SU(3) symmetry in atomic nuclei 9789811536021, 9789811536038

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V. K. B. Kota

SU(3) Symmetry in Atomic Nuclei

SU(3) Symmetry in Atomic Nuclei

V. K. B. Kota

SU(3) Symmetry in Atomic Nuclei

123

V. K. B. Kota Theoretical Physics Division Physical Research Laboratory Ahmedabad, Gujarat, India

ISBN 978-981-15-3602-1 ISBN 978-981-15-3603-8 https://doi.org/10.1007/978-981-15-3603-8

(eBook)

© Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publisher, 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 any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To all my teachers

Preface

As atomic nuclei with their constituents, protons and neutrons, exhibit both single-particle structure and collective properties, they lead to the introduction and development of nuclear shell model and geometric collective model, respectively. These two contrasting pictures of nuclei are unified by the algebraic group theoretical models or “symmetries”, in particular by the interacting boson model. Remarkably, Elliott has shown in 1958 that the nuclear shell model admits SU(3) symmetry and this will generate the collective rotational spectra in nuclei starting with the independent particle picture. As he stated in his articles in the proceedings of the volume on 40 years of SU(3)—apart from classification of states, two other crucial features of SU(3) were uncovered, the existence of an intrinsic state and the quadrupole force. During the first decade after its introduction, SU(3) was established to be good for lighter nuclei with A < 40 but not beyond. The spin–orbit force, cornerstone of the nuclear shell model, breaks the SU(3) symmetry. However, rotational structures appear in nuclear spectra all across the periodic table. Deeper scrutiny of SU(3) and rotational structures from 1968 to 2000 revived interest in SU(3) in nuclei. In particular, the pseudo-spin and pseudo-Nilsson-based pseudo-SU(3) model and the Sp(6, R) model that generates hydrodynamic structures, both within the shell model, and the interacting boson model and interacting boson–fermion model with their various extensions paved the way for the applications of SU(3) symmetry across the periodic table. Clearly, SU(3) symmetry/algebra is now one of the central themes in nuclear physics. In the last decade, renewed interest is seen from the introduction of new schemes such as the proxy-SU(3) model and the symmetry adopted no-core shell model and new applications to clustering in nuclei and quantum phase transitions to name a few. Applications of SU(3) symmetry in nuclei have become practical as the SU(3) algebra was developed in considerable detail with efforts particularly due to Hecht, Draayer, Vergados, Akiyama, Moshinsky, Biedenharn, Rowe, and many others. From Elliott’s introduction in 1958 till now, different aspects of SU(3) have been discussed briefly in different books. In addition, various technical aspects of SU(3) algebra are scattered in various research articles. In spite of its great importance, there is no single book exclusively on SU(3) symmetry in nuclei. The purpose vii

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of the present monograph is to fill this gap. Different facets of SU(3) symmetry in nuclei are covered in the present book—from various aspects of SU(3) algebra to SU(3)-based models and their applications in nuclei. An additional chapter is devoted to the applications of SU(3) in statistical nuclear physics in the context of French’s statistical nuclear spectroscopy and random matrix theory. It is appropriate to recall here that pairing plus quadrupole is the most important force among nucleons. With SU(3) generated by quadrupole force, it is but essential to have also a book on pairing symmetries, within both shell model and interacting boson model, in atomic nuclei. This book, a research monograph, should be helpful to both theoretical physicists and experimentalists in understanding and interpreting rotational structures in light to heavy nuclei with ever growing experimental data. Young researchers in nuclear physics will find it useful, with technical details described in minimal manner, in understanding the role of symmetries in nuclei and using SU(3)-based models in their work. This book may be used for course work by Ph.D. students in nuclear physics, and is appropriate for students who had one-semester course in nuclear physics with knowledge in angular momentum algebra. Over the last four decades, I have had the pleasure of collaborating and interacting with many people on topics related to SU(3) in nuclei. Firstly, I would like to express my gratitude to late Profs. J.B. French, K.T. Hecht, and J.C. Parikh from whom I have directly or indirectly learned about symmetries in general and SU(3) in particular. In the same vein, I would like to thank Profs. F. Iachello and J.P. Draayer whose papers inspired me to continue working on SU(3) in nuclei. I am also thankful to R.D. Ratna Raju for introducing me to SU(3). I would like to thank my collaborators R. Bijker, J.A. Castilho Alcarás, N.D. Chavda, Y.D. Devi, K.B.K. Mayya, Ushasi Datta Pramanik, R. Sahu, P.C. Srivastava, and Manan Vyas. Thanks to N.D. Chavda for helping in preparing some figures. I am thankful to the directors of Physical Research Laboratory (Ahmedabad, India) for facilities and support. There are many others who have directly or indirectly contributed to my research work and I sincerely thank them. Several chapters and sections in this book are based on the talks/lectures given in Kolkata, Mexico, Delhi, Puri, Roorkee, Shanghai, and Sofia. Copyright permissions for using some of the figures and tables from the American Physical Society, Elsevier Science, IOP Publishing, Springer Nature, and World Scientific are gratefully acknowledged. Thanks are also due to Profs. Dennis Bonatsos, Mark Spieker, and Philip Adsley for granting permission to use figures from their papers. Special thanks are to the editors of Springer Nature for their efforts in bringing out this book. Ahmedabad, India December 2019

V. K. B. Kota

Contents

1 5

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

SUð3Þ Algebra in Nuclei: Preliminaries . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 SUð3Þ  SOð3Þ  SOð2Þ Algebra: Quadrupole Operator . 2.3 SUð3Þ  ½SUð2Þ  SOð2Þ  Uð1Þ Algebra . . . . . . . . . . 2.4 SUð3Þ Irreps ðklÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Young Tableaux . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Kronecker Products of SUð3Þ Irreps . . . . . . . . . 2.4.3 Dimension of ðklÞ Irreps . . . . . . . . . . . . . . . . . 2.4.4 Leading SUð3Þ Irrep in Uððg þ 1Þðg þ 2Þ=2Þ . . . 2.5 SUð3Þ Quadratic and Cubic Casimir Operators . . . . . . . . 2.6 SUð3Þ  SOð3Þ  SOð2Þ States: K Label . . . . . . . . . . . . 2.7 SUð3Þ  SUð2Þ  Uð1Þ States . . . . . . . . . . . . . . . . . . . . 2.8 Preliminary Applications of SUð3Þ Symmetry . . . . . . . . . 2.8.1 SUð3Þ in Shell Model . . . . . . . . . . . . . . . . . . . . 2.8.2 SUð3Þ in Interacting Boson Model . . . . . . . . . . 2.9 SUð3Þ in Particle Physics . . . . . . . . . . . . . . . . . . . . . . . 2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3

SUð3Þ Wigner–Racah Algebra I . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 SUð3Þ Irreps for Many-Particle Systems . . . . . . . . . . . . . . 3.2.1 Plethysm Method . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Recursion Method . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Difference Method . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Method for Obtaining a Few Lower SUð3Þ Irreps

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SUð3Þ Wigner and Racah Coefficients . . . . . . . . . . . . 3.3.1 SUð3Þ  SUð2Þ  Uð1Þ Reduced Wigner Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 SUð3Þ  SOð3Þ Reduced Wigner Coefficients 3.3.3 SUð3Þ Racah or U and Z Coefficients . . . 3.4 Building Up Principle and General Comments . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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SUð3Þ Wigner–Racah Algebra II . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 SUð3Þ Tensorial Decomposition and Wigner–Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Examples from sd and sdgIBM . . . . . . . . . . . . 4.2.2 Shell Model Two-Body Interactions . . . . . . . . 4.2.3 Analytical Results for Electric Quadrupole Transition Strengths . . . . . . . . . . . . . . . . . . . . 4.3 SUð3Þ Fractional Parentage Coefficients . . . . . . . . . . . . 4.3.1 Construction of SUð3Þ Intrinsic States: IBM Examples . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 SUð3Þ Intrinsic States: Fermion Examples . . . . 4.3.3 Triple Barred SUð3Þ Reduced Matrix Elements 4.4 9  SUð3Þ Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 4.5 SUð3Þ D-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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SUð3Þ  SOð3Þ Integrity Basis Operators . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 X3 and X4 Operators and Their Matrix Elements . . . 5.3 Shape Parameters and ðklÞ Irreps Correspondence . 5.4 Integrity Basis Hamiltonian and Asymmetric Rotor . 5.5 K Quantum Number from X3 and X4 Operators . . . 5.6 Extension to KJ Quantum Number . . . . . . . . . . . . . 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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SUð3Þ in Shell Model Based Approaches and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Pseudo SUð3Þ Model with Pseudo-spin . . . . . . . . . ~sÞ Basis 6.2.1 Mapping of Operators to Pseudo-ð‘~

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Basic Results from Pseudo-spin and Pseudo-Nilsson Orbits . . . . . . . . . . . . . . . . 6.2.3 Spectroscopy with Pseudo-SUð3Þ Symmetry . . . 6.3 Proxy-SUð3Þ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Prolate Dominance over Oblate Shape . . . . . . . . 6.3.2 Results for the Deformation Parameters (b; c) and BðE2Þ’s . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Spð6; RÞ Model with SUð3Þ Subalgebra . . . . . . . . . . . . . 6.4.1 SUð3Þ Limit of Spð6; RÞ . . . . . . . . . . . . . . . . . . 6.5 Fermion Dynamical Symmetry Model with SUð3Þ Limit . 6.5.1 i Active and k Active Schemes . . . . . . . . . . 6.5.2 Fermion Dynamical Symmetry Model . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

8

SUð3Þ in Interacting Boson Models . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 SUð3Þ in sdgIBM . . . . . . . . . . . . . . . . . . . . . 7.2.1 SUsd ð3Þ  1g Limit . . . . . . . . . . . . . 7.2.2 SUsdg ð3Þ Limit . . . . . . . . . . . . . . . . . 7.2.3 DL ¼ 4 Staggering in sdgIBM . . . . . 7.3 SUð3Þ in sdpf IBM . . . . . . . . . . . . . . . . . . . . 7.3.1 Introduction . . . . . . . . . . . . . . . . . . . 7.3.2 Dynamical Symmetries of sdpf IBM and the SUð3Þ Limit . . . . . . . . . . . . . 7.3.3 Analytical Results for E1 Transitions in SUsd ð3Þ  SUpf ð3Þ Limit . . . . . . . . 7.4 SUð3Þ in Proton–Neutron IBM (IBM-2) . . . . . 7.4.1 SUð3Þ in pn  sdIBM . . . . . . . . . . . . 7.4.2 SUð3Þ in pn  sdgIBM . . . . . . . . . . . 7.5 SUð3Þ in IBM-3 and IBM-4 Models . . . . . . . 7.5.1 SUð3Þ Limit of IBM-3 . . . . . . . . . . . 7.5.2 SUð3Þ Limit of IBM-4 . . . . . . . . . . . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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SUð3Þ in Interacting Boson–Fermion Models . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 SUð3Þ  j Limit of IBFM for Odd-A Nuclei . . . 8.3 SU BF ð3Þ Limit of IBFM for Odd-A Nuclei . . . . 8.3.1 Nilsson Correspondence I . . . . . . . . . . 8.3.2 Nilsson Correspondence II . . . . . . . . . 8.3.3 Application to E2 Transition Strengths: Example . . . . . . . . . . . . . . . . . . . . . .

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8.3.4

Application to M1 Transition Strengths: 185 Re Example . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Single Nucleon Transfer: 185 W Example . . . 8.4 SUð3Þ in IBFFM for Odd–Odd Nuclei . . . . . . . . . . . 8.4.1 SU BF ð3Þ  Uð2j þ 1Þ Limit: 186 Re Example . 8.4.2 SU BFF ð3Þ Limit : 190 Ir Example . . . . . . . . . 8.5 SUð3Þ in IBF2 M for 2 Quasi-particle Excitations . . . 8.6 SUð3Þ in sdgIBFM-2 and M1 Distributions . . . . . . . 8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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Extended Applications of SUð3Þ . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Phase Transitions with SUð3Þ . . . . . . . . . . . . . . . . . . . . . 9.2.1 Uð5Þ to SUð3Þ Transition . . . . . . . . . . . . . . . . . . 9.2.2 Example of an Analytically Solvable QPT . . . . . . 9.2.3 Critical Point Xð5Þ Symmetry for Uð5Þ ! SUð3Þ Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Partial SUð3Þ Dynamical Symmetry . . . . . . . . . . . . . . . . . 9.4 SUð3Þ for Removal of Spurious c.m. States . . . . . . . . . . . 9.5 SUð3Þ for Clustering in Nuclei . . . . . . . . . . . . . . . . . . . . 9.5.1 Nuclear Vibron Model . . . . . . . . . . . . . . . . . . . . 9.5.2 Semi-microscopic Algebraic Cluster Model with SUð3Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 SUð3Þ in No-Core Shell Model . . . . . . . . . . . . . . . . . . . . 9.6.1 Symmetry Adopted SUð3Þ Based No-Core Shell Model (SA-NCSM) . . . . . . . . . . . . . . . . . . . . . . 9.6.2 No-Core Symplectic Shell Model (NCSpM) . . . . . 9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Statistical Nuclear Physics with SUð3Þ . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Preliminaries of Statistical Spectroscopy . . . . . . . . . . . 10.2.1 Averages, Traces, State Densities, and Partial Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 General Principles of Trace Propagation . . . . 10.3 SUð3Þ Energy Centroids and Goodness of SUð3Þ Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 ð2s1dÞ Shell Model Example . . . . . . . . . . . . 10.3.2 Spð6; RÞ  SUð3Þ Example . . . . . . . . . . . . . . 10.4 Application of SUð3Þ Energy Centroids: Regularities with Random Interactions . . . . . . . . . . . . . . . . . . . . . 10.4.1 Regular Structures from Random Interactions: sdpf IBM Example . . . . . . . . . . . . . . . . . . . .

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10.4.2 Regular Structures from Random Interactions: sdIBM-T Example . . . . . . . . . . . . . . . . . . . . 10.5 Partition Functions and Level Density Enhancement in Deformed Nuclei with SUð3Þ . . . . . . . . . . . . . . . . 10.6 Statistical Group Theory for SUð3Þ Multiplicities . . . . 10.7 Example of a Random Matrix Ensemble with SUð3Þ Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.1 Definition of EGUE(2)-SUð3Þ Ensemble . . . . 10.7.2 Basic Formulation for Analytical Treatment of EGUE(2)-SUð3Þ . . . . . . . . . . . . . . . . . . . . 10.7.3 Results for Lower Order Moments of Oneand Two-Point Functions . . . . . . . . . . . . . . . 10.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Multiple SUð3Þ Algebras in Interacting Boson Model and Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Four SUð3Þ Algebras in sdgIBM: Results for Quadrupole Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Structure of Intrinsic States . . . . . . . . . . . . . . . . . . 11.2.2 Large-N Limit Results for Quadrupole Moments and BðE2Þ’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Eight SUð3Þ Algebras in sdgiIBM: Results for Quadrupole Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Multiple SUð3Þ Algebras in Shell Model . . . . . . . . . . . . . . 11.4.1 ðsdgÞ6p ;2n Example . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 ðsdgiÞ6p Example . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 247 . . . 247 . . . 248 . . . 249 . . . 250 . . . 251 . . . 255 . . . 255 . . . 259 . . . 260 . . . 260

12 Summary and Future Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Appendix A: Angular Momentum Algebra . . . . . . . . . . . . . . . . . . . . . . . . 265 Appendix B: Elements of UðnÞ Lie Algebra and Its Subalgebras . . . . . . 271 Appendix C: Asymptotic Nilsson Wavefunctions . . . . . . . . . . . . . . . . . . . 279 2

Appendix D: Correspondence Between SU BF ð3Þ Irreps and 2 q.p. Nilsson Configurations for g ¼ 3 Shell . . . . . . . 283 Appendix E: Bivariate Moments, Cumulants, and Edgeworth Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

About the Author

V. K. B. Kota is an honorary faculty member at the Theoretical Physics Division, Physical Research Laboratory, Ahmedabad, India, which he joined in 1980. He completed his Ph.D. in Physics at Andhra University, Visakhapatnam, India. From 2007–2015, he was Adjunct Professor at the Department of Physics and Astronomy, Laurentian University, Sudbury, Canada. During his last 40 years of research, he made three major contributions to theoretical physics. Firstly, he identified and developed more than 10 new bosonic, fermionic and boson–fermion Lie algebraic group symmetries for collective states in atomic nuclei. In particular, he worked extensively on hexadecapole degree of freedom in nuclei and published the first review article on this topic with one of his former students. Secondly, he derived several new statistical laws using unitary group decompositions, quantum chaos and random matrices, for the spectral distributions of observables (part of statistical nuclear spectroscopy), with applications in nuclear astrophysics and neutrino physics. This work led to a number of review articles and the book Spectral Distributions in Nuclei and Statistical Spectroscopy (World Scientific, 2010), co-authored with Prof. R.U. Haq. Thirdly, he introduced and analyzed random matrix ensembles, generically called “embedded random matrix ensembles,” generated by random interactions with Lie algebraic symmetries. These were proved to be applicable to isolated finite interacting quantum systems, ranging from complex nuclei to mesoscopic devices of condensed matter to black holes. Dr. Kota wrote the first review article on this topic in 2001 and later a book, Embedded Random Matrix Ensembles in Quantum Physics (Springer, 2014).

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Chapter 1

Introduction

Atomic nuclei, with protons and neutrons, exhibit both single-particle structure and collective properties. The former seen in magic numbers, ground-state spins, magnetic moments, and so on gave birth to the nuclear shell model [1]. On the other hand, observation of rotational spectra in nuclei led to the development of collective or geometric models [2]. However, there is a third approach that unifies the shell model and collective models and this is “symmetries”or algebraic group theoretical approach. Let us recall what was written by Bohr and Mottelson in the preface of their volumes on ‘Nuclear Structure’[2, 3]: · · · in the study of a many-body system such as the nucleus with its rich variety of structural facets, the central problem appears to be the identification of the appropriate concepts and degrees of freedom that are suitable for describing the phenomena encountered. Progress in this direction has been achieved by a combination of approaches based partly on clues provided by experimental data, partly on the theoretical study of model systems, and partly on the exploration of general relations following from considerations of symmetry. Remarkably, Elliott has shown in 1958 that the nuclear shell model (SM) admits SU (3) symmetry and this will generate the collective rotational spectra in nuclei starting with the independent particle picture and a quadrupole–quadrupole force [4, 5]. Many facets of SU (3) symmetry in nuclei are described in this book. With efforts due to Elliott, Harvey, Moshinsky, Arima, and many others, by mid 60’s it was recognized that SU (3) is a reasonably good and very useful symmetry for (1 p) and (2s1d) shell nuclei (for example for 8 Be, 12 C, 20 Ne, 24 Mg, etc.) [6]. However, it will be a broken symmetry, due to strong spin–orbit force, in (2 p1 f ) shell nuclei (44 Ti, 48 Cr, etc.) and beyond; see [7, 8]. In early 70’s, Hecht and others recognized [9–11] that for heavier nuclei pseudo-SU (3), based on pseudo spin [12– 14] and pseudo Nilsson orbits [9, 15, 16], will be a useful symmetry. However, only in mid eighties due to the efforts of Draayer, pseudo-SU (3) is established to be good for heavy deformed nuclei [16–18] and in 1990, significance of pseudo-SU (3) in super deformed nuclei was established [19, 20]. Similarly, more recently, proxySU (3) scheme by Bonatsos, Casten, and others [21–23] has appeared and this scheme © Springer Nature Singapore Pte Ltd. 2020 V. K. B. Kota, SU(3) Symmetry in Atomic Nuclei, https://doi.org/10.1007/978-981-15-3603-8_1

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1 Introduction

explains in a transparent manner prolate dominance over oblate shape in heavy nuclei. In addition, there is the quasi-SU (3) scheme by Zuker [24–26], also within SM, explaining in a simple way the deformation characteristics in A = 60 − 100 nuclei. Besides all these, in the multi-shell situation of SM, again SU (3) appears within the Sp(6, R) model of Rowe and Rosensteel [27–29] and this has given rise to the ab-initio no-core-sympletic shell model [30–32]. Going beyond SM, new and more interest in SU (3) was generated by the interacting boson model (IBM) of atomic nuclei. This model is built on the recognition that interacting quadrupole bosons (d bosons or phonons) generate quadrupole vibrational spectra with U (5) symmetry and enlarging this to U (6) by addition of a scalar boson (s boson) allows one to embed SU (3) subalgebra generating rotations [33, 34]. Thus, with s and d bosons we have U (6) spectrum generating algebra (SGA) and this contains both U (5) and SU (3) as subalgebras unifying quadrupole vibrational structure and rotations of a quadrupole deformed system. Let us add that the SU (6) ⊃ SU (3) structure with quadrupole phonons was seen first in the numerical calculations due to Jolos et al [35, 36]. Going beyond sdIBM, the sdgIBM with hexadecupole bosons (g bosons) [37, 38], sdp f IBM with dipole ( p) and octupole ( f ) bosons [39, 40] contain SU (3) symmetry generating rotational spectra with many additional new structures. In addition, proton-neutron IBM or sdIBM-2 has SU (3) symmetry and this generates scissors states that are later seen for the first time in 156 Gd [34, 41]. Similarly, sdIBM-3 with isospin (T ) and sdIBM-4 with spin–isospin (ST ) degrees of freedom [34, 42] also contain U (6) ⊃ SU (3) algebra in the orbital space. In addition, in IBM-3 and IBM-4 models SU (3) also appears for isospin (T ) and spin–isospin (ST ) degrees of freedom, respectively. Similarly, for odd-A nuclei we have SU B F (3) ⊗ SU F (2) symmetry within the interacting boson–fermion model (IBFM) in correspondence with the coriolis mixed Nilsson configurations [43, 44]. This extends to SU (3) in IBFFM for odd–odd nuclei [45, 46] generating doubly decoupled bands and mixing of Nilsson quantum numbers. In addition, there is SU (3) in IBF2 M for two quasi-particle excitations in even–even nuclei with clear Nilsson correspondence [47]. Applications of SU (3) symmetry in SM and IBM became practical as the SU (3) algebra was developed in considerable detail with efforts particularly due to Hecht, Draayer, Vergados, Akiyama, Moshinsky, Rowe, and many others. For example, methods to obtain SU (3) irreducible representations (irreps) for fermions or bosons in a given oscillator shell η with U ((η + 1)(η + 2)/2) SGA, SU (3) ⊃ S O(3) reduced Wigner coefficients, SU (3) Racah coefficients, 9 − SU (3) coefficients, SU (3) coefficients of fractional parentage, SU (3) ⊃ S O(3) integrity basis operators and so on are all available now along with good computer codes for generating the same; see for example [48–63]. With SU (3) generating rotations within both SM and IBM, it is natural that many researchers looked for new perspectives in the applications of SU (3) symmetry enlarging the scope of SU (3) in nuclei. Some of these are: (i) SU (3) for clustering using ideas based on SM and IBM [64–67]; (ii) SU (3) as a partial dynamical symmetry [68]; (iii) multiple SU (3) algebras in SM and IBM spaces [69]; (iv) SU (3) in quantum phase transitions [70]; (v) statistical group theory of SU (3) [7, 71, 72].

1 Introduction

3

Fig. 1.1 (Color  online) Some typical axially symmetric nuclear  shapes. These are constructed using R(θ, φ) = R0 1 + β2 Y02 (θ, φ) + β3 Y03 (θ, φ) + β4 Y04 (θ, φ) . The shapes shown are: (a) spherical with (β2 , β3 , β4 ) = (0, 0, 0); (b) prolate spheroid with (β2 , β3 , β4 ) = (0.25, 0, 0); (c) oblate spheroid with (β2 , β3 , β4 ) = (−0.25, 0, 0); (d) octupole with (β2 , β3 , β4 ) = (0, 0.3, 0); (e) hexadecupole with (β2 , β3 , β4 ) = (0, 0, 0.3); (f) quadrupole plus hexadecupole (called β2 + β4 shape in the figure) with (β2 , β3 , β4 ) = (0.25, 0, 0.15). Figure is generated using MATHEMATICA

Clearly, SU (3) symmetry/algebra has become one of the central themes in nuclear physics as rotational structures appear in nuclear spectra all across the periodic table. As Harvey states (see p. 68 in [6]): It (SU (3)) has the important property of exhibiting simultaneously both the collective and single-particle aspects of a many-body state. This arises from the remarkable fact that SU (3) admits two subalgebras with one of them (introduced in Sects. 2.2 and 2.6) giving results directly in the laboratory frame as in the spherical shell model and the other (introduced in Sects. 2.3 and 2.7) in the intrinsic frame of a deformed nucleus as in the Nilsson model. It is well known that the nuclear collective dynamics (vibrations, rotations, etc.) arises from various nuclear shapes and some of the shapes appropriate for the low-lying nuclear levels are shown in Fig. 1.1 Most important among these is the quadrupole deformation and hence the SU (3) algebra (in the later Chapters, the relevance of SU (3) for octupole and hexadecupole deformation as well as for clustering will be described). In spite of all the above, SU (3) in nuclear structure is less appreciated and applied unlike SU (3) in particle physics. From Elliott’s introduction in 1958 till now different aspects of SU (3) are discussed briefly in different books in a Chapter or a Section or in an Appendix; see for example [7, 34, 42, 44, 63, 73–76]. However, in spite of its great importance, there is no single book exclusively on SU (3) symmetry in nuclei and the purpose of the present monograph is to fill this gap. As the literature on SU (3) in nuclear physics is vast spanning six decades, it is impractical to discuss in the book each and every development in the subject in detail. Inadvertently, some

4

1 Introduction

important topics might have been presented briefly in the book though they may deserve much larger discussion. The choices made reflect only the constraints the author has rather than on their relative importance. Literature survey for this book has ended in March 2019. Now, we will give a preview. In Chap. 2, some preliminary aspects of SU (3) algebra as introduced by Elliott in nuclear physics are presented. These include, SU (3) symmetry in an oscillator shell generated by the physically relevant angular momentum and quadrupole moment operators, two subalgebras of SU (3), irrep labels (λμ) of SU (3) and the quadratic and cubic Casimir invariants and their eigenvalues. Also, given is a brief introduction to SU (3) symmetry in SM and IBM. In addition, at the end SU (3) in particle physics is briefly described. Chapter 3 gives the methods for obtaining SU (3) irreps in an irrep of the oscillator SGA U ((η + 1)(η + 2)/2); η is the oscillator major shell number. In addition, introduced and described in some detail are SU (3) Wigner and Racah coefficients. Continuing this, in Chap. 4 described are tensor operators with respect to SU (3) for applying the Wigner–Eckart theorem, nine-SU (3) coefficients, SU (3) coefficients of fractional parentage and SU (3) D-functions. Chapter 5 is on SU (3) ⊃ S O(3) integrity basis operators that play an important role in the spectroscopy with SU (3). In Chap. 2–5 enough details are given so that a reader familiar with angular momentum algebra (say at the level of Edmonds book [77]), basic elements of group theory and Lie algebras (say at the level of the discussion in the appendices in [42, 78]) and basics of nuclear physics and nuclear models (say at the level of the first three sections of J. Lilley’s book [79]) will be able to carry out analytical and numerical calculations using SU (3) algebra and also follow the various results presented in the remaining Chapters. For easy reading, in Appendix A collected are some important results from angular momentum algebra and in Appendix B presented are some elements of group theory with particular reference to U (N ) algebra. In Chap. 6, applications of SU (3) in shell model are described with reference to pseudo-SU (3) model, proxy SU (3) model, Sp(6, R) model with SU (3) algebra that generates a nocore shell model and the fermion dynamical symmetry model (FDSM) with SU (3). Chapter 7 is on SU (3) in the interacting boson models with sd, sdg, and sdp f bosons for even–even nuclei with protons and neutrons in different shells. Discussed are also the models IBM-3 with isospin and IBM-4 with spin–isospin degrees of freedom that apply to nuclei with protons and neutrons occupying the same orbits (also IBM-4 applies to odd–odd nuclei). Similarly, Chap. 8 is on SU (3) in interacting boson– fermion models for odd-A nuclei, odd–odd nuclei and two quasi-particle excitations in even–even nuclei. Emphasized here is the correspondence between SU (3) and Nilsson model configurations. Many other applications of SU (3) in nuclei are given in Chap. 9. These include application to quantum phase transitions (QPT), partial dynamical symmetries, clustering in nuclei, and no-core shell model. In addition, described also is the application of SU (3) for removal of spurious center of mass states in SM. Chapter 10 is on statistical nuclear physics with SU (3) in the context of statistical nuclear spectroscopy and random matrix theory. Chapter 11 deals with the new paradigm of multiple SU (3) algebras in SM and IBM. Finally, Chap. 12 gives future outlook.

References

5

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1 Introduction

74. J.G. Hirsch, C.E. Vargas, G. Popa, J.P. Draayer, Pseudo+Quasi SU(3): towards a shell model description of heavy deformed nuclei, in Computational and Group Theoretical Models in Nuclear Physics, eds. by J. Escher, J.H. Hirsch, S. Pittel, O. Castansos, G. Stoicheva (World Scientific, Singapore, 2004), pp. 31–39 75. I. Talmi, Simple Models of Complex Nuclei: The Shell Model and the Interacting Boson Model (Harwood, New York, 1993) 76. P. Van Isacker, Dynamical symmetries in the structure of nuclei. Rep. Prog. Phys. 62, 1661– 1717 (1999) 77. A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton, New Jersey, 1974) 78. A. Frank, P. Van Isacker, Algebraic Methods in Molecular and Nuclear Physics (Wiley, New York, 1994) 79. J. Lilley, Nuclear Physics: Principles and Applications (Wiley, Singapore, 2003)

Chapter 2

SU(3) Algebra in Nuclei: Preliminaries

2.1 Introduction In this chapter, we will begin by introducing SU (3) in an oscillator shell and present some basic results of SU (3) algebra and preliminary applications to SM and IBM. Although the physics of SU (3) in nuclei is different from the one in particle physics, the algebraic results of SU (3) apply to both. In emphasizing this, in the end we will also present the quantum numbers for elementary particles using some simple results derived for SU (3) in nuclei.

2.2 SU(3) ⊃ S O(3) ⊃ S O(2) Algebra: Quadrupole Operator Given an oscillator shell with major shell number η, the orbital angular momentum carried by a particle (fermion or boson) in the shell takes values  = η, η − 2, . . ., 0 or 1. For simplicity, let us consider identical bosons in the shell η. With m denoting the z quantum number, the boson creation and annihilation operators for † and bm , respectively. Note that |, m  are sp states. a single particle (sp) are bm The SGA U ((η + 1)(η + 2)/2) is generated by the operators (b† b˜ )qk and significantly, it admits SU (3) subalgebra. It is well know that SU (3) will have eight generators. As nuclear levels carry angular momentum as a quantum number, it is natural to consider the three angular momentum operators L q1 and look for the remaining 5 generators. As was shown first by Elliott, the quadrupole moment oper ator Q q2  =

4π 5

r 2 Yq2 (θ, φ) + p 2 Yq2 (θ p , φ p ) with its five components complete

the set of eight generators. These in terms of (b† , b) operators, using oscillator radial wavefunctions, are easily, in the angular momentum coupled representation, given by

© Springer Nature Singapore Pte Ltd. 2020 V. K. B. Kota, SU(3) Symmetry in Atomic Nuclei, https://doi.org/10.1007/978-981-15-3603-8_2

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2 SU (3) Algebra in Nuclei: Preliminaries

10



( + 1)(2 + 1)  † ˜ 1 b b , q 3   ( + 1)(2 + 1)  † 2 b b˜  Q q2  = −(2η + 3) (2.1) q 5(2 + 3)(2 − 1)  2 2    6( + 1)( + 2)(η − )(η +  + 3)  † † b b˜+2  + b+2 + b˜  . q q 5(2 + 3)  0. Using the equivalence (λμ) → {λ + μ, μ, 0} and

2 SU (3) Algebra in Nuclei: Preliminaries

16

then applying Eq. (2.18), we have for the multiplication of two SU (3) irreps (λ1 μ1 ) × (λ2 μ2 ) the following simple rule: {λ1 + μ1 , μ1 , 0} × {λ2 + μ2 , μ2 , 0} [{λ1 + μ1 } × {μ1 } − {λ1 + μ1 + 1} × {μ1 − 1}] × {μ2 − 1}] [{λ2 + μ2 } × {μ2 } − {λ 2 + μ2 + 1}         f 1 − f 2 , f 2 − f 3 . f 1 , f 2 , f 3 −→ (λ, μ) = (2.19) Note that all the above Kronecker products follow from Eq. (2.17). For example, (λ1 μ1 ) × (λ2 μ2 ) −→ = × =

(60) × (40) → {6} × {4} = {10} + {9, 1} + {8, 2} + {7, 3} + {6, 4} = (10, 0) + (8, 1) + (6, 2) + (4, 3) + (2, 4) . Another example that gives new information is (11) × (11) → {2, 1} × {2, 1} = {2, 1} × [{2} × {1} − {3} × {0}] → (2, 2) + (0, 3) + (3, 0) + (1, 1)2 + (0, 0) . In this second example, the irrep (11) appears twice. Thus, we see that the Kronecker product of SU (3) irreps will in general give multiplicity for some irreps. In contrast, for S O(3) Kronecker products there will be no multiplicity.

