Springer Series in Nuclear

and Particle Physics

Springer Series in Nuclear and Particle Physics Editors: Mary K. Gaillard . J. Maxwell Irvine . Erich Lohrmann . Vera Liith

Achim Richter

Hasse, R. W., Myers W. D. Geometrical Relationships of Macroscopic Nuclear Physics

Belyaev, V. B. Lectures on the Theory of Few-Body Systems

Heyde, K.L.G. The Nuclear SheD Model

Gitman, D.M., Tyutin I.V. Quantization of Fields with Constraints

Sitenko, A. G. Scattering Theory

Fradkin, E. S., Gitman, D. M., Shvartsman, S. M. Quantum Electrodynamics with Unstable Vacuum

Kris L. G. Heyde

The Nuclear Shell Model

With 171 Figures

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona

Professor Dr. Kris L. G. Heyde Laboratorium voorTheoretische Fysica en Laboratorium voor Kernfysica, Rijksuniversiteit Gent Proeftuinstraat 86, B-9000 Gent, Belgium

Editor:

Professor Dr. J. Maxwell Irvine Department of Theoretical Physics The Schuster Laboratory, The University Manchester, M139PL, United Kingdom

ISBN-13: 978-3-642-97205-8 e-ISBN-13: 978-3-642-97203-4 DOl: 10.1007/978-3-642-97203-4

Library of Congress-Cataloging-in-Publication Data. Heyde, Kris L. G., 1942-. The nuclear shell model 1 Kris Heyde. p. cm.-(Springer series in nuclear and particle physics) Includes bibliographical references. 1. Nuclear shell theory. 2. Nuclear models. 3. Nuclear structure. I. TItle. II Series. QC793.3.S8H48 1990 539.1'43-dc20 90-9596

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is con­cerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act ofthe German Copyright Law.

C Springer-Verlag Berlin Heidelberg 1990 Softcover reprint of the hardcover I st edition 1990 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and there­fore free for general use.

2157/3150-543210 - Printed on acid-free paper

To Daisy, Jan, and Mieke

Preface

This book is aimed at enabling the reader to obtain a working knowledge of the nuclear shell model and to understand nuclear structure within the framework of the shell model. Attention is concentrated on a coherent, self-contained exposition of the main ideas behind the model with ample illustrations to give an idea beyond formal exposition of the concepts.

Since this text grew out of a course taught for advanced undergraduate and first-year graduate students in theoretical nuclear physics, the accents are on a detailed exposition of the material with step-by-step derivations rather than on a superficial description of a large number of topics. In this sense, the book differs from a number of books on theoretical nuclear physics by narrowing the subject to only the nuclear shell model. Most of the expressions used in many of the existing books treating the nuclear shell model are derived here in more detail, in a practitioner's way. Due to frequent student requests I have expanded the level of detail in order to take away the typical phrase " ... after some simple and straightforward algebra one finds ... ". The material could probably be treated in a one-year course (implying going through the problem sets and setting up a number of numerical studies by using the provided computer codes). The book is essentially self-contained but requires an introductory course on quantum mechanics and nuclear physics on a more general level. Because of this structure, it is not easy to pick out certain chapters for separate reading, although an experienced practitioner of the shell model could do that

After introductory but necessary chapters on angular momentum, angular mo­mentum coupling, rotations in quantum mechanics, tensor algebra and the calcu­lation of matrix elements of spherical tensor operators within angular momentum coupled states, we start the exposition of the shell model itself. Chapters 3 to 7 discuss the basic ingredients of the shell model exposing the one-particle, two­particle and three-particle aspects of the nuclear interacting shell-model picture. Mter studying electromagnetic properties (one-body and two-body moments and transition rates), a short chapter is devoted to the second quantization, or occupa­tion number representation, of the shell model. In later chapters, the elementary modes of excitation observed in closed shell nuclei (particle-hole excitations) and open shell nuclei (pairing properties) are discussed with many applications to realistic nuclei and nuclear mass regions. In Chap. 8, a state-of-the-art illustra­tion of present day possibilities within the nuclear shell model, constructing both the residual interaction and the average field properties is given. This chapter has a somewhat less pedagogical orientation than the first seven chapters. In the

VII

final chapter, some simple computer codes are included and discussed. The set of appendices constitutes an integral part of the text, as well as a number of exercises.

Several aspects of the nuclear interacting many-body system are not discussed or only briefly mentioned. This is due to the choice of developing the nuclear shell model as an in-depth example of how to approximate the interactions in a complicated many-fermion system. Having studied this text, one should be able, by using the outlined techniques, to study other fields of nuclear theory such as nuclear collective models and Hartree-Fock theory.

This book project grew out of a course taught over the past 8 years at the University of Gent on the nuclear shell model and has grown somewhat beyond the original concept. Thereby, in the initial stages of teaching, a set of unpublished lecture notes from F. Iachello on nuclear structure, taught at the "Kernfysisch Versneller Instituut" (KVI) in Groningen, were a useful guidance and influenced the first chapters in an important way. I am grateful for the many students who, by encouraging more and clearer discussions, have modified the form and content in almost every aspect. The problems given here came out of discussions with them and out of exam sets: the reader is encouraged to go through them as an essential step in mastering the content of the book.

