolvable constitute the quasi-continuous radiation. If most pathways involved decays within a single rotational band, one might expect, as did the early workers in this field, that there should be a high degree of order in the quasi-continuous radiation. For example, the coincidence gamma-spectrum generated from a single, narrow gate (say equal to the detector resolution) set on anywhere in the quasi-continuum should show “peaks” at rotational energy spacings. These peaks, resulting from a sum over very many bands, should be broadened, corresponding to dispersion in the moments of inertia and aligned spins. An equally characteristic feature in the spectrum would be a conspicuous narrow dip centered at the gate energy and reflecting the fact that in a rotational band no two transitions have the same energy. In a 2-dimensional visualisation of a coincidence matrix these features translate into a series of ridges lying parallel to the diagonal and a valley lying on the diagonal. The separation of the two ridges closest to the valley is related to the dynamical moment of inertia by:

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The ridge-valley structure in a two-dimensional spectrum may be difficult to visualise in raw data because the background is falling very rapidly with increasing gamma-ray energies. Background can be removed in 2-dimensions by applying either the COR procedure of Andersen et al.[282], or the procedure of Palamata and Waddington[283]. There are also treatments for handling higher-order coincidence arrays[284]. A good example of a ridge-valley structure is shown in Fig. 6.1[285]. The ridge is atypically strong and at low-spin, it accounts for approximately 25% of the total flux of gamma rays passing down the level scheme: discrete peaks that can be recognised as such have been subtracted out. When we analyse the data shown in Fig. 6.1, it becomes clear that the quasi-continuous radiation is not behaving in a way consistent with the premise of the above discussion. The ridge structure is not as pronounced as we expected, higher order ridges appear to be entirely missing and the valley is very shallow. With a historical perspective, we can see that prior to experiments with Compton suppressed detectors, these problems were to some extent rationalised, but with the advent of good Compton suppressed data, the fact that only a small part of the quasi-continuous radiation could be attributed to decays down well-behaved rotational bands could no longer be ignored, as was first pointed out by Love et al.[286]. Later work by Bacelar et al.[287], and by Draper et al.[288] showed that a large fraction of the gamma-ray flux passed through cascade sequences where the transition energies were only loosely correlated to spin, and were dispersed about the rotational value. Early workers represented this dispersion by a Gaussian distribution of gamma-ray energies about the rotational values with a FWHM of up to 125 keV. The term “Rotational Damping” as applied to this phenomenon came into general usage in 1986, e.g., Ref. [289]. It is now well established that there are two main contributors to the effect. First, there is the mixing of states (of the same spin and parity) which happen to fall within the interaction range. At very low level density, this happens with a low probability (c.f., Bosetti et al.[290]), and it perturbs the level energies and hence gamma-ray transition energies as shown for the case of simple 2-band mixing in Fig. 6.2. The mixed levels are each perturbed by up to V, where V is the interaction strength, typically 10 keV. Second, and more significantly, the decay can now jump between the bands. When this happens, there is potentially a large change in the transition energy away from a rotational sequence because the new band will generally have a different rotational frequency at that spin. In the language of rotational damping, the configuration mixing occurs over an energy range characteristic of the interaction and causes a dispersion in the level energies (about the rotor values) as illustrated in Fig. 6.3. The damping caused by the dispersion

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in at a given spin, which arises from variations in the moments of inertia and spin alignment between bands is denoted It should be noted that the greater disruption to the matrix is caused by the jumping between bands allowed by the mixing. The field has been mainly concerned with measuring the damping effects and extracting nuclear structure information from the results. Simulations of the decay cascades have played a major role in developing the field. The damping parameters may be expected to depend on spin and excitation energy (above yrast) as well as on the general location of the nucleus in Z and A, but probably not the individuality of the nucleus. The dependence on excitation energy, or temperature, is a major complication in the analysis since it is difficult to isolate cascades of a specific temperature. According to Broglia et al. [289]

where U is the excitation energy. The quantity

has two limiting behaviours:

The variation of with temperature is sketched in Fig. 6.4[289]. At low temperatures the damping is small, as is observed, but it goes through a maximum with increasing temperature. The tendency of to decrease with increasing

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temperature is analagous to the phenomenon of “motional narrowing” which is well understood in the field of nuclear magnetic resonance. This prediction has not been confirmed experimentally.

6.1.2. Some problems for the future Because and have different dependencies on spin and on temperature, Mottelson has pointed out[291], that there may be a regime of spin and temperature for which whilst where is the level density for states of the same spin and parity. If this situation can exist in nuclei, it would prove extremely interesting to study if the relevant region of spin and temperature could be reached. Physically it corresponds to a situation in which there is dynamical chaos in the sense that the mean-field configurations are completely mixed, but the rotational damping is so small that each state of spin I decays uniquely to a daughter of spin I-2. The term “compound bands” has been applied to describe this phenomenon. In extracting the critical parameter from experimental data it is neccesary to compare with elaborate Monte-Carlo simulations of the gamma-ray cascades. Until recently it was assumed that states of a given spin were completely mixed over the spreading width by the interaction. In the simulation it was thought permissible to randomly select transition energies independently for each step of the cascade. More recently, explicit calculations of the wavefunctions for the mixed states have become available, and it is now possible to select transition energies in a two-step decay where the structure of the three

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states involved is taken into account[292, 293, 294]. A very surprising result of this treatment is that even at high temperatures, the distribution of transition energies shows a narrow, as well as the usual broad component. The narrow component presumably arises because these two-step decays are selecting states which are related to one another despite the mixing. This correlation was of course missed in simulations that chose the transition energies independently. The term “scar” has been applied to these short two or three-step cascades predicted by these calculations. The ridge/valley structure is sensitive to scars. If they exist, however, it may be very difficult to isolate the effect experimentally. It has proven surprisingly difficult to obtain reliable values for from experiment. At low temperatures, selected by applying coincidence conditions on low energy gamma-rays at the lower-spin reaches of the gamma-ray cascades (say 700 keV in the rare-earth region) the damping width is nearly zero, or at least Progressively higher energy gamma-rays in the region of the spectrum dominated by collective rotational transitions, often called the E2 bump region, are presumed to originate from progressively higher spins, and higher temperatures. Selecting these gamma-rays as coincidence gates causes the intensity of the undamped component to decrease, but to date, there has been no convincing data on the variation of the wide component of the damping width with temperature. Bacelar et al. [287], studied the reaction at 200 MeV and measured the wide component of to vary from 75 ±20 to 110 ±40 keV with excitation energy above the yrast line of 1.25 ±0.2 to 1.0 ±0.2 MeV respectively. Draper et al.[288], studied the reaction at 185 MeV and found that the FWHM of the Gaussian describing the distribution of rotational gamma-ray energies (which is equivalent to ) varied from 90 ±15 to 125 ±20 keV over the gamma-ray energy range 960 to 1200 keV. In a later study of three reactions leading to isotopes of Hf, Er and Nd nuclei, Stephens et al.[295], concluded that the wide component of was ~300 keV. A key issue in comparing these experiments is whether or not a quantitative measure of the intensity of the “missing” transition was obtained experimentally. There is a concern that the narrow values for were obtained without such an assessment, whereas the wider value better accounts for the missing strength. Studies of superdeformed nuclei have shown that there is a quasi-continuous component in the gamma-spectra in coincidence with discrete superdeformed bands. Within the superdeformed well, states are isolated by a potential barrier from the high level density of normal-deformed states in the first well. The height of the barrier therefore limits the excitation energy in the superdeformed well and suggests that rotational damping of superdeformed states should be

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minimal. To date there have been very few measurements[296, 297]. An experiment on by Vivien et al. [296], that gave which is much narrower than damping widths typically observed in normal-deformed states. The small value for may indicate that the dispersion in between superdeformed bands is much less than it is between normal-deformed bands (c.f., identical superdeformed bands in Sect. 4). In this case, the mixing, and the speading width, might still be as large as in the normal-deformed case.

6.1.3. Fluctuation Analysis The subject of fluctuation analysis is closely related to that of rotational damping. The subject has been reviewed recently by Dossing et al. [298]. Basically, fluctuation analysis tries to measure fluctuations in the number of counts per channel in a small sector of a multi-dimensional coincidence array. As first discussed by Stephens[299], a part of these fluctuations will be statistical, and a part will be related to the number of pathways that place coincident in the selected small sector. This latter “grainyness” will persist at arbitrarily good counting statistics. In the context of n-dimensional fluctuation analysis, a pathway is the coincidence of n-gamma-rays in the small sector selected out of the total space: the number of pathways is then equal to the number of different ways that this happens. In the rare-earth region, rotational transitions are typically separated on average by approximately 60 keV and it is then appropriate to choose the ndimensional sector in which the fluctuations are to be measured to have sides of 60 keV. Empirically, the results are largely independent of this choice. The channel size should ideally be comparable to, or slightly larger than, the detector resolution, and a typical value is 5 keV. One of the most important results to come out of this analysis is a measurement of the number of paths on the ridge and in the valley made by Herskind et al.[300]. They conclude that in populated at high spin, that there are about 30 rotational bands on the ridge with very little branching between them. By contrast, they find that the number of paths in the valley is of the order and that the excitation energy dividing the two regions is approximately 750 keV.

6.1.4. Spectroscopy Components of the Spectrum In nuclei with collective rotational states at high spins, the gamma-radiation following their population by fusion reactions is generally considered to com-

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prise the following components[301]. 1. a statistical component 2. damped rotational cascades 3. discrete transitions

These are indicated in Fig. 6.5[302], and we begin this Section with a short discussion on each component of the spectrum. In the event the nucleus has a multi-particle structure, or exhibits non-collective rotation about the oblate symmetry axis, there will be a corresponding component in the spectrum. Such a component would be rather featureless and difficult to identify experimentally, nevertheless, we discuss the possibility later in this Section. Statistical gamma-rays predominantly originate in the entry region and cool the nucleus down towards the yrast line. Gamma-rays decanting low-spin states of rotational bands to the yrast line, or non-collective transitions between excited bands are generally included in the statistical component. The shape of the statistical spectrum is approximated by:

where is a temperature and in the simplest model, for dipole transitions, or for quadrupoles. However, since the strength-function could have structure, most authors have felt it better to treat as a free parameter to be fitted to data. One of the most convincing measurements of the statistical spectrum by Sie et al.[303], concluded that the parameter was in the range 3 to 5 and that seemed a reasonable choice. Generally, this equation cannot reproduce the region from to where the data tend to drop more rapidly than exponential. This problem comes from assuming just one temperature; MonteCarlo simulations which keep track of the temperature as the nucleus cools can reproduce the data[304]. The damped rotational cascades contain mainly stretched E2 transitions, but a priori, there could also be decays down strongly-coupled bands that would go by both stretched-E2’s and by stretched M1’s. In many, if not most nuclei prepared at high angular momentum, the occurence of rotational-like cascades produces a bump in the quasi-continuous spectrum. The bump has a well defined upper-edge, first characterised by Banaschik et al.[305], and illustrated in Fig. 6.6[306]. The edge of the bump arises from a correlation of with I; for a rotor this would be and its presence is an indication of either: (1) a well-defined maximum angular momentum in the reaction; or (2) a termination of the rotational-like structure at a well defined spin, for example due to fission, or to some nuclear structure effect.

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The experimental decomposition of the spectrum into its components is greatly facilitated by measuring the multiplicity, M, as pointed out by Deleplanque et al. [307]. If the spectrum is unfolded to take out the detector response, and normalised to the measured multiplicity, i.e., the integral over the spectrum is set equal to M, then the multiplicity of any component or feature in the processed spectrum is given by intergating over it. Determination of Gamma-ray Multiplicity Determination of the average multiplicity of a gamma-spectrum requires at a minimum two gamma-detectors. To lowest order:

where refers to the number of counts in coincidence with the gating detector (or in singles) selected on the whole spectrum, or on some feature of the spectrum. A number of corrections must be made, primarily to account for multiple hits on a detector as described by Hagemann et al. [308]. In the earliest experiments studying quasi-continuous radiation, gammaray multiplicity was varied by changing the bombarding energy, thereby changing the input angular momentum: the multiplicity was determined by coincidence measurements with a few detectors as described above. The introduction

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of sum gamma-ray energy spectrometers allowed selection of specific sumenergy, H, which is strongly correlated to multiplicity, M. With this method, the multiplicity was selected by a cut on the H-spectrum, and the value selected determined with coincidence techniques (c.f., for example Ref. [309, 310, 311, 312]. The advent of large gamma-arrays offered new possibilities since H and K (the number of hit detectors) could be determined on an event-by-event basis. As examples, in recent experiments with the 8PI-Spectrometer and with Gammasphere, we have used the following methodology. The K-spectrum for a given reaction is determined. Given the response function of the instrument, which is measured with radioactive sources[8], we know the distribution governing how a unique M is detected as a distribution N(K) in K, as shown in Fig. 6.7. By computation we de-convolute the measured K-spectrum of the reaction into its corresponding M-spectrum. Furthermore, we obtain by this process vectors specifying the M-content of each K-value. The centroid of such distributions is then the (M)-value as previously understood. Angular Momentum Carried by the Spectral Components In studies of the quasi-continuous radiation one frequently needs to relate the measured multiplicity of the spectrum, or a feature in the spectrum to the average spin carried by the corresponding decay path. To do this we need the decomposition as described above, namely C(collective), D(discrete), and S(statistical), measuring the average multiplicity of each component and the corresponding angular momentum carried, per gamma-ray, The angular momentum of the average pathway is then:

The angular momentum of each component can be approximately determined by experiment. Measurements of the angular distributions about the beam axis of the statistical component have shown near isotropy[302, 313, 314]. In the system linear polarisations have also been measured[314], and electron conversion co-efficients have been measured in [315]. Both these measurements indicate that the statistical transitions are predominantly electric dipole. Assuming that these transitions are pure dipoles, then near isotropy implies a decomposition of stretched and non-stretched the average spin removed per transition is then More recent measurements, e.g.,[306] have shown that the statistical spectrum is not quite isotropic, but exhibits systematic shifts at the 10% to 20% level, these are correlated with entry spin and with the mass of the residue. These effects can probably be understood as arising from changes in the