2.4.3 Dimension of (λμ) Irreps One simple application of the Kronecker products is in obtaining the formula for the number of states d(λμ) in a given SU (3) irrep (λμ). The d(λμ) is also called dimension of (λμ). Here, we will use the well-known result that  the dimension and this is d({m}) of a totally symmetric irrep {m} with respect to U (3) is m+2 2 same as the dimension of a system of m bosons in three sp states. With this, the formula for d(λμ) is (λ, μ) → {λ + μ, μ} = {λ + μ} ⊗ {μ} − {λ + μ + 1} ⊗ {μ − 1} → d(λ, μ) = d({λ + μ})d({μ}) − d({λ + μ + 1})d({μ − 1}) (λ + μ + 1)(λ + μ + 2)(μ + 1)(μ + 2) (λ + μ + 2)(λ + μ + 3)(μ)(μ + 1) − = 4 4 = (λ + 1)(μ + 1)(λ + μ + 2)/2 . (2.20) This formula is very useful in checking many of the results in Chap. 3.

2.4 SU (3) Irreps (λμ)

17

2.4.4 Leading SU(3) Irrep in U((η + 1)(η + 2)/2) Decomposition of { f } of U (N ) to (λμ) of SU (3) is a complicated problem and it is postponed to the next Chapter. Here we will consider the formula for the leading SU (3) irrep as this is simple and also important in many applications (see Sect. 2.8 and Chaps. 6 and 11). For a given { f } = { f 1 , f 2 , . . . , f N } of U (N ), the ground state or leading SU (3) irrep (λμ) = (λ H , μ H ) is the one with largest value of 2λ + μ and for this λ = λ H with largest value of μ out of all the allowed (λμ) irreps. In this section and elsewhere N = (η + 1)(η + 2)/2. For a given η, the sp orbits in (n z , n x , n y ) representation are (η − r, r − x, x) in the order with r taking values 0 to η and for each r , x takes values 0 to r ; see Eq. (2.9) for examples. Now putting f 1 number of particles in the first orbit, f 2 in the second orbit, and so on with f N in the last orbit will give the total Nz , N x , and N y (note that Ni is sum of single particle n i values). Then we have, λ H = Nz − N x and μ H = N x − N y [7]. This is converted into a simple formula that is valid for any η and { f }. The final result is { f }U (N ) → (λ H , μ H ) SU (3) with η η r r     (η − 2r + x) × f 1+x+ r (r +1) , μ H = (r − 2x) × f 1+x+ r (r +1) . λH = 2

r =0 x=0

2

r =0 x=0

(2.21) It is trivial to program Eq. (2.21). Results for shells η = 2, 3, 4, 5, 6 for identical fermions with particle number m = 2 to (η + 1)(η + 2) and spin S = 0 for even m and S = 1/2 for odd m are given in Table 2.1. They can be extended easily to η = 7, 8 as maybe needed for super heavy (SHE) nuclei. Note that { f } = {2 p } for m even and total spin S = 0 with p = m/2. Similarly, { f } = {2 p , 1} for m odd and S = 1/2 with p = (m − 1)/2. Here, we are using Eq. (2.15) for U (2N ) ⊃ U (N ) ⊗ SU (2) with { f } a irrep of U (N ) in the orbital space and spin S is generated by SU (2); note that, given a m the U (2N ) irrep is {1m }. With protons and neutrons in the same oscillator shell η, using orbital and spin– isospin SU (4) decomposition U (4N ) ⊃ U (N ) ⊗ SU (4), we need to consider four column irreps of U (N ) and find out the leading SU (3) irrep contained in this irrep; see Appendix B for more details regarding Wigner’s spin–isospin SU (4) symmetry. Given the nucleon number m and the isospin T = |TZ | (note that TZ = (m p − m n )/2 where m p is number of protons and m n is number of neutrons with m = m p + m n ), the lowest U (4) irrep {F1 , F2 , F3 , F4 } is given as follows; note that SU (4) irreps follow from U (4) irreps. For m even we have {F1 , F2 , F3 , F4 } = {F1 , F2 , F3 , F4 } =

m+2T

, m+2T , m−2T , m−2T for m2 4 4 4 4 m+2T

+2 m+2T −2 m−2T +2 m−2T −2 , , , 4 4 4 4

+ T even , for

m 2

+ T odd . (2.22)

2 SU (3) Algebra in Nuclei: Preliminaries

18

Table 2.1 Ground state or leading SU (3) irrep (λ H , μ H ) for a given number m of identical particles in a oscillator shell η. Results are shown for η = 2, 3, 4, 5, and 6. For a given m, the (λ H , μ H ) is given in the table as (λ H , μ H )m . See text for other details η=2 (4, 0)2 , (4, 1)3 , (4, 2)4 , (5, 1)5 , (6, 0)6 , (4, 2)7 , (2, 4)8 , (1, 4)9 , (0, 4)10 , (0, 2)11 , (0, 0)12 η=3 (6, 0)2 , (7, 1)3 , (8, 2)4 , (10, 1)5 , (12, 0)6 , (11, 2)7 , (10, 4)8 , (10, 4)9 , (10, 4)10 , (11, 2)11 , (12, 0)12 , (9, 3)13 , (6, 6)14 , (4, 7)15 , (2, 8)16 , (1, 7)17 , (0, 6)18 , (0, 3)19 , (0, 0)20 η=4 (8, 0)2 , (10, 1)3 , (12, 2)4 , (15, 1)5 , (18, 0)6 , (18, 2)7 , (18, 4)8 , (19, 4)9 , (20, 4)10 , (22, 2)11 , (24, 0)12 , (22, 3)13 , (20, 6)14 , (19, 7)15 , (18, 8)16 , (18, 7)17 , (18, 6)18 , (19, 3)19 , (20, 0)20 , (16, 4)21 , (12, 8)22 , (9, 10)23 , (6, 12)24 , (4, 12)25 , (2, 12)26 , (1, 10)27 , (0, 8)28 , (0, 4)29 , (0, 0)30 η=5 (10, 0)2 , (13, 1)3 , (16, 2)4 , (20, 1)5 , (24, 0)6 , (25, 2)7 , (26, 4)8 , (28, 4)9 , (30, 4)10 , (33, 2)11 , (36, 0)12 , (35, 3)13 , (34, 6)14 , (34, 7)15 , (34, 8)16 , (35, 7)17 , (36, 6)18 , (38, 3)19 , (40, 0)20 , (37, 4)21 , (34, 8)22 , (32, 10)23 , (30, 12)24 , (29, 12)25 , (28, 12)26 , (28, 10)27 , (28, 8)28 , (29, 4)29 , (30, 0)30 , (25, 5)31 , (20, 10)32 , (16, 13)33 , (12, 16)34 , (9, 17)35 , (6, 18)36 , (4, 17)37 , (2, 16)38 , (1, 13)39 , (0, 10)40 , (0, 5)41 , (0, 0)42 η=6 (12, 0)2 , (16, 1)3 , (20, 2)4 , (25, 1)5 , (30, 0)6 , (32, 2)7 , (34, 4)8 , (37, 4)9 , (40, 4)10 , (44, 2)11 , (48, 0)12 , (48, 3)13 , (48, 6)14 , (49, 7)15 , (50, 8)16 , (52, 7)17 , (54, 6)18 , (57, 3)19 , (60, 0)20 , (58, 4)21 , (56, 8)22 , (55, 10)23 , (54, 12)24 , (54, 12)25 , (54, 12)26 , (55, 10)27 , (56, 8)28 , (58, 4)29 , (60, 0)30 , (56, 5)31 , (52, 10)32 , (49, 13)33 , (46, 16)34 , (44, 17)35 , (42, 18)36 , (41, 17)37 , (40, 16)38 , (40, 13)39 , (40, 10)40 , (41, 5)41 , (42, 0)42 , (36, 6)43 , (30, 12)44 , (25, 16)45 , (20, 20)46 , (16, 22)47 , (12, 24)48 , (9, 24)49 , (6, 24)50 , (4, 22)51 , (2, 20)52 , (1, 16)53 , (0, 12)54 , (0, 6)55 , (0, 0)56

The only exception is T = 0 for m = 4r + 2 type and then  {F1 , F2 , F3 , F4 } =

m+2 m+2 m−2 m−2 , , , 4 4 4 4

 .

(2.23)

Similarly, for odd-m we have {F1 , F2 , F3 , F4 } = {F1 , F2 , F3 , F4 } =

m+2T +2 4 m+2T 4

,

−2 m−2T m−2T , m+2T , 4 , 4 4

m+2T 4

,

m−2T +2 , 4



m−2T −2 4

for for

m 2 m 2

+ T odd , + T even .

(2.24)

Equations (2.22)–(2.24) are well known and given in many papers (but with different notations). For example see [8, 9]. Using Eqs. (2.22)–(2.24), we can obtain the lowest U (4) irrep for a given m and |Tz |. Note that, with N  = (η + 1)(η + 2), values of TZ for m ≤ N  are |TZ | = m/2, m/2 − 1,…, 0 or 1/2. For m > N  , we have |TZ | = (2N  − m)/2, (2N  − m)/2 − 1,…, 0 or 1/2. Now, our task is to find the U (N ) irrep { f } that corresponds to a given {F1 , F2 , F3 , F4 }. From Eq. (2.15) it is easy to see that { f } is conjugate of {F1 , F2 , F3 , F4 } and also { f } will be {4a 3b 2c 1d } type irrep. Then, it is easy to find that a = F4 , b = F3 − F4 , c = F2 − F3 , and d = F1 − F2 . Therefore, { f } will have, in Young tableaux notation, four boxes in a (= F4 ) number

2.4 SU (3) Irreps (λμ)

19

Table 2.2 Ground state or leading SU (3) irrep (λ H , μ H ) for a given number m of nucleons in an oscillator shell η with T = |TZ | value. Note that (m, TZ ) defines uniquely the irrep { f } of U ((η + 1)(η + 2)/2). Results are shown for η = 2, 3 and 4 with m even only. For a given m and T , the (λ H , μ H ) is given in the table as (λ H , μ H )m,T η=2 (8, 0)4,0 , (6, 1)4,1 , (4, 2)4,2 , (8, 2)6,0 , (8, 2)6,1 , (7, 1)6,2 , (6, 0)6,3 , (8, 4)8,0 , (9, 2)8,1 , (10, 0)8,2 , (6, 2)8,3 , (2, 4)8,4 , (10, 2)10,0 , (10, 2)10,1 , (8, 3)10,2 , (6, 4)10,3 , (3, 4)10,4 , (0, 4)10,5 , (12, 0)12,0 , (9, 3)12,1 , (6, 6)12,2 , (5, 5)12,3 , (4, 4)12,4 , (2, 2)12,5 , (0, 0)12,6 , (8, 4)14,0 , (8, 4)14,1 , (6, 5)14,2 , (4, 6)14,3 , (4, 3)14,4 , (4, 0)14,5 , (4, 8)16,0 , (5, 6)16,1 , (6, 4)16,2 , (5, 3)16,3 , (4, 2)16,4 , (2, 8)18,0 , (2, 8)18,1 , (4, 4)18,2 , (6, 0)18,3 , (0, 8)20,0 , (1, 6)20,1 , (2, 4)20,2 , (0, 4)22,0 , (0, 4)22,1 , (0, 0)24,0 η=3 (12, 0)4,0 , (10, 1)4,1 , (8, 2)4,2 , (14, 2)6,0 , (14, 2)6,1 , (13, 1)6,2 , (12, 0)6,3 , (16, 4)8,0 , (17, 2)8,1 , (18, 0)8,2 , (14, 2)8,3 , (10, 4)8,4 , (20, 2)10,0 , (20, 2)10,1 , (18, 3)10,2 , (16, 4)10,3 , (13, 4)10,4 , (10, 4)10,5 , (24, 0)12,0 , (21, 3)12,1 , (18, 6)12,2 , (17, 5)12,3 , (16, 4)12,4 , (14, 2)12,5 , (12, 0)12,6 , (22, 4)14,0 , (22, 4)14,1 , (20, 5)14,2 , (18, 6)14,3 , (18, 3)14,4 , (18, 0)14,5 , (12, 3)14,6 , (6, 6)14,7 , (20, 8)16,0 , (21, 6)16,1 , (22, 4)16,2 , (21, 3)16,3 , (20, 2)16,4 , (16, 4)16,5 , (12, 6)16,6 , (7, 7)16,7 , (2, 8)16,8 , (20, 8)18,0 , (20, 8)18,1 , (22, 4)18,2 , (24, 0)18,3 , (19, 4)18,4 , (14, 8)18,5 , (11, 8)18,6 , (8, 8)18,7 , (4, 7)18,8 , (0, 6)18,9 , (20, 8)20,0 , (21, 6)20,1 , (22, 4)20,2 , (20, 5)20,3 , (18, 6)20,4 , (14, 8)20,5 , (10, 10)20,6 , (8, 8)20,7 , (6, 6)20,8 , (3, 3)20,9 , (0, 0)20,10 , (22, 4)22,0 , (22, 4)22,1 , (19, 7)22,2 , (16, 10)22,3 , (15, 9)22,4 , (14, 8)22,5 , (11, 8)22,6 , (8, 8)22,7 , (7, 4)22,8 , (6, 0)22,9 , (24, 0)24,0 , (20, 5)24,1 , (16, 10)24,2 , (14, 11)24,3 , (12, 12)24,4 , (12, 9)24,5 , (12, 6)24,6 , (10, 4)24,7 , (8, 2)24,8 , (18, 6)26,0 , (18, 6)26,1 , (15, 9)26,2 , (12, 12)26,3 , (11, 11)26,4 , (10, 10)26,5 , (11, 5)26,6 , (12, 0)26,7 , (12, 12)28,0 , (13, 10)28,1 , (14, 8)28,2 , (12, 9)28,3 , (10, 10)28,4 , (10, 7)28,5 , (10, 4)28,6 , (8, 14)30,0 , (8, 14)30,1 , (10, 10)30,2 , (12, 6)30,3 , (11, 5)30,4 , (10, 4)30,5 , (4, 16)32,0 , (5, 14)32,1 , (6, 12)32,2 , (9, 6)32,3 , (12, 0)32,4 , (2, 14)34,0 , (2, 14)34,1 , (4, 10)34,2 , (6, 6)34,3 , (0, 12)36,0 , (1, 10)36,1 , (2, 8)36,2 , (0, 6)38,0 , (0, 6)38,1 , (0, 0)40,0 η=4 (16, 0)4,0 , (14, 1)4,1 , (12, 2)4,2 , (20, 2)6,0 , (20, 2)6,1 , (19, 1)6,2 , (18, 0)6,3 , (24, 4)8,0 , (25, 2)8,1 , (26, 0)8,2 , (22, 2)8,3 , (18, 4)8,4 , (30, 2)10,0 , (30, 2)10,1 , (28, 3)10,2 , (26, 4)10,3 , (23, 4)10,4 , (20, 4)10,5 , (36, 0)12,0 , (33, 3)12,1 , (30, 6)12,2 , (29, 5)12,3 , (28, 4)12,4 , (26, 2)12,5 , (24, 0)12,6 , (36, 4)14,0 , (36, 4)14,1 , (34, 5)14,2 , (32, 6)14,3 , (32, 3)14,4 , (32, 0)14,5 , (26, 3)14,6 , (20, 6)14,7 , (36, 8)16,0 , (37, 6)16,1 , (38, 4)16,2 , (37, 3)16,3 , (36, 2)16,4 , (32, 4)16,5 , (28, 6)16,6 , (23, 7)16,7 , (18, 8)16,8 , (38, 8)18,0 , (38, 8)18,1 , (40, 4)18,2 , (42, 0)18,3 , (37, 4)18,4 , (32, 8)18,5 , (29, 8)18,6 , (26, 8)18,7 , (22, 7)18,8 , (18, 6)18,9 , (40, 8)20,0 , (41, 6)20,1 , (42, 4)20,2 , (40, 5)20,3 , (38, 6)20,4 , (34, 8)20,5 , (30, 10)20,6 , (28, 8)20,7 , (26, 6)20,8 , (23, 3)20,9 , (20, 0)20,10 , (44, 4)22,0 , (44, 4)22,1 , (41, 7)22,2 , (38, 10)22,3 , (37, 9)22,4 , (36, 8)22,5 , (33, 8)22,6 , (30, 8)22,7 , (29, 4)22,8 , (28, 0)22,9 , (20, 4)22,10 , (12, 8)22,11 , (48, 0)24,0 , (44, 5)24,1 , (40, 10)24,2 , (38, 11)24,3 , (36, 12)24,4 , (36, 9)24,5 , (36, 6)24,6 , (34, 4)24,7 , (32, 2)24,8 , (26, 5)24,9 , (20, 8)24,10 , (13, 10)24,11 , (6, 12)24,12 , (44, 6)26,0 , (44, 6)26,1 , (41, 9)26,2 , (38, 12)26,3 , (37, 11)26,4 , (36, 10)26,5 , (37, 5)26,6 , (38, 0)26,7 , (31, 5)26,8 , (24, 10)26,9 , (19, 11)26,10 , (14, 12)26,11 , (8, 12)26,12 , (2, 12)26,13 , (40, 12)28,0 , (41, 10)28,1 , (42, 8)28,2 , (40, 9)28,3 , (38, 10)28,4 , (38, 7)28,5 , (38, 4)28,6 , (34, 6)28,7 , (30, 8)28,8 , (24, 11)28,9 , (18, 14)28,10 , (14, 13)28,11 , (10, 12)28,12 , (5, 10)28,13 , (0, 8)28,14 , (38, 14)30,0 , (38, 14)30,1 , (40, 10)30,2 , (42, 6)30,3 , (41, 5)30,4 , (40, 4)30,5 , (35, 8)30,6 , (30, 12)30,7 , (27, 12)30,8 , (24, 12)30,9 , (19, 13)30,10 , (14, 14)30,11 , (11, 11)30,12 , (8, 8)30,13 , (4, 4)30,14 , (0, 0)30,15 , (36, 16)32,0 , (37, 14)32,1 , (38, 12)32,2 , (41, 6)32,3 , (44, 0)32,4 , (38, 6)32,5 , (32, 12)32,6 , (28, 14)32,7 , (24, 16)32,8 , (22, 14)32,9 , (20, 12)32,10 , (16, 11)32,11 , (12, 10)32,12 , (10, 5)32,13 , (8, 0)32,14 , (36, 14)34,0 , (36, 14)34,1 , (38, 10)34,2 , (40, 6)34,3 , (38, 7)34,4 , (36, 8)34,5 , (31, 12)34,6 , (26, 16)34,7 , (23, 16)34,8 , (20, 16)34,9 , (19, 12)34,10 , (18, 8)34,11 , (15, 5)34,12 , (12, 2)34,13 , (36, 12)36,0 , (37, 10)36,1 , (38, 8)36,2 , (35, 11)36,3 , (32, 14)36,4 , (31, 13)36,5 , (30, 12)36,6 , (26, 14)36,7 , (22, 16)36,8 , (20, 14)36,9 , (18, 12)36,10 , (18, 6)36,11 , (18, 0)36,12 , (38, 6)38,0 , (38, 6)38,1 , (34, 11)38,2 , (30, 16)38,3 , (28, 17)38,4 , (26, 18)38,5 , (26, 15)38,6 , (26, 12)38,7 , (23, 12)38,8 , (20, 12)38,9 , (19, 8)38,10 , (18, 4)38,11 , (40, 0)40,0 , (35, 7)40,1 , (30, 14)40,2 , (27, 17)40,3 , (24, 20)40,4 , (23, 19)40,5 , (22, 18)40,6 , (23, 13)40,7 , (24, 8)40,8 , (22, 6)40,9 , (20, 4)40,10 , (32, 8)42,0 , (32, 8)42,1 , (28, 13)42,2 , (24, 18)42,3 , (22, 19)42,4 , (20, 20)42,5 , (20, 17)42,6 , (20, 14)42,7 , (22, 7)42,8 , (24, 0)42,9 , (24, 16)44,0 , (25, 14)44,1 , (26, 12)44,2 , (23, 15)44,3 , (20, 18)44,4 , (19, 17)44,5 , (18, 16)44,6 , (19, 11)44,7 , (20, 6)44,8 , (18, 20)46,0 , (18, 20)46,1 , (20, 16)46,2 , (22, 12)46,3 , (20, 13)46,4 , (18, 14)46,5 , (18, 11)46,6 , (18, 8)46,7 , (12, 24)48,0 , (13, 22)48,1 , (14, 20)48,2 , (17, 14)48,3 , (20, 8)48,4 , (19, 7)48,5 , (18, 6)48,6 , (8, 24)50,0 , (8, 24)50,1 , (10, 20)50,2 , (12, 16)50,3 , (16, 8)50,4 , (20, 0)50,5 , (4, 24)52,0 , (5, 22)52,1 , (6, 20)52,2 , (9, 14)52,3 , (12, 8)52,4 , (2, 20)54,0 , (2, 20)54,1 , (4, 16)54,2 , (6, 12)54,3 , (0, 16)56,0 , (1, 14)56,1 , (2, 12)56,2 , (0, 8)58,0 , (0, 8)58,1 , (0, 0)60,0

2 SU (3) Algebra in Nuclei: Preliminaries

20

of rows, three boxes in b (= F3 − F4 ) number of rows, two boxes in c (= F2 − F3 ) number of rows, and one box in d (= F1 − F2 ) number of rows. With { f } thus determined, we can find the leading SU(3) irrep in a four columned { f } using Eq. (2.21). Some results are given in Tables 2.2 and their applications are discussed in Sect. 2.8 and Chap. 6.

2.5 SU(3) Quadratic and Cubic Casimir Operators  that commutes with all the generators of U (n) is called a Casimir An operator C operators in Eq. (2.10), invariant of U (n); see Appendix B. For example, using Ai j  2 = i, j Ai j A ji , and cubic 1 = i Aii , quadratic C for U (3) we have the linear C  3 = i jk Ai j A jk Aki Casimir operators (note that for SU (2) in Eq. (2.13), the C Casimir operator is Λ2 ). It is important to note that the Aii operators generate the weights of the U (3) irreps { f } = { f 1 , f 2 , f 3 } and Ai j with i < j are weights raising operators and Ai j with i > j weight lowering operators. Most important property r is that their eigenvalues depend only on f i of the irreps { f }. Therefore, it is of C r of U (3). Then we have (see for example [10] and easy to derive formulas for C Appendix B), 3  { f }   2 (U (3)) | { f }β = δαβ C 2 { f }α | C = f i ( f i + 4 − 2i) ,



i=1

 { f }  3   2 (U (3)) | { f }β = δαβ C 3 { f }α | C = fi + (7 − 3i) f i2



−( f 1 f 2 + f 1 f 3 + f 2 f 3 ) +



i

 16 − 15i + 3i 2 f i .

(2.25)

i

i

Using Ai j in Eq. (2.11) in place of Ai j will give the quadratic and cubic Casimir 1 = 0 for SU (3)). Therefore, formulas for the eigenvalues invariants of SU (3) (C follow from Eq. (2.25) by replacing f i by f i − ( f 1 + f 2 + f 3 )/3. Then, f 1 → (2λ + μ)/3, f 2 → (μ − λ)/3 and f 3 → −(2μ + λ)/3. With this we have the important formulas,     2 (SU (3)) (λμ) = 2 λ2 + μ2 + λμ + 3(λ + μ) , C 3     3 (SU (3)) (λμ) = 1 2λ3 − 2μ3 + 3λμ(λ − μ) + 18λ2 + 9λμ + 36λ + 18μ . C 9 (2.26) 3 − 3 C 2 and C3 = C  as However, it is more standard to use the operators C2 = 23 C 2 2 the quadratic and cubic Casimir operators [11]. Their eigenvalues are

2.5 SU (3) Quadratic and Cubic Casimir Operators

C2 (λμ) = C2 (λμ) = λ2 + μ2 + λμ + 3(λ + μ) , 1 C3 (λμ) = C3 (λμ) = (λ − μ)(λ + 2μ + 3)(2λ + μ + 3) . 9

21

(2.27)

Also, in terms of the SU (3) generators L q1 and Q q2  the Casimir operators are [11]  1 2 Q · Q 2 + 3L · L , 4   1 5 1 7 (Q × Q) · Q − (L × Q) · L . C3 = 36 2 4 2

C2 =

(2.28)

Significance of Eqs. (2.27) and (2.28) will be discussed briefly in Sect. 2.8 and in more detail in the later Chapters.

2.6 SU(3) ⊃ S O(3) ⊃ S O(2) States: K Label In order to classify states according to SU (3) ⊃ S O(3) ⊃ S O(2) chain, we need (λμ) SU (3) → L S O(3) reductions. Reduction of L S O(3) to M S O(2) is simply −L ≤ M ≤ L. Given an oscillator shell η, all the oscillator sp states (n z , n x , n y ) = (η, 0, 0), (η − 1, 1, 0) . . . can be generated by applying Aμν on (η, 0, 0) and therefore all the sp states belong to a single SU (3) irrep. Now, the h.w. state (see Appendix B) is clearly (η, 0, 0). Then, we have the result that all the sp states belong to the irrep (η, 0). Using Eq. (2.20), this gives correctly d(η0) = (η + 1)(η + 2)/2. Also, using the  content in a oscillator shell η, we have the important result, (η, 0) → L = η, η − 2, . . . , 0 or 1 .

(2.29)

Reduction of a general (λμ) → L follow from the results in Sect. 2.4.2 and they can be verified using Eq. (2.20) and d(L) = (2L + 1). Let us consider (λ, 1) → L. Then, firstly we have from Eq. (2.19) and SU (3) irrep to U (3) irrep equivalence, (λ, 1) → {λ + 1, 1} = {λ + 1} × {1} − {λ + 2} = (λ + 1, 0) × (1, 0) − (λ + 2, 0) .

Now, applying Eq. (2.29) will give (λ, 1) → L = (λ + 1, λ − 1, λ − 3, . . . , 0 or 1) × (1) − (λ + 2, λ, λ − 2, . . . , 0 or 1) .

Simplifying the Kronecker product of the L quantum numbers for λ even and λ odd separately will give the final result that is valid for any λ, (λ, 1) → L = 1, 2, 3, . . . , λ + 1 .

(2.30)

2 SU (3) Algebra in Nuclei: Preliminaries

22

Proceeding in a similar manner will give (λ, 2) → L. Firstly, (λ, 2) → {λ + 2, 2} = {λ + 2} × {2} − {λ + 3} ⊗ {1} = (λ + 2, 0) × (20) − (λ + 3, 0) × (10) .

Simplifying this for λ even will give, (λ2) → L = [2, 3, 4, . . . , (λ + 2)] , [0, 2, 4, . . . , (λ)] and similarly for λ odd, (λ2) → L = [2, 3, 4, . . . , (λ + 2)] , [1, 3, 5, . . . , (λ)] . Combining the even and odd λ results will finally give the result, (λ2) → L = [2, 3, 4, . . . , (λ + 2)] , [λ, λ − 2, . . . , 0 or 1] .

(2.31)

The above procedure can be continued for any (λμ) and more importantly, it is easy to program this on a computer. These will easily give the general result, (λμ) −→ L : K = min(λ, μ), min(λ, μ) − 2, . . . , 0 or 1, L = K , K + 1, K + 2, . . . , K + max(λ, μ) f or K = 0 , L = max(λ, μ), max(λ, μ) − 2, . . . , 0 or 1 f or K = 0 , (λ, μ) → L ⇐⇒ (μ, λ) → L . (2.32) The last equality in Eq. (2.32) is a consequence of p − h symmetry. Note that (λ, μ) = {λ + μ, μ} will give (λ, μ) p−h = {λ + μ, λ} = (μ, λ); here we have used Eq. (2.16) and the fact that the U (3) irrep are maximum three rowed irreps. Using the reductions in Eqs. (2.29)–(2.31) and the method of mathematical induction, one can prove that Eq. (2.32) is correct. To this end the following may be used: (i) without loss of generality assume λ ≥ μ; (ii) (λ − 1, μ + 1) = (λ, μ) × (1, 0) − (λ + 1, μ) − (λ, μ − 1); (iii) assume that Eq. (2.32) is correct for (λ, μ) and prove that it holds for (λ, μ + 1) using (ii); (iv) carryout the exercise for λ even or odd and μ even or odd. With Eq. (2.32), the states that belong to a given (λμ) irrep are labeled as |(λμ)K L M . From Eq. (2.32) we have the important result that SU (3) ⊃ S O(3) reduction gives raise to a K label (an approximate quantum number). More importantly, using Eq. (2.28) it is seen that with the quadrupole–quadrupole interaction, SU (3) ⊃ S O(3) generates rotational bands with intrinsic structure determined by (λμ)K ; the intrinsic structure is discussed in more detail in Sect. 2.7 and Chap. 3. Therefore, SU (3) describes deformed rotational nuclei. However, it should be noted that K is not a group theoretical label and hence it is an ill-defined mathematical object. This K label implies multiplicity of L in (λμ) → L and its presence makes the SU (3) ⊃ S O(3) Wigner–Racah algebra rather cumbersome unlike the well-known angular momentum S O(3) ⊃ S O(2) algebra; see Chap. 3.

2.7 SU (3) ⊃ SU (2) ⊗ U (1) States

23

2.7 SU(3) ⊃ SU(2) ⊗ U(1) States In order to classify states according to SU (3) ⊃ SU (2) ⊗ U (1) chain, first notice that the generator Q 20 of U (1), the Casimir invariant Λ2 of SU (2) and Λ0 that generates S O(2) in SU (2) commute; note that Λ2 = Λ+ Λ− + Λ20 − Λ0 . Therefore, these three operators can be diagonalized simultaneously in a basis |(λμ)α  with eigenvalues say ε, Λ(Λ + 1) and MΛ , respectively. Then, in the d(λμ) dimensional space of a (λμ) irrep, the basis states can be chosen to be |(λμ)εΛMΛ  with respect to SU (3) ⊃ SU (2) ⊗ U (1) algebra. Note that we are using the fact that the action of a generator of SU (3) on a |(λμ)α  will not change (λμ). The state with εmax = max(ε1 , ε2 , . . . , εd ), Λmax = max(Λ1 , Λ2 , . . . , Λd ) with Λmax ∈ εmax and MΛmax = Λmax is the h.w. state. Defining z, x, and y directions to be the directions # 1, 2, and 3, respectively and using the results  1 Nx − Ny , (λμ) → {λ + μ, μ, 0} 2   we have: (i) (Nz )max = λ + μ; (ii) (N x )max = μ and N y max = 0 for Nz = (Nz )max . These will give εmax = 2λ + μ, Λmax = MΛmax = μ/2. Therefore, Q 20 = 2Nz − Nx − N,y Λ0 =

  (λ, μ)εmax Λmax MΛ = |(λμ)h.w. ; max εmax = (2λ + μ), Λmax = μ/2, MΛmax = μ/2 .