I am most grateful to my colleagues at the Institute of Nuclear Physics in Gent, in particular in the theory group, in alphabetic order, C. De Coster, J. Jolie, J. Moreau, J. Ryckebusch, P. Van Isacker, D. Van Neck, J. Van Maldeghem, H. Vincx, M. Waroquier, and G. Wenes who contributed, maybe unintentionally, to the present text in an important way. More in particular, I am indebted to M. Waroquier for the generous permission to make extensive use of results obtained in his "Hoger Aggregaat" thesis about the feasability of performing shell-model calculations in a self-consistent way using Skyrme forces. I am also grateful to

C. De Coster for scrutinizing many of the formulas, reading and critizing the whole manuscript.

Also, discussions with many experimentalists, both in Gent and elsewhere, too many to cite, have kept me from "straying" from the real world of nuclei. I would like, in particular, to thank J.L. Wood, R. F. Casten and R.A. Meyer for insisting on going ahead with the project and Prof. M. Irvine for encouragement to put this manuscript in shape for the Springer Series.

Most of my shell-model roots have been laid down in the Utrecht school; I am most grateful to P.J. Brussaard, L. Dieperink, P. Endt, and P.W.M. Glaudemans for their experience and support during my extended stays in Utrecht.

Gent, March 1990 K.L.G. Heyde

VIII

Contents

Introduction 1

1. Angular Momentum in Quantum Mechanics . . . . . . . . . . . . . . . . 4 1.1 Central Force Problem and Orbital Angular Momentum ..... 4 1.2 General Definitions of Angular Momentum ............... 12

1.2.1 Matrix Representations ......................... 13 1.2.2 Example for Spin ! Particles .................... 13

1.3 Total Angular Momentum for a Spin! Particle ........... 14 1.4 Coupling of Two Angular Momenta:

Clebsch-Gordan Coefficients ........................... 17 1.5 Properties of Clebsch-Gordan Coefficients ................ 20 1.6 Racah Recoupling Coefficients:

Coupling of Three Angular Momenta .................... 22 1.7 Symmetry Properties of 6j-Symbols . . . . . . . . . . . . . . . . . . . . 23 1.8 Wigner 9j-Symbols: Coupling and Recoupling

of Four Angular Momenta ............................. 25 1.9 Classical Limit of Wigner 3j-Symbols ................... 27 Short Overview of Angular Momentum Coupling Formulas 28

2. Rotations in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1 Rotation of a Scalar Field-Rotation Group 0(3) ........... 31 2.2 General Groups of Transformations ..................... 35 2.3 Representations of the Rotation Operator ................. 37

2.3.1 The Wigner D-Matrices ........................ 37 2.3.2 The Group SU(2)-Relation with SO(3) ........... 38 2.3.3 Application: Geometric Interpretation

of Intrinsic Spin! . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Product Representations and Irreducibility ................ 43 2.5 Cartesian Tensors, Spherical Tensors, Irreducible Tensors ... 45 2.6 Tensor Product ...................................... 47 2.7 Spherical Tensor Operators: The Wigner-Eckart Theorem ... 48 2.8 Calculation of Matrix Elements ............ . . . . . . . . . . . . 49

2.8.1 Reduction Rule I .............................. 50 2.8.2 Reduction Rule IT ............................. 51

Short Overview of Rotation Properties, Tensor Operators, Matrix Elements . .. ..... ....... . ...... ... ...... . . .. ...... 52

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3. The Nuclear SheD Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1 One-particle Excitations ............................... 54

3.1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.1.2 The Radial Equation and the Single-particle Spectrum:

the Harmonic Oscillator in the SheD Model ........ 61 3.1.3 illustrative Examples of Energy Spectra ........... 67 3.1.4 Hartree-Fock Methods: A Simple Approach ........ 70

3.2 Two-particle Systems: Identical Nucleons ................ 74 3.2.1 Two-particle Wavefunctions ..................... 74 3.2.2 Two-particle Residual Interaction ................ 77 3.2.3 Calculation of Two-Body Matrix Elements ......... 87 3.2.4 Configuration Mixing:

Model Space and Model Interaction .............. 101 3.3 Three-particle Systems and Beyond ..................... 108

3.3.1 Three-particle Wave Functions .................. 108 3.3.2 Extension to n-particle Wave Functions ........... 112 3.3.3 Some Applications: Three-particle Systems ........ 115

3.4 Non-identical Particle Systems: Isospin .................. 119 3.4.1 Isospin: Introduction and Concepts ............... 119 3.4.2 Isospin Formalism ............. . . . . . . . . . . . . . . . 121 3.4.3 Two-Body Matrix Elements with Isospin .......... 130

4. Electromagnetic Properties in the Shell Model ............... 136 4.1 General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.2 Electric and Magnetic Multipole Operators ............... 137 4.3 Single-particle Estimates and Examples .................. 139 4.4 Electromagnetic Transitions in Two-particle Systems ....... 145 4.5 Quadrupole Moments................................. 149