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proportions of and dipole transitions with temperature. To date, no detailed studies have been performed but this explanation seems plausible. The average number of statistical transitions per cascade, has been measured by a number of authors in various systems. In the region, Ward et al.[302], concluded that increased slightly with increasing bombarding energy and lies in the range 4 to 6 transitions. Radford et al.[316], obtained the value 5.2 in with a high spin reaction. With a low-spin induced reaction leading to Farris et al. [317], measured 4.26 statisticals. The number of discrete transitions in the average cascade is readily obtained by subtracting away the peaks in the processed, multiplicity-normalised spectra. This quantity has been measured many times in various systems; for example typical values of in were in the range 3 to 5 transitions per cascade, but this refers to states above the 49/2 isomer and the discrete paths below that spin should be added to those numbers. Nuclei with yrast traps, such as are perhaps not typical cases since the trap attracts gamma-ray flux to the yrast line. In typical rotors, values are in the range 5 to 8 transitions per cascade. In the nucleus Radford et al., measured 9.1 discrete transitions per cascade. The spin removed by discrete transitions can be derived from the level scheme. The remaining angular momentum in the balance is that carried by the collective components, which is two units per stretched E2 transition, and one unit per stretched dipole. We have outlined above how the spectrum may be treated in order to derive the spin carried by each component. These methods have the advantage of providing explicit answers, and historically, this was the only way to proceed. With the advent of very high-speed computers, it is now possible to simulate the gamma-ray cascade in great detail with Monte-Carlo techniques. In some applications, it may be more convenient or more advantageous to use a MonteCarlo approach, and to circumvent the analysis described above by moving directly to comparison with the simulated spectrum. Features of the Spectrum Although the quasicontinuous spectrum is smooth, there are some features from which it is possible to extract nuclear properties. For example, the edge of the E2-bump generally moves to higher gamma-ray energies with increasing spin input. If we assume that the edge was caused through the spin distribution in the entry region feeding into rotationally correlated states with a near constant moment of inertia, then the mid-point of the edge can be associated with the spin at which states at the top of the cascade are on average populated 50% of the time. This spin will be close to the average entry spin minus the spin carried by the statistical cascade. From this we can extract a kinematic

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moment of inertia by:

where is the energy of gamma-rays at the mid-point of the bump edge. As we show below, the position of the bump edge can be misleading if the nuclear moment of inertia is varying with spin in the feeding region. In our example reaction the mid-point of the bump edge (c.f., Fig. 6.6) is 2410(30) keV where the estimated spin, following procedures of the previous Section, is This implies Alternatively, the gamma-ray energy associated with particular spins can be derived by subtracting the two spectra observed for two adjacent regions in entry-spin. This might be affected by: (1) changing the bombarding energy; or (2) taking two adjacent cuts either in total summed gamma-ray energy, H, or in the coincidence fold, K, at a common bombarding energy. The centroid of the peak in the difference spectrum is to be associated with the average energy of new transitions added to the top of the cascade on selecting an increased entry spin. In other words, this is another measure of the kinematic moment of inertia where is now the centroid of the bump in the difference spectra, and the associated spin, J, is the average value of the two entry spins in the subtraction. In our example, the centroid of the difference spectrum K= 35, minus K=33, shown in Fig. 6.8, lies at The average spin is estimated to be making The discrepancy between the results of these two analyses is because the centroid energy of the difference spectrum is lower than the midpoint of the edge, as can be seen directly in Fig. 6.8. This difference arises in turn from the increasing with angular momentum in the system; this is not a universal feature, for example, the does not show the effect. We conclude that the position of the bump edge is not a reliable measure of the gamma-ray energies at the top of the cascade, although other methods, such as bump-centroid differences, will give the correct results. In some instances the E2 bump has a well-defined lower edge, (in addition to an upper edge). This is believed to happen in nuclei where collective behaviour does not become established until some critical angular momentum is exceeded. The energy of the lower edge is then related to spin at which the transition to collectivity occurs, (c.f., Ref. [307]). Holzmann et al.[318], studied the isotopes populated at high spin. They found that behaves very differently from the other isotopes as shown in Fig. 6.9. The second peak observed in is attributed to the existence of a prolate to oblate shape transition at around spin More recently, new experiments on confirming this result have been reported by Ma et al. [319].

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One can experimentally determine the multiplicity of a spectrum on a point-by-point basis, and histograms of such spectra generally exhibit some features as illustrated in the top panel of Fig. 6.8, and whose origin is explained in Fig. 6.5. The first measurements of such spectra were made by Newton et al.[320], and more detailed interpretations given by Deleplanque et al.[321], and by Ward et al. [302]. The feature in the multiplicity spectrum in the example, (top panel of Fig. 6.8) centered near the peak of the E2 bump, is due both to the correlation of with J, caused by some nuclear structure effect, (usually collective rotation), and to the finite width of the spin distribution of the entry region, to Experimentally, this feature is a clear indication of: (1) a correlation of with J; and (2) that these transitions lie in the feeding region between to It is evident that transitions lying at spins

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below the average entry spin contribute very little to the multiplicity feature, and those lying below contribute not at all.The absence of a multiplicity feature means that either there are no spin-energy-correlated transitions, or that these transitions, if they exist, lie outside the spin range defined by to In the point-by-point multiplicity spectrum shown in Fig. 6.8, the centroid of the feature is which is appreciably higher than the bump-difference centroid discussed above as can be seen directly in Fig. 6.8. From inspection of Fig. 6.5, one can see qualitatively that the centroid of the multiplicity feature must always lie higher than that of the bump difference. To be quantitative, it is necessary to take account of the statistical transitions that underlie the E2 bump and dilute the multiplicity feature from the ideal depicted in Fig. 6.5. Processed spectra normalised to multiplicity, as we have described above, have the property that the number of transitions, per unit energy bin (MeV) in the yrast cascade is given directly by:

for states lying below which can be sampled in every cascade (the term full feeding was used in Ref. [312]). The method can be applied to states above by introducing a feeding correction (e.g.,[312, 311]. The rotational frequency corresponding to is just the place in the spectrum where the feeding feature in the difference spectra (c.f., Fig. 6.8) disappears on the low energy side. The same value can also be determined from where the multiplicity feature disappears (c.f., Fig. 6.8). In our example reaction at 207 MeV, the gamma-ray energy corresponding to is approximately 1700 keV or at this rotational frequency there are 1.08(4) transitions per 100 keV in the processed spectrum (after subtracting the statistical component) corresponding to

6.2.

The Nuclear Inertia at the Highest Angular Momentum

The most general application of the methods described in this chapter has been to probe the moment of inertia of the nucleus at the highest spins attainable. Here we will focus on a few experiments, each of which has studied a wide range of masses. One of the first experiments to do this with a sum spectrometer was by Folkmann et al.[310]. They studied the reactions and over a range of bombarding energies. For the case of at the highest bombarding energy they found for which is 30% higher than the rigid sphere and corresponds to an axis ratio close to 1.6:1. For the

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residues they found 2 for which is 5-15% higher than the rigid sphere. The experiments of Macchiavelli et al.[311], measured with a sumspectrometer reactions of beams on six targets ranging from to for energies in the range 170-185 MeV. At low rotational frequencies, they found close to the rigid sphere but in the case of and targets, increased to approximately 50% greater than the rigid sphere at the highest spins measured. Values for in these experiments were derived from both determinations of the E2 bump centroid in difference spectra (c.f., Sect. 6.1.4.4) versus angular momentum, and from integrating the experimental

These methods gave consistent results, and the derived values are rather close to the rigid sphere. Similar experiments on isotopes of Er, Yb, Hf, and W were performed by Deleplanque et al. [312]. In all cases except, for tungsten isotopes, the values begin to rise above the rigid sphere value beginning around and reach approximately 1.4 to 1.6 times the rigid sphere value by which is the highest frequency measured. In this experiment, the values, specifically in and are approximately 17% higher than the rigid sphere. More recently, measurements have been made with beams on targets of and using both the 8 PI Spectrometer and Gammasphere[306]. The results have a greater statistical significance and show more detail than earlier experiments, but the conclusions are similar; namely that at some spin range, the moments of inertia derived from the E2 bump centroid in difference spectra for A=98, 112 and 144 compound systems start to rise above the spherical rigid body value, and at the highest spins reach about 35% to 40% greater for the A=98 and 112 systems (I=44 and respectively), and 10% greater for the A= 144 system The A=168 compound system shows a nearly constant closely equal to the rigid sphere value up to the highest spins measured, namely Changes to can arise through effects of the pairing gap, the spin aligned to the rotation axis, i, or the deformation, The corresponding effects on may be obtained by integration as given above. We may write:

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The possibility that changes in the pairing gap could cause effects as large as those observed is remote, since the spins are so high that most of the pairing will be already quenched.

6.2.1.

Alignment gain and deformation at high spin

Alignment gains from a crossing with the intruder orbital within the valence space are commonplace and occur between spin 10 to in nearly all deformed nuclei. At much higher spin, or rotational frequency, there might be alignment gain from crossings of the intruder orbital in the next shell higher than the valence shell. At such high rotational frequency these would be presumably unpaired crossings. Macchiavelli et al.[311], have given a simple prescription to estimate the critical frequency for this crossing, namely Furthermore, since the spin of the intruder orbital is roughly then a typical alignment gain will be The data of Refs. [310,311,312,306] which cover the mass range A ~80165, all show that the moment of inertia starts to rise at sufficiently high and indeed, the onset frequency scales as but it occurs too early, near The fact that the alignment gain happens so much earlier than estimated may not rule out this effect as the main cause of the rise in the moment of inertia. All of the simplifying assumptions in deriving in Ref. [311] will tend to make it too high. The gain in aligned spin is conveniently given in terms of the experimental values by:

where is a reference value. The very pronounced increases in the moments of inertia, published in the references given above correspond to small gains in aligned spin: typical values would be in the range which is compatible with the gain expected from the higher intruder orbital. As with the case of aligned spin, modest increases in deformation with rotational frequency can lead to large changes in the moment of inertia, The moment of inertia for a rigidly rotating prolate spheroid with a deformation parameter is given by:

For in the range 0.3 to 0.6 (normal to superdeformed) we can estimate the factor entering into Eq. 6.12 and write:

High Spin Properties of Atomic Nuclei

For example, if require only that modest.

is to increase by a factor of two at increase by 0.1 per 0.1 increase in

277

we which is

6.2.2. The Jacobi Shape Transition in Nuclei In their paper on the properties of rotating liquid drops, Cohen et al.[322], discussed the evolution of fluid shapes with increasing angular momentum. They pointed out the universality of the behaviour as it pertains to rotating liquids, e.g., rotating planets, idealised stars, and atomic nuclei. A pedogogical article on this has recently been published by Ts. Dankova and G. Rosensteel[323]. For liquid drops, there is a critical rotational frequency (or angular momentum), at which there is a bifurcation between the McLaurin ellipsoids, which are oblate, and the Jacobi ellipsoids, which very quickly become very elongated, with only a slight residual triaxiality. In nuclei, depends on two parameters, namely, x which measures the ratio of the disruptive Coulomb force to the cohesive surface tension:

and y, which measures the ratio of the disruptive centrifugal forces to the surface tension: The parameter x is the fissility parameter. Cohen et al., give the square of the critical angular momentum as specified by y in the following form:

for The situation is illustrated in Fig. 6.10. In nuclei, as in other systems, the transition to Jacobi shapes is signalled by a drop in the rotational frequency at a critical angular momentum; further increases in the angular momentum result in still lower rotational frequencies, until fission is reached. For collective rotation in nuclei, there is an identity between rotational frequency and gamma-ray transition energy in a band, Eq. 1.3, therefore in nuclei, the signature of a Jacobi shape transition is a decrease in with increasing angular momentum – in other words, a strong backbending effect. Experimental problems with Jacobi shapes are in preparing nuclei at rotational frequencies (angular momenta) above the “phase transition” and in isolating and identifying the continuous gamma-radiation from those states above the phase change.

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Having produced a compound nucleus at the highest angular momentum reaction dynamics will allow, we need to consider effects that would decrease the angular momentum before the emission of rotational gamma-rays. The most limiting of these is fission. For better or for worse, the Jacobi effect and fission are tied to the same physics, and there is only a narrow zone of angular momentum where the Jacobi shape can exist without fissioning. Other factors to consider are the angular momentum removed by evaporated particles, and by statistical gamma-rays. In the mass region of interest here, the simulation code evapOR[324] predicts the average angular momentum carried away by each evaporated proton, and/or each evaporated neutron is approximately in neutron-rich systems, the neutron-value is slightly bigger than the proton-value, and vice-versa in proton-rich systems. The angular momentum carried away by ranges over We see that in the typical 4-neutron evaporation channel, when we include the spin carried by the statistical cascade, say typically there will be a loss of approximately before there is a chance to see the Jacobi signal in the rotational cascades.

High Spin Properties of Atomic Nuclei

6.2.3.

279

Oblate states at very high spin?

For rotating liquids, the Jacobi transition marks a bifurcation between oblate and prolate (slightly triaxial) states. For a nucleus, in a liquid-drop, or in a Thomas-Fermi description[325], nuclei rotate about their oblate symmetry axis as described classically by McLaurin spheroids, e.g., Ref. [323]: however, there is no direct experimental evidence that such states exist at high angular momentum. This type of non-collective behaviour in real nuclei is seen in the discrete gamma-ray spectrum most clearly in the region near [326], and in the region near [327]. In their 1981 paper, Bohr and Mottelson[328] saw this as “....nuclear matter under fascinating new conditions representing a novel type of symmetry breaking....”. The excitation energy, along the yrast line in these nuclei is proportional to the square of the angular momentum, over a wide range of spin. The relationship is not so smooth as for rotors in the collective sector, nevertheless, on average, the experimental values for the moment of inertia, are close to the rigid-body values, and sometimes exceed them by an amount consistent with a small oblate deformation. A gradually increasing oblate deformation with increasing spin is expected from the polarising effect of so many particles orbiting in the equatorial plane, and it has been proven experimentally in a few cases by measuring spectroscopic quadrupole moments of the high spin isomers[329]. In a very general discussion of nuclear shapes at very high spin, Stephens[330] pointed out that the largest moment of inertia for spheroids of comparable deformations rotating rigidly occurs for the oblate shape rotating about its symmetry axis. Therefore, at sufficiently high spin, this mode should always lie lowest in energy unless some other effects, such as shell structure, or a Jacobi transition should intervene. Unfortunately, the discrete spectrum expected for a non-collective oblate rotor is not resolvable at spins where a Jacobi transition might be expected, and it has proved difficult to examine this problem by experiment. In the nucleus rotational bands built on a normal (prolate)-deformed state, and on a superdeformed state co-exist over a wide range of spin and excitation energy with the oblate states, and all are populated in an appropriate nuclear reaction[331]. We can speculate that if non-collective oblate states lying far above the yrast line exist, and are populated at very high spins, then the associated continuous gamma-spectrum would be featureless, and difficult to identify experimentally. Nevertheless, the presence of this radiation can be inferred from the multiplicity-normalised difference-spectra (c.f., Fig 6.8). In cases where all of the normalised difference-spectrum is not concentrated in the E2-bump region, one may assume that the balance must lie in some

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featureless spectrum, plausibly characteristic of non-collective oblate rotation. There is evidence in the region that the E2-bump does in fact exhaust most of the normalised spectrum[321, 312], but in lighter nuclei, there seems to be a substantial non-collective component[306]. In some special cases such as there is evidence for both oblate and prolate shapes where the continuous radiation is concentrated into two nearly resolvable E2-bumps, but in this case they are both collective rotations[319]. What happens to nuclei at the highest angular momentum sustainable before fission remains a mystery. The large increases in the moment of inertia for collective rotation observed in the continuous radiation are suggestive of either very deformed shapes, or strong spin alignment. If the effect is one of deformation, then it occurs at lower spin than expected for a Jacobi transition, however, it might be that the transition in nuclei is more gradual and diffuse than that calculated in existing models. This possibility has been considered by Myers and Swiatecki[332].