(2.33)

Clearly, the labels (λ, μ) are defined by εmax and Λmax . With this, we will recover the result that all the sp states belong to the SU (3) irrep (η0) as h.w. corresponds to all the η number of quanta in z-direction. Now, we need to obtain the range of values ε and Λ will take for a given  (λμ). To derive this, used  are the results [12]: (i) Q 20 , A x z = −3A x z ; (ii) Λ2 , A x z = 43 A x z + 21 A x z A x x − A yy + A yz A x y ;   (iii) A x y |(λμ)εΛMΛ = Λ = 0; (iv) Λ0 , A x z = 21 A x z ; (v) with O yz = A yx A x z −       A yz A x x − A yy + 1 , (a) Q 20 , O yz = −3O yz , (b) Λ2 , O yz = −O yz 41 + 21       A x x − A yy + A yx A yz A x y , (c) Λ0 , O yz = − 21 O yz and O yz , A x z = 0. These will give the relations,   A x z |(λμ)ε, Λ, MΛ = Λ =⇒ (λμ)ε − 3, Λ + 21 , Λ + 21   O yz |(λμ)ε, Λ, MΛ = Λ =⇒ (λμ)ε − 3, Λ − 21 , Λ − 21 A yx |(λμ)ε, Λ, MΛ = Λ =⇒ |(λμ)ε, Λ, Λ − 1 .

(2.34)

Thus, A x z , O yz , and q step down operators for the ε, Λ, MΛ labels. Action  Ayxr are the O yz (A x z ) p on |(λμ)h.w. immediately gives the result of O( p, q, r ) = A yx that ε = εmax − 3( p + q), Λ = Λmax + ( p − q)/2, and MΛ = Λ − r . The range of the parameters ( p, q, r ) are determined as follows. As (λμ) −→ {λ + μ, μ, 0} partition of U (3), in the (λμ) irrep there must be μ number of boxes (in Young tableaux notation) that are antisymmetric with μ number of boxes each in z and x directions. Therefore, we can only move λ number of boxes (out of λ + μ) from z

2 SU (3) Algebra in Nuclei: Preliminaries

24

to x direction, similarly only μ number of boxes from z to y direction. Therefore, 0 ≤ p ≤ λ and 0 ≤ q ≤ μ. With these, the final result for the classification of |(λμ)α states according to the SU (3) ⊃ SU (2)⊗ U (1) chain is given by [12–14]    SU (3) ⊃ [SU (2) ⊃ S O(2)] ⊗ U (1)  ;  (λ, μ) Λ MΛ ε ε = (2λ + μ) − 3( p + q) , Λ = μ/2 + ( p − q)/2 , MΛ = Λ − r , 0 ≤ p ≤ λ, 0 ≤ q ≤ μ, 0 ≤ r ≤ 2Λ , r  q 1  |(λμ)εΛMΛ  = √ A yx O yz (A x z ) p |(λμ)h.w. . N

(2.35)

Here, N  is the normalization constant. It is important to note that the SU (3) ⊃ SU (2) ⊗ U (1) basis is multiplicity free unlike the SU (3) ⊃ S O(3) ⊃ S O(2) basis. Now we will consider some physical applications of SU (3).

2.8 Preliminary Applications of SU(3) Symmetry 2.8.1 SU(3) in Shell Model Nuclear shell model, developed based on observed magic numbers (shell structure), starts with nucleons moving in a average potential well that is approximated by a harmonic oscillator in three dimensions. Thus, shells are defined by the oscillator major shell quantum number η. The levels with different orbital angular momentum  are split by the i i2 force but more importantly there is a strong spin–orbit force splitting the j =  ± 21 levels such that the level with largest ( j) is moved far down from the rest of the levels in a major shell. Figure 2.2 gives a schematic picture of the sp spectrum with these features. Complete filling of the shells with nucleons (protons or neutrons) will give the magic numbers 2, 8, and 20 (preserving the original oscillator orbit sequence) but after this only with the spin–orbit force we will get the higher magic numbers 50, 82, 126, and 184 (there are also semi magic numbers 28, 40, 64, etc.). Thus, for light nuclei such as 8 Be, 12 C, 20 Ne, 24 Mg, and so on valence nucleons (outside the completely filled oscillator major shells) occupy all the orbits in a oscillator shell (for example, (1 p) shell for 8 Be and 12 C and (2s1d) shell for 20 Ne and 24 Mg). The valence orbits, their energies (sp energies) and a residual effective two-body interaction among the nucleons generate the lowlying states that are observed. With valence nucleons in a oscillator shell η, the shell model SGA is U (4N ) if we have valence protons and neutrons (as in 20 Ne) as we need to include the four spin–isospin (ST ) degrees of freedom. Similarly, the SGA is U (2N ) if we have only valence protons or neutrons (as in 20 O). An important observation is that light nuclei exhibit rotational features, i.e., low-lying

2.8 Preliminary Applications of SU (3) Symmetry

25

Fig. 2.2 Schematic figure showing shell model (oscillator) sp spectrum with magic numbers up to 50. Splitting  of the levels due to the i i2 force and  the spin–orbit force i i ·si are also shown

yrast levels (similarly yare and others) can be arranged into bands with the energies of the levels approximately proportional to J (J + 1). For example in (1 p) shell the lowest 0+ , 2+ , 4+ levels for 8 Be and 12 C have this pattern. Similarly, in (2s1d) shell the lowest 0+ , 2+ , 4+ , 6+ levels in 20 Ne, 24 Mg, and 28 Si have this pattern. In addition, the B(E2) values along the yrast line are much larger compared to the sp values. Therefore, these are collective rotational levels from a quadrupole deformed nucleus. See Fig. 2.3 for examples from 20 Ne, 24 Mg, and 28 Si. Assuming that L − S coupling is good for light nuclei, we will have U (4N ) ⊃ U (N ) ⊗ SU ST (4); SU (4) is the Wigner’s spin–isospin SU (4) [15]. With SU (4) symmetry being a good symmetry for light nuclei [15, 16], the SU (3) subalgebra of U (N ) provides a shell model description of the observed rotational levels with the effective interaction approximated by quadrupole–quadrupole interaction [1, 13]. For example, for 20 Ne and 24 Mg one can see from Table 2.2 that the lowest SU (3) irreps (with isospin T = 0) are (80) and (84), respectively. Thus, with SU (3) symmetry we have for 20 Ne ground K = 0+ rotational band with 0+ , 2+ , 4+ , 6+ , 8+ levels from the (80) irrep. Similarly, for 24 Mg we have K = 0+ , K = 2+ , and K = 4+ bands from the (84) irrep. These bands are in close agreement with the observed bands in these nuclei (see Fig. 2.3 and more detailed description of 20 Ne and 24 Mg levels is given in Chap. 6). Going beyond the lowest SU (3) irrep, more detailed study allowing for mixing of SU (3) irreps for (2s1d) shell nuclei have been carried out and these are

2 SU (3) Algebra in Nuclei: Preliminaries

26 Fig. 2.3 Ground band levels in 20 Ne, 24 Mg and 28 Si. The B(E2; J → J − 2) values, shown next to each arrow, are in Weisskopf units. The data are taken from [22] for 20 Ne and 28 Si and from [23, 24] for 24 Mg

14 +

8 +

12

8

16

9

10

+

Energy (MeV)

6

+

6

+

6

8

20

10.6

34

6

+

+

4

22 2

4

+

4

4

16.4

23

+

2

+

2

+

2

20.3 20

0

Ne

13.2

20.5 +

+

0

+

0 24

Mg

0 28

Si

reviewed by Harvey [17]; see also [18]. It is seen that with realistic interactions and realistic sp energies, SU (3) is a broken symmetry in this region. However, a few SU (3) irreps are found to describe the observed properties in particular for even– even nuclei. The source of breaking is clearly the strong spin–orbit force and the statistical spectral distribution method (SDM) [19], that do not require H matrix construction, was used to establish this [16, 20]. We will discuss SDM in more detail in Chap. 10. Going beyond (2s1d) shell, the SU (4) symmetry itself will be a badly broken symmetry and as a result SU (3) symmetry is not found to be useful for (2 p1 f ) shell nuclei and beyond [16, 21]. Hunting for good SU (3) for A > 40 nuclei, in early 70s it is recognized by Hecht and Arima that for heavier nuclei with protons and neutrons in different shells, there is pseudo-spin symmetry leading to pseudo-(L S) coupling and pseudo-SU (3) symmetry. This scheme is established to be good only in mid 80s by Draayer by using a Hamiltonian containing some additional S O(3) scalars in SU (3). Similarly in mid 90s Zuker suggested a quasi-SU (3) scheme for understanding deformation characteristics of nuclei in A = 60–100. More recently, there is the introduction of a proxy-SU (3) scheme by Bonatsos et al where in the high- j orbit, that is pushed to a lower shell by the spin–orbit force, the | j, ± j  states are neglected. Finally, SU (3) symmetry also plays an important role in the symplectic model of Rowe and Rosensteel and more importantly in the no-core symplectic model that is being investigated by Kristina Launey and others in the last few years. All these shell model SU (3) schemes (plus a few others) and some of the important results from

2.8 Preliminary Applications of SU (3) Symmetry

27

them will form Chap. 6. Besides SM, SU (3) also appears in a variety of interacting boson models of nuclei giving many other new results. We will discuss this briefly in the next subsection.

2.8.2 SU(3) in Interacting Boson Model The interacting boson model was introduced for unifying, in a minimal way, the wellknown vibrational quadrupole phonon U (5) and the fermionic Elliott’s quadrupole deformed rotational SU (3) algebraic descriptions. In its elementary version (called IBM or IBM-1—sometimes we use sdIBM or sdIBM-1 for the same), the model is generated by N interacting scalar (s with  = 0) and quadrupole (d with  = 2) bosons with the Hamiltonian assumed to be, as in the shell model, (1 + 2)-body preserving the boson number N (sum of s boson number Ns and d boson number Nd ) and angular momentum (L). The bosons are interpreted to be pairs of fermions (particles or holes) and as Iachello puts it “the correlations are so large that the pairs loose their memory of being fermions”. Thus the number of bosons N is half the number of valance nucleons with the p − h symmetry taken into account. The SGA for the model is U (6) and all the states of a N boson system belong to the totally symmetric irrep {N } of U (6). The U (6) algebra is generated by the 36 generators † s s, dμ† dμ , s † dμ , dμ† s; μ, μ = −2, −1, 0, 1, 2 . The U (6) admits not only U (5) and SU (3) subalgebras but also a S O(6) subalgebra [25]. Thus, sdIBM has three group chains and they are [25] U (5) : |U (6) ⊃ U (5) ⊃ S O(5) ⊃ S O(3) ⊃ S O(2) SU (3) : |U (6) ⊃ SU (3) ⊃ S O(3) ⊃ S O(2) S O(6) : |U (6) ⊃ S O(6) ⊃ S O(5) ⊃ S O(3) ⊃ S O(2)

(2.36)

It is well known that the U (5) chain generates vibrational spectra as seen for example in Cd isotopes, SU (3) chain generates rotational spectra (generated by a quadrupole deformed nucleus) as seen for example in Gd isotopes and S O(6) generates γ unstable spectrum as seen in Pt isotopes. The group chains in Eq. (2.36) lead to simple energy formulas and predictions for E2 transition probabilities and many other observables. As the focus in this book is on SU (3), in Fig. 2.4 low-lying rotational bands in 156 Gd are shown. Let us add that the SU (3) algebra in sdIBM is generated by Q 2μ =

√

 8

   √7 ˜ 2μ − ˜ 2μ ; d † s˜ + s † d) (d † d) 2

L q1 =

√ ˜ 1μ . 10(d † d)

(2.37)

In the SU (3) limit, as the U (6) irrep has to be {N }, from the discussion in Sect. 2.4 it easy to see that the lowest SU (3) irrep is (2N , 0) giving a K = 0 band with cut-off in L at L = 2N . Similarly, it is easy to see that the next lower irreps are

2 SU (3) Algebra in Nuclei: Preliminaries

28

(2N − 4, 2) with K = 0 and K = 2 bands (i.e., β and γ bands, respectively), (2N − 8, 4) with K = 0, 2, 4 bands and so on; see Chap. 3 for methods to obtain lower SU (3) irreps. Using quadratic Casimir operators, the energy formula with H = E 0 + αC2 (SU (3)) + βC2 (S O(3)) is E(N , (λμ), K L) = E 0 + α(λ2 + μ2 + λμ + 3(λ + μ)) + β L(L + 1). The degeneracy in levels with same L (but different K ) can be lifted by using some simple 3- and 4-body operators (see Chap. 5). The SU (3) irreps and the energy formula with some corrections describe quite well for example the results in Fig. 2.4 [26]. Also, physical insight into the various symmetry limits of sdIBM follow from the (projected) coherent state (CS) with standard (β, γ ) parameters [25],

  N −1/2    N  † |N ; β2 , γ  = N ! 1 + β22 |0  , s0† + β2 cos γ d0† + 2−1/2 sin γ d2† + d−2

(2.38) where β2 ≥ 0 and 0◦ ≤ γ ≤ 60◦ . Using H = −Q 2 .Q 2 and minimizing the energy functional E(N : β2 , γ ) = N , β2 , γ | H | N , β2 , γ  shows√clearly that the SU (3) limit corresponds to a prolate deformed nucleus with β20 = 2 and γ 0 = 0◦ . Also, with T E2 = Q q2 and taking N → ∞ limit we have 15(L + 1)(L + 2) B(E2; (2N , 0)L + 2 → (2N , 0)L) = , B(E2; (2N , 0)2 → (2N , 0)0) 2(2L + 3)(2L + 5) Q 2 ((2N , 0)L) 7L = . Q 2 ((2N , 0)2) 2(2L + 3)

Fig. 2.4 Experimentally observed rotational bands in 156 Gd and the corresponding SU (3) irreps as given by sdIBM-1. The data are taken from [22, 25]

2.8 Preliminary Applications of SU (3) Symmetry

29

It is important to mention that it is possible to have another SU (3) limit, denoted by SU (3) with a sign change in Q q2 in Eq. (2.37) and its generators are 2 Qμ

   √7 √  † †˜ 2 †˜ 2 (d d)μ ; = 8 d s˜ + s d)μ + 2

L q1 =

√ ˜ 1μ . 10(d † d)

(2.39)

√ 2 2 Now, minimizing the CS expectation value of H = −Q · Q will give β20 = 2 and γ 0 = 60◦ and hence an oblate shape. In fact, for SU (3) one can use (02) as the one particle SU (3) irrep and then the lowest N particle irrep is (0, 2N ). The results in Chap. 5 ahead also show that the irrep (0, 2N ) implies oblate shape. We will return to SU (3) in Chaps. 7, 9, and 11. The general IBM-1 H with U(6) SGA contains 9 free parameters, 0  1 Hsd I B M = εs nˆ s + εd nˆ d + V 0 (ssss) (s † s † )0 (˜s s˜ )0 2 0  0 √  1 ˜ 0 + h.c. + 5V 2 (sdsd) (s † d † )2 (˜s d) ˜ 2 + V 0 (ssdd) (s † s † )0 (d˜ d) 2 !1/2  0 5 ˜ 2 + h.c. + V 2 (sddd) (s † d † )2 (d˜ d) 2 0   1 ˜ L0 . (2L 0 + 1)1/2 V L 0 (dddd) (d † d † ) L 0 (d˜ d) + 2 L =0,2,4

(2.40)

0

In Eq. (2.40), εs and εd are s and d-boson energies and V 0 (ssss), V 0 (ssdd), V 2 (sdsd), V 2 (sddd), and V L 0 (dddd) are free parameters characterizing the twobody part of Hsd I B M . As this Hamiltonian is simple and the sdIBM H matrix dimensions not very large, varying the free parameters, numerical calculations are easy to perform even for large N values [27, 28]. However, most important aspect of sdIBM is that for some particular choices of the parameters, Hsd I B M reduces to a linear combination of the Casimir operators of the group chains in Eq. (2.36). Details of sdIBM, its three symmetry limits and its success in explaining experimental data and providing many new predictions are all well documented. Therefore, we will not go into any more details of sdIBM-1 and refer the readers to [25, 29]. Focusing on heavy nuclei with valence protons and neutrons occupying different shells, microscopic or SM foundations of sdIBM immediately pointed out that it is more realistic to consider proton–neutron IBM (i.e., pn − sdIBM or sdIBM-2) where proton bosons (π bosons) and neutron bosons (ν bosons) are distinguished giving Uπ (6) ⊕ Uν (6) SGA [30]. More importantly, treating π and ν bosons as projections of a fictitious (F) spin 21 boson, the SGA will be U (12). There is good evidence that F spin is a good symmetry in many situations giving the SGA U (12) ⊃ U (6) ⊗ SU (2) where SU (2) generates F-spin. Important results of this extension of IBM and other extensions such as (i) sdgIBM including hexadecupole (g) degree of

2 SU (3) Algebra in Nuclei: Preliminaries

30

freedom, (ii) sdp f IBM for octupole and E1 excited states, (iii) bosons with isospin T = 1, and (iv) bosons with (ST ) = (10) + (01)) degrees of freedom, with all these admitting SU (3) subalgebra, will be described in Chap. 7. Similarly, extension of IBM to boson–fermion models where the bosons are coupled to fermion degrees of freedom for odd-A nuclei, odd–odd nuclei, and quasi-particle excitations in even– even and odd-A nuclei, all these again admitting SU (3) algebra, will be described in Chap. 8. Many other aspects of SM and IBM where SU (3) plays an important role will be discussed in Chap. 9. Before proceeding further, it is important to emphasize that in all nuclear physics applications the SU (3) ⊃ S O(3) chain is physically relevant as all observed nuclear levels carry definite angular momentum. On the other, the SU (3) ⊃ SU (2) ⊗ U (1) chain is also important as this chain is often useful in various simplifications of SU (3) algebra and also in establishing correspondence between SU (3) irreps in the interacting boson–fermion models and the Nilsson configurations. However, in particle physics the SU (3) ⊃ SU (2) ⊗ U (1) chain is the physically relevant chain and the irrep labels here define various elementary particles. We will discuss this briefly in the following section before turning to Chap. 3.

2.9 SU(3) in Particle Physics With the recognition that SU (3) ⊃ SU (2) ⊗ U (1) is the flavor group, by identifying the SU (3) irreps and using the (λμ) → (ε, Λ, MΛ ) given by Eq. (2.35), it is easy to write the quantum numbers for quarks, baryons, and mesons. Here, we will consider some examples although all the results in this section are well known [31]. Firstly, the (u, d, s) quarks belong to the SU (3) irrep (10) that is three dimensional. Dimensions are given by Eq. (2.20). In particle physics, quantum numbers used are hyper charge Y = −ε/3, isospin I = Λ, Iz = MΛ , charge Q = MΛ + Y/2 and strangeness s = Y − B where B is baryon number (B=1/3 for quarks, 1 for baryons and 0 for mesons). Then, using 1 1 (10) → (ε, Λ, MΛ ) = (2, 0, 0) + −1, , ± 2 2

! ,

we have for the u, d, and s quarks, respectively, (Y, I, Iz , Q, B, s) = ( 13 , 21 , 21 , 23 , 1 , 0), ( 31 , 21 , − 12 , − 13 , 13 , 0) and (− 23 , 0, 0, − 13 , 13 , −1). Also, quarks are spin 21 3 particles. Proceeding further and including spin, the flavor SU (3) symmetry enlarges to SU (6) ⊃ SU (3) ⊗ SU (2). The SU (6) irreps are labeled by { f } that are maximum 6 rows and SU (2) generates total spin S. For a baryon with three quarks, the possible SU (6) irreps are {3}, {21}, and {13 } and the same irreps labels appear for U (6).

2.9 SU (3) in Particle Physics

31

With U (6) ⊃ U (3) ⊗ U (2), for the irrep {3} that is totally symmetric, using Eq. (2.14) gives immediately {3} SU (6) → (30) SU (3) (S = 23 ) + (11) SU (3) (S = 21 ). Similarly, the irrep {13 } is totally antisymmetric and therefore using Eq. (2.15) gives immediately {13 } SU (6) → (00) SU (3) (S = 23 ) + (11) SU (3) (S = 21 ). Then, by a subtraction procedure we will obtain {21} SU (6) → (30) SU (3) (S = 21 ) + (11) SU (3) (S = 1 3 , ) + (00) SU (3) (S = 21 ). It is well known that the octet with spin 21 and the decu2 2 plet with spin 23 belong to the 56 dimensional SU (6) irrep {3}. Thus, the irrep {3} is appropriate as hadrons are color singlets. Then, hadron states are antisymmetric in total flavor-spin-color space with SU (6) ⊗ SUc (3) where SUc (3) is the color SU (3) algebra [here we are again using Eq. (2.15)]. The quantum numbers (Y, I, Iz , Q, S) for the S = 21 baryon octet ( p, n, Λ, Σ + , Σ 0 , Σ − , Ξ 0 , Ξ − ) and S =

3 2

decuplet (Δ++ , Δ+ , Δ0 , Δ− , Σ +∗ , Σ 0∗ , Σ −∗ , Ξ 0∗ , Ξ −∗ , Ω − )

are given in Table 2.3 and used here is Eq. (2.35). As a last example we will consider mesons. They consist of quark antiquark pair (q q). ¯ Now, the U (6) irreps are {1} for q and {15 } for q¯ as this is hole of q; see Eq. (2.16). Then, for the q q¯ systems the U (6) irreps are {16 } and {214 } and the corresponding SU (6) irreps are {0} and {214 }. Similarly, the SU (3) irrep for q is (10) and for q¯ is (01). Thus, for q q¯ the SU (3) irreps are (10) × (01) → (11) + (00) and spin S = 0, 1. Using these, it is easy to see that {0} SU (6) → (00) SU (3) (S = 0) and {214 } → (00) SU (3) (S = 1) + (11) SU (3) (S = 0) + (11) SU (3) (S = 1). Therefore, there will be spin 0 and 1 octets of mesons and similarly singlets with spin 0 and 1. Table 2.3 gives the quantum numbers (Y, I, Iz , Q, S) for the spin S = 0 meson singlet η1 and octet 0 (π + , π 0 , π − , η8 , K + , K 0 , K , K − ) and similarly, the spin S = 1 singlet φ1 and octet 0∗

(ρ + , ρ 0 , ρ − , ω8 , K +∗ , K 0∗ , K , K −∗ ) . Note that the baryon number B = 0 for mesons.

2 SU (3) Algebra in Nuclei: Preliminaries

32

Table 2.3 Quantum numbers (Y, I, Iz , Q, s) for spin 21 hadron octet with the eight-dimensional SU (3) irrep (11) and spin 1 decuplet with the ten-dimensional irrep (30). Also, given are (Y, I, Iz , Q, s) for the spin 0 meson singlet and octet and the spin 1 meson singlet and octet. All the results in the table follow from Eq. (2.35) Spin 21 baryon octet Spin 23 baryon decuplet Particle Y I Iz Q s Particle Y I Iz Q s p n Λ + 0 − Ξ0 Ξ−

+1 +1 0 0 0 0 −1 −1

1 2 1 2

0 1 1 1 1 2 1 2

1 2

− 21 0 +1 0 −1 1 2

− 21

1 0 0 +1 0 −1 0 −1

Spin 0 meson singlet Particle Y I

Iz

Q

η1

0

0

0

Spin 0 meson octet Particle Y I

Iz

π+ π0 π− η8 K+ K0 0

K K−

0

0 0 0 0 1 1 −1 −1

1 1 1 0 1 2 1 2 1 2 1 2

0 0 −1 −1 −1 −1 −2 −2

Δ++ Δ+ Δ0 Δ− +∗ 0∗ −∗ Ξ 0∗ Ξ −∗ Ω−

1 1 1 1 0 0 0 −1 −1 −2

3 2 3 2 3 2 3 2

1 1 1 1 2 1 2

0

3 2 1 2

− 21 − 23 1 0 −1 1 2

− 21 0

2 1 0 −1 1 0 −1 0 −1 −1

0 0 0 0 −1 −1 −1 −2 −2 −3

S

Spin 1 meson singlet Particle Y I

Iz

Q

s

0

φ1

0

0

0

Q

s

Spin 1 meson octet Particle Y I

Iz

Q

s

+1 0 −1 0 + 21 − 21

+1 0 −1 0 +1 0

0 0 0 0 +1 +1

ρ+ ρ0 ρ− ω8 K +∗ K 0∗

+1 0 −1 0 + 21 − 21

+1 0 −1 0 +1 0

0 0 0 0 +1 +1

+ 21 + 21

0 −1

−1 −1

K K −∗

1 2

0 −1

−1 −1

0∗

0

0 0 0 0 +1 +1 −1 −1

0

1 1 1 0 1 2 1 2 1 2 1 2

− 21

2.10 Summary Elementary results for SU (3) algebra in nuclei are presented in Sects. 2.2–2.7. These include: (i) SU (3) ⊃ S O(3) and SU (3) ⊃ SU (2) ⊗ U (1) subalgebras; (ii) introduction to Young tableaux and Kronecker products; (iii) SU (3) irreps (λμ), dimension formula for the irreps and a formula for the leading SU (3) irrep for many-particle systems; (iv) quadratic and cubic invariants of SU (3). Preliminary discussion of applications in SM and IBM of nuclei is given in Sect. 2.8. In addition, well-known results for some elementary particles are presented in Sect. 2.9 and used here are the results from (i)–(iii). Now, we will go into more details of SU (3) algebra.

References

33

References 1. J.P. Elliott, Collective motion in the nuclear shell model I. Classification schemes for states of mixed configurations. Proc. R. Soc. (Lond.) A 245, 128–145 (1958) 2. B.G. Wybourne, Symmetry Principles and Atomic Spectroscopy (Wiley, New York, 1970) 3. M. Hamermesh, Group Theory and Its Applications to Physical Problems (Addison-Wesley, Reading, 1964) 4. D.E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, 2nd edn. (AMS Chelsea Publishing, AMS, Providence, 2006) 5. B.G. Wybourne, Classical Groups for Physicists (Wiley, New York, 1974) 6. F. Iachello, Lie Algebras and Applications, 2nd edn. (Springer, Heidelberg, 2015) 7. V.K.B. Kota, Single particle SU(3) parentage coefficients. Pramana-J. Phys. 9, 129–140 (1977) 8. M. Vyas, V.K.B. Kota, Spectral properties of embedded Gaussian unitary ensemble of random matrices with Wigner’s SU (4) symmetry. Ann. Phys. (N.Y.) 325, 2451–2485 (2010) 9. P. Van Isacker, D.D. Warner, D.S. Brenner, Wigner’s spin-isospin symmetry from double binding energy differences. Phys. Rev. Lett. 74, 4607–4610 (1995) 10. V.K.B. Kota, R. Sahu, Structure of Medium Mass Nuclei: Deformed Shell Model and SpinIsospin Interacting Boson Model (CRC Press, Taylor & Francis Group, Boca Raton, 2017) 11. J.P. Draayer, G. Rosensteel, U (3) → R(3) integrity-basis spectroscopy. Nucl. Phys. A 439, 61–85 (1985) 12. K.T. Hecht, Collective models, in Selected Topics in Nuclear Spectroscopy, ed. by B.J. Verhaar (North Holland, Amsterdam, 1964), pp. 51–105 13. J.P. Elliott, Collective motion in the nuclear shell model II. The introduction of intrinsic wavefunctions. Proc. R. Soc. (Lond.) A 245, 562–581 (1958) 14. K.T. Hecht, SU3 recoupling and fractional parentage in the 2s − 1d shell. Nucl. Phys. 62, 1–36 (1965) 15. E.P. Wigner, On the consequences of the symmetry of the nuclear Hamiltonian on the spectroscopy of nuclei. Phys. Rev. 51, 106–119 (1937) 16. J.C. Parikh, Group Symmetries in Nuclear Structure (Plenum, New York, 1978) 17. M. Harvey, The nuclear SU (3) model. Adv. Nucl. Phys. 1, 67–182 (1968) 18. Y. Akiyama, A. Arima, T. Sebe, The structure of the sd shell nuclei: (IV). 20 Ne, 21 Ne, 22 Ne, 22 Na and 24 Mg. Nucl. Phys. A 138, 273–304 (1969) 19. V.K.B. Kota, R.U. Haq, Spectral Distributions in Nuclei and Statistical Spectroscopy (World Scientific, Singapore, 2010) 20. M. Chakraborty, V.K.B. Kota, J.C. Parikh, Unitary decomposition of Hamiltonian operators: SU(4) irreducible tensors, norms and their energy variation and symmetry breaking. Ann. Phys. (N.Y.) 127, 413–435 (1980) 21. P. Vogel, W.E. Ormand, Spin-isospin SU(4) symmetry in sd- and fp-shell nuclei. Phys. Rev. C 47, 623–628 (1993) 22. http://www.nndc.bnl.gov/ensdf 23. G. Rosensteel, J.P. Draayer, K.J. Weeks, Sympletic shell-model calculations for 24 Mg. Nucl. Phys. A 419, 1–12 (1984) 24. D. Branford, A.C. McGough, I.F. Wright, Lifetime and branching measurements on the K π = 0+ and K π = 2+ 24 Mg rotational levels. Nucl. Phys. A 241, 349–364 (1975) 25. F. Iachello, A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987) 26. A. Arima, F. Iachello, Interacting boson model of collective nuclear states II: the rotational limit. Ann. Phys. (N.Y.) 111, 201–238 (1978) 27. O. Scholten, The Program Package PHINT Manual Book (National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, 1982)

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28. R.J. Casperson, Interacting boson model calculations for large system sizes. Comput. Phys. Commun. 183, 1029–1035 (2012) 29. R.F. Casten, D.D. Warner, The interacting boson approximation. Rev. Mod. Phys. 60, 389–469 (1988) 30. F. Iachello, I. Talmi, Shell-model foundations of the interacting boson model. Rev. Mod. Phys. 59, 339–362 (1987) 31. D. Griffiths, Introduction to Elementary Particles, 2nd edn. (Wiley-VCH, Weinheim, 2008)

Chapter 3

SU(3) Wigner–Racah Algebra I

3.1 Introduction Turning to the details of SU (3) algebra, first problem to be solved is enumerating the allowed SU (3) irreps for a given nucleus. There are several situations here: (i) in heavy nuclei one may recognize that in SM description, the valence protons and neutrons occupy different oscillator shells. Then, the SGA, with L − S coupling, is U (2N ) ⊃ U (N ) ⊗ SU (2) for protons in a shell η, N = (η + 1)(η + 2)/2 and similarly for neutrons in a shell η . The SU (2) here generates spin S and for a given number (m p ) of protons, the U (N ) irrep is {2r 1s } with 2r + s = m p and S = s/2 and similarly for neutrons. (ii) In lighter nuclei both protons and neutrons occupy same shell and then the SM SGA with L ST coupling is U (4N ) ⊃ U (N ) ⊗ SU (4) with U (N ) irreps of the form {4a 3b 2c 1d } and SU (4) ⊃ SU S (2) ⊗ SUT (2). (iii) In IBM description we have algebras U (r N ) ⊃ U (N ) ⊗ SU (r ) where r = 1, 2, 3 and 6 for IBM-1, IBM-2, IBM-3, and IBM-4 models. Therefore, the U (N ) irreps here are one row, two rows, three rows, and six rows, respectively. Thus, it is necessary to generate SU (3) irreps in a general { f } of U (N ). Methods and some numerical/analytical results for these are presented in the next section. Toward deriving matrix elements of operators in states with a given SU (3) irrep, we need SU (3) Wigner–Racah algebra and this will be the topic of the remaining two Sections in this Chapter.

3.2 SU(3) Irreps for Many-Particle Systems 3.2.1 Plethysm Method With protons or neutrons or nucleons in the fermionic SM or bosons in various IBM’s occupying an oscillator shell η, firstly we need the decomposition of the Hilbert space into SU (3) irreps, i.e., we need the reduction of an irrep { f } of U (N ) into SU (3) irreps (λμ). Most general theory here is the ‘Plethysm’theory based on © Springer Nature Singapore Pte Ltd. 2020 V. K. B. Kota, SU(3) Symmetry in Atomic Nuclei, https://doi.org/10.1007/978-981-15-3603-8_3

35

36

3 SU(3) Wigner–Racah Algebra I

Schur (S) functions as described, for example, in [1, 2]. Here, we will not go into all the technical details and restrict to some essential features and results. Reduction of { f }U (m1 ) to irreps of U (m2 ), with the basic association {1}U (m1 ) → {λ}U (m2 ) , is denoted by {λ} ⊗ { f } where the symbol ⊗ stands for plethysm; i.e., λ plethysm f of the corresponding S-functions {λ} and { f }; note that the  S functions are denoted by the same irrep labels { f }. The rank r of { f } is r = i f i . Then, the plethysm operation reduces the r -fold Kronecker product {λ}r into irreps of U (m2 ), catalogs them into irreps of U (m1 ) and selects those belonging to the irrep { f }. Two important theorems for plethysms are {λ} ⊗ [{μ} + {ν}] = {λ} ⊗ {μ} + {λ} ⊗ {ν} {λ} ⊗ [{μ} × {ν}] = {λ} ⊗ {μ} × {λ} ⊗ {ν} .