4.5.1 Single-particle Quadrupole Moment .............. 149 4.5.2 Two-particle Quadrupole Moment ................ 152

4.6 Magnetic Dipole Moment ............................. 153 4.6.1 Single-particle Moment: Schmidt Values .......... 153 4.6.2 Two-particle Dipole Moment .................... 156

4.7 Additivity Rules for Static Moments .................... 157

s. Second Quantization ..................................... 161 5.1 Creation and Annihilation Operators ... . . . . . . . . . . . . . . . . . 161 5.2 Operators in Second Quantization ....................... 165 5.3 Angular Momentum Coupling in Second Quantization ...... 169 5.4 Hole Operators in Second Quantization .................. 171 5.5 Normal Ordering, Contraction, Wick's Theorem ........... 174 5.6 Application to the Hartree-Fock Formalism ............... 177

x

6. Elementary Modes of Excitation: Particle-Hole Excitations at Closed Shells ... . . . . . . . . . . . . . . . . 179 6.1 General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.2 The TDA Approximation ...............•............. 181 6.3 The RPA Approximation .............................. 186 6.4 Application of the Study of 1p - 1h Excitations: 160 ....... 192

7. Pairing Correlations: Particle-Particle Excitations in Open-Shell Nuclei ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.1 Introduction ........................................ 197 7.2 Pairing in a Degenerate Single j-Shell ................... 200 7.3 Pairing in Non-Degenerate Levels: Two-Particle Systems ... 204 7.4 n Particles in Non-Degenerate Shells: BCS-Theory ........ 206 7.5 Applications of BCS ................................. 217

7.5.1 Odd-Even Mass Differences, Elqp •••••.•••••••.•• 217 7.5.2 Energy Spectra ............................... 219 7.5.3 Electromagnetic Transitions ..................... 221 7.5.4 Spectroscopic Factors .......................... 226

7.6 Broken-Pair Model ................................... 229 7.6.1 Low-Seniority Approximation to the Shell Model ... 229 7.6.2 Broken-Pair or Generalized-Seniority Scheme

for Semi-Magic Nuclei ......................... 232 7.6.3 Generalization to Both Valence Protons and Neutrons 237

7.7 Interacting Boson-Model Approximation to the Nuclear Shell Model ............................ 239

8. Self-Consistent Shell-Model Calculations . ......... ... .. . ... 254 8.1 Introduction ........................................ 254 8.2 Construction of a Nucleon-Nucleon Force: Skynne Forces 256

8.2.1 Hartree-Fock Bogoliubov (HFB) Formalism for Nucleon-Nucleon Interactions Including Three-Body Forces .................... 256

8.2.2 Application of HFB to Spherical Nuclei ........... 259 8.2.3 The Extended Skynne Force .................... 260 8.2.4 Parameterization of Extended Skynne Forces:

Nuclear Ground-State Properties ................. 265 8.3 Excited-State Properties of SkE Forces .... . . . . . . . . . . . . . . 271

8.3.1 Particle-Particle Excitations: Determination of X3 ••• 272 8.3.2 The Skynne Interaction as a Particle-Hole Interaction 279 8.3.3 Rearrangement Effects

for Density-Dependent Interactions and Applications for SkE F<;>rces ................. 291

XI

9. Some Computer Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 9.1 Oebsch-Gordan Coefficients ........................... 298 9.2 Wigner 6j-Symbol ................................... 300 9.3 Wigner 9j-Symbol ................................... 303 9.4 Calculation of Table of Slater Integrals .................. 304 9.5 Calculation of c5-Matrix Element........................ 309 9.6 Matrix Oiagonalization ............................... 316 9.7 Radial Integrals Using Hannonic Oscillator Wave Functions. 320 9.8 BCS Equations with Constant Pairing Strength ............ 323

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 A. The Angular Momentum Operator in Spherical Coordinates . 327 B. Explicit Calculation of the Transformation Coefficients for

Three-Angular Momentum Systems ..................... 328 C. Tensor Reduction Formulae for Tensor Products ........... 329 O. The Surface-Delta Interaction (SOl) ..................... 331 E. Multipole Expansion of c5(rl - rz) Interaction ............ 335 F. Calculation of Reduced Matrix Element «1/2l)jIlY kll(1/21')j')

and Some Important Angular Momentum Relations ........ 338 G. The Magnetic Multipole Operator ....................... 342 H. A Two-Group (Degenerate) RPA Model ................. 343 I. The Condon-Shortley

and Biedenharn-Rose Phase Conventions: Application to Electromagnetic Operators and BCS Theory 347 1.1 Electromagnetic Operators:

Long-Wavelength Form and Matrix Elements ........ 347 1.2 Properties of the Electromagnetic Multipole Operators

Under Parity Operation,TIme Reflection and Hermitian Conjugation ....................... 348

1.3 Phase Conventions in the BCS Formalism ........... 353

Problems .................................................. 357

References ................................................. 367

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

XII