6.2.4. Populating states at the highest angular momentum. There is no good experimental basis for calculating the distribution of angular momenta contributing to fusion, or for calculating the maximum value, Various parameterisations have been given in the literature. A treatment that has the virtue of simplicity is given by Pelte et al.[333], as follows:

where

is the centre-of-mass energy, the radius for fusion is given by:

and µ is the reduced mass:

The interaction barrier

is given by:

where: The “constants” in this treatment have been tuned to fit experimental crosssections. The parameter is smaller than to account empirically for a zone of peripheral collisions which contain the partial waves of highest angular momentum, but which do not lead to complete fusion.

High Spin Properties of Atomic Nuclei

281

A more realistic treatment of the fusion process, in terms of the nucleusnucleus potential, also limits the maximum angular momentum contributing to fusion because at some there is no longer a pocket in the radial potential. The potential is evaluated with the nuclear part given by the so-called proximity potential[334]. For heavy nuclei, the condition that there be a pocket in the potential plays no practical role, since it comes into play for such high that fission is dominant. For light nuclei e.g., the pocket condition does limit the maximum to less than However, it is possible that orbital momentum may be damped into internal degrees of freedom, e.g., heat or deformation, that allow the partners to be captured and to fuse, despite the absence of a pocket in the potential. An understanding of the limits to the angular momentum of a fused system must also consider static and dynamical shape distortions that might allow fusion to occur at a larger radius than that given by a simple prescription such as Eq. 6.20 above. Experiments on these issues have been contradictory. The reactions and to the residue were compared in Ref. [335]. Bombarding energies were chosen to match the excitation energies and maximum angular momenta (calculated in a simple way, e.g., Eq. 6.19 above.) Calculations taking into account low-lying vibrational states of the reaction partners with the code CCFUS[336] indicated that the for the near-symmetric partners would be appreciably higher. In fact enhanced population of the high spin states was observed, in good agreement with the latter calculation. Similar experiments have been performed in the mass region[148]. Here the reaction partners and produced residues with rather different distributions of angular momentum, nevertheless, the calculated distributions after fission were found to be very similar. The population of superdeformed (SD) states was found to be stronger by about a factor of five in the symmetric-partner reaction. Since the feeding-profile of the SD band was similar in both reactions, it may be that the issue here is not one of angular momentum per se, but is rather evident for a memory effect, i.e., the symmetric entrance channel looks like the SD shape. Alternatively, we may suppose that the feeding profile into the SD band is controlled and defined by how population decants out of the “continuum” precursors; in this case, the feeding profle would be determined by nuclear structure, rather than by the angular momentum input. To explain the data would then require that the angular momentum dependence of fission was somehow calculated incorrectly, and that more of the high partial waves survived the symmetric reaction. More recent measurements compared the reactions and leading to normal deformed states of Hf isotopes[337].

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The angular momentum contained in the residues was compared by observing the multiplicity, K, and summed energy, H, measured in the BGO ball of the 8 PI spectrometer and gated on specific low-spin discrete lines to select the residue. These reactions have very similar trajectories of excitation energy versus angular momentum, calculated in a simple way. The results show that the for the various residual nuclei do not depend on the mass asymmetry of the entrance channel. The typical maximum angular momentum observed** was roughly in agreement with values calculated in the statistical model code COMPLET[338] of approximately The statistical model calculations predict much too small a cross-section for the 2n and 3n reactions in this work. The problem has been with us for many years, and several explanations have been offered[339]. The simplest explanation is that the yrast line at high spin is calculated to be too low in excitation energy, but much work needs to be done to pin down the factors influencing fusion cross-sections and the angular momentum contained by residues.

ACKNOWLEDGEMENTS We are very grateful for the suggestions, discussions, figures and critical readings of the manuscript, freely given by many of our colleagues. We give special thanks to R.M. Clark, M. Cromaz, R.M. Diamond, G.D. Dracoulis, R.V.F. Janssens, G.J. Lane, I.Y. Lee, A.O. Macchiavelli, W.D. Myers, D.C. Radford, I. Ragnarsson, K. Starosta, F.S. Stephens, M.A. Stephens, W.J. Swiatecki, C. Svensson, P.M. Walker, and K. Vetter.

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Chapter 4

THE DEUTERON: STRUCTURE AND FORM FACTORS M. Garçon DAPNIA/SPhN, CEA-Saclay 91191 Gif-sur-Yvette, France and

J.W. Van Orden Old Dominion University, Norfolk, VA 23529, and Thomas Jefferson National Accelerator Facility Newport News, VA 23606, USA

1.

A Historical Introduction

294

2.

The Non-Relativistic Two-Nucleon Bound State

297

3.

Static and Low Energy Properties

301

4.

Elastic Electron-Deuteron Scattering

305

5.

Theoretical Issues

319

6.

The Nucleon Momentum Distribution in the Deuteron

358

7.

The Deuteron as a Source of "Free" Neutrons

361

8.

Prospects for the Future

361

Appendix

365

References

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294

1.

A HISTORICAL INTRODUCTION

Diplon, deuton, deuteron: under different names, the nucleus of deuterium, or diplogen, has been the subject of intense studies since its discovery in 1932. As the only two-nucleon bound state, its properties have continuously been viewed as important in nuclear theory as the hydrogen atom is in atomic theory. Yet, ambiguities remain in the relativistic description of this system and the two-nucleon picture is incomplete: meson exchange and nucleon excitation into resonances should be considered in the deuteron description. The question of rare configurations where the two nucleons overlap and loose their identity is still under debate. We are still looking for the elusive effects of quarks in the nuclear structure. In this year of the millenium, the present article will first attempt to recall the early discoveries, measurements and theories. It will then boldly jump over decades of continuous efforts, building upon these, to present not an exhaustive review but an up-to-date status of our understanding of the deuteron, with a special emphasis on its electromagnetic form factors. To do justice to some seventy years of activity in this field is an immense task which is more easily approached by quoting here several reviews along this path [2, 9, 8, 3, 11, 6, 7, 1, 31, 10, 12, 4, 5]. The important subject of electro- and photo-disintegration of the deuteron will be only partly covered, referring the reader to [36].

1.1.

Discovery of the Deuteron

The existence of the first isotope of hydrogen was suggested in 1931 by Birge and Menzel [18] in order to remove discrepancies between two different measurements of the atomic mass of hydrogen. A first estimate of an abundance ratio 1 was inferred from this hypothesis, close indeed to the actual value of 6700. The stable isotope was discovered by Urey and collaborators [59] a few months later, investigating distilled samples of natural hydrogen for the optical atomic spectrum of in a discharge tube. Isotopic separation to study the properties of deuterium quickly became an intense activity. Its mass was measured by Bainbridge [13]. While Chadwick was discovering the neutron, several tens of papers were written, in one year’s time, devoted to the study of deuterium. An illuminating summary of this early research was made by Bleakney and Gould [20]. In 1933, “deutons” were used as accelerated projectiles first at Berkeley [47], then at Caltech and at Cavendish. Chadwick and Goldhaber [22] measured the first photodisintegrations in 1934.

The Deuteron: Structure and Form Factors

1.2.

295

Early Theories

In 1932, there was no satisfactory theory of the nucleus. The nucleus was thought to be composed of protons and electrons since these were the only known charged particles and nuclei were seen to emit electrons The electrons were needed to cancel the positive charge of some of the protons in order to account for nuclei with identical charges, but with different masses, and to allow for the possibility of binding of the nucleus by means of electric forces. This was clearly unsatisfactory because the Coulomb force could not account for the binding energies of nuclei and the attempt to construct the nuclei from the incorrect number of spin-1/2 particles could not produce the correct nuclear spins. The discovery of the neutron, shortly after that of the deuteron, did not immediately eliminate the confusion since the previous model persisted by simply describing the neutron as a bound system of a proton and an electron. Based on this faulty assumption, Heisenberg produced the first model of protonneutron force [37]. Since it was not possible to actually construct a description of the neutron with the model, Heisenberg simply assumed that the force could be described by a phenomenological potential and that the neutron was a spin-1/2 object like the proton. Based on an analogy with the binding of the ion by electron sharing, Heisenberg proposed that the force must involve the exchange of both spin and charge in the form of Forces containing the remaining forms of spin and isospin operators were soon introduced by Wigner [63], Majorana [48] and Bartlett [14]. In all cases the spatial form of the potentials was to be determined phenomenologically to reproduce the deuteron properties and the available nucleon-nucleon (N N) scattering data. In 1935Bethe and Peierls [16] wrote the Hamiltonian of the “diplon” with an explicit introduction of a short range interaction. This approach became the mainstay of nuclear physics which has produced considerable success in describing nuclear systems and reactions. The model of the neutron was not completely abandoned until after the Fermi theory [33] of decay became widely accepted. The progress in discoveries and understanding was then so great that, in spite of an otherwise bleak social or political situation in many countries involved, this period is recalled as “The Happy Thirties” from a physicist’s point of view [17]. One of the other great theoretical preoccupations of the late 1920’s and the 1930’s was the development of quantum field theory starting with the first works of Dirac on quantum electrodynamics (QED) [27], the Dirac equation for the electron [26] and the Dirac hole theory [28] with field theory reaching its final modern form with Heisenberg [38]. QED at this time was very

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successful at tree-level but the calculation of finite results from loops was not really tractable until the introduction of systematic renormalization schemes in the late 1940’s. The first attempt to apply quantum field theory to the strong nuclear force was Yukawa’s suggestion [61] that the force was mediated by a new strongly coupling massive particle which became known as the pion. This started another strong thread in the theoretical approach of the nucleus by using meson-nucleon theory to obtain nuclear forces consistent with the phenomenological potential approach. The primary attraction of this approach is that a more microscopic description of the degrees of freedom of the problem is provided and that additional constraints are imposed on the theory by the necessity of simutaneously describing nucleon-nucleon and meson-nucleon scattering. Ultimately, as it became clear that the mesons and nucleons were themselves composite particles, meson-nucleon theories were replaced as fundamental field theories of the strong interactions by quantum chromodynamics (QCD). However, the meson-nucleon approach is still a strong element in nuclear physics as a basis for phenomenology and is making a potentially more rigorous comeback in the form of the effective field theories associated with chiral perturbation theory. This situation is unlikely to change until it becomes possible to at least describe the N N force and the deuteron directly from QCD.

1.3.

Spin

Breit and Rabi [21] first suggested the use of magnetic deflection of an atomic beam in an inhomogeneous field to measure nuclear spins. The coupling of electronic and nuclear (J) spins is not totally negligible compared to the coupling of the electronic spin to the external magnetic field, provided the latter is weak enough. One then observes lines with a predicted intensity pattern. The atomic and molecular beam studies were to be implemented with great success (see Sect. 3.), but the first determination of the deuteron spin used other methods. Farkas and collaborators [32] demonstrated the ortho-para conversion in the diplogen (as they called the deuterium molecule) and determined the spin and statistics of the nucleus from the equilibrium ratio between these two states at different temperatures. They concluded that the diplogen nucleus must obey Bose-Einstein statistics, that the most probable value of its spin was 1, and that its magnetic moment was about one fifth of that of the proton. Using photographic photometry, the alternating intensities in the molecular spectrum of deuterium were investigated by Murphy and Johnston [51], who concluded that indeed J = 1 for the “deuton”.

The Deuteron: Structure and Form Factors

1.4.

297

Connection with OPE

The deuteron thus quickly appeared as a loosely bound pair of nucleons with spins aligned (spin triplet state). The existence of a small quadrupole moment (see Sect. 3.1.3.) implies that these two nucleons are not in a pure S state of relative orbital angular momentum, and that the force between them is not central. Taking into account total spin and parity, an additional D wave component is allowed. Such a D wave can be generated by the tensor part of the one-pion exchange (OPE) potential [3, 144].

2.

THE NON-RELATIVISTIC TWO-NUCLEON BOUND STATE

2.1.

The Potential Model of the Deuteron

The potential model of the deuteron is described by the Hamiltonian

where is the kinetic energy operator for particle and is the two-body potential. Successful N N potentials must have several basic characteristics in order to satisfactorily describe the deuteron static properties and the N N scattering data. The long distance part of the potentials is described by one-pion exchange while the intermediate and short range parts may be either parameterized in terms of simple functional forms, or obtained from models involving meson exchanges. The very strong anticorrelation of nucleons requires that these potentials be repulsive at short distances. The potentials must have terms involving scalar, spin-spin, tensor and spin-orbit forces. The tensor force is of particular importance in producing the single spin-1, iso-singlet deuteron bound state. The long range tensor force is provided automatically by the exchange of the pseudoscalar pion. Modern phenomenological potentials also include additional nonlocalities by means of terms quadratic in the relative momentum and/or quadratic spin-orbit terms. Improved fits to scattering data also require that isospin symmetry breaking be imposed via the inclusion of electromagnetic interactions between nucleons and by additional explicit isospin symmetry breaking terms in the potential. By fitting the potentials directly to the scattering data, several phenomenological potentials have been contracted that fit the scattering database with very close to 1. A discussion of the most commonly used N N potentials may be found in the review [5]. These include the so-called Reid-SC [222], Paris [223], Bonn [219], CD-Bonn [220], Nijmegen [221], Reid93 [221] and Argonne [109] potentials.