(3.1)

Similarly, two simple plethysms are {2} ⊗ {μ} =



{12 } ⊗ {μ} =

{λ}even ,

λ 

 even . {λ}

(3.2)

λ

In Eq. (3.2), the sum is over the partitions of the integer 2μ and {λ}even means only irreps {λ} = {λ1 , λ2 , . . .} with all λi even are allowed. Similarly, tilde stands for conjugate irreps (obtained by interchanging rows with columns in the Young tableaux {λ}). Equation (3.2) gives the reductions for Usd (6) ⊃ SUsd (3), Usdg (15) ⊃ Usdg (5), Usdg (15) ⊃ Usdg (6), U p f (10) ⊃ U p f (4), etc. A significant example is for sdIBM-1 giving the complete result, {m}Usd (6) → (λμ) SUsd (3) = {2} ⊗ {m} =



(2 f 1 − 2 f 2 , 2 f 2 − 2 f 3 )

{ f1 , f2 , f3 }

(3.3)

= (2m, 0) ⊕ (2m − 4, 2) ⊕ (2m − 8, 4) ⊕ (2m − 6, 0) ⊕ . . . The sum in Eq. (3.3) is over all partitions of m with maximum three rows ( f 1 ≥ f 2 ≥ f 3 ≥ 0, f 1 + f 2 + f 3 = m). In addition, complete results for U (6) ⊃ SU (3) with η = 2 and for all U (6) irreps of maximum four columns, as needed for (2s1d) shell nuclei, are reported in [3] and here the general theory of plethysms is used. These results follow easily from Eq. (3.2) and Eqs. (B.7) and (B.8) in Appendix B. General expression for plethysms in terms S-functions is [1, 2], {g} ⊗ { f } = [r !]−1

 ρ

h ρ Cρ{ f } ({g} ⊗ S1 )ν1 ({g} ⊗ S2 )ν2 . . . ({g} ⊗ Sr )νr .

(3.4)

Here, { f } is a partition (i.e., Young tableaux) of an integer r , ρ is a class of the symmetric group of order r ! and specified by the cyclic structure (1ν1 , 2ν2 , . . . , r νr ), Si is a symmetric function (power sum; see [2]), h ρ is the order of the class ρ and {f} Cρ is the character of the class ρ corresponding to the partition { f }. It is well known that

3.2 SU (3) Irreps for Many-Particle Systems

hρ =

37

r! 1ν1 2ν2

. . . r νr ν1 !ν2 ! . . . νr !

.

(3.5)

Given a r , we can write down all allowed Young tableaux {λ1 , λ2 , . . . , λr }. Then the corresponding νi of ρ are ν1 = λ1 − λ2 , ν2 = λ2 − λ3 , . . ., νr −1 = λr −1 − λr , {f} νr = λr . In order to apply Eq. (3.4), we need the characters Cρ and they are simple for { f } totally symmetric or antisymmetric. For totally symmetric { f } = {r },all r Cρ{r } =1. Similarly, for totally antisymmetric { f } = {1r }, we have Cρ{1 } = (−1)r + νi . Secondly, we need {g} ⊗ Si . Restricting to U (N ) ⊃ SU (3), we have the association that {1}U (N ) → {η}U (3) . Therefore, here we need {η} ⊗ Si and in this plethysm we need to restrict to the irreps (S-functions) of maximum three rows as they need to be the irreps of U (3). General result for {η} ⊗ Sr restricted to irreps with maximum three rows is derived in [4] using Weyl’s reciprocity theorem [5] and the result is {η} ⊗ Sr =



[{ηr − ar, ar − br, br } − {ηr − ar, ar − br − 1, br + 1}

a,b

+ {ηr − ar − 1, ar − br − 1, br + 2} − {ηr − ar − 1, ar − br + 1, br } + {ηr − ar − 2, ar − br + 1, br + 1} − {ηr − ar − 2, ar − br, br + 2}] .

(3.6)

In Eq. (3.6), the summation is over all positive integers a and b with the constraint that all the non standard { f 1 , f 2 , f 3 } irreps to be ignored. Equation (3.4) can be used easily for totally symmetric or antisymmetric irreps { f } as the characters for these are simple. The Kronecker products of SU (3) irreps that enter into Eq. (3.4) due to the substitution of the result in Eq. (3.6) can be evaluated using an algorithm due to Chew and Sharp [6]. This algorithm gives the number of times a irrep (λ3 μ3 ) appears in the Kronecker product of any two irreps (λ1 μ1 ) and (λ2 μ2 ). Now, the reductions for a general { f } of U (N ) follow by using the expansion given by Eq. (B.7) or (B.8) combined with the U (N ) ⊃ SU (3) reductions for totally symmetric or antisymmetric irreps. A computer code using this method was developed in 1978 [7] and tabulations for U (10) ⊃ SU (3) with η = 3 and for all U (10) irreps of maximum four columns are generated [7, 8]. Similarly, using these codes results for U (15) ⊃ SU (3) with η = 4 and totally symmetric irreps {m} of U (15) with m ≤ 15 are given in [9, 10]. These codes easily extend to larger η values though the computing time increases. Some examples for { f }U (N ) −→ (λμ) SU (3) with η = 3 and η = 4 as appropriate for fermion systems are given in Tables 3.1, 3.2, and 3.3. These include complete results for antisymmetric irreps {1m } with m = 0 − 5 for η = 3 (Table 3.1) and m = 1 − 7 for η = 4 (Table 3.2). They are complete as the p − h relations gives those for [(η + 1)(η + 2)/2] − m. Given also are results for spin S = 0 for even m and S = 1/2 for odd m values for η = 3 in Table 3.1. Similarly, for η = 4, reduction for all S for m = 6 are given in Table 3.2 and for the example of {26 } irrep (i.e., for m = 12 with S = 0) in Table 3.3. The m = 12 results point out that in general the multiplicities can be very large. Further examples for this are given in the next subsection. A statistical theory for the multiplicities is presented in Chap. 10 and

38

3 SU(3) Wigner–Racah Algebra I

this is a part of ‘Statistical Group Theory’. It is important to add that the results for {1m } irreps given in the Tables together with Eq. (B.8), reductions for all two, three, and four column irreps { f }, that are needed for nuclei in SM description, can be generated for η = 3 and η = 4 shells. Similarly, it is possible to generate results for two or three rowed irreps (needed for sdIBM-2 and sdIBM-3 models) by using the reductions for totally symmetric irreps given by Eq. (3.3). For example, for the two rowed irreps {m − 1, 1} and {m − 2, 2} of U (6) with η = 2 the lowest few irreps are {m − 1, 1}U (6) , F = m/2 − 1 −→ (2m − 2, 1)m≥3 ⊕ (2m − 4, 2)m≥3 ⊕ (2m − 6, 3)m≥4 ⊕ (2m − 5, 1)m≥3 ⊕(2m − 8, 4)m≥5 ⊕ (2m − 7, 2)m≥4 ⊕ (2m − 6, 0)m≥4 ⊕ . . . {m − 2, 2}U (6) , F = m/2 − 2 −→ (2m  − 5, 1)m≥4  − 4, 2)m≥4 ⊕ (2m − 6, 3)m≥5 ⊕ (2m ⊕ (2m − 8, 4)m=4,5 , (2m − 8, 4)2m≥6  ⊕ (2m − 7, 2)m≥5 ⊕ (2m − 6, 0)m=4 , (2m − 6, 0)2m≥5 ⊕ . . .

(3.7)

3.2.2 Recursion Method A second method is due to Egecioglu et al. [11] and it is used effectively later by Castilho Alcarás et al. [12]. This uses the recursion formula [11], m 1  {η} ⊗ {m} = m k=1





 Cη,k,{ν} {ν}

({η} ⊗ {m − k}); m ≥ 2

(3.8)

ν

with initial input {η} ⊗ {1} = {η}. The sum involving {ν} includes all partitions of ηk and the coefficients Cη,k,{ν} have values 0, +1, −1. The value of Cη,k,{ν} is obtained removing, in sequence, from the Young diagram associated to {ν}, η k-border strips. If in all steps the resulting diagram represents a standard partition then Cη,k,{ν} = (−) with  =(number of lines in the removed k-border strips)−η. If in some step the resulting diagram does not represent a standard partition then Cη,k,{ν} = 0. A k−border strip of a Young diagram of a partition {ν} is a sequence of k squares in which the first of them is the last one of the first line of {ν} and the next square to a given one is the one below it, if it exists, or the one to its left, otherwise. If in all the steps of the recurrent process to obtain {η} ⊗ {m} given by Eq. (3.8) one considers only the partitions {ν} with no more than 3 rows, one obtains the U (N ) ⊃ SU (3) reductions. Using this method, tabulations are generated for {m} → (λμ) for η = 4 with m = 1 − 48, for η = 5, for m = 1 − 26, and for η = 6 with m = 1 − 19 as reported in [13]. Figure 3.1 shows the multiplicities of the irreps with λ ≤ 60 for η = 5 with m = 20 and similarly in Fig. 3.2 for η = 6 with m = 16. Clearly as

3.2 SU (3) Irreps for Many-Particle Systems

39

Table 3.1 Results for U (10) ⊃ SU (3) reductions with η = 3 for identical fermions for all antisymmetric irreps {1m } of U (10) with m ≤ 5, for {2m/2 } irreps for even m (spin S = 0) with m ≤ 10 and for {2(m−1)/2 , 1} irreps for odd m (spin S = 1/2) with m ≤ 9. Note that the p − h symmetry gives results for all other {1s } type irreps and under p − h transformation, the SU (3) irreps (λμ) will go to (μλ). Given in the table are U (10) irrep { f }, corresponding dimension (d) and the (λμ) contained in the U (10) irrep. The SU (3) irreps are given as α(λ μ) where α is the multiplicity of the SU (3) irrep (for α = 1, the number 1 is not shown) {1} ; d = 10 (3 2 0) 1 ; d = 45 (4 3 1), (0 3) 1 ; d = 120 (6 4 0), (3 3), (2 2), (0 0) 1 , ; d = 210 (5 5 2), (3 3), (0 6), (2 2), (3 0) 1 ; d = 252 (5 2), (2 5), (4 1), (1 4), (3 0), (0 3) {2} ; d = 55 (6 0), (2 2) {21} ; d = 330 (72 1), (5 2), (3 3), (4 1), (1 4), (2 2), (1 1) 2 ; d = 825 ( 8 2), (7 1), 2(4 4), (5 2), (6 0), (3 3), (4 1), (0 6), 2(2 2), (1 1) (12 4),

2 1 ; d = 3300 (10 1), (7 4), (8 2), (9 0), (5 5), 3(6 3), 2(7 1), (3 6), 3(4 4), 4(5 2), (6 0), (1 7), 3(2 5), 4(3 3), 4(4 1), 3(1 4), 3(2 2), 2(3 3 0), 2(0 3), 2(1 1) 2 ; d = 4950 (12 0), (9 3), (6 6), (7 4), 3(8 2), 2(5 5), 3(6 3), 2(7 1), (2 8), 2(3 6) 5(4 4), 3(5 2), 4(6 0), (1 7), 2(2 5), 5(3 3), 3(4 1), 3(0 6), 2(1

4), 5(2 2), (3 0), (0 3), (1 1), 2(0 0) 23 1 ; d = 13860 (11 2), (8 5), 2(9 3), (10 1), 2(6 6), 4(7 4), 4(8 2), 3(9 0), (3 9), 2(4 7), 6(5 5), 8(6 3), 6(7 1), 2(2 8), 6(3 6), 9(4 4), 11(5 2), 4(6 0), (0 9), 4(1 7), 8(2 5), 12(3 3), 8(4 1), 4(0 6), 7(1 4 4), 8(2 2), 5(3 0), 4(0 3), 3(1 1), (0 0) 2 ; d = 13860 (10 4), (12 0), (8 5), 2(9 3), (10 1), (5 8), 3(6 6), 3(7 4), 5(8 2), (9 0), (3 9), 2(4 7), 5(5 5), 7(6 3), 5(7 1), (0 12), 4(2 8), 6(3 6), 10(4 4), 7(5 2), 6(6 0), 3(1 7), 6(2 5), 9(3 3), 6(4 4 1), 5(0 6), 6(1 4), 8(2 2), 2(3 0), 2(0 3), 2(1 1), 2(0 0) 2 1 ; d = 27720 (10 4), (11 2), (7 7), 2(8 5), 3(9 3), 3(10 1), 2(5 8), 4(6 6), 7(7 4), 7(8 2), 3(9 0), 1(2 11), 2(3 9), 7(4 7), 11(5 5), 13(6 3), 10(7 1), 2(1 10), 5(2 8), 12(3 6), 17(4 4), 16(5 2), 6(6 0), 3(0 9), 8(1 7), 16(2 5), 18(3 3), 13(4 1), 4(0 6), 13(1 5 4), 12(2 2), 5(3 0), 6(0 3), 6(1 1) 2 ; d = 19404 (10 4), (12 0), (7 7), (8 5), 2(9 3), 2(10 1), (4 10), (5 8), 4(6 6), 4(7 4), 6(8 2), 2(3 9), 4(4 7), 8(5 5), 7(6 3), 6(7 1) (0 12), 2(1 10), 6(2 8), 7(3 6), 14(4 4), 9(5 2), 6(6 0), 6(1 7) 9(2 5), 11(3 3), 7(4 1), 6(0 6), 7(1 4), 11(2 2), (3 0), (0 3), 4(1 1), (0 0)

40

3 SU(3) Wigner–Racah Algebra I

Table 3.2 Results for U (15) ⊃ SU (3) reductions with η = 4 for identical fermions for antisymmetric irreps {1m } of U (15) with m ≤ 7 and for all {2r 1s } irreps for m = 6. Given in the table are U (15) irreps { f }, the corresponding dimension (d) and the (λμ) contained in the U (15) irrep. The SU (3) irreps are given as α(λ μ) where α is the multiplicity of the SU (3) irrep (for α = 1, the number 1 is not shown) {1} ; d = 15 (4 2 0) 1 ; d = 105 (6 3 1), (2 3) 1 ; d = 455 (9 4 0), (6 3), (5 2), (2 5), (3 3), (3 0), (0 3) 1 ; d = 1365 (9 2), (7 3), (4 6), (5 4), (6 2), (7 0), (3 5), (4 3), (5 1), (1 5 6), (2 4), (3 2), (1 3), (2 1) 1 ; d = 3003 (10 2), (7 5), (9 1), (5 6), 2(6 4), (7 2), (8 0), (3 7), (4 5), 2(5 3), (6 1), (0 10), 2(2 6), 2(3 4), 3(4 2), (1 5), (2 3), (36 1), (0 4), (1 2), (2 0) 1 ; d = 5005 (12 0), (8 5), (9 3), (6 6), (7 4), 2(8 2), (3 9), (4 7), 2(5 5), 3(6 3), (7 1), (2 8), 2(3 6), 3(4 4), 2(5 2), 2(6 0), (0 9), (1 7), 2(2 5), 3(3 3), 2(4 1), 2(0 6), (1 4), 2(2 2), (3 0), (0 7 3), (0 0) 1 ; d = 6435 (10 3), (7 6), (8 4), (9 2), (4 9), (5 7), 2(6 5), 2(7 3), 2(8 1), (3 8), 2(4 6), 3(5 4), 2(6 2), 2(7 0), (0 11), (1 9), 3(2 7), 3(3 5), 4(4 3), 2(5 1), 2(1 6), 2(2 4), 3(3 2), (4 0), 2(0 5), 2(1 3), (2 1), (1 0) 4

21 ; d = 40040 (14 2), (11 5), (12 3), 2(13 1), 2(9 6), (10 4), 3(11 2), 2(12 0), 3(7 7), 5(8 5), 7(9 3), 5(10 1), 1(4 10), 3(5 8), 8(6 6), 11(7 4), 11(8 2), 2(9 0), 3(3 9), 7(4 7), 13(5 5), 14(6 3), 10(7 1), 2(1 10), 7(2 8), 11(3 6), 17(4 4), 14(5 2), 7(6 0), (0 9), 8(1 7), 12(2 5), 14(3 2 2 3), 10(4 1), 6(0 6), 9(1 4), 11(2 2), 3(3 0), 2(0 3), 4(1 1) 2 1 ; d = 98280 (15 3), (16 1), 2(13 4), 2(14 2), (15 0), (10 7), 4(11 5), 6(12 3), 4(13 1), (8 8), 6(9 6), 9(10 4), 10(11 2), 4(12 0), 2(6 9), 8(7 7), 15(8 5), 17(9 3), 12(10 1), (4 10), 9(5 8), 16(6 6), 24(7 4), 21(8 2), 10(9 0), 2(2 11), 7(3 9), 18(4 7), 27(5 5), 30(6 3), 20(7 1), 4(1 10), 11(2 8), 24(3 6), 31(4 4), 28(5 2), 10(6 0), 6(0 9), 14(1 7), 25(2 5), 28(3

3), 19(4 1), 8(0 6), 18(1 4), 16(2 2), 8(3 0), 8(0 3), 7(1 1), (0 0) 23 ; d = 63700 (18 0), (15 3), (12 6), (13 4), 3(14 2), 3(11 5), 4(12 3), 2(13 1), 2(8 8), 4(9 6), 8(10 4), 5(11 2), 5(12 0), (6 9), 5(7 7), 8(8 5), 11(9 3), 7(10 1), 2(4 10), 5(5 8), 14(6 6), 12(7 4), 16(8 2), 4( 9 0), 5(3 9), 8(4 7), 16(5 5), 17(6 3), 11(7 1), 2(0 12), 2(1 10), 10(2 8), 14(3 6), 21(4 4), 14(5 2), 10(6 0), (0 9), 7(1 7), 12(2 5), 17(3 3), 8(4 1), 9(0 6), 8(1 4), 13(2 2), 3(3 0), 3(0 3), 3(1 1), 3(0 0)

3.2 SU (3) Irreps for Many-Particle Systems

41

Table 3.3 Results for U (15) ⊃ SU (3) reductions with η = 4 for the {26 } irrep. Given in the table are U (15) irrep {26 }, the corresponding dimension (d) and the (λμ) contained in this irrep. The SU (3) irreps are given as α(λ μ) where α is the multiplicity of the SU (3) irrep 6

2 ; d = 5725720 1(24 0), 1(20 5), 1(21 3), 1(16 10), 1(17 8), 4(18 6), 3(19 4), 5(20 2), 1(14 11), 5(15 9), 8(16 7), 12(17 5), 13(18 3), 7(19 1), 1(11 14), 5(12 12), 9(13 10), 23(14 8), 29(15 6), 37(16 4), 24(17 2), 15(18 0), 2(9 15), 6(10 13), 19(11 11), 38(12 9), 61(13 7), 73(14 5), 70(15 3), 43(16 1), 1(6 18), 2(7 16), 13(8 14), 30(9 12), 65(10 10), 100(11 8), 145(12 6), 137(13 4), 118(14 2), 37(15 0), 4(5 17), 13(6 15), 39(7 13), 84(8 11), 150(9 9), 208(10 7), 247(11 5), 222(12 3), 131(13 1), 1(2 20), 4(3 18), 17(4 16), 42(5 14), 106(6 12), 183(7 10), 293(8 8), 352(9 6), 375(10 4), 266(11 2), 115(12 0), 2(1 19), 9(2 17), 39(3 15), 91(4 13), 195(5 11), 319(6 9), 441(7 7), 496(8 5), 447(9 3), 25710 1), 7(0 18), 21(1 16), 76(2 14), 163(3 12), 314(4 10), 459(5 8), 594(6 6), 573(7 4), 456(8 2), 154(9 0), 22(0 15), 94(1 13), 211(2 11), 391(3 9), 544(4 7), 634(5 5), 563(6 3), 337(7 1), 93(0 12), 218(1 10), 415(2 8), 543(3 6), 590(4 4), 433(5 2), 187(6 0), 136(0 9), 314(1 7), 422(2 5), 425(3 3), 255(4 1), 179(0 6), 253(1 4), 247(2 2), 87(3 0), 88(0 3), 80(1 1), 21(0 0)

η increases and m grows, the multiplicities will become very large calling for a statistical theory (this is discussed in Chap. 10).

3.2.3 Difference Method Let us denote the number of times a (λμ) irrep of SU (3) appears in { f }U (N ) reduction  (λ,μ) to SU (3) irreps by D{η},{ f } . Going back to the n z (i), n x (i), n y (i) orbits introduced in Sect. 2.3, a difference formula for D follows. Say { f } is an irrep for m particles. Now, distributing m particles in these sp states in all possible ways, but keeping { f } structure intact, give the number of quanta F1 = Nz , F2 = N x and F3 = N y in z, x and y directions for each distribution (or configuration). Note that F1 = Nz =  N i=1 m i n z (i) where m i is number of particles in the i−th sp state. Similarly F2 (or N x ) and F3 (or N y ) can be obtained; F1 + F2 + F3 = mη. It is possible to generate

42

3 SU(3) Wigner–Racah Algebra I

η=5 shell: {20}U(21) −> (λμ)SU(3) 2

12

λ=0

λ=5

f=10

1

4

8 4

f=10

4

0 16

16

12

f=10

λ=20 Multiplicity/f

λ=10 4

8 4

f=10

4

12 8 4

2

λ=40

3

λ=60

4

f=10

2

f=10

3

1

1 0

0

10

20

μ

30

40

50

0

10

20

30

40

0

50

μ

Fig. 3.1 Multiplicities of the SU (3) irreps (λμ) versus μ for λ = 0, 5, 10, 20, 40, and 60 for the totally symmetric irrep {20} of U (21) with η = 5 and the number of bosons m = 20. Note that the multiplicities are as large as 104 − 105

all allowed configurations using a program and count the number of configurations giving the same (F1 , F2 , F3 ). Let us call this function d(F1 , F2 , F3 ). Note that this procedure is simple for identical bosons or fermions. For bosons, we can put any  number of particles in a given n z (i), n x (i), n y (i) orbit. Instead, for fermions we can put at the most one particle in a given orbit. This can be extended for a general { f } (for the leading irrep, the procedure is outlined in Sect. 2.4.4) [14]. For each SU (3) irrep (λμ), there are a set of weights (F1 , F2 , F3 ) [they are the subgroup labels in U (3) ⊃ U (1) ⊕ U (1) ⊕ U (1)] that belong to this irrep with the highest one is denoted by (F1 , F2 , F3 ) such that λ = F1 − F2 , μ = F2 − F3 and F1 + F2 + F3 = mη; see also Appendix B. Introducing the operators Oi which increase the variable p Fi by one unit so that for example O1 g(F1 , F2 , F3 ) = g(F1 + p, F2 , F3 ) etc., number of times (λμ) irrep appears in { f } of U (N ) is given by,

3.2 SU (3) Irreps for Many-Particle Systems

43

η=6 shell: {16}U(21) −> (λμ)SU(3) 30

λ=0

4

f=10

25

λ=5 4

f=10

20

4

15

2

10 5

0 45

f=10

25

λ=20 Multiplicity/f

λ=10

35

4

15 5

f=10

35

4

25 15 5 1.5

λ=40

4

f=10

λ=60

4

1

3

f=10

2

0

0.5

0

12

24

μ

36

48

0

12

24

36

48

0

μ

Fig. 3.2 Multiplicities of SU (3) irreps (λμ) versus μ for λ = 0, 5, 10, 20, 40, and 60 for the totally symmetric irrep {16} of U (28) with η = 6 and the number of bosons m = 16. Note that the multiplicities are as large as 104 − 105

(λ,μ)

D{η},{ f }



1 O −1 O −2 2 3

=

O11 1 O3−1

d(F1 , F2 , F3 )

O2 O1 1 1 2

(3.9)

This equation is exact and it is an extension of the well-known difference formula for number of times a J (L) value appears in SM (IBM) spaces. Equation (3.9) follows from the difference equation reported first in [15] and used for spin–isospin SU (4) by Bloch [16]. A computer code based on Eq. (3.9) was developed by Draayer et al. that works for any { f } and it is available in [14]. Alternatively, it is easy to generate d(F1 , F2 , F3 ) for {m} or {1m } irreps and then use the expansions for { f } in terms of symmetric (Eq. (B.7)) or antisymmetric (Eq. (B.8)) irreps to obtain the results for any irrep { f }. A program that works easily for symmetric or antisymmetric irreps is available and this is used in [13] for symmetric irreps {m} with large value for m. Using this code, as an example the reductions for {1m } irreps in η = 5 shell (then N = 21) with m ≤ 10 are obtained and the results are given in Table 3.4; p − h

44

3 SU(3) Wigner–Racah Algebra I

relation gives the reductions for m = 11 − 21. Finally, let us add that Eq. (3.9) leads (λ,μ) to statistical formulas for D{η},{ f } as discussed in Chap. 10 [13].

3.2.4 Method for Obtaining a Few Lower SU(3) Irreps Besides using the three exact methods described above, there are also methods that give quickly the leading and a few lower (with largest values for the eigenvalues of the quadratic Casimir invariant of SU (3)) SU (3) irreps in a given { f }U (N ) . In Sect. 2.4.4 we have already given a formula for the leading SU (3) irrep and here one starts with the n z (i), n x (i), n y (i) orbits. Now, using Eq. (2.35) for (ε, Λ, MΛ ) values in a SU (3) irrep, it is possible to obtain the lower SU (3) irreps. We will illustrate this using the example of U (15) ⊃ SU (3) for symmetric U (15) irreps {m}, then we have m bosons. Starting with the 15 orbits listed in (2.9), we will obtain the largest value of ε by putting all m bosons in the (400) orbit. Then, ε = 8m and this clearly gives (4m, 0) irrep. Note that the (ε, Λ, MΛ ) values for the 15 orbits are     1 3 3 1 1 , (2, 1; ±1, 0), −1, ; ± , ± , (−4, 2; ±2, ±1, 0) . (8, 0, 0), 5, ; ± 2 2 2 2 2 States with ε = 8m − 3 will be obtained by moving a boson to the (310) or (301) orbit. Then we have states with ε = 8m − 3, Λ = 1/2, MΛ = ±1/2. These two states can be generated from (4m, 0); see Eq. (2.35). Therefore, there will not be a new SU (3) irrep with ε = 8m − 3. Moving two bosons or one boson out of the (400) orbit to the other orbits, it is easy to see that there will be six configurations with ε = 4m − 6 and they will have Λ = 1 occurring twice. The (4m, 0) contains one Λ = 1 state and therefore the other Λ = 1 will correspond to a new SU (3) irrep having ε H = 8m − 6, Λ H = 1 giving (4m − 4, 2) irrep. Now let us consider all the configurations with ε = 8m − 9. There are fourteen configurations as shown in Table 3.5 and twelve of them belong to |ε = 4m − 9, Λ = 3/2 and two of them belong to |ε = 4m − 9, Λ = 1/2 state. As (4m, 0) −→ ε = 4m − 9, Λ = 3/2, (4m − 4, 2) −→ ε = 4m − 9, Λ = 3/2, 1/2, there must be a SU (3) irrep with ε H = 4m − 9, Λ H = 3/2 giving (4m − 6, 3) irrep. A further reduction in the value of ε by another three units gives the oscillator quanta distributions with ε = 8m − 12. It is easy to see that there are, with ε = 8m − 12, five Λ = 2, two Λ = 1, two Λ = 0 states. We already know that (4m, 0) −→ ε = 8m − 12, Λ = 2; (4m − 4, 2) −→ ε = 8m − 12, Λ = 2, 1, 0; (4m − 6, 3) −→ ε = 8m − 12, Λ = 2, 1. These imply that there must be two |(λμ)ε H Λ H states with (ε H = 8m − 12, Λ H = 2) and one state with (ε H = 4m − 12, Λ H = 0). These will give (λμ) = (4m − 8, 4)α=1,2 and (λμ) = (4m − 6, 0) as allowed SU (3) irreps with ε H = 8m − 12. Proceeding further, we have up to ε = 8m − 15,

3.2 SU (3) Irreps for Many-Particle Systems

45

Table 3.4 Reduction of {1m }U (21) with η = 5 into SU (3) irreps (λμ) for m ≤ 10. Given in the table are the irreps {1m }, their dimension d and the (λμ) contained in {1m } with multiplicity α. The SU (3) irreps are given as α(λ μ) in the table 2

1 ; d = 210 1( 3 8 1), 1( 4 3), 1( 0 5) 1 ; d = 1330 1(12 0), 1( 9 3), 1( 8 2), 1( 5 5), 1( 6 3), 1( 3 6)\1( 4 4), 1( 6 0), 1( 4 3 3), 1( 0 6), 1( 2 2), 1( 0 0) 1 ; d = 5985 1(13 2), 1(11 3), 1( 8 6), 1( 9 4), 1(10 2), 1(11 0), 2( 7 5), 1( 8 3), 1( 9 1), 1( 4 8), 2( 5 6), 2( 6 4), 2( 7 2), 1( 3 7), 1( 4 5), 2( 5 3), 1( 6 1), 1( 0 10), 1( 1 8), 2( 2 6), 2( 3 4), 2( 4 2), 1( 5 0), 1( 1 5), 1( 5 2 3), 1( 3 1), 1( 1 2) 1 ; d = 20349 1(15 2), 1(12 5), 1(14 1), 1(10 6), 2(11 4), 1(12 2), 1(13 0), 2( 8 7), 2( 9 5), 3(10 3), 1(11 1), 1( 5 10), 1( 6 8), 3( 7 6), 3( 8 4), 4( 9 2), 2( 4 9), 3( 5 7), 5( 6 5), 4( 7 3), 3( 8 1), 1( 2 10), 3( 3 8), 3( 4 6), 5( 5 4), 3( 6 2), 3( 7 0), 1( 0 11), 1( 1 9), 4( 2 7), 4( 3 5), 5( 4 3), 3( 6 5 1), 3( 1 6), 2( 2 4), 3( 3 2), 3( 0 5), 2( 1 3), 2( 2 1) 1 ; d = 54264 1(18 0), 1(14 5), 1(15 3), 1(12 6), 1(13 4), 2(14 2), 1( 9 9), 2(10 7), 3(11 5), 4(12 3), 1(13 1), 1( 7 10), 2( 8 8), 4( 9 6), 5(10 4), 3(11 2), 3(12 0), 1( 5 11), 4( 6 9), 5( 7 7), 7( 8 5), 7(9 3), 5(10 1), 1( 3 12), 2( 4 10), 5( 5 8), 9( 6 6), 7( 7 4), 8( 8 2), 2( 9 0), 1( 0 15), 3( 2 11), 5( 3 9), 8( 4 7), 10( 5 5), 10( 6 3), 5( 7 1), 2( 1 10), 5( 2 8), 8( 3 6), 9( 4 4), 7( 5 2), 4( 6 0), 3( 0 9), 4( 1 7), 8( 2 5), 8( 3 3), 3( 4 1), 3( 7 0 6), 3( 1 4), 5( 2 2), 2( 3 0), 2( 0 3), 1( 1 1), 1( 0 0) 1 ; d = 116280 1(17 3), 1(14 6), 1(15 4), 1(16 2), 1(11 9), 1(12 7), 3(13 5), 2(14 3), 2(15 1), 1( 9 10), 3(10 8), 4(11 6), 5(12 4), 4(13 2), 3(14 0), 2( 7 11), 3( 8 9), 7( 9 7), 7(10 5), 8(11 3), 4(12 1), 1( 4 14), 1( 5 12), 5( 6 10), 8( 7 8), 12( 8 6), 11( 9 4), 11(10 2), 3(11 0), 2( 3 13), 3( 4 11), 9( 5 9), 11( 6 7), 16( 7 5), 13( 8 3), 8( 9 1), 1( 1 14), 4( 2 12), 6( 3 10), 13( 4 8), 16( 5 6), 19( 6 4), 12( 7 2), 6( 8 0), 5( 1 11), 7( 2 9), 14( 3 7), 15( 4 5), 16( 5 3), 9( 6 1), 5( 0 10), 7( 1 8), 15( 2 6), 13( 3 4), 13( 4 2), 3( 5 0), 2( 8 0 7), 8( 1 5), 8( 2 3), 6( 3 1), 5( 0 4), 4( 1 2), 3( 2 0) 1 ; d = 203490 1(17 4), 1(19 0), 1(14 7), 1(15 5), 1(16 3), 1(17 1), 1(11 10), 2(12 8), 3(13 6), 3(14 4), 4(15 2), 1( 9 11), 3(10 9), 5(11 7), 6(12 5), 6(13 3), 4(14 1), 1( 6 14), 2( 7 12), 4( 8 10), 9(9 8), 10(10 6), 13(11 4), 7(12 2), 5(13 0), 2( 5 13), 5( 6 11), 9( 7 9), 14( 8 7), 17( 9 5), 15(10 3), 9(11 1), 1( 2 16), 2( 3 14), 6( 4 12), 11( 5 10), 16( 6 8), 23( 7 6), 21( 8 4), 18( 9 2),