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Given a potential, the resolution of the Schrödinger equation in the T = channel leads to the bound state wave function discussed hereafter.

2.2.

The Deuteron Wave Function

The tensor force requires that the deuteron wave function be a mixture of and components, so the deuteron wave function is of the form

where

are the spin-spherical harmonics. The reduced radial wave functions and correspond to the S and D waves respectively. The S and D state probability densities are defined as

The corresponding S and D state probabilities are then

and the normalization of the wave function requires that

The reduced radial wave functions for the Argonne potential are shown in Fig. 2.1. The wave functions for other modern potentials are very similar. From the wave function, a characteristic size of the deuteron is defined as the rms-half distance between the two nucleons:

A conspicuous feature of the nucleon-nucleon interaction is the short range repulsion, which leads the radial S wave function to be significantly reduced at distances smaller than approximately 1 fm (see Fig. 2.2). This introduces a distance scale in the wave function in addition to the overall deuteron size. This small distance behaviour is the subject of most of the experimental and theoretical studies which will be presented in Sect. 4.and Sect. 5..

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As a result of this dip at small the Fourier transform contains a node at approximately as seen also in Fig. 2.2. The and wave functions are given in momentum space by:

To conclude this presentation of the size and shape of the deuteron, the densities and are illustrated in Fig. 2.3.

In this simple potential model of the deuteron, the magnetic moment of the deuteron is determined entirely by the D state probability

where is the isoscalar nucleon magnetic moment. The deuteron electric quadrupole moment is also determined from the wave functions:

In both cases, these quantities are modified by extensions to the basic potential model (see Sect. 5.). In particular the direct relationship between the magnetic moment and the D state probability will be broken by such extensions and, therefore, this probability is not an observable [147].

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Since the nuclear force is of finite range, it is easy to determine the asymptotic form of the wave functions

where with being the reduced mass and the deuteron binding energy (see Ref. [150] for a relativistic definition of ). and are the asymptotic normalization factors, determined by matching the asymptotic form (2.11) to the calculated wave functions in the interior region where the potential is nonvanishing. and the ratio

are directly related to observables as discussed in the next Section.

3.

STATIC AND LOW ENERGY PROPERTIES

A review of the measured static properties of the deuteron, together with the low energy neutron proton scattering parameters, was given by Ericson and Rosa-Clot [6, 29], who studied in detail their connection with N N potential models. An updated status of the experimental information on the deuteron is given in Table 3.1 and discussed hereafter.

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Deuteron Static Properties (Experiment)

Mass and Binding Energy

In the past ten years, significant progress in the precision of the measurement of some atomic masses has been made by comparing cyclotron frequencies of different pairs of ions in a Penning trap. In this way, the deuterium atomic mass is known with a relative precision of The deuteron binding energy is best determined by measuring the energy of the gamma-rays coming from radiative capture with thermal neutrons. This energy is now measured to a relative accuracy of using a crystal diffraction spectrometer [44]. At this precision, even for such a low energy process, some of the earlier work may have to be corrected for relativistic kinematics and the Doppler effect [60]. The binding energy and the mass measurements can be combined for the most precise determination of the neutron mass [44]. The errors on the values reported in Table 3.1 include the uncertainty in the atomic mass unit . They are well beyond the accuracy of nuclear models.

3.1.2.

Magnetic Dipole Moment

The first measurement of the deuteron magnetic moment was performed by Rabi in 1934 [52], based on a principle [21] already alluded to. From the deflection of an atomic beam in an inhomogeneous magnetic field to the use of molecular beam resonance and other methods, these techniques were continuously improved [53]. Precise measurements of nuclear magnetic resonance frequencies of the deuteron and proton in the HD molecule give the ratio of deuteron to proton magnetic moments. However, the adopted value in Table 3.1 results from a simultaneous determination of the electronic and nuclear Zeeman energy levels splittings in the deuterium atom, yielding the ratio of deuteron to electron magnetic moments [50].

3.1.3.

Electric Quadrupole Moment

The deuteron was found to possess an electric quadrupole moment in 1939 [42]. This discovery had far reaching consequences: it meant that nuclear forces were not central and were more complex that previously thought. It was to become the best qualitative and quantitative evidence for the role of pions in nuclear physics [6]. In contrast to the case of the magnetic moment which is determined through

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its coupling to an external applied magnetic field, the quadrupole moment does not couple to an external electric field. One measures instead, in HD or molecules, the interaction of the deuteron quadrupole moment with the electric field gradient created along the molecular axis by the neighbouring atom. The experiment provides an electric quadrupole interaction constant [23] which must be divided by the theoretically calculated field gradient [19] to obtain the quadrupole moment.

3.1.4.

Asymptotic Ratio D/S

The ratio (2.12) is deduced from measurements of tensor analyzing powers in sub-Coulomb reactions on heavy nuclei by comparison with calculations in the distorted wave Born approximation (DWBA) [55]. The value of is then directly proportional to the analyzing powers. Other determinations based on elastic scattering rely on pole-extrapolation and are somewhat less precise.

3.1.5. Radius and Size The deuteron size may be characterized by a charge radius and by a matter radius The latter is defined from the deuteron wave function (2.7). Elastic electron scattering has been used since the early fifties [39] to measure the shape of nuclei. This topic will be discussed at length in Sect. 4. At low momentum transfer, the cross-section data yield the charge rms-radius of the target nucleus through the relation (see Sect. 4.1. for the definition of ). It was demonstrated recently that a precision extraction of the deuteron rms charge radius from electron scattering data requires taking into account the Coulomb distortion of the incoming and outgoing electrons [56]. A new analysis of the world data was then performed, yielding the value in Table 3.1. The quoted uncertainty combines quadratically the fit statistical error and the dominant systematic error, the latter coming mostly from experimental normalization uncertainties. The usual radiative corrections to electron scattering do not include the contribution of hadronic vacuum polarization, but that effect should be smaller than the present uncertainties when extracting charge radii [35]. The rms charge radius is related to the matter (or rather nucleonic) rms-radius [45] by:

where

fm is the proton charge rms-radius [233] and

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is the neutron charge ms-radius [46]. is a contribution from non-nucleonic degrees of freedom, close to 0 but with an uncertainty estimated to The quantity given by is usually defined as the deuteron radius. The last term in (3.1) is of relativistic origin [15]. Note that the above quoted value of as extracted from elastic scattering, is in slight disagreement ( difference) with recent Lamb shift measurements [41]. Finally, the theoretical uncertainty in the deuteron radius associated with the correction due to the nucleon finite size has been estimated to about 0.002 fm [30]. The nuclear-dependent correction to the Lamb shift in hydrogen and deuterium atoms is directly proportional to the nuclear mean-square radius. From the isotope shifts in the pure optical frequency of 1S – 2S two-photon transitions in atomic hydrogen and deuterium, the difference of mean-square charge radii for the deuteron and proton is accurately determined [40]. Small corrections due to the deuteron polarizability seem to be under control [49]. Then from (3.1) and the value of is extracted with a better precision than from scattering. Our quoted uncertainty is larger than in Ref. [40] because of the use of a larger uncertainty in and the addition of the uncertainty due to Taking into account additional small corrections summarized in Ref. [56], the two results given in Table 3.1 are quite compatible, in the sense that they satisfy Eq. (3.1). Furthermore, the value of follows the expectations from modern N N potentials.

3.1.6.

Electric Polarizability

The electric polarizability characterizes how the deuteron charge distribution can be stretched and acquire an electric dipole moment under the influence of an external electric field. It was determined through elastic scattering of deuterons from well below the Coulomb barrier [54] and extracted from low energy photoabsorption [34]. The two results are slightly incompatible ( difference). The value in Table 3.1 is our average.

3.2.

Low Energy

Scattering Parameters

The deuteron may also be viewed as a pole in the S-matrix describing the scattering in the coupled and channels. This S-matrix can be experimentally determined from a phase-shift analysis of the scattering data. An extrapolation to negative energies down to the measured deuteron binding energy, either by an effective range expansion [6] or a P-matrix approach [57], allows to extract and The asymptotic ratio is given by

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the extrapolated mixing parameter while the asymptotic S state normalization is essentially related to the effective range and thus to the triplet scattering length (numerical values from [58]).

3.3.

Static Properties and the NN Potential

All deuteron static properties discussed above are well reproduced by N N potential calculations such as Argonne Nijmegen II, Reid93 or CD-Bonn, with the notorious exception of the quadrupole moment, which is always a few percent too low (see also Sect. 5.1.). Meson exchange contributions, to be discussed later, must be taken into account for a better agreement with data. The binding energy is taken as a constraint in the determination of all potentials. Compilations of deuteron static properties calculated with recent NN interaction models appear in [5, 122]. Note that most potentials result in a D wave probability between 5.6 and 5.8%, except for the CD-Bonn potential where Various correlations were established between the calculated static properties, independently of the N N potential used. For example, linear relationships between and or and and and were established, the latter depending on the value of the coupling constant used in the potential calculation. For more recent potentials, linear relationships between and on one hand, and on the other hand, are illustrated in [122]. Finally the electric polarizability is directly proportional to

4. 4.1.

ELASTIC ELECTRON-DEUTERON SCATTERING Deuteron electromagnetic form factors

Invoking Lorentz invariance, current conservation, parity and time-reversal invariance, the general form of the electromagnetic current matrix element for elastic electron scattering from the spin-1 deuteron can be shown to have the general form [110]:

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where is the deuteron mass, P and are the initial and final deuteron four-momenta, is the virtual photon four-momentum, and are the polarization four-vectors for the inital and final deuteron states. The are form factors depending only upon the virtual photon fourmomentum; assuming hermiticity, they are real. Since the virtual photon fourmomentum is always spacelike for electron scattering, we use the convention The current may be expressed in terms of charge monopole, magnetic dipole and charge quadrupole form factors. These are related to the by:

with

These form factors are normalized such that

The experimental values of 25.83 (see Table 3.1).

and

are respectively 1.714 and

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307

Observables

In the Born approximation of a one-photon exchange mechanism and neglecting the electron mass, the cross-section for elastic scattering of longitudinally polarized electrons from a polarized deuteron target can be calculated from the current to give in the laboratory frame [105]:

where

is the Mott cross-section, E the electron beam energy, the electron scattering angle and the electron helicity. The are response functions and the kinematical factors are:

For elastic scattering,

Each of the response functions can be written as

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where and the are members of the set of the unique deuteron density matrix elements expressed in terms of elements of a spherical tensor This set is represented by

The nonvanishing reponse functions for elastic scattering may be written in function of the deuteron form factors:

It is conventional to write the cross-section as:

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where the unpolarized elastic structure functions fined as:

309

and

are de-

and

Note that the dependence of the cross-section on the target polarization is conventionally given by analyzing powers denoted we implicitly use here the equivalence between analyzing powers and recoil deuteron polarizations: Defining

the tensor polarization observables are

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There are only two unpolarized elastic structure functions, but three form factors. Since depends only on this form factor can be determined by a Rosenbluth separation of and or by a crosssection measurement at depends on all three form factors so that only a quadratic combination of and (the longitudinal part of A) can be determined from the unpolarized cross-section. A complete separation of the form factors therefore requires the measurement of at least one tensor polarization observable. The possible candidates are and since and depend only upon and the unpolarized structure functions. and being both proportional to are of smaller magnitude than and provide in practice a smaller “lever arm” to determine either or [246]. In addition, the measurement of requires intense polarized electron beams, which became available only recently, but offers the simplification of a vector polarization measurement. In all cases so far, this leaves as the observable of choice to extract and Note that all of the polarization observables depend upon the scattering angle through kinematical factors and Consequently, the polarization observables measured in different experiments under different kinematical conditions can only be compared if some convention is assumed. Since the first measurement [195] was performed close to it has been customary to quote the observables at that angle. For experiments not performed at 70°, the observable is extrapolated to this angle, using the known and Another convention is to use the alternate quantity:

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The choice of which is not strictly speaking an observable, has several advantages. is independent of and thus depends only on It is a purely longitudinal quantity, and as such is independent of the magnetic form factor. It has simple properties related to those of the charge and quadrupole form factors (see Sect. 4.4. and Refs. [193, 199]). In particular, the position of nodes in these form factors may be determined directly from a plot of Numerically, can be determined from and through

As illustrated in Sect. 4.3., in all measurements to date, the ratio is small, so that and are not very different from each other. For all these reasons we will use this quantity along with and for comparison of theoretical predictions to data. In closing this introduction to elastic scattering observables, we refer to App. A for a short discussion of a possible two-photon exchange contribution, especially in view of the large range of available data.

4.3. Review of elastic

data

The first experiment to measure elastic scattering of electrons on the deuterium was performed at the Stanford Mark III accelerator [182]. Since then, many cross-section data points have been measured at various accelerators over the world, with ever increasing precision and at larger and larger momentum transfers [176, 174, 179, 172, 170, 180, 184, 175, 177, 178, 168, 181, 166, 185, 173, 169, 171, 183, 167, 164]. Quite spectacular is the recent achievement of Jefferson Lab to measure up to making use of a record luminosity of about to reach cross-sections as low as New measurements of for to from the same experiment will soon be available [251]. The kinematics of all these experiments are illustrated in Fig. 4.1. Forward angle scattering yields the elastic structure function A, while backward angle scattering allows the determination of the elastic structure function B. The dashed

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lines in Fig. 4.1 indicate what fraction of the cross-section corresponds to the contribution of Other experiments measured cross-section ratios [187, 186].

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As already mentioned, the separate determination of the deuteron charge monopole and quadrupole form factors necessitates the measurement of a polarization observable. The observable of choice is which is a measure of the relative probabilities of finding the deuteron in magnetic substates M = –1, 0, +1 after the scattering. The Bates Linear Accelerator Center was the first to measure [195] and to provide an experimental evidence for the existence of a node of the charge form factor [193]. These double scattering experiments were recently brought to the (up to now) highest possible momentum transfers at Jefferson Lab. [188]. For this last experiment as well as for the previously mentioned A measurement [167], electron beams of 100 to 120 were used in conjunction with liquid deuterium targets up to 15 cm long, capable of dissipating 600 W of power deposited by the beam. The combination of a record integrated luminosity, in excess of and of a large acceptance magnetic channel focusing recoil deuterons onto the high efficiency polarimeter POLDER (see Fig. 4.2 and Appendix A.2) allowed the measurements of to be extended up to The alternative measurement of the tensor analyzing power using a polarized target intercepting a stored electron beam, was initiated at Novosibirsk [191, 198, 194], and improved at NIKHEF [192, 190]. New preliminary results from Novosibirsk [196] are also included in this review. On the other hand, the use of a solid cryogenic target in an external electron beam at Bonn resulted in a too low luminosity [189]. The kinematical settings of all these experiments are illustrated in Fig. 4.3. In all cases, the magnetic contribution to is small. A further comparison between these polarization measurements is contained in Appendix A.2. The other tensor polarization observables and (or ) were also measured [193, 188, 192]. Figure 4.4 shows a good part of the existing data. The A data at low and high will be better illustrated in the figures of Sect. 5., in particular in Figs. 5.19 and 5.17. The data appears in Fig. 5.18. In closing this Section, let us mention a few inconsistencies in this data set in the light of recent measurements.* The Cambridge [175] and Bonn [173] data are very probably too low, since both recent measurements at Jefferson Lab [167, 164] agree with the “higher” trend already given by the SLAC data [168]. Still, as apparent in Fig. 5.19, these two JLab measurements differ from each other (10-15%) in the region For a comparative discussion of these two * At the time of print of this paper, a better determination of the beam energies at JLab/Hall C

results in some corrections, within quoted errors. The values [164] should decrease by 1 to 3% with increasing while the first point [188] should move down by about one third of its error. These numbers are subject to confirmation.

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independent experiments, see [165]. The (or ) data are necessarily less precise than cross-section measurements and naturally exhibit some scatter. Although all data points are compatible with parameterizations such as discussed below, there are some trends between different data sets. No parameterization or model can accomodate both the Bates [193] and NIKHEF [190] data sets: one or the other is too low, or both are. The same Bates data are also systematically lower than both the JLab [188] and the preliminary Novosibirsk data [196], and the precise low NIKHEF point [192] is lower than many theoretical expectations. Though these scatters are compatible with the quoted experimental errors, they demonstrate (together with the theoretical models to be discussed) a need for a more accurate measurement in the region Q = 3 to 4.5 fm. Finally, the Bates data point at [193] is obviously wrong, but the authors did not find a correlation between and in their analysis .

4.4.

Empirical features of form factors

The three deuteron electromagnetic form factors may be calculated at a fixed value of from measurements of A, B and The magnetic form factor is readily available from the B measurements, while the two charge form factors and are determined from and (4.21,4.22). The resolution of these equations is most simply described in Ref. [199], together with a discussion of possible ambiguities in the choice of different solutions in the Q-regions where reaches its extrema. This procedure allows a direct comparison of theoretical models with the form factors, instead of observables, but it is limited to the domain where the three observables are measured, which is The most striking result is an experimental determination of the node of at Q = 4.21±0.08 fm [199], which corresponds to This behaviour of though expected from most models, could not have been seen with cross-section measurements only. The exact location of this node is sensitive to the strength of the N N repulsive core (in the impulse approximation, it is connected to the node of in Fig. 2.2) and to the size of relativistic corrections and of the isoscalar meson exchange contributions. A secondary maximum is also determined from the data [188]. Thus, like any other nucleus, the deuteron appears to have a charge form factor with an oscillatory diffractive pattern. Unlike other nuclei, the sign of this form factor is determined as well. The quadrupole form factor exhibits a monotonous exponential falloff, and its first node, corresponding to the yet unobserved second node of

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is expected beyond Finally the magnetic dipole form factor has a node at determined by B data [171] and in a lesser extent by data [188]. The position of this node is very sensitive to small non-nucleonic components in the deuteron wave function, to nucleon-nucleon components of relativistic origin, as well as to the description of the contribution to be discussed in the next Section. The three form factors were parameterized in three different ways: rational fractions (I), sum of Lorentzian functions in an helicity basis (II) and sum of Gaussian functions (III) [199], fitting directly the measured observables, i.e., differential cross-sections and polarizations. Figure 4.5 gives a representation of the form factors with an updated version of parameterization I, taking into account the preliminary data of Ref. [196]. The data base and the parameterizations are available in [200].

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Another representation of the experimental knowledge of the deuteron form factors is given in Fig. 4.6, where the contributions from each of them to the elastic structure function A are given, calculated using parameterization I.

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5.

319

THEORETICAL ISSUES

The different classes of theoretical models of the deuteron electromagnetic form factors are presented here, illustrated with the most recent calculations on the subject. A summary of earlier theoretical work on the subject may be found for instance in [193, 183]. In all figures of this Section, the data legend is the same as in Sect. 4.3.

5.1.

Deuteron Elastic Form Factors in the Simple Potential Model

Once the wave functions for a given potential model are obtained as discussed in Sect. 2., the electromagnetic form factors may readily be calculated in the non-relativistic impulse approximation (NRIA) using a well established formalism [103, 104]. The virtual photon can couple to any of the two nucleons, so that the isoscalar combinations of the nucleon form factors (NEMFF), defined in Appendix A.3, factorize to yield [8, 149]:

The C functions are integrals of quadratic combinations of and In this NRIA, the ratio and thus are independent of the nucleon form factors. The coupling of the virtual photon to the moving nucleon charges and to the nucleon spins both contribute the magnetization in giving rise to the two terms in (5.1). The elastic structure functions and are illustrated in Fig. 5.1 for a variety of phenomenological potentials [64, 65, 109], using the MMD parameterization of the nucleon form factors (see Appendix A.3). In all cases the calculations agree with one another and with the data up to but diverge at higher The low behaviour of the form factors is not determined with the same precision for each of them. Since the forward cross-sections at low depend mostly on the slope of this form factor at is determined to about 1% (see Sect. 3.1.5.). The slope of at the origin is determined by backward cross-sections and is known to about 5%. In contradistinction, the slope of is known to only 15%, in the absence of very precise measurements at low Still this slope is model dependent. To illustrate this point, we define the reduced quantity

and show its low

behaviour in Fig. 5.2.

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is the model calculation of the quadrupole moment. The slope is rather small, due to an approximate cancellation between the first two terms, the third one being significantly smaller. In this connection, it is instructive to list in Table 5.1 the model dependences of these various slopes. The value of the quadrupole form factor varies substantially for these calculations and is always smaller than the experimental value of 25.8 (4.4). All of the Bonn calculations and the Reid-SC calculation yield the same value for the derivative of the charge form factor and the same of is true of the derivative of the quadrupole form factor with the exception of Bonn A. Therefore the spread in for this latter set comes only from the variation in the value of the quadrupole form factor. However, this may not be the case in general. The increasing disagreement between these simple calculations and the data at higher indicates that this approach does not contain all of the necessary physical degrees of freedom and/or that since the momentum transfer approaches and then surpasses the mass of the nucleon, it is necessary to consider the impact of special relativity on the calculation of the deuteron elastic

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structure functions. We will now consider the first of these possibilities.

5.2.

Limitations of the simple potential model

For many applications in nuclear physics, the phenomenological potential approach is completely adequate and provides a good description of various phenomena. Its limitations were seen, however, by considering the calculation of electromagnetic properties of the deuteron in the preceding Section. The inclusion of electromagnetic interactions with the two-nucleon system imposes the requirement that the electromagnetic current matrix elements satisfy the continuity equation

where and are the current and charge density operators. The operators must then satisfy

Since either of the nucleons can be charged, the charge density operator is

while the current density takes the general form

where and are the charge and current densities for particle and possible additional contribution to the current density.

is a

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The current of a free charged particle must satisfy the continuity equation, which implies Using this with (5.5), (5.6) and (5.7) gives

Since is proportional to and the two-nucleon potential has terms proportional to the second term in (5.9), proportional to does not vanish and must be nonzero. In addition, since involves the coordinates of both nucleons, the current must also depend upon both sets of coordinates and is therefore a two-body operator. The physical origin of this contribution to the current comes from the fact that the two-nucleon potential contains terms corresponding to the exchange of charge between the nucleons, which must in turn give rise to an associated current. These two-body currents are called exchange or interaction currents. Equation (5.9) is a symmetry constraint on the theory. It cannot, however, be used to uniquely determine since divergenceless pieces can be added to any current satisfying (5.9) without modifying the constraint. A unique prediction of the current therefore requires that the underlying dynamics of the interaction be specified. Viewing the nuclear force in a meson-exchange model, these two-body currents are naturally associated with contributions that involve the exchange of mesons [107]. Now that it is generally accepted that quantum chromodynamics (QCD) is the correct description of the strong force at the scales of interest here, these currents must ultimately be associated with the exchange of quarks. This discussion leads to a consistent treatment in the context of a nonrelativistic treatment of the two-nucleon problem. However, any attempt to implement this approach in a manner that is applicable to the existing data, for instance to the case of elastic electron-deuteron scattering, leads directly to consistency problems. One of the most elementary indications of this problem arises from the necessity of including electromagnetic form factors for the nucleons. At low momentum transfers, these form factors differ from their static values by terms of order Since the dipole mass is of similar magnitude as the nucleon mass, this is of leading relativistic order Whenever the presence of the nucleon electromagnetic form factors has an appreciable effect on the calculations of the deuteron form factors, some effects of relativistic order are already included. It then becomes necessary to consider whether there are other relativistic effects of similar size that can appreciably modify the calculations. In fact, there are many such effects arising from Lorentz boosts and

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exchange currents that become increasingly important in the calculations of deuteron form factors as the four-momentum transfer increases. For this reason it is necessary to consider how relativity can be introduced consistently in models of the deuteron. For this purpose, we will assume that the deuteron is adequately described by nucleons and mesons.

5.3.

Construction of Relativistic Models

The requirement of any truly relativistic model is that it must satisfy Poincaré covariance: it must be covariant with respect to Lorentz boosts, spatial rotations and space-time translations. This can be imposed by requiring that the the operators which act as generators of these transformations satisfy the Lie algebra of this group:

where are the three angular momentum operators that generate rotations, are the generators of the three Lorentz boosts and is a four-vector containing the energy and momentum operators that generate space-time translations. This differs from the corresponding algebra for the Galilean transformations in only two commutators. The differing commutators for the Galilean transformations are:

and where are the generators of the Galilean “boosts” and M is the mass operator. The first of these implies that Galilean boosts commute whereas Lorentz boosts do not. The second is the source of the dynamical complexity of relativistic models and theories. While the commutators for the Galilean boosts and the three-momentum operators are proportional to the mass, the corresponding commutators for the Poincaré group are proportional to the energy operator, that is the Hamiltonian. Since the Hamiltonian contains the interactions between the constituents of the system, the first commutator of (5.14)

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implies that the Lorentz boost operators or three-momentum operators must also be dependent upon the interaction. There are two basic approaches to constructing Poincaré invariant models. We will start with the most familiar of these, quantum field theory.

5.3.1. Quantum field theory The starting point of a quantum field theory is a Lagrangian that is constructed to satisfy all of the required symmetries, including Poincaré invariance. By nature field theories have an infinite number of degrees of freedom. Canonical quantization is performed by constructing the Hamiltonian, finding the generalized position and momentum in terms of the fields, writing the fields as an expansion in terms of creation and annihilation operators, and then imposing canonical equal time commutation (anticommutation) relations on the canonical variables. This yields commutation (anticommutation) relations for the creation and annihilation operators. An immediate consequence is that the fields commute (anticommute) for all spacelike intervals, implying that events occuring at spacelike separations cannot be causally connected. This is the property of microscopic causality or microscopic locality. It should be noted here that microcausality results from imposing the commutation (anticommutation) relations on any spacelike hypersurface. The structure of field theories is very complex due to the presence of an infinite degrees of freedom. Since the Hamiltonian of an interacting system links states containing different numbers of particles, and the time-evolution operator of the system is given by the imaginary exponential of the interacting Hamiltonian, the interacting system contains contributions with any number of particles, from zero to infinity. Practical calculations in field theory, with the exception of numerical approaches such as lattice QCD, must then introduce some method of truncating the collection of configurations contributing to the calculation. The most familiar approach to this problem is Feynman perturbation theory. Here, various configurations are carefully arranged such that all quantities contributing to a process, such as free propagators and vertex functions, are individually covariant with respect to noninteracting Lorentz transformations. The truncation then requires that there be some plausible scheme for systematically organizing contributions in order of their relative importance. For example, in the classic case of quantum electrodynamics where the coupling constant is small, the terms are ordered in powers of the coupling constant and truncated at some finite order. The drawbacks of quantum field theory for constructing models such as for NN interactions are related to complexities and calculational difficulties. Most of our knowledge of field theory is based on perturbation theory and there

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is no a priori method for determining the radius of convergence (if any) for a given field theory. For strong coupling theories, it is no longer plausible to construct a perturbation scheme in terms of simple counting of coupling constants and infinite sets of contributions must be summed. In practice, the coupling of the infinite sets of states, or functions, must be truncated if there is to be any hope of calculation. Thus some physically reasonable scheme for truncation must be proposed. However, the truncation of the theory will usually violate some symmetries of the full theory such as crossing symmetries, covariance or current conservation. Local effective theories with the appropriate symmetries may also be nonrenormalizable. Model calculations may also include phenomenological elements such as form factors that are not calculated within the context of the field theory. This also leads to problems of consistency within the model and may also violate symmetries.