(continued)

46

3 SU(3) Wigner–Racah Algebra I

Table 3.4 (continued) 4(10 0), 1( 1 15), 3( 2 13), 9( 3 11), 15( 4 9), 22( 5 7), 26( 6 5), 23( 7 3), 14( 8 1), 2( 0 14), 5( 1 12), 12( 2 10), 20( 3 8), 23( 4 6), 28( 5 4), 18( 6 2), 9(7 0), 3(0 11), 11(1 9), 18( 2 7), 22( 3 5), 21( 4 3), 13( 5 1), 7(0 8), 14(1 6), 15( 2 4), 15( 3 2), 4( 4 0), 6( 0 5), 8( 1 3), 6( 9 2 1), 2(0 2), 2( 1 0) 1 ; d = 293930 1(18 3), 1(14 8), 1(15 6), 1(16 4), 1(17 2), 1(18 0), 2(12 9), 2(13 7), 4(14 5), 3(15 3),2(16 1),2(9 12), 2(10 10), 5(11 8), 7(12 6), 7(13 4), 5(14 2), 3(15 0), 1( 6 15), 1(7 13), 5( 8 11), 7(9 9), 12(10 7), 13(11 5), 13(12 3), 6(13 1), 2( 5 14), 5( 6 12), 10( 7 10), 14( 8 8), 20( 9 6), 19(10 4), 15(11 2), 5(12 0), 1( 2 17), 2( 3 15), 6( 4 13), 10( 5 11), 21( 6 9), 23(7 7), 29( 8 5), 25( 9 3), 14(10 1), 1( 1 16), 3( 2 14), 10( 3 12), 15( 4 10), 27( 5 8), 31( 6 6), 32( 7 4), 23( 8 2), 11( 9 0), 3( 0 15), 5( 1 13), 15( 2 11), 22( 3 9), 33( 4 7), 34( 5 5), 33( 6 3), 18( 7 1), 3( 0 12), 12( 1 10), 20( 2 8), 32( 3 6), 28( 4 4), 25( 5 2), 9( 6 0), 12( 0 9), 16( 1 7), 26( 2 5), 23( 3 3), 14( 4 1), 7( 0 6), 14( 1 4), 10( 10 2 2), 7( 3 0), 8( 0 3), 3( 1 1), 1( 0 0) ; d = 352716 1 1(20 0), 1(15 7), 1(16 5), 1(17 3), 1(12 10), 1(13 8), 3(14 6), 2(15 4), 3(16 2), 1( 9 13), 2(10 11), 4(11 9), 5(12 7), 7(13 5), 6(14 3), 3(15 1), 1( 7 14), 3( 8 12), 5( 9 10), 10(10 8), 10(11 6), 14(12 4), 7(13 2), 5(14 0), 1( 4 17), 2( 5 15), 4( 6 13), 9( 7 11), 14( 8 9), 19( 9 7), 20(10 5), 18(11 3), 11(12 1), 1( 3 16), 4( 4 14), 9( 5 12), 16( 6 10), 22( 7 8), 30( 8 6), 25( 9 4), 23(10 2), 6(11 0), 1( 1 17), 4( 2 15), 8( 3 13), 16( 4 11), 26( 5 9), 33( 6 7), 37( 7 5), 31( 8 3), 19( 9 1), 1( 0 16), 5( 1 14), 12( 2 12), 21( 3 10), 34( 4 8), 37( 5 6), 41( 6 4), 26( 7 2), 14( 8 0), 5( 0 13), 13( 1 11), 23( 2 9), 34( 3 7), 39( 4 5), 35( 5 3), 19( 6 1), 9( 0 10), 19( 1 8), 29( 2 6), 29( 3 4), 26( 4 2), 7( 5 0), 10( 0 7), 19( 1 5), 19( 2 3), 12( 3 1), 9( 0 4), 7( 1 2), 6( 2 0), 2( 0 1)

{m}  − 4, 2)m>1 ⊕ (4m − 6, 3)m>2  U (15) −→ (λμ) SU (3) = (4m, 0) ⊕ (4m α=1,2 α=1 ⊕ (4m − 8, 4)m=2,3 , (4m − 8, 4)m>3 ⊕ (4m − 6, 0)m>2 ⊕ (4m − 10, 5)m>4 ⊕(4m − 9, 3)m>2 ⊕ (4m − 8, 1)m>3 ⊕ · · · (3.10) It is important to note that for m ≥ 4, the (4m − 8, 4) irrep appears twice. Physical significance of this will be discussed in Chaps. 4 and 7. Using the same procedure as above, we have for {m}U (10) with η = 3 (this reduction will be useful for the sdp f IBM discussed in Chap. 7), {m}U (10) −→ (λμ) SU (3) = (3m, 0) ⊕ (3m − 4, 2)m>1 ⊕ (3m − 6, 3)m>2 ⊕(3m − 8, 4)m>3 ⊕ (3m − 6, 0)m>2 ⊕ (3m − 10, 5)m>4 ⊕(3m − 9, 3)m>2 ⊕ (3m − 8, 1)m>3 ⊕ · · ·

(3.11)

The above procedure can be continued to get further lower irreps in not only {m} but also in a general { f }U (N ) . For example, for the lowest two rowed irreps {m − 1, 1} and {m − 2, 2} of U (15) with η = 4, the lowest few SU (3) irreps are

(310)

3 2 1 − 1 1 1 − − − − − − −

(400)

m−3 m−3 m−3 m−3 m−2 m−2 m−2 m−2 m−2 m−2 m−1 m−1 m−1 m−1

#

1 2 3 4 5 6 7 8 9 10 11 12 13 14

− 1 2 3 − − − 1 1 1 − − − −

(301)

− − − − 1 − − 1 − − − − − −

(220)

− − − − − 1 − − 1 − − − − −

(211) − − − − − − 1 − − 1 − − − −

(202) − − − − − − − − − − 1 − − −

(130) − − − − − − − − − − − 1 − −

(121) − − − − − − − −− − − − − 1 −

(112) − − − − − − − − − − − − − 1

(103) − − − − − − − − − − − − − −

(040) − − − − − − − − − − − − − −

(031) − − − − − − − − − − − − − −

(022) − − − − − − − − − − − − − −

(013) − − − − − − − − − − − − − −

(004)

3/2 1/2 −1/2 −3/2 3/2 1/2 −1/2 1/2 −1/2 −3/2 3/2 1/2 −1/2 −3/2

MΛ

Table 3.5 Configurations with ε = 8m − 9 for a (sdg)m boson system. In the table, number of bosons in each (n z , n x , n y ) orbit is given below the corresponding orbit. First column gives the configuration number

3.2 SU (3) Irreps for Many-Particle Systems 47

48

3 SU(3) Wigner–Racah Algebra I   {m − 1, 1}U (15) → (4m − 2, 1) ⊕ (4m − 4, 2)m≥3 ⊕ (4m − 6, 3)2m≥4 , (4m − 6, 3)m=2,3   ⊕(4m − 5, 1)m≥3 ⊕ (4m − 8, 4)3m≥5 , (4m − 8, 4)2m=3,4   ⊕ (4m − 7, 2)2m≥4 , (4m − 7, 2)m=3 ⊕ (4m − 6, 0)m≥4 ⊕ . . . , {m− 2, 2}U (15) → (4m − 4, 2) ⊕ (4m −  6,3)m≥5 ⊕ (4m − 5, 1)  ⊕ (4m − 8, 4)3m≥6 , (4m − 8, 4)2m=4,5 ⊕ (4m − 7, 2)2m≥5 , (4m − 7, 2)m=4   ⊕ (4m − 6, 0)2m≥5 , (4m − 6, 0)m=4 ⊕ . . . .

(3.12) However, it will become cumbersome to carryout such an exercise by hand as the ε = (2λ + μ) decreases. Finally, let us add that besides the four methods described here, there are several other methods to obtain { f }U (N ) → (λμ) SU (3) ; see, for example, [17, 18].

3.3 SU(3) Wigner and Racah Coefficients In the Spectroscopic studies of quantum systems such as atomic nuclei, for deriving selection rules and analytical formulas for various observables and also for calculating numerical values, it is well known that the angular momentum (i.e., for S O(3) algebra) Wigner–Racah algebra is needed. The Wigner–Racah algebra here is well known [19] and this is briefly discussed giving all essential equations in Appendix A. For states labeled not only by angular momentum quantum number (L and/or J ) but also carry SU (3) labels (λμ), we need SU (3) Wigner and Racah coefficients, 9-SU (3) coefficients and so on. However, with SU (3) some real complications arise. In fact, SU (3) group is the first nontrivial group having all complications of a general group. The various complications are as follows. (i) The reduction (λ1 μ1 ) ⊗ (λ2 μ2 ) −→ (λ12 μ12 ) is not multiplicity free. (ii) As discussed in Chap. 2, with SU (3) we have two subgroup chains, SU (3) ⊃ [SU (2) ⊃ U (1)] ⊗ U (1) and SU (3) ⊃ S O(3) ⊃ S O(2) giving two different types of SU (3) Wigner Coefficients. max times in the reduction (λ1 μ1 ) ⊗ (λ2 μ2 ), then we have (iii) Say (λ12 μ12 ) occurs ρ12 the product basis states which are represented as max |(λ1 μ1 )(λ2 μ2 ); (λ3 , μ3 )α ρ ; ρ = 1, 2, . . . , ρ12 .

The α’s here are the subgroup labels (εΛMΛ ) or (K L M) and ρ is the multiplicity label. As there is no group theoretical (mathematical) meaning to ρ, it is impossible to derive general analytical formulas for SU (3) recoupling coefficients. (iv) In the case of SU (3) ⊃ S O(3) ⊃ S O(2) there is an additional complication due to the illdefined K quantum number which represents the multiplicity in the SU (3) ⊃ S O(3) reduction. Progress in deriving analytical expressions for SU (3) recoupling coefficients and applying the SU (3) algebra is made by: (i) addressing the question of deriving analytical expressions for SU (3) irreps of interest in Nuclear Physics appli-

3.3 SU (3) Wigner and Racah Coefficients

49

cations; (ii) developing efficient computer codes that cover most cases of interest; (iii) tabulating large number of analytical results for various special cases; (iv) by using a building up principle. The developments in this subject are due to Biedenharn, Louck, Hecht, Moshinsky, Sharp, Vergados, Akiyama, Draayer, and others; see, for example, [20–31]. In the next three subsections, we will describe these without going into too many technical details.

3.3.1 SU(3) ⊃ SU(2) ⊗ U(1) Reduced Wigner Coefficients Let us consider the states that are product of two SU (3) ⊃ SU (2) ⊗ U (1) basis states

 

(λ1 μ1 )ε1 Λ1 MΛ (λ2 μ2 )ε2 Λ2 MΛ 1 2 and the coupled basis states |(λ1 μ1 )(λ2 μ2 ); (λμ)εΛMΛ ρ for fixed (λ1 μ1 ) and (λ2 μ2 ). Both form complete set of states for fixed (λ1 μ1 ) and (λ2 μ2 ) and therefore it is possible to define a unitary transformation that takes one to the other. The matrix elements of this transforming matrix define SU (3) ⊃ SU (2) ⊗ U (1) Wigner coefficients. Firstly we have, |(λ1 μ1 )(λ 2 μ2 ); (λμ)εΛMΛ ρ   = (λ1 μ1 )ε1 Λ1 MΛ1 (λ2 μ2 )ε2 Λ2 MΛ2 | (λμ) εΛMΛ ρ ε1 (ε2 ),Λ1 ,Λ2 ,MΛ1 (MΛ2 )

  × (λ1 μ1 ) ε1 Λ1 MΛ1 (λ2 μ2 ) ε2 Λ2 MΛ2 .

(3.13)

The expansion coefficients −− | − above are SU (3) ⊃ [SU (2) ⊃ U (1)] ⊗ U (1) Wigner coefficients. Using Racah’s factorization theorem [32, 33], we can separate the coupling at the (Λ, MΛ ) level giving |(λ1 μ1 )(λ2 μ2 ); (λμ)εΛMΛ ρ =



(λ1 μ1 ) ε1 Λ1 (λ2 μ2 ) ε2 Λ2 || (λμ) εΛ ρ

ε1 (ε2 ),Λ1 ,Λ2

× |(λ1 μ1 ) ε1 Λ1 , (λ2 μ2 ) ε2 Λ2 ; ΛMΛ .

(3.14) Note that ε = ε1 + ε2 , MΛ = MΛ1 + MΛ2 , and Λ = Λ1 × Λ2 . The double-barred coefficients || in Eq. (3.14) are SU (3) ⊃ SU (2) ⊗ U (1) reduced Wigner coefficients and they will not depend on MΛ ’s. Orthonormal properties of the reduced Wigner coefficients are

50

3 SU(3) Wigner–Racah Algebra I



(λ1 μ1 ) ε1 Λ1 (λ2 μ2 ) ε2 Λ2 || (λμ) εΛ ρ    × (λ1 μ1 ) ε1 Λ1 (λ2 μ2 ) ε2 Λ2 || λ μ ε Λ ρ  = δ(λμ)(λ μ ) δε,ε δΛ,Λ δρρ  , ε1 ,Λ1 ,ε2 ,Λ2

(3.15)



(λ1 μ1 ) ε1 Λ1 (λ2 μ2 ) ε2 Λ2 || (λμ) εΛ ρ   × (λ1 μ1 ) ε1 Λ1 (λ2 μ2 ) ε2 Λ2 || (λμ) εΛ ρ = δε1 ,ε1 δΛ1 ,Λ1 δε2 ,ε2 δΛ2 ,Λ2 . (λμ),ε,Λ,ρ

Using the operator O( p, q, r ) introduced in Sect. 2.7, it is possible to derive recursion relations for the multiplicity free (λ1 μ1 ) × (λ2 , μ2 ) −→ (λ12 μ12 ) reduced Wigner coefficients. For example, we have   f [(λμ)εΛν] (λ1 μ1 )ε1 Λ1 ν 1 ; (λ2 μ2 )ε2 Λ2 ν2 | (λμ)ε − 3, Λ + 21 , ν + 21  + f [(λμ)ε , −(Λ + 1), ν] (λ μ )ε Λ ν ; (λ μ )ε Λ ν | (λμ)ε − 3, Λ − 21 , ν + 21 1 1 1  1 1 2 2 2 2 2   1 1 1 1 = f  (λ1 μ1 )ε1 + 3, Λ1 − 2 , ν1 − 2 (λ 1 μ1 )ε1 + 3, Λ1 − 2 , ν1 − 2 ; (λ2 μ2 )ε2 Λ2 ν2 | (λμ)εΛν   + f (λ1 μ1 )ε1 + 3, −Λ1 − 23 , ν1 − 21 (λ1 μ1 )ε1 + 3, Λ1 + 21 , ν1 − 21 ; (λ2 μ2 )ε2 Λ2 ν2 | (λμ)εΛν + f (λ2 μ2 )ε2 + 3, Λ2 − 21 , ν2 − 21 (λ 1 μ1 )ε1 Λ1 , ν1 ; (λ2 μ2 )ε2 + 3, Λ2 − 21 , ν2 − 21 | (λμ)εΛν  + f (λ2 μ2 )ε2 + 3, −Λ2 − 23 , ν2 − 21 (λ1 μ1 )ε1 Λ1 , ν1 ; (λ2 μ2 )ε2 + 3, Λ2 + 21 , ν2 − 21 | (λμ)εΛν ,

(3.16) where ν = MΛ and similarly ν1 and ν2 . In addition, the function f above is given by f [(λμ)εΛν] =             1 2λ + μ + 1 ε − Λ  (Λ + ν + 1) λ + 1 + 1 λ − μ − 1 ε Λ + 2 + 13 λ + 2μ − 21 ε  3 2 3 2 (2Λ + 1)(2Λ + 2)

.

(3.17) The recursion relation given by Eq. (3.16) can be solved to derive analytical results for a class of reduced Wigner coefficients. As an application, let us consider the example with λ = λ1 + λ2 , μ = μ1 = μ2 = 0, ε = 2λ giving Λ = ν = 0 and ε1 = 2λ1 giving Λ1 = ν1 = 0. Then, from Eq. (3.16) we have   f ((λ0)2λ, 0, 0) (λ1 0)2λ1 , 0, 0 ; (λ2 0)2λ2 − 3, 21 , 21 | (λ0)2λ − 3, 21 , 21 (3.18) = f ((λ2 0)2λ2 , 0, 0) (λ1 0)2λ1 , 0, 0 ; (λ2 0)2λ2 , 0, 0 | (λ0)2λ, 0, 0 . This gives immediately the result (note that the Wigner coefficient on the right side is unity) 

(λ1 0)2λ1 , 0 ; (λ2 0)2λ2 − 3, 21 | (λ1 +λ2 , 0)2λ1 + 2λ2 − 3, 21

=

f [(λ2 , 0)2λ2 , 0, 0] = f [(λ1 + λ2 , 0)2λ1 + 2λ2 , 0, 0]

λ2 . λ1 + λ2

A much more general and useful example derived by Hecht is [20],

 (3.19)

3.3 SU (3) Wigner and Racah Coefficients

51

  (λ1 μ1 ) (ε1H − 3α − 3β) Λ1H + 21 α − 21 β ; (λ2 , 0) ε2 = ε2H − 3σ + 3α + 3β, Λ2 = 21 σ − 21 α − 21 β || (λμ)ε H Λ H ⎤ 21 (λ1 − α)! (μ1 − β)! (μ1 + 1 + α − β) (λ1 + μ1 + 1 − β)! ⎢ (λ2 − σ + α + β)! (λ1 + λ2 − λ − σ )! (λ + μ − λ1 − λ2 + σ + α)! ⎥ ⎥ = (−1)φ ⎢ ⎦ ⎣ λ1 !α!β! (μ1 + 1 + α)! (λ1 + μ1 + 1)! (λ2 − σ )! (λ1 + λ2 − λ − σ − α)! (λ + μ − λ1 − λ2 + σ )! ⎡

(λ1 + λ2 + μ1 − λ + 1 − σ )! (λ1 + λ2 − λ + μ1 − μ − σ )! × (λ1 + λ2 + μ1 − λ + 1 − σ − β)! (λ1 + λ2 − λ + μ1 − μ − σ − β)!

!1 2

  × (λ1 μ1 ) ε1H Λ1H (λ2 , 0) ε2H − 3σ, Λ2 = 21 σ (λμ) ε H Λ H ; 3σ = 2λ2 + (2λ1 + μ1 ) − (2λ + μ), 0 ≤ α ≤ 13 (λ1 + λ2 − λ + μ − μ1 ) , 0 ≤ β ≤

1 3

(λ1 + λ2 − λ + 2μ1 − 2μ) .

(3.20) Note that (−1)φ is a phase factor. Using the normalization property from Eq. (3.15),

 "

(λ1 μ1 ) ε1H − 3α − 3β, Λ1H + α − β (λ2 , 0) ε2 , Λ2 = σ − α − β

2 2 2 2 2 α,β || (λμ)ε H Λ H |2 = 1

(3.21) and Eq. (3.20), formula for " (λ1 μ1 ) ε1H Λ1H (λ2 , 0) ε2H

1 − 3σ, Λ2 = σ || (λμ)ε H Λ H 2

#

was obtained in [34]. It is given by | (λ1 μ1 ) ε1H Λ1H (η0)ε0 Λ0 || (λμ) ε H Λ H |2 = ! λ1 !(λ1 + η + 1 + K − 2σ )!(λ1 + μ1 + 1)!(λ1 + μ1 + 2 + η − σ − K )! ; (λ1 − σ + K )!(λ1 + η − σ + 1)!(λ1 + μ1 + 1 − K )!(λ1 + μ1 + 2 + η − σ )! 3σ = 2λ1 + μ1 + 2η − 2λ − μ , K =

σ μ μ1 + − . 2 2 2

(3.22) Note that, σ can take values 0 to η and K from 0 to σ . Also, it is easy to see that the ε0 and Λ0 are uniquely defined. In order to determine the overall phase of the reduced Wigner coefficients in Eqs. (3.20) and (3.22), a phase convention is needed. The  μ convention chosen$ in [20, 29] is that the Wigner coefficient (λ ) 1 1 ε1H Λ1H MΛ1H  (λ2 , 0) ε2 Λ2 MΛ2 $(λμ)ε H Λ H MΛ H is positive. This phase convention leads to the following symmetry properties,

(λ1 μ1 ) ε1 Λ1 (λ2 , 0) ε2 Λ2 || (λμ)εΛ = (−1)λ1 +λ2 −λ+μ−μ1 +Λ1 +Λ2 −Λ (λ2 , 0) ε2 Λ2 (λ1 , μ1 ) ε1 Λ1 || (λμ)εΛ

(3.23)

52

and

3 SU(3) Wigner–Racah Algebra I

(λ1 μ1 ) ε1 Λ1 (λ2 , 0) ε2 Λ2 || (λ3 μ3 ) ε3 Λ3 = ! 21 μ1 −μ3 −λ1 +λ3 − 21 ε2 )+Λ3 −Λ1 (dim (λ3 μ3 )) (2Λ1 + 1) ( (−1) (dim (λ1 μ1 )) (2Λ3 + 1) × (λ3 μ3 ) ε3 Λ3 (0, λ2 ) − ε2 Λ2 || (λ1 μ1 )ε1 Λ1 . 1 3

(3.24)

Note that Eq. (3.24) is simple as the SU (3) couplings here are multiplicity free. For details regarding the phase convention and for further symmetry properties of reduced Wigner’s involving SU (3) couplings with multiplicity, see [22]. It is important to add that [22],

(λ1 μ1 )ε1 Λ1 (λ2 μ2 )ε2 Λ2 || (λ12 μ12 )ε12 Λ12 ρ and

(λ2 μ2 )ε2 Λ2 (λ1 μ1 )ε1 Λ1 || (λ12 μ12 )ε12 Λ12 ρ differ by a phase only when ρ max = 1. This restriction gives rise to certain complications in applying SU (3) algebra; see Sects. 3.3.3 and 4.4 ahead. In order to go beyond the formula given by Eq. (3.20) and derive analytical formulas for the SU (3) ⊃ SU (2) ⊗ U (1) reduced Wigner coefficients with more general (λ1 μ1 ) × (λ2 μ2 ) → (λμ), one useful approach is to use SU (3) Racah or U -coefficients and we will describe this in Sect. 3.3.3. Also, some of these Wigner coefficients will be needed in establishing the relationship between SU (3) symmetry for Bose–Fermi systems and the geometric Nilsson model; see Chap. 8. Alternatively, an algorithm to calculate general SU (3) ⊃ SU (2) ⊗ U (1) reduced Wigner coefficients was described in [22] and this is converted into an efficient computer code that is available as a publication [35]. In the later years, this code was modified to handle Wigner coefficients involving large values of (λμ) (Draayer, private communication to Kota).

3.3.2 SU(3) ⊃ S O(3) Reduced Wigner Coefficients Physically relevant chain for nuclei is the SU (3) ⊃ S O(3) chain and starting with the basis states |(λμ)K L M , we will have SU (3) ⊃ S O(3) reduced Wigner coefficients. Before introducing these recoupling coefficients, first one has to construct orthonormal states |(λμ)K L M with  K  having a physical meaning. There are several ways to define the K -label as discussed in detail in [36]. In these, physically motivated choice as appropriate for nuclei is due to Elliott [37]. Here, used is the Hill–Wheeler integral representation of |(λμ)K L M in terms of projection from the intrinsic state |(λμ)h.w. , |(λμ)K E L M = N



% D KL E M (Ω){|(λμ)h.w. }Ω dΩ

(3.25)

3.3 SU (3) Wigner and Racah Coefficients

53

where {}Ω is a rotated state through the Euler angle Ω and N  is a normalization constant; K E represents K of Eq. (2.32). The states |(λμ)K E L M defined by Eq. (3.25) are not orthogonal to each other for different K E with the same L-value. By using Schmidt’s orthonormalization, one can construct orthogonal states with a label ‘K ’such that in the limit λ → ∞ with μ fixed and for a given (L , M), K → K E . For example, for a (λ, 2) irrep K E = 0,2 and each L occurs twice (labeled by K E = 0,2) for L even if λ is even and L odd if λ is odd. Then, we can construct |(λμ)K L states such that |(λ2)K = 0, L = |(λ2)K E = 0, L , |(λ2)K = 2, L = x20 |(λ2)K E = 0, L + x22 |(λ2)K E = 2, L ;

(3.26)

x20 + x22 (λ2)K E = 0, L | (λ2)K E = 2, L = 0 . For example, for the (λ2) irrep with λ − L even, the coefficients x20 and x22 are [21] x K =2,K E =0 = −  x K =2,K E =2 =

(L − 1)(L)(L + 1)(L + 2) 4(λ + 2)(λ + 3)(λ − L + 2)(λ + L + 3)

!1/2 ,



1/2 2(λ + 2)2 − L(L + 1) 2(λ + 3)2 − L(L + 1) . 4(λ + 2)(λ + 3)(λ − L + 2)(λ + L + 3)

(3.27)

Similarly, we can construct |(λ3)K = 1, 3; L , |(λ4)K = 0, 2, 4; L states, etc. Tables for the coefficients xi j with μ ≤ 4 are given by Vergados [21] and in more general terms in [22]. With orthonormal states |(λμ)K L M , we can define SU (3) ⊃ S O(3) reduced Wigner coefficients exactly the same way as in Eq. (3.13), |(λ 1 μ1 )K 1 L 1 M1 |(λ2 μ2 )K 2 L 2 M2 =

(λ1 μ1 ) K 1 L 1 M1 (λ2 μ2 ) K 2 L 2 M2 | (λμ) K L M ρ |(λ1 μ1 )(λ2 μ2 ); (λμ) K L M ρ (λμ)K L ,ρ

=⇒ |{|(λ1 μ1 )K 1 L 1 |(λ2 μ2 )K 2 L 2 } L M

(λ1 μ1 ) K 1 L 1 (λ2 μ2 ) K 2 L 2 ||; (λμ) K L ρ |(λ1 μ1 )(λ2 μ2 ); (λμ) K L M ρ . = (λμ)K L ,ρ

(3.28) The SU (3) ⊃ S O(3) reduced Wigner coefficients || in Eq. (3.28) satisfy the orthonormal properties exactly as in Eq. (3.15). Symmetry properties of the SU (3) ⊃ S O(3) reduced Wigner coefficients are [22]

54

3 SU(3) Wigner–Racah Algebra I

(λ1 μ1 ) K 1 L 1 (λ2 μ2 ) K 2 L 2 || (λ12 μ12 ) K 12 L 12 ρ

d (λ12 μ12 ) (2L 1 + 1) d (λ1 μ1 ) (2L 12 + 1) (μ2 λ2 ) K 2 L 2 || (λ1 μ1 ) K 1 L 1 ρ ,

= (−1)(λ1 +λ2 −λ12 +μ1 +μ2 −μ12 )+(λ2 +μ2 )+L 1 +L 2 −L 12 × (λ12 μ12 ) K 12 L 12

(λ1 μ1 ) K 1 L 1 (λ2 μ2 ) K 2 L 2 || (λ12 μ12 ) K 12 L 12 ρ max = (−1)λ1 +λ2 −λ12 +μ1 +μ2 −μ12 +ρ −ρ × (μ1 λ1 ) K 1 L 1 (μ2 λ2 ) K 2 L 2 || (μ12 λ12 ) K 12 L 12 ρ ,

(3.29)

(λ1 μ1 ) K 1 L 1 (λ2 μ2 ) K 2 L 2 || (λ12 μ12 ) K 12 L 12 ρ = (−1)λ1 +λ2 −λ12 +μ1 +μ2 −μ12 +L 1 +L 2 −L 12 × (λ2 μ2 ) K 2 L 2 (λ1 μ1 ) K 1 L 1 || (λ12 μ12 ) K 12 L 12 ρ for ρ max = 1 only . Using the formulation developed by Elliott [37] based on Eq. (3.25), it is possible to write the SU (3) ⊃ S O(3) reduced Wigner coefficients in terms of Elliott’s coefficients (normalization factor in Eq. (3.25)) and SU (3) ⊃ SU (2) ⊗ U (1) reduced Wigner coefficients [21, 38]. Using this, computer codes for numerical calculations of SU (3) ⊃ S O(3) reduced coefficients are developed in [35]. These codes also use a prescription for resolving the outer multiplicity ‘ρ’ in the SU (3) ⊃ SU (2) ⊗ U (1) reduced Wigner coefficients and hence also in the SU (3) ⊃ S O(3) reduced Wigner coefficients. As (10) is the fundamental irrep of SU (3), most basic reduced Wigner coefficients are (λμ) × (10) −→ (λ μ ). Analytical formulas for these are derived in [21]. Formulas for those with μ = 0 are

(λ0)L − 1

(λ0)L + 1

(λ0)L − 1

(λ0)L + 1

! (λ + L + 2)L 1/2 , (10)1 || (λ + 1, 0)L = (λ + 1)(2L + 1) ! (λ − L + 1)(L + 1) 1/2 (10)1 || (λ + 1, 0)L = − , (λ + 1)(2L + 1) ! (λ − L + 1)(L + 1) 1/2 (10)1 || (λ − 1, 1)L = − , (λ + 1)(2L + 1) !1/2 (λ + L + 2)L (10)1 || (λ − 1, 1)L = − (λ + 1)(2L + 1)

(3.30)

for λ − L = odd and (λ0)L (10)1 || (λ − 1, 1)L = 1 for λ − L =even. Similarly, analytical formulas for μ = 1 are given Table 3.6; note that (λ, 1) × (10) = (λ0) ⊕ (λ + 1, 1) ⊕ (λ − 1, 2). Starting with these and using the symmetry properties and a building up principle, it is possible to derive analytical formulas for more general class of SU (3) ⊃ S O(3) reduced Wigner coefficients. We will briefly discuss this in Sect. 3.4 but before this we will turn to SU (3) Racah or U -coefficients and the closely related Z -coefficients.

L +1

L

L −1

λ − L = odd L1

L +1

L

L −1

L1

λ − L = even

!1 (λ + L + 2)(L + 1)(L − 1) 2 (λ + 1)L(2L + 1) !1 2 (λ + 2) (λ + 1)L(L + 1) !1 (λ − L + 2)L(L + 2) 2 − (λ + 1)(L + 1)(2L + 1)

(λ1)L 1 ; (10)1||(λ + 1, 1)L

2

!1

(λ + 2)(λ + L + 3)(L − 1)(L + 1) (λ + 1)(λ + 3)L(2L + 1) !1 (λ + L + 3)(λ − L + 2) 2 (λ + 1)(λ + 3)L(L + 1) !1 (λ + 2)(λ − L + 2)(L + 2)L 2 − (λ + 1)(λ + 3)(L + 1)(2L + 1)

(λ1)L 1 ; (10)1||(λ + 1, 1)L 2

!1

[(λ + 1)(2L + 1)φ(λ, L)] 2

1

[(λ + 1)(2L + 1)φ(λ, L)] !1 2 (λ + 2) λ (λ + 1)φ(λ, L) 1 (λ + L + 1)[(λ − L + 1)(L + 2)] 2

1 2

(λ − L)[(λ + L + 2)(L − 1)] 2

1

(λ1)L 1 ; (10)1||(λ − 1, 2)κ = 0L

−

(λ − L + 2)(L − 1) 2(λ + 3)(2L + 1) ! (λ + 2) − 2(λ + 3) !1 (λ + L + 3)(L + 2) 2 − 2(λ + 3)(2L + 1)

(λ1)L 1 ; (10)1||(λ0)L

!1 (λ − L + 2)(L + 1)(L + 2) 2 2(λ + 1)L(2L + 1) !1 (λ + 2)(L − 1)(L + 2) 2 2(λ + 1)L(L + 1) !1 2 (λ + L + 3)L(L − 1) − 2(λ + 1)(L + 1)(2L + 1)

(λ + 2)(λ − L + 1)(L + 1)(L + 2) L(2L + 1)φ(λ, L)

2

!1

2

!1

(λ − L + 1)(λ + L + 2)(L − 1)(L + 2) L(L + 1)φ(λ, L) !1 (λ + 2)(λ + L + 2)L(L − 1) 2 − (L + 1)(2L + 1)φ(λ, L)

−

(λ1)L 1 ; (10)1||(λ − 1, 2)κ = 2L

−

(λ1)L 1 ; (10)1||(λ − 1, 2)L

Table 3.6 Formulas for (λ1) × (10) → (λ μ ) SU (3) ⊃ S O(3) reduced Wigner coefficients. Results are from [21]. Note that φ(λ, L) = 2(λ + 1)2 − L(L + 1) in the table. Table taken from [21] with permission from Elsevier

3.3 SU (3) Wigner and Racah Coefficients 55

56

3 SU(3) Wigner–Racah Algebra I

3.3.3 SU(3) Racah or U− and Z− Coefficients Racah or U -coefficients for SU (3) are defined exactly as the U -coefficients for angular momentum and they will have the same meaning. Let us consider three SU (3) states (λi μi ), i = 1, 2, 3 coupled to a final SU (3) irrep (λμ) in two different ways as shown in Fig. 3.3. Unitary transformation between these two complete set of states (I) and (II) in the figure defines SU (3) U -coefficients, (I ) =



 (λ23 μ23 ),ρ23 ,ρ1,23 U (λ1 μ1 )(λ2 μ2 )(λμ)(λ3 μ3 ) ; (λ12 μ12 )ρ12 ,ρ12,3 (λ23 μ23 )ρ23 ,ρ1,23 (I I ) .

(3.31)

Note that the U -coefficient is independent of α in Fig. 3.3 where α represents subgroup labels (ε, Λ, MΛ ) or (K L M). Secondly, U -coefficient involves four multiplicity labels ρ’s and Fig. 3.3 makes their meaning clear. As the U coefficients are independent of the α label, it is convenient, due to their simplicity, to use (ε, Λ, MΛ ) labels, i.e., to use SU (3) ⊃ SU (2) ⊗ U (1) labels to write the U -coefficient in terms of four reduced Wigner coefficients,  U (λ1 μ1 ) (λ2 μ2 ) (λμ) (λ3 μ3 ) ; (λ12 μ12 )ρ12 ,ρ12,3 (λ23 μ23 )ρ23 ,ρ1,23 

(λ1 μ1 ) ε1 Λ1 (λ2 μ2 ) ε2 Λ2 || (λ12 μ12 ) ε12 Λ12 ρ12 = ε1 ,ε2 ,Λ1 ,Λ2 ,Λ3 ,Λ12 ,Λ23

× (λ12 μ12 ) ε12 Λ12 (λ3 μ3 ) ε3 Λ3 || (λμ) εΛ ρ12,3 × (λ2 μ2 ) ε2 Λ2 (λ3 μ3 ) ε3 Λ3 || (λ23 μ23 ) ε23 Λ23 ρ23 × (λ1 μ1 ) ε1 Λ1 (λ23 μ23 ) ε23 Λ23 || (λμ) εΛ ρ1,23 U (Λ1 Λ2 ΛΛ3 ; Λ12 Λ23 ) . (3.32) Note that Eq. (3.32) is valid for any (εΛ) that belong to the (λμ) irrep and therefore one can take ε = ε H = (2λ + μ) and Λ = Λ H = μ/2 in practice. Also, in the above ε is fixed and only ε1 and ε2 are free; all other ε’s are fixed by their additive property. Although we are not giving explicitly, it is easy to recognize the orthonormal properties of the U -coefficients. However, symmetry properties of U -coefficients can be written only when all the multiplicities are unity. This is because the symmetry property of SU (3) ⊃ SU (2) ⊗ U (1) Wigner coefficients under 1 ↔ 2 interchange is max for (λ1 μ1 ) × (λ2 μ2 ) −→ (λ12 μ12 ) is unity known only when the multiplicity ρ12 and this is similar to Eq. (3.29); see [22]. This difficulty led to the introduction of a new type of U -coefficient called Z -coefficient in [23] and this will transform (III) to (IV) in Fig. 3.4 giving (I I I ) =



(λ13 μ13 ),ρ13 ,ρ13,2  Z (λ2 μ2 )(λ1 μ1 )(λμ)(λ3 μ3 ) ; (λ12 μ12 )ρ12 ,ρ12,3 (λ13 μ13 )ρ13 ,ρ13,2 (I V ) .

(3.33)

The Z -coefficient in terms of SU (3) ⊃ SU (2) ⊗ U (1) reduced Wigner coefficients is given by

3.3 SU (3) Wigner and Racah Coefficients Fig. 3.3 (I) and (II) showing three coupled states in two different orderings. Transformation of (I) to (II) is given by the SU (3) U -coefficients. See text for details

Fig. 3.4 (III) and (IV) showing three coupled states in two different orderings. Transformation of (III) to (IV) is given by the SU (3) Z -coefficients. See text for details

57

58

3 SU(3) Wigner–Racah Algebra I

 Z (λ2 μ2 ) (λ1 μ1 ) (λμ) (λ3 μ3 ) ; (λ12 μ12 )ρ12 ,ρ12,3 (λ13 μ13 )ρ13 ,ρ13,2 

(λ1 μ1 ) ε1 Λ1 (λ2 μ2 ) ε2 Λ2 || (λ12 μ12 ) ε12 Λ12 ρ12 = ε1 ,ε2 ,Λ1 ,Λ2 ,Λ3 ,Λ12 ,Λ23

× (λ12 μ12 ) ε12 Λ12 (λ3 μ3 ) ε3 Λ3 || (λμ) εΛ ρ12,3 × (λ1 μ1 ) ε1 Λ1 (λ3 μ3 ) ε3 Λ3 || (λ13 μ13 ) ε13 Λ13 ρ13 × (λ13 μ13 ) ε13 Λ13 (λ2 μ2 ) ε2 Λ2 || (λμ) εΛ ρ13,2 × U (Λ2 Λ1 ΛΛ3 ; Λ12 Λ13 ) (−1)Λ1 +Λ−Λ12 −Λ13 .

(3.34)

We will discuss this further in Sect. 4.4 as the Z -coefficients are needed for 9 − SU (3) coefficients and their applications are given in later Chapters. It is important to mention that in some situations, the U -coefficients reduce to angular momentum U -coefficients just as the SU (3) ⊃ SU (2) ⊗ U (1) reduced Wigner coefficients reduce to CG coefficients in some situations. For example, U ((λμ)(Q 1 0)(λ μ )(Q 2 0); (λ μ )(Q0)) = U S O(3)



λ Q 1 λ Q 2 λ Q ; 2 2 2 2 2 2

 ,

(3.35) where λ + 2μ = λ + 2μ + Q 1 and λ + 2μ = λ + 2μ + Q 2 and Q = Q 1 + Q 2 . Similarly, 

(λ1 μ1 )ε1 = ε10 − 3σ1 , Λ1 = σ21 ; (λ2 μ2 )ε2 = ε20 − 3σ2 , Λ2 = σ22 ||  σ1 +σ2 σ 0 (λ, μ) = (λ1 + λ2 − 2x, μ1 + 3σ,  μ2 + x)ε =  Λ = 2 λ= ε −  2 λ1 λ1 λ2 λ2 λ = 2 , σ1 − μ1 − 2 ; 2 , σ2 − μ2 − 2 | 2 , σ − μ − 2 ,

(3.36)

where ε0 = 2λ + 4μ, ε10 = 2λ1 + 4μ1 and ε20 = 2λ2 + 4μ2 . See [24, 25, 29, 39] and references therein for further analytical formulas for SU (3) U -coefficients. Finally, the computer codes in [35] allow one to obtain numerical values for the U -coefficients and also for the Z -coefficients.

3.4 Building Up Principle and General Comments Most important aspect of Eq. (3.32) is, by applying orthonormal properties of the reduced Wigner coefficients, it leads to a building up principle for deriving formulas for SU (3) ⊃ SU (2) ⊗ U (1) reduced Wigner coefficients and also for the U -coefficients (similarly for the Z -coefficients). We have easily

3.4 Building Up Principle and General Comments

59

(λ1 μ1 ) ε1 Λ1 (λ23 μ23 ) ε23 Λ23 || (λμ) εΛ × U ((λ1 μ1 ) (λ2 μ2 ) (λμ) (λ3 μ3 ) ; (λ12 μ12 ) (λ23 μ23 )) = 

(λ1 μ1 ) ε1 Λ1 (λ2 μ2 ) ε2 Λ2 || (λ12 μ12 ) ε12 Λ12

ε2 ,Λ2 ,Λ3 ,Λ12

× (λ12 μ12 ) ε12 Λ12 (λ3 μ3 ) ε3 Λ3 || (λμ) εΛ × (λ2 μ2 ) ε2 Λ2 (λ3 μ3 ) ε3 Λ3 || (λ23 μ23 ) ε23 Λ23 U (Λ1 Λ2 ΛΛ3 ; Λ12 Λ23 ) . (3.37) Here, we are not showing the multiplicity labels ρ’s for simplicity. Let us consider some examples illustrating the use of Eq. (3.37). we know all the Wigner  Say coefficients for the coupling (λ1 μ1 ) ⊗ (10) −→ λ μ and choosing (λ2 μ2 ) = (λ3 μ3 ) = (10) together with the fact that (10) ⊗ (10) gives (20) will allow us to calculate (λ1 μ1 ) − −(20) − − (λμ) − − and also the U -coefficient U (−(10) − (10); −(20)). Similarly (10) ⊗ (01) −→ (11) and the symmetry properties  of SU (3) ⊃ SU (2) ⊗ U (1) Wigner coefficients will give (λ1 μ1 ) ⊗ (11) −→ λ μ Wigner coefficients. Now we can proceed to calculate (λ1 μ1 ) ⊗ (30) −→ λ μ Wigner coefficients etc. Using the orthonormal properties of the reduced Wigner coefficients, we will get U 2 and the phase convention for the SU (3) ⊃ SU (2) ⊗ U (1) Wigner coefficients and Eq. (3.37) will fix the sign of U . Note that in applying Eq. (3.37), we have the freedom in choosing (λ2 μ2 ), (λ3 μ3 ), and (λ12 μ12 ). As a specific example, let us consider deriving a formula for the U -coefficient U (λ1 , 0)(λ2 , 0)(λ1 + λ2 , 0)(11); (λ1 + λ2 , 0)(λ2 , 0)). Using Eq. (3.37) we can write the following equation,

(λ1 , 0)2λ1 , 0 ; (λ2 , 0)2λ2 , 0 || (λ1 + λ2 , 0)2λ1 + λ2 , 0 U (λ 1 , 0)(λ2 , 0)(λ1 + λ2 , 0)(11); (λ1 + λ2 , 0)(λ2 , 0)) 

(λ1 , 0)2λ1 , 0 ; (λ2 , 0)ε2 , Λ2 || (λ1 + λ2 , 0)2λ1 + ε2 , Λ2 = ε0 ,Λ0

(3.38)

(λ1 + λ2 , 0)2λ1 + ε2 , Λ2 ; (11)ε0 Λ0 || (λ1 + λ2 , 0)2λ1 + 2λ2 , 0

(λ2 , 0)ε2 Λ2 ; (11)ε0 Λ0 || (λ2 , 0)2λ2 , 0 U (0, Λ2 , 0, Λ0 ; Λ2 , 0) SU (2) . In the above Λ2 = Λ0 and ε2 = 2λ2 − ε0 . Also, it is easy to see that only (ε0 , Λ0 ) = (3, 21 ) and (0, 0) are allowed. Therefore, the sum in Eq. (3.38) will be

(λ1 , 0)2λ1 , 0 ; (λ2 , 0)2λ2 , 0 || (λ1 + λ2 , 0)2λ1 + 2λ2 , 0 × (λ1 + λ2 , 0)2λ1 + 2λ2 , 0 ; (11)0, 0 || (λ1 + λ2 , 0)2λ1 + 2λ2 , 0 ×  (λ2 , 0)2λ2 , 0 ; (11)0, 0 || (λ2 , 0)2λ2 , 0 U (0, 0, 0, 0; 0, 0) SU (2)  + (λ1 , 0)2λ1 , 0 ; (λ2 , 0)2λ2 − 3, 21 || (λ1 + λ2 , 0)2λ1 + 2λ2 − 3, 21  × (λ1 + λ2 , 0)2λ1 + 2λ2 − 3, 21 ; (11)3, 21 || (λ  1 + λ2 , 0)2λ1 + 2λ 2 , 0 × (λ2 , 0)2λ2 − 3, 21 ; (11)3, 21 || (λ2 , 0)2λ2 , 0 U 0, 21 , 0, 21 ; 21 , 0 SU (2) . Now, using Eq. (3.19) and the formulas

60

3 SU(3) Wigner–Racah Algebra I

&

(λ0)2λ, 0 ; (11)00 || (λ0)2λ, 0 = −λ/ λ2 + 3λ , &   1 1 (λ0)2λ − 3, 2 ; (11)3, 2 || (λ0)2λ, 0 = (3λ)/(λ2 + 3λ) ,

(3.39)

we will obtain finally, U (λ1 , 0)(λ2 , 0)(λ1 + λ2 , 0)(11); (λ1 + λ2 , 0)(λ2 , 0)) λ2 (λ1 + λ2 ) + 3λ2 = '  . λ22 + 3λ2 (λ1 + λ2 )2 + 3 (λ1 + λ2 )

(3.40)

This formula is useful in some examples considered in Chap. 11 ahead. In the applications of the SU (3) symmetry, in particular using shell model or various interacting boson and boson–fermion models, analytical formulas for different classes of SU (3) ⊃ S O(3) reduced matrix elements facilitate predictions for many observables; see, for example, [40–46]. The building up principle given by Eq. (3.37) with (K L)’s replacing (εΛ)’s (i.e., using SU (3) ⊃ S O(3) basis states in place of SU (3) ⊃ SU (2) ⊗ U (1) basis states) is very much useful here. Then, 

(λ1 μ1 ) K 1 L 1 (λ23 μ23 ) K 23 L 23 || (λμ) K L ρ

ρ

 ×U (λ 1 μ1 ) (λ2 μ2 ) (λμ) (λ3 μ3 ) ; (λ12 μ12 ))ρ12 ρ12,3 (λ23 μ23 )ρ23 ρ 

(λ1 μ1 ) K 1 L 1 (λ2 μ2 ) K 2 L 2 || (λ12 μ12 ) K 12 L 12 ρ12 = K 2 ,L 2 ,K 12 ,L 12 ,K 3 ,L 3

× (λ12 μ12 ) K 12 L 12 (λ3 μ3 ) K 3 L 3 || (λμ) K L ρ12,3 × (λ2 μ2 ) K 2 L 2 (λ3 μ3 ) K 3 L 3 || (λ23 μ23 ) K 23 L 23 ρ23 U (L 1 L 2 L L 3 ; L 12 L 23 ) . (3.41) Using this and starting with the formulas in Eq. (3.30), it is possible to derive analytical results for the SU (3) ⊃ O(3) reduced Wigner coefficients  (λ0)L (20) || (λ μ )K  L  . These follow by choosing (λ1 μ1 ) = (λ0), (λ2 μ2 ) = (10), (λ3 μ3 ) = (10), and (λ23 μ23 ) = (20) in Eq. (3.41). Proceeding further, it is possible to derive formulas for (λ0)L (30) || −− and then (λ0)L (40) || −− and so on. Similarly, using the symmetry properties it is possible to derive formulas for − − (0λ) || −− with λ ≤ 4 and also − − (λλ) || −− type of reduced Wigner coefficients. Some of these formulas are given in Tables 4.1 and 4.2 ahead and they are used in IBFM and sdgIBM studies [42, 43]. For illustration, let us consider the reduced Wigner coefficients (λ0)L = 0 (30) || ((λ + 3, 0) with λ even. Firstly, Eq. (3.41) gives  (λ0)0 (30) || (λ μ )k U ((λ0)(20)(λ μ )(10); (30)(λ12 μ12 ))



=

 k12 ,1

 

(λ0)0 (20)1 || (λ12 μ12 )k12 1 (λ12 μ12 )k12 1 (10)1 || (λ μ )k (3.42)

× (20)1 (10)1 || (30) .

3.4 Building Up Principle and General Comments

61

With (λ μ ) = (λ + 3, 0) and (λ12 μ12 ) = (λ + 2, 0) in Eq. (3.42) will give the relations (in the following the U -coefficient in Eq. (3.42) is called just U ),

(λ0)0 (30)3 || (λ + 3, 0)3 U

(λ + 2, 0)2 (10)1 || (λ + 3, 0)3 (20)2 (10)1 || (30)3 = (λ0)0 (20)2 || (λ + 2, 0)2     2(λ + 3)(λ + 5) 3(λ + 7) = [1] , 15(λ + 1)(λ + 2) 7(λ + 3) and

(λ0)0 (30)1 || (λ + 3, 0)1 U = (λ0)0 (20)0 || (λ + 2, 0)0 (λ + 2, 0)0 (10)1 || (λ + 3, 0)1 (20)0 (10)1 || (30)1

(λ + 2, 0)2 (10)1 || (λ + 3, 0)1 (20)2 (10)1 || (30)1 + (λ0)0 (20)2 || (λ + 2, 0)2     (  (λ + 2)(λ + 3) (λ + 5) 5 = 3(λ + 1)(λ + 2) 3(λ + 3) 9     (  2(λ + 3)(λ + 5) 2(λ + 2) 4 + − − . 15(λ + 1)(λ + 2) 3(λ + 3) 9

(3.43) In Eq. (3.43), formulas for the SU (3) ⊃ S O(3) reduced Wigner coefficients are also shown and they are from [21]. Now, using the sum rule 

| (λ + 3, 0) (03) || (λ0)0 |2 = 1

 2 (30) || (λ + 3, 0) | will give U 2 =1. Then, Eq. (3.43) gives the formulas for | (λ0)0     with  = 1, 3. The above procedure extends to all the (λ0)0 (30) || (λ , μ ) and the final formulas are given in Table 4.1 ahead (it is also possible to fix the phases of the reduced Wigner coefficients [42]). Finally, some comments about the analytical formulas for SU ((3) ⊃ S O(3) Wigner coefficients are asfollows:  (i) Vergados [21] tabulated analytical formu las for the (λμ) (λ0 μ0 ) || λ μ coefficients with (λ0 μ0 ) = (10), (01), (11), (20), (02) with μ, μ ≤ 4 (almost all the cases with this restriction are covered) and for (λ0 μ0 ) = (40) with μ, μ = 0. (ii) Using Bargmann space expansion of oscillator functions, Hecht and Suzuki [47] derived formulas for special type of Wigner coefficients (λ1 0) − (λ2 0) − || (λμ) L = 0 and (λ1 0) − (0λ2 ) − || (λμ) L = 0 . (iii) Sharp et al. [30, 48] used a polynomial basis for |(λμ)K L M states and derived analytical results for a class of SU (3) ⊃ S O(3) Wigner coefficients and most of these are also covered by Vergados [21]. (iv) Biedenharn and Louck [25–27] derived expressions for a more general class of SU (3) ⊃ S O(3) Wigner coefficients using the so-called pattern calculus (however, most of these expressions are unwieldy). (v) Using vector-coherent states, Rowe [49] derived formulas for some multiplicity-free reduced Wigner coefficients. (vi) deSwart [50] derived formulas for the coefficients that are useful in elementary particle physics. (vii) Recently, Feng Pan and collaborator developed a new projection method, starting with the canonical (Gel’fand and Zetlin) basis for SU (3), for deriving analytical results and also for numerical calculations of SU (3) ⊃ S O(3) reduced Wigner coefficients [51]. (viii) Using the building up principle given by Eq. (3.41), formulas for applying various interacting

62

3 SU(3) Wigner–Racah Algebra I

boson and boson–fermion models are derived in several papers; see for example [42, 43] and Chaps. 7–9 and 11.

3.5 Summary In this Chapter, described are several different methods for obtaining SU (3) irreps in a given irrep of the oscillator SGA U ((η + 1)(η + 2)/2) with oscillator shell number η. Using these, analytical results for low-lying irreps in many situations are obtained and several of these are given in Sect. 3.2. Similarly, complete numerical results are obtained in many examples and some of them are given in Tables 3.1 to 3.4. Going beyond this, introduced and described in some detail in Sects. 3.3 and 3.4 are SU (3) ⊃ SU (2) ⊗ U (1) and SU (3) ⊃ S O(3) reduced Wigner coefficients. Continuing this, introduced are also SU (3) Racah or U coefficients and the closely related Z -coefficients. Further details of SU (3) Wigner–Racah algebra will be given in the next two chapters.

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15. J. Bardeen, E. Feenberg, Symmetry effects in the spacing of nuclear energy levels. Phys. Rev. 54, 809–818 (1938) 16. C. Bloch, Theory of nuclear level density. Phys. Rev. 93, 1094–1106 (1954) 17. Y. Akiyama, sdg boson model in the SU (3) scheme. Nucl. Phys. A 433, 369–382 (1985) 18. A. Martinou, D. Bonatsos, N. Minkov, I. E. Assimakis, S. Sarantopoulou, S. Peroulis, Highest weight SU(3) irreducible representations for nuclei with shape coexistence. arXiv:1810.11870 [nucl-th] (2018) 19. A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton, New Jersey, 1974) 20. K.T. Hecht, SU3 recoupling and fractional parentage in the 2s − 1d shell. Nucl. Phys. 62, 1–36 (1965) 21. J.D. Vergados, SU (3) ⊃ R(3) Wigner coefficients in the 2s − 1d shell. Nucl. Phys. A 111, 681–754 (1968) 22. J.P. Draayer, Y. Akiyama, Wigner and Racah coefficients for SU3 . J. Math. Phys. 14, 1904–1912 (1973) 23. D.J. Millener, A Note on recoupling coefficients for SU (3). J. Math. Phys. 19, 1513–1514 (1978) 24. K.T. Hecht, Alpha and 8 Be cluster amplitudes and core excitations in s − d shell nuclei. Nucl. Phys. A 283, 223–252 (1977) 25. L.C. Biedenharn, J.D. Louck, E. Chacon, M. Ciftan, On the structure of the canonical tensor operators in the unitary groups. I. An extension of the pattern calculus rules and the canonical splitting in U (3). J. Math. Phys. 13,1957–1984 (1972) 26. L.C. Biedenharn, J.D. Louck, On the structure of the canonical tensor operators in the unitary groups. II. The tensor operators in U (3) characterized by maximal null space. J. Math. Phys. 13, 1985–2001 (1972) 27. J.D. Louck, L.C. Biedenharn, On the structure of the canonical tensor operators in the unitary groups. III. Further developments of the boson polynomials and their implications. J. Math. Phys. 14, 1336–1357 (1973) 28. M. Moshinsky, Wigner coefficients for the SU3 group and some applications. Rev. Mod. Phys. 34, 813–828 (1962) 29. K.T. Hecht, The use of SU (3) in the elimination of spurious center of mass states. Nucl. Phys. A 170, 34–54 (1971) 30. R.T. Sharp, H.C. Van Baeyer, S.C. Pieper, Polynomial bases and Wigner coefficients for SU (3) ⊃ R3 . Nucl. Phys. A 127, 513–524 (1969) 31. T. Sebe, A note on the SU3 coupling coefficients in the 2s–1d shell. Nucl. Phys. A 109, 65–80 (1968) 32. G. Racah, Theory of complex spectra. IV. Phys. Rev. 76, 1352–1365 (1949) 33. P.H. Butler, Coupling coefficients and tensor operators for chains of groups. Phil. Trans. R. Soc. Lond. 277, 545–585 (1975) 34. V.K.B. Kota, Single particle SU(3) parentage coefficients. Pramana-J. Phys. 9, 129–140 (1977) 35. Y. Akiyama, J.P. Draayer, A user’s guide to fortran programs for Wigner and Racah coefficients of SU3 . Comp. Phys. Commun. 5, 405–415 (1973) 36. M. Moshinsky, J. Patera, R.T. Sharp, P. Winternitz, Every thing you always wanted to know about SU (3) ⊃ O(3). Ann. Phys. (N.Y.) 95, 139–169 (1975) 37. J.P. Elliott, Collective motion in the nuclear shell model II. The introduction of intrinsic wavefunctions. Proc. Roy. Soc. (London) A245, 562–581 (1958) 38. J.P. Draayer, S.A. Williams, Coupling coefficients and matrix elements of arbitrary tensors in the Elliott projected angular momentum basis. Nucl. Phys. A 129, 647–665 (1969) 39. X. Li, J. Pladus, Relationship between S N and U (n) isoscalar factors an higher order U (n) invariants. J. Math. Phys. 31, 1589–1599 (1990) 40. F. Iachello, A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987) 41. F. Iachello, P. Van Isacker, The Interacting Boson-Fermion Model (Cambridge University Press, Cambridge, 1991)

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42. R. Bijker, V.K.B. Kota, Interacting boson fermion model of collective states : the SU (3) ⊗ U (2) limit. Ann. Phys. (N.Y.) 187, 148–197 (1988) 43. Y.D. Devi, V.K.B. Kota, sdg interacting boson model : hexadecupole degree of freedom in nuclear structure. Pramana-J. Phys. 39, 413–491 (1992) 44. G.L. Long, T.Y. Shen, H.Y. Ji, E.G. Zhao, Analytical expressions for electromagnetic transition rates in the SU (3) limit of the sdp f interacting boson model. Phys. Rev. C 57, 2301–2307 (1998) 45. H.Y. Ji, G.L. Long, E.G. Zhao, S.W. Xu, Studies of the electric dipole transition of deformed rare-earth nuclei. Nucl. Phys. A 658, 197–216 (1999) 46. J.P. Draayer, G. Rosensteel, U (3) → R(3) integrity-basis spectroscopy. Nucl. Phys. A 439, 61–85 (1985) 47. K.T. Hacht, Y. Suzuki, Some special SU (3) ⊃ R(3) Wigner coefficients and their applications. J. Math. Phys. 24, 785–792 (1983) 48. H.C. Van Baeyer, R.T. Sharp, Clebsch-Gordon coefficients for SU (3) ⊃ R3 in different bases. Nucl. Phys. A 140, 118–128 (1970) 49. D.J. Rowe, J. Repka, An algebraic algorithm for calculating Clebsch-Gordon coefficients; applications to SU (2) and SU (3). J. Math. Phys. 38, 4363–4388 (1997) 50. J.J. deSwart,The octet model and its Clebsch-Gordon coefficients. Rev. Mod. Phys. 35, 916–939 (1963); Erratum 37, 326 (1965) 51. F. Pan, S. Yuan, K.D. Launey, J.P. Draayer, A new procedure for constructing basis vectors of SU (3) ⊃ S O(3). Nucl. Phys. A 952, 70–99 (2016)

Chapter 4

SU(3) Wigner–Racah Algebra II

4.1 Introduction In the previous chapter, first given are the methods to obtain SU (3) irreps for a given number of particles (fermions or bosons) occupying an oscillator shell η and then defined and discussed in some detail the two classes of SU (3) reduced Wigner coefficients and similarly the Racah or U - (and Z -) coefficients. The next step for obtaining matrix elements of an operator that represents an observable is to (i) carry out tensorial decomposition with respect to one of the SU (3) subalgebras and (ii) define and calculate SU (3) coefficients of fractional parentage (CFP). In between often needed are also the 9 − SU (3) coefficients. We will present these three in the next three sections. These along with the Wigner–Eckart theorem extended to SU (3) will allow us to calculate many-particle matrix elements of any operator. Finally, we will also introduce SU (3) D-functions and derive these in some examples that have applications.

4.2 SU(3) Tensorial Decomposition and Wigner–Eckart Theorem μ

Λ

An operator T is a SU (3) tensor TM(λΛ0 0 )ε0 0 w.r.t. SU (3) ⊃ SU (2) ⊗ U (1) if it 0 satisfies the following commutation relations: 

 μ Λ Ai j , TM(λΛ0 0 )ε0 0  0  (λ μ )ε Λ = (λ0 μ0 ) ε0 Λ0 MΛ 0 | Ai j | (λ0 μ0 ) ε0 Λ0 MΛ0 TM 0 0 0 0 . ε0 Λ0 MΛ 0

(4.1)

Λ0

Note that Ai j are the SU (3) generators given in Eqs. (2.10) and (2.11). Also, action of Ai j on (λ0 μ0 ) will not change (λ0 μ0 ). In practical applications, more useful © Springer Nature Singapore Pte Ltd. 2020 V. K. B. Kota, SU(3) Symmetry in Atomic Nuclei, https://doi.org/10.1007/978-981-15-3603-8_4

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4 SU (3) Wigner–Racah Algebra II

66

is the tensorial decomposition w.r.t. SU (3) ⊃ S O(3) algebra. An operator T is a μ L SU (3) ⊃ S O(3) tensor TM(λ00 0 )k0 0 if it satisfies the commutation relations,  μ L G qP , TM(λ00 0 )k0 0    (λ μ )k  L  = (λ0 μ0 ) k0 L 0 M0 | G qP | (λ0 μ0 ) k0 L 0 M0 TM 0 0 0 0 , 

(4.2)

0

k0 L 0 (M0 )

where G qP = L q1 or Q q2 . It is useful to recognize that the eight generators of SU (3) are of T (11)α type, i.e., they will transform as the (11) irrep that is eight dimensional. Also, it is easy to see that L q1 = Tq(11)K =1,L=1 and Q q2 = Tq(11)K =1,L=2 . The real importance of the tensorial decomposition of an operator are: (i) we can couple two operators of definite tensor rank by using SU (3) reduced Wigner coefficients and S O(3) Wigner coefficients; (ii) we can apply Wigner–Eckart theorem for the matrix elements of operators with definite tensorial rank and also decompose the matrix elements into a SU (3) (and higher) part and a part that contains only the subgroup labels (εΛMΛ ) or (K L M). Application of the Wigner–Eckart theorem gives [1] 

 γ  (λ μ )α  | T γ0 (λ0 μ0 )α0 | γ (λμ)α      γ (λ μ ) ||| T γ0 (λ0 μ0 ) ||| γ (λμ) ρ =  ρ × (λμ)α (λ0 μ0 ) α0 | (λ μ )α  ρ .

(4.3)

Here, (α, α0 , α  ) are SU (3) subgroup labels (εΛMΛ ) or (K L M) and (γ , γ0 , γ  ) are the extra labels required to specify the states completely. Similarly, the triple barred matrix element in Eq. (4.3) is the SU (3) reduced matrix element (essentially CFP). We will now consider some examples for tensorial decomposition.