5.3.2. Hamiltonian dynamics Although Dirac was one of the founders of quantum field theory, he soon became disillusioned with its complexity and the difficulties associated with the unavoidable infinities. He continued for most of the rest of his life to seek an alternative to quantum field theory. He assumed that the problems with field theory were related to starting from an unsatisfactory relativistic classical theory. He pointed to an alternate approach, starting with a theory with a fixed number of degrees of freedom, as is done with the nonrelativistic Schrödinger equation. This led to relativistic constraint dynamics [68]. In this approach, the dynamics of a model system is determined by choosing a mass operator which contains an instantaneous interaction as in the nonrelativistic potential theory. This basic dynamics contains a finite number of particles and has a corresponding Hilbert space when quantized. Covariance is imposed by constructing a unitary representation of the inhomogenous Lorentz transformations with generators that satisfy the commutation relations of the Poincaré group. The wave functions have the same probabilistic interpretation as in nonrelativistic quantum mechanics, but microscopic causality is not respected: the theory must be constructed to respect the physical requirement of cluster separability or macroscopic locality. There are at least three different approaches for the quantization of such models. The first is the traditional method of quantizing along constant time surfaces (called instant form) where the evolution of the system is the usual time evolution which is normal to the constant-time hypersurface. The second is to quantize along spacelike surfaces with constant interval (called point form). The evolution of the system is then along a new coordinate normal to these surfaces. The third of these is to quantize along the light cone with evo-

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lution along the coordinate Since all these choices describe surfaces with spacelike separations (or the infinite momentum limit of such a surface in the light-cone case), they are also consistent with microscopic causality for field theories, which may also be quantized along these surfaces. Particularly useful in the Hamiltonian dynamics is the fact that a careful construction of the mass operator can lead to equations of motion of the same form as the two-body Schrödinger equation. It is therefore possible to use, without modification, nonrelativistic potentials that have been fitted to describe NN scattering. The drawbacks are related to the choice of the interaction without specification of any underlying dynamical content. As a result, quantities such as electromagnetic currents can be constrained by the structure of the theory, but can not be uniquely determined from the interaction dynamics. We will now proceed to a discussion of various approaches used in constructing relativistic calculations of the elastic electromagnetic form factors for the deuteron.

5.4.

expansions

This approach is actually a hybrid of field theory and Hamiltonian dynamics. It assumes that the basic dynamical content of the deuteron is nonrelativistic and that the necessary relativistic effects can be described as corrections to the nonrelativistic current matrix elements as an expansion in [98, 141, 99]. The nucleon-nucleon interaction is taken to be a standard nonrelativistic potential with parameters determined by fitting to the nucleon-nucleon scattering data and to the deuteron binding energy. It is assumed that the potential is, at least in part, represented by a one-meson-exchange model, since the meson degrees of freedom are necessary to construct two-body exchange currents from simple Feynman diagrams and for constructing corrections due to retardation of meson propagators. Examples of the required two-body interaction currents are represented in the diagrams of Fig. 5.3. Diagram (a) represents a contribution due to coupling of the photon to the current of an exchanged meson, which, because of G-parity, does not apply to isoscalar transitions such as elastic scattering. Diagram (b) is a so called pair current that arises due to the projection of the interactions onto the positive-energy space only. Contributions that couple to the negative energy part of the nucleon propagator must then be included in the current. Diagram (c) is a retardation correction. Since nonrelativistic wave functions are single-time wave functions, diagrams corresponding to the absorption of a photon while the meson is in flight are not included in the impulse approximation and must be included in the current. Retardation corrections can also be associated with retarded meson propagation

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in any of the other diagrams in this figure. Derivative couplings of the mesons to the nucleons can also give rise to contact interactions of the type shown in diagram (d), which is isovector and does not apply to elastic scattering. It is also possible for a photon that is absorbed by an exchanged meson to excite the meson. An example is the exchange current which is commonly included in calculations of elastic electron-deuteron scattering. Such a current is represented by diagram (e).

Relativity is imposed by requiring that the currents and interactions are consistent with operator commutators of Poincaré invariance to some order in The dependence of the boost operators on the interaction also gives rise to interaction currents in addition to those characterized above. This approach guarantees that the interaction model can be very well constrained by data but its application can become technically complicated. In addition, the expansion in must fail at some value of momentum transfer. The calculations shown here are from a recent paper by Arenhövel, Ritz and Wilbois (ARW) [100]. This paper is an extension of earlier work [64, 67] that included relativistic corrections for the charge and quadrupole form factors, but not the magnetic. Although there is an extensive literature on the subject, as nicely summarized in [100], only that paper and the earlier work of Tamura [151] take into account all leading order terms including the Lorentz boost of the deuteron center of mass. The need for a realistic one-bosonexchange potential leads ARW to use the Bonn OBEPQ models, which may however not have the same precision in describing AW scattering data as more recent, but more phenomenological, potentials. Finally the Galster dipole parameterization for the nucleon electromagnetic form factors was used.

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Figure 5.4 shows the observables and for this model. Three curves are shown for each observable, the impulse approximation (NRIA), the impulse approximation plus all relativistic corrections to leading order, and all of the former plus the exchange current. The exchange current is purely transverse and is not required or constrained by the interaction model, but is the longest range isoscalar current of this type.

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The elastic structure function for the impulse approximation is substantially below the data. The inclusion of the various relativistic corrections brings the calculation into much better agreement with the data, while the addition of the exchange current has only a very small effect. In the case of the magnetic structure function the NRIA again is below the data and has a diffraction minimum at lower than indicated by the data. The addition of the relativistic corrections increases the size of the calculated structure function and appears to move the diffraction minimum above the value indicated by the data. The addition of the exchange current again increases the size of the structure function to a value considerably above the data and presumably pushes the diffraction minimum to even higher values. We will return later to the question of the effect of this exchange current on the magnetic structure function, in the context of field-theoretical models. For the polarization structure function the NRIA is above the data for small below its minimum, and below at above the minimum. The addition of the relativistic corrections brings the calculation into much better agreement with the observed node of and consequently with the data. The exchange current only has a very small effect on this observable which is purely longitudinal. The examination of the figures in Ref. [100] shows that the various corrections to the NRIA are not small, they tend to be of similar magnitude, and they contribute to the deuteron form factors with varying signs. As a result, any calculation that includes some, but not all, of these corrections, is questionable. These calculations clearly indicate that relativistic effects, that is mesonexchange contributions and genuine relativistic corrections, are important even for relatively small momentum transfers. The domain of validity of this approach is thus limited, probably to though it may lead in some instances to a satisfactory description of the data up to [109]. The study of the validity of the two-nucleon description of the deuteron down to very small internucleon distances, together with new data becoming available from Jefferson Lab, make mandatory the consideration of fully relativistic models.

5.5.

Relativistic Constraint Dynamics

Figure 5.5 shows three recent calculations the deuteron elastic structure functions. Note that in this and following figures, the different observables are not shown in the same range, following the available data. This should be kept in mind for a meaningful comparison of models to data. The calculation of Allen, Klink and Polyzou (AKP) [124] uses the Argonne potential for

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an interaction and is quantized in point form. The calculation of Lev, Pace and Salm (LPS) [122, 121] uses the Nijmegen II potential and is quantized in front form. The calculation of Forest and Schiavilla (FS) [102] uses the Argonne potential and is quantized in instant form. These three calculations involve three different approaches to construct currents. AKP and LPS use different procedures for constructing currents from the single-nucleon current without including any interaction dependent two-body currents. FS assume that the potential is of one-boson-exchange origin and construct the various currents as in the expansions, but without performing this expansion. These calculations show considerable variation in all three observables. Although the FS calculation works quite well for all of them, it is clear that no consensus has been reached concerning consistent techniques for the construction of currents in this framework.

5.6.

Field Theoretical Models

5.6.1. The Bethe-Salpeter equation The oldest of the field-theoretical treatments of the two body-problem is due to Bethe and Salpeter [70]. The physical content of the Bethe-Salpeter equation can be easily understood by considering the two-body scattering matrix Figure 5.6 represents the Feynman expansion of the scattering matrix where the nucleons are represented by solid lines and the mesons are represented by the dashed lines. Note that diagrams (b), (d), (e) and (f) can be reduced to simpler diagrams by cutting across two nucleon lines as represented by the dotted lines. These diagrams are said to be two-body reducible diagrams and the remaining ones two-body irreducible diagrams. The ability to classify all of the contributing diagrams as members of one or the other of these two classes suggests that the multiple scattering series can be resummed by separating the irreducible diagrams into an interaction kernel and then using an integral equation to produce the reducible diagrams. The Feynman diagrams representing the two-nucleon irreducible kernel are shown in Fig. 5.7. The Bethe-Salpeter equation for the scattering matrix can then be written as

where P is the total four-momentum of the two-body system; and are the final, intermediate and initial relative four-momenta of the two particles; is the free two-body propagator; and V is a kernel consisting of a sum of all possible two-body irreducible diagrams. The Bethe-Salpeter equation is

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represented by the diagrams in Fig. 5.8. Since the Feynman series is organized such that all of the individual propagators and vertices are covariant with respect to the free-particle Lorentz transformations, the sum is manifestly covariant as well. The currents can be constructed by combining the free one-nucleon currents with exchange currents obtained by attaching a photon to every possible place within the irreducible interaction kernel, as shown in Fig. 5.9. These currents will then satisfy the Ward-Takahashi identities [101]. The two-body bound state is represented by the Bethe-Salpeter vertex function which satisfies

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This vertex function for the deuteron can be written [84] as

P is the deuteron four-momentum, the relative momentum of the external nucleons, the polarization four-vector for the deuteron, the Dirac charge conjugation matrix. The subscripts and are indices in the Dirac spinor space, and can be determined by basic symmetry arguments to have the general form:

where requires that

and

A generalization of the Pauli symmetry

which places constraints on the scalar functions Note that for given values of and P, the vertex function depends upon eight scalar functions whereas the

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Schrödinger wave function of the deuteron is determined by the two functions and The Bethe-Salpeter equation is a complete representation of all possible contributions to the two-body amplitudes for a given field theory. As such it retains all of the complexities of field theories in that there is an infinite number of contributions as represented by Feynman diagrams. The problem of particle dressing and renormalization is present, and for strong coupling theories there is no general scheme for organizing the renormalization program. As a practical matter, model calculations generally assume that all propagators represent dressed particles with physical masses. This still leaves an infinite number of contributions to the irreducible kernel. Again, there is no a priori means of establishing a reasonable truncation procedure in order to obtain a tractable set of contributions. However, the phenomenology of nuclear systems suggests that the importance of contributions to the nuclear force depends upon the range of the contributions due to the strong repulsion of nucleons at short distance. As a result, models of the deuteron using field-theoretical techniques usually assume that contributions can be ordered by range. In practice this reduces to the use of one-boson-exchange potentials. This is referred to as the ladder approximation to the Bethe-Salpeter equation. The ladder approximation violates the crossing symmetry of the two-body scattering amplitudes, which is a property of the full Bethe-Salpeter equation. This results from the elimination of the higher-order crossed-box contributions to the kernel. A related problem is that the two-body equation no longer reduces to a one-body wave equation at the limit of infinite mass for one of the constituents (defined as the static limit) [113]. A further difficulty for producing models of the deuteron using the fieldtheoretical methods is that phenomenology often requires the introduction of nonrenormalizable couplings such as the derivative coupling of the pion and tensor coupling of vector mesons. The loop integration of the Bethe-Salpeter equation must then be cut off to eliminate unphysically large or infinite contributions. This is usually done by introducing form factors for strong coupling vertices, which also attempts to include finite-size effects for the hadrons. In addition, the hadrons have a finite electromagnetic size so that electromagnetic form factors are also required by the phenomenology. The introduction of form factors can, however, result in violations of gauge invariance. The solution of the Bethe-Salpeter equation is also a calculational challenge since it is a four-dimensional integral equation with a complicated analytical structure. It has however been solved for the two-nucleon system in Euclidean space [71] and applied to the calculation of electron-deuteron scattering [72, 73].

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5.6.2. Quasipotential Equations The Bethe-Salpeter equation is a four-dimensional integral equation. As such it is much more difficult to solve numerically than the comparable nonrelativistic three-dimensional Lippmann-Schwinger equation. To simplify the solution of the relativistic equations, one resorts to the infinite class of quasipotential equations [3, 74, 75, 76, 77, 78, 79, 80, 90]. These equations are related to the Bethe-Salpeter equation (5.17) by replacing the free propagator by a new propagator The scattering matrix equation can now be formally rewritten as

where U is the quasipotential defined as

This pair of equations is equivalent to (5.17) and represents a re-summation of the multiple scattering series. The new propagator is usually chosen to include a constraint in the form of a delta function involving either the relative energy or time in such a way as to reduce (5.22) to three dimensions. In addition must be chosen such that it has the same residue as along the righthand elastic cut. This guarantees that the discontinuity of U produces only inelastic contributions as in the case of (5.17). Although it appears that the reduction of (5.22) to three dimensions is of great practical advantage, it should be noted that (5.23) is still a fourdimensional integral equation of difficulty comparable to (5.17). The real utility of these equations comes about when (5.23) is approximated by iteration and truncation. Then U can be obtained by quadrature of (5.23) and used in the three-dimensional equation (5.22). It would appear that we have achieved a reduction in the difficulty of solution of the problem at the expense of introducing considerable additional ambiguity since there is an infinite number of possible choices for which satisfy the above specified requirements. However, this freedom is turned to an advantage when noting that can be chosen to maximize the convergence of (5.23). In essence, the parts of the ladder and crossed box diagrams which tend to cancel are being summed in the quasipotential equation (5.23) rather than in the equation for the scattering matrix (5.22). Indeed, the lowest order calculations using a variety of quasipotential prescriptions have been calculated

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for scalar particles [112] and shown to be closer to the results of the complete Bethe-Salpeter equation than is the result from the ladder approximation to the Bethe-Salpeter equation. In addition, it has been shown that there exists an infinite number of quasipotential equations which have the correct static limit. This is also a reflection of the improved convergence of these quasipotential equations [114]. The results from three different quasipotential models are presented here. The first of these is the calculation of Hummel and Tjon (HT) [89] based on the BSLT equation [74]. In this case, the propagator contains a term which forces the relative energy of the two nucleons to zero. One can say that the two nucleons are equally off mass-shell. The vertex functions are calculated using the BSLT equation with a one-boson exchange interaction. They still have eight components. The current matrix elements are then calculated by replacing the Bethe-Salpeter vertex functions with the BSLT vertex function, assuming that it is energy independent. In the examples shown here the nucleon electromagnetic form factors are taken to be the Höhler 8.2 parameterization. The second model is that of Phillips, Wallace and Devine (PWD) [117, 118, 119] where a one-boson-exchange interaction is used with the single-time equation that introduces a constraint that the relative time be zero. This is very close to the HT approach since this choice means that the propagator is independent of the relative energy of the two nucleons. The single-time deuteron vertex functions still have eight components. A consistent treatment of the current is constructed to guarantee current conservation, but Lorentz covariance is violated. The MMD parameterization of the nucleon electromagnetic form factor are used. The third model is that of Van Orden, Devine and Gross (VODG) [86, 87, 88], based on the Gross equation [79, 80]. One of the nucleons is placed on its positive energy mass shell, which gives:

Since this constraint is itself manifestly covariant, the Gross equation amplitudes are manifestly covariant. But it is not symmetric in the nucleons and the exchange symmetry must be recovered by symmetrizing the interaction kernel. This procedure introduces unphysical singularities, which are treated in principal value to avoid unitarity problems and have little numerical effect in a weakly bound system such as the deuteron. As a result of placing one nucleon on shell in (5.20), the Gross equation deuteron vertex function has four-components that can be represented in terms of an S wave, a D wave and singlet and triplet P wave functions. A systematic procedure for constructing the effective current operators for

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the Gross equation exists [81] and has been shown to be rigorous to all orders and remarkably robust under trucation [66]. The result is that Ward identities for the Gross equation are maintained and the calculation is gauge invariant. A one-boson-exchange interaction is used, that has been fit to N N scattering data up to a lab kinetic energy of about 300 MeV [82, 83]. The fit is reasonable, but not at the level of precision obtained with modern fully phenomenological potentials. Several unique features appear in this model. All of the strong vertices are multiplied by a product of three form factors,

Here and are the initial and final four-momenta of the nucleon and is the four-momentum of the meson. The meson form factor is taken to be

and the nucleon form factor

The prescription of Gross and Riska [81] is used to construct a singlenucleon electromagnetic current which is consistent with the strong-vertex form factors and phenomenological single-nucleon electromagnetic form factors. This current is constructed in such a manner that the one- and two-body Ward identities are of the same form as for the local field theory. The simplest form that this single-nucleon current can take is

where the nucleon form factors are

The factors that depend upon the nucleon momenta are

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and

The function must be equal to one when both nucleons are on mass shell, but is otherwise unconstrained. For simplicity, the calculations shown here use The form factor must obey but is otherwise unconstrained and is chosen to be of a dipole form. Figure 5.10 shows the elastic structure functions for these models calculated in the relativistic impulse approximation (RIA). As expected from the nature of the quasipotential approximations, the calculations of HT and PWD lead to very similar results for all three observables (in spite of the fact that they use different NEMFF). The VODG calculation uses the MMD form factors. For it is systematically larger than the others due to additional contributions that are necessary to conserve the current within the context of the Gross equation. These additions yield to the so-called Complete Impulse Approximation (CIA). For the HT and PWD calculations are systematically below the data and have a minimum at a lower than is indicated by the data. The VODG calculation is higher and has a minimum at higher than the other two calculations. This difference has been shown to be due to a small P wave component of relativistic origin that contributes to the normalization of the vertex function at the level of 0.01% [88]. This clearly indicates the sensitivity of the position of the node to small components. Finally, in the case of the three models have a very similar behaviour. The sensitivity of the observables to the choice of single-nucleon form factors is shown in Fig. 5.11, using the VODG calculation in the CIA approximation. All but the VO2 parameterization are standard form factors that appear in the literature. The VO2 form factor is modeled after the MMD form factor but adjusted to fit the new data on from Jefferson Lab [234]. As expected from the discussion in App. C, and are very sensitive to the choice of single nucleon form factor while is almost totally insensitive to it. The various curves in Fig. 5.11 may be directly related to the corresponding curves in Fig. 8.3. Note that for the relativistic approaches, in contrast to the NRIA, the nucleon form factors cannot simply be factorized, so that there is no reason that be completely independent of the NEMFF. Much more accurate data on the single-nucleon form factors that will be forthcoming from experimental facilities around the world will, hopefully, provide much greater constraints on these form factors.

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Figure 5.12 shows the effect of the addition of exchange currents to the three calculations. HT were the first to implement the calculation of the contribution to the deuteron form factors in a relativistic model. They also included a exchange current, which compensates their large effect of the This last contribution is not present in the calculation presented here, because there is considerable uncertainty about the meson, and consequently about its couplings and vertex form factors (see [142] for a recent discussion of the coupling constant). Two sets of calculations for the VODG structure functions are shown, one with the MMD form factors as before, and one with the VO2 form factors. Comparing to Fig. 5.10, it is clear that this exchange current can result in considerable variation in The variation in is much smaller, in part because of the smaller range considered. The exchange current shifts the node of to lower values of for all calculations, in contrast with the non-relativistic calculations using the expansion where this current has the opposite effect. This is the result of the tensor coupling of the to the nucleon, which appears to be a higher order contribution in the expansion, but weighted by a large coefficient. In this way, expansion schemes may lead to wrong estimates without a careful examination of higherorder contributions. Finally, the addition of the contribution has a small effect on but brings the calculations in a somewhat better agreement with the available data. The variation in the contributions from the various models [89, 115, 125, 116] is associated with ambiguities in these contributions. Although the coupling constant for the vertex is constrained by the decay width little is known about the fall off of the associated form factor. Figure 5.13 shows a number of different form factors for this vertex. HT use the VMD form factor which is the hardest of the available form factors while VODG use the Rome 2 form factor which is the softest one. PWD use an intermediate parameterization. The data favour use of the softest possible form factor. In contradistinction, a recent theoretical reexamination of the form factor [152] results in a dependence close to the VMD one. Whether the contribution to the deuteron form factors is really small, or suppressed by other contributions is still an open question. Another source of ambiguity, explicitly present in the the VODG calculation, is the choice of the form factor in (5.28). In these calculations, is adjusted to optimize the fit of the calculation to the data for the deuteron structure functions. By using a very hard form factor it is possible to obtain an extremely good fit to the data. This along with ambiguities in the form factor means that no absolute predictions of the structure functions can be obtained unless some means can be found to physically constrain the ambiguities in the models.

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Light-Front Field Theory

The above approaches based on the Bethe-Salpeter equation are generally discussed in the context of Feynman perturbation theory with equal time quantization. Microcausality, however, only requires that the theory be quantized on a spacelike hypersurface. A special limiting case of such a surface is the light cone where the spacetime interval approaches zero. Quantizing field theories on the light cone has long been common practice in describing deep inelastic scattering where the large four-momenta kinematically favor contributions to scattering very near the light cone. Typically, the light-cone approach is organized as a “time-ordered” perturbation expansion in terms of the light-front time The wave functions are then Fock-space wave functions with a probabilistic interpretation as in the case of Schrödinger wave functions. A particular problem with light-front field theory is that the conventional choice of a fixed light-front orientation violates manifest covariance, although matrix elements remain covariant. One solution to this problem [95] is to de-

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scribe the quantization surface in terms of an arbitrary direction on the light cone given by a light-like four-vector and require that the quantization surface be described by This renders the theory manifestly covariant, at the expense of the additional complication that the wave functions and operators become dependent on the vector This is covariant light-front dynamics (LFD). Matrix elements and observables, in a complete calculation, do not depend on In a practical calculation, a definite procedure to eliminate the non physical terms is applied. The deuteron elastic structure functions from this approach [95, 94, 97] are shown in Fig. 5.14. In these calculations, the deuteron wave function has six components two of which ( and ) correspond to the usual S and D components in the nonrelativistic limit. These components are calculated perturbatively from the usual nonrelativistic wave function. Schematically [143],

where is the relative momentum of the two nucleons and a unit vector along the three-vector The NN interaction kernel V is of a one-boson-exchange type, calculated within the framework of LFD using the parameters (such as coupling constants) of the Bonn potential. The single-nucleon electromagnetic form factors are the MMD parameterization (the effect of various NEMFF was considered in [97]). This calculation produces a reasonable description of the data for and but the minimum in occurs well below the position indicated by the data. Given the sensitivity of the magnetic form factor to small effects, this may simply be the result of the perturbative treatment of the small components of the wave functions in this calculation. Indeed, work is in progress to improve the calculation of the interaction kernel V and of the complete wave function [153]. Meson exchange contributions are only partially included in the original calculations. The addition of recoil terms and of the contribution was also initiated: the recoil terms move the node of to higher while the contribution does not modify appreciably any observable up to [154]. A complete calculation using LFD seems at hand and very promising.

5.6.4.

Effective field theory

In principle, all these field theoretical techniques should yield the same results, given the same dynamical input. This statement applies to the calculated

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matrix elements which are the physical observables of the field theory. The identification of wave functions and operators varies from formulation to formulation and do not have unique meanings. Furthermore, within any given formulation, unitary transformations of the Lagrangian, such as field redefinitions, can also move contributions between the wave functions and operators. Such ambiguities are already present in nonrelativistic models, which may be shown to be equivalent, at least in part, up to a unitary transformation [147, 146]. None of this would be of particular concern if it were not necessary to truncate all of the approaches for reasons of practicality. Consequently, the physical content of the various formulations varies, resulting in a variety of ambiguities, some of which have been discussed above. A major problem then is that there is, in general, no organizing principle present in these calculations which indicates the relative importance of various physical contributions to guide the choice of truncation schemes. A promising development that could help to resolve this problem is the application of effective field theories to nuclear systems [131, 132, 133, 134, 135, 136, 137, 138, 139]. The basic idea of effective field theory is that at low energies the observables of a theory are largely insensitive to the details of short-range contributions to the interactions. The long-range degrees of freedom are then treated in detail and the short range pieces are replaced by contact interactions. The Lagrangian is written as a sum of terms containing contact interactions with increasing numbers of derivatives of the fields. This constitutes an expansion of the theory in terms of momenta which are small compared to some scale chosen to separate the short and long range physics. Contributions to observables are then ordered according to this small momentum, providing a well defined counting scheme that controls approximations to the theory. For nuclear systems, the appropriate effective field theory is Chiral Perturbation Theory since the long-range part of the nuclear force is associated with the pion, a Goldstone boson. A considerable amount of effort is being expended in the application of to low energy nucleon-nucleon scattering and to the deuteron. The NN system is particularly challenging in that its scattering lengths are large, although its bound state, the deuteron, is only weakly bound. This implies that there is a dynamical scale in the problem in addition to the chiral scale. The application of to the deuteron has followed two approaches. The first [131] applies the chiral counting scheme at the level of the interaction kernel. This approach appears to converge, but is cutoff dependent at each order. The second [134, 135] applies the counting scheme at the level of the scattering matrix and current matrix elements and treats the pions perturbatively while iterating the contact interactions. It gives results that are cutoff independent, but does not appear to be converging [140]. The problem of maintaining a consistent counting scheme in the infinite sums

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of diagrams necessary to describe the bound states has not yet been resolved. From the standpoint of the various models used at higher momentum transfers, it is to be hoped that effective field theory will provide some insight into the organization of the various approaches.

5.7. Deuteron models with nucleon isobar contributions In all of the approaches and models discussed to this point, the assumption has been made that the only relevant degrees of freedom are the nucleons and mesons. However, excitations of the nucleons to isobar states may produce contributions to the interactions of the same range as the heavier mesons. The deuteron wave function is in this case modified to include, in addition to the S and D wave N N components, and N N* components. A neutrino experiment, though subject to interpretation, indicates an upper limit of 0.4% to the amount of components in the deuteron [155]. Examples of two calculations including these additional isobaric components are shown here. In both cases, the basic model involves interactions due to one meson exchange which can couple the nucleon-nucleon channel with channels containing isobars. The quark model is used to relate isobarmeson couplings (for instance ) to the corresponding nucleon-meson couplings. The fully coupled (nonrelativistic) system is then solved for the deuteron bound state and the nucleon-nucleon scattering states. Figure 5.15 shows the calculations of Dymarz and Khanna (DK) [126] and Blunden et al., (BGL) [127]. In both cases presented here, only the isobar is included, and the electromagnetic form factors are assumed to be proportional to the NEMFF. The DK calculation contains components with a total probability of 0.36%. The elastic form factors include contributions from single-nucleon currents with the GK form factors, nucleon pair and exchange currents and the isobar current contributions. Two models are shown for BGL, both with only and components. Two different quark models are assumed to fix at a given radius the boundary conditions used in the determination of the wave functions. Model C’ has and yields an isobar contribution of 1.8%, while model D’ uses the Cloudy Bag Model to fix at 1.05 fm and results in an isobar contribution of 7.2%. Meson exchange contributions are included as well. Within these models, is very sensitive to the amount of components and favours the smallest probability. Although the BGL model C’ with the Höhler NEMFF seems to give an adequate description of the data, there is considerable ambiguity in these calculations due to a lack of knowledge of the isobar magnetic moments and form factors along with the difficulty of

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completely constraining the isobar contributions from the NN scattering data. It is clear, however, that isobar components must ultimately be considered in any complete calculation of deuteron electromagnetic properties in the context of hadronic models.

5.8.

5.8.1.

Quarks and gluons

Nonrelativistic quark models

Ultimately, many of the ambiguities associated with the hadronic model calculations can only be removed by describing the deuteron in terms of the fundamental quark and gluon degrees of freedom. In the absence of the capability to directly solve the QCD Lagrangian at low energies for the deuteron, we are left to explore possible QCD effects in the context of quark models. This greatly increases the difficulty of treating the NN system since what is a twobody problem in the hadronic models, becomes at least a six-body problem for quark models. Calculations for two examples of quark cluster models are presented here. These calculations are very similar in concept and are based on a quark cluster model using the resonating group method to describe the interaction of the two three-quark clusters. The quarks interact via a quadratic confining potential and a one-gluon-exchange potential. Long range interactions are also provided via and exchange between quarks. The calculation of Buchmann, Yamauchi and Faessler (BYF) [129] contains only the exchange interaction, while that of Ito and Kisslinger (IK) [128] contains both mesonexchange contributions. This approach naturally contains currents associated with the individual clusters as well as exchange currents associated with quark exchange between clusters. Since the cluster wave functions are derived using oscillator-like confining forces, the cluster form factors tend to have a Gaussian form. This is clearly not consistent with the data for single-nucleon form factors and in both cases the electromagnetic cluster form factors are replaced by phenomenological NEMFF. The results are shown in Fig. 5.16. Neither of these calculations provides an adequate description of the data, but given their necessary simplicity, they are remarkably close to the data. In the BYF calculation, the impulse approximation is not as reliable as in NN models because of the simple modeling of the intermediate range interaction, but the size of the genuine quark exchange contributions, due to the antisymmetrization of the six-quarks wave function, is significant enough to affect a comparison with the data.

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5.8.2. Perturbative QCD This subject was reviewed recently [4]. The success and shortcoming of perturbative quantum-chromodynamics (PQCD) applied to the asymptotic behaviour of form factors may be better illustrated now with recent data from Jefferson Lab. We will only recall briefly the main predictions: Dimensional scaling [156]: this property can be derived within QCD, but is more general and was established before this theory. It is based on the hypothesis that the momentum transfer is shared among all the constituent quarks of the system. In the case of a hadron composed of quarks, one would expect the leading form factor to behave asymptotically as Qualitatively, this can be viewed as the probability of having quarks within a transverse distance of of the quark struck by the virtual photon. Since this must be true both in the initial and final states, one expects and for the deuteron Logarithmic corrections to the leading amplitude, together with the factorization of the nucleon form factors in the weak binding limit, yield [157]:

where

is the nucleon form factor, an energy scale characteristic of and a calculable number much smaller than 1. An attempt to directly compute A and its normalization in PQCD was not successfull [4, 160]. Helicity conservation at each photon/gluon-quark vertex implies that the dominant contribution to elastic electron deuteron scattering should come from the configuration where the deuteron has helicity zero in both initial and final states [159]. Expressing the form factors in an helicity basis in the light-cone frame, this is equivalent to the prediction that the “+” component of the current matrix element between states of helicity 0, is the leading amplitude. It was argued [158] that this helicity conserving amplitude should dominate the scattering for Other components (single helicity flip) and (double helicity flip) should be suppressed by respectively one and two powers of

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Since the usual form factors are linear combinations of the the asymptotic behaviour of observable ratios such as B/A, and can easily be calculated. Note that the logarithmic corrections such as appearing in (5.33) may only be calculated for the dominant, helicity conserving, form factor. Equation (5.34) thus assumes the same logarithmic corrections for all helicity amplitudes. How do these predictions compare with recent data ? The A data, now extending up to [167], is suggestive of the expected behaviour, though a behaviour is still not excluded (see Fig. 5.17): a fit using the five highest data points in Ref. [167] to the dependence yields Excluding the less precise last point yields More interestingly, using the dipole form factor for in (5.33), the behaviour of A is reproduced between 2 and As for the helicity amplitudes, their behaviour is tested only up to which is the range of the available B [171] and [188] data. The pure dominance of the helicity 0 0 transition ( in (5.34) [158]) does not account for this data (see Fig. 5.18). The prescription and [162] was built to generate a node in but in order to qualitatively reproduce the data as well, one is led to and [161]. This implies that the double helicity flip amplitude is as large as the non helicity flip amplitude and contradicts the applicability of PQCD in the momentum range considered. Fits to data using these helicity amplitudes lead to the same conclusion [199, 163]. All these simple ansatz lead to a sharp increase of the ratio B/A for which may be an interesting feature for planned experiments at JLab.

5.9. Further Comparison Between Models and Data To conclude this Section, we recapitulate some of the theoretical results, in comparison with observables (Fig. 5.19) and data on separated form factors (Fig. 5.20). In the case of A and B, deviations with respect to an average representation of the data (parameterization I) are presented. A summary of remaining ambiguities and foreseable progress will be given in Sect. 8.

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THE NUCLEON MOMENTUM DISTRIBUTION IN THE DEUTERON

Several processes have been investigated for which the cross-sections, in the non-relativistic impulse approximation, are proportional to the distribution of the internal momentum in the deuteron,

with and defined by Eq. (2.8). Note that the conjugate variable of the relative coordinate is half the relative momentum between the two nucleons, so that an experiment probing a momentum may be related to elastic electron scattering at a momentum transfer These processes include quasi-free scattering on one of the nucleons, and or equivalently the detection of the spectator proton in high energy deuteron hadronic break-up, Indeed, they all are proportional to up to but deviations from the impulse approximation occur for higher values of Final state interaction, dynamical excitation of the resonance and in inclusive reactions pion production have then to be taken account, thus rendering the interpretation of the experiments less straightforward. Another interesting feature of these processes is the possibility to access the ratio with the measurement of deuteron tensor polarization observables. A simple relationship between the analyzing power and the ratio can be derived in the (nonrelativistic) impulse approximation, and again deviations are seen, or are to be expected, above

6.1.

Measurements at High Missing Momenta

The study of the single-particle properties of nuclei through reactions are the subject of an excellent review [7]. Concerning the deuteron, the (neutron) missing momentum is identified, in the plane-wave impulse approximation (PWIA), with the internal momentum The experiments reaching the highest missing momenta have been carried out at Saclay (500 MeV/c) [201], NIKHEF (700 MeV/c) [202] and MAMI (950 MeV/c) [203], but in kinematical conditions which are not always optimal to study the high momentum components in the deuteron wave function. Deviations from the PWIA are of the order of 50% at 500 MeV/c and can reach a factor 10 at 1 GeV/c (see Fig. 6.1). They can be understood, if only qualitatively at the highest missing momenta, in terms of final state interactions, meson exchange currents and excitation. Other measurements (see the reviews [204, 205] and, among the most recent experiments, Ref. [207]) do not address specifically the subject of

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the nucleon momentum distribution in the deuteron. Their theoretical understanding is however crucial for a comprehension of this subject. The first measurements using a tensor polarized target have been carried out at Novosibirsk [208] and NIKHEF [209]. They were limited to low values of missing momenta because of the available luminosity and of the detector acceptance. This situation will improve with the planned experiments using the BLAST detector in the Bates stretcher ring [197]. In the PWIA, these experiments are directly sensitive to the ratio (see below Eq.(6.2)).

6.2.

measurements and y-scaling

The concept of nuclear gives a significant insight in the determination of momentum distribution in nuclei [210]. In the inclusive electron scattering off deuterium, for sufficiently large values of momentum transfer and in the PWIA, the cross-sections are expected to be proportional to the electron-

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nucleon cross-sections and the proportionality factor represents the longitudinal (along ) momentum distribution. Properly taking into account final state interactions (a correction of about a factor 2 above 300 MeV/c), a momentum distribution was extracted in a nonrelativistic formalism up to [211]. It is remarkably close to the one calculated from the Paris potential, but the authors caution about the absence of relativistic corrections.

6.3.

Hadronic deuteron break-up at high energy

Several reactions using high energy (polarized) deuteron beams should allow a study of the vertex, provided the deuteron dissociates “cleanly”, that is with only one nucleon participating in the process. In the exclusive reactions as well as in the inclusive in some kinematical conditions, the dominant process is the one where the fast forward proton (or one of the detected protons) can be treated as a spectator. The energies of the deuteron beams available at SATURNE and at Dubna (several GeV), together with the relatively high hadronic cross-sections, have provided a handle on these processes involving high momentum components in the deuteron wave function. Moreover, the tensor analyzing power of these reactions, in the nonrelativistic impulse approximation, depends only on the D/S ratio at a given momentum:

Note the similarity between Eqs.(4.21) and (6.2). The former is a function of while the latter depends on resulting (in the impulse approximation) in similar shapes for in elastic scattering and in deuteron break-up. A relativistic treatment using the deuteron in an infinite momentum frame allows to define a new variable [212] used as the argument of and instead of for the calculation of the observables. In other words, the argument of the wave function is no longer equal to the momentum of the spectator nucleon. Once this transformation is performed, all inclusive data (see the review [214] and Refs. [215, 216]), for different beam energies and target nuclei, scale approximatively as a momentum distribution however deviates significantly from expectations within the impulse approximation as of This fact has sometimes been interpretated as the signature of non conventional components in the deuteron wave function, but this interpretation is not compatible with the behaviour of in elastic scattering. In reality, final state interactions and pion production alter significantly the interpretation of these experiments (see e.g. [213]). The polariza-

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tion transfer from vector polarized deuterons to the fast protons has also been measured up to The exclusive channels [1,214] and [217, 218] are likewise difficult to interpretate unambiguously for internal momenta larger than 300 MeV/c.

7. THE DEUTERON AS A SOURCE OF “FREE” NEUTRONS The neutron being loosely bound in the deuteron, deuterium has often been used as a substitute for a neutron target and deuteron beams as a source of neutron beams. We briefly mention the list of these applications: 1. elastic scattering to extract the neutron charge form factor 2. Quasi-elastic or to measure and to measure 3. Quasi-elastic 4. Deep inelastic scattering of leptons to extract the neutron structure functions. 5. At intermediate to high energies, the break-up of a vector polarized deuteron beam can be used to obtain a beam of polarized neutrons. A high intensity may be achieved using the inclusive reaction For a better definition of the neutron energy, the beam may be tagged by the detection of the spectator proton:

In all cases but the first one [183,145], corrections due to the deuteron structure and to the reaction mechanism were shown to be either negligible or reliably calculable (see [73] for example as an application of relativistic techniques discussed in Sect. 5.6.1.) When a polarized target is used (cases 2 and 4) or when using polarized deuteron beams (case 5), the effective neutron polarization is equal to the deuteron vector polarization multiplied by This factor accounts for the fact that, because of the deuteron D state, the neutron spin is not always aligned the deuteron spin.

8.

PROSPECTS FOR THE FUTURE

The study of the electromagnetic properties and form factors of the deuteron, from the birth of nuclear physics to the advent of hadronic physics, taking into account the internal structure of the nucleons and their excitations, has been very rich. Yet it is not completed. Ambiguities have been and still are pointed out along this path: although the descriptions of processes involved can be satisfactory, the calculations and observables do not in general allow for a unique determination of the NN interaction, of the neutron charge form

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factor, of the isoscalar meson exchange currents and of the precise dynamics of the system. As already pointed out, some characteristics like the off-shell behaviour of the N N interaction are not strictly speaking observables, and as such cannot be determined unambiguously. As for the manifestation of the underlying quark substructure in the nuclear properties, it is as elusive as ever. However the progress in experiment and theory, as summarized in this paper, is impressive and allows for promising perspectives: The nucleon electromagnetic form factors are being measured with a renewed precision. In particular, the poorly known neutron electric form factor is being determined at various laboratories with polarization techniques which make this measurement independent of the deuteron structure. In a few years, all four NEMFF will be better known up to The description of the the deuteron form factors to higher four-momentum transfers is now affected by measurements of Once these are completed, a small remaining ambiguity due to the poor knowledge of above will still be present. Yet new parameterizations of the NEMFF, guided by theoretical models, should become available soon and be taken into account in future calculations of the deuteron form factors. Modern NN interaction models are now fitted directly to the NN elastic scattering data and reach a high degree of precision. But, when compared to some models of the 1980’s, these are more empirical. The lack of knowledge (or assumption) of the underlying dynamics prevents the calculation of such effects as the Lorentz boost of the deuteron wave function. It would be highly desirable to have potentials of the one-boson exchange type brought to the degree of precision of the modern phenomenological potentials. This is also true for OBE potentials used in completely relativistic calculations. In addition, these potentials should reproduce the elastic NN scattering data up to 600 MeV of kinetic laboratory energy in order to match the of the existing elastic scattering data. There is a definite procedure to construct relativistic corrections, starting from a nonrelativistic model. Still, most calculations in the past have neglected one correction or another. From a theoretical point of view, the most satisfying success in the past twelve years is the implementation of various fully relativistic calculations. Quasi-potentials approximations to the Bethe-Salpeter equation have been applied with success to the calculation of the electromagnetic deuteron form factors. The comparative validity of each of these approximations should be studied further. Likewise, light-front dynam-

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ics yields results which compare very favourably with data and will still be improved. Concerning Hamiltonian constraint dynamics, calculations using the three different forms of quantization invoked by Dirac have been developed. The success of QPE/BSE and LFD has reaching consequences beyond the structure of the deuteron. The relevance and applicability of relativity in nuclei can now be explored in the A=3 systems. The same techniques are also applied to configurations. The isoscalar meson exchange contribution, and possibly the or other shorter range processes, still remain difficult to evaluate reliably, because of the lack of constraints on the associated form factors. Within the nucleon-meson picture of the deuteron, elastic scattering may provide a way to determine these form factors, provided all points above are addressed in a systematic way. The role of nucleon isobaric excitations is still very much model dependent. Still, within the existing models, the elastic scattering data does not favour very sizeable components in the deuteron wave function. The recent elastic scattering data from Jefferson Lab, reaching now the highest possible four-momentum transfers for the and A observables (respectively 1.7 and are still compatible with the description of the deuteron in terms of nucleons and mesons. This is somewhat surprising since internucleonic scales of 0.1-0.4 fm are being probed. These distances are smaller than the size of the nucleons themselves, and presumably of the same order of magnitude as the nucleon quark cores. Still no distinctive experimental or theoretical signature of the manifestation of quarks in this process was identified. Quarks degrees of freedom may be explicitly taken into account at intermediate four-momentum transfers via models, and at high via perturbative QCD. If the recent data seem to follow the expected from PQCD, it is not so for and albeit at lower An absolute determination of the leading PQCD amplitude would clearly be desirable but this depends on a reliable description of the soft parts of the amplitude which are not calculable in PQCD. As for quark models of the deuteron, they seem to indicate a specific role played by quark exchange processes between the nucleons at short distances. Unfortunately, these models are nonrelativistic by nature and the predicted effects occur at a scale where relativity should be taken into account. Due to the explicit appearance of the quark-gluon degrees of freedom, this a much more difficult problem than for the meson-nucleon models. Appreciable progress in quark models of the deuteron will require substantial improvements in the technology of relativistic many-body physics.

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The planned measurements of at Bates and Novosibirsk will locate more precisely the position of the node of the charge monopole form factor. while the ones of at Jefferson Lab will be performed around and beyond the node of the magnetic dipole form factor. The recent focus on intermediate and high should not be detrimental to the required precision in accounting for the low data and static properties. An updated experimental status of the latter was given. The low behaviour of any given calculation should be checked carefully, for example in such representations as in Figs 5.2 and 5.19. Going beyond elastic scattering, the same systematic expansions in or fully relativistic models, will be applied to the calculation of the electromagnetic form factors of the A=3 nuclei, and of the deuteron electrodisintegration. This will provide additional contraints on the remaining ambiguities inherent in the meson-nucleon models and will lead to a more coherent understanding of the relativistic structure of few-body nuclei.

ACKNOWLEDGEMENTS One of the authors (M.G.) gratefully acknowledges the many teachings of R. Beurtey and A. Boudard on the deuteron and on polarization. He also benefited greatly from the stimulating collaborative efforts for the Bates and Jefferson Lab experiments, in particular with E.J. Beise, J. Cameron, S. Kox and W. Turchinetz. The Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility under DOE contract DE-AC05-84ER40150.