4.2.1 Examples from sd and sd gIBM In U ((η + 1)(η + 2)/2) ⊃ SU (3) ⊃ S O(3) chain with identical bosons (η = 2 is † sdIBM and η = 4 is sdgIBM), single boson creation operators b ,m with   † † b0† → s † , b2m → dm† , b4m → gm† (η0)k =0,

. As only (η0) × (0η) gives (00) irrep, the b m operator must behave as Tm 0 (0η) b(0η) m = Tm = be a tensor of the type T (0η) . Following this, introduced is φ+ −m −m b(η0) ,−m . The phase factor (−1) follows from angular momentum (−1) algebra (see Appendix A and

[6, 7]) and the phase factor φ is determined as follows. † b must be a T (η0)(0η)(00)K 0 =0,L 0 =0 tensor. Firstly, the number operator ,m b m m Then,

4.2 SU (3) Tensorial Decomposition and Wigner–Eckart Theorem

=

67

(η0) (0η) = ,m (−1)φ+ −m bm b˜−m (−1) (η0) (0η) || (00)0

† ,m b m b m φ ,m

˜ [(η0)(0η)](00)0 . × (−1) −m  m − m | 00 (b† b) Now using the first symmetry property in Eq. (3.29), it is easy to see that (−1)φ = (−1)η . Thus, † −→ Tm(η0) , b˜ m −→ Tm(0η) = (−1)η+ −m b ,−m . b m

(4.4)

† Let us consider the number operators n s and n d of sdIBM. As b =0,2 behave as T (20) and b =0,2 behave as T(02) , the SU(3) tensorial nature of n s and n d operators is (20) ⊗ (02) −→ (λ0 μ0 ) = (00) ⊕ (11) ⊕ (22). However, as they must be rotational scalars, we need (λ0 μ0 ) −→ L 0 = 0. As (11) irrep gives L 0 = 1, 2 only, it is obvious that

(η0)(0η)(λ0 λ0 )L 0 =0 . For (λ0 μ0 ) = (22), K 0 must be K 0 = 0 as K 0 = n = λ0 =0,2 aλ0 TM=0 2 cannot give L 0 = 0. We can multiply two tensor operators using the SU(3) reduced Wigner coefficients giving † n s = s00 s00 = T (η0)0 T (0η)0 (−1)η = (−1)η

nd =

 m

† d2m d2m = (−1)η



(η0)0 (0η)0 | (λ0 λ0 ) 0 (s † s˜ )(λ0 λ0 )0

λ0 =0,2  √ ˜ (λ0 λ0 )0 . 5 (η0)2 (0η)2 || (λ0 λ0 ) 0 (d † d)

(4.5)

λ0 =0,2

Similarly, one can decompose the number operators in sdgIBM and in other interacting boson models. Also, it is possible to carry out the tensorial decomposition of a general one plus two-body H and general Q q2 and M1 operators in these models (see Chaps. 7 and 8 for examples).

4.2.2 Shell Model Two-Body Interactions Turning to SM examples, let us consider a general one plus two-body H in (2s1d) shell. Firstly, we have U (24) ⊃ [U (6) ⊃ SU (3) ⊃ S O L (3)] ⊗ [SU (4) ⊃ SU S (2) ⊗ SUT (2)] algebra. The one and two-body parts of H will be of the form a † a and a † a † aa, respectively, and they are J T scalars. Also, {1}U (6) → (20) SU (3) , {1} SU (4) → (ST ) = ( 21 , 21 ), {2}U (6) → [(40) + (02)] SU (3) , {12 }U (6) → (21) SU (3) , {2} SU (4) → [(ST ) = (00) + (11)] and {12 } SU (4) → [(ST ) = (10) + (01)]. Antisymmetry w.r.t. U (24) implies that a two-particle (a † a † ) irrep { f p } of U (6) and {F p } of U (4) must be conjugate to each other. Similarly, for two hole (aa) states { f h } and {Fh }, respectively. With these, the tensorial structure of a one-body operator is T {1}{1 }{ f0 }(λμ)k0 ,L 0 =S0 ;{1}{1 }{F0 }(S0 ,T0 =0) . 5

3

4 SU (3) Wigner–Racah Algebra II

68

As the effective interaction is J T scalar, we have L 0 = S0 = 0, 1 and T0 = 0. Note that { f 0 } = {0} + {214 } and {F0 } = {0} + {212 }. Then, { f 0 }(λμ)k0 L 0 = S0 : {F0 }(S0 , T0 = 0) takes values {0}(00)00 : {0}(0, 0), {214 }(11)11 : {212 }(1, 0),

{214 }(22)00 : {0}(0, 0) .

For the two-body part, firstly { f p }{ f h }{F p }{Fh } take values ss = {2}{25 }{12 }{12 }, aa = {12 }{14 }{2}{23 } and sa = {2}{14 }{12 }{23 } (plus as). Also, here L 0 = S0 = 0, 1, 2 and T0 = 0. For ss, the allowed { f 0 }(λμ)k0 L 0 are {0}(00)00, {214 }(11)11, 12, (22)00, 02, 22, {424 }(00)00, (11)11, 12, (22)3 00, 02, 22, (33)11, 12, (44)00, 02, 22, (60)00, 02, (06)00, 02, (41)11, 12, (14)11, 12 . Note that the irrep (22) appears 3 times in {424 }. Similarly, the allowed {F0 }(S0 T0 ) are {0}(00), {212 }(10), and {22 }(00), (20). For aa, the allowed { f 0 }(λμ)k0 L 0 are {0}(00)00, {214 }(11)11, 12, (22)00, 02, 22 {22 12 }(11)11, 12, (22)00, 02, 22, (33)11, 12, (41)11, 12, (14)11, 12, (30)01, (03)01 . Similarly, the allowed {F0 }(S0 T0 ) are {0}(00), {212 }(10), and {422 }(00); (20). Finally, for sa (results for as follow easily from the sa results) the allowed { f 0 }(λμ)k0 L 0 are {214 }(11)11, 12, (22)00, 02, 22, {313 }(11)11, 12, (22)00, 02, 22, (33)11, 12, (41)11, 12, (14)11, 12, (30)2 01, (03)01, (52)01, 22 . Note that the irrep (30) appears 2 times in {313 }. Similarly, the allowed {F0 }(S0 T0 ) are {212 }(10) and {32 2}(10). Having identified the tensorial structures, a given two-body interaction will be a linear combination of all these tensors (with J0 = 0, T0 = 0) and the strength of each of tensor part can be written in terms of U (6) ⊃ SU (3), SU (3) ⊃ S O(3) and SU (4) ⊃ SU (2) ⊗ SU (2) reduced Wigner coefficients [8]. Explicit results for various (2s1d) shell interactions are given in [8, 9]. The procedure described here can be extended to two-body interactions in η = 3 and higher shells.

4.2 SU (3) Tensorial Decomposition and Wigner–Eckart Theorem

69

4.2.3 Analytical Results for Electric Quadrupole Transition Strengths Spectrum of deformed even–even nuclei consists of K π = 0+ ground band with J π = 0+ , 2+ , 4+ , . . . states and excited β (K π = 0+ ), γ (K π = 2+ ) and other bands. Assuming that the ground band belongs to a (λμ) irrep of SU (3), it is of interest to have formulas for quadrupole moments Q 2 (J ) of various states in the ground band and electric quadrupole transition matrix elements B(E2; J → J − 2) along the ground band as these are important observables. Similarly, the (β,γ ) bands can come from a (λ μ ) irrep of SU (3). With SU (3) symmetry, the SU (3) algebra allows one write formulas for Q 2 (J ) and B(E2)’s. Firstly, they are defined by the quadrupole operator Q q2 in Eqs. (2.1) and (2.2). It is important to note that Q q2 is in b2 units where b is the oscillator length constant and for charge quadrupole moments there will be charge (e) factor. Now, Q 2 (J ) and B(E2)’s are defined by   J J 20 | J J    Q 2 (J ) = J J | Q 20 | J J = √ J || Q 2 || J ,  2J + 1 2 5  J − 2 || Q 2 || J  B(E2; J → J − 2) = . 16π (2J + 1)

(4.6)

In the above, the reduced matrix elements are w.r.t. S O(3) generating J . In the following we will ignore spin S (also isospin T ) of the nucleons and they can be put back easily. Thus, J = L in the following. As the Q 2 operator is a Tq(11) =2 tensor (L 1μ is a Tμ(11) =1 tensor) and that the Q · Q is related to C2 of SU (3) via Eq. (2.28), the reduced matrix elements of Q 2 (also L 1 ) in |(λμ)K √ L basis are proportional to a SU (3) ⊃ S O(3) reduced Wigner coefficient times C2 (λμ). Expressions for the reduced matrix elements of Q 2 and L 1 are (note that these operators do not change (λμ)) [10], 

 (λμ) K f L f || L 1 || (λμ) Ki L i C2 (λμ) (λμ)K i L i (11)1 || (λμ)K i L i ρ=1 = δ K f K i δ L f L i (−1)φ 2 3

= L(L + 1)(2L + 1) ,

(4.7)

  (λμ)K i L i (λμ) K f L f || Q 2 ||   = (−1)φ 2 C2 (λμ) (2L f + 1) (λμ) K i L i (11)2 || (λμ)K f L f ρ=1 . Here, C2 (λμ) = λ2 + μ2 + λμ + 3(λ + μ) and φ = 1 for μ = 0 and φ = 0 for μ = 0. For the (λ0) irrep, using the SU (3) ⊃ S O(3) reduced Wigner coefficients given in Table 4.1 will give the formulas (for future use, some additional Wigner coefficients are given in Table 4.2),

4 SU (3) Wigner–Racah Algebra II

70

Table 4.1 Analytical expressions for some SU (3) ⊃ S O(3) reduced Wigner coefficients. Results are for (λi μi ) × (λ0 μ0 ) → (λ f μ f ) with (λ0 μ0 ) = (11) and (30) and they are from [2, 3]. In the formulas, φ = (L 1 − L)/2

L1

1, 2

L

2

L ±2



L1

1, 2

L

2

L ±2

(λμ)

k

(2N + 3, 0)

0

(2N + 1, 1)

1

(2N − 1, 2)

0

(2N − 3, 3)

(λ0)L 1 ; (11) || (λ0)L, λ − L even  √   2λδ ,2 + 3 L(L + 1) (−1) +1    1 4λ(λ + 3) (2L − 1)(2L + 3)δ ,2 + δ ,1 2 ⎡  3 λ + φL + ⎣

+

5 2

φ 2



λ − φL −

φ 2

+

1 2



2λ(λ + 3)(2L + φ)(2L + φ + 2)

⎡ λ − φL − (−φ) ⎣

φ 2

+

1 2

 λ − φL −

φ 2

+

(2N , 0)0, 0; (30)0, L = 1 || (λμ)k, L = 1  2N + 5 5(2N + 1)  4N − 15(2N + 1)  4N (2N + 3) 15(4N 2 − 1) 0

1

−



32N (N − 1) 15(4N 2 − 1)

3

0

k

(2N + 3, 0)

0

(2N + 1, 1)

1

(2N − 1, 2)

0

(2N , 0)0, 0; (30)0, L = 3 || (λμ)k, L = 3  (2N + 5)(2N + 7) 35(2N + 1)(N + 1)  8(2N + 5) − 35(2N + 1)  3N (2N − 3)2 (2N + 3)(2N + 5) − 35(4N 2 − 1)(N + 1)(4N 2 + 4N − 5)  8N (N − 1)(2N + 3) 7(2N − 1)(4N 2 + 4N − 5)  8(N − 1)(2N + 3)(2N − 5)2 35(4N 2 − 1)(8N 2 − 8N − 15)  32N (N − 1)(N − 2) − 7(2N − 1)(8N 2 − 8N − 15)

1

3

5 2



L+

φ 2

−

1 2



(λ + 1)(λ + 3)(2L + φ)(2L + φ + 2)

(λμ)

(2N − 3, 3)

⎤ 12 ⎦

(λ0)L 1 ; (11) || (λ + 1, 1)L, λ − L even  1 2 (λ − L + 2)(λ + L + 3)(2 − 1)   − 2(λ + 1)(λ + 3) (2L − 1)(2L + 3)δ ,2 + δ ,1

2

2

(L + φ)(L + φ + 1)

L+

φ 2

+

3 2

⎤1

2

⎦

4.2 SU (3) Tensorial Decomposition and Wigner–Eckart Theorem

71

Table 4.2 Analytical expressions for some additional SU (3) ⊃ S O(3) reduced Wigner coefficients. In the table F(a, L) = [(a + L)/2]!{[(a − L)/2]! [(a + L + 1)!]}−1 . Results are from [3–5] 

(4N , 0)0, 0; (40)0, L = 0 || (4N + 4, 0)0, L = 0 = (4N , 0)0, 0; (40)0, L = 2 || (4N + 4, 0)0, L = 2 = (4N , 0)0, 0; (40)0, L = 4 || (4N + 4, 0)0, L = 4 =

 

4N + 5 5(4N + 1) 2(4N + 5)(4N + 7) 35(4N + 1)(2N + 2) 2(4N + 5)(4N + 7)(4N + 9) 315(4N + 1)(2N + 1)(2N + 2)

(2Nν , 0)0L (2Nπ , 0)0L || (2N − 2, 1) 11 =   (2N − 1)L(L + 1) 1/2 (2Nν , 0)0L (2Nπ , 0)0L || (2N , 0)00 ; N = Nπ + Nν 2(2Nπ )(2Nν ) (λπ , 0)0L (λν , 0)0L || (λπ + λν , 0)00 =  1/2 λπ ! λν ! (−1) L [(2L + 1)(λπ + λν + 1)]1/2 (λπ − L)!! (λπ + L + 1)!! (λν − L)!! (λν + L + 1)!! (2n, 0)L (0, 2m)L || (2n − 2ν, 2m − 2ν)0 =  1/2 2n + 2m − 4ν + 2)(2L + 1) (−1)min{n−ν,m−ν} F(2n, L)F(2m, L) [(2ν)!] [(2n + 2m + 2 − 2ν)!] min{n−ν,m−ν}  (n + m − 2ν − )! (2ν + 2 )! F(2ν + 2 , L) × (−1) ! (n − ν − )! (m − ν − )! =0

L (2λ + 3) , 2L + 3 5 6L(L − 1)(λ − L + 2)(λ + L + 1) . B(E2; L → L − 2) = 16π (2L − 1)(2L + 1)

Q 2 (L) = −

(4.8)

Note that here we have used Eq. (4.10) given ahead to convert the SU (3) reduced matrix elements into S O(3) reduced matrix elements. Proceeding further and using SU (3) ⊃ S O(3) reduced Wigner coefficients, it is possible to calculate quadrupole moments and B(E2)’s for β, γ and other bands with a more general (λμ) irrep labeling these bands (also the ground band as this need not belong to a (λ0) irrep for fermion systems). Equation (4.8) is useful in sdIBM and sdgIBM and also in some special situations in SM. These will be discussed in Chaps. 7, 8, and 11.

4.3 SU(3) Fractional Parentage Coefficients Matrix elements of a single creation operator will allow us to calculate the matrix elements of an operator which is a product of creation and destruction operators, by using summations over the intermediate sates and the symmetry properties of b† → b (or a † → a) matrix elements. Thus, the basic problem to be addressed is the evaluation of the many-particle matrix element of a single creation operator. Using the SU (3) tensorial decomposition and the Wigner–Eckart theorem, we can obtain   provided we know the triple barred matrix element  desired matrix element  † the b  for bosons ( a †  for fermions) appearing in Eq. (4.3). As the triple

4 SU (3) Wigner–Racah Algebra II

72

barred matrix element is independent of the subgroup labels, for their evaluation we   can use (λμ)ε H Λ H MΛ H states as the construction of these states is much easier and also the evaluation of the corresponding reduced Wigner coefficients. Let us consider a m boson system and then using the (λμ) h.w state, we have from Eq. (4.3), 

" ! † {m}β f λ f μ f ε f H Λ f H MΛ f H | b(η0)ε | {m − 1}β μ Λ M ε (λ ) i i i i H i H Λ i H 0 Λ0 M Λ 0 "  ! † ||| {m − 1}βi (λi μi ) = {m}β f λ f μ f ||| b(η0)  !   × (λi μi ) εi H Λi H (η0)ε0 Λ0 || λ f μ f ε f H Λ f H Λi H MΛi H Λ0 MΛ0 | Λ f H MΛ f H (4.9) For fermion systems, we need to replace b† by a † . Similarly, it is easy to extend the above equation when the m-particle irrep { f } is more general and in the situations with b† replaced by two or higher particle creation or annihilation operator. Before going further, it is very useful to write down the expression for the standard S O(3) reduced matrix elements in terms of the SU (3) triple barred matrix elements 

 ! { f  }α  λ μ K  L  || T { f0 }α0 (λ0 μ0 )k0 L 0 || { f }α (λμ) K L S O(3)     !  = { f }α  λ μ ||| T { f0 }α0 (λ0 μ0 ) ||| { f }α (λμ) !  √ × 2L  + 1 (λμ) K L (λ0 μ0 ) k0 L 0 || λ μ K  L 

(4.10)

Here, we have dropped the multiplicity label “ρ” for simplicity. We will now illustrate the application of Eq. (4.9)  with examples from sdIBM and sdgIBM. Here, first we need the construction of (λμ)ε H Λ H MΛ H states.

4.3.1 Construction of SU(3) Intrinsic States: IBM Examples Starting with sdIBM, first it is easy to identify that the leading SU (3) irrep for m bosons is (2m, 0) and its h.w. state is    † m |{m}(2m, 0)h.w. = {m}(2m, 0)ε H = 4m, Λ H = MΛ H = 0 = b(200) |0 . (4.11) Note that here all the bosons are in the oscillator orbit (n z , n x , n y ) = (200). Now, acting with the operator O( p, q, r ) (see Sect. 2.7) with p = 2 and q = r = 0 on |(2m, 0)h.w. gives |{m}(2m, 0)ε = 4m − 6, Λ = MΛ = 1  m−2  2 √ m−1     √ † † † † |0 . b(110) b(020) + 2(2m) b(200) (4m)(2m − 2) b(200) = (4.12)

4.3 SU (3) Fractional Parentage Coefficients

73

As this state has two components, we can construct a state orthogonal to this and this new state must be the h.w. state of (2m − 4, 2) irrep in Eq. (3.3) giving     {m}(2m − 4, 2)ε H Λ H MΛ = {m}(2m − 4, 2)ε H = 4m − 6, Λ H = MΛ = 1 H H   #  m−1   #  m−2  2 † † † † 1 |0 . b(200) b(020) + 2m−1 b(200) b(110) = − 2m−2 2m−1 (4.13) Proceeding similarly using the O( p, q, r ) operator acting on both (2m, 0) and (2m − 4, 2) h.w. states, we can construct the h.w. states of (2m − 8, 4) and (2m − 6, 0) irreps in Eq. (3.3); see [7]. These intrinsic states will be much more complex and their basic structure will be clear by taking m −→ ∞ limit (i.e., in large boson number limit). Then we have m  † |(2m, 0)h.wm→∞ = b(200) , m−1    † † |(2m − 4, 2)h.wm→∞ = b(200) b(020) , m−2  2  † † |(2m − 8, 4)h.wm→∞ = b(200) b(020) , m−2 #  2 #     † † † † 1 2 |(2m − 6, 0)h.wm→∞ = b(200) b b b . − (011) (020) (002) 3 3 (4.14) Thus, (2m − 4, 2) generates 1-phonon bands and (2m − 8, 4), (2m − 6, 0) generate two-phonon bands; the phonon structure is different in the case of (2m − !r 6, 0) irrep. The intrinsic states for |(λμ)K  bands can be constructed by using A yx operator acting on |(λμ)h.w.. It should be noted that K = 2 MΛ and for the h.w. state  −1/2 {|MΛ = K /2 + |MΛ = −K /2}. K = 2 MΛmax = μ; |K  = 2(1 + δ K ,0 ) Proceeding just as above, it is easy to construct the h.w. states that correspond to the lowest sdgIBM SU (3) irreps (4m, 0), (4m − 4, 2), (4m − 6, 3), and (4m − 8, 4)α=0,1 ; see Eq. (3.10). Firstly,    † m |{m}(4m, 0)h.w. = {m}(4m, 0)ε H = 8m, Λ H = MΛ H = 0 = b(400) |0 . (4.15) By acting on |{m}(4m, 0)h.w. with the operator O(2, 0, 0) and by constructing an orthogonal (and normalized) state we will obtain |{m}(4m − 4, 2)h.w.. Similarly, by acting with O(3, 0, 0) on |{m}(4m, 0)h.w. with ( p = 3, q = r = 0), and O(1, 0, 0) on |{m}(4m − 4, 2)h.w. we will obtain two states having three components. Then, the state orthogonal (and normalized) to these will be |{m}(4m − 6, 3)h.w.. Similarly, by acting with O(4, 0, 0) on |{m}(4m, 0)h.w., O(2, 0, 0) on |{m}(4m − 4, 2)h.w. and O(1, 0, 0) on |{m}(4m − 6, 3)h.w. we can construct |{m}α = 0, 1 (4m − 8, 4)h.w.. The final results are [11–13],

74

4 SU (3) Wigner–Racah Algebra II

 m † |{m}(4m, 0)h.w. = b(400) |0 ,  !1/2  † m−2  † 2 3 |{m}(4m − 4, 2)h.w. = − 4m−1 b(400) b(310)      m−1 !1/2 † † |0 , b(400) b(220) + 4m−4 4m−1  1/2  m−3  3 † † 3 |{m}(4m − 6, 3)h.w. = b b (400) (310) (2m−1)(m−1) 1/2  m−2     † † † 3(m−2) b(400) b(310) b(220) − (2m−1)(m−1) 1/2  m−1    † † 2(m−2)(m−1) b(400) b(130) |0 , + (2m−1)(m−1)  √  † m−4  † 4 1 |{m}(4m − 8, 4)h.w.α=0 = −3 b(310) 3 b(400) 1/2 {16m 2 −56m+51}       m−3 2 √ † † † b(310) b(220) +2 6(m − 3) b(400) m−2  2   √ † † |0 , b(220) −4 (m − 2)(m − 3) b(400) 1 |{m}(4m − 8, 4)h.w.α=1 = {(16m 2 −56m+51)(2m−3)(4m−3)(4m−5)}1/2   m−4  4 † † b(310) × −12{3(m − 2)(m − 3)}1/2 b(400)  m−3  2   † † † b(310) b(220) +8(m − 3){6(m − 2)}1/2 b(400) m−2  2  † † b(220) +3(8m − 15) b(400) !  † m−2  †   †  b(310) b(130) −(2)1/2 16m 2 − 56m + 51 b(400) m−1    ! † † 1/2 2 b(040) |0 , 16m − 56m + 51 b(400) + {2(m − 1)}

(4.16) Structure of the SUsdg (3) intrinsic states will be much more transparent in the m −→ ∞ limit giving m  † |{m}(4m, 0)h.wm→∞ = b(400) , m−1    † † |{m}(4m − 4, 2)h.wm→∞ = b(400) b(220) , m−1    † † |{m}(4m − 6, 3)h.wm→∞ = b(400) b(130) , m−2  2  † † |{m}α = 0; (4m − 8, 4)h.wm→∞ = b(400) b(220) , m−1    † † |{m}α = 1; (4m − 8, 4)h.wm→∞ = b(400) b(040) .

(4.17)

Thus, (4m − 4, 2), (4m − 6, 3), and (4m − 8, 4)α=1 have one-phonon structure and (4m − 8, 4)α=0 is of two-phonon type. Moreover, the phonon excitations in (4m − 4, 2) (giving K = 0, 2 bands) and (4m − 8, 4)α=0 (giving K = 0, 2, 4 bands) are of quadrupole type. However, the phonon excitation in (4m − 6, 3) (giving K =

4.3 SU (3) Fractional Parentage Coefficients

75

1, 3 bands) and (4m − 8, 4)α=1 (giving K = 0, 2, 4 bands) are of hexadecupole type. See Chap. 7 for further discussion.

4.3.2 SU(3) Intrinsic States: Fermion Examples Procedure described above applies to fermions as well except that for identical fermions with spin degree of freedom, an oscillator orbit can have a maximum of two fermions (if one is spin up, then the other must be spin down). Secondly, if we move a fermion from one orbit to other, there will be a phase factor for odd number of jumps. Let us consider 8 fermions in η = 4 shell with spin S = 0. Then we have U (30) ⊃ [U (15) ⊃ SU (3)] ⊗ SU (2) with SU (2) generating spin S. The U (15) irrep is {24 } for S = 0 and the leading or ground state SU (3) irrep is (18, 4). Other lower irreps are (16, 5), (17, 3), (15, 4), and so on. In terms of the sp orbits  (n z n x n y ), the {24 }(18, 4)h.w. is (with ε H = 40 and Λ H = MΛ H = 2) easy to identify  4  {2 }(18, 4)h.w. = a † 1 a † 1 a † 1 a † 1 |0 . (4.18) (400) ↑↓ (310) ↑↓ (301) ↑↓ (220) ↑↓ 2

2

2

2

    Here, 21 ↑↓ means, one fermion in  21 21 state and a second in  21 − 21 state.  fermion  Now, acting with the operator O(1, 0, 0) or A x z on the {24 }(18, 4)h.w. using the    property Ai j n j n i = n j (n i + 1) n j n i we will obtain the state √   √ A x z {24 }(18, 4)h.w. = 6 φ1 + 12 φ2 ; $ † † † † † 1 a(400) 21 ↑↓ a(310) 21 ↑↓ a(301) 21 ↑ a(220) 21 ↑↓ a(211) 21 ↓ |0 φ1 = , † |0 a† a† a† a† 2 −a(400) 1 ↑↓ (310) 1 ↑↓ (301) 1 ↓ (220) 1 ↑↓ (211) 1 ↑ 2

2

2

2

2

(4.19)

$ † † † † † 1 a(400) 21 ↑↓ a(310) 21 ↑↓ a(301) 21 ↑↓ a(220) 21 ↑ a(130) 21 ↓ |0 φ2 = . † |0 a† a† a† a† 2 −a(400) 1 ↑↓ (310) 1 ↑↓ (301) 1 ↑↓ (220) 1 ↓ (130) 1 ↑ 2

2

2

2

2

  The state orthogonal to this state will be clearly {24 }(16, 5)h.w. and it is  4  {2 }(16, 5)h.w. =



2 φ1 − 3



1 φ2 3

(4.20)

and here correctly ε H = 37 and Λ H = MΛ H = 25 . We can proceed further by acting with O( p, q, r ) on (18, 4) and (16, 5) to get the h.w. state of the next irrep and so in. This procedure also extends to fermions with spin–isospin SU (4) symmetry and it is converted into a computer code in [14, 15].

4 SU (3) Wigner–Racah Algebra II

76

4.3.3 Triple Barred SU(3) Reduced Matrix Elements Using the m and m + 1 particle SU (3) h.w states, the known analytical expression for the SU (3) ⊃ SU (2) ⊗ U (1) reduced Wigner coefficients given by Eqs. (3.20) and (3.22) and the well-known angular momentum algebra (see Appendix A), it is easy to derive expressions for the SU (3) triple barred matrix elements of a creation (or annihilation) operator by applying Eq. (4.9). For example, some sd and sdg IBM limit triple barred matrix elements for b† operator follow from the SU (3) h.w. sates given in the Sect. 4.3.1 and they are    {N    {N    {N    {N    {N    {N    {N

"2  † + 1}(2N + 2, 0) ||| b(20) ||| {N }(2N , 0)  = (N + 1) , "2sd  † + 1}(2N − 2, 2) ||| b(20) ||| {N }(2N , 0)  = 2N2N−1 , "2sd  † + 1}(4N + 4, 0) ||| b(40) ||| {N }(4N , 0)  = (N + 1) , "2 sdg  † +2 + 1}(4N , 2) ||| b(40) ||| {N }(4N , 0)  = 4N , 4N −1 sdg "2  † −2)(4N +1) + 1}(4N − 2, 3) ||| b(40) ||| {N }(4N , 0)  = (2N , (4N −1)(2N −1) sdg "2  † + 1}α = 0(4N − 4, 4) ||| b(40) ||| {N }(4N , 0)  = 0 "2sdg  † 2N (16N 2 −24N +11) + 1}α = 1(4N − 4, 4) ||| b(40) ||| {N }(4N , 0)  = (2N . −1)(4N −1)(4N −3) sdg

(4.21) It is useful to note that the above triple barred matrix elements of the b† are related to the more conventional SU (3) coefficients of fractional parentage (CFP) or SU (3) isoscalar factors by the relation, {m}(ηm,  0) {1}(η0) |}{m + 1}α (λμ) " † −1 = (m + 1) {m + 1}α (λμ) ||| b(η0) ||| {m}(ηm, 0) ,

(4.22)

where for example η = 2 for sdIBM and 4 for sdgIBM. Just as above, using h.w. states of SU (3) irreps for fermion systems, the one-particle CFP or reduced matrix elements of the a † operator can be calculated. This can be extended to products of a † and a operators such as ai† a j that give one-body operators, ai† a †j ak al that give two-body Hamiltonian operators, a † a † type operators representing two-particle transfer, a † a † a † a † type operators representing α transfer operator and so on. For these, computer codes are developed initially by Braunschweig [14, 15] but a more general and powerful code is due to Bahri and Draayer [16].

4.4 9 − SU (3) Coefficients

77

4.4 9 − SU(3) Coefficients In angular momentum algebra the role of 9 − j coefficients is well known; see Appendix A and [6]. The 9 − SU (3) (or 9 − (λμ)) coefficients serve the same purpose as the 9 − j coefficients and this is briefly described here. The 9 − SU (3) coefficients are defined by   (λ1 μ1 )(λ2 μ2 )(λ12 μ12 )ρ ; (λ3 μ3 )(λ4 μ4 )(λ34 μ34 )ρ ; (λ12 μ12 )(λ34 μ34 )(λμ)ρ , α 12 34 ⎧ ⎫ ⎪ (λ1 μ1 ) (λ2 μ2 ) (λ12 μ12 ) ρ12 ⎪ ⎪ ⎪ ⎨ ⎬  (λ3 μ3 ) (λ4 μ4 ) (λ34 μ34 ) ρ34 = χ  ⎪ (λ13 μ13 ) (λ24 μ24 ) (λμ) ρ ⎪ ⎪ ⎪ ⎩ ⎭ λ13 , μ13 , λ24 , μ24 ρ13 ρ24 ρ ρ13 , ρ24 , ρ    × (λ1 μ1 )(λ3 μ3 )(λ13 μ13 )ρ13 ; (λ2 μ2 )(λ4 μ4 )(λ24 μ24 )ρ24 ; (λ13 μ13 )(λ24 μ24 )(λμ)ρ  , α .

(4.23) ⎧ ⎫ ⎨− − −⎬ The 9 − (λμ) coefficient χ − − − is independent, as the case with SU (3) U ⎩ ⎭ −−− coefficients, of the subalgebra labels α. The meaning of the multiplicity labels ρ’s in Eq. (4.23) will be clear by writing down the 9 − (λμ) coefficients in terms of SU (3) reduced Wigner coefficients. In terms of the SU (3) ⊃ SU (2) ⊗ U (1) reduced Wigner coefficients (we can use equivalently SU (3) ⊃ S O(3) Wigner Coefficients) we have (for fixed εΛ) for example, ⎫ ⎧ (λ1 μ1 ) (λ2 μ2 ) (λ12 μ12 ) ρ12 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ (λ3 μ3 ) (λ4 μ4 ) (λ34 μ34 ) ρ34 = χ (λ13 μ13 ) (λ24 μ24 ) (λμ) ρ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ρ24 ρ ρ13

 ε1 ε2 ε3 ε4 (ε12 ε24 ε34 ε13 ) , Λ1 Λ2 Λ3 Λ4 Λ12 Λ34 Λ13 Λ24

(λ1 μ1 ) ε1 Λ1 (λ2 μ2 ) ε2 Λ2 || (λ12 μ12 ) ε12 Λ12  ρ12 × (λ3 μ3 ) ε3 Λ3 (λ4 μ4 ) ε4 Λ4 || (λ34 μ34 ) ε34 Λ34  ρ34 × (λ12 μ12 ) ε12 Λ12 (λ34 μ34 ) ε34 Λ34 || (λμ) εΛ ρ  × (λ1 μ1 ) ε1 Λ1 (λ3 μ3 ) ε3 Λ3 || (λ13 μ13 ) ε13 Λ13  ρ13 × (λ2 μ2 ) ε2 Λ2 (λ4 μ4 ) ε4 Λ4 || (λ24 μ24 ) ε24 Λ24  ρ24 × (λ13 μ13 ) ε13 Λ13 (λ24 μ24 ) ε24 Λ24

⎧ ⎫ ⎨ Λ1 Λ2 Λ12 ⎬ || (λμ) εΛ ρ χ Λ3 Λ4 Λ34 . ⎩ ⎭ Λ13 Λ24 Λ

(4.24)

4 SU (3) Wigner–Racah Algebra II

78 Fig. 4.1 a and b showing four coupled states in two different orderings. Transformation of (a) to (b) is given by the 9 − SU (3) coefficients. See text for details

Here, the last χ {− − −} is the 9 − j coefficient. Because of their position, we call ρ12 , ρ34 and ρ as column multiplicities and ρ13 , ρ24 , and ρ  as row multiplicities. We can express the 9 − SU (3) coefficients also in terms of the SU (3) ⊃ S O(3) quantum numbers. The 9 − SU (3) coefficient in Eq. (4.23) corresponds to the transformation shown in Fig. 4.1 (in the figure the α label is not shown). It is seen easily from this figure that the 9 − SU (3) coefficient reduces to a U -coefficient if the SU (3) representations in the second or fourth positions equals (00) irrep. Then we have [17] ⎫ ⎧ (λ1 μ1 ) (00) (λ1 μ1 ) ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ (λ3 μ3 ) (λ4 μ4 ) (λ34 μ34 ) ρ34 χ (λ13 μ13 ) (λ4 μ4 ) (λμ) ρ ⎪ ⎪ ⎪ ⎪ (4.25) ⎭ ⎩ ρ ρ13 = U (λ1 μ1 ) (λ3 μ3 ) (λμ) (λ4 μ4 ) ; (λ13 μ13 )ρ13 ρ (λ34 μ34 )ρ34 ρ 

!

and ⎫ ⎧ (λ1 μ1 ) (λ2 μ2 ) (λ12 μ12 ) ρ12 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ (00) (λ4 μ4 ) (λ4 μ4 ) χ (λ1 μ1 ) (λ24 μ24 ) (λμ) ρ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ρ ρ24 ! = U (λ1 μ1 ) (λ2 μ2 ) (λμ) (λ4 μ4 ) ; (λ12 μ12 )ρ12 ρ  (λ24 μ24 )ρ24 ρ .

(4.26)

4.4 9 − SU (3) Coefficients

79

If the SU (3) representation in the 5th position is zero, we can convert the corresponding 9 − SU (3) coefficient to the 9 − SU (3) coefficient given in Eq. (4.25) only by interchanging the first two rows (or first two columns). However, this interchange of rows is allowed only if the maximum multiplicities corresponding to the row multiplicities are equal to unity. Similarly if we want to interchange the first two columns, the maximum multiplicities corresponding to the two column multiplicities should be one. This is because the phase factor for this interchange of SU (3) representations in the SU (3) reduced Wigner coefficient is not known in general; see Eq. (3.29) and Ref. [18]. With multiplicities unity, the interchange of first two rows or first two columns amounts to a phase change φ where φ = i (λi + μi ) with the summation over all the SU (3) representations appearing in the 9 − SU (3) coefficient. For example if ρ  = 1 and ρ13 = 1, ⎧ ⎧ ⎫ ⎫ (λ3 μ3 ) ⎪ (λ1 μ1 ) (λ2 μ2 ) (λ12 μ12 ) ρ12 ⎪ (λ3 μ3 ) (00) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ ⎬ (λ3 μ3 ) (00) (λ3 μ3 ) (λ1 μ1 ) (λ2 μ2 ) (λ12 μ12 ) ρ12 χ = (−1)φ χ . (4.27) (λμ) ρ ⎪ (λ μ ) (λ2 μ2 ) (λμ) ρ ⎪ (λ μ ) (λ2 μ2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 13 13 ⎩ 13 13 ⎭ ⎭   ρ13 = 1 ρ13 = 1 ρ =1 ρ =1

Unlike the 9 − j coefficients, in general it is not possible to write a 9 − (λμ) coefficient in terms of the SU (3) U -coefficients alone because the phase factor for interchanging rows or columns is not known when multiplicities are present. It is here the closely related Z -coefficient defined by Eq. (3.33) will be useful. Let us add that the Z -coefficient reduces to a U -coefficient within a phase factor when some of the multiplicities are unity as in the following example:   max = 1, ρ max Z SU (3) (λ2 μ2 ) (λ1 μ1 ) (λμ) (λ3 μ3 ) , (λ12 μ12 ) ρ12 12,3 (λ13 μ13 ) ρ13 , ρ13,2 = 1 1 +λ12 +μ12 +λ+μ+λ13 +μ13 = (−1)λ1 +μ 

 max = 1, ρ max × U SU (3) (λ2 μ2 ) (λ1 μ1 ) (λμ) (λ3 μ3 ) , (λ12 μ12 ) ρ12 12,3 (λ13 μ13 ) ρ13 , ρ13,2 = 1 .

(4.28) Using the Z and U -coefficients a compact expression for 9 − (λμ) coefficients can be written ⎫ ⎧ (λ1 μ1 ) (λ2 μ2 ) (λ12 μ12 ) ρ12 ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ (λ3 μ3 ) (λ4 μ4 ) (λ34 μ34 ) ρ34 χ (λ13 μ13 ) (λ24 μ24 ) (λμ) ρ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ρ24 ρ ρ13 

U (λ13 μ13 ) (λ2 μ2 ) (λμ) (λ4 μ4 ) ; (λ0 μ0 )ρ132 ;ρ04 (λ24 μ24 )ρ24 ;ρ

!

! = (λ0 μ0 ) ρ132 , Z (λ2 μ2 ) (λ1 μ1 ) (λ0 μ0 ) (λ3 μ3 ) ; (λ12 μ12 )ρ ;ρ (λ13 μ13 )ρ ;ρ 12 123 13 132 ρ04 , ρ123 ! U (λ12 μ12 ) (λ3 μ3 ) (λμ) (λ4 μ4 ) ; (λ0 μ0 )ρ123 ;ρ04 (λ34 μ34 )ρ34 ;ρ  . (4.29)

4 SU (3) Wigner–Racah Algebra II

80

The 9 − (λμ) coefficients are needed in many applications and examples are: nuclei with protons and neutrons in different shells as is the situation with many rare-earth and actinide nuclei [19], interacting boson–fermion models [20, 21], when shell model wavefunctions are related to cluster wavefunctions [22], in the calculation of multi-nucleon transfer spectroscopic factors [23–25] and so on (see Chaps. 6–9). In all these we need the SU (3) equivalent of Eqs. (A.25), (A.27) and (A.28). For example, (λ0 μ0 )ρ0  1 1 2 2 the triple barred matrix element of a SU (3) coupled tensor R (λ0 μ0 ) × S (λ0 μ0 ) with the operators R and S operating in different spaces (say spaces #1 and #2) is given by ,

   1 1! ! ! ! ! ! 2 2 (λ0 μ0 )ρ0 f f f f λ1 μ1 λ2 μ2 λ f μ f ρ f ||| R λ0 μ0 × S λ0 μ0 ||| λi1 μi1 λi2 μi2 λi μi ρi ρ ⎧ ⎫ ! !  f f ⎪ ⎪ λi1 μi1 λ10 μ10 λ1 μ1 ρ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ i i ! 2 2!  f f  ⎬ ⎪ λ2 μ2 λ0 μ0 λ2 μ2 ρ2 = ! ! ⎪ i i ⎪ ρ1 ρ2 ⎪ ⎪ λ μ (λ0 μ0 ) λ f μ f ρ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ρ ρ ρ 0! f "   i   ! ! !" 1 1 2 2 f f f f . λ2 μ2 ||| S λ0 μ0 ||| λi2 μi2 × λ1 μ1 ||| R λ0 μ0 ||| λi1 μi1 ρ1

ρ2

(4.30) For R = 1 or S = 1, there are simplifications just as in Eqs. (A.27) and (A.28).

4.5 SU(3) D-Functions Matrix elements of the operators generating finite unitary transformations are the Dfunctions (D-matrix) of the corresponding unitary group. For SU (3), infinitesimal transformations are generated by Ai j or (L q1 , Q 2μ ) operators that are the generators of the SU (3) Lie algebra (see Appendix B). Let us say that O is one of the generators of SU (3), then a finite transformation by O is generated by exp iaO, where a is a constant. Then, the corresponding D function is DOSU (3) ((λμ), α, β : a) = (λμ)β | exp iaO | (λμ)α .

(4.31)

Here, α and β are the labels (K L M) or (εΛMΛ ). Also, note that the generators do not change the SU (3) irrep labels. The full SU (3) D-matrix, with a general (λμ) irrep, is the matrix with matrix elements  8 exp iak Ok | (λμ)α , (λμ)β | Πk=1



4.5 SU (3) D-Functions

81

I where Ok are the eight generators of SU (3) and ak are parameters. Unlike D M K (Ω) of S O(3), the full D matrix of SU (3) is quite complex. For a general discussion of SU (n) D-functions see [26] and for some asymptotic forms of SU (3) Dfunctions see [27]. However, in the context of the interacting boson models sdIBM and sdgIBM, some special D matrices of SU (3) are derived and applied [28–31]. Here, we will describe these results briefly considering a sdgIBM example. In sdgIBM, for N number of bosons, the lowest or ground state SU (3) irrep is (4N , 0) and for this the matrix elements of the type



(4N , 0)L f M | exp ia O | (4N , 0)L i = 0, Mi = 0



with O a generator of SU (3) are useful and a formula for these is obtained as follows. Firstly, the integral representation for |(4N , 0)K = 0, L M follows from Eq. (3.25), . 2L + 1 L (Ω)RΩ (φ0 ) N |0 ; dΩ D M,0 8π 2 [N (N ; L)]1/2 # # φ0 = (s † + 20 d † + 78 g0† ) , 7 0

|(4N , 0)K = 0, L M =



5 N (2L + 1)N !4N ! N (N ; L) = (4N − L)!!(4N + L + 1)!!

 .

(4.32) L In Eq. (4.32), D M0 (Ω) is the Wigner D-function and RΩ is the rotation operator. For the operator O = Q 20 with Q 20 being the quadrupole generator of SUsdg (3) [see Eq. (2.1)], action of exp ia Q 20 on (4N , 0) irrep will generate excitations only to the members of (4N , 0) but not to other excited (β, γ , K = 4+ etc.) bands. Explicit form of Q 20 as used here is Q 20 = 4

#

7 (s † d˜ 15

# 2 2 ˜ 20 + (d † d) + d † s˜ )0 − 11 21

√36 (d † g ˜ 105

˜ 0−2 + g † d) 2

#

2 33 (g † g) ˜ 0 7

(4.33)   and the matrix element (4N , 0)L M | ex p − ia Q 20 | (4N , 0)L i = 0, Mi = 0 is 2 derived by considering the action of U (0) = eia Q 0 and RΩ on (φ0 ) N . This is given simply by $ U

(0)

RΩ (φ0 ) = U N

 (0)



s + †

20  2 D (Ω)dm† + 7 m m0



 /N 8 4 † (0) −1 D  (Ω)gm  U . 7  m0 m

(4.34) −1 † Thus, we need U (0) blm U (0) and this is obtained by diagonalizing a Q 20 in the |lm basis. Then, the eigenfunction and eigenvalue matrices [ψ (m) ] and [ε(m) ], respectively, for m = 0−4 are

4 SU (3) Wigner–Racah Algebra II

82

⎛#

#

#

⎞

⎞ ⎛ 8 ⎛ †⎞ √ a 0 0 ⎟ s ⎜# # 3   ⎜ 28 5 72 ⎟ ⎝ d † ⎠ ; ε (0) = ⎜ 0 √2 a 0 ⎟ [ψ (0) ] = ⎜ 105 ⎟ − 105 ⎠ ⎝ 0 3 105 ⎠ ⎝# # # † −4 √ g a 0 0 56 40 9 0 3 − 105 105 105 7 35

20 35

8 #35

⎛# # ⎞ 8 9 6 †7 3 4 √5 a   (1)   0 d 7 7 3 # ⎠ 1† ; ε(1) = ψ = ⎝# 4 3 0 − √13 a g1 − 7 7 ⎛#

1  (2)  = ⎝#7 ψ 6 7

#

6

⎞

6

†7

#7 ⎠ d2† g2 − 17

  ; ε(2) =

8

√2 a 3

0 0 − √43 a

9

(4.35)

 (3)   †   3   1  ψ = g3 ; ε = − √3 a 

     ψ (4) = (g4† ); ε4 = − √43 a

    † † → b −m and ε[m] → ε[−m] . It is easy Note that as m → −m, we have b m −1 to prove that [a Q 20 , [ψ (m) ]] = [ε(m) ][ψ (m) ], U (0) [ψ (m) ]U (0) = [exp i[ε(m) ]][ψ (m) ] and [exp i[ε(m) ]]i j = δi j exp i[ε(m) ]ii . Now, carrying out various simplifications will give the final result in terms of a hypergeometric function [30]  " 2 (4N , 0)K = 0, L , M = 0 | eia Q 0 | (4N , 0)K i = 0, L i = 0, Mi = 0 ⎫1/2 ⎧ ⎪ ⎬ ⎨ 2 L/2 (2L + 1)(2N )!(4N + L + 1)!! ⎪ (L − 1)!! = L ⎪ ⎪ (2L + 1)!! ⎭ ⎩ (4N + 1)!!(2N − )! 2 6 6 7 7 ! L/2 L L +1 3 i6a0 −i4N a0 i6a0 , , ;L + ; 1−e ×e e −1 2 F1 − 2N − 2 2 2 (4.36) where a0 = √a3 . Generalizations of the SU (3) intrinsic states in sdgIBM are studied in [32, 33]. Using these, it is possible to generalize the SU (3) result given in † Eq. (4.36) for a general intrinsic state φ0 = l=0,2,4 xl0 bl0 generating a K π = 0+ band and for a general quadrupole plus hexadecupole transition operator O =

†˜ λ λ λ=2,4; l,l  =0,2,4 βll  (bl bl  )0 . Also, further generalizations for excitations to β, γ and k = 4+ bands is also possible. These results for SU (3) D-functions and their generalizations given in [30] can be used to study, in heavy deformed nuclei, scattering not only to GS band members but also to β, γ and one-phonon K π = 3+ , 4+ (the later two are 1g-boson type) bands which give information about the importance of g-bosons and hexadecupole deformation. In addition, one can make realistic calculations of sub-barrier fusion cross sections. For example, precise measurements

4.5 SU (3) D-Functions

83

for the fusion cross sections are available for 16 O + 154 Sm [34] and the importance of hexadecupole degree of freedom in sub-barrier fusion is well established [35]. Hence, the importance of Eq. (4.36) and its extensions given in [30].

4.6 Summary In this chapter, we have introduced SU (3) tensorial decomposition of operators, SU (3) fractional parentage coefficients and 9 − SU (3) coefficients. These will allow one to derive analytical formulas and/or perform numerical calculations, for matrix elements of operators in the SU (3) basis, using Wigner–Eckart theorem extended to SU (3) ⊃ S O(3) and SU (3) ⊃ SU (2) ⊗ U (1). These are all described using some examples from SM and IBM. In addition, SU (3) D-matrices are briefly discussed with an example from sdgIBM.

References 1. P.H. Butler, Coupling coefficients and tensor operators for chains of groups. Phil. Trans. R. Soc. Lond. 277, 545–585 (1975) 2. J.D. Vergados, SU (3) ⊃ R(3) Wigner coefficients in the 2s − 1d shell. Nucl. Phys. A 111, 681–754 (1968) 3. R. Bijker, V.K.B. Kota, Interacting boson fermion model of collective states: the SU (3) ⊗ U (2) limit. Ann. Phys. (N.Y.) 187, 148–197 (1988) 4. O. Scholten, K. Heyde, P. Van Isacker, J. Jolie, J. Moreau, M. Waroquier, Mixed-symmetry states in the neutron-proton interacting boson model. Nucl. Phys. A 438, 41–77 (1985) 5. K.T. Hacht, Y. Suzuki, Some special SU (3) ⊃ R(3) Wigner coefficients and their applications. J. Math. Phys. 24, 785–792 (1983) 6. A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1974) 7. V.K.B. Kota, Y.D. Devi, Nuclear Shell Model and the Interacting Boson Model: Lecture Notes for Practitioners (IUC-DAEF Calcutta Center, Kolkata, 1996) 8. J.P. Draayer, Structure calculations for 25 Mg - 25 Al. Nucl. Phys. A 216, 457–476 (1973) 9. M. Chakraborty, V.K.B. Kota, J.C. Parikh, Unitary decomposition of Hamiltonian operators: SU(4) irreducible tensors, norms and their energy variation and symmetry breaking. Ann. Phys. (N.Y.) 127, 413–435 (1980) 10. J.P. Draayer, G. Rosensteel, U (3) → R(3) integrity-basis spectroscopy. Nucl. Phys. A 439, 61–85 (1985) 11. Y.D. Devi, V.K.B. Kota, sdg interacting boson model: hexadecupole degree of freedom in nuclear structure. Pramana J. Phys. 39, 413–491 (1992) 12. H.C. Wu, Bosons with large angular momentum and rotation of even-even nuclei. Phys. Lett. B 110, 1–6 (1982) 13. H.C. Wu, A.E.L. Dieperink, S. Pittel, Intrinsic states for the SU(3) limit of the interacting boson model. Phys. Rev. C 34, 703–716 (1986) 14. D. Braunschweig, Reduced CFP’s. Compu. Phys. Comm. 14, 109–120 (1978) 15. D. Braunschweig, II. Reduced SU (3) matrix elements. Compu. Phys. Comm. 15, 259–273 (1978)

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16. C. Bahri, J.P. Draayer, SU (3) reduced matrix elements package. Comp. Phys. Comm. 83, 59–94 (1994) 17. V.K.B. Kota, A study of the static moments of odd-odd deformed nuclei. Prog. Theo. Phys. 59, 435–450 (1978) 18. J.P. Draayer, Y. Akiyama, Wigner and Racah coefficients for SU3 . J. Math. Phys. 14, 1904– 1912 (1973) 19. J.P. Draayer, K.J. Weeks, Towards a shell model description of the low-energy structure of deformed nuclei I. Even-Even systems. Ann. Phys. (N.Y.) 156, 41–67 (1984) 20. Y.D. Devi, V.K.B. Kota, Scissors states with and without g-bosons in the interacting boson - Fermion Model for even - odd nuclei in N = 82–126 Shell. Nucl. Phys. A541, 173–192 (1992) B F (3) and SU B F (3) × 21. Y.D. Devi, V.K.B. Kota, M1 Distributions for 163 Dy and 157 Gd in SUsdg sd 1g limits of pn-sdgIBFM. Nucl. Phys. A 600, 20–36 (1996) 22. K.T. Hecht, Relation between cluster and shell-model wave functions. Phys. Rev. C 16, 2401– 2414 (1977) 23. N. Anyas-Weiss, J.C. Cornell, P.S. Fisher, P.N. Hudson, A.M. Rocha, D.J. Millener, A.D. Panagiotou, D.K. Acott, D. Strottman, D.M. Brink, B. Buck, P.J. Ellis, T. England, Nuclear structure of light nuclei using the selectivity of high energy transfer reactions with heavy ions. Phys. Rep. 12, 201–272 (1974) 24. K.T. Hecht, D. Braunschweig, Few-nucleon SU (3) parentage coefficients and α-particle spectroscopic amplitudes for core excited states in s − d shell nuclei. Nucl. Phys. A 244, 365–434 (1975) 25. K.T. Hecht, Alpha and 8 Be cluster amplitudes and core excitations in s − d shell nuclei. Nucl. Phys. A 283, 223–252 (1977) 26. I. Dhand, B.C. Sanders, H. de Guise, Algorithms for SU (n) boson realizations and Dfunctions. J. Math. Phys. 56, 111705/1-11 (2015) 27. D.J. Rowe, H. de Guise, B.C. Sanders, Asymptotic limits of SU (2) and SU (3) Wigner functions. J. Math. Phys. 42, 2315–2342 (2001) 28. J.N. Ginocchio, Scattering theory and the group representation matrix, in Symmetries in Science V, ed. by B. Gruber, L.C. Biedenharn, H.D. Doebner (Plenum Press, New York, 1991), pp. 223–257 29. J.N. Ginocchio, T. Otsuka, R.D. Amado, D.A. Sparrow, Medium energy probes and the interacting boson model of nuclei. Phys. Rev. C 33, 247–259 (1986) 30. V.K.B. Kota, Eikonal scattering in the sdg interacting boson model: analytical results in the SUsdg (3) limit and their generalizations. Mod. Phys. Lett. A 8, 987–996 (1993) 31. A.B. Balantekin, J.R. Bennett, N. Takigawa, Description of nuclear structure effects in subbarrier fusion by the interacting boson model. Phys. Rev. C 44, 145–151 (1991) 32. S. Kuyucak, I. Morrison, Signature of g boson in the interacting-boson model from g-factor variations. Phys. Rev. Lett. 58, 315–317 (1987) 33. S. Kuyucak, I. Morrison, 1/N expansion in the interacting boson model. Ann. Phys. (N.Y.) 181, 79–119 (1988) 34. J.X. Wei, J.R. Leigh, D.J. Hinde, J.O. Newton, R.C. Lemmon, S. Elfstrom, J.X. Chen, N. Rowley, Experimental determination of the fusion-barrier distribution for the 154 Sm + 16 O reaction. Phys. Rev. Lett. 67, 3368–3371 (1991) 35. J. Fernandez Niello, M. di Tada, A.O. Macchiavelly, A.J. Pacheco, D. Abriola, M. Elgu, A. Etchegoyen, M.S. Etchegoyen, S. Gil, J.E. Tessoni, Hexadecapole deformation effects in sub-barrier fusion reactions. Phys. Rev. C 43, 2303–2306 (1991)

Chapter 5

SU(3) ⊃ S O(3) Integrity Basis Operators

5.1 Introduction In the physically important SU (3) ⊃ S O(3) ⊃ S O(2) algebra with the corresponding irrep reductions (λμ) → L → M, there is a missing label K introduced in Eq. (2.32) which is not a quantum number, i.e., K does not correspond to any group irrep label. In the attempts to resolve the resultant L degeneracy, one is led to the identification of a minimal set of S O(3) scalars in the SU (3) enveloping algebra. This set of operators is called integrity basis. Their property is that any S O(3) scalar constructed out of the SU (3) generators will be a polynomial in the integrity basis operators. It is well known now that, besides the Casimir invariants C2 and C3 of SU (3), L 2 and L Z operators, there will be a third order operator (called X 3 ) and a fourth order (called X 4 ) operator in the integrity basis. Judd et al. [1] gave a first complete discussion of the X 3 and X 4 operators and their properties. Their important applications and physical relevance was studied by Draayer et al. [2, 3]. Important among these are (i) writing the formulas for the matrix elements of X 3 and X 4 operators in terms of SU (3) ⊃ S O(3) reduced Wigner coefficients for easy calculations [2]; (ii) their role in relating the (β, γ ) parameters of the Bohr rotor to the SU (3) irreps (λμ) [4–9]; (iii) mapping of the integrity basis Hamiltonian to asymmetric rotor; (iv) constructing an operator for the K quantum number (for spin S = 0 states) that will give in an asymptotic limit the K label introduced by Elliott [4, 10]; (v) extension of this operator to the K J quantum number with J = L + S [11]; (vi) role of X 3 and X 4 in trace propagation [2, 12–15]. Topics (i)–(vi) will form the remaining part of this chapter.

© Springer Nature Singapore Pte Ltd. 2020 V. K. B. Kota, SU(3) Symmetry in Atomic Nuclei, https://doi.org/10.1007/978-981-15-3603-8_5

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5 SU (3) ⊃ S O(3) Integrity Basis Operators

86

5.2 X 3 and X 4 Operators and Their Matrix Elements Leaving group theoretical details to [1], the X 3 and X 4 operators are identified as X3 = X4 =

√ √

3[L × Q × L]0 (5.1) 3[(L × Q) × (Q × L) ] . 1

1 0

As already given in Eq. (2.28), X 3 is related in a simple manner to C3 and (Q × Q) · Q. Also, in principle, one can choose [(L × Q)k × (Q × L)k ]0 with k = 2 or 3 for X 4 . As X 3 and X 4 operators preserve (λμ), L, and M (independent of M), their matrix elements using Eq. (4.3) are given by √  (λμ)K  f L M | X 3 | (λμ)K i L M = L(L + 1) 3  1 1 2  × (λμ)K f L || Q 2 || (λμ)K i L , L L L     (−1)1+L+L (λμ)K f L M | X 4 | (λμ)K i L M = 3L(L + 1)





2

  1 2 1 (λμ)K f L || Q 2 || (λμ)K  L   L L L   × (λμ)K  L  || Q 2 || (λμ)K i L .

K  L

(5.2)

×

The Q 2 double barred matrix elements are simply related to C2 (λμ) and SU (3) ⊃ S O(3) reduced Wigner coefficient as given by Eq. (4.7). Results in Eq. (5.2) are derived by inserting complete set of states between the L and Q operators in X 3 and similarly in X 4 and using the fact that Q 2 and L 1 will not change (λμ) and L 1 will not change L also. Using the codes in [16] for evaluating the SU (3) ⊃ S O(3) reduced Wigner coefficient, it is easy to use Eq. (5.2) in numerical calculations. In this situation, the K in Eq. (5.2) is the K used in [16]. Other alternatives to calculate the matrix elements of X 3 and X 4 operators are given, for example, in [1, 17]. For a given (λμ) irrep with μ ≥ 2, the L degeneracy is lifted by X 3 and X 4 operators and therefore using a simple Hamiltonian HI nt , HI nt = a L 2 + bX 3 + cX 4

(5.3)

one can obtain good description of observed band structures in rotational nuclei as first reported by Raichev and Rusey [18] and investigated and applied in a systematic manner within the pseudo-SU (3) model by Draayer et al. [19, 20] (they have also added a (L .L)2 term in HI nt ). We will return to this integrity basis spectroscopy in Chap. 6. Before applying HI nt , it is essential to understand the physics implied by the operators X 3 and X 4 that are three- and four-body operators.

5.3 Shape Parameters and (λμ) Irreps Correspondence

87

5.3 Shape Parameters and (λμ) Irreps Correspondence N  xiα xiβ − 13 δαβ Let us consider the traceless quadrupole mass tensor Q cαβ = i=1  2 of a N particle system. Here, xiα are the Cartesian components of the γ x iγ position vector of the ith particle. The Q c (it will have five independent 

components) and three angular momentum components L q are the generators of R 5 S O(3), the symmetry group of a quantum rotor. The Casimir invariants of [R 5 ]S O(3) are a2 and a3 where  



(5.4) a2 = T r (Q c )2 , a3 = T r (Q c )3 . Note that T r [Q c ) = 0. In terms of the three principle moments λ1 , λ2 , and λ3 of Q c (they are eigenvalues of Q c and independent of the orientation of the system), the eigenvalues of a2 and a3 are 

 a2  = T r (Q c )2 = λ21 + λ22 + λ23 , 

 a3  = T r (Q c )3 = λ31 + λ32 + λ33 = 3 (λ1 λ2 λ3 )

(5.5)

and λ1 + λ2 + λ3 = 0. It is easy to see from the definition of the quadratic and cubic 2 + α2 and invariants of SU (3) given in Sect. 2.5 (see also Eq. (2.28)) that a2 ∼ α1 C 3 + α4 C 2 + α5 . Using Eqs. (2.26) and (2.27), one can fix the values of α’s a3 ∼ α3 C and derive the relationship between (λ1 , λ2 , λ3 ) and the SU (3) irrep labels (λ, μ). Strikingly, the result is [7] λ1 = (μ − λ)/3 ,

λ2 = −(λ + 2μ + 3)/3 , λ3 = (2λ + μ + 3)/3 .

(5.6)

The observables (a2  , a3 ) are related to the Bohr–Mottelson quadrupole shape parameters (βγ ) giving a2  ∝ β 2 and a3  ∝ β 3 cos 3γ [21]. Putting proper normalization factors and using Eqs. (5.5) and (5.6) will give the important result relating (λμ) to (βγ ) [7–9],   2 4π 2 β2 = 2 λ + μ + λμ + 3λ + 3μ + 3 , 5Ar 2 (λ − μ)(λ + 2μ + 3)(2λ + μ + 3) cos 3γ =  3/2 . 2 λ2 + μ2 + λμ + 3λ + 3μ + 3

(5.7)

Here, A is nucleon number and r 2 is the dimensionless mean square radius. It is well known that (r 2 )1/2 = rr.m.s = r0 A1/6 where r0 ∼ 0.9. The +3 in (λ2 + μ2 + λμ + 3λ + 3μ + 3) in Eq. (5.7) is ignored in [6]. However, in [7], the +3 is kept but for γ , the formula deduced is [7, 9],  √

3(μ + 1) γ = arctan (2λ + μ + 3)

 .

(5.8)

5 SU (3) ⊃ S O(3) Integrity Basis Operators

88

Eqautions (5.7) and (5.8) give very close values for γ except when γ is close to 0 or π/3. Thus, given the (λμ) irreps (see Tables in Chaps. 2 and 3), there will be corresponding (βγ ) values as given by Eqs. (5.7) and (5.8). As the (λμ) are discrete, the (βγ ) values will be discrete. This correspondence allows for shell model interpretation of potential energy surfaces [8]. Recently Eq. (5.7) is applied to calculate (βγ ) values for rare-earth nuclei within the proxy-SU (3) model with good agreement with experimental data (in comparison, RMF theory appear to deviate more from experiment) [22].

5.4 Integrity Basis Hamiltonian and Asymmetric Rotor Returning now to the integrity basis Hamiltonian HI nt given by Eq. (5.3), we will show using Eq. (5.6) that HI nt maps exactly to the asymmetric rotor. The Hamiltonian of the asymmetric rotor is Hr ot = A1 I12 + A2 I22 + A3 I32

(5.9)

where Ai are inertia parameters and Ii is the projection of the angular momentum of the rotor on the ith body fixed symmetry axis. Now, in terms of the three principle moments λi of Q c , we have L2 = X 3c = X 4c =

 α  αβ 

I12 + I22 + I32 , L α Q cαβ L β = λ1 I12 + λ2 I22 + λ3 I32 ,

(5.10)

L α Q cαβ Q cβγ L γ = λ21 I12 + λ22 I22 + λ23 I32 .

αβ

Note that introduced here are X 3c and X 4c similar to the X 3 and X 4 operators. With this, Eq. (5.9) can be written as Hr ot = a  L 2 + b X 3c + c X 4c

(5.11)

and therefore Hr ot is equivalent to HI nt of SU (3) defined by Eq. (5.3). We can solve for the parameters (a  , b , c ) in termsof the Ai in Eq. (5.9). The result is [7],  a = i ai Ai with ai = λ1 λ2 λ3 /Di , b = i bi Ai with bi = λi2 /Di , c = i ci Ai with ci = λi /Di and Di = 2λi3 + λ1 λ2 λ3 . Using these formulas for (a  , b , c ), the relationship between (λ1 , λ2 , λ3 ) and (λμ) as given by Eq. (5.6) and replacing (a, b, c) parameters in HI nt by (a  , b , c ), we have a map of the HI nt into Hr ot . Thus, HI nt is same as the quantum asymmetric rotor Hamiltonian. Given the (λμ) irreps for a physical system (for example, (8, 4) for 24 Mg and (30, 8) for 168 Er) and fitting the Hr ot to the observed spectrum, one can obtain the Ai parameters and by

5.4 Integrity Basis Hamiltonian and Asymmetric Rotor

89

the above correspondence (a, b, c) parameters in Eq. (5.3). Now the SU (3) spectrum generated by HI nt is found to be close to the spectrum generated by Hr ot (also experiment). See, for example, [7].

5.5

K Quantum Number from X 3 and X 4 Operators

Going further with the integrity basis operators X 3 and X 4 , it is of interest, as they mix Elliott’s K quantum number, to find a combination of these operators to define a Kˆ 2 operator whose expectation values over a |(λμ)K L state is just K 2 . A first attempt in this direction is in [4]. Following this, in [10], the operator is identified as (λ1 λ2 L 2 + λ3 X 3a + X 4a ) ; Kˆ 2 = 2 √ (2λ3 +√λ1 λ2 ) 10 30 X3 = [L × Q × L]0 , X 3a = 6 6√ 5 5 3 X 4a = − X 4 = − [(L × Q)1 × (Q × L)1 ]0 . 18 18

(5.12)

The λi dependence on λ and μ is as given by Eq. (5.6). Let us now examine X 3 (λμ)K L and X 4 (λμ)K L in the asymptotic limit defined by λ >> μ, L